LMFDB - Embedded newform 69.8.c.b.68.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,8,Mod(68,69)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("69.68");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	69.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$21.5545667584$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			68.13
Character	$\chi$	$=$	69.68
Dual form			69.8.c.b.68.35

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.0616i q^{2} +(16.6400 + 43.7048i) q^{3} +5.64089 q^{4} -185.789 q^{5} +(483.445 - 184.066i) q^{6} +711.199i q^{7} -1478.28i q^{8} +(-1633.22 + 1454.50i) q^{9} +O(q^{10})$	\(q-11.0616i q^{2} +(16.6400 + 43.7048i) q^{3} +5.64089 q^{4} -185.789 q^{5} +(483.445 - 184.066i) q^{6} +711.199i q^{7} -1478.28i q^{8} +(-1633.22 + 1454.50i) q^{9} +2055.12i q^{10} +1199.59 q^{11} +(93.8646 + 246.534i) q^{12} +4721.97 q^{13} +7867.00 q^{14} +(-3091.53 - 8119.87i) q^{15} -15630.1 q^{16} -40001.4 q^{17} +(16089.1 + 18066.0i) q^{18} +37723.6i q^{19} -1048.01 q^{20} +(-31082.8 + 11834.4i) q^{21} -13269.4i q^{22} +(-58169.5 - 4597.49i) q^{23} +(64608.0 - 24598.7i) q^{24} -43607.5 q^{25} -52232.6i q^{26} +(-90745.4 - 47176.5i) q^{27} +4011.79i q^{28} +182408. i q^{29} +(-89818.7 + 34197.3i) q^{30} +127501. q^{31} -16325.7i q^{32} +(19961.2 + 52427.9i) q^{33} +442479. i q^{34} -132133. i q^{35} +(-9212.80 + 8204.67i) q^{36} +156164. i q^{37} +417284. q^{38} +(78573.8 + 206373. i) q^{39} +274649. i q^{40} +235298. i q^{41} +(130907. + 343826. i) q^{42} +800105. i q^{43} +6766.76 q^{44} +(303434. - 270230. i) q^{45} +(-50855.6 + 643448. i) q^{46} -923838. i q^{47} +(-260086. - 683112. i) q^{48} +317739. q^{49} +482369. i q^{50} +(-665624. - 1.74825e6i) q^{51} +26636.1 q^{52} -946774. q^{53} +(-521848. + 1.00379e6i) q^{54} -222871. q^{55} +1.05135e6 q^{56} +(-1.64870e6 + 627723. i) q^{57} +2.01772e6 q^{58} +1.27840e6i q^{59} +(-17439.0 - 45803.3i) q^{60} -1.65455e6i q^{61} -1.41037e6i q^{62} +(-1.03444e6 - 1.16154e6i) q^{63} -2.18125e6 q^{64} -877289. q^{65} +(579936. - 220803. i) q^{66} -950780. i q^{67} -225643. q^{68} +(-767010. - 2.61879e6i) q^{69} -1.46160e6 q^{70} -141863. i q^{71} +(2.15016e6 + 2.41436e6i) q^{72} -2.82301e6 q^{73} +1.72743e6 q^{74} +(-725630. - 1.90586e6i) q^{75} +212795. i q^{76} +853147. i q^{77} +(2.28281e6 - 869152. i) q^{78} -4.07099e6i q^{79} +2.90391e6 q^{80} +(551834. - 4.75103e6i) q^{81} +2.60278e6 q^{82} +8.08344e6 q^{83} +(-175335. + 66756.4i) q^{84} +7.43181e6 q^{85} +8.85044e6 q^{86} +(-7.97210e6 + 3.03527e6i) q^{87} -1.77333e6i q^{88} +6.97480e6 q^{89} +(-2.98918e6 - 3.35646e6i) q^{90} +3.35826e6i q^{91} +(-328128. - 25933.9i) q^{92} +(2.12163e6 + 5.57242e6i) q^{93} -1.02191e7 q^{94} -7.00863e6i q^{95} +(713511. - 271660. i) q^{96} +6.73494e6i q^{97} -3.51471e6i q^{98} +(-1.95919e6 + 1.74480e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 24 q^{3} - 2616 q^{4} + 4416 q^{6} + 648 q^{9}+O(q^{10}) $ 48 * q + 24 * q^3 - 2616 * q^4 + 4416 * q^6 + 648 * q^9	$ 48 q + 24 q^{3} - 2616 q^{4} + 4416 q^{6} + 648 q^{9} + 16668 q^{12} - 4992 q^{13} + 76296 q^{16} + 100548 q^{18} - 620268 q^{24} + 1257120 q^{25} - 105408 q^{27} - 738480 q^{31} - 1227096 q^{36} + 3832248 q^{39} - 489768 q^{46} + 733404 q^{48} - 5569584 q^{49} - 5646984 q^{52} + 1260252 q^{54} - 305760 q^{55} + 16920336 q^{58} + 6032664 q^{64} + 4496304 q^{69} + 23272152 q^{70} - 26866080 q^{72} - 2542752 q^{73} + 487176 q^{75} - 47724612 q^{78} + 25279128 q^{81} - 68451480 q^{82} - 18046992 q^{85} + 42338184 q^{87} - 59017272 q^{93} + 106197744 q^{94} + 67778460 q^{96}+O(q^{100}) $ 48 * q + 24 * q^3 - 2616 * q^4 + 4416 * q^6 + 648 * q^9 + 16668 * q^12 - 4992 * q^13 + 76296 * q^16 + 100548 * q^18 - 620268 * q^24 + 1257120 * q^25 - 105408 * q^27 - 738480 * q^31 - 1227096 * q^36 + 3832248 * q^39 - 489768 * q^46 + 733404 * q^48 - 5569584 * q^49 - 5646984 * q^52 + 1260252 * q^54 - 305760 * q^55 + 16920336 * q^58 + 6032664 * q^64 + 4496304 * q^69 + 23272152 * q^70 - 26866080 * q^72 - 2542752 * q^73 + 487176 * q^75 - 47724612 * q^78 + 25279128 * q^81 - 68451480 * q^82 - 18046992 * q^85 + 42338184 * q^87 - 59017272 * q^93 + 106197744 * q^94 + 67778460 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	11.0616i		−	0.977717i	−0.872363	−	0.488859i	$-0.837413\pi$
$2$		−	11.0616i		−	0.977717i	0.872363	−	0.488859i	$-0.162587\pi$
$3$	16.6400	+	43.7048i	0.355820	+	0.934555i
$3$	16.6400	+	43.7048i	0.355820	+	0.934555i
$4$	5.64089			0.0440694
$5$	−185.789			−0.664699			−0.332349	−	0.943156i	$-0.607841\pi$
$5$	−185.789			−0.664699			−0.332349	+	0.943156i	$0.607841\pi$
$6$	483.445	−	184.066i	0.913730	−	0.347891i
$7$			711.199i			0.783697i	0.920030	+	0.391848i	$0.128164\pi$
$7$			711.199i			0.783697i	−0.920030	+	0.391848i	$0.871836\pi$
$8$		−	1478.28i		−	1.02080i
$9$	−1633.22	+	1454.50i	−0.746785	+	0.665066i
$10$			2055.12i			0.649887i
$11$	1199.59			0.271743			0.135872	−	0.990726i	$-0.456617\pi$
$11$	1199.59			0.271743			0.135872	+	0.990726i	$0.456617\pi$
$12$	93.8646	+	246.534i	0.0156808	+	0.0411853i
$13$	4721.97			0.596103			0.298052	−	0.954550i	$-0.403663\pi$
$13$	4721.97			0.596103			0.298052	+	0.954550i	$0.403663\pi$
$14$	7867.00			0.766233
$15$	−3091.53	−	8119.87i	−0.236513	−	0.621197i
$16$	−15630.1			−0.953988
$17$	−40001.4			−1.97471			−0.987355	−	0.158523i	$-0.949327\pi$
$17$	−40001.4			−1.97471			−0.987355	+	0.158523i	$0.949327\pi$
$18$	16089.1	+	18066.0i	0.650246	+	0.730144i
$19$			37723.6i			1.26176i	0.775881	+	0.630879i	$0.217305\pi$
$19$			37723.6i			1.26176i	−0.775881	+	0.630879i	$0.782695\pi$
$20$	−1048.01			−0.0292929
$21$	−31082.8	+	11834.4i	−0.732407	+	0.278855i
$22$		−	13269.4i		−	0.265688i
$23$	−58169.5	−	4597.49i	−0.996891	−	0.0787904i
$23$	−58169.5	−	4597.49i	−0.996891	−	0.0787904i
$24$	64608.0	−	24598.7i	0.953998	−	0.363222i
$25$	−43607.5			−0.558176
$26$		−	52232.6i		−	0.582820i
$27$	−90745.4	−	47176.5i	−0.887261	−	0.461268i
$28$			4011.79i			0.0345371i
$29$			182408.i			1.38883i	0.719573	+	0.694417i	$0.244338\pi$
$29$			182408.i			1.38883i	−0.719573	+	0.694417i	$0.755662\pi$
$30$	−89818.7	+	34197.3i	−0.607355	+	0.231243i
$31$	127501.			0.768686			0.384343	−	0.923190i	$-0.374428\pi$
$31$	127501.			0.768686			0.384343	+	0.923190i	$0.374428\pi$
$32$		−	16325.7i		−	0.0880737i
$33$	19961.2	+	52427.9i	0.0966916	+	0.253959i
$34$			442479.i			1.93071i
$35$		−	132133.i		−	0.520922i
$36$	−9212.80	+	8204.67i	−0.0329104	+	0.0293091i
$37$			156164.i			0.506845i	0.967356	+	0.253423i	$0.0815563\pi$
$37$			156164.i			0.506845i	−0.967356	+	0.253423i	$0.918444\pi$
$38$	417284.			1.23364
$39$	78573.8	+	206373.i	0.212105	+	0.557091i
$40$			274649.i			0.678527i
$41$			235298.i			0.533182i	0.963810	+	0.266591i	$0.0858972\pi$
$41$			235298.i			0.533182i	−0.963810	+	0.266591i	$0.914103\pi$
$42$	130907.	+	343826.i	0.272641	+	0.716087i
$43$			800105.i			1.53464i	0.641263	+	0.767321i	$0.278411\pi$
$43$			800105.i			1.53464i	−0.641263	+	0.767321i	$0.721589\pi$
$44$	6766.76			0.0119756
$45$	303434.	−	270230.i	0.496387	−	0.442068i
$46$	−50855.6	+	643448.i	−0.0770347	+	0.974677i
$47$		−	923838.i		−	1.29794i	−0.760815	−	0.648968i	$-0.775201\pi$
$47$		−	923838.i		−	1.29794i	0.760815	−	0.648968i	$-0.224799\pi$
$48$	−260086.	−	683112.i	−0.339448	−	0.891554i
$49$	317739.			0.385820
$50$			482369.i			0.545738i
$51$	−665624.	−	1.74825e6i	−0.702641	−	1.84547i
$52$	26636.1			0.0262699
$53$	−946774.			−0.873536			−0.436768	−	0.899574i	$-0.643877\pi$
$53$	−946774.			−0.873536			−0.436768	+	0.899574i	$0.643877\pi$
$54$	−521848.	+	1.00379e6i	−0.450989	+	0.867490i
$55$	−222871.			−0.180627
$56$	1.05135e6			0.800001
$57$	−1.64870e6	+	627723.i	−1.17918	+	0.448958i
$58$	2.01772e6			1.35789
$59$			1.27840e6i			0.810372i	0.914234	+	0.405186i	$0.132793\pi$
$59$			1.27840e6i			0.810372i	−0.914234	+	0.405186i	$0.867207\pi$
$60$	−17439.0	−	45803.3i	−0.0104230	−	0.0273758i
$61$		−	1.65455e6i		−	0.933306i	−0.884440	−	0.466653i	$-0.845460\pi$
$61$		−	1.65455e6i		−	0.933306i	0.884440	−	0.466653i	$-0.154540\pi$
$62$		−	1.41037e6i		−	0.751558i
$63$	−1.03444e6	−	1.16154e6i	−0.521210	−	0.585253i
$64$	−2.18125e6			−1.04010
$65$	−877289.			−0.396229
$66$	579936.	−	220803.i	0.248300	−	0.0945370i
$67$		−	950780.i		−	0.386205i	−0.981179	−	0.193103i	$-0.938145\pi$
$67$		−	950780.i		−	0.386205i	0.981179	−	0.193103i	$-0.0618551\pi$
$68$	−225643.			−0.0870244
$69$	−767010.	−	2.61879e6i	−0.281080	−	0.959684i
$70$	−1.46160e6			−0.509314
$71$		−	141863.i		−	0.0470397i	−0.999723	−	0.0235198i	$-0.992513\pi$
$71$		−	141863.i		−	0.0470397i	0.999723	−	0.0235198i	$-0.00748729\pi$
$72$	2.15016e6	+	2.41436e6i	0.678902	+	0.762321i
$73$	−2.82301e6			−0.849343			−0.424671	−	0.905348i	$-0.639610\pi$
$73$	−2.82301e6			−0.849343			−0.424671	+	0.905348i	$0.639610\pi$
$74$	1.72743e6			0.495551
$75$	−725630.	−	1.90586e6i	−0.198610	−	0.521646i
$76$			212795.i			0.0556050i
$77$			853147.i			0.212964i
$78$	2.28281e6	−	869152.i	0.544677	−	0.207379i
$79$		−	4.07099e6i		−	0.928977i	−0.885579	−	0.464488i	$-0.846238\pi$
$79$		−	4.07099e6i		−	0.928977i	0.885579	−	0.464488i	$-0.153762\pi$
$80$	2.90391e6			0.634115
$81$	551834.	−	4.75103e6i	0.115375	−	0.993322i
$82$	2.60278e6			0.521301
$83$	8.08344e6			1.55175			0.775877	−	0.630884i	$-0.217308\pi$
$83$	8.08344e6			1.55175			0.775877	+	0.630884i	$0.217308\pi$
$84$	−175335.	+	66756.4i	−0.0322768	+	0.0122890i
$85$	7.43181e6			1.31259
$86$	8.85044e6			1.50045
$87$	−7.97210e6	+	3.03527e6i	−1.29794	+	0.494175i
$88$		−	1.77333e6i		−	0.277397i
$89$	6.97480e6			1.04874			0.524368	−	0.851491i	$-0.324301\pi$
$89$	6.97480e6			1.04874			0.524368	+	0.851491i	$0.324301\pi$
$90$	−2.98918e6	−	3.35646e6i	−0.432218	−	0.485326i
$91$			3.35826e6i			0.467164i
$92$	−328128.	−	25933.9i	−0.0439324	−	0.00347225i
$93$	2.12163e6	+	5.57242e6i	0.273514	+	0.718379i
$94$	−1.02191e7			−1.26901
$95$		−	7.00863e6i		−	0.838688i
$96$	713511.	−	271660.i	0.0823097	−	0.0313384i
$97$			6.73494e6i			0.749260i	0.927174	+	0.374630i	$0.122230\pi$
$97$			6.73494e6i			0.749260i	−0.927174	+	0.374630i	$0.877770\pi$
$98$		−	3.51471e6i		−	0.377223i
$99$	−1.95919e6	+	1.74480e6i	−0.202934	+	0.180727i
$100$	−245985.			−0.0245985
$101$			1.69510e7i			1.63708i	0.574448	+	0.818541i	$0.305217\pi$
$101$			1.69510e7i			1.63708i	−0.574448	+	0.818541i	$0.694783\pi$
$102$	−1.93385e7	+	7.36287e6i	−1.80435	+	0.686984i
$103$		−	1.92794e7i		−	1.73845i	−0.494416	−	0.869226i	$-0.664618\pi$
$103$		−	1.92794e7i		−	1.73845i	0.494416	−	0.869226i	$-0.335382\pi$
$104$		−	6.98041e6i		−	0.608505i
$105$	5.77484e6	−	2.19870e6i	0.486830	−	0.185354i
$106$			1.04728e7i			0.854071i
$107$	−1.41929e7			−1.12003			−0.560013	−	0.828484i	$-0.689204\pi$
$107$	−1.41929e7			−1.12003			−0.560013	+	0.828484i	$0.689204\pi$
$108$	−511885.	−	266118.i	−0.0391011	−	0.0203278i
$109$		−	5.79523e6i		−	0.428625i	−0.976765	−	0.214313i	$-0.931249\pi$
$109$		−	5.79523e6i		−	0.428625i	0.976765	−	0.214313i	$-0.0687511\pi$
$110$			2.46531e6i			0.176602i
$111$	−6.82512e6	+	2.59858e6i	−0.473675	+	0.180345i
$112$		−	1.11161e7i		−	0.747637i
$113$	1.61275e7			1.05146			0.525729	−	0.850652i	$-0.323793\pi$
$113$	1.61275e7			1.05146			0.525729	+	0.850652i	$0.323793\pi$
$114$	6.94362e6	+	1.82373e7i	0.438954	+	1.15291i
$115$	1.08072e7	+	854163.i	0.662632	+	0.0523719i
$116$			1.02894e6i			0.0612052i
$117$	−7.71201e6	+	6.86810e6i	−0.445161	+	0.396448i
$118$	1.41411e7			0.792314
$119$		−	2.84489e7i		−	1.54757i
$120$	−1.20035e7	+	4.57016e6i	−0.634121	+	0.241433i
$121$	−1.80482e7			−0.926156
$122$	−1.83019e7			−0.912510
$123$	−1.02837e7	+	3.91537e6i	−0.498287	+	0.189717i
$124$	719221.			0.0338756
$125$	2.26165e7			1.03572
$126$	−1.28485e7	+	1.14425e7i	−0.572211	+	0.509596i
$127$	3.30487e7			1.43167			0.715833	−	0.698272i	$-0.246047\pi$
$127$	3.30487e7			1.43167			0.715833	+	0.698272i	$0.246047\pi$
$128$			2.20384e7i			0.928849i
$129$	−3.49684e7	+	1.33138e7i	−1.43421	+	0.546056i
$130$			9.70423e6i			0.387400i
$131$			4.68108e6i			0.181927i	0.995854	+	0.0909634i	$0.0289946\pi$
$131$			4.68108e6i			0.181927i	−0.995854	+	0.0909634i	$0.971005\pi$
$132$	112599.	+	295740.i	0.00426114	+	0.0111918i
$133$	−2.68290e7			−0.988835
$134$	−1.05172e7			−0.377600
$135$	1.68595e7	+	8.76488e6i	0.589761	+	0.306604i
$136$			5.91333e7i			2.01579i
$137$	4.81616e6			0.160022			0.0800108	−	0.996794i	$-0.474505\pi$
$137$	4.81616e6			0.160022			0.0800108	+	0.996794i	$0.474505\pi$
$138$	−2.89680e7	+	8.48436e6i	−0.938300	+	0.274816i
$139$	−3.95341e7			−1.24859			−0.624295	−	0.781189i	$-0.714614\pi$
$139$	−3.95341e7			−1.24859			−0.624295	+	0.781189i	$0.714614\pi$
$140$		−	745347.i		−	0.0229567i
$141$	4.03762e7	−	1.53727e7i	1.21299	−	0.461831i
$142$	−1.56923e6			−0.0459915
$143$	5.66443e6			0.161987
$144$	2.55274e7	−	2.27340e7i	0.712424	−	0.634465i
$145$		−	3.38893e7i		−	0.923156i
$146$			3.12271e7i			0.830417i
$147$	5.28719e6	+	1.38867e7i	0.137282	+	0.360570i
$148$			880905.i			0.0223364i
$149$	6.11401e7			1.51417			0.757085	−	0.653317i	$-0.226623\pi$
$149$	6.11401e7			1.51417			0.757085	+	0.653317i	$0.226623\pi$
$150$	−2.10818e7	+	8.02664e6i	−0.510022	+	0.194184i
$151$	−5.12211e6			−0.121068			−0.0605340	−	0.998166i	$-0.519280\pi$
$151$	−5.12211e6			−0.121068			−0.0605340	+	0.998166i	$0.519280\pi$
$152$	5.57662e7			1.28801
$153$	6.53309e7	−	5.81819e7i	1.47468	−	1.31331i
$154$	9.43718e6			0.208219
$155$	−2.36883e7			−0.510945
$156$	443226.	+	1.16413e6i	0.00934736	+	0.0245507i
$157$			4.51489e7i			0.931104i	0.885021	+	0.465552i	$0.154144\pi$
$157$			4.51489e7i			0.931104i	−0.885021	+	0.465552i	$0.845856\pi$
$158$	−4.50316e7			−0.908276
$159$	−1.57544e7	−	4.13786e7i	−0.310821	−	0.816367i
$160$			3.03313e6i			0.0585425i
$161$	3.26973e6	−	4.13701e7i	0.0617478	−	0.781260i
$162$	−5.25540e7	−	6.10417e6i	−0.971188	−	0.112804i
$163$	8.08033e7			1.46141			0.730705	−	0.682693i	$-0.239191\pi$
$163$	8.08033e7			1.46141			0.730705	+	0.682693i	$0.239191\pi$
$164$			1.32729e6i			0.0234970i
$165$	−3.70858e6	−	9.74051e6i	−0.0642707	−	0.168806i
$166$		−	8.94158e7i		−	1.51718i
$167$		−	8.28704e6i		−	0.137687i	−0.997627	−	0.0688433i	$-0.978069\pi$
$167$		−	8.28704e6i		−	0.137687i	0.997627	−	0.0688433i	$-0.0219309\pi$
$168$	1.74946e7	+	4.59492e7i	0.284656	+	0.747645i
$169$	−4.04515e7			−0.644661
$170$		−	8.22077e7i		−	1.28334i
$171$	−5.48690e7	−	6.16109e7i	−0.839152	−	0.942261i
$172$			4.51330e6i			0.0676308i
$173$		−	1.05819e7i		−	0.155382i	−0.996977	−	0.0776910i	$-0.975245\pi$
$173$		−	1.05819e7i		−	0.155382i	0.996977	−	0.0776910i	$-0.0247548\pi$
$174$	3.35750e7	+	8.81842e7i	0.483163	+	1.26902i
$175$		−	3.10136e7i		−	0.437440i
$176$	−1.87498e7			−0.259240
$177$	−5.58722e7	+	2.12726e7i	−0.757337	+	0.288346i
$178$		−	7.71525e7i		−	1.02537i
$179$			1.03729e8i			1.35181i	0.736988	+	0.675905i	$0.236247\pi$
$179$			1.03729e8i			1.35181i	−0.736988	+	0.675905i	$0.763753\pi$
$180$	1.71164e6	−	1.52434e6i	0.0218755	−	0.0194817i
$181$			1.95838e7i			0.245484i	0.992439	+	0.122742i	$0.0391687\pi$
$181$			1.95838e7i			0.245484i	−0.992439	+	0.122742i	$0.960831\pi$
$182$	3.71477e7			0.456754
$183$	7.23116e7	−	2.75317e7i	0.872226	−	0.332089i
$184$	−6.79639e6	+	8.59909e7i	−0.0804296	+	1.01763i
$185$		−	2.90136e7i		−	0.336899i
$186$	6.16400e7	−	2.34686e7i	0.702372	−	0.267419i
$187$	−4.79852e7			−0.536614
$188$		−	5.21127e6i		−	0.0571993i
$189$	3.35519e7	−	6.45380e7i	0.361494	−	0.695343i
$190$	−7.75267e7			−0.820000
$191$	−1.41631e7			−0.147076			−0.0735380	−	0.997292i	$-0.523429\pi$
$191$	−1.41631e7			−0.147076			−0.0735380	+	0.997292i	$0.523429\pi$
$192$	−3.62960e7	−	9.53310e7i	−0.370088	−	0.972030i
$193$	8.09718e7			0.810743			0.405372	−	0.914152i	$-0.367142\pi$
$193$	8.09718e7			0.810743			0.405372	+	0.914152i	$0.367142\pi$
$194$	7.44993e7			0.732564
$195$	−1.45981e7	−	3.83418e7i	−0.140986	−	0.370298i
$196$	1.79233e6			0.0170029
$197$			5.80657e7i			0.541113i	0.962704	+	0.270556i	$0.0872077\pi$
$197$			5.80657e7i			0.541113i	−0.962704	+	0.270556i	$0.912792\pi$
$198$	1.93003e7	+	2.16718e7i	0.176700	+	0.198412i
$199$		−	5.69949e7i		−	0.512685i	−0.966586	−	0.256342i	$-0.917483\pi$
$199$		−	5.69949e7i		−	0.512685i	0.966586	−	0.256342i	$-0.0825175\pi$
$200$			6.44642e7i			0.569788i
$201$	4.15537e7	−	1.58210e7i	0.360930	−	0.137420i
$202$	1.87505e8			1.60060
$203$	−1.29728e8			−1.08842
$204$	−3.75471e6	−	9.86169e6i	−0.0309650	−	0.0813291i
$205$		−	4.37158e7i		−	0.354405i
$206$	−2.13261e8			−1.69971
$207$	1.01690e8	−	7.70988e7i	0.796864	−	0.604159i
$208$	−7.38051e7			−0.568675
$209$			4.52529e7i			0.342874i
$210$	−2.43211e7	−	6.38790e7i	−0.181224	−	0.475982i
$211$	−3.93340e7			−0.288257			−0.144128	−	0.989559i	$-0.546038\pi$
$211$	−3.93340e7			−0.288257			−0.144128	+	0.989559i	$0.546038\pi$
$212$	−5.34065e6			−0.0384963
$213$	6.20009e6	−	2.36060e6i	0.0439612	−	0.0167376i
$214$			1.56996e8i			1.09507i
$215$		−	1.48651e8i		−	1.02007i
$216$	−6.97403e7	+	1.34147e8i	−0.470864	+	0.905720i
$217$			9.06789e7i			0.602417i
$218$	−6.41045e7			−0.419074
$219$	−4.69751e7	−	1.23379e8i	−0.302213	−	0.793757i
$220$	−1.25719e6			−0.00796014
$221$	−1.88885e8			−1.17713
$222$	2.87444e7	+	7.54968e7i	0.176327	+	0.463120i
$223$	−6.94396e7			−0.419315			−0.209657	−	0.977775i	$-0.567235\pi$
$223$	−6.94396e7			−0.419315			−0.209657	+	0.977775i	$0.567235\pi$
$224$	1.16108e7			0.0690231
$225$	7.12205e7	−	6.34270e7i	0.416837	−	0.371224i
$226$		−	1.78396e8i		−	1.02803i
$227$	4.77473e7			0.270931			0.135465	−	0.990782i	$-0.456747\pi$
$227$	4.77473e7			0.270931			0.135465	+	0.990782i	$0.456747\pi$
$228$	−9.30016e6	+	3.54091e6i	−0.0519659	+	0.0197853i
$229$			2.65738e8i			1.46228i	0.682229	+	0.731138i	$0.261011\pi$
$229$			2.65738e8i			1.46228i	−0.682229	+	0.731138i	$0.738989\pi$
$230$	9.44841e6	−	1.19545e8i	0.0512049	−	0.647867i
$231$	−3.72866e7	+	1.41964e7i	−0.199027	+	0.0757768i
$232$	2.69650e8			1.41773
$233$			1.77685e8i			0.920247i	0.887855	+	0.460124i	$0.152195\pi$
$233$			1.77685e8i			0.920247i	−0.887855	+	0.460124i	$0.847805\pi$
$234$	7.59722e7	+	8.53072e7i	0.387614	+	0.435241i
$235$			1.71639e8i			0.862737i
$236$			7.21131e6i			0.0357126i
$237$	1.77922e8	−	6.77414e7i	0.868179	−	0.330548i
$238$	−3.14691e8			−1.51309
$239$			1.38960e8i			0.658412i	0.944258	+	0.329206i	$0.106781\pi$
$239$			1.38960e8i			0.658412i	−0.944258	+	0.329206i	$0.893219\pi$
$240$	4.83211e7	+	1.26915e8i	0.225631	+	0.592615i
$241$		−	5.39144e7i		−	0.248111i	−0.992275	−	0.124055i	$-0.960410\pi$
$241$		−	5.39144e7i		−	0.248111i	0.992275	−	0.124055i	$-0.0395900\pi$
$242$			1.99642e8i			0.905518i
$243$	2.16825e8	−	5.49395e7i	0.969366	−	0.245620i
$244$		−	9.33311e6i		−	0.0411303i
$245$	−5.90324e7			−0.256454
$246$	4.33103e7	+	1.13754e8i	0.185489	+	0.487184i
$247$			1.78130e8i			0.752138i
$248$		−	1.88483e8i		−	0.784678i
$249$	1.34509e8	+	3.53285e8i	0.552145	+	1.45020i
$250$		−	2.50175e8i		−	1.01264i
$251$	−2.22640e8			−0.888678			−0.444339	−	0.895859i	$-0.646561\pi$
$251$	−2.22640e8			−0.888678			−0.444339	+	0.895859i	$0.646561\pi$
$252$	−5.83515e6	−	6.55213e6i	−0.0229694	−	0.0257918i
$253$	−6.97796e7	−	5.51510e6i	−0.270898	−	0.0214108i
$254$		−	3.65572e8i		−	1.39976i
$255$	1.23666e8	+	3.24806e8i	0.467044	+	1.22668i
$256$	−3.54195e7			−0.131948
$257$		−	1.36683e8i		−	0.502284i	−0.967950	−	0.251142i	$-0.919194\pi$
$257$		−	1.36683e8i		−	0.502284i	0.967950	−	0.251142i	$-0.0808061\pi$
$258$	1.47272e8	+	3.86807e8i	0.533888	+	1.40225i
$259$	−1.11064e8			−0.397213
$260$	−4.94869e6			−0.0174616
$261$	−2.65312e8	−	2.97912e8i	−0.923666	−	1.03716i
$262$	5.17803e7			0.177873
$263$	−2.29959e8			−0.779480			−0.389740	−	0.920925i	$-0.627435\pi$
$263$	−2.29959e8			−0.779480			−0.389740	+	0.920925i	$0.627435\pi$
$264$	7.75032e7	−	2.95083e7i	0.259242	−	0.0987032i
$265$	1.75900e8			0.580638
$266$			2.96772e8i			0.966801i
$267$	1.16061e8	+	3.04832e8i	0.373161	+	0.980102i
$268$		−	5.36325e6i		−	0.0170199i
$269$		−	2.79104e8i		−	0.874245i	−0.899402	−	0.437123i	$-0.855998\pi$
$269$		−	2.79104e8i		−	0.874245i	0.899402	−	0.437123i	$-0.144002\pi$
$270$	9.69536e7	−	1.86493e8i	0.299772	−	0.576620i
$271$	2.34991e8			0.717231			0.358616	−	0.933485i	$-0.383249\pi$
$271$	2.34991e8			0.717231			0.358616	+	0.933485i	$0.383249\pi$
$272$	6.25227e8			1.88385
$273$	−1.46772e8	+	5.58816e7i	−0.436590	+	0.166226i
$274$		−	5.32744e7i		−	0.156456i
$275$	−5.23111e7			−0.151680
$276$	−4.32662e6	−	1.47723e7i	−0.0123870	−	0.0422928i
$277$	−3.56961e8			−1.00912			−0.504558	−	0.863378i	$-0.668345\pi$
$277$	−3.56961e8			−1.00912			−0.504558	+	0.863378i	$0.668345\pi$
$278$			4.37310e8i			1.22077i
$279$	−2.08238e8	+	1.85451e8i	−0.574043	+	0.511227i
$280$	−1.95330e8			−0.531759
$281$	−2.07105e8			−0.556825			−0.278412	−	0.960462i	$-0.589808\pi$
$281$	−2.07105e8			−0.556825			−0.278412	+	0.960462i	$0.589808\pi$
$282$	−1.70047e8	−	4.46625e8i	−0.451540	−	1.18596i
$283$			1.09187e8i			0.286365i	0.989696	+	0.143182i	$0.0457336\pi$
$283$			1.09187e8i			0.286365i	−0.989696	+	0.143182i	$0.954266\pi$
$284$		−	800232.i		−	0.00207301i
$285$	3.06311e8	−	1.16624e8i	0.783800	−	0.298422i
$286$		−	6.26577e7i		−	0.158377i
$287$	−1.67344e8			−0.417853
$288$	2.37457e7	+	2.66634e7i	0.0585748	+	0.0657721i
$289$	1.18977e9			2.89948
$290$	−3.74871e8			−0.902586
$291$	−2.94349e8	+	1.12070e8i	−0.700225	+	0.266602i
$292$	−1.59243e7			−0.0374301
$293$	−2.37913e8			−0.552563			−0.276281	−	0.961077i	$-0.589102\pi$
$293$	−2.37913e8			−0.552563			−0.276281	+	0.961077i	$0.589102\pi$
$294$	1.53609e8	−	5.84848e7i	0.352535	−	0.134223i
$295$		−	2.37512e8i		−	0.538653i
$296$	2.30855e8			0.517390
$297$	−1.08857e8	−	5.65925e7i	−0.241107	−	0.125346i
$298$		−	6.76308e8i		−	1.48043i
$299$	−2.74674e8	−	2.17092e7i	−0.594250	−	0.0469672i
$300$	−4.09320e6	−	1.07507e7i	−0.00875263	−	0.0229886i
$301$	−5.69033e8			−1.20269
$302$			5.66587e7i			0.118370i
$303$	−7.40840e8	+	2.82065e8i	−1.52994	+	0.582506i
$304$		−	5.89626e8i		−	1.20370i
$305$			3.07396e8i			0.620367i
$306$	−6.43586e8	−	7.22665e8i	−1.28405	−	1.44182i
$307$	−4.43756e8			−0.875307			−0.437653	−	0.899144i	$-0.644190\pi$
$307$	−4.43756e8			−0.875307			−0.437653	+	0.899144i	$0.644190\pi$
$308$			4.81251e6i			0.00938521i
$309$	8.42601e8	−	3.20809e8i	1.62468	−	0.618575i
$310$			2.62031e8i			0.499559i
$311$			3.82922e8i			0.721853i	0.932594	+	0.360927i	$0.117540\pi$
$311$			3.82922e8i			0.721853i	−0.932594	+	0.360927i	$0.882460\pi$
$312$	3.05077e8	−	1.16154e8i	0.568681	−	0.216518i
$313$		−	8.04484e8i		−	1.48290i	−0.671008	−	0.741450i	$-0.734138\pi$
$313$		−	8.04484e8i		−	1.48290i	0.671008	−	0.741450i	$-0.265862\pi$
$314$	4.99419e8			0.910356
$315$	1.92187e8	+	2.15802e8i	0.346447	+	0.389017i
$316$		−	2.29640e7i		−	0.0409395i
$317$			4.52804e8i			0.798367i	0.916871	+	0.399184i	$0.130706\pi$
$317$			4.52804e8i			0.798367i	−0.916871	+	0.399184i	$0.869294\pi$
$318$	−4.57713e8	+	1.74269e8i	−0.798176	+	0.303895i
$319$			2.18815e8i			0.377406i
$320$	4.05251e8			0.691353
$321$	−2.36171e8	−	6.20298e8i	−0.398527	−	1.04673i
$322$	−4.57619e8	−	3.61685e7i	−0.763851	−	0.0603718i
$323$		−	1.50900e9i		−	2.49161i
$324$	3.11283e6	−	2.68000e7i	0.00508450	−	0.0437752i
$325$	−2.05913e8			−0.332730
$326$		−	8.93814e8i		−	1.42885i
$327$	2.53279e8	−	9.64328e7i	0.400574	−	0.152513i
$328$	3.47837e8			0.544274
$329$	6.57033e8			1.01719
$330$	−1.07746e8	+	4.10228e7i	−0.165045	+	0.0628386i
$331$	1.03898e9			1.57474			0.787368	−	0.616483i	$-0.211443\pi$
$331$	1.03898e9			1.57474			0.787368	+	0.616483i	$0.211443\pi$
$332$	4.55978e7			0.0683850
$333$	−2.27141e8	−	2.55050e8i	−0.337085	−	0.378504i
$334$	−9.16680e7			−0.134619
$335$			1.76644e8i			0.256710i
$336$	4.85829e8	−	1.84973e8i	0.698708	−	0.266024i
$337$			3.11031e8i			0.442690i	0.975196	+	0.221345i	$0.0710446\pi$
$337$			3.11031e8i			0.442690i	−0.975196	+	0.221345i	$0.928955\pi$
$338$			4.47459e8i			0.630296i
$339$	2.68362e8	+	7.04848e8i	0.374129	+	0.982645i
$340$	4.19220e7			0.0578450
$341$	1.52949e8			0.208885
$342$	−6.81516e8	+	6.06939e8i	−0.921265	+	0.820453i
$343$			8.11679e8i			1.08606i
$344$	1.18278e9			1.56657
$345$	1.42502e8	+	4.86542e8i	0.186833	+	0.637901i
$346$	−1.17052e8			−0.151920
$347$			3.93038e8i			0.504989i	0.967598	+	0.252494i	$0.0812510\pi$
$347$			3.93038e8i			0.504989i	−0.967598	+	0.252494i	$0.918749\pi$
$348$	−4.49697e7	+	1.71216e7i	−0.0571996	+	0.0217780i
$349$	−9.20996e8			−1.15976			−0.579881	−	0.814701i	$-0.696901\pi$
$349$	−9.20996e8			−1.15976			−0.579881	+	0.814701i	$0.696901\pi$
$350$	−3.43060e8			−0.427693
$351$	−4.28497e8	−	2.22766e8i	−0.528899	−	0.274963i
$352$		−	1.95841e7i		−	0.0239334i
$353$		−	1.83204e8i		−	0.221679i	−0.993838	−	0.110839i	$-0.964646\pi$
$353$		−	1.83204e8i		−	0.221679i	0.993838	−	0.110839i	$-0.0353539\pi$
$354$	2.35309e8	+	6.18036e8i	0.281921	+	0.740461i
$355$			2.63565e7i			0.0312672i
$356$	3.93441e7			0.0462173
$357$	1.24335e9	−	4.73391e8i	1.44629	−	0.550657i
$358$	1.14741e9			1.32169
$359$	7.22219e8			0.823831			0.411916	−	0.911222i	$-0.364860\pi$
$359$	7.22219e8			0.823831			0.411916	+	0.911222i	$0.364860\pi$
$360$	−3.99476e8	−	4.48561e8i	−0.451265	−	0.506714i
$361$	−5.29201e8			−0.592032
$362$	2.16629e8			0.240014
$363$	−3.00322e8	−	7.88791e8i	−0.329544	−	0.865543i
$364$			1.89436e7i			0.0205877i
$365$	5.24485e8			0.564557
$366$	−3.04545e8	−	7.99882e8i	−0.324689	−	0.852790i
$367$			8.54473e8i			0.902334i	0.892440	+	0.451167i	$0.148992\pi$
$367$			8.54473e8i			0.902334i	−0.892440	+	0.451167i	$0.851008\pi$
$368$	9.09198e8	+	7.18594e7i	0.951023	+	0.0751651i
$369$	−3.42241e8	−	3.84293e8i	−0.354601	−	0.398172i
$370$	−3.20937e8			−0.329392
$371$		−	6.73345e8i		−	0.684587i
$372$	1.19679e7	+	3.14334e7i	0.0120536	+	0.0316586i
$373$			3.16488e8i			0.315773i	0.987457	+	0.157887i	$0.0504681\pi$
$373$			3.16488e8i			0.315773i	−0.987457	+	0.157887i	$0.949532\pi$
$374$			5.30794e8i			0.524657i
$375$	3.76340e8	+	9.88451e8i	0.368529	+	0.967934i
$376$	−1.36569e9			−1.32494
$377$			8.61324e8i			0.827889i
$378$	−7.13894e8	−	3.71138e8i	−0.679849	−	0.353439i
$379$			1.68250e9i			1.58751i	0.608235	+	0.793757i	$0.291878\pi$
$379$			1.68250e9i			1.58751i	−0.608235	+	0.793757i	$0.708122\pi$
$380$		−	3.95349e7i		−	0.0369605i
$381$	5.49932e8	+	1.44439e9i	0.509415	+	1.33797i
$382$			1.56667e8i			0.143799i
$383$	−3.22258e8			−0.293094			−0.146547	−	0.989204i	$-0.546816\pi$
$383$	−3.22258e8			−0.293094			−0.146547	+	0.989204i	$0.546816\pi$
$384$	−9.63184e8	+	3.66720e8i	−0.868060	+	0.330503i
$385$		−	1.58505e8i		−	0.141557i
$386$		−	8.95678e8i		−	0.792678i
$387$	−1.16375e9	−	1.30675e9i	−1.02064	−	1.14605i
$388$			3.79911e7i			0.0330195i
$389$	−3.06377e8			−0.263895			−0.131948	−	0.991257i	$-0.542123\pi$
$389$	−3.06377e8			−0.263895			−0.131948	+	0.991257i	$0.542123\pi$
$390$	−4.24121e8	+	1.61479e8i	−0.362046	+	0.137844i
$391$	2.32686e9	+	1.83906e8i	1.96857	+	0.155588i
$392$		−	4.69708e8i		−	0.393847i
$393$	−2.04586e8	+	7.78934e7i	−0.170021	+	0.0647331i
$394$	6.42300e8			0.529055
$395$			7.56344e8i			0.617489i
$396$	−1.10516e7	+	9.84224e6i	−0.00894317	+	0.00796454i
$397$	−7.25707e8			−0.582096			−0.291048	−	0.956708i	$-0.594004\pi$
$397$	−7.25707e8			−0.582096			−0.291048	+	0.956708i	$0.594004\pi$
$398$	−6.30455e8			−0.501261
$399$	−4.46436e8	−	1.17256e9i	−0.351847	−	0.924120i
$400$	6.81591e8			0.532493
$401$	−2.33660e9			−1.80958			−0.904792	−	0.425855i	$-0.859974\pi$
$401$	−2.33660e9			−1.80958			−0.904792	+	0.425855i	$0.859974\pi$
$402$	−1.75006e8	−	4.59650e8i	−0.134357	−	0.352887i
$403$	6.02058e8			0.458216
$404$			9.56187e7i			0.0721453i
$405$	−1.02525e8	+	8.82688e8i	−0.0766894	+	0.660260i
$406$			1.43500e9i			1.06417i
$407$			1.87333e8i			0.137732i
$408$	−2.58441e9	+	9.83981e8i	−1.88387	+	0.717259i
$409$	−4.35075e8			−0.314436			−0.157218	−	0.987564i	$-0.550253\pi$
$409$	−4.35075e8			−0.314436			−0.157218	+	0.987564i	$0.550253\pi$
$410$	−4.83567e8			−0.346508
$411$	8.01410e7	+	2.10489e8i	0.0569388	+	0.149549i
$412$		−	1.08753e8i		−	0.0766126i
$413$	−9.09196e8			−0.635086
$414$	−8.52836e8	−	1.12486e9i	−0.590696	−	0.779107i
$415$	−1.50181e9			−1.03145
$416$		−	7.70894e7i		−	0.0525010i
$417$	−6.57849e8	−	1.72783e9i	−0.444273	−	1.16688i
$418$	5.00570e8			0.335234
$419$	−4.37889e7			−0.0290814			−0.0145407	−	0.999894i	$-0.504629\pi$
$419$	−4.37889e7			−0.0290814			−0.0145407	+	0.999894i	$0.504629\pi$
$420$	3.25752e7	−	1.24026e7i	0.0214543	−	0.00816846i
$421$		−	8.04693e8i		−	0.525585i	−0.964852	−	0.262792i	$-0.915357\pi$
$421$		−	8.04693e8i		−	0.525585i	0.964852	−	0.262792i	$-0.0846435\pi$
$422$			4.35097e8i			0.281834i
$423$	1.34372e9	+	1.50883e9i	0.863213	+	0.969279i
$424$			1.39960e9i			0.891710i
$425$	1.74436e9			1.10224
$426$	−2.61121e7	−	6.85829e7i	−0.0163647	−	0.0429816i
$427$	1.17671e9			0.731429
$428$	−8.00606e7			−0.0493589
$429$	9.42563e7	+	2.47563e8i	0.0576381	+	0.151386i
$430$	−1.64431e9			−0.997344
$431$	1.15215e9			0.693170			0.346585	−	0.938018i	$-0.387341\pi$
$431$	1.15215e9			0.693170			0.346585	+	0.938018i	$0.387341\pi$
$432$	1.41836e9	+	7.37376e8i	0.846437	+	0.440044i
$433$		−	2.56217e9i		−	1.51670i	−0.651846	−	0.758352i	$-0.726005\pi$
$433$		−	2.56217e9i		−	1.51670i	0.651846	−	0.758352i	$-0.273995\pi$
$434$	1.00305e9			0.588993
$435$	1.48113e9	−	5.63920e8i	0.862740	−	0.328477i
$436$		−	3.26902e7i		−	0.0188893i
$437$	1.73434e8	−	2.19436e9i	0.0994144	−	1.25784i
$438$	−1.36477e9	+	5.19620e8i	−0.776070	+	0.295479i
$439$	1.61026e9			0.908383			0.454192	−	0.890904i	$-0.349928\pi$
$439$	1.61026e9			0.908383			0.454192	+	0.890904i	$0.349928\pi$
$440$			3.29466e8i			0.184385i
$441$	−5.18937e8	+	4.62151e8i	−0.288124	+	0.256596i
$442$			2.08937e9i			1.15090i
$443$			1.26293e9i			0.690188i	0.938568	+	0.345094i	$0.112153\pi$
$443$			1.26293e9i			0.690188i	−0.938568	+	0.345094i	$0.887847\pi$
$444$	−3.84998e7	+	1.46583e7i	−0.0208746	+	0.00794773i
$445$	−1.29584e9			−0.697094
$446$			7.68113e8i			0.409971i
$447$	1.01737e9	+	2.67212e9i	0.538771	+	1.41507i
$448$		−	1.55130e9i		−	0.815122i
$449$		−	9.12707e8i		−	0.475849i	−0.971284	−	0.237925i	$-0.923533\pi$
$449$		−	9.12707e8i		−	0.475849i	0.971284	−	0.237925i	$-0.0764671\pi$
$450$	−7.01605e8	−	7.87813e8i	−0.362952	−	0.407549i
$451$			2.82262e8i			0.144888i
$452$	9.09733e7			0.0463372
$453$	−8.52321e7	−	2.23861e8i	−0.0430784	−	0.113145i
$454$		−	5.28162e8i		−	0.264894i
$455$		−	6.23927e8i		−	0.310523i
$456$	9.27952e8	+	2.43725e9i	0.458299	+	1.20371i
$457$		−	3.22166e9i		−	1.57896i	−0.613773	−	0.789482i	$-0.710349\pi$
$457$		−	3.22166e9i		−	1.57896i	0.613773	−	0.789482i	$-0.289651\pi$
$458$	2.93949e9			1.42969
$459$	3.62994e9	+	1.88713e9i	1.75208	+	0.910870i
$460$	6.09625e7	+	4.81824e6i	0.0292018	+	0.00230800i
$461$			2.63129e9i			1.25088i	0.780272	+	0.625441i	$0.215081\pi$
$461$			2.63129e9i			1.25088i	−0.780272	+	0.625441i	$0.784919\pi$
$462$	1.57035e8	+	4.12450e8i	0.0740883	+	0.194592i
$463$	−3.83677e9			−1.79652			−0.898260	−	0.439464i	$-0.855168\pi$
$463$	−3.83677e9			−1.79652			−0.898260	+	0.439464i	$0.855168\pi$
$464$		−	2.85106e9i		−	1.32493i
$465$	−3.94175e8	−	1.03529e9i	−0.181804	−	0.477506i
$466$	1.96548e9			0.899741
$467$	−1.29170e9			−0.586882			−0.293441	−	0.955977i	$-0.594800\pi$
$467$	−1.29170e9			−0.586882			−0.293441	+	0.955977i	$0.594800\pi$
$468$	−4.35026e7	+	3.87422e7i	−0.0196180	+	0.0174712i
$469$	6.76194e8			0.302668
$470$	1.89860e9			0.843512
$471$	−1.97322e9	+	7.51279e8i	−0.870167	+	0.331305i
$472$	1.88984e9			0.827231
$473$			9.59798e8i			0.417029i
$474$	−7.49328e8	−	1.96810e9i	−0.323183	−	0.848834i
$475$		−	1.64503e9i		−	0.704283i
$476$		−	1.60477e8i		−	0.0682007i
$477$	1.54629e9	−	1.37708e9i	0.652343	−	0.580959i
$478$	1.53712e9			0.643741
$479$	7.70698e7			0.0320413			0.0160206	−	0.999872i	$-0.494900\pi$
$479$	7.70698e7			0.0320413			0.0160206	+	0.999872i	$0.494900\pi$
$480$	−1.32562e8	+	5.04714e7i	−0.0547112	+	0.0208306i
$481$			7.37402e8i			0.302132i
$482$	−5.96380e8			−0.242582
$483$	1.86248e9	−	5.45497e8i	0.752101	−	0.220281i
$484$	−1.01808e8			−0.0408152
$485$		−	1.25128e9i		−	0.498032i
$486$	−6.07719e8	−	2.39844e9i	−0.240146	−	0.947766i
$487$	5.32999e7			0.0209110			0.0104555	−	0.999945i	$-0.496672\pi$
$487$	5.32999e7			0.0209110			0.0104555	+	0.999945i	$0.496672\pi$
$488$	−2.44589e9			−0.952723
$489$	1.34457e9	+	3.53149e9i	0.519999	+	1.36577i
$490$			6.52993e8i			0.250739i
$491$			1.48828e8i			0.0567412i	0.999597	+	0.0283706i	$0.00903185\pi$
$491$			1.48828e8i			0.0567412i	−0.999597	+	0.0283706i	$0.990968\pi$
$492$	−5.80090e7	+	2.20862e7i	−0.0219593	+	0.00836070i
$493$		−	7.29656e9i		−	2.74255i
$494$	1.97040e9			0.735378
$495$	3.63996e8	−	3.24165e8i	0.134890	−	0.120129i
$496$	−1.99287e9			−0.733318
$497$	1.00893e8			0.0368648
$498$	3.90790e9	−	1.48788e9i	1.41788	−	0.539841i
$499$	−5.23308e9			−1.88541			−0.942705	−	0.333628i	$-0.891727\pi$
$499$	−5.23308e9			−1.88541			−0.942705	+	0.333628i	$0.891727\pi$
$500$	1.27577e8			0.0456435
$501$	3.62183e8	−	1.37897e8i	0.128676	−	0.0489916i
$502$			2.46275e9i			0.868875i
$503$	−2.99444e9			−1.04912			−0.524562	−	0.851372i	$-0.675771\pi$
$503$	−2.99444e9			−1.04912			−0.524562	+	0.851372i	$0.675771\pi$
$504$	−1.71709e9	+	1.52919e9i	−0.597428	+	0.532053i
$505$		−	3.14931e9i		−	1.08817i
$506$	−6.10059e7	+	7.71874e8i	−0.0209337	+	0.264862i
$507$	−6.73115e8	−	1.76793e9i	−0.229383	−	0.602471i
$508$	1.86424e8			0.0630927
$509$		−	5.01978e9i		−	1.68722i	−0.536955	−	0.843611i	$-0.680425\pi$
$509$		−	5.01978e9i		−	1.68722i	0.536955	−	0.843611i	$-0.319575\pi$
$510$	3.59287e9	−	1.36794e9i	1.19935	−	0.456637i
$511$		−	2.00772e9i		−	0.665627i
$512$			3.21271e9i			1.05786i
$513$	1.77967e9	−	3.42325e9i	0.582008	−	1.11951i
$514$	−1.51194e9			−0.491091
$515$			3.58189e9i			1.15555i
$516$	−1.97253e8	+	7.51015e7i	−0.0632047	+	0.0240644i
$517$		−	1.10823e9i		−	0.352705i
$518$			1.22854e9i			0.388362i
$519$	4.62478e8	−	1.76083e8i	0.145213	−	0.0552880i
$520$			1.29688e9i			0.404472i
$521$	−5.44957e9			−1.68822			−0.844112	−	0.536168i	$-0.819871\pi$
$521$	−5.44957e9			−1.68822			−0.844112	+	0.536168i	$0.819871\pi$
$522$	−3.29538e9	+	2.93478e9i	−1.01405	+	0.903084i
$523$		−	1.68102e9i		−	0.513827i	−0.966434	−	0.256914i	$-0.917294\pi$
$523$		−	1.68102e9i		−	0.513827i	0.966434	−	0.256914i	$-0.0827056\pi$
$524$			2.64055e7i			0.00801741i
$525$	1.35544e9	−	5.16067e8i	0.408812	−	0.155650i
$526$			2.54372e9i			0.762111i
$527$	−5.10023e9			−1.51793
$528$	−3.11997e8	−	8.19455e8i	−0.0922426	−	0.242274i
$529$	3.36255e9	+	5.34867e8i	0.987584	+	0.157091i
$530$		−	1.94574e9i		−	0.567700i
$531$	−1.85943e9	−	2.08790e9i	−0.538951	−	0.605173i
$532$	−1.51339e8			−0.0435774
$533$			1.11107e9i			0.317831i
$534$	3.37193e9	−	1.28382e9i	0.958262	−	0.364846i
$535$	2.63688e9			0.744480
$536$	−1.40552e9			−0.394240
$537$	−4.53347e9	+	1.72606e9i	−1.26334	+	0.481001i
$538$	−3.08734e9			−0.854764
$539$	3.81157e8			0.104844
$540$	9.51025e7	+	4.94417e7i	0.0259904	+	0.0135119i
$541$	5.01692e9			1.36222			0.681109	−	0.732182i	$-0.261498\pi$
$541$	5.01692e9			1.36222			0.681109	+	0.732182i	$0.261498\pi$
$542$		−	2.59938e9i		−	0.701249i
$543$	−8.55908e8	+	3.25876e8i	−0.229418	+	0.0873480i
$544$			6.53049e8i			0.173920i
$545$			1.07669e9i			0.284907i
$546$	6.18140e8	+	1.62353e9i	0.162522	+	0.426862i
$547$	1.41042e9			0.368461			0.184231	−	0.982883i	$-0.441021\pi$
$547$	1.41042e9			0.368461			0.184231	+	0.982883i	$0.441021\pi$
$548$	2.71674e7			0.00705206
$549$	2.40653e9	+	2.70223e9i	0.620710	+	0.696979i
$550$			5.78645e8i			0.148301i
$551$	−6.88109e9			−1.75237
$552$	−3.87131e9	+	1.13386e9i	−0.979650	+	0.286927i
$553$	2.89528e9			0.728036
$554$			3.94856e9i			0.986630i
$555$	1.26803e9	−	4.82787e8i	0.314851	−	0.119875i
$556$	−2.23007e8			−0.0550247
$557$	4.64988e9			1.14012			0.570058	−	0.821605i	$-0.306921\pi$
$557$	4.64988e9			1.14012			0.570058	+	0.821605i	$0.306921\pi$
$558$	2.05138e9	+	2.30344e9i	0.499835	+	0.561252i
$559$			3.77807e9i			0.914805i
$560$			2.06526e9i			0.496954i
$561$	−7.98476e8	−	2.09719e9i	−0.190938	−	0.501495i
$562$			2.29091e9i			0.544417i
$563$	−2.07580e9			−0.490236			−0.245118	−	0.969493i	$-0.578827\pi$
$563$	−2.07580e9			−0.490236			−0.245118	+	0.969493i	$0.578827\pi$
$564$	2.27757e8	−	8.67157e7i	0.0534559	−	0.0203527i
$565$	−2.99631e9			−0.698902
$566$	1.20779e9			0.279984
$567$	3.37893e9	+	3.92464e8i	0.778463	+	0.0904188i
$568$	−2.09713e8			−0.0480183
$569$	−5.93889e8			−0.135149			−0.0675744	−	0.997714i	$-0.521526\pi$
$569$	−5.93889e8			−0.135149			−0.0675744	+	0.997714i	$0.521526\pi$
$570$	−1.29005e9	−	3.38829e9i	−0.291772	−	0.766335i
$571$			6.06474e9i			1.36328i	0.731687	+	0.681641i	$0.238733\pi$
$571$			6.06474e9i			1.36328i	−0.731687	+	0.681641i	$0.761267\pi$
$572$	3.19524e7			0.00713868
$573$	−2.35675e8	−	6.18995e8i	−0.0523325	−	0.137450i
$574$			1.85109e9i			0.408542i
$575$	2.53662e9	+	2.00485e8i	0.556441	+	0.0439789i
$576$	3.56245e9	−	3.17262e9i	0.776730	−	0.691735i
$577$	−1.35186e9			−0.292965			−0.146483	−	0.989213i	$-0.546795\pi$
$577$	−1.35186e9			−0.292965			−0.146483	+	0.989213i	$0.546795\pi$
$578$		−	1.31608e10i		−	2.83487i
$579$	1.34737e9	+	3.53886e9i	0.288478	+	0.757684i
$580$		−	1.91166e8i		−	0.0406830i
$581$			5.74893e9i			1.21610i
$582$	1.23967e9	+	3.25597e9i	0.260661	+	0.684622i
$583$	−1.13574e9			−0.237377
$584$			4.17321e9i			0.867013i
$585$	1.43280e9	−	1.27602e9i	0.295898	−	0.263518i
$586$			2.63170e9i			0.540250i
$587$		−	3.46372e9i		−	0.706822i	−0.935468	−	0.353411i	$-0.885022\pi$
$587$		−	3.46372e9i		−	0.706822i	0.935468	−	0.353411i	$-0.114978\pi$
$588$	2.98245e7	+	7.83335e7i	0.00604995	+	0.0158901i
$589$			4.80982e9i			0.969896i
$590$	−2.62727e9			−0.526650
$591$	−2.53775e9	+	9.66216e8i	−0.505700	+	0.192539i
$592$		−	2.44087e9i		−	0.483524i
$593$		−	8.17586e9i		−	1.61006i	−0.593234	−	0.805030i	$-0.702149\pi$
$593$		−	8.17586e9i		−	1.61006i	0.593234	−	0.805030i	$-0.297851\pi$
$594$	−6.26004e8	+	1.20414e9i	−0.122553	+	0.235735i
$595$			5.28549e9i			1.02867i
$596$	3.44885e8			0.0667286
$597$	2.49095e9	−	9.48398e8i	0.479132	−	0.182423i
$598$	−2.40139e8	+	3.03834e9i	−0.0459206	+	0.581008i
$599$			7.53755e8i			0.143297i	0.997430	+	0.0716484i	$0.0228259\pi$
$599$			7.53755e8i			0.143297i	−0.997430	+	0.0716484i	$0.977174\pi$
$600$	−2.81739e9	+	1.07269e9i	−0.532498	+	0.202742i
$601$	−4.33968e9			−0.815450			−0.407725	−	0.913105i	$-0.633678\pi$
$601$	−4.33968e9			−0.815450			−0.407725	+	0.913105i	$0.633678\pi$
$602$			6.29442e9i			1.17589i
$603$	1.38291e9	+	1.55283e9i	0.256852	+	0.288412i
$604$	−2.88933e7			−0.00533540
$605$	3.35315e9			0.615614
$606$	3.12009e9	+	8.19488e9i	0.569526	+	1.49585i
$607$	1.60399e9			0.291099			0.145550	−	0.989351i	$-0.453505\pi$
$607$	1.60399e9			0.291099			0.145550	+	0.989351i	$0.453505\pi$
$608$	6.15864e8			0.111128
$609$	−2.15868e9	−	5.66975e9i	−0.387283	−	1.01719i
$610$	3.40030e9			0.606544
$611$		−	4.36234e9i		−	0.773704i
$612$	3.68525e8	−	3.28198e8i	0.0649885	−	0.0578770i
$613$		−	3.10080e9i		−	0.543703i	−0.962339	−	0.271852i	$-0.912364\pi$
$613$		−	3.10080e9i		−	0.543703i	0.962339	−	0.271852i	$-0.0876360\pi$
$614$			4.90866e9i			0.855802i
$615$	1.91059e9	−	7.27433e8i	0.331211	−	0.126104i
$616$	1.26119e9			0.217395
$617$	−3.45170e8			−0.0591609			−0.0295804	−	0.999562i	$-0.509417\pi$
$617$	−3.45170e8			−0.0591609			−0.0295804	+	0.999562i	$0.509417\pi$
$618$	−3.54867e9	−	9.32052e9i	−0.604791	−	1.58847i
$619$		−	4.42114e9i		−	0.749233i	−0.927180	−	0.374616i	$-0.877774\pi$
$619$		−	4.42114e9i		−	0.749233i	0.927180	−	0.374616i	$-0.122226\pi$
$620$	−1.33623e8			−0.0225170
$621$	5.06172e9	+	3.16144e9i	0.848159	+	0.529741i
$622$	4.23573e9			0.705768
$623$			4.96047e9i			0.821892i
$624$	−1.22812e9	−	3.22564e9i	−0.202346	−	0.531458i
$625$	−7.95068e8			−0.130264
$626$	−8.89888e9			−1.44986
$627$	−1.97777e9	+	7.53010e8i	−0.320434	+	0.122001i
$628$			2.54680e8i			0.0410332i
$629$		−	6.24678e9i		−	1.00087i
$630$	2.38711e9	−	2.12590e9i	0.380348	−	0.338728i
$631$			1.16448e10i			1.84515i	0.385823	+	0.922573i	$0.373918\pi$
$631$			1.16448e10i			1.84515i	−0.385823	+	0.922573i	$0.626082\pi$
$632$	−6.01807e9			−0.948303
$633$	−6.54519e8	−	1.71908e9i	−0.102567	−	0.269392i
$634$	5.00874e9			0.780577
$635$	−6.14009e9			−0.951626
$636$	−8.88686e7	−	2.33412e8i	−0.0136977	−	0.0359769i
$637$	1.50035e9			0.229988
$638$	2.42044e9			0.368997
$639$	2.06339e8	+	2.31693e8i	0.0312845	+	0.0351285i
$640$		−	4.09449e9i		−	0.617405i
$641$	1.12345e10			1.68482			0.842408	−	0.538840i	$-0.181137\pi$
$641$	1.12345e10			1.68482			0.842408	+	0.538840i	$0.181137\pi$
$642$	−6.86149e9	+	2.61243e9i	−1.02340	+	0.389647i
$643$			1.04628e10i			1.55206i	0.630696	+	0.776030i	$0.282769\pi$
$643$			1.04628e10i			1.55206i	−0.630696	+	0.776030i	$0.717231\pi$
$644$	1.84442e7	−	2.33364e8i	0.00272119	−	0.0344297i
$645$	6.49674e9	−	2.47355e9i	0.953315	−	0.362963i
$646$	−1.66919e10			−2.43609
$647$			6.19843e9i			0.899739i	0.893094	+	0.449869i	$0.148529\pi$
$647$			6.19843e9i			0.899739i	−0.893094	+	0.449869i	$0.851471\pi$
$648$	−7.02336e9	−	8.15767e8i	−1.01399	−	0.117775i
$649$			1.53356e9i			0.220213i
$650$			2.27773e9i			0.325316i
$651$	−3.96310e9	+	1.50890e9i	−0.562991	+	0.214352i
$652$	4.55802e8			0.0644036
$653$			1.37964e10i			1.93896i	0.245163	+	0.969482i	$0.421158\pi$
$653$			1.37964e10i			1.93896i	−0.245163	+	0.969482i	$0.578842\pi$
$654$	−1.06670e9	−	2.80168e9i	−0.149115	−	0.391648i
$655$		−	8.69693e8i		−	0.120926i
$656$		−	3.67775e9i		−	0.508649i
$657$	4.61060e9	−	4.10607e9i	0.634276	−	0.564869i
$658$		−	7.26784e9i		−	0.994522i
$659$	8.89344e9			1.21052			0.605258	−	0.796029i	$-0.293070\pi$
$659$	8.89344e9			1.21052			0.605258	+	0.796029i	$0.293070\pi$
$660$	−2.09197e7	−	5.49452e7i	−0.00283238	−	0.00743919i
$661$			3.61008e9i			0.486197i	0.970002	+	0.243099i	$0.0781638\pi$
$661$			3.61008e9i			0.486197i	−0.970002	+	0.243099i	$0.921836\pi$
$662$		−	1.14928e10i		−	1.53965i
$663$	−3.14306e9	−	8.25519e9i	−0.418846	−	1.10009i
$664$		−	1.19496e10i		−	1.58404i
$665$	4.98453e9			0.657277
$666$	−2.82126e9	+	2.51254e9i	−0.370070	+	0.329574i
$667$	8.38618e8	−	1.06106e10i	0.109427	−	1.38452i
$668$		−	4.67463e7i		−	0.00606777i
$669$	−1.15548e9	−	3.03484e9i	−0.149200	−	0.391872i
$670$	1.95397e9			0.250990
$671$		−	1.98478e9i		−	0.253620i
$672$	1.93204e8	+	5.07448e8i	0.0245598	+	0.0645058i
$673$	−5.77186e9			−0.729900			−0.364950	−	0.931027i	$-0.618914\pi$
$673$	−5.77186e9			−0.729900			−0.364950	+	0.931027i	$0.618914\pi$
$674$	3.44050e9			0.432825
$675$	3.95718e9	+	2.05725e9i	0.495248	+	0.257468i
$676$	−2.28183e8			−0.0284099
$677$	−4.15073e9			−0.514119			−0.257060	−	0.966396i	$-0.582754\pi$
$677$	−4.15073e9			−0.514119			−0.257060	+	0.966396i	$0.582754\pi$
$678$	7.79675e9	−	2.96851e9i	0.960748	−	0.365793i
$679$	−4.78988e9			−0.587193
$680$		−	1.09863e10i		−	1.33989i
$681$	7.94517e8	+	2.08679e9i	0.0964025	+	0.253200i
$682$		−	1.69187e9i		−	0.204231i
$683$			1.45007e9i			0.174147i	0.996202	+	0.0870734i	$0.0277515\pi$
$683$			1.45007e9i			0.174147i	−0.996202	+	0.0870734i	$0.972249\pi$
$684$	−3.09510e8	−	3.47540e8i	−0.0369810	−	0.0415249i
$685$	−8.94788e8			−0.106366
$686$	8.97847e9			1.06186
$687$	−1.16140e10	+	4.42189e9i	−1.36658	+	0.520307i
$688$		−	1.25058e10i		−	1.46403i
$689$	−4.47064e9			−0.520718
$690$	5.38193e9	−	1.57630e9i	0.623687	−	0.182670i
$691$	1.29878e10			1.49748			0.748740	−	0.662864i	$-0.230659\pi$
$691$	1.29878e10			1.49748			0.748740	+	0.662864i	$0.230659\pi$
$692$		−	5.96911e7i		−	0.00684760i
$693$	−1.24090e9	−	1.39338e9i	−0.141635	−	0.159038i
$694$	4.34764e9			0.493736
$695$	7.34499e9			0.829936
$696$	4.48699e9	+	1.17850e10i	0.504456	+	1.32494i
$697$		−	9.41225e9i		−	1.05288i
$698$			1.01877e10i			1.13392i
$699$	−7.76568e9	+	2.95668e9i	−0.860021	+	0.327442i
$700$		−	1.74944e8i		−	0.0192778i
$701$	1.70840e10			1.87317			0.936585	−	0.350441i	$-0.113968\pi$
$701$	1.70840e10			1.87317			0.936585	+	0.350441i	$0.113968\pi$
$702$	−2.46415e9	+	4.73986e9i	−0.268836	+	0.517114i
$703$	−5.89108e9			−0.639516
$704$	−2.61660e9			−0.282640
$705$	−7.50144e9	+	2.85608e9i	−0.806274	+	0.306979i
$706$	−2.02653e9			−0.216739
$707$	−1.20555e10			−1.28298
$708$	−3.15169e8	+	1.19996e8i	−0.0333754	+	0.0127073i
$709$			1.99076e9i			0.209777i	0.994484	+	0.104888i	$0.0334486\pi$
$709$			1.99076e9i			0.209777i	−0.994484	+	0.104888i	$0.966551\pi$
$710$	2.91546e8			0.0305705
$711$	5.92125e9	+	6.64881e9i	0.617831	+	0.693746i
$712$		−	1.03107e10i		−	1.07056i
$713$	−7.41669e9	−	5.86187e8i	−0.766297	−	0.0605651i
$714$	−5.23647e9	−	1.37535e10i	−0.538387	−	1.41406i
$715$	−1.05239e9			−0.107672
$716$			5.85126e8i			0.0595736i
$717$	−6.07323e9	+	2.31230e9i	−0.615322	+	0.234276i
$718$		−	7.98890e9i		−	0.805474i
$719$			1.38828e10i			1.39292i	0.717596	+	0.696460i	$0.245242\pi$
$719$			1.38828e10i			1.39292i	−0.717596	+	0.696460i	$0.754758\pi$
$720$	−4.74271e9	+	4.22373e9i	−0.473547	+	0.421728i
$721$	1.37115e10			1.36242
$722$			5.85381e9i			0.578840i
$723$	2.35632e9	−	8.97138e8i	0.231873	−	0.0882826i
$724$			1.10470e8i			0.0108183i
$725$		−	7.95435e9i		−	0.775214i
$726$	−8.72529e9	+	3.32204e9i	−0.846256	+	0.322201i
$727$			1.97392e9i			0.190528i	0.995452	+	0.0952639i	$0.0303695\pi$
$727$			1.97392e9i			0.190528i	−0.995452	+	0.0952639i	$0.969631\pi$
$728$	4.96446e9			0.476883
$729$	6.00910e9	+	8.56211e9i	0.574464	+	0.818530i
$730$		−	5.80164e9i		−	0.551977i
$731$		−	3.20053e10i		−	3.03047i
$732$	4.07902e8	−	1.55303e8i	0.0384385	−	0.0146350i
$733$			4.43322e9i			0.415772i	0.978153	+	0.207886i	$0.0666583\pi$
$733$			4.43322e9i			0.415772i	−0.978153	+	0.207886i	$0.933342\pi$
$734$	9.45184e9			0.882227
$735$	−9.82302e8	−	2.58000e9i	−0.0912513	−	0.239670i
$736$	−7.50572e7	+	9.49657e8i	−0.00693937	+	0.0877999i
$737$		−	1.14055e9i		−	0.104949i
$738$	−4.25090e9	+	3.78574e9i	−0.389299	+	0.346699i
$739$	7.04118e9			0.641786			0.320893	−	0.947115i	$-0.396017\pi$
$739$	7.04118e9			0.641786			0.320893	+	0.947115i	$0.396017\pi$
$740$		−	1.63662e8i		−	0.0148470i
$741$	−7.78513e9	+	2.96409e9i	−0.702914	+	0.267625i
$742$	−7.44827e9			−0.669333
$743$	2.20218e9			0.196966			0.0984832	−	0.995139i	$-0.468601\pi$
$743$	2.20218e9			0.196966			0.0984832	+	0.995139i	$0.468601\pi$
$744$	8.23762e9	−	3.13637e9i	0.733325	−	0.279204i
$745$	−1.13592e10			−1.00647
$746$	3.50086e9			0.308737
$747$	−1.32020e10	+	1.17574e10i	−1.15883	+	1.03202i
$748$	−2.70679e8			−0.0236483
$749$		−	1.00940e10i		−	0.877761i
$750$	1.09339e10	−	4.16293e9i	0.946366	−	0.360317i
$751$		−	1.92238e9i		−	0.165615i	−0.996566	−	0.0828073i	$-0.973611\pi$
$751$		−	1.92238e9i		−	0.165615i	0.996566	−	0.0828073i	$-0.0263886\pi$
$752$			1.44397e10i			1.23822i
$753$	−3.70473e9	−	9.73042e9i	−0.316209	−	0.830518i
$754$	9.52763e9			0.809441
$755$	9.51631e8			0.0804738
$756$	1.89263e8	−	3.64052e8i	0.0159308	−	0.0306434i
$757$			1.54080e10i			1.29096i	0.763779	+	0.645478i	$0.223342\pi$
$757$			1.54080e10i			1.29096i	−0.763779	+	0.645478i	$0.776658\pi$
$758$	1.86111e10			1.55214
$759$	−9.20098e8	−	3.14147e9i	−0.0763815	−	0.260788i
$760$	−1.03607e10			−0.856137
$761$			1.56267e10i			1.28535i	0.766139	+	0.642675i	$0.222175\pi$
$761$			1.56267e10i			1.28535i	−0.766139	+	0.642675i	$0.777825\pi$
$762$	1.59772e10	−	6.08313e9i	1.30816	−	0.498064i
$763$	4.12156e9			0.335912
$764$	−7.98925e7			−0.00648155
$765$	−1.21378e10	+	1.08096e10i	−0.980220	+	0.872957i
$766$			3.56469e9i			0.286563i
$767$			6.03656e9i			0.483065i
$768$	−5.89381e8	−	1.54800e9i	−0.0469496	−	0.123312i
$769$		−	1.79051e10i		−	1.41983i	−0.704288	−	0.709914i	$-0.748734\pi$
$769$		−	1.79051e10i		−	1.41983i	0.704288	−	0.709914i	$-0.251266\pi$
$770$	−1.75332e9			−0.138403
$771$	5.97371e9	−	2.27441e9i	0.469412	−	0.178722i
$772$	4.56753e8			0.0357290
$773$	5.82816e9			0.453841			0.226920	−	0.973913i	$-0.427134\pi$
$773$	5.82816e9			0.453841			0.226920	+	0.973913i	$0.427134\pi$
$774$	−1.44547e10	+	1.28730e10i	−1.12051	+	0.997895i
$775$	−5.56002e9			−0.429062
$776$	9.95615e9			0.764848
$777$	−1.84811e9	−	4.85402e9i	−0.141336	−	0.371217i
$778$			3.38902e9i			0.258015i
$779$	−8.87631e9			−0.672746
$780$	−8.23464e7	−	2.16282e8i	−0.00621318	−	0.0163188i
$781$		−	1.70177e8i		−	0.0127827i
$782$	2.03429e9	−	2.57388e10i	0.152121	−	1.92471i
$783$	8.60537e9	−	1.65527e10i	0.640624	−	1.23226i
$784$	−4.96631e9			−0.368068
$785$		−	8.38816e9i		−	0.618903i
$786$	8.61626e8	+	2.26305e9i	0.0632907	+	0.166232i
$787$			2.59861e9i			0.190033i	0.995476	+	0.0950166i	$0.0302904\pi$
$787$			2.59861e9i			0.190033i	−0.995476	+	0.0950166i	$0.969710\pi$
$788$			3.27542e8i			0.0238465i
$789$	−3.82653e9	−	1.00503e10i	−0.277354	−	0.728467i
$790$	8.36638e9			0.603730
$791$			1.14698e10i			0.824024i
$792$	2.57931e9	+	2.89624e9i	0.184487	+	0.207156i
$793$		−	7.81271e9i		−	0.556347i
$794$			8.02748e9i			0.569125i
$795$	2.92699e9	+	7.68768e9i	0.206603	+	0.542638i
$796$		−	3.21502e8i		−	0.0225937i
$797$	1.12090e10			0.784268			0.392134	−	0.919908i	$-0.371737\pi$
$797$	1.12090e10			0.784268			0.392134	+	0.919908i	$0.371737\pi$
$798$	−1.29704e10	+	4.93830e9i	−0.903528	+	0.344007i
$799$			3.69548e10i			2.56305i
$800$			7.11922e8i			0.0491606i
$801$	−1.13914e10	+	1.01448e10i	−0.783181	+	0.697479i
$802$			2.58465e10i			1.76926i
$803$	−3.38646e9			−0.230803
$804$	2.34400e8	−	8.92446e7i	0.0159060	−	0.00605600i
$805$	−6.07479e8	+	7.68610e9i	−0.0410437	+	0.519302i
$806$		−	6.65973e9i		−	0.448006i
$807$	1.21982e10	−	4.64430e9i	0.817030	−	0.311074i
$808$	2.50584e10			1.67114
$809$			4.30844e9i			0.286088i	0.989716	+	0.143044i	$0.0456891\pi$
$809$			4.30844e9i			0.286088i	−0.989716	+	0.143044i	$0.954311\pi$
$810$	9.76395e9	+	1.13409e9i	0.645547	+	0.0749806i
$811$	−1.22808e10			−0.808449			−0.404225	−	0.914660i	$-0.632459\pi$
$811$	−1.22808e10			−0.808449			−0.404225	+	0.914660i	$0.632459\pi$
$812$	−7.31783e8			−0.0479663
$813$	3.91026e9	+	1.02702e10i	0.255205	+	0.670292i
$814$	2.07220e9			0.134663
$815$	−1.50123e10			−0.971397
$816$	1.04038e10	+	2.73254e10i	0.670311	+	1.76056i
$817$	−3.01829e10			−1.93635
$818$			4.81263e9i			0.307430i
$819$	−4.88458e9	−	5.48477e9i	−0.310695	−	0.348871i
$820$		−	2.46596e8i		−	0.0156184i
$821$			2.18098e10i			1.37547i	0.725962	+	0.687735i	$0.241395\pi$
$821$			2.18098e10i			1.37547i	−0.725962	+	0.687735i	$0.758605\pi$
$822$	2.32835e9	−	8.86488e8i	0.146217	−	0.0556701i
$823$	1.51102e10			0.944868			0.472434	−	0.881366i	$-0.343376\pi$
$823$	1.51102e10			0.944868			0.472434	+	0.881366i	$0.343376\pi$
$824$	−2.85004e10			−1.77462
$825$	−8.70459e8	−	2.28625e9i	−0.0539709	−	0.141754i
$826$			1.00572e10i			0.620934i
$827$	2.07535e10			1.27591			0.637957	−	0.770072i	$-0.279780\pi$
$827$	2.07535e10			1.27591			0.637957	+	0.770072i	$0.279780\pi$
$828$	5.73625e8	−	4.34906e8i	0.0351174	−	0.0266249i
$829$	1.88655e10			1.15008			0.575038	−	0.818127i	$-0.304987\pi$
$829$	1.88655e10			1.15008			0.575038	+	0.818127i	$0.304987\pi$
$830$			1.66125e10i			1.00847i
$831$	−5.93984e9	−	1.56009e10i	−0.359063	−	0.943074i
$832$	−1.02998e10			−0.620007
$833$	−1.27100e10			−0.761882
$834$	−1.91126e10	+	7.27686e9i	−1.14087	+	0.434373i
$835$			1.53964e9i			0.0915201i
$836$			2.55267e8i			0.0151103i
$837$	−1.15702e10	−	6.01508e9i	−0.682025	−	0.354570i
$838$			4.84376e8i			0.0284334i
$839$	−2.46228e10			−1.43937			−0.719683	−	0.694303i	$-0.755713\pi$
$839$	−2.46228e10			−1.43937			−0.719683	+	0.694303i	$0.755713\pi$
$840$	−3.25029e9	−	8.53684e9i	−0.189210	−	0.496958i
$841$	−1.60227e10			−0.928861
$842$	−8.90120e9			−0.513873
$843$	−3.44624e9	−	9.05148e9i	−0.198129	−	0.520383i
$844$	−2.21879e8			−0.0127033
$845$	7.51544e9			0.428505
$846$	1.66901e10	−	1.48637e10i	0.947681	−	0.843978i
$847$		−	1.28358e10i		−	0.725825i
$848$	1.47982e10			0.833343
$849$	−4.77201e9	+	1.81688e9i	−0.267624	+	0.101894i
$850$		−	1.92954e10i		−	1.07767i
$851$	7.17963e8	−	9.08399e9i	0.0399345	−	0.505269i
$852$	3.49740e7	−	1.33159e7i	0.00193734	−	0.000737619i
$853$	2.71122e10			1.49570			0.747848	−	0.663870i	$-0.231087\pi$
$853$	2.71122e10			1.49570			0.747848	+	0.663870i	$0.231087\pi$
$854$		−	1.30163e10i		−	0.715131i
$855$	1.01940e10	+	1.14466e10i	0.557783	+	0.626320i
$856$			2.09811e10i			1.14333i
$857$		−	1.01785e10i		−	0.552395i	−0.961101	−	0.276198i	$-0.910926\pi$
$857$		−	1.01785e10i		−	0.552395i	0.961101	−	0.276198i	$-0.0890745\pi$
$858$	2.73844e9	−	1.04263e9i	0.148012	−	0.0563538i
$859$	−2.22208e10			−1.19614			−0.598072	−	0.801442i	$-0.704066\pi$
$859$	−2.22208e10			−1.19614			−0.598072	+	0.801442i	$0.704066\pi$
$860$		−	8.38521e8i		−	0.0449541i
$861$	−2.78461e9	−	7.31373e9i	−0.148680	−	0.390506i
$862$		−	1.27447e10i		−	0.677725i
$863$		−	3.24597e10i		−	1.71912i	−0.511034	−	0.859560i	$-0.670737\pi$
$863$		−	3.24597e10i		−	1.71912i	0.511034	−	0.859560i	$-0.329263\pi$
$864$	−7.70189e8	+	1.48148e9i	−0.0406256	+	0.0781444i
$865$			1.96599e9i			0.103282i
$866$	−2.83417e10			−1.48291
$867$	1.97978e10	+	5.19986e10i	1.03169	+	2.70972i
$868$			5.11509e8i			0.0265482i
$869$		−	4.88352e9i		−	0.252443i
$870$	−6.23786e9	−	1.63836e10i	−0.321158	−	0.843516i
$871$		−	4.48956e9i		−	0.230218i
$872$	−8.56699e9			−0.437543
$873$	−9.79596e9	−	1.09996e10i	−0.498307	−	0.559536i
$874$	−2.42732e10	−	1.91846e9i	−1.22981	−	0.0971991i
$875$			1.60849e10i			0.811688i
$876$	−2.64981e8	−	6.95969e8i	−0.0133184	−	0.0349804i
$877$	3.00731e10			1.50550			0.752749	−	0.658308i	$-0.228727\pi$
$877$	3.00731e10			1.50550			0.752749	+	0.658308i	$0.228727\pi$
$878$		−	1.78120e10i		−	0.888142i
$879$	−3.95888e9	−	1.03979e10i	−0.196613	−	0.516400i
$880$	3.48350e9			0.172316
$881$	−7.65146e9			−0.376989			−0.188494	−	0.982074i	$-0.560361\pi$
$881$	−7.65146e9			−0.376989			−0.188494	+	0.982074i	$0.560361\pi$
$882$	5.11214e9	+	5.74028e9i	0.250878	+	0.281704i
$883$	−3.86170e10			−1.88763			−0.943814	−	0.330478i	$-0.892790\pi$
$883$	−3.86170e10			−1.88763			−0.943814	+	0.330478i	$0.892790\pi$
$884$	−1.06548e9			−0.0518755
$885$	1.03804e10	−	3.95222e9i	0.503401	−	0.191663i
$886$	1.39701e10			0.674808
$887$		−	5.40517e9i		−	0.260062i	−0.991510	−	0.130031i	$-0.958492\pi$
$887$		−	5.40517e9i		−	0.260062i	0.991510	−	0.130031i	$-0.0415077\pi$
$888$	3.84143e9	+	1.00895e10i	0.184097	+	0.483529i
$889$			2.35042e10i			1.12199i
$890$			1.43341e10i			0.681561i
$891$	6.61975e8	−	5.69929e9i	0.0313523	−	0.269928i
$892$	−3.91701e8			−0.0184790
$893$	3.48505e10			1.63768
$894$	2.95579e10	−	1.12538e10i	1.38354	−	0.526766i
$895$		−	1.92718e10i		−	0.898547i
$896$	−1.56737e10			−0.727936
$897$	−3.62180e9	−	1.23658e10i	−0.167552	−	0.572071i
$898$	−1.00960e10			−0.465246
$899$			2.32573e10i			1.06758i
$900$	4.01747e8	−	3.57785e8i	0.0183698	−	0.0163596i
$901$	3.78722e10			1.72498
$902$	3.12227e9			0.141660
$903$	−9.46874e9	−	2.48695e10i	−0.427942	−	1.12398i
$904$		−	2.38410e10i		−	1.07333i
$905$		−	3.63846e9i		−	0.163173i
$906$	−2.47626e9	+	9.42804e8i	−0.110623	+	0.0421185i
$907$			2.12153e10i			0.944113i	0.881568	+	0.472056i	$0.156488\pi$
$907$			2.12153e10i			0.944113i	−0.881568	+	0.472056i	$0.843512\pi$
$908$	2.69337e8			0.0119398
$909$	−2.46552e10	−	2.76847e10i	−1.08877	−	1.22255i
$910$	−6.90164e9			−0.303604
$911$	3.42893e10			1.50260			0.751302	−	0.659958i	$-0.229426\pi$
$911$	3.42893e10			1.50260			0.751302	+	0.659958i	$0.229426\pi$
$912$	2.57695e10	−	9.81140e9i	1.12493	−	0.428301i
$913$	9.69682e9			0.421679
$914$	−3.56367e10			−1.54378
$915$	−1.34347e10	+	5.11508e9i	−0.579767	+	0.220739i
$916$			1.49900e9i			0.0644417i
$917$	−3.32918e9			−0.142575
$918$	2.08746e10	−	4.01529e10i	0.890573	−	1.71304i
$919$			1.86854e10i			0.794143i	0.917788	+	0.397072i	$0.129974\pi$
$919$			1.86854e10i			0.794143i	−0.917788	+	0.397072i	$0.870026\pi$
$920$	1.26269e9	−	1.59762e10i	0.0534614	−	0.676418i
$921$	−7.38412e9	−	1.93943e10i	−0.311451	−	0.818022i
$922$	2.91063e10			1.22301
$923$		−	6.69872e8i		−	0.0280405i
$924$	−2.10330e8	+	8.00804e7i	−0.00877099	+	0.00333944i
$925$		−	6.80993e9i		−	0.282909i
$926$			4.24408e10i			1.75649i
$927$	2.80418e10	+	3.14874e10i	1.15618	+	1.29825i
$928$	2.97793e9			0.122320
$929$			3.04573e10i			1.24634i	0.782086	+	0.623170i	$0.214156\pi$
$929$			3.04573e10i			1.24634i	−0.782086	+	0.623170i	$0.785844\pi$
$930$	−1.14520e10	+	4.36021e9i	−0.466865	+	0.177753i
$931$			1.19863e10i			0.486811i
$932$			1.00230e9i			0.0405548i
$933$	−1.67355e10	+	6.37184e9i	−0.674611	+	0.256850i
$934$			1.42882e10i			0.573805i
$935$	8.91512e9			0.356687
$936$	1.01530e10	+	1.14005e10i	0.404696	+	0.454422i
$937$		−	4.00237e10i		−	1.58938i	−0.607014	−	0.794691i	$-0.707633\pi$
$937$		−	4.00237e10i		−	1.58938i	0.607014	−	0.794691i	$-0.292367\pi$
$938$		−	7.47979e9i		−	0.295924i
$939$	3.51598e10	−	1.33866e10i	1.38585	−	0.527645i
$940$			9.68196e8i			0.0380203i
$941$	−1.38644e10			−0.542421			−0.271211	−	0.962520i	$-0.587424\pi$
$941$	−1.38644e10			−0.542421			−0.271211	+	0.962520i	$0.587424\pi$
$942$	8.31035e9	+	2.18270e10i	0.323923	+	0.850778i
$943$	1.08178e9	−	1.36872e10i	0.0420096	−	0.531524i
$944$		−	1.99816e10i		−	0.773085i
$945$	−6.23357e9	+	1.19904e10i	−0.240284	+	0.462194i
$946$	1.06169e10			0.407736
$947$		−	2.65960e10i		−	1.01763i	−0.860874	−	0.508817i	$-0.830083\pi$
$947$		−	2.65960e10i		−	1.01763i	0.860874	−	0.508817i	$-0.169917\pi$
$948$	1.00364e9	−	3.82122e8i	0.0382602	−	0.0145671i
$949$	−1.33302e10			−0.506296
$950$	−1.81967e10			−0.688589
$951$	−1.97897e10	+	7.53467e9i	−0.746118	+	0.284075i
$952$	−4.20555e10			−1.57977
$953$	3.84570e10			1.43930			0.719648	−	0.694339i	$-0.244303\pi$
$953$	3.84570e10			1.43930			0.719648	+	0.694339i	$0.244303\pi$
$954$	−1.52327e10	−	1.71044e10i	−0.568014	−	0.637807i
$955$	2.63135e9			0.0977611
$956$			7.83860e8i			0.0290159i
$957$	−9.56325e9	+	3.64109e9i	−0.352707	+	0.134289i
$958$		−	8.52515e8i		−	0.0313273i
$959$			3.42524e9i			0.125408i
$960$	6.74340e9	+	1.77114e10i	0.245997	+	0.646107i
$961$	−1.12560e10			−0.409121
$962$	8.15685e9			0.295400
$963$	2.31801e10	−	2.06436e10i	0.836418	−	0.744891i
$964$		−	3.04125e8i		−	0.0109341i
$965$	−1.50437e10			−0.538900
$966$	−6.03407e9	−	2.06020e10i	−0.215373	−	0.735342i
$967$	1.86873e10			0.664590			0.332295	−	0.943175i	$-0.392177\pi$
$967$	1.86873e10			0.664590			0.332295	+	0.943175i	$0.392177\pi$
$968$			2.66803e10i			0.945424i
$969$	6.59504e10	−	2.51098e10i	2.32854	−	0.886562i
$970$	−1.38411e10			−0.486935
$971$	4.03647e9			0.141493			0.0707463	−	0.997494i	$-0.477462\pi$
$971$	4.03647e9			0.141493			0.0707463	+	0.997494i	$0.477462\pi$
$972$	1.22309e9	−	3.09908e8i	0.0427194	−	0.0108243i
$973$		−	2.81166e10i		−	0.978515i
$974$		−	5.89582e8i		−	0.0204450i
$975$	−3.42640e9	−	8.99939e9i	−0.118392	−	0.310955i
$976$			2.58608e10i			0.890364i
$977$	1.87629e10			0.643679			0.321840	−	0.946794i	$-0.395699\pi$
$977$	1.87629e10			0.643679			0.321840	+	0.946794i	$0.395699\pi$
$978$	3.90639e10	−	1.48731e10i	1.33533	−	0.508412i
$979$	8.36690e9			0.284987
$980$	−3.32995e8			−0.0113018
$981$	8.42915e9	+	9.46487e9i	0.285064	+	0.320091i
$982$	1.64627e9			0.0554768
$983$	5.01624e10			1.68438			0.842192	−	0.539178i	$-0.181265\pi$
$983$	5.01624e10			1.68438			0.842192	+	0.539178i	$0.181265\pi$
$984$	5.78803e9	+	1.52022e10i	0.193663	+	0.508654i
$985$		−	1.07880e10i		−	0.359677i
$986$	−8.07117e10			−2.68143
$987$	1.09331e10	+	2.87155e10i	0.361936	+	0.950618i
$988$			1.00481e9i			0.0331463i
$989$	3.67847e9	−	4.65417e10i	0.120915	−	1.52987i
$990$	−3.58579e9	−	4.02638e9i	−0.117452	−	0.131884i
$991$	−1.39483e9			−0.0455263			−0.0227632	−	0.999741i	$-0.507246\pi$
$991$	−1.39483e9			−0.0455263			−0.0227632	+	0.999741i	$0.507246\pi$
$992$		−	2.08155e9i		−	0.0677011i
$993$	1.72886e10	+	4.54083e10i	0.560322	+	1.47168i
$994$		−	1.11603e9i		−	0.0360434i
$995$			1.05890e10i			0.340781i
$996$	7.58749e8	+	1.99284e9i	0.0243327	+	0.0639095i
$997$	−1.75550e10			−0.561005			−0.280502	−	0.959853i	$-0.590501\pi$
$997$	−1.75550e10			−0.561005			−0.280502	+	0.959853i	$0.590501\pi$
$998$			5.78863e10i			1.84340i
$999$	7.36728e9	−	1.41712e10i	0.233791	−	0.449704i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.c.b.68.13	✓	48
3.2	odd	2	inner	69.8.c.b.68.36	yes	48
23.22	odd	2	inner	69.8.c.b.68.14	yes	48
69.68	even	2	inner	69.8.c.b.68.35	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.c.b.68.13	✓	48	1.1	even	1	trivial
69.8.c.b.68.14	yes	48	23.22	odd	2	inner
69.8.c.b.68.35	yes	48	69.68	even	2	inner
69.8.c.b.68.36	yes	48	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-11.0616i q^{2} +(16.6400 + 43.7048i) q^{3} +5.64089 q^{4} -185.789 q^{5} +(483.445 - 184.066i) q^{6} +711.199i q^{7} -1478.28i q^{8} +(-1633.22 + 1454.50i) q^{9} +O(q^{10})\)	\(q-11.0616i q^{2} +(16.6400 + 43.7048i) q^{3} +5.64089 q^{4} -185.789 q^{5} +(483.445 - 184.066i) q^{6} +711.199i q^{7} -1478.28i q^{8} +(-1633.22 + 1454.50i) q^{9} +2055.12i q^{10} +1199.59 q^{11} +(93.8646 + 246.534i) q^{12} +4721.97 q^{13} +7867.00 q^{14} +(-3091.53 - 8119.87i) q^{15} -15630.1 q^{16} -40001.4 q^{17} +(16089.1 + 18066.0i) q^{18} +37723.6i q^{19} -1048.01 q^{20} +(-31082.8 + 11834.4i) q^{21} -13269.4i q^{22} +(-58169.5 - 4597.49i) q^{23} +(64608.0 - 24598.7i) q^{24} -43607.5 q^{25} -52232.6i q^{26} +(-90745.4 - 47176.5i) q^{27} +4011.79i q^{28} +182408. i q^{29} +(-89818.7 + 34197.3i) q^{30} +127501. q^{31} -16325.7i q^{32} +(19961.2 + 52427.9i) q^{33} +442479. i q^{34} -132133. i q^{35} +(-9212.80 + 8204.67i) q^{36} +156164. i q^{37} +417284. q^{38} +(78573.8 + 206373. i) q^{39} +274649. i q^{40} +235298. i q^{41} +(130907. + 343826. i) q^{42} +800105. i q^{43} +6766.76 q^{44} +(303434. - 270230. i) q^{45} +(-50855.6 + 643448. i) q^{46} -923838. i q^{47} +(-260086. - 683112. i) q^{48} +317739. q^{49} +482369. i q^{50} +(-665624. - 1.74825e6i) q^{51} +26636.1 q^{52} -946774. q^{53} +(-521848. + 1.00379e6i) q^{54} -222871. q^{55} +1.05135e6 q^{56} +(-1.64870e6 + 627723. i) q^{57} +2.01772e6 q^{58} +1.27840e6i q^{59} +(-17439.0 - 45803.3i) q^{60} -1.65455e6i q^{61} -1.41037e6i q^{62} +(-1.03444e6 - 1.16154e6i) q^{63} -2.18125e6 q^{64} -877289. q^{65} +(579936. - 220803. i) q^{66} -950780. i q^{67} -225643. q^{68} +(-767010. - 2.61879e6i) q^{69} -1.46160e6 q^{70} -141863. i q^{71} +(2.15016e6 + 2.41436e6i) q^{72} -2.82301e6 q^{73} +1.72743e6 q^{74} +(-725630. - 1.90586e6i) q^{75} +212795. i q^{76} +853147. i q^{77} +(2.28281e6 - 869152. i) q^{78} -4.07099e6i q^{79} +2.90391e6 q^{80} +(551834. - 4.75103e6i) q^{81} +2.60278e6 q^{82} +8.08344e6 q^{83} +(-175335. + 66756.4i) q^{84} +7.43181e6 q^{85} +8.85044e6 q^{86} +(-7.97210e6 + 3.03527e6i) q^{87} -1.77333e6i q^{88} +6.97480e6 q^{89} +(-2.98918e6 - 3.35646e6i) q^{90} +3.35826e6i q^{91} +(-328128. - 25933.9i) q^{92} +(2.12163e6 + 5.57242e6i) q^{93} -1.02191e7 q^{94} -7.00863e6i q^{95} +(713511. - 271660. i) q^{96} +6.73494e6i q^{97} -3.51471e6i q^{98} +(-1.95919e6 + 1.74480e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 24 q^{3} - 2616 q^{4} + 4416 q^{6} + 648 q^{9}+O(q^{10}) \) 48 * q + 24 * q^3 - 2616 * q^4 + 4416 * q^6 + 648 * q^9	\( 48 q + 24 q^{3} - 2616 q^{4} + 4416 q^{6} + 648 q^{9} + 16668 q^{12} - 4992 q^{13} + 76296 q^{16} + 100548 q^{18} - 620268 q^{24} + 1257120 q^{25} - 105408 q^{27} - 738480 q^{31} - 1227096 q^{36} + 3832248 q^{39} - 489768 q^{46} + 733404 q^{48} - 5569584 q^{49} - 5646984 q^{52} + 1260252 q^{54} - 305760 q^{55} + 16920336 q^{58} + 6032664 q^{64} + 4496304 q^{69} + 23272152 q^{70} - 26866080 q^{72} - 2542752 q^{73} + 487176 q^{75} - 47724612 q^{78} + 25279128 q^{81} - 68451480 q^{82} - 18046992 q^{85} + 42338184 q^{87} - 59017272 q^{93} + 106197744 q^{94} + 67778460 q^{96}+O(q^{100}) \) 48 * q + 24 * q^3 - 2616 * q^4 + 4416 * q^6 + 648 * q^9 + 16668 * q^12 - 4992 * q^13 + 76296 * q^16 + 100548 * q^18 - 620268 * q^24 + 1257120 * q^25 - 105408 * q^27 - 738480 * q^31 - 1227096 * q^36 + 3832248 * q^39 - 489768 * q^46 + 733404 * q^48 - 5569584 * q^49 - 5646984 * q^52 + 1260252 * q^54 - 305760 * q^55 + 16920336 * q^58 + 6032664 * q^64 + 4496304 * q^69 + 23272152 * q^70 - 26866080 * q^72 - 2542752 * q^73 + 487176 * q^75 - 47724612 * q^78 + 25279128 * q^81 - 68451480 * q^82 - 18046992 * q^85 + 42338184 * q^87 - 59017272 * q^93 + 106197744 * q^94 + 67778460 * q^96

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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