LMFDB - Embedded newform 69.8.c.a.68.4

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:	$6$
Coefficient field:	6.0.8869743.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{3} + 8 $ x^6 - 3*x^3 + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			68.4
Root			$-0.261988 + 1.38973i$ of defining polynomial
Character	$\chi$	$=$	69.68
Dual form			69.8.c.a.68.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.95879i q^{2} +(-43.1046 + 18.1381i) q^{3} +92.4928 q^{4} +(-108.081 - 256.851i) q^{6} +1313.87i q^{8} +(1529.02 - 1563.67i) q^{9} +O(q^{10})$	\(q+5.95879i q^{2} +(-43.1046 + 18.1381i) q^{3} +92.4928 q^{4} +(-108.081 - 256.851i) q^{6} +1313.87i q^{8} +(1529.02 - 1563.67i) q^{9} +(-3986.87 + 1677.64i) q^{12} +13489.2 q^{13} +4010.01 q^{16} +(9317.58 + 9111.12i) q^{18} +58350.9i q^{23} +(-23831.1 - 56633.9i) q^{24} -78125.0 q^{25} +80379.2i q^{26} +(-37546.0 + 95134.9i) q^{27} +226129. i q^{29} -107144. q^{31} +192070. i q^{32} +(141424. - 144628. i) q^{36} +(-581447. + 244668. i) q^{39} -351517. i q^{41} -347701. q^{46} +914412. i q^{47} +(-172850. + 72733.8i) q^{48} +823543. q^{49} -465530. i q^{50} +1.24765e6 q^{52} +(-566889. - 223729. i) q^{54} -1.34746e6 q^{58} +3.15500e6i q^{59} -638447. i q^{62} -631224. q^{64} +(-1.05837e6 - 2.51519e6i) q^{69} -6.00671e6i q^{71} +(2.05446e6 + 2.00894e6i) q^{72} +6.03382e6 q^{73} +(3.36755e6 - 1.41704e6i) q^{75} +(-1.45792e6 - 3.46472e6i) q^{78} +(-107156. - 4.78177e6i) q^{81} +2.09461e6 q^{82} +(-4.10155e6 - 9.74722e6i) q^{87} +5.39704e6i q^{92} +(4.61839e6 - 1.94338e6i) q^{93} -5.44879e6 q^{94} +(-3.48378e6 - 8.27912e6i) q^{96} +4.90732e6i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 768 q^{4} - 3147 q^{6}+O(q^{10}) $ 6 * q - 768 * q^4 - 3147 * q^6	$ 6 q - 768 q^{4} - 3147 q^{6} - 21435 q^{12} + 98304 q^{16} - 72177 q^{18} + 402816 q^{24} - 468750 q^{25} - 225276 q^{27} + 1621587 q^{36} - 2196456 q^{39} - 866433 q^{48} + 4941258 q^{49} + 5680578 q^{52} - 12795270 q^{58} - 13290570 q^{64} + 9238656 q^{72} + 8715747 q^{78} + 59301462 q^{82} - 29588784 q^{87} + 30939042 q^{93} - 87793602 q^{94} - 59843631 q^{96}+O(q^{100}) $ 6 * q - 768 * q^4 - 3147 * q^6 - 21435 * q^12 + 98304 * q^16 - 72177 * q^18 + 402816 * q^24 - 468750 * q^25 - 225276 * q^27 + 1621587 * q^36 - 2196456 * q^39 - 866433 * q^48 + 4941258 * q^49 + 5680578 * q^52 - 12795270 * q^58 - 13290570 * q^64 + 9238656 * q^72 + 8715747 * q^78 + 59301462 * q^82 - 29588784 * q^87 + 30939042 * q^93 - 87793602 * q^94 - 59843631 * 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 $

$2$			5.95879i			0.526687i	0.964702	+	0.263344i	$0.0848253\pi$
$2$			5.95879i			0.526687i	−0.964702	+	0.263344i	$0.915175\pi$
$3$	−43.1046	+	18.1381i	−0.921722	+	0.387852i
$3$	−43.1046	+	18.1381i	−0.921722	+	0.387852i
$4$	92.4928			0.722600
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	−108.081	−	256.851i	−0.204277	−	0.485459i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$			1313.87i			0.907272i
$9$	1529.02	−	1563.67i	0.699141	−	0.714984i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−3986.87	+	1677.64i	−0.666036	+	0.280262i
$13$	13489.2			1.70288			0.851440	−	0.524451i	$-0.175730\pi$
$13$	13489.2			1.70288			0.851440	+	0.524451i	$0.175730\pi$
$14$		0			0
$15$		0			0
$16$	4010.01			0.244752
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	9317.58	+	9111.12i	0.376573	+	0.368229i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			58350.9i			1.00000i
$23$			58350.9i			1.00000i
$24$	−23831.1	−	56633.9i	−0.351888	−	0.836252i
$25$	−78125.0			−1.00000
$26$			80379.2i			0.896886i
$27$	−37546.0	+	95134.9i	−0.367105	+	0.930179i
$28$		0			0
$29$			226129.i			1.72172i	0.508838	+	0.860862i	$0.330075\pi$
$29$			226129.i			1.72172i	−0.508838	+	0.860862i	$0.669925\pi$
$30$		0			0
$31$	−107144.			−0.645953			−0.322977	−	0.946407i	$-0.604683\pi$
$31$	−107144.			−0.645953			−0.322977	+	0.946407i	$0.604683\pi$
$32$			192070.i			1.03618i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	141424.	−	144628.i	0.505200	−	0.516648i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	−581447.	+	244668.i	−1.56958	+	0.660466i
$40$		0			0
$41$		−	351517.i		−	0.796531i	−0.917270	−	0.398266i	$-0.869612\pi$
$41$		−	351517.i		−	0.796531i	0.917270	−	0.398266i	$-0.130388\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	−347701.			−0.526687
$47$			914412.i			1.28469i	0.766414	+	0.642347i	$0.222039\pi$
$47$			914412.i			1.28469i	−0.766414	+	0.642347i	$0.777961\pi$
$48$	−172850.	+	72733.8i	−0.225593	+	0.0949275i
$49$	823543.			1.00000
$50$		−	465530.i		−	0.526687i
$51$		0			0
$52$	1.24765e6			1.23050
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−566889.	−	223729.i	−0.489914	−	0.193350i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−1.34746e6			−0.906811
$59$			3.15500e6i			1.99994i	0.00783115	+	0.999969i	$0.497507\pi$
$59$			3.15500e6i			1.99994i	−0.00783115	+	0.999969i	$0.502493\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		−	638447.i		−	0.340215i
$63$		0			0
$64$	−631224.			−0.300991
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−1.05837e6	−	2.51519e6i	−0.387852	−	0.921722i
$70$		0			0
$71$		−	6.00671e6i		−	1.99174i	−0.0907876	−	0.995870i	$-0.528938\pi$
$71$		−	6.00671e6i		−	1.99174i	0.0907876	−	0.995870i	$-0.471062\pi$
$72$	2.05446e6	+	2.00894e6i	0.648685	+	0.634311i
$73$	6.03382e6			1.81536			0.907679	−	0.419665i	$-0.137852\pi$
$73$	6.03382e6			1.81536			0.907679	+	0.419665i	$0.137852\pi$
$74$		0			0
$75$	3.36755e6	−	1.41704e6i	0.921722	−	0.387852i
$76$		0			0
$77$		0			0
$78$	−1.45792e6	−	3.46472e6i	−0.347859	−	0.826679i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−107156.	−	4.78177e6i	−0.0224036	−	0.999749i
$82$	2.09461e6			0.419523
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−4.10155e6	−	9.74722e6i	−0.667775	−	1.58695i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$			5.39704e6i			0.722600i
$93$	4.61839e6	−	1.94338e6i	0.595389	−	0.250534i
$94$	−5.44879e6			−0.676632
$95$		0			0
$96$	−3.48378e6	−	8.27912e6i	−0.401885	−	0.955069i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$			4.90732e6i			0.526687i
$99$		0			0
$100$	−7.22600e6			−0.722600
$101$			1.08442e7i			1.04730i	0.851933	+	0.523650i	$0.175430\pi$
$101$			1.08442e7i			1.04730i	−0.851933	+	0.523650i	$0.824570\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$			1.77230e7i			1.54498i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−3.47274e6	+	8.79930e6i	−0.265270	+	0.672148i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$			2.09153e7i			1.24412i
$117$	2.06253e7	−	2.10926e7i	1.19055	−	1.21753i
$118$	−1.87999e7			−1.05334
$119$		0			0
$120$		0			0
$121$	−1.94872e7			−1.00000
$122$		0			0
$123$	6.37584e6	+	1.51520e7i	0.308936	+	0.734180i
$124$	−9.91003e6			−0.466766
$125$		0			0
$126$		0			0
$127$	4.60960e7			1.99687			0.998435	−	0.0559193i	$-0.0178090\pi$
$127$	4.60960e7			1.99687			0.998435	+	0.0559193i	$0.0178090\pi$
$128$			2.08237e7i			0.877651i
$129$		0			0
$130$		0			0
$131$			9.43094e6i			0.366527i	0.983064	+	0.183263i	$0.0586661\pi$
$131$			9.43094e6i			0.366527i	−0.983064	+	0.183263i	$0.941334\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	1.49875e7	−	6.30661e6i	0.485459	−	0.204277i
$139$	−1.83559e7			−0.579728			−0.289864	−	0.957068i	$-0.593610\pi$
$139$	−1.83559e7			−0.579728			−0.289864	+	0.957068i	$0.593610\pi$
$140$		0			0
$141$	−1.65857e7	−	3.94154e7i	−0.498272	−	1.18413i
$142$	3.57927e7			1.04902
$143$		0			0
$144$	6.13139e6	−	6.27033e6i	0.171116	−	0.174993i
$145$		0			0
$146$			3.59543e7i			0.956126i
$147$	−3.54985e7	+	1.49375e7i	−0.921722	+	0.387852i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	8.44382e6	+	2.00665e7i	0.204277	+	0.485459i
$151$	−7.30407e7			−1.72642			−0.863209	−	0.504848i	$-0.831549\pi$
$151$	−7.30407e7			−1.72642			−0.863209	+	0.504848i	$0.831549\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−5.37797e7	+	2.26300e7i	−1.13418	+	0.477253i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	2.84935e7	−	638518.i	0.526555	−	0.0117997i
$163$	−1.15460e7			−0.208821			−0.104410	−	0.994534i	$-0.533296\pi$
$163$	−1.15460e7			−0.208821			−0.104410	+	0.994534i	$0.533296\pi$
$164$		−	3.25128e7i		−	0.575574i
$165$		0			0
$166$		0			0
$167$		−	5.56023e7i		−	0.923815i	−0.886928	−	0.461907i	$-0.847165\pi$
$167$		−	5.56023e7i		−	0.923815i	0.886928	−	0.461907i	$-0.152835\pi$
$168$		0			0
$169$	1.19210e8			1.89980
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	7.26618e7i		−	1.06695i	−0.845815	−	0.533476i	$-0.820886\pi$
$173$		−	7.26618e7i		−	1.06695i	0.845815	−	0.533476i	$-0.179114\pi$
$174$	5.80816e7	−	2.44402e7i	0.835827	−	0.351709i
$175$		0			0
$176$		0			0
$177$	−5.72255e7	−	1.35995e8i	−0.775681	−	1.84339i
$178$		0			0
$179$		−	2.21761e7i		−	0.289001i	−0.989505	−	0.144500i	$-0.953843\pi$
$179$		−	2.21761e7i		−	0.289001i	0.989505	−	0.144500i	$-0.0461575\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−7.66655e7			−0.907272
$185$		0			0
$186$	1.15802e7	+	2.75200e7i	0.131953	+	0.313584i
$187$		0			0
$188$			8.45766e7i			0.928320i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	2.72087e7	−	1.14492e7i	0.277430	−	0.116740i
$193$	1.65978e8			1.66188			0.830939	−	0.556364i	$-0.187804\pi$
$193$	1.65978e8			1.66188			0.830939	+	0.556364i	$0.187804\pi$
$194$		0			0
$195$		0			0
$196$	7.61718e7			0.722600
$197$		−	1.55749e8i		−	1.45142i	−0.687998	−	0.725712i	$-0.741510\pi$
$197$		−	1.55749e8i		−	1.45142i	0.687998	−	0.725712i	$-0.258490\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		−	1.02646e8i		−	0.907272i
$201$		0			0
$202$	−6.46181e7			−0.551600
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	9.12415e7	+	8.92198e7i	0.714984	+	0.699141i
$208$	5.40918e7			0.416783
$209$		0			0
$210$		0			0
$211$	2.24635e8			1.64623			0.823113	−	0.567878i	$-0.192235\pi$
$211$	2.24635e8			1.64623			0.823113	+	0.567878i	$0.192235\pi$
$212$		0			0
$213$	1.08950e8	+	2.58917e8i	0.772501	+	1.83583i
$214$		0			0
$215$		0			0
$216$	−1.24995e8	−	4.93306e7i	−0.843926	−	0.333064i
$217$		0			0
$218$		0			0
$219$	−2.60086e8	+	1.09442e8i	−1.67325	+	0.704091i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−3.13612e8			−1.89376			−0.946881	−	0.321583i	$-0.895785\pi$
$223$	−3.13612e8			−1.89376			−0.946881	+	0.321583i	$0.895785\pi$
$224$		0			0
$225$	−1.19455e8	+	1.22162e8i	−0.699141	+	0.714984i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	−2.97104e8			−1.56207
$233$		−	3.84067e8i		−	1.98912i	−0.104149	−	0.994562i	$-0.533212\pi$
$233$		−	3.84067e8i		−	1.98912i	0.104149	−	0.994562i	$-0.466788\pi$
$234$	1.25687e8	+	1.22902e8i	0.641259	+	0.627050i
$235$		0			0
$236$			2.91814e8i			1.44516i
$237$		0			0
$238$		0			0
$239$			7.50976e7i			0.355822i	0.984047	+	0.177911i	$0.0569340\pi$
$239$			7.50976e7i			0.355822i	−0.984047	+	0.177911i	$0.943066\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		−	1.16120e8i		−	0.526687i
$243$	9.13509e7	+	2.04173e8i	0.408405	+	0.912801i
$244$		0			0
$245$		0			0
$246$	−9.02876e7	+	3.79923e7i	−0.386683	+	0.162713i
$247$		0			0
$248$		−	1.40773e8i		−	0.586055i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$			2.74676e8i			1.05173i
$255$		0			0
$256$	−2.04880e8			−0.763239
$257$			3.13972e8i			1.15379i	0.816820	+	0.576893i	$0.195735\pi$
$257$			3.13972e8i			1.15379i	−0.816820	+	0.576893i	$0.804265\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	3.53591e8	+	3.45756e8i	1.23101	+	1.20373i
$262$	−5.61970e7			−0.193045
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	4.53174e8i		−	1.41949i	−0.704459	−	0.709744i	$-0.748811\pi$
$269$		−	4.53174e8i		−	1.41949i	0.704459	−	0.709744i	$-0.251189\pi$
$270$		0			0
$271$	−2.61606e8			−0.798464			−0.399232	−	0.916850i	$-0.630723\pi$
$271$	−2.61606e8			−0.798464			−0.399232	+	0.916850i	$0.630723\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−9.78918e7	−	2.32637e8i	−0.280262	−	0.666036i
$277$	3.87080e8			1.09426			0.547131	−	0.837047i	$-0.315720\pi$
$277$	3.87080e8			1.09426			0.547131	+	0.837047i	$0.315720\pi$
$278$		−	1.09379e8i		−	0.305335i
$279$	−1.63825e8	+	1.67537e8i	−0.451612	+	0.461846i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	2.34868e8	−	9.88305e7i	0.623666	−	0.262433i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		−	5.55578e8i		−	1.43923i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	3.00334e8	+	2.93679e8i	0.740852	+	0.724436i
$289$	−4.10339e8			−1.00000
$290$		0			0
$291$		0			0
$292$	5.58085e8			1.31178
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−8.90092e7	−	2.11528e8i	−0.204277	−	0.485459i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$			7.87106e8i			1.70288i
$300$	3.11474e8	−	1.31066e8i	0.666036	−	0.280262i
$301$		0			0
$302$		−	4.35234e8i		−	0.909282i
$303$	−1.96692e8	−	4.67434e8i	−0.406198	−	0.965319i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	9.08266e8			1.79155			0.895775	−	0.444509i	$-0.146622\pi$
$307$	9.08266e8			1.79155			0.895775	+	0.444509i	$0.146622\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.02339e9i			1.92921i	0.263695	+	0.964606i	$0.415059\pi$
$311$			1.02339e9i			1.92921i	−0.263695	+	0.964606i	$0.584941\pi$
$312$	−3.21462e8	−	7.63946e8i	−0.599223	−	1.42404i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	8.45302e8i		−	1.49041i	−0.666838	−	0.745203i	$-0.732353\pi$
$317$		−	8.45302e8i		−	1.49041i	0.666838	−	0.745203i	$-0.267647\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−9.91113e6	−	4.42279e8i	−0.0161888	−	0.722419i
$325$	−1.05384e9			−1.70288
$326$		−	6.88000e7i		−	0.109983i
$327$		0			0
$328$	4.61848e8			0.722670
$329$		0			0
$330$		0			0
$331$	−6.66139e8			−1.00964			−0.504820	−	0.863225i	$-0.668441\pi$
$331$	−6.66139e8			−1.00964			−0.504820	+	0.863225i	$0.668441\pi$
$332$		0			0
$333$		0			0
$334$	3.31322e8			0.486562
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$			7.10346e8i			1.00060i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	4.32976e8			0.561950
$347$		−	1.50749e9i		−	1.93688i	−0.249251	−	0.968439i	$-0.580184\pi$
$347$		−	1.50749e9i		−	1.93688i	0.249251	−	0.968439i	$-0.419816\pi$
$348$	−3.79364e8	−	9.01548e8i	−0.482534	−	1.14673i
$349$	−2.07715e8			−0.261565			−0.130782	−	0.991411i	$-0.541749\pi$
$349$	−2.07715e8			−0.261565			−0.130782	+	0.991411i	$0.541749\pi$
$350$		0			0
$351$	−5.06465e8	+	1.28329e9i	−0.625136	+	1.58398i
$352$		0			0
$353$		−	1.41925e8i		−	0.171730i	−0.996307	−	0.0858651i	$-0.972635\pi$
$353$		−	1.41925e8i		−	0.171730i	0.996307	−	0.0858651i	$-0.0273654\pi$
$354$	8.10365e8	−	3.40995e8i	0.970889	−	0.408541i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.32143e8			0.152213
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	8.93872e8			1.00000
$362$		0			0
$363$	8.39988e8	−	3.53460e8i	0.921722	−	0.387852i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$			2.33988e8i			0.244752i
$369$	−5.49656e8	−	5.37477e8i	−0.569507	−	0.556888i
$370$		0			0
$371$		0			0
$372$	4.27168e8	−	1.79749e8i	0.430228	−	0.181036i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−1.20142e9			−1.16557
$377$			3.05030e9i			2.93189i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−1.98695e9	+	8.36091e8i	−1.84056	+	0.774491i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−3.77701e8	−	8.97596e8i	−0.340399	−	0.808950i
$385$		0			0
$386$			9.89025e8i			0.875290i
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$			1.08203e9i			0.907272i
$393$	−1.71059e8	−	4.06517e8i	−0.142158	−	0.337835i
$394$	9.28078e8			0.764447
$395$		0			0
$396$		0			0
$397$	2.35765e9			1.89110			0.945548	−	0.325484i	$-0.105527\pi$
$397$	2.35765e9			1.89110			0.945548	+	0.325484i	$0.105527\pi$
$398$		0			0
$399$		0			0
$400$	−3.13282e8			−0.244752
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−1.44528e9			−1.09998
$404$			1.00301e9i			0.756779i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.81793e9			−1.31385			−0.656924	−	0.753957i	$-0.728143\pi$
$409$	−1.81793e9			−1.31385			−0.656924	+	0.753957i	$0.728143\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−5.31642e8	+	5.43689e8i	−0.368229	+	0.376573i
$415$		0			0
$416$			2.59087e9i			1.76449i
$417$	7.91225e8	−	3.32941e8i	0.534347	−	0.224849i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$			1.33855e9i			0.867047i
$423$	1.42984e9	+	1.39816e9i	0.918535	+	0.898182i
$424$		0			0
$425$		0			0
$426$	−1.54283e9	+	6.49211e8i	−0.966909	+	0.406867i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−1.50560e8	+	3.81492e8i	−0.0898496	+	0.227663i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−6.52141e8	−	1.54980e9i	−0.370836	−	0.881282i
$439$	−3.42417e8			−0.193165			−0.0965826	−	0.995325i	$-0.530791\pi$
$439$	−3.42417e8			−0.193165			−0.0965826	+	0.995325i	$0.530791\pi$
$440$		0			0
$441$	1.25921e9	−	1.28775e9i	0.699141	−	0.714984i
$442$		0			0
$443$			1.21732e9i			0.665262i	0.943057	+	0.332631i	$0.107936\pi$
$443$			1.21732e9i			0.665262i	−0.943057	+	0.332631i	$0.892064\pi$
$444$		0			0
$445$		0			0
$446$		−	1.86875e9i		−	0.997421i
$447$		0			0
$448$		0			0
$449$		−	3.82648e9i		−	1.99498i	−0.0708395	−	0.997488i	$-0.522568\pi$
$449$		−	3.82648e9i		−	1.99498i	0.0708395	−	0.997488i	$-0.477432\pi$
$450$	−7.27936e8	−	7.11806e8i	−0.376573	−	0.368229i
$451$		0			0
$452$		0			0
$453$	3.14839e9	−	1.32482e9i	1.59128	−	0.669595i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	3.92239e9i		−	1.86465i	−0.361619	−	0.932326i	$-0.617776\pi$
$461$		−	3.92239e9i		−	1.86465i	0.361619	−	0.932326i	$-0.382224\pi$
$462$		0			0
$463$	1.71939e9			0.805085			0.402543	−	0.915401i	$-0.368126\pi$
$463$	1.71939e9			0.805085			0.402543	+	0.915401i	$0.368126\pi$
$464$			9.06780e8i			0.421395i
$465$		0			0
$466$	2.28858e9			1.04765
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	1.90769e9	−	1.95092e9i	0.860295	−	0.879789i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−4.14525e9			−1.81449
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−4.47490e8			−0.187407
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.80242e9			−0.722600
$485$		0			0
$486$	−1.21662e9	+	5.44341e8i	−0.480761	+	0.215102i
$487$	−2.89128e9			−1.13433			−0.567164	−	0.823605i	$-0.691959\pi$
$487$	−2.89128e9			−1.13433			−0.567164	+	0.823605i	$0.691959\pi$
$488$		0			0
$489$	4.97685e8	−	2.09422e8i	0.192475	−	0.0809917i
$490$		0			0
$491$		−	5.12406e9i		−	1.95357i	−0.214218	−	0.976786i	$-0.568720\pi$
$491$		−	5.12406e9i		−	1.95357i	0.214218	−	0.976786i	$-0.431280\pi$
$492$	5.89719e8	+	1.40145e9i	0.223238	+	0.530519i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−4.29648e8			−0.158098
$497$		0			0
$498$		0			0
$499$	5.35684e9			1.93000			0.964999	−	0.262255i	$-0.0844660\pi$
$499$	5.35684e9			1.93000			0.964999	+	0.262255i	$0.0844660\pi$
$500$		0			0
$501$	1.00852e9	+	2.39672e9i	0.358304	+	0.851500i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−5.13850e9	+	2.16224e9i	−1.75109	+	0.736843i
$508$	4.26355e9			1.44294
$509$		−	5.80542e9i		−	1.95129i	−0.219357	−	0.975645i	$-0.570396\pi$
$509$		−	5.80542e9i		−	1.95129i	0.219357	−	0.975645i	$-0.429604\pi$
$510$		0			0
$511$		0			0
$512$			1.44459e9i			0.475663i
$513$		0			0
$514$	−1.87089e9			−0.607685
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	1.31794e9	+	3.13206e9i	0.413820	+	0.983432i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	−2.06029e9	+	2.10698e9i	−0.633989	+	0.648355i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$			8.72294e8i			0.264852i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.40483e9			−1.00000
$530$		0			0
$531$	4.93337e9	+	4.82406e9i	1.42992	+	1.39824i
$532$		0			0
$533$		−	4.74168e9i		−	1.35640i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	4.02231e8	+	9.55892e8i	0.112090	+	0.266378i
$538$	2.70037e9			0.747627
$539$		0			0
$540$		0			0
$541$	6.67366e9			1.81207			0.906033	−	0.423208i	$-0.139096\pi$
$541$	6.67366e9			1.81207			0.906033	+	0.423208i	$0.139096\pi$
$542$		−	1.55886e9i		−	0.420541i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−7.27244e9			−1.89987			−0.949936	−	0.312445i	$-0.898852\pi$
$547$	−7.27244e9			−1.89987			−0.949936	+	0.312445i	$0.898852\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	3.30464e9	−	1.39056e9i	0.836252	−	0.351888i
$553$		0			0
$554$			2.30653e9i			0.576334i
$555$		0			0
$556$	−1.69779e9			−0.418911
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	−9.98320e8	−	9.76199e8i	−0.243248	−	0.237859i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	−1.53406e9	−	3.64564e9i	−0.360051	−	0.855653i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	7.89204e9			1.80705
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	4.55866e9i		−	1.00000i
$576$	−9.65155e8	+	9.87026e8i	−0.210435	+	0.215204i
$577$	−1.32085e8			−0.0286245			−0.0143122	−	0.999898i	$-0.504556\pi$
$577$	−1.32085e8			−0.0286245			−0.0143122	+	0.999898i	$0.504556\pi$
$578$		−	2.44512e9i		−	0.526687i
$579$	−7.15441e9	+	3.01051e9i	−1.53179	+	0.644563i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$			7.92766e9i			1.64702i
$585$		0			0
$586$		0			0
$587$			9.64879e9i			1.96897i	0.175462	+	0.984486i	$0.443858\pi$
$587$			9.64879e9i			1.96897i	−0.175462	+	0.984486i	$0.556142\pi$
$588$	−3.28336e9	+	1.38161e9i	−0.666036	+	0.280262i
$589$		0			0
$590$		0			0
$591$	2.82499e9	+	6.71352e9i	0.562939	+	1.33781i
$592$		0			0
$593$		−	6.80116e8i		−	0.133934i	−0.997755	−	0.0669671i	$-0.978668\pi$
$593$		−	6.80116e8i		−	0.133934i	0.997755	−	0.0669671i	$-0.0213323\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−4.69020e9			−0.896886
$599$		−	1.03305e10i		−	1.96393i	−0.189052	−	0.981967i	$-0.560541\pi$
$599$		−	1.03305e10i		−	1.96393i	0.189052	−	0.981967i	$-0.439459\pi$
$600$	1.86180e9	+	4.42452e9i	0.351888	+	0.836252i
$601$	−9.35197e9			−1.75729			−0.878643	−	0.477480i	$-0.841550\pi$
$601$	−9.35197e9			−1.75729			−0.878643	+	0.477480i	$0.841550\pi$
$602$		0			0
$603$		0			0
$604$	−6.75574e9			−1.24751
$605$		0			0
$606$	2.78534e9	−	1.17205e9i	0.508421	−	0.213939i
$607$	3.74889e9			0.680365			0.340183	−	0.940359i	$-0.389511\pi$
$607$	3.74889e9			0.680365			0.340183	+	0.940359i	$0.389511\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1.23347e10i			2.18768i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			5.41216e9i			0.943586i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	−5.55121e9	−	2.19084e9i	−0.930179	−	0.367105i
$622$	−6.09817e9			−1.01609
$623$		0			0
$624$	−2.33161e9	+	9.81120e8i	−0.384158	+	0.161650i
$625$	6.10352e9			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−9.68283e9	+	4.07445e9i	−1.51736	+	0.638493i
$634$	5.03698e9			0.784978
$635$		0			0
$636$		0			0
$637$	1.11089e10			1.70288
$638$		0			0
$639$	−9.39252e9	−	9.18440e9i	−1.42406	−	1.39251i
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1.28766e10i			1.86911i	0.355816	+	0.934556i	$0.384203\pi$
$647$			1.28766e10i			1.86911i	−0.355816	+	0.934556i	$0.615797\pi$
$648$	6.28262e9	−	1.40789e8i	0.907044	−	0.0203262i
$649$		0			0
$650$		−	6.27963e9i		−	0.896886i
$651$		0			0
$652$	−1.06792e9			−0.150894
$653$		−	1.30369e10i		−	1.83222i	−0.400922	−	0.916112i	$-0.631310\pi$
$653$		−	1.30369e10i		−	1.83222i	0.400922	−	0.916112i	$-0.368690\pi$
$654$		0			0
$655$		0			0
$656$		−	1.40959e9i		−	0.194952i
$657$	9.22584e9	−	9.43490e9i	1.26919	−	1.29795i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		−	3.96938e9i		−	0.531765i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.31948e10			−1.72172
$668$		−	5.14281e9i		−	0.667549i
$669$	1.35181e10	−	5.68831e9i	1.74552	−	0.734500i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.45635e10			1.84167			0.920837	−	0.389948i	$-0.127507\pi$
$673$	1.45635e10			1.84167			0.920837	+	0.389948i	$0.127507\pi$
$674$		0			0
$675$	2.93328e9	−	7.43241e9i	0.367105	−	0.930179i
$676$	1.10261e10			1.37280
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			5.24023e9i			0.629330i	0.949203	+	0.314665i	$0.101892\pi$
$683$			5.24023e9i			0.629330i	−0.949203	+	0.314665i	$0.898108\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.06109e10			−1.22343			−0.611716	−	0.791078i	$-0.709520\pi$
$691$	−1.06109e10			−1.22343			−0.611716	+	0.791078i	$0.709520\pi$
$692$		−	6.72069e9i		−	0.770979i
$693$		0			0
$694$	8.98283e9			1.02013
$695$		0			0
$696$	1.28066e10	−	5.38890e9i	1.43980	−	0.605853i
$697$		0			0
$698$		−	1.23773e9i		−	0.137763i
$699$	6.96624e9	+	1.65551e10i	0.771486	+	1.83342i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	−7.64687e9	−	3.01792e9i	−0.834265	−	0.329251i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	8.45699e8			0.0904481
$707$		0			0
$708$	−5.29295e9	−	1.25786e10i	−0.560507	−	1.33203i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	6.25193e9i		−	0.645953i
$714$		0			0
$715$		0			0
$716$		−	2.05113e9i		−	0.208832i
$717$	−1.36212e9	−	3.23705e9i	−0.138007	−	0.327969i
$718$		0			0
$719$			1.98207e10i			1.98870i	0.106173	+	0.994348i	$0.466140\pi$
$719$			1.98207e10i			1.98870i	−0.106173	+	0.994348i	$0.533860\pi$
$720$		0			0
$721$		0			0
$722$			5.32639e9i			0.526687i
$723$		0			0
$724$		0			0
$725$		−	1.76663e10i		−	1.72172i
$726$	2.10619e9	+	5.00531e9i	0.204277	+	0.485459i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−7.64095e9	−	7.14387e9i	−0.730468	−	0.682947i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	−1.12075e10			−1.03618
$737$		0			0
$738$	3.20271e9	−	3.27529e9i	0.293306	−	0.299952i
$739$	−1.80787e10			−1.64783			−0.823915	−	0.566714i	$-0.808214\pi$
$739$	−1.80787e10			−1.64783			−0.823915	+	0.566714i	$0.808214\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	2.55335e9	+	6.06797e9i	0.227303	+	0.540180i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			3.66680e9i			0.314431i
$753$		0			0
$754$	−1.81761e10			−1.54419
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			2.29017e10i			1.88374i	0.335972	+	0.941872i	$0.390935\pi$
$761$			2.29017e10i			1.88374i	−0.335972	+	0.941872i	$0.609065\pi$
$762$	−4.98209e9	−	1.18398e10i	−0.407915	−	0.969399i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			4.25583e10i			3.40566i
$768$	8.83130e9	−	3.71613e9i	0.703494	−	0.296024i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−5.69485e9	−	1.35337e10i	−0.447499	−	1.06347i
$772$	1.53517e10			1.20087
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	8.37061e9			0.645953
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−2.15128e10	−	8.49025e9i	−1.60151	−	0.632054i
$784$	3.30242e9			0.244752
$785$		0			0
$786$	2.42235e9	−	1.01930e9i	0.177934	−	0.0748729i
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		−	1.44057e10i		−	1.04880i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$			1.40488e10i			0.996016i
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		−	1.50055e10i		−	1.03618i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		−	8.61214e9i		−	0.579346i
$807$	8.21970e9	+	1.95339e10i	0.550552	+	1.30837i
$808$	−1.42478e10			−0.950186
$809$			2.68654e10i			1.78391i	0.452125	+	0.891955i	$0.350666\pi$
$809$			2.68654e10i			1.78391i	−0.452125	+	0.891955i	$0.649334\pi$
$810$		0			0
$811$	2.85023e10			1.87632			0.938161	−	0.346200i	$-0.112528\pi$
$811$	2.85023e10			1.87632			0.938161	+	0.346200i	$0.112528\pi$
$812$		0			0
$813$	1.12764e10	−	4.74503e9i	0.735962	−	0.309686i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	1.08327e10i		−	0.691987i
$819$		0			0
$820$		0			0
$821$			2.16676e10i			1.36650i	0.730185	+	0.683249i	$0.239434\pi$
$821$			2.16676e10i			1.36650i	−0.730185	+	0.683249i	$0.760566\pi$
$822$		0			0
$823$	−2.48605e10			−1.55457			−0.777285	−	0.629149i	$-0.783404\pi$
$823$	−2.48605e10			−1.55457			−0.777285	+	0.629149i	$0.783404\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	8.43919e9	+	8.25219e9i	0.516648	+	0.505200i
$829$	−7.89933e9			−0.481559			−0.240780	−	0.970580i	$-0.577403\pi$
$829$	−7.89933e9			−0.481559			−0.240780	+	0.970580i	$0.577403\pi$
$830$		0			0
$831$	−1.66849e10	+	7.02087e9i	−1.00860	+	0.424412i
$832$	−8.51471e9			−0.512552
$833$		0			0
$834$	1.98392e9	+	4.71474e9i	0.118425	+	0.281434i
$835$		0			0
$836$		0			0
$837$	4.02282e9	−	1.01931e10i	0.237133	−	0.600852i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−3.38845e10			−1.96434
$842$		0			0
$843$		0			0
$844$	2.07772e10			1.18956
$845$		0			0
$846$	−8.33132e9	+	8.52011e9i	−0.473061	+	0.483781i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	1.00771e10	+	2.39480e10i	0.558210	+	1.32657i
$853$	3.38461e10			1.86718			0.933592	−	0.358338i	$-0.116656\pi$
$853$	3.38461e10			1.86718			0.933592	+	0.358338i	$0.116656\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			2.69378e10i			1.46194i	0.682409	+	0.730970i	$0.260932\pi$
$857$			2.69378e10i			1.46194i	−0.682409	+	0.730970i	$0.739068\pi$
$858$		0			0
$859$	3.71366e10			1.99906			0.999530	−	0.0306651i	$-0.00976254\pi$
$859$	3.71366e10			1.99906			0.999530	+	0.0306651i	$0.00976254\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1.52427e10i		−	0.807279i	−0.914918	−	0.403639i	$-0.867745\pi$
$863$		−	1.52427e10i		−	0.807279i	0.914918	−	0.403639i	$-0.132255\pi$
$864$	−1.82726e10	−	7.21147e9i	−0.963833	−	0.380387i
$865$		0			0
$866$		0			0
$867$	1.76875e10	−	7.44275e9i	0.921722	−	0.387852i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−2.40561e10	+	1.01226e10i	−1.20909	+	0.508776i
$877$	−3.97530e10			−1.99008			−0.995042	−	0.0994600i	$-0.968288\pi$
$877$	−3.97530e10			−1.99008			−0.995042	+	0.0994600i	$0.968288\pi$
$878$		−	2.04039e9i		−	0.101738i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	7.67342e9	+	7.50340e9i	0.376573	+	0.368229i
$883$	3.77437e10			1.84494			0.922469	−	0.386072i	$-0.126168\pi$
$883$	3.77437e10			1.84494			0.922469	+	0.386072i	$0.126168\pi$
$884$		0			0
$885$		0			0
$886$	−7.25377e9			−0.350385
$887$		−	4.13487e10i		−	1.98943i	−0.102664	−	0.994716i	$-0.532737\pi$
$887$		−	4.13487e10i		−	1.98943i	0.102664	−	0.994716i	$-0.467263\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−2.90069e10			−1.36843
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.42766e10	−	3.39279e10i	−0.660466	−	1.56958i
$898$	2.28012e10			1.05073
$899$		−	2.42283e10i		−	1.11215i
$900$	−1.10487e10	+	1.12991e10i	−0.505200	+	0.516648i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	7.89430e9	+	1.87606e10i	0.352667	+	0.838105i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	1.69567e10	+	1.65810e10i	0.748803	+	0.732211i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−3.91505e10	+	1.64742e10i	−1.65131	+	0.694857i
$922$	2.33727e10			0.982089
$923$		−	8.10257e10i		−	3.39170i
$924$		0			0
$925$		0			0
$926$			1.02455e10i			0.424028i
$927$		0			0
$928$	−4.34327e10			−1.78402
$929$		−	3.85039e10i		−	1.57561i	−0.615922	−	0.787807i	$-0.711216\pi$
$929$		−	3.85039e10i		−	1.57561i	0.615922	−	0.787807i	$-0.288784\pi$
$930$		0			0
$931$		0			0
$932$		−	3.55235e10i		−	1.43734i
$933$	−1.85623e10	−	4.41129e10i	−0.748250	−	1.77820i
$934$		0			0
$935$		0			0
$936$	2.77130e10	+	2.70989e10i	1.10463	+	1.08016i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	2.05113e10			0.796531
$944$			1.26516e10i			0.489488i
$945$		0			0
$946$		0			0
$947$		−	1.34467e10i		−	0.514505i	−0.966344	−	0.257253i	$-0.917183\pi$
$947$		−	1.34467e10i		−	0.514505i	0.966344	−	0.257253i	$-0.0828172\pi$
$948$		0			0
$949$	8.13914e10			3.09134
$950$		0			0
$951$	1.53321e10	+	3.64364e10i	0.578057	+	1.37374i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$			6.94599e9i			0.257117i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.60328e10			−0.582745
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−3.69056e10			−1.31250			−0.656250	−	0.754544i	$-0.727858\pi$
$967$	−3.69056e10			−1.31250			−0.656250	+	0.754544i	$0.727858\pi$
$968$		−	2.56036e10i		−	0.907272i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	8.44931e9	+	1.88845e10i	0.295114	+	0.659590i
$973$		0			0
$974$		−	1.72285e10i		−	0.597436i
$975$	4.54255e10	−	1.91147e10i	1.56958	−	0.660466i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	1.24790e9	+	2.96560e9i	0.0426573	+	0.101374i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	3.05332e10			1.02892
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−1.99078e10	+	8.37702e9i	−0.666101	+	0.280289i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−6.02745e10			−1.96732			−0.983662	−	0.180023i	$-0.942383\pi$
$991$	−6.02745e10			−1.96732			−0.983662	+	0.180023i	$0.942383\pi$
$992$		−	2.05791e10i		−	0.669323i
$993$	2.87137e10	−	1.20825e10i	0.930607	−	0.391591i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.06654e10			−0.340835			−0.170417	−	0.985372i	$-0.554512\pi$
$997$	−1.06654e10			−0.340835			−0.170417	+	0.985372i	$0.554512\pi$
$998$			3.19203e10i			1.01651i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.c.a.68.4	yes	6
3.2	odd	2	inner	69.8.c.a.68.3	✓	6
23.22	odd	2	CM	69.8.c.a.68.4	yes	6
69.68	even	2	inner	69.8.c.a.68.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.c.a.68.3	✓	6	3.2	odd	2	inner
69.8.c.a.68.3	✓	6	69.68	even	2	inner
69.8.c.a.68.4	yes	6	1.1	even	1	trivial
69.8.c.a.68.4	yes	6	23.22	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+5.95879i q^{2} +(-43.1046 + 18.1381i) q^{3} +92.4928 q^{4} +(-108.081 - 256.851i) q^{6} +1313.87i q^{8} +(1529.02 - 1563.67i) q^{9} +O(q^{10})\)	\(q+5.95879i q^{2} +(-43.1046 + 18.1381i) q^{3} +92.4928 q^{4} +(-108.081 - 256.851i) q^{6} +1313.87i q^{8} +(1529.02 - 1563.67i) q^{9} +(-3986.87 + 1677.64i) q^{12} +13489.2 q^{13} +4010.01 q^{16} +(9317.58 + 9111.12i) q^{18} +58350.9i q^{23} +(-23831.1 - 56633.9i) q^{24} -78125.0 q^{25} +80379.2i q^{26} +(-37546.0 + 95134.9i) q^{27} +226129. i q^{29} -107144. q^{31} +192070. i q^{32} +(141424. - 144628. i) q^{36} +(-581447. + 244668. i) q^{39} -351517. i q^{41} -347701. q^{46} +914412. i q^{47} +(-172850. + 72733.8i) q^{48} +823543. q^{49} -465530. i q^{50} +1.24765e6 q^{52} +(-566889. - 223729. i) q^{54} -1.34746e6 q^{58} +3.15500e6i q^{59} -638447. i q^{62} -631224. q^{64} +(-1.05837e6 - 2.51519e6i) q^{69} -6.00671e6i q^{71} +(2.05446e6 + 2.00894e6i) q^{72} +6.03382e6 q^{73} +(3.36755e6 - 1.41704e6i) q^{75} +(-1.45792e6 - 3.46472e6i) q^{78} +(-107156. - 4.78177e6i) q^{81} +2.09461e6 q^{82} +(-4.10155e6 - 9.74722e6i) q^{87} +5.39704e6i q^{92} +(4.61839e6 - 1.94338e6i) q^{93} -5.44879e6 q^{94} +(-3.48378e6 - 8.27912e6i) q^{96} +4.90732e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 768 q^{4} - 3147 q^{6}+O(q^{10}) \) 6 * q - 768 * q^4 - 3147 * q^6	\( 6 q - 768 q^{4} - 3147 q^{6} - 21435 q^{12} + 98304 q^{16} - 72177 q^{18} + 402816 q^{24} - 468750 q^{25} - 225276 q^{27} + 1621587 q^{36} - 2196456 q^{39} - 866433 q^{48} + 4941258 q^{49} + 5680578 q^{52} - 12795270 q^{58} - 13290570 q^{64} + 9238656 q^{72} + 8715747 q^{78} + 59301462 q^{82} - 29588784 q^{87} + 30939042 q^{93} - 87793602 q^{94} - 59843631 q^{96}+O(q^{100}) \) 6 * q - 768 * q^4 - 3147 * q^6 - 21435 * q^12 + 98304 * q^16 - 72177 * q^18 + 402816 * q^24 - 468750 * q^25 - 225276 * q^27 + 1621587 * q^36 - 2196456 * q^39 - 866433 * q^48 + 4941258 * q^49 + 5680578 * q^52 - 12795270 * q^58 - 13290570 * q^64 + 9238656 * q^72 + 8715747 * q^78 + 59301462 * q^82 - 29588784 * q^87 + 30939042 * q^93 - 87793602 * q^94 - 59843631 * q^96

Introduction

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