LMFDB - Embedded newform 75.9.d.d.74.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,9,Mod(74,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("75.74");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	75.d (of order $2$, degree $1$, not 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:	$30.5533957546$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 3278 x^{18} + 4245491 x^{16} + 2854629536 x^{14} + 1117319469691 x^{12} + 266849787342054 x^{10} + \cdots + 89\!\cdots\!00 $ x^20 + 3278x^18 + 4245491x^16 + 2854629536x^14 + 1117319469691x^12 + 266849787342054x^10 + 38935163332201981x^8 + 3308855301815965532x^6 + 143709036159536320436x^4 + 2262672788809149782000*x^2 + 8995759125454185640000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{22}\cdot 5^{26} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			74.16
Root			$30.4704i$ of defining polynomial
Character	$\chi$	$=$	75.74
Dual form			75.9.d.d.74.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.2182 q^{2} +(40.7889 + 69.9805i) q^{3} +75.9021 q^{4} +(743.099 + 1274.92i) q^{6} -4676.68i q^{7} -3281.06 q^{8} +(-3233.54 + 5708.85i) q^{9} +O(q^{10})$	\(q+18.2182 q^{2} +(40.7889 + 69.9805i) q^{3} +75.9021 q^{4} +(743.099 + 1274.92i) q^{6} -4676.68i q^{7} -3281.06 q^{8} +(-3233.54 + 5708.85i) q^{9} -25315.1i q^{11} +(3095.96 + 5311.66i) q^{12} -600.138i q^{13} -85200.7i q^{14} -79205.8 q^{16} -73234.6 q^{17} +(-58909.2 + 104005. i) q^{18} +38440.3 q^{19} +(327277. - 190757. i) q^{21} -461196. i q^{22} +196713. q^{23} +(-133831. - 229610. i) q^{24} -10933.4i q^{26} +(-531400. + 6572.81i) q^{27} -354970. i q^{28} -504953. i q^{29} +665132. q^{31} -603035. q^{32} +(1.77157e6 - 1.03258e6i) q^{33} -1.33420e6 q^{34} +(-245432. + 433313. i) q^{36} -351425. i q^{37} +700312. q^{38} +(41997.9 - 24478.9i) q^{39} +2.88219e6i q^{41} +(5.96238e6 - 3.47524e6i) q^{42} -1.17585e6i q^{43} -1.92147e6i q^{44} +3.58375e6 q^{46} +4.20282e6 q^{47} +(-3.23071e6 - 5.54286e6i) q^{48} -1.61066e7 q^{49} +(-2.98715e6 - 5.12499e6i) q^{51} -45551.7i q^{52} -6.57665e6 q^{53} +(-9.68115e6 + 119745. i) q^{54} +1.53445e7i q^{56} +(1.56794e6 + 2.69007e6i) q^{57} -9.19932e6i q^{58} -1.46872e7i q^{59} -1.09139e7 q^{61} +1.21175e7 q^{62} +(2.66985e7 + 1.51222e7i) q^{63} +9.29049e6 q^{64} +(3.22747e7 - 1.88116e7i) q^{66} -1.82376e7i q^{67} -5.55866e6 q^{68} +(8.02369e6 + 1.37661e7i) q^{69} +1.95691e7i q^{71} +(1.06094e7 - 1.87311e7i) q^{72} -3.13857e7i q^{73} -6.40233e6i q^{74} +2.91770e6 q^{76} -1.18391e8 q^{77} +(765126. - 445962. i) q^{78} +1.63328e6 q^{79} +(-2.21352e7 - 3.69196e7i) q^{81} +5.25082e7i q^{82} +3.92795e7 q^{83} +(2.48410e7 - 1.44788e7i) q^{84} -2.14218e7i q^{86} +(3.53369e7 - 2.05965e7i) q^{87} +8.30604e7i q^{88} +1.89625e7i q^{89} -2.80665e6 q^{91} +1.49309e7 q^{92} +(2.71300e7 + 4.65462e7i) q^{93} +7.65678e7 q^{94} +(-2.45971e7 - 4.22007e7i) q^{96} +1.16684e8i q^{97} -2.93432e8 q^{98} +(1.44520e8 + 8.18574e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9}+O(q^{10}) $ 20 * q + 3108 * q^4 + 4514 * q^6 + 22414 * q^9	$ 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9} + 89268 q^{16} + 287868 q^{19} + 1346856 q^{21} + 2033718 q^{24} - 6028120 q^{31} - 9954292 q^{34} + 9001054 q^{36} + 15026564 q^{39} - 27877272 q^{46} - 17907092 q^{49} + 12418574 q^{51} + 17772544 q^{54} + 3040440 q^{61} + 9073996 q^{64} + 20930590 q^{66} - 22789956 q^{69} - 59058092 q^{76} - 17099792 q^{79} + 45225890 q^{81} + 273766584 q^{84} - 214448528 q^{91} - 712237192 q^{94} + 1050848002 q^{96} + 764842670 q^{99}+O(q^{100}) $ 20 * q + 3108 * q^4 + 4514 * q^6 + 22414 * q^9 + 89268 * q^16 + 287868 * q^19 + 1346856 * q^21 + 2033718 * q^24 - 6028120 * q^31 - 9954292 * q^34 + 9001054 * q^36 + 15026564 * q^39 - 27877272 * q^46 - 17907092 * q^49 + 12418574 * q^51 + 17772544 * q^54 + 3040440 * q^61 + 9073996 * q^64 + 20930590 * q^66 - 22789956 * q^69 - 59058092 * q^76 - 17099792 * q^79 + 45225890 * q^81 + 273766584 * q^84 - 214448528 * q^91 - 712237192 * q^94 + 1050848002 * q^96 + 764842670 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$-1$

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 $

$2$	18.2182			1.13864			0.569318	−	0.822117i	$-0.307207\pi$
$2$	18.2182			1.13864			0.569318	+	0.822117i	$0.307207\pi$
$3$	40.7889	+	69.9805i	0.503566	+	0.863957i
$3$	40.7889	+	69.9805i	0.503566	+	0.863957i
$4$	75.9021			0.296492
$5$		0			0
$5$		0			0
$6$	743.099	+	1274.92i	0.573379	+	0.983732i
$7$		−	4676.68i		−	1.94781i	−0.226964	−	0.973903i	$-0.572880\pi$
$7$		−	4676.68i		−	1.94781i	0.226964	−	0.973903i	$-0.427120\pi$
$8$	−3281.06			−0.801039
$9$	−3233.54	+	5708.85i	−0.492842	+	0.870119i
$10$		0			0
$11$		−	25315.1i		−	1.72906i	−0.502583	−	0.864529i	$-0.667617\pi$
$11$		−	25315.1i		−	1.72906i	0.502583	−	0.864529i	$-0.332383\pi$
$12$	3095.96	+	5311.66i	0.149304	+	0.256157i
$13$		−	600.138i		−	0.0210125i	−0.999945	−	0.0105062i	$-0.996656\pi$
$13$		−	600.138i		−	0.0210125i	0.999945	−	0.0105062i	$-0.00334430\pi$
$14$		−	85200.7i		−	2.21784i
$15$		0			0
$16$	−79205.8			−1.20858
$17$	−73234.6			−0.876840			−0.438420	−	0.898770i	$-0.644462\pi$
$17$	−73234.6			−0.876840			−0.438420	+	0.898770i	$0.644462\pi$
$18$	−58909.2	+	104005.i	−0.561168	+	0.990749i
$19$	38440.3			0.294966			0.147483	−	0.989065i	$-0.452883\pi$
$19$	38440.3			0.294966			0.147483	+	0.989065i	$0.452883\pi$
$20$		0			0
$21$	327277.	−	190757.i	1.68282	−	0.980849i
$22$		−	461196.i		−	1.96877i
$23$	196713.			0.702945			0.351473	−	0.936198i	$-0.385681\pi$
$23$	196713.			0.702945			0.351473	+	0.936198i	$0.385681\pi$
$24$	−133831.	−	229610.i	−0.403376	−	0.692063i
$25$		0			0
$26$		−	10933.4i		−	0.0239256i
$27$	−531400.	+	6572.81i	−0.999924	+	0.0123679i
$28$		−	354970.i		−	0.577510i
$29$		−	504953.i		−	0.713935i	−0.934117	−	0.356968i	$-0.883811\pi$
$29$		−	504953.i		−	0.713935i	0.934117	−	0.356968i	$-0.116189\pi$
$30$		0			0
$31$	665132.			0.720213			0.360106	−	0.932911i	$-0.382740\pi$
$31$	665132.			0.720213			0.360106	+	0.932911i	$0.382740\pi$
$32$	−603035.			−0.575099
$33$	1.77157e6	−	1.03258e6i	1.49383	−	0.870695i
$34$	−1.33420e6			−0.998402
$35$		0			0
$36$	−245432.	+	433313.i	−0.146124	+	0.257984i
$37$		−	351425.i		−	0.187511i	−0.995595	−	0.0937553i	$-0.970113\pi$
$37$		−	351425.i		−	0.187511i	0.995595	−	0.0937553i	$-0.0298871\pi$
$38$	700312.			0.335859
$39$	41997.9	−	24478.9i	0.0181539	−	0.0105812i
$40$		0			0
$41$			2.88219e6i			1.01997i	0.860184	+	0.509984i	$0.170349\pi$
$41$			2.88219e6i			1.01997i	−0.860184	+	0.509984i	$0.829651\pi$
$42$	5.96238e6	−	3.47524e6i	1.91612	−	1.11683i
$43$		−	1.17585e6i		−	0.343935i	−0.985103	−	0.171968i	$-0.944988\pi$
$43$		−	1.17585e6i		−	0.343935i	0.985103	−	0.171968i	$-0.0550124\pi$
$44$		−	1.92147e6i		−	0.512652i
$45$		0			0
$46$	3.58375e6			0.800399
$47$	4.20282e6			0.861291			0.430645	−	0.902521i	$-0.358286\pi$
$47$	4.20282e6			0.861291			0.430645	+	0.902521i	$0.358286\pi$
$48$	−3.23071e6	−	5.54286e6i	−0.608602	−	1.04416i
$49$	−1.61066e7			−2.79395
$50$		0			0
$51$	−2.98715e6	−	5.12499e6i	−0.441547	−	0.757552i
$52$		−	45551.7i		−	0.00623005i
$53$	−6.57665e6			−0.833491			−0.416746	−	0.909023i	$-0.636829\pi$
$53$	−6.57665e6			−0.833491			−0.416746	+	0.909023i	$0.636829\pi$
$54$	−9.68115e6	+	119745.i	−1.13855	+	0.0140825i
$55$		0			0
$56$			1.53445e7i			1.56027i
$57$	1.56794e6	+	2.69007e6i	0.148535	+	0.254838i
$58$		−	9.19932e6i		−	0.812913i
$59$		−	1.46872e7i		−	1.21208i	−0.795434	−	0.606040i	$-0.792757\pi$
$59$		−	1.46872e7i		−	1.21208i	0.795434	−	0.606040i	$-0.207243\pi$
$60$		0			0
$61$	−1.09139e7			−0.788244			−0.394122	−	0.919058i	$-0.628951\pi$
$61$	−1.09139e7			−0.788244			−0.394122	+	0.919058i	$0.628951\pi$
$62$	1.21175e7			0.820060
$63$	2.66985e7	+	1.51222e7i	1.69482	+	0.959961i
$64$	9.29049e6			0.553756
$65$		0			0
$66$	3.22747e7	−	1.88116e7i	1.70093	−	0.991405i
$67$		−	1.82376e7i		−	0.905041i	−0.891754	−	0.452521i	$-0.850525\pi$
$67$		−	1.82376e7i		−	0.905041i	0.891754	−	0.452521i	$-0.149475\pi$
$68$	−5.55866e6			−0.259977
$69$	8.02369e6	+	1.37661e7i	0.353979	+	0.607314i
$70$		0			0
$71$			1.95691e7i			0.770084i	0.922899	+	0.385042i	$0.125813\pi$
$71$			1.95691e7i			0.770084i	−0.922899	+	0.385042i	$0.874187\pi$
$72$	1.06094e7	−	1.87311e7i	0.394786	−	0.696999i
$73$		−	3.13857e7i		−	1.10520i	−0.833447	−	0.552600i	$-0.813636\pi$
$73$		−	3.13857e7i		−	1.10520i	0.833447	−	0.552600i	$-0.186364\pi$
$74$		−	6.40233e6i		−	0.213506i
$75$		0			0
$76$	2.91770e6			0.0874552
$77$	−1.18391e8			−3.36787
$78$	765126.	−	445962.i	0.0206707	−	0.0120481i
$79$	1.63328e6			0.0419327			0.0209663	−	0.999780i	$-0.493326\pi$
$79$	1.63328e6			0.0419327			0.0209663	+	0.999780i	$0.493326\pi$
$80$		0			0
$81$	−2.21352e7	−	3.69196e7i	−0.514213	−	0.857663i
$82$			5.25082e7i			1.16137i
$83$	3.92795e7			0.827663			0.413832	−	0.910353i	$-0.364190\pi$
$83$	3.92795e7			0.827663			0.413832	+	0.910353i	$0.364190\pi$
$84$	2.48410e7	−	1.44788e7i	0.498943	−	0.290814i
$85$		0			0
$86$		−	2.14218e7i		−	0.391617i
$87$	3.53369e7	−	2.05965e7i	0.616809	−	0.359514i
$88$			8.30604e7i			1.38504i
$89$			1.89625e7i			0.302229i	0.988516	+	0.151115i	$0.0482862\pi$
$89$			1.89625e7i			0.302229i	−0.988516	+	0.151115i	$0.951714\pi$
$90$		0			0
$91$	−2.80665e6			−0.0409283
$92$	1.49309e7			0.208418
$93$	2.71300e7	+	4.65462e7i	0.362675	+	0.622233i
$94$	7.65678e7			0.980697
$95$		0			0
$96$	−2.45971e7	−	4.22007e7i	−0.289600	−	0.496861i
$97$			1.16684e8i			1.31802i	0.752133	+	0.659012i	$0.229025\pi$
$97$			1.16684e8i			1.31802i	−0.752133	+	0.659012i	$0.770975\pi$
$98$	−2.93432e8			−3.18129
$99$	1.44520e8	+	8.18574e7i	1.50449	+	0.852153i
$100$		0			0
$101$			8.62148e7i			0.828507i	0.910161	+	0.414254i	$0.135957\pi$
$101$			8.62148e7i			0.828507i	−0.910161	+	0.414254i	$0.864043\pi$
$102$	−5.44205e7	−	9.33680e7i	−0.502762	−	0.862576i
$103$			9.28352e7i			0.824829i	0.910996	+	0.412415i	$0.135314\pi$
$103$			9.28352e7i			0.824829i	−0.910996	+	0.412415i	$0.864686\pi$
$104$			1.96909e6i			0.0168318i
$105$		0			0
$106$	−1.19815e8			−0.949043
$107$	1.95590e8			1.49214			0.746072	−	0.665865i	$-0.231937\pi$
$107$	1.95590e8			1.49214			0.746072	+	0.665865i	$0.231937\pi$
$108$	−4.03344e7	+	498889.i	−0.296470	+	0.00366699i
$109$	−1.62524e8			−1.15136			−0.575682	−	0.817674i	$-0.695263\pi$
$109$	−1.62524e8			−1.15136			−0.575682	+	0.817674i	$0.695263\pi$
$110$		0			0
$111$	2.45929e7	−	1.43342e7i	0.162001	−	0.0944240i
$112$			3.70420e8i			2.35409i
$113$	1.97203e8			1.20949			0.604743	−	0.796421i	$-0.293276\pi$
$113$	1.97203e8			1.20949			0.604743	+	0.796421i	$0.293276\pi$
$114$	2.85649e7	+	4.90082e7i	0.169127	+	0.290168i
$115$		0			0
$116$		−	3.83270e7i		−	0.211676i
$117$	3.42610e6	+	1.94057e6i	0.0182834	+	0.0103558i
$118$		−	2.67574e8i		−	1.38012i
$119$			3.42495e8i			1.70792i
$120$		0			0
$121$	−4.26497e8			−1.98964
$122$	−1.98831e8			−0.897523
$123$	−2.01697e8	+	1.17561e8i	−0.881209	+	0.513622i
$124$	5.04849e7			0.213538
$125$		0			0
$126$	4.86398e8	+	2.75500e8i	1.92979	+	1.09305i
$127$		−	2.85596e8i		−	1.09784i	−0.835876	−	0.548918i	$-0.815040\pi$
$127$		−	2.85596e8i		−	1.09784i	0.835876	−	0.548918i	$-0.184960\pi$
$128$	3.23633e8			1.20563
$129$	8.22863e7	−	4.79614e7i	0.297145	−	0.173194i
$130$		0			0
$131$		−	9.66770e7i		−	0.328275i	−0.986437	−	0.164138i	$-0.947516\pi$
$131$		−	9.66770e7i		−	0.328275i	0.986437	−	0.164138i	$-0.0524841\pi$
$132$	1.34465e8	−	7.83746e7i	0.442909	−	0.258154i
$133$		−	1.79773e8i		−	0.574537i
$134$		−	3.32256e8i		−	1.03051i
$135$		0			0
$136$	2.40287e8			0.702383
$137$	1.84457e8			0.523616			0.261808	−	0.965120i	$-0.415681\pi$
$137$	1.84457e8			0.523616			0.261808	+	0.965120i	$0.415681\pi$
$138$	1.46177e8	+	2.50793e8i	0.403054	+	0.691510i
$139$	1.16522e7			0.0312138			0.0156069	−	0.999878i	$-0.495032\pi$
$139$	1.16522e7			0.0312138			0.0156069	+	0.999878i	$0.495032\pi$
$140$		0			0
$141$	1.71428e8	+	2.94116e8i	0.433717	+	0.744118i
$142$			3.56514e8i			0.876845i
$143$	−1.51926e7			−0.0363318
$144$	2.56115e8	−	4.52174e8i	0.595642	−	1.05161i
$145$		0			0
$146$		−	5.71791e8i		−	1.25842i
$147$	−6.56968e8	−	1.12715e9i	−1.40694	−	2.41385i
$148$		−	2.66739e7i		−	0.0555955i
$149$		−	3.58376e8i		−	0.727099i	−0.931575	−	0.363550i	$-0.881565\pi$
$149$		−	3.58376e8i		−	0.727099i	0.931575	−	0.363550i	$-0.118435\pi$
$150$		0			0
$151$	−1.60549e8			−0.308816			−0.154408	−	0.988007i	$-0.549347\pi$
$151$	−1.60549e8			−0.308816			−0.154408	+	0.988007i	$0.549347\pi$
$152$	−1.26125e8			−0.236279
$153$	2.36807e8	−	4.18085e8i	0.432144	−	0.762955i
$154$	−2.15687e9			−3.83478
$155$		0			0
$156$	3.18773e6	−	1.85800e6i	0.00538249	−	0.00313724i
$157$			2.60289e8i			0.428408i	0.976789	+	0.214204i	$0.0687157\pi$
$157$			2.60289e8i			0.428408i	−0.976789	+	0.214204i	$0.931284\pi$
$158$	2.97554e7			0.0477461
$159$	−2.68254e8	−	4.60237e8i	−0.419718	−	0.720100i
$160$		0			0
$161$		−	9.19964e8i		−	1.36920i
$162$	−4.03263e8	−	6.72607e8i	−0.585501	−	0.976566i
$163$		−	7.22321e8i		−	1.02325i	−0.859210	−	0.511623i	$-0.829045\pi$
$163$		−	7.22321e8i		−	1.02325i	0.859210	−	0.511623i	$-0.170955\pi$
$164$			2.18764e8i			0.302413i
$165$		0			0
$166$	7.15601e8			0.942408
$167$	1.35239e9			1.73874			0.869370	−	0.494161i	$-0.164525\pi$
$167$	1.35239e9			1.73874			0.869370	+	0.494161i	$0.164525\pi$
$168$	−1.07381e9	+	6.25883e8i	−1.34801	+	0.785699i
$169$	8.15371e8			0.999558
$170$		0			0
$171$	−1.24298e8	+	2.19450e8i	−0.145372	+	0.256656i
$172$		−	8.92491e7i		−	0.101974i
$173$	1.05877e9			1.18200			0.591000	−	0.806671i	$-0.298733\pi$
$173$	1.05877e9			1.18200			0.591000	+	0.806671i	$0.298733\pi$
$174$	6.43773e8	−	3.75230e8i	0.702321	−	0.409355i
$175$		0			0
$176$			2.00511e9i			2.08971i
$177$	1.02782e9	−	5.99075e8i	1.04719	−	0.610363i
$178$			3.45463e8i			0.344129i
$179$		−	6.84450e8i		−	0.666699i	−0.942803	−	0.333349i	$-0.891821\pi$
$179$		−	6.84450e8i		−	0.666699i	0.942803	−	0.333349i	$-0.108179\pi$
$180$		0			0
$181$	−8.83562e8			−0.823233			−0.411617	−	0.911357i	$-0.635036\pi$
$181$	−8.83562e8			−0.823233			−0.411617	+	0.911357i	$0.635036\pi$
$182$	−5.11321e7			−0.0466024
$183$	−4.45166e8	−	7.63760e8i	−0.396933	−	0.681009i
$184$	−6.45426e8			−0.563087
$185$		0			0
$186$	4.94258e8	+	8.47988e8i	0.412955	+	0.708497i
$187$			1.85394e9i			1.51611i
$188$	3.19003e8			0.255366
$189$	3.07389e7	+	2.48519e9i	0.0240903	+	1.94766i
$190$		0			0
$191$		−	1.18467e9i		−	0.890148i	−0.895494	−	0.445074i	$-0.853177\pi$
$191$		−	1.18467e9i		−	0.890148i	0.895494	−	0.445074i	$-0.146823\pi$
$192$	3.78948e8	+	6.50153e8i	0.278853	+	0.478421i
$193$		−	1.65487e9i		−	1.19271i	−0.802721	−	0.596354i	$-0.796615\pi$
$193$		−	1.65487e9i		−	1.19271i	0.802721	−	0.596354i	$-0.203385\pi$
$194$			2.12576e9i			1.50075i
$195$		0			0
$196$	−1.22252e9			−0.828385
$197$	−8.60897e8			−0.571592			−0.285796	−	0.958290i	$-0.592258\pi$
$197$	−8.60897e8			−0.571592			−0.285796	+	0.958290i	$0.592258\pi$
$198$	2.63290e9	+	1.49129e9i	1.71306	+	0.970292i
$199$	−7.00975e8			−0.446982			−0.223491	−	0.974706i	$-0.571745\pi$
$199$	−7.00975e8			−0.446982			−0.223491	+	0.974706i	$0.571745\pi$
$200$		0			0
$201$	1.27628e9	−	7.43891e8i	0.781916	−	0.455748i
$202$			1.57068e9i			0.943369i
$203$	−2.36151e9			−1.39061
$204$	−2.26731e8	−	3.88997e8i	−0.130915	−	0.224608i
$205$		0			0
$206$			1.69129e9i			0.939180i
$207$	−6.36079e8	+	1.12300e9i	−0.346441	+	0.611646i
$208$			4.75344e7i			0.0253954i
$209$		−	9.73121e8i		−	0.510013i
$210$		0			0
$211$	−3.15102e9			−1.58972			−0.794862	−	0.606790i	$-0.792457\pi$
$211$	−3.15102e9			−1.58972			−0.794862	+	0.606790i	$0.792457\pi$
$212$	−4.99181e8			−0.247124
$213$	−1.36946e9	+	7.98202e8i	−0.665319	+	0.387788i
$214$	3.56329e9			1.69901
$215$		0			0
$216$	1.74355e9	−	2.15657e7i	0.800978	−	0.00990717i
$217$		−	3.11061e9i		−	1.40284i
$218$	−2.96090e9			−1.31098
$219$	2.19639e9	−	1.28019e9i	0.954845	−	0.556541i
$220$		0			0
$221$			4.39508e7i			0.0184246i
$222$	4.48038e8	−	2.61144e8i	0.184460	−	0.107515i
$223$			3.67974e9i			1.48798i	0.668191	+	0.743990i	$0.267069\pi$
$223$			3.67974e9i			1.48798i	−0.668191	+	0.743990i	$0.732931\pi$
$224$			2.82020e9i			1.12018i
$225$		0			0
$226$	3.59269e9			1.37716
$227$	−5.71593e8			−0.215270			−0.107635	−	0.994190i	$-0.534328\pi$
$227$	−5.71593e8			−0.215270			−0.107635	+	0.994190i	$0.534328\pi$
$228$	1.19009e8	+	2.04182e8i	0.0440395	+	0.0755575i
$229$	2.94346e9			1.07033			0.535164	−	0.844748i	$-0.320250\pi$
$229$	2.94346e9			1.07033			0.535164	+	0.844748i	$0.320250\pi$
$230$		0			0
$231$	−4.82903e9	−	8.28505e9i	−1.69594	−	2.90969i
$232$			1.65678e9i			0.571890i
$233$	2.11678e9			0.718210			0.359105	−	0.933297i	$-0.383082\pi$
$233$	2.11678e9			0.718210			0.359105	+	0.933297i	$0.383082\pi$
$234$	6.24172e7	+	3.53536e7i	0.0208181	+	0.0117915i
$235$		0			0
$236$		−	1.11479e9i		−	0.359373i
$237$	6.66197e7	+	1.14298e8i	0.0211159	+	0.0362280i
$238$			6.23963e9i			1.94469i
$239$		−	3.95083e9i		−	1.21087i	−0.795896	−	0.605434i	$-0.793000\pi$
$239$		−	3.95083e9i		−	1.21087i	0.795896	−	0.605434i	$-0.207000\pi$
$240$		0			0
$241$	6.74602e7			0.0199977			0.00999884	−	0.999950i	$-0.496817\pi$
$241$	6.74602e7			0.0199977			0.00999884	+	0.999950i	$0.496817\pi$
$242$	−7.77000e9			−2.26548
$243$	1.68078e9	−	3.05494e9i	0.482043	−	0.876148i
$244$	−8.28388e8			−0.233708
$245$		0			0
$246$	−3.67455e9	+	2.14175e9i	−1.00338	+	0.584828i
$247$		−	2.30695e7i		−	0.00619797i
$248$	−2.18233e9			−0.576919
$249$	1.60217e9	+	2.74880e9i	0.416783	+	0.715065i
$250$		0			0
$251$			3.35762e9i			0.845934i	0.906145	+	0.422967i	$0.139011\pi$
$251$			3.35762e9i			0.845934i	−0.906145	+	0.422967i	$0.860989\pi$
$252$	2.02647e9	+	1.14781e9i	0.502502	+	0.284621i
$253$		−	4.97981e9i		−	1.21543i
$254$		−	5.20304e9i		−	1.25004i
$255$		0			0
$256$	3.51763e9			0.819013
$257$	−5.89201e8			−0.135061			−0.0675307	−	0.997717i	$-0.521512\pi$
$257$	−5.89201e8			−0.135061			−0.0675307	+	0.997717i	$0.521512\pi$
$258$	1.49911e9	−	8.73769e8i	0.338340	−	0.197205i
$259$	−1.64350e9			−0.365235
$260$		0			0
$261$	2.88270e9	+	1.63278e9i	0.621209	+	0.351858i
$262$		−	1.76128e9i		−	0.373786i
$263$	9.46868e8			0.197909			0.0989547	−	0.995092i	$-0.468450\pi$
$263$	9.46868e8			0.197909			0.0989547	+	0.995092i	$0.468450\pi$
$264$	−5.81261e9	+	3.38794e9i	−1.19662	+	0.697461i
$265$		0			0
$266$		−	3.27514e9i		−	0.654189i
$267$	−1.32701e9	+	7.73460e8i	−0.261113	+	0.152192i
$268$		−	1.38427e9i		−	0.268338i
$269$			4.99503e9i			0.953958i	0.878915	+	0.476979i	$0.158268\pi$
$269$			4.99503e9i			0.953958i	−0.878915	+	0.476979i	$0.841732\pi$
$270$		0			0
$271$	5.27982e9			0.978908			0.489454	−	0.872029i	$-0.337196\pi$
$271$	5.27982e9			0.978908			0.489454	+	0.872029i	$0.337196\pi$
$272$	5.80060e9			1.05974
$273$	−1.14480e8	−	1.96411e8i	−0.0206101	−	0.0353603i
$274$	3.36047e9			0.596208
$275$		0			0
$276$	6.09015e8	+	1.04487e9i	0.104952	+	0.180064i
$277$		−	9.37450e9i		−	1.59231i	−0.605090	−	0.796157i	$-0.706863\pi$
$277$		−	9.37450e9i		−	1.59231i	0.605090	−	0.796157i	$-0.293137\pi$
$278$	2.12281e8			0.0355412
$279$	−2.15073e9	+	3.79714e9i	−0.354951	+	0.626671i
$280$		0			0
$281$			1.16074e10i			1.86170i	0.365396	+	0.930852i	$0.380934\pi$
$281$			1.16074e10i			1.86170i	−0.365396	+	0.930852i	$0.619066\pi$
$282$	3.12311e9	+	5.35825e9i	0.493846	+	0.847280i
$283$		−	1.65370e9i		−	0.257817i	−0.991656	−	0.128908i	$-0.958853\pi$
$283$		−	1.65370e9i		−	0.257817i	0.991656	−	0.128908i	$-0.0411473\pi$
$284$			1.48534e9i			0.228324i
$285$		0			0
$286$	−2.76781e8			−0.0413687
$287$	1.34791e10			1.98670
$288$	1.94994e9	−	3.44264e9i	0.283433	−	0.500404i
$289$	−1.61245e9			−0.231151
$290$		0			0
$291$	−8.16558e9	+	4.75939e9i	−1.13872	+	0.663712i
$292$		−	2.38224e9i		−	0.327683i
$293$	9.59915e9			1.30245			0.651227	−	0.758883i	$-0.274255\pi$
$293$	9.59915e9			1.30245			0.651227	+	0.758883i	$0.274255\pi$
$294$	−1.19688e10	−	2.05345e10i	−1.60199	−	2.74850i
$295$		0			0
$296$			1.15305e9i			0.150203i
$297$	1.66391e8	+	1.34525e10i	0.0213848	+	1.72893i
$298$		−	6.52896e9i		−	0.827901i
$299$		−	1.18055e8i		−	0.0147706i
$300$		0			0
$301$	−5.49906e9			−0.669919
$302$	−2.92491e9			−0.351629
$303$	−6.03336e9	+	3.51660e9i	−0.715795	+	0.417208i
$304$	−3.04469e9			−0.356492
$305$		0			0
$306$	4.31419e9	−	7.61675e9i	0.492055	−	0.868728i
$307$		−	8.74107e9i		−	0.984036i	−0.870585	−	0.492018i	$-0.836259\pi$
$307$		−	8.74107e9i		−	0.984036i	0.870585	−	0.492018i	$-0.163741\pi$
$308$	−8.98611e9			−0.998548
$309$	−6.49666e9	+	3.78664e9i	−0.712617	+	0.415356i
$310$		0			0
$311$		−	1.25430e10i		−	1.34079i	−0.742006	−	0.670394i	$-0.766125\pi$
$311$		−	1.25430e10i		−	1.34079i	0.742006	−	0.670394i	$-0.233875\pi$
$312$	−1.37798e8	+	8.03168e7i	−0.0145420	+	0.00847594i
$313$			3.08002e9i			0.320905i	0.987044	+	0.160452i	$0.0512953\pi$
$313$			3.08002e9i			0.320905i	−0.987044	+	0.160452i	$0.948705\pi$
$314$			4.74199e9i			0.487801i
$315$		0			0
$316$	1.23969e8			0.0124327
$317$	2.81482e9			0.278749			0.139375	−	0.990240i	$-0.455491\pi$
$317$	2.81482e9			0.278749			0.139375	+	0.990240i	$0.455491\pi$
$318$	−4.88710e9	−	8.38468e9i	−0.477906	−	0.819932i
$319$	−1.27830e10			−1.23444
$320$		0			0
$321$	7.97788e9	+	1.36875e10i	0.751393	+	1.28915i
$322$		−	1.67601e10i		−	1.55902i
$323$	−2.81516e9			−0.258638
$324$	−1.68011e9	−	2.80227e9i	−0.152460	−	0.254290i
$325$		0			0
$326$		−	1.31594e10i		−	1.16510i
$327$	−6.62918e9	−	1.13735e10i	−0.579788	−	0.994728i
$328$		−	9.45662e9i		−	0.817035i
$329$		−	1.96553e10i		−	1.67763i
$330$		0			0
$331$	2.09451e10			1.74490			0.872452	−	0.488701i	$-0.162529\pi$
$331$	2.09451e10			1.74490			0.872452	+	0.488701i	$0.162529\pi$
$332$	2.98140e9			0.245396
$333$	2.00623e9	+	1.13635e9i	0.163157	+	0.0924132i
$334$	2.46380e10			1.97979
$335$		0			0
$336$	−2.59222e10	+	1.51090e10i	−2.03383	+	1.18544i
$337$		−	1.91729e7i		−	0.00148651i	−1.00000		0.000743255i	$-0.999763\pi$
$337$		−	1.91729e7i		−	0.00148651i	1.00000		0.000743255i	$-0.000236585\pi$
$338$	1.48546e10			1.13813
$339$	8.04370e9	+	1.38004e10i	0.609056	+	1.04494i
$340$		0			0
$341$		−	1.68379e10i		−	1.24529i
$342$	−2.26449e9	+	3.99797e9i	−0.165526	+	0.292237i
$343$			4.83652e10i			3.49427i
$344$			3.85802e9i			0.275506i
$345$		0			0
$346$	1.92889e10			1.34587
$347$	−1.16312e10			−0.802247			−0.401124	−	0.916024i	$-0.631380\pi$
$347$	−1.16312e10			−0.802247			−0.401124	+	0.916024i	$0.631380\pi$
$348$	2.68214e9	−	1.56331e9i	0.182879	−	0.106593i
$349$	2.89314e9			0.195015			0.0975073	−	0.995235i	$-0.468913\pi$
$349$	2.89314e9			0.195015			0.0975073	+	0.995235i	$0.468913\pi$
$350$		0			0
$351$	3.94459e6	+	3.18913e8i	0.000259880	+	0.0210109i
$352$			1.52659e10i			0.994379i
$353$	2.73144e10			1.75911			0.879555	−	0.475798i	$-0.157841\pi$
$353$	2.73144e10			1.75911			0.879555	+	0.475798i	$0.157841\pi$
$354$	1.87250e10	−	1.09141e10i	1.19236	−	0.694981i
$355$		0			0
$356$			1.43929e9i			0.0896086i
$357$	−2.39680e10	+	1.39700e10i	−1.47556	+	0.860048i
$358$		−	1.24694e10i		−	0.759128i
$359$			4.28333e9i			0.257872i	0.991653	+	0.128936i	$0.0411562\pi$
$359$			4.28333e9i			0.257872i	−0.991653	+	0.128936i	$0.958844\pi$
$360$		0			0
$361$	−1.55059e10			−0.912995
$362$	−1.60969e10			−0.937363
$363$	−1.73963e10	−	2.98465e10i	−1.00192	−	1.71896i
$364$	−2.13031e8			−0.0121349
$365$		0			0
$366$	−8.11011e9	−	1.39143e10i	−0.451962	−	0.775421i
$367$			1.47834e10i			0.814911i	0.913225	+	0.407456i	$0.133584\pi$
$367$			1.47834e10i			0.814911i	−0.913225	+	0.407456i	$0.866416\pi$
$368$	−1.55808e10			−0.849569
$369$	−1.64540e10	−	9.31967e9i	−0.887494	−	0.502684i
$370$		0			0
$371$			3.07569e10i			1.62348i
$372$	2.05922e9	+	3.53296e9i	0.107530	+	0.184487i
$373$		−	9.06733e9i		−	0.468429i	−0.972185	−	0.234215i	$-0.924748\pi$
$373$		−	9.06733e9i		−	0.468429i	0.972185	−	0.234215i	$-0.0752518\pi$
$374$			3.37755e10i			1.72629i
$375$		0			0
$376$	−1.37897e10			−0.689928
$377$	−3.03041e8			−0.0150016
$378$	5.60007e8	+	4.52757e10i	0.0274300	+	2.21767i
$379$	3.91064e10			1.89536			0.947678	−	0.319227i	$-0.103423\pi$
$379$	3.91064e10			1.89536			0.947678	+	0.319227i	$0.103423\pi$
$380$		0			0
$381$	1.99862e10	−	1.16491e10i	0.948482	−	0.552833i
$382$		−	2.15824e10i		−	1.01355i
$383$	−1.25913e10			−0.585162			−0.292581	−	0.956241i	$-0.594514\pi$
$383$	−1.25913e10			−0.585162			−0.292581	+	0.956241i	$0.594514\pi$
$384$	1.32006e10	+	2.26480e10i	0.607112	+	1.04161i
$385$		0			0
$386$		−	3.01487e10i		−	1.35806i
$387$	6.71272e9	+	3.80214e9i	0.299264	+	0.169506i
$388$			8.85653e9i			0.390784i
$389$			1.05903e10i			0.462497i	0.972895	+	0.231249i	$0.0742811\pi$
$389$			1.05903e10i			0.462497i	−0.972895	+	0.231249i	$0.925719\pi$
$390$		0			0
$391$	−1.44062e10			−0.616371
$392$	5.28466e10			2.23806
$393$	6.76550e9	−	3.94334e9i	0.283615	−	0.165308i
$394$	−1.56840e10			−0.650836
$395$		0			0
$396$	1.09694e10	+	6.21315e9i	0.446068	+	0.252657i
$397$			2.97468e10i			1.19751i	0.800933	+	0.598753i	$0.204337\pi$
$397$			2.97468e10i			1.19751i	−0.800933	+	0.598753i	$0.795663\pi$
$398$	−1.27705e10			−0.508950
$399$	1.25806e10	−	7.33274e9i	0.496375	−	0.289317i
$400$		0			0
$401$			2.31791e10i			0.896437i	0.893924	+	0.448219i	$0.147941\pi$
$401$			2.31791e10i			0.896437i	−0.893924	+	0.448219i	$0.852059\pi$
$402$	2.32514e10	−	1.35523e10i	0.890318	−	0.518931i
$403$		−	3.99171e8i		−	0.0151335i
$404$			6.54388e9i			0.245646i
$405$		0			0
$406$	−4.30223e10			−1.58340
$407$	−8.89637e9			−0.324217
$408$	9.80102e9	+	1.68154e10i	0.353697	+	0.606829i
$409$	−2.09807e10			−0.749769			−0.374885	−	0.927071i	$-0.622318\pi$
$409$	−2.09807e10			−0.749769			−0.374885	+	0.927071i	$0.622318\pi$
$410$		0			0
$411$	7.52379e9	+	1.29084e10i	0.263675	+	0.452382i
$412$			7.04639e9i			0.244556i
$413$	−6.86875e10			−2.36090
$414$	−1.15882e10	+	2.04591e10i	−0.394470	+	0.696442i
$415$		0			0
$416$			3.61904e8i			0.0120843i
$417$	4.75278e8	+	8.15424e8i	0.0157182	+	0.0269674i
$418$		−	1.77285e10i		−	0.580720i
$419$		−	2.02730e10i		−	0.657752i	−0.944373	−	0.328876i	$-0.893330\pi$
$419$		−	2.02730e10i		−	0.657752i	0.944373	−	0.328876i	$-0.106670\pi$
$420$		0			0
$421$	−3.59575e9			−0.114462			−0.0572310	−	0.998361i	$-0.518227\pi$
$421$	−3.59575e9			−0.114462			−0.0572310	+	0.998361i	$0.518227\pi$
$422$	−5.74059e10			−1.81012
$423$	−1.35900e10	+	2.39933e10i	−0.424481	+	0.749425i
$424$	2.15784e10			0.667659
$425$		0			0
$426$	−2.49490e10	+	1.45418e10i	−0.757556	+	0.441550i
$427$			5.10409e10i			1.53535i
$428$	1.48457e10			0.442409
$429$	−6.19687e8	−	1.06318e9i	−0.0182955	−	0.0313891i
$430$		0			0
$431$		−	1.22711e10i		−	0.355610i	−0.984066	−	0.177805i	$-0.943100\pi$
$431$		−	1.22711e10i		−	0.355610i	0.984066	−	0.177805i	$-0.0568996\pi$
$432$	4.20900e10	−	5.20604e8i	1.20849	−	0.0149476i
$433$			5.23052e8i			0.0148797i	0.999972	+	0.00743983i	$0.00236819\pi$
$433$			5.23052e8i			0.0148797i	−0.999972	+	0.00743983i	$0.997632\pi$
$434$		−	5.66697e10i		−	1.59732i
$435$		0			0
$436$	−1.23359e10			−0.341371
$437$	7.56170e9			0.207345
$438$	4.00142e10	−	2.33227e10i	1.08722	−	0.633698i
$439$	−5.03940e10			−1.35682			−0.678408	−	0.734686i	$-0.737330\pi$
$439$	−5.03940e10			−1.35682			−0.678408	+	0.734686i	$0.737330\pi$
$440$		0			0
$441$	5.20812e10	−	9.19499e10i	1.37698	−	2.43107i
$442$			8.00704e8i			0.0209789i
$443$	4.23959e10			1.10080			0.550401	−	0.834901i	$-0.314475\pi$
$443$	4.23959e10			1.10080			0.550401	+	0.834901i	$0.314475\pi$
$444$	1.86665e9	−	1.08800e9i	0.0480321	−	0.0279960i
$445$		0			0
$446$			6.70381e10i			1.69427i
$447$	2.50793e10	−	1.46177e10i	0.628182	−	0.366143i
$448$		−	4.34487e10i		−	1.07861i
$449$			1.38365e9i			0.0340441i	0.999855	+	0.0170221i	$0.00541855\pi$
$449$			1.38365e9i			0.0340441i	−0.999855	+	0.0170221i	$0.994581\pi$
$450$		0			0
$451$	7.29630e10			1.76358
$452$	1.49681e10			0.358603
$453$	−6.54861e9	−	1.12353e10i	−0.155509	−	0.266804i
$454$	−1.04134e10			−0.245114
$455$		0			0
$456$	−5.14448e9	−	8.82627e9i	−0.118982	−	0.204135i
$457$		−	6.39577e10i		−	1.46632i	−0.680057	−	0.733159i	$-0.738045\pi$
$457$		−	6.39577e10i		−	1.46632i	0.680057	−	0.733159i	$-0.261955\pi$
$458$	5.36245e10			1.21871
$459$	3.89169e10	−	4.81357e8i	0.876773	−	0.0108447i
$460$		0			0
$461$			4.34582e10i			0.962206i	0.876664	+	0.481103i	$0.159764\pi$
$461$			4.34582e10i			0.962206i	−0.876664	+	0.481103i	$0.840236\pi$
$462$	−8.79761e10	−	1.50939e11i	−1.93106	−	3.31308i
$463$			1.95845e10i			0.426175i	0.977033	+	0.213088i	$0.0683520\pi$
$463$			1.95845e10i			0.426175i	−0.977033	+	0.213088i	$0.931648\pi$
$464$			3.99952e10i			0.862851i
$465$		0			0
$466$	3.85638e10			0.817780
$467$	−6.63498e10			−1.39499			−0.697497	−	0.716588i	$-0.745703\pi$
$467$	−6.63498e10			−1.39499			−0.697497	+	0.716588i	$0.745703\pi$
$468$	2.60048e8	+	1.47293e8i	0.00542088	+	0.00307043i
$469$	−8.52915e10			−1.76285
$470$		0			0
$471$	−1.82152e10	+	1.06169e10i	−0.370126	+	0.215732i
$472$			4.81896e10i			0.970924i
$473$	−2.97667e10			−0.594684
$474$	1.21369e9	+	2.08230e9i	0.0240433	+	0.0412505i
$475$		0			0
$476$			2.59961e10i			0.506384i
$477$	2.12658e10	−	3.75451e10i	0.410780	−	0.725236i
$478$		−	7.19769e10i		−	1.37874i
$479$		−	8.73284e10i		−	1.65887i	−0.558600	−	0.829437i	$-0.688661\pi$
$479$		−	8.73284e10i		−	1.65887i	0.558600	−	0.829437i	$-0.311339\pi$
$480$		0			0
$481$	−2.10904e8			−0.00394007
$482$	1.22900e9			0.0227701
$483$	6.43795e10	−	3.75243e10i	1.18293	−	0.689483i
$484$	−3.23720e10			−0.589913
$485$		0			0
$486$	3.06208e10	−	5.56554e10i	0.548872	−	0.997613i
$487$			2.65378e10i			0.471790i	0.971778	+	0.235895i	$0.0758022\pi$
$487$			2.65378e10i			0.471790i	−0.971778	+	0.235895i	$0.924198\pi$
$488$	3.58091e10			0.631414
$489$	5.05484e10	−	2.94626e10i	0.884040	−	0.515272i
$490$		0			0
$491$		−	6.17958e10i		−	1.06324i	−0.846982	−	0.531622i	$-0.821583\pi$
$491$		−	6.17958e10i		−	1.06324i	0.846982	−	0.531622i	$-0.178417\pi$
$492$	−1.53092e10	+	8.92313e9i	−0.261272	+	0.152285i
$493$			3.69800e10i			0.626007i
$494$		−	4.20284e8i		−	0.00705724i
$495$		0			0
$496$	−5.26823e10			−0.870438
$497$	9.15186e10			1.49997
$498$	2.91886e10	+	5.00781e10i	0.474565	+	0.814199i
$499$	1.22816e10			0.198086			0.0990429	−	0.995083i	$-0.468422\pi$
$499$	1.22816e10			0.198086			0.0990429	+	0.995083i	$0.468422\pi$
$500$		0			0
$501$	5.51623e10	+	9.46406e10i	0.875571	+	1.50220i
$502$			6.11697e10i			0.963211i
$503$	−6.89562e10			−1.07721			−0.538606	−	0.842558i	$-0.681049\pi$
$503$	−6.89562e10			−1.07721			−0.538606	+	0.842558i	$0.681049\pi$
$504$	−8.75992e10	−	4.96169e10i	−1.35762	−	0.768967i
$505$		0			0
$506$		−	9.07231e10i		−	1.38394i
$507$	3.32580e10	+	5.70600e10i	0.503344	+	0.863575i
$508$		−	2.16773e10i		−	0.325500i
$509$		−	8.72703e10i		−	1.30016i	−0.759868	−	0.650078i	$-0.774736\pi$
$509$		−	8.72703e10i		−	1.30016i	0.759868	−	0.650078i	$-0.225264\pi$
$510$		0			0
$511$	−1.46781e11			−2.15272
$512$	−1.87651e10			−0.273068
$513$	−2.04272e10	+	2.52660e8i	−0.294944	+	0.00364811i
$514$	−1.07342e10			−0.153786
$515$		0			0
$516$	6.24570e9	−	3.64037e9i	0.0881012	−	0.0513507i
$517$		−	1.06395e11i		−	1.48922i
$518$	−2.99417e10			−0.415869
$519$	4.31861e10	+	7.40933e10i	0.595215	+	1.02120i
$520$		0			0
$521$			2.40313e10i			0.326156i	0.986613	+	0.163078i	$0.0521423\pi$
$521$			2.40313e10i			0.326156i	−0.986613	+	0.163078i	$0.947858\pi$
$522$	5.25175e10	+	2.97464e10i	0.707331	+	0.400638i
$523$		−	5.40043e10i		−	0.721808i	−0.932603	−	0.360904i	$-0.882468\pi$
$523$		−	5.40043e10i		−	0.721808i	0.932603	−	0.360904i	$-0.117532\pi$
$524$		−	7.33798e9i		−	0.0973311i
$525$		0			0
$526$	1.72502e10			0.225347
$527$	−4.87106e10			−0.631512
$528$	−1.40318e11	+	8.17859e10i	−1.80542	+	1.05231i
$529$	−3.96150e10			−0.505868
$530$		0			0
$531$	8.38471e10	+	4.74917e10i	1.05465	+	0.597365i
$532$		−	1.36451e10i		−	0.170346i
$533$	1.72971e9			0.0214321
$534$	−2.41756e10	+	1.40910e10i	−0.297312	+	0.173292i
$535$		0			0
$536$			5.98386e10i			0.724974i
$537$	4.78982e10	−	2.79179e10i	0.575999	−	0.335727i
$538$			9.10004e10i			1.08621i
$539$			4.07740e11i			4.83090i
$540$		0			0
$541$	8.76711e9			0.102345			0.0511726	−	0.998690i	$-0.483704\pi$
$541$	8.76711e9			0.102345			0.0511726	+	0.998690i	$0.483704\pi$
$542$	9.61887e10			1.11462
$543$	−3.60395e10	−	6.18321e10i	−0.414552	−	0.711238i
$544$	4.41630e10			0.504270
$545$		0			0
$546$	−2.08562e9	−	3.57825e9i	−0.0234674	−	0.0402625i
$547$			7.10847e10i			0.794011i	0.917816	+	0.397006i	$0.129951\pi$
$547$			7.10847e10i			0.794011i	−0.917816	+	0.397006i	$0.870049\pi$
$548$	1.40007e10			0.155248
$549$	3.52905e10	−	6.23058e10i	0.388480	−	0.685866i
$550$		0			0
$551$		−	1.94105e10i		−	0.210587i
$552$	−2.63262e10	−	4.51672e10i	−0.283551	−	0.486482i
$553$		−	7.63834e9i		−	0.0816768i
$554$		−	1.70786e11i		−	1.81307i
$555$		0			0
$556$	8.84423e8			0.00925467
$557$	4.95865e10			0.515161			0.257580	−	0.966257i	$-0.417075\pi$
$557$	4.95865e10			0.515161			0.257580	+	0.966257i	$0.417075\pi$
$558$	−3.91824e10	+	6.91769e10i	−0.404160	+	0.713550i
$559$	−7.05669e8			−0.00722693
$560$		0			0
$561$	−1.29740e11	+	7.56202e10i	−1.30985	+	0.763460i
$562$			2.11466e11i			2.11980i
$563$	1.18225e11			1.17673			0.588366	−	0.808595i	$-0.299772\pi$
$563$	1.18225e11			1.17673			0.588366	+	0.808595i	$0.299772\pi$
$564$	1.30118e10	+	2.23240e10i	0.128594	+	0.220625i
$565$		0			0
$566$		−	3.01274e10i		−	0.293560i
$567$	−1.72661e11	+	1.03519e11i	−1.67056	+	1.00159i
$568$		−	6.42074e10i		−	0.616867i
$569$			4.39706e10i			0.419482i	0.977757	+	0.209741i	$0.0672620\pi$
$569$			4.39706e10i			0.419482i	−0.977757	+	0.209741i	$0.932738\pi$
$570$		0			0
$571$	−1.48977e11			−1.40144			−0.700722	−	0.713434i	$-0.747139\pi$
$571$	−1.48977e11			−1.40144			−0.700722	+	0.713434i	$0.747139\pi$
$572$	−1.15315e9			−0.0107721
$573$	8.29035e10	−	4.83211e10i	0.769049	−	0.448248i
$574$	2.45564e11			2.26213
$575$		0			0
$576$	−3.00411e10	+	5.30380e10i	−0.272914	+	0.481833i
$577$		−	1.29323e10i		−	0.116674i	−0.998297	−	0.0583368i	$-0.981420\pi$
$577$		−	1.29323e10i		−	0.116674i	0.998297	−	0.0583368i	$-0.0185797\pi$
$578$	−2.93760e10			−0.263197
$579$	1.15809e11	−	6.75002e10i	1.03045	−	0.600608i
$580$		0			0
$581$		−	1.83698e11i		−	1.61213i
$582$	−1.48762e11	+	8.67075e10i	−1.29658	+	0.755727i
$583$			1.66489e11i			1.44115i
$584$			1.02978e11i			0.885309i
$585$		0			0
$586$	1.74879e11			1.48302
$587$	8.37926e10			0.705754			0.352877	−	0.935670i	$-0.385203\pi$
$587$	8.37926e10			0.705754			0.352877	+	0.935670i	$0.385203\pi$
$588$	−4.98653e10	−	8.55527e10i	−0.417147	−	0.715689i
$589$	2.55678e10			0.212438
$590$		0			0
$591$	−3.51150e10	−	6.02460e10i	−0.287834	−	0.493831i
$592$			2.78349e10i			0.226623i
$593$	3.46484e10			0.280198			0.140099	−	0.990138i	$-0.455258\pi$
$593$	3.46484e10			0.280198			0.140099	+	0.990138i	$0.455258\pi$
$594$	3.03135e9	+	2.45079e11i	0.0243495	+	1.96862i
$595$		0			0
$596$		−	2.72015e10i		−	0.215579i
$597$	−2.85920e10	−	4.90546e10i	−0.225085	−	0.386173i
$598$		−	2.15074e9i		−	0.0168184i
$599$		−	2.01526e10i		−	0.156539i	−0.996932	−	0.0782696i	$-0.975060\pi$
$599$		−	2.01526e10i		−	0.156539i	0.996932	−	0.0782696i	$-0.0249395\pi$
$600$		0			0
$601$	4.52186e10			0.346593			0.173296	−	0.984870i	$-0.444558\pi$
$601$	4.52186e10			0.346593			0.173296	+	0.984870i	$0.444558\pi$
$602$	−1.00183e11			−0.762794
$603$	1.04116e11	+	5.89720e10i	0.787493	+	0.446043i
$604$	−1.21860e10			−0.0915616
$605$		0			0
$606$	−1.09917e11	+	6.40661e10i	−0.815030	+	0.475048i
$607$		−	2.11693e11i		−	1.55938i	−0.626167	−	0.779689i	$-0.715377\pi$
$607$		−	2.11693e11i		−	1.55938i	0.626167	−	0.779689i	$-0.284623\pi$
$608$	−2.31808e10			−0.169635
$609$	−9.63231e10	−	1.65259e11i	−0.700263	−	1.20143i
$610$		0			0
$611$		−	2.52227e9i		−	0.0180979i
$612$	1.79741e10	−	3.17335e10i	0.128127	−	0.226210i
$613$			1.83429e11i			1.29905i	0.760338	+	0.649527i	$0.225033\pi$
$613$			1.83429e11i			1.29905i	−0.760338	+	0.649527i	$0.774967\pi$
$614$		−	1.59246e11i		−	1.12046i
$615$		0			0
$616$	3.88447e11			2.69780
$617$	−1.86844e11			−1.28925			−0.644627	−	0.764497i	$-0.722988\pi$
$617$	−1.86844e11			−1.28925			−0.644627	+	0.764497i	$0.722988\pi$
$618$	−1.18357e11	+	6.89857e10i	−0.811411	+	0.472939i
$619$	2.00363e11			1.36475			0.682376	−	0.731001i	$-0.260947\pi$
$619$	2.00363e11			1.36475			0.682376	+	0.731001i	$0.260947\pi$
$620$		0			0
$621$	−1.04533e11	+	1.29296e9i	−0.702891	+	0.00869395i
$622$		−	2.28511e11i		−	1.52667i
$623$	8.86817e10			0.588684
$624$	−3.32648e9	+	1.93887e9i	−0.0219405	+	0.0127883i
$625$		0			0
$626$			5.61124e10i			0.365394i
$627$	6.80995e10	−	3.96925e10i	0.440629	−	0.256825i
$628$			1.97565e10i			0.127020i
$629$			2.57365e10i			0.164417i
$630$		0			0
$631$	−1.29595e11			−0.817469			−0.408735	−	0.912653i	$-0.634030\pi$
$631$	−1.29595e11			−0.817469			−0.408735	+	0.912653i	$0.634030\pi$
$632$	−5.35889e9			−0.0335897
$633$	−1.28527e11	−	2.20510e11i	−0.800531	−	1.37345i
$634$	5.12809e10			0.317394
$635$		0			0
$636$	−2.03610e10	−	3.49329e10i	−0.124443	−	0.213504i
$637$			9.66616e9i			0.0587079i
$638$	−2.32882e11			−1.40557
$639$	−1.11717e11	−	6.32775e10i	−0.670064	−	0.379530i
$640$		0			0
$641$		−	1.41802e11i		−	0.839945i	−0.907537	−	0.419973i	$-0.862040\pi$
$641$		−	1.41802e11i		−	0.839945i	0.907537	−	0.419973i	$-0.137960\pi$
$642$	1.45342e11	+	2.49361e11i	0.855563	+	1.46787i
$643$			2.74292e10i			0.160461i	0.996776	+	0.0802305i	$0.0255656\pi$
$643$			2.74292e10i			0.160461i	−0.996776	+	0.0802305i	$0.974434\pi$
$644$		−	6.98271e10i		−	0.405958i
$645$		0			0
$646$	−5.12870e10			−0.294495
$647$	−2.68855e11			−1.53427			−0.767134	−	0.641486i	$-0.778318\pi$
$647$	−2.68855e11			−1.53427			−0.767134	+	0.641486i	$0.778318\pi$
$648$	7.26268e10	+	1.21135e11i	0.411905	+	0.687021i
$649$	−3.71809e11			−2.09576
$650$		0			0
$651$	2.17682e11	−	1.26878e11i	1.21199	−	0.706420i
$652$		−	5.48257e10i		−	0.303385i
$653$	1.86797e11			1.02735			0.513674	−	0.857985i	$-0.328284\pi$
$653$	1.86797e11			1.02735			0.513674	+	0.857985i	$0.328284\pi$
$654$	−1.20772e11	−	2.07205e11i	−0.660167	−	1.13263i
$655$		0			0
$656$		−	2.28286e11i		−	1.23272i
$657$	1.79176e11	+	1.01487e11i	0.961655	+	0.544689i
$658$		−	3.58083e11i		−	1.91021i
$659$		−	3.39105e10i		−	0.179801i	−0.995951	−	0.0899006i	$-0.971345\pi$
$659$		−	3.39105e10i		−	0.179801i	0.995951	−	0.0899006i	$-0.0286549\pi$
$660$		0			0
$661$	−2.09742e11			−1.09870			−0.549351	−	0.835592i	$-0.685125\pi$
$661$	−2.09742e11			−1.09870			−0.549351	+	0.835592i	$0.685125\pi$
$662$	3.81582e11			1.98681
$663$	−3.07570e9	+	1.79270e9i	−0.0159181	+	0.00927801i
$664$	−1.28878e11			−0.662991
$665$		0			0
$666$	3.65499e10	+	2.07022e10i	0.185776	+	0.105225i
$667$		−	9.93308e10i		−	0.501857i
$668$	1.02649e11			0.515523
$669$	−2.57510e11	+	1.50092e11i	−1.28555	+	0.749297i
$670$		0			0
$671$			2.76287e11i			1.36292i
$672$	−1.97359e11	+	1.15033e11i	−0.967788	+	0.564086i
$673$		−	5.94010e10i		−	0.289557i	−0.989464	−	0.144778i	$-0.953753\pi$
$673$		−	5.94010e10i		−	0.289557i	0.989464	−	0.144778i	$-0.0462469\pi$
$674$		−	3.49295e8i		−	0.00169259i
$675$		0			0
$676$	6.18883e10			0.296362
$677$	1.20914e11			0.575602			0.287801	−	0.957690i	$-0.407076\pi$
$677$	1.20914e11			0.575602			0.287801	+	0.957690i	$0.407076\pi$
$678$	1.46542e11	+	2.51418e11i	0.693493	+	1.18981i
$679$	5.45693e11			2.56725
$680$		0			0
$681$	−2.33146e10	−	4.00004e10i	−0.108403	−	0.185984i
$682$		−	3.06756e11i		−	1.41793i
$683$	−4.27245e11			−1.96333			−0.981667	−	0.190605i	$-0.938955\pi$
$683$	−4.27245e11			−1.96333			−0.981667	+	0.190605i	$0.938955\pi$
$684$	−9.43448e9	+	1.66567e10i	−0.0431016	+	0.0760964i
$685$		0			0
$686$			8.81125e11i			3.97870i
$687$	1.20060e11	+	2.05985e11i	0.538981	+	0.924716i
$688$			9.31338e10i			0.415675i
$689$			3.94689e9i			0.0175137i
$690$		0			0
$691$	−1.15810e11			−0.507964			−0.253982	−	0.967209i	$-0.581740\pi$
$691$	−1.15810e11			−0.507964			−0.253982	+	0.967209i	$0.581740\pi$
$692$	8.03629e10			0.350454
$693$	3.82821e11	−	6.75875e11i	1.65983	−	2.93045i
$694$	−2.11900e11			−0.913468
$695$		0			0
$696$	−1.15942e11	+	6.75781e10i	−0.494088	+	0.287985i
$697$		−	2.11076e11i		−	0.894350i
$698$	5.27077e10			0.222051
$699$	8.63409e10	+	1.48133e11i	0.361666	+	0.620502i
$700$		0			0
$701$		−	5.27372e10i		−	0.218396i	−0.994020	−	0.109198i	$-0.965172\pi$
$701$		−	5.27372e10i		−	0.218396i	0.994020	−	0.109198i	$-0.0348283\pi$
$702$	7.18632e7	+	5.81002e9i	0.000295909	+	0.0239238i
$703$		−	1.35089e10i		−	0.0553093i
$704$		−	2.35190e11i		−	0.957476i
$705$		0			0
$706$	4.97619e11			2.00299
$707$	4.03199e11			1.61377
$708$	7.80136e10	−	4.54710e10i	0.310482	−	0.180968i
$709$	−4.17074e11			−1.65055			−0.825275	−	0.564731i	$-0.808980\pi$
$709$	−4.17074e11			−1.65055			−0.825275	+	0.564731i	$0.808980\pi$
$710$		0			0
$711$	−5.28128e9	+	9.32416e9i	−0.0206662	+	0.0364864i
$712$		−	6.22171e10i		−	0.242097i
$713$	1.30840e11			0.506270
$714$	−4.36653e11	+	2.54508e11i	−1.68013	+	0.979282i
$715$		0			0
$716$		−	5.19512e10i		−	0.197671i
$717$	2.76481e11	−	1.61150e11i	1.04614	−	0.609752i
$718$			7.80345e10i			0.293622i
$719$			2.58351e11i			0.966706i	0.875426	+	0.483353i	$0.160581\pi$
$719$			2.58351e11i			0.966706i	−0.875426	+	0.483353i	$0.839419\pi$
$720$		0			0
$721$	4.34161e11			1.60661
$722$	−2.82489e11			−1.03957
$723$	2.75162e9	+	4.72090e9i	0.0100701	+	0.0172771i
$724$	−6.70642e10			−0.244082
$725$		0			0
$726$	−3.16929e11	−	5.43748e11i	−1.14082	−	1.95727i
$727$			1.88297e9i			0.00674072i	0.999994	+	0.00337036i	$0.00107282\pi$
$727$			1.88297e9i			0.00674072i	−0.999994	+	0.00337036i	$0.998927\pi$
$728$	9.20879e9			0.0327852
$729$	2.82343e11	−	6.98558e9i	0.999694	−	0.0247339i
$730$		0			0
$731$			8.61126e10i			0.301576i
$732$	−3.37890e10	−	5.79710e10i	−0.117688	−	0.201914i
$733$		−	3.26918e11i		−	1.13246i	−0.824247	−	0.566230i	$-0.808401\pi$
$733$		−	3.26918e11i		−	1.13246i	0.824247	−	0.566230i	$-0.191599\pi$
$734$			2.69327e11i			0.927887i
$735$		0			0
$736$	−1.18625e11			−0.404263
$737$	−4.61687e11			−1.56487
$738$	−2.99761e11	−	1.69787e11i	−1.01053	−	0.572374i
$739$	1.66227e11			0.557346			0.278673	−	0.960386i	$-0.410106\pi$
$739$	1.66227e11			0.557346			0.278673	+	0.960386i	$0.410106\pi$
$740$		0			0
$741$	1.61441e9	−	9.40977e8i	0.00535478	−	0.00312109i
$742$			5.60335e11i			1.84855i
$743$	−2.38178e11			−0.781531			−0.390766	−	0.920490i	$-0.627790\pi$
$743$	−2.38178e11			−0.781531			−0.390766	+	0.920490i	$0.627790\pi$
$744$	−8.90149e10	−	1.52721e11i	−0.290517	−	0.498433i
$745$		0			0
$746$		−	1.65190e11i		−	0.533370i
$747$	−1.27012e11	+	2.24241e11i	−0.407908	+	0.720165i
$748$			1.40718e11i			0.449514i
$749$		−	9.14711e11i		−	2.90641i
$750$		0			0
$751$	−2.88947e11			−0.908361			−0.454181	−	0.890910i	$-0.650068\pi$
$751$	−2.88947e11			−0.908361			−0.454181	+	0.890910i	$0.650068\pi$
$752$	−3.32888e11			−1.04094
$753$	−2.34968e11	+	1.36953e11i	−0.730850	+	0.425984i
$754$	−5.52086e9			−0.0170813
$755$		0			0
$756$	2.33315e9	+	1.88631e11i	0.00714258	+	0.577466i
$757$			8.49379e9i			0.0258653i	0.999916	+	0.0129327i	$0.00411671\pi$
$757$			8.49379e9i			0.0258653i	−0.999916	+	0.0129327i	$0.995883\pi$
$758$	7.12447e11			2.15812
$759$	3.48490e11	−	2.03121e11i	1.05008	−	0.612051i
$760$		0			0
$761$			2.86717e11i			0.854898i	0.904039	+	0.427449i	$0.140588\pi$
$761$			2.86717e11i			0.854898i	−0.904039	+	0.427449i	$0.859412\pi$
$762$	3.64111e11	−	2.12226e11i	1.07998	−	0.629476i
$763$			7.60075e11i			2.24263i
$764$		−	8.99185e10i		−	0.263922i
$765$		0			0
$766$	−2.29391e11			−0.666286
$767$	−8.81435e9			−0.0254688
$768$	1.43480e11	+	2.46166e11i	0.412427	+	0.707592i
$769$	−1.37170e11			−0.392242			−0.196121	−	0.980580i	$-0.562834\pi$
$769$	−1.37170e11			−0.392242			−0.196121	+	0.980580i	$0.562834\pi$
$770$		0			0
$771$	−2.40328e10	−	4.12326e10i	−0.0680123	−	0.116687i
$772$		−	1.25608e11i		−	0.353629i
$773$	−5.00450e11			−1.40166			−0.700831	−	0.713328i	$-0.747187\pi$
$773$	−5.00450e11			−1.40166			−0.700831	+	0.713328i	$0.747187\pi$
$774$	1.22294e11	+	6.92681e10i	0.340753	+	0.193005i
$775$		0			0
$776$		−	3.82846e11i		−	1.05579i
$777$	−6.70367e10	−	1.15013e11i	−0.183920	−	0.315547i
$778$			1.92936e11i			0.526616i
$779$			1.10792e11i			0.300856i
$780$		0			0
$781$	4.95395e11			1.33152
$782$	−2.62454e11			−0.701822
$783$	3.31896e9	+	2.68332e11i	0.00882988	+	0.713881i
$784$	1.27573e12			3.37673
$785$		0			0
$786$	1.23255e11	−	7.18405e10i	0.322935	−	0.188226i
$787$		−	3.86758e11i		−	1.00818i	−0.863650	−	0.504092i	$-0.831827\pi$
$787$		−	3.86758e11i		−	1.00818i	0.863650	−	0.504092i	$-0.168173\pi$
$788$	−6.53439e10			−0.169473
$789$	3.86217e10	+	6.62623e10i	0.0996605	+	0.170985i
$790$		0			0
$791$		−	9.22258e11i		−	2.35584i
$792$	−4.74179e11	−	2.68579e11i	−1.20515	−	0.682608i
$793$			6.54985e9i			0.0165630i
$794$			5.41932e11i			1.36352i
$795$		0			0
$796$	−5.32054e10			−0.132527
$797$	4.69718e11			1.16414			0.582068	−	0.813140i	$-0.302244\pi$
$797$	4.69718e11			1.16414			0.582068	+	0.813140i	$0.302244\pi$
$798$	2.29196e11	−	1.33589e11i	0.565191	−	0.329427i
$799$	−3.07792e11			−0.755214
$800$		0			0
$801$	−1.08254e11	−	6.13161e10i	−0.262975	−	0.148951i
$802$			4.22282e11i			1.02072i
$803$	−7.94534e11			−1.91095
$804$	9.68720e10	−	5.64628e10i	0.231832	−	0.135126i
$805$		0			0
$806$		−	7.27216e9i		−	0.0172315i
$807$	−3.49555e11	+	2.03742e11i	−0.824178	+	0.480381i
$808$		−	2.82876e11i		−	0.663667i
$809$			6.02770e11i			1.40721i	0.710594	+	0.703603i	$0.248427\pi$
$809$			6.02770e11i			1.40721i	−0.710594	+	0.703603i	$0.751573\pi$
$810$		0			0
$811$	−2.69496e11			−0.622973			−0.311487	−	0.950251i	$-0.600827\pi$
$811$	−2.69496e11			−0.622973			−0.311487	+	0.950251i	$0.600827\pi$
$812$	−1.79243e11			−0.412305
$813$	2.15358e11	+	3.69484e11i	0.492945	+	0.845734i
$814$	−1.62076e11			−0.369165
$815$		0			0
$816$	2.36600e11	+	4.05929e11i	0.533647	+	0.915566i
$817$		−	4.51998e10i		−	0.101449i
$818$	−3.82231e11			−0.853715
$819$	9.07543e9	−	1.60228e10i	0.0201712	−	0.0356125i
$820$		0			0
$821$			4.19517e11i			0.923372i	0.887044	+	0.461686i	$0.152755\pi$
$821$			4.19517e11i			0.923372i	−0.887044	+	0.461686i	$0.847245\pi$
$822$	1.37070e11	+	2.35168e11i	0.300230	+	0.515098i
$823$		−	6.29847e11i		−	1.37289i	−0.727182	−	0.686445i	$-0.759170\pi$
$823$		−	6.29847e11i		−	1.37289i	0.727182	−	0.686445i	$-0.240830\pi$
$824$		−	3.04598e11i		−	0.660720i
$825$		0			0
$826$	−1.25136e12			−2.68820
$827$	6.51690e10			0.139322			0.0696609	−	0.997571i	$-0.477808\pi$
$827$	6.51690e10			0.139322			0.0696609	+	0.997571i	$0.477808\pi$
$828$	−4.82797e10	+	8.52383e10i	−0.102717	+	0.181348i
$829$	4.90689e11			1.03893			0.519467	−	0.854490i	$-0.326130\pi$
$829$	4.90689e11			1.03893			0.519467	+	0.854490i	$0.326130\pi$
$830$		0			0
$831$	6.56032e11	−	3.82375e11i	1.37569	−	0.801836i
$832$		−	5.57557e9i		−	0.0116358i
$833$	1.17956e12			2.44985
$834$	8.65870e9	+	1.48555e10i	0.0178973	+	0.0307061i
$835$		0			0
$836$		−	7.38619e10i		−	0.151215i
$837$	−3.53451e11	+	4.37178e9i	−0.720158	+	0.00890751i
$838$		−	3.69337e11i		−	0.748940i
$839$			7.09436e10i			0.143174i	0.997434	+	0.0715872i	$0.0228064\pi$
$839$			7.09436e10i			0.143174i	−0.997434	+	0.0715872i	$0.977194\pi$
$840$		0			0
$841$	2.45269e11			0.490296
$842$	−6.55080e10			−0.130331
$843$	−8.12294e11	+	4.73454e11i	−1.60843	+	0.937491i
$844$	−2.39169e11			−0.471341
$845$		0			0
$846$	−2.47585e11	+	4.37114e11i	−0.483329	+	0.853323i
$847$			1.99459e12i			3.87543i
$848$	5.20909e11			1.00734
$849$	1.15727e11	−	6.74526e10i	0.222743	−	0.129828i
$850$		0			0
$851$		−	6.91299e10i		−	0.131810i
$852$	−1.03945e11	+	6.05852e10i	−0.197262	+	0.114976i
$853$			3.32593e11i			0.628227i	0.949385	+	0.314113i	$0.101707\pi$
$853$			3.32593e11i			0.628227i	−0.949385	+	0.314113i	$0.898293\pi$
$854$			9.29872e11i			1.74820i
$855$		0			0
$856$	−6.41741e11			−1.19527
$857$	8.20366e11			1.52084			0.760421	−	0.649430i	$-0.224992\pi$
$857$	8.20366e11			1.52084			0.760421	+	0.649430i	$0.224992\pi$
$858$	−1.12896e10	−	1.93693e10i	−0.0208319	−	0.0357408i
$859$	−1.94847e11			−0.357867			−0.178933	−	0.983861i	$-0.557265\pi$
$859$	−1.94847e11			−0.357867			−0.178933	+	0.983861i	$0.557265\pi$
$860$		0			0
$861$	5.49796e11	+	9.43273e11i	1.00044	+	1.71642i
$862$		−	2.23557e11i		−	0.404910i
$863$	8.78219e11			1.58329			0.791643	−	0.610984i	$-0.209226\pi$
$863$	8.78219e11			1.58329			0.791643	+	0.610984i	$0.209226\pi$
$864$	3.20453e11	−	3.96363e9i	0.575055	−	0.00711276i
$865$		0			0
$866$			9.52905e9i			0.0169425i
$867$	−6.57701e10	−	1.12840e11i	−0.116400	−	0.199704i
$868$		−	2.36102e11i		−	0.415930i
$869$		−	4.13467e10i		−	0.0725040i
$870$		0			0
$871$	−1.09451e10			−0.0190172
$872$	5.33252e11			0.922287
$873$	−6.66129e11	−	3.77301e11i	−1.14684	−	0.649578i
$874$	1.37760e11			0.236091
$875$		0			0
$876$	1.66710e11	−	9.71689e10i	0.283104	−	0.165010i
$877$		−	8.56815e10i		−	0.144840i	−0.997374	−	0.0724201i	$-0.976928\pi$
$877$		−	8.56815e10i		−	0.144840i	0.997374	−	0.0724201i	$-0.0230722\pi$
$878$	−9.18087e11			−1.54492
$879$	3.91538e11	+	6.71753e11i	0.655872	+	1.12526i
$880$		0			0
$881$			1.01082e12i			1.67792i	0.544192	+	0.838961i	$0.316836\pi$
$881$			1.01082e12i			1.67792i	−0.544192	+	0.838961i	$0.683164\pi$
$882$	9.48825e11	−	1.67516e12i	1.56788	−	2.76810i
$883$			6.05032e11i			0.995257i	0.867390	+	0.497629i	$0.165796\pi$
$883$			6.05032e11i			0.995257i	−0.867390	+	0.497629i	$0.834204\pi$
$884$			3.33596e9i			0.00546275i
$885$		0			0
$886$	7.72376e11			1.25341
$887$	−7.07543e11			−1.14303			−0.571516	−	0.820591i	$-0.693645\pi$
$887$	−7.07543e11			−1.14303			−0.571516	+	0.820591i	$0.693645\pi$
$888$	−8.06907e10	+	4.70314e10i	−0.129769	+	0.0756374i
$889$	−1.33564e12			−2.13837
$890$		0			0
$891$	−9.34623e11	+	5.60355e11i	−1.48295	+	0.889104i
$892$			2.79300e11i			0.441175i
$893$	1.61558e11			0.254052
$894$	4.56899e11	−	2.66309e11i	0.715271	−	0.416903i
$895$		0			0
$896$		−	1.51353e12i		−	2.34833i
$897$	8.26154e9	−	4.81532e9i	0.0127612	−	0.00743799i
$898$			2.52077e10i			0.0387639i
$899$		−	3.35860e11i		−	0.514185i
$900$		0			0
$901$	4.81638e11			0.730839
$902$	1.32925e12			2.00808
$903$	−2.24300e11	−	3.84827e11i	−0.337349	−	0.578781i
$904$	−6.47035e11			−0.968845
$905$		0			0
$906$	−1.19304e11	−	2.04687e11i	−0.177069	−	0.303792i
$907$		−	2.21446e10i		−	0.0327220i	−0.999866	−	0.0163610i	$-0.994792\pi$
$907$		−	2.21446e10i		−	0.0327220i	0.999866	−	0.0163610i	$-0.00520810\pi$
$908$	−4.33851e10			−0.0638259
$909$	−4.92187e11	−	2.78779e11i	−0.720900	−	0.408324i
$910$		0			0
$911$		−	7.59165e11i		−	1.10221i	−0.834437	−	0.551103i	$-0.814207\pi$
$911$		−	7.59165e11i		−	1.10221i	0.834437	−	0.551103i	$-0.185793\pi$
$912$	−1.24190e11	−	2.13069e11i	−0.179517	−	0.307993i
$913$		−	9.94366e11i		−	1.43108i
$914$		−	1.16519e12i		−	1.66960i
$915$		0			0
$916$	2.23415e11			0.317344
$917$	−4.52128e11			−0.639416
$918$	7.08995e11	−	8.76944e9i	0.998326	−	0.0123481i
$919$	−2.52389e11			−0.353841			−0.176921	−	0.984225i	$-0.556614\pi$
$919$	−2.52389e11			−0.353841			−0.176921	+	0.984225i	$0.556614\pi$
$920$		0			0
$921$	6.11704e11	−	3.56538e11i	0.850165	−	0.495527i
$922$			7.91729e11i			1.09560i
$923$	1.17442e10			0.0161814
$924$	−3.66533e11	−	6.28852e11i	−0.502835	−	0.862702i
$925$		0			0
$926$			3.56794e11i			0.485259i
$927$	−5.29982e11	−	3.00186e11i	−0.717699	−	0.406511i
$928$			3.04504e11i			0.410584i
$929$		−	8.46609e11i		−	1.13663i	−0.822810	−	0.568316i	$-0.807595\pi$
$929$		−	8.46609e11i		−	1.13663i	0.822810	−	0.568316i	$-0.192405\pi$
$930$		0			0
$931$	−6.19141e11			−0.824121
$932$	1.60668e11			0.212944
$933$	8.77765e11	−	5.11615e11i	1.15838	−	0.675175i
$934$	−1.20877e12			−1.58839
$935$		0			0
$936$	−1.12412e10	−	6.36712e9i	−0.0146457	−	0.00829544i
$937$		−	1.26898e12i		−	1.64625i	−0.567858	−	0.823126i	$-0.692228\pi$
$937$		−	1.26898e12i		−	1.64625i	0.567858	−	0.823126i	$-0.307772\pi$
$938$	−1.55386e12			−2.00724
$939$	−2.15541e11	+	1.25631e11i	−0.277248	+	0.161597i
$940$		0			0
$941$			2.79702e11i			0.356728i	0.983965	+	0.178364i	$0.0570804\pi$
$941$			2.79702e11i			0.356728i	−0.983965	+	0.178364i	$0.942920\pi$
$942$	−3.31847e11	+	1.93420e11i	−0.421439	+	0.245640i
$943$			5.66963e11i			0.716982i
$944$			1.16331e12i			1.46490i
$945$		0			0
$946$	−5.42295e11			−0.677128
$947$	−1.01819e12			−1.26599			−0.632993	−	0.774158i	$-0.718174\pi$
$947$	−1.01819e12			−1.26599			−0.632993	+	0.774158i	$0.718174\pi$
$948$	5.05657e9	+	8.67544e9i	0.00626070	+	0.0107413i
$949$	−1.88358e10			−0.0232230
$950$		0			0
$951$	1.14813e11	+	1.96983e11i	0.140369	+	0.240827i
$952$		−	1.12375e12i		−	1.36811i
$953$	7.35745e11			0.891981			0.445990	−	0.895038i	$-0.352852\pi$
$953$	7.35745e11			0.891981			0.445990	+	0.895038i	$0.352852\pi$
$954$	3.87425e11	−	6.84003e11i	0.467729	−	0.825780i
$955$		0			0
$956$		−	2.99876e11i		−	0.359013i
$957$	−5.21402e11	−	8.94557e11i	−0.621620	−	1.06650i
$958$		−	1.59096e12i		−	1.88885i
$959$		−	8.62647e11i		−	1.01990i
$960$		0			0
$961$	−4.10491e11			−0.481294
$962$	−3.84228e9			−0.00448630
$963$	−6.32447e11	+	1.11659e12i	−0.735392	+	1.29834i
$964$	5.12037e9			0.00592916
$965$		0			0
$966$	1.17288e12	−	6.83624e11i	1.34693	−	0.785071i
$967$			1.31775e12i			1.50704i	0.657422	+	0.753522i	$0.271647\pi$
$967$			1.31775e12i			1.50704i	−0.657422	+	0.753522i	$0.728353\pi$
$968$	1.39936e12			1.59378
$969$	−1.14827e11	−	1.97006e11i	−0.130241	−	0.223452i
$970$		0			0
$971$			7.14621e11i			0.803895i	0.915663	+	0.401947i	$0.131666\pi$
$971$			7.14621e11i			0.803895i	−0.915663	+	0.401947i	$0.868334\pi$
$972$	1.27575e11	−	2.31876e11i	0.142922	−	0.259771i
$973$		−	5.44935e10i		−	0.0607985i
$974$			4.83470e11i			0.537198i
$975$		0			0
$976$	8.64444e11			0.952660
$977$	3.39820e11			0.372968			0.186484	−	0.982458i	$-0.440291\pi$
$977$	3.39820e11			0.372968			0.186484	+	0.982458i	$0.440291\pi$
$978$	9.20899e11	−	5.36756e11i	1.00660	−	0.586707i
$979$	4.80039e11			0.522571
$980$		0			0
$981$	5.25529e11	−	9.27827e11i	0.567441	−	1.00182i
$982$		−	1.12581e12i		−	1.21065i
$983$	1.00970e12			1.08138			0.540692	−	0.841220i	$-0.318162\pi$
$983$	1.00970e12			1.08138			0.540692	+	0.841220i	$0.318162\pi$
$984$	6.61779e11	−	3.85725e11i	0.705883	−	0.411431i
$985$		0			0
$986$			6.73709e11i			0.712795i
$987$	1.37549e12	−	8.01716e11i	1.44940	−	0.844796i
$988$		−	1.75102e9i		−	0.00183765i
$989$		−	2.31304e11i		−	0.241768i
$990$		0			0
$991$	1.10525e12			1.14595			0.572975	−	0.819573i	$-0.305789\pi$
$991$	1.10525e12			1.14595			0.572975	+	0.819573i	$0.305789\pi$
$992$	−4.01098e11			−0.414194
$993$	8.54328e11	+	1.46575e12i	0.878674	+	1.50752i
$994$	1.66730e12			1.70792
$995$		0			0
$996$	1.21608e11	+	2.08640e11i	0.123573	+	0.212011i
$997$			4.00464e11i			0.405306i	0.979251	+	0.202653i	$0.0649563\pi$
$997$			4.00464e11i			0.405306i	−0.979251	+	0.202653i	$0.935044\pi$
$998$	2.23749e11			0.225548
$999$	2.30985e9	+	1.86747e11i	0.00231911	+	0.187496i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.d.d.74.16		20
3.2	odd	2	inner	75.9.d.d.74.6		20
5.2	odd	4		75.9.c.e.26.8	yes	10
5.3	odd	4		75.9.c.f.26.3	yes	10
5.4	even	2	inner	75.9.d.d.74.5		20
15.2	even	4		75.9.c.e.26.3	✓	10
15.8	even	4		75.9.c.f.26.8	yes	10
15.14	odd	2	inner	75.9.d.d.74.15		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.c.e.26.3	✓	10	15.2	even	4
75.9.c.e.26.8	yes	10	5.2	odd	4
75.9.c.f.26.3	yes	10	5.3	odd	4
75.9.c.f.26.8	yes	10	15.8	even	4
75.9.d.d.74.5		20	5.4	even	2	inner
75.9.d.d.74.6		20	3.2	odd	2	inner
75.9.d.d.74.15		20	15.14	odd	2	inner
75.9.d.d.74.16		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+18.2182 q^{2} +(40.7889 + 69.9805i) q^{3} +75.9021 q^{4} +(743.099 + 1274.92i) q^{6} -4676.68i q^{7} -3281.06 q^{8} +(-3233.54 + 5708.85i) q^{9} +O(q^{10})\)	\(q+18.2182 q^{2} +(40.7889 + 69.9805i) q^{3} +75.9021 q^{4} +(743.099 + 1274.92i) q^{6} -4676.68i q^{7} -3281.06 q^{8} +(-3233.54 + 5708.85i) q^{9} -25315.1i q^{11} +(3095.96 + 5311.66i) q^{12} -600.138i q^{13} -85200.7i q^{14} -79205.8 q^{16} -73234.6 q^{17} +(-58909.2 + 104005. i) q^{18} +38440.3 q^{19} +(327277. - 190757. i) q^{21} -461196. i q^{22} +196713. q^{23} +(-133831. - 229610. i) q^{24} -10933.4i q^{26} +(-531400. + 6572.81i) q^{27} -354970. i q^{28} -504953. i q^{29} +665132. q^{31} -603035. q^{32} +(1.77157e6 - 1.03258e6i) q^{33} -1.33420e6 q^{34} +(-245432. + 433313. i) q^{36} -351425. i q^{37} +700312. q^{38} +(41997.9 - 24478.9i) q^{39} +2.88219e6i q^{41} +(5.96238e6 - 3.47524e6i) q^{42} -1.17585e6i q^{43} -1.92147e6i q^{44} +3.58375e6 q^{46} +4.20282e6 q^{47} +(-3.23071e6 - 5.54286e6i) q^{48} -1.61066e7 q^{49} +(-2.98715e6 - 5.12499e6i) q^{51} -45551.7i q^{52} -6.57665e6 q^{53} +(-9.68115e6 + 119745. i) q^{54} +1.53445e7i q^{56} +(1.56794e6 + 2.69007e6i) q^{57} -9.19932e6i q^{58} -1.46872e7i q^{59} -1.09139e7 q^{61} +1.21175e7 q^{62} +(2.66985e7 + 1.51222e7i) q^{63} +9.29049e6 q^{64} +(3.22747e7 - 1.88116e7i) q^{66} -1.82376e7i q^{67} -5.55866e6 q^{68} +(8.02369e6 + 1.37661e7i) q^{69} +1.95691e7i q^{71} +(1.06094e7 - 1.87311e7i) q^{72} -3.13857e7i q^{73} -6.40233e6i q^{74} +2.91770e6 q^{76} -1.18391e8 q^{77} +(765126. - 445962. i) q^{78} +1.63328e6 q^{79} +(-2.21352e7 - 3.69196e7i) q^{81} +5.25082e7i q^{82} +3.92795e7 q^{83} +(2.48410e7 - 1.44788e7i) q^{84} -2.14218e7i q^{86} +(3.53369e7 - 2.05965e7i) q^{87} +8.30604e7i q^{88} +1.89625e7i q^{89} -2.80665e6 q^{91} +1.49309e7 q^{92} +(2.71300e7 + 4.65462e7i) q^{93} +7.65678e7 q^{94} +(-2.45971e7 - 4.22007e7i) q^{96} +1.16684e8i q^{97} -2.93432e8 q^{98} +(1.44520e8 + 8.18574e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9}+O(q^{10}) \) 20 * q + 3108 * q^4 + 4514 * q^6 + 22414 * q^9	\( 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9} + 89268 q^{16} + 287868 q^{19} + 1346856 q^{21} + 2033718 q^{24} - 6028120 q^{31} - 9954292 q^{34} + 9001054 q^{36} + 15026564 q^{39} - 27877272 q^{46} - 17907092 q^{49} + 12418574 q^{51} + 17772544 q^{54} + 3040440 q^{61} + 9073996 q^{64} + 20930590 q^{66} - 22789956 q^{69} - 59058092 q^{76} - 17099792 q^{79} + 45225890 q^{81} + 273766584 q^{84} - 214448528 q^{91} - 712237192 q^{94} + 1050848002 q^{96} + 764842670 q^{99}+O(q^{100}) \) 20 * q + 3108 * q^4 + 4514 * q^6 + 22414 * q^9 + 89268 * q^16 + 287868 * q^19 + 1346856 * q^21 + 2033718 * q^24 - 6028120 * q^31 - 9954292 * q^34 + 9001054 * q^36 + 15026564 * q^39 - 27877272 * q^46 - 17907092 * q^49 + 12418574 * q^51 + 17772544 * q^54 + 3040440 * q^61 + 9073996 * q^64 + 20930590 * q^66 - 22789956 * q^69 - 59058092 * q^76 - 17099792 * q^79 + 45225890 * q^81 + 273766584 * q^84 - 214448528 * q^91 - 712237192 * q^94 + 1050848002 * q^96 + 764842670 * q^99

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