LMFDB - Embedded newform 320.10.d.c.161.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,10,Mod(161,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("320.161");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	320.d (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:	$164.811467572$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.7
Character	$\chi$	$=$	320.161
Dual form			320.10.d.c.161.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-148.894i q^{3} -625.000i q^{5} +4203.72 q^{7} -2486.48 q^{9} +O(q^{10})$	$q-148.894i q^{3} -625.000i q^{5} +4203.72 q^{7} -2486.48 q^{9} -59282.8i q^{11} -45619.8i q^{13} -93058.9 q^{15} -112666. q^{17} -231604. i q^{19} -625909. i q^{21} +1.34793e6 q^{23} -390625. q^{25} -2.56046e6i q^{27} +1.72582e6i q^{29} -6.74921e6 q^{31} -8.82686e6 q^{33} -2.62732e6i q^{35} +3.10396e6i q^{37} -6.79252e6 q^{39} +2.43801e7 q^{41} -5.22909e6i q^{43} +1.55405e6i q^{45} +1.67464e7 q^{47} -2.26824e7 q^{49} +1.67753e7i q^{51} -5.18139e7i q^{53} -3.70517e7 q^{55} -3.44845e7 q^{57} -8.25638e7i q^{59} -3.47213e7i q^{61} -1.04524e7 q^{63} -2.85124e7 q^{65} -2.24378e8i q^{67} -2.00698e8i q^{69} +2.76126e8 q^{71} -2.54000e8 q^{73} +5.81618e7i q^{75} -2.49208e8i q^{77} +2.57662e8 q^{79} -4.30179e8 q^{81} -2.69494e8i q^{83} +7.04161e7i q^{85} +2.56964e8 q^{87} +1.31468e8 q^{89} -1.91773e8i q^{91} +1.00492e9i q^{93} -1.44752e8 q^{95} -2.30989e8 q^{97} +1.47405e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 11712 q^{7} - 155768 q^{9}+O(q^{10}) $ 24 * q - 11712 * q^7 - 155768 * q^9	$ 24 q - 11712 q^{7} - 155768 q^{9} - 229152 q^{17} + 168608 q^{23} - 9375000 q^{25} + 33184704 q^{31} + 33485072 q^{33} - 61197184 q^{39} + 68442864 q^{41} - 40785728 q^{47} + 266247320 q^{49} - 26940000 q^{55} + 798839888 q^{57} - 32743904 q^{63} + 25500000 q^{65} - 597783744 q^{71} - 466601792 q^{73} + 1192034688 q^{79} + 1136218424 q^{81} + 846796000 q^{87} - 750700368 q^{89} + 527980000 q^{95} + 1385780608 q^{97}+O(q^{100}) $ 24 * q - 11712 * q^7 - 155768 * q^9 - 229152 * q^17 + 168608 * q^23 - 9375000 * q^25 + 33184704 * q^31 + 33485072 * q^33 - 61197184 * q^39 + 68442864 * q^41 - 40785728 * q^47 + 266247320 * q^49 - 26940000 * q^55 + 798839888 * q^57 - 32743904 * q^63 + 25500000 * q^65 - 597783744 * q^71 - 466601792 * q^73 + 1192034688 * q^79 + 1136218424 * q^81 + 846796000 * q^87 - 750700368 * q^89 + 527980000 * q^95 + 1385780608 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 148.894i	− 1.06129i	−0.847596	−	0.530643i	$-0.821951\pi$
$3$	− 148.894i	− 1.06129i	0.847596	−	0.530643i	$-0.178049\pi$
$4$	0	0
$5$	− 625.000i	− 0.447214i
$5$	− 625.000i	− 0.447214i
$6$	0	0
$7$	4203.72	0.661747	0.330874	−	0.943675i	$-0.392657\pi$
$7$	4203.72	0.661747	0.330874	+	0.943675i	$0.392657\pi$
$8$	0	0
$9$	−2486.48	−0.126326
$10$	0	0
$11$	− 59282.8i	− 1.22085i	−0.792075	−	0.610424i	$-0.790999\pi$
$11$	− 59282.8i	− 1.22085i	0.792075	−	0.610424i	$-0.209001\pi$
$12$	0	0
$13$	− 45619.8i	− 0.443004i	−0.975160	−	0.221502i	$-0.928904\pi$
$13$	− 45619.8i	− 0.443004i	0.975160	−	0.221502i	$-0.0710960\pi$
$14$	0	0
$15$	−93058.9	−0.474621
$16$	0	0
$17$	−112666.	−0.327169	−0.163584	−	0.986529i	$-0.552306\pi$
$17$	−112666.	−0.327169	−0.163584	+	0.986529i	$0.552306\pi$
$18$	0	0
$19$	− 231604.i	− 0.407713i	−0.979001	−	0.203857i	$-0.934652\pi$
$19$	− 231604.i	− 0.407713i	0.979001	−	0.203857i	$-0.0653476\pi$
$20$	0	0
$21$	− 625909.i	− 0.702303i
$22$	0	0
$23$	1.34793e6	1.00436	0.502182	−	0.864762i	$-0.332531\pi$
$23$	1.34793e6	1.00436	0.502182	+	0.864762i	$0.332531\pi$
$24$	0	0
$25$	−390625.	−0.200000
$26$	0	0
$27$	− 2.56046e6i	− 0.927217i
$28$	0	0
$29$	1.72582e6i	0.453110i	0.973998	+	0.226555i	$0.0727463\pi$
$29$	1.72582e6i	0.453110i	−0.973998	+	0.226555i	$0.927254\pi$
$30$	0	0
$31$	−6.74921e6	−1.31258	−0.656290	−	0.754509i	$-0.727875\pi$
$31$	−6.74921e6	−1.31258	−0.656290	+	0.754509i	$0.727875\pi$
$32$	0	0
$33$	−8.82686e6	−1.29567
$34$	0	0
$35$	− 2.62732e6i	− 0.295942i
$36$	0	0
$37$	3.10396e6i	0.272275i	0.990690	+	0.136137i	$0.0434689\pi$
$37$	3.10396e6i	0.272275i	−0.990690	+	0.136137i	$0.956531\pi$
$38$	0	0
$39$	−6.79252e6	−0.470154
$40$	0	0
$41$	2.43801e7	1.34744	0.673718	−	0.738989i	$-0.264696\pi$
$41$	2.43801e7	1.34744	0.673718	+	0.738989i	$0.264696\pi$
$42$	0	0
$43$	− 5.22909e6i	− 0.233248i	−0.993176	−	0.116624i	$-0.962793\pi$
$43$	− 5.22909e6i	− 0.233248i	0.993176	−	0.116624i	$-0.0372073\pi$
$44$	0	0
$45$	1.55405e6i	0.0564948i
$46$	0	0
$47$	1.67464e7	0.500588	0.250294	−	0.968170i	$-0.419473\pi$
$47$	1.67464e7	0.500588	0.250294	+	0.968170i	$0.419473\pi$
$48$	0	0
$49$	−2.26824e7	−0.562090
$50$	0	0
$51$	1.67753e7i	0.347219i
$52$	0	0
$53$	− 5.18139e7i	− 0.901996i	−0.892525	−	0.450998i	$-0.851068\pi$
$53$	− 5.18139e7i	− 0.901996i	0.892525	−	0.450998i	$-0.148932\pi$
$54$	0	0
$55$	−3.70517e7	−0.545979
$56$	0	0
$57$	−3.44845e7	−0.432700
$58$	0	0
$59$	− 8.25638e7i	− 0.887065i	−0.896258	−	0.443533i	$-0.853725\pi$
$59$	− 8.25638e7i	− 0.887065i	0.896258	−	0.443533i	$-0.146275\pi$
$60$	0	0
$61$	− 3.47213e7i	− 0.321079i	−0.987029	−	0.160539i	$-0.948677\pi$
$61$	− 3.47213e7i	− 0.321079i	0.987029	−	0.160539i	$-0.0513234\pi$
$62$	0	0
$63$	−1.04524e7	−0.0835960
$64$	0	0
$65$	−2.85124e7	−0.198118
$66$	0	0
$67$	− 2.24378e8i	− 1.36033i	−0.733059	−	0.680165i	$-0.761908\pi$
$67$	− 2.24378e8i	− 1.36033i	0.733059	−	0.680165i	$-0.238092\pi$
$68$	0	0
$69$	− 2.00698e8i	− 1.06592i
$70$	0	0
$71$	2.76126e8	1.28957	0.644784	−	0.764365i	$-0.276947\pi$
$71$	2.76126e8	1.28957	0.644784	+	0.764365i	$0.276947\pi$
$72$	0	0
$73$	−2.54000e8	−1.04684	−0.523421	−	0.852074i	$-0.675344\pi$
$73$	−2.54000e8	−1.04684	−0.523421	+	0.852074i	$0.675344\pi$
$74$	0	0
$75$	5.81618e7i	0.212257i
$76$	0	0
$77$	− 2.49208e8i	− 0.807892i
$78$	0	0
$79$	2.57662e8	0.744267	0.372134	−	0.928179i	$-0.378626\pi$
$79$	2.57662e8	0.744267	0.372134	+	0.928179i	$0.378626\pi$
$80$	0	0
$81$	−4.30179e8	−1.11037
$82$	0	0
$83$	− 2.69494e8i	− 0.623301i	−0.950197	−	0.311651i	$-0.899118\pi$
$83$	− 2.69494e8i	− 0.623301i	0.950197	−	0.311651i	$-0.100882\pi$
$84$	0	0
$85$	7.04161e7i	0.146314i
$86$	0	0
$87$	2.56964e8	0.480879
$88$	0	0
$89$	1.31468e8	0.222108	0.111054	−	0.993814i	$-0.464577\pi$
$89$	1.31468e8	0.222108	0.111054	+	0.993814i	$0.464577\pi$
$90$	0	0
$91$	− 1.91773e8i	− 0.293157i
$92$	0	0
$93$	1.00492e9i	1.39302i
$94$	0	0
$95$	−1.44752e8	−0.182335
$96$	0	0
$97$	−2.30989e8	−0.264922	−0.132461	−	0.991188i	$-0.542288\pi$
$97$	−2.30989e8	−0.264922	−0.132461	+	0.991188i	$0.542288\pi$
$98$	0	0
$99$	1.47405e8i	0.154225i
$100$	0	0
$101$	− 1.57117e9i	− 1.50237i	−0.660093	−	0.751184i	$-0.729483\pi$
$101$	− 1.57117e9i	− 1.50237i	0.660093	−	0.751184i	$-0.270517\pi$
$102$	0	0
$103$	−6.28409e8	−0.550142	−0.275071	−	0.961424i	$-0.588701\pi$
$103$	−6.28409e8	−0.550142	−0.275071	+	0.961424i	$0.588701\pi$
$104$	0	0
$105$	−3.91193e8	−0.314079
$106$	0	0
$107$	6.24093e8i	0.460280i	0.973158	+	0.230140i	$0.0739184\pi$
$107$	6.24093e8i	0.460280i	−0.973158	+	0.230140i	$0.926082\pi$
$108$	0	0
$109$	4.37941e8i	0.297164i	0.988900	+	0.148582i	$0.0474709\pi$
$109$	4.37941e8i	0.297164i	−0.988900	+	0.148582i	$0.952529\pi$
$110$	0	0
$111$	4.62161e8	0.288961
$112$	0	0
$113$	−2.29094e9	−1.32178	−0.660892	−	0.750481i	$-0.729822\pi$
$113$	−2.29094e9	−1.32178	−0.660892	+	0.750481i	$0.729822\pi$
$114$	0	0
$115$	− 8.42454e8i	− 0.449165i
$116$	0	0
$117$	1.13433e8i	0.0559630i
$118$	0	0
$119$	−4.73615e8	−0.216503
$120$	0	0
$121$	−1.15650e9	−0.490468
$122$	0	0
$123$	− 3.63005e9i	− 1.43001i
$124$	0	0
$125$	2.44141e8i	0.0894427i
$126$	0	0
$127$	1.27244e9	0.434031	0.217016	−	0.976168i	$-0.430368\pi$
$127$	1.27244e9	0.434031	0.217016	+	0.976168i	$0.430368\pi$
$128$	0	0
$129$	−7.78582e8	−0.247543
$130$	0	0
$131$	3.93431e9i	1.16721i	0.812039	+	0.583603i	$0.198357\pi$
$131$	3.93431e9i	1.16721i	−0.812039	+	0.583603i	$0.801643\pi$
$132$	0	0
$133$	− 9.73597e8i	− 0.269803i
$134$	0	0
$135$	−1.60029e9	−0.414664
$136$	0	0
$137$	2.91997e9	0.708168	0.354084	−	0.935214i	$-0.384793\pi$
$137$	2.91997e9	0.708168	0.354084	+	0.935214i	$0.384793\pi$
$138$	0	0
$139$	4.97270e9i	1.12986i	0.825138	+	0.564931i	$0.191097\pi$
$139$	4.97270e9i	1.12986i	−0.825138	+	0.564931i	$0.808903\pi$
$140$	0	0
$141$	− 2.49344e9i	− 0.531266i
$142$	0	0
$143$	−2.70447e9	−0.540841
$144$	0	0
$145$	1.07863e9	0.202637
$146$	0	0
$147$	3.37727e9i	0.596538i
$148$	0	0
$149$	3.17075e9i	0.527016i	0.964657	+	0.263508i	$0.0848795\pi$
$149$	3.17075e9i	0.527016i	−0.964657	+	0.263508i	$0.915121\pi$
$150$	0	0
$151$	3.24506e8	0.0507957	0.0253978	−	0.999677i	$-0.491915\pi$
$151$	3.24506e8	0.0507957	0.0253978	+	0.999677i	$0.491915\pi$
$152$	0	0
$153$	2.80141e8	0.0413300
$154$	0	0
$155$	4.21826e9i	0.587003i
$156$	0	0
$157$	− 1.13825e10i	− 1.49517i	−0.664165	−	0.747586i	$-0.731213\pi$
$157$	− 1.13825e10i	− 1.49517i	0.664165	−	0.747586i	$-0.268787\pi$
$158$	0	0
$159$	−7.71479e9	−0.957275
$160$	0	0
$161$	5.66630e9	0.664635
$162$	0	0
$163$	1.24301e10i	1.37921i	0.724186	+	0.689605i	$0.242216\pi$
$163$	1.24301e10i	1.37921i	−0.724186	+	0.689605i	$0.757784\pi$
$164$	0	0
$165$	5.51679e9i	0.579440i
$166$	0	0
$167$	−8.10266e9	−0.806127	−0.403063	−	0.915172i	$-0.632055\pi$
$167$	−8.10266e9	−0.806127	−0.403063	+	0.915172i	$0.632055\pi$
$168$	0	0
$169$	8.52333e9	0.803747
$170$	0	0
$171$	5.75878e8i	0.0515048i
$172$	0	0
$173$	− 1.38443e9i	− 0.117507i	−0.998273	−	0.0587535i	$-0.981287\pi$
$173$	− 1.38443e9i	− 0.117507i	0.998273	−	0.0587535i	$-0.0187126\pi$
$174$	0	0
$175$	−1.64208e9	−0.132349
$176$	0	0
$177$	−1.22933e10	−0.941429
$178$	0	0
$179$	1.79584e10i	1.30746i	0.756727	+	0.653731i	$0.226797\pi$
$179$	1.79584e10i	1.30746i	−0.756727	+	0.653731i	$0.773203\pi$
$180$	0	0
$181$	1.10828e10i	0.767532i	0.923430	+	0.383766i	$0.125373\pi$
$181$	1.10828e10i	0.767532i	−0.923430	+	0.383766i	$0.874627\pi$
$182$	0	0
$183$	−5.16980e9	−0.340756
$184$	0	0
$185$	1.93997e9	0.121765
$186$	0	0
$187$	6.67914e9i	0.399423i
$188$	0	0
$189$	− 1.07635e10i	− 0.613583i
$190$	0	0
$191$	−1.89241e10	−1.02888	−0.514441	−	0.857525i	$-0.672001\pi$
$191$	−1.89241e10	−1.02888	−0.514441	+	0.857525i	$0.672001\pi$
$192$	0	0
$193$	−2.32641e10	−1.20692	−0.603459	−	0.797394i	$-0.706211\pi$
$193$	−2.32641e10	−1.20692	−0.603459	+	0.797394i	$0.706211\pi$
$194$	0	0
$195$	4.24533e9i	0.210259i
$196$	0	0
$197$	1.53003e10i	0.723770i	0.932223	+	0.361885i	$0.117867\pi$
$197$	1.53003e10i	0.723770i	−0.932223	+	0.361885i	$0.882133\pi$
$198$	0	0
$199$	4.23525e9	0.191443	0.0957217	−	0.995408i	$-0.469484\pi$
$199$	4.23525e9	0.191443	0.0957217	+	0.995408i	$0.469484\pi$
$200$	0	0
$201$	−3.34086e10	−1.44370
$202$	0	0
$203$	7.25484e9i	0.299844i
$204$	0	0
$205$	− 1.52376e10i	− 0.602592i
$206$	0	0
$207$	−3.35159e9	−0.126877
$208$	0	0
$209$	−1.37301e10	−0.497755
$210$	0	0
$211$	− 1.21651e10i	− 0.422516i	−0.977430	−	0.211258i	$-0.932244\pi$
$211$	− 1.21651e10i	− 0.422516i	0.977430	−	0.211258i	$-0.0677561\pi$
$212$	0	0
$213$	− 4.11135e10i	− 1.36860i
$214$	0	0
$215$	−3.26818e9	−0.104312
$216$	0	0
$217$	−2.83718e10	−0.868596
$218$	0	0
$219$	3.78191e10i	1.11100i
$220$	0	0
$221$	5.13979e9i	0.144937i
$222$	0	0
$223$	4.22349e10	1.14367	0.571834	−	0.820369i	$-0.306232\pi$
$223$	4.22349e10	1.14367	0.571834	+	0.820369i	$0.306232\pi$
$224$	0	0
$225$	9.71280e8	0.0252652
$226$	0	0
$227$	5.44385e10i	1.36079i	0.732847	+	0.680393i	$0.238191\pi$
$227$	5.44385e10i	1.36079i	−0.732847	+	0.680393i	$0.761809\pi$
$228$	0	0
$229$	5.62096e9i	0.135068i	0.997717	+	0.0675338i	$0.0215130\pi$
$229$	5.62096e9i	0.135068i	−0.997717	+	0.0675338i	$0.978487\pi$
$230$	0	0
$231$	−3.71056e10	−0.857404
$232$	0	0
$233$	−2.60542e10	−0.579130	−0.289565	−	0.957158i	$-0.593511\pi$
$233$	−2.60542e10	−0.579130	−0.289565	+	0.957158i	$0.593511\pi$
$234$	0	0
$235$	− 1.04665e10i	− 0.223870i
$236$	0	0
$237$	− 3.83644e10i	− 0.789880i
$238$	0	0
$239$	−5.17618e10	−1.02617	−0.513084	−	0.858338i	$-0.671497\pi$
$239$	−5.17618e10	−1.02617	−0.513084	+	0.858338i	$0.671497\pi$
$240$	0	0
$241$	−8.17437e10	−1.56091	−0.780455	−	0.625212i	$-0.785012\pi$
$241$	−8.17437e10	−1.56091	−0.780455	+	0.625212i	$0.785012\pi$
$242$	0	0
$243$	1.36536e10i	0.251200i
$244$	0	0
$245$	1.41765e10i	0.251374i
$246$	0	0
$247$	−1.05657e10	−0.180619
$248$	0	0
$249$	−4.01261e10	−0.661500
$250$	0	0
$251$	1.05725e10i	0.168130i	0.996460	+	0.0840648i	$0.0267903\pi$
$251$	1.05725e10i	0.168130i	−0.996460	+	0.0840648i	$0.973210\pi$
$252$	0	0
$253$	− 7.99088e10i	− 1.22617i
$254$	0	0
$255$	1.04845e10	0.155281
$256$	0	0
$257$	6.51045e8	0.00930919	0.00465460	−	0.999989i	$-0.498518\pi$
$257$	6.51045e8	0.00930919	0.00465460	+	0.999989i	$0.498518\pi$
$258$	0	0
$259$	1.30481e10i	0.180177i
$260$	0	0
$261$	− 4.29120e9i	− 0.0572396i
$262$	0	0
$263$	3.07416e10	0.396210	0.198105	−	0.980181i	$-0.436521\pi$
$263$	3.07416e10	0.396210	0.198105	+	0.980181i	$0.436521\pi$
$264$	0	0
$265$	−3.23837e10	−0.403385
$266$	0	0
$267$	− 1.95748e10i	− 0.235720i
$268$	0	0
$269$	− 1.42503e11i	− 1.65936i	−0.558241	−	0.829679i	$-0.688524\pi$
$269$	− 1.42503e11i	− 1.65936i	0.558241	−	0.829679i	$-0.311476\pi$
$270$	0	0
$271$	−6.22787e10	−0.701419	−0.350710	−	0.936484i	$-0.614060\pi$
$271$	−6.22787e10	−0.701419	−0.350710	+	0.936484i	$0.614060\pi$
$272$	0	0
$273$	−2.85538e10	−0.311123
$274$	0	0
$275$	2.31573e10i	0.244169i
$276$	0	0
$277$	− 1.56053e11i	− 1.59262i	−0.604886	−	0.796312i	$-0.706781\pi$
$277$	− 1.56053e11i	− 1.59262i	0.604886	−	0.796312i	$-0.293219\pi$
$278$	0	0
$279$	1.67818e10	0.165813
$280$	0	0
$281$	−1.55345e11	−1.48635	−0.743173	−	0.669100i	$-0.766680\pi$
$281$	−1.55345e11	−1.48635	−0.743173	+	0.669100i	$0.766680\pi$
$282$	0	0
$283$	9.82212e10i	0.910262i	0.890425	+	0.455131i	$0.150407\pi$
$283$	9.82212e10i	0.910262i	−0.890425	+	0.455131i	$0.849593\pi$
$284$	0	0
$285$	2.15528e10i	0.193509i
$286$	0	0
$287$	1.02487e11	0.891662
$288$	0	0
$289$	−1.05894e11	−0.892961
$290$	0	0
$291$	3.43929e10i	0.281158i
$292$	0	0
$293$	7.52096e10i	0.596168i	0.954540	+	0.298084i	$0.0963476\pi$
$293$	7.52096e10i	0.596168i	−0.954540	+	0.298084i	$0.903652\pi$
$294$	0	0
$295$	−5.16024e10	−0.396708
$296$	0	0
$297$	−1.51791e11	−1.13199
$298$	0	0
$299$	− 6.14921e10i	− 0.444938i
$300$	0	0
$301$	− 2.19816e10i	− 0.154351i
$302$	0	0
$303$	−2.33938e11	−1.59444
$304$	0	0
$305$	−2.17008e10	−0.143591
$306$	0	0
$307$	1.08582e11i	0.697646i	0.937189	+	0.348823i	$0.113419\pi$
$307$	1.08582e11i	0.697646i	−0.937189	+	0.348823i	$0.886581\pi$
$308$	0	0
$309$	9.35664e10i	0.583857i
$310$	0	0
$311$	−2.41068e11	−1.46123	−0.730615	−	0.682790i	$-0.760766\pi$
$311$	−2.41068e11	−1.46123	−0.730615	+	0.682790i	$0.760766\pi$
$312$	0	0
$313$	−3.66670e9	−0.0215936	−0.0107968	−	0.999942i	$-0.503437\pi$
$313$	−3.66670e9	−0.0215936	−0.0107968	+	0.999942i	$0.503437\pi$
$314$	0	0
$315$	6.53278e9i	0.0373853i
$316$	0	0
$317$	− 7.69768e10i	− 0.428147i	−0.976818	−	0.214074i	$-0.931327\pi$
$317$	− 7.69768e10i	− 0.428147i	0.976818	−	0.214074i	$-0.0686732\pi$
$318$	0	0
$319$	1.02311e11	0.553178
$320$	0	0
$321$	9.29237e10	0.488488
$322$	0	0
$323$	2.60938e10i	0.133391i
$324$	0	0
$325$	1.78202e10i	0.0886009i
$326$	0	0
$327$	6.52068e10	0.315376
$328$	0	0
$329$	7.03970e10	0.331263
$330$	0	0
$331$	1.69150e11i	0.774544i	0.921965	+	0.387272i	$0.126583\pi$
$331$	1.69150e11i	0.774544i	−0.921965	+	0.387272i	$0.873417\pi$
$332$	0	0
$333$	− 7.71792e9i	− 0.0343954i
$334$	0	0
$335$	−1.40236e11	−0.608358
$336$	0	0
$337$	4.05514e11	1.71266	0.856330	−	0.516429i	$-0.172739\pi$
$337$	4.05514e11	1.71266	0.856330	+	0.516429i	$0.172739\pi$
$338$	0	0
$339$	3.41108e11i	1.40279i
$340$	0	0
$341$	4.00112e11i	1.60246i
$342$	0	0
$343$	−2.64985e11	−1.03371
$344$	0	0
$345$	−1.25437e11	−0.476692
$346$	0	0
$347$	4.49593e11i	1.66470i	0.554249	+	0.832351i	$0.313006\pi$
$347$	4.49593e11i	1.66470i	−0.554249	+	0.832351i	$0.686994\pi$
$348$	0	0
$349$	− 3.57913e11i	− 1.29141i	−0.763588	−	0.645704i	$-0.776564\pi$
$349$	− 3.57913e11i	− 1.29141i	0.763588	−	0.645704i	$-0.223436\pi$
$350$	0	0
$351$	−1.16808e11	−0.410761
$352$	0	0
$353$	−2.27833e11	−0.780964	−0.390482	−	0.920610i	$-0.627692\pi$
$353$	−2.27833e11	−0.780964	−0.390482	+	0.920610i	$0.627692\pi$
$354$	0	0
$355$	− 1.72579e11i	− 0.576712i
$356$	0	0
$357$	7.05185e10i	0.229771i
$358$	0	0
$359$	2.65345e11	0.843113	0.421557	−	0.906802i	$-0.361484\pi$
$359$	2.65345e11	0.843113	0.421557	+	0.906802i	$0.361484\pi$
$360$	0	0
$361$	2.69047e11	0.833770
$362$	0	0
$363$	1.72196e11i	0.520526i
$364$	0	0
$365$	1.58750e11i	0.468162i
$366$	0	0
$367$	2.28816e11	0.658398	0.329199	−	0.944260i	$-0.393221\pi$
$367$	2.28816e11	0.658398	0.329199	+	0.944260i	$0.393221\pi$
$368$	0	0
$369$	−6.06206e10	−0.170216
$370$	0	0
$371$	− 2.17811e11i	− 0.596894i
$372$	0	0
$373$	5.18029e11i	1.38568i	0.721089	+	0.692842i	$0.243642\pi$
$373$	5.18029e11i	1.38568i	−0.721089	+	0.692842i	$0.756358\pi$
$374$	0	0
$375$	3.63511e10	0.0949242
$376$	0	0
$377$	7.87313e10	0.200730
$378$	0	0
$379$	− 1.78422e10i	− 0.0444194i	−0.999753	−	0.0222097i	$-0.992930\pi$
$379$	− 1.78422e10i	− 0.0444194i	0.999753	−	0.0222097i	$-0.00707014\pi$
$380$	0	0
$381$	− 1.89459e11i	− 0.460631i
$382$	0	0
$383$	4.67453e11	1.11005	0.555026	−	0.831833i	$-0.312708\pi$
$383$	4.67453e11	1.11005	0.555026	+	0.831833i	$0.312708\pi$
$384$	0	0
$385$	−1.55755e11	−0.361300
$386$	0	0
$387$	1.30020e10i	0.0294653i
$388$	0	0
$389$	− 1.69032e11i	− 0.374280i	−0.982333	−	0.187140i	$-0.940078\pi$
$389$	− 1.69032e11i	− 0.374280i	0.982333	−	0.187140i	$-0.0599217\pi$
$390$	0	0
$391$	−1.51865e11	−0.328596
$392$	0	0
$393$	5.85795e11	1.23874
$394$	0	0
$395$	− 1.61039e11i	− 0.332847i
$396$	0	0
$397$	− 8.53342e11i	− 1.72411i	−0.506812	−	0.862056i	$-0.669176\pi$
$397$	− 8.53342e11i	− 1.72411i	0.506812	−	0.862056i	$-0.330824\pi$
$398$	0	0
$399$	−1.44963e11	−0.286338
$400$	0	0
$401$	−7.73973e11	−1.49478	−0.747388	−	0.664388i	$-0.768692\pi$
$401$	−7.73973e11	−1.49478	−0.747388	+	0.664388i	$0.768692\pi$
$402$	0	0
$403$	3.07898e11i	0.581478i
$404$	0	0
$405$	2.68862e11i	0.496572i
$406$	0	0
$407$	1.84011e11	0.332406
$408$	0	0
$409$	4.18310e11	0.739168	0.369584	−	0.929197i	$-0.379500\pi$
$409$	4.18310e11	0.739168	0.369584	+	0.929197i	$0.379500\pi$
$410$	0	0
$411$	− 4.34767e11i	− 0.751568i
$412$	0	0
$413$	− 3.47075e11i	− 0.587013i
$414$	0	0
$415$	−1.68434e11	−0.278749
$416$	0	0
$417$	7.40406e11	1.19911
$418$	0	0
$419$	− 7.44103e11i	− 1.17942i	−0.807614	−	0.589712i	$-0.799241\pi$
$419$	− 7.44103e11i	− 1.17942i	0.807614	−	0.589712i	$-0.200759\pi$
$420$	0	0
$421$	1.22303e12i	1.89743i	0.316130	+	0.948716i	$0.397617\pi$
$421$	1.22303e12i	1.89743i	−0.316130	+	0.948716i	$0.602383\pi$
$422$	0	0
$423$	−4.16395e10	−0.0632373
$424$	0	0
$425$	4.40101e10	0.0654337
$426$	0	0
$427$	− 1.45958e11i	− 0.212473i
$428$	0	0
$429$	4.02679e11i	0.573986i
$430$	0	0
$431$	−3.51500e11	−0.490657	−0.245329	−	0.969440i	$-0.578896\pi$
$431$	−3.51500e11	−0.490657	−0.245329	+	0.969440i	$0.578896\pi$
$432$	0	0
$433$	5.72847e11	0.783147	0.391573	−	0.920147i	$-0.371931\pi$
$433$	5.72847e11	0.783147	0.391573	+	0.920147i	$0.371931\pi$
$434$	0	0
$435$	− 1.60602e11i	− 0.215055i
$436$	0	0
$437$	− 3.12185e11i	− 0.409492i
$438$	0	0
$439$	2.33605e11	0.300187	0.150094	−	0.988672i	$-0.452042\pi$
$439$	2.33605e11	0.300187	0.150094	+	0.988672i	$0.452042\pi$
$440$	0	0
$441$	5.63992e10	0.0710067
$442$	0	0
$443$	− 8.49050e11i	− 1.04741i	−0.851900	−	0.523705i	$-0.824549\pi$
$443$	− 8.49050e11i	− 1.04741i	0.851900	−	0.523705i	$-0.175451\pi$
$444$	0	0
$445$	− 8.21674e10i	− 0.0993298i
$446$	0	0
$447$	4.72106e11	0.559314
$448$	0	0
$449$	−4.24174e11	−0.492534	−0.246267	−	0.969202i	$-0.579204\pi$
$449$	−4.24174e11	−0.492534	−0.246267	+	0.969202i	$0.579204\pi$
$450$	0	0
$451$	− 1.44532e12i	− 1.64501i
$452$	0	0
$453$	− 4.83171e10i	− 0.0539087i
$454$	0	0
$455$	−1.19858e11	−0.131104
$456$	0	0
$457$	−7.35290e11	−0.788562	−0.394281	−	0.918990i	$-0.629006\pi$
$457$	−7.35290e11	−0.788562	−0.394281	+	0.918990i	$0.629006\pi$
$458$	0	0
$459$	2.88476e11i	0.303356i
$460$	0	0
$461$	8.05031e11i	0.830153i	0.909786	+	0.415077i	$0.136245\pi$
$461$	8.05031e11i	0.830153i	−0.909786	+	0.415077i	$0.863755\pi$
$462$	0	0
$463$	1.16010e12	1.17323	0.586614	−	0.809866i	$-0.300460\pi$
$463$	1.16010e12	1.17323	0.586614	+	0.809866i	$0.300460\pi$
$464$	0	0
$465$	6.28074e11	0.622978
$466$	0	0
$467$	− 4.07478e11i	− 0.396441i	−0.980157	−	0.198220i	$-0.936484\pi$
$467$	− 4.07478e11i	− 0.396441i	0.980157	−	0.198220i	$-0.0635162\pi$
$468$	0	0
$469$	− 9.43222e11i	− 0.900194i
$470$	0	0
$471$	−1.69480e12	−1.58680
$472$	0	0
$473$	−3.09995e11	−0.284760
$474$	0	0
$475$	9.04703e10i	0.0815426i
$476$	0	0
$477$	1.28834e11i	0.113946i
$478$	0	0
$479$	1.74949e12	1.51846	0.759228	−	0.650825i	$-0.225577\pi$
$479$	1.74949e12	1.51846	0.759228	+	0.650825i	$0.225577\pi$
$480$	0	0
$481$	1.41602e11	0.120619
$482$	0	0
$483$	− 8.43679e11i	− 0.705367i
$484$	0	0
$485$	1.44368e11i	0.118477i
$486$	0	0
$487$	2.00468e12	1.61497	0.807486	−	0.589886i	$-0.200827\pi$
$487$	2.00468e12	1.61497	0.807486	+	0.589886i	$0.200827\pi$
$488$	0	0
$489$	1.85077e12	1.46374
$490$	0	0
$491$	− 1.93585e12i	− 1.50316i	−0.659642	−	0.751580i	$-0.729292\pi$
$491$	− 1.93585e12i	− 1.50316i	0.659642	−	0.751580i	$-0.270708\pi$
$492$	0	0
$493$	− 1.94440e11i	− 0.148243i
$494$	0	0
$495$	9.21283e10	0.0689715
$496$	0	0
$497$	1.16075e12	0.853368
$498$	0	0
$499$	− 1.43898e12i	− 1.03897i	−0.854480	−	0.519485i	$-0.826124\pi$
$499$	− 1.43898e12i	− 1.03897i	0.854480	−	0.519485i	$-0.173876\pi$
$500$	0	0
$501$	1.20644e12i	0.855530i
$502$	0	0
$503$	1.76710e12	1.23085	0.615427	−	0.788194i	$-0.288984\pi$
$503$	1.76710e12	1.23085	0.615427	+	0.788194i	$0.288984\pi$
$504$	0	0
$505$	−9.81980e11	−0.671879
$506$	0	0
$507$	− 1.26907e12i	− 0.853005i
$508$	0	0
$509$	1.29884e12i	0.857683i	0.903380	+	0.428842i	$0.141078\pi$
$509$	1.29884e12i	0.857683i	−0.903380	+	0.428842i	$0.858922\pi$
$510$	0	0
$511$	−1.06774e12	−0.692744
$512$	0	0
$513$	−5.93013e11	−0.378038
$514$	0	0
$515$	3.92756e11i	0.246031i
$516$	0	0
$517$	− 9.92771e11i	− 0.611141i
$518$	0	0
$519$	−2.06134e11	−0.124708
$520$	0	0
$521$	−3.89054e11	−0.231334	−0.115667	−	0.993288i	$-0.536901\pi$
$521$	−3.89054e11	−0.231334	−0.115667	+	0.993288i	$0.536901\pi$
$522$	0	0
$523$	− 4.00430e11i	− 0.234029i	−0.993130	−	0.117014i	$-0.962668\pi$
$523$	− 4.00430e11i	− 0.234029i	0.993130	−	0.117014i	$-0.0373324\pi$
$524$	0	0
$525$	2.44496e11i	0.140461i
$526$	0	0
$527$	7.60405e11	0.429435
$528$	0	0
$529$	1.57532e10	0.00874618
$530$	0	0
$531$	2.05293e11i	0.112060i
$532$	0	0
$533$	− 1.11221e12i	− 0.596920i
$534$	0	0
$535$	3.90058e11	0.205843
$536$	0	0
$537$	2.67390e12	1.38759
$538$	0	0
$539$	1.34467e12i	0.686226i
$540$	0	0
$541$	1.16610e12i	0.585261i	0.956226	+	0.292631i	$0.0945306\pi$
$541$	1.16610e12i	0.585261i	−0.956226	+	0.292631i	$0.905469\pi$
$542$	0	0
$543$	1.65017e12	0.814570
$544$	0	0
$545$	2.73713e11	0.132896
$546$	0	0
$547$	− 1.22980e11i	− 0.0587344i	−0.999569	−	0.0293672i	$-0.990651\pi$
$547$	− 1.22980e11i	− 0.0587344i	0.999569	−	0.0293672i	$-0.00934921\pi$
$548$	0	0
$549$	8.63337e10i	0.0405606i
$550$	0	0
$551$	3.99706e11	0.184739
$552$	0	0
$553$	1.08314e12	0.492517
$554$	0	0
$555$	− 2.88851e11i	− 0.129227i
$556$	0	0
$557$	1.38253e12i	0.608591i	0.952578	+	0.304295i	$0.0984210\pi$
$557$	1.38253e12i	0.608591i	−0.952578	+	0.304295i	$0.901579\pi$
$558$	0	0
$559$	−2.38550e11	−0.103330
$560$	0	0
$561$	9.94484e11	0.423902
$562$	0	0
$563$	1.15506e12i	0.484525i	0.970211	+	0.242262i	$0.0778895\pi$
$563$	1.15506e12i	0.484525i	−0.970211	+	0.242262i	$0.922111\pi$
$564$	0	0
$565$	1.43184e12i	0.591120i
$566$	0	0
$567$	−1.80835e12	−0.734783
$568$	0	0
$569$	4.26011e12	1.70379	0.851894	−	0.523715i	$-0.175454\pi$
$569$	4.26011e12	1.70379	0.851894	+	0.523715i	$0.175454\pi$
$570$	0	0
$571$	− 1.15223e12i	− 0.453602i	−0.973941	−	0.226801i	$-0.927173\pi$
$571$	− 1.15223e12i	− 0.453602i	0.973941	−	0.226801i	$-0.0728267\pi$
$572$	0	0
$573$	2.81769e12i	1.09194i
$574$	0	0
$575$	−5.26534e11	−0.200873
$576$	0	0
$577$	2.25328e12	0.846298	0.423149	−	0.906060i	$-0.360925\pi$
$577$	2.25328e12	0.846298	0.423149	+	0.906060i	$0.360925\pi$
$578$	0	0
$579$	3.46388e12i	1.28088i
$580$	0	0
$581$	− 1.13288e12i	− 0.412468i
$582$	0	0
$583$	−3.07167e12	−1.10120
$584$	0	0
$585$	7.08953e10	0.0250274
$586$	0	0
$587$	9.57182e11i	0.332754i	0.986062	+	0.166377i	$0.0532068\pi$
$587$	9.57182e11i	0.332754i	−0.986062	+	0.166377i	$0.946793\pi$
$588$	0	0
$589$	1.56314e12i	0.535156i
$590$	0	0
$591$	2.27812e12	0.768127
$592$	0	0
$593$	1.63092e12	0.541609	0.270805	−	0.962634i	$-0.412710\pi$
$593$	1.63092e12	0.541609	0.270805	+	0.962634i	$0.412710\pi$
$594$	0	0
$595$	2.96009e11i	0.0968231i
$596$	0	0
$597$	− 6.30604e11i	− 0.203176i
$598$	0	0
$599$	−1.80294e12	−0.572217	−0.286108	−	0.958197i	$-0.592362\pi$
$599$	−1.80294e12	−0.572217	−0.286108	+	0.958197i	$0.592362\pi$
$600$	0	0
$601$	−1.63160e12	−0.510128	−0.255064	−	0.966924i	$-0.582097\pi$
$601$	−1.63160e12	−0.510128	−0.255064	+	0.966924i	$0.582097\pi$
$602$	0	0
$603$	5.57911e11i	0.171845i
$604$	0	0
$605$	7.22811e11i	0.219344i
$606$	0	0
$607$	1.30062e12	0.388866	0.194433	−	0.980916i	$-0.437713\pi$
$607$	1.30062e12	0.388866	0.194433	+	0.980916i	$0.437713\pi$
$608$	0	0
$609$	1.08020e12	0.318220
$610$	0	0
$611$	− 7.63965e11i	− 0.221763i
$612$	0	0
$613$	1.87744e12i	0.537024i	0.963276	+	0.268512i	$0.0865319\pi$
$613$	1.87744e12i	0.537024i	−0.963276	+	0.268512i	$0.913468\pi$
$614$	0	0
$615$	−2.26878e12	−0.639521
$616$	0	0
$617$	−3.68201e12	−1.02283	−0.511414	−	0.859335i	$-0.670878\pi$
$617$	−3.68201e12	−1.02283	−0.511414	+	0.859335i	$0.670878\pi$
$618$	0	0
$619$	− 5.14607e12i	− 1.40886i	−0.709774	−	0.704430i	$-0.751203\pi$
$619$	− 5.14607e12i	− 1.40886i	0.709774	−	0.704430i	$-0.248797\pi$
$620$	0	0
$621$	− 3.45131e12i	− 0.931263i
$622$	0	0
$623$	5.52653e11	0.146979
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	2.04433e12i	0.528260i
$628$	0	0
$629$	− 3.49709e11i	− 0.0890799i
$630$	0	0
$631$	−2.96635e12	−0.744887	−0.372444	−	0.928055i	$-0.621480\pi$
$631$	−2.96635e12	−0.744887	−0.372444	+	0.928055i	$0.621480\pi$
$632$	0	0
$633$	−1.81131e12	−0.448410
$634$	0	0
$635$	− 7.95276e11i	− 0.194105i
$636$	0	0
$637$	1.03477e12i	0.249009i
$638$	0	0
$639$	−6.86580e11	−0.162906
$640$	0	0
$641$	3.07182e12	0.718678	0.359339	−	0.933207i	$-0.383002\pi$
$641$	3.07182e12	0.718678	0.359339	+	0.933207i	$0.383002\pi$
$642$	0	0
$643$	− 3.94776e12i	− 0.910755i	−0.890298	−	0.455378i	$-0.849504\pi$
$643$	− 3.94776e12i	− 0.910755i	0.890298	−	0.455378i	$-0.150496\pi$
$644$	0	0
$645$	4.86614e11i	0.110705i
$646$	0	0
$647$	−2.24909e12	−0.504589	−0.252294	−	0.967651i	$-0.581185\pi$
$647$	−2.24909e12	−0.504589	−0.252294	+	0.967651i	$0.581185\pi$
$648$	0	0
$649$	−4.89461e12	−1.08297
$650$	0	0
$651$	4.22439e12i	0.921828i
$652$	0	0
$653$	5.50784e12i	1.18542i	0.805416	+	0.592709i	$0.201942\pi$
$653$	5.50784e12i	1.18542i	−0.805416	+	0.592709i	$0.798058\pi$
$654$	0	0
$655$	2.45894e12	0.521990
$656$	0	0
$657$	6.31565e11	0.132243
$658$	0	0
$659$	2.40250e12i	0.496225i	0.968731	+	0.248113i	$0.0798103\pi$
$659$	2.40250e12i	0.496225i	−0.968731	+	0.248113i	$0.920190\pi$
$660$	0	0
$661$	− 5.03150e12i	− 1.02516i	−0.858640	−	0.512579i	$-0.828690\pi$
$661$	− 5.03150e12i	− 1.02516i	0.858640	−	0.512579i	$-0.171310\pi$
$662$	0	0
$663$	7.65284e11	0.153820
$664$	0	0
$665$	−6.08498e11	−0.120660
$666$	0	0
$667$	2.32627e12i	0.455087i
$668$	0	0
$669$	− 6.28854e12i	− 1.21376i
$670$	0	0
$671$	−2.05837e12	−0.391988
$672$	0	0
$673$	−6.68422e12	−1.25598	−0.627991	−	0.778221i	$-0.716122\pi$
$673$	−6.68422e12	−1.25598	−0.627991	+	0.778221i	$0.716122\pi$
$674$	0	0
$675$	1.00018e12i	0.185443i
$676$	0	0
$677$	− 8.94195e12i	− 1.63600i	−0.575218	−	0.818000i	$-0.695083\pi$
$677$	− 8.94195e12i	− 1.63600i	0.575218	−	0.818000i	$-0.304917\pi$
$678$	0	0
$679$	−9.71010e11	−0.175311
$680$	0	0
$681$	8.10558e12	1.44418
$682$	0	0
$683$	2.75836e12i	0.485018i	0.970149	+	0.242509i	$0.0779704\pi$
$683$	2.75836e12i	0.485018i	−0.970149	+	0.242509i	$0.922030\pi$
$684$	0	0
$685$	− 1.82498e12i	− 0.316702i
$686$	0	0
$687$	8.36929e11	0.143345
$688$	0	0
$689$	−2.36374e12	−0.399588
$690$	0	0
$691$	3.42539e12i	0.571557i	0.958296	+	0.285778i	$0.0922521\pi$
$691$	3.42539e12i	0.571557i	−0.958296	+	0.285778i	$0.907748\pi$
$692$	0	0
$693$	6.19650e11i	0.102058i
$694$	0	0
$695$	3.10794e12	0.505290
$696$	0	0
$697$	−2.74680e12	−0.440839
$698$	0	0
$699$	3.87932e12i	0.614622i
$700$	0	0
$701$	− 5.27219e12i	− 0.824632i	−0.911041	−	0.412316i	$-0.864720\pi$
$701$	− 5.27219e12i	− 0.824632i	0.911041	−	0.412316i	$-0.135280\pi$
$702$	0	0
$703$	7.18888e11	0.111010
$704$	0	0
$705$	−1.55840e12	−0.237590
$706$	0	0
$707$	− 6.60474e12i	− 0.994188i
$708$	0	0
$709$	− 4.59952e11i	− 0.0683604i	−0.999416	−	0.0341802i	$-0.989118\pi$
$709$	− 4.59952e11i	− 0.0683604i	0.999416	−	0.0341802i	$-0.0108820\pi$
$710$	0	0
$711$	−6.40672e11	−0.0940204
$712$	0	0
$713$	−9.09744e12	−1.31831
$714$	0	0
$715$	1.69029e12i	0.241871i
$716$	0	0
$717$	7.70703e12i	1.08906i
$718$	0	0
$719$	5.21179e12	0.727289	0.363645	−	0.931538i	$-0.381532\pi$
$719$	5.21179e12	0.727289	0.363645	+	0.931538i	$0.381532\pi$
$720$	0	0
$721$	−2.64165e12	−0.364055
$722$	0	0
$723$	1.21712e13i	1.65657i
$724$	0	0
$725$	− 6.74147e11i	− 0.0906219i
$726$	0	0
$727$	−1.10507e13	−1.46718	−0.733590	−	0.679593i	$-0.762157\pi$
$727$	−1.10507e13	−1.46718	−0.733590	+	0.679593i	$0.762157\pi$
$728$	0	0
$729$	−6.43427e12	−0.843773
$730$	0	0
$731$	5.89140e11i	0.0763115i
$732$	0	0
$733$	1.79109e12i	0.229166i	0.993414	+	0.114583i	$0.0365532\pi$
$733$	1.79109e12i	0.229166i	−0.993414	+	0.114583i	$0.963447\pi$
$734$	0	0
$735$	2.11080e12	0.266780
$736$	0	0
$737$	−1.33018e13	−1.66075
$738$	0	0
$739$	− 6.22160e12i	− 0.767365i	−0.923465	−	0.383682i	$-0.874656\pi$
$739$	− 6.22160e12i	− 0.767365i	0.923465	−	0.383682i	$-0.125344\pi$
$740$	0	0
$741$	1.57317e12i	0.191688i
$742$	0	0
$743$	4.88874e11	0.0588501	0.0294250	−	0.999567i	$-0.490632\pi$
$743$	4.88874e11	0.0588501	0.0294250	+	0.999567i	$0.490632\pi$
$744$	0	0
$745$	1.98172e12	0.235689
$746$	0	0
$747$	6.70091e11i	0.0787393i
$748$	0	0
$749$	2.62351e12i	0.304589i
$750$	0	0
$751$	1.47129e13	1.68779	0.843894	−	0.536510i	$-0.180258\pi$
$751$	1.47129e13	1.68779	0.843894	+	0.536510i	$0.180258\pi$
$752$	0	0
$753$	1.57418e12	0.178433
$754$	0	0
$755$	− 2.02816e11i	− 0.0227165i
$756$	0	0
$757$	− 1.00018e13i	− 1.10700i	−0.832850	−	0.553498i	$-0.813293\pi$
$757$	− 1.00018e13i	− 1.10700i	0.832850	−	0.553498i	$-0.186707\pi$
$758$	0	0
$759$	−1.18980e13	−1.30132
$760$	0	0
$761$	−9.52395e12	−1.02940	−0.514702	−	0.857369i	$-0.672098\pi$
$761$	−9.52395e12	−1.02940	−0.514702	+	0.857369i	$0.672098\pi$
$762$	0	0
$763$	1.84098e12i	0.196648i
$764$	0	0
$765$	− 1.75088e11i	− 0.0184833i
$766$	0	0
$767$	−3.76654e12	−0.392974
$768$	0	0
$769$	−1.68565e13	−1.73819	−0.869097	−	0.494642i	$-0.835299\pi$
$769$	−1.68565e13	−1.73819	−0.869097	+	0.494642i	$0.835299\pi$
$770$	0	0
$771$	− 9.69368e10i	− 0.00987971i
$772$	0	0
$773$	− 1.23900e13i	− 1.24814i	−0.781368	−	0.624070i	$-0.785478\pi$
$773$	− 1.23900e13i	− 1.24814i	0.781368	−	0.624070i	$-0.214522\pi$
$774$	0	0
$775$	2.63641e12	0.262516
$776$	0	0
$777$	1.94279e12	0.191219
$778$	0	0
$779$	− 5.64652e12i	− 0.549367i
$780$	0	0
$781$	− 1.63695e13i	− 1.57437i
$782$	0	0
$783$	4.41889e12	0.420131
$784$	0	0
$785$	−7.11409e12	−0.668661
$786$	0	0
$787$	− 1.35967e12i	− 0.126342i	−0.998003	−	0.0631711i	$-0.979879\pi$
$787$	− 1.35967e12i	− 0.126342i	0.998003	−	0.0631711i	$-0.0201214\pi$
$788$	0	0
$789$	− 4.57724e12i	− 0.420492i
$790$	0	0
$791$	−9.63046e12	−0.874687
$792$	0	0
$793$	−1.58398e12	−0.142239
$794$	0	0
$795$	4.82174e12i	0.428107i
$796$	0	0
$797$	1.51835e13i	1.33293i	0.745535	+	0.666466i	$0.232194\pi$
$797$	1.51835e13i	1.33293i	−0.745535	+	0.666466i	$0.767806\pi$
$798$	0	0
$799$	−1.88674e12	−0.163777
$800$	0	0
$801$	−3.26892e11	−0.0280581
$802$	0	0
$803$	1.50578e13i	1.27803i
$804$	0	0
$805$	− 3.54144e12i	− 0.297234i
$806$	0	0
$807$	−2.12179e13	−1.76105
$808$	0	0
$809$	−5.10573e12	−0.419073	−0.209537	−	0.977801i	$-0.567196\pi$
$809$	−5.10573e12	−0.419073	−0.209537	+	0.977801i	$0.567196\pi$
$810$	0	0
$811$	− 1.39312e12i	− 0.113082i	−0.998400	−	0.0565410i	$-0.981993\pi$
$811$	− 1.39312e12i	− 0.113082i	0.998400	−	0.0565410i	$-0.0180072\pi$
$812$	0	0
$813$	9.27293e12i	0.744406i
$814$	0	0
$815$	7.76881e12	0.616802
$816$	0	0
$817$	−1.21108e12	−0.0950983
$818$	0	0
$819$	4.76838e11i	0.0370334i
$820$	0	0
$821$	− 2.26588e13i	− 1.74057i	−0.492545	−	0.870287i	$-0.663933\pi$
$821$	− 2.26588e13i	− 1.74057i	0.492545	−	0.870287i	$-0.336067\pi$
$822$	0	0
$823$	8.79112e12	0.667952	0.333976	−	0.942582i	$-0.391610\pi$
$823$	8.79112e12	0.667952	0.333976	+	0.942582i	$0.391610\pi$
$824$	0	0
$825$	3.44799e12	0.259133
$826$	0	0
$827$	− 8.03171e12i	− 0.597081i	−0.954397	−	0.298541i	$-0.903500\pi$
$827$	− 8.03171e12i	− 0.597081i	0.954397	−	0.298541i	$-0.0964998\pi$
$828$	0	0
$829$	1.61513e13i	1.18771i	0.804570	+	0.593857i	$0.202396\pi$
$829$	1.61513e13i	1.18771i	−0.804570	+	0.593857i	$0.797604\pi$
$830$	0	0
$831$	−2.32354e13	−1.69023
$832$	0	0
$833$	2.55553e12	0.183898
$834$	0	0
$835$	5.06416e12i	0.360511i
$836$	0	0
$837$	1.72811e13i	1.21705i
$838$	0	0
$839$	−6.27585e12	−0.437264	−0.218632	−	0.975807i	$-0.570159\pi$
$839$	−6.27585e12	−0.437264	−0.218632	+	0.975807i	$0.570159\pi$
$840$	0	0
$841$	1.15287e13	0.794692
$842$	0	0
$843$	2.31300e13i	1.57744i
$844$	0	0
$845$	− 5.32708e12i	− 0.359447i
$846$	0	0
$847$	−4.86158e12	−0.324566
$848$	0	0
$849$	1.46246e13	0.966047
$850$	0	0
$851$	4.18390e12i	0.273463i
$852$	0	0
$853$	2.36087e13i	1.52687i	0.645888	+	0.763433i	$0.276487\pi$
$853$	2.36087e13i	1.52687i	−0.645888	+	0.763433i	$0.723513\pi$
$854$	0	0
$855$	3.59924e11	0.0230337
$856$	0	0
$857$	−1.55401e12	−0.0984102	−0.0492051	−	0.998789i	$-0.515669\pi$
$857$	−1.55401e12	−0.0984102	−0.0492051	+	0.998789i	$0.515669\pi$
$858$	0	0
$859$	1.98302e13i	1.24267i	0.783544	+	0.621336i	$0.213410\pi$
$859$	1.98302e13i	1.24267i	−0.783544	+	0.621336i	$0.786590\pi$
$860$	0	0
$861$	− 1.52597e13i	− 0.946308i
$862$	0	0
$863$	2.01695e13	1.23779	0.618893	−	0.785475i	$-0.287581\pi$
$863$	2.01695e13	1.23779	0.618893	+	0.785475i	$0.287581\pi$
$864$	0	0
$865$	−8.65269e11	−0.0525507
$866$	0	0
$867$	1.57670e13i	0.947686i
$868$	0	0
$869$	− 1.52749e13i	− 0.908637i
$870$	0	0
$871$	−1.02361e13	−0.602632
$872$	0	0
$873$	5.74348e11	0.0334665
$874$	0	0
$875$	1.02630e12i	0.0591885i
$876$	0	0
$877$	− 2.14517e13i	− 1.22451i	−0.790660	−	0.612255i	$-0.790262\pi$
$877$	− 2.14517e13i	− 1.22451i	0.790660	−	0.612255i	$-0.209738\pi$
$878$	0	0
$879$	1.11983e13	0.632704
$880$	0	0
$881$	2.16311e13	1.20973	0.604863	−	0.796329i	$-0.293228\pi$
$881$	2.16311e13	1.20973	0.604863	+	0.796329i	$0.293228\pi$
$882$	0	0
$883$	− 1.24113e13i	− 0.687061i	−0.939141	−	0.343531i	$-0.888377\pi$
$883$	− 1.24113e13i	− 0.687061i	0.939141	−	0.343531i	$-0.111623\pi$
$884$	0	0
$885$	7.68329e12i	0.421020i
$886$	0	0
$887$	−3.14631e13	−1.70665	−0.853327	−	0.521376i	$-0.825419\pi$
$887$	−3.14631e13	−1.70665	−0.853327	+	0.521376i	$0.825419\pi$
$888$	0	0
$889$	5.34898e12	0.287219
$890$	0	0
$891$	2.55022e13i	1.35559i
$892$	0	0
$893$	− 3.87852e12i	− 0.204096i
$894$	0	0
$895$	1.12240e13	0.584715
$896$	0	0
$897$	−9.15582e12	−0.472206
$898$	0	0
$899$	− 1.16479e13i	− 0.594742i
$900$	0	0
$901$	5.83765e12i	0.295105i
$902$	0	0
$903$	−3.27294e12	−0.163811
$904$	0	0
$905$	6.92676e12	0.343251
$906$	0	0
$907$	− 1.72741e13i	− 0.847547i	−0.905768	−	0.423773i	$-0.860705\pi$
$907$	− 1.72741e13i	− 0.847547i	0.905768	−	0.423773i	$-0.139295\pi$
$908$	0	0
$909$	3.90667e12i	0.189788i
$910$	0	0
$911$	2.81204e13	1.35266	0.676330	−	0.736599i	$-0.263569\pi$
$911$	2.81204e13	1.35266	0.676330	+	0.736599i	$0.263569\pi$
$912$	0	0
$913$	−1.59764e13	−0.760956
$914$	0	0
$915$	3.23112e12i	0.152391i
$916$	0	0
$917$	1.65387e13i	0.772395i
$918$	0	0
$919$	−4.03985e13	−1.86829	−0.934146	−	0.356890i	$-0.883837\pi$
$919$	−4.03985e13	−1.86829	−0.934146	+	0.356890i	$0.883837\pi$
$920$	0	0
$921$	1.61672e13	0.740402
$922$	0	0
$923$	− 1.25968e13i	− 0.571284i
$924$	0	0
$925$	− 1.21248e12i	− 0.0544550i
$926$	0	0
$927$	1.56252e12	0.0694973
$928$	0	0
$929$	1.14368e13	0.503771	0.251886	−	0.967757i	$-0.418949\pi$
$929$	1.14368e13	0.503771	0.251886	+	0.967757i	$0.418949\pi$
$930$	0	0
$931$	5.25333e12i	0.229172i
$932$	0	0
$933$	3.58937e13i	1.55078i
$934$	0	0
$935$	4.17446e12	0.178627
$936$	0	0
$937$	2.58590e13	1.09593	0.547966	−	0.836501i	$-0.315402\pi$
$937$	2.58590e13	1.09593	0.547966	+	0.836501i	$0.315402\pi$
$938$	0	0
$939$	5.45950e11i	0.0229170i
$940$	0	0
$941$	− 3.29355e13i	− 1.36934i	−0.728855	−	0.684669i	$-0.759947\pi$
$941$	− 3.29355e13i	− 1.36934i	0.728855	−	0.684669i	$-0.240053\pi$
$942$	0	0
$943$	3.28626e13	1.35332
$944$	0	0
$945$	−6.72716e12	−0.274403
$946$	0	0
$947$	− 2.36383e13i	− 0.955085i	−0.878609	−	0.477543i	$-0.841528\pi$
$947$	− 2.36383e13i	− 0.955085i	0.878609	−	0.477543i	$-0.158472\pi$
$948$	0	0
$949$	1.15874e13i	0.463755i
$950$	0	0
$951$	−1.14614e13	−0.454386
$952$	0	0
$953$	−4.22073e13	−1.65756	−0.828780	−	0.559575i	$-0.810965\pi$
$953$	−4.22073e13	−1.65756	−0.828780	+	0.559575i	$0.810965\pi$
$954$	0	0
$955$	1.18276e13i	0.460130i
$956$	0	0
$957$	− 1.52335e13i	− 0.587079i
$958$	0	0
$959$	1.22747e13	0.468628
$960$	0	0
$961$	1.91123e13	0.722864
$962$	0	0
$963$	− 1.55179e12i	− 0.0581454i
$964$	0	0
$965$	1.45400e13i	0.539750i
$966$	0	0
$967$	4.21727e13	1.55100	0.775501	−	0.631346i	$-0.217497\pi$
$967$	4.21727e13	1.55100	0.775501	+	0.631346i	$0.217497\pi$
$968$	0	0
$969$	3.88522e12	0.141566
$970$	0	0
$971$	− 2.69164e13i	− 0.971694i	−0.874044	−	0.485847i	$-0.838511\pi$
$971$	− 2.69164e13i	− 0.971694i	0.874044	−	0.485847i	$-0.161489\pi$
$972$	0	0
$973$	2.09038e13i	0.747683i
$974$	0	0
$975$	2.65333e12	0.0940308
$976$	0	0
$977$	2.39611e12	0.0841360	0.0420680	−	0.999115i	$-0.486605\pi$
$977$	2.39611e12	0.0841360	0.0420680	+	0.999115i	$0.486605\pi$
$978$	0	0
$979$	− 7.79377e12i	− 0.271160i
$980$	0	0
$981$	− 1.08893e12i	− 0.0375396i
$982$	0	0
$983$	−3.13055e13	−1.06937	−0.534687	−	0.845050i	$-0.679570\pi$
$983$	−3.13055e13	−1.06937	−0.534687	+	0.845050i	$0.679570\pi$
$984$	0	0
$985$	9.56266e12	0.323680
$986$	0	0
$987$	− 1.04817e13i	− 0.351564i
$988$	0	0
$989$	− 7.04843e12i	− 0.234266i
$990$	0	0
$991$	2.57591e13	0.848397	0.424199	−	0.905569i	$-0.360556\pi$
$991$	2.57591e13	0.848397	0.424199	+	0.905569i	$0.360556\pi$
$992$	0	0
$993$	2.51855e13	0.822012
$994$	0	0
$995$	− 2.64703e12i	− 0.0856161i
$996$	0	0
$997$	− 3.78990e12i	− 0.121478i	−0.998154	−	0.0607392i	$-0.980654\pi$
$997$	− 3.78990e12i	− 0.121478i	0.998154	−	0.0607392i	$-0.0193458\pi$
$998$	0	0
$999$	7.94756e12	0.252458

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.10.d.c.161.7	✓	24
4.3	odd	2		320.10.d.d.161.18	yes	24
8.3	odd	2		320.10.d.d.161.7	yes	24
8.5	even	2	inner	320.10.d.c.161.18	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.10.d.c.161.7	✓	24	1.1	even	1	trivial
320.10.d.c.161.18	yes	24	8.5	even	2	inner
320.10.d.d.161.7	yes	24	8.3	odd	2
320.10.d.d.161.18	yes	24	4.3	odd	2

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	320.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-148.894i q^{3} -625.000i q^{5} +4203.72 q^{7} -2486.48 q^{9} +O(q^{10})\)	\(q-148.894i q^{3} -625.000i q^{5} +4203.72 q^{7} -2486.48 q^{9} -59282.8i q^{11} -45619.8i q^{13} -93058.9 q^{15} -112666. q^{17} -231604. i q^{19} -625909. i q^{21} +1.34793e6 q^{23} -390625. q^{25} -2.56046e6i q^{27} +1.72582e6i q^{29} -6.74921e6 q^{31} -8.82686e6 q^{33} -2.62732e6i q^{35} +3.10396e6i q^{37} -6.79252e6 q^{39} +2.43801e7 q^{41} -5.22909e6i q^{43} +1.55405e6i q^{45} +1.67464e7 q^{47} -2.26824e7 q^{49} +1.67753e7i q^{51} -5.18139e7i q^{53} -3.70517e7 q^{55} -3.44845e7 q^{57} -8.25638e7i q^{59} -3.47213e7i q^{61} -1.04524e7 q^{63} -2.85124e7 q^{65} -2.24378e8i q^{67} -2.00698e8i q^{69} +2.76126e8 q^{71} -2.54000e8 q^{73} +5.81618e7i q^{75} -2.49208e8i q^{77} +2.57662e8 q^{79} -4.30179e8 q^{81} -2.69494e8i q^{83} +7.04161e7i q^{85} +2.56964e8 q^{87} +1.31468e8 q^{89} -1.91773e8i q^{91} +1.00492e9i q^{93} -1.44752e8 q^{95} -2.30989e8 q^{97} +1.47405e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 11712 q^{7} - 155768 q^{9}+O(q^{10}) \) 24 * q - 11712 * q^7 - 155768 * q^9	\( 24 q - 11712 q^{7} - 155768 q^{9} - 229152 q^{17} + 168608 q^{23} - 9375000 q^{25} + 33184704 q^{31} + 33485072 q^{33} - 61197184 q^{39} + 68442864 q^{41} - 40785728 q^{47} + 266247320 q^{49} - 26940000 q^{55} + 798839888 q^{57} - 32743904 q^{63} + 25500000 q^{65} - 597783744 q^{71} - 466601792 q^{73} + 1192034688 q^{79} + 1136218424 q^{81} + 846796000 q^{87} - 750700368 q^{89} + 527980000 q^{95} + 1385780608 q^{97}+O(q^{100}) \) 24 * q - 11712 * q^7 - 155768 * q^9 - 229152 * q^17 + 168608 * q^23 - 9375000 * q^25 + 33184704 * q^31 + 33485072 * q^33 - 61197184 * q^39 + 68442864 * q^41 - 40785728 * q^47 + 266247320 * q^49 - 26940000 * q^55 + 798839888 * q^57 - 32743904 * q^63 + 25500000 * q^65 - 597783744 * q^71 - 466601792 * q^73 + 1192034688 * q^79 + 1136218424 * q^81 + 846796000 * q^87 - 750700368 * q^89 + 527980000 * q^95 + 1385780608 * q^97

Introduction

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Modular forms

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Database

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