LMFDB - Embedded newform 400.6.n.g.207.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("400.143");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.n (of order $4$, degree $2$, 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:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
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} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + \cdots + 57\!\cdots\!24 $ x^20 + 271x^18 + 109637x^16 + 25993614x^14 + 5522961902x^12 + 881545050522x^10 + 133816049059481x^8 + 14779507781220031x^6 + 824105698447750789x^4 + 12044868290803250652*x^2 + 579398322543528055824
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{67}\cdot 3^{4}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			207.6
Root			$1.99079 - 10.4027i$ of defining polynomial
Character	$\chi$	$=$	400.207
Dual form			400.6.n.g.143.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.839817 + 0.839817i) q^{3} +(-99.3589 + 99.3589i) q^{7} -241.589i q^{9} +O(q^{10})$	$q+(0.839817 + 0.839817i) q^{3} +(-99.3589 + 99.3589i) q^{7} -241.589i q^{9} +637.781i q^{11} +(640.389 - 640.389i) q^{13} +(-648.760 - 648.760i) q^{17} +2506.12 q^{19} -166.887 q^{21} +(-2801.60 - 2801.60i) q^{23} +(406.966 - 406.966i) q^{27} +4955.21i q^{29} +1961.18i q^{31} +(-535.619 + 535.619i) q^{33} +(1897.88 + 1897.88i) q^{37} +1075.62 q^{39} -5828.95 q^{41} +(10692.1 + 10692.1i) q^{43} +(-8309.52 + 8309.52i) q^{47} -2937.38i q^{49} -1089.68i q^{51} +(-7437.17 + 7437.17i) q^{53} +(2104.68 + 2104.68i) q^{57} -16738.4 q^{59} +23742.5 q^{61} +(24004.1 + 24004.1i) q^{63} +(-4154.05 + 4154.05i) q^{67} -4705.66i q^{69} -12276.0i q^{71} +(-36092.8 + 36092.8i) q^{73} +(-63369.2 - 63369.2i) q^{77} +64330.4 q^{79} -58022.7 q^{81} +(62856.9 + 62856.9i) q^{83} +(-4161.47 + 4161.47i) q^{87} +24423.5i q^{89} +127257. i q^{91} +(-1647.03 + 1647.03i) q^{93} +(89666.0 + 89666.0i) q^{97} +154081. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q - 804 q^{13} + 2236 q^{17} - 4520 q^{21} + 11096 q^{33} - 44260 q^{37} - 6760 q^{41} - 182452 q^{53} + 34288 q^{57} - 41080 q^{61} - 264372 q^{73} - 399304 q^{77} - 520220 q^{81} - 713496 q^{93} - 374772 q^{97}+O(q^{100}) $ 20 * q - 804 * q^13 + 2236 * q^17 - 4520 * q^21 + 11096 * q^33 - 44260 * q^37 - 6760 * q^41 - 182452 * q^53 + 34288 * q^57 - 41080 * q^61 - 264372 * q^73 - 399304 * q^77 - 520220 * q^81 - 713496 * q^93 - 374772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$-1$

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
$3$	0.839817	+	0.839817i	0.0538743	+	0.0538743i	0.733531	−	0.679656i	$-0.237871\pi$
$3$	0.839817	+	0.839817i	0.0538743	+	0.0538743i	−0.679656	+	0.733531i	$0.737871\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−99.3589	+	99.3589i	−0.766411	+	0.766411i	−0.977473	−	0.211062i	$-0.932308\pi$
$7$	−99.3589	+	99.3589i	−0.766411	+	0.766411i	0.211062	+	0.977473i	$0.432308\pi$
$8$		0			0
$9$		−	241.589i		−	0.994195i
$10$		0			0
$11$			637.781i			1.58924i	0.607107	+	0.794620i	$0.292330\pi$
$11$			637.781i			1.58924i	−0.607107	+	0.794620i	$0.707670\pi$
$12$		0			0
$13$	640.389	−	640.389i	1.05096	−	1.05096i	0.0523281	−	0.998630i	$-0.483336\pi$
$13$	640.389	−	640.389i	1.05096	−	1.05096i	0.998630	−	0.0523281i	$-0.0166641\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−648.760	−	648.760i	−0.544455	−	0.544455i	0.380377	−	0.924832i	$-0.375794\pi$
$17$	−648.760	−	648.760i	−0.544455	−	0.544455i	−0.924832	+	0.380377i	$0.875794\pi$
$18$		0			0
$19$	2506.12			1.59264			0.796321	−	0.604874i	$-0.206776\pi$
$19$	2506.12			1.59264			0.796321	+	0.604874i	$0.206776\pi$
$20$		0			0
$21$	−166.887			−0.0825797
$22$		0			0
$23$	−2801.60	−	2801.60i	−1.10430	−	1.10430i	−0.993886	−	0.110413i	$-0.964783\pi$
$23$	−2801.60	−	2801.60i	−1.10430	−	1.10430i	−0.110413	−	0.993886i	$-0.535217\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	406.966	−	406.966i	0.107436	−	0.107436i
$28$		0			0
$29$			4955.21i			1.09413i	0.837091	+	0.547063i	$0.184254\pi$
$29$			4955.21i			1.09413i	−0.837091	+	0.547063i	$0.815746\pi$
$30$		0			0
$31$			1961.18i			0.366533i	0.983063	+	0.183267i	$0.0586672\pi$
$31$			1961.18i			0.366533i	−0.983063	+	0.183267i	$0.941333\pi$
$32$		0			0
$33$	−535.619	+	535.619i	−0.0856192	+	0.0856192i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1897.88	+	1897.88i	0.227911	+	0.227911i	0.811819	−	0.583909i	$-0.198477\pi$
$37$	1897.88	+	1897.88i	0.227911	+	0.227911i	−0.583909	+	0.811819i	$0.698477\pi$
$38$		0			0
$39$	1075.62			0.113239
$40$		0			0
$41$	−5828.95			−0.541540			−0.270770	−	0.962644i	$-0.587278\pi$
$41$	−5828.95			−0.541540			−0.270770	+	0.962644i	$0.587278\pi$
$42$		0			0
$43$	10692.1	+	10692.1i	0.881844	+	0.881844i	0.993722	−	0.111878i	$-0.0356865\pi$
$43$	10692.1	+	10692.1i	0.881844	+	0.881844i	−0.111878	+	0.993722i	$0.535686\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8309.52	+	8309.52i	−0.548695	+	0.548695i	−0.926063	−	0.377368i	$-0.876829\pi$
$47$	−8309.52	+	8309.52i	−0.548695	+	0.548695i	0.377368	+	0.926063i	$0.376829\pi$
$48$		0			0
$49$		−	2937.38i		−	0.174771i
$50$		0			0
$51$		−	1089.68i		−	0.0586642i
$52$		0			0
$53$	−7437.17	+	7437.17i	−0.363679	+	0.363679i	−0.865165	−	0.501487i	$-0.832787\pi$
$53$	−7437.17	+	7437.17i	−0.363679	+	0.363679i	0.501487	+	0.865165i	$0.332787\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2104.68	+	2104.68i	0.0858025	+	0.0858025i
$58$		0			0
$59$	−16738.4			−0.626014			−0.313007	−	0.949751i	$-0.601336\pi$
$59$	−16738.4			−0.626014			−0.313007	+	0.949751i	$0.601336\pi$
$60$		0			0
$61$	23742.5			0.816961			0.408481	−	0.912767i	$-0.366059\pi$
$61$	23742.5			0.816961			0.408481	+	0.912767i	$0.366059\pi$
$62$		0			0
$63$	24004.1	+	24004.1i	0.761962	+	0.761962i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4154.05	+	4154.05i	−0.113054	+	0.113054i	−0.761371	−	0.648317i	$-0.775473\pi$
$67$	−4154.05	+	4154.05i	−0.113054	+	0.113054i	0.648317	+	0.761371i	$0.275473\pi$
$68$		0			0
$69$		−	4705.66i		−	0.118987i
$70$		0			0
$71$		−	12276.0i		−	0.289008i	−0.989504	−	0.144504i	$-0.953841\pi$
$71$		−	12276.0i		−	0.289008i	0.989504	−	0.144504i	$-0.0461587\pi$
$72$		0			0
$73$	−36092.8	+	36092.8i	−0.792709	+	0.792709i	−0.981934	−	0.189225i	$-0.939402\pi$
$73$	−36092.8	+	36092.8i	−0.792709	+	0.792709i	0.189225	+	0.981934i	$0.439402\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−63369.2	−	63369.2i	−1.21801	−	1.21801i
$78$		0			0
$79$	64330.4			1.15971			0.579854	−	0.814721i	$-0.303110\pi$
$79$	64330.4			1.15971			0.579854	+	0.814721i	$0.303110\pi$
$80$		0			0
$81$	−58022.7			−0.982619
$82$		0			0
$83$	62856.9	+	62856.9i	1.00152	+	1.00152i	0.999999	+	0.00151672i	$0.000482787\pi$
$83$	62856.9	+	62856.9i	1.00152	+	1.00152i	0.00151672	+	0.999999i	$0.499517\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−4161.47	+	4161.47i	−0.0589453	+	0.0589453i
$88$		0			0
$89$			24423.5i			0.326838i	0.986557	+	0.163419i	$0.0522523\pi$
$89$			24423.5i			0.326838i	−0.986557	+	0.163419i	$0.947748\pi$
$90$		0			0
$91$			127257.i			1.61093i
$92$		0			0
$93$	−1647.03	+	1647.03i	−0.0197467	+	0.0197467i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	89666.0	+	89666.0i	0.967605	+	0.967605i	0.999492	−	0.0318862i	$-0.0101514\pi$
$97$	89666.0	+	89666.0i	0.967605	+	0.967605i	−0.0318862	+	0.999492i	$0.510151\pi$
$98$		0			0
$99$	154081.			1.58002
$100$		0			0
$101$	31536.0			0.307612			0.153806	−	0.988101i	$-0.450847\pi$
$101$	31536.0			0.307612			0.153806	+	0.988101i	$0.450847\pi$
$102$		0			0
$103$	−4384.20	−	4384.20i	−0.0407190	−	0.0407190i	0.686454	−	0.727173i	$-0.259166\pi$
$103$	−4384.20	−	4384.20i	−0.0407190	−	0.0407190i	−0.727173	+	0.686454i	$0.759166\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−150691.	+	150691.i	−1.27241	+	1.27241i	−0.327589	+	0.944820i	$0.606236\pi$
$107$	−150691.	+	150691.i	−1.27241	+	1.27241i	−0.944820	+	0.327589i	$0.893764\pi$
$108$		0			0
$109$			201206.i			1.62209i	0.584984	+	0.811045i	$0.301101\pi$
$109$			201206.i			1.62209i	−0.584984	+	0.811045i	$0.698899\pi$
$110$		0			0
$111$			3187.75i			0.0245571i
$112$		0			0
$113$	25930.0	−	25930.0i	0.191032	−	0.191032i	−0.605110	−	0.796142i	$-0.706871\pi$
$113$	25930.0	−	25930.0i	0.191032	−	0.191032i	0.796142	+	0.605110i	$0.206871\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−154711.	−	154711.i	−1.04486	−	1.04486i
$118$		0			0
$119$	128920.			0.834552
$120$		0			0
$121$	−245713.			−1.52569
$122$		0			0
$123$	−4895.25	−	4895.25i	−0.0291751	−	0.0291751i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−46278.8	+	46278.8i	−0.254609	+	0.254609i	−0.822857	−	0.568248i	$-0.807621\pi$
$127$	−46278.8	+	46278.8i	−0.254609	+	0.254609i	0.568248	+	0.822857i	$0.307621\pi$
$128$		0			0
$129$			17958.8i			0.0950175i
$130$		0			0
$131$			83739.0i			0.426333i	0.977016	+	0.213167i	$0.0683777\pi$
$131$			83739.0i			0.426333i	−0.977016	+	0.213167i	$0.931622\pi$
$132$		0			0
$133$	−249006.	+	249006.i	−1.22062	+	1.22062i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−29275.6	−	29275.6i	−0.133261	−	0.133261i	0.637330	−	0.770591i	$-0.280039\pi$
$137$	−29275.6	−	29275.6i	−0.133261	−	0.133261i	−0.770591	+	0.637330i	$0.780039\pi$
$138$		0			0
$139$	−56681.4			−0.248830			−0.124415	−	0.992230i	$-0.539705\pi$
$139$	−56681.4			−0.248830			−0.124415	+	0.992230i	$0.539705\pi$
$140$		0			0
$141$	−13957.0			−0.0591211
$142$		0			0
$143$	408428.	+	408428.i	1.67023	+	1.67023i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	2466.86	−	2466.86i	0.00941568	−	0.00941568i
$148$		0			0
$149$			30050.0i			0.110886i	0.998462	+	0.0554432i	$0.0176572\pi$
$149$			30050.0i			0.110886i	−0.998462	+	0.0554432i	$0.982343\pi$
$150$		0			0
$151$			255984.i			0.913631i	0.889561	+	0.456816i	$0.151010\pi$
$151$			255984.i			0.913631i	−0.889561	+	0.456816i	$0.848990\pi$
$152$		0			0
$153$	−156734.	+	156734.i	−0.541294	+	0.541294i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−30035.3	−	30035.3i	−0.0972486	−	0.0972486i	0.656809	−	0.754057i	$-0.271906\pi$
$157$	−30035.3	−	30035.3i	−0.0972486	−	0.0972486i	−0.754057	+	0.656809i	$0.771906\pi$
$158$		0			0
$159$	−12491.7			−0.0391859
$160$		0			0
$161$	556728.			1.69269
$162$		0			0
$163$	397588.	+	397588.i	1.17210	+	1.17210i	0.981708	+	0.190391i	$0.0609755\pi$
$163$	397588.	+	397588.i	1.17210	+	1.17210i	0.190391	+	0.981708i	$0.439024\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−162249.	+	162249.i	−0.450185	+	0.450185i	−0.895416	−	0.445231i	$-0.853122\pi$
$167$	−162249.	+	162249.i	−0.450185	+	0.450185i	0.445231	+	0.895416i	$0.353122\pi$
$168$		0			0
$169$		−	448903.i		−	1.20903i
$170$		0			0
$171$		−	605453.i		−	1.58340i
$172$		0			0
$173$	158564.	−	158564.i	0.402800	−	0.402800i	−0.476419	−	0.879219i	$-0.658065\pi$
$173$	158564.	−	158564.i	0.402800	−	0.402800i	0.879219	+	0.476419i	$0.158065\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−14057.2	−	14057.2i	−0.0337261	−	0.0337261i
$178$		0			0
$179$	509602.			1.18877			0.594386	−	0.804180i	$-0.297395\pi$
$179$	509602.			1.18877			0.594386	+	0.804180i	$0.297395\pi$
$180$		0			0
$181$	−356971.			−0.809910			−0.404955	−	0.914337i	$-0.632713\pi$
$181$	−356971.			−0.809910			−0.404955	+	0.914337i	$0.632713\pi$
$182$		0			0
$183$	19939.3	+	19939.3i	0.0440132	+	0.0440132i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	413767.	−	413767.i	0.865270	−	0.865270i
$188$		0			0
$189$			80871.5i			0.164680i
$190$		0			0
$191$			296067.i			0.587227i	0.955924	+	0.293614i	$0.0948579\pi$
$191$			296067.i			0.587227i	−0.955924	+	0.293614i	$0.905142\pi$
$192$		0			0
$193$	−10953.8	+	10953.8i	−0.0211676	+	0.0211676i	−0.717611	−	0.696444i	$-0.754765\pi$
$193$	−10953.8	+	10953.8i	−0.0211676	+	0.0211676i	0.696444	+	0.717611i	$0.254765\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	153834.	+	153834.i	0.282415	+	0.282415i	0.834071	−	0.551656i	$-0.186004\pi$
$197$	153834.	+	153834.i	0.282415	+	0.282415i	−0.551656	+	0.834071i	$0.686004\pi$
$198$		0			0
$199$	−415211.			−0.743252			−0.371626	−	0.928383i	$-0.621200\pi$
$199$	−415211.			−0.743252			−0.371626	+	0.928383i	$0.621200\pi$
$200$		0			0
$201$	−6977.28			−0.0121814
$202$		0			0
$203$	−492345.	−	492345.i	−0.838550	−	0.838550i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−676837.	+	676837.i	−1.09789	+	1.09789i
$208$		0			0
$209$			1.59836e6i			2.53109i
$210$		0			0
$211$			673870.i			1.04200i	0.853555	+	0.521002i	$0.174442\pi$
$211$			673870.i			1.04200i	−0.853555	+	0.521002i	$0.825558\pi$
$212$		0			0
$213$	10309.6	−	10309.6i	0.0155701	−	0.0155701i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−194861.	−	194861.i	−0.280915	−	0.280915i
$218$		0			0
$219$	−60622.7			−0.0854132
$220$		0			0
$221$	−830917.			−1.14440
$222$		0			0
$223$	−916244.	−	916244.i	−1.23381	−	1.23381i	−0.962488	−	0.271323i	$-0.912539\pi$
$223$	−916244.	−	916244.i	−1.23381	−	1.23381i	−0.271323	−	0.962488i	$-0.587461\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−317650.	+	317650.i	−0.409152	+	0.409152i	−0.881443	−	0.472291i	$-0.843427\pi$
$227$	−317650.	+	317650.i	−0.409152	+	0.409152i	0.472291	+	0.881443i	$0.343427\pi$
$228$		0			0
$229$		−	1.15003e6i		−	1.44917i	−0.689183	−	0.724587i	$-0.742030\pi$
$229$		−	1.15003e6i		−	1.44917i	0.689183	−	0.724587i	$-0.257970\pi$
$230$		0			0
$231$		−	106437.i		−	0.131239i
$232$		0			0
$233$	−290928.	+	290928.i	−0.351071	+	0.351071i	−0.860508	−	0.509437i	$-0.829854\pi$
$233$	−290928.	+	290928.i	−0.351071	+	0.351071i	0.509437	+	0.860508i	$0.329854\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	54025.7	+	54025.7i	0.0624784	+	0.0624784i
$238$		0			0
$239$	152082.			0.172220			0.0861101	−	0.996286i	$-0.472556\pi$
$239$	152082.			0.172220			0.0861101	+	0.996286i	$0.472556\pi$
$240$		0			0
$241$	43181.5			0.0478912			0.0239456	−	0.999713i	$-0.492377\pi$
$241$	43181.5			0.0478912			0.0239456	+	0.999713i	$0.492377\pi$
$242$		0			0
$243$	−147621.	−	147621.i	−0.160374	−	0.160374i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	1.60489e6	−	1.60489e6i	1.67380	−	1.67380i
$248$		0			0
$249$			105577.i			0.107912i
$250$		0			0
$251$		−	547660.i		−	0.548689i	−0.961631	−	0.274345i	$-0.911539\pi$
$251$		−	547660.i		−	0.548689i	0.961631	−	0.274345i	$-0.0884609\pi$
$252$		0			0
$253$	1.78681e6	−	1.78681e6i	1.75500	−	1.75500i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	226438.	+	226438.i	0.213853	+	0.213853i	0.805902	−	0.592049i	$-0.201681\pi$
$257$	226438.	+	226438.i	0.213853	+	0.213853i	−0.592049	+	0.805902i	$0.701681\pi$
$258$		0			0
$259$	−377143.			−0.349347
$260$		0			0
$261$	1.19713e6			1.08778
$262$		0			0
$263$	21617.3	+	21617.3i	0.0192714	+	0.0192714i	0.716677	−	0.697405i	$-0.245662\pi$
$263$	21617.3	+	21617.3i	0.0192714	+	0.0192714i	−0.697405	+	0.716677i	$0.745662\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−20511.3	+	20511.3i	−0.0176082	+	0.0176082i
$268$		0			0
$269$		−	1.78124e6i		−	1.50087i	−0.660947	−	0.750433i	$-0.729845\pi$
$269$		−	1.78124e6i		−	1.50087i	0.660947	−	0.750433i	$-0.270155\pi$
$270$		0			0
$271$		−	1.45155e6i		−	1.20063i	−0.799765	−	0.600314i	$-0.795042\pi$
$271$		−	1.45155e6i		−	1.20063i	0.799765	−	0.600314i	$-0.204958\pi$
$272$		0			0
$273$	−106872.	+	106872.i	−0.0867878	+	0.0867878i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	831788.	+	831788.i	0.651348	+	0.651348i	0.953318	−	0.301969i	$-0.0976441\pi$
$277$	831788.	+	831788.i	0.651348	+	0.651348i	−0.301969	+	0.953318i	$0.597644\pi$
$278$		0			0
$279$	473801.			0.364406
$280$		0			0
$281$	2.43146e6			1.83696			0.918482	−	0.395463i	$-0.129416\pi$
$281$	2.43146e6			1.83696			0.918482	+	0.395463i	$0.129416\pi$
$282$		0			0
$283$	−141203.	−	141203.i	−0.104804	−	0.104804i	0.652760	−	0.757565i	$-0.273611\pi$
$283$	−141203.	−	141203.i	−0.104804	−	0.104804i	−0.757565	+	0.652760i	$0.773611\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	579158.	−	579158.i	0.415042	−	0.415042i
$288$		0			0
$289$		−	578078.i		−	0.407138i
$290$		0			0
$291$			150606.i			0.104258i
$292$		0			0
$293$	−540576.	+	540576.i	−0.367864	+	0.367864i	−0.866698	−	0.498834i	$-0.833762\pi$
$293$	−540576.	+	540576.i	−0.367864	+	0.367864i	0.498834	+	0.866698i	$0.333762\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	259555.	+	259555.i	0.170741	+	0.170741i
$298$		0			0
$299$	−3.58823e6			−2.32114
$300$		0			0
$301$	−2.12471e6			−1.35171
$302$		0			0
$303$	26484.5	+	26484.5i	0.0165724	+	0.0165724i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	779170.	−	779170.i	0.471831	−	0.471831i	−0.430676	−	0.902507i	$-0.641725\pi$
$307$	779170.	−	779170.i	0.471831	−	0.471831i	0.902507	+	0.430676i	$0.141725\pi$
$308$		0			0
$309$		−	7363.86i		−	0.00438742i
$310$		0			0
$311$		−	1.56490e6i		−	0.917459i	−0.888576	−	0.458730i	$-0.848305\pi$
$311$		−	1.56490e6i		−	0.917459i	0.888576	−	0.458730i	$-0.151695\pi$
$312$		0			0
$313$	−684828.	+	684828.i	−0.395112	+	0.395112i	−0.876505	−	0.481393i	$-0.840131\pi$
$313$	−684828.	+	684828.i	−0.395112	+	0.395112i	0.481393	+	0.876505i	$0.340131\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.90957e6	−	1.90957e6i	−1.06730	−	1.06730i	−0.997565	−	0.0697372i	$-0.977784\pi$
$317$	−1.90957e6	−	1.90957e6i	−1.06730	−	1.06730i	−0.0697372	−	0.997565i	$-0.522216\pi$
$318$		0			0
$319$	−3.16034e6			−1.73883
$320$		0			0
$321$	−253105.			−0.137100
$322$		0			0
$323$	−1.62587e6	−	1.62587e6i	−0.867122	−	0.867122i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−168976.	+	168976.i	−0.0873890	+	0.0873890i
$328$		0			0
$329$		−	1.65125e6i		−	0.841052i
$330$		0			0
$331$			740425.i			0.371459i	0.982601	+	0.185730i	$0.0594649\pi$
$331$			740425.i			0.371459i	−0.982601	+	0.185730i	$0.940535\pi$
$332$		0			0
$333$	458508.	−	458508.i	0.226588	−	0.226588i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.17170e6	+	2.17170e6i	1.04166	+	1.04166i	0.999094	+	0.0425641i	$0.0135527\pi$
$337$	2.17170e6	+	2.17170e6i	1.04166	+	1.04166i	0.0425641	+	0.999094i	$0.486447\pi$
$338$		0			0
$339$	43553.0			0.0205835
$340$		0			0
$341$	−1.25080e6			−0.582510
$342$		0			0
$343$	−1.37807e6	−	1.37807e6i	−0.632464	−	0.632464i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−706412.	+	706412.i	−0.314945	+	0.314945i	−0.846822	−	0.531877i	$-0.821487\pi$
$347$	−706412.	+	706412.i	−0.314945	+	0.314945i	0.531877	+	0.846822i	$0.321487\pi$
$348$		0			0
$349$			3.03307e6i			1.33297i	0.745520	+	0.666483i	$0.232201\pi$
$349$			3.03307e6i			1.33297i	−0.745520	+	0.666483i	$0.767799\pi$
$350$		0			0
$351$		−	521234.i		−	0.225821i
$352$		0			0
$353$	2.72516e6	−	2.72516e6i	1.16401	−	1.16401i	0.180418	−	0.983590i	$-0.442255\pi$
$353$	2.72516e6	−	2.72516e6i	1.16401	−	1.16401i	0.983590	−	0.180418i	$-0.0577449\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	108269.	+	108269.i	0.0449609	+	0.0449609i
$358$		0			0
$359$	2.47478e6			1.01345			0.506723	−	0.862109i	$-0.330857\pi$
$359$	2.47478e6			1.01345			0.506723	+	0.862109i	$0.330857\pi$
$360$		0			0
$361$	3.80455e6			1.53651
$362$		0			0
$363$	−206354.	−	206354.i	−0.0821953	−	0.0821953i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	405936.	−	405936.i	0.157323	−	0.157323i	−0.624056	−	0.781379i	$-0.714516\pi$
$367$	405936.	−	405936.i	0.157323	−	0.157323i	0.781379	+	0.624056i	$0.214516\pi$
$368$		0			0
$369$			1.40821e6i			0.538397i
$370$		0			0
$371$		−	1.47790e6i		−	0.557455i
$372$		0			0
$373$	378559.	−	378559.i	0.140884	−	0.140884i	−0.633147	−	0.774031i	$-0.718237\pi$
$373$	378559.	−	378559.i	0.140884	−	0.140884i	0.774031	+	0.633147i	$0.218237\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.17326e6	+	3.17326e6i	1.14988	+	1.14988i
$378$		0			0
$379$	3.91279e6			1.39923			0.699614	−	0.714521i	$-0.253355\pi$
$379$	3.91279e6			1.39923			0.699614	+	0.714521i	$0.253355\pi$
$380$		0			0
$381$	−77731.5			−0.0274337
$382$		0			0
$383$	−2.53247e6	−	2.53247e6i	−0.882161	−	0.882161i	0.111593	−	0.993754i	$-0.464405\pi$
$383$	−2.53247e6	−	2.53247e6i	−0.882161	−	0.882161i	−0.993754	+	0.111593i	$0.964405\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.58310e6	−	2.58310e6i	0.876725	−	0.876725i
$388$		0			0
$389$		−	3.01392e6i		−	1.00985i	−0.863163	−	0.504925i	$-0.831520\pi$
$389$		−	3.01392e6i		−	1.00985i	0.863163	−	0.504925i	$-0.168480\pi$
$390$		0			0
$391$			3.63513e6i			1.20248i
$392$		0			0
$393$	−70325.4	+	70325.4i	−0.0229684	+	0.0229684i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3.43650e6	−	3.43650e6i	−1.09431	−	1.09431i	−0.995063	−	0.0992459i	$-0.968357\pi$
$397$	−3.43650e6	−	3.43650e6i	−1.09431	−	1.09431i	−0.0992459	−	0.995063i	$-0.531643\pi$
$398$		0			0
$399$	−418238.			−0.131520
$400$		0			0
$401$	−4.30306e6			−1.33634			−0.668169	−	0.744009i	$-0.732922\pi$
$401$	−4.30306e6			−1.33634			−0.668169	+	0.744009i	$0.732922\pi$
$402$		0			0
$403$	1.25592e6	+	1.25592e6i	0.385211	+	0.385211i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.21043e6	+	1.21043e6i	−0.362205	+	0.362205i
$408$		0			0
$409$		−	452926.i		−	0.133881i	−0.997757	−	0.0669405i	$-0.978676\pi$
$409$		−	452926.i		−	0.133881i	0.997757	−	0.0669405i	$-0.0213238\pi$
$410$		0			0
$411$		−	49172.2i		−	0.0143587i
$412$		0			0
$413$	1.66311e6	−	1.66311e6i	0.479784	−	0.479784i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−47602.0	−	47602.0i	−0.0134056	−	0.0134056i
$418$		0			0
$419$	−448753.			−0.124874			−0.0624370	−	0.998049i	$-0.519887\pi$
$419$	−448753.			−0.124874			−0.0624370	+	0.998049i	$0.519887\pi$
$420$		0			0
$421$	3.61031e6			0.992748			0.496374	−	0.868109i	$-0.334664\pi$
$421$	3.61031e6			0.992748			0.496374	+	0.868109i	$0.334664\pi$
$422$		0			0
$423$	2.00749e6	+	2.00749e6i	0.545510	+	0.545510i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.35903e6	+	2.35903e6i	−0.626128	+	0.626128i
$428$		0			0
$429$			686009.i			0.179964i
$430$		0			0
$431$			3.33480e6i			0.864722i	0.901701	+	0.432361i	$0.142319\pi$
$431$			3.33480e6i			0.864722i	−0.901701	+	0.432361i	$0.857681\pi$
$432$		0			0
$433$	−4.26797e6	+	4.26797e6i	−1.09396	+	1.09396i	−0.0988602	+	0.995101i	$0.531520\pi$
$433$	−4.26797e6	+	4.26797e6i	−1.09396	+	1.09396i	−0.995101	+	0.0988602i	$0.968480\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.02115e6	−	7.02115e6i	−1.75875	−	1.75875i
$438$		0			0
$439$	−985119.			−0.243965			−0.121982	−	0.992532i	$-0.538925\pi$
$439$	−985119.			−0.243965			−0.121982	+	0.992532i	$0.538925\pi$
$440$		0			0
$441$	−709641.			−0.173757
$442$		0			0
$443$	941343.	+	941343.i	0.227897	+	0.227897i	0.811814	−	0.583917i	$-0.198481\pi$
$443$	941343.	+	941343.i	0.227897	+	0.227897i	−0.583917	+	0.811814i	$0.698481\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−25236.5	+	25236.5i	−0.00597393	+	0.00597393i
$448$		0			0
$449$			2.05024e6i			0.479941i	0.970780	+	0.239971i	$0.0771378\pi$
$449$			2.05024e6i			0.479941i	−0.970780	+	0.239971i	$0.922862\pi$
$450$		0			0
$451$		−	3.71759e6i		−	0.860638i
$452$		0			0
$453$	−214980.	+	214980.i	−0.0492213	+	0.0492213i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5.54554e6	+	5.54554e6i	1.24209	+	1.24209i	0.959132	+	0.282958i	$0.0913157\pi$
$457$	5.54554e6	+	5.54554e6i	1.24209	+	1.24209i	0.282958	+	0.959132i	$0.408684\pi$
$458$		0			0
$459$	−528047.			−0.116988
$460$		0			0
$461$	−3.71258e6			−0.813624			−0.406812	−	0.913512i	$-0.633360\pi$
$461$	−3.71258e6			−0.813624			−0.406812	+	0.913512i	$0.633360\pi$
$462$		0			0
$463$	4.14889e6	+	4.14889e6i	0.899455	+	0.899455i	0.995388	−	0.0959328i	$-0.0305834\pi$
$463$	4.14889e6	+	4.14889e6i	0.899455	+	0.899455i	−0.0959328	+	0.995388i	$0.530583\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	599788.	−	599788.i	0.127264	−	0.127264i	−0.640606	−	0.767870i	$-0.721317\pi$
$467$	599788.	−	599788.i	0.127264	−	0.127264i	0.767870	+	0.640606i	$0.221317\pi$
$468$		0			0
$469$		−	825483.i		−	0.173291i
$470$		0			0
$471$		−	50448.4i		−	0.0104784i
$472$		0			0
$473$	−6.81922e6	+	6.81922e6i	−1.40146	+	1.40146i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.79674e6	+	1.79674e6i	0.361568	+	0.361568i
$478$		0			0
$479$	130460.			0.0259799			0.0129900	−	0.999916i	$-0.495865\pi$
$479$	130460.			0.0259799			0.0129900	+	0.999916i	$0.495865\pi$
$480$		0			0
$481$	2.43077e6			0.479049
$482$		0			0
$483$	467550.	+	467550.i	0.0911927	+	0.0911927i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−5.53566e6	+	5.53566e6i	−1.05766	+	1.05766i	−0.0594298	+	0.998232i	$0.518928\pi$
$487$	−5.53566e6	+	5.53566e6i	−1.05766	+	1.05766i	−0.998232	+	0.0594298i	$0.981072\pi$
$488$		0			0
$489$			667803.i			0.126292i
$490$		0			0
$491$			2.06198e6i			0.385993i	0.981199	+	0.192997i	$0.0618206\pi$
$491$			2.06198e6i			0.385993i	−0.981199	+	0.192997i	$0.938179\pi$
$492$		0			0
$493$	3.21474e6	−	3.21474e6i	0.595702	−	0.595702i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.21973e6	+	1.21973e6i	0.221499	+	0.221499i
$498$		0			0
$499$	3.96927e6			0.713608			0.356804	−	0.934179i	$-0.383866\pi$
$499$	3.96927e6			0.713608			0.356804	+	0.934179i	$0.383866\pi$
$500$		0			0
$501$	−272519.			−0.0485068
$502$		0			0
$503$	7.68593e6	+	7.68593e6i	1.35449	+	1.35449i	0.880564	+	0.473928i	$0.157164\pi$
$503$	7.68593e6	+	7.68593e6i	1.35449	+	1.35449i	0.473928	+	0.880564i	$0.342836\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	376996.	−	376996.i	0.0651354	−	0.0651354i
$508$		0			0
$509$		−	149834.i		−	0.0256340i	−0.999918	−	0.0128170i	$-0.995920\pi$
$509$		−	149834.i		−	0.0256340i	0.999918	−	0.0128170i	$-0.00407988\pi$
$510$		0			0
$511$		−	7.17229e6i		−	1.21508i
$512$		0			0
$513$	1.01991e6	−	1.01991e6i	0.171107	−	0.171107i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−5.29965e6	−	5.29965e6i	−0.872009	−	0.872009i
$518$		0			0
$519$	266330.			0.0434011
$520$		0			0
$521$	1.13714e6			0.183536			0.0917678	−	0.995780i	$-0.470748\pi$
$521$	1.13714e6			0.183536			0.0917678	+	0.995780i	$0.470748\pi$
$522$		0			0
$523$	4.86949e6	+	4.86949e6i	0.778447	+	0.778447i	0.979567	−	0.201119i	$-0.0644579\pi$
$523$	4.86949e6	+	4.86949e6i	0.778447	+	0.778447i	−0.201119	+	0.979567i	$0.564458\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.27234e6	−	1.27234e6i	0.199561	−	0.199561i
$528$		0			0
$529$			9.26159e6i			1.43895i
$530$		0			0
$531$			4.04382e6i			0.622380i
$532$		0			0
$533$	−3.73280e6	+	3.73280e6i	−0.569136	+	0.569136i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	427972.	+	427972.i	0.0640442	+	0.0640442i
$538$		0			0
$539$	1.87341e6			0.277754
$540$		0			0
$541$	−6.42771e6			−0.944198			−0.472099	−	0.881546i	$-0.656503\pi$
$541$	−6.42771e6			−0.944198			−0.472099	+	0.881546i	$0.656503\pi$
$542$		0			0
$543$	−299790.	−	299790.i	−0.0436333	−	0.0436333i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−6.88980e6	+	6.88980e6i	−0.984551	+	0.984551i	−0.999882	−	0.0153314i	$-0.995120\pi$
$547$	−6.88980e6	+	6.88980e6i	−0.984551	+	0.984551i	0.0153314	+	0.999882i	$0.495120\pi$
$548$		0			0
$549$		−	5.73593e6i		−	0.812219i
$550$		0			0
$551$			1.24184e7i			1.74255i
$552$		0			0
$553$	−6.39179e6	+	6.39179e6i	−0.888812	+	0.888812i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	626234.	+	626234.i	0.0855261	+	0.0855261i	0.748576	−	0.663049i	$-0.230738\pi$
$557$	626234.	+	626234.i	0.0855261	+	0.0855261i	−0.663049	+	0.748576i	$0.730738\pi$
$558$		0			0
$559$	1.36942e7			1.85356
$560$		0			0
$561$	694977.			0.0932316
$562$		0			0
$563$	−3.78102e6	−	3.78102e6i	−0.502733	−	0.502733i	0.409553	−	0.912286i	$-0.365685\pi$
$563$	−3.78102e6	−	3.78102e6i	−0.502733	−	0.502733i	−0.912286	+	0.409553i	$0.865685\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	5.76507e6	−	5.76507e6i	0.753090	−	0.753090i
$568$		0			0
$569$			5.86991e6i			0.760065i	0.924973	+	0.380032i	$0.124087\pi$
$569$			5.86991e6i			0.760065i	−0.924973	+	0.380032i	$0.875913\pi$
$570$		0			0
$571$			3.63228e6i			0.466218i	0.972451	+	0.233109i	$0.0748899\pi$
$571$			3.63228e6i			0.466218i	−0.972451	+	0.233109i	$0.925110\pi$
$572$		0			0
$573$	−248642.	+	248642.i	−0.0316365	+	0.0316365i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−7.85335e6	−	7.85335e6i	−0.982009	−	0.982009i	0.0178322	−	0.999841i	$-0.494324\pi$
$577$	−7.85335e6	−	7.85335e6i	−0.982009	−	0.982009i	−0.999841	+	0.0178322i	$0.994324\pi$
$578$		0			0
$579$	−18398.4			−0.00228078
$580$		0			0
$581$	−1.24908e7			−1.53514
$582$		0			0
$583$	−4.74328e6	−	4.74328e6i	−0.577973	−	0.577973i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.62105e6	+	4.62105e6i	−0.553535	+	0.553535i	−0.927459	−	0.373924i	$-0.878012\pi$
$587$	−4.62105e6	+	4.62105e6i	−0.553535	+	0.553535i	0.373924	+	0.927459i	$0.378012\pi$
$588$		0			0
$589$			4.91496e6i			0.583757i
$590$		0			0
$591$			258386.i			0.0304298i
$592$		0			0
$593$	6.65088e6	−	6.65088e6i	0.776681	−	0.776681i	−0.202584	−	0.979265i	$-0.564934\pi$
$593$	6.65088e6	−	6.65088e6i	0.776681	−	0.776681i	0.979265	+	0.202584i	$0.0649340\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−348701.	−	348701.i	−0.0400422	−	0.0400422i
$598$		0			0
$599$	4.31406e6			0.491269			0.245634	−	0.969363i	$-0.421004\pi$
$599$	4.31406e6			0.491269			0.245634	+	0.969363i	$0.421004\pi$
$600$		0			0
$601$	1.43695e7			1.62277			0.811384	−	0.584513i	$-0.198714\pi$
$601$	1.43695e7			1.62277			0.811384	+	0.584513i	$0.198714\pi$
$602$		0			0
$603$	1.00357e6	+	1.00357e6i	0.112397	+	0.112397i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−997962.	+	997962.i	−0.109937	+	0.109937i	−0.759935	−	0.649999i	$-0.774769\pi$
$607$	−997962.	+	997962.i	−0.109937	+	0.109937i	0.649999	+	0.759935i	$0.274769\pi$
$608$		0			0
$609$		−	826959.i		−	0.0903526i
$610$		0			0
$611$			1.06426e7i			1.15331i
$612$		0			0
$613$	6.18198e6	−	6.18198e6i	0.664472	−	0.664472i	−0.291959	−	0.956431i	$-0.594307\pi$
$613$	6.18198e6	−	6.18198e6i	0.664472	−	0.664472i	0.956431	+	0.291959i	$0.0943071\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	274155.	+	274155.i	0.0289924	+	0.0289924i	0.721454	−	0.692462i	$-0.243474\pi$
$617$	274155.	+	274155.i	0.0289924	+	0.0289924i	−0.692462	+	0.721454i	$0.743474\pi$
$618$		0			0
$619$	1.04395e7			1.09510			0.547549	−	0.836774i	$-0.315561\pi$
$619$	1.04395e7			1.09510			0.547549	+	0.836774i	$0.315561\pi$
$620$		0			0
$621$	−2.28031e6			−0.237283
$622$		0			0
$623$	−2.42669e6	−	2.42669e6i	−0.250492	−	0.250492i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−1.34233e6	+	1.34233e6i	−0.136361	+	0.136361i
$628$		0			0
$629$		−	2.46254e6i		−	0.248174i
$630$		0			0
$631$		−	1.95332e7i		−	1.95299i	−0.215544	−	0.976494i	$-0.569152\pi$
$631$		−	1.95332e7i		−	1.95299i	0.215544	−	0.976494i	$-0.430848\pi$
$632$		0			0
$633$	−565927.	+	565927.i	−0.0561373	+	0.0561373i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.88107e6	−	1.88107e6i	−0.183677	−	0.183677i
$638$		0			0
$639$	−2.96574e6			−0.287330
$640$		0			0
$641$	1.16310e7			1.11808			0.559038	−	0.829142i	$-0.311171\pi$
$641$	1.16310e7			1.11808			0.559038	+	0.829142i	$0.311171\pi$
$642$		0			0
$643$	8.86995e6	+	8.86995e6i	0.846045	+	0.846045i	0.989637	−	0.143592i	$-0.0458653\pi$
$643$	8.86995e6	+	8.86995e6i	0.846045	+	0.846045i	−0.143592	+	0.989637i	$0.545865\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.82351e6	+	7.82351e6i	−0.734752	+	0.734752i	−0.971557	−	0.236805i	$-0.923900\pi$
$647$	−7.82351e6	+	7.82351e6i	−0.734752	+	0.734752i	0.236805	+	0.971557i	$0.423900\pi$
$648$		0			0
$649$		−	1.06754e7i		−	0.994887i
$650$		0			0
$651$		−	327295.i		−	0.0302682i
$652$		0			0
$653$	5.80259e6	−	5.80259e6i	0.532524	−	0.532524i	−0.388799	−	0.921323i	$-0.627110\pi$
$653$	5.80259e6	−	5.80259e6i	0.532524	−	0.532524i	0.921323	+	0.388799i	$0.127110\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	8.71964e6	+	8.71964e6i	0.788107	+	0.788107i
$658$		0			0
$659$	−1.87531e7			−1.68213			−0.841065	−	0.540934i	$-0.818071\pi$
$659$	−1.87531e7			−1.68213			−0.841065	+	0.540934i	$0.818071\pi$
$660$		0			0
$661$	−1.79645e7			−1.59923			−0.799615	−	0.600513i	$-0.794963\pi$
$661$	−1.79645e7			−1.59923			−0.799615	+	0.600513i	$0.794963\pi$
$662$		0			0
$663$	−697819.	−	697819.i	−0.0616536	−	0.0616536i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.38825e7	−	1.38825e7i	1.20824	−	1.20824i
$668$		0			0
$669$		−	1.53895e6i		−	0.132941i
$670$		0			0
$671$			1.51425e7i			1.29835i
$672$		0			0
$673$	−1.23479e7	+	1.23479e7i	−1.05089	+	1.05089i	−0.0522545	+	0.998634i	$0.516641\pi$
$673$	−1.23479e7	+	1.23479e7i	−1.05089	+	1.05089i	−0.998634	+	0.0522545i	$0.983359\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5.66444e6	+	5.66444e6i	0.474991	+	0.474991i	0.903525	−	0.428535i	$-0.140970\pi$
$677$	5.66444e6	+	5.66444e6i	0.474991	+	0.474991i	−0.428535	+	0.903525i	$0.640970\pi$
$678$		0			0
$679$	−1.78182e7			−1.48317
$680$		0			0
$681$	−533536.			−0.0440855
$682$		0			0
$683$	2.21820e6	+	2.21820e6i	0.181948	+	0.181948i	0.792204	−	0.610256i	$-0.208933\pi$
$683$	2.21820e6	+	2.21820e6i	0.181948	+	0.181948i	−0.610256	+	0.792204i	$0.708933\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	965815.	−	965815.i	0.0780732	−	0.0780732i
$688$		0			0
$689$			9.52536e6i			0.764422i
$690$		0			0
$691$		−	1.63246e7i		−	1.30061i	−0.759674	−	0.650304i	$-0.774642\pi$
$691$		−	1.63246e7i		−	1.30061i	0.759674	−	0.650304i	$-0.225358\pi$
$692$		0			0
$693$	−1.53093e7	+	1.53093e7i	−1.21094	+	1.21094i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	3.78159e6	+	3.78159e6i	0.294844	+	0.294844i
$698$		0			0
$699$	−488652.			−0.0378274
$700$		0			0
$701$	1.64161e7			1.26175			0.630876	−	0.775884i	$-0.282696\pi$
$701$	1.64161e7			1.26175			0.630876	+	0.775884i	$0.282696\pi$
$702$		0			0
$703$	4.75633e6	+	4.75633e6i	0.362981	+	0.362981i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.13338e6	+	3.13338e6i	−0.235757	+	0.235757i
$708$		0			0
$709$		−	2.37201e6i		−	0.177215i	−0.996067	−	0.0886077i	$-0.971758\pi$
$709$		−	2.37201e6i		−	0.177215i	0.996067	−	0.0886077i	$-0.0282417\pi$
$710$		0			0
$711$		−	1.55415e7i		−	1.15298i
$712$		0			0
$713$	5.49445e6	−	5.49445e6i	0.404762	−	0.404762i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	127721.	+	127721.i	0.00927824	+	0.00927824i
$718$		0			0
$719$	−8.62894e6			−0.622494			−0.311247	−	0.950329i	$-0.600747\pi$
$719$	−8.62894e6			−0.622494			−0.311247	+	0.950329i	$0.600747\pi$
$720$		0			0
$721$	871219.			0.0624150
$722$		0			0
$723$	36264.6	+	36264.6i	0.00258010	+	0.00258010i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.16170e7	−	1.16170e7i	0.815187	−	0.815187i	−0.170219	−	0.985406i	$-0.554448\pi$
$727$	1.16170e7	−	1.16170e7i	0.815187	−	0.815187i	0.985406	+	0.170219i	$0.0544475\pi$
$728$		0			0
$729$			1.38516e7i			0.965339i
$730$		0			0
$731$		−	1.38732e7i		−	0.960249i
$732$		0			0
$733$	7.58933e6	−	7.58933e6i	0.521727	−	0.521727i	−0.396366	−	0.918093i	$-0.629729\pi$
$733$	7.58933e6	−	7.58933e6i	0.521727	−	0.521727i	0.918093	+	0.396366i	$0.129729\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.64937e6	−	2.64937e6i	−0.179669	−	0.179669i
$738$		0			0
$739$	−1.89099e6			−0.127373			−0.0636867	−	0.997970i	$-0.520286\pi$
$739$	−1.89099e6			−0.127373			−0.0636867	+	0.997970i	$0.520286\pi$
$740$		0			0
$741$	2.69563e6			0.180350
$742$		0			0
$743$	−1.86435e7	−	1.86435e7i	−1.23895	−	1.23895i	−0.960430	−	0.278523i	$-0.910155\pi$
$743$	−1.86435e7	−	1.86435e7i	−1.23895	−	1.23895i	−0.278523	−	0.960430i	$-0.589845\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.51856e7	−	1.51856e7i	0.995702	−	0.995702i
$748$		0			0
$749$		−	2.99449e7i		−	1.95038i
$750$		0			0
$751$			4.60302e6i			0.297813i	0.988851	+	0.148906i	$0.0475753\pi$
$751$			4.60302e6i			0.297813i	−0.988851	+	0.148906i	$0.952425\pi$
$752$		0			0
$753$	459934.	−	459934.i	0.0295602	−	0.0295602i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−6.60445e6	−	6.60445e6i	−0.418887	−	0.418887i	0.465933	−	0.884820i	$-0.345719\pi$
$757$	−6.60445e6	−	6.60445e6i	−0.418887	−	0.418887i	−0.884820	+	0.465933i	$0.845719\pi$
$758$		0			0
$759$	3.00118e6			0.189098
$760$		0			0
$761$	−7.76470e6			−0.486030			−0.243015	−	0.970023i	$-0.578136\pi$
$761$	−7.76470e6			−0.486030			−0.243015	+	0.970023i	$0.578136\pi$
$762$		0			0
$763$	−1.99916e7	−	1.99916e7i	−1.24319	−	1.24319i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.07191e7	+	1.07191e7i	−0.657915	+	0.657915i
$768$		0			0
$769$		−	1.03998e7i		−	0.634172i	−0.948397	−	0.317086i	$-0.897296\pi$
$769$		−	1.03998e7i		−	0.634172i	0.948397	−	0.317086i	$-0.102704\pi$
$770$		0			0
$771$			380332.i			0.0230424i
$772$		0			0
$773$	2.28295e7	−	2.28295e7i	1.37419	−	1.37419i	0.520068	−	0.854125i	$-0.325907\pi$
$773$	2.28295e7	−	2.28295e7i	1.37419	−	1.37419i	0.854125	−	0.520068i	$-0.174093\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−316731.	−	316731.i	−0.0188208	−	0.0188208i
$778$		0			0
$779$	−1.46081e7			−0.862480
$780$		0			0
$781$	7.82938e6			0.459303
$782$		0			0
$783$	2.01661e6	+	2.01661e6i	0.117548	+	0.117548i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	5.72096e6	−	5.72096e6i	0.329255	−	0.329255i	−0.523048	−	0.852303i	$-0.675205\pi$
$787$	5.72096e6	−	5.72096e6i	0.329255	−	0.329255i	0.852303	+	0.523048i	$0.175205\pi$
$788$		0			0
$789$			36309.2i			0.00207646i
$790$		0			0
$791$			5.15276e6i			0.292818i
$792$		0			0
$793$	1.52044e7	−	1.52044e7i	0.858592	−	0.858592i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.79102e6	−	4.79102e6i	−0.267167	−	0.267167i	0.560791	−	0.827957i	$-0.310497\pi$
$797$	−4.79102e6	−	4.79102e6i	−0.267167	−	0.267167i	−0.827957	+	0.560791i	$0.810497\pi$
$798$		0			0
$799$	1.07818e7			0.597479
$800$		0			0
$801$	5.90046e6			0.324941
$802$		0			0
$803$	−2.30193e7	−	2.30193e7i	−1.25980	−	1.25980i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.49592e6	−	1.49592e6i	0.0808580	−	0.0808580i
$808$		0			0
$809$		−	1.42552e7i		−	0.765778i	−0.923794	−	0.382889i	$-0.874929\pi$
$809$		−	1.42552e7i		−	0.765778i	0.923794	−	0.382889i	$-0.125071\pi$
$810$		0			0
$811$			2.56090e7i			1.36723i	0.729843	+	0.683614i	$0.239593\pi$
$811$			2.56090e7i			1.36723i	−0.729843	+	0.683614i	$0.760407\pi$
$812$		0			0
$813$	1.21904e6	−	1.21904e6i	0.0646830	−	0.0646830i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.67957e7	+	2.67957e7i	1.40446	+	1.40446i
$818$		0			0
$819$	3.07439e7			1.60158
$820$		0			0
$821$	2.31917e6			0.120081			0.0600406	−	0.998196i	$-0.480877\pi$
$821$	2.31917e6			0.120081			0.0600406	+	0.998196i	$0.480877\pi$
$822$		0			0
$823$	−4.90869e6	−	4.90869e6i	−0.252619	−	0.252619i	0.569425	−	0.822044i	$-0.307166\pi$
$823$	−4.90869e6	−	4.90869e6i	−0.252619	−	0.252619i	−0.822044	+	0.569425i	$0.807166\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.02142e7	+	2.02142e7i	−1.02776	+	1.02776i	−0.0281570	+	0.999604i	$0.508964\pi$
$827$	−2.02142e7	+	2.02142e7i	−1.02776	+	1.02776i	−0.999604	+	0.0281570i	$0.991036\pi$
$828$		0			0
$829$			4.54977e6i			0.229934i	0.993369	+	0.114967i	$0.0366762\pi$
$829$			4.54977e6i			0.229934i	−0.993369	+	0.114967i	$0.963324\pi$
$830$		0			0
$831$			1.39710e6i			0.0701818i
$832$		0			0
$833$	−1.90566e6	+	1.90566e6i	−0.0951551	+	0.0951551i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	798135.	+	798135.i	0.0393788	+	0.0393788i
$838$		0			0
$839$	−1.98332e7			−0.972719			−0.486360	−	0.873759i	$-0.661675\pi$
$839$	−1.98332e7			−0.972719			−0.486360	+	0.873759i	$0.661675\pi$
$840$		0			0
$841$	−4.04300e6			−0.197112
$842$		0			0
$843$	2.04198e6	+	2.04198e6i	0.0989651	+	0.0989651i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.44138e7	−	2.44138e7i	1.16930	−	1.16930i
$848$		0			0
$849$		−	237170.i		−	0.0112925i
$850$		0			0
$851$		−	1.06342e7i		−	0.503363i
$852$		0			0
$853$	693392.	−	693392.i	0.0326292	−	0.0326292i	−0.690604	−	0.723233i	$-0.742655\pi$
$853$	693392.	−	693392.i	0.0326292	−	0.0326292i	0.723233	+	0.690604i	$0.242655\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.01743e7	−	2.01743e7i	−0.938308	−	0.938308i	0.0598966	−	0.998205i	$-0.480923\pi$
$857$	−2.01743e7	−	2.01743e7i	−0.938308	−	0.938308i	−0.998205	+	0.0598966i	$0.980923\pi$
$858$		0			0
$859$	−2.16047e7			−0.999001			−0.499501	−	0.866314i	$-0.666483\pi$
$859$	−2.16047e7			−0.999001			−0.499501	+	0.866314i	$0.666483\pi$
$860$		0			0
$861$	972774.			0.0447202
$862$		0			0
$863$	−7.05264e6	−	7.05264e6i	−0.322348	−	0.322348i	0.527319	−	0.849667i	$-0.323197\pi$
$863$	−7.05264e6	−	7.05264e6i	−0.322348	−	0.322348i	−0.849667	+	0.527319i	$0.823197\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	485480.	−	485480.i	0.0219343	−	0.0219343i
$868$		0			0
$869$			4.10287e7i			1.84305i
$870$		0			0
$871$			5.32041e6i			0.237629i
$872$		0			0
$873$	2.16623e7	−	2.16623e7i	0.961988	−	0.961988i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.77065e7	+	1.77065e7i	0.777381	+	0.777381i	0.979385	−	0.202004i	$-0.0647453\pi$
$877$	1.77065e7	+	1.77065e7i	0.777381	+	0.777381i	−0.202004	+	0.979385i	$0.564745\pi$
$878$		0			0
$879$	−907969.			−0.0396368
$880$		0			0
$881$	−2.88929e7			−1.25416			−0.627078	−	0.778957i	$-0.715749\pi$
$881$	−2.88929e7			−1.25416			−0.627078	+	0.778957i	$0.715749\pi$
$882$		0			0
$883$	−4.58238e6	−	4.58238e6i	−0.197783	−	0.197783i	0.601266	−	0.799049i	$-0.294663\pi$
$883$	−4.58238e6	−	4.58238e6i	−0.197783	−	0.197783i	−0.799049	+	0.601266i	$0.794663\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.54412e6	−	9.54412e6i	0.407312	−	0.407312i	−0.473488	−	0.880800i	$-0.657005\pi$
$887$	9.54412e6	−	9.54412e6i	0.407312	−	0.407312i	0.880800	+	0.473488i	$0.157005\pi$
$888$		0			0
$889$		−	9.19643e6i		−	0.390270i
$890$		0			0
$891$		−	3.70057e7i		−	1.56162i
$892$		0			0
$893$	−2.08247e7	+	2.08247e7i	−0.873875	+	0.873875i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−3.01345e6	−	3.01345e6i	−0.125050	−	0.125050i
$898$		0			0
$899$	−9.71808e6			−0.401034
$900$		0			0
$901$	9.64987e6			0.396013
$902$		0			0
$903$	−1.78437e6	−	1.78437e6i	−0.0728224	−	0.0728224i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.57152e7	−	1.57152e7i	0.634312	−	0.634312i	−0.314835	−	0.949147i	$-0.601949\pi$
$907$	1.57152e7	−	1.57152e7i	0.634312	−	0.634312i	0.949147	+	0.314835i	$0.101949\pi$
$908$		0			0
$909$		−	7.61877e6i		−	0.305826i
$910$		0			0
$911$		−	3.37011e7i		−	1.34539i	−0.739920	−	0.672694i	$-0.765137\pi$
$911$		−	3.37011e7i		−	1.34539i	0.739920	−	0.672694i	$-0.234863\pi$
$912$		0			0
$913$	−4.00889e7	+	4.00889e7i	−1.59165	+	1.59165i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.32021e6	−	8.32021e6i	−0.326747	−	0.326747i
$918$		0			0
$919$	−1.69860e7			−0.663440			−0.331720	−	0.943378i	$-0.607629\pi$
$919$	−1.69860e7			−0.663440			−0.331720	+	0.943378i	$0.607629\pi$
$920$		0			0
$921$	1.30872e6			0.0508391
$922$		0			0
$923$	−7.86139e6	−	7.86139e6i	−0.303735	−	0.303735i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1.05918e6	+	1.05918e6i	−0.0404827	+	0.0404827i
$928$		0			0
$929$		−	2.74089e7i		−	1.04196i	−0.853568	−	0.520982i	$-0.825566\pi$
$929$		−	2.74089e7i		−	1.04196i	0.853568	−	0.520982i	$-0.174434\pi$
$930$		0			0
$931$		−	7.36144e6i		−	0.278348i
$932$		0			0
$933$	1.31423e6	−	1.31423e6i	0.0494275	−	0.0494275i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	3.46447e7	+	3.46447e7i	1.28910	+	1.28910i	0.935333	+	0.353769i	$0.115100\pi$
$937$	3.46447e7	+	3.46447e7i	1.28910	+	1.28910i	0.353769	+	0.935333i	$0.384900\pi$
$938$		0			0
$939$	−1.15026e6			−0.0425728
$940$		0			0
$941$	1.97677e7			0.727750			0.363875	−	0.931448i	$-0.381454\pi$
$941$	1.97677e7			0.727750			0.363875	+	0.931448i	$0.381454\pi$
$942$		0			0
$943$	1.63304e7	+	1.63304e7i	0.598022	+	0.598022i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.45673e7	+	1.45673e7i	−0.527842	+	0.527842i	−0.919928	−	0.392086i	$-0.871753\pi$
$947$	−1.45673e7	+	1.45673e7i	−0.527842	+	0.527842i	0.392086	+	0.919928i	$0.371753\pi$
$948$		0			0
$949$			4.62269e7i			1.66621i
$950$		0			0
$951$		−	3.20738e6i		−	0.115000i
$952$		0			0
$953$	2.23751e7	−	2.23751e7i	0.798055	−	0.798055i	−0.184734	−	0.982789i	$-0.559142\pi$
$953$	2.23751e7	−	2.23751e7i	0.798055	−	0.798055i	0.982789	+	0.184734i	$0.0591422\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−2.65411e6	−	2.65411e6i	−0.0936782	−	0.0936782i
$958$		0			0
$959$	5.81757e6			0.204266
$960$		0			0
$961$	2.47829e7			0.865653
$962$		0			0
$963$	3.64053e7	+	3.64053e7i	1.26502	+	1.26502i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−2.87842e7	+	2.87842e7i	−0.989894	+	0.989894i	−0.999949	−	0.0100555i	$-0.996799\pi$
$967$	−2.87842e7	+	2.87842e7i	−0.989894	+	0.989894i	0.0100555	+	0.999949i	$0.496799\pi$
$968$		0			0
$969$		−	2.73087e6i		−	0.0934311i
$970$		0			0
$971$			5.09828e7i			1.73530i	0.497173	+	0.867651i	$0.334371\pi$
$971$			5.09828e7i			1.73530i	−0.497173	+	0.867651i	$0.665629\pi$
$972$		0			0
$973$	5.63180e6	−	5.63180e6i	0.190706	−	0.190706i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.93752e7	−	3.93752e7i	−1.31973	−	1.31973i	−0.913984	−	0.405749i	$-0.867011\pi$
$977$	−3.93752e7	−	3.93752e7i	−1.31973	−	1.31973i	−0.405749	−	0.913984i	$-0.632989\pi$
$978$		0			0
$979$	−1.55768e7			−0.519424
$980$		0			0
$981$	4.86093e7			1.61267
$982$		0			0
$983$	2.47236e7	+	2.47236e7i	0.816070	+	0.816070i	0.985536	−	0.169466i	$-0.0542042\pi$
$983$	2.47236e7	+	2.47236e7i	0.816070	+	0.816070i	−0.169466	+	0.985536i	$0.554204\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	1.38675e6	−	1.38675e6i	0.0453111	−	0.0453111i
$988$		0			0
$989$		−	5.99100e7i		−	1.94764i
$990$		0			0
$991$			2.85549e7i			0.923627i	0.886977	+	0.461814i	$0.152801\pi$
$991$			2.85549e7i			0.923627i	−0.886977	+	0.461814i	$0.847199\pi$
$992$		0			0
$993$	−621822.	+	621822.i	−0.0200121	+	0.0200121i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.29416e7	+	1.29416e7i	0.412335	+	0.412335i	0.882551	−	0.470216i	$-0.155824\pi$
$997$	1.29416e7	+	1.29416e7i	0.412335	+	0.412335i	−0.470216	+	0.882551i	$0.655824\pi$
$998$		0			0
$999$	1.54475e6			0.0489716

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.n.g.207.6		20
4.3	odd	2	inner	400.6.n.g.207.5		20
5.2	odd	4		80.6.n.d.63.6	yes	20
5.3	odd	4	inner	400.6.n.g.143.5		20
5.4	even	2		80.6.n.d.47.5	✓	20
20.3	even	4	inner	400.6.n.g.143.6		20
20.7	even	4		80.6.n.d.63.5	yes	20
20.19	odd	2		80.6.n.d.47.6	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.n.d.47.5	✓	20	5.4	even	2
80.6.n.d.47.6	yes	20	20.19	odd	2
80.6.n.d.63.5	yes	20	20.7	even	4
80.6.n.d.63.6	yes	20	5.2	odd	4
400.6.n.g.143.5		20	5.3	odd	4	inner
400.6.n.g.143.6		20	20.3	even	4	inner
400.6.n.g.207.5		20	4.3	odd	2	inner
400.6.n.g.207.6		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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.839817 + 0.839817i) q^{3} +(-99.3589 + 99.3589i) q^{7} -241.589i q^{9} +O(q^{10})\)	\(q+(0.839817 + 0.839817i) q^{3} +(-99.3589 + 99.3589i) q^{7} -241.589i q^{9} +637.781i q^{11} +(640.389 - 640.389i) q^{13} +(-648.760 - 648.760i) q^{17} +2506.12 q^{19} -166.887 q^{21} +(-2801.60 - 2801.60i) q^{23} +(406.966 - 406.966i) q^{27} +4955.21i q^{29} +1961.18i q^{31} +(-535.619 + 535.619i) q^{33} +(1897.88 + 1897.88i) q^{37} +1075.62 q^{39} -5828.95 q^{41} +(10692.1 + 10692.1i) q^{43} +(-8309.52 + 8309.52i) q^{47} -2937.38i q^{49} -1089.68i q^{51} +(-7437.17 + 7437.17i) q^{53} +(2104.68 + 2104.68i) q^{57} -16738.4 q^{59} +23742.5 q^{61} +(24004.1 + 24004.1i) q^{63} +(-4154.05 + 4154.05i) q^{67} -4705.66i q^{69} -12276.0i q^{71} +(-36092.8 + 36092.8i) q^{73} +(-63369.2 - 63369.2i) q^{77} +64330.4 q^{79} -58022.7 q^{81} +(62856.9 + 62856.9i) q^{83} +(-4161.47 + 4161.47i) q^{87} +24423.5i q^{89} +127257. i q^{91} +(-1647.03 + 1647.03i) q^{93} +(89666.0 + 89666.0i) q^{97} +154081. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q - 804 q^{13} + 2236 q^{17} - 4520 q^{21} + 11096 q^{33} - 44260 q^{37} - 6760 q^{41} - 182452 q^{53} + 34288 q^{57} - 41080 q^{61} - 264372 q^{73} - 399304 q^{77} - 520220 q^{81} - 713496 q^{93} - 374772 q^{97}+O(q^{100}) \) 20 * q - 804 * q^13 + 2236 * q^17 - 4520 * q^21 + 11096 * q^33 - 44260 * q^37 - 6760 * q^41 - 182452 * q^53 + 34288 * q^57 - 41080 * q^61 - 264372 * q^73 - 399304 * q^77 - 520220 * q^81 - 713496 * q^93 - 374772 * q^97

Introduction

L-functions

Modular forms

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Representations

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Database

Properties

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