LMFDB - Embedded newform 4650.2.a.bu.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.a (trivial)

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:	yes
Analytic conductor:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +3.00000 q^{21} -3.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} +3.00000 q^{28} +10.0000 q^{29} +1.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +3.00000 q^{37} -1.00000 q^{39} +7.00000 q^{41} +3.00000 q^{42} -1.00000 q^{43} -3.00000 q^{44} +4.00000 q^{46} -7.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -2.00000 q^{51} -1.00000 q^{52} +9.00000 q^{53} +1.00000 q^{54} +3.00000 q^{56} +10.0000 q^{58} +7.00000 q^{61} +1.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} -2.00000 q^{67} -2.00000 q^{68} +4.00000 q^{69} +7.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +3.00000 q^{74} -9.00000 q^{77} -1.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} +7.00000 q^{82} +9.00000 q^{83} +3.00000 q^{84} -1.00000 q^{86} +10.0000 q^{87} -3.00000 q^{88} +10.0000 q^{89} -3.00000 q^{91} +4.00000 q^{92} +1.00000 q^{93} -7.00000 q^{94} +1.00000 q^{96} -2.00000 q^{97} +2.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	−3.00000	−0.639602
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	3.00000	0.566947
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	−3.00000	−0.522233
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	3.00000	0.462910
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	4.00000	0.589768
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	0	0
$51$	−2.00000	−0.280056
$52$	−1.00000	−0.138675
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	10.0000	1.31306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	1.00000	0.127000
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−2.00000	−0.242536
$69$	4.00000	0.481543
$70$	0	0
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	1.00000	0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	0	0
$77$	−9.00000	−1.02565
$78$	−1.00000	−0.113228
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	7.00000	0.773021
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	−1.00000	−0.107833
$87$	10.0000	1.07211
$88$	−3.00000	−0.319801
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	4.00000	0.417029
$93$	1.00000	0.103695
$94$	−7.00000	−0.721995
$95$	0	0
$96$	1.00000	0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	2.00000	0.202031
$99$	−3.00000	−0.301511
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−2.00000	−0.198030
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	9.00000	0.874157
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	1.00000	0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	3.00000	0.283473
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	10.0000	0.928477
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	7.00000	0.633750
$123$	7.00000	0.631169
$124$	1.00000	0.0898027
$125$	0	0
$126$	3.00000	0.267261
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	1.00000	0.0883883
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−3.00000	−0.261116
$133$	0	0
$134$	−2.00000	−0.172774
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	4.00000	0.340503
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	−7.00000	−0.589506
$142$	7.00000	0.587427
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	2.00000	0.164957
$148$	3.00000	0.246598
$149$	−20.0000	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	−9.00000	−0.725241
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−10.0000	−0.795557
$159$	9.00000	0.713746
$160$	0	0
$161$	12.0000	0.945732
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	7.00000	0.546608
$165$	0	0
$166$	9.00000	0.698535
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	3.00000	0.231455
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	10.0000	0.758098
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	10.0000	0.749532
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	0	0
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	−3.00000	−0.222375
$183$	7.00000	0.517455
$184$	4.00000	0.294884
$185$	0	0
$186$	1.00000	0.0733236
$187$	6.00000	0.438763
$188$	−7.00000	−0.510527
$189$	3.00000	0.218218
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000	0.0721688
$193$	9.00000	0.647834	0.323917	−	0.946085i	$-0.395000\pi$
$193$	9.00000	0.647834	0.323917	+	0.946085i	$0.395000\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	2.00000	0.142857
$197$	−27.0000	−1.92367	−0.961835	−	0.273629i	$-0.911776\pi$
$197$	−27.0000	−1.92367	−0.961835	+	0.273629i	$0.911776\pi$
$198$	−3.00000	−0.213201
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	2.00000	0.140720
$203$	30.0000	2.10559
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−1.00000	−0.0696733
$207$	4.00000	0.278019
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	9.00000	0.618123
$213$	7.00000	0.479632
$214$	18.0000	1.23045
$215$	0	0
$216$	1.00000	0.0680414
$217$	3.00000	0.203653
$218$	−10.0000	−0.677285
$219$	4.00000	0.270295
$220$	0	0
$221$	2.00000	0.134535
$222$	3.00000	0.201347
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−9.00000	−0.592157
$232$	10.0000	0.656532
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	0	0
$237$	−10.0000	−0.649570
$238$	−6.00000	−0.388922
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	−2.00000	−0.128565
$243$	1.00000	0.0641500
$244$	7.00000	0.448129
$245$	0	0
$246$	7.00000	0.446304
$247$	0	0
$248$	1.00000	0.0635001
$249$	9.00000	0.570352
$250$	0	0
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	3.00000	0.188982
$253$	−12.0000	−0.754434
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	−1.00000	−0.0622573
$259$	9.00000	0.559233
$260$	0	0
$261$	10.0000	0.618984
$262$	12.0000	0.741362
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	0	0
$267$	10.0000	0.611990
$268$	−2.00000	−0.122169
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−2.00000	−0.121268
$273$	−3.00000	−0.181568
$274$	−2.00000	−0.120824
$275$	0	0
$276$	4.00000	0.240772
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−5.00000	−0.299880
$279$	1.00000	0.0598684
$280$	0	0
$281$	7.00000	0.417585	0.208792	−	0.977960i	$-0.433047\pi$
$281$	7.00000	0.417585	0.208792	+	0.977960i	$0.433047\pi$
$282$	−7.00000	−0.416844
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	7.00000	0.415374
$285$	0	0
$286$	3.00000	0.177394
$287$	21.0000	1.23959
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−2.00000	−0.117242
$292$	4.00000	0.234082
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	3.00000	0.174371
$297$	−3.00000	−0.174078
$298$	−20.0000	−1.15857
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−3.00000	−0.172917
$302$	−18.0000	−1.03578
$303$	2.00000	0.114897
$304$	0	0
$305$	0	0
$306$	−2.00000	−0.114332
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	−9.00000	−0.512823
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	27.0000	1.53103	0.765515	−	0.643418i	$-0.222484\pi$
$311$	27.0000	1.53103	0.765515	+	0.643418i	$0.222484\pi$
$312$	−1.00000	−0.0566139
$313$	24.0000	1.35656	0.678280	−	0.734803i	$-0.262726\pi$
$313$	24.0000	1.35656	0.678280	+	0.734803i	$0.262726\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	−10.0000	−0.562544
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	9.00000	0.504695
$319$	−30.0000	−1.67968
$320$	0	0
$321$	18.0000	1.00466
$322$	12.0000	0.668734
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	−10.0000	−0.553001
$328$	7.00000	0.386510
$329$	−21.0000	−1.15777
$330$	0	0
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	9.00000	0.493939
$333$	3.00000	0.164399
$334$	−2.00000	−0.109435
$335$	0	0
$336$	3.00000	0.163663
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	−12.0000	−0.652714
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−3.00000	−0.162459
$342$	0	0
$343$	−15.0000	−0.809924
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	4.00000	0.215041
$347$	−7.00000	−0.375780	−0.187890	−	0.982190i	$-0.560165\pi$
$347$	−7.00000	−0.375780	−0.187890	+	0.982190i	$0.560165\pi$
$348$	10.0000	0.536056
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−3.00000	−0.159901
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	0	0
$356$	10.0000	0.529999
$357$	−6.00000	−0.317554
$358$	−5.00000	−0.264258
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−13.0000	−0.683265
$363$	−2.00000	−0.104973
$364$	−3.00000	−0.157243
$365$	0	0
$366$	7.00000	0.365896
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	4.00000	0.208514
$369$	7.00000	0.364405
$370$	0	0
$371$	27.0000	1.40177
$372$	1.00000	0.0518476
$373$	24.0000	1.24267	0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000	1.24267	0.621336	+	0.783544i	$0.286590\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	−7.00000	−0.360997
$377$	−10.0000	−0.515026
$378$	3.00000	0.154303
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	−8.00000	−0.409316
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	9.00000	0.458088
$387$	−1.00000	−0.0508329
$388$	−2.00000	−0.101535
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	2.00000	0.101015
$393$	12.0000	0.605320
$394$	−27.0000	−1.36024
$395$	0	0
$396$	−3.00000	−0.150756
$397$	28.0000	1.40528	0.702640	−	0.711546i	$-0.252005\pi$
$397$	28.0000	1.40528	0.702640	+	0.711546i	$0.252005\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	−2.00000	−0.0997509
$403$	−1.00000	−0.0498135
$404$	2.00000	0.0995037
$405$	0	0
$406$	30.0000	1.48888
$407$	−9.00000	−0.446113
$408$	−2.00000	−0.0990148
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	−1.00000	−0.0492665
$413$	0	0
$414$	4.00000	0.196589
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−5.00000	−0.244851
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	2.00000	0.0973585
$423$	−7.00000	−0.340352
$424$	9.00000	0.437079
$425$	0	0
$426$	7.00000	0.339151
$427$	21.0000	1.01626
$428$	18.0000	0.870063
$429$	3.00000	0.144841
$430$	0	0
$431$	7.00000	0.337178	0.168589	−	0.985686i	$-0.446079\pi$
$431$	7.00000	0.337178	0.168589	+	0.985686i	$0.446079\pi$
$432$	1.00000	0.0481125
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	3.00000	0.144005
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	4.00000	0.191127
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	2.00000	0.0951303
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	3.00000	0.142374
$445$	0	0
$446$	−6.00000	−0.284108
$447$	−20.0000	−0.945968
$448$	3.00000	0.141737
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	−21.0000	−0.988851
$452$	−6.00000	−0.282216
$453$	−18.0000	−0.845714
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	10.0000	0.467269
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−3.00000	−0.139724	−0.0698620	−	0.997557i	$-0.522256\pi$
$461$	−3.00000	−0.139724	−0.0698620	+	0.997557i	$0.522256\pi$
$462$	−9.00000	−0.418718
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	9.00000	0.416917
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	−1.00000	−0.0462250
$469$	−6.00000	−0.277054
$470$	0	0
$471$	−22.0000	−1.01371
$472$	0	0
$473$	3.00000	0.137940
$474$	−10.0000	−0.459315
$475$	0	0
$476$	−6.00000	−0.275010
$477$	9.00000	0.412082
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	12.0000	0.546585
$483$	12.0000	0.546019
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	1.00000	0.0453609
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	7.00000	0.316875
$489$	4.00000	0.180886
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	7.00000	0.315584
$493$	−20.0000	−0.900755
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	21.0000	0.941979
$498$	9.00000	0.403300
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	−2.00000	−0.0893534
$502$	−3.00000	−0.133897
$503$	−11.0000	−0.490466	−0.245233	−	0.969464i	$-0.578864\pi$
$503$	−11.0000	−0.490466	−0.245233	+	0.969464i	$0.578864\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	−12.0000	−0.533465
$507$	−12.0000	−0.532939
$508$	−2.00000	−0.0887357
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	1.00000	0.0441942
$513$	0	0
$514$	−7.00000	−0.308757
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	21.0000	0.923579
$518$	9.00000	0.395437
$519$	4.00000	0.175581
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	10.0000	0.437688
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−6.00000	−0.261612
$527$	−2.00000	−0.0871214
$528$	−3.00000	−0.130558
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.00000	−0.303204
$534$	10.0000	0.432742
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	−5.00000	−0.215766
$538$	−10.0000	−0.431131
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	−8.00000	−0.343629
$543$	−13.0000	−0.557883
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	−3.00000	−0.128388
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	−2.00000	−0.0854358
$549$	7.00000	0.298753
$550$	0	0
$551$	0	0
$552$	4.00000	0.170251
$553$	−30.0000	−1.27573
$554$	18.0000	0.764747
$555$	0	0
$556$	−5.00000	−0.212047
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	1.00000	0.0423334
$559$	1.00000	0.0422955
$560$	0	0
$561$	6.00000	0.253320
$562$	7.00000	0.295277
$563$	34.0000	1.43293	0.716465	−	0.697623i	$-0.245759\pi$
$563$	34.0000	1.43293	0.716465	+	0.697623i	$0.245759\pi$
$564$	−7.00000	−0.294753
$565$	0	0
$566$	24.0000	1.00880
$567$	3.00000	0.125988
$568$	7.00000	0.293713
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	3.00000	0.125436
$573$	−8.00000	−0.334205
$574$	21.0000	0.876523
$575$	0	0
$576$	1.00000	0.0416667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	−13.0000	−0.540729
$579$	9.00000	0.374027
$580$	0	0
$581$	27.0000	1.12015
$582$	−2.00000	−0.0829027
$583$	−27.0000	−1.11823
$584$	4.00000	0.165521
$585$	0	0
$586$	24.0000	0.991431
$587$	−17.0000	−0.701665	−0.350833	−	0.936438i	$-0.614101\pi$
$587$	−17.0000	−0.701665	−0.350833	+	0.936438i	$0.614101\pi$
$588$	2.00000	0.0824786
$589$	0	0
$590$	0	0
$591$	−27.0000	−1.11063
$592$	3.00000	0.123299
$593$	−1.00000	−0.0410651	−0.0205325	−	0.999789i	$-0.506536\pi$
$593$	−1.00000	−0.0410651	−0.0205325	+	0.999789i	$0.506536\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	−20.0000	−0.819232
$597$	−20.0000	−0.818546
$598$	−4.00000	−0.163572
$599$	−15.0000	−0.612883	−0.306442	−	0.951889i	$-0.599138\pi$
$599$	−15.0000	−0.612883	−0.306442	+	0.951889i	$0.599138\pi$
$600$	0	0
$601$	12.0000	0.489490	0.244745	−	0.969587i	$-0.421296\pi$
$601$	12.0000	0.489490	0.244745	+	0.969587i	$0.421296\pi$
$602$	−3.00000	−0.122271
$603$	−2.00000	−0.0814463
$604$	−18.0000	−0.732410
$605$	0	0
$606$	2.00000	0.0812444
$607$	43.0000	1.74532	0.872658	−	0.488332i	$-0.162394\pi$
$607$	43.0000	1.74532	0.872658	+	0.488332i	$0.162394\pi$
$608$	0	0
$609$	30.0000	1.21566
$610$	0	0
$611$	7.00000	0.283190
$612$	−2.00000	−0.0808452
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	−22.0000	−0.887848
$615$	0	0
$616$	−9.00000	−0.362620
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	−1.00000	−0.0402259
$619$	−5.00000	−0.200967	−0.100483	−	0.994939i	$-0.532039\pi$
$619$	−5.00000	−0.200967	−0.100483	+	0.994939i	$0.532039\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	27.0000	1.08260
$623$	30.0000	1.20192
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	24.0000	0.959233
$627$	0	0
$628$	−22.0000	−0.877896
$629$	−6.00000	−0.239236
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	−10.0000	−0.397779
$633$	2.00000	0.0794929
$634$	−22.0000	−0.873732
$635$	0	0
$636$	9.00000	0.356873
$637$	−2.00000	−0.0792429
$638$	−30.0000	−1.18771
$639$	7.00000	0.276916
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	18.0000	0.710403
$643$	−31.0000	−1.22252	−0.611260	−	0.791430i	$-0.709337\pi$
$643$	−31.0000	−1.22252	−0.611260	+	0.791430i	$0.709337\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	0	0
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	3.00000	0.117579
$652$	4.00000	0.156652
$653$	−16.0000	−0.626128	−0.313064	−	0.949732i	$-0.601356\pi$
$653$	−16.0000	−0.626128	−0.313064	+	0.949732i	$0.601356\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	7.00000	0.273304
$657$	4.00000	0.156055
$658$	−21.0000	−0.818665
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	−3.00000	−0.116598
$663$	2.00000	0.0776736
$664$	9.00000	0.349268
$665$	0	0
$666$	3.00000	0.116248
$667$	40.0000	1.54881
$668$	−2.00000	−0.0773823
$669$	−6.00000	−0.231973
$670$	0	0
$671$	−21.0000	−0.810696
$672$	3.00000	0.115728
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	−12.0000	−0.461538
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	−6.00000	−0.230429
$679$	−6.00000	−0.230259
$680$	0	0
$681$	−12.0000	−0.459841
$682$	−3.00000	−0.114876
$683$	−26.0000	−0.994862	−0.497431	−	0.867503i	$-0.665723\pi$
$683$	−26.0000	−0.994862	−0.497431	+	0.867503i	$0.665723\pi$
$684$	0	0
$685$	0	0
$686$	−15.0000	−0.572703
$687$	10.0000	0.381524
$688$	−1.00000	−0.0381246
$689$	−9.00000	−0.342873
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	4.00000	0.152057
$693$	−9.00000	−0.341882
$694$	−7.00000	−0.265716
$695$	0	0
$696$	10.0000	0.379049
$697$	−14.0000	−0.530288
$698$	0	0
$699$	9.00000	0.340411
$700$	0	0
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	−1.00000	−0.0377426
$703$	0	0
$704$	−3.00000	−0.113067
$705$	0	0
$706$	24.0000	0.903252
$707$	6.00000	0.225653
$708$	0	0
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	10.0000	0.374766
$713$	4.00000	0.149801
$714$	−6.00000	−0.224544
$715$	0	0
$716$	−5.00000	−0.186859
$717$	0	0
$718$	0	0
$719$	50.0000	1.86469	0.932343	−	0.361576i	$-0.117761\pi$
$719$	50.0000	1.86469	0.932343	+	0.361576i	$0.117761\pi$
$720$	0	0
$721$	−3.00000	−0.111726
$722$	−19.0000	−0.707107
$723$	12.0000	0.446285
$724$	−13.0000	−0.483141
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	−27.0000	−1.00137	−0.500687	−	0.865628i	$-0.666919\pi$
$727$	−27.0000	−1.00137	−0.500687	+	0.865628i	$0.666919\pi$
$728$	−3.00000	−0.111187
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.00000	0.0739727
$732$	7.00000	0.258727
$733$	−36.0000	−1.32969	−0.664845	−	0.746981i	$-0.731502\pi$
$733$	−36.0000	−1.32969	−0.664845	+	0.746981i	$0.731502\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	4.00000	0.147442
$737$	6.00000	0.221013
$738$	7.00000	0.257674
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	27.0000	0.991201
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	24.0000	0.878702
$747$	9.00000	0.329293
$748$	6.00000	0.219382
$749$	54.0000	1.97312
$750$	0	0
$751$	7.00000	0.255434	0.127717	−	0.991811i	$-0.459235\pi$
$751$	7.00000	0.255434	0.127717	+	0.991811i	$0.459235\pi$
$752$	−7.00000	−0.255264
$753$	−3.00000	−0.109326
$754$	−10.0000	−0.364179
$755$	0	0
$756$	3.00000	0.109109
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	0	0
$759$	−12.0000	−0.435572
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	−2.00000	−0.0724524
$763$	−30.0000	−1.08607
$764$	−8.00000	−0.289430
$765$	0	0
$766$	24.0000	0.867155
$767$	0	0
$768$	1.00000	0.0360844
$769$	−45.0000	−1.62274	−0.811371	−	0.584532i	$-0.801278\pi$
$769$	−45.0000	−1.62274	−0.811371	+	0.584532i	$0.801278\pi$
$770$	0	0
$771$	−7.00000	−0.252099
$772$	9.00000	0.323917
$773$	34.0000	1.22290	0.611448	−	0.791285i	$-0.290588\pi$
$773$	34.0000	1.22290	0.611448	+	0.791285i	$0.290588\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	9.00000	0.322873
$778$	10.0000	0.358517
$779$	0	0
$780$	0	0
$781$	−21.0000	−0.751439
$782$	−8.00000	−0.286079
$783$	10.0000	0.357371
$784$	2.00000	0.0714286
$785$	0	0
$786$	12.0000	0.428026
$787$	−7.00000	−0.249523	−0.124762	−	0.992187i	$-0.539817\pi$
$787$	−7.00000	−0.249523	−0.124762	+	0.992187i	$0.539817\pi$
$788$	−27.0000	−0.961835
$789$	−6.00000	−0.213606
$790$	0	0
$791$	−18.0000	−0.640006
$792$	−3.00000	−0.106600
$793$	−7.00000	−0.248577
$794$	28.0000	0.993683
$795$	0	0
$796$	−20.0000	−0.708881
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	14.0000	0.495284
$800$	0	0
$801$	10.0000	0.353333
$802$	−8.00000	−0.282490
$803$	−12.0000	−0.423471
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	−1.00000	−0.0352235
$807$	−10.0000	−0.352017
$808$	2.00000	0.0703598
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	30.0000	1.05279
$813$	−8.00000	−0.280572
$814$	−9.00000	−0.315450
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	−10.0000	−0.349642
$819$	−3.00000	−0.104828
$820$	0	0
$821$	−53.0000	−1.84971	−0.924856	−	0.380317i	$-0.875815\pi$
$821$	−53.0000	−1.84971	−0.924856	+	0.380317i	$0.875815\pi$
$822$	−2.00000	−0.0697580
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	−1.00000	−0.0348367
$825$	0	0
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	4.00000	0.139010
$829$	55.0000	1.91023	0.955114	−	0.296237i	$-0.0957318\pi$
$829$	55.0000	1.91023	0.955114	+	0.296237i	$0.0957318\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	−1.00000	−0.0346688
$833$	−4.00000	−0.138592
$834$	−5.00000	−0.173136
$835$	0	0
$836$	0	0
$837$	1.00000	0.0345651
$838$	−30.0000	−1.03633
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−28.0000	−0.964944
$843$	7.00000	0.241093
$844$	2.00000	0.0688428
$845$	0	0
$846$	−7.00000	−0.240665
$847$	−6.00000	−0.206162
$848$	9.00000	0.309061
$849$	24.0000	0.823678
$850$	0	0
$851$	12.0000	0.411355
$852$	7.00000	0.239816
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	21.0000	0.718605
$855$	0	0
$856$	18.0000	0.615227
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	3.00000	0.102418
$859$	−25.0000	−0.852989	−0.426494	−	0.904490i	$-0.640252\pi$
$859$	−25.0000	−0.852989	−0.426494	+	0.904490i	$0.640252\pi$
$860$	0	0
$861$	21.0000	0.715678
$862$	7.00000	0.238421
$863$	−46.0000	−1.56586	−0.782929	−	0.622111i	$-0.786275\pi$
$863$	−46.0000	−1.56586	−0.782929	+	0.622111i	$0.786275\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	4.00000	0.135926
$867$	−13.0000	−0.441503
$868$	3.00000	0.101827
$869$	30.0000	1.01768
$870$	0	0
$871$	2.00000	0.0677674
$872$	−10.0000	−0.338643
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	0	0
$876$	4.00000	0.135147
$877$	8.00000	0.270141	0.135070	−	0.990836i	$-0.456874\pi$
$877$	8.00000	0.270141	0.135070	+	0.990836i	$0.456874\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	0	0
$881$	32.0000	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$882$	2.00000	0.0673435
$883$	−11.0000	−0.370179	−0.185090	−	0.982722i	$-0.559258\pi$
$883$	−11.0000	−0.370179	−0.185090	+	0.982722i	$0.559258\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	−36.0000	−1.20944
$887$	13.0000	0.436497	0.218249	−	0.975893i	$-0.429966\pi$
$887$	13.0000	0.436497	0.218249	+	0.975893i	$0.429966\pi$
$888$	3.00000	0.100673
$889$	−6.00000	−0.201234
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−6.00000	−0.200895
$893$	0	0
$894$	−20.0000	−0.668900
$895$	0	0
$896$	3.00000	0.100223
$897$	−4.00000	−0.133556
$898$	20.0000	0.667409
$899$	10.0000	0.333519
$900$	0	0
$901$	−18.0000	−0.599667
$902$	−21.0000	−0.699224
$903$	−3.00000	−0.0998337
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−18.0000	−0.598010
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	−12.0000	−0.398234
$909$	2.00000	0.0663358
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	−27.0000	−0.893570
$914$	8.00000	0.264616
$915$	0	0
$916$	10.0000	0.330409
$917$	36.0000	1.18882
$918$	−2.00000	−0.0660098
$919$	25.0000	0.824674	0.412337	−	0.911031i	$-0.364713\pi$
$919$	25.0000	0.824674	0.412337	+	0.911031i	$0.364713\pi$
$920$	0	0
$921$	−22.0000	−0.724925
$922$	−3.00000	−0.0987997
$923$	−7.00000	−0.230408
$924$	−9.00000	−0.296078
$925$	0	0
$926$	14.0000	0.460069
$927$	−1.00000	−0.0328443
$928$	10.0000	0.328266
$929$	40.0000	1.31236	0.656179	−	0.754606i	$-0.272172\pi$
$929$	40.0000	1.31236	0.656179	+	0.754606i	$0.272172\pi$
$930$	0	0
$931$	0	0
$932$	9.00000	0.294805
$933$	27.0000	0.883940
$934$	18.0000	0.588978
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	−6.00000	−0.195907
$939$	24.0000	0.783210
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	−22.0000	−0.716799
$943$	28.0000	0.911805
$944$	0	0
$945$	0	0
$946$	3.00000	0.0975384
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	−10.0000	−0.324785
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−22.0000	−0.713399
$952$	−6.00000	−0.194461
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	0	0
$957$	−30.0000	−0.969762
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	1.00000	0.0322581
$962$	−3.00000	−0.0967239
$963$	18.0000	0.580042
$964$	12.0000	0.386494
$965$	0	0
$966$	12.0000	0.386094
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000	0.0320750
$973$	−15.0000	−0.480878
$974$	28.0000	0.897178
$975$	0	0
$976$	7.00000	0.224065
$977$	13.0000	0.415907	0.207953	−	0.978139i	$-0.433320\pi$
$977$	13.0000	0.415907	0.207953	+	0.978139i	$0.433320\pi$
$978$	4.00000	0.127906
$979$	−30.0000	−0.958804
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−28.0000	−0.893516
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	7.00000	0.223152
$985$	0	0
$986$	−20.0000	−0.636930
$987$	−21.0000	−0.668437
$988$	0	0
$989$	−4.00000	−0.127193
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	1.00000	0.0317500
$993$	−3.00000	−0.0952021
$994$	21.0000	0.666080
$995$	0	0
$996$	9.00000	0.285176
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	−20.0000	−0.633089
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.bu.1.1	yes	1
5.2	odd	4		4650.2.d.q.3349.2		2
5.3	odd	4		4650.2.d.q.3349.1		2
5.4	even	2		4650.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4650.2.a.b.1.1	✓	1	5.4	even	2
4650.2.a.bu.1.1	yes	1	1.1	even	1	trivial
4650.2.d.q.3349.1		2	5.3	odd	4
4650.2.d.q.3349.2		2	5.2	odd	4

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:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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