LMFDB - Embedded newform 3330.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, 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("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +5.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +6.00000 q^{19} +1.00000 q^{20} +5.00000 q^{22} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} -9.00000 q^{29} +3.00000 q^{31} +1.00000 q^{32} -1.00000 q^{34} +1.00000 q^{35} -1.00000 q^{37} +6.00000 q^{38} +1.00000 q^{40} -9.00000 q^{41} +1.00000 q^{43} +5.00000 q^{44} -6.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -9.00000 q^{53} +5.00000 q^{55} +1.00000 q^{56} -9.00000 q^{58} +10.0000 q^{59} -5.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +16.0000 q^{67} -1.00000 q^{68} +1.00000 q^{70} +12.0000 q^{73} -1.00000 q^{74} +6.00000 q^{76} +5.00000 q^{77} -12.0000 q^{79} +1.00000 q^{80} -9.00000 q^{82} -2.00000 q^{83} -1.00000 q^{85} +1.00000 q^{86} +5.00000 q^{88} +2.00000 q^{89} +2.00000 q^{91} +6.00000 q^{95} +17.0000 q^{97} -6.00000 q^{98} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	5.00000	1.06600
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	1.00000	0.188982
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	1.00000	0.169031
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	6.00000	0.973329
$39$	0	0
$40$	1.00000	0.158114
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	5.00000	0.753778
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	1.00000	0.133631
$57$	0	0
$58$	−9.00000	−1.18176
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	3.00000	0.381000
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	1.00000	0.119523
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	6.00000	0.688247
$77$	5.00000	0.569803
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−9.00000	−0.993884
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	1.00000	0.107833
$87$	0	0
$88$	5.00000	0.533002
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	1.00000	0.100000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−9.00000	−0.874157
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	5.00000	0.476731
$111$	0	0
$112$	1.00000	0.0944911
$113$	13.0000	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$113$	13.0000	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$114$	0	0
$115$	0	0
$116$	−9.00000	−0.835629
$117$	0	0
$118$	10.0000	0.920575
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	14.0000	1.27273
$122$	−5.00000	−0.452679
$123$	0	0
$124$	3.00000	0.269408
$125$	1.00000	0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	2.00000	0.175412
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	16.0000	1.38219
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	−19.0000	−1.61156	−0.805779	−	0.592216i	$-0.798253\pi$
$139$	−19.0000	−1.61156	−0.805779	+	0.592216i	$0.798253\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	0	0
$143$	10.0000	0.836242
$144$	0	0
$145$	−9.00000	−0.747409
$146$	12.0000	0.993127
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	5.00000	0.402911
$155$	3.00000	0.240966
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	−12.0000	−0.954669
$159$	0	0
$160$	1.00000	0.0790569
$161$	0	0
$162$	0	0
$163$	1.00000	0.0783260	0.0391630	−	0.999233i	$-0.487531\pi$
$163$	1.00000	0.0783260	0.0391630	+	0.999233i	$0.487531\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	−2.00000	−0.155230
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−1.00000	−0.0766965
$171$	0	0
$172$	1.00000	0.0762493
$173$	7.00000	0.532200	0.266100	−	0.963945i	$-0.414265\pi$
$173$	7.00000	0.532200	0.266100	+	0.963945i	$0.414265\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	5.00000	0.376889
$177$	0	0
$178$	2.00000	0.149906
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−5.00000	−0.365636
$188$	0	0
$189$	0	0
$190$	6.00000	0.435286
$191$	−11.0000	−0.795932	−0.397966	−	0.917400i	$-0.630284\pi$
$191$	−11.0000	−0.795932	−0.397966	+	0.917400i	$0.630284\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	17.0000	1.22053
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	12.0000	0.844317
$203$	−9.00000	−0.631676
$204$	0	0
$205$	−9.00000	−0.628587
$206$	−4.00000	−0.278693
$207$	0	0
$208$	2.00000	0.138675
$209$	30.0000	2.07514
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	−9.00000	−0.618123
$213$	0	0
$214$	−18.0000	−1.23045
$215$	1.00000	0.0681994
$216$	0	0
$217$	3.00000	0.203653
$218$	−5.00000	−0.338643
$219$	0	0
$220$	5.00000	0.337100
$221$	−2.00000	−0.134535
$222$	0	0
$223$	−5.00000	−0.334825	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	−5.00000	−0.334825	−0.167412	+	0.985887i	$0.553541\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	13.0000	0.864747
$227$	7.00000	0.464606	0.232303	−	0.972643i	$-0.425374\pi$
$227$	7.00000	0.464606	0.232303	+	0.972643i	$0.425374\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	−9.00000	−0.590879
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	0	0
$236$	10.0000	0.650945
$237$	0	0
$238$	−1.00000	−0.0648204
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	14.0000	0.899954
$243$	0	0
$244$	−5.00000	−0.320092
$245$	−6.00000	−0.383326
$246$	0	0
$247$	12.0000	0.763542
$248$	3.00000	0.190500
$249$	0	0
$250$	1.00000	0.0632456
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	2.00000	0.124035
$261$	0	0
$262$	−10.0000	−0.617802
$263$	15.0000	0.924940	0.462470	−	0.886635i	$-0.346963\pi$
$263$	15.0000	0.924940	0.462470	+	0.886635i	$0.346963\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	6.00000	0.367884
$267$	0	0
$268$	16.0000	0.977356
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	14.0000	0.845771
$275$	5.00000	0.301511
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−19.0000	−1.13954
$279$	0	0
$280$	1.00000	0.0597614
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	0	0
$285$	0	0
$286$	10.0000	0.591312
$287$	−9.00000	−0.531253
$288$	0	0
$289$	−16.0000	−0.941176
$290$	−9.00000	−0.528498
$291$	0	0
$292$	12.0000	0.702247
$293$	−19.0000	−1.10999	−0.554996	−	0.831853i	$-0.687280\pi$
$293$	−19.0000	−1.10999	−0.554996	+	0.831853i	$0.687280\pi$
$294$	0	0
$295$	10.0000	0.582223
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	−14.0000	−0.810998
$299$	0	0
$300$	0	0
$301$	1.00000	0.0576390
$302$	18.0000	1.03578
$303$	0	0
$304$	6.00000	0.344124
$305$	−5.00000	−0.286299
$306$	0	0
$307$	−30.0000	−1.71219	−0.856095	−	0.516818i	$-0.827116\pi$
$307$	−30.0000	−1.71219	−0.856095	+	0.516818i	$0.827116\pi$
$308$	5.00000	0.284901
$309$	0	0
$310$	3.00000	0.170389
$311$	25.0000	1.41762	0.708810	−	0.705399i	$-0.249232\pi$
$311$	25.0000	1.41762	0.708810	+	0.705399i	$0.249232\pi$
$312$	0	0
$313$	−18.0000	−1.01742	−0.508710	−	0.860938i	$-0.669877\pi$
$313$	−18.0000	−1.01742	−0.508710	+	0.860938i	$0.669877\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	−12.0000	−0.675053
$317$	7.00000	0.393159	0.196580	−	0.980488i	$-0.437017\pi$
$317$	7.00000	0.393159	0.196580	+	0.980488i	$0.437017\pi$
$318$	0	0
$319$	−45.0000	−2.51952
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	−6.00000	−0.333849
$324$	0	0
$325$	2.00000	0.110940
$326$	1.00000	0.0553849
$327$	0	0
$328$	−9.00000	−0.496942
$329$	0	0
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−2.00000	−0.109764
$333$	0	0
$334$	−20.0000	−1.09435
$335$	16.0000	0.874173
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	−1.00000	−0.0542326
$341$	15.0000	0.812296
$342$	0	0
$343$	−13.0000	−0.701934
$344$	1.00000	0.0539164
$345$	0	0
$346$	7.00000	0.376322
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	5.00000	0.266501
$353$	−1.00000	−0.0532246	−0.0266123	−	0.999646i	$-0.508472\pi$
$353$	−1.00000	−0.0532246	−0.0266123	+	0.999646i	$0.508472\pi$
$354$	0	0
$355$	0	0
$356$	2.00000	0.106000
$357$	0	0
$358$	16.0000	0.845626
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	14.0000	0.735824
$363$	0	0
$364$	2.00000	0.104828
$365$	12.0000	0.628109
$366$	0	0
$367$	−23.0000	−1.20059	−0.600295	−	0.799779i	$-0.704950\pi$
$367$	−23.0000	−1.20059	−0.600295	+	0.799779i	$0.704950\pi$
$368$	0	0
$369$	0	0
$370$	−1.00000	−0.0519875
$371$	−9.00000	−0.467257
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−5.00000	−0.258544
$375$	0	0
$376$	0	0
$377$	−18.0000	−0.927047
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	−11.0000	−0.562809
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	5.00000	0.254824
$386$	−2.00000	−0.101797
$387$	0	0
$388$	17.0000	0.863044
$389$	−19.0000	−0.963338	−0.481669	−	0.876353i	$-0.659969\pi$
$389$	−19.0000	−0.963338	−0.481669	+	0.876353i	$0.659969\pi$
$390$	0	0
$391$	0	0
$392$	−6.00000	−0.303046
$393$	0	0
$394$	−14.0000	−0.705310
$395$	−12.0000	−0.603786
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	−12.0000	−0.601506
$399$	0	0
$400$	1.00000	0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	12.0000	0.597022
$405$	0	0
$406$	−9.00000	−0.446663
$407$	−5.00000	−0.247841
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	−9.00000	−0.444478
$411$	0	0
$412$	−4.00000	−0.197066
$413$	10.0000	0.492068
$414$	0	0
$415$	−2.00000	−0.0981761
$416$	2.00000	0.0980581
$417$	0	0
$418$	30.0000	1.46735
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	−5.00000	−0.243396
$423$	0	0
$424$	−9.00000	−0.437079
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−5.00000	−0.241967
$428$	−18.0000	−0.870063
$429$	0	0
$430$	1.00000	0.0482243
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	0	0
$433$	36.0000	1.73005	0.865025	−	0.501729i	$-0.167303\pi$
$433$	36.0000	1.73005	0.865025	+	0.501729i	$0.167303\pi$
$434$	3.00000	0.144005
$435$	0	0
$436$	−5.00000	−0.239457
$437$	0	0
$438$	0	0
$439$	−27.0000	−1.28864	−0.644320	−	0.764756i	$-0.722859\pi$
$439$	−27.0000	−1.28864	−0.644320	+	0.764756i	$0.722859\pi$
$440$	5.00000	0.238366
$441$	0	0
$442$	−2.00000	−0.0951303
$443$	−2.00000	−0.0950229	−0.0475114	−	0.998871i	$-0.515129\pi$
$443$	−2.00000	−0.0950229	−0.0475114	+	0.998871i	$0.515129\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	−5.00000	−0.236757
$447$	0	0
$448$	1.00000	0.0472456
$449$	−24.0000	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$450$	0	0
$451$	−45.0000	−2.11897
$452$	13.0000	0.611469
$453$	0	0
$454$	7.00000	0.328526
$455$	2.00000	0.0937614
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−35.0000	−1.63011	−0.815056	−	0.579382i	$-0.803294\pi$
$461$	−35.0000	−1.63011	−0.815056	+	0.579382i	$0.803294\pi$
$462$	0	0
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	8.00000	0.370593
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	0	0
$472$	10.0000	0.460287
$473$	5.00000	0.229900
$474$	0	0
$475$	6.00000	0.275299
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	21.0000	0.960518
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	−18.0000	−0.819878
$483$	0	0
$484$	14.0000	0.636364
$485$	17.0000	0.771930
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−5.00000	−0.226339
$489$	0	0
$490$	−6.00000	−0.271052
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	12.0000	0.539906
$495$	0	0
$496$	3.00000	0.134704
$497$	0	0
$498$	0	0
$499$	−44.0000	−1.96971	−0.984855	−	0.173379i	$-0.944532\pi$
$499$	−44.0000	−1.96971	−0.984855	+	0.173379i	$0.944532\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	24.0000	1.07117
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	1.00000	0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	−4.00000	−0.176261
$516$	0	0
$517$	0	0
$518$	−1.00000	−0.0439375
$519$	0	0
$520$	2.00000	0.0877058
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−10.0000	−0.436852
$525$	0	0
$526$	15.0000	0.654031
$527$	−3.00000	−0.130682
$528$	0	0
$529$	−23.0000	−1.00000
$530$	−9.00000	−0.390935
$531$	0	0
$532$	6.00000	0.260133
$533$	−18.0000	−0.779667
$534$	0	0
$535$	−18.0000	−0.778208
$536$	16.0000	0.691095
$537$	0	0
$538$	−24.0000	−1.03471
$539$	−30.0000	−1.29219
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	−5.00000	−0.214176
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	14.0000	0.598050
$549$	0	0
$550$	5.00000	0.213201
$551$	−54.0000	−2.30048
$552$	0	0
$553$	−12.0000	−0.510292
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−19.0000	−0.805779
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	1.00000	0.0422577
$561$	0	0
$562$	2.00000	0.0843649
$563$	−13.0000	−0.547885	−0.273942	−	0.961746i	$-0.588328\pi$
$563$	−13.0000	−0.547885	−0.273942	+	0.961746i	$0.588328\pi$
$564$	0	0
$565$	13.0000	0.546914
$566$	−28.0000	−1.17693
$567$	0	0
$568$	0	0
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	0	0
$571$	35.0000	1.46470	0.732352	−	0.680926i	$-0.238422\pi$
$571$	35.0000	1.46470	0.732352	+	0.680926i	$0.238422\pi$
$572$	10.0000	0.418121
$573$	0	0
$574$	−9.00000	−0.375653
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−16.0000	−0.665512
$579$	0	0
$580$	−9.00000	−0.373705
$581$	−2.00000	−0.0829740
$582$	0	0
$583$	−45.0000	−1.86371
$584$	12.0000	0.496564
$585$	0	0
$586$	−19.0000	−0.784883
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	0	0
$589$	18.0000	0.741677
$590$	10.0000	0.411693
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	−14.0000	−0.573462
$597$	0	0
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	1.00000	0.0407570
$603$	0	0
$604$	18.0000	0.732410
$605$	14.0000	0.569181
$606$	0	0
$607$	6.00000	0.243532	0.121766	−	0.992559i	$-0.461144\pi$
$607$	6.00000	0.243532	0.121766	+	0.992559i	$0.461144\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	−5.00000	−0.202444
$611$	0	0
$612$	0	0
$613$	−23.0000	−0.928961	−0.464481	−	0.885583i	$-0.653759\pi$
$613$	−23.0000	−0.928961	−0.464481	+	0.885583i	$0.653759\pi$
$614$	−30.0000	−1.21070
$615$	0	0
$616$	5.00000	0.201456
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	0	0
$619$	−11.0000	−0.442127	−0.221064	−	0.975259i	$-0.570953\pi$
$619$	−11.0000	−0.442127	−0.221064	+	0.975259i	$0.570953\pi$
$620$	3.00000	0.120483
$621$	0	0
$622$	25.0000	1.00241
$623$	2.00000	0.0801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	−18.0000	−0.719425
$627$	0	0
$628$	7.00000	0.279330
$629$	1.00000	0.0398726
$630$	0	0
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	−12.0000	−0.477334
$633$	0	0
$634$	7.00000	0.278006
$635$	0	0
$636$	0	0
$637$	−12.0000	−0.475457
$638$	−45.0000	−1.78157
$639$	0	0
$640$	1.00000	0.0395285
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	0	0
$645$	0	0
$646$	−6.00000	−0.236067
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	50.0000	1.96267
$650$	2.00000	0.0784465
$651$	0	0
$652$	1.00000	0.0391630
$653$	12.0000	0.469596	0.234798	−	0.972044i	$-0.424557\pi$
$653$	12.0000	0.469596	0.234798	+	0.972044i	$0.424557\pi$
$654$	0	0
$655$	−10.0000	−0.390732
$656$	−9.00000	−0.351391
$657$	0	0
$658$	0	0
$659$	32.0000	1.24654	0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000	1.24654	0.623272	+	0.782006i	$0.285803\pi$
$660$	0	0
$661$	−11.0000	−0.427850	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−11.0000	−0.427850	−0.213925	+	0.976850i	$0.568625\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	−2.00000	−0.0776151
$665$	6.00000	0.232670
$666$	0	0
$667$	0	0
$668$	−20.0000	−0.773823
$669$	0	0
$670$	16.0000	0.618134
$671$	−25.0000	−0.965114
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	6.00000	0.231111
$675$	0	0
$676$	−9.00000	−0.346154
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	17.0000	0.652400
$680$	−1.00000	−0.0383482
$681$	0	0
$682$	15.0000	0.574380
$683$	−33.0000	−1.26271	−0.631355	−	0.775494i	$-0.717501\pi$
$683$	−33.0000	−1.26271	−0.631355	+	0.775494i	$0.717501\pi$
$684$	0	0
$685$	14.0000	0.534913
$686$	−13.0000	−0.496342
$687$	0	0
$688$	1.00000	0.0381246
$689$	−18.0000	−0.685745
$690$	0	0
$691$	−1.00000	−0.0380418	−0.0190209	−	0.999819i	$-0.506055\pi$
$691$	−1.00000	−0.0380418	−0.0190209	+	0.999819i	$0.506055\pi$
$692$	7.00000	0.266100
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−19.0000	−0.720711
$696$	0	0
$697$	9.00000	0.340899
$698$	0	0
$699$	0	0
$700$	1.00000	0.0377964
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	5.00000	0.188445
$705$	0	0
$706$	−1.00000	−0.0376355
$707$	12.0000	0.451306
$708$	0	0
$709$	−17.0000	−0.638448	−0.319224	−	0.947679i	$-0.603422\pi$
$709$	−17.0000	−0.638448	−0.319224	+	0.947679i	$0.603422\pi$
$710$	0	0
$711$	0	0
$712$	2.00000	0.0749532
$713$	0	0
$714$	0	0
$715$	10.0000	0.373979
$716$	16.0000	0.597948
$717$	0	0
$718$	−6.00000	−0.223918
$719$	−10.0000	−0.372937	−0.186469	−	0.982461i	$-0.559704\pi$
$719$	−10.0000	−0.372937	−0.186469	+	0.982461i	$0.559704\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	17.0000	0.632674
$723$	0	0
$724$	14.0000	0.520306
$725$	−9.00000	−0.334252
$726$	0	0
$727$	−18.0000	−0.667583	−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000	−0.667583	−0.333792	+	0.942647i	$0.608328\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	12.0000	0.444140
$731$	−1.00000	−0.0369863
$732$	0	0
$733$	−47.0000	−1.73598	−0.867992	−	0.496578i	$-0.834590\pi$
$733$	−47.0000	−1.73598	−0.867992	+	0.496578i	$0.834590\pi$
$734$	−23.0000	−0.848945
$735$	0	0
$736$	0	0
$737$	80.0000	2.94684
$738$	0	0
$739$	−7.00000	−0.257499	−0.128750	−	0.991677i	$-0.541096\pi$
$739$	−7.00000	−0.257499	−0.128750	+	0.991677i	$0.541096\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	−9.00000	−0.330400
$743$	−1.00000	−0.0366864	−0.0183432	−	0.999832i	$-0.505839\pi$
$743$	−1.00000	−0.0366864	−0.0183432	+	0.999832i	$0.505839\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	26.0000	0.951928
$747$	0	0
$748$	−5.00000	−0.182818
$749$	−18.0000	−0.657706
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	0	0
$754$	−18.0000	−0.655521
$755$	18.0000	0.655087
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	6.00000	0.217643
$761$	−33.0000	−1.19625	−0.598125	−	0.801403i	$-0.704087\pi$
$761$	−33.0000	−1.19625	−0.598125	+	0.801403i	$0.704087\pi$
$762$	0	0
$763$	−5.00000	−0.181012
$764$	−11.0000	−0.397966
$765$	0	0
$766$	0	0
$767$	20.0000	0.722158
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	5.00000	0.180187
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	−27.0000	−0.971123	−0.485561	−	0.874203i	$-0.661385\pi$
$773$	−27.0000	−0.971123	−0.485561	+	0.874203i	$0.661385\pi$
$774$	0	0
$775$	3.00000	0.107763
$776$	17.0000	0.610264
$777$	0	0
$778$	−19.0000	−0.681183
$779$	−54.0000	−1.93475
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−6.00000	−0.214286
$785$	7.00000	0.249841
$786$	0	0
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	−14.0000	−0.498729
$789$	0	0
$790$	−12.0000	−0.426941
$791$	13.0000	0.462227
$792$	0	0
$793$	−10.0000	−0.355110
$794$	18.0000	0.638796
$795$	0	0
$796$	−12.0000	−0.425329
$797$	−40.0000	−1.41687	−0.708436	−	0.705775i	$-0.750599\pi$
$797$	−40.0000	−1.41687	−0.708436	+	0.705775i	$0.750599\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	6.00000	0.211867
$803$	60.0000	2.11735
$804$	0	0
$805$	0	0
$806$	6.00000	0.211341
$807$	0	0
$808$	12.0000	0.422159
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	−9.00000	−0.315838
$813$	0	0
$814$	−5.00000	−0.175250
$815$	1.00000	0.0350285
$816$	0	0
$817$	6.00000	0.209913
$818$	6.00000	0.209785
$819$	0	0
$820$	−9.00000	−0.314294
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	10.0000	0.347945
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	0	0
$829$	−17.0000	−0.590434	−0.295217	−	0.955430i	$-0.595392\pi$
$829$	−17.0000	−0.590434	−0.295217	+	0.955430i	$0.595392\pi$
$830$	−2.00000	−0.0694210
$831$	0	0
$832$	2.00000	0.0693375
$833$	6.00000	0.207888
$834$	0	0
$835$	−20.0000	−0.692129
$836$	30.0000	1.03757
$837$	0	0
$838$	−20.0000	−0.690889
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	30.0000	1.03387
$843$	0	0
$844$	−5.00000	−0.172107
$845$	−9.00000	−0.309609
$846$	0	0
$847$	14.0000	0.481046
$848$	−9.00000	−0.309061
$849$	0	0
$850$	−1.00000	−0.0342997
$851$	0	0
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	−5.00000	−0.171096
$855$	0	0
$856$	−18.0000	−0.615227
$857$	33.0000	1.12726	0.563629	−	0.826028i	$-0.309405\pi$
$857$	33.0000	1.12726	0.563629	+	0.826028i	$0.309405\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	1.00000	0.0340997
$861$	0	0
$862$	−15.0000	−0.510902
$863$	−39.0000	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$863$	−39.0000	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$864$	0	0
$865$	7.00000	0.238007
$866$	36.0000	1.22333
$867$	0	0
$868$	3.00000	0.101827
$869$	−60.0000	−2.03536
$870$	0	0
$871$	32.0000	1.08428
$872$	−5.00000	−0.169321
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	47.0000	1.58708	0.793539	−	0.608520i	$-0.208236\pi$
$877$	47.0000	1.58708	0.793539	+	0.608520i	$0.208236\pi$
$878$	−27.0000	−0.911206
$879$	0	0
$880$	5.00000	0.168550
$881$	−41.0000	−1.38133	−0.690663	−	0.723177i	$-0.742681\pi$
$881$	−41.0000	−1.38133	−0.690663	+	0.723177i	$0.742681\pi$
$882$	0	0
$883$	−9.00000	−0.302874	−0.151437	−	0.988467i	$-0.548390\pi$
$883$	−9.00000	−0.302874	−0.151437	+	0.988467i	$0.548390\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	−2.00000	−0.0671913
$887$	35.0000	1.17518	0.587592	−	0.809157i	$-0.300076\pi$
$887$	35.0000	1.17518	0.587592	+	0.809157i	$0.300076\pi$
$888$	0	0
$889$	0	0
$890$	2.00000	0.0670402
$891$	0	0
$892$	−5.00000	−0.167412
$893$	0	0
$894$	0	0
$895$	16.0000	0.534821
$896$	1.00000	0.0334077
$897$	0	0
$898$	−24.0000	−0.800890
$899$	−27.0000	−0.900500
$900$	0	0
$901$	9.00000	0.299833
$902$	−45.0000	−1.49834
$903$	0	0
$904$	13.0000	0.432374
$905$	14.0000	0.465376
$906$	0	0
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	7.00000	0.232303
$909$	0	0
$910$	2.00000	0.0662994
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	−10.0000	−0.330952
$914$	1.00000	0.0330771
$915$	0	0
$916$	0	0
$917$	−10.0000	−0.330229
$918$	0	0
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	0	0
$921$	0	0
$922$	−35.0000	−1.15266
$923$	0	0
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	26.0000	0.854413
$927$	0	0
$928$	−9.00000	−0.295439
$929$	1.00000	0.0328089	0.0164045	−	0.999865i	$-0.494778\pi$
$929$	1.00000	0.0328089	0.0164045	+	0.999865i	$0.494778\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	8.00000	0.262049
$933$	0	0
$934$	21.0000	0.687141
$935$	−5.00000	−0.163517
$936$	0	0
$937$	56.0000	1.82944	0.914720	−	0.404088i	$-0.132411\pi$
$937$	56.0000	1.82944	0.914720	+	0.404088i	$0.132411\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	0	0
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	0	0
$943$	0	0
$944$	10.0000	0.325472
$945$	0	0
$946$	5.00000	0.162564
$947$	−1.00000	−0.0324956	−0.0162478	−	0.999868i	$-0.505172\pi$
$947$	−1.00000	−0.0324956	−0.0162478	+	0.999868i	$0.505172\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	6.00000	0.194666
$951$	0	0
$952$	−1.00000	−0.0324102
$953$	4.00000	0.129573	0.0647864	−	0.997899i	$-0.479363\pi$
$953$	4.00000	0.129573	0.0647864	+	0.997899i	$0.479363\pi$
$954$	0	0
$955$	−11.0000	−0.355952
$956$	21.0000	0.679189
$957$	0	0
$958$	24.0000	0.775405
$959$	14.0000	0.452084
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−2.00000	−0.0644826
$963$	0	0
$964$	−18.0000	−0.579741
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	36.0000	1.15768	0.578841	−	0.815440i	$-0.303505\pi$
$967$	36.0000	1.15768	0.578841	+	0.815440i	$0.303505\pi$
$968$	14.0000	0.449977
$969$	0	0
$970$	17.0000	0.545837
$971$	−29.0000	−0.930654	−0.465327	−	0.885139i	$-0.654063\pi$
$971$	−29.0000	−0.930654	−0.465327	+	0.885139i	$0.654063\pi$
$972$	0	0
$973$	−19.0000	−0.609112
$974$	2.00000	0.0640841
$975$	0	0
$976$	−5.00000	−0.160046
$977$	27.0000	0.863807	0.431903	−	0.901920i	$-0.357842\pi$
$977$	27.0000	0.863807	0.431903	+	0.901920i	$0.357842\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	−6.00000	−0.191663
$981$	0	0
$982$	36.0000	1.14881
$983$	49.0000	1.56286	0.781429	−	0.623995i	$-0.214491\pi$
$983$	49.0000	1.56286	0.781429	+	0.623995i	$0.214491\pi$
$984$	0	0
$985$	−14.0000	−0.446077
$986$	9.00000	0.286618
$987$	0	0
$988$	12.0000	0.381771
$989$	0	0
$990$	0	0
$991$	−31.0000	−0.984747	−0.492374	−	0.870384i	$-0.663871\pi$
$991$	−31.0000	−0.984747	−0.492374	+	0.870384i	$0.663871\pi$
$992$	3.00000	0.0952501
$993$	0	0
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	−44.0000	−1.39280
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.y.1.1		1
3.2	odd	2		1110.2.a.f.1.1	✓	1
12.11	even	2		8880.2.a.d.1.1		1
15.14	odd	2		5550.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.f.1.1	✓	1	3.2	odd	2
3330.2.a.y.1.1		1	1.1	even	1	trivial
5550.2.a.y.1.1		1	15.14	odd	2
8880.2.a.d.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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