LMFDB - Embedded newform 6031.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6031,2,Mod(1,6031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6031, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6031 = 37 \cdot 163 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6031.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:	$48.1577774590$
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	$\chi$	$=$	6031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -2.00000 q^{4} +4.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -2.00000 q^{4} +4.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} +1.00000 q^{11} -6.00000 q^{12} +4.00000 q^{13} +12.0000 q^{15} +4.00000 q^{16} -2.00000 q^{17} -8.00000 q^{20} +3.00000 q^{21} -8.00000 q^{23} +11.0000 q^{25} +9.00000 q^{27} -2.00000 q^{28} +2.00000 q^{31} +3.00000 q^{33} +4.00000 q^{35} -12.0000 q^{36} +1.00000 q^{37} +12.0000 q^{39} +3.00000 q^{41} +4.00000 q^{43} -2.00000 q^{44} +24.0000 q^{45} -9.00000 q^{47} +12.0000 q^{48} -6.00000 q^{49} -6.00000 q^{51} -8.00000 q^{52} +9.00000 q^{53} +4.00000 q^{55} -6.00000 q^{59} -24.0000 q^{60} -6.00000 q^{61} +6.00000 q^{63} -8.00000 q^{64} +16.0000 q^{65} +8.00000 q^{67} +4.00000 q^{68} -24.0000 q^{69} +13.0000 q^{71} +5.00000 q^{73} +33.0000 q^{75} +1.00000 q^{77} -14.0000 q^{79} +16.0000 q^{80} +9.00000 q^{81} +9.00000 q^{83} -6.00000 q^{84} -8.00000 q^{85} -12.0000 q^{89} +4.00000 q^{91} +16.0000 q^{92} +6.00000 q^{93} +12.0000 q^{97} +6.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	−2.00000	−1.00000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−6.00000	−1.73205
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	12.0000	3.09839
$16$	4.00000	1.00000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−8.00000	−1.78885
$21$	3.00000	0.654654
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	9.00000	1.73205
$28$	−2.00000	−0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	4.00000	0.676123
$36$	−12.0000	−2.00000
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	12.0000	1.92154
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−2.00000	−0.301511
$45$	24.0000	3.57771
$46$	0	0
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	12.0000	1.73205
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−6.00000	−0.840168
$52$	−8.00000	−1.10940
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−24.0000	−3.09839
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	−8.00000	−1.00000
$65$	16.0000	1.98456
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	4.00000	0.485071
$69$	−24.0000	−2.88926
$70$	0	0
$71$	13.0000	1.54282	0.771408	−	0.636341i	$-0.219553\pi$
$71$	13.0000	1.54282	0.771408	+	0.636341i	$0.219553\pi$
$72$	0	0
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	0	0
$75$	33.0000	3.81051
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	16.0000	1.78885
$81$	9.00000	1.00000
$82$	0	0
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	−6.00000	−0.654654
$85$	−8.00000	−0.867722
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	16.0000	1.66812
$93$	6.00000	0.622171
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	−22.0000	−2.20000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	12.0000	1.17108
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−18.0000	−1.73205
$109$	−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
$112$	4.00000	0.377964
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	−32.0000	−2.98402
$116$	0	0
$117$	24.0000	2.21880
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	9.00000	0.811503
$124$	−4.00000	−0.359211
$125$	24.0000	2.14663
$126$	0	0
$127$	15.0000	1.33103	0.665517	−	0.746382i	$-0.268211\pi$
$127$	15.0000	1.33103	0.665517	+	0.746382i	$0.268211\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	0	0
$135$	36.0000	3.09839
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	−8.00000	−0.676123
$141$	−27.0000	−2.27381
$142$	0	0
$143$	4.00000	0.334497
$144$	24.0000	2.00000
$145$	0	0
$146$	0	0
$147$	−18.0000	−1.48461
$148$	−2.00000	−0.164399
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	−12.0000	−0.970143
$154$	0	0
$155$	8.00000	0.642575
$156$	−24.0000	−1.92154
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	0	0
$159$	27.0000	2.14124
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−1.00000	−0.0783260
$163$	−1.00000	−0.0783260
$164$	−6.00000	−0.468521
$165$	12.0000	0.934199
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	0	0
$175$	11.0000	0.831522
$176$	4.00000	0.301511
$177$	−18.0000	−1.35296
$178$	0	0
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	−48.0000	−3.57771
$181$	−9.00000	−0.668965	−0.334482	−	0.942402i	$-0.608561\pi$
$181$	−9.00000	−0.668965	−0.334482	+	0.942402i	$0.608561\pi$
$182$	0	0
$183$	−18.0000	−1.33060
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	−2.00000	−0.146254
$188$	18.0000	1.31278
$189$	9.00000	0.654654
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−24.0000	−1.73205
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	48.0000	3.43735
$196$	12.0000	0.857143
$197$	19.0000	1.35369	0.676847	−	0.736124i	$-0.263346\pi$
$197$	19.0000	1.35369	0.676847	+	0.736124i	$0.263346\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	24.0000	1.69283
$202$	0	0
$203$	0	0
$204$	12.0000	0.840168
$205$	12.0000	0.838116
$206$	0	0
$207$	−48.0000	−3.33623
$208$	16.0000	1.10940
$209$	0	0
$210$	0	0
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	−18.0000	−1.23625
$213$	39.0000	2.67224
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	15.0000	1.01361
$220$	−8.00000	−0.539360
$221$	−8.00000	−0.538138
$222$	0	0
$223$	7.00000	0.468755	0.234377	−	0.972146i	$-0.424695\pi$
$223$	7.00000	0.468755	0.234377	+	0.972146i	$0.424695\pi$
$224$	0	0
$225$	66.0000	4.40000
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	9.00000	0.594737	0.297368	−	0.954763i	$-0.403891\pi$
$229$	9.00000	0.594737	0.297368	+	0.954763i	$0.403891\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	−36.0000	−2.34838
$236$	12.0000	0.781133
$237$	−42.0000	−2.72819
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	48.0000	3.09839
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	12.0000	0.768221
$245$	−24.0000	−1.53330
$246$	0	0
$247$	0	0
$248$	0	0
$249$	27.0000	1.71106
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	−12.0000	−0.755929
$253$	−8.00000	−0.502956
$254$	0	0
$255$	−24.0000	−1.50294
$256$	16.0000	1.00000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	−32.0000	−1.98456
$261$	0	0
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	0	0
$267$	−36.0000	−2.20316
$268$	−16.0000	−0.977356
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−17.0000	−1.03268	−0.516338	−	0.856385i	$-0.672705\pi$
$271$	−17.0000	−1.03268	−0.516338	+	0.856385i	$0.672705\pi$
$272$	−8.00000	−0.485071
$273$	12.0000	0.726273
$274$	0	0
$275$	11.0000	0.663325
$276$	48.0000	2.88926
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	12.0000	0.718421
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	−26.0000	−1.54282
$285$	0	0
$286$	0	0
$287$	3.00000	0.177084
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	36.0000	2.11036
$292$	−10.0000	−0.585206
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	0	0
$297$	9.00000	0.522233
$298$	0	0
$299$	−32.0000	−1.85061
$300$	−66.0000	−3.81051
$301$	4.00000	0.230556
$302$	0	0
$303$	−9.00000	−0.517036
$304$	0	0
$305$	−24.0000	−1.37424
$306$	0	0
$307$	31.0000	1.76926	0.884632	−	0.466290i	$-0.154410\pi$
$307$	31.0000	1.76926	0.884632	+	0.466290i	$0.154410\pi$
$308$	−2.00000	−0.113961
$309$	−42.0000	−2.38930
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	24.0000	1.35225
$316$	28.0000	1.57512
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	0	0
$320$	−32.0000	−1.78885
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	−18.0000	−1.00000
$325$	44.0000	2.44068
$326$	0	0
$327$	−30.0000	−1.65900
$328$	0	0
$329$	−9.00000	−0.496186
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	−18.0000	−0.987878
$333$	6.00000	0.328798
$334$	0	0
$335$	32.0000	1.74835
$336$	12.0000	0.654654
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	0	0
$339$	24.0000	1.30350
$340$	16.0000	0.867722
$341$	2.00000	0.108306
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−96.0000	−5.16847
$346$	0	0
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	36.0000	1.92154
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	52.0000	2.75987
$356$	24.0000	1.27200
$357$	−6.00000	−0.317554
$358$	0	0
$359$	−3.00000	−0.158334	−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000	−0.158334	−0.0791670	+	0.996861i	$0.525226\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−30.0000	−1.57459
$364$	−8.00000	−0.419314
$365$	20.0000	1.04685
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−32.0000	−1.66812
$369$	18.0000	0.937043
$370$	0	0
$371$	9.00000	0.467257
$372$	−12.0000	−0.622171
$373$	13.0000	0.673114	0.336557	−	0.941663i	$-0.390737\pi$
$373$	13.0000	0.673114	0.336557	+	0.941663i	$0.390737\pi$
$374$	0	0
$375$	72.0000	3.71806
$376$	0	0
$377$	0	0
$378$	0	0
$379$	3.00000	0.154100	0.0770498	−	0.997027i	$-0.475450\pi$
$379$	3.00000	0.154100	0.0770498	+	0.997027i	$0.475450\pi$
$380$	0	0
$381$	45.0000	2.30542
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	24.0000	1.21999
$388$	−24.0000	−1.21842
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	−48.0000	−2.42128
$394$	0	0
$395$	−56.0000	−2.81767
$396$	−12.0000	−0.603023
$397$	23.0000	1.15434	0.577168	−	0.816625i	$-0.304158\pi$
$397$	23.0000	1.15434	0.577168	+	0.816625i	$0.304158\pi$
$398$	0	0
$399$	0	0
$400$	44.0000	2.20000
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	6.00000	0.298511
$405$	36.0000	1.78885
$406$	0	0
$407$	1.00000	0.0495682
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	28.0000	1.37946
$413$	−6.00000	−0.295241
$414$	0	0
$415$	36.0000	1.76717
$416$	0	0
$417$	−36.0000	−1.76293
$418$	0	0
$419$	−29.0000	−1.41674	−0.708371	−	0.705840i	$-0.750570\pi$
$419$	−29.0000	−1.41674	−0.708371	+	0.705840i	$0.750570\pi$
$420$	−24.0000	−1.17108
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	0	0
$423$	−54.0000	−2.62557
$424$	0	0
$425$	−22.0000	−1.06716
$426$	0	0
$427$	−6.00000	−0.290360
$428$	−8.00000	−0.386695
$429$	12.0000	0.579365
$430$	0	0
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	36.0000	1.73205
$433$	35.0000	1.68199	0.840996	−	0.541041i	$-0.181970\pi$
$433$	35.0000	1.68199	0.840996	+	0.541041i	$0.181970\pi$
$434$	0	0
$435$	0	0
$436$	20.0000	0.957826
$437$	0	0
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	−1.00000	−0.0475114	−0.0237557	−	0.999718i	$-0.507562\pi$
$443$	−1.00000	−0.0475114	−0.0237557	+	0.999718i	$0.507562\pi$
$444$	−6.00000	−0.284747
$445$	−48.0000	−2.27542
$446$	0	0
$447$	−33.0000	−1.56085
$448$	−8.00000	−0.377964
$449$	−28.0000	−1.32140	−0.660701	−	0.750649i	$-0.729741\pi$
$449$	−28.0000	−1.32140	−0.660701	+	0.750649i	$0.729741\pi$
$450$	0	0
$451$	3.00000	0.141264
$452$	−16.0000	−0.752577
$453$	−24.0000	−1.12762
$454$	0	0
$455$	16.0000	0.750092
$456$	0	0
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	0	0
$459$	−18.0000	−0.840168
$460$	64.0000	2.98402
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	−10.0000	−0.464739	−0.232370	−	0.972628i	$-0.574648\pi$
$463$	−10.0000	−0.464739	−0.232370	+	0.972628i	$0.574648\pi$
$464$	0	0
$465$	24.0000	1.11297
$466$	0	0
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	−48.0000	−2.21880
$469$	8.00000	0.369406
$470$	0	0
$471$	−21.0000	−0.967629
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	0	0
$476$	4.00000	0.183340
$477$	54.0000	2.47249
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	−24.0000	−1.09204
$484$	20.0000	0.909091
$485$	48.0000	2.17957
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	−3.00000	−0.135665
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	−18.0000	−0.811503
$493$	0	0
$494$	0	0
$495$	24.0000	1.07872
$496$	8.00000	0.359211
$497$	13.0000	0.583130
$498$	0	0
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	−48.0000	−2.14663
$501$	−36.0000	−1.60836
$502$	0	0
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	9.00000	0.399704
$508$	−30.0000	−1.33103
$509$	−5.00000	−0.221621	−0.110811	−	0.993842i	$-0.535345\pi$
$509$	−5.00000	−0.221621	−0.110811	+	0.993842i	$0.535345\pi$
$510$	0	0
$511$	5.00000	0.221187
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−56.0000	−2.46765
$516$	−24.0000	−1.05654
$517$	−9.00000	−0.395820
$518$	0	0
$519$	27.0000	1.18517
$520$	0	0
$521$	21.0000	0.920027	0.460013	−	0.887912i	$-0.347845\pi$
$521$	21.0000	0.920027	0.460013	+	0.887912i	$0.347845\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	32.0000	1.39793
$525$	33.0000	1.44024
$526$	0	0
$527$	−4.00000	−0.174243
$528$	12.0000	0.522233
$529$	41.0000	1.78261
$530$	0	0
$531$	−36.0000	−1.56227
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	16.0000	0.691740
$536$	0	0
$537$	−6.00000	−0.258919
$538$	0	0
$539$	−6.00000	−0.258438
$540$	−72.0000	−3.09839
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	−27.0000	−1.15868
$544$	0	0
$545$	−40.0000	−1.71341
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−12.0000	−0.512615
$549$	−36.0000	−1.53644
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−14.0000	−0.595341
$554$	0	0
$555$	12.0000	0.509372
$556$	24.0000	1.01783
$557$	−8.00000	−0.338971	−0.169485	−	0.985533i	$-0.554211\pi$
$557$	−8.00000	−0.338971	−0.169485	+	0.985533i	$0.554211\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	16.0000	0.676123
$561$	−6.00000	−0.253320
$562$	0	0
$563$	2.00000	0.0842900	0.0421450	−	0.999112i	$-0.486581\pi$
$563$	2.00000	0.0842900	0.0421450	+	0.999112i	$0.486581\pi$
$564$	54.0000	2.27381
$565$	32.0000	1.34625
$566$	0	0
$567$	9.00000	0.377964
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	−8.00000	−0.334497
$573$	−18.0000	−0.751961
$574$	0	0
$575$	−88.0000	−3.66985
$576$	−48.0000	−2.00000
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	−18.0000	−0.748054
$580$	0	0
$581$	9.00000	0.373383
$582$	0	0
$583$	9.00000	0.372742
$584$	0	0
$585$	96.0000	3.96911
$586$	0	0
$587$	40.0000	1.65098	0.825488	−	0.564419i	$-0.190900\pi$
$587$	40.0000	1.65098	0.825488	+	0.564419i	$0.190900\pi$
$588$	36.0000	1.48461
$589$	0	0
$590$	0	0
$591$	57.0000	2.34467
$592$	4.00000	0.164399
$593$	−13.0000	−0.533846	−0.266923	−	0.963718i	$-0.586007\pi$
$593$	−13.0000	−0.533846	−0.266923	+	0.963718i	$0.586007\pi$
$594$	0	0
$595$	−8.00000	−0.327968
$596$	22.0000	0.901155
$597$	12.0000	0.491127
$598$	0	0
$599$	−9.00000	−0.367730	−0.183865	−	0.982952i	$-0.558861\pi$
$599$	−9.00000	−0.367730	−0.183865	+	0.982952i	$0.558861\pi$
$600$	0	0
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	0	0
$603$	48.0000	1.95471
$604$	16.0000	0.651031
$605$	−40.0000	−1.62623
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−36.0000	−1.45640
$612$	24.0000	0.970143
$613$	25.0000	1.00974	0.504870	−	0.863195i	$-0.331540\pi$
$613$	25.0000	1.00974	0.504870	+	0.863195i	$0.331540\pi$
$614$	0	0
$615$	36.0000	1.45166
$616$	0	0
$617$	15.0000	0.603877	0.301939	−	0.953327i	$-0.402366\pi$
$617$	15.0000	0.603877	0.301939	+	0.953327i	$0.402366\pi$
$618$	0	0
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	−16.0000	−0.642575
$621$	−72.0000	−2.88926
$622$	0	0
$623$	−12.0000	−0.480770
$624$	48.0000	1.92154
$625$	41.0000	1.64000
$626$	0	0
$627$	0	0
$628$	14.0000	0.558661
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	3.00000	0.119239
$634$	0	0
$635$	60.0000	2.38103
$636$	−54.0000	−2.14124
$637$	−24.0000	−0.950915
$638$	0	0
$639$	78.0000	3.08563
$640$	0	0
$641$	−37.0000	−1.46141	−0.730706	−	0.682692i	$-0.760809\pi$
$641$	−37.0000	−1.46141	−0.730706	+	0.682692i	$0.760809\pi$
$642$	0	0
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	16.0000	0.630488
$645$	48.0000	1.89000
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	0	0
$651$	6.00000	0.235159
$652$	2.00000	0.0783260
$653$	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
$655$	−64.0000	−2.50069
$656$	12.0000	0.468521
$657$	30.0000	1.17041
$658$	0	0
$659$	−17.0000	−0.662226	−0.331113	−	0.943591i	$-0.607424\pi$
$659$	−17.0000	−0.662226	−0.331113	+	0.943591i	$0.607424\pi$
$660$	−24.0000	−0.934199
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	24.0000	0.928588
$669$	21.0000	0.811907
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	0	0
$675$	99.0000	3.81051
$676$	−6.00000	−0.230769
$677$	−43.0000	−1.65262	−0.826312	−	0.563212i	$-0.809565\pi$
$677$	−43.0000	−1.65262	−0.826312	+	0.563212i	$0.809565\pi$
$678$	0	0
$679$	12.0000	0.460518
$680$	0	0
$681$	54.0000	2.06928
$682$	0	0
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	0	0
$687$	27.0000	1.03011
$688$	16.0000	0.609994
$689$	36.0000	1.37149
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−18.0000	−0.684257
$693$	6.00000	0.227921
$694$	0	0
$695$	−48.0000	−1.82074
$696$	0	0
$697$	−6.00000	−0.227266
$698$	0	0
$699$	66.0000	2.49635
$700$	−22.0000	−0.831522
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	0	0
$704$	−8.00000	−0.301511
$705$	−108.000	−4.06752
$706$	0	0
$707$	−3.00000	−0.112827
$708$	36.0000	1.35296
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−84.0000	−3.15025
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	16.0000	0.598366
$716$	4.00000	0.149487
$717$	18.0000	0.672222
$718$	0	0
$719$	−51.0000	−1.90198	−0.950990	−	0.309223i	$-0.899931\pi$
$719$	−51.0000	−1.90198	−0.950990	+	0.309223i	$0.899931\pi$
$720$	96.0000	3.57771
$721$	−14.0000	−0.521387
$722$	0	0
$723$	−30.0000	−1.11571
$724$	18.0000	0.668965
$725$	0	0
$726$	0	0
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−8.00000	−0.295891
$732$	36.0000	1.33060
$733$	39.0000	1.44050	0.720249	−	0.693716i	$-0.244028\pi$
$733$	39.0000	1.44050	0.720249	+	0.693716i	$0.244028\pi$
$734$	0	0
$735$	−72.0000	−2.65576
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	−29.0000	−1.06678	−0.533391	−	0.845869i	$-0.679083\pi$
$739$	−29.0000	−1.06678	−0.533391	+	0.845869i	$0.679083\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	0	0
$743$	−15.0000	−0.550297	−0.275148	−	0.961402i	$-0.588727\pi$
$743$	−15.0000	−0.550297	−0.275148	+	0.961402i	$0.588727\pi$
$744$	0	0
$745$	−44.0000	−1.61204
$746$	0	0
$747$	54.0000	1.97576
$748$	4.00000	0.146254
$749$	4.00000	0.146157
$750$	0	0
$751$	15.0000	0.547358	0.273679	−	0.961821i	$-0.411759\pi$
$751$	15.0000	0.547358	0.273679	+	0.961821i	$0.411759\pi$
$752$	−36.0000	−1.31278
$753$	6.00000	0.218652
$754$	0	0
$755$	−32.0000	−1.16460
$756$	−18.0000	−0.654654
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	0	0
$759$	−24.0000	−0.871145
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	12.0000	0.434145
$765$	−48.0000	−1.73544
$766$	0	0
$767$	−24.0000	−0.866590
$768$	48.0000	1.73205
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	0	0
$771$	−24.0000	−0.864339
$772$	12.0000	0.431889
$773$	−41.0000	−1.47467	−0.737334	−	0.675529i	$-0.763915\pi$
$773$	−41.0000	−1.47467	−0.737334	+	0.675529i	$0.763915\pi$
$774$	0	0
$775$	22.0000	0.790263
$776$	0	0
$777$	3.00000	0.107624
$778$	0	0
$779$	0	0
$780$	−96.0000	−3.43735
$781$	13.0000	0.465177
$782$	0	0
$783$	0	0
$784$	−24.0000	−0.857143
$785$	−28.0000	−0.999363
$786$	0	0
$787$	35.0000	1.24762	0.623808	−	0.781578i	$-0.285585\pi$
$787$	35.0000	1.24762	0.623808	+	0.781578i	$0.285585\pi$
$788$	−38.0000	−1.35369
$789$	−27.0000	−0.961225
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	108.000	3.83037
$796$	−8.00000	−0.283552
$797$	−16.0000	−0.566749	−0.283375	−	0.959009i	$-0.591454\pi$
$797$	−16.0000	−0.566749	−0.283375	+	0.959009i	$0.591454\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	−72.0000	−2.54399
$802$	0	0
$803$	5.00000	0.176446
$804$	−48.0000	−1.69283
$805$	−32.0000	−1.12785
$806$	0	0
$807$	−54.0000	−1.90089
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	29.0000	1.01833	0.509164	−	0.860670i	$-0.329955\pi$
$811$	29.0000	1.01833	0.509164	+	0.860670i	$0.329955\pi$
$812$	0	0
$813$	−51.0000	−1.78865
$814$	0	0
$815$	−4.00000	−0.140114
$816$	−24.0000	−0.840168
$817$	0	0
$818$	0	0
$819$	24.0000	0.838628
$820$	−24.0000	−0.838116
$821$	17.0000	0.593304	0.296652	−	0.954986i	$-0.404130\pi$
$821$	17.0000	0.593304	0.296652	+	0.954986i	$0.404130\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	33.0000	1.14891
$826$	0	0
$827$	10.0000	0.347734	0.173867	−	0.984769i	$-0.444374\pi$
$827$	10.0000	0.347734	0.173867	+	0.984769i	$0.444374\pi$
$828$	96.0000	3.33623
$829$	42.0000	1.45872	0.729360	−	0.684130i	$-0.239818\pi$
$829$	42.0000	1.45872	0.729360	+	0.684130i	$0.239818\pi$
$830$	0	0
$831$	66.0000	2.28951
$832$	−32.0000	−1.10940
$833$	12.0000	0.415775
$834$	0	0
$835$	−48.0000	−1.66111
$836$	0	0
$837$	18.0000	0.622171
$838$	0	0
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	72.0000	2.47981
$844$	−2.00000	−0.0688428
$845$	12.0000	0.412813
$846$	0	0
$847$	−10.0000	−0.343604
$848$	36.0000	1.23625
$849$	−42.0000	−1.44144
$850$	0	0
$851$	−8.00000	−0.274236
$852$	−78.0000	−2.67224
$853$	56.0000	1.91740	0.958702	−	0.284413i	$-0.0917988\pi$
$853$	56.0000	1.91740	0.958702	+	0.284413i	$0.0917988\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.0000	1.09310	0.546550	−	0.837427i	$-0.315941\pi$
$857$	32.0000	1.09310	0.546550	+	0.837427i	$0.315941\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	−32.0000	−1.09119
$861$	9.00000	0.306719
$862$	0	0
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	0	0
$865$	36.0000	1.22404
$866$	0	0
$867$	−39.0000	−1.32451
$868$	−4.00000	−0.135769
$869$	−14.0000	−0.474917
$870$	0	0
$871$	32.0000	1.08428
$872$	0	0
$873$	72.0000	2.43683
$874$	0	0
$875$	24.0000	0.811348
$876$	−30.0000	−1.01361
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	0	0
$879$	−18.0000	−0.607125
$880$	16.0000	0.539360
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	−32.0000	−1.07689	−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000	−1.07689	−0.538443	+	0.842662i	$0.680987\pi$
$884$	16.0000	0.538138
$885$	−72.0000	−2.42025
$886$	0	0
$887$	47.0000	1.57811	0.789053	−	0.614325i	$-0.210572\pi$
$887$	47.0000	1.57811	0.789053	+	0.614325i	$0.210572\pi$
$888$	0	0
$889$	15.0000	0.503084
$890$	0	0
$891$	9.00000	0.301511
$892$	−14.0000	−0.468755
$893$	0	0
$894$	0	0
$895$	−8.00000	−0.267411
$896$	0	0
$897$	−96.0000	−3.20535
$898$	0	0
$899$	0	0
$900$	−132.000	−4.40000
$901$	−18.0000	−0.599667
$902$	0	0
$903$	12.0000	0.399335
$904$	0	0
$905$	−36.0000	−1.19668
$906$	0	0
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	−36.0000	−1.19470
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	0	0
$915$	−72.0000	−2.38025
$916$	−18.0000	−0.594737
$917$	−16.0000	−0.528367
$918$	0	0
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	0	0
$921$	93.0000	3.06445
$922$	0	0
$923$	52.0000	1.71160
$924$	−6.00000	−0.197386
$925$	11.0000	0.361678
$926$	0	0
$927$	−84.0000	−2.75892
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	0	0
$932$	−44.0000	−1.44127
$933$	−24.0000	−0.785725
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	−13.0000	−0.424691	−0.212346	−	0.977195i	$-0.568110\pi$
$937$	−13.0000	−0.424691	−0.212346	+	0.977195i	$0.568110\pi$
$938$	0	0
$939$	24.0000	0.783210
$940$	72.0000	2.34838
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	−24.0000	−0.781133
$945$	36.0000	1.17108
$946$	0	0
$947$	22.0000	0.714904	0.357452	−	0.933932i	$-0.383646\pi$
$947$	22.0000	0.714904	0.357452	+	0.933932i	$0.383646\pi$
$948$	84.0000	2.72819
$949$	20.0000	0.649227
$950$	0	0
$951$	54.0000	1.75107
$952$	0	0
$953$	−9.00000	−0.291539	−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000	−0.291539	−0.145769	+	0.989319i	$0.546566\pi$
$954$	0	0
$955$	−24.0000	−0.776622
$956$	−12.0000	−0.388108
$957$	0	0
$958$	0	0
$959$	6.00000	0.193750
$960$	−96.0000	−3.09839
$961$	−27.0000	−0.870968
$962$	0	0
$963$	24.0000	0.773389
$964$	20.0000	0.644157
$965$	−24.0000	−0.772587
$966$	0	0
$967$	−54.0000	−1.73652	−0.868261	−	0.496107i	$-0.834762\pi$
$967$	−54.0000	−1.73652	−0.868261	+	0.496107i	$0.834762\pi$
$968$	0	0
$969$	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	$0.438325\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	0	0
$975$	132.000	4.22738
$976$	−24.0000	−0.768221
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	48.0000	1.53330
$981$	−60.0000	−1.91565
$982$	0	0
$983$	35.0000	1.11633	0.558163	−	0.829731i	$-0.311506\pi$
$983$	35.0000	1.11633	0.558163	+	0.829731i	$0.311506\pi$
$984$	0	0
$985$	76.0000	2.42156
$986$	0	0
$987$	−27.0000	−0.859419
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	6.00000	0.190404
$994$	0	0
$995$	16.0000	0.507234
$996$	−54.0000	−1.71106
$997$	−20.0000	−0.633406	−0.316703	−	0.948525i	$-0.602576\pi$
$997$	−20.0000	−0.633406	−0.316703	+	0.948525i	$0.602576\pi$
$998$	0	0
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6031.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6031.2.a.a.1.1	✓	1	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:	\( N \)	\(=\)	\( 6031 = 37 \cdot 163 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6031.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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