LMFDB - Embedded newform 5082.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5082,2,Mod(1,5082)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5082, 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("5082.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5082.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:	$40.5799743072$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5082.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} -1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -4.00000 q^{20} +1.00000 q^{21} -8.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} -6.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -4.00000 q^{30} +6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -2.00000 q^{38} -6.00000 q^{39} +4.00000 q^{40} -12.0000 q^{41} -1.00000 q^{42} -4.00000 q^{43} -4.00000 q^{45} +8.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -11.0000 q^{50} -4.00000 q^{51} +6.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} -2.00000 q^{57} -6.00000 q^{58} +4.00000 q^{60} -10.0000 q^{61} -6.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -24.0000 q^{65} +4.00000 q^{67} +4.00000 q^{68} +8.00000 q^{69} -4.00000 q^{70} -12.0000 q^{71} -1.00000 q^{72} +6.00000 q^{74} -11.0000 q^{75} +2.00000 q^{76} +6.00000 q^{78} +16.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} +14.0000 q^{83} +1.00000 q^{84} -16.0000 q^{85} +4.00000 q^{86} -6.00000 q^{87} -14.0000 q^{89} +4.00000 q^{90} -6.00000 q^{91} -8.00000 q^{92} -6.00000 q^{93} -6.00000 q^{94} -8.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	4.00000	1.26491
$11$	0	0
$11$	0	0
$12$	−1.00000	−0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	1.00000	0.267261
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−4.00000	−0.894427
$21$	1.00000	0.218218
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	1.00000	0.204124
$25$	11.0000	2.20000
$26$	−6.00000	−1.17670
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−4.00000	−0.730297
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.00000	−0.685994
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−2.00000	−0.324443
$39$	−6.00000	−0.960769
$40$	4.00000	0.632456
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	−1.00000	−0.154303
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−4.00000	−0.596285
$46$	8.00000	1.17954
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−11.0000	−1.55563
$51$	−4.00000	−0.560112
$52$	6.00000	0.832050
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	−2.00000	−0.264906
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	4.00000	0.516398
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−6.00000	−0.762001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−24.0000	−2.97683
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	4.00000	0.485071
$69$	8.00000	0.963087
$70$	−4.00000	−0.478091
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−1.00000	−0.117851
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	6.00000	0.697486
$75$	−11.0000	−1.27017
$76$	2.00000	0.229416
$77$	0	0
$78$	6.00000	0.679366
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	12.0000	1.32518
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	1.00000	0.109109
$85$	−16.0000	−1.73544
$86$	4.00000	0.431331
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	4.00000	0.421637
$91$	−6.00000	−0.628971
$92$	−8.00000	−0.834058
$93$	−6.00000	−0.622171
$94$	−6.00000	−0.618853
$95$	−8.00000	−0.820783
$96$	1.00000	0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	11.0000	1.10000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	4.00000	0.396059
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	−6.00000	−0.588348
$105$	−4.00000	−0.390360
$106$	−2.00000	−0.194257
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	−1.00000	−0.0962250
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	−1.00000	−0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	2.00000	0.187317
$115$	32.0000	2.98402
$116$	6.00000	0.557086
$117$	6.00000	0.554700
$118$	0	0
$119$	−4.00000	−0.366679
$120$	−4.00000	−0.365148
$121$	0	0
$122$	10.0000	0.905357
$123$	12.0000	1.08200
$124$	6.00000	0.538816
$125$	−24.0000	−2.14663
$126$	1.00000	0.0890871
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	4.00000	0.352180
$130$	24.0000	2.10494
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	−4.00000	−0.345547
$135$	4.00000	0.344265
$136$	−4.00000	−0.342997
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	−8.00000	−0.681005
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	4.00000	0.338062
$141$	−6.00000	−0.505291
$142$	12.0000	1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	−24.0000	−1.99309
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	−6.00000	−0.493197
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	11.0000	0.898146
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−2.00000	−0.162221
$153$	4.00000	0.323381
$154$	0	0
$155$	−24.0000	−1.92773
$156$	−6.00000	−0.480384
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	−16.0000	−1.27289
$159$	−2.00000	−0.158610
$160$	4.00000	0.316228
$161$	8.00000	0.630488
$162$	−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	$0.449930\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	−14.0000	−1.08661
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−1.00000	−0.0771517
$169$	23.0000	1.76923
$170$	16.0000	1.22714
$171$	2.00000	0.152944
$172$	−4.00000	−0.304997
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	6.00000	0.454859
$175$	−11.0000	−0.831522
$176$	0	0
$177$	0	0
$178$	14.0000	1.04934
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−4.00000	−0.298142
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	6.00000	0.444750
$183$	10.0000	0.739221
$184$	8.00000	0.589768
$185$	24.0000	1.76452
$186$	6.00000	0.439941
$187$	0	0
$188$	6.00000	0.437595
$189$	1.00000	0.0727393
$190$	8.00000	0.580381
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	14.0000	1.00514
$195$	24.0000	1.71868
$196$	1.00000	0.0714286
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	−11.0000	−0.777817
$201$	−4.00000	−0.282138
$202$	14.0000	0.985037
$203$	−6.00000	−0.421117
$204$	−4.00000	−0.280056
$205$	48.0000	3.35247
$206$	6.00000	0.418040
$207$	−8.00000	−0.556038
$208$	6.00000	0.416025
$209$	0	0
$210$	4.00000	0.276026
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	2.00000	0.137361
$213$	12.0000	0.822226
$214$	−8.00000	−0.546869
$215$	16.0000	1.09119
$216$	1.00000	0.0680414
$217$	−6.00000	−0.407307
$218$	6.00000	0.406371
$219$	0	0
$220$	0	0
$221$	24.0000	1.61441
$222$	−6.00000	−0.402694
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	1.00000	0.0668153
$225$	11.0000	0.733333
$226$	−14.0000	−0.931266
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	−2.00000	−0.132453
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	−32.0000	−2.11002
$231$	0	0
$232$	−6.00000	−0.393919
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	−6.00000	−0.392232
$235$	−24.0000	−1.56559
$236$	0	0
$237$	−16.0000	−1.03931
$238$	4.00000	0.259281
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	4.00000	0.258199
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	−4.00000	−0.255551
$246$	−12.0000	−0.765092
$247$	12.0000	0.763542
$248$	−6.00000	−0.381000
$249$	−14.0000	−0.887214
$250$	24.0000	1.51789
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	0	0
$255$	16.0000	1.00196
$256$	1.00000	0.0625000
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	−4.00000	−0.249029
$259$	6.00000	0.372822
$260$	−24.0000	−1.48842
$261$	6.00000	0.371391
$262$	−2.00000	−0.123560
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	2.00000	0.122628
$267$	14.0000	0.856786
$268$	4.00000	0.244339
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	−4.00000	−0.243432
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	4.00000	0.242536
$273$	6.00000	0.363137
$274$	−14.0000	−0.845771
$275$	0	0
$276$	8.00000	0.481543
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	2.00000	0.119952
$279$	6.00000	0.359211
$280$	−4.00000	−0.239046
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	6.00000	0.357295
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−12.0000	−0.712069
$285$	8.00000	0.473879
$286$	0	0
$287$	12.0000	0.708338
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	24.0000	1.40933
$291$	14.0000	0.820695
$292$	0	0
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	−10.0000	−0.579284
$299$	−48.0000	−2.77591
$300$	−11.0000	−0.635085
$301$	4.00000	0.230556
$302$	−8.00000	−0.460348
$303$	14.0000	0.804279
$304$	2.00000	0.114708
$305$	40.0000	2.29039
$306$	−4.00000	−0.228665
$307$	14.0000	0.799022	0.399511	−	0.916728i	$-0.369180\pi$
$307$	14.0000	0.799022	0.399511	+	0.916728i	$0.369180\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	24.0000	1.36311
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	6.00000	0.339683
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	4.00000	0.225733
$315$	4.00000	0.225374
$316$	16.0000	0.900070
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	2.00000	0.112154
$319$	0	0
$320$	−4.00000	−0.223607
$321$	−8.00000	−0.446516
$322$	−8.00000	−0.445823
$323$	8.00000	0.445132
$324$	1.00000	0.0555556
$325$	66.0000	3.66102
$326$	−4.00000	−0.221540
$327$	6.00000	0.331801
$328$	12.0000	0.662589
$329$	−6.00000	−0.330791
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	14.0000	0.768350
$333$	−6.00000	−0.328798
$334$	−12.0000	−0.656611
$335$	−16.0000	−0.874173
$336$	1.00000	0.0545545
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	−23.0000	−1.25104
$339$	−14.0000	−0.760376
$340$	−16.0000	−0.867722
$341$	0	0
$342$	−2.00000	−0.108148
$343$	−1.00000	−0.0539949
$344$	4.00000	0.215666
$345$	−32.0000	−1.72282
$346$	2.00000	0.107521
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	−6.00000	−0.321634
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	11.0000	0.587975
$351$	−6.00000	−0.320256
$352$	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
$355$	48.0000	2.54758
$356$	−14.0000	−0.741999
$357$	4.00000	0.211702
$358$	4.00000	0.211407
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	4.00000	0.210819
$361$	−15.0000	−0.789474
$362$	8.00000	0.420471
$363$	0	0
$364$	−6.00000	−0.314485
$365$	0	0
$366$	−10.0000	−0.522708
$367$	−34.0000	−1.77479	−0.887393	−	0.461014i	$-0.847486\pi$
$367$	−34.0000	−1.77479	−0.887393	+	0.461014i	$0.847486\pi$
$368$	−8.00000	−0.417029
$369$	−12.0000	−0.624695
$370$	−24.0000	−1.24770
$371$	−2.00000	−0.103835
$372$	−6.00000	−0.311086
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	−6.00000	−0.309426
$377$	36.0000	1.85409
$378$	−1.00000	−0.0514344
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	0	0
$383$	−22.0000	−1.12415	−0.562074	−	0.827087i	$-0.689996\pi$
$383$	−22.0000	−1.12415	−0.562074	+	0.827087i	$0.689996\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−4.00000	−0.203331
$388$	−14.0000	−0.710742
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−24.0000	−1.21529
$391$	−32.0000	−1.61831
$392$	−1.00000	−0.0505076
$393$	−2.00000	−0.100887
$394$	10.0000	0.503793
$395$	−64.0000	−3.22019
$396$	0	0
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	−2.00000	−0.100251
$399$	2.00000	0.100125
$400$	11.0000	0.550000
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	4.00000	0.199502
$403$	36.0000	1.79329
$404$	−14.0000	−0.696526
$405$	−4.00000	−0.198762
$406$	6.00000	0.297775
$407$	0	0
$408$	4.00000	0.198030
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	−48.0000	−2.37055
$411$	−14.0000	−0.690569
$412$	−6.00000	−0.295599
$413$	0	0
$414$	8.00000	0.393179
$415$	−56.0000	−2.74893
$416$	−6.00000	−0.294174
$417$	2.00000	0.0979404
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	−4.00000	−0.195180
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	−4.00000	−0.194717
$423$	6.00000	0.291730
$424$	−2.00000	−0.0971286
$425$	44.0000	2.13431
$426$	−12.0000	−0.581402
$427$	10.0000	0.483934
$428$	8.00000	0.386695
$429$	0	0
$430$	−16.0000	−0.771589
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	−1.00000	−0.0481125
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	6.00000	0.288009
$435$	24.0000	1.15071
$436$	−6.00000	−0.287348
$437$	−16.0000	−0.765384
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−24.0000	−1.14156
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	6.00000	0.284747
$445$	56.0000	2.65465
$446$	−6.00000	−0.284108
$447$	−10.0000	−0.472984
$448$	−1.00000	−0.0472456
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	−11.0000	−0.518545
$451$	0	0
$452$	14.0000	0.658505
$453$	−8.00000	−0.375873
$454$	6.00000	0.281594
$455$	24.0000	1.12514
$456$	2.00000	0.0936586
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−8.00000	−0.373815
$459$	−4.00000	−0.186704
$460$	32.0000	1.49201
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	6.00000	0.278543
$465$	24.0000	1.11297
$466$	−14.0000	−0.648537
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	6.00000	0.277350
$469$	−4.00000	−0.184703
$470$	24.0000	1.10704
$471$	4.00000	0.184310
$472$	0	0
$473$	0	0
$474$	16.0000	0.734904
$475$	22.0000	1.00943
$476$	−4.00000	−0.183340
$477$	2.00000	0.0915737
$478$	−16.0000	−0.731823
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−4.00000	−0.182574
$481$	−36.0000	−1.64146
$482$	−8.00000	−0.364390
$483$	−8.00000	−0.364013
$484$	0	0
$485$	56.0000	2.54283
$486$	1.00000	0.0453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	10.0000	0.452679
$489$	−4.00000	−0.180886
$490$	4.00000	0.180702
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	12.0000	0.541002
$493$	24.0000	1.08091
$494$	−12.0000	−0.539906
$495$	0	0
$496$	6.00000	0.269408
$497$	12.0000	0.538274
$498$	14.0000	0.627355
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	−24.0000	−1.07331
$501$	−12.0000	−0.536120
$502$	16.0000	0.714115
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	1.00000	0.0445435
$505$	56.0000	2.49197
$506$	0	0
$507$	−23.0000	−1.02147
$508$	0	0
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	−16.0000	−0.708492
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−2.00000	−0.0883022
$514$	−26.0000	−1.14681
$515$	24.0000	1.05757
$516$	4.00000	0.176090
$517$	0	0
$518$	−6.00000	−0.263625
$519$	2.00000	0.0877903
$520$	24.0000	1.05247
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−6.00000	−0.262613
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	2.00000	0.0873704
$525$	11.0000	0.480079
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	41.0000	1.78261
$530$	8.00000	0.347498
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	−72.0000	−3.11867
$534$	−14.0000	−0.605839
$535$	−32.0000	−1.38348
$536$	−4.00000	−0.172774
$537$	4.00000	0.172613
$538$	4.00000	0.172452
$539$	0	0
$540$	4.00000	0.172133
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	−4.00000	−0.171815
$543$	8.00000	0.343313
$544$	−4.00000	−0.171499
$545$	24.0000	1.02805
$546$	−6.00000	−0.256776
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	14.0000	0.598050
$549$	−10.0000	−0.426790
$550$	0	0
$551$	12.0000	0.511217
$552$	−8.00000	−0.340503
$553$	−16.0000	−0.680389
$554$	−10.0000	−0.424859
$555$	−24.0000	−1.01874
$556$	−2.00000	−0.0848189
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	−6.00000	−0.254000
$559$	−24.0000	−1.01509
$560$	4.00000	0.169031
$561$	0	0
$562$	26.0000	1.09674
$563$	26.0000	1.09577	0.547885	−	0.836554i	$-0.315433\pi$
$563$	26.0000	1.09577	0.547885	+	0.836554i	$0.315433\pi$
$564$	−6.00000	−0.252646
$565$	−56.0000	−2.35594
$566$	−14.0000	−0.588464
$567$	−1.00000	−0.0419961
$568$	12.0000	0.503509
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	−8.00000	−0.335083
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	0	0
$574$	−12.0000	−0.500870
$575$	−88.0000	−3.66985
$576$	1.00000	0.0416667
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	1.00000	0.0415945
$579$	−14.0000	−0.581820
$580$	−24.0000	−0.996546
$581$	−14.0000	−0.580818
$582$	−14.0000	−0.580319
$583$	0	0
$584$	0	0
$585$	−24.0000	−0.992278
$586$	−14.0000	−0.578335
$587$	24.0000	0.990586	0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	−1.00000	−0.0412393
$589$	12.0000	0.494451
$590$	0	0
$591$	10.0000	0.411345
$592$	−6.00000	−0.246598
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	10.0000	0.409616
$597$	−2.00000	−0.0818546
$598$	48.0000	1.96287
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	11.0000	0.449073
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	−4.00000	−0.163028
$603$	4.00000	0.162893
$604$	8.00000	0.325515
$605$	0	0
$606$	−14.0000	−0.568711
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−2.00000	−0.0811107
$609$	6.00000	0.243132
$610$	−40.0000	−1.61955
$611$	36.0000	1.45640
$612$	4.00000	0.161690
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−14.0000	−0.564994
$615$	−48.0000	−1.93555
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	−6.00000	−0.241355
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−24.0000	−0.963863
$621$	8.00000	0.321029
$622$	−10.0000	−0.400963
$623$	14.0000	0.560898
$624$	−6.00000	−0.240192
$625$	41.0000	1.64000
$626$	34.0000	1.35891
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−24.0000	−0.956943
$630$	−4.00000	−0.159364
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	−16.0000	−0.636446
$633$	−4.00000	−0.158986
$634$	6.00000	0.238290
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	6.00000	0.237729
$638$	0	0
$639$	−12.0000	−0.474713
$640$	4.00000	0.158114
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	8.00000	0.315735
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	8.00000	0.315244
$645$	−16.0000	−0.629999
$646$	−8.00000	−0.314756
$647$	26.0000	1.02217	0.511083	−	0.859532i	$-0.329245\pi$
$647$	26.0000	1.02217	0.511083	+	0.859532i	$0.329245\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−66.0000	−2.58873
$651$	6.00000	0.235159
$652$	4.00000	0.156652
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−6.00000	−0.234619
$655$	−8.00000	−0.312586
$656$	−12.0000	−0.468521
$657$	0	0
$658$	6.00000	0.233904
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	−20.0000	−0.777322
$663$	−24.0000	−0.932083
$664$	−14.0000	−0.543305
$665$	8.00000	0.310227
$666$	6.00000	0.232495
$667$	−48.0000	−1.85857
$668$	12.0000	0.464294
$669$	−6.00000	−0.231973
$670$	16.0000	0.618134
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	6.00000	0.231111
$675$	−11.0000	−0.423390
$676$	23.0000	0.884615
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	14.0000	0.537667
$679$	14.0000	0.537271
$680$	16.0000	0.613572
$681$	6.00000	0.229920
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	2.00000	0.0764719
$685$	−56.0000	−2.13965
$686$	1.00000	0.0381802
$687$	−8.00000	−0.305219
$688$	−4.00000	−0.152499
$689$	12.0000	0.457164
$690$	32.0000	1.21822
$691$	24.0000	0.913003	0.456502	−	0.889723i	$-0.349102\pi$
$691$	24.0000	0.913003	0.456502	+	0.889723i	$0.349102\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	8.00000	0.303676
$695$	8.00000	0.303457
$696$	6.00000	0.227429
$697$	−48.0000	−1.81813
$698$	6.00000	0.227103
$699$	−14.0000	−0.529529
$700$	−11.0000	−0.415761
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	6.00000	0.226455
$703$	−12.0000	−0.452589
$704$	0	0
$705$	24.0000	0.903892
$706$	6.00000	0.225813
$707$	14.0000	0.526524
$708$	0	0
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	−48.0000	−1.80141
$711$	16.0000	0.600047
$712$	14.0000	0.524672
$713$	−48.0000	−1.79761
$714$	−4.00000	−0.149696
$715$	0	0
$716$	−4.00000	−0.149487
$717$	−16.0000	−0.597531
$718$	24.0000	0.895672
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	−4.00000	−0.149071
$721$	6.00000	0.223452
$722$	15.0000	0.558242
$723$	−8.00000	−0.297523
$724$	−8.00000	−0.297318
$725$	66.0000	2.45118
$726$	0	0
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	6.00000	0.222375
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	10.0000	0.369611
$733$	−50.0000	−1.84679	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000	−1.84679	−0.923396	+	0.383849i	$0.874598\pi$
$734$	34.0000	1.25496
$735$	4.00000	0.147542
$736$	8.00000	0.294884
$737$	0	0
$738$	12.0000	0.441726
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	24.0000	0.882258
$741$	−12.0000	−0.440831
$742$	2.00000	0.0734223
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	6.00000	0.219971
$745$	−40.0000	−1.46549
$746$	−14.0000	−0.512576
$747$	14.0000	0.512233
$748$	0	0
$749$	−8.00000	−0.292314
$750$	−24.0000	−0.876356
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	6.00000	0.218797
$753$	16.0000	0.583072
$754$	−36.0000	−1.31104
$755$	−32.0000	−1.16460
$756$	1.00000	0.0363696
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	8.00000	0.290191
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	0	0
$763$	6.00000	0.217215
$764$	0	0
$765$	−16.0000	−0.578481
$766$	22.0000	0.794892
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−28.0000	−1.00971	−0.504853	−	0.863205i	$-0.668453\pi$
$769$	−28.0000	−1.00971	−0.504853	+	0.863205i	$0.668453\pi$
$770$	0	0
$771$	−26.0000	−0.936367
$772$	14.0000	0.503871
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	4.00000	0.143777
$775$	66.0000	2.37079
$776$	14.0000	0.502571
$777$	−6.00000	−0.215249
$778$	−6.00000	−0.215110
$779$	−24.0000	−0.859889
$780$	24.0000	0.859338
$781$	0	0
$782$	32.0000	1.14432
$783$	−6.00000	−0.214423
$784$	1.00000	0.0357143
$785$	16.0000	0.571064
$786$	2.00000	0.0713376
$787$	50.0000	1.78231	0.891154	−	0.453701i	$-0.149897\pi$
$787$	50.0000	1.78231	0.891154	+	0.453701i	$0.149897\pi$
$788$	−10.0000	−0.356235
$789$	0	0
$790$	64.0000	2.27702
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−60.0000	−2.13066
$794$	20.0000	0.709773
$795$	8.00000	0.283731
$796$	2.00000	0.0708881
$797$	24.0000	0.850124	0.425062	−	0.905164i	$-0.360252\pi$
$797$	24.0000	0.850124	0.425062	+	0.905164i	$0.360252\pi$
$798$	−2.00000	−0.0707992
$799$	24.0000	0.849059
$800$	−11.0000	−0.388909
$801$	−14.0000	−0.494666
$802$	26.0000	0.918092
$803$	0	0
$804$	−4.00000	−0.141069
$805$	−32.0000	−1.12785
$806$	−36.0000	−1.26805
$807$	4.00000	0.140807
$808$	14.0000	0.492518
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	4.00000	0.140546
$811$	−26.0000	−0.912983	−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000	−0.912983	−0.456492	+	0.889728i	$0.650894\pi$
$812$	−6.00000	−0.210559
$813$	−4.00000	−0.140286
$814$	0	0
$815$	−16.0000	−0.560456
$816$	−4.00000	−0.140028
$817$	−8.00000	−0.279885
$818$	4.00000	0.139857
$819$	−6.00000	−0.209657
$820$	48.0000	1.67623
$821$	26.0000	0.907406	0.453703	−	0.891153i	$-0.350103\pi$
$821$	26.0000	0.907406	0.453703	+	0.891153i	$0.350103\pi$
$822$	14.0000	0.488306
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	6.00000	0.209020
$825$	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	$0.338154\pi$
$828$	−8.00000	−0.278019
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	56.0000	1.94379
$831$	−10.0000	−0.346896
$832$	6.00000	0.208013
$833$	4.00000	0.138592
$834$	−2.00000	−0.0692543
$835$	−48.0000	−1.66111
$836$	0	0
$837$	−6.00000	−0.207390
$838$	−12.0000	−0.414533
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	4.00000	0.138013
$841$	7.00000	0.241379
$842$	−38.0000	−1.30957
$843$	26.0000	0.895488
$844$	4.00000	0.137686
$845$	−92.0000	−3.16490
$846$	−6.00000	−0.206284
$847$	0	0
$848$	2.00000	0.0686803
$849$	−14.0000	−0.480479
$850$	−44.0000	−1.50919
$851$	48.0000	1.64542
$852$	12.0000	0.411113
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	−10.0000	−0.342193
$855$	−8.00000	−0.273594
$856$	−8.00000	−0.273434
$857$	40.0000	1.36637	0.683187	−	0.730243i	$-0.260593\pi$
$857$	40.0000	1.36637	0.683187	+	0.730243i	$0.260593\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	16.0000	0.545595
$861$	−12.0000	−0.408959
$862$	−24.0000	−0.817443
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	1.00000	0.0340207
$865$	8.00000	0.272008
$866$	−26.0000	−0.883516
$867$	1.00000	0.0339618
$868$	−6.00000	−0.203653
$869$	0	0
$870$	−24.0000	−0.813676
$871$	24.0000	0.813209
$872$	6.00000	0.203186
$873$	−14.0000	−0.473828
$874$	16.0000	0.541208
$875$	24.0000	0.811348
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	−20.0000	−0.674967
$879$	−14.0000	−0.472208
$880$	0	0
$881$	−50.0000	−1.68454	−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000	−1.68454	−0.842271	+	0.539054i	$0.818782\pi$
$882$	−1.00000	−0.0336718
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	−28.0000	−0.940678
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−6.00000	−0.201347
$889$	0	0
$890$	−56.0000	−1.87712
$891$	0	0
$892$	6.00000	0.200895
$893$	12.0000	0.401565
$894$	10.0000	0.334450
$895$	16.0000	0.534821
$896$	1.00000	0.0334077
$897$	48.0000	1.60267
$898$	−14.0000	−0.467186
$899$	36.0000	1.20067
$900$	11.0000	0.366667
$901$	8.00000	0.266519
$902$	0	0
$903$	−4.00000	−0.133112
$904$	−14.0000	−0.465633
$905$	32.0000	1.06372
$906$	8.00000	0.265782
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	−6.00000	−0.199117
$909$	−14.0000	−0.464351
$910$	−24.0000	−0.795592
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	−2.00000	−0.0662266
$913$	0	0
$914$	10.0000	0.330771
$915$	−40.0000	−1.32236
$916$	8.00000	0.264327
$917$	−2.00000	−0.0660458
$918$	4.00000	0.132020
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	−32.0000	−1.05501
$921$	−14.0000	−0.461316
$922$	14.0000	0.461065
$923$	−72.0000	−2.36991
$924$	0	0
$925$	−66.0000	−2.17007
$926$	24.0000	0.788689
$927$	−6.00000	−0.197066
$928$	−6.00000	−0.196960
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	−24.0000	−0.786991
$931$	2.00000	0.0655474
$932$	14.0000	0.458585
$933$	−10.0000	−0.327385
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−6.00000	−0.196116
$937$	32.0000	1.04539	0.522697	−	0.852518i	$-0.324926\pi$
$937$	32.0000	1.04539	0.522697	+	0.852518i	$0.324926\pi$
$938$	4.00000	0.130605
$939$	34.0000	1.10955
$940$	−24.0000	−0.782794
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	−4.00000	−0.130327
$943$	96.0000	3.12619
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	−16.0000	−0.519656
$949$	0	0
$950$	−22.0000	−0.713774
$951$	6.00000	0.194563
$952$	4.00000	0.129641
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	16.0000	0.517477
$957$	0	0
$958$	24.0000	0.775405
$959$	−14.0000	−0.452084
$960$	4.00000	0.129099
$961$	5.00000	0.161290
$962$	36.0000	1.16069
$963$	8.00000	0.257796
$964$	8.00000	0.257663
$965$	−56.0000	−1.80270
$966$	8.00000	0.257396
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	−56.0000	−1.79805
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	−1.00000	−0.0320750
$973$	2.00000	0.0641171
$974$	−20.0000	−0.640841
$975$	−66.0000	−2.11369
$976$	−10.0000	−0.320092
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	4.00000	0.127906
$979$	0	0
$980$	−4.00000	−0.127775
$981$	−6.00000	−0.191565
$982$	−28.0000	−0.893516
$983$	−42.0000	−1.33959	−0.669796	−	0.742545i	$-0.733618\pi$
$983$	−42.0000	−1.33959	−0.669796	+	0.742545i	$0.733618\pi$
$984$	−12.0000	−0.382546
$985$	40.0000	1.27451
$986$	−24.0000	−0.764316
$987$	6.00000	0.190982
$988$	12.0000	0.381771
$989$	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	$0.459445\pi$
$992$	−6.00000	−0.190500
$993$	−20.0000	−0.634681
$994$	−12.0000	−0.380617
$995$	−8.00000	−0.253617
$996$	−14.0000	−0.443607
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	28.0000	0.886325
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5082.2.a.a.1.1		1
11.10	odd	2		462.2.a.e.1.1	✓	1
33.32	even	2		1386.2.a.e.1.1		1
44.43	even	2		3696.2.a.p.1.1		1
77.76	even	2		3234.2.a.v.1.1		1
231.230	odd	2		9702.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.e.1.1	✓	1	11.10	odd	2
1386.2.a.e.1.1		1	33.32	even	2
3234.2.a.v.1.1		1	77.76	even	2
3696.2.a.p.1.1		1	44.43	even	2
5082.2.a.a.1.1		1	1.1	even	1	trivial
9702.2.a.b.1.1		1	231.230	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5082.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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