LMFDB - Embedded newform 5082.2.a.s.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5082,2,Mod(1,5082)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [1,1,-1,1,0,-1,1,1,1,0,0,-1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} -1.00000 q^{21} -4.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +10.0000 q^{29} +6.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} -6.00000 q^{38} -2.00000 q^{39} +12.0000 q^{41} -1.00000 q^{42} +8.00000 q^{43} -4.00000 q^{46} +2.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -5.00000 q^{50} -4.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} +6.00000 q^{57} +10.0000 q^{58} -8.00000 q^{59} -6.00000 q^{61} +6.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} +4.00000 q^{68} +4.00000 q^{69} +1.00000 q^{72} +12.0000 q^{73} -6.00000 q^{74} +5.00000 q^{75} -6.00000 q^{76} -2.00000 q^{78} +1.00000 q^{81} +12.0000 q^{82} -14.0000 q^{83} -1.00000 q^{84} +8.00000 q^{86} -10.0000 q^{87} +10.0000 q^{89} +2.00000 q^{91} -4.00000 q^{92} -6.00000 q^{93} +2.00000 q^{94} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−1.00000	−0.288675
$13$	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
$17$	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
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	−5.00000	−1.00000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	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
$35$	0	0
$36$	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	$0.664172\pi$
$38$	−6.00000	−0.973329
$39$	−2.00000	−0.320256
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	−1.00000	−0.154303
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	−4.00000	−0.589768
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−5.00000	−0.707107
$51$	−4.00000	−0.560112
$52$	2.00000	0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	6.00000	0.794719
$58$	10.0000	1.31306
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	6.00000	0.762001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	4.00000	0.485071
$69$	4.00000	0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	−6.00000	−0.697486
$75$	5.00000	0.577350
$76$	−6.00000	−0.688247
$77$	0	0
$78$	−2.00000	−0.226455
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	12.0000	1.32518
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	8.00000	0.862662
$87$	−10.0000	−1.07211
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−4.00000	−0.417029
$93$	−6.00000	−0.622171
$94$	2.00000	0.206284
$95$	0	0
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−5.00000	−0.500000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−4.00000	−0.396059
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	−1.00000	−0.0962250
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	1.00000	0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	6.00000	0.561951
$115$	0	0
$116$	10.0000	0.928477
$117$	2.00000	0.184900
$118$	−8.00000	−0.736460
$119$	4.00000	0.366679
$120$	0	0
$121$	0	0
$122$	−6.00000	−0.543214
$123$	−12.0000	−1.08200
$124$	6.00000	0.538816
$125$	0	0
$126$	1.00000	0.0890871
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−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
$134$	−4.00000	−0.345547
$135$	0	0
$136$	4.00000	0.342997
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	4.00000	0.340503
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	12.0000	0.993127
$147$	−1.00000	−0.0824786
$148$	−6.00000	−0.493197
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	5.00000	0.408248
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−6.00000	−0.486664
$153$	4.00000	0.323381
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−4.00000	−0.315244
$162$	1.00000	0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	−14.0000	−1.08661
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	8.00000	0.609994
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	−10.0000	−0.758098
$175$	−5.00000	−0.377964
$176$	0	0
$177$	8.00000	0.601317
$178$	10.0000	0.749532
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	2.00000	0.148250
$183$	6.00000	0.443533
$184$	−4.00000	−0.294884
$185$	0	0
$186$	−6.00000	−0.439941
$187$	0	0
$188$	2.00000	0.145865
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	−5.00000	−0.353553
$201$	4.00000	0.282138
$202$	6.00000	0.422159
$203$	10.0000	0.701862
$204$	−4.00000	−0.280056
$205$	0	0
$206$	10.0000	0.696733
$207$	−4.00000	−0.278019
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−8.00000	−0.546869
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	6.00000	0.407307
$218$	−18.0000	−1.21911
$219$	−12.0000	−0.810885
$220$	0	0
$221$	8.00000	0.538138
$222$	6.00000	0.402694
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	1.00000	0.0668153
$225$	−5.00000	−0.333333
$226$	−2.00000	−0.133038
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	6.00000	0.397360
$229$	24.0000	1.58596	0.792982	−	0.609245i	$-0.208527\pi$
$229$	24.0000	1.58596	0.792982	+	0.609245i	$0.208527\pi$
$230$	0	0
$231$	0	0
$232$	10.0000	0.656532
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−8.00000	−0.520756
$237$	0	0
$238$	4.00000	0.259281
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	0	0
$246$	−12.0000	−0.765092
$247$	−12.0000	−0.763542
$248$	6.00000	0.381000
$249$	14.0000	0.887214
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−8.00000	−0.498058
$259$	−6.00000	−0.372822
$260$	0	0
$261$	10.0000	0.618984
$262$	−10.0000	−0.617802
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	0	0
$266$	−6.00000	−0.367884
$267$	−10.0000	−0.611990
$268$	−4.00000	−0.244339
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	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
$273$	−2.00000	−0.121046
$274$	−10.0000	−0.604122
$275$	0	0
$276$	4.00000	0.240772
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	14.0000	0.839664
$279$	6.00000	0.359211
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	−2.00000	−0.119098
$283$	−2.00000	−0.118888	−0.0594438	−	0.998232i	$-0.518933\pi$
$283$	−2.00000	−0.118888	−0.0594438	+	0.998232i	$0.518933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	12.0000	0.702247
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−10.0000	−0.579284
$299$	−8.00000	−0.462652
$300$	5.00000	0.288675
$301$	8.00000	0.461112
$302$	16.0000	0.920697
$303$	−6.00000	−0.344691
$304$	−6.00000	−0.344124
$305$	0	0
$306$	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
$309$	−10.0000	−0.568880
$310$	0	0
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	−2.00000	−0.113228
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	12.0000	0.677199
$315$	0	0
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	0	0
$321$	8.00000	0.446516
$322$	−4.00000	−0.222911
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	−10.0000	−0.554700
$326$	12.0000	0.664619
$327$	18.0000	0.995402
$328$	12.0000	0.662589
$329$	2.00000	0.110264
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−14.0000	−0.768350
$333$	−6.00000	−0.328798
$334$	−12.0000	−0.656611
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−9.00000	−0.489535
$339$	2.00000	0.108625
$340$	0	0
$341$	0	0
$342$	−6.00000	−0.324443
$343$	1.00000	0.0539949
$344$	8.00000	0.431331
$345$	0	0
$346$	18.0000	0.967686
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	−10.0000	−0.536056
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	−5.00000	−0.267261
$351$	−2.00000	−0.106752
$352$	0	0
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	10.0000	0.529999
$357$	−4.00000	−0.211702
$358$	20.0000	1.05703
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−8.00000	−0.420471
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	6.00000	0.313625
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	−4.00000	−0.208514
$369$	12.0000	0.624695
$370$	0	0
$371$	6.00000	0.311504
$372$	−6.00000	−0.311086
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	2.00000	0.103142
$377$	20.0000	1.03005
$378$	−1.00000	−0.0514344
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	−12.0000	−0.613973
$383$	−10.0000	−0.510976	−0.255488	−	0.966812i	$-0.582236\pi$
$383$	−10.0000	−0.510976	−0.255488	+	0.966812i	$0.582236\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	22.0000	1.11977
$387$	8.00000	0.406663
$388$	10.0000	0.507673
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	1.00000	0.0505076
$393$	10.0000	0.504433
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	10.0000	0.501255
$399$	6.00000	0.300376
$400$	−5.00000	−0.250000
$401$	−34.0000	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$402$	4.00000	0.199502
$403$	12.0000	0.597763
$404$	6.00000	0.298511
$405$	0	0
$406$	10.0000	0.496292
$407$	0	0
$408$	−4.00000	−0.198030
$409$	−24.0000	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	10.0000	0.492665
$413$	−8.00000	−0.393654
$414$	−4.00000	−0.196589
$415$	0	0
$416$	2.00000	0.0980581
$417$	−14.0000	−0.685583
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	8.00000	0.389434
$423$	2.00000	0.0972433
$424$	6.00000	0.291386
$425$	−20.0000	−0.970143
$426$	0	0
$427$	−6.00000	−0.290360
$428$	−8.00000	−0.386695
$429$	0	0
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−1.00000	−0.0481125
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	6.00000	0.288009
$435$	0	0
$436$	−18.0000	−0.862044
$437$	24.0000	1.14808
$438$	−12.0000	−0.573382
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	8.00000	0.380521
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	10.0000	0.472984
$448$	1.00000	0.0472456
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	−5.00000	−0.235702
$451$	0	0
$452$	−2.00000	−0.0940721
$453$	−16.0000	−0.751746
$454$	14.0000	0.657053
$455$	0	0
$456$	6.00000	0.280976
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	24.0000	1.12145
$459$	−4.00000	−0.186704
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	26.0000	1.20443
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	2.00000	0.0924500
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−12.0000	−0.552931
$472$	−8.00000	−0.368230
$473$	0	0
$474$	0	0
$475$	30.0000	1.37649
$476$	4.00000	0.183340
$477$	6.00000	0.274721
$478$	8.00000	0.365911
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	28.0000	1.27537
$483$	4.00000	0.182006
$484$	0	0
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	−6.00000	−0.271607
$489$	−12.0000	−0.542659
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−12.0000	−0.541002
$493$	40.0000	1.80151
$494$	−12.0000	−0.539906
$495$	0	0
$496$	6.00000	0.269408
$497$	0	0
$498$	14.0000	0.627355
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	8.00000	0.357057
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	0	0
$507$	9.00000	0.399704
$508$	16.0000	0.709885
$509$	−32.0000	−1.41838	−0.709188	−	0.705020i	$-0.750938\pi$
$509$	−32.0000	−1.41838	−0.709188	+	0.705020i	$0.750938\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	1.00000	0.0441942
$513$	6.00000	0.264906
$514$	−6.00000	−0.264649
$515$	0	0
$516$	−8.00000	−0.352180
$517$	0	0
$518$	−6.00000	−0.263625
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	10.0000	0.437688
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	−10.0000	−0.436852
$525$	5.00000	0.218218
$526$	−16.0000	−0.697633
$527$	24.0000	1.04546
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−8.00000	−0.347170
$532$	−6.00000	−0.260133
$533$	24.0000	1.03956
$534$	−10.0000	−0.432742
$535$	0	0
$536$	−4.00000	−0.172774
$537$	−20.0000	−0.863064
$538$	24.0000	1.03471
$539$	0	0
$540$	0	0
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	4.00000	0.171815
$543$	8.00000	0.343313
$544$	4.00000	0.171499
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−10.0000	−0.427179
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−60.0000	−2.55609
$552$	4.00000	0.170251
$553$	0	0
$554$	22.0000	0.934690
$555$	0	0
$556$	14.0000	0.593732
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	6.00000	0.254000
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	−30.0000	−1.26547
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	−2.00000	−0.0842152
$565$	0	0
$566$	−2.00000	−0.0840663
$567$	1.00000	0.0419961
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	12.0000	0.500870
$575$	20.0000	0.834058
$576$	1.00000	0.0416667
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	−1.00000	−0.0415945
$579$	−22.0000	−0.914289
$580$	0	0
$581$	−14.0000	−0.580818
$582$	−10.0000	−0.414513
$583$	0	0
$584$	12.0000	0.496564
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	−1.00000	−0.0412393
$589$	−36.0000	−1.48335
$590$	0	0
$591$	6.00000	0.246807
$592$	−6.00000	−0.246598
$593$	4.00000	0.164260	0.0821302	−	0.996622i	$-0.473828\pi$
$593$	4.00000	0.164260	0.0821302	+	0.996622i	$0.473828\pi$
$594$	0	0
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−10.0000	−0.409273
$598$	−8.00000	−0.327144
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	5.00000	0.204124
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	8.00000	0.326056
$603$	−4.00000	−0.162893
$604$	16.0000	0.651031
$605$	0	0
$606$	−6.00000	−0.243733
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	−6.00000	−0.243332
$609$	−10.0000	−0.405220
$610$	0	0
$611$	4.00000	0.161823
$612$	4.00000	0.161690
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	14.0000	0.564994
$615$	0	0
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	−10.0000	−0.402259
$619$	12.0000	0.482321	0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000	0.482321	0.241160	+	0.970485i	$0.422472\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	−10.0000	−0.400963
$623$	10.0000	0.400642
$624$	−2.00000	−0.0800641
$625$	25.0000	1.00000
$626$	6.00000	0.239808
$627$	0	0
$628$	12.0000	0.478852
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−44.0000	−1.75161	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000	−1.75161	−0.875806	+	0.482663i	$0.839670\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	−18.0000	−0.714871
$635$	0	0
$636$	−6.00000	−0.237915
$637$	2.00000	0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−38.0000	−1.50091	−0.750455	−	0.660922i	$-0.770166\pi$
$641$	−38.0000	−1.50091	−0.750455	+	0.660922i	$0.770166\pi$
$642$	8.00000	0.315735
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−10.0000	−0.393141	−0.196570	−	0.980490i	$-0.562980\pi$
$647$	−10.0000	−0.393141	−0.196570	+	0.980490i	$0.562980\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−10.0000	−0.392232
$651$	−6.00000	−0.235159
$652$	12.0000	0.469956
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	18.0000	0.703856
$655$	0	0
$656$	12.0000	0.468521
$657$	12.0000	0.468165
$658$	2.00000	0.0779681
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	−20.0000	−0.777322
$663$	−8.00000	−0.310694
$664$	−14.0000	−0.543305
$665$	0	0
$666$	−6.00000	−0.232495
$667$	−40.0000	−1.54881
$668$	−12.0000	−0.464294
$669$	2.00000	0.0773245
$670$	0	0
$671$	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	$0.438264\pi$
$674$	−22.0000	−0.847408
$675$	5.00000	0.192450
$676$	−9.00000	−0.346154
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	2.00000	0.0768095
$679$	10.0000	0.383765
$680$	0	0
$681$	−14.0000	−0.536481
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	1.00000	0.0381802
$687$	−24.0000	−0.915657
$688$	8.00000	0.304997
$689$	12.0000	0.457164
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−16.0000	−0.607352
$695$	0	0
$696$	−10.0000	−0.379049
$697$	48.0000	1.81813
$698$	−2.00000	−0.0757011
$699$	−26.0000	−0.983410
$700$	−5.00000	−0.188982
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	−2.00000	−0.0754851
$703$	36.0000	1.35777
$704$	0	0
$705$	0	0
$706$	10.0000	0.376355
$707$	6.00000	0.225653
$708$	8.00000	0.300658
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	0	0
$712$	10.0000	0.374766
$713$	−24.0000	−0.898807
$714$	−4.00000	−0.149696
$715$	0	0
$716$	20.0000	0.747435
$717$	−8.00000	−0.298765
$718$	−8.00000	−0.298557
$719$	38.0000	1.41716	0.708580	−	0.705630i	$-0.249336\pi$
$719$	38.0000	1.41716	0.708580	+	0.705630i	$0.249336\pi$
$720$	0	0
$721$	10.0000	0.372419
$722$	17.0000	0.632674
$723$	−28.0000	−1.04133
$724$	−8.00000	−0.297318
$725$	−50.0000	−1.85695
$726$	0	0
$727$	−34.0000	−1.26099	−0.630495	−	0.776193i	$-0.717148\pi$
$727$	−34.0000	−1.26099	−0.630495	+	0.776193i	$0.717148\pi$
$728$	2.00000	0.0741249
$729$	1.00000	0.0370370
$730$	0	0
$731$	32.0000	1.18356
$732$	6.00000	0.221766
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	−4.00000	−0.147442
$737$	0	0
$738$	12.0000	0.441726
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	6.00000	0.220267
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−6.00000	−0.219971
$745$	0	0
$746$	−22.0000	−0.805477
$747$	−14.0000	−0.512233
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	44.0000	1.60558	0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000	1.60558	0.802791	+	0.596260i	$0.203347\pi$
$752$	2.00000	0.0729325
$753$	−8.00000	−0.291536
$754$	20.0000	0.728357
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	50.0000	1.81728	0.908640	−	0.417579i	$-0.137121\pi$
$757$	50.0000	1.81728	0.908640	+	0.417579i	$0.137121\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	0	0
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	−16.0000	−0.579619
$763$	−18.0000	−0.651644
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−10.0000	−0.361315
$767$	−16.0000	−0.577727
$768$	−1.00000	−0.0360844
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	22.0000	0.791797
$773$	−28.0000	−1.00709	−0.503545	−	0.863969i	$-0.667971\pi$
$773$	−28.0000	−1.00709	−0.503545	+	0.863969i	$0.667971\pi$
$774$	8.00000	0.287554
$775$	−30.0000	−1.07763
$776$	10.0000	0.358979
$777$	6.00000	0.215249
$778$	2.00000	0.0717035
$779$	−72.0000	−2.57967
$780$	0	0
$781$	0	0
$782$	−16.0000	−0.572159
$783$	−10.0000	−0.357371
$784$	1.00000	0.0357143
$785$	0	0
$786$	10.0000	0.356688
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	−6.00000	−0.213741
$789$	16.0000	0.569615
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	−12.0000	−0.426132
$794$	−4.00000	−0.141955
$795$	0	0
$796$	10.0000	0.354441
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	6.00000	0.212398
$799$	8.00000	0.283020
$800$	−5.00000	−0.176777
$801$	10.0000	0.353333
$802$	−34.0000	−1.20058
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	12.0000	0.422682
$807$	−24.0000	−0.844840
$808$	6.00000	0.211079
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	10.0000	0.350931
$813$	−4.00000	−0.140286
$814$	0	0
$815$	0	0
$816$	−4.00000	−0.140028
$817$	−48.0000	−1.67931
$818$	−24.0000	−0.839140
$819$	2.00000	0.0698857
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	10.0000	0.348790
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	−8.00000	−0.278356
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	−4.00000	−0.139010
$829$	56.0000	1.94496	0.972480	−	0.232986i	$-0.0748495\pi$
$829$	56.0000	1.94496	0.972480	+	0.232986i	$0.0748495\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	2.00000	0.0693375
$833$	4.00000	0.138592
$834$	−14.0000	−0.484780
$835$	0	0
$836$	0	0
$837$	−6.00000	−0.207390
$838$	28.0000	0.967244
$839$	14.0000	0.483334	0.241667	−	0.970359i	$-0.422306\pi$
$839$	14.0000	0.483334	0.241667	+	0.970359i	$0.422306\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	6.00000	0.206774
$843$	30.0000	1.03325
$844$	8.00000	0.275371
$845$	0	0
$846$	2.00000	0.0687614
$847$	0	0
$848$	6.00000	0.206041
$849$	2.00000	0.0686398
$850$	−20.0000	−0.685994
$851$	24.0000	0.822709
$852$	0	0
$853$	18.0000	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$853$	18.0000	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	−8.00000	−0.273434
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	−8.00000	−0.272481
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−22.0000	−0.747590
$867$	1.00000	0.0339618
$868$	6.00000	0.203653
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−18.0000	−0.609557
$873$	10.0000	0.338449
$874$	24.0000	0.811812
$875$	0	0
$876$	−12.0000	−0.405442
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	−28.0000	−0.944954
$879$	6.00000	0.202375
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	1.00000	0.0336718
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	12.0000	0.403148
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	6.00000	0.201347
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	−2.00000	−0.0669650
$893$	−12.0000	−0.401565
$894$	10.0000	0.334450
$895$	0	0
$896$	1.00000	0.0334077
$897$	8.00000	0.267112
$898$	−2.00000	−0.0667409
$899$	60.0000	2.00111
$900$	−5.00000	−0.166667
$901$	24.0000	0.799556
$902$	0	0
$903$	−8.00000	−0.266223
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	−16.0000	−0.531564
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	14.0000	0.464606
$909$	6.00000	0.199007
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	6.00000	0.198680
$913$	0	0
$914$	6.00000	0.198462
$915$	0	0
$916$	24.0000	0.792982
$917$	−10.0000	−0.330229
$918$	−4.00000	−0.132020
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	−14.0000	−0.461316
$922$	6.00000	0.197599
$923$	0	0
$924$	0	0
$925$	30.0000	0.986394
$926$	16.0000	0.525793
$927$	10.0000	0.328443
$928$	10.0000	0.328266
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	26.0000	0.851658
$933$	10.0000	0.327385
$934$	4.00000	0.130884
$935$	0	0
$936$	2.00000	0.0653720
$937$	4.00000	0.130674	0.0653372	−	0.997863i	$-0.479188\pi$
$937$	4.00000	0.130674	0.0653372	+	0.997863i	$0.479188\pi$
$938$	−4.00000	−0.130605
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	−12.0000	−0.390981
$943$	−48.0000	−1.56310
$944$	−8.00000	−0.260378
$945$	0	0
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	30.0000	0.973329
$951$	18.0000	0.583690
$952$	4.00000	0.129641
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	8.00000	0.258468
$959$	−10.0000	−0.322917
$960$	0	0
$961$	5.00000	0.161290
$962$	−12.0000	−0.386896
$963$	−8.00000	−0.257796
$964$	28.0000	0.901819
$965$	0	0
$966$	4.00000	0.128698
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	−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
$973$	14.0000	0.448819
$974$	−28.0000	−0.897178
$975$	10.0000	0.320256
$976$	−6.00000	−0.192055
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−12.0000	−0.383718
$979$	0	0
$980$	0	0
$981$	−18.0000	−0.574696
$982$	−12.0000	−0.382935
$983$	−14.0000	−0.446531	−0.223265	−	0.974758i	$-0.571672\pi$
$983$	−14.0000	−0.446531	−0.223265	+	0.974758i	$0.571672\pi$
$984$	−12.0000	−0.382546
$985$	0	0
$986$	40.0000	1.27386
$987$	−2.00000	−0.0636607
$988$	−12.0000	−0.381771
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	6.00000	0.190500
$993$	20.0000	0.634681
$994$	0	0
$995$	0	0
$996$	14.0000	0.443607
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	4.00000	0.126618
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5082.2.a.s.1.1		1
11.10	odd	2		462.2.a.b.1.1	✓	1
33.32	even	2		1386.2.a.i.1.1		1
44.43	even	2		3696.2.a.y.1.1		1
77.76	even	2		3234.2.a.k.1.1		1
231.230	odd	2		9702.2.a.bt.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.b.1.1	✓	1	11.10	odd	2
1386.2.a.i.1.1		1	33.32	even	2
3234.2.a.k.1.1		1	77.76	even	2
3696.2.a.y.1.1		1	44.43	even	2
5082.2.a.s.1.1		1	1.1	even	1	trivial
9702.2.a.bt.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 $\F_{q}$
Belyi maps

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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