LMFDB - Embedded newform 7942.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, 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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.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:	$63.4171892853$
Analytic rank:	$1$
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	$\chi$	$=$	7942.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +4.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +5.00000 q^{13} +1.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} -4.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} -3.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -5.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} +5.00000 q^{29} +4.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +3.00000 q^{34} +4.00000 q^{35} -2.00000 q^{36} +6.00000 q^{37} +5.00000 q^{39} +4.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} -6.00000 q^{43} -1.00000 q^{44} +8.00000 q^{45} +3.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -11.0000 q^{50} -3.00000 q^{51} +5.00000 q^{52} +11.0000 q^{53} +5.00000 q^{54} +4.00000 q^{55} +1.00000 q^{56} -5.00000 q^{58} +1.00000 q^{59} -4.00000 q^{60} +6.00000 q^{61} -4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -20.0000 q^{65} +1.00000 q^{66} +13.0000 q^{67} -3.00000 q^{68} -3.00000 q^{69} -4.00000 q^{70} +12.0000 q^{71} +2.00000 q^{72} -13.0000 q^{73} -6.00000 q^{74} +11.0000 q^{75} +1.00000 q^{77} -5.00000 q^{78} +14.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +12.0000 q^{83} -1.00000 q^{84} +12.0000 q^{85} +6.00000 q^{86} +5.00000 q^{87} +1.00000 q^{88} -14.0000 q^{89} -8.00000 q^{90} -5.00000 q^{91} -3.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} -2.00000 q^{97} +6.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	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	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	4.00000	1.26491
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	1.00000	0.267261
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	2.00000	0.471405
$19$	0	0
$19$	0	0
$20$	−4.00000	−0.894427
$21$	−1.00000	−0.218218
$22$	1.00000	0.213201
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	−1.00000	−0.204124
$25$	11.0000	2.20000
$26$	−5.00000	−0.980581
$27$	−5.00000	−0.962250
$28$	−1.00000	−0.188982
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	4.00000	0.730297
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	3.00000	0.514496
$35$	4.00000	0.676123
$36$	−2.00000	−0.333333
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	5.00000	0.800641
$40$	4.00000	0.632456
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	1.00000	0.154303
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−1.00000	−0.150756
$45$	8.00000	1.19257
$46$	3.00000	0.442326
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	−11.0000	−1.55563
$51$	−3.00000	−0.420084
$52$	5.00000	0.693375
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	5.00000	0.680414
$55$	4.00000	0.539360
$56$	1.00000	0.133631
$57$	0	0
$58$	−5.00000	−0.656532
$59$	1.00000	0.130189	0.0650945	−	0.997879i	$-0.479265\pi$
$59$	1.00000	0.130189	0.0650945	+	0.997879i	$0.479265\pi$
$60$	−4.00000	−0.516398
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.00000	−0.508001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−20.0000	−2.48069
$66$	1.00000	0.123091
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	−3.00000	−0.363803
$69$	−3.00000	−0.361158
$70$	−4.00000	−0.478091
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	2.00000	0.235702
$73$	−13.0000	−1.52153	−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000	−1.52153	−0.760767	+	0.649025i	$0.775177\pi$
$74$	−6.00000	−0.697486
$75$	11.0000	1.27017
$76$	0	0
$77$	1.00000	0.113961
$78$	−5.00000	−0.566139
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−1.00000	−0.109109
$85$	12.0000	1.30158
$86$	6.00000	0.646997
$87$	5.00000	0.536056
$88$	1.00000	0.106600
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−8.00000	−0.843274
$91$	−5.00000	−0.524142
$92$	−3.00000	−0.312772
$93$	4.00000	0.414781
$94$	8.00000	0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	6.00000	0.606092
$99$	2.00000	0.201008
$100$	11.0000	1.10000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	3.00000	0.297044
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−5.00000	−0.490290
$105$	4.00000	0.390360
$106$	−11.0000	−1.06841
$107$	−15.0000	−1.45010	−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000	−1.45010	−0.725052	+	0.688694i	$0.758184\pi$
$108$	−5.00000	−0.481125
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	−4.00000	−0.381385
$111$	6.00000	0.569495
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	5.00000	0.464238
$117$	−10.0000	−0.924500
$118$	−1.00000	−0.0920575
$119$	3.00000	0.275010
$120$	4.00000	0.365148
$121$	1.00000	0.0909091
$122$	−6.00000	−0.543214
$123$	2.00000	0.180334
$124$	4.00000	0.359211
$125$	−24.0000	−2.14663
$126$	−2.00000	−0.178174
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	−1.00000	−0.0883883
$129$	−6.00000	−0.528271
$130$	20.0000	1.75412
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−13.0000	−1.12303
$135$	20.0000	1.72133
$136$	3.00000	0.257248
$137$	15.0000	1.28154	0.640768	−	0.767734i	$-0.278616\pi$
$137$	15.0000	1.28154	0.640768	+	0.767734i	$0.278616\pi$
$138$	3.00000	0.255377
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	4.00000	0.338062
$141$	−8.00000	−0.673722
$142$	−12.0000	−1.00702
$143$	−5.00000	−0.418121
$144$	−2.00000	−0.166667
$145$	−20.0000	−1.66091
$146$	13.0000	1.07589
$147$	−6.00000	−0.494872
$148$	6.00000	0.493197
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−11.0000	−0.898146
$151$	−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
$153$	6.00000	0.485071
$154$	−1.00000	−0.0805823
$155$	−16.0000	−1.28515
$156$	5.00000	0.400320
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	−14.0000	−1.11378
$159$	11.0000	0.872357
$160$	4.00000	0.316228
$161$	3.00000	0.236433
$162$	−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	$0.449930\pi$
$164$	2.00000	0.156174
$165$	4.00000	0.311400
$166$	−12.0000	−0.931381
$167$	−10.0000	−0.773823	−0.386912	−	0.922117i	$-0.626458\pi$
$167$	−10.0000	−0.773823	−0.386912	+	0.922117i	$0.626458\pi$
$168$	1.00000	0.0771517
$169$	12.0000	0.923077
$170$	−12.0000	−0.920358
$171$	0	0
$172$	−6.00000	−0.457496
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−5.00000	−0.379049
$175$	−11.0000	−0.831522
$176$	−1.00000	−0.0753778
$177$	1.00000	0.0751646
$178$	14.0000	1.04934
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	8.00000	0.596285
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	5.00000	0.370625
$183$	6.00000	0.443533
$184$	3.00000	0.221163
$185$	−24.0000	−1.76452
$186$	−4.00000	−0.293294
$187$	3.00000	0.219382
$188$	−8.00000	−0.583460
$189$	5.00000	0.363696
$190$	0	0
$191$	1.00000	0.0723575	0.0361787	−	0.999345i	$-0.488481\pi$
$191$	1.00000	0.0723575	0.0361787	+	0.999345i	$0.488481\pi$
$192$	1.00000	0.0721688
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	2.00000	0.143592
$195$	−20.0000	−1.43223
$196$	−6.00000	−0.428571
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	−2.00000	−0.142134
$199$	−3.00000	−0.212664	−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000	−0.212664	−0.106332	+	0.994331i	$0.533911\pi$
$200$	−11.0000	−0.777817
$201$	13.0000	0.916949
$202$	10.0000	0.703598
$203$	−5.00000	−0.350931
$204$	−3.00000	−0.210042
$205$	−8.00000	−0.558744
$206$	16.0000	1.11477
$207$	6.00000	0.417029
$208$	5.00000	0.346688
$209$	0	0
$210$	−4.00000	−0.276026
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	11.0000	0.755483
$213$	12.0000	0.822226
$214$	15.0000	1.02538
$215$	24.0000	1.63679
$216$	5.00000	0.340207
$217$	−4.00000	−0.271538
$218$	5.00000	0.338643
$219$	−13.0000	−0.878459
$220$	4.00000	0.269680
$221$	−15.0000	−1.00901
$222$	−6.00000	−0.402694
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	1.00000	0.0668153
$225$	−22.0000	−1.46667
$226$	6.00000	0.399114
$227$	−5.00000	−0.331862	−0.165931	−	0.986137i	$-0.553063\pi$
$227$	−5.00000	−0.331862	−0.165931	+	0.986137i	$0.553063\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−12.0000	−0.791257
$231$	1.00000	0.0657952
$232$	−5.00000	−0.328266
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	10.0000	0.653720
$235$	32.0000	2.08745
$236$	1.00000	0.0650945
$237$	14.0000	0.909398
$238$	−3.00000	−0.194461
$239$	−29.0000	−1.87585	−0.937927	−	0.346833i	$-0.887257\pi$
$239$	−29.0000	−1.87585	−0.937927	+	0.346833i	$0.887257\pi$
$240$	−4.00000	−0.258199
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	−1.00000	−0.0642824
$243$	16.0000	1.02640
$244$	6.00000	0.384111
$245$	24.0000	1.53330
$246$	−2.00000	−0.127515
$247$	0	0
$248$	−4.00000	−0.254000
$249$	12.0000	0.760469
$250$	24.0000	1.51789
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	2.00000	0.125988
$253$	3.00000	0.188608
$254$	−10.0000	−0.627456
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	−20.0000	−1.24757	−0.623783	−	0.781598i	$-0.714405\pi$
$257$	−20.0000	−1.24757	−0.623783	+	0.781598i	$0.714405\pi$
$258$	6.00000	0.373544
$259$	−6.00000	−0.372822
$260$	−20.0000	−1.24035
$261$	−10.0000	−0.618984
$262$	−4.00000	−0.247121
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	1.00000	0.0615457
$265$	−44.0000	−2.70290
$266$	0	0
$267$	−14.0000	−0.856786
$268$	13.0000	0.794101
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	−20.0000	−1.21716
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	−3.00000	−0.181902
$273$	−5.00000	−0.302614
$274$	−15.0000	−0.906183
$275$	−11.0000	−0.663325
$276$	−3.00000	−0.180579
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−8.00000	−0.479808
$279$	−8.00000	−0.478947
$280$	−4.00000	−0.239046
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	8.00000	0.476393
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	5.00000	0.295656
$287$	−2.00000	−0.118056
$288$	2.00000	0.117851
$289$	−8.00000	−0.470588
$290$	20.0000	1.17444
$291$	−2.00000	−0.117242
$292$	−13.0000	−0.760767
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	6.00000	0.349927
$295$	−4.00000	−0.232889
$296$	−6.00000	−0.348743
$297$	5.00000	0.290129
$298$	−6.00000	−0.347571
$299$	−15.0000	−0.867472
$300$	11.0000	0.635085
$301$	6.00000	0.345834
$302$	8.00000	0.460348
$303$	−10.0000	−0.574485
$304$	0	0
$305$	−24.0000	−1.37424
$306$	−6.00000	−0.342997
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	1.00000	0.0569803
$309$	−16.0000	−0.910208
$310$	16.0000	0.908739
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	−5.00000	−0.283069
$313$	−31.0000	−1.75222	−0.876112	−	0.482108i	$-0.839871\pi$
$313$	−31.0000	−1.75222	−0.876112	+	0.482108i	$0.839871\pi$
$314$	4.00000	0.225733
$315$	−8.00000	−0.450749
$316$	14.0000	0.787562
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	−11.0000	−0.616849
$319$	−5.00000	−0.279946
$320$	−4.00000	−0.223607
$321$	−15.0000	−0.837218
$322$	−3.00000	−0.167183
$323$	0	0
$324$	1.00000	0.0555556
$325$	55.0000	3.05085
$326$	−4.00000	−0.221540
$327$	−5.00000	−0.276501
$328$	−2.00000	−0.110432
$329$	8.00000	0.441054
$330$	−4.00000	−0.220193
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	12.0000	0.658586
$333$	−12.0000	−0.657596
$334$	10.0000	0.547176
$335$	−52.0000	−2.84106
$336$	−1.00000	−0.0545545
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	−12.0000	−0.652714
$339$	−6.00000	−0.325875
$340$	12.0000	0.650791
$341$	−4.00000	−0.216612
$342$	0	0
$343$	13.0000	0.701934
$344$	6.00000	0.323498
$345$	12.0000	0.646058
$346$	18.0000	0.967686
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	5.00000	0.268028
$349$	−24.0000	−1.28469	−0.642345	−	0.766415i	$-0.722038\pi$
$349$	−24.0000	−1.28469	−0.642345	+	0.766415i	$0.722038\pi$
$350$	11.0000	0.587975
$351$	−25.0000	−1.33440
$352$	1.00000	0.0533002
$353$	29.0000	1.54351	0.771757	−	0.635917i	$-0.219378\pi$
$353$	29.0000	1.54351	0.771757	+	0.635917i	$0.219378\pi$
$354$	−1.00000	−0.0531494
$355$	−48.0000	−2.54758
$356$	−14.0000	−0.741999
$357$	3.00000	0.158777
$358$	−12.0000	−0.634220
$359$	−31.0000	−1.63612	−0.818059	−	0.575135i	$-0.804950\pi$
$359$	−31.0000	−1.63612	−0.818059	+	0.575135i	$0.804950\pi$
$360$	−8.00000	−0.421637
$361$	0	0
$362$	−22.0000	−1.15629
$363$	1.00000	0.0524864
$364$	−5.00000	−0.262071
$365$	52.0000	2.72180
$366$	−6.00000	−0.313625
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	−3.00000	−0.156386
$369$	−4.00000	−0.208232
$370$	24.0000	1.24770
$371$	−11.0000	−0.571092
$372$	4.00000	0.207390
$373$	31.0000	1.60512	0.802560	−	0.596572i	$-0.203471\pi$
$373$	31.0000	1.60512	0.802560	+	0.596572i	$0.203471\pi$
$374$	−3.00000	−0.155126
$375$	−24.0000	−1.23935
$376$	8.00000	0.412568
$377$	25.0000	1.28757
$378$	−5.00000	−0.257172
$379$	−23.0000	−1.18143	−0.590715	−	0.806880i	$-0.701154\pi$
$379$	−23.0000	−1.18143	−0.590715	+	0.806880i	$0.701154\pi$
$380$	0	0
$381$	10.0000	0.512316
$382$	−1.00000	−0.0511645
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	−1.00000	−0.0510310
$385$	−4.00000	−0.203859
$386$	−4.00000	−0.203595
$387$	12.0000	0.609994
$388$	−2.00000	−0.101535
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	20.0000	1.01274
$391$	9.00000	0.455150
$392$	6.00000	0.303046
$393$	4.00000	0.201773
$394$	8.00000	0.403034
$395$	−56.0000	−2.81767
$396$	2.00000	0.100504
$397$	4.00000	0.200754	0.100377	−	0.994949i	$-0.467995\pi$
$397$	4.00000	0.200754	0.100377	+	0.994949i	$0.467995\pi$
$398$	3.00000	0.150376
$399$	0	0
$400$	11.0000	0.550000
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	−13.0000	−0.648381
$403$	20.0000	0.996271
$404$	−10.0000	−0.497519
$405$	−4.00000	−0.198762
$406$	5.00000	0.248146
$407$	−6.00000	−0.297409
$408$	3.00000	0.148522
$409$	12.0000	0.593362	0.296681	−	0.954977i	$-0.404120\pi$
$409$	12.0000	0.593362	0.296681	+	0.954977i	$0.404120\pi$
$410$	8.00000	0.395092
$411$	15.0000	0.739895
$412$	−16.0000	−0.788263
$413$	−1.00000	−0.0492068
$414$	−6.00000	−0.294884
$415$	−48.0000	−2.35623
$416$	−5.00000	−0.245145
$417$	8.00000	0.391762
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	4.00000	0.195180
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	−15.0000	−0.730189
$423$	16.0000	0.777947
$424$	−11.0000	−0.534207
$425$	−33.0000	−1.60074
$426$	−12.0000	−0.581402
$427$	−6.00000	−0.290360
$428$	−15.0000	−0.725052
$429$	−5.00000	−0.241402
$430$	−24.0000	−1.15738
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−5.00000	−0.240563
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	4.00000	0.192006
$435$	−20.0000	−0.958927
$436$	−5.00000	−0.239457
$437$	0	0
$438$	13.0000	0.621164
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	−4.00000	−0.190693
$441$	12.0000	0.571429
$442$	15.0000	0.713477
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	6.00000	0.284747
$445$	56.0000	2.65465
$446$	−2.00000	−0.0947027
$447$	6.00000	0.283790
$448$	−1.00000	−0.0472456
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	22.0000	1.03709
$451$	−2.00000	−0.0941763
$452$	−6.00000	−0.282216
$453$	−8.00000	−0.375873
$454$	5.00000	0.234662
$455$	20.0000	0.937614
$456$	0	0
$457$	−33.0000	−1.54367	−0.771837	−	0.635820i	$-0.780662\pi$
$457$	−33.0000	−1.54367	−0.771837	+	0.635820i	$0.780662\pi$
$458$	6.00000	0.280362
$459$	15.0000	0.700140
$460$	12.0000	0.559503
$461$	−8.00000	−0.372597	−0.186299	−	0.982493i	$-0.559649\pi$
$461$	−8.00000	−0.372597	−0.186299	+	0.982493i	$0.559649\pi$
$462$	−1.00000	−0.0465242
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	5.00000	0.232119
$465$	−16.0000	−0.741982
$466$	−18.0000	−0.833834
$467$	−38.0000	−1.75843	−0.879215	−	0.476425i	$-0.841932\pi$
$467$	−38.0000	−1.75843	−0.879215	+	0.476425i	$0.841932\pi$
$468$	−10.0000	−0.462250
$469$	−13.0000	−0.600284
$470$	−32.0000	−1.47605
$471$	−4.00000	−0.184310
$472$	−1.00000	−0.0460287
$473$	6.00000	0.275880
$474$	−14.0000	−0.643041
$475$	0	0
$476$	3.00000	0.137505
$477$	−22.0000	−1.00731
$478$	29.0000	1.32643
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	4.00000	0.182574
$481$	30.0000	1.36788
$482$	20.0000	0.910975
$483$	3.00000	0.136505
$484$	1.00000	0.0454545
$485$	8.00000	0.363261
$486$	−16.0000	−0.725775
$487$	−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
$489$	4.00000	0.180886
$490$	−24.0000	−1.08421
$491$	−14.0000	−0.631811	−0.315906	−	0.948791i	$-0.602308\pi$
$491$	−14.0000	−0.631811	−0.315906	+	0.948791i	$0.602308\pi$
$492$	2.00000	0.0901670
$493$	−15.0000	−0.675566
$494$	0	0
$495$	−8.00000	−0.359573
$496$	4.00000	0.179605
$497$	−12.0000	−0.538274
$498$	−12.0000	−0.537733
$499$	−2.00000	−0.0895323	−0.0447661	−	0.998997i	$-0.514254\pi$
$499$	−2.00000	−0.0895323	−0.0447661	+	0.998997i	$0.514254\pi$
$500$	−24.0000	−1.07331
$501$	−10.0000	−0.446767
$502$	28.0000	1.24970
$503$	15.0000	0.668817	0.334408	−	0.942428i	$-0.391463\pi$
$503$	15.0000	0.668817	0.334408	+	0.942428i	$0.391463\pi$
$504$	−2.00000	−0.0890871
$505$	40.0000	1.77998
$506$	−3.00000	−0.133366
$507$	12.0000	0.532939
$508$	10.0000	0.443678
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	−12.0000	−0.531369
$511$	13.0000	0.575086
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	20.0000	0.882162
$515$	64.0000	2.82018
$516$	−6.00000	−0.264135
$517$	8.00000	0.351840
$518$	6.00000	0.263625
$519$	−18.0000	−0.790112
$520$	20.0000	0.877058
$521$	−44.0000	−1.92767	−0.963837	−	0.266491i	$-0.914136\pi$
$521$	−44.0000	−1.92767	−0.963837	+	0.266491i	$0.914136\pi$
$522$	10.0000	0.437688
$523$	45.0000	1.96771	0.983856	−	0.178960i	$-0.0572733\pi$
$523$	45.0000	1.96771	0.983856	+	0.178960i	$0.0572733\pi$
$524$	4.00000	0.174741
$525$	−11.0000	−0.480079
$526$	0	0
$527$	−12.0000	−0.522728
$528$	−1.00000	−0.0435194
$529$	−14.0000	−0.608696
$530$	44.0000	1.91124
$531$	−2.00000	−0.0867926
$532$	0	0
$533$	10.0000	0.433148
$534$	14.0000	0.605839
$535$	60.0000	2.59403
$536$	−13.0000	−0.561514
$537$	12.0000	0.517838
$538$	2.00000	0.0862261
$539$	6.00000	0.258438
$540$	20.0000	0.860663
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	25.0000	1.07384
$543$	22.0000	0.944110
$544$	3.00000	0.128624
$545$	20.0000	0.856706
$546$	5.00000	0.213980
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	15.0000	0.640768
$549$	−12.0000	−0.512148
$550$	11.0000	0.469042
$551$	0	0
$552$	3.00000	0.127688
$553$	−14.0000	−0.595341
$554$	2.00000	0.0849719
$555$	−24.0000	−1.01874
$556$	8.00000	0.339276
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	8.00000	0.338667
$559$	−30.0000	−1.26886
$560$	4.00000	0.169031
$561$	3.00000	0.126660
$562$	−4.00000	−0.168730
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	−8.00000	−0.336861
$565$	24.0000	1.00969
$566$	14.0000	0.588464
$567$	−1.00000	−0.0419961
$568$	−12.0000	−0.503509
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	−5.00000	−0.209061
$573$	1.00000	0.0417756
$574$	2.00000	0.0834784
$575$	−33.0000	−1.37620
$576$	−2.00000	−0.0833333
$577$	−45.0000	−1.87337	−0.936687	−	0.350167i	$-0.886125\pi$
$577$	−45.0000	−1.87337	−0.936687	+	0.350167i	$0.886125\pi$
$578$	8.00000	0.332756
$579$	4.00000	0.166234
$580$	−20.0000	−0.830455
$581$	−12.0000	−0.497844
$582$	2.00000	0.0829027
$583$	−11.0000	−0.455573
$584$	13.0000	0.537944
$585$	40.0000	1.65380
$586$	21.0000	0.867502
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−6.00000	−0.247436
$589$	0	0
$590$	4.00000	0.164677
$591$	−8.00000	−0.329076
$592$	6.00000	0.246598
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	−5.00000	−0.205152
$595$	−12.0000	−0.491952
$596$	6.00000	0.245770
$597$	−3.00000	−0.122782
$598$	15.0000	0.613396
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	−11.0000	−0.449073
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	−6.00000	−0.244542
$603$	−26.0000	−1.05880
$604$	−8.00000	−0.325515
$605$	−4.00000	−0.162623
$606$	10.0000	0.406222
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	24.0000	0.971732
$611$	−40.0000	−1.61823
$612$	6.00000	0.242536
$613$	44.0000	1.77714	0.888572	−	0.458738i	$-0.151698\pi$
$613$	44.0000	1.77714	0.888572	+	0.458738i	$0.151698\pi$
$614$	−28.0000	−1.12999
$615$	−8.00000	−0.322591
$616$	−1.00000	−0.0402911
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	16.0000	0.643614
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	−16.0000	−0.642575
$621$	15.0000	0.601929
$622$	15.0000	0.601445
$623$	14.0000	0.560898
$624$	5.00000	0.200160
$625$	41.0000	1.64000
$626$	31.0000	1.23901
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−18.0000	−0.717707
$630$	8.00000	0.318728
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−14.0000	−0.556890
$633$	15.0000	0.596196
$634$	3.00000	0.119145
$635$	−40.0000	−1.58735
$636$	11.0000	0.436178
$637$	−30.0000	−1.18864
$638$	5.00000	0.197952
$639$	−24.0000	−0.949425
$640$	4.00000	0.158114
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	15.0000	0.592003
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	3.00000	0.118217
$645$	24.0000	0.944999
$646$	0	0
$647$	21.0000	0.825595	0.412798	−	0.910823i	$-0.364552\pi$
$647$	21.0000	0.825595	0.412798	+	0.910823i	$0.364552\pi$
$648$	−1.00000	−0.0392837
$649$	−1.00000	−0.0392534
$650$	−55.0000	−2.15728
$651$	−4.00000	−0.156772
$652$	4.00000	0.156652
$653$	−50.0000	−1.95665	−0.978326	−	0.207072i	$-0.933606\pi$
$653$	−50.0000	−1.95665	−0.978326	+	0.207072i	$0.933606\pi$
$654$	5.00000	0.195515
$655$	−16.0000	−0.625172
$656$	2.00000	0.0780869
$657$	26.0000	1.01436
$658$	−8.00000	−0.311872
$659$	1.00000	0.0389545	0.0194772	−	0.999810i	$-0.493800\pi$
$659$	1.00000	0.0389545	0.0194772	+	0.999810i	$0.493800\pi$
$660$	4.00000	0.155700
$661$	−11.0000	−0.427850	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−11.0000	−0.427850	−0.213925	+	0.976850i	$0.568625\pi$
$662$	−11.0000	−0.427527
$663$	−15.0000	−0.582552
$664$	−12.0000	−0.465690
$665$	0	0
$666$	12.0000	0.464991
$667$	−15.0000	−0.580802
$668$	−10.0000	−0.386912
$669$	2.00000	0.0773245
$670$	52.0000	2.00894
$671$	−6.00000	−0.231627
$672$	1.00000	0.0385758
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	18.0000	0.693334
$675$	−55.0000	−2.11695
$676$	12.0000	0.461538
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	6.00000	0.230429
$679$	2.00000	0.0767530
$680$	−12.0000	−0.460179
$681$	−5.00000	−0.191600
$682$	4.00000	0.153168
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	−60.0000	−2.29248
$686$	−13.0000	−0.496342
$687$	−6.00000	−0.228914
$688$	−6.00000	−0.228748
$689$	55.0000	2.09533
$690$	−12.0000	−0.456832
$691$	−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
$693$	−2.00000	−0.0759737
$694$	28.0000	1.06287
$695$	−32.0000	−1.21383
$696$	−5.00000	−0.189525
$697$	−6.00000	−0.227266
$698$	24.0000	0.908413
$699$	18.0000	0.680823
$700$	−11.0000	−0.415761
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	25.0000	0.943564
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	32.0000	1.20519
$706$	−29.0000	−1.09143
$707$	10.0000	0.376089
$708$	1.00000	0.0375823
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	48.0000	1.80141
$711$	−28.0000	−1.05008
$712$	14.0000	0.524672
$713$	−12.0000	−0.449404
$714$	−3.00000	−0.112272
$715$	20.0000	0.747958
$716$	12.0000	0.448461
$717$	−29.0000	−1.08302
$718$	31.0000	1.15691
$719$	9.00000	0.335643	0.167822	−	0.985817i	$-0.446327\pi$
$719$	9.00000	0.335643	0.167822	+	0.985817i	$0.446327\pi$
$720$	8.00000	0.298142
$721$	16.0000	0.595871
$722$	0	0
$723$	−20.0000	−0.743808
$724$	22.0000	0.817624
$725$	55.0000	2.04265
$726$	−1.00000	−0.0371135
$727$	9.00000	0.333792	0.166896	−	0.985975i	$-0.446626\pi$
$727$	9.00000	0.333792	0.166896	+	0.985975i	$0.446626\pi$
$728$	5.00000	0.185312
$729$	13.0000	0.481481
$730$	−52.0000	−1.92461
$731$	18.0000	0.665754
$732$	6.00000	0.221766
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	−4.00000	−0.147643
$735$	24.0000	0.885253
$736$	3.00000	0.110581
$737$	−13.0000	−0.478861
$738$	4.00000	0.147242
$739$	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
$741$	0	0
$742$	11.0000	0.403823
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	−4.00000	−0.146647
$745$	−24.0000	−0.879292
$746$	−31.0000	−1.13499
$747$	−24.0000	−0.878114
$748$	3.00000	0.109691
$749$	15.0000	0.548088
$750$	24.0000	0.876356
$751$	22.0000	0.802791	0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000	0.802791	0.401396	+	0.915905i	$0.368525\pi$
$752$	−8.00000	−0.291730
$753$	−28.0000	−1.02038
$754$	−25.0000	−0.910446
$755$	32.0000	1.16460
$756$	5.00000	0.181848
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	23.0000	0.835398
$759$	3.00000	0.108893
$760$	0	0
$761$	25.0000	0.906249	0.453125	−	0.891447i	$-0.350309\pi$
$761$	25.0000	0.906249	0.453125	+	0.891447i	$0.350309\pi$
$762$	−10.0000	−0.362262
$763$	5.00000	0.181012
$764$	1.00000	0.0361787
$765$	−24.0000	−0.867722
$766$	14.0000	0.505841
$767$	5.00000	0.180540
$768$	1.00000	0.0360844
$769$	−17.0000	−0.613036	−0.306518	−	0.951865i	$-0.599164\pi$
$769$	−17.0000	−0.613036	−0.306518	+	0.951865i	$0.599164\pi$
$770$	4.00000	0.144150
$771$	−20.0000	−0.720282
$772$	4.00000	0.143963
$773$	−3.00000	−0.107903	−0.0539513	−	0.998544i	$-0.517182\pi$
$773$	−3.00000	−0.107903	−0.0539513	+	0.998544i	$0.517182\pi$
$774$	−12.0000	−0.431331
$775$	44.0000	1.58053
$776$	2.00000	0.0717958
$777$	−6.00000	−0.215249
$778$	−10.0000	−0.358517
$779$	0	0
$780$	−20.0000	−0.716115
$781$	−12.0000	−0.429394
$782$	−9.00000	−0.321839
$783$	−25.0000	−0.893427
$784$	−6.00000	−0.214286
$785$	16.0000	0.571064
$786$	−4.00000	−0.142675
$787$	31.0000	1.10503	0.552515	−	0.833503i	$-0.313668\pi$
$787$	31.0000	1.10503	0.552515	+	0.833503i	$0.313668\pi$
$788$	−8.00000	−0.284988
$789$	0	0
$790$	56.0000	1.99239
$791$	6.00000	0.213335
$792$	−2.00000	−0.0710669
$793$	30.0000	1.06533
$794$	−4.00000	−0.141955
$795$	−44.0000	−1.56052
$796$	−3.00000	−0.106332
$797$	−9.00000	−0.318796	−0.159398	−	0.987214i	$-0.550955\pi$
$797$	−9.00000	−0.318796	−0.159398	+	0.987214i	$0.550955\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	−11.0000	−0.388909
$801$	28.0000	0.989331
$802$	−34.0000	−1.20058
$803$	13.0000	0.458760
$804$	13.0000	0.458475
$805$	−12.0000	−0.422944
$806$	−20.0000	−0.704470
$807$	−2.00000	−0.0704033
$808$	10.0000	0.351799
$809$	31.0000	1.08990	0.544951	−	0.838468i	$-0.316548\pi$
$809$	31.0000	1.08990	0.544951	+	0.838468i	$0.316548\pi$
$810$	4.00000	0.140546
$811$	5.00000	0.175574	0.0877869	−	0.996139i	$-0.472021\pi$
$811$	5.00000	0.175574	0.0877869	+	0.996139i	$0.472021\pi$
$812$	−5.00000	−0.175466
$813$	−25.0000	−0.876788
$814$	6.00000	0.210300
$815$	−16.0000	−0.560456
$816$	−3.00000	−0.105021
$817$	0	0
$818$	−12.0000	−0.419570
$819$	10.0000	0.349428
$820$	−8.00000	−0.279372
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	−15.0000	−0.523185
$823$	−21.0000	−0.732014	−0.366007	−	0.930612i	$-0.619275\pi$
$823$	−21.0000	−0.732014	−0.366007	+	0.930612i	$0.619275\pi$
$824$	16.0000	0.557386
$825$	−11.0000	−0.382971
$826$	1.00000	0.0347945
$827$	47.0000	1.63435	0.817175	−	0.576390i	$-0.195539\pi$
$827$	47.0000	1.63435	0.817175	+	0.576390i	$0.195539\pi$
$828$	6.00000	0.208514
$829$	−3.00000	−0.104194	−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000	−0.104194	−0.0520972	+	0.998642i	$0.516591\pi$
$830$	48.0000	1.66610
$831$	−2.00000	−0.0693792
$832$	5.00000	0.173344
$833$	18.0000	0.623663
$834$	−8.00000	−0.277017
$835$	40.0000	1.38426
$836$	0	0
$837$	−20.0000	−0.691301
$838$	−24.0000	−0.829066
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	−4.00000	−0.138013
$841$	−4.00000	−0.137931
$842$	1.00000	0.0344623
$843$	4.00000	0.137767
$844$	15.0000	0.516321
$845$	−48.0000	−1.65125
$846$	−16.0000	−0.550091
$847$	−1.00000	−0.0343604
$848$	11.0000	0.377742
$849$	−14.0000	−0.480479
$850$	33.0000	1.13189
$851$	−18.0000	−0.617032
$852$	12.0000	0.411113
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	15.0000	0.512689
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	5.00000	0.170697
$859$	−6.00000	−0.204717	−0.102359	−	0.994748i	$-0.532639\pi$
$859$	−6.00000	−0.204717	−0.102359	+	0.994748i	$0.532639\pi$
$860$	24.0000	0.818393
$861$	−2.00000	−0.0681598
$862$	0	0
$863$	28.0000	0.953131	0.476566	−	0.879139i	$-0.341881\pi$
$863$	28.0000	0.953131	0.476566	+	0.879139i	$0.341881\pi$
$864$	5.00000	0.170103
$865$	72.0000	2.44807
$866$	28.0000	0.951479
$867$	−8.00000	−0.271694
$868$	−4.00000	−0.135769
$869$	−14.0000	−0.474917
$870$	20.0000	0.678064
$871$	65.0000	2.20244
$872$	5.00000	0.169321
$873$	4.00000	0.135379
$874$	0	0
$875$	24.0000	0.811348
$876$	−13.0000	−0.439229
$877$	3.00000	0.101303	0.0506514	−	0.998716i	$-0.483870\pi$
$877$	3.00000	0.101303	0.0506514	+	0.998716i	$0.483870\pi$
$878$	22.0000	0.742464
$879$	−21.0000	−0.708312
$880$	4.00000	0.134840
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	−12.0000	−0.404061
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	−15.0000	−0.504505
$885$	−4.00000	−0.134459
$886$	−6.00000	−0.201574
$887$	−26.0000	−0.872995	−0.436497	−	0.899706i	$-0.643781\pi$
$887$	−26.0000	−0.872995	−0.436497	+	0.899706i	$0.643781\pi$
$888$	−6.00000	−0.201347
$889$	−10.0000	−0.335389
$890$	−56.0000	−1.87712
$891$	−1.00000	−0.0335013
$892$	2.00000	0.0669650
$893$	0	0
$894$	−6.00000	−0.200670
$895$	−48.0000	−1.60446
$896$	1.00000	0.0334077
$897$	−15.0000	−0.500835
$898$	0	0
$899$	20.0000	0.667037
$900$	−22.0000	−0.733333
$901$	−33.0000	−1.09939
$902$	2.00000	0.0665927
$903$	6.00000	0.199667
$904$	6.00000	0.199557
$905$	−88.0000	−2.92522
$906$	8.00000	0.265782
$907$	−1.00000	−0.0332045	−0.0166022	−	0.999862i	$-0.505285\pi$
$907$	−1.00000	−0.0332045	−0.0166022	+	0.999862i	$0.505285\pi$
$908$	−5.00000	−0.165931
$909$	20.0000	0.663358
$910$	−20.0000	−0.662994
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	33.0000	1.09154
$915$	−24.0000	−0.793416
$916$	−6.00000	−0.198246
$917$	−4.00000	−0.132092
$918$	−15.0000	−0.495074
$919$	−47.0000	−1.55039	−0.775193	−	0.631724i	$-0.782348\pi$
$919$	−47.0000	−1.55039	−0.775193	+	0.631724i	$0.782348\pi$
$920$	−12.0000	−0.395628
$921$	28.0000	0.922631
$922$	8.00000	0.263466
$923$	60.0000	1.97492
$924$	1.00000	0.0328976
$925$	66.0000	2.17007
$926$	−8.00000	−0.262896
$927$	32.0000	1.05102
$928$	−5.00000	−0.164133
$929$	−7.00000	−0.229663	−0.114831	−	0.993385i	$-0.536633\pi$
$929$	−7.00000	−0.229663	−0.114831	+	0.993385i	$0.536633\pi$
$930$	16.0000	0.524661
$931$	0	0
$932$	18.0000	0.589610
$933$	−15.0000	−0.491078
$934$	38.0000	1.24340
$935$	−12.0000	−0.392442
$936$	10.0000	0.326860
$937$	47.0000	1.53542	0.767712	−	0.640796i	$-0.221395\pi$
$937$	47.0000	1.53542	0.767712	+	0.640796i	$0.221395\pi$
$938$	13.0000	0.424465
$939$	−31.0000	−1.01165
$940$	32.0000	1.04372
$941$	25.0000	0.814977	0.407488	−	0.913210i	$-0.366405\pi$
$941$	25.0000	0.814977	0.407488	+	0.913210i	$0.366405\pi$
$942$	4.00000	0.130327
$943$	−6.00000	−0.195387
$944$	1.00000	0.0325472
$945$	−20.0000	−0.650600
$946$	−6.00000	−0.195077
$947$	−42.0000	−1.36482	−0.682408	−	0.730971i	$-0.739067\pi$
$947$	−42.0000	−1.36482	−0.682408	+	0.730971i	$0.739067\pi$
$948$	14.0000	0.454699
$949$	−65.0000	−2.10999
$950$	0	0
$951$	−3.00000	−0.0972817
$952$	−3.00000	−0.0972306
$953$	−8.00000	−0.259145	−0.129573	−	0.991570i	$-0.541361\pi$
$953$	−8.00000	−0.259145	−0.129573	+	0.991570i	$0.541361\pi$
$954$	22.0000	0.712276
$955$	−4.00000	−0.129437
$956$	−29.0000	−0.937927
$957$	−5.00000	−0.161627
$958$	−24.0000	−0.775405
$959$	−15.0000	−0.484375
$960$	−4.00000	−0.129099
$961$	−15.0000	−0.483871
$962$	−30.0000	−0.967239
$963$	30.0000	0.966736
$964$	−20.0000	−0.644157
$965$	−16.0000	−0.515058
$966$	−3.00000	−0.0965234
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−8.00000	−0.256865
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	16.0000	0.513200
$973$	−8.00000	−0.256468
$974$	28.0000	0.897178
$975$	55.0000	1.76141
$976$	6.00000	0.192055
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	−4.00000	−0.127906
$979$	14.0000	0.447442
$980$	24.0000	0.766652
$981$	10.0000	0.319275
$982$	14.0000	0.446758
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	−2.00000	−0.0637577
$985$	32.0000	1.01960
$986$	15.0000	0.477697
$987$	8.00000	0.254643
$988$	0	0
$989$	18.0000	0.572367
$990$	8.00000	0.254257
$991$	−50.0000	−1.58830	−0.794151	−	0.607720i	$-0.792084\pi$
$991$	−50.0000	−1.58830	−0.794151	+	0.607720i	$0.792084\pi$
$992$	−4.00000	−0.127000
$993$	11.0000	0.349074
$994$	12.0000	0.380617
$995$	12.0000	0.380426
$996$	12.0000	0.380235
$997$	4.00000	0.126681	0.0633406	−	0.997992i	$-0.479825\pi$
$997$	4.00000	0.126681	0.0633406	+	0.997992i	$0.479825\pi$
$998$	2.00000	0.0633089
$999$	−30.0000	−0.949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.h.1.1	✓	1
19.18	odd	2		7942.2.a.n.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7942.2.a.h.1.1	✓	1	1.1	even	1	trivial
7942.2.a.n.1.1	yes	1	19.18	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:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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