LMFDB - Embedded newform 7942.2.a.o.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:	no (minimal twist has level 418)
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)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} -4.00000 q^{10} +1.00000 q^{11} +7.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -3.00000 q^{18} -4.00000 q^{20} +1.00000 q^{22} +4.00000 q^{23} +11.0000 q^{25} +7.00000 q^{26} -4.00000 q^{28} -3.00000 q^{29} +1.00000 q^{32} +16.0000 q^{35} -3.00000 q^{36} +8.00000 q^{37} -4.00000 q^{40} -12.0000 q^{41} +7.00000 q^{43} +1.00000 q^{44} +12.0000 q^{45} +4.00000 q^{46} -5.00000 q^{47} +9.00000 q^{49} +11.0000 q^{50} +7.00000 q^{52} +6.00000 q^{53} -4.00000 q^{55} -4.00000 q^{56} -3.00000 q^{58} +6.00000 q^{59} -5.00000 q^{61} +12.0000 q^{63} +1.00000 q^{64} -28.0000 q^{65} -8.00000 q^{67} +16.0000 q^{70} -9.00000 q^{71} -3.00000 q^{72} +10.0000 q^{73} +8.00000 q^{74} -4.00000 q^{77} -6.00000 q^{79} -4.00000 q^{80} +9.00000 q^{81} -12.0000 q^{82} -3.00000 q^{83} +7.00000 q^{86} +1.00000 q^{88} -1.00000 q^{89} +12.0000 q^{90} -28.0000 q^{91} +4.00000 q^{92} -5.00000 q^{94} -5.00000 q^{97} +9.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	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	$0.852416\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	−3.00000	−1.00000
$10$	−4.00000	−1.26491
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	7.00000	1.94145	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	7.00000	1.94145	0.970725	+	0.240192i	$0.0772105\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−3.00000	−0.707107
$19$	0	0
$19$	0	0
$20$	−4.00000	−0.894427
$21$	0	0
$22$	1.00000	0.213201
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	7.00000	1.37281
$27$	0	0
$28$	−4.00000	−0.755929
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	16.0000	2.70449
$36$	−3.00000	−0.500000
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	0	0
$40$	−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	$0.886448\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	1.00000	0.150756
$45$	12.0000	1.78885
$46$	4.00000	0.589768
$47$	−5.00000	−0.729325	−0.364662	−	0.931140i	$-0.618816\pi$
$47$	−5.00000	−0.729325	−0.364662	+	0.931140i	$0.618816\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	11.0000	1.55563
$51$	0	0
$52$	7.00000	0.970725
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	−4.00000	−0.534522
$57$	0	0
$58$	−3.00000	−0.393919
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	12.0000	1.51186
$64$	1.00000	0.125000
$65$	−28.0000	−3.47297
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	0	0
$70$	16.0000	1.91237
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	−3.00000	−0.353553
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−4.00000	−0.447214
$81$	9.00000	1.00000
$82$	−12.0000	−1.32518
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	0	0
$86$	7.00000	0.754829
$87$	0	0
$88$	1.00000	0.106600
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	12.0000	1.26491
$91$	−28.0000	−2.93520
$92$	4.00000	0.417029
$93$	0	0
$94$	−5.00000	−0.515711
$95$	0	0
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	9.00000	0.909137
$99$	−3.00000	−0.301511
$100$	11.0000	1.10000
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	7.00000	0.686406
$105$	0	0
$106$	6.00000	0.582772
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	0	0
$115$	−16.0000	−1.49201
$116$	−3.00000	−0.278543
$117$	−21.0000	−1.94145
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.00000	−0.452679
$123$	0	0
$124$	0	0
$125$	−24.0000	−2.14663
$126$	12.0000	1.06904
$127$	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
$129$	0	0
$130$	−28.0000	−2.45576
$131$	−5.00000	−0.436852	−0.218426	−	0.975854i	$-0.570092\pi$
$131$	−5.00000	−0.436852	−0.218426	+	0.975854i	$0.570092\pi$
$132$	0	0
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	0	0
$137$	7.00000	0.598050	0.299025	−	0.954245i	$-0.403339\pi$
$137$	7.00000	0.598050	0.299025	+	0.954245i	$0.403339\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	16.0000	1.35225
$141$	0	0
$142$	−9.00000	−0.755263
$143$	7.00000	0.585369
$144$	−3.00000	−0.250000
$145$	12.0000	0.996546
$146$	10.0000	0.827606
$147$	0	0
$148$	8.00000	0.657596
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	0	0
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	−6.00000	−0.477334
$159$	0	0
$160$	−4.00000	−0.316228
$161$	−16.0000	−1.26098
$162$	9.00000	0.707107
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	−3.00000	−0.232845
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	36.0000	2.76923
$170$	0	0
$171$	0	0
$172$	7.00000	0.533745
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	−44.0000	−3.32609
$176$	1.00000	0.0753778
$177$	0	0
$178$	−1.00000	−0.0749532
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	12.0000	0.894427
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−28.0000	−2.07550
$183$	0	0
$184$	4.00000	0.294884
$185$	−32.0000	−2.35269
$186$	0	0
$187$	0	0
$188$	−5.00000	−0.364662
$189$	0	0
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−5.00000	−0.358979
$195$	0	0
$196$	9.00000	0.642857
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	−3.00000	−0.213201
$199$	15.0000	1.06332	0.531661	−	0.846957i	$-0.321568\pi$
$199$	15.0000	1.06332	0.531661	+	0.846957i	$0.321568\pi$
$200$	11.0000	0.777817
$201$	0	0
$202$	−15.0000	−1.05540
$203$	12.0000	0.842235
$204$	0	0
$205$	48.0000	3.35247
$206$	11.0000	0.766406
$207$	−12.0000	−0.834058
$208$	7.00000	0.485363
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−3.00000	−0.205076
$215$	−28.0000	−1.90958
$216$	0	0
$217$	0	0
$218$	−5.00000	−0.338643
$219$	0	0
$220$	−4.00000	−0.269680
$221$	0	0
$222$	0	0
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	−4.00000	−0.267261
$225$	−33.0000	−2.20000
$226$	−3.00000	−0.199557
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−16.0000	−1.05501
$231$	0	0
$232$	−3.00000	−0.196960
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−21.0000	−1.37281
$235$	20.0000	1.30466
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−5.00000	−0.320092
$245$	−36.0000	−2.29996
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−24.0000	−1.51789
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	12.0000	0.755929
$253$	4.00000	0.251478
$254$	10.0000	0.627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	−32.0000	−1.98838
$260$	−28.0000	−1.73649
$261$	9.00000	0.557086
$262$	−5.00000	−0.308901
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	−24.0000	−1.47431
$266$	0	0
$267$	0	0
$268$	−8.00000	−0.488678
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	0	0
$271$	−30.0000	−1.82237	−0.911185	−	0.411997i	$-0.864831\pi$
$271$	−30.0000	−1.82237	−0.911185	+	0.411997i	$0.864831\pi$
$272$	0	0
$273$	0	0
$274$	7.00000	0.422885
$275$	11.0000	0.663325
$276$	0	0
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	16.0000	0.956183
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	−9.00000	−0.534052
$285$	0	0
$286$	7.00000	0.413919
$287$	48.0000	2.83335
$288$	−3.00000	−0.176777
$289$	−17.0000	−1.00000
$290$	12.0000	0.704664
$291$	0	0
$292$	10.0000	0.585206
$293$	−5.00000	−0.292103	−0.146052	−	0.989277i	$-0.546657\pi$
$293$	−5.00000	−0.292103	−0.146052	+	0.989277i	$0.546657\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	8.00000	0.464991
$297$	0	0
$298$	6.00000	0.347571
$299$	28.0000	1.61928
$300$	0	0
$301$	−28.0000	−1.61389
$302$	6.00000	0.345261
$303$	0	0
$304$	0	0
$305$	20.0000	1.14520
$306$	0	0
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	0	0
$311$	−25.0000	−1.41762	−0.708810	−	0.705399i	$-0.750768\pi$
$311$	−25.0000	−1.41762	−0.708810	+	0.705399i	$0.750768\pi$
$312$	0	0
$313$	9.00000	0.508710	0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000	0.508710	0.254355	+	0.967111i	$0.418137\pi$
$314$	−14.0000	−0.790066
$315$	−48.0000	−2.70449
$316$	−6.00000	−0.337526
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	−3.00000	−0.167968
$320$	−4.00000	−0.223607
$321$	0	0
$322$	−16.0000	−0.891645
$323$	0	0
$324$	9.00000	0.500000
$325$	77.0000	4.27119
$326$	6.00000	0.332309
$327$	0	0
$328$	−12.0000	−0.662589
$329$	20.0000	1.10264
$330$	0	0
$331$	6.00000	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$332$	−3.00000	−0.164646
$333$	−24.0000	−1.31519
$334$	−16.0000	−0.875481
$335$	32.0000	1.74835
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	36.0000	1.95814
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−8.00000	−0.431959
$344$	7.00000	0.377415
$345$	0	0
$346$	2.00000	0.107521
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	13.0000	0.695874	0.347937	−	0.937518i	$-0.386882\pi$
$349$	13.0000	0.695874	0.347937	+	0.937518i	$0.386882\pi$
$350$	−44.0000	−2.35190
$351$	0	0
$352$	1.00000	0.0533002
$353$	31.0000	1.64996	0.824982	−	0.565159i	$-0.191185\pi$
$353$	31.0000	1.64996	0.824982	+	0.565159i	$0.191185\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	−16.0000	−0.845626
$359$	−2.00000	−0.105556	−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000	−0.105556	−0.0527780	+	0.998606i	$0.516808\pi$
$360$	12.0000	0.632456
$361$	0	0
$362$	−10.0000	−0.525588
$363$	0	0
$364$	−28.0000	−1.46760
$365$	−40.0000	−2.09370
$366$	0	0
$367$	35.0000	1.82699	0.913493	−	0.406855i	$-0.133375\pi$
$367$	35.0000	1.82699	0.913493	+	0.406855i	$0.133375\pi$
$368$	4.00000	0.208514
$369$	36.0000	1.87409
$370$	−32.0000	−1.66360
$371$	−24.0000	−1.24602
$372$	0	0
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	0	0
$375$	0	0
$376$	−5.00000	−0.257855
$377$	−21.0000	−1.08156
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	0	0
$382$	−3.00000	−0.153493
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	−6.00000	−0.305392
$387$	−21.0000	−1.06749
$388$	−5.00000	−0.253837
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	0	0
$392$	9.00000	0.454569
$393$	0	0
$394$	−3.00000	−0.151138
$395$	24.0000	1.20757
$396$	−3.00000	−0.150756
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	15.0000	0.751882
$399$	0	0
$400$	11.0000	0.550000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	0	0
$404$	−15.0000	−0.746278
$405$	−36.0000	−1.78885
$406$	12.0000	0.595550
$407$	8.00000	0.396545
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	48.0000	2.37055
$411$	0	0
$412$	11.0000	0.541931
$413$	−24.0000	−1.18096
$414$	−12.0000	−0.589768
$415$	12.0000	0.589057
$416$	7.00000	0.343203
$417$	0	0
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−12.0000	−0.584151
$423$	15.0000	0.729325
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	20.0000	0.967868
$428$	−3.00000	−0.145010
$429$	0	0
$430$	−28.0000	−1.35028
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	0	0
$433$	−5.00000	−0.240285	−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000	−0.240285	−0.120142	+	0.992757i	$0.538335\pi$
$434$	0	0
$435$	0	0
$436$	−5.00000	−0.239457
$437$	0	0
$438$	0	0
$439$	−36.0000	−1.71819	−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000	−1.71819	−0.859093	+	0.511819i	$0.828972\pi$
$440$	−4.00000	−0.190693
$441$	−27.0000	−1.28571
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	4.00000	0.189618
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	−4.00000	−0.188982
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	−33.0000	−1.55563
$451$	−12.0000	−0.565058
$452$	−3.00000	−0.141108
$453$	0	0
$454$	20.0000	0.938647
$455$	112.000	5.25064
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	−16.0000	−0.746004
$461$	41.0000	1.90956	0.954780	−	0.297313i	$-0.0960904\pi$
$461$	41.0000	1.90956	0.954780	+	0.297313i	$0.0960904\pi$
$462$	0	0
$463$	11.0000	0.511213	0.255607	−	0.966781i	$-0.417725\pi$
$463$	11.0000	0.511213	0.255607	+	0.966781i	$0.417725\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−18.0000	−0.833834
$467$	−34.0000	−1.57333	−0.786666	−	0.617379i	$-0.788195\pi$
$467$	−34.0000	−1.57333	−0.786666	+	0.617379i	$0.788195\pi$
$468$	−21.0000	−0.970725
$469$	32.0000	1.47762
$470$	20.0000	0.922531
$471$	0	0
$472$	6.00000	0.276172
$473$	7.00000	0.321860
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−18.0000	−0.824163
$478$	0	0
$479$	38.0000	1.73626	0.868132	−	0.496333i	$-0.165321\pi$
$479$	38.0000	1.73626	0.868132	+	0.496333i	$0.165321\pi$
$480$	0	0
$481$	56.0000	2.55338
$482$	−10.0000	−0.455488
$483$	0	0
$484$	1.00000	0.0454545
$485$	20.0000	0.908153
$486$	0	0
$487$	3.00000	0.135943	0.0679715	−	0.997687i	$-0.478347\pi$
$487$	3.00000	0.135943	0.0679715	+	0.997687i	$0.478347\pi$
$488$	−5.00000	−0.226339
$489$	0	0
$490$	−36.0000	−1.62631
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	36.0000	1.61482
$498$	0	0
$499$	18.0000	0.805791	0.402895	−	0.915246i	$-0.368004\pi$
$499$	18.0000	0.805791	0.402895	+	0.915246i	$0.368004\pi$
$500$	−24.0000	−1.07331
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−14.0000	−0.624229	−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000	−0.624229	−0.312115	+	0.950044i	$0.601037\pi$
$504$	12.0000	0.534522
$505$	60.0000	2.66996
$506$	4.00000	0.177822
$507$	0	0
$508$	10.0000	0.443678
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	−40.0000	−1.76950
$512$	1.00000	0.0441942
$513$	0	0
$514$	14.0000	0.617514
$515$	−44.0000	−1.93887
$516$	0	0
$517$	−5.00000	−0.219900
$518$	−32.0000	−1.40600
$519$	0	0
$520$	−28.0000	−1.22788
$521$	−1.00000	−0.0438108	−0.0219054	−	0.999760i	$-0.506973\pi$
$521$	−1.00000	−0.0438108	−0.0219054	+	0.999760i	$0.506973\pi$
$522$	9.00000	0.393919
$523$	13.0000	0.568450	0.284225	−	0.958758i	$-0.408264\pi$
$523$	13.0000	0.568450	0.284225	+	0.958758i	$0.408264\pi$
$524$	−5.00000	−0.218426
$525$	0	0
$526$	12.0000	0.523225
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−24.0000	−1.04249
$531$	−18.0000	−0.781133
$532$	0	0
$533$	−84.0000	−3.63844
$534$	0	0
$535$	12.0000	0.518805
$536$	−8.00000	−0.345547
$537$	0	0
$538$	−16.0000	−0.689809
$539$	9.00000	0.387657
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−30.0000	−1.28861
$543$	0	0
$544$	0	0
$545$	20.0000	0.856706
$546$	0	0
$547$	−41.0000	−1.75303	−0.876517	−	0.481371i	$-0.840139\pi$
$547$	−41.0000	−1.75303	−0.876517	+	0.481371i	$0.840139\pi$
$548$	7.00000	0.299025
$549$	15.0000	0.640184
$550$	11.0000	0.469042
$551$	0	0
$552$	0	0
$553$	24.0000	1.02058
$554$	−7.00000	−0.297402
$555$	0	0
$556$	−20.0000	−0.848189
$557$	25.0000	1.05928	0.529642	−	0.848221i	$-0.322326\pi$
$557$	25.0000	1.05928	0.529642	+	0.848221i	$0.322326\pi$
$558$	0	0
$559$	49.0000	2.07248
$560$	16.0000	0.676123
$561$	0	0
$562$	−10.0000	−0.421825
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	−32.0000	−1.34506
$567$	−36.0000	−1.51186
$568$	−9.00000	−0.377632
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	33.0000	1.38101	0.690504	−	0.723329i	$-0.257389\pi$
$571$	33.0000	1.38101	0.690504	+	0.723329i	$0.257389\pi$
$572$	7.00000	0.292685
$573$	0	0
$574$	48.0000	2.00348
$575$	44.0000	1.83493
$576$	−3.00000	−0.125000
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	12.0000	0.498273
$581$	12.0000	0.497844
$582$	0	0
$583$	6.00000	0.248495
$584$	10.0000	0.413803
$585$	84.0000	3.47297
$586$	−5.00000	−0.206548
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	0	0
$589$	0	0
$590$	−24.0000	−0.988064
$591$	0	0
$592$	8.00000	0.328798
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	28.0000	1.14501
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	32.0000	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$602$	−28.0000	−1.14119
$603$	24.0000	0.977356
$604$	6.00000	0.244137
$605$	−4.00000	−0.162623
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	0	0
$609$	0	0
$610$	20.0000	0.809776
$611$	−35.0000	−1.41595
$612$	0	0
$613$	19.0000	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$613$	19.0000	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$614$	7.00000	0.282497
$615$	0	0
$616$	−4.00000	−0.161165
$617$	−15.0000	−0.603877	−0.301939	−	0.953327i	$-0.597634\pi$
$617$	−15.0000	−0.603877	−0.301939	+	0.953327i	$0.597634\pi$
$618$	0	0
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	0	0
$621$	0	0
$622$	−25.0000	−1.00241
$623$	4.00000	0.160257
$624$	0	0
$625$	41.0000	1.64000
$626$	9.00000	0.359712
$627$	0	0
$628$	−14.0000	−0.558661
$629$	0	0
$630$	−48.0000	−1.91237
$631$	13.0000	0.517522	0.258761	−	0.965941i	$-0.416686\pi$
$631$	13.0000	0.517522	0.258761	+	0.965941i	$0.416686\pi$
$632$	−6.00000	−0.238667
$633$	0	0
$634$	12.0000	0.476581
$635$	−40.0000	−1.58735
$636$	0	0
$637$	63.0000	2.49615
$638$	−3.00000	−0.118771
$639$	27.0000	1.06810
$640$	−4.00000	−0.158114
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	0	0
$643$	24.0000	0.946468	0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000	0.946468	0.473234	+	0.880937i	$0.343087\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	0	0
$647$	−17.0000	−0.668339	−0.334169	−	0.942513i	$-0.608456\pi$
$647$	−17.0000	−0.668339	−0.334169	+	0.942513i	$0.608456\pi$
$648$	9.00000	0.353553
$649$	6.00000	0.235521
$650$	77.0000	3.02019
$651$	0	0
$652$	6.00000	0.234978
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	20.0000	0.781465
$656$	−12.0000	−0.468521
$657$	−30.0000	−1.17041
$658$	20.0000	0.779681
$659$	35.0000	1.36341	0.681703	−	0.731629i	$-0.261240\pi$
$659$	35.0000	1.36341	0.681703	+	0.731629i	$0.261240\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	6.00000	0.233197
$663$	0	0
$664$	−3.00000	−0.116423
$665$	0	0
$666$	−24.0000	−0.929981
$667$	−12.0000	−0.464642
$668$	−16.0000	−0.619059
$669$	0	0
$670$	32.0000	1.23627
$671$	−5.00000	−0.193023
$672$	0	0
$673$	−16.0000	−0.616755	−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000	−0.616755	−0.308377	+	0.951264i	$0.599786\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	36.0000	1.38462
$677$	39.0000	1.49889	0.749446	−	0.662066i	$-0.230320\pi$
$677$	39.0000	1.49889	0.749446	+	0.662066i	$0.230320\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−14.0000	−0.535695	−0.267848	−	0.963461i	$-0.586312\pi$
$683$	−14.0000	−0.535695	−0.267848	+	0.963461i	$0.586312\pi$
$684$	0	0
$685$	−28.0000	−1.06983
$686$	−8.00000	−0.305441
$687$	0	0
$688$	7.00000	0.266872
$689$	42.0000	1.60007
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	2.00000	0.0760286
$693$	12.0000	0.455842
$694$	−3.00000	−0.113878
$695$	80.0000	3.03457
$696$	0	0
$697$	0	0
$698$	13.0000	0.492057
$699$	0	0
$700$	−44.0000	−1.66304
$701$	−41.0000	−1.54855	−0.774274	−	0.632850i	$-0.781885\pi$
$701$	−41.0000	−1.54855	−0.774274	+	0.632850i	$0.781885\pi$
$702$	0	0
$703$	0	0
$704$	1.00000	0.0376889
$705$	0	0
$706$	31.0000	1.16670
$707$	60.0000	2.25653
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	36.0000	1.35106
$711$	18.0000	0.675053
$712$	−1.00000	−0.0374766
$713$	0	0
$714$	0	0
$715$	−28.0000	−1.04714
$716$	−16.0000	−0.597948
$717$	0	0
$718$	−2.00000	−0.0746393
$719$	43.0000	1.60363	0.801815	−	0.597573i	$-0.203868\pi$
$719$	43.0000	1.60363	0.801815	+	0.597573i	$0.203868\pi$
$720$	12.0000	0.447214
$721$	−44.0000	−1.63865
$722$	0	0
$723$	0	0
$724$	−10.0000	−0.371647
$725$	−33.0000	−1.22559
$726$	0	0
$727$	−19.0000	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$727$	−19.0000	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$728$	−28.0000	−1.03775
$729$	−27.0000	−1.00000
$730$	−40.0000	−1.48047
$731$	0	0
$732$	0	0
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	35.0000	1.29187
$735$	0	0
$736$	4.00000	0.147442
$737$	−8.00000	−0.294684
$738$	36.0000	1.32518
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	−32.0000	−1.17634
$741$	0	0
$742$	−24.0000	−0.881068
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	−24.0000	−0.879292
$746$	−34.0000	−1.24483
$747$	9.00000	0.329293
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−27.0000	−0.985244	−0.492622	−	0.870243i	$-0.663961\pi$
$751$	−27.0000	−0.985244	−0.492622	+	0.870243i	$0.663961\pi$
$752$	−5.00000	−0.182331
$753$	0	0
$754$	−21.0000	−0.764775
$755$	−24.0000	−0.873449
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	0	0
$761$	4.00000	0.145000	0.0724999	−	0.997368i	$-0.476902\pi$
$761$	4.00000	0.145000	0.0724999	+	0.997368i	$0.476902\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	−3.00000	−0.108536
$765$	0	0
$766$	−12.0000	−0.433578
$767$	42.0000	1.51653
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	16.0000	0.576600
$771$	0	0
$772$	−6.00000	−0.215945
$773$	−32.0000	−1.15096	−0.575480	−	0.817816i	$-0.695185\pi$
$773$	−32.0000	−1.15096	−0.575480	+	0.817816i	$0.695185\pi$
$774$	−21.0000	−0.754829
$775$	0	0
$776$	−5.00000	−0.179490
$777$	0	0
$778$	−18.0000	−0.645331
$779$	0	0
$780$	0	0
$781$	−9.00000	−0.322045
$782$	0	0
$783$	0	0
$784$	9.00000	0.321429
$785$	56.0000	1.99873
$786$	0	0
$787$	−19.0000	−0.677277	−0.338638	−	0.940917i	$-0.609966\pi$
$787$	−19.0000	−0.677277	−0.338638	+	0.940917i	$0.609966\pi$
$788$	−3.00000	−0.106871
$789$	0	0
$790$	24.0000	0.853882
$791$	12.0000	0.426671
$792$	−3.00000	−0.106600
$793$	−35.0000	−1.24289
$794$	8.00000	0.283909
$795$	0	0
$796$	15.0000	0.531661
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	0	0
$799$	0	0
$800$	11.0000	0.388909
$801$	3.00000	0.106000
$802$	−10.0000	−0.353112
$803$	10.0000	0.352892
$804$	0	0
$805$	64.0000	2.25570
$806$	0	0
$807$	0	0
$808$	−15.0000	−0.527698
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	−36.0000	−1.26491
$811$	−37.0000	−1.29925	−0.649623	−	0.760257i	$-0.725073\pi$
$811$	−37.0000	−1.29925	−0.649623	+	0.760257i	$0.725073\pi$
$812$	12.0000	0.421117
$813$	0	0
$814$	8.00000	0.280400
$815$	−24.0000	−0.840683
$816$	0	0
$817$	0	0
$818$	−10.0000	−0.349642
$819$	84.0000	2.93520
$820$	48.0000	1.67623
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	−11.0000	−0.383436	−0.191718	−	0.981450i	$-0.561406\pi$
$823$	−11.0000	−0.383436	−0.191718	+	0.981450i	$0.561406\pi$
$824$	11.0000	0.383203
$825$	0	0
$826$	−24.0000	−0.835067
$827$	27.0000	0.938882	0.469441	−	0.882964i	$-0.344455\pi$
$827$	27.0000	0.938882	0.469441	+	0.882964i	$0.344455\pi$
$828$	−12.0000	−0.417029
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	7.00000	0.242681
$833$	0	0
$834$	0	0
$835$	64.0000	2.21481
$836$	0	0
$837$	0	0
$838$	−30.0000	−1.03633
$839$	−21.0000	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$839$	−21.0000	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−10.0000	−0.344623
$843$	0	0
$844$	−12.0000	−0.413057
$845$	−144.000	−4.95375
$846$	15.0000	0.515711
$847$	−4.00000	−0.137442
$848$	6.00000	0.206041
$849$	0	0
$850$	0	0
$851$	32.0000	1.09695
$852$	0	0
$853$	50.0000	1.71197	0.855984	−	0.517003i	$-0.172952\pi$
$853$	50.0000	1.71197	0.855984	+	0.517003i	$0.172952\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	−3.00000	−0.102538
$857$	−20.0000	−0.683187	−0.341593	−	0.939848i	$-0.610967\pi$
$857$	−20.0000	−0.683187	−0.341593	+	0.939848i	$0.610967\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	−28.0000	−0.954792
$861$	0	0
$862$	10.0000	0.340601
$863$	33.0000	1.12333	0.561667	−	0.827364i	$-0.310160\pi$
$863$	33.0000	1.12333	0.561667	+	0.827364i	$0.310160\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	−5.00000	−0.169907
$867$	0	0
$868$	0	0
$869$	−6.00000	−0.203536
$870$	0	0
$871$	−56.0000	−1.89749
$872$	−5.00000	−0.169321
$873$	15.0000	0.507673
$874$	0	0
$875$	96.0000	3.24539
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	−36.0000	−1.21494
$879$	0	0
$880$	−4.00000	−0.134840
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	−27.0000	−0.909137
$883$	−30.0000	−1.00958	−0.504790	−	0.863242i	$-0.668430\pi$
$883$	−30.0000	−1.00958	−0.504790	+	0.863242i	$0.668430\pi$
$884$	0	0
$885$	0	0
$886$	−18.0000	−0.604722
$887$	−30.0000	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$888$	0	0
$889$	−40.0000	−1.34156
$890$	4.00000	0.134080
$891$	9.00000	0.301511
$892$	−1.00000	−0.0334825
$893$	0	0
$894$	0	0
$895$	64.0000	2.13928
$896$	−4.00000	−0.133631
$897$	0	0
$898$	−30.0000	−1.00111
$899$	0	0
$900$	−33.0000	−1.10000
$901$	0	0
$902$	−12.0000	−0.399556
$903$	0	0
$904$	−3.00000	−0.0997785
$905$	40.0000	1.32964
$906$	0	0
$907$	−38.0000	−1.26177	−0.630885	−	0.775877i	$-0.717308\pi$
$907$	−38.0000	−1.26177	−0.630885	+	0.775877i	$0.717308\pi$
$908$	20.0000	0.663723
$909$	45.0000	1.49256
$910$	112.000	3.71276
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	0	0
$913$	−3.00000	−0.0992855
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	14.0000	0.462573
$917$	20.0000	0.660458
$918$	0	0
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	−16.0000	−0.527504
$921$	0	0
$922$	41.0000	1.35026
$923$	−63.0000	−2.07367
$924$	0	0
$925$	88.0000	2.89342
$926$	11.0000	0.361482
$927$	−33.0000	−1.08386
$928$	−3.00000	−0.0984798
$929$	−1.00000	−0.0328089	−0.0164045	−	0.999865i	$-0.505222\pi$
$929$	−1.00000	−0.0328089	−0.0164045	+	0.999865i	$0.505222\pi$
$930$	0	0
$931$	0	0
$932$	−18.0000	−0.589610
$933$	0	0
$934$	−34.0000	−1.11251
$935$	0	0
$936$	−21.0000	−0.686406
$937$	−24.0000	−0.784046	−0.392023	−	0.919955i	$-0.628225\pi$
$937$	−24.0000	−0.784046	−0.392023	+	0.919955i	$0.628225\pi$
$938$	32.0000	1.04484
$939$	0	0
$940$	20.0000	0.652328
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	6.00000	0.195283
$945$	0	0
$946$	7.00000	0.227590
$947$	−30.0000	−0.974869	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000	−0.974869	−0.487435	+	0.873160i	$0.662067\pi$
$948$	0	0
$949$	70.0000	2.27230
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.0000	1.42530	0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000	1.42530	0.712650	+	0.701520i	$0.247495\pi$
$954$	−18.0000	−0.582772
$955$	12.0000	0.388311
$956$	0	0
$957$	0	0
$958$	38.0000	1.22772
$959$	−28.0000	−0.904167
$960$	0	0
$961$	−31.0000	−1.00000
$962$	56.0000	1.80551
$963$	9.00000	0.290021
$964$	−10.0000	−0.322078
$965$	24.0000	0.772587
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	20.0000	0.642161
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	80.0000	2.56468
$974$	3.00000	0.0961262
$975$	0	0
$976$	−5.00000	−0.160046
$977$	23.0000	0.735835	0.367918	−	0.929858i	$-0.380071\pi$
$977$	23.0000	0.735835	0.367918	+	0.929858i	$0.380071\pi$
$978$	0	0
$979$	−1.00000	−0.0319601
$980$	−36.0000	−1.14998
$981$	15.0000	0.478913
$982$	−20.0000	−0.638226
$983$	19.0000	0.606006	0.303003	−	0.952990i	$-0.402011\pi$
$983$	19.0000	0.606006	0.303003	+	0.952990i	$0.402011\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	28.0000	0.890348
$990$	12.0000	0.381385
$991$	19.0000	0.603555	0.301777	−	0.953378i	$-0.402420\pi$
$991$	19.0000	0.603555	0.301777	+	0.953378i	$0.402420\pi$
$992$	0	0
$993$	0	0
$994$	36.0000	1.14185
$995$	−60.0000	−1.90213
$996$	0	0
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	18.0000	0.569780
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.o.1.1		1
19.7	even	3		418.2.e.c.353.1	yes	2
19.11	even	3		418.2.e.c.45.1	✓	2
19.18	odd	2		7942.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.e.c.45.1	✓	2	19.11	even	3
418.2.e.c.353.1	yes	2	19.7	even	3
7942.2.a.d.1.1		1	19.18	odd	2
7942.2.a.o.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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