LMFDB - Embedded newform 429.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [429,2,Mod(1,429)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(429, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("429.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 429 = 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	429.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:	$3.42558224671$
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	$\chi$	$=$	429.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} -1.00000 q^{22} -3.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{29} -6.00000 q^{31} -5.00000 q^{32} -1.00000 q^{33} +4.00000 q^{34} -1.00000 q^{36} -6.00000 q^{37} +8.00000 q^{38} -1.00000 q^{39} +6.00000 q^{41} -2.00000 q^{43} -1.00000 q^{44} -8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +5.00000 q^{50} +4.00000 q^{51} -1.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +8.00000 q^{57} -4.00000 q^{58} -14.0000 q^{61} +6.00000 q^{62} +7.00000 q^{64} +1.00000 q^{66} +14.0000 q^{67} +4.00000 q^{68} -4.00000 q^{71} +3.00000 q^{72} +6.00000 q^{73} +6.00000 q^{74} +5.00000 q^{75} +8.00000 q^{76} +1.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} +2.00000 q^{86} -4.00000 q^{87} +3.00000 q^{88} +12.0000 q^{89} +6.00000 q^{93} +8.00000 q^{94} +5.00000 q^{96} -2.00000 q^{97} +7.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−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
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−1.00000	−0.235702
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−3.00000	−0.612372
$25$	−5.00000	−1.00000
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	−5.00000	−0.883883
$33$	−1.00000	−0.174078
$34$	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	$0.664172\pi$
$38$	8.00000	1.29777
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	0	0
$47$	−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
$49$	−7.00000	−1.00000
$50$	5.00000	0.707107
$51$	4.00000	0.560112
$52$	−1.00000	−0.138675
$53$	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
$56$	0	0
$57$	8.00000	1.05963
$58$	−4.00000	−0.525226
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	1.00000	0.123091
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	3.00000	0.353553
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	6.00000	0.697486
$75$	5.00000	0.577350
$76$	8.00000	0.917663
$77$	0	0
$78$	1.00000	0.113228
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	0	0
$86$	2.00000	0.215666
$87$	−4.00000	−0.428845
$88$	3.00000	0.319801
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.00000	0.622171
$94$	8.00000	0.825137
$95$	0	0
$96$	5.00000	0.510310
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	7.00000	0.707107
$99$	1.00000	0.100504
$100$	5.00000	0.500000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	−4.00000	−0.396059
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	1.00000	0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	−4.00000	−0.371391
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	14.0000	1.26750
$123$	−6.00000	−0.541002
$124$	6.00000	0.538816
$125$	0	0
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	3.00000	0.265165
$129$	2.00000	0.176090
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−14.0000	−1.20942
$135$	0	0
$136$	−12.0000	−1.02899
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−18.0000	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	4.00000	0.335673
$143$	1.00000	0.0836242
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	7.00000	0.577350
$148$	6.00000	0.493197
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−5.00000	−0.408248
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	−24.0000	−1.94666
$153$	−4.00000	−0.323381
$154$	0	0
$155$	0	0
$156$	1.00000	0.0800641
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	10.0000	0.795557
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	18.0000	1.40987	0.704934	−	0.709273i	$-0.250976\pi$
$163$	18.0000	1.40987	0.704934	+	0.709273i	$0.250976\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−8.00000	−0.611775
$172$	2.00000	0.152499
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−12.0000	−0.899438
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	0	0
$185$	0	0
$186$	−6.00000	−0.439941
$187$	−4.00000	−0.292509
$188$	8.00000	0.583460
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−7.00000	−0.505181
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	7.00000	0.500000
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−1.00000	−0.0710669
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	−15.0000	−1.06066
$201$	−14.0000	−0.987484
$202$	12.0000	0.844317
$203$	0	0
$204$	−4.00000	−0.280056
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−6.00000	−0.412082
$213$	4.00000	0.274075
$214$	16.0000	1.09374
$215$	0	0
$216$	−3.00000	−0.204124
$217$	0	0
$218$	−10.0000	−0.677285
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−4.00000	−0.269069
$222$	−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	$0.521332\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	−18.0000	−1.19734
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−8.00000	−0.529813
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	12.0000	0.787839
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	14.0000	0.896258
$245$	0	0
$246$	6.00000	0.382546
$247$	−8.00000	−0.509028
$248$	−18.0000	−1.14300
$249$	12.0000	0.760469
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	0	0
$254$	−10.0000	−0.627456
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	0	0
$261$	4.00000	0.247594
$262$	−12.0000	−0.741362
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	0	0
$267$	−12.0000	−0.734388
$268$	−14.0000	−0.855186
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−12.0000	−0.724947
$275$	−5.00000	−0.301511
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	18.0000	1.07957
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	−8.00000	−0.476393
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	0	0
$288$	−5.00000	−0.294628
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	2.00000	0.117242
$292$	−6.00000	−0.351123
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	−18.0000	−1.04623
$297$	−1.00000	−0.0580259
$298$	18.0000	1.04271
$299$	0	0
$300$	−5.00000	−0.288675
$301$	0	0
$302$	12.0000	0.690522
$303$	12.0000	0.689382
$304$	8.00000	0.458831
$305$	0	0
$306$	4.00000	0.228665
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	−3.00000	−0.169842
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	10.0000	0.562544
$317$	−8.00000	−0.449325	−0.224662	−	0.974437i	$-0.572128\pi$
$317$	−8.00000	−0.449325	−0.224662	+	0.974437i	$0.572128\pi$
$318$	6.00000	0.336463
$319$	4.00000	0.223957
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	32.0000	1.78053
$324$	−1.00000	−0.0555556
$325$	−5.00000	−0.277350
$326$	−18.0000	−0.996928
$327$	−10.0000	−0.553001
$328$	18.0000	0.993884
$329$	0	0
$330$	0	0
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	12.0000	0.658586
$333$	−6.00000	−0.328798
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	−1.00000	−0.0543928
$339$	−18.0000	−0.977626
$340$	0	0
$341$	−6.00000	−0.324918
$342$	8.00000	0.432590
$343$	0	0
$344$	−6.00000	−0.323498
$345$	0	0
$346$	4.00000	0.215041
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	4.00000	0.214423
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−5.00000	−0.266501
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	0	0
$356$	−12.0000	−0.635999
$357$	0	0
$358$	20.0000	1.05703
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	−2.00000	−0.105118
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	0	0
$366$	−14.0000	−0.731792
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	0	0
$372$	−6.00000	−0.311086
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−24.0000	−1.23771
$377$	4.00000	0.206010
$378$	0	0
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	0	0
$381$	−10.0000	−0.512316
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	−6.00000	−0.305392
$387$	−2.00000	−0.101666
$388$	2.00000	0.101535
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	0	0
$391$	0	0
$392$	−21.0000	−1.06066
$393$	−12.0000	−0.605320
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	12.0000	0.601506
$399$	0	0
$400$	5.00000	0.250000
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	14.0000	0.698257
$403$	−6.00000	−0.298881
$404$	12.0000	0.597022
$405$	0	0
$406$	0	0
$407$	−6.00000	−0.297409
$408$	12.0000	0.594089
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	−4.00000	−0.197066
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−5.00000	−0.245145
$417$	18.0000	0.881464
$418$	8.00000	0.391293
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	22.0000	1.07094
$423$	−8.00000	−0.388973
$424$	18.0000	0.874157
$425$	20.0000	0.970143
$426$	−4.00000	−0.193801
$427$	0	0
$428$	16.0000	0.773389
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	1.00000	0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	6.00000	0.286691
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	4.00000	0.190261
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	2.00000	0.0947027
$447$	18.0000	0.851371
$448$	0	0
$449$	−4.00000	−0.188772	−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000	−0.188772	−0.0943858	+	0.995536i	$0.530089\pi$
$450$	5.00000	0.235702
$451$	6.00000	0.282529
$452$	−18.0000	−0.846649
$453$	12.0000	0.563809
$454$	12.0000	0.563188
$455$	0	0
$456$	24.0000	1.12390
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−22.0000	−1.02799
$459$	4.00000	0.186704
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	4.00000	0.185296
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−1.00000	−0.0462250
$469$	0	0
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	−2.00000	−0.0919601
$474$	−10.0000	−0.459315
$475$	40.0000	1.83533
$476$	0	0
$477$	6.00000	0.274721
$478$	−24.0000	−1.09773
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−42.0000	−1.90125
$489$	−18.0000	−0.813988
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	6.00000	0.270501
$493$	−16.0000	−0.720604
$494$	8.00000	0.359937
$495$	0	0
$496$	6.00000	0.269408
$497$	0	0
$498$	−12.0000	−0.537733
$499$	−26.0000	−1.16392	−0.581960	−	0.813217i	$-0.697714\pi$
$499$	−26.0000	−1.16392	−0.581960	+	0.813217i	$0.697714\pi$
$500$	0	0
$501$	0	0
$502$	20.0000	0.892644
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	−10.0000	−0.443678
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	8.00000	0.353209
$514$	10.0000	0.441081
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	−8.00000	−0.351840
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	−4.00000	−0.175075
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−4.00000	−0.174408
$527$	24.0000	1.04546
$528$	1.00000	0.0435194
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.00000	0.259889
$534$	12.0000	0.519291
$535$	0	0
$536$	42.0000	1.81412
$537$	20.0000	0.863064
$538$	−18.0000	−0.776035
$539$	−7.00000	−0.301511
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	−20.0000	−0.859074
$543$	−2.00000	−0.0858282
$544$	20.0000	0.857493
$545$	0	0
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	−12.0000	−0.512615
$549$	−14.0000	−0.597505
$550$	5.00000	0.213201
$551$	−32.0000	−1.36325
$552$	0	0
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	18.0000	0.763370
$557$	−34.0000	−1.44063	−0.720313	−	0.693649i	$-0.756002\pi$
$557$	−34.0000	−1.44063	−0.720313	+	0.693649i	$0.756002\pi$
$558$	6.00000	0.254000
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	4.00000	0.168880
$562$	10.0000	0.421825
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	6.00000	0.252199
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	−1.00000	−0.0418121
$573$	0	0
$574$	0	0
$575$	0	0
$576$	7.00000	0.291667
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	1.00000	0.0415945
$579$	−6.00000	−0.249351
$580$	0	0
$581$	0	0
$582$	−2.00000	−0.0829027
$583$	6.00000	0.248495
$584$	18.0000	0.744845
$585$	0	0
$586$	−30.0000	−1.23929
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	−7.00000	−0.288675
$589$	48.0000	1.97781
$590$	0	0
$591$	−6.00000	−0.246807
$592$	6.00000	0.246598
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	18.0000	0.737309
$597$	12.0000	0.491127
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	15.0000	0.612372
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	0	0
$603$	14.0000	0.570124
$604$	12.0000	0.488273
$605$	0	0
$606$	−12.0000	−0.487467
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	40.0000	1.62221
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	4.00000	0.161690
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	0	0
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	4.00000	0.160904
$619$	−34.0000	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$619$	−34.0000	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$620$	0	0
$621$	0	0
$622$	−24.0000	−0.962312
$623$	0	0
$624$	1.00000	0.0400320
$625$	25.0000	1.00000
$626$	−6.00000	−0.239808
$627$	8.00000	0.319489
$628$	−2.00000	−0.0798087
$629$	24.0000	0.956943
$630$	0	0
$631$	−18.0000	−0.716569	−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000	−0.716569	−0.358284	+	0.933613i	$0.616638\pi$
$632$	−30.0000	−1.19334
$633$	22.0000	0.874421
$634$	8.00000	0.317721
$635$	0	0
$636$	6.00000	0.237915
$637$	−7.00000	−0.277350
$638$	−4.00000	−0.158362
$639$	−4.00000	−0.158238
$640$	0	0
$641$	50.0000	1.97488	0.987441	−	0.157991i	$-0.0505015\pi$
$641$	50.0000	1.97488	0.987441	+	0.157991i	$0.0505015\pi$
$642$	−16.0000	−0.631470
$643$	18.0000	0.709851	0.354925	−	0.934895i	$-0.384506\pi$
$643$	18.0000	0.709851	0.354925	+	0.934895i	$0.384506\pi$
$644$	0	0
$645$	0	0
$646$	−32.0000	−1.25902
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	3.00000	0.117851
$649$	0	0
$650$	5.00000	0.196116
$651$	0	0
$652$	−18.0000	−0.704934
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	−6.00000	−0.234261
$657$	6.00000	0.234082
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	14.0000	0.544125
$663$	4.00000	0.155347
$664$	−36.0000	−1.39707
$665$	0	0
$666$	6.00000	0.232495
$667$	0	0
$668$	0	0
$669$	2.00000	0.0773245
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	10.0000	0.385186
$675$	5.00000	0.192450
$676$	−1.00000	−0.0384615
$677$	−32.0000	−1.22986	−0.614930	−	0.788582i	$-0.710816\pi$
$677$	−32.0000	−1.22986	−0.614930	+	0.788582i	$0.710816\pi$
$678$	18.0000	0.691286
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	6.00000	0.229752
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	8.00000	0.305888
$685$	0	0
$686$	0	0
$687$	−22.0000	−0.839352
$688$	2.00000	0.0762493
$689$	6.00000	0.228582
$690$	0	0
$691$	42.0000	1.59776	0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000	1.59776	0.798878	+	0.601494i	$0.205427\pi$
$692$	4.00000	0.152057
$693$	0	0
$694$	4.00000	0.151838
$695$	0	0
$696$	−12.0000	−0.454859
$697$	−24.0000	−0.909065
$698$	10.0000	0.378506
$699$	4.00000	0.151294
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	1.00000	0.0377426
$703$	48.0000	1.81035
$704$	7.00000	0.263822
$705$	0	0
$706$	−24.0000	−0.903252
$707$	0	0
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	36.0000	1.34916
$713$	0	0
$714$	0	0
$715$	0	0
$716$	20.0000	0.747435
$717$	−24.0000	−0.896296
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	0	0
$722$	−45.0000	−1.67473
$723$	−2.00000	−0.0743808
$724$	−2.00000	−0.0743294
$725$	−20.0000	−0.742781
$726$	1.00000	0.0371135
$727$	−44.0000	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	−14.0000	−0.517455
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	0	0
$737$	14.0000	0.515697
$738$	−6.00000	−0.220863
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	18.0000	0.659912
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	−12.0000	−0.439057
$748$	4.00000	0.146254
$749$	0	0
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	8.00000	0.291730
$753$	20.0000	0.728841
$754$	−4.00000	−0.145671
$755$	0	0
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	−6.00000	−0.217930
$759$	0	0
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	10.0000	0.362262
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−36.0000	−1.30073
$767$	0	0
$768$	17.0000	0.613435
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	10.0000	0.360141
$772$	−6.00000	−0.215945
$773$	−16.0000	−0.575480	−0.287740	−	0.957709i	$-0.592904\pi$
$773$	−16.0000	−0.575480	−0.287740	+	0.957709i	$0.592904\pi$
$774$	2.00000	0.0718885
$775$	30.0000	1.07763
$776$	−6.00000	−0.215387
$777$	0	0
$778$	−34.0000	−1.21896
$779$	−48.0000	−1.71978
$780$	0	0
$781$	−4.00000	−0.143131
$782$	0	0
$783$	−4.00000	−0.142948
$784$	7.00000	0.250000
$785$	0	0
$786$	12.0000	0.428026
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−6.00000	−0.213741
$789$	−4.00000	−0.142404
$790$	0	0
$791$	0	0
$792$	3.00000	0.106600
$793$	−14.0000	−0.497155
$794$	22.0000	0.780751
$795$	0	0
$796$	12.0000	0.425329
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	25.0000	0.883883
$801$	12.0000	0.423999
$802$	−16.0000	−0.564980
$803$	6.00000	0.211735
$804$	14.0000	0.493742
$805$	0	0
$806$	6.00000	0.211341
$807$	−18.0000	−0.633630
$808$	−36.0000	−1.26648
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	0	0
$813$	−20.0000	−0.701431
$814$	6.00000	0.210300
$815$	0	0
$816$	−4.00000	−0.140028
$817$	16.0000	0.559769
$818$	26.0000	0.909069
$819$	0	0
$820$	0	0
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	12.0000	0.418548
$823$	−52.0000	−1.81261	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−52.0000	−1.81261	−0.906303	+	0.422628i	$0.861108\pi$
$824$	12.0000	0.418040
$825$	5.00000	0.174078
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	7.00000	0.242681
$833$	28.0000	0.970143
$834$	−18.0000	−0.623289
$835$	0	0
$836$	8.00000	0.276686
$837$	6.00000	0.207390
$838$	36.0000	1.24360
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−22.0000	−0.758170
$843$	10.0000	0.344418
$844$	22.0000	0.757271
$845$	0	0
$846$	8.00000	0.275046
$847$	0	0
$848$	−6.00000	−0.206041
$849$	6.00000	0.205919
$850$	−20.0000	−0.685994
$851$	0	0
$852$	−4.00000	−0.137038
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	0	0
$856$	−48.0000	−1.64061
$857$	−48.0000	−1.63965	−0.819824	−	0.572615i	$-0.805929\pi$
$857$	−48.0000	−1.63965	−0.819824	+	0.572615i	$0.805929\pi$
$858$	1.00000	0.0341394
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	0	0
$861$	0	0
$862$	32.0000	1.08992
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	14.0000	0.475739
$867$	1.00000	0.0339618
$868$	0	0
$869$	−10.0000	−0.339227
$870$	0	0
$871$	14.0000	0.474372
$872$	30.0000	1.01593
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	0	0
$876$	6.00000	0.202721
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	−10.0000	−0.337484
$879$	−30.0000	−1.01187
$880$	0	0
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	7.00000	0.235702
$883$	−32.0000	−1.07689	−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000	−1.07689	−0.538443	+	0.842662i	$0.680987\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	4.00000	0.134383
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	18.0000	0.604040
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	2.00000	0.0669650
$893$	64.0000	2.14168
$894$	−18.0000	−0.602010
$895$	0	0
$896$	0	0
$897$	0	0
$898$	4.00000	0.133482
$899$	−24.0000	−0.800445
$900$	5.00000	0.166667
$901$	−24.0000	−0.799556
$902$	−6.00000	−0.199778
$903$	0	0
$904$	54.0000	1.79601
$905$	0	0
$906$	−12.0000	−0.398673
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	12.0000	0.398234
$909$	−12.0000	−0.398015
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−8.00000	−0.264906
$913$	−12.0000	−0.397142
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−22.0000	−0.726900
$917$	0	0
$918$	−4.00000	−0.132020
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	−10.0000	−0.329332
$923$	−4.00000	−0.131662
$924$	0	0
$925$	30.0000	0.986394
$926$	−14.0000	−0.460069
$927$	4.00000	0.131377
$928$	−20.0000	−0.656532
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	0	0
$931$	56.0000	1.83533
$932$	4.00000	0.131024
$933$	−24.0000	−0.785725
$934$	−12.0000	−0.392652
$935$	0	0
$936$	3.00000	0.0980581
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	2.00000	0.0651635
$943$	0	0
$944$	0	0
$945$	0	0
$946$	2.00000	0.0650256
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	−10.0000	−0.324785
$949$	6.00000	0.194768
$950$	−40.0000	−1.29777
$951$	8.00000	0.259418
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−24.0000	−0.776215
$957$	−4.00000	−0.129302
$958$	32.0000	1.03387
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	6.00000	0.193448
$963$	−16.0000	−0.515593
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	3.00000	0.0964237
$969$	−32.0000	−1.02799
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−2.00000	−0.0640841
$975$	5.00000	0.160128
$976$	14.0000	0.448129
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	18.0000	0.575577
$979$	12.0000	0.383522
$980$	0	0
$981$	10.0000	0.319275
$982$	−20.0000	−0.638226
$983$	8.00000	0.255160	0.127580	−	0.991828i	$-0.459279\pi$
$983$	8.00000	0.255160	0.127580	+	0.991828i	$0.459279\pi$
$984$	−18.0000	−0.573819
$985$	0	0
$986$	16.0000	0.509544
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	0	0
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	30.0000	0.952501
$993$	14.0000	0.444277
$994$	0	0
$995$	0	0
$996$	−12.0000	−0.380235
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	26.0000	0.823016
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.a.a.1.1	✓	1
3.2	odd	2		1287.2.a.d.1.1		1
4.3	odd	2		6864.2.a.w.1.1		1
11.10	odd	2		4719.2.a.j.1.1		1
13.12	even	2		5577.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.a.1.1	✓	1	1.1	even	1	trivial
1287.2.a.d.1.1		1	3.2	odd	2
4719.2.a.j.1.1		1	11.10	odd	2
5577.2.a.f.1.1		1	13.12	even	2
6864.2.a.w.1.1		1	4.3	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:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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