LMFDB - Embedded newform 6762.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6762,2,Mod(1,6762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6762, 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("6762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6762.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:	$53.9948418468$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} -4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} +2.00000 q^{40} +6.00000 q^{41} -4.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -8.00000 q^{55} -4.00000 q^{57} -2.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} +10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} +6.00000 q^{68} -1.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +2.00000 q^{78} +2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} +12.0000 q^{85} -4.00000 q^{86} -2.00000 q^{87} -4.00000 q^{88} -2.00000 q^{89} +2.00000 q^{90} -1.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} -8.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−4.00000	−0.852803
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	2.00000	0.365148
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	2.00000	0.320256
$40$	2.00000	0.316228
$41$	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
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.00000	−0.603023
$45$	2.00000	0.298142
$46$	−1.00000	−0.147442
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−1.00000	−0.141421
$51$	6.00000	0.840168
$52$	2.00000	0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	−8.00000	−1.07872
$56$	0	0
$57$	−4.00000	−0.529813
$58$	−2.00000	−0.262613
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	2.00000	0.258199
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	−4.00000	−0.492366
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	6.00000	0.727607
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000	0.117851
$73$	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
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	0	0
$78$	2.00000	0.226455
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	−4.00000	−0.431331
$87$	−2.00000	−0.214423
$88$	−4.00000	−0.426401
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	−1.00000	−0.104257
$93$	8.00000	0.829561
$94$	8.00000	0.825137
$95$	−8.00000	−0.820783
$96$	1.00000	0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	6.00000	0.594089
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−8.00000	−0.762770
$111$	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	$0.178617\pi$
$114$	−4.00000	−0.374634
$115$	−2.00000	−0.186501
$116$	−2.00000	−0.185695
$117$	2.00000	0.184900
$118$	−4.00000	−0.368230
$119$	0	0
$120$	2.00000	0.182574
$121$	5.00000	0.454545
$122$	10.0000	0.905357
$123$	6.00000	0.541002
$124$	8.00000	0.718421
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	4.00000	0.350823
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	4.00000	0.345547
$135$	2.00000	0.172133
$136$	6.00000	0.514496
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−1.00000	−0.0851257
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	−8.00000	−0.671345
$143$	−8.00000	−0.668994
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	6.00000	0.496564
$147$	0	0
$148$	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	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−4.00000	−0.324443
$153$	6.00000	0.485071
$154$	0	0
$155$	16.0000	1.28515
$156$	2.00000	0.160128
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	2.00000	0.158114
$161$	0	0
$162$	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	$0.449930\pi$
$164$	6.00000	0.468521
$165$	−8.00000	−0.622799
$166$	12.0000	0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	12.0000	0.920358
$171$	−4.00000	−0.305888
$172$	−4.00000	−0.304997
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	−4.00000	−0.300658
$178$	−2.00000	−0.149906
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	2.00000	0.149071
$181$	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
$183$	10.0000	0.739221
$184$	−1.00000	−0.0737210
$185$	12.0000	0.882258
$186$	8.00000	0.586588
$187$	−24.0000	−1.75505
$188$	8.00000	0.583460
$189$	0	0
$190$	−8.00000	−0.580381
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	1.00000	0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−10.0000	−0.717958
$195$	4.00000	0.286446
$196$	0	0
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	−4.00000	−0.284268
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	−1.00000	−0.0707107
$201$	4.00000	0.282138
$202$	−6.00000	−0.422159
$203$	0	0
$204$	6.00000	0.420084
$205$	12.0000	0.838116
$206$	−8.00000	−0.557386
$207$	−1.00000	−0.0695048
$208$	2.00000	0.138675
$209$	16.0000	1.10674
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	6.00000	0.412082
$213$	−8.00000	−0.548151
$214$	12.0000	0.820303
$215$	−8.00000	−0.545595
$216$	1.00000	0.0680414
$217$	0	0
$218$	14.0000	0.948200
$219$	6.00000	0.405442
$220$	−8.00000	−0.539360
$221$	12.0000	0.807207
$222$	6.00000	0.402694
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	18.0000	1.19734
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−4.00000	−0.264906
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−2.00000	−0.131876
$231$	0	0
$232$	−2.00000	−0.131306
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	2.00000	0.130744
$235$	16.0000	1.04372
$236$	−4.00000	−0.260378
$237$	0	0
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	2.00000	0.129099
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	0	0
$246$	6.00000	0.382546
$247$	−8.00000	−0.509028
$248$	8.00000	0.508001
$249$	12.0000	0.760469
$250$	−12.0000	−0.758947
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	−16.0000	−1.00393
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	4.00000	0.248069
$261$	−2.00000	−0.123797
$262$	−12.0000	−0.741362
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−4.00000	−0.246183
$265$	12.0000	0.737154
$266$	0	0
$267$	−2.00000	−0.122398
$268$	4.00000	0.244339
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	2.00000	0.121716
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	4.00000	0.241209
$276$	−1.00000	−0.0601929
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	12.0000	0.719712
$279$	8.00000	0.478947
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	8.00000	0.476393
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	−8.00000	−0.474713
$285$	−8.00000	−0.473879
$286$	−8.00000	−0.473050
$287$	0	0
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	−4.00000	−0.234888
$291$	−10.0000	−0.586210
$292$	6.00000	0.351123
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	6.00000	0.348743
$297$	−4.00000	−0.232104
$298$	6.00000	0.347571
$299$	−2.00000	−0.115663
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	8.00000	0.460348
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	20.0000	1.14520
$306$	6.00000	0.342997
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	16.0000	0.908739
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	2.00000	0.113228
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	6.00000	0.336463
$319$	8.00000	0.447914
$320$	2.00000	0.111803
$321$	12.0000	0.669775
$322$	0	0
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	4.00000	0.221540
$327$	14.0000	0.774202
$328$	6.00000	0.331295
$329$	0	0
$330$	−8.00000	−0.440386
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	12.0000	0.658586
$333$	6.00000	0.328798
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	−9.00000	−0.489535
$339$	18.0000	0.977626
$340$	12.0000	0.650791
$341$	−32.0000	−1.73290
$342$	−4.00000	−0.216295
$343$	0	0
$344$	−4.00000	−0.215666
$345$	−2.00000	−0.107676
$346$	2.00000	0.107521
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	−2.00000	−0.107211
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−4.00000	−0.213201
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	−4.00000	−0.212598
$355$	−16.0000	−0.849192
$356$	−2.00000	−0.106000
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	2.00000	0.105409
$361$	−3.00000	−0.157895
$362$	2.00000	0.105118
$363$	5.00000	0.262432
$364$	0	0
$365$	12.0000	0.628109
$366$	10.0000	0.522708
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−1.00000	−0.0521286
$369$	6.00000	0.312348
$370$	12.0000	0.623850
$371$	0	0
$372$	8.00000	0.414781
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	−24.0000	−1.24101
$375$	−12.0000	−0.619677
$376$	8.00000	0.412568
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−36.0000	−1.84920	−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000	−1.84920	−0.924598	+	0.380945i	$0.875599\pi$
$380$	−8.00000	−0.410391
$381$	−16.0000	−0.819705
$382$	−16.0000	−0.818631
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	−4.00000	−0.203331
$388$	−10.0000	−0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	4.00000	0.202548
$391$	−6.00000	−0.303433
$392$	0	0
$393$	−12.0000	−0.605320
$394$	−26.0000	−1.30986
$395$	0	0
$396$	−4.00000	−0.201008
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	24.0000	1.20301
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	4.00000	0.199502
$403$	16.0000	0.797017
$404$	−6.00000	−0.298511
$405$	2.00000	0.0993808
$406$	0	0
$407$	−24.0000	−1.18964
$408$	6.00000	0.297044
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	12.0000	0.592638
$411$	−6.00000	−0.295958
$412$	−8.00000	−0.394132
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	24.0000	1.17811
$416$	2.00000	0.0980581
$417$	12.0000	0.587643
$418$	16.0000	0.782586
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.00000	0.194717
$423$	8.00000	0.388973
$424$	6.00000	0.291386
$425$	−6.00000	−0.291043
$426$	−8.00000	−0.387601
$427$	0	0
$428$	12.0000	0.580042
$429$	−8.00000	−0.386244
$430$	−8.00000	−0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	14.0000	0.670478
$437$	4.00000	0.191346
$438$	6.00000	0.286691
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	−8.00000	−0.381385
$441$	0	0
$442$	12.0000	0.570782
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	6.00000	0.284747
$445$	−4.00000	−0.189618
$446$	−24.0000	−1.13643
$447$	6.00000	0.283790
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	−1.00000	−0.0471405
$451$	−24.0000	−1.13012
$452$	18.0000	0.846649
$453$	8.00000	0.375873
$454$	12.0000	0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−14.0000	−0.654177
$459$	6.00000	0.280056
$460$	−2.00000	−0.0932505
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	−2.00000	−0.0928477
$465$	16.0000	0.741982
$466$	10.0000	0.463241
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	16.0000	0.738025
$471$	−22.0000	−1.01371
$472$	−4.00000	−0.184115
$473$	16.0000	0.735681
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	6.00000	0.274721
$478$	−16.0000	−0.731823
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	2.00000	0.0912871
$481$	12.0000	0.547153
$482$	22.0000	1.00207
$483$	0	0
$484$	5.00000	0.227273
$485$	−20.0000	−0.908153
$486$	1.00000	0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	10.0000	0.452679
$489$	4.00000	0.180886
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	6.00000	0.270501
$493$	−12.0000	−0.540453
$494$	−8.00000	−0.359937
$495$	−8.00000	−0.359573
$496$	8.00000	0.359211
$497$	0	0
$498$	12.0000	0.537733
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	4.00000	0.177822
$507$	−9.00000	−0.399704
$508$	−16.0000	−0.709885
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	12.0000	0.531369
$511$	0	0
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−18.0000	−0.793946
$515$	−16.0000	−0.705044
$516$	−4.00000	−0.176090
$517$	−32.0000	−1.40736
$518$	0	0
$519$	2.00000	0.0877903
$520$	4.00000	0.175412
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	−2.00000	−0.0875376
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−24.0000	−1.04645
$527$	48.0000	2.09091
$528$	−4.00000	−0.174078
$529$	1.00000	0.0434783
$530$	12.0000	0.521247
$531$	−4.00000	−0.173585
$532$	0	0
$533$	12.0000	0.519778
$534$	−2.00000	−0.0865485
$535$	24.0000	1.03761
$536$	4.00000	0.172774
$537$	−12.0000	−0.517838
$538$	2.00000	0.0862261
$539$	0	0
$540$	2.00000	0.0860663
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	−24.0000	−1.03089
$543$	2.00000	0.0858282
$544$	6.00000	0.257248
$545$	28.0000	1.19939
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−6.00000	−0.256307
$549$	10.0000	0.426790
$550$	4.00000	0.170561
$551$	8.00000	0.340811
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	22.0000	0.934690
$555$	12.0000	0.509372
$556$	12.0000	0.508913
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	8.00000	0.338667
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−24.0000	−1.01328
$562$	10.0000	0.421825
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	8.00000	0.336861
$565$	36.0000	1.51453
$566$	20.0000	0.840663
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	−8.00000	−0.335083
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−8.00000	−0.334497
$573$	−16.0000	−0.668410
$574$	0	0
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	19.0000	0.790296
$579$	2.00000	0.0831172
$580$	−4.00000	−0.166091
$581$	0	0
$582$	−10.0000	−0.414513
$583$	−24.0000	−0.993978
$584$	6.00000	0.248282
$585$	4.00000	0.165380
$586$	2.00000	0.0826192
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	−8.00000	−0.329355
$591$	−26.0000	−1.06950
$592$	6.00000	0.246598
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	6.00000	0.245770
$597$	24.0000	0.982255
$598$	−2.00000	−0.0817861
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	−1.00000	−0.0408248
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	8.00000	0.325515
$605$	10.0000	0.406558
$606$	−6.00000	−0.243733
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	20.0000	0.809776
$611$	16.0000	0.647291
$612$	6.00000	0.242536
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−12.0000	−0.484281
$615$	12.0000	0.483887
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	−8.00000	−0.321807
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	16.0000	0.642575
$621$	−1.00000	−0.0401286
$622$	16.0000	0.641542
$623$	0	0
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	14.0000	0.559553
$627$	16.0000	0.638978
$628$	−22.0000	−0.877896
$629$	36.0000	1.43541
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	−2.00000	−0.0794301
$635$	−32.0000	−1.26988
$636$	6.00000	0.237915
$637$	0	0
$638$	8.00000	0.316723
$639$	−8.00000	−0.316475
$640$	2.00000	0.0790569
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	12.0000	0.473602
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	−24.0000	−0.944267
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	1.00000	0.0392837
$649$	16.0000	0.628055
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	4.00000	0.156652
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	14.0000	0.547443
$655$	−24.0000	−0.937758
$656$	6.00000	0.234261
$657$	6.00000	0.234082
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	−8.00000	−0.311400
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	−4.00000	−0.155464
$663$	12.0000	0.466041
$664$	12.0000	0.465690
$665$	0	0
$666$	6.00000	0.232495
$667$	2.00000	0.0774403
$668$	0	0
$669$	−24.0000	−0.927894
$670$	8.00000	0.309067
$671$	−40.0000	−1.54418
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	−30.0000	−1.15556
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	18.0000	0.691286
$679$	0	0
$680$	12.0000	0.460179
$681$	12.0000	0.459841
$682$	−32.0000	−1.22534
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−4.00000	−0.152944
$685$	−12.0000	−0.458496
$686$	0	0
$687$	−14.0000	−0.534133
$688$	−4.00000	−0.152499
$689$	12.0000	0.457164
$690$	−2.00000	−0.0761387
$691$	52.0000	1.97817	0.989087	−	0.147335i	$-0.0470696\pi$
$691$	52.0000	1.97817	0.989087	+	0.147335i	$0.0470696\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	−36.0000	−1.36654
$695$	24.0000	0.910372
$696$	−2.00000	−0.0758098
$697$	36.0000	1.36360
$698$	2.00000	0.0757011
$699$	10.0000	0.378235
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	2.00000	0.0754851
$703$	−24.0000	−0.905177
$704$	−4.00000	−0.150756
$705$	16.0000	0.602595
$706$	30.0000	1.12906
$707$	0	0
$708$	−4.00000	−0.150329
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	−16.0000	−0.600469
$711$	0	0
$712$	−2.00000	−0.0749532
$713$	−8.00000	−0.299602
$714$	0	0
$715$	−16.0000	−0.598366
$716$	−12.0000	−0.448461
$717$	−16.0000	−0.597531
$718$	−24.0000	−0.895672
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	−3.00000	−0.111648
$723$	22.0000	0.818189
$724$	2.00000	0.0743294
$725$	2.00000	0.0742781
$726$	5.00000	0.185567
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	12.0000	0.444140
$731$	−24.0000	−0.887672
$732$	10.0000	0.369611
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	0	0
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−16.0000	−0.589368
$738$	6.00000	0.220863
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	12.0000	0.441129
$741$	−8.00000	−0.293887
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	8.00000	0.293294
$745$	12.0000	0.439646
$746$	6.00000	0.219676
$747$	12.0000	0.439057
$748$	−24.0000	−0.877527
$749$	0	0
$750$	−12.0000	−0.438178
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	8.00000	0.291730
$753$	−12.0000	−0.437304
$754$	−4.00000	−0.145671
$755$	16.0000	0.582300
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	−36.0000	−1.30758
$759$	4.00000	0.145191
$760$	−8.00000	−0.290191
$761$	38.0000	1.37750	0.688749	−	0.724999i	$-0.258160\pi$
$761$	38.0000	1.37750	0.688749	+	0.724999i	$0.258160\pi$
$762$	−16.0000	−0.579619
$763$	0	0
$764$	−16.0000	−0.578860
$765$	12.0000	0.433861
$766$	−16.0000	−0.578103
$767$	−8.00000	−0.288863
$768$	1.00000	0.0360844
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	2.00000	0.0719816
$773$	−46.0000	−1.65451	−0.827253	−	0.561830i	$-0.810097\pi$
$773$	−46.0000	−1.65451	−0.827253	+	0.561830i	$0.810097\pi$
$774$	−4.00000	−0.143777
$775$	−8.00000	−0.287368
$776$	−10.0000	−0.358979
$777$	0	0
$778$	6.00000	0.215110
$779$	−24.0000	−0.859889
$780$	4.00000	0.143223
$781$	32.0000	1.14505
$782$	−6.00000	−0.214560
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	−44.0000	−1.57043
$786$	−12.0000	−0.428026
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	−26.0000	−0.926212
$789$	−24.0000	−0.854423
$790$	0	0
$791$	0	0
$792$	−4.00000	−0.142134
$793$	20.0000	0.710221
$794$	2.00000	0.0709773
$795$	12.0000	0.425596
$796$	24.0000	0.850657
$797$	−54.0000	−1.91278	−0.956389	−	0.292096i	$-0.905647\pi$
$797$	−54.0000	−1.91278	−0.956389	+	0.292096i	$0.905647\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	−1.00000	−0.0353553
$801$	−2.00000	−0.0706665
$802$	34.0000	1.20058
$803$	−24.0000	−0.846942
$804$	4.00000	0.141069
$805$	0	0
$806$	16.0000	0.563576
$807$	2.00000	0.0704033
$808$	−6.00000	−0.211079
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	2.00000	0.0702728
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	−24.0000	−0.841200
$815$	8.00000	0.280228
$816$	6.00000	0.210042
$817$	16.0000	0.559769
$818$	−10.0000	−0.349642
$819$	0	0
$820$	12.0000	0.419058
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	−6.00000	−0.209274
$823$	56.0000	1.95204	0.976019	−	0.217687i	$-0.0698512\pi$
$823$	56.0000	1.95204	0.976019	+	0.217687i	$0.0698512\pi$
$824$	−8.00000	−0.278693
$825$	4.00000	0.139262
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−1.00000	−0.0347524
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	24.0000	0.833052
$831$	22.0000	0.763172
$832$	2.00000	0.0693375
$833$	0	0
$834$	12.0000	0.415526
$835$	0	0
$836$	16.0000	0.553372
$837$	8.00000	0.276520
$838$	12.0000	0.414533
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−10.0000	−0.344623
$843$	10.0000	0.344418
$844$	4.00000	0.137686
$845$	−18.0000	−0.619219
$846$	8.00000	0.275046
$847$	0	0
$848$	6.00000	0.206041
$849$	20.0000	0.686398
$850$	−6.00000	−0.205798
$851$	−6.00000	−0.205677
$852$	−8.00000	−0.274075
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	12.0000	0.410152
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	−8.00000	−0.273115
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	0	0
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	1.00000	0.0340207
$865$	4.00000	0.136004
$866$	−26.0000	−0.883516
$867$	19.0000	0.645274
$868$	0	0
$869$	0	0
$870$	−4.00000	−0.135613
$871$	8.00000	0.271070
$872$	14.0000	0.474100
$873$	−10.0000	−0.338449
$874$	4.00000	0.135302
$875$	0	0
$876$	6.00000	0.202721
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	16.0000	0.539974
$879$	2.00000	0.0674583
$880$	−8.00000	−0.269680
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	12.0000	0.403604
$885$	−8.00000	−0.268917
$886$	−20.0000	−0.671913
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	−4.00000	−0.134080
$891$	−4.00000	−0.134005
$892$	−24.0000	−0.803579
$893$	−32.0000	−1.07084
$894$	6.00000	0.200670
$895$	−24.0000	−0.802232
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	2.00000	0.0667409
$899$	−16.0000	−0.533630
$900$	−1.00000	−0.0333333
$901$	36.0000	1.19933
$902$	−24.0000	−0.799113
$903$	0	0
$904$	18.0000	0.598671
$905$	4.00000	0.132964
$906$	8.00000	0.265782
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	12.0000	0.398234
$909$	−6.00000	−0.199007
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−4.00000	−0.132453
$913$	−48.0000	−1.58857
$914$	−22.0000	−0.727695
$915$	20.0000	0.661180
$916$	−14.0000	−0.462573
$917$	0	0
$918$	6.00000	0.198030
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	−2.00000	−0.0659380
$921$	−12.0000	−0.395413
$922$	−30.0000	−0.987997
$923$	−16.0000	−0.526646
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	−8.00000	−0.262754
$928$	−2.00000	−0.0656532
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	16.0000	0.524661
$931$	0	0
$932$	10.0000	0.327561
$933$	16.0000	0.523816
$934$	−4.00000	−0.130884
$935$	−48.0000	−1.56977
$936$	2.00000	0.0653720
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	16.0000	0.521862
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	−22.0000	−0.716799
$943$	−6.00000	−0.195387
$944$	−4.00000	−0.130189
$945$	0	0
$946$	16.0000	0.520205
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	4.00000	0.129777
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	6.00000	0.194257
$955$	−32.0000	−1.03550
$956$	−16.0000	−0.517477
$957$	8.00000	0.258603
$958$	32.0000	1.03387
$959$	0	0
$960$	2.00000	0.0645497
$961$	33.0000	1.06452
$962$	12.0000	0.386896
$963$	12.0000	0.386695
$964$	22.0000	0.708572
$965$	4.00000	0.128765
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	5.00000	0.160706
$969$	−24.0000	−0.770991
$970$	−20.0000	−0.642161
$971$	−44.0000	−1.41203	−0.706014	−	0.708198i	$-0.749508\pi$
$971$	−44.0000	−1.41203	−0.706014	+	0.708198i	$0.749508\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	8.00000	0.256337
$975$	−2.00000	−0.0640513
$976$	10.0000	0.320092
$977$	−62.0000	−1.98356	−0.991778	−	0.127971i	$-0.959153\pi$
$977$	−62.0000	−1.98356	−0.991778	+	0.127971i	$0.959153\pi$
$978$	4.00000	0.127906
$979$	8.00000	0.255681
$980$	0	0
$981$	14.0000	0.446986
$982$	−4.00000	−0.127645
$983$	−40.0000	−1.27580	−0.637901	−	0.770118i	$-0.720197\pi$
$983$	−40.0000	−1.27580	−0.637901	+	0.770118i	$0.720197\pi$
$984$	6.00000	0.191273
$985$	−52.0000	−1.65686
$986$	−12.0000	−0.382158
$987$	0	0
$988$	−8.00000	−0.254514
$989$	4.00000	0.127193
$990$	−8.00000	−0.254257
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	8.00000	0.254000
$993$	−4.00000	−0.126936
$994$	0	0
$995$	48.0000	1.52170
$996$	12.0000	0.380235
$997$	58.0000	1.83688	0.918439	−	0.395562i	$-0.129450\pi$
$997$	58.0000	1.83688	0.918439	+	0.395562i	$0.129450\pi$
$998$	−28.0000	−0.886325
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.bl.1.1		1
7.6	odd	2		966.2.a.g.1.1	✓	1
21.20	even	2		2898.2.a.i.1.1		1
28.27	even	2		7728.2.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.g.1.1	✓	1	7.6	odd	2
2898.2.a.i.1.1		1	21.20	even	2
6762.2.a.bl.1.1		1	1.1	even	1	trivial
7728.2.a.o.1.1		1	28.27	even	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:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6762.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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