LMFDB - Embedded newform 6762.2.a.bj.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:	$1$
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	$\chi$	$=$	6762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} -4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} -2.00000 q^{39} -8.00000 q^{41} +4.00000 q^{43} -4.00000 q^{44} +1.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} -5.00000 q^{50} +2.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -6.00000 q^{57} -2.00000 q^{58} -4.00000 q^{59} +2.00000 q^{62} +1.00000 q^{64} -4.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} +1.00000 q^{69} +1.00000 q^{72} -4.00000 q^{73} -2.00000 q^{74} -5.00000 q^{75} -6.00000 q^{76} -2.00000 q^{78} -12.0000 q^{79} +1.00000 q^{81} -8.00000 q^{82} -6.00000 q^{83} +4.00000 q^{86} -2.00000 q^{87} -4.00000 q^{88} -10.0000 q^{89} +1.00000 q^{92} +2.00000 q^{93} -6.00000 q^{94} +1.00000 q^{96} +6.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−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	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	−4.00000	−0.852803
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	−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	$0.559454\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−6.00000	−0.973329
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	1.00000	0.147442
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−5.00000	−0.707107
$51$	2.00000	0.280056
$52$	−2.00000	−0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	−6.00000	−0.794719
$58$	−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	$0.583847\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.00000	0.242536
$69$	1.00000	0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−2.00000	−0.232495
$75$	−5.00000	−0.577350
$76$	−6.00000	−0.688247
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−8.00000	−0.883452
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	−2.00000	−0.214423
$88$	−4.00000	−0.426401
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	1.00000	0.104257
$93$	2.00000	0.207390
$94$	−6.00000	−0.618853
$95$	0	0
$96$	1.00000	0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	−5.00000	−0.500000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	2.00000	0.198030
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−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
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	−6.00000	−0.561951
$115$	0	0
$116$	−2.00000	−0.185695
$117$	−2.00000	−0.184900
$118$	−4.00000	−0.368230
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−8.00000	−0.721336
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	−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
$135$	0	0
$136$	2.00000	0.171499
$137$	−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
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	−2.00000	−0.164399
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	−5.00000	−0.408248
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−6.00000	−0.486664
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−12.0000	−0.954669
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−8.00000	−0.624695
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−10.0000	−0.773823	−0.386912	−	0.922117i	$-0.626458\pi$
$167$	−10.0000	−0.773823	−0.386912	+	0.922117i	$0.626458\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	4.00000	0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	−4.00000	−0.300658
$178$	−10.0000	−0.749532
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	0	0
$184$	1.00000	0.0737210
$185$	0	0
$186$	2.00000	0.146647
$187$	−8.00000	−0.585018
$188$	−6.00000	−0.437595
$189$	0	0
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	1.00000	0.0721688
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−4.00000	−0.284268
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−5.00000	−0.353553
$201$	−4.00000	−0.282138
$202$	−14.0000	−0.985037
$203$	0	0
$204$	2.00000	0.140028
$205$	0	0
$206$	4.00000	0.278693
$207$	1.00000	0.0695048
$208$	−2.00000	−0.138675
$209$	24.0000	1.66011
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	2.00000	0.135457
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−4.00000	−0.269069
$222$	−2.00000	−0.134231
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	−2.00000	−0.133038
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	−6.00000	−0.397360
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	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
$235$	0	0
$236$	−4.00000	−0.260378
$237$	−12.0000	−0.779484
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	−8.00000	−0.510061
$247$	12.0000	0.763542
$248$	2.00000	0.127000
$249$	−6.00000	−0.380235
$250$	0	0
$251$	26.0000	1.64111	0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000	1.64111	0.820553	+	0.571571i	$0.193666\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−28.0000	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$257$	−28.0000	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	−12.0000	−0.741362
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	0	0
$267$	−10.0000	−0.611990
$268$	−4.00000	−0.244339
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−30.0000	−1.82237	−0.911185	−	0.411997i	$-0.864831\pi$
$271$	−30.0000	−1.82237	−0.911185	+	0.411997i	$0.864831\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−6.00000	−0.362473
$275$	20.0000	1.20605
$276$	1.00000	0.0601929
$277$	30.0000	1.80253	0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000	1.80253	0.901263	+	0.433273i	$0.142641\pi$
$278$	8.00000	0.479808
$279$	2.00000	0.119737
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	−6.00000	−0.357295
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	0	0
$285$	0	0
$286$	8.00000	0.473050
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	6.00000	0.351726
$292$	−4.00000	−0.234082
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	−4.00000	−0.232104
$298$	10.0000	0.579284
$299$	−2.00000	−0.115663
$300$	−5.00000	−0.288675
$301$	0	0
$302$	−16.0000	−0.920697
$303$	−14.0000	−0.804279
$304$	−6.00000	−0.344124
$305$	0	0
$306$	2.00000	0.114332
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	−2.00000	−0.113228
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	6.00000	0.336463
$319$	8.00000	0.447914
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	−12.0000	−0.667698
$324$	1.00000	0.0555556
$325$	10.0000	0.554700
$326$	−8.00000	−0.443079
$327$	2.00000	0.110600
$328$	−8.00000	−0.441726
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−6.00000	−0.329293
$333$	−2.00000	−0.109599
$334$	−10.0000	−0.547176
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−9.00000	−0.489535
$339$	−2.00000	−0.108625
$340$	0	0
$341$	−8.00000	−0.433224
$342$	−6.00000	−0.324443
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	−2.00000	−0.107211
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−4.00000	−0.213201
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	−10.0000	−0.529999
$357$	0	0
$358$	12.0000	0.634220
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−16.0000	−0.840941
$363$	5.00000	0.262432
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	1.00000	0.0521286
$369$	−8.00000	−0.416463
$370$	0	0
$371$	0	0
$372$	2.00000	0.103695
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	−6.00000	−0.309426
$377$	4.00000	0.206010
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	20.0000	1.02329
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	6.00000	0.305392
$387$	4.00000	0.203331
$388$	6.00000	0.304604
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	−12.0000	−0.605320
$394$	18.0000	0.906827
$395$	0	0
$396$	−4.00000	−0.201008
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	−5.00000	−0.250000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	−4.00000	−0.199502
$403$	−4.00000	−0.199254
$404$	−14.0000	−0.696526
$405$	0	0
$406$	0	0
$407$	8.00000	0.396545
$408$	2.00000	0.0990148
$409$	−24.0000	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	4.00000	0.197066
$413$	0	0
$414$	1.00000	0.0491473
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	8.00000	0.391762
$418$	24.0000	1.17388
$419$	34.0000	1.66101	0.830504	−	0.557012i	$-0.188052\pi$
$419$	34.0000	1.66101	0.830504	+	0.557012i	$0.188052\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	−8.00000	−0.389434
$423$	−6.00000	−0.291730
$424$	6.00000	0.291386
$425$	−10.0000	−0.485071
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	8.00000	0.386244
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	1.00000	0.0481125
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	−6.00000	−0.287019
$438$	−4.00000	−0.191127
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	0	0
$442$	−4.00000	−0.190261
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	14.0000	0.662919
$447$	10.0000	0.472984
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	−5.00000	−0.235702
$451$	32.0000	1.50682
$452$	−2.00000	−0.0940721
$453$	−16.0000	−0.751746
$454$	6.00000	0.281594
$455$	0	0
$456$	−6.00000	−0.280976
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−16.0000	−0.747631
$459$	2.00000	0.0933520
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	10.0000	0.463241
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	0	0
$471$	4.00000	0.184310
$472$	−4.00000	−0.184115
$473$	−16.0000	−0.735681
$474$	−12.0000	−0.551178
$475$	30.0000	1.37649
$476$	0	0
$477$	6.00000	0.274721
$478$	8.00000	0.365911
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	−22.0000	−1.00207
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	−8.00000	−0.361773
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	−8.00000	−0.360668
$493$	−4.00000	−0.180151
$494$	12.0000	0.539906
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	−6.00000	−0.268866
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	0	0
$501$	−10.0000	−0.446767
$502$	26.0000	1.16044
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	−4.00000	−0.177822
$507$	−9.00000	−0.399704
$508$	16.0000	0.709885
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−6.00000	−0.264906
$514$	−28.0000	−1.23503
$515$	0	0
$516$	4.00000	0.176090
$517$	24.0000	1.05552
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	−2.00000	−0.0875376
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	8.00000	0.348817
$527$	4.00000	0.174243
$528$	−4.00000	−0.174078
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	16.0000	0.693037
$534$	−10.0000	−0.432742
$535$	0	0
$536$	−4.00000	−0.172774
$537$	12.0000	0.517838
$538$	−6.00000	−0.258678
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−30.0000	−1.28861
$543$	−16.0000	−0.686626
$544$	2.00000	0.0857493
$545$	0	0
$546$	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
$549$	0	0
$550$	20.0000	0.852803
$551$	12.0000	0.511217
$552$	1.00000	0.0425628
$553$	0	0
$554$	30.0000	1.27458
$555$	0	0
$556$	8.00000	0.339276
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	2.00000	0.0846668
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−8.00000	−0.337760
$562$	18.0000	0.759284
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	10.0000	0.420331
$567$	0	0
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	8.00000	0.334497
$573$	20.0000	0.835512
$574$	0	0
$575$	−5.00000	−0.208514
$576$	1.00000	0.0416667
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	−13.0000	−0.540729
$579$	6.00000	0.249351
$580$	0	0
$581$	0	0
$582$	6.00000	0.248708
$583$	−24.0000	−0.993978
$584$	−4.00000	−0.165521
$585$	0	0
$586$	16.0000	0.660954
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	18.0000	0.740421
$592$	−2.00000	−0.0821995
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	10.0000	0.409616
$597$	−4.00000	−0.163709
$598$	−2.00000	−0.0817861
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	−5.00000	−0.204124
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	−16.0000	−0.651031
$605$	0	0
$606$	−14.0000	−0.568711
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	2.00000	0.0808452
$613$	42.0000	1.69636	0.848182	−	0.529705i	$-0.177697\pi$
$613$	42.0000	1.69636	0.848182	+	0.529705i	$0.177697\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	4.00000	0.160904
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	18.0000	0.721734
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	25.0000	1.00000
$626$	−22.0000	−0.879297
$627$	24.0000	0.958468
$628$	4.00000	0.159617
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	−12.0000	−0.477334
$633$	−8.00000	−0.317971
$634$	−22.0000	−0.873732
$635$	0	0
$636$	6.00000	0.237915
$637$	0	0
$638$	8.00000	0.316723
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	12.0000	0.473602
$643$	−34.0000	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	0	0
$645$	0	0
$646$	−12.0000	−0.472134
$647$	−10.0000	−0.393141	−0.196570	−	0.980490i	$-0.562980\pi$
$647$	−10.0000	−0.393141	−0.196570	+	0.980490i	$0.562980\pi$
$648$	1.00000	0.0392837
$649$	16.0000	0.628055
$650$	10.0000	0.392232
$651$	0	0
$652$	−8.00000	−0.313304
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	−8.00000	−0.312348
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	16.0000	0.622328	0.311164	−	0.950356i	$-0.399281\pi$
$661$	16.0000	0.622328	0.311164	+	0.950356i	$0.399281\pi$
$662$	−20.0000	−0.777322
$663$	−4.00000	−0.155347
$664$	−6.00000	−0.232845
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−2.00000	−0.0774403
$668$	−10.0000	−0.386912
$669$	14.0000	0.541271
$670$	0	0
$671$	0	0
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	14.0000	0.539260
$675$	−5.00000	−0.192450
$676$	−9.00000	−0.346154
$677$	40.0000	1.53732	0.768662	−	0.639655i	$-0.220923\pi$
$677$	40.0000	1.53732	0.768662	+	0.639655i	$0.220923\pi$
$678$	−2.00000	−0.0768095
$679$	0	0
$680$	0	0
$681$	6.00000	0.229920
$682$	−8.00000	−0.306336
$683$	8.00000	0.306111	0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000	0.306111	0.153056	+	0.988218i	$0.451089\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	0	0
$687$	−16.0000	−0.610438
$688$	4.00000	0.152499
$689$	−12.0000	−0.457164
$690$	0	0
$691$	48.0000	1.82601	0.913003	−	0.407953i	$-0.133757\pi$
$691$	48.0000	1.82601	0.913003	+	0.407953i	$0.133757\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−16.0000	−0.607352
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	−16.0000	−0.606043
$698$	10.0000	0.378506
$699$	10.0000	0.378235
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−2.00000	−0.0754851
$703$	12.0000	0.452589
$704$	−4.00000	−0.150756
$705$	0	0
$706$	0	0
$707$	0	0
$708$	−4.00000	−0.150329
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	−10.0000	−0.374766
$713$	2.00000	0.0749006
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	8.00000	0.298765
$718$	−4.00000	−0.149279
$719$	−2.00000	−0.0745874	−0.0372937	−	0.999304i	$-0.511874\pi$
$719$	−2.00000	−0.0745874	−0.0372937	+	0.999304i	$0.511874\pi$
$720$	0	0
$721$	0	0
$722$	17.0000	0.632674
$723$	−22.0000	−0.818189
$724$	−16.0000	−0.594635
$725$	10.0000	0.371391
$726$	5.00000	0.185567
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	20.0000	0.738213
$735$	0	0
$736$	1.00000	0.0368605
$737$	16.0000	0.589368
$738$	−8.00000	−0.294484
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	−52.0000	−1.90769	−0.953847	−	0.300291i	$-0.902916\pi$
$743$	−52.0000	−1.90769	−0.953847	+	0.300291i	$0.902916\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	2.00000	0.0732252
$747$	−6.00000	−0.219529
$748$	−8.00000	−0.292509
$749$	0	0
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	−6.00000	−0.218797
$753$	26.0000	0.947493
$754$	4.00000	0.145671
$755$	0	0
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	4.00000	0.145287
$759$	−4.00000	−0.145191
$760$	0	0
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	16.0000	0.579619
$763$	0	0
$764$	20.0000	0.723575
$765$	0	0
$766$	24.0000	0.867155
$767$	8.00000	0.288863
$768$	1.00000	0.0360844
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	−28.0000	−1.00840
$772$	6.00000	0.215945
$773$	36.0000	1.29483	0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000	1.29483	0.647415	+	0.762138i	$0.275850\pi$
$774$	4.00000	0.143777
$775$	−10.0000	−0.359211
$776$	6.00000	0.215387
$777$	0	0
$778$	−30.0000	−1.07555
$779$	48.0000	1.71978
$780$	0	0
$781$	0	0
$782$	2.00000	0.0715199
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	0	0
$786$	−12.0000	−0.428026
$787$	46.0000	1.63972	0.819861	−	0.572562i	$-0.194050\pi$
$787$	46.0000	1.63972	0.819861	+	0.572562i	$0.194050\pi$
$788$	18.0000	0.641223
$789$	8.00000	0.284808
$790$	0	0
$791$	0	0
$792$	−4.00000	−0.142134
$793$	0	0
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−4.00000	−0.141776
$797$	28.0000	0.991811	0.495905	−	0.868377i	$-0.334836\pi$
$797$	28.0000	0.991811	0.495905	+	0.868377i	$0.334836\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	−5.00000	−0.176777
$801$	−10.0000	−0.353333
$802$	10.0000	0.353112
$803$	16.0000	0.564628
$804$	−4.00000	−0.141069
$805$	0	0
$806$	−4.00000	−0.140894
$807$	−6.00000	−0.211210
$808$	−14.0000	−0.492518
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	−30.0000	−1.05215
$814$	8.00000	0.280400
$815$	0	0
$816$	2.00000	0.0700140
$817$	−24.0000	−0.839654
$818$	−24.0000	−0.839140
$819$	0	0
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	−6.00000	−0.209274
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	4.00000	0.139347
$825$	20.0000	0.696311
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	1.00000	0.0347524
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	0	0
$831$	30.0000	1.04069
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	8.00000	0.277017
$835$	0	0
$836$	24.0000	0.830057
$837$	2.00000	0.0691301
$838$	34.0000	1.17451
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	6.00000	0.206774
$843$	18.0000	0.619953
$844$	−8.00000	−0.275371
$845$	0	0
$846$	−6.00000	−0.206284
$847$	0	0
$848$	6.00000	0.206041
$849$	10.0000	0.343199
$850$	−10.0000	−0.342997
$851$	−2.00000	−0.0685591
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	40.0000	1.36637	0.683187	−	0.730243i	$-0.260593\pi$
$857$	40.0000	1.36637	0.683187	+	0.730243i	$0.260593\pi$
$858$	8.00000	0.273115
$859$	48.0000	1.63774	0.818869	−	0.573980i	$-0.194601\pi$
$859$	48.0000	1.63774	0.818869	+	0.573980i	$0.194601\pi$
$860$	0	0
$861$	0	0
$862$	8.00000	0.272481
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	6.00000	0.203888
$867$	−13.0000	−0.441503
$868$	0	0
$869$	48.0000	1.62829
$870$	0	0
$871$	8.00000	0.271070
$872$	2.00000	0.0677285
$873$	6.00000	0.203069
$874$	−6.00000	−0.202953
$875$	0	0
$876$	−4.00000	−0.135147
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	−10.0000	−0.337484
$879$	16.0000	0.539667
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	8.00000	0.268765
$887$	22.0000	0.738688	0.369344	−	0.929293i	$-0.379582\pi$
$887$	22.0000	0.738688	0.369344	+	0.929293i	$0.379582\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	14.0000	0.468755
$893$	36.0000	1.20469
$894$	10.0000	0.334450
$895$	0	0
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	14.0000	0.467186
$899$	−4.00000	−0.133407
$900$	−5.00000	−0.166667
$901$	12.0000	0.399778
$902$	32.0000	1.06548
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	−16.0000	−0.531564
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	6.00000	0.199117
$909$	−14.0000	−0.464351
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	−6.00000	−0.198680
$913$	24.0000	0.794284
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−16.0000	−0.528655
$917$	0	0
$918$	2.00000	0.0660098
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	−6.00000	−0.197599
$923$	0	0
$924$	0	0
$925$	10.0000	0.328798
$926$	−8.00000	−0.262896
$927$	4.00000	0.131377
$928$	−2.00000	−0.0656532
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	0	0
$931$	0	0
$932$	10.0000	0.327561
$933$	18.0000	0.589294
$934$	−18.0000	−0.588978
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	4.00000	0.130327
$943$	−8.00000	−0.260516
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−16.0000	−0.520205
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	−12.0000	−0.389742
$949$	8.00000	0.259691
$950$	30.0000	0.973329
$951$	−22.0000	−0.713399
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	8.00000	0.258738
$957$	8.00000	0.258603
$958$	−4.00000	−0.129234
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	4.00000	0.128965
$963$	12.0000	0.386695
$964$	−22.0000	−0.708572
$965$	0	0
$966$	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
$969$	−12.0000	−0.385496
$970$	0	0
$971$	50.0000	1.60458	0.802288	−	0.596937i	$-0.203616\pi$
$971$	50.0000	1.60458	0.802288	+	0.596937i	$0.203616\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	16.0000	0.512673
$975$	10.0000	0.320256
$976$	0	0
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	−8.00000	−0.255812
$979$	40.0000	1.27841
$980$	0	0
$981$	2.00000	0.0638551
$982$	24.0000	0.765871
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	−8.00000	−0.255031
$985$	0	0
$986$	−4.00000	−0.127386
$987$	0	0
$988$	12.0000	0.381771
$989$	4.00000	0.127193
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	2.00000	0.0635001
$993$	−20.0000	−0.634681
$994$	0	0
$995$	0	0
$996$	−6.00000	−0.190117
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	−8.00000	−0.253236
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.bj.1.1	yes	1
7.6	odd	2		6762.2.a.z.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6762.2.a.z.1.1	✓	1	7.6	odd	2
6762.2.a.bj.1.1	yes	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:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
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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