LMFDB - Embedded newform 7644.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7644,2,Mod(1,7644)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7644, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7644.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7644.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:	$61.0376473051$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1092)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7644.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +1.00000 q^{13} -2.00000 q^{15} +4.00000 q^{17} +8.00000 q^{19} +6.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +8.00000 q^{31} -2.00000 q^{37} -1.00000 q^{39} +2.00000 q^{41} -4.00000 q^{43} +2.00000 q^{45} -2.00000 q^{47} -4.00000 q^{51} -6.00000 q^{53} -8.00000 q^{57} +6.00000 q^{59} +2.00000 q^{61} +2.00000 q^{65} -4.00000 q^{67} -6.00000 q^{69} -4.00000 q^{71} +6.00000 q^{73} +1.00000 q^{75} +1.00000 q^{81} -10.0000 q^{83} +8.00000 q^{85} -2.00000 q^{87} -10.0000 q^{89} -8.00000 q^{93} +16.0000 q^{95} +2.00000 q^{97} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−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
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	16.0000	1.64157
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	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
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.00000	−0.172133
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	16.0000	1.28515
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	8.00000	0.611775
$172$	0	0
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	14.0000	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.0000	1.01300	0.506502	−	0.862239i	$-0.330938\pi$
$191$	14.0000	1.01300	0.506502	+	0.862239i	$0.330938\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	4.00000	0.274075
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	28.0000	1.87502	0.937509	−	0.347960i	$-0.113126\pi$
$223$	28.0000	1.87502	0.937509	+	0.347960i	$0.113126\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	0	0
$238$	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
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	10.0000	0.633724
$250$	0	0
$251$	−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
$253$	0	0
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	−16.0000	−0.947758
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.00000	0.346989
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	32.0000	1.78053
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	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	$0.685239\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−12.0000	−0.646058
$346$	0	0
$347$	14.0000	0.751559	0.375780	−	0.926709i	$-0.377375\pi$
$347$	14.0000	0.751559	0.375780	+	0.926709i	$0.377375\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	22.0000	1.17094	0.585471	−	0.810693i	$-0.300910\pi$
$353$	22.0000	1.17094	0.585471	+	0.810693i	$0.300910\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	12.0000	0.628109
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	2.00000	0.103005
$378$	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
$381$	0	0
$382$	0	0
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	2.00000	0.0993808
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−20.0000	−0.981761
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	0	0
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	−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
$436$	0	0
$437$	48.0000	2.29615
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	−20.0000	−0.948091
$446$	0	0
$447$	−8.00000	−0.378387
$448$	0	0
$449$	28.0000	1.32140	0.660701	−	0.750649i	$-0.270259\pi$
$449$	28.0000	1.32140	0.660701	+	0.750649i	$0.270259\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	22.0000	1.02464	0.512321	−	0.858794i	$-0.328786\pi$
$461$	22.0000	1.02464	0.512321	+	0.858794i	$0.328786\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	0	0
$465$	−16.0000	−0.741982
$466$	0	0
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−8.00000	−0.367065
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.00000	0.181631
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−38.0000	−1.71492	−0.857458	−	0.514554i	$-0.827958\pi$
$491$	−38.0000	−1.71492	−0.857458	+	0.514554i	$0.827958\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−8.00000	−0.353209
$514$	0	0
$515$	−32.0000	−1.41009
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	0	0
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	32.0000	1.39394
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	2.00000	0.0866296
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	−14.0000	−0.604145
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	16.0000	0.681623
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	−14.0000	−0.584858
$574$	0	0
$575$	−6.00000	−0.250217
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	2.00000	0.0825488	0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000	0.0825488	0.0412744	+	0.999148i	$0.486858\pi$
$588$	0	0
$589$	64.0000	2.63707
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	−38.0000	−1.55264	−0.776319	−	0.630340i	$-0.782915\pi$
$599$	−38.0000	−1.55264	−0.776319	+	0.630340i	$0.782915\pi$
$600$	0	0
$601$	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
$604$	0	0
$605$	−22.0000	−0.894427
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.00000	−0.0809113
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	4.00000	0.161034	0.0805170	−	0.996753i	$-0.474343\pi$
$617$	4.00000	0.161034	0.0805170	+	0.996753i	$0.474343\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	44.0000	1.72982	0.864909	−	0.501928i	$-0.167376\pi$
$647$	44.0000	1.72982	0.864909	+	0.501928i	$0.167376\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	32.0000	1.25034
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	42.0000	1.63609	0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000	1.63609	0.818044	+	0.575156i	$0.195059\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0000	0.464642
$668$	0	0
$669$	−28.0000	−1.08254
$670$	0	0
$671$	0	0
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	48.0000	1.84479	0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000	1.84479	0.922395	+	0.386248i	$0.126229\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−22.0000	−0.843042
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−24.0000	−0.916993
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	22.0000	0.832116
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	−16.0000	−0.603451
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	48.0000	1.79761
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−14.0000	−0.520666
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−48.0000	−1.78022	−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000	−1.78022	−0.890111	+	0.455744i	$0.849373\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	−38.0000	−1.40356	−0.701781	−	0.712393i	$-0.747612\pi$
$733$	−38.0000	−1.40356	−0.701781	+	0.712393i	$0.747612\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	16.0000	0.586195
$746$	0	0
$747$	−10.0000	−0.365881
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	32.0000	1.16460
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.00000	0.289241
$766$	0	0
$767$	6.00000	0.216647
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	0	0
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	36.0000	1.28490
$786$	0	0
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	0	0
$789$	−30.0000	−1.06803
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	12.0000	0.425596
$796$	0	0
$797$	16.0000	0.566749	0.283375	−	0.959009i	$-0.408546\pi$
$797$	16.0000	0.566749	0.283375	+	0.959009i	$0.408546\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−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
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	0	0
$815$	8.00000	0.280228
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.0000	1.67521	0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000	1.67521	0.837606	+	0.546275i	$0.183955\pi$
$822$	0	0
$823$	−36.0000	−1.25488	−0.627441	−	0.778664i	$-0.715897\pi$
$823$	−36.0000	−1.25488	−0.627441	+	0.778664i	$0.715897\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.0000	−1.11275	−0.556375	−	0.830932i	$-0.687808\pi$
$827$	−32.0000	−1.11275	−0.556375	+	0.830932i	$0.687808\pi$
$828$	0	0
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	0	0
$831$	6.00000	0.208138
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−36.0000	−1.24583
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	20.0000	0.688837
$844$	0	0
$845$	2.00000	0.0688021
$846$	0	0
$847$	0	0
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	16.0000	0.547188
$856$	0	0
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.0000	0.953131	0.476566	−	0.879139i	$-0.341881\pi$
$863$	28.0000	0.953131	0.476566	+	0.879139i	$0.341881\pi$
$864$	0	0
$865$	24.0000	0.816024
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	0	0
$879$	30.0000	1.01187
$880$	0	0
$881$	−16.0000	−0.539054	−0.269527	−	0.962993i	$-0.586867\pi$
$881$	−16.0000	−0.539054	−0.269527	+	0.962993i	$0.586867\pi$
$882$	0	0
$883$	−56.0000	−1.88455	−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−56.0000	−1.88455	−0.942275	+	0.334840i	$0.891318\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−16.0000	−0.535420
$894$	0	0
$895$	28.0000	0.935937
$896$	0	0
$897$	−6.00000	−0.200334
$898$	0	0
$899$	16.0000	0.533630
$900$	0	0
$901$	−24.0000	−0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−36.0000	−1.19668
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	0	0
$909$	4.00000	0.132672
$910$	0	0
$911$	−10.0000	−0.331315	−0.165657	−	0.986183i	$-0.552975\pi$
$911$	−10.0000	−0.331315	−0.165657	+	0.986183i	$0.552975\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−4.00000	−0.132236
$916$	0	0
$917$	0	0
$918$	0	0
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	−4.00000	−0.131662
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	−38.0000	−1.24674	−0.623370	−	0.781927i	$-0.714237\pi$
$929$	−38.0000	−1.24674	−0.623370	+	0.781927i	$0.714237\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	12.0000	0.390774
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	0	0
$955$	28.0000	0.906059
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	6.00000	0.193347
$964$	0	0
$965$	−28.0000	−0.901352
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	0	0
$969$	−32.0000	−1.02799
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	32.0000	1.02377	0.511885	−	0.859054i	$-0.328947\pi$
$977$	32.0000	1.02377	0.511885	+	0.859054i	$0.328947\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	−32.0000	−1.01447
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7644.2.a.c.1.1		1
7.6	odd	2		1092.2.a.e.1.1	✓	1
21.20	even	2		3276.2.a.i.1.1		1
28.27	even	2		4368.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1092.2.a.e.1.1	✓	1	7.6	odd	2
3276.2.a.i.1.1		1	21.20	even	2
4368.2.a.c.1.1		1	28.27	even	2
7644.2.a.c.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7644.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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