LMFDB - Embedded newform 6776.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6776,2,Mod(1,6776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6776, 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("6776.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6776 = 2^{3} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6776.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:	$54.1066324096$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +1.00000 q^{15} +2.00000 q^{17} +2.00000 q^{19} +1.00000 q^{21} -7.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} +10.0000 q^{29} +7.00000 q^{31} +1.00000 q^{35} -9.00000 q^{37} +2.00000 q^{41} +4.00000 q^{43} +2.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} -2.00000 q^{51} +2.00000 q^{53} -2.00000 q^{57} -15.0000 q^{59} +14.0000 q^{61} +2.00000 q^{63} +3.00000 q^{67} +7.00000 q^{69} +3.00000 q^{71} -10.0000 q^{73} +4.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} -2.00000 q^{85} -10.0000 q^{87} -11.0000 q^{89} -7.00000 q^{93} -2.00000 q^{95} +7.00000 q^{97} +O(q^{100})$

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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	0	0
$39$	0	0
$40$	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	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	−15.0000	−1.95283	−0.976417	−	0.215894i	$-0.930733\pi$
$59$	−15.0000	−1.95283	−0.976417	+	0.215894i	$0.930733\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	0	0
$69$	7.00000	0.842701
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−10.0000	−1.07211
$88$	0	0
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.00000	−0.725866
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	9.00000	0.854242
$112$	0	0
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	0	0
$115$	7.00000	0.652753
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	5.00000	0.427179	0.213589	−	0.976924i	$-0.431485\pi$
$137$	5.00000	0.427179	0.213589	+	0.976924i	$0.431485\pi$
$138$	0	0
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−10.0000	−0.830455
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−7.00000	−0.562254
$156$	0	0
$157$	−9.00000	−0.718278	−0.359139	−	0.933284i	$-0.616930\pi$
$157$	−9.00000	−0.718278	−0.359139	+	0.933284i	$0.616930\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	7.00000	0.551677
$162$	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
$169$	−13.0000	−1.00000
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	15.0000	1.12747
$178$	0	0
$179$	−1.00000	−0.0747435	−0.0373718	−	0.999301i	$-0.511899\pi$
$179$	−1.00000	−0.0747435	−0.0373718	+	0.999301i	$0.511899\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	−14.0000	−1.03491
$184$	0	0
$185$	9.00000	0.661693
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−17.0000	−1.23008	−0.615038	−	0.788497i	$-0.710860\pi$
$191$	−17.0000	−1.23008	−0.615038	+	0.788497i	$0.710860\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	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
$201$	−3.00000	−0.211604
$202$	0	0
$203$	−10.0000	−0.701862
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	14.0000	0.973067
$208$	0	0
$209$	0	0
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	0	0
$213$	−3.00000	−0.205557
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	−7.00000	−0.475191
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	0	0
$222$	0	0
$223$	15.0000	1.00447	0.502237	−	0.864730i	$-0.332510\pi$
$223$	15.0000	1.00447	0.502237	+	0.864730i	$0.332510\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.00000	0.0631194	0.0315597	−	0.999502i	$-0.489953\pi$
$251$	1.00000	0.0631194	0.0315597	+	0.999502i	$0.489953\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	9.00000	0.559233
$260$	0	0
$261$	−20.0000	−1.23797
$262$	0	0
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	11.0000	0.673189
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	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
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	−14.0000	−0.838158
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−7.00000	−0.410347
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	15.0000	0.873334
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	−14.0000	−0.801638
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.0000	0.884802
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	18.0000	0.986394
$334$	0	0
$335$	−3.00000	−0.163908
$336$	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
$339$	3.00000	0.162938
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−7.00000	−0.376867
$346$	0	0
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	0	0
$355$	−3.00000	−0.159223
$356$	0	0
$357$	2.00000	0.105851
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	35.0000	1.82699	0.913493	−	0.406855i	$-0.133375\pi$
$367$	35.0000	1.82699	0.913493	+	0.406855i	$0.133375\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−3.00000	−0.154100	−0.0770498	−	0.997027i	$-0.524550\pi$
$379$	−3.00000	−0.154100	−0.0770498	+	0.997027i	$0.524550\pi$
$380$	0	0
$381$	−10.0000	−0.512316
$382$	0	0
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	−11.0000	−0.557722	−0.278861	−	0.960331i	$-0.589957\pi$
$389$	−11.0000	−0.557722	−0.278861	+	0.960331i	$0.589957\pi$
$390$	0	0
$391$	−14.0000	−0.708010
$392$	0	0
$393$	2.00000	0.100887
$394$	0	0
$395$	10.0000	0.503155
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	−5.00000	−0.246632
$412$	0	0
$413$	15.0000	0.738102
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.00000	0.293821
$418$	0	0
$419$	−40.0000	−1.95413	−0.977064	−	0.212946i	$-0.931694\pi$
$419$	−40.0000	−1.95413	−0.977064	+	0.212946i	$0.931694\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	−16.0000	−0.777947
$424$	0	0
$425$	−8.00000	−0.388057
$426$	0	0
$427$	−14.0000	−0.677507
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−29.0000	−1.39365	−0.696826	−	0.717241i	$-0.745405\pi$
$433$	−29.0000	−1.39365	−0.696826	+	0.717241i	$0.745405\pi$
$434$	0	0
$435$	10.0000	0.479463
$436$	0	0
$437$	−14.0000	−0.669711
$438$	0	0
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	23.0000	1.09276	0.546381	−	0.837536i	$-0.316005\pi$
$443$	23.0000	1.09276	0.546381	+	0.837536i	$0.316005\pi$
$444$	0	0
$445$	11.0000	0.521450
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	31.0000	1.46298	0.731490	−	0.681852i	$-0.238825\pi$
$449$	31.0000	1.46298	0.731490	+	0.681852i	$0.238825\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	10.0000	0.469841
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−40.0000	−1.87112	−0.935561	−	0.353166i	$-0.885105\pi$
$457$	−40.0000	−1.87112	−0.935561	+	0.353166i	$0.885105\pi$
$458$	0	0
$459$	10.0000	0.466760
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	23.0000	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$464$	0	0
$465$	7.00000	0.324617
$466$	0	0
$467$	9.00000	0.416470	0.208235	−	0.978079i	$-0.433228\pi$
$467$	9.00000	0.416470	0.208235	+	0.978079i	$0.433228\pi$
$468$	0	0
$469$	−3.00000	−0.138527
$470$	0	0
$471$	9.00000	0.414698
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−8.00000	−0.367065
$476$	0	0
$477$	−4.00000	−0.183147
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−7.00000	−0.318511
$484$	0	0
$485$	−7.00000	−0.317854
$486$	0	0
$487$	25.0000	1.13286	0.566429	−	0.824110i	$-0.308325\pi$
$487$	25.0000	1.13286	0.566429	+	0.824110i	$0.308325\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.00000	−0.134568
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	0	0
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	10.0000	0.441511
$514$	0	0
$515$	−12.0000	−0.528783
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−4.00000	−0.175581
$520$	0	0
$521$	−45.0000	−1.97149	−0.985743	−	0.168259i	$-0.946186\pi$
$521$	−45.0000	−1.97149	−0.985743	+	0.168259i	$0.946186\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	14.0000	0.609850
$528$	0	0
$529$	26.0000	1.13043
$530$	0	0
$531$	30.0000	1.30189
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−18.0000	−0.778208
$536$	0	0
$537$	1.00000	0.0431532
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.00000	−0.171973	−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000	−0.171973	−0.0859867	+	0.996296i	$0.527404\pi$
$542$	0	0
$543$	−5.00000	−0.214571
$544$	0	0
$545$	16.0000	0.685365
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	−28.0000	−1.19501
$550$	0	0
$551$	20.0000	0.852029
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	−9.00000	−0.382029
$556$	0	0
$557$	−10.0000	−0.423714	−0.211857	−	0.977301i	$-0.567951\pi$
$557$	−10.0000	−0.423714	−0.211857	+	0.977301i	$0.567951\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	3.00000	0.126211
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	17.0000	0.710185
$574$	0	0
$575$	28.0000	1.16768
$576$	0	0
$577$	−21.0000	−0.874241	−0.437121	−	0.899403i	$-0.644002\pi$
$577$	−21.0000	−0.874241	−0.437121	+	0.899403i	$0.644002\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	14.0000	0.576860
$590$	0	0
$591$	18.0000	0.740421
$592$	0	0
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	32.0000	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$602$	0	0
$603$	−6.00000	−0.244339
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	0	0
$609$	10.0000	0.405220
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−21.0000	−0.844061	−0.422031	−	0.906582i	$-0.638683\pi$
$619$	−21.0000	−0.844061	−0.422031	+	0.906582i	$0.638683\pi$
$620$	0	0
$621$	−35.0000	−1.40450
$622$	0	0
$623$	11.0000	0.440706
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−18.0000	−0.717707
$630$	0	0
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	0	0
$633$	−10.0000	−0.397464
$634$	0	0
$635$	−10.0000	−0.396838
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−17.0000	−0.671460	−0.335730	−	0.941958i	$-0.608983\pi$
$641$	−17.0000	−0.671460	−0.335730	+	0.941958i	$0.608983\pi$
$642$	0	0
$643$	1.00000	0.0394362	0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000	0.0394362	0.0197181	+	0.999806i	$0.493723\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	21.0000	0.825595	0.412798	−	0.910823i	$-0.364552\pi$
$647$	21.0000	0.825595	0.412798	+	0.910823i	$0.364552\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	7.00000	0.274352
$652$	0	0
$653$	−13.0000	−0.508729	−0.254365	−	0.967108i	$-0.581866\pi$
$653$	−13.0000	−0.508729	−0.254365	+	0.967108i	$0.581866\pi$
$654$	0	0
$655$	2.00000	0.0781465
$656$	0	0
$657$	20.0000	0.780274
$658$	0	0
$659$	−50.0000	−1.94772	−0.973862	−	0.227142i	$-0.927062\pi$
$659$	−50.0000	−1.94772	−0.973862	+	0.227142i	$0.927062\pi$
$660$	0	0
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	−70.0000	−2.71041
$668$	0	0
$669$	−15.0000	−0.579934
$670$	0	0
$671$	0	0
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	−20.0000	−0.769800
$676$	0	0
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	−7.00000	−0.268635
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	−5.00000	−0.191040
$686$	0	0
$687$	7.00000	0.267067
$688$	0	0
$689$	0	0
$690$	0	0
$691$	29.0000	1.10321	0.551606	−	0.834105i	$-0.314015\pi$
$691$	29.0000	1.10321	0.551606	+	0.834105i	$0.314015\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.00000	0.227593
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	2.00000	0.0756469
$700$	0	0
$701$	−16.0000	−0.604312	−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000	−0.604312	−0.302156	+	0.953259i	$0.597706\pi$
$702$	0	0
$703$	−18.0000	−0.678883
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	12.0000	0.451306
$708$	0	0
$709$	−5.00000	−0.187779	−0.0938895	−	0.995583i	$-0.529930\pi$
$709$	−5.00000	−0.187779	−0.0938895	+	0.995583i	$0.529930\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	−49.0000	−1.83506
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16.0000	−0.597531
$718$	0	0
$719$	27.0000	1.00693	0.503465	−	0.864016i	$-0.332058\pi$
$719$	27.0000	1.00693	0.503465	+	0.864016i	$0.332058\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	0	0
$723$	4.00000	0.148762
$724$	0	0
$725$	−40.0000	−1.48556
$726$	0	0
$727$	−37.0000	−1.37225	−0.686127	−	0.727482i	$-0.740691\pi$
$727$	−37.0000	−1.37225	−0.686127	+	0.727482i	$0.740691\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	44.0000	1.62518	0.812589	−	0.582838i	$-0.198058\pi$
$733$	44.0000	1.62518	0.812589	+	0.582838i	$0.198058\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	0	0
$738$	0	0
$739$	38.0000	1.39785	0.698926	−	0.715194i	$-0.253662\pi$
$739$	38.0000	1.39785	0.698926	+	0.715194i	$0.253662\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−18.0000	−0.657706
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	10.0000	0.363937
$756$	0	0
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	0	0
$763$	16.0000	0.579239
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	0	0
$768$	0	0
$769$	44.0000	1.58668	0.793340	−	0.608778i	$-0.208340\pi$
$769$	44.0000	1.58668	0.793340	+	0.608778i	$0.208340\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	0	0
$773$	−38.0000	−1.36677	−0.683383	−	0.730061i	$-0.739492\pi$
$773$	−38.0000	−1.36677	−0.683383	+	0.730061i	$0.739492\pi$
$774$	0	0
$775$	−28.0000	−1.00579
$776$	0	0
$777$	−9.00000	−0.322873
$778$	0	0
$779$	4.00000	0.143315
$780$	0	0
$781$	0	0
$782$	0	0
$783$	50.0000	1.78685
$784$	0	0
$785$	9.00000	0.321224
$786$	0	0
$787$	−10.0000	−0.356462	−0.178231	−	0.983989i	$-0.557037\pi$
$787$	−10.0000	−0.356462	−0.178231	+	0.983989i	$0.557037\pi$
$788$	0	0
$789$	−6.00000	−0.213606
$790$	0	0
$791$	3.00000	0.106668
$792$	0	0
$793$	0	0
$794$	0	0
$795$	2.00000	0.0709327
$796$	0	0
$797$	−9.00000	−0.318796	−0.159398	−	0.987214i	$-0.550955\pi$
$797$	−9.00000	−0.318796	−0.159398	+	0.987214i	$0.550955\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	22.0000	0.777332
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−7.00000	−0.246718
$806$	0	0
$807$	18.0000	0.633630
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	18.0000	0.632065	0.316033	−	0.948748i	$-0.397649\pi$
$811$	18.0000	0.632065	0.316033	+	0.948748i	$0.397649\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	−19.0000	−0.662298	−0.331149	−	0.943578i	$-0.607436\pi$
$823$	−19.0000	−0.662298	−0.331149	+	0.943578i	$0.607436\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.0000	0.834562	0.417281	−	0.908778i	$-0.362983\pi$
$827$	24.0000	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$828$	0	0
$829$	−37.0000	−1.28506	−0.642532	−	0.766259i	$-0.722116\pi$
$829$	−37.0000	−1.28506	−0.642532	+	0.766259i	$0.722116\pi$
$830$	0	0
$831$	28.0000	0.971309
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	18.0000	0.622916
$836$	0	0
$837$	35.0000	1.20978
$838$	0	0
$839$	3.00000	0.103572	0.0517858	−	0.998658i	$-0.483509\pi$
$839$	3.00000	0.103572	0.0517858	+	0.998658i	$0.483509\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	−24.0000	−0.826604
$844$	0	0
$845$	13.0000	0.447214
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	63.0000	2.15961
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	36.0000	1.22974	0.614868	−	0.788630i	$-0.289209\pi$
$857$	36.0000	1.22974	0.614868	+	0.788630i	$0.289209\pi$
$858$	0	0
$859$	−35.0000	−1.19418	−0.597092	−	0.802173i	$-0.703677\pi$
$859$	−35.0000	−1.19418	−0.597092	+	0.802173i	$0.703677\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	0	0
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	0	0
$865$	−4.00000	−0.136004
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	18.0000	0.607125
$880$	0	0
$881$	31.0000	1.04442	0.522208	−	0.852818i	$-0.325108\pi$
$881$	31.0000	1.04442	0.522208	+	0.852818i	$0.325108\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	−15.0000	−0.504219
$886$	0	0
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	0	0
$889$	−10.0000	−0.335389
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.0000	0.535420
$894$	0	0
$895$	1.00000	0.0334263
$896$	0	0
$897$	0	0
$898$	0	0
$899$	70.0000	2.33463
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	−5.00000	−0.166206
$906$	0	0
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	0	0
$909$	24.0000	0.796030
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	14.0000	0.462826
$916$	0	0
$917$	2.00000	0.0660458
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	0	0
$923$	0	0
$924$	0	0
$925$	36.0000	1.18367
$926$	0	0
$927$	−24.0000	−0.788263
$928$	0	0
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	0	0
$936$	0	0
$937$	48.0000	1.56809	0.784046	−	0.620703i	$-0.213153\pi$
$937$	48.0000	1.56809	0.784046	+	0.620703i	$0.213153\pi$
$938$	0	0
$939$	19.0000	0.620042
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	−14.0000	−0.455903
$944$	0	0
$945$	5.00000	0.162650
$946$	0	0
$947$	19.0000	0.617417	0.308709	−	0.951157i	$-0.400103\pi$
$947$	19.0000	0.617417	0.308709	+	0.951157i	$0.400103\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	3.00000	0.0972817
$952$	0	0
$953$	44.0000	1.42530	0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000	1.42530	0.712650	+	0.701520i	$0.247495\pi$
$954$	0	0
$955$	17.0000	0.550107
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.00000	−0.161458
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	−36.0000	−1.16008
$964$	0	0
$965$	−6.00000	−0.193147
$966$	0	0
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	55.0000	1.76503	0.882517	−	0.470281i	$-0.155847\pi$
$971$	55.0000	1.76503	0.882517	+	0.470281i	$0.155847\pi$
$972$	0	0
$973$	6.00000	0.192351
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.0000	−0.479893	−0.239946	−	0.970786i	$-0.577130\pi$
$977$	−15.0000	−0.479893	−0.239946	+	0.970786i	$0.577130\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	32.0000	1.02168
$982$	0	0
$983$	3.00000	0.0956851	0.0478426	−	0.998855i	$-0.484765\pi$
$983$	3.00000	0.0956851	0.0478426	+	0.998855i	$0.484765\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	−28.0000	−0.890348
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	0	0
$993$	7.00000	0.222138
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	0	0
$999$	−45.0000	−1.42374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6776.2.a.d.1.1		1
11.10	odd	2		616.2.a.b.1.1	✓	1
33.32	even	2		5544.2.a.o.1.1		1
44.43	even	2		1232.2.a.i.1.1		1
77.76	even	2		4312.2.a.h.1.1		1
88.21	odd	2		4928.2.a.ba.1.1		1
88.43	even	2		4928.2.a.k.1.1		1
308.307	odd	2		8624.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.a.b.1.1	✓	1	11.10	odd	2
1232.2.a.i.1.1		1	44.43	even	2
4312.2.a.h.1.1		1	77.76	even	2
4928.2.a.k.1.1		1	88.43	even	2
4928.2.a.ba.1.1		1	88.21	odd	2
5544.2.a.o.1.1		1	33.32	even	2
6776.2.a.d.1.1		1	1.1	even	1	trivial
8624.2.a.m.1.1		1	308.307	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6776 = 2^{3} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6776.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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