LMFDB - Embedded newform 1575.2.b.a.251.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(251,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("1575.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			251.4
Root			$2.57794i$ of defining polynomial
Character	$\chi$	$=$	1575.251
Dual form			1575.2.b.a.251.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.57794i q^{2} -4.64575 q^{4} -2.64575 q^{7} -6.82058i q^{8} +O(q^{10})$	$q+2.57794i q^{2} -4.64575 q^{4} -2.64575 q^{7} -6.82058i q^{8} +0.913230i q^{11} -6.82058i q^{14} +8.29150 q^{16} -2.35425 q^{22} -9.39851i q^{23} +12.2915 q^{28} -6.06910i q^{29} +7.73381i q^{32} +10.5830 q^{37} -5.29150 q^{43} -4.24264i q^{44} +24.2288 q^{46} +7.00000 q^{49} +14.5544i q^{53} +18.0455i q^{56} +15.6458 q^{58} -3.35425 q^{64} +4.00000 q^{67} -7.57205i q^{71} +27.2823i q^{74} -2.41618i q^{77} +8.00000 q^{79} -13.6412i q^{86} +6.22876 q^{88} +43.6631i q^{92} +18.0455i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 12 q^{16} - 20 q^{22} + 28 q^{28} + 44 q^{46} + 28 q^{49} + 52 q^{58} - 24 q^{64} + 16 q^{67} + 32 q^{79} - 28 q^{88}+O(q^{100}) $ 4 * q - 8 * q^4 + 12 * q^16 - 20 * q^22 + 28 * q^28 + 44 * q^46 + 28 * q^49 + 52 * q^58 - 24 * q^64 + 16 * q^67 + 32 * q^79 - 28 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	2.57794i	1.82288i	0.411438	+	0.911438i	$0.365027\pi$
$2$	2.57794i	1.82288i	−0.411438	+	0.911438i	$0.634973\pi$
$3$	0	0
$3$	0	0
$4$	−4.64575	−2.32288
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.64575	−1.00000
$7$	−2.64575	−1.00000
$8$	− 6.82058i	− 2.41144i
$9$	0	0
$10$	0	0
$11$	0.913230i	0.275349i	0.990478	+	0.137675i	$0.0439628\pi$
$11$	0.913230i	0.275349i	−0.990478	+	0.137675i	$0.956037\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	− 6.82058i	− 1.82288i
$15$	0	0
$16$	8.29150	2.07288
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	−2.35425	−0.501928
$23$	− 9.39851i	− 1.95973i	−0.199673	−	0.979863i	$-0.563988\pi$
$23$	− 9.39851i	− 1.95973i	0.199673	−	0.979863i	$-0.436012\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	12.2915	2.32288
$29$	− 6.06910i	− 1.12700i	−0.826115	−	0.563502i	$-0.809454\pi$
$29$	− 6.06910i	− 1.12700i	0.826115	−	0.563502i	$-0.190546\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	7.73381i	1.36716i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.5830	1.73984	0.869918	−	0.493197i	$-0.164172\pi$
$37$	10.5830	1.73984	0.869918	+	0.493197i	$0.164172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−5.29150	−0.806947	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	−5.29150	−0.806947	−0.403473	+	0.914991i	$0.632197\pi$
$44$	− 4.24264i	− 0.639602i
$45$	0	0
$46$	24.2288	3.57234
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	14.5544i	1.99920i	0.0283132	+	0.999599i	$0.490986\pi$
$53$	14.5544i	1.99920i	−0.0283132	+	0.999599i	$0.509014\pi$
$54$	0	0
$55$	0	0
$56$	18.0455i	2.41144i
$57$	0	0
$58$	15.6458	2.05439
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	−3.35425	−0.419281
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 7.57205i	− 0.898637i	−0.893372	−	0.449319i	$-0.851667\pi$
$71$	− 7.57205i	− 0.898637i	0.893372	−	0.449319i	$-0.148333\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	27.2823i	3.17150i
$75$	0	0
$76$	0	0
$77$	− 2.41618i	− 0.275349i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	− 13.6412i	− 1.47096i
$87$	0	0
$88$	6.22876	0.663988
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	43.6631i	4.55220i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	18.0455i	1.82288i
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	−37.5203	−3.64429
$107$	− 17.8838i	− 1.72889i	−0.502726	−	0.864446i	$-0.667670\pi$
$107$	− 17.8838i	− 1.72889i	0.502726	−	0.864446i	$-0.332330\pi$
$108$	0	0
$109$	−10.5830	−1.01367	−0.506834	−	0.862044i	$-0.669184\pi$
$109$	−10.5830	−1.01367	−0.506834	+	0.862044i	$0.669184\pi$
$110$	0	0
$111$	0	0
$112$	−21.9373	−2.07288
$113$	− 16.3808i	− 1.54098i	−0.637452	−	0.770490i	$-0.720012\pi$
$113$	− 16.3808i	− 1.54098i	0.637452	−	0.770490i	$-0.279988\pi$
$114$	0	0
$115$	0	0
$116$	28.1955i	2.61789i
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.1660	0.924183
$122$	0	0
$123$	0	0
$124$	0	0
$125$	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	$0.248747\pi$
$128$	6.82058i	0.602859i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	10.3117i	0.890799i
$135$	0	0
$136$	0	0
$137$	− 7.89556i	− 0.674563i	−0.941404	−	0.337282i	$-0.890493\pi$
$137$	− 7.89556i	− 0.674563i	0.941404	−	0.337282i	$-0.109507\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	19.5203	1.63810
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−49.1660	−4.04142
$149$	− 23.0397i	− 1.88748i	−0.330684	−	0.943741i	$-0.607280\pi$
$149$	− 23.0397i	− 1.88748i	0.330684	−	0.943741i	$-0.392720\pi$
$150$	0	0
$151$	5.29150	0.430616	0.215308	−	0.976546i	$-0.430924\pi$
$151$	5.29150	0.430616	0.215308	+	0.976546i	$0.430924\pi$
$152$	0	0
$153$	0	0
$154$	6.22876	0.501928
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	20.6235i	1.64072i
$159$	0	0
$160$	0	0
$161$	24.8661i	1.95973i
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	24.5830	1.87444
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	7.57205i	0.570765i
$177$	0	0
$178$	0	0
$179$	− 21.5367i	− 1.60973i	−0.593458	−	0.804865i	$-0.702238\pi$
$179$	− 21.5367i	− 1.60973i	0.593458	−	0.804865i	$-0.297762\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−64.1033	−4.72576
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 13.0514i	− 0.944369i	−0.881500	−	0.472184i	$-0.843466\pi$
$191$	− 13.0514i	− 0.944369i	0.881500	−	0.472184i	$-0.156534\pi$
$192$	0	0
$193$	−21.1660	−1.52356	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	−21.1660	−1.52356	−0.761781	+	0.647834i	$0.775675\pi$
$194$	0	0
$195$	0	0
$196$	−32.5203	−2.32288
$197$	− 10.9015i	− 0.776697i	−0.921513	−	0.388348i	$-0.873046\pi$
$197$	− 10.9015i	− 0.776697i	0.921513	−	0.388348i	$-0.126954\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	16.0573i	1.12700i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−26.4575	−1.82141	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	−26.4575	−1.82141	−0.910705	+	0.413057i	$0.864461\pi$
$212$	− 67.6160i	− 4.64389i
$213$	0	0
$214$	46.1033	3.15155
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 27.2823i	− 1.84779i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	− 20.4617i	− 1.36716i
$225$	0	0
$226$	42.2288	2.80902
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	−41.3948	−2.71770
$233$	0.589720i	0.0386338i	0.999813	+	0.0193169i	$0.00614915\pi$
$233$	0.589720i	0.0386338i	−0.999813	+	0.0193169i	$0.993851\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 30.0220i	− 1.94196i	−0.239158	−	0.970981i	$-0.576871\pi$
$239$	− 30.0220i	− 1.94196i	0.239158	−	0.970981i	$-0.423129\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	26.2073i	1.68467i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	8.58301	0.539609
$254$	41.2470i	2.58806i
$255$	0	0
$256$	−24.2915	−1.51822
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−28.0000	−1.73984
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 26.3691i	− 1.62599i	−0.582273	−	0.812993i	$-0.697836\pi$
$263$	− 26.3691i	− 1.62599i	0.582273	−	0.812993i	$-0.302164\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−18.5830	−1.13514
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	20.3542	1.22965
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	33.3514i	1.98958i	0.101955	+	0.994789i	$0.467490\pi$
$281$	33.3514i	1.98958i	−0.101955	+	0.994789i	$0.532510\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	35.1779i	2.08742i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	− 72.1822i	− 4.19550i
$297$	0	0
$298$	59.3948	3.44065
$299$	0	0
$300$	0	0
$301$	14.0000	0.806947
$302$	13.6412i	0.784960i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	11.2250i	0.639602i
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	−37.1660	−2.09075
$317$	31.5249i	1.77062i	0.465004	+	0.885309i	$0.346053\pi$
$317$	31.5249i	1.77062i	−0.465004	+	0.885309i	$0.653947\pi$
$318$	0	0
$319$	5.54249	0.310320
$320$	0	0
$321$	0	0
$322$	−64.1033	−3.57234
$323$	0	0
$324$	0	0
$325$	0	0
$326$	− 51.5587i	− 2.85557i
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5.29150	0.290847	0.145424	−	0.989369i	$-0.453545\pi$
$331$	5.29150	0.290847	0.145424	+	0.989369i	$0.453545\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−21.1660	−1.15299	−0.576493	−	0.817102i	$-0.695579\pi$
$337$	−21.1660	−1.15299	−0.576493	+	0.817102i	$0.695579\pi$
$338$	33.5132i	1.82288i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−18.5203	−1.00000
$344$	36.0911i	1.94590i
$345$	0	0
$346$	0	0
$347$	− 23.3632i	− 1.25420i	−0.778938	−	0.627100i	$-0.784242\pi$
$347$	− 23.3632i	− 1.25420i	0.778938	−	0.627100i	$-0.215758\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−7.06275	−0.376446
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	55.5203	2.93434
$359$	31.8485i	1.68090i	0.541891	+	0.840449i	$0.317708\pi$
$359$	31.8485i	1.68090i	−0.541891	+	0.840449i	$0.682292\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	− 77.9278i	− 4.06227i
$369$	0	0
$370$	0	0
$371$	− 38.5073i	− 1.99920i
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	37.0405	1.90264	0.951322	−	0.308199i	$-0.0997264\pi$
$379$	37.0405	1.90264	0.951322	+	0.308199i	$0.0997264\pi$
$380$	0	0
$381$	0	0
$382$	33.6458	1.72147
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	− 54.5646i	− 2.77727i
$387$	0	0
$388$	0	0
$389$	19.3867i	0.982947i	0.870893	+	0.491473i	$0.163542\pi$
$389$	19.3867i	0.982947i	−0.870893	+	0.491473i	$0.836458\pi$
$390$	0	0
$391$	0	0
$392$	− 47.7440i	− 2.41144i
$393$	0	0
$394$	28.1033	1.41582
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 9.07500i	− 0.453184i	−0.973990	−	0.226592i	$-0.927242\pi$
$401$	− 9.07500i	− 0.453184i	0.973990	−	0.226592i	$-0.0727584\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	−41.3948	−2.05439
$407$	9.66472i	0.479062i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	− 68.2058i	− 3.32020i
$423$	0	0
$424$	99.2693	4.82094
$425$	0	0
$426$	0	0
$427$	0	0
$428$	83.0837i	4.01600i
$429$	0	0
$430$	0	0
$431$	3.91913i	0.188778i	0.995535	+	0.0943889i	$0.0300897\pi$
$431$	3.91913i	0.188778i	−0.995535	+	0.0943889i	$0.969910\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	49.1660	2.35462
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 40.3337i	− 1.91631i	−0.286244	−	0.958157i	$-0.592407\pi$
$443$	− 40.3337i	− 1.91631i	0.286244	−	0.958157i	$-0.407593\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	8.87451	0.419281
$449$	− 28.5190i	− 1.34590i	−0.739689	−	0.672948i	$-0.765028\pi$
$449$	− 28.5190i	− 1.34590i	0.739689	−	0.672948i	$-0.234972\pi$
$450$	0	0
$451$	0	0
$452$	76.1013i	3.57951i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	42.3320	1.98021	0.990104	−	0.140334i	$-0.0448177\pi$
$457$	42.3320	1.98021	0.990104	+	0.140334i	$0.0448177\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	− 50.3220i	− 2.33614i
$465$	0	0
$466$	−1.52026	−0.0704246
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−10.5830	−0.488678
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 4.83236i	− 0.222192i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	77.3948	3.53995
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−47.2288	−2.14676
$485$	0	0
$486$	0	0
$487$	−37.0405	−1.67847	−0.839233	−	0.543772i	$-0.816996\pi$
$487$	−37.0405	−1.67847	−0.839233	+	0.543772i	$0.816996\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	34.8544i	1.57296i	0.617619	+	0.786478i	$0.288097\pi$
$491$	34.8544i	1.57296i	−0.617619	+	0.786478i	$0.711903\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.0338i	0.898637i
$498$	0	0
$499$	−26.4575	−1.18440	−0.592200	−	0.805791i	$-0.701741\pi$
$499$	−26.4575	−1.18440	−0.592200	+	0.805791i	$0.701741\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	22.1264i	0.983640i
$507$	0	0
$508$	−74.3320	−3.29795
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	− 48.9808i	− 2.16466i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	− 72.1822i	− 3.17150i
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	67.9778	2.96397
$527$	0	0
$528$	0	0
$529$	−65.3320	−2.84052
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	− 27.2823i	− 1.17842i
$537$	0	0
$538$	0	0
$539$	6.39261i	0.275349i
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	36.6808i	1.56693i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−21.1660	−0.900070
$554$	25.7794i	1.09526i
$555$	0	0
$556$	0	0
$557$	40.0102i	1.69529i	0.530566	+	0.847644i	$0.321980\pi$
$557$	40.0102i	1.69529i	−0.530566	+	0.847644i	$0.678020\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−85.9778	−3.62675
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−51.6458	−2.16701
$569$	− 45.4896i	− 1.90702i	−0.301356	−	0.953512i	$-0.597439\pi$
$569$	− 45.4896i	− 1.90702i	0.301356	−	0.953512i	$-0.402561\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 43.8249i	− 1.82288i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−13.2915	−0.550478
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	87.7490	3.60646
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	107.037i	4.38439i
$597$	0	0
$598$	0	0
$599$	48.8190i	1.99469i	0.0728143	+	0.997346i	$0.476802\pi$
$599$	48.8190i	1.99469i	−0.0728143	+	0.997346i	$0.523198\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	36.0911i	1.47096i
$603$	0	0
$604$	−24.5830	−1.00027
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	0	0
$615$	0	0
$616$	−16.4797	−0.663988
$617$	11.5485i	0.464924i	0.972605	+	0.232462i	$0.0746782\pi$
$617$	11.5485i	0.464924i	−0.972605	+	0.232462i	$0.925322\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	− 54.5646i	− 2.17046i
$633$	0	0
$634$	−81.2693	−3.22762
$635$	0	0
$636$	0	0
$637$	0	0
$638$	14.2882i	0.565674i
$639$	0	0
$640$	0	0
$641$	− 17.5603i	− 0.693589i	−0.937941	−	0.346795i	$-0.887270\pi$
$641$	− 17.5603i	− 0.693589i	0.937941	−	0.346795i	$-0.112730\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	− 115.522i	− 4.55220i
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	92.9150	3.63883
$653$	− 27.8720i	− 1.09072i	−0.838203	−	0.545358i	$-0.816394\pi$
$653$	− 27.8720i	− 1.09072i	0.838203	−	0.545358i	$-0.183606\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.4044i	0.483207i	0.970375	+	0.241604i	$0.0776734\pi$
$659$	12.4044i	0.483207i	−0.970375	+	0.241604i	$0.922327\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	13.6412i	0.530178i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−57.0405	−2.20862
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	42.3320	1.63178	0.815890	−	0.578208i	$-0.196248\pi$
$673$	42.3320	1.63178	0.815890	+	0.578208i	$0.196248\pi$
$674$	− 54.5646i	− 2.10175i
$675$	0	0
$676$	−60.3948	−2.32288
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	33.0279i	1.26378i	0.775059	+	0.631889i	$0.217720\pi$
$683$	33.0279i	1.26378i	−0.775059	+	0.631889i	$0.782280\pi$
$684$	0	0
$685$	0	0
$686$	− 47.7440i	− 1.82288i
$687$	0	0
$688$	−43.8745	−1.67270
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	60.2288	2.28625
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.8308i	1.46662i	0.679895	+	0.733309i	$0.262025\pi$
$701$	38.8308i	1.46662i	−0.679895	+	0.733309i	$0.737975\pi$
$702$	0	0
$703$	0	0
$704$	− 3.06320i	− 0.115449i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	52.9150	1.98727	0.993633	−	0.112667i	$-0.0359394\pi$
$709$	52.9150	1.98727	0.993633	+	0.112667i	$0.0359394\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	100.054i	3.73920i
$717$	0	0
$718$	−82.1033	−3.06407
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	48.9808i	1.82288i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	72.6863	2.67925
$737$	3.65292i	0.134557i
$738$	0	0
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	0	0
$741$	0	0
$742$	99.2693	3.64429
$743$	2.09267i	0.0767726i	0.999263	+	0.0383863i	$0.0122217\pi$
$743$	2.09267i	0.0767726i	−0.999263	+	0.0383863i	$0.987778\pi$
$744$	0	0
$745$	0	0
$746$	56.7146i	2.07647i
$747$	0	0
$748$	0	0
$749$	47.3161i	1.72889i
$750$	0	0
$751$	−26.4575	−0.965448	−0.482724	−	0.875772i	$-0.660353\pi$
$751$	−26.4575	−0.965448	−0.482724	+	0.875772i	$0.660353\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.5830	0.384646	0.192323	−	0.981332i	$-0.438398\pi$
$757$	10.5830	0.384646	0.192323	+	0.981332i	$0.438398\pi$
$758$	95.4881i	3.46828i
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	28.0000	1.01367
$764$	60.6337i	2.19365i
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	98.3320	3.53905
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−49.9778	−1.79179
$779$	0	0
$780$	0	0
$781$	6.91503	0.247439
$782$	0	0
$783$	0	0
$784$	58.0405	2.07288
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	50.6455i	1.80417i
$789$	0	0
$790$	0	0
$791$	43.3396i	1.54098i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	23.3948	0.826098
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 3.06320i	− 0.107696i	−0.998549	−	0.0538482i	$-0.982851\pi$
$809$	− 3.06320i	− 0.107696i	0.998549	−	0.0538482i	$-0.0171487\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	− 74.5984i	− 2.61789i
$813$	0	0
$814$	−24.9150	−0.873271
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.8602i	0.762927i	0.924384	+	0.381464i	$0.124580\pi$
$821$	21.8602i	0.762927i	−0.924384	+	0.381464i	$0.875420\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 57.3043i	− 1.99267i	−0.0855616	−	0.996333i	$-0.527268\pi$
$827$	− 57.3043i	− 1.99267i	0.0855616	−	0.996333i	$-0.472732\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−7.83399	−0.270138
$842$	67.0263i	2.30988i
$843$	0	0
$844$	122.915	4.23091
$845$	0	0
$846$	0	0
$847$	−26.8967	−0.924183
$848$	120.678i	4.14409i
$849$	0	0
$850$	0	0
$851$	− 99.4645i	− 3.40960i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−121.978	−4.16911
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	−10.1033	−0.344119
$863$	35.5014i	1.20848i	0.796802	+	0.604240i	$0.206523\pi$
$863$	35.5014i	1.20848i	−0.796802	+	0.604240i	$0.793477\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.30584i	0.247834i
$870$	0	0
$871$	0	0
$872$	72.1822i	2.44440i
$873$	0	0
$874$	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
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	58.2065	1.95881	0.979403	−	0.201916i	$-0.0647168\pi$
$883$	58.2065	1.95881	0.979403	+	0.201916i	$0.0647168\pi$
$884$	0	0
$885$	0	0
$886$	103.978	3.49320
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−42.3320	−1.41977
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	− 18.0455i	− 0.602859i
$897$	0	0
$898$	73.5203	2.45340
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−111.727	−3.71598
$905$	0	0
$906$	0	0
$907$	−5.29150	−0.175701	−0.0878507	−	0.996134i	$-0.528000\pi$
$907$	−5.29150	−0.175701	−0.0878507	+	0.996134i	$0.528000\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 52.4719i	− 1.73847i	−0.494397	−	0.869236i	$-0.664611\pi$
$911$	− 52.4719i	− 1.73847i	0.494397	−	0.869236i	$-0.335389\pi$
$912$	0	0
$913$	0	0
$914$	109.129i	3.60967i
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	37.0405	1.22185	0.610927	−	0.791687i	$-0.290797\pi$
$919$	37.0405	1.22185	0.610927	+	0.791687i	$0.290797\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	103.117i	3.38865i
$927$	0	0
$928$	46.9373	1.54079
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	− 2.73969i	− 0.0897416i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	− 27.2823i	− 0.890799i
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	12.4575	0.405029
$947$	27.0161i	0.877905i	0.898510	+	0.438953i	$0.144650\pi$
$947$	27.0161i	0.877905i	−0.898510	+	0.438953i	$0.855350\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26.0456i	0.843699i	0.906666	+	0.421849i	$0.138619\pi$
$953$	26.0456i	0.843699i	−0.906666	+	0.421849i	$0.861381\pi$
$954$	0	0
$955$	0	0
$956$	139.475i	4.51093i
$957$	0	0
$958$	0	0
$959$	20.8897i	0.674563i
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	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	$0.277624\pi$
$968$	− 69.3380i	− 2.22861i
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	− 95.4881i	− 3.05963i
$975$	0	0
$976$	0	0
$977$	62.4602i	1.99828i	0.0414892	+	0.999139i	$0.486790\pi$
$977$	62.4602i	1.99828i	−0.0414892	+	0.999139i	$0.513210\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−89.8523	−2.86730
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	49.7322i	1.58139i
$990$	0	0
$991$	−58.2065	−1.84899	−0.924496	−	0.381193i	$-0.875513\pi$
$991$	−58.2065	−1.84899	−0.924496	+	0.381193i	$0.875513\pi$
$992$	0	0
$993$	0	0
$994$	−51.6458	−1.63810
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	− 68.2058i	− 2.15902i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.b.a.251.4		4
3.2	odd	2	inner	1575.2.b.a.251.1		4
5.2	odd	4		1575.2.g.d.1574.1		8
5.3	odd	4		1575.2.g.d.1574.8		8
5.4	even	2		63.2.c.a.62.1	✓	4
7.6	odd	2	CM	1575.2.b.a.251.4		4
15.2	even	4		1575.2.g.d.1574.7		8
15.8	even	4		1575.2.g.d.1574.2		8
15.14	odd	2		63.2.c.a.62.4	yes	4
20.19	odd	2		1008.2.k.a.881.1		4
21.20	even	2	inner	1575.2.b.a.251.1		4
35.4	even	6		441.2.p.b.215.4		8
35.9	even	6		441.2.p.b.80.1		8
35.13	even	4		1575.2.g.d.1574.8		8
35.19	odd	6		441.2.p.b.80.1		8
35.24	odd	6		441.2.p.b.215.4		8
35.27	even	4		1575.2.g.d.1574.1		8
35.34	odd	2		63.2.c.a.62.1	✓	4
40.19	odd	2		4032.2.k.b.3905.2		4
40.29	even	2		4032.2.k.c.3905.3		4
45.4	even	6		567.2.o.f.188.4		8
45.14	odd	6		567.2.o.f.188.1		8
45.29	odd	6		567.2.o.f.377.4		8
45.34	even	6		567.2.o.f.377.1		8
60.59	even	2		1008.2.k.a.881.2		4
105.44	odd	6		441.2.p.b.80.4		8
105.59	even	6		441.2.p.b.215.1		8
105.62	odd	4		1575.2.g.d.1574.7		8
105.74	odd	6		441.2.p.b.215.1		8
105.83	odd	4		1575.2.g.d.1574.2		8
105.89	even	6		441.2.p.b.80.4		8
105.104	even	2		63.2.c.a.62.4	yes	4
120.29	odd	2		4032.2.k.c.3905.4		4
120.59	even	2		4032.2.k.b.3905.1		4
140.139	even	2		1008.2.k.a.881.1		4
280.69	odd	2		4032.2.k.c.3905.3		4
280.139	even	2		4032.2.k.b.3905.2		4
315.34	odd	6		567.2.o.f.377.1		8
315.104	even	6		567.2.o.f.188.1		8
315.139	odd	6		567.2.o.f.188.4		8
315.209	even	6		567.2.o.f.377.4		8
420.419	odd	2		1008.2.k.a.881.2		4
840.419	odd	2		4032.2.k.b.3905.1		4
840.629	even	2		4032.2.k.c.3905.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.c.a.62.1	✓	4	5.4	even	2
63.2.c.a.62.1	✓	4	35.34	odd	2
63.2.c.a.62.4	yes	4	15.14	odd	2
63.2.c.a.62.4	yes	4	105.104	even	2
441.2.p.b.80.1		8	35.9	even	6
441.2.p.b.80.1		8	35.19	odd	6
441.2.p.b.80.4		8	105.44	odd	6
441.2.p.b.80.4		8	105.89	even	6
441.2.p.b.215.1		8	105.59	even	6
441.2.p.b.215.1		8	105.74	odd	6
441.2.p.b.215.4		8	35.4	even	6
441.2.p.b.215.4		8	35.24	odd	6
567.2.o.f.188.1		8	45.14	odd	6
567.2.o.f.188.1		8	315.104	even	6
567.2.o.f.188.4		8	45.4	even	6
567.2.o.f.188.4		8	315.139	odd	6
567.2.o.f.377.1		8	45.34	even	6
567.2.o.f.377.1		8	315.34	odd	6
567.2.o.f.377.4		8	45.29	odd	6
567.2.o.f.377.4		8	315.209	even	6
1008.2.k.a.881.1		4	20.19	odd	2
1008.2.k.a.881.1		4	140.139	even	2
1008.2.k.a.881.2		4	60.59	even	2
1008.2.k.a.881.2		4	420.419	odd	2
1575.2.b.a.251.1		4	3.2	odd	2	inner
1575.2.b.a.251.1		4	21.20	even	2	inner
1575.2.b.a.251.4		4	1.1	even	1	trivial
1575.2.b.a.251.4		4	7.6	odd	2	CM
1575.2.g.d.1574.1		8	5.2	odd	4
1575.2.g.d.1574.1		8	35.27	even	4
1575.2.g.d.1574.2		8	15.8	even	4
1575.2.g.d.1574.2		8	105.83	odd	4
1575.2.g.d.1574.7		8	15.2	even	4
1575.2.g.d.1574.7		8	105.62	odd	4
1575.2.g.d.1574.8		8	5.3	odd	4
1575.2.g.d.1574.8		8	35.13	even	4
4032.2.k.b.3905.1		4	120.59	even	2
4032.2.k.b.3905.1		4	840.419	odd	2
4032.2.k.b.3905.2		4	40.19	odd	2
4032.2.k.b.3905.2		4	280.139	even	2
4032.2.k.c.3905.3		4	40.29	even	2
4032.2.k.c.3905.3		4	280.69	odd	2
4032.2.k.c.3905.4		4	120.29	odd	2
4032.2.k.c.3905.4		4	840.629	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.57794i q^{2} -4.64575 q^{4} -2.64575 q^{7} -6.82058i q^{8} +O(q^{10})\)	\(q+2.57794i q^{2} -4.64575 q^{4} -2.64575 q^{7} -6.82058i q^{8} +0.913230i q^{11} -6.82058i q^{14} +8.29150 q^{16} -2.35425 q^{22} -9.39851i q^{23} +12.2915 q^{28} -6.06910i q^{29} +7.73381i q^{32} +10.5830 q^{37} -5.29150 q^{43} -4.24264i q^{44} +24.2288 q^{46} +7.00000 q^{49} +14.5544i q^{53} +18.0455i q^{56} +15.6458 q^{58} -3.35425 q^{64} +4.00000 q^{67} -7.57205i q^{71} +27.2823i q^{74} -2.41618i q^{77} +8.00000 q^{79} -13.6412i q^{86} +6.22876 q^{88} +43.6631i q^{92} +18.0455i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 12 q^{16} - 20 q^{22} + 28 q^{28} + 44 q^{46} + 28 q^{49} + 52 q^{58} - 24 q^{64} + 16 q^{67} + 32 q^{79} - 28 q^{88}+O(q^{100}) \) 4 * q - 8 * q^4 + 12 * q^16 - 20 * q^22 + 28 * q^28 + 44 * q^46 + 28 * q^49 + 52 * q^58 - 24 * q^64 + 16 * q^67 + 32 * q^79 - 28 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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