LMFDB - Embedded newform 441.2.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [441,2,Mod(1,441)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("441.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	441.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,10,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.52140272914$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64575 q^{2} +5.00000 q^{4} -7.93725 q^{8} +5.29150 q^{11} +11.0000 q^{16} -14.0000 q^{22} -5.29150 q^{23} -5.00000 q^{25} +10.5830 q^{29} -13.2288 q^{32} +6.00000 q^{37} +12.0000 q^{43} +26.4575 q^{44} +14.0000 q^{46} +13.2288 q^{50} +10.5830 q^{53} -28.0000 q^{58} +13.0000 q^{64} +4.00000 q^{67} +5.29150 q^{71} -15.8745 q^{74} +8.00000 q^{79} -31.7490 q^{86} -42.0000 q^{88} -26.4575 q^{92} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{4} + 22 q^{16} - 28 q^{22} - 10 q^{25} + 12 q^{37} + 24 q^{43} + 28 q^{46} - 56 q^{58} + 26 q^{64} + 8 q^{67} + 16 q^{79} - 84 q^{88}+O(q^{100}) $ 2 * q + 10 * q^4 + 22 * q^16 - 28 * q^22 - 10 * q^25 + 12 * q^37 + 24 * q^43 + 28 * q^46 - 56 * q^58 + 26 * q^64 + 8 * q^67 + 16 * q^79 - 84 * q^88

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 $

$2$	−2.64575	−1.87083	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	−2.64575	−1.87083	−0.935414	+	0.353553i	$0.884973\pi$
$3$	0	0
$3$	0	0
$4$	5.00000	2.50000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−7.93725	−2.80624
$9$	0	0
$10$	0	0
$11$	5.29150	1.59545	0.797724	−	0.603023i	$-0.206037\pi$
$11$	5.29150	1.59545	0.797724	+	0.603023i	$0.206037\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	11.0000	2.75000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	−14.0000	−2.98481
$23$	−5.29150	−1.10335	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	−5.29150	−1.10335	−0.551677	+	0.834058i	$0.686012\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	10.5830	1.96521	0.982607	−	0.185695i	$-0.0594537\pi$
$29$	10.5830	1.96521	0.982607	+	0.185695i	$0.0594537\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−13.2288	−2.33854
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	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
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	26.4575	3.98862
$45$	0	0
$46$	14.0000	2.06419
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	13.2288	1.87083
$51$	0	0
$52$	0	0
$53$	10.5830	1.45369	0.726844	−	0.686803i	$-0.240986\pi$
$53$	10.5830	1.45369	0.726844	+	0.686803i	$0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−28.0000	−3.67658
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	13.0000	1.62500
$65$	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
$71$	5.29150	0.627986	0.313993	−	0.949425i	$-0.398333\pi$
$71$	5.29150	0.627986	0.313993	+	0.949425i	$0.398333\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−15.8745	−1.84537
$75$	0	0
$76$	0	0
$77$	0	0
$78$	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
$86$	−31.7490	−3.42358
$87$	0	0
$88$	−42.0000	−4.47722
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	−26.4575	−2.75839
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	−25.0000	−2.50000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	−28.0000	−2.71960
$107$	5.29150	0.511549	0.255774	−	0.966736i	$-0.417670\pi$
$107$	5.29150	0.511549	0.255774	+	0.966736i	$0.417670\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−21.1660	−1.99113	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	−21.1660	−1.99113	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	0	0
$115$	0	0
$116$	52.9150	4.91304
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	17.0000	1.54545
$122$	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	$0.248747\pi$
$128$	−7.93725	−0.701561
$129$	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
$134$	−10.5830	−0.914232
$135$	0	0
$136$	0	0
$137$	−21.1660	−1.80833	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	−21.1660	−1.80833	−0.904167	+	0.427179i	$0.859507\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	−14.0000	−1.17485
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	30.0000	2.46598
$149$	−10.5830	−0.866994	−0.433497	−	0.901155i	$-0.642720\pi$
$149$	−10.5830	−0.866994	−0.433497	+	0.901155i	$0.642720\pi$
$150$	0	0
$151$	24.0000	1.95309	0.976546	−	0.215308i	$-0.0690756\pi$
$151$	24.0000	1.95309	0.976546	+	0.215308i	$0.0690756\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	−21.1660	−1.68388
$159$	0	0
$160$	0	0
$161$	0	0
$162$	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
$172$	60.0000	4.57496
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	58.2065	4.38748
$177$	0	0
$178$	0	0
$179$	−26.4575	−1.97753	−0.988764	−	0.149487i	$-0.952238\pi$
$179$	−26.4575	−1.97753	−0.988764	+	0.149487i	$0.952238\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	42.0000	3.09628
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	26.4575	1.91440	0.957199	−	0.289430i	$-0.0934657\pi$
$191$	26.4575	1.91440	0.957199	+	0.289430i	$0.0934657\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.5830	−0.754008	−0.377004	−	0.926212i	$-0.623046\pi$
$197$	−10.5830	−0.754008	−0.377004	+	0.926212i	$0.623046\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	39.6863	2.80624
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	52.9150	3.63422
$213$	0	0
$214$	−14.0000	−0.957020
$215$	0	0
$216$	0	0
$217$	0	0
$218$	47.6235	3.22547
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	56.0000	3.72506
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	−84.0000	−5.51487
$233$	21.1660	1.38663	0.693316	−	0.720634i	$-0.256149\pi$
$233$	21.1660	1.38663	0.693316	+	0.720634i	$0.256149\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.4575	−1.71139	−0.855697	−	0.517477i	$-0.826871\pi$
$239$	−26.4575	−1.71139	−0.855697	+	0.517477i	$0.826871\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−44.9778	−2.89128
$243$	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
$253$	−28.0000	−1.76034
$254$	−42.3320	−2.65615
$255$	0	0
$256$	−5.00000	−0.312500
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.29150	−0.326288	−0.163144	−	0.986602i	$-0.552164\pi$
$263$	−5.29150	−0.326288	−0.163144	+	0.986602i	$0.552164\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	20.0000	1.22169
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	56.0000	3.38308
$275$	−26.4575	−1.59545
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.1660	1.26266	0.631329	−	0.775515i	$-0.282510\pi$
$281$	21.1660	1.26266	0.631329	+	0.775515i	$0.282510\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	26.4575	1.56996
$285$	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
$296$	−47.6235	−2.76806
$297$	0	0
$298$	28.0000	1.62200
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−63.4980	−3.65390
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	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		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	40.0000	2.25018
$317$	10.5830	0.594401	0.297200	−	0.954815i	$-0.403947\pi$
$317$	10.5830	0.594401	0.297200	+	0.954815i	$0.403947\pi$
$318$	0	0
$319$	56.0000	3.13540
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	52.9150	2.93069
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−36.0000	−1.97874	−0.989369	−	0.145424i	$-0.953545\pi$
$331$	−36.0000	−1.97874	−0.989369	+	0.145424i	$0.953545\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	34.3948	1.87083
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−95.2470	−5.13538
$345$	0	0
$346$	0	0
$347$	37.0405	1.98844	0.994220	−	0.107366i	$-0.0342415\pi$
$347$	37.0405	1.98844	0.994220	+	0.107366i	$0.0342415\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	−70.0000	−3.73101
$353$	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
$358$	70.0000	3.69961
$359$	−37.0405	−1.95492	−0.977462	−	0.211112i	$-0.932292\pi$
$359$	−37.0405	−1.95492	−0.977462	+	0.211112i	$0.932292\pi$
$360$	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		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−58.2065	−3.03423
$369$	0	0
$370$	0	0
$371$	0	0
$372$	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
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	0	0
$382$	−70.0000	−3.58151
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	47.6235	2.42397
$387$	0	0
$388$	0	0
$389$	−10.5830	−0.536580	−0.268290	−	0.963338i	$-0.586458\pi$
$389$	−10.5830	−0.536580	−0.268290	+	0.963338i	$0.586458\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	28.0000	1.41062
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	−55.0000	−2.75000
$401$	−21.1660	−1.05698	−0.528490	−	0.848939i	$-0.677242\pi$
$401$	−21.1660	−1.05698	−0.528490	+	0.848939i	$0.677242\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	31.7490	1.57374
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	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	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−31.7490	−1.54552
$423$	0	0
$424$	−84.0000	−4.07940
$425$	0	0
$426$	0	0
$427$	0	0
$428$	26.4575	1.27887
$429$	0	0
$430$	0	0
$431$	26.4575	1.27441	0.637207	−	0.770693i	$-0.280090\pi$
$431$	26.4575	1.27441	0.637207	+	0.770693i	$0.280090\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−90.0000	−4.31022
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	37.0405	1.75985	0.879924	−	0.475114i	$-0.157593\pi$
$443$	37.0405	1.75985	0.879924	+	0.475114i	$0.157593\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	42.3320	1.99777	0.998886	−	0.0471929i	$-0.0150276\pi$
$449$	42.3320	1.99777	0.998886	+	0.0471929i	$0.0150276\pi$
$450$	0	0
$451$	0	0
$452$	−105.830	−4.97783
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	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	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	116.413	5.40434
$465$	0	0
$466$	−56.0000	−2.59415
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	63.4980	2.91964
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	70.0000	3.20173
$479$	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
$484$	85.0000	3.86364
$485$	0	0
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.29150	−0.238802	−0.119401	−	0.992846i	$-0.538097\pi$
$491$	−5.29150	−0.238802	−0.119401	+	0.992846i	$0.538097\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	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
$506$	74.0810	3.29330
$507$	0	0
$508$	80.0000	3.54943
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	29.1033	1.28619
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	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		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	14.0000	0.610429
$527$	0	0
$528$	0	0
$529$	5.00000	0.217391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−31.7490	−1.37135
$537$	0	0
$538$	0	0
$539$	0	0
$540$	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	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	−105.830	−4.52084
$549$	0	0
$550$	70.0000	2.98481
$551$	0	0
$552$	0	0
$553$	0	0
$554$	26.4575	1.12407
$555$	0	0
$556$	0	0
$557$	−10.5830	−0.448416	−0.224208	−	0.974541i	$-0.571980\pi$
$557$	−10.5830	−0.448416	−0.224208	+	0.974541i	$0.571980\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−56.0000	−2.36222
$563$	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
$568$	−42.0000	−1.76228
$569$	−42.3320	−1.77465	−0.887325	−	0.461144i	$-0.847439\pi$
$569$	−42.3320	−1.77465	−0.887325	+	0.461144i	$0.847439\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	26.4575	1.10335
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	44.9778	1.87083
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	56.0000	2.31928
$584$	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
$592$	66.0000	2.71258
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	−52.9150	−2.16748
$597$	0	0
$598$	0	0
$599$	−37.0405	−1.51343	−0.756717	−	0.653742i	$-0.773198\pi$
$599$	−37.0405	−1.51343	−0.756717	+	0.653742i	$0.773198\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	0	0
$604$	120.000	4.88273
$605$	0	0
$606$	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
$611$	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
$616$	0	0
$617$	−42.3320	−1.70422	−0.852111	−	0.523360i	$-0.824678\pi$
$617$	−42.3320	−1.70422	−0.852111	+	0.523360i	$0.824678\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−63.4980	−2.52582
$633$	0	0
$634$	−28.0000	−1.11202
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−148.162	−5.86579
$639$	0	0
$640$	0	0
$641$	21.1660	0.836007	0.418004	−	0.908445i	$-0.362730\pi$
$641$	21.1660	0.836007	0.418004	+	0.908445i	$0.362730\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	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
$652$	−100.000	−3.91630
$653$	10.5830	0.414145	0.207072	−	0.978326i	$-0.433606\pi$
$653$	10.5830	0.414145	0.207072	+	0.978326i	$0.433606\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26.4575	1.03064	0.515319	−	0.856998i	$-0.327673\pi$
$659$	26.4575	1.03064	0.515319	+	0.856998i	$0.327673\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	95.2470	3.70188
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−56.0000	−2.16833
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	79.3725	3.05732
$675$	0	0
$676$	−65.0000	−2.50000
$677$	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
$683$	−5.29150	−0.202474	−0.101237	−	0.994862i	$-0.532280\pi$
$683$	−5.29150	−0.202474	−0.101237	+	0.994862i	$0.532280\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	132.000	5.03245
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	−98.0000	−3.72003
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−52.9150	−1.99857	−0.999286	−	0.0377695i	$-0.987975\pi$
$701$	−52.9150	−1.99857	−0.999286	+	0.0377695i	$0.987975\pi$
$702$	0	0
$703$	0	0
$704$	68.7895	2.59260
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−132.288	−4.94382
$717$	0	0
$718$	98.0000	3.65733
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	50.2693	1.87083
$723$	0	0
$724$	0	0
$725$	−52.9150	−1.96521
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	70.0000	2.58023
$737$	21.1660	0.779660
$738$	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
$742$	0	0
$743$	−37.0405	−1.35888	−0.679442	−	0.733729i	$-0.737778\pi$
$743$	−37.0405	−1.35888	−0.679442	+	0.733729i	$0.737778\pi$
$744$	0	0
$745$	0	0
$746$	−58.2065	−2.13109
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	54.0000	1.96266	0.981332	−	0.192323i	$-0.0616021\pi$
$757$	54.0000	1.96266	0.981332	+	0.192323i	$0.0616021\pi$
$758$	−31.7490	−1.15318
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	132.288	4.78600
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	−90.0000	−3.23917
$773$	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
$778$	28.0000	1.00385
$779$	0	0
$780$	0	0
$781$	28.0000	1.00192
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	−52.9150	−1.88502
$789$	0	0
$790$	0	0
$791$	0	0
$792$	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
$800$	66.1438	2.33854
$801$	0	0
$802$	56.0000	1.97743
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−42.3320	−1.48831	−0.744157	−	0.668004i	$-0.767149\pi$
$809$	−42.3320	−1.48831	−0.744157	+	0.668004i	$0.767149\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	−84.0000	−2.94420
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	52.9150	1.84675	0.923374	−	0.383903i	$-0.125420\pi$
$821$	52.9150	1.84675	0.923374	+	0.383903i	$0.125420\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37.0405	−1.28803	−0.644013	−	0.765015i	$-0.722732\pi$
$827$	−37.0405	−1.28803	−0.644013	+	0.765015i	$0.722732\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	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
$841$	83.0000	2.86207
$842$	68.7895	2.37064
$843$	0	0
$844$	60.0000	2.06529
$845$	0	0
$846$	0	0
$847$	0	0
$848$	116.413	3.99764
$849$	0	0
$850$	0	0
$851$	−31.7490	−1.08834
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−42.0000	−1.43553
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	−70.0000	−2.38421
$863$	58.2065	1.98137	0.990687	−	0.136162i	$-0.0434766\pi$
$863$	58.2065	1.98137	0.990687	+	0.136162i	$0.0434766\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	42.3320	1.43602
$870$	0	0
$871$	0	0
$872$	142.871	4.83821
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	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
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	0	0
$885$	0	0
$886$	−98.0000	−3.29237
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−112.000	−3.73749
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	168.000	5.58760
$905$	0	0
$906$	0	0
$907$	−60.0000	−1.99227	−0.996134	−	0.0878507i	$-0.972000\pi$
$907$	−60.0000	−1.99227	−0.996134	+	0.0878507i	$0.972000\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−58.2065	−1.92847	−0.964234	−	0.265052i	$-0.914611\pi$
$911$	−58.2065	−1.92847	−0.964234	+	0.265052i	$0.914611\pi$
$912$	0	0
$913$	0	0
$914$	−15.8745	−0.525082
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−30.0000	−0.986394
$926$	105.830	3.47779
$927$	0	0
$928$	−140.000	−4.59573
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	105.830	3.46658
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	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
$946$	−168.000	−5.46215
$947$	−58.2065	−1.89146	−0.945729	−	0.324956i	$-0.894650\pi$
$947$	−58.2065	−1.89146	−0.945729	+	0.324956i	$0.894650\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.1660	0.685634	0.342817	−	0.939402i	$-0.388619\pi$
$953$	21.1660	0.685634	0.342817	+	0.939402i	$0.388619\pi$
$954$	0	0
$955$	0	0
$956$	−132.288	−4.27849
$957$	0	0
$958$	0	0
$959$	0	0
$960$	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	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−134.933	−4.33692
$969$	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
$974$	−63.4980	−2.03461
$975$	0	0
$976$	0	0
$977$	−42.3320	−1.35432	−0.677161	−	0.735835i	$-0.736790\pi$
$977$	−42.3320	−1.35432	−0.677161	+	0.735835i	$0.736790\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	14.0000	0.446758
$983$	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
$989$	−63.4980	−2.01912
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	95.2470	3.01499
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.a.h.1.1	✓	2
3.2	odd	2	inner	441.2.a.h.1.2	yes	2
4.3	odd	2		7056.2.a.co.1.1		2
7.2	even	3		441.2.e.h.361.2		4
7.3	odd	6		441.2.e.h.226.2		4
7.4	even	3		441.2.e.h.226.2		4
7.5	odd	6		441.2.e.h.361.2		4
7.6	odd	2	CM	441.2.a.h.1.1	✓	2
12.11	even	2		7056.2.a.co.1.2		2
21.2	odd	6		441.2.e.h.361.1		4
21.5	even	6		441.2.e.h.361.1		4
21.11	odd	6		441.2.e.h.226.1		4
21.17	even	6		441.2.e.h.226.1		4
21.20	even	2	inner	441.2.a.h.1.2	yes	2
28.27	even	2		7056.2.a.co.1.1		2
84.83	odd	2		7056.2.a.co.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.a.h.1.1	✓	2	1.1	even	1	trivial
441.2.a.h.1.1	✓	2	7.6	odd	2	CM
441.2.a.h.1.2	yes	2	3.2	odd	2	inner
441.2.a.h.1.2	yes	2	21.20	even	2	inner
441.2.e.h.226.1		4	21.11	odd	6
441.2.e.h.226.1		4	21.17	even	6
441.2.e.h.226.2		4	7.3	odd	6
441.2.e.h.226.2		4	7.4	even	3
441.2.e.h.361.1		4	21.2	odd	6
441.2.e.h.361.1		4	21.5	even	6
441.2.e.h.361.2		4	7.2	even	3
441.2.e.h.361.2		4	7.5	odd	6
7056.2.a.co.1.1		2	4.3	odd	2
7056.2.a.co.1.1		2	28.27	even	2
7056.2.a.co.1.2		2	12.11	even	2
7056.2.a.co.1.2		2	84.83	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64575 q^{2} +5.00000 q^{4} -7.93725 q^{8} +5.29150 q^{11} +11.0000 q^{16} -14.0000 q^{22} -5.29150 q^{23} -5.00000 q^{25} +10.5830 q^{29} -13.2288 q^{32} +6.00000 q^{37} +12.0000 q^{43} +26.4575 q^{44} +14.0000 q^{46} +13.2288 q^{50} +10.5830 q^{53} -28.0000 q^{58} +13.0000 q^{64} +4.00000 q^{67} +5.29150 q^{71} -15.8745 q^{74} +8.00000 q^{79} -31.7490 q^{86} -42.0000 q^{88} -26.4575 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{4} + 22 q^{16} - 28 q^{22} - 10 q^{25} + 12 q^{37} + 24 q^{43} + 28 q^{46} - 56 q^{58} + 26 q^{64} + 8 q^{67} + 16 q^{79} - 84 q^{88}+O(q^{100}) \) 2 * q + 10 * q^4 + 22 * q^16 - 28 * q^22 - 10 * q^25 + 12 * q^37 + 24 * q^43 + 28 * q^46 - 56 * q^58 + 26 * q^64 + 8 * q^67 + 16 * q^79 - 84 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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