LMFDB - Embedded newform 225.4.b.g.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.b (of order $2$, degree $1$, not 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:	$13.2754297513$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.4.b.g.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{4} -20.0000i q^{7} +O(q^{10})$	$q+8.00000 q^{4} -20.0000i q^{7} -70.0000i q^{13} +64.0000 q^{16} -56.0000 q^{19} -160.000i q^{28} +308.000 q^{31} -110.000i q^{37} -520.000i q^{43} -57.0000 q^{49} -560.000i q^{52} +182.000 q^{61} +512.000 q^{64} +880.000i q^{67} +1190.00i q^{73} -448.000 q^{76} -884.000 q^{79} -1400.00 q^{91} +1330.00i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{4}+O(q^{10}) $ 2 * q + 16 * q^4	$ 2 q + 16 q^{4} + 128 q^{16} - 112 q^{19} + 616 q^{31} - 114 q^{49} + 364 q^{61} + 1024 q^{64} - 896 q^{76} - 1768 q^{79} - 2800 q^{91}+O(q^{100}) $ 2 * q + 16 * q^4 + 128 * q^16 - 112 * q^19 + 616 * q^31 - 114 * q^49 + 364 * q^61 + 1024 * q^64 - 896 * q^76 - 1768 * q^79 - 2800 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	0	0	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	0	0
$3$	0	0
$4$	8.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 20.0000i	− 1.07990i	−0.841698	−	0.539949i	$-0.818443\pi$
$7$	− 20.0000i	− 1.07990i	0.841698	−	0.539949i	$-0.181557\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	− 70.0000i	− 1.49342i	−0.665148	−	0.746712i	$-0.731631\pi$
$13$	− 70.0000i	− 1.49342i	0.665148	−	0.746712i	$-0.268369\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−56.0000	−0.676173	−0.338086	−	0.941115i	$-0.609780\pi$
$19$	−56.0000	−0.676173	−0.338086	+	0.941115i	$0.609780\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	− 160.000i	− 1.07990i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	308.000	1.78447	0.892233	−	0.451576i	$-0.149138\pi$
$31$	308.000	1.78447	0.892233	+	0.451576i	$0.149138\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 110.000i	− 0.488754i	−0.969680	−	0.244377i	$-0.921417\pi$
$37$	− 110.000i	− 0.488754i	0.969680	−	0.244377i	$-0.0785834\pi$
$38$	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
$43$	− 520.000i	− 1.84417i	−0.386989	−	0.922084i	$-0.626485\pi$
$43$	− 520.000i	− 1.84417i	0.386989	−	0.922084i	$-0.373515\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−57.0000	−0.166181
$50$	0	0
$51$	0	0
$52$	− 560.000i	− 1.49342i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	182.000	0.382012	0.191006	−	0.981589i	$-0.438825\pi$
$61$	182.000	0.382012	0.191006	+	0.981589i	$0.438825\pi$
$62$	0	0
$63$	0	0
$64$	512.000	1.00000
$65$	0	0
$66$	0	0
$67$	880.000i	1.60461i	0.596912	+	0.802307i	$0.296394\pi$
$67$	880.000i	1.60461i	−0.596912	+	0.802307i	$0.703606\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	1190.00i	1.90793i	0.299916	+	0.953966i	$0.403041\pi$
$73$	1190.00i	1.90793i	−0.299916	+	0.953966i	$0.596959\pi$
$74$	0	0
$75$	0	0
$76$	−448.000	−0.676173
$77$	0	0
$78$	0	0
$79$	−884.000	−1.25896	−0.629480	−	0.777017i	$-0.716732\pi$
$79$	−884.000	−1.25896	−0.629480	+	0.777017i	$0.716732\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−1400.00	−1.61275
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1330.00i	1.39218i	0.717957	+	0.696088i	$0.245078\pi$
$97$	1330.00i	1.39218i	−0.717957	+	0.696088i	$0.754922\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	1820.00i	1.74107i	0.492109	+	0.870534i	$0.336226\pi$
$103$	1820.00i	1.74107i	−0.492109	+	0.870534i	$0.663774\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	646.000	0.567666	0.283833	−	0.958874i	$-0.408394\pi$
$109$	646.000	0.567666	0.283833	+	0.958874i	$0.408394\pi$
$110$	0	0
$111$	0	0
$112$	− 1280.00i	− 1.07990i
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1331.00	−1.00000
$122$	0	0
$123$	0	0
$124$	2464.00	1.78447
$125$	0	0
$126$	0	0
$127$	− 380.000i	− 0.265508i	−0.991149	−	0.132754i	$-0.957618\pi$
$127$	− 380.000i	− 0.265508i	0.991149	−	0.132754i	$-0.0423821\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	1120.00i	0.730198i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−2576.00	−1.57190	−0.785948	−	0.618293i	$-0.787825\pi$
$139$	−2576.00	−1.57190	−0.785948	+	0.618293i	$0.787825\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	− 880.000i	− 0.488754i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	1748.00	0.942054	0.471027	−	0.882119i	$-0.343883\pi$
$151$	1748.00	0.942054	0.471027	+	0.882119i	$0.343883\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3850.00i	1.95709i	0.206028	+	0.978546i	$0.433946\pi$
$157$	3850.00i	1.95709i	−0.206028	+	0.978546i	$0.566054\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 3400.00i	− 1.63379i	−0.576783	−	0.816897i	$-0.695692\pi$
$163$	− 3400.00i	− 1.63379i	0.576783	−	0.816897i	$-0.304308\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−2703.00	−1.23031
$170$	0	0
$171$	0	0
$172$	− 4160.00i	− 1.84417i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	3458.00	1.42006	0.710031	−	0.704171i	$-0.248681\pi$
$181$	3458.00	1.42006	0.710031	+	0.704171i	$0.248681\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	− 1150.00i	− 0.428906i	−0.976734	−	0.214453i	$-0.931203\pi$
$193$	− 1150.00i	− 0.428906i	0.976734	−	0.214453i	$-0.0687968\pi$
$194$	0	0
$195$	0	0
$196$	−456.000	−0.166181
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	5236.00	1.86518	0.932588	−	0.360942i	$-0.117545\pi$
$199$	5236.00	1.86518	0.932588	+	0.360942i	$0.117545\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	− 4480.00i	− 1.49342i
$209$	0	0
$210$	0	0
$211$	6032.00	1.96806	0.984028	−	0.178011i	$-0.0569664\pi$
$211$	6032.00	1.96806	0.984028	+	0.178011i	$0.0569664\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 6160.00i	− 1.92704i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 3220.00i	− 0.966938i	−0.875362	−	0.483469i	$-0.839377\pi$
$223$	− 3220.00i	− 0.966938i	0.875362	−	0.483469i	$-0.160623\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−4466.00	−1.28874	−0.644370	−	0.764714i	$-0.722880\pi$
$229$	−4466.00	−1.28874	−0.644370	+	0.764714i	$0.722880\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−7378.00	−1.97203	−0.986014	−	0.166662i	$-0.946701\pi$
$241$	−7378.00	−1.97203	−0.986014	+	0.166662i	$0.946701\pi$
$242$	0	0
$243$	0	0
$244$	1456.00	0.382012
$245$	0	0
$246$	0	0
$247$	3920.00i	1.00981i
$248$	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
$253$	0	0
$254$	0	0
$255$	0	0
$256$	4096.00	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	−2200.00	−0.527804
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	7040.00i	1.60461i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	812.000	0.182013	0.0910064	−	0.995850i	$-0.470992\pi$
$271$	812.000	0.182013	0.0910064	+	0.995850i	$0.470992\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4030.00i	0.874149i	0.899425	+	0.437074i	$0.143985\pi$
$277$	4030.00i	0.874149i	−0.899425	+	0.437074i	$0.856015\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	5600.00i	1.17627i	0.808761	+	0.588137i	$0.200138\pi$
$283$	5600.00i	1.17627i	−0.808761	+	0.588137i	$0.799862\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4913.00	1.00000
$290$	0	0
$291$	0	0
$292$	9520.00i	1.90793i
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−10400.0	−1.99152
$302$	0	0
$303$	0	0
$304$	−3584.00	−0.676173
$305$	0	0
$306$	0	0
$307$	− 10640.0i	− 1.97804i	−0.147797	−	0.989018i	$-0.547218\pi$
$307$	− 10640.0i	− 1.97804i	0.147797	−	0.989018i	$-0.452782\pi$
$308$	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
$313$	10010.0i	1.80766i	0.427888	+	0.903832i	$0.359258\pi$
$313$	10010.0i	1.80766i	−0.427888	+	0.903832i	$0.640742\pi$
$314$	0	0
$315$	0	0
$316$	−7072.00	−1.25896
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	992.000	0.164729	0.0823644	−	0.996602i	$-0.473753\pi$
$331$	992.000	0.164729	0.0823644	+	0.996602i	$0.473753\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4930.00i	0.796897i	0.917191	+	0.398448i	$0.130451\pi$
$337$	4930.00i	0.796897i	−0.917191	+	0.398448i	$0.869549\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 5720.00i	− 0.900440i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	11914.0	1.82734	0.913670	−	0.406456i	$-0.133236\pi$
$349$	11914.0	1.82734	0.913670	+	0.406456i	$0.133236\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3723.00	−0.542790
$362$	0	0
$363$	0	0
$364$	−11200.0	−1.61275
$365$	0	0
$366$	0	0
$367$	− 4340.00i	− 0.617292i	−0.951177	−	0.308646i	$-0.900124\pi$
$367$	− 4340.00i	− 0.617292i	0.951177	−	0.308646i	$-0.0998758\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12350.0i	1.71437i	0.515011	+	0.857183i	$0.327788\pi$
$373$	12350.0i	1.71437i	−0.515011	+	0.857183i	$0.672212\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	8584.00	1.16340	0.581702	−	0.813402i	$-0.302387\pi$
$379$	8584.00	1.16340	0.581702	+	0.813402i	$0.302387\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	10640.0i	1.39218i
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1190.00i	− 0.150439i	−0.997167	−	0.0752196i	$-0.976034\pi$
$397$	− 1190.00i	− 0.150439i	0.997167	−	0.0752196i	$-0.0239658\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	− 21560.0i	− 2.66496i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−8246.00	−0.996916	−0.498458	−	0.866914i	$-0.666100\pi$
$409$	−8246.00	−0.996916	−0.498458	+	0.866914i	$0.666100\pi$
$410$	0	0
$411$	0	0
$412$	14560.0i	1.74107i
$413$	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
$421$	17138.0	1.98398	0.991989	−	0.126322i	$-0.0403172\pi$
$421$	17138.0	1.98398	0.991989	+	0.126322i	$0.0403172\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3640.00i	− 0.412534i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 2590.00i	− 0.287454i	−0.989617	−	0.143727i	$-0.954091\pi$
$433$	− 2590.00i	− 0.287454i	0.989617	−	0.143727i	$-0.0459087\pi$
$434$	0	0
$435$	0	0
$436$	5168.00	0.567666
$437$	0	0
$438$	0	0
$439$	−14924.0	−1.62251	−0.811257	−	0.584690i	$-0.801216\pi$
$439$	−14924.0	−1.62251	−0.811257	+	0.584690i	$0.801216\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	− 10240.0i	− 1.07990i
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 12710.0i	− 1.30098i	−0.759514	−	0.650491i	$-0.774563\pi$
$457$	− 12710.0i	− 1.30098i	0.759514	−	0.650491i	$-0.225437\pi$
$458$	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
$463$	− 19780.0i	− 1.98543i	−0.120482	−	0.992716i	$-0.538444\pi$
$463$	− 19780.0i	− 1.98543i	0.120482	−	0.992716i	$-0.461556\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	17600.0	1.73282
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−7700.00	−0.729916
$482$	0	0
$483$	0	0
$484$	−10648.0	−1.00000
$485$	0	0
$486$	0	0
$487$	− 20900.0i	− 1.94470i	−0.233526	−	0.972351i	$-0.575026\pi$
$487$	− 20900.0i	− 1.94470i	0.233526	−	0.972351i	$-0.424974\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	19712.0	1.78447
$497$	0	0
$498$	0	0
$499$	15136.0	1.35788	0.678938	−	0.734195i	$-0.262440\pi$
$499$	15136.0	1.35788	0.678938	+	0.734195i	$0.262440\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 3040.00i	− 0.265508i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	23800.0	2.06037
$512$	0	0
$513$	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
$523$	− 12040.0i	− 1.00664i	−0.864100	−	0.503320i	$-0.832112\pi$
$523$	− 12040.0i	− 1.00664i	0.864100	−	0.503320i	$-0.167888\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	12167.0	1.00000
$530$	0	0
$531$	0	0
$532$	8960.00i	0.730198i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22678.0	−1.80222	−0.901112	−	0.433586i	$-0.857248\pi$
$541$	−22678.0	−1.80222	−0.901112	+	0.433586i	$0.857248\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1640.00i	− 0.128193i	−0.997944	−	0.0640963i	$-0.979584\pi$
$547$	− 1640.00i	− 0.128193i	0.997944	−	0.0640963i	$-0.0204165\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	17680.0i	1.35955i
$554$	0	0
$555$	0	0
$556$	−20608.0	−1.57190
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	−36400.0	−2.75413
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	23312.0	1.70854	0.854270	−	0.519829i	$-0.174004\pi$
$571$	23312.0	1.70854	0.854270	+	0.519829i	$0.174004\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17710.0i	1.27778i	0.769300	+	0.638888i	$0.220605\pi$
$577$	17710.0i	1.27778i	−0.769300	+	0.638888i	$0.779395\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	−17248.0	−1.20661
$590$	0	0
$591$	0	0
$592$	− 7040.00i	− 0.488754i
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−29302.0	−1.98877	−0.994387	−	0.105801i	$-0.966259\pi$
$601$	−29302.0	−1.98877	−0.994387	+	0.105801i	$0.966259\pi$
$602$	0	0
$603$	0	0
$604$	13984.0	0.942054
$605$	0	0
$606$	0	0
$607$	28420.0i	1.90038i	0.311667	+	0.950191i	$0.399113\pi$
$607$	28420.0i	1.90038i	−0.311667	+	0.950191i	$0.600887\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	17390.0i	1.14580i	0.819625	+	0.572900i	$0.194182\pi$
$613$	17390.0i	1.14580i	−0.819625	+	0.572900i	$0.805818\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	26656.0	1.73085	0.865424	−	0.501040i	$-0.167049\pi$
$619$	26656.0	1.73085	0.865424	+	0.501040i	$0.167049\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	30800.0i	1.95709i
$629$	0	0
$630$	0	0
$631$	1892.00	0.119365	0.0596825	−	0.998217i	$-0.480991\pi$
$631$	1892.00	0.119365	0.0596825	+	0.998217i	$0.480991\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3990.00i	0.248178i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	13160.0i	0.807122i	0.914953	+	0.403561i	$0.132228\pi$
$643$	13160.0i	0.807122i	−0.914953	+	0.403561i	$0.867772\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 27200.0i	− 1.63379i
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−20482.0	−1.20523	−0.602615	−	0.798032i	$-0.705875\pi$
$661$	−20482.0	−1.20523	−0.602615	+	0.798032i	$0.705875\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	24050.0i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$673$	24050.0i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$674$	0	0
$675$	0	0
$676$	−21624.0	−1.23031
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	26600.0	1.50341
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	− 33280.0i	− 1.84417i
$689$	0	0
$690$	0	0
$691$	−16072.0	−0.884816	−0.442408	−	0.896814i	$-0.645876\pi$
$691$	−16072.0	−0.884816	−0.442408	+	0.896814i	$0.645876\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	6160.00i	0.330482i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−36146.0	−1.91466	−0.957328	−	0.289003i	$-0.906676\pi$
$709$	−36146.0	−1.91466	−0.957328	+	0.289003i	$0.906676\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	36400.0	1.88018
$722$	0	0
$723$	0	0
$724$	27664.0	1.42006
$725$	0	0
$726$	0	0
$727$	10780.0i	0.549942i	0.961452	+	0.274971i	$0.0886683\pi$
$727$	10780.0i	0.549942i	−0.961452	+	0.274971i	$0.911332\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	15050.0i	0.758369i	0.925321	+	0.379184i	$0.123795\pi$
$733$	15050.0i	0.758369i	−0.925321	+	0.379184i	$0.876205\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−31376.0	−1.56182	−0.780910	−	0.624644i	$-0.785244\pi$
$739$	−31376.0	−1.56182	−0.780910	+	0.624644i	$0.785244\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−23452.0	−1.13951	−0.569757	−	0.821813i	$-0.692963\pi$
$751$	−23452.0	−1.13951	−0.569757	+	0.821813i	$0.692963\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41470.0i	1.99109i	0.0943039	+	0.995543i	$0.469937\pi$
$757$	41470.0i	1.99109i	−0.0943039	+	0.995543i	$0.530063\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	− 12920.0i	− 0.613022i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	4606.00	0.215990	0.107995	−	0.994151i	$-0.465557\pi$
$769$	4606.00	0.215990	0.107995	+	0.994151i	$0.465557\pi$
$770$	0	0
$771$	0	0
$772$	− 9200.00i	− 0.428906i
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−3648.00	−0.166181
$785$	0	0
$786$	0	0
$787$	− 43400.0i	− 1.96575i	−0.184281	−	0.982874i	$-0.558996\pi$
$787$	− 43400.0i	− 1.96575i	0.184281	−	0.982874i	$-0.441004\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 12740.0i	− 0.570505i
$794$	0	0
$795$	0	0
$796$	41888.0	1.86518
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	39368.0	1.70456	0.852280	−	0.523087i	$-0.175220\pi$
$811$	39368.0	1.70456	0.852280	+	0.523087i	$0.175220\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	29120.0i	1.24698i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	− 12220.0i	− 0.517573i	−0.965935	−	0.258786i	$-0.916677\pi$
$823$	− 12220.0i	− 0.517573i	0.965935	−	0.258786i	$-0.0833226\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−17066.0	−0.714990	−0.357495	−	0.933915i	$-0.616369\pi$
$829$	−17066.0	−0.714990	−0.357495	+	0.933915i	$0.616369\pi$
$830$	0	0
$831$	0	0
$832$	− 35840.0i	− 1.49342i
$833$	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
$841$	−24389.0	−1.00000
$842$	0	0
$843$	0	0
$844$	48256.0	1.96806
$845$	0	0
$846$	0	0
$847$	26620.0i	1.07990i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 46690.0i	− 1.87413i	−0.349151	−	0.937066i	$-0.613530\pi$
$853$	− 46690.0i	− 1.87413i	0.349151	−	0.937066i	$-0.386470\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−31304.0	−1.24340	−0.621699	−	0.783256i	$-0.713557\pi$
$859$	−31304.0	−1.24340	−0.621699	+	0.783256i	$0.713557\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	− 49280.0i	− 1.92704i
$869$	0	0
$870$	0	0
$871$	61600.0	2.39637
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 50150.0i	− 1.93095i	−0.260491	−	0.965476i	$-0.583885\pi$
$877$	− 50150.0i	− 1.93095i	0.260491	−	0.965476i	$-0.416115\pi$
$878$	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
$883$	− 20680.0i	− 0.788151i	−0.919078	−	0.394076i	$-0.871065\pi$
$883$	− 20680.0i	− 0.788151i	0.919078	−	0.394076i	$-0.128935\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−7600.00	−0.286722
$890$	0	0
$891$	0	0
$892$	− 25760.0i	− 0.966938i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 44840.0i	− 1.64155i	−0.571250	−	0.820776i	$-0.693541\pi$
$907$	− 44840.0i	− 1.64155i	0.571250	−	0.820776i	$-0.306459\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−35728.0	−1.28874
$917$	0	0
$918$	0	0
$919$	−2756.00	−0.0989250	−0.0494625	−	0.998776i	$-0.515751\pi$
$919$	−2756.00	−0.0989250	−0.0494625	+	0.998776i	$0.515751\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	3192.00	0.112367
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	55510.0i	1.93536i	0.252181	+	0.967680i	$0.418852\pi$
$937$	55510.0i	1.93536i	−0.252181	+	0.967680i	$0.581148\pi$
$938$	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
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	83300.0	2.84935
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	65073.0	2.18432
$962$	0	0
$963$	0	0
$964$	−59024.0	−1.97203
$965$	0	0
$966$	0	0
$967$	50020.0i	1.66343i	0.555204	+	0.831714i	$0.312640\pi$
$967$	50020.0i	1.66343i	−0.555204	+	0.831714i	$0.687360\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	51520.0i	1.69749i
$974$	0	0
$975$	0	0
$976$	11648.0	0.382012
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	31360.0i	1.00981i
$989$	0	0
$990$	0	0
$991$	−45628.0	−1.46258	−0.731292	−	0.682064i	$-0.761082\pi$
$991$	−45628.0	−1.46258	−0.731292	+	0.682064i	$0.761082\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 28910.0i	− 0.918344i	−0.888347	−	0.459172i	$-0.848146\pi$
$997$	− 28910.0i	− 0.918344i	0.888347	−	0.459172i	$-0.151854\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.b.g.199.1		2
3.2	odd	2	CM	225.4.b.g.199.1		2
5.2	odd	4		9.4.a.a.1.1	✓	1
5.3	odd	4		225.4.a.d.1.1		1
5.4	even	2	inner	225.4.b.g.199.2		2
15.2	even	4		9.4.a.a.1.1	✓	1
15.8	even	4		225.4.a.d.1.1		1
15.14	odd	2	inner	225.4.b.g.199.2		2
20.7	even	4		144.4.a.d.1.1		1
35.2	odd	12		441.4.e.i.361.1		2
35.12	even	12		441.4.e.j.361.1		2
35.17	even	12		441.4.e.j.226.1		2
35.27	even	4		441.4.a.f.1.1		1
35.32	odd	12		441.4.e.i.226.1		2
40.27	even	4		576.4.a.l.1.1		1
40.37	odd	4		576.4.a.m.1.1		1
45.2	even	12		81.4.c.b.28.1		2
45.7	odd	12		81.4.c.b.28.1		2
45.22	odd	12		81.4.c.b.55.1		2
45.32	even	12		81.4.c.b.55.1		2
55.32	even	4		1089.4.a.g.1.1		1
60.47	odd	4		144.4.a.d.1.1		1
65.12	odd	4		1521.4.a.g.1.1		1
105.2	even	12		441.4.e.i.361.1		2
105.17	odd	12		441.4.e.j.226.1		2
105.32	even	12		441.4.e.i.226.1		2
105.47	odd	12		441.4.e.j.361.1		2
105.62	odd	4		441.4.a.f.1.1		1
120.77	even	4		576.4.a.m.1.1		1
120.107	odd	4		576.4.a.l.1.1		1
165.32	odd	4		1089.4.a.g.1.1		1
195.77	even	4		1521.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.4.a.a.1.1	✓	1	5.2	odd	4
9.4.a.a.1.1	✓	1	15.2	even	4
81.4.c.b.28.1		2	45.2	even	12
81.4.c.b.28.1		2	45.7	odd	12
81.4.c.b.55.1		2	45.22	odd	12
81.4.c.b.55.1		2	45.32	even	12
144.4.a.d.1.1		1	20.7	even	4
144.4.a.d.1.1		1	60.47	odd	4
225.4.a.d.1.1		1	5.3	odd	4
225.4.a.d.1.1		1	15.8	even	4
225.4.b.g.199.1		2	1.1	even	1	trivial
225.4.b.g.199.1		2	3.2	odd	2	CM
225.4.b.g.199.2		2	5.4	even	2	inner
225.4.b.g.199.2		2	15.14	odd	2	inner
441.4.a.f.1.1		1	35.27	even	4
441.4.a.f.1.1		1	105.62	odd	4
441.4.e.i.226.1		2	35.32	odd	12
441.4.e.i.226.1		2	105.32	even	12
441.4.e.i.361.1		2	35.2	odd	12
441.4.e.i.361.1		2	105.2	even	12
441.4.e.j.226.1		2	35.17	even	12
441.4.e.j.226.1		2	105.17	odd	12
441.4.e.j.361.1		2	35.12	even	12
441.4.e.j.361.1		2	105.47	odd	12
576.4.a.l.1.1		1	40.27	even	4
576.4.a.l.1.1		1	120.107	odd	4
576.4.a.m.1.1		1	40.37	odd	4
576.4.a.m.1.1		1	120.77	even	4
1089.4.a.g.1.1		1	55.32	even	4
1089.4.a.g.1.1		1	165.32	odd	4
1521.4.a.g.1.1		1	65.12	odd	4
1521.4.a.g.1.1		1	195.77	even	4

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.00000 q^{4} -20.0000i q^{7} +O(q^{10})\)	\(q+8.00000 q^{4} -20.0000i q^{7} -70.0000i q^{13} +64.0000 q^{16} -56.0000 q^{19} -160.000i q^{28} +308.000 q^{31} -110.000i q^{37} -520.000i q^{43} -57.0000 q^{49} -560.000i q^{52} +182.000 q^{61} +512.000 q^{64} +880.000i q^{67} +1190.00i q^{73} -448.000 q^{76} -884.000 q^{79} -1400.00 q^{91} +1330.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{4}+O(q^{10}) \) 2 * q + 16 * q^4	\( 2 q + 16 q^{4} + 128 q^{16} - 112 q^{19} + 616 q^{31} - 114 q^{49} + 364 q^{61} + 1024 q^{64} - 896 q^{76} - 1768 q^{79} - 2800 q^{91}+O(q^{100}) \) 2 * q + 16 * q^4 + 128 * q^16 - 112 * q^19 + 616 * q^31 - 114 * q^49 + 364 * q^61 + 1024 * q^64 - 896 * q^76 - 1768 * q^79 - 2800 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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