LMFDB - Embedded newform 675.3.d.c.674.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,3,Mod(674,675)] 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("675.674"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	675.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,32,0,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$18.3924178443$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			674.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	675.674
Dual form			675.3.d.c.674.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{4} -13.0000i q^{7} +1.00000i q^{13} +16.0000 q^{16} -11.0000 q^{19} +52.0000i q^{28} -46.0000 q^{31} +47.0000i q^{37} +22.0000i q^{43} -120.000 q^{49} -4.00000i q^{52} -121.000 q^{61} -64.0000 q^{64} -109.000i q^{67} +97.0000i q^{73} +44.0000 q^{76} -131.000 q^{79} +13.0000 q^{91} +167.000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 32 q^{16} - 22 q^{19} - 92 q^{31} - 240 q^{49} - 242 q^{61} - 128 q^{64} + 88 q^{76} - 262 q^{79} + 26 q^{91}+O(q^{100}) $ 2 * q - 8 * q^4 + 32 * q^16 - 22 * q^19 - 92 * q^31 - 240 * q^49 - 242 * q^61 - 128 * q^64 + 88 * q^76 - 262 * q^79 + 26 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$-1$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−4.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 13.0000i	− 1.85714i	−0.371154	−	0.928571i	$-0.621038\pi$
$7$	− 13.0000i	− 1.85714i	0.371154	−	0.928571i	$-0.378962\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	1.00000i	0.0769231i	0.999260	+	0.0384615i	$0.0122457\pi$
$13$	1.00000i	0.0769231i	−0.999260	+	0.0384615i	$0.987754\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−11.0000	−0.578947	−0.289474	−	0.957186i	$-0.593480\pi$
$19$	−11.0000	−0.578947	−0.289474	+	0.957186i	$0.593480\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	52.0000i	1.85714i
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−46.0000	−1.48387	−0.741935	−	0.670471i	$-0.766092\pi$
$31$	−46.0000	−1.48387	−0.741935	+	0.670471i	$0.766092\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	47.0000i	1.27027i	0.772401	+	0.635135i	$0.219056\pi$
$37$	47.0000i	1.27027i	−0.772401	+	0.635135i	$0.780944\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	22.0000i	0.511628i	0.966726	+	0.255814i	$0.0823435\pi$
$43$	22.0000i	0.511628i	−0.966726	+	0.255814i	$0.917657\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−120.000	−2.44898
$50$	0	0
$51$	0	0
$52$	− 4.00000i	− 0.0769231i
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−121.000	−1.98361	−0.991803	−	0.127774i	$-0.959217\pi$
$61$	−121.000	−1.98361	−0.991803	+	0.127774i	$0.959217\pi$
$62$	0	0
$63$	0	0
$64$	−64.0000	−1.00000
$65$	0	0
$66$	0	0
$67$	− 109.000i	− 1.62687i	−0.581659	−	0.813433i	$-0.697596\pi$
$67$	− 109.000i	− 1.62687i	0.581659	−	0.813433i	$-0.302404\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	97.0000i	1.32877i	0.747392	+	0.664384i	$0.231306\pi$
$73$	97.0000i	1.32877i	−0.747392	+	0.664384i	$0.768694\pi$
$74$	0	0
$75$	0	0
$76$	44.0000	0.578947
$77$	0	0
$78$	0	0
$79$	−131.000	−1.65823	−0.829114	−	0.559080i	$-0.811155\pi$
$79$	−131.000	−1.65823	−0.829114	+	0.559080i	$0.811155\pi$
$80$	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
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	13.0000	0.142857
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	167.000i	1.72165i	0.508902	+	0.860825i	$0.330052\pi$
$97$	167.000i	1.72165i	−0.508902	+	0.860825i	$0.669948\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	37.0000i	0.359223i	0.983738	+	0.179612i	$0.0574841\pi$
$103$	37.0000i	0.359223i	−0.983738	+	0.179612i	$0.942516\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	214.000	1.96330	0.981651	−	0.190684i	$-0.0610707\pi$
$109$	214.000	1.96330	0.981651	+	0.190684i	$0.0610707\pi$
$110$	0	0
$111$	0	0
$112$	− 208.000i	− 1.85714i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	184.000	1.48387
$125$	0	0
$126$	0	0
$127$	146.000i	1.14961i	0.818292	+	0.574803i	$0.194921\pi$
$127$	146.000i	1.14961i	−0.818292	+	0.574803i	$0.805079\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	143.000i	1.07519i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−251.000	−1.80576	−0.902878	−	0.429898i	$-0.858550\pi$
$139$	−251.000	−1.80576	−0.902878	+	0.429898i	$0.858550\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	− 188.000i	− 1.27027i
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	59.0000	0.390728	0.195364	−	0.980731i	$-0.437411\pi$
$151$	59.0000	0.390728	0.195364	+	0.980731i	$0.437411\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 118.000i	− 0.751592i	−0.926702	−	0.375796i	$-0.877369\pi$
$157$	− 118.000i	− 0.751592i	0.926702	−	0.375796i	$-0.122631\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 299.000i	− 1.83436i	−0.398478	−	0.917178i	$-0.630461\pi$
$163$	− 299.000i	− 1.83436i	0.398478	−	0.917178i	$-0.369539\pi$
$164$	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
$169$	168.000	0.994083
$170$	0	0
$171$	0	0
$172$	− 88.0000i	− 0.511628i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	−313.000	−1.72928	−0.864641	−	0.502390i	$-0.832454\pi$
$181$	−313.000	−1.72928	−0.864641	+	0.502390i	$0.832454\pi$
$182$	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	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	− 143.000i	− 0.740933i	−0.928846	−	0.370466i	$-0.879198\pi$
$193$	− 143.000i	− 0.740933i	0.928846	−	0.370466i	$-0.120802\pi$
$194$	0	0
$195$	0	0
$196$	480.000	2.44898
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	277.000	1.39196	0.695980	−	0.718061i	$-0.254970\pi$
$199$	277.000	1.39196	0.695980	+	0.718061i	$0.254970\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	16.0000i	0.0769231i
$209$	0	0
$210$	0	0
$211$	−253.000	−1.19905	−0.599526	−	0.800355i	$-0.704644\pi$
$211$	−253.000	−1.19905	−0.599526	+	0.800355i	$0.704644\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	598.000i	2.75576i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 338.000i	− 1.51570i	−0.652432	−	0.757848i	$-0.726251\pi$
$223$	− 338.000i	− 1.51570i	0.652432	−	0.757848i	$-0.273749\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−26.0000	−0.113537	−0.0567686	−	0.998387i	$-0.518080\pi$
$229$	−26.0000	−0.113537	−0.0567686	+	0.998387i	$0.518080\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−193.000	−0.800830	−0.400415	−	0.916334i	$-0.631134\pi$
$241$	−193.000	−0.800830	−0.400415	+	0.916334i	$0.631134\pi$
$242$	0	0
$243$	0	0
$244$	484.000	1.98361
$245$	0	0
$246$	0	0
$247$	− 11.0000i	− 0.0445344i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	256.000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	611.000	2.35907
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	436.000i	1.62687i
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	299.000	1.10332	0.551661	−	0.834069i	$-0.313994\pi$
$271$	299.000	1.10332	0.551661	+	0.834069i	$0.313994\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	122.000i	0.440433i	0.975451	+	0.220217i	$0.0706764\pi$
$277$	122.000i	0.440433i	−0.975451	+	0.220217i	$0.929324\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 458.000i	− 1.61837i	−0.587551	−	0.809187i	$-0.699908\pi$
$283$	− 458.000i	− 1.61837i	0.587551	−	0.809187i	$-0.300092\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−289.000	−1.00000
$290$	0	0
$291$	0	0
$292$	− 388.000i	− 1.32877i
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	286.000	0.950166
$302$	0	0
$303$	0	0
$304$	−176.000	−0.578947
$305$	0	0
$306$	0	0
$307$	− 358.000i	− 1.16612i	−0.812428	−	0.583062i	$-0.801855\pi$
$307$	− 358.000i	− 1.16612i	0.812428	−	0.583062i	$-0.198145\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	− 599.000i	− 1.91374i	−0.290520	−	0.956869i	$-0.593828\pi$
$313$	− 599.000i	− 1.91374i	0.290520	−	0.956869i	$-0.406172\pi$
$314$	0	0
$315$	0	0
$316$	524.000	1.65823
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	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
$331$	−661.000	−1.99698	−0.998489	−	0.0549442i	$-0.982502\pi$
$331$	−661.000	−1.99698	−0.998489	+	0.0549442i	$0.982502\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 649.000i	− 1.92582i	−0.269830	−	0.962908i	$-0.586967\pi$
$337$	− 649.000i	− 1.92582i	0.269830	−	0.962908i	$-0.413033\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	923.000i	2.69096i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−671.000	−1.92264	−0.961318	−	0.275441i	$-0.911176\pi$
$349$	−671.000	−1.92264	−0.961318	+	0.275441i	$0.911176\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	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
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−240.000	−0.664820
$362$	0	0
$363$	0	0
$364$	−52.0000	−0.142857
$365$	0	0
$366$	0	0
$367$	491.000i	1.33787i	0.743319	+	0.668937i	$0.233251\pi$
$367$	491.000i	1.33787i	−0.743319	+	0.668937i	$0.766749\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	577.000i	1.54692i	0.633847	+	0.773458i	$0.281475\pi$
$373$	577.000i	1.54692i	−0.633847	+	0.773458i	$0.718525\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−83.0000	−0.218997	−0.109499	−	0.993987i	$-0.534925\pi$
$379$	−83.0000	−0.218997	−0.109499	+	0.993987i	$0.534925\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	− 668.000i	− 1.72165i
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	362.000i	0.911839i	0.890021	+	0.455919i	$0.150689\pi$
$397$	362.000i	0.911839i	−0.890021	+	0.455919i	$0.849311\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 46.0000i	− 0.114144i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−143.000	−0.349633	−0.174817	−	0.984601i	$-0.555933\pi$
$409$	−143.000	−0.349633	−0.174817	+	0.984601i	$0.555933\pi$
$410$	0	0
$411$	0	0
$412$	− 148.000i	− 0.359223i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−481.000	−1.14252	−0.571259	−	0.820770i	$-0.693545\pi$
$421$	−481.000	−1.14252	−0.571259	+	0.820770i	$0.693545\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1573.00i	3.68384i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	862.000i	1.99076i	0.0960028	+	0.995381i	$0.469394\pi$
$433$	862.000i	1.99076i	−0.0960028	+	0.995381i	$0.530606\pi$
$434$	0	0
$435$	0	0
$436$	−856.000	−1.96330
$437$	0	0
$438$	0	0
$439$	94.0000	0.214123	0.107062	−	0.994252i	$-0.465856\pi$
$439$	94.0000	0.214123	0.107062	+	0.994252i	$0.465856\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	832.000i	1.85714i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 814.000i	− 1.78118i	−0.454805	−	0.890591i	$-0.650291\pi$
$457$	− 814.000i	− 1.78118i	0.454805	−	0.890591i	$-0.349709\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	397.000i	0.857451i	0.903435	+	0.428726i	$0.141037\pi$
$463$	397.000i	0.857451i	−0.903435	+	0.428726i	$0.858963\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−1417.00	−3.02132
$470$	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	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	−47.0000	−0.0977131
$482$	0	0
$483$	0	0
$484$	−484.000	−1.00000
$485$	0	0
$486$	0	0
$487$	− 349.000i	− 0.716632i	−0.933600	−	0.358316i	$-0.883351\pi$
$487$	− 349.000i	− 0.716632i	0.933600	−	0.358316i	$-0.116649\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−736.000	−1.48387
$497$	0	0
$498$	0	0
$499$	−26.0000	−0.0521042	−0.0260521	−	0.999661i	$-0.508294\pi$
$499$	−26.0000	−0.0521042	−0.0260521	+	0.999661i	$0.508294\pi$
$500$	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
$506$	0	0
$507$	0	0
$508$	− 584.000i	− 1.14961i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	1261.00	2.46771
$512$	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	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 803.000i	− 1.53537i	−0.640826	−	0.767686i	$-0.721408\pi$
$523$	− 803.000i	− 1.53537i	0.640826	−	0.767686i	$-0.278592\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−529.000	−1.00000
$530$	0	0
$531$	0	0
$532$	− 572.000i	− 1.07519i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−241.000	−0.445471	−0.222736	−	0.974879i	$-0.571499\pi$
$541$	−241.000	−0.445471	−0.222736	+	0.974879i	$0.571499\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1093.00i	− 1.99817i	−0.0427471	−	0.999086i	$-0.513611\pi$
$547$	− 1093.00i	− 1.99817i	0.0427471	−	0.999086i	$-0.486389\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1703.00i	3.07957i
$554$	0	0
$555$	0	0
$556$	1004.00	1.80576
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−22.0000	−0.0393560
$560$	0	0
$561$	0	0
$562$	0	0
$563$	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
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−181.000	−0.316988	−0.158494	−	0.987360i	$-0.550664\pi$
$571$	−181.000	−0.316988	−0.158494	+	0.987360i	$0.550664\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1033.00i	− 1.79029i	−0.445770	−	0.895147i	$-0.647070\pi$
$577$	− 1033.00i	− 1.79029i	0.445770	−	0.895147i	$-0.352930\pi$
$578$	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		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	506.000	0.859083
$590$	0	0
$591$	0	0
$592$	752.000i	1.27027i
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−526.000	−0.875208	−0.437604	−	0.899168i	$-0.644173\pi$
$601$	−526.000	−0.875208	−0.437604	+	0.899168i	$0.644173\pi$
$602$	0	0
$603$	0	0
$604$	−236.000	−0.390728
$605$	0	0
$606$	0	0
$607$	1187.00i	1.95552i	0.209729	+	0.977759i	$0.432742\pi$
$607$	1187.00i	1.95552i	−0.209729	+	0.977759i	$0.567258\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 983.000i	− 1.60359i	−0.597600	−	0.801794i	$-0.703879\pi$
$613$	− 983.000i	− 1.60359i	0.597600	−	0.801794i	$-0.296121\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	949.000	1.53312	0.766559	−	0.642174i	$-0.221967\pi$
$619$	949.000	1.53312	0.766559	+	0.642174i	$0.221967\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	472.000i	0.751592i
$629$	0	0
$630$	0	0
$631$	587.000	0.930269	0.465135	−	0.885240i	$-0.346006\pi$
$631$	587.000	0.930269	0.465135	+	0.885240i	$0.346006\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 120.000i	− 0.188383i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	− 314.000i	− 0.488336i	−0.969733	−	0.244168i	$-0.921485\pi$
$643$	− 314.000i	− 0.488336i	0.969733	−	0.244168i	$-0.0785148\pi$
$644$	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
$652$	1196.00i	1.83436i
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	1079.00	1.63238	0.816188	−	0.577787i	$-0.196084\pi$
$661$	1079.00	1.63238	0.816188	+	0.577787i	$0.196084\pi$
$662$	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
$673$	− 23.0000i	− 0.0341753i	−0.999854	−	0.0170877i	$-0.994561\pi$
$673$	− 23.0000i	− 0.0341753i	0.999854	−	0.0170877i	$-0.00543944\pi$
$674$	0	0
$675$	0	0
$676$	−672.000	−0.994083
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	2171.00	3.19735
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	352.000i	0.511628i
$689$	0	0
$690$	0	0
$691$	−1318.00	−1.90738	−0.953690	−	0.300790i	$-0.902750\pi$
$691$	−1318.00	−1.90738	−0.953690	+	0.300790i	$0.902750\pi$
$692$	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	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	− 517.000i	− 0.735420i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1391.00	−1.96192	−0.980959	−	0.194214i	$-0.937784\pi$
$709$	−1391.00	−1.96192	−0.980959	+	0.194214i	$0.937784\pi$
$710$	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	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	481.000	0.667129
$722$	0	0
$723$	0	0
$724$	1252.00	1.72928
$725$	0	0
$726$	0	0
$727$	482.000i	0.662999i	0.943455	+	0.331499i	$0.107554\pi$
$727$	482.000i	0.662999i	−0.943455	+	0.331499i	$0.892446\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 1034.00i	− 1.41064i	−0.708888	−	0.705321i	$-0.750803\pi$
$733$	− 1034.00i	− 1.41064i	0.708888	−	0.705321i	$-0.249197\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1222.00	1.65359	0.826793	−	0.562506i	$-0.190163\pi$
$739$	1222.00	1.65359	0.826793	+	0.562506i	$0.190163\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	179.000	0.238349	0.119174	−	0.992873i	$-0.461975\pi$
$751$	179.000	0.238349	0.119174	+	0.992873i	$0.461975\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 673.000i	− 0.889036i	−0.895770	−	0.444518i	$-0.853375\pi$
$757$	− 673.000i	− 0.889036i	0.895770	−	0.444518i	$-0.146625\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 2782.00i	− 3.64613i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−863.000	−1.12224	−0.561118	−	0.827736i	$-0.689629\pi$
$769$	−863.000	−1.12224	−0.561118	+	0.827736i	$0.689629\pi$
$770$	0	0
$771$	0	0
$772$	572.000i	0.740933i
$773$	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
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1920.00	−2.44898
$785$	0	0
$786$	0	0
$787$	− 613.000i	− 0.778907i	−0.921046	−	0.389454i	$-0.872664\pi$
$787$	− 613.000i	− 0.778907i	0.921046	−	0.389454i	$-0.127336\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 121.000i	− 0.152585i
$794$	0	0
$795$	0	0
$796$	−1108.00	−1.39196
$797$	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
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	1514.00	1.86683	0.933416	−	0.358797i	$-0.116813\pi$
$811$	1514.00	1.86683	0.933416	+	0.358797i	$0.116813\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 242.000i	− 0.296206i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1621.00i	1.96962i	0.173626	+	0.984812i	$0.444452\pi$
$823$	1621.00i	1.96962i	−0.173626	+	0.984812i	$0.555548\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	1609.00	1.94089	0.970446	−	0.241317i	$-0.0775794\pi$
$829$	1609.00	1.94089	0.970446	+	0.241317i	$0.0775794\pi$
$830$	0	0
$831$	0	0
$832$	− 64.0000i	− 0.0769231i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	1012.00	1.19905
$845$	0	0
$846$	0	0
$847$	− 1573.00i	− 1.85714i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	481.000i	0.563892i	0.959430	+	0.281946i	$0.0909799\pi$
$853$	481.000i	0.563892i	−0.959430	+	0.281946i	$0.909020\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	1549.00	1.80326	0.901630	−	0.432509i	$-0.142371\pi$
$859$	1549.00	1.80326	0.901630	+	0.432509i	$0.142371\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	− 2392.00i	− 2.75576i
$869$	0	0
$870$	0	0
$871$	109.000	0.125144
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1727.00i	1.96921i	0.174785	+	0.984607i	$0.444077\pi$
$877$	1727.00i	1.96921i	−0.174785	+	0.984607i	$0.555923\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	− 443.000i	− 0.501699i	−0.968026	−	0.250849i	$-0.919290\pi$
$883$	− 443.000i	− 0.501699i	0.968026	−	0.250849i	$-0.0807099\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	1898.00	2.13498
$890$	0	0
$891$	0	0
$892$	1352.00i	1.51570i
$893$	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
$907$	− 1453.00i	− 1.60198i	−0.598675	−	0.800992i	$-0.704306\pi$
$907$	− 1453.00i	− 1.60198i	0.598675	−	0.800992i	$-0.295694\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	104.000	0.113537
$917$	0	0
$918$	0	0
$919$	−866.000	−0.942329	−0.471164	−	0.882045i	$-0.656166\pi$
$919$	−866.000	−0.942329	−0.471164	+	0.882045i	$0.656166\pi$
$920$	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	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	1320.00	1.41783
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1847.00i	1.97118i	0.169138	+	0.985592i	$0.445902\pi$
$937$	1847.00i	1.97118i	−0.169138	+	0.985592i	$0.554098\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	−97.0000	−0.102213
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1155.00	1.20187
$962$	0	0
$963$	0	0
$964$	772.000	0.800830
$965$	0	0
$966$	0	0
$967$	− 253.000i	− 0.261634i	−0.991407	−	0.130817i	$-0.958240\pi$
$967$	− 253.000i	− 0.261634i	0.991407	−	0.130817i	$-0.0417600\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	3263.00i	3.35355i
$974$	0	0
$975$	0	0
$976$	−1936.00	−1.98361
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	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
$988$	44.0000i	0.0445344i
$989$	0	0
$990$	0	0
$991$	1739.00	1.75479	0.877397	−	0.479766i	$-0.159278\pi$
$991$	1739.00	1.75479	0.877397	+	0.479766i	$0.159278\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1894.00i	− 1.89970i	−0.312707	−	0.949850i	$-0.601236\pi$
$997$	− 1894.00i	− 1.89970i	0.312707	−	0.949850i	$-0.398764\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.3.d.c.674.1		2
3.2	odd	2	CM	675.3.d.c.674.1		2
5.2	odd	4		675.3.c.c.26.1		1
5.3	odd	4		27.3.b.a.26.1	✓	1
5.4	even	2	inner	675.3.d.c.674.2		2
15.2	even	4		675.3.c.c.26.1		1
15.8	even	4		27.3.b.a.26.1	✓	1
15.14	odd	2	inner	675.3.d.c.674.2		2
20.3	even	4		432.3.e.b.161.1		1
40.3	even	4		1728.3.e.d.1025.1		1
40.13	odd	4		1728.3.e.a.1025.1		1
45.13	odd	12		81.3.d.a.26.1		2
45.23	even	12		81.3.d.a.26.1		2
45.38	even	12		81.3.d.a.53.1		2
45.43	odd	12		81.3.d.a.53.1		2
60.23	odd	4		432.3.e.b.161.1		1
120.53	even	4		1728.3.e.a.1025.1		1
120.83	odd	4		1728.3.e.d.1025.1		1
180.23	odd	12		1296.3.q.a.593.1		2
180.43	even	12		1296.3.q.a.1025.1		2
180.83	odd	12		1296.3.q.a.1025.1		2
180.103	even	12		1296.3.q.a.593.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.b.a.26.1	✓	1	5.3	odd	4
27.3.b.a.26.1	✓	1	15.8	even	4
81.3.d.a.26.1		2	45.13	odd	12
81.3.d.a.26.1		2	45.23	even	12
81.3.d.a.53.1		2	45.38	even	12
81.3.d.a.53.1		2	45.43	odd	12
432.3.e.b.161.1		1	20.3	even	4
432.3.e.b.161.1		1	60.23	odd	4
675.3.c.c.26.1		1	5.2	odd	4
675.3.c.c.26.1		1	15.2	even	4
675.3.d.c.674.1		2	1.1	even	1	trivial
675.3.d.c.674.1		2	3.2	odd	2	CM
675.3.d.c.674.2		2	5.4	even	2	inner
675.3.d.c.674.2		2	15.14	odd	2	inner
1296.3.q.a.593.1		2	180.23	odd	12
1296.3.q.a.593.1		2	180.103	even	12
1296.3.q.a.1025.1		2	180.43	even	12
1296.3.q.a.1025.1		2	180.83	odd	12
1728.3.e.a.1025.1		1	40.13	odd	4
1728.3.e.a.1025.1		1	120.53	even	4
1728.3.e.d.1025.1		1	40.3	even	4
1728.3.e.d.1025.1		1	120.83	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	675.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000 q^{4} -13.0000i q^{7} +1.00000i q^{13} +16.0000 q^{16} -11.0000 q^{19} +52.0000i q^{28} -46.0000 q^{31} +47.0000i q^{37} +22.0000i q^{43} -120.000 q^{49} -4.00000i q^{52} -121.000 q^{61} -64.0000 q^{64} -109.000i q^{67} +97.0000i q^{73} +44.0000 q^{76} -131.000 q^{79} +13.0000 q^{91} +167.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 32 q^{16} - 22 q^{19} - 92 q^{31} - 240 q^{49} - 242 q^{61} - 128 q^{64} + 88 q^{76} - 262 q^{79} + 26 q^{91}+O(q^{100}) \) 2 * q - 8 * q^4 + 32 * q^16 - 22 * q^19 - 92 * q^31 - 240 * q^49 - 242 * q^61 - 128 * q^64 + 88 * q^76 - 262 * q^79 + 26 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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