LMFDB - Embedded newform 304.2.h.a.303.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,2,Mod(303,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, 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("304.303");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	304.h (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:	$2.42745222145$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 5 $ x^2 - x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			303.2
Root			$0.500000 - 2.17945i$ of defining polynomial
Character	$\chi$	$=$	304.303
Dual form			304.2.h.a.303.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +4.35890i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{5} +4.35890i q^{7} -3.00000 q^{9} +4.35890i q^{11} +7.00000 q^{17} -4.35890i q^{19} +8.71780i q^{23} -4.00000 q^{25} -4.35890i q^{35} -13.0767i q^{43} +3.00000 q^{45} +4.35890i q^{47} -12.0000 q^{49} -4.35890i q^{55} +15.0000 q^{61} -13.0767i q^{63} +11.0000 q^{73} -19.0000 q^{77} +9.00000 q^{81} -8.71780i q^{83} -7.00000 q^{85} +4.35890i q^{95} -13.0767i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 6 * q^9	$ 2 q - 2 q^{5} - 6 q^{9} + 14 q^{17} - 8 q^{25} + 6 q^{45} - 24 q^{49} + 30 q^{61} + 22 q^{73} - 38 q^{77} + 18 q^{81} - 14 q^{85}+O(q^{100}) $ 2 * q - 2 * q^5 - 6 * q^9 + 14 * q^17 - 8 * q^25 + 6 * q^45 - 24 * q^49 + 30 * q^61 + 22 * q^73 - 38 * q^77 + 18 * q^81 - 14 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	4.35890i	1.64751i	0.566947	+	0.823754i	$0.308125\pi$
$7$	4.35890i	1.64751i	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	4.35890i	1.31426i	0.753778	+	0.657129i	$0.228229\pi$
$11$	4.35890i	1.31426i	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	0	0
$19$	− 4.35890i	− 1.00000i
$19$	− 4.35890i	− 1.00000i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.71780i	1.81779i	0.417029	+	0.908893i	$0.363071\pi$
$23$	8.71780i	1.81779i	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 4.35890i	− 0.736788i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 13.0767i	− 1.99418i	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	− 13.0767i	− 1.99418i	0.0762493	−	0.997089i	$-0.475706\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	4.35890i	0.635811i	0.948122	+	0.317905i	$0.102979\pi$
$47$	4.35890i	0.635811i	−0.948122	+	0.317905i	$0.897021\pi$
$48$	0	0
$49$	−12.0000	−1.71429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	− 4.35890i	− 0.587754i
$56$	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
$61$	15.0000	1.92055	0.960277	−	0.279050i	$-0.0900195\pi$
$61$	15.0000	1.92055	0.960277	+	0.279050i	$0.0900195\pi$
$62$	0	0
$63$	− 13.0767i	− 1.64751i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	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
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−19.0000	−2.16525
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 8.71780i	− 0.956903i	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	− 8.71780i	− 0.956903i	0.878114	−	0.478451i	$-0.158802\pi$
$84$	0	0
$85$	−7.00000	−0.759257
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.35890i	0.447214i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	− 13.0767i	− 1.31426i
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	− 8.71780i	− 0.812939i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	30.5123i	2.79706i
$120$	0	0
$121$	−8.00000	−0.727273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	21.7945i	1.90419i	0.305796	+	0.952097i	$0.401077\pi$
$131$	21.7945i	1.90419i	−0.305796	+	0.952097i	$0.598923\pi$
$132$	0	0
$133$	19.0000	1.64751
$134$	0	0
$135$	0	0
$136$	0	0
$137$	23.0000	1.96502	0.982511	−	0.186203i	$-0.0596182\pi$
$137$	23.0000	1.96502	0.982511	+	0.186203i	$0.0596182\pi$
$138$	0	0
$139$	21.7945i	1.84858i	0.381685	+	0.924292i	$0.375344\pi$
$139$	21.7945i	1.84858i	−0.381685	+	0.924292i	$0.624656\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.0000	0.901155	0.450578	−	0.892737i	$-0.351218\pi$
$149$	11.0000	0.901155	0.450578	+	0.892737i	$0.351218\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−21.0000	−1.69775
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−38.0000	−2.99482
$162$	0	0
$163$	− 8.71780i	− 0.682831i	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	− 8.71780i	− 0.682831i	0.939913	−	0.341415i	$-0.110906\pi$
$164$	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
$171$	13.0767i	1.00000i
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	− 17.4356i	− 1.31801i
$176$	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
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	30.5123i	2.23128i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.7945i	1.57699i	0.615038	+	0.788497i	$0.289140\pi$
$191$	21.7945i	1.57699i	−0.615038	+	0.788497i	$0.710860\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	− 13.0767i	− 0.926982i	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	− 13.0767i	− 0.926982i	0.886102	−	0.463491i	$-0.153403\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 26.1534i	− 1.81779i
$208$	0	0
$209$	19.0000	1.31426
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	13.0767i	0.891823i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	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
$225$	12.0000	0.800000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−21.0000	−1.38772	−0.693860	−	0.720110i	$-0.744091\pi$
$229$	−21.0000	−1.38772	−0.693860	+	0.720110i	$0.744091\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	0	0
$235$	− 4.35890i	− 0.284343i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 30.5123i	− 1.97368i	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	− 30.5123i	− 1.97368i	0.161712	−	0.986838i	$-0.448299\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.0000	0.766652
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.7945i	1.37566i	0.725874	+	0.687828i	$0.241436\pi$
$251$	21.7945i	1.37566i	−0.725874	+	0.687828i	$0.758564\pi$
$252$	0	0
$253$	−38.0000	−2.38904
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 30.5123i	− 1.88147i	−0.339145	−	0.940734i	$-0.610138\pi$
$263$	− 30.5123i	− 1.88147i	0.339145	−	0.940734i	$-0.389862\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	− 26.1534i	− 1.58871i	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	− 26.1534i	− 1.58871i	0.607457	−	0.794353i	$-0.292190\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 17.4356i	− 1.05141i
$276$	0	0
$277$	−33.0000	−1.98278	−0.991389	−	0.130950i	$-0.958197\pi$
$277$	−33.0000	−1.98278	−0.991389	+	0.130950i	$0.958197\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 13.0767i	− 0.777329i	−0.921379	−	0.388664i	$-0.872937\pi$
$283$	− 13.0767i	− 0.777329i	0.921379	−	0.388664i	$-0.127063\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	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
$301$	57.0000	3.28543
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−15.0000	−0.858898
$306$	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
$311$	4.35890i	0.247170i	0.992334	+	0.123585i	$0.0394392\pi$
$311$	4.35890i	0.247170i	−0.992334	+	0.123585i	$0.960561\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	13.0767i	0.736788i
$316$	0	0
$317$	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
$323$	− 30.5123i	− 1.69775i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−19.0000	−1.04750
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 21.7945i	− 1.17679i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.35890i	0.233998i	0.993132	+	0.116999i	$0.0373274\pi$
$347$	4.35890i	0.233998i	−0.993132	+	0.116999i	$0.962673\pi$
$348$	0	0
$349$	35.0000	1.87351	0.936754	−	0.349990i	$-0.113815\pi$
$349$	35.0000	1.87351	0.936754	+	0.349990i	$0.113815\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.7945i	1.15027i	0.818059	+	0.575135i	$0.195050\pi$
$359$	21.7945i	1.15027i	−0.818059	+	0.575135i	$0.804950\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11.0000	−0.575766
$366$	0	0
$367$	− 26.1534i	− 1.36520i	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	− 26.1534i	− 1.36520i	0.730794	−	0.682598i	$-0.239150\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	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
$385$	19.0000	0.968330
$386$	0	0
$387$	39.2301i	1.99418i
$388$	0	0
$389$	−25.0000	−1.26755	−0.633775	−	0.773517i	$-0.718496\pi$
$389$	−25.0000	−1.26755	−0.633775	+	0.773517i	$0.718496\pi$
$390$	0	0
$391$	61.0246i	3.08615i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	0	0
$408$	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
$415$	8.71780i	0.427940i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 8.71780i	− 0.425892i	−0.977064	−	0.212946i	$-0.931694\pi$
$419$	− 8.71780i	− 0.425892i	0.977064	−	0.212946i	$-0.0683059\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	− 13.0767i	− 0.635811i
$424$	0	0
$425$	−28.0000	−1.35820
$426$	0	0
$427$	65.3835i	3.16413i
$428$	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
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	38.0000	1.81779
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	36.0000	1.71429
$442$	0	0
$443$	− 30.5123i	− 1.44968i	−0.688916	−	0.724841i	$-0.741913\pi$
$443$	− 30.5123i	− 1.44968i	0.688916	−	0.724841i	$-0.258087\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	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
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−37.0000	−1.72326	−0.861631	−	0.507535i	$-0.830557\pi$
$461$	−37.0000	−1.72326	−0.861631	+	0.507535i	$0.830557\pi$
$462$	0	0
$463$	− 13.0767i	− 0.607726i	−0.952716	−	0.303863i	$-0.901724\pi$
$463$	− 13.0767i	− 0.607726i	0.952716	−	0.303863i	$-0.0982765\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.35890i	0.201706i	0.994901	+	0.100853i	$0.0321571\pi$
$467$	4.35890i	0.201706i	−0.994901	+	0.100853i	$0.967843\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	57.0000	2.62086
$474$	0	0
$475$	17.4356i	0.800000i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	43.5890i	1.99163i	0.0913823	+	0.995816i	$0.470871\pi$
$479$	43.5890i	1.99163i	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 43.5890i	− 1.96714i	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	− 43.5890i	− 1.96714i	0.180517	−	0.983572i	$-0.442223\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	13.0767i	0.587754i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	21.7945i	0.975656i	0.872940	+	0.487828i	$0.162211\pi$
$499$	21.7945i	0.975656i	−0.872940	+	0.487828i	$0.837789\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.71780i	0.388707i	0.980932	+	0.194354i	$0.0622609\pi$
$503$	8.71780i	0.388707i	−0.980932	+	0.194354i	$0.937739\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	47.9479i	2.12109i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−19.0000	−0.835619
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−53.0000	−2.30435
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 52.3068i	− 2.25301i
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	−45.0000	−1.92055
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	47.0000	1.99145	0.995727	−	0.0923462i	$-0.0294367\pi$
$557$	47.0000	1.99145	0.995727	+	0.0923462i	$0.0294367\pi$
$558$	0	0
$559$	0	0
$560$	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
$567$	39.2301i	1.64751i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	26.1534i	1.09449i	0.836974	+	0.547243i	$0.184323\pi$
$571$	26.1534i	1.09449i	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 34.8712i	− 1.45423i
$576$	0	0
$577$	3.00000	0.124892	0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000	0.124892	0.0624458	+	0.998048i	$0.480110\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	38.0000	1.57651
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 47.9479i	− 1.97902i	−0.144460	−	0.989511i	$-0.546145\pi$
$587$	− 47.9479i	− 1.97902i	0.144460	−	0.989511i	$-0.453855\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	− 30.5123i	− 1.25088i
$596$	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
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.00000	0.325246
$606$	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
$613$	39.0000	1.57520	0.787598	−	0.616190i	$-0.211325\pi$
$613$	39.0000	1.57520	0.787598	+	0.616190i	$0.211325\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.0000	−0.523360	−0.261680	−	0.965155i	$-0.584277\pi$
$617$	−13.0000	−0.523360	−0.261680	+	0.965155i	$0.584277\pi$
$618$	0	0
$619$	− 43.5890i	− 1.75199i	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	− 43.5890i	− 1.75199i	0.482321	−	0.875995i	$-0.339794\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 47.9479i	− 1.90878i	−0.298570	−	0.954388i	$-0.596510\pi$
$631$	− 47.9479i	− 1.90878i	0.298570	−	0.954388i	$-0.403490\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	− 13.0767i	− 0.515695i	−0.966186	−	0.257847i	$-0.916987\pi$
$643$	− 13.0767i	− 0.515695i	0.966186	−	0.257847i	$-0.0830131\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 47.9479i	− 1.88503i	−0.334169	−	0.942513i	$-0.608456\pi$
$647$	− 47.9479i	− 1.88503i	0.334169	−	0.942513i	$-0.391544\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−41.0000	−1.60445	−0.802227	−	0.597019i	$-0.796352\pi$
$653$	−41.0000	−1.60445	−0.802227	+	0.597019i	$0.796352\pi$
$654$	0	0
$655$	− 21.7945i	− 0.851581i
$656$	0	0
$657$	−33.0000	−1.28745
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−19.0000	−0.736788
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	65.3835i	2.52410i
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	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
$685$	−23.0000	−0.878785
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	39.2301i	1.49238i	0.665731	+	0.746191i	$0.268120\pi$
$691$	39.2301i	1.49238i	−0.665731	+	0.746191i	$0.731880\pi$
$692$	0	0
$693$	57.0000	2.16525
$694$	0	0
$695$	− 21.7945i	− 0.826712i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 43.5890i	− 1.63933i
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	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
$719$	21.7945i	0.812798i	0.913696	+	0.406399i	$0.133216\pi$
$719$	21.7945i	0.812798i	−0.913696	+	0.406399i	$0.866784\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	39.2301i	1.45496i	0.686127	+	0.727482i	$0.259309\pi$
$727$	39.2301i	1.45496i	−0.686127	+	0.727482i	$0.740691\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	− 91.5369i	− 3.38561i
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 30.5123i	− 1.12241i	−0.827676	−	0.561206i	$-0.810337\pi$
$739$	− 30.5123i	− 1.12241i	0.827676	−	0.561206i	$-0.189663\pi$
$740$	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
$745$	−11.0000	−0.403009
$746$	0	0
$747$	26.1534i	0.956903i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	27.0000	0.981332	0.490666	−	0.871348i	$-0.336754\pi$
$757$	27.0000	0.981332	0.490666	+	0.871348i	$0.336754\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	55.0000	1.99375	0.996874	−	0.0790050i	$-0.0251743\pi$
$761$	55.0000	1.99375	0.996874	+	0.0790050i	$0.0251743\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	21.0000	0.759257
$766$	0	0
$767$	0	0
$768$	0	0
$769$	51.0000	1.83911	0.919554	−	0.392965i	$-0.128551\pi$
$769$	51.0000	1.83911	0.919554	+	0.392965i	$0.128551\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	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
$784$	0	0
$785$	18.0000	0.642448
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	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	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	30.5123i	1.07945i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	47.9479i	1.69204i
$804$	0	0
$805$	38.0000	1.33932
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.00000	−0.175791	−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000	−0.175791	−0.0878953	+	0.996130i	$0.528014\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.71780i	0.305371i
$816$	0	0
$817$	−57.0000	−1.99418
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−53.0000	−1.84971	−0.924856	−	0.380317i	$-0.875815\pi$
$821$	−53.0000	−1.84971	−0.924856	+	0.380317i	$0.875815\pi$
$822$	0	0
$823$	56.6657i	1.97524i	0.156860	+	0.987621i	$0.449863\pi$
$823$	56.6657i	1.97524i	−0.156860	+	0.987621i	$0.550137\pi$
$824$	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
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−84.0000	−2.91043
$834$	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
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−13.0000	−0.447214
$846$	0	0
$847$	− 34.8712i	− 1.19819i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	− 13.0767i	− 0.447214i
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	56.6657i	1.93341i	0.255897	+	0.966704i	$0.417629\pi$
$859$	56.6657i	1.93341i	−0.255897	+	0.966704i	$0.582371\pi$
$860$	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
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	39.2301i	1.32622i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	35.0000	1.17918	0.589590	−	0.807703i	$-0.299289\pi$
$881$	35.0000	1.17918	0.589590	+	0.807703i	$0.299289\pi$
$882$	0	0
$883$	− 30.5123i	− 1.02682i	−0.858143	−	0.513410i	$-0.828382\pi$
$883$	− 30.5123i	− 1.02682i	0.858143	−	0.513410i	$-0.171618\pi$
$884$	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
$889$	0	0
$890$	0	0
$891$	39.2301i	1.31426i
$892$	0	0
$893$	19.0000	0.635811
$894$	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
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	30.0000	0.995037
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	38.0000	1.25762
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−95.0000	−3.13718
$918$	0	0
$919$	8.71780i	0.287574i	0.989609	+	0.143787i	$0.0459280\pi$
$919$	8.71780i	0.287574i	−0.989609	+	0.143787i	$0.954072\pi$
$920$	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
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	52.3068i	1.71429i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 30.5123i	− 0.997859i
$936$	0	0
$937$	47.0000	1.53542	0.767712	−	0.640796i	$-0.221395\pi$
$937$	47.0000	1.53542	0.767712	+	0.640796i	$0.221395\pi$
$938$	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
$947$	61.0246i	1.98303i	0.129983	+	0.991516i	$0.458508\pi$
$947$	61.0246i	1.98303i	−0.129983	+	0.991516i	$0.541492\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	− 21.7945i	− 0.705253i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	100.255i	3.23739i
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 61.0246i	− 1.96242i	−0.192947	−	0.981209i	$-0.561805\pi$
$967$	− 61.0246i	− 1.96242i	0.192947	−	0.981209i	$-0.438195\pi$
$968$	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
$973$	−95.0000	−3.04556
$974$	0	0
$975$	0	0
$976$	0	0
$977$	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		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−22.0000	−0.700978
$986$	0	0
$987$	0	0
$988$	0	0
$989$	114.000	3.62499
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13.0767i	0.414559i
$996$	0	0
$997$	63.0000	1.99523	0.997615	−	0.0690239i	$-0.0219885\pi$
$997$	63.0000	1.99523	0.997615	+	0.0690239i	$0.0219885\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.h.a.303.2	yes	2
3.2	odd	2		2736.2.k.g.2431.2		2
4.3	odd	2	inner	304.2.h.a.303.1	✓	2
8.3	odd	2		1216.2.h.a.1215.1		2
8.5	even	2		1216.2.h.a.1215.2		2
12.11	even	2		2736.2.k.g.2431.1		2
19.18	odd	2	CM	304.2.h.a.303.2	yes	2
57.56	even	2		2736.2.k.g.2431.2		2
76.75	even	2	inner	304.2.h.a.303.1	✓	2
152.37	odd	2		1216.2.h.a.1215.2		2
152.75	even	2		1216.2.h.a.1215.1		2
228.227	odd	2		2736.2.k.g.2431.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.h.a.303.1	✓	2	4.3	odd	2	inner
304.2.h.a.303.1	✓	2	76.75	even	2	inner
304.2.h.a.303.2	yes	2	1.1	even	1	trivial
304.2.h.a.303.2	yes	2	19.18	odd	2	CM
1216.2.h.a.1215.1		2	8.3	odd	2
1216.2.h.a.1215.1		2	152.75	even	2
1216.2.h.a.1215.2		2	8.5	even	2
1216.2.h.a.1215.2		2	152.37	odd	2
2736.2.k.g.2431.1		2	12.11	even	2
2736.2.k.g.2431.1		2	228.227	odd	2
2736.2.k.g.2431.2		2	3.2	odd	2
2736.2.k.g.2431.2		2	57.56	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +4.35890i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{5} +4.35890i q^{7} -3.00000 q^{9} +4.35890i q^{11} +7.00000 q^{17} -4.35890i q^{19} +8.71780i q^{23} -4.00000 q^{25} -4.35890i q^{35} -13.0767i q^{43} +3.00000 q^{45} +4.35890i q^{47} -12.0000 q^{49} -4.35890i q^{55} +15.0000 q^{61} -13.0767i q^{63} +11.0000 q^{73} -19.0000 q^{77} +9.00000 q^{81} -8.71780i q^{83} -7.00000 q^{85} +4.35890i q^{95} -13.0767i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 6 * q^9	\( 2 q - 2 q^{5} - 6 q^{9} + 14 q^{17} - 8 q^{25} + 6 q^{45} - 24 q^{49} + 30 q^{61} + 22 q^{73} - 38 q^{77} + 18 q^{81} - 14 q^{85}+O(q^{100}) \) 2 * q - 2 * q^5 - 6 * q^9 + 14 * q^17 - 8 * q^25 + 6 * q^45 - 24 * q^49 + 30 * q^61 + 22 * q^73 - 38 * q^77 + 18 * q^81 - 14 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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