LMFDB - Embedded newform 2304.3.g.t.1279.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(1279,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2304.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.g (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:	$62.7794529086$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1279.1
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2304.1279
Dual form			2304.3.g.t.1279.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.79796 q^{5} -10.0000i q^{7} +O(q^{10})$	$q-9.79796 q^{5} -10.0000i q^{7} +19.5959i q^{11} +71.0000 q^{25} -29.3939 q^{29} +38.0000i q^{31} +97.9796i q^{35} -51.0000 q^{49} +48.9898 q^{53} -192.000i q^{55} +117.576i q^{59} +50.0000 q^{73} +195.959 q^{77} -58.0000i q^{79} -97.9796i q^{83} -190.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 284 q^{25} - 204 q^{49} + 200 q^{73} - 760 q^{97}+O(q^{100}) $ 4 * q + 284 * q^25 - 204 * q^49 + 200 * q^73 - 760 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$4$	0	0
$5$	−9.79796	−1.95959	−0.979796	−	0.200000i	$-0.935906\pi$
$5$	−9.79796	−1.95959	−0.979796	+	0.200000i	$0.935906\pi$
$6$	0	0
$7$	− 10.0000i	− 1.42857i	−0.699854	−	0.714286i	$-0.746752\pi$
$7$	− 10.0000i	− 1.42857i	0.699854	−	0.714286i	$-0.253248\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	19.5959i	1.78145i	0.454545	+	0.890724i	$0.349802\pi$
$11$	19.5959i	1.78145i	−0.454545	+	0.890724i	$0.650198\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	71.0000	2.84000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−29.3939	−1.01358	−0.506791	−	0.862069i	$-0.669168\pi$
$29$	−29.3939	−1.01358	−0.506791	+	0.862069i	$0.669168\pi$
$30$	0	0
$31$	38.0000i	1.22581i	0.790158	+	0.612903i	$0.209998\pi$
$31$	38.0000i	1.22581i	−0.790158	+	0.612903i	$0.790002\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	97.9796i	2.79942i
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	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
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−51.0000	−1.04082
$50$	0	0
$51$	0	0
$52$	0	0
$53$	48.9898	0.924336	0.462168	−	0.886792i	$-0.347072\pi$
$53$	48.9898	0.924336	0.462168	+	0.886792i	$0.347072\pi$
$54$	0	0
$55$	− 192.000i	− 3.49091i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	117.576i	1.99281i	0.0847458	+	0.996403i	$0.472992\pi$
$59$	117.576i	1.99281i	−0.0847458	+	0.996403i	$0.527008\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	50.0000	0.684932	0.342466	−	0.939530i	$-0.388738\pi$
$73$	50.0000	0.684932	0.342466	+	0.939530i	$0.388738\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	195.959	2.54492
$78$	0	0
$79$	− 58.0000i	− 0.734177i	−0.930186	−	0.367089i	$-0.880355\pi$
$79$	− 58.0000i	− 0.734177i	0.930186	−	0.367089i	$-0.119645\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 97.9796i	− 1.18048i	−0.807229	−	0.590238i	$-0.799034\pi$
$83$	− 97.9796i	− 1.18048i	0.807229	−	0.590238i	$-0.200966\pi$
$84$	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
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−190.000	−1.95876	−0.979381	−	0.202020i	$-0.935249\pi$
$97$	−190.000	−1.95876	−0.979381	+	0.202020i	$0.935249\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−68.5857	−0.679066	−0.339533	−	0.940594i	$-0.610269\pi$
$101$	−68.5857	−0.679066	−0.339533	+	0.940594i	$0.610269\pi$
$102$	0	0
$103$	10.0000i	0.0970874i	0.998821	+	0.0485437i	$0.0154580\pi$
$103$	10.0000i	0.0970874i	−0.998821	+	0.0485437i	$0.984542\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 195.959i	− 1.83139i	−0.401869	−	0.915697i	$-0.631639\pi$
$107$	− 195.959i	− 1.83139i	0.401869	−	0.915697i	$-0.368361\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	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
$121$	−263.000	−2.17355
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−450.706	−3.60565
$126$	0	0
$127$	− 230.000i	− 1.81102i	−0.424321	−	0.905512i	$-0.639487\pi$
$127$	− 230.000i	− 1.81102i	0.424321	−	0.905512i	$-0.360513\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 78.3837i	− 0.598349i	−0.954198	−	0.299174i	$-0.903289\pi$
$131$	− 78.3837i	− 0.598349i	0.954198	−	0.299174i	$-0.0967112\pi$
$132$	0	0
$133$	0	0
$134$	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
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	288.000	1.98621
$146$	0	0
$147$	0	0
$148$	0	0
$149$	68.5857	0.460307	0.230153	−	0.973154i	$-0.426077\pi$
$149$	68.5857	0.460307	0.230153	+	0.973154i	$0.426077\pi$
$150$	0	0
$151$	− 298.000i	− 1.97351i	−0.162218	−	0.986755i	$-0.551865\pi$
$151$	− 298.000i	− 1.97351i	0.162218	−	0.986755i	$-0.448135\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 372.322i	− 2.40208i
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−169.000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	342.929	1.98225	0.991123	−	0.132948i	$-0.0424443\pi$
$173$	342.929	1.98225	0.991123	+	0.132948i	$0.0424443\pi$
$174$	0	0
$175$	− 710.000i	− 4.05714i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 274.343i	− 1.53264i	−0.642458	−	0.766321i	$-0.722085\pi$
$179$	− 274.343i	− 1.53264i	0.642458	−	0.766321i	$-0.277915\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	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
$193$	290.000	1.50259	0.751295	−	0.659966i	$-0.229429\pi$
$193$	290.000	1.50259	0.751295	+	0.659966i	$0.229429\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−342.929	−1.74075	−0.870377	−	0.492386i	$-0.836125\pi$
$197$	−342.929	−1.74075	−0.870377	+	0.492386i	$0.836125\pi$
$198$	0	0
$199$	202.000i	1.01508i	0.861630	+	0.507538i	$0.169444\pi$
$199$	202.000i	1.01508i	−0.861630	+	0.507538i	$0.830556\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	293.939i	1.44797i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	380.000	1.75115
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 230.000i	− 1.03139i	−0.856772	−	0.515695i	$-0.827534\pi$
$223$	− 230.000i	− 1.03139i	0.856772	−	0.515695i	$-0.172466\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 293.939i	− 1.29488i	−0.762115	−	0.647442i	$-0.775839\pi$
$227$	− 293.939i	− 1.29488i	0.762115	−	0.647442i	$-0.224161\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	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
$241$	382.000	1.58506	0.792531	−	0.609831i	$-0.208763\pi$
$241$	382.000	1.58506	0.792531	+	0.609831i	$0.208763\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	499.696	2.03958
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	176.363i	0.702642i	0.936255	+	0.351321i	$0.114267\pi$
$251$	176.363i	0.702642i	−0.936255	+	0.351321i	$0.885733\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	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
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−480.000	−1.81132
$266$	0	0
$267$	0	0
$268$	0	0
$269$	323.333	1.20198	0.600990	−	0.799257i	$-0.294773\pi$
$269$	323.333	1.20198	0.600990	+	0.799257i	$0.294773\pi$
$270$	0	0
$271$	58.0000i	0.214022i	0.994258	+	0.107011i	$0.0341280\pi$
$271$	58.0000i	0.214022i	−0.994258	+	0.107011i	$0.965872\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1391.31i	5.05931i
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	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
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	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
$292$	0	0
$293$	440.908	1.50481	0.752403	−	0.658703i	$-0.228895\pi$
$293$	440.908	1.50481	0.752403	+	0.658703i	$0.228895\pi$
$294$	0	0
$295$	− 1152.00i	− 3.90508i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	530.000	1.69329	0.846645	−	0.532158i	$-0.178619\pi$
$313$	530.000	1.69329	0.846645	+	0.532158i	$0.178619\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−538.888	−1.69996	−0.849981	−	0.526814i	$-0.823386\pi$
$317$	−538.888	−1.69996	−0.849981	+	0.526814i	$0.823386\pi$
$318$	0	0
$319$	− 576.000i	− 1.80564i
$320$	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
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−190.000	−0.563798	−0.281899	−	0.959444i	$-0.590964\pi$
$337$	−190.000	−0.563798	−0.281899	+	0.959444i	$0.590964\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−744.645	−2.18371
$342$	0	0
$343$	20.0000i	0.0583090i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	685.857i	1.97653i	0.152738	+	0.988267i	$0.451191\pi$
$347$	685.857i	1.97653i	−0.152738	+	0.988267i	$0.548809\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	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
$361$	361.000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−489.898	−1.34219
$366$	0	0
$367$	710.000i	1.93460i	0.253626	+	0.967302i	$0.418377\pi$
$367$	710.000i	1.93460i	−0.253626	+	0.967302i	$0.581623\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 489.898i	− 1.32048i
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	−1920.00	−4.98701
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−754.443	−1.93944	−0.969721	−	0.244216i	$-0.921469\pi$
$389$	−754.443	−1.93944	−0.969721	+	0.244216i	$0.921469\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	568.282i	1.43869i
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−718.000	−1.75550	−0.877751	−	0.479118i	$-0.840957\pi$
$409$	−718.000	−1.75550	−0.877751	+	0.479118i	$0.840957\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1175.76	2.84686
$414$	0	0
$415$	960.000i	2.31325i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	411.514i	0.982134i	0.871122	+	0.491067i	$0.163393\pi$
$419$	411.514i	0.982134i	−0.871122	+	0.491067i	$0.836607\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	670.000	1.54734	0.773672	−	0.633586i	$-0.218418\pi$
$433$	670.000	1.54734	0.773672	+	0.633586i	$0.218418\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 278.000i	− 0.633257i	−0.948550	−	0.316629i	$-0.897449\pi$
$439$	− 278.000i	− 0.633257i	0.948550	−	0.316629i	$-0.102551\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 881.816i	− 1.99056i	−0.0970655	−	0.995278i	$-0.530946\pi$
$443$	− 881.816i	− 1.99056i	0.0970655	−	0.995278i	$-0.469054\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	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
$457$	530.000	1.15974	0.579869	−	0.814710i	$-0.303104\pi$
$457$	530.000	1.15974	0.579869	+	0.814710i	$0.303104\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−754.443	−1.63654	−0.818268	−	0.574837i	$-0.805065\pi$
$461$	−754.443	−1.63654	−0.818268	+	0.574837i	$0.805065\pi$
$462$	0	0
$463$	250.000i	0.539957i	0.962866	+	0.269978i	$0.0870166\pi$
$463$	250.000i	0.539957i	−0.962866	+	0.269978i	$0.912983\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 685.857i	− 1.46864i	−0.678801	−	0.734322i	$-0.737500\pi$
$467$	− 685.857i	− 1.46864i	0.678801	−	0.734322i	$-0.262500\pi$
$468$	0	0
$469$	0	0
$470$	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
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1861.61	3.83838
$486$	0	0
$487$	− 970.000i	− 1.99179i	−0.0905356	−	0.995893i	$-0.528858\pi$
$487$	− 970.000i	− 1.99179i	0.0905356	−	0.995893i	$-0.471142\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 862.220i	− 1.75605i	−0.478615	−	0.878025i	$-0.658861\pi$
$491$	− 862.220i	− 1.75605i	0.478615	−	0.878025i	$-0.341139\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	672.000	1.33069
$506$	0	0
$507$	0	0
$508$	0	0
$509$	127.373	0.250243	0.125121	−	0.992141i	$-0.460068\pi$
$509$	127.373	0.250243	0.125121	+	0.992141i	$0.460068\pi$
$510$	0	0
$511$	− 500.000i	− 0.978474i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 97.9796i	− 0.190252i
$516$	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	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	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
$532$	0	0
$533$	0	0
$534$	0	0
$535$	1920.00i	3.58879i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 999.392i	− 1.85416i
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−580.000	−1.04882
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−636.867	−1.14339	−0.571694	−	0.820467i	$-0.693714\pi$
$557$	−636.867	−1.14339	−0.571694	+	0.820467i	$0.693714\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 1077.78i	− 1.91434i	−0.289520	−	0.957172i	$-0.593496\pi$
$563$	− 1077.78i	− 1.91434i	0.289520	−	0.957172i	$-0.406504\pi$
$564$	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
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−290.000	−0.502600	−0.251300	−	0.967909i	$-0.580858\pi$
$577$	−290.000	−0.502600	−0.251300	+	0.967909i	$0.580858\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−979.796	−1.68640
$582$	0	0
$583$	960.000i	1.64666i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 783.837i	− 1.33533i	−0.744463	−	0.667663i	$-0.767295\pi$
$587$	− 783.837i	− 1.33533i	0.744463	−	0.667663i	$-0.232705\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	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
$601$	1198.00	1.99334	0.996672	−	0.0815138i	$-0.0259755\pi$
$601$	1198.00	1.99334	0.996672	+	0.0815138i	$0.0259755\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2576.86	4.25928
$606$	0	0
$607$	730.000i	1.20264i	0.799010	+	0.601318i	$0.205357\pi$
$607$	730.000i	1.20264i	−0.799010	+	0.601318i	$0.794643\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	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
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2641.00	4.22560
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1238.00i	1.96197i	0.194096	+	0.980983i	$0.437823\pi$
$631$	1238.00i	1.96197i	−0.194096	+	0.980983i	$0.562177\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2253.53i	3.54887i
$636$	0	0
$637$	0	0
$638$	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
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	−2304.00	−3.55008
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−832.827	−1.27539	−0.637693	−	0.770291i	$-0.720111\pi$
$653$	−832.827	−1.27539	−0.637693	+	0.770291i	$0.720111\pi$
$654$	0	0
$655$	768.000i	1.17252i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1097.37i	− 1.66521i	−0.553869	−	0.832603i	$-0.686849\pi$
$659$	− 1097.37i	− 1.66521i	0.553869	−	0.832603i	$-0.313151\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	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
$673$	190.000	0.282318	0.141159	−	0.989987i	$-0.454917\pi$
$673$	190.000	0.282318	0.141159	+	0.989987i	$0.454917\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	146.969	0.217089	0.108545	−	0.994092i	$-0.465381\pi$
$677$	146.969	0.217089	0.108545	+	0.994092i	$0.465381\pi$
$678$	0	0
$679$	1900.00i	2.79823i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 293.939i	− 0.430364i	−0.976574	−	0.215182i	$-0.930965\pi$
$683$	− 293.939i	− 0.430364i	0.976574	−	0.215182i	$-0.0690345\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1401.11	−1.99873	−0.999364	−	0.0356633i	$-0.988646\pi$
$701$	−1401.11	−1.99873	−0.999364	+	0.0356633i	$0.988646\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	685.857i	0.970095i
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	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
$721$	100.000	0.138696
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2086.97	−2.87857
$726$	0	0
$727$	− 1450.00i	− 1.99450i	−0.0741249	−	0.997249i	$-0.523616\pi$
$727$	− 1450.00i	− 1.99450i	0.0741249	−	0.997249i	$-0.476384\pi$
$728$	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
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−672.000	−0.902013
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1959.59	−2.61628
$750$	0	0
$751$	− 1402.00i	− 1.86684i	−0.358780	−	0.933422i	$-0.616807\pi$
$751$	− 1402.00i	− 1.86684i	0.358780	−	0.933422i	$-0.383193\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2919.79i	3.86727i
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	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
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−862.000	−1.12094	−0.560468	−	0.828176i	$-0.689379\pi$
$769$	−862.000	−1.12094	−0.560468	+	0.828176i	$0.689379\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1028.79	1.33090	0.665450	−	0.746442i	$-0.268240\pi$
$773$	1028.79	1.33090	0.665450	+	0.746442i	$0.268240\pi$
$774$	0	0
$775$	2698.00i	3.48129i
$776$	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
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	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
$797$	930.806	1.16789	0.583944	−	0.811794i	$-0.301509\pi$
$797$	930.806	1.16789	0.583944	+	0.811794i	$0.301509\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	979.796i	1.22017i
$804$	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
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1499.09	1.82593	0.912965	−	0.408039i	$-0.133787\pi$
$821$	1499.09	1.82593	0.912965	+	0.408039i	$0.133787\pi$
$822$	0	0
$823$	470.000i	0.571081i	0.958367	+	0.285541i	$0.0921732\pi$
$823$	470.000i	0.571081i	−0.958367	+	0.285541i	$0.907827\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 587.878i	− 0.710856i	−0.934704	−	0.355428i	$-0.884335\pi$
$827$	− 587.878i	− 0.710856i	0.934704	−	0.355428i	$-0.115665\pi$
$828$	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	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	23.0000	0.0273484
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1655.86	1.95959
$846$	0	0
$847$	2630.00i	3.10508i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	−3360.00	−3.88439
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1136.56	1.30790
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4507.06i	5.15093i
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	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
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−2300.00	−2.58718
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	2688.00i	3.00335i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1116.97i	− 1.24246i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	1920.00	2.10296
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−783.837	−0.854784
$918$	0	0
$919$	662.000i	0.720348i	0.932885	+	0.360174i	$0.117283\pi$
$919$	662.000i	0.720348i	−0.932885	+	0.360174i	$0.882717\pi$
$920$	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
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1490.00	1.59018	0.795091	−	0.606491i	$-0.207423\pi$
$937$	1490.00	1.59018	0.795091	+	0.606491i	$0.207423\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1832.22	1.94710	0.973549	−	0.228480i	$-0.0733757\pi$
$941$	1832.22	1.94710	0.973549	+	0.228480i	$0.0733757\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1371.71i	1.44848i	0.689546	+	0.724242i	$0.257810\pi$
$947$	1371.71i	1.44848i	−0.689546	+	0.724242i	$0.742190\pi$
$948$	0	0
$949$	0	0
$950$	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
$961$	−483.000	−0.502601
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2841.41	−2.94446
$966$	0	0
$967$	− 1910.00i	− 1.97518i	−0.157051	−	0.987590i	$-0.550199\pi$
$967$	− 1910.00i	− 1.97518i	0.157051	−	0.987590i	$-0.449801\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 215.555i	− 0.221993i	−0.993821	−	0.110996i	$-0.964596\pi$
$971$	− 215.555i	− 0.221993i	0.993821	−	0.110996i	$-0.0354042\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	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	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	3360.00	3.41117
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1382.00i	1.39455i	0.716803	+	0.697275i	$0.245605\pi$
$991$	1382.00i	1.39455i	−0.716803	+	0.697275i	$0.754395\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1979.19i	− 1.98913i
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.g.t.1279.1		4
3.2	odd	2	inner	2304.3.g.t.1279.3		4
4.3	odd	2	inner	2304.3.g.t.1279.2		4
8.3	odd	2	inner	2304.3.g.t.1279.4		4
8.5	even	2	inner	2304.3.g.t.1279.3		4
12.11	even	2	inner	2304.3.g.t.1279.4		4
16.3	odd	4		576.3.b.b.415.3	yes	4
16.5	even	4		576.3.b.b.415.2	yes	4
16.11	odd	4		576.3.b.b.415.1	✓	4
16.13	even	4		576.3.b.b.415.4	yes	4
24.5	odd	2	CM	2304.3.g.t.1279.1		4
24.11	even	2	inner	2304.3.g.t.1279.2		4
48.5	odd	4		576.3.b.b.415.4	yes	4
48.11	even	4		576.3.b.b.415.3	yes	4
48.29	odd	4		576.3.b.b.415.2	yes	4
48.35	even	4		576.3.b.b.415.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.3.b.b.415.1	✓	4	16.11	odd	4
576.3.b.b.415.1	✓	4	48.35	even	4
576.3.b.b.415.2	yes	4	16.5	even	4
576.3.b.b.415.2	yes	4	48.29	odd	4
576.3.b.b.415.3	yes	4	16.3	odd	4
576.3.b.b.415.3	yes	4	48.11	even	4
576.3.b.b.415.4	yes	4	16.13	even	4
576.3.b.b.415.4	yes	4	48.5	odd	4
2304.3.g.t.1279.1		4	1.1	even	1	trivial
2304.3.g.t.1279.1		4	24.5	odd	2	CM
2304.3.g.t.1279.2		4	4.3	odd	2	inner
2304.3.g.t.1279.2		4	24.11	even	2	inner
2304.3.g.t.1279.3		4	3.2	odd	2	inner
2304.3.g.t.1279.3		4	8.5	even	2	inner
2304.3.g.t.1279.4		4	8.3	odd	2	inner
2304.3.g.t.1279.4		4	12.11	even	2	inner

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.79796 q^{5} -10.0000i q^{7} +O(q^{10})\)	\(q-9.79796 q^{5} -10.0000i q^{7} +19.5959i q^{11} +71.0000 q^{25} -29.3939 q^{29} +38.0000i q^{31} +97.9796i q^{35} -51.0000 q^{49} +48.9898 q^{53} -192.000i q^{55} +117.576i q^{59} +50.0000 q^{73} +195.959 q^{77} -58.0000i q^{79} -97.9796i q^{83} -190.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 284 q^{25} - 204 q^{49} + 200 q^{73} - 760 q^{97}+O(q^{100}) \) 4 * q + 284 * q^25 - 204 * q^49 + 200 * q^73 - 760 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more