LMFDB - Embedded newform 576.3.b.c.415.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,3,Mod(415,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("576.415");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	576.b (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:	$15.6948632272$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			415.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	576.415
Dual form			576.3.b.c.415.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	− 2.00000i	− 0.285714i	−0.989743	−	0.142857i	$-0.954371\pi$
$7$	− 2.00000i	− 0.285714i	0.989743	−	0.142857i	$-0.0456289\pi$
$8$	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
$13$	− 13.8564i	− 1.06588i	−0.846154	−	0.532939i	$-0.821088\pi$
$13$	− 13.8564i	− 1.06588i	0.846154	−	0.532939i	$-0.178912\pi$
$14$	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
$19$	−27.7128	−1.45857	−0.729285	−	0.684211i	$-0.760147\pi$
$19$	−27.7128	−1.45857	−0.729285	+	0.684211i	$0.760147\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	25.0000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	− 46.0000i	− 1.48387i	−0.670471	−	0.741935i	$-0.733908\pi$
$31$	− 46.0000i	− 1.48387i	0.670471	−	0.741935i	$-0.266092\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 69.2820i	− 1.87249i	−0.351351	−	0.936244i	$-0.614278\pi$
$37$	− 69.2820i	− 1.87249i	0.351351	−	0.936244i	$-0.385722\pi$
$38$	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
$43$	−83.1384	−1.93345	−0.966726	−	0.255814i	$-0.917657\pi$
$43$	−83.1384	−1.93345	−0.966726	+	0.255814i	$0.917657\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	45.0000	0.918367
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	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
$61$	− 96.9948i	− 1.59008i	−0.606557	−	0.795040i	$-0.707450\pi$
$61$	− 96.9948i	− 1.59008i	0.606557	−	0.795040i	$-0.292550\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−55.4256	−0.827248	−0.413624	−	0.910448i	$-0.635737\pi$
$67$	−55.4256	−0.827248	−0.413624	+	0.910448i	$0.635737\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	46.0000	0.630137	0.315068	−	0.949069i	$-0.397973\pi$
$73$	46.0000	0.630137	0.315068	+	0.949069i	$0.397973\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 142.000i	− 1.79747i	−0.438494	−	0.898734i	$-0.644488\pi$
$79$	− 142.000i	− 1.79747i	0.438494	−	0.898734i	$-0.355512\pi$
$80$	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		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−27.7128	−0.304536
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000	0.0206186	0.0103093	−	0.999947i	$-0.496718\pi$
$97$	2.00000	0.0206186	0.0103093	+	0.999947i	$0.496718\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	194.000i	1.88350i	0.336321	+	0.941748i	$0.390817\pi$
$103$	194.000i	1.88350i	−0.336321	+	0.941748i	$0.609183\pi$
$104$	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
$109$	41.5692i	0.381369i	0.981651	+	0.190684i	$0.0610707\pi$
$109$	41.5692i	0.381369i	−0.981651	+	0.190684i	$0.938929\pi$
$110$	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
$121$	−121.000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 146.000i	− 1.14961i	−0.818292	−	0.574803i	$-0.805079\pi$
$127$	− 146.000i	− 1.14961i	0.818292	−	0.574803i	$-0.194921\pi$
$128$	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
$133$	55.4256i	0.416734i
$134$	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
$139$	277.128	1.99373	0.996864	−	0.0791367i	$-0.0252164\pi$
$139$	277.128	1.99373	0.996864	+	0.0791367i	$0.0252164\pi$
$140$	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
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	286.000i	1.89404i	0.321175	+	0.947020i	$0.395922\pi$
$151$	286.000i	1.89404i	−0.321175	+	0.947020i	$0.604078\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	290.985i	1.85340i	0.375796	+	0.926702i	$0.377369\pi$
$157$	290.985i	1.85340i	−0.375796	+	0.926702i	$0.622631\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	193.990	1.19012	0.595060	−	0.803681i	$-0.297128\pi$
$163$	193.990	1.19012	0.595060	+	0.803681i	$0.297128\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−23.0000	−0.136095
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	− 50.0000i	− 0.285714i
$176$	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
$181$	180.133i	0.995212i	0.867403	+	0.497606i	$0.165787\pi$
$181$	180.133i	0.995212i	−0.867403	+	0.497606i	$0.834213\pi$
$182$	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
$193$	−382.000	−1.97927	−0.989637	−	0.143590i	$-0.954135\pi$
$193$	−382.000	−1.97927	−0.989637	+	0.143590i	$0.954135\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	386.000i	1.93970i	0.243706	+	0.969849i	$0.421637\pi$
$199$	386.000i	1.93970i	−0.243706	+	0.969849i	$0.578363\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	387.979	1.83876	0.919382	−	0.393365i	$-0.128689\pi$
$211$	387.979	1.83876	0.919382	+	0.393365i	$0.128689\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−92.0000	−0.423963
$218$	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
$229$	457.261i	1.99677i	0.0567686	+	0.998387i	$0.481920\pi$
$229$	457.261i	1.99677i	−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
$241$	286.000	1.18672	0.593361	−	0.804936i	$-0.297801\pi$
$241$	286.000	1.18672	0.593361	+	0.804936i	$0.297801\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	384.000i	1.55466i
$248$	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
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−138.564	−0.534996
$260$	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
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	− 242.000i	− 0.892989i	−0.894786	−	0.446494i	$-0.852672\pi$
$271$	− 242.000i	− 0.892989i	0.894786	−	0.446494i	$-0.147328\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 540.400i	− 1.95090i	−0.220217	−	0.975451i	$-0.570676\pi$
$277$	− 540.400i	− 1.95090i	0.220217	−	0.975451i	$-0.429324\pi$
$278$	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
$283$	332.554	1.17510	0.587551	−	0.809187i	$-0.300092\pi$
$283$	332.554	1.17510	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
$292$	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
$301$	166.277i	0.552415i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−498.831	−1.62486	−0.812428	−	0.583062i	$-0.801855\pi$
$307$	−498.831	−1.62486	−0.812428	+	0.583062i	$0.801855\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	142.000	0.453674	0.226837	−	0.973933i	$-0.427162\pi$
$313$	142.000	0.453674	0.226837	+	0.973933i	$0.427162\pi$
$314$	0	0
$315$	0	0
$316$	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
$323$	0	0
$324$	0	0
$325$	− 346.410i	− 1.06588i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−554.256	−1.67449	−0.837245	−	0.546828i	$-0.815835\pi$
$331$	−554.256	−1.67449	−0.837245	+	0.546828i	$0.815835\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	482.000	1.43027	0.715134	−	0.698988i	$-0.246366\pi$
$337$	482.000	1.43027	0.715134	+	0.698988i	$0.246366\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 188.000i	− 0.548105i
$344$	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
$349$	− 484.974i	− 1.38961i	−0.719198	−	0.694805i	$-0.755490\pi$
$349$	− 484.974i	− 1.38961i	0.719198	−	0.694805i	$-0.244510\pi$
$350$	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
$361$	407.000	1.12742
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 718.000i	− 1.95640i	−0.207657	−	0.978202i	$-0.566584\pi$
$367$	− 718.000i	− 1.95640i	0.207657	−	0.978202i	$-0.433416\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	263.272i	0.705822i	0.935657	+	0.352911i	$0.114808\pi$
$373$	263.272i	0.705822i	−0.935657	+	0.352911i	$0.885192\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−304.841	−0.804330	−0.402165	−	0.915567i	$-0.631742\pi$
$379$	−304.841	−0.804330	−0.402165	+	0.915567i	$0.631742\pi$
$380$	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
$388$	0	0
$389$	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
$397$	− 706.677i	− 1.78004i	−0.455919	−	0.890021i	$-0.650689\pi$
$397$	− 706.677i	− 1.78004i	0.455919	−	0.890021i	$-0.349311\pi$
$398$	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
$403$	−637.395	−1.58162
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−626.000	−1.53056	−0.765281	−	0.643696i	$-0.777400\pi$
$409$	−626.000	−1.53056	−0.765281	+	0.643696i	$0.777400\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	− 762.102i	− 1.81022i	−0.425178	−	0.905110i	$-0.639789\pi$
$421$	− 762.102i	− 1.81022i	0.425178	−	0.905110i	$-0.360211\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−193.990	−0.454308
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	862.000	1.99076	0.995381	−	0.0960028i	$-0.0306058\pi$
$433$	862.000	1.99076	0.995381	+	0.0960028i	$0.0306058\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 94.0000i	− 0.214123i	−0.994252	−	0.107062i	$-0.965856\pi$
$439$	− 94.0000i	− 0.214123i	0.994252	−	0.107062i	$-0.0341442\pi$
$440$	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
$448$	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
$457$	814.000	1.78118	0.890591	−	0.454805i	$-0.150291\pi$
$457$	814.000	1.78118	0.890591	+	0.454805i	$0.150291\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	526.000i	1.13607i	0.823005	+	0.568035i	$0.192296\pi$
$463$	526.000i	1.13607i	−0.823005	+	0.568035i	$0.807704\pi$
$464$	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
$469$	110.851i	0.236357i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−692.820	−1.45857
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	−960.000	−1.99584
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 962.000i	− 1.97536i	−0.156489	−	0.987680i	$-0.550018\pi$
$487$	− 962.000i	− 1.97536i	0.156489	−	0.987680i	$-0.449982\pi$
$488$	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
$496$	0	0
$497$	0	0
$498$	0	0
$499$	997.661	1.99932	0.999661	−	0.0260521i	$-0.00829358\pi$
$499$	997.661	1.99932	0.999661	+	0.0260521i	$0.00829358\pi$
$500$	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
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	− 92.0000i	− 0.180039i
$512$	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
$523$	360.267	0.688846	0.344423	−	0.938815i	$-0.388075\pi$
$523$	360.267	0.688846	0.344423	+	0.938815i	$0.388075\pi$
$524$	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
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	318.697i	0.589089i	0.955638	+	0.294545i	$0.0951680\pi$
$541$	318.697i	0.589089i	−0.955638	+	0.294545i	$0.904832\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−969.948	−1.77321	−0.886607	−	0.462523i	$-0.846944\pi$
$547$	−969.948	−1.77321	−0.886607	+	0.462523i	$0.846944\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−284.000	−0.513562
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	1152.00i	2.06082i
$560$	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		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	720.533	1.26188	0.630940	−	0.775832i	$-0.282670\pi$
$571$	720.533	1.26188	0.630940	+	0.775832i	$0.282670\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−962.000	−1.66724	−0.833622	−	0.552335i	$-0.813737\pi$
$577$	−962.000	−1.66724	−0.833622	+	0.552335i	$0.813737\pi$
$578$	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
$589$	1274.79i	2.16433i
$590$	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
$601$	−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
$604$	0	0
$605$	0	0
$606$	0	0
$607$	814.000i	1.34102i	0.741900	+	0.670511i	$0.233925\pi$
$607$	814.000i	1.34102i	−0.741900	+	0.670511i	$0.766075\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	484.974i	0.791149i	0.918434	+	0.395574i	$0.129454\pi$
$613$	484.974i	0.791149i	−0.918434	+	0.395574i	$0.870546\pi$
$614$	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
$619$	1219.36	1.96989	0.984947	−	0.172859i	$-0.0553006\pi$
$619$	1219.36	1.96989	0.984947	+	0.172859i	$0.0553006\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 674.000i	− 1.06815i	−0.845438	−	0.534073i	$-0.820661\pi$
$631$	− 674.000i	− 1.06815i	0.845438	−	0.534073i	$-0.179339\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 623.538i	− 0.978867i
$638$	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
$643$	1247.08	1.93947	0.969733	−	0.244168i	$-0.0785148\pi$
$643$	1247.08	1.93947	0.969733	+	0.244168i	$0.0785148\pi$
$644$	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
$652$	0	0
$653$	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
$661$	1316.36i	1.99147i	0.0922844	+	0.995733i	$0.470583\pi$
$661$	1316.36i	1.99147i	−0.0922844	+	0.995733i	$0.529417\pi$
$662$	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
$673$	−1154.00	−1.71471	−0.857355	−	0.514725i	$-0.827894\pi$
$673$	−1154.00	−1.71471	−0.857355	+	0.514725i	$0.827894\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	− 4.00000i	− 0.00589102i
$680$	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
$688$	0	0
$689$	0	0
$690$	0	0
$691$	415.692	0.601581	0.300790	−	0.953690i	$-0.402750\pi$
$691$	415.692	0.601581	0.300790	+	0.953690i	$0.402750\pi$
$692$	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
$703$	1920.00i	2.73115i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1066.94i	1.50486i	0.658674	+	0.752428i	$0.271118\pi$
$709$	1066.94i	1.50486i	−0.658674	+	0.752428i	$0.728882\pi$
$710$	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
$721$	388.000	0.538141
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 482.000i	− 0.662999i	−0.943455	−	0.331499i	$-0.892446\pi$
$727$	− 482.000i	− 0.662999i	0.943455	−	0.331499i	$-0.107554\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1039.23i	1.41778i	0.705321	+	0.708888i	$0.250803\pi$
$733$	1039.23i	1.41778i	−0.705321	+	0.708888i	$0.749197\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	831.384	1.12501	0.562506	−	0.826793i	$-0.309837\pi$
$739$	831.384	1.12501	0.562506	+	0.826793i	$0.309837\pi$
$740$	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
$751$	1202.00i	1.60053i	0.599645	+	0.800266i	$0.295309\pi$
$751$	1202.00i	1.60053i	−0.599645	+	0.800266i	$0.704691\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1260.93i	− 1.66570i	−0.553501	−	0.832849i	$-0.686708\pi$
$757$	− 1260.93i	− 1.66570i	0.553501	−	0.832849i	$-0.313292\pi$
$758$	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
$763$	83.1384	0.108963
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1534.00	−1.99480	−0.997399	−	0.0720749i	$-0.977038\pi$
$769$	−1534.00	−1.99480	−0.997399	+	0.0720749i	$0.977038\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	− 1150.00i	− 1.48387i
$776$	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
$787$	−193.990	−0.246493	−0.123246	−	0.992376i	$-0.539331\pi$
$787$	−193.990	−0.246493	−0.123246	+	0.992376i	$0.539331\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1344.00	−1.69483
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	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
$811$	−581.969	−0.717594	−0.358797	−	0.933416i	$-0.616813\pi$
$811$	−581.969	−0.717594	−0.358797	+	0.933416i	$0.616813\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2304.00	2.82007
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 1058.00i	− 1.28554i	−0.766059	−	0.642770i	$-0.777785\pi$
$823$	− 1058.00i	− 1.28554i	0.766059	−	0.642770i	$-0.222215\pi$
$824$	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
$829$	1593.49i	1.92218i	0.276236	+	0.961090i	$0.410913\pi$
$829$	1593.49i	1.92218i	−0.276236	+	0.961090i	$0.589087\pi$
$830$	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
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	242.000i	0.285714i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 401.836i	− 0.471085i	−0.971864	−	0.235543i	$-0.924313\pi$
$853$	− 401.836i	− 0.471085i	0.971864	−	0.235543i	$-0.0756867\pi$
$854$	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
$859$	969.948	1.12916	0.564580	−	0.825378i	$-0.309038\pi$
$859$	969.948	1.12916	0.564580	+	0.825378i	$0.309038\pi$
$860$	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
$868$	0	0
$869$	0	0
$870$	0	0
$871$	768.000i	0.881745i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1648.91i	− 1.88017i	−0.340935	−	0.940087i	$-0.610744\pi$
$877$	− 1648.91i	− 1.88017i	0.340935	−	0.940087i	$-0.389256\pi$
$878$	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
$883$	−471.118	−0.533542	−0.266771	−	0.963760i	$-0.585957\pi$
$883$	−471.118	−0.533542	−0.266771	+	0.963760i	$0.585957\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−292.000	−0.328459
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	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
$907$	−1801.33	−1.98603	−0.993017	−	0.117971i	$-0.962361\pi$
$907$	−1801.33	−1.98603	−0.993017	+	0.117971i	$0.962361\pi$
$908$	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
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 866.000i	− 0.942329i	−0.882045	−	0.471164i	$-0.843834\pi$
$919$	− 866.000i	− 0.942329i	0.882045	−	0.471164i	$-0.156166\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 1732.05i	− 1.87249i
$926$	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
$931$	−1247.08	−1.33950
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1198.00	1.27855	0.639274	−	0.768979i	$-0.279235\pi$
$937$	1198.00	1.27855	0.639274	+	0.768979i	$0.279235\pi$
$938$	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
$949$	− 637.395i	− 0.671649i
$950$	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
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1534.00i	− 1.58635i	−0.608994	−	0.793175i	$-0.708427\pi$
$967$	− 1534.00i	− 1.58635i	0.608994	−	0.793175i	$-0.291573\pi$
$968$	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
$973$	− 554.256i	− 0.569636i
$974$	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
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 46.0000i	− 0.0464178i	−0.999731	−	0.0232089i	$-0.992612\pi$
$991$	− 46.0000i	− 0.0464178i	0.999731	−	0.0232089i	$-0.00738828\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 623.538i	− 0.625415i	−0.949850	−	0.312707i	$-0.898764\pi$
$997$	− 623.538i	− 0.625415i	0.949850	−	0.312707i	$-0.101236\pi$
$998$	0	0
$999$	0	0

Twists

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

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

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	576.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{7} +O(q^{10})\)	\(q-2.00000i q^{7} -13.8564i q^{13} -27.7128 q^{19} +25.0000 q^{25} -46.0000i q^{31} -69.2820i q^{37} -83.1384 q^{43} +45.0000 q^{49} -96.9948i q^{61} -55.4256 q^{67} +46.0000 q^{73} -142.000i q^{79} -27.7128 q^{91} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 100 q^{25} + 180 q^{49} + 184 q^{73} + 8 q^{97}+O(q^{100}) \) 4 * q + 100 * q^25 + 180 * q^49 + 184 * q^73 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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