LMFDB - Embedded newform 2304.2.f.f.1151.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(1151,2304)]

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, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2304.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.f (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:	$18.3975326257$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1151.3
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2304.1151
Dual form			2304.2.f.f.1151.4

$q$-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 $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

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

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
$5$	4.24264	1.89737	0.948683	−	0.316228i	$-0.102416\pi$
$5$	4.24264	1.89737	0.948683	+	0.316228i	$0.102416\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.24264i	1.02899i	0.857493	+	0.514496i	$0.172021\pi$
$17$	4.24264i	1.02899i	−0.857493	+	0.514496i	$0.827979\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	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
$25$	13.0000	2.60000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.24264	0.787839	0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264	0.787839	0.393919	+	0.919145i	$0.371119\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 12.7279i	− 1.98777i	−0.110432	−	0.993884i	$-0.535223\pi$
$41$	− 12.7279i	− 1.98777i	0.110432	−	0.993884i	$-0.464777\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	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
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.7279	1.74831	0.874157	−	0.485643i	$-0.161414\pi$
$53$	12.7279	1.74831	0.874157	+	0.485643i	$0.161414\pi$
$54$	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
$61$	10.0000i	1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$	10.0000i	1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 16.9706i	− 2.10494i
$66$	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
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	18.0000i	1.95237i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.24264i	0.449719i	0.974391	+	0.224860i	$0.0721923\pi$
$89$	4.24264i	0.449719i	−0.974391	+	0.224860i	$0.927808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.7279	−1.26648	−0.633238	−	0.773957i	$-0.718274\pi$
$101$	−12.7279	−1.26648	−0.633238	+	0.773957i	$0.718274\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	20.0000i	1.91565i	0.287348	+	0.957826i	$0.407226\pi$
$109$	20.0000i	1.91565i	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 21.2132i	− 1.99557i	−0.0665190	−	0.997785i	$-0.521189\pi$
$113$	− 21.2132i	− 1.99557i	0.0665190	−	0.997785i	$-0.478811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	33.9411	3.03579
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 21.2132i	− 1.81237i	−0.422885	−	0.906183i	$-0.638983\pi$
$137$	− 21.2132i	− 1.81237i	0.422885	−	0.906183i	$-0.361017\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	18.0000	1.49482
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.24264	−0.347571	−0.173785	−	0.984784i	$-0.555600\pi$
$149$	−4.24264	−0.347571	−0.173785	+	0.984784i	$0.555600\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.0000i	1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$	22.0000i	1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	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
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−21.2132	−1.61281	−0.806405	−	0.591364i	$-0.798590\pi$
$173$	−21.2132	−1.61281	−0.806405	+	0.591364i	$0.798590\pi$
$174$	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
$181$	− 20.0000i	− 1.48659i	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	− 20.0000i	− 1.48659i	0.668965	−	0.743294i	$-0.266738\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.48528i	0.623850i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.2132	1.51138	0.755689	−	0.654931i	$-0.227302\pi$
$197$	21.2132	1.51138	0.755689	+	0.654931i	$0.227302\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 54.0000i	− 3.77152i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	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
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.9706	1.14156
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	4.00000i	0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.00000i	0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	29.6985i	1.94561i	0.231621	+	0.972806i	$0.425597\pi$
$233$	29.6985i	1.94561i	−0.231621	+	0.972806i	$0.574403\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	29.6985	1.89737
$246$	0	0
$247$	0	0
$248$	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
$256$	0	0
$257$	21.2132i	1.32324i	0.749838	+	0.661622i	$0.230131\pi$
$257$	21.2132i	1.32324i	−0.749838	+	0.661622i	$0.769869\pi$
$258$	0	0
$259$	0	0
$260$	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
$265$	54.0000	3.31719
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.24264	−0.258678	−0.129339	−	0.991600i	$-0.541286\pi$
$269$	−4.24264	−0.258678	−0.129339	+	0.991600i	$0.541286\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.0000i	1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$	28.0000i	1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 29.6985i	− 1.77166i	−0.464007	−	0.885832i	$-0.653589\pi$
$281$	− 29.6985i	− 1.77166i	0.464007	−	0.885832i	$-0.346411\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−21.2132	−1.23929	−0.619644	−	0.784883i	$-0.712723\pi$
$293$	−21.2132	−1.23929	−0.619644	+	0.784883i	$0.712723\pi$
$294$	0	0
$295$	0	0
$296$	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
$305$	42.4264i	2.42933i
$306$	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
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.24264	−0.238290	−0.119145	−	0.992877i	$-0.538015\pi$
$317$	−4.24264	−0.238290	−0.119145	+	0.992877i	$0.538015\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 52.0000i	− 2.88444i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	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
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	10.0000i	0.535288i	0.963518	+	0.267644i	$0.0862451\pi$
$349$	10.0000i	0.535288i	−0.963518	+	0.267644i	$0.913755\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.7279i	0.677439i	0.940887	+	0.338719i	$0.109994\pi$
$353$	12.7279i	0.677439i	−0.940887	+	0.338719i	$0.890006\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−67.8823	−3.55312
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 16.9706i	− 0.874028i
$378$	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
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−38.1838	−1.93599	−0.967997	−	0.250962i	$-0.919253\pi$
$389$	−38.1838	−1.93599	−0.967997	+	0.250962i	$0.919253\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 38.0000i	− 1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	− 38.0000i	− 1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 29.6985i	− 1.48307i	−0.670913	−	0.741536i	$-0.734098\pi$
$401$	− 29.6985i	− 1.48307i	0.670913	−	0.741536i	$-0.265902\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−40.0000	−1.97787	−0.988936	−	0.148340i	$-0.952607\pi$
$409$	−40.0000	−1.97787	−0.988936	+	0.148340i	$0.952607\pi$
$410$	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	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	28.0000i	1.36464i	0.731055	+	0.682318i	$0.239028\pi$
$421$	28.0000i	1.36464i	−0.731055	+	0.682318i	$0.760972\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	55.1543i	2.67538i
$426$	0	0
$427$	0	0
$428$	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
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	18.0000i	0.853282i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 38.1838i	− 1.80200i	−0.433816	−	0.901002i	$-0.642833\pi$
$449$	− 38.1838i	− 1.80200i	0.433816	−	0.901002i	$-0.357167\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.7279	−0.592798	−0.296399	−	0.955064i	$-0.595786\pi$
$461$	−12.7279	−0.592798	−0.296399	+	0.955064i	$0.595786\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	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		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−33.9411	−1.54119
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	18.0000i	0.810679i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	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
$505$	−54.0000	−2.40297
$506$	0	0
$507$	0	0
$508$	0	0
$509$	38.1838	1.69247	0.846233	−	0.532813i	$-0.178865\pi$
$509$	38.1838	1.69247	0.846233	+	0.532813i	$0.178865\pi$
$510$	0	0
$511$	0	0
$512$	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
$521$	12.7279i	0.557620i	0.960346	+	0.278810i	$0.0899400\pi$
$521$	12.7279i	0.557620i	−0.960346	+	0.278810i	$0.910060\pi$
$522$	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
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−50.9117	−2.20523
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	20.0000i	0.859867i	0.902861	+	0.429934i	$0.141463\pi$
$541$	20.0000i	0.859867i	−0.902861	+	0.429934i	$0.858537\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	84.8528i	3.63470i
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−46.6690	−1.97743	−0.988716	−	0.149805i	$-0.952135\pi$
$557$	−46.6690	−1.97743	−0.988716	+	0.149805i	$0.952135\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	− 90.0000i	− 3.78633i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	46.6690i	1.95647i	0.207504	+	0.978234i	$0.433466\pi$
$569$	46.6690i	1.95647i	−0.207504	+	0.978234i	$0.566534\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.2132i	0.871122i	0.900159	+	0.435561i	$0.143450\pi$
$593$	21.2132i	0.871122i	−0.900159	+	0.435561i	$0.856550\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	46.6690	1.89737
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 34.0000i	− 1.37325i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	− 34.0000i	− 1.37325i	0.727013	−	0.686624i	$-0.240908\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.24264i	0.170802i	0.996347	+	0.0854011i	$0.0272172\pi$
$617$	4.24264i	0.170802i	−0.996347	+	0.0854011i	$0.972783\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	79.0000	3.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.48528	−0.338330
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 28.0000i	− 1.10940i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 29.6985i	− 1.17302i	−0.809942	−	0.586510i	$-0.800502\pi$
$641$	− 29.6985i	− 1.17302i	0.809942	−	0.586510i	$-0.199498\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	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
$652$	0	0
$653$	12.7279	0.498082	0.249041	−	0.968493i	$-0.419885\pi$
$653$	12.7279	0.498082	0.249041	+	0.968493i	$0.419885\pi$
$654$	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
$661$	50.0000i	1.94477i	0.233373	+	0.972387i	$0.425024\pi$
$661$	50.0000i	1.94477i	−0.233373	+	0.972387i	$0.574976\pi$
$662$	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
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.1838	1.46752	0.733761	−	0.679408i	$-0.237763\pi$
$677$	38.1838	1.46752	0.733761	+	0.679408i	$0.237763\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	− 90.0000i	− 3.43872i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 50.9117i	− 1.93958i
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	54.0000	2.04540
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−29.6985	−1.12170	−0.560848	−	0.827919i	$-0.689525\pi$
$701$	−29.6985	−1.12170	−0.560848	+	0.827919i	$0.689525\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 44.0000i	− 1.65245i	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	− 44.0000i	− 1.65245i	0.563337	−	0.826227i	$-0.309517\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	55.1543	2.04838
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	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
$745$	−18.0000	−0.659469
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	52.0000i	1.88997i	0.327111	+	0.944986i	$0.393925\pi$
$757$	52.0000i	1.88997i	−0.327111	+	0.944986i	$0.606075\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	55.1543i	1.99934i	0.0256326	+	0.999671i	$0.491840\pi$
$761$	55.1543i	1.99934i	−0.0256326	+	0.999671i	$0.508160\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−55.1543	−1.98376	−0.991882	−	0.127164i	$-0.959412\pi$
$773$	−55.1543	−1.98376	−0.991882	+	0.127164i	$0.959412\pi$
$774$	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
$785$	93.3381i	3.33138i
$786$	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
$793$	40.0000	1.42044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.2132	−0.751410	−0.375705	−	0.926739i	$-0.622599\pi$
$797$	−21.2132	−0.751410	−0.375705	+	0.926739i	$0.622599\pi$
$798$	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
$809$	− 46.6690i	− 1.64080i	−0.571793	−	0.820398i	$-0.693752\pi$
$809$	− 46.6690i	− 1.64080i	0.571793	−	0.820398i	$-0.306248\pi$
$810$	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
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	55.1543	1.92490	0.962450	−	0.271460i	$-0.0875065\pi$
$821$	55.1543	1.92490	0.962450	+	0.271460i	$0.0875065\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	20.0000i	0.694629i	0.937749	+	0.347314i	$0.112906\pi$
$829$	20.0000i	0.694629i	−0.937749	+	0.347314i	$0.887094\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	29.6985i	1.02899i
$834$	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
$841$	−11.0000	−0.379310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.7279	−0.437854
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 46.0000i	− 1.57501i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	− 46.0000i	− 1.57501i	0.616308	−	0.787505i	$-0.288628\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 46.6690i	− 1.59418i	−0.603858	−	0.797092i	$-0.706370\pi$
$857$	− 46.6690i	− 1.59418i	0.603858	−	0.797092i	$-0.293630\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	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
$865$	−90.0000	−3.06009
$866$	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
$875$	0	0
$876$	0	0
$877$	58.0000i	1.95852i	0.202606	+	0.979260i	$0.435059\pi$
$877$	58.0000i	1.95852i	−0.202606	+	0.979260i	$0.564941\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 12.7279i	− 0.428815i	−0.976744	−	0.214407i	$-0.931218\pi$
$881$	− 12.7279i	− 0.428815i	0.976744	−	0.214407i	$-0.0687820\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	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
$891$	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
$901$	54.0000i	1.79900i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 84.8528i	− 2.82060i
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	26.0000i	0.854875i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.24264i	0.139197i	0.997575	+	0.0695983i	$0.0221717\pi$
$929$	4.24264i	0.139197i	−0.997575	+	0.0695983i	$0.977828\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	55.1543	1.79798	0.898990	−	0.437969i	$-0.144302\pi$
$941$	55.1543	1.79798	0.898990	+	0.437969i	$0.144302\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	64.0000i	2.07753i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 21.2132i	− 0.687163i	−0.939123	−	0.343582i	$-0.888360\pi$
$953$	− 21.2132i	− 0.687163i	0.939123	−	0.343582i	$-0.111640\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	59.3970	1.91206
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 38.1838i	− 1.22161i	−0.791782	−	0.610803i	$-0.790847\pi$
$977$	− 38.1838i	− 1.22161i	0.791782	−	0.610803i	$-0.209153\pi$
$978$	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
$985$	90.0000	2.86764
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	62.0000i	1.96356i	0.190022	+	0.981780i	$0.439144\pi$
$997$	62.0000i	1.96356i	−0.190022	+	0.981780i	$0.560856\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.f.f.1151.3		4
3.2	odd	2	inner	2304.2.f.f.1151.1		4
4.3	odd	2	CM	2304.2.f.f.1151.3		4
8.3	odd	2	inner	2304.2.f.f.1151.2		4
8.5	even	2	inner	2304.2.f.f.1151.2		4
12.11	even	2	inner	2304.2.f.f.1151.1		4
16.3	odd	4		144.2.c.a.143.1	✓	2
16.5	even	4		576.2.c.a.575.2		2
16.11	odd	4		576.2.c.a.575.2		2
16.13	even	4		144.2.c.a.143.1	✓	2
24.5	odd	2	inner	2304.2.f.f.1151.4		4
24.11	even	2	inner	2304.2.f.f.1151.4		4
48.5	odd	4		576.2.c.a.575.1		2
48.11	even	4		576.2.c.a.575.1		2
48.29	odd	4		144.2.c.a.143.2	yes	2
48.35	even	4		144.2.c.a.143.2	yes	2
80.3	even	4		3600.2.o.a.3599.4		4
80.13	odd	4		3600.2.o.a.3599.4		4
80.19	odd	4		3600.2.h.b.1151.1		2
80.29	even	4		3600.2.h.b.1151.1		2
80.67	even	4		3600.2.o.a.3599.1		4
80.77	odd	4		3600.2.o.a.3599.1		4
112.13	odd	4		7056.2.h.b.4607.2		2
112.83	even	4		7056.2.h.b.4607.2		2
144.13	even	12		1296.2.s.h.431.1		4
144.29	odd	12		1296.2.s.h.863.1		4
144.61	even	12		1296.2.s.h.863.2		4
144.67	odd	12		1296.2.s.h.431.1		4
144.77	odd	12		1296.2.s.h.431.2		4
144.83	even	12		1296.2.s.h.863.1		4
144.115	odd	12		1296.2.s.h.863.2		4
144.131	even	12		1296.2.s.h.431.2		4
240.29	odd	4		3600.2.h.b.1151.2		2
240.77	even	4		3600.2.o.a.3599.2		4
240.83	odd	4		3600.2.o.a.3599.3		4
240.173	even	4		3600.2.o.a.3599.3		4
240.179	even	4		3600.2.h.b.1151.2		2
240.227	odd	4		3600.2.o.a.3599.2		4
336.83	odd	4		7056.2.h.b.4607.1		2
336.125	even	4		7056.2.h.b.4607.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.c.a.143.1	✓	2	16.3	odd	4
144.2.c.a.143.1	✓	2	16.13	even	4
144.2.c.a.143.2	yes	2	48.29	odd	4
144.2.c.a.143.2	yes	2	48.35	even	4
576.2.c.a.575.1		2	48.5	odd	4
576.2.c.a.575.1		2	48.11	even	4
576.2.c.a.575.2		2	16.5	even	4
576.2.c.a.575.2		2	16.11	odd	4
1296.2.s.h.431.1		4	144.13	even	12
1296.2.s.h.431.1		4	144.67	odd	12
1296.2.s.h.431.2		4	144.77	odd	12
1296.2.s.h.431.2		4	144.131	even	12
1296.2.s.h.863.1		4	144.29	odd	12
1296.2.s.h.863.1		4	144.83	even	12
1296.2.s.h.863.2		4	144.61	even	12
1296.2.s.h.863.2		4	144.115	odd	12
2304.2.f.f.1151.1		4	3.2	odd	2	inner
2304.2.f.f.1151.1		4	12.11	even	2	inner
2304.2.f.f.1151.2		4	8.3	odd	2	inner
2304.2.f.f.1151.2		4	8.5	even	2	inner
2304.2.f.f.1151.3		4	1.1	even	1	trivial
2304.2.f.f.1151.3		4	4.3	odd	2	CM
2304.2.f.f.1151.4		4	24.5	odd	2	inner
2304.2.f.f.1151.4		4	24.11	even	2	inner
3600.2.h.b.1151.1		2	80.19	odd	4
3600.2.h.b.1151.1		2	80.29	even	4
3600.2.h.b.1151.2		2	240.29	odd	4
3600.2.h.b.1151.2		2	240.179	even	4
3600.2.o.a.3599.1		4	80.67	even	4
3600.2.o.a.3599.1		4	80.77	odd	4
3600.2.o.a.3599.2		4	240.77	even	4
3600.2.o.a.3599.2		4	240.227	odd	4
3600.2.o.a.3599.3		4	240.83	odd	4
3600.2.o.a.3599.3		4	240.173	even	4
3600.2.o.a.3599.4		4	80.3	even	4
3600.2.o.a.3599.4		4	80.13	odd	4
7056.2.h.b.4607.1		2	336.83	odd	4
7056.2.h.b.4607.1		2	336.125	even	4
7056.2.h.b.4607.2		2	112.13	odd	4
7056.2.h.b.4607.2		2	112.83	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+4.24264 q^{5} +O(q^{10})\)	\(q+4.24264 q^{5} -4.00000i q^{13} +4.24264i q^{17} +13.0000 q^{25} +4.24264 q^{29} +2.00000i q^{37} -12.7279i q^{41} +7.00000 q^{49} +12.7279 q^{53} +10.0000i q^{61} -16.9706i q^{65} -16.0000 q^{73} +18.0000i q^{85} +4.24264i q^{89} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 52 q^{25} + 28 q^{49} - 64 q^{73} - 32 q^{97}+O(q^{100}) \) 4 * q + 52 * q^25 + 28 * q^49 - 64 * q^73 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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