LMFDB - Embedded newform 4608.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4608.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.a (trivial)

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:	yes
Analytic conductor:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Root			$-0.517638$ of defining polynomial
Character	$\chi$	$=$	4608.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.449490 q^{5} -2.04989 q^{7} +O(q^{10})$	$q-0.449490 q^{5} -2.04989 q^{7} -3.46410 q^{11} -4.79796 q^{25} +9.34847 q^{29} -3.60697 q^{31} +0.921404 q^{35} -2.79796 q^{49} +7.55051 q^{53} +1.55708 q^{55} -11.3137 q^{59} +9.79796 q^{73} +7.10102 q^{77} +17.4634 q^{79} +17.3205 q^{83} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5}+O(q^{10}) $ 4 * q + 8 * q^5	$ 4 q + 8 q^{5} + 20 q^{25} + 8 q^{29} + 28 q^{49} + 40 q^{53} + 48 q^{77} - 8 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 20 * q^25 + 8 * q^29 + 28 * q^49 + 40 * q^53 + 48 * q^77 - 8 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−0.449490	−0.201018	−0.100509	−	0.994936i	$-0.532047\pi$
$5$	−0.449490	−0.201018	−0.100509	+	0.994936i	$0.532047\pi$
$6$	0	0
$7$	−2.04989	−0.774785	−0.387392	−	0.921915i	$-0.626624\pi$
$7$	−2.04989	−0.774785	−0.387392	+	0.921915i	$0.626624\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	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		−	1.00000i	$-0.5\pi$
$19$	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
$25$	−4.79796	−0.959592
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.34847	1.73597	0.867984	−	0.496593i	$-0.165416\pi$
$29$	9.34847	1.73597	0.867984	+	0.496593i	$0.165416\pi$
$30$	0	0
$31$	−3.60697	−0.647830	−0.323915	−	0.946086i	$-0.604999\pi$
$31$	−3.60697	−0.647830	−0.323915	+	0.946086i	$0.604999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.921404	0.155746
$36$	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		−	1.00000i	$-0.5\pi$
$43$	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
$49$	−2.79796	−0.399708
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.55051	1.03714	0.518571	−	0.855034i	$-0.326464\pi$
$53$	7.55051	1.03714	0.518571	+	0.855034i	$0.326464\pi$
$54$	0	0
$55$	1.55708	0.209956
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.3137	−1.47292	−0.736460	−	0.676481i	$-0.763504\pi$
$59$	−11.3137	−1.47292	−0.736460	+	0.676481i	$0.763504\pi$
$60$	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		−	1.00000i	$-0.5\pi$
$67$	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
$73$	9.79796	1.14676	0.573382	−	0.819288i	$-0.305631\pi$
$73$	9.79796	1.14676	0.573382	+	0.819288i	$0.305631\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.10102	0.809236
$78$	0	0
$79$	17.4634	1.96478	0.982391	−	0.186834i	$-0.0598227\pi$
$79$	17.4634	1.96478	0.982391	+	0.186834i	$0.0598227\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	17.3205	1.90117	0.950586	−	0.310460i	$-0.100483\pi$
$83$	17.3205	1.90117	0.950586	+	0.310460i	$0.100483\pi$
$84$	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
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.4495	1.63679	0.818393	−	0.574659i	$-0.194865\pi$
$101$	16.4495	1.63679	0.818393	+	0.574659i	$0.194865\pi$
$102$	0	0
$103$	0.492810	0.0485580	0.0242790	−	0.999705i	$-0.492271\pi$
$103$	0.492810	0.0485580	0.0242790	+	0.999705i	$0.492271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.3137	−1.09374	−0.546869	−	0.837218i	$-0.684180\pi$
$107$	−11.3137	−1.09374	−0.546869	+	0.837218i	$0.684180\pi$
$108$	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
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.40408	0.393913
$126$	0	0
$127$	−19.0205	−1.68779	−0.843896	−	0.536507i	$-0.819744\pi$
$127$	−19.0205	−1.68779	−0.843896	+	0.536507i	$0.819744\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	22.6274	1.97697	0.988483	−	0.151330i	$-0.0483556\pi$
$131$	22.6274	1.97697	0.988483	+	0.151330i	$0.0483556\pi$
$132$	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		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.20204	−0.348961
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.1464	1.56854	0.784268	−	0.620422i	$-0.213039\pi$
$149$	19.1464	1.56854	0.784268	+	0.620422i	$0.213039\pi$
$150$	0	0
$151$	15.9063	1.29444	0.647218	−	0.762305i	$-0.275932\pi$
$151$	15.9063	1.29444	0.647218	+	0.762305i	$0.275932\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.62129	0.130225
$156$	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		−	1.00000i	$-0.5\pi$
$163$	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
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	26.2474	1.99556	0.997778	−	0.0666220i	$-0.0212222\pi$
$173$	26.2474	1.99556	0.997778	+	0.0666220i	$0.0212222\pi$
$174$	0	0
$175$	9.83528	0.743477
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.3137	0.845626	0.422813	−	0.906217i	$-0.361043\pi$
$179$	11.3137	0.845626	0.422813	+	0.906217i	$0.361043\pi$
$180$	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		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	9.79796	0.705273	0.352636	−	0.935760i	$-0.385285\pi$
$193$	9.79796	0.705273	0.352636	+	0.935760i	$0.385285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	27.1464	1.93410	0.967051	−	0.254581i	$-0.0819375\pi$
$197$	27.1464	1.93410	0.967051	+	0.254581i	$0.0819375\pi$
$198$	0	0
$199$	−27.2200	−1.92957	−0.964787	−	0.263031i	$-0.915278\pi$
$199$	−27.2200	−1.92957	−0.964787	+	0.263031i	$0.915278\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−19.1633	−1.34500
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.39388	0.501929
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−28.7771	−1.92706	−0.963528	−	0.267608i	$-0.913767\pi$
$223$	−28.7771	−1.92706	−0.963528	+	0.267608i	$0.913767\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.3923	0.689761	0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923	0.689761	0.344881	+	0.938647i	$0.387919\pi$
$228$	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		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	29.3939	1.89343	0.946713	−	0.322078i	$-0.104381\pi$
$241$	29.3939	1.89343	0.946713	+	0.322078i	$0.104381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.25765	0.0803486
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−31.1769	−1.96787	−0.983935	−	0.178529i	$-0.942866\pi$
$251$	−31.1769	−1.96787	−0.983935	+	0.178529i	$0.942866\pi$
$252$	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		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−3.39388	−0.208484
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.6515	0.893320	0.446660	−	0.894704i	$-0.352613\pi$
$269$	14.6515	0.893320	0.446660	+	0.894704i	$0.352613\pi$
$270$	0	0
$271$	32.8769	1.99713	0.998563	−	0.0535825i	$-0.0170640\pi$
$271$	32.8769	1.99713	0.998563	+	0.0535825i	$0.0170640\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	16.6206	1.00226
$276$	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		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−12.0454	−0.703700	−0.351850	−	0.936056i	$-0.614447\pi$
$293$	−12.0454	−0.703700	−0.351850	+	0.936056i	$0.614447\pi$
$294$	0	0
$295$	5.08540	0.296083
$296$	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		−	1.00000i	$-0.5\pi$
$307$	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
$313$	9.79796	0.553813	0.276907	−	0.960897i	$-0.410691\pi$
$313$	9.79796	0.553813	0.276907	+	0.960897i	$0.410691\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.2474	1.92353	0.961764	−	0.273879i	$-0.0883068\pi$
$317$	34.2474	1.92353	0.961764	+	0.273879i	$0.0883068\pi$
$318$	0	0
$319$	−32.3840	−1.81316
$320$	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		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−29.3939	−1.60119	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	−29.3939	−1.60119	−0.800593	+	0.599208i	$0.795482\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12.4949	0.676637
$342$	0	0
$343$	20.0847	1.08447
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.2487	−1.30174	−0.650870	−	0.759190i	$-0.725596\pi$
$347$	−24.2487	−1.30174	−0.650870	+	0.759190i	$0.725596\pi$
$348$	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		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.40408	−0.230520
$366$	0	0
$367$	30.3342	1.58343	0.791715	−	0.610890i	$-0.209188\pi$
$367$	30.3342	1.58343	0.791715	+	0.610890i	$0.209188\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−15.4777	−0.803562
$372$	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		−	1.00000i	$-0.5\pi$
$379$	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
$385$	−3.19184	−0.162671
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.85357	0.246086	0.123043	−	0.992401i	$-0.460735\pi$
$389$	4.85357	0.246086	0.123043	+	0.992401i	$0.460735\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.84961	−0.394957
$396$	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
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	23.1918	1.14120
$414$	0	0
$415$	−7.78539	−0.382170
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.3923	0.507697	0.253849	−	0.967244i	$-0.418303\pi$
$419$	10.3923	0.507697	0.253849	+	0.967244i	$0.418303\pi$
$420$	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		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	6.72112	0.320782	0.160391	−	0.987054i	$-0.448725\pi$
$439$	6.72112	0.320782	0.160391	+	0.987054i	$0.448725\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.1769	1.48126	0.740630	−	0.671913i	$-0.234527\pi$
$443$	31.1769	1.48126	0.740630	+	0.671913i	$0.234527\pi$
$444$	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
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.9444	−0.602880	−0.301440	−	0.953485i	$-0.597467\pi$
$461$	−12.9444	−0.602880	−0.301440	+	0.953485i	$0.597467\pi$
$462$	0	0
$463$	42.6335	1.98135	0.990673	−	0.136260i	$-0.0435083\pi$
$463$	42.6335	1.98135	0.990673	+	0.136260i	$0.0435083\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.3205	−0.801498	−0.400749	−	0.916188i	$-0.631250\pi$
$467$	−17.3205	−0.801498	−0.400749	+	0.916188i	$0.631250\pi$
$468$	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
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.898979	0.0408206
$486$	0	0
$487$	−29.7627	−1.34868	−0.674338	−	0.738422i	$-0.735571\pi$
$487$	−29.7627	−1.34868	−0.674338	+	0.738422i	$0.735571\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−22.6274	−1.02116	−0.510581	−	0.859830i	$-0.670569\pi$
$491$	−22.6274	−1.02116	−0.510581	+	0.859830i	$0.670569\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	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
$505$	−7.39388	−0.329023
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−29.8434	−1.32278	−0.661392	−	0.750040i	$-0.730034\pi$
$509$	−29.8434	−1.32278	−0.661392	+	0.750040i	$0.730034\pi$
$510$	0	0
$511$	−20.0847	−0.888496
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−0.221513	−0.00976103
$516$	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		−	1.00000i	$-0.5\pi$
$523$	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
$533$	0	0
$534$	0	0
$535$	5.08540	0.219861
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.69241	0.417482
$540$	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		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−35.7980	−1.52228
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21.8434	−0.925533	−0.462767	−	0.886480i	$-0.653143\pi$
$557$	−21.8434	−0.925533	−0.462767	+	0.886480i	$0.653143\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	38.1051	1.60594	0.802970	−	0.596020i	$-0.203252\pi$
$563$	38.1051	1.60594	0.802970	+	0.596020i	$0.203252\pi$
$564$	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		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	29.3939	1.22368	0.611842	−	0.790980i	$-0.290429\pi$
$577$	29.3939	1.22368	0.611842	+	0.790980i	$0.290429\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−35.5051	−1.47300
$582$	0	0
$583$	−26.1557	−1.08326
$584$	0	0
$585$	0	0
$586$	0	0
$587$	45.2548	1.86787	0.933933	−	0.357447i	$-0.116353\pi$
$587$	45.2548	1.86787	0.933933	+	0.357447i	$0.116353\pi$
$588$	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		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−48.9898	−1.99834	−0.999168	−	0.0407909i	$-0.987012\pi$
$601$	−48.9898	−1.99834	−0.999168	+	0.0407909i	$0.987012\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.449490	−0.0182744
$606$	0	0
$607$	−46.7333	−1.89684	−0.948422	−	0.317010i	$-0.897321\pi$
$607$	−46.7333	−1.89684	−0.948422	+	0.317010i	$0.897321\pi$
$608$	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		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	22.0102	0.880408
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	31.8912	1.26957	0.634785	−	0.772689i	$-0.281089\pi$
$631$	31.8912	1.26957	0.634785	+	0.772689i	$0.281089\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.54950	0.339276
$636$	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		−	1.00000i	$-0.5\pi$
$643$	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
$649$	39.1918	1.53841
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−21.7526	−0.851243	−0.425622	−	0.904901i	$-0.639945\pi$
$653$	−21.7526	−0.851243	−0.425622	+	0.904901i	$0.639945\pi$
$654$	0	0
$655$	−10.1708	−0.397406
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−45.2548	−1.76288	−0.881439	−	0.472298i	$-0.843425\pi$
$659$	−45.2548	−1.76288	−0.881439	+	0.472298i	$0.843425\pi$
$660$	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
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.7423	1.48899	0.744495	−	0.667628i	$-0.232690\pi$
$677$	38.7423	1.48899	0.744495	+	0.667628i	$0.232690\pi$
$678$	0	0
$679$	4.09978	0.157335
$680$	0	0
$681$	0	0
$682$	0	0
$683$	51.9615	1.98825	0.994126	−	0.108227i	$-0.0345173\pi$
$683$	51.9615	1.98825	0.994126	+	0.108227i	$0.0345173\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−52.9444	−1.99968	−0.999841	−	0.0178345i	$-0.994323\pi$
$701$	−52.9444	−1.99968	−0.999841	+	0.0178345i	$0.994323\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−33.7196	−1.26816
$708$	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		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−1.01021	−0.0376220
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−44.8536	−1.66582
$726$	0	0
$727$	−39.5193	−1.46569	−0.732845	−	0.680395i	$-0.761808\pi$
$727$	−39.5193	−1.46569	−0.732845	+	0.680395i	$0.761808\pi$
$728$	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		−	1.00000i	$-0.5\pi$
$739$	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
$745$	−8.60612	−0.315304
$746$	0	0
$747$	0	0
$748$	0	0
$749$	23.1918	0.847411
$750$	0	0
$751$	45.1762	1.64850	0.824251	−	0.566224i	$-0.191596\pi$
$751$	45.1762	1.64850	0.824251	+	0.566224i	$0.191596\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7.14972	−0.260205
$756$	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
$769$	48.9898	1.76662	0.883309	−	0.468792i	$-0.155311\pi$
$769$	48.9898	1.76662	0.883309	+	0.468792i	$0.155311\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−22.7423	−0.817985	−0.408993	−	0.912538i	$-0.634120\pi$
$773$	−22.7423	−0.817985	−0.408993	+	0.912538i	$0.634120\pi$
$774$	0	0
$775$	17.3061	0.621653
$776$	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		−	1.00000i	$-0.5\pi$
$787$	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
$797$	−50.2474	−1.77986	−0.889928	−	0.456101i	$-0.849246\pi$
$797$	−50.2474	−1.77986	−0.889928	+	0.456101i	$0.849246\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−33.9411	−1.19776
$804$	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		−	1.00000i	$-0.5\pi$
$811$	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
$821$	−11.9546	−0.417218	−0.208609	−	0.977999i	$-0.566894\pi$
$821$	−11.9546	−0.417218	−0.208609	+	0.977999i	$0.566894\pi$
$822$	0	0
$823$	−8.27820	−0.288560	−0.144280	−	0.989537i	$-0.546087\pi$
$823$	−8.27820	−0.288560	−0.144280	+	0.989537i	$0.546087\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	56.5685	1.96708	0.983540	−	0.180688i	$-0.0578324\pi$
$827$	56.5685	1.96708	0.983540	+	0.180688i	$0.0578324\pi$
$828$	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		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	58.3939	2.01358
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.84337	0.201018
$846$	0	0
$847$	−2.04989	−0.0704350
$848$	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		−	1.00000i	$-0.5\pi$
$859$	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
$865$	−11.7980	−0.401143
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−60.4949	−2.05215
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.02788	−0.305198
$876$	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		−	1.00000i	$-0.5\pi$
$883$	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
$889$	38.9898	1.30768
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−5.08540	−0.169986
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−33.7196	−1.12461
$900$	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
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−60.0000	−1.98571
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−46.3837	−1.53172
$918$	0	0
$919$	59.6041	1.96616	0.983078	−	0.183187i	$-0.0586414\pi$
$919$	59.6041	1.96616	0.983078	+	0.183187i	$0.0586414\pi$
$920$	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
$937$	−58.0000	−1.89478	−0.947389	−	0.320085i	$-0.896288\pi$
$937$	−58.0000	−1.89478	−0.947389	+	0.320085i	$0.896288\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−60.9444	−1.98673	−0.993365	−	0.115003i	$-0.963312\pi$
$941$	−60.9444	−1.98673	−0.993365	+	0.115003i	$0.963312\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−56.5685	−1.83823	−0.919115	−	0.393989i	$-0.871095\pi$
$947$	−56.5685	−1.83823	−0.919115	+	0.393989i	$0.871095\pi$
$948$	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
$961$	−17.9898	−0.580316
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.40408	−0.141772
$966$	0	0
$967$	47.3047	1.52122	0.760609	−	0.649211i	$-0.224901\pi$
$967$	47.3047	1.52122	0.760609	+	0.649211i	$0.224901\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.46410	−0.111168	−0.0555842	−	0.998454i	$-0.517702\pi$
$971$	−3.46410	−0.111168	−0.0555842	+	0.998454i	$0.517702\pi$
$972$	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		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−12.2020	−0.388789
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	23.6917	0.752591	0.376296	−	0.926500i	$-0.377198\pi$
$991$	23.6917	0.752591	0.376296	+	0.926500i	$0.377198\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	12.2351	0.387879
$996$	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	4608.2.a.bc.1.1	yes	4
3.2	odd	2		4608.2.a.s.1.3	✓	4
4.3	odd	2	inner	4608.2.a.bc.1.2	yes	4
8.3	odd	2		4608.2.a.s.1.4	yes	4
8.5	even	2		4608.2.a.s.1.3	✓	4
12.11	even	2		4608.2.a.s.1.4	yes	4
16.3	odd	4		4608.2.d.q.2305.5		8
16.5	even	4		4608.2.d.q.2305.4		8
16.11	odd	4		4608.2.d.q.2305.3		8
16.13	even	4		4608.2.d.q.2305.6		8
24.5	odd	2	CM	4608.2.a.bc.1.1	yes	4
24.11	even	2	inner	4608.2.a.bc.1.2	yes	4
48.5	odd	4		4608.2.d.q.2305.6		8
48.11	even	4		4608.2.d.q.2305.5		8
48.29	odd	4		4608.2.d.q.2305.4		8
48.35	even	4		4608.2.d.q.2305.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.a.s.1.3	✓	4	3.2	odd	2
4608.2.a.s.1.3	✓	4	8.5	even	2
4608.2.a.s.1.4	yes	4	8.3	odd	2
4608.2.a.s.1.4	yes	4	12.11	even	2
4608.2.a.bc.1.1	yes	4	1.1	even	1	trivial
4608.2.a.bc.1.1	yes	4	24.5	odd	2	CM
4608.2.a.bc.1.2	yes	4	4.3	odd	2	inner
4608.2.a.bc.1.2	yes	4	24.11	even	2	inner
4608.2.d.q.2305.3		8	16.11	odd	4
4608.2.d.q.2305.3		8	48.35	even	4
4608.2.d.q.2305.4		8	16.5	even	4
4608.2.d.q.2305.4		8	48.29	odd	4
4608.2.d.q.2305.5		8	16.3	odd	4
4608.2.d.q.2305.5		8	48.11	even	4
4608.2.d.q.2305.6		8	16.13	even	4
4608.2.d.q.2305.6		8	48.5	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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.a (trivial)

\(f(q)\)	\(=\)	\(q-0.449490 q^{5} -2.04989 q^{7} +O(q^{10})\)	\(q-0.449490 q^{5} -2.04989 q^{7} -3.46410 q^{11} -4.79796 q^{25} +9.34847 q^{29} -3.60697 q^{31} +0.921404 q^{35} -2.79796 q^{49} +7.55051 q^{53} +1.55708 q^{55} -11.3137 q^{59} +9.79796 q^{73} +7.10102 q^{77} +17.4634 q^{79} +17.3205 q^{83} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5}+O(q^{10}) \) 4 * q + 8 * q^5	\( 4 q + 8 q^{5} + 20 q^{25} + 8 q^{29} + 28 q^{49} + 40 q^{53} + 48 q^{77} - 8 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 20 * q^25 + 8 * q^29 + 28 * q^49 + 40 * q^53 + 48 * q^77 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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