LMFDB - Embedded newform 1216.3.g.a.417.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(417,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.417");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.g (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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			417.1
Root			$2.17945 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1216.417
Dual form			1216.3.g.a.417.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.35890i q^{5} -13.0767 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-4.35890i q^{5} -13.0767 q^{7} -9.00000 q^{9} +3.00000i q^{11} +15.0000 q^{17} -19.0000i q^{19} -34.8712 q^{23} +6.00000 q^{25} +57.0000i q^{35} +85.0000i q^{43} +39.2301i q^{45} +56.6657 q^{47} +122.000 q^{49} +13.0767 q^{55} +65.3835i q^{61} +117.690 q^{63} +25.0000 q^{73} -39.2301i q^{77} +81.0000 q^{81} +90.0000i q^{83} -65.3835i q^{85} -82.8191 q^{95} -27.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9}+O(q^{10}) $ 4 * q - 36 * q^9	$ 4 q - 36 q^{9} + 60 q^{17} + 24 q^{25} + 488 q^{49} + 100 q^{73} + 324 q^{81}+O(q^{100}) $ 4 * q - 36 * q^9 + 60 * q^17 + 24 * q^25 + 488 * q^49 + 100 * q^73 + 324 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	− 4.35890i	− 0.871780i	−0.900000	−	0.435890i	$-0.856434\pi$
$5$	− 4.35890i	− 0.871780i	0.900000	−	0.435890i	$-0.143566\pi$
$6$	0	0
$7$	−13.0767	−1.86810	−0.934050	−	0.357143i	$-0.883751\pi$
$7$	−13.0767	−1.86810	−0.934050	+	0.357143i	$0.883751\pi$
$8$	0	0
$9$	−9.00000	−1.00000
$10$	0	0
$11$	3.00000i	0.272727i	0.990659	+	0.136364i	$0.0435416\pi$
$11$	3.00000i	0.272727i	−0.990659	+	0.136364i	$0.956458\pi$
$12$	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
$17$	15.0000	0.882353	0.441176	−	0.897420i	$-0.354561\pi$
$17$	15.0000	0.882353	0.441176	+	0.897420i	$0.354561\pi$
$18$	0	0
$19$	− 19.0000i	− 1.00000i
$19$	− 19.0000i	− 1.00000i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−34.8712	−1.51614	−0.758069	−	0.652174i	$-0.773857\pi$
$23$	−34.8712	−1.51614	−0.758069	+	0.652174i	$0.773857\pi$
$24$	0	0
$25$	6.00000	0.240000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	57.0000i	1.62857i
$36$	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	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	85.0000i	1.97674i	0.152055	+	0.988372i	$0.451411\pi$
$43$	85.0000i	1.97674i	−0.152055	+	0.988372i	$0.548589\pi$
$44$	0	0
$45$	39.2301i	0.871780i
$46$	0	0
$47$	56.6657	1.20565	0.602826	−	0.797872i	$-0.294041\pi$
$47$	56.6657	1.20565	0.602826	+	0.797872i	$0.294041\pi$
$48$	0	0
$49$	122.000	2.48980
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	13.0767	0.237758
$56$	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
$61$	65.3835i	1.07186i	0.844262	+	0.535930i	$0.180039\pi$
$61$	65.3835i	1.07186i	−0.844262	+	0.535930i	$0.819961\pi$
$62$	0	0
$63$	117.690	1.86810
$64$	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	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	25.0000	0.342466	0.171233	−	0.985231i	$-0.445225\pi$
$73$	25.0000	0.342466	0.171233	+	0.985231i	$0.445225\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 39.2301i	− 0.509482i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	81.0000	1.00000
$82$	0	0
$83$	90.0000i	1.08434i	0.840270	+	0.542169i	$0.182397\pi$
$83$	90.0000i	1.08434i	−0.840270	+	0.542169i	$0.817603\pi$
$84$	0	0
$85$	− 65.3835i	− 0.769217i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−82.8191	−0.871780
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	− 27.0000i	− 0.272727i
$100$	0	0
$101$	− 174.356i	− 1.72630i	−0.504950	−	0.863148i	$-0.668489\pi$
$101$	− 174.356i	− 1.72630i	0.504950	−	0.863148i	$-0.331511\pi$
$102$	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		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	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	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	152.000i	1.32174i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−196.150	−1.64832
$120$	0	0
$121$	112.000	0.925620
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 135.126i	− 1.08101i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	213.000i	1.62595i	0.582296	+	0.812977i	$0.302155\pi$
$131$	213.000i	1.62595i	−0.582296	+	0.812977i	$0.697845\pi$
$132$	0	0
$133$	248.457i	1.86810i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	255.000	1.86131	0.930657	−	0.365893i	$-0.119236\pi$
$137$	255.000	1.86131	0.930657	+	0.365893i	$0.119236\pi$
$138$	0	0
$139$	197.000i	1.41727i	0.705577	+	0.708633i	$0.250688\pi$
$139$	197.000i	1.41727i	−0.705577	+	0.708633i	$0.749312\pi$
$140$	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
$149$	− 239.739i	− 1.60899i	−0.593960	−	0.804495i	$-0.702436\pi$
$149$	− 239.739i	− 1.60899i	0.593960	−	0.804495i	$-0.297564\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−135.000	−0.882353
$154$	0	0
$155$	0	0
$156$	0	0
$157$	313.841i	1.99899i	0.0318471	+	0.999493i	$0.489861\pi$
$157$	313.841i	1.99899i	−0.0318471	+	0.999493i	$0.510139\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	456.000	2.83230
$162$	0	0
$163$	− 250.000i	− 1.53374i	−0.641801	−	0.766871i	$-0.721813\pi$
$163$	− 250.000i	− 1.53374i	0.641801	−	0.766871i	$-0.278187\pi$
$164$	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
$171$	171.000i	1.00000i
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−78.4602	−0.448344
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	45.0000i	0.240642i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−370.506	−1.93982	−0.969912	−	0.243455i	$-0.921719\pi$
$191$	−370.506	−1.93982	−0.969912	+	0.243455i	$0.921719\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	383.583i	1.94712i	0.228426	+	0.973561i	$0.426642\pi$
$197$	383.583i	1.94712i	−0.228426	+	0.973561i	$0.573358\pi$
$198$	0	0
$199$	326.917	1.64280	0.821401	−	0.570352i	$-0.193193\pi$
$199$	326.917	1.64280	0.821401	+	0.570352i	$0.193193\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	313.841	1.51614
$208$	0	0
$209$	57.0000	0.272727
$210$	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
$215$	370.506	1.72329
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−54.0000	−0.240000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	457.684i	1.99862i	0.0371179	+	0.999311i	$0.488182\pi$
$229$	457.684i	1.99862i	−0.0371179	+	0.999311i	$0.511818\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−465.000	−1.99571	−0.997854	−	0.0654770i	$-0.979143\pi$
$233$	−465.000	−1.99571	−0.997854	+	0.0654770i	$0.979143\pi$
$234$	0	0
$235$	− 247.000i	− 1.05106i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−152.561	−0.638332	−0.319166	−	0.947699i	$-0.603403\pi$
$239$	−152.561	−0.638332	−0.319166	+	0.947699i	$0.603403\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 531.786i	− 2.17055i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 27.0000i	− 0.107570i	−0.998553	−	0.0537849i	$-0.982871\pi$
$251$	− 27.0000i	− 0.107570i	0.998553	−	0.0537849i	$-0.0171285\pi$
$252$	0	0
$253$	− 104.614i	− 0.413492i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	335.635	1.27618	0.638090	−	0.769962i	$-0.279725\pi$
$263$	335.635	1.27618	0.638090	+	0.769962i	$0.279725\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−523.068	−1.93014	−0.965070	−	0.261993i	$-0.915620\pi$
$271$	−523.068	−1.93014	−0.965070	+	0.261993i	$0.915620\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.0000i	0.0654545i
$276$	0	0
$277$	− 143.844i	− 0.519291i	−0.965704	−	0.259646i	$-0.916394\pi$
$277$	− 143.844i	− 0.519291i	0.965704	−	0.259646i	$-0.0836057\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 395.000i	− 1.39576i	−0.716215	−	0.697880i	$-0.754127\pi$
$283$	− 395.000i	− 1.39576i	0.716215	−	0.697880i	$-0.245873\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−64.0000	−0.221453
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 1111.52i	− 3.69275i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	285.000	0.934426
$306$	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
$311$	−152.561	−0.490551	−0.245276	−	0.969453i	$-0.578878\pi$
$311$	−152.561	−0.490551	−0.245276	+	0.969453i	$0.578878\pi$
$312$	0	0
$313$	−590.000	−1.88498	−0.942492	−	0.334229i	$-0.891524\pi$
$313$	−590.000	−1.88498	−0.942492	+	0.334229i	$0.891524\pi$
$314$	0	0
$315$	− 513.000i	− 1.62857i
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 285.000i	− 0.882353i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−741.000	−2.25228
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−954.599	−2.78309
$344$	0	0
$345$	0	0
$346$	0	0
$347$	675.000i	1.94524i	0.232391	+	0.972622i	$0.425345\pi$
$347$	675.000i	1.94524i	−0.232391	+	0.972622i	$0.574655\pi$
$348$	0	0
$349$	457.684i	1.31142i	0.755014	+	0.655708i	$0.227630\pi$
$349$	457.684i	1.31142i	−0.755014	+	0.655708i	$0.772370\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	510.000	1.44476	0.722380	−	0.691497i	$-0.243048\pi$
$353$	510.000	1.44476	0.722380	+	0.691497i	$0.243048\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	675.629	1.88198	0.940988	−	0.338440i	$-0.109899\pi$
$359$	675.629	1.88198	0.940988	+	0.338440i	$0.109899\pi$
$360$	0	0
$361$	−361.000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 108.972i	− 0.298555i
$366$	0	0
$367$	732.295	1.99535	0.997677	−	0.0681199i	$-0.0217000\pi$
$367$	732.295	1.99535	0.997677	+	0.0681199i	$0.0217000\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	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	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	−171.000	−0.444156
$386$	0	0
$387$	− 765.000i	− 1.97674i
$388$	0	0
$389$	762.807i	1.96094i	0.196658	+	0.980472i	$0.436991\pi$
$389$	762.807i	1.96094i	−0.196658	+	0.980472i	$0.563009\pi$
$390$	0	0
$391$	−523.068	−1.33777
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	274.611i	0.691714i	0.938287	+	0.345857i	$0.112412\pi$
$397$	274.611i	0.691714i	−0.938287	+	0.345857i	$0.887588\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 353.071i	− 0.871780i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	392.301	0.945303
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 762.000i	− 1.81862i	−0.416124	−	0.909308i	$-0.636612\pi$
$419$	− 762.000i	− 1.81862i	0.416124	−	0.909308i	$-0.363388\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	−509.991	−1.20565
$424$	0	0
$425$	90.0000	0.211765
$426$	0	0
$427$	− 855.000i	− 2.00234i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	662.553i	1.51614i
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−1098.00	−2.48980
$442$	0	0
$443$	− 45.0000i	− 0.101580i	−0.998709	−	0.0507901i	$-0.983826\pi$
$443$	− 45.0000i	− 0.101580i	0.998709	−	0.0507901i	$-0.0161739\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−625.000	−1.36761	−0.683807	−	0.729663i	$-0.739677\pi$
$457$	−625.000	−1.36761	−0.683807	+	0.729663i	$0.739677\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	806.396i	1.74923i	0.484816	+	0.874616i	$0.338887\pi$
$461$	806.396i	1.74923i	−0.484816	+	0.874616i	$0.661113\pi$
$462$	0	0
$463$	536.145	1.15798	0.578990	−	0.815335i	$-0.303447\pi$
$463$	536.145	1.15798	0.578990	+	0.815335i	$0.303447\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	915.000i	1.95931i	0.200677	+	0.979657i	$0.435686\pi$
$467$	915.000i	1.95931i	−0.200677	+	0.979657i	$0.564314\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−255.000	−0.539112
$474$	0	0
$475$	− 114.000i	− 0.240000i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	174.356	0.364000	0.182000	−	0.983299i	$-0.441743\pi$
$479$	174.356	0.364000	0.182000	+	0.983299i	$0.441743\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 918.000i	− 1.86965i	−0.355104	−	0.934827i	$-0.615554\pi$
$491$	− 918.000i	− 1.86965i	0.355104	−	0.934827i	$-0.384446\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−117.690	−0.237758
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 523.000i	− 1.04810i	−0.851689	−	0.524048i	$-0.824421\pi$
$499$	− 523.000i	− 1.04810i	0.851689	−	0.524048i	$-0.175579\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	383.583	0.762591	0.381295	−	0.924453i	$-0.375478\pi$
$503$	383.583	0.762591	0.381295	+	0.924453i	$0.375478\pi$
$504$	0	0
$505$	−760.000	−1.50495
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−326.917	−0.639760
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	169.997i	0.328814i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	687.000	1.29868
$530$	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
$539$	366.000i	0.679035i
$540$	0	0
$541$	− 980.752i	− 1.81285i	−0.422366	−	0.906425i	$-0.638800\pi$
$541$	− 980.752i	− 1.81285i	0.422366	−	0.906425i	$-0.361200\pi$
$542$	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
$549$	− 588.451i	− 1.07186i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	204.868i	0.367807i	0.982944	+	0.183903i	$0.0588733\pi$
$557$	204.868i	0.367807i	−0.982944	+	0.183903i	$0.941127\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1059.21	−1.86810
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	458.000i	0.802102i	0.916056	+	0.401051i	$0.131355\pi$
$571$	458.000i	0.802102i	−0.916056	+	0.401051i	$0.868645\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−209.227	−0.363873
$576$	0	0
$577$	1145.00	1.98440	0.992201	−	0.124648i	$-0.0397801\pi$
$577$	1145.00	1.98440	0.992201	+	0.124648i	$0.0397801\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1176.90i	− 2.02565i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1125.00i	1.91652i	0.285890	+	0.958262i	$0.407711\pi$
$587$	1125.00i	1.91652i	−0.285890	+	0.958262i	$0.592289\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.0000	0.0505902	0.0252951	−	0.999680i	$-0.491947\pi$
$593$	30.0000	0.0505902	0.0252951	+	0.999680i	$0.491947\pi$
$594$	0	0
$595$	855.000i	1.43697i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 488.197i	− 0.806937i
$606$	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
$613$	− 1189.98i	− 1.94124i	−0.240620	−	0.970619i	$-0.577351\pi$
$613$	− 1189.98i	− 1.94124i	0.240620	−	0.970619i	$-0.422649\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1065.00	1.72609	0.863047	−	0.505124i	$-0.168553\pi$
$617$	1065.00	1.72609	0.863047	+	0.505124i	$0.168553\pi$
$618$	0	0
$619$	− 662.000i	− 1.06947i	−0.845021	−	0.534733i	$-0.820412\pi$
$619$	− 662.000i	− 1.06947i	0.845021	−	0.534733i	$-0.179588\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−439.000	−0.702400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−719.218	−1.13981	−0.569904	−	0.821712i	$-0.693019\pi$
$631$	−719.218	−1.13981	−0.569904	+	0.821712i	$0.693019\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	− 1115.00i	− 1.73406i	−0.498257	−	0.867030i	$-0.666026\pi$
$643$	− 1115.00i	− 1.73406i	0.498257	−	0.867030i	$-0.333974\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	815.114	1.25984	0.629918	−	0.776662i	$-0.283088\pi$
$647$	815.114	1.25984	0.629918	+	0.776662i	$0.283088\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1251.00i	1.91578i	0.287136	+	0.957890i	$0.407297\pi$
$653$	1251.00i	1.91578i	−0.287136	+	0.957890i	$0.592703\pi$
$654$	0	0
$655$	928.445	1.41747
$656$	0	0
$657$	−225.000	−0.342466
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1083.00	1.62857
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−196.150	−0.292326
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	− 1111.52i	− 1.62266i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 157.000i	− 0.227207i	−0.993526	−	0.113603i	$-0.963761\pi$
$691$	− 157.000i	− 0.227207i	0.993526	−	0.113603i	$-0.0362393\pi$
$692$	0	0
$693$	353.071i	0.509482i
$694$	0	0
$695$	858.703	1.23554
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	871.780i	1.24362i	0.783167	+	0.621812i	$0.213603\pi$
$701$	871.780i	1.24362i	−0.783167	+	0.621812i	$0.786397\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2280.00i	3.22489i
$708$	0	0
$709$	523.068i	0.737754i	0.929478	+	0.368877i	$0.120258\pi$
$709$	523.068i	0.737754i	−0.929478	+	0.368877i	$0.879742\pi$
$710$	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
$719$	−1067.93	−1.48530	−0.742650	−	0.669680i	$-0.766431\pi$
$719$	−1067.93	−1.48530	−0.742650	+	0.669680i	$0.766431\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1451.51	1.99658	0.998290	−	0.0584594i	$-0.0186188\pi$
$727$	1451.51	1.99658	0.998290	+	0.0584594i	$0.0186188\pi$
$728$	0	0
$729$	−729.000	−1.00000
$730$	0	0
$731$	1275.00i	1.74419i
$732$	0	0
$733$	732.295i	0.999038i	0.866303	+	0.499519i	$0.166490\pi$
$733$	732.295i	0.999038i	−0.866303	+	0.499519i	$0.833510\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	547.000i	0.740189i	0.928994	+	0.370095i	$0.120675\pi$
$739$	547.000i	0.740189i	−0.928994	+	0.370095i	$0.879325\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−1045.00	−1.40268
$746$	0	0
$747$	− 810.000i	− 1.08434i
$748$	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
$757$	1294.59i	1.71016i	0.518494	+	0.855081i	$0.326493\pi$
$757$	1294.59i	1.71016i	−0.518494	+	0.855081i	$0.673507\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1503.00	1.97503	0.987516	−	0.157516i	$-0.0503486\pi$
$761$	1503.00	1.97503	0.987516	+	0.157516i	$0.0503486\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	588.451i	0.769217i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1063.00	−1.38231	−0.691157	−	0.722704i	$-0.742899\pi$
$769$	−1063.00	−1.38231	−0.691157	+	0.722704i	$0.742899\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	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
$785$	1368.00	1.74268
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	849.985	1.06381
$800$	0	0
$801$	0	0
$802$	0	0
$803$	75.0000i	0.0933998i
$804$	0	0
$805$	− 1987.66i	− 2.46914i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1593.00	1.96910	0.984549	−	0.175110i	$-0.0560282\pi$
$809$	1593.00	1.96910	0.984549	+	0.175110i	$0.0560282\pi$
$810$	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
$815$	−1089.72	−1.33709
$816$	0	0
$817$	1615.00	1.97674
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1155.11i	1.40695i	0.710719	+	0.703476i	$0.248370\pi$
$821$	1155.11i	1.40695i	−0.710719	+	0.703476i	$0.751630\pi$
$822$	0	0
$823$	−509.991	−0.619673	−0.309837	−	0.950790i	$-0.600274\pi$
$823$	−509.991	−0.619673	−0.309837	+	0.950790i	$0.600274\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1830.00	2.19688
$834$	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
$845$	736.654i	0.871780i
$846$	0	0
$847$	−1464.59	−1.72915
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1359.98i	1.59435i	0.603751	+	0.797173i	$0.293672\pi$
$853$	1359.98i	1.59435i	−0.603751	+	0.797173i	$0.706328\pi$
$854$	0	0
$855$	745.372	0.871780
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1493.00i	1.73807i	0.494753	+	0.869034i	$0.335259\pi$
$859$	1493.00i	1.73807i	−0.494753	+	0.869034i	$0.664741\pi$
$860$	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
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1767.00i	2.01943i
$876$	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
$881$	537.000	0.609535	0.304767	−	0.952427i	$-0.401421\pi$
$881$	537.000	0.609535	0.304767	+	0.952427i	$0.401421\pi$
$882$	0	0
$883$	835.000i	0.945640i	0.881159	+	0.472820i	$0.156764\pi$
$883$	835.000i	0.945640i	−0.881159	+	0.472820i	$0.843236\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	243.000i	0.272727i
$892$	0	0
$893$	− 1076.65i	− 1.20565i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	1569.20i	1.72630i
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−270.000	−0.295728
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2785.34i	− 3.03744i
$918$	0	0
$919$	523.068	0.569171	0.284585	−	0.958651i	$-0.408144\pi$
$919$	523.068	0.569171	0.284585	+	0.958651i	$0.408144\pi$
$920$	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
$929$	−642.000	−0.691066	−0.345533	−	0.938407i	$-0.612302\pi$
$929$	−642.000	−0.691066	−0.345533	+	0.938407i	$0.612302\pi$
$930$	0	0
$931$	− 2318.00i	− 2.48980i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	196.150	0.209787
$936$	0	0
$937$	335.000	0.357524	0.178762	−	0.983892i	$-0.442791\pi$
$937$	335.000	0.357524	0.178762	+	0.983892i	$0.442791\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1830.00i	1.93242i	0.257760	+	0.966209i	$0.417016\pi$
$947$	1830.00i	1.93242i	−0.257760	+	0.966209i	$0.582984\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	1615.00i	1.69110i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3334.56	−3.47712
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−732.295	−0.757285	−0.378643	−	0.925543i	$-0.623609\pi$
$967$	−732.295	−0.757285	−0.378643	+	0.925543i	$0.623609\pi$
$968$	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
$973$	− 2576.11i	− 2.64759i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	1672.00	1.69746
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 2964.05i	− 2.99702i
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1425.00i	− 1.43216i
$996$	0	0
$997$	274.611i	0.275437i	0.990471	+	0.137718i	$0.0439769\pi$
$997$	274.611i	0.275437i	−0.990471	+	0.137718i	$0.956023\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.g.a.417.1	✓	4
4.3	odd	2	inner	1216.3.g.a.417.2	yes	4
8.3	odd	2	inner	1216.3.g.a.417.4	yes	4
8.5	even	2	inner	1216.3.g.a.417.3	yes	4
19.18	odd	2	CM	1216.3.g.a.417.1	✓	4
76.75	even	2	inner	1216.3.g.a.417.2	yes	4
152.37	odd	2	inner	1216.3.g.a.417.3	yes	4
152.75	even	2	inner	1216.3.g.a.417.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.g.a.417.1	✓	4	1.1	even	1	trivial
1216.3.g.a.417.1	✓	4	19.18	odd	2	CM
1216.3.g.a.417.2	yes	4	4.3	odd	2	inner
1216.3.g.a.417.2	yes	4	76.75	even	2	inner
1216.3.g.a.417.3	yes	4	8.5	even	2	inner
1216.3.g.a.417.3	yes	4	152.37	odd	2	inner
1216.3.g.a.417.4	yes	4	8.3	odd	2	inner
1216.3.g.a.417.4	yes	4	152.75	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.35890i q^{5} -13.0767 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-4.35890i q^{5} -13.0767 q^{7} -9.00000 q^{9} +3.00000i q^{11} +15.0000 q^{17} -19.0000i q^{19} -34.8712 q^{23} +6.00000 q^{25} +57.0000i q^{35} +85.0000i q^{43} +39.2301i q^{45} +56.6657 q^{47} +122.000 q^{49} +13.0767 q^{55} +65.3835i q^{61} +117.690 q^{63} +25.0000 q^{73} -39.2301i q^{77} +81.0000 q^{81} +90.0000i q^{83} -65.3835i q^{85} -82.8191 q^{95} -27.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9}+O(q^{10}) \) 4 * q - 36 * q^9	\( 4 q - 36 q^{9} + 60 q^{17} + 24 q^{25} + 488 q^{49} + 100 q^{73} + 324 q^{81}+O(q^{100}) \) 4 * q - 36 * q^9 + 60 * q^17 + 24 * q^25 + 488 * q^49 + 100 * q^73 + 324 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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