LMFDB - Embedded newform 1216.2.b.c.607.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(607,1216)]

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([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("1216.607");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.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:	$9.70980888579$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			607.4
Root			$-2.17945 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1216.607
Dual form			1216.2.b.c.607.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.35890i q^{5} +3.00000i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+4.35890i q^{5} +3.00000i q^{7} +3.00000 q^{9} -4.35890 q^{11} +7.00000 q^{17} +4.35890 q^{19} +4.00000i q^{23} -14.0000 q^{25} -13.0767 q^{35} -13.0767 q^{43} +13.0767i q^{45} -13.0000i q^{47} -2.00000 q^{49} -19.0000i q^{55} +4.35890i q^{61} +9.00000i q^{63} -11.0000 q^{73} -13.0767i q^{77} +9.00000 q^{81} +8.71780 q^{83} +30.5123i q^{85} +19.0000i q^{95} -13.0767 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} + 28 q^{17} - 56 q^{25} - 8 q^{49} - 44 q^{73} + 36 q^{81}+O(q^{100}) $ 4 * q + 12 * q^9 + 28 * q^17 - 56 * q^25 - 8 * q^49 - 44 * q^73 + 36 * 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	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	4.35890i	1.94936i	0.223607	+	0.974679i	$0.428217\pi$
$5$	4.35890i	1.94936i	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−4.35890	−1.31426	−0.657129	−	0.753778i	$-0.728229\pi$
$11$	−4.35890	−1.31426	−0.657129	+	0.753778i	$0.728229\pi$
$12$	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
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	0	0
$19$	4.35890	1.00000
$19$	4.35890	1.00000
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	−14.0000	−2.80000
$26$	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		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−13.0767	−2.21037
$36$	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
$43$	−13.0767	−1.99418	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$	−13.0767	−1.99418	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	0	0
$45$	13.0767i	1.94936i
$46$	0	0
$47$	− 13.0000i	− 1.89624i	−0.317905	−	0.948122i	$-0.602979\pi$
$47$	− 13.0000i	− 1.89624i	0.317905	−	0.948122i	$-0.397021\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	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
$55$	− 19.0000i	− 2.56196i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	4.35890i	0.558100i	0.960277	+	0.279050i	$0.0900195\pi$
$61$	4.35890i	0.558100i	−0.960277	+	0.279050i	$0.909981\pi$
$62$	0	0
$63$	9.00000i	1.13389i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 13.0767i	− 1.49023i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	8.71780	0.956903	0.478451	−	0.878114i	$-0.341198\pi$
$83$	8.71780	0.956903	0.478451	+	0.878114i	$0.341198\pi$
$84$	0	0
$85$	30.5123i	3.30952i
$86$	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
$95$	19.0000i	1.94936i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	−13.0767	−1.31426
$100$	0	0
$101$	17.4356i	1.73491i	0.497519	+	0.867453i	$0.334245\pi$
$101$	17.4356i	1.73491i	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	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
$115$	−17.4356	−1.62588
$116$	0	0
$117$	0	0
$118$	0	0
$119$	21.0000i	1.92507i
$120$	0	0
$121$	8.00000	0.727273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 39.2301i	− 3.50885i
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	21.7945	1.90419	0.952097	−	0.305796i	$-0.0989227\pi$
$131$	21.7945	1.90419	0.952097	+	0.305796i	$0.0989227\pi$
$132$	0	0
$133$	13.0767i	1.13389i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	23.0000	1.96502	0.982511	−	0.186203i	$-0.0596182\pi$
$137$	23.0000	1.96502	0.982511	+	0.186203i	$0.0596182\pi$
$138$	0	0
$139$	21.7945	1.84858	0.924292	−	0.381685i	$-0.124656\pi$
$139$	21.7945	1.84858	0.924292	+	0.381685i	$0.124656\pi$
$140$	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
$149$	− 21.7945i	− 1.78547i	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	− 21.7945i	− 1.78547i	0.450578	−	0.892737i	$-0.351218\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	21.0000	1.69775
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 17.4356i	− 1.39151i	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	− 17.4356i	− 1.39151i	0.718278	−	0.695756i	$-0.244931\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	−8.71780	−0.682831	−0.341415	−	0.939913i	$-0.610906\pi$
$163$	−8.71780	−0.682831	−0.341415	+	0.939913i	$0.610906\pi$
$164$	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
$171$	13.0767	1.00000
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	− 42.0000i	− 3.17490i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	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
$187$	−30.5123	−2.23128
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.0000i	1.23008i	0.788497	+	0.615038i	$0.210860\pi$
$191$	17.0000i	1.23008i	−0.788497	+	0.615038i	$0.789140\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.4356i	1.24223i	0.783718	+	0.621117i	$0.213321\pi$
$197$	17.4356i	1.24223i	−0.783718	+	0.621117i	$0.786679\pi$
$198$	0	0
$199$	25.0000i	1.77220i	0.463491	+	0.886102i	$0.346597\pi$
$199$	25.0000i	1.77220i	−0.463491	+	0.886102i	$0.653403\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.0000i	0.834058i
$208$	0	0
$209$	−19.0000	−1.31426
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 57.0000i	− 3.88737i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−42.0000	−2.80000
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 21.7945i	− 1.44022i	−0.693860	−	0.720110i	$-0.744091\pi$
$229$	− 21.7945i	− 1.44022i	0.693860	−	0.720110i	$-0.255909\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	0	0
$235$	56.6657	3.69646
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 5.00000i	− 0.323423i	−0.986838	−	0.161712i	$-0.948299\pi$
$239$	− 5.00000i	− 0.323423i	0.986838	−	0.161712i	$-0.0517014\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 8.71780i	− 0.556960i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.7945	1.37566	0.687828	−	0.725874i	$-0.258564\pi$
$251$	21.7945	1.37566	0.687828	+	0.725874i	$0.258564\pi$
$252$	0	0
$253$	− 17.4356i	− 1.09617i
$254$	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
$263$	11.0000i	0.678289i	0.940734	+	0.339145i	$0.110138\pi$
$263$	11.0000i	0.678289i	−0.940734	+	0.339145i	$0.889862\pi$
$264$	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
$271$	− 20.0000i	− 1.21491i	−0.794353	−	0.607457i	$-0.792190\pi$
$271$	− 20.0000i	− 1.21491i	0.794353	−	0.607457i	$-0.207810\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	61.0246	3.67992
$276$	0	0
$277$	4.35890i	0.261901i	0.991389	+	0.130950i	$0.0418029\pi$
$277$	4.35890i	0.261901i	−0.991389	+	0.130950i	$0.958197\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−13.0767	−0.777329	−0.388664	−	0.921379i	$-0.627063\pi$
$283$	−13.0767	−0.777329	−0.388664	+	0.921379i	$0.627063\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	32.0000	1.88235
$290$	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
$301$	− 39.2301i	− 2.26118i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−19.0000	−1.08794
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	35.0000i	1.98467i	0.123585	+	0.992334i	$0.460561\pi$
$311$	35.0000i	1.98467i	−0.123585	+	0.992334i	$0.539439\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	−39.2301	−2.21037
$316$	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
$323$	30.5123	1.69775
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	39.0000	2.15014
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	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
$343$	15.0000i	0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.35890	−0.233998	−0.116999	−	0.993132i	$-0.537327\pi$
$347$	−4.35890	−0.233998	−0.116999	+	0.993132i	$0.537327\pi$
$348$	0	0
$349$	13.0767i	0.699980i	0.936754	+	0.349990i	$0.113815\pi$
$349$	13.0767i	0.699980i	−0.936754	+	0.349990i	$0.886185\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 31.0000i	− 1.63612i	−0.575135	−	0.818059i	$-0.695050\pi$
$359$	− 31.0000i	− 1.63612i	0.575135	−	0.818059i	$-0.304950\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 47.9479i	− 2.50971i
$366$	0	0
$367$	28.0000i	1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$	28.0000i	1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$	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	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	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
$385$	57.0000	2.90499
$386$	0	0
$387$	−39.2301	−1.99418
$388$	0	0
$389$	− 30.5123i	− 1.54703i	−0.633775	−	0.773517i	$-0.718496\pi$
$389$	− 30.5123i	− 1.54703i	0.633775	−	0.773517i	$-0.281504\pi$
$390$	0	0
$391$	28.0000i	1.41602i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	39.2301i	1.96890i	0.175660	+	0.984451i	$0.443794\pi$
$397$	39.2301i	1.96890i	−0.175660	+	0.984451i	$0.556206\pi$
$398$	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
$405$	39.2301i	1.94936i
$406$	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
$415$	38.0000i	1.86535i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−8.71780	−0.425892	−0.212946	−	0.977064i	$-0.568306\pi$
$419$	−8.71780	−0.425892	−0.212946	+	0.977064i	$0.568306\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	− 39.0000i	− 1.89624i
$424$	0	0
$425$	−98.0000	−4.75370
$426$	0	0
$427$	−13.0767	−0.632826
$428$	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
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.4356i	0.834058i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	30.5123	1.44968	0.724841	−	0.688916i	$-0.241913\pi$
$443$	30.5123	1.44968	0.724841	+	0.688916i	$0.241913\pi$
$444$	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
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 21.7945i	− 1.01507i	−0.861631	−	0.507535i	$-0.830557\pi$
$461$	− 21.7945i	− 1.01507i	0.861631	−	0.507535i	$-0.169443\pi$
$462$	0	0
$463$	41.0000i	1.90543i	0.303863	+	0.952716i	$0.401724\pi$
$463$	41.0000i	1.90543i	−0.303863	+	0.952716i	$0.598276\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.35890	−0.201706	−0.100853	−	0.994901i	$-0.532157\pi$
$467$	−4.35890	−0.201706	−0.100853	+	0.994901i	$0.532157\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	57.0000	2.62086
$474$	0	0
$475$	−61.0246	−2.80000
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 4.00000i	− 0.182765i	−0.995816	−	0.0913823i	$-0.970871\pi$
$479$	− 4.00000i	− 0.182765i	0.995816	−	0.0913823i	$-0.0291285\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	43.5890	1.96714	0.983572	−	0.180517i	$-0.0577772\pi$
$491$	43.5890	1.96714	0.983572	+	0.180517i	$0.0577772\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	− 57.0000i	− 2.56196i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	21.7945	0.975656	0.487828	−	0.872940i	$-0.337789\pi$
$499$	21.7945	0.975656	0.487828	+	0.872940i	$0.337789\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 44.0000i	− 1.96186i	−0.194354	−	0.980932i	$-0.562261\pi$
$503$	− 44.0000i	− 1.96186i	0.194354	−	0.980932i	$-0.437739\pi$
$504$	0	0
$505$	−76.0000	−3.38196
$506$	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
$511$	− 33.0000i	− 1.45983i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	56.6657i	2.49215i
$518$	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	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	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
$539$	8.71780	0.375502
$540$	0	0
$541$	39.2301i	1.68663i	0.537417	+	0.843317i	$0.319400\pi$
$541$	39.2301i	1.68663i	−0.537417	+	0.843317i	$0.680600\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	13.0767i	0.558100i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.35890i	0.184692i	0.995727	+	0.0923462i	$0.0294367\pi$
$557$	4.35890i	0.184692i	−0.995727	+	0.0923462i	$0.970563\pi$
$558$	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
$565$	0	0
$566$	0	0
$567$	27.0000i	1.13389i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−26.1534	−1.09449	−0.547243	−	0.836974i	$-0.684323\pi$
$571$	−26.1534	−1.09449	−0.547243	+	0.836974i	$0.684323\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 56.0000i	− 2.33536i
$576$	0	0
$577$	−3.00000	−0.124892	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	−3.00000	−0.124892	−0.0624458	+	0.998048i	$0.519890\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	26.1534i	1.08503i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−47.9479	−1.97902	−0.989511	−	0.144460i	$-0.953855\pi$
$587$	−47.9479	−1.97902	−0.989511	+	0.144460i	$0.953855\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	−91.5369	−3.75265
$596$	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
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	34.8712i	1.41772i
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 30.5123i	− 1.23238i	−0.787598	−	0.616190i	$-0.788675\pi$
$613$	− 30.5123i	− 1.23238i	0.787598	−	0.616190i	$-0.211325\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.0000	0.523360	0.261680	−	0.965155i	$-0.415723\pi$
$617$	13.0000	0.523360	0.261680	+	0.965155i	$0.415723\pi$
$618$	0	0
$619$	43.5890	1.75199	0.875995	−	0.482321i	$-0.160206\pi$
$619$	43.5890	1.75199	0.875995	+	0.482321i	$0.160206\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	101.000	4.04000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 15.0000i	− 0.597141i	−0.954388	−	0.298570i	$-0.903490\pi$
$631$	− 15.0000i	− 0.597141i	0.954388	−	0.298570i	$-0.0965097\pi$
$632$	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
$643$	−13.0767	−0.515695	−0.257847	−	0.966186i	$-0.583013\pi$
$643$	−13.0767	−0.515695	−0.257847	+	0.966186i	$0.583013\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.0000i	0.668339i	0.942513	+	0.334169i	$0.108456\pi$
$647$	17.0000i	0.668339i	−0.942513	+	0.334169i	$0.891544\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 30.5123i	− 1.19404i	−0.802227	−	0.597019i	$-0.796352\pi$
$653$	− 30.5123i	− 1.19404i	0.802227	−	0.597019i	$-0.203648\pi$
$654$	0	0
$655$	95.0000i	3.71196i
$656$	0	0
$657$	−33.0000	−1.28745
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	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
$665$	−57.0000	−2.21037
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 19.0000i	− 0.733487i
$672$	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	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	100.255i	3.83053i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−39.2301	−1.49238	−0.746191	−	0.665731i	$-0.768120\pi$
$691$	−39.2301	−1.49238	−0.746191	+	0.665731i	$0.768120\pi$
$692$	0	0
$693$	− 39.2301i	− 1.49023i
$694$	0	0
$695$	95.0000i	3.60356i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 17.4356i	− 0.658533i	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	− 17.4356i	− 0.658533i	0.944237	−	0.329267i	$-0.106802\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−52.3068	−1.96720
$708$	0	0
$709$	− 52.3068i	− 1.96442i	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	− 52.3068i	− 1.96442i	0.187779	−	0.982211i	$-0.439871\pi$
$710$	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
$719$	49.0000i	1.82739i	0.406399	+	0.913696i	$0.366784\pi$
$719$	49.0000i	1.82739i	−0.406399	+	0.913696i	$0.633216\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 37.0000i	− 1.37225i	−0.727482	−	0.686127i	$-0.759309\pi$
$727$	− 37.0000i	− 1.37225i	0.727482	−	0.686127i	$-0.240691\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−91.5369	−3.38561
$732$	0	0
$733$	52.3068i	1.93200i	0.258551	+	0.965998i	$0.416755\pi$
$733$	52.3068i	1.93200i	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	30.5123	1.12241	0.561206	−	0.827676i	$-0.310337\pi$
$739$	30.5123	1.12241	0.561206	+	0.827676i	$0.310337\pi$
$740$	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
$745$	95.0000	3.48053
$746$	0	0
$747$	26.1534	0.956903
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	47.9479i	1.74270i	0.490666	+	0.871348i	$0.336754\pi$
$757$	47.9479i	1.74270i	−0.490666	+	0.871348i	$0.663246\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	55.0000	1.99375	0.996874	−	0.0790050i	$-0.0251743\pi$
$761$	55.0000	1.99375	0.996874	+	0.0790050i	$0.0251743\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	91.5369i	3.30952i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−51.0000	−1.83911	−0.919554	−	0.392965i	$-0.871449\pi$
$769$	−51.0000	−1.83911	−0.919554	+	0.392965i	$0.871449\pi$
$770$	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
$785$	76.0000	2.71256
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	− 91.0000i	− 3.21935i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	47.9479	1.69204
$804$	0	0
$805$	− 52.3068i	− 1.84357i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.00000	0.175791	0.0878953	−	0.996130i	$-0.471986\pi$
$809$	5.00000	0.175791	0.0878953	+	0.996130i	$0.471986\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 38.0000i	− 1.33108i
$816$	0	0
$817$	−57.0000	−1.99418
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 21.7945i	− 0.760633i	−0.924856	−	0.380317i	$-0.875815\pi$
$821$	− 21.7945i	− 0.760633i	0.924856	−	0.380317i	$-0.124185\pi$
$822$	0	0
$823$	9.00000i	0.313720i	0.987621	+	0.156860i	$0.0501372\pi$
$823$	9.00000i	0.313720i	−0.987621	+	0.156860i	$0.949863\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−14.0000	−0.485071
$834$	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
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 56.6657i	− 1.94936i
$846$	0	0
$847$	24.0000i	0.824650i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 52.3068i	− 1.79095i	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	− 52.3068i	− 1.79095i	0.445112	−	0.895475i	$-0.353164\pi$
$854$	0	0
$855$	57.0000i	1.94936i
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	56.6657	1.93341	0.966704	−	0.255897i	$-0.0823707\pi$
$859$	56.6657	1.93341	0.966704	+	0.255897i	$0.0823707\pi$
$860$	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
$865$	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
$875$	117.690	3.97866
$876$	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
$881$	−35.0000	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$881$	−35.0000	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$882$	0	0
$883$	30.5123	1.02682	0.513410	−	0.858143i	$-0.328382\pi$
$883$	30.5123	1.02682	0.513410	+	0.858143i	$0.328382\pi$
$884$	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
$891$	−39.2301	−1.31426
$892$	0	0
$893$	− 56.6657i	− 1.89624i
$894$	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	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	52.3068i	1.73491i
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−38.0000	−1.25762
$914$	0	0
$915$	0	0
$916$	0	0
$917$	65.3835i	2.15915i
$918$	0	0
$919$	− 60.0000i	− 1.97922i	−0.143787	−	0.989609i	$-0.545928\pi$
$919$	− 60.0000i	− 1.97922i	0.143787	−	0.989609i	$-0.454072\pi$
$920$	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
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	−8.71780	−0.285714
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 133.000i	− 4.34957i
$936$	0	0
$937$	47.0000	1.53542	0.767712	−	0.640796i	$-0.221395\pi$
$937$	47.0000	1.53542	0.767712	+	0.640796i	$0.221395\pi$
$938$	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
$947$	61.0246	1.98303	0.991516	−	0.129983i	$-0.0414921\pi$
$947$	61.0246	1.98303	0.991516	+	0.129983i	$0.0414921\pi$
$948$	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
$955$	−74.1013	−2.39786
$956$	0	0
$957$	0	0
$958$	0	0
$959$	69.0000i	2.22813i
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 12.0000i	− 0.385894i	−0.981209	−	0.192947i	$-0.938195\pi$
$967$	− 12.0000i	− 0.385894i	0.981209	−	0.192947i	$-0.0618045\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	65.3835i	2.09610i
$974$	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		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−76.0000	−2.42156
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 52.3068i	− 1.66326i
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−108.972	−3.45466
$996$	0	0
$997$	4.35890i	0.138048i	0.997615	+	0.0690239i	$0.0219885\pi$
$997$	4.35890i	0.138048i	−0.997615	+	0.0690239i	$0.978012\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.b.c.607.4	yes	4
4.3	odd	2	inner	1216.2.b.c.607.3	yes	4
8.3	odd	2	inner	1216.2.b.c.607.1	✓	4
8.5	even	2	inner	1216.2.b.c.607.2	yes	4
19.18	odd	2	CM	1216.2.b.c.607.4	yes	4
76.75	even	2	inner	1216.2.b.c.607.3	yes	4
152.37	odd	2	inner	1216.2.b.c.607.2	yes	4
152.75	even	2	inner	1216.2.b.c.607.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.b.c.607.1	✓	4	8.3	odd	2	inner
1216.2.b.c.607.1	✓	4	152.75	even	2	inner
1216.2.b.c.607.2	yes	4	8.5	even	2	inner
1216.2.b.c.607.2	yes	4	152.37	odd	2	inner
1216.2.b.c.607.3	yes	4	4.3	odd	2	inner
1216.2.b.c.607.3	yes	4	76.75	even	2	inner
1216.2.b.c.607.4	yes	4	1.1	even	1	trivial
1216.2.b.c.607.4	yes	4	19.18	odd	2	CM

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.35890i q^{5} +3.00000i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+4.35890i q^{5} +3.00000i q^{7} +3.00000 q^{9} -4.35890 q^{11} +7.00000 q^{17} +4.35890 q^{19} +4.00000i q^{23} -14.0000 q^{25} -13.0767 q^{35} -13.0767 q^{43} +13.0767i q^{45} -13.0000i q^{47} -2.00000 q^{49} -19.0000i q^{55} +4.35890i q^{61} +9.00000i q^{63} -11.0000 q^{73} -13.0767i q^{77} +9.00000 q^{81} +8.71780 q^{83} +30.5123i q^{85} +19.0000i q^{95} -13.0767 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} + 28 q^{17} - 56 q^{25} - 8 q^{49} - 44 q^{73} + 36 q^{81}+O(q^{100}) \) 4 * q + 12 * q^9 + 28 * q^17 - 56 * q^25 - 8 * q^49 - 44 * q^73 + 36 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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