LMFDB - Embedded newform 1216.3.g.c.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:	$8$
Coefficient field:	8.0.2702336256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 $ x^8 + 9x^6 + 56x^4 + 225*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			417.1
Root			$-0.656712 + 2.13746i$ of defining polynomial
Character	$\chi$	$=$	1216.417
Dual form			1216.3.g.c.417.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.97368i q^{5} -2.20822 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-9.97368i q^{5} -2.20822 q^{7} -9.00000 q^{9} -20.3746i q^{11} +18.9244 q^{17} -19.0000i q^{19} +34.8712 q^{23} -74.4743 q^{25} +22.0241i q^{35} -53.8248i q^{43} +89.7631i q^{45} -36.6191 q^{47} -44.1238 q^{49} -203.210 q^{55} +121.892i q^{61} +19.8740 q^{63} +112.072 q^{73} +44.9916i q^{77} +81.0000 q^{81} +90.0000i q^{83} -188.746i q^{85} -189.500 q^{95} +183.371i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 72 q^{9}+O(q^{10}) $ 8 * q - 72 * q^9	$ 8 q - 72 q^{9} - 60 q^{17} - 324 q^{25} + 100 q^{49} - 100 q^{73} + 648 q^{81}+O(q^{100}) $ 8 * q - 72 * q^9 - 60 * q^17 - 324 * q^25 + 100 * q^49 - 100 * q^73 + 648 * 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
$5$	− 9.97368i	− 1.99474i	−0.0725083	−	0.997368i	$-0.523100\pi$
$5$	− 9.97368i	− 1.99474i	0.0725083	−	0.997368i	$-0.476900\pi$
$6$	0	0
$7$	−2.20822	−0.315460	−0.157730	−	0.987482i	$-0.550418\pi$
$7$	−2.20822	−0.315460	−0.157730	+	0.987482i	$0.550418\pi$
$8$	0	0
$9$	−9.00000	−1.00000
$10$	0	0
$11$	− 20.3746i	− 1.85224i	−0.377235	−	0.926118i	$-0.623125\pi$
$11$	− 20.3746i	− 1.85224i	0.377235	−	0.926118i	$-0.376875\pi$
$12$	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
$17$	18.9244	1.11320	0.556601	−	0.830780i	$-0.312105\pi$
$17$	18.9244	1.11320	0.556601	+	0.830780i	$0.312105\pi$
$18$	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	$-0.226143\pi$
$23$	34.8712	1.51614	0.758069	+	0.652174i	$0.226143\pi$
$24$	0	0
$25$	−74.4743	−2.97897
$26$	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
$35$	22.0241i	0.629260i
$36$	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
$43$	− 53.8248i	− 1.25174i	−0.779928	−	0.625869i	$-0.784744\pi$
$43$	− 53.8248i	− 1.25174i	0.779928	−	0.625869i	$-0.215256\pi$
$44$	0	0
$45$	89.7631i	1.99474i
$46$	0	0
$47$	−36.6191	−0.779129	−0.389564	−	0.920999i	$-0.627374\pi$
$47$	−36.6191	−0.779129	−0.389564	+	0.920999i	$0.627374\pi$
$48$	0	0
$49$	−44.1238	−0.900485
$50$	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
$55$	−203.210	−3.69472
$56$	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
$61$	121.892i	1.99824i	0.0419980	+	0.999118i	$0.486628\pi$
$61$	121.892i	1.99824i	−0.0419980	+	0.999118i	$0.513372\pi$
$62$	0	0
$63$	19.8740	0.315460
$64$	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
$73$	112.072	1.53524	0.767618	−	0.640907i	$-0.221442\pi$
$73$	112.072	1.53524	0.767618	+	0.640907i	$0.221442\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	44.9916i	0.584306i
$78$	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
$85$	− 188.746i	− 2.22054i
$86$	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
$95$	−189.500	−1.99474
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	183.371i	1.85224i
$100$	0	0
$101$	174.356i	1.72630i	0.504950	+	0.863148i	$0.331511\pi$
$101$	174.356i	1.72630i	−0.504950	+	0.863148i	$0.668489\pi$
$102$	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
$115$	− 347.794i	− 3.02430i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−41.7893	−0.351171
$120$	0	0
$121$	−294.124	−2.43077
$122$	0	0
$123$	0	0
$124$	0	0
$125$	493.440i	3.94752i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 238.622i	− 1.82154i	−0.412911	−	0.910771i	$-0.635488\pi$
$131$	− 238.622i	− 1.82154i	0.412911	−	0.910771i	$-0.364512\pi$
$132$	0	0
$133$	41.9562i	0.315460i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−40.6769	−0.296912	−0.148456	−	0.988919i	$-0.547430\pi$
$137$	−40.6769	−0.296912	−0.148456	+	0.988919i	$0.547430\pi$
$138$	0	0
$139$	71.3713i	0.513462i	0.966483	+	0.256731i	$0.0826455\pi$
$139$	71.3713i	0.513462i	−0.966483	+	0.256731i	$0.917354\pi$
$140$	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
$149$	33.4168i	0.224274i	0.993693	+	0.112137i	$0.0357695\pi$
$149$	33.4168i	0.224274i	−0.993693	+	0.112137i	$0.964231\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−170.320	−1.11320
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 313.841i	− 1.99899i	−0.0318471	−	0.999493i	$-0.510139\pi$
$157$	− 313.841i	− 1.99899i	0.0318471	−	0.999493i	$-0.489861\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−77.0033	−0.478281
$162$	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
$175$	164.456	0.939747
$176$	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
$187$	− 385.577i	− 2.06191i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−265.794	−1.39159	−0.695795	−	0.718241i	$-0.744948\pi$
$191$	−265.794	−1.39159	−0.695795	+	0.718241i	$0.744948\pi$
$192$	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	$-0.573358\pi$
$197$	− 383.583i	− 1.94712i	0.228426	−	0.973561i	$-0.426642\pi$
$198$	0	0
$199$	360.046	1.80928	0.904639	−	0.426178i	$-0.140140\pi$
$199$	360.046	1.80928	0.904639	+	0.426178i	$0.140140\pi$
$200$	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
$209$	−387.117	−1.85224
$210$	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
$215$	−536.831	−2.49689
$216$	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
$225$	670.268	2.97897
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	243.565i	1.06360i	0.846870	+	0.531800i	$0.178484\pi$
$229$	243.565i	1.06360i	−0.846870	+	0.531800i	$0.821516\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	258.924	1.11126	0.555632	−	0.831428i	$-0.312476\pi$
$233$	258.924	1.11126	0.555632	+	0.831428i	$0.312476\pi$
$234$	0	0
$235$	365.227i	1.55416i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	316.029	1.32230	0.661148	−	0.750255i	$-0.270070\pi$
$239$	316.029	1.32230	0.661148	+	0.750255i	$0.270070\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	440.076i	1.79623i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 420.615i	− 1.67576i	−0.545855	−	0.837879i	$-0.683795\pi$
$251$	− 420.615i	− 1.67576i	0.545855	−	0.837879i	$-0.316205\pi$
$252$	0	0
$253$	− 710.486i	− 2.80825i
$254$	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
$263$	518.558	1.97170	0.985852	−	0.167621i	$-0.0536085\pi$
$263$	518.558	1.97170	0.985852	+	0.167621i	$0.0536085\pi$
$264$	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	$-0.0843796\pi$
$271$	523.068	1.93014	0.965070	+	0.261993i	$0.0843796\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1517.38i	5.51775i
$276$	0	0
$277$	391.402i	1.41300i	0.707712	+	0.706501i	$0.249728\pi$
$277$	391.402i	1.41300i	−0.707712	+	0.706501i	$0.750272\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	548.567i	1.93840i	0.246274	+	0.969200i	$0.420794\pi$
$283$	548.567i	1.93840i	−0.246274	+	0.969200i	$0.579206\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	69.1337	0.239217
$290$	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
$301$	118.857i	0.394874i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1215.72	3.98595
$306$	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
$311$	−598.494	−1.92442	−0.962209	−	0.272312i	$-0.912212\pi$
$311$	−598.494	−1.92442	−0.962209	+	0.272312i	$0.912212\pi$
$312$	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
$315$	− 198.217i	− 0.629260i
$316$	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
$323$	− 359.564i	− 1.11320i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	80.8630	0.245784
$330$	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
$343$	205.638	0.599527
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 477.172i	− 1.37514i	−0.726120	−	0.687568i	$-0.758678\pi$
$347$	− 477.172i	− 1.37514i	0.726120	−	0.687568i	$-0.241322\pi$
$348$	0	0
$349$	− 227.553i	− 0.652015i	−0.945367	−	0.326007i	$-0.894297\pi$
$349$	− 227.553i	− 0.652015i	0.945367	−	0.326007i	$-0.105703\pi$
$350$	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
$359$	548.259	1.52718	0.763592	−	0.645699i	$-0.223434\pi$
$359$	548.259	1.52718	0.763592	+	0.645699i	$0.223434\pi$
$360$	0	0
$361$	−361.000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 1117.77i	− 3.06239i
$366$	0	0
$367$	−732.295	−1.99535	−0.997677	−	0.0681199i	$-0.978300\pi$
$367$	−732.295	−1.99535	−0.997677	+	0.0681199i	$0.978300\pi$
$368$	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
$385$	448.732	1.16554
$386$	0	0
$387$	484.423i	1.25174i
$388$	0	0
$389$	248.902i	0.639850i	0.947443	+	0.319925i	$0.103658\pi$
$389$	248.902i	0.639850i	−0.947443	+	0.319925i	$0.896342\pi$
$390$	0	0
$391$	659.917	1.68777
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 507.884i	− 1.27930i	−0.768665	−	0.639652i	$-0.779079\pi$
$397$	− 507.884i	− 1.27930i	0.768665	−	0.639652i	$-0.220921\pi$
$398$	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
$405$	− 807.868i	− 1.99474i
$406$	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
$415$	897.631	2.16297
$416$	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
$423$	329.572	0.779129
$424$	0	0
$425$	−1409.38	−3.31619
$426$	0	0
$427$	− 269.165i	− 0.630364i
$428$	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
$441$	397.114	0.900485
$442$	0	0
$443$	788.808i	1.78061i	0.455369	+	0.890303i	$0.349507\pi$
$443$	788.808i	1.78061i	−0.455369	+	0.890303i	$0.650493\pi$
$444$	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
$457$	−265.062	−0.580005	−0.290003	−	0.957026i	$-0.593656\pi$
$457$	−265.062	−0.580005	−0.290003	+	0.957026i	$0.593656\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16.0848i	0.0348911i	0.999848	+	0.0174455i	$0.00555337\pi$
$461$	16.0848i	0.0348911i	−0.999848	+	0.0174455i	$0.994447\pi$
$462$	0	0
$463$	921.921	1.99119	0.995596	−	0.0937525i	$-0.0298862\pi$
$463$	921.921	1.99119	0.995596	+	0.0937525i	$0.0298862\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 295.179i	− 0.632074i	−0.948747	−	0.316037i	$-0.897648\pi$
$467$	− 295.179i	− 0.632074i	0.948747	−	0.316037i	$-0.102352\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1096.66	−2.31851
$474$	0	0
$475$	1415.01i	2.97897i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−174.356	−0.364000	−0.182000	−	0.983299i	$-0.558257\pi$
$479$	−174.356	−0.364000	−0.182000	+	0.983299i	$0.558257\pi$
$480$	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
$495$	1828.89	3.69472
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 474.609i	− 0.951120i	−0.879683	−	0.475560i	$-0.842245\pi$
$499$	− 474.609i	− 0.951120i	0.879683	−	0.475560i	$-0.157755\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−383.583	−0.762591	−0.381295	−	0.924453i	$-0.624522\pi$
$503$	−383.583	−0.762591	−0.381295	+	0.924453i	$0.624522\pi$
$504$	0	0
$505$	1738.97	3.44351
$506$	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
$511$	−247.480	−0.484306
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	746.098i	1.44313i
$518$	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
$539$	899.003i	1.66791i
$540$	0	0
$541$	− 886.150i	− 1.63798i	−0.573804	−	0.818992i	$-0.694533\pi$
$541$	− 886.150i	− 1.63798i	0.573804	−	0.818992i	$-0.305467\pi$
$542$	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
$549$	− 1097.03i	− 1.99824i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1050.73i	1.88641i	0.332207	+	0.943206i	$0.392207\pi$
$557$	1050.73i	1.88641i	−0.332207	+	0.943206i	$0.607793\pi$
$558$	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
$567$	−178.866	−0.315460
$568$	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
$575$	−2597.01	−4.51653
$576$	0	0
$577$	−447.928	−0.776305	−0.388152	−	0.921595i	$-0.626887\pi$
$577$	−447.928	−0.776305	−0.388152	+	0.921595i	$0.626887\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 198.740i	− 0.342065i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 271.831i	− 0.463086i	−0.972825	−	0.231543i	$-0.925623\pi$
$587$	− 271.831i	− 0.463086i	0.972825	−	0.231543i	$-0.0743774\pi$
$588$	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
$595$	416.793i	0.700493i
$596$	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
$605$	2933.50i	4.84875i
$606$	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
$613$	− 339.512i	− 0.553854i	−0.960891	−	0.276927i	$-0.910684\pi$
$613$	− 339.512i	− 0.553854i	0.960891	−	0.276927i	$-0.0893159\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1072.31	−1.73795	−0.868973	−	0.494859i	$-0.835220\pi$
$617$	−1072.31	−1.73795	−0.868973	+	0.494859i	$0.835220\pi$
$618$	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
$625$	3059.56	4.89529
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1257.68	−1.99315	−0.996575	−	0.0826952i	$-0.973647\pi$
$631$	−1257.68	−1.99315	−0.996575	+	0.0826952i	$0.973647\pi$
$632$	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
$643$	2.58717i	0.00402359i	0.999998	+	0.00201180i	$0.000640375\pi$
$643$	2.58717i	0.00402359i	−0.999998	+	0.00201180i	$0.999360\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−462.798	−0.715299	−0.357650	−	0.933856i	$-0.616422\pi$
$647$	−462.798	−0.715299	−0.357650	+	0.933856i	$0.616422\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	950.262i	1.45522i	0.685989	+	0.727612i	$0.259370\pi$
$653$	950.262i	1.45522i	−0.685989	+	0.727612i	$0.740630\pi$
$654$	0	0
$655$	−2379.94	−3.63350
$656$	0	0
$657$	−1008.65	−1.53524
$658$	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
$665$	418.458	0.629260
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2483.51	3.70120
$672$	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
$685$	405.698i	0.592260i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1267.60i	1.83444i	0.398380	+	0.917221i	$0.369573\pi$
$691$	1267.60i	1.83444i	−0.398380	+	0.917221i	$0.630427\pi$
$692$	0	0
$693$	− 404.924i	− 0.584306i
$694$	0	0
$695$	711.834	1.02422
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 871.780i	− 1.24362i	−0.783167	−	0.621812i	$-0.786397\pi$
$701$	− 871.780i	− 1.24362i	0.783167	−	0.621812i	$-0.213603\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 385.017i	− 0.544578i
$708$	0	0
$709$	− 523.068i	− 0.737754i	−0.929478	−	0.368877i	$-0.879742\pi$
$709$	− 523.068i	− 0.737754i	0.929478	−	0.368877i	$-0.120258\pi$
$710$	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
$719$	300.017	0.417270	0.208635	−	0.977994i	$-0.433098\pi$
$719$	300.017	0.417270	0.208635	+	0.977994i	$0.433098\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	799.369	1.09954	0.549772	−	0.835315i	$-0.314715\pi$
$727$	799.369	1.09954	0.549772	+	0.835315i	$0.314715\pi$
$728$	0	0
$729$	−729.000	−1.00000
$730$	0	0
$731$	− 1018.60i	− 1.39344i
$732$	0	0
$733$	− 732.295i	− 0.999038i	−0.866303	−	0.499519i	$-0.833510\pi$
$733$	− 732.295i	− 0.999038i	0.866303	−	0.499519i	$-0.166490\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	915.599i	1.23897i	0.785008	+	0.619485i	$0.212659\pi$
$739$	915.599i	1.23897i	−0.785008	+	0.619485i	$0.787341\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	333.288	0.447367
$746$	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
$757$	1327.13i	1.75314i	0.481275	+	0.876570i	$0.340174\pi$
$757$	1327.13i	1.75314i	−0.481275	+	0.876570i	$0.659826\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−543.880	−0.714691	−0.357345	−	0.933972i	$-0.616318\pi$
$761$	−543.880	−0.714691	−0.357345	+	0.933972i	$0.616318\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1698.71i	2.22054i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−431.104	−0.560603	−0.280302	−	0.959912i	$-0.590434\pi$
$769$	−431.104	−0.560603	−0.280302	+	0.959912i	$0.590434\pi$
$770$	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
$785$	−3130.15	−3.98745
$786$	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
$799$	−692.995	−0.867327
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 2283.43i	− 2.84362i
$804$	0	0
$805$	768.006i	0.954045i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−551.130	−0.681249	−0.340624	−	0.940199i	$-0.610638\pi$
$809$	−551.130	−0.681249	−0.340624	+	0.940199i	$0.610638\pi$
$810$	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
$815$	−2493.42	−3.05941
$816$	0	0
$817$	−1022.67	−1.25174
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 433.098i	− 0.527524i	−0.964588	−	0.263762i	$-0.915037\pi$
$821$	− 433.098i	− 0.527524i	0.964588	−	0.263762i	$-0.0849634\pi$
$822$	0	0
$823$	−1610.33	−1.95665	−0.978326	−	0.207068i	$-0.933608\pi$
$823$	−1610.33	−1.95665	−0.978326	+	0.207068i	$0.933608\pi$
$824$	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
$833$	−835.017	−1.00242
$834$	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
$845$	1685.55i	1.99474i
$846$	0	0
$847$	649.490	0.766813
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 1359.98i	− 1.59435i	−0.603751	−	0.797173i	$-0.706328\pi$
$853$	− 1359.98i	− 1.59435i	0.603751	−	0.797173i	$-0.293672\pi$
$854$	0	0
$855$	1705.50	1.99474
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	− 1482.61i	− 1.72597i	−0.505229	−	0.862985i	$-0.668592\pi$
$859$	− 1482.61i	− 1.72597i	0.505229	−	0.862985i	$-0.331408\pi$
$860$	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
$875$	− 1089.63i	− 1.24529i
$876$	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
$881$	1184.84	1.34488	0.672442	−	0.740150i	$-0.265245\pi$
$881$	1184.84	1.34488	0.672442	+	0.740150i	$0.265245\pi$
$882$	0	0
$883$	− 1765.15i	− 1.99903i	−0.0311055	−	0.999516i	$-0.509903\pi$
$883$	− 1765.15i	− 1.99903i	0.0311055	−	0.999516i	$-0.490097\pi$
$884$	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
$891$	− 1650.34i	− 1.85224i
$892$	0	0
$893$	695.762i	0.779129i
$894$	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
$913$	1833.71	2.00845
$914$	0	0
$915$	0	0
$916$	0	0
$917$	526.930i	0.574624i
$918$	0	0
$919$	−523.068	−0.569171	−0.284585	−	0.958651i	$-0.591856\pi$
$919$	−523.068	−0.569171	−0.284585	+	0.958651i	$0.591856\pi$
$920$	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
$931$	838.351i	0.900485i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3845.62	−4.11297
$936$	0	0
$937$	1429.29	1.52539	0.762695	−	0.646759i	$-0.223876\pi$
$937$	1429.29	1.52539	0.762695	+	0.646759i	$0.223876\pi$
$938$	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
$955$	2650.94i	2.77585i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	89.8236	0.0936638
$960$	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	$-0.376391\pi$
$967$	732.295	0.757285	0.378643	+	0.925543i	$0.376391\pi$
$968$	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
$973$	− 157.604i	− 0.161977i
$974$	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
$985$	−3825.73	−3.88399
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1876.93i	− 1.89781i
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 3590.99i	− 3.60903i
$996$	0	0
$997$	1847.71i	1.85327i	0.375968	+	0.926633i	$0.377310\pi$
$997$	1847.71i	1.85327i	−0.375968	+	0.926633i	$0.622690\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.g.c.417.1	✓	8
4.3	odd	2	inner	1216.3.g.c.417.2	yes	8
8.3	odd	2	inner	1216.3.g.c.417.8	yes	8
8.5	even	2	inner	1216.3.g.c.417.7	yes	8
19.18	odd	2	CM	1216.3.g.c.417.1	✓	8
76.75	even	2	inner	1216.3.g.c.417.2	yes	8
152.37	odd	2	inner	1216.3.g.c.417.7	yes	8
152.75	even	2	inner	1216.3.g.c.417.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.g.c.417.1	✓	8	1.1	even	1	trivial
1216.3.g.c.417.1	✓	8	19.18	odd	2	CM
1216.3.g.c.417.2	yes	8	4.3	odd	2	inner
1216.3.g.c.417.2	yes	8	76.75	even	2	inner
1216.3.g.c.417.7	yes	8	8.5	even	2	inner
1216.3.g.c.417.7	yes	8	152.37	odd	2	inner
1216.3.g.c.417.8	yes	8	8.3	odd	2	inner
1216.3.g.c.417.8	yes	8	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)

\(f(q)\)	\(=\)	\(q-9.97368i q^{5} -2.20822 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-9.97368i q^{5} -2.20822 q^{7} -9.00000 q^{9} -20.3746i q^{11} +18.9244 q^{17} -19.0000i q^{19} +34.8712 q^{23} -74.4743 q^{25} +22.0241i q^{35} -53.8248i q^{43} +89.7631i q^{45} -36.6191 q^{47} -44.1238 q^{49} -203.210 q^{55} +121.892i q^{61} +19.8740 q^{63} +112.072 q^{73} +44.9916i q^{77} +81.0000 q^{81} +90.0000i q^{83} -188.746i q^{85} -189.500 q^{95} +183.371i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{9}+O(q^{10}) \) 8 * q - 72 * q^9	\( 8 q - 72 q^{9} - 60 q^{17} - 324 q^{25} + 100 q^{49} - 100 q^{73} + 648 q^{81}+O(q^{100}) \) 8 * q - 72 * q^9 - 60 * q^17 - 324 * q^25 + 100 * q^49 - 100 * q^73 + 648 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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