LMFDB - Embedded newform 1216.3.g.c.417.6

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			417.6
Root			$1.52274 + 1.63746i$ of defining polynomial
Character	$\chi$	$=$	1216.417
Dual form			1216.3.g.c.417.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.61478i q^{5} +10.8685 q^{7} -9.00000 q^{9} +O(q^{10})$	$q+5.61478i q^{5} +10.8685 q^{7} -9.00000 q^{9} -17.3746i q^{11} -33.9244 q^{17} +19.0000i q^{19} -34.8712 q^{23} -6.52575 q^{25} +61.0241i q^{35} +31.1752i q^{43} -50.5330i q^{45} -93.2847 q^{47} +69.1238 q^{49} +97.5545 q^{55} -56.5089i q^{61} -97.8163 q^{63} -137.072 q^{73} -188.835i q^{77} +81.0000 q^{81} -90.0000i q^{83} -190.478i q^{85} -106.681 q^{95} +156.371i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	5.61478i	1.12296i	0.827492	+	0.561478i	$0.189767\pi$
$5$	5.61478i	1.12296i	−0.827492	+	0.561478i	$0.810233\pi$
$6$	0	0
$7$	10.8685	1.55264	0.776320	−	0.630339i	$-0.217084\pi$
$7$	10.8685	1.55264	0.776320	+	0.630339i	$0.217084\pi$
$8$	0	0
$9$	−9.00000	−1.00000
$10$	0	0
$11$	− 17.3746i	− 1.57951i	−0.613424	−	0.789754i	$-0.710208\pi$
$11$	− 17.3746i	− 1.57951i	0.613424	−	0.789754i	$-0.289792\pi$
$12$	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
$17$	−33.9244	−1.99555	−0.997777	−	0.0666402i	$-0.978772\pi$
$17$	−33.9244	−1.99555	−0.997777	+	0.0666402i	$0.978772\pi$
$18$	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
$25$	−6.52575	−0.261030
$26$	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
$35$	61.0241i	1.74355i
$36$	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
$43$	31.1752i	0.725006i	0.931983	+	0.362503i	$0.118078\pi$
$43$	31.1752i	0.725006i	−0.931983	+	0.362503i	$0.881922\pi$
$44$	0	0
$45$	− 50.5330i	− 1.12296i
$46$	0	0
$47$	−93.2847	−1.98478	−0.992391	−	0.123127i	$-0.960708\pi$
$47$	−93.2847	−1.98478	−0.992391	+	0.123127i	$0.960708\pi$
$48$	0	0
$49$	69.1238	1.41069
$50$	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
$55$	97.5545	1.77372
$56$	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
$61$	− 56.5089i	− 0.926375i	−0.886260	−	0.463187i	$-0.846706\pi$
$61$	− 56.5089i	− 0.926375i	0.886260	−	0.463187i	$-0.153294\pi$
$62$	0	0
$63$	−97.8163	−1.55264
$64$	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
$73$	−137.072	−1.87770	−0.938851	−	0.344323i	$-0.888108\pi$
$73$	−137.072	−1.87770	−0.938851	+	0.344323i	$0.888108\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 188.835i	− 2.45241i
$78$	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	$-0.817603\pi$
$83$	− 90.0000i	− 1.08434i	0.840270	−	0.542169i	$-0.182397\pi$
$84$	0	0
$85$	− 190.478i	− 2.24092i
$86$	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
$95$	−106.681	−1.12296
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	156.371i	1.57951i
$100$	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
$115$	− 195.794i	− 1.70256i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−368.707	−3.09838
$120$	0	0
$121$	−180.876	−1.49484
$122$	0	0
$123$	0	0
$124$	0	0
$125$	103.729i	0.829831i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 25.6221i	− 0.195589i	−0.995207	−	0.0977943i	$-0.968821\pi$
$131$	− 25.6221i	− 0.195589i	0.995207	−	0.0977943i	$-0.0311787\pi$
$132$	0	0
$133$	206.501i	1.55264i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−214.323	−1.56440	−0.782201	−	0.623026i	$-0.785903\pi$
$137$	−214.323	−1.56440	−0.782201	+	0.623026i	$0.785903\pi$
$138$	0	0
$139$	268.371i	1.93073i	0.260906	+	0.965364i	$0.415979\pi$
$139$	268.371i	1.93073i	−0.260906	+	0.965364i	$0.584021\pi$
$140$	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
$149$	− 273.156i	− 1.83326i	−0.399733	−	0.916632i	$-0.630897\pi$
$149$	− 273.156i	− 1.83326i	0.399733	−	0.916632i	$-0.369103\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	305.320	1.99555
$154$	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
$161$	−378.997	−2.35402
$162$	0	0
$163$	250.000i	1.53374i	0.641801	+	0.766871i	$0.278187\pi$
$163$	250.000i	1.53374i	−0.641801	+	0.766871i	$0.721813\pi$
$164$	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
$175$	−70.9249	−0.405285
$176$	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
$187$	589.423i	3.15199i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	104.713	0.548235	0.274117	−	0.961696i	$-0.411614\pi$
$191$	104.713	0.548235	0.274117	+	0.961696i	$0.411614\pi$
$192$	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
$199$	33.1291	0.166478	0.0832388	−	0.996530i	$-0.473474\pi$
$199$	33.1291	0.166478	0.0832388	+	0.996530i	$0.473474\pi$
$200$	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
$209$	330.117	1.57951
$210$	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
$215$	−175.042	−0.814149
$216$	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
$225$	58.7317	0.261030
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	214.120i	0.935021i	0.883988	+	0.467510i	$0.154849\pi$
$229$	214.120i	0.935021i	−0.883988	+	0.467510i	$0.845151\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	206.076	0.884445	0.442222	−	0.896905i	$-0.354190\pi$
$233$	206.076	0.884445	0.442222	+	0.896905i	$0.354190\pi$
$234$	0	0
$235$	− 523.773i	− 2.22882i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	468.590	1.96063	0.980314	−	0.197443i	$-0.0632638\pi$
$239$	468.590	1.96063	0.980314	+	0.197443i	$0.0632638\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	388.115i	1.58414i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 447.615i	− 1.78333i	−0.452697	−	0.891664i	$-0.649538\pi$
$251$	− 447.615i	− 1.78333i	0.452697	−	0.891664i	$-0.350462\pi$
$252$	0	0
$253$	605.873i	2.39475i
$254$	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
$263$	182.923	0.695523	0.347762	−	0.937583i	$-0.386942\pi$
$263$	182.923	0.695523	0.347762	+	0.937583i	$0.386942\pi$
$264$	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
$275$	113.382i	0.412299i
$276$	0	0
$277$	− 535.245i	− 1.93229i	−0.257992	−	0.966147i	$-0.583061\pi$
$277$	− 535.245i	− 1.93229i	0.257992	−	0.966147i	$-0.416939\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	153.567i	0.542641i	0.962489	+	0.271320i	$0.0874603\pi$
$283$	153.567i	0.542641i	−0.962489	+	0.271320i	$0.912540\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	861.866	2.98224
$290$	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
$301$	338.827i	1.12567i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	317.285	1.04028
$306$	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
$311$	−445.933	−1.43387	−0.716933	−	0.697142i	$-0.754455\pi$
$311$	−445.933	−1.43387	−0.716933	+	0.697142i	$0.754455\pi$
$312$	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
$315$	− 549.217i	− 1.74355i
$316$	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
$323$	− 644.564i	− 1.99555i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1013.86	−3.08165
$330$	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
$343$	218.715	0.637652
$344$	0	0
$345$	0	0
$346$	0	0
$347$	197.828i	0.570110i	0.958511	+	0.285055i	$0.0920118\pi$
$347$	197.828i	0.570110i	−0.958511	+	0.285055i	$0.907988\pi$
$348$	0	0
$349$	685.238i	1.96343i	0.190353	+	0.981716i	$0.439037\pi$
$349$	685.238i	1.96343i	−0.190353	+	0.981716i	$0.560963\pi$
$350$	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
$359$	−127.370	−0.354792	−0.177396	−	0.984140i	$-0.556767\pi$
$359$	−127.370	−0.354792	−0.177396	+	0.984140i	$0.556767\pi$
$360$	0	0
$361$	−361.000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 769.631i	− 2.10858i
$366$	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
$385$	1060.27	2.75394
$386$	0	0
$387$	− 280.577i	− 0.725006i
$388$	0	0
$389$	513.906i	1.32109i	0.750785	+	0.660547i	$0.229676\pi$
$389$	513.906i	1.32109i	−0.750785	+	0.660547i	$0.770324\pi$
$390$	0	0
$391$	1182.98	3.02554
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	782.494i	1.97102i	0.169622	+	0.985509i	$0.445745\pi$
$397$	782.494i	1.97102i	−0.169622	+	0.985509i	$0.554255\pi$
$398$	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
$405$	454.797i	1.12296i
$406$	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
$415$	505.330	1.21766
$416$	0	0
$417$	0	0
$418$	0	0
$419$	762.000i	1.81862i	0.416124	+	0.909308i	$0.363388\pi$
$419$	762.000i	1.81862i	−0.416124	+	0.909308i	$0.636612\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	839.563	1.98478
$424$	0	0
$425$	221.382	0.520899
$426$	0	0
$427$	− 614.165i	− 1.43833i
$428$	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
$441$	−622.114	−1.41069
$442$	0	0
$443$	743.808i	1.67903i	0.543340	+	0.839513i	$0.317159\pi$
$443$	743.808i	1.67903i	−0.543340	+	0.839513i	$0.682841\pi$
$444$	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
$457$	890.062	1.94762	0.973810	−	0.227363i	$-0.0730105\pi$
$457$	890.062	1.94762	0.973810	+	0.227363i	$0.0730105\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	790.312i	1.71434i	0.515032	+	0.857171i	$0.327780\pi$
$461$	790.312i	1.71434i	−0.515032	+	0.857171i	$0.672220\pi$
$462$	0	0
$463$	385.777	0.833211	0.416606	−	0.909087i	$-0.363220\pi$
$463$	385.777	0.833211	0.416606	+	0.909087i	$0.363220\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	619.821i	1.32724i	0.748070	+	0.663620i	$0.230981\pi$
$467$	619.821i	1.32724i	−0.748070	+	0.663620i	$0.769019\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	541.657	1.14515
$474$	0	0
$475$	− 123.989i	− 0.261030i
$476$	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	$0.384446\pi$
$491$	918.000i	1.86965i	−0.355104	+	0.934827i	$0.615554\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−877.990	−1.77372
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 997.609i	− 1.99922i	−0.0279946	−	0.999608i	$-0.508912\pi$
$499$	− 997.609i	− 1.99922i	0.0279946	−	0.999608i	$-0.491088\pi$
$500$	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
$505$	−978.970	−1.93855
$506$	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
$511$	−1489.77	−2.91539
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1620.78i	3.13498i
$518$	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
$539$	− 1201.00i	− 2.22819i
$540$	0	0
$541$	− 94.6025i	− 0.174866i	−0.996170	−	0.0874330i	$-0.972134\pi$
$541$	− 94.6025i	− 0.174866i	0.996170	−	0.0874330i	$-0.0278664\pi$
$542$	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
$549$	508.580i	0.926375i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 845.864i	− 1.51861i	−0.650737	−	0.759303i	$-0.725540\pi$
$557$	− 845.864i	− 1.51861i	0.650737	−	0.759303i	$-0.274460\pi$
$558$	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
$567$	880.347	1.55264
$568$	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	$-0.868645\pi$
$571$	− 458.000i	− 0.802102i	0.916056	−	0.401051i	$-0.131355\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	227.561	0.395757
$576$	0	0
$577$	−697.072	−1.20810	−0.604049	−	0.796947i	$-0.706447\pi$
$577$	−697.072	−1.20810	−0.604049	+	0.796947i	$0.706447\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 978.163i	− 1.68358i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	853.169i	1.45344i	0.686934	+	0.726719i	$0.258956\pi$
$587$	853.169i	1.45344i	−0.686934	+	0.726719i	$0.741044\pi$
$588$	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
$595$	− 2070.21i	− 3.47934i
$596$	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
$605$	− 1015.58i	− 1.67864i
$606$	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
$613$	− 850.467i	− 1.38739i	−0.720271	−	0.693693i	$-0.755983\pi$
$613$	− 850.467i	− 1.38739i	0.720271	−	0.693693i	$-0.244017\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.31316	0.0118528	0.00592639	−	0.999982i	$-0.498114\pi$
$617$	7.31316	0.0118528	0.00592639	+	0.999982i	$0.498114\pi$
$618$	0	0
$619$	662.000i	1.06947i	0.845021	+	0.534733i	$0.179588\pi$
$619$	662.000i	1.06947i	−0.845021	+	0.534733i	$0.820412\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−745.558	−1.19289
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−538.459	−0.853343	−0.426671	−	0.904407i	$-0.640314\pi$
$631$	−538.459	−0.853343	−0.426671	+	0.904407i	$0.640314\pi$
$632$	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
$643$	− 1112.41i	− 1.73004i	−0.501741	−	0.865018i	$-0.667307\pi$
$643$	− 1112.41i	− 1.73004i	0.501741	−	0.865018i	$-0.332693\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1277.91	−1.97514	−0.987568	−	0.157194i	$-0.949755\pi$
$647$	−1277.91	−1.97514	−0.987568	+	0.157194i	$0.949755\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	300.742i	0.460555i	0.973125	+	0.230278i	$0.0739634\pi$
$653$	300.742i	0.460555i	−0.973125	+	0.230278i	$0.926037\pi$
$654$	0	0
$655$	143.862	0.219637
$656$	0	0
$657$	1233.65	1.87770
$658$	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
$665$	−1159.46	−1.74355
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−981.818	−1.46322
$672$	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
$685$	− 1203.38i	− 1.75675i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1110.60i	1.60723i	0.595147	+	0.803617i	$0.297094\pi$
$691$	1110.60i	1.60723i	−0.595147	+	0.803617i	$0.702906\pi$
$692$	0	0
$693$	1699.52i	2.45241i
$694$	0	0
$695$	−1506.85	−2.16812
$696$	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
$707$	1894.98i	2.68032i
$708$	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
$719$	1367.95	1.90257	0.951285	−	0.308313i	$-0.0997646\pi$
$719$	1367.95	1.90257	0.951285	+	0.308313i	$0.0997646\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−652.145	−0.897035	−0.448518	−	0.893774i	$-0.648048\pi$
$727$	−652.145	−0.897035	−0.448518	+	0.893774i	$0.648048\pi$
$728$	0	0
$729$	−729.000	−1.00000
$730$	0	0
$731$	− 1057.60i	− 1.44679i
$732$	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
$739$	1462.60i	1.97916i	0.143986	+	0.989580i	$0.454008\pi$
$739$	1462.60i	1.97916i	−0.143986	+	0.989580i	$0.545992\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	1533.71	2.05867
$746$	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
$757$	− 32.5334i	− 0.0429768i	−0.999769	−	0.0214884i	$-0.993160\pi$
$757$	− 32.5334i	− 0.0429768i	0.999769	−	0.0214884i	$-0.00684050\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−959.120	−1.26034	−0.630171	−	0.776456i	$-0.717015\pi$
$761$	−959.120	−1.26034	−0.630171	+	0.776456i	$0.717015\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1714.30i	2.24092i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1494.10	1.94292	0.971459	−	0.237208i	$-0.0762322\pi$
$769$	1494.10	1.94292	0.971459	+	0.237208i	$0.0762322\pi$
$770$	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
$785$	1762.15	2.24477
$786$	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
$799$	3164.63	3.96074
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2381.57i	2.96585i
$804$	0	0
$805$	− 2127.98i	− 2.64346i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1041.87	−1.28785	−0.643924	−	0.765089i	$-0.722695\pi$
$809$	−1041.87	−1.28785	−0.643924	+	0.765089i	$0.722695\pi$
$810$	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
$815$	−1403.69	−1.72232
$816$	0	0
$817$	−592.330	−0.725006
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1588.21i	1.93448i	0.253869	+	0.967239i	$0.418297\pi$
$821$	1588.21i	1.93448i	−0.253869	+	0.967239i	$0.581703\pi$
$822$	0	0
$823$	−1100.33	−1.33698	−0.668490	−	0.743721i	$-0.733059\pi$
$823$	−1100.33	−1.33698	−0.668490	+	0.743721i	$0.733059\pi$
$824$	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
$833$	−2344.98	−2.81511
$834$	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
$845$	− 948.898i	− 1.12296i
$846$	0	0
$847$	−1965.85	−2.32096
$848$	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
$855$	960.127	1.12296
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	10.3911i	0.0120968i	0.999982	+	0.00604839i	$0.00192528\pi$
$859$	10.3911i	0.0120968i	−0.999982	+	0.00604839i	$0.998075\pi$
$860$	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
$875$	1127.37i	1.28843i
$876$	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
$881$	−1721.84	−1.95442	−0.977209	−	0.212277i	$-0.931912\pi$
$881$	−1721.84	−1.95442	−0.977209	+	0.212277i	$0.931912\pi$
$882$	0	0
$883$	− 930.145i	− 1.05339i	−0.850054	−	0.526696i	$-0.823431\pi$
$883$	− 930.145i	− 1.05339i	0.850054	−	0.526696i	$-0.176569\pi$
$884$	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
$891$	− 1407.34i	− 1.57951i
$892$	0	0
$893$	− 1772.41i	− 1.98478i
$894$	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
$913$	−1563.71	−1.71272
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 278.473i	− 0.303679i
$918$	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
$931$	1313.35i	1.41069i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3309.48	−3.53955
$936$	0	0
$937$	−1764.29	−1.88291	−0.941457	−	0.337134i	$-0.890543\pi$
$937$	−1764.29	−1.88291	−0.941457	+	0.337134i	$0.890543\pi$
$938$	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	$-0.582984\pi$
$947$	− 1830.00i	− 1.93242i	0.257760	−	0.966209i	$-0.417016\pi$
$948$	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
$955$	587.939i	0.615643i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2329.37	−2.42895
$960$	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
$973$	2916.79i	2.99773i
$974$	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
$985$	2153.73	2.18653
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1087.12i	− 1.09921i
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	186.012i	0.186947i
$996$	0	0
$997$	− 1573.09i	− 1.57783i	−0.614503	−	0.788914i	$-0.710644\pi$
$997$	− 1573.09i	− 1.57783i	0.614503	−	0.788914i	$-0.289356\pi$
$998$	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.417.6	yes	8
4.3	odd	2	inner	1216.3.g.c.417.5	yes	8
8.3	odd	2	inner	1216.3.g.c.417.3	✓	8
8.5	even	2	inner	1216.3.g.c.417.4	yes	8
19.18	odd	2	CM	1216.3.g.c.417.6	yes	8
76.75	even	2	inner	1216.3.g.c.417.5	yes	8
152.37	odd	2	inner	1216.3.g.c.417.4	yes	8
152.75	even	2	inner	1216.3.g.c.417.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.g.c.417.3	✓	8	8.3	odd	2	inner
1216.3.g.c.417.3	✓	8	152.75	even	2	inner
1216.3.g.c.417.4	yes	8	8.5	even	2	inner
1216.3.g.c.417.4	yes	8	152.37	odd	2	inner
1216.3.g.c.417.5	yes	8	4.3	odd	2	inner
1216.3.g.c.417.5	yes	8	76.75	even	2	inner
1216.3.g.c.417.6	yes	8	1.1	even	1	trivial
1216.3.g.c.417.6	yes	8	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.61478i q^{5} +10.8685 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+5.61478i q^{5} +10.8685 q^{7} -9.00000 q^{9} -17.3746i q^{11} -33.9244 q^{17} +19.0000i q^{19} -34.8712 q^{23} -6.52575 q^{25} +61.0241i q^{35} +31.1752i q^{43} -50.5330i q^{45} -93.2847 q^{47} +69.1238 q^{49} +97.5545 q^{55} -56.5089i q^{61} -97.8163 q^{63} -137.072 q^{73} -188.835i q^{77} +81.0000 q^{81} -90.0000i q^{83} -190.478i q^{85} -106.681 q^{95} +156.371i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{9}+O(q^{10}) \) 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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