LMFDB - Embedded newform 1216.2.b.e.607.7

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:	$8$
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^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			607.7
Root			$0.656712 + 2.13746i$ of defining polynomial
Character	$\chi$	$=$	1216.607
Dual form			1216.2.b.e.607.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.04547i q^{5} -5.27492i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+3.04547i q^{5} -5.27492i q^{7} +3.00000 q^{9} -6.50958 q^{11} -7.27492 q^{17} -4.35890 q^{19} +4.00000i q^{23} -4.27492 q^{25} +16.0646 q^{35} -5.67232 q^{43} +9.13642i q^{45} +2.72508i q^{47} -20.8248 q^{49} -19.8248i q^{55} -10.8109i q^{61} -15.8248i q^{63} -5.82475 q^{73} +34.3375i q^{77} +9.00000 q^{81} -8.71780 q^{83} -22.1556i q^{85} -13.2749i q^{95} -19.5287 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} - 28 q^{17} - 4 q^{25} - 76 q^{49} + 44 q^{73} + 72 q^{81}+O(q^{100}) $ 8 * q + 24 * q^9 - 28 * q^17 - 4 * q^25 - 76 * q^49 + 44 * q^73 + 72 * 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
$5$	3.04547i	1.36198i	0.732294	+	0.680989i	$0.238450\pi$
$5$	3.04547i	1.36198i	−0.732294	+	0.680989i	$0.761550\pi$
$6$	0	0
$7$	− 5.27492i	− 1.99373i	−0.0791130	−	0.996866i	$-0.525209\pi$
$7$	− 5.27492i	− 1.99373i	0.0791130	−	0.996866i	$-0.474791\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−6.50958	−1.96271	−0.981356	−	0.192201i	$-0.938437\pi$
$11$	−6.50958	−1.96271	−0.981356	+	0.192201i	$0.938437\pi$
$12$	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
$17$	−7.27492	−1.76443	−0.882213	−	0.470850i	$-0.843947\pi$
$17$	−7.27492	−1.76443	−0.882213	+	0.470850i	$0.843947\pi$
$18$	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
$25$	−4.27492	−0.854983
$26$	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
$35$	16.0646	2.71542
$36$	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
$43$	−5.67232	−0.865021	−0.432511	−	0.901629i	$-0.642372\pi$
$43$	−5.67232	−0.865021	−0.432511	+	0.901629i	$0.642372\pi$
$44$	0	0
$45$	9.13642i	1.36198i
$46$	0	0
$47$	2.72508i	0.397494i	0.980051	+	0.198747i	$0.0636872\pi$
$47$	2.72508i	0.397494i	−0.980051	+	0.198747i	$0.936313\pi$
$48$	0	0
$49$	−20.8248	−2.97496
$50$	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
$55$	− 19.8248i	− 2.67317i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	− 10.8109i	− 1.38420i	−0.721803	−	0.692099i	$-0.756686\pi$
$61$	− 10.8109i	− 1.38420i	0.721803	−	0.692099i	$-0.243314\pi$
$62$	0	0
$63$	− 15.8248i	− 1.99373i
$64$	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
$73$	−5.82475	−0.681736	−0.340868	−	0.940111i	$-0.610721\pi$
$73$	−5.82475	−0.681736	−0.340868	+	0.940111i	$0.610721\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	34.3375i	3.91312i
$78$	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	$-0.658802\pi$
$83$	−8.71780	−0.956903	−0.478451	+	0.878114i	$0.658802\pi$
$84$	0	0
$85$	− 22.1556i	− 2.40311i
$86$	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
$95$	− 13.2749i	− 1.36198i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	−19.5287	−1.96271
$100$	0	0
$101$	− 17.4356i	− 1.73491i	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	− 17.4356i	− 1.73491i	0.497519	−	0.867453i	$-0.334245\pi$
$102$	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
$115$	−12.1819	−1.13597
$116$	0	0
$117$	0	0
$118$	0	0
$119$	38.3746i	3.51779i
$120$	0	0
$121$	31.3746	2.85224
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.20822i	0.197509i
$126$	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
$131$	4.83507	0.422442	0.211221	−	0.977438i	$-0.432256\pi$
$131$	4.83507	0.422442	0.211221	+	0.977438i	$0.432256\pi$
$132$	0	0
$133$	22.9928i	1.99373i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.2749	−1.30502	−0.652512	−	0.757778i	$-0.726285\pi$
$137$	−15.2749	−1.30502	−0.652512	+	0.757778i	$0.726285\pi$
$138$	0	0
$139$	18.6915	1.58539	0.792695	−	0.609618i	$-0.208677\pi$
$139$	18.6915	1.58539	0.792695	+	0.609618i	$0.208677\pi$
$140$	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
$149$	− 1.37097i	− 0.112314i	−0.998422	−	0.0561570i	$-0.982115\pi$
$149$	− 1.37097i	− 0.112314i	0.998422	−	0.0561570i	$-0.0178847\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−21.8248	−1.76443
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.4356i	1.39151i	0.718278	+	0.695756i	$0.244931\pi$
$157$	17.4356i	1.39151i	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	21.0997	1.66289
$162$	0	0
$163$	8.71780	0.682831	0.341415	−	0.939913i	$-0.389094\pi$
$163$	8.71780	0.682831	0.341415	+	0.939913i	$0.389094\pi$
$164$	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
$175$	22.5498i	1.70461i
$176$	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
$187$	47.3566	3.46306
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.3746i	0.750679i	0.926887	+	0.375339i	$0.122474\pi$
$191$	10.3746i	0.750679i	−0.926887	+	0.375339i	$0.877526\pi$
$192$	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	$-0.786679\pi$
$197$	− 17.4356i	− 1.24223i	0.783718	−	0.621117i	$-0.213321\pi$
$198$	0	0
$199$	− 23.8248i	− 1.68889i	−0.535641	−	0.844446i	$-0.679930\pi$
$199$	− 23.8248i	− 1.68889i	0.535641	−	0.844446i	$-0.320070\pi$
$200$	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
$209$	28.3746	1.96271
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 17.2749i	− 1.17814i
$216$	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
$225$	−12.8248	−0.854983
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 29.0838i	− 1.92191i	−0.276704	−	0.960955i	$-0.589242\pi$
$229$	− 29.0838i	− 1.92191i	0.276704	−	0.960955i	$-0.410758\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.9244	1.76388	0.881939	−	0.471364i	$-0.156238\pi$
$233$	26.9244	1.76388	0.881939	+	0.471364i	$0.156238\pi$
$234$	0	0
$235$	−8.29917	−0.541378
$236$	0	0
$237$	0	0
$238$	0	0
$239$	28.9244i	1.87097i	0.353373	+	0.935483i	$0.385035\pi$
$239$	28.9244i	1.87097i	−0.353373	+	0.935483i	$0.614965\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 63.4213i	− 4.05184i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.02134	−0.569422	−0.284711	−	0.958613i	$-0.591898\pi$
$251$	−9.02134	−0.569422	−0.284711	+	0.958613i	$0.591898\pi$
$252$	0	0
$253$	− 26.0383i	− 1.63701i
$254$	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
$263$	20.9244i	1.29026i	0.764075	+	0.645128i	$0.223196\pi$
$263$	20.9244i	1.29026i	−0.764075	+	0.645128i	$0.776804\pi$
$264$	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
$275$	27.8279	1.67809
$276$	0	0
$277$	30.7583i	1.84809i	0.382288	+	0.924043i	$0.375136\pi$
$277$	30.7583i	1.84809i	−0.382288	+	0.924043i	$0.624864\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−33.3851	−1.98454	−0.992270	−	0.124096i	$-0.960397\pi$
$283$	−33.3851	−1.98454	−0.992270	+	0.124096i	$0.960397\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	35.9244	2.11320
$290$	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
$301$	29.9210i	1.72462i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	32.9244	1.88525
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 21.2749i	− 1.20639i	−0.797594	−	0.603195i	$-0.793894\pi$
$311$	− 21.2749i	− 1.20639i	0.797594	−	0.603195i	$-0.206106\pi$
$312$	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
$315$	48.1939	2.71542
$316$	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
$323$	31.7106	1.76443
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	14.3746	0.792497
$330$	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
$343$	72.9244i	3.93755i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−34.2224	−1.83715	−0.918577	−	0.395242i	$-0.870661\pi$
$347$	−34.2224	−1.83715	−0.918577	+	0.395242i	$0.870661\pi$
$348$	0	0
$349$	36.8492i	1.97249i	0.165277	+	0.986247i	$0.447148\pi$
$349$	36.8492i	1.97249i	−0.165277	+	0.986247i	$0.552852\pi$
$350$	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
$359$	34.3746i	1.81422i	0.420892	+	0.907111i	$0.361717\pi$
$359$	34.3746i	1.81422i	−0.420892	+	0.907111i	$0.638283\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 17.7391i	− 0.928509i
$366$	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
$385$	−104.574	−5.32958
$386$	0	0
$387$	−17.0170	−0.865021
$388$	0	0
$389$	6.39449i	0.324213i	0.986773	+	0.162107i	$0.0518289\pi$
$389$	6.39449i	0.324213i	−0.986773	+	0.162107i	$0.948171\pi$
$390$	0	0
$391$	− 29.0997i	− 1.47163i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	13.5529i	0.680199i	0.940389	+	0.340099i	$0.110461\pi$
$397$	13.5529i	0.680199i	−0.940389	+	0.340099i	$0.889539\pi$
$398$	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
$405$	27.4093i	1.36198i
$406$	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
$415$	− 26.5498i	− 1.30328i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.71780	0.425892	0.212946	−	0.977064i	$-0.431694\pi$
$419$	8.71780	0.425892	0.212946	+	0.977064i	$0.431694\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	8.17525i	0.397494i
$424$	0	0
$425$	31.0997	1.50856
$426$	0	0
$427$	−57.0268	−2.75972
$428$	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
$441$	−62.4743	−2.97496
$442$	0	0
$443$	−9.85859	−0.468396	−0.234198	−	0.972189i	$-0.575246\pi$
$443$	−9.85859	−0.468396	−0.234198	+	0.972189i	$0.575246\pi$
$444$	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
$457$	−25.4743	−1.19164	−0.595818	−	0.803120i	$-0.703172\pi$
$457$	−25.4743	−1.19164	−0.595818	+	0.803120i	$0.703172\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 42.9402i	− 1.99992i	−0.00872311	−	0.999962i	$-0.502777\pi$
$461$	− 42.9402i	− 1.99992i	0.00872311	−	0.999962i	$-0.497223\pi$
$462$	0	0
$463$	− 31.8248i	− 1.47902i	−0.673145	−	0.739511i	$-0.735057\pi$
$463$	− 31.8248i	− 1.47902i	0.673145	−	0.739511i	$-0.264943\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.0596	1.62237	0.811183	−	0.584792i	$-0.198824\pi$
$467$	35.0596	1.62237	0.811183	+	0.584792i	$0.198824\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	36.9244	1.69779
$474$	0	0
$475$	18.6339	0.854983
$476$	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	$-0.942223\pi$
$491$	−43.5890	−1.96714	−0.983572	+	0.180517i	$0.942223\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	− 59.4743i	− 2.67317i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−22.8777	−1.02415	−0.512074	−	0.858941i	$-0.671123\pi$
$499$	−22.8777	−1.02415	−0.512074	+	0.858941i	$0.671123\pi$
$500$	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
$505$	53.0997	2.36290
$506$	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
$511$	30.7251i	1.35920i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 17.7391i	− 0.780166i
$518$	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
$539$	135.560	5.83900
$540$	0	0
$541$	41.2657i	1.77415i	0.461625	+	0.887075i	$0.347267\pi$
$541$	41.2657i	1.77415i	−0.461625	+	0.887075i	$0.652733\pi$
$542$	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
$549$	− 32.4328i	− 1.38420i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 38.5237i	− 1.63230i	−0.577838	−	0.816152i	$-0.696103\pi$
$557$	− 38.5237i	− 1.63230i	0.577838	−	0.816152i	$-0.303897\pi$
$558$	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
$567$	− 47.4743i	− 1.99373i
$568$	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	$-0.315677\pi$
$571$	26.1534	1.09449	0.547243	+	0.836974i	$0.315677\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 17.0997i	− 0.713105i
$576$	0	0
$577$	−40.0241	−1.66622	−0.833112	−	0.553104i	$-0.813443\pi$
$577$	−40.0241	−1.66622	−0.833112	+	0.553104i	$0.813443\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	45.9857i	1.90781i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.0361	−1.23972	−0.619862	−	0.784711i	$-0.712811\pi$
$587$	−30.0361	−1.23972	−0.619862	+	0.784711i	$0.712811\pi$
$588$	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
$595$	−116.869	−4.79116
$596$	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
$605$	95.5505i	3.88468i
$606$	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
$613$	− 49.0311i	− 1.98035i	−0.139837	−	0.990174i	$-0.544658\pi$
$613$	− 49.0311i	− 1.98035i	0.139837	−	0.990174i	$-0.455342\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−48.0241	−1.93338	−0.966689	−	0.255956i	$-0.917610\pi$
$617$	−48.0241	−1.93338	−0.966689	+	0.255956i	$0.917610\pi$
$618$	0	0
$619$	−43.5890	−1.75199	−0.875995	−	0.482321i	$-0.839794\pi$
$619$	−43.5890	−1.75199	−0.875995	+	0.482321i	$0.839794\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−28.0997	−1.12399
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 34.0241i	− 1.35448i	−0.735763	−	0.677239i	$-0.763176\pi$
$631$	− 34.0241i	− 1.35448i	0.735763	−	0.677239i	$-0.236824\pi$
$632$	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
$643$	35.8969	1.41564	0.707818	−	0.706395i	$-0.249680\pi$
$643$	35.8969	1.41564	0.707818	+	0.706395i	$0.249680\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 50.0241i	− 1.96665i	−0.181857	−	0.983325i	$-0.558211\pi$
$647$	− 50.0241i	− 1.96665i	0.181857	−	0.983325i	$-0.441789\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20.2509i	0.792479i	0.918147	+	0.396239i	$0.129685\pi$
$653$	20.2509i	0.792479i	−0.918147	+	0.396239i	$0.870315\pi$
$654$	0	0
$655$	14.7251i	0.575357i
$656$	0	0
$657$	−17.4743	−0.681736
$658$	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
$665$	−70.0241	−2.71542
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	70.3746i	2.71678i
$672$	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
$685$	− 46.5194i	− 1.77741i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	10.6958	0.406889	0.203445	−	0.979086i	$-0.434786\pi$
$691$	10.6958	0.406889	0.203445	+	0.979086i	$0.434786\pi$
$692$	0	0
$693$	103.012i	3.91312i
$694$	0	0
$695$	56.9244i	2.15927i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17.4356i	0.658533i	0.944237	+	0.329267i	$0.106802\pi$
$701$	17.4356i	0.658533i	−0.944237	+	0.329267i	$0.893198\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−91.9713	−3.45894
$708$	0	0
$709$	52.3068i	1.96442i	0.187779	+	0.982211i	$0.439871\pi$
$709$	52.3068i	1.96442i	−0.187779	+	0.982211i	$0.560129\pi$
$710$	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
$719$	− 5.62541i	− 0.209793i	−0.994483	−	0.104896i	$-0.966549\pi$
$719$	− 5.62541i	− 0.209793i	0.994483	−	0.104896i	$-0.0334511\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 15.4743i	− 0.573908i	−0.957944	−	0.286954i	$-0.907357\pi$
$727$	− 15.4743i	− 0.573908i	0.957944	−	0.286954i	$-0.0926427\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	41.2657	1.52627
$732$	0	0
$733$	− 52.3068i	− 1.93200i	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	− 52.3068i	− 1.93200i	0.258551	−	0.965998i	$-0.416755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−23.7150	−0.872370	−0.436185	−	0.899857i	$-0.643671\pi$
$739$	−23.7150	−0.872370	−0.436185	+	0.899857i	$0.643671\pi$
$740$	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
$745$	4.17525	0.152969
$746$	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
$757$	47.3566i	1.72121i	0.509276	+	0.860603i	$0.329913\pi$
$757$	47.3566i	1.72121i	−0.509276	+	0.860603i	$0.670087\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−31.2749	−1.13371	−0.566857	−	0.823816i	$-0.691841\pi$
$761$	−31.2749	−1.13371	−0.566857	+	0.823816i	$0.691841\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 66.4667i	− 2.40311i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	44.3746	1.60019	0.800094	−	0.599874i	$-0.204783\pi$
$769$	44.3746	1.60019	0.800094	+	0.599874i	$0.204783\pi$
$770$	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
$785$	−53.0997	−1.89521
$786$	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
$799$	− 19.8248i	− 0.701349i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	37.9167	1.33805
$804$	0	0
$805$	64.2585i	2.26481i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	46.5739	1.63745	0.818726	−	0.574184i	$-0.194681\pi$
$809$	46.5739	1.63745	0.818726	+	0.574184i	$0.194681\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	26.5498i	0.930000i
$816$	0	0
$817$	24.7251	0.865021
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 56.7966i	− 1.98221i	−0.133064	−	0.991107i	$-0.542482\pi$
$821$	− 56.7966i	− 1.98221i	0.133064	−	0.991107i	$-0.457518\pi$
$822$	0	0
$823$	44.5739i	1.55375i	0.629655	+	0.776875i	$0.283196\pi$
$823$	44.5739i	1.55375i	−0.629655	+	0.776875i	$0.716804\pi$
$824$	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
$833$	151.498	5.24911
$834$	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
$845$	− 39.5912i	− 1.36198i
$846$	0	0
$847$	− 165.498i	− 5.68659i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	52.3068i	1.79095i	0.445112	+	0.895475i	$0.353164\pi$
$853$	52.3068i	1.79095i	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	− 39.8248i	− 1.36198i
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	15.3425	0.523478	0.261739	−	0.965139i	$-0.415704\pi$
$859$	15.3425	0.523478	0.261739	+	0.965139i	$0.415704\pi$
$860$	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
$875$	11.6482	0.393781
$876$	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
$881$	−24.0241	−0.809392	−0.404696	−	0.914451i	$-0.632623\pi$
$881$	−24.0241	−0.809392	−0.404696	+	0.914451i	$0.632623\pi$
$882$	0	0
$883$	59.4234	1.99976	0.999879	−	0.0155546i	$-0.00495139\pi$
$883$	59.4234	1.99976	0.999879	+	0.0155546i	$0.00495139\pi$
$884$	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
$891$	−58.5862	−1.96271
$892$	0	0
$893$	− 11.8784i	− 0.397494i
$894$	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
$913$	56.7492	1.87812
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 25.5046i	− 0.842236i
$918$	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
$931$	90.7730	2.97496
$932$	0	0
$933$	0	0
$934$	0	0
$935$	144.223i	4.71661i
$936$	0	0
$937$	−57.4743	−1.87760	−0.938801	−	0.344460i	$-0.888062\pi$
$937$	−57.4743	−1.87760	−0.938801	+	0.344460i	$0.888062\pi$
$938$	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	$-0.958508\pi$
$947$	−61.0246	−1.98303	−0.991516	+	0.129983i	$0.958508\pi$
$948$	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
$955$	−31.5955	−1.02241
$956$	0	0
$957$	0	0
$958$	0	0
$959$	80.5739i	2.60187i
$960$	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
$973$	− 98.5960i	− 3.16084i
$974$	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
$985$	53.0997	1.69190
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 22.6893i	− 0.721478i
$990$	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
$995$	72.5577	2.30023
$996$	0	0
$997$	− 52.3802i	− 1.65890i	−0.558584	−	0.829448i	$-0.688655\pi$
$997$	− 52.3802i	− 1.65890i	0.558584	−	0.829448i	$-0.311345\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.b.e.607.7	yes	8
4.3	odd	2	inner	1216.2.b.e.607.8	yes	8
8.3	odd	2	inner	1216.2.b.e.607.2	yes	8
8.5	even	2	inner	1216.2.b.e.607.1	✓	8
19.18	odd	2	CM	1216.2.b.e.607.7	yes	8
76.75	even	2	inner	1216.2.b.e.607.8	yes	8
152.37	odd	2	inner	1216.2.b.e.607.1	✓	8
152.75	even	2	inner	1216.2.b.e.607.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.b.e.607.1	✓	8	8.5	even	2	inner
1216.2.b.e.607.1	✓	8	152.37	odd	2	inner
1216.2.b.e.607.2	yes	8	8.3	odd	2	inner
1216.2.b.e.607.2	yes	8	152.75	even	2	inner
1216.2.b.e.607.7	yes	8	1.1	even	1	trivial
1216.2.b.e.607.7	yes	8	19.18	odd	2	CM
1216.2.b.e.607.8	yes	8	4.3	odd	2	inner
1216.2.b.e.607.8	yes	8	76.75	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.04547i q^{5} -5.27492i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+3.04547i q^{5} -5.27492i q^{7} +3.00000 q^{9} -6.50958 q^{11} -7.27492 q^{17} -4.35890 q^{19} +4.00000i q^{23} -4.27492 q^{25} +16.0646 q^{35} -5.67232 q^{43} +9.13642i q^{45} +2.72508i q^{47} -20.8248 q^{49} -19.8248i q^{55} -10.8109i q^{61} -15.8248i q^{63} -5.82475 q^{73} +34.3375i q^{77} +9.00000 q^{81} -8.71780 q^{83} -22.1556i q^{85} -13.2749i q^{95} -19.5287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} - 28 q^{17} - 4 q^{25} - 76 q^{49} + 44 q^{73} + 72 q^{81}+O(q^{100}) \) 8 * q + 24 * q^9 - 28 * q^17 - 4 * q^25 - 76 * q^49 + 44 * q^73 + 72 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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