LMFDB - Embedded newform 2352.2.h.d.2255.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(2255,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2352.2255");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.h (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:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2255.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2352.2255
Dual form			2352.2.h.d.2255.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -3.00000 q^{9} +5.00000 q^{13} -8.66025i q^{19} +5.00000 q^{25} -5.19615i q^{27} -1.73205i q^{31} +11.0000 q^{37} +8.66025i q^{39} +1.73205i q^{43} +15.0000 q^{57} -14.0000 q^{61} +15.5885i q^{67} +17.0000 q^{73} +8.66025i q^{75} -5.19615i q^{79} +9.00000 q^{81} +3.00000 q^{93} +14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{9}+O(q^{10}) $ 2 * q - 6 * q^9	$ 2 q - 6 q^{9} + 10 q^{13} + 10 q^{25} + 22 q^{37} + 30 q^{57} - 28 q^{61} + 34 q^{73} + 18 q^{81} + 6 q^{93} + 28 q^{97}+O(q^{100}) $ 2 * q - 6 * q^9 + 10 * q^13 + 10 * q^25 + 22 * q^37 + 30 * q^57 - 28 * q^61 + 34 * q^73 + 18 * q^81 + 6 * q^93 + 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	1.73205i	1.00000i
$3$	1.73205i	1.00000i
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	− 8.66025i	− 1.98680i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	− 8.66025i	− 1.98680i	0.114708	−	0.993399i	$-0.463407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	− 5.19615i	− 1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	− 1.73205i	− 0.311086i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	− 1.73205i	− 0.311086i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.0000	1.80839	0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.0000	1.80839	0.904194	+	0.427121i	$0.140472\pi$
$38$	0	0
$39$	8.66025i	1.38675i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	1.73205i	0.264135i	0.991241	+	0.132068i	$0.0421616\pi$
$43$	1.73205i	0.264135i	−0.991241	+	0.132068i	$0.957838\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	15.0000	1.98680
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.5885i	1.90443i	0.305424	+	0.952217i	$0.401202\pi$
$67$	15.5885i	1.90443i	−0.305424	+	0.952217i	$0.598798\pi$
$68$	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
$73$	17.0000	1.98970	0.994850	−	0.101361i	$-0.0323196\pi$
$73$	17.0000	1.98970	0.994850	+	0.101361i	$0.0323196\pi$
$74$	0	0
$75$	8.66025i	1.00000i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 5.19615i	− 0.584613i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	− 5.19615i	− 0.584613i	0.956325	−	0.292306i	$-0.0944227\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	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
$93$	3.00000	0.311086
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 15.5885i	− 1.53598i	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	− 15.5885i	− 1.53598i	0.640464	−	0.767988i	$-0.278742\pi$
$104$	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
$109$	19.0000	1.81987	0.909935	−	0.414751i	$-0.136131\pi$
$109$	19.0000	1.81987	0.909935	+	0.414751i	$0.136131\pi$
$110$	0	0
$111$	19.0526i	1.80839i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−15.0000	−1.38675
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	22.5167i	1.99803i	0.0443678	+	0.999015i	$0.485873\pi$
$127$	22.5167i	1.99803i	−0.0443678	+	0.999015i	$0.514127\pi$
$128$	0	0
$129$	−3.00000	−0.264135
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	5.19615i	0.440732i	0.975417	+	0.220366i	$0.0707252\pi$
$139$	5.19615i	0.440732i	−0.975417	+	0.220366i	$0.929275\pi$
$140$	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
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	− 24.2487i	− 1.97333i	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	− 24.2487i	− 1.97333i	0.162758	−	0.986666i	$-0.447961\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	24.2487i	1.89931i	0.313304	+	0.949653i	$0.398564\pi$
$163$	24.2487i	1.89931i	−0.313304	+	0.949653i	$0.601436\pi$
$164$	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
$169$	12.0000	0.923077
$170$	0	0
$171$	25.9808i	1.98680i
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	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
$181$	−19.0000	−1.41226	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−19.0000	−1.41226	−0.706129	+	0.708083i	$0.749560\pi$
$182$	0	0
$183$	− 24.2487i	− 1.79252i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	23.0000	1.65558	0.827788	−	0.561041i	$-0.189599\pi$
$193$	23.0000	1.65558	0.827788	+	0.561041i	$0.189599\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	− 3.46410i	− 0.245564i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	− 3.46410i	− 0.245564i	0.992434	−	0.122782i	$-0.0391815\pi$
$200$	0	0
$201$	−27.0000	−1.90443
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	24.2487i	1.66935i	0.550743	+	0.834675i	$0.314345\pi$
$211$	24.2487i	1.66935i	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	29.4449i	1.98970i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	10.3923i	0.695920i	0.937509	+	0.347960i	$0.113126\pi$
$223$	10.3923i	0.695920i	−0.937509	+	0.347960i	$0.886874\pi$
$224$	0	0
$225$	−15.0000	−1.00000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	29.0000	1.91637	0.958187	−	0.286143i	$-0.0923732\pi$
$229$	29.0000	1.91637	0.958187	+	0.286143i	$0.0923732\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.00000	0.584613
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	15.5885i	1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 43.3013i	− 2.75519i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	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
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	− 17.3205i	− 1.05215i	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	− 17.3205i	− 1.05215i	0.850439	−	0.526073i	$-0.176336\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5.00000	−0.300421	−0.150210	−	0.988654i	$-0.547995\pi$
$277$	−5.00000	−0.300421	−0.150210	+	0.988654i	$0.547995\pi$
$278$	0	0
$279$	5.19615i	0.311086i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 22.5167i	− 1.33848i	−0.743048	−	0.669238i	$-0.766621\pi$
$283$	− 22.5167i	− 1.33848i	0.743048	−	0.669238i	$-0.233379\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	24.2487i	1.42148i
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 29.4449i	− 1.68051i	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	− 29.4449i	− 1.68051i	0.542194	−	0.840254i	$-0.317594\pi$
$308$	0	0
$309$	27.0000	1.53598
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	13.0000	0.734803	0.367402	−	0.930062i	$-0.380247\pi$
$313$	13.0000	0.734803	0.367402	+	0.930062i	$0.380247\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	25.0000	1.38675
$326$	0	0
$327$	32.9090i	1.81987i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 19.0526i	− 1.04722i	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	− 19.0526i	− 1.04722i	0.851957	−	0.523612i	$-0.175416\pi$
$332$	0	0
$333$	−33.0000	−1.80839
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−29.0000	−1.57973	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000	−1.57973	−0.789865	+	0.613280i	$0.789850\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	− 25.9808i	− 1.38675i
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−56.0000	−2.94737
$362$	0	0
$363$	− 19.0526i	− 1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 22.5167i	− 1.17536i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	− 22.5167i	− 1.17536i	0.809093	−	0.587680i	$-0.199959\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−25.0000	−1.29445	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000	−1.29445	−0.647225	+	0.762299i	$0.724071\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 25.9808i	− 1.33454i	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	− 25.9808i	− 1.33454i	0.744815	−	0.667271i	$-0.232538\pi$
$380$	0	0
$381$	−39.0000	−1.99803
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 5.19615i	− 0.264135i
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.00000	0.0501886	0.0250943	−	0.999685i	$-0.492011\pi$
$397$	1.00000	0.0501886	0.0250943	+	0.999685i	$0.492011\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 8.66025i	− 0.431398i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−31.0000	−1.53285	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	−31.0000	−1.53285	−0.766426	+	0.642333i	$0.777967\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.00000	−0.440732
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−41.0000	−1.99822	−0.999109	−	0.0422075i	$-0.986561\pi$
$421$	−41.0000	−1.99822	−0.999109	+	0.0422075i	$0.986561\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	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
$433$	37.0000	1.77811	0.889053	−	0.457804i	$-0.151364\pi$
$433$	37.0000	1.77811	0.889053	+	0.457804i	$0.151364\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 31.1769i	− 1.48799i	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	− 31.1769i	− 1.48799i	0.668184	−	0.743996i	$-0.267072\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	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
$453$	42.0000	1.97333
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.0000	1.45012	0.725059	−	0.688686i	$-0.241812\pi$
$457$	31.0000	1.45012	0.725059	+	0.688686i	$0.241812\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	1.73205i	0.0804952i	0.999190	+	0.0402476i	$0.0128147\pi$
$463$	1.73205i	0.0804952i	−0.999190	+	0.0402476i	$0.987185\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 24.2487i	− 1.11732i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 43.3013i	− 1.98680i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	55.0000	2.50778
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 39.8372i	− 1.80519i	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	− 39.8372i	− 1.80519i	0.430486	−	0.902597i	$-0.358342\pi$
$488$	0	0
$489$	−42.0000	−1.89931
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	43.3013i	1.93843i	0.246214	+	0.969216i	$0.420813\pi$
$499$	43.3013i	1.93843i	−0.246214	+	0.969216i	$0.579187\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	20.7846i	0.923077i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−45.0000	−1.98680
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 15.5885i	− 0.681636i	−0.940129	−	0.340818i	$-0.889296\pi$
$523$	− 15.5885i	− 0.681636i	0.940129	−	0.340818i	$-0.110704\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	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
$539$	0	0
$540$	0	0
$541$	−17.0000	−0.730887	−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000	−0.730887	−0.365444	+	0.930834i	$0.619083\pi$
$542$	0	0
$543$	− 32.9090i	− 1.41226i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.2487i	1.03680i	0.855138	+	0.518400i	$0.173472\pi$
$547$	24.2487i	1.03680i	−0.855138	+	0.518400i	$0.826528\pi$
$548$	0	0
$549$	42.0000	1.79252
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	8.66025i	0.366290i
$560$	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
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	8.66025i	0.362420i	0.983444	+	0.181210i	$0.0580014\pi$
$571$	8.66025i	0.362420i	−0.983444	+	0.181210i	$0.941999\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	0	0
$579$	39.8372i	1.65558i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−15.0000	−0.618064
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.00000	0.245564
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	0	0
$603$	− 46.7654i	− 1.90443i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	39.8372i	1.61694i	0.588537	+	0.808470i	$0.299704\pi$
$607$	39.8372i	1.61694i	−0.588537	+	0.808470i	$0.700296\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	46.7654i	1.87966i	0.341644	+	0.939829i	$0.389016\pi$
$619$	46.7654i	1.87966i	−0.341644	+	0.939829i	$0.610984\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 24.2487i	− 0.965326i	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	− 24.2487i	− 0.965326i	0.875806	−	0.482663i	$-0.160330\pi$
$632$	0	0
$633$	−42.0000	−1.66935
$634$	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
$643$	19.0526i	0.751360i	0.926750	+	0.375680i	$0.122591\pi$
$643$	19.0526i	0.751360i	−0.926750	+	0.375680i	$0.877409\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−51.0000	−1.98970
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−11.0000	−0.427850	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−11.0000	−0.427850	−0.213925	+	0.976850i	$0.568625\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−18.0000	−0.695920
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	0	0
$675$	− 25.9808i	− 1.00000i
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	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
$685$	0	0
$686$	0	0
$687$	50.2295i	1.91637i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	32.9090i	1.25192i	0.779857	+	0.625958i	$0.215292\pi$
$691$	32.9090i	1.25192i	−0.779857	+	0.625958i	$0.784708\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	− 95.2628i	− 3.59290i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	15.5885i	0.584613i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	24.2487i	0.901819i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	53.6936i	1.99138i	0.0927199	+	0.995692i	$0.470444\pi$
$727$	53.6936i	1.99138i	−0.0927199	+	0.995692i	$0.529556\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−43.0000	−1.58824	−0.794121	−	0.607760i	$-0.792068\pi$
$733$	−43.0000	−1.58824	−0.794121	+	0.607760i	$0.792068\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 39.8372i	− 1.46543i	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	− 39.8372i	− 1.46543i	0.680534	−	0.732717i	$-0.261748\pi$
$740$	0	0
$741$	75.0000	2.75519
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 53.6936i	− 1.95931i	−0.200698	−	0.979653i	$-0.564321\pi$
$751$	− 53.6936i	− 1.95931i	0.200698	−	0.979653i	$-0.435679\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−47.0000	−1.69486	−0.847432	−	0.530904i	$-0.821852\pi$
$769$	−47.0000	−1.69486	−0.847432	+	0.530904i	$0.821852\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	− 8.66025i	− 0.311086i
$776$	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
$785$	0	0
$786$	0	0
$787$	3.46410i	0.123482i	0.998092	+	0.0617409i	$0.0196653\pi$
$787$	3.46410i	0.123482i	−0.998092	+	0.0617409i	$0.980335\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−70.0000	−2.48577
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	− 10.3923i	− 0.364923i	−0.983213	−	0.182462i	$-0.941593\pi$
$811$	− 10.3923i	− 0.364923i	0.983213	−	0.182462i	$-0.0584065\pi$
$812$	0	0
$813$	30.0000	1.05215
$814$	0	0
$815$	0	0
$816$	0	0
$817$	15.0000	0.524784
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 24.2487i	− 0.845257i	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	− 24.2487i	− 0.845257i	0.906303	−	0.422628i	$-0.138892\pi$
$824$	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
$829$	53.0000	1.84077	0.920383	−	0.391018i	$-0.127877\pi$
$829$	53.0000	1.84077	0.920383	+	0.391018i	$0.127877\pi$
$830$	0	0
$831$	− 8.66025i	− 0.300421i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9.00000	−0.311086
$838$	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
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	39.0000	1.33848
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−23.0000	−0.787505	−0.393753	−	0.919216i	$-0.628823\pi$
$853$	−23.0000	−0.787505	−0.393753	+	0.919216i	$0.628823\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	17.3205i	0.590968i	0.955348	+	0.295484i	$0.0954809\pi$
$859$	17.3205i	0.590968i	−0.955348	+	0.295484i	$0.904519\pi$
$860$	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
$867$	29.4449i	1.00000i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	77.9423i	2.64097i
$872$	0	0
$873$	−42.0000	−1.42148
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	22.5167i	0.757746i	0.925449	+	0.378873i	$0.123688\pi$
$883$	22.5167i	0.757746i	−0.925449	+	0.378873i	$0.876312\pi$
$884$	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
$891$	0	0
$892$	0	0
$893$	0	0
$894$	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
$907$	57.1577i	1.89789i	0.315442	+	0.948945i	$0.397847\pi$
$907$	57.1577i	1.89789i	−0.315442	+	0.948945i	$0.602153\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	29.4449i	0.971296i	0.874154	+	0.485648i	$0.161416\pi$
$919$	29.4449i	0.971296i	−0.874154	+	0.485648i	$0.838584\pi$
$920$	0	0
$921$	51.0000	1.68051
$922$	0	0
$923$	0	0
$924$	0	0
$925$	55.0000	1.80839
$926$	0	0
$927$	46.7654i	1.53598i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	61.0000	1.99278	0.996392	−	0.0848755i	$-0.0270492\pi$
$937$	61.0000	1.99278	0.996392	+	0.0848755i	$0.0270492\pi$
$938$	0	0
$939$	22.5167i	0.734803i
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	85.0000	2.75922
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	28.0000	0.903226
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 46.7654i	− 1.50387i	−0.659236	−	0.751936i	$-0.729120\pi$
$967$	− 46.7654i	− 1.50387i	0.659236	−	0.751936i	$-0.270880\pi$
$968$	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
$973$	0	0
$974$	0	0
$975$	43.3013i	1.38675i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−57.0000	−1.81987
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	15.5885i	0.495184i	0.968864	+	0.247592i	$0.0796392\pi$
$991$	15.5885i	0.495184i	−0.968864	+	0.247592i	$0.920361\pi$
$992$	0	0
$993$	33.0000	1.04722
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−59.0000	−1.86855	−0.934274	−	0.356555i	$-0.883951\pi$
$997$	−59.0000	−1.86855	−0.934274	+	0.356555i	$0.883951\pi$
$998$	0	0
$999$	− 57.1577i	− 1.80839i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.h.d.2255.2		2
3.2	odd	2	CM	2352.2.h.d.2255.2		2
4.3	odd	2	inner	2352.2.h.d.2255.1		2
7.3	odd	6		336.2.bj.a.191.1	yes	2
7.5	odd	6		336.2.bj.d.95.1	yes	2
7.6	odd	2		2352.2.h.b.2255.1		2
12.11	even	2	inner	2352.2.h.d.2255.1		2
21.5	even	6		336.2.bj.d.95.1	yes	2
21.17	even	6		336.2.bj.a.191.1	yes	2
21.20	even	2		2352.2.h.b.2255.1		2
28.3	even	6		336.2.bj.d.191.1	yes	2
28.19	even	6		336.2.bj.a.95.1	✓	2
28.27	even	2		2352.2.h.b.2255.2		2
84.47	odd	6		336.2.bj.a.95.1	✓	2
84.59	odd	6		336.2.bj.d.191.1	yes	2
84.83	odd	2		2352.2.h.b.2255.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bj.a.95.1	✓	2	28.19	even	6
336.2.bj.a.95.1	✓	2	84.47	odd	6
336.2.bj.a.191.1	yes	2	7.3	odd	6
336.2.bj.a.191.1	yes	2	21.17	even	6
336.2.bj.d.95.1	yes	2	7.5	odd	6
336.2.bj.d.95.1	yes	2	21.5	even	6
336.2.bj.d.191.1	yes	2	28.3	even	6
336.2.bj.d.191.1	yes	2	84.59	odd	6
2352.2.h.b.2255.1		2	7.6	odd	2
2352.2.h.b.2255.1		2	21.20	even	2
2352.2.h.b.2255.2		2	28.27	even	2
2352.2.h.b.2255.2		2	84.83	odd	2
2352.2.h.d.2255.1		2	4.3	odd	2	inner
2352.2.h.d.2255.1		2	12.11	even	2	inner
2352.2.h.d.2255.2		2	1.1	even	1	trivial
2352.2.h.d.2255.2		2	3.2	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -3.00000 q^{9} +5.00000 q^{13} -8.66025i q^{19} +5.00000 q^{25} -5.19615i q^{27} -1.73205i q^{31} +11.0000 q^{37} +8.66025i q^{39} +1.73205i q^{43} +15.0000 q^{57} -14.0000 q^{61} +15.5885i q^{67} +17.0000 q^{73} +8.66025i q^{75} -5.19615i q^{79} +9.00000 q^{81} +3.00000 q^{93} +14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9}+O(q^{10}) \) 2 * q - 6 * q^9	\( 2 q - 6 q^{9} + 10 q^{13} + 10 q^{25} + 22 q^{37} + 30 q^{57} - 28 q^{61} + 34 q^{73} + 18 q^{81} + 6 q^{93} + 28 q^{97}+O(q^{100}) \) 2 * q - 6 * q^9 + 10 * q^13 + 10 * q^25 + 22 * q^37 + 30 * q^57 - 28 * q^61 + 34 * q^73 + 18 * q^81 + 6 * q^93 + 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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