LMFDB - Embedded newform 2352.2.h.g.2255.1

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:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 14x^{2} + 196 $ x^4 - 14*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2255.1
Root			$3.24037 - 1.87083i$ of defining polynomial
Character	$\chi$	$=$	2352.2255
Dual form			2352.2.h.g.2255.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -3.74166i q^{5} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} -3.74166i q^{5} -3.00000 q^{9} -6.48074 q^{11} -6.48074 q^{15} -3.74166i q^{17} +6.92820i q^{19} +6.48074 q^{23} -9.00000 q^{25} +5.19615i q^{27} -3.46410i q^{31} +11.2250i q^{33} -8.00000 q^{37} +3.74166i q^{41} +11.2250i q^{45} -6.48074 q^{51} +24.2487i q^{55} +12.0000 q^{57} -11.2250i q^{69} +6.48074 q^{71} +15.5885i q^{75} +9.00000 q^{81} -14.0000 q^{85} -18.7083i q^{89} -6.00000 q^{93} +25.9230 q^{95} +19.4422 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} - 36 q^{25} - 32 q^{37} + 48 q^{57} + 36 q^{81} - 56 q^{85} - 24 q^{93}+O(q^{100}) $ 4 * q - 12 * q^9 - 36 * q^25 - 32 * q^37 + 48 * q^57 + 36 * q^81 - 56 * q^85 - 24 * q^93

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
$5$	− 3.74166i	− 1.67332i	−0.547723	−	0.836660i	$-0.684505\pi$
$5$	− 3.74166i	− 1.67332i	0.547723	−	0.836660i	$-0.315495\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−6.48074	−1.95402	−0.977008	−	0.213201i	$-0.931611\pi$
$11$	−6.48074	−1.95402	−0.977008	+	0.213201i	$0.931611\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−6.48074	−1.67332
$16$	0	0
$17$	− 3.74166i	− 0.907485i	−0.891133	−	0.453743i	$-0.850089\pi$
$17$	− 3.74166i	− 0.907485i	0.891133	−	0.453743i	$-0.149911\pi$
$18$	0	0
$19$	6.92820i	1.58944i	0.606977	+	0.794719i	$0.292382\pi$
$19$	6.92820i	1.58944i	−0.606977	+	0.794719i	$0.707618\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.48074	1.35133	0.675664	−	0.737210i	$-0.263857\pi$
$23$	6.48074	1.35133	0.675664	+	0.737210i	$0.263857\pi$
$24$	0	0
$25$	−9.00000	−1.80000
$26$	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
$31$	− 3.46410i	− 0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$	− 3.46410i	− 0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$	0	0
$33$	11.2250i	1.95402i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.74166i	0.584349i	0.956365	+	0.292174i	$0.0943788\pi$
$41$	3.74166i	0.584349i	−0.956365	+	0.292174i	$0.905621\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	11.2250i	1.67332i
$46$	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
$51$	−6.48074	−0.907485
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	24.2487i	3.26970i
$56$	0	0
$57$	12.0000	1.58944
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	− 11.2250i	− 1.35133i
$70$	0	0
$71$	6.48074	0.769122	0.384561	−	0.923099i	$-0.374353\pi$
$71$	6.48074	0.769122	0.384561	+	0.923099i	$0.374353\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	15.5885i	1.80000i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	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
$85$	−14.0000	−1.51851
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 18.7083i	− 1.98307i	−0.129823	−	0.991537i	$-0.541441\pi$
$89$	− 18.7083i	− 1.98307i	0.129823	−	0.991537i	$-0.458559\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.00000	−0.622171
$94$	0	0
$95$	25.9230	2.65964
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	19.4422	1.95402
$100$	0	0
$101$	− 18.7083i	− 1.86154i	−0.365600	−	0.930772i	$-0.619136\pi$
$101$	− 18.7083i	− 1.86154i	0.365600	−	0.930772i	$-0.380864\pi$
$102$	0	0
$103$	17.3205i	1.70664i	0.521387	+	0.853320i	$0.325415\pi$
$103$	17.3205i	1.70664i	−0.521387	+	0.853320i	$0.674585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.4422	−1.87955	−0.939775	−	0.341793i	$-0.888966\pi$
$107$	−19.4422	−1.87955	−0.939775	+	0.341793i	$0.888966\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	13.8564i	1.31519i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	− 24.2487i	− 2.26120i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	31.0000	2.81818
$122$	0	0
$123$	6.48074	0.584349
$124$	0	0
$125$	14.9666i	1.33866i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	19.4422	1.67332
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	10.3923i	0.881464i	0.897639	+	0.440732i	$0.145281\pi$
$139$	10.3923i	0.881464i	−0.897639	+	0.440732i	$0.854719\pi$
$140$	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
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	11.2250i	0.907485i
$154$	0	0
$155$	−12.9615	−1.04109
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	42.0000	3.26970
$166$	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
$171$	− 20.7846i	− 1.58944i
$172$	0	0
$173$	26.1916i	1.99131i	0.0931156	+	0.995655i	$0.470317\pi$
$173$	26.1916i	1.99131i	−0.0931156	+	0.995655i	$0.529683\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−19.4422	−1.45318	−0.726590	−	0.687071i	$-0.758896\pi$
$179$	−19.4422	−1.45318	−0.726590	+	0.687071i	$0.758896\pi$
$180$	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
$185$	29.9333i	2.20074i
$186$	0	0
$187$	24.2487i	1.77324i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.48074	−0.468930	−0.234465	−	0.972125i	$-0.575334\pi$
$191$	−6.48074	−0.468930	−0.234465	+	0.972125i	$0.575334\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	27.7128i	1.96451i	0.187552	+	0.982255i	$0.439945\pi$
$199$	27.7128i	1.96451i	−0.187552	+	0.982255i	$0.560055\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	14.0000	0.977802
$206$	0	0
$207$	−19.4422	−1.35133
$208$	0	0
$209$	− 44.8999i	− 3.10579i
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	− 11.2250i	− 0.769122i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	13.8564i	0.927894i	0.885863	+	0.463947i	$0.153567\pi$
$223$	13.8564i	0.927894i	−0.885863	+	0.463947i	$0.846433\pi$
$224$	0	0
$225$	27.0000	1.80000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	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
$237$	0	0
$238$	0	0
$239$	−19.4422	−1.25761	−0.628806	−	0.777562i	$-0.716456\pi$
$239$	−19.4422	−1.25761	−0.628806	+	0.777562i	$0.716456\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	− 15.5885i	− 1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	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
$253$	−42.0000	−2.64052
$254$	0	0
$255$	24.2487i	1.51851i
$256$	0	0
$257$	3.74166i	0.233398i	0.993167	+	0.116699i	$0.0372313\pi$
$257$	3.74166i	0.233398i	−0.993167	+	0.116699i	$0.962769\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−32.4037	−1.99810	−0.999049	−	0.0436021i	$-0.986117\pi$
$263$	−32.4037	−1.99810	−0.999049	+	0.0436021i	$0.986117\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−32.4037	−1.98307
$268$	0	0
$269$	18.7083i	1.14066i	0.821414	+	0.570332i	$0.193186\pi$
$269$	18.7083i	1.14066i	−0.821414	+	0.570332i	$0.806814\pi$
$270$	0	0
$271$	− 31.1769i	− 1.89386i	−0.321436	−	0.946931i	$-0.604165\pi$
$271$	− 31.1769i	− 1.89386i	0.321436	−	0.946931i	$-0.395835\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	58.3267	3.51723
$276$	0	0
$277$	−32.0000	−1.92269	−0.961347	−	0.275340i	$-0.911209\pi$
$277$	−32.0000	−1.92269	−0.961347	+	0.275340i	$0.911209\pi$
$278$	0	0
$279$	10.3923i	0.622171i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 20.7846i	− 1.23552i	−0.786368	−	0.617758i	$-0.788041\pi$
$283$	− 20.7846i	− 1.23552i	0.786368	−	0.617758i	$-0.211959\pi$
$284$	0	0
$285$	− 44.8999i	− 2.65964i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.1916i	1.53013i	0.643953	+	0.765065i	$0.277293\pi$
$293$	26.1916i	1.53013i	−0.643953	+	0.765065i	$0.722707\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 33.6749i	− 1.95402i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−32.4037	−1.86154
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 34.6410i	− 1.97707i	−0.151001	−	0.988534i	$-0.548250\pi$
$307$	− 34.6410i	− 1.97707i	0.151001	−	0.988534i	$-0.451750\pi$
$308$	0	0
$309$	30.0000	1.70664
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	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
$321$	33.6749i	1.87955i
$322$	0	0
$323$	25.9230	1.44239
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17.3205i	0.957826i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	24.0000	1.31519
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	22.4499i	1.21573i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−42.0000	−2.26120
$346$	0	0
$347$	32.4037	1.73952	0.869761	−	0.493473i	$-0.164273\pi$
$347$	32.4037	1.73952	0.869761	+	0.493473i	$0.164273\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 26.1916i	− 1.39404i	−0.717053	−	0.697019i	$-0.754509\pi$
$353$	− 26.1916i	− 1.39404i	0.717053	−	0.697019i	$-0.245491\pi$
$354$	0	0
$355$	− 24.2487i	− 1.28699i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.4422	−1.02612	−0.513061	−	0.858352i	$-0.671488\pi$
$359$	−19.4422	−1.02612	−0.513061	+	0.858352i	$0.671488\pi$
$360$	0	0
$361$	−29.0000	−1.52632
$362$	0	0
$363$	− 53.6936i	− 2.81818i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	27.7128i	1.44660i	0.690535	+	0.723299i	$0.257375\pi$
$367$	27.7128i	1.44660i	−0.690535	+	0.723299i	$0.742625\pi$
$368$	0	0
$369$	− 11.2250i	− 0.584349i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	0	0
$375$	25.9230	1.33866
$376$	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
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	− 24.2487i	− 1.22631i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	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
$405$	− 33.6749i	− 1.67332i
$406$	0	0
$407$	51.8459	2.56991
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	18.0000	0.881464
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	40.0000	1.94948	0.974740	−	0.223341i	$-0.0716964\pi$
$421$	40.0000	1.94948	0.974740	+	0.223341i	$0.0716964\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	33.6749i	1.63347i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.48074	−0.312166	−0.156083	−	0.987744i	$-0.549887\pi$
$431$	−6.48074	−0.312166	−0.156083	+	0.987744i	$0.549887\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	44.8999i	2.14785i
$438$	0	0
$439$	− 41.5692i	− 1.98399i	−0.126275	−	0.991995i	$-0.540302\pi$
$439$	− 41.5692i	− 1.98399i	0.126275	−	0.991995i	$-0.459698\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−32.4037	−1.53955	−0.769773	−	0.638317i	$-0.779631\pi$
$443$	−32.4037	−1.53955	−0.769773	+	0.638317i	$0.779631\pi$
$444$	0	0
$445$	−70.0000	−3.31832
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	− 24.2487i	− 1.14183i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	19.4422	0.907485
$460$	0	0
$461$	− 41.1582i	− 1.91693i	−0.285210	−	0.958465i	$-0.592063\pi$
$461$	− 41.1582i	− 1.91693i	0.285210	−	0.958465i	$-0.407937\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	22.4499i	1.04109i
$466$	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
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 62.3538i	− 2.86099i
$476$	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
$481$	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
$491$	−6.48074	−0.292472	−0.146236	−	0.989250i	$-0.546716\pi$
$491$	−6.48074	−0.292472	−0.146236	+	0.989250i	$0.546716\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	− 72.7461i	− 3.26970i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	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
$505$	−70.0000	−3.11496
$506$	0	0
$507$	22.5167i	1.00000i
$508$	0	0
$509$	− 26.1916i	− 1.16092i	−0.814288	−	0.580461i	$-0.802872\pi$
$509$	− 26.1916i	− 1.16092i	0.814288	−	0.580461i	$-0.197128\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−36.0000	−1.58944
$514$	0	0
$515$	64.8074	2.85576
$516$	0	0
$517$	0	0
$518$	0	0
$519$	45.3652	1.99131
$520$	0	0
$521$	18.7083i	0.819625i	0.912170	+	0.409812i	$0.134406\pi$
$521$	18.7083i	0.819625i	−0.912170	+	0.409812i	$0.865594\pi$
$522$	0	0
$523$	17.3205i	0.757373i	0.925525	+	0.378686i	$0.123624\pi$
$523$	17.3205i	0.757373i	−0.925525	+	0.378686i	$0.876376\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.9615	−0.564611
$528$	0	0
$529$	19.0000	0.826087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	72.7461i	3.14509i
$536$	0	0
$537$	33.6749i	1.45318i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	37.4166i	1.60275i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	51.8459	2.20074
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	42.0000	1.77324
$562$	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
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	11.2250i	0.468930i
$574$	0	0
$575$	−58.3267	−2.43239
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	6.92820i	0.287926i
$580$	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
$589$	24.0000	0.988903
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 48.6415i	− 1.99747i	−0.0502942	−	0.998734i	$-0.516016\pi$
$593$	− 48.6415i	− 1.99747i	0.0502942	−	0.998734i	$-0.483984\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	48.0000	1.96451
$598$	0	0
$599$	−45.3652	−1.85357	−0.926786	−	0.375591i	$-0.877440\pi$
$599$	−45.3652	−1.85357	−0.926786	+	0.375591i	$0.877440\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 115.991i	− 4.71572i
$606$	0	0
$607$	− 41.5692i	− 1.68724i	−0.536939	−	0.843621i	$-0.680419\pi$
$607$	− 41.5692i	− 1.68724i	0.536939	−	0.843621i	$-0.319581\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	46.0000	1.85792	0.928961	−	0.370177i	$-0.120703\pi$
$613$	46.0000	1.85792	0.928961	+	0.370177i	$0.120703\pi$
$614$	0	0
$615$	− 24.2487i	− 0.977802i
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	45.0333i	1.81004i	0.425367	+	0.905021i	$0.360145\pi$
$619$	45.0333i	1.81004i	−0.425367	+	0.905021i	$0.639855\pi$
$620$	0	0
$621$	33.6749i	1.35133i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−77.7689	−3.10579
$628$	0	0
$629$	29.9333i	1.19352i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−19.4422	−0.769122
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	− 34.6410i	− 1.36611i	−0.730368	−	0.683054i	$-0.760651\pi$
$643$	− 34.6410i	− 1.36611i	0.730368	−	0.683054i	$-0.239349\pi$
$644$	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
$657$	0	0
$658$	0	0
$659$	−45.3652	−1.76718	−0.883588	−	0.468264i	$-0.844879\pi$
$659$	−45.3652	−1.76718	−0.883588	+	0.468264i	$0.844879\pi$
$660$	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
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	24.0000	0.927894
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	0	0
$675$	− 46.7654i	− 1.80000i
$676$	0	0
$677$	− 41.1582i	− 1.58184i	−0.611920	−	0.790920i	$-0.709603\pi$
$677$	− 41.1582i	− 1.58184i	0.611920	−	0.790920i	$-0.290397\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−32.4037	−1.23989	−0.619947	−	0.784644i	$-0.712846\pi$
$683$	−32.4037	−1.23989	−0.619947	+	0.784644i	$0.712846\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 31.1769i	− 1.18603i	−0.805193	−	0.593013i	$-0.797938\pi$
$691$	− 31.1769i	− 1.18603i	0.805193	−	0.593013i	$-0.202062\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	38.8844	1.47497
$696$	0	0
$697$	14.0000	0.530288
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	− 55.4256i	− 2.09042i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 22.4499i	− 0.840757i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	33.6749i	1.25761i
$718$	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
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10.3923i	0.385429i	0.981255	+	0.192715i	$0.0617292\pi$
$727$	10.3923i	0.385429i	−0.981255	+	0.192715i	$0.938271\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−32.4037	−1.18878	−0.594388	−	0.804178i	$-0.702606\pi$
$743$	−32.4037	−1.18878	−0.594388	+	0.804178i	$0.702606\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	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
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	72.7461i	2.64052i
$760$	0	0
$761$	− 41.1582i	− 1.49198i	−0.665955	−	0.745992i	$-0.731976\pi$
$761$	− 41.1582i	− 1.49198i	0.665955	−	0.745992i	$-0.268024\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	42.0000	1.51851
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	6.48074	0.233398
$772$	0	0
$773$	48.6415i	1.74951i	0.484561	+	0.874757i	$0.338979\pi$
$773$	48.6415i	1.74951i	−0.484561	+	0.874757i	$0.661021\pi$
$774$	0	0
$775$	31.1769i	1.11991i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−25.9230	−0.928787
$780$	0	0
$781$	−42.0000	−1.50288
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 51.9615i	− 1.85223i	−0.377243	−	0.926114i	$-0.623128\pi$
$787$	− 51.9615i	− 1.85223i	0.377243	−	0.926114i	$-0.376872\pi$
$788$	0	0
$789$	56.1249i	1.99810i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.74166i	0.132536i	0.997802	+	0.0662682i	$0.0211093\pi$
$797$	3.74166i	0.132536i	−0.997802	+	0.0662682i	$0.978891\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	56.1249i	1.98307i
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	32.4037	1.14066
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	− 38.1051i	− 1.33805i	−0.743239	−	0.669026i	$-0.766712\pi$
$811$	− 38.1051i	− 1.33805i	0.743239	−	0.669026i	$-0.233288\pi$
$812$	0	0
$813$	−54.0000	−1.89386
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	− 101.025i	− 3.51723i
$826$	0	0
$827$	45.3652	1.57750	0.788751	−	0.614713i	$-0.210728\pi$
$827$	45.3652	1.57750	0.788751	+	0.614713i	$0.210728\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	55.4256i	1.92269i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	18.0000	0.622171
$838$	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
$845$	48.6415i	1.67332i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−36.0000	−1.23552
$850$	0	0
$851$	−51.8459	−1.77726
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	−77.7689	−2.65964
$856$	0	0
$857$	− 41.1582i	− 1.40594i	−0.711220	−	0.702969i	$-0.751857\pi$
$857$	− 41.1582i	− 1.40594i	0.711220	−	0.702969i	$-0.248143\pi$
$858$	0	0
$859$	6.92820i	0.236387i	0.992991	+	0.118194i	$0.0377103\pi$
$859$	6.92820i	0.236387i	−0.992991	+	0.118194i	$0.962290\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	58.3267	1.98546	0.992731	−	0.120351i	$-0.0384020\pi$
$863$	58.3267	1.98546	0.992731	+	0.120351i	$0.0384020\pi$
$864$	0	0
$865$	98.0000	3.33210
$866$	0	0
$867$	− 5.19615i	− 0.176471i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	45.3652	1.53013
$880$	0	0
$881$	18.7083i	0.630298i	0.949042	+	0.315149i	$0.102055\pi$
$881$	18.7083i	0.630298i	−0.949042	+	0.315149i	$0.897945\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	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
$891$	−58.3267	−1.95402
$892$	0	0
$893$	0	0
$894$	0	0
$895$	72.7461i	2.43164i
$896$	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
$909$	56.1249i	1.86154i
$910$	0	0
$911$	58.3267	1.93245	0.966224	−	0.257702i	$-0.0829654\pi$
$911$	58.3267	1.93245	0.966224	+	0.257702i	$0.0829654\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−60.0000	−1.97707
$922$	0	0
$923$	0	0
$924$	0	0
$925$	72.0000	2.36735
$926$	0	0
$927$	− 51.9615i	− 1.70664i
$928$	0	0
$929$	48.6415i	1.59588i	0.602739	+	0.797939i	$0.294076\pi$
$929$	48.6415i	1.59588i	−0.602739	+	0.797939i	$0.705924\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	90.7304	2.96720
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 3.74166i	− 0.121975i	−0.998139	−	0.0609873i	$-0.980575\pi$
$941$	− 3.74166i	− 0.121975i	0.998139	−	0.0609873i	$-0.0194249\pi$
$942$	0	0
$943$	24.2487i	0.789647i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.4037	−1.05298	−0.526489	−	0.850182i	$-0.676492\pi$
$947$	−32.4037	−1.05298	−0.526489	+	0.850182i	$0.676492\pi$
$948$	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
$955$	24.2487i	0.784670i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	58.3267	1.87955
$964$	0	0
$965$	14.9666i	0.481793i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	− 44.8999i	− 1.44239i
$970$	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
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	121.244i	3.87496i
$980$	0	0
$981$	30.0000	0.957826
$982$	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
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	103.692	3.28725
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	− 41.5692i	− 1.31519i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.h.g.2255.1	✓	4
3.2	odd	2	inner	2352.2.h.g.2255.2	yes	4
4.3	odd	2	inner	2352.2.h.g.2255.3	yes	4
7.6	odd	2	inner	2352.2.h.g.2255.4	yes	4
12.11	even	2	inner	2352.2.h.g.2255.4	yes	4
21.20	even	2	inner	2352.2.h.g.2255.3	yes	4
28.27	even	2	inner	2352.2.h.g.2255.2	yes	4
84.83	odd	2	CM	2352.2.h.g.2255.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.2.h.g.2255.1	✓	4	1.1	even	1	trivial
2352.2.h.g.2255.1	✓	4	84.83	odd	2	CM
2352.2.h.g.2255.2	yes	4	3.2	odd	2	inner
2352.2.h.g.2255.2	yes	4	28.27	even	2	inner
2352.2.h.g.2255.3	yes	4	4.3	odd	2	inner
2352.2.h.g.2255.3	yes	4	21.20	even	2	inner
2352.2.h.g.2255.4	yes	4	7.6	odd	2	inner
2352.2.h.g.2255.4	yes	4	12.11	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -3.74166i q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} -3.74166i q^{5} -3.00000 q^{9} -6.48074 q^{11} -6.48074 q^{15} -3.74166i q^{17} +6.92820i q^{19} +6.48074 q^{23} -9.00000 q^{25} +5.19615i q^{27} -3.46410i q^{31} +11.2250i q^{33} -8.00000 q^{37} +3.74166i q^{41} +11.2250i q^{45} -6.48074 q^{51} +24.2487i q^{55} +12.0000 q^{57} -11.2250i q^{69} +6.48074 q^{71} +15.5885i q^{75} +9.00000 q^{81} -14.0000 q^{85} -18.7083i q^{89} -6.00000 q^{93} +25.9230 q^{95} +19.4422 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} - 36 q^{25} - 32 q^{37} + 48 q^{57} + 36 q^{81} - 56 q^{85} - 24 q^{93}+O(q^{100}) \) 4 * q - 12 * q^9 - 36 * q^25 - 32 * q^37 + 48 * q^57 + 36 * q^81 - 56 * q^85 - 24 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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