LMFDB - Embedded newform 4032.2.b.k.3583.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(3583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4032.3583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.b (of order $2$, degree $1$, not 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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			3583.1
Root			$-1.16372i$ of defining polynomial
Character	$\chi$	$=$	4032.3583
Dual form			4032.2.b.k.3583.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.74166i q^{5} -2.64575 q^{7} +O(q^{10})$	$q-3.74166i q^{5} -2.64575 q^{7} +1.41421i q^{11} +3.74166i q^{17} +5.29150 q^{19} +7.07107i q^{23} -9.00000 q^{25} +10.5830 q^{31} +9.89949i q^{35} -8.00000 q^{37} +3.74166i q^{41} +7.00000 q^{49} +5.29150 q^{55} +15.5563i q^{71} -3.74166i q^{77} +14.0000 q^{85} -18.7083i q^{89} -19.7990i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 36 q^{25} - 32 q^{37} + 28 q^{49} + 56 q^{85}+O(q^{100}) $ 4 * q - 36 * q^25 - 32 * q^37 + 28 * q^49 + 56 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	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
$7$	−2.64575	−1.00000
$7$	−2.64575	−1.00000
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.41421i	0.426401i	0.977008	+	0.213201i	$0.0683888\pi$
$11$	1.41421i	0.426401i	−0.977008	+	0.213201i	$0.931611\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.74166i	0.907485i	0.891133	+	0.453743i	$0.149911\pi$
$17$	3.74166i	0.907485i	−0.891133	+	0.453743i	$0.850089\pi$
$18$	0	0
$19$	5.29150	1.21395	0.606977	−	0.794719i	$-0.292382\pi$
$19$	5.29150	1.21395	0.606977	+	0.794719i	$0.292382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.07107i	1.47442i	0.675664	+	0.737210i	$0.263857\pi$
$23$	7.07107i	1.47442i	−0.675664	+	0.737210i	$0.736143\pi$
$24$	0	0
$25$	−9.00000	−1.80000
$26$	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
$31$	10.5830	1.90076	0.950382	−	0.311086i	$-0.100693\pi$
$31$	10.5830	1.90076	0.950382	+	0.311086i	$0.100693\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.89949i	1.67332i
$36$	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
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	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
$55$	5.29150	0.713506
$56$	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
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	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
$69$	0	0
$70$	0	0
$71$	15.5563i	1.84620i	0.384561	+	0.923099i	$0.374353\pi$
$71$	15.5563i	1.84620i	−0.384561	+	0.923099i	$0.625647\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.74166i	− 0.426401i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	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
$93$	0	0
$94$	0	0
$95$	− 19.7990i	− 2.03133i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.7083i	1.86154i	0.365600	+	0.930772i	$0.380864\pi$
$101$	18.7083i	1.86154i	−0.365600	+	0.930772i	$0.619136\pi$
$102$	0	0
$103$	10.5830	1.04277	0.521387	−	0.853320i	$-0.325415\pi$
$103$	10.5830	1.04277	0.521387	+	0.853320i	$0.325415\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.07107i	− 0.683586i	−0.939775	−	0.341793i	$-0.888966\pi$
$107$	− 7.07107i	− 0.683586i	0.939775	−	0.341793i	$-0.111034\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	26.4575	2.46718
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 9.89949i	− 0.907485i
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	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
$133$	−14.0000	−1.21395
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	21.1660	1.79528	0.897639	−	0.440732i	$-0.145281\pi$
$139$	21.1660	1.79528	0.897639	+	0.440732i	$0.145281\pi$
$140$	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		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 39.5980i	− 3.18059i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 18.7083i	− 1.47442i
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	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
$171$	0	0
$172$	0	0
$173$	− 26.1916i	− 1.99131i	−0.0931156	−	0.995655i	$-0.529683\pi$
$173$	− 26.1916i	− 1.99131i	0.0931156	−	0.995655i	$-0.470317\pi$
$174$	0	0
$175$	23.8118	1.80000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.3848i	1.37414i	0.726590	+	0.687071i	$0.241104\pi$
$179$	18.3848i	1.37414i	−0.726590	+	0.687071i	$0.758896\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	29.9333i	2.20074i
$186$	0	0
$187$	−5.29150	−0.386953
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 26.8701i	− 1.94425i	−0.234465	−	0.972125i	$-0.575334\pi$
$191$	− 26.8701i	− 1.94425i	0.234465	−	0.972125i	$-0.424666\pi$
$192$	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		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−5.29150	−0.375105	−0.187552	−	0.982255i	$-0.560055\pi$
$199$	−5.29150	−0.375105	−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
$207$	0	0
$208$	0	0
$209$	7.48331i	0.517632i
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−28.0000	−1.90076
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	26.4575	1.77173	0.885863	−	0.463947i	$-0.153567\pi$
$223$	26.4575	1.77173	0.885863	+	0.463947i	$0.153567\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.0416i	1.55512i	0.628806	+	0.777562i	$0.283544\pi$
$239$	24.0416i	1.55512i	−0.628806	+	0.777562i	$0.716456\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 26.1916i	− 1.67332i
$246$	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
$253$	−10.0000	−0.628695
$254$	0	0
$255$	0	0
$256$	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
$259$	21.1660	1.31519
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 1.41421i	− 0.0872041i	−0.999049	−	0.0436021i	$-0.986117\pi$
$263$	− 1.41421i	− 0.0872041i	0.999049	−	0.0436021i	$-0.0138834\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	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
$271$	10.5830	0.642872	0.321436	−	0.946931i	$-0.395835\pi$
$271$	10.5830	0.642872	0.321436	+	0.946931i	$0.395835\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 12.7279i	− 0.767523i
$276$	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
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−26.4575	−1.57274	−0.786368	−	0.617758i	$-0.788041\pi$
$283$	−26.4575	−1.57274	−0.786368	+	0.617758i	$0.788041\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 9.89949i	− 0.584349i
$288$	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	$-0.722707\pi$
$293$	− 26.1916i	− 1.53013i	0.643953	−	0.765065i	$-0.277293\pi$
$294$	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
$307$	5.29150	0.302002	0.151001	−	0.988534i	$-0.451750\pi$
$307$	5.29150	0.302002	0.151001	+	0.988534i	$0.451750\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	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
$323$	19.7990i	1.10165i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	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
$341$	14.9666i	0.810488i
$342$	0	0
$343$	−18.5203	−1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.3848i	0.986947i	0.869761	+	0.493473i	$0.164273\pi$
$347$	18.3848i	0.986947i	−0.869761	+	0.493473i	$0.835727\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.1916i	1.39404i	0.717053	+	0.697019i	$0.245491\pi$
$353$	26.1916i	1.39404i	−0.717053	+	0.697019i	$0.754509\pi$
$354$	0	0
$355$	58.2065	3.08928
$356$	0	0
$357$	0	0
$358$	0	0
$359$	32.5269i	1.71670i	0.513061	+	0.858352i	$0.328512\pi$
$359$	32.5269i	1.71670i	−0.513061	+	0.858352i	$0.671488\pi$
$360$	0	0
$361$	9.00000	0.473684
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	26.4575	1.38107	0.690535	−	0.723299i	$-0.257375\pi$
$367$	26.4575	1.38107	0.690535	+	0.723299i	$0.257375\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	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
$385$	−14.0000	−0.713506
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−26.4575	−1.33801
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 11.3137i	− 0.560800i
$408$	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
$415$	0	0
$416$	0	0
$417$	0	0
$418$	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
$431$	41.0122i	1.97549i	0.156083	+	0.987744i	$0.450113\pi$
$431$	41.0122i	1.97549i	−0.156083	+	0.987744i	$0.549887\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	37.4166i	1.78988i
$438$	0	0
$439$	−5.29150	−0.252550	−0.126275	−	0.991995i	$-0.540302\pi$
$439$	−5.29150	−0.252550	−0.126275	+	0.991995i	$0.540302\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.8701i	1.27663i	0.769773	+	0.638317i	$0.220369\pi$
$443$	26.8701i	1.27663i	−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		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	−5.29150	−0.249167
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	41.1582i	1.91693i	0.285210	+	0.958465i	$0.407937\pi$
$461$	41.1582i	1.91693i	−0.285210	+	0.958465i	$0.592063\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	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
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−47.6235	−2.18512
$476$	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
$491$	43.8406i	1.97850i	0.146236	+	0.989250i	$0.453284\pi$
$491$	43.8406i	1.97850i	−0.146236	+	0.989250i	$0.546716\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 41.1582i	− 1.84620i
$498$	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
$507$	0	0
$508$	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
$513$	0	0
$514$	0	0
$515$	− 39.5980i	− 1.74490i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 18.7083i	− 0.819625i	−0.912170	−	0.409812i	$-0.865594\pi$
$521$	− 18.7083i	− 0.819625i	0.912170	−	0.409812i	$-0.134406\pi$
$522$	0	0
$523$	−42.3320	−1.85105	−0.925525	−	0.378686i	$-0.876376\pi$
$523$	−42.3320	−1.85105	−0.925525	+	0.378686i	$0.876376\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	39.5980i	1.72492i
$528$	0	0
$529$	−27.0000	−1.17391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−26.4575	−1.14386
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.89949i	0.426401i
$540$	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
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	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
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 63.6396i	− 2.65396i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	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
$589$	56.0000	2.30744
$590$	0	0
$591$	0	0
$592$	0	0
$593$	48.6415i	1.99747i	0.0502942	+	0.998734i	$0.483984\pi$
$593$	48.6415i	1.99747i	−0.0502942	+	0.998734i	$0.516016\pi$
$594$	0	0
$595$	−37.0405	−1.51851
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 18.3848i	− 0.751182i	−0.926786	−	0.375591i	$-0.877440\pi$
$599$	− 18.3848i	− 0.751182i	0.926786	−	0.375591i	$-0.122560\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 33.6749i	− 1.36908i
$606$	0	0
$607$	26.4575	1.07388	0.536939	−	0.843621i	$-0.319581\pi$
$607$	26.4575	1.07388	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
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	21.1660	0.850734	0.425367	−	0.905021i	$-0.360145\pi$
$619$	21.1660	0.850734	0.425367	+	0.905021i	$0.360145\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	49.4975i	1.98307i
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	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
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	37.0405	1.46074	0.730368	−	0.683054i	$-0.239349\pi$
$643$	37.0405	1.46074	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		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 24.0416i	− 0.936529i	−0.883588	−	0.468264i	$-0.844879\pi$
$659$	− 24.0416i	− 0.936529i	0.883588	−	0.468264i	$-0.155121\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	52.3832i	2.03133i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	41.1582i	1.58184i	0.611920	+	0.790920i	$0.290397\pi$
$677$	41.1582i	1.58184i	−0.611920	+	0.790920i	$0.709603\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 41.0122i	− 1.56929i	−0.619947	−	0.784644i	$-0.712846\pi$
$683$	− 41.0122i	− 1.56929i	0.619947	−	0.784644i	$-0.287154\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−42.3320	−1.61039	−0.805193	−	0.593013i	$-0.797938\pi$
$691$	−42.3320	−1.61039	−0.805193	+	0.593013i	$0.797938\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 79.1960i	− 3.00407i
$696$	0	0
$697$	−14.0000	−0.530288
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−42.3320	−1.59658
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 49.4975i	− 1.86154i
$708$	0	0
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	74.8331i	2.80252i
$714$	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
$721$	−28.0000	−1.04277
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−52.9150	−1.96251	−0.981255	−	0.192715i	$-0.938271\pi$
$727$	−52.9150	−1.96251	−0.981255	+	0.192715i	$0.938271\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	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
$743$	− 43.8406i	− 1.60836i	−0.594388	−	0.804178i	$-0.702606\pi$
$743$	− 43.8406i	− 1.60836i	0.594388	−	0.804178i	$-0.297394\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	18.7083i	0.683586i
$750$	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	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	0	0
$760$	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
$763$	−26.4575	−0.957826
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 48.6415i	− 1.74951i	−0.484561	−	0.874757i	$-0.661021\pi$
$773$	− 48.6415i	− 1.74951i	0.484561	−	0.874757i	$-0.338979\pi$
$774$	0	0
$775$	−95.2470	−3.42137
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.7990i	0.709372i
$780$	0	0
$781$	−22.0000	−0.787222
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	21.1660	0.754487	0.377243	−	0.926114i	$-0.376872\pi$
$787$	21.1660	0.754487	0.377243	+	0.926114i	$0.376872\pi$
$788$	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
$797$	− 3.74166i	− 0.132536i	−0.997802	−	0.0662682i	$-0.978891\pi$
$797$	− 3.74166i	− 0.132536i	0.997802	−	0.0662682i	$-0.0211093\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−70.0000	−2.46718
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−42.3320	−1.48648	−0.743239	−	0.669026i	$-0.766712\pi$
$811$	−42.3320	−1.48648	−0.743239	+	0.669026i	$0.766712\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.3553i	1.22943i	0.788751	+	0.614713i	$0.210728\pi$
$827$	35.3553i	1.22943i	−0.788751	+	0.614713i	$0.789272\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	26.1916i	0.907485i
$834$	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
$845$	− 48.6415i	− 1.67332i
$846$	0	0
$847$	−23.8118	−0.818182
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 56.5685i	− 1.93914i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	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
$859$	−58.2065	−1.98598	−0.992991	−	0.118194i	$-0.962290\pi$
$859$	−58.2065	−1.98598	−0.992991	+	0.118194i	$0.962290\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.07107i	0.240702i	0.992731	+	0.120351i	$0.0384020\pi$
$863$	7.07107i	0.240702i	−0.992731	+	0.120351i	$0.961598\pi$
$864$	0	0
$865$	−98.0000	−3.33210
$866$	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
$875$	− 39.5980i	− 1.33866i
$876$	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
$879$	0	0
$880$	0	0
$881$	− 18.7083i	− 0.630298i	−0.949042	−	0.315149i	$-0.897945\pi$
$881$	− 18.7083i	− 0.630298i	0.949042	−	0.315149i	$-0.102055\pi$
$882$	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
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	68.7895	2.29938
$896$	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
$909$	0	0
$910$	0	0
$911$	15.5563i	0.515405i	0.966224	+	0.257702i	$0.0829654\pi$
$911$	15.5563i	0.515405i	−0.966224	+	0.257702i	$0.917035\pi$
$912$	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
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	72.0000	2.36735
$926$	0	0
$927$	0	0
$928$	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
$931$	37.0405	1.21395
$932$	0	0
$933$	0	0
$934$	0	0
$935$	19.7990i	0.647496i
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	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
$943$	−26.4575	−0.861575
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.3259i	1.70036i	0.526489	+	0.850182i	$0.323508\pi$
$947$	52.3259i	1.70036i	−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		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	−100.539	−3.25335
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	81.0000	2.61290
$962$	0	0
$963$	0	0
$964$	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
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−56.0000	−1.79528
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	26.4575	0.845586
$980$	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
$985$	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
$995$	19.7990i	0.627670i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.b.k.3583.1		4
3.2	odd	2	inner	4032.2.b.k.3583.3		4
4.3	odd	2	inner	4032.2.b.k.3583.2		4
7.6	odd	2	inner	4032.2.b.k.3583.4		4
8.3	odd	2		252.2.b.b.55.2	yes	4
8.5	even	2		252.2.b.b.55.4	yes	4
12.11	even	2	inner	4032.2.b.k.3583.4		4
21.20	even	2	inner	4032.2.b.k.3583.2		4
24.5	odd	2		252.2.b.b.55.1	✓	4
24.11	even	2		252.2.b.b.55.3	yes	4
28.27	even	2	inner	4032.2.b.k.3583.3		4
56.13	odd	2		252.2.b.b.55.3	yes	4
56.27	even	2		252.2.b.b.55.1	✓	4
84.83	odd	2	CM	4032.2.b.k.3583.1		4
168.83	odd	2		252.2.b.b.55.4	yes	4
168.125	even	2		252.2.b.b.55.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.b.b.55.1	✓	4	24.5	odd	2
252.2.b.b.55.1	✓	4	56.27	even	2
252.2.b.b.55.2	yes	4	8.3	odd	2
252.2.b.b.55.2	yes	4	168.125	even	2
252.2.b.b.55.3	yes	4	24.11	even	2
252.2.b.b.55.3	yes	4	56.13	odd	2
252.2.b.b.55.4	yes	4	8.5	even	2
252.2.b.b.55.4	yes	4	168.83	odd	2
4032.2.b.k.3583.1		4	1.1	even	1	trivial
4032.2.b.k.3583.1		4	84.83	odd	2	CM
4032.2.b.k.3583.2		4	4.3	odd	2	inner
4032.2.b.k.3583.2		4	21.20	even	2	inner
4032.2.b.k.3583.3		4	3.2	odd	2	inner
4032.2.b.k.3583.3		4	28.27	even	2	inner
4032.2.b.k.3583.4		4	7.6	odd	2	inner
4032.2.b.k.3583.4		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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.74166i q^{5} -2.64575 q^{7} +O(q^{10})\)	\(q-3.74166i q^{5} -2.64575 q^{7} +1.41421i q^{11} +3.74166i q^{17} +5.29150 q^{19} +7.07107i q^{23} -9.00000 q^{25} +10.5830 q^{31} +9.89949i q^{35} -8.00000 q^{37} +3.74166i q^{41} +7.00000 q^{49} +5.29150 q^{55} +15.5563i q^{71} -3.74166i q^{77} +14.0000 q^{85} -18.7083i q^{89} -19.7990i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 36 q^{25} - 32 q^{37} + 28 q^{49} + 56 q^{85}+O(q^{100}) \) 4 * q - 36 * q^25 - 32 * q^37 + 28 * q^49 + 56 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more