LMFDB - Embedded newform 128.4.b.c.65.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,4,Mod(65,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("128.65");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	128.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:	$7.55224448073$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
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			65.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	128.65
Dual form			128.4.b.c.65.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{5} +27.0000 q^{9} +O(q^{10})$	$q-4.00000i q^{5} +27.0000 q^{9} -92.0000i q^{13} +94.0000 q^{17} +109.000 q^{25} -284.000i q^{29} +396.000i q^{37} -230.000 q^{41} -108.000i q^{45} -343.000 q^{49} +572.000i q^{53} +468.000i q^{61} -368.000 q^{65} -1098.00 q^{73} +729.000 q^{81} -376.000i q^{85} +1670.00 q^{89} -594.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 54 q^{9}+O(q^{10}) $ 2 * q + 54 * q^9	$ 2 q + 54 q^{9} + 188 q^{17} + 218 q^{25} - 460 q^{41} - 686 q^{49} - 736 q^{65} - 2196 q^{73} + 1458 q^{81} + 3340 q^{89} - 1188 q^{97}+O(q^{100}) $ 2 * q + 54 * q^9 + 188 * q^17 + 218 * q^25 - 460 * q^41 - 686 * q^49 - 736 * q^65 - 2196 * q^73 + 1458 * q^81 + 3340 * q^89 - 1188 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$-1$	$1$

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
$5$	− 4.00000i	− 0.357771i	−0.983870	−	0.178885i	$-0.942751\pi$
$5$	− 4.00000i	− 0.357771i	0.983870	−	0.178885i	$-0.0572491\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	27.0000	1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 92.0000i	− 1.96279i	−0.192012	−	0.981393i	$-0.561501\pi$
$13$	− 92.0000i	− 1.96279i	0.192012	−	0.981393i	$-0.438499\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	94.0000	1.34108	0.670540	−	0.741874i	$-0.266063\pi$
$17$	94.0000	1.34108	0.670540	+	0.741874i	$0.266063\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	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
$25$	109.000	0.872000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 284.000i	− 1.81853i	−0.416214	−	0.909267i	$-0.636643\pi$
$29$	− 284.000i	− 1.81853i	0.416214	−	0.909267i	$-0.363357\pi$
$30$	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
$35$	0	0
$36$	0	0
$37$	396.000i	1.75951i	0.475424	+	0.879757i	$0.342295\pi$
$37$	396.000i	1.75951i	−0.475424	+	0.879757i	$0.657705\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−230.000	−0.876097	−0.438048	−	0.898951i	$-0.644330\pi$
$41$	−230.000	−0.876097	−0.438048	+	0.898951i	$0.644330\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	− 108.000i	− 0.357771i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−343.000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	572.000i	1.48246i	0.671253	+	0.741229i	$0.265757\pi$
$53$	572.000i	1.48246i	−0.671253	+	0.741229i	$0.734243\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	468.000i	0.982316i	0.871071	+	0.491158i	$0.163426\pi$
$61$	468.000i	0.982316i	−0.871071	+	0.491158i	$0.836574\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−368.000	−0.702227
$66$	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
$73$	−1098.00	−1.76043	−0.880214	−	0.474578i	$-0.842601\pi$
$73$	−1098.00	−1.76043	−0.880214	+	0.474578i	$0.842601\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	729.000	1.00000
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	− 376.000i	− 0.479799i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1670.00	1.98898	0.994492	−	0.104809i	$-0.0334231\pi$
$89$	1670.00	1.98898	0.994492	+	0.104809i	$0.0334231\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−594.000	−0.621769	−0.310884	−	0.950448i	$-0.600625\pi$
$97$	−594.000	−0.621769	−0.310884	+	0.950448i	$0.600625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1940.00i	− 1.91126i	−0.294570	−	0.955630i	$-0.595177\pi$
$101$	− 1940.00i	− 1.91126i	0.294570	−	0.955630i	$-0.404823\pi$
$102$	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
$109$	1460.00i	1.28296i	0.767140	+	0.641480i	$0.221679\pi$
$109$	1460.00i	1.28296i	−0.767140	+	0.641480i	$0.778321\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2002.00	1.66666	0.833329	−	0.552778i	$-0.186432\pi$
$113$	2002.00	1.66666	0.833329	+	0.552778i	$0.186432\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2484.00i	− 1.96279i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1331.00	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 936.000i	− 0.669747i
$126$	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
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1606.00	−1.00153	−0.500766	−	0.865583i	$-0.666948\pi$
$137$	−1606.00	−1.00153	−0.500766	+	0.865583i	$0.666948\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1136.00	−0.650618
$146$	0	0
$147$	0	0
$148$	0	0
$149$	940.000i	0.516831i	0.966034	+	0.258415i	$0.0832003\pi$
$149$	940.000i	0.516831i	−0.966034	+	0.258415i	$0.916800\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	2538.00	1.34108
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3924.00i	1.99471i	0.0726920	+	0.997354i	$0.476841\pi$
$157$	3924.00i	1.99471i	−0.0726920	+	0.997354i	$0.523159\pi$
$158$	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
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−6267.00	−2.85253
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2012.00i	− 0.884217i	−0.896962	−	0.442108i	$-0.854231\pi$
$173$	− 2012.00i	− 0.884217i	0.896962	−	0.442108i	$-0.145769\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	2860.00i	1.17449i	0.809410	+	0.587243i	$0.199787\pi$
$181$	2860.00i	1.17449i	−0.809410	+	0.587243i	$0.800213\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1584.00	0.629503
$186$	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
$193$	−5362.00	−1.99982	−0.999910	−	0.0134266i	$-0.995726\pi$
$193$	−5362.00	−1.99982	−0.999910	+	0.0134266i	$0.995726\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5404.00i	1.95441i	0.212295	+	0.977206i	$0.431906\pi$
$197$	5404.00i	1.95441i	−0.212295	+	0.977206i	$0.568094\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	920.000i	0.313442i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	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
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 8648.00i	− 2.63225i
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	2943.00	0.872000
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	2684.00i	0.774514i	0.921972	+	0.387257i	$0.126577\pi$
$229$	2684.00i	0.774514i	−0.921972	+	0.387257i	$0.873423\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	598.000	0.168139	0.0840693	−	0.996460i	$-0.473208\pi$
$233$	598.000	0.168139	0.0840693	+	0.996460i	$0.473208\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	5310.00	1.41928	0.709641	−	0.704563i	$-0.248857\pi$
$241$	5310.00	1.41928	0.709641	+	0.704563i	$0.248857\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1372.00i	0.357771i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1534.00	−0.372328	−0.186164	−	0.982519i	$-0.559606\pi$
$257$	−1534.00	−0.372328	−0.186164	+	0.982519i	$0.559606\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 7668.00i	− 1.81853i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	2288.00	0.530380
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 8140.00i	− 1.84500i	−0.385999	−	0.922499i	$-0.626143\pi$
$269$	− 8140.00i	− 1.84500i	0.385999	−	0.922499i	$-0.373857\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1316.00i	− 0.285454i	−0.989762	−	0.142727i	$-0.954413\pi$
$277$	− 1316.00i	− 0.285454i	0.989762	−	0.142727i	$-0.0455871\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7430.00	1.57735	0.788677	−	0.614807i	$-0.210766\pi$
$281$	7430.00	1.57735	0.788677	+	0.614807i	$0.210766\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3923.00	0.798494
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3452.00i	0.688287i	0.938917	+	0.344143i	$0.111831\pi$
$293$	3452.00i	0.688287i	−0.938917	+	0.344143i	$0.888169\pi$
$294$	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
$305$	1872.00	0.351444
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−6838.00	−1.23485	−0.617423	−	0.786632i	$-0.711823\pi$
$313$	−6838.00	−1.23485	−0.617423	+	0.786632i	$0.711823\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4676.00i	0.828487i	0.910166	+	0.414243i	$0.135954\pi$
$317$	4676.00i	0.828487i	−0.910166	+	0.414243i	$0.864046\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 10028.0i	− 1.71155i
$326$	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
$333$	10692.0i	1.75951i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12366.0	−1.99887	−0.999435	−	0.0336216i	$-0.989296\pi$
$337$	−12366.0	−1.99887	−0.999435	+	0.0336216i	$0.989296\pi$
$338$	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	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	8964.00i	1.37488i	0.726243	+	0.687438i	$0.241265\pi$
$349$	8964.00i	1.37488i	−0.726243	+	0.687438i	$0.758735\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3298.00	0.497266	0.248633	−	0.968598i	$-0.420019\pi$
$353$	3298.00	0.497266	0.248633	+	0.968598i	$0.420019\pi$
$354$	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
$361$	6859.00	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4392.00i	0.629830i
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−6210.00	−0.876097
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 6372.00i	− 0.884530i	−0.896884	−	0.442265i	$-0.854175\pi$
$373$	− 6372.00i	− 0.884530i	0.896884	−	0.442265i	$-0.145825\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−26128.0	−3.56939
$378$	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
$389$	15340.0i	1.99941i	0.0243735	+	0.999703i	$0.492241\pi$
$389$	15340.0i	1.99941i	−0.0243735	+	0.999703i	$0.507759\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12564.0i	1.58834i	0.607699	+	0.794168i	$0.292093\pi$
$397$	12564.0i	1.58834i	−0.607699	+	0.794168i	$0.707907\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2398.00	0.298629	0.149315	−	0.988790i	$-0.452293\pi$
$401$	2398.00	0.298629	0.149315	+	0.988790i	$0.452293\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 2916.00i	− 0.357771i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	7146.00	0.863929	0.431964	−	0.901891i	$-0.357821\pi$
$409$	7146.00	0.863929	0.431964	+	0.901891i	$0.357821\pi$
$410$	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	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	− 13412.0i	− 1.55264i	−0.630340	−	0.776319i	$-0.717084\pi$
$421$	− 13412.0i	− 1.55264i	0.630340	−	0.776319i	$-0.282916\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10246.0	1.16942
$426$	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
$433$	4862.00	0.539614	0.269807	−	0.962914i	$-0.413040\pi$
$433$	4862.00	0.539614	0.269807	+	0.962914i	$0.413040\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−9261.00	−1.00000
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	− 6680.00i	− 0.711601i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16114.0	−1.69369	−0.846845	−	0.531840i	$-0.821501\pi$
$449$	−16114.0	−1.69369	−0.846845	+	0.531840i	$0.821501\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16506.0	1.68954	0.844768	−	0.535132i	$-0.179738\pi$
$457$	16506.0	1.68954	0.844768	+	0.535132i	$0.179738\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 19660.0i	− 1.98624i	−0.117093	−	0.993121i	$-0.537358\pi$
$461$	− 19660.0i	− 1.98624i	0.117093	−	0.993121i	$-0.462642\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	15444.0i	1.48246i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	36432.0	3.45355
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2376.00i	0.222451i
$486$	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
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	− 26696.0i	− 2.43880i
$494$	0	0
$495$	0	0
$496$	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
$505$	−7760.00	−0.683793
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 17996.0i	− 1.56711i	−0.621323	−	0.783555i	$-0.713404\pi$
$509$	− 17996.0i	− 1.56711i	0.621323	−	0.783555i	$-0.286596\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	23738.0	1.99612	0.998062	−	0.0622265i	$-0.0198201\pi$
$521$	23738.0	1.99612	0.998062	+	0.0622265i	$0.0198201\pi$
$522$	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
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	21160.0i	1.71959i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 24460.0i	− 1.94384i	−0.235311	−	0.971920i	$-0.575611\pi$
$541$	− 24460.0i	− 1.94384i	0.235311	−	0.971920i	$-0.424389\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5840.00	0.459006
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	12636.0i	0.982316i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24836.0i	1.88929i	0.328093	+	0.944646i	$0.393594\pi$
$557$	24836.0i	1.88929i	−0.328093	+	0.944646i	$0.606406\pi$
$558$	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
$565$	− 8008.00i	− 0.596282i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26806.0	−1.97498	−0.987492	−	0.157669i	$-0.949602\pi$
$569$	−26806.0	−1.97498	−0.987492	+	0.157669i	$0.949602\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3454.00	−0.249206	−0.124603	−	0.992207i	$-0.539766\pi$
$577$	−3454.00	−0.249206	−0.124603	+	0.992207i	$0.539766\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−9936.00	−0.702227
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−15502.0	−1.07351	−0.536755	−	0.843738i	$-0.680350\pi$
$593$	−15502.0	−1.07351	−0.536755	+	0.843738i	$0.680350\pi$
$594$	0	0
$595$	0	0
$596$	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
$601$	17030.0	1.15585	0.577927	−	0.816089i	$-0.303862\pi$
$601$	17030.0	1.15585	0.577927	+	0.816089i	$0.303862\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 5324.00i	− 0.357771i
$606$	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
$613$	19548.0i	1.28799i	0.765031	+	0.643994i	$0.222724\pi$
$613$	19548.0i	1.28799i	−0.765031	+	0.643994i	$0.777276\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15466.0	−1.00914	−0.504569	−	0.863372i	$-0.668348\pi$
$617$	−15466.0	−1.00914	−0.504569	+	0.863372i	$0.668348\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	9881.00	0.632384
$626$	0	0
$627$	0	0
$628$	0	0
$629$	37224.0i	2.35965i
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	31556.0i	1.96279i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−28850.0	−1.77770	−0.888851	−	0.458197i	$-0.848495\pi$
$641$	−28850.0	−1.77770	−0.888851	+	0.458197i	$0.848495\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	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
$653$	1012.00i	0.0606472i	0.999540	+	0.0303236i	$0.00965378\pi$
$653$	1012.00i	0.0606472i	−0.999540	+	0.0303236i	$0.990346\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−29646.0	−1.76043
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	− 22068.0i	− 1.29856i	−0.760551	−	0.649278i	$-0.775071\pi$
$661$	− 22068.0i	− 1.29856i	0.760551	−	0.649278i	$-0.224929\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	4462.00	0.255568	0.127784	−	0.991802i	$-0.459214\pi$
$673$	4462.00	0.255568	0.127784	+	0.991802i	$0.459214\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 34996.0i	− 1.98671i	−0.115072	−	0.993357i	$-0.536710\pi$
$677$	− 34996.0i	− 1.98671i	0.115072	−	0.993357i	$-0.463290\pi$
$678$	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
$685$	6424.00i	0.358319i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	52624.0	2.90975
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−21620.0	−1.17492
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31252.0i	1.68384i	0.539602	+	0.841920i	$0.318575\pi$
$701$	31252.0i	1.68384i	−0.539602	+	0.841920i	$0.681425\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 8404.00i	− 0.445161i	−0.974914	−	0.222580i	$-0.928552\pi$
$709$	− 8404.00i	− 0.445161i	0.974914	−	0.222580i	$-0.0714479\pi$
$710$	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
$719$	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
$725$	− 30956.0i	− 1.58576i
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	19683.0	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 8732.00i	− 0.440005i	−0.975499	−	0.220003i	$-0.929393\pi$
$733$	− 8732.00i	− 0.440005i	0.975499	−	0.220003i	$-0.0706066\pi$
$734$	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
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	3760.00	0.184907
$746$	0	0
$747$	0	0
$748$	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
$757$	− 22516.0i	− 1.08105i	−0.841327	−	0.540527i	$-0.818225\pi$
$757$	− 22516.0i	− 1.08105i	0.841327	−	0.540527i	$-0.181775\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31882.0	1.51869	0.759344	−	0.650689i	$-0.225520\pi$
$761$	31882.0	1.51869	0.759344	+	0.650689i	$0.225520\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 10152.0i	− 0.479799i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−9650.00	−0.452520	−0.226260	−	0.974067i	$-0.572650\pi$
$769$	−9650.00	−0.452520	−0.226260	+	0.974067i	$0.572650\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 16852.0i	− 0.784119i	−0.919940	−	0.392060i	$-0.871763\pi$
$773$	− 16852.0i	− 0.784119i	0.919940	−	0.392060i	$-0.128237\pi$
$774$	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
$785$	15696.0	0.713649
$786$	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
$793$	43056.0	1.92807
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16276.0i	0.723370i	0.932300	+	0.361685i	$0.117798\pi$
$797$	16276.0i	0.723370i	−0.932300	+	0.361685i	$0.882202\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	45090.0	1.98898
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23270.0	−1.01129	−0.505643	−	0.862743i	$-0.668745\pi$
$809$	−23270.0	−1.01129	−0.505643	+	0.862743i	$0.668745\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 47012.0i	− 1.99845i	−0.0393212	−	0.999227i	$-0.512520\pi$
$821$	− 47012.0i	− 1.99845i	0.0393212	−	0.999227i	$-0.487480\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	− 41740.0i	− 1.74872i	−0.485276	−	0.874361i	$-0.661281\pi$
$829$	− 41740.0i	− 1.74872i	0.485276	−	0.874361i	$-0.338719\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−32242.0	−1.34108
$834$	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
$841$	−56267.0	−2.30706
$842$	0	0
$843$	0	0
$844$	0	0
$845$	25068.0i	1.02055i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	45468.0i	1.82508i	0.408986	+	0.912541i	$0.365883\pi$
$853$	45468.0i	1.82508i	−0.408986	+	0.912541i	$0.634117\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45994.0	1.83328	0.916642	−	0.399708i	$-0.130889\pi$
$857$	45994.0	1.83328	0.916642	+	0.399708i	$0.130889\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	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
$865$	−8048.00	−0.316347
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−16038.0	−0.621769
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29844.0i	1.14910i	0.818470	+	0.574550i	$0.194823\pi$
$877$	29844.0i	1.14910i	−0.818470	+	0.574550i	$0.805177\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7150.00	−0.273427	−0.136714	−	0.990611i	$-0.543654\pi$
$881$	−7150.00	−0.273427	−0.136714	+	0.990611i	$0.543654\pi$
$882$	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
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	53768.0i	1.98809i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11440.0	0.420197
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	− 52380.0i	− 1.91126i
$910$	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
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	43164.0i	1.53430i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−30866.0	−1.09008	−0.545038	−	0.838411i	$-0.683485\pi$
$929$	−30866.0	−1.09008	−0.545038	+	0.838411i	$0.683485\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−51946.0	−1.81110	−0.905551	−	0.424238i	$-0.860542\pi$
$937$	−51946.0	−1.81110	−0.905551	+	0.424238i	$0.860542\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 48460.0i	− 1.67880i	−0.543514	−	0.839400i	$-0.682907\pi$
$941$	− 48460.0i	− 1.67880i	0.543514	−	0.839400i	$-0.317093\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	101016.i	3.45534i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−56758.0	−1.92925	−0.964623	−	0.263632i	$-0.915079\pi$
$953$	−56758.0	−1.92925	−0.964623	+	0.263632i	$0.915079\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29791.0	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	21448.0i	0.715477i
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56606.0	1.85362	0.926810	−	0.375531i	$-0.122540\pi$
$977$	56606.0	1.85362	0.926810	+	0.375531i	$0.122540\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	39420.0i	1.28296i
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	21616.0	0.699232
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	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
$995$	0	0
$996$	0	0
$997$	− 34164.0i	− 1.08524i	−0.839978	−	0.542620i	$-0.817432\pi$
$997$	− 34164.0i	− 1.08524i	0.839978	−	0.542620i	$-0.182568\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.4.b.c.65.1	✓	2
3.2	odd	2		1152.4.d.d.577.2		2
4.3	odd	2	CM	128.4.b.c.65.1	✓	2
8.3	odd	2	inner	128.4.b.c.65.2	yes	2
8.5	even	2	inner	128.4.b.c.65.2	yes	2
12.11	even	2		1152.4.d.d.577.2		2
16.3	odd	4		256.4.a.d.1.1		1
16.5	even	4		256.4.a.e.1.1		1
16.11	odd	4		256.4.a.e.1.1		1
16.13	even	4		256.4.a.d.1.1		1
24.5	odd	2		1152.4.d.d.577.1		2
24.11	even	2		1152.4.d.d.577.1		2
48.5	odd	4		2304.4.a.g.1.1		1
48.11	even	4		2304.4.a.g.1.1		1
48.29	odd	4		2304.4.a.j.1.1		1
48.35	even	4		2304.4.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.b.c.65.1	✓	2	1.1	even	1	trivial
128.4.b.c.65.1	✓	2	4.3	odd	2	CM
128.4.b.c.65.2	yes	2	8.3	odd	2	inner
128.4.b.c.65.2	yes	2	8.5	even	2	inner
256.4.a.d.1.1		1	16.3	odd	4
256.4.a.d.1.1		1	16.13	even	4
256.4.a.e.1.1		1	16.5	even	4
256.4.a.e.1.1		1	16.11	odd	4
1152.4.d.d.577.1		2	24.5	odd	2
1152.4.d.d.577.1		2	24.11	even	2
1152.4.d.d.577.2		2	3.2	odd	2
1152.4.d.d.577.2		2	12.11	even	2
2304.4.a.g.1.1		1	48.5	odd	4
2304.4.a.g.1.1		1	48.11	even	4
2304.4.a.j.1.1		1	48.29	odd	4
2304.4.a.j.1.1		1	48.35	even	4

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:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	128.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{5} +27.0000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{5} +27.0000 q^{9} -92.0000i q^{13} +94.0000 q^{17} +109.000 q^{25} -284.000i q^{29} +396.000i q^{37} -230.000 q^{41} -108.000i q^{45} -343.000 q^{49} +572.000i q^{53} +468.000i q^{61} -368.000 q^{65} -1098.00 q^{73} +729.000 q^{81} -376.000i q^{85} +1670.00 q^{89} -594.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 54 q^{9}+O(q^{10}) \) 2 * q + 54 * q^9	\( 2 q + 54 q^{9} + 188 q^{17} + 218 q^{25} - 460 q^{41} - 686 q^{49} - 736 q^{65} - 2196 q^{73} + 1458 q^{81} + 3340 q^{89} - 1188 q^{97}+O(q^{100}) \) 2 * q + 54 * q^9 + 188 * q^17 + 218 * q^25 - 460 * q^41 - 686 * q^49 - 736 * q^65 - 2196 * q^73 + 1458 * q^81 + 3340 * q^89 - 1188 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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