LMFDB - Embedded newform 128.7.d.b.63.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [128,7,Mod(63,128)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("128.63"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 7, names="a")

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

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-1458] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$29.4469227033$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			63.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	128.63
Dual form			128.7.d.b.63.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-88.0000i q^{5} -729.000 q^{9} +1656.00i q^{13} +990.000 q^{17} +7881.00 q^{25} +36920.0i q^{29} +84744.0i q^{37} -84942.0 q^{41} +64152.0i q^{45} +117649. q^{49} +296296. i q^{53} +388440. i q^{61} +145728. q^{65} -427570. q^{73} +531441. q^{81} -87120.0i q^{85} -1.37896e6 q^{89} +1.47251e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 1458 q^{9} + 1980 q^{17} + 15762 q^{25} - 169884 q^{41} + 235298 q^{49} + 291456 q^{65} - 855140 q^{73} + 1062882 q^{81} - 2757924 q^{89} + 2945020 q^{97}+O(q^{100}) $ 2 * q - 1458 * q^9 + 1980 * q^17 + 15762 * q^25 - 169884 * q^41 + 235298 * q^49 + 291456 * q^65 - 855140 * q^73 + 1062882 * q^81 - 2757924 * q^89 + 2945020 * q^97

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		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	− 88.0000i	− 0.704000i	−0.936000	−	0.352000i	$-0.885502\pi$
$5$	− 88.0000i	− 0.704000i	0.936000	−	0.352000i	$-0.114498\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	−729.000	−1.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1656.00i	0.753755i	0.926263	+	0.376878i	$0.123002\pi$
$13$	1656.00i	0.753755i	−0.926263	+	0.376878i	$0.876998\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	990.000	0.201506	0.100753	−	0.994911i	$-0.467875\pi$
$17$	990.000	0.201506	0.100753	+	0.994911i	$0.467875\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	7881.00	0.504384
$26$	0	0
$27$	0	0
$28$	0	0
$29$	36920.0i	1.51380i	0.653532	+	0.756899i	$0.273286\pi$
$29$	36920.0i	1.51380i	−0.653532	+	0.756899i	$0.726714\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	84744.0i	1.67303i	0.547944	+	0.836515i	$0.315411\pi$
$37$	84744.0i	1.67303i	−0.547944	+	0.836515i	$0.684589\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−84942.0	−1.23245	−0.616227	−	0.787568i	$-0.711340\pi$
$41$	−84942.0	−1.23245	−0.616227	+	0.787568i	$0.711340\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	64152.0i	0.704000i
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	117649.	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	296296.i	1.99021i	0.0988400	+	0.995103i	$0.468487\pi$
$53$	296296.i	1.99021i	−0.0988400	+	0.995103i	$0.531513\pi$
$54$	0	0
$55$	0	0
$56$	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
$61$	388440.i	1.71133i	0.517528	+	0.855666i	$0.326852\pi$
$61$	388440.i	1.71133i	−0.517528	+	0.855666i	$0.673148\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	145728.	0.530644
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−427570.	−1.09910	−0.549552	−	0.835460i	$-0.685202\pi$
$73$	−427570.	−1.09910	−0.549552	+	0.835460i	$0.685202\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	531441.	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	− 87120.0i	− 0.141860i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.37896e6	−1.95606	−0.978030	−	0.208463i	$-0.933154\pi$
$89$	−1.37896e6	−1.95606	−0.978030	+	0.208463i	$0.933154\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.47251e6	1.61340	0.806702	−	0.590959i	$-0.201250\pi$
$97$	1.47251e6	1.61340	0.806702	+	0.590959i	$0.201250\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.16012e6i	− 1.12600i	−0.826457	−	0.563001i	$-0.809647\pi$
$101$	− 1.16012e6i	− 1.12600i	0.826457	−	0.563001i	$-0.190353\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	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
$109$	− 2.54916e6i	− 1.96842i	−0.177007	−	0.984210i	$-0.556641\pi$
$109$	− 2.54916e6i	− 1.96842i	0.177007	−	0.984210i	$-0.443359\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.12221e6	0.777748	0.388874	−	0.921291i	$-0.372864\pi$
$113$	1.12221e6	0.777748	0.388874	+	0.921291i	$0.372864\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.20722e6i	− 0.753755i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.77156e6	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 2.06853e6i	− 1.05909i
$126$	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
$135$	0	0
$136$	0	0
$137$	−2.56347e6	−0.996934	−0.498467	−	0.866909i	$-0.666104\pi$
$137$	−2.56347e6	−0.996934	−0.498467	+	0.866909i	$0.666104\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3.24896e6	1.06571
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3.30316e6i	− 0.998552i	−0.866443	−	0.499276i	$-0.833599\pi$
$149$	− 3.30316e6i	− 0.998552i	0.866443	−	0.499276i	$-0.166401\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−721710.	−0.201506
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.12226e6i	0.289999i	0.989432	+	0.144999i	$0.0463180\pi$
$157$	1.12226e6i	0.289999i	−0.989432	+	0.144999i	$0.953682\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	2.08447e6	0.431853
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.21298e6i	1.58622i	0.609080	+	0.793109i	$0.291539\pi$
$173$	8.21298e6i	1.58622i	−0.609080	+	0.793109i	$0.708461\pi$
$174$	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
$181$	1.12741e7i	1.90128i	0.310290	+	0.950642i	$0.399574\pi$
$181$	1.12741e7i	1.90128i	−0.310290	+	0.950642i	$0.600426\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.45747e6	1.17781
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	−1.43729e7	−1.99928	−0.999639	−	0.0268508i	$-0.991452\pi$
$193$	−1.43729e7	−1.99928	−0.999639	+	0.0268508i	$0.991452\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.34430e6i	0.829822i	0.909862	+	0.414911i	$0.136187\pi$
$197$	6.34430e6i	0.829822i	−0.909862	+	0.414911i	$0.863813\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	7.47490e6i	0.867648i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	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
$221$	1.63944e6i	0.151886i
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−5.74525e6	−0.504384
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	1.71508e7i	1.42816i	0.700064	+	0.714080i	$0.253155\pi$
$229$	1.71508e7i	1.42816i	−0.700064	+	0.714080i	$0.746845\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.49411e7	1.97173	0.985865	−	0.167543i	$-0.0535834\pi$
$233$	2.49411e7	1.97173	0.985865	+	0.167543i	$0.0535834\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−201058.	−0.0143638	−0.00718191	−	0.999974i	$-0.502286\pi$
$241$	−201058.	−0.0143638	−0.00718191	+	0.999974i	$0.502286\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.03531e7i	− 0.704000i
$246$	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
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.15960e7	−1.86137	−0.930686	−	0.365819i	$-0.880789\pi$
$257$	−3.15960e7	−1.86137	−0.930686	+	0.365819i	$0.880789\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 2.69147e7i	− 1.51380i
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	2.60740e7	1.40111
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 2.77248e7i	− 1.42434i	−0.702009	−	0.712168i	$-0.747714\pi$
$269$	− 2.77248e7i	− 1.42434i	0.702009	−	0.712168i	$-0.252286\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.20098e7i	− 0.565063i	−0.959258	−	0.282532i	$-0.908826\pi$
$277$	− 1.20098e7i	− 0.565063i	0.959258	−	0.282532i	$-0.0911742\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.08288e7	−0.488048	−0.244024	−	0.969769i	$-0.578467\pi$
$281$	−1.08288e7	−0.488048	−0.244024	+	0.969769i	$0.578467\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2.31575e7	−0.959395
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 3.25109e7i	− 1.29249i	−0.763131	−	0.646244i	$-0.776339\pi$
$293$	− 3.25109e7i	− 1.29249i	0.763131	−	0.646244i	$-0.223661\pi$
$294$	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
$305$	3.41827e7	1.20478
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−1.45704e7	−0.475157	−0.237578	−	0.971368i	$-0.576354\pi$
$313$	−1.45704e7	−0.475157	−0.237578	+	0.971368i	$0.576354\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4.80412e7i	− 1.50812i	−0.656805	−	0.754061i	$-0.728092\pi$
$317$	− 4.80412e7i	− 1.50812i	0.656805	−	0.754061i	$-0.271908\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1.30509e7i	0.380182i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	− 6.17784e7i	− 1.67303i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.63724e7	1.99548	0.997739	−	0.0672052i	$-0.0214082\pi$
$337$	7.63724e7	1.99548	0.997739	+	0.0672052i	$0.0214082\pi$
$338$	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
$349$	8.48891e7i	1.99699i	0.0548572	+	0.998494i	$0.482530\pi$
$349$	8.48891e7i	1.99699i	−0.0548572	+	0.998494i	$0.517470\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.70972e7	−1.75273	−0.876363	−	0.481651i	$-0.840037\pi$
$353$	−7.70972e7	−1.75273	−0.876363	+	0.481651i	$0.840037\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−4.70459e7	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.76262e7i	0.773769i
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	6.19227e7	1.23245
$370$	0	0
$371$	0	0
$372$	0	0
$373$	8.23390e7i	1.58664i	0.608803	+	0.793321i	$0.291650\pi$
$373$	8.23390e7i	1.58664i	−0.608803	+	0.793321i	$0.708350\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.11395e7	−1.14103
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.73716e6i	0.0974649i	0.998812	+	0.0487324i	$0.0155182\pi$
$389$	5.73716e6i	0.0974649i	−0.998812	+	0.0487324i	$0.984482\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.20790e8i	1.93046i	0.261404	+	0.965229i	$0.415814\pi$
$397$	1.20790e8i	1.93046i	−0.261404	+	0.965229i	$0.584186\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.23212e8	1.91082	0.955410	−	0.295282i	$-0.0954135\pi$
$401$	1.23212e8	1.91082	0.955410	+	0.295282i	$0.0954135\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 4.67668e7i	− 0.704000i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−8.57705e7	−1.25363	−0.626813	−	0.779169i	$-0.715641\pi$
$409$	−8.57705e7	−1.25363	−0.626813	+	0.779169i	$0.715641\pi$
$410$	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
$421$	1.46057e8i	1.95738i	0.205343	+	0.978690i	$0.434169\pi$
$421$	1.46057e8i	1.95738i	−0.205343	+	0.978690i	$0.565831\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.80219e6	0.101637
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	1.38726e8	1.70882	0.854408	−	0.519602i	$-0.173920\pi$
$433$	1.38726e8	1.70882	0.854408	+	0.519602i	$0.173920\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−8.57661e7	−1.00000
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	1.21349e8i	1.37707i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7.86233e7	−0.868585	−0.434292	−	0.900772i	$-0.643002\pi$
$449$	−7.86233e7	−0.868585	−0.434292	+	0.900772i	$0.643002\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.15600e7	0.854533	0.427267	−	0.904126i	$-0.359477\pi$
$457$	8.15600e7	0.854533	0.427267	+	0.904126i	$0.359477\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 4.55719e7i	− 0.465151i	−0.972578	−	0.232576i	$-0.925285\pi$
$461$	− 4.55719e7i	− 0.465151i	0.972578	−	0.232576i	$-0.0747153\pi$
$462$	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
$475$	0	0
$476$	0	0
$477$	− 2.16000e8i	− 1.99021i
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	−1.40336e8	−1.26106
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.29581e8i	− 1.13584i
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	3.65508e7i	0.305040i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−1.02091e8	−0.792705
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 2.56803e8i	− 1.94736i	−0.227916	−	0.973681i	$-0.573191\pi$
$509$	− 2.56803e8i	− 1.94736i	0.227916	−	0.973681i	$-0.426809\pi$
$510$	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
$521$	2.80651e8	1.98451	0.992256	−	0.124212i	$-0.0396402\pi$
$521$	2.80651e8	1.98451	0.992256	+	0.124212i	$0.0396402\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.48036e8	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.40664e8i	− 0.928969i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.44852e8i	0.914815i	0.889257	+	0.457407i	$0.151222\pi$
$541$	1.44852e8i	0.914815i	−0.889257	+	0.457407i	$0.848778\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.24326e8	−1.38577
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	− 2.83173e8i	− 1.71133i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.14235e8i	− 1.23973i	−0.784710	−	0.619863i	$-0.787188\pi$
$557$	− 2.14235e8i	− 1.23973i	0.784710	−	0.619863i	$-0.212812\pi$
$558$	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
$565$	− 9.87545e7i	− 0.547534i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.50122e8	1.90056	0.950281	−	0.311393i	$-0.100796\pi$
$569$	3.50122e8	1.90056	0.950281	+	0.311393i	$0.100796\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.72270e8	−1.93790	−0.968948	−	0.247264i	$-0.920469\pi$
$577$	−3.72270e8	−1.93790	−0.968948	+	0.247264i	$0.920469\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.06236e8	−0.530644
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.76744e8	−0.847578	−0.423789	−	0.905761i	$-0.639300\pi$
$593$	−1.76744e8	−0.847578	−0.423789	+	0.905761i	$0.639300\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	1.44143e8	0.664002	0.332001	−	0.943279i	$-0.392276\pi$
$601$	1.44143e8	0.664002	0.332001	+	0.943279i	$0.392276\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.55897e8i	0.704000i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	4.53944e8i	1.97070i	0.170544	+	0.985350i	$0.445447\pi$
$613$	4.53944e8i	1.97070i	−0.170544	+	0.985350i	$0.554553\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.30573e8	0.981642	0.490821	−	0.871260i	$-0.336697\pi$
$617$	2.30573e8	0.981642	0.490821	+	0.871260i	$0.336697\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−5.88898e7	−0.241213
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.38966e7i	0.337126i
$630$	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
$637$	1.94827e8i	0.753755i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.05573e8	−1.16022	−0.580111	−	0.814538i	$-0.696991\pi$
$641$	−3.05573e8	−1.16022	−0.580111	+	0.814538i	$0.696991\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.37583e7i	0.121239i	0.998161	+	0.0606193i	$0.0193076\pi$
$653$	3.37583e7i	0.121239i	−0.998161	+	0.0606193i	$0.980692\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.11699e8	1.09910
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	5.70458e8i	1.97524i	0.156876	+	0.987618i	$0.449858\pi$
$661$	5.70458e8i	1.97524i	−0.156876	+	0.987618i	$0.550142\pi$
$662$	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
$673$	5.89733e8	1.93468	0.967342	−	0.253473i	$-0.0815729\pi$
$673$	5.89733e8	1.93468	0.967342	+	0.253473i	$0.0815729\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.41874e8i	− 0.457232i	−0.973517	−	0.228616i	$-0.926580\pi$
$677$	− 1.41874e8i	− 0.457232i	0.973517	−	0.228616i	$-0.0734199\pi$
$678$	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
$685$	2.25585e8i	0.701842i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.90666e8	−1.50013
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8.40926e7	−0.248347
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.25978e8i	1.81721i	0.417658	+	0.908604i	$0.362851\pi$
$701$	6.25978e8i	1.81721i	−0.417658	+	0.908604i	$0.637149\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	3.09351e8i	0.867987i	0.900916	+	0.433993i	$0.142896\pi$
$709$	3.09351e8i	0.867987i	−0.900916	+	0.433993i	$0.857104\pi$
$710$	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	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.90967e8i	0.763535i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−3.87420e8	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 3.38086e8i	− 0.858449i	−0.903198	−	0.429225i	$-0.858787\pi$
$733$	− 3.38086e8i	− 0.858449i	0.903198	−	0.429225i	$-0.141213\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−2.90678e8	−0.702981
$746$	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
$757$	− 7.89096e8i	− 1.81904i	−0.415661	−	0.909520i	$-0.636450\pi$
$757$	− 7.89096e8i	− 1.81904i	0.415661	−	0.909520i	$-0.363550\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.35040e8	0.306413	0.153207	−	0.988194i	$-0.451040\pi$
$761$	1.35040e8	0.306413	0.153207	+	0.988194i	$0.451040\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	6.35105e7i	0.141860i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	8.16391e8	1.79523	0.897613	−	0.440785i	$-0.145300\pi$
$769$	8.16391e8	1.79523	0.897613	+	0.440785i	$0.145300\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 6.66362e8i	− 1.44269i	−0.692578	−	0.721343i	$-0.743525\pi$
$773$	− 6.66362e8i	− 1.44269i	0.692578	−	0.721343i	$-0.256475\pi$
$774$	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
$785$	9.87592e7	0.204159
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−6.43257e8	−1.28993
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6.82843e8i	− 1.34880i	−0.738368	−	0.674398i	$-0.764403\pi$
$797$	− 6.82843e8i	− 1.34880i	0.738368	−	0.674398i	$-0.235597\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	1.00526e9	1.95606
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.17457e8	−0.977302	−0.488651	−	0.872479i	$-0.662511\pi$
$809$	−5.17457e8	−0.977302	−0.488651	+	0.872479i	$0.662511\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	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
$821$	8.69722e7i	0.157163i	0.996908	+	0.0785816i	$0.0250391\pi$
$821$	8.69722e7i	0.157163i	−0.996908	+	0.0785816i	$0.974961\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	− 9.66949e8i	− 1.69723i	−0.529013	−	0.848613i	$-0.677438\pi$
$829$	− 9.66949e8i	− 1.69723i	0.529013	−	0.848613i	$-0.322562\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.16473e8	0.201506
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−7.68263e8	−1.29158
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.83434e8i	− 0.304025i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 9.26547e8i	− 1.49286i	−0.665462	−	0.746432i	$-0.731765\pi$
$853$	− 9.26547e8i	− 1.49286i	0.665462	−	0.746432i	$-0.268235\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.56602e8	1.36093	0.680467	−	0.732779i	$-0.261777\pi$
$857$	8.56602e8	1.36093	0.680467	+	0.732779i	$0.261777\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	7.22743e8	1.11670
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−1.07346e9	−1.61340
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.26879e9i	− 1.88101i	−0.339785	−	0.940503i	$-0.610354\pi$
$877$	− 1.26879e9i	− 1.88101i	0.339785	−	0.940503i	$-0.389646\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.31647e9	−1.92524	−0.962619	−	0.270860i	$-0.912692\pi$
$881$	−1.31647e9	−1.92524	−0.962619	+	0.270860i	$0.912692\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	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
$901$	2.93333e8i	0.401039i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.92123e8	1.33850
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	8.45727e8i	1.12600i
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	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
$925$	6.67867e8i	0.843850i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.50820e8	0.811734	0.405867	−	0.913932i	$-0.366970\pi$
$929$	6.50820e8	0.811734	0.405867	+	0.913932i	$0.366970\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.05307e9	−1.28009	−0.640044	−	0.768338i	$-0.721084\pi$
$937$	−1.05307e9	−1.28009	−0.640044	+	0.768338i	$0.721084\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.52058e9i	1.82490i	0.409185	+	0.912452i	$0.365813\pi$
$941$	1.52058e9i	1.82490i	−0.409185	+	0.912452i	$0.634187\pi$
$942$	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
$949$	− 7.08056e8i	− 0.828455i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.49042e9	1.72199	0.860996	−	0.508612i	$-0.169841\pi$
$953$	1.49042e9	1.72199	0.860996	+	0.508612i	$0.169841\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.87504e8	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.26482e9i	1.40749i
$966$	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
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.33909e9	−1.43591	−0.717953	−	0.696092i	$-0.754921\pi$
$977$	−1.33909e9	−1.43591	−0.717953	+	0.696092i	$0.754921\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1.85834e9i	1.96842i
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	5.58298e8	0.584194
$986$	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
$995$	0	0
$996$	0	0
$997$	− 1.80680e9i	− 1.82316i	−0.411127	−	0.911578i	$-0.634865\pi$
$997$	− 1.80680e9i	− 1.82316i	0.411127	−	0.911578i	$-0.365135\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.7.d.b.63.1	✓	2
4.3	odd	2	CM	128.7.d.b.63.1	✓	2
8.3	odd	2	inner	128.7.d.b.63.2	yes	2
8.5	even	2	inner	128.7.d.b.63.2	yes	2
16.3	odd	4		256.7.c.a.255.1		1
16.5	even	4		256.7.c.b.255.1		1
16.11	odd	4		256.7.c.b.255.1		1
16.13	even	4		256.7.c.a.255.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.7.d.b.63.1	✓	2	1.1	even	1	trivial
128.7.d.b.63.1	✓	2	4.3	odd	2	CM
128.7.d.b.63.2	yes	2	8.3	odd	2	inner
128.7.d.b.63.2	yes	2	8.5	even	2	inner
256.7.c.a.255.1		1	16.3	odd	4
256.7.c.a.255.1		1	16.13	even	4
256.7.c.b.255.1		1	16.5	even	4
256.7.c.b.255.1		1	16.11	odd	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-88.0000i q^{5} -729.000 q^{9} +1656.00i q^{13} +990.000 q^{17} +7881.00 q^{25} +36920.0i q^{29} +84744.0i q^{37} -84942.0 q^{41} +64152.0i q^{45} +117649. q^{49} +296296. i q^{53} +388440. i q^{61} +145728. q^{65} -427570. q^{73} +531441. q^{81} -87120.0i q^{85} -1.37896e6 q^{89} +1.47251e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 1458 q^{9} + 1980 q^{17} + 15762 q^{25} - 169884 q^{41} + 235298 q^{49} + 291456 q^{65} - 855140 q^{73} + 1062882 q^{81} - 2757924 q^{89} + 2945020 q^{97}+O(q^{100}) \) 2 * q - 1458 * q^9 + 1980 * q^17 + 15762 * q^25 - 169884 * q^41 + 235298 * q^49 + 291456 * q^65 - 855140 * q^73 + 1062882 * q^81 - 2757924 * q^89 + 2945020 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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