LMFDB - Embedded newform 256.3.c.d.255.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,3,Mod(255,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	256.c (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:	$6.97549476762$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			255.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.3.c.d.255.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.65685i q^{3} -23.0000 q^{9} +O(q^{10})$	$q+5.65685i q^{3} -23.0000 q^{9} +16.9706i q^{11} -2.00000 q^{17} -16.9706i q^{19} -25.0000 q^{25} -79.1960i q^{27} -96.0000 q^{33} +46.0000 q^{41} +84.8528i q^{43} +49.0000 q^{49} -11.3137i q^{51} +96.0000 q^{57} +84.8528i q^{59} +118.794i q^{67} -142.000 q^{73} -141.421i q^{75} +241.000 q^{81} +50.9117i q^{83} +146.000 q^{89} +94.0000 q^{97} -390.323i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 46 q^{9}+O(q^{10}) $ 2 * q - 46 * q^9	$ 2 q - 46 q^{9} - 4 q^{17} - 50 q^{25} - 192 q^{33} + 92 q^{41} + 98 q^{49} + 192 q^{57} - 284 q^{73} + 482 q^{81} + 292 q^{89} + 188 q^{97}+O(q^{100}) $ 2 * q - 46 * q^9 - 4 * q^17 - 50 * q^25 - 192 * q^33 + 92 * q^41 + 98 * q^49 + 192 * q^57 - 284 * q^73 + 482 * q^81 + 292 * q^89 + 188 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$1$	$-1$

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
$3$	5.65685i	1.88562i	0.333333	+	0.942809i	$0.391827\pi$
$3$	5.65685i	1.88562i	−0.333333	+	0.942809i	$0.608173\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	−23.0000	−2.55556
$10$	0	0
$11$	16.9706i	1.54278i	0.636364	+	0.771389i	$0.280438\pi$
$11$	16.9706i	1.54278i	−0.636364	+	0.771389i	$0.719562\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.117647	−0.0588235	−	0.998268i	$-0.518735\pi$
$17$	−2.00000	−0.117647	−0.0588235	+	0.998268i	$0.518735\pi$
$18$	0	0
$19$	− 16.9706i	− 0.893188i	−0.894737	−	0.446594i	$-0.852637\pi$
$19$	− 16.9706i	− 0.893188i	0.894737	−	0.446594i	$-0.147363\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−25.0000	−1.00000
$26$	0	0
$27$	− 79.1960i	− 2.93318i
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	−96.0000	−2.90909
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	46.0000	1.12195	0.560976	−	0.827832i	$-0.310426\pi$
$41$	46.0000	1.12195	0.560976	+	0.827832i	$0.310426\pi$
$42$	0	0
$43$	84.8528i	1.97332i	0.162791	+	0.986661i	$0.447950\pi$
$43$	84.8528i	1.97332i	−0.162791	+	0.986661i	$0.552050\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	49.0000	1.00000
$50$	0	0
$51$	− 11.3137i	− 0.221837i
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	96.0000	1.68421
$58$	0	0
$59$	84.8528i	1.43818i	0.694915	+	0.719092i	$0.255442\pi$
$59$	84.8528i	1.43818i	−0.694915	+	0.719092i	$0.744558\pi$
$60$	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
$67$	118.794i	1.77304i	0.462687	+	0.886522i	$0.346886\pi$
$67$	118.794i	1.77304i	−0.462687	+	0.886522i	$0.653114\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−142.000	−1.94521	−0.972603	−	0.232473i	$-0.925318\pi$
$73$	−142.000	−1.94521	−0.972603	+	0.232473i	$0.925318\pi$
$74$	0	0
$75$	− 141.421i	− 1.88562i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	241.000	2.97531
$82$	0	0
$83$	50.9117i	0.613394i	0.951807	+	0.306697i	$0.0992238\pi$
$83$	50.9117i	0.613394i	−0.951807	+	0.306697i	$0.900776\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	146.000	1.64045	0.820225	−	0.572041i	$-0.193848\pi$
$89$	146.000	1.64045	0.820225	+	0.572041i	$0.193848\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	94.0000	0.969072	0.484536	−	0.874771i	$-0.338988\pi$
$97$	94.0000	0.969072	0.484536	+	0.874771i	$0.338988\pi$
$98$	0	0
$99$	− 390.323i	− 3.94266i
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 118.794i	− 1.11022i	−0.831776	−	0.555112i	$-0.812675\pi$
$107$	− 118.794i	− 1.11022i	0.831776	−	0.555112i	$-0.187325\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	98.0000	0.867257	0.433628	−	0.901092i	$-0.357233\pi$
$113$	98.0000	0.867257	0.433628	+	0.901092i	$0.357233\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−167.000	−1.38017
$122$	0	0
$123$	260.215i	2.11557i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	−480.000	−3.72093
$130$	0	0
$131$	254.558i	1.94319i	0.236641	+	0.971597i	$0.423953\pi$
$131$	254.558i	1.94319i	−0.236641	+	0.971597i	$0.576047\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	238.000	1.73723	0.868613	−	0.495491i	$-0.165012\pi$
$137$	238.000	1.73723	0.868613	+	0.495491i	$0.165012\pi$
$138$	0	0
$139$	− 186.676i	− 1.34299i	−0.741007	−	0.671497i	$-0.765652\pi$
$139$	− 186.676i	− 1.34299i	0.741007	−	0.671497i	$-0.234348\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	277.186i	1.88562i
$148$	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
$153$	46.0000	0.300654
$154$	0	0
$155$	0	0
$156$	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
$163$	50.9117i	0.312342i	0.987730	+	0.156171i	$0.0499150\pi$
$163$	50.9117i	0.312342i	−0.987730	+	0.156171i	$0.950085\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−169.000	−1.00000
$170$	0	0
$171$	390.323i	2.28259i
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−480.000	−2.71186
$178$	0	0
$179$	− 356.382i	− 1.99096i	−0.0949721	−	0.995480i	$-0.530276\pi$
$179$	− 356.382i	− 1.99096i	0.0949721	−	0.995480i	$-0.469724\pi$
$180$	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
$185$	0	0
$186$	0	0
$187$	− 33.9411i	− 0.181503i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	−98.0000	−0.507772	−0.253886	−	0.967234i	$-0.581709\pi$
$193$	−98.0000	−0.507772	−0.253886	+	0.967234i	$0.581709\pi$
$194$	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
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	−672.000	−3.34328
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	288.000	1.37799
$210$	0	0
$211$	− 356.382i	− 1.68901i	−0.535545	−	0.844507i	$-0.679894\pi$
$211$	− 356.382i	− 1.68901i	0.535545	−	0.844507i	$-0.320106\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 803.273i	− 3.66791i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	575.000	2.55556
$226$	0	0
$227$	− 84.8528i	− 0.373801i	−0.982379	−	0.186900i	$-0.940156\pi$
$227$	− 84.8528i	− 0.373801i	0.982379	−	0.186900i	$-0.0598442\pi$
$228$	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
$233$	434.000	1.86266	0.931330	−	0.364175i	$-0.118649\pi$
$233$	434.000	1.86266	0.931330	+	0.364175i	$0.118649\pi$
$234$	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
$241$	−194.000	−0.804979	−0.402490	−	0.915425i	$-0.631855\pi$
$241$	−194.000	−0.804979	−0.402490	+	0.915425i	$0.631855\pi$
$242$	0	0
$243$	650.538i	2.67711i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−288.000	−1.15663
$250$	0	0
$251$	− 186.676i	− 0.743730i	−0.928287	−	0.371865i	$-0.878718\pi$
$251$	− 186.676i	− 0.743730i	0.928287	−	0.371865i	$-0.121282\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	386.000	1.50195	0.750973	−	0.660333i	$-0.229585\pi$
$257$	386.000	1.50195	0.750973	+	0.660333i	$0.229585\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	825.901i	3.09326i
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 424.264i	− 1.54278i
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−238.000	−0.846975	−0.423488	−	0.905902i	$-0.639194\pi$
$281$	−238.000	−0.846975	−0.423488	+	0.905902i	$0.639194\pi$
$282$	0	0
$283$	560.029i	1.97890i	0.144876	+	0.989450i	$0.453722\pi$
$283$	560.029i	1.97890i	−0.144876	+	0.989450i	$0.546278\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−285.000	−0.986159
$290$	0	0
$291$	531.744i	1.82730i
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1344.00	4.52525
$298$	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
$307$	− 288.500i	− 0.939738i	−0.882736	−	0.469869i	$-0.844301\pi$
$307$	− 288.500i	− 0.939738i	0.882736	−	0.469869i	$-0.155699\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	526.000	1.68051	0.840256	−	0.542191i	$-0.182405\pi$
$313$	526.000	1.68051	0.840256	+	0.542191i	$0.182405\pi$
$314$	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
$321$	672.000	2.09346
$322$	0	0
$323$	33.9411i	0.105081i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 661.852i	− 1.99955i	−0.0211480	−	0.999776i	$-0.506732\pi$
$331$	− 661.852i	− 1.99955i	0.0211480	−	0.999776i	$-0.493268\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−478.000	−1.41840	−0.709199	−	0.705009i	$-0.750943\pi$
$337$	−478.000	−1.41840	−0.709199	+	0.705009i	$0.750943\pi$
$338$	0	0
$339$	554.372i	1.63531i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	220.617i	0.635785i	0.948127	+	0.317892i	$0.102975\pi$
$347$	220.617i	0.635785i	−0.948127	+	0.317892i	$0.897025\pi$
$348$	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
$353$	194.000	0.549575	0.274788	−	0.961505i	$-0.411392\pi$
$353$	194.000	0.549575	0.274788	+	0.961505i	$0.411392\pi$
$354$	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
$361$	73.0000	0.202216
$362$	0	0
$363$	− 944.695i	− 2.60246i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−1058.00	−2.86721
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 322.441i	− 0.850767i	−0.905013	−	0.425383i	$-0.860139\pi$
$379$	− 322.441i	− 0.850767i	0.905013	−	0.425383i	$-0.139861\pi$
$380$	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
$387$	− 1951.61i	− 5.04293i
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−1440.00	−3.66412
$394$	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
$401$	766.000	1.91022	0.955112	−	0.296244i	$-0.0957342\pi$
$401$	766.000	1.91022	0.955112	+	0.296244i	$0.0957342\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	334.000	0.816626	0.408313	−	0.912842i	$-0.366117\pi$
$409$	334.000	0.816626	0.408313	+	0.912842i	$0.366117\pi$
$410$	0	0
$411$	1346.33i	3.27575i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1056.00	2.53237
$418$	0	0
$419$	661.852i	1.57960i	0.613365	+	0.789799i	$0.289815\pi$
$419$	661.852i	1.57960i	−0.613365	+	0.789799i	$0.710185\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	50.0000	0.117647
$426$	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
$433$	−578.000	−1.33487	−0.667436	−	0.744667i	$-0.732608\pi$
$433$	−578.000	−1.33487	−0.667436	+	0.744667i	$0.732608\pi$
$434$	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
$441$	−1127.00	−2.55556
$442$	0	0
$443$	− 118.794i	− 0.268158i	−0.990971	−	0.134079i	$-0.957192\pi$
$443$	− 118.794i	− 0.268158i	0.990971	−	0.134079i	$-0.0428076\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−866.000	−1.92873	−0.964365	−	0.264574i	$-0.914769\pi$
$449$	−866.000	−1.92873	−0.964365	+	0.264574i	$0.914769\pi$
$450$	0	0
$451$	780.646i	1.73092i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	238.000	0.520788	0.260394	−	0.965502i	$-0.416148\pi$
$457$	238.000	0.520788	0.260394	+	0.965502i	$0.416148\pi$
$458$	0	0
$459$	158.392i	0.345080i
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	933.381i	1.99867i	0.0364026	+	0.999337i	$0.488410\pi$
$467$	933.381i	1.99867i	−0.0364026	+	0.999337i	$0.511590\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1440.00	−3.04440
$474$	0	0
$475$	424.264i	0.893188i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	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
$489$	−288.000	−0.588957
$490$	0	0
$491$	− 593.970i	− 1.20971i	−0.796334	−	0.604857i	$-0.793230\pi$
$491$	− 593.970i	− 1.20971i	0.796334	−	0.604857i	$-0.206770\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	593.970i	1.19032i	0.803607	+	0.595160i	$0.202911\pi$
$499$	593.970i	1.19032i	−0.803607	+	0.595160i	$0.797089\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 956.008i	− 1.88562i
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1344.00	−2.61988
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1006.00	1.93090	0.965451	−	0.260584i	$-0.0839152\pi$
$521$	1006.00	1.93090	0.965451	+	0.260584i	$0.0839152\pi$
$522$	0	0
$523$	967.322i	1.84956i	0.380497	+	0.924782i	$0.375753\pi$
$523$	967.322i	1.84956i	−0.380497	+	0.924782i	$0.624247\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	− 1951.61i	− 3.67536i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2016.00	3.75419
$538$	0	0
$539$	831.558i	1.54278i
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	390.323i	0.713570i	0.934186	+	0.356785i	$0.116127\pi$
$547$	390.323i	0.713570i	−0.934186	+	0.356785i	$0.883873\pi$
$548$	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
$561$	192.000	0.342246
$562$	0	0
$563$	− 1103.09i	− 1.95930i	−0.200710	−	0.979651i	$-0.564325\pi$
$563$	− 1103.09i	− 1.95930i	0.200710	−	0.979651i	$-0.435675\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−626.000	−1.10018	−0.550088	−	0.835107i	$-0.685406\pi$
$569$	−626.000	−1.10018	−0.550088	+	0.835107i	$0.685406\pi$
$570$	0	0
$571$	− 933.381i	− 1.63464i	−0.576182	−	0.817321i	$-0.695458\pi$
$571$	− 933.381i	− 1.63464i	0.576182	−	0.817321i	$-0.304542\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.00346620	0.00173310	−	0.999998i	$-0.499448\pi$
$577$	2.00000	0.00346620	0.00173310	+	0.999998i	$0.499448\pi$
$578$	0	0
$579$	− 554.372i	− 0.957464i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	288.500i	0.491481i	0.969336	+	0.245741i	$0.0790312\pi$
$587$	288.500i	0.491481i	−0.969336	+	0.245741i	$0.920969\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−862.000	−1.45363	−0.726813	−	0.686836i	$-0.758999\pi$
$593$	−862.000	−1.45363	−0.726813	+	0.686836i	$0.758999\pi$
$594$	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
$601$	914.000	1.52080	0.760399	−	0.649456i	$-0.225003\pi$
$601$	914.000	1.52080	0.760399	+	0.649456i	$0.225003\pi$
$602$	0	0
$603$	− 2732.26i	− 4.53111i
$604$	0	0
$605$	0	0
$606$	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
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−334.000	−0.541329	−0.270665	−	0.962674i	$-0.587243\pi$
$617$	−334.000	−0.541329	−0.270665	+	0.962674i	$0.587243\pi$
$618$	0	0
$619$	1103.09i	1.78205i	0.453958	+	0.891023i	$0.350012\pi$
$619$	1103.09i	1.78205i	−0.453958	+	0.891023i	$0.649988\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	1629.17i	2.59836i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	2016.00	3.18483
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−482.000	−0.751950	−0.375975	−	0.926630i	$-0.622692\pi$
$641$	−482.000	−0.751950	−0.375975	+	0.926630i	$0.622692\pi$
$642$	0	0
$643$	− 424.264i	− 0.659820i	−0.944012	−	0.329910i	$-0.892982\pi$
$643$	− 424.264i	− 0.659820i	0.944012	−	0.329910i	$-0.107018\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	−1440.00	−2.21880
$650$	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
$657$	3266.00	4.97108
$658$	0	0
$659$	865.499i	1.31335i	0.754173	+	0.656676i	$0.228038\pi$
$659$	865.499i	1.31335i	−0.754173	+	0.656676i	$0.771962\pi$
$660$	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
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1246.00	1.85141	0.925706	−	0.378244i	$-0.123472\pi$
$673$	1246.00	1.85141	0.925706	+	0.378244i	$0.123472\pi$
$674$	0	0
$675$	1979.90i	2.93318i
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	480.000	0.704846
$682$	0	0
$683$	1306.73i	1.91323i	0.291362	+	0.956613i	$0.405892\pi$
$683$	1306.73i	1.91323i	−0.291362	+	0.956613i	$0.594108\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 1170.97i	− 1.69460i	−0.531114	−	0.847300i	$-0.678227\pi$
$691$	− 1170.97i	− 1.69460i	0.531114	−	0.847300i	$-0.321773\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−92.0000	−0.131994
$698$	0	0
$699$	2455.07i	3.51227i
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	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
$723$	− 1097.43i	− 1.51788i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1511.00	−2.07270
$730$	0	0
$731$	− 169.706i	− 0.232155i
$732$	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
$737$	−2016.00	−2.73541
$738$	0	0
$739$	− 1442.50i	− 1.95196i	−0.217862	−	0.975980i	$-0.569908\pi$
$739$	− 1442.50i	− 1.95196i	0.217862	−	0.975980i	$-0.430092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1170.97i	− 1.56756i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	1056.00	1.40239
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1394.00	−1.83180	−0.915900	−	0.401406i	$-0.868522\pi$
$761$	−1394.00	−1.83180	−0.915900	+	0.401406i	$0.868522\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1054.00	1.37061	0.685306	−	0.728256i	$-0.259669\pi$
$769$	1054.00	1.37061	0.685306	+	0.728256i	$0.259669\pi$
$770$	0	0
$771$	2183.55i	2.83210i
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 780.646i	− 1.00211i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1272.79i	1.61727i	0.588310	+	0.808635i	$0.299793\pi$
$787$	1272.79i	1.61727i	−0.588310	+	0.808635i	$0.700207\pi$
$788$	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
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−3358.00	−4.19226
$802$	0	0
$803$	− 2409.82i	− 3.00102i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1582.00	1.95550	0.977750	−	0.209772i	$-0.0672722\pi$
$809$	1582.00	1.95550	0.977750	+	0.209772i	$0.0672722\pi$
$810$	0	0
$811$	− 1612.20i	− 1.98792i	−0.109741	−	0.993960i	$-0.535002\pi$
$811$	− 1612.20i	− 1.98792i	0.109741	−	0.993960i	$-0.464998\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1440.00	1.76255
$818$	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
$825$	2400.00	2.90909
$826$	0	0
$827$	− 1069.15i	− 1.29280i	−0.762999	−	0.646400i	$-0.776274\pi$
$827$	− 1069.15i	− 1.29280i	0.762999	−	0.646400i	$-0.223726\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−98.0000	−0.117647
$834$	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
$841$	−841.000	−1.00000
$842$	0	0
$843$	− 1346.33i	− 1.59707i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3168.00	−3.73145
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1202.00	−1.40257	−0.701284	−	0.712882i	$-0.747389\pi$
$857$	−1202.00	−1.40257	−0.701284	+	0.712882i	$0.747389\pi$
$858$	0	0
$859$	492.146i	0.572929i	0.958091	+	0.286465i	$0.0924801\pi$
$859$	492.146i	0.572929i	−0.958091	+	0.286465i	$0.907520\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1612.20i	− 1.85952i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2162.00	−2.47652
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1438.00	−1.63224	−0.816118	−	0.577885i	$-0.803878\pi$
$881$	−1438.00	−1.63224	−0.816118	+	0.577885i	$0.803878\pi$
$882$	0	0
$883$	118.794i	0.134534i	0.997735	+	0.0672672i	$0.0214280\pi$
$883$	118.794i	0.134534i	−0.997735	+	0.0672672i	$0.978572\pi$
$884$	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
$891$	4089.91i	4.59024i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 593.970i	− 0.654873i	−0.944873	−	0.327436i	$-0.893815\pi$
$907$	− 593.970i	− 0.654873i	0.944873	−	0.327436i	$-0.106185\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−864.000	−0.946331
$914$	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
$921$	1632.00	1.77199
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1058.00	−1.13886	−0.569429	−	0.822040i	$-0.692836\pi$
$929$	−1058.00	−1.13886	−0.569429	+	0.822040i	$0.692836\pi$
$930$	0	0
$931$	− 831.558i	− 0.893188i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−718.000	−0.766275	−0.383138	−	0.923691i	$-0.625157\pi$
$937$	−718.000	−0.766275	−0.383138	+	0.923691i	$0.625157\pi$
$938$	0	0
$939$	2975.51i	3.16880i
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1612.20i	1.70243i	0.524815	+	0.851216i	$0.324134\pi$
$947$	1612.20i	1.70243i	−0.524815	+	0.851216i	$0.675866\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	142.000	0.149003	0.0745016	−	0.997221i	$-0.476263\pi$
$953$	142.000	0.149003	0.0745016	+	0.997221i	$0.476263\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	2732.26i	2.83724i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	−192.000	−0.198142
$970$	0	0
$971$	− 1680.09i	− 1.73026i	−0.501545	−	0.865132i	$-0.667235\pi$
$971$	− 1680.09i	− 1.73026i	0.501545	−	0.865132i	$-0.332765\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1918.00	1.96315	0.981576	−	0.191071i	$-0.0611960\pi$
$977$	1918.00	1.96315	0.981576	+	0.191071i	$0.0611960\pi$
$978$	0	0
$979$	2477.70i	2.53085i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	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
$993$	3744.00	3.77039
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.3.c.d.255.2		2
3.2	odd	2		2304.3.g.k.1279.1		2
4.3	odd	2	inner	256.3.c.d.255.1		2
8.3	odd	2	CM	256.3.c.d.255.2		2
8.5	even	2	inner	256.3.c.d.255.1		2
12.11	even	2		2304.3.g.k.1279.2		2
16.3	odd	4		128.3.d.b.63.2	yes	2
16.5	even	4		128.3.d.b.63.2	yes	2
16.11	odd	4		128.3.d.b.63.1	✓	2
16.13	even	4		128.3.d.b.63.1	✓	2
24.5	odd	2		2304.3.g.k.1279.2		2
24.11	even	2		2304.3.g.k.1279.1		2
48.5	odd	4		1152.3.b.c.703.2		2
48.11	even	4		1152.3.b.c.703.1		2
48.29	odd	4		1152.3.b.c.703.1		2
48.35	even	4		1152.3.b.c.703.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.d.b.63.1	✓	2	16.11	odd	4
128.3.d.b.63.1	✓	2	16.13	even	4
128.3.d.b.63.2	yes	2	16.3	odd	4
128.3.d.b.63.2	yes	2	16.5	even	4
256.3.c.d.255.1		2	4.3	odd	2	inner
256.3.c.d.255.1		2	8.5	even	2	inner
256.3.c.d.255.2		2	1.1	even	1	trivial
256.3.c.d.255.2		2	8.3	odd	2	CM
1152.3.b.c.703.1		2	48.11	even	4
1152.3.b.c.703.1		2	48.29	odd	4
1152.3.b.c.703.2		2	48.5	odd	4
1152.3.b.c.703.2		2	48.35	even	4
2304.3.g.k.1279.1		2	3.2	odd	2
2304.3.g.k.1279.1		2	24.11	even	2
2304.3.g.k.1279.2		2	12.11	even	2
2304.3.g.k.1279.2		2	24.5	odd	2

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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.65685i q^{3} -23.0000 q^{9} +O(q^{10})\)	\(q+5.65685i q^{3} -23.0000 q^{9} +16.9706i q^{11} -2.00000 q^{17} -16.9706i q^{19} -25.0000 q^{25} -79.1960i q^{27} -96.0000 q^{33} +46.0000 q^{41} +84.8528i q^{43} +49.0000 q^{49} -11.3137i q^{51} +96.0000 q^{57} +84.8528i q^{59} +118.794i q^{67} -142.000 q^{73} -141.421i q^{75} +241.000 q^{81} +50.9117i q^{83} +146.000 q^{89} +94.0000 q^{97} -390.323i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 46 q^{9}+O(q^{10}) \) 2 * q - 46 * q^9	\( 2 q - 46 q^{9} - 4 q^{17} - 50 q^{25} - 192 q^{33} + 92 q^{41} + 98 q^{49} + 192 q^{57} - 284 q^{73} + 482 q^{81} + 292 q^{89} + 188 q^{97}+O(q^{100}) \) 2 * q - 46 * q^9 - 4 * q^17 - 50 * q^25 - 192 * q^33 + 92 * q^41 + 98 * q^49 + 192 * q^57 - 284 * q^73 + 482 * q^81 + 292 * q^89 + 188 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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