LMFDB - Embedded newform 1305.2.f.g.289.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1305,2,Mod(289,1305)] 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("1305.289"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1305, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$10.4204774638$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-5}, \sqrt{-29})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 17x^{2} + 36 $ x^4 + 17*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			289.3
Root			$3.81062i$ of defining polynomial
Character	$\chi$	$=$	1305.289
Dual form			1305.2.f.g.289.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$.

$n$	$146$	$262$	$901$
$\chi(n)$	$1$	$-1$	$-1$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.38516i	− 1.62369i	−0.583874	−	0.811844i	$-0.698464\pi$
$11$	− 5.38516i	− 1.62369i	0.583874	−	0.811844i	$-0.301536\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	− 4.47214i	− 1.00000i
$21$	0	0
$22$	0	0
$23$	− 2.23607i	− 0.466252i	−0.972446	−	0.233126i	$-0.925104\pi$
$23$	− 2.23607i	− 0.466252i	0.972446	−	0.233126i	$-0.0748955\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.38516i	1.00000i
$29$	5.38516i	1.00000i
$30$	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
$37$	12.0416	1.97963	0.989813	−	0.142374i	$-0.0454735\pi$
$37$	12.0416	1.97963	0.989813	+	0.142374i	$0.0454735\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.38516i	− 0.841021i	−0.907288	−	0.420511i	$-0.861851\pi$
$41$	− 5.38516i	− 0.841021i	0.907288	−	0.420511i	$-0.138149\pi$
$42$	0	0
$43$	12.0416	1.83633	0.918163	−	0.396203i	$-0.129672\pi$
$43$	12.0416	1.83633	0.918163	+	0.396203i	$0.129672\pi$
$44$	10.7703i	1.62369i
$45$	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
$53$	11.1803i	1.53574i	0.640607	+	0.767869i	$0.278683\pi$
$53$	11.1803i	1.53574i	−0.640607	+	0.767869i	$0.721317\pi$
$54$	0	0
$55$	12.0416	1.62369
$56$	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
$64$	−8.00000	−1.00000
$65$	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
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	12.0416	1.40936	0.704681	−	0.709524i	$-0.251090\pi$
$73$	12.0416	1.40936	0.704681	+	0.709524i	$0.251090\pi$
$74$	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			$\pi$
$80$	8.94427i	1.00000i
$81$	0	0
$82$	0	0
$83$	− 15.6525i	− 1.71808i	−0.511906	−	0.859041i	$-0.671061\pi$
$83$	− 15.6525i	− 1.71808i	0.511906	−	0.859041i	$-0.328939\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.7703i	1.14165i	0.821071	+	0.570826i	$0.193377\pi$
$89$	10.7703i	1.14165i	−0.821071	+	0.570826i	$0.806623\pi$
$90$	0	0
$91$	0	0
$92$	4.47214i	0.466252i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.0416	1.22264	0.611319	−	0.791384i	$-0.290639\pi$
$97$	12.0416	1.22264	0.611319	+	0.791384i	$0.290639\pi$
$98$	0	0
$99$	0	0
$100$	10.0000	1.00000
$101$	− 5.38516i	− 0.535844i	−0.963441	−	0.267922i	$-0.913663\pi$
$101$	− 5.38516i	− 0.535844i	0.963441	−	0.267922i	$-0.0863369\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.94427i	− 0.864675i	−0.901712	−	0.432338i	$-0.857689\pi$
$107$	− 8.94427i	− 0.864675i	0.901712	−	0.432338i	$-0.142311\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	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
$115$	5.00000	0.466252
$116$	− 10.7703i	− 1.00000i
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−18.0000	−1.63636
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 1.00000i
$126$	0	0
$127$	12.0416	1.06852	0.534259	−	0.845321i	$-0.320591\pi$
$127$	12.0416	1.06852	0.534259	+	0.845321i	$0.320591\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 21.5407i	− 1.88202i	−0.338384	−	0.941008i	$-0.609880\pi$
$131$	− 21.5407i	− 1.88202i	0.338384	−	0.941008i	$-0.390120\pi$
$132$	0	0
$133$	0	0
$134$	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
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−12.0416	−1.00000
$146$	0	0
$147$	0	0
$148$	−24.0832	−1.97963
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−24.0832	−1.92205	−0.961024	−	0.276465i	$-0.910837\pi$
$157$	−24.0832	−1.92205	−0.961024	+	0.276465i	$0.910837\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.0416	0.943170	0.471585	−	0.881820i	$-0.343682\pi$
$163$	12.0416	0.943170	0.471585	+	0.881820i	$0.343682\pi$
$164$	10.7703i	0.841021i
$165$	0	0
$166$	0	0
$167$	17.8885i	1.38426i	0.721774	+	0.692129i	$0.243327\pi$
$167$	17.8885i	1.38426i	−0.721774	+	0.692129i	$0.756673\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	−24.0832	−1.83633
$173$	24.5967i	1.87006i	0.354574	+	0.935028i	$0.384626\pi$
$173$	24.5967i	1.87006i	−0.354574	+	0.935028i	$0.615374\pi$
$174$	0	0
$175$	0	0
$176$	− 21.5407i	− 1.62369i
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	26.9258i	1.97963i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 5.38516i	− 0.389657i	−0.980837	−	0.194828i	$-0.937585\pi$
$191$	− 5.38516i	− 0.389657i	0.980837	−	0.194828i	$-0.0624150\pi$
$192$	0	0
$193$	−24.0832	−1.73355	−0.866773	−	0.498703i	$-0.833810\pi$
$193$	−24.0832	−1.73355	−0.866773	+	0.498703i	$0.833810\pi$
$194$	0	0
$195$	0	0
$196$	−14.0000	−1.00000
$197$	− 2.23607i	− 0.159313i	−0.996822	−	0.0796566i	$-0.974618\pi$
$197$	− 2.23607i	− 0.159313i	0.996822	−	0.0796566i	$-0.0253824\pi$
$198$	0	0
$199$	−19.0000	−1.34687	−0.673437	−	0.739244i	$-0.735183\pi$
$199$	−19.0000	−1.34687	−0.673437	+	0.739244i	$0.735183\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	12.0416	0.841021
$206$	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			$\pi$
$212$	− 22.3607i	− 1.53574i
$213$	0	0
$214$	0	0
$215$	26.9258i	1.83633i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−24.0832	−1.62369
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.1803i	0.742065i	0.928620	+	0.371033i	$0.120996\pi$
$227$	11.1803i	0.742065i	−0.928620	+	0.371033i	$0.879004\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 29.0689i	− 1.90437i	−0.305528	−	0.952183i	$-0.598833\pi$
$233$	− 29.0689i	− 1.90437i	0.305528	−	0.952183i	$-0.401167\pi$
$234$	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
$241$	23.0000	1.48156	0.740780	−	0.671748i	$-0.234456\pi$
$241$	23.0000	1.48156	0.740780	+	0.671748i	$0.234456\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	15.6525i	1.00000i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 21.5407i	− 1.35963i	−0.733382	−	0.679817i	$-0.762059\pi$
$251$	− 21.5407i	− 1.35963i	0.733382	−	0.679817i	$-0.237941\pi$
$252$	0	0
$253$	−12.0416	−0.757049
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	− 15.6525i	− 0.976375i	−0.872739	−	0.488187i	$-0.837658\pi$
$257$	− 15.6525i	− 0.976375i	0.872739	−	0.488187i	$-0.162342\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−25.0000	−1.53574
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.7703i	0.656679i	0.944560	+	0.328339i	$0.106489\pi$
$269$	10.7703i	0.656679i	−0.944560	+	0.328339i	$0.893511\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	26.9258i	1.62369i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	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
$283$	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
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	−24.0832	−1.40936
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	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
$307$	12.0416	0.687250	0.343625	−	0.939107i	$-0.388345\pi$
$307$	12.0416	0.687250	0.343625	+	0.939107i	$0.388345\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 5.38516i	− 0.305365i	−0.988275	−	0.152682i	$-0.951209\pi$
$311$	− 5.38516i	− 0.305365i	0.988275	−	0.152682i	$-0.0487911\pi$
$312$	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
$319$	29.0000	1.62369
$320$	− 17.8885i	− 1.00000i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	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			$\pi$
$332$	31.3050i	1.71808i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−24.0832	−1.31189	−0.655947	−	0.754807i	$-0.727731\pi$
$337$	−24.0832	−1.31189	−0.655947	+	0.754807i	$0.727731\pi$
$338$	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
$347$	24.5967i	1.32042i	0.751080	+	0.660211i	$0.229533\pi$
$347$	24.5967i	1.32042i	−0.751080	+	0.660211i	$0.770467\pi$
$348$	0	0
$349$	−31.0000	−1.65939	−0.829696	−	0.558216i	$-0.811486\pi$
$349$	−31.0000	−1.65939	−0.829696	+	0.558216i	$0.811486\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.47214i	0.238028i	0.992893	+	0.119014i	$0.0379733\pi$
$353$	4.47214i	0.238028i	−0.992893	+	0.119014i	$0.962027\pi$
$354$	0	0
$355$	0	0
$356$	− 21.5407i	− 1.14165i
$357$	0	0
$358$	0	0
$359$	− 37.6962i	− 1.98953i	−0.102204	−	0.994763i	$-0.532589\pi$
$359$	− 37.6962i	− 1.98953i	0.102204	−	0.994763i	$-0.467411\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	26.9258i	1.40936i
$366$	0	0
$367$	12.0416	0.628566	0.314283	−	0.949329i	$-0.398236\pi$
$367$	12.0416	0.628566	0.314283	+	0.949329i	$0.398236\pi$
$368$	− 8.94427i	− 0.466252i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	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
$383$	38.0132i	1.94238i	0.238303	+	0.971191i	$0.423409\pi$
$383$	38.0132i	1.94238i	−0.238303	+	0.971191i	$0.576591\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	−24.0832	−1.22264
$389$	− 37.6962i	− 1.91127i	−0.294552	−	0.955635i	$-0.595170\pi$
$389$	− 37.6962i	− 1.91127i	0.294552	−	0.955635i	$-0.404830\pi$
$390$	0	0
$391$	0	0
$392$	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
$400$	−20.0000	−1.00000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	10.7703i	0.535844i
$405$	0	0
$406$	0	0
$407$	− 64.8460i	− 3.21430i
$408$	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
$415$	35.0000	1.71808
$416$	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
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	17.8885i	0.864675i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	12.0416	0.578682	0.289341	−	0.957226i	$-0.406564\pi$
$433$	12.0416	0.578682	0.289341	+	0.957226i	$0.406564\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−24.0832	−1.14165
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 37.6962i	− 1.77899i	−0.456943	−	0.889496i	$-0.651056\pi$
$449$	− 37.6962i	− 1.77899i	0.456943	−	0.889496i	$-0.348944\pi$
$450$	0	0
$451$	−29.0000	−1.36556
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	−10.0000	−0.466252
$461$	− 5.38516i	− 0.250812i	−0.992105	−	0.125406i	$-0.959977\pi$
$461$	− 5.38516i	− 0.250812i	0.992105	−	0.125406i	$-0.0400234\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	21.5407i	1.00000i
$465$	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
$473$	− 64.8460i	− 2.98162i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	43.0813i	1.96844i	0.176961	+	0.984218i	$0.443373\pi$
$479$	43.0813i	1.96844i	−0.176961	+	0.984218i	$0.556627\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	36.0000	1.63636
$485$	26.9258i	1.22264i
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 21.5407i	− 0.972116i	−0.873926	−	0.486058i	$-0.838434\pi$
$491$	− 21.5407i	− 0.972116i	0.873926	−	0.486058i	$-0.161566\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	22.3607i	1.00000i
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	12.0416	0.535844
$506$	0	0
$507$	0	0
$508$	−24.0832	−1.06852
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	43.0813i	1.88202i
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	18.0000	0.782609
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	20.0000	0.864675
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 37.6962i	− 1.62369i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 2.23607i	− 0.0957826i
$546$	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
$556$	−22.0000	−0.933008
$557$	38.0132i	1.61067i	0.592821	+	0.805335i	$0.298014\pi$
$557$	38.0132i	1.61067i	−0.592821	+	0.805335i	$0.701986\pi$
$558$	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
$569$	10.7703i	0.451516i	0.974183	+	0.225758i	$0.0724858\pi$
$569$	10.7703i	0.451516i	−0.974183	+	0.225758i	$0.927514\pi$
$570$	0	0
$571$	−43.0000	−1.79949	−0.899747	−	0.436412i	$-0.856249\pi$
$571$	−43.0000	−1.79949	−0.899747	+	0.436412i	$0.856249\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	11.1803i	0.466252i
$576$	0	0
$577$	−24.0832	−1.00260	−0.501298	−	0.865275i	$-0.667144\pi$
$577$	−24.0832	−1.00260	−0.501298	+	0.865275i	$0.667144\pi$
$578$	0	0
$579$	0	0
$580$	24.0832	1.00000
$581$	0	0
$582$	0	0
$583$	60.2080	2.49356
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44.7214i	1.84585i	0.384982	+	0.922924i	$0.374208\pi$
$587$	44.7214i	1.84585i	−0.384982	+	0.922924i	$0.625792\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	48.1664	1.97963
$593$	31.3050i	1.28554i	0.766059	+	0.642770i	$0.222215\pi$
$593$	31.3050i	1.28554i	−0.766059	+	0.642770i	$0.777785\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	43.0813i	1.76025i	0.474737	+	0.880127i	$0.342543\pi$
$599$	43.0813i	1.76025i	−0.474737	+	0.880127i	$0.657457\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	26.0000	1.05792
$605$	− 40.2492i	− 1.63636i
$606$	0	0
$607$	48.1664	1.95501	0.977506	−	0.210905i	$-0.0676412\pi$
$607$	48.1664	1.95501	0.977506	+	0.210905i	$0.0676412\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	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
$619$	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
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	48.1664	1.92205
$629$	0	0
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	26.9258i	1.06852i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 5.38516i	− 0.212701i	−0.994329	−	0.106351i	$-0.966083\pi$
$641$	− 5.38516i	− 0.212701i	0.994329	−	0.106351i	$-0.0339166\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 42.4853i	− 1.67027i	−0.550046	−	0.835135i	$-0.685390\pi$
$647$	− 42.4853i	− 1.67027i	0.550046	−	0.835135i	$-0.314610\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−24.0832	−0.943170
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	48.1664	1.88202
$656$	− 21.5407i	− 0.841021i
$657$	0	0
$658$	0	0
$659$	− 37.6962i	− 1.46843i	−0.678915	−	0.734217i	$-0.737550\pi$
$659$	− 37.6962i	− 1.46843i	0.678915	−	0.734217i	$-0.262450\pi$
$660$	0	0
$661$	47.0000	1.82809	0.914044	−	0.405615i	$-0.132943\pi$
$661$	47.0000	1.82809	0.914044	+	0.405615i	$0.132943\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0416	0.466252
$668$	− 35.7771i	− 1.38426i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	−26.0000	−1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	51.4296i	1.96790i	0.178451	+	0.983949i	$0.442891\pi$
$683$	51.4296i	1.96790i	−0.178451	+	0.983949i	$0.557109\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	48.1664	1.83633
$689$	0	0
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	− 49.1935i	− 1.87006i
$693$	0	0
$694$	0	0
$695$	24.5967i	0.933008i
$696$	0	0
$697$	0	0
$698$	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
$703$	0	0
$704$	43.0813i	1.62369i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−49.0000	−1.84023	−0.920117	−	0.391644i	$-0.871906\pi$
$709$	−49.0000	−1.84023	−0.920117	+	0.391644i	$0.871906\pi$
$710$	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
$724$	−34.0000	−1.26360
$725$	− 26.9258i	− 1.00000i
$726$	0	0
$727$	48.1664	1.78639	0.893196	−	0.449667i	$-0.148458\pi$
$727$	48.1664	1.78639	0.893196	+	0.449667i	$0.148458\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−24.0832	−0.889533	−0.444766	−	0.895647i	$-0.646713\pi$
$733$	−24.0832	−0.889533	−0.444766	+	0.895647i	$0.646713\pi$
$734$	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			$\pi$
$740$	− 53.8516i	− 1.97963i
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 29.0689i	− 1.05792i
$756$	0	0
$757$	12.0416	0.437659	0.218830	−	0.975763i	$-0.429776\pi$
$757$	12.0416	0.437659	0.218830	+	0.975763i	$0.429776\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	10.7703i	0.389657i
$765$	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
$772$	48.1664	1.73355
$773$	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
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	28.0000	1.00000
$785$	− 53.8516i	− 1.92205i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	4.47214i	0.159313i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	38.0000	1.34687
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 64.8460i	− 2.28836i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 37.6962i	− 1.32533i	−0.748918	−	0.662663i	$-0.769426\pi$
$809$	− 37.6962i	− 1.32533i	0.748918	−	0.662663i	$-0.230574\pi$
$810$	0	0
$811$	53.0000	1.86108	0.930541	−	0.366188i	$-0.119337\pi$
$811$	53.0000	1.86108	0.930541	+	0.366188i	$0.119337\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	26.9258i	0.943170i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−24.0832	−0.841021
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	48.1664	1.67897	0.839487	−	0.543379i	$-0.182856\pi$
$823$	48.1664	1.67897	0.839487	+	0.543379i	$0.182856\pi$
$824$	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
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−40.0000	−1.38426
$836$	0	0
$837$	0	0
$838$	0	0
$839$	43.0813i	1.48733i	0.668551	+	0.743666i	$0.266915\pi$
$839$	43.0813i	1.48733i	−0.668551	+	0.743666i	$0.733085\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	29.0689i	1.00000i
$846$	0	0
$847$	0	0
$848$	44.7214i	1.53574i
$849$	0	0
$850$	0	0
$851$	− 26.9258i	− 0.923005i
$852$	0	0
$853$	12.0416	0.412296	0.206148	−	0.978521i	$-0.433907\pi$
$853$	12.0416	0.412296	0.206148	+	0.978521i	$0.433907\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	51.4296i	1.75680i	0.477926	+	0.878400i	$0.341389\pi$
$857$	51.4296i	1.75680i	−0.477926	+	0.878400i	$0.658611\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	− 53.8516i	− 1.83633i
$861$	0	0
$862$	0	0
$863$	17.8885i	0.608933i	0.952523	+	0.304467i	$0.0984782\pi$
$863$	17.8885i	0.608933i	−0.952523	+	0.304467i	$0.901522\pi$
$864$	0	0
$865$	−55.0000	−1.87006
$866$	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
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	48.1664	1.62369
$881$	59.2368i	1.99574i	0.0652421	+	0.997869i	$0.479218\pi$
$881$	59.2368i	1.99574i	−0.0652421	+	0.997869i	$0.520782\pi$
$882$	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
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	38.0132i	1.26360i
$906$	0	0
$907$	−60.2080	−1.99917	−0.999586	−	0.0287559i	$-0.990845\pi$
$907$	−60.2080	−1.99917	−0.999586	+	0.0287559i	$0.990845\pi$
$908$	− 22.3607i	− 0.742065i
$909$	0	0
$910$	0	0
$911$	59.2368i	1.96260i	0.192476	+	0.981302i	$0.438348\pi$
$911$	59.2368i	1.96260i	−0.192476	+	0.981302i	$0.561652\pi$
$912$	0	0
$913$	−84.2912	−2.78963
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−60.2080	−1.97963
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	58.1378i	1.90437i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	−12.0416	−0.392128
$944$	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
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 49.1935i	− 1.59353i	−0.604287	−	0.796767i	$-0.706542\pi$
$953$	− 49.1935i	− 1.59353i	0.604287	−	0.796767i	$-0.293458\pi$
$954$	0	0
$955$	12.0416	0.389657
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	−46.0000	−1.48156
$965$	− 53.8516i	− 1.73355i
$966$	0	0
$967$	−60.2080	−1.93616	−0.968079	−	0.250645i	$-0.919357\pi$
$967$	−60.2080	−1.93616	−0.968079	+	0.250645i	$0.919357\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	59.2368i	1.90100i	0.310725	+	0.950500i	$0.399428\pi$
$971$	59.2368i	1.90100i	−0.310725	+	0.950500i	$0.600572\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 55.9017i	− 1.78845i	−0.447614	−	0.894227i	$-0.647726\pi$
$977$	− 55.9017i	− 1.78845i	0.447614	−	0.894227i	$-0.352274\pi$
$978$	0	0
$979$	58.0000	1.85369
$980$	− 31.3050i	− 1.00000i
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	5.00000	0.159313
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 26.9258i	− 0.856191i
$990$	0	0
$991$	−7.00000	−0.222362	−0.111181	−	0.993800i	$-0.535463\pi$
$991$	−7.00000	−0.222362	−0.111181	+	0.993800i	$0.535463\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 42.4853i	− 1.34687i
$996$	0	0
$997$	−60.2080	−1.90681	−0.953403	−	0.301700i	$-0.902446\pi$
$997$	−60.2080	−1.90681	−0.953403	+	0.301700i	$0.902446\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.f.g.289.3	yes	4
3.2	odd	2	inner	1305.2.f.g.289.2	yes	4
5.4	even	2	inner	1305.2.f.g.289.1	✓	4
15.14	odd	2	inner	1305.2.f.g.289.4	yes	4
29.28	even	2	inner	1305.2.f.g.289.4	yes	4
87.86	odd	2	inner	1305.2.f.g.289.1	✓	4
145.144	even	2	inner	1305.2.f.g.289.2	yes	4
435.434	odd	2	CM	1305.2.f.g.289.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1305.2.f.g.289.1	✓	4	5.4	even	2	inner
1305.2.f.g.289.1	✓	4	87.86	odd	2	inner
1305.2.f.g.289.2	yes	4	3.2	odd	2	inner
1305.2.f.g.289.2	yes	4	145.144	even	2	inner
1305.2.f.g.289.3	yes	4	1.1	even	1	trivial
1305.2.f.g.289.3	yes	4	435.434	odd	2	CM
1305.2.f.g.289.4	yes	4	15.14	odd	2	inner
1305.2.f.g.289.4	yes	4	29.28	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} +2.23607i q^{5} -5.38516i q^{11} +4.00000 q^{16} -4.47214i q^{20} -2.23607i q^{23} -5.00000 q^{25} +5.38516i q^{29} +12.0416 q^{37} -5.38516i q^{41} +12.0416 q^{43} +10.7703i q^{44} +7.00000 q^{49} +11.1803i q^{53} +12.0416 q^{55} -8.00000 q^{64} +12.0416 q^{73} +8.94427i q^{80} -15.6525i q^{83} +10.7703i q^{89} +4.47214i q^{92} +12.0416 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 28 q^{49} - 32 q^{64}+O(q^{100}) \) 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 28 * q^49 - 32 * q^64

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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