LMFDB - Embedded newform 224.3.h.a.209.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,3,Mod(209,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("224.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.h (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:	$6.10355792167$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 2 $ x^2 - x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			209.2
Root			$0.500000 - 1.32288i$ of defining polynomial
Character	$\chi$	$=$	224.209
Dual form			224.3.h.a.209.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-7.00000 q^{7} -9.00000 q^{9} +21.1660i q^{11} -18.0000 q^{23} -25.0000 q^{25} +21.1660i q^{29} +63.4980i q^{37} -63.4980i q^{43} +49.0000 q^{49} -105.830i q^{53} +63.0000 q^{63} +63.4980i q^{67} +114.000 q^{71} -148.162i q^{77} -94.0000 q^{79} +81.0000 q^{81} -190.494i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{7} - 18 q^{9}+O(q^{10}) $ 2 * q - 14 * q^7 - 18 * q^9	$ 2 q - 14 q^{7} - 18 q^{9} - 36 q^{23} - 50 q^{25} + 98 q^{49} + 126 q^{63} + 228 q^{71} - 188 q^{79} + 162 q^{81}+O(q^{100}) $ 2 * q - 14 * q^7 - 18 * q^9 - 36 * q^23 - 50 * q^25 + 98 * q^49 + 126 * q^63 + 228 * q^71 - 188 * q^79 + 162 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$-1$

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
$2$	0	0
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−7.00000	−1.00000
$7$	−7.00000	−1.00000
$8$	0	0
$9$	−9.00000	−1.00000
$10$	0	0
$11$	21.1660i	1.92418i	0.272727	+	0.962091i	$0.412074\pi$
$11$	21.1660i	1.92418i	−0.272727	+	0.962091i	$0.587926\pi$
$12$	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
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	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
$23$	−18.0000	−0.782609	−0.391304	−	0.920261i	$-0.627976\pi$
$23$	−18.0000	−0.782609	−0.391304	+	0.920261i	$0.627976\pi$
$24$	0	0
$25$	−25.0000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	21.1660i	0.729862i	0.931034	+	0.364931i	$0.118907\pi$
$29$	21.1660i	0.729862i	−0.931034	+	0.364931i	$0.881093\pi$
$30$	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
$37$	63.4980i	1.71616i	0.513514	+	0.858082i	$0.328344\pi$
$37$	63.4980i	1.71616i	−0.513514	+	0.858082i	$0.671656\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 63.4980i	− 1.47670i	−0.674419	−	0.738349i	$-0.735606\pi$
$43$	− 63.4980i	− 1.47670i	0.674419	−	0.738349i	$-0.264394\pi$
$44$	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
$51$	0	0
$52$	0	0
$53$	− 105.830i	− 1.99679i	−0.0566038	−	0.998397i	$-0.518027\pi$
$53$	− 105.830i	− 1.99679i	0.0566038	−	0.998397i	$-0.481973\pi$
$54$	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
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	63.0000	1.00000
$64$	0	0
$65$	0	0
$66$	0	0
$67$	63.4980i	0.947732i	0.880597	+	0.473866i	$0.157142\pi$
$67$	63.4980i	0.947732i	−0.880597	+	0.473866i	$0.842858\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	114.000	1.60563	0.802817	−	0.596226i	$-0.203334\pi$
$71$	114.000	1.60563	0.802817	+	0.596226i	$0.203334\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 148.162i	− 1.92418i
$78$	0	0
$79$	−94.0000	−1.18987	−0.594937	−	0.803773i	$-0.702823\pi$
$79$	−94.0000	−1.18987	−0.594937	+	0.803773i	$0.702823\pi$
$80$	0	0
$81$	81.0000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	− 190.494i	− 1.92418i
$100$	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
$107$	105.830i	0.989066i	0.869159	+	0.494533i	$0.164661\pi$
$107$	105.830i	0.989066i	−0.869159	+	0.494533i	$0.835339\pi$
$108$	0	0
$109$	190.494i	1.74765i	0.486239	+	0.873826i	$0.338369\pi$
$109$	190.494i	1.74765i	−0.486239	+	0.873826i	$0.661631\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	222.000	1.96460	0.982301	−	0.187310i	$-0.0599768\pi$
$113$	222.000	1.96460	0.982301	+	0.187310i	$0.0599768\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−327.000	−2.70248
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.00000	−0.0157480	−0.00787402	−	0.999969i	$-0.502506\pi$
$127$	−2.00000	−0.0157480	−0.00787402	+	0.999969i	$0.502506\pi$
$128$	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
$137$	−174.000	−1.27007	−0.635036	−	0.772482i	$-0.719015\pi$
$137$	−174.000	−1.27007	−0.635036	+	0.772482i	$0.719015\pi$
$138$	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
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	232.826i	1.56259i	0.624161	+	0.781296i	$0.285441\pi$
$149$	232.826i	1.56259i	−0.624161	+	0.781296i	$0.714559\pi$
$150$	0	0
$151$	−274.000	−1.81457	−0.907285	−	0.420517i	$-0.861849\pi$
$151$	−274.000	−1.81457	−0.907285	+	0.420517i	$0.861849\pi$
$152$	0	0
$153$	0	0
$154$	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
$161$	126.000	0.782609
$162$	0	0
$163$	317.490i	1.94779i	0.226994	+	0.973896i	$0.427110\pi$
$163$	317.490i	1.94779i	−0.226994	+	0.973896i	$0.572890\pi$
$164$	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
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	175.000	1.00000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 105.830i	− 0.591229i	−0.955307	−	0.295615i	$-0.904476\pi$
$179$	− 105.830i	− 0.591229i	0.955307	−	0.295615i	$-0.0955244\pi$
$180$	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
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−318.000	−1.66492	−0.832461	−	0.554084i	$-0.813069\pi$
$191$	−318.000	−1.66492	−0.832461	+	0.554084i	$0.813069\pi$
$192$	0	0
$193$	62.0000	0.321244	0.160622	−	0.987016i	$-0.448650\pi$
$193$	62.0000	0.321244	0.160622	+	0.987016i	$0.448650\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 275.158i	− 1.39674i	−0.715736	−	0.698371i	$-0.753909\pi$
$197$	− 275.158i	− 1.39674i	0.715736	−	0.698371i	$-0.246091\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 148.162i	− 0.729862i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	162.000	0.782609
$208$	0	0
$209$	0	0
$210$	0	0
$211$	317.490i	1.50469i	0.658768	+	0.752346i	$0.271078\pi$
$211$	317.490i	1.50469i	−0.658768	+	0.752346i	$0.728922\pi$
$212$	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
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	225.000	1.00000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	0.0772532	0.0386266	−	0.999254i	$-0.487702\pi$
$233$	18.0000	0.0772532	0.0386266	+	0.999254i	$0.487702\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	222.000	0.928870	0.464435	−	0.885607i	$-0.346257\pi$
$239$	222.000	0.928870	0.464435	+	0.885607i	$0.346257\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	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
$253$	− 380.988i	− 1.50588i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	− 444.486i	− 1.71616i
$260$	0	0
$261$	− 190.494i	− 0.729862i
$262$	0	0
$263$	498.000	1.89354	0.946768	−	0.321917i	$-0.104327\pi$
$263$	498.000	1.89354	0.946768	+	0.321917i	$0.104327\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	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
$275$	− 529.150i	− 1.92418i
$276$	0	0
$277$	317.490i	1.14617i	0.819495	+	0.573087i	$0.194254\pi$
$277$	317.490i	1.14617i	−0.819495	+	0.573087i	$0.805746\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	114.000	0.405694	0.202847	−	0.979210i	$-0.434981\pi$
$281$	114.000	0.405694	0.202847	+	0.979210i	$0.434981\pi$
$282$	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
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	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
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	444.486i	1.47670i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	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
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	359.822i	1.13509i	0.823344	+	0.567543i	$0.192106\pi$
$317$	359.822i	1.13509i	−0.823344	+	0.567543i	$0.807894\pi$
$318$	0	0
$319$	−448.000	−1.40439
$320$	0	0
$321$	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
$331$	190.494i	0.575511i	0.957704	+	0.287755i	$0.0929090\pi$
$331$	190.494i	0.575511i	−0.957704	+	0.287755i	$0.907091\pi$
$332$	0	0
$333$	− 571.482i	− 1.71616i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−226.000	−0.670623	−0.335312	−	0.942107i	$-0.608842\pi$
$337$	−226.000	−0.670623	−0.335312	+	0.942107i	$0.608842\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−343.000	−1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 148.162i	− 0.426980i	−0.976945	−	0.213490i	$-0.931517\pi$
$347$	− 148.162i	− 0.426980i	0.976945	−	0.213490i	$-0.0684831\pi$
$348$	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
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	654.000	1.82173	0.910864	−	0.412708i	$-0.135417\pi$
$359$	654.000	1.82173	0.910864	+	0.412708i	$0.135417\pi$
$360$	0	0
$361$	−361.000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	740.810i	1.99679i
$372$	0	0
$373$	− 698.478i	− 1.87260i	−0.351206	−	0.936298i	$-0.614228\pi$
$373$	− 698.478i	− 1.87260i	0.351206	−	0.936298i	$-0.385772\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	444.486i	1.17279i	0.810026	+	0.586393i	$0.199453\pi$
$379$	444.486i	1.17279i	−0.810026	+	0.586393i	$0.800547\pi$
$380$	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
$387$	571.482i	1.47670i
$388$	0	0
$389$	402.154i	1.03382i	0.856041	+	0.516908i	$0.172917\pi$
$389$	402.154i	1.03382i	−0.856041	+	0.516908i	$0.827083\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−354.000	−0.882793	−0.441397	−	0.897312i	$-0.645517\pi$
$401$	−354.000	−0.882793	−0.441397	+	0.897312i	$0.645517\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1344.00	−3.30221
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	825.474i	1.96075i	0.197150	+	0.980373i	$0.436832\pi$
$421$	825.474i	1.96075i	−0.197150	+	0.980373i	$0.563168\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−162.000	−0.375870	−0.187935	−	0.982181i	$-0.560179\pi$
$431$	−162.000	−0.375870	−0.187935	+	0.982181i	$0.560179\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−441.000	−1.00000
$442$	0	0
$443$	− 740.810i	− 1.67226i	−0.548533	−	0.836129i	$-0.684813\pi$
$443$	− 740.810i	− 1.67226i	0.548533	−	0.836129i	$-0.315187\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−894.000	−1.99109	−0.995546	−	0.0942807i	$-0.969945\pi$
$449$	−894.000	−1.99109	−0.995546	+	0.0942807i	$0.969945\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	878.000	1.92123	0.960613	−	0.277891i	$-0.0896353\pi$
$457$	878.000	1.92123	0.960613	+	0.277891i	$0.0896353\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	674.000	1.45572	0.727862	−	0.685724i	$-0.240514\pi$
$463$	674.000	1.45572	0.727862	+	0.685724i	$0.240514\pi$
$464$	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
$469$	− 444.486i	− 0.947732i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1344.00	2.84144
$474$	0	0
$475$	0	0
$476$	0	0
$477$	952.470i	1.99679i
$478$	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
$487$	398.000	0.817248	0.408624	−	0.912703i	$-0.366009\pi$
$487$	398.000	0.817248	0.408624	+	0.912703i	$0.366009\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 232.826i	− 0.474188i	−0.971487	−	0.237094i	$-0.923805\pi$
$491$	− 232.826i	− 0.474188i	0.971487	−	0.237094i	$-0.0761949\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−798.000	−1.60563
$498$	0	0
$499$	− 952.470i	− 1.90876i	−0.298597	−	0.954379i	$-0.596519\pi$
$499$	− 952.470i	− 1.90876i	0.298597	−	0.954379i	$-0.403481\pi$
$500$	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
$507$	0	0
$508$	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
$513$	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	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	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
$529$	−205.000	−0.387524
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1037.13i	1.92418i
$540$	0	0
$541$	− 1079.47i	− 1.99532i	−0.0683919	−	0.997659i	$-0.521787\pi$
$541$	− 1079.47i	− 1.99532i	0.0683919	−	0.997659i	$-0.478213\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 698.478i	− 1.27693i	−0.769653	−	0.638463i	$-0.779571\pi$
$547$	− 698.478i	− 1.27693i	0.769653	−	0.638463i	$-0.220429\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	658.000	1.18987
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 486.818i	− 0.874000i	−0.899461	−	0.437000i	$-0.856041\pi$
$557$	− 486.818i	− 0.874000i	0.899461	−	0.437000i	$-0.143959\pi$
$558$	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
$567$	−567.000	−1.00000
$568$	0	0
$569$	654.000	1.14938	0.574692	−	0.818369i	$-0.305122\pi$
$569$	654.000	1.14938	0.574692	+	0.818369i	$0.305122\pi$
$570$	0	0
$571$	190.494i	0.333615i	0.985989	+	0.166807i	$0.0533458\pi$
$571$	190.494i	0.333615i	−0.985989	+	0.166807i	$0.946654\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	450.000	0.782609
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2240.00	3.84220
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−174.000	−0.290484	−0.145242	−	0.989396i	$-0.546396\pi$
$599$	−174.000	−0.290484	−0.145242	+	0.989396i	$0.546396\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	− 571.482i	− 0.947732i
$604$	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
$613$	− 1206.46i	− 1.96813i	−0.177814	−	0.984064i	$-0.556903\pi$
$613$	− 1206.46i	− 1.96813i	0.177814	−	0.984064i	$-0.443097\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	558.000	0.904376	0.452188	−	0.891923i	$-0.350644\pi$
$617$	558.000	0.904376	0.452188	+	0.891923i	$0.350644\pi$
$618$	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
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1006.00	−1.59429	−0.797147	−	0.603785i	$-0.793659\pi$
$631$	−1006.00	−1.59429	−0.797147	+	0.603785i	$0.793659\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1026.00	−1.60563
$640$	0	0
$641$	−834.000	−1.30109	−0.650546	−	0.759467i	$-0.725460\pi$
$641$	−834.000	−1.30109	−0.650546	+	0.759467i	$0.725460\pi$
$642$	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
$653$	529.150i	0.810337i	0.914242	+	0.405169i	$0.132787\pi$
$653$	529.150i	0.810337i	−0.914242	+	0.405169i	$0.867213\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1164.13i	1.76651i	0.468892	+	0.883255i	$0.344653\pi$
$659$	1164.13i	1.76651i	−0.468892	+	0.883255i	$0.655347\pi$
$660$	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
$667$	− 380.988i	− 0.571197i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−446.000	−0.662704	−0.331352	−	0.943507i	$-0.607505\pi$
$673$	−446.000	−0.662704	−0.331352	+	0.943507i	$0.607505\pi$
$674$	0	0
$675$	0	0
$676$	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
$681$	0	0
$682$	0	0
$683$	275.158i	0.402867i	0.979502	+	0.201433i	$0.0645600\pi$
$683$	275.158i	0.402867i	−0.979502	+	0.201433i	$0.935440\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	1333.46i	1.92418i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	105.830i	0.150970i	0.997147	+	0.0754851i	$0.0240505\pi$
$701$	105.830i	0.150970i	−0.997147	+	0.0754851i	$0.975949\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	317.490i	0.447800i	0.974612	+	0.223900i	$0.0718789\pi$
$709$	317.490i	0.447800i	−0.974612	+	0.223900i	$0.928121\pi$
$710$	0	0
$711$	846.000	1.18987
$712$	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
$725$	− 529.150i	− 0.729862i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−729.000	−1.00000
$730$	0	0
$731$	0	0
$732$	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
$737$	−1344.00	−1.82361
$738$	0	0
$739$	825.474i	1.11702i	0.829499	+	0.558508i	$0.188626\pi$
$739$	825.474i	1.11702i	−0.829499	+	0.558508i	$0.811374\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−114.000	−0.153432	−0.0767160	−	0.997053i	$-0.524443\pi$
$743$	−114.000	−0.153432	−0.0767160	+	0.997053i	$0.524443\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 740.810i	− 0.989066i
$750$	0	0
$751$	−802.000	−1.06791	−0.533955	−	0.845513i	$-0.679295\pi$
$751$	−802.000	−1.06791	−0.533955	+	0.845513i	$0.679295\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	571.482i	0.754930i	0.926024	+	0.377465i	$0.123204\pi$
$757$	571.482i	0.754930i	−0.926024	+	0.377465i	$0.876796\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 1333.46i	− 1.74765i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	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
$781$	2412.93i	3.08953i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	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
$791$	−1554.00	−1.96460
$792$	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
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	174.000	0.215080	0.107540	−	0.994201i	$-0.465703\pi$
$809$	174.000	0.215080	0.107540	+	0.994201i	$0.465703\pi$
$810$	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
$821$	1164.13i	1.41794i	0.705238	+	0.708971i	$0.250840\pi$
$821$	1164.13i	1.41794i	−0.705238	+	0.708971i	$0.749160\pi$
$822$	0	0
$823$	−622.000	−0.755772	−0.377886	−	0.925852i	$-0.623349\pi$
$823$	−622.000	−0.755772	−0.377886	+	0.925852i	$0.623349\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1629.78i	1.97072i	0.170496	+	0.985358i	$0.445463\pi$
$827$	1629.78i	1.97072i	−0.170496	+	0.985358i	$0.554537\pi$
$828$	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
$833$	0	0
$834$	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
$841$	393.000	0.467301
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2289.00	2.70248
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1142.96i	− 1.34308i
$852$	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
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	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
$863$	−1662.00	−1.92584	−0.962920	−	0.269787i	$-0.913047\pi$
$863$	−1662.00	−1.92584	−0.962920	+	0.269787i	$0.913047\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1989.60i	− 2.28953i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1587.45i	− 1.81009i	−0.425314	−	0.905046i	$-0.639836\pi$
$877$	− 1587.45i	− 1.81009i	0.425314	−	0.905046i	$-0.360164\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	− 698.478i	− 0.791029i	−0.918460	−	0.395514i	$-0.870566\pi$
$883$	− 698.478i	− 0.791029i	0.918460	−	0.395514i	$-0.129434\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	14.0000	0.0157480
$890$	0	0
$891$	1714.45i	1.92418i
$892$	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
$907$	− 317.490i	− 0.350044i	−0.984564	−	0.175022i	$-0.944000\pi$
$907$	− 317.490i	− 0.350044i	0.984564	−	0.175022i	$-0.0559997\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1566.00	1.71899	0.859495	−	0.511144i	$-0.170778\pi$
$911$	1566.00	1.71899	0.859495	+	0.511144i	$0.170778\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	466.000	0.507073	0.253536	−	0.967326i	$-0.418406\pi$
$919$	466.000	0.507073	0.253536	+	0.967326i	$0.418406\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 1587.45i	− 1.71616i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	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
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1164.13i	1.22928i	0.788807	+	0.614641i	$0.210699\pi$
$947$	1164.13i	1.22928i	−0.788807	+	0.614641i	$0.789301\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1458.00	1.52991	0.764953	−	0.644086i	$-0.222762\pi$
$953$	1458.00	1.52991	0.764953	+	0.644086i	$0.222762\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1218.00	1.27007
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	− 952.470i	− 0.989066i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	334.000	0.345398	0.172699	−	0.984975i	$-0.444751\pi$
$967$	334.000	0.345398	0.172699	+	0.984975i	$0.444751\pi$
$968$	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
$977$	162.000	0.165814	0.0829069	−	0.996557i	$-0.473580\pi$
$977$	162.000	0.165814	0.0829069	+	0.996557i	$0.473580\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 1714.45i	− 1.74765i
$982$	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
$989$	1142.96i	1.15568i
$990$	0	0
$991$	−1406.00	−1.41877	−0.709384	−	0.704822i	$-0.751027\pi$
$991$	−1406.00	−1.41877	−0.709384	+	0.704822i	$0.751027\pi$
$992$	0	0
$993$	0	0
$994$	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	224.3.h.a.209.2		2
3.2	odd	2		2016.3.l.b.433.1		2
4.3	odd	2		56.3.h.b.13.2	yes	2
7.6	odd	2	CM	224.3.h.a.209.2		2
8.3	odd	2		56.3.h.b.13.1	✓	2
8.5	even	2	inner	224.3.h.a.209.1		2
12.11	even	2		504.3.l.b.181.1		2
21.20	even	2		2016.3.l.b.433.1		2
24.5	odd	2		2016.3.l.b.433.2		2
24.11	even	2		504.3.l.b.181.2		2
28.3	even	6		392.3.j.b.117.1		4
28.11	odd	6		392.3.j.b.117.1		4
28.19	even	6		392.3.j.b.325.2		4
28.23	odd	6		392.3.j.b.325.2		4
28.27	even	2		56.3.h.b.13.2	yes	2
56.3	even	6		392.3.j.b.117.2		4
56.11	odd	6		392.3.j.b.117.2		4
56.13	odd	2	inner	224.3.h.a.209.1		2
56.19	even	6		392.3.j.b.325.1		4
56.27	even	2		56.3.h.b.13.1	✓	2
56.51	odd	6		392.3.j.b.325.1		4
84.83	odd	2		504.3.l.b.181.1		2
168.83	odd	2		504.3.l.b.181.2		2
168.125	even	2		2016.3.l.b.433.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.h.b.13.1	✓	2	8.3	odd	2
56.3.h.b.13.1	✓	2	56.27	even	2
56.3.h.b.13.2	yes	2	4.3	odd	2
56.3.h.b.13.2	yes	2	28.27	even	2
224.3.h.a.209.1		2	8.5	even	2	inner
224.3.h.a.209.1		2	56.13	odd	2	inner
224.3.h.a.209.2		2	1.1	even	1	trivial
224.3.h.a.209.2		2	7.6	odd	2	CM
392.3.j.b.117.1		4	28.3	even	6
392.3.j.b.117.1		4	28.11	odd	6
392.3.j.b.117.2		4	56.3	even	6
392.3.j.b.117.2		4	56.11	odd	6
392.3.j.b.325.1		4	56.19	even	6
392.3.j.b.325.1		4	56.51	odd	6
392.3.j.b.325.2		4	28.19	even	6
392.3.j.b.325.2		4	28.23	odd	6
504.3.l.b.181.1		2	12.11	even	2
504.3.l.b.181.1		2	84.83	odd	2
504.3.l.b.181.2		2	24.11	even	2
504.3.l.b.181.2		2	168.83	odd	2
2016.3.l.b.433.1		2	3.2	odd	2
2016.3.l.b.433.1		2	21.20	even	2
2016.3.l.b.433.2		2	24.5	odd	2
2016.3.l.b.433.2		2	168.125	even	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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-7.00000 q^{7} -9.00000 q^{9} +21.1660i q^{11} -18.0000 q^{23} -25.0000 q^{25} +21.1660i q^{29} +63.4980i q^{37} -63.4980i q^{43} +49.0000 q^{49} -105.830i q^{53} +63.0000 q^{63} +63.4980i q^{67} +114.000 q^{71} -148.162i q^{77} -94.0000 q^{79} +81.0000 q^{81} -190.494i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{7} - 18 q^{9}+O(q^{10}) \) 2 * q - 14 * q^7 - 18 * q^9	\( 2 q - 14 q^{7} - 18 q^{9} - 36 q^{23} - 50 q^{25} + 98 q^{49} + 126 q^{63} + 228 q^{71} - 188 q^{79} + 162 q^{81}+O(q^{100}) \) 2 * q - 14 * q^7 - 18 * q^9 - 36 * q^23 - 50 * q^25 + 98 * q^49 + 126 * q^63 + 228 * q^71 - 188 * q^79 + 162 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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