LMFDB - Embedded newform 192.3.h.a.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,3,Mod(161,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, 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("192.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	192.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$5.23162107572$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			161.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	192.161
Dual form			192.3.h.a.161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -13.8564 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -13.8564 q^{7} -9.00000 q^{9} +13.8564i q^{13} -26.0000i q^{19} +41.5692i q^{21} -25.0000 q^{25} +27.0000i q^{27} -41.5692 q^{31} -69.2820i q^{37} +41.5692 q^{39} -22.0000i q^{43} +143.000 q^{49} -78.0000 q^{57} -96.9948i q^{61} +124.708 q^{63} +122.000i q^{67} -46.0000 q^{73} +75.0000i q^{75} +69.2820 q^{79} +81.0000 q^{81} -192.000i q^{91} +124.708i q^{93} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9}+O(q^{10}) $ 4 * q - 36 * q^9	$ 4 q - 36 q^{9} - 100 q^{25} + 572 q^{49} - 312 q^{57} - 184 q^{73} + 324 q^{81} - 8 q^{97}+O(q^{100}) $ 4 * q - 36 * q^9 - 100 * q^25 + 572 * q^49 - 312 * q^57 - 184 * q^73 + 324 * q^81 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	− 3.00000i	− 1.00000i
$3$	− 3.00000i	− 1.00000i
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−13.8564	−1.97949	−0.989743	−	0.142857i	$-0.954371\pi$
$7$	−13.8564	−1.97949	−0.989743	+	0.142857i	$0.954371\pi$
$8$	0	0
$9$	−9.00000	−1.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	13.8564i	1.06588i	0.846154	+	0.532939i	$0.178912\pi$
$13$	13.8564i	1.06588i	−0.846154	+	0.532939i	$0.821088\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	− 26.0000i	− 1.36842i	−0.729285	−	0.684211i	$-0.760147\pi$
$19$	− 26.0000i	− 1.36842i	0.729285	−	0.684211i	$-0.239853\pi$
$20$	0	0
$21$	41.5692i	1.97949i
$22$	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
$27$	27.0000i	1.00000i
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−41.5692	−1.34094	−0.670471	−	0.741935i	$-0.733908\pi$
$31$	−41.5692	−1.34094	−0.670471	+	0.741935i	$0.733908\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 69.2820i	− 1.87249i	−0.351351	−	0.936244i	$-0.614278\pi$
$37$	− 69.2820i	− 1.87249i	0.351351	−	0.936244i	$-0.385722\pi$
$38$	0	0
$39$	41.5692	1.06588
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 22.0000i	− 0.511628i	−0.966726	−	0.255814i	$-0.917657\pi$
$43$	− 22.0000i	− 0.511628i	0.966726	−	0.255814i	$-0.0823435\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	143.000	2.91837
$50$	0	0
$51$	0	0
$52$	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
$57$	−78.0000	−1.36842
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	− 96.9948i	− 1.59008i	−0.606557	−	0.795040i	$-0.707450\pi$
$61$	− 96.9948i	− 1.59008i	0.606557	−	0.795040i	$-0.292550\pi$
$62$	0	0
$63$	124.708	1.97949
$64$	0	0
$65$	0	0
$66$	0	0
$67$	122.000i	1.82090i	0.413624	+	0.910448i	$0.364263\pi$
$67$	122.000i	1.82090i	−0.413624	+	0.910448i	$0.635737\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−46.0000	−0.630137	−0.315068	−	0.949069i	$-0.602027\pi$
$73$	−46.0000	−0.630137	−0.315068	+	0.949069i	$0.602027\pi$
$74$	0	0
$75$	75.0000i	1.00000i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	69.2820	0.876988	0.438494	−	0.898734i	$-0.355512\pi$
$79$	69.2820	0.876988	0.438494	+	0.898734i	$0.355512\pi$
$80$	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
$91$	− 192.000i	− 2.10989i
$92$	0	0
$93$	124.708i	1.34094i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.0206186	−0.0103093	−	0.999947i	$-0.503282\pi$
$97$	−2.00000	−0.0206186	−0.0103093	+	0.999947i	$0.503282\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−69.2820	−0.672641	−0.336321	−	0.941748i	$-0.609183\pi$
$103$	−69.2820	−0.672641	−0.336321	+	0.941748i	$0.609183\pi$
$104$	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
$109$	− 41.5692i	− 0.381369i	−0.981651	−	0.190684i	$-0.938929\pi$
$109$	− 41.5692i	− 0.381369i	0.981651	−	0.190684i	$-0.0610707\pi$
$110$	0	0
$111$	−207.846	−1.87249
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 124.708i	− 1.06588i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−121.000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−207.846	−1.63658	−0.818292	−	0.574803i	$-0.805079\pi$
$127$	−207.846	−1.63658	−0.818292	+	0.574803i	$0.805079\pi$
$128$	0	0
$129$	−66.0000	−0.511628
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	360.267i	2.70877i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	22.0000i	0.158273i	0.996864	+	0.0791367i	$0.0252164\pi$
$139$	22.0000i	0.158273i	−0.996864	+	0.0791367i	$0.974784\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 429.000i	− 2.91837i
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	96.9948	0.642350	0.321175	−	0.947020i	$-0.395922\pi$
$151$	96.9948	0.642350	0.321175	+	0.947020i	$0.395922\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	290.985i	1.85340i	0.375796	+	0.926702i	$0.377369\pi$
$157$	290.985i	1.85340i	−0.375796	+	0.926702i	$0.622631\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	262.000i	1.60736i	0.595060	+	0.803681i	$0.297128\pi$
$163$	262.000i	1.60736i	−0.595060	+	0.803681i	$0.702872\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−23.0000	−0.136095
$170$	0	0
$171$	234.000i	1.36842i
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	346.410	1.97949
$176$	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
$181$	− 180.133i	− 0.995212i	−0.867403	−	0.497606i	$-0.834213\pi$
$181$	− 180.133i	− 0.995212i	0.867403	−	0.497606i	$-0.165787\pi$
$182$	0	0
$183$	−290.985	−1.59008
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 374.123i	− 1.97949i
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	−382.000	−1.97927	−0.989637	−	0.143590i	$-0.954135\pi$
$193$	−382.000	−1.97927	−0.989637	+	0.143590i	$0.954135\pi$
$194$	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
$199$	96.9948	0.487411	0.243706	−	0.969849i	$-0.421637\pi$
$199$	96.9948	0.487411	0.243706	+	0.969849i	$0.421637\pi$
$200$	0	0
$201$	366.000	1.82090
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 166.000i	− 0.786730i	−0.919382	−	0.393365i	$-0.871311\pi$
$211$	− 166.000i	− 0.786730i	0.919382	−	0.393365i	$-0.128689\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	576.000	2.65438
$218$	0	0
$219$	138.000i	0.630137i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	290.985	1.30486	0.652432	−	0.757848i	$-0.273749\pi$
$223$	290.985	1.30486	0.652432	+	0.757848i	$0.273749\pi$
$224$	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
$229$	− 457.261i	− 1.99677i	−0.0567686	−	0.998387i	$-0.518080\pi$
$229$	− 457.261i	− 1.99677i	0.0567686	−	0.998387i	$-0.481920\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 207.846i	− 0.876988i
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−286.000	−1.18672	−0.593361	−	0.804936i	$-0.702199\pi$
$241$	−286.000	−1.18672	−0.593361	+	0.804936i	$0.702199\pi$
$242$	0	0
$243$	− 243.000i	− 1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	360.267	1.45857
$248$	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
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	960.000i	3.70656i
$260$	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
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−484.974	−1.78957	−0.894786	−	0.446494i	$-0.852672\pi$
$271$	−484.974	−1.78957	−0.894786	+	0.446494i	$0.852672\pi$
$272$	0	0
$273$	−576.000	−2.10989
$274$	0	0
$275$	0	0
$276$	0	0
$277$	540.400i	1.95090i	0.220217	+	0.975451i	$0.429324\pi$
$277$	540.400i	1.95090i	−0.220217	+	0.975451i	$0.570676\pi$
$278$	0	0
$279$	374.123	1.34094
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 458.000i	− 1.61837i	−0.587551	−	0.809187i	$-0.699908\pi$
$283$	− 458.000i	− 1.61837i	0.587551	−	0.809187i	$-0.300092\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	289.000	1.00000
$290$	0	0
$291$	6.00000i	0.0206186i
$292$	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
$301$	304.841i	1.01276i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 358.000i	− 1.16612i	−0.812428	−	0.583062i	$-0.801855\pi$
$307$	− 358.000i	− 1.16612i	0.812428	−	0.583062i	$-0.198145\pi$
$308$	0	0
$309$	207.846i	0.672641i
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	142.000	0.453674	0.226837	−	0.973933i	$-0.427162\pi$
$313$	142.000	0.453674	0.226837	+	0.973933i	$0.427162\pi$
$314$	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
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 346.410i	− 1.06588i
$326$	0	0
$327$	−124.708	−0.381369
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 362.000i	− 1.09366i	−0.837245	−	0.546828i	$-0.815835\pi$
$331$	− 362.000i	− 1.09366i	0.837245	−	0.546828i	$-0.184165\pi$
$332$	0	0
$333$	623.538i	1.87249i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−482.000	−1.43027	−0.715134	−	0.698988i	$-0.753634\pi$
$337$	−482.000	−1.43027	−0.715134	+	0.698988i	$0.753634\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1302.50	−3.79738
$344$	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
$349$	− 484.974i	− 1.38961i	−0.719198	−	0.694805i	$-0.755490\pi$
$349$	− 484.974i	− 1.38961i	0.719198	−	0.694805i	$-0.244510\pi$
$350$	0	0
$351$	−374.123	−1.06588
$352$	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
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−315.000	−0.872576
$362$	0	0
$363$	363.000i	1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−152.420	−0.415315	−0.207657	−	0.978202i	$-0.566584\pi$
$367$	−152.420	−0.415315	−0.207657	+	0.978202i	$0.566584\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	263.272i	0.705822i	0.935657	+	0.352911i	$0.114808\pi$
$373$	263.272i	0.705822i	−0.935657	+	0.352911i	$0.885192\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 694.000i	− 1.83113i	−0.402165	−	0.915567i	$-0.631742\pi$
$379$	− 694.000i	− 1.83113i	0.402165	−	0.915567i	$-0.368258\pi$
$380$	0	0
$381$	623.538i	1.63658i
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	198.000i	0.511628i
$388$	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
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 706.677i	− 1.78004i	−0.455919	−	0.890021i	$-0.650689\pi$
$397$	− 706.677i	− 1.78004i	0.455919	−	0.890021i	$-0.349311\pi$
$398$	0	0
$399$	1080.80	2.70877
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 576.000i	− 1.42928i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	626.000	1.53056	0.765281	−	0.643696i	$-0.222600\pi$
$409$	626.000	1.53056	0.765281	+	0.643696i	$0.222600\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	66.0000	0.158273
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	762.102i	1.81022i	0.425178	+	0.905110i	$0.360211\pi$
$421$	762.102i	1.81022i	−0.425178	+	0.905110i	$0.639789\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1344.00i	3.14754i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	862.000	1.99076	0.995381	−	0.0960028i	$-0.0306058\pi$
$433$	862.000	1.99076	0.995381	+	0.0960028i	$0.0306058\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	872.954	1.98850	0.994252	−	0.107062i	$-0.0341442\pi$
$439$	872.954	1.98850	0.994252	+	0.107062i	$0.0341442\pi$
$440$	0	0
$441$	−1287.00	−2.91837
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 290.985i	− 0.642350i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−814.000	−1.78118	−0.890591	−	0.454805i	$-0.849709\pi$
$457$	−814.000	−1.78118	−0.890591	+	0.454805i	$0.849709\pi$
$458$	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
$463$	−762.102	−1.64601	−0.823005	−	0.568035i	$-0.807704\pi$
$463$	−762.102	−1.64601	−0.823005	+	0.568035i	$0.807704\pi$
$464$	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
$469$	− 1690.48i	− 3.60444i
$470$	0	0
$471$	872.954	1.85340
$472$	0	0
$473$	0	0
$474$	0	0
$475$	650.000i	1.36842i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	960.000	1.99584
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	152.420	0.312978	0.156489	−	0.987680i	$-0.449982\pi$
$487$	152.420	0.312978	0.156489	+	0.987680i	$0.449982\pi$
$488$	0	0
$489$	786.000	1.60736
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.0000i	0.0521042i	0.999661	+	0.0260521i	$0.00829358\pi$
$499$	26.0000i	0.0521042i	−0.999661	+	0.0260521i	$0.991706\pi$
$500$	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
$507$	69.0000i	0.136095i
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	637.395	1.24735
$512$	0	0
$513$	702.000	1.36842
$514$	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
$523$	− 982.000i	− 1.87763i	−0.344423	−	0.938815i	$-0.611925\pi$
$523$	− 982.000i	− 1.87763i	0.344423	−	0.938815i	$-0.388075\pi$
$524$	0	0
$525$	− 1039.23i	− 1.97949i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	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
$539$	0	0
$540$	0	0
$541$	− 318.697i	− 0.589089i	−0.955638	−	0.294545i	$-0.904832\pi$
$541$	− 318.697i	− 0.589089i	0.955638	−	0.294545i	$-0.0951680\pi$
$542$	0	0
$543$	−540.400	−0.995212
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 506.000i	− 0.925046i	−0.886607	−	0.462523i	$-0.846944\pi$
$547$	− 506.000i	− 0.925046i	0.886607	−	0.462523i	$-0.153056\pi$
$548$	0	0
$549$	872.954i	1.59008i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−960.000	−1.73599
$554$	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
$559$	304.841	0.545333
$560$	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
$567$	−1122.37	−1.97949
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	886.000i	1.55166i	0.630940	+	0.775832i	$0.282670\pi$
$571$	886.000i	1.55166i	−0.630940	+	0.775832i	$0.717330\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−962.000	−1.66724	−0.833622	−	0.552335i	$-0.813737\pi$
$577$	−962.000	−1.66724	−0.833622	+	0.552335i	$0.813737\pi$
$578$	0	0
$579$	1146.00i	1.97927i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	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
$589$	1080.80i	1.83497i
$590$	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
$597$	− 290.985i	− 0.487411i
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−526.000	−0.875208	−0.437604	−	0.899168i	$-0.644173\pi$
$601$	−526.000	−0.875208	−0.437604	+	0.899168i	$0.644173\pi$
$602$	0	0
$603$	− 1098.00i	− 1.82090i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	900.666	1.48380	0.741900	−	0.670511i	$-0.233925\pi$
$607$	900.666	1.48380	0.741900	+	0.670511i	$0.233925\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	484.974i	0.791149i	0.918434	+	0.395574i	$0.129454\pi$
$613$	484.974i	0.791149i	−0.918434	+	0.395574i	$0.870546\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	214.000i	0.345719i	0.984947	+	0.172859i	$0.0553006\pi$
$619$	214.000i	0.345719i	−0.984947	+	0.172859i	$0.944699\pi$
$620$	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
$631$	−1066.94	−1.69088	−0.845438	−	0.534073i	$-0.820661\pi$
$631$	−1066.94	−1.69088	−0.845438	+	0.534073i	$0.820661\pi$
$632$	0	0
$633$	−498.000	−0.786730
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1981.47i	3.11062i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	− 314.000i	− 0.488336i	−0.969733	−	0.244168i	$-0.921485\pi$
$643$	− 314.000i	− 0.488336i	0.969733	−	0.244168i	$-0.0785148\pi$
$644$	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
$651$	− 1728.00i	− 2.65438i
$652$	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
$657$	414.000	0.630137
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	1316.36i	1.99147i	0.0922844	+	0.995733i	$0.470583\pi$
$661$	1316.36i	1.99147i	−0.0922844	+	0.995733i	$0.529417\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 872.954i	− 1.30486i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1154.00	−1.71471	−0.857355	−	0.514725i	$-0.827894\pi$
$673$	−1154.00	−1.71471	−0.857355	+	0.514725i	$0.827894\pi$
$674$	0	0
$675$	− 675.000i	− 1.00000i
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	27.7128	0.0408142
$680$	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
$685$	0	0
$686$	0	0
$687$	−1371.78	−1.99677
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1318.00i	1.90738i	0.300790	+	0.953690i	$0.402750\pi$
$691$	1318.00i	1.90738i	−0.300790	+	0.953690i	$0.597250\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	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
$703$	−1801.33	−2.56235
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 1066.94i	− 1.50486i	−0.658674	−	0.752428i	$-0.728882\pi$
$709$	− 1066.94i	− 1.50486i	0.658674	−	0.752428i	$-0.271118\pi$
$710$	0	0
$711$	−623.538	−0.876988
$712$	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
$721$	960.000	1.33148
$722$	0	0
$723$	858.000i	1.18672i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1371.78	1.88691	0.943455	−	0.331499i	$-0.107554\pi$
$727$	1371.78	1.88691	0.943455	+	0.331499i	$0.107554\pi$
$728$	0	0
$729$	−729.000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 1039.23i	− 1.41778i	−0.705321	−	0.708888i	$-0.749197\pi$
$733$	− 1039.23i	− 1.41778i	0.705321	−	0.708888i	$-0.250803\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 1222.00i	− 1.65359i	−0.562506	−	0.826793i	$-0.690163\pi$
$739$	− 1222.00i	− 1.65359i	0.562506	−	0.826793i	$-0.309837\pi$
$740$	0	0
$741$	− 1080.80i	− 1.45857i
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	900.666	1.19929	0.599645	−	0.800266i	$-0.295309\pi$
$751$	900.666	1.19929	0.599645	+	0.800266i	$0.295309\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1260.93i	1.66570i	0.553501	+	0.832849i	$0.313292\pi$
$757$	1260.93i	1.66570i	−0.553501	+	0.832849i	$0.686708\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	576.000i	0.754915i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1534.00	−1.99480	−0.997399	−	0.0720749i	$-0.977038\pi$
$769$	−1534.00	−1.99480	−0.997399	+	0.0720749i	$0.977038\pi$
$770$	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
$775$	1039.23	1.34094
$776$	0	0
$777$	2880.00	3.70656
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1562.00i	− 1.98475i	−0.123246	−	0.992376i	$-0.539331\pi$
$787$	− 1562.00i	− 1.98475i	0.123246	−	0.992376i	$-0.460669\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1344.00	1.69483
$794$	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
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	1514.00i	1.86683i	0.358797	+	0.933416i	$0.383187\pi$
$811$	1514.00i	1.86683i	−0.358797	+	0.933416i	$0.616813\pi$
$812$	0	0
$813$	1454.92i	1.78957i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−572.000	−0.700122
$818$	0	0
$819$	1728.00i	2.10989i
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	1260.93	1.53212	0.766059	−	0.642770i	$-0.222215\pi$
$823$	1260.93	1.53212	0.766059	+	0.642770i	$0.222215\pi$
$824$	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
$829$	− 1593.49i	− 1.92218i	−0.276236	−	0.961090i	$-0.589087\pi$
$829$	− 1593.49i	− 1.92218i	0.276236	−	0.961090i	$-0.410913\pi$
$830$	0	0
$831$	1621.20	1.95090
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1122.37i	− 1.34094i
$838$	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
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1676.63	1.97949
$848$	0	0
$849$	−1374.00	−1.61837
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 401.836i	− 0.471085i	−0.971864	−	0.235543i	$-0.924313\pi$
$853$	− 401.836i	− 0.471085i	0.971864	−	0.235543i	$-0.0756867\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1418.00i	1.65076i	0.564580	+	0.825378i	$0.309038\pi$
$859$	1418.00i	1.65076i	−0.564580	+	0.825378i	$0.690962\pi$
$860$	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
$867$	− 867.000i	− 1.00000i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−1690.48	−1.94085
$872$	0	0
$873$	18.0000	0.0206186
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1648.91i	− 1.88017i	−0.340935	−	0.940087i	$-0.610744\pi$
$877$	− 1648.91i	− 1.88017i	0.340935	−	0.940087i	$-0.389256\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	1702.00i	1.92752i	0.266771	+	0.963760i	$0.414043\pi$
$883$	1702.00i	1.92752i	−0.266771	+	0.963760i	$0.585957\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	2880.00	3.23960
$890$	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
$903$	914.523	1.01276
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 214.000i	− 0.235943i	−0.993017	−	0.117971i	$-0.962361\pi$
$907$	− 214.000i	− 0.235943i	0.993017	−	0.117971i	$-0.0376391\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1621.20	−1.76409	−0.882045	−	0.471164i	$-0.843834\pi$
$919$	−1621.20	−1.76409	−0.882045	+	0.471164i	$0.843834\pi$
$920$	0	0
$921$	−1074.00	−1.16612
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1732.05i	1.87249i
$926$	0	0
$927$	623.538	0.672641
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	− 3718.00i	− 3.99356i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1198.00	1.27855	0.639274	−	0.768979i	$-0.279235\pi$
$937$	1198.00	1.27855	0.639274	+	0.768979i	$0.279235\pi$
$938$	0	0
$939$	− 426.000i	− 0.453674i
$940$	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
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	− 637.395i	− 0.671649i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	767.000	0.798127
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1177.79	−1.21799	−0.608994	−	0.793175i	$-0.708427\pi$
$967$	−1177.79	−1.21799	−0.608994	+	0.793175i	$0.708427\pi$
$968$	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
$973$	− 304.841i	− 0.313300i
$974$	0	0
$975$	−1039.23	−1.06588
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	374.123i	0.381369i
$982$	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
$991$	−1981.47	−1.99946	−0.999731	−	0.0232089i	$-0.992612\pi$
$991$	−1981.47	−1.99946	−0.999731	+	0.0232089i	$0.992612\pi$
$992$	0	0
$993$	−1086.00	−1.09366
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 623.538i	− 0.625415i	−0.949850	−	0.312707i	$-0.898764\pi$
$997$	− 623.538i	− 0.625415i	0.949850	−	0.312707i	$-0.101236\pi$
$998$	0	0
$999$	1870.61	1.87249

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.h.a.161.1	✓	4
3.2	odd	2	CM	192.3.h.a.161.1	✓	4
4.3	odd	2	inner	192.3.h.a.161.4	yes	4
8.3	odd	2	inner	192.3.h.a.161.2	yes	4
8.5	even	2	inner	192.3.h.a.161.3	yes	4
12.11	even	2	inner	192.3.h.a.161.4	yes	4
16.3	odd	4		768.3.e.a.257.1		2
16.5	even	4		768.3.e.a.257.2		2
16.11	odd	4		768.3.e.f.257.1		2
16.13	even	4		768.3.e.f.257.2		2
24.5	odd	2	inner	192.3.h.a.161.3	yes	4
24.11	even	2	inner	192.3.h.a.161.2	yes	4
48.5	odd	4		768.3.e.a.257.2		2
48.11	even	4		768.3.e.f.257.1		2
48.29	odd	4		768.3.e.f.257.2		2
48.35	even	4		768.3.e.a.257.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.h.a.161.1	✓	4	1.1	even	1	trivial
192.3.h.a.161.1	✓	4	3.2	odd	2	CM
192.3.h.a.161.2	yes	4	8.3	odd	2	inner
192.3.h.a.161.2	yes	4	24.11	even	2	inner
192.3.h.a.161.3	yes	4	8.5	even	2	inner
192.3.h.a.161.3	yes	4	24.5	odd	2	inner
192.3.h.a.161.4	yes	4	4.3	odd	2	inner
192.3.h.a.161.4	yes	4	12.11	even	2	inner
768.3.e.a.257.1		2	16.3	odd	4
768.3.e.a.257.1		2	48.35	even	4
768.3.e.a.257.2		2	16.5	even	4
768.3.e.a.257.2		2	48.5	odd	4
768.3.e.f.257.1		2	16.11	odd	4
768.3.e.f.257.1		2	48.11	even	4
768.3.e.f.257.2		2	16.13	even	4
768.3.e.f.257.2		2	48.29	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	192.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -13.8564 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -13.8564 q^{7} -9.00000 q^{9} +13.8564i q^{13} -26.0000i q^{19} +41.5692i q^{21} -25.0000 q^{25} +27.0000i q^{27} -41.5692 q^{31} -69.2820i q^{37} +41.5692 q^{39} -22.0000i q^{43} +143.000 q^{49} -78.0000 q^{57} -96.9948i q^{61} +124.708 q^{63} +122.000i q^{67} -46.0000 q^{73} +75.0000i q^{75} +69.2820 q^{79} +81.0000 q^{81} -192.000i q^{91} +124.708i q^{93} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9}+O(q^{10}) \) 4 * q - 36 * q^9	\( 4 q - 36 q^{9} - 100 q^{25} + 572 q^{49} - 312 q^{57} - 184 q^{73} + 324 q^{81} - 8 q^{97}+O(q^{100}) \) 4 * q - 36 * q^9 - 100 * q^25 + 572 * q^49 - 312 * q^57 - 184 * q^73 + 324 * q^81 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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