LMFDB - Embedded newform 395.3.c.b.394.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [395,3,Mod(394,395)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(395, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("395.394");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 395 = 5 \cdot 79 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	395.c (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:	yes
Analytic conductor:	$10.7629704422$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.31600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 19x^{2} + 79 $ x^4 - 19*x^2 + 79
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			394.2
Root			$2.47909$ of defining polynomial
Character	$\chi$	$=$	395.394

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53216 q^{3} +4.00000 q^{4} +5.00000 q^{5} +9.55467 q^{7} -6.65248 q^{9} +O(q^{10})$	$q-1.53216 q^{3} +4.00000 q^{4} +5.00000 q^{5} +9.55467 q^{7} -6.65248 q^{9} +20.1246 q^{11} -6.12865 q^{12} -7.66082 q^{15} +16.0000 q^{16} -33.9839 q^{17} -13.4164 q^{19} +20.0000 q^{20} -14.6393 q^{21} +25.0000 q^{25} +23.9821 q^{27} +38.2187 q^{28} -17.0000 q^{31} -30.8342 q^{33} +47.7734 q^{35} -26.6099 q^{36} -30.1108 q^{37} +82.1190 q^{43} +80.4984 q^{44} -33.2624 q^{45} -73.6494 q^{47} -24.5146 q^{48} +42.2918 q^{49} +52.0689 q^{51} +102.313 q^{53} +100.623 q^{55} +20.5561 q^{57} -30.6433 q^{60} -63.5622 q^{63} +64.0000 q^{64} -135.936 q^{68} -38.3041 q^{75} -53.6656 q^{76} +192.284 q^{77} -79.0000 q^{79} +80.0000 q^{80} +23.1277 q^{81} -58.5573 q^{84} -169.920 q^{85} -174.413 q^{89} +26.0468 q^{93} -67.0820 q^{95} -133.878 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{4} + 20 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q + 16 * q^4 + 20 * q^5 + 36 * q^9	$ 4 q + 16 q^{4} + 20 q^{5} + 36 q^{9} + 64 q^{16} + 80 q^{20} - 148 q^{21} + 100 q^{25} - 68 q^{31} + 144 q^{36} + 180 q^{45} + 196 q^{49} + 92 q^{51} + 256 q^{64} - 316 q^{79} + 320 q^{80} + 656 q^{81} - 592 q^{84} - 1260 q^{99}+O(q^{100}) $ 4 * q + 16 * q^4 + 20 * q^5 + 36 * q^9 + 64 * q^16 + 80 * q^20 - 148 * q^21 + 100 * q^25 - 68 * q^31 + 144 * q^36 + 180 * q^45 + 196 * q^49 + 92 * q^51 + 256 * q^64 - 316 * q^79 + 320 * q^80 + 656 * q^81 - 592 * q^84 - 1260 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/395\mathbb{Z}\right)^\times$.

$n$	$161$	$317$
$\chi(n)$	$-1$	$-1$

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	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	−1.53216	−0.510721	−0.255361	−	0.966846i	$-0.582194\pi$
$3$	−1.53216	−0.510721	−0.255361	+	0.966846i	$0.582194\pi$
$4$	4.00000	1.00000
$5$	5.00000	1.00000
$5$	5.00000	1.00000
$6$	0	0
$7$	9.55467	1.36495	0.682477	−	0.730907i	$-0.260903\pi$
$7$	9.55467	1.36495	0.682477	+	0.730907i	$0.260903\pi$
$8$	0	0
$9$	−6.65248	−0.739164
$10$	0	0
$11$	20.1246	1.82951	0.914755	−	0.404009i	$-0.132384\pi$
$11$	20.1246	1.82951	0.914755	+	0.404009i	$0.132384\pi$
$12$	−6.12865	−0.510721
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−7.66082	−0.510721
$16$	16.0000	1.00000
$17$	−33.9839	−1.99905	−0.999527	−	0.0307696i	$-0.990204\pi$
$17$	−33.9839	−1.99905	−0.999527	+	0.0307696i	$0.990204\pi$
$18$	0	0
$19$	−13.4164	−0.706127	−0.353063	−	0.935599i	$-0.614860\pi$
$19$	−13.4164	−0.706127	−0.353063	+	0.935599i	$0.614860\pi$
$20$	20.0000	1.00000
$21$	−14.6393	−0.697110
$22$	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
$27$	23.9821	0.888228
$28$	38.2187	1.36495
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−17.0000	−0.548387	−0.274194	−	0.961675i	$-0.588411\pi$
$31$	−17.0000	−0.548387	−0.274194	+	0.961675i	$0.588411\pi$
$32$	0	0
$33$	−30.8342	−0.934369
$34$	0	0
$35$	47.7734	1.36495
$36$	−26.6099	−0.739164
$37$	−30.1108	−0.813805	−0.406903	−	0.913472i	$-0.633391\pi$
$37$	−30.1108	−0.813805	−0.406903	+	0.913472i	$0.633391\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	82.1190	1.90974	0.954872	−	0.297019i	$-0.0959923\pi$
$43$	82.1190	1.90974	0.954872	+	0.297019i	$0.0959923\pi$
$44$	80.4984	1.82951
$45$	−33.2624	−0.739164
$46$	0	0
$47$	−73.6494	−1.56701	−0.783504	−	0.621387i	$-0.786570\pi$
$47$	−73.6494	−1.56701	−0.783504	+	0.621387i	$0.786570\pi$
$48$	−24.5146	−0.510721
$49$	42.2918	0.863098
$50$	0	0
$51$	52.0689	1.02096
$52$	0	0
$53$	102.313	1.93044	0.965221	−	0.261436i	$-0.0841961\pi$
$53$	102.313	1.93044	0.965221	+	0.261436i	$0.0841961\pi$
$54$	0	0
$55$	100.623	1.82951
$56$	0	0
$57$	20.5561	0.360634
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−30.6433	−0.510721
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−63.5622	−1.00892
$64$	64.0000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−135.936	−1.99905
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−38.3041	−0.510721
$76$	−53.6656	−0.706127
$77$	192.284	2.49720
$78$	0	0
$79$	−79.0000	−1.00000
$79$	−79.0000	−1.00000
$80$	80.0000	1.00000
$81$	23.1277	0.285527
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−58.5573	−0.697110
$85$	−169.920	−1.99905
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−174.413	−1.95970	−0.979850	−	0.199735i	$-0.935992\pi$
$89$	−174.413	−1.95970	−0.979850	+	0.199735i	$0.935992\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	26.0468	0.280073
$94$	0	0
$95$	−67.0820	−0.706127
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	−133.878	−1.35231
$100$	100.000	1.00000
$101$	−193.000	−1.91089	−0.955446	−	0.295167i	$-0.904625\pi$
$101$	−193.000	−1.91089	−0.955446	+	0.295167i	$0.904625\pi$
$102$	0	0
$103$	166.770	1.61912	0.809562	−	0.587034i	$-0.199705\pi$
$103$	166.770	1.61912	0.809562	+	0.587034i	$0.199705\pi$
$104$	0	0
$105$	−73.1966	−0.697110
$106$	0	0
$107$	1.70293	0.0159153	0.00795763	−	0.999968i	$-0.497467\pi$
$107$	1.70293	0.0159153	0.00795763	+	0.999968i	$0.497467\pi$
$108$	95.9286	0.888228
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	46.1347	0.415628
$112$	152.875	1.36495
$113$	−222.651	−1.97036	−0.985182	−	0.171514i	$-0.945134\pi$
$113$	−222.651	−1.97036	−0.985182	+	0.171514i	$0.945134\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−324.705	−2.72861
$120$	0	0
$121$	284.000	2.34711
$122$	0	0
$123$	0	0
$124$	−68.0000	−0.548387
$125$	125.000	1.00000
$126$	0	0
$127$	−167.026	−1.31516	−0.657582	−	0.753383i	$-0.728421\pi$
$127$	−167.026	−1.31516	−0.657582	+	0.753383i	$0.728421\pi$
$128$	0	0
$129$	−125.820	−0.975346
$130$	0	0
$131$	46.9574	0.358454	0.179227	−	0.983808i	$-0.442640\pi$
$131$	46.9574	0.358454	0.179227	+	0.983808i	$0.442640\pi$
$132$	−123.337	−0.934369
$133$	−128.189	−0.963830
$134$	0	0
$135$	119.911	0.888228
$136$	0	0
$137$	−37.9625	−0.277099	−0.138549	−	0.990356i	$-0.544244\pi$
$137$	−37.9625	−0.277099	−0.138549	+	0.990356i	$0.544244\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	191.093	1.36495
$141$	112.843	0.800304
$142$	0	0
$143$	0	0
$144$	−106.440	−0.739164
$145$	0	0
$146$	0	0
$147$	−64.7979	−0.440802
$148$	−120.443	−0.813805
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	−301.869	−1.99913	−0.999567	−	0.0294311i	$-0.990630\pi$
$151$	−301.869	−1.99913	−0.999567	+	0.0294311i	$0.990630\pi$
$152$	0	0
$153$	226.077	1.47763
$154$	0	0
$155$	−85.0000	−0.548387
$156$	0	0
$157$	48.2406	0.307265	0.153633	−	0.988128i	$-0.450903\pi$
$157$	48.2406	0.307265	0.153633	+	0.988128i	$0.450903\pi$
$158$	0	0
$159$	−156.761	−0.985917
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−154.171	−0.934369
$166$	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
$171$	89.2523	0.521943
$172$	328.476	1.90974
$173$	207.882	1.20163	0.600815	−	0.799388i	$-0.294843\pi$
$173$	207.882	1.20163	0.600815	+	0.799388i	$0.294843\pi$
$174$	0	0
$175$	238.867	1.36495
$176$	321.994	1.82951
$177$	0	0
$178$	0	0
$179$	−353.000	−1.97207	−0.986034	−	0.166547i	$-0.946738\pi$
$179$	−353.000	−1.97207	−0.986034	+	0.166547i	$0.946738\pi$
$180$	−133.050	−0.739164
$181$	234.787	1.29717	0.648583	−	0.761144i	$-0.275362\pi$
$181$	234.787	1.29717	0.648583	+	0.761144i	$0.275362\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−150.554	−0.813805
$186$	0	0
$187$	−683.913	−3.65729
$188$	−294.598	−1.56701
$189$	229.142	1.21239
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−98.0584	−0.510721
$193$	184.538	0.956155	0.478077	−	0.878318i	$-0.341334\pi$
$193$	184.538	0.956155	0.478077	+	0.878318i	$0.341334\pi$
$194$	0	0
$195$	0	0
$196$	169.167	0.863098
$197$	392.571	1.99274	0.996372	−	0.0851049i	$-0.0271225\pi$
$197$	392.571	1.99274	0.996372	+	0.0851049i	$0.0271225\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	208.276	1.02096
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−270.000	−1.29187
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	409.254	1.93044
$213$	0	0
$214$	0	0
$215$	410.595	1.90974
$216$	0	0
$217$	−162.429	−0.748523
$218$	0	0
$219$	0	0
$220$	402.492	1.82951
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−166.312	−0.739164
$226$	0	0
$227$	−256.529	−1.13009	−0.565043	−	0.825062i	$-0.691140\pi$
$227$	−256.529	−1.13009	−0.565043	+	0.825062i	$0.691140\pi$
$228$	82.2245	0.360634
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	−294.611	−1.27537
$232$	0	0
$233$	−399.232	−1.71344	−0.856720	−	0.515781i	$-0.827502\pi$
$233$	−399.232	−1.71344	−0.856720	+	0.515781i	$0.827502\pi$
$234$	0	0
$235$	−368.247	−1.56701
$236$	0	0
$237$	121.041	0.510721
$238$	0	0
$239$	−233.000	−0.974895	−0.487448	−	0.873152i	$-0.662072\pi$
$239$	−233.000	−0.974895	−0.487448	+	0.873152i	$0.662072\pi$
$240$	−122.573	−0.510721
$241$	−281.745	−1.16906	−0.584532	−	0.811370i	$-0.698722\pi$
$241$	−281.745	−1.16906	−0.584532	+	0.811370i	$0.698722\pi$
$242$	0	0
$243$	−251.275	−1.03405
$244$	0	0
$245$	211.459	0.863098
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−254.249	−1.00892
$253$	0	0
$254$	0	0
$255$	260.344	1.02096
$256$	256.000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	−287.699	−1.11081
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	511.567	1.93044
$266$	0	0
$267$	267.230	1.00086
$268$	0	0
$269$	395.784	1.47132	0.735658	−	0.677353i	$-0.236873\pi$
$269$	395.784	1.47132	0.735658	+	0.677353i	$0.236873\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−543.742	−1.99905
$273$	0	0
$274$	0	0
$275$	503.115	1.82951
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	113.092	0.405348
$280$	0	0
$281$	167.000	0.594306	0.297153	−	0.954830i	$-0.403963\pi$
$281$	167.000	0.594306	0.297153	+	0.954830i	$0.403963\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	102.781	0.360634
$286$	0	0
$287$	0	0
$288$	0	0
$289$	865.906	2.99621
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−578.811	−1.97547	−0.987733	−	0.156154i	$-0.950090\pi$
$293$	−578.811	−1.97547	−0.987733	+	0.156154i	$0.950090\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	482.631	1.62502
$298$	0	0
$299$	0	0
$300$	−153.216	−0.510721
$301$	784.620	2.60671
$302$	0	0
$303$	295.708	0.975932
$304$	−214.663	−0.706127
$305$	0	0
$306$	0	0
$307$	514.461	1.67577	0.837884	−	0.545849i	$-0.183793\pi$
$307$	514.461	1.67577	0.837884	+	0.545849i	$0.183793\pi$
$308$	769.136	2.49720
$309$	−255.519	−0.826921
$310$	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
$315$	−317.811	−1.00892
$316$	−316.000	−1.00000
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	320.000	1.00000
$321$	−2.60917	−0.00812826
$322$	0	0
$323$	455.942	1.41158
$324$	92.5109	0.285527
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−703.696	−2.13889
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	200.311	0.601536
$334$	0	0
$335$	0	0
$336$	−234.229	−0.697110
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	341.138	1.00631
$340$	−679.678	−1.99905
$341$	−342.118	−1.00328
$342$	0	0
$343$	−64.0947	−0.186865
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−512.079	−1.45065	−0.725325	−	0.688407i	$-0.758310\pi$
$353$	−512.079	−1.45065	−0.725325	+	0.688407i	$0.758310\pi$
$354$	0	0
$355$	0	0
$356$	−697.653	−1.95970
$357$	497.501	1.39356
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−181.000	−0.501385
$362$	0	0
$363$	−435.134	−1.19872
$364$	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
$371$	977.571	2.63496
$372$	104.187	0.280073
$373$	−354.201	−0.949601	−0.474801	−	0.880093i	$-0.657480\pi$
$373$	−354.201	−0.949601	−0.474801	+	0.880093i	$0.657480\pi$
$374$	0	0
$375$	−191.520	−0.510721
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	−268.328	−0.706127
$381$	255.911	0.671682
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	961.421	2.49720
$386$	0	0
$387$	−546.294	−1.41161
$388$	0	0
$389$	67.0000	0.172237	0.0861183	−	0.996285i	$-0.472554\pi$
$389$	67.0000	0.172237	0.0861183	+	0.996285i	$0.472554\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−71.9464	−0.183070
$394$	0	0
$395$	−395.000	−1.00000
$396$	−535.514	−1.35231
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	196.407	0.492248
$400$	400.000	1.00000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	−772.000	−1.91089
$405$	115.639	0.285527
$406$	0	0
$407$	−605.968	−1.48887
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	58.1648	0.141520
$412$	667.079	1.61912
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	−292.786	−0.697110
$421$	288.453	0.685161	0.342580	−	0.939488i	$-0.388699\pi$
$421$	288.453	0.685161	0.342580	+	0.939488i	$0.388699\pi$
$422$	0	0
$423$	489.951	1.15828
$424$	0	0
$425$	−849.598	−1.99905
$426$	0	0
$427$	0	0
$428$	6.81173	0.0159153
$429$	0	0
$430$	0	0
$431$	−718.000	−1.66589	−0.832947	−	0.553353i	$-0.813348\pi$
$431$	−718.000	−1.66589	−0.832947	+	0.553353i	$0.813348\pi$
$432$	383.714	0.888228
$433$	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
$439$	610.447	1.39054	0.695269	−	0.718749i	$-0.255285\pi$
$439$	610.447	1.39054	0.695269	+	0.718749i	$0.255285\pi$
$440$	0	0
$441$	−281.345	−0.637971
$442$	0	0
$443$	−658.971	−1.48752	−0.743760	−	0.668447i	$-0.766959\pi$
$443$	−658.971	−1.48752	−0.743760	+	0.668447i	$0.766959\pi$
$444$	184.539	0.415628
$445$	−872.067	−1.95970
$446$	0	0
$447$	0	0
$448$	611.499	1.36495
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	−890.604	−1.97036
$453$	462.513	1.02100
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	−815.007	−1.77561
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−318.514	−0.687936	−0.343968	−	0.938981i	$-0.611771\pi$
$463$	−318.514	−0.687936	−0.343968	+	0.938981i	$0.611771\pi$
$464$	0	0
$465$	130.234	0.280073
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−73.9125	−0.156927
$472$	0	0
$473$	1652.61	3.49389
$474$	0	0
$475$	−335.410	−0.706127
$476$	−1298.82	−2.72861
$477$	−680.637	−1.42691
$478$	0	0
$479$	563.000	1.17537	0.587683	−	0.809092i	$-0.300040\pi$
$479$	563.000	1.17537	0.587683	+	0.809092i	$0.300040\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	1136.00	2.34711
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−669.392	−1.35231
$496$	−272.000	−0.548387
$497$	0	0
$498$	0	0
$499$	−436.033	−0.873814	−0.436907	−	0.899507i	$-0.643926\pi$
$499$	−436.033	−0.873814	−0.436907	+	0.899507i	$0.643926\pi$
$500$	500.000	1.00000
$501$	0	0
$502$	0	0
$503$	890.860	1.77109	0.885547	−	0.464550i	$-0.153784\pi$
$503$	890.860	1.77109	0.885547	+	0.464550i	$0.153784\pi$
$504$	0	0
$505$	−965.000	−1.91089
$506$	0	0
$507$	−258.936	−0.510721
$508$	−668.104	−1.31516
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−321.754	−0.627201
$514$	0	0
$515$	833.849	1.61912
$516$	−503.279	−0.975346
$517$	−1482.17	−2.86686
$518$	0	0
$519$	−318.509	−0.613698
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	187.830	0.358454
$525$	−365.983	−0.697110
$526$	0	0
$527$	577.726	1.09625
$528$	−493.347	−0.934369
$529$	529.000	1.00000
$530$	0	0
$531$	0	0
$532$	−512.758	−0.963830
$533$	0	0
$534$	0	0
$535$	8.51466	0.0159153
$536$	0	0
$537$	540.854	1.00718
$538$	0	0
$539$	851.106	1.57905
$540$	479.643	0.888228
$541$	845.234	1.56235	0.781177	−	0.624309i	$-0.214620\pi$
$541$	845.234	1.56235	0.781177	+	0.624309i	$0.214620\pi$
$542$	0	0
$543$	−359.732	−0.662490
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−151.850	−0.277099
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−754.819	−1.36495
$554$	0	0
$555$	230.673	0.415628
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	764.374	1.36495
$561$	1047.87	1.86785
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	451.371	0.800304
$565$	−1113.26	−1.97036
$566$	0	0
$567$	220.978	0.389732
$568$	0	0
$569$	−597.030	−1.04926	−0.524631	−	0.851330i	$-0.675797\pi$
$569$	−597.030	−1.04926	−0.524631	+	0.851330i	$0.675797\pi$
$570$	0	0
$571$	1063.00	1.86165	0.930823	−	0.365470i	$-0.119092\pi$
$571$	1063.00	1.86165	0.930823	+	0.365470i	$0.119092\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−425.758	−0.739164
$577$	−50.9682	−0.0883330	−0.0441665	−	0.999024i	$-0.514063\pi$
$577$	−50.9682	−0.0883330	−0.0441665	+	0.999024i	$0.514063\pi$
$578$	0	0
$579$	−282.742	−0.488328
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2059.02	3.53176
$584$	0	0
$585$	0	0
$586$	0	0
$587$	678.910	1.15658	0.578288	−	0.815833i	$-0.303721\pi$
$587$	678.910	1.15658	0.578288	+	0.815833i	$0.303721\pi$
$588$	−259.192	−0.440802
$589$	228.079	0.387231
$590$	0	0
$591$	−601.482	−1.01774
$592$	−481.773	−0.813805
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	−1623.53	−2.72861
$596$	0	0
$597$	0	0
$598$	0	0
$599$	469.574	0.783930	0.391965	−	0.919980i	$-0.371795\pi$
$599$	469.574	0.783930	0.391965	+	0.919980i	$0.371795\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	−1207.48	−1.99913
$605$	1420.00	2.34711
$606$	0	0
$607$	255.249	0.420508	0.210254	−	0.977647i	$-0.432571\pi$
$607$	255.249	0.420508	0.210254	+	0.977647i	$0.432571\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	904.308	1.47763
$613$	795.012	1.29692	0.648460	−	0.761248i	$-0.275413\pi$
$613$	795.012	1.29692	0.648460	+	0.761248i	$0.275413\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	−340.000	−0.548387
$621$	0	0
$622$	0	0
$623$	−1666.46	−2.67490
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	413.684	0.659783
$628$	192.962	0.307265
$629$	1023.28	1.62684
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−835.130	−1.31516
$636$	−627.043	−0.985917
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1220.89	1.90467	0.952335	−	0.305055i	$-0.0986749\pi$
$641$	1220.89	1.90467	0.952335	+	0.305055i	$0.0986749\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	−629.098	−0.975346
$646$	0	0
$647$	1181.12	1.82553	0.912765	−	0.408486i	$-0.133943\pi$
$647$	1181.12	1.82553	0.912765	+	0.408486i	$0.133943\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	248.868	0.382286
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	234.787	0.358454
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	−616.684	−0.934369
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−640.947	−0.963830
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1314.48	−1.95316	−0.976580	−	0.215156i	$-0.930974\pi$
$673$	−1314.48	−1.95316	−0.976580	+	0.215156i	$0.930974\pi$
$674$	0	0
$675$	599.554	0.888228
$676$	676.000	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	393.045	0.577159
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	357.009	0.521943
$685$	−189.813	−0.277099
$686$	0	0
$687$	0	0
$688$	1313.90	1.90974
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	831.528	1.20163
$693$	−1279.17	−1.84584
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	611.688	0.875090
$700$	955.467	1.36495
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	403.979	0.574650
$704$	1287.98	1.82951
$705$	564.214	0.800304
$706$	0	0
$707$	−1844.05	−2.60828
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	525.546	0.739164
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−1412.00	−1.97207
$717$	356.994	0.497900
$718$	0	0
$719$	−142.000	−0.197497	−0.0987483	−	0.995112i	$-0.531484\pi$
$719$	−142.000	−0.197497	−0.0987483	+	0.995112i	$0.531484\pi$
$720$	−532.198	−0.739164
$721$	1593.43	2.21003
$722$	0	0
$723$	431.679	0.597066
$724$	939.149	1.29717
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	176.845	0.242585
$730$	0	0
$731$	−2790.72	−3.81768
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−323.990	−0.440802
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	−602.216	−0.813805
$741$	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
$747$	0	0
$748$	−2735.65	−3.65729
$749$	16.2710	0.0217236
$750$	0	0
$751$	362.243	0.482348	0.241174	−	0.970482i	$-0.422468\pi$
$751$	362.243	0.482348	0.241174	+	0.970482i	$0.422468\pi$
$752$	−1178.39	−1.56701
$753$	0	0
$754$	0	0
$755$	−1509.35	−1.99913
$756$	916.566	1.21239
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	717.778	0.943203	0.471602	−	0.881812i	$-0.343676\pi$
$761$	717.778	0.943203	0.471602	+	0.881812i	$0.343676\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1130.39	1.47763
$766$	0	0
$767$	0	0
$768$	−392.234	−0.510721
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	738.152	0.956155
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−425.000	−0.548387
$776$	0	0
$777$	440.802	0.567312
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	676.669	0.863098
$785$	241.203	0.307265
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	1570.28	1.99274
$789$	0	0
$790$	0	0
$791$	−2127.36	−2.68945
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−783.804	−0.985917
$796$	0	0
$797$	336.162	0.421784	0.210892	−	0.977509i	$-0.432363\pi$
$797$	336.162	0.421784	0.210892	+	0.977509i	$0.432363\pi$
$798$	0	0
$799$	2502.89	3.13253
$800$	0	0
$801$	1160.28	1.44854
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−606.406	−0.751432
$808$	0	0
$809$	1576.43	1.94861	0.974307	−	0.225226i	$-0.0723121\pi$
$809$	1576.43	1.94861	0.974307	+	0.225226i	$0.0723121\pi$
$810$	0	0
$811$	−1214.18	−1.49715	−0.748573	−	0.663053i	$-0.769261\pi$
$811$	−1214.18	−1.49715	−0.748573	+	0.663053i	$0.769261\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	833.102	1.02096
$817$	−1101.74	−1.34852
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−328.702	−0.400368	−0.200184	−	0.979758i	$-0.564154\pi$
$821$	−328.702	−0.400368	−0.200184	+	0.979758i	$0.564154\pi$
$822$	0	0
$823$	452.264	0.549532	0.274766	−	0.961511i	$-0.411400\pi$
$823$	452.264	0.549532	0.274766	+	0.961511i	$0.411400\pi$
$824$	0	0
$825$	−770.855	−0.934369
$826$	0	0
$827$	642.378	0.776757	0.388379	−	0.921500i	$-0.373035\pi$
$827$	642.378	0.776757	0.388379	+	0.921500i	$0.373035\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1437.24	−1.72538
$834$	0	0
$835$	0	0
$836$	−1080.00	−1.29187
$837$	−407.697	−0.487093
$838$	0	0
$839$	967.000	1.15256	0.576281	−	0.817251i	$-0.304503\pi$
$839$	967.000	1.15256	0.576281	+	0.817251i	$0.304503\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	−255.871	−0.303525
$844$	0	0
$845$	845.000	1.00000
$846$	0	0
$847$	2713.53	3.20369
$848$	1637.01	1.93044
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1013.41	1.18806	0.594029	−	0.804444i	$-0.297536\pi$
$853$	1013.41	1.18806	0.594029	+	0.804444i	$0.297536\pi$
$854$	0	0
$855$	446.262	0.521943
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	1642.38	1.90974
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	1039.41	1.20163
$866$	0	0
$867$	−1326.71	−1.53023
$868$	−649.718	−0.748523
$869$	−1589.84	−1.82951
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1194.33	1.36495
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	886.834	1.00891
$880$	1609.97	1.82951
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	229.584	0.260004	0.130002	−	0.991514i	$-0.458502\pi$
$883$	229.584	0.260004	0.130002	+	0.991514i	$0.458502\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−1595.88	−1.79514
$890$	0	0
$891$	465.436	0.522375
$892$	0	0
$893$	988.110	1.10651
$894$	0	0
$895$	−1765.00	−1.97207
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−665.248	−0.739164
$901$	−3477.01	−3.85905
$902$	0	0
$903$	−1202.17	−1.33130
$904$	0	0
$905$	1173.94	1.29717
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−1026.12	−1.13009
$909$	1283.93	1.41246
$910$	0	0
$911$	1427.00	1.56641	0.783205	−	0.621763i	$-0.213583\pi$
$911$	1427.00	1.56641	0.783205	+	0.621763i	$0.213583\pi$
$912$	328.898	0.360634
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	448.663	0.489273
$918$	0	0
$919$	−1717.00	−1.86834	−0.934168	−	0.356835i	$-0.883856\pi$
$919$	−1717.00	−1.86834	−0.934168	+	0.356835i	$0.883856\pi$
$920$	0	0
$921$	−788.238	−0.855850
$922$	0	0
$923$	0	0
$924$	−1178.44	−1.27537
$925$	−752.770	−0.813805
$926$	0	0
$927$	−1109.43	−1.19680
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−567.404	−0.609456
$932$	−1596.93	−1.71344
$933$	0	0
$934$	0	0
$935$	−3419.56	−3.65729
$936$	0	0
$937$	1546.98	1.65100	0.825498	−	0.564405i	$-0.190894\pi$
$937$	1546.98	1.65100	0.825498	+	0.564405i	$0.190894\pi$
$938$	0	0
$939$	0	0
$940$	−1472.99	−1.56701
$941$	−301.869	−0.320796	−0.160398	−	0.987052i	$-0.551278\pi$
$941$	−301.869	−0.320796	−0.160398	+	0.987052i	$0.551278\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	1145.71	1.21239
$946$	0	0
$947$	−1008.68	−1.06513	−0.532567	−	0.846388i	$-0.678773\pi$
$947$	−1008.68	−1.06513	−0.532567	+	0.846388i	$0.678773\pi$
$948$	484.164	0.510721
$949$	0	0
$950$	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
$956$	−932.000	−0.974895
$957$	0	0
$958$	0	0
$959$	−362.720	−0.378227
$960$	−490.292	−0.510721
$961$	−672.000	−0.699272
$962$	0	0
$963$	−11.3287	−0.0117640
$964$	−1126.98	−1.16906
$965$	922.690	0.956155
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	−698.577	−0.720926
$970$	0	0
$971$	362.000	0.372812	0.186406	−	0.982473i	$-0.440316\pi$
$971$	362.000	0.372812	0.186406	+	0.982473i	$0.440316\pi$
$972$	−1005.10	−1.03405
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1718.71	1.75917	0.879586	−	0.475740i	$-0.157820\pi$
$977$	1718.71	1.75917	0.879586	+	0.475740i	$0.157820\pi$
$978$	0	0
$979$	−3510.00	−3.58529
$980$	845.836	0.863098
$981$	0	0
$982$	0	0
$983$	1931.87	1.96528	0.982639	−	0.185529i	$-0.0593997\pi$
$983$	1931.87	1.96528	0.982639	+	0.185529i	$0.0593997\pi$
$984$	0	0
$985$	1962.85	1.99274
$986$	0	0
$987$	1078.18	1.09238
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	−722.122	−0.722845

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	395.3.c.b.394.2	✓	4
5.4	even	2	inner	395.3.c.b.394.3	yes	4
79.78	odd	2	inner	395.3.c.b.394.3	yes	4
395.394	odd	2	CM	395.3.c.b.394.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
395.3.c.b.394.2	✓	4	1.1	even	1	trivial
395.3.c.b.394.2	✓	4	395.394	odd	2	CM
395.3.c.b.394.3	yes	4	5.4	even	2	inner
395.3.c.b.394.3	yes	4	79.78	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 395 = 5 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	395.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.53216 q^{3} +4.00000 q^{4} +5.00000 q^{5} +9.55467 q^{7} -6.65248 q^{9} +O(q^{10})\)	\(q-1.53216 q^{3} +4.00000 q^{4} +5.00000 q^{5} +9.55467 q^{7} -6.65248 q^{9} +20.1246 q^{11} -6.12865 q^{12} -7.66082 q^{15} +16.0000 q^{16} -33.9839 q^{17} -13.4164 q^{19} +20.0000 q^{20} -14.6393 q^{21} +25.0000 q^{25} +23.9821 q^{27} +38.2187 q^{28} -17.0000 q^{31} -30.8342 q^{33} +47.7734 q^{35} -26.6099 q^{36} -30.1108 q^{37} +82.1190 q^{43} +80.4984 q^{44} -33.2624 q^{45} -73.6494 q^{47} -24.5146 q^{48} +42.2918 q^{49} +52.0689 q^{51} +102.313 q^{53} +100.623 q^{55} +20.5561 q^{57} -30.6433 q^{60} -63.5622 q^{63} +64.0000 q^{64} -135.936 q^{68} -38.3041 q^{75} -53.6656 q^{76} +192.284 q^{77} -79.0000 q^{79} +80.0000 q^{80} +23.1277 q^{81} -58.5573 q^{84} -169.920 q^{85} -174.413 q^{89} +26.0468 q^{93} -67.0820 q^{95} -133.878 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{4} + 20 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q + 16 * q^4 + 20 * q^5 + 36 * q^9	\( 4 q + 16 q^{4} + 20 q^{5} + 36 q^{9} + 64 q^{16} + 80 q^{20} - 148 q^{21} + 100 q^{25} - 68 q^{31} + 144 q^{36} + 180 q^{45} + 196 q^{49} + 92 q^{51} + 256 q^{64} - 316 q^{79} + 320 q^{80} + 656 q^{81} - 592 q^{84} - 1260 q^{99}+O(q^{100}) \) 4 * q + 16 * q^4 + 20 * q^5 + 36 * q^9 + 64 * q^16 + 80 * q^20 - 148 * q^21 + 100 * q^25 - 68 * q^31 + 144 * q^36 + 180 * q^45 + 196 * q^49 + 92 * q^51 + 256 * q^64 - 316 * q^79 + 320 * q^80 + 656 * q^81 - 592 * q^84 - 1260 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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