LMFDB - Embedded newform 395.3.c.b.394.4

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			394.4
Root			$3.58526$ 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)

$f(q)$	$=$	$q+5.80108 q^{3} +4.00000 q^{4} +5.00000 q^{5} -10.2327 q^{7} +24.6525 q^{9} +O(q^{10})$	$q+5.80108 q^{3} +4.00000 q^{4} +5.00000 q^{5} -10.2327 q^{7} +24.6525 q^{9} -20.1246 q^{11} +23.2043 q^{12} +29.0054 q^{15} +16.0000 q^{16} -1.04617 q^{17} +13.4164 q^{19} +20.0000 q^{20} -59.3607 q^{21} +25.0000 q^{25} +90.8012 q^{27} -40.9308 q^{28} -17.0000 q^{31} -116.744 q^{33} -51.1635 q^{35} +98.6099 q^{36} -67.5969 q^{37} -25.5436 q^{43} -80.4984 q^{44} +123.262 q^{45} -58.4104 q^{47} +92.8172 q^{48} +55.7082 q^{49} -6.06888 q^{51} +27.7122 q^{53} -100.623 q^{55} +77.8296 q^{57} +116.022 q^{60} -252.261 q^{63} +64.0000 q^{64} -4.18466 q^{68} +145.027 q^{75} +53.6656 q^{76} +205.929 q^{77} -79.0000 q^{79} +80.0000 q^{80} +304.872 q^{81} -237.443 q^{84} -5.23083 q^{85} +174.413 q^{89} -98.6183 q^{93} +67.0820 q^{95} -496.122 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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			$\pi$
$3$	5.80108	1.93369	0.966846	−	0.255361i	$-0.0821942\pi$
$3$	5.80108	1.93369	0.966846	+	0.255361i	$0.0821942\pi$
$4$	4.00000	1.00000
$5$	5.00000	1.00000
$5$	5.00000	1.00000
$6$	0	0
$7$	−10.2327	−1.46181	−0.730907	−	0.682477i	$-0.760903\pi$
$7$	−10.2327	−1.46181	−0.730907	+	0.682477i	$0.760903\pi$
$8$	0	0
$9$	24.6525	2.73916
$10$	0	0
$11$	−20.1246	−1.82951	−0.914755	−	0.404009i	$-0.867616\pi$
$11$	−20.1246	−1.82951	−0.914755	+	0.404009i	$0.867616\pi$
$12$	23.2043	1.93369
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	29.0054	1.93369
$16$	16.0000	1.00000
$17$	−1.04617	−0.0615391	−0.0307696	−	0.999527i	$-0.509796\pi$
$17$	−1.04617	−0.0615391	−0.0307696	+	0.999527i	$0.509796\pi$
$18$	0	0
$19$	13.4164	0.706127	0.353063	−	0.935599i	$-0.385140\pi$
$19$	13.4164	0.706127	0.353063	+	0.935599i	$0.385140\pi$
$20$	20.0000	1.00000
$21$	−59.3607	−2.82670
$22$	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
$27$	90.8012	3.36301
$28$	−40.9308	−1.46181
$29$	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
$33$	−116.744	−3.53771
$34$	0	0
$35$	−51.1635	−1.46181
$36$	98.6099	2.73916
$37$	−67.5969	−1.82694	−0.913472	−	0.406903i	$-0.866609\pi$
$37$	−67.5969	−1.82694	−0.913472	+	0.406903i	$0.866609\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−25.5436	−0.594037	−0.297019	−	0.954872i	$-0.595992\pi$
$43$	−25.5436	−0.594037	−0.297019	+	0.954872i	$0.595992\pi$
$44$	−80.4984	−1.82951
$45$	123.262	2.73916
$46$	0	0
$47$	−58.4104	−1.24277	−0.621387	−	0.783504i	$-0.713430\pi$
$47$	−58.4104	−1.24277	−0.621387	+	0.783504i	$0.713430\pi$
$48$	92.8172	1.93369
$49$	55.7082	1.13690
$50$	0	0
$51$	−6.06888	−0.118998
$52$	0	0
$53$	27.7122	0.522873	0.261436	−	0.965221i	$-0.415804\pi$
$53$	27.7122	0.522873	0.261436	+	0.965221i	$0.415804\pi$
$54$	0	0
$55$	−100.623	−1.82951
$56$	0	0
$57$	77.8296	1.36543
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	116.022	1.93369
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−252.261	−4.00415
$64$	64.0000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−4.18466	−0.0615391
$69$	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
$75$	145.027	1.93369
$76$	53.6656	0.706127
$77$	205.929	2.67440
$78$	0	0
$79$	−79.0000	−1.00000
$79$	−79.0000	−1.00000
$80$	80.0000	1.00000
$81$	304.872	3.76386
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−237.443	−2.82670
$85$	−5.23083	−0.0615391
$86$	0	0
$87$	0	0
$88$	0	0
$89$	174.413	1.95970	0.979850	−	0.199735i	$-0.0640080\pi$
$89$	174.413	1.95970	0.979850	+	0.199735i	$0.0640080\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−98.6183	−1.06041
$94$	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
$99$	−496.122	−5.01133
$100$	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
$103$	120.929	1.17407	0.587034	−	0.809562i	$-0.300295\pi$
$103$	120.929	1.17407	0.587034	+	0.809562i	$0.300295\pi$
$104$	0	0
$105$	−296.803	−2.82670
$106$	0	0
$107$	−213.993	−1.99994	−0.999968	−	0.00795763i	$-0.997467\pi$
$107$	−213.993	−1.99994	−0.999968	+	0.00795763i	$0.997467\pi$
$108$	363.205	3.36301
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	−392.135	−3.53274
$112$	−163.723	−1.46181
$113$	38.7622	0.343028	0.171514	−	0.985182i	$-0.445134\pi$
$113$	38.7622	0.343028	0.171514	+	0.985182i	$0.445134\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.7051	0.0899588
$120$	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
$127$	191.359	1.50677	0.753383	−	0.657582i	$-0.228421\pi$
$127$	191.359	1.50677	0.753383	+	0.657582i	$0.228421\pi$
$128$	0	0
$129$	−148.180	−1.14868
$130$	0	0
$131$	−46.9574	−0.358454	−0.179227	−	0.983808i	$-0.557360\pi$
$131$	−46.9574	−0.358454	−0.179227	+	0.983808i	$0.557360\pi$
$132$	−466.978	−3.53771
$133$	−137.286	−1.03223
$134$	0	0
$135$	454.006	3.36301
$136$	0	0
$137$	−271.357	−1.98071	−0.990356	−	0.138549i	$-0.955756\pi$
$137$	−271.357	−1.98071	−0.990356	+	0.138549i	$0.955756\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−204.654	−1.46181
$141$	−338.843	−2.40314
$142$	0	0
$143$	0	0
$144$	394.440	2.73916
$145$	0	0
$146$	0	0
$147$	323.167	2.19842
$148$	−270.388	−1.82694
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	301.869	1.99913	0.999567	−	0.0294311i	$-0.00936957\pi$
$151$	301.869	1.99913	0.999567	+	0.0294311i	$0.00936957\pi$
$152$	0	0
$153$	−25.7906	−0.168566
$154$	0	0
$155$	−85.0000	−0.548387
$156$	0	0
$157$	310.272	1.97626	0.988128	−	0.153633i	$-0.0490972\pi$
$157$	310.272	1.97626	0.988128	+	0.153633i	$0.0490972\pi$
$158$	0	0
$159$	160.761	1.01107
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−583.722	−3.53771
$166$	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
$171$	330.748	1.93420
$172$	−102.174	−0.594037
$173$	276.588	1.59878	0.799388	−	0.600815i	$-0.205157\pi$
$173$	276.588	1.59878	0.799388	+	0.600815i	$0.205157\pi$
$174$	0	0
$175$	−255.818	−1.46181
$176$	−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	$0.946738\pi$
$180$	493.050	2.73916
$181$	−234.787	−1.29717	−0.648583	−	0.761144i	$-0.724638\pi$
$181$	−234.787	−1.29717	−0.648583	+	0.761144i	$0.724638\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−337.984	−1.82694
$186$	0	0
$187$	21.0537	0.112586
$188$	−233.641	−1.24277
$189$	−929.142	−4.91609
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	371.269	1.93369
$193$	339.031	1.75664	0.878318	−	0.478077i	$-0.158666\pi$
$193$	339.031	1.75664	0.878318	+	0.478077i	$0.158666\pi$
$194$	0	0
$195$	0	0
$196$	222.833	1.13690
$197$	−33.5313	−0.170210	−0.0851049	−	0.996372i	$-0.527123\pi$
$197$	−33.5313	−0.170210	−0.0851049	+	0.996372i	$0.527123\pi$
$198$	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
$204$	−24.2755	−0.118998
$205$	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			$\pi$
$212$	110.849	0.522873
$213$	0	0
$214$	0	0
$215$	−127.718	−0.594037
$216$	0	0
$217$	173.956	0.801640
$218$	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
$225$	616.312	2.73916
$226$	0	0
$227$	374.578	1.65012	0.825062	−	0.565043i	$-0.191140\pi$
$227$	374.578	1.65012	0.825062	+	0.565043i	$0.191140\pi$
$228$	311.318	1.36543
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	1194.61	5.17147
$232$	0	0
$233$	240.354	1.03156	0.515781	−	0.856720i	$-0.327502\pi$
$233$	240.354	1.03156	0.515781	+	0.856720i	$0.327502\pi$
$234$	0	0
$235$	−292.052	−1.24277
$236$	0	0
$237$	−458.285	−1.93369
$238$	0	0
$239$	−233.000	−0.974895	−0.487448	−	0.873152i	$-0.662072\pi$
$239$	−233.000	−0.974895	−0.487448	+	0.873152i	$0.662072\pi$
$240$	464.086	1.93369
$241$	281.745	1.16906	0.584532	−	0.811370i	$-0.301278\pi$
$241$	281.745	1.16906	0.584532	+	0.811370i	$0.301278\pi$
$242$	0	0
$243$	951.376	3.91513
$244$	0	0
$245$	278.541	1.13690
$246$	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			$\pi$
$252$	−1009.05	−4.00415
$253$	0	0
$254$	0	0
$255$	−30.3444	−0.118998
$256$	256.000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	691.699	2.67065
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	138.561	0.522873
$266$	0	0
$267$	1011.78	3.78946
$268$	0	0
$269$	−395.784	−1.47132	−0.735658	−	0.677353i	$-0.763127\pi$
$269$	−395.784	−1.47132	−0.735658	+	0.677353i	$0.763127\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−16.7386	−0.0615391
$273$	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
$279$	−419.092	−1.50212
$280$	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
$285$	389.148	1.36543
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−287.906	−0.996213
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−91.5061	−0.312307	−0.156154	−	0.987733i	$-0.549910\pi$
$293$	−91.5061	−0.312307	−0.156154	+	0.987733i	$0.549910\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1827.34	−6.15266
$298$	0	0
$299$	0	0
$300$	580.108	1.93369
$301$	261.380	0.868372
$302$	0	0
$303$	−1119.61	−3.69507
$304$	214.663	0.706127
$305$	0	0
$306$	0	0
$307$	335.151	1.09170	0.545849	−	0.837884i	$-0.316207\pi$
$307$	335.151	1.09170	0.545849	+	0.837884i	$0.316207\pi$
$308$	823.717	2.67440
$309$	701.519	2.27029
$310$	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
$315$	−1261.31	−4.00415
$316$	−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
$321$	−1241.39	−3.86726
$322$	0	0
$323$	−14.0358	−0.0434544
$324$	1219.49	3.76386
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	597.696	1.81670
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	−1666.43	−5.00430
$334$	0	0
$335$	0	0
$336$	−949.771	−2.82670
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	224.862	0.663310
$340$	−20.9233	−0.0615391
$341$	342.118	1.00328
$342$	0	0
$343$	−68.6431	−0.200126
$344$	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
$353$	486.015	1.37681	0.688407	−	0.725325i	$-0.258310\pi$
$353$	486.015	1.37681	0.688407	+	0.725325i	$0.258310\pi$
$354$	0	0
$355$	0	0
$356$	697.653	1.95970
$357$	62.1011	0.173953
$358$	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
$363$	1647.51	4.53858
$364$	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
$371$	−283.571	−0.764343
$372$	−394.473	−1.06041
$373$	−656.550	−1.76019	−0.880093	−	0.474801i	$-0.842520\pi$
$373$	−656.550	−1.76019	−0.880093	+	0.474801i	$0.842520\pi$
$374$	0	0
$375$	725.134	1.93369
$376$	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
$381$	1110.09	2.91362
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	1029.65	2.67440
$386$	0	0
$387$	−629.713	−1.62717
$388$	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
$393$	−272.404	−0.693139
$394$	0	0
$395$	−395.000	−1.00000
$396$	−1984.49	−5.01133
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	−796.407	−1.99601
$400$	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
$405$	1524.36	3.76386
$406$	0	0
$407$	1360.36	3.34241
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	−1574.16	−3.83008
$412$	483.716	1.17407
$413$	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			$\pi$
$420$	−1187.21	−2.82670
$421$	−288.453	−0.685161	−0.342580	−	0.939488i	$-0.611301\pi$
$421$	−288.453	−0.685161	−0.342580	+	0.939488i	$0.611301\pi$
$422$	0	0
$423$	−1439.96	−3.40416
$424$	0	0
$425$	−26.1541	−0.0615391
$426$	0	0
$427$	0	0
$428$	−855.973	−1.99994
$429$	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	$0.813348\pi$
$432$	1452.82	3.36301
$433$	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	$-0.744715\pi$
$439$	−610.447	−1.39054	−0.695269	+	0.718749i	$0.744715\pi$
$440$	0	0
$441$	1373.35	3.11416
$442$	0	0
$443$	−592.244	−1.33689	−0.668447	−	0.743760i	$-0.733041\pi$
$443$	−592.244	−1.33689	−0.668447	+	0.743760i	$0.733041\pi$
$444$	−1568.54	−3.53274
$445$	872.067	1.95970
$446$	0	0
$447$	0	0
$448$	−654.893	−1.46181
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	155.049	0.343028
$453$	1751.17	3.86571
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	−94.9931	−0.206957
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−869.497	−1.87796	−0.938981	−	0.343968i	$-0.888229\pi$
$463$	−869.497	−1.87796	−0.938981	+	0.343968i	$0.888229\pi$
$464$	0	0
$465$	−493.091	−1.06041
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1799.91	3.82147
$472$	0	0
$473$	514.055	1.08680
$474$	0	0
$475$	335.410	0.706127
$476$	42.8204	0.0899588
$477$	683.175	1.43223
$478$	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
$495$	−2480.61	−5.01133
$496$	−272.000	−0.548387
$497$	0	0
$498$	0	0
$499$	436.033	0.873814	0.436907	−	0.899507i	$-0.356074\pi$
$499$	436.033	0.873814	0.436907	+	0.899507i	$0.356074\pi$
$500$	500.000	1.00000
$501$	0	0
$502$	0	0
$503$	−467.337	−0.929099	−0.464550	−	0.885547i	$-0.653784\pi$
$503$	−467.337	−0.929099	−0.464550	+	0.885547i	$0.653784\pi$
$504$	0	0
$505$	−965.000	−1.91089
$506$	0	0
$507$	980.382	1.93369
$508$	765.437	1.50677
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1218.23	2.37471
$514$	0	0
$515$	604.645	1.17407
$516$	−592.721	−1.14868
$517$	1175.49	2.27367
$518$	0	0
$519$	1604.51	3.09154
$520$	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
$525$	−1484.02	−2.82670
$526$	0	0
$527$	17.7848	0.0337473
$528$	−1867.91	−3.53771
$529$	529.000	1.00000
$530$	0	0
$531$	0	0
$532$	−549.144	−1.03223
$533$	0	0
$534$	0	0
$535$	−1069.97	−1.99994
$536$	0	0
$537$	−2047.78	−3.81337
$538$	0	0
$539$	−1121.11	−2.07997
$540$	1816.02	3.36301
$541$	−845.234	−1.56235	−0.781177	−	0.624309i	$-0.785380\pi$
$541$	−845.234	−1.56235	−0.781177	+	0.624309i	$0.785380\pi$
$542$	0	0
$543$	−1362.02	−2.50832
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−1085.43	−1.98071
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	808.384	1.46181
$554$	0	0
$555$	−1960.67	−3.53274
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	−818.616	−1.46181
$561$	122.134	0.217708
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−1355.37	−2.40314
$565$	193.811	0.343028
$566$	0	0
$567$	−3119.67	−5.50206
$568$	0	0
$569$	597.030	1.04926	0.524631	−	0.851330i	$-0.324203\pi$
$569$	597.030	1.04926	0.524631	+	0.851330i	$0.324203\pi$
$570$	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
$576$	1577.76	2.73916
$577$	1152.87	1.99805	0.999024	−	0.0441665i	$-0.0140632\pi$
$577$	1152.87	1.99805	0.999024	+	0.0441665i	$0.0140632\pi$
$578$	0	0
$579$	1966.74	3.39679
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−557.698	−0.956601
$584$	0	0
$585$	0	0
$586$	0	0
$587$	957.788	1.63167	0.815833	−	0.578288i	$-0.196279\pi$
$587$	957.788	1.63167	0.815833	+	0.578288i	$0.196279\pi$
$588$	1292.67	2.19842
$589$	−228.079	−0.387231
$590$	0	0
$591$	−194.518	−0.329133
$592$	−1081.55	−1.82694
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	53.5255	0.0899588
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−469.574	−0.783930	−0.391965	−	0.919980i	$-0.628205\pi$
$599$	−469.574	−0.783930	−0.391965	+	0.919980i	$0.628205\pi$
$600$	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
$607$	1186.86	1.95529	0.977647	−	0.210254i	$-0.0674292\pi$
$607$	1186.86	1.95529	0.977647	+	0.210254i	$0.0674292\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−103.162	−0.168566
$613$	933.291	1.52250	0.761248	−	0.648460i	$-0.224587\pi$
$613$	933.291	1.52250	0.761248	+	0.648460i	$0.224587\pi$
$614$	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
$623$	−1784.72	−2.86472
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	−1566.29	−2.49807
$628$	1241.09	1.97626
$629$	70.7175	0.112428
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	956.796	1.50677
$636$	643.043	1.01107
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1220.89	−1.90467	−0.952335	−	0.305055i	$-0.901325\pi$
$641$	−1220.89	−1.90467	−0.952335	+	0.305055i	$0.901325\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	−740.902	−1.14868
$646$	0	0
$647$	−528.580	−0.816971	−0.408486	−	0.912765i	$-0.633943\pi$
$647$	−528.580	−0.816971	−0.408486	+	0.912765i	$0.633943\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	1009.13	1.55013
$652$	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			$\pi$
$660$	−2334.89	−3.53771
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−686.431	−1.03223
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−289.600	−0.430312	−0.215156	−	0.976580i	$-0.569026\pi$
$673$	−289.600	−0.430312	−0.215156	+	0.976580i	$0.569026\pi$
$674$	0	0
$675$	2270.03	3.36301
$676$	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
$681$	2172.96	3.19083
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	1322.99	1.93420
$685$	−1356.79	−1.98071
$686$	0	0
$687$	0	0
$688$	−408.698	−0.594037
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	1106.35	1.59878
$693$	5076.66	7.32563
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	1394.31	1.99472
$700$	−1023.27	−1.46181
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−906.907	−1.29005
$704$	−1287.98	−1.82951
$705$	−1694.21	−2.40314
$706$	0	0
$707$	1974.91	2.79337
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	−1947.55	−2.73916
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−1412.00	−1.97207
$717$	−1351.65	−1.88515
$718$	0	0
$719$	−142.000	−0.197497	−0.0987483	−	0.995112i	$-0.531484\pi$
$719$	−142.000	−0.197497	−0.0987483	+	0.995112i	$0.531484\pi$
$720$	1972.20	2.73916
$721$	−1237.43	−1.71627
$722$	0	0
$723$	1634.42	2.26061
$724$	−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
$729$	2775.16	3.80680
$730$	0	0
$731$	26.7228	0.0365565
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	1615.84	2.19842
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	−1351.94	−1.82694
$741$	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
$748$	84.2147	0.112586
$749$	2189.73	2.92354
$750$	0	0
$751$	−362.243	−0.482348	−0.241174	−	0.970482i	$-0.577532\pi$
$751$	−362.243	−0.482348	−0.241174	+	0.970482i	$0.577532\pi$
$752$	−934.566	−1.24277
$753$	0	0
$754$	0	0
$755$	1509.35	1.99913
$756$	−3716.57	−4.91609
$757$	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	$-0.656324\pi$
$761$	−717.778	−0.943203	−0.471602	+	0.881812i	$0.656324\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−128.953	−0.168566
$766$	0	0
$767$	0	0
$768$	1485.08	1.93369
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	1356.12	1.75664
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−425.000	−0.548387
$776$	0	0
$777$	4012.60	5.16422
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	891.331	1.13690
$785$	1551.36	1.97626
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−134.125	−0.170210
$789$	0	0
$790$	0	0
$791$	−396.642	−0.501443
$792$	0	0
$793$	0	0
$794$	0	0
$795$	803.804	1.01107
$796$	0	0
$797$	−1558.15	−1.95502	−0.977509	−	0.210892i	$-0.932363\pi$
$797$	−1558.15	−1.95502	−0.977509	+	0.210892i	$0.932363\pi$
$798$	0	0
$799$	61.1069	0.0764792
$800$	0	0
$801$	4299.72	5.36794
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2295.97	−2.84507
$808$	0	0
$809$	−1576.43	−1.94861	−0.974307	−	0.225226i	$-0.927688\pi$
$809$	−1576.43	−1.94861	−0.974307	+	0.225226i	$0.927688\pi$
$810$	0	0
$811$	1214.18	1.49715	0.748573	−	0.663053i	$-0.230739\pi$
$811$	1214.18	1.49715	0.748573	+	0.663053i	$0.230739\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−97.1021	−0.118998
$817$	−342.703	−0.419466
$818$	0	0
$819$	0	0
$820$	0	0
$821$	328.702	0.400368	0.200184	−	0.979758i	$-0.435846\pi$
$821$	328.702	0.400368	0.200184	+	0.979758i	$0.435846\pi$
$822$	0	0
$823$	−1582.65	−1.92302	−0.961511	−	0.274766i	$-0.911400\pi$
$823$	−1582.65	−1.92302	−0.961511	+	0.274766i	$0.911400\pi$
$824$	0	0
$825$	−2918.61	−3.53771
$826$	0	0
$827$	−1524.16	−1.84300	−0.921500	−	0.388379i	$-0.873035\pi$
$827$	−1524.16	−1.84300	−0.921500	+	0.388379i	$0.873035\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−58.2800	−0.0699640
$834$	0	0
$835$	0	0
$836$	−1080.00	−1.29187
$837$	−1543.62	−1.84423
$838$	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
$843$	968.780	1.14920
$844$	0	0
$845$	845.000	1.00000
$846$	0	0
$847$	−2906.09	−3.43104
$848$	443.396	0.522873
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−1372.38	−1.60889	−0.804444	−	0.594029i	$-0.797536\pi$
$853$	−1372.38	−1.60889	−0.804444	+	0.594029i	$0.797536\pi$
$854$	0	0
$855$	1653.74	1.93420
$856$	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			$\pi$
$860$	−510.872	−0.594037
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	1382.94	1.59878
$866$	0	0
$867$	−1670.16	−1.92637
$868$	695.824	0.801640
$869$	1589.84	1.82951
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1279.09	−1.46181
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	−530.834	−0.603906
$880$	−1609.97	−1.82951
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	1751.01	1.98303	0.991514	−	0.130002i	$-0.0414984\pi$
$883$	1751.01	1.98303	0.991514	+	0.130002i	$0.0414984\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−1958.12	−2.20261
$890$	0	0
$891$	−6135.44	−6.88601
$892$	0	0
$893$	−783.657	−0.877556
$894$	0	0
$895$	−1765.00	−1.97207
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	2465.25	2.73916
$901$	−28.9916	−0.0321771
$902$	0	0
$903$	1516.29	1.67916
$904$	0	0
$905$	−1173.94	−1.29717
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	1498.31	1.65012
$909$	−4757.93	−5.23424
$910$	0	0
$911$	1427.00	1.56641	0.783205	−	0.621763i	$-0.213583\pi$
$911$	1427.00	1.56641	0.783205	+	0.621763i	$0.213583\pi$
$912$	1245.27	1.36543
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	480.501	0.523993
$918$	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
$921$	1944.24	2.11101
$922$	0	0
$923$	0	0
$924$	4778.44	5.17147
$925$	−1689.92	−1.82694
$926$	0	0
$927$	2981.20	3.21597
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	747.404	0.802797
$932$	961.416	1.03156
$933$	0	0
$934$	0	0
$935$	105.268	0.112586
$936$	0	0
$937$	−1057.70	−1.12881	−0.564405	−	0.825498i	$-0.690894\pi$
$937$	−1057.70	−1.12881	−0.564405	+	0.825498i	$0.690894\pi$
$938$	0	0
$939$	0	0
$940$	−1168.21	−1.24277
$941$	301.869	0.320796	0.160398	−	0.987052i	$-0.448722\pi$
$941$	301.869	0.320796	0.160398	+	0.987052i	$0.448722\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−4645.71	−4.91609
$946$	0	0
$947$	−1603.06	−1.69278	−0.846388	−	0.532567i	$-0.821227\pi$
$947$	−1603.06	−1.69278	−0.846388	+	0.532567i	$0.821227\pi$
$948$	−1833.14	−1.93369
$949$	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
$959$	2776.72	2.89543
$960$	1856.34	1.93369
$961$	−672.000	−0.699272
$962$	0	0
$963$	−5275.46	−5.47815
$964$	1126.98	1.16906
$965$	1695.15	1.75664
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	−81.4226	−0.0840275
$970$	0	0
$971$	362.000	0.372812	0.186406	−	0.982473i	$-0.440316\pi$
$971$	362.000	0.372812	0.186406	+	0.982473i	$0.440316\pi$
$972$	3805.51	3.91513
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−929.596	−0.951480	−0.475740	−	0.879586i	$-0.657820\pi$
$977$	−929.596	−0.951480	−0.475740	+	0.879586i	$0.657820\pi$
$978$	0	0
$979$	−3510.00	−3.58529
$980$	1114.16	1.13690
$981$	0	0
$982$	0	0
$983$	364.749	0.371057	0.185529	−	0.982639i	$-0.440600\pi$
$983$	364.749	0.371057	0.185529	+	0.982639i	$0.440600\pi$
$984$	0	0
$985$	−167.657	−0.170210
$986$	0	0
$987$	3467.28	3.51295
$988$	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
$999$	−6137.88	−6.14402

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	395.3.c.b.394.4	yes	4
5.4	even	2	inner	395.3.c.b.394.1	✓	4
79.78	odd	2	inner	395.3.c.b.394.1	✓	4
395.394	odd	2	CM	395.3.c.b.394.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
395.3.c.b.394.1	✓	4	5.4	even	2	inner
395.3.c.b.394.1	✓	4	79.78	odd	2	inner
395.3.c.b.394.4	yes	4	1.1	even	1	trivial
395.3.c.b.394.4	yes	4	395.394	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+5.80108 q^{3} +4.00000 q^{4} +5.00000 q^{5} -10.2327 q^{7} +24.6525 q^{9} +O(q^{10})\)	\(q+5.80108 q^{3} +4.00000 q^{4} +5.00000 q^{5} -10.2327 q^{7} +24.6525 q^{9} -20.1246 q^{11} +23.2043 q^{12} +29.0054 q^{15} +16.0000 q^{16} -1.04617 q^{17} +13.4164 q^{19} +20.0000 q^{20} -59.3607 q^{21} +25.0000 q^{25} +90.8012 q^{27} -40.9308 q^{28} -17.0000 q^{31} -116.744 q^{33} -51.1635 q^{35} +98.6099 q^{36} -67.5969 q^{37} -25.5436 q^{43} -80.4984 q^{44} +123.262 q^{45} -58.4104 q^{47} +92.8172 q^{48} +55.7082 q^{49} -6.06888 q^{51} +27.7122 q^{53} -100.623 q^{55} +77.8296 q^{57} +116.022 q^{60} -252.261 q^{63} +64.0000 q^{64} -4.18466 q^{68} +145.027 q^{75} +53.6656 q^{76} +205.929 q^{77} -79.0000 q^{79} +80.0000 q^{80} +304.872 q^{81} -237.443 q^{84} -5.23083 q^{85} +174.413 q^{89} -98.6183 q^{93} +67.0820 q^{95} -496.122 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{4} + 20 q^{5} + 36 q^{9}+O(q^{10}) \) 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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