LMFDB - Embedded newform 320.3.h.c.319.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,3,Mod(319,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("320.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	320.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$8.71936845953$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			319.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	320.319

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.47214 q^{3} -5.00000 q^{5} +13.4164 q^{7} +11.0000 q^{9} +O(q^{10})$	$q+4.47214 q^{3} -5.00000 q^{5} +13.4164 q^{7} +11.0000 q^{9} -22.3607 q^{15} +60.0000 q^{21} +13.4164 q^{23} +25.0000 q^{25} +8.94427 q^{27} +22.0000 q^{29} -67.0820 q^{35} -62.0000 q^{41} +40.2492 q^{43} -55.0000 q^{45} -93.9149 q^{47} +131.000 q^{49} -58.0000 q^{61} +147.580 q^{63} -67.0820 q^{67} +60.0000 q^{69} +111.803 q^{75} -59.0000 q^{81} +147.580 q^{83} +98.3870 q^{87} -142.000 q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{5} + 22 q^{9}+O(q^{10}) $ 2 * q - 10 * q^5 + 22 * q^9	$ 2 q - 10 q^{5} + 22 q^{9} + 120 q^{21} + 50 q^{25} + 44 q^{29} - 124 q^{41} - 110 q^{45} + 262 q^{49} - 116 q^{61} + 120 q^{69} - 118 q^{81} - 284 q^{89}+O(q^{100}) $ 2 * q - 10 * q^5 + 22 * q^9 + 120 * q^21 + 50 * q^25 + 44 * q^29 - 124 * q^41 - 110 * q^45 + 262 * q^49 - 116 * q^61 + 120 * q^69 - 118 * q^81 - 284 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	4.47214	1.49071	0.745356	−	0.666667i	$-0.232280\pi$
$3$	4.47214	1.49071	0.745356	+	0.666667i	$0.232280\pi$
$4$	0	0
$5$	−5.00000	−1.00000
$5$	−5.00000	−1.00000
$6$	0	0
$7$	13.4164	1.91663	0.958315	−	0.285714i	$-0.0922308\pi$
$7$	13.4164	1.91663	0.958315	+	0.285714i	$0.0922308\pi$
$8$	0	0
$9$	11.0000	1.22222
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−22.3607	−1.49071
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	60.0000	2.85714
$22$	0	0
$23$	13.4164	0.583322	0.291661	−	0.956522i	$-0.405792\pi$
$23$	13.4164	0.583322	0.291661	+	0.956522i	$0.405792\pi$
$24$	0	0
$25$	25.0000	1.00000
$26$	0	0
$27$	8.94427	0.331269
$28$	0	0
$29$	22.0000	0.758621	0.379310	−	0.925270i	$-0.376161\pi$
$29$	22.0000	0.758621	0.379310	+	0.925270i	$0.376161\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−67.0820	−1.91663
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−62.0000	−1.51220	−0.756098	−	0.654459i	$-0.772896\pi$
$41$	−62.0000	−1.51220	−0.756098	+	0.654459i	$0.772896\pi$
$42$	0	0
$43$	40.2492	0.936028	0.468014	−	0.883721i	$-0.344970\pi$
$43$	40.2492	0.936028	0.468014	+	0.883721i	$0.344970\pi$
$44$	0	0
$45$	−55.0000	−1.22222
$46$	0	0
$47$	−93.9149	−1.99819	−0.999094	−	0.0425532i	$-0.986451\pi$
$47$	−93.9149	−1.99819	−0.999094	+	0.0425532i	$0.986451\pi$
$48$	0	0
$49$	131.000	2.67347
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−58.0000	−0.950820	−0.475410	−	0.879764i	$-0.657700\pi$
$61$	−58.0000	−0.950820	−0.475410	+	0.879764i	$0.657700\pi$
$62$	0	0
$63$	147.580	2.34255
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−67.0820	−1.00122	−0.500612	−	0.865672i	$-0.666892\pi$
$67$	−67.0820	−1.00122	−0.500612	+	0.865672i	$0.666892\pi$
$68$	0	0
$69$	60.0000	0.869565
$70$	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
$75$	111.803	1.49071
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−59.0000	−0.728395
$82$	0	0
$83$	147.580	1.77808	0.889039	−	0.457831i	$-0.151374\pi$
$83$	147.580	1.77808	0.889039	+	0.457831i	$0.151374\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	98.3870	1.13088
$88$	0	0
$89$	−142.000	−1.59551	−0.797753	−	0.602985i	$-0.793978\pi$
$89$	−142.000	−1.59551	−0.797753	+	0.602985i	$0.793978\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−122.000	−1.20792	−0.603960	−	0.797014i	$-0.706411\pi$
$101$	−122.000	−1.20792	−0.603960	+	0.797014i	$0.706411\pi$
$102$	0	0
$103$	−201.246	−1.95385	−0.976923	−	0.213592i	$-0.931484\pi$
$103$	−201.246	−1.95385	−0.976923	+	0.213592i	$0.931484\pi$
$104$	0	0
$105$	−300.000	−2.85714
$106$	0	0
$107$	−174.413	−1.63003	−0.815015	−	0.579439i	$-0.803272\pi$
$107$	−174.413	−1.63003	−0.815015	+	0.579439i	$0.803272\pi$
$108$	0	0
$109$	38.0000	0.348624	0.174312	−	0.984690i	$-0.444230\pi$
$109$	38.0000	0.348624	0.174312	+	0.984690i	$0.444230\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−67.0820	−0.583322
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	−277.272	−2.25425
$124$	0	0
$125$	−125.000	−1.00000
$126$	0	0
$127$	−93.9149	−0.739487	−0.369744	−	0.929134i	$-0.620554\pi$
$127$	−93.9149	−0.739487	−0.369744	+	0.929134i	$0.620554\pi$
$128$	0	0
$129$	180.000	1.39535
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−44.7214	−0.331269
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−420.000	−2.97872
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−110.000	−0.758621
$146$	0	0
$147$	585.850	3.98537
$148$	0	0
$149$	278.000	1.86577	0.932886	−	0.360172i	$-0.117282\pi$
$149$	278.000	1.86577	0.932886	+	0.360172i	$0.117282\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	180.000	1.11801
$162$	0	0
$163$	−281.745	−1.72849	−0.864247	−	0.503067i	$-0.832205\pi$
$163$	−281.745	−1.72849	−0.864247	+	0.503067i	$0.832205\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	228.079	1.36574	0.682871	−	0.730539i	$-0.260731\pi$
$167$	228.079	1.36574	0.682871	+	0.730539i	$0.260731\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	335.410	1.91663
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	358.000	1.97790	0.988950	−	0.148248i	$-0.0473633\pi$
$181$	358.000	1.97790	0.988950	+	0.148248i	$0.0473633\pi$
$182$	0	0
$183$	−259.384	−1.41740
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	120.000	0.634921
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	−300.000	−1.49254
$202$	0	0
$203$	295.161	1.45399
$204$	0	0
$205$	310.000	1.51220
$206$	0	0
$207$	147.580	0.712949
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−201.246	−0.936028
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−93.9149	−0.421143	−0.210571	−	0.977578i	$-0.567532\pi$
$223$	−93.9149	−0.421143	−0.210571	+	0.977578i	$0.567532\pi$
$224$	0	0
$225$	275.000	1.22222
$226$	0	0
$227$	−281.745	−1.24117	−0.620583	−	0.784141i	$-0.713104\pi$
$227$	−281.745	−1.24117	−0.620583	+	0.784141i	$0.713104\pi$
$228$	0	0
$229$	262.000	1.14410	0.572052	−	0.820217i	$-0.306147\pi$
$229$	262.000	1.14410	0.572052	+	0.820217i	$0.306147\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	469.574	1.99819
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−302.000	−1.25311	−0.626556	−	0.779376i	$-0.715536\pi$
$241$	−302.000	−1.25311	−0.626556	+	0.779376i	$0.715536\pi$
$242$	0	0
$243$	−344.354	−1.41710
$244$	0	0
$245$	−655.000	−2.67347
$246$	0	0
$247$	0	0
$248$	0	0
$249$	660.000	2.65060
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	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
$259$	0	0
$260$	0	0
$261$	242.000	0.927203
$262$	0	0
$263$	442.741	1.68343	0.841714	−	0.539924i	$-0.181547\pi$
$263$	442.741	1.68343	0.841714	+	0.539924i	$0.181547\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−635.043	−2.37844
$268$	0	0
$269$	38.0000	0.141264	0.0706320	−	0.997502i	$-0.477498\pi$
$269$	38.0000	0.141264	0.0706320	+	0.997502i	$0.477498\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	418.000	1.48754	0.743772	−	0.668433i	$-0.233035\pi$
$281$	418.000	1.48754	0.743772	+	0.668433i	$0.233035\pi$
$282$	0	0
$283$	469.574	1.65927	0.829637	−	0.558304i	$-0.188548\pi$
$283$	469.574	1.65927	0.829637	+	0.558304i	$0.188548\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−831.817	−2.89832
$288$	0	0
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	540.000	1.79402
$302$	0	0
$303$	−545.601	−1.80066
$304$	0	0
$305$	290.000	0.950820
$306$	0	0
$307$	147.580	0.480718	0.240359	−	0.970684i	$-0.422735\pi$
$307$	147.580	0.480718	0.240359	+	0.970684i	$0.422735\pi$
$308$	0	0
$309$	−900.000	−2.91262
$310$	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
$315$	−737.902	−2.34255
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−780.000	−2.42991
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	169.941	0.519698
$328$	0	0
$329$	−1260.00	−3.82979
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	335.410	1.00122
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1100.15	3.20742
$344$	0	0
$345$	−300.000	−0.869565
$346$	0	0
$347$	684.237	1.97186	0.985932	−	0.167147i	$-0.0534554\pi$
$347$	684.237	1.97186	0.985932	+	0.167147i	$0.0534554\pi$
$348$	0	0
$349$	22.0000	0.0630372	0.0315186	−	0.999503i	$-0.489966\pi$
$349$	22.0000	0.0630372	0.0315186	+	0.999503i	$0.489966\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	541.128	1.49071
$364$	0	0
$365$	0	0
$366$	0	0
$367$	120.748	0.329013	0.164506	−	0.986376i	$-0.447397\pi$
$367$	120.748	0.329013	0.164506	+	0.986376i	$0.447397\pi$
$368$	0	0
$369$	−682.000	−1.84824
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−559.017	−1.49071
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−420.000	−1.10236
$382$	0	0
$383$	764.735	1.99670	0.998349	−	0.0574413i	$-0.0182942\pi$
$383$	764.735	1.99670	0.998349	+	0.0574413i	$0.0182942\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	442.741	1.14403
$388$	0	0
$389$	−202.000	−0.519280	−0.259640	−	0.965705i	$-0.583604\pi$
$389$	−202.000	−0.519280	−0.259640	+	0.965705i	$0.583604\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−478.000	−1.19202	−0.596010	−	0.802977i	$-0.703248\pi$
$401$	−478.000	−1.19202	−0.596010	+	0.802977i	$0.703248\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	295.000	0.728395
$406$	0	0
$407$	0	0
$408$	0	0
$409$	802.000	1.96088	0.980440	−	0.196818i	$-0.0630607\pi$
$409$	802.000	1.96088	0.980440	+	0.196818i	$0.0630607\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−737.902	−1.77808
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−778.000	−1.84798	−0.923990	−	0.382415i	$-0.875092\pi$
$421$	−778.000	−1.84798	−0.923990	+	0.382415i	$0.875092\pi$
$422$	0	0
$423$	−1033.06	−2.44223
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−778.152	−1.82237
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	−491.935	−1.13088
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	1441.00	3.26757
$442$	0	0
$443$	−389.076	−0.878275	−0.439138	−	0.898420i	$-0.644716\pi$
$443$	−389.076	−0.878275	−0.439138	+	0.898420i	$0.644716\pi$
$444$	0	0
$445$	710.000	1.59551
$446$	0	0
$447$	1243.25	2.78133
$448$	0	0
$449$	−398.000	−0.886414	−0.443207	−	0.896419i	$-0.646159\pi$
$449$	−398.000	−0.886414	−0.443207	+	0.896419i	$0.646159\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−842.000	−1.82646	−0.913232	−	0.407440i	$-0.866422\pi$
$461$	−842.000	−1.82646	−0.913232	+	0.407440i	$0.866422\pi$
$462$	0	0
$463$	−523.240	−1.13011	−0.565054	−	0.825054i	$-0.691145\pi$
$463$	−523.240	−1.13011	−0.565054	+	0.825054i	$0.691145\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−925.732	−1.98230	−0.991148	−	0.132762i	$-0.957615\pi$
$467$	−925.732	−1.98230	−0.991148	+	0.132762i	$0.957615\pi$
$468$	0	0
$469$	−900.000	−1.91898
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	804.984	1.66663
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−845.234	−1.73559	−0.867796	−	0.496920i	$-0.834464\pi$
$487$	−845.234	−1.73559	−0.867796	+	0.496920i	$0.834464\pi$
$488$	0	0
$489$	−1260.00	−2.57669
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	1020.00	2.03593
$502$	0	0
$503$	−415.909	−0.826856	−0.413428	−	0.910537i	$-0.635669\pi$
$503$	−415.909	−0.826856	−0.413428	+	0.910537i	$0.635669\pi$
$504$	0	0
$505$	610.000	1.20792
$506$	0	0
$507$	755.791	1.49071
$508$	0	0
$509$	982.000	1.92927	0.964637	−	0.263584i	$-0.0849045\pi$
$509$	982.000	1.92927	0.964637	+	0.263584i	$0.0849045\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1006.23	1.95385
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	722.000	1.38580	0.692898	−	0.721035i	$-0.256333\pi$
$521$	722.000	1.38580	0.692898	+	0.721035i	$0.256333\pi$
$522$	0	0
$523$	−1033.06	−1.97526	−0.987632	−	0.156788i	$-0.949886\pi$
$523$	−1033.06	−1.97526	−0.987632	+	0.156788i	$0.949886\pi$
$524$	0	0
$525$	1500.00	2.85714
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−349.000	−0.659735
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	872.067	1.63003
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−362.000	−0.669131	−0.334566	−	0.942372i	$-0.608590\pi$
$541$	−362.000	−0.669131	−0.334566	+	0.942372i	$0.608590\pi$
$542$	0	0
$543$	1601.02	2.94848
$544$	0	0
$545$	−190.000	−0.348624
$546$	0	0
$547$	147.580	0.269800	0.134900	−	0.990859i	$-0.456929\pi$
$547$	147.580	0.269800	0.134900	+	0.990859i	$0.456929\pi$
$548$	0	0
$549$	−638.000	−1.16211
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−67.0820	−0.119151	−0.0595755	−	0.998224i	$-0.518975\pi$
$563$	−67.0820	−0.119151	−0.0595755	+	0.998224i	$0.518975\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−791.568	−1.39606
$568$	0	0
$569$	−158.000	−0.277680	−0.138840	−	0.990315i	$-0.544337\pi$
$569$	−158.000	−0.277680	−0.138840	+	0.990315i	$0.544337\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	335.410	0.583322
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1980.00	3.40792
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	469.574	0.799956	0.399978	−	0.916525i	$-0.369018\pi$
$587$	469.574	0.799956	0.399978	+	0.916525i	$0.369018\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	418.000	0.695507	0.347754	−	0.937586i	$-0.386945\pi$
$601$	418.000	0.695507	0.347754	+	0.937586i	$0.386945\pi$
$602$	0	0
$603$	−737.902	−1.22372
$604$	0	0
$605$	−605.000	−1.00000
$606$	0	0
$607$	−737.902	−1.21565	−0.607827	−	0.794069i	$-0.707959\pi$
$607$	−737.902	−1.21565	−0.607827	+	0.794069i	$0.707959\pi$
$608$	0	0
$609$	1320.00	2.16749
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	1386.36	2.25425
$616$	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			$\pi$
$620$	0	0
$621$	120.000	0.193237
$622$	0	0
$623$	−1905.13	−3.05799
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	469.574	0.739487
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1138.00	1.77535	0.887676	−	0.460470i	$-0.152319\pi$
$641$	1138.00	1.77535	0.887676	+	0.460470i	$0.152319\pi$
$642$	0	0
$643$	1220.89	1.89875	0.949373	−	0.314152i	$-0.101720\pi$
$643$	1220.89	1.89875	0.949373	+	0.314152i	$0.101720\pi$
$644$	0	0
$645$	−900.000	−1.39535
$646$	0	0
$647$	872.067	1.34786	0.673931	−	0.738794i	$-0.264605\pi$
$647$	872.067	1.34786	0.673931	+	0.738794i	$0.264605\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−298.000	−0.450832	−0.225416	−	0.974263i	$-0.572374\pi$
$661$	−298.000	−0.450832	−0.225416	+	0.974263i	$0.572374\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	295.161	0.442520
$668$	0	0
$669$	−420.000	−0.627803
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	223.607	0.331269
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1260.00	−1.85022
$682$	0	0
$683$	−1247.73	−1.82683	−0.913416	−	0.407028i	$-0.866565\pi$
$683$	−1247.73	−1.82683	−0.913416	+	0.407028i	$0.866565\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1171.70	1.70553
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	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
$701$	902.000	1.28673	0.643367	−	0.765558i	$-0.277537\pi$
$701$	902.000	1.28673	0.643367	+	0.765558i	$0.277537\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	2100.00	2.97872
$706$	0	0
$707$	−1636.80	−2.31514
$708$	0	0
$709$	−698.000	−0.984485	−0.492243	−	0.870458i	$-0.663823\pi$
$709$	−698.000	−0.984485	−0.492243	+	0.870458i	$0.663823\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−2700.00	−3.74480
$722$	0	0
$723$	−1350.59	−1.86803
$724$	0	0
$725$	550.000	0.758621
$726$	0	0
$727$	228.079	0.313726	0.156863	−	0.987620i	$-0.449862\pi$
$727$	228.079	0.313726	0.156863	+	0.987620i	$0.449862\pi$
$728$	0	0
$729$	−1009.00	−1.38409
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−2929.25	−3.98537
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1274.56	−1.71542	−0.857711	−	0.514132i	$-0.828114\pi$
$743$	−1274.56	−1.71542	−0.857711	+	0.514132i	$0.828114\pi$
$744$	0	0
$745$	−1390.00	−1.86577
$746$	0	0
$747$	1623.39	2.17321
$748$	0	0
$749$	−2340.00	−3.12417
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	242.000	0.318003	0.159001	−	0.987278i	$-0.449173\pi$
$761$	242.000	0.318003	0.159001	+	0.987278i	$0.449173\pi$
$762$	0	0
$763$	509.823	0.668183
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1342.00	−1.74512	−0.872562	−	0.488504i	$-0.837543\pi$
$769$	−1342.00	−1.74512	−0.872562	+	0.488504i	$0.837543\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	196.774	0.251308
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1569.72	−1.99456	−0.997281	−	0.0736976i	$-0.976520\pi$
$787$	−1569.72	−1.99456	−0.997281	+	0.0736976i	$0.976520\pi$
$788$	0	0
$789$	1980.00	2.50951
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−1562.00	−1.95006
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−900.000	−1.11801
$806$	0	0
$807$	169.941	0.210584
$808$	0	0
$809$	1298.00	1.60445	0.802225	−	0.597022i	$-0.203649\pi$
$809$	1298.00	1.60445	0.802225	+	0.597022i	$0.203649\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1408.72	1.72849
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	662.000	0.806334	0.403167	−	0.915126i	$-0.367909\pi$
$821$	662.000	0.806334	0.403167	+	0.915126i	$0.367909\pi$
$822$	0	0
$823$	872.067	1.05962	0.529810	−	0.848117i	$-0.322263\pi$
$823$	872.067	1.05962	0.529810	+	0.848117i	$0.322263\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1542.89	1.86564	0.932822	−	0.360339i	$-0.117339\pi$
$827$	1542.89	1.86564	0.932822	+	0.360339i	$0.117339\pi$
$828$	0	0
$829$	1478.00	1.78287	0.891435	−	0.453148i	$-0.149699\pi$
$829$	1478.00	1.78287	0.891435	+	0.453148i	$0.149699\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1140.39	−1.36574
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−357.000	−0.424495
$842$	0	0
$843$	1869.35	2.21750
$844$	0	0
$845$	−845.000	−1.00000
$846$	0	0
$847$	1623.39	1.91663
$848$	0	0
$849$	2100.00	2.47350
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	−3720.00	−4.32056
$862$	0	0
$863$	550.073	0.637396	0.318698	−	0.947856i	$-0.396754\pi$
$863$	550.073	0.637396	0.318698	+	0.947856i	$0.396754\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1292.45	1.49071
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1677.05	−1.91663
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1618.00	1.83655	0.918275	−	0.395944i	$-0.129583\pi$
$881$	1618.00	1.83655	0.918275	+	0.395944i	$0.129583\pi$
$882$	0	0
$883$	1220.89	1.38266	0.691332	−	0.722537i	$-0.257024\pi$
$883$	1220.89	1.38266	0.691332	+	0.722537i	$0.257024\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1489.22	−1.67894	−0.839471	−	0.543405i	$-0.817135\pi$
$887$	−1489.22	−1.67894	−0.839471	+	0.543405i	$0.817135\pi$
$888$	0	0
$889$	−1260.00	−1.41732
$890$	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
$903$	2414.95	2.67437
$904$	0	0
$905$	−1790.00	−1.97790
$906$	0	0
$907$	254.912	0.281049	0.140525	−	0.990077i	$-0.455121\pi$
$907$	254.912	0.281049	0.140525	+	0.990077i	$0.455121\pi$
$908$	0	0
$909$	−1342.00	−1.47635
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	1296.92	1.41740
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	660.000	0.716612
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2213.71	−2.38803
$928$	0	0
$929$	562.000	0.604952	0.302476	−	0.953157i	$-0.402187\pi$
$929$	562.000	0.604952	0.302476	+	0.953157i	$0.402187\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	118.000	0.125399	0.0626993	−	0.998032i	$-0.480029\pi$
$941$	118.000	0.125399	0.0626993	+	0.998032i	$0.480029\pi$
$942$	0	0
$943$	−831.817	−0.882097
$944$	0	0
$945$	−600.000	−0.634921
$946$	0	0
$947$	576.906	0.609193	0.304596	−	0.952482i	$-0.401478\pi$
$947$	576.906	0.609193	0.304596	+	0.952482i	$0.401478\pi$
$948$	0	0
$949$	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
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	−1918.55	−1.99226
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1918.55	−1.98402	−0.992009	−	0.126163i	$-0.959734\pi$
$967$	−1918.55	−1.98402	−0.992009	+	0.126163i	$0.959734\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	418.000	0.426096
$982$	0	0
$983$	1945.38	1.97902	0.989511	−	0.144456i	$-0.0461431\pi$
$983$	1945.38	1.97902	0.989511	+	0.144456i	$0.0461431\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5634.89	−5.70911
$988$	0	0
$989$	540.000	0.546006
$990$	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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.3.h.c.319.2		2
4.3	odd	2	inner	320.3.h.c.319.1		2
5.2	odd	4		1600.3.b.d.1151.1		2
5.3	odd	4		1600.3.b.d.1151.2		2
5.4	even	2	inner	320.3.h.c.319.1		2
8.3	odd	2		80.3.h.a.79.2	yes	2
8.5	even	2		80.3.h.a.79.1	✓	2
16.3	odd	4		1280.3.e.e.639.2		4
16.5	even	4		1280.3.e.e.639.1		4
16.11	odd	4		1280.3.e.e.639.3		4
16.13	even	4		1280.3.e.e.639.4		4
20.3	even	4		1600.3.b.d.1151.1		2
20.7	even	4		1600.3.b.d.1151.2		2
20.19	odd	2	CM	320.3.h.c.319.2		2
24.5	odd	2		720.3.j.a.559.2		2
24.11	even	2		720.3.j.a.559.1		2
40.3	even	4		400.3.b.b.351.2		2
40.13	odd	4		400.3.b.b.351.1		2
40.19	odd	2		80.3.h.a.79.1	✓	2
40.27	even	4		400.3.b.b.351.1		2
40.29	even	2		80.3.h.a.79.2	yes	2
40.37	odd	4		400.3.b.b.351.2		2
80.19	odd	4		1280.3.e.e.639.4		4
80.29	even	4		1280.3.e.e.639.2		4
80.59	odd	4		1280.3.e.e.639.1		4
80.69	even	4		1280.3.e.e.639.3		4
120.29	odd	2		720.3.j.a.559.1		2
120.53	even	4		3600.3.e.o.3151.1		2
120.59	even	2		720.3.j.a.559.2		2
120.77	even	4		3600.3.e.o.3151.2		2
120.83	odd	4		3600.3.e.o.3151.2		2
120.107	odd	4		3600.3.e.o.3151.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.3.h.a.79.1	✓	2	8.5	even	2
80.3.h.a.79.1	✓	2	40.19	odd	2
80.3.h.a.79.2	yes	2	8.3	odd	2
80.3.h.a.79.2	yes	2	40.29	even	2
320.3.h.c.319.1		2	4.3	odd	2	inner
320.3.h.c.319.1		2	5.4	even	2	inner
320.3.h.c.319.2		2	1.1	even	1	trivial
320.3.h.c.319.2		2	20.19	odd	2	CM
400.3.b.b.351.1		2	40.13	odd	4
400.3.b.b.351.1		2	40.27	even	4
400.3.b.b.351.2		2	40.3	even	4
400.3.b.b.351.2		2	40.37	odd	4
720.3.j.a.559.1		2	24.11	even	2
720.3.j.a.559.1		2	120.29	odd	2
720.3.j.a.559.2		2	24.5	odd	2
720.3.j.a.559.2		2	120.59	even	2
1280.3.e.e.639.1		4	16.5	even	4
1280.3.e.e.639.1		4	80.59	odd	4
1280.3.e.e.639.2		4	16.3	odd	4
1280.3.e.e.639.2		4	80.29	even	4
1280.3.e.e.639.3		4	16.11	odd	4
1280.3.e.e.639.3		4	80.69	even	4
1280.3.e.e.639.4		4	16.13	even	4
1280.3.e.e.639.4		4	80.19	odd	4
1600.3.b.d.1151.1		2	5.2	odd	4
1600.3.b.d.1151.1		2	20.3	even	4
1600.3.b.d.1151.2		2	5.3	odd	4
1600.3.b.d.1151.2		2	20.7	even	4
3600.3.e.o.3151.1		2	120.53	even	4
3600.3.e.o.3151.1		2	120.107	odd	4
3600.3.e.o.3151.2		2	120.77	even	4
3600.3.e.o.3151.2		2	120.83	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	320.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.47214 q^{3} -5.00000 q^{5} +13.4164 q^{7} +11.0000 q^{9} +O(q^{10})\)	\(q+4.47214 q^{3} -5.00000 q^{5} +13.4164 q^{7} +11.0000 q^{9} -22.3607 q^{15} +60.0000 q^{21} +13.4164 q^{23} +25.0000 q^{25} +8.94427 q^{27} +22.0000 q^{29} -67.0820 q^{35} -62.0000 q^{41} +40.2492 q^{43} -55.0000 q^{45} -93.9149 q^{47} +131.000 q^{49} -58.0000 q^{61} +147.580 q^{63} -67.0820 q^{67} +60.0000 q^{69} +111.803 q^{75} -59.0000 q^{81} +147.580 q^{83} +98.3870 q^{87} -142.000 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5} + 22 q^{9}+O(q^{10}) \) 2 * q - 10 * q^5 + 22 * q^9	\( 2 q - 10 q^{5} + 22 q^{9} + 120 q^{21} + 50 q^{25} + 44 q^{29} - 124 q^{41} - 110 q^{45} + 262 q^{49} - 116 q^{61} + 120 q^{69} - 118 q^{81} - 284 q^{89}+O(q^{100}) \) 2 * q - 10 * q^5 + 22 * q^9 + 120 * q^21 + 50 * q^25 + 44 * q^29 - 124 * q^41 - 110 * q^45 + 262 * q^49 - 116 * q^61 + 120 * q^69 - 118 * q^81 - 284 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more