LMFDB - Embedded newform 320.6.f.a.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(289,320)]

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([0, 1, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("320.289");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			289.1
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	320.289
Dual form			320.6.f.a.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-55.9017 q^{5} -245.967i q^{7} -243.000 q^{9} +O(q^{10})$	$q-55.9017 q^{5} -245.967i q^{7} -243.000 q^{9} -802.000i q^{11} +245.967 q^{13} -1914.00i q^{19} +3331.74i q^{23} +3125.00 q^{25} +13750.0i q^{35} -15853.7 q^{37} +15202.0 q^{41} +13584.1 q^{45} -15361.8i q^{47} -43693.0 q^{49} -16837.6 q^{53} +44833.2i q^{55} -27986.0i q^{59} +59770.1i q^{63} -13750.0 q^{65} -197266. q^{77} +59049.0 q^{81} +128786. q^{89} -60500.0i q^{91} +106996. i q^{95} +194886. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 972 q^{9}+O(q^{10}) $ 4 * q - 972 * q^9	$ 4 q - 972 q^{9} + 12500 q^{25} + 60808 q^{41} - 174772 q^{49} - 55000 q^{65} + 236196 q^{81} + 515144 q^{89}+O(q^{100}) $ 4 * q - 972 * q^9 + 12500 * q^25 + 60808 * q^41 - 174772 * q^49 - 55000 * q^65 + 236196 * q^81 + 515144 * 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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	−55.9017	−1.00000
$5$	−55.9017	−1.00000
$6$	0	0
$7$	− 245.967i	− 1.89729i	−0.316350	−	0.948643i	$-0.602457\pi$
$7$	− 245.967i	− 1.89729i	0.316350	−	0.948643i	$-0.397543\pi$
$8$	0	0
$9$	−243.000	−1.00000
$10$	0	0
$11$	− 802.000i	− 1.99845i	−0.0393993	−	0.999224i	$-0.512544\pi$
$11$	− 802.000i	− 1.99845i	0.0393993	−	0.999224i	$-0.487456\pi$
$12$	0	0
$13$	245.967	0.403663	0.201832	−	0.979420i	$-0.435311\pi$
$13$	245.967	0.403663	0.201832	+	0.979420i	$0.435311\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	− 1914.00i	− 1.21635i	−0.793804	−	0.608174i	$-0.791902\pi$
$19$	− 1914.00i	− 1.21635i	0.793804	−	0.608174i	$-0.208098\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3331.74i	1.31326i	0.754212	+	0.656631i	$0.228019\pi$
$23$	3331.74i	1.31326i	−0.754212	+	0.656631i	$0.771981\pi$
$24$	0	0
$25$	3125.00	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	13750.0i	1.89729i
$36$	0	0
$37$	−15853.7	−1.90382	−0.951912	−	0.306371i	$-0.900885\pi$
$37$	−15853.7	−1.90382	−0.951912	+	0.306371i	$0.900885\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	15202.0	1.41235	0.706173	−	0.708039i	$-0.250420\pi$
$41$	15202.0	1.41235	0.706173	+	0.708039i	$0.250420\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	13584.1	1.00000
$46$	0	0
$47$	− 15361.8i	− 1.01437i	−0.861837	−	0.507186i	$-0.830686\pi$
$47$	− 15361.8i	− 1.01437i	0.861837	−	0.507186i	$-0.169314\pi$
$48$	0	0
$49$	−43693.0	−2.59969
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−16837.6	−0.823361	−0.411681	−	0.911328i	$-0.635058\pi$
$53$	−16837.6	−0.823361	−0.411681	+	0.911328i	$0.635058\pi$
$54$	0	0
$55$	44833.2i	1.99845i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 27986.0i	− 1.04667i	−0.852126	−	0.523336i	$-0.824687\pi$
$59$	− 27986.0i	− 1.04667i	0.852126	−	0.523336i	$-0.175313\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	59770.1i	1.89729i
$64$	0	0
$65$	−13750.0	−0.403663
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−197266.	−3.79162
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	59049.0	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	128786.	1.72343	0.861715	−	0.507393i	$-0.169391\pi$
$89$	128786.	1.72343	0.861715	+	0.507393i	$0.169391\pi$
$90$	0	0
$91$	− 60500.0i	− 0.765864i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	106996.i	1.21635i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	194886.i	1.99845i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	192950.i	1.79206i	0.443994	+	0.896030i	$0.353561\pi$
$103$	192950.i	1.79206i	−0.443994	+	0.896030i	$0.646439\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	− 186250.i	− 1.31326i
$116$	0	0
$117$	−59770.1	−0.403663
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−482153.	−2.99379
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−174693.	−1.00000
$126$	0	0
$127$	293931.i	1.61710i	0.588429	+	0.808549i	$0.299747\pi$
$127$	293931.i	1.61710i	−0.588429	+	0.808549i	$0.700253\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	66902.0i	0.340613i	0.985391	+	0.170306i	$0.0544757\pi$
$131$	66902.0i	0.340613i	−0.985391	+	0.170306i	$0.945524\pi$
$132$	0	0
$133$	−470782.	−2.30776
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	− 16786.0i	− 0.0736903i	−0.999321	−	0.0368451i	$-0.988269\pi$
$139$	− 16786.0i	− 0.0736903i	0.999321	−	0.0368451i	$-0.0117308\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 197266.i	− 0.806700i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	430555.	1.39405	0.697027	−	0.717045i	$-0.254506\pi$
$157$	430555.	1.39405	0.697027	+	0.717045i	$0.254506\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	819500.	2.49163
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	550721.i	1.52806i	0.645180	+	0.764030i	$0.276782\pi$
$167$	550721.i	1.52806i	−0.645180	+	0.764030i	$0.723218\pi$
$168$	0	0
$169$	−310793.	−0.837056
$170$	0	0
$171$	465102.i	1.21635i
$172$	0	0
$173$	353455.	0.897882	0.448941	−	0.893561i	$-0.351801\pi$
$173$	353455.	0.897882	0.448941	+	0.893561i	$0.351801\pi$
$174$	0	0
$175$	− 768648.i	− 1.89729i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 316186.i	− 0.737582i	−0.929512	−	0.368791i	$-0.879772\pi$
$179$	− 316186.i	− 0.737582i	0.929512	−	0.368791i	$-0.120228\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	886250.	1.90382
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.08349e6	1.98911	0.994553	−	0.104230i	$-0.0332377\pi$
$197$	1.08349e6	1.98911	0.994553	+	0.104230i	$0.0332377\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−849818.	−1.41235
$206$	0	0
$207$	− 809613.i	− 1.31326i
$208$	0	0
$209$	−1.53503e6	−2.43081
$210$	0	0
$211$	− 1.18270e6i	− 1.82881i	−0.404805	−	0.914403i	$-0.632660\pi$
$211$	− 1.18270e6i	− 1.82881i	0.404805	−	0.914403i	$-0.367340\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 1.31604e6i	− 1.77217i	−0.463520	−	0.886087i	$-0.653414\pi$
$223$	− 1.31604e6i	− 1.77217i	0.463520	−	0.886087i	$-0.346586\pi$
$224$	0	0
$225$	−759375.	−1.00000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	858750.i	1.01437i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	89298.0	0.0990374	0.0495187	−	0.998773i	$-0.484231\pi$
$241$	89298.0	0.0990374	0.0495187	+	0.998773i	$0.484231\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.44251e6	2.59969
$246$	0	0
$247$	− 470782.i	− 0.490995i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 620002.i	− 0.621168i	−0.950546	−	0.310584i	$-0.899475\pi$
$251$	− 620002.i	− 0.621168i	0.950546	−	0.310584i	$-0.100525\pi$
$252$	0	0
$253$	2.67206e6	2.62449
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	3.89950e6i	3.61210i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 549245.i	− 0.489640i	−0.969569	−	0.244820i	$-0.921271\pi$
$263$	− 549245.i	− 0.489640i	0.969569	−	0.244820i	$-0.0787289\pi$
$264$	0	0
$265$	941250.	0.823361
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.50625e6i	− 1.99845i
$276$	0	0
$277$	−414947.	−0.324933	−0.162466	−	0.986714i	$-0.551945\pi$
$277$	−414947.	−0.324933	−0.162466	+	0.986714i	$0.551945\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.12110e6	0.846989	0.423495	−	0.905899i	$-0.360803\pi$
$281$	1.12110e6	0.846989	0.423495	+	0.905899i	$0.360803\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.73920e6i	− 2.67962i
$288$	0	0
$289$	1.41986e6	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.79040e6	−1.21837	−0.609187	−	0.793027i	$-0.708504\pi$
$293$	−1.79040e6	−1.21837	−0.609187	+	0.793027i	$0.708504\pi$
$294$	0	0
$295$	1.56446e6i	1.04667i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	819500.i	0.530116i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	− 3.34125e6i	− 1.89729i
$316$	0	0
$317$	−2.83169e6	−1.58270	−0.791348	−	0.611366i	$-0.790620\pi$
$317$	−2.83169e6	−1.58270	−0.791348	+	0.611366i	$0.790620\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	768648.	0.403663
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.77850e6	−1.92455
$330$	0	0
$331$	− 2.09810e6i	− 1.05258i	−0.850305	−	0.526291i	$-0.823582\pi$
$331$	− 2.09810e6i	− 1.05258i	0.850305	−	0.526291i	$-0.176418\pi$
$332$	0	0
$333$	3.85245e6	1.90382
$334$	0	0
$335$	0	0
$336$	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
$343$	6.61308e6i	3.03507i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	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		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−1.18730e6	−0.479503
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	132934.i	0.0515195i	0.999668	+	0.0257598i	$0.00820049\pi$
$367$	132934.i	0.0515195i	−0.999668	+	0.0257598i	$0.991800\pi$
$368$	0	0
$369$	−3.69409e6	−1.41235
$370$	0	0
$371$	4.14150e6i	1.56215i
$372$	0	0
$373$	−5.37365e6	−1.99985	−0.999925	−	0.0122483i	$-0.996101\pi$
$373$	−5.37365e6	−1.99985	−0.999925	+	0.0122483i	$0.996101\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	4.62669e6i	1.65452i	0.561819	+	0.827260i	$0.310102\pi$
$379$	4.62669e6i	1.65452i	−0.561819	+	0.827260i	$0.689898\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 5.65025e6i	− 1.96821i	−0.177593	−	0.984104i	$-0.556831\pi$
$383$	− 5.65025e6i	− 1.96821i	0.177593	−	0.984104i	$-0.443169\pi$
$384$	0	0
$385$	1.10275e7	3.79162
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.34849e6	−1.06628	−0.533142	−	0.846026i	$-0.678989\pi$
$397$	−3.34849e6	−1.06628	−0.533142	+	0.846026i	$0.678989\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	231002.	0.0717389	0.0358695	−	0.999356i	$-0.488580\pi$
$401$	231002.	0.0717389	0.0358695	+	0.999356i	$0.488580\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−3.30094e6	−1.00000
$406$	0	0
$407$	1.27147e7i	3.80469i
$408$	0	0
$409$	−6.63601e6	−1.96155	−0.980774	−	0.195146i	$-0.937482\pi$
$409$	−6.63601e6	−1.96155	−0.980774	+	0.195146i	$0.937482\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.88365e6	−1.98584
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.11741e6i	0.589211i	0.955619	+	0.294605i	$0.0951882\pi$
$419$	2.11741e6i	0.589211i	−0.955619	+	0.294605i	$0.904812\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	3.73291e6i	1.01437i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.37695e6	1.59739
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	1.06174e7	2.59969
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−7.19936e6	−1.72343
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8.52989e6	−1.99677	−0.998383	−	0.0568371i	$-0.981898\pi$
$449$	−8.52989e6	−1.99677	−0.998383	+	0.0568371i	$0.981898\pi$
$450$	0	0
$451$	− 1.21920e7i	− 2.82250i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.38205e6i	0.765864i
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	3.67487e6i	0.796689i	0.917236	+	0.398345i	$0.130415\pi$
$463$	3.67487e6i	0.796689i	−0.917236	+	0.398345i	$0.869585\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 5.98125e6i	− 1.21635i
$476$	0	0
$477$	4.09153e6	0.823361
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−3.89950e6	−0.768504
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.51491e6i	1.43582i	0.696134	+	0.717912i	$0.254902\pi$
$487$	7.51491e6i	1.43582i	−0.696134	+	0.717912i	$0.745098\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 2.39120e6i	− 0.447623i	−0.974632	−	0.223812i	$-0.928150\pi$
$491$	− 2.39120e6i	− 0.447623i	0.974632	−	0.223812i	$-0.0718500\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	− 1.08945e7i	− 1.99845i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	6.93889e6i	1.24749i	0.781626	+	0.623747i	$0.214390\pi$
$499$	6.93889e6i	1.24749i	−0.781626	+	0.623747i	$0.785610\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 7.44027e6i	− 1.31120i	−0.755109	−	0.655600i	$-0.772416\pi$
$503$	− 7.44027e6i	− 1.31120i	0.755109	−	0.655600i	$-0.227584\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1.07863e7i	− 1.79206i
$516$	0	0
$517$	−1.23202e7	−2.02717
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.21326e7	1.95821	0.979106	−	0.203351i	$-0.0651833\pi$
$521$	1.21326e7	1.95821	0.979106	+	0.203351i	$0.0651833\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−4.66416e6	−0.724659
$530$	0	0
$531$	6.80060e6i	1.04667i
$532$	0	0
$533$	3.73920e6	0.570112
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.50418e7i	5.19534i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.45212e7	1.98319	0.991594	−	0.129387i	$-0.0413008\pi$
$557$	1.45212e7	1.98319	0.991594	+	0.129387i	$0.0413008\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.45241e7i	− 1.89729i
$568$	0	0
$569$	−3.50709e6	−0.454115	−0.227057	−	0.973881i	$-0.572911\pi$
$569$	−3.50709e6	−0.454115	−0.227057	+	0.973881i	$0.572911\pi$
$570$	0	0
$571$	− 1.45829e7i	− 1.87177i	−0.352299	−	0.935887i	$-0.614600\pi$
$571$	− 1.45829e7i	− 1.87177i	0.352299	−	0.935887i	$-0.385400\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.04117e7i	1.31326i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.35037e7i	1.64544i
$584$	0	0
$585$	3.34125e6	0.403663
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−1.60910e7	−1.81718	−0.908588	−	0.417694i	$-0.862838\pi$
$601$	−1.60910e7	−1.81718	−0.908588	+	0.417694i	$0.862838\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.69532e7	2.99379
$606$	0	0
$607$	− 1.74118e7i	− 1.91810i	−0.283233	−	0.959051i	$-0.591407\pi$
$607$	− 1.74118e7i	− 1.91810i	0.283233	−	0.959051i	$-0.408593\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 3.77850e6i	− 0.409465i
$612$	0	0
$613$	1.56408e7	1.68116	0.840579	−	0.541689i	$-0.182215\pi$
$613$	1.56408e7	1.68116	0.840579	+	0.541689i	$0.182215\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	− 7.16491e6i	− 0.751596i	−0.926702	−	0.375798i	$-0.877369\pi$
$619$	− 7.16491e6i	− 0.751596i	0.926702	−	0.375798i	$-0.122631\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 3.16772e7i	− 3.26984i
$624$	0	0
$625$	9.76562e6	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.64312e7i	− 1.61710i
$636$	0	0
$637$	−1.07471e7	−1.04940
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.08320e6	0.392515	0.196257	−	0.980552i	$-0.437121\pi$
$641$	4.08320e6	0.392515	0.196257	+	0.980552i	$0.437121\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.10606e7i	− 1.97793i	−0.148160	−	0.988963i	$-0.547335\pi$
$647$	− 2.10606e7i	− 1.97793i	0.148160	−	0.988963i	$-0.452665\pi$
$648$	0	0
$649$	−2.24448e7	−2.09172
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.24455e6	−0.848405	−0.424202	−	0.905567i	$-0.639445\pi$
$653$	−9.24455e6	−0.848405	−0.424202	+	0.905567i	$0.639445\pi$
$654$	0	0
$655$	− 3.73994e6i	− 0.340613i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1.04250e7i	− 0.935108i	−0.883964	−	0.467554i	$-0.845135\pi$
$659$	− 1.04250e7i	− 0.935108i	0.883964	−	0.467554i	$-0.154865\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.63175e7	2.30776
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.14754e6	−0.180082	−0.0900409	−	0.995938i	$-0.528700\pi$
$677$	−2.14754e6	−0.180082	−0.0900409	+	0.995938i	$0.528700\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.14150e6	−0.332361
$690$	0	0
$691$	2.50118e7i	1.99274i	0.0851509	+	0.996368i	$0.472863\pi$
$691$	2.50118e7i	1.99274i	−0.0851509	+	0.996368i	$0.527137\pi$
$692$	0	0
$693$	4.79356e7	3.79162
$694$	0	0
$695$	938366.i	0.0736903i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	3.03440e7i	2.31571i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	1.10275e7i	0.806700i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	4.74595e7	3.40005
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.14949e7i	0.806623i	0.915063	+	0.403311i	$0.132141\pi$
$727$	1.14949e7i	0.806623i	−0.915063	+	0.403311i	$0.867859\pi$
$728$	0	0
$729$	−1.43489e7	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.60595e7	−1.10401	−0.552003	−	0.833842i	$-0.686136\pi$
$733$	−1.60595e7	−1.10401	−0.552003	+	0.833842i	$0.686136\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	2.47882e7i	1.66968i	0.550490	+	0.834842i	$0.314441\pi$
$739$	2.47882e7i	1.66968i	−0.550490	+	0.834842i	$0.685559\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 2.55041e7i	− 1.69488i	−0.530893	−	0.847439i	$-0.678143\pi$
$743$	− 2.55041e7i	− 1.69488i	0.530893	−	0.847439i	$-0.321857\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.64217e6	0.231005	0.115502	−	0.993307i	$-0.463152\pi$
$757$	3.64217e6	0.231005	0.115502	+	0.993307i	$0.463152\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.83413e7	−1.77402	−0.887009	−	0.461752i	$-0.847221\pi$
$761$	−2.83413e7	−1.77402	−0.887009	+	0.461752i	$0.847221\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 6.88365e6i	− 0.422503i
$768$	0	0
$769$	−1.33834e7	−0.816114	−0.408057	−	0.912956i	$-0.633794\pi$
$769$	−1.33834e7	−0.816114	−0.408057	+	0.912956i	$0.633794\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.91839e7	−1.75669	−0.878345	−	0.478028i	$-0.841352\pi$
$773$	−2.91839e7	−1.75669	−0.878345	+	0.478028i	$0.841352\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2.90966e7i	− 1.71790i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2.40688e7	−1.39405
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.63735e7	1.47069	0.735347	−	0.677691i	$-0.237019\pi$
$797$	2.63735e7	1.47069	0.735347	+	0.677691i	$0.237019\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−3.12950e7	−1.72343
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−4.58114e7	−2.49163
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.58690e7	1.38966	0.694829	−	0.719175i	$-0.255480\pi$
$809$	2.58690e7	1.38966	0.694829	+	0.719175i	$0.255480\pi$
$810$	0	0
$811$	− 6.54480e6i	− 0.349417i	−0.984620	−	0.174709i	$-0.944102\pi$
$811$	− 6.54480e6i	− 0.349417i	0.984620	−	0.174709i	$-0.0558984\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	1.47015e7i	0.765864i
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 9.84056e6i	− 0.506431i	−0.967410	−	0.253215i	$-0.918512\pi$
$823$	− 9.84056e6i	− 0.506431i	0.967410	−	0.253215i	$-0.0814881\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 3.07862e7i	− 1.52806i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	2.05111e7	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.73739e7	0.837056
$846$	0	0
$847$	1.18594e8i	5.68007i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 5.28205e7i	− 2.50022i
$852$	0	0
$853$	−3.76003e7	−1.76937	−0.884685	−	0.466189i	$-0.845627\pi$
$853$	−3.76003e7	−1.76937	−0.884685	+	0.466189i	$0.845627\pi$
$854$	0	0
$855$	− 2.60000e7i	− 1.21635i
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	4.04040e7i	1.86828i	0.356909	+	0.934139i	$0.383831\pi$
$859$	4.04040e7i	1.86828i	−0.356909	+	0.934139i	$0.616169\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.27632e7i	1.49748i	0.662866	+	0.748738i	$0.269340\pi$
$863$	3.27632e7i	1.49748i	−0.662866	+	0.748738i	$0.730660\pi$
$864$	0	0
$865$	−1.97588e7	−0.897882
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.29688e7i	1.89729i
$876$	0	0
$877$	−2.60890e7	−1.14540	−0.572702	−	0.819763i	$-0.694105\pi$
$877$	−2.60890e7	−1.14540	−0.572702	+	0.819763i	$0.694105\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.19751e7	0.953874	0.476937	−	0.878937i	$-0.341747\pi$
$881$	2.19751e7	0.953874	0.476937	+	0.878937i	$0.341747\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.71978e7i	0.733946i	0.930232	+	0.366973i	$0.119606\pi$
$887$	1.71978e7i	0.733946i	−0.930232	+	0.366973i	$0.880394\pi$
$888$	0	0
$889$	7.22975e7	3.06810
$890$	0	0
$891$	− 4.73573e7i	− 1.99845i
$892$	0	0
$893$	−2.94025e7	−1.23383
$894$	0	0
$895$	1.76753e7i	0.737582i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.64557e7	0.646239
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−4.95429e7	−1.90382
$926$	0	0
$927$	− 4.68869e7i	− 1.79206i
$928$	0	0
$929$	9.58531e6	0.364391	0.182195	−	0.983262i	$-0.441680\pi$
$929$	9.58531e6	0.364391	0.182195	+	0.983262i	$0.441680\pi$
$930$	0	0
$931$	8.36284e7i	3.16213i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	5.06491e7i	1.85478i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	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
$961$	−2.86292e7	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.80976e7i	− 0.966280i	−0.875543	−	0.483140i	$-0.839496\pi$
$967$	− 2.80976e7i	− 0.966280i	0.875543	−	0.483140i	$-0.160504\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.13311e7i	1.74716i	0.486681	+	0.873580i	$0.338207\pi$
$971$	5.13311e7i	1.74716i	−0.486681	+	0.873580i	$0.661793\pi$
$972$	0	0
$973$	−4.12881e6	−0.139811
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	− 1.03286e8i	− 3.44418i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 2.03844e7i	− 0.672844i	−0.941711	−	0.336422i	$-0.890783\pi$
$983$	− 2.03844e7i	− 0.672844i	0.941711	−	0.336422i	$-0.109217\pi$
$984$	0	0
$985$	−6.05688e7	−1.98911
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−6.08649e7	−1.93923	−0.969614	−	0.244639i	$-0.921331\pi$
$997$	−6.08649e7	−1.93923	−0.969614	+	0.244639i	$0.921331\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.f.a.289.1	✓	4
4.3	odd	2	inner	320.6.f.a.289.2	yes	4
5.4	even	2	inner	320.6.f.a.289.4	yes	4
8.3	odd	2	inner	320.6.f.a.289.4	yes	4
8.5	even	2	inner	320.6.f.a.289.3	yes	4
20.19	odd	2	inner	320.6.f.a.289.3	yes	4
40.19	odd	2	CM	320.6.f.a.289.1	✓	4
40.29	even	2	inner	320.6.f.a.289.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.f.a.289.1	✓	4	1.1	even	1	trivial
320.6.f.a.289.1	✓	4	40.19	odd	2	CM
320.6.f.a.289.2	yes	4	4.3	odd	2	inner
320.6.f.a.289.2	yes	4	40.29	even	2	inner
320.6.f.a.289.3	yes	4	8.5	even	2	inner
320.6.f.a.289.3	yes	4	20.19	odd	2	inner
320.6.f.a.289.4	yes	4	5.4	even	2	inner
320.6.f.a.289.4	yes	4	8.3	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-55.9017 q^{5} -245.967i q^{7} -243.000 q^{9} +O(q^{10})\)	\(q-55.9017 q^{5} -245.967i q^{7} -243.000 q^{9} -802.000i q^{11} +245.967 q^{13} -1914.00i q^{19} +3331.74i q^{23} +3125.00 q^{25} +13750.0i q^{35} -15853.7 q^{37} +15202.0 q^{41} +13584.1 q^{45} -15361.8i q^{47} -43693.0 q^{49} -16837.6 q^{53} +44833.2i q^{55} -27986.0i q^{59} +59770.1i q^{63} -13750.0 q^{65} -197266. q^{77} +59049.0 q^{81} +128786. q^{89} -60500.0i q^{91} +106996. i q^{95} +194886. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 972 q^{9}+O(q^{10}) \) 4 * q - 972 * q^9	\( 4 q - 972 q^{9} + 12500 q^{25} + 60808 q^{41} - 174772 q^{49} - 55000 q^{65} + 236196 q^{81} + 515144 q^{89}+O(q^{100}) \) 4 * q - 972 * q^9 + 12500 * q^25 + 60808 * q^41 - 174772 * q^49 - 55000 * q^65 + 236196 * q^81 + 515144 * q^89

Introduction

L-functions

Modular forms

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Database

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