LMFDB - Embedded newform 576.8.d.i.289.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,8,Mod(289,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("576.289");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	576.d (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:	$179.933774679$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5249 x^{10} + 20722017 x^{8} - 34316449184 x^{6} + 42622339324672 x^{4} + \cdots + 58\!\cdots\!56 $ x^12 - 5249x^10 + 20722017x^8 - 34316449184x^6 + 42622339324672x^4 - 5239065008185344*x^2 + 588393954824749056
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{64}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.4
Root			$38.2732 - 22.0970i$ of defining polynomial
Character	$\chi$	$=$	576.289
Dual form			576.8.d.i.289.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-358.749i q^{5} +134.638 q^{7} +O(q^{10})$	$q-358.749i q^{5} +134.638 q^{7} +440.343i q^{11} -11580.0i q^{13} -36307.4 q^{17} -50244.1i q^{19} -35183.9 q^{23} -50575.9 q^{25} +150748. i q^{29} +31660.5 q^{31} -48301.3i q^{35} +300228. i q^{37} +173205. q^{41} -239859. i q^{43} -721828. q^{47} -805416. q^{49} -1.17081e6i q^{53} +157973. q^{55} +680151. i q^{59} +1.41202e6i q^{61} -4.15433e6 q^{65} -4.69502e6i q^{67} -881049. q^{71} +985879. q^{73} +59287.0i q^{77} -440546. q^{79} -3.24734e6i q^{83} +1.30252e7i q^{85} -1.45930e6 q^{89} -1.55911e6i q^{91} -1.80250e7 q^{95} +5.19710e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 4872 q^{17} - 441972 q^{25} - 356664 q^{41} + 6446076 q^{49} - 11543616 q^{65} + 26806872 q^{73} - 45367560 q^{89} + 82412136 q^{97}+O(q^{100}) $ 12 * q - 4872 * q^17 - 441972 * q^25 - 356664 * q^41 + 6446076 * q^49 - 11543616 * q^65 + 26806872 * q^73 - 45367560 * q^89 + 82412136 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$4$	0	0
$5$	− 358.749i	− 1.28350i	−0.766914	−	0.641750i	$-0.778209\pi$
$5$	− 358.749i	− 1.28350i	0.766914	−	0.641750i	$-0.221791\pi$
$6$	0	0
$7$	134.638	0.148363	0.0741814	−	0.997245i	$-0.476366\pi$
$7$	134.638	0.148363	0.0741814	+	0.997245i	$0.476366\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	440.343i	0.0997509i	0.998755	+	0.0498755i	$0.0158824\pi$
$11$	440.343i	0.0997509i	−0.998755	+	0.0498755i	$0.984118\pi$
$12$	0	0
$13$	− 11580.0i	− 1.46187i	−0.682448	−	0.730934i	$-0.739085\pi$
$13$	− 11580.0i	− 1.46187i	0.682448	−	0.730934i	$-0.260915\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−36307.4	−1.79235	−0.896177	−	0.443697i	$-0.853667\pi$
$17$	−36307.4	−1.79235	−0.896177	+	0.443697i	$0.853667\pi$
$18$	0	0
$19$	− 50244.1i	− 1.68054i	−0.542172	−	0.840268i	$-0.682398\pi$
$19$	− 50244.1i	− 1.68054i	0.542172	−	0.840268i	$-0.317602\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−35183.9	−0.602970	−0.301485	−	0.953471i	$-0.597482\pi$
$23$	−35183.9	−0.602970	−0.301485	+	0.953471i	$0.597482\pi$
$24$	0	0
$25$	−50575.9	−0.647372
$26$	0	0
$27$	0	0
$28$	0	0
$29$	150748.i	1.14778i	0.818934	+	0.573888i	$0.194566\pi$
$29$	150748.i	1.14778i	−0.818934	+	0.573888i	$0.805434\pi$
$30$	0	0
$31$	31660.5	0.190876	0.0954381	−	0.995435i	$-0.469575\pi$
$31$	31660.5	0.190876	0.0954381	+	0.995435i	$0.469575\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 48301.3i	− 0.190424i
$36$	0	0
$37$	300228.i	0.974419i	0.873285	+	0.487209i	$0.161985\pi$
$37$	300228.i	0.974419i	−0.873285	+	0.487209i	$0.838015\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	173205.	0.392479	0.196240	−	0.980556i	$-0.437127\pi$
$41$	173205.	0.392479	0.196240	+	0.980556i	$0.437127\pi$
$42$	0	0
$43$	− 239859.i	− 0.460062i	−0.973183	−	0.230031i	$-0.926117\pi$
$43$	− 239859.i	− 0.460062i	0.973183	−	0.230031i	$-0.0738828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−721828.	−1.01412	−0.507062	−	0.861910i	$-0.669269\pi$
$47$	−721828.	−1.01412	−0.507062	+	0.861910i	$0.669269\pi$
$48$	0	0
$49$	−805416.	−0.977988
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.17081e6i	− 1.08024i	−0.841589	−	0.540119i	$-0.818379\pi$
$53$	− 1.17081e6i	− 1.08024i	0.841589	−	0.540119i	$-0.181621\pi$
$54$	0	0
$55$	157973.	0.128030
$56$	0	0
$57$	0	0
$58$	0	0
$59$	680151.i	0.431145i	0.976488	+	0.215573i	$0.0691618\pi$
$59$	680151.i	0.431145i	−0.976488	+	0.215573i	$0.930838\pi$
$60$	0	0
$61$	1.41202e6i	0.796502i	0.917276	+	0.398251i	$0.130383\pi$
$61$	1.41202e6i	0.796502i	−0.917276	+	0.398251i	$0.869617\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.15433e6	−1.87631
$66$	0	0
$67$	− 4.69502e6i	− 1.90711i	−0.301217	−	0.953556i	$-0.597393\pi$
$67$	− 4.69502e6i	− 1.90711i	0.301217	−	0.953556i	$-0.402607\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−881049.	−0.292143	−0.146072	−	0.989274i	$-0.546663\pi$
$71$	−881049.	−0.292143	−0.146072	+	0.989274i	$0.546663\pi$
$72$	0	0
$73$	985879.	0.296615	0.148308	−	0.988941i	$-0.452617\pi$
$73$	985879.	0.296615	0.148308	+	0.988941i	$0.452617\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	59287.0i	0.0147993i
$78$	0	0
$79$	−440546.	−0.100530	−0.0502651	−	0.998736i	$-0.516007\pi$
$79$	−440546.	−0.100530	−0.0502651	+	0.998736i	$0.516007\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 3.24734e6i	− 0.623383i	−0.950183	−	0.311691i	$-0.899104\pi$
$83$	− 3.24734e6i	− 0.623383i	0.950183	−	0.311691i	$-0.100896\pi$
$84$	0	0
$85$	1.30252e7i	2.30049i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.45930e6	−0.219422	−0.109711	−	0.993964i	$-0.534993\pi$
$89$	−1.45930e6	−0.219422	−0.109711	+	0.993964i	$0.534993\pi$
$90$	0	0
$91$	− 1.55911e6i	− 0.216887i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.80250e7	−2.15697
$96$	0	0
$97$	5.19710e6	0.578176	0.289088	−	0.957303i	$-0.406648\pi$
$97$	5.19710e6	0.578176	0.289088	+	0.957303i	$0.406648\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.53725e7i	− 1.48463i	−0.670048	−	0.742317i	$-0.733727\pi$
$101$	− 1.53725e7i	− 1.48463i	0.670048	−	0.742317i	$-0.266273\pi$
$102$	0	0
$103$	1.78403e7	1.60869	0.804344	−	0.594163i	$-0.202517\pi$
$103$	1.78403e7	1.60869	0.804344	+	0.594163i	$0.202517\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.20117e7i	1.73705i	0.495649	+	0.868523i	$0.334930\pi$
$107$	2.20117e7i	1.73705i	−0.495649	+	0.868523i	$0.665070\pi$
$108$	0	0
$109$	− 5.44626e6i	− 0.402815i	−0.979508	−	0.201408i	$-0.935448\pi$
$109$	− 5.44626e6i	− 0.402815i	0.979508	−	0.201408i	$-0.0645516\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.07324e7	0.699714	0.349857	−	0.936803i	$-0.386230\pi$
$113$	1.07324e7	0.699714	0.349857	+	0.936803i	$0.386230\pi$
$114$	0	0
$115$	1.26222e7i	0.773912i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.88836e6	−0.265919
$120$	0	0
$121$	1.92933e7	0.990050
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.88321e6i	− 0.452598i
$126$	0	0
$127$	3.75146e7	1.62513	0.812563	−	0.582873i	$-0.198072\pi$
$127$	3.75146e7	1.62513	0.812563	+	0.582873i	$0.198072\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.66562e7i	1.03597i	0.855388	+	0.517987i	$0.173319\pi$
$131$	2.66562e7i	1.03597i	−0.855388	+	0.517987i	$0.826681\pi$
$132$	0	0
$133$	− 6.76478e6i	− 0.249329i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.80891e6	−0.226233	−0.113116	−	0.993582i	$-0.536083\pi$
$137$	−6.80891e6	−0.226233	−0.113116	+	0.993582i	$0.536083\pi$
$138$	0	0
$139$	− 2.57303e7i	− 0.812629i	−0.913733	−	0.406314i	$-0.866814\pi$
$139$	− 2.57303e7i	− 0.812629i	0.913733	−	0.406314i	$-0.133186\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.09919e6	0.145823
$144$	0	0
$145$	5.40806e7	1.47317
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.59999e7i	1.38687i	0.720520	+	0.693434i	$0.243903\pi$
$149$	5.59999e7i	1.38687i	−0.720520	+	0.693434i	$0.756097\pi$
$150$	0	0
$151$	−7.32135e7	−1.73050	−0.865250	−	0.501340i	$-0.832841\pi$
$151$	−7.32135e7	−1.73050	−0.865250	+	0.501340i	$0.832841\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.13582e7i	− 0.244990i
$156$	0	0
$157$	7.05630e7i	1.45522i	0.685991	+	0.727610i	$0.259369\pi$
$157$	7.05630e7i	1.45522i	−0.685991	+	0.727610i	$0.740631\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.73709e6	−0.0894584
$162$	0	0
$163$	9.79031e7i	1.77068i	0.464946	+	0.885339i	$0.346074\pi$
$163$	9.79031e7i	1.77068i	−0.464946	+	0.885339i	$0.653926\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.65517e7	−1.60418	−0.802088	−	0.597205i	$-0.796278\pi$
$167$	−9.65517e7	−1.60418	−0.802088	+	0.597205i	$0.796278\pi$
$168$	0	0
$169$	−7.13487e7	−1.13706
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.40397e7i	− 0.206157i	−0.994673	−	0.103078i	$-0.967131\pi$
$173$	− 1.40397e7i	− 0.206157i	0.994673	−	0.103078i	$-0.0328692\pi$
$174$	0	0
$175$	−6.80945e6	−0.0960459
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4.02743e7i	− 0.524858i	−0.964951	−	0.262429i	$-0.915476\pi$
$179$	− 4.02743e7i	− 0.524858i	0.964951	−	0.262429i	$-0.0845236\pi$
$180$	0	0
$181$	− 4.42508e7i	− 0.554684i	−0.960771	−	0.277342i	$-0.910546\pi$
$181$	− 4.42508e7i	− 0.554684i	0.960771	−	0.277342i	$-0.0894535\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.07707e8	1.25067
$186$	0	0
$187$	− 1.59877e7i	− 0.178789i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.24116e8	1.28888	0.644440	−	0.764655i	$-0.277091\pi$
$191$	1.24116e8	1.28888	0.644440	+	0.764655i	$0.277091\pi$
$192$	0	0
$193$	−9.26403e7	−0.927576	−0.463788	−	0.885946i	$-0.653510\pi$
$193$	−9.26403e7	−0.927576	−0.463788	+	0.885946i	$0.653510\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.59089e8i	1.48255i	0.671201	+	0.741275i	$0.265779\pi$
$197$	1.59089e8i	1.48255i	−0.671201	+	0.741275i	$0.734221\pi$
$198$	0	0
$199$	−1.79947e8	−1.61867	−0.809337	−	0.587344i	$-0.800174\pi$
$199$	−1.79947e8	−1.61867	−0.809337	+	0.587344i	$0.800174\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.02964e7i	0.170287i
$204$	0	0
$205$	− 6.21371e7i	− 0.503747i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.21247e7	0.167635
$210$	0	0
$211$	− 1.62163e8i	− 1.18840i	−0.804318	−	0.594199i	$-0.797469\pi$
$211$	− 1.62163e8i	− 1.18840i	0.804318	−	0.594199i	$-0.202531\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.60492e7	−0.590490
$216$	0	0
$217$	4.26271e6	0.0283189
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.20441e8i	2.62018i
$222$	0	0
$223$	−7.01893e7	−0.423842	−0.211921	−	0.977287i	$-0.567972\pi$
$223$	−7.01893e7	−0.423842	−0.211921	+	0.977287i	$0.567972\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.39202e7i	0.362700i	0.983419	+	0.181350i	$0.0580467\pi$
$227$	6.39202e7i	0.362700i	−0.983419	+	0.181350i	$0.941953\pi$
$228$	0	0
$229$	5.56480e7i	0.306214i	0.988210	+	0.153107i	$0.0489279\pi$
$229$	5.56480e7i	0.306214i	−0.988210	+	0.153107i	$0.951072\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.90650e8	1.50531	0.752653	−	0.658418i	$-0.228774\pi$
$233$	2.90650e8	1.50531	0.752653	+	0.658418i	$0.228774\pi$
$234$	0	0
$235$	2.58955e8i	1.30163i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.60260e8	−0.759334	−0.379667	−	0.925123i	$-0.623961\pi$
$239$	−1.60260e8	−0.759334	−0.379667	+	0.925123i	$0.623961\pi$
$240$	0	0
$241$	7.23258e7	0.332839	0.166419	−	0.986055i	$-0.446779\pi$
$241$	7.23258e7	0.332839	0.166419	+	0.986055i	$0.446779\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.88942e8i	1.25525i
$246$	0	0
$247$	−5.81829e8	−2.45672
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.49058e7i	0.179244i	0.995976	+	0.0896219i	$0.0285659\pi$
$251$	4.49058e7i	0.179244i	−0.995976	+	0.0896219i	$0.971434\pi$
$252$	0	0
$253$	− 1.54930e7i	− 0.0601468i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.08615e8	−0.766619	−0.383310	−	0.923620i	$-0.625216\pi$
$257$	−2.08615e8	−0.766619	−0.383310	+	0.923620i	$0.625216\pi$
$258$	0	0
$259$	4.04222e7i	0.144567i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.94811e8	−1.67724	−0.838618	−	0.544720i	$-0.816636\pi$
$263$	−4.94811e8	−1.67724	−0.838618	+	0.544720i	$0.816636\pi$
$264$	0	0
$265$	−4.20026e8	−1.38649
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.15297e7i	0.130085i	0.997883	+	0.0650423i	$0.0207182\pi$
$269$	4.15297e7i	0.130085i	−0.997883	+	0.0650423i	$0.979282\pi$
$270$	0	0
$271$	3.67290e8	1.12103	0.560515	−	0.828144i	$-0.310603\pi$
$271$	3.67290e8	1.12103	0.560515	+	0.828144i	$0.310603\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.22708e7i	− 0.0645759i
$276$	0	0
$277$	3.37071e8i	0.952889i	0.879205	+	0.476444i	$0.158075\pi$
$277$	3.37071e8i	0.952889i	−0.879205	+	0.476444i	$0.841925\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.79928e8	1.29034	0.645170	−	0.764039i	$-0.276787\pi$
$281$	4.79928e8	1.29034	0.645170	+	0.764039i	$0.276787\pi$
$282$	0	0
$283$	4.21734e8i	1.10608i	0.833155	+	0.553040i	$0.186532\pi$
$283$	4.21734e8i	1.10608i	−0.833155	+	0.553040i	$0.813468\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.33200e7	0.0582293
$288$	0	0
$289$	9.07888e8	2.21253
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.89503e8i	− 0.440128i	−0.975485	−	0.220064i	$-0.929373\pi$
$293$	− 1.89503e8i	− 0.440128i	0.975485	−	0.220064i	$-0.0706267\pi$
$294$	0	0
$295$	2.44004e8	0.553375
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.07430e8i	0.881463i
$300$	0	0
$301$	− 3.22942e7i	− 0.0682561i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.06562e8	1.02231
$306$	0	0
$307$	2.55639e8i	0.504246i	0.967695	+	0.252123i	$0.0811287\pi$
$307$	2.55639e8i	0.504246i	−0.967695	+	0.252123i	$0.918871\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.88547e7	0.129799	0.0648996	−	0.997892i	$-0.479327\pi$
$311$	6.88547e7	0.129799	0.0648996	+	0.997892i	$0.479327\pi$
$312$	0	0
$313$	7.17834e8	1.32318	0.661590	−	0.749866i	$-0.269882\pi$
$313$	7.17834e8	1.32318	0.661590	+	0.749866i	$0.269882\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.69220e8i	1.17995i	0.807423	+	0.589973i	$0.200861\pi$
$317$	6.69220e8i	1.17995i	−0.807423	+	0.589973i	$0.799139\pi$
$318$	0	0
$319$	−6.63807e7	−0.114492
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.82423e9i	3.01211i
$324$	0	0
$325$	5.85671e8i	0.946372i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.71856e7	−0.150458
$330$	0	0
$331$	− 2.63499e8i	− 0.399375i	−0.979860	−	0.199688i	$-0.936007\pi$
$331$	− 2.63499e8i	− 0.399375i	0.979860	−	0.199688i	$-0.0639927\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.68434e9	−2.44778
$336$	0	0
$337$	1.15994e9	1.65093	0.825467	−	0.564450i	$-0.190912\pi$
$337$	1.15994e9	1.65093	0.825467	+	0.564450i	$0.190912\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.39415e7i	0.0190401i
$342$	0	0
$343$	−2.19320e8	−0.293460
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.79229e8i	− 0.358763i	−0.983780	−	0.179381i	$-0.942590\pi$
$347$	− 2.79229e8i	− 0.358763i	0.983780	−	0.179381i	$-0.0574096\pi$
$348$	0	0
$349$	1.20006e9i	1.51117i	0.655049	+	0.755587i	$0.272648\pi$
$349$	1.20006e9i	1.51117i	−0.655049	+	0.755587i	$0.727352\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.35301e8	−1.13172	−0.565861	−	0.824501i	$-0.691456\pi$
$353$	−9.35301e8	−1.13172	−0.565861	+	0.824501i	$0.691456\pi$
$354$	0	0
$355$	3.16075e8i	0.374966i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.45458e9	−1.65923	−0.829616	−	0.558334i	$-0.811441\pi$
$359$	−1.45458e9	−1.65923	−0.829616	+	0.558334i	$0.811441\pi$
$360$	0	0
$361$	−1.63060e9	−1.82420
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.53683e8i	− 0.380706i
$366$	0	0
$367$	3.93164e7	0.0415186	0.0207593	−	0.999785i	$-0.493392\pi$
$367$	3.93164e7	0.0415186	0.0207593	+	0.999785i	$0.493392\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.57635e8i	− 0.160267i
$372$	0	0
$373$	− 1.23004e8i	− 0.122727i	−0.998115	−	0.0613635i	$-0.980455\pi$
$373$	− 1.23004e8i	− 0.122727i	0.998115	−	0.0613635i	$-0.0195449\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.74566e9	1.67790
$378$	0	0
$379$	− 1.56836e9i	− 1.47982i	−0.672706	−	0.739910i	$-0.734868\pi$
$379$	− 1.56836e9i	− 1.47982i	0.672706	−	0.739910i	$-0.265132\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.53912e9	−1.39983	−0.699916	−	0.714225i	$-0.746779\pi$
$383$	−1.53912e9	−1.39983	−0.699916	+	0.714225i	$0.746779\pi$
$384$	0	0
$385$	2.12692e7	0.0189949
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.21994e9i	1.05079i	0.850858	+	0.525395i	$0.176083\pi$
$389$	1.21994e9i	1.05079i	−0.850858	+	0.525395i	$0.823917\pi$
$390$	0	0
$391$	1.27743e9	1.08074
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.58046e8i	0.129031i
$396$	0	0
$397$	1.27346e9i	1.02146i	0.859742	+	0.510728i	$0.170624\pi$
$397$	1.27346e9i	1.02146i	−0.859742	+	0.510728i	$0.829376\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.92403e9	−1.49007	−0.745036	−	0.667025i	$-0.767567\pi$
$401$	−1.92403e9	−1.49007	−0.745036	+	0.667025i	$0.767567\pi$
$402$	0	0
$403$	− 3.66630e8i	− 0.279036i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.32203e8	−0.0971992
$408$	0	0
$409$	−6.14712e8	−0.444263	−0.222131	−	0.975017i	$-0.571301\pi$
$409$	−6.14712e8	−0.444263	−0.222131	+	0.975017i	$0.571301\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.15743e7i	0.0639659i
$414$	0	0
$415$	−1.16498e9	−0.800112
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.90748e9i	− 1.26681i	−0.773820	−	0.633405i	$-0.781657\pi$
$419$	− 1.90748e9i	− 1.26681i	0.773820	−	0.633405i	$-0.218343\pi$
$420$	0	0
$421$	− 1.83052e8i	− 0.119560i	−0.998212	−	0.0597801i	$-0.980960\pi$
$421$	− 1.83052e8i	− 0.119560i	0.998212	−	0.0597801i	$-0.0190399\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.83628e9	1.16032
$426$	0	0
$427$	1.90112e8i	0.118171i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.55611e9	1.53783	0.768915	−	0.639350i	$-0.220797\pi$
$431$	2.55611e9	1.53783	0.768915	+	0.639350i	$0.220797\pi$
$432$	0	0
$433$	−4.39942e7	−0.0260428	−0.0130214	−	0.999915i	$-0.504145\pi$
$433$	−4.39942e7	−0.0260428	−0.0130214	+	0.999915i	$0.504145\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.76778e9i	1.01331i
$438$	0	0
$439$	−2.24805e9	−1.26818	−0.634090	−	0.773259i	$-0.718625\pi$
$439$	−2.24805e9	−1.26818	−0.634090	+	0.773259i	$0.718625\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.88287e9i	1.02898i	0.857496	+	0.514491i	$0.172019\pi$
$443$	1.88287e9i	1.02898i	−0.857496	+	0.514491i	$0.827981\pi$
$444$	0	0
$445$	5.23523e8i	0.281628i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8.48999e8	−0.442634	−0.221317	−	0.975202i	$-0.571036\pi$
$449$	−8.48999e8	−0.442634	−0.221317	+	0.975202i	$0.571036\pi$
$450$	0	0
$451$	7.62696e7i	0.0391502i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.59331e8	−0.278374
$456$	0	0
$457$	1.99250e7	0.00976545	0.00488272	−	0.999988i	$-0.498446\pi$
$457$	1.99250e7	0.00976545	0.00488272	+	0.999988i	$0.498446\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 8.51588e8i	− 0.404833i	−0.979299	−	0.202417i	$-0.935120\pi$
$461$	− 8.51588e8i	− 0.404833i	0.979299	−	0.202417i	$-0.0648795\pi$
$462$	0	0
$463$	−1.73303e9	−0.811471	−0.405735	−	0.913991i	$-0.632985\pi$
$463$	−1.73303e9	−0.811471	−0.405735	+	0.913991i	$0.632985\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.88354e9i	1.31014i	0.755568	+	0.655070i	$0.227361\pi$
$467$	2.88354e9i	1.31014i	−0.755568	+	0.655070i	$0.772639\pi$
$468$	0	0
$469$	− 6.32129e8i	− 0.282944i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.05620e8	0.0458916
$474$	0	0
$475$	2.54114e9i	1.08793i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.84854e9	0.768518	0.384259	−	0.923225i	$-0.374457\pi$
$479$	1.84854e9	0.768518	0.384259	+	0.923225i	$0.374457\pi$
$480$	0	0
$481$	3.47665e9	1.42447
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.86445e9i	− 0.742088i
$486$	0	0
$487$	−4.89225e8	−0.191936	−0.0959682	−	0.995384i	$-0.530595\pi$
$487$	−4.89225e8	−0.191936	−0.0959682	+	0.995384i	$0.530595\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 3.57834e9i	− 1.36426i	−0.731233	−	0.682128i	$-0.761055\pi$
$491$	− 3.57834e9i	− 1.36426i	0.731233	−	0.682128i	$-0.238945\pi$
$492$	0	0
$493$	− 5.47325e9i	− 2.05722i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.18623e8	−0.0433432
$498$	0	0
$499$	1.61499e9i	0.581860i	0.956744	+	0.290930i	$0.0939647\pi$
$499$	1.61499e9i	0.581860i	−0.956744	+	0.290930i	$0.906035\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.74060e9	1.31055	0.655275	−	0.755391i	$-0.272553\pi$
$503$	3.74060e9	1.31055	0.655275	+	0.755391i	$0.272553\pi$
$504$	0	0
$505$	−5.51487e9	−1.90553
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.53603e9i	1.86074i	0.366617	+	0.930372i	$0.380516\pi$
$509$	5.53603e9i	1.86074i	−0.366617	+	0.930372i	$0.619484\pi$
$510$	0	0
$511$	1.32737e8	0.0440067
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 6.40019e9i	− 2.06475i
$516$	0	0
$517$	− 3.17852e8i	− 0.101160i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.36791e9	1.66293	0.831463	−	0.555580i	$-0.187504\pi$
$521$	5.36791e9	1.66293	0.831463	+	0.555580i	$0.187504\pi$
$522$	0	0
$523$	3.78877e7i	0.0115809i	0.999983	+	0.00579045i	$0.00184317\pi$
$523$	3.78877e7i	0.0115809i	−0.999983	+	0.00579045i	$0.998157\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.14951e9	−0.342118
$528$	0	0
$529$	−2.16692e9	−0.636427
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2.00572e9i	− 0.573753i
$534$	0	0
$535$	7.89669e9	2.22950
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 3.54659e8i	− 0.0975552i
$540$	0	0
$541$	− 3.52229e9i	− 0.956390i	−0.878254	−	0.478195i	$-0.841291\pi$
$541$	− 3.52229e9i	− 0.956390i	0.878254	−	0.478195i	$-0.158709\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.95384e9	−0.517013
$546$	0	0
$547$	1.98322e9i	0.518103i	0.965864	+	0.259052i	$0.0834099\pi$
$547$	1.98322e9i	0.518103i	−0.965864	+	0.259052i	$0.916590\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.57418e9	1.92888
$552$	0	0
$553$	−5.93143e7	−0.0149149
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3.75761e9i	− 0.921337i	−0.887572	−	0.460669i	$-0.847610\pi$
$557$	− 3.75761e9i	− 0.921337i	0.887572	−	0.460669i	$-0.152390\pi$
$558$	0	0
$559$	−2.77758e9	−0.672550
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.33516e9i	1.02382i	0.859038	+	0.511912i	$0.171063\pi$
$563$	4.33516e9i	1.02382i	−0.859038	+	0.511912i	$0.828937\pi$
$564$	0	0
$565$	− 3.85022e9i	− 0.898082i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.01866e9	−1.36964	−0.684820	−	0.728712i	$-0.740119\pi$
$569$	−6.01866e9	−1.36964	−0.684820	+	0.728712i	$0.740119\pi$
$570$	0	0
$571$	− 2.05198e9i	− 0.461262i	−0.973041	−	0.230631i	$-0.925921\pi$
$571$	− 2.05198e9i	− 0.461262i	0.973041	−	0.230631i	$-0.0740790\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.77946e9	0.390346
$576$	0	0
$577$	7.02107e9	1.52156	0.760779	−	0.649012i	$-0.224817\pi$
$577$	7.02107e9	1.52156	0.760779	+	0.649012i	$0.224817\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4.37216e8i	− 0.0924868i
$582$	0	0
$583$	5.15557e8	0.107755
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.82744e9i	− 0.372914i	−0.982463	−	0.186457i	$-0.940299\pi$
$587$	− 1.82744e9i	− 0.372914i	0.982463	−	0.186457i	$-0.0597006\pi$
$588$	0	0
$589$	− 1.59075e9i	− 0.320774i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.81688e9	1.14551	0.572755	−	0.819727i	$-0.305875\pi$
$593$	5.81688e9	1.14551	0.572755	+	0.819727i	$0.305875\pi$
$594$	0	0
$595$	1.75369e9i	0.341306i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.72073e9	−0.327129	−0.163564	−	0.986533i	$-0.552299\pi$
$599$	−1.72073e9	−0.327129	−0.163564	+	0.986533i	$0.552299\pi$
$600$	0	0
$601$	2.17942e9	0.409524	0.204762	−	0.978812i	$-0.434358\pi$
$601$	2.17942e9	0.409524	0.204762	+	0.978812i	$0.434358\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 6.92144e9i	− 1.27073i
$606$	0	0
$607$	−4.62566e9	−0.839486	−0.419743	−	0.907643i	$-0.637880\pi$
$607$	−4.62566e9	−0.839486	−0.419743	+	0.907643i	$0.637880\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.35879e9i	1.48252i
$612$	0	0
$613$	3.21758e9i	0.564181i	0.959388	+	0.282090i	$0.0910278\pi$
$613$	3.21758e9i	0.564181i	−0.959388	+	0.282090i	$0.908972\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.52140e9	−1.11775	−0.558873	−	0.829254i	$-0.688766\pi$
$617$	−6.52140e9	−1.11775	−0.558873	+	0.829254i	$0.688766\pi$
$618$	0	0
$619$	− 7.43798e9i	− 1.26049i	−0.776398	−	0.630243i	$-0.782955\pi$
$619$	− 7.43798e9i	− 1.26049i	0.776398	−	0.630243i	$-0.217045\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.96478e8	−0.0325540
$624$	0	0
$625$	−7.49684e9	−1.22828
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.09005e10i	− 1.74650i
$630$	0	0
$631$	7.00834e9	1.11048	0.555242	−	0.831689i	$-0.312626\pi$
$631$	7.00834e9	1.11048	0.555242	+	0.831689i	$0.312626\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.34583e10i	− 2.08585i
$636$	0	0
$637$	9.32674e9i	1.42969i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−9.62483e9	−1.44341	−0.721705	−	0.692200i	$-0.756641\pi$
$641$	−9.62483e9	−1.44341	−0.721705	+	0.692200i	$0.756641\pi$
$642$	0	0
$643$	− 5.22732e9i	− 0.775427i	−0.921780	−	0.387713i	$-0.873265\pi$
$643$	− 5.22732e9i	− 0.775427i	0.921780	−	0.387713i	$-0.126735\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.04814e10	1.52144	0.760721	−	0.649078i	$-0.224845\pi$
$647$	1.04814e10	1.52144	0.760721	+	0.649078i	$0.224845\pi$
$648$	0	0
$649$	−2.99500e8	−0.0430071
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 4.66858e9i	− 0.656129i	−0.944655	−	0.328064i	$-0.893604\pi$
$653$	− 4.66858e9i	− 0.656129i	0.944655	−	0.328064i	$-0.106396\pi$
$654$	0	0
$655$	9.56290e9	1.32967
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 5.50152e9i	− 0.748831i	−0.927261	−	0.374415i	$-0.877843\pi$
$659$	− 5.50152e9i	− 0.748831i	0.927261	−	0.374415i	$-0.122157\pi$
$660$	0	0
$661$	− 2.57599e9i	− 0.346928i	−0.984840	−	0.173464i	$-0.944504\pi$
$661$	− 2.57599e9i	− 0.346928i	0.984840	−	0.173464i	$-0.0554961\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.42686e9	−0.320014
$666$	0	0
$667$	− 5.30388e9i	− 0.692075i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.21774e8	−0.0794518
$672$	0	0
$673$	−9.43105e9	−1.19263	−0.596317	−	0.802749i	$-0.703370\pi$
$673$	−9.43105e9	−1.19263	−0.596317	+	0.802749i	$0.703370\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.26481e9i	0.156663i	0.996927	+	0.0783315i	$0.0249593\pi$
$677$	1.26481e9i	0.156663i	−0.996927	+	0.0783315i	$0.975041\pi$
$678$	0	0
$679$	6.99728e8	0.0857798
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 8.34630e9i	− 1.00236i	−0.865344	−	0.501178i	$-0.832900\pi$
$683$	− 8.34630e9i	− 1.00236i	0.865344	−	0.501178i	$-0.167100\pi$
$684$	0	0
$685$	2.44269e9i	0.290370i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.35580e10	−1.57917
$690$	0	0
$691$	− 8.43495e9i	− 0.972544i	−0.873808	−	0.486272i	$-0.838356\pi$
$691$	− 8.43495e9i	− 0.972544i	0.873808	−	0.486272i	$-0.161644\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.23070e9	−1.04301
$696$	0	0
$697$	−6.28862e9	−0.703462
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1.56519e10i	− 1.71615i	−0.513528	−	0.858073i	$-0.671662\pi$
$701$	− 1.56519e10i	− 1.71615i	0.513528	−	0.858073i	$-0.328338\pi$
$702$	0	0
$703$	1.50847e10	1.63755
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.06972e9i	− 0.220265i
$708$	0	0
$709$	1.23484e10i	1.30121i	0.759415	+	0.650606i	$0.225485\pi$
$709$	1.23484e10i	1.30121i	−0.759415	+	0.650606i	$0.774515\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.11394e9	−0.115093
$714$	0	0
$715$	− 1.82933e9i	− 0.187163i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.12939e9	−0.113316	−0.0566581	−	0.998394i	$-0.518045\pi$
$719$	−1.12939e9	−0.113316	−0.0566581	+	0.998394i	$0.518045\pi$
$720$	0	0
$721$	2.40199e9	0.238670
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 7.62420e9i	− 0.743038i
$726$	0	0
$727$	−5.25499e9	−0.507226	−0.253613	−	0.967306i	$-0.581619\pi$
$727$	−5.25499e9	−0.507226	−0.253613	+	0.967306i	$0.581619\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.70865e9i	0.824594i
$732$	0	0
$733$	− 4.21987e9i	− 0.395763i	−0.980226	−	0.197882i	$-0.936594\pi$
$733$	− 4.21987e9i	− 0.395763i	0.980226	−	0.197882i	$-0.0634062\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.06742e9	0.190236
$738$	0	0
$739$	3.53725e9i	0.322411i	0.986921	+	0.161205i	$0.0515382\pi$
$739$	3.53725e9i	0.322411i	−0.986921	+	0.161205i	$0.948462\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.38564e8	−0.0392259	−0.0196129	−	0.999808i	$-0.506243\pi$
$743$	−4.38564e8	−0.0392259	−0.0196129	+	0.999808i	$0.506243\pi$
$744$	0	0
$745$	2.00899e10	1.78005
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.96362e9i	0.257713i
$750$	0	0
$751$	−9.38242e9	−0.808305	−0.404153	−	0.914692i	$-0.632434\pi$
$751$	−9.38242e9	−0.808305	−0.404153	+	0.914692i	$0.632434\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.62653e10i	2.22110i
$756$	0	0
$757$	− 2.27160e10i	− 1.90325i	−0.307261	−	0.951625i	$-0.599412\pi$
$757$	− 2.27160e10i	− 1.90325i	0.307261	−	0.951625i	$-0.400588\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−9.72370e9	−0.799807	−0.399904	−	0.916557i	$-0.630956\pi$
$761$	−9.72370e9	−0.799807	−0.399904	+	0.916557i	$0.630956\pi$
$762$	0	0
$763$	− 7.33275e8i	− 0.0597628i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.87618e9	0.630277
$768$	0	0
$769$	−6.07451e9	−0.481691	−0.240846	−	0.970563i	$-0.577425\pi$
$769$	−6.07451e9	−0.481691	−0.240846	+	0.970563i	$0.577425\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.29744e10i	1.01032i	0.863025	+	0.505161i	$0.168567\pi$
$773$	1.29744e10i	1.01032i	−0.863025	+	0.505161i	$0.831433\pi$
$774$	0	0
$775$	−1.60126e9	−0.123568
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 8.70254e9i	− 0.659576i
$780$	0	0
$781$	− 3.87964e8i	− 0.0291415i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.53144e10	1.86777
$786$	0	0
$787$	− 5.03603e9i	− 0.368279i	−0.982900	−	0.184139i	$-0.941050\pi$
$787$	− 5.03603e9i	− 0.368279i	0.982900	−	0.184139i	$-0.0589498\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.44498e9	0.103811
$792$	0	0
$793$	1.63513e10	1.16438
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.15536e10i	− 0.808377i	−0.914676	−	0.404188i	$-0.867554\pi$
$797$	− 1.15536e10i	− 0.808377i	0.914676	−	0.404188i	$-0.132446\pi$
$798$	0	0
$799$	2.62077e10	1.81767
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.34125e8i	0.0295877i
$804$	0	0
$805$	1.69943e9i	0.114820i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.12561e9	−0.539556	−0.269778	−	0.962923i	$-0.586950\pi$
$809$	−8.12561e9	−0.539556	−0.269778	+	0.962923i	$0.586950\pi$
$810$	0	0
$811$	1.99340e10i	1.31227i	0.754645	+	0.656133i	$0.227809\pi$
$811$	1.99340e10i	1.31227i	−0.754645	+	0.656133i	$0.772191\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.51226e10	2.27267
$816$	0	0
$817$	−1.20515e10	−0.773151
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.17318e10i	− 0.739884i	−0.929055	−	0.369942i	$-0.879378\pi$
$821$	− 1.17318e10i	− 0.739884i	0.929055	−	0.369942i	$-0.120622\pi$
$822$	0	0
$823$	5.16344e9	0.322879	0.161440	−	0.986883i	$-0.448386\pi$
$823$	5.16344e9	0.322879	0.161440	+	0.986883i	$0.448386\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.40395e10i	− 1.47794i	−0.673739	−	0.738970i	$-0.735313\pi$
$827$	− 2.40395e10i	− 1.47794i	0.673739	−	0.738970i	$-0.264687\pi$
$828$	0	0
$829$	− 8.59651e9i	− 0.524061i	−0.965060	−	0.262030i	$-0.915608\pi$
$829$	− 8.59651e9i	− 0.524061i	0.965060	−	0.262030i	$-0.0843920\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.92425e10	1.75290
$834$	0	0
$835$	3.46378e10i	2.05896i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.53235e9	−0.498772	−0.249386	−	0.968404i	$-0.580229\pi$
$839$	−8.53235e9	−0.498772	−0.249386	+	0.968404i	$0.580229\pi$
$840$	0	0
$841$	−5.47496e9	−0.317391
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.55963e10i	1.45941i
$846$	0	0
$847$	2.59761e9	0.146887
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.05632e10i	− 0.587546i
$852$	0	0
$853$	− 1.81841e10i	− 1.00316i	−0.865112	−	0.501579i	$-0.832753\pi$
$853$	− 1.81841e10i	− 1.00316i	0.865112	−	0.501579i	$-0.167247\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.30431e9	−0.396412	−0.198206	−	0.980160i	$-0.563511\pi$
$857$	−7.30431e9	−0.396412	−0.198206	+	0.980160i	$0.563511\pi$
$858$	0	0
$859$	− 1.29771e10i	− 0.698559i	−0.937019	−	0.349279i	$-0.886426\pi$
$859$	− 1.29771e10i	− 0.698559i	0.937019	−	0.349279i	$-0.113574\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.89975e9	0.153576	0.0767879	−	0.997047i	$-0.475534\pi$
$863$	2.89975e9	0.153576	0.0767879	+	0.997047i	$0.475534\pi$
$864$	0	0
$865$	−5.03674e9	−0.264602
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1.93992e8i	− 0.0100280i
$870$	0	0
$871$	−5.43685e10	−2.78794
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1.33066e9i	− 0.0671487i
$876$	0	0
$877$	− 6.51898e9i	− 0.326348i	−0.986597	−	0.163174i	$-0.947827\pi$
$877$	− 6.51898e9i	− 0.326348i	0.986597	−	0.163174i	$-0.0521732\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.95166e9	0.293240	0.146620	−	0.989193i	$-0.453161\pi$
$881$	5.95166e9	0.293240	0.146620	+	0.989193i	$0.453161\pi$
$882$	0	0
$883$	2.76341e9i	0.135078i	0.997717	+	0.0675388i	$0.0215147\pi$
$883$	2.76341e9i	0.135078i	−0.997717	+	0.0675388i	$0.978485\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.21758e10	−0.585819	−0.292909	−	0.956140i	$-0.594623\pi$
$887$	−1.21758e10	−0.585819	−0.292909	+	0.956140i	$0.594623\pi$
$888$	0	0
$889$	5.05089e9	0.241108
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.62676e10i	1.70427i
$894$	0	0
$895$	−1.44484e10	−0.673656
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.77275e9i	0.219083i
$900$	0	0
$901$	4.25089e10i	1.93617i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.58749e10	−0.711937
$906$	0	0
$907$	− 8.44741e9i	− 0.375923i	−0.982176	−	0.187961i	$-0.939812\pi$
$907$	− 8.44741e9i	− 0.375923i	0.982176	−	0.187961i	$-0.0601879\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.30428e10	−0.571552	−0.285776	−	0.958296i	$-0.592251\pi$
$911$	−1.30428e10	−0.571552	−0.285776	+	0.958296i	$0.592251\pi$
$912$	0	0
$913$	1.42995e9	0.0621830
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.58894e9i	0.153700i
$918$	0	0
$919$	−4.51970e10	−1.92090	−0.960451	−	0.278447i	$-0.910180\pi$
$919$	−4.51970e10	−1.92090	−0.960451	+	0.278447i	$0.910180\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.02026e10i	0.427075i
$924$	0	0
$925$	− 1.51843e10i	− 0.630811i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.36989e10	0.560573	0.280286	−	0.959916i	$-0.409571\pi$
$929$	1.36989e10	0.560573	0.280286	+	0.959916i	$0.409571\pi$
$930$	0	0
$931$	4.04674e10i	1.64354i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.73558e9	−0.229476
$936$	0	0
$937$	−9.92196e9	−0.394011	−0.197006	−	0.980402i	$-0.563122\pi$
$937$	−9.92196e9	−0.394011	−0.197006	+	0.980402i	$0.563122\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.62481e10i	1.02692i	0.858115	+	0.513458i	$0.171636\pi$
$941$	2.62481e10i	1.02692i	−0.858115	+	0.513458i	$0.828364\pi$
$942$	0	0
$943$	−6.09402e9	−0.236654
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.53369e10i	1.35208i	0.736863	+	0.676042i	$0.236306\pi$
$947$	3.53369e10i	1.35208i	−0.736863	+	0.676042i	$0.763694\pi$
$948$	0	0
$949$	− 1.14165e10i	− 0.433612i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8.05048e9	0.301298	0.150649	−	0.988587i	$-0.451864\pi$
$953$	8.05048e9	0.301298	0.150649	+	0.988587i	$0.451864\pi$
$954$	0	0
$955$	− 4.45266e10i	− 1.65428i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.16739e8	−0.0335645
$960$	0	0
$961$	−2.65102e10	−0.963566
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.32346e10i	1.19054i
$966$	0	0
$967$	5.34160e10	1.89967	0.949835	−	0.312750i	$-0.101250\pi$
$967$	5.34160e10	1.89967	0.949835	+	0.312750i	$0.101250\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 3.64400e10i	− 1.27735i	−0.769476	−	0.638676i	$-0.779482\pi$
$971$	− 3.64400e10i	− 1.27735i	0.769476	−	0.638676i	$-0.220518\pi$
$972$	0	0
$973$	− 3.46427e9i	− 0.120564i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.60220e10	0.892709	0.446355	−	0.894856i	$-0.352722\pi$
$977$	2.60220e10	0.892709	0.446355	+	0.894856i	$0.352722\pi$
$978$	0	0
$979$	− 6.42593e8i	− 0.0218875i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.79073e10	0.937088	0.468544	−	0.883440i	$-0.344779\pi$
$983$	2.79073e10	0.937088	0.468544	+	0.883440i	$0.344779\pi$
$984$	0	0
$985$	5.70732e10	1.90285
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.43916e9i	0.277404i
$990$	0	0
$991$	−2.87840e10	−0.939493	−0.469747	−	0.882801i	$-0.655655\pi$
$991$	−2.87840e10	−0.939493	−0.469747	+	0.882801i	$0.655655\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.45559e10i	2.07757i
$996$	0	0
$997$	2.07759e10i	0.663936i	0.943291	+	0.331968i	$0.107713\pi$
$997$	2.07759e10i	0.663936i	−0.943291	+	0.331968i	$0.892287\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.8.d.i.289.4		12
3.2	odd	2		192.8.d.d.97.11	yes	12
4.3	odd	2	inner	576.8.d.i.289.3		12
8.3	odd	2	inner	576.8.d.i.289.9		12
8.5	even	2	inner	576.8.d.i.289.10		12
12.11	even	2		192.8.d.d.97.5	yes	12
24.5	odd	2		192.8.d.d.97.2	✓	12
24.11	even	2		192.8.d.d.97.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.8.d.d.97.2	✓	12	24.5	odd	2
192.8.d.d.97.5	yes	12	12.11	even	2
192.8.d.d.97.8	yes	12	24.11	even	2
192.8.d.d.97.11	yes	12	3.2	odd	2
576.8.d.i.289.3		12	4.3	odd	2	inner
576.8.d.i.289.4		12	1.1	even	1	trivial
576.8.d.i.289.9		12	8.3	odd	2	inner
576.8.d.i.289.10		12	8.5	even	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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	576.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-358.749i q^{5} +134.638 q^{7} +O(q^{10})\)	\(q-358.749i q^{5} +134.638 q^{7} +440.343i q^{11} -11580.0i q^{13} -36307.4 q^{17} -50244.1i q^{19} -35183.9 q^{23} -50575.9 q^{25} +150748. i q^{29} +31660.5 q^{31} -48301.3i q^{35} +300228. i q^{37} +173205. q^{41} -239859. i q^{43} -721828. q^{47} -805416. q^{49} -1.17081e6i q^{53} +157973. q^{55} +680151. i q^{59} +1.41202e6i q^{61} -4.15433e6 q^{65} -4.69502e6i q^{67} -881049. q^{71} +985879. q^{73} +59287.0i q^{77} -440546. q^{79} -3.24734e6i q^{83} +1.30252e7i q^{85} -1.45930e6 q^{89} -1.55911e6i q^{91} -1.80250e7 q^{95} +5.19710e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 4872 q^{17} - 441972 q^{25} - 356664 q^{41} + 6446076 q^{49} - 11543616 q^{65} + 26806872 q^{73} - 45367560 q^{89} + 82412136 q^{97}+O(q^{100}) \) 12 * q - 4872 * q^17 - 441972 * q^25 - 356664 * q^41 + 6446076 * q^49 - 11543616 * q^65 + 26806872 * q^73 - 45367560 * q^89 + 82412136 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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