LMFDB - Embedded newform 921.2.x.a.116.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [921,2,Mod(5,921)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(921, base_ring=CyclotomicField(306))

chi = DirichletCharacter(H, H._module([153, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("921.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 921 = 3 \cdot 307 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	921.x (of order $306$, degree $96$, 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:	$7.35422202616$
Analytic rank:	$0$
Dimension:	$96$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{306}]$

Embedding invariants

Embedding label			116.1
Character	$\chi$	$=$	921.116
Dual form			921.2.x.a.659.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.55047 + 0.772041i) q^{3} +(-1.93959 + 0.487827i) q^{4} +(0.492718 + 0.380089i) q^{7} +(1.80790 - 2.39405i) q^{9} +O(q^{10})$	\(q+(-1.55047 + 0.772041i) q^{3} +(-1.93959 + 0.487827i) q^{4} +(0.492718 + 0.380089i) q^{7} +(1.80790 - 2.39405i) q^{9} +(2.63066 - 2.25381i) q^{12} +(4.49503 + 4.01404i) q^{13} +(3.52405 - 1.89237i) q^{16} +(-0.0253129 + 0.821588i) q^{19} +(-1.05739 - 0.208917i) q^{21} +(-4.89497 + 1.01947i) q^{25} +(-0.954791 + 5.10768i) q^{27} +(-1.14109 - 0.496857i) q^{28} +(-5.62478 + 1.35344i) q^{31} +(-2.33872 + 5.52543i) q^{36} +(-3.92378 + 0.161227i) q^{37} +(-10.0684 - 2.75329i) q^{39} +(-3.60001 + 10.5607i) q^{43} +(-4.00293 + 5.65478i) q^{48} +(-1.67870 - 6.39614i) q^{49} +(-10.6767 - 5.59281i) q^{52} +(-0.595054 - 1.29339i) q^{57} +(0.0172449 + 0.839732i) q^{61} +(1.80074 - 0.492428i) q^{63} +(-5.91207 + 5.38957i) q^{64} +(-0.558146 + 0.990075i) q^{67} +(-2.30656 + 0.759934i) q^{73} +(6.80242 - 5.35977i) q^{75} +(-0.351697 - 1.60590i) q^{76} +(-13.0569 + 10.5069i) q^{79} +(-2.46297 - 8.65643i) q^{81} +(2.15282 - 0.110608i) q^{84} +(0.689090 + 3.68630i) q^{91} +(7.67613 - 6.44104i) q^{93} +(0.495106 + 16.0698i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 6 q^{4} + 18 q^{9}+O(q^{10}) $ 96 * q - 6 * q^4 + 18 * q^9	$ 96 q - 6 q^{4} + 18 q^{9} + 18 q^{12} + 12 q^{16} + 21 q^{31} + 18 q^{36} - 39 q^{43} + 36 q^{48} + 39 q^{61} + 48 q^{64} - 48 q^{67} + 51 q^{73} - 54 q^{81} + 45 q^{93}+O(q^{100}) $ 96 * q - 6 * q^4 + 18 * q^9 + 18 * q^12 + 12 * q^16 + 21 * q^31 + 18 * q^36 - 39 * q^43 + 36 * q^48 + 39 * q^61 + 48 * q^64 - 48 * q^67 + 51 * q^73 - 54 * q^81 + 45 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/921\mathbb{Z}\right)^\times$.

$n$	$308$	$619$
$\chi(n)$	$-1$	$e\left(\frac{143}{306}\right)$

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		−0.122888	−	0.992421i	$-0.539216\pi$
$2$		0			0		0.122888	+	0.992421i	$0.460784\pi$
$3$	−1.55047	+	0.772041i	−0.895163	+	0.445738i
$3$	−1.55047	+	0.772041i	−0.895163	+	0.445738i
$4$	−1.93959	+	0.487827i	−0.969797	+	0.243914i
$5$		0			0		−0.102486	−	0.994734i	$-0.532680\pi$
$5$		0			0		0.102486	+	0.994734i	$0.467320\pi$
$6$		0			0
$7$	0.492718	+	0.380089i	0.186230	+	0.143660i	0.699672	−	0.714464i	$-0.253329\pi$
$7$	0.492718	+	0.380089i	0.186230	+	0.143660i	−0.513442	+	0.858124i	$0.671630\pi$
$8$		0			0
$9$	1.80790	−	2.39405i	0.602635	−	0.798017i
$10$		0			0
$11$		0			0		0.916855	−	0.399220i	$-0.130719\pi$
$11$		0			0		−0.916855	+	0.399220i	$0.869281\pi$
$12$	2.63066	−	2.25381i	0.759405	−	0.650618i
$13$	4.49503	+	4.01404i	1.24670	+	1.11329i	0.989132	+	0.147034i	$0.0469725\pi$
$13$	4.49503	+	4.01404i	1.24670	+	1.11329i	0.257565	+	0.966261i	$0.417080\pi$
$14$		0			0
$15$		0			0
$16$	3.52405	−	1.89237i	0.881012	−	0.473094i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	−0.0253129	+	0.821588i	−0.00580717	+	0.188485i	0.992169	+	0.124901i	$0.0398612\pi$
$19$	−0.0253129	+	0.821588i	−0.00580717	+	0.188485i	−0.997976	+	0.0635848i	$0.979747\pi$
$20$		0			0
$21$	−1.05739	−	0.208917i	−0.230741	−	0.0455895i
$22$		0			0
$23$		0			0		0.454905	−	0.890540i	$-0.349673\pi$
$23$		0			0		−0.454905	+	0.890540i	$0.650327\pi$
$24$		0			0
$25$	−4.89497	+	1.01947i	−0.978993	+	0.203893i
$26$		0			0
$27$	−0.954791	+	5.10768i	−0.183750	+	0.982973i
$28$	−1.14109	−	0.496857i	−0.215646	−	0.0938972i
$29$		0			0		0.956004	−	0.293353i	$-0.0947712\pi$
$29$		0			0		−0.956004	+	0.293353i	$0.905229\pi$
$30$		0			0
$31$	−5.62478	+	1.35344i	−1.01024	+	0.243086i	−0.705013	−	0.709194i	$-0.749059\pi$
$31$	−5.62478	+	1.35344i	−1.01024	+	0.243086i	−0.305227	+	0.952280i	$0.598732\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−2.33872	+	5.52543i	−0.389786	+	0.920906i
$37$	−3.92378	+	0.161227i	−0.645066	+	0.0265055i	−0.361390	−	0.932415i	$-0.617698\pi$
$37$	−3.92378	+	0.161227i	−0.645066	+	0.0265055i	−0.283676	+	0.958920i	$0.591554\pi$
$38$		0			0
$39$	−10.0684	−	2.75329i	−1.61224	−	0.440880i
$40$		0			0
$41$		0			0		0.535133	−	0.844768i	$-0.320261\pi$
$41$		0			0		−0.535133	+	0.844768i	$0.679739\pi$
$42$		0			0
$43$	−3.60001	+	10.5607i	−0.548996	+	1.61050i	0.222964	+	0.974827i	$0.428427\pi$
$43$	−3.60001	+	10.5607i	−0.548996	+	1.61050i	−0.771960	+	0.635671i	$0.780724\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.928714	−	0.370796i	$-0.879085\pi$
$47$		0			0		0.928714	+	0.370796i	$0.120915\pi$
$48$	−4.00293	+	5.65478i	−0.577774	+	0.816197i
$49$	−1.67870	−	6.39614i	−0.239814	−	0.913734i
$50$		0			0
$51$		0			0
$52$	−10.6767	−	5.59281i	−1.48059	−	0.775583i
$53$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$53$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.595054	−	1.29339i	−0.0788168	−	0.171314i
$58$		0			0
$59$		0			0		0.822086	−	0.569364i	$-0.192810\pi$
$59$		0			0		−0.822086	+	0.569364i	$0.807190\pi$
$60$		0			0
$61$	0.0172449	+	0.839732i	0.00220798	+	0.107517i	0.999447	+	0.0332562i	$0.0105877\pi$
$61$	0.0172449	+	0.839732i	0.00220798	+	0.107517i	−0.997239	+	0.0742605i	$0.976340\pi$
$62$		0			0
$63$	1.80074	−	0.492428i	0.226872	−	0.0620401i
$64$	−5.91207	+	5.38957i	−0.739009	+	0.673696i
$65$		0			0
$66$		0			0
$67$	−0.558146	+	0.990075i	−0.0681884	+	0.120957i	−0.903105	−	0.429419i	$-0.858718\pi$
$67$	−0.558146	+	0.990075i	−0.0681884	+	0.120957i	0.834917	+	0.550376i	$0.185516\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.752685	−	0.658380i	$-0.771242\pi$
$71$		0			0		0.752685	+	0.658380i	$0.228758\pi$
$72$		0			0
$73$	−2.30656	+	0.759934i	−0.269963	+	0.0889436i	−0.444725	−	0.895667i	$-0.646699\pi$
$73$	−2.30656	+	0.759934i	−0.269963	+	0.0889436i	0.174762	+	0.984611i	$0.444084\pi$
$74$		0			0
$75$	6.80242	−	5.35977i	0.785476	−	0.618892i
$76$	−0.351697	−	1.60590i	−0.0403424	−	0.184209i
$77$		0			0
$78$		0			0
$79$	−13.0569	+	10.5069i	−1.46902	+	1.18211i	−0.525501	+	0.850793i	$0.676122\pi$
$79$	−13.0569	+	10.5069i	−1.46902	+	1.18211i	−0.943518	+	0.331322i	$0.892505\pi$
$80$		0			0
$81$	−2.46297	−	8.65643i	−0.273663	−	0.961826i
$82$		0			0
$83$		0			0		−0.999157	−	0.0410550i	$-0.986928\pi$
$83$		0			0		0.999157	+	0.0410550i	$0.0130719\pi$
$84$	2.15282	−	0.110608i	0.234892	−	0.0120684i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.0102665	−	0.999947i	$-0.503268\pi$
$89$		0			0		0.0102665	+	0.999947i	$0.496732\pi$
$90$		0			0
$91$	0.689090	+	3.68630i	0.0722362	+	0.386430i
$92$		0			0
$93$	7.67613	−	6.44104i	0.795977	−	0.667904i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.495106	+	16.0698i	0.0502704	+	1.63164i	0.602618	+	0.798030i	$0.294124\pi$
$97$	0.495106	+	16.0698i	0.0502704	+	1.63164i	−0.552347	+	0.833614i	$0.686268\pi$
$98$		0			0
$99$		0			0
$100$	8.99692	−	4.36525i	0.899692	−	0.436525i
$101$		0			0		0.213933	−	0.976848i	$-0.431373\pi$
$101$		0			0		−0.213933	+	0.976848i	$0.568627\pi$
$102$		0			0
$103$	−3.68549	+	10.4587i	−0.363143	+	1.03052i	0.608394	+	0.793635i	$0.291814\pi$
$103$	−3.68549	+	10.4587i	−0.363143	+	1.03052i	−0.971537	+	0.236889i	$0.923872\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.839229	−	0.543778i	$-0.183007\pi$
$107$		0			0		−0.839229	+	0.543778i	$0.816993\pi$
$108$	−0.639760	−	10.3726i	−0.0615609	−	0.998103i
$109$	−1.33293	+	11.7527i	−0.127672	+	1.12570i	0.755005	+	0.655719i	$0.227635\pi$
$109$	−1.33293	+	11.7527i	−0.127672	+	1.12570i	−0.882677	+	0.469981i	$0.844261\pi$
$110$		0			0
$111$	5.95923	−	3.27930i	0.565625	−	0.311257i
$112$	2.45563	+	0.407046i	0.232036	+	0.0384622i
$113$		0			0		−0.153392	−	0.988165i	$-0.549020\pi$
$113$		0			0		0.153392	+	0.988165i	$0.450980\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	17.7364	−	3.50433i	1.63973	−	0.323976i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	7.49372	−	8.05259i	0.681247	−	0.732053i
$122$		0			0
$123$		0			0
$124$	10.2495	−	5.36905i	0.920436	−	0.482155i
$125$		0			0
$126$		0			0
$127$	0.249543	−	0.813231i	0.0221434	−	0.0721626i	−0.944251	−	0.329226i	$-0.893212\pi$
$127$	0.249543	−	0.813231i	0.0221434	−	0.0721626i	0.966395	+	0.257063i	$0.0827548\pi$
$128$		0			0
$129$	−2.57163	−	19.1534i	−0.226419	−	1.68637i
$130$		0			0
$131$		0			0		−0.785476	−	0.618892i	$-0.787582\pi$
$131$		0			0		0.785476	+	0.618892i	$0.212418\pi$
$132$		0			0
$133$	−0.324749	+	0.395191i	−0.0281593	+	0.0342674i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.732053	−	0.681247i	$-0.238562\pi$
$137$		0			0		−0.732053	+	0.681247i	$0.761438\pi$
$138$		0			0
$139$	1.03865	−	0.183142i	0.0880973	−	0.0155339i	−0.129426	−	0.991589i	$-0.541313\pi$
$139$	1.03865	−	0.183142i	0.0880973	−	0.0155339i	0.217523	+	0.976055i	$0.430202\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1.84070	−	11.8580i	0.153392	−	0.988165i
$145$		0			0
$146$		0			0
$147$	7.54085	+	8.62099i	0.621959	+	0.711047i
$148$	7.53189	−	2.22684i	0.619118	−	0.183045i
$149$		0			0		0.696134	−	0.717912i	$-0.254902\pi$
$149$		0			0		−0.696134	+	0.717912i	$0.745098\pi$
$150$		0			0
$151$	0.0461659	−	2.24803i	0.00375693	−	0.182942i	−0.993718	−	0.111917i	$-0.964301\pi$
$151$	0.0461659	−	2.24803i	0.00375693	−	0.182942i	0.997475	−	0.0710253i	$-0.0226271\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	20.8718	+	0.428626i	1.67108	+	0.0343175i
$157$	13.9873	−	14.1316i	1.11631	−	1.12783i	0.125562	−	0.992086i	$-0.459926\pi$
$157$	13.9873	−	14.1316i	1.11631	−	1.12783i	0.990744	−	0.135741i	$-0.0433415\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−15.8107	−	2.45427i	−1.23839	−	0.192233i	−0.499650	−	0.866228i	$-0.666538\pi$
$163$	−15.8107	−	2.45427i	−1.23839	−	0.192233i	−0.738736	+	0.673995i	$0.764577\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.952942	−	0.303153i	$-0.901961\pi$
$167$		0			0		0.952942	+	0.303153i	$0.0980392\pi$
$168$		0			0
$169$	2.62776	+	23.1693i	0.202135	+	1.78225i
$170$		0			0
$171$	1.92116	+	1.54595i	0.146915	+	0.118222i
$172$	1.83074	−	22.2397i	0.139592	−	1.69576i
$173$		0			0		0.380311	−	0.924859i	$-0.375817\pi$
$173$		0			0		−0.380311	+	0.924859i	$0.624183\pi$
$174$		0			0
$175$	−2.79933	−	1.35821i	−0.211609	−	0.102671i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.650618	−	0.759405i	$-0.274510\pi$
$179$		0			0		−0.650618	+	0.759405i	$0.725490\pi$
$180$		0			0
$181$	24.3291	−	10.0043i	1.80837	−	0.743617i	0.824216	−	0.566275i	$-0.191616\pi$
$181$	24.3291	−	10.0043i	1.80837	−	0.743617i	0.984149	−	0.177342i	$-0.0567499\pi$
$182$		0			0
$183$	−0.675046	−	1.28866i	−0.0499008	−	0.0952608i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2.41182	+	2.15374i	−0.175434	+	0.156662i
$190$		0			0
$191$		0			0		−0.517676	−	0.855577i	$-0.673203\pi$
$191$		0			0		0.517676	+	0.855577i	$0.326797\pi$
$192$	5.00551	−	12.9207i	0.361242	−	0.932472i
$193$	3.86069	+	6.23523i	0.277898	+	0.448821i	0.959016	−	0.283353i	$-0.0914468\pi$
$193$	3.86069	+	6.23523i	0.277898	+	0.448821i	−0.681117	+	0.732174i	$0.738506\pi$
$194$		0			0
$195$		0			0
$196$	6.37620	+	11.5870i	0.455443	+	0.827643i
$197$		0			0		−0.594410	−	0.804162i	$-0.702614\pi$
$197$		0			0		0.594410	+	0.804162i	$0.297386\pi$
$198$		0			0
$199$	−23.6343	+	1.45771i	−1.67539	+	0.103335i	−0.871166	−	0.490989i	$-0.836635\pi$
$199$	−23.6343	+	1.45771i	−1.67539	+	0.103335i	−0.804226	+	0.594324i	$0.797420\pi$
$200$		0			0
$201$	0.101009	−	1.96599i	0.00712465	−	0.138670i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	23.4368	+	5.63940i	1.62505	+	0.391022i
$209$		0			0
$210$		0			0
$211$	−13.4317	+	25.0129i	−0.924673	+	1.72196i	−0.275533	+	0.961292i	$0.588854\pi$
$211$	−13.4317	+	25.0129i	−0.924673	+	1.72196i	−0.649140	+	0.760669i	$0.724871\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.28586	−	1.47105i	−0.223059	−	0.0998615i
$218$		0			0
$219$	2.98955	−	2.95902i	0.202015	−	0.199952i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.0568849	+	1.10718i	0.00380929	+	0.0741420i	0.999899	−	0.0142043i	$-0.00452152\pi$
$223$	0.0568849	+	1.10718i	0.00380929	+	0.0741420i	−0.996090	+	0.0883463i	$0.971842\pi$
$224$		0			0
$225$	−6.40898	+	13.5619i	−0.427265	+	0.904126i
$226$		0			0
$227$		0			0		0.827888	−	0.560894i	$-0.189542\pi$
$227$		0			0		−0.827888	+	0.560894i	$0.810458\pi$
$228$	1.78511	+	2.21837i	0.118222	+	0.146915i
$229$	−26.4279	−	3.82494i	−1.74641	−	0.252759i	−0.805784	−	0.592210i	$-0.798256\pi$
$229$	−26.4279	−	3.82494i	−1.74641	−	0.252759i	−0.940624	+	0.339451i	$0.889759\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.986539	−	0.163529i	$-0.0522876\pi$
$233$		0			0		−0.986539	+	0.163529i	$0.947712\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	12.1326	−	26.3711i	0.788098	−	1.71298i
$238$		0			0
$239$		0			0		0.717912	−	0.696134i	$-0.245098\pi$
$239$		0			0		−0.717912	+	0.696134i	$0.754902\pi$
$240$		0			0
$241$	11.6987	−	18.0549i	0.753578	−	1.16302i	−0.228316	−	0.973587i	$-0.573322\pi$
$241$	11.6987	−	18.0549i	0.753578	−	1.16302i	0.981894	−	0.189433i	$-0.0606650\pi$
$242$		0			0
$243$	10.5019	+	11.5200i	0.673696	+	0.739009i
$244$	−0.443092	−	1.62033i	−0.0283661	−	0.103731i
$245$		0			0
$246$		0			0
$247$	−3.41167	+	3.59146i	−0.217079	+	0.228519i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.958965	−	0.283523i	$-0.908497\pi$
$251$		0			0		0.958965	+	0.283523i	$0.0915033\pi$
$252$	−3.25249	+	1.83356i	−0.204887	+	0.115504i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	8.83784	−	13.3376i	0.552365	−	0.833602i
$257$		0			0		0.946517	−	0.322654i	$-0.104575\pi$
$257$		0			0		−0.946517	+	0.322654i	$0.895425\pi$
$258$		0			0
$259$	−1.99460	−	1.41195i	−0.123938	−	0.0877342i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.610796	−	0.791788i	$-0.709150\pi$
$263$		0			0		0.610796	+	0.791788i	$0.290850\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.599591	−	2.19262i	0.0366259	−	0.133936i
$269$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$269$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$270$		0			0
$271$	−22.6068	−	9.56863i	−1.37326	−	0.581253i	−0.426883	−	0.904307i	$-0.640388\pi$
$271$	−22.6068	−	9.56863i	−1.37326	−	0.581253i	−0.946380	+	0.323054i	$0.895290\pi$
$272$		0			0
$273$	−3.91439	−	5.18349i	−0.236910	−	0.313719i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.88220	+	2.29047i	0.113090	+	0.137621i	0.826078	−	0.563556i	$-0.190567\pi$
$277$	1.88220	+	2.29047i	0.113090	+	0.137621i	−0.712988	+	0.701177i	$0.752658\pi$
$278$		0			0
$279$	−6.92884	+	15.9129i	−0.414819	+	0.952681i
$280$		0			0
$281$		0			0		−0.822086	−	0.569364i	$-0.807190\pi$
$281$		0			0		0.822086	+	0.569364i	$0.192810\pi$
$282$		0			0
$283$	9.77170	−	10.7190i	0.580867	−	0.637181i	−0.376375	−	0.926467i	$-0.622830\pi$
$283$	9.77170	−	10.7190i	0.580867	−	0.637181i	0.957243	+	0.289286i	$0.0934179\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.50000	+	14.7224i	−0.500000	+	0.866025i
$290$		0			0
$291$	−13.1742	−	24.5335i	−0.772287	−	1.43818i
$292$	4.10308	−	2.59917i	0.240115	−	0.152105i
$293$		0			0		−0.860847	−	0.508865i	$-0.830065\pi$
$293$		0			0		0.860847	+	0.508865i	$0.169935\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−10.5793	+	13.7142i	−0.610796	+	0.791788i
$301$	−5.78781	+	3.83514i	−0.333604	+	0.221054i
$302$		0			0
$303$		0			0
$304$	1.46555	+	2.94322i	0.0840550	+	0.168805i
$305$		0			0
$306$		0			0
$307$	−8.57566	−	15.2793i	−0.489439	−	0.872038i
$307$	−8.57566	−	15.2793i	−0.489439	−	0.872038i
$308$		0			0
$309$	−2.36029	−	19.0612i	−0.134272	−	1.08435i
$310$		0			0
$311$		0			0		0.969797	−	0.243914i	$-0.0784314\pi$
$311$		0			0		−0.969797	+	0.243914i	$0.921569\pi$
$312$		0			0
$313$	18.8335	+	28.4226i	1.06453	+	1.60654i	0.755085	+	0.655627i	$0.227596\pi$
$313$	18.8335	+	28.4226i	1.06453	+	1.60654i	0.309449	+	0.950916i	$0.399855\pi$
$314$		0			0
$315$		0			0
$316$	20.1996	−	26.7486i	1.13632	−	1.50473i
$317$		0			0		0.974601	−	0.223951i	$-0.0718954\pi$
$317$		0			0		−0.974601	+	0.223951i	$0.928105\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	9.00000	+	15.5885i	0.500000	+	0.866025i
$325$	−26.0952	−	15.0661i	−1.44750	−	0.835715i
$326$		0			0
$327$	−7.00686	−	19.2512i	−0.387480	−	1.06459i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	18.3840	+	16.7592i	1.01048	+	0.921170i	0.996895	−	0.0787376i	$-0.0250889\pi$
$331$	18.3840	+	16.7592i	1.01048	+	0.921170i	0.0135802	+	0.999908i	$0.495677\pi$
$332$		0			0
$333$	−6.70784	+	9.68522i	−0.367587	+	0.530747i
$334$		0			0
$335$		0			0
$336$	−4.12164	+	1.26474i	−0.224854	+	0.0689972i
$337$	8.12790	−	6.67913i	0.442755	−	0.363835i	−0.386891	−	0.922126i	$-0.626451\pi$
$337$	8.12790	−	6.67913i	0.442755	−	0.363835i	0.829646	+	0.558290i	$0.188542\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	3.30189	−	7.80101i	0.178285	−	0.421215i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.890540	−	0.454905i	$-0.150327\pi$
$347$		0			0		−0.890540	+	0.454905i	$0.849673\pi$
$348$		0			0
$349$	−2.52489	+	35.0725i	−0.135154	+	1.87739i	0.269612	+	0.962969i	$0.413104\pi$
$349$	−2.52489	+	35.0725i	−0.135154	+	1.87739i	−0.404766	+	0.914420i	$0.632647\pi$
$350$		0			0
$351$	−24.7942	+	19.1266i	−1.32342	+	1.02090i
$352$		0			0
$353$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$353$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.885823	−	0.464024i	$-0.846405\pi$
$359$		0			0		0.885823	+	0.464024i	$0.153595\pi$
$360$		0			0
$361$	18.2896	+	1.12806i	0.962610	+	0.0593718i
$362$		0			0
$363$	−5.40184	+	18.2707i	−0.283523	+	0.958965i
$364$	−3.13483	−	6.81378i	−0.164310	−	0.357139i
$365$		0			0
$366$		0			0
$367$	26.7329	+	25.3947i	1.39545	+	1.32559i	0.884929	+	0.465727i	$0.154207\pi$
$367$	26.7329	+	25.3947i	1.39545	+	1.32559i	0.510519	+	0.859866i	$0.329453\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−11.7465	+	16.2376i	−0.609025	+	0.841881i
$373$	−23.1407	−	14.9940i	−1.19818	−	0.776359i	−0.218287	−	0.975885i	$-0.570047\pi$
$373$	−23.1407	−	14.9940i	−1.19818	−	0.776359i	−0.979893	+	0.199525i	$0.936060\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	27.1931	+	7.73710i	1.39681	+	0.397428i	0.886567	−	0.462599i	$-0.153083\pi$
$379$	27.1931	+	7.73710i	1.39681	+	0.397428i	0.510248	+	0.860028i	$0.329554\pi$
$380$		0			0
$381$	0.240940	+	1.45355i	0.0123437	+	0.0744674i
$382$		0			0
$383$		0			0		−0.213933	−	0.976848i	$-0.568627\pi$
$383$		0			0		0.213933	+	0.976848i	$0.431373\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	18.7745	+	27.7114i	0.954360	+	1.40865i
$388$	−8.79961	−	30.9274i	−0.446733	−	1.57010i
$389$		0			0		−0.904126	−	0.427265i	$-0.859477\pi$
$389$		0			0		0.904126	+	0.427265i	$0.140523\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	13.5324	−	30.2271i	0.679172	−	1.51705i	−0.168029	−	0.985782i	$-0.553740\pi$
$397$	13.5324	−	30.2271i	0.679172	−	1.51705i	0.847201	−	0.531272i	$-0.178286\pi$
$398$		0			0
$399$	0.198410	−	0.863450i	0.00993291	−	0.0432266i
$400$	−15.3209	+	12.8558i	−0.766044	+	0.642788i
$401$		0			0		0.253857	−	0.967242i	$-0.418301\pi$
$401$		0			0		−0.253857	+	0.967242i	$0.581699\pi$
$402$		0			0
$403$	−30.7163	−	16.4943i	−1.53009	−	0.821641i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−17.7240	+	35.5946i	−0.876395	+	1.76004i	−0.274234	+	0.961663i	$0.588424\pi$
$409$	−17.7240	+	35.5946i	−0.876395	+	1.76004i	−0.602161	+	0.798375i	$0.705694\pi$
$410$		0			0
$411$		0			0
$412$	2.04633	−	22.0835i	0.100816	−	1.08797i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.46900	+	1.08584i	−0.0719374	+	0.0531737i
$418$		0			0
$419$		0			0		−0.986539	−	0.163529i	$-0.947712\pi$
$419$		0			0		0.986539	+	0.163529i	$0.0522876\pi$
$420$		0			0
$421$	33.0369	−	20.4556i	1.61012	−	0.996945i	0.635780	−	0.771870i	$-0.280679\pi$
$421$	33.0369	−	20.4556i	1.61012	−	0.996945i	0.974342	−	0.225075i	$-0.0722626\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.310676	+	0.420306i	−0.0150347	+	0.0203400i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.885823	−	0.464024i	$-0.153595\pi$
$431$		0			0		−0.885823	+	0.464024i	$0.846405\pi$
$432$	6.30091	+	19.8065i	0.303153	+	0.952942i
$433$	−12.4002	−	30.1553i	−0.595913	−	1.44917i	−0.872708	−	0.488242i	$-0.837638\pi$
$433$	−12.4002	−	30.1553i	−0.595913	−	1.44917i	0.276795	−	0.960929i	$-0.410728\pi$
$434$		0			0
$435$		0			0
$436$	−3.14792	−	23.4456i	−0.150758	−	1.12284i
$437$		0			0
$438$		0			0
$439$	−12.9049	+	26.5974i	−0.615917	+	1.26943i	0.329682	+	0.944092i	$0.393059\pi$
$439$	−12.9049	+	26.5974i	−0.615917	+	1.26943i	−0.945599	+	0.325335i	$0.894523\pi$
$440$		0			0
$441$	−18.3476	−	7.54471i	−0.873696	−	0.359272i
$442$		0			0
$443$		0			0		0.626924	−	0.779081i	$-0.284314\pi$
$443$		0			0		−0.626924	+	0.779081i	$0.715686\pi$
$444$	−9.95875	+	9.26758i	−0.472621	+	0.439820i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−4.96150	+	0.408423i	−0.234409	+	0.0192962i
$449$		0			0		0.560894	−	0.827888i	$-0.310458\pi$
$449$		0			0		−0.560894	+	0.827888i	$0.689542\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.66399	+	3.52114i	0.0781813	+	0.165438i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	27.6050	+	27.3231i	1.29131	+	1.27812i	0.939712	+	0.341968i	$0.111093\pi$
$457$	27.6050	+	27.3231i	1.29131	+	1.27812i	0.351597	+	0.936151i	$0.385639\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.122888	−	0.992421i	$-0.460784\pi$
$461$		0			0		−0.122888	+	0.992421i	$0.539216\pi$
$462$		0			0
$463$	38.3004	+	0.786543i	1.77997	+	0.0365537i	0.899339	−	0.437252i	$-0.144048\pi$
$463$	38.3004	+	0.786543i	1.77997	+	0.0365537i	0.880630	+	0.473805i	$0.157120\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.752685	−	0.658380i	$-0.228758\pi$
$467$		0			0		−0.752685	+	0.658380i	$0.771242\pi$
$468$	−32.6919	+	15.4493i	−1.51118	+	0.714143i
$469$	−0.651326	+	0.275682i	−0.0300754	+	0.0127298i
$470$		0			0
$471$	−10.7766	+	32.7094i	−0.496561	+	1.50717i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.713675	−	4.04745i	−0.0327457	−	0.185710i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.0820408	−	0.996629i	$-0.473856\pi$
$479$		0			0		−0.0820408	+	0.996629i	$0.526144\pi$
$480$		0			0
$481$	−18.2847	−	15.0255i	−0.833710	−	0.685104i
$482$		0			0
$483$		0			0
$484$	−10.6065	+	19.2744i	−0.482114	+	0.876109i
$485$		0			0
$486$		0			0
$487$	−13.7565	−	4.22123i	−0.623367	−	0.191282i	−0.0343588	−	0.999410i	$-0.510939\pi$
$487$	−13.7565	−	4.22123i	−0.623367	−	0.191282i	−0.589008	+	0.808127i	$0.700481\pi$
$488$		0			0
$489$	26.4087	−	8.40122i	1.19424	−	0.379916i
$490$		0			0
$491$		0			0		−0.994734	−	0.102486i	$-0.967320\pi$
$491$		0			0		0.994734	+	0.102486i	$0.0326797\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−17.2608	+	15.4138i	−0.775032	+	0.692100i
$497$		0			0
$498$		0			0
$499$	−5.32777	+	13.7525i	−0.238504	+	0.615649i	−0.999450	−	0.0331663i	$-0.989441\pi$
$499$	−5.32777	+	13.7525i	−0.238504	+	0.615649i	0.760946	+	0.648815i	$0.224735\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.482114	−	0.876109i	$-0.660131\pi$
$503$		0			0		0.482114	+	0.876109i	$0.339869\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−21.9619	−	33.8945i	−0.975362	−	1.50531i
$508$	−0.0872955	+	1.69907i	−0.00387311	+	0.0753841i
$509$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$509$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$510$		0			0
$511$	−1.42533	−	0.502267i	−0.0630529	−	0.0222190i
$512$		0			0
$513$	−4.17224	−	0.913735i	−0.184209	−	0.0403424i
$514$		0			0
$515$		0			0
$516$	14.3315	+	35.8954i	0.630909	+	1.58021i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$521$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$522$		0			0
$523$	−14.0141	+	2.61969i	−0.612795	+	0.114551i	−0.480992	−	0.876725i	$-0.659724\pi$
$523$	−14.0141	+	2.61969i	−0.612795	+	0.114551i	−0.131803	+	0.991276i	$0.542076\pi$
$524$		0			0
$525$	5.38887	−	0.0553275i	0.235189	−	0.00241469i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−13.4808	−	18.6351i	−0.586123	−	0.810222i
$530$		0			0
$531$		0			0
$532$	0.437097	−	0.924931i	0.0189505	−	0.0401008i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−13.7072	−	41.6044i	−0.589320	−	1.78871i	−0.614361	−	0.789025i	$-0.710586\pi$
$541$	−13.7072	−	41.6044i	−0.589320	−	1.78871i	0.0250408	−	0.999686i	$-0.492028\pi$
$542$		0			0
$543$	−29.9977	+	34.2945i	−1.28732	+	1.47172i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	24.8609	+	14.0151i	1.06298	+	0.599244i	0.920575	−	0.390566i	$-0.127721\pi$
$547$	24.8609	+	14.0151i	1.06298	+	0.599244i	0.142400	+	0.989809i	$0.454518\pi$
$548$		0			0
$549$	2.04154	+	1.47687i	0.0871308	+	0.0630313i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−10.4269	+	0.214129i	−0.443398	+	0.00910570i
$554$		0			0
$555$		0			0
$556$	−1.92522	+	0.861904i	−0.0816475	+	0.0365529i
$557$		0			0		0.908465	−	0.417960i	$-0.137255\pi$
$557$		0			0		−0.908465	+	0.417960i	$0.862745\pi$
$558$		0			0
$559$	−58.5734	+	33.0202i	−2.47739	+	1.39661i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.552365	−	0.833602i	$-0.313725\pi$
$563$		0			0		−0.552365	+	0.833602i	$0.686275\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	2.07667	−	5.20133i	0.0872118	−	0.218435i
$568$		0			0
$569$		0			0		0.517676	−	0.855577i	$-0.326797\pi$
$569$		0			0		−0.517676	+	0.855577i	$0.673203\pi$
$570$		0			0
$571$	2.43772	+	0.830983i	0.102015	+	0.0347756i	0.373193	−	0.927754i	$-0.378263\pi$
$571$	2.43772	+	0.830983i	0.102015	+	0.0347756i	−0.271178	+	0.962529i	$0.587413\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	2.21444	+	23.8976i	0.0922684	+	0.995734i
$577$	0.111348	+	2.70988i	0.00463548	+	0.112814i	0.999881	+	0.0154173i	$0.00490767\pi$
$577$	0.111348	+	2.70988i	0.00463548	+	0.112814i	−0.995246	+	0.0973964i	$0.968949\pi$
$578$		0			0
$579$	−10.7997	−	6.68691i	−0.448821	−	0.277898i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.982973	−	0.183750i	$-0.941176\pi$
$587$		0			0		0.982973	+	0.183750i	$0.0588235\pi$
$588$	−18.8317	−	13.0426i	−0.776608	−	0.537867i
$589$	−0.969595	−	4.65551i	−0.0399515	−	0.191827i
$590$		0			0
$591$		0			0
$592$	−13.5225	+	7.99344i	−0.555771	+	0.328528i
$593$		0			0		0.193831	−	0.981035i	$-0.437908\pi$
$593$		0			0		−0.193831	+	0.981035i	$0.562092\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	35.5188	−	20.5068i	1.45369	−	0.839288i
$598$		0			0
$599$		0			0		0.844768	−	0.535133i	$-0.179739\pi$
$599$		0			0		−0.844768	+	0.535133i	$0.820261\pi$
$600$		0			0
$601$	−18.9896	+	21.2650i	−0.774600	+	0.867418i	−0.994130	−	0.108188i	$-0.965495\pi$
$601$	−18.9896	+	21.2650i	−0.774600	+	0.867418i	0.219530	+	0.975606i	$0.429548\pi$
$602$		0			0
$603$	1.36121	+	3.12619i	0.0554330	+	0.127308i
$604$	1.00711	+	4.38279i	0.0409786	+	0.178333i
$605$		0			0
$606$		0			0
$607$	8.02531	−	10.4034i	0.325737	−	0.422260i	−0.600252	−	0.799811i	$-0.704933\pi$
$607$	8.02531	−	10.4034i	0.325737	−	0.422260i	0.925989	+	0.377551i	$0.123234\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	44.1421i		−	1.78288i	−0.453135	−	0.891442i	$-0.649695\pi$
$613$		−	44.1421i		−	1.78288i	0.453135	−	0.891442i	$-0.350305\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$617$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$618$		0			0
$619$	1.29261	+	12.5461i	0.0519543	+	0.504271i	0.988227	+	0.152996i	$0.0488922\pi$
$619$	1.29261	+	12.5461i	0.0519543	+	0.504271i	−0.936272	+	0.351275i	$0.885748\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−40.6918	+	9.35045i	−1.62898	+	0.374318i
$625$	22.9214	−	9.98049i	0.916855	−	0.399220i
$626$		0			0
$627$		0			0
$628$	−20.2359	+	34.2330i	−0.807498	+	1.36605i
$629$		0			0
$630$		0			0
$631$	20.0555	+	34.7371i	0.798395	+	1.38286i	0.920661	+	0.390363i	$0.127651\pi$
$631$	20.0555	+	34.7371i	0.798395	+	1.38286i	−0.122266	+	0.992497i	$0.539016\pi$
$632$		0			0
$633$	1.51434	−	49.1515i	0.0601897	−	1.95360i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	18.1286	−	35.4892i	0.718280	−	1.40613i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$641$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$642$		0			0
$643$	43.5083	−	13.3507i	1.71580	−	0.526500i	0.728457	−	0.685092i	$-0.240238\pi$
$643$	43.5083	−	13.3507i	1.71580	−	0.526500i	0.987344	+	0.158592i	$0.0506953\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$647$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	6.23034	−	0.256003i	0.244186	−	0.0100335i
$652$	31.8635	−	2.95259i	1.24787	−	0.115632i
$653$		0			0		−0.964585	−	0.263774i	$-0.915033\pi$
$653$		0			0		0.964585	+	0.263774i	$0.0849673\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−2.35072	+	6.89592i	−0.0917105	+	0.269036i
$658$		0			0
$659$		0			0		−0.855577	−	0.517676i	$-0.826797\pi$
$659$		0			0		0.855577	+	0.517676i	$0.173203\pi$
$660$		0			0
$661$	12.5960	+	5.02904i	0.489927	+	0.195607i	0.602628	−	0.798022i	$-0.294120\pi$
$661$	12.5960	+	5.02904i	0.489927	+	0.195607i	−0.112701	+	0.993629i	$0.535950\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.942984	−	1.67272i	−0.0364579	−	0.0646713i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.89650	−	1.31348i	0.0731045	−	0.0506311i	−0.532249	−	0.846588i	$-0.678653\pi$
$673$	1.89650	−	1.31348i	0.0731045	−	0.0506311i	0.605353	+	0.795957i	$0.293032\pi$
$674$		0			0
$675$	−0.533433	−	25.9753i	−0.0205318	−	0.999789i
$676$	−16.3994	−	43.6571i	−0.630746	−	1.67912i
$677$		0			0		0.964585	−	0.263774i	$-0.0849673\pi$
$677$		0			0		−0.964585	+	0.263774i	$0.915033\pi$
$678$		0			0
$679$	−5.86402	+	8.10609i	−0.225041	+	0.311083i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.0102665	−	0.999947i	$-0.496732\pi$
$683$		0			0		−0.0102665	+	0.999947i	$0.503268\pi$
$684$	−4.48043	−	2.06133i	−0.171314	−	0.0788168i
$685$		0			0
$686$		0			0
$687$	43.9287	−	14.4730i	1.67598	−	0.552180i
$688$	7.29826	+	44.0291i	0.278244	+	1.67859i
$689$		0			0
$690$		0			0
$691$	49.7050	+	12.5013i	1.89087	+	0.475572i	0.999763	+	0.0217706i	$0.00693034\pi$
$691$	49.7050	+	12.5013i	1.89087	+	0.475572i	0.891103	+	0.453801i	$0.149932\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	6.09213	+	1.26880i	0.230261	+	0.0479560i
$701$		0			0		0.725021	−	0.688727i	$-0.241830\pi$
$701$		0			0		−0.725021	+	0.688727i	$0.758170\pi$
$702$		0			0
$703$	−0.0331400	−	3.22782i	−0.00124990	−	0.121739i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	10.0179	−	0.721191i	0.376229	−	0.0270849i	0.117598	−	0.993061i	$-0.462480\pi$
$709$	10.0179	−	0.721191i	0.376229	−	0.0270849i	0.258631	+	0.965976i	$0.416729\pi$
$710$		0			0
$711$	1.54832	+	50.2544i	0.0580665	+	1.88469i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$719$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$720$		0			0
$721$	−5.79114	+	3.75236i	−0.215673	+	0.139745i
$722$		0			0
$723$	−4.19927	+	37.0255i	−0.156172	+	1.37699i
$724$	−42.3081	+	31.2728i	−1.57237	+	1.16224i
$725$		0			0
$726$		0			0
$727$	−7.97784	−	51.3941i	−0.295882	−	1.90610i	−0.409025	−	0.912523i	$-0.634131\pi$
$727$	−7.97784	−	51.3941i	−0.295882	−	1.90610i	0.113144	−	0.993579i	$-0.463908\pi$
$728$		0			0
$729$	−25.1768	−	9.75352i	−0.932472	−	0.361242i
$730$		0			0
$731$		0			0
$732$	1.93796	+	2.17018i	0.0716291	+	0.0802121i
$733$	0.194322	−	1.44731i	0.00717744	−	0.0534574i	−0.987157	−	0.159751i	$-0.948931\pi$
$733$	0.194322	−	1.44731i	0.00717744	−	0.0534574i	0.994335	+	0.106293i	$0.0338983\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−16.4596	−	51.7397i	−0.605476	−	1.90328i	−0.352561	−	0.935789i	$-0.614689\pi$
$739$	−16.4596	−	51.7397i	−0.605476	−	1.90328i	−0.252915	−	0.967489i	$-0.581389\pi$
$740$		0			0
$741$	2.51693	−	8.20239i	0.0924619	−	0.301322i
$742$		0			0
$743$		0			0		−0.133070	−	0.991107i	$-0.542484\pi$
$743$		0			0		0.133070	+	0.991107i	$0.457516\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−34.6061	+	32.2044i	−1.26280	+	1.17515i	−0.286650	+	0.958035i	$0.592542\pi$
$751$	−34.6061	+	32.2044i	−1.26280	+	1.17515i	−0.976145	+	0.217119i	$0.930334\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	3.62729	−	5.35393i	0.131923	−	0.194721i
$757$	38.6474	+	12.7330i	1.40466	+	0.462789i	0.912987	−	0.407989i	$-0.133770\pi$
$757$	38.6474	+	12.7330i	1.40466	+	0.462789i	0.491676	+	0.870778i	$0.336384\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.658380	−	0.752685i	$-0.728758\pi$
$761$		0			0		0.658380	+	0.752685i	$0.271242\pi$
$762$		0			0
$763$	−5.12382	+	5.28411i	−0.185495	+	0.191298i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−3.40558	+	27.5028i	−0.122888	+	0.992421i
$769$	−13.8101	−	39.1902i	−0.498004	−	1.41323i	−0.873688	−	0.486486i	$-0.838278\pi$
$769$	−13.8101	−	39.1902i	−0.498004	−	1.41323i	0.375684	−	0.926748i	$-0.377408\pi$
$770$		0			0
$771$		0			0
$772$	−10.5299	−	10.2105i	−0.378979	−	0.367482i
$773$		0			0		−0.283523	−	0.958965i	$-0.591503\pi$
$773$		0			0		0.283523	+	0.958965i	$0.408497\pi$
$774$		0			0
$775$	26.1533	−	12.3593i	0.939455	−	0.443960i
$776$		0			0
$777$	4.18265	+	0.649267i	0.150052	+	0.0232923i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−18.0197	−	19.3636i	−0.643561	−	0.691556i
$785$		0			0
$786$		0			0
$787$	17.7896	−	43.2617i	0.634132	−	1.54211i	−0.193655	−	0.981070i	$-0.562034\pi$
$787$	17.7896	−	43.2617i	0.634132	−	1.54211i	0.827786	−	0.561043i	$-0.189600\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.29320	+	3.84384i	−0.116945	+	0.136499i
$794$		0			0
$795$		0			0
$796$	45.1298	−	14.3568i	1.59958	−	0.508865i
$797$		0			0		−0.464024	−	0.885823i	$-0.653595\pi$
$797$		0			0		0.464024	+	0.885823i	$0.346405\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.763147	+	3.86250i	0.0269141	+	0.136220i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.163529	−	0.986539i	$-0.447712\pi$
$809$		0			0		−0.163529	+	0.986539i	$0.552288\pi$
$810$		0			0
$811$	−32.7350	−	44.2863i	−1.14948	−	1.55510i	−0.779901	−	0.625903i	$-0.784731\pi$
$811$	−32.7350	−	44.2863i	−1.14948	−	1.55510i	−0.369579	−	0.929199i	$-0.620498\pi$
$812$		0			0
$813$	42.4385	−	2.61751i	1.48838	−	0.0918002i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−8.58545	−	3.22505i	−0.300367	−	0.112830i
$818$		0			0
$819$	10.0710	+	5.01477i	0.351910	+	0.175230i
$820$		0			0
$821$		0			0		−0.436525	−	0.899692i	$-0.643791\pi$
$821$		0			0		0.436525	+	0.899692i	$0.356209\pi$
$822$		0			0
$823$	14.4131	+	36.0997i	0.502408	+	1.25836i	0.934318	+	0.356440i	$0.116009\pi$
$823$	14.4131	+	36.0997i	0.502408	+	1.25836i	−0.431911	+	0.901916i	$0.642160\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.967242	−	0.253857i	$-0.918301\pi$
$827$		0			0		0.967242	+	0.253857i	$0.0816993\pi$
$828$		0			0
$829$	53.9591	+	12.3991i	1.87408	+	0.430638i	0.998613	−	0.0526472i	$-0.0167659\pi$
$829$	53.9591	+	12.3991i	1.87408	+	0.430638i	0.875463	+	0.483286i	$0.160557\pi$
$830$		0			0
$831$	−4.68663	−	2.09816i	−0.162577	−	0.0727844i
$832$	−48.2089	+	0.494961i	−1.67134	+	0.0171597i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.54247	−	30.0218i	−0.0533156	−	1.03771i
$838$		0			0
$839$		0			0		0.427265	−	0.904126i	$-0.359477\pi$
$839$		0			0		−0.427265	+	0.904126i	$0.640523\pi$
$840$		0			0
$841$	24.0087	−	16.2659i	0.827888	−	0.560894i
$842$		0			0
$843$		0			0
$844$	13.8500	−	55.0672i	0.476735	−	1.89549i
$845$		0			0
$846$		0			0
$847$	6.75300	−	1.11938i	0.232036	−	0.0384622i
$848$		0			0
$849$	−6.87516	+	24.1637i	−0.235955	+	0.829296i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−18.1428	+	17.5924i	−0.621197	+	0.602353i	−0.938209	−	0.346068i	$-0.887517\pi$
$853$	−18.1428	+	17.5924i	−0.621197	+	0.602353i	0.317012	+	0.948422i	$0.397320\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$857$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$858$		0			0
$859$	39.0861	−	14.6824i	1.33360	−	0.500955i	0.419985	−	0.907531i	$-0.362035\pi$
$859$	39.0861	−	14.6824i	1.33360	−	0.500955i	0.913617	+	0.406576i	$0.133277\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.912708	−	0.408612i	$-0.133987\pi$
$863$		0			0		−0.912708	+	0.408612i	$0.866013\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.81265	−	29.3890i	0.0615609	−	0.998103i
$868$	7.09086	+	1.25031i	0.240679	+	0.0424383i
$869$		0			0
$870$		0			0
$871$	−6.48308	+	2.20999i	−0.219671	+	0.0748827i
$872$		0			0
$873$	39.3671	+	27.8674i	1.33237	+	0.943169i
$874$		0			0
$875$		0			0
$876$	−4.35503	+	7.19768i	−0.147143	+	0.243187i
$877$	35.9221	+	46.5667i	1.21300	+	1.57244i	0.700222	+	0.713925i	$0.253084\pi$
$877$	35.9221	+	46.5667i	1.21300	+	1.57244i	0.512781	+	0.858519i	$0.328615\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.454905	−	0.890540i	$-0.650327\pi$
$881$		0			0		0.454905	+	0.890540i	$0.349673\pi$
$882$		0			0
$883$	−3.40043	−	36.6965i	−0.114434	−	1.23493i	−0.838383	−	0.545081i	$-0.816499\pi$
$883$	−3.40043	−	36.6965i	−0.114434	−	1.23493i	0.723950	−	0.689853i	$-0.242325\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$887$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$888$		0			0
$889$	0.432055	−	0.305845i	0.0144907	−	0.0102577i
$890$		0			0
$891$		0			0
$892$	−0.650445	−	2.11972i	−0.0217785	−	0.0709736i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	5.81494	−	29.4310i	0.193831	−	0.981035i
$901$		0			0
$902$		0			0
$903$	6.01293	−	10.4147i	0.200098	−	0.346580i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	8.08965	+	4.78197i	0.268612	+	0.158783i	0.636940	−	0.770914i	$-0.280200\pi$
$907$	8.08965	+	4.78197i	0.268612	+	0.158783i	−0.368327	+	0.929696i	$0.620069\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.223951	−	0.974601i	$-0.571895\pi$
$911$		0			0		0.223951	+	0.974601i	$0.428105\pi$
$912$	−4.54457	−	3.43190i	−0.150486	−	0.113642i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	53.1254	−	5.47344i	1.75531	−	0.180847i
$917$		0			0
$918$		0			0
$919$	−8.37814	+	1.03744i	−0.276369	+	0.0342219i	−0.259876	−	0.965642i	$-0.583682\pi$
$919$	−8.37814	+	1.03744i	−0.276369	+	0.0342219i	−0.0164935	+	0.999864i	$0.505250\pi$
$920$		0			0
$921$	25.0926	+	17.0694i	0.826828	+	0.562455i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	19.0424	−	4.78936i	0.626111	−	0.157473i
$926$		0			0
$927$	18.3756	+	27.7315i	0.603533	+	0.910823i
$928$		0			0
$929$		0			0		−0.361242	−	0.932472i	$-0.617647\pi$
$929$		0			0		0.361242	+	0.932472i	$0.382353\pi$
$930$		0			0
$931$	5.29749	−	1.21729i	0.173618	−	0.0398952i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	47.5910	−	25.5558i	1.55473	−	0.834872i	0.554734	−	0.832028i	$-0.312820\pi$
$937$	47.5910	−	25.5558i	1.55473	−	0.834872i	0.999996	−	0.00284454i	$-0.000905447\pi$
$938$		0			0
$939$	−51.1443	−	29.5282i	−1.66903	−	0.963615i
$940$		0			0
$941$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$941$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.569364	−	0.822086i	$-0.307190\pi$
$947$		0			0		−0.569364	+	0.822086i	$0.692810\pi$
$948$	−10.6678	+	57.0677i	−0.346474	+	1.85347i
$949$	−13.4185	−	5.84271i	−0.435582	−	0.189663i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.577774	−	0.816197i	$-0.696078\pi$
$953$		0			0		0.577774	+	0.816197i	$0.303922\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.19959	−	1.12359i	0.0709545	−	0.0362449i
$962$		0			0
$963$		0			0
$964$	−13.8830	+	40.7262i	−0.447141	+	1.31170i
$965$		0			0
$966$		0			0
$967$	55.8518	+	20.3284i	1.79607	+	0.653717i	0.998740	+	0.0501754i	$0.0159780\pi$
$967$	55.8518	+	20.3284i	1.79607	+	0.653717i	0.797332	+	0.603541i	$0.206244\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.322654	−	0.946517i	$-0.604575\pi$
$971$		0			0		0.322654	+	0.946517i	$0.395425\pi$
$972$	−25.9892	−	17.2210i	−0.833602	−	0.552365i
$973$	0.581373	+	0.304543i	0.0186380	+	0.00976319i
$974$		0			0
$975$	52.0914	+	3.21289i	1.66826	+	0.102895i
$976$	1.64986	+	2.92662i	0.0528107	+	0.0936789i
$977$		0			0		0.283523	−	0.958965i	$-0.408497\pi$
$977$		0			0		−0.283523	+	0.958965i	$0.591503\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	25.7266	+	24.4388i	0.821389	+	0.780270i
$982$		0			0
$983$		0			0		−0.351649	−	0.936132i	$-0.614379\pi$
$983$		0			0		0.351649	+	0.936132i	$0.385621\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	4.86524	−	8.63027i	0.154784	−	0.274566i
$989$		0			0
$990$		0			0
$991$	52.3546	+	24.0869i	1.66310	+	0.765146i	0.999860	+	0.0167380i	$0.00532813\pi$
$991$	52.3546	+	24.0869i	1.66310	+	0.765146i	0.663238	+	0.748408i	$0.269182\pi$
$992$		0			0
$993$	−41.4426	−	11.7914i	−1.31514	−	0.374190i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−9.74819	−	44.5116i	−0.308728	−	1.40970i	−0.830694	−	0.556730i	$-0.812056\pi$
$997$	−9.74819	−	44.5116i	−0.308728	−	1.40970i	0.521965	−	0.852967i	$-0.325199\pi$
$998$		0			0
$999$	2.92290	−	20.1954i	0.0924763	−	0.638953i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	921.2.x.a.116.1	✓	96
3.2	odd	2	CM	921.2.x.a.116.1	✓	96
307.45	odd	306	inner	921.2.x.a.659.1	yes	96
921.659	even	306	inner	921.2.x.a.659.1	yes	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
921.2.x.a.116.1	✓	96	1.1	even	1	trivial
921.2.x.a.116.1	✓	96	3.2	odd	2	CM
921.2.x.a.659.1	yes	96	307.45	odd	306	inner
921.2.x.a.659.1	yes	96	921.659	even	306	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:	\( N \)	\(=\)	\( 921 = 3 \cdot 307 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	921.x (of order \(306\), degree \(96\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.55047 + 0.772041i) q^{3} +(-1.93959 + 0.487827i) q^{4} +(0.492718 + 0.380089i) q^{7} +(1.80790 - 2.39405i) q^{9} +O(q^{10})\)	\(q+(-1.55047 + 0.772041i) q^{3} +(-1.93959 + 0.487827i) q^{4} +(0.492718 + 0.380089i) q^{7} +(1.80790 - 2.39405i) q^{9} +(2.63066 - 2.25381i) q^{12} +(4.49503 + 4.01404i) q^{13} +(3.52405 - 1.89237i) q^{16} +(-0.0253129 + 0.821588i) q^{19} +(-1.05739 - 0.208917i) q^{21} +(-4.89497 + 1.01947i) q^{25} +(-0.954791 + 5.10768i) q^{27} +(-1.14109 - 0.496857i) q^{28} +(-5.62478 + 1.35344i) q^{31} +(-2.33872 + 5.52543i) q^{36} +(-3.92378 + 0.161227i) q^{37} +(-10.0684 - 2.75329i) q^{39} +(-3.60001 + 10.5607i) q^{43} +(-4.00293 + 5.65478i) q^{48} +(-1.67870 - 6.39614i) q^{49} +(-10.6767 - 5.59281i) q^{52} +(-0.595054 - 1.29339i) q^{57} +(0.0172449 + 0.839732i) q^{61} +(1.80074 - 0.492428i) q^{63} +(-5.91207 + 5.38957i) q^{64} +(-0.558146 + 0.990075i) q^{67} +(-2.30656 + 0.759934i) q^{73} +(6.80242 - 5.35977i) q^{75} +(-0.351697 - 1.60590i) q^{76} +(-13.0569 + 10.5069i) q^{79} +(-2.46297 - 8.65643i) q^{81} +(2.15282 - 0.110608i) q^{84} +(0.689090 + 3.68630i) q^{91} +(7.67613 - 6.44104i) q^{93} +(0.495106 + 16.0698i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6 q^{4} + 18 q^{9}+O(q^{10}) \) 96 * q - 6 * q^4 + 18 * q^9	\( 96 q - 6 q^{4} + 18 q^{9} + 18 q^{12} + 12 q^{16} + 21 q^{31} + 18 q^{36} - 39 q^{43} + 36 q^{48} + 39 q^{61} + 48 q^{64} - 48 q^{67} + 51 q^{73} - 54 q^{81} + 45 q^{93}+O(q^{100}) \) 96 * q - 6 * q^4 + 18 * q^9 + 18 * q^12 + 12 * q^16 + 21 * q^31 + 18 * q^36 - 39 * q^43 + 36 * q^48 + 39 * q^61 + 48 * q^64 - 48 * q^67 + 51 * q^73 - 54 * q^81 + 45 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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