LMFDB - Embedded newform 921.2.x.a.92.1

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			92.1
Character	$\chi$	$=$	921.92
Dual form			921.2.x.a.911.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.625689 + 1.61509i) q^{3} +(1.81693 - 0.835921i) q^{4} +(0.914150 + 2.04192i) q^{7} +(-2.21703 - 2.02109i) q^{9} +O(q^{10})$	$q+(-0.625689 + 1.61509i) q^{3} +(1.81693 - 0.835921i) q^{4} +(0.914150 + 2.04192i) q^{7} +(-2.21703 - 2.02109i) q^{9} +(0.213253 + 3.45753i) q^{12} +(5.24748 - 4.31213i) q^{13} +(2.60247 - 3.03762i) q^{16} +(1.88064 + 5.33686i) q^{19} +(-3.86986 + 0.198827i) q^{21} +(1.75825 + 4.68066i) q^{25} +(4.65140 - 2.31612i) q^{27} +(3.36783 + 2.94587i) q^{28} +(0.0122284 + 0.595457i) q^{31} +(-5.71765 - 1.81892i) q^{36} +(-0.429102 - 5.96054i) q^{37} +(3.68119 + 11.1732i) q^{39} +(-6.79379 + 10.4851i) q^{43} +(3.27769 + 6.10383i) q^{48} +(1.32874 - 1.48796i) q^{49} +(5.92970 - 12.2213i) q^{52} +(-9.79620 - 0.301818i) q^{57} +(10.7550 + 10.0086i) q^{61} +(2.10020 - 6.37457i) q^{63} +(2.18930 - 7.69461i) q^{64} +(-11.0530 - 1.25359i) q^{67} +(-2.75247 - 16.6052i) q^{73} +(-8.65980 - 0.0889102i) q^{75} +(7.87818 + 8.12464i) q^{76} +(6.92546 + 16.3620i) q^{79} +(0.830415 + 8.96161i) q^{81} +(-6.86506 + 3.59615i) q^{84} +(13.6020 + 6.77299i) q^{91} +(-0.969368 - 0.352821i) q^{93} +(-5.76773 + 16.3676i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/921\mathbb{Z}\right)^\times$.

$n$	$308$	$619$
$\chi(n)$	$-1$	$e\left(\frac{59}{306}\right)$

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		0.976848	−	0.213933i	$-0.0686275\pi$
$2$		0			0		−0.976848	+	0.213933i	$0.931373\pi$
$3$	−0.625689	+	1.61509i	−0.361242	+	0.932472i
$3$	−0.625689	+	1.61509i	−0.361242	+	0.932472i
$4$	1.81693	−	0.835921i	0.908465	−	0.417960i
$5$		0			0		−0.822086	−	0.569364i	$-0.807190\pi$
$5$		0			0		0.822086	+	0.569364i	$0.192810\pi$
$6$		0			0
$7$	0.914150	+	2.04192i	0.345516	+	0.771773i	0.999878	+	0.0155920i	$0.00496330\pi$
$7$	0.914150	+	2.04192i	0.345516	+	0.771773i	−0.654362	+	0.756181i	$0.727063\pi$
$8$		0			0
$9$	−2.21703	−	2.02109i	−0.739009	−	0.673696i
$10$		0			0
$11$		0			0		0.752685	−	0.658380i	$-0.228758\pi$
$11$		0			0		−0.752685	+	0.658380i	$0.771242\pi$
$12$	0.213253	+	3.45753i	0.0615609	+	0.998103i
$13$	5.24748	−	4.31213i	1.45539	−	1.19597i	0.514390	−	0.857556i	$-0.328018\pi$
$13$	5.24748	−	4.31213i	1.45539	−	1.19597i	0.940998	−	0.338413i	$-0.109890\pi$
$14$		0			0
$15$		0			0
$16$	2.60247	−	3.03762i	0.650618	−	0.759405i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	1.88064	+	5.33686i	0.431448	+	1.22436i	0.933203	+	0.359350i	$0.117002\pi$
$19$	1.88064	+	5.33686i	0.431448	+	1.22436i	−0.501755	+	0.865010i	$0.667312\pi$
$20$		0			0
$21$	−3.86986	+	0.198827i	−0.844472	+	0.0433876i
$22$		0			0
$23$		0			0		0.0410550	−	0.999157i	$-0.486928\pi$
$23$		0			0		−0.0410550	+	0.999157i	$0.513072\pi$
$24$		0			0
$25$	1.75825	+	4.68066i	0.351649	+	0.936132i
$26$		0			0
$27$	4.65140	−	2.31612i	0.895163	−	0.445738i
$28$	3.36783	+	2.94587i	0.636460	+	0.556717i
$29$		0			0		0.610796	−	0.791788i	$-0.290850\pi$
$29$		0			0		−0.610796	+	0.791788i	$0.709150\pi$
$30$		0			0
$31$	0.0122284	+	0.595457i	0.00219629	+	0.106947i	0.999456	+	0.0329712i	$0.0104970\pi$
$31$	0.0122284	+	0.595457i	0.00219629	+	0.106947i	−0.997260	+	0.0739761i	$0.976431\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.71765	−	1.81892i	−0.952942	−	0.303153i
$37$	−0.429102	−	5.96054i	−0.0705440	−	0.979907i	−0.904058	−	0.427409i	$-0.859426\pi$
$37$	−0.429102	−	5.96054i	−0.0705440	−	0.979907i	0.833514	−	0.552498i	$-0.186325\pi$
$38$		0			0
$39$	3.68119	+	11.1732i	0.589462	+	1.78914i
$40$		0			0
$41$		0			0		−0.827888	−	0.560894i	$-0.810458\pi$
$41$		0			0		0.827888	+	0.560894i	$0.189542\pi$
$42$		0			0
$43$	−6.79379	+	10.4851i	−1.03604	+	1.59896i	−0.262798	+	0.964851i	$0.584645\pi$
$43$	−6.79379	+	10.4851i	−1.03604	+	1.59896i	−0.773245	+	0.634107i	$0.781368\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.491083	−	0.871113i	$-0.663399\pi$
$47$		0			0		0.491083	+	0.871113i	$0.336601\pi$
$48$	3.27769	+	6.10383i	0.473094	+	0.881012i
$49$	1.32874	−	1.48796i	0.189821	−	0.212566i
$50$		0			0
$51$		0			0
$52$	5.92970	−	12.2213i	0.822301	−	1.69479i
$53$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$53$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−9.79620	−	0.301818i	−1.29754	−	0.0399767i
$58$		0			0
$59$		0			0		0.785476	−	0.618892i	$-0.212418\pi$
$59$		0			0		−0.785476	+	0.618892i	$0.787582\pi$
$60$		0			0
$61$	10.7550	+	10.0086i	1.37704	+	1.28147i	0.919943	+	0.392052i	$0.128235\pi$
$61$	10.7550	+	10.0086i	1.37704	+	1.28147i	0.457097	+	0.889417i	$0.348889\pi$
$62$		0			0
$63$	2.10020	−	6.37457i	0.264601	−	0.803120i
$64$	2.18930	−	7.69461i	0.273663	−	0.961826i
$65$		0			0
$66$		0			0
$67$	−11.0530	−	1.25359i	−1.35034	−	0.153150i	−0.592491	−	0.805577i	$-0.701856\pi$
$67$	−11.0530	−	1.25359i	−1.35034	−	0.153150i	−0.757852	+	0.652427i	$0.773751\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.890540	−	0.454905i	$-0.849673\pi$
$71$		0			0		0.890540	+	0.454905i	$0.150327\pi$
$72$		0			0
$73$	−2.75247	−	16.6052i	−0.322152	−	1.94349i	−0.331287	−	0.943530i	$-0.607483\pi$
$73$	−2.75247	−	16.6052i	−0.322152	−	1.94349i	0.00913507	−	0.999958i	$-0.497092\pi$
$74$		0			0
$75$	−8.65980	−	0.0889102i	−0.999947	−	0.0102665i
$76$	7.87818	+	8.12464i	0.903689	+	0.931960i
$77$		0			0
$78$		0			0
$79$	6.92546	+	16.3620i	0.779175	+	1.84087i	0.418955	+	0.908007i	$0.362397\pi$
$79$	6.92546	+	16.3620i	0.779175	+	1.84087i	0.360220	+	0.932867i	$0.382701\pi$
$80$		0			0
$81$	0.830415	+	8.96161i	0.0922684	+	0.995734i
$82$		0			0
$83$		0			0		0.0718047	−	0.997419i	$-0.477124\pi$
$83$		0			0		−0.0718047	+	0.997419i	$0.522876\pi$
$84$	−6.86506	+	3.59615i	−0.749039	+	0.392372i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.916855	−	0.399220i	$-0.130719\pi$
$89$		0			0		−0.916855	+	0.399220i	$0.869281\pi$
$90$		0			0
$91$	13.6020	+	6.77299i	1.42588	+	0.710003i
$92$		0			0
$93$	−0.969368	−	0.352821i	−0.100519	−	0.0365858i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−5.76773	+	16.3676i	−0.585624	+	1.66188i	0.153370	+	0.988169i	$0.450987\pi$
$97$	−5.76773	+	16.3676i	−0.585624	+	1.66188i	−0.738994	+	0.673711i	$0.764699\pi$
$98$		0			0
$99$		0			0
$100$	7.10727	+	7.03468i	0.710727	+	0.703468i
$101$		0			0		0.696134	−	0.717912i	$-0.254902\pi$
$101$		0			0		−0.696134	+	0.717912i	$0.745098\pi$
$102$		0			0
$103$	0.426362	+	0.643445i	0.0420107	+	0.0634005i	0.854005	−	0.520265i	$-0.174167\pi$
$103$	0.426362	+	0.643445i	0.0420107	+	0.0634005i	−0.811994	+	0.583666i	$0.801618\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.535133	−	0.844768i	$-0.320261\pi$
$107$		0			0		−0.535133	+	0.844768i	$0.679739\pi$
$108$	6.51518	−	8.09644i	0.626924	−	0.779081i
$109$	−1.74218	−	8.81768i	−0.166871	−	0.844580i	−0.968930	−	0.247335i	$-0.920445\pi$
$109$	−1.74218	−	8.81768i	−0.166871	−	0.844580i	0.802059	−	0.597245i	$-0.203738\pi$
$110$		0			0
$111$	9.89529	+	3.03641i	0.939219	+	0.288203i
$112$	8.58163	+	2.53720i	0.810888	+	0.239743i
$113$		0			0		−0.992421	−	0.122888i	$-0.960784\pi$
$113$		0			0		0.992421	+	0.122888i	$0.0392157\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−20.3490	−	1.04550i	−1.88126	−	0.0966562i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.46378	−	10.9022i	0.133070	−	0.991107i
$122$		0			0
$123$		0			0
$124$	0.519973	+	1.07168i	0.0466950	+	0.0962399i
$125$		0			0
$126$		0			0
$127$	−1.77010	+	1.36547i	−0.157071	+	0.121166i	−0.686319	−	0.727301i	$-0.740775\pi$
$127$	−1.77010	+	1.36547i	−0.157071	+	0.121166i	0.529248	+	0.848467i	$0.322474\pi$
$128$		0			0
$129$	−12.6835	−	17.5330i	−1.11672	−	1.54369i
$130$		0			0
$131$		0			0		0.999947	−	0.0102665i	$-0.00326797\pi$
$131$		0			0		−0.999947	+	0.0102665i	$0.996732\pi$
$132$		0			0
$133$	−9.17826	+	8.71880i	−0.795856	+	0.756016i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.991107	−	0.133070i	$-0.0424837\pi$
$137$		0			0		−0.991107	+	0.133070i	$0.957516\pi$
$138$		0			0
$139$	−12.9965	−	15.4886i	−1.10235	−	1.31373i	−0.945324	−	0.326132i	$-0.894255\pi$
$139$	−12.9965	−	15.4886i	−1.10235	−	1.31373i	−0.157024	−	0.987595i	$-0.550190\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−11.9090	+	1.47466i	−0.992421	+	0.122888i
$145$		0			0
$146$		0			0
$147$	1.57181	+	3.07704i	0.129641	+	0.253790i
$148$	−5.76219	−	10.4712i	−0.473649	−	0.860727i
$149$		0			0		0.816197	−	0.577774i	$-0.196078\pi$
$149$		0			0		−0.816197	+	0.577774i	$0.803922\pi$
$150$		0			0
$151$	16.3503	−	15.2155i	1.33057	−	1.23822i	0.381033	−	0.924561i	$-0.375568\pi$
$151$	16.3503	−	15.2155i	1.33057	−	1.23822i	0.949535	−	0.313662i	$-0.101556\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	16.0284	+	17.2237i	1.28330	+	1.37900i
$157$	−5.09117	−	24.4453i	−0.406320	−	1.95095i	−0.286106	−	0.958198i	$-0.592361\pi$
$157$	−5.09117	−	24.4453i	−0.406320	−	1.95095i	−0.120213	−	0.992748i	$-0.538358\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.764137	−	6.17102i	−0.0598519	−	0.483351i	−0.992470	−	0.122487i	$-0.960913\pi$
$163$	−0.764137	−	6.17102i	−0.0598519	−	0.483351i	0.932618	−	0.360864i	$-0.117518\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.969797	−	0.243914i	$-0.0784314\pi$
$167$		0			0		−0.969797	+	0.243914i	$0.921569\pi$
$168$		0			0
$169$	6.42173	−	32.5021i	0.493979	−	2.50016i
$170$		0			0
$171$	6.61684	−	15.6329i	0.506002	−	1.19548i
$172$	−3.57916	+	24.7297i	−0.272908	+	1.88562i
$173$		0			0		−0.102486	−	0.994734i	$-0.532680\pi$
$173$		0			0		0.102486	+	0.994734i	$0.467320\pi$
$174$		0			0
$175$	−7.95023	+	7.86902i	−0.600981	+	0.594842i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.998103	−	0.0615609i	$-0.980392\pi$
$179$		0			0		0.998103	+	0.0615609i	$0.0196078\pi$
$180$		0			0
$181$	−26.1242	−	2.69154i	−1.94180	−	0.200061i	−0.948599	−	0.316482i	$-0.897498\pi$
$181$	−26.1242	−	2.69154i	−1.94180	−	0.200061i	−0.993200	+	0.116421i	$0.962858\pi$
$182$		0			0
$183$	−22.8941	+	11.1081i	−1.69238	+	0.821131i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	8.98142	+	7.38051i	0.653302	+	0.536853i
$190$		0			0
$191$		0			0		−0.974601	−	0.223951i	$-0.928105\pi$
$191$		0			0		0.974601	+	0.223951i	$0.0718954\pi$
$192$	11.0577	+	8.35035i	0.798017	+	0.602635i
$193$	0.105730	+	0.565605i	0.00761061	+	0.0407131i	0.986568	−	0.163354i	$-0.0522311\pi$
$193$	0.105730	+	0.565605i	0.00761061	+	0.0407131i	−0.978957	+	0.204067i	$0.934584\pi$
$194$		0			0
$195$		0			0
$196$	1.17042	−	3.81425i	0.0836012	−	0.272446i
$197$		0			0		−0.322654	−	0.946517i	$-0.604575\pi$
$197$		0			0		0.322654	+	0.946517i	$0.395425\pi$
$198$		0			0
$199$	−17.4025	−	14.0038i	−1.23363	−	0.992700i	−0.999791	−	0.0204246i	$-0.993498\pi$
$199$	−17.4025	−	14.0038i	−1.23363	−	0.992700i	−0.233840	−	0.972275i	$-0.575129\pi$
$200$		0			0
$201$	8.94042	−	17.0673i	0.630608	−	1.20383i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.557804	−	27.1620i	0.0386768	−	1.88335i
$209$		0			0
$210$		0			0
$211$	1.40098	−	1.20029i	0.0964474	−	0.0826310i	−0.601081	−	0.799188i	$-0.705263\pi$
$211$	1.40098	−	1.20029i	0.0964474	−	0.0826310i	0.697529	+	0.716557i	$0.254283\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.20470	+	0.569307i	−0.0817802	+	0.0386470i
$218$		0			0
$219$	28.5410	+	5.94418i	1.92862	+	0.401671i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−13.6063	−	25.9746i	−0.911148	−	1.73938i	−0.623851	−	0.781543i	$-0.714433\pi$
$223$	−13.6063	−	25.9746i	−0.911148	−	1.73938i	−0.287297	−	0.957841i	$-0.592757\pi$
$224$		0			0
$225$	5.56194	−	13.9307i	0.370796	−	0.928714i
$226$		0			0
$227$		0			0		−0.253857	−	0.967242i	$-0.581699\pi$
$227$		0			0		0.253857	+	0.967242i	$0.418301\pi$
$228$	−18.0513	+	7.64046i	−1.19548	+	0.506002i
$229$	22.9201	−	13.5485i	1.51460	−	0.895313i	0.515336	−	0.856989i	$-0.327667\pi$
$229$	22.9201	−	13.5485i	1.51460	−	0.895313i	0.999265	−	0.0383242i	$-0.0122020\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.958965	−	0.283523i	$-0.0915033\pi$
$233$		0			0		−0.958965	+	0.283523i	$0.908497\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−30.7594	+	0.947686i	−1.99803	+	0.0615588i
$238$		0			0
$239$		0			0		0.577774	−	0.816197i	$-0.303922\pi$
$239$		0			0		−0.577774	+	0.816197i	$0.696078\pi$
$240$		0			0
$241$	−25.7602	+	16.3183i	−1.65936	+	1.05115i	−0.728993	+	0.684521i	$0.760011\pi$
$241$	−25.7602	+	16.3183i	−1.65936	+	1.05115i	−0.930367	+	0.366630i	$0.880511\pi$
$242$		0			0
$243$	−14.9934	−	4.26598i	−0.961826	−	0.273663i
$244$	27.9075	+	9.19458i	1.78660	+	0.588623i
$245$		0			0
$246$		0			0
$247$	32.8818	+	19.8955i	2.09222	+	1.26592i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.482114	−	0.876109i	$-0.339869\pi$
$251$		0			0		−0.482114	+	0.876109i	$0.660131\pi$
$252$	−1.51271	−	13.3378i	−0.0952917	−	0.840199i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−2.45427	−	15.8106i	−0.153392	−	0.988165i
$257$		0			0		0.839229	−	0.543778i	$-0.183007\pi$
$257$		0			0		−0.839229	+	0.543778i	$0.816993\pi$
$258$		0			0
$259$	11.7787	−	6.32502i	0.731892	−	0.393018i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.912708	−	0.408612i	$-0.866013\pi$
$263$		0			0		0.912708	+	0.408612i	$0.133987\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−21.1305	+	6.96178i	−1.29075	+	0.425258i
$269$		0			0		0.850217	−	0.526432i	$-0.176471\pi$
$269$		0			0		−0.850217	+	0.526432i	$0.823529\pi$
$270$		0			0
$271$	−4.90519	+	15.4192i	−0.297969	+	0.936648i	0.680938	+	0.732341i	$0.261572\pi$
$271$	−4.90519	+	15.4192i	−0.297969	+	0.936648i	−0.978907	+	0.204307i	$0.934506\pi$
$272$		0			0
$273$	−19.4496	+	17.7307i	−1.17714	+	1.07311i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	17.7825	+	16.8923i	1.06844	+	1.01496i	0.999850	+	0.0173139i	$0.00551146\pi$
$277$	17.7825	+	16.8923i	1.06844	+	1.01496i	0.0685941	+	0.997645i	$0.478149\pi$
$278$		0			0
$279$	1.17636	−	1.34486i	0.0704268	−	0.0805146i
$280$		0			0
$281$		0			0		−0.785476	−	0.618892i	$-0.787582\pi$
$281$		0			0		0.785476	+	0.618892i	$0.212418\pi$
$282$		0			0
$283$	−31.5078	+	8.96474i	−1.87294	+	0.532898i	−0.874051	+	0.485834i	$0.838516\pi$
$283$	−31.5078	+	8.96474i	−1.87294	+	0.532898i	−0.998892	+	0.0470643i	$0.985013\pi$
$284$		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
$291$	−22.8264	−	19.5564i	−1.33811	−	1.14642i
$292$	−18.8817	−	27.8696i	−1.10497	−	1.63094i
$293$		0			0		0.804162	−	0.594410i	$-0.202614\pi$
$293$		0			0		−0.804162	+	0.594410i	$0.797386\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−15.8086	+	7.07736i	−0.912708	+	0.408612i
$301$	−27.6202	−	4.28745i	−1.59200	−	0.247125i
$302$		0			0
$303$		0			0
$304$	21.1057	+	8.17638i	1.21049	+	0.468947i
$305$		0			0
$306$		0			0
$307$	−16.6311	−	5.51418i	−0.949187	−	0.314711i
$307$	−16.6311	−	5.51418i	−0.949187	−	0.314711i
$308$		0			0
$309$	−1.30599	+	0.286016i	−0.0742952	+	0.0162709i
$310$		0			0
$311$		0			0		0.908465	−	0.417960i	$-0.137255\pi$
$311$		0			0		−0.908465	+	0.417960i	$0.862745\pi$
$312$		0			0
$313$	−1.92714	+	12.4148i	−0.108928	+	0.701728i	0.869314	+	0.494261i	$0.164561\pi$
$313$	−1.92714	+	12.4148i	−0.108928	+	0.701728i	−0.978242	+	0.207467i	$0.933478\pi$
$314$		0			0
$315$		0			0
$316$	26.2605	+	23.9396i	1.47727	+	1.34671i
$317$		0			0		−0.924859	−	0.380311i	$-0.875817\pi$
$317$		0			0		0.924859	+	0.380311i	$0.124183\pi$
$318$		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
$325$	29.4100	+	16.9799i	1.63137	+	0.941873i
$326$		0			0
$327$	15.3314	+	2.70334i	0.847829	+	0.149495i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.29746	+	8.07473i	0.126280	+	0.443827i	0.999015	−	0.0443660i	$-0.0141268\pi$
$331$	2.29746	+	8.07473i	0.126280	+	0.443827i	−0.872736	+	0.488193i	$0.837656\pi$
$332$		0			0
$333$	−11.0954	+	14.0819i	−0.608026	+	0.771685i
$334$		0			0
$335$		0			0
$336$	−9.46724	+	12.2726i	−0.516480	+	0.669525i
$337$	−25.2844	+	26.6168i	−1.37733	+	1.44991i	−0.678944	+	0.734190i	$0.737562\pi$
$337$	−25.2844	+	26.6168i	−1.37733	+	1.44991i	−0.698386	+	0.715721i	$0.746098\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	19.1765	+	6.10048i	1.03543	+	0.329395i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.999157	−	0.0410550i	$-0.0130719\pi$
$347$		0			0		−0.999157	+	0.0410550i	$0.986928\pi$
$348$		0			0
$349$	17.2422	−	4.71504i	0.922956	−	0.252390i	0.229850	−	0.973226i	$-0.426176\pi$
$349$	17.2422	−	4.71504i	0.922956	−	0.252390i	0.693106	+	0.720836i	$0.256242\pi$
$350$		0			0
$351$	14.4207	−	32.2113i	0.769720	−	1.71931i
$352$		0			0
$353$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$353$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.436525	−	0.899692i	$-0.356209\pi$
$359$		0			0		−0.436525	+	0.899692i	$0.643791\pi$
$360$		0			0
$361$	−10.1428	+	8.16184i	−0.533829	+	0.429571i
$362$		0			0
$363$	16.6921	+	9.18550i	0.876109	+	0.482114i
$364$	30.3756	+	0.935862i	1.59211	+	0.0490525i
$365$		0			0
$366$		0			0
$367$	11.4910	−	18.9914i	0.599823	−	0.991344i	−0.397456	−	0.917621i	$-0.630107\pi$
$367$	11.4910	−	18.9914i	0.599823	−	0.991344i	0.997279	−	0.0737229i	$-0.0234880\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−2.05620	+	0.169263i	−0.106609	+	0.00877589i
$373$	−8.74826	−	13.8101i	−0.452968	−	0.715061i	0.538895	−	0.842373i	$-0.318842\pi$
$373$	−8.74826	−	13.8101i	−0.452968	−	0.715061i	−0.991863	+	0.127313i	$0.959365\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	29.7394	+	2.75576i	1.52761	+	0.141554i	0.823000	−	0.568042i	$-0.192299\pi$
$379$	29.7394	+	2.75576i	1.52761	+	0.141554i	0.704609	+	0.709596i	$0.251122\pi$
$380$		0			0
$381$	−1.09783	−	3.71322i	−0.0562438	−	0.190234i
$382$		0			0
$383$		0			0		−0.696134	−	0.717912i	$-0.745098\pi$
$383$		0			0		0.696134	+	0.717912i	$0.254902\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	36.2532	−	9.51484i	1.84286	−	0.483666i
$388$	3.20247	+	34.5602i	0.162581	+	1.75453i
$389$		0			0		−0.928714	−	0.370796i	$-0.879085\pi$
$389$		0			0		0.928714	+	0.370796i	$0.120915\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	9.90072	+	20.9507i	0.496903	+	1.05149i	0.984005	+	0.178141i	$0.0570082\pi$
$397$	9.90072	+	20.9507i	0.496903	+	1.05149i	−0.487102	+	0.873345i	$0.661946\pi$
$398$		0			0
$399$	−8.33891	−	20.2790i	−0.417468	−	1.01522i
$400$	18.7939	+	6.84040i	0.939693	+	0.342020i
$401$		0			0		−0.666073	−	0.745886i	$-0.732026\pi$
$401$		0			0		0.666073	+	0.745886i	$0.267974\pi$
$402$		0			0
$403$	2.63186	+	3.07192i	0.131102	+	0.153023i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	20.8144	−	8.06355i	1.02921	−	0.398717i	0.213353	−	0.976975i	$-0.431561\pi$
$409$	20.8144	−	8.06355i	1.02921	−	0.398717i	0.815854	+	0.578258i	$0.196267\pi$
$410$		0			0
$411$		0			0
$412$	1.31254	+	0.812690i	0.0646642	+	0.0400384i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	33.1473	−	11.2994i	1.62323	−	0.553336i
$418$		0			0
$419$		0			0		−0.958965	−	0.283523i	$-0.908497\pi$
$419$		0			0		0.958965	+	0.283523i	$0.0915033\pi$
$420$		0			0
$421$	−22.9286	+	4.28611i	−1.11747	+	0.208892i	−0.709915	−	0.704287i	$-0.751267\pi$
$421$	−22.9286	+	4.28611i	−1.11747	+	0.208892i	−0.407559	+	0.913179i	$0.633620\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−10.6051	+	31.1103i	−0.513214	+	1.50553i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.436525	−	0.899692i	$-0.643791\pi$
$431$		0			0		0.436525	+	0.899692i	$0.356209\pi$
$432$	5.06965	−	20.1569i	0.243914	−	0.969797i
$433$	3.88718	−	37.7291i	0.186806	−	1.81315i	−0.316013	−	0.948755i	$-0.602344\pi$
$433$	3.88718	−	37.7291i	0.186806	−	1.81315i	0.502819	−	0.864392i	$-0.332296\pi$
$434$		0			0
$435$		0			0
$436$	−10.5363	−	14.5648i	−0.504598	−	0.697527i
$437$		0			0
$438$		0			0
$439$	−27.6251	−	27.9102i	−1.31847	−	1.33208i	−0.907278	−	0.420532i	$-0.861843\pi$
$439$	−27.6251	−	27.9102i	−1.31847	−	1.33208i	−0.411196	−	0.911547i	$-0.634889\pi$
$440$		0			0
$441$	−5.95316	+	0.613347i	−0.283484	+	0.0292070i
$442$		0			0
$443$		0			0		−0.920906	−	0.389786i	$-0.872549\pi$
$443$		0			0		0.920906	+	0.389786i	$0.127451\pi$
$444$	20.5172	−	2.75474i	0.973705	−	0.130734i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	17.7131	−	2.56364i	0.836866	−	0.121121i
$449$		0			0		−0.967242	−	0.253857i	$-0.918301\pi$
$449$		0			0		0.967242	+	0.253857i	$0.0816993\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	14.3443	+	35.9274i	0.673952	+	1.68802i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−40.5933	+	8.45429i	−1.89888	+	0.395475i	−0.999159	−	0.0409992i	$-0.986946\pi$
$457$	−40.5933	+	8.45429i	−1.89888	+	0.395475i	−0.899717	+	0.436474i	$0.856227\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.976848	−	0.213933i	$-0.931373\pi$
$461$		0			0		0.976848	+	0.213933i	$0.0686275\pi$
$462$		0			0
$463$	−2.12875	−	2.28751i	−0.0989316	−	0.106310i	0.680655	−	0.732604i	$-0.261695\pi$
$463$	−2.12875	−	2.28751i	−0.0989316	−	0.106310i	−0.779587	+	0.626294i	$0.784571\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.890540	−	0.454905i	$-0.150327\pi$
$467$		0			0		−0.890540	+	0.454905i	$0.849673\pi$
$468$	−37.8466	+	15.1105i	−1.74946	+	0.698485i
$469$	−7.54440	−	23.7154i	−0.348368	−	1.09507i
$470$		0			0
$471$	42.6668	+	7.07244i	1.96598	+	0.325881i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−21.6734	+	18.1861i	−0.994444	+	0.834437i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.143239	−	0.989688i	$-0.454248\pi$
$479$		0			0		−0.143239	+	0.989688i	$0.545752\pi$
$480$		0			0
$481$	−27.9543	−	29.4274i	−1.27461	−	1.34178i
$482$		0			0
$483$		0			0
$484$	−6.45377	−	21.0321i	−0.293353	−	0.956004i
$485$		0			0
$486$		0			0
$487$	16.1014	+	20.8726i	0.729624	+	0.945828i	0.999856	−	0.0169668i	$-0.00540098\pi$
$487$	16.1014	+	20.8726i	0.729624	+	0.945828i	−0.270232	+	0.962795i	$0.587100\pi$
$488$		0			0
$489$	10.4449	+	2.62699i	0.472333	+	0.118796i
$490$		0			0
$491$		0			0		−0.569364	−	0.822086i	$-0.692810\pi$
$491$		0			0		0.569364	+	0.822086i	$0.307190\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.84060	+	1.51252i	0.0826452	+	0.0679140i
$497$		0			0
$498$		0			0
$499$	13.9748	+	10.5533i	0.625597	+	0.472429i	0.867209	−	0.497944i	$-0.165911\pi$
$499$	13.9748	+	10.5533i	0.625597	+	0.472429i	−0.241612	+	0.970373i	$0.577676\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.293353	−	0.956004i	$-0.405229\pi$
$503$		0			0		−0.293353	+	0.956004i	$0.594771\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	48.4758	+	30.7079i	2.15289	+	1.36379i
$508$	−2.07471	+	3.96063i	−0.0920505	+	0.175725i
$509$		0			0		0.526432	−	0.850217i	$-0.323529\pi$
$509$		0			0		−0.526432	+	0.850217i	$0.676471\pi$
$510$		0			0
$511$	31.3903	−	20.7999i	1.38862	−	0.920135i
$512$		0			0
$513$	21.1084	+	20.4681i	0.931960	+	0.903689i
$514$		0			0
$515$		0			0
$516$	−37.7013	−	21.2538i	−1.65971	−	0.935645i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$521$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$522$		0			0
$523$	12.9236	−	25.9540i	0.565107	−	1.13489i	−0.409779	−	0.912185i	$-0.634394\pi$
$523$	12.9236	−	25.9540i	0.565107	−	1.13489i	0.974886	−	0.222704i	$-0.0714882\pi$
$524$		0			0
$525$	−7.73481	−	17.7639i	−0.337575	−	0.775280i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−22.9225	−	1.88694i	−0.996629	−	0.0820408i
$530$		0			0
$531$		0			0
$532$	−9.38803	+	23.5138i	−0.407023	+	1.01945i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	34.7161	−	5.75453i	1.49256	−	0.247407i	0.639326	−	0.768936i	$-0.279213\pi$
$541$	34.7161	−	5.75453i	1.49256	−	0.247407i	0.853233	+	0.521529i	$0.174638\pi$
$542$		0			0
$543$	20.6927	−	40.5089i	0.888010	−	1.73840i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−4.89723	+	43.1795i	−0.209390	+	1.84622i	0.262957	+	0.964808i	$0.415302\pi$
$547$	−4.89723	+	43.1795i	−0.209390	+	1.84622i	−0.472347	+	0.881412i	$0.656593\pi$
$548$		0			0
$549$	−3.61593	−	43.9262i	−0.154324	−	1.87472i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−27.0791	+	29.0986i	−1.15152	+	1.23740i
$554$		0			0
$555$		0			0
$556$	−36.5610	−	17.2777i	−1.55053	−	0.732738i
$557$		0			0		0.0307951	−	0.999526i	$-0.490196\pi$
$557$		0			0		−0.0307951	+	0.999526i	$0.509804\pi$
$558$		0			0
$559$	9.56274	+	84.3158i	0.404461	+	3.56618i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.153392	−	0.988165i	$-0.549020\pi$
$563$		0			0		0.153392	+	0.988165i	$0.450980\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−17.5398	+	9.88789i	−0.736601	+	0.415253i
$568$		0			0
$569$		0			0		0.974601	−	0.223951i	$-0.0718954\pi$
$569$		0			0		−0.974601	+	0.223951i	$0.928105\pi$
$570$		0			0
$571$	39.5847	+	25.6489i	1.65657	+	1.07337i	0.915821	+	0.401586i	$0.131541\pi$
$571$	39.5847	+	25.6489i	1.65657	+	1.07337i	0.740746	+	0.671785i	$0.234472\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−20.4052	+	12.6344i	−0.850217	+	0.526432i
$577$	−42.7843	+	3.08006i	−1.78113	+	0.128225i	−0.922901	−	0.385037i	$-0.874189\pi$
$577$	−42.7843	+	3.08006i	−1.78113	+	0.128225i	−0.858232	+	0.513262i	$0.828437\pi$
$578$		0			0
$579$	−0.979657	−	0.183130i	−0.0407131	−	0.00761061i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.445738	−	0.895163i	$-0.647059\pi$
$587$		0			0		0.445738	+	0.895163i	$0.352941\pi$
$588$	5.42804	+	4.27686i	0.223849	+	0.176375i
$589$	−3.15487	+	1.18510i	−0.129994	+	0.0488312i
$590$		0			0
$591$		0			0
$592$	−19.2226	−	14.2087i	−0.790043	−	0.583974i
$593$		0			0		−0.0513107	−	0.998683i	$-0.516340\pi$
$593$		0			0		0.0513107	+	0.998683i	$0.483660\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	33.5059	−	19.3446i	1.37130	−	0.791723i
$598$		0			0
$599$		0			0		−0.560894	−	0.827888i	$-0.689542\pi$
$599$		0			0		0.560894	+	0.827888i	$0.310458\pi$
$600$		0			0
$601$	3.68530	+	4.48468i	0.150327	+	0.182934i	0.842333	−	0.538957i	$-0.181182\pi$
$601$	3.68530	+	4.48468i	0.150327	+	0.182934i	−0.692007	+	0.721891i	$0.743273\pi$
$602$		0			0
$603$	21.9713	+	25.1184i	0.894739	+	1.02290i
$604$	16.9884	−	41.3131i	0.691247	−	1.68101i
$605$		0			0
$606$		0			0
$607$	17.2279	−	7.71277i	0.699258	−	0.313052i	−0.0278137	−	0.999613i	$-0.508855\pi$
$607$	17.2279	−	7.71277i	0.699258	−	0.313052i	0.727072	+	0.686561i	$0.240881\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	5.67802i		−	0.229333i	−0.993404	−	0.114667i	$-0.963420\pi$
$613$		−	5.67802i		−	0.229333i	0.993404	−	0.114667i	$-0.0365800\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$617$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$618$		0			0
$619$	40.0438	+	27.7338i	1.60950	+	1.11471i	0.921062	+	0.389417i	$0.127324\pi$
$619$	40.0438	+	27.7338i	1.60950	+	1.11471i	0.688436	+	0.725297i	$0.258297\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	43.5201	+	17.8959i	1.74220	+	0.716409i
$625$	−18.8171	+	16.4595i	−0.752685	+	0.658380i
$626$		0			0
$627$		0			0
$628$	−29.6846	−	40.1596i	−1.18455	−	1.60254i
$629$		0			0
$630$		0			0
$631$	1.56212	+	2.70567i	0.0621870	+	0.107711i	0.895443	−	0.445177i	$-0.146859\pi$
$631$	1.56212	+	2.70567i	0.0621870	+	0.107711i	−0.833256	+	0.552888i	$0.813526\pi$
$632$		0			0
$633$	1.06199	+	3.01371i	0.0422103	+	0.119784i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.556262	−	13.5378i	0.0220399	−	0.536386i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$641$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$642$		0			0
$643$	−3.65282	+	4.73523i	−0.144053	+	0.186739i	−0.858290	−	0.513164i	$-0.828473\pi$
$643$	−3.65282	+	4.73523i	−0.144053	+	0.186739i	0.714237	+	0.699904i	$0.246774\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$647$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.165715	−	2.30190i	−0.00649489	−	0.0902187i
$652$	−6.54687	−	10.5736i	−0.256395	−	0.414092i
$653$		0			0		−0.312920	−	0.949779i	$-0.601307\pi$
$653$		0			0		0.312920	+	0.949779i	$0.398693\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−27.4582	+	42.3771i	−1.07125	+	1.65329i
$658$		0			0
$659$		0			0		−0.223951	−	0.974601i	$-0.571895\pi$
$659$		0			0		0.223951	+	0.974601i	$0.428105\pi$
$660$		0			0
$661$	16.2311	+	28.7918i	0.631318	+	1.11987i	0.982975	+	0.183742i	$0.0588210\pi$
$661$	16.2311	+	28.7918i	0.631318	+	1.11987i	−0.351657	+	0.936129i	$0.614382\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	50.4646	−	5.72347i	1.95107	−	0.221282i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−25.0023	+	19.6998i	−0.963768	+	0.759372i	−0.970623	−	0.240604i	$-0.922654\pi$
$673$	−25.0023	+	19.6998i	−0.963768	+	0.759372i	0.00685566	+	0.999976i	$0.497818\pi$
$674$		0			0
$675$	19.0193	+	17.6993i	0.732053	+	0.681247i
$676$	−15.5014	−	64.4222i	−0.596207	−	2.47778i
$677$		0			0		0.312920	−	0.949779i	$-0.398693\pi$
$677$		0			0		−0.312920	+	0.949779i	$0.601307\pi$
$678$		0			0
$679$	−38.6939	+	3.18522i	−1.48494	+	0.122238i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.916855	−	0.399220i	$-0.869281\pi$
$683$		0			0		0.916855	+	0.399220i	$0.130719\pi$
$684$	−1.04553	−	33.9350i	−0.0399767	−	1.29754i
$685$		0			0
$686$		0			0
$687$	7.54127	+	45.4951i	0.287717	+	1.73575i
$688$	14.1690	+	47.9241i	0.540188	+	1.82709i
$689$		0			0
$690$		0			0
$691$	−5.44118	−	2.50334i	−0.206992	−	0.0952316i	0.311743	−	0.950166i	$-0.399087\pi$
$691$	−5.44118	−	2.50334i	−0.206992	−	0.0952316i	−0.518735	+	0.854935i	$0.673597\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−7.86714	+	20.9432i	−0.297350	+	0.791580i
$701$		0			0		−0.517676	−	0.855577i	$-0.673203\pi$
$701$		0			0		0.517676	+	0.855577i	$0.326797\pi$
$702$		0			0
$703$	31.0036	−	13.4997i	1.16932	−	0.509150i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−12.9088	+	47.2057i	−0.484801	+	1.77285i	0.137963	+	0.990437i	$0.455944\pi$
$709$	−12.9088	+	47.2057i	−0.484801	+	1.77285i	−0.622764	+	0.782410i	$0.713990\pi$
$710$		0			0
$711$	17.7152	−	50.2721i	0.664372	−	1.88535i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.850217	−	0.526432i	$-0.823529\pi$
$719$		0			0		0.850217	+	0.526432i	$0.176471\pi$
$720$		0			0
$721$	−0.924104	+	1.45880i	−0.0344154	+	0.0543286i
$722$		0			0
$723$	−10.2376	−	51.8152i	−0.380739	−	1.92703i
$724$	−49.7158	+	16.9474i	−1.84767	+	0.629846i
$725$		0			0
$726$		0			0
$727$	−38.0532	−	4.71200i	−1.41131	−	0.174758i	−0.619248	−	0.785195i	$-0.712563\pi$
$727$	−38.0532	−	4.71200i	−1.41131	−	0.174758i	−0.792065	+	0.610437i	$0.790994\pi$
$728$		0			0
$729$	16.2711	−	21.5465i	0.602635	−	0.798017i
$730$		0			0
$731$		0			0
$732$	−32.3115	+	39.3202i	−1.19427	+	1.45332i
$733$	−23.5359	+	32.5346i	−0.869317	+	1.20169i	0.108889	+	0.994054i	$0.465271\pi$
$733$	−23.5359	+	32.5346i	−0.869317	+	1.20169i	−0.978205	+	0.207639i	$0.933422\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	13.1451	−	52.2649i	0.483552	−	1.92259i	0.113661	−	0.993520i	$-0.463742\pi$
$739$	13.1451	−	52.2649i	0.483552	−	1.92259i	0.369891	−	0.929075i	$-0.379395\pi$
$740$		0			0
$741$	−52.7068	+	40.6587i	−1.93623	+	1.49363i
$742$		0			0
$743$		0			0		−0.586123	−	0.810222i	$-0.699346\pi$
$743$		0			0		0.586123	+	0.810222i	$0.300654\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−38.7321	+	5.20035i	−1.41335	+	0.189763i	−0.799979	−	0.600028i	$-0.795156\pi$
$751$	−38.7321	+	5.20035i	−1.41335	+	0.189763i	−0.613376	+	0.789791i	$0.710189\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	22.4881	+	5.90212i	0.817886	+	0.214658i
$757$	−5.42772	+	32.7445i	−0.197274	+	1.19012i	0.688233	+	0.725490i	$0.258387\pi$
$757$	−5.42772	+	32.7445i	−0.197274	+	1.19012i	−0.885506	+	0.464627i	$0.846188\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.454905	−	0.890540i	$-0.650327\pi$
$761$		0			0		0.454905	+	0.890540i	$0.349673\pi$
$762$		0			0
$763$	16.4124	−	11.6181i	0.594168	−	0.420603i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	27.0712	+	5.92869i	0.976848	+	0.213933i
$769$	−20.2845	+	30.6124i	−0.731479	+	1.10391i	0.258950	+	0.965891i	$0.416624\pi$
$769$	−20.2845	+	30.6124i	−0.731479	+	1.10391i	−0.990429	+	0.138022i	$0.955925\pi$
$770$		0			0
$771$		0			0
$772$	0.664905	+	0.939283i	0.0239305	+	0.0338055i
$773$		0			0		0.876109	−	0.482114i	$-0.160131\pi$
$773$		0			0		−0.876109	+	0.482114i	$0.839869\pi$
$774$		0			0
$775$	−2.76563	+	1.10420i	−0.0993444	+	0.0396640i
$776$		0			0
$777$	2.84568	+	22.9811i	0.102088	+	0.824443i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.06184	−	7.90860i	−0.0379230	−	0.282450i
$785$		0			0
$786$		0			0
$787$	−3.60413	−	34.9818i	−0.128473	−	1.24697i	−0.839327	−	0.543626i	$-0.817051\pi$
$787$	−3.60413	−	34.9818i	−0.128473	−	1.24697i	0.710854	−	0.703339i	$-0.248309\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	99.5951	+	6.14281i	3.53672	+	0.218138i
$794$		0			0
$795$		0			0
$796$	−43.3252	−	10.8967i	−1.53562	−	0.386224i
$797$		0			0		0.899692	−	0.436525i	$-0.143791\pi$
$797$		0			0		−0.899692	+	0.436525i	$0.856209\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.97722	−	38.4835i	0.0697312	−	1.35721i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.283523	−	0.958965i	$-0.408497\pi$
$809$		0			0		−0.283523	+	0.958965i	$0.591503\pi$
$810$		0			0
$811$	−18.0239	−	52.8738i	−0.632906	−	1.85665i	−0.501134	−	0.865370i	$-0.667084\pi$
$811$	−18.0239	−	52.8738i	−0.632906	−	1.85665i	−0.131772	−	0.991280i	$-0.542067\pi$
$812$		0			0
$813$	−21.8342	−	17.5699i	−0.765759	−	0.616204i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−68.7340	−	16.5389i	−2.40470	−	0.578623i
$818$		0			0
$819$	−16.4672	−	42.5067i	−0.575410	−	1.48531i
$820$		0			0
$821$		0			0		0.703468	−	0.710727i	$-0.251634\pi$
$821$		0			0		−0.703468	+	0.710727i	$0.748366\pi$
$822$		0			0
$823$	−47.2242	−	26.6223i	−1.64613	−	0.927993i	−0.983905	−	0.178690i	$-0.942814\pi$
$823$	−47.2242	−	26.6223i	−1.64613	−	0.927993i	−0.662227	−	0.749303i	$-0.730389\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.745886	−	0.666073i	$-0.232026\pi$
$827$		0			0		−0.745886	+	0.666073i	$0.767974\pi$
$828$		0			0
$829$	−9.85613	+	4.05294i	−0.342318	+	0.140764i	−0.544900	−	0.838501i	$-0.683433\pi$
$829$	−9.85613	+	4.05294i	−0.342318	+	0.140764i	0.202583	+	0.979265i	$0.435067\pi$
$830$		0			0
$831$	−38.4088	+	18.1509i	−1.33239	+	0.629649i
$832$	−21.6918	−	49.8178i	−0.752028	−	1.72712i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.43603	+	2.74139i	0.0496365	+	0.0947563i
$838$		0			0
$839$		0			0		0.370796	−	0.928714i	$-0.379085\pi$
$839$		0			0		−0.370796	+	0.928714i	$0.620915\pi$
$840$		0			0
$841$	−7.36186	−	28.0500i	−0.253857	−	0.967242i
$842$		0			0
$843$		0			0
$844$	1.54214	−	3.35194i	0.0530826	−	0.115379i
$845$		0			0
$846$		0			0
$847$	23.5995	−	6.97731i	0.810888	−	0.239743i
$848$		0			0
$849$	5.23522	−	56.4970i	0.179672	−	1.93897i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	21.5819	−	30.4878i	0.738948	−	1.04388i	−0.258025	−	0.966138i	$-0.583072\pi$
$853$	21.5819	−	30.4878i	0.738948	−	1.04388i	0.996973	−	0.0777434i	$-0.0247715\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$857$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$858$		0			0
$859$	52.3029	−	12.5852i	1.78455	−	0.429402i	0.799359	−	0.600854i	$-0.205173\pi$
$859$	52.3029	−	12.5852i	1.78455	−	0.429402i	0.985192	+	0.171452i	$0.0548459\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.904126	−	0.427265i	$-0.859477\pi$
$863$		0			0		0.904126	+	0.427265i	$0.140523\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−18.4597	−	22.9399i	−0.626924	−	0.779081i
$868$	−1.71296	+	2.04142i	−0.0581415	+	0.0692904i
$869$		0			0
$870$		0			0
$871$	−63.4062	+	41.0839i	−2.14844	+	1.39208i
$872$		0			0
$873$	45.8676	−	24.6304i	1.55238	−	0.833612i
$874$		0			0
$875$		0			0
$876$	56.8259	−	13.0579i	1.91997	−	0.441184i
$877$	52.3649	+	23.4433i	1.76824	+	0.791624i	0.985589	+	0.169160i	$0.0541053\pi$
$877$	52.3649	+	23.4433i	1.76824	+	0.791624i	0.782648	+	0.622465i	$0.213869\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.0410550	−	0.999157i	$-0.513072\pi$
$881$		0			0		0.0410550	+	0.999157i	$0.486928\pi$
$882$		0			0
$883$	24.2919	−	15.0409i	0.817486	−	0.506166i	−0.0528628	−	0.998602i	$-0.516835\pi$
$883$	24.2919	−	15.0409i	0.817486	−	0.506166i	0.870349	+	0.492436i	$0.163893\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$887$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$888$		0			0
$889$	−4.40632	−	2.36615i	−0.147783	−	0.0793580i
$890$		0			0
$891$		0			0
$892$	−46.4345	−	35.8201i	−1.55474	−	1.19935i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1.53932	−	29.9605i	−0.0513107	−	0.998683i
$901$		0			0
$902$		0			0
$903$	24.2063	−	41.9265i	0.805534	−	1.39523i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.62024	+	1.19763i	−0.0537993	+	0.0397666i	−0.620977	−	0.783829i	$-0.713264\pi$
$907$	−1.62024	+	1.19763i	−0.0537993	+	0.0397666i	0.567178	+	0.823595i	$0.308035\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.380311	−	0.924859i	$-0.375817\pi$
$911$		0			0		−0.380311	+	0.924859i	$0.624183\pi$
$912$	−26.4112	+	28.9717i	−0.874561	+	0.959347i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	30.3187	−	43.7761i	1.00176	−	1.44640i
$917$		0			0
$918$		0			0
$919$	12.6230	+	57.6383i	0.416394	+	1.90131i	0.432887	+	0.901448i	$0.357495\pi$
$919$	12.6230	+	57.6383i	0.416394	+	1.90131i	−0.0164935	+	0.999864i	$0.505250\pi$
$920$		0			0
$921$	19.3118	−	23.4106i	0.636345	−	0.771404i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	27.1448	−	12.4886i	0.892515	−	0.410622i
$926$		0			0
$927$	0.355202	−	2.28825i	0.0116664	−	0.0751560i
$928$		0			0
$929$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$929$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$930$		0			0
$931$	10.4399	+	4.29300i	0.342155	+	0.140698i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−24.0941	+	28.1228i	−0.787121	+	0.918731i	−0.998274	−	0.0587215i	$-0.981298\pi$
$937$	−24.0941	+	28.1228i	−0.787121	+	0.918731i	0.211154	+	0.977453i	$0.432278\pi$
$938$		0			0
$939$	−18.8453	−	10.8803i	−0.614992	−	0.355066i
$940$		0			0
$941$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$941$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.618892	−	0.785476i	$-0.287582\pi$
$947$		0			0		−0.618892	+	0.785476i	$0.712418\pi$
$948$	−55.0954	+	27.4343i	−1.78942	+	0.891023i
$949$	−86.0472	−	75.2662i	−2.79321	−	2.44324i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.473094	−	0.881012i	$-0.343137\pi$
$953$		0			0		−0.473094	+	0.881012i	$0.656863\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	30.6194	−	1.25814i	0.987724	−	0.0405853i
$962$		0			0
$963$		0			0
$964$	−33.1637	+	51.1826i	−1.06813	+	1.64848i
$965$		0			0
$966$		0			0
$967$	−4.30021	−	24.3877i	−0.138286	−	0.784256i	−0.972515	−	0.232839i	$-0.925198\pi$
$967$	−4.30021	−	24.3877i	−0.138286	−	0.784256i	0.834230	−	0.551417i	$-0.185913\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.543778	−	0.839229i	$-0.683007\pi$
$971$		0			0		0.543778	+	0.839229i	$0.316993\pi$
$972$	−30.8080	+	4.78228i	−0.988165	+	0.153392i
$973$	19.7458	−	40.6967i	0.633020	−	1.30468i
$974$		0			0
$975$	−45.8255	+	36.8756i	−1.46759	+	1.18096i
$976$	58.3920	−	6.62256i	1.86908	−	0.211983i
$977$		0			0		−0.876109	−	0.482114i	$-0.839869\pi$
$977$		0			0		0.876109	+	0.482114i	$0.160131\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−13.9588	+	23.0701i	−0.445671	+	0.736573i
$982$		0			0
$983$		0			0		−0.233944	−	0.972250i	$-0.575163\pi$
$983$		0			0		0.233944	+	0.972250i	$0.424837\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	76.3751	+	8.66213i	2.42981	+	0.275579i
$989$		0			0
$990$		0			0
$991$	1.66531	+	54.0516i	0.0529003	+	1.71700i	0.538335	+	0.842731i	$0.319054\pi$
$991$	1.66531	+	54.0516i	0.0529003	+	1.71700i	−0.485434	+	0.874273i	$0.661339\pi$
$992$		0			0
$993$	−14.4789	−	1.34167i	−0.459474	−	0.0425765i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−29.7865	−	30.7184i	−0.943349	−	0.972861i	0.0563247	−	0.998413i	$-0.482062\pi$
$997$	−29.7865	−	30.7184i	−0.943349	−	0.972861i	−0.999673	+	0.0255518i	$0.991866\pi$
$998$		0			0
$999$	−15.8013	−	26.7310i	−0.499930	−	0.845732i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	921.2.x.a.92.1	✓	96
3.2	odd	2	CM	921.2.x.a.92.1	✓	96
307.297	odd	306	inner	921.2.x.a.911.1	yes	96
921.911	even	306	inner	921.2.x.a.911.1	yes	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
921.2.x.a.92.1	✓	96	1.1	even	1	trivial
921.2.x.a.92.1	✓	96	3.2	odd	2	CM
921.2.x.a.911.1	yes	96	307.297	odd	306	inner
921.2.x.a.911.1	yes	96	921.911	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)

\(f(q)\)	\(=\)	\(q+(-0.625689 + 1.61509i) q^{3} +(1.81693 - 0.835921i) q^{4} +(0.914150 + 2.04192i) q^{7} +(-2.21703 - 2.02109i) q^{9} +O(q^{10})\)	\(q+(-0.625689 + 1.61509i) q^{3} +(1.81693 - 0.835921i) q^{4} +(0.914150 + 2.04192i) q^{7} +(-2.21703 - 2.02109i) q^{9} +(0.213253 + 3.45753i) q^{12} +(5.24748 - 4.31213i) q^{13} +(2.60247 - 3.03762i) q^{16} +(1.88064 + 5.33686i) q^{19} +(-3.86986 + 0.198827i) q^{21} +(1.75825 + 4.68066i) q^{25} +(4.65140 - 2.31612i) q^{27} +(3.36783 + 2.94587i) q^{28} +(0.0122284 + 0.595457i) q^{31} +(-5.71765 - 1.81892i) q^{36} +(-0.429102 - 5.96054i) q^{37} +(3.68119 + 11.1732i) q^{39} +(-6.79379 + 10.4851i) q^{43} +(3.27769 + 6.10383i) q^{48} +(1.32874 - 1.48796i) q^{49} +(5.92970 - 12.2213i) q^{52} +(-9.79620 - 0.301818i) q^{57} +(10.7550 + 10.0086i) q^{61} +(2.10020 - 6.37457i) q^{63} +(2.18930 - 7.69461i) q^{64} +(-11.0530 - 1.25359i) q^{67} +(-2.75247 - 16.6052i) q^{73} +(-8.65980 - 0.0889102i) q^{75} +(7.87818 + 8.12464i) q^{76} +(6.92546 + 16.3620i) q^{79} +(0.830415 + 8.96161i) q^{81} +(-6.86506 + 3.59615i) q^{84} +(13.6020 + 6.77299i) q^{91} +(-0.969368 - 0.352821i) q^{93} +(-5.76773 + 16.3676i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6 q^{4} + 18 q^{9}+O(q^{10}) \) 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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