LMFDB - Embedded newform 1224.2.bq.e.433.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,2,Mod(145,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(8))

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

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

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

chi := DirichletCharacter("1224.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1224 = 2^{3} \cdot 3^{2} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1224.bq (of order $8$ , degree $4$ , 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:	$9.77368920740$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 36x^{14} + 466x^{12} + 2956x^{10} + 10049x^{8} + 18032x^{6} + 14800x^{4} + 3200x^{2} + 64$ x^16 + 36x^14 + 466x^12 + 2956x^10 + 10049x^8 + 18032x^6 + 14800x^4 + 3200*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 408)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			433.3
Root			$-1.94001i$ of defining polynomial
Character	$\chi$	$=$	1224.433
Dual form			1224.2.bq.e.865.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.868455 + 0.359726i) q^{5} +(-4.01891 + 1.66469i) q^{7} +(1.34442 + 3.24572i) q^{11} -4.92143i q^{13} +(3.09219 + 2.72733i) q^{17} +(-0.636406 + 0.636406i) q^{19} +(0.803224 + 1.93916i) q^{23} +(-2.91072 - 2.91072i) q^{25} +(-8.62731 - 3.57355i) q^{29} +(-4.14466 + 10.0061i) q^{31} -4.08907 q^{35} +(-2.55888 + 6.17767i) q^{37} +(-7.89980 + 3.27221i) q^{41} +(-4.95418 - 4.95418i) q^{43} +2.49767i q^{47} +(8.43070 - 8.43070i) q^{49} +(-6.01297 + 6.01297i) q^{53} +3.30238i q^{55} +(3.18160 + 3.18160i) q^{59} +(-5.55574 + 2.30126i) q^{61} +(1.77037 - 4.27404i) q^{65} +5.32216 q^{67} +(-0.905275 + 2.18553i) q^{71} +(9.08570 + 3.76342i) q^{73} +(-10.8062 - 10.8062i) q^{77} +(-3.16068 - 7.63055i) q^{79} +(-1.42580 + 1.42580i) q^{83} +(1.70434 + 3.48090i) q^{85} +2.95863i q^{89} +(8.19264 + 19.7788i) q^{91} +(-0.781621 + 0.323758i) q^{95} +(-7.53188 - 3.11981i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 8 q^{11} - 8 q^{19} + 8 q^{23} - 8 q^{25} - 32 q^{29} + 8 q^{31} - 16 q^{35} + 8 q^{37} - 40 q^{41} - 24 q^{43} + 24 q^{49} + 24 q^{53} + 16 q^{59} + 64 q^{65} + 16 q^{71} + 24 q^{73} - 96 q^{79}+ \cdots + 56 q^{97}+O(q^{100})$ 16 * q + 8 * q^11 - 8 * q^19 + 8 * q^23 - 8 * q^25 - 32 * q^29 + 8 * q^31 - 16 * q^35 + 8 * q^37 - 40 * q^41 - 24 * q^43 + 24 * q^49 + 24 * q^53 + 16 * q^59 + 64 * q^65 + 16 * q^71 + 24 * q^73 - 96 * q^79 - 32 * q^85 + 8 * q^95 + 56 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$ .

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{8}\right)$	$1$

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
$5$	0.868455	+	0.359726i	0.388385	+	0.160874i	0.568327	−	0.822803i	$-0.307591\pi$
$5$	0.868455	+	0.359726i	0.388385	+	0.160874i	−0.179942	+	0.983677i	$0.557591\pi$
$6$		0			0
$7$	−4.01891	+	1.66469i	−1.51900	+	0.629192i	−0.977392	−	0.211436i	$-0.932186\pi$
$7$	−4.01891	+	1.66469i	−1.51900	+	0.629192i	−0.541613	+	0.840628i	$0.682186\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.34442	+	3.24572i	0.405358	+	0.978621i	0.986343	+	0.164706i	$0.0526676\pi$
$11$	1.34442	+	3.24572i	0.405358	+	0.978621i	−0.580985	+	0.813915i	$0.697332\pi$
$12$		0			0
$13$		−	4.92143i		−	1.36496i	−0.730904	−	0.682480i	$-0.760901\pi$
$13$		−	4.92143i		−	1.36496i	0.730904	−	0.682480i	$-0.239099\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.09219	+	2.72733i	0.749967	+	0.661475i
$17$	3.09219	+	2.72733i	0.749967	+	0.661475i
$18$		0			0
$19$	−0.636406	+	0.636406i	−0.146002	+	0.146002i	−0.776329	−	0.630328i	$-0.782921\pi$
$19$	−0.636406	+	0.636406i	−0.146002	+	0.146002i	0.630328	+	0.776329i	$0.282921\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.803224	+	1.93916i	0.167484	+	0.404342i	0.985230	−	0.171238i	$-0.0547766\pi$
$23$	0.803224	+	1.93916i	0.167484	+	0.404342i	−0.817746	+	0.575579i	$0.804777\pi$
$24$		0			0
$25$	−2.91072	−	2.91072i	−0.582145	−	0.582145i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.62731	−	3.57355i	−1.60205	−	0.663592i	−0.610348	−	0.792134i	$-0.708970\pi$
$29$	−8.62731	−	3.57355i	−1.60205	−	0.663592i	−0.991704	+	0.128542i	$0.958970\pi$
$30$		0			0
$31$	−4.14466	+	10.0061i	−0.744404	+	1.79715i	−0.157454	+	0.987526i	$0.550328\pi$
$31$	−4.14466	+	10.0061i	−0.744404	+	1.79715i	−0.586950	+	0.809623i	$0.699672\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.08907			−0.691179
$36$		0			0
$37$	−2.55888	+	6.17767i	−0.420676	+	1.01560i	0.561472	+	0.827496i	$0.310235\pi$
$37$	−2.55888	+	6.17767i	−0.420676	+	1.01560i	−0.982149	+	0.188107i	$0.939765\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.89980	+	3.27221i	−1.23374	+	0.511033i	−0.901753	−	0.432251i	$-0.857720\pi$
$41$	−7.89980	+	3.27221i	−1.23374	+	0.511033i	−0.331989	+	0.943283i	$0.607720\pi$
$42$		0			0
$43$	−4.95418	−	4.95418i	−0.755506	−	0.755506i	0.219995	−	0.975501i	$-0.429396\pi$
$43$	−4.95418	−	4.95418i	−0.755506	−	0.755506i	−0.975501	+	0.219995i	$0.929396\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.49767i			0.364323i	0.983269	+	0.182162i	$0.0583094\pi$
$47$			2.49767i			0.364323i	−0.983269	+	0.182162i	$0.941691\pi$
$48$		0			0
$49$	8.43070	−	8.43070i	1.20439	−	1.20439i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.01297	+	6.01297i	−0.825945	+	0.825945i	−0.986953	−	0.161008i	$-0.948525\pi$
$53$	−6.01297	+	6.01297i	−0.825945	+	0.825945i	0.161008	+	0.986953i	$0.448525\pi$
$54$		0			0
$55$			3.30238i			0.445293i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.18160	+	3.18160i	0.414208	+	0.414208i	0.883202	−	0.468993i	$-0.155383\pi$
$59$	3.18160	+	3.18160i	0.414208	+	0.414208i	−0.468993	+	0.883202i	$0.655383\pi$
$60$		0			0
$61$	−5.55574	+	2.30126i	−0.711340	+	0.294647i	−0.708859	−	0.705350i	$-0.750790\pi$
$61$	−5.55574	+	2.30126i	−0.711340	+	0.294647i	−0.00248084	+	0.999997i	$0.500790\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.77037	−	4.27404i	0.219587	−	0.530129i
$66$		0			0
$67$	5.32216			0.650205			0.325102	−	0.945679i	$-0.394601\pi$
$67$	5.32216			0.650205			0.325102	+	0.945679i	$0.394601\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.905275	+	2.18553i	−0.107436	+	0.259374i	−0.968450	−	0.249207i	$-0.919830\pi$
$71$	−0.905275	+	2.18553i	−0.107436	+	0.259374i	0.861014	+	0.508582i	$0.169830\pi$
$72$		0			0
$73$	9.08570	+	3.76342i	1.06340	+	0.440475i	0.844657	−	0.535307i	$-0.179804\pi$
$73$	9.08570	+	3.76342i	1.06340	+	0.440475i	0.218744	+	0.975782i	$0.429804\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.8062	−	10.8062i	−1.23148	−	1.23148i
$78$		0			0
$79$	−3.16068	−	7.63055i	−0.355604	−	0.858504i	−0.995907	−	0.0903815i	$-0.971191\pi$
$79$	−3.16068	−	7.63055i	−0.355604	−	0.858504i	0.640303	−	0.768122i	$-0.278809\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.42580	+	1.42580i	−0.156502	+	0.156502i	−0.781015	−	0.624512i	$-0.785298\pi$
$83$	−1.42580	+	1.42580i	−0.156502	+	0.156502i	0.624512	+	0.781015i	$0.285298\pi$
$84$		0			0
$85$	1.70434	+	3.48090i	0.184861	+	0.377557i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			2.95863i			0.313614i	0.987629	+	0.156807i	$0.0501201\pi$
$89$			2.95863i			0.313614i	−0.987629	+	0.156807i	$0.949880\pi$
$90$		0			0
$91$	8.19264	+	19.7788i	0.858822	+	2.07338i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.781621	+	0.323758i	−0.0801927	+	0.0332169i
$96$		0			0
$97$	−7.53188	−	3.11981i	−0.764746	−	0.316768i	−0.0340040	−	0.999422i	$-0.510826\pi$
$97$	−7.53188	−	3.11981i	−0.764746	−	0.316768i	−0.730742	+	0.682653i	$0.760826\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	15.1971			1.51217			0.756086	−	0.654472i	$-0.227109\pi$
$101$	15.1971			1.51217			0.756086	+	0.654472i	$0.227109\pi$
$102$		0			0
$103$	8.99999			0.886795			0.443398	−	0.896325i	$-0.353773\pi$
$103$	8.99999			0.886795			0.443398	+	0.896325i	$0.353773\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.92909	−	2.04170i	−0.476513	−	0.197378i	0.131483	−	0.991318i	$-0.458026\pi$
$107$	−4.92909	−	2.04170i	−0.476513	−	0.197378i	−0.607996	+	0.793940i	$0.708026\pi$
$108$		0			0
$109$	5.44645	−	2.25599i	0.521675	−	0.216085i	−0.106277	−	0.994337i	$-0.533893\pi$
$109$	5.44645	−	2.25599i	0.521675	−	0.216085i	0.627953	+	0.778252i	$0.283893\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.85081	+	14.1251i	0.550398	+	1.32878i	0.917181	+	0.398472i	$0.130459\pi$
$113$	5.85081	+	14.1251i	0.550398	+	1.32878i	−0.366783	+	0.930307i	$0.619541\pi$
$114$		0			0
$115$			1.97301i			0.183984i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−16.9674	−	5.81336i	−1.55540	−	0.532910i
$120$		0			0
$121$	−0.949045	+	0.949045i	−0.0862769	+	0.0862769i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.27940	−	7.91717i	−0.293318	−	0.708133i
$126$		0			0
$127$	−10.1510	−	10.1510i	−0.900758	−	0.900758i	0.0947441	−	0.995502i	$-0.469797\pi$
$127$	−10.1510	−	10.1510i	−0.900758	−	0.900758i	−0.995502	+	0.0947441i	$0.969797\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−16.3173	−	6.75885i	−1.42565	−	0.590523i	−0.469376	−	0.882998i	$-0.655521\pi$
$131$	−16.3173	−	6.75885i	−1.42565	−	0.590523i	−0.956273	+	0.292475i	$0.905521\pi$
$132$		0			0
$133$	1.49824	−	3.61707i	0.129914	−	0.313640i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.1004			0.948370			0.474185	−	0.880425i	$-0.342743\pi$
$137$	11.1004			0.948370			0.474185	+	0.880425i	$0.342743\pi$
$138$		0			0
$139$	−4.79347	+	11.5725i	−0.406577	+	0.981564i	0.579454	+	0.815005i	$0.303266\pi$
$139$	−4.79347	+	11.5725i	−0.406577	+	0.981564i	−0.986031	+	0.166559i	$0.946734\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	15.9736	−	6.61647i	1.33578	−	0.553297i
$144$		0			0
$145$	−6.20693	−	6.20693i	−0.515458	−	0.515458i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	4.87764i		−	0.399592i	−0.979837	−	0.199796i	$-0.935972\pi$
$149$		−	4.87764i		−	0.399592i	0.979837	−	0.199796i	$-0.0640280\pi$
$150$		0			0
$151$	−8.43921	+	8.43921i	−0.686773	+	0.686773i	−0.961517	−	0.274744i	$-0.911407\pi$
$151$	−8.43921	+	8.43921i	−0.686773	+	0.686773i	0.274744	+	0.961517i	$0.411407\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.19891	+	7.19891i	−0.578230	+	0.578230i
$156$		0			0
$157$		−	2.16966i		−	0.173158i	−0.996245	−	0.0865788i	$-0.972407\pi$
$157$		−	2.16966i		−	0.173158i	0.996245	−	0.0865788i	$-0.0275934\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.45617	−	6.45617i	−0.508817	−	0.508817i
$162$		0			0
$163$	12.9053	−	5.34553i	1.01082	−	0.418695i	0.185065	−	0.982726i	$-0.440750\pi$
$163$	12.9053	−	5.34553i	1.01082	−	0.418695i	0.825753	+	0.564032i	$0.190750\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.03158	+	12.1473i	−0.389355	+	0.939987i	0.600721	+	0.799459i	$0.294880\pi$
$167$	−5.03158	+	12.1473i	−0.389355	+	0.939987i	−0.990077	+	0.140529i	$0.955120\pi$
$168$		0			0
$169$	−11.2205			−0.863115
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.43489	−	3.46413i	0.109093	−	0.263373i	−0.859900	−	0.510463i	$-0.829474\pi$
$173$	1.43489	−	3.46413i	0.109093	−	0.263373i	0.968993	+	0.247090i	$0.0794742\pi$
$174$		0			0
$175$	16.5434	+	6.85249i	1.25056	+	0.517999i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.8761	+	10.8761i	0.812920	+	0.812920i	0.985071	−	0.172151i	$-0.0550716\pi$
$179$	10.8761	+	10.8761i	0.812920	+	0.812920i	−0.172151	+	0.985071i	$0.555072\pi$
$180$		0			0
$181$	−4.49806	−	10.8593i	−0.334338	−	0.807163i	−0.998238	−	0.0593418i	$-0.981100\pi$
$181$	−4.49806	−	10.8593i	−0.334338	−	0.807163i	0.663900	−	0.747822i	$-0.268900\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.44453	+	4.44453i	−0.326769	+	0.326769i
$186$		0			0
$187$	−4.69494	+	13.7031i	−0.343328	+	1.00207i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			11.2765i			0.815942i	0.912995	+	0.407971i	$0.133764\pi$
$191$			11.2765i			0.815942i	−0.912995	+	0.407971i	$0.866236\pi$
$192$		0			0
$193$	2.61738	+	6.31891i	0.188403	+	0.454845i	0.989652	−	0.143485i	$-0.0458310\pi$
$193$	2.61738	+	6.31891i	0.188403	+	0.454845i	−0.801249	+	0.598330i	$0.795831\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.8719	−	6.16015i	1.05958	−	0.438893i	0.216278	−	0.976332i	$-0.430608\pi$
$197$	14.8719	−	6.16015i	1.05958	−	0.438893i	0.843303	+	0.537439i	$0.180608\pi$
$198$		0			0
$199$	−22.1641	−	9.18068i	−1.57117	−	0.650801i	−0.584189	−	0.811618i	$-0.698587\pi$
$199$	−22.1641	−	9.18068i	−1.57117	−	0.650801i	−0.986984	+	0.160816i	$0.948587\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	40.6212			2.85105
$204$		0			0
$205$	−8.03772			−0.561378
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.92119	−	1.21000i	−0.202063	−	0.0836973i
$210$		0			0
$211$	18.2289	−	7.55068i	1.25493	−	0.519810i	0.346582	−	0.938020i	$-0.387342\pi$
$211$	18.2289	−	7.55068i	1.25493	−	0.519810i	0.908350	+	0.418210i	$0.137342\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.52034	−	6.08463i	−0.171885	−	0.414968i
$216$		0			0
$217$		−	47.1132i		−	3.19825i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	13.4224	−	15.2180i	0.902887	−	1.02367i
$222$		0			0
$223$	6.27382	−	6.27382i	0.420126	−	0.420126i	−0.465121	−	0.885247i	$-0.653989\pi$
$223$	6.27382	−	6.27382i	0.420126	−	0.420126i	0.885247	+	0.465121i	$0.153989\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.67367	+	13.6974i	0.376575	+	0.909131i	0.992603	+	0.121407i	$0.0387407\pi$
$227$	5.67367	+	13.6974i	0.376575	+	0.909131i	−0.616028	+	0.787724i	$0.711259\pi$
$228$		0			0
$229$	5.40014	+	5.40014i	0.356851	+	0.356851i	0.862651	−	0.505800i	$-0.168802\pi$
$229$	5.40014	+	5.40014i	0.356851	+	0.356851i	−0.505800	+	0.862651i	$0.668802\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	23.8414	+	9.87542i	1.56190	+	0.646960i	0.985418	−	0.170150i	$-0.0544253\pi$
$233$	23.8414	+	9.87542i	1.56190	+	0.646960i	0.576481	+	0.817110i	$0.304425\pi$
$234$		0			0
$235$	−0.898478	+	2.16912i	−0.0586102	+	0.141498i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.5008			−0.873292			−0.436646	−	0.899633i	$-0.643834\pi$
$239$	−13.5008			−0.873292			−0.436646	+	0.899633i	$0.643834\pi$
$240$		0			0
$241$	1.52678	−	3.68596i	0.0983483	−	0.237434i	−0.867046	−	0.498228i	$-0.833984\pi$
$241$	1.52678	−	3.68596i	0.0983483	−	0.237434i	0.965394	+	0.260794i	$0.0839843\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.3544	−	4.28894i	0.661519	−	0.274010i
$246$		0			0
$247$	3.13203	+	3.13203i	0.199286	+	0.199286i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	4.78794i		−	0.302212i	−0.988518	−	0.151106i	$-0.951716\pi$
$251$		−	4.78794i		−	0.302212i	0.988518	−	0.151106i	$-0.0482835\pi$
$252$		0			0
$253$	−5.21408	+	5.21408i	−0.327806	+	0.327806i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.77007	−	6.77007i	0.422305	−	0.422305i	−0.463692	−	0.885997i	$-0.653475\pi$
$257$	6.77007	−	6.77007i	0.422305	−	0.422305i	0.885997	+	0.463692i	$0.153475\pi$
$258$		0			0
$259$		−	29.0872i		−	1.80739i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.0252	−	11.0252i	−0.679845	−	0.679845i	0.280120	−	0.959965i	$-0.409626\pi$
$263$	−11.0252	−	11.0252i	−0.679845	−	0.679845i	−0.959965	+	0.280120i	$0.909626\pi$
$264$		0			0
$265$	−7.38501	+	3.05897i	−0.453657	+	0.187911i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.56471	−	15.8486i	0.400258	−	0.966307i	−0.587346	−	0.809336i	$-0.699827\pi$
$269$	6.56471	−	15.8486i	0.400258	−	0.966307i	0.987603	−	0.156971i	$-0.0501729\pi$
$270$		0			0
$271$	−6.08385			−0.369567			−0.184784	−	0.982779i	$-0.559158\pi$
$271$	−6.08385			−0.369567			−0.184784	+	0.982779i	$0.559158\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.53415	−	13.3606i	0.333722	−	0.805676i
$276$		0			0
$277$	22.1354	+	9.16879i	1.32999	+	0.550899i	0.930652	−	0.365905i	$-0.119241\pi$
$277$	22.1354	+	9.16879i	1.32999	+	0.550899i	0.399336	+	0.916804i	$0.369241\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.88621	−	8.88621i	−0.530107	−	0.530107i	0.390497	−	0.920604i	$-0.372303\pi$
$281$	−8.88621	−	8.88621i	−0.530107	−	0.530107i	−0.920604	+	0.390497i	$0.872303\pi$
$282$		0			0
$283$	−1.53709	−	3.71085i	−0.0913703	−	0.220587i	0.871587	−	0.490240i	$-0.163091\pi$
$283$	−1.53709	−	3.71085i	−0.0913703	−	0.220587i	−0.962958	+	0.269653i	$0.913091\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	26.3014	−	26.3014i	1.55252	−	1.55252i
$288$		0			0
$289$	2.12332	+	16.8669i	0.124901	+	0.992169i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			3.51320i			0.205243i	0.994720	+	0.102622i	$0.0327231\pi$
$293$			3.51320i			0.205243i	−0.994720	+	0.102622i	$0.967277\pi$
$294$		0			0
$295$	1.61857	+	3.90757i	0.0942368	+	0.227508i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.54342	−	3.95301i	0.551910	−	0.228609i
$300$		0			0
$301$	28.1576	+	11.6632i	1.62298	+	0.672258i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.65273			−0.323674
$306$		0			0
$307$	23.4675			1.33936			0.669680	−	0.742650i	$-0.266431\pi$
$307$	23.4675			1.33936			0.669680	+	0.742650i	$0.266431\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.16053	−	0.480705i	−0.0658074	−	0.0272583i	0.349537	−	0.936923i	$-0.386339\pi$
$311$	−1.16053	−	0.480705i	−0.0658074	−	0.0272583i	−0.415344	+	0.909664i	$0.636339\pi$
$312$		0			0
$313$	−19.1009	+	7.91185i	−1.07965	+	0.447204i	−0.850385	−	0.526161i	$-0.823631\pi$
$313$	−19.1009	+	7.91185i	−1.07965	+	0.447204i	−0.229261	+	0.973365i	$0.573631\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.0766	+	31.5696i	0.734453	+	1.77313i	0.627148	+	0.778900i	$0.284222\pi$
$317$	13.0766	+	31.5696i	0.734453	+	1.77313i	0.107304	+	0.994226i	$0.465778\pi$
$318$		0			0
$319$		−	32.8062i		−	1.83679i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.70358	+	0.232200i	−0.206073	+	0.0129200i
$324$		0			0
$325$	−14.3249	+	14.3249i	−0.794604	+	0.794604i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.15784	−	10.0379i	−0.229229	−	0.553409i
$330$		0			0
$331$	0.159721	+	0.159721i	0.00877908	+	0.00877908i	0.711483	−	0.702704i	$-0.248024\pi$
$331$	0.159721	+	0.159721i	0.00877908	+	0.00877908i	−0.702704	+	0.711483i	$0.748024\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.62205	+	1.91452i	0.252530	+	0.104601i
$336$		0			0
$337$	7.72983	−	18.6615i	0.421071	−	1.01655i	−0.560962	−	0.827842i	$-0.689568\pi$
$337$	7.72983	−	18.6615i	0.421071	−	1.01655i	0.982032	−	0.188713i	$-0.0604316\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−38.0492			−2.06048
$342$		0			0
$343$	−8.19492	+	19.7843i	−0.442484	+	1.06825i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.518566	−	0.214797i	0.0278381	−	0.0115309i	−0.368721	−	0.929540i	$-0.620204\pi$
$347$	0.518566	−	0.214797i	0.0278381	−	0.0115309i	0.396559	+	0.918009i	$0.370204\pi$
$348$		0			0
$349$	17.9352	+	17.9352i	0.960050	+	0.960050i	0.999232	−	0.0391825i	$-0.0124754\pi$
$349$	17.9352	+	17.9352i	0.960050	+	0.960050i	−0.0391825	+	0.999232i	$0.512475\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	14.8321i		−	0.789434i	−0.918803	−	0.394717i	$-0.870843\pi$
$353$		−	14.8321i		−	0.789434i	0.918803	−	0.394717i	$-0.129157\pi$
$354$		0			0
$355$	−1.57238	+	1.57238i	−0.0834533	+	0.0834533i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−22.2821	+	22.2821i	−1.17601	+	1.17601i	−0.195253	+	0.980753i	$0.562553\pi$
$359$	−22.2821	+	22.2821i	−1.17601	+	1.17601i	−0.980753	+	0.195253i	$0.937447\pi$
$360$		0			0
$361$			18.1900i			0.957367i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.53672	+	6.53672i	0.342148	+	0.342148i
$366$		0			0
$367$	−32.8281	+	13.5978i	−1.71361	+	0.709801i	−0.713654	+	0.700499i	$0.752961\pi$
$367$	−32.8281	+	13.5978i	−1.71361	+	0.709801i	−0.999957	+	0.00930206i	$0.997039\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	14.1559	−	34.1753i	0.734936	−	1.77429i
$372$		0			0
$373$	20.8339			1.07874			0.539370	−	0.842069i	$-0.318662\pi$
$373$	20.8339			1.07874			0.539370	+	0.842069i	$0.318662\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−17.5870	+	42.4587i	−0.905776	+	2.18674i
$378$		0			0
$379$	−10.3816	−	4.30021i	−0.533268	−	0.220887i	0.0997659	−	0.995011i	$-0.468191\pi$
$379$	−10.3816	−	4.30021i	−0.533268	−	0.220887i	−0.633034	+	0.774124i	$0.718191\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.98948	+	7.98948i	0.408244	+	0.408244i	0.881126	−	0.472882i	$-0.156786\pi$
$383$	7.98948	+	7.98948i	0.408244	+	0.408244i	−0.472882	+	0.881126i	$0.656786\pi$
$384$		0			0
$385$	−5.49743	−	13.2720i	−0.280175	−	0.676402i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.9386	+	18.9386i	−0.960223	+	0.960223i	−0.999239	−	0.0390153i	$-0.987578\pi$
$389$	−18.9386	+	18.9386i	−0.960223	+	0.960223i	0.0390153	+	0.999239i	$0.487578\pi$
$390$		0			0
$391$	−2.80499	+	8.18690i	−0.141855	+	0.414029i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	7.76376i		−	0.390637i
$396$		0			0
$397$	−5.14245	−	12.4150i	−0.258092	−	0.623090i	0.740720	−	0.671814i	$-0.234485\pi$
$397$	−5.14245	−	12.4150i	−0.258092	−	0.623090i	−0.998812	+	0.0487238i	$0.984485\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	8.96722	−	3.71434i	0.447802	−	0.185485i	−0.147375	−	0.989081i	$-0.547082\pi$
$401$	8.96722	−	3.71434i	0.447802	−	0.185485i	0.595176	+	0.803595i	$0.297082\pi$
$402$		0			0
$403$	49.2444	+	20.3977i	2.45304	+	1.01608i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−23.4912			−1.16441
$408$		0			0
$409$	6.94609			0.343462			0.171731	−	0.985144i	$-0.445064\pi$
$409$	6.94609			0.343462			0.171731	+	0.985144i	$0.445064\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−18.0829	−	7.49018i	−0.889801	−	0.368568i
$414$		0			0
$415$	−1.75115	+	0.725348i	−0.0859603	+	0.0356059i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.57653	−	20.7056i	−0.418991	−	1.01153i	−0.982641	−	0.185519i	$-0.940603\pi$
$419$	−8.57653	−	20.7056i	−0.418991	−	1.01153i	0.563650	−	0.826014i	$-0.309397\pi$
$420$		0			0
$421$		−	2.41829i		−	0.117860i	−0.998262	−	0.0589302i	$-0.981231\pi$
$421$		−	2.41829i		−	0.117860i	0.998262	−	0.0589302i	$-0.0187689\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.06201	−	16.9390i	−0.0515152	−	0.821664i
$426$		0			0
$427$	18.4971	−	18.4971i	0.895139	−	0.895139i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.34790	−	15.3252i	−0.305768	−	0.738188i	−0.999833	−	0.0182762i	$-0.994182\pi$
$431$	−6.34790	−	15.3252i	−0.305768	−	0.738188i	0.694065	−	0.719912i	$-0.255818\pi$
$432$		0			0
$433$	3.10517	+	3.10517i	0.149225	+	0.149225i	0.777772	−	0.628547i	$-0.216350\pi$
$433$	3.10517	+	3.10517i	0.149225	+	0.149225i	−0.628547	+	0.777772i	$0.716350\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.74527	−	0.722913i	−0.0834874	−	0.0345816i
$438$		0			0
$439$	9.61095	−	23.2029i	0.458706	−	1.10741i	−0.510216	−	0.860046i	$-0.670435\pi$
$439$	9.61095	−	23.2029i	0.458706	−	1.10741i	0.968922	−	0.247367i	$-0.0795655\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.9896			0.664668			0.332334	−	0.943162i	$-0.392164\pi$
$443$	13.9896			0.664668			0.332334	+	0.943162i	$0.392164\pi$
$444$		0			0
$445$	−1.06430	+	2.56944i	−0.0504525	+	0.121803i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−23.8707	+	9.88756i	−1.12653	+	0.466623i	−0.866599	−	0.499005i	$-0.833699\pi$
$449$	−23.8707	+	9.88756i	−1.12653	+	0.466623i	−0.259928	+	0.965628i	$0.583699\pi$
$450$		0			0
$451$	−21.2413	−	21.2413i	−1.00021	−	1.00021i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			20.1241i			0.943431i
$456$		0			0
$457$	−25.7263	+	25.7263i	−1.20343	+	1.20343i	−0.230307	+	0.973118i	$0.573973\pi$
$457$	−25.7263	+	25.7263i	−1.20343	+	1.20343i	−0.973118	+	0.230307i	$0.926027\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.93178	+	4.93178i	−0.229696	+	0.229696i	−0.812566	−	0.582870i	$-0.801930\pi$
$461$	−4.93178	+	4.93178i	−0.229696	+	0.229696i	0.582870	+	0.812566i	$0.301930\pi$
$462$		0			0
$463$		−	10.5659i		−	0.491039i	−0.969392	−	0.245519i	$-0.921042\pi$
$463$		−	10.5659i		−	0.491039i	0.969392	−	0.245519i	$-0.0789585\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.96529	+	2.96529i	0.137217	+	0.137217i	0.772379	−	0.635162i	$-0.219067\pi$
$467$	2.96529	+	2.96529i	0.137217	+	0.137217i	−0.635162	+	0.772379i	$0.719067\pi$
$468$		0			0
$469$	−21.3893	+	8.85972i	−0.987664	+	0.409104i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.41938	−	22.7404i	0.433103	−	1.04560i
$474$		0			0
$475$	3.70480			0.169988
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.07464	−	17.0797i	0.323249	−	0.780392i	−0.675813	−	0.737073i	$-0.736207\pi$
$479$	7.07464	−	17.0797i	0.323249	−	0.780392i	0.999061	−	0.0433180i	$-0.0137929\pi$
$480$		0			0
$481$	30.4030	+	12.5933i	1.38626	+	0.574206i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.41882	−	5.41882i	−0.246056	−	0.246056i
$486$		0			0
$487$	−3.19288	−	7.70828i	−0.144683	−	0.349296i	0.834880	−	0.550431i	$-0.185537\pi$
$487$	−3.19288	−	7.70828i	−0.144683	−	0.349296i	−0.979563	+	0.201136i	$0.935537\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15.7693	+	15.7693i	−0.711657	+	0.711657i	−0.966882	−	0.255225i	$-0.917851\pi$
$491$	−15.7693	+	15.7693i	−0.711657	+	0.711657i	0.255225	+	0.966882i	$0.417851\pi$
$492$		0			0
$493$	−16.9311	−	34.5797i	−0.762537	−	1.55739i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	10.2904i		−	0.461589i
$498$		0			0
$499$	11.3604	+	27.4263i	0.508559	+	1.22777i	0.944713	+	0.327898i	$0.106340\pi$
$499$	11.3604	+	27.4263i	0.508559	+	1.22777i	−0.436154	+	0.899872i	$0.643660\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	37.9651	−	15.7257i	1.69278	−	0.701173i	0.692976	−	0.720960i	$-0.256299\pi$
$503$	37.9651	−	15.7257i	1.69278	−	0.701173i	0.999804	+	0.0197879i	$0.00629910\pi$
$504$		0			0
$505$	13.1980	+	5.46680i	0.587305	+	0.243270i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	25.2428			1.11887			0.559434	−	0.828875i	$-0.311018\pi$
$509$	25.2428			1.11887			0.559434	+	0.828875i	$0.311018\pi$
$510$		0			0
$511$	−42.7795			−1.89245
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.81608	+	3.23753i	0.344418	+	0.142662i
$516$		0			0
$517$	−8.10675	+	3.35793i	−0.356534	+	0.147681i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.15751	+	5.20868i	0.0945221	+	0.228197i	0.964068	−	0.265656i	$-0.0855884\pi$
$521$	2.15751	+	5.20868i	0.0945221	+	0.228197i	−0.869546	+	0.493852i	$0.835588\pi$
$522$		0			0
$523$			0.995064i			0.0435111i	0.999763	+	0.0217556i	$0.00692555\pi$
$523$			0.995064i			0.0435111i	−0.999763	+	0.0217556i	$0.993074\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−40.1061	+	19.6369i	−1.74705	+	0.855399i
$528$		0			0
$529$	13.1483	−	13.1483i	0.571665	−	0.571665i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	16.1039	+	38.8783i	0.697539	+	1.68401i
$534$		0			0
$535$	−3.54624	−	3.54624i	−0.153317	−	0.153317i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	38.6981	+	16.0293i	1.66684	+	0.690429i
$540$		0			0
$541$	2.40389	−	5.80350i	0.103351	−	0.249512i	−0.863742	−	0.503934i	$-0.831885\pi$
$541$	2.40389	−	5.80350i	0.103351	−	0.249512i	0.967093	+	0.254422i	$0.0818854\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.54153			0.237373
$546$		0			0
$547$	−3.09833	+	7.48004i	−0.132475	+	0.319823i	−0.976173	−	0.216996i	$-0.930374\pi$
$547$	−3.09833	+	7.48004i	−0.132475	+	0.319823i	0.843697	+	0.536819i	$0.180374\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7.76470	−	3.21625i	0.330787	−	0.137017i
$552$		0			0
$553$	25.4049	+	25.4049i	1.08033	+	1.08033i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			36.7684i			1.55793i	0.627069	+	0.778963i	$0.284254\pi$
$557$			36.7684i			1.55793i	−0.627069	+	0.778963i	$0.715746\pi$
$558$		0			0
$559$	−24.3817	+	24.3817i	−1.03124	+	1.03124i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.6765	+	20.6765i	−0.871413	+	0.871413i	−0.992626	−	0.121214i	$-0.961321\pi$
$563$	−20.6765	+	20.6765i	−0.871413	+	0.871413i	0.121214	+	0.992626i	$0.461321\pi$
$564$		0			0
$565$			14.3717i			0.604622i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.522498	−	0.522498i	−0.0219043	−	0.0219043i	0.696070	−	0.717974i	$-0.254930\pi$
$569$	−0.522498	−	0.522498i	−0.0219043	−	0.0219043i	−0.717974	+	0.696070i	$0.754930\pi$
$570$		0			0
$571$	2.45322	−	1.01616i	0.102664	−	0.0425249i	−0.330760	−	0.943715i	$-0.607305\pi$
$571$	2.45322	−	1.01616i	0.102664	−	0.0425249i	0.433424	+	0.901190i	$0.357305\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.30638	−	7.98231i	0.137886	−	0.332885i
$576$		0			0
$577$	5.39898			0.224762			0.112381	−	0.993665i	$-0.464152\pi$
$577$	5.39898			0.224762			0.112381	+	0.993665i	$0.464152\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.35666	−	8.10370i	0.139258	−	0.336198i
$582$		0			0
$583$	−27.6004	−	11.4324i	−1.14309	−	0.473483i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.70079	−	6.70079i	−0.276571	−	0.276571i	0.555168	−	0.831739i	$-0.312654\pi$
$587$	−6.70079	−	6.70079i	−0.276571	−	0.276571i	−0.831739	+	0.555168i	$0.812654\pi$
$588$		0			0
$589$	−3.73026	−	9.00564i	−0.153703	−	0.371071i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11.3803	+	11.3803i	−0.467334	+	0.467334i	−0.901050	−	0.433715i	$-0.857202\pi$
$593$	−11.3803	+	11.3803i	−0.467334	+	0.467334i	0.433715	+	0.901050i	$0.357202\pi$
$594$		0			0
$595$	−12.6442	−	11.1522i	−0.518361	−	0.457198i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			6.05959i			0.247588i	0.992308	+	0.123794i	$0.0395062\pi$
$599$			6.05959i			0.247588i	−0.992308	+	0.123794i	$0.960494\pi$
$600$		0			0
$601$	12.0483	+	29.0871i	0.491459	+	1.18649i	0.953978	+	0.299878i	$0.0969459\pi$
$601$	12.0483	+	29.0871i	0.491459	+	1.18649i	−0.462518	+	0.886610i	$0.653054\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.16560	+	0.482807i	−0.0473883	+	0.0196289i
$606$		0			0
$607$	−5.88706	−	2.43850i	−0.238948	−	0.0989757i	0.259996	−	0.965610i	$-0.416279\pi$
$607$	−5.88706	−	2.43850i	−0.238948	−	0.0989757i	−0.498944	+	0.866634i	$0.666279\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.2921			0.497287
$612$		0			0
$613$	2.80845			0.113432			0.0567161	−	0.998390i	$-0.481937\pi$
$613$	2.80845			0.113432			0.0567161	+	0.998390i	$0.481937\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	10.8330	+	4.48716i	0.436119	+	0.180646i	0.589931	−	0.807454i	$-0.299155\pi$
$617$	10.8330	+	4.48716i	0.436119	+	0.180646i	−0.153812	+	0.988100i	$0.549155\pi$
$618$		0			0
$619$	−17.0624	+	7.06746i	−0.685794	+	0.284065i	−0.698247	−	0.715857i	$-0.746036\pi$
$619$	−17.0624	+	7.06746i	−0.685794	+	0.284065i	0.0124526	+	0.999922i	$0.496036\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.92520	−	11.8905i	−0.197324	−	0.476382i
$624$		0			0
$625$			12.5265i			0.501062i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−24.7611	+	12.1237i	−0.987290	+	0.483402i
$630$		0			0
$631$	25.3372	−	25.3372i	1.00866	−	1.00866i	0.00869728	−	0.999962i	$-0.497232\pi$
$631$	25.3372	−	25.3372i	1.00866	−	1.00866i	0.999962	−	0.00869728i	$-0.00276846\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.16412	−	12.4673i	−0.204932	−	0.494749i
$636$		0			0
$637$	−41.4911	−	41.4911i	−1.64394	−	1.64394i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−29.4216	−	12.1868i	−1.16208	−	0.481351i	−0.283516	−	0.958967i	$-0.591501\pi$
$641$	−29.4216	−	12.1868i	−1.16208	−	0.481351i	−0.878569	+	0.477616i	$0.841501\pi$
$642$		0			0
$643$	9.99336	−	24.1261i	0.394100	−	0.951440i	−0.594937	−	0.803772i	$-0.702823\pi$
$643$	9.99336	−	24.1261i	0.394100	−	0.951440i	0.989037	−	0.147668i	$-0.0471768\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−38.0869			−1.49735			−0.748675	−	0.662937i	$-0.769310\pi$
$647$	−38.0869			−1.49735			−0.748675	+	0.662937i	$0.769310\pi$
$648$		0			0
$649$	−6.04916	+	14.6040i	−0.237450	+	0.573256i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.38304	+	2.64394i	−0.249787	+	0.103465i	−0.504064	−	0.863666i	$-0.668163\pi$
$653$	−6.38304	+	2.64394i	−0.249787	+	0.103465i	0.254277	+	0.967131i	$0.418163\pi$
$654$		0			0
$655$	−11.7395	−	11.7395i	−0.458700	−	0.458700i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			40.6022i			1.58164i	0.612050	+	0.790819i	$0.290345\pi$
$659$			40.6022i			1.58164i	−0.612050	+	0.790819i	$0.709655\pi$
$660$		0			0
$661$	−21.8362	+	21.8362i	−0.849331	+	0.849331i	−0.990050	−	0.140719i	$-0.955059\pi$
$661$	−21.8362	+	21.8362i	−0.849331	+	0.849331i	0.140719	+	0.990050i	$0.455059\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.60231	−	2.60231i	0.100913	−	0.100913i
$666$		0			0
$667$		−	19.6001i		−	0.758917i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−14.9385	−	14.9385i	−0.576694	−	0.576694i
$672$		0			0
$673$	−36.1846	+	14.9881i	−1.39481	+	0.577750i	−0.948400	−	0.317077i	$-0.897299\pi$
$673$	−36.1846	+	14.9881i	−1.39481	+	0.577750i	−0.446413	+	0.894827i	$0.647299\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.82043	+	9.22334i	−0.146831	+	0.354482i	−0.980134	−	0.198334i	$-0.936447\pi$
$677$	−3.82043	+	9.22334i	−0.146831	+	0.354482i	0.833303	+	0.552816i	$0.186447\pi$
$678$		0			0
$679$	35.4634			1.36096
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.9930	+	26.5395i	−0.420637	+	1.01551i	0.561523	+	0.827461i	$0.310215\pi$
$683$	−10.9930	+	26.5395i	−0.420637	+	1.01551i	−0.982160	+	0.188046i	$0.939785\pi$
$684$		0			0
$685$	9.64018	+	3.99309i	0.368332	+	0.152568i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	29.5924	+	29.5924i	1.12738	+	1.12738i
$690$		0			0
$691$	−12.8432	−	31.0062i	−0.488578	−	1.17953i	−0.955436	−	0.295199i	$-0.904614\pi$
$691$	−12.8432	−	31.0062i	−0.488578	−	1.17953i	0.466858	−	0.884332i	$-0.345386\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.32583	+	8.32583i	−0.315817	+	0.315817i
$696$		0			0
$697$	−33.3521	−	11.4271i	−1.26330	−	0.432832i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	30.8627i		−	1.16567i	−0.812592	−	0.582833i	$-0.801944\pi$
$701$		−	30.8627i		−	1.16567i	0.812592	−	0.582833i	$-0.198056\pi$
$702$		0			0
$703$	−2.30302	−	5.55999i	−0.0868602	−	0.209699i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−61.0759	+	25.2985i	−2.29700	+	0.951447i
$708$		0			0
$709$	−43.1783	−	17.8850i	−1.62160	−	0.671687i	−0.627343	−	0.778743i	$-0.715858\pi$
$709$	−43.1783	−	17.8850i	−1.62160	−	0.671687i	−0.994253	+	0.107056i	$0.965858\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−22.7325			−0.851338
$714$		0			0
$715$	16.2524			0.607807
$716$		0			0
$717$		0			0
$718$		0			0
$719$	41.7644	+	17.2994i	1.55755	+	0.645158i	0.984662	−	0.174475i	$-0.0558230\pi$
$719$	41.7644	+	17.2994i	1.55755	+	0.645158i	0.572888	+	0.819634i	$0.305823\pi$
$720$		0			0
$721$	−36.1701	+	14.9822i	−1.34705	+	0.557965i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	14.7101	+	35.5133i	0.546320	+	1.31893i
$726$		0			0
$727$			16.0428i			0.594995i	0.954723	+	0.297497i	$0.0961519\pi$
$727$			16.0428i			0.594995i	−0.954723	+	0.297497i	$0.903848\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.80759	−	28.8310i	−0.0668562	−	1.06635i
$732$		0			0
$733$	−1.64615	+	1.64615i	−0.0608018	+	0.0608018i	−0.736854	−	0.676052i	$-0.763689\pi$
$733$	−1.64615	+	1.64615i	−0.0608018	+	0.0608018i	0.676052	+	0.736854i	$0.263689\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.15522	+	17.2742i	0.263566	+	0.636304i
$738$		0			0
$739$	18.3855	+	18.3855i	0.676321	+	0.676321i	0.959166	−	0.282845i	$-0.0912781\pi$
$739$	18.3855	+	18.3855i	0.676321	+	0.676321i	−0.282845	+	0.959166i	$0.591278\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.98172	+	2.47771i	0.219448	+	0.0908983i	0.489699	−	0.871892i	$-0.337107\pi$
$743$	5.98172	+	2.47771i	0.219448	+	0.0908983i	−0.270251	+	0.962790i	$0.587107\pi$
$744$		0			0
$745$	1.75461	−	4.23601i	0.0642841	−	0.155195i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	23.2083			0.848014
$750$		0			0
$751$	−10.3783	+	25.0555i	−0.378711	+	0.914288i	0.613498	+	0.789697i	$0.289762\pi$
$751$	−10.3783	+	25.0555i	−0.378711	+	0.914288i	−0.992208	+	0.124592i	$0.960238\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10.3649	+	4.29327i	−0.377216	+	0.156248i
$756$		0			0
$757$	18.6848	+	18.6848i	0.679110	+	0.679110i	0.959799	−	0.280689i	$-0.0905629\pi$
$757$	18.6848	+	18.6848i	0.679110	+	0.679110i	−0.280689	+	0.959799i	$0.590563\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	51.5814i		−	1.86983i	−0.354878	−	0.934913i	$-0.615478\pi$
$761$		−	51.5814i		−	1.86983i	0.354878	−	0.934913i	$-0.384522\pi$
$762$		0			0
$763$	−18.1333	+	18.1333i	−0.656468	+	0.656468i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	15.6580	−	15.6580i	0.565378	−	0.565378i
$768$		0			0
$769$			3.17588i			0.114525i	0.998359	+	0.0572625i	$0.0182372\pi$
$769$			3.17588i			0.114525i	−0.998359	+	0.0572625i	$0.981763\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.10858	−	8.10858i	−0.291645	−	0.291645i	0.546085	−	0.837730i	$-0.316118\pi$
$773$	−8.10858	−	8.10858i	−0.291645	−	0.291645i	−0.837730	+	0.546085i	$0.816118\pi$
$774$		0			0
$775$	41.1890	−	17.0610i	1.47955	−	0.612850i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.94503	−	7.10993i	0.105517	−	0.254740i
$780$		0			0
$781$	−8.31068			−0.297379
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.780482	−	1.88425i	0.0278566	−	0.0672518i
$786$		0			0
$787$	44.7759	+	18.5468i	1.59609	+	0.661121i	0.990855	−	0.134928i	$-0.0430803\pi$
$787$	44.7759	+	18.5468i	1.59609	+	0.661121i	0.605232	+	0.796049i	$0.293080\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−47.0278	−	47.0278i	−1.67211	−	1.67211i
$792$		0			0
$793$	11.3255	+	27.3422i	0.402181	+	0.970950i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−24.7797	+	24.7797i	−0.877743	+	0.877743i	−0.993301	−	0.115558i	$-0.963135\pi$
$797$	−24.7797	+	24.7797i	−0.877743	+	0.877743i	0.115558	+	0.993301i	$0.463135\pi$
$798$		0			0
$799$	−6.81199	+	7.72329i	−0.240991	+	0.273230i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			34.5493i			1.21922i
$804$		0			0
$805$	−3.28444	−	7.92934i	−0.115761	−	0.279473i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−37.6761	+	15.6059i	−1.32462	+	0.548676i	−0.929116	−	0.369788i	$-0.879430\pi$
$809$	−37.6761	+	15.6059i	−1.32462	+	0.548676i	−0.395505	+	0.918464i	$0.629430\pi$
$810$		0			0
$811$	38.5122	+	15.9523i	1.35235	+	0.560160i	0.936945	−	0.349478i	$-0.113641\pi$
$811$	38.5122	+	15.9523i	1.35235	+	0.560160i	0.415402	+	0.909638i	$0.363641\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	13.1306			0.459943
$816$		0			0
$817$	6.30574			0.220610
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−29.2465	−	12.1143i	−1.02071	−	0.422791i	−0.191358	−	0.981520i	$-0.561289\pi$
$821$	−29.2465	−	12.1143i	−1.02071	−	0.422791i	−0.829350	+	0.558729i	$0.811289\pi$
$822$		0			0
$823$	2.75434	−	1.14088i	0.0960103	−	0.0397687i	−0.334161	−	0.942516i	$-0.608453\pi$
$823$	2.75434	−	1.14088i	0.0960103	−	0.0397687i	0.430172	+	0.902747i	$0.358453\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18.7218	−	45.1985i	−0.651022	−	1.57171i	−0.811297	−	0.584634i	$-0.801238\pi$
$827$	−18.7218	−	45.1985i	−0.651022	−	1.57171i	0.160275	−	0.987072i	$-0.448762\pi$
$828$		0			0
$829$		−	10.0212i		−	0.348049i	−0.984741	−	0.174025i	$-0.944323\pi$
$829$		−	10.0212i		−	0.348049i	0.984741	−	0.174025i	$-0.0556772\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	49.0626	−	3.07604i	1.69992	−	0.106578i
$834$		0			0
$835$	−8.73940	+	8.73940i	−0.302439	+	0.302439i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	11.1965	+	27.0306i	0.386544	+	0.933201i	0.990666	+	0.136309i	$0.0435240\pi$
$839$	11.1965	+	27.0306i	0.386544	+	0.933201i	−0.604122	+	0.796892i	$0.706476\pi$
$840$		0			0
$841$	41.1542	+	41.1542i	1.41911	+	1.41911i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.74449	−	4.03630i	−0.335221	−	0.138853i
$846$		0			0
$847$	2.23426	−	5.39399i	0.0767702	−	0.185340i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−14.0348			−0.481107
$852$		0			0
$853$	−5.35358	+	12.9247i	−0.183303	+	0.442533i	−0.988644	−	0.150280i	$-0.951983\pi$
$853$	−5.35358	+	12.9247i	−0.183303	+	0.442533i	0.805340	+	0.592813i	$0.201983\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	42.4058	−	17.5651i	1.44856	−	0.600011i	0.486700	−	0.873569i	$-0.338201\pi$
$857$	42.4058	−	17.5651i	1.44856	−	0.600011i	0.961855	+	0.273558i	$0.0882006\pi$
$858$		0			0
$859$	−17.1914	−	17.1914i	−0.586562	−	0.586562i	0.350136	−	0.936699i	$-0.386135\pi$
$859$	−17.1914	−	17.1914i	−0.586562	−	0.586562i	−0.936699	+	0.350136i	$0.886135\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			7.14634i			0.243264i	0.992575	+	0.121632i	$0.0388128\pi$
$863$			7.14634i			0.243264i	−0.992575	+	0.121632i	$0.961187\pi$
$864$		0			0
$865$	2.49227	−	2.49227i	0.0847399	−	0.0847399i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	20.5173	−	20.5173i	0.696003	−	0.696003i
$870$		0			0
$871$		−	26.1926i		−	0.887503i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	26.3592	+	26.3592i	0.891103	+	0.891103i
$876$		0			0
$877$	19.2719	−	7.98267i	0.650765	−	0.269556i	−0.0327817	−	0.999463i	$-0.510437\pi$
$877$	19.2719	−	7.98267i	0.650765	−	0.269556i	0.683547	+	0.729907i	$0.260437\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.5058	+	35.0201i	−0.488712	+	1.17986i	0.466656	+	0.884439i	$0.345459\pi$
$881$	−14.5058	+	35.0201i	−0.488712	+	1.17986i	−0.955368	+	0.295417i	$0.904541\pi$
$882$		0			0
$883$	42.2764			1.42271			0.711357	−	0.702831i	$-0.248081\pi$
$883$	42.2764			1.42271			0.711357	+	0.702831i	$0.248081\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.149217	−	0.360242i	0.00501022	−	0.0120957i	−0.921355	−	0.388723i	$-0.872916\pi$
$887$	0.149217	−	0.360242i	0.00501022	−	0.0120957i	0.926365	+	0.376627i	$0.122916\pi$
$888$		0			0
$889$	57.6943	+	23.8978i	1.93500	+	0.801505i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.58954	−	1.58954i	−0.0531918	−	0.0531918i
$894$		0			0
$895$	5.53300	+	13.3578i	0.184948	+	0.446504i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	71.5146	−	71.5146i	2.38515	−	2.38515i
$900$		0			0
$901$	−34.9926	+	2.19390i	−1.16577	+	0.0730895i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	11.0489i		−	0.367276i
$906$		0			0
$907$	−2.41165	−	5.82224i	−0.0800776	−	0.193324i	0.878770	−	0.477245i	$-0.158365\pi$
$907$	−2.41165	−	5.82224i	−0.0800776	−	0.193324i	−0.958848	+	0.283921i	$0.908365\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−36.8163	+	15.2498i	−1.21978	+	0.505249i	−0.897339	−	0.441342i	$-0.854503\pi$
$911$	−36.8163	+	15.2498i	−1.21978	+	0.505249i	−0.322439	+	0.946590i	$0.604503\pi$
$912$		0			0
$913$	−6.54464	−	2.71088i	−0.216596	−	0.0897170i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	76.8291			2.53712
$918$		0			0
$919$	−25.8983			−0.854306			−0.427153	−	0.904179i	$-0.640483\pi$
$919$	−25.8983			−0.854306			−0.427153	+	0.904179i	$0.640483\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.7559	+	4.45525i	0.354036	+	0.146646i
$924$		0			0
$925$	25.4297	−	10.5333i	0.836122	−	0.346333i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.4982	−	47.0729i	−0.639716	−	1.54441i	−0.827058	−	0.562117i	$-0.809987\pi$
$929$	−19.4982	−	47.0729i	−0.639716	−	1.54441i	0.187342	−	0.982295i	$-0.440013\pi$
$930$		0			0
$931$			10.7307i			0.351684i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−9.00669	+	10.2116i	−0.294550	+	0.333955i
$936$		0			0
$937$	19.9910	−	19.9910i	0.653077	−	0.653077i	−0.300656	−	0.953733i	$-0.597206\pi$
$937$	19.9910	−	19.9910i	0.653077	−	0.653077i	0.953733	+	0.300656i	$0.0972056\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.6144	−	32.8680i	−0.443815	−	1.07147i	−0.974599	−	0.223958i	$-0.928102\pi$
$941$	−13.6144	−	32.8680i	−0.443815	−	1.07147i	0.530783	−	0.847507i	$-0.321898\pi$
$942$		0			0
$943$	−12.6906	−	12.6906i	−0.413264	−	0.413264i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.94931	+	2.46428i	0.193327	+	0.0800785i	0.477246	−	0.878770i	$-0.341635\pi$
$947$	5.94931	+	2.46428i	0.193327	+	0.0800785i	−0.283920	+	0.958848i	$0.591635\pi$
$948$		0			0
$949$	18.5214	−	44.7147i	0.601231	−	1.45150i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.04100			−0.292867			−0.146433	−	0.989221i	$-0.546779\pi$
$953$	−9.04100			−0.292867			−0.146433	+	0.989221i	$0.546779\pi$
$954$		0			0
$955$	−4.05646	+	9.79317i	−0.131264	+	0.316899i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−44.6114	+	18.4787i	−1.44058	+	0.596707i
$960$		0			0
$961$	−61.0236	−	61.0236i	−1.96850	−	1.96850i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			6.42922i			0.206964i
$966$		0			0
$967$	5.38948	−	5.38948i	0.173314	−	0.173314i	−0.615120	−	0.788434i	$-0.710892\pi$
$967$	5.38948	−	5.38948i	0.173314	−	0.173314i	0.788434	+	0.615120i	$0.210892\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	21.8198	−	21.8198i	0.700230	−	0.700230i	−0.264230	−	0.964460i	$-0.585118\pi$
$971$	21.8198	−	21.8198i	0.700230	−	0.700230i	0.964460	+	0.264230i	$0.0851178\pi$
$972$		0			0
$973$		−	54.4883i		−	1.74682i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−36.8161	−	36.8161i	−1.17785	−	1.17785i	−0.980291	−	0.197561i	$-0.936698\pi$
$977$	−36.8161	−	36.8161i	−1.17785	−	1.17785i	−0.197561	−	0.980291i	$-0.563302\pi$
$978$		0			0
$979$	−9.60289	+	3.97765i	−0.306910	+	0.127126i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.47857	−	5.98379i	0.0790541	−	0.190853i	−0.879411	−	0.476063i	$-0.842063\pi$
$983$	2.47857	−	5.98379i	0.0790541	−	0.190853i	0.958465	+	0.285210i	$0.0920633\pi$
$984$		0			0
$985$	15.1316			0.482131
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.62761	−	13.5863i	0.178948	−	0.432018i
$990$		0			0
$991$	8.51296	+	3.52618i	0.270423	+	0.112013i	0.513774	−	0.857925i	$-0.328247\pi$
$991$	8.51296	+	3.52618i	0.270423	+	0.112013i	−0.243351	+	0.969938i	$0.578247\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−15.9460	−	15.9460i	−0.505523	−	0.505523i
$996$		0			0
$997$	12.3401	+	29.7916i	0.390814	+	0.943508i	0.989763	+	0.142721i	$0.0455852\pi$
$997$	12.3401	+	29.7916i	0.390814	+	0.943508i	−0.598949	+	0.800787i	$0.704415\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1224.2.bq.e.433.3		16
3.2	odd	2		408.2.ba.a.25.3	✓	16
12.11	even	2		816.2.bq.f.433.1		16
17.15	even	8	inner	1224.2.bq.e.865.3		16
51.32	odd	8		408.2.ba.a.49.3	yes	16
51.41	even	16		6936.2.a.bl.1.6		8
51.44	even	16		6936.2.a.bo.1.3		8
204.83	even	8		816.2.bq.f.49.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.ba.a.25.3	✓	16	3.2	odd	2
408.2.ba.a.49.3	yes	16	51.32	odd	8
816.2.bq.f.49.1		16	204.83	even	8
816.2.bq.f.433.1		16	12.11	even	2
1224.2.bq.e.433.3		16	1.1	even	1	trivial
1224.2.bq.e.865.3		16	17.15	even	8	inner
6936.2.a.bl.1.6		8	51.41	even	16
6936.2.a.bo.1.3		8	51.44	even	16

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1224.2.bq.e.433.3

Level	$1224$
Weight	$2$
Character	1224.433
Analytic conductor	$9.774$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$