LMFDB - Embedded newform 240.9.c.f.209.35

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [240,9,Mod(209,240)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 9, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("240.209"); S:= CuspForms(chi, 9); N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	240.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [48,0,0,0,0,0,0,0,2528] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$97.7708664147$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.35
Character	$\chi$	$=$	240.209
Dual form			240.9.c.f.209.36

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(51.8771 - 62.2074i) q^{3} +(-254.792 - 570.707i) q^{5} +4202.69i q^{7} +(-1178.52 - 6454.29i) q^{9} -542.921i q^{11} -40742.5i q^{13} +(-48720.1 - 13756.7i) q^{15} +31206.2 q^{17} +230675. q^{19} +(261439. + 218024. i) q^{21} -465190. q^{23} +(-260787. + 290823. i) q^{25} +(-462643. - 261517. i) q^{27} +231259. i q^{29} +206459. q^{31} +(-33773.7 - 28165.2i) q^{33} +(2.39850e6 - 1.07081e6i) q^{35} -2.73444e6i q^{37} +(-2.53449e6 - 2.11360e6i) q^{39} -2.74391e6i q^{41} -638785. i q^{43} +(-3.38323e6 + 2.31709e6i) q^{45} -2.05881e6 q^{47} -1.18978e7 q^{49} +(1.61889e6 - 1.94126e6i) q^{51} -1.09780e7 q^{53} +(-309848. + 138332. i) q^{55} +(1.19668e7 - 1.43497e7i) q^{57} +5.20695e6i q^{59} -7932.68 q^{61} +(2.71254e7 - 4.95297e6i) q^{63} +(-2.32520e7 + 1.03809e7i) q^{65} +2.61289e7i q^{67} +(-2.41327e7 + 2.89383e7i) q^{69} -4.61106e7i q^{71} +3.27109e7i q^{73} +(4.56245e6 + 3.13100e7i) q^{75} +2.28173e6 q^{77} -9.49611e6 q^{79} +(-4.02689e7 + 1.52130e7i) q^{81} -6.39962e7 q^{83} +(-7.95109e6 - 1.78096e7i) q^{85} +(1.43860e7 + 1.19971e7i) q^{87} +4.56560e7i q^{89} +1.71228e8 q^{91} +(1.07105e7 - 1.28433e7i) q^{93} +(-5.87743e7 - 1.31648e8i) q^{95} +2.57539e7i q^{97} +(-3.50417e6 + 639845. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 2528 q^{9} - 132352 q^{15} + 116176 q^{21} + 56976 q^{25} - 1395648 q^{31} - 6888832 q^{39} - 4287056 q^{45} - 30813552 q^{49} + 22815168 q^{51} + 6062784 q^{55} + 14031936 q^{61} + 2522608 q^{69}+ \cdots + 21719360 q^{99}+O(q^{100}) $ 48 * q + 2528 * q^9 - 132352 * q^15 + 116176 * q^21 + 56976 * q^25 - 1395648 * q^31 - 6888832 * q^39 - 4287056 * q^45 - 30813552 * q^49 + 22815168 * q^51 + 6062784 * q^55 + 14031936 * q^61 + 2522608 * q^69 - 41684416 * q^75 - 19866432 * q^79 - 93539312 * q^81 - 116297664 * q^85 - 16826880 * q^91 + 21719360 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	51.8771	−	62.2074i	0.640459	−	0.767993i
$3$	51.8771	−	62.2074i	0.640459	−	0.767993i
$4$		0			0
$5$	−254.792	−	570.707i	−0.407667	−	0.913131i
$5$	−254.792	−	570.707i	−0.407667	−	0.913131i
$6$		0			0
$7$			4202.69i			1.75039i	0.483768	+	0.875196i	$0.339268\pi$
$7$			4202.69i			1.75039i	−0.483768	+	0.875196i	$0.660732\pi$
$8$		0			0
$9$	−1178.52	−	6454.29i	−0.179625	−	0.983735i
$10$		0			0
$11$		−	542.921i		−	0.0370822i	−0.999828	−	0.0185411i	$-0.994098\pi$
$11$		−	542.921i		−	0.0370822i	0.999828	−	0.0185411i	$-0.00590216\pi$
$12$		0			0
$13$		−	40742.5i		−	1.42651i	−0.700906	−	0.713254i	$-0.747221\pi$
$13$		−	40742.5i		−	1.42651i	0.700906	−	0.713254i	$-0.252779\pi$
$14$		0			0
$15$	−48720.1	−	13756.7i	−0.962372	−	0.271737i
$16$		0			0
$17$	31206.2			0.373633			0.186816	−	0.982395i	$-0.440183\pi$
$17$	31206.2			0.373633			0.186816	+	0.982395i	$0.440183\pi$
$18$		0			0
$19$	230675.			1.77006			0.885028	−	0.465538i	$-0.154139\pi$
$19$	230675.			1.77006			0.885028	+	0.465538i	$0.154139\pi$
$20$		0			0
$21$	261439.	+	218024.i	1.34429	+	1.12105i
$22$		0			0
$23$	−465190.			−1.66234			−0.831168	−	0.556021i	$-0.812328\pi$
$23$	−465190.			−1.66234			−0.831168	+	0.556021i	$0.812328\pi$
$24$		0			0
$25$	−260787.	+	290823.i	−0.667615	+	0.744507i
$26$		0			0
$27$	−462643.	−	261517.i	−0.870544	−	0.492091i
$28$		0			0
$29$			231259.i			0.326969i	0.986546	+	0.163485i	$0.0522734\pi$
$29$			231259.i			0.326969i	−0.986546	+	0.163485i	$0.947727\pi$
$30$		0			0
$31$	206459.			0.223556			0.111778	−	0.993733i	$-0.464345\pi$
$31$	206459.			0.223556			0.111778	+	0.993733i	$0.464345\pi$
$32$		0			0
$33$	−33773.7	−	28165.2i	−0.0284789	−	0.0237496i
$34$		0			0
$35$	2.39850e6	−	1.07081e6i	1.59834	−	0.713577i
$36$		0			0
$37$		−	2.73444e6i		−	1.45902i	−0.683970	−	0.729510i	$-0.739748\pi$
$37$		−	2.73444e6i		−	1.45902i	0.683970	−	0.729510i	$-0.260252\pi$
$38$		0			0
$39$	−2.53449e6	−	2.11360e6i	−1.09555	−	0.913619i
$40$		0			0
$41$		−	2.74391e6i		−	0.971036i	−0.874227	−	0.485518i	$-0.838631\pi$
$41$		−	2.74391e6i		−	0.971036i	0.874227	−	0.485518i	$-0.161369\pi$
$42$		0			0
$43$		−	638785.i		−	0.186845i	−0.995627	−	0.0934224i	$-0.970219\pi$
$43$		−	638785.i		−	0.186845i	0.995627	−	0.0934224i	$-0.0297807\pi$
$44$		0			0
$45$	−3.38323e6	+	2.31709e6i	−0.825051	+	0.565058i
$46$		0			0
$47$	−2.05881e6			−0.421915			−0.210957	−	0.977495i	$-0.567658\pi$
$47$	−2.05881e6			−0.421915			−0.210957	+	0.977495i	$0.567658\pi$
$48$		0			0
$49$	−1.18978e7			−2.06387
$50$		0			0
$51$	1.61889e6	−	1.94126e6i	0.239296	−	0.286947i
$52$		0			0
$53$	−1.09780e7			−1.39130			−0.695649	−	0.718382i	$-0.744883\pi$
$53$	−1.09780e7			−1.39130			−0.695649	+	0.718382i	$0.744883\pi$
$54$		0			0
$55$	−309848.	+	138332.i	−0.0338609	+	0.0151172i
$56$		0			0
$57$	1.19668e7	−	1.43497e7i	1.13365	−	1.35939i
$58$		0			0
$59$			5.20695e6i			0.429710i	0.976646	+	0.214855i	$0.0689279\pi$
$59$			5.20695e6i			0.429710i	−0.976646	+	0.214855i	$0.931072\pi$
$60$		0			0
$61$	−7932.68			−0.000572929			−0.000286464	−	1.00000i	$-0.500091\pi$
$61$	−7932.68			−0.000572929			−0.000286464		1.00000i	$0.500091\pi$
$62$		0			0
$63$	2.71254e7	−	4.95297e6i	1.72192	−	0.314415i
$64$		0			0
$65$	−2.32520e7	+	1.03809e7i	−1.30259	+	0.581540i
$66$		0			0
$67$			2.61289e7i			1.29665i	0.761364	+	0.648325i	$0.224530\pi$
$67$			2.61289e7i			1.29665i	−0.761364	+	0.648325i	$0.775470\pi$
$68$		0			0
$69$	−2.41327e7	+	2.89383e7i	−1.06466	+	1.27666i
$70$		0			0
$71$		−	4.61106e7i		−	1.81454i	−0.420546	−	0.907271i	$-0.638161\pi$
$71$		−	4.61106e7i		−	1.81454i	0.420546	−	0.907271i	$-0.361839\pi$
$72$		0			0
$73$			3.27109e7i			1.15186i	0.817497	+	0.575932i	$0.195361\pi$
$73$			3.27109e7i			1.15186i	−0.817497	+	0.575932i	$0.804639\pi$
$74$		0			0
$75$	4.56245e6	+	3.13100e7i	0.144196	+	0.989549i
$76$		0			0
$77$	2.28173e6			0.0649084
$78$		0			0
$79$	−9.49611e6			−0.243802			−0.121901	−	0.992542i	$-0.538899\pi$
$79$	−9.49611e6			−0.243802			−0.121901	+	0.992542i	$0.538899\pi$
$80$		0			0
$81$	−4.02689e7	+	1.52130e7i	−0.935469	+	0.353408i
$82$		0			0
$83$	−6.39962e7			−1.34847			−0.674236	−	0.738516i	$-0.735527\pi$
$83$	−6.39962e7			−1.34847			−0.674236	+	0.738516i	$0.735527\pi$
$84$		0			0
$85$	−7.95109e6	−	1.78096e7i	−0.152318	−	0.341176i
$86$		0			0
$87$	1.43860e7	+	1.19971e7i	0.251110	+	0.209410i
$88$		0			0
$89$			4.56560e7i			0.727675i	0.931462	+	0.363838i	$0.118534\pi$
$89$			4.56560e7i			0.727675i	−0.931462	+	0.363838i	$0.881466\pi$
$90$		0			0
$91$	1.71228e8			2.49695
$92$		0			0
$93$	1.07105e7	−	1.28433e7i	0.143178	−	0.171689i
$94$		0			0
$95$	−5.87743e7	−	1.31648e8i	−0.721594	−	1.61629i
$96$		0			0
$97$			2.57539e7i			0.290909i	0.989365	+	0.145454i	$0.0464644\pi$
$97$			2.57539e7i			0.290909i	−0.989365	+	0.145454i	$0.953536\pi$
$98$		0			0
$99$	−3.50417e6	+	639845.i	−0.0364791	+	0.00666091i
$100$		0			0
$101$		−	4.86369e7i		−	0.467391i	−0.972310	−	0.233695i	$-0.924918\pi$
$101$		−	4.86369e7i		−	0.467391i	0.972310	−	0.233695i	$-0.0750819\pi$
$102$		0			0
$103$			4.66558e7i			0.414531i	0.978285	+	0.207265i	$0.0664563\pi$
$103$			4.66558e7i			0.414531i	−0.978285	+	0.207265i	$0.933544\pi$
$104$		0			0
$105$	5.78151e7	−	2.04755e8i	0.475646	−	1.68453i
$106$		0			0
$107$	1.00639e8			0.767773			0.383887	−	0.923380i	$-0.374585\pi$
$107$	1.00639e8			0.767773			0.383887	+	0.923380i	$0.374585\pi$
$108$		0			0
$109$	−1.60517e8			−1.13714			−0.568570	−	0.822635i	$-0.692503\pi$
$109$	−1.60517e8			−1.13714			−0.568570	+	0.822635i	$0.692503\pi$
$110$		0			0
$111$	−1.70102e8	−	1.41855e8i	−1.12052	−	0.934442i
$112$		0			0
$113$	−2.56759e8			−1.57475			−0.787376	−	0.616473i	$-0.788561\pi$
$113$	−2.56759e8			−1.57475			−0.787376	+	0.616473i	$0.788561\pi$
$114$		0			0
$115$	1.18527e8	+	2.65487e8i	0.677680	+	1.51793i
$116$		0			0
$117$	−2.62964e8	+	4.80160e7i	−1.40331	+	0.256237i
$118$		0			0
$119$			1.31150e8i			0.654004i
$120$		0			0
$121$	2.14064e8			0.998625
$122$		0			0
$123$	−1.70692e8	−	1.42346e8i	−0.745748	−	0.621908i
$124$		0			0
$125$	2.32421e8	+	7.47336e7i	0.951997	+	0.306109i
$126$		0			0
$127$		−	1.27490e8i		−	0.490074i	−0.969514	−	0.245037i	$-0.921200\pi$
$127$		−	1.27490e8i		−	0.490074i	0.969514	−	0.245037i	$-0.0788001\pi$
$128$		0			0
$129$	−3.97372e7	−	3.31383e7i	−0.143495	−	0.119666i
$130$		0			0
$131$			2.57502e8i			0.874369i	0.899372	+	0.437185i	$0.144024\pi$
$131$			2.57502e8i			0.874369i	−0.899372	+	0.437185i	$0.855976\pi$
$132$		0			0
$133$			9.69458e8i			3.09829i
$134$		0			0
$135$	−3.13719e7	+	3.30666e8i	−0.0944508	+	0.995530i
$136$		0			0
$137$	−4.60197e7			−0.130636			−0.0653178	−	0.997865i	$-0.520806\pi$
$137$	−4.60197e7			−0.130636			−0.0653178	+	0.997865i	$0.520806\pi$
$138$		0			0
$139$	−4.30317e8			−1.15274			−0.576368	−	0.817190i	$-0.695530\pi$
$139$	−4.30317e8			−1.15274			−0.576368	+	0.817190i	$0.695530\pi$
$140$		0			0
$141$	−1.06805e8	+	1.28073e8i	−0.270219	+	0.324027i
$142$		0			0
$143$	−2.21199e7			−0.0528981
$144$		0			0
$145$	1.31981e8	−	5.89230e7i	0.298566	−	0.133295i
$146$		0			0
$147$	−6.17225e8	+	7.40132e8i	−1.32183	+	1.58504i
$148$		0			0
$149$		−	6.09562e6i		−	0.0123672i	−0.999981	−	0.00618362i	$-0.998032\pi$
$149$		−	6.09562e6i		−	0.0123672i	0.999981	−	0.00618362i	$-0.00196832\pi$
$150$		0			0
$151$	−6.59787e8			−1.26910			−0.634550	−	0.772882i	$-0.718815\pi$
$151$	−6.59787e8			−1.26910			−0.634550	+	0.772882i	$0.718815\pi$
$152$		0			0
$153$	−3.67772e7	−	2.01414e8i	−0.0671140	−	0.367556i
$154$		0			0
$155$	−5.26040e7	−	1.17827e8i	−0.0911364	−	0.204136i
$156$		0			0
$157$			1.45497e8i			0.239472i	0.992806	+	0.119736i	$0.0382048\pi$
$157$			1.45497e8i			0.239472i	−0.992806	+	0.119736i	$0.961795\pi$
$158$		0			0
$159$	−5.69508e8	+	6.82914e8i	−0.891069	+	1.06851i
$160$		0			0
$161$		−	1.95505e9i		−	2.90974i
$162$		0			0
$163$		−	5.62267e8i		−	0.796512i	−0.917274	−	0.398256i	$-0.869616\pi$
$163$		−	5.62267e8i		−	0.796512i	0.917274	−	0.398256i	$-0.130384\pi$
$164$		0			0
$165$	−7.46879e6	+	2.64511e7i	−0.0100766	+	0.0356869i
$166$		0			0
$167$	−1.35010e9			−1.73580			−0.867900	−	0.496739i	$-0.834531\pi$
$167$	−1.35010e9			−1.73580			−0.867900	+	0.496739i	$0.834531\pi$
$168$		0			0
$169$	−8.44221e8			−1.03493
$170$		0			0
$171$	−2.71856e8	−	1.48885e9i	−0.317947	−	1.74127i
$172$		0			0
$173$	1.33226e8			0.148733			0.0743663	−	0.997231i	$-0.476307\pi$
$173$	1.33226e8			0.148733			0.0743663	+	0.997231i	$0.476307\pi$
$174$		0			0
$175$	−1.22224e9	−	1.09601e9i	−1.30318	−	1.16859i
$176$		0			0
$177$	3.23911e8	+	2.70122e8i	0.330014	+	0.275211i
$178$		0			0
$179$		−	4.42348e8i		−	0.430875i	−0.976518	−	0.215438i	$-0.930882\pi$
$179$		−	4.42348e8i		−	0.430875i	0.976518	−	0.215438i	$-0.0691178\pi$
$180$		0			0
$181$	−2.60496e8			−0.242710			−0.121355	−	0.992609i	$-0.538724\pi$
$181$	−2.60496e8			−0.242710			−0.121355	+	0.992609i	$0.538724\pi$
$182$		0			0
$183$	−411525.	+	493472.i	−0.000366937	+	0.000440005i
$184$		0			0
$185$	−1.56056e9	+	6.96713e8i	−1.33228	+	0.594795i
$186$		0			0
$187$		−	1.69425e7i		−	0.0138551i
$188$		0			0
$189$	1.09908e9	−	1.94435e9i	0.861351	−	1.52379i
$190$		0			0
$191$		−	5.58196e8i		−	0.419424i	−0.977763	−	0.209712i	$-0.932747\pi$
$191$		−	5.58196e8i		−	0.419424i	0.977763	−	0.209712i	$-0.0672526\pi$
$192$		0			0
$193$			1.45829e9i			1.05103i	0.850784	+	0.525515i	$0.176127\pi$
$193$			1.45829e9i			1.05103i	−0.850784	+	0.525515i	$0.823873\pi$
$194$		0			0
$195$	−5.60482e8	+	1.98498e9i	−0.387635	+	1.37283i
$196$		0			0
$197$	−1.09018e9			−0.723823			−0.361911	−	0.932212i	$-0.617876\pi$
$197$	−1.09018e9			−0.723823			−0.361911	+	0.932212i	$0.617876\pi$
$198$		0			0
$199$	1.15779e9			0.738276			0.369138	−	0.929375i	$-0.379653\pi$
$199$	1.15779e9			0.738276			0.369138	+	0.929375i	$0.379653\pi$
$200$		0			0
$201$	1.62541e9	+	1.35549e9i	0.995817	+	0.830450i
$202$		0			0
$203$	−9.71911e8			−0.572325
$204$		0			0
$205$	−1.56597e9	+	6.99127e8i	−0.886682	+	0.395859i
$206$		0			0
$207$	5.48237e8	+	3.00247e9i	0.298598	+	1.63530i
$208$		0			0
$209$		−	1.25239e8i		−	0.0656376i
$210$		0			0
$211$	2.41066e9			1.21620			0.608101	−	0.793860i	$-0.291932\pi$
$211$	2.41066e9			1.21620			0.608101	+	0.793860i	$0.291932\pi$
$212$		0			0
$213$	−2.86842e9	−	2.39209e9i	−1.39356	−	1.16214i
$214$		0			0
$215$	−3.64559e8	+	1.62757e8i	−0.170614	+	0.0761704i
$216$		0			0
$217$			8.67682e8i			0.391311i
$218$		0			0
$219$	2.03486e9	+	1.69695e9i	0.884623	+	0.737721i
$220$		0			0
$221$		−	1.27142e9i		−	0.532990i
$222$		0			0
$223$			1.81420e9i			0.733610i	0.930298	+	0.366805i	$0.119548\pi$
$223$			1.81420e9i			0.733610i	−0.930298	+	0.366805i	$0.880452\pi$
$224$		0			0
$225$	2.18440e9	+	1.34045e9i	0.852318	+	0.523024i
$226$		0			0
$227$	6.98312e8			0.262994			0.131497	−	0.991317i	$-0.458022\pi$
$227$	6.98312e8			0.262994			0.131497	+	0.991317i	$0.458022\pi$
$228$		0			0
$229$	7.29412e8			0.265235			0.132618	−	0.991167i	$-0.457662\pi$
$229$	7.29412e8			0.265235			0.132618	+	0.991167i	$0.457662\pi$
$230$		0			0
$231$	1.18370e8	−	1.41940e8i	0.0415712	−	0.0498492i
$232$		0			0
$233$	6.15946e8			0.208987			0.104493	−	0.994526i	$-0.466678\pi$
$233$	6.15946e8			0.208987			0.104493	+	0.994526i	$0.466678\pi$
$234$		0			0
$235$	5.24568e8	+	1.17498e9i	0.172001	+	0.385263i
$236$		0			0
$237$	−4.92631e8	+	5.90729e8i	−0.156145	+	0.187238i
$238$		0			0
$239$		−	1.63622e9i		−	0.501477i	−0.968055	−	0.250739i	$-0.919327\pi$
$239$		−	1.63622e9i		−	0.501477i	0.968055	−	0.250739i	$-0.0806735\pi$
$240$		0			0
$241$	−3.65311e9			−1.08292			−0.541458	−	0.840727i	$-0.682128\pi$
$241$	−3.65311e9			−1.08292			−0.541458	+	0.840727i	$0.682128\pi$
$242$		0			0
$243$	−1.14267e9	+	3.29423e9i	−0.327715	+	0.944777i
$244$		0			0
$245$	3.03147e9	+	6.79016e9i	0.841373	+	1.88459i
$246$		0			0
$247$		−	9.39830e9i		−	2.52500i
$248$		0			0
$249$	−3.31994e9	+	3.98104e9i	−0.863641	+	1.03562i
$250$		0			0
$251$		−	6.67710e9i		−	1.68226i	−0.540833	−	0.841130i	$-0.681891\pi$
$251$		−	6.67710e9i		−	1.68226i	0.540833	−	0.841130i	$-0.318109\pi$
$252$		0			0
$253$			2.52561e8i			0.0616431i
$254$		0			0
$255$	−1.52037e9	−	4.29294e8i	−0.359574	−	0.101530i
$256$		0			0
$257$	−9.63439e8			−0.220847			−0.110424	−	0.993885i	$-0.535221\pi$
$257$	−9.63439e8			−0.220847			−0.110424	+	0.993885i	$0.535221\pi$
$258$		0			0
$259$	1.14920e10			2.55386
$260$		0			0
$261$	1.49261e9	−	2.72544e8i	0.321651	−	0.0587320i
$262$		0			0
$263$	2.69617e9			0.563539			0.281769	−	0.959482i	$-0.409079\pi$
$263$	2.69617e9			0.563539			0.281769	+	0.959482i	$0.409079\pi$
$264$		0			0
$265$	2.79711e9	+	6.26522e9i	0.567187	+	1.27044i
$266$		0			0
$267$	2.84014e9	+	2.36850e9i	0.558849	+	0.466046i
$268$		0			0
$269$		−	7.93905e9i		−	1.51621i	−0.652132	−	0.758105i	$-0.726125\pi$
$269$		−	7.93905e9i		−	1.51621i	0.652132	−	0.758105i	$-0.273875\pi$
$270$		0			0
$271$	−7.64024e9			−1.41654			−0.708272	−	0.705940i	$-0.750525\pi$
$271$	−7.64024e9			−1.41654			−0.708272	+	0.705940i	$0.750525\pi$
$272$		0			0
$273$	8.88283e9	−	1.06517e10i	1.59919	−	1.91764i
$274$		0			0
$275$	1.57894e8	+	1.41587e8i	0.0276080	+	0.0247566i
$276$		0			0
$277$		−	2.23691e9i		−	0.379952i	−0.981789	−	0.189976i	$-0.939159\pi$
$277$		−	2.23691e9i		−	0.379952i	0.981789	−	0.189976i	$-0.0608411\pi$
$278$		0			0
$279$	−2.43316e8	−	1.33254e9i	−0.0401564	−	0.219920i
$280$		0			0
$281$			2.15743e9i			0.346028i	0.984919	+	0.173014i	$0.0553507\pi$
$281$			2.15743e9i			0.346028i	−0.984919	+	0.173014i	$0.944649\pi$
$282$		0			0
$283$			3.23888e9i			0.504952i	0.967603	+	0.252476i	$0.0812448\pi$
$283$			3.23888e9i			0.504952i	−0.967603	+	0.252476i	$0.918755\pi$
$284$		0			0
$285$	−1.12385e10	−	3.17333e9i	−1.70345	−	0.480990i
$286$		0			0
$287$	1.15318e10			1.69969
$288$		0			0
$289$	−6.00193e9			−0.860398
$290$		0			0
$291$	1.60209e9	+	1.33604e9i	0.223416	+	0.186315i
$292$		0			0
$293$	−2.51931e8			−0.0341831			−0.0170916	−	0.999854i	$-0.505441\pi$
$293$	−2.51931e8			−0.0341831			−0.0170916	+	0.999854i	$0.505441\pi$
$294$		0			0
$295$	2.97164e9	−	1.32669e9i	0.392381	−	0.175179i
$296$		0			0
$297$	−1.41983e8	+	2.51178e8i	−0.0182478	+	0.0322817i
$298$		0			0
$299$			1.89530e10i			2.37134i
$300$		0			0
$301$	2.68462e9			0.327052
$302$		0			0
$303$	−3.02557e9	−	2.52314e9i	−0.358953	−	0.299345i
$304$		0			0
$305$	2.02118e6	+	4.52723e6i	0.000233564	+	0.000523159i
$306$		0			0
$307$		−	9.85077e9i		−	1.10896i	−0.832196	−	0.554481i	$-0.812917\pi$
$307$		−	9.85077e9i		−	1.10896i	0.832196	−	0.554481i	$-0.187083\pi$
$308$		0			0
$309$	2.90233e9	+	2.42037e9i	0.318356	+	0.265490i
$310$		0			0
$311$		−	1.01536e10i		−	1.08538i	−0.839934	−	0.542688i	$-0.817407\pi$
$311$		−	1.01536e10i		−	1.08538i	0.839934	−	0.542688i	$-0.182593\pi$
$312$		0			0
$313$		−	8.43663e9i		−	0.879005i	−0.898241	−	0.439503i	$-0.855155\pi$
$313$		−	8.43663e9i		−	0.879005i	0.898241	−	0.439503i	$-0.144845\pi$
$314$		0			0
$315$	−9.73802e9	−	1.42187e10i	−0.989073	−	1.44416i
$316$		0			0
$317$	1.97605e10			1.95687			0.978433	−	0.206564i	$-0.0662282\pi$
$317$	1.97605e10			1.95687			0.978433	+	0.206564i	$0.0662282\pi$
$318$		0			0
$319$	1.25555e8			0.0121247
$320$		0			0
$321$	5.22088e9	−	6.26052e9i	0.491727	−	0.589644i
$322$		0			0
$323$	7.19850e9			0.661351
$324$		0			0
$325$	1.18489e10	+	1.06251e10i	1.06204	+	0.952358i
$326$		0			0
$327$	−8.32714e9	+	9.98532e9i	−0.728291	+	0.873315i
$328$		0			0
$329$		−	8.65254e9i		−	0.738516i
$330$		0			0
$331$	1.59240e10			1.32660			0.663300	−	0.748354i	$-0.269156\pi$
$331$	1.59240e10			1.32660			0.663300	+	0.748354i	$0.269156\pi$
$332$		0			0
$333$	−1.76489e10	+	3.22260e9i	−1.43529	+	0.262077i
$334$		0			0
$335$	1.49120e10	−	6.65744e9i	1.18401	−	0.528601i
$336$		0			0
$337$		−	1.46724e10i		−	1.13758i	−0.822483	−	0.568790i	$-0.807412\pi$
$337$		−	1.46724e10i		−	1.13758i	0.822483	−	0.568790i	$-0.192588\pi$
$338$		0			0
$339$	−1.33199e10	+	1.59723e10i	−1.00856	+	1.20940i
$340$		0			0
$341$		−	1.12091e8i		−	0.00828995i
$342$		0			0
$343$		−	2.57752e10i		−	1.86219i
$344$		0			0
$345$	2.26641e10	+	6.39947e9i	1.59979	+	0.451718i
$346$		0			0
$347$	−3.39960e9			−0.234482			−0.117241	−	0.993103i	$-0.537405\pi$
$347$	−3.39960e9			−0.234482			−0.117241	+	0.993103i	$0.537405\pi$
$348$		0			0
$349$	−2.59863e10			−1.75163			−0.875817	−	0.482643i	$-0.839677\pi$
$349$	−2.59863e10			−1.75163			−0.875817	+	0.482643i	$0.839677\pi$
$350$		0			0
$351$	−1.06549e10	+	1.88492e10i	−0.701971	+	1.24184i
$352$		0			0
$353$	1.50600e10			0.969895			0.484948	−	0.874543i	$-0.338839\pi$
$353$	1.50600e10			0.969895			0.484948	+	0.874543i	$0.338839\pi$
$354$		0			0
$355$	−2.63156e10	+	1.17486e10i	−1.65691	+	0.739729i
$356$		0			0
$357$	8.15850e9	+	6.80369e9i	0.502270	+	0.418863i
$358$		0			0
$359$			2.73218e10i			1.64487i	0.568860	+	0.822435i	$0.307385\pi$
$359$			2.73218e10i			1.64487i	−0.568860	+	0.822435i	$0.692615\pi$
$360$		0			0
$361$	3.62276e10			2.13310
$362$		0			0
$363$	1.11050e10	−	1.33164e10i	0.639578	−	0.766937i
$364$		0			0
$365$	1.86683e10	−	8.33448e9i	1.05180	−	0.469577i
$366$		0			0
$367$			5.36389e9i			0.295675i	0.989012	+	0.147838i	$0.0472313\pi$
$367$			5.36389e9i			0.295675i	−0.989012	+	0.147838i	$0.952769\pi$
$368$		0			0
$369$	−1.77100e10	+	3.23377e9i	−0.955242	+	0.174423i
$370$		0			0
$371$		−	4.61372e10i		−	2.43532i
$372$		0			0
$373$			1.30694e10i			0.675183i	0.941293	+	0.337592i	$0.109612\pi$
$373$			1.30694e10i			0.675183i	−0.941293	+	0.337592i	$0.890388\pi$
$374$		0			0
$375$	1.67063e10	−	1.05813e10i	0.844804	−	0.535076i
$376$		0			0
$377$	9.42208e9			0.466424
$378$		0			0
$379$	−2.04087e10			−0.989140			−0.494570	−	0.869138i	$-0.664674\pi$
$379$	−2.04087e10			−0.989140			−0.494570	+	0.869138i	$0.664674\pi$
$380$		0			0
$381$	−7.93083e9	−	6.61382e9i	−0.376373	−	0.313872i
$382$		0			0
$383$	−2.41777e10			−1.12362			−0.561810	−	0.827267i	$-0.689895\pi$
$383$	−2.41777e10			−1.12362			−0.561810	+	0.827267i	$0.689895\pi$
$384$		0			0
$385$	−5.81366e8	−	1.30220e9i	−0.0264610	−	0.0592699i
$386$		0			0
$387$	−4.12290e9	+	7.52823e8i	−0.183806	+	0.0335621i
$388$		0			0
$389$		−	1.99256e10i		−	0.870187i	−0.900385	−	0.435094i	$-0.856715\pi$
$389$		−	1.99256e10i		−	0.870187i	0.900385	−	0.435094i	$-0.143285\pi$
$390$		0			0
$391$	−1.45168e10			−0.621104
$392$		0			0
$393$	1.60185e10	+	1.33585e10i	0.671509	+	0.559997i
$394$		0			0
$395$	2.41953e9	+	5.41949e9i	0.0993901	+	0.222623i
$396$		0			0
$397$			1.33592e9i			0.0537798i	0.999638	+	0.0268899i	$0.00856035\pi$
$397$			1.33592e9i			0.0537798i	−0.999638	+	0.0268899i	$0.991440\pi$
$398$		0			0
$399$	6.03075e10	+	5.02927e10i	2.37947	+	1.98433i
$400$		0			0
$401$		−	9.38673e9i		−	0.363025i	−0.983389	−	0.181513i	$-0.941901\pi$
$401$		−	9.38673e9i		−	0.363025i	0.983389	−	0.181513i	$-0.0580994\pi$
$402$		0			0
$403$		−	8.41164e9i		−	0.318904i
$404$		0			0
$405$	1.89424e10	+	1.91056e10i	0.704068	+	0.710133i
$406$		0			0
$407$	−1.48458e9			−0.0541037
$408$		0			0
$409$	2.11317e10			0.755164			0.377582	−	0.925976i	$-0.376756\pi$
$409$	2.11317e10			0.755164			0.377582	+	0.925976i	$0.376756\pi$
$410$		0			0
$411$	−2.38737e9	+	2.86277e9i	−0.0836667	+	0.100327i
$412$		0			0
$413$	−2.18832e10			−0.752161
$414$		0			0
$415$	1.63057e10	+	3.65231e10i	0.549728	+	1.23133i
$416$		0			0
$417$	−2.23236e10	+	2.67689e10i	−0.738279	+	0.885292i
$418$		0			0
$419$		−	2.46365e10i		−	0.799324i	−0.916663	−	0.399662i	$-0.869127\pi$
$419$		−	2.46365e10i		−	0.799324i	0.916663	−	0.399662i	$-0.130873\pi$
$420$		0			0
$421$	1.46468e10			0.466244			0.233122	−	0.972447i	$-0.425106\pi$
$421$	1.46468e10			0.466244			0.233122	+	0.972447i	$0.425106\pi$
$422$		0			0
$423$	2.42635e9	+	1.32881e10i	0.0757867	+	0.415052i
$424$		0			0
$425$	−8.13817e9	+	9.07548e9i	−0.249443	+	0.278172i
$426$		0			0
$427$		−	3.33386e7i		−	0.00100285i
$428$		0			0
$429$	−1.14752e9	+	1.37602e9i	−0.0338790	+	0.0406253i
$430$		0			0
$431$			3.06739e10i			0.888913i	0.895800	+	0.444457i	$0.146603\pi$
$431$			3.06739e10i			0.888913i	−0.895800	+	0.444457i	$0.853397\pi$
$432$		0			0
$433$		−	4.75808e10i		−	1.35357i	−0.736182	−	0.676784i	$-0.763373\pi$
$433$		−	4.75808e10i		−	1.35357i	0.736182	−	0.676784i	$-0.236627\pi$
$434$		0			0
$435$	3.18136e9	−	1.12670e10i	0.0888497	−	0.314666i
$436$		0			0
$437$	−1.07308e11			−2.94243
$438$		0			0
$439$	6.10565e10			1.64389			0.821947	−	0.569563i	$-0.192888\pi$
$439$	6.10565e10			1.64389			0.821947	+	0.569563i	$0.192888\pi$
$440$		0			0
$441$	1.40218e10	+	7.67919e10i	0.370724	+	2.03030i
$442$		0			0
$443$	1.70403e10			0.442448			0.221224	−	0.975223i	$-0.428995\pi$
$443$	1.70403e10			0.442448			0.221224	+	0.975223i	$0.428995\pi$
$444$		0			0
$445$	2.60562e10	−	1.16328e10i	0.664462	−	0.296649i
$446$		0			0
$447$	−3.79193e8	−	3.16223e8i	−0.00949795	−	0.00792070i
$448$		0			0
$449$		−	7.20427e10i		−	1.77257i	−0.463136	−	0.886287i	$-0.653276\pi$
$449$		−	7.20427e10i		−	1.77257i	0.463136	−	0.886287i	$-0.346724\pi$
$450$		0			0
$451$	−1.48973e9			−0.0360082
$452$		0			0
$453$	−3.42279e10	+	4.10436e10i	−0.812806	+	0.974659i
$454$		0			0
$455$	−4.36276e10	−	9.77211e10i	−1.01792	−	2.28004i
$456$		0			0
$457$		−	4.05546e10i		−	0.929770i	−0.885371	−	0.464885i	$-0.846096\pi$
$457$		−	4.05546e10i		−	0.929770i	0.885371	−	0.464885i	$-0.153904\pi$
$458$		0			0
$459$	−1.44373e10	−	8.16095e9i	−0.325264	−	0.183861i
$460$		0			0
$461$			4.20374e9i			0.0930748i	0.998917	+	0.0465374i	$0.0148187\pi$
$461$			4.20374e9i			0.0930748i	−0.998917	+	0.0465374i	$0.985181\pi$
$462$		0			0
$463$			1.04672e10i			0.227775i	0.993494	+	0.113888i	$0.0363304\pi$
$463$			1.04672e10i			0.227775i	−0.993494	+	0.113888i	$0.963670\pi$
$464$		0			0
$465$	−1.00587e10	−	2.84019e9i	−0.215144	−	0.0607484i
$466$		0			0
$467$	4.59596e10			0.966293			0.483146	−	0.875540i	$-0.339494\pi$
$467$	4.59596e10			0.966293			0.483146	+	0.875540i	$0.339494\pi$
$468$		0			0
$469$	−1.09812e11			−2.26964
$470$		0			0
$471$	9.05096e9	+	7.54795e9i	0.183912	+	0.153372i
$472$		0			0
$473$	−3.46810e8			−0.00692862
$474$		0			0
$475$	−6.01572e10	+	6.70857e10i	−1.18172	+	1.31782i
$476$		0			0
$477$	1.29378e10	+	7.08552e10i	0.249913	+	1.36867i
$478$		0			0
$479$		−	7.44346e10i		−	1.41395i	−0.707241	−	0.706973i	$-0.750060\pi$
$479$		−	7.44346e10i		−	1.41395i	0.707241	−	0.706973i	$-0.249940\pi$
$480$		0			0
$481$	−1.11408e11			−2.08131
$482$		0			0
$483$	−1.21619e11	−	1.01422e11i	−2.23466	−	1.86357i
$484$		0			0
$485$	1.46979e10	−	6.56189e9i	0.265638	−	0.118594i
$486$		0			0
$487$			7.50739e10i			1.33467i	0.744759	+	0.667334i	$0.232565\pi$
$487$			7.50739e10i			1.33467i	−0.744759	+	0.667334i	$0.767435\pi$
$488$		0			0
$489$	−3.49772e10	−	2.91688e10i	−0.611715	−	0.510133i
$490$		0			0
$491$		−	2.16105e10i		−	0.371825i	−0.982566	−	0.185913i	$-0.940476\pi$
$491$		−	2.16105e10i		−	0.371825i	0.982566	−	0.185913i	$-0.0595241\pi$
$492$		0			0
$493$			7.21672e9i			0.122167i
$494$		0			0
$495$	1.25800e9	+	1.83682e9i	0.0209536	+	0.0305947i
$496$		0			0
$497$	1.93789e11			3.17616
$498$		0			0
$499$	5.73579e10			0.925105			0.462552	−	0.886592i	$-0.346934\pi$
$499$	5.73579e10			0.925105			0.462552	+	0.886592i	$0.346934\pi$
$500$		0			0
$501$	−7.00393e10	+	8.39861e10i	−1.11171	+	1.33308i
$502$		0			0
$503$	2.74629e10			0.429017			0.214508	−	0.976722i	$-0.431185\pi$
$503$	2.74629e10			0.429017			0.214508	+	0.976722i	$0.431185\pi$
$504$		0			0
$505$	−2.77574e10	+	1.23923e10i	−0.426789	+	0.190540i
$506$		0			0
$507$	−4.37958e10	+	5.25168e10i	−0.662827	+	0.794815i
$508$		0			0
$509$		−	9.69537e10i		−	1.44442i	−0.691674	−	0.722210i	$-0.743126\pi$
$509$		−	9.69537e10i		−	1.44442i	0.691674	−	0.722210i	$-0.256874\pi$
$510$		0			0
$511$	−1.37474e11			−2.01621
$512$		0			0
$513$	−1.06720e11	−	6.03256e10i	−1.54091	−	0.871028i
$514$		0			0
$515$	2.66268e10	−	1.18875e10i	0.378521	−	0.168990i
$516$		0			0
$517$			1.11777e9i			0.0156455i
$518$		0			0
$519$	6.91141e9	−	8.28767e9i	0.0952570	−	0.114226i
$520$		0			0
$521$			8.75326e10i			1.18801i	0.804462	+	0.594003i	$0.202453\pi$
$521$			8.75326e10i			1.18801i	−0.804462	+	0.594003i	$0.797547\pi$
$522$		0			0
$523$			7.58510e10i			1.01380i	0.862003	+	0.506902i	$0.169210\pi$
$523$			7.58510e10i			1.01380i	−0.862003	+	0.506902i	$0.830790\pi$
$524$		0			0
$525$	−1.31586e11	+	1.91746e10i	−1.73210	+	0.252399i
$526$		0			0
$527$	6.44279e9			0.0835279
$528$		0			0
$529$	1.38091e11			1.76336
$530$		0			0
$531$	3.36072e10	−	6.13651e9i	0.422721	−	0.0771869i
$532$		0			0
$533$	−1.11794e11			−1.38519
$534$		0			0
$535$	−2.56421e10	−	5.74356e10i	−0.312996	−	0.701077i
$536$		0			0
$537$	−2.75173e10	−	2.29477e10i	−0.330909	−	0.275958i
$538$		0			0
$539$			6.45957e9i			0.0765330i
$540$		0			0
$541$	3.55918e10			0.415490			0.207745	−	0.978183i	$-0.433388\pi$
$541$	3.55918e10			0.415490			0.207745	+	0.978183i	$0.433388\pi$
$542$		0			0
$543$	−1.35138e10	+	1.62048e10i	−0.155445	+	0.186399i
$544$		0			0
$545$	4.08983e10	+	9.16079e10i	0.463575	+	1.03836i
$546$		0			0
$547$			4.06308e10i			0.453843i	0.973913	+	0.226922i	$0.0728661\pi$
$547$			4.06308e10i			0.453843i	−0.973913	+	0.226922i	$0.927134\pi$
$548$		0			0
$549$	9.34885e6	+	5.11998e7i	0.000102913	+	0.000563610i
$550$		0			0
$551$			5.33458e10i			0.578754i
$552$		0			0
$553$		−	3.99092e10i		−	0.426749i
$554$		0			0
$555$	−3.76168e10	+	1.33222e11i	−0.396470	+	1.40412i
$556$		0			0
$557$	−1.01469e11			−1.05417			−0.527087	−	0.849811i	$-0.676716\pi$
$557$	−1.01469e11			−1.05417			−0.527087	+	0.849811i	$0.676716\pi$
$558$		0			0
$559$	−2.60257e10			−0.266536
$560$		0			0
$561$	−1.05395e9	−	8.78928e8i	−0.0106406	−	0.00887364i
$562$		0			0
$563$	−1.88086e11			−1.87208			−0.936039	−	0.351897i	$-0.885537\pi$
$563$	−1.88086e11			−1.87208			−0.936039	+	0.351897i	$0.885537\pi$
$564$		0			0
$565$	6.54202e10	+	1.46534e11i	0.641975	+	1.43795i
$566$		0			0
$567$	−6.39357e10	−	1.69238e11i	−0.618602	−	1.63744i
$568$		0			0
$569$		−	1.25370e10i		−	0.119604i	−0.998210	−	0.0598018i	$-0.980953\pi$
$569$		−	1.25370e10i		−	0.119604i	0.998210	−	0.0598018i	$-0.0190469\pi$
$570$		0			0
$571$	5.63208e10			0.529816			0.264908	−	0.964274i	$-0.414658\pi$
$571$	5.63208e10			0.529816			0.264908	+	0.964274i	$0.414658\pi$
$572$		0			0
$573$	−3.47239e10	−	2.89576e10i	−0.322114	−	0.268624i
$574$		0			0
$575$	1.21316e11	−	1.35288e11i	1.10980	−	1.23762i
$576$		0			0
$577$			3.29296e10i			0.297087i	0.988906	+	0.148544i	$0.0474585\pi$
$577$			3.29296e10i			0.297087i	−0.988906	+	0.148544i	$0.952541\pi$
$578$		0			0
$579$	9.07166e10	+	7.56521e10i	0.807184	+	0.673142i
$580$		0			0
$581$		−	2.68957e11i		−	2.36036i
$582$		0			0
$583$			5.96019e9i			0.0515924i
$584$		0			0
$585$	9.44041e10	+	1.37841e11i	0.806060	+	1.17694i
$586$		0			0
$587$	1.97454e10			0.166308			0.0831541	−	0.996537i	$-0.473501\pi$
$587$	1.97454e10			0.166308			0.0831541	+	0.996537i	$0.473501\pi$
$588$		0			0
$589$	4.76249e10			0.395707
$590$		0			0
$591$	−5.65553e10	+	6.78171e10i	−0.463579	+	0.555891i
$592$		0			0
$593$	1.66098e11			1.34322			0.671609	−	0.740905i	$-0.265603\pi$
$593$	1.66098e11			1.34322			0.671609	+	0.740905i	$0.265603\pi$
$594$		0			0
$595$	7.48482e10	−	3.34160e10i	0.597191	−	0.266616i
$596$		0			0
$597$	6.00630e10	−	7.20233e10i	0.472835	−	0.566991i
$598$		0			0
$599$			1.01300e10i			0.0786865i	0.999226	+	0.0393433i	$0.0125266\pi$
$599$			1.01300e10i			0.0786865i	−0.999226	+	0.0393433i	$0.987473\pi$
$600$		0			0
$601$	1.99185e11			1.52672			0.763360	−	0.645974i	$-0.223549\pi$
$601$	1.99185e11			1.52672			0.763360	+	0.645974i	$0.223549\pi$
$602$		0			0
$603$	1.68644e11	−	3.07935e10i	1.27556	−	0.232911i
$604$		0			0
$605$	−5.45418e10	−	1.22168e11i	−0.407107	−	0.911875i
$606$		0			0
$607$			6.67938e10i			0.492019i	0.969267	+	0.246009i	$0.0791194\pi$
$607$			6.67938e10i			0.492019i	−0.969267	+	0.246009i	$0.920881\pi$
$608$		0			0
$609$	−5.04200e10	+	6.04601e10i	−0.366550	+	0.439541i
$610$		0			0
$611$			8.38811e10i			0.601865i
$612$		0			0
$613$		−	1.70432e11i		−	1.20700i	−0.797362	−	0.603502i	$-0.793772\pi$
$613$		−	1.70432e11i		−	1.20700i	0.797362	−	0.603502i	$-0.206228\pi$
$614$		0			0
$615$	−3.77472e10	+	1.33684e11i	−0.263866	+	0.934497i
$616$		0			0
$617$	−6.08071e10			−0.419579			−0.209790	−	0.977747i	$-0.567278\pi$
$617$	−6.08071e10			−0.419579			−0.209790	+	0.977747i	$0.567278\pi$
$618$		0			0
$619$	−8.63704e10			−0.588305			−0.294152	−	0.955759i	$-0.595037\pi$
$619$	−8.63704e10			−0.588305			−0.294152	+	0.955759i	$0.595037\pi$
$620$		0			0
$621$	2.15217e11	+	1.21655e11i	1.44714	+	0.818020i
$622$		0			0
$623$	−1.91878e11			−1.27372
$624$		0			0
$625$	−1.65680e10	−	1.51686e11i	−0.108580	−	0.994088i
$626$		0			0
$627$	−7.79076e9	−	6.49702e9i	−0.0504092	−	0.0420382i
$628$		0			0
$629$		−	8.53315e10i		−	0.545138i
$630$		0			0
$631$	2.17134e10			0.136965			0.0684826	−	0.997652i	$-0.478184\pi$
$631$	2.17134e10			0.136965			0.0684826	+	0.997652i	$0.478184\pi$
$632$		0			0
$633$	1.25058e11	−	1.49961e11i	0.778927	−	0.934034i
$634$		0			0
$635$	−7.27594e10	+	3.24834e10i	−0.447501	+	0.199787i
$636$		0			0
$637$			4.84747e11i			2.94413i
$638$		0			0
$639$	−2.97611e11	+	5.43424e10i	−1.78503	+	0.325938i
$640$		0			0
$641$			2.29392e10i			0.135877i	0.997690	+	0.0679385i	$0.0216422\pi$
$641$			2.29392e10i			0.135877i	−0.997690	+	0.0679385i	$0.978358\pi$
$642$		0			0
$643$		−	1.27837e11i		−	0.747846i	−0.927460	−	0.373923i	$-0.878012\pi$
$643$		−	1.27837e11i		−	0.747846i	0.927460	−	0.373923i	$-0.121988\pi$
$644$		0			0
$645$	−8.78756e9	+	3.11216e10i	−0.0507726	+	0.179814i
$646$		0			0
$647$	2.16440e11			1.23515			0.617576	−	0.786512i	$-0.288115\pi$
$647$	2.16440e11			1.23515			0.617576	+	0.786512i	$0.288115\pi$
$648$		0			0
$649$	2.82696e9			0.0159346
$650$		0			0
$651$	5.39763e10	+	4.50129e10i	0.300524	+	0.250618i
$652$		0			0
$653$	−2.18939e11			−1.20412			−0.602061	−	0.798450i	$-0.705654\pi$
$653$	−2.18939e11			−1.20412			−0.602061	+	0.798450i	$0.705654\pi$
$654$		0			0
$655$	1.46958e11	−	6.56094e10i	0.798414	−	0.356452i
$656$		0			0
$657$	2.11126e11	−	3.85506e10i	1.13313	−	0.206904i
$658$		0			0
$659$		−	1.82589e11i		−	0.968128i	−0.875032	−	0.484064i	$-0.839160\pi$
$659$		−	1.82589e11i		−	0.968128i	0.875032	−	0.484064i	$-0.160840\pi$
$660$		0			0
$661$	−2.06254e11			−1.08043			−0.540216	−	0.841526i	$-0.681657\pi$
$661$	−2.06254e11			−1.08043			−0.540216	+	0.841526i	$0.681657\pi$
$662$		0			0
$663$	−7.90916e10	−	6.59576e10i	−0.409333	−	0.341358i
$664$		0			0
$665$	5.53276e11	−	2.47010e11i	2.82915	−	1.26307i
$666$		0			0
$667$		−	1.07579e11i		−	0.543533i
$668$		0			0
$669$	1.12857e11	+	9.41155e10i	0.563407	+	0.469847i
$670$		0			0
$671$			4.30682e6i			2.12455e-5i
$672$		0			0
$673$		−	2.07090e11i		−	1.00948i	−0.863270	−	0.504742i	$-0.831588\pi$
$673$		−	2.07090e11i		−	1.00948i	0.863270	−	0.504742i	$-0.168412\pi$
$674$		0			0
$675$	1.96706e11	−	6.63468e10i	0.947553	−	0.319599i
$676$		0			0
$677$	−1.25282e10			−0.0596393			−0.0298196	−	0.999555i	$-0.509493\pi$
$677$	−1.25282e10			−0.0596393			−0.0298196	+	0.999555i	$0.509493\pi$
$678$		0			0
$679$	−1.08236e11			−0.509204
$680$		0			0
$681$	3.62264e10	−	4.34402e10i	0.168437	−	0.201978i
$682$		0			0
$683$	9.26965e10			0.425971			0.212986	−	0.977055i	$-0.431681\pi$
$683$	9.26965e10			0.425971			0.212986	+	0.977055i	$0.431681\pi$
$684$		0			0
$685$	1.17254e10	+	2.62637e10i	0.0532558	+	0.119287i
$686$		0			0
$687$	3.78398e10	−	4.53748e10i	0.169872	−	0.203699i
$688$		0			0
$689$			4.47272e11i			1.98470i
$690$		0			0
$691$	1.04126e11			0.456716			0.228358	−	0.973577i	$-0.426664\pi$
$691$	1.04126e11			0.456716			0.228358	+	0.973577i	$0.426664\pi$
$692$		0			0
$693$	−2.68907e9	−	1.47269e10i	−0.0116592	−	0.0638527i
$694$		0			0
$695$	1.09641e11	+	2.45585e11i	0.469932	+	1.05260i
$696$		0			0
$697$		−	8.56271e10i		−	0.362811i
$698$		0			0
$699$	3.19535e10	−	3.83164e10i	0.133847	−	0.160500i
$700$		0			0
$701$			3.27492e11i			1.35622i	0.734962	+	0.678108i	$0.237200\pi$
$701$			3.27492e11i			1.35622i	−0.734962	+	0.678108i	$0.762800\pi$
$702$		0			0
$703$		−	6.30768e11i		−	2.58255i
$704$		0			0
$705$	1.00305e11	+	2.83224e10i	0.406039	+	0.114650i
$706$		0			0
$707$	2.04406e11			0.818117
$708$		0			0
$709$	−1.62420e11			−0.642766			−0.321383	−	0.946949i	$-0.604148\pi$
$709$	−1.62420e11			−0.642766			−0.321383	+	0.946949i	$0.604148\pi$
$710$		0			0
$711$	1.11914e10	+	6.12906e10i	0.0437931	+	0.239837i
$712$		0			0
$713$	−9.60425e10			−0.371625
$714$		0			0
$715$	5.63599e9	+	1.26240e10i	0.0215648	+	0.0483029i
$716$		0			0
$717$	−1.01785e11	−	8.48826e10i	−0.385131	−	0.321175i
$718$		0			0
$719$			2.33669e11i			0.874352i	0.899376	+	0.437176i	$0.144021\pi$
$719$			2.33669e11i			0.874352i	−0.899376	+	0.437176i	$0.855979\pi$
$720$		0			0
$721$	−1.96080e11			−0.725591
$722$		0			0
$723$	−1.89513e11	+	2.27251e11i	−0.693564	+	0.831672i
$724$		0			0
$725$	−6.72555e10	−	6.03094e10i	−0.243431	−	0.218290i
$726$		0			0
$727$			2.99405e11i			1.07182i	0.844276	+	0.535909i	$0.180031\pi$
$727$			2.99405e11i			1.07182i	−0.844276	+	0.535909i	$0.819969\pi$
$728$		0			0
$729$	1.45647e11	+	2.41978e11i	0.515694	+	0.856773i
$730$		0			0
$731$		−	1.99340e10i		−	0.0698113i
$732$		0			0
$733$		−	2.22345e11i		−	0.770216i	−0.922872	−	0.385108i	$-0.874164\pi$
$733$		−	2.22345e11i		−	0.770216i	0.922872	−	0.385108i	$-0.125836\pi$
$734$		0			0
$735$	5.79662e11	+	1.63675e11i	1.98621	+	0.560831i
$736$		0			0
$737$	1.41859e10			0.0480826
$738$		0			0
$739$	−9.52594e10			−0.319396			−0.159698	−	0.987166i	$-0.551052\pi$
$739$	−9.52594e10			−0.319396			−0.159698	+	0.987166i	$0.551052\pi$
$740$		0			0
$741$	−5.84644e11	−	4.87557e11i	−1.93918	−	1.61716i
$742$		0			0
$743$	4.02306e10			0.132008			0.0660042	−	0.997819i	$-0.478975\pi$
$743$	4.02306e10			0.132008			0.0660042	+	0.997819i	$0.478975\pi$
$744$		0			0
$745$	−3.47881e9	+	1.55311e9i	−0.0112929	+	0.00504172i
$746$		0			0
$747$	7.54210e10	+	4.13050e11i	0.242220	+	1.32654i
$748$		0			0
$749$			4.22956e11i			1.34390i
$750$		0			0
$751$	3.43224e11			1.07899			0.539496	−	0.841988i	$-0.318615\pi$
$751$	3.43224e11			1.07899			0.539496	+	0.841988i	$0.318615\pi$
$752$		0			0
$753$	−4.15365e11	−	3.46389e11i	−1.29196	−	1.07742i
$754$		0			0
$755$	1.68108e11	+	3.76545e11i	0.517370	+	1.15885i
$756$		0			0
$757$			2.06609e10i			0.0629167i	0.999505	+	0.0314584i	$0.0100152\pi$
$757$			2.06609e10i			0.0629167i	−0.999505	+	0.0314584i	$0.989985\pi$
$758$		0			0
$759$	1.57112e10	+	1.31022e10i	0.0473415	+	0.0394799i
$760$		0			0
$761$		−	2.18213e11i		−	0.650643i	−0.945604	−	0.325321i	$-0.894528\pi$
$761$		−	2.18213e11i		−	0.650643i	0.945604	−	0.325321i	$-0.105472\pi$
$762$		0			0
$763$		−	6.74602e11i		−	1.99044i
$764$		0			0
$765$	−1.05578e11	+	7.23076e10i	−0.308266	+	0.211124i
$766$		0			0
$767$	2.12144e11			0.612985
$768$		0			0
$769$	−4.24678e10			−0.121438			−0.0607189	−	0.998155i	$-0.519339\pi$
$769$	−4.24678e10			−0.121438			−0.0607189	+	0.998155i	$0.519339\pi$
$770$		0			0
$771$	−4.99805e10	+	5.99330e10i	−0.141443	+	0.169609i
$772$		0			0
$773$	2.27245e11			0.636468			0.318234	−	0.948012i	$-0.396910\pi$
$773$	2.27245e11			0.636468			0.318234	+	0.948012i	$0.396910\pi$
$774$		0			0
$775$	−5.38418e10	+	6.00429e10i	−0.149249	+	0.166439i
$776$		0			0
$777$	5.96173e11	−	7.14888e11i	1.63564	−	1.96134i
$778$		0			0
$779$		−	6.32954e11i		−	1.71879i
$780$		0			0
$781$	−2.50344e10			−0.0672873
$782$		0			0
$783$	6.04782e10	−	1.06990e11i	0.160899	−	0.284641i
$784$		0			0
$785$	8.30359e10	−	3.70714e10i	0.218669	−	0.0976247i
$786$		0			0
$787$		−	3.13820e11i		−	0.818055i	−0.912522	−	0.409027i	$-0.865868\pi$
$787$		−	3.13820e11i		−	0.818055i	0.912522	−	0.409027i	$-0.134132\pi$
$788$		0			0
$789$	1.39869e11	−	1.67721e11i	0.360923	−	0.432793i
$790$		0			0
$791$		−	1.07908e12i		−	2.75643i
$792$		0			0
$793$			3.23197e8i			0.000817288i
$794$		0			0
$795$	5.34849e11	+	1.51021e11i	1.33895	+	0.378067i
$796$		0			0
$797$	2.89741e11			0.718085			0.359043	−	0.933321i	$-0.383103\pi$
$797$	2.89741e11			0.718085			0.359043	+	0.933321i	$0.383103\pi$
$798$		0			0
$799$	−6.42476e10			−0.157641
$800$		0			0
$801$	2.94677e11	−	5.38066e10i	0.715839	−	0.130709i
$802$		0			0
$803$	1.77594e10			0.0427137
$804$		0			0
$805$	−1.11576e12	+	4.98131e11i	−2.65697	+	1.18621i
$806$		0			0
$807$	−4.93868e11	−	4.11855e11i	−1.16444	−	0.971070i
$808$		0			0
$809$			3.41840e11i			0.798048i	0.916940	+	0.399024i	$0.130651\pi$
$809$			3.41840e11i			0.798048i	−0.916940	+	0.399024i	$0.869349\pi$
$810$		0			0
$811$	7.64414e10			0.176703			0.0883517	−	0.996089i	$-0.471840\pi$
$811$	7.64414e10			0.176703			0.0883517	+	0.996089i	$0.471840\pi$
$812$		0			0
$813$	−3.96354e11	+	4.75280e11i	−0.907238	+	1.08790i
$814$		0			0
$815$	−3.20890e11	+	1.43261e11i	−0.727319	+	0.324712i
$816$		0			0
$817$		−	1.47352e11i		−	0.330726i
$818$		0			0
$819$	−2.01796e11	−	1.10516e12i	−0.448516	−	2.45634i
$820$		0			0
$821$			4.25765e9i			0.00937126i	0.999989	+	0.00468563i	$0.00149149\pi$
$821$			4.25765e9i			0.00937126i	−0.999989	+	0.00468563i	$0.998509\pi$
$822$		0			0
$823$		−	5.40923e11i		−	1.17906i	−0.807746	−	0.589530i	$-0.799313\pi$
$823$		−	5.40923e11i		−	1.17906i	0.807746	−	0.589530i	$-0.200687\pi$
$824$		0			0
$825$	1.69988e10	−	2.47705e9i	0.0366947	−	0.00534710i
$826$		0			0
$827$	−9.84358e10			−0.210441			−0.105221	−	0.994449i	$-0.533555\pi$
$827$	−9.84358e10			−0.210441			−0.105221	+	0.994449i	$0.533555\pi$
$828$		0			0
$829$	7.11242e11			1.50591			0.752955	−	0.658072i	$-0.228628\pi$
$829$	7.11242e11			1.50591			0.752955	+	0.658072i	$0.228628\pi$
$830$		0			0
$831$	−1.39152e11	−	1.16044e11i	−0.291801	−	0.243344i
$832$		0			0
$833$	−3.71286e11			−0.771131
$834$		0			0
$835$	3.43994e11	+	7.70510e11i	0.707629	+	1.58501i
$836$		0			0
$837$	−9.55166e10	−	5.39925e10i	−0.194615	−	0.110010i
$838$		0			0
$839$			5.35247e11i			1.08021i	0.841599	+	0.540103i	$0.181615\pi$
$839$			5.35247e11i			1.08021i	−0.841599	+	0.540103i	$0.818385\pi$
$840$		0			0
$841$	4.46766e11			0.893091
$842$		0			0
$843$	1.34208e11	+	1.11921e11i	0.265747	+	0.221617i
$844$		0			0
$845$	2.15101e11	+	4.81802e11i	0.421905	+	0.945022i
$846$		0			0
$847$			8.99646e11i			1.74799i
$848$		0			0
$849$	2.01483e11	+	1.68024e11i	0.387799	+	0.323401i
$850$		0			0
$851$			1.27203e12i			2.42538i
$852$		0			0
$853$		−	1.89556e10i		−	0.0358048i	−0.999840	−	0.0179024i	$-0.994301\pi$
$853$		−	1.89556e10i		−	0.0358048i	0.999840	−	0.0179024i	$-0.00569882\pi$
$854$		0			0
$855$	−7.80427e11	+	5.34496e11i	−1.46039	+	1.00018i
$856$		0			0
$857$	−4.12084e11			−0.763946			−0.381973	−	0.924174i	$-0.624755\pi$
$857$	−4.12084e11			−0.763946			−0.381973	+	0.924174i	$0.624755\pi$
$858$		0			0
$859$	3.75424e11			0.689524			0.344762	−	0.938690i	$-0.387960\pi$
$859$	3.75424e11			0.689524			0.344762	+	0.938690i	$0.387960\pi$
$860$		0			0
$861$	5.98238e11	−	7.17365e11i	1.08858	−	1.30535i
$862$		0			0
$863$	−9.56584e11			−1.72457			−0.862283	−	0.506427i	$-0.830966\pi$
$863$	−9.56584e11			−1.72457			−0.862283	+	0.506427i	$0.830966\pi$
$864$		0			0
$865$	−3.39450e10	−	7.60332e10i	−0.0606334	−	0.135812i
$866$		0			0
$867$	−3.11363e11	+	3.73365e11i	−0.551050	+	0.660780i
$868$		0			0
$869$			5.15564e9i			0.00904072i
$870$		0			0
$871$	1.06456e12			1.84968
$872$		0			0
$873$	1.66223e11	−	3.03516e10i	0.286177	−	0.0522546i
$874$		0			0
$875$	−3.14082e11	+	9.76794e11i	−0.535811	+	1.66637i
$876$		0			0
$877$			1.01432e12i			1.71466i	0.514768	+	0.857329i	$0.327878\pi$
$877$			1.01432e12i			1.71466i	−0.514768	+	0.857329i	$0.672122\pi$
$878$		0			0
$879$	−1.30695e10	+	1.56720e10i	−0.0218929	+	0.0262524i
$880$		0			0
$881$			4.70566e10i			0.0781119i	0.999237	+	0.0390560i	$0.0124351\pi$
$881$			4.70566e10i			0.0781119i	−0.999237	+	0.0390560i	$0.987565\pi$
$882$		0			0
$883$		−	3.17311e11i		−	0.521965i	−0.965343	−	0.260983i	$-0.915953\pi$
$883$		−	3.17311e11i		−	0.521965i	0.965343	−	0.260983i	$-0.0840465\pi$
$884$		0			0
$885$	7.16304e10	−	2.53683e11i	0.116768	−	0.413541i
$886$		0			0
$887$	1.01566e12			1.64080			0.820399	−	0.571792i	$-0.193751\pi$
$887$	1.01566e12			1.64080			0.820399	+	0.571792i	$0.193751\pi$
$888$		0			0
$889$	5.35801e11			0.857821
$890$		0			0
$891$	8.25948e9	+	2.18628e10i	0.0131051	+	0.0346893i
$892$		0			0
$893$	−4.74917e11			−0.746813
$894$		0			0
$895$	−2.52451e11	+	1.12707e11i	−0.393445	+	0.175654i
$896$		0			0
$897$	1.17902e12	+	9.83228e11i	1.82117	+	1.51874i
$898$		0			0
$899$			4.77455e10i			0.0730960i
$900$		0			0
$901$	−3.42582e11			−0.519835
$902$		0			0
$903$	1.39270e11	−	1.67003e11i	0.209463	−	0.251173i
$904$		0			0
$905$	6.63723e10	+	1.48667e11i	0.0989447	+	0.221626i
$906$		0			0
$907$			3.14031e10i			0.0464027i	0.999731	+	0.0232013i	$0.00738588\pi$
$907$			3.14031e10i			0.0464027i	−0.999731	+	0.0232013i	$0.992614\pi$
$908$		0			0
$909$	−3.13916e11	+	5.73197e10i	−0.459789	+	0.0839553i
$910$		0			0
$911$		−	5.50154e10i		−	0.0798750i	−0.999202	−	0.0399375i	$-0.987284\pi$
$911$		−	5.50154e10i		−	0.0798750i	0.999202	−	0.0399375i	$-0.0127159\pi$
$912$		0			0
$913$			3.47449e10i			0.0500044i
$914$		0			0
$915$	3.86481e8	+	1.09127e8i	0.000551371	+	0.000155686i
$916$		0			0
$917$	−1.08220e12			−1.53049
$918$		0			0
$919$	−7.71768e11			−1.08199			−0.540997	−	0.841025i	$-0.681953\pi$
$919$	−7.71768e11			−1.08199			−0.540997	+	0.841025i	$0.681953\pi$
$920$		0			0
$921$	−6.12791e11	−	5.11030e11i	−0.851675	−	0.710245i
$922$		0			0
$923$	−1.87866e12			−2.58846
$924$		0			0
$925$	7.95238e11	+	7.13107e11i	1.08625	+	0.974064i
$926$		0			0
$927$	3.01130e11	−	5.49849e10i	0.407788	−	0.0744603i
$928$		0			0
$929$		−	5.72128e11i		−	0.768122i	−0.923308	−	0.384061i	$-0.874525\pi$
$929$		−	5.72128e11i		−	0.768122i	0.923308	−	0.384061i	$-0.125475\pi$
$930$		0			0
$931$	−2.74453e12			−3.65317
$932$		0			0
$933$	−6.31632e11	−	5.26742e11i	−0.833561	−	0.695139i
$934$		0			0
$935$	−9.66919e9	+	4.31681e9i	−0.0126515	+	0.00564828i
$936$		0			0
$937$			9.35700e11i			1.21389i	0.794745	+	0.606944i	$0.207605\pi$
$937$			9.35700e11i			1.21389i	−0.794745	+	0.606944i	$0.792395\pi$
$938$		0			0
$939$	−5.24821e11	−	4.37668e11i	−0.675070	−	0.562967i
$940$		0			0
$941$		−	1.04933e12i		−	1.33829i	−0.743130	−	0.669147i	$-0.766660\pi$
$941$		−	1.04933e12i		−	1.33829i	0.743130	−	0.669147i	$-0.233340\pi$
$942$		0			0
$943$			1.27644e12i			1.61419i
$944$		0			0
$945$	−1.38969e12	−	1.31846e11i	−1.74257	−	0.165326i
$946$		0			0
$947$	1.00880e12			1.25431			0.627156	−	0.778893i	$-0.284219\pi$
$947$	1.00880e12			1.25431			0.627156	+	0.778893i	$0.284219\pi$
$948$		0			0
$949$	1.33272e12			1.64314
$950$		0			0
$951$	1.02512e12	−	1.22925e12i	1.25329	−	1.50286i
$952$		0			0
$953$	−7.00790e11			−0.849604			−0.424802	−	0.905286i	$-0.639656\pi$
$953$	−7.00790e11			−0.849604			−0.424802	+	0.905286i	$0.639656\pi$
$954$		0			0
$955$	−3.18566e11	+	1.42224e11i	−0.382989	+	0.170985i
$956$		0			0
$957$	6.51346e9	−	7.81048e9i	0.00776540	−	0.00931172i
$958$		0			0
$959$		−	1.93407e11i		−	0.228663i
$960$		0			0
$961$	−8.10266e11			−0.950023
$962$		0			0
$963$	−1.18606e11	−	6.49555e11i	−0.137912	−	0.755285i
$964$		0			0
$965$	8.32257e11	−	3.71561e11i	0.959728	−	0.428471i
$966$		0			0
$967$		−	7.69052e11i		−	0.879528i	−0.898113	−	0.439764i	$-0.855062\pi$
$967$		−	7.69052e11i		−	0.879528i	0.898113	−	0.439764i	$-0.144938\pi$
$968$		0			0
$969$	3.73438e11	−	4.47800e11i	0.423568	−	0.507913i
$970$		0			0
$971$			3.00192e11i			0.337694i	0.985642	+	0.168847i	$0.0540043\pi$
$971$			3.00192e11i			0.337694i	−0.985642	+	0.168847i	$0.945996\pi$
$972$		0			0
$973$		−	1.80849e12i		−	2.01774i
$974$		0			0
$975$	1.27565e12	−	1.85885e11i	1.41160	−	0.205697i
$976$		0			0
$977$	−1.35705e12			−1.48942			−0.744712	−	0.667386i	$-0.767413\pi$
$977$	−1.35705e12			−1.48942			−0.744712	+	0.667386i	$0.767413\pi$
$978$		0			0
$979$	2.47876e10			0.0269838
$980$		0			0
$981$	1.89172e11	+	1.03602e12i	0.204259	+	1.11864i
$982$		0			0
$983$	−6.50185e11			−0.696342			−0.348171	−	0.937431i	$-0.613197\pi$
$983$	−6.50185e11			−0.696342			−0.348171	+	0.937431i	$0.613197\pi$
$984$		0			0
$985$	2.77768e11	+	6.22171e11i	0.295079	+	0.660945i
$986$		0			0
$987$	−5.38252e11	−	4.48869e11i	−0.567175	−	0.472989i
$988$		0			0
$989$			2.97156e11i			0.310599i
$990$		0			0
$991$	5.99394e10			0.0621467			0.0310734	−	0.999517i	$-0.490107\pi$
$991$	5.99394e10			0.0621467			0.0310734	+	0.999517i	$0.490107\pi$
$992$		0			0
$993$	8.26091e11	−	9.90590e11i	0.849632	−	1.01882i
$994$		0			0
$995$	−2.94996e11	−	6.60760e11i	−0.300971	−	0.674143i
$996$		0			0
$997$		−	1.05035e12i		−	1.06305i	−0.847042	−	0.531526i	$-0.821619\pi$
$997$		−	1.05035e12i		−	1.06305i	0.847042	−	0.531526i	$-0.178381\pi$
$998$		0			0
$999$	−7.15103e11	+	1.26507e12i	−0.717970	+	1.27014i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.9.c.f.209.35		48
3.2	odd	2	inner	240.9.c.f.209.13		48
4.3	odd	2		120.9.c.a.89.14	yes	48
5.4	even	2	inner	240.9.c.f.209.14		48
12.11	even	2		120.9.c.a.89.36	yes	48
15.14	odd	2	inner	240.9.c.f.209.36		48
20.19	odd	2		120.9.c.a.89.35	yes	48
60.59	even	2		120.9.c.a.89.13	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.9.c.a.89.13	✓	48	60.59	even	2
120.9.c.a.89.14	yes	48	4.3	odd	2
120.9.c.a.89.35	yes	48	20.19	odd	2
120.9.c.a.89.36	yes	48	12.11	even	2
240.9.c.f.209.13		48	3.2	odd	2	inner
240.9.c.f.209.14		48	5.4	even	2	inner
240.9.c.f.209.35		48	1.1	even	1	trivial
240.9.c.f.209.36		48	15.14	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	240.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(51.8771 - 62.2074i) q^{3} +(-254.792 - 570.707i) q^{5} +4202.69i q^{7} +(-1178.52 - 6454.29i) q^{9} -542.921i q^{11} -40742.5i q^{13} +(-48720.1 - 13756.7i) q^{15} +31206.2 q^{17} +230675. q^{19} +(261439. + 218024. i) q^{21} -465190. q^{23} +(-260787. + 290823. i) q^{25} +(-462643. - 261517. i) q^{27} +231259. i q^{29} +206459. q^{31} +(-33773.7 - 28165.2i) q^{33} +(2.39850e6 - 1.07081e6i) q^{35} -2.73444e6i q^{37} +(-2.53449e6 - 2.11360e6i) q^{39} -2.74391e6i q^{41} -638785. i q^{43} +(-3.38323e6 + 2.31709e6i) q^{45} -2.05881e6 q^{47} -1.18978e7 q^{49} +(1.61889e6 - 1.94126e6i) q^{51} -1.09780e7 q^{53} +(-309848. + 138332. i) q^{55} +(1.19668e7 - 1.43497e7i) q^{57} +5.20695e6i q^{59} -7932.68 q^{61} +(2.71254e7 - 4.95297e6i) q^{63} +(-2.32520e7 + 1.03809e7i) q^{65} +2.61289e7i q^{67} +(-2.41327e7 + 2.89383e7i) q^{69} -4.61106e7i q^{71} +3.27109e7i q^{73} +(4.56245e6 + 3.13100e7i) q^{75} +2.28173e6 q^{77} -9.49611e6 q^{79} +(-4.02689e7 + 1.52130e7i) q^{81} -6.39962e7 q^{83} +(-7.95109e6 - 1.78096e7i) q^{85} +(1.43860e7 + 1.19971e7i) q^{87} +4.56560e7i q^{89} +1.71228e8 q^{91} +(1.07105e7 - 1.28433e7i) q^{93} +(-5.87743e7 - 1.31648e8i) q^{95} +2.57539e7i q^{97} +(-3.50417e6 + 639845. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2528 q^{9} - 132352 q^{15} + 116176 q^{21} + 56976 q^{25} - 1395648 q^{31} - 6888832 q^{39} - 4287056 q^{45} - 30813552 q^{49} + 22815168 q^{51} + 6062784 q^{55} + 14031936 q^{61} + 2522608 q^{69}+ \cdots + 21719360 q^{99}+O(q^{100}) \) 48 * q + 2528 * q^9 - 132352 * q^15 + 116176 * q^21 + 56976 * q^25 - 1395648 * q^31 - 6888832 * q^39 - 4287056 * q^45 - 30813552 * q^49 + 22815168 * q^51 + 6062784 * q^55 + 14031936 * q^61 + 2522608 * q^69 - 41684416 * q^75 - 19866432 * q^79 - 93539312 * q^81 - 116297664 * q^85 - 16826880 * q^91 + 21719360 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more