LMFDB - Embedded newform 243.9.b.i.242.39

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [243,9,Mod(242,243)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("243.242"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 243 = 3^{5} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	243.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [48,0,0,-6144,0,0,-5538] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$98.9930022449$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			242.39
Character	$\chi$	$=$	243.242
Dual form			243.9.b.i.242.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+23.5529i q^{2} -298.740 q^{4} -538.241i q^{5} -4052.68 q^{7} -1006.66i q^{8} +12677.2 q^{10} -9903.48i q^{11} -44841.2 q^{13} -95452.4i q^{14} -52767.8 q^{16} +118686. i q^{17} -196772. q^{19} +160794. i q^{20} +233256. q^{22} -238453. i q^{23} +100921. q^{25} -1.05614e6i q^{26} +1.21070e6 q^{28} -690920. i q^{29} -393655. q^{31} -1.50054e6i q^{32} -2.79540e6 q^{34} +2.18132e6i q^{35} -815758. q^{37} -4.63456e6i q^{38} -541823. q^{40} +2.17076e6i q^{41} -2.31203e6 q^{43} +2.95857e6i q^{44} +5.61626e6 q^{46} -2.51790e6i q^{47} +1.06594e7 q^{49} +2.37699e6i q^{50} +1.33959e7 q^{52} +1.00137e7i q^{53} -5.33047e6 q^{55} +4.07965e6i q^{56} +1.62732e7 q^{58} -539181. i q^{59} -8.88226e6 q^{61} -9.27173e6i q^{62} +2.18335e7 q^{64} +2.41354e7i q^{65} +2.95954e7 q^{67} -3.54562e7i q^{68} -5.13764e7 q^{70} -2.83945e7i q^{71} +3.40772e7 q^{73} -1.92135e7i q^{74} +5.87838e7 q^{76} +4.01356e7i q^{77} +5.04348e7 q^{79} +2.84018e7i q^{80} -5.11277e7 q^{82} +8.02664e7i q^{83} +6.38816e7 q^{85} -5.44550e7i q^{86} -9.96939e6 q^{88} -2.40788e7i q^{89} +1.81727e8 q^{91} +7.12355e7i q^{92} +5.93040e7 q^{94} +1.05911e8i q^{95} -5.70595e7 q^{97} +2.51060e8i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 6144 q^{4} - 5538 q^{7} - 10110 q^{13} + 786432 q^{16} - 277662 q^{19} - 3750000 q^{25} + 2835456 q^{28} - 856704 q^{31} + 2625462 q^{34} + 1640958 q^{37} + 866250 q^{40} - 13575936 q^{43} - 12911094 q^{46}+ \cdots + 244503618 q^{97}+O(q^{100}) $ 48 * q - 6144 * q^4 - 5538 * q^7 - 10110 * q^13 + 786432 * q^16 - 277662 * q^19 - 3750000 * q^25 + 2835456 * q^28 - 856704 * q^31 + 2625462 * q^34 + 1640958 * q^37 + 866250 * q^40 - 13575936 * q^43 - 12911094 * q^46 + 40853646 * q^49 + 5176320 * q^52 - 19789776 * q^55 + 23558400 * q^58 + 10452948 * q^61 - 89377098 * q^64 - 18985620 * q^67 + 44038656 * q^70 - 10176852 * q^73 + 186967158 * q^76 - 157712718 * q^79 - 139991040 * q^82 + 205482816 * q^85 - 190726200 * q^88 - 211624044 * q^91 + 522075960 * q^94 + 244503618 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/243\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	23.5529i	1.47206i	0.676950	+	0.736029i	$0.263301\pi$
$2$	23.5529i	1.47206i	−0.676950	+	0.736029i	$0.736699\pi$
$3$	0	0
$3$	0	0
$4$	−298.740	−1.16695
$5$	− 538.241i	− 0.861186i	−0.902546	−	0.430593i	$-0.858304\pi$
$5$	− 538.241i	− 0.861186i	0.902546	−	0.430593i	$-0.141696\pi$
$6$	0	0
$7$	−4052.68	−1.68791	−0.843956	−	0.536413i	$-0.819779\pi$
$7$	−4052.68	−1.68791	−0.843956	+	0.536413i	$0.819779\pi$
$8$	− 1006.66i	− 0.245765i
$9$	0	0
$10$	12677.2	1.26772
$11$	− 9903.48i	− 0.676421i	−0.941070	−	0.338211i	$-0.890178\pi$
$11$	− 9903.48i	− 0.676421i	0.941070	−	0.338211i	$-0.109822\pi$
$12$	0	0
$13$	−44841.2	−1.57001	−0.785007	−	0.619487i	$-0.787341\pi$
$13$	−44841.2	−1.57001	−0.785007	+	0.619487i	$0.787341\pi$
$14$	− 95452.4i	− 2.48470i
$15$	0	0
$16$	−52767.8	−0.805173
$17$	118686.i	1.42103i	0.703683	+	0.710514i	$0.251538\pi$
$17$	118686.i	1.42103i	−0.703683	+	0.710514i	$0.748462\pi$
$18$	0	0
$19$	−196772.	−1.50991	−0.754953	−	0.655779i	$-0.772340\pi$
$19$	−196772.	−1.50991	−0.754953	+	0.655779i	$0.772340\pi$
$20$	160794.i	1.00496i
$21$	0	0
$22$	233256.	0.995731
$23$	− 238453.i	− 0.852101i	−0.904699	−	0.426051i	$-0.859905\pi$
$23$	− 238453.i	− 0.852101i	0.904699	−	0.426051i	$-0.140095\pi$
$24$	0	0
$25$	100921.	0.258358
$26$	− 1.05614e6i	− 2.31115i
$27$	0	0
$28$	1.21070e6	1.96971
$29$	− 690920.i	− 0.976868i	−0.872601	−	0.488434i	$-0.837568\pi$
$29$	− 690920.i	− 0.976868i	0.872601	−	0.488434i	$-0.162432\pi$
$30$	0	0
$31$	−393655.	−0.426255	−0.213127	−	0.977024i	$-0.568365\pi$
$31$	−393655.	−0.426255	−0.213127	+	0.977024i	$0.568365\pi$
$32$	− 1.50054e6i	− 1.43103i
$33$	0	0
$34$	−2.79540e6	−2.09184
$35$	2.18132e6i	1.45361i
$36$	0	0
$37$	−815758.	−0.435266	−0.217633	−	0.976031i	$-0.569834\pi$
$37$	−815758.	−0.435266	−0.217633	+	0.976031i	$0.569834\pi$
$38$	− 4.63456e6i	− 2.22267i
$39$	0	0
$40$	−541823.	−0.211650
$41$	2.17076e6i	0.768202i	0.923291	+	0.384101i	$0.125489\pi$
$41$	2.17076e6i	0.768202i	−0.923291	+	0.384101i	$0.874511\pi$
$42$	0	0
$43$	−2.31203e6	−0.676269	−0.338134	−	0.941098i	$-0.609796\pi$
$43$	−2.31203e6	−0.676269	−0.338134	+	0.941098i	$0.609796\pi$
$44$	2.95857e6i	0.789352i
$45$	0	0
$46$	5.61626e6	1.25434
$47$	− 2.51790e6i	− 0.515998i	−0.966145	−	0.257999i	$-0.916937\pi$
$47$	− 2.51790e6i	− 0.515998i	0.966145	−	0.257999i	$-0.0830631\pi$
$48$	0	0
$49$	1.06594e7	1.84905
$50$	2.37699e6i	0.380318i
$51$	0	0
$52$	1.33959e7	1.83213
$53$	1.00137e7i	1.26909i	0.772886	+	0.634545i	$0.218813\pi$
$53$	1.00137e7i	1.26909i	−0.772886	+	0.634545i	$0.781187\pi$
$54$	0	0
$55$	−5.33047e6	−0.582525
$56$	4.07965e6i	0.414830i
$57$	0	0
$58$	1.62732e7	1.43801
$59$	− 539181.i	− 0.0444966i	−0.999752	−	0.0222483i	$-0.992918\pi$
$59$	− 539181.i	− 0.0444966i	0.999752	−	0.0222483i	$-0.00708244\pi$
$60$	0	0
$61$	−8.88226e6	−0.641511	−0.320755	−	0.947162i	$-0.603937\pi$
$61$	−8.88226e6	−0.641511	−0.320755	+	0.947162i	$0.603937\pi$
$62$	− 9.27173e6i	− 0.627472i
$63$	0	0
$64$	2.18335e7	1.30138
$65$	2.41354e7i	1.35207i
$66$	0	0
$67$	2.95954e7	1.46868	0.734338	−	0.678784i	$-0.237493\pi$
$67$	2.95954e7	1.46868	0.734338	+	0.678784i	$0.237493\pi$
$68$	− 3.54562e7i	− 1.65827i
$69$	0	0
$70$	−5.13764e7	−2.13979
$71$	− 2.83945e7i	− 1.11738i	−0.829376	−	0.558691i	$-0.811304\pi$
$71$	− 2.83945e7i	− 1.11738i	0.829376	−	0.558691i	$-0.188696\pi$
$72$	0	0
$73$	3.40772e7	1.19997	0.599987	−	0.800010i	$-0.295172\pi$
$73$	3.40772e7	1.19997	0.599987	+	0.800010i	$0.295172\pi$
$74$	− 1.92135e7i	− 0.640736i
$75$	0	0
$76$	5.87838e7	1.76199
$77$	4.01356e7i	1.14174i
$78$	0	0
$79$	5.04348e7	1.29486	0.647429	−	0.762126i	$-0.275844\pi$
$79$	5.04348e7	1.29486	0.647429	+	0.762126i	$0.275844\pi$
$80$	2.84018e7i	0.693404i
$81$	0	0
$82$	−5.11277e7	−1.13084
$83$	8.02664e7i	1.69130i	0.533736	+	0.845651i	$0.320788\pi$
$83$	8.02664e7i	1.69130i	−0.533736	+	0.845651i	$0.679212\pi$
$84$	0	0
$85$	6.38816e7	1.22377
$86$	− 5.44550e7i	− 0.995507i
$87$	0	0
$88$	−9.96939e6	−0.166241
$89$	− 2.40788e7i	− 0.383773i	−0.981417	−	0.191887i	$-0.938539\pi$
$89$	− 2.40788e7i	− 0.383773i	0.981417	−	0.191887i	$-0.0614606\pi$
$90$	0	0
$91$	1.81727e8	2.65005
$92$	7.12355e7i	0.994363i
$93$	0	0
$94$	5.93040e7	0.759579
$95$	1.05911e8i	1.30031i
$96$	0	0
$97$	−5.70595e7	−0.644527	−0.322264	−	0.946650i	$-0.604444\pi$
$97$	−5.70595e7	−0.644527	−0.322264	+	0.946650i	$0.604444\pi$
$98$	2.51060e8i	2.72190i
$99$	0	0
$100$	−3.01492e7	−0.301492
$101$	1.32882e8i	1.27697i	0.769633	+	0.638486i	$0.220439\pi$
$101$	1.32882e8i	1.27697i	−0.769633	+	0.638486i	$0.779561\pi$
$102$	0	0
$103$	−1.01313e8	−0.900155	−0.450077	−	0.892990i	$-0.648604\pi$
$103$	−1.01313e8	−0.900155	−0.450077	+	0.892990i	$0.648604\pi$
$104$	4.51396e7i	0.385855i
$105$	0	0
$106$	−2.35853e8	−1.86817
$107$	1.22335e8i	0.933286i	0.884446	+	0.466643i	$0.154537\pi$
$107$	1.22335e8i	0.933286i	−0.884446	+	0.466643i	$0.845463\pi$
$108$	0	0
$109$	7.61722e7	0.539623	0.269812	−	0.962913i	$-0.413039\pi$
$109$	7.61722e7	0.539623	0.269812	+	0.962913i	$0.413039\pi$
$110$	− 1.25548e8i	− 0.857510i
$111$	0	0
$112$	2.13851e8	1.35906
$113$	− 1.15813e8i	− 0.710303i	−0.934809	−	0.355152i	$-0.884429\pi$
$113$	− 1.15813e8i	− 0.710303i	0.934809	−	0.355152i	$-0.115571\pi$
$114$	0	0
$115$	−1.28345e8	−0.733818
$116$	2.06406e8i	1.13996i
$117$	0	0
$118$	1.26993e7	0.0655015
$119$	− 4.80995e8i	− 2.39857i
$120$	0	0
$121$	1.16280e8	0.542454
$122$	− 2.09203e8i	− 0.944341i
$123$	0	0
$124$	1.17601e8	0.497420
$125$	− 2.64571e8i	− 1.08368i
$126$	0	0
$127$	−3.26554e8	−1.25528	−0.627639	−	0.778504i	$-0.715979\pi$
$127$	−3.26554e8	−1.25528	−0.627639	+	0.778504i	$0.715979\pi$
$128$	1.30105e8i	0.484680i
$129$	0	0
$130$	−5.68459e8	−1.99033
$131$	− 2.64867e8i	− 0.899380i	−0.893185	−	0.449690i	$-0.851534\pi$
$131$	− 2.64867e8i	− 0.899380i	0.893185	−	0.449690i	$-0.148466\pi$
$132$	0	0
$133$	7.97455e8	2.54859
$134$	6.97059e8i	2.16197i
$135$	0	0
$136$	1.19476e8	0.349240
$137$	6.17028e8i	1.75155i	0.482720	+	0.875775i	$0.339649\pi$
$137$	6.17028e8i	1.75155i	−0.482720	+	0.875775i	$0.660351\pi$
$138$	0	0
$139$	−3.23443e8	−0.866440	−0.433220	−	0.901288i	$-0.642623\pi$
$139$	−3.23443e8	−0.866440	−0.433220	+	0.901288i	$0.642623\pi$
$140$	− 6.51647e8i	− 1.69629i
$141$	0	0
$142$	6.68775e8	1.64485
$143$	4.44084e8i	1.06199i
$144$	0	0
$145$	−3.71882e8	−0.841266
$146$	8.02617e8i	1.76643i
$147$	0	0
$148$	2.43700e8	0.507935
$149$	− 6.88395e8i	− 1.39667i	−0.715773	−	0.698333i	$-0.753925\pi$
$149$	− 6.88395e8i	− 1.39667i	0.715773	−	0.698333i	$-0.246075\pi$
$150$	0	0
$151$	−6.89897e8	−1.32702	−0.663509	−	0.748169i	$-0.730933\pi$
$151$	−6.89897e8	−1.32702	−0.663509	+	0.748169i	$0.730933\pi$
$152$	1.98082e8i	0.371082i
$153$	0	0
$154$	−9.45311e8	−1.68071
$155$	2.11882e8i	0.367085i
$156$	0	0
$157$	−6.13844e8	−1.01032	−0.505161	−	0.863025i	$-0.668567\pi$
$157$	−6.13844e8	−1.01032	−0.505161	+	0.863025i	$0.668567\pi$
$158$	1.18789e9i	1.90610i
$159$	0	0
$160$	−8.07653e8	−1.23238
$161$	9.66372e8i	1.43827i
$162$	0	0
$163$	7.98620e7	0.113133	0.0565666	−	0.998399i	$-0.481985\pi$
$163$	7.98620e7	0.113133	0.0565666	+	0.998399i	$0.481985\pi$
$164$	− 6.48492e8i	− 0.896457i
$165$	0	0
$166$	−1.89051e9	−2.48970
$167$	4.38350e7i	0.0563580i	0.999603	+	0.0281790i	$0.00897084\pi$
$167$	4.38350e7i	0.0563580i	−0.999603	+	0.0281790i	$0.991029\pi$
$168$	0	0
$169$	1.19500e9	1.46494
$170$	1.50460e9i	1.80146i
$171$	0	0
$172$	6.90696e8	0.789175
$173$	− 1.35263e8i	− 0.151007i	−0.997146	−	0.0755033i	$-0.975944\pi$
$173$	− 1.35263e8i	− 0.151007i	0.997146	−	0.0755033i	$-0.0240563\pi$
$174$	0	0
$175$	−4.09001e8	−0.436086
$176$	5.22585e8i	0.544636i
$177$	0	0
$178$	5.67126e8	0.564936
$179$	8.17612e6i	0.00796407i	0.999992	+	0.00398204i	$0.00126752\pi$
$179$	8.17612e6i	0.00796407i	−0.999992	+	0.00398204i	$0.998732\pi$
$180$	0	0
$181$	3.14441e8	0.292971	0.146486	−	0.989213i	$-0.453204\pi$
$181$	3.14441e8	0.292971	0.146486	+	0.989213i	$0.453204\pi$
$182$	4.28020e9i	3.90102i
$183$	0	0
$184$	−2.40040e8	−0.209417
$185$	4.39075e8i	0.374845i
$186$	0	0
$187$	1.17540e9	0.961214
$188$	7.52199e8i	0.602146i
$189$	0	0
$190$	−2.49451e9	−1.91413
$191$	− 1.26785e9i	− 0.952656i	−0.879268	−	0.476328i	$-0.841968\pi$
$191$	− 1.26785e9i	− 0.952656i	0.879268	−	0.476328i	$-0.158032\pi$
$192$	0	0
$193$	1.60046e9	1.15350	0.576748	−	0.816922i	$-0.304321\pi$
$193$	1.60046e9	1.15350	0.576748	+	0.816922i	$0.304321\pi$
$194$	− 1.34392e9i	− 0.948781i
$195$	0	0
$196$	−3.18439e9	−2.15775
$197$	3.92350e8i	0.260500i	0.991481	+	0.130250i	$0.0415781\pi$
$197$	3.92350e8i	0.260500i	−0.991481	+	0.130250i	$0.958422\pi$
$198$	0	0
$199$	−1.11760e9	−0.712645	−0.356322	−	0.934363i	$-0.615970\pi$
$199$	−1.11760e9	−0.712645	−0.356322	+	0.934363i	$0.615970\pi$
$200$	− 1.01593e8i	− 0.0634955i
$201$	0	0
$202$	−3.12976e9	−1.87978
$203$	2.80008e9i	1.64887i
$204$	0	0
$205$	1.16839e9	0.661565
$206$	− 2.38622e9i	− 1.32508i
$207$	0	0
$208$	2.36617e9	1.26413
$209$	1.94873e9i	1.02133i
$210$	0	0
$211$	1.64682e9	0.830839	0.415420	−	0.909630i	$-0.363635\pi$
$211$	1.64682e9	0.830839	0.415420	+	0.909630i	$0.363635\pi$
$212$	− 2.99150e9i	− 1.48097i
$213$	0	0
$214$	−2.88134e9	−1.37385
$215$	1.24443e9i	0.582394i
$216$	0	0
$217$	1.59536e9	0.719480
$218$	1.79408e9i	0.794356i
$219$	0	0
$220$	1.59242e9	0.679779
$221$	− 5.32201e9i	− 2.23104i
$222$	0	0
$223$	3.82469e8	0.154659	0.0773297	−	0.997006i	$-0.475361\pi$
$223$	3.82469e8	0.154659	0.0773297	+	0.997006i	$0.475361\pi$
$224$	6.08120e9i	2.41545i
$225$	0	0
$226$	2.72774e9	1.04561
$227$	− 3.72889e9i	− 1.40435i	−0.712004	−	0.702175i	$-0.752212\pi$
$227$	− 3.72889e9i	− 1.40435i	0.712004	−	0.702175i	$-0.247788\pi$
$228$	0	0
$229$	−3.92728e9	−1.42807	−0.714037	−	0.700108i	$-0.753135\pi$
$229$	−3.92728e9	−1.42807	−0.714037	+	0.700108i	$0.753135\pi$
$230$	− 3.02291e9i	− 1.08022i
$231$	0	0
$232$	−6.95519e8	−0.240080
$233$	− 1.95244e9i	− 0.662452i	−0.943551	−	0.331226i	$-0.892538\pi$
$233$	− 1.95244e9i	− 0.662452i	0.943551	−	0.331226i	$-0.107462\pi$
$234$	0	0
$235$	−1.35524e9	−0.444370
$236$	1.61075e8i	0.0519255i
$237$	0	0
$238$	1.13288e10	3.53083
$239$	3.51213e9i	1.07641i	0.842813	+	0.538207i	$0.180898\pi$
$239$	3.51213e9i	1.07641i	−0.842813	+	0.538207i	$0.819102\pi$
$240$	0	0
$241$	1.15927e9	0.343649	0.171825	−	0.985128i	$-0.445034\pi$
$241$	1.15927e9	0.343649	0.171825	+	0.985128i	$0.445034\pi$
$242$	2.73873e9i	0.798524i
$243$	0	0
$244$	2.65349e9	0.748613
$245$	− 5.73732e9i	− 1.59237i
$246$	0	0
$247$	8.82350e9	2.37057
$248$	3.96275e8i	0.104759i
$249$	0	0
$250$	6.23141e9	1.59524
$251$	− 5.16023e9i	− 1.30009i	−0.759894	−	0.650046i	$-0.774749\pi$
$251$	− 5.16023e9i	− 1.30009i	0.759894	−	0.650046i	$-0.225251\pi$
$252$	0	0
$253$	−2.36151e9	−0.576379
$254$	− 7.69130e9i	− 1.84784i
$255$	0	0
$256$	2.52502e9	0.587903
$257$	− 5.10591e9i	− 1.17042i	−0.810883	−	0.585209i	$-0.801012\pi$
$257$	− 5.10591e9i	− 1.17042i	0.810883	−	0.585209i	$-0.198988\pi$
$258$	0	0
$259$	3.30600e9	0.734690
$260$	− 7.21021e9i	− 1.57781i
$261$	0	0
$262$	6.23840e9	1.32394
$263$	6.53671e9i	1.36627i	0.730292	+	0.683135i	$0.239384\pi$
$263$	6.53671e9i	1.36627i	−0.730292	+	0.683135i	$0.760616\pi$
$264$	0	0
$265$	5.38981e9	1.09292
$266$	1.87824e10i	3.75167i
$267$	0	0
$268$	−8.84135e9	−1.71388
$269$	6.54489e9i	1.24995i	0.780644	+	0.624976i	$0.214891\pi$
$269$	6.54489e9i	1.24995i	−0.780644	+	0.624976i	$0.785109\pi$
$270$	0	0
$271$	−2.86482e9	−0.531153	−0.265577	−	0.964090i	$-0.585562\pi$
$271$	−2.86482e9	−0.531153	−0.265577	+	0.964090i	$0.585562\pi$
$272$	− 6.26279e9i	− 1.14417i
$273$	0	0
$274$	−1.45328e10	−2.57838
$275$	− 9.99471e8i	− 0.174759i
$276$	0	0
$277$	−6.30864e9	−1.07156	−0.535780	−	0.844358i	$-0.679982\pi$
$277$	−6.30864e9	−1.07156	−0.535780	+	0.844358i	$0.679982\pi$
$278$	− 7.61803e9i	− 1.27545i
$279$	0	0
$280$	2.19583e9	0.357246
$281$	8.68683e9i	1.39327i	0.717424	+	0.696636i	$0.245321\pi$
$281$	8.68683e9i	1.39327i	−0.717424	+	0.696636i	$0.754679\pi$
$282$	0	0
$283$	1.10233e10	1.71857	0.859284	−	0.511498i	$-0.170909\pi$
$283$	1.10233e10	1.71857	0.859284	+	0.511498i	$0.170909\pi$
$284$	8.48259e9i	1.30393i
$285$	0	0
$286$	−1.04595e10	−1.56331
$287$	− 8.79737e9i	− 1.29666i
$288$	0	0
$289$	−7.11054e9	−1.01932
$290$	− 8.75891e9i	− 1.23839i
$291$	0	0
$292$	−1.01802e10	−1.40031
$293$	3.77481e9i	0.512182i	0.966653	+	0.256091i	$0.0824347\pi$
$293$	3.77481e9i	0.512182i	−0.966653	+	0.256091i	$0.917565\pi$
$294$	0	0
$295$	−2.90210e8	−0.0383199
$296$	8.21187e8i	0.106973i
$297$	0	0
$298$	1.62137e10	2.05597
$299$	1.06925e10i	1.33781i
$300$	0	0
$301$	9.36990e9	1.14148
$302$	− 1.62491e10i	− 1.95345i
$303$	0	0
$304$	1.03832e10	1.21573
$305$	4.78080e9i	0.552460i
$306$	0	0
$307$	6.59335e9	0.742255	0.371127	−	0.928582i	$-0.378971\pi$
$307$	6.59335e9	0.742255	0.371127	+	0.928582i	$0.378971\pi$
$308$	− 1.19901e10i	− 1.33236i
$309$	0	0
$310$	−4.99043e9	−0.540370
$311$	− 3.48676e9i	− 0.372718i	−0.982482	−	0.186359i	$-0.940331\pi$
$311$	− 3.48676e9i	− 0.372718i	0.982482	−	0.186359i	$-0.0596687\pi$
$312$	0	0
$313$	1.30101e10	1.35551	0.677757	−	0.735286i	$-0.262952\pi$
$313$	1.30101e10	1.35551	0.677757	+	0.735286i	$0.262952\pi$
$314$	− 1.44578e10i	− 1.48725i
$315$	0	0
$316$	−1.50669e10	−1.51104
$317$	1.51141e10i	1.49674i	0.663284	+	0.748368i	$0.269162\pi$
$317$	1.51141e10i	1.49674i	−0.663284	+	0.748368i	$0.730838\pi$
$318$	0	0
$319$	−6.84252e9	−0.660775
$320$	− 1.17517e10i	− 1.12073i
$321$	0	0
$322$	−2.27609e10	−2.11722
$323$	− 2.33541e10i	− 2.14562i
$324$	0	0
$325$	−4.52542e9	−0.405626
$326$	1.88098e9i	0.166538i
$327$	0	0
$328$	2.18520e9	0.188798
$329$	1.02043e10i	0.870959i
$330$	0	0
$331$	2.27268e10	1.89333	0.946666	−	0.322216i	$-0.104428\pi$
$331$	2.27268e10	1.89333	0.946666	+	0.322216i	$0.104428\pi$
$332$	− 2.39788e10i	− 1.97367i
$333$	0	0
$334$	−1.03244e9	−0.0829622
$335$	− 1.59295e10i	− 1.26480i
$336$	0	0
$337$	−2.13432e10	−1.65478	−0.827390	−	0.561627i	$-0.810176\pi$
$337$	−2.13432e10	−1.65478	−0.827390	+	0.561627i	$0.810176\pi$
$338$	2.81458e10i	2.15648i
$339$	0	0
$340$	−1.90840e10	−1.42808
$341$	3.89856e9i	0.288328i
$342$	0	0
$343$	−1.98362e10	−1.43311
$344$	2.32742e9i	0.166203i
$345$	0	0
$346$	3.18585e9	0.222290
$347$	5.00444e9i	0.345173i	0.984994	+	0.172587i	$0.0552125\pi$
$347$	5.00444e9i	0.345173i	−0.984994	+	0.172587i	$0.944788\pi$
$348$	0	0
$349$	2.46756e10	1.66328	0.831642	−	0.555313i	$-0.187401\pi$
$349$	2.46756e10	1.66328	0.831642	+	0.555313i	$0.187401\pi$
$350$	− 9.63316e9i	− 0.641943i
$351$	0	0
$352$	−1.48606e10	−0.967976
$353$	− 1.22061e10i	− 0.786101i	−0.919517	−	0.393050i	$-0.871420\pi$
$353$	− 1.22061e10i	− 0.786101i	0.919517	−	0.393050i	$-0.128580\pi$
$354$	0	0
$355$	−1.52831e10	−0.962274
$356$	7.19330e9i	0.447846i
$357$	0	0
$358$	−1.92572e8	−0.0117236
$359$	− 1.55430e10i	− 0.935743i	−0.883796	−	0.467872i	$-0.845021\pi$
$359$	− 1.55430e10i	− 0.935743i	0.883796	−	0.467872i	$-0.154979\pi$
$360$	0	0
$361$	2.17358e10	1.27981
$362$	7.40601e9i	0.431271i
$363$	0	0
$364$	−5.42891e10	−3.09248
$365$	− 1.83417e10i	− 1.03340i
$366$	0	0
$367$	1.22968e10	0.677843	0.338921	−	0.940815i	$-0.389938\pi$
$367$	1.22968e10	0.677843	0.338921	+	0.940815i	$0.389938\pi$
$368$	1.25826e10i	0.686089i
$369$	0	0
$370$	−1.03415e10	−0.551793
$371$	− 4.05824e10i	− 2.14211i
$372$	0	0
$373$	−1.50406e10	−0.777014	−0.388507	−	0.921446i	$-0.627009\pi$
$373$	−1.50406e10	−0.777014	−0.388507	+	0.921446i	$0.627009\pi$
$374$	2.76842e10i	1.41496i
$375$	0	0
$376$	−2.53466e9	−0.126814
$377$	3.09817e10i	1.53370i
$378$	0	0
$379$	−1.01950e10	−0.494119	−0.247060	−	0.969000i	$-0.579464\pi$
$379$	−1.01950e10	−0.494119	−0.247060	+	0.969000i	$0.579464\pi$
$380$	− 3.16399e10i	− 1.51740i
$381$	0	0
$382$	2.98617e10	1.40236
$383$	− 8.24876e9i	− 0.383349i	−0.981459	−	0.191674i	$-0.938608\pi$
$383$	− 8.24876e9i	− 0.383349i	0.981459	−	0.191674i	$-0.0613917\pi$
$384$	0	0
$385$	2.16026e10	0.983250
$386$	3.76956e10i	1.69801i
$387$	0	0
$388$	1.70460e10	0.752134
$389$	− 4.79075e9i	− 0.209221i	−0.994513	−	0.104610i	$-0.966640\pi$
$389$	− 4.79075e9i	− 0.209221i	0.994513	−	0.104610i	$-0.0333596\pi$
$390$	0	0
$391$	2.83010e10	1.21086
$392$	− 1.07303e10i	− 0.454431i
$393$	0	0
$394$	−9.24098e9	−0.383472
$395$	− 2.71461e10i	− 1.11511i
$396$	0	0
$397$	−3.41091e10	−1.37312	−0.686559	−	0.727074i	$-0.740880\pi$
$397$	−3.41091e10	−1.37312	−0.686559	+	0.727074i	$0.740880\pi$
$398$	− 2.63227e10i	− 1.04905i
$399$	0	0
$400$	−5.32539e9	−0.208023
$401$	− 3.34523e10i	− 1.29374i	−0.762599	−	0.646872i	$-0.776077\pi$
$401$	− 3.34523e10i	− 1.29374i	0.762599	−	0.646872i	$-0.223923\pi$
$402$	0	0
$403$	1.76520e10	0.669226
$404$	− 3.96973e10i	− 1.49017i
$405$	0	0
$406$	−6.59500e10	−2.42723
$407$	8.07884e9i	0.294423i
$408$	0	0
$409$	1.88159e10	0.672407	0.336204	−	0.941789i	$-0.390857\pi$
$409$	1.88159e10	0.672407	0.336204	+	0.941789i	$0.390857\pi$
$410$	2.75190e10i	0.973862i
$411$	0	0
$412$	3.02663e10	1.05044
$413$	2.18513e9i	0.0751063i
$414$	0	0
$415$	4.32027e10	1.45653
$416$	6.72860e10i	2.24673i
$417$	0	0
$418$	−4.58983e10	−1.50346
$419$	− 2.43950e10i	− 0.791488i	−0.918361	−	0.395744i	$-0.870487\pi$
$419$	− 2.43950e10i	− 0.791488i	0.918361	−	0.395744i	$-0.129513\pi$
$420$	0	0
$421$	−2.67893e10	−0.852771	−0.426386	−	0.904542i	$-0.640213\pi$
$421$	−2.67893e10	−0.852771	−0.426386	+	0.904542i	$0.640213\pi$
$422$	3.87875e10i	1.22304i
$423$	0	0
$424$	1.00804e10	0.311898
$425$	1.19779e10i	0.367134i
$426$	0	0
$427$	3.59969e10	1.08281
$428$	− 3.65463e10i	− 1.08910i
$429$	0	0
$430$	−2.93100e10	−0.857317
$431$	1.61308e10i	0.467461i	0.972301	+	0.233731i	$0.0750934\pi$
$431$	1.61308e10i	0.467461i	−0.972301	+	0.233731i	$0.924907\pi$
$432$	0	0
$433$	2.18440e10	0.621414	0.310707	−	0.950506i	$-0.399434\pi$
$433$	2.18440e10	0.621414	0.310707	+	0.950506i	$0.399434\pi$
$434$	3.75753e10i	1.05912i
$435$	0	0
$436$	−2.27557e10	−0.629715
$437$	4.69209e10i	1.28659i
$438$	0	0
$439$	−3.11018e10	−0.837389	−0.418694	−	0.908127i	$-0.637512\pi$
$439$	−3.11018e10	−0.837389	−0.418694	+	0.908127i	$0.637512\pi$
$440$	5.36594e9i	0.143164i
$441$	0	0
$442$	1.25349e11	3.28421
$443$	− 1.20685e10i	− 0.313357i	−0.987650	−	0.156679i	$-0.949921\pi$
$443$	− 1.20685e10i	− 0.313357i	0.987650	−	0.156679i	$-0.0500787\pi$
$444$	0	0
$445$	−1.29602e10	−0.330500
$446$	9.00825e9i	0.227667i
$447$	0	0
$448$	−8.84842e10	−2.19661
$449$	1.58142e10i	0.389101i	0.980892	+	0.194551i	$0.0623249\pi$
$449$	1.58142e10i	0.389101i	−0.980892	+	0.194551i	$0.937675\pi$
$450$	0	0
$451$	2.14980e10	0.519628
$452$	3.45980e10i	0.828891i
$453$	0	0
$454$	8.78262e10	2.06729
$455$	− 9.78129e10i	− 2.28218i
$456$	0	0
$457$	−1.32253e10	−0.303208	−0.151604	−	0.988441i	$-0.548444\pi$
$457$	−1.32253e10	−0.303208	−0.151604	+	0.988441i	$0.548444\pi$
$458$	− 9.24990e10i	− 2.10221i
$459$	0	0
$460$	3.83419e10	0.856332
$461$	5.90700e10i	1.30787i	0.756553	+	0.653933i	$0.226882\pi$
$461$	5.90700e10i	1.30787i	−0.756553	+	0.653933i	$0.773118\pi$
$462$	0	0
$463$	−6.47230e9	−0.140843	−0.0704214	−	0.997517i	$-0.522434\pi$
$463$	−6.47230e9	−0.140843	−0.0704214	+	0.997517i	$0.522434\pi$
$464$	3.64584e10i	0.786548i
$465$	0	0
$466$	4.59857e10	0.975168
$467$	− 6.99594e10i	− 1.47088i	−0.677588	−	0.735442i	$-0.736975\pi$
$467$	− 6.99594e10i	− 1.47088i	0.677588	−	0.735442i	$-0.263025\pi$
$468$	0	0
$469$	−1.19941e11	−2.47899
$470$	− 3.19199e10i	− 0.654139i
$471$	0	0
$472$	−5.42769e8	−0.0109357
$473$	2.28971e10i	0.457443i
$474$	0	0
$475$	−1.98585e10	−0.390096
$476$	1.43692e11i	2.79902i
$477$	0	0
$478$	−8.27210e10	−1.58454
$479$	4.34828e9i	0.0825991i	0.999147	+	0.0412996i	$0.0131498\pi$
$479$	4.34828e9i	0.0825991i	−0.999147	+	0.0412996i	$0.986850\pi$
$480$	0	0
$481$	3.65795e10	0.683373
$482$	2.73041e10i	0.505871i
$483$	0	0
$484$	−3.47375e10	−0.633019
$485$	3.07118e10i	0.555058i
$486$	0	0
$487$	5.82467e10	1.03551	0.517756	−	0.855528i	$-0.326767\pi$
$487$	5.82467e10	1.03551	0.517756	+	0.855528i	$0.326767\pi$
$488$	8.94137e9i	0.157661i
$489$	0	0
$490$	1.35131e11	2.34406
$491$	4.14176e9i	0.0712622i	0.999365	+	0.0356311i	$0.0113441\pi$
$491$	4.14176e9i	0.0712622i	−0.999365	+	0.0356311i	$0.988656\pi$
$492$	0	0
$493$	8.20024e10	1.38816
$494$	2.07819e11i	3.48962i
$495$	0	0
$496$	2.07723e10	0.343209
$497$	1.15074e11i	1.88604i
$498$	0	0
$499$	6.74611e10	1.08806	0.544028	−	0.839067i	$-0.316898\pi$
$499$	6.74611e10	1.08806	0.544028	+	0.839067i	$0.316898\pi$
$500$	7.90378e10i	1.26461i
$501$	0	0
$502$	1.21539e11	1.91381
$503$	6.04996e10i	0.945106i	0.881302	+	0.472553i	$0.156667\pi$
$503$	6.04996e10i	0.945106i	−0.881302	+	0.472553i	$0.843333\pi$
$504$	0	0
$505$	7.15227e10	1.09971
$506$	− 5.56206e10i	− 0.848464i
$507$	0	0
$508$	9.75548e10	1.46485
$509$	− 1.16270e11i	− 1.73220i	−0.499874	−	0.866098i	$-0.666621\pi$
$509$	− 1.16270e11i	− 1.73220i	0.499874	−	0.866098i	$-0.333379\pi$
$510$	0	0
$511$	−1.38104e11	−2.02545
$512$	9.27786e10i	1.35011i
$513$	0	0
$514$	1.20259e11	1.72292
$515$	5.45310e10i	0.775201i
$516$	0	0
$517$	−2.49360e10	−0.349032
$518$	7.78660e10i	1.08151i
$519$	0	0
$520$	2.42960e10	0.332293
$521$	− 2.97924e10i	− 0.404347i	−0.979350	−	0.202174i	$-0.935199\pi$
$521$	− 2.97924e10i	− 0.404347i	0.979350	−	0.202174i	$-0.0648005\pi$
$522$	0	0
$523$	8.16732e10	1.09162	0.545812	−	0.837908i	$-0.316221\pi$
$523$	8.16732e10	1.09162	0.545812	+	0.837908i	$0.316221\pi$
$524$	7.91265e10i	1.04954i
$525$	0	0
$526$	−1.53959e11	−2.01123
$527$	− 4.67213e10i	− 0.605720i
$528$	0	0
$529$	2.14512e10	0.273923
$530$	1.26946e11i	1.60885i
$531$	0	0
$532$	−2.38232e11	−2.97408
$533$	− 9.73393e10i	− 1.20609i
$534$	0	0
$535$	6.58457e10	0.803733
$536$	− 2.97924e10i	− 0.360949i
$537$	0	0
$538$	−1.54151e11	−1.84000
$539$	− 1.05565e11i	− 1.25073i
$540$	0	0
$541$	−3.44758e10	−0.402462	−0.201231	−	0.979544i	$-0.564494\pi$
$541$	−3.44758e10	−0.402462	−0.201231	+	0.979544i	$0.564494\pi$
$542$	− 6.74748e10i	− 0.781888i
$543$	0	0
$544$	1.78093e11	2.03353
$545$	− 4.09990e10i	− 0.464716i
$546$	0	0
$547$	2.43290e10	0.271754	0.135877	−	0.990726i	$-0.456615\pi$
$547$	2.43290e10	0.271754	0.135877	+	0.990726i	$0.456615\pi$
$548$	− 1.84331e11i	− 2.04398i
$549$	0	0
$550$	2.35405e10	0.257255
$551$	1.35954e11i	1.47498i
$552$	0	0
$553$	−2.04396e11	−2.18560
$554$	− 1.48587e11i	− 1.57740i
$555$	0	0
$556$	9.66254e10	1.01110
$557$	− 8.20305e9i	− 0.0852225i	−0.999092	−	0.0426113i	$-0.986432\pi$
$557$	− 8.20305e9i	− 0.0852225i	0.999092	−	0.0426113i	$-0.0135677\pi$
$558$	0	0
$559$	1.03674e11	1.06175
$560$	− 1.15103e11i	− 1.17040i
$561$	0	0
$562$	−2.04600e11	−2.05098
$563$	1.04123e11i	1.03637i	0.855269	+	0.518185i	$0.173392\pi$
$563$	1.04123e11i	1.03637i	−0.855269	+	0.518185i	$0.826608\pi$
$564$	0	0
$565$	−6.23354e10	−0.611703
$566$	2.59631e11i	2.52983i
$567$	0	0
$568$	−2.85835e10	−0.274614
$569$	8.11053e9i	0.0773750i	0.999251	+	0.0386875i	$0.0123177\pi$
$569$	8.11053e9i	0.0773750i	−0.999251	+	0.0386875i	$0.987682\pi$
$570$	0	0
$571$	−1.06153e11	−0.998594	−0.499297	−	0.866431i	$-0.666408\pi$
$571$	−1.06153e11	−0.998594	−0.499297	+	0.866431i	$0.666408\pi$
$572$	− 1.32666e11i	− 1.23929i
$573$	0	0
$574$	2.07204e11	1.90876
$575$	− 2.40649e10i	− 0.220147i
$576$	0	0
$577$	−1.09987e11	−0.992292	−0.496146	−	0.868239i	$-0.665252\pi$
$577$	−1.09987e11	−0.992292	−0.496146	+	0.868239i	$0.665252\pi$
$578$	− 1.67474e11i	− 1.50050i
$579$	0	0
$580$	1.11096e11	0.981718
$581$	− 3.25294e11i	− 2.85477i
$582$	0	0
$583$	9.91708e10	0.858440
$584$	− 3.43040e10i	− 0.294912i
$585$	0	0
$586$	−8.89078e10	−0.753962
$587$	− 1.23957e11i	− 1.04405i	−0.852931	−	0.522023i	$-0.825178\pi$
$587$	− 1.23957e11i	− 1.04405i	0.852931	−	0.522023i	$-0.174822\pi$
$588$	0	0
$589$	7.74605e10	0.643604
$590$	− 6.83529e9i	− 0.0564090i
$591$	0	0
$592$	4.30457e10	0.350464
$593$	− 1.61977e11i	− 1.30989i	−0.755678	−	0.654943i	$-0.772693\pi$
$593$	− 1.61977e11i	− 1.30989i	0.755678	−	0.654943i	$-0.227307\pi$
$594$	0	0
$595$	−2.58891e11	−2.06562
$596$	2.05651e11i	1.62984i
$597$	0	0
$598$	−2.51840e11	−1.96934
$599$	− 6.65098e10i	− 0.516629i	−0.966061	−	0.258314i	$-0.916833\pi$
$599$	− 6.65098e10i	− 0.516629i	0.966061	−	0.258314i	$-0.0831670\pi$
$600$	0	0
$601$	−5.22611e10	−0.400572	−0.200286	−	0.979737i	$-0.564187\pi$
$601$	−5.22611e10	−0.400572	−0.200286	+	0.979737i	$0.564187\pi$
$602$	2.20689e11i	1.68033i
$603$	0	0
$604$	2.06100e11	1.54857
$605$	− 6.25867e10i	− 0.467154i
$606$	0	0
$607$	−1.03372e11	−0.761464	−0.380732	−	0.924685i	$-0.624328\pi$
$607$	−1.03372e11	−0.761464	−0.380732	+	0.924685i	$0.624328\pi$
$608$	2.95265e11i	2.16071i
$609$	0	0
$610$	−1.12602e11	−0.813254
$611$	1.12906e11i	0.810124i
$612$	0	0
$613$	−5.30997e10	−0.376054	−0.188027	−	0.982164i	$-0.560209\pi$
$613$	−5.30997e10	−0.376054	−0.188027	+	0.982164i	$0.560209\pi$
$614$	1.55293e11i	1.09264i
$615$	0	0
$616$	4.04027e10	0.280600
$617$	2.08477e11i	1.43853i	0.694737	+	0.719264i	$0.255521\pi$
$617$	2.08477e11i	1.43853i	−0.694737	+	0.719264i	$0.744479\pi$
$618$	0	0
$619$	−2.33161e11	−1.58816	−0.794079	−	0.607815i	$-0.792046\pi$
$619$	−2.33161e11	−1.58816	−0.794079	+	0.607815i	$0.792046\pi$
$620$	− 6.32975e10i	− 0.428371i
$621$	0	0
$622$	8.21233e10	0.548662
$623$	9.75835e10i	0.647775i
$624$	0	0
$625$	−1.02981e11	−0.674893
$626$	3.06426e11i	1.99540i
$627$	0	0
$628$	1.83380e11	1.17900
$629$	− 9.68188e10i	− 0.618525i
$630$	0	0
$631$	1.64963e10	0.104057	0.0520283	−	0.998646i	$-0.483431\pi$
$631$	1.64963e10	0.104057	0.0520283	+	0.998646i	$0.483431\pi$
$632$	− 5.07704e10i	− 0.318231i
$633$	0	0
$634$	−3.55981e11	−2.20328
$635$	1.75765e11i	1.08103i
$636$	0	0
$637$	−4.77979e11	−2.90303
$638$	− 1.61161e11i	− 0.972698i
$639$	0	0
$640$	7.00281e10	0.417400
$641$	2.27185e11i	1.34570i	0.739779	+	0.672850i	$0.234930\pi$
$641$	2.27185e11i	1.34570i	−0.739779	+	0.672850i	$0.765070\pi$
$642$	0	0
$643$	−1.19364e11	−0.698282	−0.349141	−	0.937070i	$-0.613527\pi$
$643$	−1.19364e11	−0.698282	−0.349141	+	0.937070i	$0.613527\pi$
$644$	− 2.88694e11i	− 1.67840i
$645$	0	0
$646$	5.50057e11	3.15847
$647$	2.03521e11i	1.16143i	0.814108	+	0.580714i	$0.197226\pi$
$647$	2.03521e11i	1.16143i	−0.814108	+	0.580714i	$0.802774\pi$
$648$	0	0
$649$	−5.33977e9	−0.0300984
$650$	− 1.06587e11i	− 0.597105i
$651$	0	0
$652$	−2.38580e10	−0.132021
$653$	− 7.74277e10i	− 0.425837i	−0.977070	−	0.212919i	$-0.931703\pi$
$653$	− 7.74277e10i	− 0.425837i	0.977070	−	0.212919i	$-0.0682969\pi$
$654$	0	0
$655$	−1.42563e11	−0.774534
$656$	− 1.14546e11i	− 0.618536i
$657$	0	0
$658$	−2.40340e11	−1.28210
$659$	1.25263e11i	0.664173i	0.943249	+	0.332086i	$0.107753\pi$
$659$	1.25263e11i	0.664173i	−0.943249	+	0.332086i	$0.892247\pi$
$660$	0	0
$661$	−1.98619e11	−1.04044	−0.520218	−	0.854033i	$-0.674149\pi$
$661$	−1.98619e11	−1.04044	−0.520218	+	0.854033i	$0.674149\pi$
$662$	5.35283e11i	2.78709i
$663$	0	0
$664$	8.08006e10	0.415664
$665$	− 4.29223e11i	− 2.19481i
$666$	0	0
$667$	−1.64752e11	−0.832391
$668$	− 1.30953e10i	− 0.0657672i
$669$	0	0
$670$	3.75186e11	1.86186
$671$	8.79653e10i	0.433932i
$672$	0	0
$673$	2.29930e11	1.12082	0.560410	−	0.828215i	$-0.310644\pi$
$673$	2.29930e11	1.12082	0.560410	+	0.828215i	$0.310644\pi$
$674$	− 5.02695e11i	− 2.43593i
$675$	0	0
$676$	−3.56995e11	−1.70952
$677$	− 3.55228e11i	− 1.69103i	−0.533949	−	0.845517i	$-0.679292\pi$
$677$	− 3.55228e11i	− 1.69103i	0.533949	−	0.845517i	$-0.320708\pi$
$678$	0	0
$679$	2.31244e11	1.08791
$680$	− 6.43067e10i	− 0.300760i
$681$	0	0
$682$	−9.18224e10	−0.424435
$683$	1.23568e11i	0.567835i	0.958849	+	0.283918i	$0.0916342\pi$
$683$	1.23568e11i	0.567835i	−0.958849	+	0.283918i	$0.908366\pi$
$684$	0	0
$685$	3.32110e11	1.50841
$686$	− 4.67199e11i	− 2.10963i
$687$	0	0
$688$	1.22001e11	0.544513
$689$	− 4.49028e11i	− 1.99249i
$690$	0	0
$691$	7.55279e10	0.331280	0.165640	−	0.986186i	$-0.447031\pi$
$691$	7.55279e10	0.331280	0.165640	+	0.986186i	$0.447031\pi$
$692$	4.04086e10i	0.176218i
$693$	0	0
$694$	−1.17869e11	−0.508115
$695$	1.74090e11i	0.746166i
$696$	0	0
$697$	−2.57638e11	−1.09164
$698$	5.81183e11i	2.44845i
$699$	0	0
$700$	1.22185e11	0.508892
$701$	1.59198e11i	0.659273i	0.944108	+	0.329636i	$0.106926\pi$
$701$	1.59198e11i	0.659273i	−0.944108	+	0.329636i	$0.893074\pi$
$702$	0	0
$703$	1.60519e11	0.657210
$704$	− 2.16228e11i	− 0.880281i
$705$	0	0
$706$	2.87489e11	1.15719
$707$	− 5.38529e11i	− 2.15542i
$708$	0	0
$709$	7.32146e10	0.289743	0.144871	−	0.989451i	$-0.453723\pi$
$709$	7.32146e10	0.289743	0.144871	+	0.989451i	$0.453723\pi$
$710$	− 3.59962e11i	− 1.41652i
$711$	0	0
$712$	−2.42390e10	−0.0943182
$713$	9.38682e10i	0.363212i
$714$	0	0
$715$	2.39024e11	0.914572
$716$	− 2.44254e9i	− 0.00929370i
$717$	0	0
$718$	3.66083e11	1.37747
$719$	− 3.87495e11i	− 1.44994i	−0.688779	−	0.724972i	$-0.741853\pi$
$719$	− 3.87495e11i	− 1.44994i	0.688779	−	0.724972i	$-0.258147\pi$
$720$	0	0
$721$	4.10590e11	1.51938
$722$	5.11942e11i	1.88396i
$723$	0	0
$724$	−9.39362e10	−0.341884
$725$	− 6.97285e10i	− 0.252382i
$726$	0	0
$727$	1.54639e11	0.553582	0.276791	−	0.960930i	$-0.410729\pi$
$727$	1.54639e11	0.553582	0.276791	+	0.960930i	$0.410729\pi$
$728$	− 1.82936e11i	− 0.651289i
$729$	0	0
$730$	4.32002e11	1.52123
$731$	− 2.74405e11i	− 0.960998i
$732$	0	0
$733$	1.23905e10	0.0429215	0.0214607	−	0.999770i	$-0.493168\pi$
$733$	1.23905e10	0.0429215	0.0214607	+	0.999770i	$0.493168\pi$
$734$	2.89626e11i	0.997823i
$735$	0	0
$736$	−3.57808e11	−1.21938
$737$	− 2.93098e11i	− 0.993443i
$738$	0	0
$739$	1.50897e11	0.505944	0.252972	−	0.967474i	$-0.418592\pi$
$739$	1.50897e11	0.505944	0.252972	+	0.967474i	$0.418592\pi$
$740$	− 1.31169e11i	− 0.437426i
$741$	0	0
$742$	9.55834e11	3.15331
$743$	− 6.04934e11i	− 1.98497i	−0.122384	−	0.992483i	$-0.539054\pi$
$743$	− 6.04934e11i	− 1.98497i	0.122384	−	0.992483i	$-0.460946\pi$
$744$	0	0
$745$	−3.70523e11	−1.20279
$746$	− 3.54249e11i	− 1.14381i
$747$	0	0
$748$	−3.51140e11	−1.12169
$749$	− 4.95783e11i	− 1.57530i
$750$	0	0
$751$	−2.24484e11	−0.705709	−0.352854	−	0.935678i	$-0.614789\pi$
$751$	−2.24484e11	−0.705709	−0.352854	+	0.935678i	$0.614789\pi$
$752$	1.32864e11i	0.415467i
$753$	0	0
$754$	−7.29709e11	−2.25769
$755$	3.71331e11i	1.14281i
$756$	0	0
$757$	8.38107e10	0.255221	0.127610	−	0.991824i	$-0.459269\pi$
$757$	8.38107e10	0.255221	0.127610	+	0.991824i	$0.459269\pi$
$758$	− 2.40123e11i	− 0.727372i
$759$	0	0
$760$	1.06616e11	0.319571
$761$	− 2.16529e9i	− 0.00645620i	−0.999995	−	0.00322810i	$-0.998972\pi$
$761$	− 2.16529e9i	− 0.00645620i	0.999995	−	0.00322810i	$-0.00102754\pi$
$762$	0	0
$763$	−3.08701e11	−0.910836
$764$	3.78759e11i	1.11171i
$765$	0	0
$766$	1.94282e11	0.564311
$767$	2.41775e10i	0.0698603i
$768$	0	0
$769$	−3.73555e10	−0.106819	−0.0534096	−	0.998573i	$-0.517009\pi$
$769$	−3.73555e10	−0.106819	−0.0534096	+	0.998573i	$0.517009\pi$
$770$	5.08805e11i	1.44740i
$771$	0	0
$772$	−4.78122e11	−1.34608
$773$	− 5.53510e11i	− 1.55027i	−0.631796	−	0.775135i	$-0.717682\pi$
$773$	− 5.53510e11i	− 1.55027i	0.631796	−	0.775135i	$-0.282318\pi$
$774$	0	0
$775$	−3.97281e10	−0.110126
$776$	5.74393e10i	0.158403i
$777$	0	0
$778$	1.12836e11	0.307985
$779$	− 4.27145e11i	− 1.15991i
$780$	0	0
$781$	−2.81205e11	−0.755821
$782$	6.66570e11i	1.78246i
$783$	0	0
$784$	−5.62472e11	−1.48880
$785$	3.30396e11i	0.870075i
$786$	0	0
$787$	2.02266e11	0.527260	0.263630	−	0.964624i	$-0.415080\pi$
$787$	2.02266e11	0.527260	0.263630	+	0.964624i	$0.415080\pi$
$788$	− 1.17211e11i	− 0.303992i
$789$	0	0
$790$	6.39370e11	1.64151
$791$	4.69353e11i	1.19893i
$792$	0	0
$793$	3.98291e11	1.00718
$794$	− 8.03369e11i	− 2.02131i
$795$	0	0
$796$	3.33871e11	0.831624
$797$	4.43023e11i	1.09798i	0.835830	+	0.548989i	$0.184987\pi$
$797$	4.43023e11i	1.09798i	−0.835830	+	0.548989i	$0.815013\pi$
$798$	0	0
$799$	2.98839e11	0.733248
$800$	− 1.51436e11i	− 0.369717i
$801$	0	0
$802$	7.87899e11	1.90447
$803$	− 3.37483e11i	− 0.811688i
$804$	0	0
$805$	5.20142e11	1.23862
$806$	4.15755e11i	0.985139i
$807$	0	0
$808$	1.33767e11	0.313836
$809$	− 3.05754e11i	− 0.713801i	−0.934142	−	0.356901i	$-0.883833\pi$
$809$	− 3.05754e11i	− 0.713801i	0.934142	−	0.356901i	$-0.116167\pi$
$810$	0	0
$811$	4.20897e11	0.972954	0.486477	−	0.873693i	$-0.338282\pi$
$811$	4.20897e11	0.972954	0.486477	+	0.873693i	$0.338282\pi$
$812$	− 8.36495e11i	− 1.92415i
$813$	0	0
$814$	−1.90280e11	−0.433407
$815$	− 4.29850e10i	− 0.0974287i
$816$	0	0
$817$	4.54943e11	1.02110
$818$	4.43170e11i	0.989822i
$819$	0	0
$820$	−3.49045e11	−0.772016
$821$	− 6.69864e10i	− 0.147440i	−0.997279	−	0.0737198i	$-0.976513\pi$
$821$	− 6.69864e10i	− 0.147440i	0.997279	−	0.0737198i	$-0.0234870\pi$
$822$	0	0
$823$	5.08513e11	1.10842	0.554208	−	0.832378i	$-0.313021\pi$
$823$	5.08513e11	1.10842	0.554208	+	0.832378i	$0.313021\pi$
$824$	1.01987e11i	0.221227i
$825$	0	0
$826$	−5.14661e10	−0.110561
$827$	− 7.82451e11i	− 1.67277i	−0.548145	−	0.836383i	$-0.684666\pi$
$827$	− 7.82451e11i	− 1.67277i	0.548145	−	0.836383i	$-0.315334\pi$
$828$	0	0
$829$	−4.82359e11	−1.02130	−0.510648	−	0.859790i	$-0.670595\pi$
$829$	−4.82359e11	−1.02130	−0.510648	+	0.859790i	$0.670595\pi$
$830$	1.01755e12i	2.14409i
$831$	0	0
$832$	−9.79042e11	−2.04319
$833$	1.26512e12i	2.62755i
$834$	0	0
$835$	2.35938e10	0.0485347
$836$	− 5.82164e11i	− 1.19185i
$837$	0	0
$838$	5.74573e11	1.16512
$839$	3.86136e11i	0.779277i	0.920968	+	0.389639i	$0.127400\pi$
$839$	3.86136e11i	0.779277i	−0.920968	+	0.389639i	$0.872600\pi$
$840$	0	0
$841$	2.28753e10	0.0457281
$842$	− 6.30966e11i	− 1.25533i
$843$	0	0
$844$	−4.91972e11	−0.969551
$845$	− 6.43199e11i	− 1.26159i
$846$	0	0
$847$	−4.71245e11	−0.915615
$848$	− 5.28403e11i	− 1.02184i
$849$	0	0
$850$	−2.82114e11	−0.540443
$851$	1.94520e11i	0.370890i
$852$	0	0
$853$	2.57621e11	0.486615	0.243308	−	0.969949i	$-0.421768\pi$
$853$	2.57621e11	0.486615	0.243308	+	0.969949i	$0.421768\pi$
$854$	8.47832e11i	1.59396i
$855$	0	0
$856$	1.23149e11	0.229369
$857$	6.41541e11i	1.18933i	0.803975	+	0.594664i	$0.202715\pi$
$857$	6.41541e11i	1.18933i	−0.803975	+	0.594664i	$0.797285\pi$
$858$	0	0
$859$	−8.28926e11	−1.52245	−0.761225	−	0.648488i	$-0.775402\pi$
$859$	−8.28926e11	−1.52245	−0.761225	+	0.648488i	$0.775402\pi$
$860$	− 3.71761e11i	− 0.679626i
$861$	0	0
$862$	−3.79926e11	−0.688130
$863$	4.48523e11i	0.808614i	0.914623	+	0.404307i	$0.132487\pi$
$863$	4.48523e11i	0.808614i	−0.914623	+	0.404307i	$0.867513\pi$
$864$	0	0
$865$	−7.28044e10	−0.130045
$866$	5.14490e11i	0.914757i
$867$	0	0
$868$	−4.76597e11	−0.839600
$869$	− 4.99480e11i	− 0.875869i
$870$	0	0
$871$	−1.32709e12	−2.30584
$872$	− 7.66791e10i	− 0.132621i
$873$	0	0
$874$	−1.10513e12	−1.89394
$875$	1.07222e12i	1.82916i
$876$	0	0
$877$	6.69421e11	1.13162	0.565810	−	0.824535i	$-0.308563\pi$
$877$	6.69421e11	1.13162	0.565810	+	0.824535i	$0.308563\pi$
$878$	− 7.32538e11i	− 1.23268i
$879$	0	0
$880$	2.81277e11	0.469033
$881$	6.33338e11i	1.05131i	0.850697	+	0.525656i	$0.176180\pi$
$881$	6.33338e11i	1.05131i	−0.850697	+	0.525656i	$0.823820\pi$
$882$	0	0
$883$	4.65710e11	0.766077	0.383039	−	0.923732i	$-0.374878\pi$
$883$	4.65710e11	0.766077	0.383039	+	0.923732i	$0.374878\pi$
$884$	1.58990e12i	2.60351i
$885$	0	0
$886$	2.84249e11	0.461280
$887$	− 5.04766e11i	− 0.815447i	−0.913106	−	0.407723i	$-0.866323\pi$
$887$	− 5.04766e11i	− 0.815447i	0.913106	−	0.407723i	$-0.133677\pi$
$888$	0	0
$889$	1.32342e12	2.11880
$890$	− 3.05251e11i	− 0.486515i
$891$	0	0
$892$	−1.14259e11	−0.180480
$893$	4.95454e11i	0.779108i
$894$	0	0
$895$	4.40073e9	0.00685855
$896$	− 5.27275e11i	− 0.818098i
$897$	0	0
$898$	−3.72471e11	−0.572780
$899$	2.71984e11i	0.416395i
$900$	0	0
$901$	−1.18849e12	−1.80341
$902$	5.06342e11i	0.764923i
$903$	0	0
$904$	−1.16584e11	−0.174568
$905$	− 1.69245e11i	− 0.252303i
$906$	0	0
$907$	1.13679e12	1.67978	0.839890	−	0.542757i	$-0.182620\pi$
$907$	1.13679e12	1.67978	0.839890	+	0.542757i	$0.182620\pi$
$908$	1.11397e12i	1.63881i
$909$	0	0
$910$	2.30378e12	3.35950
$911$	− 7.72292e11i	− 1.12126i	−0.828065	−	0.560632i	$-0.810558\pi$
$911$	− 7.72292e11i	− 1.12126i	0.828065	−	0.560632i	$-0.189442\pi$
$912$	0	0
$913$	7.94917e11	1.14403
$914$	− 3.11495e11i	− 0.446340i
$915$	0	0
$916$	1.17324e12	1.66650
$917$	1.07342e12i	1.51807i
$918$	0	0
$919$	−6.31111e11	−0.884797	−0.442398	−	0.896819i	$-0.645872\pi$
$919$	−6.31111e11	−0.884797	−0.442398	+	0.896819i	$0.645872\pi$
$920$	1.29199e11i	0.180347i
$921$	0	0
$922$	−1.39127e12	−1.92525
$923$	1.27325e12i	1.75431i
$924$	0	0
$925$	−8.23272e10	−0.112454
$926$	− 1.52442e11i	− 0.207329i
$927$	0	0
$928$	−1.03675e12	−1.39792
$929$	8.53537e11i	1.14593i	0.819579	+	0.572967i	$0.194208\pi$
$929$	8.53537e11i	1.14593i	−0.819579	+	0.572967i	$0.805792\pi$
$930$	0	0
$931$	−2.09747e12	−2.79188
$932$	5.83273e11i	0.773051i
$933$	0	0
$934$	1.64775e12	2.16523
$935$	− 6.32650e11i	− 0.827784i
$936$	0	0
$937$	−1.41122e12	−1.83078	−0.915390	−	0.402569i	$-0.868117\pi$
$937$	−1.41122e12	−1.83078	−0.915390	+	0.402569i	$0.868117\pi$
$938$	− 2.82496e12i	− 3.64922i
$939$	0	0
$940$	4.04865e11	0.518560
$941$	4.05332e11i	0.516955i	0.966017	+	0.258477i	$0.0832207\pi$
$941$	4.05332e11i	0.516955i	−0.966017	+	0.258477i	$0.916779\pi$
$942$	0	0
$943$	5.17623e11	0.654586
$944$	2.84514e10i	0.0358274i
$945$	0	0
$946$	−5.39295e11	−0.673382
$947$	3.98605e11i	0.495613i	0.968810	+	0.247806i	$0.0797097\pi$
$947$	3.98605e11i	0.495613i	−0.968810	+	0.247806i	$0.920290\pi$
$948$	0	0
$949$	−1.52806e12	−1.88398
$950$	− 4.67725e11i	− 0.574244i
$951$	0	0
$952$	−4.84196e11	−0.589486
$953$	1.49082e11i	0.180740i	0.995908	+	0.0903700i	$0.0288050\pi$
$953$	1.49082e11i	0.180740i	−0.995908	+	0.0903700i	$0.971195\pi$
$954$	0	0
$955$	−6.82412e11	−0.820414
$956$	− 1.04921e12i	− 1.25613i
$957$	0	0
$958$	−1.02415e11	−0.121591
$959$	− 2.50061e12i	− 2.95646i
$960$	0	0
$961$	−6.97927e11	−0.818307
$962$	8.61555e11i	1.00596i
$963$	0	0
$964$	−3.46320e11	−0.401023
$965$	− 8.61435e11i	− 0.993375i
$966$	0	0
$967$	5.08196e11	0.581199	0.290600	−	0.956845i	$-0.406145\pi$
$967$	5.08196e11	0.581199	0.290600	+	0.956845i	$0.406145\pi$
$968$	− 1.17054e11i	− 0.133317i
$969$	0	0
$970$	−7.23353e11	−0.817078
$971$	1.03968e12i	1.16956i	0.811193	+	0.584778i	$0.198818\pi$
$971$	1.03968e12i	1.16956i	−0.811193	+	0.584778i	$0.801182\pi$
$972$	0	0
$973$	1.31081e12	1.46247
$974$	1.37188e12i	1.52433i
$975$	0	0
$976$	4.68697e11	0.516527
$977$	4.47633e10i	0.0491296i	0.999698	+	0.0245648i	$0.00782001\pi$
$977$	4.47633e10i	0.0491296i	−0.999698	+	0.0245648i	$0.992180\pi$
$978$	0	0
$979$	−2.38464e11	−0.259592
$980$	1.71397e12i	1.85823i
$981$	0	0
$982$	−9.75507e10	−0.104902
$983$	8.78713e11i	0.941094i	0.882375	+	0.470547i	$0.155943\pi$
$983$	8.78713e11i	0.941094i	−0.882375	+	0.470547i	$0.844057\pi$
$984$	0	0
$985$	2.11179e11	0.224339
$986$	1.93140e12i	2.04345i
$987$	0	0
$988$	−2.63594e12	−2.76635
$989$	5.51310e11i	0.576250i
$990$	0	0
$991$	3.97816e11	0.412465	0.206233	−	0.978503i	$-0.433880\pi$
$991$	3.97816e11	0.412465	0.206233	+	0.978503i	$0.433880\pi$
$992$	5.90695e11i	0.609982i
$993$	0	0
$994$	−2.71033e12	−2.77636
$995$	6.01537e11i	0.613720i
$996$	0	0
$997$	9.36739e11	0.948065	0.474032	−	0.880507i	$-0.342798\pi$
$997$	9.36739e11	0.948065	0.474032	+	0.880507i	$0.342798\pi$
$998$	1.58891e12i	1.60168i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.9.b.i.242.39	yes	48
3.2	odd	2	inner	243.9.b.i.242.10	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.9.b.i.242.10	✓	48	3.2	odd	2	inner
243.9.b.i.242.39	yes	48	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	243.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+23.5529i q^{2} -298.740 q^{4} -538.241i q^{5} -4052.68 q^{7} -1006.66i q^{8} +12677.2 q^{10} -9903.48i q^{11} -44841.2 q^{13} -95452.4i q^{14} -52767.8 q^{16} +118686. i q^{17} -196772. q^{19} +160794. i q^{20} +233256. q^{22} -238453. i q^{23} +100921. q^{25} -1.05614e6i q^{26} +1.21070e6 q^{28} -690920. i q^{29} -393655. q^{31} -1.50054e6i q^{32} -2.79540e6 q^{34} +2.18132e6i q^{35} -815758. q^{37} -4.63456e6i q^{38} -541823. q^{40} +2.17076e6i q^{41} -2.31203e6 q^{43} +2.95857e6i q^{44} +5.61626e6 q^{46} -2.51790e6i q^{47} +1.06594e7 q^{49} +2.37699e6i q^{50} +1.33959e7 q^{52} +1.00137e7i q^{53} -5.33047e6 q^{55} +4.07965e6i q^{56} +1.62732e7 q^{58} -539181. i q^{59} -8.88226e6 q^{61} -9.27173e6i q^{62} +2.18335e7 q^{64} +2.41354e7i q^{65} +2.95954e7 q^{67} -3.54562e7i q^{68} -5.13764e7 q^{70} -2.83945e7i q^{71} +3.40772e7 q^{73} -1.92135e7i q^{74} +5.87838e7 q^{76} +4.01356e7i q^{77} +5.04348e7 q^{79} +2.84018e7i q^{80} -5.11277e7 q^{82} +8.02664e7i q^{83} +6.38816e7 q^{85} -5.44550e7i q^{86} -9.96939e6 q^{88} -2.40788e7i q^{89} +1.81727e8 q^{91} +7.12355e7i q^{92} +5.93040e7 q^{94} +1.05911e8i q^{95} -5.70595e7 q^{97} +2.51060e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6144 q^{4} - 5538 q^{7} - 10110 q^{13} + 786432 q^{16} - 277662 q^{19} - 3750000 q^{25} + 2835456 q^{28} - 856704 q^{31} + 2625462 q^{34} + 1640958 q^{37} + 866250 q^{40} - 13575936 q^{43} - 12911094 q^{46}+ \cdots + 244503618 q^{97}+O(q^{100}) \) 48 * q - 6144 * q^4 - 5538 * q^7 - 10110 * q^13 + 786432 * q^16 - 277662 * q^19 - 3750000 * q^25 + 2835456 * q^28 - 856704 * q^31 + 2625462 * q^34 + 1640958 * q^37 + 866250 * q^40 - 13575936 * q^43 - 12911094 * q^46 + 40853646 * q^49 + 5176320 * q^52 - 19789776 * q^55 + 23558400 * q^58 + 10452948 * q^61 - 89377098 * q^64 - 18985620 * q^67 + 44038656 * q^70 - 10176852 * q^73 + 186967158 * q^76 - 157712718 * q^79 - 139991040 * q^82 + 205482816 * q^85 - 190726200 * q^88 - 211624044 * q^91 + 522075960 * q^94 + 244503618 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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