LMFDB - Embedded newform 252.9.bk.a.233.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(53,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("252.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.bk (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$102.659409735$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$22$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			233.21
Character	$\chi$	$=$	252.233
Dual form			252.9.bk.a.53.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1009.83 + 583.025i) q^{5} +(-1699.40 + 1696.12i) q^{7} +O(q^{10})$	$q+(1009.83 + 583.025i) q^{5} +(-1699.40 + 1696.12i) q^{7} +(-2949.92 + 1703.14i) q^{11} +21577.4 q^{13} +(-3995.22 + 2306.64i) q^{17} +(-67464.5 + 116852. i) q^{19} +(963.083 + 556.036i) q^{23} +(484524. + 839220. i) q^{25} -612721. i q^{29} +(280802. + 486363. i) q^{31} +(-2.70499e6 + 721998. i) q^{35} +(-1.43778e6 + 2.49032e6i) q^{37} +3.77743e6i q^{41} +1.31458e6 q^{43} +(-3.88499e6 - 2.24300e6i) q^{47} +(11142.3 - 5.76479e6i) q^{49} +(166235. - 95976.0i) q^{53} -3.97189e6 q^{55} +(8.80410e6 - 5.08305e6i) q^{59} +(3.83072e6 - 6.63500e6i) q^{61} +(2.17895e7 + 1.25802e7i) q^{65} +(-1.64797e7 - 2.85437e7i) q^{67} -3.07695e6i q^{71} +(-1.27557e6 - 2.20936e6i) q^{73} +(2.12438e6 - 7.89775e6i) q^{77} +(-1.67470e6 + 2.90067e6i) q^{79} -2.88063e6i q^{83} -5.37932e6 q^{85} +(8.18939e7 + 4.72815e7i) q^{89} +(-3.66688e7 + 3.65980e7i) q^{91} +(-1.36255e8 + 7.86669e7i) q^{95} +6.14470e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 1230 q^{7}+O(q^{10}) $ 44 * q + 1230 * q^7	$ 44 q + 1230 q^{7} - 101380 q^{13} + 62770 q^{19} + 2247666 q^{25} + 1389254 q^{31} - 2136026 q^{37} + 11510140 q^{43} - 3824398 q^{49} - 42646528 q^{55} + 27346232 q^{61} + 14239194 q^{67} - 64344138 q^{73} + 7061786 q^{79} - 54198208 q^{85} - 45697066 q^{91} + 476543496 q^{97}+O(q^{100}) $ 44 * q + 1230 * q^7 - 101380 * q^13 + 62770 * q^19 + 2247666 * q^25 + 1389254 * q^31 - 2136026 * q^37 + 11510140 * q^43 - 3824398 * q^49 - 42646528 * q^55 + 27346232 * q^61 + 14239194 * q^67 - 64344138 * q^73 + 7061786 * q^79 - 54198208 * q^85 - 45697066 * q^91 + 476543496 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1009.83	+	583.025i	1.61573	+	0.932840i	0.988009	+	0.154399i	$0.0493441\pi$
$5$	1009.83	+	583.025i	1.61573	+	0.932840i	0.627718	+	0.778441i	$0.283989\pi$
$6$		0			0
$7$	−1699.40	+	1696.12i	−0.707790	+	0.706423i
$7$	−1699.40	+	1696.12i	−0.707790	+	0.706423i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2949.92	+	1703.14i	−0.201484	+	0.116327i	−0.597347	−	0.801983i	$-0.703779\pi$
$11$	−2949.92	+	1703.14i	−0.201484	+	0.116327i	0.395864	+	0.918309i	$0.370445\pi$
$12$		0			0
$13$	21577.4			0.755486			0.377743	−	0.925910i	$-0.376700\pi$
$13$	21577.4			0.755486			0.377743	+	0.925910i	$0.376700\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3995.22	+	2306.64i	−0.0478349	+	0.0276175i	−0.523727	−	0.851886i	$-0.675459\pi$
$17$	−3995.22	+	2306.64i	−0.0478349	+	0.0276175i	0.475892	+	0.879504i	$0.342125\pi$
$18$		0			0
$19$	−67464.5	+	116852.i	−0.517679	+	0.896647i	0.482110	+	0.876111i	$0.339871\pi$
$19$	−67464.5	+	116852.i	−0.517679	+	0.896647i	−0.999789	+	0.0205359i	$0.993463\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	963.083	+	556.036i	0.00344154	+	0.00198697i	0.501720	−	0.865030i	$-0.332701\pi$
$23$	963.083	+	556.036i	0.00344154	+	0.00198697i	−0.498278	+	0.867017i	$0.666034\pi$
$24$		0			0
$25$	484524.	+	839220.i	1.24038	+	2.14840i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	612721.i		−	0.866305i	−0.901321	−	0.433153i	$-0.857401\pi$
$29$		−	612721.i		−	0.866305i	0.901321	−	0.433153i	$-0.142599\pi$
$30$		0			0
$31$	280802.	+	486363.i	0.304056	+	0.526640i	0.977051	−	0.213007i	$-0.0683257\pi$
$31$	280802.	+	486363.i	0.304056	+	0.526640i	−0.672995	+	0.739647i	$0.734992\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.70499e6	+	721998.i	−1.80257	+	0.481132i
$36$		0			0
$37$	−1.43778e6	+	2.49032e6i	−0.767162	+	1.32876i	0.171934	+	0.985108i	$0.444998\pi$
$37$	−1.43778e6	+	2.49032e6i	−0.767162	+	1.32876i	−0.939096	+	0.343655i	$0.888335\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.77743e6i			1.33678i	0.743809	+	0.668392i	$0.233017\pi$
$41$			3.77743e6i			1.33678i	−0.743809	+	0.668392i	$0.766983\pi$
$42$		0			0
$43$	1.31458e6			0.384515			0.192258	−	0.981345i	$-0.438419\pi$
$43$	1.31458e6			0.384515			0.192258	+	0.981345i	$0.438419\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.88499e6	−	2.24300e6i	−0.796156	−	0.459661i	0.0459693	−	0.998943i	$-0.485362\pi$
$47$	−3.88499e6	−	2.24300e6i	−0.796156	−	0.459661i	−0.842125	+	0.539282i	$0.818696\pi$
$48$		0			0
$49$	11142.3	−	5.76479e6i	0.00193281	−	0.999998i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	166235.	−	95976.0i	0.0210678	−	0.0121635i	−0.489429	−	0.872043i	$-0.662795\pi$
$53$	166235.	−	95976.0i	0.0210678	−	0.0121635i	0.510497	+	0.859880i	$0.329461\pi$
$54$		0			0
$55$	−3.97189e6			−0.434057
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.80410e6	−	5.08305e6i	0.726569	−	0.419485i	−0.0905966	−	0.995888i	$-0.528877\pi$
$59$	8.80410e6	−	5.08305e6i	0.726569	−	0.419485i	0.817166	+	0.576403i	$0.195544\pi$
$60$		0			0
$61$	3.83072e6	−	6.63500e6i	0.276669	−	0.479206i	−0.693886	−	0.720085i	$-0.744103\pi$
$61$	3.83072e6	−	6.63500e6i	0.276669	−	0.479206i	0.970555	+	0.240880i	$0.0774360\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.17895e7	+	1.25802e7i	1.22066	+	0.704748i
$66$		0			0
$67$	−1.64797e7	−	2.85437e7i	−0.817808	−	1.41648i	−0.907294	−	0.420496i	$-0.861856\pi$
$67$	−1.64797e7	−	2.85437e7i	−0.817808	−	1.41648i	0.0894868	−	0.995988i	$-0.471477\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.07695e6i		−	0.121084i	−0.998166	−	0.0605420i	$-0.980717\pi$
$71$		−	3.07695e6i		−	0.121084i	0.998166	−	0.0605420i	$-0.0192829\pi$
$72$		0			0
$73$	−1.27557e6	−	2.20936e6i	−0.0449173	−	0.0777990i	0.842693	−	0.538395i	$-0.180969\pi$
$73$	−1.27557e6	−	2.20936e6i	−0.0449173	−	0.0777990i	−0.887610	+	0.460596i	$0.847636\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.12438e6	−	7.89775e6i	0.0604323	−	0.224668i
$78$		0			0
$79$	−1.67470e6	+	2.90067e6i	−0.0429961	+	0.0744714i	−0.886723	−	0.462302i	$-0.847024\pi$
$79$	−1.67470e6	+	2.90067e6i	−0.0429961	+	0.0744714i	0.843726	+	0.536773i	$0.180357\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	2.88063e6i		−	0.0606980i	−0.999539	−	0.0303490i	$-0.990338\pi$
$83$		−	2.88063e6i		−	0.0606980i	0.999539	−	0.0303490i	$-0.00966187\pi$
$84$		0			0
$85$	−5.37932e6			−0.103051
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.18939e7	+	4.72815e7i	1.30524	+	0.753583i	0.981298	−	0.192493i	$-0.0616572\pi$
$89$	8.18939e7	+	4.72815e7i	1.30524	+	0.753583i	0.323945	+	0.946076i	$0.394991\pi$
$90$		0			0
$91$	−3.66688e7	+	3.65980e7i	−0.534725	+	0.533693i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.36255e8	+	7.86669e7i	−1.67286	+	0.965824i
$96$		0			0
$97$	6.14470e7			0.694087			0.347043	−	0.937849i	$-0.387186\pi$
$97$	6.14470e7			0.694087			0.347043	+	0.937849i	$0.387186\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.44159e8	+	8.32305e7i	−1.38534	+	0.799829i	−0.992786	−	0.119899i	$-0.961743\pi$
$101$	−1.44159e8	+	8.32305e7i	−1.38534	+	0.799829i	−0.392558	+	0.919727i	$0.628410\pi$
$102$		0			0
$103$	−8.33527e7	+	1.44371e8i	−0.740578	+	1.28272i	0.211655	+	0.977345i	$0.432115\pi$
$103$	−8.33527e7	+	1.44371e8i	−0.740578	+	1.28272i	−0.952232	+	0.305374i	$0.901219\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.84702e8	−	1.06638e8i	−1.40908	−	0.813533i	−0.413782	−	0.910376i	$-0.635792\pi$
$107$	−1.84702e8	−	1.06638e8i	−1.40908	−	0.813533i	−0.995300	+	0.0968428i	$0.969126\pi$
$108$		0			0
$109$	−6.62889e7	−	1.14816e8i	−0.469607	−	0.813383i	0.529789	−	0.848129i	$-0.322271\pi$
$109$	−6.62889e7	−	1.14816e8i	−0.469607	−	0.813383i	−0.999396	+	0.0347460i	$0.988938\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.93227e8i			1.18510i	0.805534	+	0.592549i	$0.201879\pi$
$113$			1.93227e8i			1.18510i	−0.805534	+	0.592549i	$0.798121\pi$
$114$		0			0
$115$	648366.	+	1.12300e6i	0.00370705	+	0.00642080i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.87714e6	−	1.06963e7i	0.0143474	−	0.0533391i
$120$		0			0
$121$	−1.01378e8	+	1.75592e8i	−0.472936	+	0.819150i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			6.74470e8i			2.76263i
$126$		0			0
$127$	−2.19073e8			−0.842121			−0.421061	−	0.907033i	$-0.638342\pi$
$127$	−2.19073e8			−0.842121			−0.421061	+	0.907033i	$0.638342\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.37881e7	−	2.52811e7i	−0.148686	−	0.0858440i	0.423811	−	0.905751i	$-0.360692\pi$
$131$	−4.37881e7	−	2.52811e7i	−0.148686	−	0.0858440i	−0.572497	+	0.819907i	$0.694025\pi$
$132$		0			0
$133$	−8.35457e7	−	3.13006e8i	−0.267004	−	1.00034i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.03831e8	−	2.33152e8i	1.14635	−	0.661845i	0.198355	−	0.980130i	$-0.436440\pi$
$137$	4.03831e8	−	2.33152e8i	1.14635	−	0.661845i	0.947995	+	0.318285i	$0.103107\pi$
$138$		0			0
$139$	−3.61399e8			−0.968116			−0.484058	−	0.875036i	$-0.660838\pi$
$139$	−3.61399e8			−0.968116			−0.484058	+	0.875036i	$0.660838\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.36518e7	+	3.67494e7i	−0.152218	+	0.0878832i
$144$		0			0
$145$	3.57232e8	−	6.18744e8i	0.808124	−	1.39971i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.50785e8	+	1.44791e8i	0.508811	+	0.293762i	0.732345	−	0.680934i	$-0.238426\pi$
$149$	2.50785e8	+	1.44791e8i	0.508811	+	0.293762i	−0.223534	+	0.974696i	$0.571759\pi$
$150$		0			0
$151$	−3.17031e8	−	5.49114e8i	−0.609809	−	1.05622i	−0.991272	−	0.131836i	$-0.957913\pi$
$151$	−3.17031e8	−	5.49114e8i	−0.609809	−	1.05622i	0.381463	−	0.924384i	$-0.375420\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			6.54858e8i			1.13454i
$156$		0			0
$157$	−2.47779e8	−	4.29165e8i	−0.407817	−	0.706360i	0.586828	−	0.809712i	$-0.300377\pi$
$157$	−2.47779e8	−	4.29165e8i	−0.407817	−	0.706360i	−0.994645	+	0.103352i	$0.967043\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.57977e6	+	688576.i	−0.00383953	+	0.00102482i
$162$		0			0
$163$	−4.79112e8	+	8.29847e8i	−0.678714	+	1.17557i	0.296654	+	0.954985i	$0.404129\pi$
$163$	−4.79112e8	+	8.29847e8i	−0.678714	+	1.17557i	−0.975368	+	0.220582i	$0.929204\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			1.42467e9i			1.83168i	0.401545	+	0.915839i	$0.368473\pi$
$167$			1.42467e9i			1.83168i	−0.401545	+	0.915839i	$0.631527\pi$
$168$		0			0
$169$	−3.50145e8			−0.429241
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.06210e8	+	1.76790e8i	0.341849	+	0.197367i	0.661089	−	0.750307i	$-0.270094\pi$
$173$	3.06210e8	+	1.76790e8i	0.341849	+	0.197367i	−0.319240	+	0.947674i	$0.603428\pi$
$174$		0			0
$175$	−2.24682e9	−	6.04362e8i	−2.39561	−	0.644384i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.28077e8	+	3.62621e8i	−0.611788	+	0.353216i	−0.773665	−	0.633595i	$-0.781579\pi$
$179$	−6.28077e8	+	3.62621e8i	−0.611788	+	0.353216i	0.161877	+	0.986811i	$0.448245\pi$
$180$		0			0
$181$	1.38960e9			1.29472			0.647360	−	0.762184i	$-0.275873\pi$
$181$	1.38960e9			1.29472			0.647360	+	0.762184i	$0.275873\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.90383e9	+	1.67653e9i	−2.47905	+	1.43128i
$186$		0			0
$187$	7.85706e6	−	1.36088e7i	0.00642530	−	0.0111290i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.01682e9	−	1.16441e9i	−1.51542	−	0.874928i	−0.999836	−	0.0180894i	$-0.994242\pi$
$191$	−2.01682e9	−	1.16441e9i	−1.51542	−	0.874928i	−0.515584	−	0.856839i	$-0.672425\pi$
$192$		0			0
$193$	−8.28989e8	−	1.43585e9i	−0.597475	−	1.03486i	−0.993192	−	0.116485i	$-0.962837\pi$
$193$	−8.28989e8	−	1.43585e9i	−0.597475	−	1.03486i	0.395718	−	0.918372i	$-0.370496\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2.19344e9i			1.45633i	0.685402	+	0.728165i	$0.259627\pi$
$197$			2.19344e9i			1.45633i	−0.685402	+	0.728165i	$0.740373\pi$
$198$		0			0
$199$	1.18164e9	+	2.04667e9i	0.753484	+	1.30507i	0.946124	+	0.323804i	$0.104962\pi$
$199$	1.18164e9	+	2.04667e9i	0.753484	+	1.30507i	−0.192640	+	0.981270i	$0.561705\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.03925e9	+	1.04126e9i	0.611978	+	0.613162i
$204$		0			0
$205$	−2.20234e9	+	3.81456e9i	−1.24701	+	2.15988i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	4.59606e8i		−	0.240880i
$210$		0			0
$211$	6.66470e6			0.00336241			0.00168121	−	0.999999i	$-0.499465\pi$
$211$	6.66470e6			0.00336241			0.00168121	+	0.999999i	$0.499465\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.32750e9	+	7.66433e8i	0.621271	+	0.358691i
$216$		0			0
$217$	−1.30213e9	−	3.50253e8i	−0.587238	−	0.157958i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.62066e7	+	4.97714e7i	−0.0361386	+	0.0208646i
$222$		0			0
$223$	3.94033e9			1.59336			0.796678	−	0.604404i	$-0.206589\pi$
$223$	3.94033e9			1.59336			0.796678	+	0.604404i	$0.206589\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.38305e9	−	7.98507e8i	0.520878	−	0.300729i	−0.216416	−	0.976301i	$-0.569437\pi$
$227$	1.38305e9	−	7.98507e8i	0.520878	−	0.300729i	0.737294	+	0.675572i	$0.236103\pi$
$228$		0			0
$229$	2.67450e8	−	4.63238e8i	0.0972526	−	0.168446i	−0.813294	−	0.581853i	$-0.802328\pi$
$229$	2.67450e8	−	4.63238e8i	0.0972526	−	0.168446i	0.910547	+	0.413407i	$0.135661\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.39744e9	−	1.38416e9i	−0.813438	−	0.469639i	0.0347102	−	0.999397i	$-0.488949\pi$
$233$	−2.39744e9	−	1.38416e9i	−0.813438	−	0.469639i	−0.848148	+	0.529759i	$0.822283\pi$
$234$		0			0
$235$	−2.61545e9	−	4.53009e9i	−0.857580	−	1.48537i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			2.18222e9i			0.668815i	0.942429	+	0.334408i	$0.108536\pi$
$239$			2.18222e9i			0.668815i	−0.942429	+	0.334408i	$0.891464\pi$
$240$		0			0
$241$	−6.88844e8	−	1.19311e9i	−0.204199	−	0.353682i	0.745678	−	0.666306i	$-0.232126\pi$
$241$	−6.88844e8	−	1.19311e9i	−0.204199	−	0.353682i	−0.949877	+	0.312624i	$0.898792\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.37227e9	−	5.81496e9i	0.935961	−	1.61392i
$246$		0			0
$247$	−1.45571e9	+	2.52136e9i	−0.391099	+	0.677404i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			6.99940e9i			1.76346i	0.471753	+	0.881731i	$0.343621\pi$
$251$			6.99940e9i			1.76346i	−0.471753	+	0.881731i	$0.656379\pi$
$252$		0			0
$253$	−3.78803e6			−0.000924551
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.78534e9	+	1.03077e9i	0.409250	+	0.236280i	0.690467	−	0.723364i	$-0.257405\pi$
$257$	1.78534e9	+	1.03077e9i	0.409250	+	0.236280i	−0.281218	+	0.959644i	$0.590738\pi$
$258$		0			0
$259$	−1.78050e9	−	6.67071e9i	−0.395680	−	1.48243i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.85529e9	−	3.95791e9i	1.43286	−	0.827261i	0.435520	−	0.900179i	$-0.356565\pi$
$263$	6.85529e9	−	3.95791e9i	1.43286	−	0.827261i	0.997338	+	0.0729182i	$0.0232312\pi$
$264$		0			0
$265$	2.23826e8			0.0453865
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.73924e8	+	5.62295e8i	−0.186001	+	0.107388i	−0.590109	−	0.807323i	$-0.700915\pi$
$269$	−9.73924e8	+	5.62295e8i	−0.186001	+	0.107388i	0.404108	+	0.914711i	$0.367582\pi$
$270$		0			0
$271$	4.82025e9	−	8.34893e9i	0.893702	−	1.54794i	0.0582995	−	0.998299i	$-0.481432\pi$
$271$	4.82025e9	−	8.34893e9i	0.893702	−	1.54794i	0.835403	−	0.549638i	$-0.185234\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.85862e9	−	1.65042e9i	−0.499833	−	0.288579i
$276$		0			0
$277$	8.97603e8	+	1.55469e9i	0.152463	+	0.264074i	0.932132	−	0.362118i	$-0.117946\pi$
$277$	8.97603e8	+	1.55469e9i	0.152463	+	0.264074i	−0.779669	+	0.626192i	$0.784613\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	1.21803e10i		−	1.95359i	−0.214172	−	0.976796i	$-0.568705\pi$
$281$		−	1.21803e10i		−	1.95359i	0.214172	−	0.976796i	$-0.431295\pi$
$282$		0			0
$283$	2.69485e9	+	4.66762e9i	0.420135	+	0.727696i	0.995952	−	0.0898828i	$-0.0286493\pi$
$283$	2.69485e9	+	4.66762e9i	0.420135	+	0.727696i	−0.575817	+	0.817579i	$0.695316\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.40699e9	−	6.41938e9i	−0.944335	−	0.946162i
$288$		0			0
$289$	−3.47724e9	+	6.02275e9i	−0.498475	+	0.863383i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			8.85310e9i			1.20123i	0.799540	+	0.600613i	$0.205077\pi$
$293$			8.85310e9i			1.20123i	−0.799540	+	0.600613i	$0.794923\pi$
$294$		0			0
$295$	1.18542e10			1.56525
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.07809e7	+	1.19978e7i	0.00260003	+	0.00150113i
$300$		0			0
$301$	−2.23400e9	+	2.22969e9i	−0.272156	+	0.271630i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.73675e9	−	4.46681e9i	0.894044	−	0.516177i
$306$		0			0
$307$	1.00703e10			1.13368			0.566839	−	0.823828i	$-0.308166\pi$
$307$	1.00703e10			1.13368			0.566839	+	0.823828i	$0.308166\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.12022e9	+	2.37881e9i	−0.440432	+	0.254284i	−0.703781	−	0.710417i	$-0.748506\pi$
$311$	−4.12022e9	+	2.37881e9i	−0.440432	+	0.254284i	0.263349	+	0.964701i	$0.415173\pi$
$312$		0			0
$313$	−7.58428e9	+	1.31364e10i	−0.790200	+	1.36867i	0.135643	+	0.990758i	$0.456690\pi$
$313$	−7.58428e9	+	1.31364e10i	−0.790200	+	1.36867i	−0.925843	+	0.377909i	$0.876643\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.75984e9	−	4.48015e9i	−0.768451	−	0.443665i	0.0638711	−	0.997958i	$-0.479655\pi$
$317$	−7.75984e9	−	4.48015e9i	−0.768451	−	0.443665i	−0.832322	+	0.554293i	$0.812989\pi$
$318$		0			0
$319$	1.04355e9	+	1.80748e9i	0.100774	+	0.174546i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	6.22465e8i		−	0.0571880i
$324$		0			0
$325$	1.04548e10	+	1.81082e10i	0.937091	+	1.62309i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.04066e10	−	2.77765e9i	0.888226	−	0.237080i
$330$		0			0
$331$	−1.91639e9	+	3.31928e9i	−0.159651	+	0.276524i	−0.934743	−	0.355325i	$-0.884370\pi$
$331$	−1.91639e9	+	3.31928e9i	−0.159651	+	0.276524i	0.775092	+	0.631849i	$0.217704\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	3.84324e10i		−	3.05153i
$336$		0			0
$337$	7.70765e9			0.597589			0.298794	−	0.954317i	$-0.403416\pi$
$337$	7.70765e9			0.597589			0.298794	+	0.954317i	$0.403416\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.65669e9	−	9.56490e8i	−0.122525	−	0.0707396i
$342$		0			0
$343$	9.75885e9	+	9.81560e9i	0.705054	+	0.709154i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.13003e9	+	5.27122e9i	−0.629730	+	0.363575i	−0.780647	−	0.624972i	$-0.785111\pi$
$347$	−9.13003e9	+	5.27122e9i	−0.629730	+	0.363575i	0.150918	+	0.988546i	$0.451777\pi$
$348$		0			0
$349$	1.89812e10			1.27944			0.639722	−	0.768606i	$-0.279049\pi$
$349$	1.89812e10			1.27944			0.639722	+	0.768606i	$0.279049\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.46772e10	−	1.42474e10i	1.58927	−	0.917565i	0.595841	−	0.803103i	$-0.296819\pi$
$353$	2.46772e10	−	1.42474e10i	1.58927	−	0.917565i	0.993428	−	0.114462i	$-0.0365144\pi$
$354$		0			0
$355$	1.79394e9	−	3.10719e9i	0.112952	−	0.195639i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.23971e10	+	7.15748e9i	0.746350	+	0.430906i	0.824374	−	0.566046i	$-0.191528\pi$
$359$	1.23971e10	+	7.15748e9i	0.746350	+	0.430906i	−0.0780233	+	0.996952i	$0.524861\pi$
$360$		0			0
$361$	−6.11127e8	−	1.05850e9i	−0.0359834	−	0.0623251i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	2.97476e9i		−	0.167603i
$366$		0			0
$367$	−1.31850e10	−	2.28370e10i	−0.726799	−	1.25885i	−0.958229	−	0.286002i	$-0.907674\pi$
$367$	−1.31850e10	−	2.28370e10i	−0.726799	−	1.25885i	0.231430	−	0.972852i	$-0.425660\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.19714e8	+	4.45057e8i	−0.00631901	+	0.0234920i
$372$		0			0
$373$	1.44900e10	−	2.50973e10i	0.748569	−	1.29656i	−0.199940	−	0.979808i	$-0.564075\pi$
$373$	1.44900e10	−	2.50973e10i	0.748569	−	1.29656i	0.948509	−	0.316751i	$-0.102592\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.32210e10i		−	0.654482i
$378$		0			0
$379$	−3.06511e10			−1.48556			−0.742778	−	0.669538i	$-0.766492\pi$
$379$	−3.06511e10			−1.48556			−0.742778	+	0.669538i	$0.766492\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.96490e9	−	4.59854e9i	−0.370156	−	0.213710i	0.303370	−	0.952873i	$-0.401888\pi$
$383$	−7.96490e9	−	4.59854e9i	−0.370156	−	0.213710i	−0.673527	+	0.739163i	$0.735221\pi$
$384$		0			0
$385$	6.74985e9	−	6.73681e9i	0.307221	−	0.306628i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.26837e10	−	7.32292e9i	0.553919	−	0.319805i	−0.196782	−	0.980447i	$-0.563049\pi$
$389$	1.26837e10	−	7.32292e9i	0.553919	−	0.319805i	0.750701	+	0.660642i	$0.229716\pi$
$390$		0			0
$391$	−5.13030e6			−0.000219501
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.38232e9	+	1.95278e9i	−0.138940	+	0.0802169i
$396$		0			0
$397$	1.61513e10	−	2.79750e10i	0.650200	−	1.12618i	−0.332875	−	0.942971i	$-0.608019\pi$
$397$	1.61513e10	−	2.79750e10i	0.650200	−	1.12618i	0.983074	−	0.183208i	$-0.0586481\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.70108e10	+	9.82121e9i	0.657882	+	0.379829i	0.791470	−	0.611208i	$-0.209316\pi$
$401$	1.70108e10	+	9.82121e9i	0.657882	+	0.379829i	−0.133587	+	0.991037i	$0.542650\pi$
$402$		0			0
$403$	6.05899e9	+	1.04945e10i	0.229710	+	0.397869i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	9.79499e9i		−	0.356966i
$408$		0			0
$409$	−1.07087e10	−	1.85480e10i	−0.382687	−	0.662833i	0.608758	−	0.793356i	$-0.291668\pi$
$409$	−1.07087e10	−	1.85480e10i	−0.382687	−	0.662833i	−0.991445	+	0.130522i	$0.958335\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.34025e9	+	2.35710e10i	−0.217924	+	0.810172i
$414$		0			0
$415$	1.67948e9	−	2.90894e9i	0.0566215	−	0.0980714i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	1.84846e10i		−	0.599726i	−0.953982	−	0.299863i	$-0.903059\pi$
$419$		−	1.84846e10i		−	0.599726i	0.953982	−	0.299863i	$-0.0969410\pi$
$420$		0			0
$421$	1.59856e10			0.508861			0.254431	−	0.967091i	$-0.418112\pi$
$421$	1.59856e10			0.508861			0.254431	+	0.967091i	$0.418112\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.87156e9	−	2.23524e9i	−0.118667	−	0.0685124i
$426$		0			0
$427$	4.74383e9	+	1.77729e10i	0.142698	+	0.534622i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.12038e9	+	2.37890e9i	−0.119407	+	0.0689394i	−0.558514	−	0.829495i	$-0.688628\pi$
$431$	−4.12038e9	+	2.37890e9i	−0.119407	+	0.0689394i	0.439107	+	0.898435i	$0.355295\pi$
$432$		0			0
$433$	−6.45848e10			−1.83729			−0.918647	−	0.395078i	$-0.870717\pi$
$433$	−6.45848e10			−1.83729			−0.918647	+	0.395078i	$0.870717\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.29948e8	+	7.50254e7i	−0.00356322	+	0.00205723i
$438$		0			0
$439$	−2.95978e10	+	5.12649e10i	−0.796895	+	1.38026i	0.124734	+	0.992190i	$0.460192\pi$
$439$	−2.95978e10	+	5.12649e10i	−0.796895	+	1.38026i	−0.921629	+	0.388072i	$0.873141\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.68295e10	+	1.54900e10i	0.696624	+	0.402196i	0.806089	−	0.591795i	$-0.201580\pi$
$443$	2.68295e10	+	1.54900e10i	0.696624	+	0.402196i	−0.109465	+	0.993991i	$0.534914\pi$
$444$		0			0
$445$	5.51326e10	+	9.54924e10i	1.40594	+	2.43517i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1.81243e9i		−	0.0445940i	−0.999751	−	0.0222970i	$-0.992902\pi$
$449$		−	1.81243e9i		−	0.0445940i	0.999751	−	0.0222970i	$-0.00709794\pi$
$450$		0			0
$451$	−6.43350e9	−	1.11431e10i	−0.155504	−	0.269340i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.83667e10	+	1.55789e10i	−1.36182	+	0.363488i
$456$		0			0
$457$	2.87113e10	−	4.97295e10i	0.658247	−	1.14012i	−0.322822	−	0.946460i	$-0.604632\pi$
$457$	2.87113e10	−	4.97295e10i	0.658247	−	1.14012i	0.981069	−	0.193657i	$-0.0620350\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			3.10020e10i			0.686414i	0.939260	+	0.343207i	$0.111513\pi$
$461$			3.10020e10i			0.686414i	−0.939260	+	0.343207i	$0.888487\pi$
$462$		0			0
$463$	3.10071e10			0.674742			0.337371	−	0.941372i	$-0.390462\pi$
$463$	3.10071e10			0.674742			0.337371	+	0.941372i	$0.390462\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.93878e10	−	2.85140e10i	−1.03837	−	0.599503i	−0.118998	−	0.992894i	$-0.537968\pi$
$467$	−4.93878e10	−	2.85140e10i	−1.03837	−	0.599503i	−0.919371	+	0.393392i	$0.871302\pi$
$468$		0			0
$469$	7.64194e10	+	2.05557e10i	1.57947	+	0.424855i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.87791e9	+	2.23891e9i	−0.0774736	+	0.0447294i
$474$		0			0
$475$	−1.30753e11			−2.56848
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.93970e10	+	4.00664e10i	−1.31825	+	0.761093i	−0.983447	−	0.181194i	$-0.942004\pi$
$479$	−6.93970e10	+	4.00664e10i	−1.31825	+	0.761093i	−0.334805	+	0.942287i	$0.608670\pi$
$480$		0			0
$481$	−3.10237e10	+	5.37346e10i	−0.579580	+	1.00386i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.20510e10	+	3.58251e10i	1.12145	+	0.647472i
$486$		0			0
$487$	−1.04785e10	−	1.81492e10i	−0.186287	−	0.322658i	0.757723	−	0.652577i	$-0.226312\pi$
$487$	−1.04785e10	−	1.81492e10i	−0.186287	−	0.322658i	−0.944009	+	0.329919i	$0.892979\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			6.95951e10i			1.19744i	0.800959	+	0.598719i	$0.204323\pi$
$491$			6.95951e10i			1.19744i	−0.800959	+	0.598719i	$0.795677\pi$
$492$		0			0
$493$	1.41333e9	+	2.44796e9i	0.0239252	+	0.0414396i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.21888e9	+	5.22897e9i	0.0855365	+	0.0857020i
$498$		0			0
$499$	2.47908e10	−	4.29389e10i	0.399842	−	0.692547i	−0.593864	−	0.804565i	$-0.702398\pi$
$499$	2.47908e10	−	4.29389e10i	0.399842	−	0.692547i	0.993706	+	0.112019i	$0.0357316\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			1.02162e10i			0.159594i	0.996811	+	0.0797968i	$0.0254271\pi$
$503$			1.02162e10i			0.159594i	−0.996811	+	0.0797968i	$0.974573\pi$
$504$		0			0
$505$	−1.94102e11			−2.98445
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.83594e10	+	3.94673e10i	1.01842	+	0.587985i	0.913646	−	0.406511i	$-0.133255\pi$
$509$	6.83594e10	+	3.94673e10i	1.01842	+	0.587985i	0.104774	+	0.994496i	$0.466588\pi$
$510$		0			0
$511$	5.91505e9	+	1.59106e9i	0.0867510	+	0.0233347i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.68344e11	+	9.71934e10i	−2.39314	+	1.38168i
$516$		0			0
$517$	1.52806e10			0.213883
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.38907e10	−	8.01978e9i	0.188526	−	0.108846i	−0.402766	−	0.915303i	$-0.631951\pi$
$521$	1.38907e10	−	8.01978e9i	0.188526	−	0.108846i	0.591293	+	0.806457i	$0.298618\pi$
$522$		0			0
$523$	5.37134e10	−	9.30343e10i	0.717919	−	1.24347i	−0.243903	−	0.969800i	$-0.578428\pi$
$523$	5.37134e10	−	9.30343e10i	0.717919	−	1.24347i	0.961823	−	0.273673i	$-0.0882387\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.24373e9	−	1.29542e9i	−0.0290890	−	0.0167945i
$528$		0			0
$529$	−3.91549e10	−	6.78182e10i	−0.499992	−	0.866012i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			8.15074e10i			1.00992i
$534$		0			0
$535$	−1.24345e11	−	2.15372e11i	−1.51779	−	2.62889i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.78537e9	+	1.70247e10i	0.115937	+	0.201708i
$540$		0			0
$541$	1.07445e10	−	1.86100e10i	0.125429	−	0.217249i	−0.796472	−	0.604676i	$-0.793303\pi$
$541$	1.07445e10	−	1.86100e10i	0.125429	−	0.217249i	0.921900	+	0.387427i	$0.126636\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1.54592e11i		−	1.75227i
$546$		0			0
$547$	4.60994e10			0.514927			0.257464	−	0.966288i	$-0.417113\pi$
$547$	4.60994e10			0.514927			0.257464	+	0.966288i	$0.417113\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7.15976e10	+	4.13369e10i	0.776770	+	0.448468i
$552$		0			0
$553$	−2.07389e9	−	7.76990e9i	−0.0221761	−	0.0830835i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.55361e10	+	2.62903e10i	−0.473081	+	0.273133i	−0.717529	−	0.696529i	$-0.754727\pi$
$557$	−4.55361e10	+	2.62903e10i	−0.473081	+	0.273133i	0.244448	+	0.969662i	$0.421393\pi$
$558$		0			0
$559$	2.83653e10			0.290496
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3.52554e10	−	2.03547e10i	0.350907	−	0.202596i	−0.314178	−	0.949364i	$-0.601729\pi$
$563$	3.52554e10	−	2.03547e10i	0.350907	−	0.202596i	0.665085	+	0.746768i	$0.268395\pi$
$564$		0			0
$565$	−1.12656e11	+	1.95126e11i	−1.10551	+	1.91480i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.62872e11	+	9.40342e10i	1.55381	+	0.897092i	0.997826	+	0.0658981i	$0.0209912\pi$
$569$	1.62872e11	+	9.40342e10i	1.55381	+	0.897092i	0.555983	+	0.831194i	$0.312342\pi$
$570$		0			0
$571$	−4.60776e10	−	7.98087e10i	−0.433456	−	0.750768i	0.563712	−	0.825971i	$-0.309373\pi$
$571$	−4.60776e10	−	7.98087e10i	−0.433456	−	0.750768i	−0.997168	+	0.0752031i	$0.976039\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1.07765e9i			0.00985841i
$576$		0			0
$577$	1.84010e10	+	3.18714e10i	0.166011	+	0.287540i	0.937014	−	0.349292i	$-0.113578\pi$
$577$	1.84010e10	+	3.18714e10i	0.166011	+	0.287540i	−0.771003	+	0.636832i	$0.780245\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.88589e9	+	4.89535e9i	0.0428785	+	0.0429614i
$582$		0			0
$583$	−3.26921e8	+	5.66244e8i	−0.00282988	+	0.00490150i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1.39975e11i			1.17896i	0.807784	+	0.589478i	$0.200667\pi$
$587$			1.39975e11i			1.17896i	−0.807784	+	0.589478i	$0.799333\pi$
$588$		0			0
$589$	−7.57766e10			−0.629614
$590$		0			0
$591$		0			0
$592$		0			0
$593$	2.26526e10	+	1.30785e10i	0.183189	+	0.105764i	0.588790	−	0.808286i	$-0.299604\pi$
$593$	2.26526e10	+	1.30785e10i	0.183189	+	0.105764i	−0.405601	+	0.914050i	$0.632938\pi$
$594$		0			0
$595$	9.14163e9	−	9.12398e9i	0.0729383	−	0.0727975i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5.84894e10	−	3.37689e10i	0.454328	−	0.262307i	−0.255328	−	0.966854i	$-0.582183\pi$
$599$	5.84894e10	−	3.37689e10i	0.454328	−	0.262307i	0.709656	+	0.704548i	$0.248850\pi$
$600$		0			0
$601$	3.55575e10			0.272542			0.136271	−	0.990672i	$-0.456488\pi$
$601$	3.55575e10			0.272542			0.136271	+	0.990672i	$0.456488\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.04749e11	+	1.18212e11i	−1.52827	+	0.882348i
$606$		0			0
$607$	6.08790e10	−	1.05445e11i	0.448448	−	0.776735i	−0.549837	−	0.835272i	$-0.685310\pi$
$607$	6.08790e10	−	1.05445e11i	0.448448	−	0.776735i	0.998285	+	0.0585368i	$0.0186435\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.38281e10	−	4.83982e10i	−0.601485	−	0.347267i
$612$		0			0
$613$	1.00334e11	+	1.73783e11i	0.710567	+	1.23074i	0.964645	+	0.263554i	$0.0848948\pi$
$613$	1.00334e11	+	1.73783e11i	0.710567	+	1.23074i	−0.254077	+	0.967184i	$0.581772\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.45362e11i		−	1.00302i	−0.865152	−	0.501509i	$-0.832778\pi$
$617$		−	1.45362e11i		−	1.00302i	0.865152	−	0.501509i	$-0.167222\pi$
$618$		0			0
$619$	4.44346e10	+	7.69630e10i	0.302663	+	0.524227i	0.976738	−	0.214435i	$-0.0687912\pi$
$619$	4.44346e10	+	7.69630e10i	0.302663	+	0.524227i	−0.674076	+	0.738662i	$0.735458\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.19366e11	+	5.85518e10i	−1.45619	+	0.388676i
$624$		0			0
$625$	−2.03966e11	+	3.53279e11i	−1.33671	+	2.31525i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.32658e10i		−	0.0847483i
$630$		0			0
$631$	2.04238e11			1.28831			0.644153	−	0.764897i	$-0.277210\pi$
$631$	2.04238e11			1.28831			0.644153	+	0.764897i	$0.277210\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.21227e11	−	1.27725e11i	−1.36064	−	0.785564i
$636$		0			0
$637$	2.40421e8	−	1.24389e11i	0.00146021	−	0.755485i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.25369e11	−	1.30117e11i	1.33494	−	0.770728i	0.348887	−	0.937165i	$-0.386560\pi$
$641$	2.25369e11	−	1.30117e11i	1.33494	−	0.770728i	0.986052	+	0.166437i	$0.0532263\pi$
$642$		0			0
$643$	−3.30462e10			−0.193321			−0.0966603	−	0.995317i	$-0.530816\pi$
$643$	−3.30462e10			−0.193321			−0.0966603	+	0.995317i	$0.530816\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.42517e11	+	8.22825e10i	−0.813300	+	0.469559i	−0.848101	−	0.529835i	$-0.822254\pi$
$647$	−1.42517e11	+	8.22825e10i	−0.813300	+	0.469559i	0.0348006	+	0.999394i	$0.488920\pi$
$648$		0			0
$649$	−1.73143e10	+	2.99892e10i	−0.0975946	+	0.169039i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.41617e11	+	8.17624e10i	0.778864	+	0.449677i	0.836027	−	0.548688i	$-0.184872\pi$
$653$	1.41617e11	+	8.17624e10i	0.778864	+	0.449677i	−0.0571638	+	0.998365i	$0.518206\pi$
$654$		0			0
$655$	−2.94790e10	−	5.10591e10i	−0.160157	−	0.277401i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			3.20888e11i			1.70142i	0.525636	+	0.850710i	$0.323827\pi$
$659$			3.20888e11i			1.70142i	−0.525636	+	0.850710i	$0.676173\pi$
$660$		0			0
$661$	9.20326e10	+	1.59405e11i	0.482098	+	0.835019i	0.999789	−	0.0205490i	$-0.00654140\pi$
$661$	9.20326e10	+	1.59405e11i	0.482098	+	0.835019i	−0.517690	+	0.855568i	$0.673208\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	9.81237e10	−	3.64792e11i	0.501750	−	1.86534i
$666$		0			0
$667$	3.40695e8	−	5.90101e8i	0.00172132	−	0.00298142i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			2.60970e10i			0.128736i
$672$		0			0
$673$	−1.05964e11			−0.516533			−0.258267	−	0.966074i	$-0.583151\pi$
$673$	−1.05964e11			−0.516533			−0.258267	+	0.966074i	$0.583151\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.22227e10	−	2.43773e10i	−0.200998	−	0.116046i	0.396123	−	0.918197i	$-0.370355\pi$
$677$	−4.22227e10	−	2.43773e10i	−0.200998	−	0.116046i	−0.597121	+	0.802151i	$0.703689\pi$
$678$		0			0
$679$	−1.04423e11	+	1.04222e11i	−0.491268	+	0.490319i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.95366e11	−	1.12795e11i	0.897772	−	0.518329i	0.0212953	−	0.999773i	$-0.493221\pi$
$683$	1.95366e11	−	1.12795e11i	0.897772	−	0.518329i	0.876477	+	0.481444i	$0.159888\pi$
$684$		0			0
$685$	5.43733e11			2.46958
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.58693e9	−	2.07092e9i	0.0159165	−	0.00918937i
$690$		0			0
$691$	1.83584e11	−	3.17977e11i	0.805234	−	1.39471i	−0.110898	−	0.993832i	$-0.535373\pi$
$691$	1.83584e11	−	3.17977e11i	0.805234	−	1.39471i	0.916133	−	0.400875i	$-0.131294\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.64951e11	−	2.10705e11i	−1.56421	−	0.903098i
$696$		0			0
$697$	−8.71318e9	−	1.50917e10i	−0.0369186	−	0.0639449i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1.11417e11i		−	0.461401i	−0.973025	−	0.230700i	$-0.925898\pi$
$701$		−	1.11417e11i		−	0.461401i	0.973025	−	0.230700i	$-0.0741017\pi$
$702$		0			0
$703$	−1.93999e11	−	3.36016e11i	−0.794287	−	1.37575i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.03816e11	−	3.85954e11i	0.415515	−	1.54475i
$708$		0			0
$709$	−6.89166e10	+	1.19367e11i	−0.272734	+	0.472389i	−0.969561	−	0.244850i	$-0.921261\pi$
$709$	−6.89166e10	+	1.19367e11i	−0.272734	+	0.472389i	0.696827	+	0.717239i	$0.254595\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			6.24544e8i			0.00241660i
$714$		0			0
$715$	−8.57033e10			−0.327924
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.99704e11	+	2.30769e11i	1.49563	+	0.863500i	0.999987	−	0.00502965i	$-0.00160099\pi$
$719$	3.99704e11	+	2.30769e11i	1.49563	+	0.863500i	0.495638	+	0.868529i	$0.334934\pi$
$720$		0			0
$721$	−1.03221e11	−	3.86721e11i	−0.381969	−	1.43106i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.14208e11	−	2.96878e11i	1.86117	−	1.07455i
$726$		0			0
$727$	3.63902e11			1.30271			0.651354	−	0.758774i	$-0.274201\pi$
$727$	3.63902e11			1.30271			0.651354	+	0.758774i	$0.274201\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.25204e9	+	3.03227e9i	−0.0183932	+	0.0106193i
$732$		0			0
$733$	1.02143e11	−	1.76917e11i	0.353829	−	0.612850i	−0.633088	−	0.774080i	$-0.718213\pi$
$733$	1.02143e11	−	1.76917e11i	0.353829	−	0.612850i	0.986917	+	0.161230i	$0.0515460\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.72279e10	+	5.61346e10i	0.329550	+	0.190266i
$738$		0			0
$739$	−7.11913e10	−	1.23307e11i	−0.238698	−	0.413438i	0.721643	−	0.692266i	$-0.243387\pi$
$739$	−7.11913e10	−	1.23307e11i	−0.238698	−	0.413438i	−0.960341	+	0.278828i	$0.910054\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			8.97957e10i			0.294646i	0.989088	+	0.147323i	$0.0470656\pi$
$743$			8.97957e10i			0.294646i	−0.989088	+	0.147323i	$0.952934\pi$
$744$		0			0
$745$	1.68833e11	+	2.92428e11i	0.548066	+	0.949279i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.94753e11	−	1.32056e11i	1.57203	−	0.419597i
$750$		0			0
$751$	−2.87615e11	+	4.98163e11i	−0.904173	+	1.56607i	−0.0821491	+	0.996620i	$0.526178\pi$
$751$	−2.87615e11	+	4.98163e11i	−0.904173	+	1.56607i	−0.822024	+	0.569453i	$0.807155\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	7.39348e11i		−	2.27542i
$756$		0			0
$757$	3.01398e11			0.917818			0.458909	−	0.888483i	$-0.348240\pi$
$757$	3.01398e11			0.917818			0.458909	+	0.888483i	$0.348240\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.22959e11	+	1.28725e11i	0.664792	+	0.383818i	0.794100	−	0.607787i	$-0.207942\pi$
$761$	2.22959e11	+	1.28725e11i	0.664792	+	0.383818i	−0.129308	+	0.991604i	$0.541276\pi$
$762$		0			0
$763$	3.07393e11	+	8.26842e10i	0.906976	+	0.243963i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.89970e11	−	1.09679e11i	0.548913	−	0.316915i
$768$		0			0
$769$	5.81746e11			1.66352			0.831760	−	0.555136i	$-0.187334\pi$
$769$	5.81746e11			1.66352			0.831760	+	0.555136i	$0.187334\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.05653e11	+	6.09988e10i	−0.295913	+	0.170845i	−0.640605	−	0.767870i	$-0.721317\pi$
$773$	−1.05653e11	+	6.09988e10i	−0.295913	+	0.170845i	0.344692	+	0.938716i	$0.387983\pi$
$774$		0			0
$775$	−2.72111e11	+	4.71309e11i	−0.754290	+	1.30647i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.41400e11	−	2.54843e11i	−1.19862	−	0.692025i
$780$		0			0
$781$	5.24047e9	+	9.07676e9i	0.0140853	+	0.0243965i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	5.77845e11i		−	1.52171i
$786$		0			0
$787$	−1.60415e11	−	2.77846e11i	−0.418162	−	0.724279i	0.577592	−	0.816325i	$-0.303992\pi$
$787$	−1.60415e11	−	2.77846e11i	−0.418162	−	0.724279i	−0.995755	+	0.0920468i	$0.970659\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.27737e11	−	3.28371e11i	−0.837181	−	0.838801i
$792$		0			0
$793$	8.26572e10	−	1.43166e11i	0.209020	−	0.362033i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	7.09316e10i		−	0.175795i	−0.996130	−	0.0878975i	$-0.971985\pi$
$797$		−	7.09316e10i		−	0.175795i	0.996130	−	0.0878975i	$-0.0280148\pi$
$798$		0			0
$799$	2.06952e10			0.0507787
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.52568e9	+	4.34495e9i	0.0181002	+	0.0104502i
$804$		0			0
$805$	−3.00658e9	−	8.08727e8i	−0.00715962	−	0.00192583i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.54587e11	+	2.04721e11i	−0.827805	+	0.477934i	−0.853101	−	0.521746i	$-0.825281\pi$
$809$	−3.54587e11	+	2.04721e11i	−0.827805	+	0.477934i	0.0252952	+	0.999680i	$0.491947\pi$
$810$		0			0
$811$	5.26321e11			1.21665			0.608327	−	0.793687i	$-0.291841\pi$
$811$	5.26321e11			1.21665			0.608327	+	0.793687i	$0.291841\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.67643e11	+	5.58669e11i	−2.19323	+	1.26626i
$816$		0			0
$817$	−8.86875e10	+	1.53611e11i	−0.199055	+	0.344774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	4.86513e11	+	2.80889e11i	1.07083	+	0.618246i	0.928409	−	0.371559i	$-0.121177\pi$
$821$	4.86513e11	+	2.80889e11i	1.07083	+	0.618246i	0.142425	+	0.989806i	$0.454510\pi$
$822$		0			0
$823$	−4.54562e10	−	7.87325e10i	−0.0990818	−	0.171615i	0.812223	−	0.583347i	$-0.198257\pi$
$823$	−4.54562e10	−	7.87325e10i	−0.0990818	−	0.171615i	−0.911305	+	0.411732i	$0.864924\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			7.64100e11i			1.63353i	0.576968	+	0.816767i	$0.304236\pi$
$827$			7.64100e11i			1.63353i	−0.576968	+	0.816767i	$0.695764\pi$
$828$		0			0
$829$	2.85984e11	+	4.95339e11i	0.605513	+	1.04878i	0.991970	+	0.126472i	$0.0403655\pi$
$829$	2.85984e11	+	4.95339e11i	0.605513	+	1.04878i	−0.386457	+	0.922307i	$0.626301\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.32528e10	+	2.30573e10i	0.0275250	+	0.0478882i
$834$		0			0
$835$	−8.30620e11	+	1.43868e12i	−1.70866	+	2.95949i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	3.14915e11i		−	0.635545i	−0.948167	−	0.317772i	$-0.897065\pi$
$839$		−	3.14915e11i		−	0.635545i	0.948167	−	0.317772i	$-0.102935\pi$
$840$		0			0
$841$	1.24819e11			0.249515
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.53586e11	−	2.04143e11i	−0.693535	−	0.400413i
$846$		0			0
$847$	−1.25543e11	−	4.70351e11i	−0.243927	−	0.913879i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.76941e9	+	1.59892e9i	−0.00528043	+	0.00304866i
$852$		0			0
$853$	−1.25638e11			−0.237315			−0.118658	−	0.992935i	$-0.537859\pi$
$853$	−1.25638e11			−0.237315			−0.118658	+	0.992935i	$0.537859\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5.49878e10	+	3.17472e10i	−0.101940	+	0.0588548i	−0.550103	−	0.835097i	$-0.685412\pi$
$857$	−5.49878e10	+	3.17472e10i	−0.101940	+	0.0588548i	0.448164	+	0.893952i	$0.352078\pi$
$858$		0			0
$859$	3.59865e11	−	6.23304e11i	0.660947	−	1.14479i	−0.319420	−	0.947613i	$-0.603488\pi$
$859$	3.59865e11	−	6.23304e11i	0.660947	−	1.14479i	0.980367	−	0.197181i	$-0.0631787\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	5.04334e11	+	2.91177e11i	0.909233	+	0.524946i	0.880184	−	0.474632i	$-0.157419\pi$
$863$	5.04334e11	+	2.91177e11i	0.909233	+	0.524946i	0.0290488	+	0.999578i	$0.490752\pi$
$864$		0			0
$865$	2.06146e11	+	3.57056e11i	0.368223	+	0.637781i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1.14090e10i		−	0.0200064i
$870$		0			0
$871$	−3.55591e11	−	6.15901e11i	−0.617842	−	1.07013i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.14398e12	−	1.14620e12i	−1.95158	−	1.95536i
$876$		0			0
$877$	4.80524e8	−	8.32292e8i	0.000812301	−	0.00140695i	−0.865619	−	0.500703i	$-0.833075\pi$
$877$	4.80524e8	−	8.32292e8i	0.000812301	−	0.00140695i	0.866431	+	0.499296i	$0.166408\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			6.80681e11i			1.12990i	0.825125	+	0.564950i	$0.191104\pi$
$881$			6.80681e11i			1.12990i	−0.825125	+	0.564950i	$0.808896\pi$
$882$		0			0
$883$	5.28883e11			0.869996			0.434998	−	0.900431i	$-0.356749\pi$
$883$	5.28883e11			0.869996			0.434998	+	0.900431i	$0.356749\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7.55778e9	−	4.36349e9i	−0.0122096	−	0.00704919i	0.493883	−	0.869528i	$-0.335577\pi$
$887$	−7.55778e9	−	4.36349e9i	−0.0122096	−	0.00704919i	−0.506092	+	0.862479i	$0.668911\pi$
$888$		0			0
$889$	3.72294e11	−	3.71575e11i	0.596045	−	0.594894i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.24197e11	−	3.02645e11i	0.824307	−	0.475914i
$894$		0			0
$895$	−8.45668e11			−1.31798
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.98005e11	−	1.72053e11i	0.456231	−	0.263405i
$900$		0			0
$901$	−4.42764e8	+	7.66890e8i	−0.000671852	+	0.00116368i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.40326e12	+	8.10173e11i	2.09191	+	1.20777i
$906$		0			0
$907$	3.21235e10	+	5.56395e10i	0.0474672	+	0.0822156i	0.888783	−	0.458329i	$-0.151552\pi$
$907$	3.21235e10	+	5.56395e10i	0.0474672	+	0.0822156i	−0.841316	+	0.540544i	$0.818218\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			4.11062e10i			0.0596807i	0.999555	+	0.0298404i	$0.00949989\pi$
$911$			4.11062e10i			0.0596807i	−0.999555	+	0.0298404i	$0.990500\pi$
$912$		0			0
$913$	4.90611e9	+	8.49763e9i	0.00706080	+	0.0122297i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.17293e11	−	3.13072e10i	0.165881	−	0.0442758i
$918$		0			0
$919$	4.53663e11	−	7.85768e11i	0.636021	−	1.10162i	−0.350277	−	0.936646i	$-0.613912\pi$
$919$	4.53663e11	−	7.85768e11i	0.636021	−	1.10162i	0.986298	−	0.164975i	$-0.0527543\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	6.63927e10i		−	0.0914773i
$924$		0			0
$925$	−2.78656e12			−3.80629
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.00845e11	+	4.04633e11i	0.940934	+	0.543249i	0.890253	−	0.455466i	$-0.150527\pi$
$929$	7.00845e11	+	4.04633e11i	0.940934	+	0.543249i	0.0506811	+	0.998715i	$0.483861\pi$
$930$		0			0
$931$	6.72875e11	+	3.90220e11i	0.895644	+	0.519411i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.58686e10	−	9.16173e9i	0.0207631	−	0.0119876i
$936$		0			0
$937$	3.76135e10			0.0487962			0.0243981	−	0.999702i	$-0.492233\pi$
$937$	3.76135e10			0.0487962			0.0243981	+	0.999702i	$0.492233\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.12869e12	+	6.51647e11i	−1.43951	+	0.831101i	−0.997815	−	0.0660642i	$-0.978956\pi$
$941$	−1.12869e12	+	6.51647e11i	−1.43951	+	0.831101i	−0.441694	+	0.897166i	$0.645622\pi$
$942$		0			0
$943$	−2.10039e9	+	3.63798e9i	−0.00265615	+	0.00460059i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.36835e11	+	3.67677e11i	0.791821	+	0.457158i	0.840603	−	0.541651i	$-0.182201\pi$
$947$	6.36835e11	+	3.67677e11i	0.791821	+	0.457158i	−0.0487819	+	0.998809i	$0.515534\pi$
$948$		0			0
$949$	−2.75236e10	−	4.76722e10i	−0.0339344	−	0.0587761i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			5.13996e11i			0.623143i	0.950223	+	0.311572i	$0.100855\pi$
$953$			5.13996e11i			0.623143i	−0.950223	+	0.311572i	$0.899145\pi$
$954$		0			0
$955$	−1.35776e12	−	2.35171e12i	−1.63234	−	2.82729i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.90818e11	+	1.08117e12i	−0.343832	+	1.27826i
$960$		0			0
$961$	2.68746e11	−	4.65482e11i	0.315100	−	0.545769i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.93329e12i		−	2.22939i
$966$		0			0
$967$	1.00071e12			1.14446			0.572231	−	0.820092i	$-0.306078\pi$
$967$	1.00071e12			1.14446			0.572231	+	0.820092i	$0.306078\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.62328e11	+	2.66925e11i	0.520084	+	0.300270i	0.736969	−	0.675927i	$-0.236256\pi$
$971$	4.62328e11	+	2.66925e11i	0.520084	+	0.300270i	−0.216885	+	0.976197i	$0.569590\pi$
$972$		0			0
$973$	6.14162e11	−	6.12976e11i	0.685223	−	0.683900i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.95841e11	+	3.44009e11i	−0.653961	+	0.377564i	−0.789972	−	0.613143i	$-0.789905\pi$
$977$	−5.95841e11	+	3.44009e11i	−0.653961	+	0.377564i	0.136011	+	0.990707i	$0.456572\pi$
$978$		0			0
$979$	−3.22108e11			−0.350647
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−5.61355e11	+	3.24098e11i	−0.601206	+	0.347106i	−0.769516	−	0.638628i	$-0.779502\pi$
$983$	−5.61355e11	+	3.24098e11i	−0.601206	+	0.347106i	0.168310	+	0.985734i	$0.446169\pi$
$984$		0			0
$985$	−1.27883e12	+	2.21499e12i	−1.35852	+	2.35303i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.26605e9	+	7.30954e8i	0.00132332	+	0.000764021i
$990$		0			0
$991$	3.01433e10	+	5.22098e10i	0.0312534	+	0.0541324i	0.881229	−	0.472690i	$-0.156717\pi$
$991$	3.01433e10	+	5.22098e10i	0.0312534	+	0.0541324i	−0.849976	+	0.526822i	$0.823383\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			2.75571e12i			2.81152i
$996$		0			0
$997$	−7.31592e11	−	1.26715e12i	−0.740437	−	1.28247i	−0.952296	−	0.305175i	$-0.901285\pi$
$997$	−7.31592e11	−	1.26715e12i	−0.740437	−	1.28247i	0.211859	−	0.977300i	$-0.432048\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bk.a.233.21	yes	44
3.2	odd	2	inner	252.9.bk.a.233.2	yes	44
7.4	even	3	inner	252.9.bk.a.53.2	✓	44
21.11	odd	6	inner	252.9.bk.a.53.21	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bk.a.53.2	✓	44	7.4	even	3	inner
252.9.bk.a.53.21	yes	44	21.11	odd	6	inner
252.9.bk.a.233.2	yes	44	3.2	odd	2	inner
252.9.bk.a.233.21	yes	44	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1009.83 + 583.025i) q^{5} +(-1699.40 + 1696.12i) q^{7} +O(q^{10})\)	\(q+(1009.83 + 583.025i) q^{5} +(-1699.40 + 1696.12i) q^{7} +(-2949.92 + 1703.14i) q^{11} +21577.4 q^{13} +(-3995.22 + 2306.64i) q^{17} +(-67464.5 + 116852. i) q^{19} +(963.083 + 556.036i) q^{23} +(484524. + 839220. i) q^{25} -612721. i q^{29} +(280802. + 486363. i) q^{31} +(-2.70499e6 + 721998. i) q^{35} +(-1.43778e6 + 2.49032e6i) q^{37} +3.77743e6i q^{41} +1.31458e6 q^{43} +(-3.88499e6 - 2.24300e6i) q^{47} +(11142.3 - 5.76479e6i) q^{49} +(166235. - 95976.0i) q^{53} -3.97189e6 q^{55} +(8.80410e6 - 5.08305e6i) q^{59} +(3.83072e6 - 6.63500e6i) q^{61} +(2.17895e7 + 1.25802e7i) q^{65} +(-1.64797e7 - 2.85437e7i) q^{67} -3.07695e6i q^{71} +(-1.27557e6 - 2.20936e6i) q^{73} +(2.12438e6 - 7.89775e6i) q^{77} +(-1.67470e6 + 2.90067e6i) q^{79} -2.88063e6i q^{83} -5.37932e6 q^{85} +(8.18939e7 + 4.72815e7i) q^{89} +(-3.66688e7 + 3.65980e7i) q^{91} +(-1.36255e8 + 7.86669e7i) q^{95} +6.14470e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 1230 q^{7}+O(q^{10}) \) 44 * q + 1230 * q^7	\( 44 q + 1230 q^{7} - 101380 q^{13} + 62770 q^{19} + 2247666 q^{25} + 1389254 q^{31} - 2136026 q^{37} + 11510140 q^{43} - 3824398 q^{49} - 42646528 q^{55} + 27346232 q^{61} + 14239194 q^{67} - 64344138 q^{73} + 7061786 q^{79} - 54198208 q^{85} - 45697066 q^{91} + 476543496 q^{97}+O(q^{100}) \) 44 * q + 1230 * q^7 - 101380 * q^13 + 62770 * q^19 + 2247666 * q^25 + 1389254 * q^31 - 2136026 * q^37 + 11510140 * q^43 - 3824398 * q^49 - 42646528 * q^55 + 27346232 * q^61 + 14239194 * q^67 - 64344138 * q^73 + 7061786 * q^79 - 54198208 * q^85 - 45697066 * q^91 + 476543496 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more