LMFDB - Embedded newform 252.9.bk.a.233.20

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			233.20
Character	$\chi$	$=$	252.233
Dual form			252.9.bk.a.53.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(724.791 + 418.458i) q^{5} +(613.419 - 2321.32i) q^{7} +O(q^{10})$	$q+(724.791 + 418.458i) q^{5} +(613.419 - 2321.32i) q^{7} +(18278.5 - 10553.1i) q^{11} -48972.6 q^{13} +(-57253.8 + 33055.5i) q^{17} +(-124724. + 216028. i) q^{19} +(-31876.4 - 18403.9i) q^{23} +(154902. + 268298. i) q^{25} +159982. i q^{29} +(458407. + 793984. i) q^{31} +(1.41597e6 - 1.42578e6i) q^{35} +(-670147. + 1.16073e6i) q^{37} +1.76061e6i q^{41} -2.70210e6 q^{43} +(7.31980e6 + 4.22609e6i) q^{47} +(-5.01224e6 - 2.84788e6i) q^{49} +(-4.99457e6 + 2.88361e6i) q^{53} +1.76642e7 q^{55} +(-4.25134e6 + 2.45451e6i) q^{59} +(4.36663e6 - 7.56322e6i) q^{61} +(-3.54949e7 - 2.04930e7i) q^{65} +(-1.27203e6 - 2.20322e6i) q^{67} +8.62638e6i q^{71} +(2.18886e6 + 3.79121e6i) q^{73} +(-1.32848e7 - 4.89038e7i) q^{77} +(1.66903e7 - 2.89084e7i) q^{79} +3.92393e7i q^{83} -5.53294e7 q^{85} +(9.08755e7 + 5.24670e7i) q^{89} +(-3.00407e7 + 1.13681e8i) q^{91} +(-1.80797e8 + 1.04383e8i) q^{95} +1.61722e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	724.791	+	418.458i	1.15966	+	0.669533i	0.951224	−	0.308502i	$-0.0998277\pi$
$5$	724.791	+	418.458i	1.15966	+	0.669533i	0.208441	+	0.978035i	$0.433161\pi$
$6$		0			0
$7$	613.419	−	2321.32i	0.255485	−	0.966813i
$7$	613.419	−	2321.32i	0.255485	−	0.966813i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	18278.5	−	10553.1i	1.24845	−	0.720792i	0.277649	−	0.960683i	$-0.410445\pi$
$11$	18278.5	−	10553.1i	1.24845	−	0.720792i	0.970800	+	0.239890i	$0.0771114\pi$
$12$		0			0
$13$	−48972.6			−1.71467			−0.857333	−	0.514762i	$-0.827880\pi$
$13$	−48972.6			−1.71467			−0.857333	+	0.514762i	$0.827880\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−57253.8	+	33055.5i	−0.685502	+	0.395775i	−0.801925	−	0.597425i	$-0.796191\pi$
$17$	−57253.8	+	33055.5i	−0.685502	+	0.395775i	0.116423	+	0.993200i	$0.462857\pi$
$18$		0			0
$19$	−124724.	+	216028.i	−0.957050	+	1.65766i	−0.227443	+	0.973791i	$0.573037\pi$
$19$	−124724.	+	216028.i	−0.957050	+	1.65766i	−0.729606	+	0.683867i	$0.760297\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−31876.4	−	18403.9i	−0.113909	−	0.0657655i	0.441963	−	0.897033i	$-0.354282\pi$
$23$	−31876.4	−	18403.9i	−0.113909	−	0.0657655i	−0.555872	+	0.831268i	$0.687616\pi$
$24$		0			0
$25$	154902.	+	268298.i	0.396548	+	0.686842i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			159982.i			0.226193i	0.993584	+	0.113096i	$0.0360769\pi$
$29$			159982.i			0.226193i	−0.993584	+	0.113096i	$0.963923\pi$
$30$		0			0
$31$	458407.	+	793984.i	0.496369	+	0.859736i	0.999991	−	0.00418771i	$-0.00133299\pi$
$31$	458407.	+	793984.i	0.496369	+	0.859736i	−0.503622	+	0.863924i	$0.668000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.41597e6	−	1.42578e6i	0.943590	−	0.950124i
$36$		0			0
$37$	−670147.	+	1.16073e6i	−0.357571	+	0.619332i	−0.987555	−	0.157277i	$-0.949728\pi$
$37$	−670147.	+	1.16073e6i	−0.357571	+	0.619332i	0.629983	+	0.776609i	$0.283062\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.76061e6i			0.623056i	0.950237	+	0.311528i	$0.100841\pi$
$41$			1.76061e6i			0.623056i	−0.950237	+	0.311528i	$0.899159\pi$
$42$		0			0
$43$	−2.70210e6			−0.790364			−0.395182	−	0.918603i	$-0.629318\pi$
$43$	−2.70210e6			−0.790364			−0.395182	+	0.918603i	$0.629318\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.31980e6	+	4.22609e6i	1.50006	+	0.866058i	1.00000		6.49340e-5i	$2.06691e-5\pi$
$47$	7.31980e6	+	4.22609e6i	1.50006	+	0.866058i	0.500056	+	0.865993i	$0.333313\pi$
$48$		0			0
$49$	−5.01224e6	−	2.84788e6i	−0.869455	−	0.494012i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.99457e6	+	2.88361e6i	−0.632986	+	0.365455i	−0.781908	−	0.623394i	$-0.785753\pi$
$53$	−4.99457e6	+	2.88361e6i	−0.632986	+	0.365455i	0.148921	+	0.988849i	$0.452420\pi$
$54$		0			0
$55$	1.76642e7			1.93038
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.25134e6	+	2.45451e6i	−0.350847	+	0.202561i	−0.665058	−	0.746792i	$-0.731593\pi$
$59$	−4.25134e6	+	2.45451e6i	−0.350847	+	0.202561i	0.314211	+	0.949353i	$0.398260\pi$
$60$		0			0
$61$	4.36663e6	−	7.56322e6i	0.315375	−	0.546245i	−0.664143	−	0.747606i	$-0.731203\pi$
$61$	4.36663e6	−	7.56322e6i	0.315375	−	0.546245i	0.979517	+	0.201361i	$0.0645365\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.54949e7	−	2.04930e7i	−1.98844	−	1.14803i
$66$		0			0
$67$	−1.27203e6	−	2.20322e6i	−0.0631245	−	0.109335i	0.832736	−	0.553670i	$-0.186773\pi$
$67$	−1.27203e6	−	2.20322e6i	−0.0631245	−	0.109335i	−0.895860	+	0.444335i	$0.853440\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			8.62638e6i			0.339465i	0.985490	+	0.169733i	$0.0542904\pi$
$71$			8.62638e6i			0.339465i	−0.985490	+	0.169733i	$0.945710\pi$
$72$		0			0
$73$	2.18886e6	+	3.79121e6i	0.0770772	+	0.133502i	0.901988	−	0.431762i	$-0.142108\pi$
$73$	2.18886e6	+	3.79121e6i	0.0770772	+	0.133502i	−0.824910	+	0.565263i	$0.808775\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.32848e7	−	4.89038e7i	−0.377912	−	1.39117i
$78$		0			0
$79$	1.66903e7	−	2.89084e7i	0.428505	−	0.742192i	−0.568236	−	0.822866i	$-0.692374\pi$
$79$	1.66903e7	−	2.89084e7i	0.428505	−	0.742192i	0.996741	+	0.0806739i	$0.0257072\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.92393e7i			0.826815i	0.910546	+	0.413408i	$0.135662\pi$
$83$			3.92393e7i			0.826815i	−0.910546	+	0.413408i	$0.864338\pi$
$84$		0			0
$85$	−5.53294e7			−1.05994
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.08755e7	+	5.24670e7i	1.44839	+	0.836231i	0.998386	−	0.0567936i	$-0.0180877\pi$
$89$	9.08755e7	+	5.24670e7i	1.44839	+	0.836231i	0.450008	+	0.893024i	$0.351421\pi$
$90$		0			0
$91$	−3.00407e7	+	1.13681e8i	−0.438071	+	1.65776i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.80797e8	+	1.04383e8i	−2.21971	+	1.28155i
$96$		0			0
$97$	1.61722e8			1.82676			0.913381	−	0.407107i	$-0.133462\pi$
$97$	1.61722e8			1.82676			0.913381	+	0.407107i	$0.133462\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.46479e6	+	1.42305e6i	−0.0236861	+	0.0136752i	−0.511796	−	0.859107i	$-0.671020\pi$
$101$	−2.46479e6	+	1.42305e6i	−0.0236861	+	0.0136752i	0.488110	+	0.872782i	$0.337686\pi$
$102$		0			0
$103$	−7.03776e7	+	1.21898e8i	−0.625296	+	1.08304i	0.363188	+	0.931716i	$0.381688\pi$
$103$	−7.03776e7	+	1.21898e8i	−0.625296	+	1.08304i	−0.988484	+	0.151328i	$0.951645\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.08663e8	+	6.27366e7i	0.828984	+	0.478614i	0.853505	−	0.521085i	$-0.174473\pi$
$107$	1.08663e8	+	6.27366e7i	0.828984	+	0.478614i	−0.0245206	+	0.999699i	$0.507806\pi$
$108$		0			0
$109$	8.22834e7	+	1.42519e8i	0.582916	+	1.00964i	0.995132	+	0.0985545i	$0.0314219\pi$
$109$	8.22834e7	+	1.42519e8i	0.582916	+	1.00964i	−0.412215	+	0.911087i	$0.635245\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.12894e6i			0.00692399i	0.999994	+	0.00346200i	$0.00110199\pi$
$113$			1.12894e6i			0.00692399i	−0.999994	+	0.00346200i	$0.998898\pi$
$114$		0			0
$115$	−1.54025e7	−	2.66779e7i	−0.0880643	−	0.152532i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.16118e7	+	1.53181e8i	0.207505	+	0.763867i
$120$		0			0
$121$	1.15557e8	−	2.00151e8i	0.539084	−	0.933720i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	6.76409e7i		−	0.277057i
$126$		0			0
$127$	−219505.			−0.000843781			−0.000421891	−	1.00000i	$-0.500134\pi$
$127$	−219505.			−0.000843781			−0.000421891		1.00000i	$0.500134\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.81811e8	−	1.04969e8i	−0.617355	−	0.356430i	0.158483	−	0.987362i	$-0.449340\pi$
$131$	−1.81811e8	−	1.04969e8i	−0.617355	−	0.356430i	−0.775839	+	0.630932i	$0.782673\pi$
$132$		0			0
$133$	4.24961e8	+	4.22039e8i	1.35813	+	1.34879i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.07778e8	+	2.93166e8i	−1.44142	+	0.832206i	−0.997945	−	0.0640783i	$-0.979589\pi$
$137$	−5.07778e8	+	2.93166e8i	−1.44142	+	0.832206i	−0.443479	+	0.896285i	$0.646256\pi$
$138$		0			0
$139$	−2.51220e8			−0.672968			−0.336484	−	0.941689i	$-0.609238\pi$
$139$	−2.51220e8			−0.672968			−0.336484	+	0.941689i	$0.609238\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.95148e8	+	5.16814e8i	−2.14067	+	1.23592i
$144$		0			0
$145$	−6.69457e7	+	1.15953e8i	−0.151443	+	0.262308i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.90996e8	−	4.56682e8i	−1.60483	−	0.926550i	−0.990502	−	0.137501i	$-0.956093\pi$
$149$	−7.90996e8	−	4.56682e8i	−1.60483	−	0.926550i	−0.614330	−	0.789049i	$-0.710574\pi$
$150$		0			0
$151$	2.18270e8	+	3.78056e8i	0.419843	+	0.727190i	0.995923	−	0.0902034i	$-0.0287517\pi$
$151$	2.18270e8	+	3.78056e8i	0.419843	+	0.727190i	−0.576080	+	0.817393i	$0.695418\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			7.67297e8i			1.32934i
$156$		0			0
$157$	1.29821e6	+	2.24857e6i	0.00213672	+	0.00370090i	0.867092	−	0.498148i	$-0.165987\pi$
$157$	1.29821e6	+	2.24857e6i	0.00213672	+	0.00370090i	−0.864955	+	0.501849i	$0.832653\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.22748e7	+	6.27061e7i	−0.0926849	+	0.0933268i
$162$		0			0
$163$	2.04833e8	−	3.54781e8i	0.290168	−	0.502585i	−0.683682	−	0.729780i	$-0.739622\pi$
$163$	2.04833e8	−	3.54781e8i	0.290168	−	0.502585i	0.973849	+	0.227195i	$0.0729556\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			1.09139e9i			1.40318i	0.712580	+	0.701590i	$0.247526\pi$
$167$			1.09139e9i			1.40318i	−0.712580	+	0.701590i	$0.752474\pi$
$168$		0			0
$169$	1.58258e9			1.94008
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.56035e8	−	1.47822e8i	−0.285834	−	0.165027i	0.350227	−	0.936665i	$-0.386104\pi$
$173$	−2.56035e8	−	1.47822e8i	−0.285834	−	0.165027i	−0.636062	+	0.771638i	$0.719438\pi$
$174$		0			0
$175$	7.17824e8	−	1.94997e8i	0.765360	−	0.207911i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.47767e9	−	8.53134e8i	1.43935	−	0.831008i	0.441544	−	0.897240i	$-0.354431\pi$
$179$	1.47767e9	−	8.53134e8i	1.43935	−	0.831008i	0.997804	+	0.0662313i	$0.0210975\pi$
$180$		0			0
$181$	−6.61028e8			−0.615893			−0.307946	−	0.951404i	$-0.599642\pi$
$181$	−6.61028e8			−0.615893			−0.307946	+	0.951404i	$0.599642\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9.71432e8	+	5.60856e8i	−0.829326	+	0.478812i
$186$		0			0
$187$	−6.97678e8	+	1.20841e9i	−0.570543	+	0.988210i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.40735e9	−	8.12534e8i	−1.05747	−	0.610531i	−0.132740	−	0.991151i	$-0.542377\pi$
$191$	−1.40735e9	−	8.12534e8i	−1.05747	−	0.610531i	−0.924732	+	0.380620i	$0.875711\pi$
$192$		0			0
$193$	6.11822e8	+	1.05971e9i	0.440957	+	0.763759i	0.997761	−	0.0668847i	$-0.0213060\pi$
$193$	6.11822e8	+	1.05971e9i	0.440957	+	0.763759i	−0.556804	+	0.830644i	$0.687973\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2.83505e9i			1.88233i	0.337945	+	0.941166i	$0.390268\pi$
$197$			2.83505e9i			1.88233i	−0.337945	+	0.941166i	$0.609732\pi$
$198$		0			0
$199$	1.37050e9	+	2.37377e9i	0.873908	+	1.51365i	0.857921	+	0.513781i	$0.171756\pi$
$199$	1.37050e9	+	2.37377e9i	0.873908	+	1.51365i	0.0159865	+	0.999872i	$0.494911\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.71369e8	+	9.81358e7i	0.218686	+	0.0577888i
$204$		0			0
$205$	−7.36741e8	+	1.27607e9i	−0.417157	+	0.722537i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			5.26490e9i			2.75934i
$210$		0			0
$211$	−1.66306e9			−0.839034			−0.419517	−	0.907748i	$-0.637800\pi$
$211$	−1.66306e9			−0.839034			−0.419517	+	0.907748i	$0.637800\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.95845e9	−	1.13071e9i	−0.916557	−	0.529174i
$216$		0			0
$217$	2.12429e9	−	5.77064e8i	0.958019	−	0.260247i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.80387e9	−	1.61881e9i	1.17541	−	0.678622i
$222$		0			0
$223$	−3.57119e9			−1.44409			−0.722044	−	0.691847i	$-0.756797\pi$
$223$	−3.57119e9			−1.44409			−0.722044	+	0.691847i	$0.756797\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.24886e9	+	1.29838e9i	−0.846952	+	0.488988i	−0.859621	−	0.510932i	$-0.829301\pi$
$227$	−2.24886e9	+	1.29838e9i	−0.846952	+	0.488988i	0.0126690	+	0.999920i	$0.495967\pi$
$228$		0			0
$229$	4.50800e7	−	7.80808e7i	0.0163924	−	0.0283924i	−0.857713	−	0.514129i	$-0.828115\pi$
$229$	4.50800e7	−	7.80808e7i	0.0163924	−	0.0283924i	0.874105	+	0.485737i	$0.161449\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.00270e9	−	1.73361e9i	−1.01880	−	0.588204i	−0.105043	−	0.994468i	$-0.533498\pi$
$233$	−3.00270e9	−	1.73361e9i	−1.01880	−	0.588204i	−0.913756	+	0.406264i	$0.866831\pi$
$234$		0			0
$235$	3.53688e9	+	6.12605e9i	1.15971	+	2.00867i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4.68372e9i		−	1.43549i	−0.696307	−	0.717744i	$-0.745175\pi$
$239$		−	4.68372e9i		−	1.43549i	0.696307	−	0.717744i	$-0.254825\pi$
$240$		0			0
$241$	−2.42738e9	−	4.20435e9i	−0.719565	−	1.24632i	−0.961172	−	0.275949i	$-0.911008\pi$
$241$	−2.42738e9	−	4.20435e9i	−0.719565	−	1.24632i	0.241607	−	0.970374i	$-0.422325\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.44110e9	−	4.16153e9i	−0.677520	−	1.15502i
$246$		0			0
$247$	6.10804e9	−	1.05794e10i	1.64102	−	2.84233i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	1.48552e9i		−	0.374269i	−0.982334	−	0.187134i	$-0.940080\pi$
$251$		−	1.48552e9i		−	0.374269i	0.982334	−	0.187134i	$-0.0599200\pi$
$252$		0			0
$253$	−7.76873e8			−0.189613
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.04194e9	+	3.48832e9i	1.38498	+	0.799619i	0.992744	−	0.120245i	$-0.0383679\pi$
$257$	6.04194e9	+	3.48832e9i	1.38498	+	0.799619i	0.392237	+	0.919864i	$0.371701\pi$
$258$		0			0
$259$	2.28334e9	+	2.26764e9i	0.507424	+	0.503935i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.32456e9	+	4.22883e9i	−1.53094	+	0.883889i	−0.531622	+	0.846982i	$0.678417\pi$
$263$	−7.32456e9	+	4.22883e9i	−1.53094	+	0.883889i	−0.999319	+	0.0369072i	$0.988249\pi$
$264$		0			0
$265$	−4.82669e9			−0.978736
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.46192e9	−	2.57609e9i	0.852142	−	0.491985i	−0.00923072	−	0.999957i	$-0.502938\pi$
$269$	4.46192e9	−	2.57609e9i	0.852142	−	0.491985i	0.861373	+	0.507973i	$0.169605\pi$
$270$		0			0
$271$	2.08632e9	−	3.61361e9i	0.386815	−	0.669984i	−0.605204	−	0.796071i	$-0.706908\pi$
$271$	2.08632e9	−	3.61361e9i	0.386815	−	0.669984i	0.992019	+	0.126087i	$0.0402417\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.66276e9	+	3.26939e9i	0.990141	+	0.571658i
$276$		0			0
$277$	−8.85814e8	−	1.53428e9i	−0.150461	−	0.260606i	0.780936	−	0.624611i	$-0.214742\pi$
$277$	−8.85814e8	−	1.53428e9i	−0.150461	−	0.260606i	−0.931397	+	0.364005i	$0.881409\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1.30429e9i			0.209193i	0.994515	+	0.104597i	$0.0333552\pi$
$281$			1.30429e9i			0.209193i	−0.994515	+	0.104597i	$0.966645\pi$
$282$		0			0
$283$	−3.55119e9	−	6.15084e9i	−0.553641	−	0.958934i	−0.998008	−	0.0630890i	$-0.979905\pi$
$283$	−3.55119e9	−	6.15084e9i	−0.553641	−	0.958934i	0.444367	−	0.895845i	$-0.353429\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.08693e9	+	1.07999e9i	0.602379	+	0.159181i
$288$		0			0
$289$	−1.30254e9	+	2.25607e9i	−0.186725	+	0.323416i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	4.34050e9i		−	0.588938i	−0.955661	−	0.294469i	$-0.904857\pi$
$293$		−	4.34050e9i		−	0.588938i	0.955661	−	0.294469i	$-0.0951428\pi$
$294$		0			0
$295$	−4.10844e9			−0.542486
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.56107e9	+	9.01286e8i	0.195316	+	0.112766i
$300$		0			0
$301$	−1.65752e9	+	6.27243e9i	−0.201926	+	0.764134i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.32978e9	−	3.65450e9i	0.731458	−	0.422307i
$306$		0			0
$307$	−4.58360e9			−0.516004			−0.258002	−	0.966144i	$-0.583064\pi$
$307$	−4.58360e9			−0.516004			−0.258002	+	0.966144i	$0.583064\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.28259e10	−	7.40504e9i	1.37103	−	0.791564i	0.379971	−	0.924998i	$-0.375934\pi$
$311$	1.28259e10	−	7.40504e9i	1.37103	−	0.791564i	0.991058	+	0.133435i	$0.0426006\pi$
$312$		0			0
$313$	−3.92619e9	+	6.80037e9i	−0.409067	+	0.708525i	−0.994785	−	0.101990i	$-0.967479\pi$
$313$	−3.92619e9	+	6.80037e9i	−0.409067	+	0.708525i	0.585719	+	0.810514i	$0.300812\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.40674e9	−	2.54423e9i	−0.436396	−	0.251953i	0.265672	−	0.964064i	$-0.414406\pi$
$317$	−4.40674e9	−	2.54423e9i	−0.436396	−	0.251953i	−0.702068	+	0.712110i	$0.747740\pi$
$318$		0			0
$319$	1.68831e9	+	2.92424e9i	0.163038	+	0.282390i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1.64912e10i		−	1.51510i
$324$		0			0
$325$	−7.58594e9	−	1.31392e10i	−0.679948	−	1.17771i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.43002e10	−	1.43992e10i	1.22056	−	1.22901i
$330$		0			0
$331$	4.70894e9	−	8.15612e9i	0.392294	−	0.679472i	−0.600458	−	0.799656i	$-0.705015\pi$
$331$	4.70894e9	−	8.15612e9i	0.392294	−	0.679472i	0.992752	+	0.120184i	$0.0383484\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	2.12916e9i		−	0.169056i
$336$		0			0
$337$	−1.01405e10			−0.786216			−0.393108	−	0.919492i	$-0.628600\pi$
$337$	−1.01405e10			−0.786216			−0.393108	+	0.919492i	$0.628600\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.67580e10	+	9.67525e9i	1.23938	+	0.715558i
$342$		0			0
$343$	−9.68543e9	+	9.88805e9i	−0.699749	+	0.714388i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.83864e9	−	4.52564e9i	0.540658	−	0.312149i	−0.204688	−	0.978827i	$-0.565618\pi$
$347$	7.83864e9	−	4.52564e9i	0.540658	−	0.312149i	0.745346	+	0.666678i	$0.232284\pi$
$348$		0			0
$349$	2.27218e10			1.53159			0.765794	−	0.643086i	$-0.222346\pi$
$349$	2.27218e10			1.53159			0.765794	+	0.643086i	$0.222346\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.37058e10	+	7.91307e9i	−0.882687	+	0.509620i	−0.871543	−	0.490318i	$-0.836881\pi$
$353$	−1.37058e10	+	7.91307e9i	−0.882687	+	0.509620i	−0.0111437	+	0.999938i	$0.503547\pi$
$354$		0			0
$355$	−3.60978e9	+	6.25232e9i	−0.227283	+	0.393666i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.05475e10	−	6.08960e9i	−0.634997	−	0.366616i	0.147688	−	0.989034i	$-0.452817\pi$
$359$	−1.05475e10	−	6.08960e9i	−0.634997	−	0.366616i	−0.782685	+	0.622418i	$0.786150\pi$
$360$		0			0
$361$	−2.26202e10	−	3.91793e10i	−1.33189	−	2.30690i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			3.66378e9i			0.206423i
$366$		0			0
$367$	6.23886e9	+	1.08060e10i	0.343907	+	0.595664i	0.985155	−	0.171670i	$-0.0549162\pi$
$367$	6.23886e9	+	1.08060e10i	0.343907	+	0.595664i	−0.641248	+	0.767334i	$0.721583\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.63003e9	+	1.33628e10i	0.191608	+	0.705348i
$372$		0			0
$373$	−4.61759e9	+	7.99790e9i	−0.238550	+	0.413181i	−0.960299	−	0.278974i	$-0.910006\pi$
$373$	−4.61759e9	+	7.99790e9i	−0.238550	+	0.413181i	0.721748	+	0.692156i	$0.243339\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	7.83473e9i		−	0.387845i
$378$		0			0
$379$	−8.45607e9			−0.409838			−0.204919	−	0.978779i	$-0.565693\pi$
$379$	−8.45607e9			−0.409838			−0.204919	+	0.978779i	$0.565693\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.65796e10	−	1.53458e10i	−1.23525	−	0.713170i	−0.267128	−	0.963661i	$-0.586075\pi$
$383$	−2.65796e10	−	1.53458e10i	−1.23525	−	0.713170i	−0.968119	+	0.250491i	$0.919408\pi$
$384$		0			0
$385$	1.08355e10	−	4.10041e10i	0.493182	−	1.86631i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.61072e10	−	2.08465e10i	1.57687	−	0.910406i	0.581578	−	0.813491i	$-0.302436\pi$
$389$	3.61072e10	−	2.08465e10i	1.57687	−	0.910406i	0.995293	−	0.0969156i	$-0.0308977\pi$
$390$		0			0
$391$	2.43340e9			0.104113
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.41939e10	−	1.39684e10i	0.993844	−	0.573796i
$396$		0			0
$397$	1.55350e10	−	2.69073e10i	0.625386	−	1.08320i	−0.363081	−	0.931758i	$-0.618275\pi$
$397$	1.55350e10	−	2.69073e10i	0.625386	−	1.08320i	0.988466	−	0.151442i	$-0.0483916\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.39090e10	+	8.03037e9i	0.537921	+	0.310569i	0.744236	−	0.667917i	$-0.232814\pi$
$401$	1.39090e10	+	8.03037e9i	0.537921	+	0.310569i	−0.206315	+	0.978486i	$0.566147\pi$
$402$		0			0
$403$	−2.24494e10	−	3.88835e10i	−0.851107	−	1.47416i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.82886e10i			1.03094i
$408$		0			0
$409$	2.40215e9	+	4.16065e9i	0.0858434	+	0.148685i	0.905750	−	0.423812i	$-0.139308\pi$
$409$	2.40215e9	+	4.16065e9i	0.0858434	+	0.148685i	−0.819907	+	0.572497i	$0.805975\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.08985e9	+	1.13743e10i	0.106203	+	0.390955i
$414$		0			0
$415$	−1.64200e10	+	2.84403e10i	−0.553580	+	0.958829i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	4.03712e10i		−	1.30983i	−0.755702	−	0.654915i	$-0.772704\pi$
$419$		−	4.03712e10i		−	1.30983i	0.755702	−	0.654915i	$-0.227296\pi$
$420$		0			0
$421$	4.59449e10			1.46254			0.731271	−	0.682087i	$-0.238927\pi$
$421$	4.59449e10			1.46254			0.731271	+	0.682087i	$0.238927\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.77374e10	−	1.02407e10i	−0.543670	−	0.313888i
$426$		0			0
$427$	−1.48781e10	−	1.47757e10i	−0.447543	−	0.444465i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.51461e10	+	2.02916e10i	−1.01852	+	0.588041i	−0.913674	−	0.406448i	$-0.866767\pi$
$431$	−3.51461e10	+	2.02916e10i	−1.01852	+	0.588041i	−0.104843	+	0.994489i	$0.533434\pi$
$432$		0			0
$433$	1.67077e10			0.475297			0.237648	−	0.971351i	$-0.423623\pi$
$433$	1.67077e10			0.475297			0.237648	+	0.971351i	$0.423623\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.95149e9	−	4.59080e9i	0.218033	−	0.125882i
$438$		0			0
$439$	−2.49785e10	+	4.32640e10i	−0.672524	+	1.16485i	0.304661	+	0.952461i	$0.401457\pi$
$439$	−2.49785e10	+	4.32640e10i	−0.672524	+	1.16485i	−0.977186	+	0.212386i	$0.931877\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.84406e9	−	4.52877e9i	−0.203670	−	0.117589i	0.394696	−	0.918812i	$-0.370850\pi$
$443$	−7.84406e9	−	4.52877e9i	−0.203670	−	0.117589i	−0.598366	+	0.801223i	$0.704183\pi$
$444$		0			0
$445$	4.39105e10	+	7.60552e10i	1.11977	+	1.93949i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1.89084e10i		−	0.465231i	−0.972569	−	0.232615i	$-0.925272\pi$
$449$		−	1.89084e10i		−	0.465231i	0.972569	−	0.232615i	$-0.0747284\pi$
$450$		0			0
$451$	1.85799e10	+	3.21814e10i	0.449094	+	0.777854i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.93439e10	+	6.98241e10i	−1.61794	+	1.62915i
$456$		0			0
$457$	−3.46303e10	+	5.99814e10i	−0.793946	+	1.37516i	0.129560	+	0.991572i	$0.458644\pi$
$457$	−3.46303e10	+	5.99814e10i	−0.793946	+	1.37516i	−0.923506	+	0.383584i	$0.874690\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			2.87667e10i			0.636922i	0.947936	+	0.318461i	$0.103166\pi$
$461$			2.87667e10i			0.636922i	−0.947936	+	0.318461i	$0.896834\pi$
$462$		0			0
$463$	−1.38852e10			−0.302154			−0.151077	−	0.988522i	$-0.548274\pi$
$463$	−1.38852e10			−0.302154			−0.151077	+	0.988522i	$0.548274\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.31919e10	−	7.61637e9i	−0.277358	−	0.160133i	0.354869	−	0.934916i	$-0.384526\pi$
$467$	−1.31919e10	−	7.61637e9i	−0.277358	−	0.160133i	−0.632227	+	0.774783i	$0.717859\pi$
$468$		0			0
$469$	−5.89466e9	+	1.60129e9i	−0.121834	+	0.0330962i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.93904e10	+	2.85156e10i	−0.986729	+	0.569688i
$474$		0			0
$475$	−7.72796e10			−1.51807
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.30795e10	−	2.48720e10i	0.818331	−	0.472464i	−0.0315095	−	0.999503i	$-0.510031\pi$
$479$	4.30795e10	−	2.48720e10i	0.818331	−	0.472464i	0.849841	+	0.527040i	$0.176698\pi$
$480$		0			0
$481$	3.28188e10	−	5.68439e10i	0.613116	−	1.06195i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.17214e11	+	6.76738e10i	2.11843	+	1.22308i
$486$		0			0
$487$	3.22486e10	+	5.58563e10i	0.573318	+	0.993016i	0.996222	+	0.0868413i	$0.0276773\pi$
$487$	3.22486e10	+	5.58563e10i	0.573318	+	0.993016i	−0.422904	+	0.906174i	$0.638989\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	7.21476e9i		−	0.124135i	−0.998072	−	0.0620677i	$-0.980231\pi$
$491$		−	7.21476e9i		−	0.124135i	0.998072	−	0.0620677i	$-0.0197695\pi$
$492$		0			0
$493$	−5.28828e9	−	9.15957e9i	−0.0895214	−	0.155056i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.00246e10	+	5.29158e9i	0.328199	+	0.0867281i
$498$		0			0
$499$	−4.75926e9	+	8.24328e9i	−0.0767604	+	0.132953i	−0.901850	−	0.432048i	$-0.857791\pi$
$499$	−4.75926e9	+	8.24328e9i	−0.0767604	+	0.132953i	0.825090	+	0.565001i	$0.191124\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			2.93889e10i			0.459104i	0.973296	+	0.229552i	$0.0737260\pi$
$503$			2.93889e10i			0.459104i	−0.973296	+	0.229552i	$0.926274\pi$
$504$		0			0
$505$	−2.38194e9			−0.0366239
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.43159e10	−	1.40388e10i	−0.362259	−	0.209150i	0.307812	−	0.951447i	$-0.400403\pi$
$509$	−2.43159e10	−	1.40388e10i	−0.362259	−	0.209150i	−0.670071	+	0.742297i	$0.733736\pi$
$510$		0			0
$511$	1.01433e10	−	2.75543e9i	0.148763	−	0.0404116i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.02018e11	+	5.89001e10i	−1.45027	+	0.837312i
$516$		0			0
$517$	1.78394e11			2.49699
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8.67202e10	+	5.00680e10i	−1.17698	+	0.679530i	−0.955314	−	0.295592i	$-0.904483\pi$
$521$	−8.67202e10	+	5.00680e10i	−1.17698	+	0.679530i	−0.221667	+	0.975122i	$0.571150\pi$
$522$		0			0
$523$	3.91195e10	−	6.77569e10i	0.522861	−	0.905622i	−0.476785	−	0.879020i	$-0.658198\pi$
$523$	3.91195e10	−	6.77569e10i	0.522861	−	0.905622i	0.999646	−	0.0266019i	$-0.00846865\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.24911e10	−	3.03058e10i	−0.680524	−	0.392901i
$528$		0			0
$529$	−3.84781e10	−	6.66460e10i	−0.491350	−	0.851043i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	8.62216e10i		−	1.06833i
$534$		0			0
$535$	5.25052e10	+	9.09417e10i	0.640896	+	1.11006i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.21670e11	+	8.39655e8i	−1.44155	+	0.00994823i
$540$		0			0
$541$	−4.48762e10	+	7.77278e10i	−0.523874	+	0.907376i	0.475740	+	0.879586i	$0.342180\pi$
$541$	−4.48762e10	+	7.77278e10i	−0.523874	+	0.907376i	−0.999614	+	0.0277901i	$0.991153\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1.37729e11i			1.56113i
$546$		0			0
$547$	1.66942e11			1.86473			0.932366	−	0.361516i	$-0.117741\pi$
$547$	1.66942e11			1.86473			0.932366	+	0.361516i	$0.117741\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.45605e10	−	1.99535e10i	−0.374950	−	0.216478i
$552$		0			0
$553$	−5.68675e10	−	5.64764e10i	−0.608084	−	0.603902i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7.62662e10	+	4.40323e10i	−0.792340	+	0.457458i	−0.840786	−	0.541368i	$-0.817907\pi$
$557$	−7.62662e10	+	4.40323e10i	−0.792340	+	0.457458i	0.0484458	+	0.998826i	$0.484573\pi$
$558$		0			0
$559$	1.32329e11			1.35521
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.58868e10	+	1.49458e10i	−0.257659	+	0.148759i	−0.623266	−	0.782010i	$-0.714195\pi$
$563$	−2.58868e10	+	1.49458e10i	−0.257659	+	0.148759i	0.365607	+	0.930769i	$0.380861\pi$
$564$		0			0
$565$	−4.72413e8	+	8.18244e8i	−0.00463584	+	0.00802951i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.42108e10	−	2.55251e10i	−0.421774	−	0.243511i	0.274062	−	0.961712i	$-0.411633\pi$
$569$	−4.42108e10	−	2.55251e10i	−0.421774	−	0.243511i	−0.695836	+	0.718201i	$0.744966\pi$
$570$		0			0
$571$	4.00852e10	+	6.94296e10i	0.377085	+	0.653131i	0.990637	−	0.136524i	$-0.0435930\pi$
$571$	4.00852e10	+	6.94296e10i	0.377085	+	0.653131i	−0.613551	+	0.789655i	$0.710260\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1.14032e10i		−	0.104317i
$576$		0			0
$577$	−1.03022e11	−	1.78440e11i	−0.929453	−	1.60986i	−0.784238	−	0.620460i	$-0.786946\pi$
$577$	−1.03022e11	−	1.78440e11i	−0.929453	−	1.60986i	−0.145215	−	0.989400i	$-0.546387\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	9.10868e10	+	2.40701e10i	0.799376	+	0.211239i
$582$		0			0
$583$	−6.08623e10	+	1.05417e11i	−0.526834	+	0.912504i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			8.93891e10i			0.752891i	0.926439	+	0.376446i	$0.122854\pi$
$587$			8.93891e10i			0.752891i	−0.926439	+	0.376446i	$0.877146\pi$
$588$		0			0
$589$	−2.28697e11			−1.90020
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.30044e11	+	7.50812e10i	1.05165	+	0.607173i	0.923112	−	0.384531i	$-0.125637\pi$
$593$	1.30044e11	+	7.50812e10i	1.05165	+	0.607173i	0.128542	+	0.991704i	$0.458970\pi$
$594$		0			0
$595$	−3.39401e10	+	1.28437e11i	−0.270798	+	1.02476i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.89741e10	+	5.13692e10i	−0.691124	+	0.399021i	−0.804033	−	0.594585i	$-0.797317\pi$
$599$	−8.89741e10	+	5.13692e10i	−0.691124	+	0.399021i	0.112909	+	0.993605i	$0.463983\pi$
$600$		0			0
$601$	−9.05848e9			−0.0694316			−0.0347158	−	0.999397i	$-0.511053\pi$
$601$	−9.05848e9			−0.0694316			−0.0347158	+	0.999397i	$0.511053\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.67510e11	−	9.67118e10i	1.25031	−	0.721868i
$606$		0			0
$607$	−8.29946e10	+	1.43751e11i	−0.611357	+	1.05890i	0.379654	+	0.925128i	$0.376043\pi$
$607$	−8.29946e10	+	1.43751e11i	−0.611357	+	1.05890i	−0.991012	+	0.133774i	$0.957290\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.58469e11	−	2.06962e11i	−2.57210	−	1.48500i
$612$		0			0
$613$	1.93003e10	+	3.34291e10i	0.136685	+	0.236746i	0.926240	−	0.376934i	$-0.123022\pi$
$613$	1.93003e10	+	3.34291e10i	0.136685	+	0.236746i	−0.789555	+	0.613680i	$0.789688\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			2.21113e11i			1.52571i	0.646568	+	0.762856i	$0.276204\pi$
$617$			2.21113e11i			1.52571i	−0.646568	+	0.762856i	$0.723796\pi$
$618$		0			0
$619$	6.39023e10	+	1.10682e11i	0.435265	+	0.753902i	0.997317	−	0.0732002i	$-0.0233212\pi$
$619$	6.39023e10	+	1.10682e11i	0.435265	+	0.753902i	−0.562052	+	0.827102i	$0.689988\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.77537e11	−	1.78767e11i	1.17852	−	1.18668i
$624$		0			0
$625$	8.88133e10	−	1.53829e11i	0.582047	−	1.00814i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	8.86081e10i		−	0.566071i
$630$		0			0
$631$	−1.26478e11			−0.797806			−0.398903	−	0.916993i	$-0.630609\pi$
$631$	−1.26478e11			−0.797806			−0.398903	+	0.916993i	$0.630609\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.59095e8	−	9.18537e7i	−0.000978504	−	0.000564939i
$636$		0			0
$637$	2.45462e11	+	1.39468e11i	1.49083	+	0.847066i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.09854e11	−	1.21159e11i	1.24304	−	0.717669i	0.273328	−	0.961921i	$-0.411876\pi$
$641$	2.09854e11	−	1.21159e11i	1.24304	−	0.717669i	0.969712	+	0.244252i	$0.0785423\pi$
$642$		0			0
$643$	1.70210e11			0.995730			0.497865	−	0.867255i	$-0.334117\pi$
$643$	1.70210e11			0.995730			0.497865	+	0.867255i	$0.334117\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5.28533e10	+	3.05149e10i	−0.301617	+	0.174138i	−0.643169	−	0.765724i	$-0.722381\pi$
$647$	−5.28533e10	+	3.05149e10i	−0.301617	+	0.174138i	0.341552	+	0.939863i	$0.389047\pi$
$648$		0			0
$649$	−5.18055e10	+	8.97298e10i	−0.292010	+	0.505775i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.38832e11	+	8.01548e10i	0.763550	+	0.440836i	0.830569	−	0.556916i	$-0.188016\pi$
$653$	1.38832e11	+	8.01548e10i	0.763550	+	0.440836i	−0.0670191	+	0.997752i	$0.521349\pi$
$654$		0			0
$655$	−8.78499e10	−	1.52161e11i	−0.477283	−	0.826679i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1.13989e11i			0.604393i	0.953246	+	0.302197i	$0.0977199\pi$
$659$			1.13989e11i			0.604393i	−0.953246	+	0.302197i	$0.902280\pi$
$660$		0			0
$661$	−5.48808e10	−	9.50563e10i	−0.287484	−	0.497938i	0.685724	−	0.727861i	$-0.259486\pi$
$661$	−5.48808e10	−	9.50563e10i	−0.287484	−	0.497938i	−0.973209	+	0.229924i	$0.926152\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.31402e11	+	4.83718e11i	0.671919	+	2.47347i
$666$		0			0
$667$	2.94429e9	−	5.09965e9i	0.0148757	−	0.0257654i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1.84326e11i		−	0.909278i
$672$		0			0
$673$	3.43979e11			1.67676			0.838382	−	0.545083i	$-0.183502\pi$
$673$	3.43979e11			1.67676			0.838382	+	0.545083i	$0.183502\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.16116e9	+	1.82510e9i	0.0150484	+	0.00868823i	0.507505	−	0.861649i	$-0.330568\pi$
$677$	3.16116e9	+	1.82510e9i	0.0150484	+	0.00868823i	−0.492457	+	0.870337i	$0.663901\pi$
$678$		0			0
$679$	9.92032e10	−	3.75408e11i	0.466709	−	1.76614i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	2.03296e11	−	1.17373e11i	0.934213	−	0.539368i	0.0460716	−	0.998938i	$-0.485330\pi$
$683$	2.03296e11	−	1.17373e11i	0.934213	−	0.539368i	0.888142	+	0.459570i	$0.151996\pi$
$684$		0			0
$685$	−4.90710e11			−2.22876
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.44597e11	−	1.41218e11i	1.08536	−	0.626633i
$690$		0			0
$691$	−1.53270e11	+	2.65472e11i	−0.672273	+	1.16441i	0.304985	+	0.952357i	$0.401348\pi$
$691$	−1.53270e11	+	2.65472e11i	−0.672273	+	1.16441i	−0.977258	+	0.212053i	$0.931985\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.82082e11	−	1.05125e11i	−0.780417	−	0.450574i
$696$		0			0
$697$	−5.81978e10	−	1.00802e11i	−0.246590	−	0.427107i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	4.37987e11i		−	1.81380i	−0.421347	−	0.906900i	$-0.638443\pi$
$701$		−	4.37987e11i		−	1.81380i	0.421347	−	0.906900i	$-0.361557\pi$
$702$		0			0
$703$	−1.67166e11	−	2.89540e11i	−0.684427	−	1.18546i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.79140e9	+	6.59448e9i	0.00716991	+	0.0263938i
$708$		0			0
$709$	2.20872e11	−	3.82561e11i	0.874089	−	1.51397i	0.0163580	−	0.999866i	$-0.494793\pi$
$709$	2.20872e11	−	3.82561e11i	0.874089	−	1.51397i	0.857731	−	0.514100i	$-0.171874\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	3.37459e10i		−	0.130576i
$714$		0			0
$715$	−8.65060e11			−3.30995
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.51748e11	−	8.76116e10i	−0.567815	−	0.327828i	0.188461	−	0.982081i	$-0.439650\pi$
$719$	−1.51748e11	−	8.76116e10i	−0.567815	−	0.327828i	−0.756276	+	0.654253i	$0.772983\pi$
$720$		0			0
$721$	2.39792e11	+	2.38143e11i	0.887348	+	0.881245i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.29228e10	+	2.47815e10i	−0.155359	+	0.0896964i
$726$		0			0
$727$	−3.98686e10			−0.142723			−0.0713614	−	0.997451i	$-0.522734\pi$
$727$	−3.98686e10			−0.142723			−0.0713614	+	0.997451i	$0.522734\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.54705e11	−	8.93192e10i	0.541796	−	0.312806i
$732$		0			0
$733$	1.30870e11	−	2.26673e11i	0.453339	−	0.785206i	−0.545252	−	0.838272i	$-0.683566\pi$
$733$	1.30870e11	−	2.26673e11i	0.453339	−	0.785206i	0.998591	+	0.0530658i	$0.0168993\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.65017e10	−	2.68478e10i	−0.157615	−	0.0909993i
$738$		0			0
$739$	1.42907e11	+	2.47523e11i	0.479155	+	0.829921i	0.999714	−	0.0239043i	$-0.00760970\pi$
$739$	1.42907e11	+	2.47523e11i	0.479155	+	0.829921i	−0.520559	+	0.853826i	$0.674276\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	3.26170e11i		−	1.07026i	−0.844770	−	0.535130i	$-0.820263\pi$
$743$		−	3.26170e11i		−	1.07026i	0.844770	−	0.535130i	$-0.179737\pi$
$744$		0			0
$745$	−3.82204e11	−	6.61998e11i	−1.24071	−	2.14897i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.12287e11	−	2.13757e11i	0.674523	−	0.679194i
$750$		0			0
$751$	1.19607e11	−	2.07165e11i	0.376007	−	0.651263i	−0.614470	−	0.788940i	$-0.710630\pi$
$751$	1.19607e11	−	2.07165e11i	0.376007	−	0.651263i	0.990477	+	0.137677i	$0.0439636\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			3.65348e11i			1.12440i
$756$		0			0
$757$	5.21959e11			1.58947			0.794736	−	0.606955i	$-0.207609\pi$
$757$	5.21959e11			1.58947			0.794736	+	0.606955i	$0.207609\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2.55548e11	−	1.47541e11i	−0.761963	−	0.439920i	0.0680370	−	0.997683i	$-0.478326\pi$
$761$	−2.55548e11	−	1.47541e11i	−0.761963	−	0.439920i	−0.830000	+	0.557763i	$0.811660\pi$
$762$		0			0
$763$	3.81306e11	−	1.03582e11i	1.12506	−	0.305624i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.08199e11	−	1.20204e11i	0.601585	−	0.347325i
$768$		0			0
$769$	−2.58950e11			−0.740475			−0.370237	−	0.928937i	$-0.620724\pi$
$769$	−2.58950e11			−0.740475			−0.370237	+	0.928937i	$0.620724\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.21275e11	−	3.00958e11i	1.45999	−	0.842924i	0.460978	−	0.887412i	$-0.347499\pi$
$773$	5.21275e11	−	3.00958e11i	1.45999	−	0.842924i	0.999010	+	0.0444875i	$0.0141655\pi$
$774$		0			0
$775$	−1.42016e11	+	2.45979e11i	−0.393669	+	0.681854i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.80340e11	−	2.19590e11i	−1.03281	−	0.596296i
$780$		0			0
$781$	9.10352e10	+	1.57678e11i	0.244684	+	0.423805i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			2.17299e9i			0.00572240i
$786$		0			0
$787$	1.00655e10	+	1.74339e10i	0.0262382	+	0.0454460i	0.878846	−	0.477105i	$-0.158314\pi$
$787$	1.00655e10	+	1.74339e10i	0.0262382	+	0.0454460i	−0.852608	+	0.522551i	$0.824980\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.62063e9	+	6.92512e8i	0.00669421	+	0.00176897i
$792$		0			0
$793$	−2.13845e11	+	3.70390e11i	−0.540762	+	0.936628i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			5.96631e11i			1.47868i	0.673335	+	0.739338i	$0.264861\pi$
$797$			5.96631e11i			1.47868i	−0.673335	+	0.739338i	$0.735139\pi$
$798$		0			0
$799$	−5.58782e11			−1.37106
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8.00182e10	+	4.61985e10i	0.192454	+	0.111113i
$804$		0			0
$805$	−7.13761e10	+	1.93894e10i	−0.169969	+	0.0461722i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.39044e11	−	3.11217e11i	1.25843	−	0.726557i	0.285664	−	0.958330i	$-0.407786\pi$
$809$	5.39044e11	−	3.11217e11i	1.25843	−	0.726557i	0.972770	+	0.231773i	$0.0744527\pi$
$810$		0			0
$811$	−4.09734e11			−0.947151			−0.473575	−	0.880753i	$-0.657037\pi$
$811$	−4.09734e11			−0.947151			−0.473575	+	0.880753i	$0.657037\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.96922e11	−	1.71428e11i	0.672994	−	0.388553i
$816$		0			0
$817$	3.37015e11	−	5.83728e11i	0.756417	−	1.31015i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.95045e10	−	3.43549e10i	−0.130972	−	0.0756165i	0.433083	−	0.901354i	$-0.357426\pi$
$821$	−5.95045e10	−	3.43549e10i	−0.130972	−	0.0756165i	−0.564054	+	0.825738i	$0.690759\pi$
$822$		0			0
$823$	3.63746e11	+	6.30026e11i	0.792864	+	1.37328i	0.924187	+	0.381941i	$0.124744\pi$
$823$	3.63746e11	+	6.30026e11i	0.792864	+	1.37328i	−0.131323	+	0.991340i	$0.541922\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	2.93292e11i		−	0.627016i	−0.949586	−	0.313508i	$-0.898496\pi$
$827$		−	2.93292e11i		−	0.627016i	0.949586	−	0.313508i	$-0.101504\pi$
$828$		0			0
$829$	−3.61801e11	−	6.26659e11i	−0.766041	−	1.32682i	−0.939694	−	0.342016i	$-0.888890\pi$
$829$	−3.61801e11	−	6.26659e11i	−0.766041	−	1.32682i	0.173653	−	0.984807i	$-0.444443\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.81108e11	−	2.63005e9i	0.791531	−	0.00546240i
$834$		0			0
$835$	−4.56700e11	+	7.91028e11i	−0.939476	+	1.62722i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			9.46491e11i			1.91015i	0.296356	+	0.955077i	$0.404228\pi$
$839$			9.46491e11i			1.91015i	−0.296356	+	0.955077i	$0.595772\pi$
$840$		0			0
$841$	4.74652e11			0.948837
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.14704e12	+	6.62245e11i	2.24985	+	1.29895i
$846$		0			0
$847$	−3.93730e11	−	3.91022e11i	−0.765005	−	0.759744i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.27238e10	−	2.46666e10i	0.0814613	−	0.0470317i
$852$		0			0
$853$	1.74752e11			0.330086			0.165043	−	0.986286i	$-0.447224\pi$
$853$	1.74752e11			0.330086			0.165043	+	0.986286i	$0.447224\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5.00768e11	−	2.89118e11i	0.928353	−	0.535985i	0.0420626	−	0.999115i	$-0.486607\pi$
$857$	5.00768e11	−	2.89118e11i	0.928353	−	0.535985i	0.886290	+	0.463130i	$0.153274\pi$
$858$		0			0
$859$	−1.02744e10	+	1.77957e10i	−0.0188704	+	0.0326846i	−0.875306	−	0.483569i	$-0.839340\pi$
$859$	−1.02744e10	+	1.77957e10i	−0.0188704	+	0.0326846i	0.856436	+	0.516253i	$0.172674\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−5.06769e11	−	2.92583e11i	−0.913623	−	0.527481i	−0.0320281	−	0.999487i	$-0.510197\pi$
$863$	−5.06769e11	−	2.92583e11i	−0.913623	−	0.527481i	−0.881595	+	0.472006i	$0.843530\pi$
$864$		0			0
$865$	−1.23714e11	−	2.14280e11i	−0.220981	−	0.382751i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	7.04539e11i		−	1.23545i
$870$		0			0
$871$	6.22946e10	+	1.07897e11i	0.108237	+	0.187473i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.57016e11	−	4.14922e10i	−0.267862	−	0.0707838i
$876$		0			0
$877$	−5.83203e10	+	1.01014e11i	−0.0985874	+	0.170758i	−0.911100	−	0.412185i	$-0.864766\pi$
$877$	−5.83203e10	+	1.01014e11i	−0.0985874	+	0.170758i	0.812513	+	0.582943i	$0.198099\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			3.98286e11i			0.661138i	0.943782	+	0.330569i	$0.107241\pi$
$881$			3.98286e11i			0.661138i	−0.943782	+	0.330569i	$0.892759\pi$
$882$		0			0
$883$	−5.18954e11			−0.853663			−0.426831	−	0.904331i	$-0.640370\pi$
$883$	−5.18954e11			−0.853663			−0.426831	+	0.904331i	$0.640370\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4.34306e11	−	2.50747e11i	−0.701620	−	0.405080i	0.106331	−	0.994331i	$-0.466090\pi$
$887$	−4.34306e11	−	2.50747e11i	−0.701620	−	0.405080i	−0.807950	+	0.589251i	$0.799423\pi$
$888$		0			0
$889$	−1.34649e8	+	5.09541e8i	−0.000215573	+	0.000815779i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.82590e12	+	1.05419e12i	−2.87126	+	1.65772i
$894$		0			0
$895$	1.42800e12			2.22555
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.27023e11	+	7.33368e10i	−0.194466	+	0.112275i
$900$		0			0
$901$	1.90639e11	−	3.30196e11i	0.289276	−	0.501040i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.79106e11	−	2.76612e11i	−0.714229	−	0.412361i
$906$		0			0
$907$	4.02133e11	+	6.96515e11i	0.594211	+	1.02920i	0.993658	+	0.112447i	$0.0358688\pi$
$907$	4.02133e11	+	6.96515e11i	0.594211	+	1.02920i	−0.399447	+	0.916756i	$0.630798\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			3.63450e11i			0.527681i	0.964566	+	0.263841i	$0.0849893\pi$
$911$			3.63450e11i			0.527681i	−0.964566	+	0.263841i	$0.915011\pi$
$912$		0			0
$913$	4.14097e11	+	7.17237e11i	0.595962	+	1.03224i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.55192e11	+	3.57652e11i	−0.502326	+	0.505805i
$918$		0			0
$919$	−2.39421e11	+	4.14689e11i	−0.335660	+	0.581380i	−0.983611	−	0.180302i	$-0.942293\pi$
$919$	−2.39421e11	+	4.14689e11i	−0.335660	+	0.581380i	0.647952	+	0.761682i	$0.275626\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	4.22456e11i		−	0.582070i
$924$		0			0
$925$	−4.15227e11			−0.567178
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.20034e12	−	6.93019e11i	−1.61155	−	0.930427i	−0.989012	−	0.147835i	$-0.952770\pi$
$929$	−1.20034e12	−	6.93019e11i	−1.61155	−	0.930427i	−0.622535	−	0.782592i	$-0.713897\pi$
$930$		0			0
$931$	1.24037e12	−	7.27584e11i	1.65101	−	0.968466i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.01134e12	+	5.83898e11i	−1.32328	+	0.763995i
$936$		0			0
$937$	1.23299e11			0.159957			0.0799784	−	0.996797i	$-0.474515\pi$
$937$	1.23299e11			0.159957			0.0799784	+	0.996797i	$0.474515\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.85157e11	−	3.37841e11i	0.746301	−	0.430877i	−0.0780549	−	0.996949i	$-0.524871\pi$
$941$	5.85157e11	−	3.37841e11i	0.746301	−	0.430877i	0.824356	+	0.566072i	$0.191538\pi$
$942$		0			0
$943$	3.24020e10	−	5.61219e10i	0.0409756	−	0.0709718i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.68682e11	−	5.59269e11i	−1.20443	−	0.695377i	−0.242892	−	0.970053i	$-0.578096\pi$
$947$	−9.68682e11	−	5.59269e11i	−1.20443	−	0.695377i	−0.961537	+	0.274676i	$0.911429\pi$
$948$		0			0
$949$	−1.07194e11	−	1.85665e11i	−0.132162	−	0.228911i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1.21309e12i		−	1.47069i	−0.677694	−	0.735344i	$-0.737021\pi$
$953$		−	1.21309e12i		−	1.47069i	0.677694	−	0.735344i	$-0.262979\pi$
$954$		0			0
$955$	−6.80023e11	−	1.17783e12i	−0.817542	−	1.41602i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.69051e11	+	1.35855e12i	0.436327	+	1.60620i
$960$		0			0
$961$	6.17129e9	−	1.06890e10i	0.00723573	−	0.0125326i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1.02409e12i			1.18094i
$966$		0			0
$967$	−4.18069e11			−0.478125			−0.239063	−	0.971004i	$-0.576840\pi$
$967$	−4.18069e11			−0.478125			−0.239063	+	0.971004i	$0.576840\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.68325e11	+	2.70388e11i	0.526830	+	0.304165i	0.739725	−	0.672910i	$-0.234956\pi$
$971$	4.68325e11	+	2.70388e11i	0.526830	+	0.304165i	−0.212895	+	0.977075i	$0.568289\pi$
$972$		0			0
$973$	−1.54103e11	+	5.83161e11i	−0.171933	+	0.650634i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.40623e11	−	1.38924e11i	0.264095	−	0.152475i	−0.362106	−	0.932137i	$-0.617942\pi$
$977$	2.40623e11	−	1.38924e11i	0.264095	−	0.152475i	0.626201	+	0.779662i	$0.284609\pi$
$978$		0			0
$979$	2.21476e12			2.41100
$980$		0			0
$981$		0			0
$982$		0			0
$983$	7.07349e11	−	4.08388e11i	0.757565	−	0.437380i	−0.0708561	−	0.997487i	$-0.522573\pi$
$983$	7.07349e11	−	4.08388e11i	0.757565	−	0.437380i	0.828421	+	0.560106i	$0.189240\pi$
$984$		0			0
$985$	−1.18635e12	+	2.05482e12i	−1.26028	+	2.18287i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.61332e10	+	4.97290e10i	0.0900296	+	0.0519786i
$990$		0			0
$991$	−3.77809e11	−	6.54384e11i	−0.391721	−	0.678481i	0.600955	−	0.799283i	$-0.294787\pi$
$991$	−3.77809e11	−	6.54384e11i	−0.391721	−	0.678481i	−0.992677	+	0.120801i	$0.961454\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			2.29398e12i			2.34044i
$996$		0			0
$997$	−2.76497e11	−	4.78908e11i	−0.279840	−	0.484698i	0.691504	−	0.722372i	$-0.256948\pi$
$997$	−2.76497e11	−	4.78908e11i	−0.279840	−	0.484698i	−0.971345	+	0.237674i	$0.923615\pi$
$998$		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.233.20	yes	44
3.2	odd	2	inner	252.9.bk.a.233.3	yes	44
7.4	even	3	inner	252.9.bk.a.53.3	✓	44
21.11	odd	6	inner	252.9.bk.a.53.20	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bk.a.53.3	✓	44	7.4	even	3	inner
252.9.bk.a.53.20	yes	44	21.11	odd	6	inner
252.9.bk.a.233.3	yes	44	3.2	odd	2	inner
252.9.bk.a.233.20	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)

\(f(q)\)	\(=\)	\(q+(724.791 + 418.458i) q^{5} +(613.419 - 2321.32i) q^{7} +O(q^{10})\)	\(q+(724.791 + 418.458i) q^{5} +(613.419 - 2321.32i) q^{7} +(18278.5 - 10553.1i) q^{11} -48972.6 q^{13} +(-57253.8 + 33055.5i) q^{17} +(-124724. + 216028. i) q^{19} +(-31876.4 - 18403.9i) q^{23} +(154902. + 268298. i) q^{25} +159982. i q^{29} +(458407. + 793984. i) q^{31} +(1.41597e6 - 1.42578e6i) q^{35} +(-670147. + 1.16073e6i) q^{37} +1.76061e6i q^{41} -2.70210e6 q^{43} +(7.31980e6 + 4.22609e6i) q^{47} +(-5.01224e6 - 2.84788e6i) q^{49} +(-4.99457e6 + 2.88361e6i) q^{53} +1.76642e7 q^{55} +(-4.25134e6 + 2.45451e6i) q^{59} +(4.36663e6 - 7.56322e6i) q^{61} +(-3.54949e7 - 2.04930e7i) q^{65} +(-1.27203e6 - 2.20322e6i) q^{67} +8.62638e6i q^{71} +(2.18886e6 + 3.79121e6i) q^{73} +(-1.32848e7 - 4.89038e7i) q^{77} +(1.66903e7 - 2.89084e7i) q^{79} +3.92393e7i q^{83} -5.53294e7 q^{85} +(9.08755e7 + 5.24670e7i) q^{89} +(-3.00407e7 + 1.13681e8i) q^{91} +(-1.80797e8 + 1.04383e8i) q^{95} +1.61722e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 1230 q^{7}+O(q^{10}) \) 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

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Database

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