LMFDB - Embedded newform 252.8.k.e.109.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,8,Mod(37,252)]

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, 0, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, 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:	$78.7210264220$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 89566 x^{14} - 207320 x^{13} + 5161603375 x^{12} - 17143558340 x^{11} + 178819626045814 x^{10} + \cdots + 34\!\cdots\!16 $ x^16 + 89566x^14 - 207320x^13 + 5161603375x^12 - 17143558340x^11 + 178819626045814x^10 - 993183560713460x^9 + 4531523101103784409x^8 - 25038421018919762700x^7 + 77467785793463920028824x^6 - 352418565495722251364640x^5 + 967025470523891143640719680x^4 - 1651641278575894088053178880x^3 + 7200793191683961412674394484736x^2 + 21939035854717669821876654243840x + 34494438285389383231614325978300416
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{13}\cdot 7^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.6
Root			$62.2223 + 107.772i$ of defining polynomial
Character	$\chi$	$=$	252.109
Dual form			252.8.k.e.37.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(135.624 + 234.907i) q^{5} +(87.0016 - 903.313i) q^{7} +(2753.41 - 4769.04i) q^{11} +8802.63 q^{13} +(-3420.94 + 5925.24i) q^{17} +(13231.5 + 22917.6i) q^{19} +(-41328.3 - 71582.8i) q^{23} +(2274.86 - 3940.17i) q^{25} -233335. q^{29} +(19823.7 - 34335.7i) q^{31} +(223994. - 102073. i) q^{35} +(-115873. - 200699. i) q^{37} +403925. q^{41} -122932. q^{43} +(52674.9 + 91235.6i) q^{47} +(-808404. - 157179. i) q^{49} +(-665248. + 1.15224e6i) q^{53} +1.49371e6 q^{55} +(429413. - 743764. i) q^{59} +(525507. + 910205. i) q^{61} +(1.19385e6 + 2.06780e6i) q^{65} +(2.21180e6 - 3.83096e6i) q^{67} -154446. q^{71} +(1.07857e6 - 1.86813e6i) q^{73} +(-4.06838e6 - 2.90210e6i) q^{77} +(-954414. - 1.65309e6i) q^{79} +8.56018e6 q^{83} -1.85584e6 q^{85} +(-5.54046e6 - 9.59637e6i) q^{89} +(765843. - 7.95153e6i) q^{91} +(-3.58900e6 + 6.21634e6i) q^{95} +1.13502e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 1680 q^{7} - 28280 q^{13} + 42224 q^{19} - 80460 q^{25} + 164752 q^{31} - 647980 q^{37} + 1341440 q^{43} + 230104 q^{49} - 323120 q^{55} - 4319336 q^{61} - 3905760 q^{67} + 6471780 q^{73} - 6093104 q^{79}+ \cdots - 27141240 q^{97}+O(q^{100}) $ 16 * q + 1680 * q^7 - 28280 * q^13 + 42224 * q^19 - 80460 * q^25 + 164752 * q^31 - 647980 * q^37 + 1341440 * q^43 + 230104 * q^49 - 323120 * q^55 - 4319336 * q^61 - 3905760 * q^67 + 6471780 * q^73 - 6093104 * q^79 + 456400 * q^85 + 15969856 * q^91 - 27141240 * q^97

Character values

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

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	135.624	+	234.907i	0.485223	+	0.840430i	0.999856	−	0.0169803i	$-0.00540526\pi$
$5$	135.624	+	234.907i	0.485223	+	0.840430i	−0.514633	+	0.857410i	$0.672072\pi$
$6$		0			0
$7$	87.0016	−	903.313i	0.0958703	−	0.995394i
$7$	87.0016	−	903.313i	0.0958703	−	0.995394i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2753.41	−	4769.04i	0.623729	−	1.08033i	−0.365056	−	0.930985i	$-0.618950\pi$
$11$	2753.41	−	4769.04i	0.623729	−	1.08033i	0.988785	−	0.149345i	$-0.0477164\pi$
$12$		0			0
$13$	8802.63			1.11125			0.555624	−	0.831434i	$-0.312479\pi$
$13$	8802.63			1.11125			0.555624	+	0.831434i	$0.312479\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3420.94	+	5925.24i	−0.168878	+	0.292506i	−0.938026	−	0.346565i	$-0.887348\pi$
$17$	−3420.94	+	5925.24i	−0.168878	+	0.292506i	0.769147	+	0.639071i	$0.220681\pi$
$18$		0			0
$19$	13231.5	+	22917.6i	0.442558	+	0.766533i	0.997879	−	0.0651034i	$-0.0207377\pi$
$19$	13231.5	+	22917.6i	0.442558	+	0.766533i	−0.555320	+	0.831636i	$0.687404\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−41328.3	−	71582.8i	−0.708273	−	1.22676i	−0.965497	−	0.260413i	$-0.916141\pi$
$23$	−41328.3	−	71582.8i	−0.708273	−	1.22676i	0.257224	−	0.966352i	$-0.417192\pi$
$24$		0			0
$25$	2274.86	−	3940.17i	0.0291182	−	0.0504342i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−233335.			−1.77659			−0.888295	−	0.459274i	$-0.848110\pi$
$29$	−233335.			−1.77659			−0.888295	+	0.459274i	$0.848110\pi$
$30$		0			0
$31$	19823.7	−	34335.7i	0.119514	−	0.207005i	−0.800061	−	0.599919i	$-0.795200\pi$
$31$	19823.7	−	34335.7i	0.119514	−	0.207005i	0.919575	+	0.392914i	$0.128533\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	223994.	−	102073.i	0.883077	−	0.402415i
$36$		0			0
$37$	−115873.	−	200699.i	−0.376078	−	0.651386i	0.614410	−	0.788987i	$-0.289394\pi$
$37$	−115873.	−	200699.i	−0.376078	−	0.651386i	−0.990488	+	0.137601i	$0.956061\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	403925.			0.915287			0.457643	−	0.889136i	$-0.348694\pi$
$41$	403925.			0.915287			0.457643	+	0.889136i	$0.348694\pi$
$42$		0			0
$43$	−122932.			−0.235791			−0.117895	−	0.993026i	$-0.537615\pi$
$43$	−122932.			−0.235791			−0.117895	+	0.993026i	$0.537615\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	52674.9	+	91235.6i	0.0740050	+	0.128180i	0.900653	−	0.434539i	$-0.143089\pi$
$47$	52674.9	+	91235.6i	0.0740050	+	0.128180i	−0.826648	+	0.562719i	$0.809755\pi$
$48$		0			0
$49$	−808404.	−	157179.i	−0.981618	−	0.190857i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−665248.	+	1.15224e6i	−0.613788	+	1.06311i	0.376808	+	0.926291i	$0.377022\pi$
$53$	−665248.	+	1.15224e6i	−0.613788	+	1.06311i	−0.990596	+	0.136820i	$0.956312\pi$
$54$		0			0
$55$	1.49371e6			1.21059
$56$		0			0
$57$		0			0
$58$		0			0
$59$	429413.	−	743764.i	0.272203	−	0.471469i	−0.697223	−	0.716855i	$-0.745581\pi$
$59$	429413.	−	743764.i	0.272203	−	0.471469i	0.969426	+	0.245385i	$0.0789145\pi$
$60$		0			0
$61$	525507.	+	910205.i	0.296431	+	0.513434i	0.975317	−	0.220810i	$-0.0708701\pi$
$61$	525507.	+	910205.i	0.296431	+	0.513434i	−0.678886	+	0.734244i	$0.737537\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.19385e6	+	2.06780e6i	0.539202	+	0.933926i
$66$		0			0
$67$	2.21180e6	−	3.83096e6i	0.898432	−	1.55613i	0.0689321	−	0.997621i	$-0.478041\pi$
$67$	2.21180e6	−	3.83096e6i	0.898432	−	1.55613i	0.829499	−	0.558508i	$-0.188626\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−154446.			−0.0512121			−0.0256060	−	0.999672i	$-0.508152\pi$
$71$	−154446.			−0.0512121			−0.0256060	+	0.999672i	$0.508152\pi$
$72$		0			0
$73$	1.07857e6	−	1.86813e6i	0.324502	−	0.562054i	−0.656910	−	0.753969i	$-0.728137\pi$
$73$	1.07857e6	−	1.86813e6i	0.324502	−	0.562054i	0.981411	+	0.191916i	$0.0614700\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.06838e6	−	2.90210e6i	−1.01556	−	0.724427i
$78$		0			0
$79$	−954414.	−	1.65309e6i	−0.217792	−	0.377227i	0.736341	−	0.676611i	$-0.236552\pi$
$79$	−954414.	−	1.65309e6i	−0.217792	−	0.377227i	−0.954133	+	0.299384i	$0.903219\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.56018e6			1.64327			0.821637	−	0.570012i	$-0.193061\pi$
$83$	8.56018e6			1.64327			0.821637	+	0.570012i	$0.193061\pi$
$84$		0			0
$85$	−1.85584e6			−0.327774
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.54046e6	−	9.59637e6i	−0.833069	−	1.44292i	−0.895593	−	0.444875i	$-0.853248\pi$
$89$	−5.54046e6	−	9.59637e6i	−0.833069	−	1.44292i	0.0625233	−	0.998044i	$-0.480085\pi$
$90$		0			0
$91$	765843.	−	7.95153e6i	0.106536	−	1.10613i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.58900e6	+	6.21634e6i	−0.429478	+	0.743878i
$96$		0			0
$97$	1.13502e6			0.126270			0.0631352	−	0.998005i	$-0.479890\pi$
$97$	1.13502e6			0.126270			0.0631352	+	0.998005i	$0.479890\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.16132e6	+	7.20762e6i	−0.401889	+	0.696092i	−0.993954	−	0.109798i	$-0.964980\pi$
$101$	−4.16132e6	+	7.20762e6i	−0.401889	+	0.696092i	0.592065	+	0.805890i	$0.298313\pi$
$102$		0			0
$103$	−5.20374e6	−	9.01315e6i	−0.469230	−	0.812730i	0.530151	−	0.847903i	$-0.322135\pi$
$103$	−5.20374e6	−	9.01315e6i	−0.469230	−	0.812730i	−0.999381	+	0.0351731i	$0.988802\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.54590e6	−	1.30699e7i	−0.595481	−	1.03140i	−0.993479	−	0.114017i	$-0.963628\pi$
$107$	−7.54590e6	−	1.30699e7i	−0.595481	−	1.03140i	0.397998	−	0.917386i	$-0.369705\pi$
$108$		0			0
$109$	1.33676e7	−	2.31534e7i	0.988694	−	1.71247i	0.364489	−	0.931208i	$-0.381244\pi$
$109$	1.33676e7	−	2.31534e7i	0.988694	−	1.71247i	0.624205	−	0.781261i	$-0.285423\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−54643.3			−0.00356256			−0.00178128	−	0.999998i	$-0.500567\pi$
$113$	−54643.3			−0.00356256			−0.00178128	+	0.999998i	$0.500567\pi$
$114$		0			0
$115$	1.12102e7	−	1.94167e7i	0.687340	−	1.19051i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.05472e6	+	3.60568e6i	0.274968	+	0.196143i
$120$		0			0
$121$	−5.41890e6	−	9.38582e6i	−0.278075	−	0.481641i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.24253e7			1.02696
$126$		0			0
$127$	2.99538e7			1.29760			0.648798	−	0.760961i	$-0.275272\pi$
$127$	2.99538e7			1.29760			0.648798	+	0.760961i	$0.275272\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.29309e7	+	3.97174e7i	0.891191	+	1.54359i	0.838449	+	0.544980i	$0.183463\pi$
$131$	2.29309e7	+	3.97174e7i	0.891191	+	1.54359i	0.0527416	+	0.998608i	$0.483204\pi$
$132$		0			0
$133$	2.18529e7	−	9.95828e6i	0.805431	−	0.367032i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.37558e7	+	2.38257e7i	−0.457049	+	0.791631i	−0.998803	−	0.0489051i	$-0.984427\pi$
$137$	−1.37558e7	+	2.38257e7i	−0.457049	+	0.791631i	0.541755	+	0.840537i	$0.317760\pi$
$138$		0			0
$139$	1.41540e7			0.447021			0.223510	−	0.974702i	$-0.428248\pi$
$139$	1.41540e7			0.447021			0.223510	+	0.974702i	$0.428248\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.42372e7	−	4.19801e7i	0.693117	−	1.20051i
$144$		0			0
$145$	−3.16458e7	−	5.48121e7i	−0.862041	−	1.49310i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.57625e7	−	2.73014e7i	−0.390367	−	0.676135i	0.602131	−	0.798397i	$-0.294319\pi$
$149$	−1.57625e7	−	2.73014e7i	−0.390367	−	0.676135i	−0.992498	+	0.122262i	$0.960985\pi$
$150$		0			0
$151$	−5.45008e6	+	9.43982e6i	−0.128820	+	0.223123i	−0.923220	−	0.384272i	$-0.874452\pi$
$151$	−5.45008e6	+	9.43982e6i	−0.128820	+	0.223123i	0.794400	+	0.607396i	$0.207786\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.07543e7			0.231964
$156$		0			0
$157$	4.48305e7	−	7.76486e7i	0.924537	−	1.60135i	0.132233	−	0.991219i	$-0.457785\pi$
$157$	4.48305e7	−	7.76486e7i	0.924537	−	1.60135i	0.792304	−	0.610127i	$-0.208881\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.82573e7	+	3.11046e7i	−1.28902	+	0.587400i
$162$		0			0
$163$	−4.56018e7	−	7.89846e7i	−0.824755	−	1.42852i	−0.902106	−	0.431514i	$-0.857980\pi$
$163$	−4.56018e7	−	7.89846e7i	−0.824755	−	1.42852i	0.0773509	−	0.997004i	$-0.475354\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.72822e7			−0.785579			−0.392789	−	0.919628i	$-0.628490\pi$
$167$	−4.72822e7			−0.785579			−0.392789	+	0.919628i	$0.628490\pi$
$168$		0			0
$169$	1.47378e7			0.234872
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.07168e7	+	1.85621e7i	0.157364	+	0.272562i	0.933917	−	0.357489i	$-0.116367\pi$
$173$	1.07168e7	+	1.85621e7i	0.157364	+	0.272562i	−0.776553	+	0.630051i	$0.783034\pi$
$174$		0			0
$175$	−3.36129e6	−	2.39771e6i	−0.0474103	−	0.0338192i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.11093e7	−	3.65624e7i	0.275099	−	0.476485i	−0.695061	−	0.718951i	$-0.744623\pi$
$179$	2.11093e7	−	3.65624e7i	0.275099	−	0.476485i	0.970160	+	0.242465i	$0.0779561\pi$
$180$		0			0
$181$	4.62466e6			0.0579702			0.0289851	−	0.999580i	$-0.490772\pi$
$181$	4.62466e6			0.0579702			0.0289851	+	0.999580i	$0.490772\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.14304e7	−	5.44390e7i	0.364963	−	0.632134i
$186$		0			0
$187$	1.88385e7	+	3.26292e7i	0.210669	+	0.364889i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.94101e7	−	1.54863e8i	−0.928474	−	1.60816i	−0.785877	−	0.618383i	$-0.787788\pi$
$191$	−8.94101e7	−	1.54863e8i	−0.928474	−	1.60816i	−0.142597	−	0.989781i	$-0.545545\pi$
$192$		0			0
$193$	2.05863e7	−	3.56566e7i	0.206124	−	0.357017i	−0.744366	−	0.667771i	$-0.767248\pi$
$193$	2.05863e7	−	3.56566e7i	0.206124	−	0.357017i	0.950490	+	0.310754i	$0.100582\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.97492e7			0.836370			0.418185	−	0.908362i	$-0.362666\pi$
$197$	8.97492e7			0.836370			0.418185	+	0.908362i	$0.362666\pi$
$198$		0			0
$199$	−1.51247e7	+	2.61968e7i	−0.136051	+	0.235647i	−0.925998	−	0.377527i	$-0.876774\pi$
$199$	−1.51247e7	+	2.61968e7i	−0.136051	+	0.235647i	0.789948	+	0.613174i	$0.210108\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.03005e7	+	2.10775e8i	−0.170322	+	1.76841i
$204$		0			0
$205$	5.47819e7	+	9.48850e7i	0.444118	+	0.769235i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.45726e8			1.10415
$210$		0			0
$211$	4.15724e7			0.304661			0.152331	−	0.988330i	$-0.451322\pi$
$211$	4.15724e7			0.304661			0.152331	+	0.988330i	$0.451322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.66726e7	−	2.88777e7i	−0.114411	−	0.198166i
$216$		0			0
$217$	−2.92912e7	−	2.08943e7i	−0.194593	−	0.138809i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.01133e7	+	5.21577e7i	−0.187666	+	0.325047i
$222$		0			0
$223$	1.48057e7			0.0894052			0.0447026	−	0.999000i	$-0.485766\pi$
$223$	1.48057e7			0.0894052			0.0447026	+	0.999000i	$0.485766\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.63163e7	+	1.66825e8i	−0.546524	+	0.946607i	0.451985	+	0.892025i	$0.350716\pi$
$227$	−9.63163e7	+	1.66825e8i	−0.546524	+	0.946607i	−0.998509	+	0.0545820i	$0.982617\pi$
$228$		0			0
$229$	1.58823e7	+	2.75090e7i	0.0873956	+	0.151374i	0.906409	−	0.422400i	$-0.138812\pi$
$229$	1.58823e7	+	2.75090e7i	0.0873956	+	0.151374i	−0.819014	+	0.573774i	$0.805479\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.65294e6	−	4.59502e6i	−0.0137398	−	0.0237981i	0.859074	−	0.511852i	$-0.171040\pi$
$233$	−2.65294e6	−	4.59502e6i	−0.0137398	−	0.0237981i	−0.872814	+	0.488054i	$0.837707\pi$
$234$		0			0
$235$	−1.42879e7	+	2.47474e7i	−0.0718178	+	0.124392i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.84527e7			−0.276957			−0.138478	−	0.990365i	$-0.544221\pi$
$239$	−5.84527e7			−0.276957			−0.138478	+	0.990365i	$0.544221\pi$
$240$		0			0
$241$	−1.49460e8	+	2.58872e8i	−0.687803	+	1.19131i	0.284743	+	0.958604i	$0.408092\pi$
$241$	−1.49460e8	+	2.58872e8i	−0.687803	+	1.19131i	−0.972547	+	0.232707i	$0.925242\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7.27163e7	−	2.11217e8i	−0.315901	−	0.917589i
$246$		0			0
$247$	1.16472e8	+	2.01735e8i	0.491792	+	0.851808i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.54841e8			1.01721			0.508606	−	0.860999i	$-0.330161\pi$
$251$	2.54841e8			1.01721			0.508606	+	0.860999i	$0.330161\pi$
$252$		0			0
$253$	−4.55175e8			−1.76708
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.67755e8	+	2.90560e8i	0.616467	+	1.06775i	0.990125	+	0.140185i	$0.0447698\pi$
$257$	1.67755e8	+	2.90560e8i	0.616467	+	1.06775i	−0.373659	+	0.927566i	$0.621897\pi$
$258$		0			0
$259$	−1.91375e8	+	8.72088e7i	−0.684440	+	0.311897i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.52202e8	+	2.63622e8i	−0.515912	+	0.893585i	0.483918	+	0.875114i	$0.339213\pi$
$263$	−1.52202e8	+	2.63622e8i	−0.515912	+	0.893585i	−0.999829	+	0.0184718i	$0.994120\pi$
$264$		0			0
$265$	−3.60894e8			−1.19129
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.75579e8	−	4.77317e8i	0.863204	−	1.49511i	−0.00561523	−	0.999984i	$-0.501787\pi$
$269$	2.75579e8	−	4.77317e8i	0.863204	−	1.49511i	0.868819	−	0.495129i	$-0.164879\pi$
$270$		0			0
$271$	3.05360e8	+	5.28899e8i	0.932008	+	1.61429i	0.779884	+	0.625924i	$0.215278\pi$
$271$	3.05360e8	+	5.28899e8i	0.932008	+	1.61429i	0.152124	+	0.988361i	$0.451389\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.25272e7	−	2.16978e7i	−0.0363237	−	0.0629146i
$276$		0			0
$277$	−8.96395e7	+	1.55260e8i	−0.253408	+	0.438915i	−0.964462	−	0.264222i	$-0.914885\pi$
$277$	−8.96395e7	+	1.55260e8i	−0.253408	+	0.438915i	0.711054	+	0.703137i	$0.248218\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.22014e8			−0.328048			−0.164024	−	0.986456i	$-0.552447\pi$
$281$	−1.22014e8			−0.328048			−0.164024	+	0.986456i	$0.552447\pi$
$282$		0			0
$283$	−1.24497e8	+	2.15635e8i	−0.326518	+	0.565546i	−0.981818	−	0.189823i	$-0.939209\pi$
$283$	−1.24497e8	+	2.15635e8i	−0.326518	+	0.565546i	0.655300	+	0.755368i	$0.272542\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.51421e7	−	3.64871e8i	0.0877488	−	0.911071i
$288$		0			0
$289$	1.81764e8	+	3.14824e8i	0.442960	+	0.767229i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.74504e8			−1.10205			−0.551027	−	0.834487i	$-0.685764\pi$
$293$	−4.74504e8			−1.10205			−0.551027	+	0.834487i	$0.685764\pi$
$294$		0			0
$295$	2.32954e8			0.528316
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.63798e8	−	6.30117e8i	−0.787067	−	1.36324i
$300$		0			0
$301$	−1.06953e7	+	1.11046e8i	−0.0226053	+	0.234705i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.42543e8	+	2.46891e8i	−0.287670	+	0.498260i
$306$		0			0
$307$	3.83066e8			0.755595			0.377798	−	0.925888i	$-0.376682\pi$
$307$	3.83066e8			0.755595			0.377798	+	0.925888i	$0.376682\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.36877e8	+	4.10282e8i	−0.446540	+	0.773431i	−0.998158	−	0.0606664i	$-0.980677\pi$
$311$	−2.36877e8	+	4.10282e8i	−0.446540	+	0.773431i	0.551618	+	0.834097i	$0.314011\pi$
$312$		0			0
$313$	−1.10694e8	−	1.91728e8i	−0.204042	−	0.353411i	0.745785	−	0.666187i	$-0.232075\pi$
$313$	−1.10694e8	−	1.91728e8i	−0.204042	−	0.353411i	−0.949827	+	0.312776i	$0.898741\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.36623e8	+	4.09843e8i	0.417205	+	0.722620i	0.995657	−	0.0930963i	$-0.0296764\pi$
$317$	2.36623e8	+	4.09843e8i	0.417205	+	0.722620i	−0.578452	+	0.815716i	$0.696343\pi$
$318$		0			0
$319$	−6.42466e8	+	1.11278e9i	−1.10811	+	1.91930i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.81056e8			−0.298954
$324$		0			0
$325$	2.00248e7	−	3.46839e7i	0.0323575	−	0.0560449i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.69970e7	−	3.96442e7i	0.134685	−	0.0613754i
$330$		0			0
$331$	−4.58900e7	−	7.94838e7i	−0.0695537	−	0.120471i	0.829151	−	0.559024i	$-0.188824\pi$
$331$	−4.58900e7	−	7.94838e7i	−0.0695537	−	0.120471i	−0.898705	+	0.438554i	$0.855491\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.19989e9			1.74376
$336$		0			0
$337$	4.12466e8			0.587061			0.293530	−	0.955950i	$-0.405170\pi$
$337$	4.12466e8			0.587061			0.293530	+	0.955950i	$0.405170\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.09166e8	−	1.89080e8i	−0.149089	−	0.258230i
$342$		0			0
$343$	−2.12315e8	+	7.16567e8i	−0.284086	+	0.958799i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.94162e8	−	8.55913e8i	0.634916	−	1.09971i	−0.351617	−	0.936144i	$-0.614368\pi$
$347$	4.94162e8	−	8.55913e8i	0.634916	−	1.09971i	0.986533	−	0.163562i	$-0.0522985\pi$
$348$		0			0
$349$	−1.13371e9			−1.42762			−0.713812	−	0.700337i	$-0.753033\pi$
$349$	−1.13371e9			−1.42762			−0.713812	+	0.700337i	$0.753033\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.88180e8	+	1.01876e9i	−0.711703	+	1.23271i	0.252514	+	0.967593i	$0.418743\pi$
$353$	−5.88180e8	+	1.01876e9i	−0.711703	+	1.23271i	−0.964217	+	0.265113i	$0.914591\pi$
$354$		0			0
$355$	−2.09466e7	−	3.62805e7i	−0.0248493	−	0.0430402i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3.65381e8	+	6.32858e8i	0.416788	+	0.721898i	0.995614	−	0.0935532i	$-0.0298225\pi$
$359$	3.65381e8	+	6.32858e8i	0.416788	+	0.721898i	−0.578827	+	0.815451i	$0.696489\pi$
$360$		0			0
$361$	9.67926e7	−	1.67650e8i	0.108285	−	0.187555i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.85118e8			0.629822
$366$		0			0
$367$	3.06230e8	−	5.30405e8i	0.323382	−	0.560114i	−0.657802	−	0.753191i	$-0.728513\pi$
$367$	3.06230e8	−	5.30405e8i	0.323382	−	0.560114i	0.981184	+	0.193077i	$0.0618468\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.82959e8	+	7.01174e8i	0.999371	+	0.712882i
$372$		0			0
$373$	−2.75024e8	−	4.76356e8i	−0.274404	−	0.475281i	0.695581	−	0.718448i	$-0.255147\pi$
$373$	−2.75024e8	−	4.76356e8i	−0.274404	−	0.475281i	−0.969985	+	0.243167i	$0.921814\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.05396e9			−1.97423
$378$		0			0
$379$	1.77173e8			0.167171			0.0835853	−	0.996501i	$-0.473363\pi$
$379$	1.77173e8			0.167171			0.0835853	+	0.996501i	$0.473363\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.41656e8	+	2.45356e8i	0.128837	+	0.223152i	0.923226	−	0.384257i	$-0.125542\pi$
$383$	1.41656e8	+	2.45356e8i	0.128837	+	0.223152i	−0.794389	+	0.607409i	$0.792209\pi$
$384$		0			0
$385$	1.29955e8	−	1.34929e9i	0.116060	−	1.20501i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.25354e8	−	1.25635e9i	0.624779	−	1.08215i	−0.363805	−	0.931475i	$-0.618523\pi$
$389$	7.25354e8	−	1.25635e9i	0.624779	−	1.08215i	0.988584	−	0.150673i	$-0.0481442\pi$
$390$		0			0
$391$	5.65527e8			0.478448
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.58882e8	−	4.48398e8i	0.211355	−	0.366078i
$396$		0			0
$397$	−5.93238e8	−	1.02752e9i	−0.475842	−	0.824182i	0.523775	−	0.851856i	$-0.324523\pi$
$397$	−5.93238e8	−	1.02752e9i	−0.475842	−	0.824182i	−0.999617	+	0.0276745i	$0.991190\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.92674e8	+	5.06926e8i	0.226662	+	0.392590i	0.956817	−	0.290692i	$-0.0938854\pi$
$401$	2.92674e8	+	5.06926e8i	0.226662	+	0.392590i	−0.730155	+	0.683282i	$0.760552\pi$
$402$		0			0
$403$	1.74501e8	−	3.02245e8i	0.132810	−	0.230034i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.27619e9			−0.938283
$408$		0			0
$409$	1.05156e9	−	1.82135e9i	0.759978	−	1.31632i	−0.182882	−	0.983135i	$-0.558543\pi$
$409$	1.05156e9	−	1.82135e9i	0.759978	−	1.31632i	0.942861	−	0.333187i	$-0.108124\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.34492e8	−	4.52603e8i	−0.443201	−	0.316149i
$414$		0			0
$415$	1.16096e9	+	2.01085e9i	0.797353	+	1.38106i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.71830e9			−1.80530			−0.902648	−	0.430380i	$-0.858380\pi$
$419$	−2.71830e9			−1.80530			−0.902648	+	0.430380i	$0.858380\pi$
$420$		0			0
$421$	5.29822e8			0.346053			0.173027	−	0.984917i	$-0.444645\pi$
$421$	5.29822e8			0.346053			0.173027	+	0.984917i	$0.444645\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.55643e7	+	2.69582e7i	0.00983488	+	0.0170345i
$426$		0			0
$427$	8.67919e8	−	3.95508e8i	0.539488	−	0.245843i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.01224e8	−	1.75325e8i	0.0608995	−	0.105481i	−0.833968	−	0.551812i	$-0.813936\pi$
$431$	1.01224e8	−	1.75325e8i	0.0608995	−	0.105481i	0.894868	+	0.446331i	$0.147270\pi$
$432$		0			0
$433$	−6.56435e8			−0.388584			−0.194292	−	0.980944i	$-0.562241\pi$
$433$	−6.56435e8			−0.388584			−0.194292	+	0.980944i	$0.562241\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.09367e9	−	1.89429e9i	0.626904	−	1.08583i
$438$		0			0
$439$	4.19575e8	+	7.26724e8i	0.236692	+	0.409962i	0.959763	−	0.280811i	$-0.0906035\pi$
$439$	4.19575e8	+	7.26724e8i	0.236692	+	0.409962i	−0.723071	+	0.690774i	$0.757270\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.11299e7	−	1.05880e8i	−0.0334072	−	0.0578630i	0.848838	−	0.528652i	$-0.177303\pi$
$443$	−6.11299e7	−	1.05880e8i	−0.0334072	−	0.0578630i	−0.882246	+	0.470789i	$0.843969\pi$
$444$		0			0
$445$	1.50284e9	−	2.60299e9i	0.808448	−	1.40027i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.95736e9			1.02049			0.510244	−	0.860030i	$-0.329555\pi$
$449$	1.95736e9			1.02049			0.510244	+	0.860030i	$0.329555\pi$
$450$		0			0
$451$	1.11217e9	−	1.92633e9i	0.570891	−	0.988812i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.97174e9	−	8.98515e8i	0.981318	−	0.447183i
$456$		0			0
$457$	−1.37930e9	−	2.38901e9i	−0.676007	−	1.17088i	−0.976174	−	0.216991i	$-0.930376\pi$
$457$	−1.37930e9	−	2.38901e9i	−0.676007	−	1.17088i	0.300167	−	0.953887i	$-0.402958\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.05133e9			−0.499788			−0.249894	−	0.968273i	$-0.580396\pi$
$461$	−1.05133e9			−0.499788			−0.249894	+	0.968273i	$0.580396\pi$
$462$		0			0
$463$	1.67036e9			0.782127			0.391063	−	0.920364i	$-0.372107\pi$
$463$	1.67036e9			0.782127			0.391063	+	0.920364i	$0.372107\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.73863e8	+	1.51357e9i	0.397040	+	0.687693i	0.993359	−	0.115054i	$-0.0367041\pi$
$467$	8.73863e8	+	1.51357e9i	0.397040	+	0.687693i	−0.596319	+	0.802747i	$0.703371\pi$
$468$		0			0
$469$	−3.26812e9	−	2.33125e9i	−1.46283	−	1.04348i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.38483e8	+	5.86269e8i	−0.147070	+	0.254732i
$474$		0			0
$475$	1.20399e8			0.0515460
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.61974e8	+	1.66619e9i	−0.399935	+	0.692707i	−0.993717	−	0.111918i	$-0.964301\pi$
$479$	−9.61974e8	+	1.66619e9i	−0.399935	+	0.692707i	0.593783	+	0.804625i	$0.297634\pi$
$480$		0			0
$481$	−1.01999e9	−	1.76668e9i	−0.417916	−	0.723851i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.53935e8	+	2.66624e8i	0.0612692	+	0.106121i
$486$		0			0
$487$	1.71220e9	−	2.96562e9i	0.671743	−	1.16349i	−0.305666	−	0.952139i	$-0.598879\pi$
$487$	1.71220e9	−	2.96562e9i	0.671743	−	1.16349i	0.977409	−	0.211355i	$-0.0677875\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.92888e9			−1.11665			−0.558325	−	0.829622i	$-0.688556\pi$
$491$	−2.92888e9			−1.11665			−0.558325	+	0.829622i	$0.688556\pi$
$492$		0			0
$493$	7.98225e8	−	1.38257e9i	0.300028	−	0.519663i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.34370e7	+	1.39513e8i	−0.00490972	+	0.0509762i
$498$		0			0
$499$	1.54754e9	+	2.68042e9i	0.557559	+	0.965721i	0.997699	+	0.0677918i	$0.0215953\pi$
$499$	1.54754e9	+	2.68042e9i	0.557559	+	0.965721i	−0.440140	+	0.897929i	$0.645071\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2.54605e9			0.892029			0.446014	−	0.895026i	$-0.352843\pi$
$503$	2.54605e9			0.892029			0.446014	+	0.895026i	$0.352843\pi$
$504$		0			0
$505$	−2.25750e9			−0.780023
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.01304e8	−	1.75463e8i	−0.0340496	−	0.0589757i	0.848498	−	0.529198i	$-0.177507\pi$
$509$	−1.01304e8	−	1.75463e8i	−0.0340496	−	0.0589757i	−0.882548	+	0.470222i	$0.844174\pi$
$510$		0			0
$511$	−1.59367e9	−	1.13681e9i	−0.528355	−	0.376891i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.41150e9	−	2.44480e9i	0.455362	−	0.788710i
$516$		0			0
$517$	5.80141e8			0.184636
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.74740e9	+	4.75864e9i	−0.851119	+	1.47418i	0.0290803	+	0.999577i	$0.490742\pi$
$521$	−2.74740e9	+	4.75864e9i	−0.851119	+	1.47418i	−0.880199	+	0.474604i	$0.842591\pi$
$522$		0			0
$523$	1.84283e9	+	3.19187e9i	0.563286	+	0.975640i	0.997207	+	0.0746890i	$0.0237964\pi$
$523$	1.84283e9	+	3.19187e9i	0.563286	+	0.975640i	−0.433921	+	0.900951i	$0.642870\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.35632e8	+	2.34921e8i	0.0403668	+	0.0699173i
$528$		0			0
$529$	−1.71365e9	+	2.96813e9i	−0.503301	+	0.871743i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.55560e9			1.01711
$534$		0			0
$535$	2.04681e9	−	3.54518e9i	0.577882	−	1.00092i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.97546e9	+	3.42253e9i	−0.818452	+	0.941428i
$540$		0			0
$541$	−7.06594e8	−	1.22386e9i	−0.191858	−	0.332308i	0.754008	−	0.656865i	$-0.228118\pi$
$541$	−7.06594e8	−	1.22386e9i	−0.191858	−	0.332308i	−0.945866	+	0.324557i	$0.894785\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.25188e9			1.91895
$546$		0			0
$547$	5.40167e9			1.41115			0.705574	−	0.708637i	$-0.250689\pi$
$547$	5.40167e9			1.41115			0.705574	+	0.708637i	$0.250689\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.08736e9	−	5.34747e9i	−0.786244	−	1.36181i
$552$		0			0
$553$	−1.57630e9	+	7.18312e8i	−0.396369	+	0.180624i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.85131e9	+	4.93861e9i	−0.699118	+	1.21091i	0.269654	+	0.962957i	$0.413091\pi$
$557$	−2.85131e9	+	4.93861e9i	−0.699118	+	1.21091i	−0.968772	+	0.247951i	$0.920243\pi$
$558$		0			0
$559$	−1.08213e9			−0.262022
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.78880e9	−	4.83035e9i	0.658626	−	1.14077i	−0.322346	−	0.946622i	$-0.604471\pi$
$563$	2.78880e9	−	4.83035e9i	0.658626	−	1.14077i	0.980972	−	0.194151i	$-0.0621952\pi$
$564$		0			0
$565$	−7.41094e6	−	1.28361e7i	−0.00172864	−	0.00299408i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.19424e9	−	2.06848e9i	−0.271767	−	0.470714i	0.697547	−	0.716539i	$-0.254275\pi$
$569$	−1.19424e9	−	2.06848e9i	−0.271767	−	0.470714i	−0.969314	+	0.245824i	$0.920941\pi$
$570$		0			0
$571$	−2.78817e9	+	4.82925e9i	−0.626747	+	1.08556i	0.361453	+	0.932390i	$0.382281\pi$
$571$	−2.78817e9	+	4.82925e9i	−0.626747	+	1.08556i	−0.988200	+	0.153168i	$0.951053\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.76065e8			−0.0824946
$576$		0			0
$577$	3.88721e9	−	6.73285e9i	0.842409	−	1.45909i	−0.0454440	−	0.998967i	$-0.514470\pi$
$577$	3.88721e9	−	6.73285e9i	0.842409	−	1.45909i	0.887853	−	0.460128i	$-0.152196\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.44750e8	−	7.73252e9i	0.157541	−	1.63570i
$582$		0			0
$583$	3.66340e9	+	6.34519e9i	0.765675	+	1.32619i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.89217e8			−0.120238			−0.0601190	−	0.998191i	$-0.519148\pi$
$587$	−5.89217e8			−0.120238			−0.0601190	+	0.998191i	$0.519148\pi$
$588$		0			0
$589$	1.04919e9			0.211568
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8.58290e8	−	1.48660e9i	−0.169022	−	0.292754i	0.769054	−	0.639183i	$-0.220727\pi$
$593$	−8.58290e8	−	1.48660e9i	−0.169022	−	0.292754i	−0.938076	+	0.346429i	$0.887394\pi$
$594$		0			0
$595$	−1.61461e8	+	1.67641e9i	−0.0314238	+	0.326265i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.84213e9	+	6.65477e9i	−0.730430	+	1.26514i	0.226270	+	0.974065i	$0.427347\pi$
$599$	−3.84213e9	+	6.65477e9i	−0.730430	+	1.26514i	−0.956700	+	0.291076i	$0.905987\pi$
$600$		0			0
$601$	−1.01577e10			−1.90869			−0.954344	−	0.298709i	$-0.903444\pi$
$601$	−1.01577e10			−1.90869			−0.954344	+	0.298709i	$0.903444\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.46986e9	−	2.54588e9i	0.269857	−	0.467406i
$606$		0			0
$607$	1.12929e9	+	1.95598e9i	0.204948	+	0.354981i	0.950116	−	0.311896i	$-0.100964\pi$
$607$	1.12929e9	+	1.95598e9i	0.204948	+	0.354981i	−0.745168	+	0.666877i	$0.767631\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.63678e8	+	8.03113e8i	0.0822379	+	0.142440i
$612$		0			0
$613$	−3.16951e9	+	5.48975e9i	−0.555752	+	0.962590i	0.442093	+	0.896969i	$0.354236\pi$
$613$	−3.16951e9	+	5.48975e9i	−0.555752	+	0.962590i	−0.997845	+	0.0656207i	$0.979097\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.51460e9			−1.11658			−0.558290	−	0.829646i	$-0.688542\pi$
$617$	−6.51460e9			−1.11658			−0.558290	+	0.829646i	$0.688542\pi$
$618$		0			0
$619$	−4.99392e9	+	8.64972e9i	−0.846299	+	1.46583i	0.0381887	+	0.999271i	$0.487841\pi$
$619$	−4.99392e9	+	8.64972e9i	−0.846299	+	1.46583i	−0.884488	+	0.466563i	$0.845492\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−9.15055e9	+	4.16987e9i	−1.51614	+	0.690899i
$624$		0			0
$625$	2.86368e9	+	4.96005e9i	0.469186	+	0.812654i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.58558e9			0.254046
$630$		0			0
$631$	−6.90293e8			−0.109378			−0.0546891	−	0.998503i	$-0.517417\pi$
$631$	−6.90293e8			−0.109378			−0.0546891	+	0.998503i	$0.517417\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.06245e9	+	7.03637e9i	0.629622	+	1.09054i
$636$		0			0
$637$	−7.11609e9	−	1.38359e9i	−1.09082	−	0.212090i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.03158e8	−	6.98291e8i	0.0604606	−	0.104721i	−0.834211	−	0.551446i	$-0.814076\pi$
$641$	4.03158e8	−	6.98291e8i	0.0604606	−	0.104721i	0.894671	+	0.446725i	$0.147410\pi$
$642$		0			0
$643$	−7.24540e9			−1.07479			−0.537395	−	0.843331i	$-0.680592\pi$
$643$	−7.24540e9			−1.07479			−0.537395	+	0.843331i	$0.680592\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.06148e9	−	8.76675e9i	0.734705	−	1.27255i	−0.220148	−	0.975466i	$-0.570654\pi$
$647$	5.06148e9	−	8.76675e9i	0.734705	−	1.27255i	0.954853	−	0.297079i	$-0.0960126\pi$
$648$		0			0
$649$	−2.36469e9	−	4.09577e9i	−0.339562	−	0.588138i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.23757e9	−	9.07174e9i	−0.736095	−	1.27495i	−0.954241	−	0.299038i	$-0.903334\pi$
$653$	−5.23757e9	−	9.07174e9i	−0.736095	−	1.27495i	0.218146	−	0.975916i	$-0.429999\pi$
$654$		0			0
$655$	−6.21994e9	+	1.07733e10i	−0.864852	+	1.49797i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5.86587e9			0.798424			0.399212	−	0.916859i	$-0.369284\pi$
$659$	5.86587e9			0.798424			0.399212	+	0.916859i	$0.369284\pi$
$660$		0			0
$661$	3.88047e9	−	6.72116e9i	0.522611	−	0.905189i	−0.477043	−	0.878880i	$-0.658291\pi$
$661$	3.88047e9	−	6.72116e9i	0.522611	−	0.905189i	0.999654	−	0.0263091i	$-0.00837541\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.30305e9	+	3.78282e9i	0.699278	+	0.498816i
$666$		0			0
$667$	9.64335e9	+	1.67028e10i	1.25831	+	2.17946i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.78774e9			0.739571
$672$		0			0
$673$	3.37850e9			0.427239			0.213620	−	0.976917i	$-0.431475\pi$
$673$	3.37850e9			0.427239			0.213620	+	0.976917i	$0.431475\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.21681e9	+	1.24999e10i	0.893891	+	1.54827i	0.835171	+	0.549991i	$0.185369\pi$
$677$	7.21681e9	+	1.24999e10i	0.893891	+	1.54827i	0.0587208	+	0.998274i	$0.481298\pi$
$678$		0			0
$679$	9.87483e7	−	1.02528e9i	0.0121056	−	0.125689i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.49427e9	−	2.58816e9i	0.179456	−	0.310827i	−0.762238	−	0.647296i	$-0.775900\pi$
$683$	1.49427e9	−	2.58816e9i	0.179456	−	0.310827i	0.941694	+	0.336470i	$0.109233\pi$
$684$		0			0
$685$	−7.46243e9			−0.887081
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.85594e9	+	1.01428e10i	−0.682070	+	1.18138i
$690$		0			0
$691$	5.72440e9	+	9.91495e9i	0.660019	+	1.14319i	0.980610	+	0.195969i	$0.0627852\pi$
$691$	5.72440e9	+	9.91495e9i	0.660019	+	1.14319i	−0.320591	+	0.947218i	$0.603881\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.91962e9	+	3.32488e9i	0.216904	+	0.375690i
$696$		0			0
$697$	−1.38180e9	+	2.39335e9i	−0.154572	+	0.267727i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	2.77015e9			0.303732			0.151866	−	0.988401i	$-0.451472\pi$
$701$	2.77015e9			0.303732			0.151866	+	0.988401i	$0.451472\pi$
$702$		0			0
$703$	3.06635e9	−	5.31107e9i	0.332873	−	0.576552i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.14869e9	+	4.38605e9i	0.654357	+	0.466773i
$708$		0			0
$709$	9.01249e8	+	1.56101e9i	0.0949692	+	0.164492i	0.909596	−	0.415494i	$-0.136391\pi$
$709$	9.01249e8	+	1.56101e9i	0.0949692	+	0.164492i	−0.814627	+	0.579986i	$0.803058\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.27713e9			−0.338595
$714$		0			0
$715$	1.31486e10			1.34526
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.56454e9	−	7.90602e9i	−0.457979	−	0.793244i	0.540875	−	0.841103i	$-0.318093\pi$
$719$	−4.56454e9	−	7.90602e9i	−0.457979	−	0.793244i	−0.998854	+	0.0478595i	$0.984760\pi$
$720$		0			0
$721$	−8.59442e9	+	3.91645e9i	−0.853972	+	0.389152i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.30805e8	+	9.19381e8i	−0.0517311	+	0.0896009i
$726$		0			0
$727$	−1.21081e10			−1.16871			−0.584354	−	0.811499i	$-0.698652\pi$
$727$	−1.21081e10			−1.16871			−0.584354	+	0.811499i	$0.698652\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.20544e8	−	7.28404e8i	0.0398200	−	0.0689702i
$732$		0			0
$733$	6.82100e9	+	1.18143e10i	0.639711	+	1.10801i	0.985496	+	0.169699i	$0.0542795\pi$
$733$	6.82100e9	+	1.18143e10i	0.639711	+	1.10801i	−0.345785	+	0.938314i	$0.612387\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.21800e10	−	2.10964e10i	−1.12076	−	1.94121i
$738$		0			0
$739$	−2.37583e9	+	4.11506e9i	−0.216551	+	0.375077i	−0.953751	−	0.300597i	$-0.902814\pi$
$739$	−2.37583e9	+	4.11506e9i	−0.216551	+	0.375077i	0.737200	+	0.675674i	$0.236147\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−4.47115e9			−0.399907			−0.199953	−	0.979805i	$-0.564079\pi$
$743$	−4.47115e9			−0.399907			−0.199953	+	0.979805i	$0.564079\pi$
$744$		0			0
$745$	4.27554e9	−	7.40545e9i	0.378830	−	0.656152i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.24627e10	+	5.67921e9i	−1.08374	+	0.493857i
$750$		0			0
$751$	7.08632e9	+	1.22739e10i	0.610494	+	1.05741i	0.991157	+	0.132693i	$0.0423623\pi$
$751$	7.08632e9	+	1.22739e10i	0.610494	+	1.05741i	−0.380663	+	0.924714i	$0.624304\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.95664e9			−0.250026
$756$		0			0
$757$	−1.37310e10			−1.15045			−0.575223	−	0.817997i	$-0.695085\pi$
$757$	−1.37310e10			−1.15045			−0.575223	+	0.817997i	$0.695085\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.17090e8	−	5.49216e8i	−0.0260817	−	0.0451749i	0.852690	−	0.522417i	$-0.174970\pi$
$761$	−3.17090e8	−	5.49216e8i	−0.0260817	−	0.0451749i	−0.878772	+	0.477242i	$0.841636\pi$
$762$		0			0
$763$	−1.97518e10	−	1.40895e10i	−1.60979	−	1.14831i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.77996e9	−	6.54709e9i	0.302485	−	0.523919i
$768$		0			0
$769$	8.38079e9			0.664573			0.332287	−	0.943178i	$-0.392180\pi$
$769$	8.38079e9			0.664573			0.332287	+	0.943178i	$0.392180\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3.91233e9	+	6.77635e9i	−0.304654	+	0.527676i	−0.977184	−	0.212393i	$-0.931874\pi$
$773$	−3.91233e9	+	6.77635e9i	−0.304654	+	0.527676i	0.672530	+	0.740070i	$0.265207\pi$
$774$		0			0
$775$	−9.01925e7	−	1.56218e8i	−0.00696008	−	0.0120552i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.34452e9	+	9.25698e9i	0.405068	+	0.701598i
$780$		0			0
$781$	−4.25252e8	+	7.36559e8i	−0.0319425	+	0.0553260i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.43203e10			1.79443
$786$		0			0
$787$	1.28524e10	−	2.22609e10i	0.939878	−	1.62792i	0.174181	−	0.984714i	$-0.444272\pi$
$787$	1.28524e10	−	2.22609e10i	0.939878	−	1.62792i	0.765696	−	0.643202i	$-0.222395\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.75406e6	+	4.93600e7i	−0.000341544	+	0.00354615i
$792$		0			0
$793$	4.62585e9	+	8.01220e9i	0.329409	+	0.570552i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.31351e10			−1.61870			−0.809352	−	0.587324i	$-0.800182\pi$
$797$	−2.31351e10			−1.61870			−0.809352	+	0.587324i	$0.800182\pi$
$798$		0			0
$799$	−7.20790e8			−0.0499914
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5.93947e9	−	1.02875e10i	−0.404802	−	0.701138i
$804$		0			0
$805$	−1.65640e10	−	1.18156e10i	−1.11913	−	0.798308i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9.55541e9	−	1.65504e10i	0.634497	−	1.09898i	−0.352125	−	0.935953i	$-0.614541\pi$
$809$	9.55541e9	−	1.65504e10i	0.634497	−	1.09898i	0.986622	−	0.163027i	$-0.0521259\pi$
$810$		0			0
$811$	−2.86864e9			−0.188844			−0.0944219	−	0.995532i	$-0.530100\pi$
$811$	−2.86864e9			−0.188844			−0.0944219	+	0.995532i	$0.530100\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.23694e10	−	2.14244e10i	0.800380	−	1.38630i
$816$		0			0
$817$	−1.62658e9	−	2.81731e9i	−0.104351	−	0.180741i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.05325e10	+	1.82429e10i	0.664250	+	1.15051i	0.979488	+	0.201502i	$0.0645821\pi$
$821$	1.05325e10	+	1.82429e10i	0.664250	+	1.15051i	−0.315239	+	0.949012i	$0.602085\pi$
$822$		0			0
$823$	1.06065e10	−	1.83710e10i	0.663243	−	1.14877i	−0.316515	−	0.948588i	$-0.602513\pi$
$823$	1.06065e10	−	1.83710e10i	0.663243	−	1.14877i	0.979758	−	0.200184i	$-0.0641539\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.41630e10			−0.870735			−0.435368	−	0.900253i	$-0.643382\pi$
$827$	−1.41630e10			−0.870735			−0.435368	+	0.900253i	$0.643382\pi$
$828$		0			0
$829$	1.10354e10	−	1.91138e10i	0.672738	−	1.16522i	−0.304387	−	0.952548i	$-0.598452\pi$
$829$	1.10354e10	−	1.91138e10i	0.672738	−	1.16522i	0.977125	−	0.212667i	$-0.0682150\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.69683e9	−	4.25229e9i	0.221601	−	0.254897i
$834$		0			0
$835$	−6.41259e9	−	1.11069e10i	−0.381180	−	0.660224i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.97333e8			0.0115354			0.00576771	−	0.999983i	$-0.498164\pi$
$839$	1.97333e8			0.0115354			0.00576771	+	0.999983i	$0.498164\pi$
$840$		0			0
$841$	3.71954e10			2.15627
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.99880e9	+	3.46203e9i	0.113965	+	0.197393i
$846$		0			0
$847$	−8.94978e9	+	4.07838e9i	−0.506081	+	0.230620i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−9.57771e9	+	1.65891e10i	−0.532732	+	0.922718i
$852$		0			0
$853$	−2.85016e10			−1.57234			−0.786171	−	0.618009i	$-0.787940\pi$
$853$	−2.85016e10			−1.57234			−0.786171	+	0.618009i	$0.787940\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.52798e9	+	1.47709e10i	−0.462821	+	0.801630i	−0.999100	−	0.0424108i	$-0.986496\pi$
$857$	−8.52798e9	+	1.47709e10i	−0.462821	+	0.801630i	0.536279	+	0.844041i	$0.319830\pi$
$858$		0			0
$859$	3.70608e8	+	6.41913e8i	0.0199498	+	0.0345541i	0.875828	−	0.482623i	$-0.160316\pi$
$859$	3.70608e8	+	6.41913e8i	0.0199498	+	0.0345541i	−0.855878	+	0.517178i	$0.826983\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.07828e10	+	1.86764e10i	0.571078	+	0.989136i	0.996456	+	0.0841198i	$0.0268078\pi$
$863$	1.07828e10	+	1.86764e10i	0.571078	+	0.989136i	−0.425378	+	0.905016i	$0.639859\pi$
$864$		0			0
$865$	−2.90691e9	+	5.03492e9i	−0.152713	+	0.264506i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.05116e10			−0.543373
$870$		0			0
$871$	1.94697e10	−	3.37225e10i	0.998380	−	1.72924i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.95104e9	−	2.02571e10i	0.0984550	−	1.02223i
$876$		0			0
$877$	1.91711e10	+	3.32053e10i	0.959728	+	1.66230i	0.723158	+	0.690683i	$0.242690\pi$
$877$	1.91711e10	+	3.32053e10i	0.959728	+	1.66230i	0.236570	+	0.971614i	$0.423977\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.03208e10			1.00121			0.500605	−	0.865676i	$-0.333111\pi$
$881$	2.03208e10			1.00121			0.500605	+	0.865676i	$0.333111\pi$
$882$		0			0
$883$	2.54529e10			1.24415			0.622077	−	0.782956i	$-0.286289\pi$
$883$	2.54529e10			1.24415			0.622077	+	0.782956i	$0.286289\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.37821e10	+	2.38713e10i	0.663105	+	1.14853i	0.979795	+	0.200002i	$0.0640950\pi$
$887$	1.37821e10	+	2.38713e10i	0.663105	+	1.14853i	−0.316691	+	0.948529i	$0.602572\pi$
$888$		0			0
$889$	2.60603e9	−	2.70577e10i	0.124401	−	1.29162i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.39393e9	+	2.41436e9i	−0.0655030	+	0.113455i
$894$		0			0
$895$	1.14517e10			0.533937
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.62557e9	+	8.01173e9i	−0.212328	+	0.367762i
$900$		0			0
$901$	−4.55155e9	−	7.88352e9i	−0.207311	−	0.359073i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.27214e8	+	1.08637e9i	0.0281284	+	0.0487199i
$906$		0			0
$907$	−4.18519e9	+	7.24896e9i	−0.186247	+	0.322589i	−0.943996	−	0.329957i	$-0.892966\pi$
$907$	−4.18519e9	+	7.24896e9i	−0.186247	+	0.322589i	0.757749	+	0.652546i	$0.226299\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.69466e10			−1.61905			−0.809526	−	0.587085i	$-0.800276\pi$
$911$	−3.69466e10			−1.61905			−0.809526	+	0.587085i	$0.800276\pi$
$912$		0			0
$913$	2.35697e10	−	4.08238e10i	1.02496	−	1.77528i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.78723e10	−	1.72583e10i	1.62192	−	0.739102i
$918$		0			0
$919$	2.32414e9	+	4.02553e9i	0.0987774	+	0.171087i	0.911179	−	0.412011i	$-0.135173\pi$
$919$	2.32414e9	+	4.02553e9i	0.0987774	+	0.171087i	−0.812401	+	0.583099i	$0.801840\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.35953e9			−0.0569093
$924$		0			0
$925$	−1.05438e9			−0.0438029
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.94340e10	−	3.36606e10i	−0.795256	−	1.37742i	−0.922677	−	0.385575i	$-0.874003\pi$
$929$	−1.94340e10	−	3.36606e10i	−0.795256	−	1.37742i	0.127421	−	0.991849i	$-0.459330\pi$
$930$		0			0
$931$	−7.09421e9	−	2.06064e10i	−0.288124	−	0.836908i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.10989e9	+	8.85059e9i	−0.204442	+	0.354105i
$936$		0			0
$937$	−3.56802e10			−1.41690			−0.708448	−	0.705763i	$-0.750604\pi$
$937$	−3.56802e10			−1.41690			−0.708448	+	0.705763i	$0.750604\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.53266e9	+	6.11875e9i	−0.138210	+	0.239386i	−0.926819	−	0.375508i	$-0.877468\pi$
$941$	−3.53266e9	+	6.11875e9i	−0.138210	+	0.239386i	0.788609	+	0.614895i	$0.210801\pi$
$942$		0			0
$943$	−1.66936e10	−	2.89141e10i	−0.648273	−	1.12284i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.83189e9	+	1.01011e10i	0.223144	+	0.386496i	0.955761	−	0.294145i	$-0.0950347\pi$
$947$	5.83189e9	+	1.01011e10i	0.223144	+	0.386496i	−0.732617	+	0.680641i	$0.761701\pi$
$948$		0			0
$949$	9.49423e9	−	1.64445e10i	0.360602	−	0.624581i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.29165e10			0.857675			0.428837	−	0.903382i	$-0.358923\pi$
$953$	2.29165e10			0.857675			0.428837	+	0.903382i	$0.358923\pi$
$954$		0			0
$955$	2.42523e10	−	4.20062e10i	0.901033	−	1.56063i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.03253e10	+	1.44986e10i	0.744168	+	0.530837i
$960$		0			0
$961$	1.29703e10	+	2.24653e10i	0.471433	+	0.816545i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.11680e10			0.400064
$966$		0			0
$967$	2.42924e9			0.0863928			0.0431964	−	0.999067i	$-0.486246\pi$
$967$	2.42924e9			0.0863928			0.0431964	+	0.999067i	$0.486246\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2.48392e10	−	4.30227e10i	−0.870703	−	1.50810i	−0.861271	−	0.508146i	$-0.830331\pi$
$971$	−2.48392e10	−	4.30227e10i	−0.870703	−	1.50810i	−0.00943237	−	0.999956i	$-0.503002\pi$
$972$		0			0
$973$	1.23142e9	−	1.27855e10i	0.0428560	−	0.444962i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	8.26462e9	−	1.43147e10i	0.283525	−	0.491080i	−0.688725	−	0.725022i	$-0.741829\pi$
$977$	8.26462e9	−	1.43147e10i	0.283525	−	0.491080i	0.972250	+	0.233942i	$0.0751627\pi$
$978$		0			0
$979$	−6.10206e10			−2.07844
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.02453e10	+	1.77454e10i	−0.344023	+	0.595864i	−0.985176	−	0.171548i	$-0.945123\pi$
$983$	−1.02453e10	+	1.77454e10i	−0.344023	+	0.595864i	0.641153	+	0.767413i	$0.278456\pi$
$984$		0			0
$985$	1.21721e10	+	2.10827e10i	0.405826	+	0.702911i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.08059e9	+	8.79985e9i	0.167004	+	0.289260i
$990$		0			0
$991$	−1.87248e10	+	3.24322e10i	−0.611165	+	1.05857i	0.379879	+	0.925036i	$0.375966\pi$
$991$	−1.87248e10	+	3.24322e10i	−0.611165	+	1.05857i	−0.991044	+	0.133533i	$0.957368\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.20508e9			−0.264060
$996$		0			0
$997$	−1.57909e9	+	2.73507e9i	−0.0504632	+	0.0874049i	−0.890154	−	0.455661i	$-0.849403\pi$
$997$	−1.57909e9	+	2.73507e9i	−0.0504632	+	0.0874049i	0.839690	+	0.543065i	$0.182736\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.8.k.e.109.6	yes	16
3.2	odd	2	inner	252.8.k.e.109.3	yes	16
7.2	even	3	inner	252.8.k.e.37.6	yes	16
21.2	odd	6	inner	252.8.k.e.37.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.8.k.e.37.3	✓	16	21.2	odd	6	inner
252.8.k.e.37.6	yes	16	7.2	even	3	inner
252.8.k.e.109.3	yes	16	3.2	odd	2	inner
252.8.k.e.109.6	yes	16	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(135.624 + 234.907i) q^{5} +(87.0016 - 903.313i) q^{7} +(2753.41 - 4769.04i) q^{11} +8802.63 q^{13} +(-3420.94 + 5925.24i) q^{17} +(13231.5 + 22917.6i) q^{19} +(-41328.3 - 71582.8i) q^{23} +(2274.86 - 3940.17i) q^{25} -233335. q^{29} +(19823.7 - 34335.7i) q^{31} +(223994. - 102073. i) q^{35} +(-115873. - 200699. i) q^{37} +403925. q^{41} -122932. q^{43} +(52674.9 + 91235.6i) q^{47} +(-808404. - 157179. i) q^{49} +(-665248. + 1.15224e6i) q^{53} +1.49371e6 q^{55} +(429413. - 743764. i) q^{59} +(525507. + 910205. i) q^{61} +(1.19385e6 + 2.06780e6i) q^{65} +(2.21180e6 - 3.83096e6i) q^{67} -154446. q^{71} +(1.07857e6 - 1.86813e6i) q^{73} +(-4.06838e6 - 2.90210e6i) q^{77} +(-954414. - 1.65309e6i) q^{79} +8.56018e6 q^{83} -1.85584e6 q^{85} +(-5.54046e6 - 9.59637e6i) q^{89} +(765843. - 7.95153e6i) q^{91} +(-3.58900e6 + 6.21634e6i) q^{95} +1.13502e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 1680 q^{7} - 28280 q^{13} + 42224 q^{19} - 80460 q^{25} + 164752 q^{31} - 647980 q^{37} + 1341440 q^{43} + 230104 q^{49} - 323120 q^{55} - 4319336 q^{61} - 3905760 q^{67} + 6471780 q^{73} - 6093104 q^{79}+ \cdots - 27141240 q^{97}+O(q^{100}) \) 16 * q + 1680 * q^7 - 28280 * q^13 + 42224 * q^19 - 80460 * q^25 + 164752 * q^31 - 647980 * q^37 + 1341440 * q^43 + 230104 * q^49 - 323120 * q^55 - 4319336 * q^61 - 3905760 * q^67 + 6471780 * q^73 - 6093104 * q^79 + 456400 * q^85 + 15969856 * q^91 - 27141240 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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