LMFDB - Embedded newform 45.12.b.c.19.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,12,Mod(19,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45.19"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	45.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$34.5754431252$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2 x^{7} - 5706 x^{6} + 91460 x^{5} + 8073323 x^{4} - 237717036 x^{3} + 1329030846 x^{2} + \cdots + 2557326711753 $ x^8 - 2x^7 - 5706x^6 + 91460x^5 + 8073323x^4 - 237717036x^3 + 1329030846x^2 - 44751025314*x + 2557326711753
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{12}\cdot 5^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.6
Root			$-9.06433 - 17.3199i$ of defining polynomial
Character	$\chi$	$=$	45.19
Dual form			45.12.b.c.19.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+22.0579i q^{2} +1561.45 q^{4} +(5411.49 + 4420.85i) q^{5} +55546.4i q^{7} +79616.9i q^{8} +(-97514.6 + 119366. i) q^{10} -390374. q^{11} +294278. i q^{13} -1.22524e6 q^{14} +1.44167e6 q^{16} +1.34966e6i q^{17} +4.87782e6 q^{19} +(8.44977e6 + 6.90293e6i) q^{20} -8.61083e6i q^{22} -2.15035e7i q^{23} +(9.74030e6 + 4.78468e7i) q^{25} -6.49116e6 q^{26} +8.67329e7i q^{28} -1.06704e8 q^{29} +8.85723e7 q^{31} +1.94855e8i q^{32} -2.97707e7 q^{34} +(-2.45562e8 + 3.00589e8i) q^{35} -7.15174e8i q^{37} +1.07594e8i q^{38} +(-3.51974e8 + 4.30846e8i) q^{40} -1.01025e9 q^{41} +6.84508e7i q^{43} -6.09549e8 q^{44} +4.74321e8 q^{46} +1.18367e9i q^{47} -1.10807e9 q^{49} +(-1.05540e9 + 2.14851e8i) q^{50} +4.59501e8i q^{52} +5.58230e9i q^{53} +(-2.11250e9 - 1.72578e9i) q^{55} -4.42243e9 q^{56} -2.35366e9i q^{58} -8.83822e9 q^{59} +9.40678e9 q^{61} +1.95372e9i q^{62} -1.34557e9 q^{64} +(-1.30096e9 + 1.59248e9i) q^{65} +1.67537e10i q^{67} +2.10743e9i q^{68} +(-6.63035e9 - 5.41659e9i) q^{70} +9.80178e9 q^{71} +7.18622e9i q^{73} +1.57752e10 q^{74} +7.61646e9 q^{76} -2.16839e10i q^{77} +2.66926e9 q^{79} +(7.80157e9 + 6.37340e9i) q^{80} -2.22841e10i q^{82} -4.33938e10i q^{83} +(-5.96665e9 + 7.30368e9i) q^{85} -1.50988e9 q^{86} -3.10803e10i q^{88} +5.89906e10 q^{89} -1.63461e10 q^{91} -3.35766e10i q^{92} -2.61093e10 q^{94} +(2.63963e10 + 2.15641e10i) q^{95} -9.59614e10i q^{97} -2.44418e10i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8976 q^{4} - 1155800 q^{10} + 15740992 q^{16} + 61220032 q^{19} + 184187000 q^{25} + 279698464 q^{31} + 309623120 q^{34} - 62144800 q^{40} - 13964841760 q^{46} - 1596098056 q^{49} + 15819804000 q^{55}+ \cdots - 460752353440 q^{94}+O(q^{100}) $ 8 * q - 8976 * q^4 - 1155800 * q^10 + 15740992 * q^16 + 61220032 * q^19 + 184187000 * q^25 + 279698464 * q^31 + 309623120 * q^34 - 62144800 * q^40 - 13964841760 * q^46 - 1596098056 * q^49 + 15819804000 * q^55 - 4923969584 * q^61 + 40933220096 * q^64 + 44585352000 * q^70 - 128254720704 * q^76 + 87780965728 * q^79 - 9845896400 * q^85 + 71755992000 * q^91 - 460752353440 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$			22.0579i			0.487415i	0.969849	+	0.243708i	$0.0783637\pi$
$2$			22.0579i			0.487415i	−0.969849	+	0.243708i	$0.921636\pi$
$3$		0			0
$3$		0			0
$4$	1561.45			0.762426
$5$	5411.49	+	4420.85i	0.774429	+	0.632661i
$5$	5411.49	+	4420.85i	0.774429	+	0.632661i
$6$		0			0
$7$			55546.4i			1.24916i	0.780963	+	0.624578i	$0.214729\pi$
$7$			55546.4i			1.24916i	−0.780963	+	0.624578i	$0.785271\pi$
$8$			79616.9i			0.859033i
$9$		0			0
$10$	−97514.6	+	119366.i	−0.308368	+	0.377469i
$11$	−390374.			−0.730838			−0.365419	−	0.930843i	$-0.619074\pi$
$11$	−390374.			−0.730838			−0.365419	+	0.930843i	$0.619074\pi$
$12$		0			0
$13$			294278.i			0.219821i	0.993941	+	0.109911i	$0.0350565\pi$
$13$			294278.i			0.219821i	−0.993941	+	0.109911i	$0.964944\pi$
$14$	−1.22524e6			−0.608858
$15$		0			0
$16$	1.44167e6			0.343720
$17$			1.34966e6i			0.230545i	0.993334	+	0.115272i	$0.0367741\pi$
$17$			1.34966e6i			0.230545i	−0.993334	+	0.115272i	$0.963226\pi$
$18$		0			0
$19$	4.87782e6			0.451940			0.225970	−	0.974134i	$-0.427445\pi$
$19$	4.87782e6			0.451940			0.225970	+	0.974134i	$0.427445\pi$
$20$	8.44977e6	+	6.90293e6i	0.590445	+	0.482357i
$21$		0			0
$22$		−	8.61083e6i		−	0.356221i
$23$		−	2.15035e7i		−	0.696635i	−0.937377	−	0.348318i	$-0.886753\pi$
$23$		−	2.15035e7i		−	0.696635i	0.937377	−	0.348318i	$-0.113247\pi$
$24$		0			0
$25$	9.74030e6	+	4.78468e7i	0.199481	+	0.979902i
$26$	−6.49116e6			−0.107144
$27$		0			0
$28$			8.67329e7i			0.952389i
$29$	−1.06704e8			−0.966030			−0.483015	−	0.875612i	$-0.660458\pi$
$29$	−1.06704e8			−0.966030			−0.483015	+	0.875612i	$0.660458\pi$
$30$		0			0
$31$	8.85723e7			0.555659			0.277830	−	0.960630i	$-0.410385\pi$
$31$	8.85723e7			0.555659			0.277830	+	0.960630i	$0.410385\pi$
$32$			1.94855e8i			1.02657i
$33$		0			0
$34$	−2.97707e7			−0.112371
$35$	−2.45562e8	+	3.00589e8i	−0.790292	+	0.967383i
$36$		0			0
$37$		−	7.15174e8i		−	1.69552i	−0.530382	−	0.847759i	$-0.677951\pi$
$37$		−	7.15174e8i		−	1.69552i	0.530382	−	0.847759i	$-0.322049\pi$
$38$			1.07594e8i			0.220282i
$39$		0			0
$40$	−3.51974e8	+	4.30846e8i	−0.543477	+	0.665261i
$41$	−1.01025e9			−1.36182			−0.680909	−	0.732368i	$-0.738415\pi$
$41$	−1.01025e9			−1.36182			−0.680909	+	0.732368i	$0.738415\pi$
$42$		0			0
$43$			6.84508e7i			0.0710072i	0.999370	+	0.0355036i	$0.0113035\pi$
$43$			6.84508e7i			0.0710072i	−0.999370	+	0.0355036i	$0.988696\pi$
$44$	−6.09549e8			−0.557210
$45$		0			0
$46$	4.74321e8			0.339551
$47$			1.18367e9i			0.752822i	0.926453	+	0.376411i	$0.122842\pi$
$47$			1.18367e9i			0.752822i	−0.926453	+	0.376411i	$0.877158\pi$
$48$		0			0
$49$	−1.10807e9			−0.560390
$50$	−1.05540e9	+	2.14851e8i	−0.477619	+	0.0972303i
$51$		0			0
$52$			4.59501e8i			0.167598i
$53$			5.58230e9i			1.83356i	0.399389	+	0.916781i	$0.369222\pi$
$53$			5.58230e9i			1.83356i	−0.399389	+	0.916781i	$0.630778\pi$
$54$		0			0
$55$	−2.11250e9	−	1.72578e9i	−0.565982	−	0.462372i
$56$	−4.42243e9			−1.07307
$57$		0			0
$58$		−	2.35366e9i		−	0.470858i
$59$	−8.83822e9			−1.60945			−0.804727	−	0.593645i	$-0.797688\pi$
$59$	−8.83822e9			−1.60945			−0.804727	+	0.593645i	$0.797688\pi$
$60$		0			0
$61$	9.40678e9			1.42602			0.713012	−	0.701152i	$-0.247330\pi$
$61$	9.40678e9			1.42602			0.713012	+	0.701152i	$0.247330\pi$
$62$			1.95372e9i			0.270837i
$63$		0			0
$64$	−1.34557e9			−0.156644
$65$	−1.30096e9	+	1.59248e9i	−0.139072	+	0.170236i
$66$		0			0
$67$			1.67537e10i			1.51600i	0.652253	+	0.758002i	$0.273824\pi$
$67$			1.67537e10i			1.51600i	−0.652253	+	0.758002i	$0.726176\pi$
$68$			2.10743e9i			0.175774i
$69$		0			0
$70$	−6.63035e9	−	5.41659e9i	−0.471517	−	0.385200i
$71$	9.80178e9			0.644739			0.322370	−	0.946614i	$-0.395521\pi$
$71$	9.80178e9			0.644739			0.322370	+	0.946614i	$0.395521\pi$
$72$		0			0
$73$			7.18622e9i			0.405718i	0.979208	+	0.202859i	$0.0650234\pi$
$73$			7.18622e9i			0.405718i	−0.979208	+	0.202859i	$0.934977\pi$
$74$	1.57752e10			0.826421
$75$		0			0
$76$	7.61646e9			0.344571
$77$		−	2.16839e10i		−	0.912930i
$78$		0			0
$79$	2.66926e9			0.0975983			0.0487991	−	0.998809i	$-0.484461\pi$
$79$	2.66926e9			0.0975983			0.0487991	+	0.998809i	$0.484461\pi$
$80$	7.80157e9	+	6.37340e9i	0.266187	+	0.217458i
$81$		0			0
$82$		−	2.22841e10i		−	0.663771i
$83$		−	4.33938e10i		−	1.20920i	−0.796529	−	0.604600i	$-0.793333\pi$
$83$		−	4.33938e10i		−	1.20920i	0.796529	−	0.604600i	$-0.206667\pi$
$84$		0			0
$85$	−5.96665e9	+	7.30368e9i	−0.145857	+	0.178541i
$86$	−1.50988e9			−0.0346100
$87$		0			0
$88$		−	3.10803e10i		−	0.627814i
$89$	5.89906e10			1.11979			0.559896	−	0.828563i	$-0.310841\pi$
$89$	5.89906e10			1.11979			0.559896	+	0.828563i	$0.310841\pi$
$90$		0			0
$91$	−1.63461e10			−0.274591
$92$		−	3.35766e10i		−	0.531133i
$93$		0			0
$94$	−2.61093e10			−0.366937
$95$	2.63963e10	+	2.15641e10i	0.349996	+	0.285925i
$96$		0			0
$97$		−	9.59614e10i		−	1.13462i	−0.823503	−	0.567312i	$-0.807983\pi$
$97$		−	9.59614e10i		−	1.13462i	0.823503	−	0.567312i	$-0.192017\pi$
$98$		−	2.44418e10i		−	0.273143i
$99$		0			0
$100$	1.52090e10	+	7.47103e10i	0.152090	+	0.747103i
$101$	1.58971e11			1.50505			0.752524	−	0.658565i	$-0.228836\pi$
$101$	1.58971e11			1.50505			0.752524	+	0.658565i	$0.228836\pi$
$102$		0			0
$103$			1.21843e11i			1.03561i	0.855499	+	0.517805i	$0.173251\pi$
$103$			1.21843e11i			1.03561i	−0.855499	+	0.517805i	$0.826749\pi$
$104$	−2.34295e10			−0.188834
$105$		0			0
$106$	−1.23134e11			−0.893707
$107$		−	2.64305e11i		−	1.82178i	−0.412653	−	0.910888i	$-0.635398\pi$
$107$		−	2.64305e11i		−	1.82178i	0.412653	−	0.910888i	$-0.364602\pi$
$108$		0			0
$109$	1.84618e11			1.14929			0.574644	−	0.818403i	$-0.305140\pi$
$109$	1.84618e11			1.14929			0.574644	+	0.818403i	$0.305140\pi$
$110$	3.80672e10	−	4.65974e10i	0.225367	−	0.275868i
$111$		0			0
$112$			8.00794e10i			0.429360i
$113$			2.06690e11i			1.05533i	0.849453	+	0.527664i	$0.176932\pi$
$113$			2.06690e11i			1.05533i	−0.849453	+	0.527664i	$0.823068\pi$
$114$		0			0
$115$	9.50636e10	−	1.16366e11i	0.440733	−	0.539495i
$116$	−1.66613e11			−0.736527
$117$		0			0
$118$		−	1.94952e11i		−	0.784472i
$119$	−7.49688e10			−0.287987
$120$		0			0
$121$	−1.32920e11			−0.465876
$122$			2.07494e11i			0.695066i
$123$		0			0
$124$	1.38301e11			0.423649
$125$	−1.58814e11	+	3.01983e11i	−0.465461	+	0.885068i
$126$		0			0
$127$			3.53722e11i			0.950039i	0.879975	+	0.475020i	$0.157559\pi$
$127$			3.53722e11i			0.950039i	−0.879975	+	0.475020i	$0.842441\pi$
$128$			3.69384e11i			0.950217i
$129$		0			0
$130$	−3.51269e10	−	2.86965e10i	−0.0829757	−	0.0677860i
$131$	−4.97397e11			−1.12645			−0.563223	−	0.826305i	$-0.690439\pi$
$131$	−4.97397e11			−1.12645			−0.563223	+	0.826305i	$0.690439\pi$
$132$		0			0
$133$			2.70945e11i			0.564543i
$134$	−3.69552e11			−0.738923
$135$		0			0
$136$	−1.07456e11			−0.198046
$137$		−	3.10911e11i		−	0.550393i	−0.961388	−	0.275196i	$-0.911257\pi$
$137$		−	3.10911e11i		−	0.550393i	0.961388	−	0.275196i	$-0.0887429\pi$
$138$		0			0
$139$	1.80237e11			0.294620			0.147310	−	0.989090i	$-0.452938\pi$
$139$	1.80237e11			0.294620			0.147310	+	0.989090i	$0.452938\pi$
$140$	−3.83433e11	+	4.69354e11i	−0.602539	+	0.737558i
$141$		0			0
$142$			2.16207e11i			0.314256i
$143$		−	1.14879e11i		−	0.160654i
$144$		0			0
$145$	−5.77426e11	−	4.71721e11i	−0.748122	−	0.611169i
$146$	−1.58513e11			−0.197753
$147$		0			0
$148$		−	1.11671e12i		−	1.29271i
$149$	2.91760e11			0.325463			0.162731	−	0.986670i	$-0.447970\pi$
$149$	2.91760e11			0.325463			0.162731	+	0.986670i	$0.447970\pi$
$150$		0			0
$151$	−2.33328e11			−0.241876			−0.120938	−	0.992660i	$-0.538590\pi$
$151$	−2.33328e11			−0.241876			−0.120938	+	0.992660i	$0.538590\pi$
$152$			3.88356e11i			0.388232i
$153$		0			0
$154$	4.78300e11			0.444976
$155$	4.79308e11	+	3.91565e11i	0.430319	+	0.351544i
$156$		0			0
$157$		−	1.54856e12i		−	1.29563i	−0.761800	−	0.647813i	$-0.775684\pi$
$157$		−	1.54856e12i		−	1.29563i	0.761800	−	0.647813i	$-0.224316\pi$
$158$			5.88783e10i			0.0475709i
$159$		0			0
$160$	−8.61427e11	+	1.05446e12i	−0.649469	+	0.795004i
$161$	1.19444e12			0.870206
$162$		0			0
$163$			1.69870e12i			1.15634i	0.815917	+	0.578168i	$0.196232\pi$
$163$			1.69870e12i			1.15634i	−0.815917	+	0.578168i	$0.803768\pi$
$164$	−1.57746e12			−1.03829
$165$		0			0
$166$	9.57176e11			0.589383
$167$		−	1.59377e12i		−	0.949481i	−0.880126	−	0.474740i	$-0.842542\pi$
$167$		−	1.59377e12i		−	0.949481i	0.880126	−	0.474740i	$-0.157458\pi$
$168$		0			0
$169$	1.70556e12			0.951679
$170$	−1.61104e11	−	1.31612e11i	−0.0870235	−	0.0710928i
$171$		0			0
$172$			1.06883e11i			0.0541378i
$173$		−	1.16978e12i		−	0.573917i	−0.957943	−	0.286958i	$-0.907356\pi$
$173$		−	1.16978e12i		−	0.573917i	0.957943	−	0.286958i	$-0.0926441\pi$
$174$		0			0
$175$	−2.65771e12	+	5.41039e11i	−1.22405	+	0.249183i
$176$	−5.62789e11			−0.251204
$177$		0			0
$178$			1.30121e12i			0.545804i
$179$	4.37250e12			1.77844			0.889218	−	0.457484i	$-0.151249\pi$
$179$	4.37250e12			1.77844			0.889218	+	0.457484i	$0.151249\pi$
$180$		0			0
$181$	5.31067e11			0.203197			0.101599	−	0.994825i	$-0.467604\pi$
$181$	5.31067e11			0.203197			0.101599	+	0.994825i	$0.467604\pi$
$182$		−	3.60561e11i		−	0.133840i
$183$		0			0
$184$	1.71204e12			0.598433
$185$	3.16168e12	−	3.87016e12i	1.07269	−	1.31306i
$186$		0			0
$187$		−	5.26872e11i		−	0.168491i
$188$			1.84824e12i			0.573971i
$189$		0			0
$190$	−4.75659e11	+	5.82246e11i	−0.139364	+	0.170593i
$191$	4.26612e12			1.21437			0.607183	−	0.794562i	$-0.292300\pi$
$191$	4.26612e12			1.21437			0.607183	+	0.794562i	$0.292300\pi$
$192$		0			0
$193$		−	3.58148e12i		−	0.962713i	−0.876525	−	0.481356i	$-0.840144\pi$
$193$		−	3.58148e12i		−	0.962713i	0.876525	−	0.481356i	$-0.159856\pi$
$194$	2.11671e12			0.553033
$195$		0			0
$196$	−1.73020e12			−0.427256
$197$		−	3.14971e12i		−	0.756322i	−0.925740	−	0.378161i	$-0.876557\pi$
$197$		−	3.14971e12i		−	0.756322i	0.925740	−	0.378161i	$-0.123443\pi$
$198$		0			0
$199$	7.71437e12			1.75230			0.876150	−	0.482039i	$-0.160104\pi$
$199$	7.71437e12			1.75230			0.876150	+	0.482039i	$0.160104\pi$
$200$	−3.80941e12	+	7.75492e11i	−0.841768	+	0.171361i
$201$		0			0
$202$			3.50656e12i			0.733583i
$203$		−	5.92701e12i		−	1.20672i
$204$		0			0
$205$	−5.46698e12	−	4.46618e12i	−1.05463	−	0.861569i
$206$	−2.68760e12			−0.504772
$207$		0			0
$208$			4.24252e11i			0.0755571i
$209$	−1.90417e12			−0.330295
$210$		0			0
$211$	−1.06063e13			−1.74586			−0.872931	−	0.487843i	$-0.837784\pi$
$211$	−1.06063e13			−1.74586			−0.872931	+	0.487843i	$0.837784\pi$
$212$			8.71648e12i			1.39796i
$213$		0			0
$214$	5.83002e12			0.887962
$215$	−3.02611e11	+	3.70421e11i	−0.0449234	+	0.0549900i
$216$		0			0
$217$			4.91987e12i			0.694105i
$218$			4.07229e12i			0.560181i
$219$		0			0
$220$	−3.29857e12	−	2.69472e12i	−0.431520	−	0.352525i
$221$	−3.97176e11			−0.0506787
$222$		0			0
$223$		−	1.19680e12i		−	0.145326i	−0.997357	−	0.0726630i	$-0.976850\pi$
$223$		−	1.19680e12i		−	0.145326i	0.997357	−	0.0726630i	$-0.0231497\pi$
$224$	−1.08235e13			−1.28234
$225$		0			0
$226$	−4.55914e12			−0.514383
$227$			1.43614e13i			1.58145i	0.612172	+	0.790724i	$0.290296\pi$
$227$			1.43614e13i			1.58145i	−0.612172	+	0.790724i	$0.709704\pi$
$228$		0			0
$229$	1.73779e13			1.82349			0.911744	−	0.410758i	$-0.134736\pi$
$229$	1.73779e13			1.82349			0.911744	+	0.410758i	$0.134736\pi$
$230$	2.56678e12	+	2.09690e12i	0.262958	+	0.214820i
$231$		0			0
$232$		−	8.49542e12i		−	0.829852i
$233$		−	4.48575e12i		−	0.427935i	−0.976841	−	0.213968i	$-0.931361\pi$
$233$		−	4.48575e12i		−	0.427935i	0.976841	−	0.213968i	$-0.0686387\pi$
$234$		0			0
$235$	−5.23282e12	+	6.40541e12i	−0.476281	+	0.583007i
$236$	−1.38004e13			−1.22709
$237$		0			0
$238$		−	1.65365e12i		−	0.140369i
$239$	−7.64513e10			−0.00634157			−0.00317078	−	0.999995i	$-0.501009\pi$
$239$	−7.64513e10			−0.00634157			−0.00317078	+	0.999995i	$0.501009\pi$
$240$		0			0
$241$	−1.06281e13			−0.842097			−0.421049	−	0.907038i	$-0.638338\pi$
$241$	−1.06281e13			−0.842097			−0.421049	+	0.907038i	$0.638338\pi$
$242$		−	2.93193e12i		−	0.227075i
$243$		0			0
$244$	1.46882e13			1.08724
$245$	−5.99633e12	−	4.89863e12i	−0.433983	−	0.354537i
$246$		0			0
$247$			1.43544e12i			0.0993461i
$248$			7.05184e12i			0.477330i
$249$		0			0
$250$	−6.66110e12	−	3.50310e12i	−0.431396	−	0.226873i
$251$	4.24494e12			0.268946			0.134473	−	0.990917i	$-0.457066\pi$
$251$	4.24494e12			0.268946			0.134473	+	0.990917i	$0.457066\pi$
$252$		0			0
$253$			8.39439e12i			0.509127i
$254$	−7.80236e12			−0.463064
$255$		0			0
$256$	−1.09035e13			−0.619795
$257$			1.26005e13i			0.701059i	0.936552	+	0.350529i	$0.113998\pi$
$257$			1.26005e13i			0.701059i	−0.936552	+	0.350529i	$0.886002\pi$
$258$		0			0
$259$	3.97253e13			2.11797
$260$	−2.03138e12	+	2.48658e12i	−0.106032	+	0.129793i
$261$		0			0
$262$		−	1.09715e13i		−	0.549047i
$263$			7.46619e12i			0.365883i	0.983124	+	0.182942i	$0.0585620\pi$
$263$			7.46619e12i			0.365883i	−0.983124	+	0.182942i	$0.941438\pi$
$264$		0			0
$265$	−2.46785e13	+	3.02086e13i	−1.16002	+	1.41996i
$266$	−5.97648e12			−0.275167
$267$		0			0
$268$			2.61601e13i			1.15584i
$269$	1.69203e13			0.732435			0.366218	−	0.930529i	$-0.380653\pi$
$269$	1.69203e13			0.732435			0.366218	+	0.930529i	$0.380653\pi$
$270$		0			0
$271$	2.59160e12			0.107705			0.0538526	−	0.998549i	$-0.482850\pi$
$271$	2.59160e12			0.107705			0.0538526	+	0.998549i	$0.482850\pi$
$272$			1.94576e12i			0.0792430i
$273$		0			0
$274$	6.85804e12			0.268270
$275$	−3.80236e12	−	1.86781e13i	−0.145788	−	0.716149i
$276$		0			0
$277$		−	8.12272e12i		−	0.299270i	−0.988741	−	0.149635i	$-0.952190\pi$
$277$		−	8.12272e12i		−	0.299270i	0.988741	−	0.149635i	$-0.0478098\pi$
$278$			3.97565e12i			0.143602i
$279$		0			0
$280$	−2.39319e13	−	1.95509e13i	−0.831014	−	0.678887i
$281$	1.40507e13			0.478425			0.239213	−	0.970967i	$-0.423111\pi$
$281$	1.40507e13			0.478425			0.239213	+	0.970967i	$0.423111\pi$
$282$		0			0
$283$		−	5.56339e13i		−	1.82186i	−0.412563	−	0.910929i	$-0.635366\pi$
$283$		−	5.56339e13i		−	1.82186i	0.412563	−	0.910929i	$-0.364634\pi$
$284$	1.53050e13			0.491566
$285$		0			0
$286$	2.53398e12			0.0783051
$287$		−	5.61159e13i		−	1.70112i
$288$		0			0
$289$	3.24503e13			0.946849
$290$	1.04052e13	−	1.27368e13i	0.297893	−	0.364646i
$291$		0			0
$292$			1.12209e13i			0.309330i
$293$		−	3.50276e13i		−	0.947631i	−0.880624	−	0.473815i	$-0.842876\pi$
$293$		−	3.50276e13i		−	0.947631i	0.880624	−	0.473815i	$-0.157124\pi$
$294$		0			0
$295$	−4.78279e13	−	3.90724e13i	−1.24641	−	1.01824i
$296$	5.69399e13			1.45651
$297$		0			0
$298$			6.43561e12i			0.158635i
$299$	6.32801e12			0.153135
$300$		0			0
$301$	−3.80220e12			−0.0886990
$302$		−	5.14672e12i		−	0.117894i
$303$		0			0
$304$	7.03219e12			0.155341
$305$	5.09047e13	+	4.15860e13i	1.10435	+	0.902189i
$306$		0			0
$307$			4.17912e13i			0.874628i	0.899309	+	0.437314i	$0.144070\pi$
$307$			4.17912e13i			0.874628i	−0.899309	+	0.437314i	$0.855930\pi$
$308$		−	3.38582e13i		−	0.696042i
$309$		0			0
$310$	−8.63709e12	+	1.05725e13i	−0.171348	+	0.209744i
$311$	−1.46541e13			−0.285612			−0.142806	−	0.989751i	$-0.545612\pi$
$311$	−1.46541e13			−0.285612			−0.142806	+	0.989751i	$0.545612\pi$
$312$		0			0
$313$			4.36887e13i			0.822007i	0.911634	+	0.411004i	$0.134822\pi$
$313$			4.36887e13i			0.822007i	−0.911634	+	0.411004i	$0.865178\pi$
$314$	3.41579e13			0.631507
$315$		0			0
$316$	4.16792e12			0.0744115
$317$			8.62787e12i			0.151383i	0.997131	+	0.0756916i	$0.0241164\pi$
$317$			8.62787e12i			0.151383i	−0.997131	+	0.0756916i	$0.975884\pi$
$318$		0			0
$319$	4.16544e13			0.706011
$320$	−7.28151e12	−	5.94854e12i	−0.121310	−	0.0991028i
$321$		0			0
$322$			2.63468e13i			0.424151i
$323$			6.58340e12i			0.104192i
$324$		0			0
$325$	−1.40803e13	+	2.86636e12i	−0.215403	+	0.0438503i
$326$	−3.74697e13			−0.563616
$327$		0			0
$328$		−	8.04332e13i		−	1.16985i
$329$	−6.57486e13			−0.940392
$330$		0			0
$331$	−2.29938e13			−0.318095			−0.159048	−	0.987271i	$-0.550842\pi$
$331$	−2.29938e13			−0.318095			−0.159048	+	0.987271i	$0.550842\pi$
$332$		−	6.77572e13i		−	0.921926i
$333$		0			0
$334$	3.51553e13			0.462791
$335$	−7.40657e13	+	9.06626e13i	−0.959115	+	1.17404i
$336$		0			0
$337$		−	2.07560e13i		−	0.260123i	−0.991506	−	0.130062i	$-0.958483\pi$
$337$		−	2.07560e13i		−	0.260123i	0.991506	−	0.130062i	$-0.0415175\pi$
$338$			3.76211e13i			0.463863i
$339$		0			0
$340$	−9.31662e12	+	1.14043e13i	−0.111205	+	0.136124i
$341$	−3.45763e13			−0.406097
$342$		0			0
$343$			4.82838e13i			0.549141i
$344$	−5.44984e12			−0.0609975
$345$		0			0
$346$	2.58028e13			0.279736
$347$			4.77409e13i			0.509423i	0.967017	+	0.254711i	$0.0819805\pi$
$347$			4.77409e13i			0.509423i	−0.967017	+	0.254711i	$0.918020\pi$
$348$		0			0
$349$	3.89716e13			0.402910			0.201455	−	0.979498i	$-0.435433\pi$
$349$	3.89716e13			0.402910			0.201455	+	0.979498i	$0.435433\pi$
$350$	−1.19342e13	−	5.86236e13i	−0.121456	−	0.596620i
$351$		0			0
$352$		−	7.60665e13i		−	0.750255i
$353$			1.70145e14i			1.65218i	0.563536	+	0.826091i	$0.309441\pi$
$353$			1.70145e14i			1.65218i	−0.563536	+	0.826091i	$0.690559\pi$
$354$		0			0
$355$	5.30422e13	+	4.33322e13i	0.499305	+	0.407901i
$356$	9.21108e13			0.853760
$357$		0			0
$358$			9.64481e13i			0.866837i
$359$	−2.00471e13			−0.177432			−0.0887160	−	0.996057i	$-0.528276\pi$
$359$	−2.00471e13			−0.177432			−0.0887160	+	0.996057i	$0.528276\pi$
$360$		0			0
$361$	−9.26972e13			−0.795750
$362$			1.17142e13i			0.0990413i
$363$		0			0
$364$	−2.55236e13			−0.209356
$365$	−3.17692e13	+	3.88882e13i	−0.256682	+	0.314200i
$366$		0			0
$367$		−	1.03112e14i		−	0.808433i	−0.914663	−	0.404217i	$-0.867544\pi$
$367$		−	1.03112e14i		−	0.808433i	0.914663	−	0.404217i	$-0.132456\pi$
$368$		−	3.10009e13i		−	0.239448i
$369$		0			0
$370$	8.53675e13	+	6.97399e13i	0.640005	+	0.522844i
$371$	−3.10077e14			−2.29041
$372$		0			0
$373$		−	1.04079e13i		−	0.0746386i	−0.999303	−	0.0373193i	$-0.988118\pi$
$373$		−	1.04079e13i		−	0.0746386i	0.999303	−	0.0373193i	$-0.0118819\pi$
$374$	1.16217e13			0.0821251
$375$		0			0
$376$	−9.42400e13			−0.646699
$377$		−	3.14006e13i		−	0.212354i
$378$		0			0
$379$	−2.25424e14			−1.48076			−0.740380	−	0.672188i	$-0.765355\pi$
$379$	−2.25424e14			−1.48076			−0.740380	+	0.672188i	$0.765355\pi$
$380$	4.12164e13	+	3.36712e13i	0.266846	+	0.217996i
$381$		0			0
$382$			9.41016e13i			0.591900i
$383$			5.14914e13i			0.319258i	0.987177	+	0.159629i	$0.0510298\pi$
$383$			5.14914e13i			0.319258i	−0.987177	+	0.159629i	$0.948970\pi$
$384$		0			0
$385$	9.58611e13	−	1.17342e14i	0.577575	−	0.707000i
$386$	7.89998e13			0.469241
$387$		0			0
$388$		−	1.49839e14i		−	0.865068i
$389$	−3.01596e14			−1.71673			−0.858366	−	0.513037i	$-0.828520\pi$
$389$	−3.01596e14			−1.71673			−0.858366	+	0.513037i	$0.828520\pi$
$390$		0			0
$391$	2.90224e13			0.160606
$392$		−	8.82214e13i		−	0.481394i
$393$		0			0
$394$	6.94760e13			0.368643
$395$	1.44447e13	+	1.18004e13i	0.0755829	+	0.0617466i
$396$		0			0
$397$		−	1.38344e14i		−	0.704065i	−0.935988	−	0.352033i	$-0.885491\pi$
$397$		−	1.38344e14i		−	0.704065i	0.935988	−	0.352033i	$-0.114509\pi$
$398$			1.70163e14i			0.854097i
$399$		0			0
$400$	1.40423e13	+	6.89791e13i	0.0685658	+	0.336812i
$401$	2.15849e14			1.03958			0.519788	−	0.854295i	$-0.326011\pi$
$401$	2.15849e14			1.03958			0.519788	+	0.854295i	$0.326011\pi$
$402$		0			0
$403$			2.60649e13i			0.122146i
$404$	2.48225e14			1.14749
$405$		0			0
$406$	1.30737e14			0.588175
$407$			2.79185e14i			1.23915i
$408$		0			0
$409$	−2.13616e14			−0.922901			−0.461450	−	0.887166i	$-0.652671\pi$
$409$	−2.13616e14			−0.922901			−0.461450	+	0.887166i	$0.652671\pi$
$410$	9.85145e13	−	1.20590e14i	0.419942	−	0.514044i
$411$		0			0
$412$			1.90252e14i			0.789576i
$413$		−	4.90931e14i		−	2.01046i
$414$		0			0
$415$	1.91837e14	−	2.34825e14i	0.765013	−	0.936440i
$416$	−5.73418e13			−0.225662
$417$		0			0
$418$		−	4.20020e13i		−	0.160991i
$419$	3.09233e14			1.16979			0.584896	−	0.811108i	$-0.301135\pi$
$419$	3.09233e14			1.16979			0.584896	+	0.811108i	$0.301135\pi$
$420$		0			0
$421$	2.18306e14			0.804476			0.402238	−	0.915535i	$-0.368232\pi$
$421$	2.18306e14			0.804476			0.402238	+	0.915535i	$0.368232\pi$
$422$		−	2.33953e14i		−	0.850960i
$423$		0			0
$424$	−4.44445e14			−1.57509
$425$	−6.45769e13	+	1.31461e13i	−0.225911	+	0.0459894i
$426$		0			0
$427$			5.22513e14i			1.78133i
$428$		−	4.12699e14i		−	1.38897i
$429$		0			0
$430$	−8.17071e12	−	6.67496e12i	−0.0268030	−	0.0218964i
$431$	4.05282e14			1.31260			0.656300	−	0.754500i	$-0.272121\pi$
$431$	4.05282e14			1.31260			0.656300	+	0.754500i	$0.272121\pi$
$432$		0			0
$433$			1.06068e14i			0.334888i	0.985882	+	0.167444i	$0.0535513\pi$
$433$			1.06068e14i			0.334888i	−0.985882	+	0.167444i	$0.946449\pi$
$434$	−1.08522e14			−0.338317
$435$		0			0
$436$	2.88272e14			0.876248
$437$		−	1.04890e14i		−	0.314837i
$438$		0			0
$439$	−2.34311e14			−0.685864			−0.342932	−	0.939360i	$-0.611420\pi$
$439$	−2.34311e14			−0.685864			−0.342932	+	0.939360i	$0.611420\pi$
$440$	1.37401e14	−	1.68191e14i	0.397193	−	0.486198i
$441$		0			0
$442$		−	8.76087e12i		−	0.0247016i
$443$		−	1.59155e13i		−	0.0443199i	−0.999754	−	0.0221600i	$-0.992946\pi$
$443$		−	1.59155e13i		−	0.0443199i	0.999754	−	0.0221600i	$-0.00705431\pi$
$444$		0			0
$445$	3.19227e14	+	2.60789e14i	0.867200	+	0.708449i
$446$	2.63988e13			0.0708341
$447$		0			0
$448$		−	7.47413e13i		−	0.195673i
$449$	5.35002e14			1.38357			0.691784	−	0.722104i	$-0.256825\pi$
$449$	5.35002e14			1.38357			0.691784	+	0.722104i	$0.256825\pi$
$450$		0			0
$451$	3.94377e14			0.995268
$452$			3.22735e14i			0.804610i
$453$		0			0
$454$	−3.16783e14			−0.770822
$455$	−8.84568e13	−	7.22637e13i	−0.212652	−	0.173723i
$456$		0			0
$457$		−	5.44568e14i		−	1.27795i	−0.769229	−	0.638973i	$-0.779359\pi$
$457$		−	5.44568e14i		−	1.27795i	0.769229	−	0.638973i	$-0.220641\pi$
$458$			3.83321e14i			0.888796i
$459$		0			0
$460$	1.48437e14	−	1.81699e14i	0.336027	−	0.411325i
$461$	−3.08085e14			−0.689154			−0.344577	−	0.938758i	$-0.611978\pi$
$461$	−3.08085e14			−0.689154			−0.344577	+	0.938758i	$0.611978\pi$
$462$		0			0
$463$			1.40262e14i			0.306370i	0.988198	+	0.153185i	$0.0489530\pi$
$463$			1.40262e14i			0.306370i	−0.988198	+	0.153185i	$0.951047\pi$
$464$	−1.53831e14			−0.332044
$465$		0			0
$466$	9.89463e13			0.208582
$467$			3.04077e14i			0.633491i	0.948511	+	0.316746i	$0.102590\pi$
$467$			3.04077e14i			0.633491i	−0.948511	+	0.316746i	$0.897410\pi$
$468$		0			0
$469$	−9.30609e14			−1.89372
$470$	−1.41290e14	−	1.15425e14i	−0.284167	−	0.232146i
$471$		0			0
$472$		−	7.03671e14i		−	1.38257i
$473$		−	2.67214e13i		−	0.0518947i
$474$		0			0
$475$	4.75114e13	+	2.33388e14i	0.0901536	+	0.442857i
$476$	−1.17060e14			−0.219569
$477$		0			0
$478$		−	1.68635e12i		−	0.00309098i
$479$	−8.80901e14			−1.59618			−0.798090	−	0.602538i	$-0.794156\pi$
$479$	−8.80901e14			−1.59618			−0.798090	+	0.602538i	$0.794156\pi$
$480$		0			0
$481$	2.10460e14			0.372711
$482$		−	2.34434e14i		−	0.410451i
$483$		0			0
$484$	−2.07548e14			−0.355196
$485$	4.24231e14	−	5.19294e14i	0.717832	−	0.878687i
$486$		0			0
$487$			3.64694e14i			0.603280i	0.953422	+	0.301640i	$0.0975340\pi$
$487$			3.64694e14i			0.603280i	−0.953422	+	0.301640i	$0.902466\pi$
$488$			7.48938e14i			1.22500i
$489$		0			0
$490$	1.08053e14	−	1.32266e14i	0.172807	−	0.211530i
$491$	−2.81758e14			−0.445582			−0.222791	−	0.974866i	$-0.571517\pi$
$491$	−2.81758e14			−0.445582			−0.222791	+	0.974866i	$0.571517\pi$
$492$		0			0
$493$		−	1.44014e14i		−	0.222713i
$494$	−3.16627e13			−0.0484228
$495$		0			0
$496$	1.27692e14			0.190991
$497$			5.44454e14i			0.805380i
$498$		0			0
$499$	−5.04088e14			−0.729379			−0.364690	−	0.931129i	$-0.618825\pi$
$499$	−5.04088e14			−0.729379			−0.364690	+	0.931129i	$0.618825\pi$
$500$	−2.47980e14	+	4.71531e14i	−0.354880	+	0.674800i
$501$		0			0
$502$			9.36344e13i			0.131089i
$503$		−	7.09290e14i		−	0.982200i	−0.871103	−	0.491100i	$-0.836595\pi$
$503$		−	7.09290e14i		−	0.982200i	0.871103	−	0.491100i	$-0.163405\pi$
$504$		0			0
$505$	8.60270e14	+	7.02787e14i	1.16555	+	0.952184i
$506$	−1.85163e14			−0.248156
$507$		0			0
$508$			5.52319e14i			0.724335i
$509$	−3.89311e12			−0.00505067			−0.00252534	−	0.999997i	$-0.500804\pi$
$509$	−3.89311e12			−0.00505067			−0.00252534	+	0.999997i	$0.500804\pi$
$510$		0			0
$511$	−3.99169e14			−0.506806
$512$			5.15989e14i			0.648120i
$513$		0			0
$514$	−2.77940e14			−0.341707
$515$	−5.38650e14	+	6.59352e14i	−0.655189	+	0.802006i
$516$		0			0
$517$		−	4.62074e14i		−	0.550191i
$518$			8.76257e14i			1.03233i
$519$		0			0
$520$	−1.26789e14	−	1.03578e14i	−0.146239	−	0.119468i
$521$	6.67006e14			0.761241			0.380621	−	0.924731i	$-0.375710\pi$
$521$	6.67006e14			0.761241			0.380621	+	0.924731i	$0.375710\pi$
$522$		0			0
$523$		−	9.54669e14i		−	1.06683i	−0.845855	−	0.533413i	$-0.820909\pi$
$523$		−	9.54669e14i		−	1.06683i	0.845855	−	0.533413i	$-0.179091\pi$
$524$	−7.76660e14			−0.858833
$525$		0			0
$526$	−1.64689e14			−0.178337
$527$			1.19543e14i			0.128104i
$528$		0			0
$529$	4.90411e14			0.514700
$530$	−6.66337e14	−	5.44356e14i	−0.692112	−	0.565413i
$531$		0			0
$532$			4.23067e14i			0.430423i
$533$		−	2.97296e14i		−	0.299357i
$534$		0			0
$535$	1.16845e15	−	1.43029e15i	1.15257	−	1.41084i
$536$	−1.33388e15			−1.30230
$537$		0			0
$538$			3.73225e14i			0.357000i
$539$	4.32563e14			0.409554
$540$		0			0
$541$	−2.01083e14			−0.186548			−0.0932739	−	0.995640i	$-0.529733\pi$
$541$	−2.01083e14			−0.186548			−0.0932739	+	0.995640i	$0.529733\pi$
$542$			5.71653e13i			0.0524972i
$543$		0			0
$544$	−2.62989e14			−0.236670
$545$	9.99060e14	+	8.16170e14i	0.890043	+	0.727110i
$546$		0			0
$547$			1.97509e15i			1.72447i	0.506507	+	0.862236i	$0.330936\pi$
$547$			1.97509e15i			1.72447i	−0.506507	+	0.862236i	$0.669064\pi$
$548$		−	4.85472e14i		−	0.419634i
$549$		0			0
$550$	4.12000e14	−	8.38720e13i	0.349062	−	0.0710595i
$551$	−5.20481e14			−0.436588
$552$		0			0
$553$			1.48268e14i			0.121915i
$554$	1.79170e14			0.145869
$555$		0			0
$556$	2.81431e14			0.224626
$557$		−	1.73969e15i		−	1.37489i	−0.726238	−	0.687444i	$-0.758733\pi$
$557$		−	1.73969e15i		−	1.37489i	0.726238	−	0.687444i	$-0.241267\pi$
$558$		0			0
$559$	−2.01436e13			−0.0156089
$560$	−3.54019e14	+	4.33349e14i	−0.271639	+	0.332509i
$561$		0			0
$562$			3.09929e14i			0.233192i
$563$		−	7.00557e14i		−	0.521972i	−0.965343	−	0.260986i	$-0.915952\pi$
$563$		−	7.00557e14i		−	0.521972i	0.965343	−	0.260986i	$-0.0840477\pi$
$564$		0			0
$565$	−9.13744e14	+	1.11850e15i	−0.667664	+	0.817276i
$566$	1.22717e15			0.888001
$567$		0			0
$568$			7.80387e14i			0.553853i
$569$	2.47841e15			1.74203			0.871014	−	0.491257i	$-0.163463\pi$
$569$	2.47841e15			1.74203			0.871014	+	0.491257i	$0.163463\pi$
$570$		0			0
$571$	−1.29540e15			−0.893111			−0.446556	−	0.894756i	$-0.647350\pi$
$571$	−1.29540e15			−0.893111			−0.446556	+	0.894756i	$0.647350\pi$
$572$		−	1.79377e14i		−	0.122487i
$573$		0			0
$574$	1.23780e15			0.829153
$575$	1.02887e15	−	2.09450e14i	0.682634	−	0.138966i
$576$		0			0
$577$		−	8.93465e14i		−	0.581582i	−0.956787	−	0.290791i	$-0.906082\pi$
$577$		−	8.93465e14i		−	0.581582i	0.956787	−	0.290791i	$-0.0939184\pi$
$578$			7.15786e14i			0.461509i
$579$		0			0
$580$	−9.01622e14	−	7.36569e14i	−0.570388	−	0.465972i
$581$	2.41037e15			1.51048
$582$		0			0
$583$		−	2.17918e15i		−	1.34004i
$584$	−5.72144e14			−0.348526
$585$		0			0
$586$	7.72636e14			0.461890
$587$		−	2.15415e15i		−	1.27575i	−0.770138	−	0.637877i	$-0.779813\pi$
$587$		−	2.15415e15i		−	1.27575i	0.770138	−	0.637877i	$-0.220187\pi$
$588$		0			0
$589$	4.32039e14			0.251125
$590$	8.61855e14	−	1.05498e15i	0.496305	−	0.607518i
$591$		0			0
$592$		−	1.03104e15i		−	0.582784i
$593$		−	1.51707e15i		−	0.849582i	−0.905291	−	0.424791i	$-0.860348\pi$
$593$		−	1.51707e15i		−	0.849582i	0.905291	−	0.424791i	$-0.139652\pi$
$594$		0			0
$595$	−4.05693e14	−	3.31426e14i	−0.223025	−	0.182198i
$596$	4.55568e14			0.248141
$597$		0			0
$598$			1.39583e14i			0.0746405i
$599$	−4.59566e14			−0.243501			−0.121750	−	0.992561i	$-0.538851\pi$
$599$	−4.59566e14			−0.243501			−0.121750	+	0.992561i	$0.538851\pi$
$600$		0			0
$601$	−1.61074e15			−0.837944			−0.418972	−	0.907999i	$-0.637610\pi$
$601$	−1.61074e15			−0.837944			−0.418972	+	0.907999i	$0.637610\pi$
$602$		−	8.38685e13i		−	0.0432333i
$603$		0			0
$604$	−3.64330e14			−0.184413
$605$	−7.19295e14	−	5.87619e14i	−0.360788	−	0.294742i
$606$		0			0
$607$		−	2.27195e15i		−	1.11908i	−0.828803	−	0.559541i	$-0.810978\pi$
$607$		−	2.27195e15i		−	1.11908i	0.828803	−	0.559541i	$-0.189022\pi$
$608$			9.50469e14i			0.463947i
$609$		0			0
$610$	−9.17299e14	+	1.12285e15i	−0.439741	+	0.538279i
$611$	−3.48328e14			−0.165486
$612$		0			0
$613$			2.51698e15i			1.17448i	0.809411	+	0.587242i	$0.199786\pi$
$613$			2.51698e15i			1.17448i	−0.809411	+	0.587242i	$0.800214\pi$
$614$	−9.21825e14			−0.426307
$615$		0			0
$616$	1.72640e15			0.784238
$617$		−	2.36230e15i		−	1.06357i	−0.846879	−	0.531785i	$-0.821521\pi$
$617$		−	2.36230e15i		−	1.06357i	0.846879	−	0.531785i	$-0.178479\pi$
$618$		0			0
$619$	3.01613e15			1.33399			0.666993	−	0.745064i	$-0.267581\pi$
$619$	3.01613e15			1.33399			0.666993	+	0.745064i	$0.267581\pi$
$620$	7.48415e14	+	6.11408e14i	0.328086	+	0.268026i
$621$		0			0
$622$		−	3.23238e14i		−	0.139211i
$623$			3.27671e15i			1.39880i
$624$		0			0
$625$	−2.19444e15	+	9.32084e14i	−0.920414	+	0.390944i
$626$	−9.63681e14			−0.400659
$627$		0			0
$628$		−	2.41799e15i		−	0.987819i
$629$	9.65243e14			0.390893
$630$		0			0
$631$	4.31590e14			0.171755			0.0858776	−	0.996306i	$-0.472631\pi$
$631$	4.31590e14			0.171755			0.0858776	+	0.996306i	$0.472631\pi$
$632$			2.12518e14i			0.0838402i
$633$		0			0
$634$	−1.90313e14			−0.0737865
$635$	−1.56375e15	+	1.91416e15i	−0.601052	+	0.735738i
$636$		0			0
$637$		−	3.26083e14i		−	0.123186i
$638$			9.18807e14i			0.344121i
$639$		0			0
$640$	−1.63299e15	+	1.99892e15i	−0.601165	+	0.735876i
$641$	6.67595e13			0.0243666			0.0121833	−	0.999926i	$-0.496122\pi$
$641$	6.67595e13			0.0243666			0.0121833	+	0.999926i	$0.496122\pi$
$642$		0			0
$643$			3.74464e15i			1.34354i	0.740760	+	0.671769i	$0.234465\pi$
$643$			3.74464e15i			1.34354i	−0.740760	+	0.671769i	$0.765535\pi$
$644$	1.86506e15			0.663468
$645$		0			0
$646$	−1.45216e14			−0.0507850
$647$			3.70369e15i			1.28428i	0.766585	+	0.642142i	$0.221954\pi$
$647$			3.70369e15i			1.28428i	−0.766585	+	0.642142i	$0.778046\pi$
$648$		0			0
$649$	3.45021e15			1.17625
$650$	−6.32259e13	−	3.10581e14i	−0.0213733	−	0.104991i
$651$		0			0
$652$			2.65243e15i			0.881622i
$653$		−	4.04155e15i		−	1.33207i	−0.745922	−	0.666033i	$-0.767991\pi$
$653$		−	4.04155e15i		−	1.33207i	0.745922	−	0.666033i	$-0.232009\pi$
$654$		0			0
$655$	−2.69166e15	−	2.19892e15i	−0.872353	−	0.712658i
$656$	−1.45645e15			−0.468085
$657$		0			0
$658$		−	1.45027e15i		−	0.458361i
$659$	−4.16521e15			−1.30547			−0.652735	−	0.757586i	$-0.726378\pi$
$659$	−4.16521e15			−1.30547			−0.652735	+	0.757586i	$0.726378\pi$
$660$		0			0
$661$	−1.23316e15			−0.380111			−0.190056	−	0.981773i	$-0.560867\pi$
$661$	−1.23316e15			−0.380111			−0.190056	+	0.981773i	$0.560867\pi$
$662$		−	5.07195e14i		−	0.155045i
$663$		0			0
$664$	3.45488e15			1.03874
$665$	−1.19781e15	+	1.46622e15i	−0.357164	+	0.437199i
$666$		0			0
$667$			2.29450e15i			0.672971i
$668$		−	2.48860e15i		−	0.723909i
$669$		0			0
$670$	−1.99983e15	−	1.63373e15i	−0.572244	−	0.467487i
$671$	−3.67216e15			−1.04219
$672$		0			0
$673$		−	7.13397e14i		−	0.199181i	−0.995029	−	0.0995905i	$-0.968247\pi$
$673$		−	7.13397e14i		−	0.199181i	0.995029	−	0.0995905i	$-0.0317533\pi$
$674$	4.57833e14			0.126788
$675$		0			0
$676$	2.66315e15			0.725585
$677$			3.93889e15i			1.06448i	0.846594	+	0.532239i	$0.178649\pi$
$677$			3.93889e15i			1.06448i	−0.846594	+	0.532239i	$0.821351\pi$
$678$		0			0
$679$	5.33031e15			1.41732
$680$	−5.81496e14	−	4.75046e14i	−0.153373	−	0.125296i
$681$		0			0
$682$		−	7.62680e14i		−	0.197938i
$683$			5.84741e15i			1.50539i	0.658369	+	0.752696i	$0.271247\pi$
$683$			5.84741e15i			1.50539i	−0.658369	+	0.752696i	$0.728753\pi$
$684$		0			0
$685$	1.37449e15	−	1.68249e15i	0.348212	−	0.426240i
$686$	−1.06504e15			−0.267660
$687$		0			0
$688$			9.86834e13i			0.0244066i
$689$	−1.64275e15			−0.403056
$690$		0			0
$691$	−6.16517e15			−1.48873			−0.744364	−	0.667774i	$-0.767247\pi$
$691$	−6.16517e15			−1.48873			−0.744364	+	0.667774i	$0.767247\pi$
$692$		−	1.82654e15i		−	0.437569i
$693$		0			0
$694$	−1.05306e15			−0.248301
$695$	9.75351e14	+	7.96801e14i	0.228163	+	0.186395i
$696$		0			0
$697$		−	1.36350e15i		−	0.313960i
$698$			8.59632e14i			0.196385i
$699$		0			0
$700$	−4.14989e15	+	8.44804e14i	−0.933248	+	0.189984i
$701$	−1.34268e15			−0.299587			−0.149793	−	0.988717i	$-0.547861\pi$
$701$	−1.34268e15			−0.299587			−0.149793	+	0.988717i	$0.547861\pi$
$702$		0			0
$703$		−	3.48849e15i		−	0.766272i
$704$	5.25274e14			0.114482
$705$		0			0
$706$	−3.75304e15			−0.805299
$707$			8.83026e15i			1.88004i
$708$		0			0
$709$	6.27775e15			1.31598			0.657991	−	0.753026i	$-0.271406\pi$
$709$	6.27775e15			1.31598			0.657991	+	0.753026i	$0.271406\pi$
$710$	−9.55817e14	+	1.17000e15i	−0.198817	+	0.243369i
$711$		0			0
$712$			4.69665e15i			0.961940i
$713$		−	1.90461e15i		−	0.387092i
$714$		0			0
$715$	5.07861e14	−	6.21664e14i	0.101639	−	0.124415i
$716$	6.82744e15			1.35593
$717$		0			0
$718$		−	4.42197e14i		−	0.0864830i
$719$	−8.75562e15			−1.69933			−0.849665	−	0.527322i	$-0.823196\pi$
$719$	−8.75562e15			−1.69933			−0.849665	+	0.527322i	$0.823196\pi$
$720$		0			0
$721$	−6.76794e15			−1.29364
$722$		−	2.04470e15i		−	0.387861i
$723$		0			0
$724$	8.29235e14			0.154923
$725$	−1.03933e15	−	5.10543e15i	−0.192705	−	0.946615i
$726$		0			0
$727$			4.60975e15i			0.841857i	0.907094	+	0.420929i	$0.138296\pi$
$727$			4.60975e15i			0.841857i	−0.907094	+	0.420929i	$0.861704\pi$
$728$		−	1.30143e15i		−	0.235883i
$729$		0			0
$730$	−8.57791e14	−	7.00762e14i	−0.153146	−	0.125111i
$731$	−9.23854e13			−0.0163704
$732$		0			0
$733$			7.71163e14i			0.134609i	0.997732	+	0.0673046i	$0.0214399\pi$
$733$			7.71163e14i			0.134609i	−0.997732	+	0.0673046i	$0.978560\pi$
$734$	2.27443e15			0.394043
$735$		0			0
$736$	4.19007e15			0.715143
$737$		−	6.54022e15i		−	1.10795i
$738$		0			0
$739$	2.51328e15			0.419466			0.209733	−	0.977759i	$-0.432741\pi$
$739$	2.51328e15			0.419466			0.209733	+	0.977759i	$0.432741\pi$
$740$	4.93680e15	−	6.04305e15i	0.817845	−	1.00111i
$741$		0			0
$742$		−	6.83964e15i		−	1.11638i
$743$		−	1.08955e15i		−	0.176526i	−0.996097	−	0.0882631i	$-0.971868\pi$
$743$		−	1.08955e15i		−	0.176526i	0.996097	−	0.0882631i	$-0.0281316\pi$
$744$		0			0
$745$	1.57886e15	+	1.28983e15i	0.252048	+	0.205907i
$746$	2.29576e14			0.0363800
$747$		0			0
$748$		−	8.22685e14i		−	0.128462i
$749$	1.46812e16			2.27568
$750$		0			0
$751$	−3.61242e15			−0.551795			−0.275898	−	0.961187i	$-0.588975\pi$
$751$	−3.61242e15			−0.551795			−0.275898	+	0.961187i	$0.588975\pi$
$752$			1.70646e15i			0.258760i
$753$		0			0
$754$	6.92632e14			0.103505
$755$	−1.26265e15	−	1.03151e15i	−0.187316	−	0.153026i
$756$		0			0
$757$		−	1.16985e16i		−	1.71041i	−0.518287	−	0.855207i	$-0.673430\pi$
$757$		−	1.16985e16i		−	1.71041i	0.518287	−	0.855207i	$-0.326570\pi$
$758$		−	4.97238e15i		−	0.721745i
$759$		0			0
$760$	−1.71687e15	+	2.10159e15i	−0.245619	+	0.300658i
$761$	3.29015e15			0.467304			0.233652	−	0.972320i	$-0.424932\pi$
$761$	3.29015e15			0.467304			0.233652	+	0.972320i	$0.424932\pi$
$762$		0			0
$763$			1.02549e16i			1.43564i
$764$	6.66133e15			0.925864
$765$		0			0
$766$	−1.13579e15			−0.155611
$767$		−	2.60090e15i		−	0.353793i
$768$		0			0
$769$	6.00040e15			0.804610			0.402305	−	0.915506i	$-0.368209\pi$
$769$	6.00040e15			0.804610			0.402305	+	0.915506i	$0.368209\pi$
$770$	2.58832e15	+	2.11449e15i	0.344602	+	0.281519i
$771$		0			0
$772$		−	5.59229e15i		−	0.733998i
$773$			3.98905e15i			0.519856i	0.965628	+	0.259928i	$0.0836988\pi$
$773$			3.98905e15i			0.519856i	−0.965628	+	0.259928i	$0.916301\pi$
$774$		0			0
$775$	8.62720e14	+	4.23790e15i	0.110844	+	0.544491i
$776$	7.64015e15			0.974681
$777$		0			0
$778$		−	6.65257e15i		−	0.836762i
$779$	−4.92783e15			−0.615460
$780$		0			0
$781$	−3.82636e15			−0.471200
$782$			6.40173e14i			0.0782817i
$783$		0			0
$784$	−1.59748e15			−0.192618
$785$	6.84594e15	−	8.38000e15i	0.819691	−	1.00337i
$786$		0			0
$787$			1.17707e16i			1.38976i	0.719123	+	0.694882i	$0.244544\pi$
$787$			1.17707e16i			1.38976i	−0.719123	+	0.694882i	$0.755456\pi$
$788$		−	4.91812e15i		−	0.576640i
$789$		0			0
$790$	−2.60292e14	+	3.18619e14i	−0.0300962	+	0.0368403i
$791$	−1.14809e16			−1.31827
$792$		0			0
$793$			2.76821e15i			0.313471i
$794$	3.05158e15			0.343172
$795$		0			0
$796$	1.20456e16			1.33600
$797$			6.56353e15i			0.722965i	0.932379	+	0.361482i	$0.117729\pi$
$797$			6.56353e15i			0.722965i	−0.932379	+	0.361482i	$0.882271\pi$
$798$		0			0
$799$	−1.59755e15			−0.173559
$800$	−9.32320e15	+	1.89795e15i	−1.00594	+	0.204781i
$801$		0			0
$802$			4.76118e15i			0.506705i
$803$		−	2.80531e15i		−	0.296514i
$804$		0			0
$805$	6.46370e15	+	5.28044e15i	0.673913	+	0.550545i
$806$	−5.74937e14			−0.0595357
$807$		0			0
$808$			1.26568e16i			1.29289i
$809$	−1.26091e15			−0.127928			−0.0639641	−	0.997952i	$-0.520374\pi$
$809$	−1.26091e15			−0.127928			−0.0639641	+	0.997952i	$0.520374\pi$
$810$		0			0
$811$	−1.13759e15			−0.113860			−0.0569302	−	0.998378i	$-0.518131\pi$
$811$	−1.13759e15			−0.113860			−0.0569302	+	0.998378i	$0.518131\pi$
$812$		−	9.25472e15i		−	0.920037i
$813$		0			0
$814$	−6.15824e15			−0.603980
$815$	−7.50969e15	+	9.19249e15i	−0.731569	+	0.895501i
$816$		0			0
$817$			3.33891e14i			0.0320910i
$818$		−	4.71191e15i		−	0.449836i
$819$		0			0
$820$	−8.53640e15	−	6.97371e15i	−0.804079	−	0.656883i
$821$	1.31809e16			1.23326			0.616632	−	0.787251i	$-0.288497\pi$
$821$	1.31809e16			1.23326			0.616632	+	0.787251i	$0.288497\pi$
$822$		0			0
$823$		−	1.98574e16i		−	1.83326i	−0.399742	−	0.916628i	$-0.630900\pi$
$823$		−	1.98574e16i		−	1.83326i	0.399742	−	0.916628i	$-0.369100\pi$
$824$	−9.70076e15			−0.889623
$825$		0			0
$826$	1.08289e16			0.979928
$827$			1.25548e16i			1.12857i	0.825580	+	0.564285i	$0.190848\pi$
$827$			1.25548e16i			1.12857i	−0.825580	+	0.564285i	$0.809152\pi$
$828$		0			0
$829$	−1.26916e16			−1.12581			−0.562905	−	0.826521i	$-0.690316\pi$
$829$	−1.26916e16			−1.12581			−0.562905	+	0.826521i	$0.690316\pi$
$830$	5.17975e15	+	4.23153e15i	0.456435	+	0.372879i
$831$		0			0
$832$		−	3.95971e14i		−	0.0344338i
$833$		−	1.49553e15i		−	0.129195i
$834$		0			0
$835$	7.04584e15	−	8.62469e15i	0.600699	−	0.735306i
$836$	−2.97327e15			−0.251825
$837$		0			0
$838$			6.82103e15i			0.570174i
$839$	−1.32436e15			−0.109981			−0.0549903	−	0.998487i	$-0.517513\pi$
$839$	−1.32436e15			−0.109981			−0.0549903	+	0.998487i	$0.517513\pi$
$840$		0			0
$841$	−8.14816e14			−0.0667854
$842$			4.81536e15i			0.392114i
$843$		0			0
$844$	−1.65612e16			−1.33109
$845$	9.22962e15	+	7.54003e15i	0.737008	+	0.602089i
$846$		0			0
$847$		−	7.38322e15i		−	0.581952i
$848$			8.04782e15i			0.630233i
$849$		0			0
$850$	−2.89975e14	−	1.42443e15i	−0.0224159	−	0.110113i
$851$	−1.53787e16			−1.18116
$852$		0			0
$853$		−	1.99538e16i		−	1.51289i	−0.654060	−	0.756443i	$-0.726936\pi$
$853$		−	1.99538e16i		−	1.51289i	0.654060	−	0.756443i	$-0.273064\pi$
$854$	−1.15255e16			−0.868245
$855$		0			0
$856$	2.10432e16			1.56497
$857$		−	2.21687e16i		−	1.63812i	−0.573709	−	0.819060i	$-0.694496\pi$
$857$		−	2.21687e16i		−	1.63812i	0.573709	−	0.819060i	$-0.305504\pi$
$858$		0			0
$859$	−9.18375e15			−0.669973			−0.334987	−	0.942223i	$-0.608732\pi$
$859$	−9.18375e15			−0.669973			−0.334987	+	0.942223i	$0.608732\pi$
$860$	−4.72512e14	+	5.78394e14i	−0.0342508	+	0.0419259i
$861$		0			0
$862$			8.93967e15i			0.639781i
$863$			2.31329e16i			1.64502i	0.568750	+	0.822511i	$0.307427\pi$
$863$			2.31329e16i			1.64502i	−0.568750	+	0.822511i	$0.692573\pi$
$864$		0			0
$865$	5.17140e15	−	6.33023e15i	0.363094	−	0.444458i
$866$	−2.33963e15			−0.163229
$867$		0			0
$868$			7.68213e15i			0.529204i
$869$	−1.04201e15			−0.0713285
$870$		0			0
$871$	−4.93026e15			−0.333250
$872$			1.46987e16i			0.987277i
$873$		0			0
$874$	2.31365e15			0.153456
$875$	−1.67740e16	−	8.82153e15i	−1.10559	−	0.581433i
$876$		0			0
$877$			2.28779e16i			1.48908i	0.667577	+	0.744541i	$0.267331\pi$
$877$			2.28779e16i			1.48908i	−0.667577	+	0.744541i	$0.732669\pi$
$878$		−	5.16841e15i		−	0.334301i
$879$		0			0
$880$	−3.04553e15	−	2.48801e15i	−0.194540	−	0.158927i
$881$	−3.00048e15			−0.190469			−0.0952344	−	0.995455i	$-0.530360\pi$
$881$	−3.00048e15			−0.190469			−0.0952344	+	0.995455i	$0.530360\pi$
$882$		0			0
$883$		−	1.43409e15i		−	0.0899066i	−0.998989	−	0.0449533i	$-0.985686\pi$
$883$		−	1.43409e15i		−	0.0899066i	0.998989	−	0.0449533i	$-0.0143139\pi$
$884$	−6.20171e14			−0.0386388
$885$		0			0
$886$	3.51062e14			0.0216022
$887$		−	4.51694e15i		−	0.276226i	−0.990416	−	0.138113i	$-0.955896\pi$
$887$		−	4.51694e15i		−	0.276226i	0.990416	−	0.138113i	$-0.0441037\pi$
$888$		0			0
$889$	−1.96480e16			−1.18675
$890$	−5.75245e15	+	7.04147e15i	−0.345309	+	0.422687i
$891$		0			0
$892$		−	1.86873e15i		−	0.110800i
$893$			5.77372e15i			0.340230i
$894$		0			0
$895$	2.36617e16	+	1.93302e16i	1.37727	+	1.12515i
$896$	−2.05179e16			−1.18697
$897$		0			0
$898$			1.18010e16i			0.674373i
$899$	−9.45099e15			−0.536783
$900$		0			0
$901$	−7.53421e15			−0.422719
$902$			8.69912e15i			0.485109i
$903$		0			0
$904$	−1.64560e16			−0.906562
$905$	2.87386e15	+	2.34777e15i	0.157362	+	0.128555i
$906$		0			0
$907$		−	3.33072e16i		−	1.80176i	−0.434064	−	0.900882i	$-0.642921\pi$
$907$		−	3.33072e16i		−	1.80176i	0.434064	−	0.900882i	$-0.357079\pi$
$908$			2.24246e16i			1.20574i
$909$		0			0
$910$	1.59398e15	−	1.95117e15i	0.0846753	−	0.103650i
$911$	9.75985e15			0.515338			0.257669	−	0.966233i	$-0.417046\pi$
$911$	9.75985e15			0.515338			0.257669	+	0.966233i	$0.417046\pi$
$912$		0			0
$913$			1.69398e16i			0.883729i
$914$	1.20120e16			0.622891
$915$		0			0
$916$	2.71348e16			1.39028
$917$		−	2.76286e16i		−	1.40711i
$918$		0			0
$919$	−8.87018e15			−0.446372			−0.223186	−	0.974776i	$-0.571646\pi$
$919$	−8.87018e15			−0.446372			−0.223186	+	0.974776i	$0.571646\pi$
$920$	9.26468e15	+	7.56866e15i	0.463444	+	0.378605i
$921$		0			0
$922$		−	6.79572e15i		−	0.335904i
$923$			2.88445e15i			0.141728i
$924$		0			0
$925$	3.42188e16	−	6.96601e15i	1.66144	−	0.338224i
$926$	−3.09389e15			−0.149329
$927$		0			0
$928$		−	2.07918e16i		−	0.991696i
$929$	2.79034e16			1.32303			0.661517	−	0.749930i	$-0.269913\pi$
$929$	2.79034e16			1.32303			0.661517	+	0.749930i	$0.269913\pi$
$930$		0			0
$931$	−5.40499e15			−0.253263
$932$		−	7.00428e15i		−	0.326269i
$933$		0			0
$934$	−6.70730e15			−0.308773
$935$	2.32922e15	−	2.85116e15i	0.106598	−	0.130484i
$936$		0			0
$937$			1.42598e16i			0.644978i	0.946573	+	0.322489i	$0.104520\pi$
$937$			1.42598e16i			0.644978i	−0.946573	+	0.322489i	$0.895480\pi$
$938$		−	2.05273e16i		−	0.923030i
$939$		0			0
$940$	−8.17079e15	+	1.00017e16i	−0.363129	+	0.444500i
$941$	4.11724e16			1.81913			0.909563	−	0.415566i	$-0.136416\pi$
$941$	4.11724e16			1.81913			0.909563	+	0.415566i	$0.136416\pi$
$942$		0			0
$943$			2.17239e16i			0.948690i
$944$	−1.27418e16			−0.553202
$945$		0			0
$946$	5.89418e14			0.0252943
$947$		−	2.03073e16i		−	0.866419i	−0.901293	−	0.433209i	$-0.857381\pi$
$947$		−	2.03073e16i		−	0.866419i	0.901293	−	0.433209i	$-0.142619\pi$
$948$		0			0
$949$	−2.11475e15			−0.0891856
$950$	−5.14804e15	+	1.04800e15i	−0.215855	+	0.0439422i
$951$		0			0
$952$		−	5.96878e15i		−	0.247390i
$953$		−	3.65143e16i		−	1.50471i	−0.658760	−	0.752353i	$-0.728919\pi$
$953$		−	3.65143e16i		−	1.50471i	0.658760	−	0.752353i	$-0.271081\pi$
$954$		0			0
$955$	2.30861e16	+	1.88599e16i	0.940440	+	0.768281i
$956$	−1.19375e14			−0.00483498
$957$		0			0
$958$		−	1.94308e16i		−	0.778003i
$959$	1.72700e16			0.687527
$960$		0			0
$961$	−1.75634e16			−0.691243
$962$			4.64231e15i			0.181665i
$963$		0			0
$964$	−1.65953e16			−0.642037
$965$	1.58332e16	−	1.93811e16i	0.609070	−	0.745553i
$966$		0			0
$967$		−	2.33229e16i		−	0.887027i	−0.896268	−	0.443513i	$-0.853732\pi$
$967$		−	2.33229e16i		−	0.887027i	0.896268	−	0.443513i	$-0.146268\pi$
$968$		−	1.05827e16i		−	0.400203i
$969$		0			0
$970$	1.14545e16	+	9.35765e15i	0.428285	+	0.349882i
$971$	1.97670e16			0.734910			0.367455	−	0.930041i	$-0.380229\pi$
$971$	1.97670e16			0.734910			0.367455	+	0.930041i	$0.380229\pi$
$972$		0			0
$973$			1.00115e16i			0.368027i
$974$	−8.04437e15			−0.294048
$975$		0			0
$976$	1.35615e16			0.490154
$977$		−	1.51393e16i		−	0.544109i	−0.962282	−	0.272055i	$-0.912297\pi$
$977$		−	1.51393e16i		−	0.544109i	0.962282	−	0.272055i	$-0.0877032\pi$
$978$		0			0
$979$	−2.30284e16			−0.818387
$980$	−9.36297e15	−	7.64896e15i	−0.330880	−	0.270308i
$981$		0			0
$982$		−	6.21498e15i		−	0.217183i
$983$			1.59131e16i			0.552982i	0.961017	+	0.276491i	$0.0891716\pi$
$983$			1.59131e16i			0.552982i	−0.961017	+	0.276491i	$0.910828\pi$
$984$		0			0
$985$	1.39244e16	−	1.70446e16i	0.478495	−	0.585718i
$986$	3.17664e15			0.108554
$987$		0			0
$988$			2.24136e15i			0.0757441i
$989$	1.47193e15			0.0494661
$990$		0			0
$991$	−3.31820e16			−1.10280			−0.551402	−	0.834240i	$-0.685907\pi$
$991$	−3.31820e16			−1.10280			−0.551402	+	0.834240i	$0.685907\pi$
$992$			1.72588e16i			0.570422i
$993$		0			0
$994$	−1.20095e16			−0.392554
$995$	4.17462e16	+	3.41041e16i	1.35703	+	1.10861i
$996$		0			0
$997$			3.89617e16i			1.25261i	0.779579	+	0.626303i	$0.215433\pi$
$997$			3.89617e16i			1.25261i	−0.779579	+	0.626303i	$0.784567\pi$
$998$		−	1.11191e16i		−	0.355511i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.12.b.c.19.6	yes	8
3.2	odd	2	inner	45.12.b.c.19.3	✓	8
5.2	odd	4		225.12.a.z.1.3		8
5.3	odd	4		225.12.a.z.1.6		8
5.4	even	2	inner	45.12.b.c.19.4	yes	8
15.2	even	4		225.12.a.z.1.5		8
15.8	even	4		225.12.a.z.1.4		8
15.14	odd	2	inner	45.12.b.c.19.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.12.b.c.19.3	✓	8	3.2	odd	2	inner
45.12.b.c.19.4	yes	8	5.4	even	2	inner
45.12.b.c.19.5	yes	8	15.14	odd	2	inner
45.12.b.c.19.6	yes	8	1.1	even	1	trivial
225.12.a.z.1.3		8	5.2	odd	4
225.12.a.z.1.4		8	15.8	even	4
225.12.a.z.1.5		8	15.2	even	4
225.12.a.z.1.6		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	45.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+22.0579i q^{2} +1561.45 q^{4} +(5411.49 + 4420.85i) q^{5} +55546.4i q^{7} +79616.9i q^{8} +(-97514.6 + 119366. i) q^{10} -390374. q^{11} +294278. i q^{13} -1.22524e6 q^{14} +1.44167e6 q^{16} +1.34966e6i q^{17} +4.87782e6 q^{19} +(8.44977e6 + 6.90293e6i) q^{20} -8.61083e6i q^{22} -2.15035e7i q^{23} +(9.74030e6 + 4.78468e7i) q^{25} -6.49116e6 q^{26} +8.67329e7i q^{28} -1.06704e8 q^{29} +8.85723e7 q^{31} +1.94855e8i q^{32} -2.97707e7 q^{34} +(-2.45562e8 + 3.00589e8i) q^{35} -7.15174e8i q^{37} +1.07594e8i q^{38} +(-3.51974e8 + 4.30846e8i) q^{40} -1.01025e9 q^{41} +6.84508e7i q^{43} -6.09549e8 q^{44} +4.74321e8 q^{46} +1.18367e9i q^{47} -1.10807e9 q^{49} +(-1.05540e9 + 2.14851e8i) q^{50} +4.59501e8i q^{52} +5.58230e9i q^{53} +(-2.11250e9 - 1.72578e9i) q^{55} -4.42243e9 q^{56} -2.35366e9i q^{58} -8.83822e9 q^{59} +9.40678e9 q^{61} +1.95372e9i q^{62} -1.34557e9 q^{64} +(-1.30096e9 + 1.59248e9i) q^{65} +1.67537e10i q^{67} +2.10743e9i q^{68} +(-6.63035e9 - 5.41659e9i) q^{70} +9.80178e9 q^{71} +7.18622e9i q^{73} +1.57752e10 q^{74} +7.61646e9 q^{76} -2.16839e10i q^{77} +2.66926e9 q^{79} +(7.80157e9 + 6.37340e9i) q^{80} -2.22841e10i q^{82} -4.33938e10i q^{83} +(-5.96665e9 + 7.30368e9i) q^{85} -1.50988e9 q^{86} -3.10803e10i q^{88} +5.89906e10 q^{89} -1.63461e10 q^{91} -3.35766e10i q^{92} -2.61093e10 q^{94} +(2.63963e10 + 2.15641e10i) q^{95} -9.59614e10i q^{97} -2.44418e10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8976 q^{4} - 1155800 q^{10} + 15740992 q^{16} + 61220032 q^{19} + 184187000 q^{25} + 279698464 q^{31} + 309623120 q^{34} - 62144800 q^{40} - 13964841760 q^{46} - 1596098056 q^{49} + 15819804000 q^{55}+ \cdots - 460752353440 q^{94}+O(q^{100}) \) 8 * q - 8976 * q^4 - 1155800 * q^10 + 15740992 * q^16 + 61220032 * q^19 + 184187000 * q^25 + 279698464 * q^31 + 309623120 * q^34 - 62144800 * q^40 - 13964841760 * q^46 - 1596098056 * q^49 + 15819804000 * q^55 - 4923969584 * q^61 + 40933220096 * q^64 + 44585352000 * q^70 - 128254720704 * q^76 + 87780965728 * q^79 - 9845896400 * q^85 + 71755992000 * q^91 - 460752353440 * q^94

Introduction

L-functions

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Database

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