LMFDB - Embedded newform 45.18.b.c.19.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,18,Mod(19,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("45.19");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	45.b (of order $2$, degree $1$, 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:	$82.4499393051$
Analytic rank:	$0$
Dimension:	$16$
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} - 2 x^{15} - 1682025200 x^{14} - 16813165927404 x^{13} + \cdots + 36\!\cdots\!64 $ x^16 - 2x^15 - 1682025200x^14 - 16813165927404x^13 + 796193499528494077x^12 + 16301244096602769412534x^11 + 22468988706757383219149998x^10 - 2031337430834232484626265965288x^9 - 18965778077394737678963728161357692x^8 + 25704126016361292501091646945485563832x^7 + 1644988399708750269938578582195442295700744x^6 + 14232521872815010732766246857708462237763067840x^5 + 67152285951153667387295269789516112222302378490976x^4 + 200750173873523679964327426765877619483061237847842560x^3 + 397851771611595237118165695936875382658256196788975559168x^2 + 518025736280135169275335244745754264186455072537349946738688*x + 369441263867398942845365267821438853432252606507544720063780864
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{73}\cdot 3^{42}\cdot 5^{34} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.4
Root			$10898.0 - 3526.36i$ of defining polynomial
Character	$\chi$	$=$	45.19
Dual form			45.18.b.c.19.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-507.394i q^{2} -126376. q^{4} +(567939. + 663615. i) q^{5} -7.16981e6i q^{7} -2.38261e6i q^{8} +O(q^{10})$	\(q-507.394i q^{2} -126376. q^{4} +(567939. + 663615. i) q^{5} -7.16981e6i q^{7} -2.38261e6i q^{8} +(3.36714e8 - 2.88169e8i) q^{10} +1.34198e9 q^{11} -4.11363e8i q^{13} -3.63792e9 q^{14} -1.77733e10 q^{16} +3.37388e10i q^{17} +4.51529e10 q^{19} +(-7.17740e10 - 8.38652e10i) q^{20} -6.80910e11i q^{22} +4.67187e11i q^{23} +(-1.17830e11 + 7.53786e11i) q^{25} -2.08723e11 q^{26} +9.06094e11i q^{28} -3.39882e12 q^{29} -1.73362e12 q^{31} +8.70577e12i q^{32} +1.71188e13 q^{34} +(4.75799e12 - 4.07201e12i) q^{35} +3.71322e13i q^{37} -2.29103e13i q^{38} +(1.58114e12 - 1.35318e12i) q^{40} +9.02744e13 q^{41} +4.31572e13i q^{43} -1.69594e14 q^{44} +2.37048e14 q^{46} -1.79468e14i q^{47} +1.81224e14 q^{49} +(3.82466e14 + 5.97864e13i) q^{50} +5.19865e13i q^{52} +2.79714e14i q^{53} +(7.62161e14 + 8.90556e14i) q^{55} -1.70829e13 q^{56} +1.72454e15i q^{58} -1.18567e15 q^{59} +1.54968e15 q^{61} +8.79626e14i q^{62} +2.08767e15 q^{64} +(2.72986e14 - 2.33629e14i) q^{65} +2.66250e15i q^{67} -4.26378e15i q^{68} +(-2.06611e15 - 2.41418e15i) q^{70} +1.19400e14 q^{71} -4.39676e15i q^{73} +1.88407e16 q^{74} -5.70625e15 q^{76} -9.62172e15i q^{77} -9.50086e15 q^{79} +(-1.00942e16 - 1.17946e16i) q^{80} -4.58047e16i q^{82} -5.86719e15i q^{83} +(-2.23896e16 + 1.91616e16i) q^{85} +2.18977e16 q^{86} -3.19741e15i q^{88} +2.12662e16 q^{89} -2.94939e15 q^{91} -5.90414e16i q^{92} -9.10611e16 q^{94} +(2.56441e16 + 2.99641e16i) q^{95} +4.45337e16i q^{97} -9.19521e16i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 1473168 q^{4}+O(q^{10}) $ 16 * q - 1473168 * q^4	$ 16 q - 1473168 q^{4} - 670199320 q^{10} + 35028482624 q^{16} + 236279498336 q^{19} + 2507484007120 q^{25} - 12501422403232 q^{31} - 36533616605360 q^{34} - 44259047245280 q^{40} - 28675167752480 q^{46} - 430197530548688 q^{49} - 16\!\cdots\!00 q^{55}+ \cdots - 20\!\cdots\!80 q^{94}+O(q^{100}) $ 16 * q - 1473168 * q^4 - 670199320 * q^10 + 35028482624 * q^16 + 236279498336 * q^19 + 2507484007120 * q^25 - 12501422403232 * q^31 - 36533616605360 * q^34 - 44259047245280 * q^40 - 28675167752480 * q^46 - 430197530548688 * q^49 - 1615648303533600 * q^55 - 6459270800401888 * q^61 - 4539898356653312 * q^64 + 1120556748124800 * q^70 + 72078801859831872 * q^76 - 2212561840444576 * q^79 - 75398163530251360 * q^85 + 201977821610712000 * q^91 - 20802101586267680 * q^94

Display 100 coefficients 10 coefficients

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 $

$2$		−	507.394i		−	1.40149i	−0.713412	−	0.700745i	$-0.752851\pi$
$2$		−	507.394i		−	1.40149i	0.713412	−	0.700745i	$-0.247149\pi$
$3$		0			0
$3$		0			0
$4$	−126376.			−0.964174
$5$	567939.	+	663615.i	0.650214	+	0.759751i
$5$	567939.	+	663615.i	0.650214	+	0.759751i
$6$		0			0
$7$		−	7.16981e6i		−	0.470083i	−0.971985	−	0.235041i	$-0.924477\pi$
$7$		−	7.16981e6i		−	0.470083i	0.971985	−	0.235041i	$-0.0755226\pi$
$8$		−	2.38261e6i		−	0.0502098i
$9$		0			0
$10$	3.36714e8	−	2.88169e8i	1.06478	−	0.911269i
$11$	1.34198e9			1.88759			0.943794	−	0.330535i	$-0.107229\pi$
$11$	1.34198e9			1.88759			0.943794	+	0.330535i	$0.107229\pi$
$12$		0			0
$13$		−	4.11363e8i		−	0.139864i	−0.997552	−	0.0699321i	$-0.977722\pi$
$13$		−	4.11363e8i		−	0.139864i	0.997552	−	0.0699321i	$-0.0222783\pi$
$14$	−3.63792e9			−0.658816
$15$		0			0
$16$	−1.77733e10			−1.03454
$17$			3.37388e10i			1.17304i	0.809934	+	0.586521i	$0.199503\pi$
$17$			3.37388e10i			1.17304i	−0.809934	+	0.586521i	$0.800497\pi$
$18$		0			0
$19$	4.51529e10			0.609930			0.304965	−	0.952364i	$-0.401355\pi$
$19$	4.51529e10			0.609930			0.304965	+	0.952364i	$0.401355\pi$
$20$	−7.17740e10	−	8.38652e10i	−0.626920	−	0.732532i
$21$		0			0
$22$		−	6.80910e11i		−	2.64543i
$23$			4.67187e11i			1.24395i	0.783035	+	0.621977i	$0.213670\pi$
$23$			4.67187e11i			1.24395i	−0.783035	+	0.621977i	$0.786330\pi$
$24$		0			0
$25$	−1.17830e11	+	7.53786e11i	−0.154443	+	0.988002i
$26$	−2.08723e11			−0.196018
$27$		0			0
$28$			9.06094e11i			0.453242i
$29$	−3.39882e12			−1.26167			−0.630834	−	0.775918i	$-0.717287\pi$
$29$	−3.39882e12			−1.26167			−0.630834	+	0.775918i	$0.717287\pi$
$30$		0			0
$31$	−1.73362e12			−0.365072			−0.182536	−	0.983199i	$-0.558431\pi$
$31$	−1.73362e12			−0.365072			−0.182536	+	0.983199i	$0.558431\pi$
$32$			8.70577e12i			1.39969i
$33$		0			0
$34$	1.71188e13			1.64401
$35$	4.75799e12	−	4.07201e12i	0.357146	−	0.305655i
$36$		0			0
$37$			3.71322e13i			1.73795i	0.494859	+	0.868973i	$0.335220\pi$
$37$			3.71322e13i			1.73795i	−0.494859	+	0.868973i	$0.664780\pi$
$38$		−	2.29103e13i		−	0.854811i
$39$		0			0
$40$	1.58114e12	−	1.35318e12i	0.0381470	−	0.0326472i
$41$	9.02744e13			1.76564			0.882819	−	0.469713i	$-0.155642\pi$
$41$	9.02744e13			1.76564			0.882819	+	0.469713i	$0.155642\pi$
$42$		0			0
$43$			4.31572e13i			0.563081i	0.959549	+	0.281541i	$0.0908454\pi$
$43$			4.31572e13i			0.563081i	−0.959549	+	0.281541i	$0.909155\pi$
$44$	−1.69594e14			−1.81996
$45$		0			0
$46$	2.37048e14			1.74339
$47$		−	1.79468e14i		−	1.09940i	−0.835362	−	0.549701i	$-0.814742\pi$
$47$		−	1.79468e14i		−	1.09940i	0.835362	−	0.549701i	$-0.185258\pi$
$48$		0			0
$49$	1.81224e14			0.779022
$50$	3.82466e14	+	5.97864e13i	1.38467	+	0.216450i
$51$		0			0
$52$			5.19865e13i			0.134853i
$53$			2.79714e14i			0.617119i	0.951205	+	0.308560i	$0.0998470\pi$
$53$			2.79714e14i			0.617119i	−0.951205	+	0.308560i	$0.900153\pi$
$54$		0			0
$55$	7.62161e14	+	8.90556e14i	1.22734	+	1.43410i
$56$	−1.70829e13			−0.0236028
$57$		0			0
$58$			1.72454e15i			1.76821i
$59$	−1.18567e15			−1.05129			−0.525644	−	0.850705i	$-0.676176\pi$
$59$	−1.18567e15			−1.05129			−0.525644	+	0.850705i	$0.676176\pi$
$60$		0			0
$61$	1.54968e15			1.03500			0.517499	−	0.855684i	$-0.326863\pi$
$61$	1.54968e15			1.03500			0.517499	+	0.855684i	$0.326863\pi$
$62$			8.79626e14i			0.511645i
$63$		0			0
$64$	2.08767e15			0.927110
$65$	2.72986e14	−	2.33629e14i	0.106262	−	0.0909417i
$66$		0			0
$67$			2.66250e15i			0.801038i	0.916288	+	0.400519i	$0.131170\pi$
$67$			2.66250e15i			0.801038i	−0.916288	+	0.400519i	$0.868830\pi$
$68$		−	4.26378e15i		−	1.13102i
$69$		0			0
$70$	−2.06611e15	−	2.41418e15i	−0.428372	−	0.500536i
$71$	1.19400e14			0.0219437			0.0109718	−	0.999940i	$-0.496507\pi$
$71$	1.19400e14			0.0219437			0.0109718	+	0.999940i	$0.496507\pi$
$72$		0			0
$73$		−	4.39676e15i		−	0.638100i	−0.947738	−	0.319050i	$-0.896636\pi$
$73$		−	4.39676e15i		−	0.638100i	0.947738	−	0.319050i	$-0.103364\pi$
$74$	1.88407e16			2.43571
$75$		0			0
$76$	−5.70625e15			−0.588079
$77$		−	9.62172e15i		−	0.887322i
$78$		0			0
$79$	−9.50086e15			−0.704584			−0.352292	−	0.935890i	$-0.614598\pi$
$79$	−9.50086e15			−0.704584			−0.352292	+	0.935890i	$0.614598\pi$
$80$	−1.00942e16	−	1.17946e16i	−0.672674	−	0.785995i
$81$		0			0
$82$		−	4.58047e16i		−	2.47452i
$83$		−	5.86719e15i		−	0.285934i	−0.989727	−	0.142967i	$-0.954336\pi$
$83$		−	5.86719e15i		−	0.285934i	0.989727	−	0.142967i	$-0.0456644\pi$
$84$		0			0
$85$	−2.23896e16	+	1.91616e16i	−0.891220	+	0.762729i
$86$	2.18977e16			0.789153
$87$		0			0
$88$		−	3.19741e15i		−	0.0947755i
$89$	2.12662e16			0.572631			0.286316	−	0.958135i	$-0.407569\pi$
$89$	2.12662e16			0.572631			0.286316	+	0.958135i	$0.407569\pi$
$90$		0			0
$91$	−2.94939e15			−0.0657477
$92$		−	5.90414e16i		−	1.19939i
$93$		0			0
$94$	−9.10611e16			−1.54080
$95$	2.56441e16	+	2.99641e16i	0.396585	+	0.463395i
$96$		0			0
$97$			4.45337e16i			0.576938i	0.957489	+	0.288469i	$0.0931462\pi$
$97$			4.45337e16i			0.576938i	−0.957489	+	0.288469i	$0.906854\pi$
$98$		−	9.19521e16i		−	1.09179i
$99$		0			0
$100$	1.48910e16	−	9.52606e16i	0.148910	−	0.952606i
$101$	9.71898e16			0.893077			0.446539	−	0.894764i	$-0.352657\pi$
$101$	9.71898e16			0.893077			0.446539	+	0.894764i	$0.352657\pi$
$102$		0			0
$103$		−	8.80492e16i		−	0.684871i	−0.939541	−	0.342436i	$-0.888748\pi$
$103$		−	8.80492e16i		−	0.684871i	0.939541	−	0.342436i	$-0.111252\pi$
$104$	−9.80119e14			−0.00702256
$105$		0			0
$106$	1.41925e17			0.864887
$107$		−	1.05069e17i		−	0.591169i	−0.955317	−	0.295584i	$-0.904486\pi$
$107$		−	1.05069e17i		−	0.591169i	0.955317	−	0.295584i	$-0.0955143\pi$
$108$		0			0
$109$	1.88087e17			0.904136			0.452068	−	0.891983i	$-0.350686\pi$
$109$	1.88087e17			0.904136			0.452068	+	0.891983i	$0.350686\pi$
$110$	4.51862e17	−	3.86715e17i	2.00987	−	1.72010i
$111$		0			0
$112$			1.27431e17i			0.486321i
$113$		−	3.77644e17i		−	1.33634i	−0.744010	−	0.668169i	$-0.767078\pi$
$113$		−	3.77644e17i		−	1.33634i	0.744010	−	0.668169i	$-0.232922\pi$
$114$		0			0
$115$	−3.10033e17	+	2.65334e17i	−0.945095	+	0.808837i
$116$	4.29530e17			1.21647
$117$		0			0
$118$			6.01601e17i			1.47337i
$119$	2.41901e17			0.551427
$120$		0			0
$121$	1.29545e18			2.56299
$122$		−	7.86300e17i		−	1.45054i
$123$		0			0
$124$	2.19088e17			0.351993
$125$	−5.67144e17	+	3.49910e17i	−0.851056	+	0.525075i
$126$		0			0
$127$			5.65338e17i			0.741271i	0.928778	+	0.370635i	$0.120860\pi$
$127$			5.65338e17i			0.741271i	−0.928778	+	0.370635i	$0.879140\pi$
$128$			8.18135e16i			0.100355i
$129$		0			0
$130$	−1.18542e17	−	1.38512e17i	−0.127454	−	0.148925i
$131$	1.33183e17			0.134166			0.0670828	−	0.997747i	$-0.478631\pi$
$131$	1.33183e17			0.134166			0.0670828	+	0.997747i	$0.478631\pi$
$132$		0			0
$133$		−	3.23738e17i		−	0.286718i
$134$	1.35093e18			1.12265
$135$		0			0
$136$	8.03865e16			0.0588982
$137$			2.00258e16i			0.0137869i	0.999976	+	0.00689344i	$0.00219427\pi$
$137$			2.00258e16i			0.0137869i	−0.999976	+	0.00689344i	$0.997806\pi$
$138$		0			0
$139$	−2.14474e18			−1.30542			−0.652708	−	0.757610i	$-0.726367\pi$
$139$	−2.14474e18			−1.30542			−0.652708	+	0.757610i	$0.726367\pi$
$140$	−6.01297e17	+	5.14606e17i	−0.344351	+	0.294704i
$141$		0			0
$142$		−	6.05829e16i		−	0.0307538i
$143$		−	5.52039e17i		−	0.264006i
$144$		0			0
$145$	−1.93032e18	−	2.25551e18i	−0.820354	−	0.958553i
$146$	−2.23089e18			−0.894291
$147$		0			0
$148$		−	4.69263e18i		−	1.67568i
$149$	8.65395e17			0.291831			0.145915	−	0.989297i	$-0.453387\pi$
$149$	8.65395e17			0.291831			0.145915	+	0.989297i	$0.453387\pi$
$150$		0			0
$151$	2.44951e18			0.737523			0.368762	−	0.929524i	$-0.379782\pi$
$151$	2.44951e18			0.737523			0.368762	+	0.929524i	$0.379782\pi$
$152$		−	1.07582e17i		−	0.0306245i
$153$		0			0
$154$	−4.88200e18			−1.24357
$155$	−9.84589e17	−	1.15045e18i	−0.237375	−	0.277364i
$156$		0			0
$157$		−	5.74960e18i		−	1.24306i	−0.783392	−	0.621528i	$-0.786512\pi$
$157$		−	5.74960e18i		−	1.24306i	0.783392	−	0.621528i	$-0.213488\pi$
$158$			4.82068e18i			0.987467i
$159$		0			0
$160$	−5.77728e18	+	4.94434e18i	−1.06342	+	0.910099i
$161$	3.34965e18			0.584762
$162$		0			0
$163$			1.03122e19i			1.62091i	0.585802	+	0.810454i	$0.300780\pi$
$163$			1.03122e19i			1.62091i	−0.585802	+	0.810454i	$0.699220\pi$
$164$	−1.14085e19			−1.70238
$165$		0			0
$166$	−2.97698e18			−0.400734
$167$			3.75546e18i			0.480367i	0.970727	+	0.240184i	$0.0772077\pi$
$167$			3.75546e18i			0.480367i	−0.970727	+	0.240184i	$0.922792\pi$
$168$		0			0
$169$	8.48120e18			0.980438
$170$	9.72245e18	+	1.13603e19i	1.06896	+	1.24904i
$171$		0			0
$172$		−	5.45404e18i		−	0.542908i
$173$			9.23135e18i			0.874728i	0.899284	+	0.437364i	$0.144088\pi$
$173$			9.23135e18i			0.874728i	−0.899284	+	0.437364i	$0.855912\pi$
$174$		0			0
$175$	5.40450e18	+	8.44821e17i	0.464443	+	0.0726008i
$176$	−2.38514e19			−1.95279
$177$		0			0
$178$		−	1.07903e19i		−	0.802537i
$179$	−2.19249e19			−1.55484			−0.777422	−	0.628979i	$-0.783473\pi$
$179$	−2.19249e19			−1.55484			−0.777422	+	0.628979i	$0.783473\pi$
$180$		0			0
$181$	1.81678e19			1.17229			0.586145	−	0.810206i	$-0.300645\pi$
$181$	1.81678e19			1.17229			0.586145	+	0.810206i	$0.300645\pi$
$182$			1.49650e18i			0.0921448i
$183$		0			0
$184$	1.11313e18			0.0624588
$185$	−2.46415e19	+	2.10888e19i	−1.32041	+	1.13004i
$186$		0			0
$187$			4.52766e19i			2.21422i
$188$			2.26805e19i			1.06001i
$189$		0			0
$190$	1.52036e19	−	1.30116e19i	0.649443	−	0.555810i
$191$	3.05805e19			1.24928			0.624641	−	0.780912i	$-0.285245\pi$
$191$	3.05805e19			1.24928			0.624641	+	0.780912i	$0.285245\pi$
$192$		0			0
$193$			2.60930e19i			0.975635i	0.872946	+	0.487817i	$0.162207\pi$
$193$			2.60930e19i			0.975635i	−0.872946	+	0.487817i	$0.837793\pi$
$194$	2.25961e19			0.808573
$195$		0			0
$196$	−2.29024e19			−0.751113
$197$		−	5.53665e19i		−	1.73894i	−0.493987	−	0.869469i	$-0.664461\pi$
$197$		−	5.53665e19i		−	1.73894i	0.493987	−	0.869469i	$-0.335539\pi$
$198$		0			0
$199$	3.08681e19			0.889732			0.444866	−	0.895597i	$-0.353251\pi$
$199$	3.08681e19			0.889732			0.444866	+	0.895597i	$0.353251\pi$
$200$	1.79598e18	+	2.80744e17i	0.0496074	+	0.00775454i
$201$		0			0
$202$		−	4.93135e19i		−	1.25164i
$203$			2.43689e19i			0.593088i
$204$		0			0
$205$	5.12703e19	+	5.99075e19i	1.14804	+	1.34145i
$206$	−4.46756e19			−0.959840
$207$		0			0
$208$			7.31128e18i			0.144695i
$209$	6.05941e19			1.15130
$210$		0			0
$211$	−3.80653e19			−0.667004			−0.333502	−	0.942749i	$-0.608230\pi$
$211$	−3.80653e19			−0.667004			−0.333502	+	0.942749i	$0.608230\pi$
$212$		−	3.53492e19i		−	0.595010i
$213$		0			0
$214$	−5.33112e19			−0.828517
$215$	−2.86397e19	+	2.45106e19i	−0.427801	+	0.366124i
$216$		0			0
$217$			1.24297e19i			0.171614i
$218$		−	9.54342e19i		−	1.26714i
$219$		0			0
$220$	−9.63190e19	−	1.12545e20i	−1.18337	−	1.38272i
$221$	1.38789e19			0.164067
$222$		0			0
$223$		−	1.25630e20i		−	1.37563i	−0.725884	−	0.687817i	$-0.758569\pi$
$223$		−	1.25630e20i		−	1.37563i	0.725884	−	0.687817i	$-0.241431\pi$
$224$	6.24187e19			0.657971
$225$		0			0
$226$	−1.91614e20			−1.87286
$227$		−	1.95914e20i		−	1.84436i	−0.386761	−	0.922180i	$-0.626406\pi$
$227$		−	1.95914e20i		−	1.84436i	0.386761	−	0.922180i	$-0.373594\pi$
$228$		0			0
$229$	8.02319e19			0.701044			0.350522	−	0.936554i	$-0.386004\pi$
$229$	8.02319e19			0.701044			0.350522	+	0.936554i	$0.386004\pi$
$230$	1.34629e20	+	1.57309e20i	1.13358	+	1.32454i
$231$		0			0
$232$			8.09807e18i			0.0633481i
$233$			1.53861e20i			1.16039i	0.814478	+	0.580194i	$0.197023\pi$
$233$			1.53861e20i			1.16039i	−0.814478	+	0.580194i	$0.802977\pi$
$234$		0			0
$235$	1.19098e20	−	1.01927e20i	0.835271	−	0.714847i
$236$	1.49841e20			1.01362
$237$		0			0
$238$		−	1.22739e20i		−	0.772819i
$239$	−2.61381e20			−1.58815			−0.794076	−	0.607818i	$-0.792045\pi$
$239$	−2.61381e20			−1.58815			−0.794076	+	0.607818i	$0.792045\pi$
$240$		0			0
$241$	−8.95825e19			−0.507082			−0.253541	−	0.967325i	$-0.581595\pi$
$241$	−8.95825e19			−0.507082			−0.253541	+	0.967325i	$0.581595\pi$
$242$		−	6.57305e20i		−	3.59200i
$243$		0			0
$244$	−1.95843e20			−0.997919
$245$	1.02924e20	+	1.20263e20i	0.506531	+	0.591863i
$246$		0			0
$247$		−	1.85742e19i		−	0.0853073i
$248$			4.13054e18i			0.0183302i
$249$		0			0
$250$	1.77542e20	+	2.87765e20i	0.735887	+	1.19275i
$251$	3.18933e20			1.27783			0.638915	−	0.769278i	$-0.279384\pi$
$251$	3.18933e20			1.27783			0.638915	+	0.769278i	$0.279384\pi$
$252$		0			0
$253$			6.26954e20i			2.34807i
$254$	2.86849e20			1.03888
$255$		0			0
$256$	3.15146e20			1.06776
$257$		−	2.96444e20i		−	0.971653i	−0.874055	−	0.485827i	$-0.838519\pi$
$257$		−	2.96444e20i		−	0.971653i	0.874055	−	0.485827i	$-0.161481\pi$
$258$		0			0
$259$	2.66231e20			0.816979
$260$	−3.44990e19	+	2.95251e19i	−0.102455	+	0.0876836i
$261$		0			0
$262$		−	6.75760e19i		−	0.188032i
$263$		−	2.00822e20i		−	0.540989i	−0.962722	−	0.270495i	$-0.912813\pi$
$263$		−	2.00822e20i		−	0.540989i	0.962722	−	0.270495i	$-0.0871872\pi$
$264$		0			0
$265$	−1.85622e20	+	1.58860e20i	−0.468857	+	0.401260i
$266$	−1.64262e20			−0.401832
$267$		0			0
$268$		−	3.36476e20i		−	0.772340i
$269$	1.36640e20			0.303867			0.151933	−	0.988391i	$-0.451450\pi$
$269$	1.36640e20			0.303867			0.151933	+	0.988391i	$0.451450\pi$
$270$		0			0
$271$	8.65078e20			1.80641			0.903205	−	0.429210i	$-0.141208\pi$
$271$	8.65078e20			1.80641			0.903205	+	0.429210i	$0.141208\pi$
$272$		−	5.99650e20i		−	1.21356i
$273$		0			0
$274$	1.01610e19			0.0193222
$275$	−1.58126e20	+	1.01156e21i	−0.291524	+	1.86494i
$276$		0			0
$277$			1.21966e20i			0.211427i	0.994397	+	0.105714i	$0.0337127\pi$
$277$			1.21966e20i			0.211427i	−0.994397	+	0.105714i	$0.966287\pi$
$278$			1.08823e21i			1.82953i
$279$		0			0
$280$	−9.70204e18	−	1.13365e19i	−0.0153469	−	0.0179322i
$281$	−9.37640e20			−1.43891			−0.719453	−	0.694541i	$-0.755608\pi$
$281$	−9.37640e20			−1.43891			−0.719453	+	0.694541i	$0.755608\pi$
$282$		0			0
$283$			1.89858e20i			0.274311i	0.990550	+	0.137156i	$0.0437960\pi$
$283$			1.89858e20i			0.274311i	−0.990550	+	0.137156i	$0.956204\pi$
$284$	−1.50894e19			−0.0211575
$285$		0			0
$286$	−2.80101e20			−0.370002
$287$		−	6.47250e20i		−	0.829996i
$288$		0			0
$289$	−3.11065e20			−0.376027
$290$	−1.14443e21	+	9.79432e20i	−1.34340	+	1.14972i
$291$		0			0
$292$			5.55646e20i			0.615239i
$293$			1.18697e21i			1.27663i	0.769776	+	0.638315i	$0.220368\pi$
$293$			1.18697e21i			1.27663i	−0.769776	+	0.638315i	$0.779632\pi$
$294$		0			0
$295$	−6.73388e20	−	7.86829e20i	−0.683563	−	0.798717i
$296$	8.84718e19			0.0872620
$297$		0			0
$298$		−	4.39096e20i		−	0.408998i
$299$	1.92183e20			0.173985
$300$		0			0
$301$	3.09429e20			0.264695
$302$		−	1.24287e21i		−	1.03363i
$303$		0			0
$304$	−8.02516e20			−0.630998
$305$	8.80126e20	+	1.02839e21i	0.672971	+	0.786341i
$306$		0			0
$307$		−	1.18679e21i		−	0.858417i	−0.903205	−	0.429208i	$-0.858793\pi$
$307$		−	1.18679e21i		−	0.858417i	0.903205	−	0.429208i	$-0.141207\pi$
$308$			1.21596e21i			0.855533i
$309$		0			0
$310$	−5.83733e20	+	4.99574e20i	−0.388723	+	0.332679i
$311$	−2.30195e21			−1.49153			−0.745766	−	0.666208i	$-0.767916\pi$
$311$	−2.30195e21			−1.49153			−0.745766	+	0.666208i	$0.767916\pi$
$312$		0			0
$313$		−	4.34167e20i		−	0.266397i	−0.991089	−	0.133199i	$-0.957475\pi$
$313$		−	4.34167e20i		−	0.266397i	0.991089	−	0.133199i	$-0.0425248\pi$
$314$	−2.91731e21			−1.74213
$315$		0			0
$316$	1.20068e21			0.679342
$317$		−	1.78839e21i		−	0.985053i	−0.870298	−	0.492526i	$-0.836074\pi$
$317$		−	1.78839e21i		−	0.985053i	0.870298	−	0.492526i	$-0.163926\pi$
$318$		0			0
$319$	−4.56113e21			−2.38151
$320$	1.18567e21	+	1.38541e21i	0.602820	+	0.704373i
$321$		0			0
$322$		−	1.69959e21i		−	0.819538i
$323$			1.52340e21i			0.715473i
$324$		0			0
$325$	3.10079e20	+	4.84710e19i	0.138186	+	0.0216010i
$326$	5.23237e21			2.27169
$327$		0			0
$328$		−	2.15089e20i		−	0.0886524i
$329$	−1.28676e21			−0.516810
$330$		0			0
$331$	−1.51587e21			−0.578262			−0.289131	−	0.957290i	$-0.593366\pi$
$331$	−1.51587e21			−0.578262			−0.289131	+	0.957290i	$0.593366\pi$
$332$			7.41474e20i			0.275691i
$333$		0			0
$334$	1.90550e21			0.673230
$335$	−1.76687e21	+	1.51213e21i	−0.608589	+	0.520846i
$336$		0			0
$337$			1.48344e21i			0.485754i	0.970057	+	0.242877i	$0.0780912\pi$
$337$			1.48344e21i			0.485754i	−0.970057	+	0.242877i	$0.921909\pi$
$338$		−	4.30330e21i		−	1.37407i
$339$		0			0
$340$	2.82951e21	−	2.42157e21i	0.859291	−	0.735403i
$341$	−2.32647e21			−0.689106
$342$		0			0
$343$		−	2.96726e21i		−	0.836288i
$344$	1.02827e20			0.0282722
$345$		0			0
$346$	4.68393e21			1.22592
$347$			1.82303e21i			0.465580i	0.972527	+	0.232790i	$0.0747855\pi$
$347$			1.82303e21i			0.465580i	−0.972527	+	0.232790i	$0.925215\pi$
$348$		0			0
$349$	2.68425e21			0.652839			0.326420	−	0.945225i	$-0.394158\pi$
$349$	2.68425e21			0.652839			0.326420	+	0.945225i	$0.394158\pi$
$350$	4.28657e20	−	2.74221e21i	0.101749	−	0.650912i
$351$		0			0
$352$			1.16829e22i			2.64204i
$353$			5.94400e21i			1.31218i	0.754682	+	0.656091i	$0.227791\pi$
$353$			5.94400e21i			1.31218i	−0.754682	+	0.656091i	$0.772209\pi$
$354$		0			0
$355$	6.78120e19	+	7.92358e19i	0.0142681	+	0.0166717i
$356$	−2.68754e21			−0.552116
$357$		0			0
$358$			1.11246e22i			2.17910i
$359$	3.86028e21			0.738441			0.369220	−	0.929342i	$-0.379625\pi$
$359$	3.86028e21			0.738441			0.369220	+	0.929342i	$0.379625\pi$
$360$		0			0
$361$	−3.44160e21			−0.627986
$362$		−	9.21824e21i		−	1.64295i
$363$		0			0
$364$	3.72733e20			0.0633923
$365$	2.91776e21	−	2.49709e21i	0.484797	−	0.414902i
$366$		0			0
$367$		−	9.02294e21i		−	1.43115i	−0.698534	−	0.715577i	$-0.746164\pi$
$367$		−	9.02294e21i		−	1.43115i	0.698534	−	0.715577i	$-0.253836\pi$
$368$		−	8.30346e21i		−	1.28692i
$369$		0			0
$370$	1.07003e22	+	1.25029e22i	1.58374	+	1.85054i
$371$	2.00550e21			0.290097
$372$		0			0
$373$		−	2.62603e21i		−	0.362889i	−0.983401	−	0.181445i	$-0.941923\pi$
$373$		−	2.62603e21i		−	0.362889i	0.983401	−	0.181445i	$-0.0580773\pi$
$374$	2.29731e22			3.10321
$375$		0			0
$376$	−4.27604e20			−0.0552008
$377$			1.39815e21i			0.176462i
$378$		0			0
$379$	1.72311e21			0.207912			0.103956	−	0.994582i	$-0.466850\pi$
$379$	1.72311e21			0.207912			0.103956	+	0.994582i	$0.466850\pi$
$380$	−3.24080e21	−	3.78675e21i	−0.382377	−	0.446793i
$381$		0			0
$382$		−	1.55163e22i		−	1.75086i
$383$		−	9.21422e21i		−	1.01688i	−0.861098	−	0.508439i	$-0.830223\pi$
$383$		−	9.21422e21i		−	1.01688i	0.861098	−	0.508439i	$-0.169777\pi$
$384$		0			0
$385$	6.38512e21	−	5.46455e21i	0.674144	−	0.576950i
$386$	1.32394e22			1.36734
$387$		0			0
$388$		−	5.62800e21i		−	0.556269i
$389$	−1.03250e22			−0.998436			−0.499218	−	0.866476i	$-0.666379\pi$
$389$	−1.03250e22			−0.998436			−0.499218	+	0.866476i	$0.666379\pi$
$390$		0			0
$391$	−1.57623e22			−1.45921
$392$		−	4.31788e20i		−	0.0391146i
$393$		0			0
$394$	−2.80926e22			−2.43711
$395$	−5.39591e21	−	6.30492e21i	−0.458131	−	0.535308i
$396$		0			0
$397$			7.07422e21i			0.575386i	0.957723	+	0.287693i	$0.0928883\pi$
$397$			7.07422e21i			0.575386i	−0.957723	+	0.287693i	$0.907112\pi$
$398$		−	1.56623e22i		−	1.24695i
$399$		0			0
$400$	2.09423e21	−	1.33973e22i	0.159777	−	1.02213i
$401$	−5.01418e21			−0.374518			−0.187259	−	0.982311i	$-0.559960\pi$
$401$	−5.01418e21			−0.374518			−0.187259	+	0.982311i	$0.559960\pi$
$402$		0			0
$403$			7.13146e20i			0.0510605i
$404$	−1.22825e22			−0.861082
$405$		0			0
$406$	1.23646e22			0.831207
$407$			4.98306e22i			3.28053i
$408$		0			0
$409$	−2.31062e22			−1.45908			−0.729541	−	0.683937i	$-0.760266\pi$
$409$	−2.31062e22			−1.45908			−0.729541	+	0.683937i	$0.760266\pi$
$410$	3.03967e22	−	2.60142e22i	1.88002	−	1.60897i
$411$		0			0
$412$			1.11273e22i			0.660335i
$413$			8.50103e21i			0.494193i
$414$		0			0
$415$	3.89356e21	−	3.33221e21i	0.217239	−	0.185919i
$416$	3.58123e21			0.195767
$417$		0			0
$418$		−	3.07451e22i		−	1.61353i
$419$	−2.42934e22			−1.24931			−0.624653	−	0.780902i	$-0.714760\pi$
$419$	−2.42934e22			−1.24931			−0.624653	+	0.780902i	$0.714760\pi$
$420$		0			0
$421$	3.52897e20			0.0174281			0.00871406	−	0.999962i	$-0.497226\pi$
$421$	3.52897e20			0.0174281			0.00871406	+	0.999962i	$0.497226\pi$
$422$			1.93141e22i			0.934799i
$423$		0			0
$424$	6.66450e20			0.0309855
$425$	−2.54318e22	−	3.97545e21i	−1.15897	−	0.181168i
$426$		0			0
$427$		−	1.11109e22i		−	0.486535i
$428$			1.32782e22i			0.569989i
$429$		0			0
$430$	1.24365e22	+	1.45316e22i	0.513118	+	0.599559i
$431$	5.48634e21			0.221935			0.110968	−	0.993824i	$-0.464605\pi$
$431$	5.48634e21			0.221935			0.110968	+	0.993824i	$0.464605\pi$
$432$		0			0
$433$			5.04872e22i			1.96351i	0.190141	+	0.981757i	$0.439106\pi$
$433$			5.04872e22i			1.96351i	−0.190141	+	0.981757i	$0.560894\pi$
$434$	6.30675e21			0.240516
$435$		0			0
$436$	−2.37697e22			−0.871745
$437$			2.10949e22i			0.758725i
$438$		0			0
$439$	5.34678e21			0.184988			0.0924941	−	0.995713i	$-0.470516\pi$
$439$	5.34678e21			0.184988			0.0924941	+	0.995713i	$0.470516\pi$
$440$	2.12185e21	−	1.81593e21i	0.0720057	−	0.0616244i
$441$		0			0
$442$		−	7.04205e21i		−	0.229938i
$443$		−	1.21434e22i		−	0.388963i	−0.980906	−	0.194482i	$-0.937698\pi$
$443$		−	1.21434e22i		−	0.388963i	0.980906	−	0.194482i	$-0.0623024\pi$
$444$		0			0
$445$	1.20779e22	+	1.41126e22i	0.372333	+	0.435057i
$446$	−6.37440e22			−1.92794
$447$		0			0
$448$		−	1.49682e22i		−	0.435819i
$449$	−1.53675e22			−0.439044			−0.219522	−	0.975608i	$-0.570450\pi$
$449$	−1.53675e22			−0.439044			−0.219522	+	0.975608i	$0.570450\pi$
$450$		0			0
$451$	1.21146e23			3.33280
$452$			4.77253e22i			1.28846i
$453$		0			0
$454$	−9.94055e22			−2.58485
$455$	−1.67507e21	−	1.95726e21i	−0.0427501	−	0.0499519i
$456$		0			0
$457$			4.56261e22i			1.12183i	0.827875	+	0.560913i	$0.189550\pi$
$457$			4.56261e22i			1.12183i	−0.827875	+	0.560913i	$0.810450\pi$
$458$		−	4.07091e22i		−	0.982506i
$459$		0			0
$460$	3.91807e22	−	3.35319e22i	0.911236	−	0.779860i
$461$	2.39376e22			0.546542			0.273271	−	0.961937i	$-0.411894\pi$
$461$	2.39376e22			0.546542			0.273271	+	0.961937i	$0.411894\pi$
$462$		0			0
$463$		−	3.50789e22i		−	0.771982i	−0.922503	−	0.385991i	$-0.873860\pi$
$463$		−	3.50789e22i		−	0.771982i	0.922503	−	0.385991i	$-0.126140\pi$
$464$	6.04082e22			1.30525
$465$		0			0
$466$	7.80681e22			1.62627
$467$			9.51264e22i			1.94584i	0.231137	+	0.972921i	$0.425755\pi$
$467$			9.51264e22i			1.94584i	−0.231137	+	0.972921i	$0.574245\pi$
$468$		0			0
$469$	1.90896e22			0.376554
$470$	−5.17172e22	−	6.04295e22i	−1.00185	−	1.17062i
$471$		0			0
$472$			2.82499e21i			0.0527850i
$473$			5.79159e22i			1.06287i
$474$		0			0
$475$	−5.32038e21	+	3.40356e22i	−0.0941992	+	0.602612i
$476$	−3.05705e22			−0.531671
$477$		0			0
$478$			1.32623e23i			2.22578i
$479$	−2.22622e22			−0.367043			−0.183521	−	0.983016i	$-0.558750\pi$
$479$	−2.22622e22			−0.367043			−0.183521	+	0.983016i	$0.558750\pi$
$480$		0			0
$481$	1.52748e22			0.243076
$482$			4.54536e22i			0.710671i
$483$		0			0
$484$	−1.63715e23			−2.47116
$485$	−2.95532e22	+	2.52924e22i	−0.438329	+	0.375133i
$486$		0			0
$487$			8.95812e22i			1.28298i	0.767130	+	0.641491i	$0.221684\pi$
$487$			8.95812e22i			1.28298i	−0.767130	+	0.641491i	$0.778316\pi$
$488$		−	3.69230e21i		−	0.0519671i
$489$		0			0
$490$	6.10208e22	−	5.22231e22i	0.829490	−	0.709899i
$491$	−1.20397e23			−1.60850			−0.804252	−	0.594289i	$-0.797434\pi$
$491$	−1.20397e23			−1.60850			−0.804252	+	0.594289i	$0.797434\pi$
$492$		0			0
$493$		−	1.14672e23i		−	1.47999i
$494$	−9.42444e21			−0.119557
$495$		0			0
$496$	3.08121e22			0.377683
$497$		−	8.56077e20i		−	0.0103153i
$498$		0			0
$499$	−1.23494e23			−1.43811			−0.719054	−	0.694954i	$-0.755425\pi$
$499$	−1.23494e23			−1.43811			−0.719054	+	0.694954i	$0.755425\pi$
$500$	7.16735e22	−	4.42203e22i	0.820566	−	0.506264i
$501$		0			0
$502$		−	1.61825e23i		−	1.79086i
$503$		−	1.74448e22i		−	0.189818i	−0.995486	−	0.0949090i	$-0.969744\pi$
$503$		−	1.74448e22i		−	0.189818i	0.995486	−	0.0949090i	$-0.0302560\pi$
$504$		0			0
$505$	5.51979e22	+	6.44966e22i	0.580692	+	0.678516i
$506$	3.18113e23			3.29080
$507$		0			0
$508$		−	7.14453e22i		−	0.714714i
$509$	−8.27894e22			−0.814468			−0.407234	−	0.913324i	$-0.633507\pi$
$509$	−8.27894e22			−0.814468			−0.407234	+	0.913324i	$0.633507\pi$
$510$		0			0
$511$	−3.15240e22			−0.299960
$512$		−	1.49180e23i		−	1.39610i
$513$		0			0
$514$	−1.50414e23			−1.36176
$515$	5.84308e22	−	5.00066e22i	0.520332	−	0.445313i
$516$		0			0
$517$		−	2.40842e23i		−	2.07522i
$518$		−	1.35084e23i		−	1.14499i
$519$		0			0
$520$	−5.56647e20	−	6.50421e20i	−0.00456617	−	0.00533539i
$521$	1.07844e22			0.0870316			0.0435158	−	0.999053i	$-0.486144\pi$
$521$	1.07844e22			0.0870316			0.0435158	+	0.999053i	$0.486144\pi$
$522$		0			0
$523$			8.09901e22i			0.632656i	0.948650	+	0.316328i	$0.102450\pi$
$523$			8.09901e22i			0.632656i	−0.948650	+	0.316328i	$0.897550\pi$
$524$	−1.68311e22			−0.129359
$525$		0			0
$526$	−1.01896e23			−0.758191
$527$		−	5.84901e22i		−	0.428245i
$528$		0			0
$529$	−7.72140e22			−0.547423
$530$	8.06047e22	+	9.41836e22i	0.562362	+	0.657098i
$531$		0			0
$532$			4.09127e22i			0.276446i
$533$		−	3.71355e22i		−	0.246950i
$534$		0			0
$535$	6.97252e22	−	5.96726e22i	0.449141	−	0.384386i
$536$	6.34370e21			0.0402200
$537$		0			0
$538$		−	6.93302e22i		−	0.425866i
$539$	2.43199e23			1.47047
$540$		0			0
$541$	7.17171e22			0.420190			0.210095	−	0.977681i	$-0.432623\pi$
$541$	7.17171e22			0.420190			0.210095	+	0.977681i	$0.432623\pi$
$542$		−	4.38935e23i		−	2.53166i
$543$		0			0
$544$	−2.93722e23			−1.64190
$545$	1.06822e23	+	1.24817e23i	0.587882	+	0.686918i
$546$		0			0
$547$			2.58207e23i			1.37745i	0.725022	+	0.688725i	$0.241829\pi$
$547$			2.58207e23i			1.37745i	−0.725022	+	0.688725i	$0.758171\pi$
$548$		−	2.53079e21i		−	0.0132929i
$549$		0			0
$550$	5.13260e23	+	8.02319e22i	2.61369	+	0.408568i
$551$	−1.53466e23			−0.769528
$552$		0			0
$553$			6.81194e22i			0.331213i
$554$	6.18848e22			0.296313
$555$		0			0
$556$	2.71044e23			1.25865
$557$		−	1.34677e23i		−	0.615920i	−0.951399	−	0.307960i	$-0.900354\pi$
$557$		−	1.34677e23i		−	0.615920i	0.951399	−	0.307960i	$-0.0996463\pi$
$558$		0			0
$559$	1.77533e22			0.0787549
$560$	−8.45653e22	+	7.23732e22i	−0.369483	+	0.316213i
$561$		0			0
$562$			4.75753e23i			2.01661i
$563$			2.55092e23i			1.06506i	0.846410	+	0.532532i	$0.178760\pi$
$563$			2.55092e23i			1.06506i	−0.846410	+	0.532532i	$0.821240\pi$
$564$		0			0
$565$	2.50610e23	−	2.14479e23i	1.01528	−	0.868906i
$566$	9.63326e22			0.384444
$567$		0			0
$568$		−	2.84485e20i		−	0.00110179i
$569$	5.11608e22			0.195201			0.0976007	−	0.995226i	$-0.468883\pi$
$569$	5.11608e22			0.195201			0.0976007	+	0.995226i	$0.468883\pi$
$570$		0			0
$571$	−1.90762e23			−0.706457			−0.353228	−	0.935537i	$-0.614916\pi$
$571$	−1.90762e23			−0.706457			−0.353228	+	0.935537i	$0.614916\pi$
$572$			6.97646e22i			0.254548i
$573$		0			0
$574$	−3.28411e23			−1.16323
$575$	−3.52159e23	−	5.50489e22i	−1.22903	−	0.192120i
$576$		0			0
$577$		−	9.22934e22i		−	0.312735i	−0.987699	−	0.156367i	$-0.950022\pi$
$577$		−	9.22934e22i		−	0.312735i	0.987699	−	0.156367i	$-0.0499784\pi$
$578$			1.57832e23i			0.526998i
$579$		0			0
$580$	2.43947e23	+	2.85042e23i	0.790964	+	0.924211i
$581$	−4.20667e22			−0.134413
$582$		0			0
$583$			3.75369e23i			1.16487i
$584$	−1.04758e22			−0.0320389
$585$		0			0
$586$	6.02261e23			1.78918
$587$		−	3.82858e23i		−	1.12102i	−0.828147	−	0.560511i	$-0.810605\pi$
$587$		−	3.82858e23i		−	1.12102i	0.828147	−	0.560511i	$-0.189395\pi$
$588$		0			0
$589$	−7.82778e22			−0.222669
$590$	−3.99232e23	+	3.41673e23i	−1.11939	+	0.958006i
$591$		0			0
$592$		−	6.59962e23i		−	1.79798i
$593$			3.51443e22i			0.0943822i	0.998886	+	0.0471911i	$0.0150270\pi$
$593$			3.51443e22i			0.0943822i	−0.998886	+	0.0471911i	$0.984973\pi$
$594$		0			0
$595$	1.37385e23	+	1.60529e23i	0.358546	+	0.418947i
$596$	−1.09365e23			−0.281376
$597$		0			0
$598$		−	9.75127e22i		−	0.243838i
$599$	2.37562e23			0.585665			0.292832	−	0.956164i	$-0.405402\pi$
$599$	2.37562e23			0.585665			0.292832	+	0.956164i	$0.405402\pi$
$600$		0			0
$601$	−6.50859e23			−1.55975			−0.779873	−	0.625938i	$-0.784716\pi$
$601$	−6.50859e23			−1.55975			−0.779873	+	0.625938i	$0.784716\pi$
$602$		−	1.57002e23i		−	0.370967i
$603$		0			0
$604$	−3.09560e23			−0.711101
$605$	7.35739e23	+	8.59683e23i	1.66649	+	1.94723i
$606$		0			0
$607$		−	2.46962e23i		−	0.543910i	−0.962310	−	0.271955i	$-0.912330\pi$
$607$		−	2.46962e23i		−	0.543910i	0.962310	−	0.271955i	$-0.0876702\pi$
$608$			3.93090e23i			0.853713i
$609$		0			0
$610$	5.21800e23	−	4.46570e23i	1.10205	−	0.943162i
$611$	−7.38266e22			−0.153767
$612$		0			0
$613$		−	7.66264e23i		−	1.55226i	−0.630573	−	0.776130i	$-0.717180\pi$
$613$		−	7.66264e23i		−	1.55226i	0.630573	−	0.776130i	$-0.282820\pi$
$614$	−6.02170e23			−1.20306
$615$		0			0
$616$	−2.29248e22			−0.0445523
$617$		−	8.97264e23i		−	1.71987i	−0.510402	−	0.859936i	$-0.670503\pi$
$617$		−	8.97264e23i		−	1.71987i	0.510402	−	0.859936i	$-0.329497\pi$
$618$		0			0
$619$	−6.29052e23			−1.17305			−0.586525	−	0.809931i	$-0.699504\pi$
$619$	−6.29052e23			−1.17305			−0.586525	+	0.809931i	$0.699504\pi$
$620$	1.24429e23	+	1.45390e23i	0.228871	+	0.267427i
$621$		0			0
$622$			1.16799e24i			2.09037i
$623$		−	1.52475e23i		−	0.269184i
$624$		0			0
$625$	−5.54309e23	−	1.77638e23i	−0.952295	−	0.305179i
$626$	−2.20293e23			−0.373353
$627$		0			0
$628$			7.26612e23i			1.19852i
$629$	−1.25280e24			−2.03868
$630$		0			0
$631$	2.88063e23			0.456287			0.228143	−	0.973628i	$-0.426735\pi$
$631$	2.88063e23			0.456287			0.228143	+	0.973628i	$0.426735\pi$
$632$			2.26369e22i			0.0353770i
$633$		0			0
$634$	−9.07419e23			−1.38054
$635$	−3.75167e23	+	3.21078e23i	−0.563181	+	0.481985i
$636$		0			0
$637$		−	7.45489e22i		−	0.108957i
$638$			2.31429e24i			3.33766i
$639$		0			0
$640$	−5.42926e22	+	4.64650e22i	−0.0762450	+	0.0652524i
$641$	2.86914e23			0.397611			0.198806	−	0.980039i	$-0.436294\pi$
$641$	2.86914e23			0.397611			0.198806	+	0.980039i	$0.436294\pi$
$642$		0			0
$643$		−	1.37702e24i		−	1.85843i	−0.369534	−	0.929217i	$-0.620483\pi$
$643$		−	1.37702e24i		−	1.85843i	0.369534	−	0.929217i	$-0.379517\pi$
$644$	−4.23315e23			−0.563812
$645$		0			0
$646$	7.72965e23			1.00273
$647$			4.32243e23i			0.553403i	0.960956	+	0.276701i	$0.0892412\pi$
$647$			4.32243e23i			0.553403i	−0.960956	+	0.276701i	$0.910759\pi$
$648$		0			0
$649$	−1.59114e24			−1.98440
$650$	2.45939e22	−	1.57332e23i	0.0302736	−	0.193666i
$651$		0			0
$652$		−	1.30322e24i		−	1.56284i
$653$		−	1.02897e24i		−	1.21798i	−0.793179	−	0.608989i	$-0.791575\pi$
$653$		−	1.02897e24i		−	1.21798i	0.793179	−	0.608989i	$-0.208425\pi$
$654$		0			0
$655$	7.56395e22	+	8.83819e22i	0.0872364	+	0.101932i
$656$	−1.60447e24			−1.82663
$657$		0			0
$658$			6.52891e23i			0.724304i
$659$	1.11269e24			1.21856			0.609280	−	0.792955i	$-0.291459\pi$
$659$	1.11269e24			1.21856			0.609280	+	0.792955i	$0.291459\pi$
$660$		0			0
$661$	1.75632e24			1.87453			0.937263	−	0.348624i	$-0.113351\pi$
$661$	1.75632e24			1.87453			0.937263	+	0.348624i	$0.113351\pi$
$662$			7.69144e23i			0.810429i
$663$		0			0
$664$	−1.39793e22			−0.0143567
$665$	2.14837e23	−	1.83863e23i	0.217834	−	0.186428i
$666$		0			0
$667$		−	1.58788e24i		−	1.56946i
$668$		−	4.74601e23i		−	0.463158i
$669$		0			0
$670$	7.67247e23	+	8.96499e23i	0.729961	+	0.852932i
$671$	2.07964e24			1.95365
$672$		0			0
$673$			1.46558e24i			1.34240i	0.741277	+	0.671199i	$0.234220\pi$
$673$			1.46558e24i			1.34240i	−0.741277	+	0.671199i	$0.765780\pi$
$674$	7.52689e23			0.680780
$675$		0			0
$676$	−1.07182e24			−0.945313
$677$		−	1.08985e23i		−	0.0949209i	−0.998873	−	0.0474604i	$-0.984887\pi$
$677$		−	1.08985e23i		−	0.0949209i	0.998873	−	0.0474604i	$-0.0151128\pi$
$678$		0			0
$679$	3.19298e23			0.271209
$680$	4.56546e22	+	5.33457e22i	0.0382965	+	0.0447480i
$681$		0			0
$682$			1.18044e24i			0.965775i
$683$		−	1.09633e24i		−	0.885862i	−0.896556	−	0.442931i	$-0.853939\pi$
$683$		−	1.09633e24i		−	0.885862i	0.896556	−	0.442931i	$-0.146061\pi$
$684$		0			0
$685$	−1.32894e22	+	1.13735e22i	−0.0104746	+	0.00896442i
$686$	−1.50557e24			−1.17205
$687$		0			0
$688$		−	7.67046e23i		−	0.582532i
$689$	1.15064e23			0.0863129
$690$		0			0
$691$	8.45376e23			0.618709			0.309355	−	0.950947i	$-0.399887\pi$
$691$	8.45376e23			0.618709			0.309355	+	0.950947i	$0.399887\pi$
$692$		−	1.16662e24i		−	0.843390i
$693$		0			0
$694$	9.24996e23			0.652506
$695$	−1.21808e24	−	1.42328e24i	−0.848800	−	0.991791i
$696$		0			0
$697$			3.04575e24i			2.07117i
$698$		−	1.36197e24i		−	0.914947i
$699$		0			0
$700$	−6.83000e23	−	1.06765e23i	−0.447804	−	0.0699998i
$701$	1.46775e24			0.950713			0.475356	−	0.879793i	$-0.342319\pi$
$701$	1.46775e24			0.950713			0.475356	+	0.879793i	$0.342319\pi$
$702$		0			0
$703$			1.67663e24i			1.06003i
$704$	2.80160e24			1.75000
$705$		0			0
$706$	3.01595e24			1.83901
$707$		−	6.96833e23i		−	0.419820i
$708$		0			0
$709$	2.30539e24			1.35597			0.677986	−	0.735075i	$-0.262853\pi$
$709$	2.30539e24			1.35597			0.677986	+	0.735075i	$0.262853\pi$
$710$	4.02037e22	−	3.44074e22i	0.0233652	−	0.0199966i
$711$		0			0
$712$		−	5.06692e22i		−	0.0287517i
$713$		−	8.09924e23i		−	0.454133i
$714$		0			0
$715$	3.66341e23	−	3.13524e23i	0.200579	−	0.171660i
$716$	2.77079e24			1.49914
$717$		0			0
$718$		−	1.95868e24i		−	1.03492i
$719$	1.89798e24			0.991051			0.495526	−	0.868593i	$-0.334975\pi$
$719$	1.89798e24			0.991051			0.495526	+	0.868593i	$0.334975\pi$
$720$		0			0
$721$	−6.31296e23			−0.321946
$722$			1.74625e24i			0.880115i
$723$		0			0
$724$	−2.29598e24			−1.13029
$725$	4.00484e23	−	2.56198e24i	0.194855	−	1.24653i
$726$		0			0
$727$			1.05546e24i			0.501649i	0.968033	+	0.250825i	$0.0807017\pi$
$727$			1.05546e24i			0.501649i	−0.968033	+	0.250825i	$0.919298\pi$
$728$			7.02727e21i			0.00330118i
$729$		0			0
$730$	−1.26701e24	−	1.48045e24i	−0.581481	−	0.679438i
$731$	−1.45607e24			−0.660518
$732$		0			0
$733$			2.43831e24i			1.08070i	0.841441	+	0.540349i	$0.181708\pi$
$733$			2.43831e24i			1.08070i	−0.841441	+	0.540349i	$0.818292\pi$
$734$	−4.57818e24			−2.00575
$735$		0			0
$736$	−4.06722e24			−1.74115
$737$			3.57301e24i			1.51203i
$738$		0			0
$739$	3.05341e23			0.126272			0.0631360	−	0.998005i	$-0.479890\pi$
$739$	3.05341e23			0.126272			0.0631360	+	0.998005i	$0.479890\pi$
$740$	3.11410e24	−	2.66513e24i	1.27310	−	1.08955i
$741$		0			0
$742$		−	1.01758e24i		−	0.406568i
$743$		−	1.22011e24i		−	0.481943i	−0.970532	−	0.240971i	$-0.922534\pi$
$743$		−	1.22011e24i		−	0.481943i	0.970532	−	0.240971i	$-0.0774660\pi$
$744$		0			0
$745$	4.91491e23	+	5.74289e23i	0.189753	+	0.221719i
$746$	−1.33243e24			−0.508585
$747$		0			0
$748$		−	5.72189e24i		−	2.13489i
$749$	−7.53323e23			−0.277898
$750$		0			0
$751$	−2.60928e24			−0.940983			−0.470492	−	0.882404i	$-0.655923\pi$
$751$	−2.60928e24			−0.940983			−0.470492	+	0.882404i	$0.655923\pi$
$752$			3.18975e24i			1.13738i
$753$		0			0
$754$	7.09411e23			0.247310
$755$	1.39117e24	+	1.62553e24i	0.479548	+	0.560334i
$756$		0			0
$757$		−	5.45847e24i		−	1.83974i	−0.392224	−	0.919870i	$-0.628294\pi$
$757$		−	5.45847e24i		−	1.83974i	0.392224	−	0.919870i	$-0.371706\pi$
$758$		−	8.74295e23i		−	0.291387i
$759$		0			0
$760$	7.13930e22	−	6.10999e22i	0.0232670	−	0.0199125i
$761$	1.46965e24			0.473636			0.236818	−	0.971554i	$-0.423895\pi$
$761$	1.46965e24			0.473636			0.236818	+	0.971554i	$0.423895\pi$
$762$		0			0
$763$		−	1.34855e24i		−	0.425019i
$764$	−3.86464e24			−1.20452
$765$		0			0
$766$	−4.67523e24			−1.42514
$767$			4.87741e23i			0.147038i
$768$		0			0
$769$	3.22356e24			0.950522			0.475261	−	0.879845i	$-0.342354\pi$
$769$	3.22356e24			0.950522			0.475261	+	0.879845i	$0.342354\pi$
$770$	−2.77268e24	−	3.23977e24i	−0.808589	−	0.944806i
$771$		0			0
$772$		−	3.29754e24i		−	0.940682i
$773$		−	5.84112e24i		−	1.64805i	−0.566553	−	0.824025i	$-0.691723\pi$
$773$		−	5.84112e24i		−	1.64805i	0.566553	−	0.824025i	$-0.308277\pi$
$774$		0			0
$775$	2.04273e23	−	1.30678e24i	0.0563827	−	0.360692i
$776$	1.06107e23			0.0289680
$777$		0			0
$778$			5.23886e24i			1.39930i
$779$	4.07615e24			1.07692
$780$		0			0
$781$	1.60232e23			0.0414206
$782$			7.99771e24i			2.04507i
$783$		0			0
$784$	−3.22096e24			−0.805932
$785$	3.81552e24	−	3.26542e24i	0.944412	−	0.808252i
$786$		0			0
$787$			6.89254e24i			1.66953i	0.550606	+	0.834765i	$0.314397\pi$
$787$			6.89254e24i			1.66953i	−0.550606	+	0.834765i	$0.685603\pi$
$788$			6.99701e24i			1.67664i
$789$		0			0
$790$	−3.19907e24	+	2.73785e24i	−0.750229	+	0.642065i
$791$	−2.70764e24			−0.628189
$792$		0			0
$793$		−	6.37482e23i		−	0.144759i
$794$	3.58941e24			0.806398
$795$		0			0
$796$	−3.90100e24			−0.857856
$797$			6.02149e24i			1.31011i	0.755581	+	0.655055i	$0.227355\pi$
$797$			6.02149e24i			1.31011i	−0.755581	+	0.655055i	$0.772645\pi$
$798$		0			0
$799$	6.05505e24			1.28964
$800$	−6.56228e24	−	1.02580e24i	−1.38290	−	0.216172i
$801$		0			0
$802$			2.54416e24i			0.524883i
$803$		−	5.90035e24i		−	1.20447i
$804$		0			0
$805$	1.90239e24	+	2.22287e24i	0.380220	+	0.444273i
$806$	3.61845e23			0.0715608
$807$		0			0
$808$		−	2.31566e23i		−	0.0448413i
$809$	−8.51194e24			−1.63105			−0.815523	−	0.578724i	$-0.803551\pi$
$809$	−8.51194e24			−1.63105			−0.815523	+	0.578724i	$0.803551\pi$
$810$		0			0
$811$	−3.37937e24			−0.634101			−0.317050	−	0.948409i	$-0.602692\pi$
$811$	−3.37937e24			−0.634101			−0.317050	+	0.948409i	$0.602692\pi$
$812$		−	3.07965e24i		−	0.571840i
$813$		0			0
$814$	2.52837e25			4.59762
$815$	−6.84336e24	+	5.85672e24i	−1.23149	+	1.05394i
$816$		0			0
$817$			1.94867e24i			0.343440i
$818$			1.17239e25i			2.04489i
$819$		0			0
$820$	−6.47935e24	−	7.57088e24i	−1.10691	−	1.29339i
$821$	−7.85056e23			−0.132735			−0.0663673	−	0.997795i	$-0.521141\pi$
$821$	−7.85056e23			−0.132735			−0.0663673	+	0.997795i	$0.521141\pi$
$822$		0			0
$823$		−	1.90020e24i		−	0.314703i	−0.987543	−	0.157352i	$-0.949704\pi$
$823$		−	1.90020e24i		−	0.314703i	0.987543	−	0.157352i	$-0.0502956\pi$
$824$	−2.09787e23			−0.0343873
$825$		0			0
$826$	4.31337e24			0.692606
$827$			2.70889e24i			0.430522i	0.976557	+	0.215261i	$0.0690602\pi$
$827$			2.70889e24i			0.430522i	−0.976557	+	0.215261i	$0.930940\pi$
$828$		0			0
$829$	−4.30135e24			−0.669717			−0.334859	−	0.942268i	$-0.608689\pi$
$829$	−4.30135e24			−0.669717			−0.334859	+	0.942268i	$0.608689\pi$
$830$	−1.69074e24	−	1.97557e24i	−0.260563	−	0.304458i
$831$		0			0
$832$		−	8.58788e23i		−	0.129670i
$833$			6.11429e24i			0.913826i
$834$		0			0
$835$	−2.49218e24	+	2.13287e24i	−0.364960	+	0.312342i
$836$	−7.65765e24			−1.11005
$837$		0			0
$838$			1.23263e25i			1.75089i
$839$	1.24382e25			1.74897			0.874483	−	0.485056i	$-0.161201\pi$
$839$	1.24382e25			1.74897			0.874483	+	0.485056i	$0.161201\pi$
$840$		0			0
$841$	4.29481e24			0.591804
$842$		−	1.79058e23i		−	0.0244253i
$843$		0			0
$844$	4.81055e24			0.643108
$845$	4.81680e24	+	5.62825e24i	0.637495	+	0.744889i
$846$		0			0
$847$		−	9.28816e24i		−	1.20482i
$848$		−	4.97144e24i		−	0.638436i
$849$		0			0
$850$	−2.01712e24	+	1.29039e25i	−0.253905	+	1.62428i
$851$	−1.73477e25			−2.16193
$852$		0			0
$853$		−	1.88913e24i		−	0.230778i	−0.993320	−	0.115389i	$-0.963188\pi$
$853$		−	1.88913e24i		−	0.230778i	0.993320	−	0.115389i	$-0.0368115\pi$
$854$	−5.63762e24			−0.681874
$855$		0			0
$856$	−2.50338e23			−0.0296825
$857$			2.15497e24i			0.252990i	0.991967	+	0.126495i	$0.0403728\pi$
$857$			2.15497e24i			0.252990i	−0.991967	+	0.126495i	$0.959627\pi$
$858$		0			0
$859$	−1.10633e24			−0.127334			−0.0636669	−	0.997971i	$-0.520280\pi$
$859$	−1.10633e24			−0.127334			−0.0636669	+	0.997971i	$0.520280\pi$
$860$	3.61938e24	−	3.09756e24i	0.412475	−	0.353007i
$861$		0			0
$862$		−	2.78373e24i		−	0.311040i
$863$		−	7.62129e24i		−	0.843212i	−0.906779	−	0.421606i	$-0.861466\pi$
$863$		−	7.62129e24i		−	0.843212i	0.906779	−	0.421606i	$-0.138534\pi$
$864$		0			0
$865$	−6.12606e24	+	5.24284e24i	−0.664576	+	0.568761i
$866$	2.56169e25			2.75184
$867$		0			0
$868$		−	1.57082e24i		−	0.165466i
$869$	−1.27499e25			−1.32996
$870$		0			0
$871$	1.09525e24			0.112036
$872$		−	4.48139e23i		−	0.0453965i
$873$		0			0
$874$	1.07034e25			1.06335
$875$	2.50879e24	+	4.06631e24i	0.246829	+	0.400067i
$876$		0			0
$877$			5.89453e24i			0.568791i	0.958707	+	0.284396i	$0.0917929\pi$
$877$			5.89453e24i			0.568791i	−0.958707	+	0.284396i	$0.908207\pi$
$878$		−	2.71292e24i		−	0.259259i
$879$		0			0
$880$	−1.35461e25	−	1.58281e25i	−1.26973	−	1.48363i
$881$	9.93638e24			0.922429			0.461214	−	0.887289i	$-0.347414\pi$
$881$	9.93638e24			0.922429			0.461214	+	0.887289i	$0.347414\pi$
$882$		0			0
$883$		−	1.62270e25i		−	1.47765i	−0.673896	−	0.738826i	$-0.735381\pi$
$883$		−	1.62270e25i		−	1.47765i	0.673896	−	0.738826i	$-0.264619\pi$
$884$	−1.75396e24			−0.158189
$885$		0			0
$886$	−6.16148e24			−0.545128
$887$		−	5.38254e24i		−	0.471668i	−0.971793	−	0.235834i	$-0.924218\pi$
$887$		−	5.38254e24i		−	0.471668i	0.971793	−	0.235834i	$-0.0757821\pi$
$888$		0			0
$889$	4.05337e24			0.348459
$890$	7.16063e24	−	6.12825e24i	0.609728	−	0.521821i
$891$		0			0
$892$			1.58767e25i			1.32635i
$893$		−	8.10352e24i		−	0.670558i
$894$		0			0
$895$	−1.24520e25	−	1.45497e25i	−1.01098	−	1.18129i
$896$	5.86587e23			0.0471753
$897$		0			0
$898$			7.79735e24i			0.615316i
$899$	5.89225e24			0.460600
$900$		0			0
$901$	−9.43721e24			−0.723907
$902$		−	6.14688e25i		−	4.67088i
$903$		0			0
$904$	−8.99781e23			−0.0670973
$905$	1.03182e25	+	1.20565e25i	0.762240	+	0.890649i
$906$		0			0
$907$		−	3.47532e23i		−	0.0251961i	−0.999921	−	0.0125980i	$-0.995990\pi$
$907$		−	3.47532e23i		−	0.0251961i	0.999921	−	0.0125980i	$-0.00401019\pi$
$908$			2.47589e25i			1.77828i
$909$		0			0
$910$	−9.93102e23	+	8.49922e23i	−0.0700071	+	0.0599139i
$911$	1.01314e25			0.707557			0.353779	−	0.935329i	$-0.384897\pi$
$911$	1.01314e25			0.707557			0.353779	+	0.935329i	$0.384897\pi$
$912$		0			0
$913$		−	7.87363e24i		−	0.539726i
$914$	2.31504e25			1.57223
$915$		0			0
$916$	−1.01394e25			−0.675928
$917$		−	9.54894e23i		−	0.0630690i
$918$		0			0
$919$	−8.36927e24			−0.542633			−0.271316	−	0.962490i	$-0.587459\pi$
$919$	−8.36927e24			−0.542633			−0.271316	+	0.962490i	$0.587459\pi$
$920$	6.32188e23	+	7.38688e23i	0.0406116	+	0.0474531i
$921$		0			0
$922$		−	1.21458e25i		−	0.765973i
$923$		−	4.91168e22i		−	0.00306913i
$924$		0			0
$925$	−2.79897e25	−	4.37530e24i	−1.71709	−	0.268413i
$926$	−1.77988e25			−1.08192
$927$		0			0
$928$		−	2.95893e25i		−	1.76594i
$929$	−1.96799e25			−1.16383			−0.581913	−	0.813251i	$-0.697696\pi$
$929$	−1.96799e25			−1.16383			−0.581913	+	0.813251i	$0.697696\pi$
$930$		0			0
$931$	8.18280e24			0.475149
$932$		−	1.94444e25i		−	1.11882i
$933$		0			0
$934$	4.82665e25			2.72708
$935$	−3.00463e25	+	2.57144e25i	−1.68225	+	1.43972i
$936$		0			0
$937$		−	2.48771e25i		−	1.36777i	−0.729590	−	0.683885i	$-0.760289\pi$
$937$		−	2.48771e25i		−	1.36777i	0.729590	−	0.683885i	$-0.239711\pi$
$938$		−	9.68593e24i		−	0.527737i
$939$		0			0
$940$	−1.50512e25	+	1.28812e25i	−0.805347	+	0.689237i
$941$	1.69158e23			0.00896974			0.00448487	−	0.999990i	$-0.498572\pi$
$941$	1.69158e23			0.00896974			0.00448487	+	0.999990i	$0.498572\pi$
$942$		0			0
$943$			4.21751e25i			2.19637i
$944$	2.10733e25			1.08760
$945$		0			0
$946$	2.93862e25			1.48959
$947$		−	7.24672e24i		−	0.364055i	−0.983293	−	0.182027i	$-0.941734\pi$
$947$		−	7.24672e24i		−	0.364055i	0.983293	−	0.182027i	$-0.0582660\pi$
$948$		0			0
$949$	−1.80866e24			−0.0892473
$950$	1.72694e25	+	2.69953e24i	0.844554	+	0.132019i
$951$		0			0
$952$		−	5.76356e23i		−	0.0276871i
$953$			6.99427e24i			0.333006i	0.986041	+	0.166503i	$0.0532476\pi$
$953$			6.99427e24i			0.333006i	−0.986041	+	0.166503i	$0.946752\pi$
$954$		0			0
$955$	1.73678e25	+	2.02936e25i	0.812301	+	0.949143i
$956$	3.30324e25			1.53126
$957$		0			0
$958$			1.12957e25i			0.514407i
$959$	1.43581e23			0.00648097
$960$		0			0
$961$	−1.95447e25			−0.866722
$962$		−	7.75034e24i		−	0.340669i
$963$		0			0
$964$	1.13211e25			0.488916
$965$	−1.73157e25	+	1.48192e25i	−0.741239	+	0.634372i
$966$		0			0
$967$			1.84240e25i			0.774924i	0.921886	+	0.387462i	$0.126648\pi$
$967$			1.84240e25i			0.774924i	−0.921886	+	0.387462i	$0.873352\pi$
$968$		−	3.08657e24i		−	0.128687i
$969$		0			0
$970$	1.28332e25	+	1.49951e25i	0.525746	+	0.614314i
$971$	−5.07475e24			−0.206087			−0.103044	−	0.994677i	$-0.532858\pi$
$971$	−5.07475e24			−0.206087			−0.103044	+	0.994677i	$0.532858\pi$
$972$		0			0
$973$			1.53774e25i			0.613653i
$974$	4.54529e25			1.79809
$975$		0			0
$976$	−2.75430e25			−1.07075
$977$		−	1.16146e25i		−	0.447611i	−0.974634	−	0.223805i	$-0.928152\pi$
$977$		−	1.16146e25i		−	0.447611i	0.974634	−	0.223805i	$-0.0718480\pi$
$978$		0			0
$979$	2.85388e25			1.08089
$980$	−1.30072e25	−	1.51984e25i	−0.488384	−	0.570659i
$981$		0			0
$982$			6.10885e25i			2.25430i
$983$		−	5.71703e24i		−	0.209154i	−0.994517	−	0.104577i	$-0.966651\pi$
$983$		−	5.71703e24i		−	0.209154i	0.994517	−	0.104577i	$-0.0333488\pi$
$984$		0			0
$985$	3.67421e25	−	3.14448e25i	1.32116	−	1.13068i
$986$	−5.81838e25			−2.07419
$987$		0			0
$988$			2.34734e24i			0.0822511i
$989$	−2.01625e25			−0.700447
$990$		0			0
$991$	1.22688e25			0.418964			0.209482	−	0.977813i	$-0.432822\pi$
$991$	1.22688e25			0.418964			0.209482	+	0.977813i	$0.432822\pi$
$992$		−	1.50925e25i		−	0.510988i
$993$		0			0
$994$	−4.34368e23			−0.0144568
$995$	1.75312e25	+	2.04846e25i	0.578516	+	0.675974i
$996$		0			0
$997$		−	6.87845e23i		−	0.0223142i	−0.999938	−	0.0111571i	$-0.996449\pi$
$997$		−	6.87845e23i		−	0.0223142i	0.999938	−	0.0111571i	$-0.00355149\pi$
$998$			6.26601e25i			2.01549i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.18.b.c.19.4	yes	16
3.2	odd	2	inner	45.18.b.c.19.13	yes	16
5.4	even	2	inner	45.18.b.c.19.14	yes	16
15.14	odd	2	inner	45.18.b.c.19.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.18.b.c.19.3	✓	16	15.14	odd	2	inner
45.18.b.c.19.4	yes	16	1.1	even	1	trivial
45.18.b.c.19.13	yes	16	3.2	odd	2	inner
45.18.b.c.19.14	yes	16	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	45.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-507.394i q^{2} -126376. q^{4} +(567939. + 663615. i) q^{5} -7.16981e6i q^{7} -2.38261e6i q^{8} +O(q^{10})\)	\(q-507.394i q^{2} -126376. q^{4} +(567939. + 663615. i) q^{5} -7.16981e6i q^{7} -2.38261e6i q^{8} +(3.36714e8 - 2.88169e8i) q^{10} +1.34198e9 q^{11} -4.11363e8i q^{13} -3.63792e9 q^{14} -1.77733e10 q^{16} +3.37388e10i q^{17} +4.51529e10 q^{19} +(-7.17740e10 - 8.38652e10i) q^{20} -6.80910e11i q^{22} +4.67187e11i q^{23} +(-1.17830e11 + 7.53786e11i) q^{25} -2.08723e11 q^{26} +9.06094e11i q^{28} -3.39882e12 q^{29} -1.73362e12 q^{31} +8.70577e12i q^{32} +1.71188e13 q^{34} +(4.75799e12 - 4.07201e12i) q^{35} +3.71322e13i q^{37} -2.29103e13i q^{38} +(1.58114e12 - 1.35318e12i) q^{40} +9.02744e13 q^{41} +4.31572e13i q^{43} -1.69594e14 q^{44} +2.37048e14 q^{46} -1.79468e14i q^{47} +1.81224e14 q^{49} +(3.82466e14 + 5.97864e13i) q^{50} +5.19865e13i q^{52} +2.79714e14i q^{53} +(7.62161e14 + 8.90556e14i) q^{55} -1.70829e13 q^{56} +1.72454e15i q^{58} -1.18567e15 q^{59} +1.54968e15 q^{61} +8.79626e14i q^{62} +2.08767e15 q^{64} +(2.72986e14 - 2.33629e14i) q^{65} +2.66250e15i q^{67} -4.26378e15i q^{68} +(-2.06611e15 - 2.41418e15i) q^{70} +1.19400e14 q^{71} -4.39676e15i q^{73} +1.88407e16 q^{74} -5.70625e15 q^{76} -9.62172e15i q^{77} -9.50086e15 q^{79} +(-1.00942e16 - 1.17946e16i) q^{80} -4.58047e16i q^{82} -5.86719e15i q^{83} +(-2.23896e16 + 1.91616e16i) q^{85} +2.18977e16 q^{86} -3.19741e15i q^{88} +2.12662e16 q^{89} -2.94939e15 q^{91} -5.90414e16i q^{92} -9.10611e16 q^{94} +(2.56441e16 + 2.99641e16i) q^{95} +4.45337e16i q^{97} -9.19521e16i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 1473168 q^{4}+O(q^{10}) \) 16 * q - 1473168 * q^4	\( 16 q - 1473168 q^{4} - 670199320 q^{10} + 35028482624 q^{16} + 236279498336 q^{19} + 2507484007120 q^{25} - 12501422403232 q^{31} - 36533616605360 q^{34} - 44259047245280 q^{40} - 28675167752480 q^{46} - 430197530548688 q^{49} - 16\!\cdots\!00 q^{55}+ \cdots - 20\!\cdots\!80 q^{94}+O(q^{100}) \) 16 * q - 1473168 * q^4 - 670199320 * q^10 + 35028482624 * q^16 + 236279498336 * q^19 + 2507484007120 * q^25 - 12501422403232 * q^31 - 36533616605360 * q^34 - 44259047245280 * q^40 - 28675167752480 * q^46 - 430197530548688 * q^49 - 1615648303533600 * q^55 - 6459270800401888 * q^61 - 4539898356653312 * q^64 + 1120556748124800 * q^70 + 72078801859831872 * q^76 - 2212561840444576 * q^79 - 75398163530251360 * q^85 + 201977821610712000 * q^91 - 20802101586267680 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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