LMFDB - Embedded newform 45.18.b.c.19.12

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			19.12
Root			$-595.928 - 2068.52i$ of defining polynomial
Character	$\chi$	$=$	45.19
Dual form			45.18.b.c.19.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+406.510i q^{2} -34178.7 q^{4} +(228254. + 843113. i) q^{5} +2.76153e7i q^{7} +3.93881e7i q^{8} +O(q^{10})$	\(q+406.510i q^{2} -34178.7 q^{4} +(228254. + 843113. i) q^{5} +2.76153e7i q^{7} +3.93881e7i q^{8} +(-3.42734e8 + 9.27876e7i) q^{10} -5.76916e8 q^{11} -3.01577e9i q^{13} -1.12259e10 q^{14} -2.04916e10 q^{16} -2.06445e10i q^{17} -1.38783e10 q^{19} +(-7.80142e9 - 2.88165e10i) q^{20} -2.34522e11i q^{22} +5.15758e11i q^{23} +(-6.58740e11 + 3.84888e11i) q^{25} +1.22594e12 q^{26} -9.43853e11i q^{28} -5.86433e11 q^{29} -2.42601e12 q^{31} -3.16735e12i q^{32} +8.39218e12 q^{34} +(-2.32828e13 + 6.30330e12i) q^{35} +1.73827e13i q^{37} -5.64168e12i q^{38} +(-3.32087e13 + 8.99051e12i) q^{40} +8.77960e13 q^{41} -1.31458e14i q^{43} +1.97182e13 q^{44} -2.09661e14 q^{46} -7.43721e12i q^{47} -5.29973e14 q^{49} +(-1.56461e14 - 2.67784e14i) q^{50} +1.03075e14i q^{52} +3.75793e14i q^{53} +(-1.31683e14 - 4.86405e14i) q^{55} -1.08771e15 q^{56} -2.38391e14i q^{58} +1.43476e15 q^{59} -9.34695e14 q^{61} -9.86197e14i q^{62} -1.39831e15 q^{64} +(2.54264e15 - 6.88363e14i) q^{65} +6.23567e15i q^{67} +7.05600e14i q^{68} +(-2.56236e15 - 9.46470e15i) q^{70} +5.20405e14 q^{71} -3.30547e15i q^{73} -7.06626e15 q^{74} +4.74343e14 q^{76} -1.59317e16i q^{77} +1.77897e16 q^{79} +(-4.67728e15 - 1.72767e16i) q^{80} +3.56900e16i q^{82} -4.82164e15i q^{83} +(1.74056e16 - 4.71218e15i) q^{85} +5.34389e16 q^{86} -2.27236e16i q^{88} +5.87120e15 q^{89} +8.32815e16 q^{91} -1.76279e16i q^{92} +3.02330e15 q^{94} +(-3.16778e15 - 1.17010e16i) q^{95} +8.56736e16i q^{97} -2.15440e17i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			406.510i			1.12284i	0.827532	+	0.561418i	$0.189744\pi$
$2$			406.510i			1.12284i	−0.827532	+	0.561418i	$0.810256\pi$
$3$		0			0
$3$		0			0
$4$	−34178.7			−0.260762
$5$	228254.	+	843113.i	0.261321	+	0.965252i
$5$	228254.	+	843113.i	0.261321	+	0.965252i
$6$		0			0
$7$			2.76153e7i			1.81057i	0.424801	+	0.905287i	$0.360344\pi$
$7$			2.76153e7i			1.81057i	−0.424801	+	0.905287i	$0.639656\pi$
$8$			3.93881e7i			0.830043i
$9$		0			0
$10$	−3.42734e8	+	9.27876e7i	−1.08382	+	0.293420i
$11$	−5.76916e8			−0.811474			−0.405737	−	0.913990i	$-0.632985\pi$
$11$	−5.76916e8			−0.811474			−0.405737	+	0.913990i	$0.632985\pi$
$12$		0			0
$13$		−	3.01577e9i		−	1.02537i	−0.858577	−	0.512685i	$-0.828651\pi$
$13$		−	3.01577e9i		−	1.02537i	0.858577	−	0.512685i	$-0.171349\pi$
$14$	−1.12259e10			−2.03298
$15$		0			0
$16$	−2.04916e10			−1.19277
$17$		−	2.06445e10i		−	0.717774i	−0.933381	−	0.358887i	$-0.883156\pi$
$17$		−	2.06445e10i		−	0.717774i	0.933381	−	0.358887i	$-0.116844\pi$
$18$		0			0
$19$	−1.38783e10			−0.187470			−0.0937349	−	0.995597i	$-0.529881\pi$
$19$	−1.38783e10			−0.187470			−0.0937349	+	0.995597i	$0.529881\pi$
$20$	−7.80142e9	−	2.88165e10i	−0.0681426	−	0.251702i
$21$		0			0
$22$		−	2.34522e11i		−	0.911153i
$23$			5.15758e11i			1.37328i	0.726998	+	0.686640i	$0.240915\pi$
$23$			5.15758e11i			1.37328i	−0.726998	+	0.686640i	$0.759085\pi$
$24$		0			0
$25$	−6.58740e11	+	3.84888e11i	−0.863423	+	0.504480i
$26$	1.22594e12			1.15132
$27$		0			0
$28$		−	9.43853e11i		−	0.472130i
$29$	−5.86433e11			−0.217689			−0.108844	−	0.994059i	$-0.534715\pi$
$29$	−5.86433e11			−0.217689			−0.108844	+	0.994059i	$0.534715\pi$
$30$		0			0
$31$	−2.42601e12			−0.510878			−0.255439	−	0.966825i	$-0.582220\pi$
$31$	−2.42601e12			−0.510878			−0.255439	+	0.966825i	$0.582220\pi$
$32$		−	3.16735e12i		−	0.509238i
$33$		0			0
$34$	8.39218e12			0.805943
$35$	−2.32828e13	+	6.30330e12i	−1.74766	+	0.473140i
$36$		0			0
$37$			1.73827e13i			0.813586i	0.913520	+	0.406793i	$0.133353\pi$
$37$			1.73827e13i			0.813586i	−0.913520	+	0.406793i	$0.866647\pi$
$38$		−	5.64168e12i		−	0.210498i
$39$		0			0
$40$	−3.32087e13	+	8.99051e12i	−0.801201	+	0.216907i
$41$	8.77960e13			1.71716			0.858582	−	0.512676i	$-0.171346\pi$
$41$	8.77960e13			1.71716			0.858582	+	0.512676i	$0.171346\pi$
$42$		0			0
$43$		−	1.31458e14i		−	1.71516i	−0.514352	−	0.857579i	$-0.671968\pi$
$43$		−	1.31458e14i		−	1.71516i	0.514352	−	0.857579i	$-0.328032\pi$
$44$	1.97182e13			0.211602
$45$		0			0
$46$	−2.09661e14			−1.54197
$47$		−	7.43721e12i		−	0.0455594i	−0.999741	−	0.0227797i	$-0.992748\pi$
$47$		−	7.43721e12i		−	0.0455594i	0.999741	−	0.0227797i	$-0.00725164\pi$
$48$		0			0
$49$	−5.29973e14			−2.27818
$50$	−1.56461e14	−	2.67784e14i	−0.566449	−	0.969483i
$51$		0			0
$52$			1.03075e14i			0.267378i
$53$			3.75793e14i			0.829093i	0.910028	+	0.414546i	$0.136060\pi$
$53$			3.75793e14i			0.829093i	−0.910028	+	0.414546i	$0.863940\pi$
$54$		0			0
$55$	−1.31683e14	−	4.86405e14i	−0.212055	−	0.783277i
$56$	−1.08771e15			−1.50285
$57$		0			0
$58$		−	2.38391e14i		−	0.244429i
$59$	1.43476e15			1.27215			0.636074	−	0.771628i	$-0.280557\pi$
$59$	1.43476e15			1.27215			0.636074	+	0.771628i	$0.280557\pi$
$60$		0			0
$61$	−9.34695e14			−0.624261			−0.312131	−	0.950039i	$-0.601043\pi$
$61$	−9.34695e14			−0.624261			−0.312131	+	0.950039i	$0.601043\pi$
$62$		−	9.86197e14i		−	0.573633i
$63$		0			0
$64$	−1.39831e15			−0.620974
$65$	2.54264e15	−	6.88363e14i	0.989740	−	0.267950i
$66$		0			0
$67$			6.23567e15i			1.87606i	0.346552	+	0.938031i	$0.387352\pi$
$67$			6.23567e15i			1.87606i	−0.346552	+	0.938031i	$0.612648\pi$
$68$			7.05600e14i			0.187168i
$69$		0			0
$70$	−2.56236e15	−	9.46470e15i	−0.531259	−	1.96234i
$71$	5.20405e14			0.0956414			0.0478207	−	0.998856i	$-0.484772\pi$
$71$	5.20405e14			0.0956414			0.0478207	+	0.998856i	$0.484772\pi$
$72$		0			0
$73$		−	3.30547e15i		−	0.479721i	−0.970807	−	0.239861i	$-0.922898\pi$
$73$		−	3.30547e15i		−	0.479721i	0.970807	−	0.239861i	$-0.0771018\pi$
$74$	−7.06626e15			−0.913524
$75$		0			0
$76$	4.74343e14			0.0488851
$77$		−	1.59317e16i		−	1.46923i
$78$		0			0
$79$	1.77897e16			1.31928			0.659641	−	0.751581i	$-0.270708\pi$
$79$	1.77897e16			1.31928			0.659641	+	0.751581i	$0.270708\pi$
$80$	−4.67728e15	−	1.72767e16i	−0.311694	−	1.15132i
$81$		0			0
$82$			3.56900e16i			1.92810i
$83$		−	4.82164e15i		−	0.234980i	−0.993074	−	0.117490i	$-0.962515\pi$
$83$		−	4.82164e15i		−	0.234980i	0.993074	−	0.117490i	$-0.0374848\pi$
$84$		0			0
$85$	1.74056e16	−	4.71218e15i	0.692833	−	0.187569i
$86$	5.34389e16			1.92584
$87$		0			0
$88$		−	2.27236e16i		−	0.673558i
$89$	5.87120e15			0.158093			0.0790464	−	0.996871i	$-0.474812\pi$
$89$	5.87120e15			0.158093			0.0790464	+	0.996871i	$0.474812\pi$
$90$		0			0
$91$	8.32815e16			1.85651
$92$		−	1.76279e16i		−	0.358100i
$93$		0			0
$94$	3.02330e15			0.0511558
$95$	−3.16778e15	−	1.17010e16i	−0.0489897	−	0.180956i
$96$		0			0
$97$			8.56736e16i			1.10991i	0.831881	+	0.554954i	$0.187264\pi$
$97$			8.56736e16i			1.10991i	−0.831881	+	0.554954i	$0.812736\pi$
$98$		−	2.15440e17i		−	2.55802i
$99$		0			0
$100$	2.25148e16	−	1.31550e16i	0.225148	−	0.131550i
$101$	−1.19416e17			−1.09732			−0.548659	−	0.836046i	$-0.684861\pi$
$101$	−1.19416e17			−1.09732			−0.548659	+	0.836046i	$0.684861\pi$
$102$		0			0
$103$		−	3.36777e16i		−	0.261954i	−0.991385	−	0.130977i	$-0.958189\pi$
$103$		−	3.36777e16i		−	0.261954i	0.991385	−	0.130977i	$-0.0418115\pi$
$104$	1.18786e17			0.851101
$105$		0			0
$106$	−1.52764e17			−0.930936
$107$		−	3.12377e17i		−	1.75759i	−0.477201	−	0.878794i	$-0.658349\pi$
$107$		−	3.12377e17i		−	1.75759i	0.477201	−	0.878794i	$-0.341651\pi$
$108$		0			0
$109$	−1.72558e17			−0.829486			−0.414743	−	0.909939i	$-0.636129\pi$
$109$	−1.72558e17			−0.829486			−0.414743	+	0.909939i	$0.636129\pi$
$110$	1.97729e17	−	5.35306e16i	0.879492	−	0.238103i
$111$		0			0
$112$		−	5.65880e17i		−	2.15959i
$113$		−	3.26931e17i		−	1.15688i	−0.815724	−	0.578441i	$-0.803661\pi$
$113$		−	3.26931e17i		−	1.15688i	0.815724	−	0.578441i	$-0.196339\pi$
$114$		0			0
$115$	−4.34842e17	+	1.17724e17i	−1.32556	+	0.358866i
$116$	2.00435e16			0.0567650
$117$		0			0
$118$			5.83245e17i			1.42841i
$119$	5.70102e17			1.29958
$120$		0			0
$121$	−1.72615e17			−0.341510
$122$		−	3.79963e17i		−	0.700944i
$123$		0			0
$124$	8.29177e16			0.133218
$125$	−4.74864e17	−	4.67540e17i	−0.712581	−	0.701590i
$126$		0			0
$127$		−	9.88589e17i		−	1.29624i	−0.761540	−	0.648118i	$-0.775556\pi$
$127$		−	9.88589e17i		−	1.29624i	0.761540	−	0.648118i	$-0.224444\pi$
$128$		−	9.83578e17i		−	1.20649i
$129$		0			0
$130$	2.79827e17	+	1.03361e18i	0.300864	+	1.11132i
$131$	1.16109e18			1.16966			0.584829	−	0.811156i	$-0.301162\pi$
$131$	1.16109e18			1.16966			0.584829	+	0.811156i	$0.301162\pi$
$132$		0			0
$133$		−	3.83254e17i		−	0.339428i
$134$	−2.53486e18			−2.10651
$135$		0			0
$136$	8.13147e17			0.595783
$137$			5.68166e17i			0.391156i	0.980688	+	0.195578i	$0.0626583\pi$
$137$			5.68166e17i			0.391156i	−0.980688	+	0.195578i	$0.937342\pi$
$138$		0			0
$139$	−1.75129e18			−1.06594			−0.532970	−	0.846134i	$-0.678924\pi$
$139$	−1.75129e18			−1.06594			−0.532970	+	0.846134i	$0.678924\pi$
$140$	7.95775e17	−	2.15438e17i	0.455724	−	0.123377i
$141$		0			0
$142$			2.11550e17i			0.107390i
$143$			1.73985e18i			0.832060i
$144$		0			0
$145$	−1.33856e17	−	4.94430e17i	−0.0568865	−	0.210124i
$146$	1.34371e18			0.538649
$147$		0			0
$148$		−	5.94118e17i		−	0.212153i
$149$	3.57309e18			1.20493			0.602463	−	0.798147i	$-0.294186\pi$
$149$	3.57309e18			1.20493			0.602463	+	0.798147i	$0.294186\pi$
$150$		0			0
$151$	3.80045e18			1.14428			0.572138	−	0.820158i	$-0.306114\pi$
$151$	3.80045e18			1.14428			0.572138	+	0.820158i	$0.306114\pi$
$152$		−	5.46641e17i		−	0.155608i
$153$		0			0
$154$	6.47639e18			1.64971
$155$	−5.53746e17	−	2.04540e18i	−0.133503	−	0.493127i
$156$		0			0
$157$			7.04103e18i			1.52226i	0.648599	+	0.761130i	$0.275355\pi$
$157$			7.04103e18i			1.52226i	−0.648599	+	0.761130i	$0.724645\pi$
$158$			7.23168e18i			1.48134i
$159$		0			0
$160$	2.67043e18	−	7.22960e17i	0.491543	−	0.133074i
$161$	−1.42428e19			−2.48642
$162$		0			0
$163$		−	7.70935e17i		−	0.121178i	−0.998163	−	0.0605889i	$-0.980702\pi$
$163$		−	7.70935e17i		−	0.121178i	0.998163	−	0.0605889i	$-0.0192979\pi$
$164$	−3.00075e18			−0.447772
$165$		0			0
$166$	1.96005e18			0.263844
$167$			1.34073e19i			1.71496i	0.514521	+	0.857478i	$0.327970\pi$
$167$			1.34073e19i			1.71496i	−0.514521	+	0.857478i	$0.672030\pi$
$168$		0			0
$169$	−4.44482e17			−0.0513827
$170$	1.91555e18	+	7.07556e18i	0.210609	+	0.777938i
$171$		0			0
$172$			4.49305e18i			0.447249i
$173$		−	7.18352e17i		−	0.0680684i	−0.999421	−	0.0340342i	$-0.989164\pi$
$173$		−	7.18352e17i		−	0.0680684i	0.999421	−	0.0340342i	$-0.0108355\pi$
$174$		0			0
$175$	−1.06288e19	−	1.81913e19i	−0.913399	−	1.56329i
$176$	1.18219e19			0.967898
$177$		0			0
$178$			2.38671e18i			0.177512i
$179$	6.38506e18			0.452808			0.226404	−	0.974033i	$-0.427303\pi$
$179$	6.38506e18			0.452808			0.226404	+	0.974033i	$0.427303\pi$
$180$		0			0
$181$	−2.44646e19			−1.57860			−0.789298	−	0.614010i	$-0.789555\pi$
$181$	−2.44646e19			−1.57860			−0.789298	+	0.614010i	$0.789555\pi$
$182$			3.38548e19i			2.08455i
$183$		0			0
$184$	−2.03147e19			−1.13988
$185$	−1.46556e19	+	3.96768e18i	−0.785315	+	0.212607i
$186$		0			0
$187$			1.19101e19i			0.582455i
$188$			2.54194e17i			0.0118802i
$189$		0			0
$190$	4.75658e18	−	1.28774e18i	0.203184	−	0.0550075i
$191$	5.68053e18			0.232063			0.116031	−	0.993246i	$-0.462983\pi$
$191$	5.68053e18			0.232063			0.116031	+	0.993246i	$0.462983\pi$
$192$		0			0
$193$		−	3.05325e19i		−	1.14163i	−0.821078	−	0.570815i	$-0.806627\pi$
$193$		−	3.05325e19i		−	1.14163i	0.821078	−	0.570815i	$-0.193373\pi$
$194$	−3.48272e19			−1.24625
$195$		0			0
$196$	1.81138e19			0.594063
$197$			2.61548e19i			0.821462i	0.911756	+	0.410731i	$0.134727\pi$
$197$			2.61548e19i			0.821462i	−0.911756	+	0.410731i	$0.865273\pi$
$198$		0			0
$199$	−3.21379e19			−0.926331			−0.463165	−	0.886272i	$-0.653286\pi$
$199$	−3.21379e19			−0.926331			−0.463165	+	0.886272i	$0.653286\pi$
$200$	−1.51600e19	−	2.59465e19i	−0.418740	−	0.716678i
$201$		0			0
$202$		−	4.85440e19i		−	1.23211i
$203$		−	1.61945e19i		−	0.394141i
$204$		0			0
$205$	2.00398e19	+	7.40219e19i	0.448730	+	1.65750i
$206$	1.36903e19			0.294132
$207$		0			0
$208$			6.17979e19i			1.22303i
$209$	8.00662e18			0.152127
$210$		0			0
$211$	−4.63855e19			−0.812797			−0.406398	−	0.913696i	$-0.633215\pi$
$211$	−4.63855e19			−0.812797			−0.406398	+	0.913696i	$0.633215\pi$
$212$		−	1.28441e19i		−	0.216196i
$213$		0			0
$214$	1.26985e20			1.97348
$215$	1.10834e20	−	3.00058e19i	1.65556	−	0.448206i
$216$		0			0
$217$		−	6.69948e19i		−	0.924983i
$218$		−	7.01465e19i		−	0.931378i
$219$		0			0
$220$	4.50076e18	+	1.66247e19i	0.0552959	+	0.204249i
$221$	−6.22590e19			−0.735983
$222$		0			0
$223$		−	3.37347e19i		−	0.369390i	−0.982796	−	0.184695i	$-0.940870\pi$
$223$		−	3.37347e19i		−	0.369390i	0.982796	−	0.184695i	$-0.0591297\pi$
$224$	8.74671e19			0.922012
$225$		0			0
$226$	1.32901e20			1.29899
$227$			1.21995e20i			1.14848i	0.818687	+	0.574240i	$0.194702\pi$
$227$			1.21995e20i			1.14848i	−0.818687	+	0.574240i	$0.805298\pi$
$228$		0			0
$229$	−1.07669e18			−0.00940781			−0.00470390	−	0.999989i	$-0.501497\pi$
$229$	−1.07669e18			−0.00940781			−0.00470390	+	0.999989i	$0.501497\pi$
$230$	−4.78559e19	−	1.76768e20i	−0.402948	−	1.48839i
$231$		0			0
$232$		−	2.30985e19i		−	0.180691i
$233$		−	1.45399e20i		−	1.09657i	−0.836291	−	0.548286i	$-0.815281\pi$
$233$		−	1.45399e20i		−	1.09657i	0.836291	−	0.548286i	$-0.184719\pi$
$234$		0			0
$235$	6.27041e18	−	1.69757e18i	0.0439763	−	0.0119056i
$236$	−4.90382e19			−0.331728
$237$		0			0
$238$			2.31752e20i			1.45922i
$239$	−5.65041e19			−0.343319			−0.171660	−	0.985156i	$-0.554913\pi$
$239$	−5.65041e19			−0.343319			−0.171660	+	0.985156i	$0.554913\pi$
$240$		0			0
$241$	−1.58771e20			−0.898723			−0.449362	−	0.893350i	$-0.648349\pi$
$241$	−1.58771e20			−0.898723			−0.449362	+	0.893350i	$0.648349\pi$
$242$		−	7.01700e19i		−	0.383460i
$243$		0			0
$244$	3.19466e19			0.162784
$245$	−1.20969e20	−	4.46827e20i	−0.595334	−	2.19901i
$246$		0			0
$247$			4.18539e19i			0.192226i
$248$		−	9.55559e19i		−	0.424051i
$249$		0			0
$250$	1.90060e20	−	1.93037e20i	0.787771	−	0.800112i
$251$	3.89569e20			1.56084			0.780419	−	0.625257i	$-0.215006\pi$
$251$	3.89569e20			1.56084			0.780419	+	0.625257i	$0.215006\pi$
$252$		0			0
$253$		−	2.97549e20i		−	1.11438i
$254$	4.01872e20			1.45546
$255$		0			0
$256$	2.16555e20			0.733718
$257$			4.60902e20i			1.51070i	0.655323	+	0.755349i	$0.272532\pi$
$257$			4.60902e20i			1.51070i	−0.655323	+	0.755349i	$0.727468\pi$
$258$		0			0
$259$	−4.80029e20			−1.47306
$260$	−8.69040e19	+	2.35273e19i	−0.258087	+	0.0698713i
$261$		0			0
$262$			4.71994e20i			1.31334i
$263$		−	3.43800e20i		−	0.926152i	−0.886319	−	0.463076i	$-0.846746\pi$
$263$		−	3.43800e20i		−	0.926152i	0.886319	−	0.463076i	$-0.153254\pi$
$264$		0			0
$265$	−3.16836e20	+	8.57762e19i	−0.800284	+	0.216659i
$266$	1.55797e20			0.381122
$267$		0			0
$268$		−	2.13127e20i		−	0.489206i
$269$	−8.38824e20			−1.86542			−0.932709	−	0.360630i	$-0.882562\pi$
$269$	−8.38824e20			−1.86542			−0.932709	+	0.360630i	$0.882562\pi$
$270$		0			0
$271$	−4.68360e20			−0.978004			−0.489002	−	0.872283i	$-0.662639\pi$
$271$	−4.68360e20			−0.978004			−0.489002	+	0.872283i	$0.662639\pi$
$272$			4.23037e20i			0.856136i
$273$		0			0
$274$	−2.30965e20			−0.439204
$275$	3.80037e20	−	2.22048e20i	0.700645	−	0.409373i
$276$		0			0
$277$			2.95652e20i			0.512511i	0.966609	+	0.256255i	$0.0824888\pi$
$277$			2.95652e20i			0.512511i	−0.966609	+	0.256255i	$0.917511\pi$
$278$		−	7.11919e20i		−	1.19688i
$279$		0			0
$280$	−2.48275e20	−	9.17066e20i	−0.392727	−	1.45063i
$281$	−8.72471e20			−1.33890			−0.669449	−	0.742858i	$-0.733470\pi$
$281$	−8.72471e20			−1.33890			−0.669449	+	0.742858i	$0.733470\pi$
$282$		0			0
$283$		−	8.61943e20i		−	1.24536i	−0.782478	−	0.622679i	$-0.786044\pi$
$283$		−	8.61943e20i		−	1.24536i	0.782478	−	0.622679i	$-0.213956\pi$
$284$	−1.77868e19			−0.0249397
$285$		0			0
$286$	−7.07266e20			−0.934268
$287$			2.42451e21i			3.10905i
$288$		0			0
$289$	4.01047e20			0.484801
$290$	2.00991e20	−	5.44138e19i	0.235935	−	0.0638743i
$291$		0			0
$292$			1.12977e20i			0.125093i
$293$			9.43270e20i			1.01452i	0.861793	+	0.507261i	$0.169342\pi$
$293$			9.43270e20i			1.01452i	−0.861793	+	0.507261i	$0.830658\pi$
$294$		0			0
$295$	3.27490e20	+	1.20967e21i	0.332438	+	1.22794i
$296$	−6.84673e20			−0.675311
$297$		0			0
$298$			1.45250e21i			1.35294i
$299$	1.55541e21			1.40812
$300$		0			0
$301$	3.63024e21			3.10542
$302$			1.54492e21i			1.28483i
$303$		0			0
$304$	2.84388e20			0.223608
$305$	−2.13348e20	−	7.88054e20i	−0.163132	−	0.602570i
$306$		0			0
$307$		−	2.17966e20i		−	0.157656i	−0.996888	−	0.0788282i	$-0.974882\pi$
$307$		−	2.17966e20i		−	0.157656i	0.996888	−	0.0788282i	$-0.0251179\pi$
$308$			5.44524e20i			0.383121i
$309$		0			0
$310$	8.31475e20	−	2.25103e20i	0.553701	−	0.149902i
$311$	9.25823e20			0.599881			0.299940	−	0.953958i	$-0.403033\pi$
$311$	9.25823e20			0.599881			0.299940	+	0.953958i	$0.403033\pi$
$312$		0			0
$313$			8.69108e20i			0.533269i	0.963798	+	0.266635i	$0.0859117\pi$
$313$			8.69108e20i			0.533269i	−0.963798	+	0.266635i	$0.914088\pi$
$314$	−2.86225e21			−1.70925
$315$		0			0
$316$	−6.08027e20			−0.344019
$317$			2.17713e21i			1.19917i	0.800311	+	0.599585i	$0.204668\pi$
$317$			2.17713e21i			1.19917i	−0.800311	+	0.599585i	$0.795332\pi$
$318$		0			0
$319$	3.38323e20			0.176649
$320$	−3.19170e20	−	1.17893e21i	−0.162273	−	0.599397i
$321$		0			0
$322$		−	5.78984e21i		−	2.79185i
$323$			2.86510e20i			0.134561i
$324$		0			0
$325$	1.16074e21	+	1.98661e21i	0.517279	+	0.885328i
$326$	3.13393e20			0.136063
$327$		0			0
$328$			3.45812e21i			1.42532i
$329$	2.05381e20			0.0824887
$330$		0			0
$331$	−6.33411e20			−0.241628			−0.120814	−	0.992675i	$-0.538550\pi$
$331$	−6.33411e20			−0.241628			−0.120814	+	0.992675i	$0.538550\pi$
$332$			1.64797e20i			0.0612739i
$333$		0			0
$334$	−5.45022e21			−1.92561
$335$	−5.25737e21	+	1.42332e21i	−1.81087	+	0.490253i
$336$		0			0
$337$			2.43586e21i			0.797624i	0.917033	+	0.398812i	$0.130577\pi$
$337$			2.43586e21i			0.797624i	−0.917033	+	0.398812i	$0.869423\pi$
$338$		−	1.80686e20i		−	0.0576944i
$339$		0			0
$340$	−5.94900e20	+	1.61056e20i	−0.180665	+	0.0489110i
$341$	1.39960e21			0.414564
$342$		0			0
$343$		−	8.21120e21i		−	2.31423i
$344$	5.17787e21			1.42366
$345$		0			0
$346$	2.92017e20			0.0764296
$347$		−	3.46482e20i		−	0.0884872i	−0.999021	−	0.0442436i	$-0.985912\pi$
$347$		−	3.46482e20i		−	0.0884872i	0.999021	−	0.0442436i	$-0.0140878\pi$
$348$		0			0
$349$	1.55028e21			0.377045			0.188523	−	0.982069i	$-0.439630\pi$
$349$	1.55028e21			0.377045			0.188523	+	0.982069i	$0.439630\pi$
$350$	7.39494e21	−	4.32071e21i	1.75532	−	1.02560i
$351$		0			0
$352$			1.82729e21i			0.413233i
$353$			2.37357e21i			0.523983i	0.965070	+	0.261991i	$0.0843792\pi$
$353$			2.37357e21i			0.523983i	−0.965070	+	0.261991i	$0.915621\pi$
$354$		0			0
$355$	1.18785e20	+	4.38760e20i	0.0249931	+	0.0923180i
$356$	−2.00670e20			−0.0412247
$357$		0			0
$358$			2.59559e21i			0.508430i
$359$	−3.60101e21			−0.688845			−0.344422	−	0.938815i	$-0.611925\pi$
$359$	−3.60101e21			−0.688845			−0.344422	+	0.938815i	$0.611925\pi$
$360$		0			0
$361$	−5.28778e21			−0.964855
$362$		−	9.94513e21i		−	1.77251i
$363$		0			0
$364$	−2.84645e21			−0.484107
$365$	2.78689e21	−	7.54487e20i	0.463052	−	0.125361i
$366$		0			0
$367$			3.76857e21i			0.597744i	0.954293	+	0.298872i	$0.0966103\pi$
$367$			3.76857e21i			0.597744i	−0.954293	+	0.298872i	$0.903390\pi$
$368$		−	1.05687e22i		−	1.63800i
$369$		0			0
$370$	−1.61290e21	−	5.95765e21i	−0.238723	−	0.881781i
$371$	−1.03776e22			−1.50113
$372$		0			0
$373$		−	1.76930e21i		−	0.244498i	−0.992499	−	0.122249i	$-0.960989\pi$
$373$		−	1.76930e21i		−	0.244498i	0.992499	−	0.122249i	$-0.0390107\pi$
$374$	−4.84158e21			−0.654001
$375$		0			0
$376$	2.92938e20			0.0378163
$377$			1.76855e21i			0.223211i
$378$		0			0
$379$	−9.86999e21			−1.19092			−0.595461	−	0.803384i	$-0.703031\pi$
$379$	−9.86999e21			−1.19092			−0.595461	+	0.803384i	$0.703031\pi$
$380$	1.08271e20	+	3.99924e20i	0.0127747	+	0.0471864i
$381$		0			0
$382$			2.30919e21i			0.260569i
$383$		−	7.21179e21i		−	0.795891i	−0.917409	−	0.397945i	$-0.869723\pi$
$383$		−	7.21179e21i		−	0.795891i	0.917409	−	0.397945i	$-0.130277\pi$
$384$		0			0
$385$	1.34322e22	−	3.63647e21i	1.41818	−	0.383941i
$386$	1.24118e22			1.28187
$387$		0			0
$388$		−	2.92821e21i		−	0.289423i
$389$	1.09989e22			1.06360			0.531801	−	0.846869i	$-0.321515\pi$
$389$	1.09989e22			1.06360			0.531801	+	0.846869i	$0.321515\pi$
$390$		0			0
$391$	1.06475e22			0.985704
$392$		−	2.08747e22i		−	1.89098i
$393$		0			0
$394$	−1.06322e22			−0.922368
$395$	4.06056e21	+	1.49987e22i	0.344755	+	1.27344i
$396$		0			0
$397$			1.36745e22i			1.11222i	0.831109	+	0.556110i	$0.187707\pi$
$397$			1.36745e22i			1.11222i	−0.831109	+	0.556110i	$0.812293\pi$
$398$		−	1.30644e22i		−	1.04012i
$399$		0			0
$400$	1.34986e22	−	7.88695e21i	1.02986	−	0.601727i
$401$	1.56760e22			1.17087			0.585433	−	0.810721i	$-0.300925\pi$
$401$	1.56760e22			1.17087			0.585433	+	0.810721i	$0.300925\pi$
$402$		0			0
$403$			7.31629e21i			0.523839i
$404$	4.08149e21			0.286139
$405$		0			0
$406$	6.58324e21			0.442556
$407$		−	1.00284e22i		−	0.660203i
$408$		0			0
$409$	1.18697e22			0.749533			0.374766	−	0.927119i	$-0.377723\pi$
$409$	1.18697e22			0.749533			0.374766	+	0.927119i	$0.377723\pi$
$410$	−3.00907e22	+	8.14638e21i	−1.86110	+	0.503851i
$411$		0			0
$412$			1.15106e21i			0.0683078i
$413$			3.96213e22i			2.30332i
$414$		0			0
$415$	4.06518e21	−	1.10056e21i	0.226815	−	0.0614051i
$416$	−9.55200e21			−0.522157
$417$		0			0
$418$			3.25477e21i			0.170814i
$419$	−2.22648e22			−1.14498			−0.572491	−	0.819911i	$-0.694023\pi$
$419$	−2.22648e22			−1.14498			−0.572491	+	0.819911i	$0.694023\pi$
$420$		0			0
$421$	2.10465e22			1.03940			0.519698	−	0.854350i	$-0.326044\pi$
$421$	2.10465e22			1.03940			0.519698	+	0.854350i	$0.326044\pi$
$422$		−	1.88562e22i		−	0.912638i
$423$		0			0
$424$	−1.48018e22			−0.688183
$425$	7.94580e21	+	1.35993e22i	0.362103	+	0.619742i
$426$		0			0
$427$		−	2.58119e22i		−	1.13027i
$428$			1.06766e22i			0.458313i
$429$		0			0
$430$	1.21977e22	+	4.50550e22i	0.503262	+	1.85892i
$431$	−2.49277e22			−1.00838			−0.504192	−	0.863592i	$-0.668209\pi$
$431$	−2.49277e22			−1.00838			−0.504192	+	0.863592i	$0.668209\pi$
$432$		0			0
$433$			9.04908e21i			0.351931i	0.984396	+	0.175966i	$0.0563047\pi$
$433$			9.04908e21i			0.351931i	−0.984396	+	0.175966i	$0.943695\pi$
$434$	2.72341e22			1.03860
$435$		0			0
$436$	5.89779e21			0.216299
$437$		−	7.15785e21i		−	0.257449i
$438$		0			0
$439$	−3.79196e22			−1.31194			−0.655971	−	0.754786i	$-0.727741\pi$
$439$	−3.79196e22			−1.31194			−0.655971	+	0.754786i	$0.727741\pi$
$440$	1.91586e22	−	5.18676e21i	0.650153	−	0.176015i
$441$		0			0
$442$		−	2.53089e22i		−	0.826389i
$443$			5.78605e22i			1.85332i	0.375901	+	0.926660i	$0.377333\pi$
$443$			5.78605e22i			1.85332i	−0.375901	+	0.926660i	$0.622667\pi$
$444$		0			0
$445$	1.34013e21	+	4.95009e21i	0.0413129	+	0.152599i
$446$	1.37135e22			0.414765
$447$		0			0
$448$		−	3.86147e22i		−	1.12432i
$449$	−2.41383e22			−0.689626			−0.344813	−	0.938671i	$-0.612058\pi$
$449$	−2.41383e22			−0.689626			−0.344813	+	0.938671i	$0.612058\pi$
$450$		0			0
$451$	−5.06509e22			−1.39343
$452$			1.11741e22i			0.301671i
$453$		0			0
$454$	−4.95924e22			−1.28956
$455$	1.90093e22	+	7.02157e22i	0.485143	+	1.79200i
$456$		0			0
$457$		−	6.01994e22i		−	1.48015i	−0.672527	−	0.740073i	$-0.734791\pi$
$457$		−	6.01994e22i		−	1.48015i	0.672527	−	0.740073i	$-0.265209\pi$
$458$		−	4.37685e20i		−	0.0105634i
$459$		0			0
$460$	1.48623e22	−	4.02364e21i	0.345657	−	0.0935788i
$461$	−4.47482e22			−1.02169			−0.510843	−	0.859674i	$-0.670667\pi$
$461$	−4.47482e22			−1.02169			−0.510843	+	0.859674i	$0.670667\pi$
$462$		0			0
$463$			5.17509e22i			1.13888i	0.822032	+	0.569442i	$0.192841\pi$
$463$			5.17509e22i			1.13888i	−0.822032	+	0.569442i	$0.807159\pi$
$464$	1.20169e22			0.259651
$465$		0			0
$466$	5.91063e22			1.23127
$467$		−	1.60093e22i		−	0.327476i	−0.986504	−	0.163738i	$-0.947645\pi$
$467$		−	1.60093e22i		−	0.327476i	0.986504	−	0.163738i	$-0.0523551\pi$
$468$		0			0
$469$	−1.72200e23			−3.39675
$470$	6.90081e20	+	2.54899e21i	0.0133681	+	0.0493782i
$471$		0			0
$472$			5.65126e22i			1.05594i
$473$			7.58400e22i			1.39181i
$474$		0			0
$475$	9.14220e21	−	5.34160e21i	0.161866	−	0.0945749i
$476$	−1.94853e22			−0.338882
$477$		0			0
$478$		−	2.29695e22i		−	0.385491i
$479$	−5.80933e22			−0.957799			−0.478900	−	0.877870i	$-0.658964\pi$
$479$	−5.80933e22			−0.957799			−0.478900	+	0.877870i	$0.658964\pi$
$480$		0			0
$481$	5.24224e22			0.834226
$482$		−	6.45420e22i		−	1.00912i
$483$		0			0
$484$	5.89976e21			0.0890531
$485$	−7.22325e22	+	1.95553e22i	−1.07134	+	0.290042i
$486$		0			0
$487$			4.89874e21i			0.0701597i	0.999385	+	0.0350799i	$0.0111686\pi$
$487$			4.89874e21i			0.0701597i	−0.999385	+	0.0350799i	$0.988831\pi$
$488$		−	3.68159e22i		−	0.518164i
$489$		0			0
$490$	1.81640e23	−	4.91750e22i	2.46913	−	0.668463i
$491$	1.07129e23			1.43124			0.715622	−	0.698487i	$-0.246143\pi$
$491$	1.07129e23			1.43124			0.715622	+	0.698487i	$0.246143\pi$
$492$		0			0
$493$			1.21066e22i			0.156251i
$494$	−1.70140e22			−0.215838
$495$		0			0
$496$	4.97126e22			0.609358
$497$			1.43711e22i			0.173166i
$498$		0			0
$499$	2.77173e22			0.322772			0.161386	−	0.986891i	$-0.448404\pi$
$499$	2.77173e22			0.322772			0.161386	+	0.986891i	$0.448404\pi$
$500$	1.62302e22	+	1.59799e22i	0.185814	+	0.182948i
$501$		0			0
$502$			1.58364e23i			1.75257i
$503$		−	1.29566e22i		−	0.140981i	−0.997512	−	0.0704907i	$-0.977543\pi$
$503$		−	1.29566e22i		−	0.140981i	0.997512	−	0.0704907i	$-0.0224565\pi$
$504$		0			0
$505$	−2.72573e22	−	1.00682e23i	−0.286752	−	1.05919i
$506$	1.20957e23			1.25127
$507$		0			0
$508$			3.37886e22i			0.338010i
$509$	1.02047e23			1.00392			0.501962	−	0.864890i	$-0.332612\pi$
$509$	1.02047e23			1.00392			0.501962	+	0.864890i	$0.332612\pi$
$510$		0			0
$511$	9.12815e22			0.868571
$512$		−	4.08876e22i		−	0.382645i
$513$		0			0
$514$	−1.87362e23			−1.69627
$515$	2.83941e22	−	7.68706e21i	0.252852	−	0.0684540i
$516$		0			0
$517$			4.29064e21i			0.0369703i
$518$		−	1.95137e23i		−	1.65400i
$519$		0			0
$520$	2.71133e22	+	1.00150e23i	0.222410	+	0.821527i
$521$	−1.08882e23			−0.878687			−0.439343	−	0.898319i	$-0.644789\pi$
$521$	−1.08882e23			−0.878687			−0.439343	+	0.898319i	$0.644789\pi$
$522$		0			0
$523$		−	8.32948e22i		−	0.650659i	−0.945601	−	0.325330i	$-0.894525\pi$
$523$		−	8.32948e22i		−	0.650659i	0.945601	−	0.325330i	$-0.105475\pi$
$524$	−3.96844e22			−0.305003
$525$		0			0
$526$	1.39758e23			1.03992
$527$			5.00836e22i			0.366695i
$528$		0			0
$529$	−1.24956e23			−0.885897
$530$	−3.48689e22	−	1.28797e23i	−0.243273	−	0.898588i
$531$		0			0
$532$			1.30991e22i			0.0885101i
$533$		−	2.64773e23i		−	1.76073i
$534$		0			0
$535$	2.63369e23	−	7.13014e22i	1.69652	−	0.459294i
$536$	−2.45611e23			−1.55721
$537$		0			0
$538$		−	3.40990e23i		−	2.09456i
$539$	3.05750e23			1.84868
$540$		0			0
$541$	−1.32060e23			−0.773738			−0.386869	−	0.922135i	$-0.626443\pi$
$541$	−1.32060e23			−0.773738			−0.386869	+	0.922135i	$0.626443\pi$
$542$		−	1.90393e23i		−	1.09814i
$543$		0			0
$544$	−6.53881e22			−0.365518
$545$	−3.93870e22	−	1.45486e23i	−0.216762	−	0.800663i
$546$		0			0
$547$			1.89705e23i			1.01201i	0.862529	+	0.506007i	$0.168879\pi$
$547$			1.89705e23i			1.01201i	−0.862529	+	0.506007i	$0.831121\pi$
$548$		−	1.94191e22i		−	0.101999i
$549$		0			0
$550$	9.02648e22	+	1.54489e23i	0.459659	+	0.786710i
$551$	8.13871e21			0.0408101
$552$		0			0
$553$			4.91267e23i			2.38866i
$554$	−1.20186e23			−0.575466
$555$		0			0
$556$	5.98569e22			0.277957
$557$			4.23248e23i			1.93564i	0.251634	+	0.967822i	$0.419032\pi$
$557$			4.23248e23i			1.93564i	−0.251634	+	0.967822i	$0.580968\pi$
$558$		0			0
$559$	−3.96447e23			−1.75867
$560$	4.77101e23	−	1.29164e23i	2.08455	−	0.564345i
$561$		0			0
$562$		−	3.54668e23i		−	1.50336i
$563$			2.54744e23i			1.06361i	0.846866	+	0.531806i	$0.178486\pi$
$563$			2.54744e23i			1.06361i	−0.846866	+	0.531806i	$0.821514\pi$
$564$		0			0
$565$	2.75640e23	−	7.46233e22i	1.11668	−	0.302317i
$566$	3.50389e23			1.39833
$567$		0			0
$568$			2.04978e22i			0.0793865i
$569$	−3.72274e23			−1.42039			−0.710196	−	0.704004i	$-0.751393\pi$
$569$	−3.72274e23			−1.42039			−0.710196	+	0.704004i	$0.751393\pi$
$570$		0			0
$571$	1.97336e23			0.730803			0.365401	−	0.930850i	$-0.380932\pi$
$571$	1.97336e23			0.730803			0.365401	+	0.930850i	$0.380932\pi$
$572$		−	5.94657e22i		−	0.216970i
$573$		0			0
$574$	−9.85589e23			−3.49096
$575$	−1.98509e23	−	3.39750e23i	−0.692793	−	1.18572i
$576$		0			0
$577$		−	4.79665e23i		−	1.62534i	−0.582725	−	0.812669i	$-0.698014\pi$
$577$		−	4.79665e23i		−	1.62534i	0.582725	−	0.812669i	$-0.301986\pi$
$578$			1.63030e23i			0.544352i
$579$		0			0
$580$	4.57501e21	+	1.68989e22i	0.0148339	+	0.0547926i
$581$	1.33151e23			0.425448
$582$		0			0
$583$		−	2.16801e23i		−	0.672787i
$584$	1.30196e23			0.398189
$585$		0			0
$586$	−3.83449e23			−1.13914
$587$		−	2.27706e22i		−	0.0666732i	−0.999444	−	0.0333366i	$-0.989387\pi$
$587$		−	2.27706e22i		−	0.0666732i	0.999444	−	0.0333366i	$-0.0106133\pi$
$588$		0			0
$589$	3.36689e22			0.0957743
$590$	−4.91742e23	+	1.33128e23i	−1.37878	+	0.373274i
$591$		0			0
$592$		−	3.56199e23i		−	0.970417i
$593$		−	3.58690e23i		−	0.963285i	−0.876368	−	0.481642i	$-0.840040\pi$
$593$		−	3.58690e23i		−	0.963285i	0.876368	−	0.481642i	$-0.159960\pi$
$594$		0			0
$595$	1.30128e23	+	4.80661e23i	0.339607	+	1.25442i
$596$	−1.22123e23			−0.314200
$597$		0			0
$598$			6.32290e23i			1.58109i
$599$	−2.42668e23			−0.598252			−0.299126	−	0.954214i	$-0.596695\pi$
$599$	−2.42668e23			−0.598252			−0.299126	+	0.954214i	$0.596695\pi$
$600$		0			0
$601$	1.07797e23			0.258328			0.129164	−	0.991623i	$-0.458771\pi$
$601$	1.07797e23			0.258328			0.129164	+	0.991623i	$0.458771\pi$
$602$			1.47573e24i			3.48688i
$603$		0			0
$604$	−1.29894e23			−0.298384
$605$	−3.94002e22	−	1.45534e23i	−0.0892437	−	0.329644i
$606$		0			0
$607$			1.37475e23i			0.302775i	0.988475	+	0.151387i	$0.0483741\pi$
$607$			1.37475e23i			0.302775i	−0.988475	+	0.151387i	$0.951626\pi$
$608$			4.39574e22i			0.0954667i
$609$		0			0
$610$	3.20352e23	−	8.67282e22i	0.676587	−	0.183171i
$611$	−2.24289e22			−0.0467152
$612$		0			0
$613$		−	2.80694e23i		−	0.568616i	−0.958733	−	0.284308i	$-0.908236\pi$
$613$		−	2.80694e23i		−	0.568616i	0.958733	−	0.284308i	$-0.0917639\pi$
$614$	8.86053e22			0.177022
$615$		0			0
$616$	6.27520e23			1.21953
$617$			4.17822e23i			0.800880i	0.916323	+	0.400440i	$0.131143\pi$
$617$			4.17822e23i			0.800880i	−0.916323	+	0.400440i	$0.868857\pi$
$618$		0			0
$619$	5.32143e23			0.992335			0.496167	−	0.868227i	$-0.334740\pi$
$619$	5.32143e23			0.992335			0.496167	+	0.868227i	$0.334740\pi$
$620$	1.89263e22	+	6.99090e22i	0.0348126	+	0.128589i
$621$		0			0
$622$			3.76357e23i			0.673568i
$623$			1.62135e23i			0.286239i
$624$		0			0
$625$	2.85799e23	−	5.07082e23i	0.490999	−	0.871160i
$626$	−3.53301e23			−0.598775
$627$		0			0
$628$		−	2.40653e23i		−	0.396948i
$629$	3.58857e23			0.583970
$630$		0			0
$631$	1.07303e24			1.69965			0.849827	−	0.527062i	$-0.176707\pi$
$631$	1.07303e24			1.69965			0.849827	+	0.527062i	$0.176707\pi$
$632$			7.00702e23i			1.09506i
$633$		0			0
$634$	−8.85027e23			−1.34647
$635$	8.33492e23	−	2.25649e23i	1.25119	−	0.338733i
$636$		0			0
$637$			1.59828e24i			2.33597i
$638$			1.37532e23i			0.198348i
$639$		0			0
$640$	8.29267e23	−	2.24506e23i	1.16457	−	0.315281i
$641$	1.05579e22			0.0146314			0.00731568	−	0.999973i	$-0.497671\pi$
$641$	1.05579e22			0.0146314			0.00731568	+	0.999973i	$0.497671\pi$
$642$		0			0
$643$		−	4.81305e23i		−	0.649572i	−0.945788	−	0.324786i	$-0.894708\pi$
$643$		−	4.81305e23i		−	0.649572i	0.945788	−	0.324786i	$-0.105292\pi$
$644$	4.86799e23			0.648366
$645$		0			0
$646$	−1.16469e23			−0.151090
$647$			3.51093e23i			0.449506i	0.974416	+	0.224753i	$0.0721575\pi$
$647$			3.51093e23i			0.449506i	−0.974416	+	0.224753i	$0.927843\pi$
$648$		0			0
$649$	−8.27736e23			−1.03231
$650$	−8.07578e23	+	4.71851e23i	−0.994079	+	0.580820i
$651$		0			0
$652$			2.63495e22i			0.0315986i
$653$			2.41714e23i			0.286114i	0.989714	+	0.143057i	$0.0456932\pi$
$653$			2.41714e23i			0.286114i	−0.989714	+	0.143057i	$0.954307\pi$
$654$		0			0
$655$	2.65023e23	+	9.78928e23i	0.305656	+	1.12902i
$656$	−1.79908e24			−2.04817
$657$		0			0
$658$			8.34893e22i			0.0926213i
$659$	−6.59216e23			−0.721941			−0.360970	−	0.932577i	$-0.617554\pi$
$659$	−6.59216e23			−0.721941			−0.360970	+	0.932577i	$0.617554\pi$
$660$		0			0
$661$	1.91029e23			0.203886			0.101943	−	0.994790i	$-0.467494\pi$
$661$	1.91029e23			0.203886			0.101943	+	0.994790i	$0.467494\pi$
$662$		−	2.57488e23i		−	0.271309i
$663$		0			0
$664$	1.89915e23			0.195043
$665$	3.23126e23	−	8.74792e22i	0.327633	−	0.0886995i
$666$		0			0
$667$		−	3.02457e23i		−	0.298947i
$668$		−	4.58245e23i		−	0.447196i
$669$		0			0
$670$	−5.78593e23	−	2.13718e24i	−0.550475	−	2.03331i
$671$	5.39240e23			0.506572
$672$		0			0
$673$		−	1.46782e24i		−	1.34445i	−0.740346	−	0.672226i	$-0.765338\pi$
$673$		−	1.46782e24i		−	1.34445i	0.740346	−	0.672226i	$-0.234662\pi$
$674$	−9.90202e23			−0.895601
$675$		0			0
$676$	1.51918e22			0.0133987
$677$		−	1.35936e24i		−	1.18394i	−0.805959	−	0.591971i	$-0.798350\pi$
$677$		−	1.35936e24i		−	1.18394i	0.805959	−	0.591971i	$-0.201650\pi$
$678$		0			0
$679$	−2.36590e24			−2.00957
$680$	1.85604e23	+	6.85575e23i	0.155690	+	0.575081i
$681$		0			0
$682$			5.68952e23i			0.465488i
$683$		−	1.20522e24i		−	0.973849i	−0.873444	−	0.486925i	$-0.838119\pi$
$683$		−	1.20522e24i		−	0.973849i	0.873444	−	0.486925i	$-0.161881\pi$
$684$		0			0
$685$	−4.79028e23	+	1.29686e23i	−0.377564	+	0.102217i
$686$	3.33794e24			2.59850
$687$		0			0
$688$			2.69377e24i			2.04578i
$689$	1.13331e24			0.850127
$690$		0			0
$691$	−1.07458e24			−0.786459			−0.393230	−	0.919440i	$-0.628642\pi$
$691$	−1.07458e24			−0.786459			−0.393230	+	0.919440i	$0.628642\pi$
$692$			2.45523e22i			0.0177497i
$693$		0			0
$694$	1.40849e23			0.0993567
$695$	−3.99740e23	−	1.47654e24i	−0.278552	−	1.02890i
$696$		0			0
$697$		−	1.81250e24i		−	1.23254i
$698$			6.30204e23i			0.423361i
$699$		0			0
$700$	3.63278e23	+	6.21753e23i	0.238180	+	0.407648i
$701$	5.52035e23			0.357572			0.178786	−	0.983888i	$-0.442783\pi$
$701$	5.52035e23			0.357572			0.178786	+	0.983888i	$0.442783\pi$
$702$		0			0
$703$		−	2.41243e23i		−	0.152523i
$704$	8.06707e23			0.503904
$705$		0			0
$706$	−9.64881e23			−0.588347
$707$		−	3.29772e24i		−	1.98677i
$708$		0			0
$709$	1.00912e24			0.593541			0.296770	−	0.954949i	$-0.404090\pi$
$709$	1.00912e24			0.593541			0.296770	+	0.954949i	$0.404090\pi$
$710$	−1.78361e23	+	4.82872e22i	−0.103658	+	0.0280631i
$711$		0			0
$712$			2.31256e23i			0.131224i
$713$		−	1.25123e24i		−	0.701579i
$714$		0			0
$715$	−1.46689e24	+	3.97127e23i	−0.803148	+	0.217434i
$716$	−2.18233e23			−0.118075
$717$		0			0
$718$		−	1.46385e24i		−	0.773460i
$719$	−2.57522e24			−1.34468			−0.672341	−	0.740242i	$-0.734711\pi$
$719$	−2.57522e24			−1.34468			−0.672341	+	0.740242i	$0.734711\pi$
$720$		0			0
$721$	9.30018e23			0.474287
$722$		−	2.14954e24i		−	1.08337i
$723$		0			0
$724$	8.36169e23			0.411639
$725$	3.86307e23	−	2.25711e23i	0.187957	−	0.109820i
$726$		0			0
$727$		−	1.13745e24i		−	0.540618i	−0.962774	−	0.270309i	$-0.912874\pi$
$727$		−	1.13745e24i		−	0.540618i	0.962774	−	0.270309i	$-0.0871258\pi$
$728$			3.28030e24i			1.54098i
$729$		0			0
$730$	3.06707e23	+	1.13290e24i	0.140760	+	0.519932i
$731$	−2.71387e24			−1.23110
$732$		0			0
$733$			2.98752e24i			1.32412i	0.749452	+	0.662059i	$0.230317\pi$
$733$			2.98752e24i			1.32412i	−0.749452	+	0.662059i	$0.769683\pi$
$734$	−1.53196e24			−0.671168
$735$		0			0
$736$	1.63358e24			0.699326
$737$		−	3.59745e24i		−	1.52237i
$738$		0			0
$739$	2.33260e24			0.964635			0.482318	−	0.875996i	$-0.339795\pi$
$739$	2.33260e24			0.964635			0.482318	+	0.875996i	$0.339795\pi$
$740$	5.00909e23	−	1.35610e23i	0.204781	−	0.0554398i
$741$		0			0
$742$		−	4.21861e24i		−	1.68553i
$743$		−	3.28616e22i		−	0.0129803i	−0.999979	−	0.00649013i	$-0.997934\pi$
$743$		−	3.28616e22i		−	0.0129803i	0.999979	−	0.00649013i	$-0.00206589\pi$
$744$		0			0
$745$	8.15572e23	+	3.01252e24i	0.314872	+	1.16306i
$746$	7.19238e23			0.274532
$747$		0			0
$748$		−	4.07071e23i		−	0.151882i
$749$	8.62638e24			3.18224
$750$		0			0
$751$	−3.79989e24			−1.37035			−0.685174	−	0.728379i	$-0.740274\pi$
$751$	−3.79989e24			−1.37035			−0.685174	+	0.728379i	$0.740274\pi$
$752$			1.52400e23i			0.0543417i
$753$		0			0
$754$	−7.18934e23			−0.250630
$755$	8.67467e23	+	3.20421e24i	0.299023	+	1.10451i
$756$		0			0
$757$			2.22264e24i			0.749123i	0.927202	+	0.374562i	$0.122207\pi$
$757$			2.22264e24i			0.749123i	−0.927202	+	0.374562i	$0.877793\pi$
$758$		−	4.01225e24i		−	1.33721i
$759$		0			0
$760$	4.60881e23	−	1.24773e23i	0.150201	−	0.0406636i
$761$	−2.61895e24			−0.844028			−0.422014	−	0.906589i	$-0.638677\pi$
$761$	−2.61895e24			−0.844028			−0.422014	+	0.906589i	$0.638677\pi$
$762$		0			0
$763$		−	4.76523e24i		−	1.50185i
$764$	−1.94153e23			−0.0605132
$765$		0			0
$766$	2.93167e24			0.893656
$767$		−	4.32692e24i		−	1.30442i
$768$		0			0
$769$	9.71387e23			0.286430			0.143215	−	0.989692i	$-0.454256\pi$
$769$	9.71387e23			0.286430			0.143215	+	0.989692i	$0.454256\pi$
$770$	1.47826e24	+	5.46033e24i	0.431103	+	1.59238i
$771$		0			0
$772$			1.04356e24i			0.297695i
$773$			5.61109e24i			1.58315i	0.611072	+	0.791575i	$0.290738\pi$
$773$			5.61109e24i			1.58315i	−0.611072	+	0.791575i	$0.709262\pi$
$774$		0			0
$775$	1.59811e24	−	9.33741e23i	0.441104	−	0.257728i
$776$	−3.37452e24			−0.921272
$777$		0			0
$778$			4.47118e24i			1.19425i
$779$	−1.21846e24			−0.321917
$780$		0			0
$781$	−3.00230e23			−0.0776105
$782$			4.32833e24i			1.10678i
$783$		0			0
$784$	1.08600e25			2.71733
$785$	−5.93638e24	+	1.60714e24i	−1.46937	+	0.397798i
$786$		0			0
$787$			7.78702e23i			0.188619i	0.995543	+	0.0943097i	$0.0300644\pi$
$787$			7.78702e23i			0.188619i	−0.995543	+	0.0943097i	$0.969936\pi$
$788$		−	8.93934e23i		−	0.214207i
$789$		0			0
$790$	−6.09713e24	+	1.65066e24i	−1.42986	+	0.387104i
$791$	9.02828e24			2.09462
$792$		0			0
$793$			2.81883e24i			0.640099i
$794$	−5.55881e24			−1.24884
$795$		0			0
$796$	1.09843e24			0.241552
$797$		−	2.98087e24i		−	0.648555i	−0.945962	−	0.324278i	$-0.894879\pi$
$797$		−	2.98087e24i		−	0.648555i	0.945962	−	0.324278i	$-0.105121\pi$
$798$		0			0
$799$	−1.53537e23			−0.0327014
$800$	1.21907e24	+	2.08646e24i	0.256900	+	0.439688i
$801$		0			0
$802$			6.37244e24i			1.31469i
$803$			1.90698e24i			0.389281i
$804$		0			0
$805$	−3.25097e24	−	1.20083e25i	−0.649753	−	2.40003i
$806$	−2.97415e24			−0.588186
$807$		0			0
$808$		−	4.70359e24i		−	0.910821i
$809$	−8.38770e24			−1.60724			−0.803619	−	0.595144i	$-0.797095\pi$
$809$	−8.38770e24			−1.60724			−0.803619	+	0.595144i	$0.797095\pi$
$810$		0			0
$811$	−5.93252e24			−1.11317			−0.556585	−	0.830790i	$-0.687889\pi$
$811$	−5.93252e24			−1.11317			−0.556585	+	0.830790i	$0.687889\pi$
$812$			5.53507e23i			0.102777i
$813$		0			0
$814$	4.07663e24			0.741301
$815$	6.49985e23	−	1.75969e23i	0.116967	−	0.0316662i
$816$		0			0
$817$			1.82441e24i			0.321540i
$818$			4.82514e24i			0.841603i
$819$		0			0
$820$	−6.84933e23	−	2.52997e24i	−0.117012	−	0.432213i
$821$	5.48179e24			0.926842			0.463421	−	0.886138i	$-0.346622\pi$
$821$	5.48179e24			0.926842			0.463421	+	0.886138i	$0.346622\pi$
$822$		0			0
$823$			1.56562e22i			0.00259291i	0.999999	+	0.00129646i	$0.000412675\pi$
$823$			1.56562e22i			0.00259291i	−0.999999	+	0.00129646i	$0.999587\pi$
$824$	1.32650e24			0.217433
$825$		0			0
$826$	−1.61065e25			−2.58625
$827$			7.60751e24i			1.20905i	0.796584	+	0.604527i	$0.206638\pi$
$827$			7.60751e24i			1.20905i	−0.796584	+	0.604527i	$0.793362\pi$
$828$		0			0
$829$	1.32131e24			0.205727			0.102863	−	0.994695i	$-0.467200\pi$
$829$	1.32131e24			0.205727			0.102863	+	0.994695i	$0.467200\pi$
$830$	4.47388e23	+	1.65254e24i	0.0689479	+	0.254676i
$831$		0			0
$832$			4.21699e24i			0.636728i
$833$			1.09410e25i			1.63521i
$834$		0			0
$835$	−1.13039e25	+	3.06028e24i	−1.65536	+	0.448153i
$836$	−2.73656e23			−0.0396690
$837$		0			0
$838$		−	9.05085e24i		−	1.28563i
$839$	5.31605e24			0.747502			0.373751	−	0.927529i	$-0.378071\pi$
$839$	5.31605e24			0.747502			0.373751	+	0.927529i	$0.378071\pi$
$840$		0			0
$841$	−6.91324e24			−0.952612
$842$			8.55560e24i			1.16707i
$843$		0			0
$844$	1.58540e24			0.211947
$845$	−1.01455e23	−	3.74748e23i	−0.0134274	−	0.0495972i
$846$		0			0
$847$		−	4.76682e24i		−	0.618330i
$848$		−	7.70057e24i		−	0.988913i
$849$		0			0
$850$	−5.52826e24	+	3.23005e24i	−0.695870	+	0.406582i
$851$	−8.96527e24			−1.11728
$852$		0			0
$853$			3.42826e24i			0.418801i	0.977830	+	0.209400i	$0.0671512\pi$
$853$			3.42826e24i			0.418801i	−0.977830	+	0.209400i	$0.932849\pi$
$854$	1.04928e25			1.26911
$855$		0			0
$856$	1.23040e25			1.45887
$857$		−	1.10696e25i		−	1.29956i	−0.760122	−	0.649780i	$-0.774861\pi$
$857$		−	1.10696e25i		−	1.29956i	0.760122	−	0.649780i	$-0.225139\pi$
$858$		0			0
$859$	7.30612e24			0.840901			0.420450	−	0.907316i	$-0.361872\pi$
$859$	7.30612e24			0.840901			0.420450	+	0.907316i	$0.361872\pi$
$860$	−3.78815e24	+	1.02556e24i	−0.431708	+	0.116875i
$861$		0			0
$862$		−	1.01334e25i		−	1.13225i
$863$			4.59279e24i			0.508141i	0.967186	+	0.254071i	$0.0817696\pi$
$863$			4.59279e24i			0.508141i	−0.967186	+	0.254071i	$0.918230\pi$
$864$		0			0
$865$	6.05652e23	−	1.63967e23i	0.0657031	−	0.0177877i
$866$	−3.67855e24			−0.395161
$867$		0			0
$868$			2.28979e24i			0.241201i
$869$	−1.02631e25			−1.07056
$870$		0			0
$871$	1.88054e25			1.92366
$872$		−	6.79673e24i		−	0.688509i
$873$		0			0
$874$	2.90974e24			0.289073
$875$	1.29112e25	−	1.31135e25i	1.27028	−	1.29018i
$876$		0			0
$877$			5.95372e24i			0.574503i	0.957855	+	0.287251i	$0.0927415\pi$
$877$			5.95372e24i			0.574503i	−0.957855	+	0.287251i	$0.907258\pi$
$878$		−	1.54147e25i		−	1.47310i
$879$		0			0
$880$	2.69840e24	+	9.96720e24i	0.252932	+	0.934265i
$881$	−2.12837e25			−1.97585			−0.987923	−	0.154949i	$-0.950479\pi$
$881$	−2.12837e25			−1.97585			−0.987923	+	0.154949i	$0.950479\pi$
$882$		0			0
$883$			1.17761e25i			1.07235i	0.844106	+	0.536176i	$0.180132\pi$
$883$			1.17761e25i			1.07235i	−0.844106	+	0.536176i	$0.819868\pi$
$884$	2.12793e24			0.191917
$885$		0			0
$886$	−2.35209e25			−2.08098
$887$			1.58250e25i			1.38673i	0.720586	+	0.693366i	$0.243873\pi$
$887$			1.58250e25i			1.38673i	−0.720586	+	0.693366i	$0.756127\pi$
$888$		0			0
$889$	2.73002e25			2.34693
$890$	−2.01226e24	+	5.44775e23i	−0.171344	+	0.0463876i
$891$		0			0
$892$			1.15301e24i			0.0963231i
$893$			1.03216e23i			0.00854102i
$894$		0			0
$895$	1.45742e24	+	5.38333e24i	0.118328	+	0.437074i
$896$	2.71618e25			2.18444
$897$		0			0
$898$		−	9.81249e24i		−	0.774337i
$899$	1.42269e24			0.111212
$900$		0			0
$901$	7.75803e24			0.595101
$902$		−	2.05901e25i		−	1.56460i
$903$		0			0
$904$	1.28772e25			0.960262
$905$	−5.58416e24	−	2.06265e25i	−0.412520	−	1.52374i
$906$		0			0
$907$		−	1.26063e25i		−	0.913954i	−0.889479	−	0.456977i	$-0.848932\pi$
$907$		−	1.26063e25i		−	0.913954i	0.889479	−	0.456977i	$-0.151068\pi$
$908$		−	4.16964e24i		−	0.299480i
$909$		0			0
$910$	−2.85434e25	+	7.72749e24i	−2.01212	+	0.544737i
$911$	2.37643e25			1.65966			0.829828	−	0.558019i	$-0.188439\pi$
$911$	2.37643e25			1.65966			0.829828	+	0.558019i	$0.188439\pi$
$912$		0			0
$913$			2.78168e24i			0.190680i
$914$	2.44717e25			1.66196
$915$		0			0
$916$	3.67998e22			0.00245320
$917$			3.20638e25i			2.11775i
$918$		0			0
$919$	1.89052e25			1.22574			0.612870	−	0.790183i	$-0.290015\pi$
$919$	1.89052e25			1.22574			0.612870	+	0.790183i	$0.290015\pi$
$920$	−4.63692e24	−	1.71276e25i	−0.297874	−	1.10027i
$921$		0			0
$922$		−	1.81906e25i		−	1.14719i
$923$		−	1.56943e24i		−	0.0980677i
$924$		0			0
$925$	−6.69040e24	−	1.14507e25i	−0.410438	−	0.702469i
$926$	−2.10373e25			−1.27878
$927$		0			0
$928$			1.85744e24i			0.110855i
$929$	−6.68973e24			−0.395617			−0.197808	−	0.980241i	$-0.563382\pi$
$929$	−6.68973e24			−0.395617			−0.197808	+	0.980241i	$0.563382\pi$
$930$		0			0
$931$	7.35514e24			0.427089
$932$			4.96956e24i			0.285945i
$933$		0			0
$934$	6.50795e24			0.367702
$935$	−1.00416e25	+	2.71853e24i	−0.562215	+	0.152207i
$936$		0			0
$937$		−	7.19515e24i		−	0.395597i	−0.980243	−	0.197799i	$-0.936621\pi$
$937$		−	7.19515e24i		−	0.395597i	0.980243	−	0.197799i	$-0.0633792\pi$
$938$		−	7.00010e25i		−	3.81399i
$939$		0			0
$940$	−2.14314e23	+	5.80208e22i	−0.0114674	+	0.00310454i
$941$	2.21212e25			1.17299			0.586497	−	0.809951i	$-0.300506\pi$
$941$	2.21212e25			1.17299			0.586497	+	0.809951i	$0.300506\pi$
$942$		0			0
$943$			4.52814e25i			2.35815i
$944$	−2.94005e25			−1.51737
$945$		0			0
$946$	−3.08297e25			−1.56277
$947$		−	1.99520e23i		−	0.0100233i	−0.999987	−	0.00501166i	$-0.998405\pi$
$947$		−	1.99520e23i		−	0.0100233i	0.999987	−	0.00501166i	$-0.00159527\pi$
$948$		0			0
$949$	−9.96856e24			−0.491892
$950$	2.17142e24	+	3.71640e24i	0.106192	+	0.181749i
$951$		0			0
$952$			2.24553e25i			1.07871i
$953$			1.42481e25i			0.678370i	0.940720	+	0.339185i	$0.110151\pi$
$953$			1.42481e25i			0.678370i	−0.940720	+	0.339185i	$0.889849\pi$
$954$		0			0
$955$	1.29660e24	+	4.78933e24i	0.0606427	+	0.223999i
$956$	1.93123e24			0.0895247
$957$		0			0
$958$		−	2.36155e25i		−	1.07545i
$959$	−1.56900e25			−0.708217
$960$		0			0
$961$	−1.66646e25			−0.739003
$962$			2.13102e25i			0.936699i
$963$		0			0
$964$	5.42658e24			0.234353
$965$	2.57424e25	−	6.96918e24i	1.10196	−	0.298332i
$966$		0			0
$967$		−	1.07659e25i		−	0.452822i	−0.974032	−	0.226411i	$-0.927301\pi$
$967$		−	1.07659e25i		−	0.452822i	0.974032	−	0.226411i	$-0.0726992\pi$
$968$		−	6.79900e24i		−	0.283468i
$969$		0			0
$970$	−7.94945e24	−	2.93633e25i	−0.325670	−	1.20294i
$971$	−1.66161e25			−0.674786			−0.337393	−	0.941364i	$-0.609545\pi$
$971$	−1.66161e25			−0.674786			−0.337393	+	0.941364i	$0.609545\pi$
$972$		0			0
$973$		−	4.83625e25i		−	1.92996i
$974$	−1.99139e24			−0.0787779
$975$		0			0
$976$	1.91534e25			0.744597
$977$			2.38066e25i			0.917475i	0.888572	+	0.458737i	$0.151698\pi$
$977$			2.38066e25i			0.917475i	−0.888572	+	0.458737i	$0.848302\pi$
$978$		0			0
$979$	−3.38719e24			−0.128288
$980$	4.13454e24	+	1.52720e25i	0.155241	+	0.573420i
$981$		0			0
$982$			4.35490e25i			1.60705i
$983$			1.66009e25i			0.607333i	0.952778	+	0.303667i	$0.0982109\pi$
$983$			1.66009e25i			0.607333i	−0.952778	+	0.303667i	$0.901789\pi$
$984$		0			0
$985$	−2.20514e25	+	5.96993e24i	−0.792918	+	0.214665i
$986$	−4.92146e24			−0.175445
$987$		0			0
$988$		−	1.43051e24i		−	0.0501253i
$989$	6.78003e25			2.35539
$990$		0			0
$991$	−3.53873e25			−1.20843			−0.604215	−	0.796821i	$-0.706513\pi$
$991$	−3.53873e25			−1.20843			−0.604215	+	0.796821i	$0.706513\pi$
$992$			7.68400e24i			0.260159i
$993$		0			0
$994$	−5.84202e24			−0.194437
$995$	−7.33560e24	−	2.70959e25i	−0.242069	−	0.894143i
$996$		0			0
$997$			5.16783e25i			1.67648i	0.545300	+	0.838241i	$0.316416\pi$
$997$			5.16783e25i			1.67648i	−0.545300	+	0.838241i	$0.683584\pi$
$998$			1.12674e25i			0.362420i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.18.b.c.19.12	yes	16
3.2	odd	2	inner	45.18.b.c.19.5	✓	16
5.4	even	2	inner	45.18.b.c.19.6	yes	16
15.14	odd	2	inner	45.18.b.c.19.11	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.18.b.c.19.5	✓	16	3.2	odd	2	inner
45.18.b.c.19.6	yes	16	5.4	even	2	inner
45.18.b.c.19.11	yes	16	15.14	odd	2	inner
45.18.b.c.19.12	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+406.510i q^{2} -34178.7 q^{4} +(228254. + 843113. i) q^{5} +2.76153e7i q^{7} +3.93881e7i q^{8} +O(q^{10})\)	\(q+406.510i q^{2} -34178.7 q^{4} +(228254. + 843113. i) q^{5} +2.76153e7i q^{7} +3.93881e7i q^{8} +(-3.42734e8 + 9.27876e7i) q^{10} -5.76916e8 q^{11} -3.01577e9i q^{13} -1.12259e10 q^{14} -2.04916e10 q^{16} -2.06445e10i q^{17} -1.38783e10 q^{19} +(-7.80142e9 - 2.88165e10i) q^{20} -2.34522e11i q^{22} +5.15758e11i q^{23} +(-6.58740e11 + 3.84888e11i) q^{25} +1.22594e12 q^{26} -9.43853e11i q^{28} -5.86433e11 q^{29} -2.42601e12 q^{31} -3.16735e12i q^{32} +8.39218e12 q^{34} +(-2.32828e13 + 6.30330e12i) q^{35} +1.73827e13i q^{37} -5.64168e12i q^{38} +(-3.32087e13 + 8.99051e12i) q^{40} +8.77960e13 q^{41} -1.31458e14i q^{43} +1.97182e13 q^{44} -2.09661e14 q^{46} -7.43721e12i q^{47} -5.29973e14 q^{49} +(-1.56461e14 - 2.67784e14i) q^{50} +1.03075e14i q^{52} +3.75793e14i q^{53} +(-1.31683e14 - 4.86405e14i) q^{55} -1.08771e15 q^{56} -2.38391e14i q^{58} +1.43476e15 q^{59} -9.34695e14 q^{61} -9.86197e14i q^{62} -1.39831e15 q^{64} +(2.54264e15 - 6.88363e14i) q^{65} +6.23567e15i q^{67} +7.05600e14i q^{68} +(-2.56236e15 - 9.46470e15i) q^{70} +5.20405e14 q^{71} -3.30547e15i q^{73} -7.06626e15 q^{74} +4.74343e14 q^{76} -1.59317e16i q^{77} +1.77897e16 q^{79} +(-4.67728e15 - 1.72767e16i) q^{80} +3.56900e16i q^{82} -4.82164e15i q^{83} +(1.74056e16 - 4.71218e15i) q^{85} +5.34389e16 q^{86} -2.27236e16i q^{88} +5.87120e15 q^{89} +8.32815e16 q^{91} -1.76279e16i q^{92} +3.02330e15 q^{94} +(-3.16778e15 - 1.17010e16i) q^{95} +8.56736e16i q^{97} -2.15440e17i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 1473168 q^{4}+O(q^{10}) \) 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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