LMFDB - Embedded newform 12.17.c.b.5.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [12,17,Mod(5,12)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(12, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 17, names="a")

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

chi := DirichletCharacter("12.5");

S:= CuspForms(chi, 17);

N := Newforms(S);

Level:	$ N $	$=$	$ 12 = 2^{2} \cdot 3 $
Weight:	$ k $	$=$	$ 17 $
Character orbit:	$[\chi]$	$=$	12.c (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:	$19.4789452628$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 15630x^{2} + 12922000 $ x^4 + 15630*x^2 + 12922000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{11}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5.2
Root			$-29.5942i$ of defining polynomial
Character	$\chi$	$=$	12.5
Dual form			12.17.c.b.5.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6363.40 + 1598.09i) q^{3} +746888. i q^{5} +4.25357e6 q^{7} +(3.79389e7 - 2.03386e7i) q^{9} +O(q^{10})$	\(q+(-6363.40 + 1598.09i) q^{3} +746888. i q^{5} +4.25357e6 q^{7} +(3.79389e7 - 2.03386e7i) q^{9} +2.12654e8i q^{11} -7.06137e8 q^{13} +(-1.19359e9 - 4.75274e9i) q^{15} +8.78643e9i q^{17} -8.43718e9 q^{19} +(-2.70672e10 + 6.79758e9i) q^{21} -1.19825e11i q^{23} -4.05253e11 q^{25} +(-2.08918e11 + 1.90052e11i) q^{27} -3.07746e11i q^{29} +2.63113e10 q^{31} +(-3.39841e11 - 1.35320e12i) q^{33} +3.17694e12i q^{35} -3.10657e11 q^{37} +(4.49343e12 - 1.12847e12i) q^{39} -5.28023e12i q^{41} +1.92635e13 q^{43} +(1.51906e13 + 2.83361e13i) q^{45} +2.19814e12i q^{47} -1.51401e13 q^{49} +(-1.40415e13 - 5.59116e13i) q^{51} -1.73729e13i q^{53} -1.58829e14 q^{55} +(5.36891e13 - 1.34834e13i) q^{57} +1.68820e14i q^{59} -2.49059e14 q^{61} +(1.61376e14 - 8.65115e13i) q^{63} -5.27405e14i q^{65} -1.53606e14 q^{67} +(1.91491e14 + 7.62494e14i) q^{69} +1.82810e14i q^{71} -4.12396e14 q^{73} +(2.57879e15 - 6.47631e14i) q^{75} +9.04541e14i q^{77} +1.16547e15 q^{79} +(1.02571e15 - 1.54325e15i) q^{81} +2.11590e15i q^{83} -6.56248e15 q^{85} +(4.91805e14 + 1.95831e15i) q^{87} -5.45154e15i q^{89} -3.00360e15 q^{91} +(-1.67430e14 + 4.20479e13i) q^{93} -6.30163e15i q^{95} -9.42108e15 q^{97} +(4.32509e15 + 8.06789e15i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12420 q^{3} - 1050280 q^{7} - 10121436 q^{9}+O(q^{10}) $ 4 * q - 12420 * q^3 - 1050280 * q^7 - 10121436 * q^9	$ 4 q - 12420 q^{3} - 1050280 q^{7} - 10121436 q^{9} - 274863160 q^{13} - 1065273120 q^{15} - 41348739784 q^{19} - 55600393464 q^{21} - 525638756540 q^{25} - 457426227780 q^{27} - 1712336487784 q^{31} - 953298720 q^{33} + 5258670683720 q^{37} + 9161332213176 q^{39} + 18970169729720 q^{43} + 30786788111040 q^{45} - 51073893429300 q^{49} + 65388537498240 q^{51} - 328085114005440 q^{55} + 103623957688680 q^{57} - 984197053056184 q^{61} + 733717575221400 q^{63} - 798011457001480 q^{67} + 20\!\cdots\!80 q^{69}+ \cdots + 88\!\cdots\!40 q^{99}+O(q^{100}) $ 4 * q - 12420 * q^3 - 1050280 * q^7 - 10121436 * q^9 - 274863160 * q^13 - 1065273120 * q^15 - 41348739784 * q^19 - 55600393464 * q^21 - 525638756540 * q^25 - 457426227780 * q^27 - 1712336487784 * q^31 - 953298720 * q^33 + 5258670683720 * q^37 + 9161332213176 * q^39 + 18970169729720 * q^43 + 30786788111040 * q^45 - 51073893429300 * q^49 + 65388537498240 * q^51 - 328085114005440 * q^55 + 103623957688680 * q^57 - 984197053056184 * q^61 + 733717575221400 * q^63 - 798011457001480 * q^67 + 2092913995855680 * q^69 - 2128148319514360 * q^73 + 5201271725969340 * q^75 - 1949813936692072 * q^79 + 5741255080330884 * q^81 - 14560936135384320 * q^85 + 10006046679515040 * q^87 - 11442553772585936 * q^91 - 605599832708280 * q^93 - 31851968157642040 * q^97 + 8858401034408640 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/12\mathbb{Z}\right)^\times$.

$n$	$5$	$7$
$\chi(n)$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−6363.40	+	1598.09i	−0.969882	+	0.243574i
$3$	−6363.40	+	1598.09i	−0.969882	+	0.243574i
$4$		0			0
$5$			746888.i			1.91203i	0.293314	+	0.956016i	$0.405242\pi$
$5$			746888.i			1.91203i	−0.293314	+	0.956016i	$0.594758\pi$
$6$		0			0
$7$	4.25357e6			0.737852			0.368926	−	0.929459i	$-0.379726\pi$
$7$	4.25357e6			0.737852			0.368926	+	0.929459i	$0.379726\pi$
$8$		0			0
$9$	3.79389e7	−	2.03386e7i	0.881343	−	0.472476i
$10$		0			0
$11$			2.12654e8i			0.992049i	0.868309	+	0.496024i	$0.165207\pi$
$11$			2.12654e8i			0.992049i	−0.868309	+	0.496024i	$0.834793\pi$
$12$		0			0
$13$	−7.06137e8			−0.865649			−0.432825	−	0.901478i	$-0.642483\pi$
$13$	−7.06137e8			−0.865649			−0.432825	+	0.901478i	$0.642483\pi$
$14$		0			0
$15$	−1.19359e9	−	4.75274e9i	−0.465721	−	1.85445i
$16$		0			0
$17$			8.78643e9i			1.25957i	0.776771	+	0.629783i	$0.216856\pi$
$17$			8.78643e9i			1.25957i	−0.776771	+	0.629783i	$0.783144\pi$
$18$		0			0
$19$	−8.43718e9			−0.496785			−0.248393	−	0.968659i	$-0.579902\pi$
$19$	−8.43718e9			−0.496785			−0.248393	+	0.968659i	$0.579902\pi$
$20$		0			0
$21$	−2.70672e10	+	6.79758e9i	−0.715629	+	0.179722i
$22$		0			0
$23$		−	1.19825e11i		−	1.53012i	−0.643960	−	0.765059i	$-0.722710\pi$
$23$		−	1.19825e11i		−	1.53012i	0.643960	−	0.765059i	$-0.277290\pi$
$24$		0			0
$25$	−4.05253e11			−2.65587
$26$		0			0
$27$	−2.08918e11	+	1.90052e11i	−0.739716	+	0.672919i
$28$		0			0
$29$		−	3.07746e11i		−	0.615188i	−0.951518	−	0.307594i	$-0.900476\pi$
$29$		−	3.07746e11i		−	0.615188i	0.951518	−	0.307594i	$-0.0995239\pi$
$30$		0			0
$31$	2.63113e10			0.0308496			0.0154248	−	0.999881i	$-0.495090\pi$
$31$	2.63113e10			0.0308496			0.0154248	+	0.999881i	$0.495090\pi$
$32$		0			0
$33$	−3.39841e11	−	1.35320e12i	−0.241637	−	0.962171i
$34$		0			0
$35$			3.17694e12i			1.41080i
$36$		0			0
$37$	−3.10657e11			−0.0884438			−0.0442219	−	0.999022i	$-0.514081\pi$
$37$	−3.10657e11			−0.0884438			−0.0442219	+	0.999022i	$0.514081\pi$
$38$		0			0
$39$	4.49343e12	−	1.12847e12i	0.839578	−	0.210850i
$40$		0			0
$41$		−	5.28023e12i		−	0.661274i	−0.943758	−	0.330637i	$-0.892736\pi$
$41$		−	5.28023e12i		−	0.661274i	0.943758	−	0.330637i	$-0.107264\pi$
$42$		0			0
$43$	1.92635e13			1.64812			0.824059	−	0.566503i	$-0.191704\pi$
$43$	1.92635e13			1.64812			0.824059	+	0.566503i	$0.191704\pi$
$44$		0			0
$45$	1.51906e13	+	2.83361e13i	0.903390	+	1.68516i
$46$		0			0
$47$			2.19814e12i			0.0923152i	0.998934	+	0.0461576i	$0.0146976\pi$
$47$			2.19814e12i			0.0923152i	−0.998934	+	0.0461576i	$0.985302\pi$
$48$		0			0
$49$	−1.51401e13			−0.455575
$50$		0			0
$51$	−1.40415e13	−	5.59116e13i	−0.306798	−	1.22163i
$52$		0			0
$53$		−	1.73729e13i		−	0.279039i	−0.990219	−	0.139519i	$-0.955444\pi$
$53$		−	1.73729e13i		−	0.279039i	0.990219	−	0.139519i	$-0.0445557\pi$
$54$		0			0
$55$	−1.58829e14			−1.89683
$56$		0			0
$57$	5.36891e13	−	1.34834e13i	0.481823	−	0.121004i
$58$		0			0
$59$			1.68820e14i			1.14976i	0.818237	+	0.574881i	$0.194952\pi$
$59$			1.68820e14i			1.14976i	−0.818237	+	0.574881i	$0.805048\pi$
$60$		0			0
$61$	−2.49059e14			−1.29916			−0.649581	−	0.760292i	$-0.725056\pi$
$61$	−2.49059e14			−1.29916			−0.649581	+	0.760292i	$0.725056\pi$
$62$		0			0
$63$	1.61376e14	−	8.65115e13i	0.650301	−	0.348618i
$64$		0			0
$65$		−	5.27405e14i		−	1.65515i
$66$		0			0
$67$	−1.53606e14			−0.378276			−0.189138	−	0.981951i	$-0.560569\pi$
$67$	−1.53606e14			−0.378276			−0.189138	+	0.981951i	$0.560569\pi$
$68$		0			0
$69$	1.91491e14	+	7.62494e14i	0.372697	+	1.48403i
$70$		0			0
$71$			1.82810e14i			0.283095i	0.989931	+	0.141548i	$0.0452078\pi$
$71$			1.82810e14i			0.283095i	−0.989931	+	0.141548i	$0.954792\pi$
$72$		0			0
$73$	−4.12396e14			−0.511365			−0.255683	−	0.966761i	$-0.582300\pi$
$73$	−4.12396e14			−0.511365			−0.255683	+	0.966761i	$0.582300\pi$
$74$		0			0
$75$	2.57879e15	−	6.47631e14i	2.57588	−	0.646900i
$76$		0			0
$77$			9.04541e14i			0.731985i
$78$		0			0
$79$	1.16547e15			0.768220			0.384110	−	0.923287i	$-0.374508\pi$
$79$	1.16547e15			0.768220			0.384110	+	0.923287i	$0.374508\pi$
$80$		0			0
$81$	1.02571e15	−	1.54325e15i	0.553532	−	0.832828i
$82$		0			0
$83$			2.11590e15i			0.939443i	0.882815	+	0.469721i	$0.155646\pi$
$83$			2.11590e15i			0.939443i	−0.882815	+	0.469721i	$0.844354\pi$
$84$		0			0
$85$	−6.56248e15			−2.40833
$86$		0			0
$87$	4.91805e14	+	1.95831e15i	0.149844	+	0.596660i
$88$		0			0
$89$		−	5.45154e15i		−	1.38484i	−0.721495	−	0.692420i	$-0.756545\pi$
$89$		−	5.45154e15i		−	1.38484i	0.721495	−	0.692420i	$-0.243455\pi$
$90$		0			0
$91$	−3.00360e15			−0.638721
$92$		0			0
$93$	−1.67430e14	+	4.20479e13i	−0.0299205	+	0.00751416i
$94$		0			0
$95$		−	6.30163e15i		−	0.949869i
$96$		0			0
$97$	−9.42108e15			−1.20206			−0.601031	−	0.799226i	$-0.705243\pi$
$97$	−9.42108e15			−1.20206			−0.601031	+	0.799226i	$0.705243\pi$
$98$		0			0
$99$	4.32509e15	+	8.06789e15i	0.468720	+	0.874336i
$100$		0			0
$101$			1.60798e16i			1.48494i	0.669879	+	0.742470i	$0.266346\pi$
$101$			1.60798e16i			1.48494i	−0.669879	+	0.742470i	$0.733654\pi$
$102$		0			0
$103$	7.93748e15			0.626592			0.313296	−	0.949656i	$-0.398567\pi$
$103$	7.93748e15			0.626592			0.313296	+	0.949656i	$0.398567\pi$
$104$		0			0
$105$	−5.07703e15	−	2.02161e16i	−0.343633	−	1.36831i
$106$		0			0
$107$		−	1.36742e16i		−	0.795849i	−0.917418	−	0.397925i	$-0.869731\pi$
$107$		−	1.36742e16i		−	0.795849i	0.917418	−	0.397925i	$-0.130269\pi$
$108$		0			0
$109$	4.80176e15			0.240984			0.120492	−	0.992714i	$-0.461553\pi$
$109$	4.80176e15			0.240984			0.120492	+	0.992714i	$0.461553\pi$
$110$		0			0
$111$	1.97683e15	−	4.96458e14i	0.0857801	−	0.0215426i
$112$		0			0
$113$			2.81793e16i			1.05999i	0.848000	+	0.529996i	$0.177807\pi$
$113$			2.81793e16i			1.05999i	−0.848000	+	0.529996i	$0.822193\pi$
$114$		0			0
$115$	8.94958e16			2.92563
$116$		0			0
$117$	−2.67901e16	+	1.43618e16i	−0.762934	+	0.408999i
$118$		0			0
$119$			3.73737e16i			0.929374i
$120$		0			0
$121$	7.27808e14			0.0158392
$122$		0			0
$123$	8.43827e15	+	3.36002e16i	0.161069	+	0.641358i
$124$		0			0
$125$		−	1.88713e17i		−	3.16607i
$126$		0			0
$127$	−2.22418e16			−0.328655			−0.164327	−	0.986406i	$-0.552545\pi$
$127$	−2.22418e16			−0.328655			−0.164327	+	0.986406i	$0.552545\pi$
$128$		0			0
$129$	−1.22582e17	+	3.07849e16i	−1.59848	+	0.401439i
$130$		0			0
$131$			9.36844e16i			1.08018i	0.841606	+	0.540091i	$0.181610\pi$
$131$			9.36844e16i			1.08018i	−0.841606	+	0.540091i	$0.818390\pi$
$132$		0			0
$133$	−3.58881e16			−0.366554
$134$		0			0
$135$	−1.41948e17	−	1.56038e17i	−1.28664	−	1.41436i
$136$		0			0
$137$			1.66939e17i			1.34522i	0.739995	+	0.672612i	$0.234828\pi$
$137$			1.66939e17i			1.34522i	−0.739995	+	0.672612i	$0.765172\pi$
$138$		0			0
$139$	4.40771e16			0.316296			0.158148	−	0.987415i	$-0.449448\pi$
$139$	4.40771e16			0.316296			0.158148	+	0.987415i	$0.449448\pi$
$140$		0			0
$141$	−3.51283e15	−	1.39877e16i	−0.0224856	−	0.0895349i
$142$		0			0
$143$		−	1.50163e17i		−	0.858766i
$144$		0			0
$145$	2.29852e17			1.17626
$146$		0			0
$147$	9.63423e16	−	2.41952e16i	0.441854	−	0.110966i
$148$		0			0
$149$			6.18777e16i			0.254709i	0.991857	+	0.127354i	$0.0406486\pi$
$149$			6.18777e16i			0.254709i	−0.991857	+	0.127354i	$0.959351\pi$
$150$		0			0
$151$	−2.88732e17			−1.06827			−0.534133	−	0.845400i	$-0.679362\pi$
$151$	−2.88732e17			−1.06827			−0.534133	+	0.845400i	$0.679362\pi$
$152$		0			0
$153$	1.78703e17	+	3.33348e17i	0.595115	+	1.11011i
$154$		0			0
$155$			1.96516e16i			0.0589854i
$156$		0			0
$157$	3.58661e17			0.971599			0.485799	−	0.874070i	$-0.338529\pi$
$157$	3.58661e17			0.971599			0.485799	+	0.874070i	$0.338529\pi$
$158$		0			0
$159$	2.77634e16	+	1.10550e17i	0.0679666	+	0.270635i
$160$		0			0
$161$		−	5.09684e17i		−	1.12900i
$162$		0			0
$163$	−3.54469e16			−0.0711340			−0.0355670	−	0.999367i	$-0.511324\pi$
$163$	−3.54469e16			−0.0711340			−0.0355670	+	0.999367i	$0.511324\pi$
$164$		0			0
$165$	1.01069e18	−	2.53823e17i	1.83970	−	0.462018i
$166$		0			0
$167$			3.52766e17i			0.583116i	0.956553	+	0.291558i	$0.0941736\pi$
$167$			3.52766e17i			0.583116i	−0.956553	+	0.291558i	$0.905826\pi$
$168$		0			0
$169$	−1.66788e17			−0.250652
$170$		0			0
$171$	−3.20098e17	+	1.71600e17i	−0.437838	+	0.234719i
$172$		0			0
$173$			1.31046e18i			1.63326i	0.577160	+	0.816631i	$0.304161\pi$
$173$			1.31046e18i			1.63326i	−0.577160	+	0.816631i	$0.695839\pi$
$174$		0			0
$175$	−1.72377e18			−1.95964
$176$		0			0
$177$	−2.69789e17	−	1.07427e18i	−0.280052	−	1.11513i
$178$		0			0
$179$		−	1.31831e18i		−	1.25082i	−0.780297	−	0.625409i	$-0.784932\pi$
$179$		−	1.31831e18i		−	1.25082i	0.780297	−	0.625409i	$-0.215068\pi$
$180$		0			0
$181$	−1.66781e18			−1.44784			−0.723918	−	0.689886i	$-0.757660\pi$
$181$	−1.66781e18			−1.44784			−0.723918	+	0.689886i	$0.757660\pi$
$182$		0			0
$183$	1.58486e18	−	3.98018e17i	1.26003	−	0.316442i
$184$		0			0
$185$		−	2.32026e17i		−	0.169107i
$186$		0			0
$187$	−1.86847e18			−1.24955
$188$		0			0
$189$	−8.88646e17	+	8.08400e17i	−0.545801	+	0.496514i
$190$		0			0
$191$		−	1.09254e18i		−	0.616839i	−0.951251	−	0.308419i	$-0.900200\pi$
$191$		−	1.09254e18i		−	0.616839i	0.951251	−	0.308419i	$-0.0998000\pi$
$192$		0			0
$193$	2.54466e18			1.32182			0.660908	−	0.750467i	$-0.270171\pi$
$193$	2.54466e18			1.32182			0.660908	+	0.750467i	$0.270171\pi$
$194$		0			0
$195$	8.42840e17	+	3.35609e18i	0.403151	+	1.60530i
$196$		0			0
$197$			1.55165e18i			0.684013i	0.939698	+	0.342006i	$0.111106\pi$
$197$			1.55165e18i			0.684013i	−0.939698	+	0.342006i	$0.888894\pi$
$198$		0			0
$199$	1.96681e18			0.799721			0.399860	−	0.916576i	$-0.369059\pi$
$199$	1.96681e18			0.799721			0.399860	+	0.916576i	$0.369059\pi$
$200$		0			0
$201$	9.77453e17	−	2.45475e17i	0.366883	−	0.0921381i
$202$		0			0
$203$		−	1.30902e18i		−	0.453918i
$204$		0			0
$205$	3.94373e18			1.26438
$206$		0			0
$207$	−2.43707e18	−	4.54603e18i	−0.722944	−	1.34856i
$208$		0			0
$209$		−	1.79420e18i		−	0.492835i
$210$		0			0
$211$	−3.58859e17			−0.0913408			−0.0456704	−	0.998957i	$-0.514542\pi$
$211$	−3.58859e17			−0.0913408			−0.0456704	+	0.998957i	$0.514542\pi$
$212$		0			0
$213$	−2.92146e17	−	1.16329e18i	−0.0689547	−	0.274569i
$214$		0			0
$215$			1.43877e19i			3.15126i
$216$		0			0
$217$	1.11917e17			0.0227624
$218$		0			0
$219$	2.62424e18	−	6.59045e17i	0.495964	−	0.124555i
$220$		0			0
$221$		−	6.20442e18i		−	1.09034i
$222$		0			0
$223$	9.91315e18			1.62096			0.810481	−	0.585765i	$-0.199206\pi$
$223$	9.91315e18			1.62096			0.810481	+	0.585765i	$0.199206\pi$
$224$		0			0
$225$	−1.53749e19	+	8.24226e18i	−2.34073	+	1.25483i
$226$		0			0
$227$		−	1.30954e19i		−	1.85742i	−0.370804	−	0.928711i	$-0.620918\pi$
$227$		−	1.30954e19i		−	1.85742i	0.370804	−	0.928711i	$-0.379082\pi$
$228$		0			0
$229$	−4.57797e18			−0.605326			−0.302663	−	0.953098i	$-0.597876\pi$
$229$	−4.57797e18			−0.605326			−0.302663	+	0.953098i	$0.597876\pi$
$230$		0			0
$231$	−1.44554e18	−	5.75595e18i	−0.178293	−	0.709939i
$232$		0			0
$233$			1.24583e18i			0.143420i	0.997426	+	0.0717102i	$0.0228457\pi$
$233$			1.24583e18i			0.143420i	−0.997426	+	0.0717102i	$0.977154\pi$
$234$		0			0
$235$	−1.64177e18			−0.176510
$236$		0			0
$237$	−7.41637e18	+	1.86253e18i	−0.745083	+	0.187118i
$238$		0			0
$239$			4.45351e18i			0.418330i	0.977880	+	0.209165i	$0.0670746\pi$
$239$			4.45351e18i			0.418330i	−0.977880	+	0.209165i	$0.932925\pi$
$240$		0			0
$241$	−1.25387e19			−1.10184			−0.550918	−	0.834559i	$-0.685722\pi$
$241$	−1.25387e19			−1.10184			−0.550918	+	0.834559i	$0.685722\pi$
$242$		0			0
$243$	−4.06073e18	+	1.14595e19i	−0.334006	+	0.942571i
$244$		0			0
$245$		−	1.13079e19i		−	0.871073i
$246$		0			0
$247$	5.95780e18			0.430042
$248$		0			0
$249$	−3.38140e18	−	1.34643e19i	−0.228824	−	0.911149i
$250$		0			0
$251$			9.80510e18i			0.622389i	0.950346	+	0.311195i	$0.100729\pi$
$251$			9.80510e18i			0.622389i	−0.950346	+	0.311195i	$0.899271\pi$
$252$		0			0
$253$	2.54813e19			1.51795
$254$		0			0
$255$	4.17596e19	−	1.04874e19i	2.33580	−	0.586607i
$256$		0			0
$257$			2.05769e19i			1.08122i	0.841272	+	0.540612i	$0.181807\pi$
$257$			2.05769e19i			1.08122i	−0.841272	+	0.540612i	$0.818193\pi$
$258$		0			0
$259$	−1.32140e18			−0.0652584
$260$		0			0
$261$	−6.25911e18	−	1.16756e19i	−0.290662	−	0.542192i
$262$		0			0
$263$			1.66023e19i			0.725309i	0.931924	+	0.362654i	$0.118129\pi$
$263$			1.66023e19i			0.725309i	−0.931924	+	0.362654i	$0.881871\pi$
$264$		0			0
$265$	1.29756e19			0.533531
$266$		0			0
$267$	8.71206e18	+	3.46903e19i	0.337311	+	1.34313i
$268$		0			0
$269$			4.58122e18i			0.167095i	0.996504	+	0.0835474i	$0.0266250\pi$
$269$			4.58122e18i			0.167095i	−0.996504	+	0.0835474i	$0.973375\pi$
$270$		0			0
$271$	−5.33941e19			−1.83543			−0.917717	−	0.397236i	$-0.869970\pi$
$271$	−5.33941e19			−1.83543			−0.917717	+	0.397236i	$0.869970\pi$
$272$		0			0
$273$	1.91131e19	−	4.80002e18i	0.619484	−	0.155576i
$274$		0			0
$275$		−	8.61789e19i		−	2.63475i
$276$		0			0
$277$	1.97883e19			0.570914			0.285457	−	0.958392i	$-0.407855\pi$
$277$	1.97883e19			0.570914			0.285457	+	0.958392i	$0.407855\pi$
$278$		0			0
$279$	9.98224e17	−	5.35135e17i	0.0271891	−	0.0145757i
$280$		0			0
$281$			1.55596e19i			0.400265i	0.979769	+	0.200133i	$0.0641373\pi$
$281$			1.55596e19i			0.400265i	−0.979769	+	0.200133i	$0.935863\pi$
$282$		0			0
$283$	−6.46024e19			−1.57021			−0.785105	−	0.619363i	$-0.787391\pi$
$283$	−6.46024e19			−1.57021			−0.785105	+	0.619363i	$0.787391\pi$
$284$		0			0
$285$	1.00706e19	+	4.00997e19i	0.231363	+	0.921261i
$286$		0			0
$287$		−	2.24598e19i		−	0.487922i
$288$		0			0
$289$	−2.85402e19			−0.586508
$290$		0			0
$291$	5.99501e19	−	1.50557e19i	1.16586	−	0.292791i
$292$		0			0
$293$			7.33404e19i			1.35021i	0.737720	+	0.675106i	$0.235902\pi$
$293$			7.33404e19i			1.35021i	−0.737720	+	0.675106i	$0.764098\pi$
$294$		0			0
$295$	−1.26090e20			−2.19838
$296$		0			0
$297$	−4.04154e19	−	4.44273e19i	−0.667568	−	0.733835i
$298$		0			0
$299$			8.46128e19i			1.32454i
$300$		0			0
$301$	8.19388e19			1.21607
$302$		0			0
$303$	−2.56969e19	−	1.02322e20i	−0.361693	−	1.44022i
$304$		0			0
$305$		−	1.86019e20i		−	2.48404i
$306$		0			0
$307$	9.68120e18			0.122694			0.0613468	−	0.998117i	$-0.480460\pi$
$307$	9.68120e18			0.122694			0.0613468	+	0.998117i	$0.480460\pi$
$308$		0			0
$309$	−5.05093e19	+	1.26848e19i	−0.607720	+	0.152622i
$310$		0			0
$311$			1.00466e20i			1.14799i	0.818859	+	0.573995i	$0.194607\pi$
$311$			1.00466e20i			1.14799i	−0.818859	+	0.573995i	$0.805393\pi$
$312$		0			0
$313$	3.94054e19			0.427761			0.213880	−	0.976860i	$-0.431390\pi$
$313$	3.94054e19			0.427761			0.213880	+	0.976860i	$0.431390\pi$
$314$		0			0
$315$	6.46143e19	+	1.20530e20i	0.666568	+	1.24340i
$316$		0			0
$317$			5.15025e19i			0.505073i	0.967587	+	0.252537i	$0.0812648\pi$
$317$			5.15025e19i			0.505073i	−0.967587	+	0.252537i	$0.918735\pi$
$318$		0			0
$319$	6.54435e19			0.610297
$320$		0			0
$321$	2.18525e19	+	8.70142e19i	0.193848	+	0.771880i
$322$		0			0
$323$		−	7.41327e19i		−	0.625734i
$324$		0			0
$325$	2.86164e20			2.29905
$326$		0			0
$327$	−3.05555e19	+	7.67364e18i	−0.233726	+	0.0586975i
$328$		0			0
$329$			9.34996e18i			0.0681150i
$330$		0			0
$331$	7.61281e19			0.528349			0.264175	−	0.964475i	$-0.414900\pi$
$331$	7.61281e19			0.528349			0.264175	+	0.964475i	$0.414900\pi$
$332$		0			0
$333$	−1.17860e19	+	6.31832e18i	−0.0779494	+	0.0417876i
$334$		0			0
$335$		−	1.14726e20i		−	0.723275i
$336$		0			0
$337$	−1.40478e19			−0.0844441			−0.0422220	−	0.999108i	$-0.513444\pi$
$337$	−1.40478e19			−0.0844441			−0.0422220	+	0.999108i	$0.513444\pi$
$338$		0			0
$339$	−4.50331e19	−	1.79316e20i	−0.258187	−	1.02807i
$340$		0			0
$341$			5.59522e18i			0.0306043i
$342$		0			0
$343$	−2.05758e20			−1.07400
$344$		0			0
$345$	−5.69497e20	+	1.43022e20i	−2.83752	+	0.712609i
$346$		0			0
$347$			1.77682e20i			0.845293i	0.906295	+	0.422647i	$0.138899\pi$
$347$			1.77682e20i			0.845293i	−0.906295	+	0.422647i	$0.861101\pi$
$348$		0			0
$349$	2.15644e20			0.979793			0.489896	−	0.871781i	$-0.337035\pi$
$349$	2.15644e20			0.979793			0.489896	+	0.871781i	$0.337035\pi$
$350$		0			0
$351$	1.47524e20	−	1.34203e20i	0.640335	−	0.582512i
$352$		0			0
$353$			1.21434e20i			0.503668i	0.967770	+	0.251834i	$0.0810337\pi$
$353$			1.21434e20i			0.503668i	−0.967770	+	0.251834i	$0.918966\pi$
$354$		0			0
$355$	−1.36538e20			−0.541287
$356$		0			0
$357$	−5.97265e19	−	2.37824e20i	−0.226371	−	0.901383i
$358$		0			0
$359$			2.07868e19i			0.0753411i	0.999290	+	0.0376706i	$0.0119938\pi$
$359$			2.07868e19i			0.0753411i	−0.999290	+	0.0376706i	$0.988006\pi$
$360$		0			0
$361$	−2.17255e20			−0.753205
$362$		0			0
$363$	−4.63133e18	+	1.16310e18i	−0.0153622	+	0.00385803i
$364$		0			0
$365$		−	3.08013e20i		−	0.977747i
$366$		0			0
$367$	2.18091e20			0.662689			0.331345	−	0.943510i	$-0.392498\pi$
$367$	2.18091e20			0.662689			0.331345	+	0.943510i	$0.392498\pi$
$368$		0			0
$369$	−1.07392e20	−	2.00326e20i	−0.312436	−	0.582810i
$370$		0			0
$371$		−	7.38967e19i		−	0.205889i
$372$		0			0
$373$	2.01515e19			0.0537820			0.0268910	−	0.999638i	$-0.491439\pi$
$373$	2.01515e19			0.0537820			0.0268910	+	0.999638i	$0.491439\pi$
$374$		0			0
$375$	3.01579e20	+	1.20085e21i	0.771173	+	3.07072i
$376$		0			0
$377$			2.17311e20i			0.532537i
$378$		0			0
$379$	−2.34050e20			−0.549789			−0.274894	−	0.961474i	$-0.588643\pi$
$379$	−2.34050e20			−0.549789			−0.274894	+	0.961474i	$0.588643\pi$
$380$		0			0
$381$	1.41533e20	−	3.55444e19i	0.318757	−	0.0800518i
$382$		0			0
$383$		−	6.84934e20i		−	1.47931i	−0.672988	−	0.739653i	$-0.734989\pi$
$383$		−	6.84934e20i		−	1.47931i	0.672988	−	0.739653i	$-0.265011\pi$
$384$		0			0
$385$	−6.75590e20			−1.39958
$386$		0			0
$387$	7.30838e20	−	3.91793e20i	1.45256	−	0.778697i
$388$		0			0
$389$			2.48083e19i			0.0473152i	0.999720	+	0.0236576i	$0.00753115\pi$
$389$			2.48083e19i			0.0473152i	−0.999720	+	0.0236576i	$0.992469\pi$
$390$		0			0
$391$	1.05283e21			1.92728
$392$		0			0
$393$	−1.49716e20	−	5.96151e20i	−0.263104	−	1.04765i
$394$		0			0
$395$			8.70477e20i			1.46886i
$396$		0			0
$397$	4.59136e20			0.744076			0.372038	−	0.928218i	$-0.378659\pi$
$397$	4.59136e20			0.744076			0.372038	+	0.928218i	$0.378659\pi$
$398$		0			0
$399$	2.28370e20	−	5.73524e19i	0.355514	−	0.0892830i
$400$		0			0
$401$			5.45469e20i			0.815859i	0.913013	+	0.407930i	$0.133749\pi$
$401$			5.45469e20i			0.815859i	−0.913013	+	0.407930i	$0.866251\pi$
$402$		0			0
$403$	−1.85794e19			−0.0267049
$404$		0			0
$405$	1.15263e21	+	7.66087e20i	1.59239	+	1.05837i
$406$		0			0
$407$		−	6.60626e19i		−	0.0877406i
$408$		0			0
$409$	−1.31557e21			−1.68007			−0.840033	−	0.542536i	$-0.817464\pi$
$409$	−1.31557e21			−1.68007			−0.840033	+	0.542536i	$0.817464\pi$
$410$		0			0
$411$	−2.66784e20	−	1.06230e21i	−0.327662	−	1.30471i
$412$		0			0
$413$			7.18088e20i			0.848354i
$414$		0			0
$415$	−1.58034e21			−1.79624
$416$		0			0
$417$	−2.80480e20	+	7.04391e19i	−0.306770	+	0.0770416i
$418$		0			0
$419$		−	5.18730e20i		−	0.546046i	−0.962008	−	0.273023i	$-0.911977\pi$
$419$		−	5.18730e20i		−	0.546046i	0.962008	−	0.273023i	$-0.0880235\pi$
$420$		0			0
$421$	−3.31992e20			−0.336411			−0.168206	−	0.985752i	$-0.553797\pi$
$421$	−3.31992e20			−0.336411			−0.168206	+	0.985752i	$0.553797\pi$
$422$		0			0
$423$	4.47071e19	+	8.33953e19i	0.0436168	+	0.0813614i
$424$		0			0
$425$		−	3.56073e21i		−	3.34524i
$426$		0			0
$427$	−1.05939e21			−0.958589
$428$		0			0
$429$	2.39974e20	+	9.55548e20i	0.209173	+	0.832902i
$430$		0			0
$431$			7.04947e20i			0.592023i	0.955184	+	0.296011i	$0.0956567\pi$
$431$			7.04947e20i			0.592023i	−0.955184	+	0.296011i	$0.904343\pi$
$432$		0			0
$433$	8.45661e20			0.684374			0.342187	−	0.939632i	$-0.388832\pi$
$433$	8.45661e20			0.684374			0.342187	+	0.939632i	$0.388832\pi$
$434$		0			0
$435$	−1.46264e21	+	3.67323e20i	−1.14083	+	0.286506i
$436$		0			0
$437$			1.01099e21i			0.760140i
$438$		0			0
$439$	−1.26457e21			−0.916700			−0.458350	−	0.888772i	$-0.651559\pi$
$439$	−1.26457e21			−0.916700			−0.458350	+	0.888772i	$0.651559\pi$
$440$		0			0
$441$	−5.74399e20	+	3.07927e20i	−0.401518	+	0.215248i
$442$		0			0
$443$			1.57262e21i			1.06022i	0.847930	+	0.530109i	$0.177849\pi$
$443$			1.57262e21i			1.06022i	−0.847930	+	0.530109i	$0.822151\pi$
$444$		0			0
$445$	4.07169e21			2.64786
$446$		0			0
$447$	−9.88861e19	−	3.93752e20i	−0.0620405	−	0.247038i
$448$		0			0
$449$			4.67371e19i			0.0282938i	0.999900	+	0.0141469i	$0.00450325\pi$
$449$			4.67371e19i			0.0282938i	−0.999900	+	0.0141469i	$0.995497\pi$
$450$		0			0
$451$	1.12286e21			0.656016
$452$		0			0
$453$	1.83732e21	−	4.61420e20i	1.03609	−	0.260202i
$454$		0			0
$455$		−	2.24335e21i		−	1.22125i
$456$		0			0
$457$	−1.31734e21			−0.692419			−0.346210	−	0.938157i	$-0.612531\pi$
$457$	−1.31734e21			−0.692419			−0.346210	+	0.938157i	$0.612531\pi$
$458$		0			0
$459$	−1.66988e21	−	1.83564e21i	−0.847586	−	0.931722i
$460$		0			0
$461$			2.16306e21i			1.06038i	0.847879	+	0.530189i	$0.177879\pi$
$461$			2.16306e21i			1.06038i	−0.847879	+	0.530189i	$0.822121\pi$
$462$		0			0
$463$	4.06356e21			1.92424			0.962119	−	0.272629i	$-0.0878933\pi$
$463$	4.06356e21			1.92424			0.962119	+	0.272629i	$0.0878933\pi$
$464$		0			0
$465$	−3.14050e19	−	1.25051e20i	−0.0143673	−	0.0572089i
$466$		0			0
$467$		−	1.02632e20i		−	0.0453679i	−0.999743	−	0.0226840i	$-0.992779\pi$
$467$		−	1.02632e20i		−	0.0453679i	0.999743	−	0.0226840i	$-0.00722115\pi$
$468$		0			0
$469$	−6.53372e20			−0.279111
$470$		0			0
$471$	−2.28230e21	+	5.73172e20i	−0.942336	+	0.236656i
$472$		0			0
$473$			4.09648e21i			1.63501i
$474$		0			0
$475$	3.41919e21			1.31940
$476$		0			0
$477$	−3.53339e20	−	6.59108e20i	−0.131839	−	0.245929i
$478$		0			0
$479$			1.09124e21i			0.393763i	0.980427	+	0.196882i	$0.0630815\pi$
$479$			1.09124e21i			0.393763i	−0.980427	+	0.196882i	$0.936918\pi$
$480$		0			0
$481$	2.19366e20			0.0765613
$482$		0			0
$483$	8.14521e20	+	3.24332e21i	0.274995	+	1.09500i
$484$		0			0
$485$		−	7.03649e21i		−	2.29838i
$486$		0			0
$487$	−3.18496e20			−0.100663			−0.0503317	−	0.998733i	$-0.516028\pi$
$487$	−3.18496e20			−0.100663			−0.0503317	+	0.998733i	$0.516028\pi$
$488$		0			0
$489$	2.25562e20	−	5.66473e19i	0.0689916	−	0.0173264i
$490$		0			0
$491$		−	3.97065e21i		−	1.17547i	−0.809055	−	0.587733i	$-0.800021\pi$
$491$		−	3.97065e21i		−	1.17547i	0.809055	−	0.587733i	$-0.199979\pi$
$492$		0			0
$493$	2.70399e21			0.774871
$494$		0			0
$495$	−6.02580e21	+	3.23035e21i	−1.67176	+	0.896207i
$496$		0			0
$497$			7.77594e20i			0.208882i
$498$		0			0
$499$	−2.34699e21			−0.610530			−0.305265	−	0.952267i	$-0.598745\pi$
$499$	−2.34699e21			−0.610530			−0.305265	+	0.952267i	$0.598745\pi$
$500$		0			0
$501$	−5.63752e20	−	2.24479e21i	−0.142032	−	0.565554i
$502$		0			0
$503$		−	1.64122e21i		−	0.400519i	−0.979743	−	0.200260i	$-0.935821\pi$
$503$		−	1.64122e21i		−	0.400519i	0.979743	−	0.200260i	$-0.0641786\pi$
$504$		0			0
$505$	−1.20098e22			−2.83925
$506$		0			0
$507$	1.06134e21	−	2.66542e20i	0.243103	−	0.0610522i
$508$		0			0
$509$			5.48809e21i			1.21809i	0.793136	+	0.609045i	$0.208447\pi$
$509$			5.48809e21i			1.21809i	−0.793136	+	0.609045i	$0.791553\pi$
$510$		0			0
$511$	−1.75415e21			−0.377312
$512$		0			0
$513$	1.76268e21	−	1.60350e21i	0.367480	−	0.334296i
$514$		0			0
$515$			5.92840e21i			1.19806i
$516$		0			0
$517$	−4.67445e20			−0.0915812
$518$		0			0
$519$	−2.09424e21	−	8.33899e21i	−0.397820	−	1.58407i
$520$		0			0
$521$		−	4.67179e21i		−	0.860561i	−0.902695	−	0.430281i	$-0.858415\pi$
$521$		−	4.67179e21i		−	0.860561i	0.902695	−	0.430281i	$-0.141585\pi$
$522$		0			0
$523$	6.28304e21			1.12242			0.561211	−	0.827673i	$-0.310336\pi$
$523$	6.28304e21			1.12242			0.561211	+	0.827673i	$0.310336\pi$
$524$		0			0
$525$	1.09690e22	−	2.75474e21i	1.90062	−	0.477317i
$526$		0			0
$527$			2.31183e20i			0.0388571i
$528$		0			0
$529$	−8.22542e21			−1.34126
$530$		0			0
$531$	3.43356e21	+	6.40485e21i	0.543235	+	1.01333i
$532$		0			0
$533$			3.72856e21i			0.572431i
$534$		0			0
$535$	1.02131e22			1.52169
$536$		0			0
$537$	2.10678e21	+	8.38894e21i	0.304667	+	1.21315i
$538$		0			0
$539$		−	3.21961e21i		−	0.451952i
$540$		0			0
$541$	−2.42895e21			−0.331009			−0.165505	−	0.986209i	$-0.552925\pi$
$541$	−2.42895e21			−0.331009			−0.165505	+	0.986209i	$0.552925\pi$
$542$		0			0
$543$	1.06130e22	−	2.66532e21i	1.40423	−	0.352655i
$544$		0			0
$545$			3.58637e21i			0.460769i
$546$		0			0
$547$	3.08354e21			0.384726			0.192363	−	0.981324i	$-0.438385\pi$
$547$	3.08354e21			0.384726			0.192363	+	0.981324i	$0.438385\pi$
$548$		0			0
$549$	−9.44903e21	+	5.06550e21i	−1.14501	+	0.613823i
$550$		0			0
$551$			2.59651e21i			0.305616i
$552$		0			0
$553$	4.95742e21			0.566832
$554$		0			0
$555$	3.70798e20	+	1.47647e21i	0.0411902	+	0.164014i
$556$		0			0
$557$		−	1.18853e22i		−	1.28282i	−0.767197	−	0.641412i	$-0.778349\pi$
$557$		−	1.18853e22i		−	1.28282i	0.767197	−	0.641412i	$-0.221651\pi$
$558$		0			0
$559$	−1.36027e22			−1.42669
$560$		0			0
$561$	1.18898e22	−	2.98599e21i	1.21192	−	0.304358i
$562$		0			0
$563$		−	9.84524e21i		−	0.975347i	−0.873026	−	0.487673i	$-0.837846\pi$
$563$		−	9.84524e21i		−	0.975347i	0.873026	−	0.487673i	$-0.162154\pi$
$564$		0			0
$565$	−2.10468e22			−2.02674
$566$		0			0
$567$	4.36291e21	−	6.56431e21i	0.408425	−	0.614504i
$568$		0			0
$569$		−	1.88869e22i		−	1.71895i	−0.511181	−	0.859473i	$-0.670792\pi$
$569$		−	1.88869e22i		−	1.71895i	0.511181	−	0.859473i	$-0.329208\pi$
$570$		0			0
$571$	1.17564e22			1.04036			0.520182	−	0.854055i	$-0.325864\pi$
$571$	1.17564e22			1.04036			0.520182	+	0.854055i	$0.325864\pi$
$572$		0			0
$573$	1.74598e21	+	6.95228e21i	0.150246	+	0.598261i
$574$		0			0
$575$			4.85595e22i			4.06379i
$576$		0			0
$577$	−1.30486e21			−0.106208			−0.0531039	−	0.998589i	$-0.516911\pi$
$577$	−1.30486e21			−0.106208			−0.0531039	+	0.998589i	$0.516911\pi$
$578$		0			0
$579$	−1.61927e22	+	4.06659e21i	−1.28201	+	0.321960i
$580$		0			0
$581$			9.00012e21i			0.693169i
$582$		0			0
$583$	3.69442e21			0.276820
$584$		0			0
$585$	−1.07266e22	−	2.00092e22i	−0.782019	−	1.45875i
$586$		0			0
$587$		−	1.53907e22i		−	1.09183i	−0.837842	−	0.545913i	$-0.816183\pi$
$587$		−	1.53907e22i		−	1.09183i	0.837842	−	0.545913i	$-0.183817\pi$
$588$		0			0
$589$	−2.21994e20			−0.0153256
$590$		0			0
$591$	−2.47968e21	−	9.87377e21i	−0.166608	−	0.663412i
$592$		0			0
$593$			7.29950e21i			0.477371i	0.971097	+	0.238685i	$0.0767165\pi$
$593$			7.29950e21i			0.477371i	−0.971097	+	0.238685i	$0.923284\pi$
$594$		0			0
$595$	−2.79139e22			−1.77699
$596$		0			0
$597$	−1.25156e22	+	3.14314e21i	−0.775635	+	0.194791i
$598$		0			0
$599$		−	3.06584e22i		−	1.84984i	−0.380157	−	0.924922i	$-0.624130\pi$
$599$		−	3.06584e22i		−	1.84984i	0.380157	−	0.924922i	$-0.375870\pi$
$600$		0			0
$601$	1.97824e22			1.16220			0.581102	−	0.813831i	$-0.302622\pi$
$601$	1.97824e22			1.16220			0.581102	+	0.813831i	$0.302622\pi$
$602$		0			0
$603$	−5.82763e21	+	3.12411e21i	−0.333391	+	0.178726i
$604$		0			0
$605$			5.43591e20i			0.0302851i
$606$		0			0
$607$	3.19954e22			1.73611			0.868055	−	0.496467i	$-0.165370\pi$
$607$	3.19954e22			1.73611			0.868055	+	0.496467i	$0.165370\pi$
$608$		0			0
$609$	2.09193e21	+	8.32980e21i	0.110563	+	0.440247i
$610$		0			0
$611$		−	1.55219e21i		−	0.0799126i
$612$		0			0
$613$	8.92366e21			0.447568			0.223784	−	0.974639i	$-0.428159\pi$
$613$	8.92366e21			0.447568			0.223784	+	0.974639i	$0.428159\pi$
$614$		0			0
$615$	−2.50956e22	+	6.30244e21i	−1.22630	+	0.307970i
$616$		0			0
$617$			2.96808e22i			1.41317i	0.707628	+	0.706585i	$0.249765\pi$
$617$			2.96808e22i			1.41317i	−0.707628	+	0.706585i	$0.750235\pi$
$618$		0			0
$619$	−3.30189e22			−1.53192			−0.765961	−	0.642887i	$-0.777737\pi$
$619$	−3.30189e22			−1.53192			−0.765961	+	0.642887i	$0.777737\pi$
$620$		0			0
$621$	2.27730e22	+	2.50336e22i	1.02964	+	1.13185i
$622$		0			0
$623$		−	2.31885e22i		−	1.02181i
$624$		0			0
$625$	7.91103e22			3.39776
$626$		0			0
$627$	2.86730e21	+	1.14172e22i	0.120042	+	0.477992i
$628$		0			0
$629$		−	2.72957e21i		−	0.111401i
$630$		0			0
$631$	1.79182e22			0.712952			0.356476	−	0.934304i	$-0.383978\pi$
$631$	1.79182e22			0.712952			0.356476	+	0.934304i	$0.383978\pi$
$632$		0			0
$633$	2.28356e21	−	5.73489e20i	0.0885898	−	0.0222482i
$634$		0			0
$635$		−	1.66121e22i		−	0.628399i
$636$		0			0
$637$	1.06910e22			0.394368
$638$		0			0
$639$	3.71809e21	+	6.93561e21i	0.133756	+	0.249504i
$640$		0			0
$641$			2.46019e22i			0.863186i	0.902069	+	0.431593i	$0.142048\pi$
$641$			2.46019e22i			0.863186i	−0.902069	+	0.431593i	$0.857952\pi$
$642$		0			0
$643$	2.70680e22			0.926334			0.463167	−	0.886271i	$-0.346713\pi$
$643$	2.70680e22			0.926334			0.463167	+	0.886271i	$0.346713\pi$
$644$		0			0
$645$	−2.29928e22	−	9.15547e22i	−0.767564	−	3.05635i
$646$		0			0
$647$			3.19827e22i			1.04155i	0.853693	+	0.520776i	$0.174357\pi$
$647$			3.19827e22i			1.04155i	−0.853693	+	0.520776i	$0.825643\pi$
$648$		0			0
$649$	−3.59003e22			−1.14062
$650$		0			0
$651$	−7.12173e20	+	1.78854e20i	−0.0220769	+	0.00554434i
$652$		0			0
$653$			1.49704e21i			0.0452824i	0.999744	+	0.0226412i	$0.00720753\pi$
$653$			1.49704e21i			0.0452824i	−0.999744	+	0.0226412i	$0.992792\pi$
$654$		0			0
$655$	−6.99717e22			−2.06534
$656$		0			0
$657$	−1.56459e22	+	8.38754e21i	−0.450689	+	0.241608i
$658$		0			0
$659$			3.97007e22i			1.11613i	0.829797	+	0.558065i	$0.188456\pi$
$659$			3.97007e22i			1.11613i	−0.829797	+	0.558065i	$0.811544\pi$
$660$		0			0
$661$	6.63002e21			0.181930			0.0909648	−	0.995854i	$-0.471005\pi$
$661$	6.63002e21			0.181930			0.0909648	+	0.995854i	$0.471005\pi$
$662$		0			0
$663$	9.91522e21	+	3.94812e22i	0.265579	+	1.05750i
$664$		0			0
$665$		−	2.68044e22i		−	0.700863i
$666$		0			0
$667$	−3.68756e22			−0.941311
$668$		0			0
$669$	−6.30813e22	+	1.58421e22i	−1.57214	+	0.394824i
$670$		0			0
$671$		−	5.29635e22i		−	1.28883i
$672$		0			0
$673$	4.96705e22			1.18026			0.590130	−	0.807308i	$-0.299077\pi$
$673$	4.96705e22			1.18026			0.590130	+	0.807308i	$0.299077\pi$
$674$		0			0
$675$	8.46646e22	−	7.70192e22i	1.96459	−	1.78718i
$676$		0			0
$677$		−	5.83670e22i		−	1.32269i	−0.750082	−	0.661345i	$-0.769986\pi$
$677$		−	5.83670e22i		−	1.32269i	0.750082	−	0.661345i	$-0.230014\pi$
$678$		0			0
$679$	−4.00732e22			−0.886944
$680$		0			0
$681$	2.09276e22	+	8.33310e22i	0.452420	+	1.80148i
$682$		0			0
$683$			1.55012e22i			0.327340i	0.986515	+	0.163670i	$0.0523333\pi$
$683$			1.55012e22i			0.327340i	−0.986515	+	0.163670i	$0.947667\pi$
$684$		0			0
$685$	−1.24685e23			−2.57211
$686$		0			0
$687$	2.91314e22	−	7.31601e21i	0.587095	−	0.147442i
$688$		0			0
$689$			1.22676e22i			0.241550i
$690$		0			0
$691$	−1.24102e22			−0.238757			−0.119378	−	0.992849i	$-0.538090\pi$
$691$	−1.24102e22			−0.238757			−0.119378	+	0.992849i	$0.538090\pi$
$692$		0			0
$693$	1.83970e22	+	3.43173e22i	0.345846	+	0.645130i
$694$		0			0
$695$			3.29206e22i			0.604769i
$696$		0			0
$697$	4.63943e22			0.832919
$698$		0			0
$699$	−1.99094e21	−	7.92770e21i	−0.0349335	−	0.139101i
$700$		0			0
$701$		−	4.06748e22i		−	0.697559i	−0.937205	−	0.348779i	$-0.886596\pi$
$701$		−	4.06748e22i		−	0.697559i	0.937205	−	0.348779i	$-0.113404\pi$
$702$		0			0
$703$	2.62107e21			0.0439376
$704$		0			0
$705$	1.04472e22	−	2.62369e21i	0.171194	−	0.0429932i
$706$		0			0
$707$			6.83965e22i			1.09567i
$708$		0			0
$709$	−7.92786e22			−1.24161			−0.620807	−	0.783964i	$-0.713195\pi$
$709$	−7.92786e22			−1.24161			−0.620807	+	0.783964i	$0.713195\pi$
$710$		0			0
$711$	4.42168e22	−	2.37040e22i	0.677065	−	0.362966i
$712$		0			0
$713$		−	3.15276e21i		−	0.0472035i
$714$		0			0
$715$	1.12155e23			1.64199
$716$		0			0
$717$	−7.11711e21	−	2.83395e22i	−0.101894	−	0.405731i
$718$		0			0
$719$		−	1.32052e23i		−	1.84891i	−0.381294	−	0.924454i	$-0.624521\pi$
$719$		−	1.32052e23i		−	1.84891i	0.381294	−	0.924454i	$-0.375479\pi$
$720$		0			0
$721$	3.37626e22			0.462332
$722$		0			0
$723$	7.97889e22	−	2.00380e22i	1.06865	−	0.268379i
$724$		0			0
$725$			1.24715e23i			1.63386i
$726$		0			0
$727$	8.53915e21			0.109431			0.0547154	−	0.998502i	$-0.482575\pi$
$727$	8.53915e21			0.109431			0.0547154	+	0.998502i	$0.482575\pi$
$728$		0			0
$729$	7.52680e21	−	7.94105e22i	0.0943605	−	0.995538i
$730$		0			0
$731$			1.69258e23i			2.07592i
$732$		0			0
$733$	1.52541e23			1.83044			0.915219	−	0.402958i	$-0.132018\pi$
$733$	1.52541e23			1.83044			0.915219	+	0.402958i	$0.132018\pi$
$734$		0			0
$735$	1.80711e22	+	7.19569e22i	0.212171	+	0.844839i
$736$		0			0
$737$		−	3.26649e22i		−	0.375268i
$738$		0			0
$739$	−1.67218e23			−1.87987			−0.939933	−	0.341359i	$-0.889113\pi$
$739$	−1.67218e23			−1.87987			−0.939933	+	0.341359i	$0.889113\pi$
$740$		0			0
$741$	−3.79119e22	+	9.52110e21i	−0.417090	+	0.104747i
$742$		0			0
$743$			5.36483e22i			0.577624i	0.957386	+	0.288812i	$0.0932602\pi$
$743$			5.36483e22i			0.577624i	−0.957386	+	0.288812i	$0.906740\pi$
$744$		0			0
$745$	−4.62157e22			−0.487011
$746$		0			0
$747$	4.30343e22	+	8.02750e22i	0.443864	+	0.827971i
$748$		0			0
$749$		−	5.81640e22i		−	0.587219i
$750$		0			0
$751$	7.08954e22			0.700646			0.350323	−	0.936629i	$-0.386072\pi$
$751$	7.08954e22			0.700646			0.350323	+	0.936629i	$0.386072\pi$
$752$		0			0
$753$	−1.56694e22	−	6.23937e22i	−0.151598	−	0.603645i
$754$		0			0
$755$		−	2.15651e23i		−	2.04256i
$756$		0			0
$757$	−1.35846e23			−1.25974			−0.629868	−	0.776702i	$-0.716891\pi$
$757$	−1.35846e23			−1.25974			−0.629868	+	0.776702i	$0.716891\pi$
$758$		0			0
$759$	−1.62148e23	+	4.07214e22i	−1.47223	+	0.369734i
$760$		0			0
$761$		−	1.71366e23i		−	1.52352i	−0.647859	−	0.761760i	$-0.724335\pi$
$761$		−	1.71366e23i		−	1.52352i	0.647859	−	0.761760i	$-0.275665\pi$
$762$		0			0
$763$	2.04246e22			0.177811
$764$		0			0
$765$	−2.48973e23	+	1.33471e23i	−2.12257	+	1.13788i
$766$		0			0
$767$		−	1.19210e23i		−	0.995290i
$768$		0			0
$769$	5.38746e22			0.440528			0.220264	−	0.975440i	$-0.429308\pi$
$769$	5.38746e22			0.440528			0.220264	+	0.975440i	$0.429308\pi$
$770$		0			0
$771$	−3.28838e22	−	1.30939e23i	−0.263358	−	1.04866i
$772$		0			0
$773$			1.77834e23i			1.39502i	0.716576	+	0.697509i	$0.245708\pi$
$773$			1.77834e23i			1.39502i	−0.716576	+	0.697509i	$0.754292\pi$
$774$		0			0
$775$	−1.06628e22			−0.0819324
$776$		0			0
$777$	8.40860e21	−	2.11172e21i	0.0632930	−	0.0158953i
$778$		0			0
$779$			4.45502e22i			0.328511i
$780$		0			0
$781$	−3.88753e22			−0.280844
$782$		0			0
$783$	5.84878e22	+	6.42936e22i	0.413972	+	0.455065i
$784$		0			0
$785$			2.67879e23i			1.85773i
$786$		0			0
$787$	1.56618e23			1.06425			0.532125	−	0.846666i	$-0.321394\pi$
$787$	1.56618e23			1.06425			0.532125	+	0.846666i	$0.321394\pi$
$788$		0			0
$789$	−2.65320e22	−	1.05647e23i	−0.176666	−	0.703464i
$790$		0			0
$791$			1.19863e23i			0.782117i
$792$		0			0
$793$	1.75870e23			1.12462
$794$		0			0
$795$	−8.25687e22	+	2.07361e22i	−0.517462	+	0.129954i
$796$		0			0
$797$		−	1.69116e23i		−	1.03876i	−0.854542	−	0.519382i	$-0.826162\pi$
$797$		−	1.69116e23i		−	1.03876i	0.854542	−	0.519382i	$-0.173838\pi$
$798$		0			0
$799$	−1.93138e22			−0.116277
$800$		0			0
$801$	−1.10877e23	−	2.06826e23i	−0.654304	−	1.22052i
$802$		0			0
$803$		−	8.76978e22i		−	0.507299i
$804$		0			0
$805$	3.80677e23			2.15868
$806$		0			0
$807$	−7.32120e21	−	2.91521e22i	−0.0406999	−	0.162062i
$808$		0			0
$809$			1.92445e23i			1.04886i	0.851453	+	0.524431i	$0.175722\pi$
$809$			1.92445e23i			1.04886i	−0.851453	+	0.524431i	$0.824278\pi$
$810$		0			0
$811$	−1.17036e23			−0.625391			−0.312696	−	0.949853i	$-0.601232\pi$
$811$	−1.17036e23			−0.625391			−0.312696	+	0.949853i	$0.601232\pi$
$812$		0			0
$813$	3.39768e23	−	8.53285e22i	1.78015	−	0.447064i
$814$		0			0
$815$		−	2.64748e22i		−	0.136010i
$816$		0			0
$817$	−1.62530e23			−0.818761
$818$		0			0
$819$	−1.13953e23	+	6.10889e22i	−0.562932	+	0.301780i
$820$		0			0
$821$		−	1.69300e23i		−	0.820185i	−0.912044	−	0.410092i	$-0.865496\pi$
$821$		−	1.69300e23i		−	0.820185i	0.912044	−	0.410092i	$-0.134504\pi$
$822$		0			0
$823$	1.30845e23			0.621668			0.310834	−	0.950464i	$-0.399392\pi$
$823$	1.30845e23			0.621668			0.310834	+	0.950464i	$0.399392\pi$
$824$		0			0
$825$	1.37722e23	+	5.48391e23i	0.641757	+	2.55540i
$826$		0			0
$827$			1.76300e23i			0.805762i	0.915252	+	0.402881i	$0.131991\pi$
$827$			1.76300e23i			0.805762i	−0.915252	+	0.402881i	$0.868009\pi$
$828$		0			0
$829$	1.67836e23			0.752401			0.376201	−	0.926538i	$-0.377230\pi$
$829$	1.67836e23			0.752401			0.376201	+	0.926538i	$0.377230\pi$
$830$		0			0
$831$	−1.25921e23	+	3.16235e22i	−0.553719	+	0.139060i
$832$		0			0
$833$		−	1.33027e23i		−	0.573827i
$834$		0			0
$835$	−2.63477e23			−1.11494
$836$		0			0
$837$	−5.49691e21	+	5.00053e21i	−0.0228199	+	0.0207593i
$838$		0			0
$839$		−	1.56408e23i		−	0.637036i	−0.947917	−	0.318518i	$-0.896815\pi$
$839$		−	1.56408e23i		−	0.637036i	0.947917	−	0.318518i	$-0.103185\pi$
$840$		0			0
$841$	1.55539e23			0.621543
$842$		0			0
$843$	−2.48656e22	−	9.90120e22i	−0.0974943	−	0.388210i
$844$		0			0
$845$		−	1.24572e23i		−	0.479254i
$846$		0			0
$847$	3.09578e21			0.0116870
$848$		0			0
$849$	4.11091e23	−	1.03240e23i	1.52292	−	0.382462i
$850$		0			0
$851$			3.72245e22i			0.135329i
$852$		0			0
$853$	2.12527e22			0.0758266			0.0379133	−	0.999281i	$-0.487929\pi$
$853$	2.12527e22			0.0758266			0.0379133	+	0.999281i	$0.487929\pi$
$854$		0			0
$855$	−1.28166e23	−	2.39077e23i	−0.448791	−	0.837161i
$856$		0			0
$857$		−	3.24417e22i		−	0.111495i	−0.998445	−	0.0557476i	$-0.982246\pi$
$857$		−	3.24417e22i		−	0.111495i	0.998445	−	0.0557476i	$-0.0177542\pi$
$858$		0			0
$859$	−3.36058e23			−1.13362			−0.566812	−	0.823847i	$-0.691823\pi$
$859$	−3.36058e23			−1.13362			−0.566812	+	0.823847i	$0.691823\pi$
$860$		0			0
$861$	3.58928e22	+	1.42921e23i	0.118845	+	0.473227i
$862$		0			0
$863$			3.75326e23i			1.21990i	0.792442	+	0.609948i	$0.208810\pi$
$863$			3.75326e23i			1.21990i	−0.792442	+	0.609948i	$0.791190\pi$
$864$		0			0
$865$	−9.78768e23			−3.12285
$866$		0			0
$867$	1.81612e23	−	4.56098e22i	0.568844	−	0.142858i
$868$		0			0
$869$			2.47843e23i			0.762111i
$870$		0			0
$871$	1.08466e23			0.327454
$872$		0			0
$873$	−3.57426e23	+	1.91611e23i	−1.05943	+	0.567946i
$874$		0			0
$875$		−	8.02702e23i		−	2.33609i
$876$		0			0
$877$	−6.41168e23			−1.83221			−0.916105	−	0.400939i	$-0.868684\pi$
$877$	−6.41168e23			−1.83221			−0.916105	+	0.400939i	$0.868684\pi$
$878$		0			0
$879$	−1.17205e23	−	4.66694e23i	−0.328877	−	1.30955i
$880$		0			0
$881$			2.46174e23i			0.678320i	0.940729	+	0.339160i	$0.110143\pi$
$881$			2.46174e23i			0.678320i	−0.940729	+	0.339160i	$0.889857\pi$
$882$		0			0
$883$	−4.12946e23			−1.11740			−0.558698	−	0.829371i	$-0.688699\pi$
$883$	−4.12946e23			−1.11740			−0.558698	+	0.829371i	$0.688699\pi$
$884$		0			0
$885$	8.02358e23	−	2.01502e23i	2.13217	−	0.535469i
$886$		0			0
$887$			6.55637e23i			1.71110i	0.517724	+	0.855548i	$0.326779\pi$
$887$			6.55637e23i			1.71110i	−0.517724	+	0.855548i	$0.673221\pi$
$888$		0			0
$889$	−9.46070e22			−0.242499
$890$		0			0
$891$	3.28178e23	+	2.18121e23i	0.826206	+	0.549131i
$892$		0			0
$893$		−	1.85461e22i		−	0.0458608i
$894$		0			0
$895$	9.84631e23			2.39160
$896$		0			0
$897$	−1.35219e23	−	5.38425e23i	−0.322625	−	1.28465i
$898$		0			0
$899$		−	8.09720e21i		−	0.0189783i
$900$		0			0
$901$	1.52645e23			0.351468
$902$		0			0
$903$	−5.21409e23	+	1.30946e23i	−1.17944	+	0.296203i
$904$		0			0
$905$		−	1.24567e24i		−	2.76831i
$906$		0			0
$907$	7.10051e23			1.55035			0.775177	−	0.631744i	$-0.217660\pi$
$907$	7.10051e23			1.55035			0.775177	+	0.631744i	$0.217660\pi$
$908$		0			0
$909$	3.27040e23	+	6.10050e23i	0.701599	+	1.30874i
$910$		0			0
$911$			5.89006e23i			1.24157i	0.783979	+	0.620787i	$0.213187\pi$
$911$			5.89006e23i			1.24157i	−0.783979	+	0.620787i	$0.786813\pi$
$912$		0			0
$913$	−4.49955e23			−0.931973
$914$		0			0
$915$	2.97275e23	+	1.18371e24i	0.605048	+	2.40923i
$916$		0			0
$917$			3.98493e23i			0.797015i
$918$		0			0
$919$	4.56281e23			0.896827			0.448414	−	0.893826i	$-0.351989\pi$
$919$	4.56281e23			0.896827			0.448414	+	0.893826i	$0.351989\pi$
$920$		0			0
$921$	−6.16053e22	+	1.54714e22i	−0.118998	+	0.0298850i
$922$		0			0
$923$		−	1.29089e23i		−	0.245061i
$924$		0			0
$925$	1.25895e23			0.234895
$926$		0			0
$927$	3.01140e23	−	1.61437e23i	0.552243	−	0.296050i
$928$		0			0
$929$			3.40711e23i			0.614131i	0.951688	+	0.307065i	$0.0993470\pi$
$929$			3.40711e23i			0.614131i	−0.951688	+	0.307065i	$0.900653\pi$
$930$		0			0
$931$	1.27740e23			0.226323
$932$		0			0
$933$	−1.60554e23	−	6.39308e23i	−0.279620	−	1.11341i
$934$		0			0
$935$		−	1.39554e24i		−	2.38918i
$936$		0			0
$937$	3.05708e23			0.514505			0.257253	−	0.966344i	$-0.417183\pi$
$937$	3.05708e23			0.514505			0.257253	+	0.966344i	$0.417183\pi$
$938$		0			0
$939$	−2.50752e23	+	6.29733e22i	−0.414878	+	0.104191i
$940$		0			0
$941$			3.89419e23i			0.633433i	0.948520	+	0.316716i	$0.102580\pi$
$941$			3.89419e23i			0.633433i	−0.948520	+	0.316716i	$0.897420\pi$
$942$		0			0
$943$	−6.32703e23			−1.01183
$944$		0			0
$945$	−6.03784e23	−	6.63719e23i	−0.949352	−	1.04359i
$946$		0			0
$947$			3.22588e23i			0.498711i	0.968412	+	0.249355i	$0.0802187\pi$
$947$			3.22588e23i			0.498711i	−0.968412	+	0.249355i	$0.919781\pi$
$948$		0			0
$949$	2.91208e23			0.442663
$950$		0			0
$951$	−8.23056e22	−	3.27731e23i	−0.123023	−	0.489862i
$952$		0			0
$953$			7.16681e22i			0.105337i	0.998612	+	0.0526687i	$0.0167727\pi$
$953$			7.16681e22i			0.105337i	−0.998612	+	0.0526687i	$0.983227\pi$
$954$		0			0
$955$	8.16007e23			1.17942
$956$		0			0
$957$	−4.16443e23	+	1.04585e23i	−0.591916	+	0.148653i
$958$		0			0
$959$			7.10089e23i			0.992576i
$960$		0			0
$961$	−7.26731e23			−0.999048
$962$		0			0
$963$	−2.78113e23	−	5.18784e23i	−0.376020	−	0.701416i
$964$		0			0
$965$			1.90057e24i			2.52735i
$966$		0			0
$967$	9.50017e22			0.124257			0.0621283	−	0.998068i	$-0.480211\pi$
$967$	9.50017e22			0.124257			0.0621283	+	0.998068i	$0.480211\pi$
$968$		0			0
$969$	1.18471e23	+	4.71736e23i	0.152413	+	0.606888i
$970$		0			0
$971$		−	1.07363e24i		−	1.35862i	−0.733850	−	0.679312i	$-0.762279\pi$
$971$		−	1.07363e24i		−	1.35862i	0.733850	−	0.679312i	$-0.237721\pi$
$972$		0			0
$973$	1.87485e23			0.233380
$974$		0			0
$975$	−1.82098e24	+	4.57316e23i	−2.22981	+	0.559989i
$976$		0			0
$977$		−	4.39983e23i		−	0.530004i	−0.964248	−	0.265002i	$-0.914627\pi$
$977$		−	4.39983e23i		−	0.530004i	0.964248	−	0.265002i	$-0.0853726\pi$
$978$		0			0
$979$	1.15930e24			1.37383
$980$		0			0
$981$	1.82174e23	−	9.76609e22i	0.212390	−	0.113859i
$982$		0			0
$983$		−	8.60862e23i		−	0.987427i	−0.869625	−	0.493714i	$-0.835639\pi$
$983$		−	8.60862e23i		−	0.987427i	0.869625	−	0.493714i	$-0.164361\pi$
$984$		0			0
$985$	−1.15891e24			−1.30785
$986$		0			0
$987$	−1.49421e22	−	5.94975e22i	−0.0165910	−	0.0660635i
$988$		0			0
$989$		−	2.30825e24i		−	2.52182i
$990$		0			0
$991$	−4.93879e23			−0.530923			−0.265461	−	0.964121i	$-0.585524\pi$
$991$	−4.93879e23			−0.530923			−0.265461	+	0.964121i	$0.585524\pi$
$992$		0			0
$993$	−4.84433e23	+	1.21660e23i	−0.512437	+	0.128692i
$994$		0			0
$995$			1.46899e24i			1.52909i
$996$		0			0
$997$	4.73874e23			0.485402			0.242701	−	0.970101i	$-0.421967\pi$
$997$	4.73874e23			0.485402			0.242701	+	0.970101i	$0.421967\pi$
$998$		0			0
$999$	6.49018e22	−	5.90410e22i	0.0654233	−	0.0595155i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.17.c.b.5.2	yes	4
3.2	odd	2	inner	12.17.c.b.5.1	✓	4
4.3	odd	2		48.17.e.c.17.3		4
12.11	even	2		48.17.e.c.17.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.17.c.b.5.1	✓	4	3.2	odd	2	inner
12.17.c.b.5.2	yes	4	1.1	even	1	trivial
48.17.e.c.17.3		4	4.3	odd	2
48.17.e.c.17.4		4	12.11	even	2

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:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 17 \)
Character orbit:	\([\chi]\)	\(=\)	12.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-6363.40 + 1598.09i) q^{3} +746888. i q^{5} +4.25357e6 q^{7} +(3.79389e7 - 2.03386e7i) q^{9} +O(q^{10})\)	\(q+(-6363.40 + 1598.09i) q^{3} +746888. i q^{5} +4.25357e6 q^{7} +(3.79389e7 - 2.03386e7i) q^{9} +2.12654e8i q^{11} -7.06137e8 q^{13} +(-1.19359e9 - 4.75274e9i) q^{15} +8.78643e9i q^{17} -8.43718e9 q^{19} +(-2.70672e10 + 6.79758e9i) q^{21} -1.19825e11i q^{23} -4.05253e11 q^{25} +(-2.08918e11 + 1.90052e11i) q^{27} -3.07746e11i q^{29} +2.63113e10 q^{31} +(-3.39841e11 - 1.35320e12i) q^{33} +3.17694e12i q^{35} -3.10657e11 q^{37} +(4.49343e12 - 1.12847e12i) q^{39} -5.28023e12i q^{41} +1.92635e13 q^{43} +(1.51906e13 + 2.83361e13i) q^{45} +2.19814e12i q^{47} -1.51401e13 q^{49} +(-1.40415e13 - 5.59116e13i) q^{51} -1.73729e13i q^{53} -1.58829e14 q^{55} +(5.36891e13 - 1.34834e13i) q^{57} +1.68820e14i q^{59} -2.49059e14 q^{61} +(1.61376e14 - 8.65115e13i) q^{63} -5.27405e14i q^{65} -1.53606e14 q^{67} +(1.91491e14 + 7.62494e14i) q^{69} +1.82810e14i q^{71} -4.12396e14 q^{73} +(2.57879e15 - 6.47631e14i) q^{75} +9.04541e14i q^{77} +1.16547e15 q^{79} +(1.02571e15 - 1.54325e15i) q^{81} +2.11590e15i q^{83} -6.56248e15 q^{85} +(4.91805e14 + 1.95831e15i) q^{87} -5.45154e15i q^{89} -3.00360e15 q^{91} +(-1.67430e14 + 4.20479e13i) q^{93} -6.30163e15i q^{95} -9.42108e15 q^{97} +(4.32509e15 + 8.06789e15i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12420 q^{3} - 1050280 q^{7} - 10121436 q^{9}+O(q^{10}) \) 4 * q - 12420 * q^3 - 1050280 * q^7 - 10121436 * q^9	\( 4 q - 12420 q^{3} - 1050280 q^{7} - 10121436 q^{9} - 274863160 q^{13} - 1065273120 q^{15} - 41348739784 q^{19} - 55600393464 q^{21} - 525638756540 q^{25} - 457426227780 q^{27} - 1712336487784 q^{31} - 953298720 q^{33} + 5258670683720 q^{37} + 9161332213176 q^{39} + 18970169729720 q^{43} + 30786788111040 q^{45} - 51073893429300 q^{49} + 65388537498240 q^{51} - 328085114005440 q^{55} + 103623957688680 q^{57} - 984197053056184 q^{61} + 733717575221400 q^{63} - 798011457001480 q^{67} + 20\!\cdots\!80 q^{69}+ \cdots + 88\!\cdots\!40 q^{99}+O(q^{100}) \) 4 * q - 12420 * q^3 - 1050280 * q^7 - 10121436 * q^9 - 274863160 * q^13 - 1065273120 * q^15 - 41348739784 * q^19 - 55600393464 * q^21 - 525638756540 * q^25 - 457426227780 * q^27 - 1712336487784 * q^31 - 953298720 * q^33 + 5258670683720 * q^37 + 9161332213176 * q^39 + 18970169729720 * q^43 + 30786788111040 * q^45 - 51073893429300 * q^49 + 65388537498240 * q^51 - 328085114005440 * q^55 + 103623957688680 * q^57 - 984197053056184 * q^61 + 733717575221400 * q^63 - 798011457001480 * q^67 + 2092913995855680 * q^69 - 2128148319514360 * q^73 + 5201271725969340 * q^75 - 1949813936692072 * q^79 + 5741255080330884 * q^81 - 14560936135384320 * q^85 + 10006046679515040 * q^87 - 11442553772585936 * q^91 - 605599832708280 * q^93 - 31851968157642040 * q^97 + 8858401034408640 * q^99

Introduction

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