LMFDB - Embedded newform 12.15.c.b.5.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [12,15,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, 15, names="a")

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

chi := DirichletCharacter("12.5");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$ N $	$=$	$ 12 = 2^{2} \cdot 3 $
Weight:	$ k $	$=$	$ 15 $
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:	$14.9194761782$
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} + 440x^{2} + 48015 $ x^4 + 440*x^2 + 48015
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{9}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5.3
Root			$-15.4797i$ of defining polynomial
Character	$\chi$	$=$	12.5
Dual form			12.15.c.b.5.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1714.29 - 1358.01i) q^{3} -97311.2i q^{5} -526279. q^{7} +(1.09458e6 - 4.65604e6i) q^{9} +O(q^{10})$	\(q+(1714.29 - 1358.01i) q^{3} -97311.2i q^{5} -526279. q^{7} +(1.09458e6 - 4.65604e6i) q^{9} +1.93610e7i q^{11} -3.32937e7 q^{13} +(-1.32150e8 - 1.66819e8i) q^{15} -5.75994e8i q^{17} -1.45320e9 q^{19} +(-9.02192e8 + 7.14693e8i) q^{21} -4.29912e9i q^{23} -3.36595e9 q^{25} +(-4.44654e9 - 9.46823e9i) q^{27} +1.71636e10i q^{29} +3.40275e10 q^{31} +(2.62924e10 + 3.31902e10i) q^{33} +5.12128e10i q^{35} +8.31195e10 q^{37} +(-5.70749e10 + 4.52133e10i) q^{39} -2.11971e11i q^{41} +1.48530e11 q^{43} +(-4.53085e11 - 1.06515e11i) q^{45} +4.40890e11i q^{47} -4.01254e11 q^{49} +(-7.82207e11 - 9.87418e11i) q^{51} -1.70820e12i q^{53} +1.88404e12 q^{55} +(-2.49121e12 + 1.97347e12i) q^{57} +1.79612e12i q^{59} +5.48983e12 q^{61} +(-5.76053e11 + 2.45037e12i) q^{63} +3.23985e12i q^{65} +1.01831e13 q^{67} +(-5.83825e12 - 7.36992e12i) q^{69} +1.61449e12i q^{71} -8.08871e12 q^{73} +(-5.77020e12 + 4.57100e12i) q^{75} -1.01893e13i q^{77} +1.32951e13 q^{79} +(-2.04806e13 - 1.01928e13i) q^{81} -8.28432e12i q^{83} -5.60507e13 q^{85} +(2.33084e13 + 2.94234e13i) q^{87} -5.62661e13i q^{89} +1.75218e13 q^{91} +(5.83328e13 - 4.62098e13i) q^{93} +1.41413e14i q^{95} -1.34231e12 q^{97} +(9.01454e13 + 2.11921e13i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2148 q^{3} + 1709288 q^{7} - 5736924 q^{9}+O(q^{10}) $ 4 * q + 2148 * q^3 + 1709288 * q^7 - 5736924 * q^9	$ 4 q + 2148 q^{3} + 1709288 q^{7} - 5736924 q^{9} - 93505048 q^{13} + 24235200 q^{15} - 1349200696 q^{19} - 3572752344 q^{21} - 4043764700 q^{25} + 7381750212 q^{27} + 46585213736 q^{31} - 60689217600 q^{33} + 300873217064 q^{37} - 96914866776 q^{39} + 246448758152 q^{43} - 1275658243200 q^{45} + 1654942475340 q^{49} - 4102471929600 q^{51} + 7505039836800 q^{55} - 5979467701752 q^{57} + 8008332264296 q^{61} - 12097408557528 q^{63} + 38294908213448 q^{67} - 27868623868800 q^{69} + 12721406693576 q^{73} - 13261586187900 q^{75} - 29803403331928 q^{79} - 23892053776956 q^{81} - 28369656691200 q^{85} + 58743848116800 q^{87} - 2127612761456 q^{91} + 130412478704232 q^{93} - 301625619131512 q^{97} + 325346192265600 q^{99}+O(q^{100}) $ 4 * q + 2148 * q^3 + 1709288 * q^7 - 5736924 * q^9 - 93505048 * q^13 + 24235200 * q^15 - 1349200696 * q^19 - 3572752344 * q^21 - 4043764700 * q^25 + 7381750212 * q^27 + 46585213736 * q^31 - 60689217600 * q^33 + 300873217064 * q^37 - 96914866776 * q^39 + 246448758152 * q^43 - 1275658243200 * q^45 + 1654942475340 * q^49 - 4102471929600 * q^51 + 7505039836800 * q^55 - 5979467701752 * q^57 + 8008332264296 * q^61 - 12097408557528 * q^63 + 38294908213448 * q^67 - 27868623868800 * q^69 + 12721406693576 * q^73 - 13261586187900 * q^75 - 29803403331928 * q^79 - 23892053776956 * q^81 - 28369656691200 * q^85 + 58743848116800 * q^87 - 2127612761456 * q^91 + 130412478704232 * q^93 - 301625619131512 * q^97 + 325346192265600 * 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
$3$	1714.29	−	1358.01i	0.783852	−	0.620947i
$3$	1714.29	−	1358.01i	0.783852	−	0.620947i
$4$		0			0
$5$		−	97311.2i		−	1.24558i	−0.782388	−	0.622792i	$-0.785998\pi$
$5$		−	97311.2i		−	1.24558i	0.782388	−	0.622792i	$-0.214002\pi$
$6$		0			0
$7$	−526279.			−0.639042			−0.319521	−	0.947579i	$-0.603522\pi$
$7$	−526279.			−0.639042			−0.319521	+	0.947579i	$0.603522\pi$
$8$		0			0
$9$	1.09458e6	−	4.65604e6i	0.228849	−	0.973462i
$10$		0			0
$11$			1.93610e7i			0.993524i	0.867887	+	0.496762i	$0.165478\pi$
$11$			1.93610e7i			0.993524i	−0.867887	+	0.496762i	$0.834522\pi$
$12$		0			0
$13$	−3.32937e7			−0.530590			−0.265295	−	0.964167i	$-0.585469\pi$
$13$	−3.32937e7			−0.530590			−0.265295	+	0.964167i	$0.585469\pi$
$14$		0			0
$15$	−1.32150e8	−	1.66819e8i	−0.773442	−	0.976353i
$16$		0			0
$17$		−	5.75994e8i		−	1.40370i	−0.712323	−	0.701852i	$-0.752357\pi$
$17$		−	5.75994e8i		−	1.40370i	0.712323	−	0.701852i	$-0.247643\pi$
$18$		0			0
$19$	−1.45320e9			−1.62574			−0.812870	−	0.582444i	$-0.802096\pi$
$19$	−1.45320e9			−1.62574			−0.812870	+	0.582444i	$0.802096\pi$
$20$		0			0
$21$	−9.02192e8	+	7.14693e8i	−0.500915	+	0.396812i
$22$		0			0
$23$		−	4.29912e9i		−	1.26265i	−0.775516	−	0.631327i	$-0.782510\pi$
$23$		−	4.29912e9i		−	1.26265i	0.775516	−	0.631327i	$-0.217490\pi$
$24$		0			0
$25$	−3.36595e9			−0.551478
$26$		0			0
$27$	−4.44654e9	−	9.46823e9i	−0.425085	−	0.905154i
$28$		0			0
$29$			1.71636e10i			0.995001i	0.867464	+	0.497500i	$0.165749\pi$
$29$			1.71636e10i			0.995001i	−0.867464	+	0.497500i	$0.834251\pi$
$30$		0			0
$31$	3.40275e10			1.23680			0.618398	−	0.785865i	$-0.287782\pi$
$31$	3.40275e10			1.23680			0.618398	+	0.785865i	$0.287782\pi$
$32$		0			0
$33$	2.62924e10	+	3.31902e10i	0.616926	+	0.778776i
$34$		0			0
$35$			5.12128e10i			0.795981i
$36$		0			0
$37$	8.31195e10			0.875570			0.437785	−	0.899080i	$-0.355763\pi$
$37$	8.31195e10			0.875570			0.437785	+	0.899080i	$0.355763\pi$
$38$		0			0
$39$	−5.70749e10	+	4.52133e10i	−0.415904	+	0.329468i
$40$		0			0
$41$		−	2.11971e11i		−	1.08840i	−0.838955	−	0.544201i	$-0.816833\pi$
$41$		−	2.11971e11i		−	1.08840i	0.838955	−	0.544201i	$-0.183167\pi$
$42$		0			0
$43$	1.48530e11			0.546429			0.273215	−	0.961953i	$-0.411913\pi$
$43$	1.48530e11			0.546429			0.273215	+	0.961953i	$0.411913\pi$
$44$		0			0
$45$	−4.53085e11	−	1.06515e11i	−1.21253	−	0.285050i
$46$		0			0
$47$			4.40890e11i			0.870252i	0.900370	+	0.435126i	$0.143296\pi$
$47$			4.40890e11i			0.870252i	−0.900370	+	0.435126i	$0.856704\pi$
$48$		0			0
$49$	−4.01254e11			−0.591625
$50$		0			0
$51$	−7.82207e11	−	9.87418e11i	−0.871626	−	1.10030i
$52$		0			0
$53$		−	1.70820e12i		−	1.45414i	−0.686563	−	0.727071i	$-0.740881\pi$
$53$		−	1.70820e12i		−	1.45414i	0.686563	−	0.727071i	$-0.259119\pi$
$54$		0			0
$55$	1.88404e12			1.23752
$56$		0			0
$57$	−2.49121e12	+	1.97347e12i	−1.27434	+	1.00950i
$58$		0			0
$59$			1.79612e12i			0.721723i	0.932619	+	0.360861i	$0.117517\pi$
$59$			1.79612e12i			0.721723i	−0.932619	+	0.360861i	$0.882483\pi$
$60$		0			0
$61$	5.48983e12			1.74683			0.873414	−	0.486978i	$-0.161901\pi$
$61$	5.48983e12			1.74683			0.873414	+	0.486978i	$0.161901\pi$
$62$		0			0
$63$	−5.76053e11	+	2.45037e12i	−0.146244	+	0.622083i
$64$		0			0
$65$			3.23985e12i			0.660894i
$66$		0			0
$67$	1.01831e13			1.68018			0.840092	−	0.542443i	$-0.182501\pi$
$67$	1.01831e13			1.68018			0.840092	+	0.542443i	$0.182501\pi$
$68$		0			0
$69$	−5.83825e12	−	7.36992e12i	−0.784042	−	0.989735i
$70$		0			0
$71$			1.61449e12i			0.177511i	0.996053	+	0.0887557i	$0.0282890\pi$
$71$			1.61449e12i			0.177511i	−0.996053	+	0.0887557i	$0.971711\pi$
$72$		0			0
$73$	−8.08871e12			−0.732182			−0.366091	−	0.930579i	$-0.619304\pi$
$73$	−8.08871e12			−0.732182			−0.366091	+	0.930579i	$0.619304\pi$
$74$		0			0
$75$	−5.77020e12	+	4.57100e12i	−0.432277	+	0.342439i
$76$		0			0
$77$		−	1.01893e13i		−	0.634904i
$78$		0			0
$79$	1.32951e13			0.692314			0.346157	−	0.938177i	$-0.387487\pi$
$79$	1.32951e13			0.692314			0.346157	+	0.938177i	$0.387487\pi$
$80$		0			0
$81$	−2.04806e13	−	1.01928e13i	−0.895256	−	0.445551i
$82$		0			0
$83$		−	8.28432e12i		−	0.305288i	−0.988281	−	0.152644i	$-0.951221\pi$
$83$		−	8.28432e12i		−	0.305288i	0.988281	−	0.152644i	$-0.0487788\pi$
$84$		0			0
$85$	−5.60507e13			−1.74843
$86$		0			0
$87$	2.33084e13	+	2.94234e13i	0.617843	+	0.779934i
$88$		0			0
$89$		−	5.62661e13i		−	1.27209i	−0.771654	−	0.636043i	$-0.780570\pi$
$89$		−	5.62661e13i		−	1.27209i	0.771654	−	0.636043i	$-0.219430\pi$
$90$		0			0
$91$	1.75218e13			0.339069
$92$		0			0
$93$	5.83328e13	−	4.62098e13i	0.969466	−	0.767985i
$94$		0			0
$95$			1.41413e14i			2.02500i
$96$		0			0
$97$	−1.34231e12			−0.0166131			−0.00830657	−	0.999965i	$-0.502644\pi$
$97$	−1.34231e12			−0.0166131			−0.00830657	+	0.999965i	$0.502644\pi$
$98$		0			0
$99$	9.01454e13	+	2.11921e13i	0.967158	+	0.227367i
$100$		0			0
$101$			1.36092e14i			1.26935i	0.772778	+	0.634677i	$0.218867\pi$
$101$			1.36092e14i			1.26935i	−0.772778	+	0.634677i	$0.781133\pi$
$102$		0			0
$103$	−7.93677e12			−0.0645332			−0.0322666	−	0.999479i	$-0.510273\pi$
$103$	−7.93677e12			−0.0645332			−0.0322666	+	0.999479i	$0.510273\pi$
$104$		0			0
$105$	6.95476e13	+	8.77934e13i	0.494262	+	0.623931i
$106$		0			0
$107$		−	2.90162e14i		−	1.80698i	−0.428608	−	0.903490i	$-0.640996\pi$
$107$		−	2.90162e14i		−	1.80698i	0.428608	−	0.903490i	$-0.359004\pi$
$108$		0			0
$109$	−1.80217e14			−0.985849			−0.492924	−	0.870072i	$-0.664072\pi$
$109$	−1.80217e14			−0.985849			−0.492924	+	0.870072i	$0.664072\pi$
$110$		0			0
$111$	1.42490e14	−	1.12877e14i	0.686317	−	0.543683i
$112$		0			0
$113$		−	1.75226e14i		−	0.744817i	−0.928069	−	0.372409i	$-0.878532\pi$
$113$		−	1.75226e14i		−	0.744817i	0.928069	−	0.372409i	$-0.121468\pi$
$114$		0			0
$115$	−4.18352e14			−1.57274
$116$		0			0
$117$	−3.64425e13	+	1.55017e14i	−0.121425	+	0.516509i
$118$		0			0
$119$			3.03133e14i			0.897026i
$120$		0			0
$121$	4.90274e12			0.0129104
$122$		0			0
$123$	−2.87859e14	−	3.63379e14i	−0.675841	−	0.853147i
$124$		0			0
$125$		−	2.66395e14i		−	0.558672i
$126$		0			0
$127$	1.16420e14			0.218474			0.109237	−	0.994016i	$-0.465159\pi$
$127$	1.16420e14			0.218474			0.109237	+	0.994016i	$0.465159\pi$
$128$		0			0
$129$	2.54622e14	−	2.01705e14i	0.428320	−	0.339304i
$130$		0			0
$131$			5.57275e13i			0.0841725i	0.999114	+	0.0420863i	$0.0134004\pi$
$131$			5.57275e13i			0.0841725i	−0.999114	+	0.0420863i	$0.986600\pi$
$132$		0			0
$133$	7.64790e14			1.03892
$134$		0			0
$135$	−9.21364e14	+	4.32698e14i	−1.12744	+	0.529479i
$136$		0			0
$137$			9.83751e14i			1.08603i	0.839724	+	0.543014i	$0.182717\pi$
$137$			9.83751e14i			1.08603i	−0.839724	+	0.543014i	$0.817283\pi$
$138$		0			0
$139$	1.03585e15			1.03322			0.516612	−	0.856219i	$-0.327193\pi$
$139$	1.03585e15			1.03322			0.516612	+	0.856219i	$0.327193\pi$
$140$		0			0
$141$	5.98733e14	+	7.55811e14i	0.540380	+	0.682149i
$142$		0			0
$143$		−	6.44598e14i		−	0.527153i
$144$		0			0
$145$	1.67021e15			1.23936
$146$		0			0
$147$	−6.87863e14	+	5.44907e14i	−0.463747	+	0.367368i
$148$		0			0
$149$			1.07073e15i			0.656715i	0.944554	+	0.328357i	$0.106495\pi$
$149$			1.07073e15i			0.656715i	−0.944554	+	0.328357i	$0.893505\pi$
$150$		0			0
$151$	3.90828e13			0.0218347			0.0109173	−	0.999940i	$-0.496525\pi$
$151$	3.90828e13			0.0218347			0.0109173	+	0.999940i	$0.496525\pi$
$152$		0			0
$153$	−2.68185e15	−	6.30470e14i	−1.36645	−	0.321236i
$154$		0			0
$155$		−	3.31126e15i		−	1.54053i
$156$		0			0
$157$	−4.60184e14			−0.195720			−0.0978598	−	0.995200i	$-0.531200\pi$
$157$	−4.60184e14			−0.195720			−0.0978598	+	0.995200i	$0.531200\pi$
$158$		0			0
$159$	−2.31975e15	−	2.92833e15i	−0.902945	−	1.13983i
$160$		0			0
$161$			2.26254e15i			0.806890i
$162$		0			0
$163$	−1.20118e15			−0.392911			−0.196455	−	0.980513i	$-0.562943\pi$
$163$	−1.20118e15			−0.392911			−0.196455	+	0.980513i	$0.562943\pi$
$164$		0			0
$165$	3.22978e15	−	2.55855e15i	0.970030	−	0.768433i
$166$		0			0
$167$			1.80816e14i			0.0499138i	0.999689	+	0.0249569i	$0.00794485\pi$
$167$			1.80816e14i			0.0499138i	−0.999689	+	0.0249569i	$0.992055\pi$
$168$		0			0
$169$	−2.82891e15			−0.718475
$170$		0			0
$171$	−1.59064e15	+	6.76617e15i	−0.372049	+	1.58260i
$172$		0			0
$173$		−	5.02762e15i		−	1.08403i	−0.840370	−	0.542013i	$-0.817662\pi$
$173$		−	5.02762e15i		−	1.08403i	0.840370	−	0.542013i	$-0.182338\pi$
$174$		0			0
$175$	1.77143e15			0.352418
$176$		0			0
$177$	2.43915e15	+	3.07906e15i	0.448152	+	0.565724i
$178$		0			0
$179$			6.86369e15i			1.16570i	0.812580	+	0.582849i	$0.198062\pi$
$179$			6.86369e15i			1.16570i	−0.812580	+	0.582849i	$0.801938\pi$
$180$		0			0
$181$	−3.40537e15			−0.535074			−0.267537	−	0.963548i	$-0.586210\pi$
$181$	−3.40537e15			−0.535074			−0.267537	+	0.963548i	$0.586210\pi$
$182$		0			0
$183$	9.41114e15	−	7.45526e15i	1.36926	−	1.08469i
$184$		0			0
$185$		−	8.08845e15i		−	1.09059i
$186$		0			0
$187$	1.11518e16			1.39461
$188$		0			0
$189$	2.34012e15	+	4.98293e15i	0.271647	+	0.578431i
$190$		0			0
$191$			1.44955e16i			1.56315i	0.623813	+	0.781573i	$0.285582\pi$
$191$			1.44955e16i			1.56315i	−0.623813	+	0.781573i	$0.714418\pi$
$192$		0			0
$193$	−1.79374e16			−1.79828			−0.899140	−	0.437662i	$-0.855807\pi$
$193$	−1.79374e16			−1.79828			−0.899140	+	0.437662i	$0.855807\pi$
$194$		0			0
$195$	4.39976e15	+	5.55403e15i	0.410380	+	0.518043i
$196$		0			0
$197$			9.68902e15i			0.841427i	0.907194	+	0.420714i	$0.138220\pi$
$197$			9.68902e15i			0.841427i	−0.907194	+	0.420714i	$0.861780\pi$
$198$		0			0
$199$	5.06502e14			0.0409835			0.0204918	−	0.999790i	$-0.493477\pi$
$199$	5.06502e14			0.0409835			0.0204918	+	0.999790i	$0.493477\pi$
$200$		0			0
$201$	1.74568e16	−	1.38288e16i	1.31702	−	1.04331i
$202$		0			0
$203$		−	9.03286e15i		−	0.635848i
$204$		0			0
$205$	−2.06272e16			−1.35570
$206$		0			0
$207$	−2.00169e16	−	4.70572e15i	−1.22915	−	0.288957i
$208$		0			0
$209$		−	2.81354e16i		−	1.61521i
$210$		0			0
$211$	−8.03611e14			−0.0431587			−0.0215794	−	0.999767i	$-0.506869\pi$
$211$	−8.03611e14			−0.0431587			−0.0215794	+	0.999767i	$0.506869\pi$
$212$		0			0
$213$	2.19249e15	+	2.76769e15i	0.110225	+	0.139143i
$214$		0			0
$215$		−	1.44536e16i		−	0.680623i
$216$		0			0
$217$	−1.79080e16			−0.790365
$218$		0			0
$219$	−1.38664e16	+	1.09846e16i	−0.573923	+	0.454647i
$220$		0			0
$221$			1.91770e16i			0.744791i
$222$		0			0
$223$	2.62423e16			0.956904			0.478452	−	0.878114i	$-0.341198\pi$
$223$	2.62423e16			0.956904			0.478452	+	0.878114i	$0.341198\pi$
$224$		0			0
$225$	−3.68430e15	+	1.56720e16i	−0.126205	+	0.536843i
$226$		0			0
$227$			4.14545e15i			0.133472i	0.997771	+	0.0667361i	$0.0212586\pi$
$227$			4.14545e15i			0.133472i	−0.997771	+	0.0667361i	$0.978741\pi$
$228$		0			0
$229$	3.96515e16			1.20064			0.600319	−	0.799761i	$-0.295040\pi$
$229$	3.96515e16			1.20064			0.600319	+	0.799761i	$0.295040\pi$
$230$		0			0
$231$	−1.38371e16	−	1.74673e16i	−0.394242	−	0.497671i
$232$		0			0
$233$			2.60867e16i			0.699725i	0.936801	+	0.349862i	$0.113772\pi$
$233$			2.60867e16i			0.699725i	−0.936801	+	0.349862i	$0.886228\pi$
$234$		0			0
$235$	4.29035e16			1.08397
$236$		0			0
$237$	2.27916e16	−	1.80549e16i	0.542672	−	0.429890i
$238$		0			0
$239$			3.03783e16i			0.681991i	0.940065	+	0.340996i	$0.110764\pi$
$239$			3.03783e16i			0.681991i	−0.940065	+	0.340996i	$0.889236\pi$
$240$		0			0
$241$	−3.53056e15			−0.0747695			−0.0373848	−	0.999301i	$-0.511903\pi$
$241$	−3.53056e15			−0.0747695			−0.0373848	+	0.999301i	$0.511903\pi$
$242$		0			0
$243$	−4.89515e16	+	1.03395e16i	−0.978413	+	0.206661i
$244$		0			0
$245$			3.90465e16i			0.736918i
$246$		0			0
$247$	4.83826e16			0.862601
$248$		0			0
$249$	−1.12502e16	−	1.42017e16i	−0.189568	−	0.239301i
$250$		0			0
$251$		−	1.00472e17i		−	1.60077i	−0.599488	−	0.800384i	$-0.704629\pi$
$251$		−	1.00472e17i		−	1.60077i	0.599488	−	0.800384i	$-0.295371\pi$
$252$		0			0
$253$	8.32351e16			1.25448
$254$		0			0
$255$	−9.60868e16	+	7.61175e16i	−1.37051	+	1.08568i
$256$		0			0
$257$		−	1.63242e16i		−	0.220446i	−0.993907	−	0.110223i	$-0.964844\pi$
$257$		−	1.63242e16i		−	0.220446i	0.993907	−	0.110223i	$-0.0351564\pi$
$258$		0			0
$259$	−4.37440e16			−0.559526
$260$		0			0
$261$	7.99146e16	+	1.87869e16i	0.968596	+	0.227705i
$262$		0			0
$263$			2.15956e16i			0.248128i	0.992274	+	0.124064i	$0.0395928\pi$
$263$			2.15956e16i			0.248128i	−0.992274	+	0.124064i	$0.960407\pi$
$264$		0			0
$265$	−1.66227e17			−1.81125
$266$		0			0
$267$	−7.64100e16	−	9.64561e16i	−0.789898	−	0.997128i
$268$		0			0
$269$		−	6.66206e16i		−	0.653646i	−0.945086	−	0.326823i	$-0.894022\pi$
$269$		−	6.66206e16i		−	0.653646i	0.945086	−	0.326823i	$-0.105978\pi$
$270$		0			0
$271$	−5.39247e16			−0.502346			−0.251173	−	0.967942i	$-0.580816\pi$
$271$	−5.39247e16			−0.502346			−0.251173	+	0.967942i	$0.580816\pi$
$272$		0			0
$273$	3.00373e16	−	2.37948e16i	0.265780	−	0.210544i
$274$		0			0
$275$		−	6.51681e16i		−	0.547906i
$276$		0			0
$277$	1.00496e16			0.0803136			0.0401568	−	0.999193i	$-0.487214\pi$
$277$	1.00496e16			0.0803136			0.0401568	+	0.999193i	$0.487214\pi$
$278$		0			0
$279$	3.72457e16	−	1.58433e17i	0.283040	−	1.20397i
$280$		0			0
$281$			5.32956e16i			0.385254i	0.981272	+	0.192627i	$0.0617008\pi$
$281$			5.32956e16i			0.385254i	−0.981272	+	0.192627i	$0.938299\pi$
$282$		0			0
$283$	5.16678e16			0.355398			0.177699	−	0.984085i	$-0.443135\pi$
$283$	5.16678e16			0.355398			0.177699	+	0.984085i	$0.443135\pi$
$284$		0			0
$285$	1.92041e17	+	2.42422e17i	1.25742	+	1.58730i
$286$		0			0
$287$			1.11556e17i			0.695536i
$288$		0			0
$289$	−1.63391e17			−0.970385
$290$		0			0
$291$	−2.30111e15	+	1.82288e15i	−0.0130223	+	0.0103159i
$292$		0			0
$293$			2.53787e17i			1.36898i	0.729023	+	0.684489i	$0.239975\pi$
$293$			2.53787e17i			1.36898i	−0.729023	+	0.684489i	$0.760025\pi$
$294$		0			0
$295$	1.74782e17			0.898966
$296$		0			0
$297$	1.83314e17	−	8.60893e16i	0.899292	−	0.422332i
$298$		0			0
$299$			1.43134e17i			0.669952i
$300$		0			0
$301$	−7.81680e16			−0.349191
$302$		0			0
$303$	1.84814e17	+	2.33300e17i	0.788202	+	0.994986i
$304$		0			0
$305$		−	5.34222e17i		−	2.17582i
$306$		0			0
$307$	2.42813e16			0.0944721			0.0472360	−	0.998884i	$-0.484959\pi$
$307$	2.42813e16			0.0944721			0.0472360	+	0.998884i	$0.484959\pi$
$308$		0			0
$309$	−1.36059e16	+	1.07782e16i	−0.0505845	+	0.0400717i
$310$		0			0
$311$		−	2.02491e17i		−	0.719587i	−0.933032	−	0.359794i	$-0.882847\pi$
$311$		−	2.02491e17i		−	0.719587i	0.933032	−	0.359794i	$-0.117153\pi$
$312$		0			0
$313$	−3.18003e17			−1.08049			−0.540245	−	0.841508i	$-0.681669\pi$
$313$	−3.18003e17			−1.08049			−0.540245	+	0.841508i	$0.681669\pi$
$314$		0			0
$315$	2.38449e17	+	5.60564e16i	0.774857	+	0.182159i
$316$		0			0
$317$		−	4.03946e15i		−	0.0125576i	−0.999980	−	0.00627882i	$-0.998001\pi$
$317$		−	4.03946e15i		−	0.0125576i	0.999980	−	0.00627882i	$-0.00199862\pi$
$318$		0			0
$319$	−3.32305e17			−0.988557
$320$		0			0
$321$	−3.94043e17	−	4.97420e17i	−1.12204	−	1.41641i
$322$		0			0
$323$			8.37037e17i			2.28206i
$324$		0			0
$325$	1.12065e17			0.292608
$326$		0			0
$327$	−3.08943e17	+	2.44737e17i	−0.772760	+	0.612160i
$328$		0			0
$329$		−	2.32031e17i		−	0.556128i
$330$		0			0
$331$	4.05257e17			0.930967			0.465484	−	0.885056i	$-0.345880\pi$
$331$	4.05257e17			0.930967			0.465484	+	0.885056i	$0.345880\pi$
$332$		0			0
$333$	9.09807e16	−	3.87007e17i	0.200373	−	0.852334i
$334$		0			0
$335$		−	9.90931e17i		−	2.09281i
$336$		0			0
$337$	−5.12082e17			−1.03736			−0.518681	−	0.854968i	$-0.673577\pi$
$337$	−5.12082e17			−1.03736			−0.518681	+	0.854968i	$0.673577\pi$
$338$		0			0
$339$	−2.37959e17	−	3.00388e17i	−0.462492	−	0.583827i
$340$		0			0
$341$			6.58805e17i			1.22879i
$342$		0			0
$343$	5.68106e17			1.01712
$344$		0			0
$345$	−7.17175e17	+	5.68128e17i	−1.23280	+	0.976590i
$346$		0			0
$347$		−	2.12481e17i		−	0.350764i	−0.984500	−	0.175382i	$-0.943884\pi$
$347$		−	2.12481e17i		−	0.350764i	0.984500	−	0.175382i	$-0.0561160\pi$
$348$		0			0
$349$	5.51417e17			0.874384			0.437192	−	0.899368i	$-0.355973\pi$
$349$	5.51417e17			0.874384			0.437192	+	0.899368i	$0.355973\pi$
$350$		0			0
$351$	1.48042e17	+	3.15232e17i	0.225546	+	0.480265i
$352$		0			0
$353$		−	2.65610e17i		−	0.388886i	−0.980914	−	0.194443i	$-0.937710\pi$
$353$		−	2.65610e17i		−	0.388886i	0.980914	−	0.194443i	$-0.0622899\pi$
$354$		0			0
$355$	1.57108e17			0.221105
$356$		0			0
$357$	4.11659e17	+	5.19657e17i	0.557006	+	0.703136i
$358$		0			0
$359$			3.98246e17i			0.518192i	0.965852	+	0.259096i	$0.0834246\pi$
$359$			3.98246e17i			0.518192i	−0.965852	+	0.259096i	$0.916575\pi$
$360$		0			0
$361$	1.31279e18			1.64303
$362$		0			0
$363$	8.40469e15	−	6.65797e15i	0.0101199	−	0.00801670i
$364$		0			0
$365$			7.87122e17i			0.911994i
$366$		0			0
$367$	2.22675e17			0.248319			0.124159	−	0.992262i	$-0.460377\pi$
$367$	2.22675e17			0.248319			0.124159	+	0.992262i	$0.460377\pi$
$368$		0			0
$369$	−9.86946e17	−	2.32019e17i	−1.05952	−	0.249080i
$370$		0			0
$371$			8.98987e17i			0.929258i
$372$		0			0
$373$	1.36397e18			1.35782			0.678912	−	0.734220i	$-0.262452\pi$
$373$	1.36397e18			1.35782			0.678912	+	0.734220i	$0.262452\pi$
$374$		0			0
$375$	−3.61768e17	−	4.56678e17i	−0.346906	−	0.437916i
$376$		0			0
$377$		−	5.71441e17i		−	0.527937i
$378$		0			0
$379$	−1.33205e18			−1.18589			−0.592947	−	0.805241i	$-0.702036\pi$
$379$	−1.33205e18			−1.18589			−0.592947	+	0.805241i	$0.702036\pi$
$380$		0			0
$381$	1.99576e17	−	1.58099e17i	0.171251	−	0.135661i
$382$		0			0
$383$		−	6.72019e17i		−	0.555892i	−0.960597	−	0.277946i	$-0.910346\pi$
$383$		−	6.72019e17i		−	0.555892i	0.960597	−	0.277946i	$-0.0896536\pi$
$384$		0			0
$385$	−9.91530e17			−0.790826
$386$		0			0
$387$	1.62577e17	−	6.91560e17i	0.125050	−	0.531928i
$388$		0			0
$389$		−	8.09758e17i		−	0.600770i	−0.953818	−	0.300385i	$-0.902885\pi$
$389$		−	8.09758e17i		−	0.600770i	0.953818	−	0.300385i	$-0.0971152\pi$
$390$		0			0
$391$	−2.47627e18			−1.77239
$392$		0			0
$393$	7.56786e16	+	9.55328e16i	0.0522667	+	0.0659788i
$394$		0			0
$395$		−	1.29376e18i		−	0.862334i
$396$		0			0
$397$	1.41709e18			0.911725			0.455862	−	0.890050i	$-0.349331\pi$
$397$	1.41709e18			0.911725			0.455862	+	0.890050i	$0.349331\pi$
$398$		0			0
$399$	1.31107e18	−	1.03859e18i	0.814358	−	0.645113i
$400$		0			0
$401$			6.68770e17i			0.401112i	0.979682	+	0.200556i	$0.0642749\pi$
$401$			6.68770e17i			0.401112i	−0.979682	+	0.200556i	$0.935725\pi$
$402$		0			0
$403$	−1.13290e18			−0.656231
$404$		0			0
$405$	−9.91872e17	+	1.99299e18i	−0.554971	+	1.11512i
$406$		0			0
$407$			1.60927e18i			0.869899i
$408$		0			0
$409$	−2.96473e18			−1.54854			−0.774269	−	0.632856i	$-0.781882\pi$
$409$	−2.96473e18			−1.54854			−0.774269	+	0.632856i	$0.781882\pi$
$410$		0			0
$411$	1.33595e18	+	1.68643e18i	0.674366	+	0.851286i
$412$		0			0
$413$		−	9.45258e17i		−	0.461211i
$414$		0			0
$415$	−8.06157e17			−0.380262
$416$		0			0
$417$	1.77575e18	−	1.40670e18i	0.809896	−	0.641578i
$418$		0			0
$419$		−	3.46840e18i		−	1.52979i	−0.644156	−	0.764894i	$-0.722791\pi$
$419$		−	3.46840e18i		−	1.52979i	0.644156	−	0.764894i	$-0.277209\pi$
$420$		0			0
$421$	3.98720e18			1.70096			0.850479	−	0.526009i	$-0.176312\pi$
$421$	3.98720e18			1.70096			0.850479	+	0.526009i	$0.176312\pi$
$422$		0			0
$423$	2.05280e18	+	4.82588e17i	0.847157	+	0.199156i
$424$		0			0
$425$			1.93877e18i			0.774112i
$426$		0			0
$427$	−2.88918e18			−1.11630
$428$		0			0
$429$	−8.75372e17	−	1.10503e18i	−0.327334	−	0.413210i
$430$		0			0
$431$			2.47761e18i			0.896791i	0.893835	+	0.448396i	$0.148004\pi$
$431$			2.47761e18i			0.896791i	−0.893835	+	0.448396i	$0.851996\pi$
$432$		0			0
$433$	4.29385e18			1.50464			0.752318	−	0.658800i	$-0.228936\pi$
$433$	4.29385e18			1.50464			0.752318	+	0.658800i	$0.228936\pi$
$434$		0			0
$435$	2.86322e18	−	2.26817e18i	0.971473	−	0.769575i
$436$		0			0
$437$			6.24750e18i			2.05275i
$438$		0			0
$439$	−1.32175e18			−0.420627			−0.210313	−	0.977634i	$-0.567448\pi$
$439$	−1.32175e18			−0.420627			−0.210313	+	0.977634i	$0.567448\pi$
$440$		0			0
$441$	−4.39203e17	+	1.86825e18i	−0.135393	+	0.575924i
$442$		0			0
$443$			1.78110e18i			0.531938i	0.963982	+	0.265969i	$0.0856919\pi$
$443$			1.78110e18i			0.531938i	−0.963982	+	0.265969i	$0.914308\pi$
$444$		0			0
$445$	−5.47532e18			−1.58449
$446$		0			0
$447$	1.45407e18	+	1.83554e18i	0.407785	+	0.514767i
$448$		0			0
$449$		−	5.41979e18i		−	1.47319i	−0.676335	−	0.736594i	$-0.736433\pi$
$449$		−	5.41979e18i		−	1.47319i	0.676335	−	0.736594i	$-0.263567\pi$
$450$		0			0
$451$	4.10397e18			1.08135
$452$		0			0
$453$	6.69991e16	−	5.30749e16i	0.0171152	−	0.0135582i
$454$		0			0
$455$		−	1.70507e18i		−	0.422339i
$456$		0			0
$457$	1.89327e18			0.454777			0.227389	−	0.973804i	$-0.426981\pi$
$457$	1.89327e18			0.454777			0.227389	+	0.973804i	$0.426981\pi$
$458$		0			0
$459$	−5.45364e18	+	2.56118e18i	−1.27057	+	0.596693i
$460$		0			0
$461$		−	6.44977e18i		−	1.45760i	−0.684728	−	0.728798i	$-0.740079\pi$
$461$		−	6.44977e18i		−	1.45760i	0.684728	−	0.728798i	$-0.259921\pi$
$462$		0			0
$463$	−7.30354e17			−0.160128			−0.0800638	−	0.996790i	$-0.525512\pi$
$463$	−7.30354e17			−0.160128			−0.0800638	+	0.996790i	$0.525512\pi$
$464$		0			0
$465$	−4.49673e18	−	5.67644e18i	−0.956590	−	1.20755i
$466$		0			0
$467$		−	4.57416e18i		−	0.944262i	−0.881528	−	0.472131i	$-0.843485\pi$
$467$		−	4.57416e18i		−	0.944262i	0.881528	−	0.472131i	$-0.156515\pi$
$468$		0			0
$469$	−5.35916e18			−1.07371
$470$		0			0
$471$	−7.88887e17	+	6.24936e17i	−0.153415	+	0.121532i
$472$		0			0
$473$			2.87568e18i			0.542891i
$474$		0			0
$475$	4.89142e18			0.896560
$476$		0			0
$477$	−7.95343e18	−	1.86975e18i	−1.41555	−	0.332779i
$478$		0			0
$479$			2.97052e18i			0.513434i	0.966487	+	0.256717i	$0.0826408\pi$
$479$			2.97052e18i			0.513434i	−0.966487	+	0.256717i	$0.917359\pi$
$480$		0			0
$481$	−2.76736e18			−0.464568
$482$		0			0
$483$	3.07255e18	+	3.87863e18i	0.501036	+	0.632483i
$484$		0			0
$485$			1.30622e17i			0.0206931i
$486$		0			0
$487$	9.81081e18			1.51009			0.755044	−	0.655674i	$-0.227616\pi$
$487$	9.81081e18			1.51009			0.755044	+	0.655674i	$0.227616\pi$
$488$		0			0
$489$	−2.05916e18	+	1.63121e18i	−0.307984	+	0.243977i
$490$		0			0
$491$			5.92929e18i			0.861852i	0.902387	+	0.430926i	$0.141813\pi$
$491$			5.92929e18i			0.861852i	−0.902387	+	0.430926i	$0.858187\pi$
$492$		0			0
$493$	9.88616e18			1.39669
$494$		0			0
$495$	2.06223e18	−	8.77216e18i	0.283204	−	1.20468i
$496$		0			0
$497$		−	8.49671e17i		−	0.113437i
$498$		0			0
$499$	−9.92074e18			−1.28778			−0.643888	−	0.765120i	$-0.722680\pi$
$499$	−9.92074e18			−1.28778			−0.643888	+	0.765120i	$0.722680\pi$
$500$		0			0
$501$	2.45550e17	+	3.09969e17i	0.0309938	+	0.0391250i
$502$		0			0
$503$			1.52007e19i			1.86590i	0.360003	+	0.932951i	$0.382776\pi$
$503$			1.52007e19i			1.86590i	−0.360003	+	0.932951i	$0.617224\pi$
$504$		0			0
$505$	1.32433e19			1.58109
$506$		0			0
$507$	−4.84955e18	+	3.84169e18i	−0.563178	+	0.446135i
$508$		0			0
$509$		−	1.57343e19i		−	1.77756i	−0.458334	−	0.888780i	$-0.651554\pi$
$509$		−	1.57343e19i		−	1.77756i	0.458334	−	0.888780i	$-0.348446\pi$
$510$		0			0
$511$	4.25692e18			0.467896
$512$		0			0
$513$	6.46173e18	+	1.37593e19i	0.691078	+	1.47155i
$514$		0			0
$515$			7.72336e17i			0.0803815i
$516$		0			0
$517$	−8.53605e18			−0.864616
$518$		0			0
$519$	−6.82756e18	−	8.61877e18i	−0.673123	−	0.849716i
$520$		0			0
$521$			1.11020e19i			1.06546i	0.846285	+	0.532731i	$0.178834\pi$
$521$			1.11020e19i			1.06546i	−0.846285	+	0.532731i	$0.821166\pi$
$522$		0			0
$523$	1.83229e19			1.71192			0.855958	−	0.517045i	$-0.172968\pi$
$523$	1.83229e19			1.71192			0.855958	+	0.517045i	$0.172968\pi$
$524$		0			0
$525$	3.03674e18	−	2.40562e18i	0.276243	−	0.218833i
$526$		0			0
$527$		−	1.95996e19i		−	1.73610i
$528$		0			0
$529$	−6.88959e18			−0.594297
$530$		0			0
$531$	8.36279e18	+	1.96599e18i	0.702570	+	0.165165i
$532$		0			0
$533$			7.05731e18i			0.577495i
$534$		0			0
$535$	−2.82360e19			−2.25075
$536$		0			0
$537$	9.32097e18	+	1.17663e19i	0.723837	+	0.913735i
$538$		0			0
$539$		−	7.76866e18i		−	0.587793i
$540$		0			0
$541$	−1.26656e19			−0.933781			−0.466891	−	0.884315i	$-0.654626\pi$
$541$	−1.26656e19			−0.933781			−0.466891	+	0.884315i	$0.654626\pi$
$542$		0			0
$543$	−5.83777e18	+	4.62453e18i	−0.419419	+	0.332253i
$544$		0			0
$545$			1.75371e19i			1.22796i
$546$		0			0
$547$	−7.84853e18			−0.535645			−0.267823	−	0.963468i	$-0.586304\pi$
$547$	−7.84853e18			−0.535645			−0.267823	+	0.963468i	$0.586304\pi$
$548$		0			0
$549$	6.00905e18	−	2.55609e19i	0.399760	−	1.70047i
$550$		0			0
$551$		−	2.49423e19i		−	1.61761i
$552$		0			0
$553$	−6.99694e18			−0.442418
$554$		0			0
$555$	−1.09842e19	−	1.38659e19i	−0.677202	−	0.854865i
$556$		0			0
$557$		−	2.28579e18i		−	0.137420i	−0.997637	−	0.0687100i	$-0.978112\pi$
$557$		−	2.28579e18i		−	0.137420i	0.997637	−	0.0687100i	$-0.0218883\pi$
$558$		0			0
$559$	−4.94510e18			−0.289930
$560$		0			0
$561$	1.91174e19	−	1.51443e19i	1.09317	−	0.865981i
$562$		0			0
$563$			1.35717e19i			0.756965i	0.925608	+	0.378483i	$0.123554\pi$
$563$			1.35717e19i			0.756965i	−0.925608	+	0.378483i	$0.876446\pi$
$564$		0			0
$565$	−1.70515e19			−0.927732
$566$		0			0
$567$	1.07785e19	+	5.36425e18i	0.572107	+	0.284726i
$568$		0			0
$569$		−	9.42699e18i		−	0.488188i	−0.969752	−	0.244094i	$-0.921509\pi$
$569$		−	9.42699e18i		−	0.488188i	0.969752	−	0.244094i	$-0.0784905\pi$
$570$		0			0
$571$	−8.52277e18			−0.430654			−0.215327	−	0.976542i	$-0.569082\pi$
$571$	−8.52277e18			−0.430654			−0.215327	+	0.976542i	$0.569082\pi$
$572$		0			0
$573$	1.96851e19	+	2.48494e19i	0.970632	+	1.22528i
$574$		0			0
$575$			1.44706e19i			0.696326i
$576$		0			0
$577$	3.59609e19			1.68889			0.844443	−	0.535645i	$-0.179932\pi$
$577$	3.59609e19			1.68889			0.844443	+	0.535645i	$0.179932\pi$
$578$		0			0
$579$	−3.07497e19	+	2.43591e19i	−1.40959	+	1.11664i
$580$		0			0
$581$			4.35986e18i			0.195092i
$582$		0			0
$583$	3.30723e19			1.44472
$584$		0			0
$585$	1.50849e19	+	3.54627e18i	0.643355	+	0.151245i
$586$		0			0
$587$			1.07859e19i			0.449147i	0.974457	+	0.224574i	$0.0720990\pi$
$587$			1.07859e19i			0.449147i	−0.974457	+	0.224574i	$0.927901\pi$
$588$		0			0
$589$	−4.94489e19			−2.01071
$590$		0			0
$591$	1.31578e19	+	1.66097e19i	0.522482	+	0.659555i
$592$		0			0
$593$			3.05082e18i			0.118313i	0.998249	+	0.0591567i	$0.0188412\pi$
$593$			3.05082e18i			0.118313i	−0.998249	+	0.0591567i	$0.981159\pi$
$594$		0			0
$595$	2.94983e19			1.11732
$596$		0			0
$597$	8.68288e17	−	6.87835e17i	0.0321250	−	0.0254486i
$598$		0			0
$599$		−	1.56052e19i		−	0.564004i	−0.959414	−	0.282002i	$-0.909002\pi$
$599$		−	1.56052e19i		−	0.564004i	0.959414	−	0.282002i	$-0.0909985\pi$
$600$		0			0
$601$	3.79279e19			1.33918			0.669588	−	0.742733i	$-0.266471\pi$
$601$	3.79279e19			1.33918			0.669588	+	0.742733i	$0.266471\pi$
$602$		0			0
$603$	1.11462e19	−	4.74130e19i	0.384508	−	1.63560i
$604$		0			0
$605$		−	4.77091e17i		−	0.0160810i
$606$		0			0
$607$	−2.58464e19			−0.851293			−0.425646	−	0.904890i	$-0.639953\pi$
$607$	−2.58464e19			−0.851293			−0.425646	+	0.904890i	$0.639953\pi$
$608$		0			0
$609$	−1.22667e19	−	1.54849e19i	−0.394828	−	0.498411i
$610$		0			0
$611$		−	1.46789e19i		−	0.461747i
$612$		0			0
$613$	3.83140e17			0.0117797			0.00588985	−	0.999983i	$-0.498125\pi$
$613$	3.83140e17			0.0117797			0.00588985	+	0.999983i	$0.498125\pi$
$614$		0			0
$615$	−3.53608e19	+	2.80119e19i	−1.06267	+	0.841816i
$616$		0			0
$617$			2.71591e19i			0.797845i	0.916985	+	0.398923i	$0.130616\pi$
$617$			2.71591e19i			0.797845i	−0.916985	+	0.398923i	$0.869384\pi$
$618$		0			0
$619$	3.23907e19			0.930220			0.465110	−	0.885253i	$-0.346015\pi$
$619$	3.23907e19			0.930220			0.465110	+	0.885253i	$0.346015\pi$
$620$		0			0
$621$	−4.07050e19	+	1.91162e19i	−1.14290	+	0.536735i
$622$		0			0
$623$			2.96116e19i			0.812917i
$624$		0			0
$625$	−4.64674e19			−1.24735
$626$		0			0
$627$	−3.82083e19	−	4.82322e19i	−1.00296	−	1.26609i
$628$		0			0
$629$		−	4.78763e19i		−	1.22904i
$630$		0			0
$631$	4.65653e19			1.16911			0.584557	−	0.811353i	$-0.301268\pi$
$631$	4.65653e19			1.16911			0.584557	+	0.811353i	$0.301268\pi$
$632$		0			0
$633$	−1.37762e18	+	1.09131e18i	−0.0338301	+	0.0267993i
$634$		0			0
$635$		−	1.13289e19i		−	0.272128i
$636$		0			0
$637$	1.33592e19			0.313910
$638$		0			0
$639$	7.51712e18	+	1.76718e18i	0.172801	+	0.0406233i
$640$		0			0
$641$			1.36879e19i			0.307843i	0.988083	+	0.153922i	$0.0491903\pi$
$641$			1.36879e19i			0.307843i	−0.988083	+	0.153922i	$0.950810\pi$
$642$		0			0
$643$	3.13485e18			0.0689826			0.0344913	−	0.999405i	$-0.489019\pi$
$643$	3.13485e18			0.0689826			0.0344913	+	0.999405i	$0.489019\pi$
$644$		0			0
$645$	−1.96282e19	−	2.47776e19i	−0.422631	−	0.533508i
$646$		0			0
$647$		−	1.54689e19i		−	0.325933i	−0.986632	−	0.162966i	$-0.947894\pi$
$647$		−	1.54689e19i		−	0.325933i	0.986632	−	0.162966i	$-0.0521062\pi$
$648$		0			0
$649$	−3.47746e19			−0.717049
$650$		0			0
$651$	−3.06993e19	+	2.43192e19i	−0.619530	+	0.490775i
$652$		0			0
$653$		−	7.75981e19i		−	1.53271i	−0.642419	−	0.766353i	$-0.722069\pi$
$653$		−	7.75981e19i		−	1.53271i	0.642419	−	0.766353i	$-0.277931\pi$
$654$		0			0
$655$	5.42291e18			0.104844
$656$		0			0
$657$	−8.85372e18	+	3.76614e19i	−0.167559	+	0.712752i
$658$		0			0
$659$			4.80538e19i			0.890287i	0.895459	+	0.445144i	$0.146847\pi$
$659$			4.80538e19i			0.890287i	−0.895459	+	0.445144i	$0.853153\pi$
$660$		0			0
$661$	−1.28461e18			−0.0233003			−0.0116502	−	0.999932i	$-0.503708\pi$
$661$	−1.28461e18			−0.0233003			−0.0116502	+	0.999932i	$0.503708\pi$
$662$		0			0
$663$	2.60426e19	+	3.28748e19i	0.462476	+	0.583806i
$664$		0			0
$665$		−	7.44227e19i		−	1.29406i
$666$		0			0
$667$	7.37885e19			1.25634
$668$		0			0
$669$	4.49869e19	−	3.56374e19i	0.750072	−	0.594187i
$670$		0			0
$671$			1.06289e20i			1.73552i
$672$		0			0
$673$	6.68601e18			0.106921			0.0534604	−	0.998570i	$-0.482975\pi$
$673$	6.68601e18			0.106921			0.0534604	+	0.998570i	$0.482975\pi$
$674$		0			0
$675$	1.49668e19	+	3.18696e19i	0.234425	+	0.499172i
$676$		0			0
$677$			7.53120e19i			1.15543i	0.816239	+	0.577715i	$0.196055\pi$
$677$			7.53120e19i			1.15543i	−0.816239	+	0.577715i	$0.803945\pi$
$678$		0			0
$679$	7.06431e17			0.0106165
$680$		0			0
$681$	5.62957e18	+	7.10649e18i	0.0828792	+	0.104623i
$682$		0			0
$683$			3.00235e19i			0.433028i	0.976280	+	0.216514i	$0.0694687\pi$
$683$			3.00235e19i			0.433028i	−0.976280	+	0.216514i	$0.930531\pi$
$684$		0			0
$685$	9.57300e19			1.35274
$686$		0			0
$687$	6.79741e19	−	5.38473e19i	0.941122	−	0.745532i
$688$		0			0
$689$			5.68722e19i			0.771552i
$690$		0			0
$691$	−1.98335e19			−0.263666			−0.131833	−	0.991272i	$-0.542086\pi$
$691$	−1.98335e19			−0.263666			−0.131833	+	0.991272i	$0.542086\pi$
$692$		0			0
$693$	−4.74416e19	−	1.11529e19i	−0.618055	−	0.145297i
$694$		0			0
$695$		−	1.00800e20i		−	1.28697i
$696$		0			0
$697$	−1.22094e20			−1.52780
$698$		0			0
$699$	3.54260e19	+	4.47200e19i	0.434492	+	0.548481i
$700$		0			0
$701$			1.35604e20i			1.63022i	0.579308	+	0.815109i	$0.303323\pi$
$701$			1.35604e20i			1.63022i	−0.579308	+	0.815109i	$0.696677\pi$
$702$		0			0
$703$	−1.20790e20			−1.42345
$704$		0			0
$705$	7.35488e19	−	5.82635e19i	0.849673	−	0.673089i
$706$		0			0
$707$		−	7.16223e19i		−	0.811171i
$708$		0			0
$709$	−2.65821e19			−0.295166			−0.147583	−	0.989050i	$-0.547149\pi$
$709$	−2.65821e19			−0.295166			−0.147583	+	0.989050i	$0.547149\pi$
$710$		0			0
$711$	1.45525e19	−	6.19026e19i	0.158435	−	0.673941i
$712$		0			0
$713$		−	1.46288e20i		−	1.56165i
$714$		0			0
$715$	−6.27267e19			−0.656614
$716$		0			0
$717$	4.12541e19	+	5.20771e19i	0.423480	+	0.534580i
$718$		0			0
$719$		−	1.08961e20i		−	1.09691i	−0.836182	−	0.548453i	$-0.815217\pi$
$719$		−	1.08961e20i		−	1.09691i	0.836182	−	0.548453i	$-0.184783\pi$
$720$		0			0
$721$	4.17695e18			0.0412394
$722$		0			0
$723$	−6.05239e18	+	4.79455e18i	−0.0586083	+	0.0464279i
$724$		0			0
$725$		−	5.77720e19i		−	0.548721i
$726$		0			0
$727$	−1.84427e20			−1.71824			−0.859121	−	0.511773i	$-0.828989\pi$
$727$	−1.84427e20			−1.71824			−0.859121	+	0.511773i	$0.828989\pi$
$728$		0			0
$729$	−6.98756e19	+	8.42016e19i	−0.638606	+	0.769534i
$730$		0			0
$731$		−	8.55522e19i		−	0.767025i
$732$		0			0
$733$	5.08472e19			0.447239			0.223619	−	0.974677i	$-0.428213\pi$
$733$	5.08472e19			0.447239			0.223619	+	0.974677i	$0.428213\pi$
$734$		0			0
$735$	5.30256e19	+	6.69368e19i	0.457587	+	0.577635i
$736$		0			0
$737$			1.97155e20i			1.66930i
$738$		0			0
$739$	−1.08986e20			−0.905438			−0.452719	−	0.891653i	$-0.649546\pi$
$739$	−1.08986e20			−0.905438			−0.452719	+	0.891653i	$0.649546\pi$
$740$		0			0
$741$	8.29415e19	−	6.57041e19i	0.676152	−	0.535630i
$742$		0			0
$743$			6.62880e19i			0.530289i	0.964209	+	0.265145i	$0.0854197\pi$
$743$			6.62880e19i			0.530289i	−0.964209	+	0.265145i	$0.914580\pi$
$744$		0			0
$745$	1.04194e20			0.817993
$746$		0			0
$747$	−3.85721e19	−	9.06783e18i	−0.297187	−	0.0698649i
$748$		0			0
$749$			1.52706e20i			1.15474i
$750$		0			0
$751$	−8.99078e19			−0.667295			−0.333647	−	0.942698i	$-0.608280\pi$
$751$	−8.99078e19			−0.667295			−0.333647	+	0.942698i	$0.608280\pi$
$752$		0			0
$753$	−1.36442e20	−	1.72237e20i	−0.993992	−	1.25477i
$754$		0			0
$755$		−	3.80319e18i		−	0.0271969i
$756$		0			0
$757$	2.34422e20			1.64561			0.822804	−	0.568325i	$-0.192408\pi$
$757$	2.34422e20			1.64561			0.822804	+	0.568325i	$0.192408\pi$
$758$		0			0
$759$	1.42689e20	−	1.13034e20i	0.983325	−	0.778965i
$760$		0			0
$761$		−	2.61008e20i		−	1.76588i	−0.469483	−	0.882942i	$-0.655560\pi$
$761$		−	2.61008e20i		−	1.76588i	0.469483	−	0.882942i	$-0.344440\pi$
$762$		0			0
$763$	9.48444e19			0.629999
$764$		0			0
$765$	−6.13518e19	+	2.60974e20i	−0.400126	+	1.70203i
$766$		0			0
$767$		−	5.97994e19i		−	0.382939i
$768$		0			0
$769$	2.37938e20			1.49617			0.748083	−	0.663605i	$-0.230975\pi$
$769$	2.37938e20			1.49617			0.748083	+	0.663605i	$0.230975\pi$
$770$		0			0
$771$	−2.21685e19	−	2.79844e19i	−0.136885	−	0.172797i
$772$		0			0
$773$			2.04502e20i			1.24006i	0.784580	+	0.620028i	$0.212879\pi$
$773$			2.04502e20i			1.24006i	−0.784580	+	0.620028i	$0.787121\pi$
$774$		0			0
$775$	−1.14535e20			−0.682066
$776$		0			0
$777$	−7.49897e19	+	5.94049e19i	−0.438586	+	0.347436i
$778$		0			0
$779$			3.08037e20i			1.76946i
$780$		0			0
$781$	−3.12580e19			−0.176362
$782$		0			0
$783$	1.62509e20	−	7.63188e19i	0.900629	−	0.422960i
$784$		0			0
$785$			4.47811e19i			0.243785i
$786$		0			0
$787$	2.83866e20			1.51807			0.759034	−	0.651052i	$-0.225672\pi$
$787$	2.83866e20			1.51807			0.759034	+	0.651052i	$0.225672\pi$
$788$		0			0
$789$	2.93271e19	+	3.70211e19i	0.154074	+	0.194496i
$790$		0			0
$791$			9.22178e19i			0.475970i
$792$		0			0
$793$	−1.82777e20			−0.926849
$794$		0			0
$795$	−2.84960e20	+	2.25738e20i	−1.41976	+	1.12469i
$796$		0			0
$797$		−	2.68944e20i		−	1.31660i	−0.752755	−	0.658300i	$-0.771276\pi$
$797$		−	2.68944e20i		−	1.31660i	0.752755	−	0.658300i	$-0.228724\pi$
$798$		0			0
$799$	2.53950e20			1.22158
$800$		0			0
$801$	−2.61977e20	−	6.15876e19i	−1.23833	−	0.291116i
$802$		0			0
$803$		−	1.56605e20i		−	0.727441i
$804$		0			0
$805$	2.20170e20			1.00505
$806$		0			0
$807$	−9.04715e19	−	1.14207e20i	−0.405880	−	0.512362i
$808$		0			0
$809$		−	3.75068e20i		−	1.65375i	−0.562384	−	0.826876i	$-0.690116\pi$
$809$		−	3.75068e20i		−	1.65375i	0.562384	−	0.826876i	$-0.309884\pi$
$810$		0			0
$811$	−8.03173e17			−0.00348067			−0.00174034	−	0.999998i	$-0.500554\pi$
$811$	−8.03173e17			−0.00348067			−0.00174034	+	0.999998i	$0.500554\pi$
$812$		0			0
$813$	−9.24424e19	+	7.32304e19i	−0.393765	+	0.311931i
$814$		0			0
$815$			1.16888e20i			0.489403i
$816$		0			0
$817$	−2.15844e20			−0.888353
$818$		0			0
$819$	1.91789e19	−	8.15821e19i	0.0775956	−	0.330071i
$820$		0			0
$821$		−	1.05033e20i		−	0.417757i	−0.977942	−	0.208879i	$-0.933019\pi$
$821$		−	1.05033e20i		−	0.417757i	0.977942	−	0.208879i	$-0.0669814\pi$
$822$		0			0
$823$	−4.56811e20			−1.78623			−0.893116	−	0.449826i	$-0.851486\pi$
$823$	−4.56811e20			−1.78623			−0.893116	+	0.449826i	$0.851486\pi$
$824$		0			0
$825$	−8.84991e19	−	1.11717e20i	−0.340221	−	0.429478i
$826$		0			0
$827$		−	2.92556e20i		−	1.10578i	−0.833253	−	0.552892i	$-0.813524\pi$
$827$		−	2.92556e20i		−	1.10578i	0.833253	−	0.552892i	$-0.186476\pi$
$828$		0			0
$829$	1.04881e20			0.389774			0.194887	−	0.980826i	$-0.437566\pi$
$829$	1.04881e20			0.389774			0.194887	+	0.980826i	$0.437566\pi$
$830$		0			0
$831$	1.72278e19	−	1.36474e19i	0.0629540	−	0.0498705i
$832$		0			0
$833$			2.31120e20i			0.830466i
$834$		0			0
$835$	1.75954e19			0.0621718
$836$		0			0
$837$	−1.51305e20	−	3.22180e20i	−0.525743	−	1.11949i
$838$		0			0
$839$			5.20635e20i			1.77910i	0.456840	+	0.889549i	$0.348981\pi$
$839$			5.20635e20i			1.77910i	−0.456840	+	0.889549i	$0.651019\pi$
$840$		0			0
$841$	2.96760e18			0.00997317
$842$		0			0
$843$	7.23761e19	+	9.13639e19i	0.239223	+	0.301982i
$844$		0			0
$845$			2.75284e20i			0.894920i
$846$		0			0
$847$	−2.58021e18			−0.00825031
$848$		0			0
$849$	8.85734e19	−	7.01655e19i	0.278580	−	0.220683i
$850$		0			0
$851$		−	3.57340e20i		−	1.10554i
$852$		0			0
$853$	−5.25898e19			−0.160051			−0.0800255	−	0.996793i	$-0.525500\pi$
$853$	−5.25898e19			−0.160051			−0.0800255	+	0.996793i	$0.525500\pi$
$854$		0			0
$855$	6.58424e20	+	1.54787e20i	1.97126	+	0.463418i
$856$		0			0
$857$			2.16570e20i			0.637870i	0.947777	+	0.318935i	$0.103325\pi$
$857$			2.16570e20i			0.637870i	−0.947777	+	0.318935i	$0.896675\pi$
$858$		0			0
$859$	6.65847e20			1.92940			0.964701	−	0.263349i	$-0.0848269\pi$
$859$	6.65847e20			1.92940			0.964701	+	0.263349i	$0.0848269\pi$
$860$		0			0
$861$	1.51494e20	+	1.91239e20i	0.431891	+	0.545197i
$862$		0			0
$863$		−	3.16647e20i		−	0.888177i	−0.895983	−	0.444088i	$-0.853528\pi$
$863$		−	3.16647e20i		−	0.888177i	0.895983	−	0.444088i	$-0.146472\pi$
$864$		0			0
$865$	−4.89244e20			−1.35024
$866$		0			0
$867$	−2.80099e20	+	2.21887e20i	−0.760638	+	0.602558i
$868$		0			0
$869$			2.57407e20i			0.687830i
$870$		0			0
$871$	−3.39034e20			−0.891489
$872$		0			0
$873$	−1.46927e18	+	6.24987e18i	−0.00380190	+	0.0161723i
$874$		0			0
$875$			1.40198e20i			0.357015i
$876$		0			0
$877$	3.00017e20			0.751881			0.375941	−	0.926644i	$-0.377320\pi$
$877$	3.00017e20			0.751881			0.375941	+	0.926644i	$0.377320\pi$
$878$		0			0
$879$	3.44646e20	+	4.35064e20i	0.850063	+	1.07308i
$880$		0			0
$881$		−	3.56485e20i		−	0.865385i	−0.901542	−	0.432692i	$-0.857564\pi$
$881$		−	3.56485e20i		−	0.865385i	0.901542	−	0.432692i	$-0.142436\pi$
$882$		0			0
$883$	−1.83193e20			−0.437707			−0.218854	−	0.975758i	$-0.570232\pi$
$883$	−1.83193e20			−0.437707			−0.218854	+	0.975758i	$0.570232\pi$
$884$		0			0
$885$	2.99627e20	−	2.37356e20i	0.704657	−	0.558210i
$886$		0			0
$887$			9.05340e19i			0.209578i	0.994494	+	0.104789i	$0.0334168\pi$
$887$			9.05340e19i			0.209578i	−0.994494	+	0.104789i	$0.966583\pi$
$888$		0			0
$889$	−6.12691e19			−0.139614
$890$		0			0
$891$	1.97342e20	−	3.96524e20i	0.442666	−	0.889459i
$892$		0			0
$893$		−	6.40703e20i		−	1.41480i
$894$		0			0
$895$	6.67913e20			1.45197
$896$		0			0
$897$	1.94377e20	+	2.45372e20i	0.416005	+	0.525143i
$898$		0			0
$899$			5.84036e20i			1.23061i
$900$		0			0
$901$	−9.83911e20			−2.04118
$902$		0			0
$903$	−1.34002e20	+	1.06153e20i	−0.273715	+	0.216830i
$904$		0			0
$905$			3.31380e20i			0.666479i
$906$		0			0
$907$	−5.72079e20			−1.13294			−0.566468	−	0.824084i	$-0.691690\pi$
$907$	−5.72079e20			−1.13294			−0.566468	+	0.824084i	$0.691690\pi$
$908$		0			0
$909$	6.33649e20	+	1.48963e20i	1.23567	+	0.290490i
$910$		0			0
$911$			1.94255e20i			0.373029i	0.982452	+	0.186515i	$0.0597192\pi$
$911$			1.94255e20i			0.373029i	−0.982452	+	0.186515i	$0.940281\pi$
$912$		0			0
$913$	1.60392e20			0.303311
$914$		0			0
$915$	−7.25480e20	−	9.15809e20i	−1.35107	−	1.70552i
$916$		0			0
$917$		−	2.93282e19i		−	0.0537898i
$918$		0			0
$919$	−8.80650e20			−1.59072			−0.795362	−	0.606135i	$-0.792719\pi$
$919$	−8.80650e20			−1.59072			−0.795362	+	0.606135i	$0.792719\pi$
$920$		0			0
$921$	4.16251e19	−	3.29743e19i	0.0740521	−	0.0586622i
$922$		0			0
$923$		−	5.37523e19i		−	0.0941857i
$924$		0			0
$925$	−2.79776e20			−0.482857
$926$		0			0
$927$	−8.68741e18	+	3.69539e19i	−0.0147683	+	0.0628206i
$928$		0			0
$929$		−	2.78100e20i		−	0.465684i	−0.972515	−	0.232842i	$-0.925197\pi$
$929$		−	2.78100e20i		−	0.465684i	0.972515	−	0.232842i	$-0.0748025\pi$
$930$		0			0
$931$	5.83103e20			0.961829
$932$		0			0
$933$	−2.74985e20	−	3.47128e20i	−0.446826	−	0.564050i
$934$		0			0
$935$		−	1.08520e21i		−	1.73711i
$936$		0			0
$937$	3.96119e20			0.624667			0.312333	−	0.949973i	$-0.398889\pi$
$937$	3.96119e20			0.624667			0.312333	+	0.949973i	$0.398889\pi$
$938$		0			0
$939$	−5.45148e20	+	4.31852e20i	−0.846945	+	0.670928i
$940$		0			0
$941$		−	1.88721e19i		−	0.0288863i	−0.999896	−	0.0144432i	$-0.995402\pi$
$941$		−	1.88721e19i		−	0.0288863i	0.999896	−	0.0144432i	$-0.00459756\pi$
$942$		0			0
$943$	−9.11289e20			−1.37428
$944$		0			0
$945$	4.84895e20	−	2.27720e20i	0.720485	−	0.338359i
$946$		0			0
$947$		−	1.74408e20i		−	0.255339i	−0.991817	−	0.127670i	$-0.959250\pi$
$947$		−	1.74408e20i		−	0.255339i	0.991817	−	0.127670i	$-0.0407497\pi$
$948$		0			0
$949$	2.69303e20			0.388488
$950$		0			0
$951$	−5.48563e18	−	6.92478e18i	−0.00779764	−	0.00984334i
$952$		0			0
$953$			1.01548e20i			0.142240i	0.997468	+	0.0711202i	$0.0226574\pi$
$953$			1.01548e20i			0.142240i	−0.997468	+	0.0711202i	$0.977343\pi$
$954$		0			0
$955$	1.41057e21			1.94703
$956$		0			0
$957$	−5.69665e20	+	4.51274e20i	−0.774883	+	0.613842i
$958$		0			0
$959$		−	5.17727e20i		−	0.694018i
$960$		0			0
$961$	4.00927e20			0.529666
$962$		0			0
$963$	−1.35100e21	−	3.17604e20i	−1.75903	−	0.413526i
$964$		0			0
$965$			1.74551e21i			2.23991i
$966$		0			0
$967$	8.06676e20			1.02027			0.510133	−	0.860095i	$-0.329596\pi$
$967$	8.06676e20			1.02027			0.510133	+	0.860095i	$0.329596\pi$
$968$		0			0
$969$	1.13671e21	+	1.43492e21i	1.41704	+	1.78880i
$970$		0			0
$971$			1.36954e21i			1.68283i	0.540387	+	0.841417i	$0.318278\pi$
$971$			1.36954e21i			1.68283i	−0.540387	+	0.841417i	$0.681722\pi$
$972$		0			0
$973$	−5.45148e20			−0.660274
$974$		0			0
$975$	1.92112e20	−	1.52186e20i	0.229362	−	0.181694i
$976$		0			0
$977$			7.87163e19i			0.0926409i	0.998927	+	0.0463204i	$0.0147495\pi$
$977$			7.87163e19i			0.0926409i	−0.998927	+	0.0463204i	$0.985250\pi$
$978$		0			0
$979$	1.08937e21			1.26385
$980$		0			0
$981$	−1.97261e20	+	8.39097e20i	−0.225610	+	0.959686i
$982$		0			0
$983$		−	1.43475e21i		−	1.61771i	−0.588009	−	0.808854i	$-0.700088\pi$
$983$		−	1.43475e21i		−	1.61771i	0.588009	−	0.808854i	$-0.299912\pi$
$984$		0			0
$985$	9.42851e20			1.04807
$986$		0			0
$987$	−3.15101e20	−	3.97767e20i	−0.345326	−	0.435922i
$988$		0			0
$989$		−	6.38547e20i		−	0.689952i
$990$		0			0
$991$	−7.37056e20			−0.785208			−0.392604	−	0.919708i	$-0.628426\pi$
$991$	−7.37056e20			−0.785208			−0.392604	+	0.919708i	$0.628426\pi$
$992$		0			0
$993$	6.94726e20	−	5.50344e20i	0.729741	−	0.578082i
$994$		0			0
$995$		−	4.92883e19i		−	0.0510484i
$996$		0			0
$997$	7.34171e20			0.749776			0.374888	−	0.927070i	$-0.377681\pi$
$997$	7.34171e20			0.749776			0.374888	+	0.927070i	$0.377681\pi$
$998$		0			0
$999$	−3.69594e20	−	7.86994e20i	−0.372191	−	0.792525i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.15.c.b.5.3	✓	4
3.2	odd	2	inner	12.15.c.b.5.4	yes	4
4.3	odd	2		48.15.e.c.17.2		4
12.11	even	2		48.15.e.c.17.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.15.c.b.5.3	✓	4	1.1	even	1	trivial
12.15.c.b.5.4	yes	4	3.2	odd	2	inner
48.15.e.c.17.1		4	12.11	even	2
48.15.e.c.17.2		4	4.3	odd	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:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	12.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1714.29 - 1358.01i) q^{3} -97311.2i q^{5} -526279. q^{7} +(1.09458e6 - 4.65604e6i) q^{9} +O(q^{10})\)	\(q+(1714.29 - 1358.01i) q^{3} -97311.2i q^{5} -526279. q^{7} +(1.09458e6 - 4.65604e6i) q^{9} +1.93610e7i q^{11} -3.32937e7 q^{13} +(-1.32150e8 - 1.66819e8i) q^{15} -5.75994e8i q^{17} -1.45320e9 q^{19} +(-9.02192e8 + 7.14693e8i) q^{21} -4.29912e9i q^{23} -3.36595e9 q^{25} +(-4.44654e9 - 9.46823e9i) q^{27} +1.71636e10i q^{29} +3.40275e10 q^{31} +(2.62924e10 + 3.31902e10i) q^{33} +5.12128e10i q^{35} +8.31195e10 q^{37} +(-5.70749e10 + 4.52133e10i) q^{39} -2.11971e11i q^{41} +1.48530e11 q^{43} +(-4.53085e11 - 1.06515e11i) q^{45} +4.40890e11i q^{47} -4.01254e11 q^{49} +(-7.82207e11 - 9.87418e11i) q^{51} -1.70820e12i q^{53} +1.88404e12 q^{55} +(-2.49121e12 + 1.97347e12i) q^{57} +1.79612e12i q^{59} +5.48983e12 q^{61} +(-5.76053e11 + 2.45037e12i) q^{63} +3.23985e12i q^{65} +1.01831e13 q^{67} +(-5.83825e12 - 7.36992e12i) q^{69} +1.61449e12i q^{71} -8.08871e12 q^{73} +(-5.77020e12 + 4.57100e12i) q^{75} -1.01893e13i q^{77} +1.32951e13 q^{79} +(-2.04806e13 - 1.01928e13i) q^{81} -8.28432e12i q^{83} -5.60507e13 q^{85} +(2.33084e13 + 2.94234e13i) q^{87} -5.62661e13i q^{89} +1.75218e13 q^{91} +(5.83328e13 - 4.62098e13i) q^{93} +1.41413e14i q^{95} -1.34231e12 q^{97} +(9.01454e13 + 2.11921e13i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2148 q^{3} + 1709288 q^{7} - 5736924 q^{9}+O(q^{10}) \) 4 * q + 2148 * q^3 + 1709288 * q^7 - 5736924 * q^9	\( 4 q + 2148 q^{3} + 1709288 q^{7} - 5736924 q^{9} - 93505048 q^{13} + 24235200 q^{15} - 1349200696 q^{19} - 3572752344 q^{21} - 4043764700 q^{25} + 7381750212 q^{27} + 46585213736 q^{31} - 60689217600 q^{33} + 300873217064 q^{37} - 96914866776 q^{39} + 246448758152 q^{43} - 1275658243200 q^{45} + 1654942475340 q^{49} - 4102471929600 q^{51} + 7505039836800 q^{55} - 5979467701752 q^{57} + 8008332264296 q^{61} - 12097408557528 q^{63} + 38294908213448 q^{67} - 27868623868800 q^{69} + 12721406693576 q^{73} - 13261586187900 q^{75} - 29803403331928 q^{79} - 23892053776956 q^{81} - 28369656691200 q^{85} + 58743848116800 q^{87} - 2127612761456 q^{91} + 130412478704232 q^{93} - 301625619131512 q^{97} + 325346192265600 q^{99}+O(q^{100}) \) 4 * q + 2148 * q^3 + 1709288 * q^7 - 5736924 * q^9 - 93505048 * q^13 + 24235200 * q^15 - 1349200696 * q^19 - 3572752344 * q^21 - 4043764700 * q^25 + 7381750212 * q^27 + 46585213736 * q^31 - 60689217600 * q^33 + 300873217064 * q^37 - 96914866776 * q^39 + 246448758152 * q^43 - 1275658243200 * q^45 + 1654942475340 * q^49 - 4102471929600 * q^51 + 7505039836800 * q^55 - 5979467701752 * q^57 + 8008332264296 * q^61 - 12097408557528 * q^63 + 38294908213448 * q^67 - 27868623868800 * q^69 + 12721406693576 * q^73 - 13261586187900 * q^75 - 29803403331928 * q^79 - 23892053776956 * q^81 - 28369656691200 * q^85 + 58743848116800 * q^87 - 2127612761456 * q^91 + 130412478704232 * q^93 - 301625619131512 * q^97 + 325346192265600 * q^99

Introduction

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