LMFDB - Embedded newform 12.15.c.b.5.1

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			5.1
Root			$14.1555i$ of defining polynomial
Character	$\chi$	$=$	12.5
Dual form			12.15.c.b.5.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-640.285 - 2091.17i) q^{3} +68988.7i q^{5} +1.38092e6 q^{7} +(-3.96304e6 + 2.67789e6i) q^{9} +O(q^{10})$	\(q+(-640.285 - 2091.17i) q^{3} +68988.7i q^{5} +1.38092e6 q^{7} +(-3.96304e6 + 2.67789e6i) q^{9} -2.70839e7i q^{11} -1.34588e7 q^{13} +(1.44267e8 - 4.41724e7i) q^{15} -6.06850e8i q^{17} +7.78604e8 q^{19} +(-8.84184e8 - 2.88775e9i) q^{21} -3.87154e9i q^{23} +1.34407e9 q^{25} +(8.13741e9 + 6.57279e9i) q^{27} +2.89957e9i q^{29} -1.07349e10 q^{31} +(-5.66370e10 + 1.73414e10i) q^{33} +9.52681e10i q^{35} +6.73171e10 q^{37} +(8.61748e9 + 2.81447e10i) q^{39} +7.05568e10i q^{41} -2.53053e10 q^{43} +(-1.84744e11 - 2.73405e11i) q^{45} -9.91845e11i q^{47} +1.22872e12 q^{49} +(-1.26903e12 + 3.88557e11i) q^{51} -3.88078e11i q^{53} +1.86848e12 q^{55} +(-4.98528e11 - 1.62819e12i) q^{57} +4.01476e12i q^{59} -1.48567e12 q^{61} +(-5.47265e12 + 3.69796e12i) q^{63} -9.28506e11i q^{65} +8.96434e12 q^{67} +(-8.09606e12 + 2.47889e12i) q^{69} +1.00451e13i q^{71} +1.44494e13 q^{73} +(-8.60589e11 - 2.81069e12i) q^{75} -3.74007e13i q^{77} -2.81968e13 q^{79} +(8.53457e12 - 2.12252e13i) q^{81} +1.17334e13i q^{83} +4.18658e13 q^{85} +(6.06350e12 - 1.85655e12i) q^{87} +1.95724e12i q^{89} -1.85856e13 q^{91} +(6.87340e12 + 2.24485e13i) q^{93} +5.37149e13i q^{95} -1.49470e14 q^{97} +(7.25277e13 + 1.07334e14i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−640.285	−	2091.17i	−0.292769	−	0.956183i
$3$	−640.285	−	2091.17i	−0.292769	−	0.956183i
$4$		0			0
$5$			68988.7i			0.883056i	0.897248	+	0.441528i	$0.145563\pi$
$5$			68988.7i			0.883056i	−0.897248	+	0.441528i	$0.854437\pi$
$6$		0			0
$7$	1.38092e6			1.67681			0.838404	−	0.545050i	$-0.183489\pi$
$7$	1.38092e6			1.67681			0.838404	+	0.545050i	$0.183489\pi$
$8$		0			0
$9$	−3.96304e6	+	2.67789e6i	−0.828573	+	0.559881i
$10$		0			0
$11$		−	2.70839e7i		−	1.38983i	−0.719092	−	0.694915i	$-0.755442\pi$
$11$		−	2.70839e7i		−	1.38983i	0.719092	−	0.694915i	$-0.244558\pi$
$12$		0			0
$13$	−1.34588e7			−0.214488			−0.107244	−	0.994233i	$-0.534203\pi$
$13$	−1.34588e7			−0.214488			−0.107244	+	0.994233i	$0.534203\pi$
$14$		0			0
$15$	1.44267e8	−	4.41724e7i	0.844363	−	0.258531i
$16$		0			0
$17$		−	6.06850e8i		−	1.47890i	−0.673211	−	0.739451i	$-0.735085\pi$
$17$		−	6.06850e8i		−	1.47890i	0.673211	−	0.739451i	$-0.264915\pi$
$18$		0			0
$19$	7.78604e8			0.871046			0.435523	−	0.900178i	$-0.356563\pi$
$19$	7.78604e8			0.871046			0.435523	+	0.900178i	$0.356563\pi$
$20$		0			0
$21$	−8.84184e8	−	2.88775e9i	−0.490917	−	1.60334i
$22$		0			0
$23$		−	3.87154e9i		−	1.13707i	−0.822658	−	0.568537i	$-0.807510\pi$
$23$		−	3.87154e9i		−	1.13707i	0.822658	−	0.568537i	$-0.192490\pi$
$24$		0			0
$25$	1.34407e9			0.220213
$26$		0			0
$27$	8.13741e9	+	6.57279e9i	0.777929	+	0.628352i
$28$		0			0
$29$			2.89957e9i			0.168092i	0.996462	+	0.0840460i	$0.0267843\pi$
$29$			2.89957e9i			0.168092i	−0.996462	+	0.0840460i	$0.973216\pi$
$30$		0			0
$31$	−1.07349e10			−0.390181			−0.195091	−	0.980785i	$-0.562500\pi$
$31$	−1.07349e10			−0.390181			−0.195091	+	0.980785i	$0.562500\pi$
$32$		0			0
$33$	−5.66370e10	+	1.73414e10i	−1.32893	+	0.406899i
$34$		0			0
$35$			9.52681e10i			1.48071i
$36$		0			0
$37$	6.73171e10			0.709110			0.354555	−	0.935035i	$-0.384632\pi$
$37$	6.73171e10			0.709110			0.354555	+	0.935035i	$0.384632\pi$
$38$		0			0
$39$	8.61748e9	+	2.81447e10i	0.0627954	+	0.205090i
$40$		0			0
$41$			7.05568e10i			0.362286i	0.983457	+	0.181143i	$0.0579797\pi$
$41$			7.05568e10i			0.362286i	−0.983457	+	0.181143i	$0.942020\pi$
$42$		0			0
$43$	−2.53053e10			−0.0930962			−0.0465481	−	0.998916i	$-0.514822\pi$
$43$	−2.53053e10			−0.0930962			−0.0465481	+	0.998916i	$0.514822\pi$
$44$		0			0
$45$	−1.84744e11	−	2.73405e11i	−0.494406	−	0.731676i
$46$		0			0
$47$		−	9.91845e11i		−	1.95776i	−0.204443	−	0.978878i	$-0.565538\pi$
$47$		−	9.91845e11i		−	1.95776i	0.204443	−	0.978878i	$-0.434462\pi$
$48$		0			0
$49$	1.22872e12			1.81168
$50$		0			0
$51$	−1.26903e12	+	3.88557e11i	−1.41410	+	0.432976i
$52$		0			0
$53$		−	3.88078e11i		−	0.330360i	−0.986263	−	0.165180i	$-0.947179\pi$
$53$		−	3.88078e11i		−	0.330360i	0.986263	−	0.165180i	$-0.0528205\pi$
$54$		0			0
$55$	1.86848e12			1.22730
$56$		0			0
$57$	−4.98528e11	−	1.62819e12i	−0.255015	−	0.832880i
$58$		0			0
$59$			4.01476e12i			1.61323i	0.591079	+	0.806614i	$0.298702\pi$
$59$			4.01476e12i			1.61323i	−0.591079	+	0.806614i	$0.701298\pi$
$60$		0			0
$61$	−1.48567e12			−0.472730			−0.236365	−	0.971664i	$-0.575956\pi$
$61$	−1.48567e12			−0.472730			−0.236365	+	0.971664i	$0.575956\pi$
$62$		0			0
$63$	−5.47265e12	+	3.69796e12i	−1.38936	+	0.938812i
$64$		0			0
$65$		−	9.28506e11i		−	0.189405i
$66$		0			0
$67$	8.96434e12			1.47909			0.739545	−	0.673107i	$-0.235041\pi$
$67$	8.96434e12			1.47909			0.739545	+	0.673107i	$0.235041\pi$
$68$		0			0
$69$	−8.09606e12	+	2.47889e12i	−1.08725	+	0.332900i
$70$		0			0
$71$			1.00451e13i			1.10445i	0.833694	+	0.552227i	$0.186222\pi$
$71$			1.00451e13i			1.10445i	−0.833694	+	0.552227i	$0.813778\pi$
$72$		0			0
$73$	1.44494e13			1.30795			0.653974	−	0.756517i	$-0.273101\pi$
$73$	1.44494e13			1.30795			0.653974	+	0.756517i	$0.273101\pi$
$74$		0			0
$75$	−8.60589e11	−	2.81069e12i	−0.0644714	−	0.210564i
$76$		0			0
$77$		−	3.74007e13i		−	2.33048i
$78$		0			0
$79$	−2.81968e13			−1.46829			−0.734143	−	0.678995i	$-0.762416\pi$
$79$	−2.81968e13			−1.46829			−0.734143	+	0.678995i	$0.762416\pi$
$80$		0			0
$81$	8.53457e12	−	2.12252e13i	0.373067	−	0.927805i
$82$		0			0
$83$			1.17334e13i			0.432392i	0.976350	+	0.216196i	$0.0693649\pi$
$83$			1.17334e13i			0.432392i	−0.976350	+	0.216196i	$0.930635\pi$
$84$		0			0
$85$	4.18658e13			1.30595
$86$		0			0
$87$	6.06350e12	−	1.85655e12i	0.160727	−	0.0492121i
$88$		0			0
$89$			1.95724e12i			0.0442500i	0.999755	+	0.0221250i	$0.00704318\pi$
$89$			1.95724e12i			0.0442500i	−0.999755	+	0.0221250i	$0.992957\pi$
$90$		0			0
$91$	−1.85856e13			−0.359655
$92$		0			0
$93$	6.87340e12	+	2.24485e13i	0.114233	+	0.373085i
$94$		0			0
$95$			5.37149e13i			0.769182i
$96$		0			0
$97$	−1.49470e14			−1.84992			−0.924961	−	0.380062i	$-0.875902\pi$
$97$	−1.49470e14			−1.84992			−0.924961	+	0.380062i	$0.875902\pi$
$98$		0			0
$99$	7.25277e13	+	1.07334e14i	0.778139	+	1.15158i
$100$		0			0
$101$		−	3.42108e13i		−	0.319090i	−0.987191	−	0.159545i	$-0.948997\pi$
$101$		−	3.42108e13i		−	0.319090i	0.987191	−	0.159545i	$-0.0510027\pi$
$102$		0			0
$103$	−1.10454e14			−0.898089			−0.449044	−	0.893509i	$-0.648235\pi$
$103$	−1.10454e14			−0.898089			−0.449044	+	0.893509i	$0.648235\pi$
$104$		0			0
$105$	1.99222e14	−	6.09987e13i	1.41583	−	0.433507i
$106$		0			0
$107$			2.01954e14i			1.25767i	0.777540	+	0.628834i	$0.216467\pi$
$107$			2.01954e14i			1.25767i	−0.777540	+	0.628834i	$0.783533\pi$
$108$		0			0
$109$	−6.52716e13			−0.357058			−0.178529	−	0.983935i	$-0.557134\pi$
$109$	−6.52716e13			−0.357058			−0.178529	+	0.983935i	$0.557134\pi$
$110$		0			0
$111$	−4.31022e13	−	1.40772e14i	−0.207605	−	0.678039i
$112$		0			0
$113$		−	9.53710e13i		−	0.405385i	−0.979242	−	0.202692i	$-0.935031\pi$
$113$		−	9.53710e13i		−	0.405385i	0.979242	−	0.202692i	$-0.0649691\pi$
$114$		0			0
$115$	2.67093e14			1.00410
$116$		0			0
$117$	5.33378e13	−	3.60413e13i	0.177719	−	0.120088i
$118$		0			0
$119$		−	8.38014e14i		−	2.47983i
$120$		0			0
$121$	−3.53786e14			−0.931628
$122$		0			0
$123$	1.47546e14	−	4.51765e13i	0.346412	−	0.106066i
$124$		0			0
$125$			5.13800e14i			1.07752i
$126$		0			0
$127$	−1.75936e14			−0.330163			−0.165082	−	0.986280i	$-0.552789\pi$
$127$	−1.75936e14			−0.330163			−0.165082	+	0.986280i	$0.552789\pi$
$128$		0			0
$129$	1.62026e13	+	5.29177e13i	0.0272557	+	0.0890171i
$130$		0			0
$131$		−	1.46193e14i		−	0.220814i	−0.993886	−	0.110407i	$-0.964785\pi$
$131$		−	1.46193e14i		−	0.220814i	0.993886	−	0.110407i	$-0.0352155\pi$
$132$		0			0
$133$	1.07519e15			1.46058
$134$		0			0
$135$	−4.53448e14	+	5.61390e14i	−0.554870	+	0.686955i
$136$		0			0
$137$			1.26878e15i			1.40069i	0.713805	+	0.700344i	$0.246970\pi$
$137$			1.26878e15i			1.40069i	−0.713805	+	0.700344i	$0.753030\pi$
$138$		0			0
$139$	2.80462e14			0.279750			0.139875	−	0.990169i	$-0.455330\pi$
$139$	2.80462e14			0.279750			0.139875	+	0.990169i	$0.455330\pi$
$140$		0			0
$141$	−2.07412e15	+	6.35063e14i	−1.87197	+	0.573170i
$142$		0			0
$143$			3.64517e14i			0.298102i
$144$		0			0
$145$	−2.00037e14			−0.148435
$146$		0			0
$147$	−7.86734e14	−	2.56948e15i	−0.530404	−	1.73230i
$148$		0			0
$149$			2.24281e14i			0.137559i	0.997632	+	0.0687795i	$0.0219105\pi$
$149$			2.24281e14i			0.137559i	−0.997632	+	0.0687795i	$0.978089\pi$
$150$		0			0
$151$	2.08375e15			1.16414			0.582072	−	0.813137i	$-0.302242\pi$
$151$	2.08375e15			1.16414			0.582072	+	0.813137i	$0.302242\pi$
$152$		0			0
$153$	1.62508e15	+	2.40497e15i	0.828009	+	1.22538i
$154$		0			0
$155$		−	7.40587e14i		−	0.344552i
$156$		0			0
$157$	−9.86555e13			−0.0419589			−0.0209794	−	0.999780i	$-0.506678\pi$
$157$	−9.86555e13			−0.0419589			−0.0209794	+	0.999780i	$0.506678\pi$
$158$		0			0
$159$	−8.11537e14	+	2.48480e14i	−0.315885	+	0.0967191i
$160$		0			0
$161$		−	5.34630e15i		−	1.90665i
$162$		0			0
$163$	3.36626e15			1.10112			0.550559	−	0.834796i	$-0.314415\pi$
$163$	3.36626e15			1.10112			0.550559	+	0.834796i	$0.314415\pi$
$164$		0			0
$165$	−1.19636e15	−	3.90732e15i	−0.359314	−	1.17352i
$166$		0			0
$167$		−	1.34309e15i		−	0.370758i	−0.982667	−	0.185379i	$-0.940649\pi$
$167$		−	1.34309e15i		−	0.370758i	0.982667	−	0.185379i	$-0.0593513\pi$
$168$		0			0
$169$	−3.75624e15			−0.953995
$170$		0			0
$171$	−3.08564e15	+	2.08502e15i	−0.721725	+	0.487682i
$172$		0			0
$173$			4.49540e15i			0.969272i	0.874716	+	0.484636i	$0.161048\pi$
$173$			4.49540e15i			0.969272i	−0.874716	+	0.484636i	$0.838952\pi$
$174$		0			0
$175$	1.85606e15			0.369254
$176$		0			0
$177$	8.39556e15	−	2.57059e15i	1.54254	−	0.472302i
$178$		0			0
$179$			5.21605e15i			0.885872i	0.896553	+	0.442936i	$0.146063\pi$
$179$			5.21605e15i			0.885872i	−0.896553	+	0.442936i	$0.853937\pi$
$180$		0			0
$181$	−4.34290e15			−0.682385			−0.341193	−	0.939993i	$-0.610831\pi$
$181$	−4.34290e15			−0.682385			−0.341193	+	0.939993i	$0.610831\pi$
$182$		0			0
$183$	9.51251e14	+	3.10679e15i	0.138400	+	0.452016i
$184$		0			0
$185$			4.64412e15i			0.626184i
$186$		0			0
$187$	−1.64359e16			−2.05542
$188$		0			0
$189$	1.12371e16	+	9.07651e15i	1.30444	+	1.05363i
$190$		0			0
$191$			3.52605e15i			0.380237i	0.981761	+	0.190119i	$0.0608873\pi$
$191$			3.52605e15i			0.380237i	−0.981761	+	0.190119i	$0.939113\pi$
$192$		0			0
$193$	−2.54539e14			−0.0255184			−0.0127592	−	0.999919i	$-0.504061\pi$
$193$	−2.54539e14			−0.0255184			−0.0127592	+	0.999919i	$0.504061\pi$
$194$		0			0
$195$	−1.94167e15	+	5.94509e14i	−0.181106	+	0.0554518i
$196$		0			0
$197$		−	1.33824e16i		−	1.16217i	−0.813843	−	0.581085i	$-0.802628\pi$
$197$		−	1.33824e16i		−	1.16217i	0.813843	−	0.581085i	$-0.197372\pi$
$198$		0			0
$199$	2.45124e15			0.198342			0.0991711	−	0.995070i	$-0.468381\pi$
$199$	2.45124e15			0.198342			0.0991711	+	0.995070i	$0.468381\pi$
$200$		0			0
$201$	−5.73973e15	−	1.87460e16i	−0.433031	−	1.41428i
$202$		0			0
$203$			4.00408e15i			0.281858i
$204$		0			0
$205$	−4.86762e15			−0.319919
$206$		0			0
$207$	1.03676e16	+	1.53431e16i	0.636626	+	0.942149i
$208$		0			0
$209$		−	2.10876e16i		−	1.21061i
$210$		0			0
$211$	1.26632e16			0.680088			0.340044	−	0.940410i	$-0.389558\pi$
$211$	1.26632e16			0.680088			0.340044	+	0.940410i	$0.389558\pi$
$212$		0			0
$213$	2.10061e16	−	6.43176e15i	1.05606	−	0.323350i
$214$		0			0
$215$		−	1.74578e15i		−	0.0822091i
$216$		0			0
$217$	−1.48241e16			−0.654258
$218$		0			0
$219$	−9.25174e15	−	3.02162e16i	−0.382926	−	1.25064i
$220$		0			0
$221$			8.16749e15i			0.317207i
$222$		0			0
$223$	2.59561e16			0.946466			0.473233	−	0.880937i	$-0.343087\pi$
$223$	2.59561e16			0.946466			0.473233	+	0.880937i	$0.343087\pi$
$224$		0			0
$225$	−5.32661e15	+	3.59928e15i	−0.182462	+	0.123293i
$226$		0			0
$227$			1.07499e16i			0.346117i	0.984912	+	0.173058i	$0.0553649\pi$
$227$			1.07499e16i			0.346117i	−0.984912	+	0.173058i	$0.944635\pi$
$228$		0			0
$229$	−3.84804e16			−1.16517			−0.582587	−	0.812768i	$-0.697960\pi$
$229$	−3.84804e16			−1.16517			−0.582587	+	0.812768i	$0.697960\pi$
$230$		0			0
$231$	−7.82114e16	+	2.39471e16i	−2.22836	+	0.682291i
$232$		0			0
$233$			1.43134e16i			0.383930i	0.981402	+	0.191965i	$0.0614860\pi$
$233$			1.43134e16i			0.383930i	−0.981402	+	0.191965i	$0.938514\pi$
$234$		0			0
$235$	6.84261e16			1.72881
$236$		0			0
$237$	1.80540e16	+	5.89644e16i	0.429868	+	1.40395i
$238$		0			0
$239$		−	1.96828e16i		−	0.441877i	−0.975288	−	0.220938i	$-0.929088\pi$
$239$		−	1.96828e16i		−	0.441877i	0.975288	−	0.220938i	$-0.0709119\pi$
$240$		0			0
$241$	6.49775e16			1.37608			0.688040	−	0.725672i	$-0.258471\pi$
$241$	6.49775e16			1.37608			0.688040	+	0.725672i	$0.258471\pi$
$242$		0			0
$243$	−4.98501e16	−	4.25708e15i	−0.996373	−	0.0850880i
$244$		0			0
$245$			8.47682e16i			1.59982i
$246$		0			0
$247$	−1.04791e16			−0.186829
$248$		0			0
$249$	2.45366e16	−	7.51272e15i	0.413446	−	0.126591i
$250$		0			0
$251$			1.01654e16i			0.161961i	0.996716	+	0.0809805i	$0.0258052\pi$
$251$			1.01654e16i			0.161961i	−0.996716	+	0.0809805i	$0.974195\pi$
$252$		0			0
$253$	−1.04856e17			−1.58034
$254$		0			0
$255$	−2.68061e16	−	8.75487e16i	−0.382342	−	1.24873i
$256$		0			0
$257$		−	7.68554e16i		−	1.03787i	−0.854814	−	0.518935i	$-0.826329\pi$
$257$		−	7.68554e16i		−	1.03787i	0.854814	−	0.518935i	$-0.173671\pi$
$258$		0			0
$259$	9.29598e16			1.18904
$260$		0			0
$261$	−7.76473e15	−	1.14911e16i	−0.0941115	−	0.139277i
$262$		0			0
$263$		−	1.41566e17i		−	1.62656i	−0.581875	−	0.813278i	$-0.697681\pi$
$263$		−	1.41566e17i		−	1.62656i	0.581875	−	0.813278i	$-0.302319\pi$
$264$		0			0
$265$	2.67730e16			0.291726
$266$		0			0
$267$	4.09292e15	−	1.25319e15i	0.0423111	−	0.0129550i
$268$		0			0
$269$			9.75053e16i			0.956670i	0.878178	+	0.478335i	$0.158759\pi$
$269$			9.75053e16i			0.956670i	−0.878178	+	0.478335i	$0.841241\pi$
$270$		0			0
$271$	1.31104e17			1.22133			0.610664	−	0.791890i	$-0.290903\pi$
$271$	1.31104e17			1.22133			0.610664	+	0.791890i	$0.290903\pi$
$272$		0			0
$273$	1.19001e16	+	3.88657e16i	0.105296	+	0.343896i
$274$		0			0
$275$		−	3.64026e16i		−	0.306058i
$276$		0			0
$277$	−8.65715e16			−0.691858			−0.345929	−	0.938261i	$-0.612436\pi$
$277$	−8.65715e16			−0.691858			−0.345929	+	0.938261i	$0.612436\pi$
$278$		0			0
$279$	4.25428e16	−	2.87469e16i	0.323293	−	0.218455i
$280$		0			0
$281$		−	1.28286e17i		−	0.927335i	−0.886009	−	0.463667i	$-0.846533\pi$
$281$		−	1.28286e17i		−	0.927335i	0.886009	−	0.463667i	$-0.153467\pi$
$282$		0			0
$283$	−1.43803e17			−0.989155			−0.494577	−	0.869134i	$-0.664677\pi$
$283$	−1.43803e17			−0.989155			−0.494577	+	0.869134i	$0.664677\pi$
$284$		0			0
$285$	1.12327e17	−	3.43928e16i	0.735479	−	0.225192i
$286$		0			0
$287$			9.74335e16i			0.607484i
$288$		0			0
$289$	−1.99890e17			−1.18715
$290$		0			0
$291$	9.57037e16	+	3.12569e17i	0.541599	+	1.76886i
$292$		0			0
$293$			2.82287e17i			1.52271i	0.648336	+	0.761355i	$0.275465\pi$
$293$			2.82287e17i			1.52271i	−0.648336	+	0.761355i	$0.724535\pi$
$294$		0			0
$295$	−2.76973e17			−1.42457
$296$		0			0
$297$	1.78016e17	−	2.20393e17i	0.873303	−	1.08119i
$298$		0			0
$299$			5.21063e16i			0.243889i
$300$		0			0
$301$	−3.49446e16			−0.156104
$302$		0			0
$303$	−7.15406e16	+	2.19046e16i	−0.305109	+	0.0934196i
$304$		0			0
$305$		−	1.02494e17i		−	0.417447i
$306$		0			0
$307$	2.91381e17			1.13368			0.566842	−	0.823826i	$-0.308165\pi$
$307$	2.91381e17			1.13368			0.566842	+	0.823826i	$0.308165\pi$
$308$		0			0
$309$	7.07218e16	+	2.30978e17i	0.262932	+	0.858737i
$310$		0			0
$311$			3.39433e17i			1.20623i	0.797653	+	0.603117i	$0.206075\pi$
$311$			3.39433e17i			1.20623i	−0.797653	+	0.603117i	$0.793925\pi$
$312$		0			0
$313$	−1.00517e17			−0.341529			−0.170765	−	0.985312i	$-0.554624\pi$
$313$	−1.00517e17			−0.341529			−0.170765	+	0.985312i	$0.554624\pi$
$314$		0			0
$315$	−2.55118e17	−	3.77551e17i	−0.829024	−	1.22688i
$316$		0			0
$317$		−	3.05957e16i		−	0.0951144i	−0.998869	−	0.0475572i	$-0.984856\pi$
$317$		−	3.05957e16i		−	0.0951144i	0.998869	−	0.0475572i	$-0.0151436\pi$
$318$		0			0
$319$	7.85315e16			0.233619
$320$		0			0
$321$	4.22321e17	−	1.29308e17i	1.20256	−	0.368206i
$322$		0			0
$323$		−	4.72496e17i		−	1.28819i
$324$		0			0
$325$	−1.80896e16			−0.0472330
$326$		0			0
$327$	4.17924e16	+	1.36494e17i	0.104535	+	0.341413i
$328$		0			0
$329$		−	1.36966e18i		−	3.28278i
$330$		0			0
$331$	−3.66084e17			−0.840979			−0.420490	−	0.907297i	$-0.638142\pi$
$331$	−3.66084e17			−0.840979			−0.420490	+	0.907297i	$0.638142\pi$
$332$		0			0
$333$	−2.66781e17	+	1.80268e17i	−0.587549	+	0.397017i
$334$		0			0
$335$			6.18438e17i			1.30612i
$336$		0			0
$337$	2.55887e17			0.518368			0.259184	−	0.965828i	$-0.416546\pi$
$337$	2.55887e17			0.518368			0.259184	+	0.965828i	$0.416546\pi$
$338$		0			0
$339$	−1.99437e17	+	6.10646e16i	−0.387622	+	0.118684i
$340$		0			0
$341$			2.90742e17i			0.542285i
$342$		0			0
$343$	7.60201e17			1.36104
$344$		0			0
$345$	−1.71015e17	−	5.58537e17i	−0.293969	−	0.960103i
$346$		0			0
$347$			6.67030e17i			1.10113i	0.834792	+	0.550565i	$0.185588\pi$
$347$			6.67030e17i			1.10113i	−0.834792	+	0.550565i	$0.814412\pi$
$348$		0			0
$349$	1.23010e18			1.95057			0.975285	−	0.220953i	$-0.0709166\pi$
$349$	1.23010e18			1.95057			0.975285	+	0.220953i	$0.0709166\pi$
$350$		0			0
$351$	−1.09520e17	−	8.84619e16i	−0.166857	−	0.134774i
$352$		0			0
$353$			6.70227e17i			0.981294i	0.871358	+	0.490647i	$0.163240\pi$
$353$			6.70227e17i			0.981294i	−0.871358	+	0.490647i	$0.836760\pi$
$354$		0			0
$355$	−6.93002e17			−0.975295
$356$		0			0
$357$	−1.75243e18	+	5.36568e17i	−2.37117	+	0.726017i
$358$		0			0
$359$		−	6.07433e17i		−	0.790382i	−0.918599	−	0.395191i	$-0.870678\pi$
$359$		−	6.07433e17i		−	0.790382i	0.918599	−	0.395191i	$-0.129322\pi$
$360$		0			0
$361$	−1.92783e17			−0.241279
$362$		0			0
$363$	2.26524e17	+	7.39827e17i	0.272751	+	0.890807i
$364$		0			0
$365$			9.96847e17i			1.15499i
$366$		0			0
$367$	2.92124e17			0.325765			0.162883	−	0.986645i	$-0.447921\pi$
$367$	2.92124e17			0.325765			0.162883	+	0.986645i	$0.447921\pi$
$368$		0			0
$369$	−1.88944e17	−	2.79619e17i	−0.202837	−	0.300181i
$370$		0			0
$371$		−	5.35905e17i		−	0.553950i
$372$		0			0
$373$	−1.04551e18			−1.04080			−0.520399	−	0.853923i	$-0.674217\pi$
$373$	−1.04551e18			−1.04080			−0.520399	+	0.853923i	$0.674217\pi$
$374$		0			0
$375$	1.07444e18	−	3.28978e17i	1.03030	−	0.315463i
$376$		0			0
$377$		−	3.90247e16i		−	0.0360537i
$378$		0			0
$379$	5.91356e17			0.526471			0.263236	−	0.964732i	$-0.415210\pi$
$379$	5.91356e17			0.526471			0.263236	+	0.964732i	$0.415210\pi$
$380$		0			0
$381$	1.12649e17	+	3.67913e17i	0.0966615	+	0.315697i
$382$		0			0
$383$		−	3.90949e16i		−	0.0323392i	−0.999869	−	0.0161696i	$-0.994853\pi$
$383$		−	3.90949e16i		−	0.0323392i	0.999869	−	0.0161696i	$-0.00514716\pi$
$384$		0			0
$385$	2.58023e18			2.05794
$386$		0			0
$387$	1.00286e17	−	6.77649e16i	0.0771370	−	0.0521228i
$388$		0			0
$389$		−	7.21106e17i		−	0.534997i	−0.963558	−	0.267499i	$-0.913803\pi$
$389$		−	7.21106e17i		−	0.534997i	0.963558	−	0.267499i	$-0.0861971\pi$
$390$		0			0
$391$	−2.34945e18			−1.68162
$392$		0			0
$393$	−3.05715e17	+	9.36051e16i	−0.211139	+	0.0646475i
$394$		0			0
$395$		−	1.94526e18i		−	1.29658i
$396$		0			0
$397$	1.54741e17			0.0995571			0.0497786	−	0.998760i	$-0.484148\pi$
$397$	1.54741e17			0.0995571			0.0497786	+	0.998760i	$0.484148\pi$
$398$		0			0
$399$	−6.88429e17	−	2.24841e18i	−0.427611	−	1.39658i
$400$		0			0
$401$		−	4.78610e17i		−	0.287059i	−0.989646	−	0.143530i	$-0.954155\pi$
$401$		−	4.78610e17i		−	0.287059i	0.989646	−	0.143530i	$-0.0458452\pi$
$402$		0			0
$403$	1.44479e17			0.0836892
$404$		0			0
$405$	1.46430e18	+	5.88789e17i	0.819303	+	0.329439i
$406$		0			0
$407$		−	1.82321e18i		−	0.985543i
$408$		0			0
$409$	−1.72580e18			−0.901422			−0.450711	−	0.892670i	$-0.648829\pi$
$409$	−1.72580e18			−0.901422			−0.450711	+	0.892670i	$0.648829\pi$
$410$		0			0
$411$	2.65323e18	−	8.12379e17i	1.33932	−	0.410078i
$412$		0			0
$413$			5.54407e18i			2.70507i
$414$		0			0
$415$	−8.09472e17			−0.381826
$416$		0			0
$417$	−1.79576e17	−	5.86494e17i	−0.0819020	−	0.267492i
$418$		0			0
$419$			4.39335e17i			0.193775i	0.995295	+	0.0968875i	$0.0308887\pi$
$419$			4.39335e17i			0.193775i	−0.995295	+	0.0968875i	$0.969111\pi$
$420$		0			0
$421$	−3.08023e18			−1.31404			−0.657020	−	0.753873i	$-0.728183\pi$
$421$	−3.08023e18			−1.31404			−0.657020	+	0.753873i	$0.728183\pi$
$422$		0			0
$423$	2.65605e18	+	3.93072e18i	1.09611	+	1.62214i
$424$		0			0
$425$		−	8.15650e17i		−	0.325673i
$426$		0			0
$427$	−2.05159e18			−0.792677
$428$		0			0
$429$	7.62267e17	−	2.33395e17i	0.285040	−	0.0872749i
$430$		0			0
$431$			2.37699e18i			0.860373i	0.902740	+	0.430186i	$0.141552\pi$
$431$			2.37699e18i			0.860373i	−0.902740	+	0.430186i	$0.858448\pi$
$432$		0			0
$433$	3.02947e18			1.06158			0.530788	−	0.847505i	$-0.321896\pi$
$433$	3.02947e18			1.06158			0.530788	+	0.847505i	$0.321896\pi$
$434$		0			0
$435$	1.28081e17	+	4.18313e17i	0.0434570	+	0.141931i
$436$		0			0
$437$		−	3.01439e18i		−	0.990444i
$438$		0			0
$439$	−1.91418e18			−0.609160			−0.304580	−	0.952487i	$-0.598516\pi$
$439$	−1.91418e18			−0.609160			−0.304580	+	0.952487i	$0.598516\pi$
$440$		0			0
$441$	−4.86948e18	+	3.29039e18i	−1.50111	+	1.01433i
$442$		0			0
$443$			5.12040e18i			1.52925i	0.644477	+	0.764624i	$0.277075\pi$
$443$			5.12040e18i			1.52925i	−0.644477	+	0.764624i	$0.722925\pi$
$444$		0			0
$445$	−1.35027e17			−0.0390752
$446$		0			0
$447$	4.69011e17	−	1.43604e17i	0.131532	−	0.0402730i
$448$		0			0
$449$		−	4.64483e18i		−	1.26254i	−0.775563	−	0.631270i	$-0.782534\pi$
$449$		−	4.64483e18i		−	1.26254i	0.775563	−	0.631270i	$-0.217466\pi$
$450$		0			0
$451$	1.91095e18			0.503516
$452$		0			0
$453$	−1.33419e18	−	4.35748e18i	−0.340825	−	1.11313i
$454$		0			0
$455$		−	1.28220e18i		−	0.317596i
$456$		0			0
$457$	2.22480e18			0.534413			0.267207	−	0.963639i	$-0.413899\pi$
$457$	2.22480e18			0.534413			0.267207	+	0.963639i	$0.413899\pi$
$458$		0			0
$459$	3.98870e18	−	4.93819e18i	0.929271	−	1.15048i
$460$		0			0
$461$		−	6.65714e17i		−	0.150446i	−0.997167	−	0.0752231i	$-0.976033\pi$
$461$		−	6.65714e17i		−	0.150446i	0.997167	−	0.0752231i	$-0.0239669\pi$
$462$		0			0
$463$	−5.49092e18			−1.20387			−0.601933	−	0.798546i	$-0.705603\pi$
$463$	−5.49092e18			−1.20387			−0.601933	+	0.798546i	$0.705603\pi$
$464$		0			0
$465$	−1.54870e18	+	4.74187e17i	−0.329454	+	0.100874i
$466$		0			0
$467$			2.59003e18i			0.534670i	0.963604	+	0.267335i	$0.0861431\pi$
$467$			2.59003e18i			0.534670i	−0.963604	+	0.267335i	$0.913857\pi$
$468$		0			0
$469$	1.23791e19			2.48015
$470$		0			0
$471$	6.31676e16	+	2.06306e17i	0.0122842	+	0.0401204i
$472$		0			0
$473$			6.85365e17i			0.129388i
$474$		0			0
$475$	1.04650e18			0.191815
$476$		0			0
$477$	1.03923e18	+	1.53797e18i	0.184962	+	0.273727i
$478$		0			0
$479$		−	9.62912e18i		−	1.66432i	−0.554532	−	0.832162i	$-0.687103\pi$
$479$		−	9.62912e18i		−	1.66432i	0.554532	−	0.832162i	$-0.312897\pi$
$480$		0			0
$481$	−9.06009e17			−0.152096
$482$		0			0
$483$	−1.11800e19	+	3.42315e18i	−1.82311	+	0.558209i
$484$		0			0
$485$		−	1.03118e19i		−	1.63358i
$486$		0			0
$487$	3.64749e18			0.561424			0.280712	−	0.959792i	$-0.409429\pi$
$487$	3.64749e18			0.561424			0.280712	+	0.959792i	$0.409429\pi$
$488$		0			0
$489$	−2.15536e18	−	7.03943e18i	−0.322373	−	1.05287i
$490$		0			0
$491$			7.94505e18i			1.15485i	0.816443	+	0.577426i	$0.195943\pi$
$491$			7.94505e18i			1.15485i	−0.816443	+	0.577426i	$0.804057\pi$
$492$		0			0
$493$	1.75960e18			0.248592
$494$		0			0
$495$	−7.40486e18	+	5.00359e18i	−1.01691	+	0.687140i
$496$		0			0
$497$			1.38716e19i			1.85196i
$498$		0			0
$499$	−2.03402e18			−0.264029			−0.132014	−	0.991248i	$-0.542144\pi$
$499$	−2.03402e18			−0.264029			−0.132014	+	0.991248i	$0.542144\pi$
$500$		0			0
$501$	−2.80864e18	+	8.59962e17i	−0.354513	+	0.108546i
$502$		0			0
$503$			9.75970e18i			1.19801i	0.800745	+	0.599005i	$0.204437\pi$
$503$			9.75970e18i			1.19801i	−0.800745	+	0.599005i	$0.795563\pi$
$504$		0			0
$505$	2.36016e18			0.281774
$506$		0			0
$507$	2.40506e18	+	7.85494e18i	0.279300	+	0.912194i
$508$		0			0
$509$		−	6.39781e18i		−	0.722781i	−0.932415	−	0.361390i	$-0.882302\pi$
$509$		−	6.39781e18i		−	0.722781i	0.932415	−	0.361390i	$-0.117698\pi$
$510$		0			0
$511$	1.99535e19			2.19318
$512$		0			0
$513$	6.33582e18	+	5.11759e18i	0.677612	+	0.547324i
$514$		0			0
$515$		−	7.62005e18i		−	0.793062i
$516$		0			0
$517$	−2.68630e19			−2.72095
$518$		0			0
$519$	9.40066e18	−	2.87834e18i	0.926802	−	0.283773i
$520$		0			0
$521$			1.54487e19i			1.48261i	0.671168	+	0.741305i	$0.265793\pi$
$521$			1.54487e19i			1.48261i	−0.671168	+	0.741305i	$0.734207\pi$
$522$		0			0
$523$	−5.39022e18			−0.503610			−0.251805	−	0.967778i	$-0.581024\pi$
$523$	−5.39022e18			−0.503610			−0.251805	+	0.967778i	$0.581024\pi$
$524$		0			0
$525$	−1.18841e18	−	3.88134e18i	−0.108106	−	0.353075i
$526$		0			0
$527$			6.51448e18i			0.577039i
$528$		0			0
$529$	−3.39598e18			−0.292938
$530$		0			0
$531$	−1.07511e19	−	1.59107e19i	−0.903215	−	1.33668i
$532$		0			0
$533$		−	9.49611e17i		−	0.0777061i
$534$		0			0
$535$	−1.39325e19			−1.11059
$536$		0			0
$537$	1.09077e19	−	3.33976e18i	0.847056	−	0.259356i
$538$		0			0
$539$		−	3.32786e19i		−	2.51793i
$540$		0			0
$541$	2.35219e19			1.73417			0.867086	−	0.498158i	$-0.165990\pi$
$541$	2.35219e19			1.73417			0.867086	+	0.498158i	$0.165990\pi$
$542$		0			0
$543$	2.78069e18	+	9.08175e18i	0.199781	+	0.652485i
$544$		0			0
$545$		−	4.50300e18i		−	0.315302i
$546$		0			0
$547$	2.11047e19			1.44035			0.720176	−	0.693792i	$-0.244061\pi$
$547$	2.11047e19			1.44035			0.720176	+	0.693792i	$0.244061\pi$
$548$		0			0
$549$	5.88776e18	−	3.97846e18i	0.391691	−	0.264672i
$550$		0			0
$551$			2.25761e18i			0.146416i
$552$		0			0
$553$	−3.89376e19			−2.46203
$554$		0			0
$555$	9.71167e18	−	2.97356e18i	0.598746	−	0.183327i
$556$		0			0
$557$		−	1.57942e19i		−	0.949538i	−0.880110	−	0.474769i	$-0.842532\pi$
$557$		−	1.57942e19i		−	0.949538i	0.880110	−	0.474769i	$-0.157468\pi$
$558$		0			0
$559$	3.40579e17			0.0199680
$560$		0			0
$561$	1.05236e19	+	3.43702e19i	0.601763	+	1.96536i
$562$		0			0
$563$			5.39943e18i			0.301155i	0.988598	+	0.150577i	$0.0481132\pi$
$563$			5.39943e18i			0.301155i	−0.988598	+	0.150577i	$0.951887\pi$
$564$		0			0
$565$	6.57953e18			0.357977
$566$		0			0
$567$	1.17856e19	−	2.93104e19i	0.625561	−	1.55575i
$568$		0			0
$569$			1.17179e19i			0.606825i	0.952859	+	0.303413i	$0.0981260\pi$
$569$			1.17179e19i			0.606825i	−0.952859	+	0.303413i	$0.901874\pi$
$570$		0			0
$571$	−3.21736e19			−1.62572			−0.812862	−	0.582456i	$-0.802092\pi$
$571$	−3.21736e19			−1.62572			−0.812862	+	0.582456i	$0.802092\pi$
$572$		0			0
$573$	7.37358e18	−	2.25768e18i	0.363577	−	0.111322i
$574$		0			0
$575$		−	5.20363e18i		−	0.250398i
$576$		0			0
$577$	−1.16837e19			−0.548717			−0.274358	−	0.961627i	$-0.588465\pi$
$577$	−1.16837e19			−0.548717			−0.274358	+	0.961627i	$0.588465\pi$
$578$		0			0
$579$	1.62977e17	+	5.32284e17i	0.00747097	+	0.0244002i
$580$		0			0
$581$			1.62029e19i			0.725037i
$582$		0			0
$583$	−1.05106e19			−0.459144
$584$		0			0
$585$	2.48644e18	+	3.67971e18i	0.106044	+	0.156936i
$586$		0			0
$587$			2.46586e19i			1.02684i	0.858139	+	0.513418i	$0.171621\pi$
$587$			2.46586e19i			1.02684i	−0.858139	+	0.513418i	$0.828379\pi$
$588$		0			0
$589$	−8.35823e18			−0.339866
$590$		0			0
$591$	−2.79849e19	+	8.56854e18i	−1.11125	+	0.340247i
$592$		0			0
$593$			4.87788e19i			1.89168i	0.324630	+	0.945841i	$0.394760\pi$
$593$			4.87788e19i			1.89168i	−0.324630	+	0.945841i	$0.605240\pi$
$594$		0			0
$595$	5.78135e19			2.18983
$596$		0			0
$597$	−1.56950e18	−	5.12598e18i	−0.0580684	−	0.189652i
$598$		0			0
$599$		−	2.50684e19i		−	0.906022i	−0.891505	−	0.453011i	$-0.850350\pi$
$599$		−	2.50684e19i		−	0.906022i	0.891505	−	0.453011i	$-0.149650\pi$
$600$		0			0
$601$	−1.80139e19			−0.636042			−0.318021	−	0.948084i	$-0.603018\pi$
$601$	−1.80139e19			−0.636042			−0.318021	+	0.948084i	$0.603018\pi$
$602$		0			0
$603$	−3.55260e19	+	2.40055e19i	−1.22553	+	0.828114i
$604$		0			0
$605$		−	2.44072e19i		−	0.822679i
$606$		0			0
$607$	1.93861e19			0.638513			0.319256	−	0.947668i	$-0.396567\pi$
$607$	1.93861e19			0.638513			0.319256	+	0.947668i	$0.396567\pi$
$608$		0			0
$609$	8.37322e18	−	2.56375e18i	0.269508	−	0.0825192i
$610$		0			0
$611$			1.33491e19i			0.419916i
$612$		0			0
$613$	2.46685e19			0.758436			0.379218	−	0.925307i	$-0.376193\pi$
$613$	2.46685e19			0.758436			0.379218	+	0.925307i	$0.376193\pi$
$614$		0			0
$615$	3.11667e18	+	1.01790e19i	0.0936623	+	0.305901i
$616$		0			0
$617$		−	3.13779e19i		−	0.921780i	−0.887457	−	0.460890i	$-0.847530\pi$
$617$		−	3.13779e19i		−	0.921780i	0.887457	−	0.460890i	$-0.152470\pi$
$618$		0			0
$619$	2.72248e19			0.781861			0.390931	−	0.920420i	$-0.372153\pi$
$619$	2.72248e19			0.781861			0.390931	+	0.920420i	$0.372153\pi$
$620$		0			0
$621$	2.54468e19	−	3.15043e19i	0.714483	−	0.884563i
$622$		0			0
$623$			2.70279e18i			0.0741987i
$624$		0			0
$625$	−2.72428e19			−0.731294
$626$		0			0
$627$	−4.40978e19	+	1.35021e19i	−1.15756	+	0.354428i
$628$		0			0
$629$		−	4.08514e19i		−	1.04870i
$630$		0			0
$631$	2.02403e18			0.0508174			0.0254087	−	0.999677i	$-0.491911\pi$
$631$	2.02403e18			0.0508174			0.0254087	+	0.999677i	$0.491911\pi$
$632$		0			0
$633$	−8.10803e18	−	2.64809e19i	−0.199108	−	0.650289i
$634$		0			0
$635$		−	1.21376e19i		−	0.291553i
$636$		0			0
$637$	−1.65372e19			−0.388584
$638$		0			0
$639$	−2.68998e19	−	3.98093e19i	−0.618363	−	0.915121i
$640$		0			0
$641$		−	7.98787e19i		−	1.79649i	−0.439495	−	0.898245i	$-0.644843\pi$
$641$		−	7.98787e19i		−	1.79649i	0.439495	−	0.898245i	$-0.355157\pi$
$642$		0			0
$643$	3.79361e19			0.834788			0.417394	−	0.908726i	$-0.362944\pi$
$643$	3.79361e19			0.834788			0.417394	+	0.908726i	$0.362944\pi$
$644$		0			0
$645$	−3.65073e18	+	1.11780e18i	−0.0786070	+	0.0240683i
$646$		0			0
$647$			4.84341e19i			1.02052i	0.860021	+	0.510259i	$0.170451\pi$
$647$			4.84341e19i			1.02052i	−0.860021	+	0.510259i	$0.829549\pi$
$648$		0			0
$649$	1.08735e20			2.24211
$650$		0			0
$651$	9.49163e18	+	3.09997e19i	0.191546	+	0.625591i
$652$		0			0
$653$			4.77183e19i			0.942524i	0.881993	+	0.471262i	$0.156201\pi$
$653$			4.77183e19i			0.942524i	−0.881993	+	0.471262i	$0.843799\pi$
$654$		0			0
$655$	1.00857e19			0.194991
$656$		0			0
$657$	−5.72636e19	+	3.86940e19i	−1.08373	+	0.732295i
$658$		0			0
$659$			1.16539e19i			0.215911i	0.994156	+	0.107956i	$0.0344304\pi$
$659$			1.16539e19i			0.215911i	−0.994156	+	0.107956i	$0.965570\pi$
$660$		0			0
$661$	−4.06800e19			−0.737854			−0.368927	−	0.929458i	$-0.620275\pi$
$661$	−4.06800e19			−0.737854			−0.368927	+	0.929458i	$0.620275\pi$
$662$		0			0
$663$	1.70796e19	−	5.22952e18i	0.303308	−	0.0928682i
$664$		0			0
$665$			7.41761e19i			1.28977i
$666$		0			0
$667$	1.12258e19			0.191133
$668$		0			0
$669$	−1.66193e19	−	5.42786e19i	−0.277096	−	0.904995i
$670$		0			0
$671$			4.02376e19i			0.657014i
$672$		0			0
$673$	6.17111e19			0.986865			0.493432	−	0.869784i	$-0.335742\pi$
$673$	6.17111e19			0.986865			0.493432	+	0.869784i	$0.335742\pi$
$674$		0			0
$675$	1.09373e19	+	8.83429e18i	0.171310	+	0.138371i
$676$		0			0
$677$		−	1.68368e19i		−	0.258308i	−0.991625	−	0.129154i	$-0.958774\pi$
$677$		−	1.68368e19i		−	0.258308i	0.991625	−	0.129154i	$-0.0412261\pi$
$678$		0			0
$679$	−2.06407e20			−3.10196
$680$		0			0
$681$	2.24798e19	−	6.88298e18i	0.330951	−	0.101332i
$682$		0			0
$683$		−	4.05494e19i		−	0.584843i	−0.956290	−	0.292421i	$-0.905539\pi$
$683$		−	4.05494e19i		−	0.584843i	0.956290	−	0.292421i	$-0.0944610\pi$
$684$		0			0
$685$	−8.75314e19			−1.23689
$686$		0			0
$687$	2.46384e19	+	8.04691e19i	0.341126	+	1.11412i
$688$		0			0
$689$			5.22306e18i			0.0708583i
$690$		0			0
$691$	−4.30948e19			−0.572900			−0.286450	−	0.958095i	$-0.592475\pi$
$691$	−4.30948e19			−0.572900			−0.286450	+	0.958095i	$0.592475\pi$
$692$		0			0
$693$	1.00155e20	+	1.48221e20i	1.30479	+	1.93097i
$694$		0			0
$695$			1.93487e19i			0.247035i
$696$		0			0
$697$	4.28174e19			0.535786
$698$		0			0
$699$	2.99318e19	−	9.16467e18i	0.367107	−	0.112403i
$700$		0			0
$701$		−	2.96929e19i		−	0.356966i	−0.983943	−	0.178483i	$-0.942881\pi$
$701$		−	2.96929e19i		−	0.356966i	0.983943	−	0.178483i	$-0.0571190\pi$
$702$		0			0
$703$	5.24134e19			0.617668
$704$		0			0
$705$	−4.38122e19	−	1.43091e20i	−0.506141	−	1.65306i
$706$		0			0
$707$		−	4.72424e19i		−	0.535053i
$708$		0			0
$709$	9.03569e19			1.00332			0.501658	−	0.865066i	$-0.332724\pi$
$709$	9.03569e19			1.00332			0.501658	+	0.865066i	$0.332724\pi$
$710$		0			0
$711$	1.11745e20	−	7.55081e19i	1.21658	−	0.822065i
$712$		0			0
$713$			4.15606e19i			0.443665i
$714$		0			0
$715$	−2.51475e19			−0.263241
$716$		0			0
$717$	−4.11601e19	+	1.26026e19i	−0.422515	+	0.129368i
$718$		0			0
$719$		−	3.52487e19i		−	0.354847i	−0.984135	−	0.177424i	$-0.943224\pi$
$719$		−	3.52487e19i		−	0.354847i	0.984135	−	0.177424i	$-0.0567763\pi$
$720$		0			0
$721$	−1.52528e20			−1.50592
$722$		0			0
$723$	−4.16041e19	−	1.35879e20i	−0.402873	−	1.31579i
$724$		0			0
$725$			3.89723e18i			0.0370160i
$726$		0			0
$727$	6.84164e19			0.637411			0.318705	−	0.947854i	$-0.396752\pi$
$727$	6.84164e19			0.637411			0.318705	+	0.947854i	$0.396752\pi$
$728$		0			0
$729$	2.30160e19	+	1.06971e20i	0.210347	+	0.977627i
$730$		0			0
$731$			1.53565e19i			0.137680i
$732$		0			0
$733$	−1.32414e20			−1.16468			−0.582341	−	0.812944i	$-0.697863\pi$
$733$	−1.32414e20			−1.16468			−0.582341	+	0.812944i	$0.697863\pi$
$734$		0			0
$735$	1.77265e20	−	5.42758e19i	1.52972	−	0.468376i
$736$		0			0
$737$		−	2.42789e20i		−	2.05568i
$738$		0			0
$739$	1.21190e19			0.100683			0.0503413	−	0.998732i	$-0.483969\pi$
$739$	1.21190e19			0.100683			0.0503413	+	0.998732i	$0.483969\pi$
$740$		0			0
$741$	6.70960e18	+	2.19136e19i	0.0546977	+	0.178643i
$742$		0			0
$743$		−	1.67371e20i		−	1.33893i	−0.742844	−	0.669465i	$-0.766524\pi$
$743$		−	1.67371e20i		−	1.33893i	0.742844	−	0.669465i	$-0.233476\pi$
$744$		0			0
$745$	−1.54729e19			−0.121472
$746$		0			0
$747$	−3.14208e19	−	4.64999e19i	−0.242088	−	0.358268i
$748$		0			0
$749$			2.78883e20i			2.10887i
$750$		0			0
$751$	−3.05322e19			−0.226610			−0.113305	−	0.993560i	$-0.536144\pi$
$751$	−3.05322e19			−0.226610			−0.113305	+	0.993560i	$0.536144\pi$
$752$		0			0
$753$	2.12577e19	−	6.50878e18i	0.154864	−	0.0474171i
$754$		0			0
$755$			1.43755e20i			1.02800i
$756$		0			0
$757$	3.88353e19			0.272618			0.136309	−	0.990666i	$-0.456476\pi$
$757$	3.88353e19			0.272618			0.136309	+	0.990666i	$0.456476\pi$
$758$		0			0
$759$	6.71379e19	+	2.19272e20i	0.462674	+	1.51109i
$760$		0			0
$761$			7.43742e19i			0.503187i	0.967833	+	0.251594i	$0.0809547\pi$
$761$			7.43742e19i			0.503187i	−0.967833	+	0.251594i	$0.919045\pi$
$762$		0			0
$763$	−9.01350e19			−0.598717
$764$		0			0
$765$	−1.65916e20	+	1.12112e20i	−1.08208	+	0.731178i
$766$		0			0
$767$		−	5.40339e19i		−	0.346018i
$768$		0			0
$769$	2.13892e20			1.34496			0.672481	−	0.740115i	$-0.265229\pi$
$769$	2.13892e20			1.34496			0.672481	+	0.740115i	$0.265229\pi$
$770$		0			0
$771$	−1.60718e20	+	4.92094e19i	−0.992393	+	0.303856i
$772$		0			0
$773$			1.15894e20i			0.702756i	0.936234	+	0.351378i	$0.114287\pi$
$773$			1.15894e20i			0.702756i	−0.936234	+	0.351378i	$0.885713\pi$
$774$		0			0
$775$	−1.44285e19			−0.0859228
$776$		0			0
$777$	−5.95208e19	−	1.94395e20i	−0.348114	−	1.13694i
$778$		0			0
$779$			5.49358e19i			0.315568i
$780$		0			0
$781$	2.72061e20			1.53500
$782$		0			0
$783$	−1.90582e19	+	2.35950e19i	−0.105621	+	0.130764i
$784$		0			0
$785$		−	6.80612e18i		−	0.0370520i
$786$		0			0
$787$	1.00050e20			0.535051			0.267526	−	0.963551i	$-0.413794\pi$
$787$	1.00050e20			0.535051			0.267526	+	0.963551i	$0.413794\pi$
$788$		0			0
$789$	−2.96039e20	+	9.06427e19i	−1.55529	+	0.476205i
$790$		0			0
$791$		−	1.31700e20i		−	0.679752i
$792$		0			0
$793$	1.99953e19			0.101395
$794$		0			0
$795$	−1.71423e19	−	5.59869e19i	−0.0854083	−	0.278944i
$796$		0			0
$797$			3.77847e20i			1.84973i	0.380300	+	0.924863i	$0.375821\pi$
$797$			3.77847e20i			1.84973i	−0.380300	+	0.924863i	$0.624179\pi$
$798$		0			0
$799$	−6.01902e20			−2.89533
$800$		0			0
$801$	−5.24127e18	−	7.75661e18i	−0.0247747	−	0.0366644i
$802$		0			0
$803$		−	3.91346e20i		−	1.81782i
$804$		0			0
$805$	3.68834e20			1.68368
$806$		0			0
$807$	2.03900e20	−	6.24312e19i	0.914752	−	0.280083i
$808$		0			0
$809$		−	2.21152e20i		−	0.975104i	−0.873094	−	0.487552i	$-0.837890\pi$
$809$		−	2.21152e20i		−	0.975104i	0.873094	−	0.487552i	$-0.162110\pi$
$810$		0			0
$811$	3.29020e20			1.42586			0.712929	−	0.701236i	$-0.247368\pi$
$811$	3.29020e20			1.42586			0.712929	+	0.701236i	$0.247368\pi$
$812$		0			0
$813$	−8.39442e19	−	2.74162e20i	−0.357567	−	1.16781i
$814$		0			0
$815$			2.32234e20i			0.972349i
$816$		0			0
$817$	−1.97028e19			−0.0810911
$818$		0			0
$819$	7.36554e19	−	4.97702e19i	0.298001	−	0.201364i
$820$		0			0
$821$		−	3.23812e20i		−	1.28792i	−0.765057	−	0.643962i	$-0.777289\pi$
$821$		−	3.23812e20i		−	1.28792i	0.765057	−	0.643962i	$-0.222711\pi$
$822$		0			0
$823$	3.05331e20			1.19391			0.596956	−	0.802274i	$-0.296377\pi$
$823$	3.05331e20			1.19391			0.596956	+	0.802274i	$0.296377\pi$
$824$		0			0
$825$	−7.61242e19	+	2.33081e19i	−0.292648	+	0.0896043i
$826$		0			0
$827$			4.32658e20i			1.63533i	0.575692	+	0.817666i	$0.304733\pi$
$827$			4.32658e20i			1.63533i	−0.575692	+	0.817666i	$0.695267\pi$
$828$		0			0
$829$	−1.40970e20			−0.523897			−0.261949	−	0.965082i	$-0.584365\pi$
$829$	−1.40970e20			−0.523897			−0.261949	+	0.965082i	$0.584365\pi$
$830$		0			0
$831$	5.54305e19	+	1.81036e20i	0.202554	+	0.661543i
$832$		0			0
$833$		−	7.45652e20i		−	2.67930i
$834$		0			0
$835$	9.26582e19			0.327400
$836$		0			0
$837$	−8.73543e19	−	7.05582e19i	−0.303533	−	0.245171i
$838$		0			0
$839$			3.14846e20i			1.07588i	0.842982	+	0.537941i	$0.180798\pi$
$839$			3.14846e20i			1.07588i	−0.842982	+	0.537941i	$0.819202\pi$
$840$		0			0
$841$	2.89151e20			0.971745
$842$		0			0
$843$	−2.68269e20	+	8.21399e19i	−0.886702	+	0.271494i
$844$		0			0
$845$		−	2.59138e20i		−	0.842431i
$846$		0			0
$847$	−4.88551e20			−1.56216
$848$		0			0
$849$	9.20752e19	+	3.00718e20i	0.289593	+	0.945813i
$850$		0			0
$851$		−	2.60621e20i		−	0.806311i
$852$		0			0
$853$	−4.02131e20			−1.22384			−0.611919	−	0.790920i	$-0.709602\pi$
$853$	−4.02131e20			−1.22384			−0.611919	+	0.790920i	$0.709602\pi$
$854$		0			0
$855$	−1.43843e20	−	2.12874e20i	−0.430651	−	0.637324i
$856$		0			0
$857$			1.24590e20i			0.366960i	0.983023	+	0.183480i	$0.0587362\pi$
$857$			1.24590e20i			0.366960i	−0.983023	+	0.183480i	$0.941264\pi$
$858$		0			0
$859$	2.48728e20			0.720731			0.360365	−	0.932811i	$-0.382652\pi$
$859$	2.48728e20			0.720731			0.360365	+	0.932811i	$0.382652\pi$
$860$		0			0
$861$	2.03750e20	−	6.23852e19i	0.580866	−	0.177852i
$862$		0			0
$863$			2.83129e20i			0.794162i	0.917784	+	0.397081i	$0.129977\pi$
$863$			2.83129e20i			0.794162i	−0.917784	+	0.397081i	$0.870023\pi$
$864$		0			0
$865$	−3.10132e20			−0.855922
$866$		0			0
$867$	1.27986e20	+	4.18004e20i	0.347560	+	1.13513i
$868$		0			0
$869$			7.63679e20i			2.04067i
$870$		0			0
$871$	−1.20649e20			−0.317247
$872$		0			0
$873$	5.92357e20	−	4.00266e20i	1.53280	−	1.03574i
$874$		0			0
$875$			7.09518e20i			1.80679i
$876$		0			0
$877$	−1.14867e20			−0.287870			−0.143935	−	0.989587i	$-0.545976\pi$
$877$	−1.14867e20			−0.287870			−0.143935	+	0.989587i	$0.545976\pi$
$878$		0			0
$879$	5.90310e20	−	1.80744e20i	1.45599	−	0.445801i
$880$		0			0
$881$			6.03422e20i			1.46484i	0.680853	+	0.732420i	$0.261609\pi$
$881$			6.03422e20i			1.46484i	−0.680853	+	0.732420i	$0.738391\pi$
$882$		0			0
$883$	−3.17121e20			−0.757706			−0.378853	−	0.925457i	$-0.623681\pi$
$883$	−3.17121e20			−0.757706			−0.378853	+	0.925457i	$0.623681\pi$
$884$		0			0
$885$	1.77342e20	+	5.79199e20i	0.417069	+	1.36215i
$886$		0			0
$887$		−	6.63312e20i		−	1.53551i	−0.640744	−	0.767754i	$-0.721374\pi$
$887$		−	6.63312e20i		−	1.53551i	0.640744	−	0.767754i	$-0.278626\pi$
$888$		0			0
$889$	−2.42954e20			−0.553620
$890$		0			0
$891$	−5.74860e20	−	2.31149e20i	−1.28949	−	0.518499i
$892$		0			0
$893$		−	7.72254e20i		−	1.70530i
$894$		0			0
$895$	−3.59849e20			−0.782274
$896$		0			0
$897$	1.08963e20	−	3.33629e19i	0.233203	−	0.0714030i
$898$		0			0
$899$		−	3.11266e19i		−	0.0655863i
$900$		0			0
$901$	−2.35505e20			−0.488570
$902$		0			0
$903$	2.23745e19	+	7.30753e19i	0.0457025	+	0.149264i
$904$		0			0
$905$		−	2.99611e20i		−	0.602584i
$906$		0			0
$907$	7.54603e20			1.49440			0.747201	−	0.664598i	$-0.231397\pi$
$907$	7.54603e20			1.49440			0.747201	+	0.664598i	$0.231397\pi$
$908$		0			0
$909$	9.16128e19	+	1.35579e20i	0.178652	+	0.264389i
$910$		0			0
$911$		−	2.26952e20i		−	0.435819i	−0.975969	−	0.217909i	$-0.930076\pi$
$911$		−	2.26952e20i		−	0.435819i	0.975969	−	0.217909i	$-0.0699237\pi$
$912$		0			0
$913$	3.17786e20			0.600951
$914$		0			0
$915$	−2.14333e20	+	6.56256e19i	−0.399156	+	0.122215i
$916$		0			0
$917$		−	2.01881e20i		−	0.370263i
$918$		0			0
$919$	−7.30333e20			−1.31921			−0.659603	−	0.751614i	$-0.729276\pi$
$919$	−7.30333e20			−1.31921			−0.659603	+	0.751614i	$0.729276\pi$
$920$		0			0
$921$	−1.86567e20	−	6.09327e20i	−0.331907	−	1.08401i
$922$		0			0
$923$		−	1.35196e20i		−	0.236892i
$924$		0			0
$925$	9.04791e19			0.156155
$926$		0			0
$927$	4.37732e20	−	2.95783e20i	0.744132	−	0.502823i
$928$		0			0
$929$			5.36475e20i			0.898337i	0.893447	+	0.449169i	$0.148280\pi$
$929$			5.36475e20i			0.898337i	−0.893447	+	0.449169i	$0.851720\pi$
$930$		0			0
$931$	9.56690e20			1.57806
$932$		0			0
$933$	7.09814e20	−	2.17334e20i	1.15338	−	0.353148i
$934$		0			0
$935$		−	1.13389e21i		−	1.81505i
$936$		0			0
$937$	−3.70668e20			−0.584531			−0.292266	−	0.956337i	$-0.594409\pi$
$937$	−3.70668e20			−0.584531			−0.292266	+	0.956337i	$0.594409\pi$
$938$		0			0
$939$	6.43593e19	+	2.10198e20i	0.0999890	+	0.326564i
$940$		0			0
$941$			2.25213e20i			0.344720i	0.985034	+	0.172360i	$0.0551393\pi$
$941$			2.25213e20i			0.344720i	−0.985034	+	0.172360i	$0.944861\pi$
$942$		0			0
$943$	2.73163e20			0.411946
$944$		0			0
$945$	−6.26177e20	+	7.75236e20i	−0.930410	+	1.15189i
$946$		0			0
$947$		−	1.02524e21i		−	1.50099i	−0.660877	−	0.750494i	$-0.729816\pi$
$947$		−	1.02524e21i		−	1.50099i	0.660877	−	0.750494i	$-0.270184\pi$
$948$		0			0
$949$	−1.94472e20			−0.280539
$950$		0			0
$951$	−6.39810e19	+	1.95900e19i	−0.0909468	+	0.0278465i
$952$		0			0
$953$			1.14513e21i			1.60400i	0.597324	+	0.802000i	$0.296231\pi$
$953$			1.14513e21i			1.60400i	−0.597324	+	0.802000i	$0.703769\pi$
$954$		0			0
$955$	−2.43258e20			−0.335771
$956$		0			0
$957$	−5.02825e19	−	1.64223e20i	−0.0683964	−	0.223383i
$958$		0			0
$959$			1.75208e21i			2.34868i
$960$		0			0
$961$	−6.41706e20			−0.847759
$962$		0			0
$963$	−5.40811e20	−	8.00351e20i	−0.704144	−	1.04207i
$964$		0			0
$965$		−	1.75603e19i		−	0.0225341i
$966$		0			0
$967$	1.47587e21			1.86665			0.933327	−	0.359026i	$-0.116891\pi$
$967$	1.47587e21			1.86665			0.933327	+	0.359026i	$0.116891\pi$
$968$		0			0
$969$	−9.88071e20	+	3.02532e20i	−1.23175	+	0.377142i
$970$		0			0
$971$		−	7.59504e20i		−	0.933245i	−0.884457	−	0.466623i	$-0.845471\pi$
$971$		−	7.59504e20i		−	0.933245i	0.884457	−	0.466623i	$-0.154529\pi$
$972$		0			0
$973$	3.87296e20			0.469087
$974$		0			0
$975$	1.15825e19	+	3.78285e19i	0.0138283	+	0.0451634i
$976$		0			0
$977$		−	5.70330e18i		−	0.00671219i	−0.999994	−	0.00335610i	$-0.998932\pi$
$977$		−	5.70330e18i		−	0.00671219i	0.999994	−	0.00335610i	$-0.00106828\pi$
$978$		0			0
$979$	5.30095e19			0.0615000
$980$		0			0
$981$	2.58674e20	−	1.74790e20i	0.295848	−	0.199910i
$982$		0			0
$983$			1.23409e21i			1.39147i	0.718299	+	0.695735i	$0.244921\pi$
$983$			1.23409e21i			1.39147i	−0.718299	+	0.695735i	$0.755079\pi$
$984$		0			0
$985$	9.23233e20			1.02626
$986$		0			0
$987$	−2.86420e21	+	8.76974e20i	−3.13894	+	0.961095i
$988$		0			0
$989$			9.79704e19i			0.105857i
$990$		0			0
$991$	−3.93368e20			−0.419067			−0.209533	−	0.977802i	$-0.567194\pi$
$991$	−3.93368e20			−0.419067			−0.209533	+	0.977802i	$0.567194\pi$
$992$		0			0
$993$	2.34398e20	+	7.65546e20i	0.246212	+	0.804131i
$994$		0			0
$995$			1.69108e20i			0.175147i
$996$		0			0
$997$	−9.62481e20			−0.982938			−0.491469	−	0.870895i	$-0.663540\pi$
$997$	−9.62481e20			−0.982938			−0.491469	+	0.870895i	$0.663540\pi$
$998$		0			0
$999$	5.47787e20	+	4.42461e20i	0.551637	+	0.445571i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.15.c.b.5.1	✓	4
3.2	odd	2	inner	12.15.c.b.5.2	yes	4
4.3	odd	2		48.15.e.c.17.4		4
12.11	even	2		48.15.e.c.17.3		4

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

\(f(q)\)	\(=\)	\(q+(-640.285 - 2091.17i) q^{3} +68988.7i q^{5} +1.38092e6 q^{7} +(-3.96304e6 + 2.67789e6i) q^{9} +O(q^{10})\)	\(q+(-640.285 - 2091.17i) q^{3} +68988.7i q^{5} +1.38092e6 q^{7} +(-3.96304e6 + 2.67789e6i) q^{9} -2.70839e7i q^{11} -1.34588e7 q^{13} +(1.44267e8 - 4.41724e7i) q^{15} -6.06850e8i q^{17} +7.78604e8 q^{19} +(-8.84184e8 - 2.88775e9i) q^{21} -3.87154e9i q^{23} +1.34407e9 q^{25} +(8.13741e9 + 6.57279e9i) q^{27} +2.89957e9i q^{29} -1.07349e10 q^{31} +(-5.66370e10 + 1.73414e10i) q^{33} +9.52681e10i q^{35} +6.73171e10 q^{37} +(8.61748e9 + 2.81447e10i) q^{39} +7.05568e10i q^{41} -2.53053e10 q^{43} +(-1.84744e11 - 2.73405e11i) q^{45} -9.91845e11i q^{47} +1.22872e12 q^{49} +(-1.26903e12 + 3.88557e11i) q^{51} -3.88078e11i q^{53} +1.86848e12 q^{55} +(-4.98528e11 - 1.62819e12i) q^{57} +4.01476e12i q^{59} -1.48567e12 q^{61} +(-5.47265e12 + 3.69796e12i) q^{63} -9.28506e11i q^{65} +8.96434e12 q^{67} +(-8.09606e12 + 2.47889e12i) q^{69} +1.00451e13i q^{71} +1.44494e13 q^{73} +(-8.60589e11 - 2.81069e12i) q^{75} -3.74007e13i q^{77} -2.81968e13 q^{79} +(8.53457e12 - 2.12252e13i) q^{81} +1.17334e13i q^{83} +4.18658e13 q^{85} +(6.06350e12 - 1.85655e12i) q^{87} +1.95724e12i q^{89} -1.85856e13 q^{91} +(6.87340e12 + 2.24485e13i) q^{93} +5.37149e13i q^{95} -1.49470e14 q^{97} +(7.25277e13 + 1.07334e14i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2148 q^{3} + 1709288 q^{7} - 5736924 q^{9}+O(q^{10}) \) 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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