LMFDB - Embedded newform 48.16.c.d.47.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,16,Mod(47,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	48.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:	$68.4928824480$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 10 x^{19} + 8885809 x^{18} - 79971996 x^{17} + 21106062365235 x^{16} - 168846686224596 x^{15} + \cdots + 85\!\cdots\!61 $ x^20 - 10x^19 + 8885809x^18 - 79971996x^17 + 21106062365235x^16 - 168846686224596x^15 + 6104796603193796014x^14 - 42730621411692080224x^13 + 612830101834948128969517x^12 - 3676425120613837434937014x^11 + 23515317858743883397971890679x^10 - 117542889748506812960062582524x^9 + 442824580967063105118845295045457x^8 - 1770593106968691891463936968183484x^7 + 4516272979574514115212279965662378678x^6 - 13542622356488850024523824403808515248x^5 + 25368182055011473290428060306984553604575x^4 - 50713795138219024400871126679601580299346x^3 + 73555006160055781174118405609684271136003209x^2 - 73529651519295334152034903037741123700144732*x + 85774090629870699032700202010530622843520083061
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{194}\cdot 3^{63}\cdot 5^{6}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.6
Root			$0.500000 - 123.126i$ of defining polynomial
Character	$\chi$	$=$	48.47
Dual form			48.16.c.d.47.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2222.78 + 3067.27i) q^{3} +265238. i q^{5} -2.85832e6i q^{7} +(-4.46743e6 - 1.36357e7i) q^{9} +O(q^{10})$	\(q+(-2222.78 + 3067.27i) q^{3} +265238. i q^{5} -2.85832e6i q^{7} +(-4.46743e6 - 1.36357e7i) q^{9} -1.31314e7 q^{11} +3.54218e8 q^{13} +(-8.13557e8 - 5.89565e8i) q^{15} +3.16150e9i q^{17} +6.73405e9i q^{19} +(8.76725e9 + 6.35341e9i) q^{21} +1.63863e10 q^{23} -3.98336e10 q^{25} +(5.17546e10 + 1.66064e10i) q^{27} +4.06607e9i q^{29} -1.12231e11i q^{31} +(2.91882e10 - 4.02777e10i) q^{33} +7.58135e11 q^{35} -4.93505e11 q^{37} +(-7.87347e11 + 1.08648e12i) q^{39} +1.89518e12i q^{41} -7.30853e11i q^{43} +(3.61671e12 - 1.18493e12i) q^{45} -2.27279e12 q^{47} -3.42243e12 q^{49} +(-9.69718e12 - 7.02730e12i) q^{51} -5.07028e12i q^{53} -3.48295e12i q^{55} +(-2.06552e13 - 1.49683e13i) q^{57} +3.32877e13 q^{59} +1.64456e12 q^{61} +(-3.89753e13 + 1.27693e13i) q^{63} +9.39520e13i q^{65} -1.08829e13i q^{67} +(-3.64230e13 + 5.02611e13i) q^{69} -3.38204e13 q^{71} -3.67887e13 q^{73} +(8.85411e13 - 1.22180e14i) q^{75} +3.75338e13i q^{77} -4.12245e12i q^{79} +(-1.65975e14 + 1.21833e14i) q^{81} -1.60341e14 q^{83} -8.38549e14 q^{85} +(-1.24717e13 - 9.03797e12i) q^{87} +7.70790e13i q^{89} -1.01247e15i q^{91} +(3.44244e14 + 2.49465e14i) q^{93} -1.78613e15 q^{95} -5.27251e14 q^{97} +(5.86637e13 + 1.79057e14i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2271972 q^{9}+O(q^{10}) $ 20 * q + 2271972 * q^9	$ 20 q + 2271972 q^{9} + 318771128 q^{13} + 13145285784 q^{21} - 57334310012 q^{25} + 628079136192 q^{33} - 1811120039336 q^{37} + 7518335948928 q^{45} - 8329580497444 q^{49} - 36365149089912 q^{57} + 46120845287032 q^{61} - 117111587094144 q^{69} + 83221863805064 q^{73} + 73507522500468 q^{81} - 12\!\cdots\!52 q^{85}+ \cdots - 12\!\cdots\!12 q^{97}+O(q^{100}) $ 20 * q + 2271972 * q^9 + 318771128 * q^13 + 13145285784 * q^21 - 57334310012 * q^25 + 628079136192 * q^33 - 1811120039336 * q^37 + 7518335948928 * q^45 - 8329580497444 * q^49 - 36365149089912 * q^57 + 46120845287032 * q^61 - 117111587094144 * q^69 + 83221863805064 * q^73 + 73507522500468 * q^81 - 1297131567063552 * q^85 + 1162806897994488 * q^93 - 1219036801794712 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−2222.78	+	3067.27i	−0.586795	+	0.809735i
$3$	−2222.78	+	3067.27i	−0.586795	+	0.809735i
$4$		0			0
$5$			265238.i			1.51831i	0.650910	+	0.759155i	$0.274388\pi$
$5$			265238.i			1.51831i	−0.650910	+	0.759155i	$0.725612\pi$
$6$		0			0
$7$		−	2.85832e6i		−	1.31182i	−0.754837	−	0.655912i	$-0.772284\pi$
$7$		−	2.85832e6i		−	1.31182i	0.754837	−	0.655912i	$-0.227716\pi$
$8$		0			0
$9$	−4.46743e6	−	1.36357e7i	−0.311343	−	0.950298i
$10$		0			0
$11$	−1.31314e7			−0.203173			−0.101587	−	0.994827i	$-0.532392\pi$
$11$	−1.31314e7			−0.203173			−0.101587	+	0.994827i	$0.532392\pi$
$12$		0			0
$13$	3.54218e8			1.56565			0.782826	−	0.622241i	$-0.213777\pi$
$13$	3.54218e8			1.56565			0.782826	+	0.622241i	$0.213777\pi$
$14$		0			0
$15$	−8.13557e8	−	5.89565e8i	−1.22943	−	0.890937i
$16$		0			0
$17$			3.16150e9i			1.86864i	0.356434	+	0.934321i	$0.383992\pi$
$17$			3.16150e9i			1.86864i	−0.356434	+	0.934321i	$0.616008\pi$
$18$		0			0
$19$			6.73405e9i			1.72832i	0.503216	+	0.864161i	$0.332150\pi$
$19$			6.73405e9i			1.72832i	−0.503216	+	0.864161i	$0.667850\pi$
$20$		0			0
$21$	8.76725e9	+	6.35341e9i	1.06223	+	0.769772i
$22$		0			0
$23$	1.63863e10			1.00351			0.501754	−	0.865010i	$-0.332688\pi$
$23$	1.63863e10			1.00351			0.501754	+	0.865010i	$0.332688\pi$
$24$		0			0
$25$	−3.98336e10			−1.30527
$26$		0			0
$27$	5.17546e10	+	1.66064e10i	0.952184	+	0.305525i
$28$		0			0
$29$			4.06607e9i			0.0437713i	0.999760	+	0.0218857i	$0.00696698\pi$
$29$			4.06607e9i			0.0437713i	−0.999760	+	0.0218857i	$0.993033\pi$
$30$		0			0
$31$		−	1.12231e11i		−	0.732657i	−0.930486	−	0.366328i	$-0.880615\pi$
$31$		−	1.12231e11i		−	0.732657i	0.930486	−	0.366328i	$-0.119385\pi$
$32$		0			0
$33$	2.91882e10	−	4.02777e10i	0.119221	−	0.164517i
$34$		0			0
$35$	7.58135e11			1.99176
$36$		0			0
$37$	−4.93505e11			−0.854632			−0.427316	−	0.904102i	$-0.640541\pi$
$37$	−4.93505e11			−0.854632			−0.427316	+	0.904102i	$0.640541\pi$
$38$		0			0
$39$	−7.87347e11	+	1.08648e12i	−0.918717	+	1.26776i
$40$		0			0
$41$			1.89518e12i			1.51975i	0.650070	+	0.759874i	$0.274739\pi$
$41$			1.89518e12i			1.51975i	−0.650070	+	0.759874i	$0.725261\pi$
$42$		0			0
$43$		−	7.30853e11i		−	0.410031i	−0.978759	−	0.205015i	$-0.934276\pi$
$43$		−	7.30853e11i		−	0.410031i	0.978759	−	0.205015i	$-0.0657244\pi$
$44$		0			0
$45$	3.61671e12	−	1.18493e12i	1.44285	−	0.472715i
$46$		0			0
$47$	−2.27279e12			−0.654374			−0.327187	−	0.944960i	$-0.606101\pi$
$47$	−2.27279e12			−0.654374			−0.327187	+	0.944960i	$0.606101\pi$
$48$		0			0
$49$	−3.42243e12			−0.720883
$50$		0			0
$51$	−9.69718e12	−	7.02730e12i	−1.51310	−	1.09651i
$52$		0			0
$53$		−	5.07028e12i		−	0.592874i	−0.955052	−	0.296437i	$-0.904201\pi$
$53$		−	5.07028e12i		−	0.592874i	0.955052	−	0.296437i	$-0.0957985\pi$
$54$		0			0
$55$		−	3.48295e12i		−	0.308480i
$56$		0			0
$57$	−2.06552e13	−	1.49683e13i	−1.39948	−	1.01417i
$58$		0			0
$59$	3.32877e13			1.74138			0.870690	−	0.491833i	$-0.163673\pi$
$59$	3.32877e13			1.74138			0.870690	+	0.491833i	$0.163673\pi$
$60$		0			0
$61$	1.64456e12			0.0670002			0.0335001	−	0.999439i	$-0.489335\pi$
$61$	1.64456e12			0.0670002			0.0335001	+	0.999439i	$0.489335\pi$
$62$		0			0
$63$	−3.89753e13	+	1.27693e13i	−1.24662	+	0.408427i
$64$		0			0
$65$			9.39520e13i			2.37715i
$66$		0			0
$67$		−	1.08829e13i		−	0.219374i	−0.993966	−	0.109687i	$-0.965015\pi$
$67$		−	1.08829e13i		−	0.219374i	0.993966	−	0.109687i	$-0.0349848\pi$
$68$		0			0
$69$	−3.64230e13	+	5.02611e13i	−0.588854	+	0.812576i
$70$		0			0
$71$	−3.38204e13			−0.441308			−0.220654	−	0.975352i	$-0.570819\pi$
$71$	−3.38204e13			−0.441308			−0.220654	+	0.975352i	$0.570819\pi$
$72$		0			0
$73$	−3.67887e13			−0.389756			−0.194878	−	0.980827i	$-0.562431\pi$
$73$	−3.67887e13			−0.389756			−0.194878	+	0.980827i	$0.562431\pi$
$74$		0			0
$75$	8.85411e13	−	1.22180e14i	0.765924	−	1.05692i
$76$		0			0
$77$			3.75338e13i			0.266528i
$78$		0			0
$79$		−	4.12245e12i		−	0.0241519i	−0.999927	−	0.0120760i	$-0.996156\pi$
$79$		−	4.12245e12i		−	0.0241519i	0.999927	−	0.0120760i	$-0.00384399\pi$
$80$		0			0
$81$	−1.65975e14	+	1.21833e14i	−0.806131	+	0.591737i
$82$		0			0
$83$	−1.60341e14			−0.648571			−0.324286	−	0.945959i	$-0.605124\pi$
$83$	−1.60341e14			−0.648571			−0.324286	+	0.945959i	$0.605124\pi$
$84$		0			0
$85$	−8.38549e14			−2.83718
$86$		0			0
$87$	−1.24717e13	−	9.03797e12i	−0.0354432	−	0.0256848i
$88$		0			0
$89$			7.70790e13i			0.184719i	0.995726	+	0.0923593i	$0.0294408\pi$
$89$			7.70790e13i			0.184719i	−0.995726	+	0.0923593i	$0.970559\pi$
$90$		0			0
$91$		−	1.01247e15i		−	2.05386i
$92$		0			0
$93$	3.44244e14	+	2.49465e14i	0.593258	+	0.429920i
$94$		0			0
$95$	−1.78613e15			−2.62413
$96$		0			0
$97$	−5.27251e14			−0.662566			−0.331283	−	0.943531i	$-0.607481\pi$
$97$	−5.27251e14			−0.662566			−0.331283	+	0.943531i	$0.607481\pi$
$98$		0			0
$99$	5.86637e13	+	1.79057e14i	0.0632566	+	0.193075i
$100$		0			0
$101$			4.02972e14i			0.373994i	0.982360	+	0.186997i	$0.0598755\pi$
$101$			4.02972e14i			0.373994i	−0.982360	+	0.186997i	$0.940125\pi$
$102$		0			0
$103$		−	8.53605e14i		−	0.683877i	−0.939722	−	0.341938i	$-0.888917\pi$
$103$		−	8.53605e14i		−	0.683877i	0.939722	−	0.341938i	$-0.111083\pi$
$104$		0			0
$105$	−1.68516e15	+	2.32541e15i	−1.16875	+	1.61280i
$106$		0			0
$107$	−4.52585e14			−0.272472			−0.136236	−	0.990676i	$-0.543501\pi$
$107$	−4.52585e14			−0.272472			−0.136236	+	0.990676i	$0.543501\pi$
$108$		0			0
$109$	−8.06355e14			−0.422501			−0.211250	−	0.977432i	$-0.567754\pi$
$109$	−8.06355e14			−0.422501			−0.211250	+	0.977432i	$0.567754\pi$
$110$		0			0
$111$	1.09695e15	−	1.51372e15i	0.501494	−	0.692026i
$112$		0			0
$113$			3.53494e15i			1.41349i	0.707467	+	0.706746i	$0.249838\pi$
$113$			3.53494e15i			1.41349i	−0.707467	+	0.706746i	$0.750162\pi$
$114$		0			0
$115$			4.34626e15i			1.52364i
$116$		0			0
$117$	−1.58244e15	−	4.83002e15i	−0.487455	−	1.48784i
$118$		0			0
$119$	9.03657e15			2.45133
$120$		0			0
$121$	−4.00481e15			−0.958721
$122$		0			0
$123$	−5.81304e15	−	4.21256e15i	−1.23059	−	0.891781i
$124$		0			0
$125$		−	2.47095e15i		−	0.463489i
$126$		0			0
$127$		−	6.44998e15i		−	1.07407i	−0.843562	−	0.537033i	$-0.819545\pi$
$127$		−	6.44998e15i		−	1.07407i	0.843562	−	0.537033i	$-0.180455\pi$
$128$		0			0
$129$	2.24173e15	+	1.62452e15i	0.332016	+	0.240604i
$130$		0			0
$131$	−3.96873e15			−0.523741			−0.261871	−	0.965103i	$-0.584339\pi$
$131$	−3.96873e15			−0.523741			−0.261871	+	0.965103i	$0.584339\pi$
$132$		0			0
$133$	1.92481e16			2.26725
$134$		0			0
$135$	−4.40464e15	+	1.37273e16i	−0.463881	+	1.44571i
$136$		0			0
$137$			3.71165e15i			0.350076i	0.984562	+	0.175038i	$0.0560048\pi$
$137$			3.71165e15i			0.350076i	−0.984562	+	0.175038i	$0.943995\pi$
$138$		0			0
$139$			1.48458e16i			1.25601i	0.778211	+	0.628003i	$0.216128\pi$
$139$			1.48458e16i			1.25601i	−0.778211	+	0.628003i	$0.783872\pi$
$140$		0			0
$141$	5.05191e15	−	6.97128e15i	0.383983	−	0.529870i
$142$		0			0
$143$	−4.65139e15			−0.318099
$144$		0			0
$145$	−1.07848e15			−0.0664585
$146$		0			0
$147$	7.60731e15	−	1.04975e16i	0.423010	−	0.583724i
$148$		0			0
$149$		−	3.19715e16i		−	1.60645i	−0.595679	−	0.803223i	$-0.703117\pi$
$149$		−	3.19715e16i		−	1.60645i	0.595679	−	0.803223i	$-0.296883\pi$
$150$		0			0
$151$		−	1.10052e15i		−	0.0500346i	−0.999687	−	0.0250173i	$-0.992036\pi$
$151$		−	1.10052e15i		−	0.0500346i	0.999687	−	0.0250173i	$-0.00796408\pi$
$152$		0			0
$153$	4.31093e16	−	1.41238e16i	1.77577	−	0.581788i
$154$		0			0
$155$	2.97680e16			1.11240
$156$		0			0
$157$	−1.72070e16			−0.584061			−0.292031	−	0.956409i	$-0.594331\pi$
$157$	−1.72070e16			−0.584061			−0.292031	+	0.956409i	$0.594331\pi$
$158$		0			0
$159$	1.55519e16	+	1.12701e16i	0.480071	+	0.347896i
$160$		0			0
$161$		−	4.68372e16i		−	1.31643i
$162$		0			0
$163$			2.87122e16i			0.735630i	0.929899	+	0.367815i	$0.119894\pi$
$163$			2.87122e16i			0.735630i	−0.929899	+	0.367815i	$0.880106\pi$
$164$		0			0
$165$	1.06832e16	+	7.74183e15i	0.249787	+	0.181015i
$166$		0			0
$167$	−2.17903e16			−0.465469			−0.232734	−	0.972540i	$-0.574767\pi$
$167$	−2.17903e16			−0.465469			−0.232734	+	0.972540i	$0.574767\pi$
$168$		0			0
$169$	7.42844e16			1.45127
$170$		0			0
$171$	9.18237e16	−	3.00839e16i	1.64242	−	0.538101i
$172$		0			0
$173$			8.08758e16i			1.32578i	0.748715	+	0.662892i	$0.230671\pi$
$173$			8.08758e16i			1.32578i	−0.748715	+	0.662892i	$0.769329\pi$
$174$		0			0
$175$			1.13857e17i			1.71228i
$176$		0			0
$177$	−7.39911e16	+	1.02102e17i	−1.02183	+	1.41006i
$178$		0			0
$179$	−3.74113e16			−0.474904			−0.237452	−	0.971399i	$-0.576312\pi$
$179$	−3.74113e16			−0.474904			−0.237452	+	0.971399i	$0.576312\pi$
$180$		0			0
$181$	4.73789e16			0.553345			0.276673	−	0.960964i	$-0.410768\pi$
$181$	4.73789e16			0.553345			0.276673	+	0.960964i	$0.410768\pi$
$182$		0			0
$183$	−3.65549e15	+	5.04432e15i	−0.0393154	+	0.0542525i
$184$		0			0
$185$		−	1.30896e17i		−	1.29760i
$186$		0			0
$187$		−	4.15150e16i		−	0.379658i
$188$		0			0
$189$	4.74663e16	−	1.47931e17i	0.400795	−	1.24910i
$190$		0			0
$191$	−7.91527e15			−0.0617611			−0.0308806	−	0.999523i	$-0.509831\pi$
$191$	−7.91527e15			−0.0617611			−0.0308806	+	0.999523i	$0.509831\pi$
$192$		0			0
$193$	−1.44372e17			−1.04185			−0.520924	−	0.853603i	$-0.674413\pi$
$193$	−1.44372e17			−1.04185			−0.520924	+	0.853603i	$0.674413\pi$
$194$		0			0
$195$	−2.88176e17	−	2.08834e17i	−1.92486	−	1.39490i
$196$		0			0
$197$			2.11986e17i			1.31162i	0.754924	+	0.655812i	$0.227674\pi$
$197$			2.11986e17i			1.31162i	−0.754924	+	0.655812i	$0.772326\pi$
$198$		0			0
$199$		−	1.69799e17i		−	0.973949i	−0.873416	−	0.486974i	$-0.838100\pi$
$199$		−	1.69799e17i		−	0.973949i	0.873416	−	0.486974i	$-0.161900\pi$
$200$		0			0
$201$	3.33809e16	+	2.41903e16i	0.177635	+	0.128727i
$202$		0			0
$203$	1.16221e16			0.0574203
$204$		0			0
$205$	−5.02674e17			−2.30745
$206$		0			0
$207$	−7.32045e16	−	2.23439e17i	−0.312435	−	0.953631i
$208$		0			0
$209$		−	8.84277e16i		−	0.351149i
$210$		0			0
$211$			4.06924e17i			1.50451i	0.658872	+	0.752255i	$0.271034\pi$
$211$			4.06924e17i			1.50451i	−0.658872	+	0.752255i	$0.728966\pi$
$212$		0			0
$213$	7.51753e16	−	1.03737e17i	0.258957	−	0.357343i
$214$		0			0
$215$	1.93850e17			0.622554
$216$		0			0
$217$	−3.20793e17			−0.961117
$218$		0			0
$219$	8.17731e16	−	1.12841e17i	0.228707	−	0.315599i
$220$		0			0
$221$			1.11986e18i			2.92564i
$222$		0			0
$223$			4.69180e16i			0.114565i	0.998358	+	0.0572827i	$0.0182436\pi$
$223$			4.69180e16i			0.114565i	−0.998358	+	0.0572827i	$0.981756\pi$
$224$		0			0
$225$	1.77954e17	+	5.43160e17i	0.406385	+	1.24039i
$226$		0			0
$227$	−4.25067e17			−0.908372			−0.454186	−	0.890907i	$-0.650070\pi$
$227$	−4.25067e17			−0.908372			−0.454186	+	0.890907i	$0.650070\pi$
$228$		0			0
$229$	−2.97232e17			−0.594744			−0.297372	−	0.954762i	$-0.596110\pi$
$229$	−2.97232e17			−0.594744			−0.297372	+	0.954762i	$0.596110\pi$
$230$		0			0
$231$	−1.15127e17	−	8.34293e16i	−0.215817	−	0.156397i
$232$		0			0
$233$		−	3.29822e17i		−	0.579575i	−0.957091	−	0.289788i	$-0.906415\pi$
$233$		−	3.29822e17i		−	0.579575i	0.957091	−	0.289788i	$-0.0935847\pi$
$234$		0			0
$235$		−	6.02831e17i		−	0.993542i
$236$		0			0
$237$	1.26447e16	+	9.16328e15i	0.0195567	+	0.0141722i
$238$		0			0
$239$	6.90050e17			1.00207			0.501033	−	0.865428i	$-0.332954\pi$
$239$	6.90050e17			1.00207			0.501033	+	0.865428i	$0.332954\pi$
$240$		0			0
$241$	−3.50839e17			−0.478608			−0.239304	−	0.970945i	$-0.576919\pi$
$241$	−3.50839e17			−0.478608			−0.239304	+	0.970945i	$0.576919\pi$
$242$		0			0
$243$	−4.77034e15	−	7.79900e17i	−0.00611648	−	0.999981i
$244$		0			0
$245$		−	9.07759e17i		−	1.09452i
$246$		0			0
$247$			2.38532e18i			2.70595i
$248$		0			0
$249$	3.56401e17	−	4.91808e17i	0.380578	−	0.525171i
$250$		0			0
$251$	−6.24796e17			−0.628327			−0.314163	−	0.949369i	$-0.601724\pi$
$251$	−6.24796e17			−0.628327			−0.314163	+	0.949369i	$0.601724\pi$
$252$		0			0
$253$	−2.15175e17			−0.203886
$254$		0			0
$255$	1.86391e18	−	2.57206e18i	1.66484	−	2.29736i
$256$		0			0
$257$		−	1.01441e18i		−	0.854502i	−0.904133	−	0.427251i	$-0.859482\pi$
$257$		−	1.01441e18i		−	0.854502i	0.904133	−	0.427251i	$-0.140518\pi$
$258$		0			0
$259$			1.41060e18i			1.12113i
$260$		0			0
$261$	5.54438e16	−	1.81649e16i	0.0415958	−	0.0136279i
$262$		0			0
$263$	−2.59605e18			−1.83927			−0.919635	−	0.392774i	$-0.871516\pi$
$263$	−2.59605e18			−1.83927			−0.919635	+	0.392774i	$0.871516\pi$
$264$		0			0
$265$	1.34483e18			0.900167
$266$		0			0
$267$	−2.36422e17	−	1.71329e17i	−0.149573	−	0.108392i
$268$		0			0
$269$		−	1.31267e18i		−	0.785257i	−0.919697	−	0.392629i	$-0.871566\pi$
$269$		−	1.31267e18i		−	0.785257i	0.919697	−	0.392629i	$-0.128434\pi$
$270$		0			0
$271$			3.43658e18i			1.94472i	0.233485	+	0.972360i	$0.424987\pi$
$271$			3.43658e18i			1.94472i	−0.233485	+	0.972360i	$0.575013\pi$
$272$		0			0
$273$	3.10552e18	+	2.25049e18i	1.66308	+	1.20520i
$274$		0			0
$275$	5.23072e17			0.265195
$276$		0			0
$277$	3.43682e18			1.65028			0.825142	−	0.564925i	$-0.191095\pi$
$277$	3.43682e18			1.65028			0.825142	+	0.564925i	$0.191095\pi$
$278$		0			0
$279$	−1.53035e18	+	5.01385e17i	−0.696242	+	0.228108i
$280$		0			0
$281$			9.27753e17i			0.400069i	0.979789	+	0.200034i	$0.0641054\pi$
$281$			9.27753e17i			0.400069i	−0.979789	+	0.200034i	$0.935895\pi$
$282$		0			0
$283$		−	3.14874e18i		−	1.28747i	−0.765246	−	0.643737i	$-0.777383\pi$
$283$		−	3.14874e18i		−	1.28747i	0.765246	−	0.643737i	$-0.222617\pi$
$284$		0			0
$285$	3.97016e18	−	5.47854e18i	1.53983	−	2.12485i
$286$		0			0
$287$	5.41703e18			1.99364
$288$		0			0
$289$	−7.13264e18			−2.49182
$290$		0			0
$291$	1.17196e18	−	1.61722e18i	0.388791	−	0.536503i
$292$		0			0
$293$		−	1.01570e17i		−	0.0320081i	−0.999872	−	0.0160040i	$-0.994906\pi$
$293$		−	1.01570e17i		−	0.0320081i	0.999872	−	0.0160040i	$-0.00509446\pi$
$294$		0			0
$295$			8.82915e18i			2.64395i
$296$		0			0
$297$	−6.79612e17	−	2.18065e17i	−0.193458	−	0.0620745i
$298$		0			0
$299$	5.80430e18			1.57114
$300$		0			0
$301$	−2.08901e18			−0.537888
$302$		0			0
$303$	−1.23603e18	−	8.95717e17i	−0.302836	−	0.219458i
$304$		0			0
$305$			4.36200e17i			0.101727i
$306$		0			0
$307$			2.48324e18i			0.551418i	0.961241	+	0.275709i	$0.0889126\pi$
$307$			2.48324e18i			0.551418i	−0.961241	+	0.275709i	$0.911087\pi$
$308$		0			0
$309$	2.61824e18	+	1.89737e18i	0.553759	+	0.401296i
$310$		0			0
$311$	3.94556e18			0.795070			0.397535	−	0.917587i	$-0.369866\pi$
$311$	3.94556e18			0.795070			0.397535	+	0.917587i	$0.369866\pi$
$312$		0			0
$313$	5.58739e18			1.07307			0.536533	−	0.843879i	$-0.319734\pi$
$313$	5.58739e18			1.07307			0.536533	+	0.843879i	$0.319734\pi$
$314$		0			0
$315$	−3.38692e18	−	1.03377e19i	−0.620119	−	1.89276i
$316$		0			0
$317$		−	4.66386e17i		−	0.0814332i	−0.999171	−	0.0407166i	$-0.987036\pi$
$317$		−	4.66386e17i		−	0.0814332i	0.999171	−	0.0407166i	$-0.0129641\pi$
$318$		0			0
$319$		−	5.33933e16i		−	0.00889317i
$320$		0			0
$321$	1.00600e18	−	1.38820e18i	0.159885	−	0.220630i
$322$		0			0
$323$	−2.12897e19			−3.22961
$324$		0			0
$325$	−1.41098e19			−2.04359
$326$		0			0
$327$	1.79235e18	−	2.47331e18i	0.247921	−	0.342114i
$328$		0			0
$329$			6.49637e18i			0.858423i
$330$		0			0
$331$			4.31036e18i			0.544257i	0.962261	+	0.272128i	$0.0877276\pi$
$331$			4.31036e18i			0.544257i	−0.962261	+	0.272128i	$0.912272\pi$
$332$		0			0
$333$	2.20470e18	+	6.72931e18i	0.266084	+	0.812154i
$334$		0			0
$335$	2.88657e18			0.333078
$336$		0			0
$337$	6.00174e18			0.662298			0.331149	−	0.943579i	$-0.392564\pi$
$337$	6.00174e18			0.662298			0.331149	+	0.943579i	$0.392564\pi$
$338$		0			0
$339$	−1.08426e19	−	7.85738e18i	−1.14455	−	0.829430i
$340$		0			0
$341$			1.47376e18i			0.148856i
$342$		0			0
$343$		−	3.78764e18i		−	0.366153i
$344$		0			0
$345$	−1.33312e19	−	9.66076e18i	−1.23374	−	0.894063i
$346$		0			0
$347$	−4.71734e18			−0.418048			−0.209024	−	0.977911i	$-0.567029\pi$
$347$	−4.71734e18			−0.418048			−0.209024	+	0.977911i	$0.567029\pi$
$348$		0			0
$349$	−4.92550e18			−0.418080			−0.209040	−	0.977907i	$-0.567034\pi$
$349$	−4.92550e18			−0.418080			−0.209040	+	0.977907i	$0.567034\pi$
$350$		0			0
$351$	1.83324e19	+	5.88227e18i	1.49079	+	0.478345i
$352$		0			0
$353$			2.72307e18i			0.212202i	0.994355	+	0.106101i	$0.0338366\pi$
$353$			2.72307e18i			0.212202i	−0.994355	+	0.106101i	$0.966163\pi$
$354$		0			0
$355$		−	8.97046e18i		−	0.670042i
$356$		0			0
$357$	−2.00863e19	+	2.77176e19i	−1.43843	+	1.98493i
$358$		0			0
$359$	−4.43785e18			−0.304764			−0.152382	−	0.988322i	$-0.548694\pi$
$359$	−4.43785e18			−0.304764			−0.152382	+	0.988322i	$0.548694\pi$
$360$		0			0
$361$	−3.01663e19			−1.98709
$362$		0			0
$363$	8.90181e18	−	1.22839e19i	0.562573	−	0.776310i
$364$		0			0
$365$		−	9.75776e18i		−	0.591771i
$366$		0			0
$367$			9.50439e18i			0.553260i	0.960977	+	0.276630i	$0.0892176\pi$
$367$			9.50439e18i			0.553260i	−0.960977	+	0.276630i	$0.910782\pi$
$368$		0			0
$369$	2.58422e19	−	8.46659e18i	1.44421	−	0.473163i
$370$		0			0
$371$	−1.44925e19			−0.777747
$372$		0			0
$373$	3.08165e18			0.158843			0.0794213	−	0.996841i	$-0.474693\pi$
$373$	3.08165e18			0.158843			0.0794213	+	0.996841i	$0.474693\pi$
$374$		0			0
$375$	7.57908e18	+	5.49237e18i	0.375303	+	0.271973i
$376$		0			0
$377$			1.44027e18i			0.0685307i
$378$		0			0
$379$		−	3.18315e19i		−	1.45567i	−0.685752	−	0.727835i	$-0.740527\pi$
$379$		−	3.18315e19i		−	1.45567i	0.685752	−	0.727835i	$-0.259473\pi$
$380$		0			0
$381$	1.97839e19	+	1.43369e19i	0.869709	+	0.630256i
$382$		0			0
$383$	2.93823e19			1.24192			0.620962	−	0.783841i	$-0.286742\pi$
$383$	2.93823e19			1.24192			0.620962	+	0.783841i	$0.286742\pi$
$384$		0			0
$385$	−9.95539e18			−0.404672
$386$		0			0
$387$	−9.96571e18	+	3.26503e18i	−0.389651	+	0.127660i
$388$		0			0
$389$			2.46147e19i			0.925918i	0.886380	+	0.462959i	$0.153212\pi$
$389$			2.46147e19i			0.925918i	−0.886380	+	0.462959i	$0.846788\pi$
$390$		0			0
$391$			5.18051e19i			1.87520i
$392$		0			0
$393$	8.82160e18	−	1.21732e19i	0.307329	−	0.424092i
$394$		0			0
$395$	1.09343e18			0.0366701
$396$		0			0
$397$	2.34981e19			0.758758			0.379379	−	0.925241i	$-0.376138\pi$
$397$	2.34981e19			0.758758			0.379379	+	0.925241i	$0.376138\pi$
$398$		0			0
$399$	−4.27842e19	+	5.90391e19i	−1.33041	+	1.83588i
$400$		0			0
$401$			4.07901e19i			1.22172i	0.791738	+	0.610860i	$0.209176\pi$
$401$			4.07901e19i			1.22172i	−0.791738	+	0.610860i	$0.790824\pi$
$402$		0			0
$403$		−	3.97543e19i		−	1.14709i
$404$		0			0
$405$	−3.23148e19	−	4.40229e19i	−0.898440	−	1.22396i
$406$		0			0
$407$	6.48043e18			0.173638
$408$		0			0
$409$	6.87173e19			1.77477			0.887383	−	0.461033i	$-0.152521\pi$
$409$	6.87173e19			1.77477			0.887383	+	0.461033i	$0.152521\pi$
$410$		0			0
$411$	−1.13846e19	−	8.25016e18i	−0.283469	−	0.205423i
$412$		0			0
$413$		−	9.51469e19i		−	2.28438i
$414$		0			0
$415$		−	4.25284e19i		−	0.984732i
$416$		0			0
$417$	−4.55361e19	−	3.29988e19i	−1.01703	−	0.737018i
$418$		0			0
$419$	7.47869e19			1.61146			0.805732	−	0.592281i	$-0.201772\pi$
$419$	7.47869e19			1.61146			0.805732	+	0.592281i	$0.201772\pi$
$420$		0			0
$421$	8.28694e19			1.72297			0.861486	−	0.507781i	$-0.169534\pi$
$421$	8.28694e19			1.72297			0.861486	+	0.507781i	$0.169534\pi$
$422$		0			0
$423$	1.01535e19	+	3.09912e19i	0.203735	+	0.621850i
$424$		0			0
$425$		−	1.25934e20i		−	2.43907i
$426$		0			0
$427$		−	4.70068e18i		−	0.0878925i
$428$		0			0
$429$	1.03390e19	−	1.42671e19i	0.186659	−	0.257576i
$430$		0			0
$431$	−3.38681e19			−0.590489			−0.295245	−	0.955422i	$-0.595401\pi$
$431$	−3.38681e19			−0.590489			−0.295245	+	0.955422i	$0.595401\pi$
$432$		0			0
$433$	3.50108e19			0.589580			0.294790	−	0.955562i	$-0.404750\pi$
$433$	3.50108e19			0.589580			0.294790	+	0.955562i	$0.404750\pi$
$434$		0			0
$435$	2.39721e18	−	3.30798e18i	0.0389975	−	0.0538138i
$436$		0			0
$437$			1.10346e20i			1.73438i
$438$		0			0
$439$			4.54338e19i			0.690074i	0.938589	+	0.345037i	$0.112134\pi$
$439$			4.54338e19i			0.690074i	−0.938589	+	0.345037i	$0.887866\pi$
$440$		0			0
$441$	1.52895e19	+	4.66674e19i	0.224442	+	0.685053i
$442$		0			0
$443$	−1.12301e20			−1.59351			−0.796754	−	0.604304i	$-0.793451\pi$
$443$	−1.12301e20			−1.59351			−0.796754	+	0.604304i	$0.793451\pi$
$444$		0			0
$445$	−2.04443e19			−0.280460
$446$		0			0
$447$	9.80653e19	+	7.10655e19i	1.30080	+	0.942654i
$448$		0			0
$449$			9.87480e19i			1.26672i	0.773857	+	0.633360i	$0.218325\pi$
$449$			9.87480e19i			1.26672i	−0.773857	+	0.633360i	$0.781675\pi$
$450$		0			0
$451$		−	2.48864e19i		−	0.308772i
$452$		0			0
$453$	3.37559e18	+	2.44621e18i	0.0405148	+	0.0293600i
$454$		0			0
$455$	2.68545e20			3.11840
$456$		0			0
$457$	1.11843e20			1.25672			0.628361	−	0.777922i	$-0.283726\pi$
$457$	1.11843e20			1.25672			0.628361	+	0.777922i	$0.283726\pi$
$458$		0			0
$459$	−5.25010e19	+	1.63622e20i	−0.570916	+	1.77929i
$460$		0			0
$461$		−	9.35080e19i		−	0.984220i	−0.870533	−	0.492110i	$-0.836226\pi$
$461$		−	9.35080e19i		−	0.984220i	0.870533	−	0.492110i	$-0.163774\pi$
$462$		0			0
$463$			6.25066e18i			0.0636896i	0.999493	+	0.0318448i	$0.0101382\pi$
$463$			6.25066e18i			0.0636896i	−0.999493	+	0.0318448i	$0.989862\pi$
$464$		0			0
$465$	−6.61675e19	+	9.13065e19i	−0.652751	+	0.900750i
$466$		0			0
$467$	−1.28661e20			−1.22905			−0.614524	−	0.788898i	$-0.710652\pi$
$467$	−1.28661e20			−1.22905			−0.614524	+	0.788898i	$0.710652\pi$
$468$		0			0
$469$	−3.11069e19			−0.287780
$470$		0			0
$471$	3.82474e19	−	5.27787e19i	0.342724	−	0.472935i
$472$		0			0
$473$			9.59714e18i			0.0833073i
$474$		0			0
$475$		−	2.68241e20i		−	2.25592i
$476$		0			0
$477$	−6.91369e19	+	2.26511e19i	−0.563407	+	0.184587i
$478$		0			0
$479$	−2.15251e20			−1.69992			−0.849962	−	0.526843i	$-0.823375\pi$
$479$	−2.15251e20			−1.69992			−0.849962	+	0.526843i	$0.823375\pi$
$480$		0			0
$481$	−1.74808e20			−1.33806
$482$		0			0
$483$	1.43662e20	+	1.04109e20i	1.06596	+	0.772473i
$484$		0			0
$485$		−	1.39847e20i		−	1.00598i
$486$		0			0
$487$		−	2.40847e20i		−	1.67986i	−0.542692	−	0.839931i	$-0.682595\pi$
$487$		−	2.40847e20i		−	1.67986i	0.542692	−	0.839931i	$-0.317405\pi$
$488$		0			0
$489$	−8.80681e19	−	6.38208e19i	−0.595665	−	0.431664i
$490$		0			0
$491$	−3.64884e19			−0.239356			−0.119678	−	0.992813i	$-0.538186\pi$
$491$	−3.64884e19			−0.239356			−0.119678	+	0.992813i	$0.538186\pi$
$492$		0			0
$493$	−1.28549e19			−0.0817929
$494$		0			0
$495$	−4.74926e19	+	1.55598e19i	−0.293148	+	0.0960431i
$496$		0			0
$497$			9.66697e19i			0.578918i
$498$		0			0
$499$		−	2.29014e20i		−	1.33079i	−0.746493	−	0.665393i	$-0.768264\pi$
$499$		−	2.29014e20i		−	1.33079i	0.746493	−	0.665393i	$-0.231736\pi$
$500$		0			0
$501$	4.84351e19	−	6.68370e19i	0.273135	−	0.376907i
$502$		0			0
$503$	−2.69943e20			−1.47745			−0.738724	−	0.674008i	$-0.764571\pi$
$503$	−2.69943e20			−1.47745			−0.738724	+	0.674008i	$0.764571\pi$
$504$		0			0
$505$	−1.06883e20			−0.567839
$506$		0			0
$507$	−1.65118e20	+	2.27850e20i	−0.851596	+	1.17514i
$508$		0			0
$509$			2.83187e20i			1.41804i	0.705187	+	0.709022i	$0.250863\pi$
$509$			2.83187e20i			1.41804i	−0.705187	+	0.709022i	$0.749137\pi$
$510$		0			0
$511$			1.05154e20i			0.511292i
$512$		0			0
$513$	−1.11828e20	+	3.48518e20i	−0.528045	+	1.64568i
$514$		0			0
$515$	2.26408e20			1.03834
$516$		0			0
$517$	2.98450e19			0.132951
$518$		0			0
$519$	−2.48068e20	−	1.79769e20i	−1.07353	−	0.777964i
$520$		0			0
$521$		−	1.00854e20i		−	0.424043i	−0.977265	−	0.212022i	$-0.931995\pi$
$521$		−	1.00854e20i		−	0.424043i	0.977265	−	0.212022i	$-0.0680047\pi$
$522$		0			0
$523$			2.70261e20i			1.10413i	0.833800	+	0.552067i	$0.186161\pi$
$523$			2.70261e20i			1.10413i	−0.833800	+	0.552067i	$0.813839\pi$
$524$		0			0
$525$	−3.49231e20	−	2.53079e20i	−1.38649	−	1.00476i
$526$		0			0
$527$	3.54818e20			1.36907
$528$		0			0
$529$	1.87407e18			0.00702861
$530$		0			0
$531$	−1.48710e20	−	4.53902e20i	−0.542166	−	1.65483i
$532$		0			0
$533$			6.71307e20i			2.37940i
$534$		0			0
$535$		−	1.20043e20i		−	0.413697i
$536$		0			0
$537$	8.31571e19	−	1.14751e20i	0.278671	−	0.384546i
$538$		0			0
$539$	4.49415e19			0.146464
$540$		0			0
$541$	4.50730e20			1.42869			0.714343	−	0.699796i	$-0.246726\pi$
$541$	4.50730e20			1.42869			0.714343	+	0.699796i	$0.246726\pi$
$542$		0			0
$543$	−1.05313e20	+	1.45324e20i	−0.324700	+	0.448063i
$544$		0			0
$545$		−	2.13876e20i		−	0.641487i
$546$		0			0
$547$			1.39655e20i			0.407521i	0.979021	+	0.203761i	$0.0653164\pi$
$547$			1.39655e20i			0.407521i	−0.979021	+	0.203761i	$0.934684\pi$
$548$		0			0
$549$	−7.34697e18	−	2.24248e19i	−0.0208601	−	0.0636702i
$550$		0			0
$551$	−2.73811e19			−0.0756509
$552$		0			0
$553$	−1.17833e19			−0.0316831
$554$		0			0
$555$	4.01495e20	+	2.90953e20i	1.05071	+	0.761423i
$556$		0			0
$557$		−	2.42202e20i		−	0.616969i	−0.951229	−	0.308484i	$-0.900178\pi$
$557$		−	2.42202e20i		−	0.616969i	0.951229	−	0.308484i	$-0.0998218\pi$
$558$		0			0
$559$		−	2.58881e20i		−	0.641966i
$560$		0			0
$561$	1.27338e20	+	9.22785e19i	0.307423	+	0.222782i
$562$		0			0
$563$	6.76943e20			1.59125			0.795626	−	0.605788i	$-0.207142\pi$
$563$	6.76943e20			1.59125			0.795626	+	0.605788i	$0.207142\pi$
$564$		0			0
$565$	−9.37599e20			−2.14612
$566$		0			0
$567$	3.48239e20	+	4.74410e20i	0.776255	+	1.05750i
$568$		0			0
$569$		−	8.30962e20i		−	1.80401i	−0.431724	−	0.902006i	$-0.642095\pi$
$569$		−	8.30962e20i		−	1.80401i	0.431724	−	0.902006i	$-0.357905\pi$
$570$		0			0
$571$			1.23859e20i			0.261912i	0.991388	+	0.130956i	$0.0418047\pi$
$571$			1.23859e20i			0.261912i	−0.991388	+	0.130956i	$0.958195\pi$
$572$		0			0
$573$	1.75939e19	−	2.42783e19i	0.0362411	−	0.0500102i
$574$		0			0
$575$	−6.52723e20			−1.30985
$576$		0			0
$577$	5.61397e20			1.09762			0.548810	−	0.835947i	$-0.315081\pi$
$577$	5.61397e20			1.09762			0.548810	+	0.835947i	$0.315081\pi$
$578$		0			0
$579$	3.20908e20	−	4.42830e20i	0.611351	−	0.843621i
$580$		0			0
$581$			4.58305e20i			0.850811i
$582$		0			0
$583$			6.65800e19i			0.120456i
$584$		0			0
$585$	1.28110e21	−	4.19724e20i	2.25900	−	0.740107i
$586$		0			0
$587$	4.26686e19			0.0733370			0.0366685	−	0.999327i	$-0.488325\pi$
$587$	4.26686e19			0.0733370			0.0366685	+	0.999327i	$0.488325\pi$
$588$		0			0
$589$	7.55770e20			1.26627
$590$		0			0
$591$	−6.50218e20	−	4.71197e20i	−1.06207	−	0.769655i
$592$		0			0
$593$		−	4.87110e20i		−	0.775741i	−0.921714	−	0.387871i	$-0.873211\pi$
$593$		−	4.87110e20i		−	0.775741i	0.921714	−	0.387871i	$-0.126789\pi$
$594$		0			0
$595$			2.39684e21i			3.72188i
$596$		0			0
$597$	5.20819e20	+	3.77425e20i	0.788641	+	0.571509i
$598$		0			0
$599$	−7.65145e20			−1.12991			−0.564953	−	0.825123i	$-0.691106\pi$
$599$	−7.65145e20			−1.12991			−0.564953	+	0.825123i	$0.691106\pi$
$600$		0			0
$601$	−3.29245e20			−0.474199			−0.237100	−	0.971485i	$-0.576197\pi$
$601$	−3.29245e20			−0.474199			−0.237100	+	0.971485i	$0.576197\pi$
$602$		0			0
$603$	−1.48397e20	+	4.86187e19i	−0.208470	+	0.0683005i
$604$		0			0
$605$		−	1.06223e21i		−	1.45564i
$606$		0			0
$607$		−	9.50006e20i		−	1.27002i	−0.772503	−	0.635011i	$-0.780996\pi$
$607$		−	9.50006e20i		−	1.27002i	0.772503	−	0.635011i	$-0.219004\pi$
$608$		0			0
$609$	−2.58334e19	+	3.56483e19i	−0.0336940	+	0.0464953i
$610$		0			0
$611$	−8.05064e20			−1.02452
$612$		0			0
$613$	5.84462e20			0.725776			0.362888	−	0.931833i	$-0.381791\pi$
$613$	5.84462e20			0.725776			0.362888	+	0.931833i	$0.381791\pi$
$614$		0			0
$615$	1.11733e21	−	1.54184e21i	1.35400	−	1.86842i
$616$		0			0
$617$			1.05579e21i			1.24864i	0.781169	+	0.624320i	$0.214624\pi$
$617$			1.05579e21i			1.24864i	−0.781169	+	0.624320i	$0.785376\pi$
$618$		0			0
$619$			1.18439e20i			0.136714i	0.997661	+	0.0683571i	$0.0217757\pi$
$619$			1.18439e20i			0.136714i	−0.997661	+	0.0683571i	$0.978224\pi$
$620$		0			0
$621$	8.48064e20	+	2.72116e20i	0.955525	+	0.306596i
$622$		0			0
$623$	2.20316e20			0.242318
$624$		0			0
$625$	−5.60234e20			−0.601546
$626$		0			0
$627$	2.71232e20	+	1.96555e20i	0.284338	+	0.206052i
$628$		0			0
$629$		−	1.56022e21i		−	1.59700i
$630$		0			0
$631$			1.07242e21i			1.07188i	0.844257	+	0.535938i	$0.180042\pi$
$631$			1.07242e21i			1.07188i	−0.844257	+	0.535938i	$0.819958\pi$
$632$		0			0
$633$	−1.24815e21	−	9.04502e20i	−1.21826	−	0.882839i
$634$		0			0
$635$	1.71078e21			1.63076
$636$		0			0
$637$	−1.21229e21			−1.12865
$638$		0			0
$639$	1.51090e20	+	4.61167e20i	0.137398	+	0.419374i
$640$		0			0
$641$			6.09243e20i			0.541197i	0.962692	+	0.270598i	$0.0872215\pi$
$641$			6.09243e20i			0.541197i	−0.962692	+	0.270598i	$0.912778\pi$
$642$		0			0
$643$			1.66388e21i			1.44391i	0.691939	+	0.721956i	$0.256757\pi$
$643$			1.66388e21i			1.44391i	−0.691939	+	0.721956i	$0.743243\pi$
$644$		0			0
$645$	−4.30885e20	+	5.94591e20i	−0.365312	+	0.504104i
$646$		0			0
$647$	1.68486e21			1.39566			0.697832	−	0.716261i	$-0.254148\pi$
$647$	1.68486e21			1.39566			0.697832	+	0.716261i	$0.254148\pi$
$648$		0			0
$649$	−4.37115e20			−0.353802
$650$		0			0
$651$	7.13050e20	−	9.83959e20i	0.563979	−	0.778251i
$652$		0			0
$653$		−	2.42205e20i		−	0.187212i	−0.995609	−	0.0936060i	$-0.970161\pi$
$653$		−	2.42205e20i		−	0.187212i	0.995609	−	0.0936060i	$-0.0298394\pi$
$654$		0			0
$655$		−	1.05266e21i		−	0.795202i
$656$		0			0
$657$	1.64351e20	+	5.01641e20i	0.121348	+	0.370384i
$658$		0			0
$659$	−1.28172e21			−0.925020			−0.462510	−	0.886614i	$-0.653051\pi$
$659$	−1.28172e21			−0.925020			−0.462510	+	0.886614i	$0.653051\pi$
$660$		0			0
$661$	−2.64099e20			−0.186319			−0.0931594	−	0.995651i	$-0.529697\pi$
$661$	−2.64099e20			−0.186319			−0.0931594	+	0.995651i	$0.529697\pi$
$662$		0			0
$663$	−3.43491e21	−	2.48920e21i	−2.36900	−	1.71675i
$664$		0			0
$665$			5.10532e21i			3.44239i
$666$		0			0
$667$			6.66276e19i			0.0439249i
$668$		0			0
$669$	−1.43910e20	−	1.04288e20i	−0.0927677	−	0.0672264i
$670$		0			0
$671$	−2.15954e19			−0.0136127
$672$		0			0
$673$	2.59542e20			0.159991			0.0799953	−	0.996795i	$-0.474509\pi$
$673$	2.59542e20			0.159991			0.0799953	+	0.996795i	$0.474509\pi$
$674$		0			0
$675$	−2.06157e21	−	6.61490e20i	−1.24285	−	0.398791i
$676$		0			0
$677$			1.67716e21i			0.988914i	0.869202	+	0.494457i	$0.164633\pi$
$677$			1.67716e21i			0.988914i	−0.869202	+	0.494457i	$0.835367\pi$
$678$		0			0
$679$			1.50705e21i			0.869170i
$680$		0			0
$681$	9.44830e20	−	1.30380e21i	0.533029	−	0.735541i
$682$		0			0
$683$	7.68531e20			0.424137			0.212068	−	0.977255i	$-0.431980\pi$
$683$	7.68531e20			0.424137			0.212068	+	0.977255i	$0.431980\pi$
$684$		0			0
$685$	−9.84469e20			−0.531524
$686$		0			0
$687$	6.60681e20	−	9.11693e20i	0.348993	−	0.481585i
$688$		0			0
$689$		−	1.79598e21i		−	0.928235i
$690$		0			0
$691$			3.04921e20i			0.154206i	0.997023	+	0.0771031i	$0.0245671\pi$
$691$			3.04921e20i			0.154206i	−0.997023	+	0.0771031i	$0.975433\pi$
$692$		0			0
$693$	5.11801e20	−	1.67680e20i	0.253281	−	0.0829815i
$694$		0			0
$695$	−3.93766e21			−1.90701
$696$		0			0
$697$	−5.99161e21			−2.83986
$698$		0			0
$699$	1.01165e21	+	7.33120e20i	0.469303	+	0.340092i
$700$		0			0
$701$		−	1.41966e21i		−	0.644612i	−0.946636	−	0.322306i	$-0.895542\pi$
$701$		−	1.41966e21i		−	0.644612i	0.946636	−	0.322306i	$-0.104458\pi$
$702$		0			0
$703$		−	3.32329e21i		−	1.47708i
$704$		0			0
$705$	1.84905e21	+	1.33996e21i	0.804507	+	0.583006i
$706$		0			0
$707$	1.15182e21			0.490614
$708$		0			0
$709$	−1.11904e21			−0.466659			−0.233329	−	0.972398i	$-0.574962\pi$
$709$	−1.11904e21			−0.466659			−0.233329	+	0.972398i	$0.574962\pi$
$710$		0			0
$711$	−5.62126e19	+	1.84167e19i	−0.0229515	+	0.00751953i
$712$		0			0
$713$		−	1.83905e21i		−	0.735227i
$714$		0			0
$715$		−	1.23372e21i		−	0.482973i
$716$		0			0
$717$	−1.53383e21	+	2.11657e21i	−0.588007	+	0.811408i
$718$		0			0
$719$	−2.12915e21			−0.799356			−0.399678	−	0.916656i	$-0.630878\pi$
$719$	−2.12915e21			−0.799356			−0.399678	+	0.916656i	$0.630878\pi$
$720$		0			0
$721$	−2.43988e21			−0.897126
$722$		0			0
$723$	7.79836e20	−	1.07612e21i	0.280845	−	0.387546i
$724$		0			0
$725$		−	1.61966e20i		−	0.0571332i
$726$		0			0
$727$			3.68931e21i			1.27478i	0.770540	+	0.637392i	$0.219987\pi$
$727$			3.68931e21i			1.27478i	−0.770540	+	0.637392i	$0.780013\pi$
$728$		0			0
$729$	2.40277e21	+	1.71891e21i	0.813309	+	0.581831i
$730$		0			0
$731$	2.31059e21			0.766200
$732$		0			0
$733$	3.40333e21			1.10567			0.552834	−	0.833291i	$-0.313546\pi$
$733$	3.40333e21			1.10567			0.552834	+	0.833291i	$0.313546\pi$
$734$		0			0
$735$	2.78435e21	+	2.01775e21i	0.886275	+	0.642261i
$736$		0			0
$737$			1.42908e20i			0.0445709i
$738$		0			0
$739$		−	5.60817e21i		−	1.71391i	−0.515391	−	0.856955i	$-0.672353\pi$
$739$		−	5.60817e21i		−	1.71391i	0.515391	−	0.856955i	$-0.327647\pi$
$740$		0			0
$741$	−7.31643e21	−	5.30204e21i	−2.19110	−	1.58784i
$742$		0			0
$743$	−3.84487e21			−1.12841			−0.564203	−	0.825636i	$-0.690817\pi$
$743$	−3.84487e21			−1.12841			−0.564203	+	0.825636i	$0.690817\pi$
$744$		0			0
$745$	8.48005e21			2.43908
$746$		0			0
$747$	7.16310e20	+	2.18636e21i	0.201928	+	0.616336i
$748$		0			0
$749$			1.29363e21i			0.357435i
$750$		0			0
$751$			3.67079e21i			0.994168i	0.867703	+	0.497084i	$0.165596\pi$
$751$			3.67079e21i			0.994168i	−0.867703	+	0.497084i	$0.834404\pi$
$752$		0			0
$753$	1.38878e21	−	1.91642e21i	0.368699	−	0.508779i
$754$		0			0
$755$	2.91899e20			0.0759680
$756$		0			0
$757$	5.72399e21			1.46043			0.730213	−	0.683219i	$-0.239421\pi$
$757$	5.72399e21			1.46043			0.730213	+	0.683219i	$0.239421\pi$
$758$		0			0
$759$	4.78286e20	−	6.60000e20i	0.119639	−	0.165094i
$760$		0			0
$761$		−	3.49082e20i		−	0.0856135i	−0.999083	−	0.0428067i	$-0.986370\pi$
$761$		−	3.49082e20i		−	0.0856135i	0.999083	−	0.0428067i	$-0.0136300\pi$
$762$		0			0
$763$			2.30482e21i			0.554247i
$764$		0			0
$765$	3.74616e21	+	1.14342e22i	0.883335	+	2.69616i
$766$		0			0
$767$	1.17911e22			2.72639
$768$		0			0
$769$	−2.11183e21			−0.478863			−0.239432	−	0.970913i	$-0.576961\pi$
$769$	−2.11183e21			−0.478863			−0.239432	+	0.970913i	$0.576961\pi$
$770$		0			0
$771$	3.11146e21	+	2.25480e21i	0.691921	+	0.501418i
$772$		0			0
$773$		−	2.19464e20i		−	0.0478648i	−0.999714	−	0.0239324i	$-0.992381\pi$
$773$		−	2.19464e20i		−	0.0478648i	0.999714	−	0.0239324i	$-0.00761865\pi$
$774$		0			0
$775$			4.47057e21i			0.956312i
$776$		0			0
$777$	−4.32669e21	−	3.13544e21i	−0.907816	−	0.657872i
$778$		0			0
$779$	−1.27622e22			−2.62661
$780$		0			0
$781$	4.44111e20			0.0896620
$782$		0			0
$783$	−6.75226e19	+	2.10438e20i	−0.0133732	+	0.0416784i
$784$		0			0
$785$		−	4.56396e21i		−	0.886786i
$786$		0			0
$787$			2.92876e21i			0.558307i	0.960247	+	0.279153i	$0.0900538\pi$
$787$			2.92876e21i			0.558307i	−0.960247	+	0.279153i	$0.909946\pi$
$788$		0			0
$789$	5.77045e21	−	7.96281e21i	1.07927	−	1.48932i
$790$		0			0
$791$	1.01040e22			1.85425
$792$		0			0
$793$	5.82533e20			0.104899
$794$		0			0
$795$	−2.98926e21	+	4.12496e21i	−0.528214	+	0.728897i
$796$		0			0
$797$		−	6.36696e21i		−	1.10407i	−0.833822	−	0.552033i	$-0.813852\pi$
$797$		−	6.36696e21i		−	1.10407i	0.833822	−	0.552033i	$-0.186148\pi$
$798$		0			0
$799$		−	7.18543e21i		−	1.22279i
$800$		0			0
$801$	1.05103e21	−	3.44345e20i	0.175538	−	0.0575109i
$802$		0			0
$803$	4.83088e20			0.0791881
$804$		0			0
$805$	1.24230e22			1.99874
$806$		0			0
$807$	4.02630e21	+	2.91776e21i	0.635851	+	0.460785i
$808$		0			0
$809$		−	4.51829e21i		−	0.700423i	−0.936671	−	0.350211i	$-0.886110\pi$
$809$		−	4.51829e21i		−	0.700423i	0.936671	−	0.350211i	$-0.113890\pi$
$810$		0			0
$811$		−	1.15787e22i		−	1.76199i	−0.473129	−	0.880993i	$-0.656876\pi$
$811$		−	1.15787e22i		−	1.76199i	0.473129	−	0.880993i	$-0.343124\pi$
$812$		0			0
$813$	−1.05409e22	−	7.63876e21i	−1.57471	−	1.14115i
$814$		0			0
$815$	−7.61556e21			−1.11691
$816$		0			0
$817$	4.92160e21			0.708665
$818$		0			0
$819$	−1.38057e22	+	4.52313e21i	−1.95178	+	0.639455i
$820$		0			0
$821$			9.64272e21i			1.33852i	0.743027	+	0.669262i	$0.233389\pi$
$821$			9.64272e21i			1.33852i	−0.743027	+	0.669262i	$0.766611\pi$
$822$		0			0
$823$		−	1.10927e22i		−	1.51195i	−0.654600	−	0.755976i	$-0.727163\pi$
$823$		−	1.10927e22i		−	1.51195i	0.654600	−	0.755976i	$-0.272837\pi$
$824$		0			0
$825$	−1.16267e21	+	1.60440e21i	−0.155615	+	0.214738i
$826$		0			0
$827$	9.93415e21			1.30569			0.652844	−	0.757492i	$-0.273576\pi$
$827$	9.93415e21			1.30569			0.652844	+	0.757492i	$0.273576\pi$
$828$		0			0
$829$	−1.15367e22			−1.48910			−0.744548	−	0.667569i	$-0.767335\pi$
$829$	−1.15367e22			−1.48910			−0.744548	+	0.667569i	$0.767335\pi$
$830$		0			0
$831$	−7.63929e21	+	1.05417e22i	−0.968379	+	1.33629i
$832$		0			0
$833$		−	1.08200e22i		−	1.34707i
$834$		0			0
$835$		−	5.77962e21i		−	0.706726i
$836$		0			0
$837$	1.86375e21	−	5.80848e21i	0.223845	−	0.697624i
$838$		0			0
$839$	9.93183e21			1.17169			0.585847	−	0.810421i	$-0.300762\pi$
$839$	9.93183e21			1.17169			0.585847	+	0.810421i	$0.300762\pi$
$840$		0			0
$841$	8.61266e21			0.998084
$842$		0			0
$843$	−2.84567e21	−	2.06219e21i	−0.323950	−	0.234759i
$844$		0			0
$845$			1.97030e22i			2.20347i
$846$		0			0
$847$			1.14470e22i			1.25767i
$848$		0			0
$849$	9.65805e21	+	6.99895e21i	1.04251	+	0.755484i
$850$		0			0
$851$	−8.08670e21			−0.857630
$852$		0			0
$853$	−8.94130e20			−0.0931715			−0.0465858	−	0.998914i	$-0.514834\pi$
$853$	−8.94130e20			−0.0931715			−0.0465858	+	0.998914i	$0.514834\pi$
$854$		0			0
$855$	7.97939e21	+	2.43551e22i	0.817004	+	2.49370i
$856$		0			0
$857$			1.82633e22i			1.83749i	0.394856	+	0.918743i	$0.370795\pi$
$857$			1.82633e22i			1.83749i	−0.394856	+	0.918743i	$0.629205\pi$
$858$		0			0
$859$		−	5.82547e21i		−	0.575946i	−0.957639	−	0.287973i	$-0.907019\pi$
$859$		−	5.82547e21i		−	0.575946i	0.957639	−	0.287973i	$-0.0929814\pi$
$860$		0			0
$861$	−1.20409e22	+	1.66155e22i	−1.16986	+	1.61432i
$862$		0			0
$863$	−5.79395e21			−0.553215			−0.276607	−	0.960983i	$-0.589210\pi$
$863$	−5.79395e21			−0.553215			−0.276607	+	0.960983i	$0.589210\pi$
$864$		0			0
$865$	−2.14513e22			−2.01295
$866$		0			0
$867$	1.58543e22	−	2.18778e22i	1.46219	−	2.01771i
$868$		0			0
$869$			5.41336e19i			0.00490703i
$870$		0			0
$871$		−	3.85493e21i		−	0.343463i
$872$		0			0
$873$	2.35546e21	+	7.18945e21i	0.206285	+	0.629635i
$874$		0			0
$875$	−7.06277e21			−0.608016
$876$		0			0
$877$	8.91585e21			0.754511			0.377256	−	0.926109i	$-0.376868\pi$
$877$	8.91585e21			0.754511			0.377256	+	0.926109i	$0.376868\pi$
$878$		0			0
$879$	3.11544e20	+	2.25768e20i	0.0259181	+	0.0187822i
$880$		0			0
$881$		−	1.63857e22i		−	1.34013i	−0.742303	−	0.670064i	$-0.766266\pi$
$881$		−	1.63857e22i		−	1.34013i	0.742303	−	0.670064i	$-0.233734\pi$
$882$		0			0
$883$		−	1.37759e22i		−	1.10768i	−0.832622	−	0.553841i	$-0.813161\pi$
$883$		−	1.37759e22i		−	1.10768i	0.832622	−	0.553841i	$-0.186839\pi$
$884$		0			0
$885$	−2.70814e22	−	1.96252e22i	−2.14090	−	1.55146i
$886$		0			0
$887$	2.99476e20			0.0232774			0.0116387	−	0.999932i	$-0.496295\pi$
$887$	2.99476e20			0.0232774			0.0116387	+	0.999932i	$0.496295\pi$
$888$		0			0
$889$	−1.84361e22			−1.40898
$890$		0			0
$891$	2.17949e21	−	1.59985e21i	0.163784	−	0.120225i
$892$		0			0
$893$		−	1.53051e22i		−	1.13097i
$894$		0			0
$895$		−	9.92291e21i		−	0.721051i
$896$		0			0
$897$	−1.29017e22	+	1.78034e22i	−0.921940	+	1.27221i
$898$		0			0
$899$	4.56340e20			0.0320694
$900$		0			0
$901$	1.60297e22			1.10787
$902$		0			0
$903$	4.64341e21	−	6.40757e21i	0.315630	−	0.435547i
$904$		0			0
$905$			1.25667e22i			0.840149i
$906$		0			0
$907$			6.76966e21i			0.445156i	0.974915	+	0.222578i	$0.0714471\pi$
$907$			6.76966e21i			0.445156i	−0.974915	+	0.222578i	$0.928553\pi$
$908$		0			0
$909$	5.49482e21	−	1.80025e21i	0.355406	−	0.116440i
$910$		0			0
$911$	−1.97929e22			−1.25928			−0.629640	−	0.776887i	$-0.716798\pi$
$911$	−1.97929e22			−1.25928			−0.629640	+	0.776887i	$0.716798\pi$
$912$		0			0
$913$	2.10550e21			0.131772
$914$		0			0
$915$	−1.33795e21	−	9.69576e20i	−0.0823721	−	0.0596930i
$916$		0			0
$917$			1.13439e22i			0.687056i
$918$		0			0
$919$		−	1.22200e20i		−	0.00728123i	−0.999993	−	0.00364061i	$-0.998841\pi$
$919$		−	1.22200e20i		−	0.00728123i	0.999993	−	0.00364061i	$-0.00115885\pi$
$920$		0			0
$921$	−7.61678e21	−	5.51969e21i	−0.446503	−	0.323570i
$922$		0			0
$923$	−1.19798e22			−0.690935
$924$		0			0
$925$	1.96581e22			1.11552
$926$		0			0
$927$	−1.16395e22	+	3.81342e21i	−0.649887	+	0.212920i
$928$		0			0
$929$			2.55676e21i			0.140466i	0.997531	+	0.0702331i	$0.0223743\pi$
$929$			2.55676e21i			0.140466i	−0.997531	+	0.0702331i	$0.977626\pi$
$930$		0			0
$931$		−	2.30469e22i		−	1.24592i
$932$		0			0
$933$	−8.77010e21	+	1.21021e22i	−0.466543	+	0.643797i
$934$		0			0
$935$	1.10113e22			0.576439
$936$		0			0
$937$	−2.63453e22			−1.35724			−0.678620	−	0.734489i	$-0.737422\pi$
$937$	−2.63453e22			−1.35724			−0.678620	+	0.734489i	$0.737422\pi$
$938$		0			0
$939$	−1.24195e22	+	1.71381e22i	−0.629670	+	0.868900i
$940$		0			0
$941$		−	2.31671e22i		−	1.15598i	−0.816044	−	0.577989i	$-0.803837\pi$
$941$		−	2.31671e22i		−	1.15598i	0.816044	−	0.577989i	$-0.196163\pi$
$942$		0			0
$943$			3.10549e22i			1.52508i
$944$		0			0
$945$	3.92370e22	+	1.25899e22i	1.89652	+	0.608531i
$946$		0			0
$947$	−1.82837e22			−0.869838			−0.434919	−	0.900470i	$-0.643223\pi$
$947$	−1.82837e22			−0.869838			−0.434919	+	0.900470i	$0.643223\pi$
$948$		0			0
$949$	−1.30312e22			−0.610222
$950$		0			0
$951$	1.43053e21	+	1.03667e21i	0.0659393	+	0.0477846i
$952$		0			0
$953$			3.69996e22i			1.67880i	0.543511	+	0.839402i	$0.317095\pi$
$953$			3.69996e22i			1.67880i	−0.543511	+	0.839402i	$0.682905\pi$
$954$		0			0
$955$		−	2.09943e21i		−	0.0937726i
$956$		0			0
$957$	1.63772e20	+	1.18681e20i	0.00720112	+	0.00521847i
$958$		0			0
$959$	1.06091e22			0.459238
$960$		0			0
$961$	1.08694e22			0.463214
$962$		0			0
$963$	2.02189e21	+	6.17133e21i	0.0848322	+	0.258929i
$964$		0			0
$965$		−	3.82930e22i		−	1.58185i
$966$		0			0
$967$			8.31210e21i			0.338074i	0.985610	+	0.169037i	$0.0540658\pi$
$967$			8.31210e21i			0.338074i	−0.985610	+	0.169037i	$0.945934\pi$
$968$		0			0
$969$	4.73222e22	−	6.53013e22i	1.89512	−	2.61513i
$970$		0			0
$971$	1.32751e22			0.523472			0.261736	−	0.965139i	$-0.415705\pi$
$971$	1.32751e22			0.523472			0.261736	+	0.965139i	$0.415705\pi$
$972$		0			0
$973$	4.24340e22			1.64766
$974$		0			0
$975$	3.13628e22	−	4.32785e22i	1.19917	−	1.65477i
$976$		0			0
$977$		−	3.04277e21i		−	0.114567i	−0.998358	−	0.0572836i	$-0.981756\pi$
$977$		−	3.04277e21i		−	0.114567i	0.998358	−	0.0572836i	$-0.0182439\pi$
$978$		0			0
$979$		−	1.01216e21i		−	0.0375299i
$980$		0			0
$981$	3.60233e21	+	1.09952e22i	0.131543	+	0.401501i
$982$		0			0
$983$	3.80951e22			1.36999			0.684996	−	0.728547i	$-0.259804\pi$
$983$	3.80951e22			1.36999			0.684996	+	0.728547i	$0.259804\pi$
$984$		0			0
$985$	−5.62266e22			−1.99145
$986$		0			0
$987$	−1.99261e22	−	1.44400e22i	−0.695096	−	0.503719i
$988$		0			0
$989$		−	1.19759e22i		−	0.411469i
$990$		0			0
$991$			2.06460e22i			0.698689i	0.936994	+	0.349344i	$0.113596\pi$
$991$			2.06460e22i			0.698689i	−0.936994	+	0.349344i	$0.886404\pi$
$992$		0			0
$993$	−1.32211e22	−	9.58098e21i	−0.440704	−	0.319367i
$994$		0			0
$995$	4.50370e22			1.47876
$996$		0			0
$997$	−1.02491e22			−0.331490			−0.165745	−	0.986169i	$-0.553003\pi$
$997$	−1.02491e22			−0.331490			−0.165745	+	0.986169i	$0.553003\pi$
$998$		0			0
$999$	−2.55412e22	−	8.19533e21i	−0.813767	−	0.261111i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.16.c.d.47.6	yes	20
3.2	odd	2	inner	48.16.c.d.47.16	yes	20
4.3	odd	2	inner	48.16.c.d.47.15	yes	20
12.11	even	2	inner	48.16.c.d.47.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.16.c.d.47.5	✓	20	12.11	even	2	inner
48.16.c.d.47.6	yes	20	1.1	even	1	trivial
48.16.c.d.47.15	yes	20	4.3	odd	2	inner
48.16.c.d.47.16	yes	20	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	48.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2222.78 + 3067.27i) q^{3} +265238. i q^{5} -2.85832e6i q^{7} +(-4.46743e6 - 1.36357e7i) q^{9} +O(q^{10})\)	\(q+(-2222.78 + 3067.27i) q^{3} +265238. i q^{5} -2.85832e6i q^{7} +(-4.46743e6 - 1.36357e7i) q^{9} -1.31314e7 q^{11} +3.54218e8 q^{13} +(-8.13557e8 - 5.89565e8i) q^{15} +3.16150e9i q^{17} +6.73405e9i q^{19} +(8.76725e9 + 6.35341e9i) q^{21} +1.63863e10 q^{23} -3.98336e10 q^{25} +(5.17546e10 + 1.66064e10i) q^{27} +4.06607e9i q^{29} -1.12231e11i q^{31} +(2.91882e10 - 4.02777e10i) q^{33} +7.58135e11 q^{35} -4.93505e11 q^{37} +(-7.87347e11 + 1.08648e12i) q^{39} +1.89518e12i q^{41} -7.30853e11i q^{43} +(3.61671e12 - 1.18493e12i) q^{45} -2.27279e12 q^{47} -3.42243e12 q^{49} +(-9.69718e12 - 7.02730e12i) q^{51} -5.07028e12i q^{53} -3.48295e12i q^{55} +(-2.06552e13 - 1.49683e13i) q^{57} +3.32877e13 q^{59} +1.64456e12 q^{61} +(-3.89753e13 + 1.27693e13i) q^{63} +9.39520e13i q^{65} -1.08829e13i q^{67} +(-3.64230e13 + 5.02611e13i) q^{69} -3.38204e13 q^{71} -3.67887e13 q^{73} +(8.85411e13 - 1.22180e14i) q^{75} +3.75338e13i q^{77} -4.12245e12i q^{79} +(-1.65975e14 + 1.21833e14i) q^{81} -1.60341e14 q^{83} -8.38549e14 q^{85} +(-1.24717e13 - 9.03797e12i) q^{87} +7.70790e13i q^{89} -1.01247e15i q^{91} +(3.44244e14 + 2.49465e14i) q^{93} -1.78613e15 q^{95} -5.27251e14 q^{97} +(5.86637e13 + 1.79057e14i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2271972 q^{9}+O(q^{10}) \) 20 * q + 2271972 * q^9	\( 20 q + 2271972 q^{9} + 318771128 q^{13} + 13145285784 q^{21} - 57334310012 q^{25} + 628079136192 q^{33} - 1811120039336 q^{37} + 7518335948928 q^{45} - 8329580497444 q^{49} - 36365149089912 q^{57} + 46120845287032 q^{61} - 117111587094144 q^{69} + 83221863805064 q^{73} + 73507522500468 q^{81} - 12\!\cdots\!52 q^{85}+ \cdots - 12\!\cdots\!12 q^{97}+O(q^{100}) \) 20 * q + 2271972 * q^9 + 318771128 * q^13 + 13145285784 * q^21 - 57334310012 * q^25 + 628079136192 * q^33 - 1811120039336 * q^37 + 7518335948928 * q^45 - 8329580497444 * q^49 - 36365149089912 * q^57 + 46120845287032 * q^61 - 117111587094144 * q^69 + 83221863805064 * q^73 + 73507522500468 * q^81 - 1297131567063552 * q^85 + 1162806897994488 * q^93 - 1219036801794712 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more