LMFDB - Embedded newform 48.16.c.d.47.13

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			47.13
Root			$0.500000 + 2068.99i$ of defining polynomial
Character	$\chi$	$=$	48.47
Dual form			48.16.c.d.47.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2123.98 - 3136.50i) q^{3} -150623. i q^{5} -1.78114e6i q^{7} +(-5.32633e6 - 1.33237e7i) q^{9} +O(q^{10})$	\(q+(2123.98 - 3136.50i) q^{3} -150623. i q^{5} -1.78114e6i q^{7} +(-5.32633e6 - 1.33237e7i) q^{9} -5.09536e7 q^{11} -1.77193e8 q^{13} +(-4.72429e8 - 3.19921e8i) q^{15} -1.92281e8i q^{17} +1.92921e9i q^{19} +(-5.58655e9 - 3.78311e9i) q^{21} -1.25600e10 q^{23} +7.83023e9 q^{25} +(-5.31028e10 - 1.15933e10i) q^{27} +1.46187e11i q^{29} -1.90823e11i q^{31} +(-1.08224e11 + 1.59816e11i) q^{33} -2.68282e11 q^{35} +1.34369e11 q^{37} +(-3.76354e11 + 5.55765e11i) q^{39} -1.28878e12i q^{41} +2.35180e11i q^{43} +(-2.00686e12 + 8.02268e11i) q^{45} +4.90443e12 q^{47} +1.57509e12 q^{49} +(-6.03088e11 - 4.08400e11i) q^{51} -1.28694e12i q^{53} +7.67479e12i q^{55} +(6.05095e12 + 4.09760e12i) q^{57} +3.14681e13 q^{59} -2.73330e13 q^{61} +(-2.37315e13 + 9.48695e12i) q^{63} +2.66894e13i q^{65} +9.65692e12i q^{67} +(-2.66772e13 + 3.93944e13i) q^{69} -9.63560e13 q^{71} -1.81985e14 q^{73} +(1.66313e13 - 2.45595e13i) q^{75} +9.07557e13i q^{77} +1.62240e14i q^{79} +(-1.49152e14 + 1.41933e14i) q^{81} +2.86085e14 q^{83} -2.89619e13 q^{85} +(4.58514e14 + 3.10498e14i) q^{87} +7.12277e14i q^{89} +3.15606e14i q^{91} +(-5.98516e14 - 4.05305e14i) q^{93} +2.90583e14 q^{95} -1.36420e15 q^{97} +(2.71395e14 + 6.78891e14i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2123.98	−	3136.50i	0.560713	−	0.828010i
$3$	2123.98	−	3136.50i	0.560713	−	0.828010i
$4$		0			0
$5$		−	150623.i		−	0.862217i	−0.902300	−	0.431109i	$-0.858123\pi$
$5$		−	150623.i		−	0.862217i	0.902300	−	0.431109i	$-0.141877\pi$
$6$		0			0
$7$		−	1.78114e6i		−	0.817455i	−0.912657	−	0.408727i	$-0.865973\pi$
$7$		−	1.78114e6i		−	0.817455i	0.912657	−	0.408727i	$-0.134027\pi$
$8$		0			0
$9$	−5.32633e6	−	1.33237e7i	−0.371201	−	0.928553i
$10$		0			0
$11$	−5.09536e7			−0.788369			−0.394185	−	0.919031i	$-0.628973\pi$
$11$	−5.09536e7			−0.788369			−0.394185	+	0.919031i	$0.628973\pi$
$12$		0			0
$13$	−1.77193e8			−0.783198			−0.391599	−	0.920136i	$-0.628078\pi$
$13$	−1.77193e8			−0.783198			−0.391599	+	0.920136i	$0.628078\pi$
$14$		0			0
$15$	−4.72429e8	−	3.19921e8i	−0.713925	−	0.483457i
$16$		0			0
$17$		−	1.92281e8i		−	0.113650i	−0.998384	−	0.0568249i	$-0.981902\pi$
$17$		−	1.92281e8i		−	0.113650i	0.998384	−	0.0568249i	$-0.0180977\pi$
$18$		0			0
$19$			1.92921e9i			0.495139i	0.968870	+	0.247569i	$0.0796318\pi$
$19$			1.92921e9i			0.495139i	−0.968870	+	0.247569i	$0.920368\pi$
$20$		0			0
$21$	−5.58655e9	−	3.78311e9i	−0.676861	−	0.458358i
$22$		0			0
$23$	−1.25600e10			−0.769185			−0.384593	−	0.923086i	$-0.625658\pi$
$23$	−1.25600e10			−0.769185			−0.384593	+	0.923086i	$0.625658\pi$
$24$		0			0
$25$	7.83023e9			0.256581
$26$		0			0
$27$	−5.31028e10	−	1.15933e10i	−0.976988	−	0.213294i
$28$		0			0
$29$			1.46187e11i			1.57370i	0.617142	+	0.786852i	$0.288291\pi$
$29$			1.46187e11i			1.57370i	−0.617142	+	0.786852i	$0.711709\pi$
$30$		0			0
$31$		−	1.90823e11i		−	1.24571i	−0.782336	−	0.622857i	$-0.785972\pi$
$31$		−	1.90823e11i		−	1.24571i	0.782336	−	0.622857i	$-0.214028\pi$
$32$		0			0
$33$	−1.08224e11	+	1.59816e11i	−0.442049	+	0.652777i
$34$		0			0
$35$	−2.68282e11			−0.704824
$36$		0			0
$37$	1.34369e11			0.232695			0.116347	−	0.993209i	$-0.462881\pi$
$37$	1.34369e11			0.232695			0.116347	+	0.993209i	$0.462881\pi$
$38$		0			0
$39$	−3.76354e11	+	5.55765e11i	−0.439149	+	0.648495i
$40$		0			0
$41$		−	1.28878e12i		−	1.03347i	−0.856145	−	0.516736i	$-0.827147\pi$
$41$		−	1.28878e12i		−	1.03347i	0.856145	−	0.516736i	$-0.172853\pi$
$42$		0			0
$43$			2.35180e11i			0.131943i	0.997822	+	0.0659716i	$0.0210147\pi$
$43$			2.35180e11i			0.131943i	−0.997822	+	0.0659716i	$0.978985\pi$
$44$		0			0
$45$	−2.00686e12	+	8.02268e11i	−0.800614	+	0.320056i
$46$		0			0
$47$	4.90443e12			1.41207			0.706033	−	0.708179i	$-0.250483\pi$
$47$	4.90443e12			1.41207			0.706033	+	0.708179i	$0.250483\pi$
$48$		0			0
$49$	1.57509e12			0.331768
$50$		0			0
$51$	−6.03088e11	−	4.08400e11i	−0.0941031	−	0.0637250i
$52$		0			0
$53$		−	1.28694e12i		−	0.150483i	−0.997165	−	0.0752417i	$-0.976027\pi$
$53$		−	1.28694e12i		−	0.150483i	0.997165	−	0.0752417i	$-0.0239728\pi$
$54$		0			0
$55$			7.67479e12i			0.679746i
$56$		0			0
$57$	6.05095e12	+	4.09760e12i	0.409980	+	0.277631i
$58$		0			0
$59$	3.14681e13			1.64619			0.823096	−	0.567902i	$-0.192245\pi$
$59$	3.14681e13			1.64619			0.823096	+	0.567902i	$0.192245\pi$
$60$		0			0
$61$	−2.73330e13			−1.11356			−0.556781	−	0.830660i	$-0.687964\pi$
$61$	−2.73330e13			−1.11356			−0.556781	+	0.830660i	$0.687964\pi$
$62$		0			0
$63$	−2.37315e13	+	9.48695e12i	−0.759050	+	0.303440i
$64$		0			0
$65$			2.66894e13i			0.675287i
$66$		0			0
$67$			9.65692e12i			0.194660i	0.995252	+	0.0973302i	$0.0310303\pi$
$67$			9.65692e12i			0.194660i	−0.995252	+	0.0973302i	$0.968970\pi$
$68$		0			0
$69$	−2.66772e13	+	3.93944e13i	−0.431293	+	0.636893i
$70$		0			0
$71$	−9.63560e13			−1.25731			−0.628654	−	0.777685i	$-0.716394\pi$
$71$	−9.63560e13			−1.25731			−0.628654	+	0.777685i	$0.716394\pi$
$72$		0			0
$73$	−1.81985e14			−1.92803			−0.964017	−	0.265842i	$-0.914350\pi$
$73$	−1.81985e14			−1.92803			−0.964017	+	0.265842i	$0.914350\pi$
$74$		0			0
$75$	1.66313e13	−	2.45595e13i	0.143868	−	0.212452i
$76$		0			0
$77$			9.07557e13i			0.644456i
$78$		0			0
$79$			1.62240e14i			0.950506i	0.879849	+	0.475253i	$0.157643\pi$
$79$			1.62240e14i			0.950506i	−0.879849	+	0.475253i	$0.842357\pi$
$80$		0			0
$81$	−1.49152e14	+	1.41933e14i	−0.724420	+	0.689359i
$82$		0			0
$83$	2.86085e14			1.15720			0.578600	−	0.815611i	$-0.303599\pi$
$83$	2.86085e14			1.15720			0.578600	+	0.815611i	$0.303599\pi$
$84$		0			0
$85$	−2.89619e13			−0.0979908
$86$		0			0
$87$	4.58514e14	+	3.10498e14i	1.30304	+	0.882397i
$88$		0			0
$89$			7.12277e14i			1.70696i	0.521124	+	0.853481i	$0.325513\pi$
$89$			7.12277e14i			1.70696i	−0.521124	+	0.853481i	$0.674487\pi$
$90$		0			0
$91$			3.15606e14i			0.640229i
$92$		0			0
$93$	−5.98516e14	−	4.05305e14i	−1.03146	−	0.698489i
$94$		0			0
$95$	2.90583e14			0.426917
$96$		0			0
$97$	−1.36420e15			−1.71431			−0.857154	−	0.515061i	$-0.827769\pi$
$97$	−1.36420e15			−1.71431			−0.857154	+	0.515061i	$0.827769\pi$
$98$		0			0
$99$	2.71395e14	+	6.78891e14i	0.292643	+	0.732042i
$100$		0			0
$101$		−	1.37145e14i		−	0.127283i	−0.997973	−	0.0636416i	$-0.979729\pi$
$101$		−	1.37145e14i		−	0.127283i	0.997973	−	0.0636416i	$-0.0202715\pi$
$102$		0			0
$103$			5.20079e14i			0.416668i	0.978058	+	0.208334i	$0.0668041\pi$
$103$			5.20079e14i			0.416668i	−0.978058	+	0.208334i	$0.933196\pi$
$104$		0			0
$105$	−5.69825e14	+	8.41464e14i	−0.395204	+	0.583601i
$106$		0			0
$107$	−2.22978e14			−0.134240			−0.0671202	−	0.997745i	$-0.521381\pi$
$107$	−2.22978e14			−0.134240			−0.0671202	+	0.997745i	$0.521381\pi$
$108$		0			0
$109$	−1.43763e15			−0.753268			−0.376634	−	0.926362i	$-0.622919\pi$
$109$	−1.43763e15			−0.753268			−0.376634	+	0.926362i	$0.622919\pi$
$110$		0			0
$111$	2.85397e14	−	4.21448e14i	0.130475	−	0.192673i
$112$		0			0
$113$			3.78993e15i			1.51545i	0.652573	+	0.757726i	$0.273690\pi$
$113$			3.78993e15i			1.51545i	−0.652573	+	0.757726i	$0.726310\pi$
$114$		0			0
$115$			1.89183e15i			0.663205i
$116$		0			0
$117$	9.43788e14	+	2.36087e15i	0.290724	+	0.727240i
$118$		0			0
$119$	−3.42479e14			−0.0929035
$120$		0			0
$121$	−1.58098e15			−0.378474
$122$		0			0
$123$	−4.04224e15	−	2.73733e15i	−0.855724	−	0.579481i
$124$		0			0
$125$		−	5.77607e15i		−	1.08345i
$126$		0			0
$127$		−	5.61979e15i		−	0.935820i	−0.883776	−	0.467910i	$-0.845007\pi$
$127$		−	5.61979e15i		−	0.935820i	0.883776	−	0.467910i	$-0.154993\pi$
$128$		0			0
$129$	7.37641e14	+	4.99517e14i	0.109250	+	0.0739823i
$130$		0			0
$131$	3.43372e15			0.453137			0.226569	−	0.973995i	$-0.427249\pi$
$131$	3.43372e15			0.453137			0.226569	+	0.973995i	$0.427249\pi$
$132$		0			0
$133$	3.43620e15			0.404753
$134$		0			0
$135$	−1.74622e15	+	7.99851e15i	−0.183906	+	0.842376i
$136$		0			0
$137$		−	4.69433e15i		−	0.442761i	−0.975188	−	0.221380i	$-0.928944\pi$
$137$		−	4.69433e15i		−	0.442761i	0.975188	−	0.221380i	$-0.0710563\pi$
$138$		0			0
$139$			2.48509e15i			0.210247i	0.994459	+	0.105124i	$0.0335238\pi$
$139$			2.48509e15i			0.210247i	−0.994459	+	0.105124i	$0.966476\pi$
$140$		0			0
$141$	1.04169e16	−	1.53827e16i	0.791764	−	1.16920i
$142$		0			0
$143$	9.02862e15			0.617449
$144$		0			0
$145$	2.20191e16			1.35688
$146$		0			0
$147$	3.34546e15	−	4.94026e15i	0.186027	−	0.274707i
$148$		0			0
$149$		−	2.60926e16i		−	1.31105i	−0.755173	−	0.655526i	$-0.772447\pi$
$149$		−	2.60926e16i		−	1.31105i	0.755173	−	0.655526i	$-0.227553\pi$
$150$		0			0
$151$		−	2.92596e16i		−	1.33027i	−0.746721	−	0.665137i	$-0.768373\pi$
$151$		−	2.92596e16i		−	1.33027i	0.746721	−	0.665137i	$-0.231627\pi$
$152$		0			0
$153$	−2.56189e15	+	1.02415e15i	−0.105530	+	0.0421869i
$154$		0			0
$155$	−2.87424e16			−1.07408
$156$		0			0
$157$	4.23131e16			1.43624			0.718122	−	0.695918i	$-0.245002\pi$
$157$	4.23131e16			1.43624			0.718122	+	0.695918i	$0.245002\pi$
$158$		0			0
$159$	−4.03648e15	−	2.73343e15i	−0.124602	−	0.0843781i
$160$		0			0
$161$			2.23712e16i			0.628774i
$162$		0			0
$163$		−	3.52064e16i		−	0.902018i	−0.892519	−	0.451009i	$-0.851064\pi$
$163$		−	3.52064e16i		−	0.902018i	0.892519	−	0.451009i	$-0.148936\pi$
$164$		0			0
$165$	2.40720e16	+	1.63011e16i	0.562836	+	0.381143i
$166$		0			0
$167$	−7.62355e16			−1.62849			−0.814243	−	0.580525i	$-0.802848\pi$
$167$	−7.62355e16			−1.62849			−0.814243	+	0.580525i	$0.802848\pi$
$168$		0			0
$169$	−1.97885e16			−0.386601
$170$		0			0
$171$	2.57042e16	−	1.02756e16i	0.459762	−	0.183796i
$172$		0			0
$173$		−	1.77301e16i		−	0.290647i	−0.989384	−	0.145323i	$-0.953578\pi$
$173$		−	1.77301e16i		−	0.290647i	0.989384	−	0.145323i	$-0.0464223\pi$
$174$		0			0
$175$		−	1.39468e16i		−	0.209743i
$176$		0			0
$177$	6.68376e16	−	9.86997e16i	0.923042	−	1.36306i
$178$		0			0
$179$	5.39830e15			0.0685266			0.0342633	−	0.999413i	$-0.489092\pi$
$179$	5.39830e15			0.0685266			0.0342633	+	0.999413i	$0.489092\pi$
$180$		0			0
$181$	−8.81542e16			−1.02957			−0.514783	−	0.857321i	$-0.672127\pi$
$181$	−8.81542e16			−1.02957			−0.514783	+	0.857321i	$0.672127\pi$
$182$		0			0
$183$	−5.80548e16	+	8.57300e16i	−0.624389	+	0.922040i
$184$		0			0
$185$		−	2.02391e16i		−	0.200633i
$186$		0			0
$187$			9.79739e15i			0.0895980i
$188$		0			0
$189$	−2.06493e16	+	9.45837e16i	−0.174358	+	0.798643i
$190$		0			0
$191$	1.35305e17			1.05576			0.527879	−	0.849319i	$-0.322987\pi$
$191$	1.35305e17			1.05576			0.527879	+	0.849319i	$0.322987\pi$
$192$		0			0
$193$	−3.74038e16			−0.269921			−0.134960	−	0.990851i	$-0.543091\pi$
$193$	−3.74038e16			−0.269921			−0.134960	+	0.990851i	$0.543091\pi$
$194$		0			0
$195$	8.37111e16	+	5.66877e16i	0.559144	+	0.378642i
$196$		0			0
$197$			1.65953e17i			1.02680i	0.858148	+	0.513402i	$0.171615\pi$
$197$			1.65953e17i			1.02680i	−0.858148	+	0.513402i	$0.828385\pi$
$198$		0			0
$199$		−	9.38133e16i		−	0.538104i	−0.963126	−	0.269052i	$-0.913290\pi$
$199$		−	9.38133e16i		−	0.538104i	0.963126	−	0.269052i	$-0.0867104\pi$
$200$		0			0
$201$	3.02889e16	+	2.05111e16i	0.161181	+	0.109149i
$202$		0			0
$203$	2.60380e17			1.28643
$204$		0			0
$205$	−1.94119e17			−0.891077
$206$		0			0
$207$	6.68987e16	+	1.67346e17i	0.285522	+	0.714229i
$208$		0			0
$209$		−	9.83000e16i		−	0.390352i
$210$		0			0
$211$			2.09131e17i			0.773215i	0.922244	+	0.386608i	$0.126353\pi$
$211$			2.09131e17i			0.773215i	−0.922244	+	0.386608i	$0.873647\pi$
$212$		0			0
$213$	−2.04658e17	+	3.02221e17i	−0.704989	+	1.04106i
$214$		0			0
$215$	3.54235e16			0.113764
$216$		0			0
$217$	−3.39884e17			−1.01831
$218$		0			0
$219$	−3.86533e17	+	5.70796e17i	−1.08107	+	1.59643i
$220$		0			0
$221$			3.40708e16i			0.0890102i
$222$		0			0
$223$		−	4.79469e17i		−	1.17078i	−0.810753	−	0.585389i	$-0.800942\pi$
$223$		−	4.79469e17i		−	1.17078i	0.810753	−	0.585389i	$-0.199058\pi$
$224$		0			0
$225$	−4.17064e16	−	1.04328e17i	−0.0952431	−	0.238249i
$226$		0			0
$227$	−3.62602e17			−0.774883			−0.387442	−	0.921894i	$-0.626641\pi$
$227$	−3.62602e17			−0.774883			−0.387442	+	0.921894i	$0.626641\pi$
$228$		0			0
$229$	−6.95132e17			−1.39092			−0.695459	−	0.718566i	$-0.744799\pi$
$229$	−6.95132e17			−1.39092			−0.695459	+	0.718566i	$0.744799\pi$
$230$		0			0
$231$	2.84655e17	+	1.92763e17i	0.533616	+	0.361355i
$232$		0			0
$233$			8.87704e17i			1.55991i	0.625837	+	0.779954i	$0.284757\pi$
$233$			8.87704e17i			1.55991i	−0.625837	+	0.779954i	$0.715243\pi$
$234$		0			0
$235$		−	7.38721e17i		−	1.21751i
$236$		0			0
$237$	5.08865e17	+	3.44595e17i	0.787028	+	0.532961i
$238$		0			0
$239$	−2.52185e17			−0.366215			−0.183107	−	0.983093i	$-0.558616\pi$
$239$	−2.52185e17			−0.366215			−0.183107	+	0.983093i	$0.558616\pi$
$240$		0			0
$241$	−4.20375e17			−0.573468			−0.286734	−	0.958010i	$-0.592570\pi$
$241$	−4.20375e17			−0.573468			−0.286734	+	0.958010i	$0.592570\pi$
$242$		0			0
$243$	1.28377e17	+	7.69276e17i	0.164604	+	0.986360i
$244$		0			0
$245$		−	2.37245e17i		−	0.286056i
$246$		0			0
$247$		−	3.41842e17i		−	0.387791i
$248$		0			0
$249$	6.07638e17	−	8.97303e17i	0.648858	−	0.958174i
$250$		0			0
$251$	1.05433e18			1.06029			0.530145	−	0.847907i	$-0.322137\pi$
$251$	1.05433e18			1.06029			0.530145	+	0.847907i	$0.322137\pi$
$252$		0			0
$253$	6.39977e17			0.606402
$254$		0			0
$255$	−6.15145e16	+	9.08390e16i	−0.0549448	+	0.0811374i
$256$		0			0
$257$		−	2.35514e17i		−	0.198389i	−0.995068	−	0.0991945i	$-0.968373\pi$
$257$		−	2.35514e17i		−	0.198389i	0.995068	−	0.0991945i	$-0.0316266\pi$
$258$		0			0
$259$		−	2.39331e17i		−	0.190217i
$260$		0			0
$261$	1.94775e18	−	7.78638e17i	1.46127	−	0.584160i
$262$		0			0
$263$	−1.39462e18			−0.988073			−0.494037	−	0.869441i	$-0.664479\pi$
$263$	−1.39462e18			−0.988073			−0.494037	+	0.869441i	$0.664479\pi$
$264$		0			0
$265$	−1.93843e17			−0.129749
$266$		0			0
$267$	2.23406e18	+	1.51286e18i	1.41338	+	0.957117i
$268$		0			0
$269$		−	2.63946e18i		−	1.57897i	−0.613773	−	0.789483i	$-0.710349\pi$
$269$		−	2.63946e18i		−	1.57897i	0.613773	−	0.789483i	$-0.289651\pi$
$270$		0			0
$271$		−	2.98899e18i		−	1.69143i	−0.533633	−	0.845716i	$-0.679174\pi$
$271$		−	2.98899e18i		−	1.69143i	0.533633	−	0.845716i	$-0.320826\pi$
$272$		0			0
$273$	9.89898e17	+	6.70341e17i	0.530116	+	0.358985i
$274$		0			0
$275$	−3.98979e17			−0.202281
$276$		0			0
$277$	−4.28774e17			−0.205887			−0.102944	−	0.994687i	$-0.532826\pi$
$277$	−4.28774e17			−0.205887			−0.102944	+	0.994687i	$0.532826\pi$
$278$		0			0
$279$	−2.54247e18	+	1.01639e18i	−1.15671	+	0.462410i
$280$		0			0
$281$		−	5.21785e17i		−	0.225006i	−0.993651	−	0.112503i	$-0.964113\pi$
$281$		−	5.21785e17i		−	0.225006i	0.993651	−	0.112503i	$-0.0358868\pi$
$282$		0			0
$283$		−	4.50905e18i		−	1.84368i	−0.387565	−	0.921842i	$-0.626684\pi$
$283$		−	4.50905e18i		−	1.84368i	0.387565	−	0.921842i	$-0.373316\pi$
$284$		0			0
$285$	6.17193e17	−	9.11414e17i	0.239378	−	0.353492i
$286$		0			0
$287$	−2.29549e18			−0.844816
$288$		0			0
$289$	2.82545e18			0.987084
$290$		0			0
$291$	−2.89752e18	+	4.27880e18i	−0.961235	+	1.41946i
$292$		0			0
$293$		−	7.02881e17i		−	0.221500i	−0.993848	−	0.110750i	$-0.964675\pi$
$293$		−	7.02881e17i		−	0.221500i	0.993848	−	0.110750i	$-0.0353254\pi$
$294$		0			0
$295$		−	4.73983e18i		−	1.41938i
$296$		0			0
$297$	2.70578e18	+	5.90720e17i	0.770227	+	0.168154i
$298$		0			0
$299$	2.22554e18			0.602424
$300$		0			0
$301$	4.18889e17			0.107858
$302$		0			0
$303$	−4.30156e17	−	2.91294e17i	−0.105392	−	0.0713694i
$304$		0			0
$305$			4.11699e18i			0.960132i
$306$		0			0
$307$			6.88752e17i			0.152941i	0.997072	+	0.0764707i	$0.0243652\pi$
$307$			6.88752e17i			0.152941i	−0.997072	+	0.0764707i	$0.975635\pi$
$308$		0			0
$309$	1.63123e18	+	1.10464e18i	0.345005	+	0.233631i
$310$		0			0
$311$	1.98406e18			0.399808			0.199904	−	0.979815i	$-0.435937\pi$
$311$	1.98406e18			0.399808			0.199904	+	0.979815i	$0.435937\pi$
$312$		0			0
$313$	−7.23625e18			−1.38973			−0.694866	−	0.719139i	$-0.744536\pi$
$313$	−7.23625e18			−1.38973			−0.694866	+	0.719139i	$0.744536\pi$
$314$		0			0
$315$	1.42895e18	+	3.57451e18i	0.261631	+	0.654466i
$316$		0			0
$317$		−	9.53989e17i		−	0.166571i	−0.996526	−	0.0832854i	$-0.973459\pi$
$317$		−	9.53989e17i		−	0.166571i	0.996526	−	0.0832854i	$-0.0265413\pi$
$318$		0			0
$319$		−	7.44874e18i		−	1.24066i
$320$		0			0
$321$	−4.73600e17	+	6.99369e17i	−0.0752704	+	0.111152i
$322$		0			0
$323$	3.70949e17			0.0562724
$324$		0			0
$325$	−1.38746e18			−0.200954
$326$		0			0
$327$	−3.05350e18	+	4.50913e18i	−0.422367	+	0.623713i
$328$		0			0
$329$		−	8.73550e18i		−	1.15430i
$330$		0			0
$331$			1.46775e19i			1.85328i	0.375951	+	0.926639i	$0.377316\pi$
$331$			1.46775e19i			1.85328i	−0.375951	+	0.926639i	$0.622684\pi$
$332$		0			0
$333$	−7.15693e17	−	1.79030e18i	−0.0863764	−	0.216069i
$334$		0			0
$335$	1.45456e18			0.167840
$336$		0			0
$337$	−1.56986e19			−1.73235			−0.866177	−	0.499737i	$-0.833430\pi$
$337$	−1.56986e19			−1.73235			−0.866177	+	0.499737i	$0.833430\pi$
$338$		0			0
$339$	1.18871e19	+	8.04973e18i	1.25481	+	0.849735i
$340$		0			0
$341$			9.72312e18i			0.982083i
$342$		0			0
$343$		−	1.12615e19i		−	1.08866i
$344$		0			0
$345$	5.93371e18	+	4.01820e18i	0.549140	+	0.371868i
$346$		0			0
$347$	1.44993e19			1.28492			0.642461	−	0.766318i	$-0.277913\pi$
$347$	1.44993e19			1.28492			0.642461	+	0.766318i	$0.277913\pi$
$348$		0			0
$349$	−1.38970e19			−1.17959			−0.589793	−	0.807554i	$-0.700791\pi$
$349$	−1.38970e19			−1.17959			−0.589793	+	0.807554i	$0.700791\pi$
$350$		0			0
$351$	9.40945e18	+	2.05425e18i	0.765175	+	0.167051i
$352$		0			0
$353$			1.30895e19i			1.02003i	0.860165	+	0.510015i	$0.170360\pi$
$353$			1.30895e19i			1.02003i	−0.860165	+	0.510015i	$0.829640\pi$
$354$		0			0
$355$			1.45135e19i			1.08407i
$356$		0			0
$357$	−7.27419e17	+	1.07419e18i	−0.0520923	+	0.0769250i
$358$		0			0
$359$	−2.12879e19			−1.46192			−0.730962	−	0.682418i	$-0.760928\pi$
$359$	−2.12879e19			−1.46192			−0.730962	+	0.682418i	$0.760928\pi$
$360$		0			0
$361$	1.14593e19			0.754838
$362$		0			0
$363$	−3.35797e18	+	4.95874e18i	−0.212216	+	0.313380i
$364$		0			0
$365$			2.74112e19i			1.66238i
$366$		0			0
$367$		−	1.75552e19i		−	1.02191i	−0.859609	−	0.510953i	$-0.829293\pi$
$367$		−	1.75552e19i		−	1.02191i	0.859609	−	0.510953i	$-0.170707\pi$
$368$		0			0
$369$	−1.71713e19	+	6.86444e18i	−0.959632	+	0.383625i
$370$		0			0
$371$	−2.29222e18			−0.123013
$372$		0			0
$373$	−1.68001e19			−0.865957			−0.432979	−	0.901404i	$-0.642537\pi$
$373$	−1.68001e19			−0.865957			−0.432979	+	0.901404i	$0.642537\pi$
$374$		0			0
$375$	−1.81166e19	−	1.22683e19i	−0.897104	−	0.607503i
$376$		0			0
$377$		−	2.59033e19i		−	1.23252i
$378$		0			0
$379$			8.93471e18i			0.408589i	0.978910	+	0.204294i	$0.0654900\pi$
$379$			8.93471e18i			0.408589i	−0.978910	+	0.204294i	$0.934510\pi$
$380$		0			0
$381$	−1.76265e19	−	1.19363e19i	−0.774868	−	0.524727i
$382$		0			0
$383$	−1.10124e19			−0.465471			−0.232736	−	0.972540i	$-0.574768\pi$
$383$	−1.10124e19			−0.465471			−0.232736	+	0.972540i	$0.574768\pi$
$384$		0			0
$385$	1.36699e19			0.555661
$386$		0			0
$387$	3.13347e18	−	1.25264e18i	0.122516	−	0.0489774i
$388$		0			0
$389$			1.50117e19i			0.564687i	0.959313	+	0.282343i	$0.0911117\pi$
$389$			1.50117e19i			0.564687i	−0.959313	+	0.282343i	$0.908888\pi$
$390$		0			0
$391$			2.41505e18i			0.0874177i
$392$		0			0
$393$	7.29314e18	−	1.07698e19i	0.254080	−	0.375202i
$394$		0			0
$395$	2.44371e19			0.819543
$396$		0			0
$397$	3.55260e19			1.14714			0.573572	−	0.819155i	$-0.305557\pi$
$397$	3.55260e19			1.14714			0.573572	+	0.819155i	$0.305557\pi$
$398$		0			0
$399$	7.29841e18	−	1.07776e19i	0.226951	−	0.335140i
$400$		0			0
$401$		−	4.46681e19i		−	1.33787i	−0.743319	−	0.668937i	$-0.766750\pi$
$401$		−	4.46681e19i		−	1.33787i	0.743319	−	0.668937i	$-0.233250\pi$
$402$		0			0
$403$			3.38125e19i			0.975640i
$404$		0			0
$405$	2.13784e19	+	2.24657e19i	0.594377	+	0.624607i
$406$		0			0
$407$	−6.84659e18			−0.183449
$408$		0			0
$409$	3.93110e19			1.01529			0.507645	−	0.861567i	$-0.330516\pi$
$409$	3.93110e19			1.01529			0.507645	+	0.861567i	$0.330516\pi$
$410$		0			0
$411$	−1.47238e19	−	9.97066e18i	−0.366610	−	0.248262i
$412$		0			0
$413$		−	5.60492e19i		−	1.34569i
$414$		0			0
$415$		−	4.30910e19i		−	0.997759i
$416$		0			0
$417$	7.79447e18	+	5.27827e18i	0.174087	+	0.117889i
$418$		0			0
$419$	7.47087e19			1.60978			0.804889	−	0.593426i	$-0.202225\pi$
$419$	7.47087e19			1.60978			0.804889	+	0.593426i	$0.202225\pi$
$420$		0			0
$421$	2.20853e19			0.459185			0.229592	−	0.973287i	$-0.426261\pi$
$421$	2.20853e19			0.459185			0.229592	+	0.973287i	$0.426261\pi$
$422$		0			0
$423$	−2.61226e19	−	6.53453e19i	−0.524160	−	1.31118i
$424$		0			0
$425$		−	1.50560e18i		−	0.0291604i
$426$		0			0
$427$			4.86841e19i			0.910286i
$428$		0			0
$429$	1.91766e19	−	2.83182e19i	0.346212	−	0.511254i
$430$		0			0
$431$	−7.94180e19			−1.38465			−0.692324	−	0.721586i	$-0.743413\pi$
$431$	−7.94180e19			−1.38465			−0.692324	+	0.721586i	$0.743413\pi$
$432$		0			0
$433$	2.54604e19			0.428752			0.214376	−	0.976751i	$-0.431228\pi$
$433$	2.54604e19			0.428752			0.214376	+	0.976751i	$0.431228\pi$
$434$		0			0
$435$	4.67682e19	−	6.90629e19i	0.760818	−	1.12351i
$436$		0			0
$437$		−	2.42309e19i		−	0.380853i
$438$		0			0
$439$			1.80028e19i			0.273436i	0.990610	+	0.136718i	$0.0436554\pi$
$439$			1.80028e19i			0.273436i	−0.990610	+	0.136718i	$0.956345\pi$
$440$		0			0
$441$	−8.38943e18	−	2.09860e19i	−0.123152	−	0.308064i
$442$		0			0
$443$	−6.05876e19			−0.859717			−0.429858	−	0.902896i	$-0.641437\pi$
$443$	−6.05876e19			−0.859717			−0.429858	+	0.902896i	$0.641437\pi$
$444$		0			0
$445$	1.07285e20			1.47177
$446$		0			0
$447$	−8.18392e19	−	5.54201e19i	−1.08556	−	0.735124i
$448$		0			0
$449$			1.25020e20i			1.60374i	0.597500	+	0.801869i	$0.296161\pi$
$449$			1.25020e20i			1.60374i	−0.597500	+	0.801869i	$0.703839\pi$
$450$		0			0
$451$			6.56677e19i			0.814757i
$452$		0			0
$453$	−9.17726e19	−	6.21468e19i	−1.10148	−	0.745903i
$454$		0			0
$455$	4.75376e19			0.552016
$456$		0			0
$457$	−3.33516e19			−0.374753			−0.187376	−	0.982288i	$-0.559998\pi$
$457$	−3.33516e19			−0.374753			−0.187376	+	0.982288i	$0.559998\pi$
$458$		0			0
$459$	−2.22917e18	+	1.02106e19i	−0.0242408	+	0.111034i
$460$		0			0
$461$		−	8.99553e19i		−	0.946825i	−0.880841	−	0.473413i	$-0.843022\pi$
$461$		−	8.99553e19i		−	0.946825i	0.880841	−	0.473413i	$-0.156978\pi$
$462$		0			0
$463$			6.49328e19i			0.661617i	0.943698	+	0.330809i	$0.107321\pi$
$463$			6.49328e19i			0.661617i	−0.943698	+	0.330809i	$0.892679\pi$
$464$		0			0
$465$	−6.10483e19	+	9.01504e19i	−0.602249	+	0.889346i
$466$		0			0
$467$	−1.52133e20			−1.45327			−0.726636	−	0.687023i	$-0.758917\pi$
$467$	−1.52133e20			−1.45327			−0.726636	+	0.687023i	$0.758917\pi$
$468$		0			0
$469$	1.72004e19			0.159126
$470$		0			0
$471$	8.98723e19	−	1.32715e20i	0.805321	−	1.18922i
$472$		0			0
$473$		−	1.19833e19i		−	0.104020i
$474$		0			0
$475$			1.51061e19i			0.127043i
$476$		0			0
$477$	−1.71468e19	+	6.85465e18i	−0.139732	+	0.0558596i
$478$		0			0
$479$	2.55085e18			0.0201450			0.0100725	−	0.999949i	$-0.496794\pi$
$479$	2.55085e18			0.0201450			0.0100725	+	0.999949i	$0.496794\pi$
$480$		0			0
$481$	−2.38093e19			−0.182246
$482$		0			0
$483$	7.01671e19	+	4.75159e19i	0.520631	+	0.352562i
$484$		0			0
$485$			2.05480e20i			1.47811i
$486$		0			0
$487$			8.08671e19i			0.564033i	0.959410	+	0.282016i	$0.0910033\pi$
$487$			8.08671e19i			0.564033i	−0.959410	+	0.282016i	$0.908997\pi$
$488$		0			0
$489$	−1.10425e20	−	7.47778e19i	−0.746880	−	0.505774i
$490$		0			0
$491$	1.73996e20			1.14137			0.570687	−	0.821168i	$-0.306677\pi$
$491$	1.73996e20			1.14137			0.570687	+	0.821168i	$0.306677\pi$
$492$		0			0
$493$	2.81089e19			0.178851
$494$		0			0
$495$	1.02257e20	−	4.08784e19i	0.631180	−	0.252322i
$496$		0			0
$497$			1.71624e20i			1.02779i
$498$		0			0
$499$			1.44476e20i			0.839542i	0.907630	+	0.419771i	$0.137890\pi$
$499$			1.44476e20i			0.839542i	−0.907630	+	0.419771i	$0.862110\pi$
$500$		0			0
$501$	−1.61923e20	+	2.39113e20i	−0.913114	+	1.34840i
$502$		0			0
$503$	2.15006e20			1.17677			0.588385	−	0.808581i	$-0.299764\pi$
$503$	2.15006e20			1.17677			0.588385	+	0.808581i	$0.299764\pi$
$504$		0			0
$505$	−2.06573e19			−0.109746
$506$		0			0
$507$	−4.20305e19	+	6.20667e19i	−0.216773	+	0.320110i
$508$		0			0
$509$		−	3.21744e20i		−	1.61112i	−0.592517	−	0.805558i	$-0.701866\pi$
$509$		−	3.21744e20i		−	1.61112i	0.592517	−	0.805558i	$-0.298134\pi$
$510$		0			0
$511$			3.24142e20i			1.57608i
$512$		0			0
$513$	2.23659e19	−	1.02446e20i	0.105610	−	0.483745i
$514$		0			0
$515$	7.83359e19			0.359258
$516$		0			0
$517$	−2.49898e20			−1.11323
$518$		0			0
$519$	−5.56104e19	−	3.76584e19i	−0.240658	−	0.162970i
$520$		0			0
$521$		−	2.87151e19i		−	0.120733i	−0.998176	−	0.0603666i	$-0.980773\pi$
$521$		−	2.87151e19i		−	0.120733i	0.998176	−	0.0603666i	$-0.0192270\pi$
$522$		0			0
$523$			1.49088e20i			0.609088i	0.952498	+	0.304544i	$0.0985040\pi$
$523$			1.49088e20i			0.609088i	−0.952498	+	0.304544i	$0.901496\pi$
$524$		0			0
$525$	−4.37440e19	−	2.96227e19i	−0.173670	−	0.117606i
$526$		0			0
$527$	−3.66916e19			−0.141575
$528$		0			0
$529$	−1.08881e20			−0.408354
$530$		0			0
$531$	−1.67609e20	−	4.19272e20i	−0.611068	−	1.52858i
$532$		0			0
$533$			2.28362e20i			0.809412i
$534$		0			0
$535$			3.35856e19i			0.115744i
$536$		0			0
$537$	1.14659e19	−	1.69317e19i	0.0384238	−	0.0567407i
$538$		0			0
$539$	−8.02564e19			−0.261556
$540$		0			0
$541$	−2.44980e20			−0.776517			−0.388259	−	0.921550i	$-0.626923\pi$
$541$	−2.44980e20			−0.776517			−0.388259	+	0.921550i	$0.626923\pi$
$542$		0			0
$543$	−1.87238e20	+	2.76496e20i	−0.577291	+	0.852490i
$544$		0			0
$545$			2.16541e20i			0.649481i
$546$		0			0
$547$		−	4.68673e20i		−	1.36762i	−0.729661	−	0.683810i	$-0.760322\pi$
$547$		−	4.68673e20i		−	1.36762i	0.729661	−	0.683810i	$-0.239678\pi$
$548$		0			0
$549$	1.45585e20	+	3.64178e20i	0.413355	+	1.03400i
$550$		0			0
$551$	−2.82025e20			−0.779202
$552$		0			0
$553$	2.88973e20			0.776996
$554$		0			0
$555$	−6.34799e19	−	4.29874e19i	−0.166126	−	0.112498i
$556$		0			0
$557$			2.53853e20i			0.646648i	0.946288	+	0.323324i	$0.104800\pi$
$557$			2.53853e20i			0.646648i	−0.946288	+	0.323324i	$0.895200\pi$
$558$		0			0
$559$		−	4.16722e19i		−	0.103338i
$560$		0			0
$561$	3.07295e19	+	2.08094e19i	0.0741880	+	0.0502388i
$562$		0			0
$563$	3.02808e20			0.711794			0.355897	−	0.934525i	$-0.384175\pi$
$563$	3.02808e20			0.711794			0.355897	+	0.934525i	$0.384175\pi$
$564$		0			0
$565$	5.70851e20			1.30665
$566$		0			0
$567$	2.52803e20	+	2.65661e20i	0.563520	+	0.592180i
$568$		0			0
$569$		−	8.45867e20i		−	1.83637i	−0.396153	−	0.918185i	$-0.629655\pi$
$569$		−	8.45867e20i		−	1.83637i	0.396153	−	0.918185i	$-0.370345\pi$
$570$		0			0
$571$		−	7.64059e20i		−	1.61568i	−0.589400	−	0.807841i	$-0.700636\pi$
$571$		−	7.64059e20i		−	1.61568i	0.589400	−	0.807841i	$-0.299364\pi$
$572$		0			0
$573$	2.87386e20	−	4.24385e20i	0.591978	−	0.874179i
$574$		0			0
$575$	−9.83478e19			−0.197358
$576$		0			0
$577$	−1.03880e20			−0.203102			−0.101551	−	0.994830i	$-0.532380\pi$
$577$	−1.03880e20			−0.203102			−0.101551	+	0.994830i	$0.532380\pi$
$578$		0			0
$579$	−7.94449e19	+	1.17317e20i	−0.151348	+	0.223497i
$580$		0			0
$581$		−	5.09558e20i		−	0.945959i
$582$		0			0
$583$			6.55741e19i			0.118636i
$584$		0			0
$585$	3.55602e20	−	1.42156e20i	0.627039	−	0.250667i
$586$		0			0
$587$	−9.53417e20			−1.63869			−0.819346	−	0.573299i	$-0.805663\pi$
$587$	−9.53417e20			−1.63869			−0.819346	+	0.573299i	$0.805663\pi$
$588$		0			0
$589$	3.68137e20			0.616801
$590$		0			0
$591$	5.20511e20	+	3.52480e20i	0.850205	+	0.575743i
$592$		0			0
$593$		−	1.80245e20i		−	0.287047i	−0.989647	−	0.143524i	$-0.954157\pi$
$593$		−	1.80245e20i		−	0.287047i	0.989647	−	0.143524i	$-0.0458433\pi$
$594$		0			0
$595$			5.15853e19i			0.0801030i
$596$		0			0
$597$	−2.94245e20	−	1.99258e20i	−0.445556	−	0.301722i
$598$		0			0
$599$	7.66398e20			1.13176			0.565879	−	0.824489i	$-0.308537\pi$
$599$	7.66398e20			1.13176			0.565879	+	0.824489i	$0.308537\pi$
$600$		0			0
$601$	5.50070e20			0.792245			0.396122	−	0.918198i	$-0.370356\pi$
$601$	5.50070e20			0.792245			0.396122	+	0.918198i	$0.370356\pi$
$602$		0			0
$603$	1.28666e20	−	5.14359e19i	0.180752	−	0.0722581i
$604$		0			0
$605$			2.38132e20i			0.326327i
$606$		0			0
$607$		−	9.30697e20i		−	1.24421i	−0.782934	−	0.622104i	$-0.786278\pi$
$607$		−	9.30697e20i		−	1.24421i	0.782934	−	0.622104i	$-0.213722\pi$
$608$		0			0
$609$	5.53041e20	−	8.16680e20i	0.721320	−	1.06518i
$610$		0			0
$611$	−8.69031e20			−1.10593
$612$		0			0
$613$	−2.45116e20			−0.304381			−0.152191	−	0.988351i	$-0.548633\pi$
$613$	−2.45116e20			−0.304381			−0.152191	+	0.988351i	$0.548633\pi$
$614$		0			0
$615$	−4.12306e20	+	6.08855e20i	−0.499639	+	0.737820i
$616$		0			0
$617$		−	4.19105e19i		−	0.0495661i	−0.999693	−	0.0247830i	$-0.992111\pi$
$617$		−	4.19105e19i		−	0.0495661i	0.999693	−	0.0247830i	$-0.00788949\pi$
$618$		0			0
$619$		−	1.30977e21i		−	1.51187i	−0.654646	−	0.755935i	$-0.727182\pi$
$619$		−	1.30977e21i		−	1.51187i	0.654646	−	0.755935i	$-0.272818\pi$
$620$		0			0
$621$	6.66972e20	+	1.45612e20i	0.751485	+	0.164063i
$622$		0			0
$623$	1.26867e21			1.39536
$624$		0			0
$625$	−6.31050e20			−0.677585
$626$		0			0
$627$	−3.08318e20	−	2.08787e20i	−0.323215	−	0.218876i
$628$		0			0
$629$		−	2.58366e19i		−	0.0264457i
$630$		0			0
$631$			4.45301e20i			0.445075i	0.974924	+	0.222538i	$0.0714340\pi$
$631$			4.45301e20i			0.445075i	−0.974924	+	0.222538i	$0.928566\pi$
$632$		0			0
$633$	6.55939e20	+	4.44190e20i	0.640230	+	0.433552i
$634$		0			0
$635$	−8.46471e20			−0.806880
$636$		0			0
$637$	−2.79095e20			−0.259840
$638$		0			0
$639$	5.13224e20	+	1.28382e21i	0.466713	+	1.16748i
$640$		0			0
$641$			9.43639e20i			0.838244i	0.907930	+	0.419122i	$0.137662\pi$
$641$			9.43639e20i			0.838244i	−0.907930	+	0.419122i	$0.862338\pi$
$642$		0			0
$643$		−	4.89954e20i		−	0.425180i	−0.977141	−	0.212590i	$-0.931810\pi$
$643$		−	4.89954e20i		−	0.425180i	0.977141	−	0.212590i	$-0.0681899\pi$
$644$		0			0
$645$	7.52389e19	−	1.11106e20i	0.0637888	−	0.0941974i
$646$		0			0
$647$	−1.02118e21			−0.845903			−0.422951	−	0.906152i	$-0.639006\pi$
$647$	−1.02118e21			−0.845903			−0.422951	+	0.906152i	$0.639006\pi$
$648$		0			0
$649$	−1.60341e21			−1.29781
$650$		0			0
$651$	−7.21906e20	+	1.06604e21i	−0.570983	+	0.843175i
$652$		0			0
$653$			5.00646e20i			0.386974i	0.981103	+	0.193487i	$0.0619797\pi$
$653$			5.00646e20i			0.386974i	−0.981103	+	0.193487i	$0.938020\pi$
$654$		0			0
$655$		−	5.17197e20i		−	0.390703i
$656$		0			0
$657$	9.69312e20	+	2.42472e21i	0.715688	+	1.79028i
$658$		0			0
$659$	5.25069e20			0.378945			0.189472	−	0.981886i	$-0.439322\pi$
$659$	5.25069e20			0.378945			0.189472	+	0.981886i	$0.439322\pi$
$660$		0			0
$661$	−8.92703e20			−0.629790			−0.314895	−	0.949126i	$-0.601969\pi$
$661$	−8.92703e20			−0.629790			−0.314895	+	0.949126i	$0.601969\pi$
$662$		0			0
$663$	1.06863e20	+	7.23656e19i	0.0737014	+	0.0499092i
$664$		0			0
$665$		−	5.17571e20i		−	0.348985i
$666$		0			0
$667$		−	1.83611e21i		−	1.21047i
$668$		0			0
$669$	−1.50385e21	−	1.01838e21i	−0.969415	−	0.656471i
$670$		0			0
$671$	1.39272e21			0.877897
$672$		0			0
$673$	−2.52206e21			−1.55469			−0.777343	−	0.629077i	$-0.783433\pi$
$673$	−2.52206e21			−1.55469			−0.777343	+	0.629077i	$0.783433\pi$
$674$		0			0
$675$	−4.15807e20	−	9.07782e19i	−0.250677	−	0.0547272i
$676$		0			0
$677$		−	2.25901e20i		−	0.133200i	−0.997780	−	0.0665998i	$-0.978785\pi$
$677$		−	2.25901e20i		−	0.133200i	0.997780	−	0.0665998i	$-0.0212151\pi$
$678$		0			0
$679$			2.42983e21i			1.40137i
$680$		0			0
$681$	−7.70159e20	+	1.13730e21i	−0.434487	+	0.641611i
$682$		0			0
$683$	1.83490e21			1.01264			0.506321	−	0.862345i	$-0.331005\pi$
$683$	1.83490e21			1.01264			0.506321	+	0.862345i	$0.331005\pi$
$684$		0			0
$685$	−7.07075e20			−0.381756
$686$		0			0
$687$	−1.47645e21	+	2.18028e21i	−0.779906	+	1.15169i
$688$		0			0
$689$			2.28036e20i			0.117858i
$690$		0			0
$691$			6.51879e20i			0.329672i	0.986321	+	0.164836i	$0.0527095\pi$
$691$			6.51879e20i			0.329672i	−0.986321	+	0.164836i	$0.947291\pi$
$692$		0			0
$693$	1.20920e21	−	4.83394e20i	0.598411	−	0.239223i
$694$		0			0
$695$	3.74312e20			0.181279
$696$		0			0
$697$	−2.47807e20			−0.117454
$698$		0			0
$699$	2.78428e21	+	1.88547e21i	1.29162	+	0.874661i
$700$		0			0
$701$			1.74353e19i			0.00791670i	0.999992	+	0.00395835i	$0.00125999\pi$
$701$			1.74353e19i			0.00791670i	−0.999992	+	0.00395835i	$0.998740\pi$
$702$		0			0
$703$			2.59226e20i			0.115216i
$704$		0			0
$705$	−2.31700e21	−	1.56903e21i	−1.00811	−	0.682673i
$706$		0			0
$707$	−2.44276e20			−0.104048
$708$		0			0
$709$	1.86605e21			0.778176			0.389088	−	0.921201i	$-0.372790\pi$
$709$	1.86605e21			0.778176			0.389088	+	0.921201i	$0.372790\pi$
$710$		0			0
$711$	2.16164e21	−	8.64143e20i	0.882595	−	0.352829i
$712$		0			0
$713$			2.39674e21i			0.958185i
$714$		0			0
$715$		−	1.35992e21i		−	0.532375i
$716$		0			0
$717$	−5.35637e20	+	7.90979e20i	−0.205342	+	0.303229i
$718$		0			0
$719$	2.74752e21			1.03151			0.515755	−	0.856736i	$-0.327511\pi$
$719$	2.74752e21			1.03151			0.515755	+	0.856736i	$0.327511\pi$
$720$		0			0
$721$	9.26335e20			0.340607
$722$		0			0
$723$	−8.92869e20	+	1.31851e21i	−0.321551	+	0.474838i
$724$		0			0
$725$			1.14468e21i			0.403783i
$726$		0			0
$727$		−	6.68169e20i		−	0.230876i	−0.993315	−	0.115438i	$-0.963173\pi$
$727$		−	6.68169e20i		−	0.230876i	0.993315	−	0.115438i	$-0.0368271\pi$
$728$		0			0
$729$	2.68550e21	+	1.23127e21i	0.909011	+	0.416771i
$730$		0			0
$731$	4.52205e19			0.0149953
$732$		0			0
$733$	−4.98378e20			−0.161912			−0.0809559	−	0.996718i	$-0.525797\pi$
$733$	−4.98378e20			−0.161912			−0.0809559	+	0.996718i	$0.525797\pi$
$734$		0			0
$735$	−7.44118e20	−	5.03903e20i	−0.236857	−	0.160395i
$736$		0			0
$737$		−	4.92055e20i		−	0.153464i
$738$		0			0
$739$		−	1.35181e21i		−	0.413127i	−0.978433	−	0.206564i	$-0.933772\pi$
$739$		−	1.35181e21i		−	0.413127i	0.978433	−	0.206564i	$-0.0662280\pi$
$740$		0			0
$741$	−1.07219e21	−	7.26066e20i	−0.321095	−	0.217440i
$742$		0			0
$743$	−6.65059e21			−1.95184			−0.975920	−	0.218131i	$-0.930004\pi$
$743$	−6.65059e21			−1.95184			−0.975920	+	0.218131i	$0.930004\pi$
$744$		0			0
$745$	−3.93014e21			−1.13041
$746$		0			0
$747$	−1.52378e21	−	3.81171e21i	−0.429554	−	1.07452i
$748$		0			0
$749$			3.97155e20i			0.109735i
$750$		0			0
$751$			3.72358e20i			0.100846i	0.998728	+	0.0504232i	$0.0160570\pi$
$751$			3.72358e20i			0.100846i	−0.998728	+	0.0504232i	$0.983943\pi$
$752$		0			0
$753$	2.23938e21	−	3.30691e21i	0.594519	−	0.877931i
$754$		0			0
$755$	−4.40717e21			−1.14699
$756$		0			0
$757$	1.17974e21			0.301001			0.150501	−	0.988610i	$-0.451911\pi$
$757$	1.17974e21			0.301001			0.150501	+	0.988610i	$0.451911\pi$
$758$		0			0
$759$	1.35930e21	−	2.00729e21i	0.340018	−	0.502107i
$760$		0			0
$761$		−	2.62003e21i		−	0.642572i	−0.946982	−	0.321286i	$-0.895885\pi$
$761$		−	2.62003e21i		−	0.642572i	0.946982	−	0.321286i	$-0.104115\pi$
$762$		0			0
$763$			2.56063e21i			0.615762i
$764$		0			0
$765$	1.54261e20	+	3.85880e20i	0.0363743	+	0.0909896i
$766$		0			0
$767$	−5.57593e21			−1.28929
$768$		0			0
$769$	7.47130e21			1.69414			0.847069	−	0.531483i	$-0.178365\pi$
$769$	7.47130e21			1.69414			0.847069	+	0.531483i	$0.178365\pi$
$770$		0			0
$771$	−7.38688e20	−	5.00226e20i	−0.164268	−	0.111239i
$772$		0			0
$773$		−	4.61341e21i		−	1.00618i	−0.864234	−	0.503090i	$-0.832197\pi$
$773$		−	4.61341e21i		−	1.00618i	0.864234	−	0.503090i	$-0.167803\pi$
$774$		0			0
$775$		−	1.49419e21i		−	0.319627i
$776$		0			0
$777$	−7.50660e20	−	5.08333e20i	−0.157502	−	0.106657i
$778$		0			0
$779$	2.48632e21			0.511712
$780$		0			0
$781$	4.90969e21			0.991222
$782$		0			0
$783$	1.69479e21	−	7.76293e21i	0.335662	−	1.53749i
$784$		0			0
$785$		−	6.37334e21i		−	1.23835i
$786$		0			0
$787$			5.71532e21i			1.08951i	0.838596	+	0.544753i	$0.183377\pi$
$787$			5.71532e21i			1.08951i	−0.838596	+	0.544753i	$0.816623\pi$
$788$		0			0
$789$	−2.96215e21	+	4.37424e21i	−0.554026	+	0.818134i
$790$		0			0
$791$	6.75040e21			1.23881
$792$		0			0
$793$	4.84322e21			0.872139
$794$		0			0
$795$	−4.11718e20	+	6.07987e20i	−0.0727523	+	0.107434i
$796$		0			0
$797$			9.61551e21i			1.66738i	0.552232	+	0.833691i	$0.313776\pi$
$797$			9.61551e21i			1.66738i	−0.552232	+	0.833691i	$0.686224\pi$
$798$		0			0
$799$		−	9.43027e20i		−	0.160481i
$800$		0			0
$801$	9.49018e21	−	3.79382e21i	1.58500	−	0.633626i
$802$		0			0
$803$	9.27280e21			1.52000
$804$		0			0
$805$	3.36962e21			0.542140
$806$		0			0
$807$	−8.27865e21	−	5.60616e21i	−1.30740	−	0.885347i
$808$		0			0
$809$		−	1.10432e21i		−	0.171191i	−0.996330	−	0.0855955i	$-0.972721\pi$
$809$		−	1.10432e21i		−	0.171191i	0.996330	−	0.0855955i	$-0.0272793\pi$
$810$		0			0
$811$			3.10967e21i			0.473214i	0.971605	+	0.236607i	$0.0760354\pi$
$811$			3.10967e21i			0.473214i	−0.971605	+	0.236607i	$0.923965\pi$
$812$		0			0
$813$	−9.37495e21	−	6.34855e21i	−1.40052	−	0.948409i
$814$		0			0
$815$	−5.30291e21			−0.777735
$816$		0			0
$817$	−4.53711e20			−0.0653301
$818$		0			0
$819$	4.20505e21	−	1.68102e21i	0.594486	−	0.237653i
$820$		0			0
$821$		−	1.11643e22i		−	1.54974i	−0.632122	−	0.774869i	$-0.717816\pi$
$821$		−	1.11643e22i		−	1.54974i	0.632122	−	0.774869i	$-0.282184\pi$
$822$		0			0
$823$		−	5.21263e21i		−	0.710491i	−0.934773	−	0.355245i	$-0.884397\pi$
$823$		−	5.21263e21i		−	0.710491i	0.934773	−	0.355245i	$-0.115603\pi$
$824$		0			0
$825$	−8.47422e20	+	1.25140e21i	−0.113421	+	0.167490i
$826$		0			0
$827$	−1.43684e21			−0.188850			−0.0944250	−	0.995532i	$-0.530101\pi$
$827$	−1.43684e21			−0.188850			−0.0944250	+	0.995532i	$0.530101\pi$
$828$		0			0
$829$	1.36226e22			1.75833			0.879167	−	0.476514i	$-0.158100\pi$
$829$	1.36226e22			1.75833			0.879167	+	0.476514i	$0.158100\pi$
$830$		0			0
$831$	−9.10706e20	+	1.34485e21i	−0.115444	+	0.170477i
$832$		0			0
$833$		−	3.02859e20i		−	0.0377053i
$834$		0			0
$835$			1.14828e22i			1.40411i
$836$		0			0
$837$	−2.21227e21	+	1.01332e22i	−0.265703	+	1.21705i
$838$		0			0
$839$	6.66937e21			0.786811			0.393405	−	0.919365i	$-0.371297\pi$
$839$	6.66937e21			0.786811			0.393405	+	0.919365i	$0.371297\pi$
$840$		0			0
$841$	−1.27414e22			−1.47654
$842$		0			0
$843$	−1.63658e21	−	1.10826e21i	−0.186307	−	0.126164i
$844$		0			0
$845$			2.98061e21i			0.333334i
$846$		0			0
$847$			2.81595e21i			0.309385i
$848$		0			0
$849$	−1.41426e22	−	9.57713e21i	−1.52659	−	1.03378i
$850$		0			0
$851$	−1.68768e21			−0.178985
$852$		0			0
$853$	3.11672e21			0.324773			0.162387	−	0.986727i	$-0.448081\pi$
$853$	3.11672e21			0.324773			0.162387	+	0.986727i	$0.448081\pi$
$854$		0			0
$855$	−1.54774e21	−	3.87165e21i	−0.158472	−	0.396415i
$856$		0			0
$857$			9.64432e21i			0.970321i	0.874425	+	0.485161i	$0.161239\pi$
$857$			9.64432e21i			0.970321i	−0.874425	+	0.485161i	$0.838761\pi$
$858$		0			0
$859$			1.39397e22i			1.37818i	0.724676	+	0.689090i	$0.241989\pi$
$859$			1.39397e22i			1.37818i	−0.724676	+	0.689090i	$0.758011\pi$
$860$		0			0
$861$	−4.87558e21	+	7.19981e21i	−0.473700	+	0.699516i
$862$		0			0
$863$	8.15452e20			0.0778606			0.0389303	−	0.999242i	$-0.487605\pi$
$863$	8.15452e20			0.0778606			0.0389303	+	0.999242i	$0.487605\pi$
$864$		0			0
$865$	−2.67057e21			−0.250601
$866$		0			0
$867$	6.00120e21	−	8.86202e21i	0.553471	−	0.817315i
$868$		0			0
$869$		−	8.26671e21i		−	0.749350i
$870$		0			0
$871$		−	1.71114e21i		−	0.152458i
$872$		0			0
$873$	7.26615e21	+	1.81762e22i	0.636352	+	1.59182i
$874$		0			0
$875$	−1.02880e22			−0.885668
$876$		0			0
$877$	−7.95130e21			−0.672886			−0.336443	−	0.941704i	$-0.609224\pi$
$877$	−7.95130e21			−0.672886			−0.336443	+	0.941704i	$0.609224\pi$
$878$		0			0
$879$	−2.20458e21	−	1.49290e21i	−0.183405	−	0.124198i
$880$		0			0
$881$			2.15402e22i			1.76169i	0.473404	+	0.880845i	$0.343025\pi$
$881$			2.15402e22i			1.76169i	−0.473404	+	0.880845i	$0.656975\pi$
$882$		0			0
$883$			1.43035e22i			1.15011i	0.818116	+	0.575054i	$0.195019\pi$
$883$			1.43035e22i			1.15011i	−0.818116	+	0.575054i	$0.804981\pi$
$884$		0			0
$885$	−1.48665e22	−	1.00673e22i	−1.17526	−	0.795863i
$886$		0			0
$887$	−2.20968e22			−1.71752			−0.858762	−	0.512375i	$-0.828766\pi$
$887$	−2.20968e22			−1.71752			−0.858762	+	0.512375i	$0.828766\pi$
$888$		0			0
$889$	−1.00097e22			−0.764990
$890$		0			0
$891$	7.59981e21	−	7.23199e21i	0.571110	−	0.543469i
$892$		0			0
$893$			9.46167e21i			0.699168i
$894$		0			0
$895$		−	8.13109e20i		−	0.0590848i
$896$		0			0
$897$	4.72701e21	−	6.98042e21i	0.337787	−	0.498813i
$898$		0			0
$899$	2.78958e22			1.96039
$900$		0			0
$901$	−2.47453e20			−0.0171024
$902$		0			0
$903$	8.89712e20	−	1.31384e21i	0.0604772	−	0.0893071i
$904$		0			0
$905$			1.32781e22i			0.887709i
$906$		0			0
$907$			8.30476e21i			0.546101i	0.962000	+	0.273050i	$0.0880325\pi$
$907$			8.30476e21i			0.546101i	−0.962000	+	0.273050i	$0.911967\pi$
$908$		0			0
$909$	−1.82729e21	+	7.30481e20i	−0.118189	+	0.0472476i
$910$		0			0
$911$	9.32002e19			0.00592965			0.00296483	−	0.999996i	$-0.499056\pi$
$911$	9.32002e19			0.00592965			0.00296483	+	0.999996i	$0.499056\pi$
$912$		0			0
$913$	−1.45770e22			−0.912301
$914$		0			0
$915$	1.29129e22	+	8.74440e21i	0.794999	+	0.538359i
$916$		0			0
$917$		−	6.11594e21i		−	0.370419i
$918$		0			0
$919$		−	5.62410e20i		−	0.0335109i	−0.999860	−	0.0167555i	$-0.994666\pi$
$919$		−	5.62410e20i		−	0.0335109i	0.999860	−	0.0167555i	$-0.00533368\pi$
$920$		0			0
$921$	2.16027e21	+	1.46289e21i	0.126637	+	0.0857562i
$922$		0			0
$923$	1.70736e22			0.984720
$924$		0			0
$925$	1.05214e21			0.0597050
$926$		0			0
$927$	6.92938e21	−	2.77011e21i	0.386898	−	0.154667i
$928$		0			0
$929$		−	1.09996e22i		−	0.604308i	−0.953259	−	0.302154i	$-0.902294\pi$
$929$		−	1.09996e22i		−	0.604308i	0.953259	−	0.302154i	$-0.0977057\pi$
$930$		0			0
$931$			3.03867e21i			0.164271i
$932$		0			0
$933$	4.21410e21	−	6.22300e21i	0.224178	−	0.331045i
$934$		0			0
$935$	1.47571e21			0.0772529
$936$		0			0
$937$	−1.66662e21			−0.0858595			−0.0429298	−	0.999078i	$-0.513669\pi$
$937$	−1.66662e21			−0.0858595			−0.0429298	+	0.999078i	$0.513669\pi$
$938$		0			0
$939$	−1.53697e22	+	2.26965e22i	−0.779241	+	1.15071i
$940$		0			0
$941$		−	1.63299e21i		−	0.0814821i	−0.999170	−	0.0407410i	$-0.987028\pi$
$941$		−	1.63299e21i		−	0.0814821i	0.999170	−	0.0407410i	$-0.0129719\pi$
$942$		0			0
$943$			1.61870e22i			0.794931i
$944$		0			0
$945$	1.42465e22	+	3.11027e21i	0.688604	+	0.150335i
$946$		0			0
$947$	−1.84506e21			−0.0877781			−0.0438891	−	0.999036i	$-0.513975\pi$
$947$	−1.84506e21			−0.0877781			−0.0438891	+	0.999036i	$0.513975\pi$
$948$		0			0
$949$	3.22465e22			1.51003
$950$		0			0
$951$	−2.99218e21	−	2.02625e21i	−0.137922	−	0.0933985i
$952$		0			0
$953$		−	3.45734e22i		−	1.56872i	−0.620307	−	0.784359i	$-0.712992\pi$
$953$		−	3.45734e22i		−	1.56872i	0.620307	−	0.784359i	$-0.287008\pi$
$954$		0			0
$955$		−	2.03801e22i		−	0.910294i
$956$		0			0
$957$	−2.33630e22	−	1.58210e22i	−1.02728	−	0.695655i
$958$		0			0
$959$	−8.36128e21			−0.361937
$960$		0			0
$961$	−1.29482e22			−0.551804
$962$		0			0
$963$	1.18765e21	+	2.97089e21i	0.0498301	+	0.124649i
$964$		0			0
$965$			5.63388e21i			0.232730i
$966$		0			0
$967$		−	3.90998e22i		−	1.59029i	−0.606421	−	0.795144i	$-0.707395\pi$
$967$		−	3.90998e22i		−	1.59029i	0.606421	−	0.795144i	$-0.292605\pi$
$968$		0			0
$969$	7.87888e20	−	1.16348e21i	0.0315527	−	0.0465941i
$970$		0			0
$971$	−2.15169e22			−0.848467			−0.424234	−	0.905553i	$-0.639456\pi$
$971$	−2.15169e22			−0.848467			−0.424234	+	0.905553i	$0.639456\pi$
$972$		0			0
$973$	4.42630e21			0.171868
$974$		0			0
$975$	−2.94694e21	+	4.35177e21i	−0.112677	+	0.166392i
$976$		0			0
$977$			7.50230e21i			0.282478i	0.989976	+	0.141239i	$0.0451087\pi$
$977$			7.50230e21i			0.282478i	−0.989976	+	0.141239i	$0.954891\pi$
$978$		0			0
$979$		−	3.62931e22i		−	1.34572i
$980$		0			0
$981$	7.65730e21	+	1.91546e22i	0.279614	+	0.699449i
$982$		0			0
$983$	1.14976e22			0.413480			0.206740	−	0.978396i	$-0.433715\pi$
$983$	1.14976e22			0.413480			0.206740	+	0.978396i	$0.433715\pi$
$984$		0			0
$985$	2.49963e22			0.885329
$986$		0			0
$987$	−2.73989e22	−	1.85540e22i	−0.955771	−	0.647231i
$988$		0			0
$989$		−	2.95386e21i		−	0.101489i
$990$		0			0
$991$		−	5.24451e21i		−	0.177481i	−0.996055	−	0.0887406i	$-0.971716\pi$
$991$		−	5.24451e21i		−	0.177481i	0.996055	−	0.0887406i	$-0.0282842\pi$
$992$		0			0
$993$	4.60358e22	+	3.11746e22i	1.53453	+	1.03916i
$994$		0			0
$995$	−1.41305e22			−0.463963
$996$		0			0
$997$	3.60420e22			1.16572			0.582861	−	0.812572i	$-0.301933\pi$
$997$	3.60420e22			1.16572			0.582861	+	0.812572i	$0.301933\pi$
$998$		0			0
$999$	−7.13537e21	−	1.55778e21i	−0.227340	−	0.0496324i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.16.c.d.47.13	yes	20
3.2	odd	2	inner	48.16.c.d.47.7	✓	20
4.3	odd	2	inner	48.16.c.d.47.8	yes	20
12.11	even	2	inner	48.16.c.d.47.14	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.16.c.d.47.7	✓	20	3.2	odd	2	inner
48.16.c.d.47.8	yes	20	4.3	odd	2	inner
48.16.c.d.47.13	yes	20	1.1	even	1	trivial
48.16.c.d.47.14	yes	20	12.11	even	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)

\(f(q)\)	\(=\)	\(q+(2123.98 - 3136.50i) q^{3} -150623. i q^{5} -1.78114e6i q^{7} +(-5.32633e6 - 1.33237e7i) q^{9} +O(q^{10})\)	\(q+(2123.98 - 3136.50i) q^{3} -150623. i q^{5} -1.78114e6i q^{7} +(-5.32633e6 - 1.33237e7i) q^{9} -5.09536e7 q^{11} -1.77193e8 q^{13} +(-4.72429e8 - 3.19921e8i) q^{15} -1.92281e8i q^{17} +1.92921e9i q^{19} +(-5.58655e9 - 3.78311e9i) q^{21} -1.25600e10 q^{23} +7.83023e9 q^{25} +(-5.31028e10 - 1.15933e10i) q^{27} +1.46187e11i q^{29} -1.90823e11i q^{31} +(-1.08224e11 + 1.59816e11i) q^{33} -2.68282e11 q^{35} +1.34369e11 q^{37} +(-3.76354e11 + 5.55765e11i) q^{39} -1.28878e12i q^{41} +2.35180e11i q^{43} +(-2.00686e12 + 8.02268e11i) q^{45} +4.90443e12 q^{47} +1.57509e12 q^{49} +(-6.03088e11 - 4.08400e11i) q^{51} -1.28694e12i q^{53} +7.67479e12i q^{55} +(6.05095e12 + 4.09760e12i) q^{57} +3.14681e13 q^{59} -2.73330e13 q^{61} +(-2.37315e13 + 9.48695e12i) q^{63} +2.66894e13i q^{65} +9.65692e12i q^{67} +(-2.66772e13 + 3.93944e13i) q^{69} -9.63560e13 q^{71} -1.81985e14 q^{73} +(1.66313e13 - 2.45595e13i) q^{75} +9.07557e13i q^{77} +1.62240e14i q^{79} +(-1.49152e14 + 1.41933e14i) q^{81} +2.86085e14 q^{83} -2.89619e13 q^{85} +(4.58514e14 + 3.10498e14i) q^{87} +7.12277e14i q^{89} +3.15606e14i q^{91} +(-5.98516e14 - 4.05305e14i) q^{93} +2.90583e14 q^{95} -1.36420e15 q^{97} +(2.71395e14 + 6.78891e14i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2271972 q^{9}+O(q^{10}) \) 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

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