LMFDB - Embedded newform 48.16.c.d.47.18

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			47.18
Root			$0.500000 + 86.2495i$ of defining polynomial
Character	$\chi$	$=$	48.47
Dual form			48.16.c.d.47.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3468.03 + 1523.70i) q^{3} +104758. i q^{5} -2.34829e6i q^{7} +(9.70559e6 + 1.05685e7i) q^{9} +O(q^{10})$	\(q+(3468.03 + 1523.70i) q^{3} +104758. i q^{5} -2.34829e6i q^{7} +(9.70559e6 + 1.05685e7i) q^{9} -5.92633e7 q^{11} +1.86301e8 q^{13} +(-1.59619e8 + 3.63303e8i) q^{15} +2.96192e7i q^{17} -1.60508e9i q^{19} +(3.57809e9 - 8.14395e9i) q^{21} +2.10506e10 q^{23} +1.95434e10 q^{25} +(1.75561e10 + 5.14402e10i) q^{27} +2.65850e7i q^{29} +1.58426e11i q^{31} +(-2.05527e11 - 9.02995e10i) q^{33} +2.46002e11 q^{35} +8.36572e11 q^{37} +(6.46098e11 + 2.83867e11i) q^{39} -4.46787e11i q^{41} -1.80685e12i q^{43} +(-1.10713e12 + 1.01674e12i) q^{45} -4.90433e12 q^{47} -7.66916e11 q^{49} +(-4.51308e10 + 1.02721e11i) q^{51} +1.50184e13i q^{53} -6.20829e12i q^{55} +(2.44566e12 - 5.56648e12i) q^{57} +7.28757e12 q^{59} -7.96889e12 q^{61} +(2.48179e13 - 2.27916e13i) q^{63} +1.95165e13i q^{65} +5.25193e13i q^{67} +(7.30043e13 + 3.20749e13i) q^{69} +5.59670e13 q^{71} +1.05153e14 q^{73} +(6.77771e13 + 2.97783e13i) q^{75} +1.39168e14i q^{77} -1.43999e14i q^{79} +(-1.74943e13 + 2.05147e14i) q^{81} +3.35783e14 q^{83} -3.10285e12 q^{85} +(-4.05076e10 + 9.21978e10i) q^{87} +7.03539e14i q^{89} -4.37489e14i q^{91} +(-2.41394e14 + 5.49427e14i) q^{93} +1.68145e14 q^{95} +1.41944e15 q^{97} +(-5.75186e14 - 6.26323e14i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3468.03	+	1523.70i	0.915532	+	0.402244i
$3$	3468.03	+	1523.70i	0.915532	+	0.402244i
$4$		0			0
$5$			104758.i			0.599668i	0.953991	+	0.299834i	$0.0969313\pi$
$5$			104758.i			0.599668i	−0.953991	+	0.299834i	$0.903069\pi$
$6$		0			0
$7$		−	2.34829e6i		−	1.07775i	−0.842387	−	0.538874i	$-0.818850\pi$
$7$		−	2.34829e6i		−	1.07775i	0.842387	−	0.538874i	$-0.181150\pi$
$8$		0			0
$9$	9.70559e6	+	1.05685e7i	0.676399	+	0.736535i
$10$		0			0
$11$	−5.92633e7			−0.916940			−0.458470	−	0.888710i	$-0.651602\pi$
$11$	−5.92633e7			−0.916940			−0.458470	+	0.888710i	$0.651602\pi$
$12$		0			0
$13$	1.86301e8			0.823455			0.411728	−	0.911307i	$-0.364925\pi$
$13$	1.86301e8			0.823455			0.411728	+	0.911307i	$0.364925\pi$
$14$		0			0
$15$	−1.59619e8	+	3.63303e8i	−0.241213	+	0.549016i
$16$		0			0
$17$			2.96192e7i			0.0175068i	0.999962	+	0.00875341i	$0.00278633\pi$
$17$			2.96192e7i			0.0175068i	−0.999962	+	0.00875341i	$0.997214\pi$
$18$		0			0
$19$		−	1.60508e9i		−	0.411951i	−0.978557	−	0.205975i	$-0.933963\pi$
$19$		−	1.60508e9i		−	0.411951i	0.978557	−	0.205975i	$-0.0660367\pi$
$20$		0			0
$21$	3.57809e9	−	8.14395e9i	0.433518	−	0.986712i
$22$		0			0
$23$	2.10506e10			1.28916			0.644580	−	0.764537i	$-0.277032\pi$
$23$	2.10506e10			1.28916			0.644580	+	0.764537i	$0.277032\pi$
$24$		0			0
$25$	1.95434e10			0.640398
$26$		0			0
$27$	1.75561e10	+	5.14402e10i	0.322998	+	0.946400i
$28$		0			0
$29$			2.65850e7i			0.000286189i	1.00000		0.000143094i	$4.55483e-5\pi$
$29$			2.65850e7i			0.000286189i	−1.00000		0.000143094i	$0.999954\pi$
$30$		0			0
$31$			1.58426e11i			1.03422i	0.855918	+	0.517112i	$0.172993\pi$
$31$			1.58426e11i			1.03422i	−0.855918	+	0.517112i	$0.827007\pi$
$32$		0			0
$33$	−2.05527e11	−	9.02995e10i	−0.839488	−	0.368834i
$34$		0			0
$35$	2.46002e11			0.646291
$36$		0			0
$37$	8.36572e11			1.44874			0.724370	−	0.689411i	$-0.242131\pi$
$37$	8.36572e11			1.44874			0.724370	+	0.689411i	$0.242131\pi$
$38$		0			0
$39$	6.46098e11	+	2.83867e11i	0.753900	+	0.331230i
$40$		0			0
$41$		−	4.46787e11i		−	0.358280i	−0.983824	−	0.179140i	$-0.942669\pi$
$41$		−	4.46787e11i		−	0.358280i	0.983824	−	0.179140i	$-0.0573314\pi$
$42$		0			0
$43$		−	1.80685e12i		−	1.01370i	−0.862035	−	0.506849i	$-0.830810\pi$
$43$		−	1.80685e12i		−	1.01370i	0.862035	−	0.506849i	$-0.169190\pi$
$44$		0			0
$45$	−1.10713e12	+	1.01674e12i	−0.441677	+	0.405615i
$46$		0			0
$47$	−4.90433e12			−1.41204			−0.706018	−	0.708194i	$-0.749510\pi$
$47$	−4.90433e12			−1.41204			−0.706018	+	0.708194i	$0.749510\pi$
$48$		0			0
$49$	−7.66916e11			−0.161539
$50$		0			0
$51$	−4.51308e10	+	1.02721e11i	−0.00704202	+	0.0160281i
$52$		0			0
$53$			1.50184e13i			1.75612i	0.478548	+	0.878061i	$0.341163\pi$
$53$			1.50184e13i			1.75612i	−0.478548	+	0.878061i	$0.658837\pi$
$54$		0			0
$55$		−	6.20829e12i		−	0.549860i
$56$		0			0
$57$	2.44566e12	−	5.56648e12i	0.165705	−	0.377154i
$58$		0			0
$59$	7.28757e12			0.381235			0.190618	−	0.981664i	$-0.438951\pi$
$59$	7.28757e12			0.381235			0.190618	+	0.981664i	$0.438951\pi$
$60$		0			0
$61$	−7.96889e12			−0.324657			−0.162328	−	0.986737i	$-0.551900\pi$
$61$	−7.96889e12			−0.324657			−0.162328	+	0.986737i	$0.551900\pi$
$62$		0			0
$63$	2.48179e13	−	2.27916e13i	0.793799	−	0.728987i
$64$		0			0
$65$			1.95165e13i			0.493800i
$66$		0			0
$67$			5.25193e13i			1.05866i	0.848415	+	0.529332i	$0.177557\pi$
$67$			5.25193e13i			1.05866i	−0.848415	+	0.529332i	$0.822443\pi$
$68$		0			0
$69$	7.30043e13	+	3.20749e13i	1.18027	+	0.518557i
$70$		0			0
$71$	5.59670e13			0.730289			0.365144	−	0.930951i	$-0.381020\pi$
$71$	5.59670e13			0.730289			0.365144	+	0.930951i	$0.381020\pi$
$72$		0			0
$73$	1.05153e14			1.11404			0.557020	−	0.830499i	$-0.311945\pi$
$73$	1.05153e14			1.11404			0.557020	+	0.830499i	$0.311945\pi$
$74$		0			0
$75$	6.77771e13	+	2.97783e13i	0.586305	+	0.257596i
$76$		0			0
$77$			1.39168e14i			0.988230i
$78$		0			0
$79$		−	1.43999e14i		−	0.843638i	−0.906680	−	0.421819i	$-0.861392\pi$
$79$		−	1.43999e14i		−	0.843638i	0.906680	−	0.421819i	$-0.138608\pi$
$80$		0			0
$81$	−1.74943e13	+	2.05147e14i	−0.0849687	+	0.996384i
$82$		0			0
$83$	3.35783e14			1.35823			0.679114	−	0.734033i	$-0.262364\pi$
$83$	3.35783e14			1.35823			0.679114	+	0.734033i	$0.262364\pi$
$84$		0			0
$85$	−3.10285e12			−0.0104983
$86$		0			0
$87$	−4.05076e10	+	9.21978e10i	−0.000115118	+	0.000262015i
$88$		0			0
$89$			7.03539e14i			1.68602i	0.537898	+	0.843010i	$0.319219\pi$
$89$			7.03539e14i			1.68602i	−0.537898	+	0.843010i	$0.680781\pi$
$90$		0			0
$91$		−	4.37489e14i		−	0.887477i
$92$		0			0
$93$	−2.41394e14	+	5.49427e14i	−0.416011	+	0.946865i
$94$		0			0
$95$	1.68145e14			0.247034
$96$		0			0
$97$	1.41944e15			1.78372			0.891862	−	0.452308i	$-0.149399\pi$
$97$	1.41944e15			1.78372			0.891862	+	0.452308i	$0.149399\pi$
$98$		0			0
$99$	−5.75186e14	−	6.26323e14i	−0.620217	−	0.675359i
$100$		0			0
$101$			2.00409e15i			1.85997i	0.367593	+	0.929987i	$0.380182\pi$
$101$			2.00409e15i			1.85997i	−0.367593	+	0.929987i	$0.619818\pi$
$102$		0			0
$103$		−	6.58638e14i		−	0.527676i	−0.964567	−	0.263838i	$-0.915012\pi$
$103$		−	6.58638e14i		−	0.527676i	0.964567	−	0.263838i	$-0.0849885\pi$
$104$		0			0
$105$	8.53142e14	+	3.74833e14i	0.591700	+	0.259967i
$106$		0			0
$107$	2.15242e15			1.29583			0.647916	−	0.761712i	$-0.275641\pi$
$107$	2.15242e15			1.29583			0.647916	+	0.761712i	$0.275641\pi$
$108$		0			0
$109$	−8.22674e14			−0.431051			−0.215526	−	0.976498i	$-0.569147\pi$
$109$	−8.22674e14			−0.431051			−0.215526	+	0.976498i	$0.569147\pi$
$110$		0			0
$111$	2.90126e15	+	1.27468e15i	1.32637	+	0.582748i
$112$		0			0
$113$		−	2.22564e15i		−	0.889951i	−0.895543	−	0.444975i	$-0.853212\pi$
$113$		−	2.22564e15i		−	0.889951i	0.895543	−	0.444975i	$-0.146788\pi$
$114$		0			0
$115$			2.20522e15i			0.773068i
$116$		0			0
$117$	1.80816e15	+	1.96892e15i	0.556984	+	0.606504i
$118$		0			0
$119$	6.95547e13			0.0188679
$120$		0			0
$121$	−6.65104e14			−0.159221
$122$		0			0
$123$	6.80770e14	−	1.54947e15i	0.144116	−	0.328017i
$124$		0			0
$125$			5.24427e15i			0.983695i
$126$		0			0
$127$			3.79715e14i			0.0632309i	0.999500	+	0.0316155i	$0.0100652\pi$
$127$			3.79715e14i			0.0632309i	−0.999500	+	0.0316155i	$0.989935\pi$
$128$		0			0
$129$	2.75310e15	−	6.26622e15i	0.407755	−	0.928074i
$130$		0			0
$131$	−7.40897e15			−0.977739			−0.488869	−	0.872357i	$-0.662591\pi$
$131$	−7.40897e15			−0.977739			−0.488869	+	0.872357i	$0.662591\pi$
$132$		0			0
$133$	−3.76920e15			−0.443979
$134$		0			0
$135$	−5.38876e15	+	1.83914e15i	−0.567526	+	0.193692i
$136$		0			0
$137$		−	1.07531e16i		−	1.01421i	−0.861885	−	0.507104i	$-0.830716\pi$
$137$		−	1.07531e16i		−	1.01421i	0.861885	−	0.507104i	$-0.169284\pi$
$138$		0			0
$139$		−	5.46664e15i		−	0.462498i	−0.972895	−	0.231249i	$-0.925719\pi$
$139$		−	5.46664e15i		−	0.462498i	0.972895	−	0.231249i	$-0.0742812\pi$
$140$		0			0
$141$	−1.70084e16	−	7.47273e15i	−1.29276	−	0.567984i
$142$		0			0
$143$	−1.10408e16			−0.755059
$144$		0			0
$145$	−2.78499e12			−0.000171618
$146$		0			0
$147$	−2.65969e15	−	1.16855e15i	−0.147894	−	0.0649781i
$148$		0			0
$149$		−	1.61598e16i		−	0.811970i	−0.913880	−	0.405985i	$-0.866929\pi$
$149$		−	1.61598e16i		−	0.811970i	0.913880	−	0.405985i	$-0.133071\pi$
$150$		0			0
$151$		−	1.53774e16i		−	0.699126i	−0.936913	−	0.349563i	$-0.886330\pi$
$151$		−	1.53774e16i		−	0.699126i	0.936913	−	0.349563i	$-0.113670\pi$
$152$		0			0
$153$	−3.13030e14	+	2.87472e14i	−0.0128944	+	0.0118416i
$154$		0			0
$155$	−1.65964e16			−0.620191
$156$		0			0
$157$	−2.46806e16			−0.837738			−0.418869	−	0.908047i	$-0.637573\pi$
$157$	−2.46806e16			−0.837738			−0.418869	+	0.908047i	$0.637573\pi$
$158$		0			0
$159$	−2.28835e16	+	5.20843e16i	−0.706390	+	1.60779i
$160$		0			0
$161$		−	4.94331e16i		−	1.38939i
$162$		0			0
$163$			4.90523e16i			1.25676i	0.777906	+	0.628380i	$0.216282\pi$
$163$			4.90523e16i			1.25676i	−0.777906	+	0.628380i	$0.783718\pi$
$164$		0			0
$165$	9.45957e15	−	2.15306e16i	0.221178	−	0.503415i
$166$		0			0
$167$	2.04146e16			0.436082			0.218041	−	0.975940i	$-0.430033\pi$
$167$	2.04146e16			0.436082			0.218041	+	0.975940i	$0.430033\pi$
$168$		0			0
$169$	−1.64778e16			−0.321921
$170$		0			0
$171$	1.69633e16	−	1.55783e16i	0.303416	−	0.278643i
$172$		0			0
$173$		−	6.97946e16i		−	1.14413i	−0.820208	−	0.572066i	$-0.806142\pi$
$173$		−	6.97946e16i		−	1.14413i	0.820208	−	0.572066i	$-0.193858\pi$
$174$		0			0
$175$		−	4.58936e16i		−	0.690187i
$176$		0			0
$177$	2.52735e16	+	1.11041e16i	0.349033	+	0.153350i
$178$		0			0
$179$	4.36032e16			0.553504			0.276752	−	0.960941i	$-0.410742\pi$
$179$	4.36032e16			0.553504			0.276752	+	0.960941i	$0.410742\pi$
$180$		0			0
$181$	1.61264e16			0.188343			0.0941714	−	0.995556i	$-0.469980\pi$
$181$	1.61264e16			0.188343			0.0941714	+	0.995556i	$0.469980\pi$
$182$		0			0
$183$	−2.76364e16	−	1.21422e16i	−0.297234	−	0.130591i
$184$		0			0
$185$			8.76374e16i			0.868764i
$186$		0			0
$187$		−	1.75534e15i		−	0.0160527i
$188$		0			0
$189$	1.20797e17	−	4.12269e16i	1.01998	−	0.348110i
$190$		0			0
$191$	−7.85694e16			−0.613060			−0.306530	−	0.951861i	$-0.599168\pi$
$191$	−7.85694e16			−0.613060			−0.306530	+	0.951861i	$0.599168\pi$
$192$		0			0
$193$	−1.69068e17			−1.22006			−0.610031	−	0.792378i	$-0.708843\pi$
$193$	−1.69068e17			−1.22006			−0.610031	+	0.792378i	$0.708843\pi$
$194$		0			0
$195$	−2.97372e16	+	6.76838e16i	−0.198628	+	0.452090i
$196$		0			0
$197$			3.07024e17i			1.89966i	0.312773	+	0.949828i	$0.398742\pi$
$197$			3.07024e17i			1.89966i	−0.312773	+	0.949828i	$0.601258\pi$
$198$		0			0
$199$		−	2.11412e17i		−	1.21264i	−0.795222	−	0.606318i	$-0.792646\pi$
$199$		−	2.11412e17i		−	1.21264i	0.795222	−	0.606318i	$-0.207354\pi$
$200$		0			0
$201$	−8.00237e16	+	1.82139e17i	−0.425842	+	0.969241i
$202$		0			0
$203$	6.24295e13			0.000308439
$204$		0			0
$205$	4.68044e16			0.214849
$206$		0			0
$207$	2.04309e17	+	2.22473e17i	0.871986	+	0.949512i
$208$		0			0
$209$			9.51226e16i			0.377734i
$210$		0			0
$211$		−	5.26872e17i		−	1.94799i	−0.226567	−	0.973996i	$-0.572750\pi$
$211$		−	5.26872e17i		−	1.94799i	0.226567	−	0.973996i	$-0.427250\pi$
$212$		0			0
$213$	1.94095e17	+	8.52769e16i	0.668603	+	0.293755i
$214$		0			0
$215$	1.89282e17			0.607883
$216$		0			0
$217$	3.72031e17			1.11463
$218$		0			0
$219$	3.64675e17	+	1.60222e17i	1.01994	+	0.448116i
$220$		0			0
$221$			5.51810e15i			0.0144161i
$222$		0			0
$223$			1.54274e17i			0.376708i	0.982101	+	0.188354i	$0.0603152\pi$
$223$			1.54274e17i			0.376708i	−0.982101	+	0.188354i	$0.939685\pi$
$224$		0			0
$225$	1.89680e17	+	2.06544e17i	0.433165	+	0.471676i
$226$		0			0
$227$	4.81676e17			1.02935			0.514673	−	0.857387i	$-0.327913\pi$
$227$	4.81676e17			1.02935			0.514673	+	0.857387i	$0.327913\pi$
$228$		0			0
$229$	1.81185e17			0.362541			0.181270	−	0.983433i	$-0.441979\pi$
$229$	1.81185e17			0.362541			0.181270	+	0.983433i	$0.441979\pi$
$230$		0			0
$231$	−2.12050e17	+	4.82638e17i	−0.397510	+	0.904756i
$232$		0			0
$233$			6.14480e17i			1.07979i	0.841733	+	0.539894i	$0.181536\pi$
$233$			6.14480e17i			1.07979i	−0.841733	+	0.539894i	$0.818464\pi$
$234$		0			0
$235$		−	5.13767e17i		−	0.846753i
$236$		0			0
$237$	2.19411e17	−	4.99393e17i	0.339349	−	0.772378i
$238$		0			0
$239$	−2.49448e17			−0.362239			−0.181119	−	0.983461i	$-0.557972\pi$
$239$	−2.49448e17			−0.362239			−0.181119	+	0.983461i	$0.557972\pi$
$240$		0			0
$241$	−1.07289e18			−1.46362			−0.731809	−	0.681510i	$-0.761324\pi$
$241$	−1.07289e18			−1.46362			−0.731809	+	0.681510i	$0.761324\pi$
$242$		0			0
$243$	−3.73252e17	+	6.84799e17i	−0.478581	+	0.878043i
$244$		0			0
$245$		−	8.03404e16i		−	0.0968698i
$246$		0			0
$247$		−	2.99029e17i		−	0.339223i
$248$		0			0
$249$	1.16451e18	+	5.11632e17i	1.24350	+	0.546339i
$250$		0			0
$251$	8.62418e17			0.867291			0.433646	−	0.901083i	$-0.357227\pi$
$251$	8.62418e17			0.867291			0.433646	+	0.901083i	$0.357227\pi$
$252$		0			0
$253$	−1.24753e18			−1.18208
$254$		0			0
$255$	−1.07608e16	−	4.72780e15i	−0.00961152	−	0.00422287i
$256$		0			0
$257$		−	1.67445e18i		−	1.41050i	−0.708956	−	0.705252i	$-0.750834\pi$
$257$		−	1.67445e18i		−	1.41050i	0.708956	−	0.705252i	$-0.249166\pi$
$258$		0			0
$259$		−	1.96452e18i		−	1.56138i
$260$		0			0
$261$	−2.80963e14	+	2.58023e14i	−0.000210788	+	0.000193578i
$262$		0			0
$263$	−2.35593e18			−1.66915			−0.834574	−	0.550896i	$-0.814286\pi$
$263$	−2.35593e18			−1.66915			−0.834574	+	0.550896i	$0.814286\pi$
$264$		0			0
$265$	−1.57329e18			−1.05309
$266$		0			0
$267$	−1.07198e18	+	2.43989e18i	−0.678192	+	1.54361i
$268$		0			0
$269$		−	1.10877e17i		−	0.0663282i	−0.999450	−	0.0331641i	$-0.989442\pi$
$269$		−	1.10877e17i		−	0.0663282i	0.999450	−	0.0331641i	$-0.0105584\pi$
$270$		0			0
$271$		−	1.47573e18i		−	0.835099i	−0.908654	−	0.417550i	$-0.862889\pi$
$271$		−	1.47573e18i		−	0.835099i	0.908654	−	0.417550i	$-0.137111\pi$
$272$		0			0
$273$	6.66602e17	−	1.51723e18i	0.356982	−	0.812514i
$274$		0			0
$275$	−1.15821e18			−0.587207
$276$		0			0
$277$	−1.87454e18			−0.900111			−0.450056	−	0.893001i	$-0.648596\pi$
$277$	−1.87454e18			−0.900111			−0.450056	+	0.893001i	$0.648596\pi$
$278$		0			0
$279$	−1.67432e18	+	1.53762e18i	−0.761742	+	0.699548i
$280$		0			0
$281$		−	3.96042e18i		−	1.70783i	−0.520415	−	0.853914i	$-0.674223\pi$
$281$		−	3.96042e18i		−	1.70783i	0.520415	−	0.853914i	$-0.325777\pi$
$282$		0			0
$283$		−	2.40376e17i		−	0.0982865i	−0.998792	−	0.0491432i	$-0.984351\pi$
$283$		−	2.40376e17i		−	0.0982865i	0.998792	−	0.0491432i	$-0.0156491\pi$
$284$		0			0
$285$	5.83132e17	+	2.56202e17i	0.226168	+	0.0993680i
$286$		0			0
$287$	−1.04919e18			−0.386135
$288$		0			0
$289$	2.86155e18			0.999694
$290$		0			0
$291$	4.92265e18	+	2.16279e18i	1.63306	+	0.717493i
$292$		0			0
$293$		−	4.32866e16i		−	0.0136410i	−0.999977	−	0.00682050i	$-0.997829\pi$
$293$		−	4.32866e16i		−	0.0136410i	0.999977	−	0.00682050i	$-0.00217105\pi$
$294$		0			0
$295$			7.63430e17i			0.228615i
$296$		0			0
$297$	−1.04043e18	−	3.04852e18i	−0.296170	−	0.867792i
$298$		0			0
$299$	3.92176e18			1.06157
$300$		0			0
$301$	−4.24301e18			−1.09251
$302$		0			0
$303$	−3.05363e18	+	6.95025e18i	−0.748164	+	1.70287i
$304$		0			0
$305$		−	8.34803e17i		−	0.194686i
$306$		0			0
$307$		−	8.89513e18i		−	1.97522i	−0.156940	−	0.987608i	$-0.550163\pi$
$307$		−	8.89513e18i		−	1.97522i	0.156940	−	0.987608i	$-0.449837\pi$
$308$		0			0
$309$	1.00357e18	−	2.28418e18i	0.212255	−	0.483105i
$310$		0			0
$311$	−3.23166e18			−0.651212			−0.325606	−	0.945505i	$-0.605568\pi$
$311$	−3.23166e18			−0.651212			−0.325606	+	0.945505i	$0.605568\pi$
$312$		0			0
$313$	−4.38472e18			−0.842092			−0.421046	−	0.907039i	$-0.638337\pi$
$313$	−4.38472e18			−0.842092			−0.421046	+	0.907039i	$0.638337\pi$
$314$		0			0
$315$	2.38759e18	+	2.59986e18i	0.437150	+	0.476016i
$316$		0			0
$317$			4.47059e18i			0.780586i	0.920691	+	0.390293i	$0.127626\pi$
$317$			4.47059e18i			0.780586i	−0.920691	+	0.390293i	$0.872374\pi$
$318$		0			0
$319$		−	1.57552e15i		−	0.000262418i
$320$		0			0
$321$	7.46466e18	+	3.27964e18i	1.18638	+	0.521241i
$322$		0			0
$323$	4.75413e16			0.00721195
$324$		0			0
$325$	3.64095e18			0.527339
$326$		0			0
$327$	−2.85306e18	−	1.25351e18i	−0.394642	−	0.173388i
$328$		0			0
$329$			1.15168e19i			1.52182i
$330$		0			0
$331$		−	7.26070e18i		−	0.916788i	−0.888749	−	0.458394i	$-0.848425\pi$
$331$		−	7.26070e18i		−	0.916788i	0.888749	−	0.458394i	$-0.151575\pi$
$332$		0			0
$333$	8.11943e18	+	8.84130e18i	0.979927	+	1.06705i
$334$		0			0
$335$	−5.50181e18			−0.634847
$336$		0			0
$337$	−3.32875e18			−0.367330			−0.183665	−	0.982989i	$-0.558796\pi$
$337$	−3.32875e18			−0.367330			−0.183665	+	0.982989i	$0.558796\pi$
$338$		0			0
$339$	3.39120e18	−	7.71858e18i	0.357978	−	0.814779i
$340$		0			0
$341$		−	9.38887e18i		−	0.948321i
$342$		0			0
$343$		−	9.34772e18i		−	0.903649i
$344$		0			0
$345$	−3.36009e18	+	7.64777e18i	−0.310962	+	0.707769i
$346$		0			0
$347$	2.07708e19			1.84070			0.920349	−	0.391098i	$-0.127905\pi$
$347$	2.07708e19			1.84070			0.920349	+	0.391098i	$0.127905\pi$
$348$		0			0
$349$	−7.99055e18			−0.678244			−0.339122	−	0.940742i	$-0.610130\pi$
$349$	−7.99055e18			−0.678244			−0.339122	+	0.940742i	$0.610130\pi$
$350$		0			0
$351$	3.27072e18	+	9.58337e18i	0.265975	+	0.779318i
$352$		0			0
$353$			1.74354e19i			1.35869i	0.733818	+	0.679346i	$0.237737\pi$
$353$			1.74354e19i			1.35869i	−0.733818	+	0.679346i	$0.762263\pi$
$354$		0			0
$355$			5.86298e18i			0.437931i
$356$		0			0
$357$	2.41218e17	+	1.05980e17i	0.0172742	+	0.00758951i
$358$		0			0
$359$	8.33468e17			0.0572375			0.0286187	−	0.999590i	$-0.490889\pi$
$359$	8.33468e17			0.0572375			0.0286187	+	0.999590i	$0.490889\pi$
$360$		0			0
$361$	1.26048e19			0.830296
$362$		0			0
$363$	−2.30660e18	−	1.01342e18i	−0.145772	−	0.0640456i
$364$		0			0
$365$			1.10156e19i			0.668055i
$366$		0			0
$367$			9.97718e18i			0.580781i	0.956908	+	0.290390i	$0.0937851\pi$
$367$			9.97718e18i			0.580781i	−0.956908	+	0.290390i	$0.906215\pi$
$368$		0			0
$369$	4.72186e18	−	4.33633e18i	0.263886	−	0.242340i
$370$		0			0
$371$	3.52676e19			1.89266
$372$		0			0
$373$	2.68650e19			1.38475			0.692374	−	0.721539i	$-0.256565\pi$
$373$	2.68650e19			1.38475			0.692374	+	0.721539i	$0.256565\pi$
$374$		0			0
$375$	−7.99070e18	+	1.81873e19i	−0.395686	+	0.900604i
$376$		0			0
$377$			4.95282e15i			0.000235664i
$378$		0			0
$379$		−	1.45511e19i		−	0.665430i	−0.943027	−	0.332715i	$-0.892035\pi$
$379$		−	1.45511e19i		−	0.665430i	0.943027	−	0.332715i	$-0.107965\pi$
$380$		0			0
$381$	−5.78571e17	+	1.31686e18i	−0.0254343	+	0.0578899i
$382$		0			0
$383$	−4.18470e19			−1.76878			−0.884389	−	0.466751i	$-0.845424\pi$
$383$	−4.18470e19			−1.76878			−0.884389	+	0.466751i	$0.845424\pi$
$384$		0			0
$385$	−1.45789e19			−0.592610
$386$		0			0
$387$	1.90957e19	−	1.75365e19i	0.746625	−	0.685665i
$388$		0			0
$389$		−	2.40177e19i		−	0.903460i	−0.892155	−	0.451730i	$-0.850807\pi$
$389$		−	2.40177e19i		−	0.903460i	0.892155	−	0.451730i	$-0.149193\pi$
$390$		0			0
$391$			6.23504e17i			0.0225691i
$392$		0			0
$393$	−2.56945e19	−	1.12890e19i	−0.895152	−	0.393290i
$394$		0			0
$395$	1.50850e19			0.505903
$396$		0			0
$397$	2.61249e19			0.843578			0.421789	−	0.906694i	$-0.361402\pi$
$397$	2.61249e19			0.843578			0.421789	+	0.906694i	$0.361402\pi$
$398$		0			0
$399$	−1.30717e19	−	5.74313e18i	−0.406477	−	0.178588i
$400$		0			0
$401$		−	1.84584e19i		−	0.552855i	−0.961035	−	0.276427i	$-0.910849\pi$
$401$		−	1.84584e19i		−	0.552855i	0.961035	−	0.276427i	$-0.0891505\pi$
$402$		0			0
$403$			2.95150e19i			0.851637i
$404$		0			0
$405$	−2.14907e19	−	1.83266e18i	−0.597500	−	0.0509530i
$406$		0			0
$407$	−4.95781e19			−1.32841
$408$		0			0
$409$	−4.83453e19			−1.24862			−0.624309	−	0.781177i	$-0.714619\pi$
$409$	−4.83453e19			−1.24862			−0.624309	+	0.781177i	$0.714619\pi$
$410$		0			0
$411$	1.63844e19	−	3.72919e19i	0.407960	−	0.928541i
$412$		0			0
$413$		−	1.71134e19i		−	0.410875i
$414$		0			0
$415$			3.51758e19i			0.814486i
$416$		0			0
$417$	8.32952e18	−	1.89585e19i	0.186037	−	0.423432i
$418$		0			0
$419$	−5.55177e19			−1.19626			−0.598131	−	0.801398i	$-0.704090\pi$
$419$	−5.55177e19			−1.19626			−0.598131	+	0.801398i	$0.704090\pi$
$420$		0			0
$421$	−3.28002e19			−0.681964			−0.340982	−	0.940070i	$-0.610760\pi$
$421$	−3.28002e19			−0.681964			−0.340982	+	0.940070i	$0.610760\pi$
$422$		0			0
$423$	−4.75994e19	−	5.18313e19i	−0.955100	−	1.04001i
$424$		0			0
$425$			5.78861e17i			0.0112113i
$426$		0			0
$427$			1.87133e19i			0.349898i
$428$		0			0
$429$	−3.82899e19	−	1.68229e19i	−0.691281	−	0.303718i
$430$		0			0
$431$	−4.21666e19			−0.735171			−0.367586	−	0.929990i	$-0.619816\pi$
$431$	−4.21666e19			−0.735171			−0.367586	+	0.929990i	$0.619816\pi$
$432$		0			0
$433$	6.16089e19			1.03749			0.518745	−	0.854929i	$-0.326399\pi$
$433$	6.16089e19			1.03749			0.518745	+	0.854929i	$0.326399\pi$
$434$		0			0
$435$	−9.65843e15	−	4.24349e15i	−0.000157122	−	6.90325e-5i
$436$		0			0
$437$		−	3.37880e19i		−	0.531070i
$438$		0			0
$439$			1.17329e20i			1.78206i	0.453947	+	0.891029i	$0.350015\pi$
$439$			1.17329e20i			1.78206i	−0.453947	+	0.891029i	$0.649985\pi$
$440$		0			0
$441$	−7.44337e18	−	8.10514e18i	−0.109265	−	0.118979i
$442$		0			0
$443$	7.29132e19			1.03461			0.517307	−	0.855800i	$-0.326935\pi$
$443$	7.29132e19			1.03461			0.517307	+	0.855800i	$0.326935\pi$
$444$		0			0
$445$	−7.37011e19			−1.01105
$446$		0			0
$447$	2.46227e19	−	5.60428e19i	0.326610	−	0.743385i
$448$		0			0
$449$		−	1.15623e20i		−	1.48319i	−0.670849	−	0.741594i	$-0.734070\pi$
$449$		−	1.15623e20i		−	1.48319i	0.670849	−	0.741594i	$-0.265930\pi$
$450$		0			0
$451$			2.64781e19i			0.328521i
$452$		0			0
$453$	2.34305e19	−	5.33293e19i	0.281219	−	0.640072i
$454$		0			0
$455$	4.58304e19			0.532192
$456$		0			0
$457$	1.16330e20			1.30714			0.653568	−	0.756868i	$-0.273271\pi$
$457$	1.16330e20			1.30714			0.653568	+	0.756868i	$0.273271\pi$
$458$		0			0
$459$	−1.52362e18	+	5.19999e17i	−0.0165684	+	0.00565467i
$460$		0			0
$461$			2.96219e19i			0.311786i	0.987774	+	0.155893i	$0.0498254\pi$
$461$			2.96219e19i			0.311786i	−0.987774	+	0.155893i	$0.950175\pi$
$462$		0			0
$463$			4.12948e19i			0.420763i	0.977619	+	0.210382i	$0.0674706\pi$
$463$			4.12948e19i			0.420763i	−0.977619	+	0.210382i	$0.932529\pi$
$464$		0			0
$465$	−5.75568e19	−	2.52879e19i	−0.567805	−	0.249468i
$466$		0			0
$467$	1.04532e20			0.998553			0.499276	−	0.866443i	$-0.333599\pi$
$467$	1.04532e20			0.998553			0.499276	+	0.866443i	$0.333599\pi$
$468$		0			0
$469$	1.23331e20			1.14097
$470$		0			0
$471$	−8.55931e19	−	3.76058e19i	−0.766977	−	0.336975i
$472$		0			0
$473$			1.07080e20i			0.929501i
$474$		0			0
$475$		−	3.13688e19i		−	0.263812i
$476$		0			0
$477$	−1.58722e20	+	1.45762e20i	−1.29345	+	1.18784i
$478$		0			0
$479$	−2.68669e19			−0.212179			−0.106089	−	0.994357i	$-0.533833\pi$
$479$	−2.68669e19			−0.212179			−0.106089	+	0.994357i	$0.533833\pi$
$480$		0			0
$481$	1.55854e20			1.19297
$482$		0			0
$483$	7.53211e19	−	1.71435e20i	0.558873	−	1.27203i
$484$		0			0
$485$			1.48697e20i			1.06964i
$486$		0			0
$487$			2.15219e20i			1.50112i	0.660805	+	0.750558i	$0.270215\pi$
$487$			2.15219e20i			1.50112i	−0.660805	+	0.750558i	$0.729785\pi$
$488$		0			0
$489$	−7.47410e19	+	1.70115e20i	−0.505525	+	1.15061i
$490$		0			0
$491$	−1.49686e20			−0.981908			−0.490954	−	0.871185i	$-0.663352\pi$
$491$	−1.49686e20			−0.981908			−0.490954	+	0.871185i	$0.663352\pi$
$492$		0			0
$493$	−7.87429e14			−5.01025e−6
$494$		0			0
$495$	6.56122e19	−	6.02551e19i	0.404991	−	0.371925i
$496$		0			0
$497$		−	1.31427e20i		−	0.787067i
$498$		0			0
$499$		−	3.90528e19i		−	0.226933i	−0.993542	−	0.113467i	$-0.963805\pi$
$499$		−	3.90528e19i		−	0.226933i	0.993542	−	0.113467i	$-0.0361955\pi$
$500$		0			0
$501$	7.07986e19	+	3.11058e19i	0.399247	+	0.175412i
$502$		0			0
$503$	1.88983e20			1.03434			0.517170	−	0.855883i	$-0.326985\pi$
$503$	1.88983e20			1.03434			0.517170	+	0.855883i	$0.326985\pi$
$504$		0			0
$505$	−2.09944e20			−1.11537
$506$		0			0
$507$	−5.71456e19	−	2.51072e19i	−0.294729	−	0.129491i
$508$		0			0
$509$			3.37368e20i			1.68936i	0.535275	+	0.844678i	$0.320208\pi$
$509$			3.37368e20i			1.68936i	−0.535275	+	0.844678i	$0.679792\pi$
$510$		0			0
$511$		−	2.46930e20i		−	1.20065i
$512$		0			0
$513$	8.25658e19	−	2.81790e19i	0.389870	−	0.133059i
$514$		0			0
$515$	6.89974e19			0.316431
$516$		0			0
$517$	2.90647e20			1.29475
$518$		0			0
$519$	1.06346e20	−	2.42050e20i	0.460220	−	1.04749i
$520$		0			0
$521$			3.11034e20i			1.30775i	0.756603	+	0.653875i	$0.226858\pi$
$521$			3.11034e20i			1.30775i	−0.756603	+	0.653875i	$0.773142\pi$
$522$		0			0
$523$			4.65670e18i			0.0190246i	0.999955	+	0.00951231i	$0.00302791\pi$
$523$			4.65670e18i			0.0190246i	−0.999955	+	0.00951231i	$0.996972\pi$
$524$		0			0
$525$	6.99281e19	−	1.59160e20i	0.277624	−	0.631889i
$526$		0			0
$527$	−4.69247e18			−0.0181060
$528$		0			0
$529$	1.76494e20			0.661932
$530$		0			0
$531$	7.07302e19	+	7.70185e19i	0.257867	+	0.280793i
$532$		0			0
$533$		−	8.32370e19i		−	0.295027i
$534$		0			0
$535$			2.25483e20i			0.777069i
$536$		0			0
$537$	1.51217e20	+	6.64382e19i	0.506751	+	0.222644i
$538$		0			0
$539$	4.54500e19			0.148122
$540$		0			0
$541$	−1.98266e20			−0.628447			−0.314224	−	0.949349i	$-0.601744\pi$
$541$	−1.98266e20			−0.628447			−0.314224	+	0.949349i	$0.601744\pi$
$542$		0			0
$543$	5.59270e19	+	2.45718e19i	0.172434	+	0.0757598i
$544$		0			0
$545$		−	8.61815e19i		−	0.258488i
$546$		0			0
$547$			5.98981e20i			1.74787i	0.486047	+	0.873933i	$0.338438\pi$
$547$			5.98981e20i			1.74787i	−0.486047	+	0.873933i	$0.661562\pi$
$548$		0			0
$549$	−7.73428e19	−	8.42190e19i	−0.219597	−	0.239121i
$550$		0			0
$551$	4.26712e16			0.000117896
$552$		0			0
$553$	−3.38152e20			−0.909229
$554$		0			0
$555$	−1.33533e20	+	3.03929e20i	−0.349455	+	0.795381i
$556$		0			0
$557$		−	4.05809e20i		−	1.03373i	−0.856067	−	0.516865i	$-0.827099\pi$
$557$		−	4.05809e20i		−	1.03373i	0.856067	−	0.516865i	$-0.172901\pi$
$558$		0			0
$559$		−	3.36618e20i		−	0.834736i
$560$		0			0
$561$	2.67460e18	−	6.08756e18i	0.00645711	−	0.0146968i
$562$		0			0
$563$	−3.54919e20			−0.834289			−0.417145	−	0.908840i	$-0.636969\pi$
$563$	−3.54919e20			−0.834289			−0.417145	+	0.908840i	$0.636969\pi$
$564$		0			0
$565$	2.33153e20			0.533675
$566$		0			0
$567$	4.81744e20	+	4.10817e19i	1.07385	+	0.0915748i
$568$		0			0
$569$		−	7.73602e19i		−	0.167948i	−0.996468	−	0.0839741i	$-0.973239\pi$
$569$		−	7.73602e19i		−	0.167948i	0.996468	−	0.0839741i	$-0.0267613\pi$
$570$		0			0
$571$		−	9.68629e19i		−	0.204827i	−0.994742	−	0.102413i	$-0.967344\pi$
$571$		−	9.68629e19i		−	0.204827i	0.994742	−	0.102413i	$-0.0326564\pi$
$572$		0			0
$573$	−2.72481e20	−	1.19716e20i	−0.561276	−	0.246600i
$574$		0			0
$575$	4.11401e20			0.825575
$576$		0			0
$577$	−3.99103e20			−0.780309			−0.390154	−	0.920749i	$-0.627578\pi$
$577$	−3.99103e20			−0.780309			−0.390154	+	0.920749i	$0.627578\pi$
$578$		0			0
$579$	−5.86333e20	−	2.57609e20i	−1.11701	−	0.490763i
$580$		0			0
$581$		−	7.88516e20i		−	1.46383i
$582$		0			0
$583$		−	8.90041e20i		−	1.61026i
$584$		0			0
$585$	−2.06259e20	+	1.89419e20i	−0.363701	+	0.334006i
$586$		0			0
$587$	−5.88495e20			−1.01148			−0.505739	−	0.862686i	$-0.668780\pi$
$587$	−5.88495e20			−1.01148			−0.505739	+	0.862686i	$0.668780\pi$
$588$		0			0
$589$	2.54287e20			0.426049
$590$		0			0
$591$	−4.67812e20	+	1.06477e21i	−0.764126	+	1.73920i
$592$		0			0
$593$		−	1.48932e20i		−	0.237180i	−0.992943	−	0.118590i	$-0.962163\pi$
$593$		−	1.48932e20i		−	0.237180i	0.992943	−	0.118590i	$-0.0378374\pi$
$594$		0			0
$595$			7.28639e18i			0.0113145i
$596$		0			0
$597$	3.22128e20	−	7.33182e20i	0.487776	−	1.11021i
$598$		0			0
$599$	−6.12435e20			−0.904397			−0.452198	−	0.891917i	$-0.649360\pi$
$599$	−6.12435e20			−0.904397			−0.452198	+	0.891917i	$0.649360\pi$
$600$		0			0
$601$	−4.60503e20			−0.663245			−0.331623	−	0.943412i	$-0.607596\pi$
$601$	−4.60503e20			−0.663245			−0.331623	+	0.943412i	$0.607596\pi$
$602$		0			0
$603$	−5.55049e20	+	5.09731e20i	−0.779744	+	0.716079i
$604$		0			0
$605$		−	6.96748e19i		−	0.0954796i
$606$		0			0
$607$		−	5.67654e20i		−	0.758872i	−0.925218	−	0.379436i	$-0.876118\pi$
$607$		−	5.67654e20i		−	0.758872i	0.925218	−	0.379436i	$-0.123882\pi$
$608$		0			0
$609$	2.16507e17	+	9.51237e16i	0.000282386	+	0.000124068i
$610$		0			0
$611$	−9.13682e20			−1.16275
$612$		0			0
$613$	−1.29501e21			−1.60812			−0.804062	−	0.594546i	$-0.797332\pi$
$613$	−1.29501e21			−1.60812			−0.804062	+	0.594546i	$0.797332\pi$
$614$		0			0
$615$	1.62319e20	+	7.13159e19i	0.196701	+	0.0864217i
$616$		0			0
$617$		−	2.05210e20i		−	0.242695i	−0.992610	−	0.121347i	$-0.961278\pi$
$617$		−	2.05210e20i		−	0.242695i	0.992610	−	0.121347i	$-0.0387215\pi$
$618$		0			0
$619$		−	7.90563e20i		−	0.912549i	−0.889839	−	0.456275i	$-0.849183\pi$
$619$		−	7.90563e20i		−	0.912549i	0.889839	−	0.456275i	$-0.150817\pi$
$620$		0			0
$621$	3.69567e20	+	1.08285e21i	0.416396	+	1.22006i
$622$		0			0
$623$	1.65211e21			1.81710
$624$		0			0
$625$	4.70387e19			0.0505074
$626$		0			0
$627$	−1.44938e20	+	3.29888e20i	−0.151941	+	0.345828i
$628$		0			0
$629$			2.47786e19i			0.0253628i
$630$		0			0
$631$			3.53483e20i			0.353304i	0.984273	+	0.176652i	$0.0565266\pi$
$631$			3.53483e20i			0.353304i	−0.984273	+	0.176652i	$0.943473\pi$
$632$		0			0
$633$	8.02795e20	−	1.82721e21i	0.783568	−	1.78345i
$634$		0			0
$635$	−3.97780e19			−0.0379176
$636$		0			0
$637$	−1.42877e20			−0.133020
$638$		0			0
$639$	5.43193e20	+	5.91486e20i	0.493967	+	0.537884i
$640$		0			0
$641$		−	2.36821e20i		−	0.210371i	−0.994453	−	0.105185i	$-0.966456\pi$
$641$		−	2.36821e20i		−	0.210371i	0.994453	−	0.105185i	$-0.0335436\pi$
$642$		0			0
$643$			5.53838e20i			0.480618i	0.970696	+	0.240309i	$0.0772488\pi$
$643$			5.53838e20i			0.480618i	−0.970696	+	0.240309i	$0.922751\pi$
$644$		0			0
$645$	6.56435e20	+	2.88408e20i	0.556537	+	0.244517i
$646$		0			0
$647$	1.57531e21			1.30492			0.652461	−	0.757822i	$-0.273737\pi$
$647$	1.57531e21			1.30492			0.652461	+	0.757822i	$0.273737\pi$
$648$		0			0
$649$	−4.31886e20			−0.349570
$650$		0			0
$651$	1.29022e21	+	5.66864e20i	1.02048	+	0.448354i
$652$		0			0
$653$			1.44817e21i			1.11936i	0.828709	+	0.559680i	$0.189076\pi$
$653$			1.44817e21i			1.11936i	−0.828709	+	0.559680i	$0.810924\pi$
$654$		0			0
$655$		−	7.76147e20i		−	0.586319i
$656$		0			0
$657$	1.02057e21	+	1.11131e21i	0.753536	+	0.820530i
$658$		0			0
$659$	−1.12670e21			−0.813143			−0.406571	−	0.913619i	$-0.633276\pi$
$659$	−1.12670e21			−0.813143			−0.406571	+	0.913619i	$0.633276\pi$
$660$		0			0
$661$	−2.43494e20			−0.171782			−0.0858909	−	0.996305i	$-0.527374\pi$
$661$	−2.43494e20			−0.171782			−0.0858909	+	0.996305i	$0.527374\pi$
$662$		0			0
$663$	−8.40792e18	+	1.91369e19i	−0.00579879	+	0.0131984i
$664$		0			0
$665$		−	3.94853e20i		−	0.266240i
$666$		0			0
$667$			5.59632e17i			0.000368943i
$668$		0			0
$669$	−2.35066e20	+	5.35026e20i	−0.151529	+	0.344888i
$670$		0			0
$671$	4.72263e20			0.297691
$672$		0			0
$673$	7.05371e20			0.434815			0.217408	−	0.976081i	$-0.430240\pi$
$673$	7.05371e20			0.434815			0.217408	+	0.976081i	$0.430240\pi$
$674$		0			0
$675$	3.43106e20	+	1.00532e21i	0.206847	+	0.606072i
$676$		0			0
$677$			1.49697e21i			0.882668i	0.897343	+	0.441334i	$0.145495\pi$
$677$			1.49697e21i			0.882668i	−0.897343	+	0.441334i	$0.854505\pi$
$678$		0			0
$679$		−	3.33325e21i		−	1.92240i
$680$		0			0
$681$	1.67047e21	+	7.33930e20i	0.942399	+	0.414048i
$682$		0			0
$683$	−2.05391e21			−1.13351			−0.566757	−	0.823885i	$-0.691802\pi$
$683$	−2.05391e21			−1.13351			−0.566757	+	0.823885i	$0.691802\pi$
$684$		0			0
$685$	1.12647e21			0.608189
$686$		0			0
$687$	6.28356e20	+	2.76072e20i	0.331918	+	0.145830i
$688$		0			0
$689$			2.79794e21i			1.44609i
$690$		0			0
$691$		−	2.10182e21i		−	1.06294i	−0.847076	−	0.531471i	$-0.821639\pi$
$691$		−	2.10182e21i		−	1.06294i	0.847076	−	0.531471i	$-0.178361\pi$
$692$		0			0
$693$	−1.47079e21	+	1.35070e21i	−0.727866	+	0.668438i
$694$		0			0
$695$	5.72673e20			0.277345
$696$		0			0
$697$	1.32335e19			0.00627233
$698$		0			0
$699$	−9.36283e20	+	2.13104e21i	−0.434339	+	0.988581i
$700$		0			0
$701$		−	3.44224e20i		−	0.156299i	−0.996942	−	0.0781495i	$-0.975099\pi$
$701$		−	3.44224e20i		−	0.156299i	0.996942	−	0.0781495i	$-0.0249011\pi$
$702$		0			0
$703$		−	1.34277e21i		−	0.596810i
$704$		0			0
$705$	7.82826e20	−	1.78176e21i	0.340602	−	0.775230i
$706$		0			0
$707$	4.70619e21			2.00458
$708$		0			0
$709$	−2.04344e20			−0.0852150			−0.0426075	−	0.999092i	$-0.513566\pi$
$709$	−2.04344e20			−0.0852150			−0.0426075	+	0.999092i	$0.513566\pi$
$710$		0			0
$711$	1.52185e21	−	1.39759e21i	0.621369	−	0.570636i
$712$		0			0
$713$			3.33497e21i			1.33328i
$714$		0			0
$715$		−	1.15661e21i		−	0.452785i
$716$		0			0
$717$	−8.65092e20	−	3.80083e20i	−0.331641	−	0.145709i
$718$		0			0
$719$	−1.47491e21			−0.553733			−0.276866	−	0.960908i	$-0.589296\pi$
$719$	−1.47491e21			−0.553733			−0.276866	+	0.960908i	$0.589296\pi$
$720$		0			0
$721$	−1.54667e21			−0.568701
$722$		0			0
$723$	−3.72082e21	−	1.63476e21i	−1.33999	−	0.588732i
$724$		0			0
$725$			5.19562e17i			0.000183275i
$726$		0			0
$727$			5.31019e21i			1.83486i	0.397900	+	0.917429i	$0.369739\pi$
$727$			5.31019e21i			1.83486i	−0.397900	+	0.917429i	$0.630261\pi$
$728$		0			0
$729$	−2.33788e21	+	1.80618e21i	−0.791345	+	0.611370i
$730$		0			0
$731$	5.35176e19			0.0177466
$732$		0			0
$733$	2.29570e20			0.0745823			0.0372911	−	0.999304i	$-0.488127\pi$
$733$	2.29570e20			0.0745823			0.0372911	+	0.999304i	$0.488127\pi$
$734$		0			0
$735$	1.22415e20	−	2.78623e20i	0.0389653	−	0.0886874i
$736$		0			0
$737$		−	3.11247e21i		−	0.970732i
$738$		0			0
$739$		−	3.37549e20i		−	0.103158i	−0.998669	−	0.0515791i	$-0.983575\pi$
$739$		−	3.37549e20i		−	0.103158i	0.998669	−	0.0515791i	$-0.0164254\pi$
$740$		0			0
$741$	4.55630e20	−	1.03704e21i	0.136451	−	0.310570i
$742$		0			0
$743$	−2.47551e21			−0.726522			−0.363261	−	0.931687i	$-0.618337\pi$
$743$	−2.47551e21			−0.726522			−0.363261	+	0.931687i	$0.618337\pi$
$744$		0			0
$745$	1.69287e21			0.486912
$746$		0			0
$747$	3.25897e21	+	3.54871e21i	0.918704	+	1.00038i
$748$		0			0
$749$		−	5.05451e21i		−	1.39658i
$750$		0			0
$751$			2.45765e21i			0.665611i	0.942995	+	0.332806i	$0.107995\pi$
$751$			2.45765e21i			0.665611i	−0.942995	+	0.332806i	$0.892005\pi$
$752$		0			0
$753$	2.99089e21	+	1.31407e21i	0.794033	+	0.348863i
$754$		0			0
$755$	1.61090e21			0.419244
$756$		0			0
$757$	5.59107e21			1.42651			0.713257	−	0.700902i	$-0.247219\pi$
$757$	5.59107e21			1.42651			0.713257	+	0.700902i	$0.247219\pi$
$758$		0			0
$759$	−4.32648e21	−	1.90086e21i	−1.08223	−	0.475486i
$760$		0			0
$761$		−	6.35983e21i		−	1.55977i	−0.625922	−	0.779886i	$-0.715277\pi$
$761$		−	6.35983e21i		−	1.55977i	0.625922	−	0.779886i	$-0.284723\pi$
$762$		0			0
$763$			1.93188e21i			0.464564i
$764$		0			0
$765$	−3.01149e19	−	3.27924e19i	−0.00710103	−	0.00773236i
$766$		0			0
$767$	1.35768e21			0.313930
$768$		0			0
$769$	−8.49239e21			−1.92567			−0.962836	−	0.270088i	$-0.912947\pi$
$769$	−8.49239e21			−1.92567			−0.962836	+	0.270088i	$0.912947\pi$
$770$		0			0
$771$	2.55136e21	−	5.80706e21i	0.567367	−	1.29136i
$772$		0			0
$773$			2.45347e21i			0.535101i	0.963544	+	0.267550i	$0.0862141\pi$
$773$			2.45347e21i			0.535101i	−0.963544	+	0.267550i	$0.913786\pi$
$774$		0			0
$775$			3.09619e21i			0.662315i
$776$		0			0
$777$	2.99333e21	−	6.81301e21i	0.628055	−	1.42949i
$778$		0			0
$779$	−7.17131e20			−0.147594
$780$		0			0
$781$	−3.31679e21			−0.669631
$782$		0			0
$783$	−1.36754e18	+	4.66730e17i	−0.000270849	+	9.24384e-5i
$784$		0			0
$785$		−	2.58548e21i		−	0.502365i
$786$		0			0
$787$			4.85453e21i			0.925415i	0.886511	+	0.462707i	$0.153122\pi$
$787$			4.85453e21i			0.925415i	−0.886511	+	0.462707i	$0.846878\pi$
$788$		0			0
$789$	−8.17045e21	−	3.58973e21i	−1.52816	−	0.671405i
$790$		0			0
$791$	−5.22645e21			−0.959142
$792$		0			0
$793$	−1.48461e21			−0.267340
$794$		0			0
$795$	−5.45624e21	−	2.39723e21i	−0.964139	−	0.423600i
$796$		0			0
$797$		−	8.35320e21i		−	1.44849i	−0.689543	−	0.724245i	$-0.742188\pi$
$797$		−	8.35320e21i		−	1.44849i	0.689543	−	0.724245i	$-0.257812\pi$
$798$		0			0
$799$		−	1.45263e20i		−	0.0247203i
$800$		0			0
$801$	−7.43533e21	+	6.82825e21i	−1.24181	+	1.14042i
$802$		0			0
$803$	−6.23173e21			−1.02151
$804$		0			0
$805$	5.17850e21			0.833172
$806$		0			0
$807$	1.68943e20	−	3.84524e20i	0.0266801	−	0.0607256i
$808$		0			0
$809$			8.62691e21i			1.33734i	0.743560	+	0.668669i	$0.233136\pi$
$809$			8.62691e21i			1.33734i	−0.743560	+	0.668669i	$0.766864\pi$
$810$		0			0
$811$		−	1.49666e21i		−	0.227754i	−0.993495	−	0.113877i	$-0.963673\pi$
$811$		−	1.49666e21i		−	0.227754i	0.993495	−	0.113877i	$-0.0363270\pi$
$812$		0			0
$813$	2.24857e21	−	5.11789e21i	0.335914	−	0.764561i
$814$		0			0
$815$	−5.13861e21			−0.753640
$816$		0			0
$817$	−2.90015e21			−0.417594
$818$		0			0
$819$	4.62360e21	−	4.24609e21i	0.653658	−	0.600288i
$820$		0			0
$821$			6.56202e21i			0.910886i	0.890265	+	0.455443i	$0.150519\pi$
$821$			6.56202e21i			0.910886i	−0.890265	+	0.455443i	$0.849481\pi$
$822$		0			0
$823$			6.73485e21i			0.917971i	0.888444	+	0.458985i	$0.151787\pi$
$823$			6.73485e21i			0.917971i	−0.888444	+	0.458985i	$0.848213\pi$
$824$		0			0
$825$	−4.01670e21	−	1.76476e21i	−0.537607	−	0.236201i
$826$		0			0
$827$	4.00080e21			0.525843			0.262921	−	0.964817i	$-0.415314\pi$
$827$	4.00080e21			0.525843			0.262921	+	0.964817i	$0.415314\pi$
$828$		0			0
$829$	1.51557e22			1.95621			0.978106	−	0.208106i	$-0.0667298\pi$
$829$	1.51557e22			1.95621			0.978106	+	0.208106i	$0.0667298\pi$
$830$		0			0
$831$	−6.50096e21	−	2.85623e21i	−0.824081	−	0.362065i
$832$		0			0
$833$		−	2.27155e19i		−	0.00282803i
$834$		0			0
$835$			2.13859e21i			0.261505i
$836$		0			0
$837$	−8.14948e21	+	2.78135e21i	−0.978789	+	0.334052i
$838$		0			0
$839$	−7.62193e21			−0.899187			−0.449594	−	0.893233i	$-0.648431\pi$
$839$	−7.62193e21			−0.899187			−0.449594	+	0.893233i	$0.648431\pi$
$840$		0			0
$841$	8.62919e21			1.00000
$842$		0			0
$843$	6.03449e21	−	1.37349e22i	0.686964	−	1.56357i
$844$		0			0
$845$		−	1.72618e21i		−	0.193046i
$846$		0			0
$847$			1.56186e21i			0.171600i
$848$		0			0
$849$	3.66262e20	−	8.33633e20i	0.0395352	−	0.0899844i
$850$		0			0
$851$	1.76104e22			1.86766
$852$		0			0
$853$	−7.53319e21			−0.784985			−0.392493	−	0.919755i	$-0.628387\pi$
$853$	−7.53319e21			−0.784985			−0.392493	+	0.919755i	$0.628387\pi$
$854$		0			0
$855$	1.63194e21	+	1.77704e21i	0.167093	+	0.181949i
$856$		0			0
$857$		−	7.20004e21i		−	0.724401i	−0.932100	−	0.362200i	$-0.882026\pi$
$857$		−	7.20004e21i		−	0.724401i	0.932100	−	0.362200i	$-0.117974\pi$
$858$		0			0
$859$			1.95339e21i			0.193126i	0.995327	+	0.0965630i	$0.0307849\pi$
$859$			1.95339e21i			0.193126i	−0.995327	+	0.0965630i	$0.969215\pi$
$860$		0			0
$861$	−3.63862e21	−	1.59865e21i	−0.353519	−	0.155321i
$862$		0			0
$863$	1.67003e21			0.159457			0.0797285	−	0.996817i	$-0.474595\pi$
$863$	1.67003e21			0.159457			0.0797285	+	0.996817i	$0.474595\pi$
$864$		0			0
$865$	7.31152e21			0.686099
$866$		0			0
$867$	9.92393e21	+	4.36014e21i	0.915252	+	0.402121i
$868$		0			0
$869$			8.53386e21i			0.773566i
$870$		0			0
$871$			9.78441e21i			0.871763i
$872$		0			0
$873$	1.37765e22	+	1.50013e22i	1.20651	+	1.31378i
$874$		0			0
$875$	1.23151e22			1.06017
$876$		0			0
$877$	1.06061e22			0.897546			0.448773	−	0.893646i	$-0.351861\pi$
$877$	1.06061e22			0.897546			0.448773	+	0.893646i	$0.351861\pi$
$878$		0			0
$879$	6.59557e19	−	1.50119e20i	0.00548701	−	0.0124888i
$880$		0			0
$881$			1.01946e22i			0.833782i	0.908956	+	0.416891i	$0.136880\pi$
$881$			1.01946e22i			0.833782i	−0.908956	+	0.416891i	$0.863120\pi$
$882$		0			0
$883$			1.77014e21i			0.142332i	0.997464	+	0.0711659i	$0.0226720\pi$
$883$			1.77014e21i			0.142332i	−0.997464	+	0.0711659i	$0.977328\pi$
$884$		0			0
$885$	−1.16324e21	+	2.64760e21i	−0.0919589	+	0.209304i
$886$		0			0
$887$	4.12547e21			0.320661			0.160331	−	0.987063i	$-0.448744\pi$
$887$	4.12547e21			0.320661			0.160331	+	0.987063i	$0.448744\pi$
$888$		0			0
$889$	8.91681e20			0.0681469
$890$		0			0
$891$	1.03677e21	−	1.21577e22i	0.0779112	−	0.913624i
$892$		0			0
$893$			7.87186e21i			0.581690i
$894$		0			0
$895$			4.56777e21i			0.331919i
$896$		0			0
$897$	1.36008e22	+	5.97558e21i	0.971898	+	0.427009i
$898$		0			0
$899$	−4.21177e18			−0.000295983
$900$		0			0
$901$	−4.44834e20			−0.0307441
$902$		0			0
$903$	−1.47149e22	−	6.46508e21i	−1.00023	−	0.439456i
$904$		0			0
$905$			1.68937e21i			0.112943i
$906$		0			0
$907$			7.79004e21i			0.512254i	0.966643	+	0.256127i	$0.0824465\pi$
$907$			7.79004e21i			0.512254i	−0.966643	+	0.256127i	$0.917554\pi$
$908$		0			0
$909$	−2.11802e22	+	1.94509e22i	−1.36994	+	1.25808i
$910$		0			0
$911$	−2.83742e21			−0.180525			−0.0902623	−	0.995918i	$-0.528771\pi$
$911$	−2.83742e21			−0.180525			−0.0902623	+	0.995918i	$0.528771\pi$
$912$		0			0
$913$	−1.98996e22			−1.24541
$914$		0			0
$915$	1.27199e21	−	2.89512e21i	0.0783114	−	0.178242i
$916$		0			0
$917$			1.73984e22i			1.05376i
$918$		0			0
$919$		−	1.88532e22i		−	1.12336i	−0.827355	−	0.561680i	$-0.810155\pi$
$919$		−	1.88532e22i		−	1.12336i	0.827355	−	0.561680i	$-0.189845\pi$
$920$		0			0
$921$	1.35535e22	−	3.08486e22i	0.794519	−	1.80837i
$922$		0			0
$923$	1.04267e22			0.601360
$924$		0			0
$925$	1.63495e22			0.927770
$926$		0			0
$927$	6.96080e21	−	6.39247e21i	0.388652	−	0.356920i
$928$		0			0
$929$		−	2.04199e21i		−	0.112185i	−0.998426	−	0.0560926i	$-0.982136\pi$
$929$		−	2.04199e21i		−	0.112185i	0.998426	−	0.0560926i	$-0.0178642\pi$
$930$		0			0
$931$			1.23096e21i			0.0665461i
$932$		0			0
$933$	−1.12075e22	−	4.92408e21i	−0.596206	−	0.261946i
$934$		0			0
$935$	1.83885e20			0.00962630
$936$		0			0
$937$	−1.32127e22			−0.680680			−0.340340	−	0.940302i	$-0.610542\pi$
$937$	−1.32127e22			−0.680680			−0.340340	+	0.940302i	$0.610542\pi$
$938$		0			0
$939$	−1.52064e22	−	6.68100e21i	−0.770963	−	0.338727i
$940$		0			0
$941$			8.09140e21i			0.403740i	0.979412	+	0.201870i	$0.0647018\pi$
$941$			8.09140e21i			0.403740i	−0.979412	+	0.201870i	$0.935298\pi$
$942$		0			0
$943$		−	9.40516e21i		−	0.461880i
$944$		0			0
$945$	4.31883e21	+	1.26544e22i	0.208751	+	0.611649i
$946$		0			0
$947$	−2.14670e21			−0.102128			−0.0510642	−	0.998695i	$-0.516261\pi$
$947$	−2.14670e21			−0.102128			−0.0510642	+	0.998695i	$0.516261\pi$
$948$		0			0
$949$	1.95901e22			0.917363
$950$		0			0
$951$	−6.81184e21	+	1.55042e22i	−0.313986	+	0.714651i
$952$		0			0
$953$		−	2.31794e22i		−	1.05173i	−0.850567	−	0.525867i	$-0.823741\pi$
$953$		−	2.31794e22i		−	1.05173i	0.850567	−	0.525867i	$-0.176259\pi$
$954$		0			0
$955$		−	8.23075e21i		−	0.367633i
$956$		0			0
$957$	2.40062e18	−	5.46395e18i	0.000105556	−	0.000240252i
$958$		0			0
$959$	−2.52513e22			−1.09306
$960$		0			0
$961$	−1.63361e21			−0.0696183
$962$		0			0
$963$	2.08905e22	+	2.27478e22i	0.876499	+	0.954426i
$964$		0			0
$965$		−	1.77112e22i		−	0.731632i
$966$		0			0
$967$			9.55164e20i			0.0388489i	0.999811	+	0.0194245i	$0.00618339\pi$
$967$			9.55164e20i			0.0388489i	−0.999811	+	0.0194245i	$0.993817\pi$
$968$		0			0
$969$	1.64875e20	+	7.24387e19i	0.00660277	+	0.00290096i
$970$		0			0
$971$	−2.99143e22			−1.17960			−0.589799	−	0.807550i	$-0.700793\pi$
$971$	−2.99143e22			−1.17960			−0.589799	+	0.807550i	$0.700793\pi$
$972$		0			0
$973$	−1.28373e22			−0.498456
$974$		0			0
$975$	1.26269e22	+	5.54772e21i	0.482796	+	0.212119i
$976$		0			0
$977$		−	1.14277e22i		−	0.430279i	−0.976583	−	0.215140i	$-0.930979\pi$
$977$		−	1.14277e22i		−	0.430279i	0.976583	−	0.215140i	$-0.0690207\pi$
$978$		0			0
$979$		−	4.16940e22i		−	1.54598i
$980$		0			0
$981$	−7.98453e21	−	8.69441e21i	−0.291563	−	0.317485i
$982$		0			0
$983$	−1.26209e22			−0.453877			−0.226939	−	0.973909i	$-0.572872\pi$
$983$	−1.26209e22			−0.453877			−0.226939	+	0.973909i	$0.572872\pi$
$984$		0			0
$985$	−3.21631e22			−1.13916
$986$		0			0
$987$	−1.75481e22	+	3.99406e22i	−0.612143	+	1.39327i
$988$		0			0
$989$		−	3.80354e22i		−	1.30682i
$990$		0			0
$991$			4.61738e22i			1.56258i	0.624166	+	0.781291i	$0.285439\pi$
$991$			4.61738e22i			1.56258i	−0.624166	+	0.781291i	$0.714561\pi$
$992$		0			0
$993$	1.10631e22	−	2.51804e22i	0.368773	−	0.839349i
$994$		0			0
$995$	2.21470e22			0.727180
$996$		0			0
$997$	3.34158e21			0.108078			0.0540391	−	0.998539i	$-0.482790\pi$
$997$	3.34158e21			0.108078			0.0540391	+	0.998539i	$0.482790\pi$
$998$		0			0
$999$	1.46870e22	+	4.30335e22i	0.467940	+	1.37109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.16.c.d.47.18	yes	20
3.2	odd	2	inner	48.16.c.d.47.4	yes	20
4.3	odd	2	inner	48.16.c.d.47.3	✓	20
12.11	even	2	inner	48.16.c.d.47.17	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.16.c.d.47.3	✓	20	4.3	odd	2	inner
48.16.c.d.47.4	yes	20	3.2	odd	2	inner
48.16.c.d.47.17	yes	20	12.11	even	2	inner
48.16.c.d.47.18	yes	20	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(3468.03 + 1523.70i) q^{3} +104758. i q^{5} -2.34829e6i q^{7} +(9.70559e6 + 1.05685e7i) q^{9} +O(q^{10})\)	\(q+(3468.03 + 1523.70i) q^{3} +104758. i q^{5} -2.34829e6i q^{7} +(9.70559e6 + 1.05685e7i) q^{9} -5.92633e7 q^{11} +1.86301e8 q^{13} +(-1.59619e8 + 3.63303e8i) q^{15} +2.96192e7i q^{17} -1.60508e9i q^{19} +(3.57809e9 - 8.14395e9i) q^{21} +2.10506e10 q^{23} +1.95434e10 q^{25} +(1.75561e10 + 5.14402e10i) q^{27} +2.65850e7i q^{29} +1.58426e11i q^{31} +(-2.05527e11 - 9.02995e10i) q^{33} +2.46002e11 q^{35} +8.36572e11 q^{37} +(6.46098e11 + 2.83867e11i) q^{39} -4.46787e11i q^{41} -1.80685e12i q^{43} +(-1.10713e12 + 1.01674e12i) q^{45} -4.90433e12 q^{47} -7.66916e11 q^{49} +(-4.51308e10 + 1.02721e11i) q^{51} +1.50184e13i q^{53} -6.20829e12i q^{55} +(2.44566e12 - 5.56648e12i) q^{57} +7.28757e12 q^{59} -7.96889e12 q^{61} +(2.48179e13 - 2.27916e13i) q^{63} +1.95165e13i q^{65} +5.25193e13i q^{67} +(7.30043e13 + 3.20749e13i) q^{69} +5.59670e13 q^{71} +1.05153e14 q^{73} +(6.77771e13 + 2.97783e13i) q^{75} +1.39168e14i q^{77} -1.43999e14i q^{79} +(-1.74943e13 + 2.05147e14i) q^{81} +3.35783e14 q^{83} -3.10285e12 q^{85} +(-4.05076e10 + 9.21978e10i) q^{87} +7.03539e14i q^{89} -4.37489e14i q^{91} +(-2.41394e14 + 5.49427e14i) q^{93} +1.68145e14 q^{95} +1.41944e15 q^{97} +(-5.75186e14 - 6.26323e14i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2271972 q^{9}+O(q^{10}) \) 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

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Database

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