LMFDB - Embedded newform 48.16.c.d.47.19

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			47.19
Root			$0.500000 - 65.9777i$ of defining polynomial
Character	$\chi$	$=$	48.47
Dual form			48.16.c.d.47.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3773.75 - 328.241i) q^{3} +91619.9i q^{5} -2.91696e6i q^{7} +(1.41334e7 - 2.47740e6i) q^{9} +O(q^{10})$	\(q+(3773.75 - 328.241i) q^{3} +91619.9i q^{5} -2.91696e6i q^{7} +(1.41334e7 - 2.47740e6i) q^{9} +1.20504e8 q^{11} -2.22828e8 q^{13} +(3.00734e7 + 3.45750e8i) q^{15} +2.08506e9i q^{17} -6.74224e9i q^{19} +(-9.57464e8 - 1.10079e10i) q^{21} -1.31392e10 q^{23} +2.21234e10 q^{25} +(5.25228e10 - 1.39882e10i) q^{27} +9.45712e10i q^{29} -2.50013e11i q^{31} +(4.54753e11 - 3.95545e10i) q^{33} +2.67251e11 q^{35} -5.54788e11 q^{37} +(-8.40896e11 + 7.31413e10i) q^{39} -1.24374e12i q^{41} -4.38035e11i q^{43} +(2.26979e11 + 1.29490e12i) q^{45} +2.15000e12 q^{47} -3.76107e12 q^{49} +(6.84401e11 + 7.86847e12i) q^{51} +2.56397e12i q^{53} +1.10406e13i q^{55} +(-2.21308e12 - 2.54435e13i) q^{57} +2.44400e12 q^{59} +2.11888e13 q^{61} +(-7.22646e12 - 4.12266e13i) q^{63} -2.04155e13i q^{65} -5.84632e12i q^{67} +(-4.95842e13 + 4.31284e12i) q^{69} +1.30114e14 q^{71} +8.46179e13 q^{73} +(8.34880e13 - 7.26180e12i) q^{75} -3.51506e14i q^{77} -1.42970e14i q^{79} +(1.93616e14 - 7.00282e13i) q^{81} +2.01259e13 q^{83} -1.91033e14 q^{85} +(3.10421e13 + 3.56888e14i) q^{87} -4.59438e14i q^{89} +6.49979e14i q^{91} +(-8.20644e13 - 9.43484e14i) q^{93} +6.17723e14 q^{95} -5.20584e14 q^{97} +(1.70314e15 - 2.98537e14i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3773.75	−	328.241i	0.996239	−	0.0866530i
$3$	3773.75	−	328.241i	0.996239	−	0.0866530i
$4$		0			0
$5$			91619.9i			0.524463i	0.965005	+	0.262232i	$0.0844584\pi$
$5$			91619.9i			0.524463i	−0.965005	+	0.262232i	$0.915542\pi$
$6$		0			0
$7$		−	2.91696e6i		−	1.33873i	−0.742932	−	0.669367i	$-0.766565\pi$
$7$		−	2.91696e6i		−	1.33873i	0.742932	−	0.669367i	$-0.233435\pi$
$8$		0			0
$9$	1.41334e7	−	2.47740e6i	0.984983	−	0.172654i
$10$		0			0
$11$	1.20504e8			1.86448			0.932240	−	0.361841i	$-0.117852\pi$
$11$	1.20504e8			1.86448			0.932240	+	0.361841i	$0.117852\pi$
$12$		0			0
$13$	−2.22828e8			−0.984906			−0.492453	−	0.870339i	$-0.663900\pi$
$13$	−2.22828e8			−0.984906			−0.492453	+	0.870339i	$0.663900\pi$
$14$		0			0
$15$	3.00734e7	+	3.45750e8i	0.0454463	+	0.522490i
$16$		0			0
$17$			2.08506e9i			1.23240i	0.787591	+	0.616199i	$0.211328\pi$
$17$			2.08506e9i			1.23240i	−0.787591	+	0.616199i	$0.788672\pi$
$18$		0			0
$19$		−	6.74224e9i		−	1.73042i	−0.501408	−	0.865211i	$-0.667185\pi$
$19$		−	6.74224e9i		−	1.73042i	0.501408	−	0.865211i	$-0.332815\pi$
$20$		0			0
$21$	−9.57464e8	−	1.10079e10i	−0.116005	−	1.33370i
$22$		0			0
$23$	−1.31392e10			−0.804659			−0.402329	−	0.915495i	$-0.631799\pi$
$23$	−1.31392e10			−0.804659			−0.402329	+	0.915495i	$0.631799\pi$
$24$		0			0
$25$	2.21234e10			0.724938
$26$		0			0
$27$	5.25228e10	−	1.39882e10i	0.966317	−	0.257356i
$28$		0			0
$29$			9.45712e10i			1.01806i	0.860748	+	0.509031i	$0.169996\pi$
$29$			9.45712e10i			1.01806i	−0.860748	+	0.509031i	$0.830004\pi$
$30$		0			0
$31$		−	2.50013e11i		−	1.63211i	−0.577975	−	0.816054i	$-0.696157\pi$
$31$		−	2.50013e11i		−	1.63211i	0.577975	−	0.816054i	$-0.303843\pi$
$32$		0			0
$33$	4.54753e11	−	3.95545e10i	1.85747	−	0.161563i
$34$		0			0
$35$	2.67251e11			0.702117
$36$		0			0
$37$	−5.54788e11			−0.960758			−0.480379	−	0.877061i	$-0.659501\pi$
$37$	−5.54788e11			−0.960758			−0.480379	+	0.877061i	$0.659501\pi$
$38$		0			0
$39$	−8.40896e11	+	7.31413e10i	−0.981201	+	0.0853450i
$40$		0			0
$41$		−	1.24374e12i		−	0.997357i	−0.866787	−	0.498679i	$-0.833819\pi$
$41$		−	1.24374e12i		−	0.997357i	0.866787	−	0.498679i	$-0.166181\pi$
$42$		0			0
$43$		−	4.38035e11i		−	0.245751i	−0.992422	−	0.122876i	$-0.960788\pi$
$43$		−	4.38035e11i		−	0.245751i	0.992422	−	0.122876i	$-0.0392116\pi$
$44$		0			0
$45$	2.26979e11	+	1.29490e12i	0.0905507	+	0.516587i
$46$		0			0
$47$	2.15000e12			0.619021			0.309510	−	0.950896i	$-0.399835\pi$
$47$	2.15000e12			0.619021			0.309510	+	0.950896i	$0.399835\pi$
$48$		0			0
$49$	−3.76107e12			−0.792210
$50$		0			0
$51$	6.84401e11	+	7.86847e12i	0.106791	+	1.22776i
$52$		0			0
$53$			2.56397e12i			0.299808i	0.988701	+	0.149904i	$0.0478965\pi$
$53$			2.56397e12i			0.299808i	−0.988701	+	0.149904i	$0.952103\pi$
$54$		0			0
$55$			1.10406e13i			0.977851i
$56$		0			0
$57$	−2.21308e12	−	2.54435e13i	−0.149946	−	1.72391i
$58$		0			0
$59$	2.44400e12			0.127853			0.0639264	−	0.997955i	$-0.479638\pi$
$59$	2.44400e12			0.127853			0.0639264	+	0.997955i	$0.479638\pi$
$60$		0			0
$61$	2.11888e13			0.863241			0.431620	−	0.902055i	$-0.357942\pi$
$61$	2.11888e13			0.863241			0.431620	+	0.902055i	$0.357942\pi$
$62$		0			0
$63$	−7.22646e12	−	4.12266e13i	−0.231138	−	1.31863i
$64$		0			0
$65$		−	2.04155e13i		−	0.516547i
$66$		0			0
$67$		−	5.84632e12i		−	0.117848i	−0.998262	−	0.0589239i	$-0.981233\pi$
$67$		−	5.84632e12i		−	0.117848i	0.998262	−	0.0589239i	$-0.0187669\pi$
$68$		0			0
$69$	−4.95842e13	+	4.31284e12i	−0.801632	+	0.0697261i
$70$		0			0
$71$	1.30114e14			1.69780			0.848898	−	0.528556i	$-0.177266\pi$
$71$	1.30114e14			1.69780			0.848898	+	0.528556i	$0.177266\pi$
$72$		0			0
$73$	8.46179e13			0.896480			0.448240	−	0.893913i	$-0.352051\pi$
$73$	8.46179e13			0.896480			0.448240	+	0.893913i	$0.352051\pi$
$74$		0			0
$75$	8.34880e13	−	7.26180e12i	0.722212	−	0.0628181i
$76$		0			0
$77$		−	3.51506e14i		−	2.49604i
$78$		0			0
$79$		−	1.42970e14i		−	0.837608i	−0.908077	−	0.418804i	$-0.862449\pi$
$79$		−	1.42970e14i		−	0.837608i	0.908077	−	0.418804i	$-0.137551\pi$
$80$		0			0
$81$	1.93616e14	−	7.00282e13i	0.940381	−	0.340122i
$82$		0			0
$83$	2.01259e13			0.0814083			0.0407042	−	0.999171i	$-0.487040\pi$
$83$	2.01259e13			0.0814083			0.0407042	+	0.999171i	$0.487040\pi$
$84$		0			0
$85$	−1.91033e14			−0.646347
$86$		0			0
$87$	3.10421e13	+	3.56888e14i	0.0882180	+	1.01423i
$88$		0			0
$89$		−	4.59438e14i		−	1.10104i	−0.834823	−	0.550519i	$-0.814430\pi$
$89$		−	4.59438e14i		−	1.10104i	0.834823	−	0.550519i	$-0.185570\pi$
$90$		0			0
$91$			6.49979e14i			1.31853i
$92$		0			0
$93$	−8.20644e13	−	9.43484e14i	−0.141427	−	1.62597i
$94$		0			0
$95$	6.17723e14			0.907542
$96$		0			0
$97$	−5.20584e14			−0.654189			−0.327094	−	0.944992i	$-0.606070\pi$
$97$	−5.20584e14			−0.654189			−0.327094	+	0.944992i	$0.606070\pi$
$98$		0			0
$99$	1.70314e15	−	2.98537e14i	1.83648	−	0.321910i
$100$		0			0
$101$		−	1.36026e15i		−	1.26244i	−0.775603	−	0.631221i	$-0.782554\pi$
$101$		−	1.36026e15i		−	1.26244i	0.775603	−	0.631221i	$-0.217446\pi$
$102$		0			0
$103$		−	4.92884e14i		−	0.394880i	−0.980315	−	0.197440i	$-0.936737\pi$
$103$		−	4.92884e14i		−	0.394880i	0.980315	−	0.197440i	$-0.0632628\pi$
$104$		0			0
$105$	1.00854e15	−	8.77228e13i	0.699476	−	0.0608405i
$106$		0			0
$107$	−1.86968e14			−0.112561			−0.0562805	−	0.998415i	$-0.517924\pi$
$107$	−1.86968e14			−0.112561			−0.0562805	+	0.998415i	$0.517924\pi$
$108$		0			0
$109$	2.56133e14			0.134204			0.0671022	−	0.997746i	$-0.478625\pi$
$109$	2.56133e14			0.134204			0.0671022	+	0.997746i	$0.478625\pi$
$110$		0			0
$111$	−2.09363e15	+	1.82104e14i	−0.957144	+	0.0832525i
$112$		0			0
$113$		−	3.67406e15i		−	1.46912i	−0.678542	−	0.734562i	$-0.737388\pi$
$113$		−	3.67406e15i		−	1.46912i	0.678542	−	0.734562i	$-0.262612\pi$
$114$		0			0
$115$		−	1.20382e15i		−	0.422014i
$116$		0			0
$117$	−3.14932e15	+	5.52033e14i	−0.970115	+	0.170048i
$118$		0			0
$119$	6.08201e15			1.64985
$120$		0			0
$121$	1.03441e16			2.47628
$122$		0			0
$123$	−4.08247e14	−	4.69356e15i	−0.0864240	−	0.993606i
$124$		0			0
$125$			4.82296e15i			0.904667i
$126$		0			0
$127$			2.37612e15i			0.395677i	0.980235	+	0.197839i	$0.0633922\pi$
$127$			2.37612e15i			0.395677i	−0.980235	+	0.197839i	$0.936608\pi$
$128$		0			0
$129$	−1.43781e14	−	1.65303e15i	−0.0212951	−	0.244827i
$130$		0			0
$131$	−5.37388e15			−0.709175			−0.354588	−	0.935023i	$-0.615379\pi$
$131$	−5.37388e15			−0.709175			−0.354588	+	0.935023i	$0.615379\pi$
$132$		0			0
$133$	−1.96668e16			−2.31658
$134$		0			0
$135$	1.28160e15	+	4.81213e15i	0.134974	+	0.506797i
$136$		0			0
$137$			1.58771e16i			1.49750i	0.662854	+	0.748749i	$0.269345\pi$
$137$			1.58771e16i			1.49750i	−0.662854	+	0.748749i	$0.730655\pi$
$138$		0			0
$139$			1.08203e16i			0.915439i	0.889097	+	0.457720i	$0.151334\pi$
$139$			1.08203e16i			0.915439i	−0.889097	+	0.457720i	$0.848666\pi$
$140$		0			0
$141$	8.11357e15	−	7.05719e14i	0.616692	−	0.0536400i
$142$		0			0
$143$	−2.68517e16			−1.83634
$144$		0			0
$145$	−8.66461e15			−0.533936
$146$		0			0
$147$	−1.41933e16	+	1.23454e15i	−0.789231	+	0.0686474i
$148$		0			0
$149$			1.13166e16i			0.568614i	0.958733	+	0.284307i	$0.0917635\pi$
$149$			1.13166e16i			0.568614i	−0.958733	+	0.284307i	$0.908236\pi$
$150$		0			0
$151$		−	1.71632e16i		−	0.780316i	−0.920748	−	0.390158i	$-0.872420\pi$
$151$		−	1.71632e16i		−	0.780316i	0.920748	−	0.390158i	$-0.127580\pi$
$152$		0			0
$153$	5.16551e15	+	2.94690e16i	0.212778	+	1.21389i
$154$		0			0
$155$	2.29061e16			0.855981
$156$		0			0
$157$	1.00953e16			0.342668			0.171334	−	0.985213i	$-0.445192\pi$
$157$	1.00953e16			0.342668			0.171334	+	0.985213i	$0.445192\pi$
$158$		0			0
$159$	8.41600e14	+	9.67577e15i	0.0259793	+	0.298681i
$160$		0			0
$161$			3.83266e16i			1.07722i
$162$		0			0
$163$			1.83997e16i			0.471416i	0.971824	+	0.235708i	$0.0757409\pi$
$163$			1.83997e16i			0.471416i	−0.971824	+	0.235708i	$0.924259\pi$
$164$		0			0
$165$	3.62398e15	+	4.16644e16i	0.0847337	+	0.974173i
$166$		0			0
$167$	−5.03318e16			−1.07515			−0.537575	−	0.843216i	$-0.680659\pi$
$167$	−5.03318e16			−1.07515			−0.537575	+	0.843216i	$0.680659\pi$
$168$		0			0
$169$	−1.53358e15			−0.0299610
$170$		0			0
$171$	−1.67032e16	−	9.52909e16i	−0.298764	−	1.70444i
$172$		0			0
$173$		−	1.75166e15i		−	0.0287147i	−0.999897	−	0.0143574i	$-0.995430\pi$
$173$		−	1.75166e15i		−	0.0287147i	0.999897	−	0.0143574i	$-0.00457025\pi$
$174$		0			0
$175$		−	6.45329e16i		−	0.970500i
$176$		0			0
$177$	9.22302e15	−	8.02220e14i	0.127372	−	0.0110788i
$178$		0			0
$179$	−5.36164e16			−0.680613			−0.340306	−	0.940315i	$-0.610531\pi$
$179$	−5.36164e16			−0.680613			−0.340306	+	0.940315i	$0.610531\pi$
$180$		0			0
$181$	1.26268e16			0.147470			0.0737351	−	0.997278i	$-0.476508\pi$
$181$	1.26268e16			0.147470			0.0737351	+	0.997278i	$0.476508\pi$
$182$		0			0
$183$	7.99611e16	−	6.95502e15i	0.859994	−	0.0748024i
$184$		0			0
$185$		−	5.08296e16i		−	0.503882i
$186$		0			0
$187$			2.51258e17i			2.29778i
$188$		0			0
$189$	−4.08031e16	−	1.53207e17i	−0.344532	−	1.29364i
$190$		0			0
$191$	8.21144e16			0.640721			0.320361	−	0.947296i	$-0.396196\pi$
$191$	8.21144e16			0.640721			0.320361	+	0.947296i	$0.396196\pi$
$192$		0			0
$193$	−9.08429e16			−0.655558			−0.327779	−	0.944754i	$-0.606300\pi$
$193$	−9.08429e16			−0.655558			−0.327779	+	0.944754i	$0.606300\pi$
$194$		0			0
$195$	−6.70120e15	−	7.70429e16i	−0.0447603	−	0.514604i
$196$		0			0
$197$			1.84764e17i			1.14320i	0.820533	+	0.571599i	$0.193677\pi$
$197$			1.84764e17i			1.14320i	−0.820533	+	0.571599i	$0.806323\pi$
$198$		0			0
$199$			1.94422e17i			1.11519i	0.830114	+	0.557594i	$0.188275\pi$
$199$			1.94422e17i			1.11519i	−0.830114	+	0.557594i	$0.811725\pi$
$200$		0			0
$201$	−1.91900e15	−	2.20625e16i	−0.0102119	−	0.117405i
$202$		0			0
$203$	2.75860e17			1.36291
$204$		0			0
$205$	1.13951e17			0.523077
$206$		0			0
$207$	−1.85703e17	+	3.25511e16i	−0.792575	+	0.138928i
$208$		0			0
$209$		−	8.12469e17i		−	3.22634i
$210$		0			0
$211$		−	2.44613e17i		−	0.904401i	−0.891916	−	0.452201i	$-0.850639\pi$
$211$		−	2.44613e17i		−	0.904401i	0.891916	−	0.452201i	$-0.149361\pi$
$212$		0			0
$213$	4.91017e17	−	4.27087e16i	1.69141	−	0.147119i
$214$		0			0
$215$	4.01328e16			0.128887
$216$		0			0
$217$	−7.29275e17			−2.18496
$218$		0			0
$219$	3.19326e17	−	2.77751e16i	0.893108	−	0.0776827i
$220$		0			0
$221$		−	4.64609e17i		−	1.21379i
$222$		0			0
$223$			3.26598e17i			0.797494i	0.917061	+	0.398747i	$0.130555\pi$
$223$			3.26598e17i			0.797494i	−0.917061	+	0.398747i	$0.869445\pi$
$224$		0			0
$225$	3.12679e17	−	5.48084e16i	0.714052	−	0.125164i
$226$		0			0
$227$	3.27486e17			0.699840			0.349920	−	0.936780i	$-0.386209\pi$
$227$	3.27486e17			0.699840			0.349920	+	0.936780i	$0.386209\pi$
$228$		0			0
$229$	1.03248e17			0.206593			0.103296	−	0.994651i	$-0.467061\pi$
$229$	1.03248e17			0.206593			0.103296	+	0.994651i	$0.467061\pi$
$230$		0			0
$231$	−1.15379e17	−	1.32649e18i	−0.216290	−	2.48665i
$232$		0			0
$233$			9.99582e17i			1.75650i	0.478199	+	0.878251i	$0.341290\pi$
$233$			9.99582e17i			1.75650i	−0.478199	+	0.878251i	$0.658710\pi$
$234$		0			0
$235$			1.96983e17i			0.324654i
$236$		0			0
$237$	−4.69285e16	−	5.39532e17i	−0.0725812	−	0.834458i
$238$		0			0
$239$	2.83454e17			0.411622			0.205811	−	0.978592i	$-0.434017\pi$
$239$	2.83454e17			0.411622			0.205811	+	0.978592i	$0.434017\pi$
$240$		0			0
$241$	7.71721e17			1.05277			0.526384	−	0.850247i	$-0.323547\pi$
$241$	7.71721e17			1.05277			0.526384	+	0.850247i	$0.323547\pi$
$242$		0			0
$243$	7.07672e17	−	3.27821e17i	0.907371	−	0.420330i
$244$		0			0
$245$		−	3.44589e17i		−	0.415485i
$246$		0			0
$247$			1.50236e18i			1.70430i
$248$		0			0
$249$	7.59499e16	−	6.60613e15i	0.0811021	−	0.00705427i
$250$		0			0
$251$	−1.43019e18			−1.43827			−0.719137	−	0.694868i	$-0.755463\pi$
$251$	−1.43019e18			−1.43827			−0.719137	+	0.694868i	$0.755463\pi$
$252$		0			0
$253$	−1.58334e18			−1.50027
$254$		0			0
$255$	−7.20909e17	+	6.27048e16i	−0.643916	+	0.0560079i
$256$		0			0
$257$		−	2.19887e17i		−	0.185226i	−0.995702	−	0.0926128i	$-0.970478\pi$
$257$		−	2.19887e17i		−	0.185226i	0.995702	−	0.0926128i	$-0.0295219\pi$
$258$		0			0
$259$			1.61829e18i			1.28620i
$260$		0			0
$261$	2.34290e17	+	1.33661e18i	0.175772	+	1.00277i
$262$		0			0
$263$	−1.77718e18			−1.25911			−0.629553	−	0.776957i	$-0.716762\pi$
$263$	−1.77718e18			−1.25911			−0.629553	+	0.776957i	$0.716762\pi$
$264$		0			0
$265$	−2.34911e17			−0.157238
$266$		0			0
$267$	−1.50807e17	−	1.73380e18i	−0.0954082	−	1.09690i
$268$		0			0
$269$			2.21535e18i			1.32526i	0.748948	+	0.662629i	$0.230559\pi$
$269$			2.21535e18i			1.32526i	−0.748948	+	0.662629i	$0.769441\pi$
$270$		0			0
$271$			8.40929e17i			0.475871i	0.971281	+	0.237936i	$0.0764707\pi$
$271$			8.40929e17i			0.475871i	−0.971281	+	0.237936i	$0.923529\pi$
$272$		0			0
$273$	2.13350e17	+	2.45286e18i	0.114254	+	1.31357i
$274$		0			0
$275$	2.66596e18			1.35163
$276$		0			0
$277$	−2.37475e18			−1.14030			−0.570150	−	0.821541i	$-0.693115\pi$
$277$	−2.37475e18			−1.14030			−0.570150	+	0.821541i	$0.693115\pi$
$278$		0			0
$279$	−6.19380e17	−	3.53353e18i	−0.281790	−	1.60760i
$280$		0			0
$281$			8.29636e17i			0.357759i	0.983871	+	0.178879i	$0.0572472\pi$
$281$			8.29636e17i			0.357759i	−0.983871	+	0.178879i	$0.942753\pi$
$282$		0			0
$283$			2.44669e18i			1.00042i	0.865905	+	0.500209i	$0.166743\pi$
$283$			2.44669e18i			1.00042i	−0.865905	+	0.500209i	$0.833257\pi$
$284$		0			0
$285$	2.33113e18	−	2.02762e17i	0.904129	−	0.0786412i
$286$		0			0
$287$	−3.62794e18			−1.33520
$288$		0			0
$289$	−1.48503e18			−0.518803
$290$		0			0
$291$	−1.96455e18	+	1.70877e17i	−0.651728	+	0.0566874i
$292$		0			0
$293$			3.91289e18i			1.23308i	0.787324	+	0.616539i	$0.211466\pi$
$293$			3.91289e18i			1.23308i	−0.787324	+	0.616539i	$0.788534\pi$
$294$		0			0
$295$			2.23919e17i			0.0670541i
$296$		0			0
$297$	6.32922e18	−	1.68564e18i	1.80168	−	0.479836i
$298$		0			0
$299$	2.92779e18			0.792513
$300$		0			0
$301$	−1.27773e18			−0.328996
$302$		0			0
$303$	−4.46493e17	−	5.13328e18i	−0.109394	−	1.25769i
$304$		0			0
$305$			1.94131e18i			0.452738i
$306$		0			0
$307$		−	4.11807e18i		−	0.914441i	−0.889353	−	0.457220i	$-0.848845\pi$
$307$		−	4.11807e18i		−	0.914441i	0.889353	−	0.457220i	$-0.151155\pi$
$308$		0			0
$309$	−1.61785e17	−	1.86002e18i	−0.0342175	−	0.393395i
$310$		0			0
$311$	−8.36299e18			−1.68523			−0.842614	−	0.538518i	$-0.818984\pi$
$311$	−8.36299e18			−1.68523			−0.842614	+	0.538518i	$0.818984\pi$
$312$		0			0
$313$	−3.51721e17			−0.0675485			−0.0337743	−	0.999429i	$-0.510753\pi$
$313$	−3.51721e17			−0.0675485			−0.0337743	+	0.999429i	$0.510753\pi$
$314$		0			0
$315$	3.77718e18	−	6.62088e17i	0.691573	−	0.121223i
$316$		0			0
$317$		−	1.22031e18i		−	0.213072i	−0.994309	−	0.106536i	$-0.966024\pi$
$317$		−	1.22031e18i		−	0.213072i	0.994309	−	0.106536i	$-0.0339760\pi$
$318$		0			0
$319$			1.13962e19i			1.89815i
$320$		0			0
$321$	−7.05568e17	+	6.13704e16i	−0.112138	+	0.00975374i
$322$		0			0
$323$	1.40579e19			2.13257
$324$		0			0
$325$	−4.92970e18			−0.713996
$326$		0			0
$327$	9.66581e17	−	8.40733e16i	0.133700	−	0.0116292i
$328$		0			0
$329$		−	6.27146e18i		−	0.828705i
$330$		0			0
$331$			1.43269e19i			1.80901i	0.426461	+	0.904506i	$0.359760\pi$
$331$			1.43269e19i			1.80901i	−0.426461	+	0.904506i	$0.640240\pi$
$332$		0			0
$333$	−7.84105e18	+	1.37443e18i	−0.946330	+	0.165879i
$334$		0			0
$335$	5.35639e17			0.0618068
$336$		0			0
$337$	9.54302e17			0.105308			0.0526540	−	0.998613i	$-0.483232\pi$
$337$	9.54302e17			0.105308			0.0526540	+	0.998613i	$0.483232\pi$
$338$		0			0
$339$	−1.20598e18	−	1.38650e19i	−0.127304	−	1.46360i
$340$		0			0
$341$		−	3.01276e19i		−	3.04303i
$342$		0			0
$343$		−	2.87756e18i		−	0.278175i
$344$		0			0
$345$	−3.95142e17	−	4.54290e18i	−0.0365688	−	0.420426i
$346$		0			0
$347$	6.14589e18			0.544645			0.272322	−	0.962206i	$-0.412208\pi$
$347$	6.14589e18			0.544645			0.272322	+	0.962206i	$0.412208\pi$
$348$		0			0
$349$	1.53448e19			1.30248			0.651238	−	0.758874i	$-0.274250\pi$
$349$	1.53448e19			1.30248			0.651238	+	0.758874i	$0.274250\pi$
$350$		0			0
$351$	−1.17035e19	+	3.11697e18i	−0.951731	+	0.253472i
$352$		0			0
$353$			5.81472e18i			0.453126i	0.973997	+	0.226563i	$0.0727489\pi$
$353$			5.81472e18i			0.453126i	−0.973997	+	0.226563i	$0.927251\pi$
$354$		0			0
$355$			1.19210e19i			0.890432i
$356$		0			0
$357$	2.29520e19	−	1.99637e18i	1.64365	−	0.142965i
$358$		0			0
$359$	−8.32500e18			−0.571710			−0.285855	−	0.958273i	$-0.592278\pi$
$359$	−8.32500e18			−0.571710			−0.285855	+	0.958273i	$0.592278\pi$
$360$		0			0
$361$	−3.02766e19			−1.99436
$362$		0			0
$363$	3.90358e19	−	3.39534e18i	2.46697	−	0.214577i
$364$		0			0
$365$			7.75268e18i			0.470171i
$366$		0			0
$367$		−	2.06582e19i		−	1.20253i	−0.799049	−	0.601265i	$-0.794663\pi$
$367$		−	2.06582e19i		−	1.20253i	0.799049	−	0.601265i	$-0.205337\pi$
$368$		0			0
$369$	−3.08124e18	−	1.75783e19i	−0.172198	−	0.982380i
$370$		0			0
$371$	7.47898e18			0.401364
$372$		0			0
$373$	1.76828e19			0.911453			0.455727	−	0.890120i	$-0.349379\pi$
$373$	1.76828e19			0.911453			0.455727	+	0.890120i	$0.349379\pi$
$374$		0			0
$375$	1.58309e18	+	1.82006e19i	0.0783920	+	0.901264i
$376$		0			0
$377$		−	2.10731e19i		−	1.00269i
$378$		0			0
$379$		−	2.53073e19i		−	1.15731i	−0.815571	−	0.578657i	$-0.803577\pi$
$379$		−	2.53073e19i		−	1.15731i	0.815571	−	0.578657i	$-0.196423\pi$
$380$		0			0
$381$	7.79941e17	+	8.96689e18i	0.0342866	+	0.394189i
$382$		0			0
$383$	−1.30319e18			−0.0550829			−0.0275414	−	0.999621i	$-0.508768\pi$
$383$	−1.30319e18			−0.0550829			−0.0275414	+	0.999621i	$0.508768\pi$
$384$		0			0
$385$	3.22049e19			1.30908
$386$		0			0
$387$	−1.08519e18	−	6.19094e18i	−0.0424299	−	0.242061i
$388$		0			0
$389$			3.05944e19i			1.15085i	0.817853	+	0.575427i	$0.195164\pi$
$389$			3.05944e19i			1.15085i	−0.817853	+	0.575427i	$0.804836\pi$
$390$		0			0
$391$		−	2.73961e19i		−	0.991659i
$392$		0			0
$393$	−2.02797e19	+	1.76393e18i	−0.706508	+	0.0614521i
$394$		0			0
$395$	1.30989e19			0.439295
$396$		0			0
$397$	−5.83722e19			−1.88485			−0.942426	−	0.334414i	$-0.891462\pi$
$397$	−5.83722e19			−1.88485			−0.942426	+	0.334414i	$0.891462\pi$
$398$		0			0
$399$	−7.42175e19	+	6.45545e18i	−2.30786	+	0.200738i
$400$		0			0
$401$			2.14385e19i			0.642112i	0.947060	+	0.321056i	$0.104038\pi$
$401$			2.14385e19i			0.642112i	−0.947060	+	0.321056i	$0.895962\pi$
$402$		0			0
$403$			5.57098e19i			1.60747i
$404$		0			0
$405$	6.41598e18	+	1.77391e19i	0.178382	+	0.493195i
$406$		0			0
$407$	−6.68543e19			−1.79131
$408$		0			0
$409$	−1.09849e19			−0.283709			−0.141854	−	0.989888i	$-0.545306\pi$
$409$	−1.09849e19			−0.283709			−0.141854	+	0.989888i	$0.545306\pi$
$410$		0			0
$411$	5.21151e18	+	5.99160e19i	0.129763	+	1.49186i
$412$		0			0
$413$		−	7.12903e18i		−	0.171161i
$414$		0			0
$415$			1.84393e18i			0.0426957i
$416$		0			0
$417$	3.55168e18	+	4.08332e19i	0.0793255	+	0.911996i
$418$		0			0
$419$	7.10414e19			1.53076			0.765378	−	0.643581i	$-0.222552\pi$
$419$	7.10414e19			1.53076			0.765378	+	0.643581i	$0.222552\pi$
$420$		0			0
$421$	7.02493e19			1.46058			0.730291	−	0.683136i	$-0.239384\pi$
$421$	7.02493e19			1.46058			0.730291	+	0.683136i	$0.239384\pi$
$422$		0			0
$423$	3.03869e19	−	5.32641e18i	0.609725	−	0.106876i
$424$		0			0
$425$			4.61284e19i			0.893412i
$426$		0			0
$427$		−	6.18067e19i		−	1.15565i
$428$		0			0
$429$	−1.01332e20	+	8.81384e18i	−1.82943	+	0.159124i
$430$		0			0
$431$	−5.48889e19			−0.956985			−0.478492	−	0.878092i	$-0.658817\pi$
$431$	−5.48889e19			−0.956985			−0.478492	+	0.878092i	$0.658817\pi$
$432$		0			0
$433$	6.36612e19			1.07205			0.536026	−	0.844202i	$-0.319925\pi$
$433$	6.36612e19			1.07205			0.536026	+	0.844202i	$0.319925\pi$
$434$		0			0
$435$	−3.26980e19	+	2.84408e18i	−0.531927	+	0.0462671i
$436$		0			0
$437$			8.85879e19i			1.39240i
$438$		0			0
$439$			1.34028e18i			0.0203570i	0.999948	+	0.0101785i	$0.00323997\pi$
$439$			1.34028e18i			0.0203570i	−0.999948	+	0.0101785i	$0.996760\pi$
$440$		0			0
$441$	−5.31568e19	+	9.31766e18i	−0.780313	+	0.136778i
$442$		0			0
$443$	−6.44524e19			−0.914558			−0.457279	−	0.889323i	$-0.651176\pi$
$443$	−6.44524e19			−0.914558			−0.457279	+	0.889323i	$0.651176\pi$
$444$		0			0
$445$	4.20937e19			0.577454
$446$		0			0
$447$	3.71456e18	+	4.27058e19i	0.0492721	+	0.566475i
$448$		0			0
$449$			7.19613e19i			0.923105i	0.887113	+	0.461553i	$0.152707\pi$
$449$			7.19613e19i			0.923105i	−0.887113	+	0.461553i	$0.847293\pi$
$450$		0			0
$451$		−	1.49876e20i		−	1.85955i
$452$		0			0
$453$	−5.63366e18	−	6.47695e19i	−0.0676167	−	0.777381i
$454$		0			0
$455$	−5.95511e19			−0.691519
$456$		0			0
$457$	−2.39947e19			−0.269615			−0.134808	−	0.990872i	$-0.543042\pi$
$457$	−2.39947e19			−0.269615			−0.134808	+	0.990872i	$0.543042\pi$
$458$		0			0
$459$	2.91663e19	+	1.09513e20i	0.317165	+	1.19089i
$460$		0			0
$461$		−	1.24275e20i		−	1.30805i	−0.756471	−	0.654027i	$-0.773078\pi$
$461$		−	1.24275e20i		−	1.30805i	0.756471	−	0.654027i	$-0.226922\pi$
$462$		0			0
$463$		−	1.28962e20i		−	1.31403i	−0.753879	−	0.657014i	$-0.771819\pi$
$463$		−	1.28962e20i		−	1.31403i	0.753879	−	0.657014i	$-0.228181\pi$
$464$		0			0
$465$	8.64419e19	−	7.51873e18i	0.852761	−	0.0741733i
$466$		0			0
$467$	9.07767e19			0.867157			0.433579	−	0.901116i	$-0.357251\pi$
$467$	9.07767e19			0.867157			0.433579	+	0.901116i	$0.357251\pi$
$468$		0			0
$469$	−1.70534e19			−0.157767
$470$		0			0
$471$	3.80973e19	−	3.31371e18i	0.341379	−	0.0296932i
$472$		0			0
$473$		−	5.27851e19i		−	0.458198i
$474$		0			0
$475$		−	1.49161e20i		−	1.25445i
$476$		0			0
$477$	6.35197e18	+	3.62377e19i	0.0517631	+	0.295306i
$478$		0			0
$479$	8.21775e19			0.648988			0.324494	−	0.945888i	$-0.394806\pi$
$479$	8.21775e19			0.648988			0.324494	+	0.945888i	$0.394806\pi$
$480$		0			0
$481$	1.23622e20			0.946256
$482$		0			0
$483$	1.25804e19	+	1.44635e20i	0.0933447	+	1.07317i
$484$		0			0
$485$		−	4.76959e19i		−	0.343098i
$486$		0			0
$487$			1.84649e20i			1.28789i	0.765071	+	0.643946i	$0.222704\pi$
$487$			1.84649e20i			1.28789i	−0.765071	+	0.643946i	$0.777296\pi$
$488$		0			0
$489$	6.03955e18	+	6.94360e19i	0.0408496	+	0.469643i
$490$		0			0
$491$	1.65994e20			1.08888			0.544442	−	0.838799i	$-0.316741\pi$
$491$	1.65994e20			1.08888			0.544442	+	0.838799i	$0.316741\pi$
$492$		0			0
$493$	−1.97186e20			−1.25466
$494$		0			0
$495$	2.73520e19	+	1.56041e20i	0.168830	+	0.963166i
$496$		0			0
$497$		−	3.79536e20i		−	2.27290i
$498$		0			0
$499$			1.51240e20i			0.878844i	0.898281	+	0.439422i	$0.144817\pi$
$499$			1.51240e20i			0.878844i	−0.898281	+	0.439422i	$0.855183\pi$
$500$		0			0
$501$	−1.89939e20	+	1.65210e19i	−1.07111	+	0.0931649i
$502$		0			0
$503$	−2.04561e20			−1.11960			−0.559800	−	0.828628i	$-0.689122\pi$
$503$	−2.04561e20			−1.11960			−0.559800	+	0.828628i	$0.689122\pi$
$504$		0			0
$505$	1.24627e20			0.662104
$506$		0			0
$507$	−5.78735e18	+	5.03385e17i	−0.0298483	+	0.00259621i
$508$		0			0
$509$		−	1.16643e20i		−	0.584082i	−0.956406	−	0.292041i	$-0.905666\pi$
$509$		−	1.16643e20i		−	0.584082i	0.956406	−	0.292041i	$-0.0943344\pi$
$510$		0			0
$511$		−	2.46827e20i		−	1.20015i
$512$		0			0
$513$	−9.43120e19	−	3.54121e20i	−0.445335	−	1.67214i
$514$		0			0
$515$	4.51580e19			0.207100
$516$		0			0
$517$	2.59085e20			1.15415
$518$		0			0
$519$	−5.74968e17	−	6.61033e18i	−0.00248822	−	0.0286067i
$520$		0			0
$521$			9.40159e19i			0.395292i	0.980273	+	0.197646i	$0.0633297\pi$
$521$			9.40159e19i			0.395292i	−0.980273	+	0.197646i	$0.936670\pi$
$522$		0			0
$523$		−	1.02555e20i		−	0.418980i	−0.977811	−	0.209490i	$-0.932820\pi$
$523$		−	1.02555e20i		−	0.418980i	0.977811	−	0.209490i	$-0.0671803\pi$
$524$		0			0
$525$	−2.11823e19	−	2.43531e20i	−0.0840967	−	0.966850i
$526$		0			0
$527$	5.21290e20			2.01141
$528$		0			0
$529$	−9.39954e19			−0.352524
$530$		0			0
$531$	3.45420e19	−	6.05475e18i	0.125933	−	0.0220743i
$532$		0			0
$533$			2.77140e20i			0.982303i
$534$		0			0
$535$		−	1.71300e19i		−	0.0590341i
$536$		0			0
$537$	−2.02335e20	+	1.75991e19i	−0.678053	+	0.0589771i
$538$		0			0
$539$	−4.53225e20			−1.47706
$540$		0			0
$541$	3.70899e20			1.17565			0.587823	−	0.808989i	$-0.299985\pi$
$541$	3.70899e20			1.17565			0.587823	+	0.808989i	$0.299985\pi$
$542$		0			0
$543$	4.76504e19	−	4.14464e18i	0.146916	−	0.0127787i
$544$		0			0
$545$			2.34669e19i			0.0703852i
$546$		0			0
$547$		−	5.95393e19i		−	0.173740i	−0.996220	−	0.0868698i	$-0.972314\pi$
$547$		−	5.95393e19i		−	0.173740i	0.996220	−	0.0868698i	$-0.0276864\pi$
$548$		0			0
$549$	2.99470e20	−	5.24930e19i	0.850277	−	0.149042i
$550$		0			0
$551$	6.37621e20			1.76168
$552$		0			0
$553$	−4.17036e20			−1.12134
$554$		0			0
$555$	−1.66844e19	−	1.91818e20i	−0.0436629	−	0.501987i
$556$		0			0
$557$			1.19190e20i			0.303616i	0.988410	+	0.151808i	$0.0485096\pi$
$557$			1.19190e20i			0.303616i	−0.988410	+	0.151808i	$0.951490\pi$
$558$		0			0
$559$			9.76065e19i			0.242042i
$560$		0			0
$561$	8.24733e19	+	9.48185e20i	0.199109	+	2.28914i
$562$		0			0
$563$	−7.55853e20			−1.77674			−0.888371	−	0.459127i	$-0.848162\pi$
$563$	−7.55853e20			−1.77674			−0.888371	+	0.459127i	$0.848162\pi$
$564$		0			0
$565$	3.36618e20			0.770501
$566$		0			0
$567$	−2.04269e20	−	5.64770e20i	−0.455334	−	1.25892i
$568$		0			0
$569$		−	1.96195e20i		−	0.425938i	−0.977059	−	0.212969i	$-0.931687\pi$
$569$		−	1.96195e20i		−	0.425938i	0.977059	−	0.212969i	$-0.0683133\pi$
$570$		0			0
$571$			1.61066e20i			0.340592i	0.985393	+	0.170296i	$0.0544723\pi$
$571$			1.61066e20i			0.340592i	−0.985393	+	0.170296i	$0.945528\pi$
$572$		0			0
$573$	3.09879e20	−	2.69533e19i	0.638311	−	0.0555204i
$574$		0			0
$575$	−2.90684e20			−0.583328
$576$		0			0
$577$	1.00141e20			0.195791			0.0978954	−	0.995197i	$-0.468789\pi$
$577$	1.00141e20			0.195791			0.0978954	+	0.995197i	$0.468789\pi$
$578$		0			0
$579$	−3.42818e20	+	2.98184e19i	−0.653092	+	0.0568061i
$580$		0			0
$581$		−	5.87062e19i		−	0.108984i
$582$		0			0
$583$			3.08969e20i			0.558987i
$584$		0			0
$585$	−5.05773e19	−	2.88541e20i	−0.0891839	−	0.508789i
$586$		0			0
$587$	−3.48320e20			−0.598677			−0.299339	−	0.954147i	$-0.596766\pi$
$587$	−3.48320e20			−0.598677			−0.299339	+	0.954147i	$0.596766\pi$
$588$		0			0
$589$	−1.68564e21			−2.82424
$590$		0			0
$591$	6.06473e19	+	6.97254e20i	0.0990616	+	1.13890i
$592$		0			0
$593$			9.93907e20i			1.58284i	0.611276	+	0.791418i	$0.290657\pi$
$593$			9.93907e20i			1.58284i	−0.611276	+	0.791418i	$0.709343\pi$
$594$		0			0
$595$			5.57234e20i			0.865287i
$596$		0			0
$597$	6.38174e19	+	7.33701e20i	0.0966344	+	1.11099i
$598$		0			0
$599$	4.71732e20			0.696617			0.348308	−	0.937380i	$-0.386756\pi$
$599$	4.71732e20			0.696617			0.348308	+	0.937380i	$0.386756\pi$
$600$		0			0
$601$	−1.09217e21			−1.57301			−0.786503	−	0.617587i	$-0.788110\pi$
$601$	−1.09217e21			−1.57301			−0.786503	+	0.617587i	$0.788110\pi$
$602$		0			0
$603$	−1.44837e19	−	8.26285e19i	−0.0203469	−	0.116078i
$604$		0			0
$605$			9.47721e20i			1.29872i
$606$		0			0
$607$		−	4.13358e20i		−	0.552600i	−0.961071	−	0.276300i	$-0.910892\pi$
$607$		−	4.13358e20i		−	0.552600i	0.961071	−	0.276300i	$-0.0891084\pi$
$608$		0			0
$609$	1.04103e21	−	9.05486e19i	1.35779	−	0.118101i
$610$		0			0
$611$	−4.79081e20			−0.609677
$612$		0			0
$613$	9.63118e20			1.19599			0.597993	−	0.801502i	$-0.295965\pi$
$613$	9.63118e20			1.19599			0.597993	+	0.801502i	$0.295965\pi$
$614$		0			0
$615$	4.30024e20	−	3.74035e19i	0.521110	−	0.0453262i
$616$		0			0
$617$			7.09668e20i			0.839299i	0.907686	+	0.419650i	$0.137847\pi$
$617$			7.09668e20i			0.839299i	−0.907686	+	0.419650i	$0.862153\pi$
$618$		0			0
$619$			6.48141e20i			0.748152i	0.927398	+	0.374076i	$0.122040\pi$
$619$			6.48141e20i			0.748152i	−0.927398	+	0.374076i	$0.877960\pi$
$620$		0			0
$621$	−6.90110e20	+	1.83795e20i	−0.777555	+	0.207084i
$622$		0			0
$623$	−1.34016e21			−1.47400
$624$		0			0
$625$	2.33272e20			0.250474
$626$		0			0
$627$	−2.66686e20	−	3.06605e21i	−0.279572	−	3.21420i
$628$		0			0
$629$		−	1.15676e21i		−	1.18404i
$630$		0			0
$631$			1.26945e21i			1.26881i	0.773001	+	0.634405i	$0.218755\pi$
$631$			1.26945e21i			1.26881i	−0.773001	+	0.634405i	$0.781245\pi$
$632$		0			0
$633$	−8.02920e19	−	9.23108e20i	−0.0783690	−	0.900999i
$634$		0			0
$635$	−2.17700e20			−0.207518
$636$		0			0
$637$	8.38071e20			0.780253
$638$		0			0
$639$	1.83895e21	−	3.22344e20i	1.67230	−	0.293132i
$640$		0			0
$641$		−	7.33282e20i		−	0.651382i	−0.945476	−	0.325691i	$-0.894403\pi$
$641$		−	7.33282e20i		−	0.651382i	0.945476	−	0.325691i	$-0.105597\pi$
$642$		0			0
$643$			1.36465e21i			1.18424i	0.805850	+	0.592119i	$0.201709\pi$
$643$			1.36465e21i			1.18424i	−0.805850	+	0.592119i	$0.798291\pi$
$644$		0			0
$645$	1.51451e20	−	1.31732e19i	0.128403	−	0.0111685i
$646$		0			0
$647$	2.20674e21			1.82797			0.913987	−	0.405744i	$-0.132987\pi$
$647$	2.20674e21			1.82797			0.913987	+	0.405744i	$0.132987\pi$
$648$		0			0
$649$	2.94512e20			0.238379
$650$		0			0
$651$	−2.75210e21	+	2.39378e20i	−2.17674	+	0.189333i
$652$		0			0
$653$			3.50941e20i			0.271260i	0.990760	+	0.135630i	$0.0433058\pi$
$653$			3.50941e20i			0.271260i	−0.990760	+	0.135630i	$0.956694\pi$
$654$		0			0
$655$		−	4.92355e20i		−	0.371936i
$656$		0			0
$657$	1.19594e21	−	2.09632e20i	0.883017	−	0.154781i
$658$		0			0
$659$	−6.05839e19			−0.0437237			−0.0218618	−	0.999761i	$-0.506959\pi$
$659$	−6.05839e19			−0.0437237			−0.0218618	+	0.999761i	$0.506959\pi$
$660$		0			0
$661$	−8.53243e20			−0.601952			−0.300976	−	0.953632i	$-0.597312\pi$
$661$	−8.53243e20			−0.601952			−0.300976	+	0.953632i	$0.597312\pi$
$662$		0			0
$663$	−1.52504e20	−	1.75332e21i	−0.105179	−	1.20923i
$664$		0			0
$665$		−	1.80187e21i		−	1.21496i
$666$		0			0
$667$		−	1.24259e21i		−	0.819192i
$668$		0			0
$669$	1.07203e20	+	1.23250e21i	0.0691052	+	0.794494i
$670$		0			0
$671$	2.55334e21			1.60949
$672$		0			0
$673$	−1.72542e21			−1.06361			−0.531803	−	0.846868i	$-0.678485\pi$
$673$	−1.72542e21			−1.06361			−0.531803	+	0.846868i	$0.678485\pi$
$674$		0			0
$675$	1.16198e21	−	3.09467e20i	0.700520	−	0.186567i
$676$		0			0
$677$		−	9.87672e20i		−	0.582368i	−0.956667	−	0.291184i	$-0.905951\pi$
$677$		−	9.87672e20i		−	0.582368i	0.956667	−	0.291184i	$-0.0940492\pi$
$678$		0			0
$679$			1.51852e21i			0.875785i
$680$		0			0
$681$	1.23585e21	−	1.07494e20i	0.697208	−	0.0606432i
$682$		0			0
$683$	2.20150e21			1.21496			0.607482	−	0.794333i	$-0.292180\pi$
$683$	2.20150e21			1.21496			0.607482	+	0.794333i	$0.292180\pi$
$684$		0			0
$685$	−1.45466e21			−0.785382
$686$		0			0
$687$	3.89631e20	−	3.38902e19i	0.205816	−	0.0179019i
$688$		0			0
$689$		−	5.71324e20i		−	0.295283i
$690$		0			0
$691$			1.24816e21i			0.631228i	0.948888	+	0.315614i	$0.102210\pi$
$691$			1.24816e21i			0.631228i	−0.948888	+	0.315614i	$0.897790\pi$
$692$		0			0
$693$	−8.70820e20	−	4.96798e21i	−0.430952	−	2.45856i
$694$		0			0
$695$	−9.91358e20			−0.480114
$696$		0			0
$697$	2.59327e21			1.22914
$698$		0			0
$699$	3.28104e20	+	3.77217e21i	0.152206	+	1.74990i
$700$		0			0
$701$		−	4.24355e21i		−	1.92683i	−0.268009	−	0.963416i	$-0.586366\pi$
$701$		−	4.24355e21i		−	1.92683i	0.268009	−	0.963416i	$-0.413634\pi$
$702$		0			0
$703$			3.74051e21i			1.66252i
$704$		0			0
$705$	6.46580e19	+	7.43365e20i	0.0281322	+	0.323432i
$706$		0			0
$707$	−3.96782e21			−1.69008
$708$		0			0
$709$	−3.57825e21			−1.49219			−0.746094	−	0.665841i	$-0.768073\pi$
$709$	−3.57825e21			−1.49219			−0.746094	+	0.665841i	$0.768073\pi$
$710$		0			0
$711$	−3.54193e20	−	2.02065e21i	−0.144616	−	0.825029i
$712$		0			0
$713$			3.28498e21i			1.31329i
$714$		0			0
$715$		−	2.46015e21i		−	0.963091i
$716$		0			0
$717$	1.06968e21	−	9.30413e19i	0.410074	−	0.0356683i
$718$		0			0
$719$	−3.97493e20			−0.149232			−0.0746162	−	0.997212i	$-0.523773\pi$
$719$	−3.97493e20			−0.149232			−0.0746162	+	0.997212i	$0.523773\pi$
$720$		0			0
$721$	−1.43772e21			−0.528640
$722$		0			0
$723$	2.91228e21	−	2.53311e20i	1.04881	−	0.0912255i
$724$		0			0
$725$			2.09223e21i			0.738032i
$726$		0			0
$727$		−	1.33381e21i		−	0.460877i	−0.973087	−	0.230438i	$-0.925984\pi$
$727$		−	1.33381e21i		−	0.460877i	0.973087	−	0.230438i	$-0.0740160\pi$
$728$		0			0
$729$	2.56297e21	−	1.46940e21i	0.867535	−	0.497375i
$730$		0			0
$731$	9.13328e20			0.302863
$732$		0			0
$733$	4.44650e21			1.44457			0.722285	−	0.691596i	$-0.243092\pi$
$733$	4.44650e21			1.44457			0.722285	+	0.691596i	$0.243092\pi$
$734$		0			0
$735$	−1.13108e20	−	1.30039e21i	−0.0360030	−	0.413922i
$736$		0			0
$737$		−	7.04507e20i		−	0.219725i
$738$		0			0
$739$			5.39016e20i			0.164728i	0.996602	+	0.0823642i	$0.0262471\pi$
$739$			5.39016e20i			0.164728i	−0.996602	+	0.0823642i	$0.973753\pi$
$740$		0			0
$741$	4.93136e20	+	5.66952e21i	0.147683	+	1.69789i
$742$		0			0
$743$	1.73352e21			0.508759			0.254380	−	0.967104i	$-0.418129\pi$
$743$	1.73352e21			0.508759			0.254380	+	0.967104i	$0.418129\pi$
$744$		0			0
$745$	−1.03682e21			−0.298217
$746$		0			0
$747$	2.84447e20	−	4.98598e19i	0.0801858	−	0.0140555i
$748$		0			0
$749$			5.45376e20i			0.150689i
$750$		0			0
$751$		−	2.43093e21i		−	0.658375i	−0.944265	−	0.329188i	$-0.893225\pi$
$751$		−	2.43093e21i		−	0.658375i	0.944265	−	0.329188i	$-0.106775\pi$
$752$		0			0
$753$	−5.39718e21	+	4.69448e20i	−1.43286	+	0.124631i
$754$		0			0
$755$	1.57249e21			0.409247
$756$		0			0
$757$	1.58554e21			0.404538			0.202269	−	0.979330i	$-0.435169\pi$
$757$	1.58554e21			0.404538			0.202269	+	0.979330i	$0.435169\pi$
$758$		0			0
$759$	−5.97511e21	+	5.19716e20i	−1.49463	+	0.130003i
$760$		0			0
$761$		−	2.50338e21i		−	0.613964i	−0.951715	−	0.306982i	$-0.900681\pi$
$761$		−	2.50338e21i		−	0.613964i	0.951715	−	0.306982i	$-0.0993191\pi$
$762$		0			0
$763$		−	7.47128e20i		−	0.179664i
$764$		0			0
$765$	−2.69995e21	+	4.73264e20i	−0.636640	+	0.111594i
$766$		0			0
$767$	−5.44591e20			−0.125923
$768$		0			0
$769$	−1.95350e21			−0.442961			−0.221481	−	0.975165i	$-0.571089\pi$
$769$	−1.95350e21			−0.442961			−0.221481	+	0.975165i	$0.571089\pi$
$770$		0			0
$771$	−7.21759e19	−	8.29798e20i	−0.0160504	−	0.184529i
$772$		0			0
$773$			8.08728e21i			1.76383i	0.471410	+	0.881914i	$0.343745\pi$
$773$			8.08728e21i			1.76383i	−0.471410	+	0.881914i	$0.656255\pi$
$774$		0			0
$775$		−	5.53112e21i		−	1.18318i
$776$		0			0
$777$	5.31189e20	+	6.10702e21i	0.111453	+	1.28136i
$778$		0			0
$779$	−8.38559e21			−1.72585
$780$		0			0
$781$	1.56793e22			3.16551
$782$		0			0
$783$	1.32288e21	+	4.96714e21i	0.262004	+	0.983770i
$784$		0			0
$785$			9.24935e20i			0.179717i
$786$		0			0
$787$		−	3.39098e21i		−	0.646420i	−0.946327	−	0.323210i	$-0.895238\pi$
$787$		−	3.39098e21i		−	0.646420i	0.946327	−	0.323210i	$-0.104762\pi$
$788$		0			0
$789$	−6.70662e21	+	5.83343e20i	−1.25437	+	0.109105i
$790$		0			0
$791$	−1.07171e22			−1.96677
$792$		0			0
$793$	−4.72145e21			−0.850211
$794$		0			0
$795$	−8.86493e20	+	7.71073e19i	−0.156647	+	0.0136252i
$796$		0			0
$797$			7.65124e20i			0.132677i	0.997797	+	0.0663383i	$0.0211317\pi$
$797$			7.65124e20i			0.132677i	−0.997797	+	0.0663383i	$0.978868\pi$
$798$		0			0
$799$			4.48288e21i			0.762879i
$800$		0			0
$801$	−1.13821e21	−	6.49344e21i	−0.190099	−	1.08450i
$802$		0			0
$803$	1.01968e22			1.67147
$804$		0			0
$805$	−3.51148e21			−0.564965
$806$		0			0
$807$	7.27169e20	+	8.36017e21i	0.114838	+	1.32027i
$808$		0			0
$809$			1.27093e22i			1.97019i	0.172018	+	0.985094i	$0.444971\pi$
$809$			1.27093e22i			1.97019i	−0.172018	+	0.985094i	$0.555029\pi$
$810$		0			0
$811$		−	1.54863e21i		−	0.235662i	−0.993034	−	0.117831i	$-0.962406\pi$
$811$		−	1.54863e21i		−	0.235662i	0.993034	−	0.117831i	$-0.0375942\pi$
$812$		0			0
$813$	2.76027e20	+	3.17345e21i	0.0412357	+	0.474081i
$814$		0			0
$815$	−1.68578e21			−0.247240
$816$		0			0
$817$	−2.95334e21			−0.425253
$818$		0			0
$819$	1.61026e21	+	9.18643e21i	0.227649	+	1.29873i
$820$		0			0
$821$			3.11147e21i			0.431908i	0.976404	+	0.215954i	$0.0692862\pi$
$821$			3.11147e21i			0.431908i	−0.976404	+	0.215954i	$0.930714\pi$
$822$		0			0
$823$			3.90247e21i			0.531913i	0.963985	+	0.265957i	$0.0856878\pi$
$823$			3.90247e21i			0.531913i	−0.963985	+	0.265957i	$0.914312\pi$
$824$		0			0
$825$	1.00607e22	−	8.75078e20i	1.34655	−	0.117123i
$826$		0			0
$827$	8.97423e21			1.17952			0.589761	−	0.807578i	$-0.299222\pi$
$827$	8.97423e21			1.17952			0.589761	+	0.807578i	$0.299222\pi$
$828$		0			0
$829$	−8.22258e21			−1.06133			−0.530664	−	0.847583i	$-0.678057\pi$
$829$	−8.22258e21			−1.06133			−0.530664	+	0.847583i	$0.678057\pi$
$830$		0			0
$831$	−8.96169e21	+	7.79489e20i	−1.13601	+	0.0988104i
$832$		0			0
$833$		−	7.84204e21i		−	0.976318i
$834$		0			0
$835$		−	4.61140e21i		−	0.563876i
$836$		0			0
$837$	−3.49723e21	−	1.31314e22i	−0.420033	−	1.57713i
$838$		0			0
$839$	−5.94628e21			−0.701505			−0.350753	−	0.936468i	$-0.614074\pi$
$839$	−5.94628e21			−0.701505			−0.350753	+	0.936468i	$0.614074\pi$
$840$		0			0
$841$	−3.14524e20			−0.0364488
$842$		0			0
$843$	2.72321e20	+	3.13084e21i	0.0310008	+	0.356413i
$844$		0			0
$845$		−	1.40507e20i		−	0.0157135i
$846$		0			0
$847$		−	3.01731e22i		−	3.31509i
$848$		0			0
$849$	8.03105e20	+	9.23320e21i	0.0866892	+	0.996655i
$850$		0			0
$851$	7.28949e21			0.773082
$852$		0			0
$853$	1.86722e22			1.94571			0.972855	−	0.231415i	$-0.0743355\pi$
$853$	1.86722e22			1.94571			0.972855	+	0.231415i	$0.0743355\pi$
$854$		0			0
$855$	8.73054e21	−	1.53035e21i	0.893913	−	0.156691i
$856$		0			0
$857$		−	1.20113e22i		−	1.20847i	−0.796806	−	0.604235i	$-0.793479\pi$
$857$		−	1.20113e22i		−	1.20847i	0.796806	−	0.604235i	$-0.206521\pi$
$858$		0			0
$859$			1.77679e21i			0.175665i	0.996135	+	0.0878327i	$0.0279941\pi$
$859$			1.77679e21i			0.175665i	−0.996135	+	0.0878327i	$0.972006\pi$
$860$		0			0
$861$	−1.36909e22	+	1.19084e21i	−1.33017	+	0.115699i
$862$		0			0
$863$	−9.70387e21			−0.926540			−0.463270	−	0.886217i	$-0.653324\pi$
$863$	−9.70387e21			−0.926540			−0.463270	+	0.886217i	$0.653324\pi$
$864$		0			0
$865$	1.60487e20			0.0150598
$866$		0			0
$867$	−5.60414e21	+	4.87449e20i	−0.516852	+	0.0449558i
$868$		0			0
$869$		−	1.72285e22i		−	1.56170i
$870$		0			0
$871$			1.30272e21i			0.116069i
$872$		0			0
$873$	−7.35764e21	+	1.28969e21i	−0.644365	+	0.112948i
$874$		0			0
$875$	1.40684e22			1.21111
$876$		0			0
$877$	1.66780e22			1.41139			0.705693	−	0.708517i	$-0.250636\pi$
$877$	1.66780e22			1.41139			0.705693	+	0.708517i	$0.250636\pi$
$878$		0			0
$879$	1.28437e21	+	1.47663e22i	0.106850	+	1.22844i
$880$		0			0
$881$		−	4.95738e21i		−	0.405446i	−0.979236	−	0.202723i	$-0.935021\pi$
$881$		−	4.95738e21i		−	0.405446i	0.979236	−	0.202723i	$-0.0649791\pi$
$882$		0			0
$883$		−	2.65431e20i		−	0.0213425i	−0.999943	−	0.0106713i	$-0.996603\pi$
$883$		−	2.65431e20i		−	0.0213425i	0.999943	−	0.0106713i	$-0.00339683\pi$
$884$		0			0
$885$	7.34993e19	+	8.45013e20i	0.00581044	+	0.0668019i
$886$		0			0
$887$	2.55057e21			0.198249			0.0991244	−	0.995075i	$-0.468396\pi$
$887$	2.55057e21			0.198249			0.0991244	+	0.995075i	$0.468396\pi$
$888$		0			0
$889$	6.93105e21			0.529707
$890$		0			0
$891$	2.33316e22	−	8.43870e21i	1.75332	−	0.634151i
$892$		0			0
$893$		−	1.44958e22i		−	1.07117i
$894$		0			0
$895$		−	4.91233e21i		−	0.356956i
$896$		0			0
$897$	1.10487e22	−	9.61022e20i	0.789532	−	0.0686736i
$898$		0			0
$899$	2.36440e22			1.66159
$900$		0			0
$901$	−5.34602e21			−0.369483
$902$		0			0
$903$	−4.82183e21	+	4.19403e20i	−0.327758	+	0.0285084i
$904$		0			0
$905$			1.15687e21i			0.0773427i
$906$		0			0
$907$		−	1.22756e21i		−	0.0807210i	−0.999185	−	0.0403605i	$-0.987149\pi$
$907$		−	1.22756e21i		−	0.0807210i	0.999185	−	0.0403605i	$-0.0128506\pi$
$908$		0			0
$909$	−3.36990e21	−	1.92251e22i	−0.217966	−	1.24348i
$910$		0			0
$911$	−2.39285e21			−0.152239			−0.0761197	−	0.997099i	$-0.524253\pi$
$911$	−2.39285e21			−0.152239			−0.0761197	+	0.997099i	$0.524253\pi$
$912$		0			0
$913$	2.42525e21			0.151784
$914$		0			0
$915$	6.37219e20	+	7.32603e21i	0.0392311	+	0.451035i
$916$		0			0
$917$			1.56754e22i			0.949397i
$918$		0			0
$919$			5.70944e21i			0.340194i	0.985427	+	0.170097i	$0.0544082\pi$
$919$			5.70944e21i			0.340194i	−0.985427	+	0.170097i	$0.945592\pi$
$920$		0			0
$921$	−1.35172e21	−	1.55405e22i	−0.0792390	−	0.911001i
$922$		0			0
$923$	−2.89930e22			−1.67217
$924$		0			0
$925$	−1.22738e22			−0.696490
$926$		0			0
$927$	−1.22107e21	−	6.96613e21i	−0.0681776	−	0.388950i
$928$		0			0
$929$		−	5.33097e21i		−	0.292879i	−0.989220	−	0.146440i	$-0.953219\pi$
$929$		−	5.33097e21i		−	0.292879i	0.989220	−	0.146440i	$-0.0467814\pi$
$930$		0			0
$931$			2.53580e22i			1.37086i
$932$		0			0
$933$	−3.15598e22	+	2.74508e21i	−1.67889	+	0.146030i
$934$		0			0
$935$	−2.30203e22			−1.20510
$936$		0			0
$937$	8.68116e21			0.447230			0.223615	−	0.974678i	$-0.428214\pi$
$937$	8.68116e21			0.447230			0.223615	+	0.974678i	$0.428214\pi$
$938$		0			0
$939$	−1.32731e21	+	1.15449e20i	−0.0672945	+	0.00585328i
$940$		0			0
$941$			1.42716e22i			0.712117i	0.934464	+	0.356059i	$0.115880\pi$
$941$			1.42716e22i			0.712117i	−0.934464	+	0.356059i	$0.884120\pi$
$942$		0			0
$943$			1.63418e22i			0.802532i
$944$		0			0
$945$	1.40368e22	−	3.73837e21i	0.678467	−	0.180694i
$946$		0			0
$947$	−1.61533e21			−0.0768489			−0.0384244	−	0.999262i	$-0.512234\pi$
$947$	−1.61533e21			−0.0768489			−0.0384244	+	0.999262i	$0.512234\pi$
$948$		0			0
$949$	−1.88552e22			−0.882948
$950$		0			0
$951$	−4.00557e20	−	4.60516e21i	−0.0184634	−	0.212271i
$952$		0			0
$953$		−	1.78749e22i		−	0.811049i	−0.914084	−	0.405524i	$-0.867089\pi$
$953$		−	1.78749e22i		−	0.811049i	0.914084	−	0.405524i	$-0.132911\pi$
$954$		0			0
$955$			7.52332e21i			0.336035i
$956$		0			0
$957$	3.74071e21	+	4.30065e22i	0.164481	+	1.89101i
$958$		0			0
$959$	4.63127e22			2.00475
$960$		0			0
$961$	−3.90410e22			−1.66378
$962$		0			0
$963$	−2.64249e21	+	4.63193e20i	−0.110871	+	0.0194341i
$964$		0			0
$965$		−	8.32302e21i		−	0.343816i
$966$		0			0
$967$		−	3.09074e22i		−	1.25708i	−0.777776	−	0.628542i	$-0.783652\pi$
$967$		−	3.09074e22i		−	1.25708i	0.777776	−	0.628542i	$-0.216348\pi$
$968$		0			0
$969$	5.30511e22	−	4.61439e21i	2.12455	−	0.184793i
$970$		0			0
$971$	4.32111e21			0.170393			0.0851964	−	0.996364i	$-0.472848\pi$
$971$	4.32111e21			0.170393			0.0851964	+	0.996364i	$0.472848\pi$
$972$		0			0
$973$	3.15624e22			1.22553
$974$		0			0
$975$	−1.86035e22	+	1.61813e21i	−0.711310	+	0.0618699i
$976$		0			0
$977$			4.14164e22i			1.55942i	0.626140	+	0.779710i	$0.284634\pi$
$977$			4.14164e22i			1.55942i	−0.626140	+	0.779710i	$0.715366\pi$
$978$		0			0
$979$		−	5.53643e22i		−	2.05286i
$980$		0			0
$981$	3.62003e21	−	6.34543e20i	0.132189	−	0.0231709i
$982$		0			0
$983$	−2.15037e22			−0.773325			−0.386662	−	0.922221i	$-0.626372\pi$
$983$	−2.15037e22			−0.773325			−0.386662	+	0.922221i	$0.626372\pi$
$984$		0			0
$985$	−1.69281e22			−0.599566
$986$		0			0
$987$	−2.05855e21	−	2.36669e22i	−0.0718097	−	0.825587i
$988$		0			0
$989$			5.75545e21i			0.197746i
$990$		0			0
$991$			4.90628e22i			1.66035i	0.557503	+	0.830175i	$0.311760\pi$
$991$			4.90628e22i			1.66035i	−0.557503	+	0.830175i	$0.688240\pi$
$992$		0			0
$993$	4.70267e21	+	5.40660e22i	0.156756	+	1.80221i
$994$		0			0
$995$	−1.78130e22			−0.584875
$996$		0			0
$997$	1.79757e22			0.581396			0.290698	−	0.956815i	$-0.406113\pi$
$997$	1.79757e22			0.581396			0.290698	+	0.956815i	$0.406113\pi$
$998$		0			0
$999$	−2.91390e22	+	7.76050e21i	−0.928396	+	0.247257i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.16.c.d.47.19	yes	20
3.2	odd	2	inner	48.16.c.d.47.1	✓	20
4.3	odd	2	inner	48.16.c.d.47.2	yes	20
12.11	even	2	inner	48.16.c.d.47.20	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.16.c.d.47.1	✓	20	3.2	odd	2	inner
48.16.c.d.47.2	yes	20	4.3	odd	2	inner
48.16.c.d.47.19	yes	20	1.1	even	1	trivial
48.16.c.d.47.20	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)

\(f(q)\)	\(=\)	\(q+(3773.75 - 328.241i) q^{3} +91619.9i q^{5} -2.91696e6i q^{7} +(1.41334e7 - 2.47740e6i) q^{9} +O(q^{10})\)	\(q+(3773.75 - 328.241i) q^{3} +91619.9i q^{5} -2.91696e6i q^{7} +(1.41334e7 - 2.47740e6i) q^{9} +1.20504e8 q^{11} -2.22828e8 q^{13} +(3.00734e7 + 3.45750e8i) q^{15} +2.08506e9i q^{17} -6.74224e9i q^{19} +(-9.57464e8 - 1.10079e10i) q^{21} -1.31392e10 q^{23} +2.21234e10 q^{25} +(5.25228e10 - 1.39882e10i) q^{27} +9.45712e10i q^{29} -2.50013e11i q^{31} +(4.54753e11 - 3.95545e10i) q^{33} +2.67251e11 q^{35} -5.54788e11 q^{37} +(-8.40896e11 + 7.31413e10i) q^{39} -1.24374e12i q^{41} -4.38035e11i q^{43} +(2.26979e11 + 1.29490e12i) q^{45} +2.15000e12 q^{47} -3.76107e12 q^{49} +(6.84401e11 + 7.86847e12i) q^{51} +2.56397e12i q^{53} +1.10406e13i q^{55} +(-2.21308e12 - 2.54435e13i) q^{57} +2.44400e12 q^{59} +2.11888e13 q^{61} +(-7.22646e12 - 4.12266e13i) q^{63} -2.04155e13i q^{65} -5.84632e12i q^{67} +(-4.95842e13 + 4.31284e12i) q^{69} +1.30114e14 q^{71} +8.46179e13 q^{73} +(8.34880e13 - 7.26180e12i) q^{75} -3.51506e14i q^{77} -1.42970e14i q^{79} +(1.93616e14 - 7.00282e13i) q^{81} +2.01259e13 q^{83} -1.91033e14 q^{85} +(3.10421e13 + 3.56888e14i) q^{87} -4.59438e14i q^{89} +6.49979e14i q^{91} +(-8.20644e13 - 9.43484e14i) q^{93} +6.17723e14 q^{95} -5.20584e14 q^{97} +(1.70314e15 - 2.98537e14i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2271972 q^{9}+O(q^{10}) \) 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

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