LMFDB - Embedded newform 48.22.c.c.47.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,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, 22, names="a")

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
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:	$134.149125258$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.1
Character	$\chi$	$=$	48.47
Dual form			48.22.c.c.47.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-102247. - 2435.85i) q^{3} -1.44607e7i q^{5} +1.26694e9i q^{7} +(1.04485e10 + 4.98117e8i) q^{9} +O(q^{10})$	\(q+(-102247. - 2435.85i) q^{3} -1.44607e7i q^{5} +1.26694e9i q^{7} +(1.04485e10 + 4.98117e8i) q^{9} -6.25605e10 q^{11} -3.73946e11 q^{13} +(-3.52241e10 + 1.47856e12i) q^{15} -7.29673e12i q^{17} -1.62545e12i q^{19} +(3.08608e12 - 1.29541e14i) q^{21} -3.19324e14 q^{23} +2.67726e14 q^{25} +(-1.06711e15 - 7.63819e13i) q^{27} +6.11800e14i q^{29} +5.24424e15i q^{31} +(6.39661e15 + 1.52388e14i) q^{33} +1.83208e16 q^{35} +1.14778e15 q^{37} +(3.82348e16 + 9.10878e14i) q^{39} +5.72121e16i q^{41} +1.59500e17i q^{43} +(7.20311e15 - 1.51092e17i) q^{45} -5.37480e17 q^{47} -1.04659e18 q^{49} +(-1.77738e16 + 7.46067e17i) q^{51} -4.19972e16i q^{53} +9.04667e17i q^{55} +(-3.95935e15 + 1.66197e17i) q^{57} +5.79173e18 q^{59} -7.14941e18 q^{61} +(-6.31084e17 + 1.32376e19i) q^{63} +5.40752e18i q^{65} +1.96789e18i q^{67} +(3.26498e19 + 7.77825e17i) q^{69} -2.21539e19 q^{71} -3.72720e19 q^{73} +(-2.73741e19 - 6.52141e17i) q^{75} -7.92603e19i q^{77} -1.09813e20i q^{79} +(1.08923e20 + 1.04091e19i) q^{81} +1.78006e20 q^{83} -1.05516e20 q^{85} +(1.49026e18 - 6.25546e19i) q^{87} +5.72354e20i q^{89} -4.73767e20i q^{91} +(1.27742e19 - 5.36207e20i) q^{93} -2.35051e19 q^{95} +8.41397e20 q^{97} +(-6.53662e20 - 3.11624e19i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 109254828 q^{9}+O(q^{10}) $ 28 * q + 109254828 * q^9	$ 28 q + 109254828 q^{9} + 285248048392 q^{13} + 247146979606248 q^{21} - 31\!\cdots\!84 q^{25}+ \cdots + 16\!\cdots\!20 q^{97}+O(q^{100}) $ 28 * q + 109254828 * q^9 + 285248048392 * q^13 + 247146979606248 * q^21 - 3168527686741684 * q^25 - 7429429202649408 * q^33 + 33089121948065192 * q^37 - 121806830473453440 * q^45 - 569345069725368716 * q^49 + 10547854228195302552 * q^57 - 8280751575178739512 * q^61 - 22754055866538936960 * q^69 - 98548109928833394536 * q^73 + 563980976909609989500 * q^81 - 93841573689641488896 * q^85 - 1293830212233248374968 * q^93 + 1670728237822569097720 * 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
$3$	−102247.	−	2435.85i	−0.999716	−	0.0238165i
$3$	−102247.	−	2435.85i	−0.999716	−	0.0238165i
$4$		0			0
$5$		−	1.44607e7i		−	0.662222i	−0.943592	−	0.331111i	$-0.892576\pi$
$5$		−	1.44607e7i		−	0.662222i	0.943592	−	0.331111i	$-0.107424\pi$
$6$		0			0
$7$			1.26694e9i			1.69522i	0.530618	+	0.847611i	$0.321960\pi$
$7$			1.26694e9i			1.69522i	−0.530618	+	0.847611i	$0.678040\pi$
$8$		0			0
$9$	1.04485e10	+	4.98117e8i	0.998866	+	0.0476195i
$10$		0			0
$11$	−6.25605e10			−0.727238			−0.363619	−	0.931548i	$-0.618459\pi$
$11$	−6.25605e10			−0.727238			−0.363619	+	0.931548i	$0.618459\pi$
$12$		0			0
$13$	−3.73946e11			−0.752322			−0.376161	−	0.926554i	$-0.622756\pi$
$13$	−3.73946e11			−0.752322			−0.376161	+	0.926554i	$0.622756\pi$
$14$		0			0
$15$	−3.52241e10	+	1.47856e12i	−0.0157718	+	0.662035i
$16$		0			0
$17$		−	7.29673e12i		−	0.877838i	−0.898527	−	0.438919i	$-0.855361\pi$
$17$		−	7.29673e12i		−	0.877838i	0.898527	−	0.438919i	$-0.144639\pi$
$18$		0			0
$19$		−	1.62545e12i		−	0.0608219i	−0.999537	−	0.0304110i	$-0.990318\pi$
$19$		−	1.62545e12i		−	0.0608219i	0.999537	−	0.0304110i	$-0.00968160\pi$
$20$		0			0
$21$	3.08608e12	−	1.29541e14i	0.0403742	−	1.69474i
$22$		0			0
$23$	−3.19324e14			−1.60727			−0.803635	−	0.595123i	$-0.797103\pi$
$23$	−3.19324e14			−1.60727			−0.803635	+	0.595123i	$0.797103\pi$
$24$		0			0
$25$	2.67726e14			0.561461
$26$		0			0
$27$	−1.06711e15	−	7.63819e13i	−0.997448	−	0.0713955i
$28$		0			0
$29$			6.11800e14i			0.270042i	0.990843	+	0.135021i	$0.0431101\pi$
$29$			6.11800e14i			0.270042i	−0.990843	+	0.135021i	$0.956890\pi$
$30$		0			0
$31$			5.24424e15i			1.14917i	0.818445	+	0.574586i	$0.194837\pi$
$31$			5.24424e15i			1.14917i	−0.818445	+	0.574586i	$0.805163\pi$
$32$		0			0
$33$	6.39661e15	+	1.52388e14i	0.727032	+	0.0173203i
$34$		0			0
$35$	1.83208e16			1.12261
$36$		0			0
$37$	1.14778e15			0.0392412			0.0196206	−	0.999807i	$-0.493754\pi$
$37$	1.14778e15			0.0392412			0.0196206	+	0.999807i	$0.493754\pi$
$38$		0			0
$39$	3.82348e16	+	9.10878e14i	0.752109	+	0.0179177i
$40$		0			0
$41$			5.72121e16i			0.665668i	0.942985	+	0.332834i	$0.108005\pi$
$41$			5.72121e16i			0.665668i	−0.942985	+	0.332834i	$0.891995\pi$
$42$		0			0
$43$			1.59500e17i			1.12549i	0.826631	+	0.562744i	$0.190254\pi$
$43$			1.59500e17i			1.12549i	−0.826631	+	0.562744i	$0.809746\pi$
$44$		0			0
$45$	7.20311e15	−	1.51092e17i	0.0315347	−	0.661471i
$46$		0			0
$47$	−5.37480e17			−1.49051			−0.745255	−	0.666780i	$-0.767672\pi$
$47$	−5.37480e17			−1.49051			−0.745255	+	0.666780i	$0.767672\pi$
$48$		0			0
$49$	−1.04659e18			−1.87378
$50$		0			0
$51$	−1.77738e16	+	7.46067e17i	−0.0209070	+	0.877589i
$52$		0			0
$53$		−	4.19972e16i		−	0.0329856i	−0.999864	−	0.0164928i	$-0.994750\pi$
$53$		−	4.19972e16i		−	0.0329856i	0.999864	−	0.0164928i	$-0.00525006\pi$
$54$		0			0
$55$			9.04667e17i			0.481594i
$56$		0			0
$57$	−3.95935e15	+	1.66197e17i	−0.00144857	+	0.0608047i
$58$		0			0
$59$	5.79173e18			1.47524			0.737620	−	0.675216i	$-0.235950\pi$
$59$	5.79173e18			1.47524			0.737620	+	0.675216i	$0.235950\pi$
$60$		0			0
$61$	−7.14941e18			−1.28324			−0.641618	−	0.767024i	$-0.721737\pi$
$61$	−7.14941e18			−1.28324			−0.641618	+	0.767024i	$0.721737\pi$
$62$		0			0
$63$	−6.31084e17	+	1.32376e19i	−0.0807256	+	1.69330i
$64$		0			0
$65$			5.40752e18i			0.498205i
$66$		0			0
$67$			1.96789e18i			0.131891i	0.997823	+	0.0659454i	$0.0210063\pi$
$67$			1.96789e18i			0.131891i	−0.997823	+	0.0659454i	$0.978994\pi$
$68$		0			0
$69$	3.26498e19	+	7.77825e17i	1.60681	+	0.0382795i
$70$		0			0
$71$	−2.21539e19			−0.807676			−0.403838	−	0.914830i	$-0.632324\pi$
$71$	−2.21539e19			−0.807676			−0.403838	+	0.914830i	$0.632324\pi$
$72$		0			0
$73$	−3.72720e19			−1.01506			−0.507530	−	0.861634i	$-0.669442\pi$
$73$	−3.72720e19			−1.01506			−0.507530	+	0.861634i	$0.669442\pi$
$74$		0			0
$75$	−2.73741e19	−	6.52141e17i	−0.561302	−	0.0133720i
$76$		0			0
$77$		−	7.92603e19i		−	1.23283i
$78$		0			0
$79$		−	1.09813e20i		−	1.30487i	−0.757844	−	0.652436i	$-0.773747\pi$
$79$		−	1.09813e20i		−	1.30487i	0.757844	−	0.652436i	$-0.226253\pi$
$80$		0			0
$81$	1.08923e20	+	1.04091e19i	0.995465	+	0.0951309i
$82$		0			0
$83$	1.78006e20			1.25926			0.629628	−	0.776897i	$-0.283207\pi$
$83$	1.78006e20			1.25926			0.629628	+	0.776897i	$0.283207\pi$
$84$		0			0
$85$	−1.05516e20			−0.581324
$86$		0			0
$87$	1.49026e18	−	6.25546e19i	0.00643145	−	0.269965i
$88$		0			0
$89$			5.72354e20i			1.94567i	0.231496	+	0.972836i	$0.425638\pi$
$89$			5.72354e20i			1.94567i	−0.231496	+	0.972836i	$0.574362\pi$
$90$		0			0
$91$		−	4.73767e20i		−	1.27535i
$92$		0			0
$93$	1.27742e19	−	5.36207e20i	0.0273692	−	1.14885i
$94$		0			0
$95$	−2.35051e19			−0.0402776
$96$		0			0
$97$	8.41397e20			1.15850			0.579252	−	0.815149i	$-0.303345\pi$
$97$	8.41397e20			1.15850			0.579252	+	0.815149i	$0.303345\pi$
$98$		0			0
$99$	−6.53662e20	−	3.11624e19i	−0.726413	−	0.0346307i
$100$		0			0
$101$		−	1.59031e21i		−	1.43254i	−0.697822	−	0.716271i	$-0.745847\pi$
$101$		−	1.59031e21i		−	1.43254i	0.697822	−	0.716271i	$-0.254153\pi$
$102$		0			0
$103$			1.97301e21i			1.44657i	0.690551	+	0.723284i	$0.257368\pi$
$103$			1.97301e21i			1.44657i	−0.690551	+	0.723284i	$0.742632\pi$
$104$		0			0
$105$	−1.87325e21	−	4.46268e19i	−1.12230	−	0.0267367i
$106$		0			0
$107$	1.22412e21			0.601580			0.300790	−	0.953690i	$-0.402750\pi$
$107$	1.22412e21			0.601580			0.300790	+	0.953690i	$0.402750\pi$
$108$		0			0
$109$	1.06497e21			0.430885			0.215442	−	0.976517i	$-0.430881\pi$
$109$	1.06497e21			0.430885			0.215442	+	0.976517i	$0.430881\pi$
$110$		0			0
$111$	−1.17357e20	−	2.79583e18i	−0.0392300	−	0.000934587i
$112$		0			0
$113$			1.86312e21i			0.516317i	0.966103	+	0.258159i	$0.0831157\pi$
$113$			1.86312e21i			0.516317i	−0.966103	+	0.258159i	$0.916884\pi$
$114$		0			0
$115$			4.61764e21i			1.06437i
$116$		0			0
$117$	−3.90717e21	−	1.86269e20i	−0.751469	−	0.0358252i
$118$		0			0
$119$	9.24451e21			1.48813
$120$		0			0
$121$	−3.48644e21			−0.471124
$122$		0			0
$123$	1.39360e20	−	5.84976e21i	0.0158539	−	0.665479i
$124$		0			0
$125$		−	1.07669e22i		−	1.03403i
$126$		0			0
$127$			1.42745e21i			0.116044i	0.998315	+	0.0580219i	$0.0184793\pi$
$127$			1.42745e21i			0.116044i	−0.998315	+	0.0580219i	$0.981521\pi$
$128$		0			0
$129$	3.88518e20	−	1.63083e22i	0.0268052	−	1.12517i
$130$		0			0
$131$	−7.28941e21			−0.427901			−0.213951	−	0.976844i	$-0.568633\pi$
$131$	−7.28941e21			−0.427901			−0.213951	+	0.976844i	$0.568633\pi$
$132$		0			0
$133$	2.05934e21			0.103107
$134$		0			0
$135$	−1.10453e21	+	1.54312e22i	−0.0472797	+	0.660532i
$136$		0			0
$137$		−	4.27704e22i		−	1.56884i	−0.620232	−	0.784418i	$-0.712962\pi$
$137$		−	4.27704e22i		−	1.56884i	0.620232	−	0.784418i	$-0.287038\pi$
$138$		0			0
$139$		−	6.06635e22i		−	1.91105i	−0.294911	−	0.955525i	$-0.595290\pi$
$139$		−	6.06635e22i		−	1.91105i	0.294911	−	0.955525i	$-0.404710\pi$
$140$		0			0
$141$	5.49556e22	+	1.30922e21i	1.49009	+	0.0354987i
$142$		0			0
$143$	2.33942e22			0.547118
$144$		0			0
$145$	8.84705e21			0.178828
$146$		0			0
$147$	1.07011e23	+	2.54934e21i	1.87324	+	0.0446268i
$148$		0			0
$149$		−	2.88081e22i		−	0.437581i	−0.975772	−	0.218791i	$-0.929789\pi$
$149$		−	2.88081e22i		−	0.437581i	0.975772	−	0.218791i	$-0.0702111\pi$
$150$		0			0
$151$		−	1.35106e22i		−	0.178409i	−0.996013	−	0.0892044i	$-0.971568\pi$
$151$		−	1.35106e22i		−	0.178409i	0.996013	−	0.0892044i	$-0.0284324\pi$
$152$		0			0
$153$	3.63462e21	−	7.62398e22i	0.0418022	−	0.876842i
$154$		0			0
$155$	7.58353e22			0.761007
$156$		0			0
$157$	−1.26266e23			−1.10749			−0.553746	−	0.832685i	$-0.686802\pi$
$157$	−1.26266e23			−1.10749			−0.553746	+	0.832685i	$0.686802\pi$
$158$		0			0
$159$	−1.02299e20	+	4.29409e21i	−0.000785601	+	0.0329762i
$160$		0			0
$161$		−	4.04564e23i		−	2.72468i
$162$		0			0
$163$			9.59406e22i			0.567587i	0.958885	+	0.283794i	$0.0915931\pi$
$163$			9.59406e22i			0.567587i	−0.958885	+	0.283794i	$0.908407\pi$
$164$		0			0
$165$	2.20364e21	−	9.24994e22i	0.0114699	−	0.481457i
$166$		0			0
$167$	1.08195e23			0.496230			0.248115	−	0.968731i	$-0.420189\pi$
$167$	1.08195e23			0.496230			0.248115	+	0.968731i	$0.420189\pi$
$168$		0			0
$169$	−1.07229e23			−0.434011
$170$		0			0
$171$	8.09663e20	−	1.69835e22i	0.00289631	−	0.0607529i
$172$		0			0
$173$		−	5.23999e23i		−	1.65900i	−0.558506	−	0.829500i	$-0.688625\pi$
$173$		−	5.23999e23i		−	1.65900i	0.558506	−	0.829500i	$-0.311375\pi$
$174$		0			0
$175$			3.39192e23i			0.951802i
$176$		0			0
$177$	−5.92187e23	−	1.41078e22i	−1.47482	−	0.0351350i
$178$		0			0
$179$	3.56488e23			0.789019			0.394510	−	0.918892i	$-0.370914\pi$
$179$	3.56488e23			0.789019			0.394510	+	0.918892i	$0.370914\pi$
$180$		0			0
$181$	8.43340e23			1.66103			0.830516	−	0.556995i	$-0.188046\pi$
$181$	8.43340e23			1.66103			0.830516	+	0.556995i	$0.188046\pi$
$182$		0			0
$183$	7.31004e23	+	1.74149e22i	1.28287	+	0.0305622i
$184$		0			0
$185$		−	1.65977e22i		−	0.0259864i
$186$		0			0
$187$			4.56487e23i			0.638398i
$188$		0			0
$189$	9.67712e22	−	1.35197e24i	0.121031	−	1.69090i
$190$		0			0
$191$	1.21395e24			1.35941			0.679705	−	0.733486i	$-0.262108\pi$
$191$	1.21395e24			1.35941			0.679705	+	0.733486i	$0.262108\pi$
$192$		0			0
$193$	−1.49461e22			−0.0150029			−0.00750145	−	0.999972i	$-0.502388\pi$
$193$	−1.49461e22			−0.0150029			−0.00750145	+	0.999972i	$0.502388\pi$
$194$		0			0
$195$	1.31719e22	−	5.52902e23i	0.0118655	−	0.498063i
$196$		0			0
$197$		−	1.61435e24i		−	1.30648i	−0.757153	−	0.653238i	$-0.773410\pi$
$197$		−	1.61435e24i		−	1.30648i	0.757153	−	0.653238i	$-0.226590\pi$
$198$		0			0
$199$		−	2.54198e24i		−	1.85019i	−0.379740	−	0.925093i	$-0.623987\pi$
$199$		−	2.54198e24i		−	1.85019i	0.379740	−	0.925093i	$-0.376013\pi$
$200$		0			0
$201$	4.79348e21	−	2.01210e23i	0.00314118	−	0.131853i
$202$		0			0
$203$	−7.75114e23			−0.457780
$204$		0			0
$205$	8.27327e23			0.440820
$206$		0			0
$207$	−3.33645e24	−	1.59060e23i	−1.60545	−	0.0765374i
$208$		0			0
$209$			1.01689e23i			0.0442320i
$210$		0			0
$211$			4.41939e23i			0.173939i	0.996211	+	0.0869695i	$0.0277183\pi$
$211$			4.41939e23i			0.173939i	−0.996211	+	0.0869695i	$0.972282\pi$
$212$		0			0
$213$	2.26517e24	+	5.39636e22i	0.807447	+	0.0192360i
$214$		0			0
$215$	2.30647e24			0.745323
$216$		0			0
$217$	−6.64413e24			−1.94810
$218$		0			0
$219$	3.81094e24	+	9.07890e22i	1.01477	+	0.0241752i
$220$		0			0
$221$			2.72858e24i			0.660417i
$222$		0			0
$223$		−	1.66796e24i		−	0.367270i	−0.982995	−	0.183635i	$-0.941214\pi$
$223$		−	1.66796e24i		−	0.367270i	0.982995	−	0.183635i	$-0.0587864\pi$
$224$		0			0
$225$	2.79733e24	+	1.33359e23i	0.560825	+	0.0267365i
$226$		0			0
$227$	1.72728e24			0.315568			0.157784	−	0.987474i	$-0.449565\pi$
$227$	1.72728e24			0.315568			0.157784	+	0.987474i	$0.449565\pi$
$228$		0			0
$229$	3.85511e24			0.642340			0.321170	−	0.947022i	$-0.395924\pi$
$229$	3.85511e24			0.642340			0.321170	+	0.947022i	$0.395924\pi$
$230$		0			0
$231$	−1.93067e23	+	8.10412e24i	−0.0293617	+	1.23248i
$232$		0			0
$233$			1.35383e24i			0.188074i	0.995569	+	0.0940369i	$0.0299772\pi$
$233$			1.35383e24i			0.188074i	−0.995569	+	0.0940369i	$0.970023\pi$
$234$		0			0
$235$			7.77233e24i			0.987049i
$236$		0			0
$237$	−2.67488e23	+	1.12280e25i	−0.0310775	+	1.30450i
$238$		0			0
$239$	−1.63175e25			−1.73570			−0.867850	−	0.496826i	$-0.834499\pi$
$239$	−1.63175e25			−1.73570			−0.867850	+	0.496826i	$0.834499\pi$
$240$		0			0
$241$	−6.31456e24			−0.615409			−0.307704	−	0.951482i	$-0.599561\pi$
$241$	−6.31456e24			−0.615409			−0.307704	+	0.951482i	$0.599561\pi$
$242$		0			0
$243$	−1.11117e25	−	1.32962e24i	−0.992917	−	0.118812i
$244$		0			0
$245$			1.51344e25i			1.24086i
$246$		0			0
$247$			6.07830e23i			0.0457577i
$248$		0			0
$249$	−1.82005e25	−	4.33596e23i	−1.25890	−	0.0299911i
$250$		0			0
$251$	−1.71290e25			−1.08933			−0.544663	−	0.838655i	$-0.683343\pi$
$251$	−1.71290e25			−1.08933			−0.544663	+	0.838655i	$0.683343\pi$
$252$		0			0
$253$	1.99770e25			1.16887
$254$		0			0
$255$	1.07886e25	+	2.57021e23i	0.581159	+	0.0138451i
$256$		0			0
$257$			2.00498e25i			0.994977i	0.867471	+	0.497488i	$0.165744\pi$
$257$			2.00498e25i			0.994977i	−0.867471	+	0.497488i	$0.834256\pi$
$258$		0			0
$259$			1.45417e24i			0.0665225i
$260$		0			0
$261$	−3.04748e23	+	6.39239e24i	−0.0128592	+	0.269735i
$262$		0			0
$263$	−1.36777e24			−0.0532693			−0.0266347	−	0.999645i	$-0.508479\pi$
$263$	−1.36777e24			−0.0532693			−0.0266347	+	0.999645i	$0.508479\pi$
$264$		0			0
$265$	−6.07309e23			−0.0218438
$266$		0			0
$267$	1.39417e24	−	5.85214e25i	0.0463391	−	1.94512i
$268$		0			0
$269$			4.06762e25i			1.25009i	0.780589	+	0.625045i	$0.214919\pi$
$269$			4.06762e25i			1.25009i	−0.780589	+	0.625045i	$0.785081\pi$
$270$		0			0
$271$		−	3.09348e25i		−	0.879569i	−0.898103	−	0.439784i	$-0.855055\pi$
$271$		−	3.09348e25i		−	0.879569i	0.898103	−	0.439784i	$-0.144945\pi$
$272$		0			0
$273$	−1.15403e24	+	4.84412e25i	−0.0303744	+	1.27499i
$274$		0			0
$275$	−1.67490e25			−0.408316
$276$		0			0
$277$	4.46430e25			1.00859			0.504296	−	0.863531i	$-0.331752\pi$
$277$	4.46430e25			1.00859			0.504296	+	0.863531i	$0.331752\pi$
$278$		0			0
$279$	−2.61224e24	+	5.47944e25i	−0.0547229	+	1.14787i
$280$		0			0
$281$			1.09703e25i			0.213208i	0.994302	+	0.106604i	$0.0339977\pi$
$281$			1.09703e25i			0.213208i	−0.994302	+	0.106604i	$0.966002\pi$
$282$		0			0
$283$			3.28057e25i			0.591823i	0.955215	+	0.295911i	$0.0956233\pi$
$283$			3.28057e25i			0.591823i	−0.955215	+	0.295911i	$0.904377\pi$
$284$		0			0
$285$	2.40332e24	+	5.72550e22i	0.0402662	+	0.000959273i
$286$		0			0
$287$	−7.24843e25			−1.12845
$288$		0			0
$289$	1.58497e25			0.229400
$290$		0			0
$291$	−8.60301e25	−	2.04952e24i	−1.15818	−	0.0275915i
$292$		0			0
$293$		−	8.37957e25i		−	1.04981i	−0.851160	−	0.524906i	$-0.824100\pi$
$293$		−	8.37957e25i		−	1.04981i	0.851160	−	0.524906i	$-0.175900\pi$
$294$		0			0
$295$		−	8.37525e25i		−	0.976937i
$296$		0			0
$297$	6.67590e25	+	4.77848e24i	0.725383	+	0.0519215i
$298$		0			0
$299$	1.19410e26			1.20918
$300$		0			0
$301$	−2.02076e26			−1.90795
$302$		0			0
$303$	−3.87376e24	+	1.62604e26i	−0.0341182	+	1.43214i
$304$		0			0
$305$			1.03385e26i			0.849788i
$306$		0			0
$307$			5.28274e25i			0.405421i	0.979239	+	0.202710i	$0.0649750\pi$
$307$			5.28274e25i			0.405421i	−0.979239	+	0.202710i	$0.935025\pi$
$308$		0			0
$309$	4.80597e24	−	2.01734e26i	0.0344522	−	1.44616i
$310$		0			0
$311$	1.79352e26			1.20150			0.600748	−	0.799439i	$-0.294870\pi$
$311$	1.79352e26			1.20150			0.600748	+	0.799439i	$0.294870\pi$
$312$		0			0
$313$	−1.92967e25			−0.120856			−0.0604279	−	0.998173i	$-0.519247\pi$
$313$	−1.92967e25			−0.120856			−0.0604279	+	0.998173i	$0.519247\pi$
$314$		0			0
$315$	1.91425e26	+	9.12591e24i	1.12134	+	0.0534583i
$316$		0			0
$317$			8.54659e25i			0.468458i	0.972181	+	0.234229i	$0.0752566\pi$
$317$			8.54659e25i			0.468458i	−0.972181	+	0.234229i	$0.924743\pi$
$318$		0			0
$319$		−	3.82745e25i		−	0.196385i
$320$		0			0
$321$	−1.25162e26	−	2.98177e24i	−0.601410	−	0.0143275i
$322$		0			0
$323$	−1.18604e25			−0.0533918
$324$		0			0
$325$	−1.00115e26			−0.422400
$326$		0			0
$327$	−1.08890e26	−	2.59412e24i	−0.430763	−	0.0102622i
$328$		0			0
$329$		−	6.80955e26i		−	2.52674i
$330$		0			0
$331$			1.21850e26i			0.424261i	0.977241	+	0.212130i	$0.0680401\pi$
$331$			1.21850e26i			0.424261i	−0.977241	+	0.212130i	$0.931960\pi$
$332$		0			0
$333$	1.19926e25	+	5.71730e23i	0.0391966	+	0.00186864i
$334$		0			0
$335$	2.84570e25			0.0873411
$336$		0			0
$337$	5.94137e26			1.71306			0.856531	−	0.516096i	$-0.172615\pi$
$337$	5.94137e26			1.71306			0.856531	+	0.516096i	$0.172615\pi$
$338$		0			0
$339$	4.53828e24	−	1.90498e26i	0.0122969	−	0.516171i
$340$		0			0
$341$		−	3.28082e26i		−	0.835721i
$342$		0			0
$343$		−	6.18322e26i		−	1.48124i
$344$		0			0
$345$	1.12479e25	−	4.72139e26i	0.0253496	−	1.06407i
$346$		0			0
$347$	2.11155e26			0.447859			0.223930	−	0.974605i	$-0.428111\pi$
$347$	2.11155e26			0.447859			0.223930	+	0.974605i	$0.428111\pi$
$348$		0			0
$349$	4.17434e26			0.833528			0.416764	−	0.909015i	$-0.363164\pi$
$349$	4.17434e26			0.833528			0.416764	+	0.909015i	$0.363164\pi$
$350$		0			0
$351$	3.99042e26	+	2.85627e25i	0.750402	+	0.0537124i
$352$		0			0
$353$		−	6.96060e26i		−	1.23314i	−0.787300	−	0.616570i	$-0.788522\pi$
$353$		−	6.96060e26i		−	1.23314i	0.787300	−	0.616570i	$-0.211478\pi$
$354$		0			0
$355$			3.20360e26i			0.534861i
$356$		0			0
$357$	−9.45222e26	−	2.25183e25i	−1.48771	−	0.0354421i
$358$		0			0
$359$	5.46587e26			0.811274			0.405637	−	0.914034i	$-0.367050\pi$
$359$	5.46587e26			0.811274			0.405637	+	0.914034i	$0.367050\pi$
$360$		0			0
$361$	7.11567e26			0.996301
$362$		0			0
$363$	3.56477e26	+	8.49245e24i	0.470991	+	0.0112205i
$364$		0			0
$365$			5.38978e26i			0.672196i
$366$		0			0
$367$		−	6.69850e26i		−	0.788831i	−0.918932	−	0.394415i	$-0.870947\pi$
$367$		−	6.69850e26i		−	0.788831i	0.918932	−	0.394415i	$-0.129053\pi$
$368$		0			0
$369$	−2.84983e25	+	5.97780e26i	−0.0316988	+	0.664913i
$370$		0			0
$371$	5.32080e25			0.0559179
$372$		0			0
$373$	−8.40144e26			−0.834470			−0.417235	−	0.908799i	$-0.637001\pi$
$373$	−8.40144e26			−0.834470			−0.417235	+	0.908799i	$0.637001\pi$
$374$		0			0
$375$	−2.62266e25	+	1.10088e27i	−0.0246271	+	1.03374i
$376$		0			0
$377$		−	2.28780e26i		−	0.203158i
$378$		0			0
$379$		−	1.36114e27i		−	1.14338i	−0.820470	−	0.571689i	$-0.806288\pi$
$379$		−	1.36114e27i		−	1.14338i	0.820470	−	0.571689i	$-0.193712\pi$
$380$		0			0
$381$	3.47706e24	−	1.45952e26i	0.00276376	−	0.116011i
$382$		0			0
$383$	−1.27335e27			−0.957993			−0.478996	−	0.877817i	$-0.658999\pi$
$383$	−1.27335e27			−0.957993			−0.478996	+	0.877817i	$0.658999\pi$
$384$		0			0
$385$	−1.14616e27			−0.816408
$386$		0			0
$387$	−7.94494e25	+	1.66653e27i	−0.0535952	+	1.12421i
$388$		0			0
$389$		−	1.86447e27i		−	1.19147i	−0.803179	−	0.595737i	$-0.796860\pi$
$389$		−	1.86447e27i		−	1.19147i	0.803179	−	0.595737i	$-0.203140\pi$
$390$		0			0
$391$			2.33002e27i			1.41092i
$392$		0			0
$393$	7.45319e26	+	1.77559e25i	0.427780	+	0.0101911i
$394$		0			0
$395$	−1.58797e27			−0.864116
$396$		0			0
$397$	−3.15817e26			−0.162980			−0.0814901	−	0.996674i	$-0.525968\pi$
$397$	−3.15817e26			−0.162980			−0.0814901	+	0.996674i	$0.525968\pi$
$398$		0			0
$399$	−2.10561e26	−	5.01626e24i	−0.103077	−	0.00245564i
$400$		0			0
$401$			4.18021e27i			1.94170i	0.239687	+	0.970850i	$0.422955\pi$
$401$			4.18021e27i			1.94170i	−0.239687	+	0.970850i	$0.577045\pi$
$402$		0			0
$403$		−	1.96106e27i		−	0.864547i
$404$		0			0
$405$	1.50523e26	−	1.57510e27i	0.0629978	−	0.659219i
$406$		0			0
$407$	−7.18059e25			−0.0285377
$408$		0			0
$409$	1.49621e26			0.0564803			0.0282402	−	0.999601i	$-0.491010\pi$
$409$	1.49621e26			0.0564803			0.0282402	+	0.999601i	$0.491010\pi$
$410$		0			0
$411$	−1.04183e26	+	4.37314e27i	−0.0373642	+	1.56839i
$412$		0			0
$413$			7.33778e27i			2.50086i
$414$		0			0
$415$		−	2.57408e27i		−	0.833907i
$416$		0			0
$417$	−1.47767e26	+	6.20265e27i	−0.0455145	+	1.91051i
$418$		0			0
$419$	3.60551e27			1.05614			0.528068	−	0.849202i	$-0.322917\pi$
$419$	3.60551e27			1.05614			0.528068	+	0.849202i	$0.322917\pi$
$420$		0			0
$421$	6.38111e27			1.77801			0.889005	−	0.457897i	$-0.151397\pi$
$421$	6.38111e27			1.77801			0.889005	+	0.457897i	$0.151397\pi$
$422$		0			0
$423$	−5.61585e27	−	2.67728e26i	−1.48882	−	0.0709773i
$424$		0			0
$425$		−	1.95352e27i		−	0.492872i
$426$		0			0
$427$		−	9.05787e27i		−	2.17537i
$428$		0			0
$429$	−2.39199e27	−	5.69850e25i	−0.546962	−	0.0130304i
$430$		0			0
$431$	−2.61301e27			−0.569022			−0.284511	−	0.958673i	$-0.591831\pi$
$431$	−2.61301e27			−0.569022			−0.284511	+	0.958673i	$0.591831\pi$
$432$		0			0
$433$	−1.89110e26			−0.0392275			−0.0196138	−	0.999808i	$-0.506244\pi$
$433$	−1.89110e26			−0.0392275			−0.0196138	+	0.999808i	$0.506244\pi$
$434$		0			0
$435$	−9.04583e26	−	2.15501e25i	−0.178777	−	0.00425905i
$436$		0			0
$437$			5.19044e26i			0.0977572i
$438$		0			0
$439$		−	2.23329e26i		−	0.0400929i	−0.999799	−	0.0200464i	$-0.993619\pi$
$439$		−	2.23329e26i		−	0.0400929i	0.999799	−	0.0200464i	$-0.00638141\pi$
$440$		0			0
$441$	−1.09353e28	−	5.21324e26i	−1.87165	−	0.0892283i
$442$		0			0
$443$	−6.22802e27			−1.01651			−0.508254	−	0.861207i	$-0.669709\pi$
$443$	−6.22802e27			−1.01651			−0.508254	+	0.861207i	$0.669709\pi$
$444$		0			0
$445$	8.27663e27			1.28847
$446$		0			0
$447$	−7.01723e25	+	2.94554e27i	−0.0104217	+	0.437457i
$448$		0			0
$449$			2.25192e27i			0.319129i	0.987187	+	0.159565i	$0.0510090\pi$
$449$			2.25192e27i			0.319129i	−0.987187	+	0.159565i	$0.948991\pi$
$450$		0			0
$451$		−	3.57922e27i		−	0.484099i
$452$		0			0
$453$	−3.29098e25	+	1.38142e27i	−0.00424908	+	0.178358i
$454$		0			0
$455$	−6.85100e27			−0.844567
$456$		0			0
$457$	7.61688e27			0.896721			0.448360	−	0.893853i	$-0.352008\pi$
$457$	7.61688e27			0.896721			0.448360	+	0.893853i	$0.352008\pi$
$458$		0			0
$459$	−5.57338e26	+	7.78642e27i	−0.0626737	+	0.875598i
$460$		0			0
$461$			1.41281e28i			1.51783i	0.651190	+	0.758915i	$0.274270\pi$
$461$			1.41281e28i			1.51783i	−0.651190	+	0.758915i	$0.725730\pi$
$462$		0			0
$463$			2.98295e26i			0.0306229i	0.999883	+	0.0153114i	$0.00487397\pi$
$463$			2.98295e26i			0.0306229i	−0.999883	+	0.0153114i	$0.995126\pi$
$464$		0			0
$465$	−7.75392e27	−	1.84724e26i	−0.760791	−	0.0181245i
$466$		0			0
$467$	8.39959e27			0.787827			0.393913	−	0.919148i	$-0.371121\pi$
$467$	8.39959e27			0.787827			0.393913	+	0.919148i	$0.371121\pi$
$468$		0			0
$469$	−2.49319e27			−0.223584
$470$		0			0
$471$	1.29103e28	+	3.07566e26i	1.10718	+	0.0263766i
$472$		0			0
$473$		−	9.97837e27i		−	0.818498i
$474$		0			0
$475$		−	4.35174e26i		−	0.0341492i
$476$		0			0
$477$	2.09195e25	−	4.38808e26i	0.00157076	−	0.0329482i
$478$		0			0
$479$	6.88687e27			0.494879			0.247440	−	0.968903i	$-0.420411\pi$
$479$	6.88687e27			0.494879			0.247440	+	0.968903i	$0.420411\pi$
$480$		0			0
$481$	−4.29209e26			−0.0295220
$482$		0			0
$483$	−9.85458e26	+	4.13654e28i	−0.0648923	+	2.72390i
$484$		0			0
$485$		−	1.21672e28i		−	0.767187i
$486$		0			0
$487$		−	1.23081e28i		−	0.743255i	−0.928382	−	0.371628i	$-0.878800\pi$
$487$		−	1.23081e28i		−	0.743255i	0.928382	−	0.371628i	$-0.121200\pi$
$488$		0			0
$489$	2.33697e26	−	9.80963e27i	0.0135179	−	0.567426i
$490$		0			0
$491$	1.79744e28			0.996092			0.498046	−	0.867151i	$-0.334051\pi$
$491$	1.79744e28			0.996092			0.498046	+	0.867151i	$0.334051\pi$
$492$		0			0
$493$	4.46414e27			0.237053
$494$		0			0
$495$	−4.50630e26	+	9.45240e27i	−0.0229332	+	0.481047i
$496$		0			0
$497$		−	2.80676e28i		−	1.36919i
$498$		0			0
$499$		−	2.99831e28i		−	1.40223i	−0.713046	−	0.701117i	$-0.752685\pi$
$499$		−	2.99831e28i		−	1.40223i	0.713046	−	0.701117i	$-0.247315\pi$
$500$		0			0
$501$	−1.10626e28	−	2.63546e26i	−0.496089	−	0.0118185i
$502$		0			0
$503$	−4.39197e28			−1.88884			−0.944421	−	0.328737i	$-0.893377\pi$
$503$	−4.39197e28			−1.88884			−0.944421	+	0.328737i	$0.893377\pi$
$504$		0			0
$505$	−2.29970e28			−0.948662
$506$		0			0
$507$	1.09638e28	+	2.61194e26i	0.433888	+	0.0103366i
$508$		0			0
$509$		−	4.95379e28i		−	1.88105i	−0.339722	−	0.940526i	$-0.610333\pi$
$509$		−	4.95379e28i		−	1.88105i	0.339722	−	0.940526i	$-0.389667\pi$
$510$		0			0
$511$		−	4.72213e28i		−	1.72075i
$512$		0			0
$513$	−1.24155e26	+	1.73453e27i	−0.00434241	+	0.0606667i
$514$		0			0
$515$	2.85311e28			0.957950
$516$		0			0
$517$	3.36250e28			1.08396
$518$		0			0
$519$	−1.27639e27	+	5.35773e28i	−0.0395116	+	1.65853i
$520$		0			0
$521$			2.53526e28i			0.753748i	0.926265	+	0.376874i	$0.123001\pi$
$521$			2.53526e28i			0.753748i	−0.926265	+	0.376874i	$0.876999\pi$
$522$		0			0
$523$		−	6.34446e28i		−	1.81187i	−0.423418	−	0.905935i	$-0.639170\pi$
$523$		−	6.34446e28i		−	1.81187i	0.423418	−	0.905935i	$-0.360830\pi$
$524$		0			0
$525$	8.26223e26	−	3.46813e28i	0.0226686	−	0.951532i
$526$		0			0
$527$	3.82658e28			1.00879
$528$		0			0
$529$	6.24959e28			1.58331
$530$		0			0
$531$	6.05149e28	+	2.88496e27i	1.47357	+	0.0702502i
$532$		0			0
$533$		−	2.13943e28i		−	0.500797i
$534$		0			0
$535$		−	1.77016e28i		−	0.398380i
$536$		0			0
$537$	−3.64497e28	−	8.68352e26i	−0.788796	−	0.0187917i
$538$		0			0
$539$	6.54752e28			1.36268
$540$		0			0
$541$	−2.66328e28			−0.533146			−0.266573	−	0.963815i	$-0.585891\pi$
$541$	−2.66328e28			−0.533146			−0.266573	+	0.963815i	$0.585891\pi$
$542$		0			0
$543$	−8.62289e28	−	2.05425e27i	−1.66056	−	0.0395600i
$544$		0			0
$545$		−	1.54003e28i		−	0.285342i
$546$		0			0
$547$			7.88590e28i			1.40600i	0.711191	+	0.702999i	$0.248156\pi$
$547$			7.88590e28i			1.40600i	−0.711191	+	0.702999i	$0.751844\pi$
$548$		0			0
$549$	−7.47005e28	−	3.56124e27i	−1.28178	−	0.0611071i
$550$		0			0
$551$	9.94449e26			0.0164244
$552$		0			0
$553$	1.39126e29			2.21205
$554$		0			0
$555$	−4.04297e25	+	1.69707e27i	−0.000618905	+	0.0259790i
$556$		0			0
$557$		−	2.48332e28i		−	0.366060i	−0.983107	−	0.183030i	$-0.941409\pi$
$557$		−	2.48332e28i		−	0.366060i	0.983107	−	0.183030i	$-0.0585906\pi$
$558$		0			0
$559$		−	5.96443e28i		−	0.846730i
$560$		0			0
$561$	1.11193e27	−	4.66743e28i	0.0152044	−	0.638217i
$562$		0			0
$563$	−9.32629e28			−1.22849			−0.614244	−	0.789116i	$-0.710539\pi$
$563$	−9.32629e28			−1.22849			−0.614244	+	0.789116i	$0.710539\pi$
$564$		0			0
$565$	2.69420e28			0.341917
$566$		0			0
$567$	−1.31877e28	+	1.37999e29i	−0.161268	+	1.68753i
$568$		0			0
$569$			8.51564e28i			1.00355i	0.864999	+	0.501774i	$0.167319\pi$
$569$			8.51564e28i			1.00355i	−0.864999	+	0.501774i	$0.832681\pi$
$570$		0			0
$571$		−	2.03823e28i		−	0.231512i	−0.993278	−	0.115756i	$-0.963071\pi$
$571$		−	2.03823e28i		−	0.231512i	0.993278	−	0.115756i	$-0.0369291\pi$
$572$		0			0
$573$	−1.24123e29	−	2.95700e27i	−1.35902	−	0.0323764i
$574$		0			0
$575$	−8.54911e28			−0.902420
$576$		0			0
$577$	−1.49693e29			−1.52355			−0.761775	−	0.647842i	$-0.775672\pi$
$577$	−1.49693e29			−1.52355			−0.761775	+	0.647842i	$0.775672\pi$
$578$		0			0
$579$	1.52819e27	+	3.64064e25i	0.0149986	+	0.000357316i
$580$		0			0
$581$			2.25522e29i			2.13472i
$582$		0			0
$583$			2.62737e27i			0.0239884i
$584$		0			0
$585$	−2.69358e27	+	5.65004e28i	−0.0237243	+	0.497639i
$586$		0			0
$587$	1.22519e29			1.04113			0.520564	−	0.853822i	$-0.325722\pi$
$587$	1.22519e29			1.04113			0.520564	+	0.853822i	$0.325722\pi$
$588$		0			0
$589$	8.52424e27			0.0698948
$590$		0			0
$591$	−3.93231e27	+	1.65062e29i	−0.0311157	+	1.30611i
$592$		0			0
$593$		−	1.53304e29i		−	1.17079i	−0.810747	−	0.585397i	$-0.800939\pi$
$593$		−	1.53304e29i		−	1.17079i	0.810747	−	0.585397i	$-0.199061\pi$
$594$		0			0
$595$		−	1.33682e29i		−	0.985473i
$596$		0			0
$597$	−6.19190e27	+	2.59910e29i	−0.0440650	+	1.84966i
$598$		0			0
$599$	−1.10571e29			−0.759733			−0.379867	−	0.925041i	$-0.624030\pi$
$599$	−1.10571e29			−0.759733			−0.379867	+	0.925041i	$0.624030\pi$
$600$		0			0
$601$	2.20855e29			1.46530			0.732650	−	0.680606i	$-0.238283\pi$
$601$	2.20855e29			1.46530			0.732650	+	0.680606i	$0.238283\pi$
$602$		0			0
$603$	−9.80237e26	+	2.05614e28i	−0.00628058	+	0.131741i
$604$		0			0
$605$			5.04163e28i			0.311989i
$606$		0			0
$607$		−	2.01277e29i		−	1.20313i	−0.798822	−	0.601567i	$-0.794543\pi$
$607$		−	2.01277e29i		−	1.20313i	0.798822	−	0.601567i	$-0.205457\pi$
$608$		0			0
$609$	7.92530e28	+	1.88806e27i	0.457650	+	0.0109027i
$610$		0			0
$611$	2.00989e29			1.12134
$612$		0			0
$613$	−2.14701e29			−1.15744			−0.578722	−	0.815525i	$-0.696448\pi$
$613$	−2.14701e29			−1.15744			−0.578722	+	0.815525i	$0.696448\pi$
$614$		0			0
$615$	−8.45915e28	−	2.01525e27i	−0.440695	−	0.0104988i
$616$		0			0
$617$			3.00995e29i			1.51553i	0.652526	+	0.757766i	$0.273709\pi$
$617$			3.00995e29i			1.51553i	−0.652526	+	0.757766i	$0.726291\pi$
$618$		0			0
$619$		−	1.47678e26i		−	0.000718726i	−1.00000		0.000359363i	$-0.999886\pi$
$619$		−	1.47678e26i		−	0.000718726i	1.00000		0.000359363i	$-0.000114389\pi$
$620$		0			0
$621$	3.40754e29	+	2.43905e28i	1.60317	+	0.114752i
$622$		0			0
$623$	−7.25138e29			−3.29834
$624$		0			0
$625$	−2.80351e28			−0.123300
$626$		0			0
$627$	2.47699e26	−	1.03974e28i	0.00105345	−	0.0442195i
$628$		0			0
$629$		−	8.37506e27i		−	0.0344474i
$630$		0			0
$631$		−	2.80848e29i		−	1.11728i	−0.829409	−	0.558641i	$-0.811323\pi$
$631$		−	2.80848e29i		−	1.11728i	0.829409	−	0.558641i	$-0.188677\pi$
$632$		0			0
$633$	1.07650e27	−	4.51869e28i	0.00414262	−	0.173890i
$634$		0			0
$635$	2.06419e28			0.0768468
$636$		0			0
$637$	3.91368e29			1.40968
$638$		0			0
$639$	−2.31475e29	−	1.10352e28i	−0.806760	−	0.0384611i
$640$		0			0
$641$			3.39624e29i			1.14548i	0.819736	+	0.572742i	$0.194120\pi$
$641$			3.39624e29i			1.14548i	−0.819736	+	0.572742i	$0.805880\pi$
$642$		0			0
$643$			3.08480e29i			1.00696i	0.864007	+	0.503479i	$0.167947\pi$
$643$			3.08480e29i			1.00696i	−0.864007	+	0.503479i	$0.832053\pi$
$644$		0			0
$645$	−2.35830e29	−	5.61823e27i	−0.745112	−	0.0177510i
$646$		0			0
$647$	−2.11901e29			−0.648093			−0.324047	−	0.946041i	$-0.605043\pi$
$647$	−2.11901e29			−0.648093			−0.324047	+	0.946041i	$0.605043\pi$
$648$		0			0
$649$	−3.62334e29			−1.07285
$650$		0			0
$651$	6.79342e29	+	1.61841e28i	1.94755	+	0.0463969i
$652$		0			0
$653$		−	3.78004e29i		−	1.04932i	−0.851312	−	0.524660i	$-0.824192\pi$
$653$		−	3.78004e29i		−	1.04932i	0.851312	−	0.524660i	$-0.175808\pi$
$654$		0			0
$655$			1.05410e29i			0.283366i
$656$		0			0
$657$	−3.89435e29	−	1.85658e28i	−1.01391	−	0.0483367i
$658$		0			0
$659$	6.26637e29			1.58022			0.790112	−	0.612962i	$-0.210022\pi$
$659$	6.26637e29			1.58022			0.790112	+	0.612962i	$0.210022\pi$
$660$		0			0
$661$	1.72827e29			0.422180			0.211090	−	0.977467i	$-0.432299\pi$
$661$	1.72827e29			0.422180			0.211090	+	0.977467i	$0.432299\pi$
$662$		0			0
$663$	6.64643e27	−	2.78989e29i	0.0157288	−	0.660230i
$664$		0			0
$665$		−	2.97795e28i		−	0.0682795i
$666$		0			0
$667$		−	1.95362e29i		−	0.434030i
$668$		0			0
$669$	−4.06291e27	+	1.70544e29i	−0.00874708	+	0.367166i
$670$		0			0
$671$	4.47270e29			0.933219
$672$		0			0
$673$	1.35828e29			0.274682			0.137341	−	0.990524i	$-0.456144\pi$
$673$	1.35828e29			0.274682			0.137341	+	0.990524i	$0.456144\pi$
$674$		0			0
$675$	−2.85693e29	−	2.04494e28i	−0.560029	−	0.0400858i
$676$		0			0
$677$		−	4.47879e29i		−	0.851097i	−0.904936	−	0.425549i	$-0.860081\pi$
$677$		−	4.47879e29i		−	0.851097i	0.904936	−	0.425549i	$-0.139919\pi$
$678$		0			0
$679$			1.06600e30i			1.96392i
$680$		0			0
$681$	−1.76609e29	−	4.20741e27i	−0.315478	−	0.00751571i
$682$		0			0
$683$	9.20490e28			0.159442			0.0797208	−	0.996817i	$-0.474597\pi$
$683$	9.20490e28			0.159442			0.0797208	+	0.996817i	$0.474597\pi$
$684$		0			0
$685$	−6.18490e29			−1.03892
$686$		0			0
$687$	−3.94173e29	−	9.39049e27i	−0.642157	−	0.0152983i
$688$		0			0
$689$			1.57047e28i			0.0248158i
$690$		0			0
$691$			6.70754e29i			1.02812i	0.857754	+	0.514060i	$0.171859\pi$
$691$			6.70754e29i			1.02812i	−0.857754	+	0.514060i	$0.828141\pi$
$692$		0			0
$693$	3.94809e28	−	8.28150e29i	0.0587068	−	1.23143i
$694$		0			0
$695$	−8.77236e29			−1.26554
$696$		0			0
$697$	4.17461e29			0.584349
$698$		0			0
$699$	3.29774e27	−	1.38425e29i	0.00447926	−	0.188020i
$700$		0			0
$701$		−	5.24791e29i		−	0.691746i	−0.938281	−	0.345873i	$-0.887583\pi$
$701$		−	5.24791e29i		−	0.691746i	0.938281	−	0.345873i	$-0.112417\pi$
$702$		0			0
$703$		−	1.86566e27i		−	0.00238672i
$704$		0			0
$705$	1.89323e28	−	7.94696e29i	0.0235081	−	0.986769i
$706$		0			0
$707$	2.01483e30			2.42848
$708$		0			0
$709$	8.19847e29			0.959285			0.479642	−	0.877464i	$-0.340766\pi$
$709$	8.19847e29			0.959285			0.479642	+	0.877464i	$0.340766\pi$
$710$		0			0
$711$	5.46995e28	−	1.14738e30i	0.0621373	−	1.30339i
$712$		0			0
$713$		−	1.67461e30i		−	1.84703i
$714$		0			0
$715$		−	3.38297e29i		−	0.362314i
$716$		0			0
$717$	1.66841e30	+	3.97470e28i	1.73521	+	0.0413383i
$718$		0			0
$719$	5.47030e28			0.0552533			0.0276266	−	0.999618i	$-0.491205\pi$
$719$	5.47030e28			0.0552533			0.0276266	+	0.999618i	$0.491205\pi$
$720$		0			0
$721$	−2.49969e30			−2.45225
$722$		0			0
$723$	6.45643e29	+	1.53813e28i	0.615234	+	0.0146569i
$724$		0			0
$725$			1.63795e29i			0.151618i
$726$		0			0
$727$			7.67790e29i			0.690448i	0.938520	+	0.345224i	$0.112197\pi$
$727$			7.67790e29i			0.690448i	−0.938520	+	0.345224i	$0.887803\pi$
$728$		0			0
$729$	1.13289e30	+	1.63016e29i	0.989805	+	0.142427i
$730$		0			0
$731$	1.16383e30			0.987996
$732$		0			0
$733$	−1.91588e28			−0.0158044			−0.00790218	−	0.999969i	$-0.502515\pi$
$733$	−1.91588e28			−0.0158044			−0.00790218	+	0.999969i	$0.502515\pi$
$734$		0			0
$735$	3.68652e28	−	1.54745e30i	0.0295529	−	1.24050i
$736$		0			0
$737$		−	1.23112e29i		−	0.0959161i
$738$		0			0
$739$		−	2.42674e30i		−	1.83763i	−0.394693	−	0.918813i	$-0.629149\pi$
$739$		−	2.42674e30i		−	1.83763i	0.394693	−	0.918813i	$-0.370851\pi$
$740$		0			0
$741$	1.48058e27	−	6.21487e28i	0.00108979	−	0.0457447i
$742$		0			0
$743$	−1.25572e30			−0.898485			−0.449242	−	0.893410i	$-0.648306\pi$
$743$	−1.25572e30			−0.898485			−0.449242	+	0.893410i	$0.648306\pi$
$744$		0			0
$745$	−4.16585e29			−0.289776
$746$		0			0
$747$	1.85989e30	+	8.86676e28i	1.25783	+	0.0599651i
$748$		0			0
$749$			1.55088e30i			1.01981i
$750$		0			0
$751$			2.21444e30i			1.41594i	0.706243	+	0.707970i	$0.250389\pi$
$751$			2.21444e30i			1.41594i	−0.706243	+	0.707970i	$0.749611\pi$
$752$		0			0
$753$	1.75139e30	+	4.17238e28i	1.08902	+	0.0259440i
$754$		0			0
$755$	−1.95372e29			−0.118146
$756$		0			0
$757$	−2.28676e30			−1.34497			−0.672486	−	0.740110i	$-0.734773\pi$
$757$	−2.28676e30			−1.34497			−0.672486	+	0.740110i	$0.734773\pi$
$758$		0			0
$759$	−2.04259e30	−	4.86611e28i	−1.16854	−	0.0278383i
$760$		0			0
$761$		−	1.81334e30i		−	1.00911i	−0.863378	−	0.504557i	$-0.831656\pi$
$761$		−	1.81334e30i		−	1.00911i	0.863378	−	0.504557i	$-0.168344\pi$
$762$		0			0
$763$			1.34926e30i			0.730445i
$764$		0			0
$765$	−1.10248e30	−	5.25591e28i	−0.580665	−	0.0276824i
$766$		0			0
$767$	−2.16580e30			−1.10986
$768$		0			0
$769$	−1.10801e29			−0.0552483			−0.0276241	−	0.999618i	$-0.508794\pi$
$769$	−1.10801e29			−0.0552483			−0.0276241	+	0.999618i	$0.508794\pi$
$770$		0			0
$771$	4.88384e28	−	2.05003e30i	0.0236969	−	0.994694i
$772$		0			0
$773$		−	3.95428e28i		−	0.0186716i	−0.999956	−	0.00933582i	$-0.997028\pi$
$773$		−	3.95428e28i		−	0.0186716i	0.999956	−	0.00933582i	$-0.00297173\pi$
$774$		0			0
$775$			1.40402e30i			0.645215i
$776$		0			0
$777$	3.54215e27	−	1.48685e29i	0.00158433	−	0.0665036i
$778$		0			0
$779$	9.29953e28			0.0404872
$780$		0			0
$781$	1.38596e30			0.587373
$782$		0			0
$783$	4.67304e28	−	6.52859e29i	0.0192797	−	0.269352i
$784$		0			0
$785$			1.82590e30i			0.733407i
$786$		0			0
$787$			3.93608e30i			1.53932i	0.638453	+	0.769661i	$0.279575\pi$
$787$			3.93608e30i			1.53932i	−0.638453	+	0.769661i	$0.720425\pi$
$788$		0			0
$789$	1.39850e29	+	3.33168e27i	0.0532542	+	0.00126869i
$790$		0			0
$791$	−2.36046e30			−0.875272
$792$		0			0
$793$	2.67349e30			0.965407
$794$		0			0
$795$	6.20954e28	+	1.47932e27i	0.0218376	+	0.000520243i
$796$		0			0
$797$			2.56655e30i			0.879098i	0.898219	+	0.439549i	$0.144862\pi$
$797$			2.56655e30i			0.879098i	−0.898219	+	0.439549i	$0.855138\pi$
$798$		0			0
$799$			3.92185e30i			1.30843i
$800$		0			0
$801$	−2.85099e29	+	5.98023e30i	−0.0926519	+	1.94346i
$802$		0			0
$803$	2.33175e30			0.738191
$804$		0			0
$805$	−5.85027e30			−1.80434
$806$		0			0
$807$	9.90812e28	−	4.15901e30i	0.0297728	−	1.24974i
$808$		0			0
$809$			3.45727e30i			1.01222i	0.862469	+	0.506109i	$0.168917\pi$
$809$			3.45727e30i			1.01222i	−0.862469	+	0.506109i	$0.831083\pi$
$810$		0			0
$811$			5.79196e30i			1.65237i	0.563400	+	0.826184i	$0.309493\pi$
$811$			5.79196e30i			1.65237i	−0.563400	+	0.826184i	$0.690507\pi$
$812$		0			0
$813$	−7.53526e28	+	3.16299e30i	−0.0209482	+	0.879319i
$814$		0			0
$815$	1.38737e30			0.375869
$816$		0			0
$817$	2.59258e29			0.0684543
$818$		0			0
$819$	2.35991e29	−	4.95015e30i	0.0607317	−	1.27391i
$820$		0			0
$821$		−	1.68701e30i		−	0.423171i	−0.977360	−	0.211585i	$-0.932137\pi$
$821$		−	1.68701e30i		−	0.423171i	0.977360	−	0.211585i	$-0.0678626\pi$
$822$		0			0
$823$		−	3.39383e30i		−	0.829836i	−0.909859	−	0.414918i	$-0.863810\pi$
$823$		−	3.39383e30i		−	0.829836i	0.909859	−	0.414918i	$-0.136190\pi$
$824$		0			0
$825$	1.71254e30	+	4.07982e28i	0.408201	+	0.00972467i
$826$		0			0
$827$	−7.04347e30			−1.63674			−0.818368	−	0.574694i	$-0.805121\pi$
$827$	−7.04347e30			−1.63674			−0.818368	+	0.574694i	$0.805121\pi$
$828$		0			0
$829$	1.05175e30			0.238282			0.119141	−	0.992877i	$-0.461986\pi$
$829$	1.05175e30			0.238282			0.119141	+	0.992877i	$0.461986\pi$
$830$		0			0
$831$	−4.56461e30	−	1.08744e29i	−1.00831	−	0.0240212i
$832$		0			0
$833$			7.63668e30i			1.64487i
$834$		0			0
$835$		−	1.56457e30i		−	0.328614i
$836$		0			0
$837$	4.00565e29	−	5.59619e30i	0.0820456	−	1.14624i
$838$		0			0
$839$	−3.33296e30			−0.665778			−0.332889	−	0.942966i	$-0.608024\pi$
$839$	−3.33296e30			−0.665778			−0.332889	+	0.942966i	$0.608024\pi$
$840$		0			0
$841$	4.75854e30			0.927078
$842$		0			0
$843$	2.67221e28	−	1.12168e30i	0.00507787	−	0.213148i
$844$		0			0
$845$			1.55060e30i			0.287412i
$846$		0			0
$847$		−	4.41711e30i		−	0.798660i
$848$		0			0
$849$	7.99098e28	−	3.35428e30i	0.0140951	−	0.591655i
$850$		0			0
$851$	−3.66514e29			−0.0630711
$852$		0			0
$853$	−8.04716e30			−1.35107			−0.675535	−	0.737328i	$-0.736087\pi$
$853$	−8.04716e30			−1.35107			−0.675535	+	0.737328i	$0.736087\pi$
$854$		0			0
$855$	−2.45593e29	−	1.17083e28i	−0.0402319	−	0.00191800i
$856$		0			0
$857$		−	2.83304e30i		−	0.452849i	−0.974029	−	0.226425i	$-0.927296\pi$
$857$		−	2.83304e30i		−	0.452849i	0.974029	−	0.226425i	$-0.0727037\pi$
$858$		0			0
$859$		−	4.17407e30i		−	0.651075i	−0.945529	−	0.325538i	$-0.894455\pi$
$859$		−	4.17407e30i		−	0.651075i	0.945529	−	0.325538i	$-0.105545\pi$
$860$		0			0
$861$	7.41129e30	+	1.76561e29i	1.12813	+	0.0268758i
$862$		0			0
$863$	1.39339e30			0.206995			0.103497	−	0.994630i	$-0.466997\pi$
$863$	1.39339e30			0.206995			0.103497	+	0.994630i	$0.466997\pi$
$864$		0			0
$865$	−7.57739e30			−1.09863
$866$		0			0
$867$	−1.62058e30	−	3.86075e28i	−0.229335	−	0.00546351i
$868$		0			0
$869$			6.86993e30i			0.948953i
$870$		0			0
$871$		−	7.35884e29i		−	0.0992244i
$872$		0			0
$873$	8.79132e30	+	4.19114e29i	1.15719	+	0.0551674i
$874$		0			0
$875$	1.36410e31			1.75292
$876$		0			0
$877$	7.36988e30			0.924623			0.462311	−	0.886718i	$-0.347020\pi$
$877$	7.36988e30			0.924623			0.462311	+	0.886718i	$0.347020\pi$
$878$		0			0
$879$	−2.04114e29	+	8.56784e30i	−0.0250029	+	1.04951i
$880$		0			0
$881$		−	1.14429e31i		−	1.36864i	−0.729183	−	0.684319i	$-0.760100\pi$
$881$		−	1.14429e31i		−	1.36864i	0.729183	−	0.684319i	$-0.239900\pi$
$882$		0			0
$883$			1.36011e31i			1.58850i	0.607594	+	0.794248i	$0.292135\pi$
$883$			1.36011e31i			1.58850i	−0.607594	+	0.794248i	$0.707865\pi$
$884$		0			0
$885$	−2.04009e29	+	8.56343e30i	−0.0232672	+	0.976660i
$886$		0			0
$887$	8.68721e30			0.967570			0.483785	−	0.875187i	$-0.339262\pi$
$887$	8.68721e30			0.967570			0.483785	+	0.875187i	$0.339262\pi$
$888$		0			0
$889$	−1.80849e30			−0.196720
$890$		0			0
$891$	−6.81426e30	−	6.51200e29i	−0.723940	−	0.0691829i
$892$		0			0
$893$			8.73646e29i			0.0906557i
$894$		0			0
$895$		−	5.15506e30i		−	0.522506i
$896$		0			0
$897$	−1.22093e31	−	2.90865e29i	−1.20884	−	0.0287985i
$898$		0			0
$899$	−3.20843e30			−0.310324
$900$		0			0
$901$	−3.06442e29			−0.0289560
$902$		0			0
$903$	2.06617e31	+	4.92228e29i	1.90741	+	0.0454407i
$904$		0			0
$905$		−	1.21953e31i		−	1.09997i
$906$		0			0
$907$			8.74800e29i			0.0770961i	0.999257	+	0.0385481i	$0.0122733\pi$
$907$			8.74800e29i			0.0770961i	−0.999257	+	0.0385481i	$0.987727\pi$
$908$		0			0
$909$	7.92160e29	−	1.66163e31i	0.0682169	−	1.43092i
$910$		0			0
$911$	−1.25842e31			−1.05896			−0.529482	−	0.848321i	$-0.677614\pi$
$911$	−1.25842e31			−1.05896			−0.529482	+	0.848321i	$0.677614\pi$
$912$		0			0
$913$	−1.11361e31			−0.915779
$914$		0			0
$915$	2.51832e29	−	1.05708e31i	0.0202390	−	0.849547i
$916$		0			0
$917$		−	9.23524e30i		−	0.725387i
$918$		0			0
$919$			1.09348e31i			0.839459i	0.907649	+	0.419729i	$0.137875\pi$
$919$			1.09348e31i			0.839459i	−0.907649	+	0.419729i	$0.862125\pi$
$920$		0			0
$921$	1.28680e29	−	5.40143e30i	0.00965571	−	0.405306i
$922$		0			0
$923$	8.28436e30			0.607633
$924$		0			0
$925$	3.07291e29			0.0220324
$926$		0			0
$927$	−9.82790e29	+	2.06150e31i	−0.0688849	+	1.44493i
$928$		0			0
$929$		−	2.57400e31i		−	1.76378i	−0.471457	−	0.881889i	$-0.656272\pi$
$929$		−	2.57400e31i		−	1.76378i	0.471457	−	0.881889i	$-0.343728\pi$
$930$		0			0
$931$			1.70118e30i			0.113967i
$932$		0			0
$933$	−1.83382e31	−	4.36875e29i	−1.20115	−	0.0286154i
$934$		0			0
$935$	6.60111e30			0.422761
$936$		0			0
$937$	2.26989e31			1.42148			0.710738	−	0.703457i	$-0.248361\pi$
$937$	2.26989e31			1.42148			0.710738	+	0.703457i	$0.248361\pi$
$938$		0			0
$939$	1.97303e30	+	4.70040e28i	0.120822	+	0.00287836i
$940$		0			0
$941$		−	1.84816e31i		−	1.10675i	−0.832933	−	0.553374i	$-0.813340\pi$
$941$		−	1.84816e31i		−	1.10675i	0.832933	−	0.553374i	$-0.186660\pi$
$942$		0			0
$943$		−	1.82692e31i		−	1.06991i
$944$		0			0
$945$	−1.95504e31	−	1.39938e30i	−1.11975	−	0.0801495i
$946$		0			0
$947$	−1.96335e31			−1.09982			−0.549912	−	0.835223i	$-0.685339\pi$
$947$	−1.96335e31			−1.09982			−0.549912	+	0.835223i	$0.685339\pi$
$948$		0			0
$949$	1.39377e31			0.763653
$950$		0			0
$951$	2.08183e29	−	8.73862e30i	0.0111570	−	0.468325i
$952$		0			0
$953$			3.48094e31i			1.82483i	0.409271	+	0.912413i	$0.365783\pi$
$953$			3.48094e31i			1.82483i	−0.409271	+	0.912413i	$0.634217\pi$
$954$		0			0
$955$		−	1.75546e31i		−	0.900232i
$956$		0			0
$957$	−9.32311e28	+	3.91345e30i	−0.00467719	+	0.196329i
$958$		0			0
$959$	5.41876e31			2.65953
$960$		0			0
$961$	−6.67654e30			−0.320594
$962$		0			0
$963$	1.27902e31	+	6.09754e29i	0.600898	+	0.0286469i
$964$		0			0
$965$			2.16130e29i			0.00993525i
$966$		0			0
$967$		−	5.53917e30i		−	0.249153i	−0.992210	−	0.124577i	$-0.960243\pi$
$967$		−	5.53917e30i		−	0.249153i	0.992210	−	0.124577i	$-0.0397573\pi$
$968$		0			0
$969$	1.21269e30	+	2.88903e28i	0.0533767	+	0.00127161i
$970$		0			0
$971$	−2.70404e31			−1.16469			−0.582346	−	0.812941i	$-0.697865\pi$
$971$	−2.70404e31			−1.16469			−0.582346	+	0.812941i	$0.697865\pi$
$972$		0			0
$973$	7.68570e31			3.23965
$974$		0			0
$975$	1.02364e31	+	2.43865e29i	0.422280	+	0.0100601i
$976$		0			0
$977$			7.61350e30i			0.307391i	0.988118	+	0.153696i	$0.0491175\pi$
$977$			7.61350e30i			0.307391i	−0.988118	+	0.153696i	$0.950882\pi$
$978$		0			0
$979$		−	3.58067e31i		−	1.41497i
$980$		0			0
$981$	1.11274e31	+	5.30482e29i	0.430396	+	0.0205185i
$982$		0			0
$983$	6.27122e30			0.237433			0.118716	−	0.992928i	$-0.462122\pi$
$983$	6.27122e30			0.237433			0.118716	+	0.992928i	$0.462122\pi$
$984$		0			0
$985$	−2.33446e31			−0.865178
$986$		0			0
$987$	−1.65871e30	+	6.96255e31i	−0.0601782	+	2.52603i
$988$		0			0
$989$		−	5.09320e31i		−	1.80896i
$990$		0			0
$991$		−	1.23668e31i		−	0.430015i	−0.976612	−	0.215008i	$-0.931022\pi$
$991$		−	1.23668e31i		−	0.430015i	0.976612	−	0.215008i	$-0.0689776\pi$
$992$		0			0
$993$	2.96809e29	−	1.24588e31i	0.0101044	−	0.424140i
$994$		0			0
$995$	−3.67588e31			−1.22523
$996$		0			0
$997$	−4.92593e31			−1.60764			−0.803821	−	0.594871i	$-0.797203\pi$
$997$	−4.92593e31			−1.60764			−0.803821	+	0.594871i	$0.797203\pi$
$998$		0			0
$999$	−1.22481e30	−	8.76698e28i	−0.0391410	−	0.00280164i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.22.c.c.47.1	✓	28
3.2	odd	2	inner	48.22.c.c.47.27	yes	28
4.3	odd	2	inner	48.22.c.c.47.28	yes	28
12.11	even	2	inner	48.22.c.c.47.2	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.22.c.c.47.1	✓	28	1.1	even	1	trivial
48.22.c.c.47.2	yes	28	12.11	even	2	inner
48.22.c.c.47.27	yes	28	3.2	odd	2	inner
48.22.c.c.47.28	yes	28	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-102247. - 2435.85i) q^{3} -1.44607e7i q^{5} +1.26694e9i q^{7} +(1.04485e10 + 4.98117e8i) q^{9} +O(q^{10})\)	\(q+(-102247. - 2435.85i) q^{3} -1.44607e7i q^{5} +1.26694e9i q^{7} +(1.04485e10 + 4.98117e8i) q^{9} -6.25605e10 q^{11} -3.73946e11 q^{13} +(-3.52241e10 + 1.47856e12i) q^{15} -7.29673e12i q^{17} -1.62545e12i q^{19} +(3.08608e12 - 1.29541e14i) q^{21} -3.19324e14 q^{23} +2.67726e14 q^{25} +(-1.06711e15 - 7.63819e13i) q^{27} +6.11800e14i q^{29} +5.24424e15i q^{31} +(6.39661e15 + 1.52388e14i) q^{33} +1.83208e16 q^{35} +1.14778e15 q^{37} +(3.82348e16 + 9.10878e14i) q^{39} +5.72121e16i q^{41} +1.59500e17i q^{43} +(7.20311e15 - 1.51092e17i) q^{45} -5.37480e17 q^{47} -1.04659e18 q^{49} +(-1.77738e16 + 7.46067e17i) q^{51} -4.19972e16i q^{53} +9.04667e17i q^{55} +(-3.95935e15 + 1.66197e17i) q^{57} +5.79173e18 q^{59} -7.14941e18 q^{61} +(-6.31084e17 + 1.32376e19i) q^{63} +5.40752e18i q^{65} +1.96789e18i q^{67} +(3.26498e19 + 7.77825e17i) q^{69} -2.21539e19 q^{71} -3.72720e19 q^{73} +(-2.73741e19 - 6.52141e17i) q^{75} -7.92603e19i q^{77} -1.09813e20i q^{79} +(1.08923e20 + 1.04091e19i) q^{81} +1.78006e20 q^{83} -1.05516e20 q^{85} +(1.49026e18 - 6.25546e19i) q^{87} +5.72354e20i q^{89} -4.73767e20i q^{91} +(1.27742e19 - 5.36207e20i) q^{93} -2.35051e19 q^{95} +8.41397e20 q^{97} +(-6.53662e20 - 3.11624e19i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 109254828 q^{9}+O(q^{10}) \) 28 * q + 109254828 * q^9	\( 28 q + 109254828 q^{9} + 285248048392 q^{13} + 247146979606248 q^{21} - 31\!\cdots\!84 q^{25}+ \cdots + 16\!\cdots\!20 q^{97}+O(q^{100}) \) 28 * q + 109254828 * q^9 + 285248048392 * q^13 + 247146979606248 * q^21 - 3168527686741684 * q^25 - 7429429202649408 * q^33 + 33089121948065192 * q^37 - 121806830473453440 * q^45 - 569345069725368716 * q^49 + 10547854228195302552 * q^57 - 8280751575178739512 * q^61 - 22754055866538936960 * q^69 - 98548109928833394536 * q^73 + 563980976909609989500 * q^81 - 93841573689641488896 * q^85 - 1293830212233248374968 * q^93 + 1670728237822569097720 * q^97

Introduction

L-functions

Modular forms

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