LMFDB - Embedded newform 48.22.c.c.47.11

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			47.11
Character	$\chi$	$=$	48.47
Dual form			48.22.c.c.47.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-31002.5 - 97463.8i) q^{3} -2.82711e7i q^{5} +1.02999e8i q^{7} +(-8.53805e9 + 6.04324e9i) q^{9} +O(q^{10})$	\(q+(-31002.5 - 97463.8i) q^{3} -2.82711e7i q^{5} +1.02999e8i q^{7} +(-8.53805e9 + 6.04324e9i) q^{9} -1.32765e11 q^{11} -9.58265e11 q^{13} +(-2.75541e12 + 8.76472e11i) q^{15} -2.11447e12i q^{17} -5.68648e12i q^{19} +(1.00386e13 - 3.19321e12i) q^{21} +9.10497e13 q^{23} -3.22416e14 q^{25} +(8.53698e14 + 6.44796e14i) q^{27} +3.40505e15i q^{29} -2.15156e15i q^{31} +(4.11604e15 + 1.29398e16i) q^{33} +2.91188e15 q^{35} -2.84050e16 q^{37} +(2.97086e16 + 9.33962e16i) q^{39} -2.99640e16i q^{41} -6.56209e16i q^{43} +(1.70849e17 + 2.41380e17i) q^{45} -6.12232e17 q^{47} +5.47937e17 q^{49} +(-2.06084e17 + 6.55537e16i) q^{51} -1.81846e18i q^{53} +3.75340e18i q^{55} +(-5.54227e17 + 1.76295e17i) q^{57} -4.67013e18 q^{59} +6.14363e18 q^{61} +(-6.22446e17 - 8.79408e17i) q^{63} +2.70912e19i q^{65} +1.06076e19i q^{67} +(-2.82276e18 - 8.87405e18i) q^{69} +2.58108e19 q^{71} -1.97090e19 q^{73} +(9.99568e18 + 3.14239e19i) q^{75} -1.36746e19i q^{77} -1.51982e20i q^{79} +(3.63775e19 - 1.03195e20i) q^{81} -1.89392e20 q^{83} -5.97782e19 q^{85} +(3.31869e20 - 1.05565e20i) q^{87} -2.33145e20i q^{89} -9.87001e19i q^{91} +(-2.09699e20 + 6.67037e19i) q^{93} -1.60763e20 q^{95} +4.64481e20 q^{97} +(1.13355e21 - 8.02329e20i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−31002.5	−	97463.8i	−0.303126	−	0.952951i
$3$	−31002.5	−	97463.8i	−0.303126	−	0.952951i
$4$		0			0
$5$		−	2.82711e7i		−	1.29466i	−0.762208	−	0.647332i	$-0.775885\pi$
$5$		−	2.82711e7i		−	1.29466i	0.762208	−	0.647332i	$-0.224115\pi$
$6$		0			0
$7$			1.02999e8i			0.137817i	0.997623	+	0.0689084i	$0.0219516\pi$
$7$			1.02999e8i			0.137817i	−0.997623	+	0.0689084i	$0.978048\pi$
$8$		0			0
$9$	−8.53805e9	+	6.04324e9i	−0.816229	+	0.577728i
$10$		0			0
$11$	−1.32765e11			−1.54333			−0.771667	−	0.636027i	$-0.780577\pi$
$11$	−1.32765e11			−1.54333			−0.771667	+	0.636027i	$0.780577\pi$
$12$		0			0
$13$	−9.58265e11			−1.92788			−0.963941	−	0.266115i	$-0.914260\pi$
$13$	−9.58265e11			−1.92788			−0.963941	+	0.266115i	$0.914260\pi$
$14$		0			0
$15$	−2.75541e12	+	8.76472e11i	−1.23375	+	0.392446i
$16$		0			0
$17$		−	2.11447e12i		−	0.254383i	−0.991878	−	0.127191i	$-0.959404\pi$
$17$		−	2.11447e12i		−	0.254383i	0.991878	−	0.127191i	$-0.0405962\pi$
$18$		0			0
$19$		−	5.68648e12i		−	0.212780i	−0.994324	−	0.106390i	$-0.966071\pi$
$19$		−	5.68648e12i		−	0.212780i	0.994324	−	0.106390i	$-0.0339292\pi$
$20$		0			0
$21$	1.00386e13	−	3.19321e12i	0.131333	−	0.0417759i
$22$		0			0
$23$	9.10497e13			0.458286			0.229143	−	0.973393i	$-0.426408\pi$
$23$	9.10497e13			0.458286			0.229143	+	0.973393i	$0.426408\pi$
$24$		0			0
$25$	−3.22416e14			−0.676155
$26$		0			0
$27$	8.53698e14	+	6.44796e14i	0.797966	+	0.602702i
$28$		0			0
$29$			3.40505e15i			1.50295i	0.659761	+	0.751475i	$0.270657\pi$
$29$			3.40505e15i			1.50295i	−0.659761	+	0.751475i	$0.729343\pi$
$30$		0			0
$31$		−	2.15156e15i		−	0.471472i	−0.971817	−	0.235736i	$-0.924250\pi$
$31$		−	2.15156e15i		−	0.471472i	0.971817	−	0.235736i	$-0.0757500\pi$
$32$		0			0
$33$	4.11604e15	+	1.29398e16i	0.467824	+	1.47072i
$34$		0			0
$35$	2.91188e15			0.178427
$36$		0			0
$37$	−2.84050e16			−0.971130			−0.485565	−	0.874201i	$-0.661386\pi$
$37$	−2.84050e16			−0.971130			−0.485565	+	0.874201i	$0.661386\pi$
$38$		0			0
$39$	2.97086e16	+	9.33962e16i	0.584391	+	1.83718i
$40$		0			0
$41$		−	2.99640e16i		−	0.348634i	−0.984690	−	0.174317i	$-0.944228\pi$
$41$		−	2.99640e16i		−	0.348634i	0.984690	−	0.174317i	$-0.0557717\pi$
$42$		0			0
$43$		−	6.56209e16i		−	0.463046i	−0.972829	−	0.231523i	$-0.925629\pi$
$43$		−	6.56209e16i		−	0.463046i	0.972829	−	0.231523i	$-0.0743708\pi$
$44$		0			0
$45$	1.70849e17	+	2.41380e17i	0.747963	+	1.05674i
$46$		0			0
$47$	−6.12232e17			−1.69781			−0.848904	−	0.528547i	$-0.822737\pi$
$47$	−6.12232e17			−1.69781			−0.848904	+	0.528547i	$0.822737\pi$
$48$		0			0
$49$	5.47937e17			0.981007
$50$		0			0
$51$	−2.06084e17	+	6.55537e16i	−0.242414	+	0.0771099i
$52$		0			0
$53$		−	1.81846e18i		−	1.42826i	−0.700013	−	0.714130i	$-0.746822\pi$
$53$		−	1.81846e18i		−	1.42826i	0.700013	−	0.714130i	$-0.253178\pi$
$54$		0			0
$55$			3.75340e18i			1.99810i
$56$		0			0
$57$	−5.54227e17	+	1.76295e17i	−0.202769	+	0.0644992i
$58$		0			0
$59$	−4.67013e18			−1.18955			−0.594775	−	0.803892i	$-0.702759\pi$
$59$	−4.67013e18			−1.18955			−0.594775	+	0.803892i	$0.702759\pi$
$60$		0			0
$61$	6.14363e18			1.10271			0.551356	−	0.834270i	$-0.314111\pi$
$61$	6.14363e18			1.10271			0.551356	+	0.834270i	$0.314111\pi$
$62$		0			0
$63$	−6.22446e17	−	8.79408e17i	−0.0796206	−	0.112490i
$64$		0			0
$65$			2.70912e19i			2.49596i
$66$		0			0
$67$			1.06076e19i			0.710938i	0.934688	+	0.355469i	$0.115679\pi$
$67$			1.06076e19i			0.710938i	−0.934688	+	0.355469i	$0.884321\pi$
$68$		0			0
$69$	−2.82276e18	−	8.87405e18i	−0.138918	−	0.436723i
$70$		0			0
$71$	2.58108e19			0.940998			0.470499	−	0.882400i	$-0.344074\pi$
$71$	2.58108e19			0.940998			0.470499	+	0.882400i	$0.344074\pi$
$72$		0			0
$73$	−1.97090e19			−0.536752			−0.268376	−	0.963314i	$-0.586487\pi$
$73$	−1.97090e19			−0.536752			−0.268376	+	0.963314i	$0.586487\pi$
$74$		0			0
$75$	9.99568e18	+	3.14239e19i	0.204960	+	0.644342i
$76$		0			0
$77$		−	1.36746e19i		−	0.212697i
$78$		0			0
$79$		−	1.51982e20i		−	1.80596i	−0.429681	−	0.902981i	$-0.641374\pi$
$79$		−	1.51982e20i		−	1.80596i	0.429681	−	0.902981i	$-0.358626\pi$
$80$		0			0
$81$	3.63775e19	−	1.03195e20i	0.332461	−	0.943117i
$82$		0			0
$83$	−1.89392e20			−1.33980			−0.669902	−	0.742450i	$-0.733664\pi$
$83$	−1.89392e20			−1.33980			−0.669902	+	0.742450i	$0.733664\pi$
$84$		0			0
$85$	−5.97782e19			−0.329340
$86$		0			0
$87$	3.31869e20	−	1.05565e20i	1.43224	−	0.455583i
$88$		0			0
$89$		−	2.33145e20i		−	0.792559i	−0.918130	−	0.396280i	$-0.870301\pi$
$89$		−	2.33145e20i		−	0.792559i	0.918130	−	0.396280i	$-0.129699\pi$
$90$		0			0
$91$		−	9.87001e19i		−	0.265695i
$92$		0			0
$93$	−2.09699e20	+	6.67037e19i	−0.449289	+	0.142915i
$94$		0			0
$95$	−1.60763e20			−0.275479
$96$		0			0
$97$	4.64481e20			0.639535			0.319768	−	0.947496i	$-0.396395\pi$
$97$	4.64481e20			0.639535			0.319768	+	0.947496i	$0.396395\pi$
$98$		0			0
$99$	1.13355e21	−	8.02329e20i	1.25971	−	0.891627i
$100$		0			0
$101$			1.47323e21i			1.32707i	0.748144	+	0.663537i	$0.230945\pi$
$101$			1.47323e21i			1.32707i	−0.748144	+	0.663537i	$0.769055\pi$
$102$		0			0
$103$			1.44249e21i			1.05760i	0.848746	+	0.528800i	$0.177358\pi$
$103$			1.44249e21i			1.05760i	−0.848746	+	0.528800i	$0.822642\pi$
$104$		0			0
$105$	−9.02755e19	−	2.83803e20i	−0.0540857	−	0.170032i
$106$		0			0
$107$	−2.97389e21			−1.46149			−0.730743	−	0.682653i	$-0.760826\pi$
$107$	−2.97389e21			−1.46149			−0.730743	+	0.682653i	$0.760826\pi$
$108$		0			0
$109$	1.34548e20			0.0544376			0.0272188	−	0.999630i	$-0.491335\pi$
$109$	1.34548e20			0.0544376			0.0272188	+	0.999630i	$0.491335\pi$
$110$		0			0
$111$	8.80626e20	+	2.76847e21i	0.294375	+	0.925439i
$112$		0			0
$113$			5.17323e21i			1.43363i	0.697262	+	0.716817i	$0.254402\pi$
$113$			5.17323e21i			1.43363i	−0.697262	+	0.716817i	$0.745598\pi$
$114$		0			0
$115$		−	2.57407e21i		−	0.593326i
$116$		0			0
$117$	8.18172e21	−	5.79103e21i	1.57359	−	1.11379i
$118$		0			0
$119$	2.17787e20			0.0350582
$120$		0			0
$121$	1.02262e22			1.38188
$122$		0			0
$123$	−2.92041e21	+	9.28959e20i	−0.332231	+	0.105680i
$124$		0			0
$125$		−	4.36566e21i		−	0.419271i
$126$		0			0
$127$		−	6.15270e21i		−	0.500180i	−0.968223	−	0.250090i	$-0.919540\pi$
$127$		−	6.15270e21i		−	0.500180i	0.968223	−	0.250090i	$-0.0804602\pi$
$128$		0			0
$129$	−6.39567e21	+	2.03441e21i	−0.441260	+	0.140361i
$130$		0			0
$131$	−1.36342e22			−0.800351			−0.400175	−	0.916439i	$-0.631051\pi$
$131$	−1.36342e22			−0.800351			−0.400175	+	0.916439i	$0.631051\pi$
$132$		0			0
$133$	5.85700e20			0.0293247
$134$		0			0
$135$	1.82291e22	−	2.41349e22i	0.780297	−	1.03310i
$136$		0			0
$137$			1.82719e21i			0.0670220i	0.999438	+	0.0335110i	$0.0106689\pi$
$137$			1.82719e21i			0.0670220i	−0.999438	+	0.0335110i	$0.989331\pi$
$138$		0			0
$139$		−	2.58923e22i		−	0.815672i	−0.913055	−	0.407836i	$-0.866283\pi$
$139$		−	2.58923e22i		−	0.815672i	0.913055	−	0.407836i	$-0.133717\pi$
$140$		0			0
$141$	1.89807e22	+	5.96705e22i	0.514650	+	1.61793i
$142$		0			0
$143$	1.27224e23			2.97537
$144$		0			0
$145$	9.62644e22			1.94582
$146$		0			0
$147$	−1.69874e22	−	5.34041e22i	−0.297368	−	0.934851i
$148$		0			0
$149$		−	9.86155e21i		−	0.149792i	−0.997191	−	0.0748961i	$-0.976137\pi$
$149$		−	9.86155e21i		−	0.149792i	0.997191	−	0.0748961i	$-0.0238625\pi$
$150$		0			0
$151$		−	6.10341e22i		−	0.805963i	−0.915208	−	0.402981i	$-0.867974\pi$
$151$		−	6.10341e22i		−	0.805963i	0.915208	−	0.402981i	$-0.132026\pi$
$152$		0			0
$153$	1.27782e22	+	1.80534e22i	0.146964	+	0.207634i
$154$		0			0
$155$	−6.08269e22			−0.610398
$156$		0			0
$157$	4.60041e22			0.403506			0.201753	−	0.979436i	$-0.435336\pi$
$157$	4.60041e22			0.403506			0.201753	+	0.979436i	$0.435336\pi$
$158$		0			0
$159$	−1.77234e23	+	5.63768e22i	−1.36106	+	0.432943i
$160$		0			0
$161$			9.37800e21i			0.0631595i
$162$		0			0
$163$		−	2.18729e23i		−	1.29401i	−0.762488	−	0.647003i	$-0.776022\pi$
$163$		−	2.18729e23i		−	1.29401i	0.762488	−	0.647003i	$-0.223978\pi$
$164$		0			0
$165$	3.65821e23	−	1.16365e23i	1.90409	−	0.605675i
$166$		0			0
$167$	2.28734e23			1.04908			0.524539	−	0.851386i	$-0.324238\pi$
$167$	2.28734e23			1.04908			0.524539	+	0.851386i	$0.324238\pi$
$168$		0			0
$169$	6.71208e23			2.71673
$170$		0			0
$171$	3.43648e22	+	4.85515e22i	0.122929	+	0.173677i
$172$		0			0
$173$			2.75361e23i			0.871804i	0.899994	+	0.435902i	$0.143571\pi$
$173$			2.75361e23i			0.871804i	−0.899994	+	0.435902i	$0.856429\pi$
$174$		0			0
$175$		−	3.32084e22i		−	0.0931855i
$176$		0			0
$177$	1.44786e23	+	4.55169e23i	0.360584	+	1.13358i
$178$		0			0
$179$	−1.90244e22			−0.0421070			−0.0210535	−	0.999778i	$-0.506702\pi$
$179$	−1.90244e22			−0.0421070			−0.0210535	+	0.999778i	$0.506702\pi$
$180$		0			0
$181$	2.28573e23			0.450194			0.225097	−	0.974336i	$-0.427730\pi$
$181$	2.28573e23			0.450194			0.225097	+	0.974336i	$0.427730\pi$
$182$		0			0
$183$	−1.90468e23	−	5.98782e23i	−0.334260	−	1.05083i
$184$		0			0
$185$			8.03041e23i			1.25729i
$186$		0			0
$187$			2.80727e23i			0.392597i
$188$		0			0
$189$	−6.64131e22	+	8.79298e22i	−0.0830625	+	0.109973i
$190$		0			0
$191$	−3.32680e23			−0.372543			−0.186272	−	0.982498i	$-0.559640\pi$
$191$	−3.32680e23			−0.372543			−0.186272	+	0.982498i	$0.559640\pi$
$192$		0			0
$193$	1.45667e24			1.46221			0.731103	−	0.682267i	$-0.239006\pi$
$193$	1.45667e24			1.46221			0.731103	+	0.682267i	$0.239006\pi$
$194$		0			0
$195$	2.64041e24	−	8.39893e23i	2.37853	−	0.756590i
$196$		0			0
$197$		−	1.20235e24i		−	0.973055i	−0.873665	−	0.486527i	$-0.838263\pi$
$197$		−	1.20235e24i		−	0.973055i	0.873665	−	0.486527i	$-0.161737\pi$
$198$		0			0
$199$		−	1.98535e24i		−	1.44504i	−0.691350	−	0.722520i	$-0.742984\pi$
$199$		−	1.98535e24i		−	1.44504i	0.691350	−	0.722520i	$-0.257016\pi$
$200$		0			0
$201$	1.03386e24	−	3.28862e23i	0.677489	−	0.215504i
$202$		0			0
$203$	−3.50716e23			−0.207132
$204$		0			0
$205$	−8.47115e23			−0.451364
$206$		0			0
$207$	−7.77387e23	+	5.50235e23i	−0.374066	+	0.264764i
$208$		0			0
$209$			7.54965e23i			0.328391i
$210$		0			0
$211$		−	3.13097e24i		−	1.23229i	−0.787633	−	0.616144i	$-0.788694\pi$
$211$		−	3.13097e24i		−	1.23229i	0.787633	−	0.616144i	$-0.211306\pi$
$212$		0			0
$213$	−8.00198e23	−	2.51562e24i	−0.285241	−	0.896725i
$214$		0			0
$215$	−1.85517e24			−0.599488
$216$		0			0
$217$	2.21608e23			0.0649768
$218$		0			0
$219$	6.11027e23	+	1.92091e24i	0.162704	+	0.511498i
$220$		0			0
$221$			2.02622e24i			0.490420i
$222$		0			0
$223$			3.11866e24i			0.686699i	0.939208	+	0.343349i	$0.111562\pi$
$223$			3.11866e24i			0.686699i	−0.939208	+	0.343349i	$0.888438\pi$
$224$		0			0
$225$	2.75280e24	−	1.94844e24i	0.551897	−	0.390634i
$226$		0			0
$227$	1.65570e24			0.302488			0.151244	−	0.988496i	$-0.451672\pi$
$227$	1.65570e24			0.302488			0.151244	+	0.988496i	$0.451672\pi$
$228$		0			0
$229$	−3.46226e24			−0.576883			−0.288441	−	0.957498i	$-0.593137\pi$
$229$	−3.46226e24			−0.576883			−0.288441	+	0.957498i	$0.593137\pi$
$230$		0			0
$231$	−1.33278e24	+	4.23946e23i	−0.202690	+	0.0644741i
$232$		0			0
$233$			6.12899e24i			0.851435i	0.904856	+	0.425718i	$0.139978\pi$
$233$			6.12899e24i			0.851435i	−0.904856	+	0.425718i	$0.860022\pi$
$234$		0			0
$235$			1.73085e25i			2.19809i
$236$		0			0
$237$	−1.48128e25	+	4.71182e24i	−1.72099	+	0.547434i
$238$		0			0
$239$	2.51297e24			0.267306			0.133653	−	0.991028i	$-0.457329\pi$
$239$	2.51297e24			0.267306			0.133653	+	0.991028i	$0.457329\pi$
$240$		0			0
$241$	−4.20676e24			−0.409986			−0.204993	−	0.978763i	$-0.565717\pi$
$241$	−4.20676e24			−0.409986			−0.204993	+	0.978763i	$0.565717\pi$
$242$		0			0
$243$	−1.11856e25	−	3.46199e23i	−0.999521	−	0.0309357i
$244$		0			0
$245$		−	1.54908e25i		−	1.27007i
$246$		0			0
$247$			5.44916e24i			0.410215i
$248$		0			0
$249$	5.87161e24	+	1.84588e25i	0.406129	+	1.27677i
$250$		0			0
$251$	−9.99727e24			−0.635781			−0.317890	−	0.948128i	$-0.602974\pi$
$251$	−9.99727e24			−0.635781			−0.317890	+	0.948128i	$0.602974\pi$
$252$		0			0
$253$	−1.20882e25			−0.707287
$254$		0			0
$255$	1.85327e24	+	5.82622e24i	0.0998314	+	0.313845i
$256$		0			0
$257$		−	3.50375e25i		−	1.73874i	−0.494161	−	0.869370i	$-0.664525\pi$
$257$		−	3.50375e25i		−	1.73874i	0.494161	−	0.869370i	$-0.335475\pi$
$258$		0			0
$259$		−	2.92568e24i		−	0.133838i
$260$		0			0
$261$	−2.05775e25	−	2.90725e25i	−0.868296	−	1.22675i
$262$		0			0
$263$	8.42834e24			0.328251			0.164126	−	0.986439i	$-0.447520\pi$
$263$	8.42834e24			0.328251			0.164126	+	0.986439i	$0.447520\pi$
$264$		0			0
$265$	−5.14098e25			−1.84912
$266$		0			0
$267$	−2.27232e25	+	7.22808e24i	−0.755270	+	0.240245i
$268$		0			0
$269$		−	2.84505e25i		−	0.874361i	−0.899374	−	0.437180i	$-0.855977\pi$
$269$		−	2.84505e25i		−	0.874361i	0.899374	−	0.437180i	$-0.144023\pi$
$270$		0			0
$271$			1.27050e25i			0.361241i	0.983553	+	0.180620i	$0.0578105\pi$
$271$			1.27050e25i			0.361241i	−0.983553	+	0.180620i	$0.942189\pi$
$272$		0			0
$273$	−9.61969e24	+	3.05995e24i	−0.253194	+	0.0805389i
$274$		0			0
$275$	4.28055e25			1.04353
$276$		0			0
$277$	−5.84703e25			−1.32098			−0.660492	−	0.750833i	$-0.729652\pi$
$277$	−5.84703e25			−1.32098			−0.660492	+	0.750833i	$0.729652\pi$
$278$		0			0
$279$	1.30024e25	+	1.83701e25i	0.272382	+	0.384829i
$280$		0			0
$281$			6.75662e25i			1.31315i	0.754263	+	0.656573i	$0.227995\pi$
$281$			6.75662e25i			1.31315i	−0.754263	+	0.656573i	$0.772005\pi$
$282$		0			0
$283$		−	5.15465e25i		−	0.929911i	−0.885334	−	0.464956i	$-0.846070\pi$
$283$		−	5.15465e25i		−	0.929911i	0.885334	−	0.464956i	$-0.153930\pi$
$284$		0			0
$285$	4.98405e24	+	1.56686e25i	0.0835047	+	0.262518i
$286$		0			0
$287$	3.08626e24			0.0480476
$288$		0			0
$289$	6.46210e25			0.935290
$290$		0			0
$291$	−1.44000e25	−	4.52701e25i	−0.193860	−	0.609446i
$292$		0			0
$293$			1.04908e26i			1.31431i	0.753755	+	0.657156i	$0.228241\pi$
$293$			1.04908e26i			1.31431i	−0.753755	+	0.657156i	$0.771759\pi$
$294$		0			0
$295$			1.32030e26i			1.54007i
$296$		0			0
$297$	−1.13341e26	−	8.56062e25i	−1.23153	−	0.930170i
$298$		0			0
$299$	−8.72498e25			−0.883521
$300$		0			0
$301$	6.75887e24			0.0638155
$302$		0			0
$303$	1.43586e26	−	4.56736e25i	1.26464	−	0.402270i
$304$		0			0
$305$		−	1.73687e26i		−	1.42764i
$306$		0			0
$307$			2.17670e26i			1.67050i	0.549871	+	0.835250i	$0.314677\pi$
$307$			2.17670e26i			1.67050i	−0.549871	+	0.835250i	$0.685323\pi$
$308$		0			0
$309$	1.40591e26	−	4.47207e25i	1.00784	−	0.320586i
$310$		0			0
$311$	−2.61347e26			−1.75079			−0.875394	−	0.483410i	$-0.839398\pi$
$311$	−2.61347e26			−1.75079			−0.875394	+	0.483410i	$0.839398\pi$
$312$		0			0
$313$	2.60748e26			1.63307			0.816536	−	0.577294i	$-0.195891\pi$
$313$	2.60748e26			1.63307			0.816536	+	0.577294i	$0.195891\pi$
$314$		0			0
$315$	−2.48618e25	+	1.75972e25i	−0.145637	+	0.103082i
$316$		0			0
$317$			1.66287e26i			0.911456i	0.890119	+	0.455728i	$0.150621\pi$
$317$			1.66287e26i			0.911456i	−0.890119	+	0.455728i	$0.849379\pi$
$318$		0			0
$319$		−	4.52071e26i		−	2.31955i
$320$		0			0
$321$	9.21978e25	+	2.89846e26i	0.443014	+	1.39272i
$322$		0			0
$323$	−1.20239e25			−0.0541275
$324$		0			0
$325$	3.08960e26			1.30355
$326$		0			0
$327$	−4.17132e24	−	1.31136e25i	−0.0165014	−	0.0518763i
$328$		0			0
$329$		−	6.30591e25i		−	0.233987i
$330$		0			0
$331$		−	1.22020e26i		−	0.424851i	−0.977177	−	0.212426i	$-0.931864\pi$
$331$		−	1.22020e26i		−	0.424851i	0.977177	−	0.212426i	$-0.0681363\pi$
$332$		0			0
$333$	2.42524e26	−	1.71658e26i	0.792665	−	0.561049i
$334$		0			0
$335$	2.99888e26			0.920426
$336$		0			0
$337$	−3.49704e26			−1.00829			−0.504147	−	0.863618i	$-0.668193\pi$
$337$	−3.49704e26			−1.00829			−0.504147	+	0.863618i	$0.668193\pi$
$338$		0			0
$339$	5.04203e26	−	1.60383e26i	1.36618	−	0.434571i
$340$		0			0
$341$			2.85652e26i			0.727638i
$342$		0			0
$343$			1.13966e26i			0.273016i
$344$		0			0
$345$	−2.50879e26	+	7.98025e25i	−0.565410	+	0.179852i
$346$		0			0
$347$	7.68084e26			1.62911			0.814553	−	0.580089i	$-0.196982\pi$
$347$	7.68084e26			1.62911			0.814553	+	0.580089i	$0.196982\pi$
$348$		0			0
$349$	8.33514e26			1.66435			0.832177	−	0.554510i	$-0.187094\pi$
$349$	8.33514e26			1.66435			0.832177	+	0.554510i	$0.187094\pi$
$350$		0			0
$351$	−8.18069e26	−	6.17885e26i	−1.53839	−	1.16194i
$352$		0			0
$353$			2.22202e26i			0.393654i	0.980438	+	0.196827i	$0.0630637\pi$
$353$			2.22202e26i			0.393654i	−0.980438	+	0.196827i	$0.936936\pi$
$354$		0			0
$355$		−	7.29699e26i		−	1.21828i
$356$		0			0
$357$	−6.75194e24	−	2.12264e25i	−0.0106270	−	0.0334087i
$358$		0			0
$359$	1.65969e26			0.246340			0.123170	−	0.992386i	$-0.460694\pi$
$359$	1.65969e26			0.246340			0.123170	+	0.992386i	$0.460694\pi$
$360$		0			0
$361$	6.81873e26			0.954725
$362$		0			0
$363$	−3.17039e26	−	9.96689e26i	−0.418883	−	1.31686i
$364$		0			0
$365$			5.57194e26i			0.694914i
$366$		0			0
$367$			1.02252e27i			1.20414i	0.798442	+	0.602072i	$0.205658\pi$
$367$			1.02252e27i			1.20414i	−0.798442	+	0.602072i	$0.794342\pi$
$368$		0			0
$369$	1.81080e26	+	2.55834e26i	0.201416	+	0.284565i
$370$		0			0
$371$	1.87299e26			0.196838
$372$		0			0
$373$	6.52613e26			0.648206			0.324103	−	0.946022i	$-0.394938\pi$
$373$	6.52613e26			0.648206			0.324103	+	0.946022i	$0.394938\pi$
$374$		0			0
$375$	−4.25494e26	+	1.35346e26i	−0.399544	+	0.127092i
$376$		0			0
$377$		−	3.26294e27i		−	2.89751i
$378$		0			0
$379$			1.42847e27i			1.19994i	0.800024	+	0.599968i	$0.204820\pi$
$379$			1.42847e27i			1.19994i	−0.800024	+	0.599968i	$0.795180\pi$
$380$		0			0
$381$	−5.99666e26	+	1.90749e26i	−0.476647	+	0.151618i
$382$		0			0
$383$	8.63330e25			0.0649516			0.0324758	−	0.999473i	$-0.489661\pi$
$383$	8.63330e25			0.0649516			0.0324758	+	0.999473i	$0.489661\pi$
$384$		0			0
$385$	−3.86596e26			−0.275372
$386$		0			0
$387$	3.96563e26	+	5.60275e26i	0.267514	+	0.377951i
$388$		0			0
$389$			1.31047e27i			0.837447i	0.908114	+	0.418723i	$0.137522\pi$
$389$			1.31047e27i			0.837447i	−0.908114	+	0.418723i	$0.862478\pi$
$390$		0			0
$391$		−	1.92522e26i		−	0.116580i
$392$		0			0
$393$	4.22693e26	+	1.32884e27i	0.242607	+	0.762694i
$394$		0			0
$395$	−4.29670e27			−2.33811
$396$		0			0
$397$	7.44555e25			0.0384234			0.0192117	−	0.999815i	$-0.493884\pi$
$397$	7.44555e25			0.0384234			0.0192117	+	0.999815i	$0.493884\pi$
$398$		0			0
$399$	−1.81582e25	−	5.70846e25i	−0.00888907	−	0.0279450i
$400$		0			0
$401$			1.74900e27i			0.812408i	0.913783	+	0.406204i	$0.133148\pi$
$401$			1.74900e27i			0.812408i	−0.913783	+	0.406204i	$0.866852\pi$
$402$		0			0
$403$			2.06177e27i			0.908942i
$404$		0			0
$405$	−2.91743e27	−	1.02843e27i	−1.22102	−	0.430425i
$406$		0			0
$407$	3.77119e27			1.49878
$408$		0			0
$409$	−3.02987e27			−1.14375			−0.571873	−	0.820342i	$-0.693783\pi$
$409$	−3.02987e27			−1.14375			−0.571873	+	0.820342i	$0.693783\pi$
$410$		0			0
$411$	1.78085e26	−	5.66473e25i	0.0638687	−	0.0203161i
$412$		0			0
$413$		−	4.81017e26i		−	0.163940i
$414$		0			0
$415$			5.35430e27i			1.73459i
$416$		0			0
$417$	−2.52357e27	+	8.02726e26i	−0.777295	+	0.247251i
$418$		0			0
$419$	2.11973e26			0.0620917			0.0310459	−	0.999518i	$-0.490116\pi$
$419$	2.11973e26			0.0620917			0.0310459	+	0.999518i	$0.490116\pi$
$420$		0			0
$421$	−1.85950e27			−0.518123			−0.259062	−	0.965861i	$-0.583413\pi$
$421$	−1.85950e27			−0.518123			−0.259062	+	0.965861i	$0.583413\pi$
$422$		0			0
$423$	5.22727e27	−	3.69987e27i	1.38580	−	0.980871i
$424$		0			0
$425$			6.81737e26i			0.172002i
$426$		0			0
$427$			6.32786e26i			0.151972i
$428$		0			0
$429$	−3.94425e27	−	1.23997e28i	−0.901910	−	2.83538i
$430$		0			0
$431$	−7.39954e27			−1.61136			−0.805682	−	0.592348i	$-0.798201\pi$
$431$	−7.39954e27			−1.61136			−0.805682	+	0.592348i	$0.798201\pi$
$432$		0			0
$433$	1.39690e27			0.289762			0.144881	−	0.989449i	$-0.453720\pi$
$433$	1.39690e27			0.289762			0.144881	+	0.989449i	$0.453720\pi$
$434$		0			0
$435$	−2.98443e27	−	9.38230e27i	−0.589827	−	1.85427i
$436$		0			0
$437$		−	5.17753e26i		−	0.0975141i
$438$		0			0
$439$			7.42003e27i			1.33207i	0.745919	+	0.666037i	$0.232011\pi$
$439$			7.42003e27i			1.33207i	−0.745919	+	0.666037i	$0.767989\pi$
$440$		0			0
$441$	−4.67831e27	+	3.31131e27i	−0.800726	+	0.566755i
$442$		0			0
$443$	1.50828e27			0.246174			0.123087	−	0.992396i	$-0.460721\pi$
$443$	1.50828e27			0.246174			0.123087	+	0.992396i	$0.460721\pi$
$444$		0			0
$445$	−6.59127e27			−1.02610
$446$		0			0
$447$	−9.61144e26	+	3.05732e26i	−0.142745	+	0.0454059i
$448$		0			0
$449$		−	1.26145e28i		−	1.78765i	−0.448416	−	0.893825i	$-0.648012\pi$
$449$		−	1.26145e28i		−	1.78765i	0.448416	−	0.893825i	$-0.351988\pi$
$450$		0			0
$451$			3.97817e27i			0.538058i
$452$		0			0
$453$	−5.94862e27	+	1.89221e27i	−0.768043	+	0.244308i
$454$		0			0
$455$	−2.79036e27			−0.343985
$456$		0			0
$457$	8.36070e27			0.984289			0.492145	−	0.870513i	$-0.336213\pi$
$457$	8.36070e27			0.984289			0.492145	+	0.870513i	$0.336213\pi$
$458$		0			0
$459$	1.36340e27	−	1.80512e27i	0.153317	−	0.202989i
$460$		0			0
$461$			9.97423e27i			1.07157i	0.844355	+	0.535784i	$0.179984\pi$
$461$			9.97423e27i			1.07157i	−0.844355	+	0.535784i	$0.820016\pi$
$462$		0			0
$463$			1.10785e28i			1.13732i	0.822574	+	0.568658i	$0.192537\pi$
$463$			1.10785e28i			1.13732i	−0.822574	+	0.568658i	$0.807463\pi$
$464$		0			0
$465$	1.88578e27	+	5.92842e27i	0.185027	+	0.581679i
$466$		0			0
$467$	1.46064e28			1.36999			0.684993	−	0.728550i	$-0.259805\pi$
$467$	1.46064e28			1.36999			0.684993	+	0.728550i	$0.259805\pi$
$468$		0			0
$469$	−1.09257e27			−0.0979793
$470$		0			0
$471$	−1.42624e27	−	4.48373e27i	−0.122313	−	0.384521i
$472$		0			0
$473$			8.71215e27i			0.714634i
$474$		0			0
$475$			1.83341e27i			0.143872i
$476$		0			0
$477$	1.09894e28	+	1.55261e28i	0.825146	+	1.16579i
$478$		0			0
$479$	−1.52456e28			−1.09553			−0.547763	−	0.836634i	$-0.684520\pi$
$479$	−1.52456e28			−1.09553			−0.547763	+	0.836634i	$0.684520\pi$
$480$		0			0
$481$	2.72196e28			1.87222
$482$		0			0
$483$	9.14016e26	−	2.90741e26i	0.0601879	−	0.0191453i
$484$		0			0
$485$		−	1.31314e28i		−	0.827983i
$486$		0			0
$487$			1.11194e28i			0.671472i	0.941956	+	0.335736i	$0.108985\pi$
$487$			1.11194e28i			0.671472i	−0.941956	+	0.335736i	$0.891015\pi$
$488$		0			0
$489$	−2.13182e28	+	6.78113e27i	−1.23312	+	0.392247i
$490$		0			0
$491$	7.63475e27			0.423096			0.211548	−	0.977368i	$-0.432149\pi$
$491$	7.63475e27			0.423096			0.211548	+	0.977368i	$0.432149\pi$
$492$		0			0
$493$	7.19987e27			0.382324
$494$		0			0
$495$	−2.26827e28	−	3.20467e28i	−1.15436	−	1.63091i
$496$		0			0
$497$			2.65848e27i			0.129685i
$498$		0			0
$499$			3.25737e28i			1.52339i	0.647936	+	0.761695i	$0.275632\pi$
$499$			3.25737e28i			1.52339i	−0.647936	+	0.761695i	$0.724368\pi$
$500$		0			0
$501$	−7.09132e27	−	2.22933e28i	−0.318003	−	0.999720i
$502$		0			0
$503$	1.92019e28			0.825812			0.412906	−	0.910774i	$-0.364514\pi$
$503$	1.92019e28			0.825812			0.412906	+	0.910774i	$0.364514\pi$
$504$		0			0
$505$	4.16497e28			1.71811
$506$		0			0
$507$	−2.08091e28	−	6.54185e28i	−0.823512	−	2.58891i
$508$		0			0
$509$		−	6.25570e26i		−	0.0237541i	−0.999929	−	0.0118771i	$-0.996219\pi$
$509$		−	6.25570e26i		−	0.0237541i	0.999929	−	0.0118771i	$-0.00378068\pi$
$510$		0			0
$511$		−	2.03000e27i		−	0.0739735i
$512$		0			0
$513$	3.66662e27	−	4.85454e27i	0.128243	−	0.169791i
$514$		0			0
$515$	4.07807e28			1.36924
$516$		0			0
$517$	8.12829e28			2.62028
$518$		0			0
$519$	2.68378e28	−	8.53688e27i	0.830786	−	0.264266i
$520$		0			0
$521$			1.09254e28i			0.324820i	0.986723	+	0.162410i	$0.0519267\pi$
$521$			1.09254e28i			0.324820i	−0.986723	+	0.162410i	$0.948073\pi$
$522$		0			0
$523$		−	6.60756e28i		−	1.88701i	−0.331365	−	0.943503i	$-0.607509\pi$
$523$		−	6.60756e28i		−	1.88701i	0.331365	−	0.943503i	$-0.392491\pi$
$524$		0			0
$525$	−3.23662e27	+	1.02954e27i	−0.0888012	+	0.0282469i
$526$		0			0
$527$	−4.54940e27			−0.119934
$528$		0			0
$529$	−3.11815e28			−0.789974
$530$		0			0
$531$	3.98738e28	−	2.82227e28i	0.970946	−	0.687237i
$532$		0			0
$533$			2.87135e28i			0.672125i
$534$		0			0
$535$			8.40749e28i			1.89213i
$536$		0			0
$537$	5.89804e26	+	1.85419e27i	0.0127637	+	0.0401259i
$538$		0			0
$539$	−7.27468e28			−1.51402
$540$		0			0
$541$	−8.43619e27			−0.168879			−0.0844394	−	0.996429i	$-0.526910\pi$
$541$	−8.43619e27			−0.168879			−0.0844394	+	0.996429i	$0.526910\pi$
$542$		0			0
$543$	−7.08632e27	−	2.22776e28i	−0.136465	−	0.429012i
$544$		0			0
$545$		−	3.80381e27i		−	0.0704784i
$546$		0			0
$547$		−	1.65125e28i		−	0.294406i	−0.989106	−	0.147203i	$-0.952973\pi$
$547$		−	1.65125e28i		−	0.294406i	0.989106	−	0.147203i	$-0.0470270\pi$
$548$		0			0
$549$	−5.24546e28	+	3.71274e28i	−0.900065	+	0.637067i
$550$		0			0
$551$	1.93628e28			0.319798
$552$		0			0
$553$	1.56540e28			0.248892
$554$		0			0
$555$	7.82674e28	−	2.48962e28i	1.19813	−	0.381116i
$556$		0			0
$557$			4.22102e28i			0.622211i	0.950375	+	0.311106i	$0.100699\pi$
$557$			4.22102e28i			0.622211i	−0.950375	+	0.311106i	$0.899301\pi$
$558$		0			0
$559$			6.28823e28i			0.892697i
$560$		0			0
$561$	2.73607e28	−	8.70322e27i	0.374126	−	0.119006i
$562$		0			0
$563$	−4.42765e28			−0.583224			−0.291612	−	0.956537i	$-0.594192\pi$
$563$	−4.42765e28			−0.583224			−0.291612	+	0.956537i	$0.594192\pi$
$564$		0			0
$565$	1.46253e29			1.85607
$566$		0			0
$567$	1.06289e28	+	3.74684e27i	0.129977	+	0.0458187i
$568$		0			0
$569$		−	5.43934e28i		−	0.641013i	−0.947246	−	0.320507i	$-0.896147\pi$
$569$		−	5.43934e28i		−	0.641013i	0.947246	−	0.320507i	$-0.103853\pi$
$570$		0			0
$571$			1.11713e29i			1.26889i	0.772967	+	0.634447i	$0.218772\pi$
$571$			1.11713e29i			1.26889i	−0.772967	+	0.634447i	$0.781228\pi$
$572$		0			0
$573$	1.03139e28	+	3.24243e28i	0.112927	+	0.355015i
$574$		0			0
$575$	−2.93559e28			−0.309872
$576$		0			0
$577$	−8.86684e28			−0.902450			−0.451225	−	0.892410i	$-0.649013\pi$
$577$	−8.86684e28			−0.902450			−0.451225	+	0.892410i	$0.649013\pi$
$578$		0			0
$579$	−4.51603e28	−	1.41972e29i	−0.443232	−	1.39341i
$580$		0			0
$581$		−	1.95071e28i		−	0.184647i
$582$		0			0
$583$			2.41428e29i			2.20428i
$584$		0			0
$585$	−1.63718e29	−	2.31306e29i	−1.44199	−	2.03728i
$586$		0			0
$587$	−1.90260e29			−1.61677			−0.808385	−	0.588654i	$-0.799658\pi$
$587$	−1.90260e29			−1.61677			−0.808385	+	0.588654i	$0.799658\pi$
$588$		0			0
$589$	−1.22348e28			−0.100320
$590$		0			0
$591$	−1.17186e29	+	3.72760e28i	−0.927273	+	0.294958i
$592$		0			0
$593$			1.06979e29i			0.817001i	0.912758	+	0.408501i	$0.133948\pi$
$593$			1.06979e29i			0.817001i	−0.912758	+	0.408501i	$0.866052\pi$
$594$		0			0
$595$		−	6.15708e27i		−	0.0453886i
$596$		0			0
$597$	−1.93500e29	+	6.15507e28i	−1.37705	+	0.438029i
$598$		0			0
$599$	7.75529e27			0.0532864			0.0266432	−	0.999645i	$-0.491518\pi$
$599$	7.75529e27			0.0532864			0.0266432	+	0.999645i	$0.491518\pi$
$600$		0			0
$601$	−8.50419e28			−0.564224			−0.282112	−	0.959382i	$-0.591035\pi$
$601$	−8.50419e28			−0.564224			−0.282112	+	0.959382i	$0.591035\pi$
$602$		0			0
$603$	−6.41043e28	−	9.05683e28i	−0.410729	−	0.580289i
$604$		0			0
$605$		−	2.89107e29i		−	1.78907i
$606$		0			0
$607$			1.82510e29i			1.09095i	0.838127	+	0.545475i	$0.183651\pi$
$607$			1.82510e29i			1.09095i	−0.838127	+	0.545475i	$0.816349\pi$
$608$		0			0
$609$	1.08731e28	+	3.41821e28i	0.0627870	+	0.197386i
$610$		0			0
$611$	5.86681e29			3.27318
$612$		0			0
$613$	−9.95618e28			−0.536732			−0.268366	−	0.963317i	$-0.586484\pi$
$613$	−9.95618e28			−0.536732			−0.268366	+	0.963317i	$0.586484\pi$
$614$		0			0
$615$	2.62626e28	+	8.25631e28i	0.136820	+	0.430127i
$616$		0			0
$617$			9.54357e28i			0.480526i	0.970708	+	0.240263i	$0.0772336\pi$
$617$			9.54357e28i			0.480526i	−0.970708	+	0.240263i	$0.922766\pi$
$618$		0			0
$619$		−	2.02344e29i		−	0.984776i	−0.870376	−	0.492388i	$-0.836124\pi$
$619$		−	2.02344e29i		−	0.984776i	0.870376	−	0.492388i	$-0.163876\pi$
$620$		0			0
$621$	7.77289e28	+	5.87084e28i	0.365696	+	0.276210i
$622$		0			0
$623$	2.40137e28			0.109228
$624$		0			0
$625$	−2.77162e29			−1.21897
$626$		0			0
$627$	7.35818e28	−	2.34058e28i	0.312940	−	0.0995437i
$628$		0			0
$629$			6.00615e28i			0.247039i
$630$		0			0
$631$			1.11148e29i			0.442176i	0.975254	+	0.221088i	$0.0709608\pi$
$631$			1.11148e29i			0.442176i	−0.975254	+	0.221088i	$0.929039\pi$
$632$		0			0
$633$	−3.05156e29	+	9.70676e28i	−1.17431	+	0.373539i
$634$		0			0
$635$	−1.73943e29			−0.647565
$636$		0			0
$637$	−5.25069e29			−1.89127
$638$		0			0
$639$	−2.20374e29	+	1.55981e29i	−0.768071	+	0.543641i
$640$		0			0
$641$		−	1.71859e29i		−	0.579646i	−0.957080	−	0.289823i	$-0.906404\pi$
$641$		−	1.71859e29i		−	0.579646i	0.957080	−	0.289823i	$-0.0935964\pi$
$642$		0			0
$643$		−	4.05229e29i		−	1.32277i	−0.750045	−	0.661386i	$-0.769968\pi$
$643$		−	4.05229e29i		−	1.32277i	0.750045	−	0.661386i	$-0.230032\pi$
$644$		0			0
$645$	5.75150e28	+	1.80812e29i	0.181720	+	0.571283i
$646$		0			0
$647$	−2.72497e29			−0.833426			−0.416713	−	0.909038i	$-0.636818\pi$
$647$	−2.72497e29			−0.833426			−0.416713	+	0.909038i	$0.636818\pi$
$648$		0			0
$649$	6.20029e29			1.83587
$650$		0			0
$651$	−6.87039e27	−	2.15988e28i	−0.0196961	−	0.0619196i
$652$		0			0
$653$		−	5.82209e29i		−	1.61618i	−0.589056	−	0.808092i	$-0.700500\pi$
$653$		−	5.82209e29i		−	1.61618i	0.589056	−	0.808092i	$-0.299500\pi$
$654$		0			0
$655$			3.85453e29i			1.03618i
$656$		0			0
$657$	1.68276e29	−	1.19106e29i	0.438113	−	0.310097i
$658$		0			0
$659$	4.06748e29			1.02572			0.512860	−	0.858472i	$-0.328586\pi$
$659$	4.06748e29			1.02572			0.512860	+	0.858472i	$0.328586\pi$
$660$		0			0
$661$	5.30340e28			0.129551			0.0647753	−	0.997900i	$-0.479367\pi$
$661$	5.30340e28			0.129551			0.0647753	+	0.997900i	$0.479367\pi$
$662$		0			0
$663$	1.97483e29	−	6.28178e28i	0.467346	−	0.148659i
$664$		0			0
$665$		−	1.65584e28i		−	0.0379656i
$666$		0			0
$667$			3.10029e29i			0.688780i
$668$		0			0
$669$	3.03956e29	−	9.66860e28i	0.654390	−	0.208156i
$670$		0			0
$671$	−8.15658e29			−1.70185
$672$		0			0
$673$	6.99829e29			1.41525			0.707626	−	0.706588i	$-0.249766\pi$
$673$	6.99829e29			1.41525			0.707626	+	0.706588i	$0.249766\pi$
$674$		0			0
$675$	−2.75246e29	−	2.07892e29i	−0.539549	−	0.407520i
$676$		0			0
$677$			2.22239e29i			0.422318i	0.977452	+	0.211159i	$0.0677238\pi$
$677$			2.22239e29i			0.422318i	−0.977452	+	0.211159i	$0.932276\pi$
$678$		0			0
$679$			4.78409e28i			0.0881388i
$680$		0			0
$681$	−5.13306e28	−	1.61370e29i	−0.0916921	−	0.288257i
$682$		0			0
$683$	5.53816e29			0.959286			0.479643	−	0.877464i	$-0.340766\pi$
$683$	5.53816e29			0.959286			0.479643	+	0.877464i	$0.340766\pi$
$684$		0			0
$685$	5.16566e28			0.0867710
$686$		0			0
$687$	1.07339e29	+	3.37446e29i	0.174868	+	0.549741i
$688$		0			0
$689$			1.74257e30i			2.75352i
$690$		0			0
$691$		−	9.14133e29i		−	1.40117i	−0.713570	−	0.700583i	$-0.752923\pi$
$691$		−	9.14133e29i		−	1.40117i	0.713570	−	0.700583i	$-0.247077\pi$
$692$		0			0
$693$	8.26389e28	+	1.16754e29i	0.122881	+	0.173610i
$694$		0			0
$695$	−7.32004e29			−1.05602
$696$		0			0
$697$	−6.33579e28			−0.0886864
$698$		0			0
$699$	5.97355e29	−	1.90014e29i	0.811375	−	0.258092i
$700$		0			0
$701$			9.60460e29i			1.26602i	0.774144	+	0.633009i	$0.218180\pi$
$701$			9.60460e29i			1.26602i	−0.774144	+	0.633009i	$0.781820\pi$
$702$		0			0
$703$			1.61525e29i			0.206637i
$704$		0			0
$705$	1.68695e30	−	5.36605e29i	2.09467	−	0.666298i
$706$		0			0
$707$	−1.51740e29			−0.182893
$708$		0			0
$709$	4.96656e29			0.581126			0.290563	−	0.956856i	$-0.406157\pi$
$709$	4.96656e29			0.581126			0.290563	+	0.956856i	$0.406157\pi$
$710$		0			0
$711$	9.18465e29	+	1.29763e30i	1.04335	+	1.47408i
$712$		0			0
$713$		−	1.95899e29i		−	0.216069i
$714$		0			0
$715$		−	3.59676e30i		−	3.85210i
$716$		0			0
$717$	−7.79081e28	−	2.44923e29i	−0.0810273	−	0.254729i
$718$		0			0
$719$	1.73758e30			1.75506			0.877529	−	0.479524i	$-0.159191\pi$
$719$	1.73758e30			1.75506			0.877529	+	0.479524i	$0.159191\pi$
$720$		0			0
$721$	−1.48575e29			−0.145755
$722$		0			0
$723$	1.30420e29	+	4.10007e29i	0.124277	+	0.390696i
$724$		0			0
$725$		−	1.09784e30i		−	1.01623i
$726$		0			0
$727$			2.34382e29i			0.210772i	0.994431	+	0.105386i	$0.0336078\pi$
$727$			2.34382e29i			0.210772i	−0.994431	+	0.105386i	$0.966392\pi$
$728$		0			0
$729$	3.13038e29	+	1.10092e30i	0.273501	+	0.961872i
$730$		0			0
$731$	−1.38753e29			−0.117791
$732$		0			0
$733$	−7.87339e29			−0.649486			−0.324743	−	0.945802i	$-0.605278\pi$
$733$	−7.87339e29			−0.649486			−0.324743	+	0.945802i	$0.605278\pi$
$734$		0			0
$735$	−1.50979e30	+	4.80252e29i	−1.21032	+	0.384992i
$736$		0			0
$737$		−	1.40832e30i		−	1.09722i
$738$		0			0
$739$		−	1.66863e30i		−	1.26355i	−0.775150	−	0.631777i	$-0.782326\pi$
$739$		−	1.66863e30i		−	1.26355i	0.775150	−	0.631777i	$-0.217674\pi$
$740$		0			0
$741$	5.31096e29	−	1.68937e29i	0.390915	−	0.124347i
$742$		0			0
$743$	−1.22564e30			−0.876962			−0.438481	−	0.898740i	$-0.644483\pi$
$743$	−1.22564e30			−0.876962			−0.438481	+	0.898740i	$0.644483\pi$
$744$		0			0
$745$	−2.78796e29			−0.193931
$746$		0			0
$747$	1.61704e30	−	1.14454e30i	1.09359	−	0.774042i
$748$		0			0
$749$		−	3.06306e29i		−	0.201417i
$750$		0			0
$751$		−	6.25060e29i		−	0.399671i	−0.979829	−	0.199835i	$-0.935959\pi$
$751$		−	6.25060e29i		−	0.399671i	0.979829	−	0.199835i	$-0.0640408\pi$
$752$		0			0
$753$	3.09940e29	+	9.74372e29i	0.192722	+	0.605867i
$754$		0			0
$755$	−1.72550e30			−1.04345
$756$		0			0
$757$	2.38779e30			1.40440			0.702198	−	0.711982i	$-0.252202\pi$
$757$	2.38779e30			1.40440			0.702198	+	0.711982i	$0.252202\pi$
$758$		0			0
$759$	3.74764e29	+	1.17816e30i	0.214397	+	0.674010i
$760$		0			0
$761$			1.26950e28i			0.00706472i	0.999994	+	0.00353236i	$0.00112439\pi$
$761$			1.26950e28i			0.00706472i	−0.999994	+	0.00353236i	$0.998876\pi$
$762$		0			0
$763$			1.38583e28i			0.00750242i
$764$		0			0
$765$	5.10389e29	−	3.61254e29i	0.268817	−	0.190269i
$766$		0			0
$767$	4.47522e30			2.29331
$768$		0			0
$769$	−1.61066e30			−0.803116			−0.401558	−	0.915834i	$-0.631531\pi$
$769$	−1.61066e30			−0.803116			−0.401558	+	0.915834i	$0.631531\pi$
$770$		0			0
$771$	−3.41488e30	+	1.08625e30i	−1.65693	+	0.527057i
$772$		0			0
$773$			5.92505e29i			0.279774i	0.990167	+	0.139887i	$0.0446739\pi$
$773$			5.92505e29i			0.279774i	−0.990167	+	0.139887i	$0.955326\pi$
$774$		0			0
$775$			6.93697e29i			0.318788i
$776$		0			0
$777$	−2.85148e29	+	9.07034e28i	−0.127541	+	0.0405698i
$778$		0			0
$779$	−1.70390e29			−0.0741824
$780$		0			0
$781$	−3.42677e30			−1.45227
$782$		0			0
$783$	−2.19556e30	+	2.90688e30i	−0.905831	+	1.19930i
$784$		0			0
$785$		−	1.30058e30i		−	0.522404i
$786$		0			0
$787$			1.28957e30i			0.504323i	0.967685	+	0.252162i	$0.0811414\pi$
$787$			1.28957e30i			0.504323i	−0.967685	+	0.252162i	$0.918859\pi$
$788$		0			0
$789$	−2.61299e29	−	8.21458e29i	−0.0995015	−	0.312807i
$790$		0			0
$791$	−5.32836e29			−0.197579
$792$		0			0
$793$	−5.88723e30			−2.12590
$794$		0			0
$795$	1.59383e30	+	5.01060e30i	0.560515	+	1.76212i
$796$		0			0
$797$			5.80733e30i			1.98913i	0.104098	+	0.994567i	$0.466804\pi$
$797$			5.80733e30i			1.98913i	−0.104098	+	0.994567i	$0.533196\pi$
$798$		0			0
$799$			1.29454e30i			0.431893i
$800$		0			0
$801$	1.40895e30	+	1.99061e30i	0.457883	+	0.646910i
$802$		0			0
$803$	2.61666e30			0.828388
$804$		0			0
$805$	2.65126e29			0.0817703
$806$		0			0
$807$	−2.77289e30	+	8.82035e29i	−0.833222	+	0.265041i
$808$		0			0
$809$		−	6.48011e30i		−	1.89724i	−0.316412	−	0.948622i	$-0.602478\pi$
$809$		−	6.48011e30i		−	1.89724i	0.316412	−	0.948622i	$-0.397522\pi$
$810$		0			0
$811$		−	5.34568e30i		−	1.52505i	−0.646959	−	0.762525i	$-0.723960\pi$
$811$		−	5.34568e30i		−	1.52505i	0.646959	−	0.762525i	$-0.276040\pi$
$812$		0			0
$813$	1.23828e30	−	3.93886e29i	0.344245	−	0.109501i
$814$		0			0
$815$	−6.18370e30			−1.67530
$816$		0			0
$817$	−3.73152e29			−0.0985269
$818$		0			0
$819$	5.96468e29	+	8.42706e29i	0.153499	+	0.216868i
$820$		0			0
$821$			4.00066e30i			1.00353i	0.865005	+	0.501763i	$0.167315\pi$
$821$			4.00066e30i			1.00353i	−0.865005	+	0.501763i	$0.832685\pi$
$822$		0			0
$823$			5.40650e29i			0.132196i	0.997813	+	0.0660979i	$0.0210550\pi$
$823$			5.40650e29i			0.132196i	−0.997813	+	0.0660979i	$0.978945\pi$
$824$		0			0
$825$	−1.32707e30	−	4.17199e30i	−0.316322	−	0.994435i
$826$		0			0
$827$	3.75851e30			0.873389			0.436694	−	0.899610i	$-0.356149\pi$
$827$	3.75851e30			0.873389			0.436694	+	0.899610i	$0.356149\pi$
$828$		0			0
$829$	−6.13973e30			−1.39100			−0.695500	−	0.718526i	$-0.744817\pi$
$829$	−6.13973e30			−1.39100			−0.695500	+	0.718526i	$0.744817\pi$
$830$		0			0
$831$	1.81272e30	+	5.69874e30i	0.400424	+	1.25883i
$832$		0			0
$833$		−	1.15860e30i		−	0.249551i
$834$		0			0
$835$		−	6.46655e30i		−	1.35820i
$836$		0			0
$837$	1.38732e30	−	1.83678e30i	0.284157	−	0.376219i
$838$		0			0
$839$	−6.51398e30			−1.30120			−0.650602	−	0.759419i	$-0.725483\pi$
$839$	−6.51398e30			−1.30120			−0.650602	+	0.759419i	$0.725483\pi$
$840$		0			0
$841$	−6.46153e30			−1.25886
$842$		0			0
$843$	6.58526e30	−	2.09472e30i	1.25136	−	0.398048i
$844$		0			0
$845$		−	1.89758e31i		−	3.51725i
$846$		0			0
$847$			1.05329e30i			0.190446i
$848$		0			0
$849$	−5.02392e30	+	1.59807e30i	−0.886159	+	0.281880i
$850$		0			0
$851$	−2.58627e30			−0.445055
$852$		0			0
$853$	−3.48795e30			−0.585606			−0.292803	−	0.956173i	$-0.594588\pi$
$853$	−3.48795e30			−0.585606			−0.292803	+	0.956173i	$0.594588\pi$
$854$		0			0
$855$	1.37260e30	−	9.71529e29i	0.224854	−	0.159152i
$856$		0			0
$857$			2.16850e30i			0.346626i	0.984867	+	0.173313i	$0.0554472\pi$
$857$			2.16850e30i			0.346626i	−0.984867	+	0.173313i	$0.944553\pi$
$858$		0			0
$859$			9.79610e30i			1.52801i	0.645213	+	0.764003i	$0.276769\pi$
$859$			9.79610e30i			1.52801i	−0.645213	+	0.764003i	$0.723231\pi$
$860$		0			0
$861$	−9.56815e28	−	3.00798e29i	−0.0145645	−	0.0457870i
$862$		0			0
$863$	3.39459e30			0.504282			0.252141	−	0.967690i	$-0.418865\pi$
$863$	3.39459e30			0.504282			0.252141	+	0.967690i	$0.418865\pi$
$864$		0			0
$865$	7.78475e30			1.12869
$866$		0			0
$867$	−2.00341e30	−	6.29821e30i	−0.283510	−	0.891285i
$868$		0			0
$869$			2.01779e31i			2.78720i
$870$		0			0
$871$		−	1.01649e31i		−	1.37061i
$872$		0			0
$873$	−3.96576e30	+	2.80697e30i	−0.522008	+	0.369477i
$874$		0			0
$875$	4.49657e29			0.0577826
$876$		0			0
$877$	7.62526e30			0.956663			0.478332	−	0.878179i	$-0.341242\pi$
$877$	7.62526e30			0.956663			0.478332	+	0.878179i	$0.341242\pi$
$878$		0			0
$879$	1.02247e31	−	3.25240e30i	1.25247	−	0.398402i
$880$		0			0
$881$			8.22474e30i			0.983729i	0.870672	+	0.491864i	$0.163684\pi$
$881$			8.22474e30i			0.983729i	−0.870672	+	0.491864i	$0.836316\pi$
$882$		0			0
$883$		−	1.65510e30i		−	0.193302i	−0.995318	−	0.0966510i	$-0.969187\pi$
$883$		−	1.65510e30i		−	0.193302i	0.995318	−	0.0966510i	$-0.0308131\pi$
$884$		0			0
$885$	1.28681e31	−	4.09324e30i	1.46761	−	0.466835i
$886$		0			0
$887$	−9.11120e30			−1.01479			−0.507397	−	0.861713i	$-0.669392\pi$
$887$	−9.11120e30			−1.01479			−0.507397	+	0.861713i	$0.669392\pi$
$888$		0			0
$889$	6.33720e29			0.0689333
$890$		0			0
$891$	−4.82966e30	+	1.37007e31i	−0.513098	+	1.45554i
$892$		0			0
$893$			3.48145e30i			0.361260i
$894$		0			0
$895$			5.37841e29i			0.0545145i
$896$		0			0
$897$	2.70496e30	+	8.50370e30i	0.267818	+	0.841952i
$898$		0			0
$899$	7.32617e30			0.708599
$900$		0			0
$901$	−3.84508e30			−0.363325
$902$		0			0
$903$	−2.09542e29	−	6.58746e29i	−0.0193441	−	0.0608130i
$904$		0			0
$905$		−	6.46199e30i		−	0.582849i
$906$		0			0
$907$			6.08788e30i			0.536524i	0.963346	+	0.268262i	$0.0864494\pi$
$907$			6.08788e30i			0.536524i	−0.963346	+	0.268262i	$0.913551\pi$
$908$		0			0
$909$	−8.90306e30	−	1.25785e31i	−0.766687	−	1.08320i
$910$		0			0
$911$	2.25131e31			1.89449			0.947244	−	0.320515i	$-0.103856\pi$
$911$	2.25131e31			1.89449			0.947244	+	0.320515i	$0.103856\pi$
$912$		0			0
$913$	2.51446e31			2.06776
$914$		0			0
$915$	−1.69282e31	+	5.38472e30i	−1.36047	+	0.432755i
$916$		0			0
$917$		−	1.40430e30i		−	0.110302i
$918$		0			0
$919$		−	9.98793e30i		−	0.766766i	−0.923589	−	0.383383i	$-0.874759\pi$
$919$		−	9.98793e30i		−	0.766766i	0.923589	−	0.383383i	$-0.125241\pi$
$920$		0			0
$921$	2.12150e31	−	6.74831e30i	1.59190	−	0.506371i
$922$		0			0
$923$	−2.47336e31			−1.81413
$924$		0			0
$925$	9.15823e30			0.656634
$926$		0			0
$927$	−8.71731e30	−	1.23160e31i	−0.611005	−	0.863245i
$928$		0			0
$929$		−	3.09724e30i		−	0.212231i	−0.994354	−	0.106116i	$-0.966159\pi$
$929$		−	3.09724e30i		−	0.212231i	0.994354	−	0.106116i	$-0.0338414\pi$
$930$		0			0
$931$		−	3.11584e30i		−	0.208739i
$932$		0			0
$933$	8.10240e30	+	2.54719e31i	0.530709	+	1.66841i
$934$		0			0
$935$	7.93644e30			0.508281
$936$		0			0
$937$	2.72021e30			0.170348			0.0851738	−	0.996366i	$-0.472855\pi$
$937$	2.72021e30			0.170348			0.0851738	+	0.996366i	$0.472855\pi$
$938$		0			0
$939$	−8.08383e30	−	2.54135e31i	−0.495027	−	1.55624i
$940$		0			0
$941$		−	2.37293e30i		−	0.142100i	−0.997473	−	0.0710500i	$-0.977365\pi$
$941$		−	2.37293e30i		−	0.142100i	0.997473	−	0.0710500i	$-0.0226350\pi$
$942$		0			0
$943$		−	2.72822e30i		−	0.159774i
$944$		0			0
$945$	2.48587e30	+	1.87757e30i	0.142378	+	0.107538i
$946$		0			0
$947$	7.97325e30			0.446643			0.223322	−	0.974745i	$-0.428310\pi$
$947$	7.97325e30			0.446643			0.223322	+	0.974745i	$0.428310\pi$
$948$		0			0
$949$	1.88864e31			1.03480
$950$		0			0
$951$	1.62070e31	−	5.15530e30i	0.868572	−	0.276286i
$952$		0			0
$953$			2.54500e30i			0.133417i	0.997772	+	0.0667086i	$0.0212498\pi$
$953$			2.54500e30i			0.133417i	−0.997772	+	0.0667086i	$0.978750\pi$
$954$		0			0
$955$			9.40522e30i			0.482318i
$956$		0			0
$957$	−4.40606e31	+	1.40153e31i	−2.21042	+	0.703117i
$958$		0			0
$959$	−1.88198e29			−0.00923676
$960$		0			0
$961$	1.61963e31			0.777714
$962$		0			0
$963$	2.53912e31	−	1.79719e31i	1.19291	−	0.844341i
$964$		0			0
$965$		−	4.11815e31i		−	1.89307i
$966$		0			0
$967$			2.23490e31i			1.00527i	0.864500	+	0.502633i	$0.167635\pi$
$967$			2.23490e31i			1.00527i	−0.864500	+	0.502633i	$0.832365\pi$
$968$		0			0
$969$	3.72770e29	+	1.17189e30i	0.0164075	+	0.0515809i
$970$		0			0
$971$	3.69855e31			1.59305			0.796524	−	0.604607i	$-0.206670\pi$
$971$	3.69855e31			1.59305			0.796524	+	0.604607i	$0.206670\pi$
$972$		0			0
$973$	2.66688e30			0.112413
$974$		0			0
$975$	−9.57852e30	−	3.01124e31i	−0.395139	−	1.24222i
$976$		0			0
$977$			4.23844e31i			1.71125i	0.517597	+	0.855624i	$0.326827\pi$
$977$			4.23844e31i			1.71125i	−0.517597	+	0.855624i	$0.673173\pi$
$978$		0			0
$979$			3.09535e31i			1.22318i
$980$		0			0
$981$	−1.14878e30	+	8.13105e29i	−0.0444336	+	0.0314501i
$982$		0			0
$983$	2.57484e31			0.974851			0.487425	−	0.873165i	$-0.337936\pi$
$983$	2.57484e31			0.974851			0.487425	+	0.873165i	$0.337936\pi$
$984$		0			0
$985$	−3.39918e31			−1.25978
$986$		0			0
$987$	−6.14598e30	+	1.95499e30i	−0.222978	+	0.0709274i
$988$		0			0
$989$		−	5.97477e30i		−	0.212207i
$990$		0			0
$991$		−	3.23067e31i		−	1.12336i	−0.827355	−	0.561680i	$-0.810155\pi$
$991$		−	3.23067e31i		−	1.12336i	0.827355	−	0.561680i	$-0.189845\pi$
$992$		0			0
$993$	−1.18925e31	+	3.78292e30i	−0.404862	+	0.128783i
$994$		0			0
$995$	−5.61279e31			−1.87084
$996$		0			0
$997$	4.41308e30			0.144027			0.0720133	−	0.997404i	$-0.477058\pi$
$997$	4.41308e30			0.144027			0.0720133	+	0.997404i	$0.477058\pi$
$998$		0			0
$999$	−2.42493e31	−	1.83155e31i	−0.774929	−	0.585302i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.22.c.c.47.11	✓	28
3.2	odd	2	inner	48.22.c.c.47.17	yes	28
4.3	odd	2	inner	48.22.c.c.47.18	yes	28
12.11	even	2	inner	48.22.c.c.47.12	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.22.c.c.47.11	✓	28	1.1	even	1	trivial
48.22.c.c.47.12	yes	28	12.11	even	2	inner
48.22.c.c.47.17	yes	28	3.2	odd	2	inner
48.22.c.c.47.18	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)

\(f(q)\)	\(=\)	\(q+(-31002.5 - 97463.8i) q^{3} -2.82711e7i q^{5} +1.02999e8i q^{7} +(-8.53805e9 + 6.04324e9i) q^{9} +O(q^{10})\)	\(q+(-31002.5 - 97463.8i) q^{3} -2.82711e7i q^{5} +1.02999e8i q^{7} +(-8.53805e9 + 6.04324e9i) q^{9} -1.32765e11 q^{11} -9.58265e11 q^{13} +(-2.75541e12 + 8.76472e11i) q^{15} -2.11447e12i q^{17} -5.68648e12i q^{19} +(1.00386e13 - 3.19321e12i) q^{21} +9.10497e13 q^{23} -3.22416e14 q^{25} +(8.53698e14 + 6.44796e14i) q^{27} +3.40505e15i q^{29} -2.15156e15i q^{31} +(4.11604e15 + 1.29398e16i) q^{33} +2.91188e15 q^{35} -2.84050e16 q^{37} +(2.97086e16 + 9.33962e16i) q^{39} -2.99640e16i q^{41} -6.56209e16i q^{43} +(1.70849e17 + 2.41380e17i) q^{45} -6.12232e17 q^{47} +5.47937e17 q^{49} +(-2.06084e17 + 6.55537e16i) q^{51} -1.81846e18i q^{53} +3.75340e18i q^{55} +(-5.54227e17 + 1.76295e17i) q^{57} -4.67013e18 q^{59} +6.14363e18 q^{61} +(-6.22446e17 - 8.79408e17i) q^{63} +2.70912e19i q^{65} +1.06076e19i q^{67} +(-2.82276e18 - 8.87405e18i) q^{69} +2.58108e19 q^{71} -1.97090e19 q^{73} +(9.99568e18 + 3.14239e19i) q^{75} -1.36746e19i q^{77} -1.51982e20i q^{79} +(3.63775e19 - 1.03195e20i) q^{81} -1.89392e20 q^{83} -5.97782e19 q^{85} +(3.31869e20 - 1.05565e20i) q^{87} -2.33145e20i q^{89} -9.87001e19i q^{91} +(-2.09699e20 + 6.67037e19i) q^{93} -1.60763e20 q^{95} +4.64481e20 q^{97} +(1.13355e21 - 8.02329e20i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 109254828 q^{9}+O(q^{10}) \) 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

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