LMFDB - Embedded newform 48.16.c.b.47.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,16,Mod(47,48)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	48.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$68.4928824480$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 32461x^{2} + 32462x + 263623935 $ x^4 - 2x^3 - 32461x^2 + 32462*x + 263623935
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.1
Root			$127.913 + 1.65831i$ of defining polynomial
Character	$\chi$	$=$	48.47
Dual form			48.16.c.b.47.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2293.43 - 3014.81i) q^{3} +76064.4i q^{5} -1.82913e6i q^{7} +(-3.82928e6 + 1.38285e7i) q^{9} +O(q^{10})$	\(q+(-2293.43 - 3014.81i) q^{3} +76064.4i q^{5} -1.82913e6i q^{7} +(-3.82928e6 + 1.38285e7i) q^{9} -7.64125e7 q^{11} +2.54445e8 q^{13} +(2.29320e8 - 1.74448e8i) q^{15} +7.38677e8i q^{17} -5.32265e8i q^{19} +(-5.51447e9 + 4.19497e9i) q^{21} -1.70562e10 q^{23} +2.47318e10 q^{25} +(5.04725e10 - 2.01702e10i) q^{27} +1.64361e11i q^{29} +1.19565e11i q^{31} +(1.75247e11 + 2.30369e11i) q^{33} +1.39131e11 q^{35} -1.78908e11 q^{37} +(-5.83552e11 - 7.67105e11i) q^{39} -5.29655e11i q^{41} +5.71398e11i q^{43} +(-1.05186e12 - 2.91272e11i) q^{45} +1.37657e12 q^{47} +1.40186e12 q^{49} +(2.22697e12 - 1.69410e12i) q^{51} -1.30462e13i q^{53} -5.81227e12i q^{55} +(-1.60468e12 + 1.22071e12i) q^{57} +6.31592e11 q^{59} -1.44698e13 q^{61} +(2.52941e13 + 7.00422e12i) q^{63} +1.93542e13i q^{65} -8.74692e13i q^{67} +(3.91172e13 + 5.14212e13i) q^{69} +8.26841e13 q^{71} +6.67870e13 q^{73} +(-5.67206e13 - 7.45617e13i) q^{75} +1.39768e14i q^{77} -1.93474e14i q^{79} +(-1.76564e14 - 1.05906e14i) q^{81} +4.13488e14 q^{83} -5.61870e13 q^{85} +(4.95518e14 - 3.76951e14i) q^{87} -5.61280e14i q^{89} -4.65412e14i q^{91} +(3.60466e14 - 2.74214e14i) q^{93} +4.04865e13 q^{95} -3.75171e14 q^{97} +(2.92604e14 - 1.05667e15i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 15317100 q^{9}+O(q^{10}) $ 4 * q - 15317100 * q^9	$ 4 q - 15317100 q^{9} + 1017781336 q^{13} - 22057873992 q^{21} + 98927122100 q^{25} + 700986197952 q^{33} - 715633821064 q^{37} - 4207432014720 q^{45} + 5607449975596 q^{49} - 6418720383768 q^{57} - 57879101335528 q^{61} + 156468753896832 q^{69} + 267147849446632 q^{73} - 706257752173596 q^{81} - 224748150612480 q^{85} + 14\!\cdots\!64 q^{93}+ \cdots - 15\!\cdots\!20 q^{97}+O(q^{100}) $ 4 * q - 15317100 * q^9 + 1017781336 * q^13 - 22057873992 * q^21 + 98927122100 * q^25 + 700986197952 * q^33 - 715633821064 * q^37 - 4207432014720 * q^45 + 5607449975596 * q^49 - 6418720383768 * q^57 - 57879101335528 * q^61 + 156468753896832 * q^69 + 267147849446632 * q^73 - 706257752173596 * q^81 - 224748150612480 * q^85 + 1441865040435864 * q^93 - 1500683175752120 * 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
$3$	−2293.43	−	3014.81i	−0.605447	−	0.795886i
$3$	−2293.43	−	3014.81i	−0.605447	−	0.795886i
$4$		0			0
$5$			76064.4i			0.435418i	0.976014	+	0.217709i	$0.0698584\pi$
$5$			76064.4i			0.435418i	−0.976014	+	0.217709i	$0.930142\pi$
$6$		0			0
$7$		−	1.82913e6i		−	0.839476i	−0.907645	−	0.419738i	$-0.862122\pi$
$7$		−	1.82913e6i		−	0.839476i	0.907645	−	0.419738i	$-0.137878\pi$
$8$		0			0
$9$	−3.82928e6	+	1.38285e7i	−0.266869	+	0.963733i
$10$		0			0
$11$	−7.64125e7			−1.18228			−0.591138	−	0.806570i	$-0.701321\pi$
$11$	−7.64125e7			−1.18228			−0.591138	+	0.806570i	$0.701321\pi$
$12$		0			0
$13$	2.54445e8			1.12466			0.562328	−	0.826915i	$-0.309906\pi$
$13$	2.54445e8			1.12466			0.562328	+	0.826915i	$0.309906\pi$
$14$		0			0
$15$	2.29320e8	−	1.74448e8i	0.346543	−	0.263622i
$16$		0			0
$17$			7.38677e8i			0.436604i	0.975881	+	0.218302i	$0.0700518\pi$
$17$			7.38677e8i			0.436604i	−0.975881	+	0.218302i	$0.929948\pi$
$18$		0			0
$19$		−	5.32265e8i		−	0.136608i	−0.997665	−	0.0683040i	$-0.978241\pi$
$19$		−	5.32265e8i		−	0.136608i	0.997665	−	0.0683040i	$-0.0217588\pi$
$20$		0			0
$21$	−5.51447e9	+	4.19497e9i	−0.668127	+	0.508258i
$22$		0			0
$23$	−1.70562e10			−1.04454			−0.522268	−	0.852781i	$-0.674914\pi$
$23$	−1.70562e10			−1.04454			−0.522268	+	0.852781i	$0.674914\pi$
$24$		0			0
$25$	2.47318e10			0.810411
$26$		0			0
$27$	5.04725e10	−	2.01702e10i	0.928596	−	0.371092i
$28$		0			0
$29$			1.64361e11i			1.76935i	0.466206	+	0.884676i	$0.345620\pi$
$29$			1.64361e11i			1.76935i	−0.466206	+	0.884676i	$0.654380\pi$
$30$		0			0
$31$			1.19565e11i			0.780534i	0.920702	+	0.390267i	$0.127617\pi$
$31$			1.19565e11i			0.780534i	−0.920702	+	0.390267i	$0.872383\pi$
$32$		0			0
$33$	1.75247e11	+	2.30369e11i	0.715805	+	0.940957i
$34$		0			0
$35$	1.39131e11			0.365523
$36$		0			0
$37$	−1.78908e11			−0.309826			−0.154913	−	0.987928i	$-0.549510\pi$
$37$	−1.78908e11			−0.309826			−0.154913	+	0.987928i	$0.549510\pi$
$38$		0			0
$39$	−5.83552e11	−	7.67105e11i	−0.680919	−	0.895097i
$40$		0			0
$41$		−	5.29655e11i		−	0.424731i	−0.977190	−	0.212365i	$-0.931883\pi$
$41$		−	5.29655e11i		−	0.424731i	0.977190	−	0.212365i	$-0.0681167\pi$
$42$		0			0
$43$			5.71398e11i			0.320572i	0.987071	+	0.160286i	$0.0512416\pi$
$43$			5.71398e11i			0.320572i	−0.987071	+	0.160286i	$0.948758\pi$
$44$		0			0
$45$	−1.05186e12	−	2.91272e11i	−0.419627	−	0.116200i
$46$		0			0
$47$	1.37657e12			0.396337			0.198169	−	0.980168i	$-0.436501\pi$
$47$	1.37657e12			0.396337			0.198169	+	0.980168i	$0.436501\pi$
$48$		0			0
$49$	1.40186e12			0.295281
$50$		0			0
$51$	2.22697e12	−	1.69410e12i	0.347487	−	0.264340i
$52$		0			0
$53$		−	1.30462e13i		−	1.52551i	−0.646685	−	0.762757i	$-0.723845\pi$
$53$		−	1.30462e13i		−	1.52551i	0.646685	−	0.762757i	$-0.276155\pi$
$54$		0			0
$55$		−	5.81227e12i		−	0.514785i
$56$		0			0
$57$	−1.60468e12	+	1.22071e12i	−0.108724	+	0.0827089i
$58$		0			0
$59$	6.31592e11			0.0330405			0.0165202	−	0.999864i	$-0.494741\pi$
$59$	6.31592e11			0.0330405			0.0165202	+	0.999864i	$0.494741\pi$
$60$		0			0
$61$	−1.44698e13			−0.589506			−0.294753	−	0.955574i	$-0.595237\pi$
$61$	−1.44698e13			−0.589506			−0.294753	+	0.955574i	$0.595237\pi$
$62$		0			0
$63$	2.52941e13	+	7.00422e12i	0.809030	+	0.224030i
$64$		0			0
$65$			1.93542e13i			0.489695i
$66$		0			0
$67$		−	8.74692e13i		−	1.76317i	−0.472025	−	0.881585i	$-0.656477\pi$
$67$		−	8.74692e13i		−	1.76317i	0.472025	−	0.881585i	$-0.343523\pi$
$68$		0			0
$69$	3.91172e13	+	5.14212e13i	0.632411	+	0.831332i
$70$		0			0
$71$	8.26841e13			1.07891			0.539454	−	0.842015i	$-0.318631\pi$
$71$	8.26841e13			1.07891			0.539454	+	0.842015i	$0.318631\pi$
$72$		0			0
$73$	6.67870e13			0.707571			0.353786	−	0.935327i	$-0.384894\pi$
$73$	6.67870e13			0.707571			0.353786	+	0.935327i	$0.384894\pi$
$74$		0			0
$75$	−5.67206e13	−	7.45617e13i	−0.490661	−	0.644995i
$76$		0			0
$77$			1.39768e14i			0.992492i
$78$		0			0
$79$		−	1.93474e14i		−	1.13349i	−0.823892	−	0.566747i	$-0.808202\pi$
$79$		−	1.93474e14i		−	1.13349i	0.823892	−	0.566747i	$-0.191798\pi$
$80$		0			0
$81$	−1.76564e14	−	1.05906e14i	−0.857562	−	0.514380i
$82$		0			0
$83$	4.13488e14			1.67254			0.836271	−	0.548316i	$-0.184731\pi$
$83$	4.13488e14			1.67254			0.836271	+	0.548316i	$0.184731\pi$
$84$		0			0
$85$	−5.61870e13			−0.190105
$86$		0			0
$87$	4.95518e14	−	3.76951e14i	1.40820	−	1.07125i
$88$		0			0
$89$		−	5.61280e14i		−	1.34510i	−0.740051	−	0.672550i	$-0.765199\pi$
$89$		−	5.61280e14i		−	1.34510i	0.740051	−	0.672550i	$-0.234801\pi$
$90$		0			0
$91$		−	4.65412e14i		−	0.944121i
$92$		0			0
$93$	3.60466e14	−	2.74214e14i	0.621216	−	0.472572i
$94$		0			0
$95$	4.04865e13			0.0594816
$96$		0			0
$97$	−3.75171e14			−0.471456			−0.235728	−	0.971819i	$-0.575747\pi$
$97$	−3.75171e14			−0.471456			−0.235728	+	0.971819i	$0.575747\pi$
$98$		0			0
$99$	2.92604e14	−	1.05667e15i	0.315513	−	1.13940i
$100$		0			0
$101$			8.90919e14i			0.826852i	0.910538	+	0.413426i	$0.135668\pi$
$101$			8.90919e14i			0.826852i	−0.910538	+	0.413426i	$0.864332\pi$
$102$		0			0
$103$		−	1.87003e15i		−	1.49820i	−0.662458	−	0.749099i	$-0.730487\pi$
$103$		−	1.87003e15i		−	1.49820i	0.662458	−	0.749099i	$-0.269513\pi$
$104$		0			0
$105$	−3.19088e14	−	4.19455e14i	−0.221305	−	0.290915i
$106$		0			0
$107$	2.30400e15			1.38709			0.693544	−	0.720414i	$-0.256048\pi$
$107$	2.30400e15			1.38709			0.693544	+	0.720414i	$0.256048\pi$
$108$		0			0
$109$	8.68707e14			0.455171			0.227585	−	0.973758i	$-0.426917\pi$
$109$	8.68707e14			0.455171			0.227585	+	0.973758i	$0.426917\pi$
$110$		0			0
$111$	4.10314e14	+	5.39375e14i	0.187583	+	0.246586i
$112$		0			0
$113$		−	1.92630e14i		−	0.0770259i	−0.999258	−	0.0385129i	$-0.987738\pi$
$113$		−	1.92630e14i		−	0.0770259i	0.999258	−	0.0385129i	$-0.0122621\pi$
$114$		0			0
$115$		−	1.29737e15i		−	0.454810i
$116$		0			0
$117$	−9.74341e14	+	3.51860e15i	−0.300135	+	1.08387i
$118$		0			0
$119$	1.35113e15			0.366518
$120$		0			0
$121$	1.66162e15			0.397778
$122$		0			0
$123$	−1.59681e15	+	1.21473e15i	−0.338037	+	0.257152i
$124$		0			0
$125$			4.20251e15i			0.788286i
$126$		0			0
$127$		−	1.64302e15i		−	0.273599i	−0.990599	−	0.136799i	$-0.956318\pi$
$127$		−	1.64302e15i		−	0.273599i	0.990599	−	0.136799i	$-0.0436816\pi$
$128$		0			0
$129$	1.72266e15	−	1.31046e15i	0.255138	−	0.194089i
$130$		0			0
$131$	−5.77321e15			−0.761873			−0.380937	−	0.924601i	$-0.624398\pi$
$131$	−5.77321e15			−0.761873			−0.380937	+	0.924601i	$0.624398\pi$
$132$		0			0
$133$	−9.73580e14			−0.114679
$134$		0			0
$135$	1.53423e15	+	3.83916e15i	0.161580	+	0.404328i
$136$		0			0
$137$		−	1.98302e16i		−	1.87035i	−0.354185	−	0.935175i	$-0.615242\pi$
$137$		−	1.98302e16i		−	1.87035i	0.354185	−	0.935175i	$-0.384758\pi$
$138$		0			0
$139$			1.91399e16i			1.61930i	0.586912	+	0.809651i	$0.300343\pi$
$139$			1.91399e16i			1.61930i	−0.586912	+	0.809651i	$0.699657\pi$
$140$		0			0
$141$	−3.15707e15	−	4.15010e15i	−0.239961	−	0.315439i
$142$		0			0
$143$	−1.94428e16			−1.32965
$144$		0			0
$145$	−1.25020e16			−0.770408
$146$		0			0
$147$	−3.21507e15	−	4.22635e15i	−0.178777	−	0.235010i
$148$		0			0
$149$			6.29842e15i			0.316472i	0.987401	+	0.158236i	$0.0505806\pi$
$149$			6.29842e15i			0.316472i	−0.987401	+	0.158236i	$0.949419\pi$
$150$		0			0
$151$			2.59510e16i			1.17985i	0.807458	+	0.589925i	$0.200843\pi$
$151$			2.59510e16i			1.17985i	−0.807458	+	0.589925i	$0.799157\pi$
$152$		0			0
$153$	−1.02148e16	−	2.82860e15i	−0.420770	−	0.116516i
$154$		0			0
$155$	−9.09465e15			−0.339859
$156$		0			0
$157$	4.78146e16			1.62298			0.811490	−	0.584366i	$-0.198657\pi$
$157$	4.78146e16			1.62298			0.811490	+	0.584366i	$0.198657\pi$
$158$		0			0
$159$	−3.93319e16	+	2.99206e16i	−1.21413	+	0.923617i
$160$		0			0
$161$			3.11979e16i			0.876863i
$162$		0			0
$163$			5.01923e16i			1.28597i	0.765880	+	0.642984i	$0.222304\pi$
$163$			5.01923e16i			1.28597i	−0.765880	+	0.642984i	$0.777696\pi$
$164$		0			0
$165$	−1.75229e16	+	1.33300e16i	−0.409710	+	0.311675i
$166$		0			0
$167$	2.24657e16			0.479896			0.239948	−	0.970786i	$-0.422870\pi$
$167$	2.24657e16			0.479896			0.239948	+	0.970786i	$0.422870\pi$
$168$		0			0
$169$	1.35565e16			0.264849
$170$		0			0
$171$	7.36044e15	+	2.03819e15i	0.131654	+	0.0364564i
$172$		0			0
$173$			3.73031e16i			0.611504i	0.952111	+	0.305752i	$0.0989079\pi$
$173$			3.73031e16i			0.611504i	−0.952111	+	0.305752i	$0.901092\pi$
$174$		0			0
$175$		−	4.52375e16i		−	0.680320i
$176$		0			0
$177$	−1.44851e15	−	1.90413e15i	−0.0200043	−	0.0262965i
$178$		0			0
$179$	−1.37575e17			−1.74639			−0.873197	−	0.487367i	$-0.837958\pi$
$179$	−1.37575e17			−1.74639			−0.873197	+	0.487367i	$0.837958\pi$
$180$		0			0
$181$	−6.36349e16			−0.743201			−0.371600	−	0.928393i	$-0.621191\pi$
$181$	−6.36349e16			−0.743201			−0.371600	+	0.928393i	$0.621191\pi$
$182$		0			0
$183$	3.31854e16	+	4.36237e16i	0.356914	+	0.469179i
$184$		0			0
$185$		−	1.36086e16i		−	0.134904i
$186$		0			0
$187$		−	5.64441e16i		−	0.516187i
$188$		0			0
$189$	−3.68938e16	−	9.23206e16i	−0.311523	−	0.779534i
$190$		0			0
$191$	1.30203e17			1.01595			0.507973	−	0.861373i	$-0.330395\pi$
$191$	1.30203e17			1.01595			0.507973	+	0.861373i	$0.330395\pi$
$192$		0			0
$193$	2.33515e17			1.68513			0.842566	−	0.538593i	$-0.181044\pi$
$193$	2.33515e17			1.68513			0.842566	+	0.538593i	$0.181044\pi$
$194$		0			0
$195$	5.83494e16	−	4.43876e16i	0.389742	−	0.296484i
$196$		0			0
$197$		−	2.42634e17i		−	1.50126i	−0.660724	−	0.750629i	$-0.729751\pi$
$197$		−	2.42634e17i		−	1.50126i	0.660724	−	0.750629i	$-0.270249\pi$
$198$		0			0
$199$		−	1.14963e17i		−	0.659419i	−0.944082	−	0.329710i	$-0.893049\pi$
$199$		−	1.14963e17i		−	0.659419i	0.944082	−	0.329710i	$-0.106951\pi$
$200$		0			0
$201$	−2.63703e17	+	2.00604e17i	−1.40328	+	1.06751i
$202$		0			0
$203$	3.00637e17			1.48533
$204$		0			0
$205$	4.02879e16			0.184936
$206$		0			0
$207$	6.53129e16	−	2.35862e17i	0.278754	−	1.00665i
$208$		0			0
$209$			4.06717e16i			0.161508i
$210$		0			0
$211$		−	1.14655e17i		−	0.423913i	−0.977279	−	0.211956i	$-0.932017\pi$
$211$		−	1.14655e17i		−	0.423913i	0.977279	−	0.211956i	$-0.0679834\pi$
$212$		0			0
$213$	−1.89630e17	−	2.49277e17i	−0.653221	−	0.858688i
$214$		0			0
$215$	−4.34630e16			−0.139583
$216$		0			0
$217$	2.18700e17			0.655239
$218$		0			0
$219$	−1.53171e17	−	2.01350e17i	−0.428397	−	0.563146i
$220$		0			0
$221$			1.87953e17i			0.491029i
$222$		0			0
$223$		−	4.46157e17i		−	1.08943i	−0.838620	−	0.544717i	$-0.816637\pi$
$223$		−	4.46157e17i		−	1.08943i	0.838620	−	0.544717i	$-0.183363\pi$
$224$		0			0
$225$	−9.47048e16	+	3.42004e17i	−0.216273	+	0.781020i
$226$		0			0
$227$	5.24609e17			1.12109			0.560547	−	0.828123i	$-0.310591\pi$
$227$	5.24609e17			1.12109			0.560547	+	0.828123i	$0.310591\pi$
$228$		0			0
$229$	6.48369e17			1.29735			0.648674	−	0.761067i	$-0.275324\pi$
$229$	6.48369e17			1.29735			0.648674	+	0.761067i	$0.275324\pi$
$230$		0			0
$231$	4.21374e17	−	3.20548e17i	0.789911	−	0.600901i
$232$		0			0
$233$			4.01531e17i			0.705585i	0.935702	+	0.352793i	$0.114768\pi$
$233$			4.01531e17i			0.705585i	−0.935702	+	0.352793i	$0.885232\pi$
$234$		0			0
$235$			1.04708e17i			0.172572i
$236$		0			0
$237$	−5.83288e17	+	4.43719e17i	−0.902132	+	0.686271i
$238$		0			0
$239$	−1.21318e17			−0.176173			−0.0880867	−	0.996113i	$-0.528075\pi$
$239$	−1.21318e17			−0.176173			−0.0880867	+	0.996113i	$0.528075\pi$
$240$		0			0
$241$	3.65127e17			0.498100			0.249050	−	0.968491i	$-0.419882\pi$
$241$	3.65127e17			0.498100			0.249050	+	0.968491i	$0.419882\pi$
$242$		0			0
$243$	8.56502e16	+	7.75197e17i	0.109820	+	0.993951i
$244$		0			0
$245$			1.06632e17i			0.128571i
$246$		0			0
$247$		−	1.35432e17i		−	0.153637i
$248$		0			0
$249$	−9.48305e17	−	1.24659e18i	−1.01263	−	1.33115i
$250$		0			0
$251$	−1.39376e18			−1.40164			−0.700819	−	0.713339i	$-0.747182\pi$
$251$	−1.39376e18			−1.40164			−0.700819	+	0.713339i	$0.747182\pi$
$252$		0			0
$253$	1.30331e18			1.23493
$254$		0			0
$255$	1.28861e17	+	1.69393e17i	0.115099	+	0.151302i
$256$		0			0
$257$		−	1.58617e18i		−	1.33614i	−0.744099	−	0.668070i	$-0.767121\pi$
$257$		−	1.58617e18i		−	1.33614i	0.744099	−	0.668070i	$-0.232879\pi$
$258$		0			0
$259$			3.27246e17i			0.260091i
$260$		0			0
$261$	−2.27287e18	−	6.29384e17i	−1.70518	−	0.472185i
$262$		0			0
$263$	1.02896e18			0.729001			0.364501	−	0.931203i	$-0.381240\pi$
$263$	1.02896e18			0.729001			0.364501	+	0.931203i	$0.381240\pi$
$264$		0			0
$265$	9.92354e17			0.664236
$266$		0			0
$267$	−1.69215e18	+	1.28726e18i	−1.07055	+	0.814387i
$268$		0			0
$269$			9.01288e17i			0.539165i	0.962977	+	0.269582i	$0.0868857\pi$
$269$			9.01288e17i			0.539165i	−0.962977	+	0.269582i	$0.913114\pi$
$270$		0			0
$271$			2.19619e18i			1.24280i	0.783494	+	0.621399i	$0.213435\pi$
$271$			2.19619e18i			1.24280i	−0.783494	+	0.621399i	$0.786565\pi$
$272$		0			0
$273$	−1.40313e18	+	1.06739e18i	−0.751412	+	0.571615i
$274$		0			0
$275$	−1.88982e18			−0.958130
$276$		0			0
$277$	2.49263e18			1.19690			0.598451	−	0.801159i	$-0.295783\pi$
$277$	2.49263e18			1.19690			0.598451	+	0.801159i	$0.295783\pi$
$278$		0			0
$279$	−1.65341e18	−	4.57848e17i	−0.752226	−	0.208300i
$280$		0			0
$281$		−	1.55724e17i		−	0.0671520i	−0.999436	−	0.0335760i	$-0.989310\pi$
$281$		−	1.55724e17i		−	0.0671520i	0.999436	−	0.0335760i	$-0.0106896\pi$
$282$		0			0
$283$			2.54289e18i			1.03975i	0.854242	+	0.519875i	$0.174022\pi$
$283$			2.54289e18i			1.03975i	−0.854242	+	0.519875i	$0.825978\pi$
$284$		0			0
$285$	−9.28528e16	−	1.22059e17i	−0.0360129	−	0.0473406i
$286$		0			0
$287$	−9.68805e17			−0.356551
$288$		0			0
$289$	2.31678e18			0.809377
$290$		0			0
$291$	8.60428e17	+	1.13107e18i	0.285441	+	0.375225i
$292$		0			0
$293$			2.17295e18i			0.684765i	0.939561	+	0.342383i	$0.111234\pi$
$293$			2.17295e18i			0.684765i	−0.939561	+	0.342383i	$0.888766\pi$
$294$		0			0
$295$			4.80417e16i			0.0143864i
$296$		0			0
$297$	−3.85673e18	+	1.54125e18i	−1.09786	+	0.438733i
$298$		0			0
$299$	−4.33987e18			−1.17474
$300$		0			0
$301$	1.04516e18			0.269112
$302$		0			0
$303$	2.68595e18	−	2.04326e18i	0.658080	−	0.500615i
$304$		0			0
$305$		−	1.10064e18i		−	0.256682i
$306$		0			0
$307$			7.66257e18i			1.70152i	0.525555	+	0.850759i	$0.323858\pi$
$307$			7.66257e18i			1.70152i	−0.525555	+	0.850759i	$0.676142\pi$
$308$		0			0
$309$	−5.63779e18	+	4.28878e18i	−1.19240	+	0.907079i
$310$		0			0
$311$	4.84682e18			0.976685			0.488342	−	0.872652i	$-0.337602\pi$
$311$	4.84682e18			0.976685			0.488342	+	0.872652i	$0.337602\pi$
$312$		0			0
$313$	8.77812e18			1.68585			0.842926	−	0.538030i	$-0.180831\pi$
$313$	8.77812e18			1.68585			0.842926	+	0.538030i	$0.180831\pi$
$314$		0			0
$315$	−5.32772e17	+	1.92398e18i	−0.0975467	+	0.352267i
$316$		0			0
$317$		−	6.53956e17i		−	0.114184i	−0.998369	−	0.0570918i	$-0.981817\pi$
$317$		−	6.53956e17i		−	0.114184i	0.998369	−	0.0570918i	$-0.0181828\pi$
$318$		0			0
$319$		−	1.25592e19i		−	2.09186i
$320$		0			0
$321$	−5.28406e18	−	6.94612e18i	−0.839808	−	1.10396i
$322$		0			0
$323$	3.93172e17			0.0596436
$324$		0			0
$325$	6.29289e18			0.911433
$326$		0			0
$327$	−1.99232e18	−	2.61899e18i	−0.275582	−	0.362264i
$328$		0			0
$329$		−	2.51792e18i		−	0.332715i
$330$		0			0
$331$		−	7.41780e18i		−	0.936623i	−0.883563	−	0.468312i	$-0.844862\pi$
$331$		−	7.41780e18i		−	0.936623i	0.883563	−	0.468312i	$-0.155138\pi$
$332$		0			0
$333$	6.85090e17	−	2.47404e18i	0.0826829	−	0.298590i
$334$		0			0
$335$	6.65330e18			0.767716
$336$		0			0
$337$	−7.56506e18			−0.834811			−0.417405	−	0.908720i	$-0.637060\pi$
$337$	−7.56506e18			−0.834811			−0.417405	+	0.908720i	$0.637060\pi$
$338$		0			0
$339$	−5.80744e17	+	4.41784e17i	−0.0613038	+	0.0466350i
$340$		0			0
$341$		−	9.13626e18i		−	0.922807i
$342$		0			0
$343$		−	1.12481e19i		−	1.08736i
$344$		0			0
$345$	−3.91133e18	+	2.97543e18i	−0.361977	+	0.275363i
$346$		0			0
$347$	5.19618e18			0.460483			0.230241	−	0.973134i	$-0.426048\pi$
$347$	5.19618e18			0.460483			0.230241	+	0.973134i	$0.426048\pi$
$348$		0			0
$349$	−3.25826e18			−0.276563			−0.138282	−	0.990393i	$-0.544158\pi$
$349$	−3.25826e18			−0.276563			−0.138282	+	0.990393i	$0.544158\pi$
$350$		0			0
$351$	1.28425e19	−	5.13221e18i	1.04435	−	0.417350i
$352$		0			0
$353$			1.05498e19i			0.822121i	0.911608	+	0.411060i	$0.134841\pi$
$353$			1.05498e19i			0.822121i	−0.911608	+	0.411060i	$0.865159\pi$
$354$		0			0
$355$			6.28932e18i			0.469776i
$356$		0			0
$357$	−3.09873e18	−	4.07341e18i	−0.221907	−	0.291707i
$358$		0			0
$359$	2.60617e19			1.78976			0.894880	−	0.446308i	$-0.147261\pi$
$359$	2.60617e19			1.78976			0.894880	+	0.446308i	$0.147261\pi$
$360$		0			0
$361$	1.48978e19			0.981338
$362$		0			0
$363$	−3.81080e18	−	5.00946e18i	−0.240833	−	0.316586i
$364$		0			0
$365$			5.08011e18i			0.308089i
$366$		0			0
$367$		−	3.03458e19i		−	1.76646i	−0.468942	−	0.883229i	$-0.655365\pi$
$367$		−	3.03458e19i		−	1.76646i	0.468942	−	0.883229i	$-0.344635\pi$
$368$		0			0
$369$	7.32434e18	+	2.02819e18i	0.409327	+	0.113347i
$370$		0			0
$371$	−2.38632e19			−1.28063
$372$		0			0
$373$	9.93056e18			0.511868			0.255934	−	0.966694i	$-0.417617\pi$
$373$	9.93056e18			0.511868			0.255934	+	0.966694i	$0.417617\pi$
$374$		0			0
$375$	1.26698e19	−	9.63816e18i	0.627386	−	0.477265i
$376$		0			0
$377$			4.18209e19i			1.98991i
$378$		0			0
$379$			4.08110e19i			1.86631i	0.359480	+	0.933153i	$0.382954\pi$
$379$			4.08110e19i			1.86631i	−0.359480	+	0.933153i	$0.617046\pi$
$380$		0			0
$381$	−4.95339e18	+	3.76814e18i	−0.217753	+	0.165650i
$382$		0			0
$383$	−2.12130e19			−0.896626			−0.448313	−	0.893877i	$-0.647975\pi$
$383$	−2.12130e19			−0.896626			−0.448313	+	0.893877i	$0.647975\pi$
$384$		0			0
$385$	−1.06314e19			−0.432149
$386$		0			0
$387$	−7.90158e18	−	2.18804e18i	−0.308945	−	0.0855505i
$388$		0			0
$389$		−	3.25171e19i		−	1.22318i	−0.791176	−	0.611588i	$-0.790531\pi$
$389$		−	3.25171e19i		−	1.22318i	0.791176	−	0.611588i	$-0.209469\pi$
$390$		0			0
$391$		−	1.25990e19i		−	0.456049i
$392$		0			0
$393$	1.32404e19	+	1.74051e19i	0.461273	+	0.606364i
$394$		0			0
$395$	1.47165e19			0.493544
$396$		0			0
$397$	−3.17360e19			−1.02476			−0.512382	−	0.858758i	$-0.671237\pi$
$397$	−3.17360e19			−1.02476			−0.512382	+	0.858758i	$0.671237\pi$
$398$		0			0
$399$	2.23284e18	+	2.93516e18i	0.0694321	+	0.0912715i
$400$		0			0
$401$		−	5.30666e19i		−	1.58942i	−0.606989	−	0.794710i	$-0.707623\pi$
$401$		−	5.30666e19i		−	1.58942i	0.606989	−	0.794710i	$-0.292377\pi$
$402$		0			0
$403$			3.04228e19i			0.877831i
$404$		0			0
$405$	8.05571e18	−	1.34303e19i	0.223971	−	0.373398i
$406$		0			0
$407$	1.36708e19			0.366300
$408$		0			0
$409$	−2.32775e19			−0.601189			−0.300595	−	0.953752i	$-0.597185\pi$
$409$	−2.32775e19			−0.601189			−0.300595	+	0.953752i	$0.597185\pi$
$410$		0			0
$411$	−5.97844e19	+	4.54792e19i	−1.48859	+	1.13240i
$412$		0			0
$413$		−	1.15526e18i		−	0.0277367i
$414$		0			0
$415$			3.14517e19i			0.728255i
$416$		0			0
$417$	5.77031e19	−	4.38959e19i	1.28878	−	0.980401i
$418$		0			0
$419$	−6.13766e19			−1.32251			−0.661253	−	0.750163i	$-0.729975\pi$
$419$	−6.13766e19			−1.32251			−0.661253	+	0.750163i	$0.729975\pi$
$420$		0			0
$421$	−3.49520e19			−0.726703			−0.363351	−	0.931652i	$-0.618368\pi$
$421$	−3.49520e19			−0.726703			−0.363351	+	0.931652i	$0.618368\pi$
$422$		0			0
$423$	−5.27127e18	+	1.90359e19i	−0.105770	+	0.381963i
$424$		0			0
$425$			1.82688e19i			0.353829i
$426$		0			0
$427$			2.64670e19i			0.494876i
$428$		0			0
$429$	4.45907e19	+	5.86164e19i	0.805034	+	1.05825i
$430$		0			0
$431$	−6.15689e17			−0.0107345			−0.00536725	−	0.999986i	$-0.501708\pi$
$431$	−6.15689e17			−0.0107345			−0.00536725	+	0.999986i	$0.501708\pi$
$432$		0			0
$433$	−6.85272e19			−1.15399			−0.576997	−	0.816746i	$-0.695776\pi$
$433$	−6.85272e19			−1.15399			−0.576997	+	0.816746i	$0.695776\pi$
$434$		0			0
$435$	2.86725e19	+	3.76913e19i	0.466441	+	0.613157i
$436$		0			0
$437$			9.07843e18i			0.142692i
$438$		0			0
$439$		−	7.27308e18i		−	0.110467i	−0.998473	−	0.0552337i	$-0.982410\pi$
$439$		−	7.27308e18i		−	0.110467i	0.998473	−	0.0552337i	$-0.0175904\pi$
$440$		0			0
$441$	−5.36812e18	+	1.93857e19i	−0.0788011	+	0.284572i
$442$		0			0
$443$	−8.96453e19			−1.27204			−0.636018	−	0.771674i	$-0.719420\pi$
$443$	−8.96453e19			−1.27204			−0.636018	+	0.771674i	$0.719420\pi$
$444$		0			0
$445$	4.26935e19			0.585681
$446$		0			0
$447$	1.89886e19	−	1.44450e19i	0.251875	−	0.191607i
$448$		0			0
$449$		−	2.97759e19i		−	0.381959i	−0.981594	−	0.190980i	$-0.938834\pi$
$449$		−	2.97759e19i		−	0.381959i	0.981594	−	0.190980i	$-0.0611665\pi$
$450$		0			0
$451$			4.04722e19i			0.502149i
$452$		0			0
$453$	7.82374e19	−	5.95168e19i	0.939026	−	0.714336i
$454$		0			0
$455$	3.54013e19			0.411087
$456$		0			0
$457$	−5.35657e19			−0.601887			−0.300944	−	0.953642i	$-0.597302\pi$
$457$	−5.35657e19			−0.601887			−0.300944	+	0.953642i	$0.597302\pi$
$458$		0			0
$459$	1.48992e19	+	3.72829e19i	0.162020	+	0.405429i
$460$		0			0
$461$			1.88829e20i			1.98752i	0.111527	+	0.993761i	$0.464426\pi$
$461$			1.88829e20i			1.98752i	−0.111527	+	0.993761i	$0.535574\pi$
$462$		0			0
$463$		−	2.55431e19i		−	0.260265i	−0.991497	−	0.130133i	$-0.958460\pi$
$463$		−	2.55431e19i		−	0.260265i	0.991497	−	0.130133i	$-0.0415403\pi$
$464$		0			0
$465$	2.08579e19	+	2.74187e19i	0.205766	+	0.270489i
$466$		0			0
$467$	−5.46072e19			−0.521642			−0.260821	−	0.965387i	$-0.583993\pi$
$467$	−5.46072e19			−0.521642			−0.260821	+	0.965387i	$0.583993\pi$
$468$		0			0
$469$	−1.59992e20			−1.48014
$470$		0			0
$471$	−1.09659e20	−	1.44152e20i	−0.982628	−	1.29171i
$472$		0			0
$473$		−	4.36619e19i		−	0.379004i
$474$		0			0
$475$		−	1.31639e19i		−	0.110709i
$476$		0			0
$477$	1.80410e20	+	4.99576e19i	1.47019	+	0.407112i
$478$		0			0
$479$	1.65419e20			1.30638			0.653188	−	0.757196i	$-0.273431\pi$
$479$	1.65419e20			1.30638			0.653188	+	0.757196i	$0.273431\pi$
$480$		0			0
$481$	−4.55224e19			−0.348447
$482$		0			0
$483$	9.40559e19	−	7.15502e19i	0.697883	−	0.530894i
$484$		0			0
$485$		−	2.85372e19i		−	0.205280i
$486$		0			0
$487$		−	5.27556e19i		−	0.367960i	−0.982930	−	0.183980i	$-0.941102\pi$
$487$		−	5.27556e19i		−	0.367960i	0.982930	−	0.183980i	$-0.0588982\pi$
$488$		0			0
$489$	1.51320e20	−	1.15112e20i	1.02348	−	0.778585i
$490$		0			0
$491$	2.76893e20			1.81636			0.908178	−	0.418584i	$-0.137474\pi$
$491$	2.76893e20			1.81636			0.908178	+	0.418584i	$0.137474\pi$
$492$		0			0
$493$	−1.21410e20			−0.772506
$494$		0			0
$495$	8.03751e19	+	2.22568e19i	0.496115	+	0.137380i
$496$		0			0
$497$		−	1.51240e20i		−	0.905717i
$498$		0			0
$499$		−	1.21401e20i		−	0.705453i	−0.935726	−	0.352727i	$-0.885255\pi$
$499$		−	1.21401e20i		−	0.705453i	0.935726	−	0.352727i	$-0.114745\pi$
$500$		0			0
$501$	−5.15235e19	−	6.77299e19i	−0.290551	−	0.381942i
$502$		0			0
$503$	−1.34169e20			−0.734330			−0.367165	−	0.930156i	$-0.619672\pi$
$503$	−1.34169e20			−0.734330			−0.367165	+	0.930156i	$0.619672\pi$
$504$		0			0
$505$	−6.77672e19			−0.360026
$506$		0			0
$507$	−3.10909e19	−	4.08704e19i	−0.160352	−	0.210790i
$508$		0			0
$509$		−	1.59029e20i		−	0.796328i	−0.917314	−	0.398164i	$-0.869647\pi$
$509$		−	1.59029e20i		−	0.796328i	0.917314	−	0.398164i	$-0.130353\pi$
$510$		0			0
$511$		−	1.22162e20i		−	0.593989i
$512$		0			0
$513$	−1.07359e19	−	2.68648e19i	−0.0506941	−	0.126854i
$514$		0			0
$515$	1.42243e20			0.652343
$516$		0			0
$517$	−1.05187e20			−0.468580
$518$		0			0
$519$	1.12462e20	−	8.55521e19i	0.486688	−	0.370233i
$520$		0			0
$521$			2.22638e20i			0.936087i	0.883706	+	0.468043i	$0.155041\pi$
$521$			2.22638e20i			0.936087i	−0.883706	+	0.468043i	$0.844959\pi$
$522$		0			0
$523$		−	4.79645e20i		−	1.95955i	−0.200093	−	0.979777i	$-0.564124\pi$
$523$		−	4.79645e20i		−	1.95955i	0.200093	−	0.979777i	$-0.435876\pi$
$524$		0			0
$525$	−1.36383e20	+	1.03749e20i	−0.541457	+	0.411898i
$526$		0			0
$527$	−8.83200e19			−0.340784
$528$		0			0
$529$	2.42788e19			0.0910563
$530$		0			0
$531$	−2.41854e18	+	8.73398e18i	−0.00881747	+	0.0318422i
$532$		0			0
$533$		−	1.34768e20i		−	0.477676i
$534$		0			0
$535$			1.75252e20i			0.603963i
$536$		0			0
$537$	3.15519e20	+	4.14763e20i	1.05735	+	1.38993i
$538$		0			0
$539$	−1.07120e20			−0.349103
$540$		0			0
$541$	4.62548e20			1.46615			0.733074	−	0.680149i	$-0.238085\pi$
$541$	4.62548e20			1.46615			0.733074	+	0.680149i	$0.238085\pi$
$542$		0			0
$543$	1.45942e20	+	1.91847e20i	0.449968	+	0.591503i
$544$		0			0
$545$			6.60777e19i			0.198190i
$546$		0			0
$547$			1.52374e20i			0.444638i	0.974974	+	0.222319i	$0.0713626\pi$
$547$			1.52374e20i			0.444638i	−0.974974	+	0.222319i	$0.928637\pi$
$548$		0			0
$549$	5.54087e19	−	2.00095e20i	0.157321	−	0.568126i
$550$		0			0
$551$	8.74838e19			0.241708
$552$		0			0
$553$	−3.53888e20			−0.951541
$554$		0			0
$555$	−4.10273e19	+	3.12103e19i	−0.107368	+	0.0816771i
$556$		0			0
$557$			5.19363e20i			1.32299i	0.749949	+	0.661495i	$0.230078\pi$
$557$			5.19363e20i			1.32299i	−0.749949	+	0.661495i	$0.769922\pi$
$558$		0			0
$559$			1.45389e20i			0.360532i
$560$		0			0
$561$	−1.70168e20	+	1.29451e20i	−0.410826	+	0.312523i
$562$		0			0
$563$	1.10226e20			0.259102			0.129551	−	0.991573i	$-0.458646\pi$
$563$	1.10226e20			0.259102			0.129551	+	0.991573i	$0.458646\pi$
$564$		0			0
$565$	1.46523e19			0.0335385
$566$		0			0
$567$	−1.93716e20	+	3.22958e20i	−0.431810	+	0.719903i
$568$		0			0
$569$			1.11868e20i			0.242864i	0.992600	+	0.121432i	$0.0387486\pi$
$569$			1.11868e20i			0.242864i	−0.992600	+	0.121432i	$0.961251\pi$
$570$		0			0
$571$			2.44274e20i			0.516543i	0.966072	+	0.258271i	$0.0831529\pi$
$571$			2.44274e20i			0.516543i	−0.966072	+	0.258271i	$0.916847\pi$
$572$		0			0
$573$	−2.98611e20	−	3.92537e20i	−0.615101	−	0.808577i
$574$		0			0
$575$	−4.21830e20			−0.846504
$576$		0			0
$577$	2.24168e20			0.438284			0.219142	−	0.975693i	$-0.429674\pi$
$577$	2.24168e20			0.438284			0.219142	+	0.975693i	$0.429674\pi$
$578$		0			0
$579$	−5.35549e20	−	7.04002e20i	−1.02026	−	1.34117i
$580$		0			0
$581$		−	7.56321e20i		−	1.40406i
$582$		0			0
$583$			9.96895e20i			1.80358i
$584$		0			0
$585$	−2.67640e20	−	7.41127e19i	−0.471935	−	0.130684i
$586$		0			0
$587$	−1.36189e19			−0.0234075			−0.0117038	−	0.999932i	$-0.503726\pi$
$587$	−1.36189e19			−0.0234075			−0.0117038	+	0.999932i	$0.503726\pi$
$588$		0			0
$589$	6.36404e19			0.106627
$590$		0			0
$591$	−7.31497e20	+	5.56464e20i	−1.19483	+	0.908932i
$592$		0			0
$593$		−	5.06250e20i		−	0.806222i	−0.915151	−	0.403111i	$-0.867929\pi$
$593$		−	5.06250e20i		−	0.806222i	0.915151	−	0.403111i	$-0.132071\pi$
$594$		0			0
$595$			1.02773e20i			0.159589i
$596$		0			0
$597$	−3.46593e20	+	2.63661e20i	−0.524823	+	0.399243i
$598$		0			0
$599$	−1.02091e21			−1.50760			−0.753802	−	0.657102i	$-0.771782\pi$
$599$	−1.02091e21			−1.50760			−0.753802	+	0.657102i	$0.771782\pi$
$600$		0			0
$601$	1.06150e21			1.52884			0.764421	−	0.644717i	$-0.223025\pi$
$601$	1.06150e21			1.52884			0.764421	+	0.644717i	$0.223025\pi$
$602$		0			0
$603$	1.20957e21	+	3.34944e20i	1.69922	+	0.470535i
$604$		0			0
$605$			1.26390e20i			0.173200i
$606$		0			0
$607$			4.51118e20i			0.603080i	0.953453	+	0.301540i	$0.0975007\pi$
$607$			4.51118e20i			0.603080i	−0.953453	+	0.301540i	$0.902499\pi$
$608$		0			0
$609$	−6.89490e20	−	9.06364e20i	−0.899287	−	1.18215i
$610$		0			0
$611$	3.50262e20			0.445743
$612$		0			0
$613$	4.84225e20			0.601304			0.300652	−	0.953734i	$-0.402796\pi$
$613$	4.84225e20			0.601304			0.300652	+	0.953734i	$0.402796\pi$
$614$		0			0
$615$	−9.23974e19	−	1.21460e20i	−0.111969	−	0.147188i
$616$		0			0
$617$			1.04009e21i			1.23008i	0.788497	+	0.615038i	$0.210859\pi$
$617$			1.04009e21i			1.23008i	−0.788497	+	0.615038i	$0.789141\pi$
$618$		0			0
$619$		−	5.15454e19i		−	0.0594990i	−0.999557	−	0.0297495i	$-0.990529\pi$
$619$		−	5.15454e19i		−	0.0594990i	0.999557	−	0.0297495i	$-0.00947096\pi$
$620$		0			0
$621$	−8.60870e20	+	3.44026e20i	−0.969953	+	0.387619i
$622$		0			0
$623$	−1.02665e21			−1.12918
$624$		0			0
$625$	4.35092e20			0.467177
$626$		0			0
$627$	1.22618e20	−	9.32777e19i	0.128542	−	0.0977848i
$628$		0			0
$629$		−	1.32156e20i		−	0.135271i
$630$		0			0
$631$		−	2.22096e20i		−	0.221984i	−0.993821	−	0.110992i	$-0.964597\pi$
$631$		−	2.22096e20i		−	0.221984i	0.993821	−	0.110992i	$-0.0354028\pi$
$632$		0			0
$633$	−3.45665e20	+	2.62954e20i	−0.337386	+	0.256656i
$634$		0			0
$635$	1.24975e20			0.119130
$636$		0			0
$637$	3.56697e20			0.332089
$638$		0			0
$639$	−3.16620e20	+	1.14340e21i	−0.287927	+	1.03978i
$640$		0			0
$641$			5.19914e19i			0.0461845i	0.999733	+	0.0230922i	$0.00735114\pi$
$641$			5.19914e19i			0.0461845i	−0.999733	+	0.0230922i	$0.992649\pi$
$642$		0			0
$643$		−	6.41865e20i		−	0.557008i	−0.960435	−	0.278504i	$-0.910161\pi$
$643$		−	6.41865e20i		−	0.557008i	0.960435	−	0.278504i	$-0.0898385\pi$
$644$		0			0
$645$	9.96794e19	+	1.31033e20i	0.0845099	+	0.111092i
$646$		0			0
$647$	1.59499e21			1.32122			0.660611	−	0.750729i	$-0.270297\pi$
$647$	1.59499e21			1.32122			0.660611	+	0.750729i	$0.270297\pi$
$648$		0			0
$649$	−4.82615e19			−0.0390630
$650$		0			0
$651$	−5.01572e20	−	6.59338e20i	−0.396712	−	0.521496i
$652$		0			0
$653$		−	3.61572e17i		−	0.000279477i	−1.00000		0.000139739i	$-0.999956\pi$
$653$		−	3.61572e17i		−	0.000279477i	1.00000		0.000139739i	$-4.44802e-5\pi$
$654$		0			0
$655$		−	4.39136e20i		−	0.331733i
$656$		0			0
$657$	−2.55746e20	+	9.23564e20i	−0.188829	+	0.681910i
$658$		0			0
$659$	9.68333e20			0.698851			0.349425	−	0.936964i	$-0.386377\pi$
$659$	9.68333e20			0.698851			0.349425	+	0.936964i	$0.386377\pi$
$660$		0			0
$661$	1.92498e21			1.35805			0.679025	−	0.734115i	$-0.262403\pi$
$661$	1.92498e21			1.35805			0.679025	+	0.734115i	$0.262403\pi$
$662$		0			0
$663$	5.66643e20	−	4.31057e20i	0.390803	−	0.297292i
$664$		0			0
$665$		−	7.40548e19i		−	0.0499334i
$666$		0			0
$667$		−	2.80338e21i		−	1.84815i
$668$		0			0
$669$	−1.34508e21	+	1.02323e21i	−0.867065	+	0.659594i
$670$		0			0
$671$	1.10567e21			0.696959
$672$		0			0
$673$	5.09103e20			0.313829			0.156914	−	0.987612i	$-0.449845\pi$
$673$	5.09103e20			0.313829			0.156914	+	0.987612i	$0.449845\pi$
$674$		0			0
$675$	1.24828e21	−	4.98844e20i	0.752545	−	0.300737i
$676$		0			0
$677$			1.96477e21i			1.15850i	0.815148	+	0.579252i	$0.196655\pi$
$677$			1.96477e21i			1.15850i	−0.815148	+	0.579252i	$0.803345\pi$
$678$		0			0
$679$			6.86234e20i			0.395776i
$680$		0			0
$681$	−1.20315e21	−	1.58160e21i	−0.678762	−	0.892262i
$682$		0			0
$683$	2.40991e20			0.132998			0.0664991	−	0.997786i	$-0.478817\pi$
$683$	2.40991e20			0.132998			0.0664991	+	0.997786i	$0.478817\pi$
$684$		0			0
$685$	1.50837e21			0.814385
$686$		0			0
$687$	−1.48699e21	−	1.95471e21i	−0.785474	−	1.03254i
$688$		0			0
$689$		−	3.31955e21i		−	1.71568i
$690$		0			0
$691$		−	9.40696e20i		−	0.475734i	−0.971298	−	0.237867i	$-0.923552\pi$
$691$		−	9.40696e20i		−	0.475734i	0.971298	−	0.237867i	$-0.0764482\pi$
$692$		0			0
$693$	−1.93278e21	−	5.35210e20i	−0.956498	−	0.264865i
$694$		0			0
$695$	−1.45586e21			−0.705073
$696$		0			0
$697$	3.91244e20			0.185439
$698$		0			0
$699$	1.21054e21	−	9.20883e20i	0.561565	−	0.427194i
$700$		0			0
$701$		−	2.13672e21i		−	0.970201i	−0.874458	−	0.485101i	$-0.838783\pi$
$701$		−	2.13672e21i		−	0.970201i	0.874458	−	0.485101i	$-0.161217\pi$
$702$		0			0
$703$			9.52268e19i			0.0423247i
$704$		0			0
$705$	3.15675e20	−	2.40141e20i	0.137348	−	0.104483i
$706$		0			0
$707$	1.62960e21			0.694122
$708$		0			0
$709$	−5.05315e20			−0.210724			−0.105362	−	0.994434i	$-0.533600\pi$
$709$	−5.05315e20			−0.210724			−0.105362	+	0.994434i	$0.533600\pi$
$710$		0			0
$711$	2.67546e21	+	7.40865e20i	1.09239	+	0.302494i
$712$		0			0
$713$		−	2.03933e21i		−	0.815296i
$714$		0			0
$715$		−	1.47891e21i		−	0.578955i
$716$		0			0
$717$	2.78234e20	+	3.65750e20i	0.106664	+	0.140214i
$718$		0			0
$719$	1.96314e21			0.737030			0.368515	−	0.929622i	$-0.379866\pi$
$719$	1.96314e21			0.737030			0.368515	+	0.929622i	$0.379866\pi$
$720$		0			0
$721$	−3.42052e21			−1.25770
$722$		0			0
$723$	−8.37393e20	−	1.10079e21i	−0.301573	−	0.396430i
$724$		0			0
$725$			4.06494e21i			1.43390i
$726$		0			0
$727$			1.15913e21i			0.400519i	0.979743	+	0.200259i	$0.0641785\pi$
$727$			1.15913e21i			0.400519i	−0.979743	+	0.200259i	$0.935822\pi$
$728$		0			0
$729$	2.14064e21	−	2.03608e21i	0.724582	−	0.689189i
$730$		0			0
$731$	−4.22078e20			−0.139963
$732$		0			0
$733$	2.01215e21			0.653703			0.326852	−	0.945076i	$-0.394012\pi$
$733$	2.01215e21			0.653703			0.326852	+	0.945076i	$0.394012\pi$
$734$		0			0
$735$	3.21475e20	−	2.44553e20i	0.102327	−	0.0778426i
$736$		0			0
$737$			6.68374e21i			2.08455i
$738$		0			0
$739$			3.78711e21i			1.15738i	0.815549	+	0.578688i	$0.196435\pi$
$739$			3.78711e21i			1.15738i	−0.815549	+	0.578688i	$0.803565\pi$
$740$		0			0
$741$	−4.08303e20	+	3.10605e20i	−0.122277	+	0.0930190i
$742$		0			0
$743$	−3.73122e21			−1.09505			−0.547525	−	0.836789i	$-0.684430\pi$
$743$	−3.73122e21			−1.09505			−0.547525	+	0.836789i	$0.684430\pi$
$744$		0			0
$745$	−4.79086e20			−0.137798
$746$		0			0
$747$	−1.58336e21	+	5.71792e21i	−0.446349	+	1.61188i
$748$		0			0
$749$		−	4.21430e21i		−	1.16443i
$750$		0			0
$751$		−	1.59970e21i		−	0.433251i	−0.976255	−	0.216626i	$-0.930495\pi$
$751$		−	1.59970e21i		−	0.433251i	0.976255	−	0.216626i	$-0.0695051\pi$
$752$		0			0
$753$	3.19650e21	+	4.20193e21i	0.848617	+	1.11554i
$754$		0			0
$755$	−1.97395e21			−0.513728
$756$		0			0
$757$	1.89582e21			0.483702			0.241851	−	0.970313i	$-0.422245\pi$
$757$	1.89582e21			0.483702			0.241851	+	0.970313i	$0.422245\pi$
$758$		0			0
$759$	−2.98904e21	−	3.92922e21i	−0.747685	−	0.982864i
$760$		0			0
$761$			2.60427e21i			0.638706i	0.947636	+	0.319353i	$0.103466\pi$
$761$			2.60427e21i			0.638706i	−0.947636	+	0.319353i	$0.896534\pi$
$762$		0			0
$763$		−	1.58897e21i		−	0.382105i
$764$		0			0
$765$	2.15156e20	−	7.76983e20i	0.0507332	−	0.183211i
$766$		0			0
$767$	1.60706e20			0.0371592
$768$		0			0
$769$	−1.38301e21			−0.313601			−0.156801	−	0.987630i	$-0.550118\pi$
$769$	−1.38301e21			−0.313601			−0.156801	+	0.987630i	$0.550118\pi$
$770$		0			0
$771$	−4.78201e21	+	3.63777e21i	−1.06341	+	0.808961i
$772$		0			0
$773$		−	2.47373e21i		−	0.539519i	−0.962928	−	0.269760i	$-0.913056\pi$
$773$		−	2.47373e21i		−	0.539519i	0.962928	−	0.269760i	$-0.0869442\pi$
$774$		0			0
$775$			2.95706e21i			0.632553i
$776$		0			0
$777$	9.86585e20	−	7.50515e20i	0.207003	−	0.157471i
$778$		0			0
$779$	−2.81917e20			−0.0580216
$780$		0			0
$781$	−6.31810e21			−1.27557
$782$		0			0
$783$	3.31519e21	+	8.29572e21i	0.656592	+	1.64301i
$784$		0			0
$785$			3.63699e21i			0.706675i
$786$		0			0
$787$			6.69372e21i			1.27602i	0.770028	+	0.638010i	$0.220242\pi$
$787$			6.69372e21i			1.27602i	−0.770028	+	0.638010i	$0.779758\pi$
$788$		0			0
$789$	−2.35984e21	−	3.10211e21i	−0.441371	−	0.580202i
$790$		0			0
$791$	−3.52345e20			−0.0646613
$792$		0			0
$793$	−3.68177e21			−0.662991
$794$		0			0
$795$	−2.27589e21	−	2.99176e21i	−0.402160	−	0.528656i
$796$		0			0
$797$		−	3.88279e20i		−	0.0673296i	−0.999433	−	0.0336648i	$-0.989282\pi$
$797$		−	3.88279e20i		−	0.0673296i	0.999433	−	0.0336648i	$-0.0107179\pi$
$798$		0			0
$799$			1.01684e21i			0.173042i
$800$		0			0
$801$	7.76167e21	+	2.14930e21i	1.29632	+	0.358965i
$802$		0			0
$803$	−5.10336e21			−0.836545
$804$		0			0
$805$	−2.37305e21			−0.381802
$806$		0			0
$807$	2.71721e21	−	2.06704e21i	0.429114	−	0.326436i
$808$		0			0
$809$		−	1.22179e22i		−	1.89400i	−0.321226	−	0.947002i	$-0.604095\pi$
$809$		−	1.22179e22i		−	1.89400i	0.321226	−	0.947002i	$-0.395905\pi$
$810$		0			0
$811$			1.00119e22i			1.52356i	0.647835	+	0.761781i	$0.275675\pi$
$811$			1.00119e22i			1.52356i	−0.647835	+	0.761781i	$0.724325\pi$
$812$		0			0
$813$	6.62111e21	−	5.03681e21i	0.989126	−	0.752448i
$814$		0			0
$815$	−3.81785e21			−0.559934
$816$		0			0
$817$	3.04135e20			0.0437926
$818$		0			0
$819$	6.43596e21	+	1.78219e21i	0.909880	+	0.251956i
$820$		0			0
$821$		−	2.50781e21i		−	0.348113i	−0.984736	−	0.174057i	$-0.944312\pi$
$821$		−	2.50781e21i		−	0.348113i	0.984736	−	0.174057i	$-0.0556876\pi$
$822$		0			0
$823$		−	8.65240e21i		−	1.17934i	−0.807646	−	0.589668i	$-0.799259\pi$
$823$		−	8.65240e21i		−	1.17934i	0.807646	−	0.589668i	$-0.200741\pi$
$824$		0			0
$825$	4.33416e21	+	5.69744e21i	0.580097	+	0.762562i
$826$		0			0
$827$	−9.91132e21			−1.30269			−0.651344	−	0.758783i	$-0.725794\pi$
$827$	−9.91132e21			−1.30269			−0.651344	+	0.758783i	$0.725794\pi$
$828$		0			0
$829$	−6.33882e21			−0.818181			−0.409090	−	0.912494i	$-0.634154\pi$
$829$	−6.33882e21			−0.818181			−0.409090	+	0.912494i	$0.634154\pi$
$830$		0			0
$831$	−5.71666e21	−	7.51480e21i	−0.724660	−	0.952598i
$832$		0			0
$833$			1.03552e21i			0.128921i
$834$		0			0
$835$			1.70884e21i			0.208955i
$836$		0			0
$837$	2.41165e21	+	6.03475e21i	0.289650	+	0.724801i
$838$		0			0
$839$	1.55967e22			1.84000			0.919998	−	0.391922i	$-0.128190\pi$
$839$	1.55967e22			1.84000			0.919998	+	0.391922i	$0.128190\pi$
$840$		0			0
$841$	−1.83854e22			−2.13061
$842$		0			0
$843$	−4.69479e20	+	3.57143e20i	−0.0534453	+	0.0406569i
$844$		0			0
$845$			1.03117e21i			0.115320i
$846$		0			0
$847$		−	3.03930e21i		−	0.333925i
$848$		0			0
$849$	7.66634e21	−	5.83194e21i	0.827523	−	0.629514i
$850$		0			0
$851$	3.05150e21			0.323625
$852$		0			0
$853$	3.08832e21			0.321814			0.160907	−	0.986970i	$-0.448558\pi$
$853$	3.08832e21			0.321814			0.160907	+	0.986970i	$0.448558\pi$
$854$		0			0
$855$	−1.55034e20	+	5.59868e20i	−0.0158738	+	0.0573244i
$856$		0			0
$857$			5.02229e21i			0.505295i	0.967558	+	0.252648i	$0.0813014\pi$
$857$			5.02229e21i			0.505295i	−0.967558	+	0.252648i	$0.918699\pi$
$858$		0			0
$859$			3.68380e21i			0.364206i	0.983279	+	0.182103i	$0.0582905\pi$
$859$			3.68380e21i			0.364206i	−0.983279	+	0.182103i	$0.941709\pi$
$860$		0			0
$861$	2.22188e21	+	2.92076e21i	0.215873	+	0.283774i
$862$		0			0
$863$	7.57484e21			0.723257			0.361628	−	0.932322i	$-0.382221\pi$
$863$	7.57484e21			0.723257			0.361628	+	0.932322i	$0.382221\pi$
$864$		0			0
$865$	−2.83744e21			−0.266260
$866$		0			0
$867$	−5.31337e21	−	6.98465e21i	−0.490035	−	0.644172i
$868$		0			0
$869$			1.47838e22i			1.34010i
$870$		0			0
$871$		−	2.22561e22i		−	1.98296i
$872$		0			0
$873$	1.43663e21	−	5.18805e21i	0.125817	−	0.454357i
$874$		0			0
$875$	7.68692e21			0.661747
$876$		0			0
$877$	9.26987e21			0.784471			0.392235	−	0.919865i	$-0.371702\pi$
$877$	9.26987e21			0.784471			0.392235	+	0.919865i	$0.371702\pi$
$878$		0			0
$879$	6.55102e21	−	4.98350e21i	0.544995	−	0.414589i
$880$		0			0
$881$			4.46199e20i			0.0364930i	0.999834	+	0.0182465i	$0.00580836\pi$
$881$			4.46199e20i			0.0364930i	−0.999834	+	0.0182465i	$0.994192\pi$
$882$		0			0
$883$		−	5.92769e21i		−	0.476628i	−0.971188	−	0.238314i	$-0.923405\pi$
$883$		−	5.92769e21i		−	0.476628i	0.971188	−	0.238314i	$-0.0765948\pi$
$884$		0			0
$885$	1.44837e20	−	1.10180e20i	0.0114500	−	0.00871022i
$886$		0			0
$887$	1.48935e22			1.15763			0.578815	−	0.815459i	$-0.303515\pi$
$887$	1.48935e22			1.15763			0.578815	+	0.815459i	$0.303515\pi$
$888$		0			0
$889$	−3.00528e21			−0.229680
$890$		0			0
$891$	1.34917e22	+	8.09257e21i	1.01388	+	0.608140i
$892$		0			0
$893$		−	7.32701e20i		−	0.0541428i
$894$		0			0
$895$		−	1.04646e22i		−	0.760412i
$896$		0			0
$897$	9.95319e21	+	1.30839e22i	0.711244	+	0.934962i
$898$		0			0
$899$	−1.96519e22			−1.38104
$900$		0			0
$901$	9.63695e21			0.666045
$902$		0			0
$903$	−2.39699e21	−	3.15095e21i	−0.162933	−	0.214182i
$904$		0			0
$905$		−	4.84035e21i		−	0.323603i
$906$		0			0
$907$		−	4.94918e21i		−	0.325446i	−0.986672	−	0.162723i	$-0.947972\pi$
$907$		−	4.94918e21i		−	0.325446i	0.986672	−	0.162723i	$-0.0520276\pi$
$908$		0			0
$909$	−1.23201e22	−	3.41157e21i	−0.796864	−	0.220661i
$910$		0			0
$911$	1.18206e21			0.0752061			0.0376031	−	0.999293i	$-0.488028\pi$
$911$	1.18206e21			0.0752061			0.0376031	+	0.999293i	$0.488028\pi$
$912$		0			0
$913$	−3.15956e22			−1.97741
$914$		0			0
$915$	−3.31821e21	+	2.52423e21i	−0.204289	+	0.155407i
$916$		0			0
$917$			1.05599e22i			0.639574i
$918$		0			0
$919$		−	4.78002e21i		−	0.284815i	−0.989808	−	0.142408i	$-0.954516\pi$
$919$		−	4.78002e21i		−	0.284815i	0.989808	−	0.142408i	$-0.0454844\pi$
$920$		0			0
$921$	2.31012e22	−	1.75736e22i	1.35421	−	1.03018i
$922$		0			0
$923$	2.10386e22			1.21340
$924$		0			0
$925$	−4.42472e21			−0.251086
$926$		0			0
$927$	2.58597e22	+	7.16086e21i	1.44386	+	0.399822i
$928$		0			0
$929$		−	2.64184e22i		−	1.45140i	−0.688010	−	0.725702i	$-0.741515\pi$
$929$		−	2.64184e22i		−	1.45140i	0.688010	−	0.725702i	$-0.258485\pi$
$930$		0			0
$931$		−	7.46163e20i		−	0.0403377i
$932$		0			0
$933$	−1.11158e22	−	1.46123e22i	−0.591330	−	0.777329i
$934$		0			0
$935$	4.29339e21			0.224757
$936$		0			0
$937$	−4.04724e21			−0.208503			−0.104251	−	0.994551i	$-0.533245\pi$
$937$	−4.04724e21			−0.208503			−0.104251	+	0.994551i	$0.533245\pi$
$938$		0			0
$939$	−2.01320e22	−	2.64644e22i	−1.02069	−	1.34175i
$940$		0			0
$941$		−	1.81515e22i		−	0.905712i	−0.891584	−	0.452856i	$-0.850405\pi$
$941$		−	1.81515e22i		−	0.905712i	0.891584	−	0.452856i	$-0.149595\pi$
$942$		0			0
$943$			9.03390e21i			0.443647i
$944$		0			0
$945$	7.02231e21	−	2.80630e21i	0.339423	−	0.135643i
$946$		0			0
$947$	−2.53746e22			−1.20719			−0.603593	−	0.797292i	$-0.706265\pi$
$947$	−2.53746e22			−1.20719			−0.603593	+	0.797292i	$0.706265\pi$
$948$		0			0
$949$	1.69936e22			0.795774
$950$		0			0
$951$	−1.97155e21	+	1.49980e21i	−0.0908771	+	0.0691321i
$952$		0			0
$953$			1.18394e21i			0.0537196i	0.999639	+	0.0268598i	$0.00855077\pi$
$953$			1.18394e21i			0.0537196i	−0.999639	+	0.0268598i	$0.991449\pi$
$954$		0			0
$955$			9.90381e21i			0.442361i
$956$		0			0
$957$	−3.78638e22	+	2.88037e22i	−1.66488	+	1.26651i
$958$		0			0
$959$	−3.62719e22			−1.57011
$960$		0			0
$961$	9.16945e21			0.390767
$962$		0			0
$963$	−8.82265e21	+	3.18609e22i	−0.370170	+	1.33678i
$964$		0			0
$965$			1.77621e22i			0.733737i
$966$		0			0
$967$			7.31766e21i			0.297628i	0.988865	+	0.148814i	$0.0475456\pi$
$967$			7.31766e21i			0.297628i	−0.988865	+	0.148814i	$0.952454\pi$
$968$		0			0
$969$	−9.01712e20	−	1.18534e21i	−0.0361110	−	0.0474695i
$970$		0			0
$971$	2.17549e22			0.857851			0.428926	−	0.903340i	$-0.358892\pi$
$971$	2.17549e22			0.857851			0.428926	+	0.903340i	$0.358892\pi$
$972$		0			0
$973$	3.50092e22			1.35936
$974$		0			0
$975$	−1.44323e22	−	1.89719e22i	−0.551824	−	0.725397i
$976$		0			0
$977$			2.25901e22i			0.850569i	0.905060	+	0.425285i	$0.139826\pi$
$977$			2.25901e22i			0.850569i	−0.905060	+	0.425285i	$0.860174\pi$
$978$		0			0
$979$			4.28888e22i			1.59028i
$980$		0			0
$981$	−3.32652e21	+	1.20129e22i	−0.121471	+	0.438663i
$982$		0			0
$983$	3.81448e21			0.137178			0.0685890	−	0.997645i	$-0.478150\pi$
$983$	3.81448e21			0.137178			0.0685890	+	0.997645i	$0.478150\pi$
$984$		0			0
$985$	1.84558e22			0.653675
$986$		0			0
$987$	−7.59106e21	+	5.77467e21i	−0.264803	+	0.201441i
$988$		0			0
$989$		−	9.74587e21i		−	0.334849i
$990$		0			0
$991$			3.21358e22i			1.08752i	0.839241	+	0.543759i	$0.183000\pi$
$991$			3.21358e22i			1.08752i	−0.839241	+	0.543759i	$0.817000\pi$
$992$		0			0
$993$	−2.23633e22	+	1.70122e22i	−0.745445	+	0.567076i
$994$		0			0
$995$	8.74463e21			0.287123
$996$		0			0
$997$	1.57764e22			0.510264			0.255132	−	0.966906i	$-0.417881\pi$
$997$	1.57764e22			0.510264			0.255132	+	0.966906i	$0.417881\pi$
$998$		0			0
$999$	−9.02996e21	+	3.60861e21i	−0.287703	+	0.114974i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.16.c.b.47.1	✓	4
3.2	odd	2	inner	48.16.c.b.47.3	yes	4
4.3	odd	2	inner	48.16.c.b.47.4	yes	4
12.11	even	2	inner	48.16.c.b.47.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.16.c.b.47.1	✓	4	1.1	even	1	trivial
48.16.c.b.47.2	yes	4	12.11	even	2	inner
48.16.c.b.47.3	yes	4	3.2	odd	2	inner
48.16.c.b.47.4	yes	4	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:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	48.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2293.43 - 3014.81i) q^{3} +76064.4i q^{5} -1.82913e6i q^{7} +(-3.82928e6 + 1.38285e7i) q^{9} +O(q^{10})\)	\(q+(-2293.43 - 3014.81i) q^{3} +76064.4i q^{5} -1.82913e6i q^{7} +(-3.82928e6 + 1.38285e7i) q^{9} -7.64125e7 q^{11} +2.54445e8 q^{13} +(2.29320e8 - 1.74448e8i) q^{15} +7.38677e8i q^{17} -5.32265e8i q^{19} +(-5.51447e9 + 4.19497e9i) q^{21} -1.70562e10 q^{23} +2.47318e10 q^{25} +(5.04725e10 - 2.01702e10i) q^{27} +1.64361e11i q^{29} +1.19565e11i q^{31} +(1.75247e11 + 2.30369e11i) q^{33} +1.39131e11 q^{35} -1.78908e11 q^{37} +(-5.83552e11 - 7.67105e11i) q^{39} -5.29655e11i q^{41} +5.71398e11i q^{43} +(-1.05186e12 - 2.91272e11i) q^{45} +1.37657e12 q^{47} +1.40186e12 q^{49} +(2.22697e12 - 1.69410e12i) q^{51} -1.30462e13i q^{53} -5.81227e12i q^{55} +(-1.60468e12 + 1.22071e12i) q^{57} +6.31592e11 q^{59} -1.44698e13 q^{61} +(2.52941e13 + 7.00422e12i) q^{63} +1.93542e13i q^{65} -8.74692e13i q^{67} +(3.91172e13 + 5.14212e13i) q^{69} +8.26841e13 q^{71} +6.67870e13 q^{73} +(-5.67206e13 - 7.45617e13i) q^{75} +1.39768e14i q^{77} -1.93474e14i q^{79} +(-1.76564e14 - 1.05906e14i) q^{81} +4.13488e14 q^{83} -5.61870e13 q^{85} +(4.95518e14 - 3.76951e14i) q^{87} -5.61280e14i q^{89} -4.65412e14i q^{91} +(3.60466e14 - 2.74214e14i) q^{93} +4.04865e13 q^{95} -3.75171e14 q^{97} +(2.92604e14 - 1.05667e15i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 15317100 q^{9}+O(q^{10}) \) 4 * q - 15317100 * q^9	\( 4 q - 15317100 q^{9} + 1017781336 q^{13} - 22057873992 q^{21} + 98927122100 q^{25} + 700986197952 q^{33} - 715633821064 q^{37} - 4207432014720 q^{45} + 5607449975596 q^{49} - 6418720383768 q^{57} - 57879101335528 q^{61} + 156468753896832 q^{69} + 267147849446632 q^{73} - 706257752173596 q^{81} - 224748150612480 q^{85} + 14\!\cdots\!64 q^{93}+ \cdots - 15\!\cdots\!20 q^{97}+O(q^{100}) \) 4 * q - 15317100 * q^9 + 1017781336 * q^13 - 22057873992 * q^21 + 98927122100 * q^25 + 700986197952 * q^33 - 715633821064 * q^37 - 4207432014720 * q^45 + 5607449975596 * q^49 - 6418720383768 * q^57 - 57879101335528 * q^61 + 156468753896832 * q^69 + 267147849446632 * q^73 - 706257752173596 * q^81 - 224748150612480 * q^85 + 1441865040435864 * q^93 - 1500683175752120 * q^97

Introduction

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