LMFDB - Embedded newform 117.16.b.f.64.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,16,Mod(64,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("117.64");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	117.b (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:	$166.951400967$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.12
Character	$\chi$	$=$	117.64
Dual form			117.16.b.f.64.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-141.961i q^{2} +12615.2 q^{4} -269224. i q^{5} +1.58849e6i q^{7} -6.44262e6i q^{8} +O(q^{10})$	\(q-141.961i q^{2} +12615.2 q^{4} -269224. i q^{5} +1.58849e6i q^{7} -6.44262e6i q^{8} -3.82193e7 q^{10} +8.17038e7i q^{11} +(-6.49559e7 + 2.16718e8i) q^{13} +2.25503e8 q^{14} -5.01226e8 q^{16} -2.01236e9 q^{17} +6.31189e9i q^{19} -3.39631e9i q^{20} +1.15987e10 q^{22} -2.44145e9 q^{23} -4.19641e10 q^{25} +(3.07654e10 + 9.22118e9i) q^{26} +2.00391e10i q^{28} +1.75153e11 q^{29} -2.52980e10i q^{31} -1.39958e11i q^{32} +2.85676e11i q^{34} +4.27660e11 q^{35} -1.06109e12i q^{37} +8.96040e11 q^{38} -1.73451e12 q^{40} -1.72604e12i q^{41} +6.66320e11 q^{43} +1.03071e12i q^{44} +3.46590e11i q^{46} -5.73554e11i q^{47} +2.22426e12 q^{49} +5.95725e12i q^{50} +(-8.19429e11 + 2.73393e12i) q^{52} +5.50139e12 q^{53} +2.19966e13 q^{55} +1.02340e13 q^{56} -2.48648e13i q^{58} -3.69490e13i q^{59} -2.65290e13 q^{61} -3.59133e12 q^{62} -3.62926e13 q^{64} +(5.83457e13 + 1.74877e13i) q^{65} +6.41643e13i q^{67} -2.53862e13 q^{68} -6.07109e13i q^{70} +6.55183e13i q^{71} -1.34473e14i q^{73} -1.50633e14 q^{74} +7.96256e13i q^{76} -1.29786e14 q^{77} +1.46783e14 q^{79} +1.34942e14i q^{80} -2.45029e14 q^{82} -1.34487e14i q^{83} +5.41776e14i q^{85} -9.45912e13i q^{86} +5.26387e14 q^{88} -1.76064e14i q^{89} +(-3.44254e14 - 1.03182e14i) q^{91} -3.07993e13 q^{92} -8.14221e13 q^{94} +1.69931e15 q^{95} -6.00061e14i q^{97} -3.15757e14i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 458752 q^{4}+O(q^{10}) $ 28 * q - 458752 * q^4	$ 28 q - 458752 q^{4} - 228861840 q^{10} + 803257676 q^{13} - 7451384960 q^{16} - 30544905456 q^{22} - 238197886180 q^{25} + 3510668297280 q^{40} + 10830121503776 q^{43} - 19674845564108 q^{49} + 30846257429728 q^{52} + 53012410974240 q^{55} - 23758117088504 q^{61} + 327062953083392 q^{64} - 829003210390736 q^{79} - 13\!\cdots\!56 q^{82}+ \cdots + 81131101347024 q^{94}+O(q^{100}) $ 28 * q - 458752 * q^4 - 228861840 * q^10 + 803257676 * q^13 - 7451384960 * q^16 - 30544905456 * q^22 - 238197886180 * q^25 + 3510668297280 * q^40 + 10830121503776 * q^43 - 19674845564108 * q^49 + 30846257429728 * q^52 + 53012410974240 * q^55 - 23758117088504 * q^61 + 327062953083392 * q^64 - 829003210390736 * q^79 - 1396925338271856 * q^82 + 1057698205097664 * q^88 - 1934316425142192 * q^91 + 81131101347024 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	141.961i		−	0.784229i	−0.919916	−	0.392115i	$-0.871744\pi$
$2$		−	141.961i		−	0.784229i	0.919916	−	0.392115i	$-0.128256\pi$
$3$		0			0
$3$		0			0
$4$	12615.2			0.384984
$5$		−	269224.i		−	1.54113i	−0.637362	−	0.770565i	$-0.719974\pi$
$5$		−	269224.i		−	1.54113i	0.637362	−	0.770565i	$-0.280026\pi$
$6$		0			0
$7$			1.58849e6i			0.729037i	0.931196	+	0.364518i	$0.118766\pi$
$7$			1.58849e6i			0.729037i	−0.931196	+	0.364518i	$0.881234\pi$
$8$		−	6.44262e6i		−	1.08615i
$9$		0			0
$10$	−3.82193e7			−1.20860
$11$			8.17038e7i			1.26415i	0.774909	+	0.632073i	$0.217796\pi$
$11$			8.17038e7i			1.26415i	−0.774909	+	0.632073i	$0.782204\pi$
$12$		0			0
$13$	−6.49559e7	+	2.16718e8i	−0.287107	+	0.957899i
$13$	−6.49559e7	+	2.16718e8i	−0.287107	+	0.957899i
$14$	2.25503e8			0.571732
$15$		0			0
$16$	−5.01226e8			−0.466803
$17$	−2.01236e9			−1.18943			−0.594714	−	0.803937i	$-0.702735\pi$
$17$	−2.01236e9			−1.18943			−0.594714	+	0.803937i	$0.702735\pi$
$18$		0			0
$19$			6.31189e9i			1.61997i	0.586449	+	0.809986i	$0.300525\pi$
$19$			6.31189e9i			1.61997i	−0.586449	+	0.809986i	$0.699475\pi$
$20$		−	3.39631e9i		−	0.593311i
$21$		0			0
$22$	1.15987e10			0.991380
$23$	−2.44145e9			−0.149516			−0.0747582	−	0.997202i	$-0.523818\pi$
$23$	−2.44145e9			−0.149516			−0.0747582	+	0.997202i	$0.523818\pi$
$24$		0			0
$25$	−4.19641e10			−1.37508
$26$	3.07654e10	+	9.22118e9i	0.751212	+	0.225158i
$27$		0			0
$28$			2.00391e10i			0.280668i
$29$	1.75153e11			1.88552			0.942761	−	0.333468i	$-0.108219\pi$
$29$	1.75153e11			1.88552			0.942761	+	0.333468i	$0.108219\pi$
$30$		0			0
$31$		−	2.52980e10i		−	0.165148i	−0.996585	−	0.0825742i	$-0.973686\pi$
$31$		−	2.52980e10i		−	0.165148i	0.996585	−	0.0825742i	$-0.0263141\pi$
$32$		−	1.39958e11i		−	0.720065i
$33$		0			0
$34$			2.85676e11i			0.932785i
$35$	4.27660e11			1.12354
$36$		0			0
$37$		−	1.06109e12i		−	1.83755i	−0.394780	−	0.918776i	$-0.629179\pi$
$37$		−	1.06109e12i		−	1.83755i	0.394780	−	0.918776i	$-0.370821\pi$
$38$	8.96040e11			1.27043
$39$		0			0
$40$	−1.73451e12			−1.67389
$41$		−	1.72604e12i		−	1.38411i	−0.721844	−	0.692056i	$-0.756705\pi$
$41$		−	1.72604e12i		−	1.38411i	0.721844	−	0.692056i	$-0.243295\pi$
$42$		0			0
$43$	6.66320e11			0.373826			0.186913	−	0.982376i	$-0.440152\pi$
$43$	6.66320e11			0.373826			0.186913	+	0.982376i	$0.440152\pi$
$44$			1.03071e12i			0.486676i
$45$		0			0
$46$			3.46590e11i			0.117255i
$47$		−	5.73554e11i		−	0.165136i	−0.996585	−	0.0825678i	$-0.973688\pi$
$47$		−	5.73554e11i		−	0.165136i	0.996585	−	0.0825678i	$-0.0263121\pi$
$48$		0			0
$49$	2.22426e12			0.468506
$50$			5.95725e12i			1.07838i
$51$		0			0
$52$	−8.19429e11	+	2.73393e12i	−0.110532	+	0.368776i
$53$	5.50139e12			0.643285			0.321643	−	0.946861i	$-0.395765\pi$
$53$	5.50139e12			0.643285			0.321643	+	0.946861i	$0.395765\pi$
$54$		0			0
$55$	2.19966e13			1.94821
$56$	1.02340e13			0.791840
$57$		0			0
$58$		−	2.48648e13i		−	1.47868i
$59$		−	3.69490e13i		−	1.93292i	−0.256825	−	0.966458i	$-0.582676\pi$
$59$		−	3.69490e13i		−	1.93292i	0.256825	−	0.966458i	$-0.417324\pi$
$60$		0			0
$61$	−2.65290e13			−1.08080			−0.540401	−	0.841407i	$-0.681728\pi$
$61$	−2.65290e13			−1.08080			−0.540401	+	0.841407i	$0.681728\pi$
$62$	−3.59133e12			−0.129514
$63$		0			0
$64$	−3.62926e13			−1.03150
$65$	5.83457e13	+	1.74877e13i	1.47625	+	0.442469i
$66$		0			0
$67$			6.41643e13i			1.29340i	0.762745	+	0.646699i	$0.223851\pi$
$67$			6.41643e13i			1.29340i	−0.762745	+	0.646699i	$0.776149\pi$
$68$	−2.53862e13			−0.457911
$69$		0			0
$70$		−	6.07109e13i		−	0.881113i
$71$			6.55183e13i			0.854919i	0.904035	+	0.427459i	$0.140591\pi$
$71$			6.55183e13i			0.854919i	−0.904035	+	0.427459i	$0.859409\pi$
$72$		0			0
$73$		−	1.34473e14i		−	1.42467i	−0.701842	−	0.712333i	$-0.747639\pi$
$73$		−	1.34473e14i		−	1.42467i	0.701842	−	0.712333i	$-0.252361\pi$
$74$	−1.50633e14			−1.44106
$75$		0			0
$76$			7.96256e13i			0.623664i
$77$	−1.29786e14			−0.921609
$78$		0			0
$79$	1.46783e14			0.859946			0.429973	−	0.902842i	$-0.358523\pi$
$79$	1.46783e14			0.859946			0.429973	+	0.902842i	$0.358523\pi$
$80$			1.34942e14i			0.719403i
$81$		0			0
$82$	−2.45029e14			−1.08546
$83$		−	1.34487e14i		−	0.543996i	−0.962298	−	0.271998i	$-0.912316\pi$
$83$		−	1.34487e14i		−	0.543996i	0.962298	−	0.271998i	$-0.0876844\pi$
$84$		0			0
$85$			5.41776e14i			1.83306i
$86$		−	9.45912e13i		−	0.293165i
$87$		0			0
$88$	5.26387e14			1.37305
$89$		−	1.76064e14i		−	0.421935i	−0.977493	−	0.210968i	$-0.932339\pi$
$89$		−	1.76064e14i		−	0.421935i	0.977493	−	0.210968i	$-0.0676615\pi$
$90$		0			0
$91$	−3.44254e14	−	1.03182e14i	−0.698343	−	0.209311i
$92$	−3.07993e13			−0.0575615
$93$		0			0
$94$	−8.14221e13			−0.129504
$95$	1.69931e15			2.49659
$96$		0			0
$97$		−	6.00061e14i		−	0.754063i	−0.926200	−	0.377032i	$-0.876945\pi$
$97$		−	6.00061e14i		−	0.754063i	0.926200	−	0.377032i	$-0.123055\pi$
$98$		−	3.15757e14i		−	0.367416i
$99$		0			0
$100$	−5.29384e14			−0.529384
$101$	1.57592e15			1.46259			0.731297	−	0.682059i	$-0.238915\pi$
$101$	1.57592e15			1.46259			0.731297	+	0.682059i	$0.238915\pi$
$102$		0			0
$103$	1.80457e15			1.44576			0.722878	−	0.690976i	$-0.242819\pi$
$103$	1.80457e15			1.44576			0.722878	+	0.690976i	$0.242819\pi$
$104$	1.39623e15	+	4.18486e14i	1.04042	+	0.311840i
$105$		0			0
$106$		−	7.80981e14i		−	0.504483i
$107$	1.85666e15			1.11777			0.558887	−	0.829244i	$-0.311228\pi$
$107$	1.85666e15			1.11777			0.558887	+	0.829244i	$0.311228\pi$
$108$		0			0
$109$			7.79314e14i			0.408332i	0.978936	+	0.204166i	$0.0654482\pi$
$109$			7.79314e14i			0.408332i	−0.978936	+	0.204166i	$0.934552\pi$
$110$		−	3.12266e15i		−	1.52785i
$111$		0			0
$112$		−	7.96192e14i		−	0.340316i
$113$	−3.58550e14			−0.143371			−0.0716855	−	0.997427i	$-0.522838\pi$
$113$	−3.58550e14			−0.143371			−0.0716855	+	0.997427i	$0.522838\pi$
$114$		0			0
$115$			6.57297e14i			0.230424i
$116$	2.20958e15			0.725897
$117$		0			0
$118$	−5.24531e15			−1.51585
$119$		−	3.19661e15i		−	0.867137i
$120$		0			0
$121$	−2.49826e15			−0.598065
$122$			3.76607e15i			0.847597i
$123$		0			0
$124$		−	3.19139e14i		−	0.0635795i
$125$			3.08168e15i			0.578047i
$126$		0			0
$127$	9.04525e15			1.50623			0.753117	−	0.657886i	$-0.228549\pi$
$127$	9.04525e15			1.50623			0.753117	+	0.657886i	$0.228549\pi$
$128$			5.65996e14i			0.0888666i
$129$		0			0
$130$	2.48257e15	−	8.28279e15i	0.346997	−	1.15772i
$131$	−9.24204e15			−1.21964			−0.609822	−	0.792539i	$-0.708759\pi$
$131$	−9.24204e15			−1.21964			−0.609822	+	0.792539i	$0.708759\pi$
$132$		0			0
$133$	−1.00264e16			−1.18102
$134$	9.10881e15			1.01432
$135$		0			0
$136$			1.29649e16i			1.29189i
$137$		−	2.06025e15i		−	0.194319i	−0.995269	−	0.0971594i	$-0.969024\pi$
$137$		−	2.06025e15i		−	0.194319i	0.995269	−	0.0971594i	$-0.0309757\pi$
$138$		0			0
$139$	−9.45441e15			−0.799878			−0.399939	−	0.916542i	$-0.630969\pi$
$139$	−9.45441e15			−0.799878			−0.399939	+	0.916542i	$0.630969\pi$
$140$	5.39500e15			0.432545
$141$		0			0
$142$	9.30102e15			0.670452
$143$	−1.77067e16	−	5.30714e15i	−1.21092	−	0.362945i
$144$		0			0
$145$		−	4.71553e16i		−	2.90583i
$146$	−1.90898e16			−1.11726
$147$		0			0
$148$		−	1.33858e16i		−	0.707429i
$149$			1.41857e16i			0.712779i	0.934337	+	0.356390i	$0.115992\pi$
$149$			1.41857e16i			0.712779i	−0.934337	+	0.356390i	$0.884008\pi$
$150$		0			0
$151$		−	1.87340e16i		−	0.851734i	−0.904786	−	0.425867i	$-0.859969\pi$
$151$		−	1.87340e16i		−	0.851734i	0.904786	−	0.425867i	$-0.140031\pi$
$152$	4.06651e16			1.75952
$153$		0			0
$154$			1.84245e16i			0.722753i
$155$	−6.81085e15			−0.254515
$156$		0			0
$157$	4.93734e16			1.67589			0.837945	−	0.545754i	$-0.183757\pi$
$157$	4.93734e16			1.67589			0.837945	+	0.545754i	$0.183757\pi$
$158$		−	2.08373e16i		−	0.674395i
$159$		0			0
$160$	−3.76800e16			−1.10971
$161$		−	3.87822e15i		−	0.109003i
$162$		0			0
$163$			5.60000e16i			1.43477i	0.696679	+	0.717383i	$0.254660\pi$
$163$			5.60000e16i			1.43477i	−0.696679	+	0.717383i	$0.745340\pi$
$164$		−	2.17743e16i		−	0.532862i
$165$		0			0
$166$	−1.90919e16			−0.426617
$167$			4.99017e16i			1.06596i	0.846127	+	0.532981i	$0.178928\pi$
$167$			4.99017e16i			1.06596i	−0.846127	+	0.532981i	$0.821072\pi$
$168$		0			0
$169$	−4.27474e16	−	2.81542e16i	−0.835139	−	0.550038i
$170$	7.69108e16			1.43754
$171$		0			0
$172$	8.40573e15			0.143917
$173$	5.64896e15			0.0926025			0.0463013	−	0.998928i	$-0.485257\pi$
$173$	5.64896e15			0.0926025			0.0463013	+	0.998928i	$0.485257\pi$
$174$		0			0
$175$		−	6.66596e16i		−	1.00248i
$176$		−	4.09520e16i		−	0.590107i
$177$		0			0
$178$	−2.49942e16			−0.330894
$179$	−4.30578e14			−0.00546580			−0.00273290	−	0.999996i	$-0.500870\pi$
$179$	−4.30578e14			−0.00546580			−0.00273290	+	0.999996i	$0.500870\pi$
$180$		0			0
$181$	−5.86902e16			−0.685451			−0.342726	−	0.939436i	$-0.611350\pi$
$181$	−5.86902e16			−0.685451			−0.342726	+	0.939436i	$0.611350\pi$
$182$	−1.46478e16	+	4.88706e16i	−0.164148	+	0.547661i
$183$		0			0
$184$			1.57293e16i			0.162396i
$185$	−2.85671e17			−2.83190
$186$		0			0
$187$		−	1.64417e17i		−	1.50361i
$188$		−	7.23548e15i		−	0.0635746i
$189$		0			0
$190$		−	2.41236e17i		−	1.95790i
$191$	−4.56593e16			−0.356270			−0.178135	−	0.984006i	$-0.557006\pi$
$191$	−4.56593e16			−0.356270			−0.178135	+	0.984006i	$0.557006\pi$
$192$		0			0
$193$		−	4.87882e16i		−	0.352075i	−0.984383	−	0.176037i	$-0.943672\pi$
$193$		−	4.87882e16i		−	0.352075i	0.984383	−	0.176037i	$-0.0563279\pi$
$194$	−8.51851e16			−0.591358
$195$		0			0
$196$	2.80594e16			0.180367
$197$		−	6.15425e16i		−	0.380784i	−0.981708	−	0.190392i	$-0.939024\pi$
$197$		−	6.15425e16i		−	0.380784i	0.981708	−	0.190392i	$-0.0609758\pi$
$198$		0			0
$199$	2.63408e17			1.51088			0.755440	−	0.655217i	$-0.227423\pi$
$199$	2.63408e17			1.51088			0.755440	+	0.655217i	$0.227423\pi$
$200$			2.70359e17i			1.49354i
$201$		0			0
$202$		−	2.23719e17i		−	1.14701i
$203$			2.78228e17i			1.37462i
$204$		0			0
$205$	−4.64691e17			−2.13310
$206$		−	2.56178e17i		−	1.13380i
$207$		0			0
$208$	3.25575e16	−	1.08625e17i	0.134022	−	0.447150i
$209$	−5.15705e17			−2.04788
$210$		0			0
$211$	−3.09018e17			−1.14253			−0.571263	−	0.820767i	$-0.693546\pi$
$211$	−3.09018e17			−1.14253			−0.571263	+	0.820767i	$0.693546\pi$
$212$	6.94010e16			0.247655
$213$		0			0
$214$		−	2.63573e17i		−	0.876592i
$215$		−	1.79389e17i		−	0.576114i
$216$		0			0
$217$	4.01857e16			0.120399
$218$	1.10632e17			0.320226
$219$		0			0
$220$	2.77491e17			0.750031
$221$	1.30714e17	−	4.36114e17i	0.341493	−	1.13935i
$222$		0			0
$223$		−	5.37325e17i		−	1.31205i	−0.754739	−	0.656025i	$-0.772236\pi$
$223$		−	5.37325e17i		−	1.31205i	0.754739	−	0.656025i	$-0.227764\pi$
$224$	2.22321e17			0.524954
$225$		0			0
$226$			5.09000e16i			0.112436i
$227$			8.61724e17i			1.84151i	0.390139	+	0.920756i	$0.372427\pi$
$227$			8.61724e17i			1.84151i	−0.390139	+	0.920756i	$0.627573\pi$
$228$		0			0
$229$			4.13888e16i			0.0828165i	0.999142	+	0.0414082i	$0.0131844\pi$
$229$			4.13888e16i			0.0828165i	−0.999142	+	0.0414082i	$0.986816\pi$
$230$	9.33103e16			0.180705
$231$		0			0
$232$		−	1.12844e18i		−	2.04795i
$233$	4.27098e17			0.750513			0.375257	−	0.926921i	$-0.377555\pi$
$233$	4.27098e17			0.750513			0.375257	+	0.926921i	$0.377555\pi$
$234$		0			0
$235$	−1.54415e17			−0.254495
$236$		−	4.66118e17i		−	0.744142i
$237$		0			0
$238$	−4.53793e17			−0.680034
$239$		−	1.96160e16i		−	0.0284857i	−0.999899	−	0.0142428i	$-0.995466\pi$
$239$		−	1.96160e16i		−	0.0284857i	0.999899	−	0.0142428i	$-0.00453379\pi$
$240$		0			0
$241$		−	1.15034e18i		−	1.56927i	−0.619955	−	0.784637i	$-0.712849\pi$
$241$		−	1.15034e18i		−	1.56927i	0.619955	−	0.784637i	$-0.287151\pi$
$242$			3.54655e17i			0.469020i
$243$		0			0
$244$	−3.34667e17			−0.416092
$245$		−	5.98824e17i		−	0.722028i
$246$		0			0
$247$	−1.36790e18	−	4.09994e17i	−1.55177	−	0.465105i
$248$	−1.62986e17			−0.179375
$249$		0			0
$250$	4.37478e17			0.453321
$251$	1.20029e18			1.20707			0.603537	−	0.797335i	$-0.293758\pi$
$251$	1.20029e18			1.20707			0.603537	+	0.797335i	$0.293758\pi$
$252$		0			0
$253$		−	1.99476e17i		−	0.189010i
$254$		−	1.28407e18i		−	1.18123i
$255$		0			0
$256$	−1.10889e18			−0.961807
$257$	1.71728e17			0.144658			0.0723290	−	0.997381i	$-0.476957\pi$
$257$	1.71728e17			0.144658			0.0723290	+	0.997381i	$0.476957\pi$
$258$		0			0
$259$	1.68553e18			1.33964
$260$	7.36041e17	+	2.20610e17i	0.568332	+	0.170343i
$261$		0			0
$262$			1.31201e18i			0.956480i
$263$	1.87034e18			1.32511			0.662555	−	0.749014i	$-0.269472\pi$
$263$	1.87034e18			1.32511			0.662555	+	0.749014i	$0.269472\pi$
$264$		0			0
$265$		−	1.48111e18i		−	0.991386i
$266$			1.42335e18i			0.926190i
$267$		0			0
$268$			8.09443e17i			0.497938i
$269$	5.70746e17			0.341429			0.170715	−	0.985321i	$-0.445392\pi$
$269$	5.70746e17			0.341429			0.170715	+	0.985321i	$0.445392\pi$
$270$		0			0
$271$		−	2.12642e18i		−	1.20332i	−0.798753	−	0.601659i	$-0.794507\pi$
$271$		−	2.12642e18i		−	1.20332i	0.798753	−	0.601659i	$-0.205493\pi$
$272$	1.00865e18			0.555228
$273$		0			0
$274$	−2.92474e17			−0.152391
$275$		−	3.42863e18i		−	1.73830i
$276$		0			0
$277$	2.94236e18			1.41286			0.706428	−	0.707785i	$-0.250305\pi$
$277$	2.94236e18			1.41286			0.706428	+	0.707785i	$0.250305\pi$
$278$			1.34215e18i			0.627288i
$279$		0			0
$280$		−	2.75525e18i		−	1.22033i
$281$			5.21616e17i			0.224933i	0.993655	+	0.112467i	$0.0358752\pi$
$281$			5.21616e17i			0.224933i	−0.993655	+	0.112467i	$0.964125\pi$
$282$		0			0
$283$	4.49428e17			0.183764			0.0918822	−	0.995770i	$-0.470712\pi$
$283$	4.49428e17			0.183764			0.0918822	+	0.995770i	$0.470712\pi$
$284$			8.26524e17i			0.329130i
$285$		0			0
$286$	−7.53406e17	+	2.51365e18i	−0.284632	+	0.949642i
$287$	2.74179e18			1.00907
$288$		0			0
$289$	1.18716e18			0.414740
$290$	−6.69420e18			−2.27884
$291$		0			0
$292$		−	1.69640e18i		−	0.548474i
$293$		−	6.32579e17i		−	0.199346i	−0.995020	−	0.0996731i	$-0.968220\pi$
$293$		−	6.32579e17i		−	0.199346i	0.995020	−	0.0996731i	$-0.0317797\pi$
$294$		0			0
$295$	−9.94758e18			−2.97887
$296$	−6.83621e18			−1.99585
$297$		0			0
$298$	2.01382e18			0.558982
$299$	1.58586e17	−	5.29105e17i	0.0429271	−	0.143221i
$300$		0			0
$301$			1.05844e18i			0.272533i
$302$	−2.65950e18			−0.667955
$303$		0			0
$304$		−	3.16368e18i		−	0.756207i
$305$			7.14224e18i			1.66566i
$306$		0			0
$307$		−	3.88414e18i		−	0.862496i	−0.902233	−	0.431248i	$-0.858073\pi$
$307$		−	3.88414e18i		−	0.862496i	0.902233	−	0.431248i	$-0.141927\pi$
$308$	−1.63727e18			−0.354805
$309$		0			0
$310$			9.66872e17i			0.199598i
$311$	9.39943e17			0.189408			0.0947041	−	0.995505i	$-0.469810\pi$
$311$	9.39943e17			0.189408			0.0947041	+	0.995505i	$0.469810\pi$
$312$		0			0
$313$	1.90519e18			0.365895			0.182948	−	0.983123i	$-0.441436\pi$
$313$	1.90519e18			0.365895			0.182948	+	0.983123i	$0.441436\pi$
$314$		−	7.00908e18i		−	1.31428i
$315$		0			0
$316$	1.85169e18			0.331066
$317$			3.49238e18i			0.609786i	0.952387	+	0.304893i	$0.0986207\pi$
$317$			3.49238e18i			0.609786i	−0.952387	+	0.304893i	$0.901379\pi$
$318$		0			0
$319$			1.43106e19i			2.38358i
$320$			9.77086e18i			1.58967i
$321$		0			0
$322$	−5.50554e17			−0.0854833
$323$		−	1.27018e19i		−	1.92684i
$324$		0			0
$325$	2.72582e18	−	9.09437e18i	0.394795	−	1.31719i
$326$	7.94980e18			1.12519
$327$		0			0
$328$	−1.11202e19			−1.50335
$329$	9.11086e17			0.120390
$330$		0			0
$331$			1.13573e19i			1.43405i	0.697045	+	0.717027i	$0.254498\pi$
$331$			1.13573e19i			1.43405i	−0.697045	+	0.717027i	$0.745502\pi$
$332$		−	1.69658e18i		−	0.209430i
$333$		0			0
$334$	7.08407e18			0.835958
$335$	1.72746e19			1.99330
$336$		0			0
$337$	1.18014e19			1.30230			0.651148	−	0.758951i	$-0.274288\pi$
$337$	1.18014e19			1.30230			0.651148	+	0.758951i	$0.274288\pi$
$338$	−3.99679e18	+	6.06844e18i	−0.431356	+	0.654941i
$339$		0			0
$340$			6.83459e18i			0.705701i
$341$	2.06695e18			0.208772
$342$		0			0
$343$			1.10747e19i			1.07059i
$344$		−	4.29285e18i		−	0.406029i
$345$		0			0
$346$		−	8.01930e17i		−	0.0726216i
$347$	−6.70836e18			−0.594491			−0.297245	−	0.954801i	$-0.596068\pi$
$347$	−6.70836e18			−0.594491			−0.297245	+	0.954801i	$0.596068\pi$
$348$		0			0
$349$			1.12618e19i			0.955913i	0.878383	+	0.477957i	$0.158622\pi$
$349$			1.12618e19i			0.955913i	−0.878383	+	0.477957i	$0.841378\pi$
$350$	−9.46304e18			−0.786177
$351$		0			0
$352$	1.14351e19			0.910267
$353$		−	1.56700e19i		−	1.22112i	−0.791968	−	0.610562i	$-0.790944\pi$
$353$		−	1.56700e19i		−	1.22112i	0.791968	−	0.610562i	$-0.209056\pi$
$354$		0			0
$355$	1.76391e19			1.31754
$356$		−	2.22108e18i		−	0.162439i
$357$		0			0
$358$			6.11251e16i			0.00428644i
$359$			1.12879e19i			0.775185i	0.921831	+	0.387592i	$0.126693\pi$
$359$			1.12879e19i			0.775185i	−0.921831	+	0.387592i	$0.873307\pi$
$360$		0			0
$361$	−2.46588e19			−1.62431
$362$			8.33171e18i			0.537551i
$363$		0			0
$364$	−4.34283e18	−	1.30166e18i	−0.268851	−	0.0805816i
$365$	−3.62033e19			−2.19559
$366$		0			0
$367$	1.60597e19			0.934853			0.467426	−	0.884032i	$-0.345181\pi$
$367$	1.60597e19			0.934853			0.467426	+	0.884032i	$0.345181\pi$
$368$	1.22372e18			0.0697946
$369$		0			0
$370$			4.05541e19i			2.22086i
$371$			8.73891e18i			0.468979i
$372$		0			0
$373$	−8.71777e18			−0.449355			−0.224677	−	0.974433i	$-0.572133\pi$
$373$	−8.71777e18			−0.449355			−0.224677	+	0.974433i	$0.572133\pi$
$374$	−2.33408e19			−1.17918
$375$		0			0
$376$	−3.69519e18			−0.179361
$377$	−1.13772e19	+	3.79587e19i	−0.541346	+	1.80614i
$378$		0			0
$379$		−	8.65764e18i		−	0.395918i	−0.980210	−	0.197959i	$-0.936569\pi$
$379$		−	8.65764e18i		−	0.395918i	0.980210	−	0.197959i	$-0.0634313\pi$
$380$	2.14371e19			0.961147
$381$		0			0
$382$			6.48182e18i			0.279397i
$383$			1.21669e19i			0.514268i	0.966376	+	0.257134i	$0.0827782\pi$
$383$			1.21669e19i			0.514268i	−0.966376	+	0.257134i	$0.917222\pi$
$384$		0			0
$385$			3.49415e19i			1.42032i
$386$	−6.92600e18			−0.276107
$387$		0			0
$388$		−	7.56988e18i		−	0.290303i
$389$	−2.49905e19			−0.940054			−0.470027	−	0.882652i	$-0.655756\pi$
$389$	−2.49905e19			−0.940054			−0.470027	+	0.882652i	$0.655756\pi$
$390$		0			0
$391$	4.91307e18			0.177839
$392$		−	1.43301e19i		−	0.508865i
$393$		0			0
$394$	−8.73661e18			−0.298622
$395$		−	3.95174e19i		−	1.32529i
$396$		0			0
$397$			5.50755e19i			1.77840i	0.457519	+	0.889200i	$0.348738\pi$
$397$			5.50755e19i			1.77840i	−0.457519	+	0.889200i	$0.651262\pi$
$398$		−	3.73935e19i		−	1.18488i
$399$		0			0
$400$	2.10335e19			0.641891
$401$		−	4.29204e19i		−	1.28553i	−0.766065	−	0.642763i	$-0.777788\pi$
$401$		−	4.29204e19i		−	1.28553i	0.766065	−	0.642763i	$-0.222212\pi$
$402$		0			0
$403$	5.48254e18	+	1.64326e18i	0.158195	+	0.0474152i
$404$	1.98805e19			0.563076
$405$		0			0
$406$	3.94975e19			1.07801
$407$	8.66951e19			2.32293
$408$		0			0
$409$			2.01533e19i			0.520500i	0.965541	+	0.260250i	$0.0838049\pi$
$409$			2.01533e19i			0.520500i	−0.965541	+	0.260250i	$0.916195\pi$
$410$			6.59679e19i			1.67284i
$411$		0			0
$412$	2.27650e19			0.556593
$413$	5.86932e19			1.40917
$414$		0			0
$415$	−3.62072e19			−0.838368
$416$	3.03313e19	+	9.09107e18i	0.689749	+	0.206736i
$417$		0			0
$418$			7.32099e19i			1.60601i
$419$	−6.29868e19			−1.35720			−0.678601	−	0.734507i	$-0.737414\pi$
$419$	−6.29868e19			−1.35720			−0.678601	+	0.734507i	$0.737414\pi$
$420$		0			0
$421$		−	5.97337e19i		−	1.24195i	−0.783831	−	0.620974i	$-0.786737\pi$
$421$		−	5.97337e19i		−	1.24195i	0.783831	−	0.620974i	$-0.213263\pi$
$422$			4.38684e19i			0.896002i
$423$		0			0
$424$		−	3.54434e19i		−	0.698701i
$425$	8.44468e19			1.63556
$426$		0			0
$427$		−	4.21410e19i		−	0.787945i
$428$	2.34221e19			0.430326
$429$		0			0
$430$	−2.54662e19			−0.451805
$431$		−	4.65563e19i		−	0.811706i	−0.913938	−	0.405853i	$-0.866975\pi$
$431$		−	4.65563e19i		−	0.811706i	0.913938	−	0.405853i	$-0.133025\pi$
$432$		0			0
$433$	−4.75213e19			−0.800256			−0.400128	−	0.916459i	$-0.631034\pi$
$433$	−4.75213e19			−0.800256			−0.400128	+	0.916459i	$0.631034\pi$
$434$		−	5.70479e18i		−	0.0944206i
$435$		0			0
$436$			9.83117e18i			0.157201i
$437$		−	1.54102e19i		−	0.242212i
$438$		0			0
$439$	6.01663e19			0.913839			0.456919	−	0.889508i	$-0.348953\pi$
$439$	6.01663e19			0.913839			0.456919	+	0.889508i	$0.348953\pi$
$440$		−	1.41716e20i		−	2.11604i
$441$		0			0
$442$	−6.19110e19	−	1.85563e19i	−0.893513	−	0.267809i
$443$	1.13380e20			1.60882			0.804412	−	0.594072i	$-0.202481\pi$
$443$	1.13380e20			1.60882			0.804412	+	0.594072i	$0.202481\pi$
$444$		0			0
$445$	−4.74008e19			−0.650257
$446$	−7.62790e19			−1.02895
$447$		0			0
$448$		−	5.76505e19i		−	0.752000i
$449$			1.06765e20i			1.36956i	0.728748	+	0.684782i	$0.240103\pi$
$449$			1.06765e20i			1.36956i	−0.728748	+	0.684782i	$0.759897\pi$
$450$		0			0
$451$	1.41024e20			1.74972
$452$	−4.52317e18			−0.0551956
$453$		0			0
$454$	1.22331e20			1.44417
$455$	−2.77790e19	+	9.26816e19i	−0.322576	+	1.07624i
$456$		0			0
$457$			1.12729e20i			1.26668i	0.773875	+	0.633338i	$0.218316\pi$
$457$			1.12729e20i			1.26668i	−0.773875	+	0.633338i	$0.781684\pi$
$458$	5.87558e18			0.0649471
$459$		0			0
$460$			8.29191e18i			0.0887096i
$461$		−	1.31348e20i		−	1.38250i	−0.722614	−	0.691252i	$-0.757059\pi$
$461$		−	1.31348e20i		−	1.38250i	0.722614	−	0.691252i	$-0.242941\pi$
$462$		0			0
$463$			5.31460e19i			0.541518i	0.962647	+	0.270759i	$0.0872746\pi$
$463$			5.31460e19i			0.541518i	−0.962647	+	0.270759i	$0.912725\pi$
$464$	−8.77910e19			−0.880167
$465$		0			0
$466$		−	6.06311e19i		−	0.588574i
$467$	1.05638e20			1.00912			0.504559	−	0.863377i	$-0.331655\pi$
$467$	1.05638e20			1.00912			0.504559	+	0.863377i	$0.331655\pi$
$468$		0			0
$469$	−1.01924e20			−0.942935
$470$			2.19208e19i			0.199583i
$471$		0			0
$472$	−2.38049e20			−2.09943
$473$			5.44409e19i			0.472570i
$474$		0			0
$475$		−	2.64873e20i		−	2.22759i
$476$		−	4.03258e19i		−	0.333834i
$477$		0			0
$478$	−2.78470e18			−0.0223393
$479$		−	4.53323e19i		−	0.358007i	−0.983848	−	0.179004i	$-0.942713\pi$
$479$		−	4.53323e19i		−	0.358007i	0.983848	−	0.179004i	$-0.0572874\pi$
$480$		0			0
$481$	2.29957e20	+	6.89241e19i	1.76019	+	0.527573i
$482$	−1.63303e20			−1.23067
$483$		0			0
$484$	−3.15160e19			−0.230246
$485$	−1.61551e20			−1.16211
$486$		0			0
$487$			1.68685e20i			1.17655i	0.808662	+	0.588273i	$0.200192\pi$
$487$			1.68685e20i			1.17655i	−0.808662	+	0.588273i	$0.799808\pi$
$488$			1.70916e20i			1.17391i
$489$		0			0
$490$	−8.50095e19			−0.566235
$491$	−1.17394e20			−0.770080			−0.385040	−	0.922900i	$-0.625812\pi$
$491$	−1.17394e20			−0.770080			−0.385040	+	0.922900i	$0.625812\pi$
$492$		0			0
$493$	−3.52470e20			−2.24269
$494$	−5.82031e19	+	1.94188e20i	−0.364749	+	1.21694i
$495$		0			0
$496$			1.26800e19i			0.0770917i
$497$	−1.04075e20			−0.623267
$498$		0			0
$499$		−	5.93136e19i		−	0.344668i	−0.985039	−	0.172334i	$-0.944869\pi$
$499$		−	5.93136e19i		−	0.344668i	0.985039	−	0.172334i	$-0.0551308\pi$
$500$			3.88760e19i			0.222539i
$501$		0			0
$502$		−	1.70394e20i		−	0.946623i
$503$	7.71988e19			0.422523			0.211262	−	0.977430i	$-0.432243\pi$
$503$	7.71988e19			0.422523			0.211262	+	0.977430i	$0.432243\pi$
$504$		0			0
$505$		−	4.24276e20i		−	2.25405i
$506$	−2.83177e19			−0.148228
$507$		0			0
$508$	1.14107e20			0.579877
$509$			1.24384e19i			0.0622847i	0.999515	+	0.0311424i	$0.00991453\pi$
$509$			1.24384e19i			0.0622847i	−0.999515	+	0.0311424i	$0.990085\pi$
$510$		0			0
$511$	2.13609e20			1.03863
$512$			1.75965e20i			0.843144i
$513$		0			0
$514$		−	2.43786e19i		−	0.113445i
$515$		−	4.85834e20i		−	2.22810i
$516$		0			0
$517$	4.68616e19			0.208755
$518$		−	2.39279e20i		−	1.05059i
$519$		0			0
$520$	1.12667e20	−	3.75899e20i	0.480585	−	1.60342i
$521$	−9.96615e19			−0.419029			−0.209515	−	0.977806i	$-0.567188\pi$
$521$	−9.96615e19			−0.419029			−0.209515	+	0.977806i	$0.567188\pi$
$522$		0			0
$523$	1.01263e20			0.413701			0.206851	−	0.978373i	$-0.433679\pi$
$523$	1.01263e20			0.413701			0.206851	+	0.978373i	$0.433679\pi$
$524$	−1.16590e20			−0.469544
$525$		0			0
$526$		−	2.65514e20i		−	1.03919i
$527$			5.09087e19i			0.196432i
$528$		0			0
$529$	−2.60675e20			−0.977645
$530$	−2.10259e20			−0.777474
$531$		0			0
$532$	−1.26484e20			−0.454674
$533$	3.74063e20	+	1.12116e20i	1.32584	+	0.397388i
$534$		0			0
$535$		−	4.99858e20i		−	1.72264i
$536$	4.13387e20			1.40482
$537$		0			0
$538$		−	8.10235e19i		−	0.267759i
$539$			1.81730e20i			0.592259i
$540$		0			0
$541$		−	1.98925e18i		−	0.00630537i	−0.999995	−	0.00315269i	$-0.998996\pi$
$541$		−	1.98925e18i		−	0.00630537i	0.999995	−	0.00315269i	$-0.00100353\pi$
$542$	−3.01869e20			−0.943677
$543$		0			0
$544$			2.81645e20i			0.856466i
$545$	2.09810e20			0.629293
$546$		0			0
$547$	−1.10254e19			−0.0321727			−0.0160864	−	0.999871i	$-0.505121\pi$
$547$	−1.10254e19			−0.0321727			−0.0160864	+	0.999871i	$0.505121\pi$
$548$		−	2.59903e19i		−	0.0748097i
$549$		0			0
$550$	−4.86730e20			−1.36323
$551$			1.10554e21i			3.05449i
$552$		0			0
$553$			2.33163e20i			0.626932i
$554$		−	4.17700e20i		−	1.10800i
$555$		0			0
$556$	−1.19269e20			−0.307940
$557$		−	6.95297e20i		−	1.77116i	−0.464491	−	0.885578i	$-0.653763\pi$
$557$		−	6.95297e20i		−	1.77116i	0.464491	−	0.885578i	$-0.346237\pi$
$558$		0			0
$559$	−4.32814e19	+	1.44403e20i	−0.107328	+	0.358087i
$560$	−2.14354e20			−0.524471
$561$		0			0
$562$	7.40490e19			0.176399
$563$	5.82981e20			1.37038			0.685190	−	0.728364i	$-0.259719\pi$
$563$	5.82981e20			1.37038			0.685190	+	0.728364i	$0.259719\pi$
$564$		0			0
$565$			9.65303e19i			0.220953i
$566$		−	6.38011e19i		−	0.144113i
$567$		0			0
$568$	4.22110e20			0.928566
$569$	−8.71389e19			−0.189178			−0.0945889	−	0.995516i	$-0.530154\pi$
$569$	−8.71389e19			−0.189178			−0.0945889	+	0.995516i	$0.530154\pi$
$570$		0			0
$571$	−2.77001e20			−0.585749			−0.292874	−	0.956151i	$-0.594612\pi$
$571$	−2.77001e20			−0.585749			−0.292874	+	0.956151i	$0.594612\pi$
$572$	−2.23373e20	−	6.69505e19i	−0.466187	−	0.139728i
$573$		0			0
$574$		−	3.89227e20i		−	0.791341i
$575$	1.02453e20			0.205597
$576$		0			0
$577$		−	7.90363e20i		−	1.54528i	−0.634843	−	0.772641i	$-0.718935\pi$
$577$		−	7.90363e20i		−	1.54528i	0.634843	−	0.772641i	$-0.281065\pi$
$578$		−	1.68530e20i		−	0.325251i
$579$		0			0
$580$		−	5.94873e20i		−	1.11870i
$581$	2.13632e20			0.396593
$582$		0			0
$583$			4.49485e20i			0.813206i
$584$	−8.66357e20			−1.54739
$585$		0			0
$586$	−8.98014e19			−0.156333
$587$		−	7.13914e20i		−	1.22704i	−0.789678	−	0.613522i	$-0.789752\pi$
$587$		−	7.13914e20i		−	1.22704i	0.789678	−	0.613522i	$-0.210248\pi$
$588$		0			0
$589$	1.59678e20			0.267536
$590$			1.41216e21i			2.33612i
$591$		0			0
$592$			5.31846e20i			0.857774i
$593$		−	1.02402e21i		−	1.63079i	−0.578903	−	0.815397i	$-0.696519\pi$
$593$		−	1.02402e21i		−	1.63079i	0.578903	−	0.815397i	$-0.303481\pi$
$594$		0			0
$595$	−8.60605e20			−1.33637
$596$			1.78956e20i			0.274409i
$597$		0			0
$598$	−7.51121e19	−	2.25130e19i	−0.112318	−	0.0336647i
$599$	4.51645e20			0.666954			0.333477	−	0.942758i	$-0.391778\pi$
$599$	4.51645e20			0.666954			0.333477	+	0.942758i	$0.391778\pi$
$600$		0			0
$601$	4.52750e20			0.652078			0.326039	−	0.945356i	$-0.394286\pi$
$601$	4.52750e20			0.652078			0.326039	+	0.945356i	$0.394286\pi$
$602$	1.50257e20			0.213728
$603$		0			0
$604$		−	2.36333e20i		−	0.327904i
$605$			6.72593e20i			0.921695i
$606$		0			0
$607$	2.30992e20			0.308803			0.154401	−	0.988008i	$-0.450655\pi$
$607$	2.30992e20			0.308803			0.154401	+	0.988008i	$0.450655\pi$
$608$	8.83397e20			1.16648
$609$		0			0
$610$	1.01392e21			1.30626
$611$	1.24299e20	+	3.72557e19i	0.158183	+	0.0474115i
$612$		0			0
$613$			5.11436e20i			0.635093i	0.948243	+	0.317547i	$0.102859\pi$
$613$			5.11436e20i			0.635093i	−0.948243	+	0.317547i	$0.897141\pi$
$614$	−5.51395e20			−0.676395
$615$		0			0
$616$			8.36161e20i			1.00100i
$617$		−	7.17125e19i		−	0.0848117i	−0.999100	−	0.0424059i	$-0.986498\pi$
$617$		−	7.17125e19i		−	0.0848117i	0.999100	−	0.0424059i	$-0.0135023\pi$
$618$		0			0
$619$			7.50519e20i			0.866326i	0.901315	+	0.433163i	$0.142603\pi$
$619$			7.50519e20i			0.866326i	−0.901315	+	0.433163i	$0.857397\pi$
$620$	−8.59200e19			−0.0979843
$621$		0			0
$622$		−	1.33435e20i		−	0.148539i
$623$	2.79676e20			0.307606
$624$		0			0
$625$	−4.50979e20			−0.484235
$626$		−	2.70463e20i		−	0.286946i
$627$		0			0
$628$	6.22854e20			0.645192
$629$			2.13529e21i			2.18564i
$630$		0			0
$631$			4.57217e20i			0.456985i	0.973546	+	0.228493i	$0.0733797\pi$
$631$			4.57217e20i			0.456985i	−0.973546	+	0.228493i	$0.926620\pi$
$632$		−	9.45665e20i		−	0.934026i
$633$		0			0
$634$	4.95781e20			0.478212
$635$		−	2.43520e21i		−	2.32130i
$636$		0			0
$637$	−1.44479e20	+	4.82037e20i	−0.134511	+	0.448781i
$638$	2.03155e21			1.86927
$639$		0			0
$640$	1.52380e20			0.136955
$641$	−1.81335e21			−1.61082			−0.805411	−	0.592717i	$-0.798055\pi$
$641$	−1.81335e21			−1.61082			−0.805411	+	0.592717i	$0.798055\pi$
$642$		0			0
$643$		−	2.69253e19i		−	0.0233657i	−0.999932	−	0.0116828i	$-0.996281\pi$
$643$		−	2.69253e19i		−	0.0233657i	0.999932	−	0.0116828i	$-0.00371885\pi$
$644$		−	4.89244e19i		−	0.0419644i
$645$		0			0
$646$	−1.80315e21			−1.51108
$647$	7.71732e20			0.639270			0.319635	−	0.947541i	$-0.396440\pi$
$647$	7.71732e20			0.639270			0.319635	+	0.947541i	$0.396440\pi$
$648$		0			0
$649$	3.01888e21			2.44349
$650$	−1.29104e21	−	3.86959e20i	−1.03298	−	0.309610i
$651$		0			0
$652$			7.06449e20i			0.552362i
$653$	−2.62007e20			−0.202518			−0.101259	−	0.994860i	$-0.532287\pi$
$653$	−2.62007e20			−0.202518			−0.101259	+	0.994860i	$0.532287\pi$
$654$		0			0
$655$			2.48818e21i			1.87963i
$656$			8.65134e20i			0.646107i
$657$		0			0
$658$		−	1.29338e20i		−	0.0944133i
$659$	7.06669e20			0.510006			0.255003	−	0.966940i	$-0.417924\pi$
$659$	7.06669e20			0.510006			0.255003	+	0.966940i	$0.417924\pi$
$660$		0			0
$661$			2.78756e21i			1.96659i	0.182025	+	0.983294i	$0.441735\pi$
$661$			2.78756e21i			1.96659i	−0.182025	+	0.983294i	$0.558265\pi$
$662$	1.61229e21			1.12463
$663$		0			0
$664$	−8.66451e20			−0.590859
$665$			2.69934e21i			1.82010i
$666$		0			0
$667$	−4.27626e20			−0.281916
$668$			6.29518e20i			0.410378i
$669$		0			0
$670$		−	2.45231e21i		−	1.56320i
$671$		−	2.16752e21i		−	1.36629i
$672$		0			0
$673$	−7.99627e20			−0.492918			−0.246459	−	0.969153i	$-0.579267\pi$
$673$	−7.99627e20			−0.492918			−0.246459	+	0.969153i	$0.579267\pi$
$674$		−	1.67534e21i		−	1.02130i
$675$		0			0
$676$	−5.39265e20	−	3.55170e20i	−0.321516	−	0.211756i
$677$	−2.42949e21			−1.43252			−0.716258	−	0.697835i	$-0.754147\pi$
$677$	−2.42949e21			−1.43252			−0.716258	+	0.697835i	$0.754147\pi$
$678$		0			0
$679$	9.53192e20			0.549740
$680$	3.49046e21			1.99097
$681$		0			0
$682$		−	2.93425e20i		−	0.163725i
$683$		−	1.49218e21i		−	0.823504i	−0.911296	−	0.411752i	$-0.864917\pi$
$683$		−	1.49218e21i		−	0.823504i	0.911296	−	0.411752i	$-0.135083\pi$
$684$		0			0
$685$	−5.54668e20			−0.299470
$686$	1.57217e21			0.839592
$687$		0			0
$688$	−3.33976e20			−0.174503
$689$	−3.57348e20	+	1.19225e21i	−0.184692	+	0.616202i
$690$		0			0
$691$		−	1.99590e21i		−	1.00938i	−0.863301	−	0.504689i	$-0.831607\pi$
$691$		−	1.99590e21i		−	1.00938i	0.863301	−	0.504689i	$-0.168393\pi$
$692$	7.12626e19			0.0356505
$693$		0			0
$694$			9.52323e20i			0.466217i
$695$			2.54536e21i			1.23272i
$696$		0			0
$697$			3.47341e21i			1.64630i
$698$	1.59874e21			0.749655
$699$		0			0
$700$		−	8.40922e20i		−	0.385941i
$701$	−3.15599e21			−1.43301			−0.716507	−	0.697580i	$-0.754260\pi$
$701$	−3.15599e21			−1.43301			−0.716507	+	0.697580i	$0.754260\pi$
$702$		0			0
$703$	6.69749e21			2.97678
$704$		−	2.96525e21i		−	1.30396i
$705$		0			0
$706$	−2.22453e21			−0.957641
$707$			2.50333e21i			1.06628i
$708$		0			0
$709$			7.21089e20i			0.300706i	0.988632	+	0.150353i	$0.0480410\pi$
$709$			7.21089e20i			0.300706i	−0.988632	+	0.150353i	$0.951959\pi$
$710$		−	2.50406e21i		−	1.03325i
$711$		0			0
$712$	−1.13432e21			−0.458283
$713$			6.17638e19i			0.0246924i
$714$		0			0
$715$	−1.42881e21	+	4.76707e21i	−0.559345	+	1.86619i
$716$	−5.43181e18			−0.00210425
$717$		0			0
$718$	1.60244e21			0.607923
$719$	−1.47998e21			−0.555634			−0.277817	−	0.960634i	$-0.589611\pi$
$719$	−1.47998e21			−0.555634			−0.277817	+	0.960634i	$0.589611\pi$
$720$		0			0
$721$			2.86654e21i			1.05401i
$722$			3.50058e21i			1.27383i
$723$		0			0
$724$	−7.40387e20			−0.263888
$725$	−7.35013e21			−2.59274
$726$		0			0
$727$	3.53706e20			0.122218			0.0611089	−	0.998131i	$-0.480536\pi$
$727$	3.53706e20			0.122218			0.0611089	+	0.998131i	$0.480536\pi$
$728$	−6.64762e20	+	2.21790e21i	−0.227343	+	0.758502i
$729$		0			0
$730$			5.13945e21i			1.72185i
$731$	−1.34087e21			−0.444639
$732$		0			0
$733$		−	1.17727e21i		−	0.382469i	−0.981544	−	0.191234i	$-0.938751\pi$
$733$		−	1.17727e21i		−	0.382469i	0.981544	−	0.191234i	$-0.0612491\pi$
$734$		−	2.27985e21i		−	0.733139i
$735$		0			0
$736$			3.41699e20i			0.107661i
$737$	−5.24247e21			−1.63504
$738$		0			0
$739$		−	1.74625e21i		−	0.533670i	−0.963742	−	0.266835i	$-0.914022\pi$
$739$		−	1.74625e21i		−	0.533670i	0.963742	−	0.266835i	$-0.0859778\pi$
$740$	−3.60379e21			−1.09024
$741$		0			0
$742$	1.24058e21			0.367787
$743$		−	2.86345e20i		−	0.0840377i	−0.999117	−	0.0420188i	$-0.986621\pi$
$743$		−	2.86345e20i		−	0.0840377i	0.999117	−	0.0420188i	$-0.0133790\pi$
$744$		0			0
$745$	3.81915e21			1.09849
$746$			1.23758e21i			0.352397i
$747$		0			0
$748$		−	2.07415e21i		−	0.578867i
$749$			2.94929e21i			0.814899i
$750$		0			0
$751$	−9.03010e20			−0.244564			−0.122282	−	0.992495i	$-0.539021\pi$
$751$	−9.03010e20			−0.244564			−0.122282	+	0.992495i	$0.539021\pi$
$752$			2.87480e20i			0.0770857i
$753$		0			0
$754$	5.38864e21	+	1.61511e21i	1.41643	+	0.424540i
$755$	−5.04365e21			−1.31263
$756$		0			0
$757$	−5.67308e21			−1.44744			−0.723718	−	0.690096i	$-0.757569\pi$
$757$	−5.67308e21			−1.44744			−0.723718	+	0.690096i	$0.757569\pi$
$758$	−1.22904e21			−0.310491
$759$		0			0
$760$		−	1.09480e22i		−	2.71166i
$761$			3.21333e21i			0.788080i	0.919093	+	0.394040i	$0.128923\pi$
$761$			3.21333e21i			0.788080i	−0.919093	+	0.394040i	$0.871077\pi$
$762$		0			0
$763$	−1.23793e21			−0.297689
$764$	−5.76000e20			−0.137158
$765$		0			0
$766$	1.72722e21			0.403304
$767$	8.00752e21	+	2.40006e21i	1.85154	+	0.554953i
$768$		0			0
$769$		−	1.91918e21i		−	0.435179i	−0.976040	−	0.217589i	$-0.930181\pi$
$769$		−	1.91918e21i		−	0.435179i	0.976040	−	0.217589i	$-0.0698194\pi$
$770$	4.96031e21			1.11386
$771$		0			0
$772$		−	6.15471e20i		−	0.135543i
$773$		−	4.85315e21i		−	1.05847i	−0.848476	−	0.529233i	$-0.822480\pi$
$773$		−	4.85315e21i		−	1.05847i	0.848476	−	0.529233i	$-0.177520\pi$
$774$		0			0
$775$			1.06161e21i			0.227092i
$776$	−3.86597e21			−0.819022
$777$		0			0
$778$			3.54767e21i			0.737218i
$779$	1.08946e22			2.24222
$780$		0			0
$781$	−5.35309e21			−1.08074
$782$		−	6.97462e20i		−	0.139467i
$783$		0			0
$784$	−1.11486e21			−0.218700
$785$		−	1.32925e22i		−	2.58276i
$786$		0			0
$787$			7.30773e21i			1.39307i	0.717525	+	0.696533i	$0.245275\pi$
$787$			7.30773e21i			1.39307i	−0.717525	+	0.696533i	$0.754725\pi$
$788$		−	7.76368e20i		−	0.146596i
$789$		0			0
$790$	−5.60992e21			−1.03933
$791$		−	5.69553e20i		−	0.104523i
$792$		0			0
$793$	1.72321e21	−	5.74930e21i	0.310306	−	1.03530i
$794$	7.81855e21			1.39467
$795$		0			0
$796$	3.32293e21			0.581666
$797$	2.60278e20			0.0451337			0.0225668	−	0.999745i	$-0.492816\pi$
$797$	2.60278e20			0.0451337			0.0225668	+	0.999745i	$0.492816\pi$
$798$		0			0
$799$			1.15420e21i			0.196417i
$800$			5.87320e21i			0.990147i
$801$		0			0
$802$	−6.09300e21			−1.00815
$803$	1.09869e22			1.80098
$804$		0			0
$805$	−1.04411e21			−0.167988
$806$	2.33278e20	−	7.78305e20i	0.0371844	−	0.124061i
$807$		0			0
$808$		−	1.01531e22i		−	1.58859i
$809$	−1.22136e21			−0.189334			−0.0946672	−	0.995509i	$-0.530179\pi$
$809$	−1.22136e21			−0.189334			−0.0946672	+	0.995509i	$0.530179\pi$
$810$		0			0
$811$			7.49803e20i			0.114101i	0.998371	+	0.0570507i	$0.0181697\pi$
$811$			7.49803e20i			0.114101i	−0.998371	+	0.0570507i	$0.981830\pi$
$812$			3.50990e21i			0.529205i
$813$		0			0
$814$		−	1.23073e22i		−	1.82171i
$815$	1.50766e22			2.21116
$816$		0			0
$817$			4.20574e21i			0.605587i
$818$	2.86097e21			0.408191
$819$		0			0
$820$	−5.86216e21			−0.821209
$821$		−	7.94418e20i		−	0.110275i	−0.998479	−	0.0551373i	$-0.982440\pi$
$821$		−	7.94418e20i		−	0.110275i	0.998479	−	0.0551373i	$-0.0175596\pi$
$822$		0			0
$823$	−1.00396e22			−1.36842			−0.684208	−	0.729287i	$-0.739852\pi$
$823$	−1.00396e22			−1.36842			−0.684208	+	0.729287i	$0.739852\pi$
$824$		−	1.16262e22i		−	1.57030i
$825$		0			0
$826$		−	8.33213e21i		−	1.10511i
$827$			3.30964e21i			0.435000i	0.976060	+	0.217500i	$0.0697902\pi$
$827$			3.30964e21i			0.435000i	−0.976060	+	0.217500i	$0.930210\pi$
$828$		0			0
$829$	4.10936e21			0.530414			0.265207	−	0.964191i	$-0.414560\pi$
$829$	4.10936e21			0.530414			0.265207	+	0.964191i	$0.414560\pi$
$830$			5.14000e21i			0.657473i
$831$		0			0
$832$	2.35742e21	−	7.86526e21i	0.296150	−	0.988071i
$833$	−4.47601e21			−0.557254
$834$		0			0
$835$	1.34347e22			1.64278
$836$	−6.50571e21			−0.788402
$837$		0			0
$838$			8.94165e21i			1.06436i
$839$		−	1.16812e22i		−	1.37807i	−0.724727	−	0.689036i	$-0.758034\pi$
$839$		−	1.16812e22i		−	1.37807i	0.724727	−	0.689036i	$-0.241966\pi$
$840$		0			0
$841$	2.20493e22			2.55520
$842$	−8.47983e21			−0.973972
$843$		0			0
$844$	−3.89832e21			−0.439854
$845$	−7.57979e21	+	1.15086e22i	−0.847680	+	1.28706i
$846$		0			0
$847$		−	3.96847e21i		−	0.436011i
$848$	−2.75744e21			−0.300287
$849$		0			0
$850$		−	1.19881e22i		−	1.28265i
$851$			2.59060e21i			0.274744i
$852$		0			0
$853$		−	3.85437e21i		−	0.401639i	−0.979628	−	0.200819i	$-0.935640\pi$
$853$		−	3.85437e21i		−	0.401639i	0.979628	−	0.200819i	$-0.0643604\pi$
$854$	−5.98237e21			−0.617930
$855$		0			0
$856$		−	1.19618e22i		−	1.21407i
$857$	7.63919e20			0.0768584			0.0384292	−	0.999261i	$-0.487765\pi$
$857$	7.63919e20			0.0768584			0.0384292	+	0.999261i	$0.487765\pi$
$858$		0			0
$859$	−1.12354e22			−1.11081			−0.555404	−	0.831581i	$-0.687436\pi$
$859$	−1.12354e22			−1.11081			−0.555404	+	0.831581i	$0.687436\pi$
$860$		−	2.26303e21i		−	0.221795i
$861$		0			0
$862$	−6.60916e21			−0.636564
$863$		−	3.48334e21i		−	0.332595i	−0.986076	−	0.166297i	$-0.946819\pi$
$863$		−	3.48334e21i		−	0.332595i	0.986076	−	0.166297i	$-0.0531811\pi$
$864$		0			0
$865$		−	1.52084e21i		−	0.142712i
$866$			6.74616e21i			0.627584i
$867$		0			0
$868$	5.06949e20			0.0463518
$869$			1.19927e22i			1.08710i
$870$		0			0
$871$	−1.39055e22	−	4.16785e21i	−1.23894	−	0.371344i
$872$	5.02083e21			0.443508
$873$		0			0
$874$	−2.18764e21			−0.189950
$875$	−4.89523e21			−0.421417
$876$		0			0
$877$		−	4.35289e21i		−	0.368367i	−0.982892	−	0.184184i	$-0.941036\pi$
$877$		−	4.35289e21i		−	0.368367i	0.982892	−	0.184184i	$-0.0589641\pi$
$878$		−	8.54125e21i		−	0.716659i
$879$		0			0
$880$	−1.10253e22			−0.909431
$881$	−3.18781e20			−0.0260720			−0.0130360	−	0.999915i	$-0.504150\pi$
$881$	−3.18781e20			−0.0260720			−0.0130360	+	0.999915i	$0.504150\pi$
$882$		0			0
$883$	−9.12242e21			−0.733508			−0.366754	−	0.930318i	$-0.619531\pi$
$883$	−9.12242e21			−0.733508			−0.366754	+	0.930318i	$0.619531\pi$
$884$	1.64899e21	−	5.50165e21i	0.131469	−	0.438633i
$885$		0			0
$886$		−	1.60955e22i		−	1.26169i
$887$	−3.98657e21			−0.309865			−0.154932	−	0.987925i	$-0.549516\pi$
$887$	−3.98657e21			−0.309865			−0.154932	+	0.987925i	$0.549516\pi$
$888$		0			0
$889$			1.43683e22i			1.09810i
$890$			6.72904e21i			0.509951i
$891$		0			0
$892$		−	6.77845e21i		−	0.505119i
$893$	3.62021e21			0.267515
$894$		0			0
$895$			1.15922e20i			0.00842350i
$896$	−8.99079e20			−0.0647870
$897$		0			0
$898$	1.51565e22			1.07405
$899$		−	4.43102e21i		−	0.311391i
$900$		0			0
$901$	−1.10708e22			−0.765142
$902$		−	2.00198e22i		−	1.37218i
$903$		0			0
$904$			2.31000e21i			0.155722i
$905$			1.58008e22i			1.05637i
$906$		0			0
$907$	9.05373e21			0.595351			0.297675	−	0.954667i	$-0.403789\pi$
$907$	9.05373e21			0.595351			0.297675	+	0.954667i	$0.403789\pi$
$908$			1.08708e22i			0.708953i
$909$		0			0
$910$	1.31571e22	+	3.94353e21i	0.844017	+	0.252973i
$911$	1.71425e22			1.09065			0.545327	−	0.838223i	$-0.316405\pi$
$911$	1.71425e22			1.09065			0.545327	+	0.838223i	$0.316405\pi$
$912$		0			0
$913$	1.09881e22			0.687690
$914$	1.60031e22			0.993365
$915$		0			0
$916$			5.22127e20i			0.0318831i
$917$		−	1.46809e22i		−	0.889165i
$918$		0			0
$919$	1.08944e22			0.649140			0.324570	−	0.945862i	$-0.394780\pi$
$919$	1.08944e22			0.649140			0.324570	+	0.945862i	$0.394780\pi$
$920$	4.23472e21			0.250274
$921$		0			0
$922$	−1.86462e22			−1.08420
$923$	−1.41990e22	−	4.25580e21i	−0.818925	−	0.245453i
$924$		0			0
$925$			4.45277e22i			2.52678i
$926$	7.54464e21			0.424674
$927$		0			0
$928$		−	2.45140e22i		−	1.35770i
$929$			9.48991e21i			0.521368i	0.965424	+	0.260684i	$0.0839480\pi$
$929$			9.48991e21i			0.521368i	−0.965424	+	0.260684i	$0.916052\pi$
$930$		0			0
$931$			1.40393e22i			0.758966i
$932$	5.38792e21			0.288936
$933$		0			0
$934$		−	1.49964e22i		−	0.791379i
$935$	−4.42651e22			−2.31726
$936$		0			0
$937$	−2.45169e22			−1.26304			−0.631522	−	0.775358i	$-0.717569\pi$
$937$	−2.45169e22			−1.26304			−0.631522	+	0.775358i	$0.717569\pi$
$938$			1.44693e22i			0.739477i
$939$		0			0
$940$	−1.94797e21			−0.0979767
$941$		−	2.54883e22i		−	1.27180i	−0.771772	−	0.635900i	$-0.780629\pi$
$941$		−	2.54883e22i		−	1.27180i	0.771772	−	0.635900i	$-0.219371\pi$
$942$		0			0
$943$			4.21403e21i			0.206947i
$944$			1.85198e22i			0.902290i
$945$		0			0
$946$	7.72846e21			0.370603
$947$			2.00878e21i			0.0955671i	0.998858	+	0.0477835i	$0.0152158\pi$
$947$			2.00878e21i			0.0955671i	−0.998858	+	0.0477835i	$0.984784\pi$
$948$		0			0
$949$	2.91426e22	+	8.73480e21i	1.36468	+	0.409031i
$950$	−3.76015e22			−1.74694
$951$		0			0
$952$	−2.05946e22			−0.941837
$953$	2.76739e22			1.25566			0.627831	−	0.778349i	$-0.283943\pi$
$953$	2.76739e22			1.25566			0.627831	+	0.778349i	$0.283943\pi$
$954$		0			0
$955$			1.22926e22i			0.549058i
$956$		−	2.47459e20i		−	0.0109665i
$957$		0			0
$958$	−6.43541e21			−0.280760
$959$	3.27268e21			0.141666
$960$		0			0
$961$	2.28253e22			0.972726
$962$	9.78451e21	−	3.26449e22i	0.413739	−	1.38039i
$963$		0			0
$964$		−	1.45117e22i		−	0.604146i
$965$	−1.31350e22			−0.542593
$966$		0			0
$967$		−	1.19108e22i		−	0.484442i	−0.970221	−	0.242221i	$-0.922124\pi$
$967$		−	1.19108e22i		−	0.484442i	0.970221	−	0.242221i	$-0.0778759\pi$
$968$			1.60954e22i			0.649585i
$969$		0			0
$970$			2.29339e22i			0.911360i
$971$	−2.02133e22			−0.797063			−0.398531	−	0.917155i	$-0.630480\pi$
$971$	−2.02133e22			−0.797063			−0.398531	+	0.917155i	$0.630480\pi$
$972$		0			0
$973$		−	1.50182e22i		−	0.583140i
$974$	2.39466e22			0.922682
$975$		0			0
$976$	1.32970e22			0.504522
$977$			2.91356e22i			1.09702i	0.836144	+	0.548510i	$0.184805\pi$
$977$			2.91356e22i			1.09702i	−0.836144	+	0.548510i	$0.815195\pi$
$978$		0			0
$979$	1.43851e22			0.533388
$980$		−	7.55427e21i		−	0.277969i
$981$		0			0
$982$			1.66654e22i			0.603919i
$983$			2.23625e22i			0.804210i	0.915594	+	0.402105i	$0.131721\pi$
$983$			2.23625e22i			0.804210i	−0.915594	+	0.402105i	$0.868279\pi$
$984$		0			0
$985$	−1.65687e22			−0.586837
$986$			5.00369e22i			1.75879i
$987$		0			0
$988$	−1.72563e22	−	5.17215e21i	−0.597407	−	0.179058i
$989$	−1.62678e21			−0.0558931
$990$		0			0
$991$	−1.95068e21			−0.0660135			−0.0330068	−	0.999455i	$-0.510508\pi$
$991$	−1.95068e21			−0.0660135			−0.0330068	+	0.999455i	$0.510508\pi$
$992$	−3.54065e21			−0.118918
$993$		0			0
$994$			1.47746e22i			0.488784i
$995$		−	7.09157e22i		−	2.32846i
$996$		0			0
$997$	1.10543e21			0.0357536			0.0178768	−	0.999840i	$-0.494309\pi$
$997$	1.10543e21			0.0357536			0.0178768	+	0.999840i	$0.494309\pi$
$998$	−8.42020e21			−0.270298
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.16.b.f.64.12	yes	28
3.2	odd	2	inner	117.16.b.f.64.18	yes	28
13.12	even	2	inner	117.16.b.f.64.17	yes	28
39.38	odd	2	inner	117.16.b.f.64.11	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.16.b.f.64.11	✓	28	39.38	odd	2	inner
117.16.b.f.64.12	yes	28	1.1	even	1	trivial
117.16.b.f.64.17	yes	28	13.12	even	2	inner
117.16.b.f.64.18	yes	28	3.2	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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	117.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-141.961i q^{2} +12615.2 q^{4} -269224. i q^{5} +1.58849e6i q^{7} -6.44262e6i q^{8} +O(q^{10})\)	\(q-141.961i q^{2} +12615.2 q^{4} -269224. i q^{5} +1.58849e6i q^{7} -6.44262e6i q^{8} -3.82193e7 q^{10} +8.17038e7i q^{11} +(-6.49559e7 + 2.16718e8i) q^{13} +2.25503e8 q^{14} -5.01226e8 q^{16} -2.01236e9 q^{17} +6.31189e9i q^{19} -3.39631e9i q^{20} +1.15987e10 q^{22} -2.44145e9 q^{23} -4.19641e10 q^{25} +(3.07654e10 + 9.22118e9i) q^{26} +2.00391e10i q^{28} +1.75153e11 q^{29} -2.52980e10i q^{31} -1.39958e11i q^{32} +2.85676e11i q^{34} +4.27660e11 q^{35} -1.06109e12i q^{37} +8.96040e11 q^{38} -1.73451e12 q^{40} -1.72604e12i q^{41} +6.66320e11 q^{43} +1.03071e12i q^{44} +3.46590e11i q^{46} -5.73554e11i q^{47} +2.22426e12 q^{49} +5.95725e12i q^{50} +(-8.19429e11 + 2.73393e12i) q^{52} +5.50139e12 q^{53} +2.19966e13 q^{55} +1.02340e13 q^{56} -2.48648e13i q^{58} -3.69490e13i q^{59} -2.65290e13 q^{61} -3.59133e12 q^{62} -3.62926e13 q^{64} +(5.83457e13 + 1.74877e13i) q^{65} +6.41643e13i q^{67} -2.53862e13 q^{68} -6.07109e13i q^{70} +6.55183e13i q^{71} -1.34473e14i q^{73} -1.50633e14 q^{74} +7.96256e13i q^{76} -1.29786e14 q^{77} +1.46783e14 q^{79} +1.34942e14i q^{80} -2.45029e14 q^{82} -1.34487e14i q^{83} +5.41776e14i q^{85} -9.45912e13i q^{86} +5.26387e14 q^{88} -1.76064e14i q^{89} +(-3.44254e14 - 1.03182e14i) q^{91} -3.07993e13 q^{92} -8.14221e13 q^{94} +1.69931e15 q^{95} -6.00061e14i q^{97} -3.15757e14i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 458752 q^{4}+O(q^{10}) \) 28 * q - 458752 * q^4	\( 28 q - 458752 q^{4} - 228861840 q^{10} + 803257676 q^{13} - 7451384960 q^{16} - 30544905456 q^{22} - 238197886180 q^{25} + 3510668297280 q^{40} + 10830121503776 q^{43} - 19674845564108 q^{49} + 30846257429728 q^{52} + 53012410974240 q^{55} - 23758117088504 q^{61} + 327062953083392 q^{64} - 829003210390736 q^{79} - 13\!\cdots\!56 q^{82}+ \cdots + 81131101347024 q^{94}+O(q^{100}) \) 28 * q - 458752 * q^4 - 228861840 * q^10 + 803257676 * q^13 - 7451384960 * q^16 - 30544905456 * q^22 - 238197886180 * q^25 + 3510668297280 * q^40 + 10830121503776 * q^43 - 19674845564108 * q^49 + 30846257429728 * q^52 + 53012410974240 * q^55 - 23758117088504 * q^61 + 327062953083392 * q^64 - 829003210390736 * q^79 - 1396925338271856 * q^82 + 1057698205097664 * q^88 - 1934316425142192 * q^91 + 81131101347024 * q^94

Introduction

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