LMFDB - Embedded newform 192.8.c.b.191.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,8,Mod(191,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("192.191");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	192.c (of order $2$, degree $1$, not 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:	$59.9779248930$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + x^{2} + 4 $ x^4 + x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.1
Root			$-0.866025 - 1.11803i$ of defining polynomial
Character	$\chi$	$=$	192.191
Dual form			192.8.c.b.191.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-31.1769 - 34.8569i) q^{3} -205.718i q^{5} -999.230i q^{7} +(-243.000 + 2173.46i) q^{9} +O(q^{10})$	\(q+(-31.1769 - 34.8569i) q^{3} -205.718i q^{5} -999.230i q^{7} +(-243.000 + 2173.46i) q^{9} +1974.54 q^{11} +12730.0 q^{13} +(-7170.69 + 6413.66i) q^{15} +18854.5i q^{17} +23911.8i q^{19} +(-34830.0 + 31152.9i) q^{21} +72372.0 q^{23} +35805.0 q^{25} +(83335.9 - 59291.5i) q^{27} +106106. i q^{29} +90372.2i q^{31} +(-61560.0 - 68826.2i) q^{33} -205560. q^{35} -43310.0 q^{37} +(-396882. - 443728. i) q^{39} +785790. i q^{41} -70805.9i q^{43} +(447120. + 49989.5i) q^{45} +563762. q^{47} -174917. q^{49} +(657209. - 587826. i) q^{51} +348943. i q^{53} -406198. i q^{55} +(833490. - 745496. i) q^{57} +758119. q^{59} -314198. q^{61} +(2.17178e6 + 242813. i) q^{63} -2.61879e6i q^{65} -811351. i q^{67} +(-2.25634e6 - 2.52266e6i) q^{69} +265004. q^{71} -259270. q^{73} +(-1.11629e6 - 1.24805e6i) q^{75} -1.97302e6i q^{77} -5.23482e6i q^{79} +(-4.66487e6 - 1.05630e6i) q^{81} +1.01017e7 q^{83} +3.87872e6 q^{85} +(3.69852e6 - 3.30805e6i) q^{87} +3.88455e6i q^{89} -1.27202e7i q^{91} +(3.15009e6 - 2.81753e6i) q^{93} +4.91909e6 q^{95} +7.24301e6 q^{97} +(-479813. + 4.29158e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 972 q^{9}+O(q^{10}) $ 4 * q - 972 * q^9	$ 4 q - 972 q^{9} + 50920 q^{13} - 139320 q^{21} + 143220 q^{25} - 246240 q^{33} - 173240 q^{37} + 1788480 q^{45} - 699668 q^{49} + 3333960 q^{57} - 1256792 q^{61} - 9025344 q^{69} - 1037080 q^{73} - 18659484 q^{81} + 15514880 q^{85} + 12600360 q^{93} + 28972040 q^{97}+O(q^{100}) $ 4 * q - 972 * q^9 + 50920 * q^13 - 139320 * q^21 + 143220 * q^25 - 246240 * q^33 - 173240 * q^37 + 1788480 * q^45 - 699668 * q^49 + 3333960 * q^57 - 1256792 * q^61 - 9025344 * q^69 - 1037080 * q^73 - 18659484 * q^81 + 15514880 * q^85 + 12600360 * q^93 + 28972040 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−31.1769	−	34.8569i	−0.666667	−	0.745356i
$3$	−31.1769	−	34.8569i	−0.666667	−	0.745356i
$4$		0			0
$5$		−	205.718i		−	0.736000i	−0.929826	−	0.368000i	$-0.880043\pi$
$5$		−	205.718i		−	0.736000i	0.929826	−	0.368000i	$-0.119957\pi$
$6$		0			0
$7$		−	999.230i		−	1.10109i	−0.834806	−	0.550544i	$-0.814420\pi$
$7$		−	999.230i		−	1.10109i	0.834806	−	0.550544i	$-0.185580\pi$
$8$		0			0
$9$	−243.000	+	2173.46i	−0.111111	+	0.993808i
$10$		0			0
$11$	1974.54			0.447292			0.223646	−	0.974670i	$-0.428204\pi$
$11$	1974.54			0.447292			0.223646	+	0.974670i	$0.428204\pi$
$12$		0			0
$13$	12730.0			1.60704			0.803520	−	0.595278i	$-0.202958\pi$
$13$	12730.0			1.60704			0.803520	+	0.595278i	$0.202958\pi$
$14$		0			0
$15$	−7170.69	+	6413.66i	−0.548582	+	0.490667i
$16$		0			0
$17$			18854.5i			0.930774i	0.885107	+	0.465387i	$0.154085\pi$
$17$			18854.5i			0.930774i	−0.885107	+	0.465387i	$0.845915\pi$
$18$		0			0
$19$			23911.8i			0.799788i	0.916561	+	0.399894i	$0.130953\pi$
$19$			23911.8i			0.799788i	−0.916561	+	0.399894i	$0.869047\pi$
$20$		0			0
$21$	−34830.0	+	31152.9i	−0.820703	+	0.734059i
$22$		0			0
$23$	72372.0			1.24029			0.620145	−	0.784487i	$-0.287074\pi$
$23$	72372.0			1.24029			0.620145	+	0.784487i	$0.287074\pi$
$24$		0			0
$25$	35805.0			0.458304
$26$		0			0
$27$	83335.9	−	59291.5i	0.814815	−	0.579721i
$28$		0			0
$29$			106106.i			0.807879i	0.914786	+	0.403940i	$0.132359\pi$
$29$			106106.i			0.807879i	−0.914786	+	0.403940i	$0.867641\pi$
$30$		0			0
$31$			90372.2i			0.544840i	0.962178	+	0.272420i	$0.0878240\pi$
$31$			90372.2i			0.544840i	−0.962178	+	0.272420i	$0.912176\pi$
$32$		0			0
$33$	−61560.0	−	68826.2i	−0.298195	−	0.333392i
$34$		0			0
$35$	−205560.			−0.810401
$36$		0			0
$37$	−43310.0			−0.140567			−0.0702833	−	0.997527i	$-0.522390\pi$
$37$	−43310.0			−0.140567			−0.0702833	+	0.997527i	$0.522390\pi$
$38$		0			0
$39$	−396882.	−	443728.i	−1.07136	−	1.19782i
$40$		0			0
$41$			785790.i			1.78059i	0.455388	+	0.890293i	$0.349501\pi$
$41$			785790.i			1.78059i	−0.455388	+	0.890293i	$0.650499\pi$
$42$		0			0
$43$		−	70805.9i		−	0.135809i	−0.997692	−	0.0679047i	$-0.978369\pi$
$43$		−	70805.9i		−	0.135809i	0.997692	−	0.0679047i	$-0.0216314\pi$
$44$		0			0
$45$	447120.	+	49989.5i	0.731443	+	0.0817778i
$46$		0			0
$47$	563762.			0.792051			0.396025	−	0.918240i	$-0.370389\pi$
$47$	563762.			0.792051			0.396025	+	0.918240i	$0.370389\pi$
$48$		0			0
$49$	−174917.			−0.212396
$50$		0			0
$51$	657209.	−	587826.i	0.693758	−	0.620516i
$52$		0			0
$53$			348943.i			0.321950i	0.986958	+	0.160975i	$0.0514639\pi$
$53$			348943.i			0.321950i	−0.986958	+	0.160975i	$0.948536\pi$
$54$		0			0
$55$		−	406198.i		−	0.329207i
$56$		0			0
$57$	833490.	−	745496.i	0.596126	−	0.533192i
$58$		0			0
$59$	758119.			0.480568			0.240284	−	0.970703i	$-0.422759\pi$
$59$	758119.			0.480568			0.240284	+	0.970703i	$0.422759\pi$
$60$		0			0
$61$	−314198.			−0.177235			−0.0886174	−	0.996066i	$-0.528245\pi$
$61$	−314198.			−0.177235			−0.0886174	+	0.996066i	$0.528245\pi$
$62$		0			0
$63$	2.17178e6	+	242813.i	1.09427	+	0.122343i
$64$		0			0
$65$		−	2.61879e6i		−	1.18278i
$66$		0			0
$67$		−	811351.i		−	0.329570i	−0.986330	−	0.164785i	$-0.947307\pi$
$67$		−	811351.i		−	0.329570i	0.986330	−	0.164785i	$-0.0526930\pi$
$68$		0			0
$69$	−2.25634e6	−	2.52266e6i	−0.826860	−	0.924458i
$70$		0			0
$71$	265004.			0.0878715			0.0439357	−	0.999034i	$-0.486010\pi$
$71$	265004.			0.0878715			0.0439357	+	0.999034i	$0.486010\pi$
$72$		0			0
$73$	−259270.			−0.0780050			−0.0390025	−	0.999239i	$-0.512418\pi$
$73$	−259270.			−0.0780050			−0.0390025	+	0.999239i	$0.512418\pi$
$74$		0			0
$75$	−1.11629e6	−	1.24805e6i	−0.305536	−	0.341600i
$76$		0			0
$77$		−	1.97302e6i		−	0.492508i
$78$		0			0
$79$		−	5.23482e6i		−	1.19456i	−0.802033	−	0.597279i	$-0.796248\pi$
$79$		−	5.23482e6i		−	1.19456i	0.802033	−	0.597279i	$-0.203752\pi$
$80$		0			0
$81$	−4.66487e6	−	1.05630e6i	−0.975309	−	0.220846i
$82$		0			0
$83$	1.01017e7			1.93919			0.969594	−	0.244718i	$-0.0786955\pi$
$83$	1.01017e7			1.93919			0.969594	+	0.244718i	$0.0786955\pi$
$84$		0			0
$85$	3.87872e6			0.685050
$86$		0			0
$87$	3.69852e6	−	3.30805e6i	0.602158	−	0.538586i
$88$		0			0
$89$			3.88455e6i			0.584085i	0.956405	+	0.292042i	$0.0943348\pi$
$89$			3.88455e6i			0.584085i	−0.956405	+	0.292042i	$0.905665\pi$
$90$		0			0
$91$		−	1.27202e7i		−	1.76949i
$92$		0			0
$93$	3.15009e6	−	2.81753e6i	0.406100	−	0.363227i
$94$		0			0
$95$	4.91909e6			0.588644
$96$		0			0
$97$	7.24301e6			0.805783			0.402891	−	0.915248i	$-0.368005\pi$
$97$	7.24301e6			0.805783			0.402891	+	0.915248i	$0.368005\pi$
$98$		0			0
$99$	−479813.	+	4.29158e6i	−0.0496991	+	0.444522i
$100$		0			0
$101$			1.12157e7i			1.08318i	0.840642	+	0.541591i	$0.182178\pi$
$101$			1.12157e7i			1.08318i	−0.840642	+	0.541591i	$0.817822\pi$
$102$		0			0
$103$		−	1.15817e7i		−	1.04434i	−0.852843	−	0.522168i	$-0.825123\pi$
$103$		−	1.15817e7i		−	1.04434i	0.852843	−	0.522168i	$-0.174877\pi$
$104$		0			0
$105$	6.40872e6	+	7.16517e6i	0.540267	+	0.604037i
$106$		0			0
$107$	−218675.			−0.0172566			−0.00862831	−	0.999963i	$-0.502747\pi$
$107$	−218675.			−0.0172566			−0.00862831	+	0.999963i	$0.502747\pi$
$108$		0			0
$109$	−9.80701e6			−0.725344			−0.362672	−	0.931917i	$-0.618135\pi$
$109$	−9.80701e6			−0.725344			−0.362672	+	0.931917i	$0.618135\pi$
$110$		0			0
$111$	1.35027e6	+	1.50965e6i	0.0937111	+	0.104772i
$112$		0			0
$113$		−	5.41533e6i		−	0.353061i	−0.984295	−	0.176531i	$-0.943513\pi$
$113$		−	5.41533e6i		−	0.353061i	0.984295	−	0.176531i	$-0.0564875\pi$
$114$		0			0
$115$		−	1.48882e7i		−	0.912853i
$116$		0			0
$117$	−3.09339e6	+	2.76681e7i	−0.178560	+	1.59709i
$118$		0			0
$119$	1.88400e7			1.02486
$120$		0			0
$121$	−1.55884e7			−0.799930
$122$		0			0
$123$	2.73902e7	−	2.44985e7i	1.32717	−	1.18706i
$124$		0			0
$125$		−	2.34375e7i		−	1.07331i
$126$		0			0
$127$		−	1.65498e7i		−	0.716936i	−0.933542	−	0.358468i	$-0.883299\pi$
$127$		−	1.65498e7i		−	0.716936i	0.933542	−	0.358468i	$-0.116701\pi$
$128$		0			0
$129$	−2.46807e6	+	2.20751e6i	−0.101226	+	0.0905396i
$130$		0			0
$131$	−2.54476e6			−0.0989004			−0.0494502	−	0.998777i	$-0.515747\pi$
$131$	−2.54476e6			−0.0989004			−0.0494502	+	0.998777i	$0.515747\pi$
$132$		0			0
$133$	2.38934e7			0.880637
$134$		0			0
$135$	−1.21973e7	−	1.71437e7i	−0.426675	−	0.599704i
$136$		0			0
$137$			2.54632e7i			0.846040i	0.906120	+	0.423020i	$0.139030\pi$
$137$			2.54632e7i			0.846040i	−0.906120	+	0.423020i	$0.860970\pi$
$138$		0			0
$139$			3.98426e7i			1.25833i	0.777270	+	0.629167i	$0.216604\pi$
$139$			3.98426e7i			1.25833i	−0.777270	+	0.629167i	$0.783396\pi$
$140$		0			0
$141$	−1.75764e7	−	1.96510e7i	−0.528034	−	0.590360i
$142$		0			0
$143$	2.51359e7			0.718816
$144$		0			0
$145$	2.18279e7			0.594599
$146$		0			0
$147$	5.45337e6	+	6.09706e6i	0.141597	+	0.158310i
$148$		0			0
$149$		−	2.47666e6i		−	0.0613359i	−0.999530	−	0.0306679i	$-0.990237\pi$
$149$		−	2.47666e6i		−	0.0613359i	0.999530	−	0.0306679i	$-0.00976344\pi$
$150$		0			0
$151$			5.67175e7i			1.34060i	0.742092	+	0.670298i	$0.233834\pi$
$151$			5.67175e7i			1.34060i	−0.742092	+	0.670298i	$0.766166\pi$
$152$		0			0
$153$	−4.09795e7	−	4.58165e6i	−0.925011	−	0.103419i
$154$		0			0
$155$	1.85912e7			0.401002
$156$		0			0
$157$	2.54237e7			0.524312			0.262156	−	0.965026i	$-0.415566\pi$
$157$	2.54237e7			0.524312			0.262156	+	0.965026i	$0.415566\pi$
$158$		0			0
$159$	1.21630e7	−	1.08790e7i	0.239968	−	0.214634i
$160$		0			0
$161$		−	7.23163e7i		−	1.36567i
$162$		0			0
$163$		−	4.04251e7i		−	0.731130i	−0.930786	−	0.365565i	$-0.880876\pi$
$163$		−	4.04251e7i		−	0.731130i	0.930786	−	0.365565i	$-0.119124\pi$
$164$		0			0
$165$	−1.41588e7	+	1.26640e7i	−0.245376	+	0.219471i
$166$		0			0
$167$	−5.30738e7			−0.881805			−0.440902	−	0.897555i	$-0.645342\pi$
$167$	−5.30738e7			−0.881805			−0.440902	+	0.897555i	$0.645342\pi$
$168$		0			0
$169$	9.93044e7			1.58258
$170$		0			0
$171$	−5.19713e7	−	5.81057e6i	−0.794835	−	0.0888653i
$172$		0			0
$173$			7.80723e6i			0.114640i	0.998356	+	0.0573199i	$0.0182555\pi$
$173$			7.80723e6i			0.114640i	−0.998356	+	0.0573199i	$0.981744\pi$
$174$		0			0
$175$		−	3.57774e7i		−	0.504633i
$176$		0			0
$177$	−2.36358e7	−	2.64256e7i	−0.320379	−	0.358194i
$178$		0			0
$179$	−7.52215e7			−0.980294			−0.490147	−	0.871640i	$-0.663057\pi$
$179$	−7.52215e7			−0.980294			−0.490147	+	0.871640i	$0.663057\pi$
$180$		0			0
$181$	−3.63087e7			−0.455130			−0.227565	−	0.973763i	$-0.573076\pi$
$181$	−3.63087e7			−0.455130			−0.227565	+	0.973763i	$0.573076\pi$
$182$		0			0
$183$	9.79572e6	+	1.09520e7i	0.118157	+	0.132103i
$184$		0			0
$185$			8.90966e6i			0.103457i
$186$		0			0
$187$			3.72290e7i			0.416328i
$188$		0			0
$189$	−5.92458e7	−	8.32717e7i	−0.638324	−	0.897183i
$190$		0			0
$191$	−1.29538e8			−1.34518			−0.672590	−	0.740015i	$-0.734818\pi$
$191$	−1.29538e8			−1.34518			−0.672590	+	0.740015i	$0.734818\pi$
$192$		0			0
$193$	−1.19989e8			−1.20141			−0.600703	−	0.799472i	$-0.705112\pi$
$193$	−1.19989e8			−1.20141			−0.600703	+	0.799472i	$0.705112\pi$
$194$		0			0
$195$	−9.12829e7	+	8.16459e7i	−0.881593	+	0.788521i
$196$		0			0
$197$		−	3.80955e7i		−	0.355011i	−0.984120	−	0.177506i	$-0.943197\pi$
$197$		−	3.80955e7i		−	0.355011i	0.984120	−	0.177506i	$-0.0568028\pi$
$198$		0			0
$199$			1.00917e8i			0.907775i	0.891059	+	0.453888i	$0.149963\pi$
$199$			1.00917e8i			0.907775i	−0.891059	+	0.453888i	$0.850037\pi$
$200$		0			0
$201$	−2.82812e7	+	2.52954e7i	−0.245647	+	0.219713i
$202$		0			0
$203$	1.06024e8			0.889547
$204$		0			0
$205$	1.61651e8			1.31051
$206$		0			0
$207$	−1.75864e7	+	1.57298e8i	−0.137810	+	1.23261i
$208$		0			0
$209$			4.72148e7i			0.357739i
$210$		0			0
$211$			2.02240e8i			1.48210i	0.671448	+	0.741052i	$0.265673\pi$
$211$			2.02240e8i			1.48210i	−0.671448	+	0.741052i	$0.734327\pi$
$212$		0			0
$213$	−8.26200e6	−	9.23720e6i	−0.0585810	−	0.0654955i
$214$		0			0
$215$	−1.45661e7			−0.0999557
$216$		0			0
$217$	9.03026e7			0.599917
$218$		0			0
$219$	8.08324e6	+	9.03734e6i	0.0520033	+	0.0581415i
$220$		0			0
$221$			2.40018e8i			1.49579i
$222$		0			0
$223$		−	4.63341e7i		−	0.279791i	−0.990166	−	0.139896i	$-0.955323\pi$
$223$		−	4.63341e7i		−	0.279791i	0.990166	−	0.139896i	$-0.0446767\pi$
$224$		0			0
$225$	−8.70062e6	+	7.78207e7i	−0.0509227	+	0.455466i
$226$		0			0
$227$	−1.52540e8			−0.865554			−0.432777	−	0.901501i	$-0.642466\pi$
$227$	−1.52540e8			−0.865554			−0.432777	+	0.901501i	$0.642466\pi$
$228$		0			0
$229$	4.39128e7			0.241639			0.120819	−	0.992675i	$-0.461448\pi$
$229$	4.39128e7			0.241639			0.120819	+	0.992675i	$0.461448\pi$
$230$		0			0
$231$	−6.87732e7	+	6.15126e7i	−0.367094	+	0.328339i
$232$		0			0
$233$			2.42578e8i			1.25634i	0.778077	+	0.628168i	$0.216195\pi$
$233$			2.42578e8i			1.25634i	−0.778077	+	0.628168i	$0.783805\pi$
$234$		0			0
$235$		−	1.15976e8i		−	0.582949i
$236$		0			0
$237$	−1.82470e8	+	1.63206e8i	−0.890371	+	0.796372i
$238$		0			0
$239$	1.34211e8			0.635909			0.317954	−	0.948106i	$-0.397004\pi$
$239$	1.34211e8			0.635909			0.317954	+	0.948106i	$0.397004\pi$
$240$		0			0
$241$	4.19220e8			1.92922			0.964612	−	0.263674i	$-0.0849343\pi$
$241$	4.19220e8			1.92922			0.964612	+	0.263674i	$0.0849343\pi$
$242$		0			0
$243$	1.08617e8	+	1.95535e8i	0.485597	+	0.874183i
$244$		0			0
$245$			3.59836e7i			0.156323i
$246$		0			0
$247$			3.04397e8i			1.28529i
$248$		0			0
$249$	−3.14939e8	−	3.52113e8i	−1.29279	−	1.44539i
$250$		0			0
$251$	−4.44551e8			−1.77445			−0.887225	−	0.461337i	$-0.847370\pi$
$251$	−4.44551e8			−1.77445			−0.887225	+	0.461337i	$0.847370\pi$
$252$		0			0
$253$	1.42901e8			0.554772
$254$		0			0
$255$	−1.20927e8	−	1.35200e8i	−0.456700	−	0.510606i
$256$		0			0
$257$		−	3.24979e7i		−	0.119423i	−0.998216	−	0.0597117i	$-0.980982\pi$
$257$		−	3.24979e7i		−	0.119423i	0.998216	−	0.0597117i	$-0.0190181\pi$
$258$		0			0
$259$			4.32766e7i			0.154776i
$260$		0			0
$261$	−2.30617e8	−	2.57837e7i	−0.802877	−	0.0897644i
$262$		0			0
$263$	−4.26351e8			−1.44518			−0.722590	−	0.691277i	$-0.757048\pi$
$263$	−4.26351e8			−1.44518			−0.722590	+	0.691277i	$0.757048\pi$
$264$		0			0
$265$	7.17839e7			0.236955
$266$		0			0
$267$	1.35403e8	−	1.21108e8i	0.435351	−	0.389390i
$268$		0			0
$269$		−	3.00361e8i		−	0.940829i	−0.882445	−	0.470415i	$-0.844104\pi$
$269$		−	3.00361e8i		−	0.940829i	0.882445	−	0.470415i	$-0.155896\pi$
$270$		0			0
$271$		−	1.68618e8i		−	0.514648i	−0.966325	−	0.257324i	$-0.917159\pi$
$271$		−	1.68618e8i		−	0.514648i	0.966325	−	0.257324i	$-0.0828408\pi$
$272$		0			0
$273$	−4.43386e8	+	3.96576e8i	−1.31890	+	1.17966i
$274$		0			0
$275$	7.06983e7			0.204996
$276$		0			0
$277$	4.20256e8			1.18805			0.594026	−	0.804446i	$-0.297538\pi$
$277$	4.20256e8			1.18805			0.594026	+	0.804446i	$0.297538\pi$
$278$		0			0
$279$	−1.96420e8	−	2.19604e7i	−0.541466	−	0.0605378i
$280$		0			0
$281$		−	5.09954e8i		−	1.37107i	−0.728041	−	0.685533i	$-0.759569\pi$
$281$		−	5.09954e8i		−	1.37107i	0.728041	−	0.685533i	$-0.240431\pi$
$282$		0			0
$283$		−	1.54638e7i		−	0.0405567i	−0.999794	−	0.0202784i	$-0.993545\pi$
$283$		−	1.54638e7i		−	0.0405567i	0.999794	−	0.0202784i	$-0.00645525\pi$
$284$		0			0
$285$	−1.53362e8	−	1.71464e8i	−0.392429	−	0.438749i
$286$		0			0
$287$	7.85185e8			1.96058
$288$		0			0
$289$	5.48456e7			0.133659
$290$		0			0
$291$	−2.25815e8	−	2.52469e8i	−0.537189	−	0.600595i
$292$		0			0
$293$		−	7.93186e8i		−	1.84221i	−0.389318	−	0.921104i	$-0.627289\pi$
$293$		−	7.93186e8i		−	1.84221i	0.389318	−	0.921104i	$-0.372711\pi$
$294$		0			0
$295$		−	1.55959e8i		−	0.353698i
$296$		0			0
$297$	1.64550e8	−	1.17073e8i	0.364460	−	0.259305i
$298$		0			0
$299$	9.21296e8			1.99320
$300$		0			0
$301$	−7.07513e7			−0.149538
$302$		0			0
$303$	3.90944e8	−	3.49671e8i	0.807356	−	0.722121i
$304$		0			0
$305$			6.46363e7i			0.130445i
$306$		0			0
$307$			5.07669e8i			1.00137i	0.865628	+	0.500687i	$0.166919\pi$
$307$			5.07669e8i			1.00137i	−0.865628	+	0.500687i	$0.833081\pi$
$308$		0			0
$309$	−4.03700e8	+	3.61080e8i	−0.778402	+	0.696224i
$310$		0			0
$311$	−2.79184e8			−0.526296			−0.263148	−	0.964756i	$-0.584761\pi$
$311$	−2.79184e8			−0.526296			−0.263148	+	0.964756i	$0.584761\pi$
$312$		0			0
$313$	−6.92840e7			−0.127711			−0.0638554	−	0.997959i	$-0.520340\pi$
$313$	−6.92840e7			−0.127711			−0.0638554	+	0.997959i	$0.520340\pi$
$314$		0			0
$315$	4.99510e7	−	4.46776e8i	0.0900446	−	0.805383i
$316$		0			0
$317$			8.97465e8i			1.58238i	0.611572	+	0.791189i	$0.290537\pi$
$317$			8.97465e8i			1.58238i	−0.611572	+	0.791189i	$0.709463\pi$
$318$		0			0
$319$			2.09510e8i			0.361358i
$320$		0			0
$321$	6.81761e6	+	7.62232e6i	0.0115044	+	0.0128623i
$322$		0			0
$323$	−4.50846e8			−0.744422
$324$		0			0
$325$	4.55798e8			0.736513
$326$		0			0
$327$	3.05752e8	+	3.41842e8i	0.483563	+	0.540640i
$328$		0			0
$329$		−	5.63327e8i		−	0.872118i
$330$		0			0
$331$		−	1.19136e9i		−	1.80570i	−0.429957	−	0.902849i	$-0.641471\pi$
$331$		−	1.19136e9i		−	1.80570i	0.429957	−	0.902849i	$-0.358529\pi$
$332$		0			0
$333$	1.05243e7	−	9.41325e7i	0.0156185	−	0.139696i
$334$		0			0
$335$	−1.66910e8			−0.242563
$336$		0			0
$337$	1.15595e9			1.64526			0.822630	−	0.568577i	$-0.192506\pi$
$337$	1.15595e9			1.64526			0.822630	+	0.568577i	$0.192506\pi$
$338$		0			0
$339$	−1.88761e8	+	1.68833e8i	−0.263156	+	0.235374i
$340$		0			0
$341$			1.78443e8i			0.243702i
$342$		0			0
$343$		−	6.48126e8i		−	0.867222i
$344$		0			0
$345$	−5.18957e8	+	4.64170e8i	−0.680401	+	0.608569i
$346$		0			0
$347$	1.31673e9			1.69178			0.845892	−	0.533355i	$-0.179069\pi$
$347$	1.31673e9			1.69178			0.845892	+	0.533355i	$0.179069\pi$
$348$		0			0
$349$	2.79115e8			0.351474			0.175737	−	0.984437i	$-0.443769\pi$
$349$	2.79115e8			0.351474			0.175737	+	0.984437i	$0.443769\pi$
$350$		0			0
$351$	1.06087e9	−	7.54781e8i	1.30944	−	0.931635i
$352$		0			0
$353$			6.39614e8i			0.773938i	0.922093	+	0.386969i	$0.126478\pi$
$353$			6.39614e8i			0.773938i	−0.922093	+	0.386969i	$0.873522\pi$
$354$		0			0
$355$		−	5.45161e7i		−	0.0646734i
$356$		0			0
$357$	−5.87373e8	−	6.56703e8i	−0.683243	−	0.763889i
$358$		0			0
$359$	7.06233e8			0.805596			0.402798	−	0.915289i	$-0.368038\pi$
$359$	7.06233e8			0.805596			0.402798	+	0.915289i	$0.368038\pi$
$360$		0			0
$361$	3.22098e8			0.360340
$362$		0			0
$363$	4.85997e8	+	5.43362e8i	0.533287	+	0.596233i
$364$		0			0
$365$			5.33366e7i			0.0574117i
$366$		0			0
$367$		−	1.52201e9i		−	1.60726i	−0.595126	−	0.803632i	$-0.702898\pi$
$367$		−	1.52201e9i		−	1.60726i	0.595126	−	0.803632i	$-0.297102\pi$
$368$		0			0
$369$	−1.70788e9	−	1.90947e8i	−1.76956	−	0.197843i
$370$		0			0
$371$	3.48674e8			0.354496
$372$		0			0
$373$	4.19200e8			0.418254			0.209127	−	0.977888i	$-0.432938\pi$
$373$	4.19200e8			0.418254			0.209127	+	0.977888i	$0.432938\pi$
$374$		0			0
$375$	−8.16957e8	+	7.30708e8i	−0.799999	+	0.715541i
$376$		0			0
$377$			1.35073e9i			1.29829i
$378$		0			0
$379$		−	1.84045e9i		−	1.73655i	−0.496082	−	0.868275i	$-0.665229\pi$
$379$		−	1.84045e9i		−	1.73655i	0.496082	−	0.868275i	$-0.334771\pi$
$380$		0			0
$381$	−5.76875e8	+	5.15972e8i	−0.534372	+	0.477957i
$382$		0			0
$383$	1.87842e8			0.170843			0.0854216	−	0.996345i	$-0.472776\pi$
$383$	1.87842e8			0.170843			0.0854216	+	0.996345i	$0.472776\pi$
$384$		0			0
$385$	−4.05886e8			−0.362486
$386$		0			0
$387$	1.53894e8	+	1.72058e7i	0.134968	+	0.0150899i
$388$		0			0
$389$			3.22401e8i			0.277698i	0.990314	+	0.138849i	$0.0443403\pi$
$389$			3.22401e8i			0.277698i	−0.990314	+	0.138849i	$0.955660\pi$
$390$		0			0
$391$			1.36454e9i			1.15443i
$392$		0			0
$393$	7.93379e7	+	8.87024e7i	0.0659336	+	0.0737160i
$394$		0			0
$395$	−1.07690e9			−0.879195
$396$		0			0
$397$	1.34603e9			1.07966			0.539832	−	0.841773i	$-0.318488\pi$
$397$	1.34603e9			1.07966			0.539832	+	0.841773i	$0.318488\pi$
$398$		0			0
$399$	−7.44922e8	−	8.32848e8i	−0.587091	−	0.656388i
$400$		0			0
$401$		−	8.73134e8i		−	0.676201i	−0.941110	−	0.338100i	$-0.890216\pi$
$401$		−	8.73134e8i		−	0.676201i	0.941110	−	0.338100i	$-0.109784\pi$
$402$		0			0
$403$			1.15044e9i			0.875579i
$404$		0			0
$405$	−2.17300e8	+	9.59649e8i	−0.162543	+	0.717827i
$406$		0			0
$407$	−8.55172e7			−0.0628743
$408$		0			0
$409$	1.61913e9			1.17017			0.585086	−	0.810971i	$-0.301061\pi$
$409$	1.61913e9			1.17017			0.585086	+	0.810971i	$0.301061\pi$
$410$		0			0
$411$	8.87566e8	−	7.93863e8i	0.630601	−	0.564026i
$412$		0			0
$413$		−	7.57535e8i		−	0.529148i
$414$		0			0
$415$		−	2.07810e9i		−	1.42724i
$416$		0			0
$417$	1.38879e9	−	1.24217e9i	0.937907	−	0.838889i
$418$		0			0
$419$	−1.83155e9			−1.21638			−0.608192	−	0.793790i	$-0.708105\pi$
$419$	−1.83155e9			−1.21638			−0.608192	+	0.793790i	$0.708105\pi$
$420$		0			0
$421$	−1.29973e9			−0.848918			−0.424459	−	0.905447i	$-0.639536\pi$
$421$	−1.29973e9			−0.848918			−0.424459	+	0.905447i	$0.639536\pi$
$422$		0			0
$423$	−1.36994e8	+	1.22531e9i	−0.0880057	+	0.787147i
$424$		0			0
$425$			6.75086e8i			0.426578i
$426$		0			0
$427$			3.13956e8i			0.195151i
$428$		0			0
$429$	−7.83659e8	−	8.76157e8i	−0.479211	−	0.535774i
$430$		0			0
$431$	2.25303e9			1.35549			0.677746	−	0.735296i	$-0.262957\pi$
$431$	2.25303e9			1.35549			0.677746	+	0.735296i	$0.262957\pi$
$432$		0			0
$433$	−3.38794e8			−0.200552			−0.100276	−	0.994960i	$-0.531973\pi$
$433$	−3.38794e8			−0.200552			−0.100276	+	0.994960i	$0.531973\pi$
$434$		0			0
$435$	−6.80527e8	−	7.60853e8i	−0.396399	−	0.443188i
$436$		0			0
$437$			1.73054e9i			0.991968i
$438$		0			0
$439$			1.72596e9i			0.973657i	0.873498	+	0.486828i	$0.161846\pi$
$439$			1.72596e9i			0.973657i	−0.873498	+	0.486828i	$0.838154\pi$
$440$		0			0
$441$	4.25048e7	−	3.80175e8i	0.0235995	−	0.211081i
$442$		0			0
$443$	1.73582e9			0.948616			0.474308	−	0.880359i	$-0.342698\pi$
$443$	1.73582e9			0.948616			0.474308	+	0.880359i	$0.342698\pi$
$444$		0			0
$445$	7.99123e8			0.429886
$446$		0			0
$447$	−8.63286e7	+	7.72146e7i	−0.0457170	+	0.0408906i
$448$		0			0
$449$		−	9.57315e8i		−	0.499106i	−0.968361	−	0.249553i	$-0.919716\pi$
$449$		−	9.57315e8i		−	0.499106i	0.968361	−	0.249553i	$-0.0802836\pi$
$450$		0			0
$451$			1.55157e9i			0.796442i
$452$		0			0
$453$	1.97699e9	−	1.76828e9i	0.999221	−	0.893730i
$454$		0			0
$455$	−2.61678e9			−1.30235
$456$		0			0
$457$	−2.22751e9			−1.09172			−0.545862	−	0.837875i	$-0.683798\pi$
$457$	−2.22751e9			−1.09172			−0.545862	+	0.837875i	$0.683798\pi$
$458$		0			0
$459$	1.11791e9	+	1.57126e9i	0.539590	+	0.758409i
$460$		0			0
$461$		−	2.47517e9i		−	1.17666i	−0.808620	−	0.588331i	$-0.799785\pi$
$461$		−	2.47517e9i		−	1.17666i	0.808620	−	0.588331i	$-0.200215\pi$
$462$		0			0
$463$			2.48039e9i			1.16141i	0.814113	+	0.580706i	$0.197223\pi$
$463$			2.48039e9i			1.16141i	−0.814113	+	0.580706i	$0.802777\pi$
$464$		0			0
$465$	−5.79617e8	−	6.48031e8i	−0.267335	−	0.298889i
$466$		0			0
$467$	−1.07555e9			−0.488677			−0.244339	−	0.969690i	$-0.578571\pi$
$467$	−1.07555e9			−0.488677			−0.244339	+	0.969690i	$0.578571\pi$
$468$		0			0
$469$	−8.10726e8			−0.362885
$470$		0			0
$471$	−7.92632e8	−	8.86190e8i	−0.349541	−	0.390799i
$472$		0			0
$473$		−	1.39809e8i		−	0.0607464i
$474$		0			0
$475$			8.56162e8i			0.366546i
$476$		0			0
$477$	−7.58413e8	−	8.47931e7i	−0.319957	−	0.0357723i
$478$		0			0
$479$	−3.09105e8			−0.128508			−0.0642541	−	0.997934i	$-0.520467\pi$
$479$	−3.09105e8			−0.128508			−0.0642541	+	0.997934i	$0.520467\pi$
$480$		0			0
$481$	−5.51336e8			−0.225896
$482$		0			0
$483$	−2.52072e9	+	2.25460e9i	−1.01791	+	0.910446i
$484$		0			0
$485$		−	1.49002e9i		−	0.593056i
$486$		0			0
$487$			2.10568e9i			0.826115i	0.910705	+	0.413058i	$0.135539\pi$
$487$			2.10568e9i			0.826115i	−0.910705	+	0.413058i	$0.864461\pi$
$488$		0			0
$489$	−1.40909e9	+	1.26033e9i	−0.544952	+	0.487420i
$490$		0			0
$491$	5.94413e8			0.226622			0.113311	−	0.993560i	$-0.463854\pi$
$491$	5.94413e8			0.226622			0.113311	+	0.993560i	$0.463854\pi$
$492$		0			0
$493$	−2.00058e9			−0.751953
$494$		0			0
$495$	8.82855e8	+	9.87062e7i	0.327168	+	0.0365785i
$496$		0			0
$497$		−	2.64800e8i		−	0.0967542i
$498$		0			0
$499$			3.39922e9i			1.22469i	0.790590	+	0.612346i	$0.209774\pi$
$499$			3.39922e9i			1.22469i	−0.790590	+	0.612346i	$0.790226\pi$
$500$		0			0
$501$	1.65468e9	+	1.84999e9i	0.587870	+	0.657259i
$502$		0			0
$503$	−4.02401e9			−1.40985			−0.704923	−	0.709284i	$-0.749018\pi$
$503$	−4.02401e9			−1.40985			−0.704923	+	0.709284i	$0.749018\pi$
$504$		0			0
$505$	2.30727e9			0.797222
$506$		0			0
$507$	−3.09600e9	−	3.46144e9i	−1.05505	−	1.17958i
$508$		0			0
$509$		−	1.18155e9i		−	0.397136i	−0.980087	−	0.198568i	$-0.936371\pi$
$509$		−	1.18155e9i		−	0.397136i	0.980087	−	0.198568i	$-0.0636291\pi$
$510$		0			0
$511$			2.59070e8i			0.0858904i
$512$		0			0
$513$	1.41777e9	+	1.99271e9i	0.463654	+	0.651679i
$514$		0			0
$515$	−2.38256e9			−0.768631
$516$		0			0
$517$	1.11317e9			0.354278
$518$		0			0
$519$	2.72135e8	−	2.43405e8i	0.0854475	−	0.0764266i
$520$		0			0
$521$		−	3.54418e9i		−	1.09795i	−0.835838	−	0.548976i	$-0.815018\pi$
$521$		−	3.54418e9i		−	1.09795i	0.835838	−	0.548976i	$-0.184982\pi$
$522$		0			0
$523$		−	2.88980e9i		−	0.883307i	−0.897186	−	0.441654i	$-0.854392\pi$
$523$		−	2.88980e9i		−	0.883307i	0.897186	−	0.441654i	$-0.145608\pi$
$524$		0			0
$525$	−1.24709e9	+	1.11543e9i	−0.376131	+	0.336422i
$526$		0			0
$527$	−1.70392e9			−0.507123
$528$		0			0
$529$	1.83288e9			0.538319
$530$		0			0
$531$	−1.84223e8	+	1.64774e9i	−0.0533965	+	0.477593i
$532$		0			0
$533$			1.00031e10i			2.86147i
$534$		0			0
$535$			4.49854e7i			0.0127009i
$536$		0			0
$537$	2.34517e9	+	2.62198e9i	0.653529	+	0.730668i
$538$		0			0
$539$	−3.45380e8			−0.0950029
$540$		0			0
$541$	4.59048e8			0.124643			0.0623215	−	0.998056i	$-0.480150\pi$
$541$	4.59048e8			0.124643			0.0623215	+	0.998056i	$0.480150\pi$
$542$		0			0
$543$	1.13199e9	+	1.26561e9i	0.303420	+	0.339234i
$544$		0			0
$545$			2.01748e9i			0.533853i
$546$		0			0
$547$			6.87269e7i			0.0179544i	0.999960	+	0.00897720i	$0.00285757\pi$
$547$			6.87269e7i			0.0179544i	−0.999960	+	0.00897720i	$0.997142\pi$
$548$		0			0
$549$	7.63501e7	−	6.82896e8i	0.0196928	−	0.176137i
$550$		0			0
$551$	−2.53718e9			−0.646132
$552$		0			0
$553$	−5.23079e9			−1.31531
$554$		0			0
$555$	3.10563e8	−	2.77776e8i	0.0771123	−	0.0689713i
$556$		0			0
$557$		−	1.71176e9i		−	0.419710i	−0.977732	−	0.209855i	$-0.932701\pi$
$557$		−	1.71176e9i		−	0.419710i	0.977732	−	0.209855i	$-0.0672992\pi$
$558$		0			0
$559$		−	9.01359e8i		−	0.218251i
$560$		0			0
$561$	1.29768e9	−	1.16068e9i	0.310312	−	0.277552i
$562$		0			0
$563$	−1.29253e9			−0.305255			−0.152627	−	0.988284i	$-0.548773\pi$
$563$	−1.29253e9			−0.305255			−0.152627	+	0.988284i	$0.548773\pi$
$564$		0			0
$565$	−1.11403e9			−0.259853
$566$		0			0
$567$	−1.05549e9	+	4.66128e9i	−0.243171	+	1.07390i
$568$		0			0
$569$			5.52303e9i			1.25685i	0.777869	+	0.628426i	$0.216301\pi$
$569$			5.52303e9i			1.25685i	−0.777869	+	0.628426i	$0.783699\pi$
$570$		0			0
$571$			6.19314e9i			1.39215i	0.717971	+	0.696073i	$0.245071\pi$
$571$			6.19314e9i			1.39215i	−0.717971	+	0.696073i	$0.754929\pi$
$572$		0			0
$573$	4.03860e9	+	4.51529e9i	0.896787	+	1.00264i
$574$		0			0
$575$	2.59128e9			0.568430
$576$		0			0
$577$	−2.63098e9			−0.570167			−0.285083	−	0.958503i	$-0.592021\pi$
$577$	−2.63098e9			−0.570167			−0.285083	+	0.958503i	$0.592021\pi$
$578$		0			0
$579$	3.74088e9	+	4.18243e9i	0.800937	+	0.895475i
$580$		0			0
$581$		−	1.00939e10i		−	2.13522i
$582$		0			0
$583$			6.89001e8i			0.144006i
$584$		0			0
$585$	5.69184e9	+	6.36367e8i	1.17546	+	0.131420i
$586$		0			0
$587$	−6.81888e8			−0.139149			−0.0695744	−	0.997577i	$-0.522164\pi$
$587$	−6.81888e8			−0.139149			−0.0695744	+	0.997577i	$0.522164\pi$
$588$		0			0
$589$	−2.16096e9			−0.435756
$590$		0			0
$591$	−1.32789e9	+	1.18770e9i	−0.264610	+	0.236674i
$592$		0			0
$593$			8.90525e8i			0.175370i	0.996148	+	0.0876849i	$0.0279468\pi$
$593$			8.90525e8i			0.175370i	−0.996148	+	0.0876849i	$0.972053\pi$
$594$		0			0
$595$		−	3.87573e9i		−	0.754300i
$596$		0			0
$597$	3.51765e9	−	3.14628e9i	0.676616	−	0.605183i
$598$		0			0
$599$	8.68984e9			1.65203			0.826015	−	0.563648i	$-0.190602\pi$
$599$	8.68984e9			1.65203			0.826015	+	0.563648i	$0.190602\pi$
$600$		0			0
$601$	−6.48769e9			−1.21907			−0.609536	−	0.792758i	$-0.708644\pi$
$601$	−6.48769e9			−1.21907			−0.609536	+	0.792758i	$0.708644\pi$
$602$		0			0
$603$	1.76344e9	+	1.97158e8i	0.327529	+	0.0366188i
$604$		0			0
$605$			3.20681e9i			0.588748i
$606$		0			0
$607$		−	7.67054e9i		−	1.39209i	−0.718001	−	0.696043i	$-0.754942\pi$
$607$		−	7.67054e9i		−	1.39209i	0.718001	−	0.696043i	$-0.245058\pi$
$608$		0			0
$609$	−3.30551e9	−	3.69567e9i	−0.593031	−	0.663029i
$610$		0			0
$611$	7.17669e9			1.27286
$612$		0			0
$613$	−1.01986e10			−1.78825			−0.894123	−	0.447821i	$-0.852200\pi$
$613$	−1.01986e10			−1.78825			−0.894123	+	0.447821i	$0.852200\pi$
$614$		0			0
$615$	−5.03979e9	−	5.63466e9i	−0.873674	−	0.976798i
$616$		0			0
$617$		−	7.66633e9i		−	1.31398i	−0.753898	−	0.656991i	$-0.771829\pi$
$617$		−	7.66633e9i		−	1.31398i	0.753898	−	0.656991i	$-0.228171\pi$
$618$		0			0
$619$		−	7.72256e9i		−	1.30871i	−0.756187	−	0.654356i	$-0.772940\pi$
$619$		−	7.72256e9i		−	1.30871i	0.756187	−	0.654356i	$-0.227060\pi$
$620$		0			0
$621$	6.03119e9	−	4.29105e9i	1.01061	−	0.719023i
$622$		0			0
$623$	3.88156e9			0.643129
$624$		0			0
$625$	−2.02425e9			−0.331653
$626$		0			0
$627$	1.64576e9	−	1.47201e9i	0.266643	−	0.238492i
$628$		0			0
$629$		−	8.16589e8i		−	0.130836i
$630$		0			0
$631$			3.86511e9i			0.612433i	0.951962	+	0.306217i	$0.0990632\pi$
$631$			3.86511e9i			0.612433i	−0.951962	+	0.306217i	$0.900937\pi$
$632$		0			0
$633$	7.04945e9	−	6.30522e9i	1.10470	−	0.988069i
$634$		0			0
$635$	−3.40460e9			−0.527665
$636$		0			0
$637$	−2.22669e9			−0.341328
$638$		0			0
$639$	−6.43959e7	+	5.75975e8i	−0.00976350	+	0.0873274i
$640$		0			0
$641$			3.14005e9i			0.470905i	0.971886	+	0.235453i	$0.0756573\pi$
$641$			3.14005e9i			0.470905i	−0.971886	+	0.235453i	$0.924343\pi$
$642$		0			0
$643$		−	7.73104e9i		−	1.14683i	−0.819265	−	0.573416i	$-0.805618\pi$
$643$		−	7.73104e9i		−	1.14683i	0.819265	−	0.573416i	$-0.194382\pi$
$644$		0			0
$645$	4.54125e8	+	5.07727e8i	0.0666371	+	0.0745026i
$646$		0			0
$647$	−2.19275e9			−0.318290			−0.159145	−	0.987255i	$-0.550874\pi$
$647$	−2.19275e9			−0.318290			−0.159145	+	0.987255i	$0.550874\pi$
$648$		0			0
$649$	1.49693e9			0.214954
$650$		0			0
$651$	−2.81536e9	−	3.14766e9i	−0.399945	−	0.447152i
$652$		0			0
$653$			6.45493e9i			0.907184i	0.891209	+	0.453592i	$0.149858\pi$
$653$			6.45493e9i			0.907184i	−0.891209	+	0.453592i	$0.850142\pi$
$654$		0			0
$655$			5.23504e8i			0.0727907i
$656$		0			0
$657$	6.30026e7	−	5.63512e8i	0.00866722	−	0.0775220i
$658$		0			0
$659$	−7.43719e9			−1.01230			−0.506151	−	0.862445i	$-0.668932\pi$
$659$	−7.43719e9			−1.01230			−0.506151	+	0.862445i	$0.668932\pi$
$660$		0			0
$661$	−2.66092e9			−0.358365			−0.179183	−	0.983816i	$-0.557345\pi$
$661$	−2.66092e9			−0.358365			−0.179183	+	0.983816i	$0.557345\pi$
$662$		0			0
$663$	8.36628e9	−	7.48302e9i	1.11490	−	0.997194i
$664$		0			0
$665$		−	4.91530e9i		−	0.648149i
$666$		0			0
$667$			7.67910e9i			1.00200i
$668$		0			0
$669$	−1.61506e9	+	1.44455e9i	−0.208544	+	0.186527i
$670$		0			0
$671$	−6.20396e8			−0.0792757
$672$		0			0
$673$	9.42499e9			1.19187			0.595934	−	0.803033i	$-0.296782\pi$
$673$	9.42499e9			1.19187			0.595934	+	0.803033i	$0.296782\pi$
$674$		0			0
$675$	2.98384e9	−	2.12293e9i	0.373433	−	0.265689i
$676$		0			0
$677$		−	1.08473e10i		−	1.34357i	−0.740746	−	0.671785i	$-0.765528\pi$
$677$		−	1.08473e10i		−	1.34357i	0.740746	−	0.671785i	$-0.234472\pi$
$678$		0			0
$679$		−	7.23743e9i		−	0.887238i
$680$		0			0
$681$	4.75574e9	+	5.31707e9i	0.577036	+	0.645146i
$682$		0			0
$683$	1.18751e10			1.42615			0.713075	−	0.701088i	$-0.247302\pi$
$683$	1.18751e10			1.42615			0.713075	+	0.701088i	$0.247302\pi$
$684$		0			0
$685$	5.23824e9			0.622685
$686$		0			0
$687$	−1.36907e9	−	1.53066e9i	−0.161093	−	0.180107i
$688$		0			0
$689$			4.44204e9i			0.517387i
$690$		0			0
$691$			9.50025e8i			0.109537i	0.998499	+	0.0547686i	$0.0174421\pi$
$691$			9.50025e8i			0.109537i	−0.998499	+	0.0547686i	$0.982558\pi$
$692$		0			0
$693$	4.28827e9	+	4.79443e8i	0.489458	+	0.0547231i
$694$		0			0
$695$	8.19635e9			0.926134
$696$		0			0
$697$	−1.48157e10			−1.65732
$698$		0			0
$699$	8.45551e9	−	7.56284e9i	0.936418	−	0.837558i
$700$		0			0
$701$			7.66122e9i			0.840011i	0.907522	+	0.420006i	$0.137972\pi$
$701$			7.66122e9i			0.840011i	−0.907522	+	0.420006i	$0.862028\pi$
$702$		0			0
$703$		−	1.03562e9i		−	0.112423i
$704$		0			0
$705$	−4.04256e9	+	3.61578e9i	−0.434505	+	0.388633i
$706$		0			0
$707$	1.12071e10			1.19268
$708$		0			0
$709$	6.07576e9			0.640234			0.320117	−	0.947378i	$-0.396278\pi$
$709$	6.07576e9			0.640234			0.320117	+	0.947378i	$0.396278\pi$
$710$		0			0
$711$	1.13777e10	+	1.27206e9i	1.18716	+	0.132729i
$712$		0			0
$713$			6.54042e9i			0.675759i
$714$		0			0
$715$		−	5.17091e9i		−	0.529049i
$716$		0			0
$717$	−4.18428e9	−	4.67817e9i	−0.423939	−	0.473978i
$718$		0			0
$719$	−1.50425e10			−1.50928			−0.754639	−	0.656140i	$-0.772188\pi$
$719$	−1.50425e10			−1.50928			−0.754639	+	0.656140i	$0.772188\pi$
$720$		0			0
$721$	−1.15727e10			−1.14991
$722$		0			0
$723$	−1.30700e10	−	1.46127e10i	−1.28615	−	1.43796i
$724$		0			0
$725$			3.79912e9i			0.370254i
$726$		0			0
$727$		−	9.39530e9i		−	0.906860i	−0.891292	−	0.453430i	$-0.850200\pi$
$727$		−	9.39530e9i		−	0.906860i	0.891292	−	0.453430i	$-0.149800\pi$
$728$		0			0
$729$	3.42939e9	−	9.88222e9i	0.327846	−	0.944731i
$730$		0			0
$731$	1.33501e9			0.126408
$732$		0			0
$733$	−7.71856e9			−0.723890			−0.361945	−	0.932200i	$-0.617887\pi$
$733$	−7.71856e9			−0.723890			−0.361945	+	0.932200i	$0.617887\pi$
$734$		0			0
$735$	1.25428e9	−	1.12186e9i	0.116516	−	0.104215i
$736$		0			0
$737$		−	1.60204e9i		−	0.147414i
$738$		0			0
$739$		−	3.96381e8i		−	0.0361291i	−0.999837	−	0.0180646i	$-0.994250\pi$
$739$		−	3.96381e8i		−	0.0361291i	0.999837	−	0.0180646i	$-0.00575044\pi$
$740$		0			0
$741$	1.06103e10	−	9.49017e9i	0.957999	−	0.856860i
$742$		0			0
$743$	4.86457e9			0.435094			0.217547	−	0.976050i	$-0.430194\pi$
$743$	4.86457e9			0.435094			0.217547	+	0.976050i	$0.430194\pi$
$744$		0			0
$745$	−5.09494e8			−0.0451432
$746$		0			0
$747$	−2.45471e9	+	2.19556e10i	−0.215465	+	1.92718i
$748$		0			0
$749$			2.18506e8i			0.0190011i
$750$		0			0
$751$			8.71438e9i			0.750753i	0.926872	+	0.375376i	$0.122487\pi$
$751$			8.71438e9i			0.750753i	−0.926872	+	0.375376i	$0.877513\pi$
$752$		0			0
$753$	1.38597e10	+	1.54957e10i	1.18297	+	1.32260i
$754$		0			0
$755$	1.16678e10			0.986678
$756$		0			0
$757$	−1.67972e10			−1.40735			−0.703673	−	0.710524i	$-0.748458\pi$
$757$	−1.67972e10			−1.40735			−0.703673	+	0.710524i	$0.748458\pi$
$758$		0			0
$759$	−4.45522e9	−	4.98109e9i	−0.369848	−	0.413502i
$760$		0			0
$761$		−	4.88336e8i		−	0.0401673i	−0.999798	−	0.0200836i	$-0.993607\pi$
$761$		−	4.88336e8i		−	0.0401673i	0.999798	−	0.0200836i	$-0.00639325\pi$
$762$		0			0
$763$			9.79946e9i			0.798668i
$764$		0			0
$765$	−9.42529e8	+	8.43024e9i	−0.0761167	+	0.680808i
$766$		0			0
$767$	9.65085e9			0.772292
$768$		0			0
$769$	9.73330e9			0.771824			0.385912	−	0.922536i	$-0.373887\pi$
$769$	9.73330e9			0.771824			0.385912	+	0.922536i	$0.373887\pi$
$770$		0			0
$771$	−1.13277e9	+	1.01318e9i	−0.0890129	+	0.0796156i
$772$		0			0
$773$			2.10035e10i			1.63555i	0.575540	+	0.817774i	$0.304792\pi$
$773$			2.10035e10i			1.63555i	−0.575540	+	0.817774i	$0.695208\pi$
$774$		0			0
$775$			3.23578e9i			0.249702i
$776$		0			0
$777$	1.50849e9	−	1.34923e9i	0.115363	−	0.103184i
$778$		0			0
$779$	−1.87897e10			−1.42409
$780$		0			0
$781$	5.23260e8			0.0393042
$782$		0			0
$783$	6.29118e9	+	8.84243e9i	0.468345	+	0.658272i
$784$		0			0
$785$		−	5.23012e9i		−	0.385894i
$786$		0			0
$787$		−	2.27878e10i		−	1.66644i	−0.552939	−	0.833222i	$-0.686494\pi$
$787$		−	2.27878e10i		−	1.66644i	0.552939	−	0.833222i	$-0.313506\pi$
$788$		0			0
$789$	1.32923e10	+	1.48612e10i	0.963453	+	1.07717i
$790$		0			0
$791$	−5.41116e9			−0.388752
$792$		0			0
$793$	−3.99974e9			−0.284823
$794$		0			0
$795$	−2.23800e9	−	2.50216e9i	−0.157970	−	0.176616i
$796$		0			0
$797$		−	1.22577e10i		−	0.857638i	−0.903391	−	0.428819i	$-0.858930\pi$
$797$		−	1.22577e10i		−	0.857638i	0.903391	−	0.428819i	$-0.141070\pi$
$798$		0			0
$799$			1.06295e10i			0.737221i
$800$		0			0
$801$	−8.44291e9	−	9.43946e8i	−0.580468	−	0.0648983i
$802$		0			0
$803$	−5.11938e8			−0.0348910
$804$		0			0
$805$	−1.48768e10			−1.00513
$806$		0			0
$807$	−1.04696e10	+	9.36434e9i	−0.701253	+	0.627220i
$808$		0			0
$809$			3.89938e9i			0.258926i	0.991584	+	0.129463i	$0.0413253\pi$
$809$			3.89938e9i			0.258926i	−0.991584	+	0.129463i	$0.958675\pi$
$810$		0			0
$811$			9.83894e9i			0.647702i	0.946108	+	0.323851i	$0.104978\pi$
$811$			9.83894e9i			0.647702i	−0.946108	+	0.323851i	$0.895022\pi$
$812$		0			0
$813$	−5.87748e9	+	5.25698e9i	−0.383596	+	0.343099i
$814$		0			0
$815$	−8.31618e9			−0.538111
$816$		0			0
$817$	1.69310e9			0.108619
$818$		0			0
$819$	2.76468e10	+	3.09101e9i	1.75854	+	0.196610i
$820$		0			0
$821$		−	1.76060e10i		−	1.11035i	−0.831733	−	0.555176i	$-0.812651\pi$
$821$		−	1.76060e10i		−	1.11035i	0.831733	−	0.555176i	$-0.187349\pi$
$822$		0			0
$823$		−	1.59689e10i		−	0.998564i	−0.866439	−	0.499282i	$-0.833597\pi$
$823$		−	1.59689e10i		−	0.998564i	0.866439	−	0.499282i	$-0.166403\pi$
$824$		0			0
$825$	−2.20416e9	−	2.46432e9i	−0.136664	−	0.152795i
$826$		0			0
$827$	−1.38691e10			−0.852666			−0.426333	−	0.904566i	$-0.640195\pi$
$827$	−1.38691e10			−0.852666			−0.426333	+	0.904566i	$0.640195\pi$
$828$		0			0
$829$	3.03083e10			1.84766			0.923828	−	0.382808i	$-0.125043\pi$
$829$	3.03083e10			1.84766			0.923828	+	0.382808i	$0.125043\pi$
$830$		0			0
$831$	−1.31023e10	−	1.46488e10i	−0.792034	−	0.885521i
$832$		0			0
$833$		−	3.29798e9i		−	0.197692i
$834$		0			0
$835$			1.09182e10i			0.649008i
$836$		0			0
$837$	5.35830e9	+	7.53125e9i	0.315855	+	0.443944i
$838$		0			0
$839$	1.76156e10			1.02975			0.514873	−	0.857266i	$-0.327839\pi$
$839$	1.76156e10			1.02975			0.514873	+	0.857266i	$0.327839\pi$
$840$		0			0
$841$	5.99141e9			0.347331
$842$		0			0
$843$	−1.77754e10	+	1.58988e10i	−1.02193	+	0.914044i
$844$		0			0
$845$		−	2.04287e10i		−	1.16478i
$846$		0			0
$847$			1.55764e10i			0.880794i
$848$		0			0
$849$	−5.39019e8	+	4.82113e8i	−0.0302292	+	0.0270378i
$850$		0			0
$851$	−3.13443e9			−0.174343
$852$		0			0
$853$	7.77452e9			0.428896			0.214448	−	0.976735i	$-0.431205\pi$
$853$	7.77452e9			0.428896			0.214448	+	0.976735i	$0.431205\pi$
$854$		0			0
$855$	−1.19534e9	+	1.06914e10i	−0.0654049	+	0.584999i
$856$		0			0
$857$			3.40944e9i			0.185033i	0.995711	+	0.0925166i	$0.0294911\pi$
$857$			3.40944e9i			0.185033i	−0.995711	+	0.0925166i	$0.970509\pi$
$858$		0			0
$859$		−	1.01843e9i		−	0.0548221i	−0.999624	−	0.0274111i	$-0.991274\pi$
$859$		−	1.01843e9i		−	0.0548221i	0.999624	−	0.0274111i	$-0.00872631\pi$
$860$		0			0
$861$	−2.44796e10	−	2.73691e10i	−1.30706	−	1.46133i
$862$		0			0
$863$	−2.23663e10			−1.18456			−0.592278	−	0.805734i	$-0.701771\pi$
$863$	−2.23663e10			−1.18456			−0.592278	+	0.805734i	$0.701771\pi$
$864$		0			0
$865$	1.60609e9			0.0843749
$866$		0			0
$867$	−1.70992e9	−	1.91174e9i	−0.0891062	−	0.0996237i
$868$		0			0
$869$		−	1.03364e10i		−	0.534316i
$870$		0			0
$871$		−	1.03285e10i		−	0.529631i
$872$		0			0
$873$	−1.76005e9	+	1.57424e10i	−0.0895314	+	0.800793i
$874$		0			0
$875$	−2.34194e10			−1.18181
$876$		0			0
$877$	7.42860e9			0.371885			0.185942	−	0.982561i	$-0.440466\pi$
$877$	7.42860e9			0.371885			0.185942	+	0.982561i	$0.440466\pi$
$878$		0			0
$879$	−2.76480e10	+	2.47291e10i	−1.37310	+	1.22814i
$880$		0			0
$881$		−	2.83256e10i		−	1.39561i	−0.716288	−	0.697804i	$-0.754160\pi$
$881$		−	2.83256e10i		−	1.39561i	0.716288	−	0.697804i	$-0.245840\pi$
$882$		0			0
$883$			2.16342e10i			1.05750i	0.848779	+	0.528748i	$0.177338\pi$
$883$			2.16342e10i			1.05750i	−0.848779	+	0.528748i	$0.822662\pi$
$884$		0			0
$885$	−5.43623e9	+	4.86232e9i	−0.263631	+	0.235799i
$886$		0			0
$887$	2.48940e9			0.119774			0.0598869	−	0.998205i	$-0.480926\pi$
$887$	2.48940e9			0.119774			0.0598869	+	0.998205i	$0.480926\pi$
$888$		0			0
$889$	−1.65371e10			−0.789410
$890$		0			0
$891$	−9.21096e9	−	2.08571e9i	−0.436248	−	0.0987827i
$892$		0			0
$893$			1.34806e10i			0.633472i
$894$		0			0
$895$			1.54744e10i			0.721496i
$896$		0			0
$897$	−2.87232e10	−	3.21135e10i	−1.32880	−	1.48564i
$898$		0			0
$899$	−9.58902e9			−0.440165
$900$		0			0
$901$	−6.57915e9			−0.299663
$902$		0			0
$903$	2.20581e9	+	2.46617e9i	0.0996921	+	0.111459i
$904$		0			0
$905$			7.46936e9i			0.334976i
$906$		0			0
$907$			6.03196e8i			0.0268431i	0.999910	+	0.0134216i	$0.00427234\pi$
$907$			6.03196e8i			0.0268431i	−0.999910	+	0.0134216i	$0.995728\pi$
$908$		0			0
$909$	−2.43768e10	−	2.72541e9i	−1.07647	−	0.120354i
$910$		0			0
$911$	−2.68931e10			−1.17849			−0.589246	−	0.807954i	$-0.700575\pi$
$911$	−2.68931e10			−1.17849			−0.589246	+	0.807954i	$0.700575\pi$
$912$		0			0
$913$	1.99461e10			0.867383
$914$		0			0
$915$	2.25302e9	−	2.01516e9i	0.0972278	−	0.0869632i
$916$		0			0
$917$			2.54280e9i			0.108898i
$918$		0			0
$919$		−	1.44011e10i		−	0.612056i	−0.952022	−	0.306028i	$-0.901000\pi$
$919$		−	1.44011e10i		−	0.612056i	0.952022	−	0.306028i	$-0.0990002\pi$
$920$		0			0
$921$	1.76957e10	−	1.58276e10i	0.746380	−	0.667583i
$922$		0			0
$923$	3.37350e9			0.141213
$924$		0			0
$925$	−1.55071e9			−0.0644222
$926$		0			0
$927$	2.51722e10	+	2.81434e9i	1.03787	+	0.116037i
$928$		0			0
$929$		−	8.33324e9i		−	0.341003i	−0.985357	−	0.170502i	$-0.945461\pi$
$929$		−	8.33324e9i		−	0.341003i	0.985357	−	0.170502i	$-0.0545388\pi$
$930$		0			0
$931$		−	4.18258e9i		−	0.169871i
$932$		0			0
$933$	8.70411e9	+	9.73149e9i	0.350864	+	0.392278i
$934$		0			0
$935$	7.65868e9			0.306417
$936$		0			0
$937$	2.24458e10			0.891346			0.445673	−	0.895196i	$-0.352964\pi$
$937$	2.24458e10			0.891346			0.445673	+	0.895196i	$0.352964\pi$
$938$		0			0
$939$	2.16006e9	+	2.41502e9i	0.0851405	+	0.0951900i
$940$		0			0
$941$			2.64833e10i			1.03612i	0.855345	+	0.518059i	$0.173345\pi$
$941$			2.64833e10i			1.03612i	−0.855345	+	0.518059i	$0.826655\pi$
$942$		0			0
$943$			5.68692e10i			2.20844i
$944$		0			0
$945$	−1.71305e10	+	1.21879e10i	−0.660327	+	0.469807i
$946$		0			0
$947$	2.26400e10			0.866266			0.433133	−	0.901330i	$-0.357408\pi$
$947$	2.26400e10			0.866266			0.433133	+	0.901330i	$0.357408\pi$
$948$		0			0
$949$	−3.30051e9			−0.125357
$950$		0			0
$951$	3.12828e10	−	2.79802e10i	1.17943	−	1.05492i
$952$		0			0
$953$			4.67746e10i			1.75059i	0.483586	+	0.875297i	$0.339334\pi$
$953$			4.67746e10i			1.75059i	−0.483586	+	0.875297i	$0.660666\pi$
$954$		0			0
$955$			2.66483e10i			0.990052i
$956$		0			0
$957$	7.30286e9	−	6.53188e9i	0.269340	−	0.240905i
$958$		0			0
$959$	2.54436e10			0.931564
$960$		0			0
$961$	1.93455e10			0.703149
$962$		0			0
$963$	5.31380e7	−	4.75281e8i	0.00191740	−	0.0171498i
$964$		0			0
$965$			2.46839e10i			0.884235i
$966$		0			0
$967$			1.66727e10i			0.592942i	0.955042	+	0.296471i	$0.0958098\pi$
$967$			1.66727e10i			0.592942i	−0.955042	+	0.296471i	$0.904190\pi$
$968$		0			0
$969$	1.40560e10	+	1.57151e10i	0.496281	+	0.554859i
$970$		0			0
$971$	1.55445e9			0.0544889			0.0272445	−	0.999629i	$-0.491327\pi$
$971$	1.55445e9			0.0544889			0.0272445	+	0.999629i	$0.491327\pi$
$972$		0			0
$973$	3.98119e10			1.38554
$974$		0			0
$975$	−1.42104e10	−	1.58877e10i	−0.491009	−	0.548964i
$976$		0			0
$977$			7.71063e9i			0.264520i	0.991215	+	0.132260i	$0.0422234\pi$
$977$			7.71063e9i			0.264520i	−0.991215	+	0.132260i	$0.957777\pi$
$978$		0			0
$979$			7.67019e9i			0.261256i
$980$		0			0
$981$	2.38310e9	−	2.13151e10i	0.0805938	−	0.720853i
$982$		0			0
$983$	−5.24036e10			−1.75964			−0.879820	−	0.475307i	$-0.842337\pi$
$983$	−5.24036e10			−1.75964			−0.879820	+	0.475307i	$0.842337\pi$
$984$		0			0
$985$	−7.83694e9			−0.261288
$986$		0			0
$987$	−1.96358e10	+	1.75628e10i	−0.650038	+	0.581412i
$988$		0			0
$989$		−	5.12436e9i		−	0.168443i
$990$		0			0
$991$		−	6.02219e9i		−	0.196561i	−0.995159	−	0.0982803i	$-0.968666\pi$
$991$		−	6.02219e9i		−	0.196561i	0.995159	−	0.0982803i	$-0.0313342\pi$
$992$		0			0
$993$	−4.15271e10	+	3.71430e10i	−1.34589	+	1.20380i
$994$		0			0
$995$	2.07605e10			0.668122
$996$		0			0
$997$	−9.88566e9			−0.315917			−0.157958	−	0.987446i	$-0.550491\pi$
$997$	−9.88566e9			−0.315917			−0.157958	+	0.987446i	$0.550491\pi$
$998$		0			0
$999$	−3.60928e9	+	2.56791e9i	−0.114536	+	0.0814895i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.8.c.b.191.1		4
3.2	odd	2	inner	192.8.c.b.191.3		4
4.3	odd	2	inner	192.8.c.b.191.4		4
8.3	odd	2		12.8.b.a.11.1	✓	4
8.5	even	2		12.8.b.a.11.3	yes	4
12.11	even	2	inner	192.8.c.b.191.2		4
24.5	odd	2		12.8.b.a.11.2	yes	4
24.11	even	2		12.8.b.a.11.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.8.b.a.11.1	✓	4	8.3	odd	2
12.8.b.a.11.2	yes	4	24.5	odd	2
12.8.b.a.11.3	yes	4	8.5	even	2
12.8.b.a.11.4	yes	4	24.11	even	2
192.8.c.b.191.1		4	1.1	even	1	trivial
192.8.c.b.191.2		4	12.11	even	2	inner
192.8.c.b.191.3		4	3.2	odd	2	inner
192.8.c.b.191.4		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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	192.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-31.1769 - 34.8569i) q^{3} -205.718i q^{5} -999.230i q^{7} +(-243.000 + 2173.46i) q^{9} +O(q^{10})\)	\(q+(-31.1769 - 34.8569i) q^{3} -205.718i q^{5} -999.230i q^{7} +(-243.000 + 2173.46i) q^{9} +1974.54 q^{11} +12730.0 q^{13} +(-7170.69 + 6413.66i) q^{15} +18854.5i q^{17} +23911.8i q^{19} +(-34830.0 + 31152.9i) q^{21} +72372.0 q^{23} +35805.0 q^{25} +(83335.9 - 59291.5i) q^{27} +106106. i q^{29} +90372.2i q^{31} +(-61560.0 - 68826.2i) q^{33} -205560. q^{35} -43310.0 q^{37} +(-396882. - 443728. i) q^{39} +785790. i q^{41} -70805.9i q^{43} +(447120. + 49989.5i) q^{45} +563762. q^{47} -174917. q^{49} +(657209. - 587826. i) q^{51} +348943. i q^{53} -406198. i q^{55} +(833490. - 745496. i) q^{57} +758119. q^{59} -314198. q^{61} +(2.17178e6 + 242813. i) q^{63} -2.61879e6i q^{65} -811351. i q^{67} +(-2.25634e6 - 2.52266e6i) q^{69} +265004. q^{71} -259270. q^{73} +(-1.11629e6 - 1.24805e6i) q^{75} -1.97302e6i q^{77} -5.23482e6i q^{79} +(-4.66487e6 - 1.05630e6i) q^{81} +1.01017e7 q^{83} +3.87872e6 q^{85} +(3.69852e6 - 3.30805e6i) q^{87} +3.88455e6i q^{89} -1.27202e7i q^{91} +(3.15009e6 - 2.81753e6i) q^{93} +4.91909e6 q^{95} +7.24301e6 q^{97} +(-479813. + 4.29158e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 972 q^{9}+O(q^{10}) \) 4 * q - 972 * q^9	\( 4 q - 972 q^{9} + 50920 q^{13} - 139320 q^{21} + 143220 q^{25} - 246240 q^{33} - 173240 q^{37} + 1788480 q^{45} - 699668 q^{49} + 3333960 q^{57} - 1256792 q^{61} - 9025344 q^{69} - 1037080 q^{73} - 18659484 q^{81} + 15514880 q^{85} + 12600360 q^{93} + 28972040 q^{97}+O(q^{100}) \) 4 * q - 972 * q^9 + 50920 * q^13 - 139320 * q^21 + 143220 * q^25 - 246240 * q^33 - 173240 * q^37 + 1788480 * q^45 - 699668 * q^49 + 3333960 * q^57 - 1256792 * q^61 - 9025344 * q^69 - 1037080 * q^73 - 18659484 * q^81 + 15514880 * q^85 + 12600360 * q^93 + 28972040 * q^97

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