LMFDB - Embedded newform 192.12.c.d.191.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,12,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, 12, names="a")

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

chi := DirichletCharacter("192.191");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 12 $
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:	$147.521890667$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 247 x^{18} - 23916 x^{16} + 14713536 x^{14} - 45723119616 x^{12} + 40864324780032 x^{10} + \cdots + 12\!\cdots\!24 $ x^20 - 247x^18 - 23916x^16 + 14713536x^14 - 45723119616x^12 + 40864324780032x^10 - 11986041468616704x^8 + 1011106494856298496x^6 - 430832354752771129344x^4 - 1166424521268802367782912*x^2 + 1237940039285380274899124224
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{194}\cdot 3^{42}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.10
Root			$-16.0196 + 15.9804i$ of defining polynomial
Character	$\chi$	$=$	192.191
Dual form			192.12.c.d.191.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-70.6484 + 414.917i) q^{3} -1842.04i q^{5} +16302.4i q^{7} +(-167165. - 58626.4i) q^{9} +O(q^{10})$	\(q+(-70.6484 + 414.917i) q^{3} -1842.04i q^{5} +16302.4i q^{7} +(-167165. - 58626.4i) q^{9} +485828. q^{11} +1.36551e6 q^{13} +(764291. + 130137. i) q^{15} +8.90746e6i q^{17} +107974. i q^{19} +(-6.76412e6 - 1.15174e6i) q^{21} -5.58155e7 q^{23} +4.54350e7 q^{25} +(3.61350e7 - 6.52175e7i) q^{27} -1.27942e8i q^{29} -1.30377e8i q^{31} +(-3.43230e7 + 2.01578e8i) q^{33} +3.00295e7 q^{35} +3.44489e8 q^{37} +(-9.64708e7 + 5.66571e8i) q^{39} -9.40493e8i q^{41} +9.39748e8i q^{43} +(-1.07992e8 + 3.07923e8i) q^{45} +9.85584e8 q^{47} +1.71156e9 q^{49} +(-3.69585e9 - 6.29298e8i) q^{51} +1.84952e9i q^{53} -8.94912e8i q^{55} +(-4.48001e7 - 7.62817e6i) q^{57} +5.76202e9 q^{59} +7.38482e9 q^{61} +(9.55749e8 - 2.72518e9i) q^{63} -2.51531e9i q^{65} -1.05400e10i q^{67} +(3.94328e9 - 2.31588e10i) q^{69} -1.58034e10 q^{71} +1.99415e10 q^{73} +(-3.20991e9 + 1.88517e10i) q^{75} +7.92014e9i q^{77} +1.66155e10i q^{79} +(2.45069e10 + 1.96005e10i) q^{81} +3.12525e10 q^{83} +1.64079e10 q^{85} +(5.30852e10 + 9.03889e9i) q^{87} -5.55310e10i q^{89} +2.22610e10i q^{91} +(5.40955e10 + 9.21092e9i) q^{93} +1.98891e8 q^{95} -4.52802e10 q^{97} +(-8.12132e10 - 2.84823e10i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 103620 q^{9}+O(q^{10}) $ 20 * q + 103620 * q^9	$ 20 q + 103620 q^{9} - 1543864 q^{13} + 12211752 q^{21} - 141128700 q^{25} + 229769760 q^{33} + 44517800 q^{37} - 1020227520 q^{45} - 1018138084 q^{49} + 1438636392 q^{57} - 6873199864 q^{61} + 13308470976 q^{69} - 12426469112 q^{73} - 42462874764 q^{81} - 17135502080 q^{85} - 114387515256 q^{93} + 75764383528 q^{97}+O(q^{100}) $ 20 * q + 103620 * q^9 - 1543864 * q^13 + 12211752 * q^21 - 141128700 * q^25 + 229769760 * q^33 + 44517800 * q^37 - 1020227520 * q^45 - 1018138084 * q^49 + 1438636392 * q^57 - 6873199864 * q^61 + 13308470976 * q^69 - 12426469112 * q^73 - 42462874764 * q^81 - 17135502080 * q^85 - 114387515256 * q^93 + 75764383528 * 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
$3$	−70.6484	+	414.917i	−0.167855	+	0.985812i
$3$	−70.6484	+	414.917i	−0.167855	+	0.985812i
$4$		0			0
$5$		−	1842.04i		−	0.263611i	−0.991276	−	0.131805i	$-0.957923\pi$
$5$		−	1842.04i		−	0.263611i	0.991276	−	0.131805i	$-0.0420774\pi$
$6$		0			0
$7$			16302.4i			0.366616i	0.983056	+	0.183308i	$0.0586805\pi$
$7$			16302.4i			0.366616i	−0.983056	+	0.183308i	$0.941319\pi$
$8$		0			0
$9$	−167165.	−	58626.4i	−0.943649	−	0.330948i
$10$		0			0
$11$	485828.			0.909542			0.454771	−	0.890608i	$-0.349721\pi$
$11$	485828.			0.909542			0.454771	+	0.890608i	$0.349721\pi$
$12$		0			0
$13$	1.36551e6			1.02001			0.510006	−	0.860171i	$-0.329643\pi$
$13$	1.36551e6			1.02001			0.510006	+	0.860171i	$0.329643\pi$
$14$		0			0
$15$	764291.	+	130137.i	0.259871	+	0.0442485i
$16$		0			0
$17$			8.90746e6i			1.52154i	0.649019	+	0.760772i	$0.275180\pi$
$17$			8.90746e6i			1.52154i	−0.649019	+	0.760772i	$0.724820\pi$
$18$		0			0
$19$			107974.i			0.0100040i	0.999987	+	0.00500199i	$0.00159219\pi$
$19$			107974.i			0.0100040i	−0.999987	+	0.00500199i	$0.998408\pi$
$20$		0			0
$21$	−6.76412e6	−	1.15174e6i	−0.361414	−	0.0615385i
$22$		0			0
$23$	−5.58155e7			−1.80822			−0.904112	−	0.427296i	$-0.859466\pi$
$23$	−5.58155e7			−1.80822			−0.904112	+	0.427296i	$0.859466\pi$
$24$		0			0
$25$	4.54350e7			0.930509
$26$		0			0
$27$	3.61350e7	−	6.52175e7i	0.484649	−	0.874709i
$28$		0			0
$29$		−	1.27942e8i		−	1.15831i	−0.815218	−	0.579154i	$-0.803383\pi$
$29$		−	1.27942e8i		−	1.15831i	0.815218	−	0.579154i	$-0.196617\pi$
$30$		0			0
$31$		−	1.30377e8i		−	0.817921i	−0.912552	−	0.408960i	$-0.865891\pi$
$31$		−	1.30377e8i		−	0.817921i	0.912552	−	0.408960i	$-0.134109\pi$
$32$		0			0
$33$	−3.43230e7	+	2.01578e8i	−0.152672	+	0.896637i
$34$		0			0
$35$	3.00295e7			0.0966439
$36$		0			0
$37$	3.44489e8			0.816705			0.408353	−	0.912824i	$-0.366103\pi$
$37$	3.44489e8			0.816705			0.408353	+	0.912824i	$0.366103\pi$
$38$		0			0
$39$	−9.64708e7	+	5.66571e8i	−0.171215	+	1.00554i
$40$		0			0
$41$		−	9.40493e8i		−	1.26778i	−0.773423	−	0.633891i	$-0.781457\pi$
$41$		−	9.40493e8i		−	1.26778i	0.773423	−	0.633891i	$-0.218543\pi$
$42$		0			0
$43$			9.39748e8i			0.974844i	0.873167	+	0.487422i	$0.162063\pi$
$43$			9.39748e8i			0.974844i	−0.873167	+	0.487422i	$0.837937\pi$
$44$		0			0
$45$	−1.07992e8	+	3.07923e8i	−0.0872414	+	0.248756i
$46$		0			0
$47$	9.85584e8			0.626838			0.313419	−	0.949615i	$-0.398526\pi$
$47$	9.85584e8			0.626838			0.313419	+	0.949615i	$0.398526\pi$
$48$		0			0
$49$	1.71156e9			0.865593
$50$		0			0
$51$	−3.69585e9	−	6.29298e8i	−1.49996	−	0.255400i
$52$		0			0
$53$			1.84952e9i			0.607495i	0.952753	+	0.303748i	$0.0982379\pi$
$53$			1.84952e9i			0.607495i	−0.952753	+	0.303748i	$0.901762\pi$
$54$		0			0
$55$		−	8.94912e8i		−	0.239765i
$56$		0			0
$57$	−4.48001e7	−	7.62817e6i	−0.00986204	−	0.00167922i
$58$		0			0
$59$	5.76202e9			1.04927			0.524637	−	0.851326i	$-0.324201\pi$
$59$	5.76202e9			1.04927			0.524637	+	0.851326i	$0.324201\pi$
$60$		0			0
$61$	7.38482e9			1.11950			0.559752	−	0.828660i	$-0.310896\pi$
$61$	7.38482e9			1.11950			0.559752	+	0.828660i	$0.310896\pi$
$62$		0			0
$63$	9.55749e8	−	2.72518e9i	0.121331	−	0.345957i
$64$		0			0
$65$		−	2.51531e9i		−	0.268886i
$66$		0			0
$67$		−	1.05400e10i		−	0.953735i	−0.878975	−	0.476868i	$-0.841772\pi$
$67$		−	1.05400e10i		−	0.953735i	0.878975	−	0.476868i	$-0.158228\pi$
$68$		0			0
$69$	3.94328e9	−	2.31588e10i	0.303520	−	1.78257i
$70$		0			0
$71$	−1.58034e10			−1.03951			−0.519755	−	0.854315i	$-0.673977\pi$
$71$	−1.58034e10			−1.03951			−0.519755	+	0.854315i	$0.673977\pi$
$72$		0			0
$73$	1.99415e10			1.12585			0.562927	−	0.826506i	$-0.309675\pi$
$73$	1.99415e10			1.12585			0.562927	+	0.826506i	$0.309675\pi$
$74$		0			0
$75$	−3.20991e9	+	1.88517e10i	−0.156191	+	0.917307i
$76$		0			0
$77$			7.92014e9i			0.333452i
$78$		0			0
$79$			1.66155e10i			0.607525i	0.952748	+	0.303762i	$0.0982429\pi$
$79$			1.66155e10i			0.607525i	−0.952748	+	0.303762i	$0.901757\pi$
$80$		0			0
$81$	2.45069e10	+	1.96005e10i	0.780947	+	0.624597i
$82$		0			0
$83$	3.12525e10			0.870873			0.435436	−	0.900219i	$-0.356594\pi$
$83$	3.12525e10			0.870873			0.435436	+	0.900219i	$0.356594\pi$
$84$		0			0
$85$	1.64079e10			0.401096
$86$		0			0
$87$	5.30852e10	+	9.03889e9i	1.14187	+	0.194428i
$88$		0			0
$89$		−	5.55310e10i		−	1.05412i	−0.849828	−	0.527061i	$-0.823294\pi$
$89$		−	5.55310e10i		−	1.05412i	0.849828	−	0.527061i	$-0.176706\pi$
$90$		0			0
$91$			2.22610e10i			0.373952i
$92$		0			0
$93$	5.40955e10	+	9.21092e9i	0.806316	+	0.137292i
$94$		0			0
$95$	1.98891e8			0.00263716
$96$		0			0
$97$	−4.52802e10			−0.535382			−0.267691	−	0.963505i	$-0.586261\pi$
$97$	−4.52802e10			−0.535382			−0.267691	+	0.963505i	$0.586261\pi$
$98$		0			0
$99$	−8.12132e10	−	2.84823e10i	−0.858288	−	0.301011i
$100$		0			0
$101$			7.04743e10i			0.667211i	0.942713	+	0.333606i	$0.108265\pi$
$101$			7.04743e10i			0.667211i	−0.942713	+	0.333606i	$0.891735\pi$
$102$		0			0
$103$		−	8.56630e10i		−	0.728096i	−0.931380	−	0.364048i	$-0.881394\pi$
$103$		−	8.56630e10i		−	0.728096i	0.931380	−	0.364048i	$-0.118606\pi$
$104$		0			0
$105$	−2.12154e9	+	1.24598e10i	−0.0162222	+	0.0952727i
$106$		0			0
$107$	−6.87116e10			−0.473608			−0.236804	−	0.971557i	$-0.576100\pi$
$107$	−6.87116e10			−0.473608			−0.236804	+	0.971557i	$0.576100\pi$
$108$		0			0
$109$	−1.08394e11			−0.674777			−0.337388	−	0.941366i	$-0.609544\pi$
$109$	−1.08394e11			−0.674777			−0.337388	+	0.941366i	$0.609544\pi$
$110$		0			0
$111$	−2.43376e10	+	1.42934e11i	−0.137088	+	0.805117i
$112$		0			0
$113$			2.57009e10i			0.131225i	0.997845	+	0.0656125i	$0.0209001\pi$
$113$			2.57009e10i			0.131225i	−0.997845	+	0.0656125i	$0.979100\pi$
$114$		0			0
$115$			1.02814e11i			0.476667i
$116$		0			0
$117$	−2.28264e11	−	8.00547e10i	−0.962533	−	0.337571i
$118$		0			0
$119$	−1.45213e11			−0.557822
$120$		0			0
$121$	−4.92830e10			−0.172734
$122$		0			0
$123$	3.90226e11	+	6.64443e10i	1.24979	+	0.212804i
$124$		0			0
$125$		−	1.73636e11i		−	0.508903i
$126$		0			0
$127$			2.98675e11i			0.802191i	0.916036	+	0.401096i	$0.131370\pi$
$127$			2.98675e11i			0.802191i	−0.916036	+	0.401096i	$0.868630\pi$
$128$		0			0
$129$	−3.89917e11	−	6.63917e10i	−0.961012	−	0.163633i
$130$		0			0
$131$	−2.12063e10			−0.0480255			−0.0240127	−	0.999712i	$-0.507644\pi$
$131$	−2.12063e10			−0.0480255			−0.0240127	+	0.999712i	$0.507644\pi$
$132$		0			0
$133$	−1.76022e9			−0.00366762
$134$		0			0
$135$	−1.20133e11	−	6.65619e10i	−0.230583	−	0.127759i
$136$		0			0
$137$			1.01885e12i			1.80363i	0.432117	+	0.901817i	$0.357767\pi$
$137$			1.01885e12i			1.80363i	−0.432117	+	0.901817i	$0.642233\pi$
$138$		0			0
$139$			2.22475e11i			0.363663i	0.983330	+	0.181832i	$0.0582026\pi$
$139$			2.22475e11i			0.363663i	−0.983330	+	0.181832i	$0.941797\pi$
$140$		0			0
$141$	−6.96300e10	+	4.08935e11i	−0.105218	+	0.617944i
$142$		0			0
$143$	6.63401e11			0.927743
$144$		0			0
$145$	−2.35674e11			−0.305342
$146$		0			0
$147$	−1.20919e11	+	7.10155e11i	−0.145294	+	0.853311i
$148$		0			0
$149$			4.83320e11i			0.539150i	0.962979	+	0.269575i	$0.0868833\pi$
$149$			4.83320e11i			0.539150i	−0.962979	+	0.269575i	$0.913117\pi$
$150$		0			0
$151$			1.38396e12i			1.43466i	0.696733	+	0.717331i	$0.254636\pi$
$151$			1.38396e12i			1.43466i	−0.696733	+	0.717331i	$0.745364\pi$
$152$		0			0
$153$	5.22212e11	−	1.48901e12i	0.503552	−	1.43580i
$154$		0			0
$155$	−2.40159e11			−0.215613
$156$		0			0
$157$	−5.51215e11			−0.461183			−0.230592	−	0.973051i	$-0.574066\pi$
$157$	−5.51215e11			−0.461183			−0.230592	+	0.973051i	$0.574066\pi$
$158$		0			0
$159$	−7.67398e11	−	1.30666e11i	−0.598876	−	0.101971i
$160$		0			0
$161$		−	9.09925e11i		−	0.662923i
$162$		0			0
$163$			1.35112e12i			0.919735i	0.887988	+	0.459867i	$0.152103\pi$
$163$			1.35112e12i			0.919735i	−0.887988	+	0.459867i	$0.847897\pi$
$164$		0			0
$165$	3.71314e11	+	6.32241e10i	0.236363	+	0.0402459i
$166$		0			0
$167$	2.76988e12			1.65014			0.825070	−	0.565030i	$-0.191135\pi$
$167$	2.76988e12			1.65014			0.825070	+	0.565030i	$0.191135\pi$
$168$		0			0
$169$	7.24460e10			0.0404239
$170$		0			0
$171$	6.33011e9	−	1.80494e10i	0.00331080	−	0.00944025i
$172$		0			0
$173$		−	1.53462e12i		−	0.752915i	−0.926434	−	0.376458i	$-0.877142\pi$
$173$		−	1.53462e12i		−	0.752915i	0.926434	−	0.376458i	$-0.122858\pi$
$174$		0			0
$175$			7.40698e11i			0.341139i
$176$		0			0
$177$	−4.07077e11	+	2.39076e12i	−0.176126	+	1.03439i
$178$		0			0
$179$	1.08982e12			0.443263			0.221632	−	0.975130i	$-0.428862\pi$
$179$	1.08982e12			0.443263			0.221632	+	0.975130i	$0.428862\pi$
$180$		0			0
$181$	2.29429e12			0.877841			0.438921	−	0.898526i	$-0.355361\pi$
$181$	2.29429e12			0.877841			0.438921	+	0.898526i	$0.355361\pi$
$182$		0			0
$183$	−5.21726e11	+	3.06408e12i	−0.187915	+	1.10362i
$184$		0			0
$185$		−	6.34560e11i		−	0.215292i
$186$		0			0
$187$			4.32749e12i			1.38391i
$188$		0			0
$189$	1.06320e12	+	5.89085e11i	0.320682	+	0.177680i
$190$		0			0
$191$	−6.77238e12			−1.92778			−0.963891	−	0.266299i	$-0.914199\pi$
$191$	−6.77238e12			−1.92778			−0.963891	+	0.266299i	$0.914199\pi$
$192$		0			0
$193$	2.41050e12			0.647950			0.323975	−	0.946066i	$-0.394981\pi$
$193$	2.41050e12			0.647950			0.323975	+	0.946066i	$0.394981\pi$
$194$		0			0
$195$	1.04364e12	+	1.77703e11i	0.265071	+	0.0451340i
$196$		0			0
$197$		−	1.77642e12i		−	0.426562i	−0.976991	−	0.213281i	$-0.931585\pi$
$197$		−	1.77642e12i		−	0.426562i	0.976991	−	0.213281i	$-0.0684150\pi$
$198$		0			0
$199$			4.72841e12i			1.07405i	0.843567	+	0.537024i	$0.180451\pi$
$199$			4.72841e12i			1.07405i	−0.843567	+	0.537024i	$0.819549\pi$
$200$		0			0
$201$	4.37321e12	+	7.44632e11i	0.940203	+	0.160090i
$202$		0			0
$203$	2.08576e12			0.424654
$204$		0			0
$205$	−1.73242e12			−0.334201
$206$		0			0
$207$	9.33038e12	+	3.27226e12i	1.70633	+	0.598427i
$208$		0			0
$209$			5.24566e10i			0.00909904i
$210$		0			0
$211$			8.68943e12i			1.43033i	0.698953	+	0.715167i	$0.253650\pi$
$211$			8.68943e12i			1.43033i	−0.698953	+	0.715167i	$0.746350\pi$
$212$		0			0
$213$	1.11648e12	−	6.55708e12i	0.174487	−	1.02476i
$214$		0			0
$215$	1.73105e12			0.256979
$216$		0			0
$217$	2.12545e12			0.299863
$218$		0			0
$219$	−1.40884e12	+	8.27406e12i	−0.188981	+	1.10988i
$220$		0			0
$221$			1.21632e13i			1.55199i
$222$		0			0
$223$			1.25001e13i			1.51788i	0.651160	+	0.758941i	$0.274283\pi$
$223$			1.25001e13i			1.51788i	−0.651160	+	0.758941i	$0.725717\pi$
$224$		0			0
$225$	−7.59513e12	−	2.66369e12i	−0.878074	−	0.307950i
$226$		0			0
$227$	−7.81442e11			−0.0860507			−0.0430253	−	0.999074i	$-0.513700\pi$
$227$	−7.81442e11			−0.0860507			−0.0430253	+	0.999074i	$0.513700\pi$
$228$		0			0
$229$	1.14635e13			1.20288			0.601439	−	0.798919i	$-0.294594\pi$
$229$	1.14635e13			1.20288			0.601439	+	0.798919i	$0.294594\pi$
$230$		0			0
$231$	−3.28620e12	−	5.59545e11i	−0.328721	−	0.0559718i
$232$		0			0
$233$		−	1.35109e13i		−	1.28892i	−0.764638	−	0.644460i	$-0.777082\pi$
$233$		−	1.35109e13i		−	1.28892i	0.764638	−	0.644460i	$-0.222918\pi$
$234$		0			0
$235$		−	1.81548e12i		−	0.165241i
$236$		0			0
$237$	−6.89404e12	−	1.17386e12i	−0.598905	−	0.101976i
$238$		0			0
$239$	6.11384e12			0.507137			0.253569	−	0.967317i	$-0.418396\pi$
$239$	6.11384e12			0.507137			0.253569	+	0.967317i	$0.418396\pi$
$240$		0			0
$241$	−6.42898e12			−0.509387			−0.254694	−	0.967022i	$-0.581975\pi$
$241$	−6.42898e12			−0.509387			−0.254694	+	0.967022i	$0.581975\pi$
$242$		0			0
$243$	−9.86396e12	+	8.78360e12i	−0.746821	+	0.665025i
$244$		0			0
$245$		−	3.15276e12i		−	0.228180i
$246$		0			0
$247$			1.47439e11i			0.0102042i
$248$		0			0
$249$	−2.20794e12	+	1.29672e13i	−0.146181	+	0.858517i
$250$		0			0
$251$	6.62584e12			0.419793			0.209897	−	0.977724i	$-0.432687\pi$
$251$	6.62584e12			0.419793			0.209897	+	0.977724i	$0.432687\pi$
$252$		0			0
$253$	−2.71167e13			−1.64465
$254$		0			0
$255$	−1.15919e12	+	6.80789e12i	−0.0673261	+	0.395405i
$256$		0			0
$257$			2.03446e13i			1.13193i	0.824431	+	0.565963i	$0.191495\pi$
$257$			2.03446e13i			1.13193i	−0.824431	+	0.565963i	$0.808505\pi$
$258$		0			0
$259$			5.61598e12i			0.299417i
$260$		0			0
$261$	−7.50077e12	+	2.13874e13i	−0.383339	+	1.09304i
$262$		0			0
$263$	−2.41081e12			−0.118142			−0.0590712	−	0.998254i	$-0.518814\pi$
$263$	−2.41081e12			−0.118142			−0.0590712	+	0.998254i	$0.518814\pi$
$264$		0			0
$265$	3.40689e12			0.160142
$266$		0			0
$267$	2.30407e13	+	3.92318e12i	1.03917	+	0.176940i
$268$		0			0
$269$		−	1.94916e13i		−	0.843745i	−0.906655	−	0.421872i	$-0.861373\pi$
$269$		−	1.94916e13i		−	0.843745i	0.906655	−	0.421872i	$-0.138627\pi$
$270$		0			0
$271$			2.17704e13i			0.904766i	0.891824	+	0.452383i	$0.149426\pi$
$271$			2.17704e13i			0.904766i	−0.891824	+	0.452383i	$0.850574\pi$
$272$		0			0
$273$	−9.23645e12	−	1.57270e12i	−0.368647	−	0.0627700i
$274$		0			0
$275$	2.20736e13			0.846337
$276$		0			0
$277$	1.17163e13			0.431669			0.215835	−	0.976430i	$-0.430753\pi$
$277$	1.17163e13			0.431669			0.215835	+	0.976430i	$0.430753\pi$
$278$		0			0
$279$	−7.64353e12	+	2.17944e13i	−0.270689	+	0.771830i
$280$		0			0
$281$			9.99836e12i			0.340443i	0.985406	+	0.170221i	$0.0544483\pi$
$281$			9.99836e12i			0.340443i	−0.985406	+	0.170221i	$0.945552\pi$
$282$		0			0
$283$			4.71859e13i			1.54521i	0.634889	+	0.772603i	$0.281046\pi$
$283$			4.71859e13i			1.54521i	−0.634889	+	0.772603i	$0.718954\pi$
$284$		0			0
$285$	−1.40514e10	+	8.25233e10i	−0.000442661	+	0.00259974i
$286$		0			0
$287$	1.53323e13			0.464789
$288$		0			0
$289$	−4.50709e13			−1.31510
$290$		0			0
$291$	3.19898e12	−	1.87875e13i	0.0898669	−	0.527786i
$292$		0			0
$293$		−	2.18955e13i		−	0.592357i	−0.955133	−	0.296178i	$-0.904288\pi$
$293$		−	2.18955e13i		−	0.592357i	0.955133	−	0.296178i	$-0.0957123\pi$
$294$		0			0
$295$		−	1.06138e13i		−	0.276600i
$296$		0			0
$297$	1.75554e13	−	3.16845e13i	0.440808	−	0.795584i
$298$		0			0
$299$	−7.62165e13			−1.84441
$300$		0			0
$301$	−1.53201e13			−0.357393
$302$		0			0
$303$	−2.92410e13	−	4.97890e12i	−0.657745	−	0.111995i
$304$		0			0
$305$		−	1.36031e13i		−	0.295113i
$306$		0			0
$307$		−	2.68610e13i		−	0.562162i	−0.959684	−	0.281081i	$-0.909307\pi$
$307$		−	2.68610e13i		−	0.562162i	0.959684	−	0.281081i	$-0.0906929\pi$
$308$		0			0
$309$	3.55430e13	+	6.05196e12i	0.717766	+	0.122215i
$310$		0			0
$311$	3.44085e13			0.670630			0.335315	−	0.942106i	$-0.391157\pi$
$311$	3.44085e13			0.670630			0.335315	+	0.942106i	$0.391157\pi$
$312$		0			0
$313$	9.40692e13			1.76992			0.884960	−	0.465668i	$-0.154186\pi$
$313$	9.40692e13			1.76992			0.884960	+	0.465668i	$0.154186\pi$
$314$		0			0
$315$	−5.01988e12	−	1.76052e12i	−0.0911979	−	0.0319841i
$316$		0			0
$317$			7.46947e13i			1.31058i	0.755377	+	0.655291i	$0.227454\pi$
$317$			7.46947e13i			1.31058i	−0.755377	+	0.655291i	$0.772546\pi$
$318$		0			0
$319$		−	6.21577e13i		−	1.05353i
$320$		0			0
$321$	4.85436e12	−	2.85096e13i	0.0794977	−	0.466888i
$322$		0			0
$323$	−9.61771e11			−0.0152215
$324$		0			0
$325$	6.20418e13			0.949130
$326$		0			0
$327$	7.65788e12	−	4.49745e13i	0.113265	−	0.665203i
$328$		0			0
$329$			1.60673e13i			0.229809i
$330$		0			0
$331$		−	9.71996e13i		−	1.34465i	−0.740254	−	0.672327i	$-0.765295\pi$
$331$		−	9.71996e13i		−	1.34465i	0.740254	−	0.672327i	$-0.234705\pi$
$332$		0			0
$333$	−5.75863e13	−	2.01961e13i	−0.770683	−	0.270287i
$334$		0			0
$335$	−1.94150e13			−0.251415
$336$		0			0
$337$	−3.36820e12			−0.0422117			−0.0211059	−	0.999777i	$-0.506719\pi$
$337$	−3.36820e12			−0.0422117			−0.0211059	+	0.999777i	$0.506719\pi$
$338$		0			0
$339$	−1.06637e13	−	1.81573e12i	−0.129363	−	0.0220268i
$340$		0			0
$341$		−	6.33407e13i		−	0.743933i
$342$		0			0
$343$			6.01376e13i			0.683956i
$344$		0			0
$345$	−4.26593e13	−	7.26366e12i	−0.469904	−	0.0800112i
$346$		0			0
$347$	8.18457e13			0.873340			0.436670	−	0.899622i	$-0.356158\pi$
$347$	8.18457e13			0.873340			0.436670	+	0.899622i	$0.356158\pi$
$348$		0			0
$349$	1.19377e14			1.23419			0.617095	−	0.786889i	$-0.288310\pi$
$349$	1.19377e14			1.23419			0.617095	+	0.786889i	$0.288310\pi$
$350$		0			0
$351$	4.93425e13	−	8.90549e13i	0.494347	−	0.892213i
$352$		0			0
$353$			4.57746e13i			0.444491i	0.974991	+	0.222246i	$0.0713387\pi$
$353$			4.57746e13i			0.444491i	−0.974991	+	0.222246i	$0.928661\pi$
$354$		0			0
$355$			2.91104e13i			0.274026i
$356$		0			0
$357$	1.02590e13	−	6.02511e13i	0.0936335	−	0.549908i
$358$		0			0
$359$	−1.78716e14			−1.58177			−0.790886	−	0.611964i	$-0.790380\pi$
$359$	−1.78716e14			−1.58177			−0.790886	+	0.611964i	$0.790380\pi$
$360$		0			0
$361$	1.16479e14			0.999900
$362$		0			0
$363$	3.48177e12	−	2.04483e13i	0.0289943	−	0.170283i
$364$		0			0
$365$		−	3.67330e13i		−	0.296787i
$366$		0			0
$367$		−	2.55478e12i		−	0.0200304i	−0.999950	−	0.0100152i	$-0.996812\pi$
$367$		−	2.55478e12i		−	0.0200304i	0.999950	−	0.0100152i	$-0.00318799\pi$
$368$		0			0
$369$	−5.51377e13	+	1.57217e14i	−0.419569	+	1.19634i
$370$		0			0
$371$	−3.01516e13			−0.222717
$372$		0			0
$373$	−7.10115e13			−0.509249			−0.254625	−	0.967040i	$-0.581952\pi$
$373$	−7.10115e13			−0.509249			−0.254625	+	0.967040i	$0.581952\pi$
$374$		0			0
$375$	7.20445e13	+	1.22671e13i	0.501683	+	0.0854222i
$376$		0			0
$377$		−	1.74705e14i		−	1.18149i
$378$		0			0
$379$		−	1.80425e14i		−	1.18517i	−0.805507	−	0.592587i	$-0.798107\pi$
$379$		−	1.80425e14i		−	1.18517i	0.805507	−	0.592587i	$-0.201893\pi$
$380$		0			0
$381$	−1.23925e14	−	2.11009e13i	−0.790809	−	0.134652i
$382$		0			0
$383$	−3.90572e12			−0.0242163			−0.0121082	−	0.999927i	$-0.503854\pi$
$383$	−3.90572e12			−0.0242163			−0.0121082	+	0.999927i	$0.503854\pi$
$384$		0			0
$385$	1.45892e13			0.0879016
$386$		0			0
$387$	5.50941e13	−	1.57093e14i	0.322622	−	0.919910i
$388$		0			0
$389$			1.85772e14i			1.05744i	0.848795	+	0.528722i	$0.177328\pi$
$389$			1.85772e14i			1.05744i	−0.848795	+	0.528722i	$0.822672\pi$
$390$		0			0
$391$		−	4.97175e14i		−	2.75129i
$392$		0			0
$393$	1.49819e12	−	8.79883e12i	0.00806134	−	0.0473441i
$394$		0			0
$395$	3.06063e13			0.160150
$396$		0			0
$397$	−2.02762e13			−0.103190			−0.0515951	−	0.998668i	$-0.516431\pi$
$397$	−2.02762e13			−0.103190			−0.0515951	+	0.998668i	$0.516431\pi$
$398$		0			0
$399$	1.24357e11	−	7.30347e11i	0.000615630	−	0.00361558i
$400$		0			0
$401$		−	1.37435e14i		−	0.661915i	−0.943646	−	0.330957i	$-0.892628\pi$
$401$		−	1.37435e14i		−	0.661915i	0.943646	−	0.330957i	$-0.107372\pi$
$402$		0			0
$403$		−	1.78030e14i		−	0.834289i
$404$		0			0
$405$	3.61049e13	−	4.51427e13i	0.164651	−	0.205866i
$406$		0			0
$407$	1.67362e14			0.742827
$408$		0			0
$409$	−2.48502e14			−1.07362			−0.536812	−	0.843702i	$-0.680372\pi$
$409$	−2.48502e14			−1.07362			−0.536812	+	0.843702i	$0.680372\pi$
$410$		0			0
$411$	−4.22739e14	−	7.19804e13i	−1.77804	−	0.302750i
$412$		0			0
$413$			9.39345e13i			0.384680i
$414$		0			0
$415$		−	5.75682e13i		−	0.229571i
$416$		0			0
$417$	−9.23085e13	−	1.57175e13i	−0.358504	−	0.0610429i
$418$		0			0
$419$	−1.45011e14			−0.548558			−0.274279	−	0.961650i	$-0.588439\pi$
$419$	−1.45011e14			−0.548558			−0.274279	+	0.961650i	$0.588439\pi$
$420$		0			0
$421$	6.30248e13			0.232252			0.116126	−	0.993234i	$-0.462952\pi$
$421$	6.30248e13			0.232252			0.116126	+	0.993234i	$0.462952\pi$
$422$		0			0
$423$	−1.64755e14	−	5.77812e13i	−0.591515	−	0.207451i
$424$		0			0
$425$			4.04711e14i			1.41581i
$426$		0			0
$427$			1.20390e14i			0.410428i
$428$		0			0
$429$	−4.68682e13	+	2.75256e14i	−0.155727	+	0.914580i
$430$		0			0
$431$	−1.07798e14			−0.349128			−0.174564	−	0.984646i	$-0.555852\pi$
$431$	−1.07798e14			−0.349128			−0.174564	+	0.984646i	$0.555852\pi$
$432$		0			0
$433$	1.20967e14			0.381930			0.190965	−	0.981597i	$-0.438838\pi$
$433$	1.20967e14			0.381930			0.190965	+	0.981597i	$0.438838\pi$
$434$		0			0
$435$	1.66500e13	−	9.77849e13i	0.0512534	−	0.301010i
$436$		0			0
$437$		−	6.02661e12i		−	0.0180894i
$438$		0			0
$439$		−	3.29980e13i		−	0.0965901i	−0.998833	−	0.0482950i	$-0.984621\pi$
$439$		−	3.29980e13i		−	0.0965901i	0.998833	−	0.0482950i	$-0.0153788\pi$
$440$		0			0
$441$	−2.86112e14	−	1.00343e14i	−0.816816	−	0.286466i
$442$		0			0
$443$	3.24790e14			0.904445			0.452222	−	0.891905i	$-0.350631\pi$
$443$	3.24790e14			0.904445			0.452222	+	0.891905i	$0.350631\pi$
$444$		0			0
$445$	−1.02290e14			−0.277878
$446$		0			0
$447$	−2.00537e14	−	3.41458e13i	−0.531501	−	0.0904993i
$448$		0			0
$449$		−	9.66778e13i		−	0.250018i	−0.992156	−	0.125009i	$-0.960104\pi$
$449$		−	9.66778e13i		−	0.250018i	0.992156	−	0.125009i	$-0.0398960\pi$
$450$		0			0
$451$		−	4.56918e14i		−	1.15310i
$452$		0			0
$453$	−5.74227e14	−	9.77744e13i	−1.41431	−	0.240816i
$454$		0			0
$455$	4.10055e13			0.0985779
$456$		0			0
$457$	2.56726e14			0.602463			0.301231	−	0.953551i	$-0.402602\pi$
$457$	2.56726e14			0.602463			0.301231	+	0.953551i	$0.402602\pi$
$458$		0			0
$459$	5.80922e14	+	3.21871e14i	1.33091	+	0.737415i
$460$		0			0
$461$		−	6.32926e14i		−	1.41579i	−0.706319	−	0.707894i	$-0.749646\pi$
$461$		−	6.32926e14i		−	1.41579i	0.706319	−	0.707894i	$-0.250354\pi$
$462$		0			0
$463$			1.88075e14i			0.410804i	0.978678	+	0.205402i	$0.0658502\pi$
$463$			1.88075e14i			0.410804i	−0.978678	+	0.205402i	$0.934150\pi$
$464$		0			0
$465$	1.69668e13	−	9.96459e13i	0.0361918	−	0.212553i
$466$		0			0
$467$	−3.69467e14			−0.769720			−0.384860	−	0.922975i	$-0.625750\pi$
$467$	−3.69467e14			−0.769720			−0.384860	+	0.922975i	$0.625750\pi$
$468$		0			0
$469$	1.71826e14			0.349654
$470$		0			0
$471$	3.89425e13	−	2.28708e14i	0.0774121	−	0.454640i
$472$		0			0
$473$			4.56556e14i			0.886661i
$474$		0			0
$475$			4.90578e12i			0.00930880i
$476$		0			0
$477$	1.08431e14	−	3.09175e14i	0.201049	−	0.573262i
$478$		0			0
$479$	2.87674e14			0.521260			0.260630	−	0.965439i	$-0.416070\pi$
$479$	2.87674e14			0.521260			0.260630	+	0.965439i	$0.416070\pi$
$480$		0			0
$481$	4.70401e14			0.833049
$482$		0			0
$483$	3.77543e14	+	6.42848e13i	0.653517	+	0.111275i
$484$		0			0
$485$			8.34078e13i			0.141133i
$486$		0			0
$487$			2.50643e13i			0.0414616i	0.999785	+	0.0207308i	$0.00659929\pi$
$487$			2.50643e13i			0.0414616i	−0.999785	+	0.0207308i	$0.993401\pi$
$488$		0			0
$489$	−5.60603e14	−	9.54546e13i	−0.906685	−	0.154383i
$490$		0			0
$491$	−5.60964e14			−0.887129			−0.443565	−	0.896242i	$-0.646286\pi$
$491$	−5.60964e14			−0.887129			−0.443565	+	0.896242i	$0.646286\pi$
$492$		0			0
$493$	1.13964e15			1.76242
$494$		0			0
$495$	−5.24655e13	+	1.49598e14i	−0.0793497	+	0.226254i
$496$		0			0
$497$		−	2.57632e14i		−	0.381101i
$498$		0			0
$499$		−	4.82841e13i		−	0.0698636i	−0.999390	−	0.0349318i	$-0.988879\pi$
$499$		−	4.82841e13i		−	0.0698636i	0.999390	−	0.0349318i	$-0.0111214\pi$
$500$		0			0
$501$	−1.95688e14	+	1.14927e15i	−0.276985	+	1.62673i
$502$		0			0
$503$	−5.99429e14			−0.830068			−0.415034	−	0.909806i	$-0.636230\pi$
$503$	−5.99429e14			−0.830068			−0.415034	+	0.909806i	$0.636230\pi$
$504$		0			0
$505$	1.29816e14			0.175884
$506$		0			0
$507$	−5.11820e12	+	3.00591e13i	−0.00678537	+	0.0398503i
$508$		0			0
$509$			2.04513e14i			0.265322i	0.991161	+	0.132661i	$0.0423522\pi$
$509$			2.04513e14i			0.265322i	−0.991161	+	0.132661i	$0.957648\pi$
$510$		0			0
$511$			3.25094e14i			0.412756i
$512$		0			0
$513$	7.04177e12	+	3.90163e12i	0.00875057	+	0.00484842i
$514$		0			0
$515$	−1.57794e14			−0.191934
$516$		0			0
$517$	4.78824e14			0.570136
$518$		0			0
$519$	6.36737e14	+	1.08418e14i	0.742232	+	0.126381i
$520$		0			0
$521$		−	1.15393e15i		−	1.31695i	−0.752601	−	0.658477i	$-0.771201\pi$
$521$		−	1.15393e15i		−	1.31695i	0.752601	−	0.658477i	$-0.228799\pi$
$522$		0			0
$523$			1.32762e15i			1.48359i	0.670628	+	0.741794i	$0.266025\pi$
$523$			1.32762e15i			1.48359i	−0.670628	+	0.741794i	$0.733975\pi$
$524$		0			0
$525$	−3.07328e14	−	5.23291e13i	−0.336299	−	0.0572621i
$526$		0			0
$527$	1.16133e15			1.24450
$528$		0			0
$529$	2.16256e15			2.26967
$530$		0			0
$531$	−9.63206e14	−	3.37806e14i	−0.990146	−	0.347255i
$532$		0			0
$533$		−	1.28425e15i		−	1.29315i
$534$		0			0
$535$			1.26569e14i			0.124848i
$536$		0			0
$537$	−7.69938e13	+	4.52183e14i	−0.0744042	+	0.436974i
$538$		0			0
$539$	8.31523e14			0.787293
$540$		0			0
$541$	9.52340e14			0.883501			0.441750	−	0.897138i	$-0.354358\pi$
$541$	9.52340e14			0.883501			0.441750	+	0.897138i	$0.354358\pi$
$542$		0			0
$543$	−1.62088e14	+	9.51939e14i	−0.147350	+	0.865386i
$544$		0			0
$545$			1.99666e14i			0.177878i
$546$		0			0
$547$			1.34526e15i			1.17456i	0.809382	+	0.587282i	$0.199802\pi$
$547$			1.34526e15i			1.17456i	−0.809382	+	0.587282i	$0.800198\pi$
$548$		0			0
$549$	−1.23448e15	−	4.32945e14i	−1.05642	−	0.370497i
$550$		0			0
$551$	1.38144e13			0.0115877
$552$		0			0
$553$	−2.70872e14			−0.222728
$554$		0			0
$555$	2.63290e14	+	4.48307e13i	0.212238	+	0.0361380i
$556$		0			0
$557$			1.99341e15i			1.57541i	0.616053	+	0.787705i	$0.288731\pi$
$557$			1.99341e15i			1.57541i	−0.616053	+	0.787705i	$0.711269\pi$
$558$		0			0
$559$			1.28323e15i			0.994352i
$560$		0			0
$561$	−1.79555e15	−	3.05730e14i	−1.36427	−	0.232297i
$562$		0			0
$563$	2.09224e15			1.55889			0.779444	−	0.626472i	$-0.215502\pi$
$563$	2.09224e15			1.55889			0.779444	+	0.626472i	$0.215502\pi$
$564$		0			0
$565$	4.73420e13			0.0345923
$566$		0			0
$567$	−3.19535e14	+	3.99521e14i	−0.228987	+	0.286308i
$568$		0			0
$569$			8.04249e14i			0.565292i	0.959224	+	0.282646i	$0.0912122\pi$
$569$			8.04249e14i			0.565292i	−0.959224	+	0.282646i	$0.908788\pi$
$570$		0			0
$571$			3.99575e14i			0.275486i	0.990468	+	0.137743i	$0.0439848\pi$
$571$			3.99575e14i			0.275486i	−0.990468	+	0.137743i	$0.956015\pi$
$572$		0			0
$573$	4.78458e14	−	2.80997e15i	0.323589	−	1.90043i
$574$		0			0
$575$	−2.53598e15			−1.68257
$576$		0			0
$577$	−2.75977e14			−0.179641			−0.0898205	−	0.995958i	$-0.528629\pi$
$577$	−2.75977e14			−0.179641			−0.0898205	+	0.995958i	$0.528629\pi$
$578$		0			0
$579$	−1.70298e14	+	1.00016e15i	−0.108762	+	0.638757i
$580$		0			0
$581$			5.09489e14i			0.319276i
$582$		0			0
$583$			8.98550e14i			0.552542i
$584$		0			0
$585$	−1.47464e14	+	4.20471e14i	−0.0889872	+	0.253734i
$586$		0			0
$587$	−1.45165e15			−0.859712			−0.429856	−	0.902898i	$-0.641436\pi$
$587$	−1.45165e15			−0.859712			−0.429856	+	0.902898i	$0.641436\pi$
$588$		0			0
$589$	1.40773e13			0.00818246
$590$		0			0
$591$	7.37068e14	+	1.25502e14i	0.420510	+	0.0716008i
$592$		0			0
$593$			6.13996e14i			0.343846i	0.985110	+	0.171923i	$0.0549981\pi$
$593$			6.13996e14i			0.343846i	−0.985110	+	0.171923i	$0.945002\pi$
$594$		0			0
$595$			2.67487e14i			0.147048i
$596$		0			0
$597$	−1.96190e15	−	3.34055e14i	−1.05881	−	0.180285i
$598$		0			0
$599$	1.34920e15			0.714872			0.357436	−	0.933938i	$-0.383651\pi$
$599$	1.34920e15			0.714872			0.357436	+	0.933938i	$0.383651\pi$
$600$		0			0
$601$	−6.98822e14			−0.363544			−0.181772	−	0.983341i	$-0.558183\pi$
$601$	−6.98822e14			−0.363544			−0.181772	+	0.983341i	$0.558183\pi$
$602$		0			0
$603$	−6.17920e14	+	1.76191e15i	−0.315637	+	0.899991i
$604$		0			0
$605$			9.07811e13i			0.0455345i
$606$		0			0
$607$			3.22822e15i			1.59010i	0.606541	+	0.795052i	$0.292557\pi$
$607$			3.22822e15i			1.59010i	−0.606541	+	0.795052i	$0.707443\pi$
$608$		0			0
$609$	−1.47355e14	+	8.65415e14i	−0.0712805	+	0.418629i
$610$		0			0
$611$	1.34582e15			0.639382
$612$		0			0
$613$	−1.30407e15			−0.608510			−0.304255	−	0.952591i	$-0.598407\pi$
$613$	−1.30407e15			−0.608510			−0.304255	+	0.952591i	$0.598407\pi$
$614$		0			0
$615$	1.22393e14	−	7.18811e14i	0.0560974	−	0.329459i
$616$		0			0
$617$		−	3.21229e14i		−	0.144626i	−0.997382	−	0.0723131i	$-0.976962\pi$
$617$		−	3.21229e14i		−	0.144626i	0.997382	−	0.0723131i	$-0.0230381\pi$
$618$		0			0
$619$			1.82215e15i			0.805906i	0.915221	+	0.402953i	$0.132016\pi$
$619$			1.82215e15i			0.805906i	−0.915221	+	0.402953i	$0.867984\pi$
$620$		0			0
$621$	−2.01689e15	+	3.64015e15i	−0.876353	+	1.58167i
$622$		0			0
$623$	9.05287e14			0.386458
$624$		0			0
$625$	1.89866e15			0.796357
$626$		0			0
$627$	−2.17651e13	−	3.70598e12i	−0.00896994	−	0.00152732i
$628$		0			0
$629$			3.06852e15i			1.24265i
$630$		0			0
$631$		−	2.15030e15i		−	0.855731i	−0.903843	−	0.427865i	$-0.859266\pi$
$631$		−	2.15030e15i		−	0.855731i	0.903843	−	0.427865i	$-0.140734\pi$
$632$		0			0
$633$	−3.60539e15	−	6.13894e14i	−1.41004	−	0.240090i
$634$		0			0
$635$	5.50169e14			0.211466
$636$		0			0
$637$	2.33715e15			0.882915
$638$		0			0
$639$	2.64176e15	+	9.26494e14i	0.980932	+	0.344023i
$640$		0			0
$641$		−	1.81435e15i		−	0.662219i	−0.943592	−	0.331109i	$-0.892577\pi$
$641$		−	1.81435e15i		−	0.662219i	0.943592	−	0.331109i	$-0.107423\pi$
$642$		0			0
$643$		−	5.04028e15i		−	1.80840i	−0.427111	−	0.904199i	$-0.640468\pi$
$643$		−	5.04028e15i		−	1.80840i	0.427111	−	0.904199i	$-0.359532\pi$
$644$		0			0
$645$	−1.22296e14	+	7.18242e14i	−0.0431354	+	0.253333i
$646$		0			0
$647$	−3.96904e14			−0.137630			−0.0688149	−	0.997629i	$-0.521922\pi$
$647$	−3.96904e14			−0.137630			−0.0688149	+	0.997629i	$0.521922\pi$
$648$		0			0
$649$	2.79935e15			0.954358
$650$		0			0
$651$	−1.50160e14	+	8.81885e14i	−0.0503336	+	0.295608i
$652$		0			0
$653$		−	5.63137e15i		−	1.85606i	−0.372507	−	0.928030i	$-0.621502\pi$
$653$		−	5.63137e15i		−	1.85606i	0.372507	−	0.928030i	$-0.378498\pi$
$654$		0			0
$655$			3.90627e13i			0.0126600i
$656$		0			0
$657$	−3.33351e15	−	1.16910e15i	−1.06241	−	0.372599i
$658$		0			0
$659$	−5.78078e15			−1.81183			−0.905914	−	0.423463i	$-0.860815\pi$
$659$	−5.78078e15			−1.81183			−0.905914	+	0.423463i	$0.860815\pi$
$660$		0			0
$661$	5.65482e15			1.74306			0.871528	−	0.490347i	$-0.163130\pi$
$661$	5.65482e15			1.74306			0.871528	+	0.490347i	$0.163130\pi$
$662$		0			0
$663$	−5.04671e15	−	8.59310e14i	−1.52997	−	0.260511i
$664$		0			0
$665$			3.24240e12i			0.000966824i
$666$		0			0
$667$			7.14115e15i			2.09448i
$668$		0			0
$669$	−5.18651e15	−	8.83114e14i	−1.49634	−	0.254785i
$670$		0			0
$671$	3.58775e15			1.01824
$672$		0			0
$673$	2.96214e13			0.00827032			0.00413516	−	0.999991i	$-0.498684\pi$
$673$	2.96214e13			0.00827032			0.00413516	+	0.999991i	$0.498684\pi$
$674$		0			0
$675$	1.64179e15	−	2.96316e15i	0.450970	−	0.813925i
$676$		0			0
$677$		−	1.99030e15i		−	0.537874i	−0.963158	−	0.268937i	$-0.913328\pi$
$677$		−	1.99030e15i		−	0.537874i	0.963158	−	0.268937i	$-0.0866723\pi$
$678$		0			0
$679$		−	7.38175e14i		−	0.196280i
$680$		0			0
$681$	5.52076e13	−	3.24233e14i	0.0144441	−	0.0848298i
$682$		0			0
$683$	5.07970e15			1.30775			0.653874	−	0.756603i	$-0.273143\pi$
$683$	5.07970e15			1.30775			0.653874	+	0.756603i	$0.273143\pi$
$684$		0			0
$685$	1.87677e15			0.475458
$686$		0			0
$687$	−8.09877e14	+	4.75639e15i	−0.201910	+	1.18581i
$688$		0			0
$689$			2.52554e15i			0.619652i
$690$		0			0
$691$		−	2.69916e15i		−	0.651777i	−0.945408	−	0.325889i	$-0.894337\pi$
$691$		−	2.69916e15i		−	0.651777i	0.945408	−	0.325889i	$-0.105663\pi$
$692$		0			0
$693$	4.64329e14	−	1.32397e15i	0.110355	−	0.314662i
$694$		0			0
$695$	4.09807e14			0.0958656
$696$		0			0
$697$	8.37740e15			1.92899
$698$		0			0
$699$	5.60589e15	+	9.54522e14i	1.27063	+	0.216352i
$700$		0			0
$701$			4.18730e15i			0.934298i	0.884179	+	0.467149i	$0.154719\pi$
$701$			4.18730e15i			0.934298i	−0.884179	+	0.467149i	$0.845281\pi$
$702$		0			0
$703$			3.71957e13i			0.00817030i
$704$		0			0
$705$	7.53274e14	+	1.28261e14i	0.162897	+	0.0277367i
$706$		0			0
$707$	−1.14890e15			−0.244610
$708$		0			0
$709$	3.09885e14			0.0649601			0.0324801	−	0.999472i	$-0.489659\pi$
$709$	3.09885e14			0.0649601			0.0324801	+	0.999472i	$0.489659\pi$
$710$		0			0
$711$	9.74106e14	−	2.77752e15i	0.201059	−	0.573290i
$712$		0			0
$713$			7.27705e15i			1.47898i
$714$		0			0
$715$		−	1.22201e15i		−	0.244563i
$716$		0			0
$717$	−4.31933e14	+	2.53673e15i	−0.0851258	+	0.499942i
$718$		0			0
$719$	−6.87880e15			−1.33507			−0.667534	−	0.744579i	$-0.732650\pi$
$719$	−6.87880e15			−1.33507			−0.667534	+	0.744579i	$0.732650\pi$
$720$		0			0
$721$	1.39651e15			0.266932
$722$		0			0
$723$	4.54197e14	−	2.66749e15i	0.0855035	−	0.502160i
$724$		0			0
$725$		−	5.81304e15i		−	1.07782i
$726$		0			0
$727$		−	9.49131e15i		−	1.73335i	−0.498871	−	0.866676i	$-0.666252\pi$
$727$		−	9.49131e15i		−	1.73335i	0.498871	−	0.866676i	$-0.333748\pi$
$728$		0			0
$729$	−2.94759e15	−	4.71327e15i	−0.530231	−	0.847853i
$730$		0			0
$731$	−8.37077e15			−1.48327
$732$		0			0
$733$	1.98406e15			0.346324			0.173162	−	0.984893i	$-0.444602\pi$
$733$	1.98406e15			0.346324			0.173162	+	0.984893i	$0.444602\pi$
$734$		0			0
$735$	1.30813e15	+	2.22737e14i	0.224942	+	0.0383012i
$736$		0			0
$737$		−	5.12061e15i		−	0.867462i
$738$		0			0
$739$		−	8.24660e15i		−	1.37636i	−0.725542	−	0.688178i	$-0.758411\pi$
$739$		−	8.24660e15i		−	1.37636i	0.725542	−	0.688178i	$-0.241589\pi$
$740$		0			0
$741$	−6.11747e13	−	1.04163e13i	−0.0100594	−	0.00171283i
$742$		0			0
$743$	−2.64416e15			−0.428399			−0.214199	−	0.976790i	$-0.568714\pi$
$743$	−2.64416e15			−0.428399			−0.214199	+	0.976790i	$0.568714\pi$
$744$		0			0
$745$	8.90292e14			0.142126
$746$		0			0
$747$	−5.22431e15	−	1.83222e15i	−0.821798	−	0.288213i
$748$		0			0
$749$		−	1.12016e15i		−	0.173632i
$750$		0			0
$751$			2.39438e15i			0.365742i	0.983137	+	0.182871i	$0.0585390\pi$
$751$			2.39438e15i			0.365742i	−0.983137	+	0.182871i	$0.941461\pi$
$752$		0			0
$753$	−4.68105e14	+	2.74917e15i	−0.0704646	+	0.413837i
$754$		0			0
$755$	2.54930e15			0.378192
$756$		0			0
$757$	1.10647e16			1.61775			0.808875	−	0.587981i	$-0.200077\pi$
$757$	1.10647e16			1.61775			0.808875	+	0.587981i	$0.200077\pi$
$758$		0			0
$759$	1.91575e15	−	1.12512e16i	0.276064	−	1.62132i
$760$		0			0
$761$		−	7.90096e15i		−	1.12218i	−0.827753	−	0.561092i	$-0.810381\pi$
$761$		−	7.90096e15i		−	1.12218i	0.827753	−	0.561092i	$-0.189619\pi$
$762$		0			0
$763$		−	1.76708e15i		−	0.247384i
$764$		0			0
$765$	−2.74281e15	−	9.61934e14i	−0.378493	−	0.132742i
$766$		0			0
$767$	7.86807e15			1.07027
$768$		0			0
$769$	−4.46095e15			−0.598181			−0.299091	−	0.954225i	$-0.596683\pi$
$769$	−4.46095e15			−0.598181			−0.299091	+	0.954225i	$0.596683\pi$
$770$		0			0
$771$	−8.44133e15	−	1.43732e15i	−1.11587	−	0.190000i
$772$		0			0
$773$			7.87880e15i			1.02677i	0.858159	+	0.513385i	$0.171609\pi$
$773$			7.87880e15i			1.02677i	−0.858159	+	0.513385i	$0.828391\pi$
$774$		0			0
$775$		−	5.92368e15i		−	0.761083i
$776$		0			0
$777$	−2.33016e15	−	3.96760e14i	−0.295169	−	0.0502588i
$778$		0			0
$779$	1.01548e14			0.0126829
$780$		0			0
$781$	−7.67771e15			−0.945477
$782$		0			0
$783$	−8.34405e15	−	4.62318e15i	−1.01318	−	0.561372i
$784$		0			0
$785$			1.01536e15i			0.121573i
$786$		0			0
$787$		−	9.38324e15i		−	1.10788i	−0.832557	−	0.553939i	$-0.813124\pi$
$787$		−	9.38324e15i		−	1.10788i	0.832557	−	0.553939i	$-0.186876\pi$
$788$		0			0
$789$	1.70320e14	−	1.00028e15i	0.0198308	−	0.116466i
$790$		0			0
$791$	−4.18985e14			−0.0481092
$792$		0			0
$793$	1.00840e16			1.14191
$794$		0			0
$795$	−2.40691e14	+	1.41358e15i	−0.0268808	+	0.157870i
$796$		0			0
$797$			5.92155e15i			0.652251i	0.945327	+	0.326125i	$0.105743\pi$
$797$			5.92155e15i			0.652251i	−0.945327	+	0.326125i	$0.894257\pi$
$798$		0			0
$799$			8.77905e15i			0.953762i
$800$		0			0
$801$	−3.25558e15	+	9.28282e15i	−0.348859	+	0.994721i
$802$		0			0
$803$	9.68814e15			1.02401
$804$		0			0
$805$	−1.67611e15			−0.174754
$806$		0			0
$807$	8.08741e15	+	1.37705e15i	0.831773	+	0.141627i
$808$		0			0
$809$		−	1.37849e16i		−	1.39858i	−0.714837	−	0.699291i	$-0.753499\pi$
$809$		−	1.37849e16i		−	1.39858i	0.714837	−	0.699291i	$-0.246501\pi$
$810$		0			0
$811$			3.77518e15i			0.377853i	0.981991	+	0.188926i	$0.0605008\pi$
$811$			3.77518e15i			0.377853i	−0.981991	+	0.188926i	$0.939499\pi$
$812$		0			0
$813$	−9.03292e15	−	1.53805e15i	−0.891928	−	0.151870i
$814$		0			0
$815$	2.48882e15			0.242452
$816$		0			0
$817$	−1.01468e14			−0.00975232
$818$		0			0
$819$	1.30508e15	−	3.72125e15i	0.123759	−	0.352880i
$820$		0			0
$821$		−	8.31426e15i		−	0.777922i	−0.921254	−	0.388961i	$-0.872834\pi$
$821$		−	8.31426e15i		−	0.777922i	0.921254	−	0.388961i	$-0.127166\pi$
$822$		0			0
$823$		−	1.53630e16i		−	1.41833i	−0.705042	−	0.709166i	$-0.749072\pi$
$823$		−	1.53630e16i		−	1.41833i	0.705042	−	0.709166i	$-0.250928\pi$
$824$		0			0
$825$	−1.55946e15	+	9.15870e15i	−0.142062	+	0.834329i
$826$		0			0
$827$	1.57568e16			1.41640			0.708201	−	0.706011i	$-0.249507\pi$
$827$	1.57568e16			1.41640			0.708201	+	0.706011i	$0.249507\pi$
$828$		0			0
$829$	4.80421e15			0.426159			0.213079	−	0.977035i	$-0.431651\pi$
$829$	4.80421e15			0.426159			0.213079	+	0.977035i	$0.431651\pi$
$830$		0			0
$831$	−8.27737e14	+	4.86128e15i	−0.0724581	+	0.425545i
$832$		0			0
$833$			1.52456e16i			1.31704i
$834$		0			0
$835$		−	5.10223e15i		−	0.434995i
$836$		0			0
$837$	−8.50285e15	−	4.71117e15i	−0.715442	−	0.396404i
$838$		0			0
$839$	−1.39625e16			−1.15950			−0.579752	−	0.814793i	$-0.696850\pi$
$839$	−1.39625e16			−1.15950			−0.579752	+	0.814793i	$0.696850\pi$
$840$		0			0
$841$	−4.16863e15			−0.341676
$842$		0			0
$843$	−4.14848e15	−	7.06368e14i	−0.335612	−	0.0571452i
$844$		0			0
$845$		−	1.33448e14i		−	0.0106562i
$846$		0			0
$847$		−	8.03429e14i		−	0.0633270i
$848$		0			0
$849$	−1.95782e16	−	3.33361e15i	−1.52328	−	0.259371i
$850$		0			0
$851$	−1.92278e16			−1.47679
$852$		0			0
$853$	3.75114e15			0.284409			0.142205	−	0.989837i	$-0.454581\pi$
$853$	3.75114e15			0.284409			0.142205	+	0.989837i	$0.454581\pi$
$854$		0			0
$855$	−3.32476e13	−	1.16603e13i	−0.00248855	−	0.000872761i
$856$		0			0
$857$			1.49867e16i			1.10742i	0.832709	+	0.553710i	$0.186789\pi$
$857$			1.49867e16i			1.10742i	−0.832709	+	0.553710i	$0.813211\pi$
$858$		0			0
$859$			5.52120e15i			0.402783i	0.979511	+	0.201392i	$0.0645464\pi$
$859$			5.52120e15i			0.402783i	−0.979511	+	0.201392i	$0.935454\pi$
$860$		0			0
$861$	−1.08320e15	+	6.36161e15i	−0.0780173	+	0.458194i
$862$		0			0
$863$	−1.49995e16			−1.06664			−0.533320	−	0.845913i	$-0.679056\pi$
$863$	−1.49995e16			−1.06664			−0.533320	+	0.845913i	$0.679056\pi$
$864$		0			0
$865$	−2.82682e15			−0.198477
$866$		0			0
$867$	3.18419e15	−	1.87007e16i	0.220746	−	1.29644i
$868$		0			0
$869$			8.07227e15i			0.552569i
$870$		0			0
$871$		−	1.43924e16i		−	0.972821i
$872$		0			0
$873$	7.56925e15	+	2.65462e15i	0.505213	+	0.177184i
$874$		0			0
$875$	2.83068e15			0.186572
$876$		0			0
$877$	−5.40949e15			−0.352094			−0.176047	−	0.984382i	$-0.556331\pi$
$877$	−5.40949e15			−0.352094			−0.176047	+	0.984382i	$0.556331\pi$
$878$		0			0
$879$	9.08481e15	+	1.54688e15i	0.583952	+	0.0994303i
$880$		0			0
$881$		−	1.56262e16i		−	0.991939i	−0.868340	−	0.495970i	$-0.834813\pi$
$881$		−	1.56262e16i		−	0.991939i	0.868340	−	0.495970i	$-0.165187\pi$
$882$		0			0
$883$		−	1.67545e16i		−	1.05038i	−0.850985	−	0.525191i	$-0.823994\pi$
$883$		−	1.67545e16i		−	1.05038i	0.850985	−	0.525191i	$-0.176006\pi$
$884$		0			0
$885$	4.40386e15	+	7.49851e14i	0.272675	+	0.0464288i
$886$		0			0
$887$	−1.40157e14			−0.00857104			−0.00428552	−	0.999991i	$-0.501364\pi$
$887$	−1.40157e14			−0.00857104			−0.00428552	+	0.999991i	$0.501364\pi$
$888$		0			0
$889$	−4.86910e15			−0.294096
$890$		0			0
$891$	1.19062e16	+	9.52248e15i	0.710304	+	0.568097i
$892$		0			0
$893$			1.06417e14i			0.00627088i
$894$		0			0
$895$		−	2.00748e15i		−	0.116849i
$896$		0			0
$897$	5.38457e15	−	3.16235e16i	0.309594	−	1.81824i
$898$		0			0
$899$	−1.66807e16			−0.947404
$900$		0			0
$901$	−1.64746e16			−0.924331
$902$		0			0
$903$	1.08234e15	−	6.35657e15i	0.0599904	−	0.352322i
$904$		0			0
$905$		−	4.22616e15i		−	0.231408i
$906$		0			0
$907$			2.50422e16i			1.35467i	0.735676	+	0.677334i	$0.236865\pi$
$907$			2.50422e16i			1.35467i	−0.735676	+	0.677334i	$0.763135\pi$
$908$		0			0
$909$	4.13166e15	−	1.17808e16i	0.220812	−	0.629613i
$910$		0			0
$911$	3.70211e16			1.95478			0.977390	−	0.211445i	$-0.0678168\pi$
$911$	3.70211e16			1.95478			0.977390	+	0.211445i	$0.0678168\pi$
$912$		0			0
$913$	1.51833e16			0.792095
$914$		0			0
$915$	5.64415e15	+	9.61038e14i	0.290926	+	0.0495364i
$916$		0			0
$917$		−	3.45712e14i		−	0.0176069i
$918$		0			0
$919$			1.66848e16i			0.839625i	0.907611	+	0.419813i	$0.137904\pi$
$919$			1.66848e16i			0.839625i	−0.907611	+	0.419813i	$0.862096\pi$
$920$		0			0
$921$	1.11451e16	+	1.89769e15i	0.554186	+	0.0943619i
$922$		0			0
$923$	−2.15796e16			−1.06031
$924$		0			0
$925$	1.56518e16			0.759952
$926$		0			0
$927$	−5.02212e15	+	1.43198e16i	−0.240962	+	0.687067i
$928$		0			0
$929$			1.97824e16i			0.937977i	0.883204	+	0.468989i	$0.155381\pi$
$929$			1.97824e16i			0.937977i	−0.883204	+	0.468989i	$0.844619\pi$
$930$		0			0
$931$			1.84803e14i			0.00865937i
$932$		0			0
$933$	−2.43090e15	+	1.42766e16i	−0.112569	+	0.661115i
$934$		0			0
$935$	7.97139e15			0.364813
$936$		0			0
$937$	−7.46411e15			−0.337606			−0.168803	−	0.985650i	$-0.553990\pi$
$937$	−7.46411e15			−0.337606			−0.168803	+	0.985650i	$0.553990\pi$
$938$		0			0
$939$	−6.64584e15	+	3.90309e16i	−0.297091	+	1.74481i
$940$		0			0
$941$		−	1.75071e14i		−	0.00773518i	−0.999993	−	0.00386759i	$-0.998769\pi$
$941$		−	1.75071e14i		−	0.00773518i	0.999993	−	0.00386759i	$-0.00123110\pi$
$942$		0			0
$943$			5.24941e16i			2.29243i
$944$		0			0
$945$	1.08512e15	−	1.95845e15i	0.0468383	−	0.0845353i
$946$		0			0
$947$	8.61225e15			0.367444			0.183722	−	0.982978i	$-0.441185\pi$
$947$	8.61225e15			0.367444			0.183722	+	0.982978i	$0.441185\pi$
$948$		0			0
$949$	2.72303e16			1.14838
$950$		0			0
$951$	−3.09921e16	−	5.27706e15i	−1.29199	−	0.219988i
$952$		0			0
$953$		−	9.31560e15i		−	0.383884i	−0.981406	−	0.191942i	$-0.938521\pi$
$953$		−	9.31560e15i		−	0.383884i	0.981406	−	0.191942i	$-0.0614785\pi$
$954$		0			0
$955$			1.24750e16i			0.508184i
$956$		0			0
$957$	2.57903e16	+	4.39135e15i	1.03858	+	0.176841i
$958$		0			0
$959$	−1.66097e16			−0.661241
$960$		0			0
$961$	8.41035e15			0.331006
$962$		0			0
$963$	1.14861e16	+	4.02831e15i	0.446920	+	0.156740i
$964$		0			0
$965$		−	4.44023e15i		−	0.170807i
$966$		0			0
$967$		−	4.03279e16i		−	1.53377i	−0.641785	−	0.766885i	$-0.721806\pi$
$967$		−	4.03279e16i		−	1.53377i	0.641785	−	0.766885i	$-0.278194\pi$
$968$		0			0
$969$	6.79476e13	−	3.99055e14i	0.00255501	−	0.0150055i
$970$		0			0
$971$	3.67060e16			1.36468			0.682341	−	0.731034i	$-0.260962\pi$
$971$	3.67060e16			1.36468			0.682341	+	0.731034i	$0.260962\pi$
$972$		0			0
$973$	−3.62687e15			−0.133325
$974$		0			0
$975$	−4.38315e15	+	2.57422e16i	−0.159317	+	0.935664i
$976$		0			0
$977$			1.96544e16i			0.706383i	0.935551	+	0.353192i	$0.114904\pi$
$977$			1.96544e16i			0.706383i	−0.935551	+	0.353192i	$0.885096\pi$
$978$		0			0
$979$		−	2.69785e16i		−	0.958767i
$980$		0			0
$981$	1.81197e16	+	6.35476e15i	0.636753	+	0.223316i
$982$		0			0
$983$	2.22734e16			0.774003			0.387002	−	0.922079i	$-0.373511\pi$
$983$	2.22734e16			0.774003			0.387002	+	0.922079i	$0.373511\pi$
$984$		0			0
$985$	−3.27224e15			−0.112446
$986$		0			0
$987$	−6.66661e15	−	1.13513e15i	−0.226548	−	0.0385747i
$988$		0			0
$989$		−	5.24526e16i		−	1.76274i
$990$		0			0
$991$			5.30529e16i			1.76321i	0.471988	+	0.881605i	$0.343537\pi$
$991$			5.30529e16i			1.76321i	−0.471988	+	0.881605i	$0.656463\pi$
$992$		0			0
$993$	4.03297e16	+	6.86700e15i	1.32558	+	0.225707i
$994$		0			0
$995$	8.70990e15			0.283130
$996$		0			0
$997$	−4.29174e16			−1.37978			−0.689891	−	0.723913i	$-0.742342\pi$
$997$	−4.29174e16			−1.37978			−0.689891	+	0.723913i	$0.742342\pi$
$998$		0			0
$999$	1.24481e16	−	2.24667e16i	0.395815	−	0.714379i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.12.c.d.191.10		20
3.2	odd	2	inner	192.12.c.d.191.12		20
4.3	odd	2	inner	192.12.c.d.191.11		20
8.3	odd	2		12.12.b.a.11.16	yes	20
8.5	even	2		12.12.b.a.11.6	yes	20
12.11	even	2	inner	192.12.c.d.191.9		20
24.5	odd	2		12.12.b.a.11.15	yes	20
24.11	even	2		12.12.b.a.11.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.12.b.a.11.5	✓	20	24.11	even	2
12.12.b.a.11.6	yes	20	8.5	even	2
12.12.b.a.11.15	yes	20	24.5	odd	2
12.12.b.a.11.16	yes	20	8.3	odd	2
192.12.c.d.191.9		20	12.11	even	2	inner
192.12.c.d.191.10		20	1.1	even	1	trivial
192.12.c.d.191.11		20	4.3	odd	2	inner
192.12.c.d.191.12		20	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	192.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-70.6484 + 414.917i) q^{3} -1842.04i q^{5} +16302.4i q^{7} +(-167165. - 58626.4i) q^{9} +O(q^{10})\)	\(q+(-70.6484 + 414.917i) q^{3} -1842.04i q^{5} +16302.4i q^{7} +(-167165. - 58626.4i) q^{9} +485828. q^{11} +1.36551e6 q^{13} +(764291. + 130137. i) q^{15} +8.90746e6i q^{17} +107974. i q^{19} +(-6.76412e6 - 1.15174e6i) q^{21} -5.58155e7 q^{23} +4.54350e7 q^{25} +(3.61350e7 - 6.52175e7i) q^{27} -1.27942e8i q^{29} -1.30377e8i q^{31} +(-3.43230e7 + 2.01578e8i) q^{33} +3.00295e7 q^{35} +3.44489e8 q^{37} +(-9.64708e7 + 5.66571e8i) q^{39} -9.40493e8i q^{41} +9.39748e8i q^{43} +(-1.07992e8 + 3.07923e8i) q^{45} +9.85584e8 q^{47} +1.71156e9 q^{49} +(-3.69585e9 - 6.29298e8i) q^{51} +1.84952e9i q^{53} -8.94912e8i q^{55} +(-4.48001e7 - 7.62817e6i) q^{57} +5.76202e9 q^{59} +7.38482e9 q^{61} +(9.55749e8 - 2.72518e9i) q^{63} -2.51531e9i q^{65} -1.05400e10i q^{67} +(3.94328e9 - 2.31588e10i) q^{69} -1.58034e10 q^{71} +1.99415e10 q^{73} +(-3.20991e9 + 1.88517e10i) q^{75} +7.92014e9i q^{77} +1.66155e10i q^{79} +(2.45069e10 + 1.96005e10i) q^{81} +3.12525e10 q^{83} +1.64079e10 q^{85} +(5.30852e10 + 9.03889e9i) q^{87} -5.55310e10i q^{89} +2.22610e10i q^{91} +(5.40955e10 + 9.21092e9i) q^{93} +1.98891e8 q^{95} -4.52802e10 q^{97} +(-8.12132e10 - 2.84823e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 103620 q^{9}+O(q^{10}) \) 20 * q + 103620 * q^9	\( 20 q + 103620 q^{9} - 1543864 q^{13} + 12211752 q^{21} - 141128700 q^{25} + 229769760 q^{33} + 44517800 q^{37} - 1020227520 q^{45} - 1018138084 q^{49} + 1438636392 q^{57} - 6873199864 q^{61} + 13308470976 q^{69} - 12426469112 q^{73} - 42462874764 q^{81} - 17135502080 q^{85} - 114387515256 q^{93} + 75764383528 q^{97}+O(q^{100}) \) 20 * q + 103620 * q^9 - 1543864 * q^13 + 12211752 * q^21 - 141128700 * q^25 + 229769760 * q^33 + 44517800 * q^37 - 1020227520 * q^45 - 1018138084 * q^49 + 1438636392 * q^57 - 6873199864 * q^61 + 13308470976 * q^69 - 12426469112 * q^73 - 42462874764 * q^81 - 17135502080 * q^85 - 114387515256 * q^93 + 75764383528 * q^97

Introduction

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