LMFDB - Embedded newform 342.10.b.b.341.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,10,Mod(341,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("342.341");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	342.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$176.142255968$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			341.9
Character	$\chi$	$=$	342.341
Dual form			342.10.b.b.341.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.0000 q^{2} +256.000 q^{4} -1273.82i q^{5} +10689.6 q^{7} +4096.00 q^{8} +O(q^{10})$	\(q+16.0000 q^{2} +256.000 q^{4} -1273.82i q^{5} +10689.6 q^{7} +4096.00 q^{8} -20381.1i q^{10} -9759.86i q^{11} +8653.09i q^{13} +171033. q^{14} +65536.0 q^{16} +33425.4i q^{17} +(380758. + 421557. i) q^{19} -326098. i q^{20} -156158. i q^{22} +2.00994e6i q^{23} +330510. q^{25} +138449. i q^{26} +2.73653e6 q^{28} +4.16504e6 q^{29} +6.52534e6i q^{31} +1.04858e6 q^{32} +534806. i q^{34} -1.36166e7i q^{35} +1.41147e7i q^{37} +(6.09214e6 + 6.74492e6i) q^{38} -5.21756e6i q^{40} -1.68962e7 q^{41} +3.65295e7 q^{43} -2.49852e6i q^{44} +3.21591e7i q^{46} +1.20331e6i q^{47} +7.39133e7 q^{49} +5.28816e6 q^{50} +2.21519e6i q^{52} +8.17096e6 q^{53} -1.24323e7 q^{55} +4.37845e7 q^{56} +6.66406e7 q^{58} -3.94654e7 q^{59} -1.25698e8 q^{61} +1.04405e8i q^{62} +1.67772e7 q^{64} +1.10225e7 q^{65} +7.55895e7i q^{67} +8.55690e6i q^{68} -2.17865e8i q^{70} -2.64976e8 q^{71} +8.58471e6 q^{73} +2.25835e8i q^{74} +(9.74742e7 + 1.07919e8i) q^{76} -1.04329e8i q^{77} +9.51474e7i q^{79} -8.34810e7i q^{80} -2.70339e8 q^{82} +9.37632e7i q^{83} +4.25779e7 q^{85} +5.84471e8 q^{86} -3.99764e7i q^{88} -4.40222e8 q^{89} +9.24978e7i q^{91} +5.14545e8i q^{92} +1.92530e7i q^{94} +(5.36988e8 - 4.85017e8i) q^{95} -1.34369e9i q^{97} +1.18261e9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 480 q^{2} + 7680 q^{4} + 1596 q^{7} + 122880 q^{8}+O(q^{10}) $ 30 * q + 480 * q^2 + 7680 * q^4 + 1596 * q^7 + 122880 * q^8	$ 30 q + 480 q^{2} + 7680 q^{4} + 1596 q^{7} + 122880 q^{8} + 25536 q^{14} + 1966080 q^{16} - 323202 q^{19} - 5473830 q^{25} + 408576 q^{28} - 8475660 q^{29} + 31457280 q^{32} - 5171232 q^{38} + 20345076 q^{41} - 60729780 q^{43} + 214984890 q^{49} - 87581280 q^{50} - 181795212 q^{53} - 316864944 q^{55} + 6537216 q^{56} - 135610560 q^{58} + 197198784 q^{59} + 53410728 q^{61} + 503316480 q^{64} + 640935936 q^{65} - 335924712 q^{71} + 136407840 q^{73} - 82739712 q^{76} + 325521216 q^{82} + 269215776 q^{85} - 971676480 q^{86} - 1186853004 q^{89} + 506121804 q^{95} + 3439758240 q^{98}+O(q^{100}) $ 30 * q + 480 * q^2 + 7680 * q^4 + 1596 * q^7 + 122880 * q^8 + 25536 * q^14 + 1966080 * q^16 - 323202 * q^19 - 5473830 * q^25 + 408576 * q^28 - 8475660 * q^29 + 31457280 * q^32 - 5171232 * q^38 + 20345076 * q^41 - 60729780 * q^43 + 214984890 * q^49 - 87581280 * q^50 - 181795212 * q^53 - 316864944 * q^55 + 6537216 * q^56 - 135610560 * q^58 + 197198784 * q^59 + 53410728 * q^61 + 503316480 * q^64 + 640935936 * q^65 - 335924712 * q^71 + 136407840 * q^73 - 82739712 * q^76 + 325521216 * q^82 + 269215776 * q^85 - 971676480 * q^86 - 1186853004 * q^89 + 506121804 * q^95 + 3439758240 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	16.0000			0.707107
$2$	16.0000			0.707107
$3$		0			0
$3$		0			0
$4$	256.000			0.500000
$5$		−	1273.82i		−	0.911471i	−0.890115	−	0.455735i	$-0.849376\pi$
$5$		−	1273.82i		−	0.911471i	0.890115	−	0.455735i	$-0.150624\pi$
$6$		0			0
$7$	10689.6			1.68275			0.841374	−	0.540453i	$-0.181747\pi$
$7$	10689.6			1.68275			0.841374	+	0.540453i	$0.181747\pi$
$8$	4096.00			0.353553
$9$		0			0
$10$		−	20381.1i		−	0.644507i
$11$		−	9759.86i		−	0.200991i	−0.994938	−	0.100495i	$-0.967957\pi$
$11$		−	9759.86i		−	0.200991i	0.994938	−	0.100495i	$-0.0320428\pi$
$12$		0			0
$13$			8653.09i			0.0840284i	0.999117	+	0.0420142i	$0.0133775\pi$
$13$			8653.09i			0.0840284i	−0.999117	+	0.0420142i	$0.986623\pi$
$14$	171033.			1.18988
$15$		0			0
$16$	65536.0			0.250000
$17$			33425.4i			0.0970635i	0.998822	+	0.0485318i	$0.0154542\pi$
$17$			33425.4i			0.0970635i	−0.998822	+	0.0485318i	$0.984546\pi$
$18$		0			0
$19$	380758.	+	421557.i	0.670283	+	0.742105i
$19$	380758.	+	421557.i	0.670283	+	0.742105i
$20$		−	326098.i		−	0.455735i
$21$		0			0
$22$		−	156158.i		−	0.142122i
$23$			2.00994e6i			1.49764i	0.662771	+	0.748822i	$0.269380\pi$
$23$			2.00994e6i			1.49764i	−0.662771	+	0.748822i	$0.730620\pi$
$24$		0			0
$25$	330510.			0.169221
$26$			138449.i			0.0594170i
$27$		0			0
$28$	2.73653e6			0.841374
$29$	4.16504e6			1.09352			0.546762	−	0.837288i	$-0.315860\pi$
$29$	4.16504e6			1.09352			0.546762	+	0.837288i	$0.315860\pi$
$30$		0			0
$31$			6.52534e6i			1.26904i	0.772907	+	0.634520i	$0.218802\pi$
$31$			6.52534e6i			1.26904i	−0.772907	+	0.634520i	$0.781198\pi$
$32$	1.04858e6			0.176777
$33$		0			0
$34$			534806.i			0.0686343i
$35$		−	1.36166e7i		−	1.53378i
$36$		0			0
$37$			1.41147e7i			1.23812i	0.785343	+	0.619061i	$0.212486\pi$
$37$			1.41147e7i			1.23812i	−0.785343	+	0.619061i	$0.787514\pi$
$38$	6.09214e6	+	6.74492e6i	0.473962	+	0.524748i
$39$		0			0
$40$		−	5.21756e6i		−	0.322254i
$41$	−1.68962e7			−0.933816			−0.466908	−	0.884306i	$-0.654632\pi$
$41$	−1.68962e7			−0.933816			−0.466908	+	0.884306i	$0.654632\pi$
$42$		0			0
$43$	3.65295e7			1.62943			0.814714	−	0.579863i	$-0.196894\pi$
$43$	3.65295e7			1.62943			0.814714	+	0.579863i	$0.196894\pi$
$44$		−	2.49852e6i		−	0.100495i
$45$		0			0
$46$			3.21591e7i			1.05899i
$47$			1.20331e6i			0.0359698i	0.999838	+	0.0179849i	$0.00572509\pi$
$47$			1.20331e6i			0.0359698i	−0.999838	+	0.0179849i	$0.994275\pi$
$48$		0			0
$49$	7.39133e7			1.83164
$50$	5.28816e6			0.119657
$51$		0			0
$52$			2.21519e6i			0.0420142i
$53$	8.17096e6			0.142243			0.0711216	−	0.997468i	$-0.477342\pi$
$53$	8.17096e6			0.142243			0.0711216	+	0.997468i	$0.477342\pi$
$54$		0			0
$55$	−1.24323e7			−0.183197
$56$	4.37845e7			0.594941
$57$		0			0
$58$	6.66406e7			0.773238
$59$	−3.94654e7			−0.424016			−0.212008	−	0.977268i	$-0.568000\pi$
$59$	−3.94654e7			−0.424016			−0.212008	+	0.977268i	$0.568000\pi$
$60$		0			0
$61$	−1.25698e8			−1.16237			−0.581184	−	0.813772i	$-0.697410\pi$
$61$	−1.25698e8			−1.16237			−0.581184	+	0.813772i	$0.697410\pi$
$62$			1.04405e8i			0.897347i
$63$		0			0
$64$	1.67772e7			0.125000
$65$	1.10225e7			0.0765894
$66$		0			0
$67$			7.55895e7i			0.458274i	0.973394	+	0.229137i	$0.0735903\pi$
$67$			7.55895e7i			0.458274i	−0.973394	+	0.229137i	$0.926410\pi$
$68$			8.55690e6i			0.0485318i
$69$		0			0
$70$		−	2.17865e8i		−	1.08454i
$71$	−2.64976e8			−1.23750			−0.618748	−	0.785590i	$-0.712360\pi$
$71$	−2.64976e8			−1.23750			−0.618748	+	0.785590i	$0.712360\pi$
$72$		0			0
$73$	8.58471e6			0.0353812			0.0176906	−	0.999844i	$-0.494369\pi$
$73$	8.58471e6			0.0353812			0.0176906	+	0.999844i	$0.494369\pi$
$74$			2.25835e8i			0.875484i
$75$		0			0
$76$	9.74742e7	+	1.07919e8i	0.335142	+	0.371053i
$77$		−	1.04329e8i		−	0.338217i
$78$		0			0
$79$			9.51474e7i			0.274837i	0.990513	+	0.137418i	$0.0438805\pi$
$79$			9.51474e7i			0.274837i	−0.990513	+	0.137418i	$0.956119\pi$
$80$		−	8.34810e7i		−	0.227868i
$81$		0			0
$82$	−2.70339e8			−0.660308
$83$			9.37632e7i			0.216861i	0.994104	+	0.108430i	$0.0345825\pi$
$83$			9.37632e7i			0.216861i	−0.994104	+	0.108430i	$0.965418\pi$
$84$		0			0
$85$	4.25779e7			0.0884706
$86$	5.84471e8			1.15218
$87$		0			0
$88$		−	3.99764e7i		−	0.0710610i
$89$	−4.40222e8			−0.743733			−0.371867	−	0.928286i	$-0.621282\pi$
$89$	−4.40222e8			−0.743733			−0.371867	+	0.928286i	$0.621282\pi$
$90$		0			0
$91$			9.24978e7i			0.141399i
$92$			5.14545e8i			0.748822i
$93$		0			0
$94$			1.92530e7i			0.0254345i
$95$	5.36988e8	−	4.85017e8i	0.676407	−	0.610944i
$96$		0			0
$97$		−	1.34369e9i		−	1.54108i	−0.637390	−	0.770542i	$-0.719986\pi$
$97$		−	1.34369e9i		−	1.54108i	0.637390	−	0.770542i	$-0.280014\pi$
$98$	1.18261e9			1.29517
$99$		0			0
$100$	8.46106e7			0.0846106
$101$		−	4.68799e8i		−	0.448271i	−0.974558	−	0.224136i	$-0.928044\pi$
$101$		−	4.68799e8i		−	0.448271i	0.974558	−	0.224136i	$-0.0719559\pi$
$102$		0			0
$103$			1.15326e9i			1.00962i	0.863230	+	0.504810i	$0.168438\pi$
$103$			1.15326e9i			1.00962i	−0.863230	+	0.504810i	$0.831562\pi$
$104$			3.54430e7i			0.0297085i
$105$		0			0
$106$	1.30735e8			0.100581
$107$	6.91006e8			0.509630			0.254815	−	0.966990i	$-0.417985\pi$
$107$	6.91006e8			0.509630			0.254815	+	0.966990i	$0.417985\pi$
$108$		0			0
$109$		−	2.01842e9i		−	1.36959i	−0.728735	−	0.684796i	$-0.759891\pi$
$109$		−	2.01842e9i		−	1.36959i	0.728735	−	0.684796i	$-0.240109\pi$
$110$	−1.98917e8			−0.129540
$111$		0			0
$112$	7.00552e8			0.420687
$113$	3.08358e9			1.77911			0.889553	−	0.456831i	$-0.151016\pi$
$113$	3.08358e9			1.77911			0.889553	+	0.456831i	$0.151016\pi$
$114$		0			0
$115$	2.56030e9			1.36506
$116$	1.06625e9			0.546762
$117$		0			0
$118$	−6.31446e8			−0.299825
$119$			3.57303e8i			0.163333i
$120$		0			0
$121$	2.26269e9			0.959603
$122$	−2.01117e9			−0.821918
$123$		0			0
$124$			1.67049e9i			0.634520i
$125$		−	2.90894e9i		−	1.06571i
$126$		0			0
$127$		−	2.26090e9i		−	0.771196i	−0.922667	−	0.385598i	$-0.873995\pi$
$127$		−	2.26090e9i		−	0.771196i	0.922667	−	0.385598i	$-0.126005\pi$
$128$	2.68435e8			0.0883883
$129$		0			0
$130$	1.76359e8			0.0541569
$131$			2.82619e9i			0.838455i	0.907881	+	0.419228i	$0.137699\pi$
$131$			2.82619e9i			0.838455i	−0.907881	+	0.419228i	$0.862301\pi$
$132$		0			0
$133$	4.07014e9	+	4.50627e9i	1.12792	+	1.24878i
$134$			1.20943e9i			0.324049i
$135$		0			0
$136$			1.36910e8i			0.0343171i
$137$			2.24927e9i			0.545506i	0.962084	+	0.272753i	$0.0879342\pi$
$137$			2.24927e9i			0.545506i	−0.962084	+	0.272753i	$0.912066\pi$
$138$		0			0
$139$	−5.45583e9			−1.23964			−0.619819	−	0.784745i	$-0.712794\pi$
$139$	−5.45583e9			−1.23964			−0.619819	+	0.784745i	$0.712794\pi$
$140$		−	3.48584e9i		−	0.766888i
$141$		0			0
$142$	−4.23961e9			−0.875042
$143$	8.44529e7			0.0168889
$144$		0			0
$145$		−	5.30550e9i		−	0.996714i
$146$	1.37355e8			0.0250183
$147$		0			0
$148$			3.61336e9i			0.619061i
$149$			1.15080e9i			0.191276i	0.995416	+	0.0956380i	$0.0304891\pi$
$149$			1.15080e9i			0.191276i	−0.995416	+	0.0956380i	$0.969511\pi$
$150$		0			0
$151$		−	3.32084e9i		−	0.519818i	−0.965633	−	0.259909i	$-0.916307\pi$
$151$		−	3.32084e9i		−	0.519818i	0.965633	−	0.259909i	$-0.0836926\pi$
$152$	1.55959e9	+	1.72670e9i	0.236981	+	0.262374i
$153$		0			0
$154$		−	1.66926e9i		−	0.239156i
$155$	8.31210e9			1.15669
$156$		0			0
$157$	−3.28231e7			−0.00431153			−0.00215577	−	0.999998i	$-0.500686\pi$
$157$	−3.28231e7			−0.00431153			−0.00215577	+	0.999998i	$0.500686\pi$
$158$			1.52236e9i			0.194339i
$159$		0			0
$160$		−	1.33570e9i		−	0.161127i
$161$			2.14854e10i			2.52016i
$162$		0			0
$163$	8.38764e9			0.930669			0.465335	−	0.885135i	$-0.345934\pi$
$163$	8.38764e9			0.930669			0.465335	+	0.885135i	$0.345934\pi$
$164$	−4.32542e9			−0.466908
$165$		0			0
$166$			1.50021e9i			0.153344i
$167$	−1.34936e10			−1.34247			−0.671233	−	0.741247i	$-0.734235\pi$
$167$	−1.34936e10			−1.34247			−0.671233	+	0.741247i	$0.734235\pi$
$168$		0			0
$169$	1.05296e10			0.992939
$170$	6.81246e8			0.0625581
$171$		0			0
$172$	9.35154e9			0.814714
$173$	1.65983e10			1.40882			0.704412	−	0.709791i	$-0.251211\pi$
$173$	1.65983e10			1.40882			0.704412	+	0.709791i	$0.251211\pi$
$174$		0			0
$175$	3.53301e9			0.284756
$176$		−	6.39622e8i		−	0.0502477i
$177$		0			0
$178$	−7.04356e9			−0.525899
$179$	−1.20476e10			−0.877127			−0.438563	−	0.898700i	$-0.644512\pi$
$179$	−1.20476e10			−0.877127			−0.438563	+	0.898700i	$0.644512\pi$
$180$		0			0
$181$			1.22486e10i			0.848266i	0.905600	+	0.424133i	$0.139421\pi$
$181$			1.22486e10i			0.848266i	−0.905600	+	0.424133i	$0.860579\pi$
$182$			1.47996e9i			0.0999839i
$183$		0			0
$184$			8.23273e9i			0.529497i
$185$	1.79795e10			1.12851
$186$		0			0
$187$	3.26227e8			0.0195089
$188$			3.08048e8i			0.0179849i
$189$		0			0
$190$	8.59181e9	−	7.76028e9i	0.478292	−	0.432002i
$191$		−	2.69976e10i		−	1.46783i	−0.679243	−	0.733913i	$-0.737692\pi$
$191$		−	2.69976e10i		−	1.46783i	0.679243	−	0.733913i	$-0.262308\pi$
$192$		0			0
$193$		−	8.03207e9i		−	0.416696i	−0.978055	−	0.208348i	$-0.933191\pi$
$193$		−	8.03207e9i		−	0.416696i	0.978055	−	0.208348i	$-0.0668087\pi$
$194$		−	2.14990e10i		−	1.08971i
$195$		0			0
$196$	1.89218e10			0.915820
$197$		−	1.92581e10i		−	0.910993i	−0.890238	−	0.455496i	$-0.849462\pi$
$197$		−	1.92581e10i		−	0.910993i	0.890238	−	0.455496i	$-0.150538\pi$
$198$		0			0
$199$	2.24995e10			1.01703			0.508516	−	0.861052i	$-0.330194\pi$
$199$	2.24995e10			1.01703			0.508516	+	0.861052i	$0.330194\pi$
$200$	1.35377e9			0.0598287
$201$		0			0
$202$		−	7.50079e9i		−	0.316976i
$203$	4.45225e10			1.84012
$204$		0			0
$205$			2.15227e10i			0.851146i
$206$			1.84521e10i			0.713910i
$207$		0			0
$208$			5.67089e8i			0.0210071i
$209$	4.11434e9	−	3.71615e9i	0.149156	−	0.134721i
$210$		0			0
$211$			4.22554e10i			1.46761i	0.679359	+	0.733806i	$0.262258\pi$
$211$			4.22554e10i			1.46761i	−0.679359	+	0.733806i	$0.737742\pi$
$212$	2.09176e9			0.0711216
$213$		0			0
$214$	1.10561e10			0.360363
$215$		−	4.65319e10i		−	1.48518i
$216$		0			0
$217$			6.97530e10i			2.13547i
$218$		−	3.22947e10i		−	0.968448i
$219$		0			0
$220$	−3.18267e9			−0.0915987
$221$	−2.89233e8			−0.00815609
$222$		0			0
$223$			5.40847e10i			1.46454i	0.681012	+	0.732272i	$0.261540\pi$
$223$			5.40847e10i			1.46454i	−0.681012	+	0.732272i	$0.738460\pi$
$224$	1.12088e10			0.297471
$225$		0			0
$226$	4.93372e10			1.25802
$227$	−1.00757e10			−0.251859			−0.125929	−	0.992039i	$-0.540191\pi$
$227$	−1.00757e10			−0.251859			−0.125929	+	0.992039i	$0.540191\pi$
$228$		0			0
$229$	9.24816e9			0.222226			0.111113	−	0.993808i	$-0.464558\pi$
$229$	9.24816e9			0.222226			0.111113	+	0.993808i	$0.464558\pi$
$230$	4.09649e10			0.965242
$231$		0			0
$232$	1.70600e10			0.386619
$233$		−	1.66145e10i		−	0.369306i	−0.982804	−	0.184653i	$-0.940884\pi$
$233$		−	1.66145e10i		−	0.369306i	0.982804	−	0.184653i	$-0.0591161\pi$
$234$		0			0
$235$	1.53280e9			0.0327855
$236$	−1.01031e10			−0.212008
$237$		0			0
$238$			5.71685e9i			0.115494i
$239$		−	3.72140e10i		−	0.737761i	−0.929477	−	0.368880i	$-0.879741\pi$
$239$		−	3.72140e10i		−	0.737761i	0.929477	−	0.368880i	$-0.120259\pi$
$240$		0			0
$241$		−	6.77768e10i		−	1.29421i	−0.762401	−	0.647105i	$-0.775980\pi$
$241$		−	6.77768e10i		−	1.29421i	0.762401	−	0.647105i	$-0.224020\pi$
$242$	3.62031e10			0.678542
$243$		0			0
$244$	−3.21786e10			−0.581184
$245$		−	9.41522e10i		−	1.66949i
$246$		0			0
$247$	−3.64777e9	+	3.29474e9i	−0.0623579	+	0.0563228i
$248$			2.67278e10i			0.448673i
$249$		0			0
$250$		−	4.65430e10i		−	0.753571i
$251$			4.56494e10i			0.725944i	0.931800	+	0.362972i	$0.118238\pi$
$251$			4.56494e10i			0.725944i	−0.931800	+	0.362972i	$0.881762\pi$
$252$		0			0
$253$	1.96168e10			0.301013
$254$		−	3.61744e10i		−	0.545318i
$255$		0			0
$256$	4.29497e9			0.0625000
$257$	−5.16413e10			−0.738410			−0.369205	−	0.929348i	$-0.620370\pi$
$257$	−5.16413e10			−0.738410			−0.369205	+	0.929348i	$0.620370\pi$
$258$		0			0
$259$			1.50880e11i			2.08345i
$260$	2.82175e9			0.0382947
$261$		0			0
$262$			4.52190e10i			0.592877i
$263$		−	1.04758e11i		−	1.35016i	−0.737742	−	0.675082i	$-0.764108\pi$
$263$		−	1.04758e11i		−	1.35016i	0.737742	−	0.675082i	$-0.235892\pi$
$264$		0			0
$265$		−	1.04083e10i		−	0.129651i
$266$	6.51223e10	+	7.21003e10i	0.797558	+	0.883018i
$267$		0			0
$268$			1.93509e10i			0.229137i
$269$	8.39466e9			0.0977502			0.0488751	−	0.998805i	$-0.484436\pi$
$269$	8.39466e9			0.0977502			0.0488751	+	0.998805i	$0.484436\pi$
$270$		0			0
$271$	−6.09060e10			−0.685959			−0.342979	−	0.939343i	$-0.611436\pi$
$271$	−6.09060e10			−0.685959			−0.342979	+	0.939343i	$0.611436\pi$
$272$			2.19057e9i			0.0242659i
$273$		0			0
$274$			3.59884e10i			0.385731i
$275$		−	3.22573e9i		−	0.0340119i
$276$		0			0
$277$	1.71436e10			0.174962			0.0874810	−	0.996166i	$-0.472118\pi$
$277$	1.71436e10			0.174962			0.0874810	+	0.996166i	$0.472118\pi$
$278$	−8.72933e10			−0.876556
$279$		0			0
$280$		−	5.57735e10i		−	0.542271i
$281$	−7.00380e10			−0.670124			−0.335062	−	0.942196i	$-0.608757\pi$
$281$	−7.00380e10			−0.670124			−0.335062	+	0.942196i	$0.608757\pi$
$282$		0			0
$283$	−1.52547e11			−1.41372			−0.706862	−	0.707352i	$-0.749890\pi$
$283$	−1.52547e11			−1.41372			−0.706862	+	0.707352i	$0.749890\pi$
$284$	−6.78338e10			−0.618748
$285$		0			0
$286$	1.35125e9			0.0119423
$287$	−1.80613e11			−1.57138
$288$		0			0
$289$	1.17471e11			0.990579
$290$		−	8.48881e10i		−	0.704783i
$291$		0			0
$292$	2.19769e9			0.0176906
$293$	1.17557e11			0.931844			0.465922	−	0.884826i	$-0.345723\pi$
$293$	1.17557e11			0.931844			0.465922	+	0.884826i	$0.345723\pi$
$294$		0			0
$295$			5.02718e10i			0.386478i
$296$			5.78137e10i			0.437742i
$297$		0			0
$298$			1.84127e10i			0.135253i
$299$	−1.73922e10			−0.125845
$300$		0			0
$301$	3.90484e11			2.74192
$302$		−	5.31334e10i		−	0.367567i
$303$		0			0
$304$	2.49534e10	+	2.76272e10i	0.167571	+	0.185526i
$305$			1.60116e11i			1.05946i
$306$		0			0
$307$			2.20913e11i			1.41938i	0.704513	+	0.709691i	$0.251166\pi$
$307$			2.20913e11i			1.41938i	−0.704513	+	0.709691i	$0.748834\pi$
$308$		−	2.67081e10i		−	0.169109i
$309$		0			0
$310$	1.32994e11			0.817905
$311$		−	2.35392e11i		−	1.42682i	−0.700746	−	0.713411i	$-0.747149\pi$
$311$		−	2.35392e11i		−	1.42682i	0.700746	−	0.713411i	$-0.252851\pi$
$312$		0			0
$313$	1.11313e11			0.655535			0.327768	−	0.944758i	$-0.393704\pi$
$313$	1.11313e11			0.655535			0.327768	+	0.944758i	$0.393704\pi$
$314$	−5.25170e8			−0.00304871
$315$		0			0
$316$			2.43577e10i			0.137418i
$317$	2.96822e11			1.65093			0.825467	−	0.564451i	$-0.190912\pi$
$317$	2.96822e11			1.65093			0.825467	+	0.564451i	$0.190912\pi$
$318$		0			0
$319$		−	4.06502e10i		−	0.219788i
$320$		−	2.13711e10i		−	0.113934i
$321$		0			0
$322$			3.43767e11i			1.78202i
$323$	−1.40907e10	+	1.27270e10i	−0.0720314	+	0.0650601i
$324$		0			0
$325$			2.85993e9i			0.0142194i
$326$	1.34202e11			0.658082
$327$		0			0
$328$	−6.92068e10			−0.330154
$329$			1.28629e10i			0.0605282i
$330$		0			0
$331$			8.79336e10i			0.402651i	0.979524	+	0.201325i	$0.0645249\pi$
$331$			8.79336e10i			0.402651i	−0.979524	+	0.201325i	$0.935475\pi$
$332$			2.40034e10i			0.108430i
$333$		0			0
$334$	−2.15897e11			−0.949266
$335$	9.62874e10			0.417703
$336$		0			0
$337$		−	1.28802e11i		−	0.543988i	−0.962299	−	0.271994i	$-0.912317\pi$
$337$		−	1.28802e11i		−	0.543988i	0.962299	−	0.271994i	$-0.0876831\pi$
$338$	1.68474e11			0.702114
$339$		0			0
$340$	1.08999e10			0.0442353
$341$	6.36864e10			0.255065
$342$		0			0
$343$	3.58739e11			1.39944
$344$	1.49625e11			0.576090
$345$		0			0
$346$	2.65573e11			0.996189
$347$		−	4.15127e11i		−	1.53708i	−0.639799	−	0.768542i	$-0.720982\pi$
$347$		−	4.15127e11i		−	1.53708i	0.639799	−	0.768542i	$-0.279018\pi$
$348$		0			0
$349$	1.58855e11			0.573174			0.286587	−	0.958054i	$-0.407479\pi$
$349$	1.58855e11			0.573174			0.286587	+	0.958054i	$0.407479\pi$
$350$	5.65282e10			0.201353
$351$		0			0
$352$		−	1.02340e10i		−	0.0355305i
$353$		−	5.13861e11i		−	1.76141i	−0.473669	−	0.880703i	$-0.657071\pi$
$353$		−	5.13861e11i		−	1.76141i	0.473669	−	0.880703i	$-0.342929\pi$
$354$		0			0
$355$			3.37531e11i			1.12794i
$356$	−1.12697e11			−0.371867
$357$		0			0
$358$	−1.92762e11			−0.620222
$359$			1.80466e11i			0.573417i	0.958018	+	0.286708i	$0.0925611\pi$
$359$			1.80466e11i			0.573417i	−0.958018	+	0.286708i	$0.907439\pi$
$360$		0			0
$361$	−3.27337e10	+	3.21023e11i	−0.101441	+	0.994842i
$362$			1.95977e11i			0.599815i
$363$		0			0
$364$			2.36794e10i			0.0706993i
$365$		−	1.09354e10i		−	0.0322489i
$366$		0			0
$367$	−4.17170e10			−0.120037			−0.0600185	−	0.998197i	$-0.519116\pi$
$367$	−4.17170e10			−0.120037			−0.0600185	+	0.998197i	$0.519116\pi$
$368$			1.31724e11i			0.374411i
$369$		0			0
$370$	2.87673e11			0.797978
$371$	8.73440e10			0.239359
$372$		0			0
$373$		−	2.07691e11i		−	0.555557i	−0.960645	−	0.277778i	$-0.910402\pi$
$373$		−	2.07691e11i		−	0.555557i	0.960645	−	0.277778i	$-0.0895981\pi$
$374$	5.21963e9			0.0137949
$375$		0			0
$376$			4.92877e9i			0.0127173i
$377$			3.60404e10i			0.0918870i
$378$		0			0
$379$			1.03626e11i			0.257983i	0.991646	+	0.128991i	$0.0411739\pi$
$379$			1.03626e11i			0.257983i	−0.991646	+	0.128991i	$0.958826\pi$
$380$	1.37469e11	−	1.24164e11i	0.338204	−	0.305472i
$381$		0			0
$382$		−	4.31961e11i		−	1.03791i
$383$	−4.55554e11			−1.08180			−0.540898	−	0.841088i	$-0.681915\pi$
$383$	−4.55554e11			−1.08180			−0.540898	+	0.841088i	$0.681915\pi$
$384$		0			0
$385$	−1.32896e11			−0.308275
$386$		−	1.28513e11i		−	0.294649i
$387$		0			0
$388$		−	3.43984e11i		−	0.770542i
$389$			3.40569e10i			0.0754104i	0.999289	+	0.0377052i	$0.0120048\pi$
$389$			3.40569e10i			0.0754104i	−0.999289	+	0.0377052i	$0.987995\pi$
$390$		0			0
$391$	−6.71831e10			−0.145367
$392$	3.02749e11			0.647583
$393$		0			0
$394$		−	3.08129e11i		−	0.644169i
$395$	1.21201e11			0.250506
$396$		0			0
$397$	−5.40748e11			−1.09254			−0.546270	−	0.837609i	$-0.683953\pi$
$397$	−5.40748e11			−1.09254			−0.546270	+	0.837609i	$0.683953\pi$
$398$	3.59993e11			0.719150
$399$		0			0
$400$	2.16603e10			0.0423053
$401$	6.33848e11			1.22415			0.612076	−	0.790799i	$-0.290334\pi$
$401$	6.33848e11			1.22415			0.612076	+	0.790799i	$0.290334\pi$
$402$		0			0
$403$	−5.64643e10			−0.106635
$404$		−	1.20013e11i		−	0.224136i
$405$		0			0
$406$	7.12359e11			1.30116
$407$	1.37757e11			0.248851
$408$		0			0
$409$		−	9.96205e11i		−	1.76033i	−0.474669	−	0.880165i	$-0.657432\pi$
$409$		−	9.96205e11i		−	1.76033i	0.474669	−	0.880165i	$-0.342568\pi$
$410$			3.44363e11i			0.601851i
$411$		0			0
$412$			2.95234e11i			0.504810i
$413$	−4.21868e11			−0.713512
$414$		0			0
$415$	1.19437e11			0.197662
$416$			9.07342e9i			0.0148543i
$417$		0			0
$418$	6.58295e10	−	5.94584e10i	0.105470	−	0.0952620i
$419$			6.13818e10i			0.0972919i	0.998816	+	0.0486460i	$0.0154906\pi$
$419$			6.13818e10i			0.0972919i	−0.998816	+	0.0486460i	$0.984509\pi$
$420$		0			0
$421$		−	7.02486e11i		−	1.08985i	−0.838484	−	0.544927i	$-0.816557\pi$
$421$		−	7.02486e11i		−	1.08985i	0.838484	−	0.544927i	$-0.183443\pi$
$422$			6.76086e11i			1.03776i
$423$		0			0
$424$	3.34682e10			0.0502906
$425$			1.10474e10i			0.0164252i
$426$		0			0
$427$	−1.34366e12			−1.95597
$428$	1.76898e11			0.254815
$429$		0			0
$430$		−	7.44511e11i		−	1.05018i
$431$	−7.88334e11			−1.10043			−0.550215	−	0.835023i	$-0.685454\pi$
$431$	−7.88334e11			−1.10043			−0.550215	+	0.835023i	$0.685454\pi$
$432$		0			0
$433$			7.79096e10i			0.106511i	0.998581	+	0.0532556i	$0.0169598\pi$
$433$			7.79096e10i			0.106511i	−0.998581	+	0.0532556i	$0.983040\pi$
$434$			1.11605e12i			1.51001i
$435$		0			0
$436$		−	5.16714e11i		−	0.684796i
$437$	−8.47307e11	+	7.65303e11i	−1.11141	+	1.00385i
$438$		0			0
$439$			9.31736e11i			1.19730i	0.801011	+	0.598650i	$0.204296\pi$
$439$			9.31736e11i			1.19730i	−0.801011	+	0.598650i	$0.795704\pi$
$440$	−5.09227e10			−0.0647700
$441$		0			0
$442$	−4.62772e9			−0.00576723
$443$			4.20445e11i			0.518671i	0.965787	+	0.259336i	$0.0835036\pi$
$443$			4.20445e11i			0.518671i	−0.965787	+	0.259336i	$0.916496\pi$
$444$		0			0
$445$			5.60764e11i			0.677891i
$446$			8.65355e11i			1.03559i
$447$		0			0
$448$	1.79341e11			0.210343
$449$	−8.98235e8			−0.00104299			−0.000521497	−	1.00000i	$-0.500166\pi$
$449$	−8.98235e8			−0.00104299			−0.000521497		1.00000i	$0.500166\pi$
$450$		0			0
$451$			1.64904e11i			0.187689i
$452$	7.89396e11			0.889553
$453$		0			0
$454$	−1.61210e11			−0.178091
$455$	1.17825e11			0.128881
$456$		0			0
$457$	9.75334e11			1.04600			0.522998	−	0.852334i	$-0.324813\pi$
$457$	9.75334e11			1.04600			0.522998	+	0.852334i	$0.324813\pi$
$458$	1.47971e11			0.157138
$459$		0			0
$460$	6.55438e11			0.682529
$461$		−	5.39372e11i		−	0.556204i	−0.960552	−	0.278102i	$-0.910295\pi$
$461$		−	5.39372e11i		−	0.556204i	0.960552	−	0.278102i	$-0.0897053\pi$
$462$		0			0
$463$	−1.67831e12			−1.69730			−0.848648	−	0.528959i	$-0.822583\pi$
$463$	−1.67831e12			−1.69730			−0.848648	+	0.528959i	$0.822583\pi$
$464$	2.72960e11			0.273381
$465$		0			0
$466$		−	2.65832e11i		−	0.261139i
$467$		−	2.84893e10i		−	0.0277176i	−0.999904	−	0.0138588i	$-0.995588\pi$
$467$		−	2.84893e10i		−	0.0277176i	0.999904	−	0.0138588i	$-0.00441154\pi$
$468$		0			0
$469$			8.08020e11i			0.771159i
$470$	2.45249e10			0.0231828
$471$		0			0
$472$	−1.61650e11			−0.149912
$473$		−	3.56522e11i		−	0.327500i
$474$		0			0
$475$	1.25844e11	+	1.39329e11i	0.113426	+	0.125580i
$476$			9.14695e10i			0.0816667i
$477$		0			0
$478$		−	5.95423e11i		−	0.521675i
$479$		−	1.01944e12i		−	0.884812i	−0.896815	−	0.442406i	$-0.854125\pi$
$479$		−	1.01944e12i		−	0.884812i	0.896815	−	0.442406i	$-0.145875\pi$
$480$		0			0
$481$	−1.22135e11			−0.104037
$482$		−	1.08443e12i		−	0.915144i
$483$		0			0
$484$	5.79249e11			0.479801
$485$	−1.71162e12			−1.40465
$486$		0			0
$487$			9.68124e10i			0.0779921i	0.999239	+	0.0389960i	$0.0124160\pi$
$487$			9.68124e10i			0.0779921i	−0.999239	+	0.0389960i	$0.987584\pi$
$488$	−5.14858e11			−0.410959
$489$		0			0
$490$		−	1.50643e12i		−	1.18051i
$491$			2.27796e10i			0.0176880i	0.999961	+	0.00884400i	$0.00281517\pi$
$491$			2.27796e10i			0.0176880i	−0.999961	+	0.00884400i	$0.997185\pi$
$492$		0			0
$493$			1.39218e11i			0.106141i
$494$	−5.83644e10	+	5.27158e10i	−0.0440937	+	0.0398262i
$495$		0			0
$496$			4.27644e11i			0.317260i
$497$	−2.83248e12			−2.08239
$498$		0			0
$499$	−1.86722e12			−1.34817			−0.674084	−	0.738655i	$-0.735461\pi$
$499$	−1.86722e12			−1.34817			−0.674084	+	0.738655i	$0.735461\pi$
$500$		−	7.44688e11i		−	0.532855i
$501$		0			0
$502$			7.30390e11i			0.513320i
$503$			2.66841e12i			1.85865i	0.369264	+	0.929325i	$0.379610\pi$
$503$			2.66841e12i			1.85865i	−0.369264	+	0.929325i	$0.620390\pi$
$504$		0			0
$505$	−5.97166e11			−0.408586
$506$	3.13868e11			0.212848
$507$		0			0
$508$		−	5.78791e11i		−	0.385598i
$509$	−1.83671e12			−1.21286			−0.606429	−	0.795138i	$-0.707398\pi$
$509$	−1.83671e12			−1.21286			−0.606429	+	0.795138i	$0.707398\pi$
$510$		0			0
$511$	9.17669e10			0.0595377
$512$	6.87195e10			0.0441942
$513$		0			0
$514$	−8.26260e11			−0.522135
$515$	1.46904e12			0.920240
$516$		0			0
$517$	1.17442e10			0.00722961
$518$			2.41408e12i			1.47322i
$519$		0			0
$520$	4.51480e10			0.0270784
$521$	1.46582e12			0.871590			0.435795	−	0.900046i	$-0.356467\pi$
$521$	1.46582e12			0.871590			0.435795	+	0.900046i	$0.356467\pi$
$522$		0			0
$523$		−	1.71718e12i		−	1.00359i	−0.864985	−	0.501797i	$-0.832672\pi$
$523$		−	1.71718e12i		−	1.00359i	0.864985	−	0.501797i	$-0.167328\pi$
$524$			7.23504e11i			0.419228i
$525$		0			0
$526$		−	1.67613e12i		−	0.954711i
$527$	−2.18112e11			−0.123177
$528$		0			0
$529$	−2.23872e12			−1.24294
$530$		−	1.66533e11i		−	0.0916767i
$531$		0			0
$532$	1.04196e12	+	1.15360e12i	0.563959	+	0.624388i
$533$		−	1.46204e11i		−	0.0784670i
$534$		0			0
$535$		−	8.80217e11i		−	0.464513i
$536$			3.09615e11i			0.162024i
$537$		0			0
$538$	1.34315e11			0.0691198
$539$		−	7.21383e11i		−	0.368143i
$540$		0			0
$541$	−2.62198e12			−1.31596			−0.657978	−	0.753037i	$-0.728588\pi$
$541$	−2.62198e12			−1.31596			−0.657978	+	0.753037i	$0.728588\pi$
$542$	−9.74495e11			−0.485046
$543$		0			0
$544$			3.50490e10i			0.0171586i
$545$	−2.57110e12			−1.24834
$546$		0			0
$547$		−	3.21191e12i		−	1.53398i	−0.641657	−	0.766992i	$-0.721753\pi$
$547$		−	3.21191e12i		−	1.53398i	0.641657	−	0.766992i	$-0.278247\pi$
$548$			5.75814e11i			0.272753i
$549$		0			0
$550$		−	5.16117e10i		−	0.0240501i
$551$	1.58587e12	+	1.75580e12i	0.732970	+	0.811509i
$552$		0			0
$553$			1.01709e12i			0.462481i
$554$	2.74298e11			0.123717
$555$		0			0
$556$	−1.39669e12			−0.619819
$557$		−	4.05907e12i		−	1.78681i	−0.449253	−	0.893404i	$-0.648310\pi$
$557$		−	4.05907e12i		−	1.78681i	0.449253	−	0.893404i	$-0.351690\pi$
$558$		0			0
$559$			3.16093e11i			0.136918i
$560$		−	8.92376e11i		−	0.383444i
$561$		0			0
$562$	−1.12061e12			−0.473849
$563$	−1.57325e12			−0.659950			−0.329975	−	0.943990i	$-0.607040\pi$
$563$	−1.57325e12			−0.659950			−0.329975	+	0.943990i	$0.607040\pi$
$564$		0			0
$565$		−	3.92792e12i		−	1.62160i
$566$	−2.44075e12			−0.999653
$567$		0			0
$568$	−1.08534e12			−0.437521
$569$	−2.62777e12			−1.05095			−0.525476	−	0.850809i	$-0.676113\pi$
$569$	−2.62777e12			−1.05095			−0.525476	+	0.850809i	$0.676113\pi$
$570$		0			0
$571$	−3.55201e12			−1.39834			−0.699168	−	0.714957i	$-0.746446\pi$
$571$	−3.55201e12			−1.39834			−0.699168	+	0.714957i	$0.746446\pi$
$572$	2.16199e10			0.00844447
$573$		0			0
$574$	−2.88981e12			−1.11113
$575$			6.64306e11i			0.253433i
$576$		0			0
$577$	8.69241e11			0.326474			0.163237	−	0.986587i	$-0.447806\pi$
$577$	8.69241e11			0.326474			0.163237	+	0.986587i	$0.447806\pi$
$578$	1.87953e12			0.700445
$579$		0			0
$580$		−	1.35821e12i		−	0.498357i
$581$			1.00229e12i			0.364922i
$582$		0			0
$583$		−	7.97474e10i		−	0.0285896i
$584$	3.51630e10			0.0125091
$585$		0			0
$586$	1.88091e12			0.658913
$587$		−	8.63203e10i		−	0.0300083i	−0.999887	−	0.0150042i	$-0.995224\pi$
$587$		−	8.63203e10i		−	0.0300083i	0.999887	−	0.0150042i	$-0.00477615\pi$
$588$		0			0
$589$	−2.75080e12	+	2.48458e12i	−0.941761	+	0.850616i
$590$			8.04348e11i			0.273281i
$591$		0			0
$592$			9.25019e11i			0.309530i
$593$		−	2.73601e12i		−	0.908597i	−0.890850	−	0.454298i	$-0.849890\pi$
$593$		−	2.73601e12i		−	0.908597i	0.890850	−	0.454298i	$-0.150110\pi$
$594$		0			0
$595$	4.55139e11			0.148874
$596$			2.94604e11i			0.0956380i
$597$		0			0
$598$	−2.78275e11			−0.0889855
$599$	−4.73911e12			−1.50410			−0.752049	−	0.659108i	$-0.770934\pi$
$599$	−4.73911e12			−1.50410			−0.752049	+	0.659108i	$0.770934\pi$
$600$		0			0
$601$		−	2.31066e12i		−	0.722437i	−0.932481	−	0.361219i	$-0.882361\pi$
$601$		−	2.31066e12i		−	0.722437i	0.932481	−	0.361219i	$-0.117639\pi$
$602$	6.24775e12			1.93883
$603$		0			0
$604$		−	8.50134e11i		−	0.259909i
$605$		−	2.88226e12i		−	0.874650i
$606$		0			0
$607$			6.21018e12i			1.85676i	0.371637	+	0.928378i	$0.378797\pi$
$607$			6.21018e12i			1.85676i	−0.371637	+	0.928378i	$0.621203\pi$
$608$	3.99254e11	+	4.42035e11i	0.118490	+	0.131187i
$609$		0			0
$610$			2.56186e12i			0.749154i
$611$	−1.04124e10			−0.00302249
$612$		0			0
$613$	−1.31931e12			−0.377376			−0.188688	−	0.982037i	$-0.560424\pi$
$613$	−1.31931e12			−0.377376			−0.188688	+	0.982037i	$0.560424\pi$
$614$			3.53461e12i			1.00365i
$615$		0			0
$616$		−	4.27330e11i		−	0.119578i
$617$		−	1.59210e12i		−	0.442271i	−0.975243	−	0.221135i	$-0.929024\pi$
$617$		−	1.59210e12i		−	0.442271i	0.975243	−	0.221135i	$-0.0709763\pi$
$618$		0			0
$619$	2.70155e12			0.739613			0.369806	−	0.929109i	$-0.379424\pi$
$619$	2.70155e12			0.739613			0.369806	+	0.929109i	$0.379424\pi$
$620$	2.12790e12			0.578346
$621$		0			0
$622$		−	3.76627e12i		−	1.00892i
$623$	−4.70579e12			−1.25152
$624$		0			0
$625$	−3.05993e12			−0.802143
$626$	1.78101e12			0.463533
$627$		0			0
$628$	−8.40272e9			−0.00215577
$629$	−4.71788e11			−0.120176
$630$		0			0
$631$	−2.97095e12			−0.746041			−0.373020	−	0.927823i	$-0.621678\pi$
$631$	−2.97095e12			−0.746041			−0.373020	+	0.927823i	$0.621678\pi$
$632$			3.89724e11i			0.0971695i
$633$		0			0
$634$	4.74915e12			1.16739
$635$	−2.87998e12			−0.702923
$636$		0			0
$637$			6.39578e11i			0.153910i
$638$		−	6.50403e11i		−	0.155414i
$639$		0			0
$640$		−	3.41938e11i		−	0.0805634i
$641$	−1.69775e12			−0.397203			−0.198601	−	0.980080i	$-0.563640\pi$
$641$	−1.69775e12			−0.397203			−0.198601	+	0.980080i	$0.563640\pi$
$642$		0			0
$643$	4.17000e12			0.962025			0.481013	−	0.876714i	$-0.340269\pi$
$643$	4.17000e12			0.962025			0.481013	+	0.876714i	$0.340269\pi$
$644$			5.50027e12i			1.26008i
$645$		0			0
$646$	−2.25451e11	+	2.03632e11i	−0.0509339	+	0.0460044i
$647$		−	7.07814e12i		−	1.58800i	−0.607920	−	0.793998i	$-0.707996\pi$
$647$		−	7.07814e12i		−	1.58800i	0.607920	−	0.793998i	$-0.292004\pi$
$648$		0			0
$649$			3.85177e11i			0.0852234i
$650$			4.57589e10i			0.0100546i
$651$		0			0
$652$	2.14723e12			0.465335
$653$			5.70950e12i			1.22882i	0.788987	+	0.614411i	$0.210606\pi$
$653$			5.70950e12i			1.22882i	−0.788987	+	0.614411i	$0.789394\pi$
$654$		0			0
$655$	3.60005e12			0.764227
$656$	−1.10731e12			−0.233454
$657$		0			0
$658$			2.05806e11i			0.0427999i
$659$	5.00097e12			1.03293			0.516464	−	0.856309i	$-0.327248\pi$
$659$	5.00097e12			1.03293			0.516464	+	0.856309i	$0.327248\pi$
$660$		0			0
$661$			4.10337e12i			0.836053i	0.908435	+	0.418026i	$0.137278\pi$
$661$			4.10337e12i			0.836053i	−0.908435	+	0.418026i	$0.862722\pi$
$662$			1.40694e12i			0.284717i
$663$		0			0
$664$			3.84054e11i			0.0766719i
$665$	5.74017e12	−	5.18463e12i	1.13822	−	1.02806i
$666$		0			0
$667$			8.37149e12i			1.63771i
$668$	−3.45436e12			−0.671233
$669$		0			0
$670$	1.54060e12			0.295361
$671$			1.22679e12i			0.233625i
$672$		0			0
$673$			4.76026e12i			0.894464i	0.894418	+	0.447232i	$0.147590\pi$
$673$			4.76026e12i			0.894464i	−0.894418	+	0.447232i	$0.852410\pi$
$674$		−	2.06084e12i		−	0.384658i
$675$		0			0
$676$	2.69558e12			0.496470
$677$	−7.26053e12			−1.32837			−0.664186	−	0.747568i	$-0.731222\pi$
$677$	−7.26053e12			−1.32837			−0.664186	+	0.747568i	$0.731222\pi$
$678$		0			0
$679$		−	1.43635e13i		−	2.59325i
$680$	1.74399e11			0.0312791
$681$		0			0
$682$	1.01898e12			0.180359
$683$	6.99535e11			0.123003			0.0615016	−	0.998107i	$-0.480411\pi$
$683$	6.99535e11			0.123003			0.0615016	+	0.998107i	$0.480411\pi$
$684$		0			0
$685$	2.86517e12			0.497213
$686$	5.73982e12			0.989554
$687$		0			0
$688$	2.39399e12			0.407357
$689$			7.07040e10i			0.0119525i
$690$		0			0
$691$	2.01218e12			0.335749			0.167875	−	0.985808i	$-0.446310\pi$
$691$	2.01218e12			0.335749			0.167875	+	0.985808i	$0.446310\pi$
$692$	4.24917e12			0.704412
$693$		0			0
$694$		−	6.64202e12i		−	1.08688i
$695$			6.94975e12i			1.12989i
$696$		0			0
$697$		−	5.64761e11i		−	0.0906395i
$698$	2.54168e12			0.405296
$699$		0			0
$700$	9.04450e11			0.142378
$701$		−	2.83831e12i		−	0.443944i	−0.975053	−	0.221972i	$-0.928751\pi$
$701$		−	2.83831e12i		−	0.443944i	0.975053	−	0.221972i	$-0.0712494\pi$
$702$		0			0
$703$	−5.95015e12	+	5.37428e12i	−0.918816	+	0.829892i
$704$		−	1.63743e11i		−	0.0251239i
$705$		0			0
$706$		−	8.22178e12i		−	1.24550i
$707$		−	5.01126e12i		−	0.754328i
$708$		0			0
$709$	−1.18969e12			−0.176818			−0.0884090	−	0.996084i	$-0.528178\pi$
$709$	−1.18969e12			−0.176818			−0.0884090	+	0.996084i	$0.528178\pi$
$710$			5.40050e12i			0.797575i
$711$		0			0
$712$	−1.80315e12			−0.262949
$713$	−1.31156e13			−1.90057
$714$		0			0
$715$		−	1.07578e11i		−	0.0153938i
$716$	−3.08419e12			−0.438563
$717$		0			0
$718$			2.88746e12i			0.405467i
$719$			4.31868e12i			0.602659i	0.953520	+	0.301329i	$0.0974304\pi$
$719$			4.31868e12i			0.602659i	−0.953520	+	0.301329i	$0.902570\pi$
$720$		0			0
$721$			1.23278e13i			1.69894i
$722$	−5.23738e11	+	5.13637e12i	−0.0717294	+	0.703459i
$723$		0			0
$724$			3.13564e12i			0.424133i
$725$	1.37659e12			0.185047
$726$		0			0
$727$	1.10421e13			1.46604			0.733020	−	0.680207i	$-0.238110\pi$
$727$	1.10421e13			1.46604			0.733020	+	0.680207i	$0.238110\pi$
$728$			3.78871e11i			0.0499919i
$729$		0			0
$730$		−	1.74966e11i		−	0.0228034i
$731$			1.22101e12i			0.158158i
$732$		0			0
$733$	−8.60855e12			−1.10144			−0.550721	−	0.834689i	$-0.685647\pi$
$733$	−8.60855e12			−1.10144			−0.550721	+	0.834689i	$0.685647\pi$
$734$	−6.67471e11			−0.0848790
$735$		0			0
$736$			2.10758e12i			0.264749i
$737$	7.37743e11			0.0921089
$738$		0			0
$739$	−3.01902e12			−0.372362			−0.186181	−	0.982515i	$-0.559611\pi$
$739$	−3.01902e12			−0.372362			−0.186181	+	0.982515i	$0.559611\pi$
$740$	4.60276e12			0.564256
$741$		0			0
$742$	1.39750e12			0.169253
$743$	1.62805e13			1.95982			0.979912	−	0.199433i	$-0.0639099\pi$
$743$	1.62805e13			1.95982			0.979912	+	0.199433i	$0.0639099\pi$
$744$		0			0
$745$	1.46591e12			0.174342
$746$		−	3.32306e12i		−	0.392838i
$747$		0			0
$748$	8.35141e10			0.00975445
$749$	7.38656e12			0.857578
$750$		0			0
$751$		−	1.09778e13i		−	1.25932i	−0.776872	−	0.629659i	$-0.783195\pi$
$751$		−	1.09778e13i		−	1.25932i	0.776872	−	0.629659i	$-0.216805\pi$
$752$			7.88604e10i			0.00899246i
$753$		0			0
$754$			5.76647e11i			0.0649739i
$755$	−4.23015e12			−0.473799
$756$		0			0
$757$	−4.62956e12			−0.512399			−0.256200	−	0.966624i	$-0.582470\pi$
$757$	−4.62956e12			−0.512399			−0.256200	+	0.966624i	$0.582470\pi$
$758$			1.65801e12i			0.182421i
$759$		0			0
$760$	2.19950e12	−	1.98663e12i	0.239146	−	0.216001i
$761$			1.26662e13i			1.36903i	0.728997	+	0.684517i	$0.239987\pi$
$761$			1.26662e13i			1.36903i	−0.728997	+	0.684517i	$0.760013\pi$
$762$		0			0
$763$		−	2.15760e13i		−	2.30468i
$764$		−	6.91138e12i		−	0.733913i
$765$		0			0
$766$	−7.28886e12			−0.764945
$767$		−	3.41497e11i		−	0.0356294i
$768$		0			0
$769$	−5.14278e11			−0.0530309			−0.0265155	−	0.999648i	$-0.508441\pi$
$769$	−5.14278e11			−0.0530309			−0.0265155	+	0.999648i	$0.508441\pi$
$770$	−2.12633e12			−0.217983
$771$		0			0
$772$		−	2.05621e12i		−	0.208348i
$773$	1.42623e13			1.43676			0.718378	−	0.695653i	$-0.244885\pi$
$773$	1.42623e13			1.43676			0.718378	+	0.695653i	$0.244885\pi$
$774$		0			0
$775$			2.15669e12i			0.214748i
$776$		−	5.50375e12i		−	0.544855i
$777$		0			0
$778$			5.44910e11i			0.0533232i
$779$	−6.43337e12	−	7.12271e12i	−0.625921	−	0.692990i
$780$		0			0
$781$			2.58613e12i			0.248725i
$782$	−1.07493e12			−0.102790
$783$		0			0
$784$	4.84398e12			0.457910
$785$			4.18107e10i			0.00392984i
$786$		0			0
$787$			1.11177e13i			1.03307i	0.856266	+	0.516535i	$0.172778\pi$
$787$			1.11177e13i			1.03307i	−0.856266	+	0.516535i	$0.827222\pi$
$788$		−	4.93007e12i		−	0.455496i
$789$		0			0
$790$	1.93921e12			0.177134
$791$	3.29621e13			2.99379
$792$		0			0
$793$		−	1.08767e12i		−	0.0976719i
$794$	−8.65196e12			−0.772543
$795$		0			0
$796$	5.75988e12			0.508516
$797$	1.47326e13			1.29335			0.646675	−	0.762765i	$-0.276159\pi$
$797$	1.47326e13			1.29335			0.646675	+	0.762765i	$0.276159\pi$
$798$		0			0
$799$	−4.02212e10			−0.00349136
$800$	3.46565e11			0.0299144
$801$		0			0
$802$	1.01416e13			0.865607
$803$		−	8.37856e10i		−	0.00711130i
$804$		0			0
$805$	2.73685e13			2.29705
$806$	−9.03429e11			−0.0754026
$807$		0			0
$808$		−	1.92020e12i		−	0.158488i
$809$		−	3.32343e12i		−	0.272784i	−0.990655	−	0.136392i	$-0.956449\pi$
$809$		−	3.32343e12i		−	0.272784i	0.990655	−	0.136392i	$-0.0435507\pi$
$810$		0			0
$811$			2.06895e13i			1.67941i	0.543044	+	0.839704i	$0.317272\pi$
$811$			2.06895e13i			1.67941i	−0.543044	+	0.839704i	$0.682728\pi$
$812$	1.13978e13			0.920062
$813$		0			0
$814$	2.20412e12			0.175964
$815$		−	1.06843e13i		−	0.848278i
$816$		0			0
$817$	1.39089e13	+	1.53993e13i	1.09218	+	1.20921i
$818$		−	1.59393e13i		−	1.24474i
$819$		0			0
$820$			5.50981e12i			0.425573i
$821$		−	4.94597e12i		−	0.379933i	−0.981791	−	0.189967i	$-0.939162\pi$
$821$		−	4.94597e12i		−	0.379933i	0.981791	−	0.189967i	$-0.0608380\pi$
$822$		0			0
$823$	1.55374e13			1.18053			0.590266	−	0.807209i	$-0.299023\pi$
$823$	1.55374e13			1.18053			0.590266	+	0.807209i	$0.299023\pi$
$824$			4.72374e12i			0.356955i
$825$		0			0
$826$	−6.74989e12			−0.504529
$827$	−8.58096e12			−0.637912			−0.318956	−	0.947769i	$-0.603332\pi$
$827$	−8.58096e12			−0.637912			−0.318956	+	0.947769i	$0.603332\pi$
$828$		0			0
$829$		−	5.61542e12i		−	0.412940i	−0.978453	−	0.206470i	$-0.933802\pi$
$829$		−	5.61542e12i		−	0.412940i	0.978453	−	0.206470i	$-0.0661976\pi$
$830$	1.91100e12			0.139768
$831$		0			0
$832$			1.45175e11i			0.0105035i
$833$			2.47058e12i			0.177785i
$834$		0			0
$835$			1.71884e13i			1.22362i
$836$	1.05327e12	−	9.51334e11i	0.0745782	−	0.0673604i
$837$		0			0
$838$			9.82109e11i			0.0687958i
$839$	−2.89279e12			−0.201553			−0.100776	−	0.994909i	$-0.532133\pi$
$839$	−2.89279e12			−0.201553			−0.100776	+	0.994909i	$0.532133\pi$
$840$		0			0
$841$	2.84039e12			0.195793
$842$		−	1.12398e13i		−	0.770643i
$843$		0			0
$844$			1.08174e13i			0.733806i
$845$		−	1.34128e13i		−	0.905035i
$846$		0			0
$847$	2.41872e13			1.61477
$848$	5.35492e11			0.0355608
$849$		0			0
$850$			1.76759e11i			0.0116144i
$851$	−2.83697e13			−1.85426
$852$		0			0
$853$	7.01713e12			0.453825			0.226913	−	0.973915i	$-0.427137\pi$
$853$	7.01713e12			0.453825			0.226913	+	0.973915i	$0.427137\pi$
$854$	−2.14985e13			−1.38308
$855$		0			0
$856$	2.83036e12			0.180181
$857$	2.70209e13			1.71114			0.855571	−	0.517686i	$-0.173206\pi$
$857$	2.70209e13			1.71114			0.855571	+	0.517686i	$0.173206\pi$
$858$		0			0
$859$	−1.11382e13			−0.697985			−0.348993	−	0.937125i	$-0.613476\pi$
$859$	−1.11382e13			−0.697985			−0.348993	+	0.937125i	$0.613476\pi$
$860$		−	1.19122e13i		−	0.742588i
$861$		0			0
$862$	−1.26133e13			−0.778121
$863$	2.74516e13			1.68468			0.842342	−	0.538943i	$-0.181176\pi$
$863$	2.74516e13			1.68468			0.842342	+	0.538943i	$0.181176\pi$
$864$		0			0
$865$		−	2.11433e13i		−	1.28410i
$866$			1.24655e12i			0.0753148i
$867$		0			0
$868$			1.78568e13i			1.06774i
$869$	9.28625e11			0.0552397
$870$		0			0
$871$	−6.54083e11			−0.0385080
$872$		−	8.26743e12i		−	0.484224i
$873$		0			0
$874$	−1.35569e13	+	1.22448e13i	−0.785885	+	0.709826i
$875$		−	3.10953e13i		−	1.79332i
$876$		0			0
$877$			3.15626e13i			1.80167i	0.434167	+	0.900833i	$0.357043\pi$
$877$			3.15626e13i			1.80167i	−0.434167	+	0.900833i	$0.642957\pi$
$878$			1.49078e13i			0.846618i
$879$		0			0
$880$	−8.14763e11			−0.0457993
$881$		−	4.08978e12i		−	0.228722i	−0.993439	−	0.114361i	$-0.963518\pi$
$881$		−	4.08978e12i		−	0.228722i	0.993439	−	0.114361i	$-0.0364820\pi$
$882$		0			0
$883$	1.57265e13			0.870583			0.435292	−	0.900290i	$-0.356645\pi$
$883$	1.57265e13			0.870583			0.435292	+	0.900290i	$0.356645\pi$
$884$	−7.40435e10			−0.00407805
$885$		0			0
$886$			6.72712e12i			0.366756i
$887$	−7.48291e12			−0.405895			−0.202948	−	0.979190i	$-0.565052\pi$
$887$	−7.48291e12			−0.405895			−0.202948	+	0.979190i	$0.565052\pi$
$888$		0			0
$889$		−	2.41681e13i		−	1.29773i
$890$			8.97222e12i			0.479341i
$891$		0			0
$892$			1.38457e13i			0.732272i
$893$	−5.07266e11	+	4.58172e11i	−0.0266934	+	0.0241100i
$894$		0			0
$895$			1.53465e13i			0.799475i
$896$	2.86946e12			0.148735
$897$		0			0
$898$	−1.43718e10			−0.000737508
$899$			2.71783e13i			1.38772i
$900$		0			0
$901$			2.73117e11i			0.0138066i
$902$			2.63847e12i			0.132716i
$903$		0			0
$904$	1.26303e13			0.629009
$905$	1.56025e13			0.773169
$906$		0			0
$907$		−	1.15082e13i		−	0.564643i	−0.959320	−	0.282322i	$-0.908895\pi$
$907$		−	1.15082e13i		−	0.564643i	0.959320	−	0.282322i	$-0.0911045\pi$
$908$	−2.57937e12			−0.125929
$909$		0			0
$910$	1.88521e12			0.0911324
$911$	−2.96792e13			−1.42764			−0.713822	−	0.700327i	$-0.753038\pi$
$911$	−2.96792e13			−1.42764			−0.713822	+	0.700327i	$0.753038\pi$
$912$		0			0
$913$	9.15116e11			0.0435871
$914$	1.56053e13			0.739631
$915$		0			0
$916$	2.36753e12			0.111113
$917$			3.02107e13i			1.41091i
$918$		0			0
$919$	−3.50713e13			−1.62193			−0.810964	−	0.585096i	$-0.801057\pi$
$919$	−3.50713e13			−1.62193			−0.810964	+	0.585096i	$0.801057\pi$
$920$	1.04870e13			0.482621
$921$		0			0
$922$		−	8.62995e12i		−	0.393296i
$923$		−	2.29286e12i		−	0.103985i
$924$		0			0
$925$			4.66504e12i			0.209516i
$926$	−2.68529e13			−1.20017
$927$		0			0
$928$	4.36736e12			0.193309
$929$		−	2.83618e12i		−	0.124929i	−0.998047	−	0.0624644i	$-0.980104\pi$
$929$		−	2.83618e12i		−	0.124929i	0.998047	−	0.0624644i	$-0.0198960\pi$
$930$		0			0
$931$	2.81431e13	+	3.11587e13i	1.22772	+	1.35927i
$932$		−	4.25332e12i		−	0.184653i
$933$		0			0
$934$		−	4.55829e11i		−	0.0195993i
$935$		−	4.15554e11i		−	0.0177818i
$936$		0			0
$937$	1.58933e13			0.673576			0.336788	−	0.941580i	$-0.390659\pi$
$937$	1.58933e13			0.673576			0.336788	+	0.941580i	$0.390659\pi$
$938$			1.29283e13i			0.545292i
$939$		0			0
$940$	3.92398e11			0.0163927
$941$	−2.09831e13			−0.872401			−0.436201	−	0.899849i	$-0.643676\pi$
$941$	−2.09831e13			−0.872401			−0.436201	+	0.899849i	$0.643676\pi$
$942$		0			0
$943$		−	3.39604e13i		−	1.39852i
$944$	−2.58640e12			−0.106004
$945$		0			0
$946$		−	5.70436e12i		−	0.231578i
$947$		−	3.11613e13i		−	1.25904i	−0.776982	−	0.629522i	$-0.783251\pi$
$947$		−	3.11613e13i		−	1.25904i	0.776982	−	0.629522i	$-0.216749\pi$
$948$		0			0
$949$			7.42842e10i			0.00297303i
$950$	2.01351e12	+	2.22926e12i	0.0802044	+	0.0887984i
$951$		0			0
$952$			1.46351e12i			0.0577471i
$953$	−3.07047e13			−1.20583			−0.602916	−	0.797804i	$-0.705995\pi$
$953$	−3.07047e13			−1.20583			−0.602916	+	0.797804i	$0.705995\pi$
$954$		0			0
$955$	−3.43900e13			−1.33788
$956$		−	9.52678e12i		−	0.368880i
$957$		0			0
$958$		−	1.63110e13i		−	0.625657i
$959$			2.40438e13i			0.917949i
$960$		0			0
$961$	−1.61404e13			−0.610462
$962$	−1.95417e12			−0.0735655
$963$		0			0
$964$		−	1.73509e13i		−	0.647105i
$965$	−1.02314e13			−0.379807
$966$		0			0
$967$	2.46909e13			0.908065			0.454033	−	0.890985i	$-0.349985\pi$
$967$	2.46909e13			0.908065			0.454033	+	0.890985i	$0.349985\pi$
$968$	9.26799e12			0.339271
$969$		0			0
$970$	−2.73859e13			−0.993239
$971$	−5.37258e13			−1.93953			−0.969764	−	0.244044i	$-0.921526\pi$
$971$	−5.37258e13			−1.93953			−0.969764	+	0.244044i	$0.921526\pi$
$972$		0			0
$973$	−5.83205e13			−2.08600
$974$			1.54900e12i			0.0551487i
$975$		0			0
$976$	−8.23773e12			−0.290592
$977$	3.46804e13			1.21775			0.608875	−	0.793266i	$-0.291621\pi$
$977$	3.46804e13			1.21775			0.608875	+	0.793266i	$0.291621\pi$
$978$		0			0
$979$			4.29651e12i			0.149484i
$980$		−	2.41030e13i		−	0.834743i
$981$		0			0
$982$			3.64473e11i			0.0125073i
$983$	2.46138e13			0.840792			0.420396	−	0.907341i	$-0.361891\pi$
$983$	2.46138e13			0.840792			0.420396	+	0.907341i	$0.361891\pi$
$984$		0			0
$985$	−2.45313e13			−0.830343
$986$			2.22749e12i			0.0750532i
$987$		0			0
$988$	−9.33830e11	+	8.43452e11i	−0.0311790	+	0.0281614i
$989$			7.34221e13i			2.44030i
$990$		0			0
$991$		−	2.77473e13i		−	0.913881i	−0.889497	−	0.456941i	$-0.848945\pi$
$991$		−	2.77473e13i		−	0.913881i	0.889497	−	0.456941i	$-0.151055\pi$
$992$			6.84231e12i			0.224337i
$993$		0			0
$994$	−4.53196e13			−1.47247
$995$		−	2.86603e13i		−	0.926995i
$996$		0			0
$997$	3.63661e13			1.16565			0.582825	−	0.812598i	$-0.301947\pi$
$997$	3.63661e13			1.16565			0.582825	+	0.812598i	$0.301947\pi$
$998$	−2.98756e13			−0.953298
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.10.b.b.341.9	yes	30
3.2	odd	2		342.10.b.a.341.22	yes	30
19.18	odd	2		342.10.b.a.341.9	✓	30
57.56	even	2	inner	342.10.b.b.341.22	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.10.b.a.341.9	✓	30	19.18	odd	2
342.10.b.a.341.22	yes	30	3.2	odd	2
342.10.b.b.341.9	yes	30	1.1	even	1	trivial
342.10.b.b.341.22	yes	30	57.56	even	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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	342.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+16.0000 q^{2} +256.000 q^{4} -1273.82i q^{5} +10689.6 q^{7} +4096.00 q^{8} +O(q^{10})\)	\(q+16.0000 q^{2} +256.000 q^{4} -1273.82i q^{5} +10689.6 q^{7} +4096.00 q^{8} -20381.1i q^{10} -9759.86i q^{11} +8653.09i q^{13} +171033. q^{14} +65536.0 q^{16} +33425.4i q^{17} +(380758. + 421557. i) q^{19} -326098. i q^{20} -156158. i q^{22} +2.00994e6i q^{23} +330510. q^{25} +138449. i q^{26} +2.73653e6 q^{28} +4.16504e6 q^{29} +6.52534e6i q^{31} +1.04858e6 q^{32} +534806. i q^{34} -1.36166e7i q^{35} +1.41147e7i q^{37} +(6.09214e6 + 6.74492e6i) q^{38} -5.21756e6i q^{40} -1.68962e7 q^{41} +3.65295e7 q^{43} -2.49852e6i q^{44} +3.21591e7i q^{46} +1.20331e6i q^{47} +7.39133e7 q^{49} +5.28816e6 q^{50} +2.21519e6i q^{52} +8.17096e6 q^{53} -1.24323e7 q^{55} +4.37845e7 q^{56} +6.66406e7 q^{58} -3.94654e7 q^{59} -1.25698e8 q^{61} +1.04405e8i q^{62} +1.67772e7 q^{64} +1.10225e7 q^{65} +7.55895e7i q^{67} +8.55690e6i q^{68} -2.17865e8i q^{70} -2.64976e8 q^{71} +8.58471e6 q^{73} +2.25835e8i q^{74} +(9.74742e7 + 1.07919e8i) q^{76} -1.04329e8i q^{77} +9.51474e7i q^{79} -8.34810e7i q^{80} -2.70339e8 q^{82} +9.37632e7i q^{83} +4.25779e7 q^{85} +5.84471e8 q^{86} -3.99764e7i q^{88} -4.40222e8 q^{89} +9.24978e7i q^{91} +5.14545e8i q^{92} +1.92530e7i q^{94} +(5.36988e8 - 4.85017e8i) q^{95} -1.34369e9i q^{97} +1.18261e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 480 q^{2} + 7680 q^{4} + 1596 q^{7} + 122880 q^{8}+O(q^{10}) \) 30 * q + 480 * q^2 + 7680 * q^4 + 1596 * q^7 + 122880 * q^8	\( 30 q + 480 q^{2} + 7680 q^{4} + 1596 q^{7} + 122880 q^{8} + 25536 q^{14} + 1966080 q^{16} - 323202 q^{19} - 5473830 q^{25} + 408576 q^{28} - 8475660 q^{29} + 31457280 q^{32} - 5171232 q^{38} + 20345076 q^{41} - 60729780 q^{43} + 214984890 q^{49} - 87581280 q^{50} - 181795212 q^{53} - 316864944 q^{55} + 6537216 q^{56} - 135610560 q^{58} + 197198784 q^{59} + 53410728 q^{61} + 503316480 q^{64} + 640935936 q^{65} - 335924712 q^{71} + 136407840 q^{73} - 82739712 q^{76} + 325521216 q^{82} + 269215776 q^{85} - 971676480 q^{86} - 1186853004 q^{89} + 506121804 q^{95} + 3439758240 q^{98}+O(q^{100}) \) 30 * q + 480 * q^2 + 7680 * q^4 + 1596 * q^7 + 122880 * q^8 + 25536 * q^14 + 1966080 * q^16 - 323202 * q^19 - 5473830 * q^25 + 408576 * q^28 - 8475660 * q^29 + 31457280 * q^32 - 5171232 * q^38 + 20345076 * q^41 - 60729780 * q^43 + 214984890 * q^49 - 87581280 * q^50 - 181795212 * q^53 - 316864944 * q^55 + 6537216 * q^56 - 135610560 * q^58 + 197198784 * q^59 + 53410728 * q^61 + 503316480 * q^64 + 640935936 * q^65 - 335924712 * q^71 + 136407840 * q^73 - 82739712 * q^76 + 325521216 * q^82 + 269215776 * q^85 - 971676480 * q^86 - 1186853004 * q^89 + 506121804 * q^95 + 3439758240 * q^98

Introduction

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