LMFDB - Embedded newform 84.12.k.b.5.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,12,Mod(5,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 5]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("84.5");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	84.k (of order $6$, degree $2$, 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:	$64.5408271670$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$28$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.18
Character	$\chi$	$=$	84.5
Dual form			84.12.k.b.17.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(119.823 + 403.472i) q^{3} +(1893.47 - 3279.58i) q^{5} +(44455.1 + 1035.29i) q^{7} +(-148432. + 96690.7i) q^{9} +O(q^{10})$	\(q+(119.823 + 403.472i) q^{3} +(1893.47 - 3279.58i) q^{5} +(44455.1 + 1035.29i) q^{7} +(-148432. + 96690.7i) q^{9} +(-289993. + 167428. i) q^{11} -710965. i q^{13} +(1.55010e6 + 370990. i) q^{15} +(-4.43857e6 - 7.68782e6i) q^{17} +(-1.41707e7 - 8.18146e6i) q^{19} +(4.90905e6 + 1.80604e7i) q^{21} +(2.71256e7 + 1.56610e7i) q^{23} +(1.72436e7 + 2.98668e7i) q^{25} +(-5.67976e7 - 4.83022e7i) q^{27} +1.22965e8i q^{29} +(-2.71270e8 + 1.56618e8i) q^{31} +(-1.02300e8 - 9.69423e7i) q^{33} +(8.75696e7 - 1.43834e8i) q^{35} +(-8.64684e7 + 1.49768e8i) q^{37} +(2.86854e8 - 8.51902e7i) q^{39} -9.54564e8 q^{41} -4.41738e8 q^{43} +(3.60546e7 + 6.69875e8i) q^{45} +(3.52402e8 - 6.10379e8i) q^{47} +(1.97518e9 + 9.20482e7i) q^{49} +(2.56997e9 - 2.71202e9i) q^{51} +(-2.00992e9 + 1.16042e9i) q^{53} +1.26808e9i q^{55} +(1.60300e9 - 6.69781e9i) q^{57} +(-3.65791e9 - 6.33569e9i) q^{59} +(1.82043e9 + 1.05103e9i) q^{61} +(-6.69865e9 + 4.14472e9i) q^{63} +(-2.33167e9 - 1.34619e9i) q^{65} +(-5.32355e9 - 9.22065e9i) q^{67} +(-3.06848e9 + 1.28210e10i) q^{69} -2.07970e10i q^{71} +(8.13793e9 - 4.69844e9i) q^{73} +(-9.98423e9 + 1.05361e10i) q^{75} +(-1.30650e10 + 7.14279e9i) q^{77} +(-1.42637e10 + 2.47055e10i) q^{79} +(1.26829e10 - 2.87039e10i) q^{81} -5.42123e10 q^{83} -3.36171e10 q^{85} +(-4.96130e10 + 1.47341e10i) q^{87} +(4.60716e10 - 7.97983e10i) q^{89} +(7.36058e8 - 3.16060e10i) q^{91} +(-9.56952e10 - 9.06832e10i) q^{93} +(-5.36636e10 + 3.09827e10i) q^{95} +8.51782e10i q^{97} +(2.68555e10 - 5.28913e10i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 729 q^{3} - 27194 q^{7} - 172347 q^{9}+O(q^{10}) $ 56 * q - 729 * q^3 - 27194 * q^7 - 172347 * q^9	$ 56 q - 729 q^{3} - 27194 q^{7} - 172347 q^{9} - 4853058 q^{15} + 28700520 q^{19} - 11325429 q^{21} - 316601194 q^{25} - 1368416388 q^{31} + 40874949 q^{33} - 87435712 q^{37} + 1177474410 q^{39} - 3055078348 q^{43} + 4109921793 q^{45} - 742582522 q^{49} - 694793715 q^{51} + 14605100370 q^{57} + 72584834058 q^{61} - 7310837811 q^{63} + 6131679148 q^{67} - 74402605464 q^{73} - 161115157854 q^{75} + 52181713528 q^{79} + 44948282337 q^{81} + 4658488716 q^{85} + 243101263104 q^{87} - 85311757146 q^{91} - 256628211777 q^{93} + 157345775874 q^{99}+O(q^{100}) $ 56 * q - 729 * q^3 - 27194 * q^7 - 172347 * q^9 - 4853058 * q^15 + 28700520 * q^19 - 11325429 * q^21 - 316601194 * q^25 - 1368416388 * q^31 + 40874949 * q^33 - 87435712 * q^37 + 1177474410 * q^39 - 3055078348 * q^43 + 4109921793 * q^45 - 742582522 * q^49 - 694793715 * q^51 + 14605100370 * q^57 + 72584834058 * q^61 - 7310837811 * q^63 + 6131679148 * q^67 - 74402605464 * q^73 - 161115157854 * q^75 + 52181713528 * q^79 + 44948282337 * q^81 + 4658488716 * q^85 + 243101263104 * q^87 - 85311757146 * q^91 - 256628211777 * q^93 + 157345775874 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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
$3$	119.823	+	403.472i	0.284692	+	0.958619i
$3$	119.823	+	403.472i	0.284692	+	0.958619i
$4$		0			0
$5$	1893.47	−	3279.58i	0.270971	−	0.469336i	−0.698140	−	0.715962i	$-0.745989\pi$
$5$	1893.47	−	3279.58i	0.270971	−	0.469336i	0.969111	+	0.246626i	$0.0793219\pi$
$6$		0			0
$7$	44455.1	+	1035.29i	0.999729	+	0.0232822i
$7$	44455.1	+	1035.29i	0.999729	+	0.0232822i
$8$		0			0
$9$	−148432.	+	96690.7i	−0.837901	+	0.545822i
$10$		0			0
$11$	−289993.	+	167428.i	−0.542911	+	0.313450i	−0.746258	−	0.665657i	$-0.768151\pi$
$11$	−289993.	+	167428.i	−0.542911	+	0.313450i	0.203347	+	0.979107i	$0.434818\pi$
$12$		0			0
$13$		−	710965.i		−	0.531079i	−0.964100	−	0.265540i	$-0.914450\pi$
$13$		−	710965.i		−	0.531079i	0.964100	−	0.265540i	$-0.0855501\pi$
$14$		0			0
$15$	1.55010e6	+	370990.i	0.527057	+	0.126142i
$16$		0			0
$17$	−4.43857e6	−	7.68782e6i	−0.758182	−	1.31321i	−0.943777	−	0.330584i	$-0.892754\pi$
$17$	−4.43857e6	−	7.68782e6i	−0.758182	−	1.31321i	0.185594	−	0.982626i	$-0.440579\pi$
$18$		0			0
$19$	−1.41707e7	−	8.18146e6i	−1.31295	−	0.758030i	−0.330363	−	0.943854i	$-0.607171\pi$
$19$	−1.41707e7	−	8.18146e6i	−1.31295	−	0.758030i	−0.982583	+	0.185824i	$0.940504\pi$
$20$		0			0
$21$	4.90905e6	+	1.80604e7i	0.262296	+	0.964988i
$22$		0			0
$23$	2.71256e7	+	1.56610e7i	0.878773	+	0.507360i	0.870254	−	0.492604i	$-0.163955\pi$
$23$	2.71256e7	+	1.56610e7i	0.878773	+	0.507360i	0.00851947	+	0.999964i	$0.497288\pi$
$24$		0			0
$25$	1.72436e7	+	2.98668e7i	0.353149	+	0.611673i
$26$		0			0
$27$	−5.67976e7	−	4.83022e7i	−0.761779	−	0.647837i
$28$		0			0
$29$			1.22965e8i			1.11325i	0.830763	+	0.556626i	$0.187904\pi$
$29$			1.22965e8i			1.11325i	−0.830763	+	0.556626i	$0.812096\pi$
$30$		0			0
$31$	−2.71270e8	+	1.56618e8i	−1.70181	+	0.982543i	−0.757888	+	0.652385i	$0.773769\pi$
$31$	−2.71270e8	+	1.56618e8i	−1.70181	+	0.982543i	−0.943926	+	0.330158i	$0.892898\pi$
$32$		0			0
$33$	−1.02300e8	−	9.69423e7i	−0.455041	−	0.431208i
$34$		0			0
$35$	8.75696e7	−	1.43834e8i	0.281825	−	0.462900i
$36$		0			0
$37$	−8.64684e7	+	1.49768e8i	−0.204997	+	0.355065i	−0.950132	−	0.311849i	$-0.899052\pi$
$37$	−8.64684e7	+	1.49768e8i	−0.204997	+	0.355065i	0.745135	+	0.666914i	$0.232385\pi$
$38$		0			0
$39$	2.86854e8	−	8.51902e7i	0.509103	−	0.151194i
$40$		0			0
$41$	−9.54564e8			−1.28675			−0.643375	−	0.765551i	$-0.722466\pi$
$41$	−9.54564e8			−1.28675			−0.643375	+	0.765551i	$0.722466\pi$
$42$		0			0
$43$	−4.41738e8			−0.458235			−0.229118	−	0.973399i	$-0.573584\pi$
$43$	−4.41738e8			−0.458235			−0.229118	+	0.973399i	$0.573584\pi$
$44$		0			0
$45$	3.60546e7	+	6.69875e8i	0.0291267	+	0.541159i
$46$		0			0
$47$	3.52402e8	−	6.10379e8i	0.224130	−	0.388205i	−0.731928	−	0.681382i	$-0.761379\pi$
$47$	3.52402e8	−	6.10379e8i	0.224130	−	0.388205i	0.956058	+	0.293177i	$0.0947125\pi$
$48$		0			0
$49$	1.97518e9	+	9.20482e7i	0.998916	+	0.0465519i
$50$		0			0
$51$	2.56997e9	−	2.71202e9i	1.04302	−	1.10067i
$52$		0			0
$53$	−2.00992e9	+	1.16042e9i	−0.660177	+	0.381153i	−0.792344	−	0.610074i	$-0.791140\pi$
$53$	−2.00992e9	+	1.16042e9i	−0.660177	+	0.381153i	0.132167	+	0.991227i	$0.457806\pi$
$54$		0			0
$55$			1.26808e9i			0.339743i
$56$		0			0
$57$	1.60300e9	−	6.69781e9i	0.352877	−	1.47442i
$58$		0			0
$59$	−3.65791e9	−	6.33569e9i	−0.666111	−	1.15374i	−0.978983	−	0.203943i	$-0.934624\pi$
$59$	−3.65791e9	−	6.33569e9i	−0.666111	−	1.15374i	0.312871	−	0.949796i	$-0.398709\pi$
$60$		0			0
$61$	1.82043e9	+	1.05103e9i	0.275969	+	0.159331i	0.631597	−	0.775297i	$-0.282400\pi$
$61$	1.82043e9	+	1.05103e9i	0.275969	+	0.159331i	−0.355628	+	0.934628i	$0.615733\pi$
$62$		0			0
$63$	−6.69865e9	+	4.14472e9i	−0.850382	+	0.526166i
$64$		0			0
$65$	−2.33167e9	−	1.34619e9i	−0.249254	−	0.143907i
$66$		0			0
$67$	−5.32355e9	−	9.22065e9i	−0.481714	−	0.834354i	0.518065	−	0.855341i	$-0.326652\pi$
$67$	−5.32355e9	−	9.22065e9i	−0.481714	−	0.834354i	−0.999780	+	0.0209873i	$0.993319\pi$
$68$		0			0
$69$	−3.06848e9	+	1.28210e10i	−0.236185	+	0.986850i
$70$		0			0
$71$		−	2.07970e10i		−	1.36798i	−0.729491	−	0.683990i	$-0.760243\pi$
$71$		−	2.07970e10i		−	1.36798i	0.729491	−	0.683990i	$-0.239757\pi$
$72$		0			0
$73$	8.13793e9	−	4.69844e9i	0.459450	−	0.265264i	−0.252363	−	0.967633i	$-0.581208\pi$
$73$	8.13793e9	−	4.69844e9i	0.459450	−	0.265264i	0.711813	+	0.702369i	$0.247874\pi$
$74$		0			0
$75$	−9.98423e9	+	1.05361e10i	−0.485822	+	0.512674i
$76$		0			0
$77$	−1.30650e10	+	7.14279e9i	−0.550061	+	0.300724i
$78$		0			0
$79$	−1.42637e10	+	2.47055e10i	−0.521536	+	0.903326i	0.478150	+	0.878278i	$0.341307\pi$
$79$	−1.42637e10	+	2.47055e10i	−0.521536	+	0.903326i	−0.999686	+	0.0250485i	$0.992026\pi$
$80$		0			0
$81$	1.26829e10	−	2.87039e10i	0.404157	−	0.914690i
$82$		0			0
$83$	−5.42123e10			−1.51067			−0.755333	−	0.655341i	$-0.772525\pi$
$83$	−5.42123e10			−1.51067			−0.755333	+	0.655341i	$0.772525\pi$
$84$		0			0
$85$	−3.36171e10			−0.821782
$86$		0			0
$87$	−4.96130e10	+	1.47341e10i	−1.06718	+	0.316934i
$88$		0			0
$89$	4.60716e10	−	7.97983e10i	0.874557	−	1.51478i	0.0173224	−	0.999850i	$-0.494486\pi$
$89$	4.60716e10	−	7.97983e10i	0.874557	−	1.51478i	0.857234	−	0.514927i	$-0.172181\pi$
$90$		0			0
$91$	7.36058e8	−	3.16060e10i	0.0123647	−	0.530935i
$92$		0			0
$93$	−9.56952e10	−	9.06832e10i	−1.42638	−	1.35167i
$94$		0			0
$95$	−5.36636e10	+	3.09827e10i	−0.711541	+	0.410808i
$96$		0			0
$97$			8.51782e10i			1.00713i	0.863959	+	0.503563i	$0.167978\pi$
$97$			8.51782e10i			1.00713i	−0.863959	+	0.503563i	$0.832022\pi$
$98$		0			0
$99$	2.68555e10	−	5.28913e10i	0.283818	−	0.558972i
$100$		0			0
$101$	−7.81170e10	−	1.35303e11i	−0.739568	−	1.28097i	−0.952690	−	0.303944i	$-0.901696\pi$
$101$	−7.81170e10	−	1.35303e11i	−0.739568	−	1.28097i	0.213122	−	0.977026i	$-0.431637\pi$
$102$		0			0
$103$	−1.32217e11	−	7.63353e10i	−1.12378	−	0.648815i	−0.181417	−	0.983406i	$-0.558068\pi$
$103$	−1.32217e11	−	7.63353e10i	−1.12378	−	0.648815i	−0.942363	+	0.334592i	$0.891402\pi$
$104$		0			0
$105$	6.85257e10	+	1.80972e10i	0.523978	+	0.138379i
$106$		0			0
$107$	1.74363e10	+	1.00669e10i	0.120183	+	0.0693878i	0.558887	−	0.829244i	$-0.311229\pi$
$107$	1.74363e10	+	1.00669e10i	0.120183	+	0.0693878i	−0.438703	+	0.898632i	$0.644562\pi$
$108$		0			0
$109$	1.24387e11	+	2.15445e11i	0.774337	+	1.34119i	0.935166	+	0.354209i	$0.115250\pi$
$109$	1.24387e11	+	2.15445e11i	0.774337	+	1.34119i	−0.160829	+	0.986982i	$0.551417\pi$
$110$		0			0
$111$	−7.07879e10	−	1.69419e10i	−0.398733	−	0.0954299i
$112$		0			0
$113$		−	1.10541e11i		−	0.564404i	−0.959355	−	0.282202i	$-0.908935\pi$
$113$		−	1.10541e11i		−	0.564404i	0.959355	−	0.282202i	$-0.0910649\pi$
$114$		0			0
$115$	1.02723e11	−	5.93072e10i	0.476244	−	0.274960i
$116$		0			0
$117$	6.87437e10	+	1.05530e11i	0.289875	+	0.444992i
$118$		0			0
$119$	−1.89358e11	−	3.46358e11i	−0.727402	−	1.33051i
$120$		0			0
$121$	−8.65917e10	+	1.49981e11i	−0.303499	+	0.525675i
$122$		0			0
$123$	−1.14379e11	−	3.85140e11i	−0.366327	−	1.23350i
$124$		0			0
$125$	3.15510e11			0.924715
$126$		0			0
$127$	3.84394e11			1.03242			0.516210	−	0.856462i	$-0.327342\pi$
$127$	3.84394e11			1.03242			0.516210	+	0.856462i	$0.327342\pi$
$128$		0			0
$129$	−5.29306e10	−	1.78229e11i	−0.130456	−	0.439273i
$130$		0			0
$131$	−8.31004e10	+	1.43934e11i	−0.188196	+	0.325965i	−0.944649	−	0.328083i	$-0.893597\pi$
$131$	−8.31004e10	+	1.43934e11i	−0.188196	+	0.325965i	0.756453	+	0.654048i	$0.226931\pi$
$132$		0			0
$133$	−6.21490e11	−	3.78379e11i	−1.29494	−	0.788393i
$134$		0			0
$135$	−2.65955e11	+	9.48137e10i	−0.510473	+	0.181985i
$136$		0			0
$137$	−5.50434e11	+	3.17793e11i	−0.974411	+	0.562577i	−0.900578	−	0.434694i	$-0.856857\pi$
$137$	−5.50434e11	+	3.17793e11i	−0.974411	+	0.562577i	−0.0738330	+	0.997271i	$0.523523\pi$
$138$		0			0
$139$			3.31599e11i			0.542040i	0.962574	+	0.271020i	$0.0873610\pi$
$139$			3.31599e11i			0.542040i	−0.962574	+	0.271020i	$0.912639\pi$
$140$		0			0
$141$	2.88497e11	+	6.90467e10i	0.435949	+	0.104337i
$142$		0			0
$143$	1.19035e11	+	2.06175e11i	0.166467	+	0.288329i
$144$		0			0
$145$	4.03275e11	+	2.32831e11i	0.522489	+	0.301659i
$146$		0			0
$147$	1.99534e11	+	8.07960e11i	0.239758	+	0.970833i
$148$		0			0
$149$	7.50228e11	+	4.33144e11i	0.836891	+	0.483179i	0.856206	−	0.516634i	$-0.172815\pi$
$149$	7.50228e11	+	4.33144e11i	0.836891	+	0.483179i	−0.0193154	+	0.999813i	$0.506149\pi$
$150$		0			0
$151$	−6.98239e11	−	1.20939e12i	−0.723821	−	1.25369i	−0.959458	−	0.281853i	$-0.909051\pi$
$151$	−6.98239e11	−	1.20939e12i	−0.723821	−	1.25369i	0.235637	−	0.971841i	$-0.424282\pi$
$152$		0			0
$153$	1.40217e12	+	7.11948e11i	1.35206	+	0.686508i
$154$		0			0
$155$			1.18620e12i			1.06496i
$156$		0			0
$157$	−3.95692e11	+	2.28453e11i	−0.331062	+	0.191139i	−0.656312	−	0.754489i	$-0.727885\pi$
$157$	−3.95692e11	+	2.28453e11i	−0.331062	+	0.191139i	0.325251	+	0.945628i	$0.394551\pi$
$158$		0			0
$159$	−7.09034e11	−	6.71898e11i	−0.553328	−	0.524347i
$160$		0			0
$161$	1.18966e12	+	7.24294e11i	0.866723	+	0.527682i
$162$		0			0
$163$	−4.40923e11	+	7.63701e11i	−0.300145	+	0.519866i	−0.976169	−	0.217014i	$-0.930368\pi$
$163$	−4.40923e11	+	7.63701e11i	−0.300145	+	0.519866i	0.676024	+	0.736880i	$0.263702\pi$
$164$		0			0
$165$	−5.11633e11	+	1.51945e11i	−0.325684	+	0.0967220i
$166$		0			0
$167$	−4.82125e11			−0.287223			−0.143612	−	0.989634i	$-0.545872\pi$
$167$	−4.82125e11			−0.287223			−0.143612	+	0.989634i	$0.545872\pi$
$168$		0			0
$169$	1.28669e12			0.717955
$170$		0			0
$171$	2.89445e12	−	1.55788e11i	1.51387	−	0.0814808i
$172$		0			0
$173$	−4.03342e11	+	6.98608e11i	−0.197888	+	0.342752i	−0.947843	−	0.318736i	$-0.896742\pi$
$173$	−4.03342e11	+	6.98608e11i	−0.197888	+	0.342752i	0.749955	+	0.661488i	$0.230075\pi$
$174$		0			0
$175$	7.35646e11	+	1.34558e12i	0.338813	+	0.619729i
$176$		0			0
$177$	2.11797e12	−	2.23503e12i	0.916360	−	0.967007i
$178$		0			0
$179$	1.74973e12	−	1.01021e12i	0.711673	−	0.410885i	−0.100007	−	0.994987i	$-0.531887\pi$
$179$	1.74973e12	−	1.01021e12i	0.711673	−	0.410885i	0.811680	+	0.584102i	$0.198553\pi$
$180$		0			0
$181$			1.77822e12i			0.680384i	0.940356	+	0.340192i	$0.110492\pi$
$181$			1.77822e12i			0.680384i	−0.940356	+	0.340192i	$0.889508\pi$
$182$		0			0
$183$	−2.05929e11	+	8.60431e11i	−0.0741715	+	0.309909i
$184$		0			0
$185$	3.27450e11	+	5.67160e11i	0.111097	+	0.192425i
$186$		0			0
$187$	2.57431e12	+	1.48628e12i	0.823251	+	0.475304i
$188$		0			0
$189$	−2.47493e12	−	2.20608e12i	−0.746489	−	0.665397i
$190$		0			0
$191$	3.08153e12	+	1.77912e12i	0.877169	+	0.506434i	0.869724	−	0.493539i	$-0.164297\pi$
$191$	3.08153e12	+	1.77912e12i	0.877169	+	0.506434i	0.00744480	+	0.999972i	$0.497630\pi$
$192$		0			0
$193$	−1.35502e12	−	2.34697e12i	−0.364235	−	0.630874i	0.624418	−	0.781090i	$-0.285336\pi$
$193$	−1.35502e12	−	2.34697e12i	−0.364235	−	0.630874i	−0.988653	+	0.150216i	$0.952003\pi$
$194$		0			0
$195$	2.63760e11	−	1.10207e12i	0.0669914	−	0.279909i
$196$		0			0
$197$		−	2.84714e12i		−	0.683668i	−0.939760	−	0.341834i	$-0.888952\pi$
$197$		−	2.84714e12i		−	0.683668i	0.939760	−	0.341834i	$-0.111048\pi$
$198$		0			0
$199$	1.81239e12	−	1.04639e12i	0.411681	−	0.237684i	−0.279831	−	0.960049i	$-0.590278\pi$
$199$	1.81239e12	−	1.04639e12i	0.411681	−	0.237684i	0.691512	+	0.722365i	$0.256945\pi$
$200$		0			0
$201$	3.08239e12	−	3.25275e12i	0.662687	−	0.699314i
$202$		0			0
$203$	−1.27305e11	+	5.46643e12i	−0.0259190	+	1.11295i
$204$		0			0
$205$	−1.80744e12	+	3.13057e12i	−0.348672	+	0.603917i
$206$		0			0
$207$	−5.54058e12	+	2.98210e11i	−1.01325	+	0.0545362i
$208$		0			0
$209$	5.47922e12			0.950416
$210$		0			0
$211$	−2.62985e12			−0.432890			−0.216445	−	0.976295i	$-0.569446\pi$
$211$	−2.62985e12			−0.432890			−0.216445	+	0.976295i	$0.569446\pi$
$212$		0			0
$213$	8.39100e12	−	2.49197e12i	1.31137	−	0.389453i
$214$		0			0
$215$	−8.36417e11	+	1.44872e12i	−0.124168	+	0.215066i
$216$		0			0
$217$	−1.22215e13	+	6.68161e12i	−1.72423	+	0.942654i
$218$		0			0
$219$	2.87080e12	+	2.72044e12i	0.385089	+	0.364919i
$220$		0			0
$221$	−5.46577e12	+	3.15566e12i	−0.697419	+	0.402655i
$222$		0			0
$223$			1.44430e13i			1.75380i	0.480672	+	0.876900i	$0.340393\pi$
$223$			1.44430e13i			1.75380i	−0.480672	+	0.876900i	$0.659607\pi$
$224$		0			0
$225$	−5.44735e12	−	2.76589e12i	−0.629769	−	0.319765i
$226$		0			0
$227$	−8.82885e12	−	1.52920e13i	−0.972214	−	1.68392i	−0.688839	−	0.724915i	$-0.741879\pi$
$227$	−8.82885e12	−	1.52920e13i	−0.972214	−	1.68392i	−0.283375	−	0.959009i	$-0.591454\pi$
$228$		0			0
$229$	−2.46228e12	−	1.42160e12i	−0.258370	−	0.149170i	0.365221	−	0.930921i	$-0.380993\pi$
$229$	−2.46228e12	−	1.42160e12i	−0.258370	−	0.149170i	−0.623591	+	0.781751i	$0.714327\pi$
$230$		0			0
$231$	−4.44741e12	−	4.41549e12i	−0.444878	−	0.441685i
$232$		0			0
$233$	−7.08722e12	−	4.09181e12i	−0.676112	−	0.390353i	0.122277	−	0.992496i	$-0.460981\pi$
$233$	−7.08722e12	−	4.09181e12i	−0.676112	−	0.390353i	−0.798389	+	0.602143i	$0.794314\pi$
$234$		0			0
$235$	−1.33453e12	−	2.31147e12i	−0.121466	−	0.210385i
$236$		0			0
$237$	−1.16771e13	−	2.79471e12i	−1.01442	−	0.242785i
$238$		0			0
$239$		−	1.01995e12i		−	0.0846036i	−0.999105	−	0.0423018i	$-0.986531\pi$
$239$		−	1.01995e12i		−	0.0846036i	0.999105	−	0.0423018i	$-0.0134691\pi$
$240$		0			0
$241$	9.95408e12	−	5.74699e12i	0.788692	−	0.455351i	−0.0508100	−	0.998708i	$-0.516180\pi$
$241$	9.95408e12	−	5.74699e12i	0.788692	−	0.455351i	0.839502	+	0.543357i	$0.182847\pi$
$242$		0			0
$243$	1.31009e13	+	1.67777e12i	0.991899	+	0.127028i
$244$		0			0
$245$	4.04182e12	−	6.30348e12i	0.292526	−	0.456213i
$246$		0			0
$247$	−5.81673e12	+	1.00749e13i	−0.402574	+	0.697279i
$248$		0			0
$249$	−6.49591e12	−	2.18731e13i	−0.430074	−	1.44815i
$250$		0			0
$251$	−5.63091e12			−0.356758			−0.178379	−	0.983962i	$-0.557085\pi$
$251$	−5.63091e12			−0.356758			−0.178379	+	0.983962i	$0.557085\pi$
$252$		0			0
$253$	−1.04883e13			−0.636127
$254$		0			0
$255$	−4.02812e12	−	1.35636e13i	−0.233955	−	0.787776i
$256$		0			0
$257$	4.16559e12	−	7.21502e12i	0.231763	−	0.401426i	−0.726564	−	0.687099i	$-0.758884\pi$
$257$	4.16559e12	−	7.21502e12i	0.231763	−	0.401426i	0.958327	+	0.285673i	$0.0922172\pi$
$258$		0			0
$259$	−3.99901e12	+	6.56841e12i	−0.213208	+	0.350196i
$260$		0			0
$261$	−1.18896e13	−	1.82519e13i	−0.607637	−	0.932795i
$262$		0			0
$263$	−8.48505e12	+	4.89885e12i	−0.415813	+	0.240070i	−0.693284	−	0.720664i	$-0.743837\pi$
$263$	−8.48505e12	+	4.89885e12i	−0.415813	+	0.240070i	0.277472	+	0.960734i	$0.410504\pi$
$264$		0			0
$265$			8.78891e12i			0.413126i
$266$		0			0
$267$	3.77168e13	+	9.02686e12i	1.70107	+	0.407122i
$268$		0			0
$269$	−1.25769e13	−	2.17838e13i	−0.544422	−	0.942966i	−0.998643	−	0.0520771i	$-0.983416\pi$
$269$	−1.25769e13	−	2.17838e13i	−0.544422	−	0.942966i	0.454221	−	0.890889i	$-0.349917\pi$
$270$		0			0
$271$	2.40119e13	+	1.38633e13i	0.997920	+	0.576149i	0.907632	−	0.419766i	$-0.137888\pi$
$271$	2.40119e13	+	1.38633e13i	0.997920	+	0.576149i	0.0902878	+	0.995916i	$0.471221\pi$
$272$		0			0
$273$	1.28403e13	−	3.49016e12i	0.512485	−	0.139300i
$274$		0			0
$275$	−1.00011e13	−	5.77412e12i	−0.383457	−	0.221389i
$276$		0			0
$277$	1.63939e13	+	2.83950e13i	0.604008	+	1.04617i	0.992207	+	0.124597i	$0.0397639\pi$
$277$	1.63939e13	+	2.83950e13i	0.604008	+	1.04617i	−0.388199	+	0.921575i	$0.626903\pi$
$278$		0			0
$279$	2.51215e13	−	4.94763e13i	0.889658	−	1.75216i
$280$		0			0
$281$		−	1.37058e13i		−	0.466680i	−0.972395	−	0.233340i	$-0.925035\pi$
$281$		−	1.37058e13i		−	0.466680i	0.972395	−	0.233340i	$-0.0749655\pi$
$282$		0			0
$283$	−3.21933e13	+	1.85868e13i	−1.05424	+	0.608667i	−0.923834	−	0.382794i	$-0.874962\pi$
$283$	−3.21933e13	+	1.85868e13i	−1.05424	+	0.608667i	−0.130408	+	0.991460i	$0.541629\pi$
$284$		0			0
$285$	−1.89308e13	−	1.79393e13i	−0.596378	−	0.565143i
$286$		0			0
$287$	−4.24352e13	−	9.88255e11i	−1.28640	−	0.0299584i
$288$		0			0
$289$	−2.22658e13	+	3.85655e13i	−0.649681	+	1.12528i
$290$		0			0
$291$	−3.43670e13	+	1.02063e13i	−0.965450	+	0.286720i
$292$		0			0
$293$	1.06497e13			0.288116			0.144058	−	0.989569i	$-0.453985\pi$
$293$	1.06497e13			0.288116			0.144058	+	0.989569i	$0.453985\pi$
$294$		0			0
$295$	−2.77045e13			−0.721988
$296$		0			0
$297$	2.45580e13	+	4.49782e12i	0.616642	+	0.112938i
$298$		0			0
$299$	1.11344e13	−	1.92854e13i	0.269448	−	0.466698i
$300$		0			0
$301$	−1.96375e13	−	4.57329e11i	−0.458111	−	0.0106687i
$302$		0			0
$303$	4.52305e13	−	4.77304e13i	1.01741	−	1.07365i
$304$		0			0
$305$	6.89386e12	−	3.98017e12i	0.149559	−	0.0863481i
$306$		0			0
$307$			6.18899e13i			1.29527i	0.761952	+	0.647633i	$0.224241\pi$
$307$			6.18899e13i			1.29527i	−0.761952	+	0.647633i	$0.775759\pi$
$308$		0			0
$309$	1.49565e13	−	6.24924e13i	0.302035	−	1.26199i
$310$		0			0
$311$	4.75481e13	+	8.23557e13i	0.926726	+	1.60514i	0.788762	+	0.614698i	$0.210722\pi$
$311$	4.75481e13	+	8.23557e13i	0.926726	+	1.60514i	0.137963	+	0.990437i	$0.455944\pi$
$312$		0			0
$313$	−1.34824e13	−	7.78405e12i	−0.253672	−	0.146457i	0.367773	−	0.929916i	$-0.380120\pi$
$313$	−1.34824e13	−	7.78405e12i	−0.253672	−	0.146457i	−0.621444	+	0.783458i	$0.713454\pi$
$314$		0			0
$315$	9.09292e11	+	2.98167e13i	0.0165194	+	0.541690i
$316$		0			0
$317$	7.51582e13	+	4.33926e13i	1.31871	+	0.761359i	0.983522	−	0.180790i	$-0.0578656\pi$
$317$	7.51582e13	+	4.33926e13i	1.31871	+	0.761359i	0.335192	+	0.942150i	$0.391199\pi$
$318$		0			0
$319$	−2.05878e13	−	3.56591e13i	−0.348948	−	0.604396i
$320$		0			0
$321$	−1.97241e12	+	8.24131e12i	−0.0323013	+	0.134964i
$322$		0			0
$323$			1.45256e14i			2.29890i
$324$		0			0
$325$	2.12343e13	−	1.22596e13i	0.324847	−	0.187550i
$326$		0			0
$327$	−7.20215e13	+	7.60021e13i	−1.06524	+	1.12412i
$328$		0			0
$329$	1.62980e13	−	2.67696e13i	0.233108	−	0.382882i
$330$		0			0
$331$	−2.41501e12	+	4.18291e12i	−0.0334091	+	0.0578662i	−0.882246	−	0.470788i	$-0.843970\pi$
$331$	−2.41501e12	+	4.18291e12i	−0.0334091	+	0.0578662i	0.848837	+	0.528654i	$0.177303\pi$
$332$		0			0
$333$	−1.64649e12	−	3.05909e13i	−0.0220352	−	0.409402i
$334$		0			0
$335$	−4.03198e13			−0.522122
$336$		0			0
$337$	1.19226e13			0.149420			0.0747099	−	0.997205i	$-0.476197\pi$
$337$	1.19226e13			0.149420			0.0747099	+	0.997205i	$0.476197\pi$
$338$		0			0
$339$	4.46000e13	−	1.32454e13i	0.541049	−	0.160681i
$340$		0			0
$341$	5.24443e13	−	9.08362e13i	0.615955	−	1.06687i
$342$		0			0
$343$	8.77116e13	+	6.13691e12i	0.997561	+	0.0697962i
$344$		0			0
$345$	3.62374e13	+	3.43394e13i	0.399164	+	0.378258i
$346$		0			0
$347$	4.04536e13	−	2.33559e13i	0.431663	−	0.249221i	−0.268392	−	0.963310i	$-0.586492\pi$
$347$	4.04536e13	−	2.33559e13i	0.431663	−	0.249221i	0.700055	+	0.714089i	$0.253159\pi$
$348$		0			0
$349$			5.21249e13i			0.538896i	0.963015	+	0.269448i	$0.0868413\pi$
$349$			5.21249e13i			0.538896i	−0.963015	+	0.269448i	$0.913159\pi$
$350$		0			0
$351$	−3.43411e13	+	4.03810e13i	−0.344053	+	0.404565i
$352$		0			0
$353$	−1.22614e13	−	2.12374e13i	−0.119064	−	0.206225i	0.800333	−	0.599556i	$-0.204656\pi$
$353$	−1.22614e13	−	2.12374e13i	−0.119064	−	0.206225i	−0.919397	+	0.393331i	$0.871323\pi$
$354$		0			0
$355$	−6.82055e13	−	3.93784e13i	−0.642042	−	0.370683i
$356$		0			0
$357$	1.17056e14	−	1.17902e14i	1.06836	−	1.07609i
$358$		0			0
$359$	1.11719e14	+	6.45009e13i	0.988797	+	0.570882i	0.904914	−	0.425594i	$-0.139935\pi$
$359$	1.11719e14	+	6.45009e13i	0.988797	+	0.570882i	0.0838824	+	0.996476i	$0.473268\pi$
$360$		0			0
$361$	7.56276e13	+	1.30991e14i	0.649218	+	1.12448i
$362$		0			0
$363$	−7.08889e13	−	1.69660e13i	−0.590326	−	0.141284i
$364$		0			0
$365$		−	3.55854e13i		−	0.287515i
$366$		0			0
$367$	−1.29569e14	+	7.48068e13i	−1.01587	+	0.586513i	−0.912905	−	0.408172i	$-0.866166\pi$
$367$	−1.29569e14	+	7.48068e13i	−1.01587	+	0.586513i	−0.102965	+	0.994685i	$0.532833\pi$
$368$		0			0
$369$	1.41688e14	−	9.22975e13i	1.07817	−	0.702336i
$370$		0			0
$371$	−9.05523e13	+	4.95059e13i	−0.668872	+	0.365680i
$372$		0			0
$373$	7.56276e13	−	1.30991e14i	0.542353	−	0.939382i	−0.456416	−	0.889767i	$-0.650867\pi$
$373$	7.56276e13	−	1.30991e14i	0.542353	−	0.939382i	0.998768	−	0.0496156i	$-0.0157996\pi$
$374$		0			0
$375$	3.78055e13	+	1.27299e14i	0.263259	+	0.886450i
$376$		0			0
$377$	8.74239e13			0.591225
$378$		0			0
$379$	1.57656e14			1.03560			0.517802	−	0.855500i	$-0.326750\pi$
$379$	1.57656e14			1.03560			0.517802	+	0.855500i	$0.326750\pi$
$380$		0			0
$381$	4.60595e13	+	1.55092e14i	0.293922	+	0.989698i
$382$		0			0
$383$	−1.30737e14	+	2.26444e14i	−0.810600	+	1.40400i	0.101844	+	0.994800i	$0.467526\pi$
$383$	−1.30737e14	+	2.26444e14i	−0.810600	+	1.40400i	−0.912445	+	0.409200i	$0.865808\pi$
$384$		0			0
$385$	−1.31283e12	+	5.63724e13i	−0.00790998	+	0.339651i
$386$		0			0
$387$	6.55679e13	−	4.27120e13i	0.383956	−	0.250115i
$388$		0			0
$389$	4.46410e13	−	2.57735e13i	0.254104	−	0.146707i	−0.367538	−	0.930008i	$-0.619799\pi$
$389$	4.46410e13	−	2.57735e13i	0.254104	−	0.146707i	0.621642	+	0.783302i	$0.286466\pi$
$390$		0			0
$391$		−	2.78049e14i		−	1.53869i
$392$		0			0
$393$	−6.80307e13	−	1.62820e13i	−0.366055	−	0.0876088i
$394$		0			0
$395$	5.40158e13	+	9.35581e13i	0.282642	+	0.489551i
$396$		0			0
$397$	−1.21735e14	−	7.02840e13i	−0.619540	−	0.357692i	0.157150	−	0.987575i	$-0.449769\pi$
$397$	−1.21735e14	−	7.02840e13i	−0.619540	−	0.357692i	−0.776690	+	0.629883i	$0.783103\pi$
$398$		0			0
$399$	7.81959e13	−	2.96092e14i	0.387109	−	1.46580i
$400$		0			0
$401$	−1.55102e14	−	8.95480e13i	−0.747003	−	0.431283i	0.0776069	−	0.996984i	$-0.475272\pi$
$401$	−1.55102e14	−	8.95480e13i	−0.747003	−	0.431283i	−0.824610	+	0.565702i	$0.808605\pi$
$402$		0			0
$403$	1.11350e14	+	1.92863e14i	0.521808	+	0.903798i
$404$		0			0
$405$	−7.01223e13	−	9.59445e13i	−0.319782	−	0.437540i
$406$		0			0
$407$		−	5.79088e13i		−	0.257025i
$408$		0			0
$409$	−3.31735e14	+	1.91527e14i	−1.43322	+	0.827470i	−0.997365	−	0.0725475i	$-0.976887\pi$
$409$	−3.31735e14	+	1.91527e14i	−1.43322	+	0.827470i	−0.435854	+	0.900017i	$0.643554\pi$
$410$		0			0
$411$	−1.94176e14	−	1.84005e14i	−0.816704	−	0.773928i
$412$		0			0
$413$	−1.56053e14	−	2.85440e14i	−0.639069	−	1.16893i
$414$		0			0
$415$	−1.02649e14	+	1.77794e14i	−0.409347	+	0.709009i
$416$		0			0
$417$	−1.33791e14	+	3.97333e13i	−0.519610	+	0.154314i
$418$		0			0
$419$	−1.82839e14			−0.691659			−0.345829	−	0.938297i	$-0.612402\pi$
$419$	−1.82839e14			−0.691659			−0.345829	+	0.938297i	$0.612402\pi$
$420$		0			0
$421$	−1.16909e14			−0.430821			−0.215411	−	0.976524i	$-0.569109\pi$
$421$	−1.16909e14			−0.430821			−0.215411	+	0.976524i	$0.569109\pi$
$422$		0			0
$423$	6.71029e12	+	1.24674e14i	0.0240918	+	0.447613i
$424$		0			0
$425$	1.53074e14	−	2.65132e14i	0.535503	−	0.927519i
$426$		0			0
$427$	7.98394e13	+	4.86082e13i	0.272185	+	0.165713i
$428$		0			0
$429$	−6.89226e13	+	7.27319e13i	−0.229006	+	0.241663i
$430$		0			0
$431$	−1.66202e14	+	9.59570e13i	−0.538285	+	0.310779i	−0.744384	−	0.667752i	$-0.767257\pi$
$431$	−1.66202e14	+	9.59570e13i	−0.538285	+	0.310779i	0.206099	+	0.978531i	$0.433923\pi$
$432$		0			0
$433$		−	9.52094e13i		−	0.300605i	−0.988640	−	0.150303i	$-0.951975\pi$
$433$		−	9.52094e13i		−	0.300605i	0.988640	−	0.150303i	$-0.0480248\pi$
$434$		0			0
$435$	−4.56188e13	+	1.90608e14i	−0.140428	+	0.586747i
$436$		0			0
$437$	−2.56260e14	−	4.43855e14i	−0.769188	−	1.33227i
$438$		0			0
$439$	1.12739e14	+	6.50898e13i	0.330003	+	0.190528i	0.655843	−	0.754898i	$-0.272313\pi$
$439$	1.12739e14	+	6.50898e13i	0.330003	+	0.190528i	−0.325839	+	0.945425i	$0.605647\pi$
$440$		0			0
$441$	−3.02080e14	+	1.77319e14i	−0.862402	+	0.506224i
$442$		0			0
$443$	8.35272e13	+	4.82245e13i	0.232599	+	0.134291i	0.611770	−	0.791035i	$-0.290458\pi$
$443$	8.35272e13	+	4.82245e13i	0.232599	+	0.134291i	−0.379172	+	0.925326i	$0.623791\pi$
$444$		0			0
$445$	−1.74470e14	−	3.02191e14i	−0.473959	−	0.820921i
$446$		0			0
$447$	−8.48665e13	+	3.54597e14i	−0.224929	+	0.939817i
$448$		0			0
$449$			3.09922e14i			0.801488i	0.916190	+	0.400744i	$0.131248\pi$
$449$			3.09922e14i			0.801488i	−0.916190	+	0.400744i	$0.868752\pi$
$450$		0			0
$451$	2.76817e14	−	1.59821e14i	0.698590	−	0.403331i
$452$		0			0
$453$	4.04287e14	−	4.26633e14i	0.995750	−	1.05079i
$454$		0			0
$455$	−1.02261e14	−	6.22589e13i	−0.245836	−	0.149671i
$456$		0			0
$457$	2.05612e14	−	3.56130e14i	0.482513	−	0.835737i	−0.517286	−	0.855813i	$-0.673058\pi$
$457$	2.05612e14	−	3.56130e14i	0.482513	−	0.835737i	0.999798	+	0.0200761i	$0.00639084\pi$
$458$		0			0
$459$	−1.19239e14	+	6.51042e14i	−0.273179	+	1.49155i
$460$		0			0
$461$	−1.39703e14			−0.312500			−0.156250	−	0.987718i	$-0.549941\pi$
$461$	−1.39703e14			−0.312500			−0.156250	+	0.987718i	$0.549941\pi$
$462$		0			0
$463$	1.33928e14			0.292535			0.146267	−	0.989245i	$-0.453274\pi$
$463$	1.33928e14			0.292535			0.146267	+	0.989245i	$0.453274\pi$
$464$		0			0
$465$	−4.78599e14	+	1.42135e14i	−1.02089	+	0.303186i
$466$		0			0
$467$	2.88740e14	−	5.00112e14i	0.601539	−	1.04190i	−0.391049	−	0.920370i	$-0.627888\pi$
$467$	2.88740e14	−	5.00112e14i	0.601539	−	1.04190i	0.992588	−	0.121527i	$-0.0387790\pi$
$468$		0			0
$469$	−2.27113e14	−	4.15416e14i	−0.462158	−	0.845343i
$470$		0			0
$471$	−1.39587e14	−	1.32276e14i	−0.277480	−	0.262947i
$472$		0			0
$473$	1.28101e14	−	7.39592e13i	0.248781	−	0.143634i
$474$		0			0
$475$		−	5.64312e14i		−	1.07079i
$476$		0			0
$477$	1.86133e14	−	3.66584e14i	0.345121	−	0.679708i
$478$		0			0
$479$	−7.23020e13	−	1.25231e14i	−0.131010	−	0.226916i	0.793056	−	0.609149i	$-0.208489\pi$
$479$	−7.23020e13	−	1.25231e14i	−0.131010	−	0.226916i	−0.924066	+	0.382232i	$0.875155\pi$
$480$		0			0
$481$	1.06479e14	+	6.14759e13i	0.188568	+	0.108870i
$482$		0			0
$483$	−1.49683e14	+	5.66781e14i	−0.259097	+	0.981084i
$484$		0			0
$485$	2.79349e14	+	1.61282e14i	0.472680	+	0.272902i
$486$		0			0
$487$	2.84557e14	+	4.92868e14i	0.470718	+	0.815307i	0.999439	−	0.0334885i	$-0.0106617\pi$
$487$	2.84557e14	+	4.92868e14i	0.470718	+	0.815307i	−0.528721	+	0.848795i	$0.677328\pi$
$488$		0			0
$489$	−3.60964e14	−	8.63906e13i	−0.583802	−	0.139723i
$490$		0			0
$491$			1.87388e14i			0.296342i	0.988962	+	0.148171i	$0.0473386\pi$
$491$			1.87388e14i			0.296342i	−0.988962	+	0.148171i	$0.952661\pi$
$492$		0			0
$493$	9.45335e14	−	5.45789e14i	1.46193	−	0.844048i
$494$		0			0
$495$	−1.22611e14	−	1.88223e14i	−0.185439	−	0.284671i
$496$		0			0
$497$	2.15310e13	−	9.24532e14i	0.0318496	−	1.36761i
$498$		0			0
$499$	2.70886e14	−	4.69188e14i	0.391952	−	0.678881i	−0.600755	−	0.799433i	$-0.705133\pi$
$499$	2.70886e14	−	4.69188e14i	0.391952	−	0.678881i	0.992707	+	0.120552i	$0.0384665\pi$
$500$		0			0
$501$	−5.77699e13	−	1.94524e14i	−0.0817700	−	0.275338i
$502$		0			0
$503$	−9.42273e14			−1.30483			−0.652413	−	0.757863i	$-0.726243\pi$
$503$	−9.42273e14			−1.30483			−0.652413	+	0.757863i	$0.726243\pi$
$504$		0			0
$505$	−5.91648e14			−0.801606
$506$		0			0
$507$	1.54176e14	+	5.19143e14i	0.204396	+	0.688245i
$508$		0			0
$509$	−1.07535e14	+	1.86257e14i	−0.139509	+	0.241637i	−0.927311	−	0.374292i	$-0.877886\pi$
$509$	−1.07535e14	+	1.86257e14i	−0.139509	+	0.241637i	0.787802	+	0.615929i	$0.211219\pi$
$510$		0			0
$511$	3.66637e14	−	2.00444e14i	0.465502	−	0.254495i
$512$		0			0
$513$	4.09679e14	+	1.14916e15i	0.509095	+	1.42803i
$514$		0			0
$515$	−5.00696e14	+	2.89077e14i	−0.609023	+	0.351620i
$516$		0			0
$517$			2.36008e14i			0.281014i
$518$		0			0
$519$	−3.30198e14	−	7.90273e13i	−0.384906	−	0.0921205i
$520$		0			0
$521$	5.35404e14	+	9.27347e14i	0.611047	+	1.05836i	0.991064	+	0.133385i	$0.0425845\pi$
$521$	5.35404e14	+	9.27347e14i	0.611047	+	1.05836i	−0.380018	+	0.924979i	$0.624082\pi$
$522$		0			0
$523$	−7.19744e14	−	4.15545e14i	−0.804302	−	0.464364i	0.0406715	−	0.999173i	$-0.487050\pi$
$523$	−7.19744e14	−	4.15545e14i	−0.804302	−	0.464364i	−0.844973	+	0.534809i	$0.820384\pi$
$524$		0			0
$525$	−4.54758e14	+	4.58045e14i	−0.497627	+	0.501224i
$526$		0			0
$527$	2.40810e15	+	1.39032e15i	2.58057	+	1.48989i
$528$		0			0
$529$	1.41285e13	+	2.44712e13i	0.0148282	+	0.0256832i
$530$		0			0
$531$	1.15555e15	+	5.86730e14i	1.18787	+	0.603141i
$532$		0			0
$533$			6.78661e14i			0.683366i
$534$		0			0
$535$	6.60302e13	−	3.81226e13i	0.0651324	−	0.0376042i
$536$		0			0
$537$	6.17250e14	+	5.84921e14i	0.596489	+	0.565248i
$538$		0			0
$539$	−5.88201e14	+	3.04007e14i	−0.556914	+	0.287836i
$540$		0			0
$541$	5.67430e14	−	9.82817e14i	0.526414	−	0.911775i	−0.473113	−	0.881002i	$-0.656870\pi$
$541$	5.67430e14	−	9.82817e14i	0.526414	−	0.911775i	0.999526	−	0.0307734i	$-0.00979701\pi$
$542$		0			0
$543$	−7.17462e14	+	2.13073e14i	−0.652229	+	0.193700i
$544$		0			0
$545$	9.42093e14			0.839292
$546$		0			0
$547$	−5.77690e14			−0.504387			−0.252194	−	0.967677i	$-0.581152\pi$
$547$	−5.77690e14			−0.504387			−0.252194	+	0.967677i	$0.581152\pi$
$548$		0			0
$549$	−3.71834e14	+	2.00132e13i	−0.318201	+	0.0171265i
$550$		0			0
$551$	1.00604e15	−	1.74250e15i	0.843878	−	1.46164i
$552$		0			0
$553$	−6.59673e14	+	1.08352e15i	−0.542426	+	0.890939i
$554$		0			0
$555$	−1.89597e14	+	2.00076e14i	−0.152834	+	0.161281i
$556$		0			0
$557$	−1.69856e15	+	9.80664e14i	−1.34238	+	0.775026i	−0.987157	−	0.159753i	$-0.948930\pi$
$557$	−1.69856e15	+	9.80664e14i	−1.34238	+	0.775026i	−0.355228	+	0.934780i	$0.615597\pi$
$558$		0			0
$559$			3.14060e14i			0.243359i
$560$		0			0
$561$	−2.91209e14	+	1.21675e15i	−0.221263	+	0.924499i
$562$		0			0
$563$	1.44132e14	+	2.49644e14i	0.107390	+	0.186005i	0.914712	−	0.404106i	$-0.132417\pi$
$563$	1.44132e14	+	2.49644e14i	0.107390	+	0.186005i	−0.807322	+	0.590111i	$0.799084\pi$
$564$		0			0
$565$	−3.62527e14	−	2.09305e14i	−0.264895	−	0.152937i
$566$		0			0
$567$	5.93535e14	−	1.26291e15i	0.425343	−	0.905032i
$568$		0			0
$569$	2.12803e15	+	1.22862e15i	1.49575	+	0.863573i	0.999988	−	0.00488432i	$-0.00155473\pi$
$569$	2.12803e15	+	1.22862e15i	1.49575	+	0.863573i	0.495764	+	0.868457i	$0.334888\pi$
$570$		0			0
$571$	−1.25480e15	−	2.17338e15i	−0.865119	−	1.49843i	−0.866929	−	0.498432i	$-0.833909\pi$
$571$	−1.25480e15	−	2.17338e15i	−0.865119	−	1.49843i	0.00180981	−	0.999998i	$-0.499424\pi$
$572$		0			0
$573$	−3.48586e14	+	1.45649e15i	−0.235754	+	0.985048i
$574$		0			0
$575$			1.08021e15i			0.716695i
$576$		0			0
$577$	2.37213e15	−	1.36955e15i	1.54408	−	0.891477i	0.545509	−	0.838105i	$-0.316336\pi$
$577$	2.37213e15	−	1.36955e15i	1.54408	−	0.891477i	0.998575	−	0.0533726i	$-0.0169971\pi$
$578$		0			0
$579$	7.84573e14	−	8.27936e14i	0.501073	−	0.528768i
$580$		0			0
$581$	−2.41001e15	−	5.61257e13i	−1.51026	−	0.0351717i
$582$		0			0
$583$	3.88575e14	−	6.73031e14i	0.238945	−	0.413864i
$584$		0			0
$585$	4.76257e14	−	2.56335e13i	0.287398	−	0.0154686i
$586$		0			0
$587$	−3.92949e14			−0.232716			−0.116358	−	0.993207i	$-0.537122\pi$
$587$	−3.92949e14			−0.232716			−0.116358	+	0.993207i	$0.537122\pi$
$588$		0			0
$589$	5.12545e15			2.97919
$590$		0			0
$591$	1.14874e15	−	3.41155e14i	0.655377	−	0.194635i
$592$		0			0
$593$	1.76073e15	−	3.04968e15i	0.986035	−	1.70786i	0.348791	−	0.937200i	$-0.386592\pi$
$593$	1.76073e15	−	3.04968e15i	0.986035	−	1.70786i	0.637244	−	0.770662i	$-0.280075\pi$
$594$		0			0
$595$	−1.49445e15	−	3.48036e13i	−0.821559	−	0.0191329i
$596$		0			0
$597$	6.39355e14	+	6.05868e14i	0.345051	+	0.326979i
$598$		0			0
$599$	−2.43865e15	+	1.40795e15i	−1.29212	+	0.746003i	−0.979029	−	0.203721i	$-0.934696\pi$
$599$	−2.43865e15	+	1.40795e15i	−1.29212	+	0.746003i	−0.313087	+	0.949725i	$0.601363\pi$
$600$		0			0
$601$			1.64799e15i			0.857325i	0.903465	+	0.428662i	$0.141015\pi$
$601$			1.64799e15i			0.857325i	−0.903465	+	0.428662i	$0.858985\pi$
$602$		0			0
$603$	1.68173e15	+	8.53899e14i	0.859038	+	0.436176i
$604$		0			0
$605$	3.27917e14	+	5.67969e14i	0.164479	+	0.284885i
$606$		0			0
$607$	3.27564e14	+	1.89119e14i	0.161346	+	0.0931531i	0.578499	−	0.815683i	$-0.303639\pi$
$607$	3.27564e14	+	1.89119e14i	0.161346	+	0.0931531i	−0.417153	+	0.908836i	$0.636972\pi$
$608$		0			0
$609$	−2.22080e15	+	6.03643e14i	−1.07427	+	0.292001i
$610$		0			0
$611$	−4.33958e14	−	2.50546e14i	−0.206168	−	0.119031i
$612$		0			0
$613$	−1.04914e15	−	1.81717e15i	−0.489555	−	0.847934i	0.510373	−	0.859953i	$-0.329507\pi$
$613$	−1.04914e15	−	1.81717e15i	−0.489555	−	0.847934i	−0.999928	+	0.0120195i	$0.996174\pi$
$614$		0			0
$615$	−1.47967e15	−	3.54133e14i	−0.678191	−	0.162313i
$616$		0			0
$617$			1.27139e15i			0.572413i	0.958168	+	0.286206i	$0.0923943\pi$
$617$			1.27139e15i			0.572413i	−0.958168	+	0.286206i	$0.907606\pi$
$618$		0			0
$619$	−5.65218e14	+	3.26329e14i	−0.249987	+	0.144330i	−0.619758	−	0.784793i	$-0.712769\pi$
$619$	−5.65218e14	+	3.26329e14i	−0.249987	+	0.144330i	0.369771	+	0.929123i	$0.379436\pi$
$620$		0			0
$621$	−7.84210e14	−	2.19973e15i	−0.340744	−	0.955798i
$622$		0			0
$623$	2.13073e15	−	3.49974e15i	0.909587	−	1.49400i
$624$		0			0
$625$	−2.44566e14	+	4.23601e14i	−0.102578	+	0.177671i
$626$		0			0
$627$	6.56539e14	+	2.21071e15i	0.270576	+	0.911087i
$628$		0			0
$629$	1.53518e15			0.621701
$630$		0			0
$631$	−1.80030e14			−0.0716445			−0.0358222	−	0.999358i	$-0.511405\pi$
$631$	−1.80030e14			−0.0716445			−0.0358222	+	0.999358i	$0.511405\pi$
$632$		0			0
$633$	−3.15118e14	−	1.06107e15i	−0.123240	−	0.414977i
$634$		0			0
$635$	7.27838e14	−	1.26065e15i	0.279756	−	0.484551i
$636$		0			0
$637$	6.54430e13	−	1.40429e15i	0.0247227	−	0.530504i
$638$		0			0
$639$	2.01088e15	+	3.08693e15i	0.746674	+	1.14623i
$640$		0			0
$641$	−3.85669e15	+	2.22666e15i	−1.40765	+	0.812710i	−0.995162	−	0.0982515i	$-0.968675\pi$
$641$	−3.85669e15	+	2.22666e15i	−1.40765	+	0.812710i	−0.412493	+	0.910961i	$0.635342\pi$
$642$		0			0
$643$		−	1.11892e15i		−	0.401456i	−0.979647	−	0.200728i	$-0.935669\pi$
$643$		−	1.11892e15i		−	0.401456i	0.979647	−	0.200728i	$-0.0643307\pi$
$644$		0			0
$645$	−6.84738e14	−	1.63880e14i	−0.241516	−	0.0578027i
$646$		0			0
$647$	−1.40207e15	−	2.42846e15i	−0.486180	−	0.842088i	0.513694	−	0.857973i	$-0.328277\pi$
$647$	−1.40207e15	−	2.42846e15i	−0.486180	−	0.842088i	−0.999874	+	0.0158856i	$0.994943\pi$
$648$		0			0
$649$	2.12154e15	+	1.22487e15i	0.723278	+	0.417585i
$650$		0			0
$651$	−4.16026e15	−	4.13040e15i	−1.39452	−	1.38451i
$652$		0			0
$653$	−3.97213e13	−	2.29331e13i	−0.0130919	−	0.00755859i	0.493440	−	0.869780i	$-0.335739\pi$
$653$	−3.97213e13	−	2.29331e13i	−0.0130919	−	0.00755859i	−0.506532	+	0.862221i	$0.669073\pi$
$654$		0			0
$655$	3.14696e14	+	5.45069e14i	0.101991	+	0.176654i
$656$		0			0
$657$	−7.53632e14	+	1.48426e15i	−0.240187	+	0.473043i
$658$		0			0
$659$		−	3.31415e15i		−	1.03873i	−0.854553	−	0.519364i	$-0.826169\pi$
$659$		−	3.31415e15i		−	1.03873i	0.854553	−	0.519364i	$-0.173831\pi$
$660$		0			0
$661$	−4.85628e15	+	2.80378e15i	−1.49691	+	0.864242i	−0.999994	−	0.00355565i	$-0.998868\pi$
$661$	−4.85628e15	+	2.80378e15i	−1.49691	+	0.864242i	−0.496918	+	0.867798i	$0.665535\pi$
$662$		0			0
$663$	−1.92815e15	−	1.82716e15i	−0.584542	−	0.553927i
$664$		0			0
$665$	−2.41769e15	+	1.32178e15i	−0.720912	+	0.394131i
$666$		0			0
$667$	−1.92576e15	+	3.33551e15i	−0.564819	+	0.978296i
$668$		0			0
$669$	−5.82734e15	+	1.73061e15i	−1.68123	+	0.499293i
$670$		0			0
$671$	−7.03885e14			−0.199769
$672$		0			0
$673$	−5.01825e15			−1.40110			−0.700550	−	0.713603i	$-0.747062\pi$
$673$	−5.01825e15			−1.40110			−0.700550	+	0.713603i	$0.747062\pi$
$674$		0			0
$675$	4.63237e14	−	2.52927e15i	0.127242	−	0.694743i
$676$		0			0
$677$	1.12390e15	−	1.94664e15i	0.303730	−	0.526076i	−0.673248	−	0.739417i	$-0.735101\pi$
$677$	1.12390e15	−	1.94664e15i	0.303730	−	0.526076i	0.976978	+	0.213341i	$0.0684345\pi$
$678$		0			0
$679$	−8.81845e13	+	3.78660e15i	−0.0234481	+	1.00685i
$680$		0			0
$681$	5.11199e15	−	5.39453e15i	1.33746	−	1.41138i
$682$		0			0
$683$	2.20717e15	−	1.27431e15i	0.568227	−	0.328066i	−0.188214	−	0.982128i	$-0.560270\pi$
$683$	2.20717e15	−	1.27431e15i	0.568227	−	0.328066i	0.756441	+	0.654062i	$0.226936\pi$
$684$		0			0
$685$			2.40693e15i			0.609768i
$686$		0			0
$687$	2.78535e14	−	1.16380e15i	0.0694414	−	0.290146i
$688$		0			0
$689$	8.25021e14	+	1.42898e15i	0.202423	+	0.350606i
$690$		0			0
$691$	−5.81448e15	−	3.35699e15i	−1.40405	−	0.810627i	−0.409241	−	0.912426i	$-0.634207\pi$
$691$	−5.81448e15	−	3.35699e15i	−1.40405	−	0.810627i	−0.994805	+	0.101800i	$0.967540\pi$
$692$		0			0
$693$	1.24862e15	−	2.32348e15i	0.296755	−	0.552213i
$694$		0			0
$695$	1.08751e15	+	6.27872e14i	0.254399	+	0.146877i
$696$		0			0
$697$	4.23690e15	+	7.33852e15i	0.975591	+	1.68977i
$698$		0			0
$699$	8.01714e14	−	3.34979e15i	0.181717	−	0.759264i
$700$		0			0
$701$			2.85288e15i			0.636552i	0.947998	+	0.318276i	$0.103104\pi$
$701$			2.85288e15i			0.636552i	−0.947998	+	0.318276i	$0.896896\pi$
$702$		0			0
$703$	2.45064e15	−	1.41488e15i	0.538300	−	0.310788i
$704$		0			0
$705$	7.72703e14	−	8.15411e14i	0.167098	−	0.176334i
$706$		0			0
$707$	−3.33262e15	−	6.09577e15i	−0.709544	−	1.29784i
$708$		0			0
$709$	1.64130e15	−	2.84281e15i	0.344059	−	0.595927i	−0.641124	−	0.767438i	$-0.721531\pi$
$709$	1.64130e15	−	2.84281e15i	0.344059	−	0.595927i	0.985182	+	0.171511i	$0.0548648\pi$
$710$		0			0
$711$	−2.71604e14	−	5.04625e15i	−0.0560600	−	1.04156i
$712$		0			0
$713$	−9.81115e15			−1.99401
$714$		0			0
$715$	9.01557e14			0.180431
$716$		0			0
$717$	4.11519e14	−	1.22213e14i	0.0811026	−	0.0240859i
$718$		0			0
$719$	4.51079e15	−	7.81291e15i	0.875475	−	1.51637i	0.0192181	−	0.999815i	$-0.493882\pi$
$719$	4.51079e15	−	7.81291e15i	0.875475	−	1.51637i	0.856256	−	0.516551i	$-0.172784\pi$
$720$		0			0
$721$	−5.79867e15	−	3.53037e15i	−1.10837	−	0.674803i
$722$		0			0
$723$	3.51148e15	+	3.32756e15i	0.661043	+	0.626420i
$724$		0			0
$725$	−3.67258e15	+	2.12037e15i	−0.680946	+	0.393144i
$726$		0			0
$727$			5.67859e15i			1.03705i	0.855061	+	0.518527i	$0.173519\pi$
$727$			5.67859e15i			1.03705i	−0.855061	+	0.518527i	$0.826481\pi$
$728$		0			0
$729$	8.92865e14	+	5.48689e15i	0.160614	+	0.987017i
$730$		0			0
$731$	1.96068e15	+	3.39600e15i	0.347426	+	0.601759i
$732$		0			0
$733$	4.26385e15	+	2.46174e15i	0.744270	+	0.429704i	0.823620	−	0.567143i	$-0.191951\pi$
$733$	4.26385e15	+	2.46174e15i	0.744270	+	0.429704i	−0.0793501	+	0.996847i	$0.525284\pi$
$734$		0			0
$735$	3.02758e15	+	8.75456e14i	0.520614	+	0.150541i
$736$		0			0
$737$	3.08759e15	+	1.78262e15i	0.523056	+	0.301986i
$738$		0			0
$739$	2.81543e15	+	4.87647e15i	0.469895	+	0.813882i	0.999407	−	0.0344201i	$-0.0109584\pi$
$739$	2.81543e15	+	4.87647e15i	0.469895	+	0.813882i	−0.529512	+	0.848302i	$0.677625\pi$
$740$		0			0
$741$	−4.76191e15	−	1.13968e15i	−0.783034	−	0.187406i
$742$		0			0
$743$			3.15277e14i			0.0510803i	0.999674	+	0.0255401i	$0.00813056\pi$
$743$			3.15277e14i			0.0510803i	−0.999674	+	0.0255401i	$0.991869\pi$
$744$		0			0
$745$	2.84106e15	−	1.64029e15i	0.453546	−	0.261855i
$746$		0			0
$747$	8.04682e15	−	5.24183e15i	1.26579	−	0.824555i
$748$		0			0
$749$	7.64711e14	+	4.65575e14i	0.118535	+	0.0721672i
$750$		0			0
$751$	−7.57015e14	+	1.31119e15i	−0.115634	+	0.200284i	−0.918033	−	0.396504i	$-0.870223\pi$
$751$	−7.57015e14	+	1.31119e15i	−0.115634	+	0.200284i	0.802399	+	0.596788i	$0.203557\pi$
$752$		0			0
$753$	−6.74715e14	−	2.27191e15i	−0.101566	−	0.341995i
$754$		0			0
$755$	−5.28837e15			−0.784538
$756$		0			0
$757$	−1.57255e15			−0.229920			−0.114960	−	0.993370i	$-0.536674\pi$
$757$	−1.57255e15			−0.229920			−0.114960	+	0.993370i	$0.536674\pi$
$758$		0			0
$759$	−1.25675e15	−	4.23175e15i	−0.181100	−	0.609804i
$760$		0			0
$761$	1.73874e15	−	3.01159e15i	0.246956	−	0.427740i	−0.715724	−	0.698383i	$-0.753903\pi$
$761$	1.73874e15	−	3.01159e15i	0.246956	−	0.427740i	0.962680	+	0.270644i	$0.0872365\pi$
$762$		0			0
$763$	5.30660e15	+	9.70641e15i	0.742901	+	1.35886i
$764$		0			0
$765$	4.98985e15	−	3.25046e15i	0.688572	−	0.448547i
$766$		0			0
$767$	−4.50445e15	+	2.60064e15i	−0.612727	+	0.353758i
$768$		0			0
$769$		−	7.06918e15i		−	0.947925i	−0.880545	−	0.473963i	$-0.842823\pi$
$769$		−	7.06918e15i		−	0.947925i	0.880545	−	0.473963i	$-0.157177\pi$
$770$		0			0
$771$	3.41019e15	+	8.16170e14i	0.450795	+	0.107890i
$772$		0			0
$773$	6.09457e15	+	1.05561e16i	0.794247	+	1.37568i	0.923316	+	0.384041i	$0.125468\pi$
$773$	6.09457e15	+	1.05561e16i	0.794247	+	1.37568i	−0.129069	+	0.991636i	$0.541199\pi$
$774$		0			0
$775$	−9.35535e15	−	5.40131e15i	−1.20199	−	0.693969i
$776$		0			0
$777$	−3.12934e15	−	8.26438e14i	−0.396404	−	0.104687i
$778$		0			0
$779$	1.35269e16	+	7.80973e15i	1.68943	+	0.975394i
$780$		0			0
$781$	3.48200e15	+	6.03099e15i	0.428793	+	0.742691i
$782$		0			0
$783$	5.93949e15	−	6.98412e15i	0.721206	−	0.848052i
$784$		0			0
$785$			1.73027e15i			0.207172i
$786$		0			0
$787$	−6.79026e15	+	3.92036e15i	−0.801724	+	0.462876i	−0.844074	−	0.536227i	$-0.819849\pi$
$787$	−6.79026e15	+	3.92036e15i	−0.801724	+	0.462876i	0.0423494	+	0.999103i	$0.486516\pi$
$788$		0			0
$789$	−2.99325e15	−	2.83648e15i	−0.348514	−	0.330260i
$790$		0			0
$791$	1.14442e14	−	4.91409e15i	0.0131406	−	0.564251i
$792$		0			0
$793$	7.47243e14	−	1.29426e15i	0.0846173	−	0.146562i
$794$		0			0
$795$	−3.54607e15	+	1.05312e15i	−0.396031	+	0.117614i
$796$		0			0
$797$	8.00214e15			0.881425			0.440712	−	0.897648i	$-0.354726\pi$
$797$	8.00214e15			0.881425			0.440712	+	0.897648i	$0.354726\pi$
$798$		0			0
$799$	−6.25665e15			−0.679727
$800$		0			0
$801$	8.77275e14	+	1.62993e16i	0.0940063	+	1.74659i
$802$		0			0
$803$	−1.57330e15	+	2.72503e15i	−0.166294	+	0.288029i
$804$		0			0
$805$	4.62796e15	−	2.53016e15i	0.482517	−	0.263797i
$806$		0			0
$807$	7.28214e15	−	7.68462e15i	0.748953	−	0.790348i
$808$		0			0
$809$	−3.36452e15	+	1.94251e15i	−0.341355	+	0.197081i	−0.660871	−	0.750499i	$-0.729813\pi$
$809$	−3.36452e15	+	1.94251e15i	−0.341355	+	0.197081i	0.319516	+	0.947581i	$0.396480\pi$
$810$		0			0
$811$		−	8.79138e15i		−	0.879918i	−0.898018	−	0.439959i	$-0.854993\pi$
$811$		−	8.79138e15i		−	0.879918i	0.898018	−	0.439959i	$-0.145007\pi$
$812$		0			0
$813$	−2.71625e15	+	1.13493e16i	−0.268208	+	1.12065i
$814$		0			0
$815$	1.66975e15	+	2.89208e15i	0.162661	+	0.281737i
$816$		0			0
$817$	6.25974e15	+	3.61406e15i	0.601638	+	0.347356i
$818$		0			0
$819$	2.94675e15	+	4.76250e15i	0.279436	+	0.451620i
$820$		0			0
$821$	5.31044e15	+	3.06599e15i	0.496871	+	0.286868i	0.727420	−	0.686192i	$-0.240719\pi$
$821$	5.31044e15	+	3.06599e15i	0.496871	+	0.286868i	−0.230550	+	0.973061i	$0.574052\pi$
$822$		0			0
$823$	2.02457e15	+	3.50666e15i	0.186911	+	0.323739i	0.944219	−	0.329319i	$-0.106819\pi$
$823$	2.02457e15	+	3.50666e15i	0.186911	+	0.323739i	−0.757308	+	0.653058i	$0.773486\pi$
$824$		0			0
$825$	1.13133e15	−	4.72702e15i	0.103061	−	0.430617i
$826$		0			0
$827$			1.92695e16i			1.73217i	0.499898	+	0.866085i	$0.333371\pi$
$827$			1.92695e16i			1.73217i	−0.499898	+	0.866085i	$0.666629\pi$
$828$		0			0
$829$	−1.25146e16	+	7.22532e15i	−1.11011	+	0.640925i	−0.938859	−	0.344302i	$-0.888116\pi$
$829$	−1.25146e16	+	7.22532e15i	−1.11011	+	0.640925i	−0.171255	+	0.985227i	$0.554782\pi$
$830$		0			0
$831$	−9.49221e15	+	1.00168e16i	−0.830925	+	0.876850i
$832$		0			0
$833$	−8.05933e15	−	1.55934e16i	−0.696228	−	1.34708i
$834$		0			0
$835$	−9.12889e14	+	1.58117e15i	−0.0778291	+	0.134804i
$836$		0			0
$837$	2.29724e16	+	4.20741e15i	1.93293	+	0.354018i
$838$		0			0
$839$	8.12961e15			0.675117			0.337559	−	0.941305i	$-0.390399\pi$
$839$	8.12961e15			0.675117			0.337559	+	0.941305i	$0.390399\pi$
$840$		0			0
$841$	−2.91994e15			−0.239329
$842$		0			0
$843$	5.52989e15	−	1.64227e15i	0.447368	−	0.132860i
$844$		0			0
$845$	2.43631e15	−	4.21980e15i	0.194545	−	0.336962i
$846$		0			0
$847$	−4.00472e15	+	6.57778e15i	−0.315655	+	0.518467i
$848$		0			0
$849$	−1.13568e16	−	1.07619e16i	−0.883614	−	0.837334i
$850$		0			0
$851$	−4.69102e15	+	2.70836e15i	−0.360292	+	0.208015i
$852$		0			0
$853$			4.68796e15i			0.355439i	0.984081	+	0.177719i	$0.0568719\pi$
$853$			4.68796e15i			0.355439i	−0.984081	+	0.177719i	$0.943128\pi$
$854$		0			0
$855$	4.96964e15	−	9.78758e15i	0.371973	−	0.732591i
$856$		0			0
$857$	5.47192e15	+	9.47765e15i	0.404339	+	0.700335i	0.994244	−	0.107137i	$-0.0341683\pi$
$857$	5.47192e15	+	9.47765e15i	0.404339	+	0.700335i	−0.589905	+	0.807472i	$0.700835\pi$
$858$		0			0
$859$	−1.71174e15	−	9.88273e14i	−0.124875	−	0.0720965i	0.436261	−	0.899820i	$-0.356302\pi$
$859$	−1.71174e15	−	9.88273e14i	−0.124875	−	0.0720965i	−0.561136	+	0.827723i	$0.689636\pi$
$860$		0			0
$861$	−4.68600e15	−	1.72398e16i	−0.337509	−	1.24170i
$862$		0			0
$863$	5.21383e15	+	3.01021e15i	0.370764	+	0.214061i	0.673792	−	0.738921i	$-0.264664\pi$
$863$	5.21383e15	+	3.01021e15i	0.370764	+	0.214061i	−0.303028	+	0.952982i	$0.597998\pi$
$864$		0			0
$865$	1.52743e15	+	2.64558e15i	0.107244	+	0.185752i
$866$		0			0
$867$	−1.82281e16	−	4.36257e15i	−1.26367	−	0.302439i
$868$		0			0
$869$		−	9.55258e15i		−	0.653901i
$870$		0			0
$871$	−6.55556e15	+	3.78485e15i	−0.443108	+	0.255829i
$872$		0			0
$873$	−8.23594e15	−	1.26431e16i	−0.549711	−	0.843872i
$874$		0			0
$875$	1.40260e16	+	3.26646e14i	0.924464	+	0.0215294i
$876$		0			0
$877$	−2.79145e15	+	4.83493e15i	−0.181690	+	0.314697i	−0.942456	−	0.334330i	$-0.891490\pi$
$877$	−2.79145e15	+	4.83493e15i	−0.181690	+	0.314697i	0.760766	+	0.649026i	$0.224823\pi$
$878$		0			0
$879$	1.27609e15	+	4.29687e15i	0.0820242	+	0.276193i
$880$		0			0
$881$	−1.28479e16			−0.815576			−0.407788	−	0.913077i	$-0.633700\pi$
$881$	−1.28479e16			−0.815576			−0.407788	+	0.913077i	$0.633700\pi$
$882$		0			0
$883$	1.61249e16			1.01091			0.505455	−	0.862853i	$-0.331325\pi$
$883$	1.61249e16			1.01091			0.505455	+	0.862853i	$0.331325\pi$
$884$		0			0
$885$	−3.31965e15	−	1.11780e16i	−0.205544	−	0.692111i
$886$		0			0
$887$	2.84974e15	−	4.93589e15i	0.174271	−	0.301846i	−0.765638	−	0.643272i	$-0.777577\pi$
$887$	2.84974e15	−	4.93589e15i	0.174271	−	0.301846i	0.939909	+	0.341426i	$0.110910\pi$
$888$		0			0
$889$	1.70883e16	+	3.97961e14i	1.03214	+	0.0240371i
$890$		0			0
$891$	1.12789e15	+	1.04474e16i	0.0672881	+	0.623278i
$892$		0			0
$893$	−9.98759e15	+	5.76634e15i	−0.588542	+	0.339795i
$894$		0			0
$895$		−	7.65120e15i		−	0.445351i
$896$		0			0
$897$	9.11526e15	+	2.18158e15i	0.524096	+	0.125433i
$898$		0			0
$899$	−1.92585e16	−	3.33567e16i	−1.09382	−	1.89455i
$900$		0			0
$901$	1.78423e16	+	1.03012e16i	1.00107	+	0.577968i
$902$		0			0
$903$	−2.16851e15	−	7.97797e15i	−0.120193	−	0.442191i
$904$		0			0
$905$	5.83183e15	+	3.36701e15i	0.319328	+	0.184364i
$906$		0			0
$907$	−7.95495e15	−	1.37784e16i	−0.430326	−	0.745346i	0.566575	−	0.824010i	$-0.308268\pi$
$907$	−7.95495e15	−	1.37784e16i	−0.430326	−	0.745346i	−0.996901	+	0.0786636i	$0.974935\pi$
$908$		0			0
$909$	2.46776e16	+	1.25300e16i	1.31887	+	0.669653i
$910$		0			0
$911$			4.75922e15i			0.251295i	0.992075	+	0.125648i	$0.0401009\pi$
$911$			4.75922e15i			0.251295i	−0.992075	+	0.125648i	$0.959899\pi$
$912$		0			0
$913$	1.57212e16	−	9.07665e15i	0.820156	−	0.473518i
$914$		0			0
$915$	2.43193e15	+	2.30456e15i	0.125353	+	0.118788i
$916$		0			0
$917$	−3.84325e15	+	6.31257e15i	−0.195734	+	0.321495i
$918$		0			0
$919$	3.44810e14	−	5.97229e14i	0.0173518	−	0.0300542i	−0.857219	−	0.514952i	$-0.827810\pi$
$919$	3.44810e14	−	5.97229e14i	0.0173518	−	0.0300542i	0.874571	+	0.484898i	$0.161143\pi$
$920$		0			0
$921$	−2.49708e16	+	7.41586e15i	−1.24167	+	0.368752i
$922$		0			0
$923$	−1.47859e16			−0.726506
$924$		0			0
$925$	−5.96411e15			−0.289578
$926$		0			0
$927$	2.70060e16	−	1.45354e15i	1.29575	−	0.0697412i
$928$		0			0
$929$	1.22195e16	−	2.11648e16i	0.579385	−	1.00352i	−0.416165	−	0.909289i	$-0.636626\pi$
$929$	1.22195e16	−	2.11648e16i	0.579385	−	1.00352i	0.995550	−	0.0942352i	$-0.0300406\pi$
$930$		0			0
$931$	−2.72367e16	−	1.74643e16i	−1.27623	−	0.818328i
$932$		0			0
$933$	−2.75308e16	+	2.90525e16i	−1.27488	+	1.34535i
$934$		0			0
$935$	9.74875e15	−	5.62844e15i	0.446154	−	0.257587i
$936$		0			0
$937$			1.33972e16i			0.605965i	0.952996	+	0.302983i	$0.0979824\pi$
$937$			1.33972e16i			0.605965i	−0.952996	+	0.302983i	$0.902018\pi$
$938$		0			0
$939$	1.52514e15	−	6.37246e15i	0.0681787	−	0.284870i
$940$		0			0
$941$	−1.17399e16	−	2.03341e16i	−0.518707	−	0.898426i	−0.999764	−	0.0217370i	$-0.993080\pi$
$941$	−1.17399e16	−	2.03341e16i	−0.518707	−	0.898426i	0.481057	−	0.876689i	$-0.340253\pi$
$942$		0			0
$943$	−2.58932e16	−	1.49494e16i	−1.13076	−	0.652845i
$944$		0			0
$945$	−1.19212e16	+	3.93961e15i	−0.514572	+	0.170051i
$946$		0			0
$947$	−3.58370e16	−	2.06905e16i	−1.52900	−	0.882768i	−0.999404	−	0.0345179i	$-0.989010\pi$
$947$	−3.58370e16	−	2.06905e16i	−1.52900	−	0.882768i	−0.529595	−	0.848250i	$-0.677656\pi$
$948$		0			0
$949$	−3.34042e15	−	5.78578e15i	−0.140876	−	0.244005i
$950$		0			0
$951$	−8.50197e15	+	3.55236e16i	−0.354427	+	1.48090i
$952$		0			0
$953$		−	3.13932e16i		−	1.29367i	−0.762628	−	0.646837i	$-0.776091\pi$
$953$		−	3.13932e16i		−	1.29367i	0.762628	−	0.646837i	$-0.223909\pi$
$954$		0			0
$955$	1.16696e16	−	6.73742e15i	0.475375	−	0.274458i
$956$		0			0
$957$	1.19205e16	−	1.25794e16i	0.480043	−	0.506575i
$958$		0			0
$959$	−2.47986e16	+	1.35577e16i	−0.987245	+	0.539738i
$960$		0			0
$961$	3.63539e16	−	6.29669e16i	1.43078	−	2.47818i
$962$		0			0
$963$	−3.56147e15	+	1.91689e14i	−0.138575	+	0.00745851i
$964$		0			0
$965$	−1.02628e16			−0.394789
$966$		0			0
$967$	−1.80324e16			−0.685817			−0.342908	−	0.939369i	$-0.611412\pi$
$967$	−1.80324e16			−0.685817			−0.342908	+	0.939369i	$0.611412\pi$
$968$		0			0
$969$	−5.86066e16	+	1.74051e16i	−2.20377	+	0.654478i
$970$		0			0
$971$	−2.37428e16	+	4.11237e16i	−0.882725	+	1.52892i	−0.0344260	+	0.999407i	$0.510960\pi$
$971$	−2.37428e16	+	4.11237e16i	−0.882725	+	1.52892i	−0.848299	+	0.529517i	$0.822373\pi$
$972$		0			0
$973$	−3.43303e14	+	1.47413e16i	−0.0126199	+	0.541893i
$974$		0			0
$975$	7.49076e15	+	7.09843e15i	0.272271	+	0.258010i
$976$		0			0
$977$	4.59389e16	−	2.65229e16i	1.65105	−	0.953235i	0.674412	−	0.738355i	$-0.264397\pi$
$977$	4.59389e16	−	2.65229e16i	1.65105	−	0.953235i	0.976640	−	0.214880i	$-0.0689361\pi$
$978$		0			0
$979$			3.08546e16i			1.09652i
$980$		0			0
$981$	−3.92945e16	−	1.99518e16i	−1.38087	−	0.701136i
$982$		0			0
$983$	2.24638e16	+	3.89084e16i	0.780619	+	1.35207i	0.931582	+	0.363532i	$0.118429\pi$
$983$	2.24638e16	+	3.89084e16i	0.780619	+	1.35207i	−0.150963	+	0.988539i	$0.548237\pi$
$984$		0			0
$985$	−9.33744e15	−	5.39097e15i	−0.320870	−	0.185254i
$986$		0			0
$987$	1.27537e16	+	3.36815e15i	0.433402	+	0.114458i
$988$		0			0
$989$	−1.19824e16	−	6.91806e15i	−0.402685	−	0.232490i
$990$		0			0
$991$	−1.11539e16	−	1.93191e16i	−0.370699	−	0.642069i	0.618974	−	0.785411i	$-0.287549\pi$
$991$	−1.11539e16	−	1.93191e16i	−0.370699	−	0.642069i	−0.989673	+	0.143342i	$0.954215\pi$
$992$		0			0
$993$	−1.97706e15	−	4.73175e14i	−0.0649829	−	0.0155525i
$994$		0			0
$995$		−	7.92519e15i		−	0.257622i
$996$		0			0
$997$	4.57109e15	−	2.63912e15i	0.146959	−	0.0848469i	−0.424717	−	0.905326i	$-0.639627\pi$
$997$	4.57109e15	−	2.63912e15i	0.146959	−	0.0848469i	0.571677	+	0.820479i	$0.306293\pi$
$998$		0			0
$999$	1.21453e16	−	4.32983e15i	0.386187	−	0.137677i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.k.b.5.18	yes	56
3.2	odd	2	inner	84.12.k.b.5.8	✓	56
7.3	odd	6	inner	84.12.k.b.17.8	yes	56
21.17	even	6	inner	84.12.k.b.17.18	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.k.b.5.8	✓	56	3.2	odd	2	inner
84.12.k.b.5.18	yes	56	1.1	even	1	trivial
84.12.k.b.17.8	yes	56	7.3	odd	6	inner
84.12.k.b.17.18	yes	56	21.17	even	6	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:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	84.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(119.823 + 403.472i) q^{3} +(1893.47 - 3279.58i) q^{5} +(44455.1 + 1035.29i) q^{7} +(-148432. + 96690.7i) q^{9} +O(q^{10})\)	\(q+(119.823 + 403.472i) q^{3} +(1893.47 - 3279.58i) q^{5} +(44455.1 + 1035.29i) q^{7} +(-148432. + 96690.7i) q^{9} +(-289993. + 167428. i) q^{11} -710965. i q^{13} +(1.55010e6 + 370990. i) q^{15} +(-4.43857e6 - 7.68782e6i) q^{17} +(-1.41707e7 - 8.18146e6i) q^{19} +(4.90905e6 + 1.80604e7i) q^{21} +(2.71256e7 + 1.56610e7i) q^{23} +(1.72436e7 + 2.98668e7i) q^{25} +(-5.67976e7 - 4.83022e7i) q^{27} +1.22965e8i q^{29} +(-2.71270e8 + 1.56618e8i) q^{31} +(-1.02300e8 - 9.69423e7i) q^{33} +(8.75696e7 - 1.43834e8i) q^{35} +(-8.64684e7 + 1.49768e8i) q^{37} +(2.86854e8 - 8.51902e7i) q^{39} -9.54564e8 q^{41} -4.41738e8 q^{43} +(3.60546e7 + 6.69875e8i) q^{45} +(3.52402e8 - 6.10379e8i) q^{47} +(1.97518e9 + 9.20482e7i) q^{49} +(2.56997e9 - 2.71202e9i) q^{51} +(-2.00992e9 + 1.16042e9i) q^{53} +1.26808e9i q^{55} +(1.60300e9 - 6.69781e9i) q^{57} +(-3.65791e9 - 6.33569e9i) q^{59} +(1.82043e9 + 1.05103e9i) q^{61} +(-6.69865e9 + 4.14472e9i) q^{63} +(-2.33167e9 - 1.34619e9i) q^{65} +(-5.32355e9 - 9.22065e9i) q^{67} +(-3.06848e9 + 1.28210e10i) q^{69} -2.07970e10i q^{71} +(8.13793e9 - 4.69844e9i) q^{73} +(-9.98423e9 + 1.05361e10i) q^{75} +(-1.30650e10 + 7.14279e9i) q^{77} +(-1.42637e10 + 2.47055e10i) q^{79} +(1.26829e10 - 2.87039e10i) q^{81} -5.42123e10 q^{83} -3.36171e10 q^{85} +(-4.96130e10 + 1.47341e10i) q^{87} +(4.60716e10 - 7.97983e10i) q^{89} +(7.36058e8 - 3.16060e10i) q^{91} +(-9.56952e10 - 9.06832e10i) q^{93} +(-5.36636e10 + 3.09827e10i) q^{95} +8.51782e10i q^{97} +(2.68555e10 - 5.28913e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 729 q^{3} - 27194 q^{7} - 172347 q^{9}+O(q^{10}) \) 56 * q - 729 * q^3 - 27194 * q^7 - 172347 * q^9	\( 56 q - 729 q^{3} - 27194 q^{7} - 172347 q^{9} - 4853058 q^{15} + 28700520 q^{19} - 11325429 q^{21} - 316601194 q^{25} - 1368416388 q^{31} + 40874949 q^{33} - 87435712 q^{37} + 1177474410 q^{39} - 3055078348 q^{43} + 4109921793 q^{45} - 742582522 q^{49} - 694793715 q^{51} + 14605100370 q^{57} + 72584834058 q^{61} - 7310837811 q^{63} + 6131679148 q^{67} - 74402605464 q^{73} - 161115157854 q^{75} + 52181713528 q^{79} + 44948282337 q^{81} + 4658488716 q^{85} + 243101263104 q^{87} - 85311757146 q^{91} - 256628211777 q^{93} + 157345775874 q^{99}+O(q^{100}) \) 56 * q - 729 * q^3 - 27194 * q^7 - 172347 * q^9 - 4853058 * q^15 + 28700520 * q^19 - 11325429 * q^21 - 316601194 * q^25 - 1368416388 * q^31 + 40874949 * q^33 - 87435712 * q^37 + 1177474410 * q^39 - 3055078348 * q^43 + 4109921793 * q^45 - 742582522 * q^49 - 694793715 * q^51 + 14605100370 * q^57 + 72584834058 * q^61 - 7310837811 * q^63 + 6131679148 * q^67 - 74402605464 * q^73 - 161115157854 * q^75 + 52181713528 * q^79 + 44948282337 * q^81 + 4658488716 * q^85 + 243101263104 * q^87 - 85311757146 * q^91 - 256628211777 * q^93 + 157345775874 * q^99

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