LMFDB - Embedded newform 84.12.k.b.5.8

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			5.8
Character	$\chi$	$=$	84.5
Dual form			84.12.k.b.17.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-289.505 - 305.506i) q^{3} +(-1893.47 + 3279.58i) q^{5} +(44455.1 + 1035.29i) q^{7} +(-9520.78 + 176891. i) q^{9} +O(q^{10})$	\(q+(-289.505 - 305.506i) q^{3} +(-1893.47 + 3279.58i) q^{5} +(44455.1 + 1035.29i) q^{7} +(-9520.78 + 176891. i) q^{9} +(289993. - 167428. i) q^{11} -710965. i q^{13} +(1.55010e6 - 370990. i) q^{15} +(4.43857e6 + 7.68782e6i) q^{17} +(-1.41707e7 - 8.18146e6i) q^{19} +(-1.25537e7 - 1.38810e7i) q^{21} +(-2.71256e7 - 1.56610e7i) q^{23} +(1.72436e7 + 2.98668e7i) q^{25} +(5.67976e7 - 4.83022e7i) q^{27} -1.22965e8i q^{29} +(-2.71270e8 + 1.56618e8i) q^{31} +(-1.35105e8 - 4.01235e7i) q^{33} +(-8.75696e7 + 1.43834e8i) q^{35} +(-8.64684e7 + 1.49768e8i) q^{37} +(-2.17204e8 + 2.05828e8i) q^{39} +9.54564e8 q^{41} -4.41738e8 q^{43} +(-5.62101e8 - 3.66161e8i) q^{45} +(-3.52402e8 + 6.10379e8i) q^{47} +(1.97518e9 + 9.20482e7i) q^{49} +(1.06369e9 - 3.58167e9i) q^{51} +(2.00992e9 - 1.16042e9i) q^{53} +1.26808e9i q^{55} +(1.60300e9 + 6.69781e9i) q^{57} +(3.65791e9 + 6.33569e9i) q^{59} +(1.82043e9 + 1.05103e9i) q^{61} +(-6.06382e8 + 7.85385e9i) q^{63} +(2.33167e9 + 1.34619e9i) q^{65} +(-5.32355e9 - 9.22065e9i) q^{67} +(3.06848e9 + 1.28210e10i) q^{69} +2.07970e10i q^{71} +(8.13793e9 - 4.69844e9i) q^{73} +(4.13238e9 - 1.39146e10i) q^{75} +(1.30650e10 - 7.14279e9i) q^{77} +(-1.42637e10 + 2.47055e10i) q^{79} +(-3.11998e10 - 3.36828e9i) q^{81} +5.42123e10 q^{83} -3.36171e10 q^{85} +(-3.75666e10 + 3.55990e10i) q^{87} +(-4.60716e10 + 7.97983e10i) q^{89} +(7.36058e8 - 3.16060e10i) q^{91} +(1.26382e11 + 3.75329e10i) q^{93} +(5.36636e10 - 3.09827e10i) q^{95} +8.51782e10i q^{97} +(2.68555e10 + 5.28913e10i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−289.505	−	305.506i	−0.687843	−	0.725860i
$3$	−289.505	−	305.506i	−0.687843	−	0.725860i
$4$		0			0
$5$	−1893.47	+	3279.58i	−0.270971	+	0.469336i	−0.969111	−	0.246626i	$-0.920678\pi$
$5$	−1893.47	+	3279.58i	−0.270971	+	0.469336i	0.698140	+	0.715962i	$0.254011\pi$
$6$		0			0
$7$	44455.1	+	1035.29i	0.999729	+	0.0232822i
$7$	44455.1	+	1035.29i	0.999729	+	0.0232822i
$8$		0			0
$9$	−9520.78	+	176891.i	−0.0537451	+	0.998555i
$10$		0			0
$11$	289993.	−	167428.i	0.542911	−	0.313450i	−0.203347	−	0.979107i	$-0.565182\pi$
$11$	289993.	−	167428.i	0.542911	−	0.313450i	0.746258	+	0.665657i	$0.231849\pi$
$12$		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	$0.107246\pi$
$17$	4.43857e6	+	7.68782e6i	0.758182	+	1.31321i	−0.185594	+	0.982626i	$0.559421\pi$
$18$		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
$21$	−1.25537e7	−	1.38810e7i	−0.670757	−	0.741678i
$22$		0			0
$23$	−2.71256e7	−	1.56610e7i	−0.878773	−	0.507360i	−0.00851947	−	0.999964i	$-0.502712\pi$
$23$	−2.71256e7	−	1.56610e7i	−0.878773	−	0.507360i	−0.870254	+	0.492604i	$0.836045\pi$
$24$		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	$-0.812096\pi$
$29$		−	1.22965e8i		−	1.11325i	0.830763	−	0.556626i	$-0.187904\pi$
$30$		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
$33$	−1.35105e8	−	4.01235e7i	−0.600958	−	0.178473i
$34$		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
$39$	−2.17204e8	+	2.05828e8i	−0.385489	+	0.365299i
$40$		0			0
$41$	9.54564e8			1.28675			0.643375	−	0.765551i	$-0.277534\pi$
$41$	9.54564e8			1.28675			0.643375	+	0.765551i	$0.277534\pi$
$42$		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
$45$	−5.62101e8	−	3.66161e8i	−0.454094	−	0.295804i
$46$		0			0
$47$	−3.52402e8	+	6.10379e8i	−0.224130	+	0.388205i	−0.956058	−	0.293177i	$-0.905287\pi$
$47$	−3.52402e8	+	6.10379e8i	−0.224130	+	0.388205i	0.731928	+	0.681382i	$0.238621\pi$
$48$		0			0
$49$	1.97518e9	+	9.20482e7i	0.998916	+	0.0465519i
$50$		0			0
$51$	1.06369e9	−	3.58167e9i	0.431697	−	1.45362i
$52$		0			0
$53$	2.00992e9	−	1.16042e9i	0.660177	−	0.381153i	−0.132167	−	0.991227i	$-0.542194\pi$
$53$	2.00992e9	−	1.16042e9i	0.660177	−	0.381153i	0.792344	+	0.610074i	$0.208860\pi$
$54$		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	$0.0653758\pi$
$59$	3.65791e9	+	6.33569e9i	0.666111	+	1.15374i	−0.312871	+	0.949796i	$0.601291\pi$
$60$		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
$63$	−6.06382e8	+	7.85385e9i	−0.0769791	+	0.997033i
$64$		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	$0.239757\pi$
$71$			2.07970e10i			1.36798i	−0.729491	+	0.683990i	$0.760243\pi$
$72$		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
$75$	4.13238e9	−	1.39146e10i	0.201077	−	0.677072i
$76$		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
$81$	−3.11998e10	−	3.36828e9i	−0.994223	−	0.107335i
$82$		0			0
$83$	5.42123e10			1.51067			0.755333	−	0.655341i	$-0.227475\pi$
$83$	5.42123e10			1.51067			0.755333	+	0.655341i	$0.227475\pi$
$84$		0			0
$85$	−3.36171e10			−0.821782
$86$		0			0
$87$	−3.75666e10	+	3.55990e10i	−0.808065	+	0.765742i
$88$		0			0
$89$	−4.60716e10	+	7.97983e10i	−0.874557	+	1.51478i	−0.0173224	+	0.999850i	$0.505514\pi$
$89$	−4.60716e10	+	7.97983e10i	−0.874557	+	1.51478i	−0.857234	+	0.514927i	$0.827819\pi$
$90$		0			0
$91$	7.36058e8	−	3.16060e10i	0.0123647	−	0.530935i
$92$		0			0
$93$	1.26382e11	+	3.75329e10i	1.88377	+	0.559444i
$94$		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	$0.0983036\pi$
$101$	7.81170e10	+	1.35303e11i	0.739568	+	1.28097i	−0.213122	+	0.977026i	$0.568363\pi$
$102$		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
$105$	6.92939e10	−	1.48876e10i	0.529851	−	0.113837i
$106$		0			0
$107$	−1.74363e10	−	1.00669e10i	−0.120183	−	0.0693878i	0.438703	−	0.898632i	$-0.355438\pi$
$107$	−1.74363e10	−	1.00669e10i	−0.120183	−	0.0693878i	−0.558887	+	0.829244i	$0.688771\pi$
$108$		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	$0.0910649\pi$
$113$			1.10541e11i			0.564404i	−0.959355	+	0.282202i	$0.908935\pi$
$114$		0			0
$115$	1.02723e11	−	5.93072e10i	0.476244	−	0.274960i
$116$		0			0
$117$	1.25763e11	+	6.76894e9i	0.530312	+	0.0285429i
$118$		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
$123$	−2.76351e11	−	2.91625e11i	−0.885081	−	0.934000i
$124$		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
$129$	1.27885e11	+	1.34954e11i	0.315194	+	0.332614i
$130$		0			0
$131$	8.31004e10	−	1.43934e11i	0.188196	−	0.325965i	−0.756453	−	0.654048i	$-0.773069\pi$
$131$	8.31004e10	−	1.43934e11i	0.188196	−	0.325965i	0.944649	+	0.328083i	$0.106403\pi$
$132$		0			0
$133$	−6.21490e11	−	3.78379e11i	−1.29494	−	0.788393i
$134$		0			0
$135$	5.08665e10	+	2.77731e11i	0.0976330	+	0.533075i
$136$		0			0
$137$	5.50434e11	−	3.17793e11i	0.974411	−	0.562577i	0.0738330	−	0.997271i	$-0.476477\pi$
$137$	5.50434e11	−	3.17793e11i	0.974411	−	0.562577i	0.900578	+	0.434694i	$0.143143\pi$
$138$		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
$147$	−5.43704e11	−	6.30079e11i	−0.653307	−	0.757093i
$148$		0			0
$149$	−7.50228e11	−	4.33144e11i	−0.836891	−	0.483179i	0.0193154	−	0.999813i	$-0.493851\pi$
$149$	−7.50228e11	−	4.33144e11i	−0.836891	−	0.483179i	−0.856206	+	0.516634i	$0.827185\pi$
$150$		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
$159$	−9.36397e11	−	2.78092e11i	−0.730762	−	0.217022i
$160$		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
$165$	3.87405e11	−	3.67114e11i	0.246606	−	0.233690i
$166$		0			0
$167$	4.82125e11			0.287223			0.143612	−	0.989634i	$-0.454128\pi$
$167$	4.82125e11			0.287223			0.143612	+	0.989634i	$0.454128\pi$
$168$		0			0
$169$	1.28669e12			0.717955
$170$		0			0
$171$	1.58214e12	−	2.42878e12i	0.827499	−	1.27031i
$172$		0			0
$173$	4.03342e11	−	6.98608e11i	0.197888	−	0.342752i	−0.749955	−	0.661488i	$-0.769925\pi$
$173$	4.03342e11	−	6.98608e11i	0.197888	−	0.342752i	0.947843	+	0.318736i	$0.103258\pi$
$174$		0			0
$175$	7.35646e11	+	1.34558e12i	0.338813	+	0.619729i
$176$		0			0
$177$	8.76607e11	−	2.95173e12i	0.379273	−	1.27709i
$178$		0			0
$179$	−1.74973e12	+	1.01021e12i	−0.711673	+	0.410885i	−0.811680	−	0.584102i	$-0.801447\pi$
$179$	−1.74973e12	+	1.01021e12i	−0.711673	+	0.410885i	0.100007	+	0.994987i	$0.468113\pi$
$180$		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
$189$	2.57495e12	−	2.08847e12i	0.776656	−	0.629926i
$190$		0			0
$191$	−3.08153e12	−	1.77912e12i	−0.877169	−	0.506434i	−0.00744480	−	0.999972i	$-0.502370\pi$
$191$	−3.08153e12	−	1.77912e12i	−0.877169	−	0.506434i	−0.869724	+	0.493539i	$0.835703\pi$
$192$		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	$0.111048\pi$
$197$			2.84714e12i			0.683668i	−0.939760	+	0.341834i	$0.888952\pi$
$198$		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
$201$	−1.27577e12	+	4.29580e12i	−0.274280	+	0.923561i
$202$		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
$207$	3.02855e12	−	4.64917e12i	0.553856	−	0.850235i
$208$		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
$213$	6.35361e12	−	6.02083e12i	0.992962	−	0.940955i
$214$		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
$219$	−3.79137e12	−	1.12597e12i	−0.508574	−	0.151037i
$220$		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	$0.258121\pi$
$227$	8.82885e12	+	1.52920e13i	0.972214	+	1.68392i	0.283375	+	0.959009i	$0.408546\pi$
$228$		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
$231$	−5.96455e12	−	1.92357e12i	−0.596639	−	0.192416i
$232$		0			0
$233$	7.08722e12	+	4.09181e12i	0.676112	+	0.390353i	0.798389	−	0.602143i	$-0.205686\pi$
$233$	7.08722e12	+	4.09181e12i	0.676112	+	0.390353i	−0.122277	+	0.992496i	$0.539019\pi$
$234$		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	$0.0134691\pi$
$239$			1.01995e12i			0.0846036i	−0.999105	+	0.0423018i	$0.986531\pi$
$240$		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
$243$	8.00346e12	+	1.05068e13i	0.605959	+	0.795496i
$244$		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
$249$	−1.56947e13	−	1.65622e13i	−1.03910	−	1.09653i
$250$		0			0
$251$	5.63091e12			0.356758			0.178379	−	0.983962i	$-0.442915\pi$
$251$	5.63091e12			0.356758			0.178379	+	0.983962i	$0.442915\pi$
$252$		0			0
$253$	−1.04883e13			−0.636127
$254$		0			0
$255$	9.73232e12	+	1.02702e13i	0.565257	+	0.596498i
$256$		0			0
$257$	−4.16559e12	+	7.21502e12i	−0.231763	+	0.401426i	−0.958327	−	0.285673i	$-0.907783\pi$
$257$	−4.16559e12	+	7.21502e12i	−0.231763	+	0.401426i	0.726564	+	0.687099i	$0.241116\pi$
$258$		0			0
$259$	−3.99901e12	+	6.56841e12i	−0.213208	+	0.350196i
$260$		0			0
$261$	2.17514e13	+	1.17073e12i	1.11164	+	0.0598318i
$262$		0			0
$263$	8.48505e12	−	4.89885e12i	0.415813	−	0.240070i	−0.277472	−	0.960734i	$-0.589496\pi$
$263$	8.48505e12	−	4.89885e12i	0.415813	−	0.240070i	0.693284	+	0.720664i	$0.256163\pi$
$264$		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	$0.0165842\pi$
$269$	1.25769e13	+	2.17838e13i	0.544422	+	0.942966i	−0.454221	+	0.890889i	$0.650083\pi$
$270$		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
$273$	−9.86891e12	+	8.92522e12i	−0.393890	+	0.356225i
$274$		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	$0.0749655\pi$
$281$			1.37058e13i			0.466680i	−0.972395	+	0.233340i	$0.925035\pi$
$282$		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
$285$	−2.50013e13	−	7.42490e12i	−0.787617	−	0.233907i
$286$		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
$291$	2.60224e13	−	2.46595e13i	0.731032	−	0.692744i
$292$		0			0
$293$	−1.06497e13			−0.288116			−0.144058	−	0.989569i	$-0.546015\pi$
$293$	−1.06497e13			−0.288116			−0.144058	+	0.989569i	$0.546015\pi$
$294$		0			0
$295$	−2.77045e13			−0.721988
$296$		0			0
$297$	8.38380e12	−	2.35168e13i	0.210514	−	0.590497i
$298$		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
$303$	1.87205e13	−	6.30360e13i	0.421098	−	1.41793i
$304$		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	$-0.789278\pi$
$311$	−4.75481e13	−	8.23557e13i	−0.926726	−	1.60514i	−0.137963	−	0.990437i	$-0.544056\pi$
$312$		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
$315$	−2.46092e13	−	1.68597e13i	−0.447084	−	0.306296i
$316$		0			0
$317$	−7.51582e13	−	4.33926e13i	−1.31871	−	0.761359i	−0.335192	−	0.942150i	$-0.608801\pi$
$317$	−7.51582e13	−	4.33926e13i	−1.31871	−	0.761359i	−0.983522	+	0.180790i	$0.942134\pi$
$318$		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
$327$	2.98090e13	−	1.00373e14i	0.440895	−	1.48459i
$328$		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
$333$	−2.56693e13	−	1.67214e13i	−0.343535	−	0.223784i
$334$		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
$339$	3.37708e13	−	3.20020e13i	0.409678	−	0.388221i
$340$		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
$345$	−4.78575e13	−	1.42128e13i	−0.527163	−	0.156558i
$346$		0			0
$347$	−4.04536e13	+	2.33559e13i	−0.431663	+	0.249221i	−0.700055	−	0.714089i	$-0.746841\pi$
$347$	−4.04536e13	+	2.33559e13i	−0.431663	+	0.249221i	0.268392	+	0.963310i	$0.413508\pi$
$348$		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	0.919397	−	0.393331i	$-0.128677\pi$
$353$	1.22614e13	+	2.12374e13i	0.119064	+	0.206225i	−0.800333	+	0.599556i	$0.795344\pi$
$354$		0			0
$355$	−6.82055e13	−	3.93784e13i	−0.642042	−	0.370683i
$356$		0			0
$357$	5.09945e13	−	1.58122e14i	0.465423	−	1.44317i
$358$		0			0
$359$	−1.11719e14	−	6.45009e13i	−0.988797	−	0.570882i	−0.0838824	−	0.996476i	$-0.526732\pi$
$359$	−1.11719e14	−	6.45009e13i	−0.988797	−	0.570882i	−0.904914	+	0.425594i	$0.860065\pi$
$360$		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
$369$	−9.08820e12	+	1.68854e14i	−0.0691565	+	1.28489i
$370$		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
$375$	9.13417e13	+	9.63902e13i	0.636058	+	0.671214i
$376$		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
$381$	−1.11284e14	−	1.17435e14i	−0.710143	−	0.749392i
$382$		0			0
$383$	1.30737e14	−	2.26444e14i	0.810600	−	1.40400i	−0.101844	−	0.994800i	$-0.532474\pi$
$383$	1.30737e14	−	2.26444e14i	0.810600	−	1.40400i	0.912445	−	0.409200i	$-0.134192\pi$
$384$		0			0
$385$	−1.31283e12	+	5.63724e13i	−0.00790998	+	0.339651i
$386$		0			0
$387$	4.20569e12	−	7.81395e13i	0.0246279	−	0.457573i
$388$		0			0
$389$	−4.46410e13	+	2.57735e13i	−0.254104	+	0.146707i	−0.621642	−	0.783302i	$-0.713534\pi$
$389$	−4.46410e13	+	2.57735e13i	−0.254104	+	0.146707i	0.367538	+	0.930008i	$0.380201\pi$
$390$		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
$399$	6.43275e13	+	2.99411e14i	0.318453	+	1.48224i
$400$		0			0
$401$	1.55102e14	+	8.95480e13i	0.747003	+	0.431283i	0.824610	−	0.565702i	$-0.191395\pi$
$401$	1.55102e14	+	8.95480e13i	0.747003	+	0.431283i	−0.0776069	+	0.996984i	$0.524728\pi$
$402$		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
$411$	−2.56441e14	−	7.61582e13i	−1.07859	−	0.320322i
$412$		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
$417$	1.01305e14	−	9.59995e13i	0.393445	−	0.372838i
$418$		0			0
$419$	1.82839e14			0.691659			0.345829	−	0.938297i	$-0.387598\pi$
$419$	1.82839e14			0.691659			0.345829	+	0.938297i	$0.387598\pi$
$420$		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
$423$	−1.04615e14	−	6.81481e13i	−0.375598	−	0.244671i
$424$		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
$429$	−2.85264e13	+	9.60547e13i	−0.0947834	+	0.319156i
$430$		0			0
$431$	1.66202e14	−	9.59570e13i	0.538285	−	0.310779i	−0.206099	−	0.978531i	$-0.566077\pi$
$431$	1.66202e14	−	9.59570e13i	0.538285	−	0.310779i	0.744384	+	0.667752i	$0.232743\pi$
$432$		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
$441$	−3.50878e13	+	3.48516e14i	−0.100171	+	0.994970i
$442$		0			0
$443$	−8.35272e13	−	4.82245e13i	−0.232599	−	0.134291i	0.379172	−	0.925326i	$-0.376209\pi$
$443$	−8.35272e13	−	4.82245e13i	−0.232599	−	0.134291i	−0.611770	+	0.791035i	$0.709542\pi$
$444$		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	$-0.868752\pi$
$449$		−	3.09922e14i		−	0.801488i	0.916190	−	0.400744i	$-0.131248\pi$
$450$		0			0
$451$	2.76817e14	−	1.59821e14i	0.698590	−	0.403331i
$452$		0			0
$453$	−1.67331e14	+	5.63440e14i	−0.412132	+	1.38774i
$454$		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
$459$	6.23438e14	+	2.22257e14i	1.42831	+	0.509197i
$460$		0			0
$461$	1.39703e14			0.312500			0.156250	−	0.987718i	$-0.450059\pi$
$461$	1.39703e14			0.312500			0.156250	+	0.987718i	$0.450059\pi$
$462$		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
$465$	−3.62392e14	+	3.43411e14i	−0.773013	+	0.732526i
$466$		0			0
$467$	−2.88740e14	+	5.00112e14i	−0.601539	+	1.04190i	0.391049	+	0.920370i	$0.372112\pi$
$467$	−2.88740e14	+	5.00112e14i	−0.601539	+	1.04190i	−0.992588	+	0.121527i	$0.961221\pi$
$468$		0			0
$469$	−2.27113e14	−	4.15416e14i	−0.462158	−	0.845343i
$470$		0			0
$471$	1.84348e14	+	5.47480e13i	0.366458	+	0.108831i
$472$		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	0.924066	−	0.382232i	$-0.124845\pi$
$479$	7.23020e13	+	1.25231e14i	0.131010	+	0.226916i	−0.793056	+	0.609149i	$0.791511\pi$
$480$		0			0
$481$	1.06479e14	+	6.14759e13i	0.188568	+	0.108870i
$482$		0			0
$483$	1.23136e14	+	5.73134e14i	0.213145	+	0.992081i
$484$		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	$-0.952661\pi$
$491$		−	1.87388e14i		−	0.296342i	0.988962	−	0.148171i	$-0.0473386\pi$
$492$		0			0
$493$	9.45335e14	−	5.45789e14i	1.46193	−	0.844048i
$494$		0			0
$495$	−2.24311e14	−	1.20731e13i	−0.339252	−	0.0182595i
$496$		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
$501$	−1.39578e14	−	1.47292e14i	−0.197564	−	0.208484i
$502$		0			0
$503$	9.42273e14			1.30483			0.652413	−	0.757863i	$-0.273757\pi$
$503$	9.42273e14			1.30483			0.652413	+	0.757863i	$0.273757\pi$
$504$		0			0
$505$	−5.91648e14			−0.801606
$506$		0			0
$507$	−3.72503e14	−	3.93091e14i	−0.493840	−	0.521134i
$508$		0			0
$509$	1.07535e14	−	1.86257e14i	0.139509	−	0.241637i	−0.787802	−	0.615929i	$-0.788781\pi$
$509$	1.07535e14	−	1.86257e14i	0.139509	−	0.241637i	0.927311	+	0.374292i	$0.122114\pi$
$510$		0			0
$511$	3.66637e14	−	2.00444e14i	0.465502	−	0.254495i
$512$		0			0
$513$	−1.20004e15	+	2.19789e14i	−1.49125	+	0.273124i
$514$		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	$-0.957415\pi$
$521$	−5.35404e14	−	9.27347e14i	−0.611047	−	1.05836i	0.380018	−	0.924979i	$-0.375918\pi$
$522$		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
$525$	1.98111e14	−	6.14298e14i	0.216787	−	0.672207i
$526$		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
$537$	8.15182e14	+	2.42094e14i	0.787764	+	0.233951i
$538$		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
$543$	5.43258e14	−	5.14804e14i	0.493863	−	0.467997i
$544$		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
$549$	−2.03249e14	+	3.12011e14i	−0.173933	+	0.267007i
$550$		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
$555$	−7.84724e13	+	2.64234e14i	−0.0632565	+	0.212999i
$556$		0			0
$557$	1.69856e15	−	9.80664e14i	1.34238	−	0.775026i	0.355228	−	0.934780i	$-0.384403\pi$
$557$	1.69856e15	−	9.80664e14i	1.34238	−	0.775026i	0.987157	+	0.159753i	$0.0510699\pi$
$558$		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	0.807322	−	0.590111i	$-0.200916\pi$
$563$	−1.44132e14	−	2.49644e14i	−0.107390	−	0.186005i	−0.914712	+	0.404106i	$0.867583\pi$
$564$		0			0
$565$	−3.62527e14	−	2.09305e14i	−0.264895	−	0.152937i
$566$		0			0
$567$	−1.38350e15	−	1.82038e14i	−0.991454	−	0.130453i
$568$		0			0
$569$	−2.12803e15	−	1.22862e15i	−1.49575	−	0.863573i	−0.495764	−	0.868457i	$-0.665112\pi$
$569$	−2.12803e15	−	1.22862e15i	−1.49575	−	0.863573i	−0.999988	+	0.00488432i	$0.998445\pi$
$570$		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
$579$	−3.24727e14	+	1.09343e15i	−0.207390	+	0.698326i
$580$		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
$585$	−2.60328e14	+	3.99634e14i	−0.157095	+	0.241160i
$586$		0			0
$587$	3.92949e14			0.232716			0.116358	−	0.993207i	$-0.462878\pi$
$587$	3.92949e14			0.232716			0.116358	+	0.993207i	$0.462878\pi$
$588$		0			0
$589$	5.12545e15			2.97919
$590$		0			0
$591$	8.69819e14	−	8.24262e14i	0.496247	−	0.470256i
$592$		0			0
$593$	−1.76073e15	+	3.04968e15i	−0.986035	+	1.70786i	−0.348791	+	0.937200i	$0.613408\pi$
$593$	−1.76073e15	+	3.04968e15i	−0.986035	+	1.70786i	−0.637244	+	0.770662i	$0.719925\pi$
$594$		0			0
$595$	−1.49445e15	−	3.48036e13i	−0.821559	−	0.0191329i
$596$		0			0
$597$	−8.44374e14	−	2.50763e14i	−0.455697	−	0.135333i
$598$		0			0
$599$	2.43865e15	−	1.40795e15i	1.29212	−	0.746003i	0.313087	−	0.949725i	$-0.398637\pi$
$599$	2.43865e15	−	1.40795e15i	1.29212	−	0.746003i	0.979029	+	0.203721i	$0.0653036\pi$
$600$		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
$609$	−1.70688e15	+	1.54367e15i	−0.825674	+	0.746721i
$610$		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	$-0.907606\pi$
$617$		−	1.27139e15i		−	0.572413i	0.958168	−	0.286206i	$-0.0923943\pi$
$618$		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
$621$	−2.29713e15	+	4.20720e14i	−0.998117	+	0.182806i
$622$		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
$627$	1.58626e15	+	1.67393e15i	0.653737	+	0.689869i
$628$		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
$633$	7.61355e14	+	8.03436e14i	0.297760	+	0.314218i
$634$		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
$639$	−3.67880e15	−	1.98004e14i	−1.36600	−	0.0735222i
$640$		0			0
$641$	3.85669e15	−	2.22666e15i	1.40765	−	0.812710i	0.412493	−	0.910961i	$-0.364658\pi$
$641$	3.85669e15	−	2.22666e15i	1.40765	−	0.812710i	0.995162	+	0.0982515i	$0.0313250\pi$
$642$		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	0.999874	−	0.0158856i	$-0.00505677\pi$
$647$	1.40207e15	+	2.42846e15i	0.486180	+	0.842088i	−0.513694	+	0.857973i	$0.671723\pi$
$648$		0			0
$649$	2.12154e15	+	1.22487e15i	0.723278	+	0.417585i
$650$		0			0
$651$	5.57945e15	+	1.79937e15i	1.87023	+	0.603150i
$652$		0			0
$653$	3.97213e13	+	2.29331e13i	0.0130919	+	0.00755859i	0.506532	−	0.862221i	$-0.330927\pi$
$653$	3.97213e13	+	2.29331e13i	0.0130919	+	0.00755859i	−0.493440	+	0.869780i	$0.664261\pi$
$654$		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	$0.173831\pi$
$659$			3.31415e15i			1.03873i	−0.854553	+	0.519364i	$0.826169\pi$
$660$		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
$663$	−2.54644e15	−	7.56245e14i	−0.771986	−	0.229265i
$664$		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
$669$	4.41242e15	−	4.18132e15i	1.27301	−	1.20634i
$670$		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
$675$	2.42203e15	+	8.63459e14i	0.665286	+	0.237176i
$676$		0			0
$677$	−1.12390e15	+	1.94664e15i	−0.303730	+	0.526076i	−0.976978	−	0.213341i	$-0.931565\pi$
$677$	−1.12390e15	+	1.94664e15i	−0.303730	+	0.526076i	0.673248	+	0.739417i	$0.264899\pi$
$678$		0			0
$679$	−8.81845e13	+	3.78660e15i	−0.0234481	+	1.00685i
$680$		0			0
$681$	2.11581e15	−	7.12438e15i	0.553563	−	1.86397i
$682$		0			0
$683$	−2.20717e15	+	1.27431e15i	−0.568227	+	0.328066i	−0.756441	−	0.654062i	$-0.773064\pi$
$683$	−2.20717e15	+	1.27431e15i	−0.568227	+	0.328066i	0.188214	+	0.982128i	$0.439730\pi$
$684$		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
$693$	1.13911e15	+	2.37909e15i	0.270727	+	0.565429i
$694$		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	$-0.896896\pi$
$701$		−	2.85288e15i		−	0.636552i	0.947998	−	0.318276i	$-0.103104\pi$
$702$		0			0
$703$	2.45064e15	−	1.41488e15i	0.538300	−	0.310788i
$704$		0			0
$705$	−3.19815e14	+	1.07689e15i	−0.0691605	+	0.232879i
$706$		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
$711$	−4.23438e15	−	2.75834e15i	−0.873991	−	0.569331i
$712$		0			0
$713$	9.81115e15			1.99401
$714$		0			0
$715$	9.01557e14			0.180431
$716$		0			0
$717$	3.11600e14	−	2.95279e14i	0.0614104	−	0.0581940i
$718$		0			0
$719$	−4.51079e15	+	7.81291e15i	−0.875475	+	1.51637i	−0.0192181	+	0.999815i	$0.506118\pi$
$719$	−4.51079e15	+	7.81291e15i	−0.875475	+	1.51637i	−0.856256	+	0.516551i	$0.827216\pi$
$720$		0			0
$721$	−5.79867e15	−	3.53037e15i	−1.10837	−	0.674803i
$722$		0			0
$723$	−4.63749e15	−	1.37725e15i	−0.873017	−	0.259270i
$724$		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
$735$	3.09588e15	−	5.90088e14i	0.532358	−	0.101470i
$736$		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	$-0.991869\pi$
$743$		−	3.15277e14i		−	0.0510803i	0.999674	−	0.0255401i	$-0.00813056\pi$
$744$		0			0
$745$	2.84106e15	−	1.64029e15i	0.453546	−	0.261855i
$746$		0			0
$747$	−5.16144e14	+	9.58967e15i	−0.0811909	+	1.50848i
$748$		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
$753$	−1.63018e15	−	1.72028e15i	−0.245393	−	0.258956i
$754$		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
$759$	3.03643e15	+	3.20425e15i	0.437555	+	0.461739i
$760$		0			0
$761$	−1.73874e15	+	3.01159e15i	−0.246956	+	0.427740i	−0.962680	−	0.270644i	$-0.912764\pi$
$761$	−1.73874e15	+	3.01159e15i	−0.246956	+	0.427740i	0.715724	+	0.698383i	$0.246097\pi$
$762$		0			0
$763$	5.30660e15	+	9.70641e15i	0.742901	+	1.35886i
$764$		0			0
$765$	3.20061e14	−	5.94657e15i	0.0441667	−	0.820594i
$766$		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	$-0.874532\pi$
$773$	−6.09457e15	−	1.05561e16i	−0.794247	−	1.37568i	0.129069	−	0.991636i	$-0.458801\pi$
$774$		0			0
$775$	−9.35535e15	−	5.40131e15i	−1.20199	−	0.693969i
$776$		0			0
$777$	3.16442e15	−	6.79865e14i	0.400847	−	0.0861207i
$778$		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
$789$	−3.95309e15	−	1.17399e15i	−0.460271	−	0.136692i
$790$		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
$795$	2.68506e15	−	2.54443e15i	0.299872	−	0.284166i
$796$		0			0
$797$	−8.00214e15			−0.881425			−0.440712	−	0.897648i	$-0.645274\pi$
$797$	−8.00214e15			−0.881425			−0.440712	+	0.897648i	$0.645274\pi$
$798$		0			0
$799$	−6.25665e15			−0.679727
$800$		0			0
$801$	−1.36770e16	−	8.90939e15i	−1.46558	−	0.954704i
$802$		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
$807$	3.01401e15	−	1.01488e16i	0.309985	−	1.04379i
$808$		0			0
$809$	3.36452e15	−	1.94251e15i	0.341355	−	0.197081i	−0.319516	−	0.947581i	$-0.603520\pi$
$809$	3.36452e15	−	1.94251e15i	0.341355	−	0.197081i	0.660871	+	0.750499i	$0.270187\pi$
$810$		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
$819$	5.58381e15	+	4.31116e14i	0.529504	+	0.0408820i
$820$		0			0
$821$	−5.31044e15	−	3.06599e15i	−0.496871	−	0.286868i	0.230550	−	0.973061i	$-0.425948\pi$
$821$	−5.31044e15	−	3.06599e15i	−0.496871	−	0.286868i	−0.727420	+	0.686192i	$0.759281\pi$
$822$		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	$-0.666629\pi$
$827$		−	1.92695e16i		−	1.73217i	0.499898	−	0.866085i	$-0.333371\pi$
$828$		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
$831$	3.92874e15	−	1.32289e16i	0.343912	−	1.15803i
$832$		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
$837$	−7.84249e15	+	2.19984e16i	−0.659878	+	1.85098i
$838$		0			0
$839$	−8.12961e15			−0.675117			−0.337559	−	0.941305i	$-0.609601\pi$
$839$	−8.12961e15			−0.675117			−0.337559	+	0.941305i	$0.609601\pi$
$840$		0			0
$841$	−2.91994e15			−0.239329
$842$		0			0
$843$	4.18720e15	−	3.96789e15i	0.338744	−	0.321002i
$844$		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
$849$	1.49985e16	+	4.45427e15i	1.16696	+	0.346565i
$850$		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	0.589905	−	0.807472i	$-0.299165\pi$
$857$	−5.47192e15	−	9.47765e15i	−0.404339	−	0.700335i	−0.994244	+	0.107137i	$0.965832\pi$
$858$		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
$861$	−1.19833e16	−	1.32503e16i	−0.863096	−	0.954353i
$862$		0			0
$863$	−5.21383e15	−	3.01021e15i	−0.370764	−	0.214061i	0.303028	−	0.952982i	$-0.402002\pi$
$863$	−5.21383e15	−	3.01021e15i	−0.370764	−	0.214061i	−0.673792	+	0.738921i	$0.735336\pi$
$864$		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
$873$	−1.50672e16	−	8.10963e14i	−1.00567	−	0.0541281i
$874$		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
$879$	3.08315e15	+	3.25356e15i	0.198178	+	0.209132i
$880$		0			0
$881$	1.28479e16			0.815576			0.407788	−	0.913077i	$-0.366300\pi$
$881$	1.28479e16			0.815576			0.407788	+	0.913077i	$0.366300\pi$
$882$		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
$885$	8.02060e15	+	8.46390e15i	0.496614	+	0.524062i
$886$		0			0
$887$	−2.84974e15	+	4.93589e15i	−0.174271	+	0.301846i	−0.939909	−	0.341426i	$-0.889090\pi$
$887$	−2.84974e15	+	4.93589e15i	−0.174271	+	0.301846i	0.765638	+	0.643272i	$0.222423\pi$
$888$		0			0
$889$	1.70883e16	+	3.97961e14i	1.03214	+	0.0240371i
$890$		0			0
$891$	−9.61167e15	+	4.24693e15i	−0.573418	+	0.253366i
$892$		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
$903$	5.54544e15	+	6.13177e15i	0.307364	+	0.339863i
$904$		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	$-0.959899\pi$
$911$		−	4.75922e15i		−	0.251295i	0.992075	−	0.125648i	$-0.0401009\pi$
$912$		0			0
$913$	1.57212e16	−	9.07665e15i	0.820156	−	0.473518i
$914$		0			0
$915$	3.21177e15	+	9.53836e14i	0.165550	+	0.0491652i
$916$		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
$921$	1.89077e16	−	1.79174e16i	0.940181	−	0.890939i
$922$		0			0
$923$	1.47859e16			0.726506
$924$		0			0
$925$	−5.96411e15			−0.289578
$926$		0			0
$927$	1.47618e16	−	2.26611e16i	0.708274	−	1.08728i
$928$		0			0
$929$	−1.22195e16	+	2.11648e16i	−0.579385	+	1.00352i	0.416165	+	0.909289i	$0.363374\pi$
$929$	−1.22195e16	+	2.11648e16i	−0.579385	+	1.00352i	−0.995550	+	0.0942352i	$0.969959\pi$
$930$		0			0
$931$	−2.72367e16	−	1.74643e16i	−1.27623	−	0.818328i
$932$		0			0
$933$	−1.13948e16	+	3.83686e16i	−0.527662	+	1.77675i
$934$		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	$0.00691964\pi$
$941$	1.17399e16	+	2.03341e16i	0.518707	+	0.898426i	−0.481057	+	0.876689i	$0.659747\pi$
$942$		0			0
$943$	−2.58932e16	−	1.49494e16i	−1.13076	−	0.652845i
$944$		0			0
$945$	1.97374e15	+	1.23992e16i	0.0851953	+	0.535204i
$946$		0			0
$947$	3.58370e16	+	2.06905e16i	1.52900	+	0.882768i	0.999404	+	0.0345179i	$0.0109896\pi$
$947$	3.58370e16	+	2.06905e16i	1.52900	+	0.882768i	0.529595	+	0.848250i	$0.322344\pi$
$948$		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	$0.223909\pi$
$953$			3.13932e16i			1.29367i	−0.762628	+	0.646837i	$0.776091\pi$
$954$		0			0
$955$	1.16696e16	−	6.73742e15i	0.475375	−	0.274458i
$956$		0			0
$957$	−4.93380e15	+	1.66132e16i	−0.198685	+	0.669017i
$958$		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
$963$	1.94674e15	−	2.98848e15i	0.0757468	−	0.116280i
$964$		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
$969$	−4.43765e16	+	4.20523e16i	−1.66868	+	1.58128i
$970$		0			0
$971$	2.37428e16	−	4.11237e16i	0.882725	−	1.52892i	0.0344260	−	0.999407i	$-0.489040\pi$
$971$	2.37428e16	−	4.11237e16i	0.882725	−	1.52892i	0.848299	−	0.529517i	$-0.177627\pi$
$972$		0			0
$973$	−3.43303e14	+	1.47413e16i	−0.0126199	+	0.541893i
$974$		0			0
$975$	−9.89281e15	−	2.93798e15i	−0.359579	−	0.106788i
$976$		0			0
$977$	−4.59389e16	+	2.65229e16i	−1.65105	+	0.953235i	−0.674412	+	0.738355i	$0.735603\pi$
$977$	−4.59389e16	+	2.65229e16i	−1.65105	+	0.953235i	−0.976640	+	0.214880i	$0.931064\pi$
$978$		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	$-0.881571\pi$
$983$	−2.24638e16	−	3.89084e16i	−0.780619	−	1.35207i	0.150963	−	0.988539i	$-0.451763\pi$
$984$		0			0
$985$	−9.33744e15	−	5.39097e15i	−0.320870	−	0.185254i
$986$		0			0
$987$	1.28966e16	−	2.77080e15i	0.438260	−	0.0941587i
$988$		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
$999$	2.32291e15	+	1.26830e16i	0.0738620	+	0.403286i

Twists

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

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

\(f(q)\)	\(=\)	\(q+(-289.505 - 305.506i) q^{3} +(-1893.47 + 3279.58i) q^{5} +(44455.1 + 1035.29i) q^{7} +(-9520.78 + 176891. i) q^{9} +O(q^{10})\)	\(q+(-289.505 - 305.506i) q^{3} +(-1893.47 + 3279.58i) q^{5} +(44455.1 + 1035.29i) q^{7} +(-9520.78 + 176891. i) q^{9} +(289993. - 167428. i) q^{11} -710965. i q^{13} +(1.55010e6 - 370990. i) q^{15} +(4.43857e6 + 7.68782e6i) q^{17} +(-1.41707e7 - 8.18146e6i) q^{19} +(-1.25537e7 - 1.38810e7i) q^{21} +(-2.71256e7 - 1.56610e7i) q^{23} +(1.72436e7 + 2.98668e7i) q^{25} +(5.67976e7 - 4.83022e7i) q^{27} -1.22965e8i q^{29} +(-2.71270e8 + 1.56618e8i) q^{31} +(-1.35105e8 - 4.01235e7i) q^{33} +(-8.75696e7 + 1.43834e8i) q^{35} +(-8.64684e7 + 1.49768e8i) q^{37} +(-2.17204e8 + 2.05828e8i) q^{39} +9.54564e8 q^{41} -4.41738e8 q^{43} +(-5.62101e8 - 3.66161e8i) q^{45} +(-3.52402e8 + 6.10379e8i) q^{47} +(1.97518e9 + 9.20482e7i) q^{49} +(1.06369e9 - 3.58167e9i) q^{51} +(2.00992e9 - 1.16042e9i) q^{53} +1.26808e9i q^{55} +(1.60300e9 + 6.69781e9i) q^{57} +(3.65791e9 + 6.33569e9i) q^{59} +(1.82043e9 + 1.05103e9i) q^{61} +(-6.06382e8 + 7.85385e9i) q^{63} +(2.33167e9 + 1.34619e9i) q^{65} +(-5.32355e9 - 9.22065e9i) q^{67} +(3.06848e9 + 1.28210e10i) q^{69} +2.07970e10i q^{71} +(8.13793e9 - 4.69844e9i) q^{73} +(4.13238e9 - 1.39146e10i) q^{75} +(1.30650e10 - 7.14279e9i) q^{77} +(-1.42637e10 + 2.47055e10i) q^{79} +(-3.11998e10 - 3.36828e9i) q^{81} +5.42123e10 q^{83} -3.36171e10 q^{85} +(-3.75666e10 + 3.55990e10i) q^{87} +(-4.60716e10 + 7.97983e10i) q^{89} +(7.36058e8 - 3.16060e10i) q^{91} +(1.26382e11 + 3.75329e10i) q^{93} +(5.36636e10 - 3.09827e10i) q^{95} +8.51782e10i q^{97} +(2.68555e10 + 5.28913e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 729 q^{3} - 27194 q^{7} - 172347 q^{9}+O(q^{10}) \) 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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