LMFDB - Embedded newform 90.13.g.b.37.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [90,13,Mod(37,90)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 13, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("90.37"); S:= CuspForms(chi, 13); N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 13 $
Character orbit:	$[\chi]$	$=$	90.g (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,192,0,0,16260] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$82.2594435549$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 2385x^{4} + 1422264x^{2} + 490000 $ x^6 + 2385x^4 + 1422264x^2 + 490000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2}\cdot 5^{6} $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			37.1
Root			$34.3230i$ of defining polynomial
Character	$\chi$	$=$	90.37
Dual form			90.13.g.b.73.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(32.0000 + 32.0000i) q^{2} +2048.00i q^{4} +(-3818.66 + 15151.2i) q^{5} +(-59325.1 - 59325.1i) q^{7} +(-65536.0 + 65536.0i) q^{8} +(-607035. + 362641. i) q^{10} +2.34646e6 q^{11} +(-4.56702e6 + 4.56702e6i) q^{13} -3.79681e6i q^{14} -4.19430e6 q^{16} +(1.56434e7 + 1.56434e7i) q^{17} +1.45864e7i q^{19} +(-3.10296e7 - 7.82062e6i) q^{20} +(7.50867e7 + 7.50867e7i) q^{22} +(-1.10378e8 + 1.10378e8i) q^{23} +(-2.14976e8 - 1.15715e8i) q^{25} -2.92289e8 q^{26} +(1.21498e8 - 1.21498e8i) q^{28} +5.34534e7i q^{29} -1.35539e9 q^{31} +(-1.34218e8 - 1.34218e8i) q^{32} +1.00118e9i q^{34} +(1.12539e9 - 6.72303e8i) q^{35} +(7.96598e8 + 7.96598e8i) q^{37} +(-4.66766e8 + 4.66766e8i) q^{38} +(-7.42688e8 - 1.24321e9i) q^{40} -5.38572e8 q^{41} +(7.82281e9 - 7.82281e9i) q^{43} +4.80555e9i q^{44} -7.06422e9 q^{46} +(-5.73384e9 - 5.73384e9i) q^{47} -6.80235e9i q^{49} +(-3.17637e9 - 1.05821e10i) q^{50} +(-9.35326e9 - 9.35326e9i) q^{52} +(1.01663e10 - 1.01663e10i) q^{53} +(-8.96034e9 + 3.55517e10i) q^{55} +7.77586e9 q^{56} +(-1.71051e9 + 1.71051e9i) q^{58} +5.72907e10i q^{59} -9.47423e10 q^{61} +(-4.33726e10 - 4.33726e10i) q^{62} -8.58993e9i q^{64} +(-5.17559e10 - 8.66357e10i) q^{65} +(-4.17965e10 - 4.17965e10i) q^{67} +(-3.20376e10 + 3.20376e10i) q^{68} +(5.75261e10 + 1.44987e10i) q^{70} +3.51901e10 q^{71} +(1.08806e11 - 1.08806e11i) q^{73} +5.09823e10i q^{74} -2.98730e10 q^{76} +(-1.39204e11 - 1.39204e11i) q^{77} -1.31009e11i q^{79} +(1.60166e10 - 6.35487e10i) q^{80} +(-1.72343e10 - 1.72343e10i) q^{82} +(-1.61869e11 + 1.61869e11i) q^{83} +(-2.96752e11 + 1.77279e11i) q^{85} +5.00660e11 q^{86} +(-1.53778e11 + 1.53778e11i) q^{88} -8.98485e11i q^{89} +5.41878e11 q^{91} +(-2.26055e11 - 2.26055e11i) q^{92} -3.66966e11i q^{94} +(-2.21002e11 - 5.57007e10i) q^{95} +(-3.98581e11 - 3.98581e11i) q^{97} +(2.17675e11 - 2.17675e11i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 192 q^{2} + 16260 q^{5} - 45336 q^{7} - 393216 q^{8} + 886080 q^{10} + 3418008 q^{11} + 8106834 q^{13} - 25165824 q^{16} - 10772514 q^{17} + 23408640 q^{20} + 109376256 q^{22} - 241146864 q^{23} - 520941150 q^{25}+ \cdots + 1960900383168 q^{98}+O(q^{100}) $ 6 * q + 192 * q^2 + 16260 * q^5 - 45336 * q^7 - 393216 * q^8 + 886080 * q^10 + 3418008 * q^11 + 8106834 * q^13 - 25165824 * q^16 - 10772514 * q^17 + 23408640 * q^20 + 109376256 * q^22 - 241146864 * q^23 - 520941150 * q^25 + 518837376 * q^26 + 92848128 * q^28 - 1709108208 * q^31 - 805306368 * q^32 + 3823952160 * q^35 - 742614906 * q^37 - 3927847680 * q^38 - 316538880 * q^40 + 24748899528 * q^41 + 14532965304 * q^43 - 15433399296 * q^46 - 8427287424 * q^47 - 4302859200 * q^50 + 16602796032 * q^52 - 1297014534 * q^53 + 75773173080 * q^55 + 5942280192 * q^56 + 576729600 * q^58 - 201276821928 * q^61 - 54691462656 * q^62 - 146156357430 * q^65 + 108303977544 * q^67 + 22062108672 * q^68 + 132252737280 * q^70 + 570401667408 * q^71 + 710242228134 * q^73 - 251382251520 * q^76 - 9201523848 * q^77 - 68199383040 * q^80 + 791964784896 * q^82 - 727867488144 * q^83 - 1459335929310 * q^85 + 930109779456 * q^86 - 224002572288 * q^88 + 1841466543792 * q^91 - 493868777472 * q^92 + 972100512600 * q^95 - 2207840630346 * q^97 + 1960900383168 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	32.0000	+	32.0000i	0.500000	+	0.500000i
$2$	32.0000	+	32.0000i	0.500000	+	0.500000i
$3$		0			0
$3$		0			0
$4$			2048.00i			0.500000i
$5$	−3818.66	+	15151.2i	−0.244394	+	0.969676i
$5$	−3818.66	+	15151.2i	−0.244394	+	0.969676i
$6$		0			0
$7$	−59325.1	−	59325.1i	−0.504255	−	0.504255i	0.408502	−	0.912757i	$-0.366051\pi$
$7$	−59325.1	−	59325.1i	−0.504255	−	0.504255i	−0.912757	+	0.408502i	$0.866051\pi$
$8$	−65536.0	+	65536.0i	−0.250000	+	0.250000i
$9$		0			0
$10$	−607035.	+	362641.i	−0.607035	+	0.362641i
$11$	2.34646e6			1.32452			0.662258	−	0.749276i	$-0.269598\pi$
$11$	2.34646e6			1.32452			0.662258	+	0.749276i	$0.269598\pi$
$12$		0			0
$13$	−4.56702e6	+	4.56702e6i	−0.946178	+	0.946178i	−0.998624	−	0.0524458i	$-0.983298\pi$
$13$	−4.56702e6	+	4.56702e6i	−0.946178	+	0.946178i	0.0524458	+	0.998624i	$0.483298\pi$
$14$		−	3.79681e6i		−	0.504255i
$15$		0			0
$16$	−4.19430e6			−0.250000
$17$	1.56434e7	+	1.56434e7i	0.648092	+	0.648092i	0.952532	−	0.304440i	$-0.0984692\pi$
$17$	1.56434e7	+	1.56434e7i	0.648092	+	0.648092i	−0.304440	+	0.952532i	$0.598469\pi$
$18$		0			0
$19$			1.45864e7i			0.310047i	0.987911	+	0.155023i	$0.0495453\pi$
$19$			1.45864e7i			0.310047i	−0.987911	+	0.155023i	$0.950455\pi$
$20$	−3.10296e7	−	7.82062e6i	−0.484838	−	0.122197i
$21$		0			0
$22$	7.50867e7	+	7.50867e7i	0.662258	+	0.662258i
$23$	−1.10378e8	+	1.10378e8i	−0.745619	+	0.745619i	−0.973653	−	0.228034i	$-0.926770\pi$
$23$	−1.10378e8	+	1.10378e8i	−0.745619	+	0.745619i	0.228034	+	0.973653i	$0.426770\pi$
$24$		0			0
$25$	−2.14976e8	−	1.15715e8i	−0.880543	−	0.473967i
$26$	−2.92289e8			−0.946178
$27$		0			0
$28$	1.21498e8	−	1.21498e8i	0.252128	−	0.252128i
$29$			5.34534e7i			0.0898643i	0.998990	+	0.0449322i	$0.0143072\pi$
$29$			5.34534e7i			0.0898643i	−0.998990	+	0.0449322i	$0.985693\pi$
$30$		0			0
$31$	−1.35539e9			−1.52720			−0.763600	−	0.645690i	$-0.776570\pi$
$31$	−1.35539e9			−1.52720			−0.763600	+	0.645690i	$0.776570\pi$
$32$	−1.34218e8	−	1.34218e8i	−0.125000	−	0.125000i
$33$		0			0
$34$			1.00118e9i			0.648092i
$35$	1.12539e9	−	6.72303e8i	0.612201	−	0.365727i
$36$		0			0
$37$	7.96598e8	+	7.96598e8i	0.310477	+	0.310477i	0.845094	−	0.534618i	$-0.179544\pi$
$37$	7.96598e8	+	7.96598e8i	0.310477	+	0.310477i	−0.534618	+	0.845094i	$0.679544\pi$
$38$	−4.66766e8	+	4.66766e8i	−0.155023	+	0.155023i
$39$		0			0
$40$	−7.42688e8	−	1.24321e9i	−0.181320	−	0.303518i
$41$	−5.38572e8			−0.113381			−0.0566905	−	0.998392i	$-0.518055\pi$
$41$	−5.38572e8			−0.113381			−0.0566905	+	0.998392i	$0.518055\pi$
$42$		0			0
$43$	7.82281e9	−	7.82281e9i	1.23752	−	1.23752i	0.276508	−	0.961012i	$-0.410823\pi$
$43$	7.82281e9	−	7.82281e9i	1.23752	−	1.23752i	0.961012	−	0.276508i	$-0.0891773\pi$
$44$			4.80555e9i			0.662258i
$45$		0			0
$46$	−7.06422e9			−0.745619
$47$	−5.73384e9	−	5.73384e9i	−0.531935	−	0.531935i	0.389213	−	0.921148i	$-0.372747\pi$
$47$	−5.73384e9	−	5.73384e9i	−0.531935	−	0.531935i	−0.921148	+	0.389213i	$0.872747\pi$
$48$		0			0
$49$		−	6.80235e9i		−	0.491453i
$50$	−3.17637e9	−	1.05821e10i	−0.203288	−	0.677255i
$51$		0			0
$52$	−9.35326e9	−	9.35326e9i	−0.473089	−	0.473089i
$53$	1.01663e10	−	1.01663e10i	0.458679	−	0.458679i	−0.439543	−	0.898222i	$-0.644860\pi$
$53$	1.01663e10	−	1.01663e10i	0.458679	−	0.458679i	0.898222	+	0.439543i	$0.144860\pi$
$54$		0			0
$55$	−8.96034e9	+	3.55517e10i	−0.323704	+	1.28435i
$56$	7.77586e9			0.252128
$57$		0			0
$58$	−1.71051e9	+	1.71051e9i	−0.0449322	+	0.0449322i
$59$			5.72907e10i			1.35823i	0.734034	+	0.679113i	$0.237635\pi$
$59$			5.72907e10i			1.35823i	−0.734034	+	0.679113i	$0.762365\pi$
$60$		0			0
$61$	−9.47423e10			−1.83893			−0.919464	−	0.393174i	$-0.871377\pi$
$61$	−9.47423e10			−1.83893			−0.919464	+	0.393174i	$0.871377\pi$
$62$	−4.33726e10	−	4.33726e10i	−0.763600	−	0.763600i
$63$		0			0
$64$		−	8.58993e9i		−	0.125000i
$65$	−5.17559e10	−	8.66357e10i	−0.686245	−	1.14873i
$66$		0			0
$67$	−4.17965e10	−	4.17965e10i	−0.462052	−	0.462052i	0.437276	−	0.899327i	$-0.355943\pi$
$67$	−4.17965e10	−	4.17965e10i	−0.462052	−	0.462052i	−0.899327	+	0.437276i	$0.855943\pi$
$68$	−3.20376e10	+	3.20376e10i	−0.324046	+	0.324046i
$69$		0			0
$70$	5.75261e10	+	1.44987e10i	0.488964	+	0.123237i
$71$	3.51901e10			0.274708			0.137354	−	0.990522i	$-0.456140\pi$
$71$	3.51901e10			0.274708			0.137354	+	0.990522i	$0.456140\pi$
$72$		0			0
$73$	1.08806e11	−	1.08806e11i	0.718980	−	0.718980i	−0.249416	−	0.968396i	$-0.580239\pi$
$73$	1.08806e11	−	1.08806e11i	0.718980	−	0.718980i	0.968396	+	0.249416i	$0.0802387\pi$
$74$			5.09823e10i			0.310477i
$75$		0			0
$76$	−2.98730e10			−0.155023
$77$	−1.39204e11	−	1.39204e11i	−0.667894	−	0.667894i
$78$		0			0
$79$		−	1.31009e11i		−	0.538939i	−0.963009	−	0.269470i	$-0.913152\pi$
$79$		−	1.31009e11i		−	0.538939i	0.963009	−	0.269470i	$-0.0868484\pi$
$80$	1.60166e10	−	6.35487e10i	0.0610986	−	0.242419i
$81$		0			0
$82$	−1.72343e10	−	1.72343e10i	−0.0566905	−	0.0566905i
$83$	−1.61869e11	+	1.61869e11i	−0.495102	+	0.495102i	−0.909909	−	0.414807i	$-0.863849\pi$
$83$	−1.61869e11	+	1.61869e11i	−0.495102	+	0.495102i	0.414807	+	0.909909i	$0.363849\pi$
$84$		0			0
$85$	−2.96752e11	+	1.77279e11i	−0.786829	+	0.470049i
$86$	5.00660e11			1.23752
$87$		0			0
$88$	−1.53778e11	+	1.53778e11i	−0.331129	+	0.331129i
$89$		−	8.98485e11i		−	1.80789i	−0.427654	−	0.903943i	$-0.640660\pi$
$89$		−	8.98485e11i		−	1.80789i	0.427654	−	0.903943i	$-0.359340\pi$
$90$		0			0
$91$	5.41878e11			0.954230
$92$	−2.26055e11	−	2.26055e11i	−0.372810	−	0.372810i
$93$		0			0
$94$		−	3.66966e11i		−	0.531935i
$95$	−2.21002e11	−	5.57007e10i	−0.300645	−	0.0757738i
$96$		0			0
$97$	−3.98581e11	−	3.98581e11i	−0.478505	−	0.478505i	0.426149	−	0.904653i	$-0.359870\pi$
$97$	−3.98581e11	−	3.98581e11i	−0.478505	−	0.478505i	−0.904653	+	0.426149i	$0.859870\pi$
$98$	2.17675e11	−	2.17675e11i	0.245727	−	0.245727i
$99$		0			0
$100$	2.36983e11	−	4.40271e11i	0.236983	−	0.440271i
$101$	−1.57852e12			−1.48704			−0.743520	−	0.668714i	$-0.766845\pi$
$101$	−1.57852e12			−1.48704			−0.743520	+	0.668714i	$0.766845\pi$
$102$		0			0
$103$	5.59456e11	−	5.59456e11i	0.468536	−	0.468536i	−0.432904	−	0.901440i	$-0.642511\pi$
$103$	5.59456e11	−	5.59456e11i	0.468536	−	0.468536i	0.901440	+	0.432904i	$0.142511\pi$
$104$		−	5.98608e11i		−	0.473089i
$105$		0			0
$106$	6.50644e11			0.458679
$107$	8.66153e11	+	8.66153e11i	0.577154	+	0.577154i	0.934118	−	0.356964i	$-0.116188\pi$
$107$	8.66153e11	+	8.66153e11i	0.577154	+	0.577154i	−0.356964	+	0.934118i	$0.616188\pi$
$108$		0			0
$109$			2.03012e12i			1.21049i	0.796037	+	0.605247i	$0.206926\pi$
$109$			2.03012e12i			1.21049i	−0.796037	+	0.605247i	$0.793074\pi$
$110$	−1.42438e12	+	8.50922e11i	−0.804028	+	0.480323i
$111$		0			0
$112$	2.48828e11	+	2.48828e11i	0.126064	+	0.126064i
$113$	2.69479e12	−	2.69479e12i	1.29436	−	1.29436i	0.362290	−	0.932065i	$-0.381995\pi$
$113$	2.69479e12	−	2.69479e12i	1.29436	−	1.29436i	0.932065	−	0.362290i	$-0.118005\pi$
$114$		0			0
$115$	−1.25087e12	−	2.09386e12i	−0.540784	−	0.905235i
$116$	−1.09473e11			−0.0449322
$117$		0			0
$118$	−1.83330e12	+	1.83330e12i	−0.679113	+	0.679113i
$119$		−	1.85609e12i		−	0.653608i
$120$		0			0
$121$	2.36745e12			0.754342
$122$	−3.03175e12	−	3.03175e12i	−0.919464	−	0.919464i
$123$		0			0
$124$		−	2.77585e12i		−	0.763600i
$125$	2.57414e12	−	2.81527e12i	0.674794	−	0.738006i
$126$		0			0
$127$	−3.64991e12	−	3.64991e12i	−0.869880	−	0.869880i	0.122579	−	0.992459i	$-0.460884\pi$
$127$	−3.64991e12	−	3.64991e12i	−0.869880	−	0.869880i	−0.992459	+	0.122579i	$0.960884\pi$
$128$	2.74878e11	−	2.74878e11i	0.0625000	−	0.0625000i
$129$		0			0
$130$	1.11615e12	−	4.42853e12i	0.231241	−	0.917486i
$131$	−7.99465e12			−1.58187			−0.790936	−	0.611899i	$-0.790406\pi$
$131$	−7.99465e12			−1.58187			−0.790936	+	0.611899i	$0.790406\pi$
$132$		0			0
$133$	8.65342e11	−	8.65342e11i	0.156343	−	0.156343i
$134$		−	2.67497e12i		−	0.462052i
$135$		0			0
$136$	−2.05041e12			−0.324046
$137$	−4.94921e12	−	4.94921e12i	−0.748535	−	0.748535i	0.225669	−	0.974204i	$-0.427543\pi$
$137$	−4.94921e12	−	4.94921e12i	−0.748535	−	0.748535i	−0.974204	+	0.225669i	$0.927543\pi$
$138$		0			0
$139$			5.34137e12i			0.740567i	0.928919	+	0.370283i	$0.120739\pi$
$139$			5.34137e12i			0.740567i	−0.928919	+	0.370283i	$0.879261\pi$
$140$	1.37688e12	+	2.30480e12i	0.182863	+	0.306101i
$141$		0			0
$142$	1.12608e12	+	1.12608e12i	0.137354	+	0.137354i
$143$	−1.07163e13	+	1.07163e13i	−1.25323	+	1.25323i
$144$		0			0
$145$	−8.09882e11	−	2.04121e11i	−0.0871393	−	0.0219623i
$146$	6.96360e12			0.718980
$147$		0			0
$148$	−1.63143e12	+	1.63143e12i	−0.155238	+	0.155238i
$149$		−	1.91205e12i		−	0.174736i	−0.996176	−	0.0873679i	$-0.972154\pi$
$149$		−	1.91205e12i		−	0.174736i	0.996176	−	0.0873679i	$-0.0278456\pi$
$150$		0			0
$151$	−1.03879e13			−0.876327			−0.438164	−	0.898895i	$-0.644371\pi$
$151$	−1.03879e13			−0.876327			−0.438164	+	0.898895i	$0.644371\pi$
$152$	−9.55937e11	−	9.55937e11i	−0.0775117	−	0.0775117i
$153$		0			0
$154$		−	8.90906e12i		−	0.667894i
$155$	5.17580e12	−	2.05358e13i	0.373239	−	1.48089i
$156$		0			0
$157$	−5.32378e12	−	5.32378e12i	−0.355486	−	0.355486i	0.506660	−	0.862146i	$-0.330880\pi$
$157$	−5.32378e12	−	5.32378e12i	−0.355486	−	0.355486i	−0.862146	+	0.506660i	$0.830880\pi$
$158$	4.19230e12	−	4.19230e12i	0.269470	−	0.269470i
$159$		0			0
$160$	2.54609e12	−	1.52103e12i	0.151759	−	0.0906602i
$161$	1.30964e13			0.751965
$162$		0			0
$163$	−1.54050e13	+	1.54050e13i	−0.821365	+	0.821365i	−0.986304	−	0.164938i	$-0.947257\pi$
$163$	−1.54050e13	+	1.54050e13i	−0.821365	+	0.821365i	0.164938	+	0.986304i	$0.447257\pi$
$164$		−	1.10299e12i		−	0.0566905i
$165$		0			0
$166$	−1.03596e13			−0.495102
$167$	4.16031e12	+	4.16031e12i	0.191790	+	0.191790i	0.796469	−	0.604679i	$-0.206699\pi$
$167$	4.16031e12	+	4.16031e12i	0.191790	+	0.191790i	−0.604679	+	0.796469i	$0.706699\pi$
$168$		0			0
$169$		−	1.84173e13i		−	0.790505i
$170$	−1.51690e13	−	3.82315e12i	−0.628439	−	0.158390i
$171$		0			0
$172$	1.60211e13	+	1.60211e13i	0.618760	+	0.618760i
$173$	−1.69171e13	+	1.69171e13i	−0.631030	+	0.631030i	−0.948326	−	0.317297i	$-0.897225\pi$
$173$	−1.69171e13	+	1.69171e13i	−0.631030	+	0.631030i	0.317297	+	0.948326i	$0.397225\pi$
$174$		0			0
$175$	5.88871e12	+	1.96183e13i	0.205018	+	0.683019i
$176$	−9.84177e12			−0.331129
$177$		0			0
$178$	2.87515e13	−	2.87515e13i	0.903943	−	0.903943i
$179$			4.51000e13i			1.37107i	0.728041	+	0.685533i	$0.240431\pi$
$179$			4.51000e13i			1.37107i	−0.728041	+	0.685533i	$0.759569\pi$
$180$		0			0
$181$	1.58853e13			0.451777			0.225889	−	0.974153i	$-0.427471\pi$
$181$	1.58853e13			0.451777			0.225889	+	0.974153i	$0.427471\pi$
$182$	1.73401e13	+	1.73401e13i	0.477115	+	0.477115i
$183$		0			0
$184$		−	1.44675e13i		−	0.372810i
$185$	−1.51113e13	+	9.02746e12i	−0.376940	+	0.225183i
$186$		0			0
$187$	3.67065e13	+	3.67065e13i	0.858408	+	0.858408i
$188$	1.17429e13	−	1.17429e13i	0.265968	−	0.265968i
$189$		0			0
$190$	−5.28963e12	−	8.85448e12i	−0.112436	−	0.188209i
$191$	3.46025e13			0.712701			0.356350	−	0.934352i	$-0.384021\pi$
$191$	3.46025e13			0.712701			0.356350	+	0.934352i	$0.384021\pi$
$192$		0			0
$193$	−7.29983e12	+	7.29983e12i	−0.141244	+	0.141244i	−0.774193	−	0.632949i	$-0.781844\pi$
$193$	−7.29983e12	+	7.29983e12i	−0.141244	+	0.141244i	0.632949	+	0.774193i	$0.281844\pi$
$194$		−	2.55092e13i		−	0.478505i
$195$		0			0
$196$	1.39312e13			0.245727
$197$	4.95012e13	+	4.95012e13i	0.846873	+	0.846873i	0.989742	−	0.142868i	$-0.0456326\pi$
$197$	4.95012e13	+	4.95012e13i	0.846873	+	0.846873i	−0.142868	+	0.989742i	$0.545633\pi$
$198$		0			0
$199$			1.16610e14i			1.87766i	0.344386	+	0.938828i	$0.388087\pi$
$199$			1.16610e14i			1.87766i	−0.344386	+	0.938828i	$0.611913\pi$
$200$	2.16722e13	−	6.50521e12i	0.338627	−	0.101644i
$201$		0			0
$202$	−5.05127e13	−	5.05127e13i	−0.743520	−	0.743520i
$203$	3.17113e12	−	3.17113e12i	0.0453146	−	0.0453146i
$204$		0			0
$205$	2.05662e12	−	8.16000e12i	0.0277097	−	0.109943i
$206$	3.58052e13			0.468536
$207$		0			0
$208$	1.91555e13	−	1.91555e13i	0.236544	−	0.236544i
$209$			3.42265e13i			0.410662i
$210$		0			0
$211$	3.42858e13			0.388525			0.194263	−	0.980950i	$-0.437769\pi$
$211$	3.42858e13			0.388525			0.194263	+	0.980950i	$0.437769\pi$
$212$	2.08206e13	+	2.08206e13i	0.229339	+	0.229339i
$213$		0			0
$214$			5.54338e13i			0.577154i
$215$	8.86522e13	+	1.48398e14i	0.897550	+	1.50244i
$216$		0			0
$217$	8.04090e13	+	8.04090e13i	0.770098	+	0.770098i
$218$	−6.49639e13	+	6.49639e13i	−0.605247	+	0.605247i
$219$		0			0
$220$	−7.28098e13	−	1.83508e13i	−0.642175	−	0.161852i
$221$	−1.42887e14			−1.22642
$222$		0			0
$223$	−6.17679e13	+	6.17679e13i	−0.502265	+	0.502265i	−0.912141	−	0.409876i	$-0.865572\pi$
$223$	−6.17679e13	+	6.17679e13i	−0.502265	+	0.502265i	0.409876	+	0.912141i	$0.365572\pi$
$224$			1.59250e13i			0.126064i
$225$		0			0
$226$	1.72466e14			1.29436
$227$	5.96352e13	+	5.96352e13i	0.435861	+	0.435861i	0.890616	−	0.454756i	$-0.150273\pi$
$227$	5.96352e13	+	5.96352e13i	0.435861	+	0.435861i	−0.454756	+	0.890616i	$0.650273\pi$
$228$		0			0
$229$			6.56031e13i			0.454895i	0.973790	+	0.227448i	$0.0730381\pi$
$229$			6.56031e13i			0.454895i	−0.973790	+	0.227448i	$0.926962\pi$
$230$	2.69759e13	−	1.07031e14i	0.182225	−	0.723009i
$231$		0			0
$232$	−3.50312e12	−	3.50312e12i	−0.0224661	−	0.0224661i
$233$	7.61404e13	−	7.61404e13i	0.475860	−	0.475860i	−0.427945	−	0.903805i	$-0.640762\pi$
$233$	7.61404e13	−	7.61404e13i	0.475860	−	0.475860i	0.903805	+	0.427945i	$0.140762\pi$
$234$		0			0
$235$	1.08770e14	−	6.49789e13i	0.645807	−	0.385803i
$236$	−1.17331e14			−0.679113
$237$		0			0
$238$	5.93949e13	−	5.93949e13i	0.326804	−	0.326804i
$239$			4.35416e13i			0.233624i	0.993154	+	0.116812i	$0.0372674\pi$
$239$			4.35416e13i			0.233624i	−0.993154	+	0.116812i	$0.962733\pi$
$240$		0			0
$241$	−3.05059e14			−1.55697			−0.778486	−	0.627662i	$-0.784012\pi$
$241$	−3.05059e14			−1.55697			−0.778486	+	0.627662i	$0.784012\pi$
$242$	7.57583e13	+	7.57583e13i	0.377171	+	0.377171i
$243$		0			0
$244$		−	1.94032e14i		−	0.919464i
$245$	1.03064e14	+	2.59759e13i	0.476550	+	0.120108i
$246$		0			0
$247$	−6.66165e13	−	6.66165e13i	−0.293360	−	0.293360i
$248$	8.88272e13	−	8.88272e13i	0.381800	−	0.381800i
$249$		0			0
$250$	1.72461e14	−	7.71631e12i	0.706400	−	0.0316060i
$251$	−6.24088e13			−0.249577			−0.124788	−	0.992183i	$-0.539825\pi$
$251$	−6.24088e13			−0.249577			−0.124788	+	0.992183i	$0.539825\pi$
$252$		0			0
$253$	−2.58999e14	+	2.58999e14i	−0.987585	+	0.987585i
$254$		−	2.33594e14i		−	0.869880i
$255$		0			0
$256$	1.75922e13			0.0625000
$257$	−5.89058e13	−	5.89058e13i	−0.204437	−	0.204437i	0.597461	−	0.801898i	$-0.296176\pi$
$257$	−5.89058e13	−	5.89058e13i	−0.204437	−	0.204437i	−0.801898	+	0.597461i	$0.796176\pi$
$258$		0			0
$259$		−	9.45165e13i		−	0.313119i
$260$	1.77430e14	−	1.05996e14i	0.574363	−	0.343123i
$261$		0			0
$262$	−2.55829e14	−	2.55829e14i	−0.790936	−	0.790936i
$263$	−3.16432e14	+	3.16432e14i	−0.956193	+	0.956193i	−0.999080	−	0.0428873i	$-0.986344\pi$
$263$	−3.16432e14	+	3.16432e14i	−0.956193	+	0.956193i	0.0428873	+	0.999080i	$0.486344\pi$
$264$		0			0
$265$	1.15210e14	+	1.92854e14i	0.332671	+	0.556868i
$266$	5.53819e13			0.156343
$267$		0			0
$268$	8.55991e13	−	8.55991e13i	0.231026	−	0.231026i
$269$		−	6.27369e13i		−	0.165581i	−0.996567	−	0.0827903i	$-0.973617\pi$
$269$		−	6.27369e13i		−	0.165581i	0.996567	−	0.0827903i	$-0.0263832\pi$
$270$		0			0
$271$	−2.92212e14			−0.737705			−0.368853	−	0.929488i	$-0.620249\pi$
$271$	−2.92212e14			−0.737705			−0.368853	+	0.929488i	$0.620249\pi$
$272$	−6.56130e13	−	6.56130e13i	−0.162023	−	0.162023i
$273$		0			0
$274$		−	3.16749e14i		−	0.748535i
$275$	−5.04433e14	−	2.71520e14i	−1.16629	−	0.627777i
$276$		0			0
$277$	4.26675e14	+	4.26675e14i	0.944535	+	0.944535i	0.998541	−	0.0540053i	$-0.0171988\pi$
$277$	4.26675e14	+	4.26675e14i	0.944535	+	0.944535i	−0.0540053	+	0.998541i	$0.517199\pi$
$278$	−1.70924e14	+	1.70924e14i	−0.370283	+	0.370283i
$279$		0			0
$280$	−2.96934e13	+	1.17814e14i	−0.0616186	+	0.244482i
$281$	−3.02979e14			−0.615424			−0.307712	−	0.951480i	$-0.599563\pi$
$281$	−3.02979e14			−0.615424			−0.307712	+	0.951480i	$0.599563\pi$
$282$		0			0
$283$	3.57301e14	−	3.57301e14i	0.695529	−	0.695529i	−0.267914	−	0.963443i	$-0.586334\pi$
$283$	3.57301e14	−	3.57301e14i	0.695529	−	0.695529i	0.963443	+	0.267914i	$0.0863343\pi$
$284$			7.20694e13i			0.137354i
$285$		0			0
$286$	−6.85845e14			−1.25323
$287$	3.19508e13	+	3.19508e13i	0.0571730	+	0.0571730i
$288$		0			0
$289$		−	9.31923e13i		−	0.159953i
$290$	−1.93844e13	−	3.24481e13i	−0.0325885	−	0.0545508i
$291$		0			0
$292$	2.22835e14	+	2.22835e14i	0.359490	+	0.359490i
$293$	−5.09282e13	+	5.09282e13i	−0.0804920	+	0.0804920i	−0.746207	−	0.665715i	$-0.768127\pi$
$293$	−5.09282e13	+	5.09282e13i	−0.0804920	+	0.0804920i	0.665715	+	0.746207i	$0.268127\pi$
$294$		0			0
$295$	−8.68022e14	−	2.18774e14i	−1.31704	−	0.331943i
$296$	−1.04412e14			−0.155238
$297$		0			0
$298$	6.11856e13	−	6.11856e13i	0.0873679	−	0.0873679i
$299$		−	1.00820e15i		−	1.41098i
$300$		0			0
$301$	−9.28179e14			−1.24805
$302$	−3.32413e14	−	3.32413e14i	−0.438164	−	0.438164i
$303$		0			0
$304$		−	6.11799e13i		−	0.0775117i
$305$	3.61789e14	−	1.43546e15i	0.449424	−	1.78316i
$306$		0			0
$307$	2.54088e14	+	2.54088e14i	0.303496	+	0.303496i	0.842380	−	0.538884i	$-0.181154\pi$
$307$	2.54088e14	+	2.54088e14i	0.303496	+	0.303496i	−0.538884	+	0.842380i	$0.681154\pi$
$308$	2.85090e14	−	2.85090e14i	0.333947	−	0.333947i
$309$		0			0
$310$	8.22772e14	−	4.91521e14i	0.927064	−	0.553825i
$311$	6.59240e14			0.728587			0.364293	−	0.931284i	$-0.381311\pi$
$311$	6.59240e14			0.728587			0.364293	+	0.931284i	$0.381311\pi$
$312$		0			0
$313$	4.43077e14	−	4.43077e14i	0.471209	−	0.471209i	−0.431097	−	0.902306i	$-0.641873\pi$
$313$	4.43077e14	−	4.43077e14i	0.471209	−	0.471209i	0.902306	+	0.431097i	$0.141873\pi$
$314$		−	3.40722e14i		−	0.355486i
$315$		0			0
$316$	2.68307e14			0.269470
$317$	1.11031e15	+	1.11031e15i	1.09418	+	1.09418i	0.995077	+	0.0991001i	$0.0315964\pi$
$317$	1.11031e15	+	1.11031e15i	1.09418	+	1.09418i	0.0991001	+	0.995077i	$0.468404\pi$
$318$		0			0
$319$			1.25426e14i			0.119027i
$320$	1.30148e14	+	3.28021e13i	0.121209	+	0.0305493i
$321$		0			0
$322$	4.19086e14	+	4.19086e14i	0.375983	+	0.375983i
$323$	−2.28181e14	+	2.28181e14i	−0.200939	+	0.200939i
$324$		0			0
$325$	1.51027e15	−	4.53330e14i	1.28161	−	0.384693i
$326$	−9.85921e14			−0.821365
$327$		0			0
$328$	3.52958e13	−	3.52958e13i	0.0283453	−	0.0283453i
$329$			6.80322e14i			0.536462i
$330$		0			0
$331$	2.05156e15			1.55997			0.779985	−	0.625798i	$-0.215227\pi$
$331$	2.05156e15			1.55997			0.779985	+	0.625798i	$0.215227\pi$
$332$	−3.31508e14	−	3.31508e14i	−0.247551	−	0.247551i
$333$		0			0
$334$			2.66260e14i			0.191790i
$335$	7.92872e14	−	4.73659e14i	0.560963	−	0.335118i
$336$		0			0
$337$	−1.38553e15	−	1.38553e15i	−0.945884	−	0.945884i	0.0527249	−	0.998609i	$-0.483209\pi$
$337$	−1.38553e15	−	1.38553e15i	−0.945884	−	0.945884i	−0.998609	+	0.0527249i	$0.983209\pi$
$338$	5.89352e14	−	5.89352e14i	0.395253	−	0.395253i
$339$		0			0
$340$	−3.63067e14	−	6.07749e14i	−0.235025	−	0.393415i
$341$	−3.18038e15			−2.02280
$342$		0			0
$343$	−1.22469e15	+	1.22469e15i	−0.752073	+	0.752073i
$344$			1.02535e15i			0.618760i
$345$		0			0
$346$	−1.08270e15			−0.631030
$347$	−5.07241e14	−	5.07241e14i	−0.290561	−	0.290561i	0.546741	−	0.837302i	$-0.315868\pi$
$347$	−5.07241e14	−	5.07241e14i	−0.290561	−	0.290561i	−0.837302	+	0.546741i	$0.815868\pi$
$348$		0			0
$349$			5.27073e14i			0.291688i	0.989308	+	0.145844i	$0.0465898\pi$
$349$			5.27073e14i			0.291688i	−0.989308	+	0.145844i	$0.953410\pi$
$350$	−4.39346e14	+	8.16223e14i	−0.239000	+	0.444018i
$351$		0			0
$352$	−3.14937e14	−	3.14937e14i	−0.165564	−	0.165564i
$353$	2.96169e13	−	2.96169e13i	0.0153070	−	0.0153070i	−0.699412	−	0.714719i	$-0.746555\pi$
$353$	2.96169e13	−	2.96169e13i	0.0153070	−	0.0153070i	0.714719	+	0.699412i	$0.246555\pi$
$354$		0			0
$355$	−1.34379e14	+	5.33172e14i	−0.0671370	+	0.266377i
$356$	1.84010e15			0.903943
$357$		0			0
$358$	−1.44320e15	+	1.44320e15i	−0.685533	+	0.685533i
$359$		−	2.28760e15i		−	1.06859i	−0.845297	−	0.534297i	$-0.820577\pi$
$359$		−	2.28760e15i		−	1.06859i	0.845297	−	0.534297i	$-0.179423\pi$
$360$		0			0
$361$	2.00055e15			0.903871
$362$	5.08330e14	+	5.08330e14i	0.225889	+	0.225889i
$363$		0			0
$364$			1.10977e15i			0.477115i
$365$	1.23305e15	+	2.06404e15i	0.521463	+	0.872892i
$366$		0			0
$367$	−2.81458e15	−	2.81458e15i	−1.15191	−	1.15191i	−0.986170	−	0.165739i	$-0.946999\pi$
$367$	−2.81458e15	−	2.81458e15i	−1.15191	−	1.15191i	−0.165739	−	0.986170i	$-0.553001\pi$
$368$	4.62961e14	−	4.62961e14i	0.186405	−	0.186405i
$369$		0			0
$370$	−7.72442e14	−	1.94684e14i	−0.301062	−	0.0758788i
$371$	−1.20624e15			−0.462582
$372$		0			0
$373$	−7.01497e14	+	7.01497e14i	−0.260479	+	0.260479i	−0.825249	−	0.564770i	$-0.808965\pi$
$373$	−7.01497e14	+	7.01497e14i	−0.260479	+	0.260479i	0.564770	+	0.825249i	$0.308965\pi$
$374$			2.34922e15i			0.858408i
$375$		0			0
$376$	7.51546e14			0.265968
$377$	−2.44123e14	−	2.44123e14i	−0.0850276	−	0.0850276i
$378$		0			0
$379$			3.39861e15i			1.14674i	0.819295	+	0.573372i	$0.194365\pi$
$379$			3.39861e15i			1.14674i	−0.819295	+	0.573372i	$0.805635\pi$
$380$	1.14075e14	−	4.52612e14i	0.0378869	−	0.150323i
$381$		0			0
$382$	1.10728e15	+	1.10728e15i	0.356350	+	0.356350i
$383$	−1.42429e14	+	1.42429e14i	−0.0451238	+	0.0451238i	−0.729309	−	0.684185i	$-0.760158\pi$
$383$	−1.42429e14	+	1.42429e14i	−0.0451238	+	0.0451238i	0.684185	+	0.729309i	$0.260158\pi$
$384$		0			0
$385$	2.64068e15	−	1.57753e15i	0.810870	−	0.484411i
$386$	−4.67189e14			−0.141244
$387$		0			0
$388$	8.16294e14	−	8.16294e14i	0.239252	−	0.239252i
$389$			5.00987e15i			1.44587i	0.690916	+	0.722935i	$0.257207\pi$
$389$			5.00987e15i			1.44587i	−0.690916	+	0.722935i	$0.742793\pi$
$390$		0			0
$391$	−3.45338e15			−0.966460
$392$	4.45799e14	+	4.45799e14i	0.122863	+	0.122863i
$393$		0			0
$394$			3.16808e15i			0.846873i
$395$	1.98495e15	+	5.00281e14i	0.522597	+	0.131714i
$396$		0			0
$397$	−3.66883e14	−	3.66883e14i	−0.0937097	−	0.0937097i	0.658698	−	0.752408i	$-0.271108\pi$
$397$	−3.66883e14	−	3.66883e14i	−0.0937097	−	0.0937097i	−0.752408	+	0.658698i	$0.771108\pi$
$398$	−3.73151e15	+	3.73151e15i	−0.938828	+	0.938828i
$399$		0			0
$400$	9.01676e14	+	4.85342e14i	0.220136	+	0.118492i
$401$	2.47959e15			0.596368			0.298184	−	0.954508i	$-0.403619\pi$
$401$	2.47959e15			0.596368			0.298184	+	0.954508i	$0.403619\pi$
$402$		0			0
$403$	6.19012e15	−	6.19012e15i	1.44500	−	1.44500i
$404$		−	3.23282e15i		−	0.743520i
$405$		0			0
$406$	2.02952e14			0.0453146
$407$	1.86919e15	+	1.86919e15i	0.411231	+	0.411231i
$408$		0			0
$409$		−	1.83428e15i		−	0.391855i	−0.980618	−	0.195927i	$-0.937228\pi$
$409$		−	1.83428e15i		−	0.391855i	0.980618	−	0.195927i	$-0.0627717\pi$
$410$	3.26932e14	−	1.95308e14i	0.0688263	−	0.0411166i
$411$		0			0
$412$	1.14577e15	+	1.14577e15i	0.234268	+	0.234268i
$413$	3.39878e15	−	3.39878e15i	0.684892	−	0.684892i
$414$		0			0
$415$	−1.83438e15	−	3.07063e15i	−0.359088	−	0.601089i
$416$	1.22595e15			0.236544
$417$		0			0
$418$	−1.09525e15	+	1.09525e15i	−0.205331	+	0.205331i
$419$			1.64549e15i			0.304097i	0.988373	+	0.152048i	$0.0485870\pi$
$419$			1.64549e15i			0.304097i	−0.988373	+	0.152048i	$0.951413\pi$
$420$		0			0
$421$	−4.22560e15			−0.758919			−0.379459	−	0.925208i	$-0.623890\pi$
$421$	−4.22560e15			−0.758919			−0.379459	+	0.925208i	$0.623890\pi$
$422$	1.09714e15	+	1.09714e15i	0.194263	+	0.194263i
$423$		0			0
$424$			1.33252e15i			0.229339i
$425$	−1.55279e15	−	5.17312e15i	−0.263499	−	0.877847i
$426$		0			0
$427$	5.62060e15	+	5.62060e15i	0.927289	+	0.927289i
$428$	−1.77388e15	+	1.77388e15i	−0.288577	+	0.288577i
$429$		0			0
$430$	−1.91185e15	+	7.58559e15i	−0.302443	+	1.19999i
$431$	1.10167e16			1.71866			0.859329	−	0.511423i	$-0.170882\pi$
$431$	1.10167e16			1.71866			0.859329	+	0.511423i	$0.170882\pi$
$432$		0			0
$433$	−3.63481e15	+	3.63481e15i	−0.551512	+	0.551512i	−0.926877	−	0.375365i	$-0.877517\pi$
$433$	−3.63481e15	+	3.63481e15i	−0.551512	+	0.551512i	0.375365	+	0.926877i	$0.377517\pi$
$434$			5.14617e15i			0.770098i
$435$		0			0
$436$	−4.15769e15			−0.605247
$437$	−1.61003e15	−	1.61003e15i	−0.231177	−	0.231177i
$438$		0			0
$439$		−	1.02927e15i		−	0.143794i	−0.997412	−	0.0718969i	$-0.977095\pi$
$439$		−	1.02927e15i		−	0.143794i	0.997412	−	0.0718969i	$-0.0229053\pi$
$440$	−1.74269e15	−	2.91714e15i	−0.240162	−	0.402014i
$441$		0			0
$442$	−4.57239e15	−	4.57239e15i	−0.613210	−	0.613210i
$443$	−5.98471e15	+	5.98471e15i	−0.791810	+	0.791810i	−0.981788	−	0.189979i	$-0.939158\pi$
$443$	−5.98471e15	+	5.98471e15i	−0.791810	+	0.791810i	0.189979	+	0.981788i	$0.439158\pi$
$444$		0			0
$445$	1.36131e16	+	3.43101e15i	1.75306	+	0.441837i
$446$	−3.95314e15			−0.502265
$447$		0			0
$448$	−5.09599e14	+	5.09599e14i	−0.0630319	+	0.0630319i
$449$			5.85542e15i			0.714628i	0.933984	+	0.357314i	$0.116307\pi$
$449$			5.85542e15i			0.714628i	−0.933984	+	0.357314i	$0.883693\pi$
$450$		0			0
$451$	−1.26374e15			−0.150175
$452$	5.51892e15	+	5.51892e15i	0.647178	+	0.647178i
$453$		0			0
$454$			3.81665e15i			0.435861i
$455$	−2.06925e15	+	8.21010e15i	−0.233209	+	0.925294i
$456$		0			0
$457$	−7.20985e14	−	7.20985e14i	−0.0791460	−	0.0791460i	0.666426	−	0.745572i	$-0.267823\pi$
$457$	−7.20985e14	−	7.20985e14i	−0.0791460	−	0.0791460i	−0.745572	+	0.666426i	$0.767823\pi$
$458$	−2.09930e15	+	2.09930e15i	−0.227448	+	0.227448i
$459$		0			0
$460$	4.28823e15	−	2.56177e15i	0.452617	−	0.270392i
$461$	−6.66525e15			−0.694402			−0.347201	−	0.937791i	$-0.612868\pi$
$461$	−6.66525e15			−0.694402			−0.347201	+	0.937791i	$0.612868\pi$
$462$		0			0
$463$	1.04215e16	−	1.04215e16i	1.05789	−	1.05789i	0.0596773	−	0.998218i	$-0.480993\pi$
$463$	1.04215e16	−	1.04215e16i	1.05789	−	1.05789i	0.998218	−	0.0596773i	$-0.0190072\pi$
$464$		−	2.24200e14i		−	0.0224661i
$465$		0			0
$466$	4.87298e15			0.475860
$467$	−1.42258e16	−	1.42258e16i	−1.37143	−	1.37143i	−0.858325	−	0.513107i	$-0.828494\pi$
$467$	−1.42258e16	−	1.42258e16i	−1.37143	−	1.37143i	−0.513107	−	0.858325i	$-0.671506\pi$
$468$		0			0
$469$			4.95916e15i			0.465984i
$470$	5.55997e15	+	1.40132e15i	0.515805	+	0.130002i
$471$		0			0
$472$	−3.75460e15	−	3.75460e15i	−0.339556	−	0.339556i
$473$	1.83559e16	−	1.83559e16i	1.63911	−	1.63911i
$474$		0			0
$475$	1.68786e15	−	3.13574e15i	0.146952	−	0.273010i
$476$	3.80127e15			0.326804
$477$		0			0
$478$	−1.39333e15	+	1.39333e15i	−0.116812	+	0.116812i
$479$			1.03779e16i			0.859202i	0.903019	+	0.429601i	$0.141346\pi$
$479$			1.03779e16i			0.859202i	−0.903019	+	0.429601i	$0.858654\pi$
$480$		0			0
$481$	−7.27616e15			−0.587532
$482$	−9.76188e15	−	9.76188e15i	−0.778486	−	0.778486i
$483$		0			0
$484$			4.84853e15i			0.377171i
$485$	7.56102e15	−	4.51693e15i	0.580938	−	0.347050i
$486$		0			0
$487$	−1.20111e16	−	1.20111e16i	−0.900347	−	0.900347i	0.0951192	−	0.995466i	$-0.469677\pi$
$487$	−1.20111e16	−	1.20111e16i	−0.900347	−	0.900347i	−0.995466	+	0.0951192i	$0.969677\pi$
$488$	6.20903e15	−	6.20903e15i	0.459732	−	0.459732i
$489$		0			0
$490$	2.46681e15	+	4.12926e15i	0.178221	+	0.298329i
$491$	−6.06257e15			−0.432681			−0.216341	−	0.976318i	$-0.569412\pi$
$491$	−6.06257e15			−0.432681			−0.216341	+	0.976318i	$0.569412\pi$
$492$		0			0
$493$	−8.36191e14	+	8.36191e14i	−0.0582404	+	0.0582404i
$494$		−	4.26346e15i		−	0.293360i
$495$		0			0
$496$	5.68494e15			0.381800
$497$	−2.08766e15	−	2.08766e15i	−0.138523	−	0.138523i
$498$		0			0
$499$			2.32445e16i			1.50562i	0.658235	+	0.752812i	$0.271303\pi$
$499$			2.32445e16i			1.50562i	−0.658235	+	0.752812i	$0.728697\pi$
$500$	5.76567e15	+	5.27183e15i	0.369003	+	0.337397i
$501$		0			0
$502$	−1.99708e15	−	1.99708e15i	−0.124788	−	0.124788i
$503$	5.32163e15	−	5.32163e15i	0.328576	−	0.328576i	−0.523469	−	0.852045i	$-0.675362\pi$
$503$	5.32163e15	−	5.32163e15i	0.328576	−	0.328576i	0.852045	+	0.523469i	$0.175362\pi$
$504$		0			0
$505$	6.02785e15	−	2.39165e16i	0.363424	−	1.44195i
$506$	−1.65759e16			−0.987585
$507$		0			0
$508$	7.47501e15	−	7.47501e15i	0.434940	−	0.434940i
$509$		−	1.17956e16i		−	0.678287i	−0.940735	−	0.339144i	$-0.889863\pi$
$509$		−	1.17956e16i		−	0.678287i	0.940735	−	0.339144i	$-0.110137\pi$
$510$		0			0
$511$	−1.29099e16			−0.725099
$512$	5.62950e14	+	5.62950e14i	0.0312500	+	0.0312500i
$513$		0			0
$514$		−	3.76997e15i		−	0.204437i
$515$	6.34005e15	+	1.06128e16i	0.339820	+	0.568835i
$516$		0			0
$517$	−1.34542e16	−	1.34542e16i	−0.704556	−	0.704556i
$518$	3.02453e15	−	3.02453e15i	0.156559	−	0.156559i
$519$		0			0
$520$	9.06963e15	+	2.28588e15i	0.458743	+	0.115620i
$521$	−2.57819e16			−1.28911			−0.644553	−	0.764560i	$-0.722956\pi$
$521$	−2.57819e16			−1.28911			−0.644553	+	0.764560i	$0.722956\pi$
$522$		0			0
$523$	1.24797e16	−	1.24797e16i	0.609812	−	0.609812i	−0.333085	−	0.942897i	$-0.608090\pi$
$523$	1.24797e16	−	1.24797e16i	0.609812	−	0.609812i	0.942897	+	0.333085i	$0.108090\pi$
$524$		−	1.63730e16i		−	0.790936i
$525$		0			0
$526$	−2.02516e16			−0.956193
$527$	−2.12029e16	−	2.12029e16i	−0.989766	−	0.989766i
$528$		0			0
$529$		−	2.45218e15i		−	0.111897i
$530$	−2.48459e15	+	9.85803e15i	−0.112099	+	0.444770i
$531$		0			0
$532$	1.77222e15	+	1.77222e15i	0.0781714	+	0.0781714i
$533$	2.45967e15	−	2.45967e15i	0.107279	−	0.107279i
$534$		0			0
$535$	−1.64308e16	+	9.81570e15i	−0.700706	+	0.418599i
$536$	5.47834e15			0.231026
$537$		0			0
$538$	2.00758e15	−	2.00758e15i	0.0827903	−	0.0827903i
$539$		−	1.59614e16i		−	0.650938i
$540$		0			0
$541$	1.45220e16			0.579218			0.289609	−	0.957145i	$-0.406475\pi$
$541$	1.45220e16			0.579218			0.289609	+	0.957145i	$0.406475\pi$
$542$	−9.35080e15	−	9.35080e15i	−0.368853	−	0.368853i
$543$		0			0
$544$		−	4.19923e15i		−	0.162023i
$545$	−3.07587e16	−	7.75235e15i	−1.17379	−	0.295838i
$546$		0			0
$547$	−2.64412e16	−	2.64412e16i	−0.987089	−	0.987089i	0.0128284	−	0.999918i	$-0.495916\pi$
$547$	−2.64412e16	−	2.64412e16i	−0.987089	−	0.987089i	−0.999918	+	0.0128284i	$0.995916\pi$
$548$	1.01360e16	−	1.01360e16i	0.374267	−	0.374267i
$549$		0			0
$550$	−7.45323e15	−	2.48305e16i	−0.269258	−	0.897035i
$551$	−7.79694e14			−0.0278622
$552$		0			0
$553$	−7.77215e15	+	7.77215e15i	−0.271763	+	0.271763i
$554$			2.73072e16i			0.944535i
$555$		0			0
$556$	−1.09391e16			−0.370283
$557$	1.22128e16	+	1.22128e16i	0.408962	+	0.408962i	0.881377	−	0.472414i	$-0.156617\pi$
$557$	1.22128e16	+	1.22128e16i	0.408962	+	0.408962i	−0.472414	+	0.881377i	$0.656617\pi$
$558$		0			0
$559$			7.14539e16i			2.34183i
$560$	−4.72022e15	+	2.81984e15i	−0.153050	+	0.0914317i
$561$		0			0
$562$	−9.69532e15	−	9.69532e15i	−0.307712	−	0.307712i
$563$	−3.08347e15	+	3.08347e15i	−0.0968253	+	0.0968253i	−0.753860	−	0.657035i	$-0.771810\pi$
$563$	−3.08347e15	+	3.08347e15i	−0.0968253	+	0.0968253i	0.657035	+	0.753860i	$0.271810\pi$
$564$		0			0
$565$	3.05387e16	+	5.11197e16i	0.938772	+	1.57144i
$566$	2.28672e16			0.695529
$567$		0			0
$568$	−2.30622e15	+	2.30622e15i	−0.0686769	+	0.0686769i
$569$			4.92549e16i			1.45136i	0.688031	+	0.725681i	$0.258475\pi$
$569$			4.92549e16i			1.45136i	−0.688031	+	0.725681i	$0.741525\pi$
$570$		0			0
$571$	−3.71286e16			−1.07125			−0.535626	−	0.844455i	$-0.679924\pi$
$571$	−3.71286e16			−1.07125			−0.535626	+	0.844455i	$0.679924\pi$
$572$	−2.19470e16	−	2.19470e16i	−0.626614	−	0.626614i
$573$		0			0
$574$			2.04485e15i			0.0571730i
$575$	3.65011e16	−	1.09563e16i	1.00995	−	0.303151i
$576$		0			0
$577$	−1.58541e16	−	1.58541e16i	−0.429623	−	0.429623i	0.458877	−	0.888500i	$-0.348252\pi$
$577$	−1.58541e16	−	1.58541e16i	−0.429623	−	0.429623i	−0.888500	+	0.458877i	$0.848252\pi$
$578$	2.98215e15	−	2.98215e15i	0.0799766	−	0.0799766i
$579$		0			0
$580$	4.18039e14	−	1.65864e15i	0.0109812	−	0.0435696i
$581$	1.92058e16			0.499316
$582$		0			0
$583$	2.38549e16	−	2.38549e16i	0.607527	−	0.607527i
$584$			1.42615e16i			0.359490i
$585$		0			0
$586$	−3.25941e15			−0.0804920
$587$	−2.04641e16	−	2.04641e16i	−0.500224	−	0.500224i	0.411284	−	0.911507i	$-0.365081\pi$
$587$	−2.04641e16	−	2.04641e16i	−0.500224	−	0.500224i	−0.911507	+	0.411284i	$0.865081\pi$
$588$		0			0
$589$		−	1.97704e16i		−	0.473504i
$590$	−2.07759e16	−	3.47775e16i	−0.492548	−	0.824491i
$591$		0			0
$592$	−3.34117e15	−	3.34117e15i	−0.0776191	−	0.0776191i
$593$	−6.50583e15	+	6.50583e15i	−0.149615	+	0.149615i	−0.777946	−	0.628331i	$-0.783738\pi$
$593$	−6.50583e15	+	6.50583e15i	−0.149615	+	0.149615i	0.628331	+	0.777946i	$0.283738\pi$
$594$		0			0
$595$	2.81220e16	+	7.08778e15i	0.633788	+	0.159738i
$596$	3.91588e15			0.0873679
$597$		0			0
$598$	3.22624e16	−	3.22624e16i	0.705489	−	0.705489i
$599$			1.83187e16i			0.396583i	0.980143	+	0.198292i	$0.0635393\pi$
$599$			1.83187e16i			0.396583i	−0.980143	+	0.198292i	$0.936461\pi$
$600$		0			0
$601$	2.90777e16			0.617041			0.308520	−	0.951218i	$-0.400166\pi$
$601$	2.90777e16			0.617041			0.308520	+	0.951218i	$0.400166\pi$
$602$	−2.97017e16	−	2.97017e16i	−0.624026	−	0.624026i
$603$		0			0
$604$		−	2.12744e16i		−	0.438164i
$605$	−9.04049e15	+	3.58696e16i	−0.184357	+	0.731467i
$606$		0			0
$607$	5.05207e15	+	5.05207e15i	0.101004	+	0.101004i	0.755803	−	0.654799i	$-0.227247\pi$
$607$	5.05207e15	+	5.05207e15i	0.101004	+	0.101004i	−0.654799	+	0.755803i	$0.727247\pi$
$608$	1.95776e15	−	1.95776e15i	0.0387559	−	0.0387559i
$609$		0			0
$610$	5.75119e16	−	3.43574e16i	1.11629	−	0.666870i
$611$	5.23732e16			1.00661
$612$		0			0
$613$	1.21687e16	−	1.21687e16i	0.229340	−	0.229340i	−0.583077	−	0.812417i	$-0.698151\pi$
$613$	1.21687e16	−	1.21687e16i	0.229340	−	0.229340i	0.812417	+	0.583077i	$0.198151\pi$
$614$			1.62616e16i			0.303496i
$615$		0			0
$616$	1.82458e16			0.333947
$617$	6.86609e16	+	6.86609e16i	1.24451	+	1.24451i	0.958110	+	0.286400i	$0.0924587\pi$
$617$	6.86609e16	+	6.86609e16i	1.24451	+	1.24451i	0.286400	+	0.958110i	$0.407541\pi$
$618$		0			0
$619$			5.30269e16i			0.942655i	0.881958	+	0.471327i	$0.156225\pi$
$619$			5.30269e16i			0.942655i	−0.881958	+	0.471327i	$0.843775\pi$
$620$	4.20574e16	+	1.06000e16i	0.740444	+	0.186620i
$621$		0			0
$622$	2.10957e16	+	2.10957e16i	0.364293	+	0.364293i
$623$	−5.33027e16	+	5.33027e16i	−0.911635	+	0.911635i
$624$		0			0
$625$	3.28249e16	+	4.97518e16i	0.550711	+	0.834696i
$626$	2.83570e16			0.471209
$627$		0			0
$628$	1.09031e16	−	1.09031e16i	0.177743	−	0.177743i
$629$			2.49230e16i			0.402435i
$630$		0			0
$631$	−7.10226e16			−1.12518			−0.562588	−	0.826738i	$-0.690194\pi$
$631$	−7.10226e16			−1.12518			−0.562588	+	0.826738i	$0.690194\pi$
$632$	8.58583e15	+	8.58583e15i	0.134735	+	0.134735i
$633$		0			0
$634$			7.10597e16i			1.09418i
$635$	6.92382e16	−	4.13626e16i	1.05610	−	0.630908i
$636$		0			0
$637$	3.10665e16	+	3.10665e16i	0.465002	+	0.465002i
$638$	−4.01364e15	+	4.01364e15i	−0.0595134	+	0.0595134i
$639$		0			0
$640$	3.11506e15	+	5.21439e15i	0.0453301	+	0.0758794i
$641$	−1.46335e16			−0.210960			−0.105480	−	0.994421i	$-0.533638\pi$
$641$	−1.46335e16			−0.210960			−0.105480	+	0.994421i	$0.533638\pi$
$642$		0			0
$643$	−5.21919e16	+	5.21919e16i	−0.738478	+	0.738478i	−0.972283	−	0.233805i	$-0.924882\pi$
$643$	−5.21919e16	+	5.21919e16i	−0.738478	+	0.738478i	0.233805	+	0.972283i	$0.424882\pi$
$644$			2.68215e16i			0.375983i
$645$		0			0
$646$	−1.46036e16			−0.200939
$647$	−3.33260e16	−	3.33260e16i	−0.454315	−	0.454315i	0.442469	−	0.896784i	$-0.354103\pi$
$647$	−3.33260e16	−	3.33260e16i	−0.454315	−	0.454315i	−0.896784	+	0.442469i	$0.854103\pi$
$648$		0			0
$649$			1.34430e17i			1.79899i
$650$	6.28353e16	+	3.38221e16i	0.833150	+	0.448457i
$651$		0			0
$652$	−3.15495e16	−	3.15495e16i	−0.410683	−	0.410683i
$653$	6.47983e16	−	6.47983e16i	0.835766	−	0.835766i	−0.152532	−	0.988298i	$-0.548743\pi$
$653$	6.47983e16	−	6.47983e16i	0.835766	−	0.835766i	0.988298	+	0.152532i	$0.0487428\pi$
$654$		0			0
$655$	3.05289e16	−	1.21128e17i	0.386601	−	1.53390i
$656$	2.25893e15			0.0283453
$657$		0			0
$658$	−2.17703e16	+	2.17703e16i	−0.268231	+	0.268231i
$659$			1.34385e17i			1.64073i	0.571841	+	0.820365i	$0.306229\pi$
$659$			1.34385e17i			1.64073i	−0.571841	+	0.820365i	$0.693771\pi$
$660$		0			0
$661$	−4.46690e16			−0.535547			−0.267773	−	0.963482i	$-0.586288\pi$
$661$	−4.46690e16			−0.535547			−0.267773	+	0.963482i	$0.586288\pi$
$662$	6.56499e16	+	6.56499e16i	0.779985	+	0.779985i
$663$		0			0
$664$		−	2.12165e16i		−	0.247551i
$665$	9.80651e15	+	1.64154e16i	0.113393	+	0.189811i
$666$		0			0
$667$	−5.90010e15	−	5.90010e15i	−0.0670046	−	0.0670046i
$668$	−8.52032e15	+	8.52032e15i	−0.0958952	+	0.0958952i
$669$		0			0
$670$	4.05290e16	+	1.02148e16i	0.448040	+	0.112923i
$671$	−2.22309e17			−2.43569
$672$		0			0
$673$	1.25981e16	−	1.25981e16i	0.135586	−	0.135586i	−0.636056	−	0.771642i	$-0.719435\pi$
$673$	1.25981e16	−	1.25981e16i	0.135586	−	0.135586i	0.771642	+	0.636056i	$0.219435\pi$
$674$		−	8.86742e16i		−	0.945884i
$675$		0			0
$676$	3.77186e16			0.395253
$677$	8.47278e15	+	8.47278e15i	0.0880022	+	0.0880022i	0.749738	−	0.661735i	$-0.230180\pi$
$677$	8.47278e15	+	8.47278e15i	0.0880022	+	0.0880022i	−0.661735	+	0.749738i	$0.730180\pi$
$678$		0			0
$679$			4.72917e16i			0.482577i
$680$	7.82982e15	−	3.10661e16i	0.0791951	−	0.314220i
$681$		0			0
$682$	−1.01772e17	−	1.01772e17i	−1.01140	−	1.01140i
$683$	4.43256e15	−	4.43256e15i	0.0436647	−	0.0436647i	−0.684937	−	0.728602i	$-0.740170\pi$
$683$	4.43256e15	−	4.43256e15i	0.0436647	−	0.0436647i	0.728602	+	0.684937i	$0.240170\pi$
$684$		0			0
$685$	9.38857e16	−	5.60870e16i	0.908774	−	0.542898i
$686$	−7.83799e16			−0.752073
$687$		0			0
$688$	−3.28113e16	+	3.28113e16i	−0.309380	+	0.309380i
$689$			9.28596e16i			0.867983i
$690$		0			0
$691$	4.70857e16			0.432535			0.216267	−	0.976334i	$-0.430612\pi$
$691$	4.70857e16			0.432535			0.216267	+	0.976334i	$0.430612\pi$
$692$	−3.46463e16	−	3.46463e16i	−0.315515	−	0.315515i
$693$		0			0
$694$		−	3.24634e16i		−	0.290561i
$695$	−8.09281e16	−	2.03969e16i	−0.718109	−	0.180990i
$696$		0			0
$697$	−8.42507e15	−	8.42507e15i	−0.0734813	−	0.0734813i
$698$	−1.68663e16	+	1.68663e16i	−0.145844	+	0.145844i
$699$		0			0
$700$	−4.01782e16	+	1.20601e16i	−0.341509	+	0.102509i
$701$	−8.06515e15			−0.0679680			−0.0339840	−	0.999422i	$-0.510820\pi$
$701$	−8.06515e15			−0.0679680			−0.0339840	+	0.999422i	$0.510820\pi$
$702$		0			0
$703$	−1.16195e16	+	1.16195e16i	−0.0962623	+	0.0962623i
$704$		−	2.01559e16i		−	0.165564i
$705$		0			0
$706$	1.89548e15			0.0153070
$707$	9.36461e16	+	9.36461e16i	0.749848	+	0.749848i
$708$		0			0
$709$			1.43911e17i			1.13297i	0.824073	+	0.566484i	$0.191697\pi$
$709$			1.43911e17i			1.13297i	−0.824073	+	0.566484i	$0.808303\pi$
$710$	−2.13616e16	+	1.27614e16i	−0.166757	+	0.0996202i
$711$		0			0
$712$	5.88831e16	+	5.88831e16i	0.451971	+	0.451971i
$713$	1.49606e17	−	1.49606e17i	1.13871	−	1.13871i
$714$		0			0
$715$	−1.21443e17	−	2.03287e17i	−0.908943	−	1.52151i
$716$	−9.23648e16			−0.685533
$717$		0			0
$718$	7.32030e16	−	7.32030e16i	0.534297	−	0.534297i
$719$		−	9.19905e16i		−	0.665840i	−0.942955	−	0.332920i	$-0.891966\pi$
$719$		−	9.19905e16i		−	0.665840i	0.942955	−	0.332920i	$-0.108034\pi$
$720$		0			0
$721$	−6.63796e16			−0.472523
$722$	6.40176e16	+	6.40176e16i	0.451935	+	0.451935i
$723$		0			0
$724$			3.25331e16i			0.225889i
$725$	6.18534e15	−	1.14912e16i	0.0425927	−	0.0791294i
$726$		0			0
$727$	−3.54829e16	−	3.54829e16i	−0.240333	−	0.240333i	0.576655	−	0.816988i	$-0.304358\pi$
$727$	−3.54829e16	−	3.54829e16i	−0.240333	−	0.240333i	−0.816988	+	0.576655i	$0.804358\pi$
$728$	−3.55125e16	+	3.55125e16i	−0.238558	+	0.238558i
$729$		0			0
$730$	−2.65917e16	+	1.05507e17i	−0.175715	+	0.697178i
$731$	2.44750e17			1.60405
$732$		0			0
$733$	−2.35715e15	+	2.35715e15i	−0.0151972	+	0.0151972i	−0.714665	−	0.699467i	$-0.753421\pi$
$733$	−2.35715e15	+	2.35715e15i	−0.0151972	+	0.0151972i	0.699467	+	0.714665i	$0.253421\pi$
$734$		−	1.80133e17i		−	1.15191i
$735$		0			0
$736$	2.96295e16			0.186405
$737$	−9.80737e16	−	9.80737e16i	−0.611995	−	0.611995i
$738$		0			0
$739$		−	1.74369e17i		−	1.07054i	−0.844680	−	0.535271i	$-0.820209\pi$
$739$		−	1.74369e17i		−	1.07054i	0.844680	−	0.535271i	$-0.179791\pi$
$740$	−1.84882e16	−	3.09480e16i	−0.112591	−	0.188470i
$741$		0			0
$742$	−3.85996e16	−	3.85996e16i	−0.231291	−	0.231291i
$743$	−4.60511e16	+	4.60511e16i	−0.273720	+	0.273720i	−0.830596	−	0.556876i	$-0.812000\pi$
$743$	−4.60511e16	+	4.60511e16i	−0.273720	+	0.273720i	0.556876	+	0.830596i	$0.312000\pi$
$744$		0			0
$745$	2.89698e16	+	7.30148e15i	0.169437	+	0.0427045i
$746$	−4.48958e16			−0.260479
$747$		0			0
$748$	−7.51750e16	+	7.51750e16i	−0.429204	+	0.429204i
$749$		−	1.02769e17i		−	0.582066i
$750$		0			0
$751$	−2.46982e17			−1.37666			−0.688328	−	0.725400i	$-0.741655\pi$
$751$	−2.46982e17			−1.37666			−0.688328	+	0.725400i	$0.741655\pi$
$752$	2.40495e16	+	2.40495e16i	0.132984	+	0.132984i
$753$		0			0
$754$		−	1.56239e16i		−	0.0850276i
$755$	3.96679e16	−	1.57389e17i	0.214169	−	0.849753i
$756$		0			0
$757$	−8.81831e15	−	8.81831e15i	−0.0468608	−	0.0468608i	0.683288	−	0.730149i	$-0.260549\pi$
$757$	−8.81831e15	−	8.81831e15i	−0.0468608	−	0.0468608i	−0.730149	+	0.683288i	$0.760549\pi$
$758$	−1.08756e17	+	1.08756e17i	−0.573372	+	0.573372i
$759$		0			0
$760$	1.81340e16	−	1.08332e16i	0.0941047	−	0.0562178i
$761$	−9.59846e16			−0.494189			−0.247095	−	0.968991i	$-0.579476\pi$
$761$	−9.59846e16			−0.494189			−0.247095	+	0.968991i	$0.579476\pi$
$762$		0			0
$763$	1.20437e17	−	1.20437e17i	0.610398	−	0.610398i
$764$			7.08659e16i			0.356350i
$765$		0			0
$766$	−9.11546e15			−0.0451238
$767$	−2.61648e17	−	2.61648e17i	−1.28512	−	1.28512i
$768$		0			0
$769$		−	9.31936e16i		−	0.450638i	−0.974285	−	0.225319i	$-0.927658\pi$
$769$		−	9.31936e16i		−	0.450638i	0.974285	−	0.225319i	$-0.0723424\pi$
$770$	1.34983e17	+	3.40207e16i	0.647641	+	0.163230i
$771$		0			0
$772$	−1.49501e16	−	1.49501e16i	−0.0706218	−	0.0706218i
$773$	−1.71053e17	+	1.71053e17i	−0.801777	+	0.801777i	−0.983373	−	0.181596i	$-0.941874\pi$
$773$	−1.71053e17	+	1.71053e17i	−0.801777	+	0.801777i	0.181596	+	0.983373i	$0.441874\pi$
$774$		0			0
$775$	2.91378e17	+	1.56839e17i	1.34476	+	0.723842i
$776$	5.22428e16			0.239252
$777$		0			0
$778$	−1.60316e17	+	1.60316e17i	−0.722935	+	0.722935i
$779$		−	7.85584e15i		−	0.0351534i
$780$		0			0
$781$	8.25722e16			0.363855
$782$	−1.10508e17	−	1.10508e17i	−0.483230	−	0.483230i
$783$		0			0
$784$			2.85311e16i			0.122863i
$785$	1.00991e17	−	6.03319e16i	0.431585	−	0.257827i
$786$		0			0
$787$	−7.61701e16	−	7.61701e16i	−0.320580	−	0.320580i	0.528410	−	0.848990i	$-0.322789\pi$
$787$	−7.61701e16	−	7.61701e16i	−0.320580	−	0.320580i	−0.848990	+	0.528410i	$0.822789\pi$
$788$	−1.01378e17	+	1.01378e17i	−0.423437	+	0.423437i
$789$		0			0
$790$	4.75093e16	+	7.95273e16i	0.195441	+	0.327155i
$791$	−3.19737e17			−1.30537
$792$		0			0
$793$	4.32690e17	−	4.32690e17i	1.73995	−	1.73995i
$794$		−	2.34805e16i		−	0.0937097i
$795$		0			0
$796$	−2.38817e17			−0.938828
$797$	5.12571e16	+	5.12571e16i	0.199988	+	0.199988i	0.799995	−	0.600007i	$-0.204835\pi$
$797$	5.12571e16	+	5.12571e16i	0.199988	+	0.199988i	−0.600007	+	0.799995i	$0.704835\pi$
$798$		0			0
$799$		−	1.79393e17i		−	0.689486i
$800$	1.33227e16	+	4.43846e16i	0.0508220	+	0.169314i
$801$		0			0
$802$	7.93470e16	+	7.93470e16i	0.298184	+	0.298184i
$803$	2.55310e17	−	2.55310e17i	0.952300	−	0.952300i
$804$		0			0
$805$	−5.00109e16	+	1.98426e17i	−0.183776	+	0.729162i
$806$	3.96167e17			1.44500
$807$		0			0
$808$	1.03450e17	−	1.03450e17i	0.371760	−	0.371760i
$809$		−	1.73173e17i		−	0.617717i	−0.951108	−	0.308859i	$-0.900053\pi$
$809$		−	1.73173e17i		−	0.617717i	0.951108	−	0.308859i	$-0.0999470\pi$
$810$		0			0
$811$	−3.57991e17			−1.25819			−0.629095	−	0.777328i	$-0.716575\pi$
$811$	−3.57991e17			−1.25819			−0.629095	+	0.777328i	$0.716575\pi$
$812$	6.49447e15	+	6.49447e15i	0.0226573	+	0.0226573i
$813$		0			0
$814$			1.19628e17i			0.411231i
$815$	−1.74578e17	−	2.92231e17i	−0.595721	−	0.997195i
$816$		0			0
$817$	1.14107e17	+	1.14107e17i	0.383689	+	0.383689i
$818$	5.86969e16	−	5.86969e16i	0.195927	−	0.195927i
$819$		0			0
$820$	1.67117e16	+	4.21197e15i	0.0549714	+	0.0138548i
$821$	−1.62633e17			−0.531068			−0.265534	−	0.964102i	$-0.585548\pi$
$821$	−1.62633e17			−0.531068			−0.265534	+	0.964102i	$0.585548\pi$
$822$		0			0
$823$	−3.78884e17	+	3.78884e17i	−1.21929	+	1.21929i	−0.251409	+	0.967881i	$0.580894\pi$
$823$	−3.78884e17	+	3.78884e17i	−1.21929	+	1.21929i	−0.967881	+	0.251409i	$0.919106\pi$
$824$			7.33290e16i			0.234268i
$825$		0			0
$826$	2.17522e17			0.684892
$827$	3.48092e17	+	3.48092e17i	1.08808	+	1.08808i	0.995726	+	0.0923537i	$0.0294390\pi$
$827$	3.48092e17	+	3.48092e17i	1.08808	+	1.08808i	0.0923537	+	0.995726i	$0.470561\pi$
$828$		0			0
$829$		−	3.69157e17i		−	1.13732i	−0.822572	−	0.568661i	$-0.807461\pi$
$829$		−	3.69157e17i		−	1.13732i	0.822572	−	0.568661i	$-0.192539\pi$
$830$	3.95599e16	−	1.56960e17i	0.121000	−	0.480089i
$831$		0			0
$832$	3.92304e16	+	3.92304e16i	0.118272	+	0.118272i
$833$	1.06412e17	−	1.06412e17i	0.318507	−	0.318507i
$834$		0			0
$835$	−7.89205e16	+	4.71468e16i	−0.232847	+	0.139102i
$836$	−7.00958e16			−0.205331
$837$		0			0
$838$	−5.26558e16	+	5.26558e16i	−0.152048	+	0.152048i
$839$			2.28480e17i			0.655054i	0.944842	+	0.327527i	$0.106215\pi$
$839$			2.28480e17i			0.655054i	−0.944842	+	0.327527i	$0.893785\pi$
$840$		0			0
$841$	3.50958e17			0.991924
$842$	−1.35219e17	−	1.35219e17i	−0.379459	−	0.379459i
$843$		0			0
$844$			7.02172e16i			0.194263i
$845$	2.79043e17	+	7.03293e16i	0.766534	+	0.193195i
$846$		0			0
$847$	−1.40449e17	−	1.40449e17i	−0.380381	−	0.380381i
$848$	−4.26406e16	+	4.26406e16i	−0.114670	+	0.114670i
$849$		0			0
$850$	1.15851e17	−	2.15229e17i	0.307174	−	0.570673i
$851$	−1.75854e17			−0.462995
$852$		0			0
$853$	3.95302e17	−	3.95302e17i	1.02621	−	1.02621i	0.0265604	−	0.999647i	$-0.491545\pi$
$853$	3.95302e17	−	3.95302e17i	1.02621	−	1.02621i	0.999647	−	0.0265604i	$-0.00845542\pi$
$854$			3.59718e17i			0.927289i
$855$		0			0
$856$	−1.13528e17			−0.288577
$857$	1.52003e17	+	1.52003e17i	0.383680	+	0.383680i	0.872426	−	0.488746i	$-0.162546\pi$
$857$	1.52003e17	+	1.52003e17i	0.383680	+	0.383680i	−0.488746	+	0.872426i	$0.662546\pi$
$858$		0			0
$859$			1.83511e17i			0.456775i	0.973570	+	0.228388i	$0.0733453\pi$
$859$			1.83511e17i			0.456775i	−0.973570	+	0.228388i	$0.926655\pi$
$860$	−3.03918e17	+	1.81560e17i	−0.751218	+	0.448775i
$861$		0			0
$862$	3.52536e17	+	3.52536e17i	0.859329	+	0.859329i
$863$	−7.21471e16	+	7.21471e16i	−0.174644	+	0.174644i	−0.789016	−	0.614372i	$-0.789409\pi$
$863$	−7.21471e16	+	7.21471e16i	−0.174644	+	0.174644i	0.614372	+	0.789016i	$0.289409\pi$
$864$		0			0
$865$	−1.91714e17	−	3.20915e17i	−0.457674	−	0.766115i
$866$	−2.32628e17			−0.551512
$867$		0			0
$868$	−1.64678e17	+	1.64678e17i	−0.385049	+	0.385049i
$869$		−	3.07408e17i		−	0.713834i
$870$		0			0
$871$	3.81770e17			0.874366
$872$	−1.33046e17	−	1.33046e17i	−0.302624	−	0.302624i
$873$		0			0
$874$		−	1.03042e17i		−	0.231177i
$875$	−3.19727e17	+	1.43053e16i	−0.712412	+	0.0318750i
$876$		0			0
$877$	2.24375e17	+	2.24375e17i	0.493147	+	0.493147i	0.909296	−	0.416149i	$-0.136621\pi$
$877$	2.24375e17	+	2.24375e17i	0.493147	+	0.493147i	−0.416149	+	0.909296i	$0.636621\pi$
$878$	3.29365e16	−	3.29365e16i	0.0718969	−	0.0718969i
$879$		0			0
$880$	3.75824e16	−	1.49114e17i	0.0809261	−	0.321088i
$881$	4.52418e17			0.967575			0.483788	−	0.875185i	$-0.339261\pi$
$881$	4.52418e17			0.967575			0.483788	+	0.875185i	$0.339261\pi$
$882$		0			0
$883$	−2.62955e17	+	2.62955e17i	−0.554776	+	0.554776i	−0.927815	−	0.373040i	$-0.878315\pi$
$883$	−2.62955e17	+	2.62955e17i	−0.554776	+	0.554776i	0.373040	+	0.927815i	$0.378315\pi$
$884$		−	2.92633e17i		−	0.613210i
$885$		0			0
$886$	−3.83021e17			−0.791810
$887$	2.84621e17	+	2.84621e17i	0.584421	+	0.584421i	0.936115	−	0.351694i	$-0.114394\pi$
$887$	2.84621e17	+	2.84621e17i	0.584421	+	0.584421i	−0.351694	+	0.936115i	$0.614394\pi$
$888$		0			0
$889$			4.33062e17i			0.877283i
$890$	3.25827e17	+	5.45412e17i	0.655613	+	1.09745i
$891$		0			0
$892$	−1.26501e17	−	1.26501e17i	−0.251133	−	0.251133i
$893$	8.36363e16	−	8.36363e16i	0.164925	−	0.164925i
$894$		0			0
$895$	−6.83319e17	−	1.72222e17i	−1.32949	−	0.335081i
$896$	−3.26143e16			−0.0630319
$897$		0			0
$898$	−1.87373e17	+	1.87373e17i	−0.357314	+	0.357314i
$899$		−	7.24505e16i		−	0.137241i
$900$		0			0
$901$	3.18071e17			0.594532
$902$	−4.04396e16	−	4.04396e16i	−0.0750875	−	0.0750875i
$903$		0			0
$904$			3.53211e17i			0.647178i
$905$	−6.06607e16	+	2.40681e17i	−0.110412	+	0.438078i
$906$		0			0
$907$	−1.04107e17	−	1.04107e17i	−0.186997	−	0.186997i	0.607400	−	0.794396i	$-0.292213\pi$
$907$	−1.04107e17	−	1.04107e17i	−0.186997	−	0.186997i	−0.794396	+	0.607400i	$0.792213\pi$
$908$	−1.22133e17	+	1.22133e17i	−0.217930	+	0.217930i
$909$		0			0
$910$	−3.28939e17	+	1.96507e17i	−0.579251	+	0.346043i
$911$	−3.22588e17			−0.564337			−0.282168	−	0.959365i	$-0.591054\pi$
$911$	−3.22588e17			−0.564337			−0.282168	+	0.959365i	$0.591054\pi$
$912$		0			0
$913$	−3.79819e17	+	3.79819e17i	−0.655771	+	0.655771i
$914$		−	4.61430e16i		−	0.0791460i
$915$		0			0
$916$	−1.34355e17			−0.227448
$917$	4.74283e17	+	4.74283e17i	0.797667	+	0.797667i
$918$		0			0
$919$		−	3.75318e17i		−	0.623027i	−0.950242	−	0.311513i	$-0.899164\pi$
$919$		−	3.75318e17i		−	0.623027i	0.950242	−	0.311513i	$-0.100836\pi$
$920$	2.19200e17	+	5.52466e16i	0.361505	+	0.0911127i
$921$		0			0
$922$	−2.13288e17	−	2.13288e17i	−0.347201	−	0.347201i
$923$	−1.60714e17	+	1.60714e17i	−0.259922	+	0.259922i
$924$		0			0
$925$	−7.90716e16	−	2.63428e17i	−0.126232	−	0.420544i
$926$	6.66973e17			1.05789
$927$		0			0
$928$	7.17439e15	−	7.17439e15i	0.0112330	−	0.0112330i
$929$			1.22101e16i			0.0189943i	0.999955	+	0.00949717i	$0.00302309\pi$
$929$			1.22101e16i			0.0189943i	−0.999955	+	0.00949717i	$0.996977\pi$
$930$		0			0
$931$	9.92220e16			0.152374
$932$	1.55935e17	+	1.55935e17i	0.237930	+	0.237930i
$933$		0			0
$934$		−	9.10449e17i		−	1.37143i
$935$	−6.96318e17	+	4.15978e17i	−1.04217	+	0.622587i
$936$		0			0
$937$	6.76414e17	+	6.76414e17i	0.999482	+	0.999482i	1.00000		0.000518293i	$-0.000164978\pi$
$937$	6.76414e17	+	6.76414e17i	0.999482	+	0.999482i	−0.000518293		1.00000i	$0.500165\pi$
$938$	−1.58693e17	+	1.58693e17i	−0.232992	+	0.232992i
$939$		0			0
$940$	1.33077e17	+	2.22761e17i	0.192901	+	0.322903i
$941$	6.59259e17			0.949551			0.474775	−	0.880107i	$-0.342529\pi$
$941$	6.59259e17			0.949551			0.474775	+	0.880107i	$0.342529\pi$
$942$		0			0
$943$	5.94467e16	−	5.94467e16i	0.0845391	−	0.0845391i
$944$		−	2.40294e17i		−	0.339556i
$945$		0			0
$946$	1.17478e18			1.63911
$947$	1.50923e17	+	1.50923e17i	0.209245	+	0.209245i	0.803946	−	0.594702i	$-0.202730\pi$
$947$	1.50923e17	+	1.50923e17i	0.209245	+	0.209245i	−0.594702	+	0.803946i	$0.702730\pi$
$948$		0			0
$949$			9.93841e17i			1.36057i
$950$	1.54355e17	−	4.63320e16i	0.209981	−	0.0630288i
$951$		0			0
$952$	1.21641e17	+	1.21641e17i	0.163402	+	0.163402i
$953$	−2.51624e17	+	2.51624e17i	−0.335888	+	0.335888i	−0.854817	−	0.518929i	$-0.826331\pi$
$953$	−2.51624e17	+	2.51624e17i	−0.335888	+	0.335888i	0.518929	+	0.854817i	$0.326331\pi$
$954$		0			0
$955$	−1.32135e17	+	5.24269e17i	−0.174180	+	0.691089i
$956$	−8.91731e16			−0.116812
$957$		0			0
$958$	−3.32092e17	+	3.32092e17i	−0.429601	+	0.429601i
$959$			5.87224e17i			0.754905i
$960$		0			0
$961$	1.04943e18			1.33234
$962$	−2.32837e17	−	2.32837e17i	−0.293766	−	0.293766i
$963$		0			0
$964$		−	6.24760e17i		−	0.778486i
$965$	−8.27255e16	−	1.38477e17i	−0.102441	−	0.171480i
$966$		0			0
$967$	−5.27730e17	−	5.27730e17i	−0.645436	−	0.645436i	0.306451	−	0.951887i	$-0.400859\pi$
$967$	−5.27730e17	−	5.27730e17i	−0.645436	−	0.645436i	−0.951887	+	0.306451i	$0.900859\pi$
$968$	−1.55153e17	+	1.55153e17i	−0.188585	+	0.188585i
$969$		0			0
$970$	3.86494e17	+	9.74110e16i	0.463994	+	0.116944i
$971$	−3.09306e16			−0.0369040			−0.0184520	−	0.999830i	$-0.505874\pi$
$971$	−3.09306e16			−0.0369040			−0.0184520	+	0.999830i	$0.505874\pi$
$972$		0			0
$973$	3.16878e17	−	3.16878e17i	0.373435	−	0.373435i
$974$		−	7.68712e17i		−	0.900347i
$975$		0			0
$976$	3.97378e17			0.459732
$977$	−3.95237e17	−	3.95237e17i	−0.454455	−	0.454455i	0.442375	−	0.896830i	$-0.354136\pi$
$977$	−3.95237e17	−	3.95237e17i	−0.454455	−	0.454455i	−0.896830	+	0.442375i	$0.854136\pi$
$978$		0			0
$979$		−	2.10826e18i		−	2.39457i
$980$	−5.31986e16	+	2.11074e17i	−0.0600542	+	0.238275i
$981$		0			0
$982$	−1.94002e17	−	1.94002e17i	−0.216341	−	0.216341i
$983$	1.05857e18	−	1.05857e18i	1.17327	−	1.17327i	0.191842	−	0.981426i	$-0.438554\pi$
$983$	1.05857e18	−	1.05857e18i	1.17327	−	1.17327i	0.981426	−	0.191842i	$-0.0614460\pi$
$984$		0			0
$985$	−9.39031e17	+	5.60974e17i	−1.02816	+	0.614221i
$986$	−5.35162e16			−0.0582404
$987$		0			0
$988$	1.36431e17	−	1.36431e17i	0.146680	−	0.146680i
$989$			1.72694e18i			1.84544i
$990$		0			0
$991$	−1.13103e18			−1.19408			−0.597040	−	0.802211i	$-0.703657\pi$
$991$	−1.13103e18			−1.19408			−0.597040	+	0.802211i	$0.703657\pi$
$992$	1.81918e17	+	1.81918e17i	0.190900	+	0.190900i
$993$		0			0
$994$		−	1.33610e17i		−	0.138523i
$995$	−1.76677e18	−	4.45293e17i	−1.82072	−	0.458889i
$996$		0			0
$997$	1.23508e18	+	1.23508e18i	1.25755	+	1.25755i	0.952258	+	0.305293i	$0.0987544\pi$
$997$	1.23508e18	+	1.23508e18i	1.25755	+	1.25755i	0.305293	+	0.952258i	$0.401246\pi$
$998$	−7.43824e17	+	7.43824e17i	−0.752812	+	0.752812i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.13.g.b.37.1		6
3.2	odd	2		10.13.c.a.7.2	yes	6
5.3	odd	4	inner	90.13.g.b.73.1		6
12.11	even	2		80.13.p.b.17.2		6
15.2	even	4		50.13.c.d.43.2		6
15.8	even	4		10.13.c.a.3.2	✓	6
15.14	odd	2		50.13.c.d.7.2		6
60.23	odd	4		80.13.p.b.33.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.13.c.a.3.2	✓	6	15.8	even	4
10.13.c.a.7.2	yes	6	3.2	odd	2
50.13.c.d.7.2		6	15.14	odd	2
50.13.c.d.43.2		6	15.2	even	4
80.13.p.b.17.2		6	12.11	even	2
80.13.p.b.33.2		6	60.23	odd	4
90.13.g.b.37.1		6	1.1	even	1	trivial
90.13.g.b.73.1		6	5.3	odd	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	90.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(32.0000 + 32.0000i) q^{2} +2048.00i q^{4} +(-3818.66 + 15151.2i) q^{5} +(-59325.1 - 59325.1i) q^{7} +(-65536.0 + 65536.0i) q^{8} +(-607035. + 362641. i) q^{10} +2.34646e6 q^{11} +(-4.56702e6 + 4.56702e6i) q^{13} -3.79681e6i q^{14} -4.19430e6 q^{16} +(1.56434e7 + 1.56434e7i) q^{17} +1.45864e7i q^{19} +(-3.10296e7 - 7.82062e6i) q^{20} +(7.50867e7 + 7.50867e7i) q^{22} +(-1.10378e8 + 1.10378e8i) q^{23} +(-2.14976e8 - 1.15715e8i) q^{25} -2.92289e8 q^{26} +(1.21498e8 - 1.21498e8i) q^{28} +5.34534e7i q^{29} -1.35539e9 q^{31} +(-1.34218e8 - 1.34218e8i) q^{32} +1.00118e9i q^{34} +(1.12539e9 - 6.72303e8i) q^{35} +(7.96598e8 + 7.96598e8i) q^{37} +(-4.66766e8 + 4.66766e8i) q^{38} +(-7.42688e8 - 1.24321e9i) q^{40} -5.38572e8 q^{41} +(7.82281e9 - 7.82281e9i) q^{43} +4.80555e9i q^{44} -7.06422e9 q^{46} +(-5.73384e9 - 5.73384e9i) q^{47} -6.80235e9i q^{49} +(-3.17637e9 - 1.05821e10i) q^{50} +(-9.35326e9 - 9.35326e9i) q^{52} +(1.01663e10 - 1.01663e10i) q^{53} +(-8.96034e9 + 3.55517e10i) q^{55} +7.77586e9 q^{56} +(-1.71051e9 + 1.71051e9i) q^{58} +5.72907e10i q^{59} -9.47423e10 q^{61} +(-4.33726e10 - 4.33726e10i) q^{62} -8.58993e9i q^{64} +(-5.17559e10 - 8.66357e10i) q^{65} +(-4.17965e10 - 4.17965e10i) q^{67} +(-3.20376e10 + 3.20376e10i) q^{68} +(5.75261e10 + 1.44987e10i) q^{70} +3.51901e10 q^{71} +(1.08806e11 - 1.08806e11i) q^{73} +5.09823e10i q^{74} -2.98730e10 q^{76} +(-1.39204e11 - 1.39204e11i) q^{77} -1.31009e11i q^{79} +(1.60166e10 - 6.35487e10i) q^{80} +(-1.72343e10 - 1.72343e10i) q^{82} +(-1.61869e11 + 1.61869e11i) q^{83} +(-2.96752e11 + 1.77279e11i) q^{85} +5.00660e11 q^{86} +(-1.53778e11 + 1.53778e11i) q^{88} -8.98485e11i q^{89} +5.41878e11 q^{91} +(-2.26055e11 - 2.26055e11i) q^{92} -3.66966e11i q^{94} +(-2.21002e11 - 5.57007e10i) q^{95} +(-3.98581e11 - 3.98581e11i) q^{97} +(2.17675e11 - 2.17675e11i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 192 q^{2} + 16260 q^{5} - 45336 q^{7} - 393216 q^{8} + 886080 q^{10} + 3418008 q^{11} + 8106834 q^{13} - 25165824 q^{16} - 10772514 q^{17} + 23408640 q^{20} + 109376256 q^{22} - 241146864 q^{23} - 520941150 q^{25}+ \cdots + 1960900383168 q^{98}+O(q^{100}) \) 6 * q + 192 * q^2 + 16260 * q^5 - 45336 * q^7 - 393216 * q^8 + 886080 * q^10 + 3418008 * q^11 + 8106834 * q^13 - 25165824 * q^16 - 10772514 * q^17 + 23408640 * q^20 + 109376256 * q^22 - 241146864 * q^23 - 520941150 * q^25 + 518837376 * q^26 + 92848128 * q^28 - 1709108208 * q^31 - 805306368 * q^32 + 3823952160 * q^35 - 742614906 * q^37 - 3927847680 * q^38 - 316538880 * q^40 + 24748899528 * q^41 + 14532965304 * q^43 - 15433399296 * q^46 - 8427287424 * q^47 - 4302859200 * q^50 + 16602796032 * q^52 - 1297014534 * q^53 + 75773173080 * q^55 + 5942280192 * q^56 + 576729600 * q^58 - 201276821928 * q^61 - 54691462656 * q^62 - 146156357430 * q^65 + 108303977544 * q^67 + 22062108672 * q^68 + 132252737280 * q^70 + 570401667408 * q^71 + 710242228134 * q^73 - 251382251520 * q^76 - 9201523848 * q^77 - 68199383040 * q^80 + 791964784896 * q^82 - 727867488144 * q^83 - 1459335929310 * q^85 + 930109779456 * q^86 - 224002572288 * q^88 + 1841466543792 * q^91 - 493868777472 * q^92 + 972100512600 * q^95 - 2207840630346 * q^97 + 1960900383168 * q^98

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