LMFDB - Embedded newform 90.12.c.b.19.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [90,12,Mod(19,90)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 12, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("90.19"); S:= CuspForms(chi, 12); N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	90.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-6144,-530] 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:	$69.1508862504$
Analytic rank:	$0$
Dimension:	$6$
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} - 2x^{5} + 2x^{4} + 198x^{3} + 3568321x^{2} - 6762620x + 6408200 $ x^6 - 2x^5 + 2x^4 + 198x^3 + 3568321x^2 - 6762620*x + 6408200
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{2}\cdot 5^{5} $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.6
Root			$30.7598 + 30.7598i$ of defining polynomial
Character	$\chi$	$=$	90.19
Dual form			90.12.c.b.19.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+32.0000i q^{2} -1024.00 q^{4} +(4156.64 + 5616.98i) q^{5} -73603.8i q^{7} -32768.0i q^{8} +(-179743. + 133012. i) q^{10} +276114. q^{11} -726988. i q^{13} +2.35532e6 q^{14} +1.04858e6 q^{16} -3.21222e6i q^{17} -1.55879e7 q^{19} +(-4.25640e6 - 5.75179e6i) q^{20} +8.83564e6i q^{22} +3.25041e7i q^{23} +(-1.42729e7 + 4.66955e7i) q^{25} +2.32636e7 q^{26} +7.53703e7i q^{28} -4.53029e7 q^{29} -1.06580e8 q^{31} +3.35544e7i q^{32} +1.02791e8 q^{34} +(4.13431e8 - 3.05944e8i) q^{35} +1.65530e8i q^{37} -4.98812e8i q^{38} +(1.84057e8 - 1.36205e8i) q^{40} -2.36566e8 q^{41} +7.28874e8i q^{43} -2.82740e8 q^{44} -1.04013e9 q^{46} +1.41793e9i q^{47} -3.44020e9 q^{49} +(-1.49426e9 - 4.56731e8i) q^{50} +7.44435e8i q^{52} +3.08706e9i q^{53} +(1.14770e9 + 1.55093e9i) q^{55} -2.41185e9 q^{56} -1.44969e9i q^{58} -7.61532e9 q^{59} +1.45582e9 q^{61} -3.41054e9i q^{62} -1.07374e9 q^{64} +(4.08348e9 - 3.02182e9i) q^{65} -1.32563e10i q^{67} +3.28931e9i q^{68} +(9.79022e9 + 1.32298e10i) q^{70} -9.49012e8 q^{71} -1.30829e10i q^{73} -5.29697e9 q^{74} +1.59620e10 q^{76} -2.03230e10i q^{77} -5.30163e10 q^{79} +(4.35855e9 + 5.88983e9i) q^{80} -7.57010e9i q^{82} -2.15490e10i q^{83} +(1.80430e10 - 1.33520e10i) q^{85} -2.33240e10 q^{86} -9.04769e9i q^{88} -5.64241e10 q^{89} -5.35091e10 q^{91} -3.32842e10i q^{92} -4.53739e10 q^{94} +(-6.47932e10 - 8.75569e10i) q^{95} +4.40060e10i q^{97} -1.10086e11i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6144 q^{4} - 530 q^{5} - 385920 q^{10} + 642728 q^{11} + 2125952 q^{14} + 6291456 q^{16} - 24109080 q^{19} + 542720 q^{20} - 181718850 q^{25} - 89251584 q^{26} + 256409820 q^{29} + 458481792 q^{31}+ \cdots - 104896380600 q^{95}+O(q^{100}) $ 6 * q - 6144 * q^4 - 530 * q^5 - 385920 * q^10 + 642728 * q^11 + 2125952 * q^14 + 6291456 * q^16 - 24109080 * q^19 + 542720 * q^20 - 181718850 * q^25 - 89251584 * q^26 + 256409820 * q^29 + 458481792 * q^31 - 225288192 * q^34 + 697136360 * q^35 + 395182080 * q^40 + 164768948 * q^41 - 658153472 * q^44 - 2956208256 * q^46 - 675514158 * q^49 - 3912262400 * q^50 + 3688644360 * q^55 - 2176974848 * q^56 - 17663962360 * q^59 - 5020792428 * q^61 - 6442450944 * q^64 + 19996916880 * q^65 + 24956826240 * q^70 - 56788418832 * q^71 + 64135292672 * q^74 + 24687697920 * q^76 + 2602550880 * q^79 - 555745280 * q^80 - 85024210560 * q^85 - 111995790464 * q^86 - 249448412540 * q^89 - 184446766128 * q^91 - 337749482112 * q^94 - 104896380600 * q^95

Character values

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

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$			32.0000i			0.707107i
$2$			32.0000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1024.00			−0.500000
$5$	4156.64	+	5616.98i	0.594850	+	0.803837i
$5$	4156.64	+	5616.98i	0.594850	+	0.803837i
$6$		0			0
$7$		−	73603.8i		−	1.65524i	−0.561288	−	0.827620i	$-0.689694\pi$
$7$		−	73603.8i		−	1.65524i	0.561288	−	0.827620i	$-0.310306\pi$
$8$		−	32768.0i		−	0.353553i
$9$		0			0
$10$	−179743.	+	133012.i	−0.568399	+	0.420622i
$11$	276114.			0.516926			0.258463	−	0.966021i	$-0.416784\pi$
$11$	276114.			0.516926			0.258463	+	0.966021i	$0.416784\pi$
$12$		0			0
$13$		−	726988.i		−	0.543049i	−0.962432	−	0.271524i	$-0.912472\pi$
$13$		−	726988.i		−	0.543049i	0.962432	−	0.271524i	$-0.0875277\pi$
$14$	2.35532e6			1.17043
$15$		0			0
$16$	1.04858e6			0.250000
$17$		−	3.21222e6i		−	0.548701i	−0.961630	−	0.274351i	$-0.911537\pi$
$17$		−	3.21222e6i		−	0.548701i	0.961630	−	0.274351i	$-0.0884629\pi$
$18$		0			0
$19$	−1.55879e7			−1.44425			−0.722125	−	0.691762i	$-0.756835\pi$
$19$	−1.55879e7			−1.44425			−0.722125	+	0.691762i	$0.756835\pi$
$20$	−4.25640e6	−	5.75179e6i	−0.297425	−	0.401919i
$21$		0			0
$22$			8.83564e6i			0.365522i
$23$			3.25041e7i			1.05302i	0.850170	+	0.526508i	$0.176499\pi$
$23$			3.25041e7i			1.05302i	−0.850170	+	0.526508i	$0.823501\pi$
$24$		0			0
$25$	−1.42729e7	+	4.66955e7i	−0.292308	+	0.956324i
$26$	2.32636e7			0.383993
$27$		0			0
$28$			7.53703e7i			0.827620i
$29$	−4.53029e7			−0.410144			−0.205072	−	0.978747i	$-0.565743\pi$
$29$	−4.53029e7			−0.410144			−0.205072	+	0.978747i	$0.565743\pi$
$30$		0			0
$31$	−1.06580e8			−0.668628			−0.334314	−	0.942462i	$-0.608505\pi$
$31$	−1.06580e8			−0.668628			−0.334314	+	0.942462i	$0.608505\pi$
$32$			3.35544e7i			0.176777i
$33$		0			0
$34$	1.02791e8			0.387990
$35$	4.13431e8	−	3.05944e8i	1.33054	−	0.984619i
$36$		0			0
$37$			1.65530e8i			0.392435i	0.980560	+	0.196218i	$0.0628659\pi$
$37$			1.65530e8i			0.392435i	−0.980560	+	0.196218i	$0.937134\pi$
$38$		−	4.98812e8i		−	1.02124i
$39$		0			0
$40$	1.84057e8	−	1.36205e8i	0.284199	−	0.210311i
$41$	−2.36566e8			−0.318890			−0.159445	−	0.987207i	$-0.550970\pi$
$41$	−2.36566e8			−0.318890			−0.159445	+	0.987207i	$0.550970\pi$
$42$		0			0
$43$			7.28874e8i			0.756095i	0.925786	+	0.378047i	$0.123404\pi$
$43$			7.28874e8i			0.756095i	−0.925786	+	0.378047i	$0.876596\pi$
$44$	−2.82740e8			−0.258463
$45$		0			0
$46$	−1.04013e9			−0.744595
$47$			1.41793e9i			0.901816i	0.892570	+	0.450908i	$0.148900\pi$
$47$			1.41793e9i			0.901816i	−0.892570	+	0.450908i	$0.851100\pi$
$48$		0			0
$49$	−3.44020e9			−1.73982
$50$	−1.49426e9	−	4.56731e8i	−0.676223	−	0.206693i
$51$		0			0
$52$			7.44435e8i			0.271524i
$53$			3.08706e9i			1.01397i	0.861953	+	0.506987i	$0.169241\pi$
$53$			3.08706e9i			1.01397i	−0.861953	+	0.506987i	$0.830759\pi$
$54$		0			0
$55$	1.14770e9	+	1.55093e9i	0.307493	+	0.415524i
$56$	−2.41185e9			−0.585216
$57$		0			0
$58$		−	1.44969e9i		−	0.290016i
$59$	−7.61532e9			−1.38676			−0.693381	−	0.720571i	$-0.743880\pi$
$59$	−7.61532e9			−1.38676			−0.693381	+	0.720571i	$0.743880\pi$
$60$		0			0
$61$	1.45582e9			0.220695			0.110347	−	0.993893i	$-0.464804\pi$
$61$	1.45582e9			0.220695			0.110347	+	0.993893i	$0.464804\pi$
$62$		−	3.41054e9i		−	0.472791i
$63$		0			0
$64$	−1.07374e9			−0.125000
$65$	4.08348e9	−	3.02182e9i	0.436523	−	0.323032i
$66$		0			0
$67$		−	1.32563e10i		−	1.19953i	−0.800177	−	0.599764i	$-0.795261\pi$
$67$		−	1.32563e10i		−	1.19953i	0.800177	−	0.599764i	$-0.204739\pi$
$68$			3.28931e9i			0.274351i
$69$		0			0
$70$	9.79022e9	+	1.32298e10i	0.696231	+	0.940837i
$71$	−9.49012e8			−0.0624239			−0.0312119	−	0.999513i	$-0.509937\pi$
$71$	−9.49012e8			−0.0624239			−0.0312119	+	0.999513i	$0.509937\pi$
$72$		0			0
$73$		−	1.30829e10i		−	0.738634i	−0.929303	−	0.369317i	$-0.879592\pi$
$73$		−	1.30829e10i		−	0.738634i	0.929303	−	0.369317i	$-0.120408\pi$
$74$	−5.29697e9			−0.277494
$75$		0			0
$76$	1.59620e10			0.722125
$77$		−	2.03230e10i		−	0.855637i
$78$		0			0
$79$	−5.30163e10			−1.93848			−0.969238	−	0.246124i	$-0.920843\pi$
$79$	−5.30163e10			−1.93848			−0.969238	+	0.246124i	$0.920843\pi$
$80$	4.35855e9	+	5.88983e9i	0.148712	+	0.200959i
$81$		0			0
$82$		−	7.57010e9i		−	0.225489i
$83$		−	2.15490e10i		−	0.600479i	−0.953864	−	0.300239i	$-0.902933\pi$
$83$		−	2.15490e10i		−	0.600479i	0.953864	−	0.300239i	$-0.0970666\pi$
$84$		0			0
$85$	1.80430e10	−	1.33520e10i	0.441066	−	0.326395i
$86$	−2.33240e10			−0.534640
$87$		0			0
$88$		−	9.04769e9i		−	0.182761i
$89$	−5.64241e10			−1.07107			−0.535537	−	0.844511i	$-0.679891\pi$
$89$	−5.64241e10			−1.07107			−0.535537	+	0.844511i	$0.679891\pi$
$90$		0			0
$91$	−5.35091e10			−0.898876
$92$		−	3.32842e10i		−	0.526508i
$93$		0			0
$94$	−4.53739e10			−0.637680
$95$	−6.47932e10	−	8.75569e10i	−0.859112	−	1.16094i
$96$		0			0
$97$			4.40060e10i			0.520316i	0.965566	+	0.260158i	$0.0837746\pi$
$97$			4.40060e10i			0.520316i	−0.965566	+	0.260158i	$0.916225\pi$
$98$		−	1.10086e11i		−	1.23024i
$99$		0			0
$100$	1.46154e10	−	4.78162e10i	0.146154	−	0.478162i
$101$	9.78679e8			0.00926558			0.00463279	−	0.999989i	$-0.498525\pi$
$101$	9.78679e8			0.00926558			0.00463279	+	0.999989i	$0.498525\pi$
$102$		0			0
$103$		−	9.00212e10i		−	0.765139i	−0.923927	−	0.382569i	$-0.875039\pi$
$103$		−	9.00212e10i		−	0.765139i	0.923927	−	0.382569i	$-0.124961\pi$
$104$	−2.38219e10			−0.191997
$105$		0			0
$106$	−9.87858e10			−0.716988
$107$		−	1.75119e11i		−	1.20705i	−0.797346	−	0.603523i	$-0.793763\pi$
$107$		−	1.75119e11i		−	1.20705i	0.797346	−	0.603523i	$-0.206237\pi$
$108$		0			0
$109$	2.69261e11			1.67620			0.838102	−	0.545513i	$-0.183665\pi$
$109$	2.69261e11			1.67620			0.838102	+	0.545513i	$0.183665\pi$
$110$	−4.96296e10	+	3.67265e10i	−0.293820	+	0.217430i
$111$		0			0
$112$		−	7.71792e10i		−	0.413810i
$113$			2.51799e11i			1.28565i	0.766013	+	0.642825i	$0.222238\pi$
$113$			2.51799e11i			1.28565i	−0.766013	+	0.642825i	$0.777762\pi$
$114$		0			0
$115$	−1.82575e11	+	1.35108e11i	−0.846454	+	0.626386i
$116$	4.63901e10			0.205072
$117$		0			0
$118$		−	2.43690e11i		−	0.980589i
$119$	−2.36431e11			−0.908232
$120$		0			0
$121$	−2.09073e11			−0.732788
$122$			4.65861e10i			0.156055i
$123$		0			0
$124$	1.09137e11			0.334314
$125$	−3.21615e11	+	1.13926e11i	−0.942608	+	0.333901i
$126$		0			0
$127$		−	5.68757e11i		−	1.52759i	−0.645459	−	0.763795i	$-0.723334\pi$
$127$		−	5.68757e11i		−	1.52759i	0.645459	−	0.763795i	$-0.276666\pi$
$128$		−	3.43597e10i		−	0.0883883i
$129$		0			0
$130$	9.66984e10	+	1.30671e11i	0.228418	+	0.308668i
$131$	−3.29083e11			−0.745270			−0.372635	−	0.927978i	$-0.621546\pi$
$131$	−3.29083e11			−0.745270			−0.372635	+	0.927978i	$0.621546\pi$
$132$		0			0
$133$			1.14733e12i			2.39058i
$134$	4.24201e11			0.848194
$135$		0			0
$136$	−1.05258e11			−0.193995
$137$		−	3.26164e11i		−	0.577394i	−0.957420	−	0.288697i	$-0.906778\pi$
$137$		−	3.26164e11i		−	0.577394i	0.957420	−	0.288697i	$-0.0932221\pi$
$138$		0			0
$139$	−2.17139e10			−0.0354942			−0.0177471	−	0.999843i	$-0.505649\pi$
$139$	−2.17139e10			−0.0354942			−0.0177471	+	0.999843i	$0.505649\pi$
$140$	−4.23354e11	+	3.13287e11i	−0.665272	+	0.492310i
$141$		0			0
$142$		−	3.03684e10i		−	0.0441403i
$143$		−	2.00731e11i		−	0.280716i
$144$		0			0
$145$	−1.88308e11	−	2.54465e11i	−0.243974	−	0.329689i
$146$	4.18654e11			0.522293
$147$		0			0
$148$		−	1.69503e11i		−	0.196218i
$149$	−9.81939e10			−0.109537			−0.0547684	−	0.998499i	$-0.517442\pi$
$149$	−9.81939e10			−0.109537			−0.0547684	+	0.998499i	$0.517442\pi$
$150$		0			0
$151$	−1.60820e12			−1.66712			−0.833558	−	0.552432i	$-0.813700\pi$
$151$	−1.60820e12			−1.66712			−0.833558	+	0.552432i	$0.813700\pi$
$152$			5.10784e11i			0.510620i
$153$		0			0
$154$	6.50337e11			0.605026
$155$	−4.43012e11	−	5.98655e11i	−0.397733	−	0.537468i
$156$		0			0
$157$			9.79921e11i			0.819867i	0.912116	+	0.409933i	$0.134448\pi$
$157$			9.79921e11i			0.819867i	−0.912116	+	0.409933i	$0.865552\pi$
$158$		−	1.69652e12i		−	1.37071i
$159$		0			0
$160$	−1.88475e11	+	1.39474e11i	−0.142100	+	0.105156i
$161$	2.39243e12			1.74300
$162$		0			0
$163$			8.12475e11i			0.553068i	0.961004	+	0.276534i	$0.0891858\pi$
$163$			8.12475e11i			0.553068i	−0.961004	+	0.276534i	$0.910814\pi$
$164$	2.42243e11			0.159445
$165$		0			0
$166$	6.89568e11			0.424603
$167$			2.27106e12i			1.35297i	0.736458	+	0.676484i	$0.236497\pi$
$167$			2.27106e12i			1.35297i	−0.736458	+	0.676484i	$0.763503\pi$
$168$		0			0
$169$	1.26365e12			0.705098
$170$	4.27265e11	+	5.77375e11i	0.230796	+	0.311881i
$171$		0			0
$172$		−	7.46367e11i		−	0.378047i
$173$			3.16932e12i			1.55494i	0.628921	+	0.777469i	$0.283497\pi$
$173$			3.16932e12i			1.55494i	−0.628921	+	0.777469i	$0.716503\pi$
$174$		0			0
$175$	3.43697e12	+	1.05054e12i	1.58295	+	0.483840i
$176$	2.89526e11			0.129231
$177$		0			0
$178$		−	1.80557e12i		−	0.757364i
$179$	4.56681e12			1.85747			0.928734	−	0.370747i	$-0.120898\pi$
$179$	4.56681e12			1.85747			0.928734	+	0.370747i	$0.120898\pi$
$180$		0			0
$181$	−1.55390e12			−0.594553			−0.297276	−	0.954791i	$-0.596078\pi$
$181$	−1.55390e12			−0.594553			−0.297276	+	0.954791i	$0.596078\pi$
$182$		−	1.71229e12i		−	0.635601i
$183$		0			0
$184$	1.06509e12			0.372297
$185$	−9.29781e11	+	6.88050e11i	−0.315454	+	0.233440i
$186$		0			0
$187$		−	8.86937e11i		−	0.283638i
$188$		−	1.45196e12i		−	0.450908i
$189$		0			0
$190$	2.80182e12	−	2.07338e12i	0.820910	−	0.607484i
$191$	2.09841e12			0.597319			0.298659	−	0.954360i	$-0.403461\pi$
$191$	2.09841e12			0.597319			0.298659	+	0.954360i	$0.403461\pi$
$192$		0			0
$193$			1.49038e12i			0.400620i	0.979733	+	0.200310i	$0.0641950\pi$
$193$			1.49038e12i			0.400620i	−0.979733	+	0.200310i	$0.935805\pi$
$194$	−1.40819e12			−0.367919
$195$		0			0
$196$	3.52276e12			0.869911
$197$			1.54638e12i			0.371322i	0.982614	+	0.185661i	$0.0594426\pi$
$197$			1.54638e12i			0.371322i	−0.982614	+	0.185661i	$0.940557\pi$
$198$		0			0
$199$	−3.26839e12			−0.742407			−0.371204	−	0.928552i	$-0.621055\pi$
$199$	−3.26839e12			−0.742407			−0.371204	+	0.928552i	$0.621055\pi$
$200$	1.53012e12	+	4.67693e11i	0.338112	+	0.103347i
$201$		0			0
$202$			3.13177e10i			0.00655176i
$203$			3.33446e12i			0.678888i
$204$		0			0
$205$	−9.83317e11	−	1.32878e12i	−0.189691	−	0.256335i
$206$	2.88068e12			0.541035
$207$		0			0
$208$		−	7.62302e11i		−	0.135762i
$209$	−4.30403e12			−0.746570
$210$		0			0
$211$	−2.09010e12			−0.344043			−0.172021	−	0.985093i	$-0.555030\pi$
$211$	−2.09010e12			−0.344043			−0.172021	+	0.985093i	$0.555030\pi$
$212$		−	3.16114e12i		−	0.506987i
$213$		0			0
$214$	5.60382e12			0.853510
$215$	−4.09407e12	+	3.02967e12i	−0.607777	+	0.449763i
$216$		0			0
$217$			7.84466e12i			1.10674i
$218$			8.61634e12i			1.18526i
$219$		0			0
$220$	−1.17525e12	−	1.58815e12i	−0.153747	−	0.207762i
$221$	−2.33524e12			−0.297971
$222$		0			0
$223$			1.89689e12i			0.230338i	0.993346	+	0.115169i	$0.0367409\pi$
$223$			1.89689e12i			0.230338i	−0.993346	+	0.115169i	$0.963259\pi$
$224$	2.46973e12			0.292608
$225$		0			0
$226$	−8.05757e12			−0.909092
$227$			2.22297e12i			0.244788i	0.992482	+	0.122394i	$0.0390572\pi$
$227$			2.22297e12i			0.244788i	−0.992482	+	0.122394i	$0.960943\pi$
$228$		0			0
$229$	−1.10056e12			−0.115483			−0.0577415	−	0.998332i	$-0.518390\pi$
$229$	−1.10056e12			−0.115483			−0.0577415	+	0.998332i	$0.518390\pi$
$230$	−4.32345e12	−	5.84240e12i	−0.442922	−	0.598533i
$231$		0			0
$232$			1.48448e12i			0.145008i
$233$		−	7.26117e12i		−	0.692706i	−0.938104	−	0.346353i	$-0.887420\pi$
$233$		−	7.26117e12i		−	0.692706i	0.938104	−	0.346353i	$-0.112580\pi$
$234$		0			0
$235$	−7.96451e12	+	5.89384e12i	−0.724913	+	0.536445i
$236$	7.79808e12			0.693381
$237$		0			0
$238$		−	7.56581e12i		−	0.642217i
$239$	1.69859e13			1.40896			0.704482	−	0.709722i	$-0.251179\pi$
$239$	1.69859e13			1.40896			0.704482	+	0.709722i	$0.251179\pi$
$240$		0			0
$241$	3.71937e12			0.294697			0.147349	−	0.989085i	$-0.452926\pi$
$241$	3.71937e12			0.294697			0.147349	+	0.989085i	$0.452926\pi$
$242$		−	6.69033e12i		−	0.518159i
$243$		0			0
$244$	−1.49076e12			−0.110347
$245$	−1.42996e13	−	1.93235e13i	−1.03493	−	1.39853i
$246$		0			0
$247$			1.13322e13i			0.784298i
$248$			3.49240e12i			0.236396i
$249$		0			0
$250$	−3.64563e12	−	1.02917e13i	−0.236104	−	0.666525i
$251$	−1.95302e13			−1.23737			−0.618687	−	0.785637i	$-0.712335\pi$
$251$	−1.95302e13			−1.23737			−0.618687	+	0.785637i	$0.712335\pi$
$252$		0			0
$253$			8.97483e12i			0.544331i
$254$	1.82002e13			1.08017
$255$		0			0
$256$	1.09951e12			0.0625000
$257$			9.96454e12i			0.554402i	0.960812	+	0.277201i	$0.0894068\pi$
$257$			9.96454e12i			0.554402i	−0.960812	+	0.277201i	$0.910593\pi$
$258$		0			0
$259$	1.21837e13			0.649575
$260$	−4.18148e12	+	3.09435e12i	−0.218261	+	0.161516i
$261$		0			0
$262$		−	1.05307e13i		−	0.526985i
$263$		−	1.85332e13i		−	0.908225i	−0.890944	−	0.454112i	$-0.849956\pi$
$263$		−	1.85332e13i		−	0.908225i	0.890944	−	0.454112i	$-0.150044\pi$
$264$		0			0
$265$	−1.73399e13	+	1.28318e13i	−0.815070	+	0.603162i
$266$	−3.67145e13			−1.69040
$267$		0			0
$268$			1.35744e13i			0.599764i
$269$	−1.34102e13			−0.580493			−0.290247	−	0.956952i	$-0.593737\pi$
$269$	−1.34102e13			−0.580493			−0.290247	+	0.956952i	$0.593737\pi$
$270$		0			0
$271$	−2.99278e13			−1.24378			−0.621890	−	0.783105i	$-0.713635\pi$
$271$	−2.99278e13			−1.24378			−0.621890	+	0.783105i	$0.713635\pi$
$272$		−	3.36825e12i		−	0.137175i
$273$		0			0
$274$	1.04372e13			0.408279
$275$	−3.94093e12	+	1.28933e13i	−0.151102	+	0.494349i
$276$		0			0
$277$		−	4.40434e13i		−	1.62271i	−0.584551	−	0.811357i	$-0.698729\pi$
$277$		−	4.40434e13i		−	1.62271i	0.584551	−	0.811357i	$-0.301271\pi$
$278$		−	6.94846e11i		−	0.0250982i
$279$		0			0
$280$	−1.00252e13	−	1.35473e13i	−0.348115	−	0.470418i
$281$	−3.84310e13			−1.30857			−0.654286	−	0.756247i	$-0.727031\pi$
$281$	−3.84310e13			−1.30857			−0.654286	+	0.756247i	$0.727031\pi$
$282$		0			0
$283$		−	1.73740e13i		−	0.568951i	−0.958683	−	0.284475i	$-0.908181\pi$
$283$		−	1.73740e13i		−	0.568951i	0.958683	−	0.284475i	$-0.0918194\pi$
$284$	9.71788e11			0.0312119
$285$		0			0
$286$	6.42340e12			0.198496
$287$			1.74121e13i			0.527839i
$288$		0			0
$289$	2.39536e13			0.698927
$290$	8.14289e12	−	6.02584e12i	0.233125	−	0.172516i
$291$		0			0
$292$			1.33969e13i			0.369317i
$293$		−	2.71688e13i		−	0.735019i	−0.930020	−	0.367510i	$-0.880210\pi$
$293$		−	2.71688e13i		−	0.735019i	0.930020	−	0.367510i	$-0.119790\pi$
$294$		0			0
$295$	−3.16541e13	−	4.27751e13i	−0.824915	−	1.11473i
$296$	5.42410e12			0.138747
$297$		0			0
$298$		−	3.14220e12i		−	0.0774542i
$299$	2.36301e13			0.571839
$300$		0			0
$301$	5.36479e13			1.25152
$302$		−	5.14623e13i		−	1.17883i
$303$		0			0
$304$	−1.63451e13			−0.361063
$305$	6.05130e12	+	8.17729e12i	0.131280	+	0.177403i
$306$		0			0
$307$		−	7.68468e13i		−	1.60829i	−0.594432	−	0.804146i	$-0.702623\pi$
$307$		−	7.68468e13i		−	1.60829i	0.594432	−	0.804146i	$-0.297377\pi$
$308$			2.08108e13i			0.427818i
$309$		0			0
$310$	1.91570e13	−	1.41764e13i	0.380047	−	0.281240i
$311$	4.58065e12			0.0892781			0.0446390	−	0.999003i	$-0.485786\pi$
$311$	4.58065e12			0.0892781			0.0446390	+	0.999003i	$0.485786\pi$
$312$		0			0
$313$			6.56768e13i			1.23571i	0.786290	+	0.617857i	$0.211999\pi$
$313$			6.56768e13i			1.23571i	−0.786290	+	0.617857i	$0.788001\pi$
$314$	−3.13575e13			−0.579733
$315$		0			0
$316$	5.42887e13			0.969238
$317$		−	6.53941e13i		−	1.14739i	−0.819067	−	0.573697i	$-0.805509\pi$
$317$		−	6.53941e13i		−	1.14739i	0.819067	−	0.573697i	$-0.194491\pi$
$318$		0			0
$319$	−1.25087e13			−0.212014
$320$	−4.46316e12	−	6.03119e12i	−0.0743562	−	0.100480i
$321$		0			0
$322$			7.65576e13i			1.23248i
$323$			5.00717e13i			0.792462i
$324$		0			0
$325$	3.39471e13	+	1.03762e13i	0.519330	+	0.158737i
$326$	−2.59992e13			−0.391078
$327$		0			0
$328$			7.75178e12i			0.112744i
$329$	1.04365e14			1.49272
$330$		0			0
$331$	1.82867e13			0.252977			0.126488	−	0.991968i	$-0.459629\pi$
$331$	1.82867e13			0.252977			0.126488	+	0.991968i	$0.459629\pi$
$332$			2.20662e13i			0.300239i
$333$		0			0
$334$	−7.26738e13			−0.956693
$335$	7.44603e13	−	5.51015e13i	0.964225	−	0.713538i
$336$		0			0
$337$		−	6.32083e12i		−	0.0792155i	−0.999215	−	0.0396077i	$-0.987389\pi$
$337$		−	6.32083e12i		−	0.0792155i	0.999215	−	0.0396077i	$-0.0126108\pi$
$338$			4.04368e13i			0.498580i
$339$		0			0
$340$	−1.84760e13	+	1.36725e13i	−0.220533	+	0.163197i
$341$	−2.94281e13			−0.345631
$342$		0			0
$343$			1.07673e14i			1.22458i
$344$	2.38838e13			0.267320
$345$		0			0
$346$	−1.01418e14			−1.09951
$347$			9.38390e13i			1.00132i	0.865645	+	0.500658i	$0.166909\pi$
$347$			9.38390e13i			1.00132i	−0.865645	+	0.500658i	$0.833091\pi$
$348$		0			0
$349$	−1.24898e14			−1.29127			−0.645634	−	0.763647i	$-0.723407\pi$
$349$	−1.24898e14			−1.29127			−0.645634	+	0.763647i	$0.723407\pi$
$350$	−3.36172e13	+	1.09983e14i	−0.342127	+	1.11931i
$351$		0			0
$352$			9.26484e12i			0.0913804i
$353$			4.95093e13i			0.480757i	0.970679	+	0.240378i	$0.0772715\pi$
$353$			4.95093e13i			0.480757i	−0.970679	+	0.240378i	$0.922728\pi$
$354$		0			0
$355$	−3.94470e12	−	5.33058e12i	−0.0371328	−	0.0501786i
$356$	5.77783e13			0.535537
$357$		0			0
$358$			1.46138e14i			1.31343i
$359$	−8.82048e13			−0.780680			−0.390340	−	0.920671i	$-0.627642\pi$
$359$	−8.82048e13			−0.780680			−0.390340	+	0.920671i	$0.627642\pi$
$360$		0			0
$361$	1.26492e14			1.08586
$362$		−	4.97248e13i		−	0.420412i
$363$		0			0
$364$	5.47933e13			0.449438
$365$	7.34866e13	−	5.43810e13i	0.593742	−	0.439376i
$366$		0			0
$367$		−	9.89014e13i		−	0.775423i	−0.921781	−	0.387712i	$-0.873266\pi$
$367$		−	9.89014e13i		−	0.775423i	0.921781	−	0.387712i	$-0.126734\pi$
$368$			3.40830e13i			0.263254i
$369$		0			0
$370$	−2.20176e13	−	2.97530e13i	−0.165067	−	0.223060i
$371$	2.27219e14			1.67837
$372$		0			0
$373$			3.15437e13i			0.226211i	0.993583	+	0.113105i	$0.0360798\pi$
$373$			3.15437e13i			0.226211i	−0.993583	+	0.113105i	$0.963920\pi$
$374$	2.83820e13			0.200562
$375$		0			0
$376$	4.64629e13			0.318840
$377$			3.29346e13i			0.222728i
$378$		0			0
$379$	1.45966e14			0.958815			0.479408	−	0.877592i	$-0.340852\pi$
$379$	1.45966e14			0.958815			0.479408	+	0.877592i	$0.340852\pi$
$380$	6.63482e13	+	8.96583e13i	0.429556	+	0.580471i
$381$		0			0
$382$			6.71490e13i			0.422368i
$383$			9.39255e13i			0.582358i	0.956669	+	0.291179i	$0.0940476\pi$
$383$			9.39255e13i			0.582358i	−0.956669	+	0.291179i	$0.905952\pi$
$384$		0			0
$385$	1.14154e14	−	8.44754e13i	0.687792	−	0.508975i
$386$	−4.76923e13			−0.283281
$387$		0			0
$388$		−	4.50621e13i		−	0.260158i
$389$	2.77248e13			0.157814			0.0789071	−	0.996882i	$-0.474857\pi$
$389$	2.77248e13			0.157814			0.0789071	+	0.996882i	$0.474857\pi$
$390$		0			0
$391$	1.04410e14			0.577791
$392$			1.12728e14i			0.615120i
$393$		0			0
$394$	−4.94840e13			−0.262564
$395$	−2.20370e14	−	2.97792e14i	−1.15310	−	1.55822i
$396$		0			0
$397$		−	1.80795e14i		−	0.920106i	−0.887892	−	0.460053i	$-0.847830\pi$
$397$		−	1.80795e14i		−	0.920106i	0.887892	−	0.460053i	$-0.152170\pi$
$398$		−	1.04589e14i		−	0.524961i
$399$		0			0
$400$	−1.49662e13	+	4.89638e13i	−0.0730770	+	0.239081i
$401$	−1.34767e14			−0.649065			−0.324533	−	0.945874i	$-0.605207\pi$
$401$	−1.34767e14			−0.649065			−0.324533	+	0.945874i	$0.605207\pi$
$402$		0			0
$403$			7.74820e13i			0.363097i
$404$	−1.00217e12			−0.00463279
$405$		0			0
$406$	−1.06703e14			−0.480046
$407$			4.57052e13i			0.202860i
$408$		0			0
$409$	1.61853e14			0.699264			0.349632	−	0.936887i	$-0.386307\pi$
$409$	1.61853e14			0.699264			0.349632	+	0.936887i	$0.386307\pi$
$410$	4.25211e13	−	3.14662e13i	0.181256	−	0.134132i
$411$		0			0
$412$			9.21817e13i			0.382569i
$413$			5.60516e14i			2.29542i
$414$		0			0
$415$	1.21040e14	−	8.95714e13i	0.482687	−	0.357195i
$416$	2.43937e13			0.0959983
$417$		0			0
$418$		−	1.37729e14i		−	0.527905i
$419$	−4.46107e13			−0.168757			−0.0843785	−	0.996434i	$-0.526890\pi$
$419$	−4.46107e13			−0.168757			−0.0843785	+	0.996434i	$0.526890\pi$
$420$		0			0
$421$	−3.96897e14			−1.46260			−0.731301	−	0.682055i	$-0.761086\pi$
$421$	−3.96897e14			−1.46260			−0.731301	+	0.682055i	$0.761086\pi$
$422$		−	6.68831e13i		−	0.243275i
$423$		0			0
$424$	1.01157e14			0.358494
$425$	1.49996e14	+	4.58475e13i	0.524736	+	0.160390i
$426$		0			0
$427$		−	1.07154e14i		−	0.365303i
$428$			1.79322e14i			0.603523i
$429$		0			0
$430$	−9.69493e13	−	1.31010e14i	−0.318030	−	0.429763i
$431$	4.73312e14			1.53293			0.766465	−	0.642286i	$-0.222014\pi$
$431$	4.73312e14			1.53293			0.766465	+	0.642286i	$0.222014\pi$
$432$		0			0
$433$			6.09516e14i			1.92443i	0.272297	+	0.962213i	$0.412217\pi$
$433$			6.09516e14i			1.92443i	−0.272297	+	0.962213i	$0.587783\pi$
$434$	−2.51029e14			−0.782583
$435$		0			0
$436$	−2.75723e14			−0.838102
$437$		−	5.06670e14i		−	1.52082i
$438$		0			0
$439$	3.89020e14			1.13872			0.569360	−	0.822088i	$-0.307191\pi$
$439$	3.89020e14			1.13872			0.569360	+	0.822088i	$0.307191\pi$
$440$	5.08207e13	−	3.76080e13i	0.146910	−	0.108715i
$441$		0			0
$442$		−	7.47278e13i		−	0.210698i
$443$		−	6.78395e14i		−	1.88913i	−0.328322	−	0.944566i	$-0.606483\pi$
$443$		−	6.78395e14i		−	1.88913i	0.328322	−	0.944566i	$-0.393517\pi$
$444$		0			0
$445$	−2.34535e14	−	3.16933e14i	−0.637128	−	0.860970i
$446$	−6.07005e13			−0.162873
$447$		0			0
$448$			7.90315e13i			0.206905i
$449$	−2.75571e14			−0.712653			−0.356327	−	0.934361i	$-0.615971\pi$
$449$	−2.75571e14			−0.712653			−0.356327	+	0.934361i	$0.615971\pi$
$450$		0			0
$451$	−6.53190e13			−0.164842
$452$		−	2.57842e14i		−	0.642825i
$453$		0			0
$454$	−7.11349e13			−0.173091
$455$	−2.22418e14	−	3.00560e14i	−0.534696	−	0.722550i
$456$		0			0
$457$		−	5.15690e14i		−	1.21018i	−0.796158	−	0.605089i	$-0.793137\pi$
$457$		−	5.15690e14i		−	1.21018i	0.796158	−	0.605089i	$-0.206863\pi$
$458$		−	3.52179e13i		−	0.0816588i
$459$		0			0
$460$	1.86957e14	−	1.38350e14i	0.423227	−	0.313193i
$461$	1.71680e14			0.384030			0.192015	−	0.981392i	$-0.438498\pi$
$461$	1.71680e14			0.384030			0.192015	+	0.981392i	$0.438498\pi$
$462$		0			0
$463$			2.98973e14i			0.653034i	0.945191	+	0.326517i	$0.105875\pi$
$463$			2.98973e14i			0.653034i	−0.945191	+	0.326517i	$0.894125\pi$
$464$	−4.75035e13			−0.102536
$465$		0			0
$466$	2.32357e14			0.489817
$467$			9.49939e13i			0.197903i	0.995092	+	0.0989516i	$0.0315489\pi$
$467$			9.49939e13i			0.197903i	−0.995092	+	0.0989516i	$0.968451\pi$
$468$		0			0
$469$	−9.75713e14			−1.98551
$470$	−1.88603e14	−	2.54864e14i	−0.379324	−	0.512591i
$471$		0			0
$472$			2.49539e14i			0.490294i
$473$			2.01252e14i			0.390845i
$474$		0			0
$475$	2.22484e14	−	7.27885e14i	0.422166	−	1.38117i
$476$	2.42106e14			0.454116
$477$		0			0
$478$			5.43549e14i			0.996288i
$479$	−3.78434e14			−0.685717			−0.342859	−	0.939387i	$-0.611395\pi$
$479$	−3.78434e14			−0.685717			−0.342859	+	0.939387i	$0.611395\pi$
$480$		0			0
$481$	1.20339e14			0.213112
$482$			1.19020e14i			0.208382i
$483$		0			0
$484$	2.14091e14			0.366394
$485$	−2.47181e14	+	1.82917e14i	−0.418249	+	0.309510i
$486$		0			0
$487$			6.10112e14i			1.00925i	0.863337	+	0.504627i	$0.168370\pi$
$487$			6.10112e14i			1.00925i	−0.863337	+	0.504627i	$0.831630\pi$
$488$		−	4.77042e13i		−	0.0780274i
$489$		0			0
$490$	6.18353e14	−	4.57589e14i	0.988912	−	0.731808i
$491$	−1.13713e15			−1.79830			−0.899152	−	0.437637i	$-0.855815\pi$
$491$	−1.13713e15			−1.79830			−0.899152	+	0.437637i	$0.855815\pi$
$492$		0			0
$493$			1.45523e14i			0.225047i
$494$	−3.62631e14			−0.554583
$495$		0			0
$496$	−1.11757e14			−0.167157
$497$			6.98509e13i			0.103327i
$498$		0			0
$499$	−8.96225e14			−1.29677			−0.648387	−	0.761311i	$-0.724556\pi$
$499$	−8.96225e14			−1.29677			−0.648387	+	0.761311i	$0.724556\pi$
$500$	3.29334e14	−	1.16660e14i	0.471304	−	0.166950i
$501$		0			0
$502$		−	6.24966e14i		−	0.874956i
$503$			4.50342e14i			0.623618i	0.950145	+	0.311809i	$0.100935\pi$
$503$			4.50342e14i			0.623618i	−0.950145	+	0.311809i	$0.899065\pi$
$504$		0			0
$505$	4.06801e12	+	5.49722e12i	0.00551163	+	0.00744802i
$506$	−2.87194e14			−0.384900
$507$		0			0
$508$			5.82408e14i			0.763795i
$509$	−1.35840e15			−1.76230			−0.881151	−	0.472834i	$-0.843231\pi$
$509$	−1.35840e15			−1.76230			−0.881151	+	0.472834i	$0.843231\pi$
$510$		0			0
$511$	−9.62954e14			−1.22262
$512$			3.51844e13i			0.0441942i
$513$		0			0
$514$	−3.18865e14			−0.392022
$515$	5.05648e14	−	3.74186e14i	0.615047	−	0.455142i
$516$		0			0
$517$			3.91511e14i			0.466172i
$518$			3.89877e14i			0.459319i
$519$		0			0
$520$	−9.90192e13	−	1.33807e14i	−0.114209	−	0.154334i
$521$	1.60516e15			1.83194			0.915970	−	0.401247i	$-0.131423\pi$
$521$	1.60516e15			1.83194			0.915970	+	0.401247i	$0.131423\pi$
$522$		0			0
$523$			9.61728e13i			0.107471i	0.998555	+	0.0537357i	$0.0171129\pi$
$523$			9.61728e13i			0.107471i	−0.998555	+	0.0537357i	$0.982887\pi$
$524$	3.36981e14			0.372635
$525$		0			0
$526$	5.93062e14			0.642212
$527$			3.42357e14i			0.366877i
$528$		0			0
$529$	−1.03707e14			−0.108843
$530$	−4.10617e14	−	5.54878e14i	−0.426500	−	0.576342i
$531$		0			0
$532$		−	1.17486e15i		−	1.19529i
$533$			1.71980e14i			0.173173i
$534$		0			0
$535$	9.83643e14	−	7.27908e14i	0.970268	−	0.718010i
$536$	−4.34382e14			−0.424097
$537$		0			0
$538$		−	4.29126e14i		−	0.410471i
$539$	−9.49885e14			−0.899359
$540$		0			0
$541$	2.13640e14			0.198197			0.0990986	−	0.995078i	$-0.468404\pi$
$541$	2.13640e14			0.198197			0.0990986	+	0.995078i	$0.468404\pi$
$542$		−	9.57689e14i		−	0.879485i
$543$		0			0
$544$	1.07784e14			0.0969976
$545$	1.11922e15	+	1.51243e15i	0.997090	+	1.34740i
$546$		0			0
$547$			4.37872e13i			0.0382311i	0.999817	+	0.0191156i	$0.00608504\pi$
$547$			4.37872e13i			0.0382311i	−0.999817	+	0.0191156i	$0.993915\pi$
$548$			3.33992e14i			0.288697i
$549$		0			0
$550$	−4.12585e14	−	1.26110e14i	−0.349557	−	0.106845i
$551$	7.06176e14			0.592351
$552$		0			0
$553$			3.90220e15i			3.20865i
$554$	1.40939e15			1.14743
$555$		0			0
$556$	2.22351e13			0.0177471
$557$		−	1.57027e14i		−	0.124100i	−0.998073	−	0.0620500i	$-0.980236\pi$
$557$		−	1.57027e14i		−	0.124100i	0.998073	−	0.0620500i	$-0.0197638\pi$
$558$		0			0
$559$	5.29883e14			0.410596
$560$	4.33514e14	−	3.20806e14i	0.332636	−	0.246155i
$561$		0			0
$562$		−	1.22979e15i		−	0.925300i
$563$			4.81449e14i			0.358719i	0.983784	+	0.179359i	$0.0574025\pi$
$563$			4.81449e14i			0.358719i	−0.983784	+	0.179359i	$0.942598\pi$
$564$		0			0
$565$	−1.41435e15	+	1.04664e15i	−1.03345	+	0.764768i
$566$	5.55968e14			0.402309
$567$		0			0
$568$			3.10972e13i			0.0220702i
$569$	1.74463e15			1.22627			0.613134	−	0.789979i	$-0.289908\pi$
$569$	1.74463e15			1.22627			0.613134	+	0.789979i	$0.289908\pi$
$570$		0			0
$571$	2.52262e15			1.73921			0.869606	−	0.493747i	$-0.164373\pi$
$571$	2.52262e15			1.73921			0.869606	+	0.493747i	$0.164373\pi$
$572$			2.05549e14i			0.140358i
$573$		0			0
$574$	−5.57188e14			−0.373238
$575$	−1.51780e15	−	4.63926e14i	−1.00702	−	0.307805i
$576$		0			0
$577$		−	1.34881e15i		−	0.877978i	−0.898492	−	0.438989i	$-0.855337\pi$
$577$		−	1.34881e15i		−	0.877978i	0.898492	−	0.438989i	$-0.144663\pi$
$578$			7.66514e14i			0.494216i
$579$		0			0
$580$	1.92827e14	+	2.60573e14i	0.121987	+	0.164845i
$581$	−1.58609e15			−0.993937
$582$		0			0
$583$			8.52378e14i			0.524150i
$584$	−4.28702e14			−0.261147
$585$		0			0
$586$	8.69402e14			0.519737
$587$		−	7.29536e14i		−	0.432053i	−0.976387	−	0.216027i	$-0.930690\pi$
$587$		−	7.29536e14i		−	0.432053i	0.976387	−	0.216027i	$-0.0693098\pi$
$588$		0			0
$589$	1.66135e15			0.965666
$590$	1.36880e15	−	1.01293e15i	0.788234	−	0.583303i
$591$		0			0
$592$			1.73571e14i			0.0981089i
$593$			5.23392e14i			0.293107i	0.989203	+	0.146554i	$0.0468181\pi$
$593$			5.23392e14i			0.293107i	−0.989203	+	0.146554i	$0.953182\pi$
$594$		0			0
$595$	−9.82760e14	−	1.32803e15i	−0.540262	−	0.730071i
$596$	1.00551e14			0.0547684
$597$		0			0
$598$			7.56163e14i			0.404351i
$599$	3.40071e15			1.80186			0.900932	−	0.433961i	$-0.142884\pi$
$599$	3.40071e15			1.80186			0.900932	+	0.433961i	$0.142884\pi$
$600$		0			0
$601$	−6.34979e14			−0.330331			−0.165166	−	0.986266i	$-0.552816\pi$
$601$	−6.34979e14			−0.330331			−0.165166	+	0.986266i	$0.552816\pi$
$602$			1.71673e15i			0.884957i
$603$		0			0
$604$	1.64679e15			0.833558
$605$	−8.69040e14	−	1.17436e15i	−0.435898	−	0.589042i
$606$		0			0
$607$			3.98915e14i			0.196491i	0.995162	+	0.0982455i	$0.0313230\pi$
$607$			3.98915e14i			0.196491i	−0.995162	+	0.0982455i	$0.968677\pi$
$608$		−	5.23043e14i		−	0.255310i
$609$		0			0
$610$	−2.61673e14	+	1.93642e14i	−0.125443	+	0.0928292i
$611$	1.03082e15			0.489730
$612$		0			0
$613$		−	2.02942e15i		−	0.946977i	−0.880800	−	0.473489i	$-0.842994\pi$
$613$		−	2.02942e15i		−	0.946977i	0.880800	−	0.473489i	$-0.157006\pi$
$614$	2.45910e15			1.13723
$615$		0			0
$616$	−6.65945e14			−0.302513
$617$		−	2.44138e15i		−	1.09917i	−0.835436	−	0.549587i	$-0.814785\pi$
$617$		−	2.44138e15i		−	1.09917i	0.835436	−	0.549587i	$-0.185215\pi$
$618$		0			0
$619$	5.08622e14			0.224955			0.112478	−	0.993654i	$-0.464121\pi$
$619$	5.08622e14			0.224955			0.112478	+	0.993654i	$0.464121\pi$
$620$	4.53645e14	+	6.13023e14i	0.198866	+	0.268734i
$621$		0			0
$622$			1.46581e14i			0.0631291i
$623$			4.15303e15i			1.77289i
$624$		0			0
$625$	−1.97676e15	−	1.33296e15i	−0.829112	−	0.559083i
$626$	−2.10166e15			−0.873782
$627$		0			0
$628$		−	1.00344e15i		−	0.409933i
$629$	5.31720e14			0.215330
$630$		0			0
$631$	5.49054e14			0.218501			0.109251	−	0.994014i	$-0.465155\pi$
$631$	5.49054e14			0.218501			0.109251	+	0.994014i	$0.465155\pi$
$632$			1.73724e15i			0.685355i
$633$		0			0
$634$	2.09261e15			0.811330
$635$	3.19470e15	−	2.36412e15i	1.22793	−	0.908686i
$636$		0			0
$637$			2.50098e15i			0.944808i
$638$		−	4.00280e14i		−	0.149917i
$639$		0			0
$640$	1.92998e14	−	1.42821e14i	0.0710498	−	0.0525778i
$641$	−4.07205e15			−1.48626			−0.743129	−	0.669149i	$-0.766659\pi$
$641$	−4.07205e15			−1.48626			−0.743129	+	0.669149i	$0.766659\pi$
$642$		0			0
$643$			1.05002e15i			0.376738i	0.982098	+	0.188369i	$0.0603200\pi$
$643$			1.05002e15i			0.376738i	−0.982098	+	0.188369i	$0.939680\pi$
$644$	−2.44984e15			−0.871498
$645$		0			0
$646$	−1.60229e15			−0.560355
$647$			1.33847e15i			0.464125i	0.972701	+	0.232063i	$0.0745475\pi$
$647$			1.33847e15i			0.464125i	−0.972701	+	0.232063i	$0.925453\pi$
$648$		0			0
$649$	−2.10269e15			−0.716853
$650$	−3.32038e14	+	1.08631e15i	−0.112244	+	0.367222i
$651$		0			0
$652$		−	8.31975e14i		−	0.276534i
$653$			5.09952e15i			1.68076i	0.541995	+	0.840382i	$0.317669\pi$
$653$			5.09952e15i			1.68076i	−0.541995	+	0.840382i	$0.682331\pi$
$654$		0			0
$655$	−1.36788e15	−	1.84845e15i	−0.443323	−	0.599075i
$656$	−2.48057e14			−0.0797224
$657$		0			0
$658$			3.33969e15i			1.05551i
$659$	1.69518e15			0.531307			0.265654	−	0.964069i	$-0.414412\pi$
$659$	1.69518e15			0.531307			0.265654	+	0.964069i	$0.414412\pi$
$660$		0			0
$661$	2.86480e15			0.883051			0.441526	−	0.897249i	$-0.354437\pi$
$661$	2.86480e15			0.883051			0.441526	+	0.897249i	$0.354437\pi$
$662$			5.85173e14i			0.178881i
$663$		0			0
$664$	−7.06118e14			−0.212301
$665$	−6.44452e15	+	4.76903e15i	−1.92164	+	1.42204i
$666$		0			0
$667$		−	1.47253e15i		−	0.431889i
$668$		−	2.32556e15i		−	0.676484i
$669$		0			0
$670$	1.76325e15	+	2.38273e15i	0.504548	+	0.681810i
$671$	4.01971e14			0.114083
$672$		0			0
$673$			2.65808e15i			0.742139i	0.928605	+	0.371070i	$0.121009\pi$
$673$			2.65808e15i			0.742139i	−0.928605	+	0.371070i	$0.878991\pi$
$674$	2.02267e14			0.0560138
$675$		0			0
$676$	−1.29398e15			−0.352549
$677$		−	4.00056e15i		−	1.08114i	−0.841298	−	0.540572i	$-0.818208\pi$
$677$		−	4.00056e15i		−	1.08114i	0.841298	−	0.540572i	$-0.181792\pi$
$678$		0			0
$679$	3.23901e15			0.861248
$680$	−4.37519e14	−	5.91232e14i	−0.115398	−	0.155941i
$681$		0			0
$682$		−	9.41698e14i		−	0.244398i
$683$		−	4.87594e15i		−	1.25529i	−0.778499	−	0.627646i	$-0.784019\pi$
$683$		−	4.87594e15i		−	1.25529i	0.778499	−	0.627646i	$-0.215981\pi$
$684$		0			0
$685$	1.83206e15	−	1.35574e15i	0.464131	−	0.343463i
$686$	−3.44553e15			−0.865911
$687$		0			0
$688$			7.64280e14i			0.189024i
$689$	2.24425e15			0.550638
$690$		0			0
$691$	9.81621e14			0.237036			0.118518	−	0.992952i	$-0.462186\pi$
$691$	9.81621e14			0.237036			0.118518	+	0.992952i	$0.462186\pi$
$692$		−	3.24539e15i		−	0.777469i
$693$		0			0
$694$	−3.00285e15			−0.708038
$695$	−9.02570e13	−	1.21967e14i	−0.0211137	−	0.0285315i
$696$		0			0
$697$			7.59900e14i			0.174975i
$698$		−	3.99675e15i		−	0.913065i
$699$		0			0
$700$	−3.51946e15	−	1.07575e15i	−0.791473	−	0.241920i
$701$	5.96272e15			1.33044			0.665220	−	0.746648i	$-0.268338\pi$
$701$	5.96272e15			1.33044			0.665220	+	0.746648i	$0.268338\pi$
$702$		0			0
$703$		−	2.58027e15i		−	0.566775i
$704$	−2.96475e14			−0.0646157
$705$		0			0
$706$	−1.58430e15			−0.339946
$707$		−	7.20345e13i		−	0.0153368i
$708$		0			0
$709$	−7.33103e15			−1.53678			−0.768388	−	0.639984i	$-0.778941\pi$
$709$	−7.33103e15			−1.53678			−0.768388	+	0.639984i	$0.778941\pi$
$710$	1.70579e14	−	1.26230e14i	0.0354816	−	0.0262569i
$711$		0			0
$712$			1.84891e15i			0.378682i
$713$		−	3.46427e15i		−	0.704076i
$714$		0			0
$715$	1.12750e15	−	8.34367e14i	0.225650	−	0.166984i
$716$	−4.67641e15			−0.928734
$717$		0			0
$718$		−	2.82255e15i		−	0.552024i
$719$	−5.64957e15			−1.09649			−0.548247	−	0.836317i	$-0.684705\pi$
$719$	−5.64957e15			−1.09649			−0.548247	+	0.836317i	$0.684705\pi$
$720$		0			0
$721$	−6.62591e15			−1.26649
$722$			4.04774e15i			0.767818i
$723$		0			0
$724$	1.59119e15			0.297276
$725$	6.46601e14	−	2.11544e15i	0.119888	−	0.392231i
$726$		0			0
$727$		−	1.03387e16i		−	1.88811i	−0.329790	−	0.944054i	$-0.606978\pi$
$727$		−	1.03387e16i		−	1.88811i	0.329790	−	0.944054i	$-0.393022\pi$
$728$			1.75339e15i			0.317801i
$729$		0			0
$730$	1.74019e15	+	2.35157e15i	0.310686	+	0.419839i
$731$	2.34130e15			0.414870
$732$		0			0
$733$			4.98348e15i			0.869883i	0.900459	+	0.434942i	$0.143231\pi$
$733$			4.98348e15i			0.869883i	−0.900459	+	0.434942i	$0.856769\pi$
$734$	3.16484e15			0.548307
$735$		0			0
$736$	−1.09066e15			−0.186149
$737$		−	3.66024e15i		−	0.620067i
$738$		0			0
$739$	−2.94537e15			−0.491582			−0.245791	−	0.969323i	$-0.579048\pi$
$739$	−2.94537e15			−0.491582			−0.245791	+	0.969323i	$0.579048\pi$
$740$	9.52096e14	−	7.04563e14i	0.157727	−	0.116720i
$741$		0			0
$742$			7.27101e15i			1.18679i
$743$		−	5.61294e15i		−	0.909393i	−0.890646	−	0.454697i	$-0.849748\pi$
$743$		−	5.61294e15i		−	0.909393i	0.890646	−	0.454697i	$-0.150252\pi$
$744$		0			0
$745$	−4.08156e14	−	5.51553e14i	−0.0651579	−	0.0880497i
$746$	−1.00940e15			−0.159955
$747$		0			0
$748$			9.08224e14i			0.141819i
$749$	−1.28895e16			−1.99795
$750$		0			0
$751$	5.13668e15			0.784626			0.392313	−	0.919832i	$-0.371675\pi$
$751$	5.13668e15			0.784626			0.392313	+	0.919832i	$0.371675\pi$
$752$			1.48681e15i			0.225454i
$753$		0			0
$754$	−1.05391e15			−0.157493
$755$	−6.68469e15	−	9.03321e15i	−0.991683	−	1.34009i
$756$		0			0
$757$			9.14554e15i			1.33716i	0.743642	+	0.668578i	$0.233097\pi$
$757$			9.14554e15i			1.33716i	−0.743642	+	0.668578i	$0.766903\pi$
$758$			4.67090e15i			0.677985i
$759$		0			0
$760$	−2.86906e15	+	2.12314e15i	−0.410455	+	0.303742i
$761$	−2.23954e15			−0.318084			−0.159042	−	0.987272i	$-0.550841\pi$
$761$	−2.23954e15			−0.318084			−0.159042	+	0.987272i	$0.550841\pi$
$762$		0			0
$763$		−	1.98186e16i		−	2.77452i
$764$	−2.14877e15			−0.298659
$765$		0			0
$766$	−3.00562e15			−0.411789
$767$			5.53624e15i			0.753079i
$768$		0			0
$769$	1.08968e16			1.46118			0.730588	−	0.682819i	$-0.239246\pi$
$769$	1.08968e16			1.46118			0.730588	+	0.682819i	$0.239246\pi$
$770$	2.70321e15	+	3.65293e15i	0.359900	+	0.486343i
$771$		0			0
$772$		−	1.52615e15i		−	0.200310i
$773$		−	4.35156e15i		−	0.567098i	−0.958958	−	0.283549i	$-0.908488\pi$
$773$		−	4.35156e15i		−	0.567098i	0.958958	−	0.283549i	$-0.0915119\pi$
$774$		0			0
$775$	1.52119e15	−	4.97679e15i	0.195445	−	0.639425i
$776$	1.44199e15			0.183959
$777$		0			0
$778$			8.87195e14i			0.111592i
$779$	3.68756e15			0.460556
$780$		0			0
$781$	−2.62035e14			−0.0322685
$782$			3.34113e15i			0.408560i
$783$		0			0
$784$	−3.60731e15			−0.434955
$785$	−5.50420e15	+	4.07318e15i	−0.659039	+	0.487697i
$786$		0			0
$787$			1.61926e15i			0.191186i	0.995420	+	0.0955930i	$0.0304747\pi$
$787$			1.61926e15i			0.191186i	−0.995420	+	0.0955930i	$0.969525\pi$
$788$		−	1.58349e15i		−	0.185661i
$789$		0			0
$790$	9.52933e15	−	7.05183e15i	1.10183	−	0.815366i
$791$	1.85334e16			2.12806
$792$		0			0
$793$		−	1.05836e15i		−	0.119848i
$794$	5.78543e15			0.650613
$795$		0			0
$796$	3.34683e15			0.371204
$797$			8.03974e15i			0.885567i	0.896629	+	0.442783i	$0.146009\pi$
$797$			8.03974e15i			0.885567i	−0.896629	+	0.442783i	$0.853991\pi$
$798$		0			0
$799$	4.55471e15			0.494827
$800$	−1.56684e15	−	4.78918e14i	−0.169056	−	0.0516733i
$801$		0			0
$802$		−	4.31253e15i		−	0.458959i
$803$		−	3.61238e15i		−	0.381819i
$804$		0			0
$805$	9.94445e15	+	1.34382e16i	1.03682	+	1.40108i
$806$	−2.47942e15			−0.256749
$807$		0			0
$808$		−	3.20694e13i		−	0.00327588i
$809$	1.14828e16			1.16501			0.582506	−	0.812826i	$-0.302072\pi$
$809$	1.14828e16			1.16501			0.582506	+	0.812826i	$0.302072\pi$
$810$		0			0
$811$	5.96668e15			0.597198			0.298599	−	0.954379i	$-0.403481\pi$
$811$	5.96668e15			0.597198			0.298599	+	0.954379i	$0.403481\pi$
$812$		−	3.41449e15i		−	0.339444i
$813$		0			0
$814$	−1.46257e15			−0.143444
$815$	−4.56366e15	+	3.37717e15i	−0.444576	+	0.328992i
$816$		0			0
$817$		−	1.13616e16i		−	1.09199i
$818$			5.17928e15i			0.494454i
$819$		0			0
$820$	1.00692e15	+	1.36068e15i	0.0948456	+	0.128168i
$821$	1.58047e15			0.147876			0.0739381	−	0.997263i	$-0.476443\pi$
$821$	1.58047e15			0.147876			0.0739381	+	0.997263i	$0.476443\pi$
$822$		0			0
$823$		−	1.03090e16i		−	0.951734i	−0.879517	−	0.475867i	$-0.842134\pi$
$823$		−	1.03090e16i		−	0.951734i	0.879517	−	0.475867i	$-0.157866\pi$
$824$	−2.94982e15			−0.270517
$825$		0			0
$826$	−1.79365e16			−1.62311
$827$		−	1.03944e16i		−	0.934374i	−0.884158	−	0.467187i	$-0.845267\pi$
$827$		−	1.03944e16i		−	0.934374i	0.884158	−	0.467187i	$-0.154733\pi$
$828$		0			0
$829$	1.52772e16			1.35517			0.677584	−	0.735445i	$-0.263027\pi$
$829$	1.52772e16			1.35517			0.677584	+	0.735445i	$0.263027\pi$
$830$	2.86629e15	+	3.87329e15i	0.252575	+	0.341311i
$831$		0			0
$832$			7.80597e14i			0.0678811i
$833$			1.10507e16i			0.954642i
$834$		0			0
$835$	−1.27565e16	+	9.43996e15i	−1.08757	+	0.804812i
$836$	4.40733e15			0.373285
$837$		0			0
$838$		−	1.42754e15i		−	0.119329i
$839$	4.62607e13			0.00384168			0.00192084	−	0.999998i	$-0.499389\pi$
$839$	4.62607e13			0.00384168			0.00192084	+	0.999998i	$0.499389\pi$
$840$		0			0
$841$	−1.01482e16			−0.831782
$842$		−	1.27007e16i		−	1.03422i
$843$		0			0
$844$	2.14026e15			0.172021
$845$	5.25253e15	+	7.09789e15i	0.419427	+	0.566784i
$846$		0			0
$847$			1.53886e16i			1.21294i
$848$			3.23701e15i			0.253494i
$849$		0			0
$850$	−1.46712e15	+	4.79988e15i	−0.113413	+	0.371045i
$851$	−5.38042e15			−0.413241
$852$		0			0
$853$			1.39374e16i			1.05672i	0.849019	+	0.528362i	$0.177194\pi$
$853$			1.39374e16i			1.05672i	−0.849019	+	0.528362i	$0.822806\pi$
$854$	3.42892e15			0.258308
$855$		0			0
$856$	−5.73831e15			−0.426755
$857$			2.60789e16i			1.92706i	0.267605	+	0.963529i	$0.413768\pi$
$857$			2.60789e16i			1.92706i	−0.267605	+	0.963529i	$0.586232\pi$
$858$		0			0
$859$	4.60719e15			0.336104			0.168052	−	0.985778i	$-0.446252\pi$
$859$	4.60719e15			0.336104			0.168052	+	0.985778i	$0.446252\pi$
$860$	4.19233e15	−	3.10238e15i	0.303889	−	0.224881i
$861$		0			0
$862$			1.51460e16i			1.08395i
$863$		−	5.99532e15i		−	0.426337i	−0.977015	−	0.213169i	$-0.931622\pi$
$863$		−	5.99532e15i		−	0.426337i	0.977015	−	0.213169i	$-0.0683784\pi$
$864$		0			0
$865$	−1.78020e16	+	1.31737e16i	−1.24992	+	0.924954i
$866$	−1.95045e16			−1.36078
$867$		0			0
$868$		−	8.03293e15i		−	0.553370i
$869$	−1.46385e16			−1.00205
$870$		0			0
$871$	−9.63715e15			−0.651402
$872$		−	8.82313e15i		−	0.592628i
$873$		0			0
$874$	1.62135e16			1.07538
$875$	8.38539e15	+	2.36721e16i	0.552686	+	1.56024i
$876$		0			0
$877$			2.57912e16i			1.67870i	0.543592	+	0.839350i	$0.317064\pi$
$877$			2.57912e16i			1.67870i	−0.543592	+	0.839350i	$0.682936\pi$
$878$			1.24486e16i			0.805197i
$879$		0			0
$880$	1.20346e15	+	1.62626e15i	0.0768733	+	0.103881i
$881$	2.80787e16			1.78242			0.891209	−	0.453592i	$-0.149858\pi$
$881$	2.80787e16			1.78242			0.891209	+	0.453592i	$0.149858\pi$
$882$		0			0
$883$		−	2.68062e16i		−	1.68055i	−0.542162	−	0.840274i	$-0.682394\pi$
$883$		−	2.68062e16i		−	1.68055i	0.542162	−	0.840274i	$-0.317606\pi$
$884$	2.39129e15			0.148986
$885$		0			0
$886$	2.17086e16			1.33582
$887$		−	7.52556e15i		−	0.460213i	−0.973165	−	0.230106i	$-0.926093\pi$
$887$		−	7.52556e15i		−	0.460213i	0.973165	−	0.230106i	$-0.0739074\pi$
$888$		0			0
$889$	−4.18627e16			−2.52853
$890$	1.01419e16	−	7.50511e15i	0.608798	−	0.450518i
$891$		0			0
$892$		−	1.94241e15i		−	0.115169i
$893$		−	2.21026e16i		−	1.30245i
$894$		0			0
$895$	1.89826e16	+	2.56517e16i	1.10491	+	1.49310i
$896$	−2.52901e15			−0.146304
$897$		0			0
$898$		−	8.81827e15i		−	0.503922i
$899$	4.82836e15			0.274234
$900$		0			0
$901$	9.91629e15			0.556369
$902$		−	2.09021e15i		−	0.116561i
$903$		0			0
$904$	8.25095e15			0.454546
$905$	−6.45899e15	−	8.72822e15i	−0.353670	−	0.477924i
$906$		0			0
$907$		−	1.54682e16i		−	0.836756i	−0.908273	−	0.418378i	$-0.862599\pi$
$907$		−	1.54682e16i		−	0.836756i	0.908273	−	0.418378i	$-0.137401\pi$
$908$		−	2.27632e15i		−	0.122394i
$909$		0			0
$910$	9.61791e15	−	7.11737e15i	0.510920	−	0.378087i
$911$	2.54945e16			1.34616			0.673079	−	0.739570i	$-0.264971\pi$
$911$	2.54945e16			1.34616			0.673079	+	0.739570i	$0.264971\pi$
$912$		0			0
$913$		−	5.94998e15i		−	0.310403i
$914$	1.65021e16			0.855725
$915$		0			0
$916$	1.12697e15			0.0577415
$917$			2.42218e16i			1.23360i
$918$		0			0
$919$	−1.50582e16			−0.757771			−0.378885	−	0.925444i	$-0.623693\pi$
$919$	−1.50582e16			−0.757771			−0.378885	+	0.925444i	$0.623693\pi$
$920$	4.42721e15	+	5.98262e15i	0.221461	+	0.299267i
$921$		0			0
$922$			5.49376e15i			0.271550i
$923$			6.89920e14i			0.0338992i
$924$		0			0
$925$	−7.72953e15	−	2.36259e15i	−0.375296	−	0.114712i
$926$	−9.56712e15			−0.461765
$927$		0			0
$928$		−	1.52011e15i		−	0.0725040i
$929$	−2.90569e16			−1.37773			−0.688864	−	0.724891i	$-0.741890\pi$
$929$	−2.90569e16			−1.37773			−0.688864	+	0.724891i	$0.741890\pi$
$930$		0			0
$931$	5.36254e16			2.51274
$932$			7.43544e15i			0.346353i
$933$		0			0
$934$	−3.03981e15			−0.139939
$935$	4.98191e15	−	3.68668e15i	0.227999	−	0.168722i
$936$		0			0
$937$		−	1.97723e16i		−	0.894313i	−0.894456	−	0.447156i	$-0.852437\pi$
$937$		−	1.97723e16i		−	0.894313i	0.894456	−	0.447156i	$-0.147563\pi$
$938$		−	3.12228e16i		−	1.40397i
$939$		0			0
$940$	8.15566e15	−	6.03529e15i	0.362456	−	0.268222i
$941$	1.84496e16			0.815160			0.407580	−	0.913169i	$-0.366373\pi$
$941$	1.84496e16			0.815160			0.407580	+	0.913169i	$0.366373\pi$
$942$		0			0
$943$		−	7.68935e15i		−	0.335796i
$944$	−7.98524e15			−0.346690
$945$		0			0
$946$	−6.44007e15			−0.276369
$947$			1.21152e16i			0.516897i	0.966025	+	0.258449i	$0.0832113\pi$
$947$			1.21152e16i			0.516897i	−0.966025	+	0.258449i	$0.916789\pi$
$948$		0			0
$949$	−9.51114e15			−0.401114
$950$	2.32923e16	+	7.11948e15i	0.976636	+	0.298516i
$951$		0			0
$952$			7.74739e15i			0.321109i
$953$		−	1.02683e16i		−	0.423143i	−0.977362	−	0.211572i	$-0.932142\pi$
$953$		−	1.02683e16i		−	0.423143i	0.977362	−	0.211572i	$-0.0678582\pi$
$954$		0			0
$955$	8.72231e15	+	1.17867e16i	0.355315	+	0.480147i
$956$	−1.73936e16			−0.704482
$957$		0			0
$958$		−	1.21099e16i		−	0.484875i
$959$	−2.40069e16			−0.955726
$960$		0			0
$961$	−1.40493e16			−0.552937
$962$			3.85083e15i			0.150693i
$963$		0			0
$964$	−3.80864e15			−0.147349
$965$	−8.37146e15	+	6.19499e15i	−0.322034	+	0.238309i
$966$		0			0
$967$		−	6.61638e15i		−	0.251637i	−0.992053	−	0.125819i	$-0.959844\pi$
$967$		−	6.61638e15i		−	0.251637i	0.992053	−	0.125819i	$-0.0401557\pi$
$968$			6.85090e15i			0.259080i
$969$		0			0
$970$	−5.85334e15	−	7.90979e15i	−0.218856	−	0.295747i
$971$	−4.23912e16			−1.57605			−0.788024	−	0.615644i	$-0.788896\pi$
$971$	−4.23912e16			−1.57605			−0.788024	+	0.615644i	$0.788896\pi$
$972$		0			0
$973$			1.59823e15i			0.0587514i
$974$	−1.95236e16			−0.713650
$975$		0			0
$976$	1.52653e15			0.0551737
$977$			3.85191e15i			0.138438i	0.997601	+	0.0692191i	$0.0220507\pi$
$977$			3.85191e15i			0.138438i	−0.997601	+	0.0692191i	$0.977949\pi$
$978$		0			0
$979$	−1.55795e16			−0.553666
$980$	1.46428e16	+	1.97873e16i	0.517466	+	0.699267i
$981$		0			0
$982$		−	3.63882e16i		−	1.27159i
$983$			1.22656e16i			0.426230i	0.977027	+	0.213115i	$0.0683610\pi$
$983$			1.22656e16i			0.426230i	−0.977027	+	0.213115i	$0.931639\pi$
$984$		0			0
$985$	−8.68597e15	+	6.42772e15i	−0.298483	+	0.220881i
$986$	−4.65673e15			−0.159132
$987$		0			0
$988$		−	1.16042e16i		−	0.392149i
$989$	−2.36914e16			−0.796180
$990$		0			0
$991$	1.86075e16			0.618420			0.309210	−	0.950994i	$-0.399935\pi$
$991$	1.86075e16			0.618420			0.309210	+	0.950994i	$0.399935\pi$
$992$		−	3.57621e15i		−	0.118198i
$993$		0			0
$994$	−2.23523e15			−0.0730629
$995$	−1.35855e16	−	1.83585e16i	−0.441621	−	0.596774i
$996$		0			0
$997$		−	1.75336e16i		−	0.563701i	−0.959458	−	0.281850i	$-0.909052\pi$
$997$		−	1.75336e16i		−	0.563701i	0.959458	−	0.281850i	$-0.0909482\pi$
$998$		−	2.86792e16i		−	0.916957i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.12.c.b.19.6		6
3.2	odd	2		10.12.b.a.9.3	✓	6
5.4	even	2	inner	90.12.c.b.19.3		6
12.11	even	2		80.12.c.c.49.1		6
15.2	even	4		50.12.a.j.1.3		3
15.8	even	4		50.12.a.i.1.1		3
15.14	odd	2		10.12.b.a.9.4	yes	6
60.59	even	2		80.12.c.c.49.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.12.b.a.9.3	✓	6	3.2	odd	2
10.12.b.a.9.4	yes	6	15.14	odd	2
50.12.a.i.1.1		3	15.8	even	4
50.12.a.j.1.3		3	15.2	even	4
80.12.c.c.49.1		6	12.11	even	2
80.12.c.c.49.6		6	60.59	even	2
90.12.c.b.19.3		6	5.4	even	2	inner
90.12.c.b.19.6		6	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	90.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+32.0000i q^{2} -1024.00 q^{4} +(4156.64 + 5616.98i) q^{5} -73603.8i q^{7} -32768.0i q^{8} +(-179743. + 133012. i) q^{10} +276114. q^{11} -726988. i q^{13} +2.35532e6 q^{14} +1.04858e6 q^{16} -3.21222e6i q^{17} -1.55879e7 q^{19} +(-4.25640e6 - 5.75179e6i) q^{20} +8.83564e6i q^{22} +3.25041e7i q^{23} +(-1.42729e7 + 4.66955e7i) q^{25} +2.32636e7 q^{26} +7.53703e7i q^{28} -4.53029e7 q^{29} -1.06580e8 q^{31} +3.35544e7i q^{32} +1.02791e8 q^{34} +(4.13431e8 - 3.05944e8i) q^{35} +1.65530e8i q^{37} -4.98812e8i q^{38} +(1.84057e8 - 1.36205e8i) q^{40} -2.36566e8 q^{41} +7.28874e8i q^{43} -2.82740e8 q^{44} -1.04013e9 q^{46} +1.41793e9i q^{47} -3.44020e9 q^{49} +(-1.49426e9 - 4.56731e8i) q^{50} +7.44435e8i q^{52} +3.08706e9i q^{53} +(1.14770e9 + 1.55093e9i) q^{55} -2.41185e9 q^{56} -1.44969e9i q^{58} -7.61532e9 q^{59} +1.45582e9 q^{61} -3.41054e9i q^{62} -1.07374e9 q^{64} +(4.08348e9 - 3.02182e9i) q^{65} -1.32563e10i q^{67} +3.28931e9i q^{68} +(9.79022e9 + 1.32298e10i) q^{70} -9.49012e8 q^{71} -1.30829e10i q^{73} -5.29697e9 q^{74} +1.59620e10 q^{76} -2.03230e10i q^{77} -5.30163e10 q^{79} +(4.35855e9 + 5.88983e9i) q^{80} -7.57010e9i q^{82} -2.15490e10i q^{83} +(1.80430e10 - 1.33520e10i) q^{85} -2.33240e10 q^{86} -9.04769e9i q^{88} -5.64241e10 q^{89} -5.35091e10 q^{91} -3.32842e10i q^{92} -4.53739e10 q^{94} +(-6.47932e10 - 8.75569e10i) q^{95} +4.40060e10i q^{97} -1.10086e11i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6144 q^{4} - 530 q^{5} - 385920 q^{10} + 642728 q^{11} + 2125952 q^{14} + 6291456 q^{16} - 24109080 q^{19} + 542720 q^{20} - 181718850 q^{25} - 89251584 q^{26} + 256409820 q^{29} + 458481792 q^{31}+ \cdots - 104896380600 q^{95}+O(q^{100}) \) 6 * q - 6144 * q^4 - 530 * q^5 - 385920 * q^10 + 642728 * q^11 + 2125952 * q^14 + 6291456 * q^16 - 24109080 * q^19 + 542720 * q^20 - 181718850 * q^25 - 89251584 * q^26 + 256409820 * q^29 + 458481792 * q^31 - 225288192 * q^34 + 697136360 * q^35 + 395182080 * q^40 + 164768948 * q^41 - 658153472 * q^44 - 2956208256 * q^46 - 675514158 * q^49 - 3912262400 * q^50 + 3688644360 * q^55 - 2176974848 * q^56 - 17663962360 * q^59 - 5020792428 * q^61 - 6442450944 * q^64 + 19996916880 * q^65 + 24956826240 * q^70 - 56788418832 * q^71 + 64135292672 * q^74 + 24687697920 * q^76 + 2602550880 * q^79 - 555745280 * q^80 - 85024210560 * q^85 - 111995790464 * q^86 - 249448412540 * q^89 - 184446766128 * q^91 - 337749482112 * q^94 - 104896380600 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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