LMFDB - Embedded newform 192.9.l.a.175.21

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [192,9,Mod(79,192)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("192.79"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(192, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	192.l (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$78.2166931317$
Analytic rank:	$0$
Dimension:	$64$
Relative dimension:	$32$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			175.21
Character	$\chi$	$=$	192.175
Dual form			192.9.l.a.79.21

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(33.0681 - 33.0681i) q^{3} +(-349.282 + 349.282i) q^{5} +4640.58 q^{7} -2187.00i q^{9} +(549.148 + 549.148i) q^{11} +(-4126.26 - 4126.26i) q^{13} +23100.2i q^{15} +121373. q^{17} +(-16510.6 + 16510.6i) q^{19} +(153455. - 153455. i) q^{21} +104834. q^{23} +146628. i q^{25} +(-72320.0 - 72320.0i) q^{27} +(-781240. - 781240. i) q^{29} +340243. i q^{31} +36318.6 q^{33} +(-1.62088e6 + 1.62088e6i) q^{35} +(1.29420e6 - 1.29420e6i) q^{37} -272895. q^{39} +4.04263e6i q^{41} +(-1.64022e6 - 1.64022e6i) q^{43} +(763881. + 763881. i) q^{45} +2.89021e6i q^{47} +1.57702e7 q^{49} +(4.01357e6 - 4.01357e6i) q^{51} +(-3.17202e6 + 3.17202e6i) q^{53} -383616. q^{55} +1.09195e6i q^{57} +(5.50700e6 + 5.50700e6i) q^{59} +(1.51527e7 + 1.51527e7i) q^{61} -1.01490e7i q^{63} +2.88246e6 q^{65} +(1.20675e7 - 1.20675e7i) q^{67} +(3.46667e6 - 3.46667e6i) q^{69} -3.40128e7 q^{71} -3.77315e7i q^{73} +(4.84873e6 + 4.84873e6i) q^{75} +(2.54837e6 + 2.54837e6i) q^{77} -3.12659e6i q^{79} -4.78297e6 q^{81} +(6.95385e6 - 6.95385e6i) q^{83} +(-4.23934e7 + 4.23934e7i) q^{85} -5.16682e7 q^{87} -4.53098e7i q^{89} +(-1.91483e7 - 1.91483e7i) q^{91} +(1.12512e7 + 1.12512e7i) q^{93} -1.15337e7i q^{95} +1.53165e8 q^{97} +(1.20099e6 - 1.20099e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q - 39552 q^{11} + 167552 q^{19} - 1691136 q^{23} - 2132352 q^{29} + 2415744 q^{35} - 4720512 q^{37} + 7244672 q^{43} + 52706752 q^{49} - 13862016 q^{51} - 5358720 q^{53} + 46326784 q^{55} - 44938752 q^{59}+ \cdots - 86500224 q^{99}+O(q^{100}) $ 64 * q - 39552 * q^11 + 167552 * q^19 - 1691136 * q^23 - 2132352 * q^29 + 2415744 * q^35 - 4720512 * q^37 + 7244672 * q^43 + 52706752 * q^49 - 13862016 * q^51 - 5358720 * q^53 + 46326784 * q^55 - 44938752 * q^59 + 24476032 * q^61 + 29941632 * q^65 + 44244736 * q^67 - 8636544 * q^69 - 159664128 * q^71 - 12918528 * q^75 - 94964352 * q^77 - 306110016 * q^81 - 209328000 * q^83 + 106960000 * q^85 + 45401472 * q^91 - 86500224 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	33.0681	−	33.0681i	0.408248	−	0.408248i
$3$	33.0681	−	33.0681i	0.408248	−	0.408248i
$4$		0			0
$5$	−349.282	+	349.282i	−0.558852	+	0.558852i	−0.928981	−	0.370129i	$-0.879314\pi$
$5$	−349.282	+	349.282i	−0.558852	+	0.558852i	0.370129	+	0.928981i	$0.379314\pi$
$6$		0			0
$7$	4640.58			1.93277			0.966386	−	0.257096i	$-0.0827657\pi$
$7$	4640.58			1.93277			0.966386	+	0.257096i	$0.0827657\pi$
$8$		0			0
$9$		−	2187.00i		−	0.333333i
$10$		0			0
$11$	549.148	+	549.148i	0.0375076	+	0.0375076i	0.725612	−	0.688104i	$-0.241557\pi$
$11$	549.148	+	549.148i	0.0375076	+	0.0375076i	−0.688104	+	0.725612i	$0.741557\pi$
$12$		0			0
$13$	−4126.26	−	4126.26i	−0.144472	−	0.144472i	0.631171	−	0.775643i	$-0.282574\pi$
$13$	−4126.26	−	4126.26i	−0.144472	−	0.144472i	−0.775643	+	0.631171i	$0.782574\pi$
$14$		0			0
$15$			23100.2i			0.456301i
$16$		0			0
$17$	121373.			1.45320			0.726600	−	0.687061i	$-0.241099\pi$
$17$	121373.			1.45320			0.726600	+	0.687061i	$0.241099\pi$
$18$		0			0
$19$	−16510.6	+	16510.6i	−0.126692	+	0.126692i	−0.767610	−	0.640918i	$-0.778554\pi$
$19$	−16510.6	+	16510.6i	−0.126692	+	0.126692i	0.640918	+	0.767610i	$0.278554\pi$
$20$		0			0
$21$	153455.	−	153455.i	0.789051	−	0.789051i
$22$		0			0
$23$	104834.			0.374620			0.187310	−	0.982301i	$-0.440023\pi$
$23$	104834.			0.374620			0.187310	+	0.982301i	$0.440023\pi$
$24$		0			0
$25$			146628.i			0.375369i
$26$		0			0
$27$	−72320.0	−	72320.0i	−0.136083	−	0.136083i
$28$		0			0
$29$	−781240.	−	781240.i	−1.10457	−	1.10457i	−0.993852	−	0.110715i	$-0.964686\pi$
$29$	−781240.	−	781240.i	−1.10457	−	1.10457i	−0.110715	−	0.993852i	$-0.535314\pi$
$30$		0			0
$31$			340243.i			0.368419i	0.982887	+	0.184210i	$0.0589726\pi$
$31$			340243.i			0.368419i	−0.982887	+	0.184210i	$0.941027\pi$
$32$		0			0
$33$	36318.6			0.0306248
$34$		0			0
$35$	−1.62088e6	+	1.62088e6i	−1.08013	+	1.08013i
$36$		0			0
$37$	1.29420e6	−	1.29420e6i	0.690549	−	0.690549i	−0.271804	−	0.962353i	$-0.587620\pi$
$37$	1.29420e6	−	1.29420e6i	0.690549	−	0.690549i	0.962353	+	0.271804i	$0.0876202\pi$
$38$		0			0
$39$	−272895.			−0.117961
$40$		0			0
$41$			4.04263e6i			1.43063i	0.698801	+	0.715316i	$0.253717\pi$
$41$			4.04263e6i			1.43063i	−0.698801	+	0.715316i	$0.746283\pi$
$42$		0			0
$43$	−1.64022e6	−	1.64022e6i	−0.479766	−	0.479766i	0.425291	−	0.905057i	$-0.360172\pi$
$43$	−1.64022e6	−	1.64022e6i	−0.479766	−	0.479766i	−0.905057	+	0.425291i	$0.860172\pi$
$44$		0			0
$45$	763881.	+	763881.i	0.186284	+	0.186284i
$46$		0			0
$47$			2.89021e6i			0.592294i	0.955142	+	0.296147i	$0.0957018\pi$
$47$			2.89021e6i			0.592294i	−0.955142	+	0.296147i	$0.904298\pi$
$48$		0			0
$49$	1.57702e7			2.73561
$50$		0			0
$51$	4.01357e6	−	4.01357e6i	0.593267	−	0.593267i
$52$		0			0
$53$	−3.17202e6	+	3.17202e6i	−0.402006	+	0.402006i	−0.878939	−	0.476934i	$-0.841748\pi$
$53$	−3.17202e6	+	3.17202e6i	−0.402006	+	0.402006i	0.476934	+	0.878939i	$0.341748\pi$
$54$		0			0
$55$	−383616.			−0.0419224
$56$		0			0
$57$			1.09195e6i			0.103443i
$58$		0			0
$59$	5.50700e6	+	5.50700e6i	0.454472	+	0.454472i	0.896836	−	0.442364i	$-0.145860\pi$
$59$	5.50700e6	+	5.50700e6i	0.454472	+	0.454472i	−0.442364	+	0.896836i	$0.645860\pi$
$60$		0			0
$61$	1.51527e7	+	1.51527e7i	1.09438	+	1.09438i	0.995055	+	0.0993283i	$0.0316694\pi$
$61$	1.51527e7	+	1.51527e7i	1.09438	+	1.09438i	0.0993283	+	0.995055i	$0.468331\pi$
$62$		0			0
$63$		−	1.01490e7i		−	0.644257i
$64$		0			0
$65$	2.88246e6			0.161477
$66$		0			0
$67$	1.20675e7	−	1.20675e7i	0.598851	−	0.598851i	−0.341156	−	0.940007i	$-0.610818\pi$
$67$	1.20675e7	−	1.20675e7i	0.598851	−	0.598851i	0.940007	+	0.341156i	$0.110818\pi$
$68$		0			0
$69$	3.46667e6	−	3.46667e6i	0.152938	−	0.152938i
$70$		0			0
$71$	−3.40128e7			−1.33847			−0.669236	−	0.743050i	$-0.733379\pi$
$71$	−3.40128e7			−1.33847			−0.669236	+	0.743050i	$0.733379\pi$
$72$		0			0
$73$		−	3.77315e7i		−	1.32865i	−0.747442	−	0.664327i	$-0.768718\pi$
$73$		−	3.77315e7i		−	1.32865i	0.747442	−	0.664327i	$-0.231282\pi$
$74$		0			0
$75$	4.84873e6	+	4.84873e6i	0.153244	+	0.153244i
$76$		0			0
$77$	2.54837e6	+	2.54837e6i	0.0724936	+	0.0724936i
$78$		0			0
$79$		−	3.12659e6i		−	0.0802717i	−0.999194	−	0.0401359i	$-0.987221\pi$
$79$		−	3.12659e6i		−	0.0802717i	0.999194	−	0.0401359i	$-0.0127791\pi$
$80$		0			0
$81$	−4.78297e6			−0.111111
$82$		0			0
$83$	6.95385e6	−	6.95385e6i	0.146525	−	0.146525i	−0.630038	−	0.776564i	$-0.716961\pi$
$83$	6.95385e6	−	6.95385e6i	0.146525	−	0.146525i	0.776564	+	0.630038i	$0.216961\pi$
$84$		0			0
$85$	−4.23934e7	+	4.23934e7i	−0.812124	+	0.812124i
$86$		0			0
$87$	−5.16682e7			−0.901875
$88$		0			0
$89$		−	4.53098e7i		−	0.722158i	−0.932535	−	0.361079i	$-0.882409\pi$
$89$		−	4.53098e7i		−	0.722158i	0.932535	−	0.361079i	$-0.117591\pi$
$90$		0			0
$91$	−1.91483e7	−	1.91483e7i	−0.279231	−	0.279231i
$92$		0			0
$93$	1.12512e7	+	1.12512e7i	0.150407	+	0.150407i
$94$		0			0
$95$		−	1.15337e7i		−	0.141604i
$96$		0			0
$97$	1.53165e8			1.73010			0.865052	−	0.501683i	$-0.167286\pi$
$97$	1.53165e8			1.73010			0.865052	+	0.501683i	$0.167286\pi$
$98$		0			0
$99$	1.20099e6	−	1.20099e6i	0.0125025	−	0.0125025i
$100$		0			0
$101$	1.29544e8	−	1.29544e8i	1.24489	−	1.24489i	0.286943	−	0.957948i	$-0.407361\pi$
$101$	1.29544e8	−	1.29544e8i	1.24489	−	1.24489i	0.957948	−	0.286943i	$-0.0926391\pi$
$102$		0			0
$103$	1.92950e8			1.71434			0.857168	−	0.515036i	$-0.172222\pi$
$103$	1.92950e8			1.71434			0.857168	+	0.515036i	$0.172222\pi$
$104$		0			0
$105$			1.07199e8i			0.881925i
$106$		0			0
$107$	1.42084e8	+	1.42084e8i	1.08395	+	1.08395i	0.996137	+	0.0878170i	$0.0279891\pi$
$107$	1.42084e8	+	1.42084e8i	1.08395	+	1.08395i	0.0878170	+	0.996137i	$0.472011\pi$
$108$		0			0
$109$	8.57719e7	+	8.57719e7i	0.607630	+	0.607630i	0.942326	−	0.334696i	$-0.108634\pi$
$109$	8.57719e7	+	8.57719e7i	0.607630	+	0.607630i	−0.334696	+	0.942326i	$0.608634\pi$
$110$		0			0
$111$		−	8.55935e7i		−	0.563831i
$112$		0			0
$113$	6.15982e7			0.377793			0.188897	−	0.981997i	$-0.439509\pi$
$113$	6.15982e7			0.377793			0.188897	+	0.981997i	$0.439509\pi$
$114$		0			0
$115$	−3.66167e7	+	3.66167e7i	−0.209357	+	0.209357i
$116$		0			0
$117$	−9.02414e6	+	9.02414e6i	−0.0481573	+	0.0481573i
$118$		0			0
$119$	5.63240e8			2.80870
$120$		0			0
$121$		−	2.13756e8i		−	0.997186i
$122$		0			0
$123$	1.33682e8	+	1.33682e8i	0.584053	+	0.584053i
$124$		0			0
$125$	−1.87653e8	−	1.87653e8i	−0.768628	−	0.768628i
$126$		0			0
$127$			4.09831e8i			1.57540i	0.616061	+	0.787699i	$0.288728\pi$
$127$			4.09831e8i			1.57540i	−0.616061	+	0.787699i	$0.711272\pi$
$128$		0			0
$129$	−1.08478e8			−0.391727
$130$		0			0
$131$	1.85539e8	−	1.85539e8i	0.630013	−	0.630013i	−0.318058	−	0.948071i	$-0.603031\pi$
$131$	1.85539e8	−	1.85539e8i	0.630013	−	0.630013i	0.948071	+	0.318058i	$0.103031\pi$
$132$		0			0
$133$	−7.66189e7	+	7.66189e7i	−0.244867	+	0.244867i
$134$		0			0
$135$	5.05202e7			0.152100
$136$		0			0
$137$			5.35882e8i			1.52120i	0.649220	+	0.760601i	$0.275096\pi$
$137$			5.35882e8i			1.52120i	−0.649220	+	0.760601i	$0.724904\pi$
$138$		0			0
$139$	−2.10569e8	−	2.10569e8i	−0.564072	−	0.564072i	0.366389	−	0.930462i	$-0.380594\pi$
$139$	−2.10569e8	−	2.10569e8i	−0.564072	−	0.564072i	−0.930462	+	0.366389i	$0.880594\pi$
$140$		0			0
$141$	9.55736e7	+	9.55736e7i	0.241803	+	0.241803i
$142$		0			0
$143$		−	4.53186e6i		−	0.0108376i
$144$		0			0
$145$	5.45747e8			1.23458
$146$		0			0
$147$	5.21492e8	−	5.21492e8i	1.11681	−	1.11681i
$148$		0			0
$149$	−2.80503e8	+	2.80503e8i	−0.569106	+	0.569106i	−0.931878	−	0.362772i	$-0.881830\pi$
$149$	−2.80503e8	+	2.80503e8i	−0.569106	+	0.569106i	0.362772	+	0.931878i	$0.381830\pi$
$150$		0			0
$151$	−1.68382e8			−0.323884			−0.161942	−	0.986800i	$-0.551776\pi$
$151$	−1.68382e8			−0.323884			−0.161942	+	0.986800i	$0.551776\pi$
$152$		0			0
$153$		−	2.65442e8i		−	0.484400i
$154$		0			0
$155$	−1.18841e8	−	1.18841e8i	−0.205892	−	0.205892i
$156$		0			0
$157$	3.86067e7	+	3.86067e7i	0.0635425	+	0.0635425i	0.738164	−	0.674621i	$-0.235693\pi$
$157$	3.86067e7	+	3.86067e7i	0.0635425	+	0.0635425i	−0.674621	+	0.738164i	$0.735693\pi$
$158$		0			0
$159$			2.09785e8i			0.328236i
$160$		0			0
$161$	4.86492e8			0.724056
$162$		0			0
$163$	5.58902e8	−	5.58902e8i	0.791745	−	0.791745i	−0.190033	−	0.981778i	$-0.560859\pi$
$163$	5.58902e8	−	5.58902e8i	0.791745	−	0.791745i	0.981778	+	0.190033i	$0.0608594\pi$
$164$		0			0
$165$	−1.26855e7	+	1.26855e7i	−0.0171147	+	0.0171147i
$166$		0			0
$167$	−2.25264e8			−0.289619			−0.144809	−	0.989460i	$-0.546257\pi$
$167$	−2.25264e8			−0.289619			−0.144809	+	0.989460i	$0.546257\pi$
$168$		0			0
$169$		−	7.81679e8i		−	0.958256i
$170$		0			0
$171$	3.61087e7	+	3.61087e7i	0.0422306	+	0.0422306i
$172$		0			0
$173$	1.20004e8	+	1.20004e8i	0.133971	+	0.133971i	0.770912	−	0.636941i	$-0.219801\pi$
$173$	1.20004e8	+	1.20004e8i	0.133971	+	0.133971i	−0.636941	+	0.770912i	$0.719801\pi$
$174$		0			0
$175$			6.80442e8i			0.725502i
$176$		0			0
$177$	3.64212e8			0.371075
$178$		0			0
$179$	−1.63238e8	+	1.63238e8i	−0.159004	+	0.159004i	−0.782125	−	0.623121i	$-0.785864\pi$
$179$	−1.63238e8	+	1.63238e8i	−0.159004	+	0.159004i	0.623121	+	0.782125i	$0.285864\pi$
$180$		0			0
$181$	1.06929e9	−	1.06929e9i	0.996283	−	0.996283i	−0.00371014	−	0.999993i	$-0.501181\pi$
$181$	1.06929e9	−	1.06929e9i	0.996283	−	0.996283i	0.999993	+	0.00371014i	$0.00118098\pi$
$182$		0			0
$183$	1.00214e9			0.893560
$184$		0			0
$185$			9.04083e8i			0.771829i
$186$		0			0
$187$	6.66517e7	+	6.66517e7i	0.0545060	+	0.0545060i
$188$		0			0
$189$	−3.35607e8	−	3.35607e8i	−0.263017	−	0.263017i
$190$		0			0
$191$			2.17451e9i			1.63391i	0.576701	+	0.816955i	$0.304340\pi$
$191$			2.17451e9i			1.63391i	−0.576701	+	0.816955i	$0.695660\pi$
$192$		0			0
$193$	1.80281e9			1.29934			0.649668	−	0.760218i	$-0.274908\pi$
$193$	1.80281e9			1.29934			0.649668	+	0.760218i	$0.274908\pi$
$194$		0			0
$195$	9.53176e7	−	9.53176e7i	0.0659226	−	0.0659226i
$196$		0			0
$197$	−1.25683e9	+	1.25683e9i	−0.834474	+	0.834474i	−0.988125	−	0.153651i	$-0.950897\pi$
$197$	−1.25683e9	+	1.25683e9i	−0.834474	+	0.834474i	0.153651	+	0.988125i	$0.450897\pi$
$198$		0			0
$199$	−1.91242e9			−1.21947			−0.609733	−	0.792607i	$-0.708723\pi$
$199$	−1.91242e9			−1.21947			−0.609733	+	0.792607i	$0.708723\pi$
$200$		0			0
$201$		−	7.98100e8i		−	0.488960i
$202$		0			0
$203$	−3.62541e9	−	3.62541e9i	−2.13488	−	2.13488i
$204$		0			0
$205$	−1.41202e9	−	1.41202e9i	−0.799512	−	0.799512i
$206$		0			0
$207$		−	2.29272e8i		−	0.124873i
$208$		0			0
$209$	−1.81336e7			−0.00950381
$210$		0			0
$211$	−2.42091e9	+	2.42091e9i	−1.22137	+	1.22137i	−0.254231	+	0.967143i	$0.581822\pi$
$211$	−2.42091e9	+	2.42091e9i	−1.22137	+	1.22137i	−0.967143	+	0.254231i	$0.918178\pi$
$212$		0			0
$213$	−1.12474e9	+	1.12474e9i	−0.546429	+	0.546429i
$214$		0			0
$215$	1.14580e9			0.536236
$216$		0			0
$217$			1.57893e9i			0.712071i
$218$		0			0
$219$	−1.24771e9	−	1.24771e9i	−0.542421	−	0.542421i
$220$		0			0
$221$	−5.00816e8	−	5.00816e8i	−0.209947	−	0.209947i
$222$		0			0
$223$			1.17635e8i			0.0475683i	0.999717	+	0.0237842i	$0.00757145\pi$
$223$			1.17635e8i			0.0475683i	−0.999717	+	0.0237842i	$0.992429\pi$
$224$		0			0
$225$	3.20676e8			0.125123
$226$		0			0
$227$	1.20260e8	−	1.20260e8i	0.0452915	−	0.0452915i	−0.684098	−	0.729390i	$-0.739804\pi$
$227$	1.20260e8	−	1.20260e8i	0.0452915	−	0.0452915i	0.729390	+	0.684098i	$0.239804\pi$
$228$		0			0
$229$	7.99469e8	−	7.99469e8i	0.290710	−	0.290710i	−0.546651	−	0.837361i	$-0.684097\pi$
$229$	7.99469e8	−	7.99469e8i	0.290710	−	0.290710i	0.837361	+	0.546651i	$0.184097\pi$
$230$		0			0
$231$	1.68540e8			0.0591908
$232$		0			0
$233$		−	7.89356e8i		−	0.267824i	−0.990993	−	0.133912i	$-0.957246\pi$
$233$		−	7.89356e8i		−	0.267824i	0.990993	−	0.133912i	$-0.0427539\pi$
$234$		0			0
$235$	−1.00950e9	−	1.00950e9i	−0.331005	−	0.331005i
$236$		0			0
$237$	−1.03390e8	−	1.03390e8i	−0.0327708	−	0.0327708i
$238$		0			0
$239$		−	1.65241e9i		−	0.506436i	−0.967409	−	0.253218i	$-0.918511\pi$
$239$		−	1.65241e9i		−	0.506436i	0.967409	−	0.253218i	$-0.0814891\pi$
$240$		0			0
$241$	−2.71108e7			−0.00803664			−0.00401832	−	0.999992i	$-0.501279\pi$
$241$	−2.71108e7			−0.00803664			−0.00401832	+	0.999992i	$0.501279\pi$
$242$		0			0
$243$	−1.58164e8	+	1.58164e8i	−0.0453609	+	0.0453609i
$244$		0			0
$245$	−5.50826e9	+	5.50826e9i	−1.52880	+	1.52880i
$246$		0			0
$247$	1.36254e8			0.0366068
$248$		0			0
$249$		−	4.59901e8i		−	0.119638i
$250$		0			0
$251$	1.72776e9	+	1.72776e9i	0.435299	+	0.435299i	0.890426	−	0.455127i	$-0.150406\pi$
$251$	1.72776e9	+	1.72776e9i	0.435299	+	0.435299i	−0.455127	+	0.890426i	$0.650406\pi$
$252$		0			0
$253$	5.75695e7	+	5.75695e7i	0.0140511	+	0.0140511i
$254$		0			0
$255$			2.80374e9i			0.663096i
$256$		0			0
$257$	1.40666e9			0.322446			0.161223	−	0.986918i	$-0.448456\pi$
$257$	1.40666e9			0.322446			0.161223	+	0.986918i	$0.448456\pi$
$258$		0			0
$259$	6.00584e9	−	6.00584e9i	1.33467	−	1.33467i
$260$		0			0
$261$	−1.70857e9	+	1.70857e9i	−0.368189	+	0.368189i
$262$		0			0
$263$	−5.15945e9			−1.07840			−0.539201	−	0.842177i	$-0.681274\pi$
$263$	−5.15945e9			−1.07840			−0.539201	+	0.842177i	$0.681274\pi$
$264$		0			0
$265$		−	2.21586e9i		−	0.449323i
$266$		0			0
$267$	−1.49831e9	−	1.49831e9i	−0.294820	−	0.294820i
$268$		0			0
$269$	−4.42147e9	−	4.42147e9i	−0.844419	−	0.844419i	0.145011	−	0.989430i	$-0.453678\pi$
$269$	−4.42147e9	−	4.42147e9i	−0.844419	−	0.844419i	−0.989430	+	0.145011i	$0.953678\pi$
$270$		0			0
$271$			3.85251e9i			0.714276i	0.934052	+	0.357138i	$0.116247\pi$
$271$			3.85251e9i			0.714276i	−0.934052	+	0.357138i	$0.883753\pi$
$272$		0			0
$273$	−1.26639e9			−0.227991
$274$		0			0
$275$	−8.05208e7	+	8.05208e7i	−0.0140792	+	0.0140792i
$276$		0			0
$277$	−6.72321e9	+	6.72321e9i	−1.14198	+	1.14198i	−0.153890	+	0.988088i	$0.549180\pi$
$277$	−6.72321e9	+	6.72321e9i	−1.14198	+	1.14198i	−0.988088	+	0.153890i	$0.950820\pi$
$278$		0			0
$279$	7.44112e8			0.122806
$280$		0			0
$281$		−	6.67183e9i		−	1.07009i	−0.844824	−	0.535044i	$-0.820295\pi$
$281$		−	6.67183e9i		−	1.07009i	0.844824	−	0.535044i	$-0.179705\pi$
$282$		0			0
$283$	−1.00632e9	−	1.00632e9i	−0.156888	−	0.156888i	0.624298	−	0.781186i	$-0.285385\pi$
$283$	−1.00632e9	−	1.00632e9i	−0.156888	−	0.156888i	−0.781186	+	0.624298i	$0.785385\pi$
$284$		0			0
$285$	−3.81399e8	−	3.81399e8i	−0.0578096	−	0.0578096i
$286$		0			0
$287$			1.87602e10i			2.76509i
$288$		0			0
$289$	7.75558e9			1.11179
$290$		0			0
$291$	5.06487e9	−	5.06487e9i	0.706312	−	0.706312i
$292$		0			0
$293$	−8.19417e9	+	8.19417e9i	−1.11182	+	1.11182i	−0.118915	+	0.992904i	$0.537942\pi$
$293$	−8.19417e9	+	8.19417e9i	−1.11182	+	1.11182i	−0.992904	+	0.118915i	$0.962058\pi$
$294$		0			0
$295$	−3.84700e9			−0.507965
$296$		0			0
$297$		−	7.94288e7i		−	0.0102083i
$298$		0			0
$299$	−4.32573e8	−	4.32573e8i	−0.0541221	−	0.0541221i
$300$		0			0
$301$	−7.61160e9	−	7.61160e9i	−0.927278	−	0.927278i
$302$		0			0
$303$		−	8.56754e9i		−	1.01645i
$304$		0			0
$305$	−1.05851e10			−1.22320
$306$		0			0
$307$	−4.32070e9	+	4.32070e9i	−0.486408	+	0.486408i	−0.907171	−	0.420763i	$-0.861762\pi$
$307$	−4.32070e9	+	4.32070e9i	−0.486408	+	0.486408i	0.420763	+	0.907171i	$0.361762\pi$
$308$		0			0
$309$	6.38049e9	−	6.38049e9i	0.699875	−	0.699875i
$310$		0			0
$311$	−1.13313e10			−1.21126			−0.605631	−	0.795746i	$-0.707079\pi$
$311$	−1.13313e10			−1.21126			−0.605631	+	0.795746i	$0.707079\pi$
$312$		0			0
$313$		−	7.72644e9i		−	0.805011i	−0.915418	−	0.402506i	$-0.868139\pi$
$313$		−	7.72644e9i		−	0.805011i	0.915418	−	0.402506i	$-0.131861\pi$
$314$		0			0
$315$	3.54485e9	+	3.54485e9i	0.360044	+	0.360044i
$316$		0			0
$317$	−1.03658e9	−	1.03658e9i	−0.102651	−	0.102651i	0.653916	−	0.756567i	$-0.273125\pi$
$317$	−1.03658e9	−	1.03658e9i	−0.102651	−	0.102651i	−0.756567	+	0.653916i	$0.773125\pi$
$318$		0			0
$319$		−	8.58033e8i		−	0.0828593i
$320$		0			0
$321$	9.39691e9			0.885044
$322$		0			0
$323$	−2.00394e9	+	2.00394e9i	−0.184109	+	0.184109i
$324$		0			0
$325$	6.05028e8	−	6.05028e8i	0.0542303	−	0.0542303i
$326$		0			0
$327$	5.67263e9			0.496128
$328$		0			0
$329$			1.34122e10i			1.14477i
$330$		0			0
$331$	−1.58849e10	−	1.58849e10i	−1.32334	−	1.32334i	−0.911053	−	0.412289i	$-0.864729\pi$
$331$	−1.58849e10	−	1.58849e10i	−1.32334	−	1.32334i	−0.412289	−	0.911053i	$-0.635271\pi$
$332$		0			0
$333$	−2.83041e9	−	2.83041e9i	−0.230183	−	0.230183i
$334$		0			0
$335$			8.42995e9i			0.669338i
$336$		0			0
$337$	1.38780e10			1.07599			0.537995	−	0.842948i	$-0.319182\pi$
$337$	1.38780e10			1.07599			0.537995	+	0.842948i	$0.319182\pi$
$338$		0			0
$339$	2.03694e9	−	2.03694e9i	0.154234	−	0.154234i
$340$		0			0
$341$	−1.86844e8	+	1.86844e8i	−0.0138185	+	0.0138185i
$342$		0			0
$343$	4.64310e10			3.35453
$344$		0			0
$345$			2.42169e9i			0.170940i
$346$		0			0
$347$	1.49255e9	+	1.49255e9i	0.102946	+	0.102946i	0.756704	−	0.653758i	$-0.226808\pi$
$347$	1.49255e9	+	1.49255e9i	0.102946	+	0.102946i	−0.653758	+	0.756704i	$0.726808\pi$
$348$		0			0
$349$	−5.44079e9	−	5.44079e9i	−0.366742	−	0.366742i	0.499546	−	0.866287i	$-0.333500\pi$
$349$	−5.44079e9	−	5.44079e9i	−0.366742	−	0.366742i	−0.866287	+	0.499546i	$0.833500\pi$
$350$		0			0
$351$			5.96822e8i			0.0393203i
$352$		0			0
$353$	1.23319e8			0.00794203			0.00397101	−	0.999992i	$-0.498736\pi$
$353$	1.23319e8			0.00794203			0.00397101	+	0.999992i	$0.498736\pi$
$354$		0			0
$355$	1.18801e10	−	1.18801e10i	0.748008	−	0.748008i
$356$		0			0
$357$	1.86253e10	−	1.86253e10i	1.14665	−	1.14665i
$358$		0			0
$359$	−1.27174e10			−0.765635			−0.382817	−	0.923824i	$-0.625046\pi$
$359$	−1.27174e10			−0.765635			−0.382817	+	0.923824i	$0.625046\pi$
$360$		0			0
$361$			1.64384e10i			0.967898i
$362$		0			0
$363$	−7.06850e9	−	7.06850e9i	−0.407100	−	0.407100i
$364$		0			0
$365$	1.31789e10	+	1.31789e10i	0.742521	+	0.742521i
$366$		0			0
$367$		−	2.23031e10i		−	1.22942i	−0.788752	−	0.614711i	$-0.789272\pi$
$367$		−	2.23031e10i		−	1.22942i	0.788752	−	0.614711i	$-0.210728\pi$
$368$		0			0
$369$	8.84122e9			0.476878
$370$		0			0
$371$	−1.47200e10	+	1.47200e10i	−0.776985	+	0.776985i
$372$		0			0
$373$	−1.28562e10	+	1.28562e10i	−0.664168	+	0.664168i	−0.956360	−	0.292192i	$-0.905616\pi$
$373$	−1.28562e10	+	1.28562e10i	−0.664168	+	0.664168i	0.292192	+	0.956360i	$0.405616\pi$
$374$		0			0
$375$	−1.24107e10			−0.627582
$376$		0			0
$377$			6.44720e9i			0.319158i
$378$		0			0
$379$	−4.44623e9	−	4.44623e9i	−0.215494	−	0.215494i	0.591103	−	0.806596i	$-0.298693\pi$
$379$	−4.44623e9	−	4.44623e9i	−0.215494	−	0.215494i	−0.806596	+	0.591103i	$0.798693\pi$
$380$		0			0
$381$	1.35523e10	+	1.35523e10i	0.643153	+	0.643153i
$382$		0			0
$383$			3.76186e9i			0.174827i	0.996172	+	0.0874134i	$0.0278601\pi$
$383$			3.76186e9i			0.174827i	−0.996172	+	0.0874134i	$0.972140\pi$
$384$		0			0
$385$	−1.78020e9			−0.0810264
$386$		0			0
$387$	−3.58717e9	+	3.58717e9i	−0.159922	+	0.159922i
$388$		0			0
$389$	1.83324e10	−	1.83324e10i	0.800611	−	0.800611i	−0.182580	−	0.983191i	$-0.558445\pi$
$389$	1.83324e10	−	1.83324e10i	0.800611	−	0.800611i	0.983191	+	0.182580i	$0.0584450\pi$
$390$		0			0
$391$	1.27240e10			0.544398
$392$		0			0
$393$		−	1.22708e10i		−	0.514403i
$394$		0			0
$395$	1.09206e9	+	1.09206e9i	0.0448600	+	0.0448600i
$396$		0			0
$397$	1.45854e10	+	1.45854e10i	0.587161	+	0.587161i	0.936862	−	0.349700i	$-0.113717\pi$
$397$	1.45854e10	+	1.45854e10i	0.587161	+	0.587161i	−0.349700	+	0.936862i	$0.613717\pi$
$398$		0			0
$399$			5.06729e9i			0.199933i
$400$		0			0
$401$	6.59308e9			0.254983			0.127491	−	0.991840i	$-0.459307\pi$
$401$	6.59308e9			0.254983			0.127491	+	0.991840i	$0.459307\pi$
$402$		0			0
$403$	1.40393e9	−	1.40393e9i	0.0532263	−	0.0532263i
$404$		0			0
$405$	1.67061e9	−	1.67061e9i	0.0620947	−	0.0620947i
$406$		0			0
$407$	1.42142e9			0.0518016
$408$		0			0
$409$		−	1.51172e10i		−	0.540230i	−0.962828	−	0.270115i	$-0.912938\pi$
$409$		−	1.51172e10i		−	0.540230i	0.962828	−	0.270115i	$-0.0870617\pi$
$410$		0			0
$411$	1.77206e10	+	1.77206e10i	0.621028	+	0.621028i
$412$		0			0
$413$	2.55557e10	+	2.55557e10i	0.878390	+	0.878390i
$414$		0			0
$415$			4.85772e9i			0.163772i
$416$		0			0
$417$	−1.39262e10			−0.460563
$418$		0			0
$419$	8.17978e9	−	8.17978e9i	0.265391	−	0.265391i	−0.561849	−	0.827240i	$-0.689910\pi$
$419$	8.17978e9	−	8.17978e9i	0.265391	−	0.265391i	0.827240	+	0.561849i	$0.189910\pi$
$420$		0			0
$421$	2.56878e10	−	2.56878e10i	0.817709	−	0.817709i	−0.168066	−	0.985776i	$-0.553752\pi$
$421$	2.56878e10	−	2.56878e10i	0.817709	−	0.817709i	0.985776	+	0.168066i	$0.0537523\pi$
$422$		0			0
$423$	6.32088e9			0.197431
$424$		0			0
$425$			1.77967e10i			0.545486i
$426$		0			0
$427$	7.03172e10	+	7.03172e10i	2.11519	+	2.11519i
$428$		0			0
$429$	−1.49860e8	−	1.49860e8i	−0.00442442	−	0.00442442i
$430$		0			0
$431$		−	1.66830e10i		−	0.483466i	−0.970343	−	0.241733i	$-0.922284\pi$
$431$		−	1.66830e10i		−	0.483466i	0.970343	−	0.241733i	$-0.0777159\pi$
$432$		0			0
$433$	7.67816e9			0.218427			0.109213	−	0.994018i	$-0.465167\pi$
$433$	7.67816e9			0.218427			0.109213	+	0.994018i	$0.465167\pi$
$434$		0			0
$435$	1.80468e10	−	1.80468e10i	0.504015	−	0.504015i
$436$		0			0
$437$	−1.73088e9	+	1.73088e9i	−0.0474614	+	0.0474614i
$438$		0			0
$439$	−1.34964e10			−0.363379			−0.181690	−	0.983356i	$-0.558157\pi$
$439$	−1.34964e10			−0.363379			−0.181690	+	0.983356i	$0.558157\pi$
$440$		0			0
$441$		−	3.44895e10i		−	0.911869i
$442$		0			0
$443$	−1.43417e9	−	1.43417e9i	−0.0372380	−	0.0372380i	0.688243	−	0.725481i	$-0.258382\pi$
$443$	−1.43417e9	−	1.43417e9i	−0.0372380	−	0.0372380i	−0.725481	+	0.688243i	$0.758382\pi$
$444$		0			0
$445$	1.58259e10	+	1.58259e10i	0.403579	+	0.403579i
$446$		0			0
$447$			1.85514e10i			0.464673i
$448$		0			0
$449$	3.58489e9			0.0882045			0.0441023	−	0.999027i	$-0.485957\pi$
$449$	3.58489e9			0.0882045			0.0441023	+	0.999027i	$0.485957\pi$
$450$		0			0
$451$	−2.22000e9	+	2.22000e9i	−0.0536596	+	0.0536596i
$452$		0			0
$453$	−5.56809e9	+	5.56809e9i	−0.132225	+	0.132225i
$454$		0			0
$455$	1.33763e10			0.312098
$456$		0			0
$457$			9.01706e9i			0.206729i	0.994644	+	0.103364i	$0.0329607\pi$
$457$			9.01706e9i			0.206729i	−0.994644	+	0.103364i	$0.967039\pi$
$458$		0			0
$459$	−8.77767e9	−	8.77767e9i	−0.197756	−	0.197756i
$460$		0			0
$461$	−3.51252e9	−	3.51252e9i	−0.0777705	−	0.0777705i	0.667152	−	0.744922i	$-0.267513\pi$
$461$	−3.51252e9	−	3.51252e9i	−0.0777705	−	0.0777705i	−0.744922	+	0.667152i	$0.767513\pi$
$462$		0			0
$463$		−	5.97409e10i		−	1.30001i	−0.759929	−	0.650006i	$-0.774766\pi$
$463$		−	5.97409e10i		−	1.30001i	0.759929	−	0.650006i	$-0.225234\pi$
$464$		0			0
$465$	−7.85969e9			−0.168110
$466$		0			0
$467$	−2.93856e10	+	2.93856e10i	−0.617827	+	0.617827i	−0.944974	−	0.327147i	$-0.893913\pi$
$467$	−2.93856e10	+	2.93856e10i	−0.617827	+	0.617827i	0.327147	+	0.944974i	$0.393913\pi$
$468$		0			0
$469$	5.60004e10	−	5.60004e10i	1.15744	−	1.15744i
$470$		0			0
$471$	2.55330e9			0.0518823
$472$		0			0
$473$		−	1.80145e9i		−	0.0359897i
$474$		0			0
$475$	−2.42093e9	−	2.42093e9i	−0.0475562	−	0.0475562i
$476$		0			0
$477$	6.93720e9	+	6.93720e9i	0.134002	+	0.134002i
$478$		0			0
$479$			7.02776e9i			0.133498i	0.997770	+	0.0667490i	$0.0212627\pi$
$479$			7.02776e9i			0.133498i	−0.997770	+	0.0667490i	$0.978737\pi$
$480$		0			0
$481$	−1.06804e10			−0.199530
$482$		0			0
$483$	1.60874e10	−	1.60874e10i	0.295594	−	0.295594i
$484$		0			0
$485$	−5.34978e10	+	5.34978e10i	−0.966872	+	0.966872i
$486$		0			0
$487$	3.09001e10			0.549344			0.274672	−	0.961538i	$-0.411431\pi$
$487$	3.09001e10			0.549344			0.274672	+	0.961538i	$0.411431\pi$
$488$		0			0
$489$		−	3.69637e10i		−	0.646457i
$490$		0			0
$491$	−1.14690e9	−	1.14690e9i	−0.0197333	−	0.0197333i	0.697171	−	0.716905i	$-0.254442\pi$
$491$	−1.14690e9	−	1.14690e9i	−0.0197333	−	0.0197333i	−0.716905	+	0.697171i	$0.754442\pi$
$492$		0			0
$493$	−9.48212e10	−	9.48212e10i	−1.60516	−	1.60516i
$494$		0			0
$495$			8.38968e8i			0.0139741i
$496$		0			0
$497$	−1.57839e11			−2.58696
$498$		0			0
$499$	5.15755e9	−	5.15755e9i	0.0831843	−	0.0831843i	−0.664290	−	0.747475i	$-0.731266\pi$
$499$	5.15755e9	−	5.15755e9i	0.0831843	−	0.0831843i	0.747475	+	0.664290i	$0.231266\pi$
$500$		0			0
$501$	−7.44907e9	+	7.44907e9i	−0.118236	+	0.118236i
$502$		0			0
$503$	−5.78499e10			−0.903713			−0.451857	−	0.892091i	$-0.649238\pi$
$503$	−5.78499e10			−0.903713			−0.451857	+	0.892091i	$0.649238\pi$
$504$		0			0
$505$			9.04948e10i			1.39142i
$506$		0			0
$507$	−2.58486e10	−	2.58486e10i	−0.391206	−	0.391206i
$508$		0			0
$509$	−5.10199e10	−	5.10199e10i	−0.760097	−	0.760097i	0.216243	−	0.976340i	$-0.430620\pi$
$509$	−5.10199e10	−	5.10199e10i	−0.760097	−	0.760097i	−0.976340	+	0.216243i	$0.930620\pi$
$510$		0			0
$511$		−	1.75096e11i		−	2.56799i
$512$		0			0
$513$	2.38809e9			0.0344812
$514$		0			0
$515$	−6.73941e10	+	6.73941e10i	−0.958060	+	0.958060i
$516$		0			0
$517$	−1.58715e9	+	1.58715e9i	−0.0222155	+	0.0222155i
$518$		0			0
$519$	7.93659e9			0.109387
$520$		0			0
$521$			6.53646e10i			0.887139i	0.896240	+	0.443569i	$0.146288\pi$
$521$			6.53646e10i			0.887139i	−0.896240	+	0.443569i	$0.853712\pi$
$522$		0			0
$523$	8.70195e10	+	8.70195e10i	1.16308	+	1.16308i	0.983798	+	0.179283i	$0.0573777\pi$
$523$	8.70195e10	+	8.70195e10i	1.16308	+	1.16308i	0.179283	+	0.983798i	$0.442622\pi$
$524$		0			0
$525$	2.25009e10	+	2.25009e10i	0.296185	+	0.296185i
$526$		0			0
$527$			4.12962e10i			0.535387i
$528$		0			0
$529$	−6.73208e10			−0.859660
$530$		0			0
$531$	1.20438e10	−	1.20438e10i	0.151491	−	0.151491i
$532$		0			0
$533$	1.66809e10	−	1.66809e10i	0.206686	−	0.206686i
$534$		0			0
$535$	−9.92551e10			−1.21154
$536$		0			0
$537$			1.07960e10i			0.129827i
$538$		0			0
$539$	8.66020e9	+	8.66020e9i	0.102606	+	0.102606i
$540$		0			0
$541$	2.92281e10	+	2.92281e10i	0.341202	+	0.341202i	0.856819	−	0.515617i	$-0.172437\pi$
$541$	2.92281e10	+	2.92281e10i	0.341202	+	0.341202i	−0.515617	+	0.856819i	$0.672437\pi$
$542$		0			0
$543$		−	7.07190e10i		−	0.813462i
$544$		0			0
$545$	−5.99173e10			−0.679150
$546$		0			0
$547$	−1.10860e11	+	1.10860e11i	−1.23830	+	1.23830i	−0.277610	+	0.960694i	$0.589542\pi$
$547$	−1.10860e11	+	1.10860e11i	−1.23830	+	1.23830i	−0.960694	+	0.277610i	$0.910458\pi$
$548$		0			0
$549$	3.31389e10	−	3.31389e10i	0.364794	−	0.364794i
$550$		0			0
$551$	2.57975e10			0.279879
$552$		0			0
$553$		−	1.45092e10i		−	0.155147i
$554$		0			0
$555$	2.98963e10	+	2.98963e10i	0.315098	+	0.315098i
$556$		0			0
$557$	9.54682e10	+	9.54682e10i	0.991831	+	0.991831i	0.999967	−	0.00813548i	$-0.00258963\pi$
$557$	9.54682e10	+	9.54682e10i	0.991831	+	0.991831i	−0.00813548	+	0.999967i	$0.502590\pi$
$558$		0			0
$559$			1.35360e10i			0.138625i
$560$		0			0
$561$	4.40809e9			0.0445040
$562$		0			0
$563$	7.33771e10	−	7.33771e10i	0.730343	−	0.730343i	−0.240345	−	0.970688i	$-0.577260\pi$
$563$	7.33771e10	−	7.33771e10i	0.730343	−	0.730343i	0.970688	+	0.240345i	$0.0772604\pi$
$564$		0			0
$565$	−2.15152e10	+	2.15152e10i	−0.211131	+	0.211131i
$566$		0			0
$567$	−2.21958e10			−0.214752
$568$		0			0
$569$		−	1.46591e11i		−	1.39849i	−0.714883	−	0.699244i	$-0.753520\pi$
$569$		−	1.46591e11i		−	1.39849i	0.714883	−	0.699244i	$-0.246480\pi$
$570$		0			0
$571$	−4.69788e10	−	4.69788e10i	−0.441934	−	0.441934i	0.450727	−	0.892662i	$-0.351165\pi$
$571$	−4.69788e10	−	4.69788e10i	−0.441934	−	0.441934i	−0.892662	+	0.450727i	$0.851165\pi$
$572$		0			0
$573$	7.19070e10	+	7.19070e10i	0.667041	+	0.667041i
$574$		0			0
$575$			1.53717e10i			0.140621i
$576$		0			0
$577$	1.24179e11			1.12032			0.560162	−	0.828383i	$-0.310739\pi$
$577$	1.24179e11			1.12032			0.560162	+	0.828383i	$0.310739\pi$
$578$		0			0
$579$	5.96156e10	−	5.96156e10i	0.530452	−	0.530452i
$580$		0			0
$581$	3.22699e10	−	3.22699e10i	0.283200	−	0.283200i
$582$		0			0
$583$	−3.48382e9			−0.0301565
$584$		0			0
$585$		−	6.30394e9i		−	0.0538256i
$586$		0			0
$587$	1.02901e11	+	1.02901e11i	0.866696	+	0.866696i	0.992105	−	0.125409i	$-0.0400244\pi$
$587$	1.02901e11	+	1.02901e11i	0.866696	+	0.866696i	−0.125409	+	0.992105i	$0.540024\pi$
$588$		0			0
$589$	−5.61762e9	−	5.61762e9i	−0.0466757	−	0.0466757i
$590$		0			0
$591$			8.31222e10i			0.681345i
$592$		0			0
$593$	−1.17548e11			−0.950597			−0.475298	−	0.879825i	$-0.657660\pi$
$593$	−1.17548e11			−0.950597			−0.475298	+	0.879825i	$0.657660\pi$
$594$		0			0
$595$	−1.96730e11	+	1.96730e11i	−1.56965	+	1.56965i
$596$		0			0
$597$	−6.32400e10	+	6.32400e10i	−0.497845	+	0.497845i
$598$		0			0
$599$	−1.63720e11			−1.27173			−0.635863	−	0.771802i	$-0.719356\pi$
$599$	−1.63720e11			−1.27173			−0.635863	+	0.771802i	$0.719356\pi$
$600$		0			0
$601$		−	5.57589e10i		−	0.427382i	−0.976901	−	0.213691i	$-0.931451\pi$
$601$		−	5.57589e10i		−	0.427382i	0.976901	−	0.213691i	$-0.0685485\pi$
$602$		0			0
$603$	−2.63917e10	−	2.63917e10i	−0.199617	−	0.199617i
$604$		0			0
$605$	7.46611e10	+	7.46611e10i	0.557280	+	0.557280i
$606$		0			0
$607$		−	4.32872e10i		−	0.318863i	−0.987209	−	0.159432i	$-0.949034\pi$
$607$		−	4.32872e10i		−	0.318863i	0.987209	−	0.159432i	$-0.0509661\pi$
$608$		0			0
$609$	−2.39771e11			−1.74312
$610$		0			0
$611$	1.19257e10	−	1.19257e10i	0.0855698	−	0.0855698i
$612$		0			0
$613$	1.36205e10	−	1.36205e10i	0.0964610	−	0.0964610i	−0.657230	−	0.753691i	$-0.728272\pi$
$613$	1.36205e10	−	1.36205e10i	0.0964610	−	0.0964610i	0.753691	+	0.657230i	$0.228272\pi$
$614$		0			0
$615$	−9.33856e10			−0.652799
$616$		0			0
$617$		−	6.25456e9i		−	0.0431575i	−0.999767	−	0.0215787i	$-0.993131\pi$
$617$		−	6.25456e9i		−	0.0431575i	0.999767	−	0.0215787i	$-0.00686926\pi$
$618$		0			0
$619$	−1.57733e11	−	1.57733e11i	−1.07439	−	1.07439i	−0.997001	−	0.0773863i	$-0.975343\pi$
$619$	−1.57733e11	−	1.57733e11i	−1.07439	−	1.07439i	−0.0773863	−	0.997001i	$-0.524657\pi$
$620$		0			0
$621$	−7.58160e9	−	7.58160e9i	−0.0509794	−	0.0509794i
$622$		0			0
$623$		−	2.10264e11i		−	1.39577i
$624$		0			0
$625$	7.38112e10			0.483729
$626$		0			0
$627$	−5.99643e8	+	5.99643e8i	−0.00387992	+	0.00387992i
$628$		0			0
$629$	1.57081e11	−	1.57081e11i	1.00351	−	1.00351i
$630$		0			0
$631$	−2.08602e11			−1.31583			−0.657917	−	0.753091i	$-0.728562\pi$
$631$	−2.08602e11			−1.31583			−0.657917	+	0.753091i	$0.728562\pi$
$632$		0			0
$633$			1.60110e11i			0.997248i
$634$		0			0
$635$	−1.43147e11	−	1.43147e11i	−0.880414	−	0.880414i
$636$		0			0
$637$	−6.50721e10	−	6.50721e10i	−0.395218	−	0.395218i
$638$		0			0
$639$			7.43861e10i			0.446158i
$640$		0			0
$641$	1.21135e11			0.717525			0.358762	−	0.933429i	$-0.383199\pi$
$641$	1.21135e11			0.717525			0.358762	+	0.933429i	$0.383199\pi$
$642$		0			0
$643$	−1.88623e11	+	1.88623e11i	−1.10345	+	1.10345i	−0.109455	+	0.993992i	$0.534910\pi$
$643$	−1.88623e11	+	1.88623e11i	−1.10345	+	1.10345i	−0.993992	+	0.109455i	$0.965090\pi$
$644$		0			0
$645$	3.78895e10	−	3.78895e10i	0.218917	−	0.218917i
$646$		0			0
$647$	1.15579e11			0.659569			0.329784	−	0.944056i	$-0.393024\pi$
$647$	1.15579e11			0.659569			0.329784	+	0.944056i	$0.393024\pi$
$648$		0			0
$649$			6.04832e9i			0.0340923i
$650$		0			0
$651$	5.22121e10	+	5.22121e10i	0.290702	+	0.290702i
$652$		0			0
$653$	−7.72771e10	−	7.72771e10i	−0.425009	−	0.425009i	0.461915	−	0.886924i	$-0.347162\pi$
$653$	−7.72771e10	−	7.72771e10i	−0.425009	−	0.425009i	−0.886924	+	0.461915i	$0.847162\pi$
$654$		0			0
$655$			1.29611e11i			0.704168i
$656$		0			0
$657$	−8.25187e10			−0.442885
$658$		0			0
$659$	1.10931e11	−	1.10931e11i	0.588182	−	0.588182i	−0.348957	−	0.937139i	$-0.613464\pi$
$659$	1.10931e11	−	1.10931e11i	0.588182	−	0.588182i	0.937139	+	0.348957i	$0.113464\pi$
$660$		0			0
$661$	−1.22962e11	+	1.22962e11i	−0.644117	+	0.644117i	−0.951565	−	0.307448i	$-0.900525\pi$
$661$	−1.22962e11	+	1.22962e11i	−0.644117	+	0.644117i	0.307448	+	0.951565i	$0.400525\pi$
$662$		0			0
$663$	−3.31221e10			−0.171421
$664$		0			0
$665$		−	5.35233e10i		−	0.273688i
$666$		0			0
$667$	−8.19006e10	−	8.19006e10i	−0.413793	−	0.413793i
$668$		0			0
$669$	3.88998e9	+	3.88998e9i	0.0194197	+	0.0194197i
$670$		0			0
$671$			1.66421e10i			0.0820953i
$672$		0			0
$673$	4.31551e10			0.210364			0.105182	−	0.994453i	$-0.466457\pi$
$673$	4.31551e10			0.210364			0.105182	+	0.994453i	$0.466457\pi$
$674$		0			0
$675$	1.06042e10	−	1.06042e10i	0.0510812	−	0.0510812i
$676$		0			0
$677$	−3.94181e10	+	3.94181e10i	−0.187647	+	0.187647i	−0.794678	−	0.607031i	$-0.792360\pi$
$677$	−3.94181e10	+	3.94181e10i	−0.187647	+	0.187647i	0.607031	+	0.794678i	$0.292360\pi$
$678$		0			0
$679$	7.10774e11			3.34389
$680$		0			0
$681$		−	7.95352e9i		−	0.0369803i
$682$		0			0
$683$	1.51155e11	+	1.51155e11i	0.694607	+	0.694607i	0.963242	−	0.268635i	$-0.0865724\pi$
$683$	1.51155e11	+	1.51155e11i	0.694607	+	0.694607i	−0.268635	+	0.963242i	$0.586572\pi$
$684$		0			0
$685$	−1.87174e11	−	1.87174e11i	−0.850127	−	0.850127i
$686$		0			0
$687$		−	5.28739e10i		−	0.237364i
$688$		0			0
$689$	2.61771e10			0.116157
$690$		0			0
$691$	2.45807e9	−	2.45807e9i	0.0107816	−	0.0107816i	−0.701695	−	0.712477i	$-0.747573\pi$
$691$	2.45807e9	−	2.45807e9i	0.0107816	−	0.0107816i	0.712477	+	0.701695i	$0.247573\pi$
$692$		0			0
$693$	5.57329e9	−	5.57329e9i	0.0241645	−	0.0241645i
$694$		0			0
$695$	1.47096e11			0.630466
$696$		0			0
$697$			4.90665e11i			2.07900i
$698$		0			0
$699$	−2.61025e10	−	2.61025e10i	−0.109339	−	0.109339i
$700$		0			0
$701$	−6.09190e10	−	6.09190e10i	−0.252279	−	0.252279i	0.569626	−	0.821904i	$-0.307088\pi$
$701$	−6.09190e10	−	6.09190e10i	−0.252279	−	0.252279i	−0.821904	+	0.569626i	$0.807088\pi$
$702$		0			0
$703$			4.27361e10i			0.174974i
$704$		0			0
$705$	−6.67644e10			−0.270264
$706$		0			0
$707$	6.01159e11	−	6.01159e11i	2.40609	−	2.40609i
$708$		0			0
$709$	1.24268e10	−	1.24268e10i	0.0491785	−	0.0491785i	−0.682090	−	0.731268i	$-0.738929\pi$
$709$	1.24268e10	−	1.24268e10i	0.0491785	−	0.0491785i	0.731268	+	0.682090i	$0.238929\pi$
$710$		0			0
$711$	−6.83785e9			−0.0267572
$712$		0			0
$713$			3.56691e10i			0.138017i
$714$		0			0
$715$	1.58290e9	+	1.58290e9i	0.00605660	+	0.00605660i
$716$		0			0
$717$	−5.46419e10	−	5.46419e10i	−0.206752	−	0.206752i
$718$		0			0
$719$			3.85268e11i			1.44161i	0.693139	+	0.720804i	$0.256227\pi$
$719$			3.85268e11i			1.44161i	−0.693139	+	0.720804i	$0.743773\pi$
$720$		0			0
$721$	8.95401e11			3.31342
$722$		0			0
$723$	−8.96504e8	+	8.96504e8i	−0.00328094	+	0.00328094i
$724$		0			0
$725$	1.14552e11	−	1.14552e11i	0.414620	−	0.414620i
$726$		0			0
$727$	2.31720e11			0.829517			0.414758	−	0.909932i	$-0.363866\pi$
$727$	2.31720e11			0.829517			0.414758	+	0.909932i	$0.363866\pi$
$728$		0			0
$729$			1.04604e10i			0.0370370i
$730$		0			0
$731$	−1.99078e11	−	1.99078e11i	−0.697196	−	0.697196i
$732$		0			0
$733$	−3.46561e11	−	3.46561e11i	−1.20051	−	1.20051i	−0.974013	−	0.226492i	$-0.927274\pi$
$733$	−3.46561e11	−	3.46561e11i	−1.20051	−	1.20051i	−0.226492	−	0.974013i	$-0.572726\pi$
$734$		0			0
$735$			3.64296e11i			1.24826i
$736$		0			0
$737$	1.32537e10			0.0449229
$738$		0			0
$739$	−1.05521e11	+	1.05521e11i	−0.353803	+	0.353803i	−0.861523	−	0.507719i	$-0.830489\pi$
$739$	−1.05521e11	+	1.05521e11i	−0.353803	+	0.353803i	0.507719	+	0.861523i	$0.330489\pi$
$740$		0			0
$741$	4.50567e9	−	4.50567e9i	0.0149447	−	0.0149447i
$742$		0			0
$743$	−3.02342e11			−0.992072			−0.496036	−	0.868302i	$-0.665212\pi$
$743$	−3.02342e11			−0.992072			−0.496036	+	0.868302i	$0.665212\pi$
$744$		0			0
$745$		−	1.95950e11i		−	0.636092i
$746$		0			0
$747$	−1.52081e10	−	1.52081e10i	−0.0488418	−	0.0488418i
$748$		0			0
$749$	6.59354e11	+	6.59354e11i	2.09503	+	2.09503i
$750$		0			0
$751$		−	3.24940e11i		−	1.02151i	−0.859725	−	0.510757i	$-0.829365\pi$
$751$		−	3.24940e11i		−	1.02151i	0.859725	−	0.510757i	$-0.170635\pi$
$752$		0			0
$753$	1.14267e11			0.355420
$754$		0			0
$755$	5.88130e10	−	5.88130e10i	0.181003	−	0.181003i
$756$		0			0
$757$	−1.75367e11	+	1.75367e11i	−0.534027	+	0.534027i	−0.921768	−	0.387741i	$-0.873255\pi$
$757$	−1.75367e11	+	1.75367e11i	−0.534027	+	0.534027i	0.387741	+	0.921768i	$0.373255\pi$
$758$		0			0
$759$	3.80743e9			0.0114727
$760$		0			0
$761$		−	3.45616e11i		−	1.03052i	−0.857035	−	0.515259i	$-0.827696\pi$
$761$		−	3.45616e11i		−	1.03052i	0.857035	−	0.515259i	$-0.172304\pi$
$762$		0			0
$763$	3.98032e11	+	3.98032e11i	1.17441	+	1.17441i
$764$		0			0
$765$	9.27143e10	+	9.27143e10i	0.270708	+	0.270708i
$766$		0			0
$767$		−	4.54466e10i		−	0.131317i
$768$		0			0
$769$	−1.85498e11			−0.530438			−0.265219	−	0.964188i	$-0.585444\pi$
$769$	−1.85498e11			−0.530438			−0.265219	+	0.964188i	$0.585444\pi$
$770$		0			0
$771$	4.65156e10	−	4.65156e10i	0.131638	−	0.131638i
$772$		0			0
$773$	−4.08694e11	+	4.08694e11i	−1.14467	+	1.14467i	−0.157087	+	0.987585i	$0.550210\pi$
$773$	−4.08694e11	+	4.08694e11i	−1.14467	+	1.14467i	−0.987585	+	0.157087i	$0.949790\pi$
$774$		0			0
$775$	−4.98893e10			−0.138293
$776$		0			0
$777$		−	3.97204e11i		−	1.08976i
$778$		0			0
$779$	−6.67462e10	−	6.67462e10i	−0.181250	−	0.181250i
$780$		0			0
$781$	−1.86781e10	−	1.86781e10i	−0.0502029	−	0.0502029i
$782$		0			0
$783$			1.12998e11i			0.300625i
$784$		0			0
$785$	−2.69693e10			−0.0710218
$786$		0			0
$787$	2.43389e11	−	2.43389e11i	0.634456	−	0.634456i	−0.314726	−	0.949183i	$-0.601913\pi$
$787$	2.43389e11	−	2.43389e11i	0.634456	−	0.634456i	0.949183	+	0.314726i	$0.101913\pi$
$788$		0			0
$789$	−1.70613e11	+	1.70613e11i	−0.440255	+	0.440255i
$790$		0			0
$791$	2.85852e11			0.730189
$792$		0			0
$793$		−	1.25048e11i		−	0.316215i
$794$		0			0
$795$	−7.32743e10	−	7.32743e10i	−0.183435	−	0.183435i
$796$		0			0
$797$	−4.03719e11	−	4.03719e11i	−1.00057	−	1.00057i	−1.00000		0.000566047i	$-0.999820\pi$
$797$	−4.03719e11	−	4.03719e11i	−1.00057	−	1.00057i	−0.000566047	−	1.00000i	$-0.500180\pi$
$798$		0			0
$799$			3.50792e11i			0.860722i
$800$		0			0
$801$	−9.90925e10			−0.240719
$802$		0			0
$803$	2.07202e10	−	2.07202e10i	0.0498346	−	0.0498346i
$804$		0			0
$805$	−1.69923e11	+	1.69923e11i	−0.404640	+	0.404640i
$806$		0			0
$807$	−2.92420e11			−0.689465
$808$		0			0
$809$		−	6.29891e11i		−	1.47052i	−0.677784	−	0.735261i	$-0.737060\pi$
$809$		−	6.29891e11i		−	1.47052i	0.677784	−	0.735261i	$-0.262940\pi$
$810$		0			0
$811$	3.55543e11	+	3.55543e11i	0.821880	+	0.821880i	0.986378	−	0.164497i	$-0.0526002\pi$
$811$	3.55543e11	+	3.55543e11i	0.821880	+	0.821880i	−0.164497	+	0.986378i	$0.552600\pi$
$812$		0			0
$813$	1.27395e11	+	1.27395e11i	0.291602	+	0.291602i
$814$		0			0
$815$			3.90429e11i			0.884937i
$816$		0			0
$817$	5.41622e10			0.121565
$818$		0			0
$819$	−4.18773e10	+	4.18773e10i	−0.0930771	+	0.0930771i
$820$		0			0
$821$	4.52982e11	−	4.52982e11i	0.997031	−	0.997031i	−0.00296475	−	0.999996i	$-0.500944\pi$
$821$	4.52982e11	−	4.52982e11i	0.997031	−	0.997031i	0.999996	+	0.00296475i	$0.000943711\pi$
$822$		0			0
$823$	6.69757e10			0.145988			0.0729941	−	0.997332i	$-0.476745\pi$
$823$	6.69757e10			0.145988			0.0729941	+	0.997332i	$0.476745\pi$
$824$		0			0
$825$			5.32534e9i			0.0114956i
$826$		0			0
$827$	1.77018e11	+	1.77018e11i	0.378440	+	0.378440i	0.870539	−	0.492099i	$-0.163770\pi$
$827$	1.77018e11	+	1.77018e11i	0.378440	+	0.378440i	−0.492099	+	0.870539i	$0.663770\pi$
$828$		0			0
$829$	1.85563e10	+	1.85563e10i	0.0392892	+	0.0392892i	0.726478	−	0.687189i	$-0.241156\pi$
$829$	1.85563e10	+	1.85563e10i	0.0392892	+	0.0392892i	−0.687189	+	0.726478i	$0.741156\pi$
$830$		0			0
$831$			4.44648e11i			0.932421i
$832$		0			0
$833$	1.91408e12			3.97538
$834$		0			0
$835$	7.86809e10	−	7.86809e10i	0.161854	−	0.161854i
$836$		0			0
$837$	2.46064e10	−	2.46064e10i	0.0501355	−	0.0501355i
$838$		0			0
$839$	2.97050e11			0.599490			0.299745	−	0.954019i	$-0.403098\pi$
$839$	2.97050e11			0.599490			0.299745	+	0.954019i	$0.403098\pi$
$840$		0			0
$841$			7.20424e11i			1.44014i
$842$		0			0
$843$	−2.20625e11	−	2.20625e11i	−0.436862	−	0.436862i
$844$		0			0
$845$	2.73027e11	+	2.73027e11i	0.535523	+	0.535523i
$846$		0			0
$847$		−	9.91952e11i		−	1.92733i
$848$		0			0
$849$	−6.65541e10			−0.128098
$850$		0			0
$851$	1.35676e11	−	1.35676e11i	0.258694	−	0.258694i
$852$		0			0
$853$	−2.40559e11	+	2.40559e11i	−0.454386	+	0.454386i	−0.896807	−	0.442421i	$-0.854120\pi$
$853$	−2.40559e11	+	2.40559e11i	−0.454386	+	0.454386i	0.442421	+	0.896807i	$0.354120\pi$
$854$		0			0
$855$	−2.52243e10			−0.0472013
$856$		0			0
$857$		−	5.39842e11i		−	1.00079i	−0.865797	−	0.500395i	$-0.833188\pi$
$857$		−	5.39842e11i		−	1.00079i	0.865797	−	0.500395i	$-0.166812\pi$
$858$		0			0
$859$	−3.59477e11	−	3.59477e11i	−0.660234	−	0.660234i	0.295201	−	0.955435i	$-0.404613\pi$
$859$	−3.59477e11	−	3.59477e11i	−0.660234	−	0.660234i	−0.955435	+	0.295201i	$0.904613\pi$
$860$		0			0
$861$	6.20363e11	+	6.20363e11i	1.12884	+	1.12884i
$862$		0			0
$863$			3.07520e11i			0.554409i	0.960811	+	0.277204i	$0.0894079\pi$
$863$			3.07520e11i			0.554409i	−0.960811	+	0.277204i	$0.910592\pi$
$864$		0			0
$865$	−8.38304e10			−0.149740
$866$		0			0
$867$	2.56463e11	−	2.56463e11i	0.453887	−	0.453887i
$868$		0			0
$869$	1.71696e9	−	1.71696e9i	0.00301080	−	0.00301080i
$870$		0			0
$871$	−9.95875e10			−0.173034
$872$		0			0
$873$		−	3.34971e11i		−	0.576701i
$874$		0			0
$875$	−8.70821e11	−	8.70821e11i	−1.48558	−	1.48558i
$876$		0			0
$877$	−7.10933e11	−	7.10933e11i	−1.20179	−	1.20179i	−0.973620	−	0.228173i	$-0.926725\pi$
$877$	−7.10933e11	−	7.10933e11i	−1.20179	−	1.20179i	−0.228173	−	0.973620i	$-0.573275\pi$
$878$		0			0
$879$			5.41931e11i			0.907797i
$880$		0			0
$881$	−2.06616e11			−0.342973			−0.171486	−	0.985187i	$-0.554857\pi$
$881$	−2.06616e11			−0.342973			−0.171486	+	0.985187i	$0.554857\pi$
$882$		0			0
$883$	−7.60639e11	+	7.60639e11i	−1.25123	+	1.25123i	−0.296055	+	0.955171i	$0.595671\pi$
$883$	−7.60639e11	+	7.60639e11i	−1.25123	+	1.25123i	−0.955171	+	0.296055i	$0.904329\pi$
$884$		0			0
$885$	−1.27213e11	+	1.27213e11i	−0.207376	+	0.207376i
$886$		0			0
$887$	5.30110e11			0.856390			0.428195	−	0.903686i	$-0.359150\pi$
$887$	5.30110e11			0.856390			0.428195	+	0.903686i	$0.359150\pi$
$888$		0			0
$889$			1.90186e12i			3.04488i
$890$		0			0
$891$	−2.62656e9	−	2.62656e9i	−0.00416751	−	0.00416751i
$892$		0			0
$893$	−4.77191e10	−	4.77191e10i	−0.0750388	−	0.0750388i
$894$		0			0
$895$		−	1.14032e11i		−	0.177720i
$896$		0			0
$897$	−2.86087e10			−0.0441905
$898$		0			0
$899$	2.65811e11	−	2.65811e11i	0.406944	−	0.406944i
$900$		0			0
$901$	−3.84996e11	+	3.84996e11i	−0.584195	+	0.584195i
$902$		0			0
$903$	−5.03402e11			−0.757119
$904$		0			0
$905$			7.46971e11i			1.11355i
$906$		0			0
$907$	−3.02031e11	−	3.02031e11i	−0.446296	−	0.446296i	0.447825	−	0.894121i	$-0.352199\pi$
$907$	−3.02031e11	−	3.02031e11i	−0.446296	−	0.446296i	−0.894121	+	0.447825i	$0.852199\pi$
$908$		0			0
$909$	−2.83312e11	−	2.83312e11i	−0.414964	−	0.414964i
$910$		0			0
$911$		−	1.29875e12i		−	1.88561i	−0.333351	−	0.942803i	$-0.608179\pi$
$911$		−	1.29875e12i		−	1.88561i	0.333351	−	0.942803i	$-0.391821\pi$
$912$		0			0
$913$	7.63739e9			0.0109916
$914$		0			0
$915$	−3.50030e11	+	3.50030e11i	−0.499368	+	0.499368i
$916$		0			0
$917$	8.61008e11	−	8.61008e11i	1.21767	−	1.21767i
$918$		0			0
$919$	−8.43775e11			−1.18295			−0.591473	−	0.806325i	$-0.701453\pi$
$919$	−8.43775e11			−1.18295			−0.591473	+	0.806325i	$0.701453\pi$
$920$		0			0
$921$			2.85755e11i			0.397150i
$922$		0			0
$923$	1.40346e11	+	1.40346e11i	0.193372	+	0.193372i
$924$		0			0
$925$	1.89767e11	+	1.89767e11i	0.259211	+	0.259211i
$926$		0			0
$927$		−	4.21982e11i		−	0.571445i
$928$		0			0
$929$	−1.17440e12			−1.57671			−0.788357	−	0.615219i	$-0.789068\pi$
$929$	−1.17440e12			−1.57671			−0.788357	+	0.615219i	$0.789068\pi$
$930$		0			0
$931$	−2.60376e11	+	2.60376e11i	−0.346579	+	0.346579i
$932$		0			0
$933$	−3.74705e11	+	3.74705e11i	−0.494496	+	0.494496i
$934$		0			0
$935$	−4.65605e10			−0.0609216
$936$		0			0
$937$		−	1.36352e12i		−	1.76890i	−0.466639	−	0.884448i	$-0.654535\pi$
$937$		−	1.36352e12i		−	1.76890i	0.466639	−	0.884448i	$-0.345465\pi$
$938$		0			0
$939$	−2.55499e11	−	2.55499e11i	−0.328644	−	0.328644i
$940$		0			0
$941$	−9.62141e11	−	9.62141e11i	−1.22710	−	1.22710i	−0.965056	−	0.262045i	$-0.915603\pi$
$941$	−9.62141e11	−	9.62141e11i	−1.22710	−	1.22710i	−0.262045	−	0.965056i	$-0.584397\pi$
$942$		0			0
$943$			4.23805e11i			0.535944i
$944$		0			0
$945$	2.34443e11			0.293975
$946$		0			0
$947$	−5.13392e11	+	5.13392e11i	−0.638336	+	0.638336i	−0.950145	−	0.311809i	$-0.899065\pi$
$947$	−5.13392e11	+	5.13392e11i	−0.638336	+	0.638336i	0.311809	+	0.950145i	$0.399065\pi$
$948$		0			0
$949$	−1.55690e11	+	1.55690e11i	−0.191953	+	0.191953i
$950$		0			0
$951$	−6.85554e10			−0.0838145
$952$		0			0
$953$		−	2.21524e11i		−	0.268565i	−0.990943	−	0.134283i	$-0.957127\pi$
$953$		−	2.21524e11i		−	0.268565i	0.990943	−	0.134283i	$-0.0428730\pi$
$954$		0			0
$955$	−7.59519e11	−	7.59519e11i	−0.913114	−	0.913114i
$956$		0			0
$957$	−2.83735e10	−	2.83735e10i	−0.0338272	−	0.0338272i
$958$		0			0
$959$			2.48681e12i			2.94014i
$960$		0			0
$961$	7.37126e11			0.864267
$962$		0			0
$963$	3.10738e11	−	3.10738e11i	0.361318	−	0.361318i
$964$		0			0
$965$	−6.29691e11	+	6.29691e11i	−0.726137	+	0.726137i
$966$		0			0
$967$	−1.19611e11			−0.136793			−0.0683967	−	0.997658i	$-0.521788\pi$
$967$	−1.19611e11			−0.136793			−0.0683967	+	0.997658i	$0.521788\pi$
$968$		0			0
$969$			1.32533e11i			0.150324i
$970$		0			0
$971$	−3.26614e11	−	3.26614e11i	−0.367416	−	0.367416i	0.499118	−	0.866534i	$-0.333657\pi$
$971$	−3.26614e11	−	3.26614e11i	−0.367416	−	0.367416i	−0.866534	+	0.499118i	$0.833657\pi$
$972$		0			0
$973$	−9.77162e11	−	9.77162e11i	−1.09022	−	1.09022i
$974$		0			0
$975$		−	4.00142e10i		−	0.0442788i
$976$		0			0
$977$	−1.04861e12			−1.15090			−0.575448	−	0.817838i	$-0.695172\pi$
$977$	−1.04861e12			−1.15090			−0.575448	+	0.817838i	$0.695172\pi$
$978$		0			0
$979$	2.48818e10	−	2.48818e10i	0.0270864	−	0.0270864i
$980$		0			0
$981$	1.87583e11	−	1.87583e11i	0.202543	−	0.202543i
$982$		0			0
$983$	−1.01847e12			−1.09078			−0.545389	−	0.838183i	$-0.683618\pi$
$983$	−1.01847e12			−1.09078			−0.545389	+	0.838183i	$0.683618\pi$
$984$		0			0
$985$		−	8.77979e11i		−	0.932694i
$986$		0			0
$987$	4.43518e11	+	4.43518e11i	0.467350	+	0.467350i
$988$		0			0
$989$	−1.71951e11	−	1.71951e11i	−0.179730	−	0.179730i
$990$		0			0
$991$		−	6.45658e11i		−	0.669435i	−0.942319	−	0.334717i	$-0.891359\pi$
$991$		−	6.45658e11i		−	0.669435i	0.942319	−	0.334717i	$-0.108641\pi$
$992$		0			0
$993$	−1.05057e12			−1.08050
$994$		0			0
$995$	6.67973e11	−	6.67973e11i	0.681501	−	0.681501i
$996$		0			0
$997$	5.11002e10	−	5.11002e10i	0.0517180	−	0.0517180i	−0.680775	−	0.732493i	$-0.738357\pi$
$997$	5.11002e10	−	5.11002e10i	0.0517180	−	0.0517180i	0.732493	+	0.680775i	$0.238357\pi$
$998$		0			0
$999$	−1.87193e11			−0.187944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.9.l.a.175.21		64
4.3	odd	2		48.9.l.a.19.14	✓	64
16.5	even	4		48.9.l.a.43.14	yes	64
16.11	odd	4	inner	192.9.l.a.79.21		64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.9.l.a.19.14	✓	64	4.3	odd	2
48.9.l.a.43.14	yes	64	16.5	even	4
192.9.l.a.79.21		64	16.11	odd	4	inner
192.9.l.a.175.21		64	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	192.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(33.0681 - 33.0681i) q^{3} +(-349.282 + 349.282i) q^{5} +4640.58 q^{7} -2187.00i q^{9} +(549.148 + 549.148i) q^{11} +(-4126.26 - 4126.26i) q^{13} +23100.2i q^{15} +121373. q^{17} +(-16510.6 + 16510.6i) q^{19} +(153455. - 153455. i) q^{21} +104834. q^{23} +146628. i q^{25} +(-72320.0 - 72320.0i) q^{27} +(-781240. - 781240. i) q^{29} +340243. i q^{31} +36318.6 q^{33} +(-1.62088e6 + 1.62088e6i) q^{35} +(1.29420e6 - 1.29420e6i) q^{37} -272895. q^{39} +4.04263e6i q^{41} +(-1.64022e6 - 1.64022e6i) q^{43} +(763881. + 763881. i) q^{45} +2.89021e6i q^{47} +1.57702e7 q^{49} +(4.01357e6 - 4.01357e6i) q^{51} +(-3.17202e6 + 3.17202e6i) q^{53} -383616. q^{55} +1.09195e6i q^{57} +(5.50700e6 + 5.50700e6i) q^{59} +(1.51527e7 + 1.51527e7i) q^{61} -1.01490e7i q^{63} +2.88246e6 q^{65} +(1.20675e7 - 1.20675e7i) q^{67} +(3.46667e6 - 3.46667e6i) q^{69} -3.40128e7 q^{71} -3.77315e7i q^{73} +(4.84873e6 + 4.84873e6i) q^{75} +(2.54837e6 + 2.54837e6i) q^{77} -3.12659e6i q^{79} -4.78297e6 q^{81} +(6.95385e6 - 6.95385e6i) q^{83} +(-4.23934e7 + 4.23934e7i) q^{85} -5.16682e7 q^{87} -4.53098e7i q^{89} +(-1.91483e7 - 1.91483e7i) q^{91} +(1.12512e7 + 1.12512e7i) q^{93} -1.15337e7i q^{95} +1.53165e8 q^{97} +(1.20099e6 - 1.20099e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 39552 q^{11} + 167552 q^{19} - 1691136 q^{23} - 2132352 q^{29} + 2415744 q^{35} - 4720512 q^{37} + 7244672 q^{43} + 52706752 q^{49} - 13862016 q^{51} - 5358720 q^{53} + 46326784 q^{55} - 44938752 q^{59}+ \cdots - 86500224 q^{99}+O(q^{100}) \) 64 * q - 39552 * q^11 + 167552 * q^19 - 1691136 * q^23 - 2132352 * q^29 + 2415744 * q^35 - 4720512 * q^37 + 7244672 * q^43 + 52706752 * q^49 - 13862016 * q^51 - 5358720 * q^53 + 46326784 * q^55 - 44938752 * q^59 + 24476032 * q^61 + 29941632 * q^65 + 44244736 * q^67 - 8636544 * q^69 - 159664128 * q^71 - 12918528 * q^75 - 94964352 * q^77 - 306110016 * q^81 - 209328000 * q^83 + 106960000 * q^85 + 45401472 * q^91 - 86500224 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more