LMFDB - Embedded newform 112.10.i.b.81.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(65,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.65"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,233] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$57.6840136504$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
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} - x^{5} + 1116x^{4} - 3085x^{3} + 1245325x^{2} - 2341500x + 4410000 $ x^6 - x^5 + 1116x^4 - 3085x^3 + 1245325x^2 - 2341500x + 4410000
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.3
Root			$16.4632 + 28.5151i$ of defining polynomial
Character	$\chi$	$=$	112.81
Dual form			112.10.i.b.65.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(135.410 - 234.536i) q^{3} +(-684.785 - 1186.08i) q^{5} +(-6327.81 + 558.972i) q^{7} +(-26830.1 - 46471.0i) q^{9} +(-5462.17 + 9460.75i) q^{11} -7968.27 q^{13} -370906. q^{15} +(162006. - 280602. i) q^{17} +(-115939. - 200812. i) q^{19} +(-725747. + 1.55979e6i) q^{21} +(1.05331e6 + 1.82439e6i) q^{23} +(38701.4 - 67032.7i) q^{25} -9.20167e6 q^{27} -5.46332e6 q^{29} +(-989336. + 1.71358e6i) q^{31} +(1.47926e6 + 2.56215e6i) q^{33} +(4.99618e6 + 7.12253e6i) q^{35} +(-2.38266e6 - 4.12689e6i) q^{37} +(-1.07898e6 + 1.86885e6i) q^{39} +5.57398e6 q^{41} +3.11341e7 q^{43} +(-3.67457e7 + 6.36454e7i) q^{45} +(2.15595e6 + 3.73422e6i) q^{47} +(3.97287e7 - 7.07414e6i) q^{49} +(-4.38743e7 - 7.59926e7i) q^{51} +(4.51452e7 - 7.81937e7i) q^{53} +1.49616e7 q^{55} -6.27970e7 q^{57} +(-3.64555e7 + 6.31429e7i) q^{59} +(3.71748e6 + 6.43887e6i) q^{61} +(1.95752e8 + 2.79063e8i) q^{63} +(5.45655e6 + 9.45102e6i) q^{65} +(5.28936e7 - 9.16144e7i) q^{67} +5.70514e8 q^{69} -1.10248e8 q^{71} +(-1.02225e8 + 1.77058e8i) q^{73} +(-1.04811e7 - 1.81538e7i) q^{75} +(2.92752e7 - 6.29190e7i) q^{77} +(4.43923e7 + 7.68897e7i) q^{79} +(-7.17899e8 + 1.24344e9i) q^{81} -2.41708e8 q^{83} -4.43757e8 q^{85} +(-7.39786e8 + 1.28135e9i) q^{87} +(-2.74345e8 - 4.75180e8i) q^{89} +(5.04217e7 - 4.45404e6i) q^{91} +(2.67931e8 + 4.64071e8i) q^{93} +(-1.58787e8 + 2.75026e8i) q^{95} -1.16746e9 q^{97} +5.86201e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 233 q^{3} - 733 q^{5} - 5012 q^{7} - 15058 q^{9} - 7339 q^{11} + 197036 q^{13} - 738238 q^{15} - 306665 q^{17} + 377991 q^{19} - 1585927 q^{21} + 2267255 q^{23} - 142612 q^{25} - 21348358 q^{27} - 13085956 q^{29}+ \cdots + 1256218868 q^{99}+O(q^{100}) $ 6 * q + 233 * q^3 - 733 * q^5 - 5012 * q^7 - 15058 * q^9 - 7339 * q^11 + 197036 * q^13 - 738238 * q^15 - 306665 * q^17 + 377991 * q^19 - 1585927 * q^21 + 2267255 * q^23 - 142612 * q^25 - 21348358 * q^27 - 13085956 * q^29 + 6654517 * q^31 + 3747977 * q^33 - 22864807 * q^35 - 22287969 * q^37 - 3151974 * q^39 + 68096196 * q^41 + 125648280 * q^43 - 86300638 * q^45 + 52703019 * q^47 + 57596070 * q^49 - 97736333 * q^51 - 12091125 * q^53 - 377435398 * q^55 - 158517670 * q^57 + 12949897 * q^59 - 160252153 * q^61 + 356021666 * q^63 + 334191270 * q^65 + 480890225 * q^67 + 1131233162 * q^69 + 74421440 * q^71 + 251382283 * q^73 - 16322324 * q^75 - 527045155 * q^77 - 286494785 * q^79 - 2165894587 * q^81 - 2295182344 * q^83 - 2389185074 * q^85 - 1412199358 * q^87 - 901243845 * q^89 - 1026798920 * q^91 + 710456889 * q^93 + 887366177 * q^95 + 629707876 * q^97 + 1256218868 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	135.410	−	234.536i	0.965171	−	1.67172i	0.256015	−	0.966673i	$-0.417590\pi$
$3$	135.410	−	234.536i	0.965171	−	1.67172i	0.709156	−	0.705052i	$-0.249076\pi$
$4$		0			0
$5$	−684.785	−	1186.08i	−0.489992	−	0.848692i	0.509941	−	0.860209i	$-0.329667\pi$
$5$	−684.785	−	1186.08i	−0.489992	−	0.848692i	−0.999934	+	0.0115176i	$0.996334\pi$
$6$		0			0
$7$	−6327.81	+	558.972i	−0.996121	+	0.0879932i
$7$	−6327.81	+	558.972i	−0.996121	+	0.0879932i
$8$		0			0
$9$	−26830.1	−	46471.0i	−1.36311	−	2.36097i
$10$		0			0
$11$	−5462.17	+	9460.75i	−0.112486	+	0.194831i	−0.916772	−	0.399411i	$-0.869215\pi$
$11$	−5462.17	+	9460.75i	−0.112486	+	0.194831i	0.804286	+	0.594242i	$0.202548\pi$
$12$		0			0
$13$	−7968.27			−0.0773783			−0.0386891	−	0.999251i	$-0.512318\pi$
$13$	−7968.27			−0.0773783			−0.0386891	+	0.999251i	$0.512318\pi$
$14$		0			0
$15$	−370906.			−1.89170
$16$		0			0
$17$	162006.	−	280602.i	0.470447	−	0.814838i	−0.528982	−	0.848633i	$-0.677426\pi$
$17$	162006.	−	280602.i	0.470447	−	0.814838i	0.999429	+	0.0337953i	$0.0107594\pi$
$18$		0			0
$19$	−115939.	−	200812.i	−0.204098	−	0.353508i	0.745747	−	0.666229i	$-0.232093\pi$
$19$	−115939.	−	200812.i	−0.204098	−	0.353508i	−0.949845	+	0.312721i	$0.898759\pi$
$20$		0			0
$21$	−725747.	+	1.55979e6i	−0.814326	+	1.75017i
$22$		0			0
$23$	1.05331e6	+	1.82439e6i	0.784841	+	1.35938i	0.929094	+	0.369843i	$0.120589\pi$
$23$	1.05331e6	+	1.82439e6i	0.784841	+	1.35938i	−0.144253	+	0.989541i	$0.546078\pi$
$24$		0			0
$25$	38701.4	−	67032.7i	0.0198151	−	0.0343208i
$26$		0			0
$27$	−9.20167e6			−3.33219
$28$		0			0
$29$	−5.46332e6			−1.43438			−0.717192	−	0.696876i	$-0.754573\pi$
$29$	−5.46332e6			−1.43438			−0.717192	+	0.696876i	$0.754573\pi$
$30$		0			0
$31$	−989336.	+	1.71358e6i	−0.192405	+	0.333255i	−0.946047	−	0.324030i	$-0.894962\pi$
$31$	−989336.	+	1.71358e6i	−0.192405	+	0.333255i	0.753642	+	0.657285i	$0.228295\pi$
$32$		0			0
$33$	1.47926e6	+	2.56215e6i	0.217136	+	0.376091i
$34$		0			0
$35$	4.99618e6	+	7.12253e6i	0.562771	+	0.802284i
$36$		0			0
$37$	−2.38266e6	−	4.12689e6i	−0.209004	−	0.362006i	0.742397	−	0.669960i	$-0.233689\pi$
$37$	−2.38266e6	−	4.12689e6i	−0.209004	−	0.362006i	−0.951401	+	0.307955i	$0.900356\pi$
$38$		0			0
$39$	−1.07898e6	+	1.86885e6i	−0.0746832	+	0.129355i
$40$		0			0
$41$	5.57398e6			0.308062			0.154031	−	0.988066i	$-0.450774\pi$
$41$	5.57398e6			0.308062			0.154031	+	0.988066i	$0.450774\pi$
$42$		0			0
$43$	3.11341e7			1.38876			0.694382	−	0.719607i	$-0.255678\pi$
$43$	3.11341e7			1.38876			0.694382	+	0.719607i	$0.255678\pi$
$44$		0			0
$45$	−3.67457e7	+	6.36454e7i	−1.33583	+	2.31372i
$46$		0			0
$47$	2.15595e6	+	3.73422e6i	0.0644464	+	0.111624i	0.896448	−	0.443148i	$-0.146139\pi$
$47$	2.15595e6	+	3.73422e6i	0.0644464	+	0.111624i	−0.832002	+	0.554773i	$0.812805\pi$
$48$		0			0
$49$	3.97287e7	−	7.07414e6i	0.984514	−	0.175304i
$50$		0			0
$51$	−4.38743e7	−	7.59926e7i	−0.908123	−	1.57291i
$52$		0			0
$53$	4.51452e7	−	7.81937e7i	0.785905	−	1.36123i	−0.142553	−	0.989787i	$-0.545531\pi$
$53$	4.51452e7	−	7.81937e7i	0.785905	−	1.36123i	0.928457	−	0.371439i	$-0.121136\pi$
$54$		0			0
$55$	1.49616e7			0.220469
$56$		0			0
$57$	−6.27970e7			−0.787957
$58$		0			0
$59$	−3.64555e7	+	6.31429e7i	−0.391678	+	0.678407i	−0.992671	−	0.120848i	$-0.961439\pi$
$59$	−3.64555e7	+	6.31429e7i	−0.391678	+	0.678407i	0.600993	+	0.799255i	$0.294772\pi$
$60$		0			0
$61$	3.71748e6	+	6.43887e6i	0.0343768	+	0.0595423i	0.882702	−	0.469933i	$-0.155722\pi$
$61$	3.71748e6	+	6.43887e6i	0.0343768	+	0.0595423i	−0.848325	+	0.529476i	$0.822389\pi$
$62$		0			0
$63$	1.95752e8	+	2.79063e8i	1.56557	+	2.23187i
$64$		0			0
$65$	5.45655e6	+	9.45102e6i	0.0379147	+	0.0656703i
$66$		0			0
$67$	5.28936e7	−	9.16144e7i	0.320676	−	0.555427i	−0.659952	−	0.751308i	$-0.729423\pi$
$67$	5.28936e7	−	9.16144e7i	0.320676	−	0.555427i	0.980628	+	0.195881i	$0.0627566\pi$
$68$		0			0
$69$	5.70514e8			3.03002
$70$		0			0
$71$	−1.10248e8			−0.514884			−0.257442	−	0.966294i	$-0.582880\pi$
$71$	−1.10248e8			−0.514884			−0.257442	+	0.966294i	$0.582880\pi$
$72$		0			0
$73$	−1.02225e8	+	1.77058e8i	−0.421311	+	0.729731i	−0.996068	−	0.0885924i	$-0.971763\pi$
$73$	−1.02225e8	+	1.77058e8i	−0.421311	+	0.729731i	0.574757	+	0.818324i	$0.305096\pi$
$74$		0			0
$75$	−1.04811e7	−	1.81538e7i	−0.0382499	−	0.0662508i
$76$		0			0
$77$	2.92752e7	−	6.29190e7i	0.0949057	−	0.203973i
$78$		0			0
$79$	4.43923e7	+	7.68897e7i	0.128229	+	0.222099i	0.922990	−	0.384823i	$-0.125737\pi$
$79$	4.43923e7	+	7.68897e7i	0.128229	+	0.222099i	−0.794762	+	0.606922i	$0.792404\pi$
$80$		0			0
$81$	−7.17899e8	+	1.24344e9i	−1.85302	+	3.20953i
$82$		0			0
$83$	−2.41708e8			−0.559035			−0.279517	−	0.960141i	$-0.590174\pi$
$83$	−2.41708e8			−0.559035			−0.279517	+	0.960141i	$0.590174\pi$
$84$		0			0
$85$	−4.43757e8			−0.922061
$86$		0			0
$87$	−7.39786e8	+	1.28135e9i	−1.38443	+	2.39790i
$88$		0			0
$89$	−2.74345e8	−	4.75180e8i	−0.463492	−	0.802792i	0.535640	−	0.844446i	$-0.320070\pi$
$89$	−2.74345e8	−	4.75180e8i	−0.463492	−	0.802792i	−0.999132	+	0.0416548i	$0.986737\pi$
$90$		0			0
$91$	5.04217e7	−	4.45404e6i	0.0770781	−	0.00680876i
$92$		0			0
$93$	2.67931e8	+	4.64071e8i	0.371407	+	0.643296i
$94$		0			0
$95$	−1.58787e8	+	2.75026e8i	−0.200013	+	0.346432i
$96$		0			0
$97$	−1.16746e9			−1.33897			−0.669483	−	0.742828i	$-0.733484\pi$
$97$	−1.16746e9			−1.33897			−0.669483	+	0.742828i	$0.733484\pi$
$98$		0			0
$99$	5.86201e8			0.613322
$100$		0			0
$101$	−6.94928e8	+	1.20365e9i	−0.664498	+	1.15094i	0.314924	+	0.949117i	$0.398021\pi$
$101$	−6.94928e8	+	1.20365e9i	−0.664498	+	1.15094i	−0.979421	+	0.201827i	$0.935312\pi$
$102$		0			0
$103$	4.69404e8	+	8.13031e8i	0.410941	+	0.711770i	0.994993	−	0.0999461i	$-0.0318670\pi$
$103$	4.69404e8	+	8.13031e8i	0.410941	+	0.711770i	−0.584052	+	0.811716i	$0.698534\pi$
$104$		0			0
$105$	2.34702e9	−	2.07326e8i	1.88437	−	0.166457i
$106$		0			0
$107$	1.29644e8	+	2.24550e8i	0.0956149	+	0.165610i	0.909865	−	0.414904i	$-0.136185\pi$
$107$	1.29644e8	+	2.24550e8i	0.0956149	+	0.165610i	−0.814250	+	0.580514i	$0.802852\pi$
$108$		0			0
$109$	4.12893e7	−	7.15152e7i	0.0280168	−	0.0485265i	−0.851677	−	0.524067i	$-0.824414\pi$
$109$	4.12893e7	−	7.15152e7i	0.0280168	−	0.0485265i	0.879694	+	0.475540i	$0.157747\pi$
$110$		0			0
$111$	−1.29054e9			−0.806899
$112$		0			0
$113$	1.17279e9			0.676654			0.338327	−	0.941029i	$-0.390139\pi$
$113$	1.17279e9			0.676654			0.338327	+	0.941029i	$0.390139\pi$
$114$		0			0
$115$	1.44258e9	−	2.49863e9i	0.769132	−	1.33218i
$116$		0			0
$117$	2.13789e8	+	3.70294e8i	0.105475	+	0.182688i
$118$		0			0
$119$	−8.68293e8	+	1.86615e9i	−0.396922	+	0.853073i
$120$		0			0
$121$	1.11930e9	+	1.93869e9i	0.474694	+	0.822194i
$122$		0			0
$123$	7.54770e8	−	1.30730e9i	0.297332	−	0.514994i
$124$		0			0
$125$	−2.78095e9			−1.01882
$126$		0			0
$127$	2.85883e9			0.975150			0.487575	−	0.873081i	$-0.337882\pi$
$127$	2.85883e9			0.975150			0.487575	+	0.873081i	$0.337882\pi$
$128$		0			0
$129$	4.21586e9	−	7.30208e9i	1.34039	−	2.32163i
$130$		0			0
$131$	−2.58550e9	−	4.47822e9i	−0.767050	−	1.32857i	−0.939156	−	0.343492i	$-0.888390\pi$
$131$	−2.58550e9	−	4.47822e9i	−0.767050	−	1.32857i	0.172105	−	0.985079i	$-0.444943\pi$
$132$		0			0
$133$	8.45888e8	+	1.20589e9i	0.234412	+	0.334177i
$134$		0			0
$135$	6.30116e9	+	1.09139e10i	1.63275	+	2.82800i
$136$		0			0
$137$	8.48240e8	−	1.46920e9i	0.205720	−	0.356317i	−0.744642	−	0.667464i	$-0.767380\pi$
$137$	8.48240e8	−	1.46920e9i	0.205720	−	0.356317i	0.950362	+	0.311147i	$0.100713\pi$
$138$		0			0
$139$	−5.00914e9			−1.13814			−0.569071	−	0.822288i	$-0.692697\pi$
$139$	−5.00914e9			−1.13814			−0.569071	+	0.822288i	$0.692697\pi$
$140$		0			0
$141$	1.16775e9			0.248807
$142$		0			0
$143$	4.35240e7	−	7.53858e7i	0.00870396	−	0.0150757i
$144$		0			0
$145$	3.74120e9	+	6.47995e9i	0.702837	+	1.21735i
$146$		0			0
$147$	3.72051e9	−	1.02757e10i	0.657165	−	1.81503i
$148$		0			0
$149$	−8.39970e8	−	1.45487e9i	−0.139613	−	0.241817i	0.787737	−	0.616011i	$-0.211252\pi$
$149$	−8.39970e8	−	1.45487e9i	−0.139613	−	0.241817i	−0.927350	+	0.374195i	$0.877919\pi$
$150$		0			0
$151$	−3.40730e9	+	5.90162e9i	−0.533353	+	0.923794i	0.465889	+	0.884843i	$0.345735\pi$
$151$	−3.40730e9	+	5.90162e9i	−0.533353	+	0.923794i	−0.999241	+	0.0389503i	$0.987599\pi$
$152$		0			0
$153$	−1.73865e10			−2.56508
$154$		0			0
$155$	2.70993e9			0.377108
$156$		0			0
$157$	−4.78331e8	+	8.28493e8i	−0.0628318	+	0.108828i	−0.895730	−	0.444598i	$-0.853347\pi$
$157$	−4.78331e8	+	8.28493e8i	−0.0628318	+	0.108828i	0.832898	+	0.553426i	$0.186680\pi$
$158$		0			0
$159$	−1.22262e10	−	2.11764e10i	−1.51706	−	2.62763i
$160$		0			0
$161$	−7.68494e9	−	1.09556e10i	−0.901413	−	1.28505i
$162$		0			0
$163$	−3.10977e9	−	5.38627e9i	−0.345051	−	0.597646i	0.640312	−	0.768115i	$-0.278805\pi$
$163$	−3.10977e9	−	5.38627e9i	−0.345051	−	0.597646i	−0.985363	+	0.170469i	$0.945472\pi$
$164$		0			0
$165$	2.02595e9	−	3.50905e9i	0.212790	−	0.368563i
$166$		0			0
$167$	3.33475e9			0.331771			0.165886	−	0.986145i	$-0.446952\pi$
$167$	3.33475e9			0.331771			0.165886	+	0.986145i	$0.446952\pi$
$168$		0			0
$169$	−1.05410e10			−0.994013
$170$		0			0
$171$	−6.22130e9	+	1.07756e10i	−0.556415	+	0.963739i
$172$		0			0
$173$	−5.65009e9	−	9.78625e9i	−0.479566	−	0.830632i	0.520160	−	0.854069i	$-0.325872\pi$
$173$	−5.65009e9	−	9.78625e9i	−0.479566	−	0.830632i	−0.999725	+	0.0234370i	$0.992539\pi$
$174$		0			0
$175$	−2.07425e8	+	4.45803e8i	−0.0167182	+	0.0359312i
$176$		0			0
$177$	9.87287e9	+	1.71003e10i	0.756073	+	1.30956i
$178$		0			0
$179$	9.27568e8	−	1.60660e9i	0.0675316	−	0.116968i	−0.830283	−	0.557343i	$-0.811821\pi$
$179$	9.27568e8	−	1.60660e9i	0.0675316	−	0.116968i	0.897814	+	0.440374i	$0.145154\pi$
$180$		0			0
$181$	1.64314e9			0.113794			0.0568972	−	0.998380i	$-0.481879\pi$
$181$	1.64314e9			0.113794			0.0568972	+	0.998380i	$0.481879\pi$
$182$		0			0
$183$	2.01353e9			0.132718
$184$		0			0
$185$	−3.26322e9	+	5.65207e9i	−0.204821	+	0.354760i
$186$		0			0
$187$	1.76981e9	+	3.06539e9i	0.105837	+	0.183315i
$188$		0			0
$189$	5.82264e10	−	5.14348e9i	3.31926	−	0.293210i
$190$		0			0
$191$	5.68436e9	+	9.84559e9i	0.309052	+	0.535293i	0.978155	−	0.207876i	$-0.0666551\pi$
$191$	5.68436e9	+	9.84559e9i	0.309052	+	0.535293i	−0.669104	+	0.743169i	$0.733322\pi$
$192$		0			0
$193$	1.90511e10	−	3.29975e10i	0.988354	−	1.71188i	0.362389	−	0.932027i	$-0.381961\pi$
$193$	1.90511e10	−	3.29975e10i	0.988354	−	1.71188i	0.625964	−	0.779852i	$-0.284705\pi$
$194$		0			0
$195$	2.95548e9			0.146377
$196$		0			0
$197$	−1.71349e10			−0.810558			−0.405279	−	0.914193i	$-0.632826\pi$
$197$	−1.71349e10			−0.810558			−0.405279	+	0.914193i	$0.632826\pi$
$198$		0			0
$199$	−1.15090e10	+	1.99341e10i	−0.520232	+	0.901069i	0.479491	+	0.877547i	$0.340821\pi$
$199$	−1.15090e10	+	1.99341e10i	−0.520232	+	0.901069i	−0.999723	+	0.0235222i	$0.992512\pi$
$200$		0			0
$201$	−1.43246e10	−	2.48110e10i	−0.619014	−	1.07216i
$202$		0			0
$203$	3.45708e10	−	3.05384e9i	1.42882	−	0.126216i
$204$		0			0
$205$	−3.81698e9	−	6.61119e9i	−0.150948	−	0.261449i
$206$		0			0
$207$	5.65208e10	−	9.78970e10i	2.13965	−	3.70598i
$208$		0			0
$209$	2.53311e9			0.0918324
$210$		0			0
$211$	−4.95907e10			−1.72238			−0.861190	−	0.508283i	$-0.830280\pi$
$211$	−4.95907e10			−1.72238			−0.861190	+	0.508283i	$0.830280\pi$
$212$		0			0
$213$	−1.49287e10	+	2.58572e10i	−0.496951	+	0.860744i
$214$		0			0
$215$	−2.13202e10	−	3.69276e10i	−0.680483	−	1.17863i
$216$		0			0
$217$	5.30248e9	−	1.13962e10i	0.162334	−	0.348893i
$218$		0			0
$219$	2.76844e10	+	4.79508e10i	0.813273	+	1.40863i
$220$		0			0
$221$	−1.29091e9	+	2.23591e9i	−0.0364024	+	0.0630507i
$222$		0			0
$223$	−5.54434e10			−1.50134			−0.750669	−	0.660679i	$-0.770269\pi$
$223$	−5.54434e10			−1.50134			−0.750669	+	0.660679i	$0.770269\pi$
$224$		0			0
$225$	−4.15344e9			−0.108041
$226$		0			0
$227$	−1.76917e9	+	3.06430e9i	−0.0442236	+	0.0765974i	−0.887290	−	0.461212i	$-0.847415\pi$
$227$	−1.76917e9	+	3.06430e9i	−0.0442236	+	0.0765974i	0.843066	+	0.537810i	$0.180748\pi$
$228$		0			0
$229$	1.52164e10	+	2.63556e10i	0.365639	+	0.633305i	0.988878	−	0.148726i	$-0.0475171\pi$
$229$	1.52164e10	+	2.63556e10i	0.365639	+	0.633305i	−0.623239	+	0.782031i	$0.714184\pi$
$230$		0			0
$231$	−1.07927e10	−	1.53860e10i	−0.249387	−	0.355525i
$232$		0			0
$233$	−3.08048e9	−	5.33554e9i	−0.0684725	−	0.118598i	0.829757	−	0.558125i	$-0.188479\pi$
$233$	−3.08048e9	−	5.33554e9i	−0.0684725	−	0.118598i	−0.898229	+	0.439528i	$0.855146\pi$
$234$		0			0
$235$	2.95273e9	−	5.11427e9i	0.0631565	−	0.109390i
$236$		0			0
$237$	2.40446e10			0.495051
$238$		0			0
$239$	6.33526e10			1.25595			0.627977	−	0.778232i	$-0.283883\pi$
$239$	6.33526e10			1.25595			0.627977	+	0.778232i	$0.283883\pi$
$240$		0			0
$241$	−3.75764e10	+	6.50842e10i	−0.717528	+	1.24279i	0.244449	+	0.969662i	$0.421393\pi$
$241$	−3.75764e10	+	6.50842e10i	−0.717528	+	1.24279i	−0.961977	+	0.273132i	$0.911940\pi$
$242$		0			0
$243$	1.03863e11	+	1.79895e11i	1.91087	+	3.30973i
$244$		0			0
$245$	−3.55961e10	−	4.22773e10i	−0.631183	−	0.749652i
$246$		0			0
$247$	9.23833e8	+	1.60013e9i	0.0157927	+	0.0273538i
$248$		0			0
$249$	−3.27296e10	+	5.66892e10i	−0.539564	+	0.934552i
$250$		0			0
$251$	−1.01701e11			−1.61731			−0.808653	−	0.588286i	$-0.799803\pi$
$251$	−1.01701e11			−1.61731			−0.808653	+	0.588286i	$0.799803\pi$
$252$		0			0
$253$	−2.30135e10			−0.353134
$254$		0			0
$255$	−6.00890e10	+	1.04077e11i	−0.889946	+	1.54143i
$256$		0			0
$257$	−2.75846e10	−	4.77780e10i	−0.394429	−	0.683170i	0.598600	−	0.801048i	$-0.295724\pi$
$257$	−2.75846e10	−	4.77780e10i	−0.394429	−	0.683170i	−0.993028	+	0.117878i	$0.962391\pi$
$258$		0			0
$259$	1.73839e10	+	2.47823e10i	0.240047	+	0.342211i
$260$		0			0
$261$	1.46581e11	+	2.53886e11i	1.95522	+	3.38654i
$262$		0			0
$263$	−1.12230e10	+	1.94389e10i	−0.144647	+	0.250536i	−0.929241	−	0.369474i	$-0.879538\pi$
$263$	−1.12230e10	+	1.94389e10i	−0.144647	+	0.250536i	0.784594	+	0.620010i	$0.212871\pi$
$264$		0			0
$265$	−1.23659e11			−1.54035
$266$		0			0
$267$	−1.48596e11			−1.78940
$268$		0			0
$269$	−1.17607e9	+	2.03702e9i	−0.0136946	+	0.0237198i	−0.872791	−	0.488093i	$-0.837693\pi$
$269$	−1.17607e9	+	2.03702e9i	−0.0136946	+	0.0237198i	0.859097	+	0.511813i	$0.171026\pi$
$270$		0			0
$271$	−7.47028e10	−	1.29389e11i	−0.841346	−	1.45725i	−0.888757	−	0.458379i	$-0.848430\pi$
$271$	−7.47028e10	−	1.29389e11i	−0.841346	−	1.45725i	0.0474103	−	0.998875i	$-0.484903\pi$
$272$		0			0
$273$	5.78295e9	−	1.24288e10i	0.0630112	−	0.135425i
$274$		0			0
$275$	4.22787e8	+	7.32288e8i	0.00445784	+	0.00772120i
$276$		0			0
$277$	7.21993e10	−	1.25053e11i	0.736842	−	1.27625i	−0.217069	−	0.976156i	$-0.569650\pi$
$277$	7.21993e10	−	1.25053e11i	0.736842	−	1.27625i	0.953911	−	0.300091i	$-0.0970170\pi$
$278$		0			0
$279$	1.06176e11			1.04908
$280$		0			0
$281$	−7.04301e10			−0.673876			−0.336938	−	0.941527i	$-0.609391\pi$
$281$	−7.04301e10			−0.673876			−0.336938	+	0.941527i	$0.609391\pi$
$282$		0			0
$283$	1.02279e11	−	1.77152e11i	0.947868	−	1.64175i	0.197962	−	0.980210i	$-0.436568\pi$
$283$	1.02279e11	−	1.77152e11i	0.947868	−	1.64175i	0.749905	−	0.661545i	$-0.230099\pi$
$284$		0			0
$285$	4.30025e10	+	7.44825e10i	0.386093	+	0.668732i
$286$		0			0
$287$	−3.52710e10	+	3.11570e9i	−0.306867	+	0.0271073i
$288$		0			0
$289$	6.80216e9	+	1.17817e10i	0.0573596	+	0.0993498i
$290$		0			0
$291$	−1.58085e11	+	2.73812e11i	−1.29233	+	2.23838i
$292$		0			0
$293$	1.08283e11			0.858335			0.429168	−	0.903225i	$-0.358807\pi$
$293$	1.08283e11			0.858335			0.429168	+	0.903225i	$0.358807\pi$
$294$		0			0
$295$	9.98569e10			0.767677
$296$		0			0
$297$	5.02610e10	−	8.70547e10i	0.374824	−	0.649214i
$298$		0			0
$299$	−8.39307e9	−	1.45372e10i	−0.0607296	−	0.105187i
$300$		0			0
$301$	−1.97011e11	+	1.74031e10i	−1.38338	+	0.122202i
$302$		0			0
$303$	1.88200e11	+	3.25972e11i	1.28271	+	2.22171i
$304$		0			0
$305$	5.09136e9	−	8.81849e9i	0.0336887	−	0.0583505i
$306$		0			0
$307$	1.17650e11			0.755912			0.377956	−	0.925824i	$-0.376627\pi$
$307$	1.17650e11			0.755912			0.377956	+	0.925824i	$0.376627\pi$
$308$		0			0
$309$	2.54247e11			1.58651
$310$		0			0
$311$	−8.86373e9	+	1.53524e10i	−0.0537273	+	0.0930584i	−0.891638	−	0.452749i	$-0.850443\pi$
$311$	−8.86373e9	+	1.53524e10i	−0.0537273	+	0.0930584i	0.837911	+	0.545807i	$0.183777\pi$
$312$		0			0
$313$	−5.33183e10	−	9.23500e10i	−0.313998	−	0.543861i	0.665226	−	0.746642i	$-0.268335\pi$
$313$	−5.33183e10	−	9.23500e10i	−0.313998	−	0.543861i	−0.979224	+	0.202781i	$0.935002\pi$
$314$		0			0
$315$	1.96944e11	−	4.23275e11i	1.12705	−	2.42229i
$316$		0			0
$317$	1.65551e10	+	2.86742e10i	0.0920797	+	0.159487i	0.908386	−	0.418132i	$-0.137315\pi$
$317$	1.65551e10	+	2.86742e10i	0.0920797	+	0.159487i	−0.816306	+	0.577619i	$0.803982\pi$
$318$		0			0
$319$	2.98416e10	−	5.16871e10i	0.161348	−	0.279463i
$320$		0			0
$321$	7.02203e10			0.369139
$322$		0			0
$323$	−7.51311e10			−0.384068
$324$		0			0
$325$	−3.08383e8	+	5.34135e8i	−0.00153326	+	0.00265568i
$326$		0			0
$327$	−1.11819e10	−	1.93677e10i	−0.0540819	−	0.0936727i
$328$		0			0
$329$	−1.57298e10	−	2.24243e10i	−0.0740186	−	0.105521i
$330$		0			0
$331$	−1.10008e11	−	1.90539e11i	−0.503730	−	0.872486i	−0.999991	−	0.00431221i	$-0.998627\pi$
$331$	−1.10008e11	−	1.90539e11i	−0.503730	−	0.872486i	0.496261	−	0.868173i	$-0.334706\pi$
$332$		0			0
$333$	−1.27854e11	+	2.21450e11i	−0.569791	+	0.986906i
$334$		0			0
$335$	−1.44883e11			−0.628515
$336$		0			0
$337$	1.58927e11			0.671217			0.335609	−	0.942001i	$-0.391058\pi$
$337$	1.58927e11			0.671217			0.335609	+	0.942001i	$0.391058\pi$
$338$		0			0
$339$	1.58807e11	−	2.75061e11i	0.653086	−	1.13118i
$340$		0			0
$341$	−1.08078e10	−	1.87197e10i	−0.0432857	−	0.0749730i
$342$		0			0
$343$	−2.47441e11	+	6.69710e10i	−0.965270	+	0.261254i
$344$		0			0
$345$	−3.90680e11	−	6.76677e11i	−1.48469	−	2.57155i
$346$		0			0
$347$	4.52947e10	−	7.84527e10i	0.167712	−	0.290486i	−0.769903	−	0.638161i	$-0.779695\pi$
$347$	4.52947e10	−	7.84527e10i	0.167712	−	0.290486i	0.937615	+	0.347675i	$0.113029\pi$
$348$		0			0
$349$	−1.82702e11			−0.659219			−0.329609	−	0.944117i	$-0.606917\pi$
$349$	−1.82702e11			−0.659219			−0.329609	+	0.944117i	$0.606917\pi$
$350$		0			0
$351$	7.33214e10			0.257839
$352$		0			0
$353$	−1.44574e11	+	2.50409e11i	−0.495568	+	0.858349i	−0.999987	−	0.00511018i	$-0.998373\pi$
$353$	−1.44574e11	+	2.50409e11i	−0.495568	+	0.858349i	0.504419	+	0.863459i	$0.331707\pi$
$354$		0			0
$355$	7.54963e10	+	1.30764e11i	0.252289	+	0.436977i
$356$		0			0
$357$	3.20106e11	+	4.56342e11i	1.04301	+	1.48691i
$358$		0			0
$359$	−1.64524e11	−	2.84963e11i	−0.522761	−	0.905449i	−0.999649	−	0.0264852i	$-0.991569\pi$
$359$	−1.64524e11	−	2.84963e11i	−0.522761	−	0.905449i	0.476888	−	0.878964i	$-0.341765\pi$
$360$		0			0
$361$	1.34460e11	−	2.32892e11i	0.416688	−	0.721725i
$362$		0			0
$363$	6.06258e11			1.83264
$364$		0			0
$365$	2.80007e11			0.825756
$366$		0			0
$367$	−2.08227e11	+	3.60660e11i	−0.599156	+	1.03777i	0.393790	+	0.919200i	$0.371164\pi$
$367$	−2.08227e11	+	3.60660e11i	−0.599156	+	1.03777i	−0.992946	+	0.118568i	$0.962170\pi$
$368$		0			0
$369$	−1.49550e11	−	2.59028e11i	−0.419922	−	0.727326i
$370$		0			0
$371$	−2.41962e11	+	5.20030e11i	−0.663077	+	1.42510i
$372$		0			0
$373$	−5.38148e10	−	9.32099e10i	−0.143950	−	0.249329i	0.785031	−	0.619457i	$-0.212647\pi$
$373$	−5.38148e10	−	9.32099e10i	−0.143950	−	0.249329i	−0.928981	+	0.370128i	$0.879314\pi$
$374$		0			0
$375$	−3.76568e11	+	6.52234e11i	−0.983337	+	1.70319i
$376$		0			0
$377$	4.35332e10			0.110990
$378$		0			0
$379$	−5.34243e11			−1.33003			−0.665016	−	0.746829i	$-0.731575\pi$
$379$	−5.34243e11			−1.33003			−0.665016	+	0.746829i	$0.731575\pi$
$380$		0			0
$381$	3.87113e11	−	6.70500e11i	0.941186	−	1.63018i
$382$		0			0
$383$	−1.53952e11	−	2.66652e11i	−0.365586	−	0.633214i	0.623284	−	0.781996i	$-0.285798\pi$
$383$	−1.53952e11	−	2.66652e11i	−0.365586	−	0.633214i	−0.988870	+	0.148782i	$0.952465\pi$
$384$		0			0
$385$	−9.46744e10	+	8.36314e9i	−0.219614	+	0.0193998i
$386$		0			0
$387$	−8.35330e11	−	1.44683e12i	−1.89304	−	3.27883i
$388$		0			0
$389$	−2.93605e11	+	5.08539e11i	−0.650115	+	1.12603i	0.332979	+	0.942934i	$0.391946\pi$
$389$	−2.93605e11	+	5.08539e11i	−0.650115	+	1.12603i	−0.983095	+	0.183098i	$0.941387\pi$
$390$		0			0
$391$	6.82570e11			1.47690
$392$		0			0
$393$	−1.40041e12			−2.96134
$394$		0			0
$395$	6.07983e10	−	1.05306e11i	0.125662	−	0.217653i
$396$		0			0
$397$	−2.96801e11	−	5.14075e11i	−0.599665	−	1.03865i	−0.992870	−	0.119199i	$-0.961967\pi$
$397$	−2.96801e11	−	5.14075e11i	−0.599665	−	1.03865i	0.393205	−	0.919451i	$-0.371366\pi$
$398$		0			0
$399$	3.97368e11	−	3.51018e10i	0.784900	−	0.0693348i
$400$		0			0
$401$	−2.25093e11	−	3.89873e11i	−0.434723	−	0.752962i	0.562550	−	0.826763i	$-0.309820\pi$
$401$	−2.25093e11	−	3.89873e11i	−0.434723	−	0.752962i	−0.997273	+	0.0738012i	$0.976487\pi$
$402$		0			0
$403$	7.88330e9	−	1.36543e10i	0.0148880	−	0.0257867i
$404$		0			0
$405$	1.96643e12			3.63187
$406$		0			0
$407$	5.20580e10			0.0940400
$408$		0			0
$409$	−2.17320e11	+	3.76409e11i	−0.384012	+	0.665128i	−0.991632	−	0.129100i	$-0.958791\pi$
$409$	−2.17320e11	+	3.76409e11i	−0.384012	+	0.665128i	0.607620	+	0.794228i	$0.292124\pi$
$410$		0			0
$411$	−2.29720e11	−	3.97887e11i	−0.397110	−	0.687814i
$412$		0			0
$413$	1.95389e11	−	4.19934e11i	0.330464	−	0.710240i
$414$		0			0
$415$	1.65518e11	+	2.86685e11i	0.273923	+	0.474448i
$416$		0			0
$417$	−6.78286e11	+	1.17483e12i	−1.09850	+	1.90266i
$418$		0			0
$419$	4.94533e11			0.783848			0.391924	−	0.919998i	$-0.371810\pi$
$419$	4.94533e11			0.783848			0.391924	+	0.919998i	$0.371810\pi$
$420$		0			0
$421$	6.78076e11			1.05198			0.525992	−	0.850490i	$-0.323694\pi$
$421$	6.78076e11			1.05198			0.525992	+	0.850490i	$0.323694\pi$
$422$		0			0
$423$	1.15689e11	−	2.00379e11i	0.175695	−	0.304313i
$424$		0			0
$425$	−1.25397e10	−	2.17194e10i	−0.0186439	−	0.0322922i
$426$		0			0
$427$	−2.71227e10	−	3.86660e10i	−0.0394827	−	0.0562864i
$428$		0			0
$429$	−1.17871e10	−	2.04159e10i	−0.0168016	−	0.0291012i
$430$		0			0
$431$	5.71308e11	−	9.89535e11i	0.797485	−	1.38128i	−0.123764	−	0.992312i	$-0.539497\pi$
$431$	5.71308e11	−	9.89535e11i	0.797485	−	1.38128i	0.921249	−	0.388973i	$-0.127170\pi$
$432$		0			0
$433$	9.94314e11			1.35934			0.679670	−	0.733518i	$-0.262123\pi$
$433$	9.94314e11			1.35934			0.679670	+	0.733518i	$0.262123\pi$
$434$		0			0
$435$	2.02638e12			2.71343
$436$		0			0
$437$	2.44240e11	−	4.23035e11i	0.320368	−	0.554894i
$438$		0			0
$439$	1.08471e11	+	1.87877e11i	0.139387	+	0.241426i	0.927265	−	0.374406i	$-0.122153\pi$
$439$	1.08471e11	+	1.87877e11i	0.139387	+	0.241426i	−0.787877	+	0.615832i	$0.788820\pi$
$440$		0			0
$441$	−1.39467e12	−	1.65643e12i	−1.75589	−	2.08545i
$442$		0			0
$443$	5.34760e11	+	9.26231e11i	0.659693	+	1.14262i	0.980695	+	0.195544i	$0.0626472\pi$
$443$	5.34760e11	+	9.26231e11i	0.659693	+	1.14262i	−0.321002	+	0.947079i	$0.604020\pi$
$444$		0			0
$445$	−3.75735e11	+	6.50792e11i	−0.454215	+	0.786723i
$446$		0			0
$447$	−4.54960e11			−0.539001
$448$		0			0
$449$	1.65225e12			1.91853			0.959264	−	0.282510i	$-0.0911669\pi$
$449$	1.65225e12			1.91853			0.959264	+	0.282510i	$0.0911669\pi$
$450$		0			0
$451$	−3.04460e10	+	5.27340e10i	−0.0346526	+	0.0600200i
$452$		0			0
$453$	9.22763e11	+	1.59827e12i	1.02955	+	1.78324i
$454$		0			0
$455$	−3.98109e10	−	5.67542e10i	−0.0435462	−	0.0620793i
$456$		0			0
$457$	3.73254e10	+	6.46496e10i	0.0400297	+	0.0693334i	0.885346	−	0.464933i	$-0.153921\pi$
$457$	3.73254e10	+	6.46496e10i	0.0400297	+	0.0693334i	−0.845316	+	0.534266i	$0.820588\pi$
$458$		0			0
$459$	−1.49072e12	+	2.58201e12i	−1.56762	+	2.71519i
$460$		0			0
$461$	5.63966e11			0.581565			0.290783	−	0.956789i	$-0.406084\pi$
$461$	5.63966e11			0.581565			0.290783	+	0.956789i	$0.406084\pi$
$462$		0			0
$463$	−9.11377e11			−0.921687			−0.460844	−	0.887481i	$-0.652453\pi$
$463$	−9.11377e11			−0.921687			−0.460844	+	0.887481i	$0.652453\pi$
$464$		0			0
$465$	3.66951e11	−	6.35577e11i	0.363973	−	0.630420i
$466$		0			0
$467$	−9.58921e10	−	1.66090e11i	−0.0932946	−	0.161591i	0.815601	−	0.578615i	$-0.196407\pi$
$467$	−9.58921e10	−	1.66090e11i	−0.0932946	−	0.161591i	−0.908896	+	0.417024i	$0.863073\pi$
$468$		0			0
$469$	−2.83491e11	+	6.09285e11i	−0.270558	+	0.581490i
$470$		0			0
$471$	1.29541e11	+	2.24372e11i	0.121287	+	0.210075i
$472$		0			0
$473$	−1.70060e11	+	2.94552e11i	−0.156216	+	0.270575i
$474$		0			0
$475$	−1.79480e10			−0.0161769
$476$		0			0
$477$	−4.84499e12			−4.28509
$478$		0			0
$479$	7.45288e11	−	1.29088e12i	0.646866	−	1.12040i	−0.337001	−	0.941504i	$-0.609413\pi$
$479$	7.45288e11	−	1.29088e12i	0.646866	−	1.12040i	0.983867	−	0.178900i	$-0.0572540\pi$
$480$		0			0
$481$	1.89857e10	+	3.28842e10i	0.0161724	+	0.0280114i
$482$		0			0
$483$	−3.61011e12	+	3.18902e11i	−3.01827	+	0.266621i
$484$		0			0
$485$	7.99459e11	+	1.38470e12i	0.656083	+	1.13637i
$486$		0			0
$487$	−6.34738e11	+	1.09940e12i	−0.511345	+	0.885675i	0.488569	+	0.872525i	$0.337519\pi$
$487$	−6.34738e11	+	1.09940e12i	−0.511345	+	0.885675i	−0.999914	+	0.0131499i	$0.995814\pi$
$488$		0			0
$489$	−1.68437e12			−1.33213
$490$		0			0
$491$	7.11119e11			0.552173			0.276087	−	0.961133i	$-0.410962\pi$
$491$	7.11119e11			0.552173			0.276087	+	0.961133i	$0.410962\pi$
$492$		0			0
$493$	−8.85089e11	+	1.53302e12i	−0.674801	+	1.16879i
$494$		0			0
$495$	−4.01422e11	−	6.95283e11i	−0.300523	−	0.520521i
$496$		0			0
$497$	6.97630e11	−	6.16257e10i	0.512886	−	0.0453063i
$498$		0			0
$499$	8.01313e11	+	1.38791e12i	0.578562	+	1.00210i	0.995645	+	0.0932299i	$0.0297192\pi$
$499$	8.01313e11	+	1.38791e12i	0.578562	+	1.00210i	−0.417083	+	0.908868i	$0.636947\pi$
$500$		0			0
$501$	4.51557e11	−	7.82120e11i	0.320216	−	0.554630i
$502$		0			0
$503$	6.48944e11			0.452013			0.226007	−	0.974126i	$-0.427433\pi$
$503$	6.48944e11			0.452013			0.226007	+	0.974126i	$0.427433\pi$
$504$		0			0
$505$	1.90350e12			1.30239
$506$		0			0
$507$	−1.42735e12	+	2.47225e12i	−0.959392	+	1.66172i
$508$		0			0
$509$	1.90962e10	+	3.30757e10i	0.0126101	+	0.0218413i	0.872262	−	0.489040i	$-0.162653\pi$
$509$	1.90962e10	+	3.30757e10i	0.0126101	+	0.0218413i	−0.859652	+	0.510881i	$0.829319\pi$
$510$		0			0
$511$	5.47887e11	−	1.17753e12i	0.355465	−	0.763973i
$512$		0			0
$513$	1.06683e12	+	1.84781e12i	0.680092	+	1.17795i
$514$		0			0
$515$	6.42881e11	−	1.11350e12i	0.402715	−	0.697524i
$516$		0			0
$517$	−4.71047e10			−0.0289972
$518$		0			0
$519$	−3.06031e12			−1.85145
$520$		0			0
$521$	5.84088e11	−	1.01167e12i	0.347303	−	0.601546i	−0.638466	−	0.769650i	$-0.720431\pi$
$521$	5.84088e11	−	1.01167e12i	0.347303	−	0.601546i	0.985769	+	0.168103i	$0.0537642\pi$
$522$		0			0
$523$	7.59331e11	+	1.31520e12i	0.443786	+	0.768660i	0.997967	−	0.0637364i	$-0.0203017\pi$
$523$	7.59331e11	+	1.31520e12i	0.443786	+	0.768660i	−0.554181	+	0.832396i	$0.686968\pi$
$524$		0			0
$525$	7.64697e10	+	1.09015e11i	0.0439312	+	0.0626281i
$526$		0			0
$527$	3.20556e11	+	5.55220e11i	0.181033	+	0.313558i
$528$		0			0
$529$	−1.31835e12	+	2.28346e12i	−0.731950	+	1.26777i
$530$		0			0
$531$	3.91242e12			2.13560
$532$		0			0
$533$	−4.44149e10			−0.0238373
$534$		0			0
$535$	1.77557e11	−	3.07537e11i	0.0937012	−	0.162295i
$536$		0			0
$537$	−2.51203e11	−	4.35097e11i	−0.130359	−	0.225789i
$538$		0			0
$539$	−1.50078e11	+	4.14504e11i	−0.0765893	+	0.211533i
$540$		0			0
$541$	−5.97210e11	−	1.03440e12i	−0.299736	−	0.519159i	0.676339	−	0.736590i	$-0.263565\pi$
$541$	−5.97210e11	−	1.03440e12i	−0.299736	−	0.519159i	−0.976076	+	0.217432i	$0.930232\pi$
$542$		0			0
$543$	2.22497e11	−	3.85376e11i	0.109831	−	0.190233i
$544$		0			0
$545$	−1.13097e11			−0.0549120
$546$		0			0
$547$	3.52503e11			0.168353			0.0841764	−	0.996451i	$-0.473174\pi$
$547$	3.52503e11			0.168353			0.0841764	+	0.996451i	$0.473174\pi$
$548$		0			0
$549$	1.99481e11	−	3.45511e11i	0.0937185	−	0.162325i
$550$		0			0
$551$	6.33411e11	+	1.09710e12i	0.292755	+	0.507066i
$552$		0			0
$553$	−3.23885e11	−	4.61729e11i	−0.147275	−	0.209954i
$554$		0			0
$555$	8.83744e11	+	1.53069e12i	0.395374	+	0.684808i
$556$		0			0
$557$	9.05992e11	−	1.56922e12i	0.398819	−	0.690775i	−0.594761	−	0.803902i	$-0.702753\pi$
$557$	9.05992e11	−	1.56922e12i	0.398819	−	0.690775i	0.993580	+	0.113127i	$0.0360868\pi$
$558$		0			0
$559$	−2.48085e11			−0.107460
$560$		0			0
$561$	9.58595e11			0.408604
$562$		0			0
$563$	−1.41540e12	+	2.45154e12i	−0.593733	+	1.02838i	0.399991	+	0.916519i	$0.369013\pi$
$563$	−1.41540e12	+	2.45154e12i	−0.593733	+	1.02838i	−0.993724	+	0.111857i	$0.964320\pi$
$564$		0			0
$565$	−8.03107e11	−	1.39102e12i	−0.331555	−	0.574270i
$566$		0			0
$567$	3.84768e12	−	8.26952e12i	1.56342	−	3.36013i
$568$		0			0
$569$	−1.76736e12	−	3.06115e12i	−0.706837	−	1.22428i	−0.966025	−	0.258450i	$-0.916788\pi$
$569$	−1.76736e12	−	3.06115e12i	−0.706837	−	1.22428i	0.259188	−	0.965827i	$-0.416545\pi$
$570$		0			0
$571$	5.31965e11	−	9.21390e11i	0.209421	−	0.362728i	−0.742111	−	0.670277i	$-0.766175\pi$
$571$	5.31965e11	−	9.21390e11i	0.209421	−	0.362728i	0.951532	+	0.307549i	$0.0995087\pi$
$572$		0			0
$573$	3.07887e12			1.19315
$574$		0			0
$575$	1.63058e11			0.0622068
$576$		0			0
$577$	−1.77763e12	+	3.07894e12i	−0.667652	+	1.15641i	0.310907	+	0.950440i	$0.399367\pi$
$577$	−1.77763e12	+	3.07894e12i	−0.667652	+	1.15641i	−0.978559	+	0.205966i	$0.933966\pi$
$578$		0			0
$579$	−5.15941e12	−	8.93636e12i	−1.90786	−	3.30451i
$580$		0			0
$581$	1.52948e12	−	1.35108e11i	0.556866	−	0.0491913i
$582$		0			0
$583$	4.93181e11	+	8.54214e11i	0.176806	+	0.306237i
$584$		0			0
$585$	2.92799e11	−	5.07143e11i	0.103364	−	0.179031i
$586$		0			0
$587$	5.01945e12			1.74496			0.872478	−	0.488653i	$-0.162512\pi$
$587$	5.01945e12			1.74496			0.872478	+	0.488653i	$0.162512\pi$
$588$		0			0
$589$	4.58810e11			0.157078
$590$		0			0
$591$	−2.32023e12	+	4.01876e12i	−0.782327	+	1.35503i
$592$		0			0
$593$	1.80719e12	+	3.13014e12i	0.600146	+	1.03948i	0.992798	+	0.119797i	$0.0382243\pi$
$593$	1.80719e12	+	3.13014e12i	0.600146	+	1.03948i	−0.392652	+	0.919687i	$0.628442\pi$
$594$		0			0
$595$	2.80801e12	−	2.48048e11i	0.918485	−	0.0811351i
$596$		0			0
$597$	3.11685e12	+	5.39854e12i	1.00423	+	1.73937i
$598$		0			0
$599$	−4.48277e11	+	7.76438e11i	−0.142274	+	0.246426i	−0.928353	−	0.371701i	$-0.878775\pi$
$599$	−4.48277e11	+	7.76438e11i	−0.142274	+	0.246426i	0.786079	+	0.618127i	$0.212108\pi$
$600$		0			0
$601$	2.12154e11			0.0663310			0.0331655	−	0.999450i	$-0.489441\pi$
$601$	2.12154e11			0.0663310			0.0331655	+	0.999450i	$0.489441\pi$
$602$		0			0
$603$	−5.67656e12			−1.74847
$604$		0			0
$605$	1.53296e12	−	2.65517e12i	0.465193	−	0.805737i
$606$		0			0
$607$	9.87826e11	+	1.71096e12i	0.295346	+	0.511554i	0.975065	−	0.221918i	$-0.0712317\pi$
$607$	9.87826e11	+	1.71096e12i	0.295346	+	0.511554i	−0.679719	+	0.733472i	$0.737898\pi$
$608$		0			0
$609$	3.96499e12	−	8.52164e12i	1.16806	−	2.51041i
$610$		0			0
$611$	−1.71792e10	−	2.97553e10i	−0.00498675	−	0.00863730i
$612$		0			0
$613$	1.92970e12	−	3.34235e12i	0.551974	−	0.956047i	−0.446158	−	0.894954i	$-0.647208\pi$
$613$	1.92970e12	−	3.34235e12i	0.551974	−	0.956047i	0.998132	−	0.0610931i	$-0.0194586\pi$
$614$		0			0
$615$	−2.06742e12			−0.582762
$616$		0			0
$617$	−3.19932e12			−0.888738			−0.444369	−	0.895844i	$-0.646572\pi$
$617$	−3.19932e12			−0.888738			−0.444369	+	0.895844i	$0.646572\pi$
$618$		0			0
$619$	2.86605e12	−	4.96414e12i	0.784649	−	1.35905i	−0.144560	−	0.989496i	$-0.546177\pi$
$619$	2.86605e12	−	4.96414e12i	0.784649	−	1.35905i	0.929209	−	0.369556i	$-0.120490\pi$
$620$		0			0
$621$	−9.69222e12	−	1.67874e13i	−2.61524	−	4.52972i
$622$		0			0
$623$	2.00162e12	+	2.85349e12i	0.532334	+	0.758894i
$624$		0			0
$625$	1.82876e12	+	3.16751e12i	0.479400	+	0.830345i
$626$		0			0
$627$	3.43008e11	−	5.94107e11i	0.0886340	−	0.153519i
$628$		0			0
$629$	−1.54402e12			−0.393301
$630$		0			0
$631$	−6.65474e12			−1.67109			−0.835543	−	0.549425i	$-0.814847\pi$
$631$	−6.65474e12			−1.67109			−0.835543	+	0.549425i	$0.814847\pi$
$632$		0			0
$633$	−6.71506e12	+	1.16308e13i	−1.66239	+	2.87935i
$634$		0			0
$635$	−1.95768e12	−	3.39081e12i	−0.477816	−	0.827602i
$636$		0			0
$637$	−3.16569e11	+	5.63686e10i	−0.0761800	+	0.0135647i
$638$		0			0
$639$	2.95797e12	+	5.12335e12i	0.701842	+	1.21563i
$640$		0			0
$641$	2.30160e12	−	3.98649e12i	0.538479	−	0.932673i	−0.460507	−	0.887656i	$-0.652333\pi$
$641$	2.30160e12	−	3.98649e12i	0.538479	−	0.932673i	0.998986	−	0.0450169i	$-0.0143342\pi$
$642$		0			0
$643$	−6.55681e12			−1.51267			−0.756333	−	0.654186i	$-0.773011\pi$
$643$	−6.55681e12			−1.51267			−0.756333	+	0.654186i	$0.773011\pi$
$644$		0			0
$645$	−1.15478e13			−2.62713
$646$		0			0
$647$	8.97874e11	−	1.55516e12i	0.201440	−	0.348905i	−0.747552	−	0.664203i	$-0.768771\pi$
$647$	8.97874e11	−	1.55516e12i	0.201440	−	0.348905i	0.948993	+	0.315298i	$0.102104\pi$
$648$		0			0
$649$	−3.98253e11	−	6.89794e11i	−0.0881165	−	0.152622i
$650$		0			0
$651$	−1.95482e12	−	2.78678e12i	−0.426572	−	0.608119i
$652$		0			0
$653$	3.15320e12	+	5.46151e12i	0.678645	+	1.17545i	0.975389	+	0.220491i	$0.0707659\pi$
$653$	3.15320e12	+	5.46151e12i	0.678645	+	1.17545i	−0.296744	+	0.954957i	$0.595901\pi$
$654$		0			0
$655$	−3.54102e12	+	6.13323e12i	−0.751697	+	1.30198i
$656$		0			0
$657$	1.09708e13			2.29717
$658$		0			0
$659$	6.63478e12			1.37038			0.685191	−	0.728363i	$-0.259719\pi$
$659$	6.63478e12			1.37038			0.685191	+	0.728363i	$0.259719\pi$
$660$		0			0
$661$	1.19222e11	−	2.06499e11i	0.0242913	−	0.0420737i	−0.853624	−	0.520889i	$-0.825600\pi$
$661$	1.19222e11	−	2.06499e11i	0.0242913	−	0.0420737i	0.877915	+	0.478816i	$0.158934\pi$
$662$		0			0
$663$	3.49602e11	+	6.05529e11i	0.0702690	+	0.121709i
$664$		0			0
$665$	8.51039e11	−	1.82907e12i	0.168753	−	0.362688i
$666$		0			0
$667$	−5.75457e12	−	9.96722e12i	−1.12576	−	1.94988i
$668$		0			0
$669$	−7.50758e12	+	1.30035e13i	−1.44905	+	2.50982i
$670$		0			0
$671$	−8.12221e10			−0.0154676
$672$		0			0
$673$	7.84956e12			1.47495			0.737475	−	0.675374i	$-0.236018\pi$
$673$	7.84956e12			1.47495			0.737475	+	0.675374i	$0.236018\pi$
$674$		0			0
$675$	−3.56117e11	+	6.16813e11i	−0.0660277	+	0.114363i
$676$		0			0
$677$	3.63691e11	+	6.29932e11i	0.0665401	+	0.115251i	0.897376	−	0.441266i	$-0.145471\pi$
$677$	3.63691e11	+	6.29932e11i	0.0665401	+	0.115251i	−0.830836	+	0.556517i	$0.812137\pi$
$678$		0			0
$679$	7.38747e12	−	6.52578e11i	1.33377	−	0.117820i
$680$		0			0
$681$	4.79126e11	+	8.29871e11i	0.0853666	+	0.147859i
$682$		0			0
$683$	−1.01626e11	+	1.76022e11i	−0.0178695	+	0.0309509i	−0.874822	−	0.484445i	$-0.839022\pi$
$683$	−1.01626e11	+	1.76022e11i	−0.0178695	+	0.0309509i	0.856952	+	0.515396i	$0.172355\pi$
$684$		0			0
$685$	−2.32345e12			−0.403205
$686$		0			0
$687$	8.24180e12			1.41162
$688$		0			0
$689$	−3.59729e11	+	6.23069e11i	−0.0608119	+	0.105329i
$690$		0			0
$691$	2.33350e12	+	4.04173e12i	0.389364	+	0.674398i	0.992364	−	0.123343i	$-0.0393614\pi$
$691$	2.33350e12	+	4.04173e12i	0.389364	+	0.674398i	−0.603000	+	0.797741i	$0.706028\pi$
$692$		0			0
$693$	−3.70937e12	+	3.27670e11i	−0.610943	+	0.0539681i
$694$		0			0
$695$	3.43018e12	+	5.94125e12i	0.557681	+	0.965932i
$696$		0			0
$697$	9.03017e11	−	1.56407e12i	0.144927	−	0.251020i
$698$		0			0
$699$	−1.66851e12			−0.264351
$700$		0			0
$701$	−1.00377e12			−0.157001			−0.0785003	−	0.996914i	$-0.525013\pi$
$701$	−1.00377e12			−0.157001			−0.0785003	+	0.996914i	$0.525013\pi$
$702$		0			0
$703$	−5.52487e11	+	9.56935e11i	−0.0853145	+	0.147769i
$704$		0			0
$705$	−7.99656e11	−	1.38504e12i	−0.121914	−	0.211160i
$706$		0			0
$707$	3.72456e12	−	8.00491e12i	0.560645	−	1.20495i
$708$		0			0
$709$	−1.51043e12	−	2.61615e12i	−0.224488	−	0.388825i	0.731678	−	0.681651i	$-0.238738\pi$
$709$	−1.51043e12	−	2.61615e12i	−0.224488	−	0.388825i	−0.956166	+	0.292826i	$0.905404\pi$
$710$		0			0
$711$	2.38210e12	−	4.12591e12i	0.349580	−	0.605490i
$712$		0			0
$713$	−4.16832e12			−0.604029
$714$		0			0
$715$	−1.19218e11			−0.0170595
$716$		0			0
$717$	8.57855e12	−	1.48585e13i	1.21221	−	2.09961i
$718$		0			0
$719$	−5.47763e12	−	9.48754e12i	−0.764386	−	1.32396i	−0.940570	−	0.339599i	$-0.889709\pi$
$719$	−5.47763e12	−	9.48754e12i	−0.764386	−	1.32396i	0.176184	−	0.984357i	$-0.443625\pi$
$720$		0			0
$721$	−3.42476e12	−	4.88232e12i	−0.471978	−	0.672849i
$722$		0			0
$723$	1.01764e13	+	1.76261e13i	1.38507	+	2.39902i
$724$		0			0
$725$	−2.11438e11	+	3.66221e11i	−0.0284225	+	0.0492292i
$726$		0			0
$727$	−1.28144e13			−1.70135			−0.850674	−	0.525694i	$-0.823806\pi$
$727$	−1.28144e13			−1.70135			−0.850674	+	0.525694i	$0.823806\pi$
$728$		0			0
$729$	2.79953e13			3.67122
$730$		0			0
$731$	5.04391e12	−	8.73630e12i	0.653339	−	1.13162i
$732$		0			0
$733$	−3.56009e12	−	6.16625e12i	−0.455505	−	0.788957i	0.543212	−	0.839595i	$-0.317208\pi$
$733$	−3.56009e12	−	6.16625e12i	−0.455505	−	0.788957i	−0.998717	+	0.0506382i	$0.983874\pi$
$734$		0			0
$735$	−1.47356e13	+	2.62384e12i	−1.86241	+	0.331623i
$736$		0			0
$737$	5.77828e11	+	1.00083e12i	0.0721431	+	0.124955i
$738$		0			0
$739$	−2.76680e12	+	4.79224e12i	−0.341254	+	0.591070i	−0.984666	−	0.174451i	$-0.944185\pi$
$739$	−2.76680e12	+	4.79224e12i	−0.341254	+	0.591070i	0.643412	+	0.765520i	$0.277518\pi$
$740$		0			0
$741$	5.00384e11			0.0609707
$742$		0			0
$743$	−2.58825e12			−0.311571			−0.155785	−	0.987791i	$-0.549791\pi$
$743$	−2.58825e12			−0.311571			−0.155785	+	0.987791i	$0.549791\pi$
$744$		0			0
$745$	−1.15040e12	+	1.99255e12i	−0.136819	+	0.236977i
$746$		0			0
$747$	6.48503e12	+	1.12324e13i	0.762025	+	1.31987i
$748$		0			0
$749$	−9.45880e11	−	1.34844e12i	−0.109817	−	0.156554i
$750$		0			0
$751$	−2.82166e12	−	4.88725e12i	−0.323686	−	0.560641i	0.657559	−	0.753403i	$-0.271589\pi$
$751$	−2.82166e12	−	4.88725e12i	−0.323686	−	0.560641i	−0.981246	+	0.192762i	$0.938256\pi$
$752$		0			0
$753$	−1.37713e13	+	2.38525e13i	−1.56098	+	2.70369i
$754$		0			0
$755$	9.33308e12			1.04535
$756$		0			0
$757$	−1.51570e13			−1.67757			−0.838786	−	0.544462i	$-0.816734\pi$
$757$	−1.51570e13			−1.67757			−0.838786	+	0.544462i	$0.816734\pi$
$758$		0			0
$759$	−3.11624e12	+	5.39749e12i	−0.340834	+	0.590343i
$760$		0			0
$761$	4.02856e12	+	6.97767e12i	0.435430	+	0.754187i	0.997331	−	0.0730176i	$-0.0232629\pi$
$761$	4.02856e12	+	6.97767e12i	0.435430	+	0.754187i	−0.561900	+	0.827205i	$0.689930\pi$
$762$		0			0
$763$	−2.21296e11	+	4.75614e11i	−0.0236381	+	0.0508035i
$764$		0			0
$765$	1.19060e13	+	2.06218e13i	1.25687	+	2.17696i
$766$		0			0
$767$	2.90488e11	−	5.03139e11i	0.0303074	−	0.0524939i
$768$		0			0
$769$	−2.31836e12			−0.239063			−0.119531	−	0.992830i	$-0.538139\pi$
$769$	−2.31836e12			−0.239063			−0.119531	+	0.992830i	$0.538139\pi$
$770$		0			0
$771$	−1.49409e13			−1.52276
$772$		0			0
$773$	8.54611e12	−	1.48023e13i	0.860916	−	1.49115i	−0.0101300	−	0.999949i	$-0.503225\pi$
$773$	8.54611e12	−	1.48023e13i	0.860916	−	1.49115i	0.871046	−	0.491201i	$-0.163442\pi$
$774$		0			0
$775$	7.65773e10	+	1.32636e11i	0.00762505	+	0.0132070i
$776$		0			0
$777$	8.16631e12	−	7.21378e11i	0.803769	−	0.0710016i
$778$		0			0
$779$	−6.46241e11	−	1.11932e12i	−0.0628747	−	0.108902i
$780$		0			0
$781$	6.02194e11	−	1.04303e12i	0.0579171	−	0.100315i
$782$		0			0
$783$	5.02716e13			4.77964
$784$		0			0
$785$	1.31021e12			0.123148
$786$		0			0
$787$	−2.09412e12	+	3.62713e12i	−0.194588	+	0.337036i	−0.946765	−	0.321925i	$-0.895670\pi$
$787$	−2.09412e12	+	3.62713e12i	−0.194588	+	0.337036i	0.752178	+	0.658961i	$0.229004\pi$
$788$		0			0
$789$	3.03942e12	+	5.26442e12i	0.279218	+	0.483620i
$790$		0			0
$791$	−7.42118e12	+	6.55556e11i	−0.674029	+	0.0595409i
$792$		0			0
$793$	−2.96219e10	−	5.13067e10i	−0.00266001	−	0.00460728i
$794$		0			0
$795$	−1.67446e13	+	2.90025e13i	−1.48670	+	2.57504i
$796$		0			0
$797$	1.78707e13			1.56884			0.784421	−	0.620229i	$-0.212960\pi$
$797$	1.78707e13			1.56884			0.784421	+	0.620229i	$0.212960\pi$
$798$		0			0
$799$	1.39711e12			0.121274
$800$		0			0
$801$	−1.47214e13	+	2.54982e13i	−1.26358	+	2.18858i
$802$		0			0
$803$	−1.11674e12	−	1.93424e12i	−0.0947830	−	0.164169i
$804$		0			0
$805$	−7.73173e12	+	1.66172e13i	−0.648926	+	1.39469i
$806$		0			0
$807$	3.18504e11	+	5.51665e11i	0.0264353	+	0.0457872i
$808$		0			0
$809$	1.59521e12	−	2.76299e12i	0.130934	−	0.226783i	−0.793103	−	0.609087i	$-0.791536\pi$
$809$	1.59521e12	−	2.76299e12i	0.130934	−	0.226783i	0.924037	+	0.382304i	$0.124869\pi$
$810$		0			0
$811$	1.37637e12			0.111722			0.0558612	−	0.998439i	$-0.482210\pi$
$811$	1.37637e12			0.111722			0.0558612	+	0.998439i	$0.482210\pi$
$812$		0			0
$813$	−4.04619e13			−3.24817
$814$		0			0
$815$	−4.25904e12	+	7.37688e12i	−0.338145	+	0.585684i
$816$		0			0
$817$	−3.60966e12	−	6.25211e12i	−0.283444	−	0.490939i
$818$		0			0
$819$	−1.55980e12	−	2.22365e12i	−0.121141	−	0.172698i
$820$		0			0
$821$	−1.05084e13	−	1.82011e13i	−0.807223	−	1.39815i	−0.914780	−	0.403952i	$-0.867636\pi$
$821$	−1.05084e13	−	1.82011e13i	−0.807223	−	1.39815i	0.107557	−	0.994199i	$-0.465697\pi$
$822$		0			0
$823$	−6.80472e12	+	1.17861e13i	−0.517024	+	0.895512i	0.482781	+	0.875741i	$0.339627\pi$
$823$	−6.80472e12	+	1.17861e13i	−0.517024	+	0.895512i	−0.999805	+	0.0197705i	$0.993706\pi$
$824$		0			0
$825$	2.28998e11			0.0172103
$826$		0			0
$827$	−1.02944e13			−0.765288			−0.382644	−	0.923896i	$-0.624986\pi$
$827$	−1.02944e13			−0.765288			−0.382644	+	0.923896i	$0.624986\pi$
$828$		0			0
$829$	−1.29564e13	+	2.24411e13i	−0.952769	+	1.65024i	−0.213377	+	0.976970i	$0.568446\pi$
$829$	−1.29564e13	+	2.24411e13i	−0.952769	+	1.65024i	−0.739392	+	0.673275i	$0.764887\pi$
$830$		0			0
$831$	−1.95530e13	−	3.38667e13i	−1.42236	−	2.46359i
$832$		0			0
$833$	4.45126e12	−	1.22940e13i	0.320318	−	0.884691i
$834$		0			0
$835$	−2.28359e12	−	3.95529e12i	−0.162565	−	0.281572i
$836$		0			0
$837$	9.10354e12	−	1.57678e13i	0.641130	−	1.11047i
$838$		0			0
$839$	5.20978e12			0.362986			0.181493	−	0.983392i	$-0.441907\pi$
$839$	5.20978e12			0.362986			0.181493	+	0.983392i	$0.441907\pi$
$840$		0			0
$841$	1.53407e13			1.05746
$842$		0			0
$843$	−9.53692e12	+	1.65184e13i	−0.650405	+	1.12653i
$844$		0			0
$845$	7.21832e12	+	1.25025e13i	0.487059	+	0.843610i
$846$		0			0
$847$	−8.16641e12	−	1.16420e13i	−0.545200	−	0.777235i
$848$		0			0
$849$	−2.76991e13	−	4.79763e13i	−1.82971	−	3.16915i
$850$		0			0
$851$	5.01937e12	−	8.69381e12i	0.328070	−	0.568234i
$852$		0			0
$853$	−2.58714e12			−0.167321			−0.0836604	−	0.996494i	$-0.526661\pi$
$853$	−2.58714e12			−0.167321			−0.0836604	+	0.996494i	$0.526661\pi$
$854$		0			0
$855$	1.70410e13			1.09056
$856$		0			0
$857$	9.56688e12	−	1.65703e13i	0.605838	−	1.04934i	−0.386080	−	0.922465i	$-0.626171\pi$
$857$	9.56688e12	−	1.65703e13i	0.605838	−	1.04934i	0.991918	−	0.126877i	$-0.0404954\pi$
$858$		0			0
$859$	1.33238e13	+	2.30774e13i	0.834945	+	1.44617i	0.894075	+	0.447917i	$0.147834\pi$
$859$	1.33238e13	+	2.30774e13i	0.834945	+	1.44617i	−0.0591305	+	0.998250i	$0.518833\pi$
$860$		0			0
$861$	−4.04530e12	+	8.69424e12i	−0.250863	+	0.539160i
$862$		0			0
$863$	−1.24167e13	−	2.15063e13i	−0.762003	−	1.31983i	−0.941817	−	0.336127i	$-0.890883\pi$
$863$	−1.24167e13	−	2.15063e13i	−0.762003	−	1.31983i	0.179814	−	0.983701i	$-0.442450\pi$
$864$		0			0
$865$	−7.73820e12	+	1.34030e13i	−0.469967	+	0.814007i
$866$		0			0
$867$	3.68431e12			0.221447
$868$		0			0
$869$	−9.69912e11			−0.0576957
$870$		0			0
$871$	−4.21471e11	+	7.30008e11i	−0.0248134	+	0.0429780i
$872$		0			0
$873$	3.13230e13	+	5.42531e13i	1.82516	+	3.16126i
$874$		0			0
$875$	1.75973e13	−	1.55447e12i	1.01487	−	0.0896494i
$876$		0			0
$877$	−5.91911e12	−	1.02522e13i	−0.337876	−	0.585219i	0.646157	−	0.763205i	$-0.276375\pi$
$877$	−5.91911e12	−	1.02522e13i	−0.337876	−	0.585219i	−0.984033	+	0.177986i	$0.943042\pi$
$878$		0			0
$879$	1.46626e13	−	2.53964e13i	0.828440	−	1.43490i
$880$		0			0
$881$	2.57086e13			1.43776			0.718882	−	0.695133i	$-0.244654\pi$
$881$	2.57086e13			1.43776			0.718882	+	0.695133i	$0.244654\pi$
$882$		0			0
$883$	1.32530e13			0.733656			0.366828	−	0.930289i	$-0.380444\pi$
$883$	1.32530e13			0.733656			0.366828	+	0.930289i	$0.380444\pi$
$884$		0			0
$885$	1.35216e13	−	2.34201e13i	0.740940	−	1.28335i
$886$		0			0
$887$	−1.57245e11	−	2.72355e11i	−0.00852941	−	0.0147734i	0.861729	−	0.507369i	$-0.169382\pi$
$887$	−1.57245e11	−	2.72355e11i	−0.00852941	−	0.0147734i	−0.870259	+	0.492595i	$0.836048\pi$
$888$		0			0
$889$	−1.80901e13	+	1.59801e12i	−0.971367	+	0.0858066i
$890$		0			0
$891$	−7.84257e12	−	1.35837e13i	−0.416878	−	0.722053i
$892$		0			0
$893$	4.99918e11	−	8.65883e11i	0.0263067	−	0.0455646i
$894$		0			0
$895$	−2.54074e12			−0.132360
$896$		0			0
$897$	−4.54601e12			−0.234458
$898$		0			0
$899$	5.40506e12	−	9.36183e12i	0.275983	−	0.478016i
$900$		0			0
$901$	−1.46276e13	−	2.53357e13i	−0.739453	−	1.28077i
$902$		0			0
$903$	−2.25955e13	+	4.85627e13i	−1.13091	+	2.43057i
$904$		0			0
$905$	−1.12520e12	−	1.94890e12i	−0.0557583	−	0.0965763i
$906$		0			0
$907$	1.84071e12	−	3.18820e12i	0.0903133	−	0.156427i	−0.817330	−	0.576170i	$-0.804546\pi$
$907$	1.84071e12	−	3.18820e12i	0.0903133	−	0.156427i	0.907643	+	0.419743i	$0.137880\pi$
$908$		0			0
$909$	7.45798e13			3.62313
$910$		0			0
$911$	2.58776e13			1.24478			0.622388	−	0.782709i	$-0.286163\pi$
$911$	2.58776e13			1.24478			0.622388	+	0.782709i	$0.286163\pi$
$912$		0			0
$913$	1.32025e12	−	2.28674e12i	0.0628835	−	0.108917i
$914$		0			0
$915$	−1.37884e12	−	2.38822e12i	−0.0650307	−	0.112636i
$916$		0			0
$917$	1.88638e13	+	2.68921e13i	0.880980	+	1.25592i
$918$		0			0
$919$	−6.11222e12	−	1.05867e13i	−0.282670	−	0.489598i	0.689372	−	0.724408i	$-0.257887\pi$
$919$	−6.11222e12	−	1.05867e13i	−0.282670	−	0.489598i	−0.972041	+	0.234809i	$0.924553\pi$
$920$		0			0
$921$	1.59310e13	−	2.75933e13i	0.729584	−	1.26368i
$922$		0			0
$923$	8.78488e11			0.0398408
$924$		0			0
$925$	−3.68849e11			−0.0165657
$926$		0			0
$927$	2.51883e13	−	4.36274e13i	1.12031	−	1.94044i
$928$		0			0
$929$	−1.39115e13	−	2.40955e13i	−0.612780	−	1.06137i	−0.990770	−	0.135556i	$-0.956718\pi$
$929$	−1.39115e13	−	2.40955e13i	−0.612780	−	1.06137i	0.377990	−	0.925810i	$-0.376615\pi$
$930$		0			0
$931$	−6.02668e12	−	7.15784e12i	−0.262908	−	0.312254i
$932$		0			0
$933$	2.40047e12	+	4.15774e12i	0.103712	+	0.179634i
$934$		0			0
$935$	2.42387e12	−	4.19827e12i	0.103719	−	0.179646i
$936$		0			0
$937$	−3.45042e13			−1.46232			−0.731162	−	0.682204i	$-0.761022\pi$
$937$	−3.45042e13			−1.46232			−0.731162	+	0.682204i	$0.761022\pi$
$938$		0			0
$939$	−2.88793e13			−1.21225
$940$		0			0
$941$	2.09282e13	−	3.62487e13i	0.870119	−	1.50709i	0.00824588	−	0.999966i	$-0.497375\pi$
$941$	2.09282e13	−	3.62487e13i	0.870119	−	1.50709i	0.861873	−	0.507124i	$-0.169291\pi$
$942$		0			0
$943$	5.87113e12	+	1.01691e13i	0.241779	+	0.418774i
$944$		0			0
$945$	−4.59731e13	−	6.55391e13i	−1.87526	−	2.67336i
$946$		0			0
$947$	−8.29779e12	−	1.43722e13i	−0.335265	−	0.580695i	0.648271	−	0.761410i	$-0.275492\pi$
$947$	−8.29779e12	−	1.43722e13i	−0.335265	−	0.580695i	−0.983536	+	0.180714i	$0.942159\pi$
$948$		0			0
$949$	8.14553e11	−	1.41085e12i	0.0326003	−	0.0564653i
$950$		0			0
$951$	8.96686e12			0.355491
$952$		0			0
$953$	1.49663e13			0.587756			0.293878	−	0.955843i	$-0.405054\pi$
$953$	1.49663e13			0.587756			0.293878	+	0.955843i	$0.405054\pi$
$954$		0			0
$955$	7.78512e12	−	1.34842e13i	0.302866	−	0.524579i
$956$		0			0
$957$	−8.08167e12	−	1.39979e13i	−0.311457	−	0.539459i
$958$		0			0
$959$	−4.54626e12	+	9.77093e12i	−0.173568	+	0.373037i
$960$		0			0
$961$	1.12622e13	+	1.95068e13i	0.425961	+	0.737786i
$962$		0			0
$963$	6.95672e12	−	1.20494e13i	0.260667	−	0.451489i
$964$		0			0
$965$	−5.21837e13			−1.93714
$966$		0			0
$967$	−2.21214e13			−0.813568			−0.406784	−	0.913524i	$-0.633350\pi$
$967$	−2.21214e13			−0.813568			−0.406784	+	0.913524i	$0.633350\pi$
$968$		0			0
$969$	−1.01735e13	+	1.76210e13i	−0.370692	+	0.642057i
$970$		0			0
$971$	−5.86828e12	−	1.01642e13i	−0.211848	−	0.366931i	0.740445	−	0.672117i	$-0.234615\pi$
$971$	−5.86828e12	−	1.01642e13i	−0.211848	−	0.366931i	−0.952293	+	0.305186i	$0.901281\pi$
$972$		0			0
$973$	3.16969e13	−	2.79997e12i	1.13373	−	0.100149i
$974$		0			0
$975$	8.35161e10	+	1.44654e11i	0.00295971	+	0.00512637i
$976$		0			0
$977$	−2.79376e13	+	4.83894e13i	−0.980989	+	1.69912i	−0.322429	+	0.946594i	$0.604499\pi$
$977$	−2.79376e13	+	4.83894e13i	−0.980989	+	1.69912i	−0.658560	+	0.752528i	$0.728834\pi$
$978$		0			0
$979$	5.99408e12			0.208545
$980$		0			0
$981$	−4.43118e12			−0.152760
$982$		0			0
$983$	−1.93378e13	+	3.34941e13i	−0.660567	+	1.14414i	0.319899	+	0.947452i	$0.396351\pi$
$983$	−1.93378e13	+	3.34941e13i	−0.660567	+	1.14414i	−0.980467	+	0.196685i	$0.936982\pi$
$984$		0			0
$985$	1.17337e13	+	2.03234e13i	0.397167	+	0.687914i
$986$		0			0
$987$	−7.38928e12	+	6.52738e11i	−0.247842	+	0.0218933i
$988$		0			0
$989$	3.27939e13	+	5.68007e13i	1.08996	+	1.88786i
$990$		0			0
$991$	−1.24956e13	+	2.16430e13i	−0.411553	+	0.712831i	−0.995060	−	0.0992775i	$-0.968347\pi$
$991$	−1.24956e13	+	2.16430e13i	−0.411553	+	0.712831i	0.583507	+	0.812108i	$0.301680\pi$
$992$		0			0
$993$	−5.95845e13			−1.94474
$994$		0			0
$995$	3.15247e13			1.01964
$996$		0			0
$997$	−6.76021e12	+	1.17090e13i	−0.216687	+	0.375312i	−0.953793	−	0.300464i	$-0.902858\pi$
$997$	−6.76021e12	+	1.17090e13i	−0.216687	+	0.375312i	0.737106	+	0.675777i	$0.236192\pi$
$998$		0			0
$999$	2.19245e13	+	3.79743e13i	0.696441	+	1.20627i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.i.b.81.3		6
4.3	odd	2		14.10.c.a.11.1	yes	6
7.2	even	3	inner	112.10.i.b.65.3		6
12.11	even	2		126.10.g.f.109.3		6
28.3	even	6		98.10.a.i.1.1		3
28.11	odd	6		98.10.a.j.1.3		3
28.19	even	6		98.10.c.k.79.3		6
28.23	odd	6		14.10.c.a.9.1	✓	6
28.27	even	2		98.10.c.k.67.3		6
84.23	even	6		126.10.g.f.37.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.10.c.a.9.1	✓	6	28.23	odd	6
14.10.c.a.11.1	yes	6	4.3	odd	2
98.10.a.i.1.1		3	28.3	even	6
98.10.a.j.1.3		3	28.11	odd	6
98.10.c.k.67.3		6	28.27	even	2
98.10.c.k.79.3		6	28.19	even	6
112.10.i.b.65.3		6	7.2	even	3	inner
112.10.i.b.81.3		6	1.1	even	1	trivial
126.10.g.f.37.3		6	84.23	even	6
126.10.g.f.109.3		6	12.11	even	2

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(135.410 - 234.536i) q^{3} +(-684.785 - 1186.08i) q^{5} +(-6327.81 + 558.972i) q^{7} +(-26830.1 - 46471.0i) q^{9} +(-5462.17 + 9460.75i) q^{11} -7968.27 q^{13} -370906. q^{15} +(162006. - 280602. i) q^{17} +(-115939. - 200812. i) q^{19} +(-725747. + 1.55979e6i) q^{21} +(1.05331e6 + 1.82439e6i) q^{23} +(38701.4 - 67032.7i) q^{25} -9.20167e6 q^{27} -5.46332e6 q^{29} +(-989336. + 1.71358e6i) q^{31} +(1.47926e6 + 2.56215e6i) q^{33} +(4.99618e6 + 7.12253e6i) q^{35} +(-2.38266e6 - 4.12689e6i) q^{37} +(-1.07898e6 + 1.86885e6i) q^{39} +5.57398e6 q^{41} +3.11341e7 q^{43} +(-3.67457e7 + 6.36454e7i) q^{45} +(2.15595e6 + 3.73422e6i) q^{47} +(3.97287e7 - 7.07414e6i) q^{49} +(-4.38743e7 - 7.59926e7i) q^{51} +(4.51452e7 - 7.81937e7i) q^{53} +1.49616e7 q^{55} -6.27970e7 q^{57} +(-3.64555e7 + 6.31429e7i) q^{59} +(3.71748e6 + 6.43887e6i) q^{61} +(1.95752e8 + 2.79063e8i) q^{63} +(5.45655e6 + 9.45102e6i) q^{65} +(5.28936e7 - 9.16144e7i) q^{67} +5.70514e8 q^{69} -1.10248e8 q^{71} +(-1.02225e8 + 1.77058e8i) q^{73} +(-1.04811e7 - 1.81538e7i) q^{75} +(2.92752e7 - 6.29190e7i) q^{77} +(4.43923e7 + 7.68897e7i) q^{79} +(-7.17899e8 + 1.24344e9i) q^{81} -2.41708e8 q^{83} -4.43757e8 q^{85} +(-7.39786e8 + 1.28135e9i) q^{87} +(-2.74345e8 - 4.75180e8i) q^{89} +(5.04217e7 - 4.45404e6i) q^{91} +(2.67931e8 + 4.64071e8i) q^{93} +(-1.58787e8 + 2.75026e8i) q^{95} -1.16746e9 q^{97} +5.86201e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 233 q^{3} - 733 q^{5} - 5012 q^{7} - 15058 q^{9} - 7339 q^{11} + 197036 q^{13} - 738238 q^{15} - 306665 q^{17} + 377991 q^{19} - 1585927 q^{21} + 2267255 q^{23} - 142612 q^{25} - 21348358 q^{27} - 13085956 q^{29}+ \cdots + 1256218868 q^{99}+O(q^{100}) \) 6 * q + 233 * q^3 - 733 * q^5 - 5012 * q^7 - 15058 * q^9 - 7339 * q^11 + 197036 * q^13 - 738238 * q^15 - 306665 * q^17 + 377991 * q^19 - 1585927 * q^21 + 2267255 * q^23 - 142612 * q^25 - 21348358 * q^27 - 13085956 * q^29 + 6654517 * q^31 + 3747977 * q^33 - 22864807 * q^35 - 22287969 * q^37 - 3151974 * q^39 + 68096196 * q^41 + 125648280 * q^43 - 86300638 * q^45 + 52703019 * q^47 + 57596070 * q^49 - 97736333 * q^51 - 12091125 * q^53 - 377435398 * q^55 - 158517670 * q^57 + 12949897 * q^59 - 160252153 * q^61 + 356021666 * q^63 + 334191270 * q^65 + 480890225 * q^67 + 1131233162 * q^69 + 74421440 * q^71 + 251382283 * q^73 - 16322324 * q^75 - 527045155 * q^77 - 286494785 * q^79 - 2165894587 * q^81 - 2295182344 * q^83 - 2389185074 * q^85 - 1412199358 * q^87 - 901243845 * q^89 - 1026798920 * q^91 + 710456889 * q^93 + 887366177 * q^95 + 629707876 * q^97 + 1256218868 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more