LMFDB - Embedded newform 117.5.j.b.73.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,5,Mod(73,117)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("117.73");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	117.j (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.0942856808$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 $ x^20 + 5446x^16 - 1452x^15 + 106320x^13 + 8376897x^12 - 1643220x^11 + 1054152x^10 + 232107192x^9 + 3337033804x^8 + 3105930144x^7 + 2297014848x^6 + 21034904256x^5 + 158658808624x^4 + 235144077504x^3 + 177846480000x^2 + 27353289600*x + 2103506496
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			73.8
Root			$4.05053 + 4.05053i$ of defining polynomial
Character	$\chi$	$=$	117.73
Dual form			117.5.j.b.109.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.05053 + 4.05053i) q^{2} +16.8136i q^{4} +(-33.5573 - 33.5573i) q^{5} +(23.7132 - 23.7132i) q^{7} +(-3.29565 + 3.29565i) q^{8} -271.850i q^{10} +(-32.2545 + 32.2545i) q^{11} +(72.8001 - 152.516i) q^{13} +192.102 q^{14} +242.320 q^{16} -354.222i q^{17} +(-168.478 - 168.478i) q^{19} +(564.221 - 564.221i) q^{20} -261.296 q^{22} +182.161i q^{23} +1627.19i q^{25} +(912.650 - 322.892i) q^{26} +(398.704 + 398.704i) q^{28} -913.769 q^{29} +(-876.095 - 876.095i) q^{31} +(1034.25 + 1034.25i) q^{32} +(1434.79 - 1434.79i) q^{34} -1591.50 q^{35} +(112.687 - 112.687i) q^{37} -1364.85i q^{38} +221.186 q^{40} +(1133.48 + 1133.48i) q^{41} +411.247i q^{43} +(-542.315 - 542.315i) q^{44} +(-737.848 + 737.848i) q^{46} +(3000.97 - 3000.97i) q^{47} +1276.37i q^{49} +(-6590.99 + 6590.99i) q^{50} +(2564.35 + 1224.03i) q^{52} +4228.78 q^{53} +2164.75 q^{55} +156.301i q^{56} +(-3701.25 - 3701.25i) q^{58} +(-1690.19 + 1690.19i) q^{59} -3760.65 q^{61} -7097.31i q^{62} +4501.45i q^{64} +(-7561.01 + 2675.06i) q^{65} +(-2631.54 - 2631.54i) q^{67} +5955.76 q^{68} +(-6446.43 - 6446.43i) q^{70} +(-297.661 - 297.661i) q^{71} +(2616.29 - 2616.29i) q^{73} +912.887 q^{74} +(2832.73 - 2832.73i) q^{76} +1529.71i q^{77} +7885.52 q^{79} +(-8131.61 - 8131.61i) q^{80} +9182.36i q^{82} +(4078.45 + 4078.45i) q^{83} +(-11886.7 + 11886.7i) q^{85} +(-1665.77 + 1665.77i) q^{86} -212.599i q^{88} +(-1722.86 + 1722.86i) q^{89} +(-1890.32 - 5342.96i) q^{91} -3062.78 q^{92} +24311.0 q^{94} +11307.4i q^{95} +(1560.33 + 1560.33i) q^{97} +(-5169.99 + 5169.99i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) $ 20 * q - 24 * q^5 - 24 * q^7 - 372 * q^11 - 224 * q^13 - 480 * q^14 - 2328 * q^16 - 840 * q^19 - 228 * q^20 + 3536 * q^22 + 828 * q^26 - 1984 * q^28 + 5064 * q^29 + 1712 * q^31 + 7260 * q^32 + 8040 * q^34 - 5976 * q^35 - 868 * q^37 + 6336 * q^40 + 6000 * q^41 - 20868 * q^44 - 1992 * q^46 - 1692 * q^47 - 8952 * q^50 - 12416 * q^52 + 18816 * q^53 - 3280 * q^55 + 1256 * q^58 + 6588 * q^59 - 5512 * q^61 - 33048 * q^65 - 4704 * q^67 + 28296 * q^68 - 31096 * q^70 - 9756 * q^71 + 13692 * q^73 + 28296 * q^74 - 17072 * q^76 - 5056 * q^79 - 42228 * q^80 - 24708 * q^83 - 4752 * q^85 + 77424 * q^86 - 36000 * q^89 + 19104 * q^91 - 12912 * q^92 + 112832 * q^94 + 50012 * q^97 - 11544 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$

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 $

$2$	4.05053	+	4.05053i	1.01263	+	1.01263i	0.999919	+	0.0127141i	$0.00404712\pi$
$2$	4.05053	+	4.05053i	1.01263	+	1.01263i	0.0127141	+	0.999919i	$0.495953\pi$
$3$		0			0
$3$		0			0
$4$			16.8136i			1.05085i
$5$	−33.5573	−	33.5573i	−1.34229	−	1.34229i	−0.893772	−	0.448521i	$-0.851951\pi$
$5$	−33.5573	−	33.5573i	−1.34229	−	1.34229i	−0.448521	−	0.893772i	$-0.648049\pi$
$6$		0			0
$7$	23.7132	−	23.7132i	0.483942	−	0.483942i	−0.422446	−	0.906388i	$-0.638828\pi$
$7$	23.7132	−	23.7132i	0.483942	−	0.483942i	0.906388	+	0.422446i	$0.138828\pi$
$8$	−3.29565	+	3.29565i	−0.0514945	+	0.0514945i
$9$		0			0
$10$		−	271.850i		−	2.71850i
$11$	−32.2545	+	32.2545i	−0.266566	+	0.266566i	−0.827715	−	0.561149i	$-0.810359\pi$
$11$	−32.2545	+	32.2545i	−0.266566	+	0.266566i	0.561149	+	0.827715i	$0.310359\pi$
$12$		0			0
$13$	72.8001	−	152.516i	0.430770	−	0.902462i
$13$	72.8001	−	152.516i	0.430770	−	0.902462i
$14$	192.102			0.980111
$15$		0			0
$16$	242.320			0.946562
$17$		−	354.222i		−	1.22568i	−0.790206	−	0.612841i	$-0.790027\pi$
$17$		−	354.222i		−	1.22568i	0.790206	−	0.612841i	$-0.209973\pi$
$18$		0			0
$19$	−168.478	−	168.478i	−0.466699	−	0.466699i	0.434145	−	0.900843i	$-0.357051\pi$
$19$	−168.478	−	168.478i	−0.466699	−	0.466699i	−0.900843	+	0.434145i	$0.857051\pi$
$20$	564.221	−	564.221i	1.41055	−	1.41055i
$21$		0			0
$22$	−261.296			−0.539867
$23$			182.161i			0.344349i	0.985066	+	0.172175i	$0.0550793\pi$
$23$			182.161i			0.344349i	−0.985066	+	0.172175i	$0.944921\pi$
$24$		0			0
$25$			1627.19i			2.60350i
$26$	912.650	−	322.892i	1.35007	−	0.477651i
$27$		0			0
$28$	398.704	+	398.704i	0.508551	+	0.508551i
$29$	−913.769			−1.08653			−0.543264	−	0.839562i	$-0.682812\pi$
$29$	−913.769			−1.08653			−0.543264	+	0.839562i	$0.682812\pi$
$30$		0			0
$31$	−876.095	−	876.095i	−0.911650	−	0.911650i	0.0847522	−	0.996402i	$-0.472990\pi$
$31$	−876.095	−	876.095i	−0.911650	−	0.911650i	−0.996402	+	0.0847522i	$0.972990\pi$
$32$	1034.25	+	1034.25i	1.01001	+	1.01001i
$33$		0			0
$34$	1434.79	−	1434.79i	1.24117	−	1.24117i
$35$	−1591.50			−1.29918
$36$		0			0
$37$	112.687	−	112.687i	0.0823135	−	0.0823135i	−0.664751	−	0.747065i	$-0.731462\pi$
$37$	112.687	−	112.687i	0.0823135	−	0.0823135i	0.747065	+	0.664751i	$0.231462\pi$
$38$		−	1364.85i		−	0.945189i
$39$		0			0
$40$	221.186			0.138242
$41$	1133.48	+	1133.48i	0.674287	+	0.674287i	0.958701	−	0.284415i	$-0.0917993\pi$
$41$	1133.48	+	1133.48i	0.674287	+	0.674287i	−0.284415	+	0.958701i	$0.591799\pi$
$42$		0			0
$43$			411.247i			0.222416i	0.993797	+	0.111208i	$0.0354720\pi$
$43$			411.247i			0.222416i	−0.993797	+	0.111208i	$0.964528\pi$
$44$	−542.315	−	542.315i	−0.280121	−	0.280121i
$45$		0			0
$46$	−737.848	+	737.848i	−0.348700	+	0.348700i
$47$	3000.97	−	3000.97i	1.35852	−	1.35852i	0.482772	−	0.875746i	$-0.339630\pi$
$47$	3000.97	−	3000.97i	1.35852	−	1.35852i	0.875746	−	0.482772i	$-0.160370\pi$
$48$		0			0
$49$			1276.37i			0.531600i
$50$	−6590.99	+	6590.99i	−2.63639	+	2.63639i
$51$		0			0
$52$	2564.35	+	1224.03i	0.948354	+	0.452675i
$53$	4228.78			1.50544			0.752720	−	0.658341i	$-0.228741\pi$
$53$	4228.78			1.50544			0.752720	+	0.658341i	$0.228741\pi$
$54$		0			0
$55$	2164.75			0.715619
$56$			156.301i			0.0498407i
$57$		0			0
$58$	−3701.25	−	3701.25i	−1.10025	−	1.10025i
$59$	−1690.19	+	1690.19i	−0.485547	+	0.485547i	−0.906898	−	0.421350i	$-0.861556\pi$
$59$	−1690.19	+	1690.19i	−0.485547	+	0.485547i	0.421350	+	0.906898i	$0.361556\pi$
$60$		0			0
$61$	−3760.65			−1.01066			−0.505328	−	0.862928i	$-0.668628\pi$
$61$	−3760.65			−1.01066			−0.505328	+	0.862928i	$0.668628\pi$
$62$		−	7097.31i		−	1.84633i
$63$		0			0
$64$			4501.45i			1.09899i
$65$	−7561.01	+	2675.06i	−1.78959	+	0.633149i
$66$		0			0
$67$	−2631.54	−	2631.54i	−0.586219	−	0.586219i	0.350386	−	0.936605i	$-0.386050\pi$
$67$	−2631.54	−	2631.54i	−0.586219	−	0.586219i	−0.936605	+	0.350386i	$0.886050\pi$
$68$	5955.76			1.28801
$69$		0			0
$70$	−6446.43	−	6446.43i	−1.31560	−	1.31560i
$71$	−297.661	−	297.661i	−0.0590479	−	0.0590479i	0.676966	−	0.736014i	$-0.263294\pi$
$71$	−297.661	−	297.661i	−0.0590479	−	0.0590479i	−0.736014	+	0.676966i	$0.763294\pi$
$72$		0			0
$73$	2616.29	−	2616.29i	0.490953	−	0.490953i	−0.417653	−	0.908606i	$-0.637147\pi$
$73$	2616.29	−	2616.29i	0.490953	−	0.490953i	0.908606	+	0.417653i	$0.137147\pi$
$74$	912.887			0.166707
$75$		0			0
$76$	2832.73	−	2832.73i	0.490431	−	0.490431i
$77$			1529.71i			0.258005i
$78$		0			0
$79$	7885.52			1.26350			0.631752	−	0.775171i	$-0.282336\pi$
$79$	7885.52			1.26350			0.631752	+	0.775171i	$0.282336\pi$
$80$	−8131.61	−	8131.61i	−1.27056	−	1.27056i
$81$		0			0
$82$			9182.36i			1.36561i
$83$	4078.45	+	4078.45i	0.592024	+	0.592024i	0.938178	−	0.346154i	$-0.112512\pi$
$83$	4078.45	+	4078.45i	0.592024	+	0.592024i	−0.346154	+	0.938178i	$0.612512\pi$
$84$		0			0
$85$	−11886.7	+	11886.7i	−1.64522	+	1.64522i
$86$	−1665.77	+	1665.77i	−0.225226	+	0.225226i
$87$		0			0
$88$		−	212.599i		−	0.0274534i
$89$	−1722.86	+	1722.86i	−0.217505	+	0.217505i	−0.807446	−	0.589941i	$-0.799151\pi$
$89$	−1722.86	+	1722.86i	−0.217505	+	0.217505i	0.589941	+	0.807446i	$0.299151\pi$
$90$		0			0
$91$	−1890.32	−	5342.96i	−0.228272	−	0.645207i
$92$	−3062.78			−0.361860
$93$		0			0
$94$	24311.0			2.75136
$95$			11307.4i			1.25289i
$96$		0			0
$97$	1560.33	+	1560.33i	0.165834	+	0.165834i	0.785146	−	0.619311i	$-0.212588\pi$
$97$	1560.33	+	1560.33i	0.165834	+	0.165834i	−0.619311	+	0.785146i	$0.712588\pi$
$98$	−5169.99	+	5169.99i	−0.538316	+	0.538316i
$99$		0			0
$100$	−27359.0			−2.73590
$101$			9982.36i			0.978567i	0.872125	+	0.489283i	$0.162742\pi$
$101$			9982.36i			0.978567i	−0.872125	+	0.489283i	$0.837258\pi$
$102$		0			0
$103$			6.56148i			0.000618483i	1.00000		0.000309241i	$9.84346e-5\pi$
$103$			6.56148i			0.000618483i	−1.00000		0.000309241i	$0.999902\pi$
$104$	262.716	+	742.563i	0.0242896	+	0.0686541i
$105$		0			0
$106$	17128.8	+	17128.8i	1.52446	+	1.52446i
$107$	6183.10			0.540056			0.270028	−	0.962853i	$-0.412967\pi$
$107$	6183.10			0.540056			0.270028	+	0.962853i	$0.412967\pi$
$108$		0			0
$109$	3694.46	+	3694.46i	0.310956	+	0.310956i	0.845280	−	0.534324i	$-0.179434\pi$
$109$	3694.46	+	3694.46i	0.310956	+	0.310956i	−0.534324	+	0.845280i	$0.679434\pi$
$110$	8768.39	+	8768.39i	0.724660	+	0.724660i
$111$		0			0
$112$	5746.17	−	5746.17i	0.458081	−	0.458081i
$113$	21267.9			1.66559			0.832794	−	0.553583i	$-0.186740\pi$
$113$	21267.9			1.66559			0.832794	+	0.553583i	$0.186740\pi$
$114$		0			0
$115$	6112.83	−	6112.83i	0.462218	−	0.462218i
$116$		−	15363.8i		−	1.14178i
$117$		0			0
$118$	−13692.3			−0.983363
$119$	−8399.72	−	8399.72i	−0.593159	−	0.593159i
$120$		0			0
$121$			12560.3i			0.857885i
$122$	−15232.6	−	15232.6i	−1.02342	−	1.02342i
$123$		0			0
$124$	14730.3	−	14730.3i	0.958009	−	0.958009i
$125$	33630.8	−	33630.8i	2.15237	−	2.15237i
$126$		0			0
$127$		−	26466.7i		−	1.64094i	−0.571693	−	0.820468i	$-0.693713\pi$
$127$		−	26466.7i		−	1.64094i	0.571693	−	0.820468i	$-0.306287\pi$
$128$	−1685.19	+	1685.19i	−0.102856	+	0.102856i
$129$		0			0
$130$	−41461.5	−	19790.7i	−2.45334	−	1.17105i
$131$	−21491.7			−1.25236			−0.626178	−	0.779680i	$-0.715382\pi$
$131$	−21491.7			−1.25236			−0.626178	+	0.779680i	$0.715382\pi$
$132$		0			0
$133$	−7990.30			−0.451710
$134$		−	21318.3i		−	1.18725i
$135$		0			0
$136$	1167.39	+	1167.39i	0.0631159	+	0.0631159i
$137$	5030.14	−	5030.14i	0.268002	−	0.268002i	−0.560292	−	0.828295i	$-0.689311\pi$
$137$	5030.14	−	5030.14i	0.268002	−	0.268002i	0.828295	+	0.560292i	$0.189311\pi$
$138$		0			0
$139$	12182.9			0.630554			0.315277	−	0.949000i	$-0.397903\pi$
$139$	12182.9			0.630554			0.315277	+	0.949000i	$0.397903\pi$
$140$		−	26758.9i		−	1.36525i
$141$		0			0
$142$		−	2411.37i		−	0.119588i
$143$	2571.20	+	7267.45i	0.125737	+	0.355394i
$144$		0			0
$145$	30663.7	+	30663.7i	1.45844	+	1.45844i
$146$	21194.7			0.994311
$147$		0			0
$148$	1894.68	+	1894.68i	0.0864994	+	0.0864994i
$149$	10032.3	+	10032.3i	0.451887	+	0.451887i	0.895980	−	0.444094i	$-0.146474\pi$
$149$	10032.3	+	10032.3i	0.451887	+	0.451887i	−0.444094	+	0.895980i	$0.646474\pi$
$150$		0			0
$151$	10057.5	−	10057.5i	0.441098	−	0.441098i	−0.451283	−	0.892381i	$-0.649033\pi$
$151$	10057.5	−	10057.5i	0.441098	−	0.441098i	0.892381	+	0.451283i	$0.149033\pi$
$152$	1110.49			0.0480649
$153$		0			0
$154$	−6196.14	+	6196.14i	−0.261264	+	0.261264i
$155$			58798.9i			2.44740i
$156$		0			0
$157$	8554.44			0.347050			0.173525	−	0.984829i	$-0.444484\pi$
$157$	8554.44			0.347050			0.173525	+	0.984829i	$0.444484\pi$
$158$	31940.6	+	31940.6i	1.27947	+	1.27947i
$159$		0			0
$160$		−	69413.7i		−	2.71147i
$161$	4319.61	+	4319.61i	0.166645	+	0.166645i
$162$		0			0
$163$	−1164.17	+	1164.17i	−0.0438170	+	0.0438170i	−0.728676	−	0.684859i	$-0.759864\pi$
$163$	−1164.17	+	1164.17i	−0.0438170	+	0.0438170i	0.684859	+	0.728676i	$0.259864\pi$
$164$	−19057.8	+	19057.8i	−0.708576	+	0.708576i
$165$		0			0
$166$			33039.8i			1.19901i
$167$	−7534.77	+	7534.77i	−0.270170	+	0.270170i	−0.829169	−	0.558999i	$-0.811186\pi$
$167$	−7534.77	+	7534.77i	−0.270170	+	0.270170i	0.558999	+	0.829169i	$0.311186\pi$
$168$		0			0
$169$	−17961.3	−	22206.4i	−0.628875	−	0.777506i
$170$	−96295.3			−3.33202
$171$		0			0
$172$	−6914.56			−0.233726
$173$		−	39158.0i		−	1.30836i	−0.756337	−	0.654182i	$-0.773013\pi$
$173$		−	39158.0i		−	1.30836i	0.756337	−	0.654182i	$-0.226987\pi$
$174$		0			0
$175$	38585.8	+	38585.8i	1.25994	+	1.25994i
$176$	−7815.90	+	7815.90i	−0.252321	+	0.252321i
$177$		0			0
$178$	−13957.0			−0.440505
$179$		−	37765.1i		−	1.17865i	−0.807897	−	0.589324i	$-0.799394\pi$
$179$		−	37765.1i		−	1.17865i	0.807897	−	0.589324i	$-0.200606\pi$
$180$		0			0
$181$			9590.57i			0.292744i	0.989230	+	0.146372i	$0.0467596\pi$
$181$			9590.57i			0.292744i	−0.989230	+	0.146372i	$0.953240\pi$
$182$	13985.0	−	29298.6i	0.422202	−	0.884513i
$183$		0			0
$184$	−600.338	−	600.338i	−0.0177321	−	0.0177321i
$185$	−7562.97			−0.220978
$186$		0			0
$187$	11425.2	+	11425.2i	0.326725	+	0.326725i
$188$	50457.2	+	50457.2i	1.42760	+	1.42760i
$189$		0			0
$190$	−45800.8	+	45800.8i	−1.26872	+	1.26872i
$191$	−46392.7			−1.27170			−0.635848	−	0.771814i	$-0.719349\pi$
$191$	−46392.7			−1.27170			−0.635848	+	0.771814i	$0.719349\pi$
$192$		0			0
$193$	9357.45	−	9357.45i	0.251213	−	0.251213i	−0.570255	−	0.821468i	$-0.693155\pi$
$193$	9357.45	−	9357.45i	0.251213	−	0.251213i	0.821468	+	0.570255i	$0.193155\pi$
$194$			12640.4i			0.335858i
$195$		0			0
$196$	−21460.5			−0.558633
$197$	4463.00	+	4463.00i	0.114999	+	0.114999i	0.762265	−	0.647266i	$-0.224088\pi$
$197$	4463.00	+	4463.00i	0.114999	+	0.114999i	−0.647266	+	0.762265i	$0.724088\pi$
$198$		0			0
$199$			71326.4i			1.80113i	0.434726	+	0.900563i	$0.356845\pi$
$199$			71326.4i			1.80113i	−0.434726	+	0.900563i	$0.643155\pi$
$200$	−5362.65	−	5362.65i	−0.134066	−	0.134066i
$201$		0			0
$202$	−40433.9	+	40433.9i	−0.990929	+	0.990929i
$203$	−21668.4	+	21668.4i	−0.525816	+	0.525816i
$204$		0			0
$205$		−	76072.9i		−	1.81018i
$206$	−26.5775	+	26.5775i	−0.000626296	+	0.000626296i
$207$		0			0
$208$	17640.9	−	36957.7i	0.407750	−	0.854236i
$209$	10868.4			0.248812
$210$		0			0
$211$	−45146.1			−1.01404			−0.507020	−	0.861934i	$-0.669253\pi$
$211$	−45146.1			−1.01404			−0.507020	+	0.861934i	$0.669253\pi$
$212$			71101.2i			1.58199i
$213$		0			0
$214$	25044.8	+	25044.8i	0.546878	+	0.546878i
$215$	13800.4	−	13800.4i	0.298548	−	0.298548i
$216$		0			0
$217$	−41550.0			−0.882371
$218$			29929.1i			0.629768i
$219$		0			0
$220$			36397.3i			0.752010i
$221$	−54024.5	−	25787.4i	−1.10613	−	0.527986i
$222$		0			0
$223$	22521.9	+	22521.9i	0.452892	+	0.452892i	0.896313	−	0.443421i	$-0.146235\pi$
$223$	22521.9	+	22521.9i	0.452892	+	0.452892i	−0.443421	+	0.896313i	$0.646235\pi$
$224$	49050.9			0.977577
$225$		0			0
$226$	86146.3	+	86146.3i	1.68663	+	1.68663i
$227$	−33678.9	−	33678.9i	−0.653592	−	0.653592i	0.300264	−	0.953856i	$-0.402925\pi$
$227$	−33678.9	−	33678.9i	−0.653592	−	0.653592i	−0.953856	+	0.300264i	$0.902925\pi$
$228$		0			0
$229$	55067.0	−	55067.0i	1.05008	−	1.05008i	0.0513970	−	0.998678i	$-0.483633\pi$
$229$	55067.0	−	55067.0i	1.05008	−	1.05008i	0.998678	−	0.0513970i	$-0.0163674\pi$
$230$	49520.4			0.936114
$231$		0			0
$232$	3011.46	−	3011.46i	0.0559502	−	0.0559502i
$233$			6668.81i			0.122839i	0.998112	+	0.0614195i	$0.0195627\pi$
$233$			6668.81i			0.122839i	−0.998112	+	0.0614195i	$0.980437\pi$
$234$		0			0
$235$	−201409.			−3.64706
$236$	−28418.2	−	28418.2i	−0.510239	−	0.510239i
$237$		0			0
$238$		−	68046.7i		−	1.20130i
$239$	24924.8	+	24924.8i	0.436350	+	0.436350i	0.890782	−	0.454431i	$-0.150158\pi$
$239$	24924.8	+	24924.8i	0.436350	+	0.436350i	−0.454431	+	0.890782i	$0.650158\pi$
$240$		0			0
$241$	4314.21	−	4314.21i	0.0742792	−	0.0742792i	−0.668991	−	0.743270i	$-0.733274\pi$
$241$	4314.21	−	4314.21i	0.0742792	−	0.0742792i	0.743270	+	0.668991i	$0.233274\pi$
$242$	−50875.9	+	50875.9i	−0.868723	+	0.868723i
$243$		0			0
$244$		−	63230.2i		−	1.06205i
$245$	42831.7	−	42831.7i	0.713564	−	0.713564i
$246$		0			0
$247$	−37960.9	+	13430.4i	−0.622217	+	0.220138i
$248$	5774.61			0.0938900
$249$		0			0
$250$	272445.			4.35913
$251$			100235.i			1.59100i	0.605954	+	0.795500i	$0.292791\pi$
$251$			100235.i			1.59100i	−0.605954	+	0.795500i	$0.707209\pi$
$252$		0			0
$253$	−5875.50	−	5875.50i	−0.0917918	−	0.0917918i
$254$	107204.	−	107204.i	1.66167	−	1.66167i
$255$		0			0
$256$	58371.4			0.890676
$257$		−	14954.5i		−	0.226415i	−0.993571	−	0.113208i	$-0.963887\pi$
$257$		−	14954.5i		−	0.226415i	0.993571	−	0.113208i	$-0.0361125\pi$
$258$		0			0
$259$		−	5344.34i		−	0.0796700i
$260$	−44977.4	−	127128.i	−0.665346	−	1.88059i
$261$		0			0
$262$	−87052.7	−	87052.7i	−1.26818	−	1.26818i
$263$	7574.07			0.109501			0.0547505	−	0.998500i	$-0.482564\pi$
$263$	7574.07			0.109501			0.0547505	+	0.998500i	$0.482564\pi$
$264$		0			0
$265$	−141907.	−	141907.i	−2.02074	−	2.02074i
$266$	−32365.0	−	32365.0i	−0.457417	−	0.457417i
$267$		0			0
$268$	44245.7	−	44245.7i	0.616030	−	0.616030i
$269$	−100457.			−1.38827			−0.694136	−	0.719844i	$-0.744213\pi$
$269$	−100457.			−1.38827			−0.694136	+	0.719844i	$0.744213\pi$
$270$		0			0
$271$	66620.4	−	66620.4i	0.907128	−	0.907128i	−0.0889115	−	0.996040i	$-0.528339\pi$
$271$	66620.4	−	66620.4i	0.907128	−	0.907128i	0.996040	+	0.0889115i	$0.0283388\pi$
$272$		−	85835.0i		−	1.16018i
$273$		0			0
$274$	40749.5			0.542776
$275$	−52484.2	−	52484.2i	−0.694005	−	0.694005i
$276$		0			0
$277$			70180.3i			0.914652i	0.889299	+	0.457326i	$0.151193\pi$
$277$			70180.3i			0.914652i	−0.889299	+	0.457326i	$0.848807\pi$
$278$	49347.3	+	49347.3i	0.638519	+	0.638519i
$279$		0			0
$280$	5245.03	−	5245.03i	0.0669009	−	0.0669009i
$281$	20515.7	−	20515.7i	0.259820	−	0.259820i	−0.565161	−	0.824981i	$-0.691186\pi$
$281$	20515.7	−	20515.7i	0.259820	−	0.259820i	0.824981	+	0.565161i	$0.191186\pi$
$282$		0			0
$283$		−	30170.2i		−	0.376708i	−0.982101	−	0.188354i	$-0.939685\pi$
$283$		−	30170.2i		−	0.376708i	0.982101	−	0.188354i	$-0.0603153\pi$
$284$	5004.76	−	5004.76i	0.0620507	−	0.0620507i
$285$		0			0
$286$	−19022.3	+	39851.8i	−0.232558	+	0.487209i
$287$	53756.6			0.652631
$288$		0			0
$289$	−41952.2			−0.502295
$290$			248408.i			2.95373i
$291$		0			0
$292$	43989.3	+	43989.3i	0.515919	+	0.515919i
$293$	67660.9	−	67660.9i	0.788139	−	0.788139i	−0.193050	−	0.981189i	$-0.561838\pi$
$293$	67660.9	−	67660.9i	0.788139	−	0.788139i	0.981189	+	0.193050i	$0.0618380\pi$
$294$		0			0
$295$	113437.			1.30349
$296$			742.755i			0.00847739i
$297$		0			0
$298$			81272.6i			0.915191i
$299$	27782.4	+	13261.3i	0.310762	+	0.148335i
$300$		0			0
$301$	9751.97	+	9751.97i	0.107636	+	0.107636i
$302$	81476.3			0.893341
$303$		0			0
$304$	−40825.6	−	40825.6i	−0.441759	−	0.441759i
$305$	126197.	+	126197.i	1.35660	+	1.35660i
$306$		0			0
$307$	−96671.1	+	96671.1i	−1.02570	+	1.02570i	−0.0260377	+	0.999661i	$0.508289\pi$
$307$	−96671.1	+	96671.1i	−1.02570	+	1.02570i	−0.999661	+	0.0260377i	$0.991711\pi$
$308$	−25720.0			−0.271125
$309$		0			0
$310$	−238167.	+	238167.i	−2.47832	+	2.47832i
$311$		−	70642.4i		−	0.730373i	−0.930934	−	0.365186i	$-0.881005\pi$
$311$		−	70642.4i		−	0.730373i	0.930934	−	0.365186i	$-0.118995\pi$
$312$		0			0
$313$	68524.0			0.699446			0.349723	−	0.936853i	$-0.386276\pi$
$313$	68524.0			0.699446			0.349723	+	0.936853i	$0.386276\pi$
$314$	34650.0	+	34650.0i	0.351435	+	0.351435i
$315$		0			0
$316$			132584.i			1.32775i
$317$	−140417.	−	140417.i	−1.39734	−	1.39734i	−0.807578	−	0.589760i	$-0.799222\pi$
$317$	−140417.	−	140417.i	−1.39734	−	1.39734i	−0.589760	−	0.807578i	$-0.700778\pi$
$318$		0			0
$319$	29473.2	−	29473.2i	0.289631	−	0.289631i
$320$	151057.	−	151057.i	1.47516	−	1.47516i
$321$		0			0
$322$			34993.4i			0.337501i
$323$	−59678.7	+	59678.7i	−0.572024	+	0.572024i
$324$		0			0
$325$	248173.	+	118460.i	2.34956	+	1.12151i
$326$	−9431.05			−0.0887411
$327$		0			0
$328$	−7471.08			−0.0694441
$329$		−	142325.i		−	1.31489i
$330$		0			0
$331$	−73424.6	−	73424.6i	−0.670171	−	0.670171i	0.287585	−	0.957755i	$-0.407148\pi$
$331$	−73424.6	−	73424.6i	−0.670171	−	0.670171i	−0.957755	+	0.287585i	$0.907148\pi$
$332$	−68573.6	+	68573.6i	−0.622130	+	0.622130i
$333$		0			0
$334$	−61039.7			−0.547166
$335$			176615.i			1.57376i
$336$		0			0
$337$		−	14312.2i		−	0.126022i	−0.998013	−	0.0630111i	$-0.979930\pi$
$337$		−	14312.2i		−	0.126022i	0.998013	−	0.0630111i	$-0.0200703\pi$
$338$	17194.8	−	162700.i	0.150509	−	1.42415i
$339$		0			0
$340$	−199859.	−	199859.i	−1.72889	−	1.72889i
$341$	56516.0			0.486030
$342$		0			0
$343$	87202.1	+	87202.1i	0.741206	+	0.741206i
$344$	−1355.33	−	1355.33i	−0.0114532	−	0.0114532i
$345$		0			0
$346$	158611.	−	158611.i	1.32489	−	1.32489i
$347$	−216160.			−1.79521			−0.897607	−	0.440797i	$-0.854696\pi$
$347$	−216160.			−1.79521			−0.897607	+	0.440797i	$0.854696\pi$
$348$		0			0
$349$	−20673.5	+	20673.5i	−0.169732	+	0.169732i	−0.786862	−	0.617130i	$-0.788295\pi$
$349$	−20673.5	+	20673.5i	−0.169732	+	0.169732i	0.617130	+	0.786862i	$0.288295\pi$
$350$			312586.i			2.55172i
$351$		0			0
$352$	−66718.7			−0.538471
$353$	70477.9	+	70477.9i	0.565592	+	0.565592i	0.930890	−	0.365298i	$-0.119033\pi$
$353$	70477.9	+	70477.9i	0.565592	+	0.565592i	−0.365298	+	0.930890i	$0.619033\pi$
$354$		0			0
$355$			19977.4i			0.158519i
$356$	−28967.5	−	28967.5i	−0.228565	−	0.228565i
$357$		0			0
$358$	152969.	−	152969.i	1.19354	−	1.19354i
$359$	90432.4	−	90432.4i	0.701674	−	0.701674i	−0.263096	−	0.964770i	$-0.584744\pi$
$359$	90432.4	−	90432.4i	0.701674	−	0.701674i	0.964770	+	0.263096i	$0.0847436\pi$
$360$		0			0
$361$		−	73551.2i		−	0.564385i
$362$	−38846.9	+	38846.9i	−0.296442	+	0.296442i
$363$		0			0
$364$	89834.5	−	31783.1i	0.678017	−	0.239880i
$365$	−175591.			−1.31801
$366$		0			0
$367$	−156339.			−1.16074			−0.580369	−	0.814354i	$-0.697091\pi$
$367$	−156339.			−1.16074			−0.580369	+	0.814354i	$0.697091\pi$
$368$			44141.2i			0.325948i
$369$		0			0
$370$	−30634.1	−	30634.1i	−0.223770	−	0.223770i
$371$	100278.	−	100278.i	0.728546	−	0.728546i
$372$		0			0
$373$	12154.0			0.0873580			0.0436790	−	0.999046i	$-0.486092\pi$
$373$	12154.0			0.0873580			0.0436790	+	0.999046i	$0.486092\pi$
$374$			92556.7i			0.661705i
$375$		0			0
$376$			19780.3i			0.139913i
$377$	−66522.5	+	139364.i	−0.468043	+	0.980549i
$378$		0			0
$379$	56764.9	+	56764.9i	0.395186	+	0.395186i	0.876531	−	0.481345i	$-0.159852\pi$
$379$	56764.9	+	56764.9i	0.395186	+	0.395186i	−0.481345	+	0.876531i	$0.659852\pi$
$380$	−190118.			−1.31661
$381$		0			0
$382$	−187915.	−	187915.i	−1.28776	−	1.28776i
$383$	−14158.7	−	14158.7i	−0.0965218	−	0.0965218i	0.657197	−	0.753719i	$-0.271742\pi$
$383$	−14158.7	−	14158.7i	−0.0965218	−	0.0965218i	−0.753719	+	0.657197i	$0.771742\pi$
$384$		0			0
$385$	51333.0	−	51333.0i	0.346318	−	0.346318i
$386$	75805.3			0.508774
$387$		0			0
$388$	−26234.9	+	26234.9i	−0.174267	+	0.174267i
$389$			14487.6i			0.0957405i	0.998854	+	0.0478703i	$0.0152434\pi$
$389$			14487.6i			0.0957405i	−0.998854	+	0.0478703i	$0.984757\pi$
$390$		0			0
$391$	64525.4			0.422063
$392$	−4206.48	−	4206.48i	−0.0273745	−	0.0273745i
$393$		0			0
$394$			36155.0i			0.232904i
$395$	−264617.	−	264617.i	−1.69599	−	1.69599i
$396$		0			0
$397$	120415.	−	120415.i	0.764013	−	0.764013i	−0.213032	−	0.977045i	$-0.568334\pi$
$397$	120415.	−	120415.i	0.764013	−	0.764013i	0.977045	+	0.213032i	$0.0683339\pi$
$398$	−288910.	+	288910.i	−1.82388	+	1.82388i
$399$		0			0
$400$			394300.i			2.46438i
$401$	−31598.4	+	31598.4i	−0.196506	+	0.196506i	−0.798500	−	0.601994i	$-0.794373\pi$
$401$	−31598.4	+	31598.4i	−0.196506	+	0.196506i	0.601994	+	0.798500i	$0.294373\pi$
$402$		0			0
$403$	−197398.	+	69838.8i	−1.21544	+	0.430018i
$404$	−167840.			−1.02833
$405$		0			0
$406$	−175537.			−1.06492
$407$			7269.34i			0.0438840i
$408$		0			0
$409$	16690.9	+	16690.9i	0.0997776	+	0.0997776i	0.755233	−	0.655456i	$-0.227523\pi$
$409$	16690.9	+	16690.9i	0.0997776	+	0.0997776i	−0.655456	+	0.755233i	$0.727523\pi$
$410$	308136.	−	308136.i	1.83305	−	1.83305i
$411$		0			0
$412$	−110.322			−0.000649934
$413$			80159.5i			0.469954i
$414$		0			0
$415$		−	273724.i		−	1.58934i
$416$	233034.	−	82446.6i	1.34658	−	0.476416i
$417$		0			0
$418$	44022.6	+	44022.6i	0.251955	+	0.251955i
$419$	45247.7			0.257732			0.128866	−	0.991662i	$-0.458866\pi$
$419$	45247.7			0.257732			0.128866	+	0.991662i	$0.458866\pi$
$420$		0			0
$421$	95387.7	+	95387.7i	0.538181	+	0.538181i	0.922994	−	0.384814i	$-0.125734\pi$
$421$	95387.7	+	95387.7i	0.538181	+	0.538181i	−0.384814	+	0.922994i	$0.625734\pi$
$422$	−182866.	−	182866.i	−1.02685	−	1.02685i
$423$		0			0
$424$	−13936.6	+	13936.6i	−0.0775219	+	0.0775219i
$425$	576386.			3.19107
$426$		0			0
$427$	−89176.8	+	89176.8i	−0.489098	+	0.489098i
$428$			103960.i			0.567519i
$429$		0			0
$430$	111798.			0.604639
$431$	120110.	+	120110.i	0.646582	+	0.646582i	0.952165	−	0.305583i	$-0.0988514\pi$
$431$	120110.	+	120110.i	0.646582	+	0.646582i	−0.305583	+	0.952165i	$0.598851\pi$
$432$		0			0
$433$			104909.i			0.559549i	0.960066	+	0.279774i	$0.0902596\pi$
$433$			104909.i			0.559549i	−0.960066	+	0.279774i	$0.909740\pi$
$434$	−168300.	−	168300.i	−0.893518	−	0.893518i
$435$		0			0
$436$	−62117.4	+	62117.4i	−0.326768	+	0.326768i
$437$	30690.1	−	30690.1i	0.160707	−	0.160707i
$438$		0			0
$439$			189251.i			0.981997i	0.871160	+	0.490998i	$0.163368\pi$
$439$			189251.i			0.981997i	−0.871160	+	0.490998i	$0.836632\pi$
$440$	−7134.25	+	7134.25i	−0.0368505	+	0.0368505i
$441$		0			0
$442$	−114376.	−	323281.i	−0.585448	−	1.65476i
$443$	154643.			0.787995			0.393998	−	0.919111i	$-0.371092\pi$
$443$	154643.			0.787995			0.393998	+	0.919111i	$0.371092\pi$
$444$		0			0
$445$	115629.			0.583910
$446$			182451.i			0.917227i
$447$		0			0
$448$	106744.	+	106744.i	0.531846	+	0.531846i
$449$	−54131.2	+	54131.2i	−0.268507	+	0.268507i	−0.828498	−	0.559992i	$-0.810804\pi$
$449$	−54131.2	+	54131.2i	−0.268507	+	0.268507i	0.559992	+	0.828498i	$0.310804\pi$
$450$		0			0
$451$	−73119.3			−0.359484
$452$			357591.i			1.75029i
$453$		0			0
$454$		−	272835.i		−	1.32370i
$455$	−115861.	+	242729.i	−0.559649	+	1.17246i
$456$		0			0
$457$	−44160.2	−	44160.2i	−0.211446	−	0.211446i	0.593436	−	0.804881i	$-0.297771\pi$
$457$	−44160.2	−	44160.2i	−0.211446	−	0.211446i	−0.804881	+	0.593436i	$0.797771\pi$
$458$	446101.			2.12668
$459$		0			0
$460$	102779.	+	102779.i	0.485723	+	0.485723i
$461$	145494.	+	145494.i	0.684610	+	0.684610i	0.961035	−	0.276425i	$-0.0891499\pi$
$461$	145494.	+	145494.i	0.684610	+	0.684610i	−0.276425	+	0.961035i	$0.589150\pi$
$462$		0			0
$463$	−168589.	+	168589.i	−0.786443	+	0.786443i	−0.980909	−	0.194467i	$-0.937702\pi$
$463$	−168589.	+	168589.i	−0.786443	+	0.786443i	0.194467	+	0.980909i	$0.437702\pi$
$464$	−221424.			−1.02847
$465$		0			0
$466$	−27012.2	+	27012.2i	−0.124391	+	0.124391i
$467$		−	164176.i		−	0.752793i	−0.926459	−	0.376397i	$-0.877163\pi$
$467$		−	164176.i		−	0.752793i	0.926459	−	0.376397i	$-0.122837\pi$
$468$		0			0
$469$	−124804.			−0.567392
$470$	−815813.	−	815813.i	−3.69313	−	3.69313i
$471$		0			0
$472$		−	11140.6i		−	0.0500061i
$473$	−13264.6	−	13264.6i	−0.0592885	−	0.0592885i
$474$		0			0
$475$	274146.	−	274146.i	1.21505	−	1.21505i
$476$	141230.	−	141230.i	0.623322	−	0.623322i
$477$		0			0
$478$			201917.i			0.883726i
$479$	−10906.2	+	10906.2i	−0.0475337	+	0.0475337i	−0.730474	−	0.682940i	$-0.760701\pi$
$479$	−10906.2	+	10906.2i	−0.0475337	+	0.0475337i	0.682940	+	0.730474i	$0.260701\pi$
$480$		0			0
$481$	−8982.97	−	25390.3i	−0.0388267	−	0.109743i
$482$	34949.7			0.150435
$483$		0			0
$484$	−211184.			−0.901510
$485$		−	104721.i		−	0.445196i
$486$		0			0
$487$	225502.	+	225502.i	0.950808	+	0.950808i	0.998846	−	0.0480379i	$-0.0152968\pi$
$487$	225502.	+	225502.i	0.950808	+	0.950808i	−0.0480379	+	0.998846i	$0.515297\pi$
$488$	12393.8	−	12393.8i	0.0520432	−	0.0520432i
$489$		0			0
$490$	346982.			1.44516
$491$			311585.i			1.29245i	0.763147	+	0.646225i	$0.223653\pi$
$491$			311585.i			1.29245i	−0.763147	+	0.646225i	$0.776347\pi$
$492$		0			0
$493$			323677.i			1.33174i
$494$	−208162.	−	99361.4i	−0.852997	−	0.407159i
$495$		0			0
$496$	−212295.	−	212295.i	−0.862933	−	0.862933i
$497$	−14116.9			−0.0571516
$498$		0			0
$499$	1928.12	+	1928.12i	0.00774341	+	0.00774341i	0.710968	−	0.703224i	$-0.248257\pi$
$499$	1928.12	+	1928.12i	0.00774341	+	0.00774341i	−0.703224	+	0.710968i	$0.748257\pi$
$500$	565456.	+	565456.i	2.26183	+	2.26183i
$501$		0			0
$502$	−406003.	+	406003.i	−1.61110	+	1.61110i
$503$	185299.			0.732382			0.366191	−	0.930540i	$-0.380662\pi$
$503$	185299.			0.732382			0.366191	+	0.930540i	$0.380662\pi$
$504$		0			0
$505$	334981.	−	334981.i	1.31352	−	1.31352i
$506$		−	47597.8i		−	0.185903i
$507$		0			0
$508$	445001.			1.72438
$509$	−103850.	−	103850.i	−0.400841	−	0.400841i	0.477688	−	0.878529i	$-0.341475\pi$
$509$	−103850.	−	103850.i	−0.400841	−	0.400841i	−0.878529	+	0.477688i	$0.841475\pi$
$510$		0			0
$511$		−	124081.i		−	0.475186i
$512$	263398.	+	263398.i	1.00478	+	1.00478i
$513$		0			0
$514$	60573.7	−	60573.7i	0.229276	−	0.229276i
$515$	220.186	−	220.186i	0.000830185	−	0.000830185i
$516$		0			0
$517$			193589.i			0.724269i
$518$	21647.4	−	21647.4i	0.0806765	−	0.0806765i
$519$		0			0
$520$	16102.4	−	33734.5i	0.0595503	−	0.124758i
$521$	−200788.			−0.739712			−0.369856	−	0.929089i	$-0.620593\pi$
$521$	−200788.			−0.739712			−0.369856	+	0.929089i	$0.620593\pi$
$522$		0			0
$523$	−9411.84			−0.0344089			−0.0172045	−	0.999852i	$-0.505477\pi$
$523$	−9411.84			−0.0344089			−0.0172045	+	0.999852i	$0.505477\pi$
$524$		−	361353.i		−	1.31604i
$525$		0			0
$526$	30679.0	+	30679.0i	0.110884	+	0.110884i
$527$	−310332.	+	310332.i	−1.11739	+	1.11739i
$528$		0			0
$529$	246658.			0.881424
$530$		−	1.14959e6i		−	4.09254i
$531$		0			0
$532$		−	134346.i		−	0.474681i
$533$	255390.	−	90356.1i	0.898980	−	0.318056i
$534$		0			0
$535$	−207488.	−	207488.i	−0.724913	−	0.724913i
$536$	17345.3			0.0603742
$537$		0			0
$538$	−406904.	−	406904.i	−1.40581	−	1.40581i
$539$	−41168.7	−	41168.7i	−0.141707	−	0.141707i
$540$		0			0
$541$	−16197.1	+	16197.1i	−0.0553406	+	0.0553406i	−0.734235	−	0.678895i	$-0.762459\pi$
$541$	−16197.1	+	16197.1i	−0.0553406	+	0.0553406i	0.678895	+	0.734235i	$0.262459\pi$
$542$	539696.			1.83718
$543$		0			0
$544$	366356.	−	366356.i	1.23796	−	1.23796i
$545$		−	247953.i		−	0.834788i
$546$		0			0
$547$	−211628.			−0.707292			−0.353646	−	0.935379i	$-0.615058\pi$
$547$	−211628.			−0.707292			−0.353646	+	0.935379i	$0.615058\pi$
$548$	84574.9	+	84574.9i	0.281631	+	0.281631i
$549$		0			0
$550$		−	425178.i		−	1.40555i
$551$	153950.	+	153950.i	0.507081	+	0.507081i
$552$		0			0
$553$	186991.	−	186991.i	0.611462	−	0.611462i
$554$	−284268.	+	284268.i	−0.926207	+	0.926207i
$555$		0			0
$556$			204839.i			0.662619i
$557$	143135.	−	143135.i	0.461354	−	0.461354i	−0.437745	−	0.899099i	$-0.644223\pi$
$557$	143135.	−	143135.i	0.461354	−	0.461354i	0.899099	+	0.437745i	$0.144223\pi$
$558$		0			0
$559$	62721.8	+	29938.8i	0.200722	+	0.0958101i
$560$	−385652.			−1.22976
$561$		0			0
$562$	166199.			0.526206
$563$			230941.i			0.728592i	0.931283	+	0.364296i	$0.118690\pi$
$563$			230941.i			0.728592i	−0.931283	+	0.364296i	$0.881310\pi$
$564$		0			0
$565$	−713694.	−	713694.i	−2.23571	−	2.23571i
$566$	122205.	−	122205.i	0.381467	−	0.381467i
$567$		0			0
$568$	1961.97			0.00608129
$569$			243018.i			0.750609i	0.926902	+	0.375305i	$0.122462\pi$
$569$			243018.i			0.750609i	−0.926902	+	0.375305i	$0.877538\pi$
$570$		0			0
$571$		−	229921.i		−	0.705189i	−0.935776	−	0.352595i	$-0.885299\pi$
$571$		−	229921.i		−	0.705189i	0.935776	−	0.352595i	$-0.114701\pi$
$572$	−122192.	+	43231.2i	−0.373467	+	0.132131i
$573$		0			0
$574$	217743.	+	217743.i	0.660876	+	0.660876i
$575$	−296410.			−0.896515
$576$		0			0
$577$	−45748.5	−	45748.5i	−0.137412	−	0.137412i	0.635055	−	0.772467i	$-0.280977\pi$
$577$	−45748.5	−	45748.5i	−0.137412	−	0.137412i	−0.772467	+	0.635055i	$0.780977\pi$
$578$	−169929.	−	169929.i	−0.508641	−	0.508641i
$579$		0			0
$580$	−515568.	+	515568.i	−1.53260	+	1.53260i
$581$	193426.			0.573010
$582$		0			0
$583$	−136397.	+	136397.i	−0.401299	+	0.401299i
$584$			17244.7i			0.0505628i
$585$		0			0
$586$	548126.			1.59619
$587$	183355.	+	183355.i	0.532127	+	0.532127i	0.921205	−	0.389078i	$-0.127206\pi$
$587$	183355.	+	183355.i	0.532127	+	0.532127i	−0.389078	+	0.921205i	$0.627206\pi$
$588$		0			0
$589$			295206.i			0.850931i
$590$	459479.	+	459479.i	1.31996	+	1.31996i
$591$		0			0
$592$	27306.4	−	27306.4i	0.0779149	−	0.0779149i
$593$	83397.8	−	83397.8i	0.237162	−	0.237162i	−0.578512	−	0.815674i	$-0.696366\pi$
$593$	83397.8	−	83397.8i	0.237162	−	0.237162i	0.815674	+	0.578512i	$0.196366\pi$
$594$		0			0
$595$			563745.i			1.59239i
$596$	−168680.	+	168680.i	−0.474866	+	0.474866i
$597$		0			0
$598$	58818.3	+	166249.i	0.164479	+	0.464897i
$599$	217649.			0.606601			0.303300	−	0.952895i	$-0.401911\pi$
$599$	217649.			0.606601			0.303300	+	0.952895i	$0.401911\pi$
$600$		0			0
$601$	−441379.			−1.22197			−0.610987	−	0.791640i	$-0.709227\pi$
$601$	−441379.			−1.22197			−0.610987	+	0.791640i	$0.709227\pi$
$602$			79001.4i			0.217993i
$603$		0			0
$604$	169103.	+	169103.i	0.463529	+	0.463529i
$605$	421490.	−	421490.i	1.15153	−	1.15153i
$606$		0			0
$607$	−283335.			−0.768993			−0.384497	−	0.923126i	$-0.625625\pi$
$607$	−283335.			−0.768993			−0.384497	+	0.923126i	$0.625625\pi$
$608$		−	348499.i		−	0.942745i
$609$		0			0
$610$			1.02233e6i			2.74747i
$611$	−239225.	−	676166.i	−0.640802	−	1.81122i
$612$		0			0
$613$	482403.	+	482403.i	1.28377	+	1.28377i	0.938503	+	0.345272i	$0.112213\pi$
$613$	482403.	+	482403.i	1.28377	+	1.28377i	0.345272	+	0.938503i	$0.387787\pi$
$614$	−783139.			−2.07731
$615$		0			0
$616$	−5041.39	−	5041.39i	−0.0132858	−	0.0132858i
$617$	−136398.	−	136398.i	−0.358293	−	0.358293i	0.504890	−	0.863184i	$-0.331533\pi$
$617$	−136398.	−	136398.i	−0.358293	−	0.358293i	−0.863184	+	0.504890i	$0.831533\pi$
$618$		0			0
$619$	130340.	−	130340.i	0.340170	−	0.340170i	−0.516261	−	0.856431i	$-0.672677\pi$
$619$	130340.	−	130340.i	0.340170	−	0.340170i	0.856431	+	0.516261i	$0.172677\pi$
$620$	−988623.			−2.57186
$621$		0			0
$622$	286139.	−	286139.i	0.739600	−	0.739600i
$623$			81708.7i			0.210519i
$624$		0			0
$625$	−1.24013e6			−3.17473
$626$	277559.	+	277559.i	0.708282	+	0.708282i
$627$		0			0
$628$			143831.i			0.364698i
$629$	−39916.3	−	39916.3i	−0.100890	−	0.100890i
$630$		0			0
$631$	−145847.	+	145847.i	−0.366302	+	0.366302i	−0.866127	−	0.499825i	$-0.833398\pi$
$631$	−145847.	+	145847.i	−0.366302	+	0.366302i	0.499825	+	0.866127i	$0.333398\pi$
$632$	−25987.9	+	25987.9i	−0.0650635	+	0.0650635i
$633$		0			0
$634$		−	1.13753e6i		−	2.82998i
$635$	−888150.	+	888150.i	−2.20262	+	2.20262i
$636$		0			0
$637$	194667.	+	92920.0i	0.479749	+	0.228997i
$638$	238764.			0.586580
$639$		0			0
$640$	113101.			0.276126
$641$			207974.i			0.506165i	0.967445	+	0.253083i	$0.0814445\pi$
$641$			207974.i			0.506165i	−0.967445	+	0.253083i	$0.918556\pi$
$642$		0			0
$643$	−30795.0	−	30795.0i	−0.0744831	−	0.0744831i	0.668884	−	0.743367i	$-0.266772\pi$
$643$	−30795.0	−	30795.0i	−0.0744831	−	0.0744831i	−0.743367	+	0.668884i	$0.766772\pi$
$644$	−72628.3	+	72628.3i	−0.175119	+	0.175119i
$645$		0			0
$646$	−483461.			−1.15850
$647$			216754.i			0.517795i	0.965905	+	0.258898i	$0.0833592\pi$
$647$			216754.i			0.517795i	−0.965905	+	0.258898i	$0.916641\pi$
$648$		0			0
$649$		−	109032.i		−	0.258861i
$650$	525407.	+	1.48506e6i	1.24357	+	3.51492i
$651$		0			0
$652$	−19574.0	−	19574.0i	−0.0460452	−	0.0460452i
$653$	−622399.			−1.45963			−0.729815	−	0.683645i	$-0.760394\pi$
$653$	−622399.			−1.45963			−0.729815	+	0.683645i	$0.760394\pi$
$654$		0			0
$655$	721203.	+	721203.i	1.68103	+	1.68103i
$656$	274664.	+	274664.i	0.638254	+	0.638254i
$657$		0			0
$658$	576491.	−	576491.i	1.33150	−	1.33150i
$659$	−127164.			−0.292814			−0.146407	−	0.989224i	$-0.546771\pi$
$659$	−127164.			−0.292814			−0.146407	+	0.989224i	$0.546771\pi$
$660$		0			0
$661$	512043.	−	512043.i	1.17194	−	1.17194i	0.190187	−	0.981748i	$-0.439091\pi$
$661$	512043.	−	512043.i	1.17194	−	1.17194i	0.981748	−	0.190187i	$-0.0609095\pi$
$662$		−	594817.i		−	1.35727i
$663$		0			0
$664$	−26882.3			−0.0609720
$665$	268133.	+	268133.i	0.606328	+	0.606328i
$666$		0			0
$667$		−	166453.i		−	0.374145i
$668$	−126687.	−	126687.i	−0.283909	−	0.283909i
$669$		0			0
$670$	−715384.	+	715384.i	−1.59364	+	1.59364i
$671$	121298.	−	121298.i	0.269406	−	0.269406i
$672$		0			0
$673$			70363.7i			0.155353i	0.996979	+	0.0776763i	$0.0247501\pi$
$673$			70363.7i			0.155353i	−0.996979	+	0.0776763i	$0.975250\pi$
$674$	57972.1	−	57972.1i	0.127614	−	0.127614i
$675$		0			0
$676$	373370.	−	301995.i	0.817044	−	0.660855i
$677$	−289817.			−0.632334			−0.316167	−	0.948704i	$-0.602396\pi$
$677$	−289817.			−0.632334			−0.316167	+	0.948704i	$0.602396\pi$
$678$		0			0
$679$	74000.9			0.160508
$680$		−	78349.1i		−	0.169440i
$681$		0			0
$682$	228920.	+	228920.i	0.492170	+	0.492170i
$683$	−126812.	+	126812.i	−0.271844	+	0.271844i	−0.829842	−	0.557998i	$-0.811570\pi$
$683$	−126812.	+	126812.i	−0.271844	+	0.271844i	0.557998	+	0.829842i	$0.311570\pi$
$684$		0			0
$685$	−337596.			−0.719476
$686$			706430.i			1.50114i
$687$		0			0
$688$			99653.4i			0.210531i
$689$	307856.	−	644957.i	0.648498	−	1.35860i
$690$		0			0
$691$	512067.	+	512067.i	1.07243	+	1.07243i	0.997163	+	0.0752718i	$0.0239824\pi$
$691$	512067.	+	512067.i	1.07243	+	1.07243i	0.0752718	+	0.997163i	$0.476018\pi$
$692$	658388.			1.37490
$693$		0			0
$694$	−875563.	−	875563.i	−1.81789	−	1.81789i
$695$	−408827.	−	408827.i	−0.846388	−	0.846388i
$696$		0			0
$697$	401502.	−	401502.i	0.826461	−	0.826461i
$698$	−167478.			−0.343753
$699$		0			0
$700$	−648768.	+	648768.i	−1.32402	+	1.32402i
$701$			718609.i			1.46237i	0.682181	+	0.731184i	$0.261032\pi$
$701$			718609.i			1.46237i	−0.682181	+	0.731184i	$0.738968\pi$
$702$		0			0
$703$	−37970.7			−0.0768312
$704$	−145192.	−	145192.i	−0.292952	−	0.292952i
$705$		0			0
$706$			570946.i			1.14547i
$707$	236713.	+	236713.i	0.473570	+	0.473570i
$708$		0			0
$709$	−360756.	+	360756.i	−0.717664	+	0.717664i	−0.968126	−	0.250462i	$-0.919417\pi$
$709$	−360756.	+	360756.i	−0.717664	+	0.717664i	0.250462	+	0.968126i	$0.419417\pi$
$710$	−80919.1	+	80919.1i	−0.160522	+	0.160522i
$711$		0			0
$712$		−	11355.9i		−	0.0224006i
$713$	159590.	−	159590.i	0.313926	−	0.313926i
$714$		0			0
$715$	157594.	−	330159.i	0.308267	−	0.645819i
$716$	634968.			1.23858
$717$		0			0
$718$	732599.			1.42108
$719$		−	756213.i		−	1.46280i	−0.681946	−	0.731402i	$-0.738866\pi$
$719$		−	756213.i		−	1.46280i	0.681946	−	0.731402i	$-0.261134\pi$
$720$		0			0
$721$	155.593	+	155.593i	0.000299310	+	0.000299310i
$722$	297921.	−	297921.i	0.571515	−	0.571515i
$723$		0			0
$724$	−161252.			−0.307630
$725$		−	1.48688e6i		−	2.82878i
$726$		0			0
$727$		−	2488.50i		−	0.00470835i	−0.999997	−	0.00235417i	$-0.999251\pi$
$727$		−	2488.50i		−	0.00470835i	0.999997	−	0.00235417i	$-0.000749357\pi$
$728$	23838.3	+	11378.7i	0.0449794	+	0.0214699i
$729$		0			0
$730$	−711239.	−	711239.i	−1.33466	−	1.33466i
$731$	145673.			0.272611
$732$		0			0
$733$	269312.	+	269312.i	0.501243	+	0.501243i	0.911824	−	0.410581i	$-0.134674\pi$
$733$	269312.	+	269312.i	0.501243	+	0.501243i	−0.410581	+	0.911824i	$0.634674\pi$
$734$	−633254.	−	633254.i	−1.17540	−	1.17540i
$735$		0			0
$736$	−188401.	+	188401.i	−0.347798	+	0.347798i
$737$	169758.			0.312532
$738$		0			0
$739$	−710849.	+	710849.i	−1.30163	+	1.30163i	−0.374342	+	0.927291i	$0.622131\pi$
$739$	−710849.	+	710849.i	−1.30163	+	1.30163i	−0.927291	+	0.374342i	$0.877869\pi$
$740$		−	127161.i		−	0.232215i
$741$		0			0
$742$	812357.			1.47550
$743$	516575.	+	516575.i	0.935741	+	0.935741i	0.998057	−	0.0623154i	$-0.0198485\pi$
$743$	516575.	+	516575.i	0.935741	+	0.935741i	−0.0623154	+	0.998057i	$0.519848\pi$
$744$		0			0
$745$		−	673317.i		−	1.21313i
$746$	49230.3	+	49230.3i	0.0884616	+	0.0884616i
$747$		0			0
$748$	−192100.	+	192100.i	−0.343340	+	0.343340i
$749$	146621.	−	146621.i	0.261356	−	0.261356i
$750$		0			0
$751$		−	628582.i		−	1.11450i	−0.830343	−	0.557252i	$-0.811856\pi$
$751$		−	628582.i		−	1.11450i	0.830343	−	0.557252i	$-0.188144\pi$
$752$	727194.	−	727194.i	1.28592	−	1.28592i
$753$		0			0
$754$	−833952.	+	295049.i	−1.46689	+	0.518981i
$755$	−675004.			−1.18417
$756$		0			0
$757$	874918.			1.52678			0.763389	−	0.645940i	$-0.223534\pi$
$757$	874918.			1.52678			0.763389	+	0.645940i	$0.223534\pi$
$758$			459856.i			0.800357i
$759$		0			0
$760$	−37265.1	−	37265.1i	−0.0645171	−	0.0645171i
$761$	−176909.	+	176909.i	−0.305479	+	0.305479i	−0.843153	−	0.537674i	$-0.819303\pi$
$761$	−176909.	+	176909.i	−0.305479	+	0.305479i	0.537674	+	0.843153i	$0.319303\pi$
$762$		0			0
$763$	175215.			0.300969
$764$		−	780031.i		−	1.33636i
$765$		0			0
$766$		−	114700.i		−	0.195482i
$767$	134735.	+	380827.i	0.229029	+	0.647347i
$768$		0			0
$769$	64761.1	+	64761.1i	0.109512	+	0.109512i	0.759740	−	0.650228i	$-0.225326\pi$
$769$	64761.1	+	64761.1i	0.109512	+	0.109512i	−0.650228	+	0.759740i	$0.725326\pi$
$770$	415852.			0.701387
$771$		0			0
$772$	157333.	+	157333.i	0.263988	+	0.263988i
$773$	−65728.0	−	65728.0i	−0.110000	−	0.110000i	0.649965	−	0.759964i	$-0.274784\pi$
$773$	−65728.0	−	65728.0i	−0.110000	−	0.110000i	−0.759964	+	0.649965i	$0.774784\pi$
$774$		0			0
$775$	1.42557e6	−	1.42557e6i	2.37348	−	2.37348i
$776$	−10284.6			−0.0170791
$777$		0			0
$778$	−58682.3	+	58682.3i	−0.0969501	+	0.0969501i
$779$		−	381932.i		−	0.629377i
$780$		0			0
$781$	19201.8			0.0314803
$782$	261362.	+	261362.i	0.427395	+	0.427395i
$783$		0			0
$784$			309290.i			0.503193i
$785$	−287064.	−	287064.i	−0.465843	−	0.465843i
$786$		0			0
$787$	853121.	−	853121.i	1.37740	−	1.37740i	0.528420	−	0.848983i	$-0.322785\pi$
$787$	853121.	−	853121.i	1.37740	−	1.37740i	0.848983	−	0.528420i	$-0.177215\pi$
$788$	−75039.2	+	75039.2i	−0.120847	+	0.120847i
$789$		0			0
$790$		−	2.14368e6i		−	3.43484i
$791$	504329.	−	504329.i	0.806048	−	0.806048i
$792$		0			0
$793$	−273775.	+	573559.i	−0.435360	+	0.912078i
$794$	975493.			1.54733
$795$		0			0
$796$	−1.19926e6			−1.89272
$797$			264563.i			0.416498i	0.978076	+	0.208249i	$0.0667765\pi$
$797$			264563.i			0.416498i	−0.978076	+	0.208249i	$0.933224\pi$
$798$		0			0
$799$	−1.06301e6	−	1.06301e6i	−1.66511	−	1.66511i
$800$	−1.68293e6	+	1.68293e6i	−2.62958	+	2.62958i
$801$		0			0
$802$	−255980.			−0.397977
$803$			168774.i			0.261743i
$804$		0			0
$805$		−	289909.i		−	0.447373i
$806$	−1.08245e6	−	516685.i	−1.66625	−	0.795345i
$807$		0			0
$808$	−32898.4	−	32898.4i	−0.0503908	−	0.0503908i
$809$	903066.			1.37982			0.689910	−	0.723895i	$-0.257650\pi$
$809$	903066.			1.37982			0.689910	+	0.723895i	$0.257650\pi$
$810$		0			0
$811$	158801.	+	158801.i	0.241442	+	0.241442i	0.817446	−	0.576005i	$-0.195389\pi$
$811$	158801.	+	158801.i	0.241442	+	0.241442i	−0.576005	+	0.817446i	$0.695389\pi$
$812$	−364324.	−	364324.i	−0.552555	−	0.552555i
$813$		0			0
$814$	−29444.7	+	29444.7i	−0.0444384	+	0.0444384i
$815$	78133.2			0.117631
$816$		0			0
$817$	69286.2	−	69286.2i	0.103801	−	0.103801i
$818$			135214.i			0.202076i
$819$		0			0
$820$	1.27906e6			1.90223
$821$	−566290.	−	566290.i	−0.840142	−	0.840142i	0.148735	−	0.988877i	$-0.452480\pi$
$821$	−566290.	−	566290.i	−0.840142	−	0.840142i	−0.988877	+	0.148735i	$0.952480\pi$
$822$		0			0
$823$			664034.i			0.980371i	0.871618	+	0.490186i	$0.163071\pi$
$823$			664034.i			0.980371i	−0.871618	+	0.490186i	$0.836929\pi$
$824$	−21.6244	−	21.6244i	−3.18485e−5	−	3.18485e-5i
$825$		0			0
$826$	−324689.	+	324689.i	−0.475891	+	0.475891i
$827$	694781.	−	694781.i	1.01587	−	1.01587i	0.0159954	−	0.999872i	$-0.494908\pi$
$827$	694781.	−	694781.i	1.01587	−	1.01587i	0.999872	−	0.0159954i	$-0.00509171\pi$
$828$		0			0
$829$			1.07123e6i			1.55874i	0.626564	+	0.779370i	$0.284461\pi$
$829$			1.07123e6i			1.55874i	−0.626564	+	0.779370i	$0.715539\pi$
$830$	1.10873e6	−	1.10873e6i	1.60942	−	1.60942i
$831$		0			0
$832$	686543.	+	327706.i	0.991794	+	0.473410i
$833$	452119.			0.651573
$834$		0			0
$835$	505694.			0.725295
$836$			182737.i			0.261465i
$837$		0			0
$838$	183277.	+	183277.i	0.260988	+	0.260988i
$839$	−710646.	+	710646.i	−1.00955	+	1.00955i	−0.00959972	+	0.999954i	$0.503056\pi$
$839$	−710646.	+	710646.i	−1.00955	+	1.00955i	−0.999954	+	0.00959972i	$0.996944\pi$
$840$		0			0
$841$	127693.			0.180541
$842$			772742.i			1.08996i
$843$		0			0
$844$		−	759070.i		−	1.06561i
$845$	−142453.	+	1.34792e6i	−0.199507	+	1.88778i
$846$		0			0
$847$	297844.	+	297844.i	0.415167	+	0.415167i
$848$	1.02472e6			1.42499
$849$		0			0
$850$	2.33467e6	+	2.33467e6i	3.23138	+	3.23138i
$851$	20527.2	+	20527.2i	0.0283446	+	0.0283446i
$852$		0			0
$853$	−719087.	+	719087.i	−0.988288	+	0.988288i	−0.999932	−	0.0116445i	$-0.996293\pi$
$853$	−719087.	+	719087.i	−0.988288	+	0.988288i	0.0116445	+	0.999932i	$0.496293\pi$
$854$	−722427.			−0.990555
$855$		0			0
$856$	−20377.3	+	20377.3i	−0.0278099	+	0.0278099i
$857$			625779.i			0.852038i	0.904714	+	0.426019i	$0.140084\pi$
$857$			625779.i			0.852038i	−0.904714	+	0.426019i	$0.859916\pi$
$858$		0			0
$859$	−1.38660e6			−1.87917			−0.939585	−	0.342315i	$-0.888789\pi$
$859$	−1.38660e6			−1.87917			−0.939585	+	0.342315i	$0.888789\pi$
$860$	232034.	+	232034.i	0.313729	+	0.313729i
$861$		0			0
$862$			973017.i			1.30950i
$863$	251871.	+	251871.i	0.338187	+	0.338187i	0.855685	−	0.517497i	$-0.173136\pi$
$863$	251871.	+	251871.i	0.338187	+	0.338187i	−0.517497	+	0.855685i	$0.673136\pi$
$864$		0			0
$865$	−1.31404e6	+	1.31404e6i	−1.75621	+	1.75621i
$866$	−424938.	+	424938.i	−0.566618	+	0.566618i
$867$		0			0
$868$		−	698606.i		−	0.927242i
$869$	−254343.	+	254343.i	−0.336807	+	0.336807i
$870$		0			0
$871$	−592928.	+	209776.i	−0.781566	+	0.276515i
$872$	−24351.3			−0.0320250
$873$		0			0
$874$	248623.			0.325475
$875$		−	1.59499e6i		−	2.08325i
$876$		0			0
$877$	248488.	+	248488.i	0.323077	+	0.323077i	0.849946	−	0.526869i	$-0.176634\pi$
$877$	248488.	+	248488.i	0.323077	+	0.323077i	−0.526869	+	0.849946i	$0.676634\pi$
$878$	−766569.	+	766569.i	−0.994403	+	0.994403i
$879$		0			0
$880$	524562.			0.677378
$881$			1.01415e6i			1.30662i	0.757089	+	0.653312i	$0.226621\pi$
$881$			1.01415e6i			1.30662i	−0.757089	+	0.653312i	$0.773379\pi$
$882$		0			0
$883$		−	1.21924e6i		−	1.56375i	−0.623433	−	0.781877i	$-0.714263\pi$
$883$		−	1.21924e6i		−	1.56375i	0.623433	−	0.781877i	$-0.285737\pi$
$884$	433580.	−	908349.i	0.554836	−	1.16238i
$885$		0			0
$886$	626388.	+	626388.i	0.797950	+	0.797950i
$887$	−904300.			−1.14938			−0.574692	−	0.818370i	$-0.694878\pi$
$887$	−904300.			−1.14938			−0.574692	+	0.818370i	$0.694878\pi$
$888$		0			0
$889$	−627608.	−	627608.i	−0.794118	−	0.794118i
$890$	468359.	+	468359.i	0.591287	+	0.591287i
$891$		0			0
$892$	−378674.	+	378674.i	−0.475923	+	0.475923i
$893$	−1.01120e6			−1.26804
$894$		0			0
$895$	−1.26729e6	+	1.26729e6i	−1.58209	+	1.58209i
$896$			79922.4i			0.0995526i
$897$		0			0
$898$	−438520.			−0.543797
$899$	800549.	+	800549.i	0.990532	+	0.990532i
$900$		0			0
$901$		−	1.49793e6i		−	1.84519i
$902$	−296172.	−	296172.i	−0.364025	−	0.364025i
$903$		0			0
$904$	−70091.5	+	70091.5i	−0.0857687	+	0.0857687i
$905$	321834.	−	321834.i	0.392948	−	0.392948i
$906$		0			0
$907$		−	1.40623e6i		−	1.70940i	−0.519124	−	0.854699i	$-0.673742\pi$
$907$		−	1.40623e6i		−	1.70940i	0.519124	−	0.854699i	$-0.326258\pi$
$908$	566265.	−	566265.i	0.686828	−	0.686828i
$909$		0			0
$910$	−1.45248e6	+	513883.i	−1.75400	+	0.620557i
$911$	320928.			0.386697			0.193348	−	0.981130i	$-0.438065\pi$
$911$	320928.			0.386697			0.193348	+	0.981130i	$0.438065\pi$
$912$		0			0
$913$	−263097.			−0.315627
$914$		−	357745.i		−	0.428234i
$915$		0			0
$916$	925876.	+	925876.i	1.10347	+	1.10347i
$917$	−509635.	+	509635.i	−0.606067	+	0.606067i
$918$		0			0
$919$	−824931.			−0.976757			−0.488378	−	0.872632i	$-0.662411\pi$
$919$	−824931.			−0.976757			−0.488378	+	0.872632i	$0.662411\pi$
$920$			40291.5i			0.0476034i
$921$		0			0
$922$			1.17866e6i			1.38652i
$923$	−67067.8	+	23728.3i	−0.0787246	+	0.0278525i
$924$		0			0
$925$	183364.	+	183364.i	0.214304	+	0.214304i
$926$	−1.36575e6			−1.59276
$927$		0			0
$928$	−945070.	−	945070.i	−1.09741	−	1.09741i
$929$	−678405.	−	678405.i	−0.786064	−	0.786064i	0.194783	−	0.980846i	$-0.437600\pi$
$929$	−678405.	−	678405.i	−0.786064	−	0.786064i	−0.980846	+	0.194783i	$0.937600\pi$
$930$		0			0
$931$	215041.	−	215041.i	0.248097	−	0.248097i
$932$	−112127.			−0.129086
$933$		0			0
$934$	665000.	−	665000.i	0.762303	−	0.762303i
$935$		−	766802.i		−	0.877122i
$936$		0			0
$937$	1.11931e6			1.27488			0.637441	−	0.770499i	$-0.279993\pi$
$937$	1.11931e6			1.27488			0.637441	+	0.770499i	$0.279993\pi$
$938$	−505523.	−	505523.i	−0.574560	−	0.574560i
$939$		0			0
$940$		−	3.38642e6i		−	3.83252i
$941$	−498975.	−	498975.i	−0.563507	−	0.563507i	0.366795	−	0.930302i	$-0.380455\pi$
$941$	−498975.	−	498975.i	−0.563507	−	0.563507i	−0.930302	+	0.366795i	$0.880455\pi$
$942$		0			0
$943$	−206475.	+	206475.i	−0.232190	+	0.232190i
$944$	−409567.	+	409567.i	−0.459601	+	0.459601i
$945$		0			0
$946$		−	107457.i		−	0.120075i
$947$	672812.	−	672812.i	0.750228	−	0.750228i	−0.224293	−	0.974522i	$-0.572007\pi$
$947$	672812.	−	672812.i	0.750228	−	0.750228i	0.974522	+	0.224293i	$0.0720073\pi$
$948$		0			0
$949$	−208560.	−	589492.i	−0.231579	−	0.654554i
$950$	2.22087e6			2.46080
$951$		0			0
$952$	55365.1			0.0610889
$953$			226250.i			0.249117i	0.992212	+	0.124558i	$0.0397514\pi$
$953$			226250.i			0.249117i	−0.992212	+	0.124558i	$0.960249\pi$
$954$		0			0
$955$	1.55682e6	+	1.55682e6i	1.70699	+	1.70699i
$956$	−419076.	+	419076.i	−0.458540	+	0.458540i
$957$		0			0
$958$	−88351.7			−0.0962684
$959$		−	238561.i		−	0.259395i
$960$		0			0
$961$			611566.i			0.662211i
$962$	66458.2	−	139230.i	0.0718123	−	0.150447i
$963$		0			0
$964$	72537.6	+	72537.6i	0.0780565	+	0.0780565i
$965$	−628022.			−0.674404
$966$		0			0
$967$	−652291.	−	652291.i	−0.697571	−	0.697571i	0.266315	−	0.963886i	$-0.414194\pi$
$967$	−652291.	−	652291.i	−0.697571	−	0.697571i	−0.963886	+	0.266315i	$0.914194\pi$
$968$	−41394.3	−	41394.3i	−0.0441764	−	0.0441764i
$969$		0			0
$970$	424177.	−	424177.i	0.450821	−	0.450821i
$971$	−330964.			−0.351028			−0.175514	−	0.984477i	$-0.556159\pi$
$971$	−330964.			−0.351028			−0.175514	+	0.984477i	$0.556159\pi$
$972$		0			0
$973$	288896.	−	288896.i	0.305151	−	0.305151i
$974$			1.82681e6i			1.92564i
$975$		0			0
$976$	−911280.			−0.956648
$977$	812569.	+	812569.i	0.851278	+	0.851278i	0.990291	−	0.139013i	$-0.0443930\pi$
$977$	812569.	+	812569.i	0.851278	+	0.851278i	−0.139013	+	0.990291i	$0.544393\pi$
$978$		0			0
$979$		−	111140.i		−	0.115959i
$980$	720156.	+	720156.i	0.749850	+	0.749850i
$981$		0			0
$982$	−1.26209e6	+	1.26209e6i	−1.30878	+	1.30878i
$983$	−1.32394e6	+	1.32394e6i	−1.37013	+	1.37013i	−0.509898	+	0.860235i	$0.670317\pi$
$983$	−1.32394e6	+	1.32394e6i	−1.37013	+	1.37013i	−0.860235	+	0.509898i	$0.829683\pi$
$984$		0			0
$985$		−	299533.i		−	0.308725i
$986$	−1.31107e6	+	1.31107e6i	−1.34856	+	1.34856i
$987$		0			0
$988$	−225814.	−	638260.i	−0.231333	−	0.653858i
$989$	−74913.1			−0.0765888
$990$		0			0
$991$	−424185.			−0.431925			−0.215962	−	0.976402i	$-0.569289\pi$
$991$	−424185.			−0.431925			−0.215962	+	0.976402i	$0.569289\pi$
$992$		−	1.81221e6i		−	1.84156i
$993$		0			0
$994$	−57181.2	−	57181.2i	−0.0578736	−	0.0578736i
$995$	2.39352e6	−	2.39352e6i	2.41764	−	2.41764i
$996$		0			0
$997$	1.53487e6			1.54412			0.772060	−	0.635549i	$-0.219226\pi$
$997$	1.53487e6			1.54412			0.772060	+	0.635549i	$0.219226\pi$
$998$			15619.8i			0.0156825i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.5.j.b.73.8		20
3.2	odd	2		39.5.g.a.34.3	yes	20
13.5	odd	4	inner	117.5.j.b.109.8		20
39.5	even	4		39.5.g.a.31.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.3	✓	20	39.5	even	4
39.5.g.a.34.3	yes	20	3.2	odd	2
117.5.j.b.73.8		20	1.1	even	1	trivial
117.5.j.b.109.8		20	13.5	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	117.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.05053 + 4.05053i) q^{2} +16.8136i q^{4} +(-33.5573 - 33.5573i) q^{5} +(23.7132 - 23.7132i) q^{7} +(-3.29565 + 3.29565i) q^{8} -271.850i q^{10} +(-32.2545 + 32.2545i) q^{11} +(72.8001 - 152.516i) q^{13} +192.102 q^{14} +242.320 q^{16} -354.222i q^{17} +(-168.478 - 168.478i) q^{19} +(564.221 - 564.221i) q^{20} -261.296 q^{22} +182.161i q^{23} +1627.19i q^{25} +(912.650 - 322.892i) q^{26} +(398.704 + 398.704i) q^{28} -913.769 q^{29} +(-876.095 - 876.095i) q^{31} +(1034.25 + 1034.25i) q^{32} +(1434.79 - 1434.79i) q^{34} -1591.50 q^{35} +(112.687 - 112.687i) q^{37} -1364.85i q^{38} +221.186 q^{40} +(1133.48 + 1133.48i) q^{41} +411.247i q^{43} +(-542.315 - 542.315i) q^{44} +(-737.848 + 737.848i) q^{46} +(3000.97 - 3000.97i) q^{47} +1276.37i q^{49} +(-6590.99 + 6590.99i) q^{50} +(2564.35 + 1224.03i) q^{52} +4228.78 q^{53} +2164.75 q^{55} +156.301i q^{56} +(-3701.25 - 3701.25i) q^{58} +(-1690.19 + 1690.19i) q^{59} -3760.65 q^{61} -7097.31i q^{62} +4501.45i q^{64} +(-7561.01 + 2675.06i) q^{65} +(-2631.54 - 2631.54i) q^{67} +5955.76 q^{68} +(-6446.43 - 6446.43i) q^{70} +(-297.661 - 297.661i) q^{71} +(2616.29 - 2616.29i) q^{73} +912.887 q^{74} +(2832.73 - 2832.73i) q^{76} +1529.71i q^{77} +7885.52 q^{79} +(-8131.61 - 8131.61i) q^{80} +9182.36i q^{82} +(4078.45 + 4078.45i) q^{83} +(-11886.7 + 11886.7i) q^{85} +(-1665.77 + 1665.77i) q^{86} -212.599i q^{88} +(-1722.86 + 1722.86i) q^{89} +(-1890.32 - 5342.96i) q^{91} -3062.78 q^{92} +24311.0 q^{94} +11307.4i q^{95} +(1560.33 + 1560.33i) q^{97} +(-5169.99 + 5169.99i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) 20 * q - 24 * q^5 - 24 * q^7 - 372 * q^11 - 224 * q^13 - 480 * q^14 - 2328 * q^16 - 840 * q^19 - 228 * q^20 + 3536 * q^22 + 828 * q^26 - 1984 * q^28 + 5064 * q^29 + 1712 * q^31 + 7260 * q^32 + 8040 * q^34 - 5976 * q^35 - 868 * q^37 + 6336 * q^40 + 6000 * q^41 - 20868 * q^44 - 1992 * q^46 - 1692 * q^47 - 8952 * q^50 - 12416 * q^52 + 18816 * q^53 - 3280 * q^55 + 1256 * q^58 + 6588 * q^59 - 5512 * q^61 - 33048 * q^65 - 4704 * q^67 + 28296 * q^68 - 31096 * q^70 - 9756 * q^71 + 13692 * q^73 + 28296 * q^74 - 17072 * q^76 - 5056 * q^79 - 42228 * q^80 - 24708 * q^83 - 4752 * q^85 + 77424 * q^86 - 36000 * q^89 + 19104 * q^91 - 12912 * q^92 + 112832 * q^94 + 50012 * q^97 - 11544 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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