LMFDB - Embedded newform 76.5.j.a.21.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,5,Mod(13,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("76.13");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	76.j (of order $18$, degree $6$, 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:	$7.85611719437$
Analytic rank:	$0$
Dimension:	$42$
Relative dimension:	$7$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			21.5
Character	$\chi$	$=$	76.21
Dual form			76.5.j.a.29.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.92113 - 3.48127i) q^{3} +(20.0929 - 7.31321i) q^{5} +(-36.0108 - 62.3725i) q^{7} +(10.4793 + 59.4309i) q^{9} +O(q^{10})$	\(q+(2.92113 - 3.48127i) q^{3} +(20.0929 - 7.31321i) q^{5} +(-36.0108 - 62.3725i) q^{7} +(10.4793 + 59.4309i) q^{9} +(55.1380 - 95.5017i) q^{11} +(-18.3793 - 21.9036i) q^{13} +(33.2347 - 91.3116i) q^{15} +(66.2932 - 375.967i) q^{17} +(-20.3946 - 360.423i) q^{19} +(-322.328 - 56.8351i) q^{21} +(225.549 + 82.0931i) q^{23} +(-128.537 + 107.855i) q^{25} +(556.293 + 321.176i) q^{27} +(1172.95 - 206.823i) q^{29} +(-806.771 + 465.789i) q^{31} +(-171.402 - 470.923i) q^{33} +(-1179.70 - 989.888i) q^{35} +1391.16i q^{37} -129.941 q^{39} +(-1665.47 + 1984.83i) q^{41} +(2036.86 - 741.355i) q^{43} +(645.190 + 1117.50i) q^{45} +(593.018 + 3363.17i) q^{47} +(-1393.05 + 2412.84i) q^{49} +(-1115.19 - 1329.03i) q^{51} +(-799.720 + 2197.21i) q^{53} +(409.456 - 2322.14i) q^{55} +(-1314.31 - 981.846i) q^{57} +(-186.660 - 32.9132i) q^{59} +(-1412.65 - 514.161i) q^{61} +(3329.49 - 2793.77i) q^{63} +(-529.478 - 305.695i) q^{65} +(7401.57 - 1305.10i) q^{67} +(944.647 - 545.392i) q^{69} +(-1338.30 - 3676.95i) q^{71} +(2645.33 + 2219.70i) q^{73} +762.533i q^{75} -7942.24 q^{77} +(685.145 - 816.524i) q^{79} +(-1850.27 + 673.444i) q^{81} +(-2624.90 - 4546.46i) q^{83} +(-1417.51 - 8039.08i) q^{85} +(2706.35 - 4687.53i) q^{87} +(-333.507 - 397.458i) q^{89} +(-704.329 + 1935.13i) q^{91} +(-735.146 + 4169.22i) q^{93} +(-3045.64 - 7092.79i) q^{95} +(13798.9 + 2433.11i) q^{97} +(6253.56 + 2276.11i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9}+O(q^{10}) $ 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9	$ 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9} - 45 q^{11} + 33 q^{13} - 393 q^{15} + 909 q^{17} + 1242 q^{19} + 1107 q^{21} - 360 q^{23} - 810 q^{25} - 7056 q^{27} - 2889 q^{29} + 2808 q^{31} + 10875 q^{33} + 6741 q^{35} - 3480 q^{39} - 3060 q^{41} - 8079 q^{43} - 4320 q^{45} - 2655 q^{47} - 474 q^{49} - 12222 q^{51} - 6705 q^{53} + 4623 q^{55} - 8022 q^{57} + 24309 q^{59} + 7104 q^{61} + 12063 q^{63} + 25245 q^{65} + 15573 q^{67} - 10881 q^{69} - 25506 q^{71} + 3036 q^{73} + 12924 q^{77} - 16839 q^{79} - 2208 q^{81} - 6363 q^{83} - 37890 q^{85} - 21924 q^{87} - 22644 q^{89} + 17418 q^{91} + 8184 q^{93} - 82413 q^{95} + 13383 q^{97} + 23565 q^{99}+O(q^{100}) $ 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9 - 45 * q^11 + 33 * q^13 - 393 * q^15 + 909 * q^17 + 1242 * q^19 + 1107 * q^21 - 360 * q^23 - 810 * q^25 - 7056 * q^27 - 2889 * q^29 + 2808 * q^31 + 10875 * q^33 + 6741 * q^35 - 3480 * q^39 - 3060 * q^41 - 8079 * q^43 - 4320 * q^45 - 2655 * q^47 - 474 * q^49 - 12222 * q^51 - 6705 * q^53 + 4623 * q^55 - 8022 * q^57 + 24309 * q^59 + 7104 * q^61 + 12063 * q^63 + 25245 * q^65 + 15573 * q^67 - 10881 * q^69 - 25506 * q^71 + 3036 * q^73 + 12924 * q^77 - 16839 * q^79 - 2208 * q^81 - 6363 * q^83 - 37890 * q^85 - 21924 * q^87 - 22644 * q^89 + 17418 * q^91 + 8184 * q^93 - 82413 * q^95 + 13383 * q^97 + 23565 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{1}{18}\right)$	$1$

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
$3$	2.92113	−	3.48127i	0.324570	−	0.386808i	−0.578943	−	0.815368i	$-0.696535\pi$
$3$	2.92113	−	3.48127i	0.324570	−	0.386808i	0.903513	+	0.428560i	$0.140979\pi$
$4$		0			0
$5$	20.0929	−	7.31321i	0.803715	−	0.292528i	0.0926900	−	0.995695i	$-0.470453\pi$
$5$	20.0929	−	7.31321i	0.803715	−	0.292528i	0.711025	+	0.703167i	$0.248231\pi$
$6$		0			0
$7$	−36.0108	−	62.3725i	−0.734914	−	1.27291i	−0.954761	−	0.297374i	$-0.903889\pi$
$7$	−36.0108	−	62.3725i	−0.734914	−	1.27291i	0.219847	−	0.975534i	$-0.429444\pi$
$8$		0			0
$9$	10.4793	+	59.4309i	0.129374	+	0.733715i
$10$		0			0
$11$	55.1380	−	95.5017i	0.455686	−	0.789271i	−0.543042	−	0.839706i	$-0.682727\pi$
$11$	55.1380	−	95.5017i	0.455686	−	0.789271i	0.998727	+	0.0504351i	$0.0160608\pi$
$12$		0			0
$13$	−18.3793	−	21.9036i	−0.108753	−	0.129607i	0.708922	−	0.705287i	$-0.249182\pi$
$13$	−18.3793	−	21.9036i	−0.108753	−	0.129607i	−0.817675	+	0.575680i	$0.804737\pi$
$14$		0			0
$15$	33.2347	−	91.3116i	0.147710	−	0.405829i
$16$		0			0
$17$	66.2932	−	375.967i	0.229388	−	1.30092i	−0.624728	−	0.780842i	$-0.714790\pi$
$17$	66.2932	−	375.967i	0.229388	−	1.30092i	0.854116	−	0.520082i	$-0.174099\pi$
$18$		0			0
$19$	−20.3946	−	360.423i	−0.0564948	−	0.998403i
$19$	−20.3946	−	360.423i	−0.0564948	−	0.998403i
$20$		0			0
$21$	−322.328	−	56.8351i	−0.730902	−	0.128878i
$22$		0			0
$23$	225.549	+	82.0931i	0.426369	+	0.155185i	0.546287	−	0.837598i	$-0.316041\pi$
$23$	225.549	+	82.0931i	0.426369	+	0.155185i	−0.119918	+	0.992784i	$0.538263\pi$
$24$		0			0
$25$	−128.537	+	107.855i	−0.205659	+	0.172569i
$26$		0			0
$27$	556.293	+	321.176i	0.763090	+	0.440570i
$28$		0			0
$29$	1172.95	−	206.823i	1.39471	−	0.245926i	0.574744	−	0.818333i	$-0.305102\pi$
$29$	1172.95	−	206.823i	1.39471	−	0.245926i	0.819969	+	0.572407i	$0.193990\pi$
$30$		0			0
$31$	−806.771	+	465.789i	−0.839512	+	0.484692i	−0.857098	−	0.515153i	$-0.827735\pi$
$31$	−806.771	+	465.789i	−0.839512	+	0.484692i	0.0175864	+	0.999845i	$0.494402\pi$
$32$		0			0
$33$	−171.402	−	470.923i	−0.157394	−	0.432436i
$34$		0			0
$35$	−1179.70	−	989.888i	−0.963023	−	0.808072i
$36$		0			0
$37$			1391.16i			1.01619i	0.861302	+	0.508094i	$0.169650\pi$
$37$			1391.16i			1.01619i	−0.861302	+	0.508094i	$0.830350\pi$
$38$		0			0
$39$	−129.941			−0.0854311
$40$		0			0
$41$	−1665.47	+	1984.83i	−0.990760	+	1.18074i	−0.00723414	+	0.999974i	$0.502303\pi$
$41$	−1665.47	+	1984.83i	−0.990760	+	1.18074i	−0.983526	+	0.180768i	$0.942142\pi$
$42$		0			0
$43$	2036.86	−	741.355i	1.10160	−	0.400949i	0.273693	−	0.961817i	$-0.411755\pi$
$43$	2036.86	−	741.355i	1.10160	−	0.400949i	0.827905	+	0.560868i	$0.189532\pi$
$44$		0			0
$45$	645.190	+	1117.50i	0.318612	+	0.551852i
$46$		0			0
$47$	593.018	+	3363.17i	0.268456	+	1.52249i	0.759012	+	0.651077i	$0.225683\pi$
$47$	593.018	+	3363.17i	0.268456	+	1.52249i	−0.490556	+	0.871410i	$0.663206\pi$
$48$		0			0
$49$	−1393.05	+	2412.84i	−0.580196	+	1.00493i
$50$		0			0
$51$	−1115.19	−	1329.03i	−0.428755	−	0.510971i
$52$		0			0
$53$	−799.720	+	2197.21i	−0.284699	+	0.782205i	0.712086	+	0.702092i	$0.247750\pi$
$53$	−799.720	+	2197.21i	−0.284699	+	0.782205i	−0.996786	+	0.0801131i	$0.974472\pi$
$54$		0			0
$55$	409.456	−	2322.14i	0.135357	−	0.767650i
$56$		0			0
$57$	−1314.31	−	981.846i	−0.404527	−	0.302199i
$58$		0			0
$59$	−186.660	−	32.9132i	−0.0536225	−	0.00945509i	0.146773	−	0.989170i	$-0.453111\pi$
$59$	−186.660	−	32.9132i	−0.0536225	−	0.00945509i	−0.200395	+	0.979715i	$0.564223\pi$
$60$		0			0
$61$	−1412.65	−	514.161i	−0.379641	−	0.138178i	0.145149	−	0.989410i	$-0.453634\pi$
$61$	−1412.65	−	514.161i	−0.379641	−	0.138178i	−0.524790	+	0.851232i	$0.675856\pi$
$62$		0			0
$63$	3329.49	−	2793.77i	0.838873	−	0.703898i
$64$		0			0
$65$	−529.478	−	305.695i	−0.125320	−	0.0723537i
$66$		0			0
$67$	7401.57	−	1305.10i	1.64882	−	0.290732i	0.729425	−	0.684061i	$-0.239788\pi$
$67$	7401.57	−	1305.10i	1.64882	−	0.290732i	0.919398	+	0.393329i	$0.128677\pi$
$68$		0			0
$69$	944.647	−	545.392i	0.198414	−	0.114554i
$70$		0			0
$71$	−1338.30	−	3676.95i	−0.265483	−	0.729409i	−0.998774	−	0.0494941i	$-0.984239\pi$
$71$	−1338.30	−	3676.95i	−0.265483	−	0.729409i	0.733291	−	0.679915i	$-0.237983\pi$
$72$		0			0
$73$	2645.33	+	2219.70i	0.496403	+	0.416532i	0.856314	−	0.516455i	$-0.172749\pi$
$73$	2645.33	+	2219.70i	0.496403	+	0.416532i	−0.359911	+	0.932987i	$0.617193\pi$
$74$		0			0
$75$			762.533i			0.135561i
$76$		0			0
$77$	−7942.24			−1.33956
$78$		0			0
$79$	685.145	−	816.524i	0.109781	−	0.130832i	−0.708356	−	0.705856i	$-0.750563\pi$
$79$	685.145	−	816.524i	0.109781	−	0.130832i	0.818137	+	0.575024i	$0.195007\pi$
$80$		0			0
$81$	−1850.27	+	673.444i	−0.282011	+	0.102643i
$82$		0			0
$83$	−2624.90	−	4546.46i	−0.381028	−	0.659960i	0.610182	−	0.792262i	$-0.291096\pi$
$83$	−2624.90	−	4546.46i	−0.381028	−	0.659960i	−0.991209	+	0.132302i	$0.957763\pi$
$84$		0			0
$85$	−1417.51	−	8039.08i	−0.196195	−	1.11268i
$86$		0			0
$87$	2706.35	−	4687.53i	0.357557	−	0.619306i
$88$		0			0
$89$	−333.507	−	397.458i	−0.0421042	−	0.0501778i	0.744581	−	0.667532i	$-0.232649\pi$
$89$	−333.507	−	397.458i	−0.0421042	−	0.0501778i	−0.786686	+	0.617354i	$0.788205\pi$
$90$		0			0
$91$	−704.329	+	1935.13i	−0.0850536	+	0.233683i
$92$		0			0
$93$	−735.146	+	4169.22i	−0.0849978	+	0.482047i
$94$		0			0
$95$	−3045.64	−	7092.79i	−0.337467	−	0.785905i
$96$		0			0
$97$	13798.9	+	2433.11i	1.46656	+	0.258594i	0.849193	−	0.528082i	$-0.177089\pi$
$97$	13798.9	+	2433.11i	1.46656	+	0.258594i	0.617366	+	0.786676i	$0.288200\pi$
$98$		0			0
$99$	6253.56	+	2276.11i	0.638053	+	0.232232i
$100$		0			0
$101$	5816.06	−	4880.25i	0.570146	−	0.478409i	−0.311548	−	0.950230i	$-0.600848\pi$
$101$	5816.06	−	4880.25i	0.570146	−	0.478409i	0.881694	+	0.471821i	$0.156403\pi$
$102$		0			0
$103$	−8868.49	−	5120.23i	−0.835941	−	0.482631i	0.0199417	−	0.999801i	$-0.493652\pi$
$103$	−8868.49	−	5120.23i	−0.835941	−	0.482631i	−0.855882	+	0.517171i	$0.826985\pi$
$104$		0			0
$105$	−6892.14	+	1215.27i	−0.625137	+	0.110229i
$106$		0			0
$107$	−849.798	+	490.631i	−0.0742247	+	0.0428536i	−0.536653	−	0.843803i	$-0.680312\pi$
$107$	−849.798	+	490.631i	−0.0742247	+	0.0428536i	0.462428	+	0.886657i	$0.346978\pi$
$108$		0			0
$109$	6094.78	+	16745.3i	0.512985	+	1.40942i	0.878111	+	0.478457i	$0.158804\pi$
$109$	6094.78	+	16745.3i	0.512985	+	1.40942i	−0.365126	+	0.930958i	$0.618974\pi$
$110$		0			0
$111$	4843.01	+	4063.77i	0.393069	+	0.329824i
$112$		0			0
$113$		−	10436.4i		−	0.817324i	−0.912686	−	0.408662i	$-0.865996\pi$
$113$		−	10436.4i		−	0.817324i	0.912686	−	0.408662i	$-0.134004\pi$
$114$		0			0
$115$	5132.29			0.388075
$116$		0			0
$117$	1109.15	−	1321.83i	0.0810248	−	0.0965616i
$118$		0			0
$119$	−25837.3	+	9404.00i	−1.82454	+	0.664077i
$120$		0			0
$121$	1240.11	+	2147.94i	0.0847013	+	0.146707i
$122$		0			0
$123$	2044.67	+	11595.9i	0.135149	+	0.766467i
$124$		0			0
$125$	−8475.91	+	14680.7i	−0.542458	+	0.939565i
$126$		0			0
$127$	2218.85	+	2644.32i	0.137569	+	0.163948i	0.830430	−	0.557123i	$-0.188095\pi$
$127$	2218.85	+	2644.32i	0.137569	+	0.163948i	−0.692861	+	0.721071i	$0.743650\pi$
$128$		0			0
$129$	3369.07	−	9256.44i	0.202456	−	0.556243i
$130$		0			0
$131$	−2311.17	+	13107.3i	−0.134676	+	0.763784i	0.840409	+	0.541952i	$0.182315\pi$
$131$	−2311.17	+	13107.3i	−0.134676	+	0.763784i	−0.975085	+	0.221832i	$0.928796\pi$
$132$		0			0
$133$	−21746.1	+	14251.2i	−1.22936	+	0.805653i
$134$		0			0
$135$	13526.3	+	2385.06i	0.742186	+	0.130867i
$136$		0			0
$137$	504.996	+	183.804i	0.0269059	+	0.00979293i	0.355438	−	0.934700i	$-0.384332\pi$
$137$	504.996	+	183.804i	0.0269059	+	0.00979293i	−0.328532	+	0.944493i	$0.606554\pi$
$138$		0			0
$139$	−7577.41	+	6358.20i	−0.392185	+	0.329083i	−0.817464	−	0.575980i	$-0.804621\pi$
$139$	−7577.41	+	6358.20i	−0.392185	+	0.329083i	0.425278	+	0.905063i	$0.360176\pi$
$140$		0			0
$141$	13440.4	+	7759.82i	0.676042	+	0.390313i
$142$		0			0
$143$	−3105.23	+	547.535i	−0.151852	+	0.0267757i
$144$		0			0
$145$	22055.5	−	12733.7i	1.04901	−	0.605647i
$146$		0			0
$147$	4330.45	+	11897.8i	0.200400	+	0.550595i
$148$		0			0
$149$	−21693.6	−	18203.1i	−0.977146	−	0.819923i	0.00651023	−	0.999979i	$-0.497928\pi$
$149$	−21693.6	−	18203.1i	−0.977146	−	0.819923i	−0.983656	+	0.180056i	$0.942372\pi$
$150$		0			0
$151$		−	17781.4i		−	0.779850i	−0.920847	−	0.389925i	$-0.872501\pi$
$151$		−	17781.4i		−	0.779850i	0.920847	−	0.389925i	$-0.127499\pi$
$152$		0			0
$153$	23038.8			0.984185
$154$		0			0
$155$	−12803.9	+	15259.1i	−0.532942	+	0.635136i
$156$		0			0
$157$	−30266.2	+	11016.0i	−1.22789	+	0.446915i	−0.872874	−	0.487946i	$-0.837746\pi$
$157$	−30266.2	+	11016.0i	−1.22789	+	0.446915i	−0.355015	+	0.934861i	$0.615524\pi$
$158$		0			0
$159$	5313.01	+	9202.39i	0.210158	+	0.364004i
$160$		0			0
$161$	−3001.84	−	17024.3i	−0.115807	−	0.656776i
$162$		0			0
$163$	20771.9	−	35978.0i	0.781810	−	1.35414i	−0.149076	−	0.988826i	$-0.547630\pi$
$163$	20771.9	−	35978.0i	0.781810	−	1.35414i	0.930886	−	0.365309i	$-0.119037\pi$
$164$		0			0
$165$	−6887.92	−	8208.71i	−0.253000	−	0.301513i
$166$		0			0
$167$	−8320.45	+	22860.3i	−0.298342	+	0.819687i	0.696436	+	0.717619i	$0.254768\pi$
$167$	−8320.45	+	22860.3i	−0.298342	+	0.819687i	−0.994777	+	0.102068i	$0.967454\pi$
$168$		0			0
$169$	4817.60	−	27321.9i	0.168677	−	0.956617i
$170$		0			0
$171$	21206.6	−	4989.05i	0.725234	−	0.170618i
$172$		0			0
$173$	−7993.40	−	1409.45i	−0.267079	−	0.0470932i	0.0385050	−	0.999258i	$-0.487740\pi$
$173$	−7993.40	−	1409.45i	−0.267079	−	0.0470932i	−0.305584	+	0.952165i	$0.598852\pi$
$174$		0			0
$175$	11355.9	+	4133.22i	0.370806	+	0.134962i
$176$		0			0
$177$	−659.838	+	553.670i	−0.0210616	+	0.0176728i
$178$		0			0
$179$	−17021.7	−	9827.49i	−0.531248	−	0.306716i	0.210277	−	0.977642i	$-0.432564\pi$
$179$	−17021.7	−	9827.49i	−0.531248	−	0.306716i	−0.741525	+	0.670926i	$0.765897\pi$
$180$		0			0
$181$	52617.3	−	9277.85i	1.60610	−	0.283198i	0.702533	−	0.711651i	$-0.252052\pi$
$181$	52617.3	−	9277.85i	1.60610	−	0.283198i	0.903564	+	0.428453i	$0.140941\pi$
$182$		0			0
$183$	−5916.46	+	3415.87i	−0.176669	+	0.102000i
$184$		0			0
$185$	10173.9	+	27952.4i	0.297264	+	0.816726i
$186$		0			0
$187$	−32250.2	−	27061.2i	−0.922253	−	0.773862i
$188$		0			0
$189$		−	46263.1i		−	1.29512i
$190$		0			0
$191$	−3031.85			−0.0831075			−0.0415538	−	0.999136i	$-0.513231\pi$
$191$	−3031.85			−0.0831075			−0.0415538	+	0.999136i	$0.513231\pi$
$192$		0			0
$193$	43770.8	−	52164.1i	1.17509	−	1.40041i	0.276845	−	0.960915i	$-0.410711\pi$
$193$	43770.8	−	52164.1i	1.17509	−	1.40041i	0.898242	−	0.439500i	$-0.144844\pi$
$194$		0			0
$195$	−2610.88	+	950.283i	−0.0686623	+	0.0249910i
$196$		0			0
$197$	−3966.59	−	6870.34i	−0.102208	−	0.177030i	0.810386	−	0.585896i	$-0.199257\pi$
$197$	−3966.59	−	6870.34i	−0.102208	−	0.177030i	−0.912594	+	0.408867i	$0.865924\pi$
$198$		0			0
$199$	10936.9	+	62026.3i	0.276178	+	1.56628i	0.735195	+	0.677855i	$0.237090\pi$
$199$	10936.9	+	62026.3i	0.276178	+	1.56628i	−0.459018	+	0.888427i	$0.651799\pi$
$200$		0			0
$201$	17077.6	−	29579.2i	0.422702	−	0.732141i
$202$		0			0
$203$	−55139.1	−	65712.2i	−1.33803	−	1.59461i
$204$		0			0
$205$	−18948.6	+	52060.8i	−0.450888	+	1.23881i
$206$		0			0
$207$	−2515.28	+	14264.9i	−0.0587010	+	0.332910i
$208$		0			0
$209$	−35545.6	−	17925.3i	−0.813754	−	0.410368i
$210$		0			0
$211$	−40436.8	−	7130.10i	−0.908264	−	0.160151i	−0.300049	−	0.953924i	$-0.597003\pi$
$211$	−40436.8	−	7130.10i	−0.908264	−	0.160151i	−0.608215	+	0.793773i	$0.708114\pi$
$212$		0			0
$213$	−16709.8	−	6081.87i	−0.368309	−	0.134053i
$214$		0			0
$215$	35504.6	−	29791.9i	0.768082	−	0.644498i
$216$		0			0
$217$	58104.9	+	33546.9i	1.23394	+	0.712414i
$218$		0			0
$219$	15454.7	−	2725.09i	0.322236	−	0.0568188i
$220$		0			0
$221$	−9453.45	+	5457.95i	−0.193556	+	0.111749i
$222$		0			0
$223$	20272.3	+	55697.7i	0.407656	+	1.12003i	0.958419	+	0.285364i	$0.0921144\pi$
$223$	20272.3	+	55697.7i	0.407656	+	1.12003i	−0.550763	+	0.834661i	$0.685663\pi$
$224$		0			0
$225$	−7756.93	−	6508.83i	−0.153223	−	0.128570i
$226$		0			0
$227$			87779.7i			1.70350i	0.523947	+	0.851751i	$0.324459\pi$
$227$			87779.7i			1.70350i	−0.523947	+	0.851751i	$0.675541\pi$
$228$		0			0
$229$	−30764.0			−0.586640			−0.293320	−	0.956014i	$-0.594760\pi$
$229$	−30764.0			−0.586640			−0.293320	+	0.956014i	$0.594760\pi$
$230$		0			0
$231$	−23200.3	+	27649.1i	−0.434781	+	0.518152i
$232$		0			0
$233$	26105.0	−	9501.45i	0.480853	−	0.175016i	−0.0902096	−	0.995923i	$-0.528754\pi$
$233$	26105.0	−	9501.45i	0.480853	−	0.175016i	0.571062	+	0.820907i	$0.306531\pi$
$234$		0			0
$235$	36511.0	+	63239.0i	0.661132	+	1.14511i
$236$		0			0
$237$	−841.141	−	4770.35i	−0.0149752	−	0.0849285i
$238$		0			0
$239$	42411.9	−	73459.6i	0.742492	−	1.28603i	−0.208865	−	0.977945i	$-0.566977\pi$
$239$	42411.9	−	73459.6i	0.742492	−	1.28603i	0.951357	−	0.308090i	$-0.0996898\pi$
$240$		0			0
$241$	−28331.8	−	33764.5i	−0.487797	−	0.581334i	0.464859	−	0.885385i	$-0.346105\pi$
$241$	−28331.8	−	33764.5i	−0.487797	−	0.581334i	−0.952656	+	0.304051i	$0.901661\pi$
$242$		0			0
$243$	−20855.9	+	57301.1i	−0.353197	+	0.970400i
$244$		0			0
$245$	−10344.8	+	58668.5i	−0.172342	+	0.977401i
$246$		0			0
$247$	−7519.73	+	7071.04i	−0.123256	+	0.115902i
$248$		0			0
$249$	−23495.1	−	4142.83i	−0.378948	−	0.0668187i
$250$		0			0
$251$	−34909.5	−	12706.0i	−0.554111	−	0.201680i	0.0497613	−	0.998761i	$-0.484154\pi$
$251$	−34909.5	−	12706.0i	−0.554111	−	0.201680i	−0.603872	+	0.797081i	$0.706376\pi$
$252$		0			0
$253$	20276.3	−	17013.9i	0.316773	−	0.265804i
$254$		0			0
$255$	−32126.9	−	18548.5i	−0.494070	−	0.285252i
$256$		0			0
$257$	64595.5	−	11389.9i	0.977993	−	0.172447i	0.338267	−	0.941050i	$-0.390159\pi$
$257$	64595.5	−	11389.9i	0.977993	−	0.172447i	0.639725	+	0.768603i	$0.279048\pi$
$258$		0			0
$259$	86770.2	−	50096.8i	1.29351	−	0.746811i
$260$		0			0
$261$	24583.4	+	67542.4i	0.360879	+	0.991506i
$262$		0			0
$263$	−59033.7	−	49535.1i	−0.853470	−	0.716146i	0.107081	−	0.994250i	$-0.465850\pi$
$263$	−59033.7	−	49535.1i	−0.853470	−	0.716146i	−0.960551	+	0.278104i	$0.910294\pi$
$264$		0			0
$265$			49996.9i			0.711952i
$266$		0			0
$267$	−2357.88			−0.0330749
$268$		0			0
$269$	−69947.8	+	83360.5i	−0.966650	+	1.15201i	0.0216924	+	0.999765i	$0.493095\pi$
$269$	−69947.8	+	83360.5i	−0.966650	+	1.15201i	−0.988343	+	0.152245i	$0.951350\pi$
$270$		0			0
$271$	−82365.7	+	29978.7i	−1.12152	+	0.408201i	−0.835208	−	0.549935i	$-0.814653\pi$
$271$	−82365.7	+	29978.7i	−1.12152	+	0.408201i	−0.286315	+	0.958136i	$0.592430\pi$
$272$		0			0
$273$	4679.26	+	8104.72i	0.0627845	+	0.108746i
$274$		0			0
$275$	3213.11	+	18222.4i	0.0424874	+	0.240958i
$276$		0			0
$277$	3982.46	−	6897.82i	0.0519029	−	0.0898985i	−0.838907	−	0.544275i	$-0.816805\pi$
$277$	3982.46	−	6897.82i	0.0519029	−	0.0898985i	0.890810	+	0.454377i	$0.150138\pi$
$278$		0			0
$279$	−36136.7	−	43066.0i	−0.464237	−	0.553256i
$280$		0			0
$281$	−34569.7	+	94979.4i	−0.437807	+	1.20286i	0.503109	+	0.864223i	$0.332190\pi$
$281$	−34569.7	+	94979.4i	−0.437807	+	1.20286i	−0.940916	+	0.338641i	$0.890033\pi$
$282$		0			0
$283$	−6296.57	+	35709.6i	−0.0786196	+	0.445874i	0.919932	+	0.392077i	$0.128243\pi$
$283$	−6296.57	+	35709.6i	−0.0786196	+	0.445874i	−0.998552	+	0.0537968i	$0.982868\pi$
$284$		0			0
$285$	−33588.6	−	10116.3i	−0.413526	−	0.124547i
$286$		0			0
$287$	183773.	+	32404.2i	2.23110	+	0.393403i
$288$		0			0
$289$	−58472.5	−	21282.2i	−0.700093	−	0.254813i
$290$		0			0
$291$	48778.6	−	40930.1i	0.576028	−	0.483345i
$292$		0			0
$293$	86673.3	+	50040.9i	1.00960	+	0.582894i	0.911075	−	0.412241i	$-0.135254\pi$
$293$	86673.3	+	50040.9i	1.00960	+	0.582894i	0.0985267	+	0.995134i	$0.468587\pi$
$294$		0			0
$295$	−3991.24	+	703.763i	−0.0458631	+	0.00808690i
$296$		0			0
$297$	61345.7	−	35417.9i	0.695458	−	0.401523i
$298$		0			0
$299$	−2347.30	−	6449.15i	−0.0262558	−	0.0721373i
$300$		0			0
$301$	−119589.	−	100347.i	−1.31995	−	1.10757i
$302$		0			0
$303$		−	34503.1i		−	0.375814i
$304$		0			0
$305$	−32144.3			−0.345544
$306$		0			0
$307$	19427.7	−	23153.0i	0.206132	−	0.245658i	−0.653067	−	0.757300i	$-0.726518\pi$
$307$	19427.7	−	23153.0i	0.206132	−	0.245658i	0.859199	+	0.511642i	$0.170963\pi$
$308$		0			0
$309$	−43730.9	+	15916.8i	−0.458007	+	0.166701i
$310$		0			0
$311$	−89682.7	−	155335.i	−0.927231	−	1.60601i	−0.787934	−	0.615760i	$-0.788849\pi$
$311$	−89682.7	−	155335.i	−0.927231	−	1.60601i	−0.139297	−	0.990251i	$-0.544484\pi$
$312$		0			0
$313$	−13398.2	−	75984.8i	−0.136759	−	0.775600i	−0.973618	−	0.228182i	$-0.926722\pi$
$313$	−13398.2	−	75984.8i	−0.136759	−	0.775600i	0.836859	−	0.547418i	$-0.184389\pi$
$314$		0			0
$315$	46467.5	−	80484.2i	0.468305	−	0.811128i
$316$		0			0
$317$	33428.4	+	39838.4i	0.332657	+	0.396445i	0.906283	−	0.422672i	$-0.138908\pi$
$317$	33428.4	+	39838.4i	0.332657	+	0.396445i	−0.573625	+	0.819118i	$0.694463\pi$
$318$		0			0
$319$	44922.3	−	123423.i	0.441449	−	1.21287i
$320$		0			0
$321$	−774.353	+	4391.58i	−0.00751500	+	0.0426197i
$322$		0			0
$323$	−136859.	−	16225.9i	−1.31181	−	0.155526i
$324$		0			0
$325$	4724.84	+	833.118i	0.0447323	+	0.00788750i
$326$		0			0
$327$	76098.4	+	27697.6i	0.711672	+	0.259028i
$328$		0			0
$329$	188414.	−	158098.i	1.74069	−	1.46062i
$330$		0			0
$331$	134064.	+	77401.8i	1.22365	+	0.706473i	0.965693	−	0.259685i	$-0.0836187\pi$
$331$	134064.	+	77401.8i	1.22365	+	0.706473i	0.257953	+	0.966157i	$0.416952\pi$
$332$		0			0
$333$	−82678.0	+	14578.4i	−0.745593	+	0.131468i
$334$		0			0
$335$	139174.	−	80352.3i	1.24014	−	0.715993i
$336$		0			0
$337$	56873.9	+	156260.i	0.500788	+	1.37590i	0.890507	+	0.454969i	$0.150350\pi$
$337$	56873.9	+	156260.i	0.500788	+	1.37590i	−0.389720	+	0.920934i	$0.627428\pi$
$338$		0			0
$339$	−36331.9	−	30486.1i	−0.316147	−	0.265279i
$340$		0			0
$341$			102731.i			0.883469i
$342$		0			0
$343$	27735.7			0.235750
$344$		0			0
$345$	14992.1	−	17866.9i	0.125958	−	0.150110i
$346$		0			0
$347$	−97436.4	+	35463.9i	−0.809212	+	0.294529i	−0.713298	−	0.700861i	$-0.752799\pi$
$347$	−97436.4	+	35463.9i	−0.809212	+	0.294529i	−0.0959134	+	0.995390i	$0.530577\pi$
$348$		0			0
$349$	−91112.1	−	157811.i	−0.748041	−	1.29564i	−0.948761	−	0.315996i	$-0.897661\pi$
$349$	−91112.1	−	157811.i	−0.748041	−	1.29564i	0.200720	−	0.979649i	$-0.435672\pi$
$350$		0			0
$351$	−3189.37	−	18087.8i	−0.0258875	−	0.146815i
$352$		0			0
$353$	79186.0	−	137154.i	0.635476	−	1.10068i	−0.350938	−	0.936399i	$-0.614137\pi$
$353$	79186.0	−	137154.i	0.635476	−	1.10068i	0.986414	−	0.164278i	$-0.0525295\pi$
$354$		0			0
$355$	−53780.6	−	64093.2i	−0.426745	−	0.508575i
$356$		0			0
$357$	−42736.2	+	117417.i	−0.335320	+	0.921285i
$358$		0			0
$359$	−31561.2	+	178993.i	−0.244886	+	1.38882i	0.575870	+	0.817542i	$0.304664\pi$
$359$	−31561.2	+	178993.i	−0.244886	+	1.38882i	−0.820756	+	0.571279i	$0.806448\pi$
$360$		0			0
$361$	−129489.	+	14701.4i	−0.993617	+	0.112809i
$362$		0			0
$363$	11100.1	+	1957.24i	0.0842390	+	0.0148536i
$364$		0			0
$365$	69385.5	+	25254.2i	0.520814	+	0.189561i
$366$		0			0
$367$	−88559.0	+	74309.9i	−0.657508	+	0.551714i	−0.909339	−	0.416057i	$-0.863412\pi$
$367$	−88559.0	+	74309.9i	−0.657508	+	0.551714i	0.251831	+	0.967771i	$0.418967\pi$
$368$		0			0
$369$	−135413.	−	78180.7i	−0.994506	−	0.574179i
$370$		0			0
$371$	165844.	−	29242.8i	1.20490	−	0.212457i
$372$		0			0
$373$	−158975.	+	91784.3i	−1.14265	+	0.659706i	−0.947084	−	0.320985i	$-0.895986\pi$
$373$	−158975.	+	91784.3i	−1.14265	+	0.659706i	−0.195561	+	0.980692i	$0.562653\pi$
$374$		0			0
$375$	26348.2	+	72391.2i	0.187365	+	0.514782i
$376$		0			0
$377$	−26088.2	−	21890.6i	−0.183553	−	0.154020i
$378$		0			0
$379$		−	227813.i		−	1.58599i	−0.609228	−	0.792995i	$-0.708521\pi$
$379$		−	227813.i		−	1.58599i	0.609228	−	0.792995i	$-0.291479\pi$
$380$		0			0
$381$	15687.1			0.108067
$382$		0			0
$383$	11702.7	−	13946.7i	0.0797789	−	0.0950768i	−0.724679	−	0.689087i	$-0.758012\pi$
$383$	11702.7	−	13946.7i	0.0797789	−	0.0950768i	0.804457	+	0.594010i	$0.202456\pi$
$384$		0			0
$385$	−159582.	+	58083.3i	−1.07662	+	0.391859i
$386$		0			0
$387$	65404.2	+	113283.i	0.436700	+	0.756387i
$388$		0			0
$389$	19572.2	+	111000.i	0.129342	+	0.733537i	0.978634	+	0.205612i	$0.0659186\pi$
$389$	19572.2	+	111000.i	0.129342	+	0.733537i	−0.849291	+	0.527925i	$0.822970\pi$
$390$		0			0
$391$	45816.7	−	79356.8i	0.299688	−	0.519076i
$392$		0			0
$393$	38878.8	+	46334.0i	0.251726	+	0.299995i
$394$		0			0
$395$	7795.12	−	21416.9i	0.0499607	−	0.137266i
$396$		0			0
$397$	−32755.9	+	185768.i	−0.207830	+	1.17866i	0.685094	+	0.728455i	$0.259761\pi$
$397$	−32755.9	+	185768.i	−0.207830	+	1.17866i	−0.892924	+	0.450207i	$0.851350\pi$
$398$		0			0
$399$	−13910.9	+	117334.i	−0.0873798	+	0.737015i
$400$		0			0
$401$	252009.	+	44436.0i	1.56721	+	0.276341i	0.888784	−	0.458327i	$-0.151551\pi$
$401$	252009.	+	44436.0i	1.56721	+	0.276341i	0.678427	+	0.734668i	$0.262662\pi$
$402$		0			0
$403$	25030.3	+	9110.30i	0.154119	+	0.0560948i
$404$		0			0
$405$	−32252.2	+	27062.8i	−0.196630	+	0.164992i
$406$		0			0
$407$	132858.	+	76705.8i	0.802047	+	0.463062i
$408$		0			0
$409$	198698.	−	35035.8i	1.18781	−	0.209443i	0.455386	−	0.890294i	$-0.349501\pi$
$409$	198698.	−	35035.8i	1.18781	−	0.209443i	0.732422	+	0.680851i	$0.238390\pi$
$410$		0			0
$411$	2115.03	−	1221.11i	0.0125208	−	0.00722890i
$412$		0			0
$413$	4668.89	+	12827.7i	0.0273724	+	0.0752052i
$414$		0			0
$415$	−85991.0	−	72155.0i	−0.499295	−	0.418958i
$416$		0			0
$417$			44952.2i			0.258511i
$418$		0			0
$419$	−248905.			−1.41777			−0.708884	−	0.705325i	$-0.750801\pi$
$419$	−248905.			−1.41777			−0.708884	+	0.705325i	$0.750801\pi$
$420$		0			0
$421$	−33536.2	+	39966.9i	−0.189212	+	0.225494i	−0.852308	−	0.523040i	$-0.824798\pi$
$421$	−33536.2	+	39966.9i	−0.189212	+	0.225494i	0.663096	+	0.748535i	$0.269242\pi$
$422$		0			0
$423$	−193662.	+	70487.2i	−1.08234	+	0.393940i
$424$		0			0
$425$	32029.0	+	55475.8i	0.177323	+	0.307133i
$426$		0			0
$427$	18801.0	+	106626.i	0.103116	+	0.584797i
$428$		0			0
$429$	−7164.66	+	12409.6i	−0.0389297	+	0.0674282i
$430$		0			0
$431$	−75081.8	−	89479.0i	−0.404185	−	0.481689i	0.525106	−	0.851037i	$-0.324026\pi$
$431$	−75081.8	−	89479.0i	−0.404185	−	0.481689i	−0.929291	+	0.369348i	$0.879581\pi$
$432$		0			0
$433$	−95539.5	+	262493.i	−0.509574	+	1.40004i	0.372104	+	0.928191i	$0.378636\pi$
$433$	−95539.5	+	262493.i	−0.509574	+	1.40004i	−0.881678	+	0.471851i	$0.843586\pi$
$434$		0			0
$435$	20097.4	−	113978.i	0.106209	−	0.602341i
$436$		0			0
$437$	24988.3	−	82967.4i	0.130850	−	0.434455i
$438$		0			0
$439$	−104136.	−	18362.1i	−0.540348	−	0.0952780i	−0.103188	−	0.994662i	$-0.532904\pi$
$439$	−104136.	−	18362.1i	−0.540348	−	0.0952780i	−0.437160	+	0.899384i	$0.644016\pi$
$440$		0			0
$441$	−157995.	−	57505.6i	−0.812394	−	0.295687i
$442$		0			0
$443$	45631.2	−	38289.2i	0.232517	−	0.195105i	−0.519083	−	0.854724i	$-0.673727\pi$
$443$	45631.2	−	38289.2i	0.232517	−	0.195105i	0.751600	+	0.659619i	$0.229282\pi$
$444$		0			0
$445$	−9607.82	−	5547.08i	−0.0485182	−	0.0280120i
$446$		0			0
$447$	−126740.	+	22347.7i	−0.634305	+	0.111845i
$448$		0			0
$449$	278912.	−	161030.i	1.38349	−	0.798756i	0.390916	−	0.920426i	$-0.372158\pi$
$449$	278912.	−	161030.i	1.38349	−	0.798756i	0.992571	+	0.121670i	$0.0388250\pi$
$450$		0			0
$451$	97723.9	+	268494.i	0.480450	+	1.32002i
$452$		0			0
$453$	−61901.7	−	51941.7i	−0.301652	−	0.253116i
$454$		0			0
$455$			44033.2i			0.212695i
$456$		0			0
$457$	259740.			1.24367			0.621837	−	0.783146i	$-0.286386\pi$
$457$	259740.			1.24367			0.621837	+	0.783146i	$0.286386\pi$
$458$		0			0
$459$	157630.	−	187856.i	0.748192	−	0.891661i
$460$		0			0
$461$	197505.	−	71885.8i	0.929342	−	0.338253i	0.167393	−	0.985890i	$-0.446465\pi$
$461$	197505.	−	71885.8i	0.929342	−	0.338253i	0.761949	+	0.647638i	$0.224243\pi$
$462$		0			0
$463$	121468.	+	210389.i	0.566631	+	0.981433i	0.996896	+	0.0787306i	$0.0250867\pi$
$463$	121468.	+	210389.i	0.566631	+	0.981433i	−0.430265	+	0.902703i	$0.641580\pi$
$464$		0			0
$465$	15719.2	+	89147.9i	0.0726983	+	0.412292i
$466$		0			0
$467$	−79005.0	+	136841.i	−0.362261	+	0.627454i	−0.988333	−	0.152312i	$-0.951328\pi$
$467$	−79005.0	+	136841.i	−0.362261	+	0.627454i	0.626072	+	0.779765i	$0.284662\pi$
$468$		0			0
$469$	−347938.	−	414657.i	−1.58182	−	1.88514i
$470$		0			0
$471$	−50062.0	+	137544.i	−0.225666	+	0.620012i
$472$		0			0
$473$	41507.4	−	235400.i	0.185525	−	1.05217i
$474$		0			0
$475$	41495.1	+	44128.1i	0.183912	+	0.195582i
$476$		0			0
$477$	−138963.	−	24502.9i	−0.610748	−	0.107691i
$478$		0			0
$479$	−190477.	−	69328.0i	−0.830179	−	0.302161i	−0.108247	−	0.994124i	$-0.534524\pi$
$479$	−190477.	−	69328.0i	−0.830179	−	0.302161i	−0.721932	+	0.691964i	$0.756746\pi$
$480$		0			0
$481$	30471.4	−	25568.6i	0.131705	−	0.110514i
$482$		0			0
$483$	−68034.9	−	39280.0i	−0.291634	−	0.168375i
$484$		0			0
$485$	295053.	−	52025.7i	1.25434	−	0.221174i
$486$		0			0
$487$	−99429.3	+	57405.5i	−0.419234	+	0.242045i	−0.694750	−	0.719252i	$-0.744485\pi$
$487$	−99429.3	+	57405.5i	−0.419234	+	0.242045i	0.275516	+	0.961297i	$0.411151\pi$
$488$		0			0
$489$	−64571.7	−	177409.i	−0.270038	−	0.741922i
$490$		0			0
$491$	125947.	+	105682.i	0.522428	+	0.438369i	0.865477	−	0.500949i	$-0.167015\pi$
$491$	125947.	+	105682.i	0.522428	+	0.438369i	−0.343050	+	0.939317i	$0.611460\pi$
$492$		0			0
$493$		−	454703.i		−	1.87083i
$494$		0			0
$495$	142298.			0.580748
$496$		0			0
$497$	−181147.	+	215883.i	−0.733363	+	0.873988i
$498$		0			0
$499$	−63335.8	+	23052.3i	−0.254359	+	0.0925793i	−0.466053	−	0.884757i	$-0.654324\pi$
$499$	−63335.8	+	23052.3i	−0.254359	+	0.0925793i	0.211693	+	0.977336i	$0.432102\pi$
$500$		0			0
$501$	55277.6	+	95743.6i	0.220229	+	0.381447i
$502$		0			0
$503$	−25707.1	−	145792.i	−0.101606	−	0.576233i	−0.992522	−	0.122066i	$-0.961048\pi$
$503$	−25707.1	−	145792.i	−0.101606	−	0.576233i	0.890916	−	0.454167i	$-0.150063\pi$
$504$		0			0
$505$	81171.1	−	140592.i	0.318287	−	0.551289i
$506$		0			0
$507$	−81042.2	−	96582.4i	−0.315279	−	0.375735i
$508$		0			0
$509$	18261.1	−	50171.8i	0.0704839	−	0.193653i	−0.899449	−	0.437026i	$-0.856032\pi$
$509$	18261.1	−	50171.8i	0.0704839	−	0.193653i	0.969933	+	0.243373i	$0.0782539\pi$
$510$		0			0
$511$	43187.6	−	244929.i	0.165393	−	0.937991i
$512$		0			0
$513$	104414.	−	207051.i	0.396756	−	0.786761i
$514$		0			0
$515$	−215639.	−	38022.9i	−0.813041	−	0.143361i
$516$		0			0
$517$	353887.	+	128804.i	1.32399	+	0.481891i
$518$		0			0
$519$	−28256.5	+	23710.0i	−0.104902	+	0.0880231i
$520$		0			0
$521$	−309057.	−	178434.i	−1.13858	−	0.657360i	−0.192502	−	0.981297i	$-0.561660\pi$
$521$	−309057.	−	178434.i	−1.13858	−	0.657360i	−0.946079	+	0.323937i	$0.894994\pi$
$522$		0			0
$523$	112140.	−	19773.3i	0.409974	−	0.0722896i	0.0351424	−	0.999382i	$-0.488812\pi$
$523$	112140.	−	19773.3i	0.409974	−	0.0722896i	0.374832	+	0.927093i	$0.377700\pi$
$524$		0			0
$525$	47561.1	−	27459.4i	0.172557	−	0.0996259i
$526$		0			0
$527$	121638.	+	334198.i	0.437974	+	1.20332i
$528$		0			0
$529$	−170238.	−	142846.i	−0.608337	−	0.510455i
$530$		0			0
$531$		−	11438.3i		−	0.0405669i
$532$		0			0
$533$	74085.0			0.260781
$534$		0			0
$535$	−13486.8	+	16072.9i	−0.0471196	+	0.0561549i
$536$		0			0
$537$	−83934.8	+	30549.8i	−0.291068	+	0.105940i
$538$		0			0
$539$	153620.	+	266078.i	0.528774	+	0.915864i
$540$		0			0
$541$	36704.4	+	208161.i	0.125407	+	0.711221i	0.981065	+	0.193678i	$0.0620419\pi$
$541$	36704.4	+	208161.i	0.125407	+	0.711221i	−0.855658	+	0.517542i	$0.826847\pi$
$542$		0			0
$543$	121403.	−	210277.i	0.411748	−	0.713168i
$544$		0			0
$545$	244923.	+	291888.i	0.824588	+	0.982705i
$546$		0			0
$547$	−2504.61	+	6881.36i	−0.00837077	+	0.0229985i	−0.943808	−	0.330495i	$-0.892784\pi$
$547$	−2504.61	+	6881.36i	−0.00837077	+	0.0229985i	0.935437	+	0.353494i	$0.115006\pi$
$548$		0			0
$549$	15753.6	−	89342.8i	0.0522678	−	0.296425i
$550$		0			0
$551$	−98466.0	−	418542.i	−0.324327	−	1.37859i
$552$		0			0
$553$	−75601.2	−	13330.5i	−0.247217	−	0.0435910i
$554$		0			0
$555$	127029.	+	46234.8i	0.412399	+	0.150101i
$556$		0			0
$557$	330651.	−	277449.i	1.06576	−	0.894280i	0.0710994	−	0.997469i	$-0.477349\pi$
$557$	330651.	−	277449.i	1.06576	−	0.894280i	0.994662	+	0.103189i	$0.0329048\pi$
$558$		0			0
$559$	−53674.3	−	30988.9i	−0.171768	−	0.0991704i
$560$		0			0
$561$	−188415.	+	33222.6i	−0.598672	+	0.105562i
$562$		0			0
$563$	−227258.	+	131207.i	−0.716971	+	0.413943i	−0.813637	−	0.581373i	$-0.802516\pi$
$563$	−227258.	+	131207.i	−0.716971	+	0.413943i	0.0966658	+	0.995317i	$0.469182\pi$
$564$		0			0
$565$	−76323.6	−	209697.i	−0.239090	−	0.656895i
$566$		0			0
$567$	108634.	+	91154.8i	0.337909	+	0.283539i
$568$		0			0
$569$			592913.i			1.83133i	0.401943	+	0.915665i	$0.368335\pi$
$569$			592913.i			1.83133i	−0.401943	+	0.915665i	$0.631665\pi$
$570$		0			0
$571$	−37331.0			−0.114498			−0.0572489	−	0.998360i	$-0.518233\pi$
$571$	−37331.0			−0.114498			−0.0572489	+	0.998360i	$0.518233\pi$
$572$		0			0
$573$	−8856.42	+	10554.7i	−0.0269742	+	0.0321466i
$574$		0			0
$575$	−37845.6	+	13774.7i	−0.114467	+	0.0416625i
$576$		0			0
$577$	−121380.	−	210236.i	−0.364581	−	0.631473i	0.624128	−	0.781322i	$-0.285454\pi$
$577$	−121380.	−	210236.i	−0.364581	−	0.631473i	−0.988709	+	0.149849i	$0.952121\pi$
$578$		0			0
$579$	−53736.7	−	304756.i	−0.160293	−	0.909066i
$580$		0			0
$581$	−189049.	+	327443.i	−0.560045	+	0.970027i
$582$		0			0
$583$	165743.	+	197525.i	0.487638	+	0.581144i
$584$		0			0
$585$	12619.2	−	34670.9i	0.0368739	−	0.101310i
$586$		0			0
$587$	18243.6	−	103464.i	0.0529460	−	0.300272i	−0.946823	−	0.321754i	$-0.895727\pi$
$587$	18243.6	−	103464.i	0.0529460	−	0.300272i	0.999769	+	0.0214825i	$0.00683861\pi$
$588$		0			0
$589$	184335.	+	281280.i	0.531346	+	0.810789i
$590$		0			0
$591$	−35504.5	−	6260.40i	−0.101650	−	0.0179237i
$592$		0			0
$593$	−177281.	−	64525.0i	−0.504142	−	0.183493i	0.0774143	−	0.996999i	$-0.475334\pi$
$593$	−177281.	−	64525.0i	−0.504142	−	0.183493i	−0.581556	+	0.813506i	$0.697556\pi$
$594$		0			0
$595$	−450372.	+	377907.i	−1.27215	+	1.06746i
$596$		0			0
$597$	247879.	+	143113.i	0.695489	+	0.401541i
$598$		0			0
$599$	−345357.	+	60895.7i	−0.962530	+	0.169720i	−0.632766	−	0.774343i	$-0.718080\pi$
$599$	−345357.	+	60895.7i	−0.962530	+	0.169720i	−0.329764	+	0.944063i	$0.606969\pi$
$600$		0			0
$601$	−184895.	+	106749.i	−0.511889	+	0.295539i	−0.733610	−	0.679571i	$-0.762166\pi$
$601$	−184895.	+	106749.i	−0.511889	+	0.295539i	0.221721	+	0.975110i	$0.428833\pi$
$602$		0			0
$603$	155126.	+	426205.i	0.426629	+	1.17215i
$604$		0			0
$605$	40625.7	+	34089.0i	0.110992	+	0.0931331i
$606$		0			0
$607$		−	94696.2i		−	0.257013i	−0.991709	−	0.128507i	$-0.958982\pi$
$607$		−	94696.2i		−	0.257013i	0.991709	−	0.128507i	$-0.0410183\pi$
$608$		0			0
$609$	−389830.			−1.05109
$610$		0			0
$611$	62766.3	−	74802.0i	0.168130	−	0.200369i
$612$		0			0
$613$	−81985.0	+	29840.1i	−0.218179	+	0.0794108i	−0.448797	−	0.893634i	$-0.648147\pi$
$613$	−81985.0	+	29840.1i	−0.218179	+	0.0794108i	0.230618	+	0.973044i	$0.425925\pi$
$614$		0			0
$615$	125886.	+	218042.i	0.332835	+	0.576486i
$616$		0			0
$617$	28590.9	+	162147.i	0.0751031	+	0.425931i	0.999057	+	0.0434262i	$0.0138274\pi$
$617$	28590.9	+	162147.i	0.0751031	+	0.425931i	−0.923953	+	0.382505i	$0.875062\pi$
$618$		0			0
$619$	−161150.	+	279120.i	−0.420580	+	0.728466i	−0.995996	−	0.0893948i	$-0.971507\pi$
$619$	−161150.	+	279120.i	−0.420580	+	0.728466i	0.575416	+	0.817861i	$0.304840\pi$
$620$		0			0
$621$	99104.9	+	118109.i	0.256988	+	0.306266i
$622$		0			0
$623$	−12780.6	+	35114.5i	−0.0329288	+	0.0904711i
$624$		0			0
$625$	−44731.7	+	253686.i	−0.114513	+	0.649436i
$626$		0			0
$627$	−166236.	+	71381.6i	−0.422854	+	0.181573i
$628$		0			0
$629$	523031.	+	92224.5i	1.32198	+	0.233101i
$630$		0			0
$631$	146067.	+	53164.2i	0.366855	+	0.133524i	0.518868	−	0.854854i	$-0.326353\pi$
$631$	146067.	+	53164.2i	0.366855	+	0.133524i	−0.152013	+	0.988378i	$0.548576\pi$
$632$		0			0
$633$	−142943.	+	119943.i	−0.356743	+	0.299343i
$634$		0			0
$635$	63921.5	+	36905.1i	0.158526	+	0.0915248i
$636$		0			0
$637$	78453.1	−	13833.4i	0.193344	−	0.0340918i
$638$		0			0
$639$	204500.	−	118068.i	0.500832	−	0.289155i
$640$		0			0
$641$	−221071.	−	607387.i	−0.538041	−	1.47826i	−0.849290	−	0.527927i	$-0.822969\pi$
$641$	−221071.	−	607387.i	−0.538041	−	1.47826i	0.311248	−	0.950329i	$-0.399253\pi$
$642$		0			0
$643$	375051.	+	314705.i	0.907127	+	0.761170i	0.971570	−	0.236752i	$-0.0760828\pi$
$643$	375051.	+	314705.i	0.907127	+	0.761170i	−0.0644434	+	0.997921i	$0.520527\pi$
$644$		0			0
$645$		−	210627.i		−	0.506285i
$646$		0			0
$647$	230175.			0.549858			0.274929	−	0.961465i	$-0.411346\pi$
$647$	230175.			0.549858			0.274929	+	0.961465i	$0.411346\pi$
$648$		0			0
$649$	−13435.3	+	16011.6i	−0.0318976	+	0.0380141i
$650$		0			0
$651$	286518.	−	104284.i	0.676067	−	0.246068i
$652$		0			0
$653$	−361571.	−	626259.i	−0.847944	−	1.46868i	−0.883040	−	0.469298i	$-0.844507\pi$
$653$	−361571.	−	626259.i	−0.847944	−	1.46868i	0.0350959	−	0.999384i	$-0.488826\pi$
$654$		0			0
$655$	49418.4	+	280265.i	0.115188	+	0.653261i
$656$		0			0
$657$	−104198.	+	180475.i	−0.241394	+	0.418107i
$658$		0			0
$659$	−200214.	−	238606.i	−0.461025	−	0.549428i	0.484580	−	0.874747i	$-0.338973\pi$
$659$	−200214.	−	238606.i	−0.461025	−	0.549428i	−0.945604	+	0.325320i	$0.894528\pi$
$660$		0			0
$661$	255188.	−	701123.i	0.584060	−	1.60469i	−0.197117	−	0.980380i	$-0.563158\pi$
$661$	255188.	−	701123.i	0.584060	−	1.60469i	0.781176	−	0.624311i	$-0.214620\pi$
$662$		0			0
$663$	−8614.18	+	48853.4i	−0.0195969	+	0.111139i
$664$		0			0
$665$	−332719.	+	445381.i	−0.752376	+	1.00714i
$666$		0			0
$667$	281537.	+	49642.6i	0.632826	+	0.111584i
$668$		0			0
$669$	253117.	+	92127.0i	0.565547	+	0.205842i
$670$		0			0
$671$	−126994.	+	106560.i	−0.282057	+	0.236674i
$672$		0			0
$673$	−166906.	−	96363.3i	−0.368504	−	0.212756i	0.304301	−	0.952576i	$-0.401577\pi$
$673$	−166906.	−	96363.3i	−0.368504	−	0.212756i	−0.672805	+	0.739820i	$0.734911\pi$
$674$		0			0
$675$	−106145.	+	18716.2i	−0.232965	+	0.0410781i
$676$		0			0
$677$	−48977.6	+	28277.2i	−0.106861	+	0.0616963i	−0.552478	−	0.833527i	$-0.686318\pi$
$677$	−48977.6	+	28277.2i	−0.106861	+	0.0616963i	0.445617	+	0.895224i	$0.352984\pi$
$678$		0			0
$679$	−345148.	−	948287.i	−0.748628	−	2.05684i
$680$		0			0
$681$	305585.	+	256416.i	0.658928	+	0.552906i
$682$		0			0
$683$		−	603564.i		−	1.29384i	−0.762556	−	0.646922i	$-0.776056\pi$
$683$		−	603564.i		−	1.29384i	0.762556	−	0.646922i	$-0.223944\pi$
$684$		0			0
$685$	11491.0			0.0244894
$686$		0			0
$687$	−89865.7	+	107098.i	−0.190406	+	0.226917i
$688$		0			0
$689$	62825.2	−	22866.5i	0.132341	−	0.0481683i
$690$		0			0
$691$	−345267.	−	598020.i	−0.723100	−	1.25245i	−0.959751	−	0.280852i	$-0.909383\pi$
$691$	−345267.	−	598020.i	−0.723100	−	1.25245i	0.236651	−	0.971595i	$-0.423950\pi$
$692$		0			0
$693$	−83228.9	−	472015.i	−0.173304	−	0.982854i
$694$		0			0
$695$	−105753.	+	183170.i	−0.218939	+	0.379214i
$696$		0			0
$697$	635821.	+	757742.i	1.30879	+	1.55975i
$698$		0			0
$699$	43179.1	−	118634.i	0.0883729	−	0.242803i
$700$		0			0
$701$	−105203.	+	596633.i	−0.214087	+	1.21415i	0.668397	+	0.743805i	$0.266981\pi$
$701$	−105203.	+	596633.i	−0.214087	+	1.21415i	−0.882484	+	0.470343i	$0.844130\pi$
$702$		0			0
$703$	501407.	−	28372.2i	1.01457	−	0.0574093i
$704$		0			0
$705$	326805.	+	57624.6i	0.657523	+	0.115939i
$706$		0			0
$707$	−513834.	−	187020.i	−1.02798	−	0.374154i
$708$		0			0
$709$	304300.	−	255338.i	0.605353	−	0.507952i	−0.287808	−	0.957688i	$-0.592927\pi$
$709$	304300.	−	255338.i	0.605353	−	0.507952i	0.893161	+	0.449736i	$0.148482\pi$
$710$		0			0
$711$	55706.6	+	32162.2i	0.110196	+	0.0636219i
$712$		0			0
$713$	−220205.	+	38828.0i	−0.433159	+	0.0763776i
$714$		0			0
$715$	−58388.7	+	33710.7i	−0.114213	+	0.0659411i
$716$		0			0
$717$	−131842.	−	362232.i	−0.256457	−	0.704610i
$718$		0			0
$719$	−186996.	−	156909.i	−0.361722	−	0.303521i	0.443754	−	0.896148i	$-0.353646\pi$
$719$	−186996.	−	156909.i	−0.361722	−	0.303521i	−0.805477	+	0.592627i	$0.798091\pi$
$720$		0			0
$721$			737533.i			1.41877i
$722$		0			0
$723$	−200304.			−0.383189
$724$		0			0
$725$	−128461.	+	153094.i	−0.244397	+	0.291261i
$726$		0			0
$727$	633293.	−	230500.i	1.19822	−	0.436116i	0.335616	−	0.941999i	$-0.391056\pi$
$727$	633293.	−	230500.i	1.19822	−	0.436116i	0.862602	+	0.505883i	$0.168833\pi$
$728$		0			0
$729$	58812.7	+	101867.i	0.110666	+	0.191680i
$730$		0			0
$731$	−143695.	−	814938.i	−0.268911	−	1.52507i
$732$		0			0
$733$	−63729.5	+	110383.i	−0.118613	+	0.205444i	−0.919218	−	0.393748i	$-0.871178\pi$
$733$	−63729.5	+	110383.i	−0.118613	+	0.205444i	0.800605	+	0.599192i	$0.204511\pi$
$734$		0			0
$735$	174022.	+	207392.i	0.322129	+	0.383899i
$736$		0			0
$737$	283468.	−	778823.i	0.521879	−	1.43385i
$738$		0			0
$739$	−43423.8	+	246269.i	−0.0795132	+	0.450942i	0.918893	+	0.394507i	$0.129085\pi$
$739$	−43423.8	+	246269.i	−0.0795132	+	0.450942i	−0.998406	+	0.0564352i	$0.982027\pi$
$740$		0			0
$741$	2650.09	+	46833.7i	0.00482641	+	0.0852946i
$742$		0			0
$743$	−592950.	−	104553.i	−1.07409	−	0.189391i	−0.391489	−	0.920183i	$-0.628040\pi$
$743$	−592950.	−	104553.i	−1.07409	−	0.189391i	−0.682600	+	0.730792i	$0.739151\pi$
$744$		0			0
$745$	−569010.	−	207103.i	−1.02520	−	0.373141i
$746$		0			0
$747$	242693.	−	203644.i	0.434927	−	0.364947i
$748$		0			0
$749$	61203.8	+	35336.0i	0.109097	+	0.0629874i
$750$		0			0
$751$	184265.	−	32491.0i	0.326711	−	0.0576080i	−0.00788703	−	0.999969i	$-0.502511\pi$
$751$	184265.	−	32491.0i	0.326711	−	0.0576080i	0.334598	+	0.942361i	$0.391399\pi$
$752$		0			0
$753$	−146208.	+	84413.5i	−0.257859	+	0.148875i
$754$		0			0
$755$	−130039.	−	357278.i	−0.228128	−	0.626777i
$756$		0			0
$757$	156846.	+	131610.i	0.273705	+	0.229666i	0.769300	−	0.638888i	$-0.220605\pi$
$757$	156846.	+	131610.i	0.273705	+	0.229666i	−0.495595	+	0.868554i	$0.665050\pi$
$758$		0			0
$759$		−	120287.i		−	0.208803i
$760$		0			0
$761$	−976866.			−1.68681			−0.843404	−	0.537280i	$-0.819452\pi$
$761$	−976866.			−1.68681			−0.843404	+	0.537280i	$0.819452\pi$
$762$		0			0
$763$	824966.	−	983156.i	1.41706	−	1.68878i
$764$		0			0
$765$	462915.	−	168487.i	0.791004	−	0.287902i
$766$		0			0
$767$	2709.76	+	4693.44i	0.00460617	+	0.00797813i
$768$		0			0
$769$	40758.6	+	231154.i	0.0689235	+	0.390884i	0.999681	+	0.0252479i	$0.00803752\pi$
$769$	40758.6	+	231154.i	0.0689235	+	0.390884i	−0.930758	+	0.365636i	$0.880851\pi$
$770$		0			0
$771$	149040.	−	258146.i	0.250724	−	0.434266i
$772$		0			0
$773$	−91616.0	−	109184.i	−0.153325	−	0.182725i	0.683914	−	0.729562i	$-0.260276\pi$
$773$	−91616.0	−	109184.i	−0.153325	−	0.182725i	−0.837239	+	0.546837i	$0.815832\pi$
$774$		0			0
$775$	53462.1	−	146886.i	0.0890108	−	0.244555i
$776$		0			0
$777$	79066.8	−	448410.i	0.130964	−	0.742734i
$778$		0			0
$779$	749345.	+	559794.i	1.23483	+	0.922472i
$780$		0			0
$781$	−424946.	−	74929.5i	−0.696677	−	0.122843i
$782$		0			0
$783$	718932.	+	261670.i	1.17264	+	0.426806i
$784$		0			0
$785$	−527573.	+	442686.i	−0.856137	+	0.718384i
$786$		0			0
$787$	−230126.	−	132863.i	−0.371549	−	0.214514i	0.302586	−	0.953122i	$-0.402150\pi$
$787$	−230126.	−	132863.i	−0.371549	−	0.214514i	−0.674135	+	0.738608i	$0.735483\pi$
$788$		0			0
$789$	−344890.	+	60813.5i	−0.554022	+	0.0976890i
$790$		0			0
$791$	−650945.	+	375823.i	−1.04038	+	0.600662i
$792$		0			0
$793$	14701.5	+	40391.9i	0.0233784	+	0.0642315i
$794$		0			0
$795$	174053.	+	146047.i	0.275389	+	0.231079i
$796$		0			0
$797$			926703.i			1.45889i	0.684037	+	0.729447i	$0.260223\pi$
$797$			926703.i			1.45889i	−0.684037	+	0.729447i	$0.739777\pi$
$798$		0			0
$799$	1.30376e6			2.04222
$800$		0			0
$801$	20126.4	−	23985.7i	0.0313690	−	0.0373842i
$802$		0			0
$803$	357843.	−	130244.i	0.554960	−	0.201989i
$804$		0			0
$805$	−184818.	−	320114.i	−0.285202	−	0.493984i
$806$		0			0
$807$	85873.8	+	487014.i	0.131860	+	0.747816i
$808$		0			0
$809$	−547564.	+	948408.i	−0.836638	+	1.44910i	0.0560523	+	0.998428i	$0.482149\pi$
$809$	−547564.	+	948408.i	−0.836638	+	1.44910i	−0.892690	+	0.450671i	$0.851185\pi$
$810$		0			0
$811$	577529.	+	688272.i	0.878076	+	1.04645i	0.998555	+	0.0537344i	$0.0171124\pi$
$811$	577529.	+	688272.i	0.878076	+	1.04645i	−0.120479	+	0.992716i	$0.538443\pi$
$812$		0			0
$813$	−136237.	+	374309.i	−0.206118	+	0.566303i
$814$		0			0
$815$	154253.	−	874811.i	0.232230	−	1.31704i
$816$		0			0
$817$	−308743.	−	719011.i	−0.462543	−	1.07719i
$818$		0			0
$819$	−122387.	−	21580.2i	−0.182460	−	0.0321727i
$820$		0			0
$821$	681125.	+	247909.i	1.01051	+	0.367795i	0.793628	−	0.608403i	$-0.208189\pi$
$821$	681125.	+	247909.i	1.01051	+	0.367795i	0.216881	+	0.976198i	$0.430412\pi$
$822$		0			0
$823$	−906038.	+	760256.i	−1.33766	+	1.12243i	−0.355443	+	0.934698i	$0.615670\pi$
$823$	−906038.	+	760256.i	−1.33766	+	1.12243i	−0.982220	+	0.187734i	$0.939886\pi$
$824$		0			0
$825$	72823.2	+	42044.5i	0.106995	+	0.0617733i
$826$		0			0
$827$	−1.24093e6	+	218809.i	−1.81441	+	0.319930i	−0.974771	−	0.223208i	$-0.928347\pi$
$827$	−1.24093e6	+	218809.i	−1.81441	+	0.319930i	−0.839643	+	0.543138i	$0.817236\pi$
$828$		0			0
$829$	−232054.	+	133976.i	−0.337660	+	0.194948i	−0.659237	−	0.751935i	$-0.729120\pi$
$829$	−232054.	+	133976.i	−0.337660	+	0.194948i	0.321577	+	0.946883i	$0.395787\pi$
$830$		0			0
$831$	−12379.9	−	34013.5i	−0.0179273	−	0.0492548i
$832$		0			0
$833$	814797.	+	683696.i	1.17425	+	0.985311i
$834$		0			0
$835$			520178.i			0.746068i
$836$		0			0
$837$	−598401.			−0.854164
$838$		0			0
$839$	−551640.	+	657418.i	−0.783667	+	0.933938i	−0.999093	−	0.0425824i	$-0.986441\pi$
$839$	−551640.	+	657418.i	−0.783667	+	0.933938i	0.215426	+	0.976520i	$0.430886\pi$
$840$		0			0
$841$	668419.	−	243284.i	0.945054	−	0.343971i
$842$		0			0
$843$	229666.	+	397794.i	0.323178	+	0.559761i
$844$		0			0
$845$	−103012.	−	584209.i	−0.144269	−	0.818191i
$846$		0			0
$847$	89314.8	−	154698.i	0.124496	−	0.215634i
$848$		0			0
$849$	105922.	+	126233.i	0.146950	+	0.175128i
$850$		0			0
$851$	−114205.	+	313775.i	−0.157698	+	0.433271i
$852$		0			0
$853$	43477.0	−	246571.i	0.0597533	−	0.338878i	−0.940245	−	0.340498i	$-0.889404\pi$
$853$	43477.0	−	246571.i	0.0597533	−	0.338878i	0.999999	+	0.00161981i	$0.000515602\pi$
$854$		0			0
$855$	389615.	−	255332.i	0.532971	−	0.349280i
$856$		0			0
$857$	−1.34106e6	−	236466.i	−1.82595	−	0.321963i	−0.847868	−	0.530207i	$-0.822114\pi$
$857$	−1.34106e6	−	236466.i	−1.82595	−	0.321963i	−0.978077	+	0.208243i	$0.933225\pi$
$858$		0			0
$859$	−1.01299e6	−	368699.i	−1.37284	−	0.499673i	−0.452840	−	0.891592i	$-0.649589\pi$
$859$	−1.01299e6	−	368699.i	−1.37284	−	0.499673i	−0.920000	+	0.391919i	$0.871811\pi$
$860$		0			0
$861$	649634.	−	545108.i	0.876320	−	0.735319i
$862$		0			0
$863$	−391822.	−	226219.i	−0.526099	−	0.303743i	0.213327	−	0.976981i	$-0.431570\pi$
$863$	−391822.	−	226219.i	−0.526099	−	0.303743i	−0.739426	+	0.673237i	$0.764903\pi$
$864$		0			0
$865$	−170918.	+	30137.5i	−0.228431	+	0.0402786i
$866$		0			0
$867$	−244895.	+	141390.i	−0.325793	+	0.188097i
$868$		0			0
$869$	−40201.9	−	110454.i	−0.0532363	−	0.146265i
$870$		0			0
$871$	−164622.	−	138134.i	−0.216996	−	0.182081i
$872$		0			0
$873$			845576.i			1.10949i
$874$		0			0
$875$	1.22090e6			1.59464
$876$		0			0
$877$	−679353.	+	809621.i	−0.883276	+	1.05265i	0.114966	+	0.993369i	$0.463324\pi$
$877$	−679353.	+	809621.i	−0.883276	+	1.05265i	−0.998242	+	0.0592775i	$0.981120\pi$
$878$		0			0
$879$	427390.	−	155557.i	0.553155	−	0.201332i
$880$		0			0
$881$	394829.	+	683865.i	0.508695	+	0.881086i	0.999949	+	0.0100696i	$0.00320531\pi$
$881$	394829.	+	683865.i	0.508695	+	0.881086i	−0.491254	+	0.871016i	$0.663461\pi$
$882$		0			0
$883$	85348.0	+	484033.i	0.109464	+	0.620802i	0.989343	+	0.145604i	$0.0465125\pi$
$883$	85348.0	+	484033.i	0.109464	+	0.620802i	−0.879879	+	0.475198i	$0.842376\pi$
$884$		0			0
$885$	−9208.94	+	15950.4i	−0.0117577	+	0.0203650i
$886$		0			0
$887$	−113784.	−	135603.i	−0.144622	−	0.172354i	0.688871	−	0.724884i	$-0.258107\pi$
$887$	−113784.	−	135603.i	−0.144622	−	0.172354i	−0.833493	+	0.552530i	$0.813662\pi$
$888$		0			0
$889$	85030.4	−	233619.i	0.107590	−	0.295600i
$890$		0			0
$891$	−37705.1	+	213836.i	−0.0474947	+	0.269356i
$892$		0			0
$893$	1.20007e6	−	282328.i	1.50489	−	0.354039i
$894$		0			0
$895$	−413886.	−	72979.2i	−0.516695	−	0.0911073i
$896$		0			0
$897$	−29308.0	−	10667.2i	−0.0364251	−	0.0132577i
$898$		0			0
$899$	−849969.	+	713209.i	−1.05168	+	0.882465i
$900$		0			0
$901$	773064.	+	446329.i	0.952283	+	0.549801i
$902$		0			0
$903$	−698670.	+	123194.i	−0.856834	+	0.151083i
$904$		0			0
$905$	989383.	−	571220.i	1.20800	−	0.697439i
$906$		0			0
$907$	−278572.	−	765369.i	−0.338628	−	0.930371i	−0.985784	−	0.168015i	$-0.946264\pi$
$907$	−278572.	−	765369.i	−0.338628	−	0.930371i	0.647157	−	0.762357i	$-0.275958\pi$
$908$		0			0
$909$	350986.	+	294512.i	0.424778	+	0.356431i
$910$		0			0
$911$		−	1.12646e6i		−	1.35731i	−0.734457	−	0.678656i	$-0.762563\pi$
$911$		−	1.12646e6i		−	1.35731i	0.734457	−	0.678656i	$-0.237437\pi$
$912$		0			0
$913$	−578927.			−0.694516
$914$		0			0
$915$	−93897.7	+	111903.i	−0.112153	+	0.133659i
$916$		0			0
$917$	900762.	−	327851.i	1.07120	−	0.389886i
$918$		0			0
$919$	737353.	+	1.27713e6i	0.873060	+	1.51218i	0.858815	+	0.512286i	$0.171201\pi$
$919$	737353.	+	1.27713e6i	0.873060	+	1.51218i	0.0142453	+	0.999899i	$0.495465\pi$
$920$		0			0
$921$	−23851.1	−	135266.i	−0.0281183	−	0.159467i
$922$		0			0
$923$	−55941.4	+	96893.3i	−0.0656644	+	0.113734i
$924$		0			0
$925$	−150044.	−	178816.i	−0.175362	−	0.208989i
$926$		0			0
$927$	211364.	−	580719.i	0.245965	−	0.675782i
$928$		0			0
$929$	124449.	−	705786.i	0.144198	−	0.817790i	−0.823809	−	0.566868i	$-0.808155\pi$
$929$	124449.	−	705786.i	0.144198	−	0.817790i	0.968007	−	0.250922i	$-0.0807339\pi$
$930$		0			0
$931$	898053.	+	452880.i	1.03610	+	0.522496i
$932$		0			0
$933$	−802738.	−	141544.i	−0.922169	−	0.162603i
$934$		0			0
$935$	−845904.	−	307884.i	−0.967605	−	0.352179i
$936$		0			0
$937$	−351489.	+	294934.i	−0.400343	+	0.335927i	−0.820626	−	0.571465i	$-0.806375\pi$
$937$	−351489.	+	294934.i	−0.400343	+	0.335927i	0.420283	+	0.907393i	$0.361931\pi$
$938$		0			0
$939$	−303661.	−	175319.i	−0.344396	−	0.198837i
$940$		0			0
$941$	−53702.3	+	9469.16i	−0.0606475	+	0.0106938i	−0.203890	−	0.978994i	$-0.565358\pi$
$941$	−53702.3	+	9469.16i	−0.0606475	+	0.0106938i	0.143242	+	0.989688i	$0.454247\pi$
$942$		0			0
$943$	−538585.	+	310952.i	−0.605663	+	0.349680i
$944$		0			0
$945$	−338332.	−	929559.i	−0.378861	−	1.04091i
$946$		0			0
$947$	−149800.	−	125697.i	−0.167036	−	0.140160i	0.555438	−	0.831558i	$-0.312551\pi$
$947$	−149800.	−	125697.i	−0.167036	−	0.140160i	−0.722474	+	0.691398i	$0.756995\pi$
$948$		0			0
$949$		−	98738.8i		−	0.109637i
$950$		0			0
$951$	236337.			0.261319
$952$		0			0
$953$	1.08819e6	−	1.29686e6i	1.19817	−	1.42793i	0.321456	−	0.946925i	$-0.395828\pi$
$953$	1.08819e6	−	1.29686e6i	1.19817	−	1.42793i	0.876718	−	0.481004i	$-0.159728\pi$
$954$		0			0
$955$	−60918.5	+	22172.5i	−0.0667948	+	0.0243113i
$956$		0			0
$957$	−298445.	−	516921.i	−0.325867	−	0.564418i
$958$		0			0
$959$	−6721.01	−	38116.8i	−0.00730798	−	0.0414456i
$960$		0			0
$961$	−27840.9	+	48221.8i	−0.0301465	+	0.0522152i
$962$		0			0
$963$	−38063.9	−	45362.8i	−0.0410451	−	0.0489156i
$964$		0			0
$965$	497995.	−	1.36823e6i	0.534775	−	1.46928i
$966$		0			0
$967$	193856.	−	1.09941e6i	0.207313	−	1.17573i	−0.686445	−	0.727182i	$-0.740830\pi$
$967$	193856.	−	1.09941e6i	0.207313	−	1.17573i	0.893758	−	0.448549i	$-0.148059\pi$
$968$		0			0
$969$	−456271.	+	429047.i	−0.485932	+	0.456938i
$970$		0			0
$971$	−88596.1	−	15621.9i	−0.0939671	−	0.0165689i	0.126467	−	0.991971i	$-0.459636\pi$
$971$	−88596.1	−	15621.9i	−0.0939671	−	0.0165689i	−0.220434	+	0.975402i	$0.570747\pi$
$972$		0			0
$973$	669445.	+	243658.i	0.707114	+	0.257368i
$974$		0			0
$975$	16702.2	−	14014.8i	0.0175697	−	0.0147427i
$976$		0			0
$977$	−51203.2	−	29562.2i	−0.0536424	−	0.0309704i	0.472939	−	0.881095i	$-0.343193\pi$
$977$	−51203.2	−	29562.2i	−0.0536424	−	0.0309704i	−0.526581	+	0.850125i	$0.676526\pi$
$978$		0			0
$979$	−56346.9	+	9935.47i	−0.0587901	+	0.0103663i
$980$		0			0
$981$	−931317.	+	537696.i	−0.967742	+	0.558726i
$982$		0			0
$983$	244561.	+	671927.i	0.253093	+	0.695368i	0.999552	+	0.0299364i	$0.00953046\pi$
$983$	244561.	+	671927.i	0.253093	+	0.695368i	−0.746458	+	0.665432i	$0.768247\pi$
$984$		0			0
$985$	−129945.	−	109036.i	−0.133932	−	0.112383i
$986$		0			0
$987$		−	1.11775e6i		−	1.14739i
$988$		0			0
$989$	520271.			0.531909
$990$		0			0
$991$	214603.	−	255754.i	0.218518	−	0.260420i	−0.645638	−	0.763644i	$-0.723408\pi$
$991$	214603.	−	255754.i	0.218518	−	0.260420i	0.864156	+	0.503224i	$0.167853\pi$
$992$		0			0
$993$	661075.	−	240612.i	0.670428	−	0.244016i
$994$		0			0
$995$	673366.	+	1.16630e6i	0.680150	+	1.17805i
$996$		0			0
$997$	−231991.	−	1.31569e6i	−0.233389	−	1.32362i	−0.845979	−	0.533216i	$-0.820983\pi$
$997$	−231991.	−	1.31569e6i	−0.233389	−	1.32362i	0.612590	−	0.790401i	$-0.290128\pi$
$998$		0			0
$999$	−446807.	+	773893.i	−0.447702	+	0.775443i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.j.a.21.5	✓	42
19.10	odd	18	inner	76.5.j.a.29.5	yes	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.j.a.21.5	✓	42	1.1	even	1	trivial
76.5.j.a.29.5	yes	42	19.10	odd	18	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.j (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(2.92113 - 3.48127i) q^{3} +(20.0929 - 7.31321i) q^{5} +(-36.0108 - 62.3725i) q^{7} +(10.4793 + 59.4309i) q^{9} +O(q^{10})\)	\(q+(2.92113 - 3.48127i) q^{3} +(20.0929 - 7.31321i) q^{5} +(-36.0108 - 62.3725i) q^{7} +(10.4793 + 59.4309i) q^{9} +(55.1380 - 95.5017i) q^{11} +(-18.3793 - 21.9036i) q^{13} +(33.2347 - 91.3116i) q^{15} +(66.2932 - 375.967i) q^{17} +(-20.3946 - 360.423i) q^{19} +(-322.328 - 56.8351i) q^{21} +(225.549 + 82.0931i) q^{23} +(-128.537 + 107.855i) q^{25} +(556.293 + 321.176i) q^{27} +(1172.95 - 206.823i) q^{29} +(-806.771 + 465.789i) q^{31} +(-171.402 - 470.923i) q^{33} +(-1179.70 - 989.888i) q^{35} +1391.16i q^{37} -129.941 q^{39} +(-1665.47 + 1984.83i) q^{41} +(2036.86 - 741.355i) q^{43} +(645.190 + 1117.50i) q^{45} +(593.018 + 3363.17i) q^{47} +(-1393.05 + 2412.84i) q^{49} +(-1115.19 - 1329.03i) q^{51} +(-799.720 + 2197.21i) q^{53} +(409.456 - 2322.14i) q^{55} +(-1314.31 - 981.846i) q^{57} +(-186.660 - 32.9132i) q^{59} +(-1412.65 - 514.161i) q^{61} +(3329.49 - 2793.77i) q^{63} +(-529.478 - 305.695i) q^{65} +(7401.57 - 1305.10i) q^{67} +(944.647 - 545.392i) q^{69} +(-1338.30 - 3676.95i) q^{71} +(2645.33 + 2219.70i) q^{73} +762.533i q^{75} -7942.24 q^{77} +(685.145 - 816.524i) q^{79} +(-1850.27 + 673.444i) q^{81} +(-2624.90 - 4546.46i) q^{83} +(-1417.51 - 8039.08i) q^{85} +(2706.35 - 4687.53i) q^{87} +(-333.507 - 397.458i) q^{89} +(-704.329 + 1935.13i) q^{91} +(-735.146 + 4169.22i) q^{93} +(-3045.64 - 7092.79i) q^{95} +(13798.9 + 2433.11i) q^{97} +(6253.56 + 2276.11i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9}+O(q^{10}) \) 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9	\( 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9} - 45 q^{11} + 33 q^{13} - 393 q^{15} + 909 q^{17} + 1242 q^{19} + 1107 q^{21} - 360 q^{23} - 810 q^{25} - 7056 q^{27} - 2889 q^{29} + 2808 q^{31} + 10875 q^{33} + 6741 q^{35} - 3480 q^{39} - 3060 q^{41} - 8079 q^{43} - 4320 q^{45} - 2655 q^{47} - 474 q^{49} - 12222 q^{51} - 6705 q^{53} + 4623 q^{55} - 8022 q^{57} + 24309 q^{59} + 7104 q^{61} + 12063 q^{63} + 25245 q^{65} + 15573 q^{67} - 10881 q^{69} - 25506 q^{71} + 3036 q^{73} + 12924 q^{77} - 16839 q^{79} - 2208 q^{81} - 6363 q^{83} - 37890 q^{85} - 21924 q^{87} - 22644 q^{89} + 17418 q^{91} + 8184 q^{93} - 82413 q^{95} + 13383 q^{97} + 23565 q^{99}+O(q^{100}) \) 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9 - 45 * q^11 + 33 * q^13 - 393 * q^15 + 909 * q^17 + 1242 * q^19 + 1107 * q^21 - 360 * q^23 - 810 * q^25 - 7056 * q^27 - 2889 * q^29 + 2808 * q^31 + 10875 * q^33 + 6741 * q^35 - 3480 * q^39 - 3060 * q^41 - 8079 * q^43 - 4320 * q^45 - 2655 * q^47 - 474 * q^49 - 12222 * q^51 - 6705 * q^53 + 4623 * q^55 - 8022 * q^57 + 24309 * q^59 + 7104 * q^61 + 12063 * q^63 + 25245 * q^65 + 15573 * q^67 - 10881 * q^69 - 25506 * q^71 + 3036 * q^73 + 12924 * q^77 - 16839 * q^79 - 2208 * q^81 - 6363 * q^83 - 37890 * q^85 - 21924 * q^87 - 22644 * q^89 + 17418 * q^91 + 8184 * q^93 - 82413 * q^95 + 13383 * q^97 + 23565 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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