LMFDB - Embedded newform 378.4.k.a.269.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(215,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.k (of order $6$, 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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 11x^{10} + 88x^{8} - 331x^{6} + 913x^{4} - 528x^{2} + 256 $ x^12 - 11x^10 + 88x^8 - 331x^6 + 913x^4 - 528*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.1
Root			$-0.669321 - 0.386433i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.4.k.a.215.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-4.16156 - 7.20804i) q^{5} +(-16.6528 - 8.10467i) q^{7} +8.00000i q^{8} +O(q^{10})$	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-4.16156 - 7.20804i) q^{5} +(-16.6528 - 8.10467i) q^{7} +8.00000i q^{8} +(14.4161 + 8.32312i) q^{10} +(-20.0413 - 11.5708i) q^{11} -39.4467i q^{13} +(36.9481 - 2.61505i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(-50.8277 + 88.0362i) q^{17} +(28.7060 - 16.5734i) q^{19} -33.2925 q^{20} +46.2834 q^{22} +(-25.0303 + 14.4512i) q^{23} +(27.8628 - 48.2598i) q^{25} +(39.4467 + 68.3236i) q^{26} +(-61.3809 + 41.4775i) q^{28} +177.789i q^{29} +(135.306 + 78.1189i) q^{31} +(27.7128 + 16.0000i) q^{32} -203.311i q^{34} +(10.8827 + 153.762i) q^{35} +(-9.16181 - 15.8687i) q^{37} +(-33.1469 + 57.4120i) q^{38} +(57.6643 - 33.2925i) q^{40} +123.758 q^{41} +242.188 q^{43} +(-80.1652 + 46.2834i) q^{44} +(28.9025 - 50.0605i) q^{46} +(31.7746 + 55.0352i) q^{47} +(211.629 + 269.930i) q^{49} +111.451i q^{50} +(-136.647 - 78.8933i) q^{52} +(-54.9048 - 31.6993i) q^{53} +192.611i q^{55} +(64.8374 - 133.222i) q^{56} +(-177.789 - 307.939i) q^{58} +(-215.903 + 373.955i) q^{59} +(-749.448 + 432.694i) q^{61} -312.476 q^{62} -64.0000 q^{64} +(-284.333 + 164.160i) q^{65} +(93.3127 - 161.622i) q^{67} +(203.311 + 352.145i) q^{68} +(-172.611 - 255.440i) q^{70} +1168.93i q^{71} +(-667.923 - 385.625i) q^{73} +(31.7375 + 18.3236i) q^{74} -132.587i q^{76} +(239.965 + 355.114i) q^{77} +(323.607 + 560.504i) q^{79} +(-66.5850 + 115.329i) q^{80} +(-214.356 + 123.758i) q^{82} +231.713 q^{83} +846.090 q^{85} +(-419.482 + 242.188i) q^{86} +(92.5667 - 160.330i) q^{88} +(129.606 + 224.485i) q^{89} +(-319.702 + 656.896i) q^{91} +115.610i q^{92} +(-110.070 - 63.5492i) q^{94} +(-238.924 - 137.943i) q^{95} +1357.81i q^{97} +(-636.482 - 255.904i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 24 q^{4} - 42 q^{7}+O(q^{10}) $ 12 * q + 24 * q^4 - 42 * q^7	$ 12 q + 24 q^{4} - 42 q^{7} - 48 q^{10} - 96 q^{16} - 66 q^{19} + 240 q^{22} + 366 q^{25} - 168 q^{28} + 630 q^{31} + 348 q^{37} - 192 q^{40} - 252 q^{43} + 600 q^{46} - 1218 q^{49} + 192 q^{52} - 144 q^{58} + 114 q^{61} - 768 q^{64} + 108 q^{67} - 1344 q^{70} - 978 q^{73} - 744 q^{79} + 240 q^{82} + 7248 q^{85} + 480 q^{88} - 5544 q^{91} - 1920 q^{94}+O(q^{100}) $ 12 * q + 24 * q^4 - 42 * q^7 - 48 * q^10 - 96 * q^16 - 66 * q^19 + 240 * q^22 + 366 * q^25 - 168 * q^28 + 630 * q^31 + 348 * q^37 - 192 * q^40 - 252 * q^43 + 600 * q^46 - 1218 * q^49 + 192 * q^52 - 144 * q^58 + 114 * q^61 - 768 * q^64 + 108 * q^67 - 1344 * q^70 - 978 * q^73 - 744 * q^79 + 240 * q^82 + 7248 * q^85 + 480 * q^88 - 5544 * q^91 - 1920 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$	−1.73205	+	1.00000i	−0.612372	+	0.353553i
$2$	−1.73205	+	1.00000i	−0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	2.00000	−	3.46410i	0.250000	−	0.433013i
$5$	−4.16156	−	7.20804i	−0.372221	−	0.644706i	0.617686	−	0.786425i	$-0.288070\pi$
$5$	−4.16156	−	7.20804i	−0.372221	−	0.644706i	−0.989907	+	0.141719i	$0.954737\pi$
$6$		0			0
$7$	−16.6528	−	8.10467i	−0.899164	−	0.437611i
$7$	−16.6528	−	8.10467i	−0.899164	−	0.437611i
$8$			8.00000i			0.353553i
$9$		0			0
$10$	14.4161	+	8.32312i	0.455876	+	0.263200i
$11$	−20.0413	−	11.5708i	−0.549334	−	0.317158i	0.199519	−	0.979894i	$-0.436062\pi$
$11$	−20.0413	−	11.5708i	−0.549334	−	0.317158i	−0.748853	+	0.662736i	$0.769395\pi$
$12$		0			0
$13$		−	39.4467i		−	0.841580i	−0.907158	−	0.420790i	$-0.861753\pi$
$13$		−	39.4467i		−	0.841580i	0.907158	−	0.420790i	$-0.138247\pi$
$14$	36.9481	−	2.61505i	0.705342	−	0.0499216i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	−50.8277	+	88.0362i	−0.725149	+	1.25599i	0.233764	+	0.972293i	$0.424896\pi$
$17$	−50.8277	+	88.0362i	−0.725149	+	1.25599i	−0.958913	+	0.283701i	$0.908438\pi$
$18$		0			0
$19$	28.7060	−	16.5734i	0.346611	−	0.200116i	−0.316580	−	0.948566i	$-0.602535\pi$
$19$	28.7060	−	16.5734i	0.346611	−	0.200116i	0.663192	+	0.748450i	$0.269201\pi$
$20$	−33.2925			−0.372221
$21$		0			0
$22$	46.2834			0.448529
$23$	−25.0303	+	14.4512i	−0.226920	+	0.131013i	−0.609151	−	0.793055i	$-0.708489\pi$
$23$	−25.0303	+	14.4512i	−0.226920	+	0.131013i	0.382230	+	0.924067i	$0.375156\pi$
$24$		0			0
$25$	27.8628	−	48.2598i	0.222902	−	0.386078i
$26$	39.4467	+	68.3236i	0.297543	+	0.515360i
$27$		0			0
$28$	−61.3809	+	41.4775i	−0.414282	+	0.279947i
$29$			177.789i			1.13843i	0.822187	+	0.569217i	$0.192754\pi$
$29$			177.789i			1.13843i	−0.822187	+	0.569217i	$0.807246\pi$
$30$		0			0
$31$	135.306	+	78.1189i	0.783924	+	0.452599i	0.837819	−	0.545948i	$-0.183830\pi$
$31$	135.306	+	78.1189i	0.783924	+	0.452599i	−0.0538948	+	0.998547i	$0.517164\pi$
$32$	27.7128	+	16.0000i	0.153093	+	0.0883883i
$33$		0			0
$34$		−	203.311i		−	1.02552i
$35$	10.8827	+	153.762i	0.0525575	+	0.742585i
$36$		0			0
$37$	−9.16181	−	15.8687i	−0.0407079	−	0.0705082i	0.844954	−	0.534840i	$-0.179628\pi$
$37$	−9.16181	−	15.8687i	−0.0407079	−	0.0705082i	−0.885662	+	0.464331i	$0.846295\pi$
$38$	−33.1469	+	57.4120i	−0.141503	+	0.245091i
$39$		0			0
$40$	57.6643	−	33.2925i	0.227938	−	0.131600i
$41$	123.758			0.471410			0.235705	−	0.971825i	$-0.424260\pi$
$41$	123.758			0.471410			0.235705	+	0.971825i	$0.424260\pi$
$42$		0			0
$43$	242.188			0.858914			0.429457	−	0.903087i	$-0.358705\pi$
$43$	242.188			0.858914			0.429457	+	0.903087i	$0.358705\pi$
$44$	−80.1652	+	46.2834i	−0.274667	+	0.158579i
$45$		0			0
$46$	28.9025	−	50.0605i	0.0926399	−	0.160457i
$47$	31.7746	+	55.0352i	0.0986128	+	0.170802i	0.911111	−	0.412162i	$-0.135226\pi$
$47$	31.7746	+	55.0352i	0.0986128	+	0.170802i	−0.812498	+	0.582964i	$0.801893\pi$
$48$		0			0
$49$	211.629	+	269.930i	0.616993	+	0.786969i
$50$			111.451i			0.315232i
$51$		0			0
$52$	−136.647	−	78.8933i	−0.364415	−	0.210395i
$53$	−54.9048	−	31.6993i	−0.142297	−	0.0821554i	0.427161	−	0.904176i	$-0.359514\pi$
$53$	−54.9048	−	31.6993i	−0.142297	−	0.0821554i	−0.569458	+	0.822020i	$0.692847\pi$
$54$		0			0
$55$			192.611i			0.472212i
$56$	64.8374	−	133.222i	0.154719	−	0.317903i
$57$		0			0
$58$	−177.789	−	307.939i	−0.402497	−	0.697145i
$59$	−215.903	+	373.955i	−0.476409	+	0.825165i	−0.999635	−	0.0270293i	$-0.991395\pi$
$59$	−215.903	+	373.955i	−0.476409	+	0.825165i	0.523225	+	0.852194i	$0.324729\pi$
$60$		0			0
$61$	−749.448	+	432.694i	−1.57307	+	0.908210i	−0.577276	+	0.816549i	$0.695884\pi$
$61$	−749.448	+	432.694i	−1.57307	+	0.908210i	−0.995790	+	0.0916609i	$0.970782\pi$
$62$	−312.476			−0.640072
$63$		0			0
$64$	−64.0000			−0.125000
$65$	−284.333	+	164.160i	−0.542572	+	0.313254i
$66$		0			0
$67$	93.3127	−	161.622i	0.170149	−	0.294706i	−0.768323	−	0.640062i	$-0.778909\pi$
$67$	93.3127	−	161.622i	0.170149	−	0.294706i	0.938472	+	0.345356i	$0.112242\pi$
$68$	203.311	+	352.145i	0.362574	+	0.627997i
$69$		0			0
$70$	−172.611	−	255.440i	−0.294728	−	0.436157i
$71$			1168.93i			1.95389i	0.213495	+	0.976944i	$0.431515\pi$
$71$			1168.93i			1.95389i	−0.213495	+	0.976944i	$0.568485\pi$
$72$		0			0
$73$	−667.923	−	385.625i	−1.07088	−	0.618274i	−0.142460	−	0.989801i	$-0.545501\pi$
$73$	−667.923	−	385.625i	−1.07088	−	0.618274i	−0.928422	+	0.371526i	$0.878835\pi$
$74$	31.7375	+	18.3236i	0.0498568	+	0.0287848i
$75$		0			0
$76$		−	132.587i		−	0.200116i
$77$	239.965	+	355.114i	0.355150	+	0.525572i
$78$		0			0
$79$	323.607	+	560.504i	0.460869	+	0.798248i	0.999004	−	0.0446099i	$-0.0142045\pi$
$79$	323.607	+	560.504i	0.460869	+	0.798248i	−0.538136	+	0.842858i	$0.680871\pi$
$80$	−66.5850	+	115.329i	−0.0930554	+	0.161177i
$81$		0			0
$82$	−214.356	+	123.758i	−0.288679	+	0.166669i
$83$	231.713			0.306431			0.153216	−	0.988193i	$-0.451037\pi$
$83$	231.713			0.306431			0.153216	+	0.988193i	$0.451037\pi$
$84$		0			0
$85$	846.090			1.07966
$86$	−419.482	+	242.188i	−0.525976	+	0.303672i
$87$		0			0
$88$	92.5667	−	160.330i	0.112132	−	0.194219i
$89$	129.606	+	224.485i	0.154362	+	0.267364i	0.932827	−	0.360325i	$-0.117334\pi$
$89$	129.606	+	224.485i	0.154362	+	0.267364i	−0.778464	+	0.627689i	$0.784001\pi$
$90$		0			0
$91$	−319.702	+	656.896i	−0.368285	+	0.756718i
$92$			115.610i			0.131013i
$93$		0			0
$94$	−110.070	−	63.5492i	−0.120776	−	0.0697298i
$95$	−238.924	−	137.943i	−0.258032	−	0.148975i
$96$		0			0
$97$			1357.81i			1.42128i	0.703555	+	0.710641i	$0.251595\pi$
$97$			1357.81i			1.42128i	−0.703555	+	0.710641i	$0.748405\pi$
$98$	−636.482	−	255.904i	−0.656065	−	0.263778i
$99$		0			0
$100$	−111.451	−	193.039i	−0.111451	−	0.193039i
$101$	622.990	−	1079.05i	0.613761	−	1.06306i	−0.376840	−	0.926278i	$-0.622989\pi$
$101$	622.990	−	1079.05i	0.613761	−	1.06306i	0.990601	−	0.136786i	$-0.0436773\pi$
$102$		0			0
$103$	280.587	−	161.997i	0.268418	−	0.154971i	−0.359750	−	0.933049i	$-0.617138\pi$
$103$	280.587	−	161.997i	0.268418	−	0.154971i	0.628169	+	0.778077i	$0.283805\pi$
$104$	315.573			0.297543
$105$		0			0
$106$	126.797			0.116185
$107$	−1497.33	+	864.484i	−1.35283	+	0.781055i	−0.988644	−	0.150273i	$-0.951985\pi$
$107$	−1497.33	+	864.484i	−1.35283	+	0.781055i	−0.364182	+	0.931328i	$0.618651\pi$
$108$		0			0
$109$	−823.869	+	1426.98i	−0.723966	+	1.25395i	0.235432	+	0.971891i	$0.424349\pi$
$109$	−823.869	+	1426.98i	−0.723966	+	1.25395i	−0.959398	+	0.282055i	$0.908984\pi$
$110$	−192.611	−	333.612i	−0.166952	−	0.289170i
$111$		0			0
$112$	20.9204	+	295.585i	0.0176499	+	0.249376i
$113$		−	1054.26i		−	0.877669i	−0.898568	−	0.438834i	$-0.855392\pi$
$113$		−	1054.26i		−	0.877669i	0.898568	−	0.438834i	$-0.144608\pi$
$114$		0			0
$115$	208.330	+	120.279i	0.168929	+	0.0975314i
$116$	615.879	+	355.578i	0.492956	+	0.284608i
$117$		0			0
$118$		−	863.611i		−	0.673744i
$119$	1559.93	−	1054.10i	1.20166	−	0.812012i
$120$		0			0
$121$	−397.731	−	688.891i	−0.298821	−	0.517574i
$122$	865.388	−	1498.90i	0.642202	−	1.11233i
$123$		0			0
$124$	541.224	−	312.476i	0.391962	−	0.226300i
$125$	−1504.20			−1.07632
$126$		0			0
$127$	1867.27			1.30467			0.652337	−	0.757929i	$-0.273789\pi$
$127$	1867.27			1.30467			0.652337	+	0.757929i	$0.273789\pi$
$128$	110.851	−	64.0000i	0.0765466	−	0.0441942i
$129$		0			0
$130$	328.319	−	568.666i	0.221504	−	0.383656i
$131$	−1095.33	−	1897.16i	−0.730528	−	1.26531i	−0.956658	−	0.291214i	$-0.905941\pi$
$131$	−1095.33	−	1897.16i	−0.730528	−	1.26531i	0.226130	−	0.974097i	$-0.427393\pi$
$132$		0			0
$133$	−612.357	+	43.3404i	−0.399233	+	0.0282563i
$134$			373.251i			0.240627i
$135$		0			0
$136$	−704.289	−	406.622i	−0.444061	−	0.256379i
$137$	1555.53	+	898.084i	0.970056	+	0.560062i	0.899253	−	0.437428i	$-0.144111\pi$
$137$	1555.53	+	898.084i	0.970056	+	0.560062i	0.0708029	+	0.997490i	$0.477444\pi$
$138$		0			0
$139$			269.664i			0.164551i	0.996610	+	0.0822757i	$0.0262188\pi$
$139$			269.664i			0.164551i	−0.996610	+	0.0822757i	$0.973781\pi$
$140$	554.412	+	269.825i	0.334688	+	0.162888i
$141$		0			0
$142$	−1168.93	−	2024.64i	−0.690804	−	1.19651i
$143$	−456.431	+	790.562i	−0.266914	+	0.462308i
$144$		0			0
$145$	1281.51	−	739.880i	0.733955	−	0.423749i
$146$	1542.50			0.874372
$147$		0			0
$148$	−73.2945			−0.0407079
$149$	2000.93	−	1155.24i	1.10015	−	0.635172i	0.163889	−	0.986479i	$-0.447596\pi$
$149$	2000.93	−	1155.24i	1.10015	−	0.635172i	0.936260	+	0.351307i	$0.114263\pi$
$150$		0			0
$151$	−336.013	+	581.992i	−0.181088	+	0.313654i	−0.942251	−	0.334906i	$-0.891295\pi$
$151$	−336.013	+	581.992i	−0.181088	+	0.313654i	0.761163	+	0.648561i	$0.224629\pi$
$152$	132.587	+	229.648i	0.0707517	+	0.122546i
$153$		0			0
$154$	−770.746	−	375.112i	−0.403302	−	0.196281i
$155$		−	1300.39i		−	0.673868i
$156$		0			0
$157$	946.238	+	546.311i	0.481007	+	0.277709i	0.720836	−	0.693106i	$-0.243758\pi$
$157$	946.238	+	546.311i	0.481007	+	0.277709i	−0.239829	+	0.970815i	$0.577091\pi$
$158$	−1121.01	−	647.214i	−0.564447	−	0.325884i
$159$		0			0
$160$		−	266.340i		−	0.131600i
$161$	533.945	−	37.7907i	0.261371	−	0.0184989i
$162$		0			0
$163$	−93.0142	−	161.105i	−0.0446959	−	0.0774156i	0.842812	−	0.538208i	$-0.180899\pi$
$163$	−93.0142	−	161.105i	−0.0446959	−	0.0774156i	−0.887508	+	0.460793i	$0.847565\pi$
$164$	247.517	−	428.712i	0.117853	−	0.204127i
$165$		0			0
$166$	−401.339	+	231.713i	−0.187650	+	0.108340i
$167$	1250.77			0.579568			0.289784	−	0.957092i	$-0.406417\pi$
$167$	1250.77			0.579568			0.289784	+	0.957092i	$0.406417\pi$
$168$		0			0
$169$	640.961			0.291744
$170$	−1465.47	+	846.090i	−0.661156	+	0.381719i
$171$		0			0
$172$	484.376	−	838.964i	0.214729	−	0.371921i
$173$	−1959.52	−	3393.99i	−0.861154	−	1.49156i	−0.870816	−	0.491609i	$-0.836409\pi$
$173$	−1959.52	−	3393.99i	−0.861154	−	1.49156i	0.00966227	−	0.999953i	$-0.496924\pi$
$174$		0			0
$175$	−855.122	+	577.840i	−0.369378	+	0.249603i
$176$			370.267i			0.158579i
$177$		0			0
$178$	−448.970	−	259.213i	−0.189055	−	0.109151i
$179$	−1677.56	−	968.538i	−0.700483	−	0.404424i	0.107044	−	0.994254i	$-0.465861\pi$
$179$	−1677.56	−	968.538i	−0.700483	−	0.404424i	−0.807527	+	0.589830i	$0.799195\pi$
$180$		0			0
$181$			822.430i			0.337739i	0.985638	+	0.168870i	$0.0540117\pi$
$181$			822.430i			0.337739i	−0.985638	+	0.168870i	$0.945988\pi$
$182$	−103.155	−	1457.48i	−0.0420130	−	0.593602i
$183$		0			0
$184$	−115.610	−	200.242i	−0.0463199	−	0.0802285i
$185$	−76.2549	+	132.077i	−0.0303047	+	0.0524893i
$186$		0			0
$187$	2037.31	−	1176.24i	0.796698	−	0.459974i
$188$	254.197			0.0986128
$189$		0			0
$190$	551.771			0.210682
$191$	1590.75	−	918.417i	0.602630	−	0.347928i	−0.167446	−	0.985881i	$-0.553552\pi$
$191$	1590.75	−	918.417i	0.602630	−	0.347928i	0.770075	+	0.637953i	$0.220219\pi$
$192$		0			0
$193$	1809.83	−	3134.73i	0.674999	−	1.16913i	−0.301470	−	0.953476i	$-0.597477\pi$
$193$	1809.83	−	3134.73i	0.674999	−	1.16913i	0.976469	−	0.215657i	$-0.0691892\pi$
$194$	−1357.81	−	2351.79i	−0.502499	−	0.870354i
$195$		0			0
$196$	1358.32	−	193.242i	0.495016	−	0.0704236i
$197$			670.900i			0.242638i	0.992614	+	0.121319i	$0.0387123\pi$
$197$			670.900i			0.242638i	−0.992614	+	0.121319i	$0.961288\pi$
$198$		0			0
$199$	708.411	+	409.002i	0.252352	+	0.145695i	0.620841	−	0.783937i	$-0.286791\pi$
$199$	708.411	+	409.002i	0.252352	+	0.145695i	−0.368489	+	0.929632i	$0.620125\pi$
$200$	386.078	+	222.902i	0.136499	+	0.0788079i
$201$		0			0
$202$			2491.96i			0.867988i
$203$	1440.92	−	2960.68i	0.498191	−	1.02364i
$204$		0			0
$205$	−515.029	−	892.056i	−0.175469	−	0.303921i
$206$	−323.994	+	561.175i	−0.109581	+	0.189800i
$207$		0			0
$208$	−546.589	+	315.573i	−0.182207	+	0.105197i
$209$	−767.074			−0.253874
$210$		0			0
$211$	−705.692			−0.230246			−0.115123	−	0.993351i	$-0.536726\pi$
$211$	−705.692			−0.230246			−0.115123	+	0.993351i	$0.536726\pi$
$212$	−219.619	+	126.797i	−0.0711486	+	0.0410777i
$213$		0			0
$214$	1728.97	−	2994.66i	0.552289	−	0.956593i
$215$	−1007.88	−	1745.70i	−0.319706	−	0.553748i
$216$		0			0
$217$	−1620.09	−	2397.51i	−0.506815	−	0.750015i
$218$		−	3295.47i		−	1.02384i
$219$		0			0
$220$	667.224	+	385.222i	0.204474	+	0.118053i
$221$	3472.73	+	2004.98i	1.05702	+	0.610270i
$222$		0			0
$223$			6345.19i			1.90541i	0.303904	+	0.952703i	$0.401710\pi$
$223$			6345.19i			1.90541i	−0.303904	+	0.952703i	$0.598290\pi$
$224$	−331.820	−	491.047i	−0.0989761	−	0.146471i
$225$		0			0
$226$	1054.26	+	1826.03i	0.310303	+	0.537460i
$227$	875.012	−	1515.57i	0.255844	−	0.443135i	−0.709280	−	0.704926i	$-0.750980\pi$
$227$	875.012	−	1515.57i	0.255844	−	0.443135i	0.965124	+	0.261792i	$0.0843133\pi$
$228$		0			0
$229$	−3514.42	+	2029.05i	−1.01414	+	0.585517i	−0.912403	−	0.409294i	$-0.865775\pi$
$229$	−3514.42	+	2029.05i	−1.01414	+	0.585517i	−0.101742	+	0.994811i	$0.532442\pi$
$230$	−481.117			−0.137930
$231$		0			0
$232$	−1422.31			−0.402497
$233$	−5089.91	+	2938.66i	−1.43112	+	0.826258i	−0.997206	−	0.0746957i	$-0.976201\pi$
$233$	−5089.91	+	2938.66i	−1.43112	+	0.826258i	−0.433915	+	0.900954i	$0.642868\pi$
$234$		0			0
$235$	264.464	−	458.065i	0.0734116	−	0.127153i
$236$	863.611	+	1495.82i	0.238205	+	0.412583i
$237$		0			0
$238$	−1647.77	+	3385.69i	−0.448777	+	0.922107i
$239$			5691.65i			1.54043i	0.637786	+	0.770214i	$0.279850\pi$
$239$			5691.65i			1.54043i	−0.637786	+	0.770214i	$0.720150\pi$
$240$		0			0
$241$	−3589.60	−	2072.46i	−0.959447	−	0.553937i	−0.0634443	−	0.997985i	$-0.520209\pi$
$241$	−3589.60	−	2072.46i	−0.959447	−	0.553937i	−0.896003	+	0.444048i	$0.853542\pi$
$242$	1377.78	+	795.462i	0.365980	+	0.211299i
$243$		0			0
$244$			3461.55i			0.908210i
$245$	1064.96	−	2648.76i	0.277706	−	0.690706i
$246$		0			0
$247$	−653.767	−	1132.36i	−0.168414	−	0.291701i
$248$	−624.951	+	1082.45i	−0.160018	+	0.277159i
$249$		0			0
$250$	2605.35	−	1504.20i	0.659108	−	0.380536i
$251$	5103.36			1.28335			0.641676	−	0.766976i	$-0.278239\pi$
$251$	5103.36			1.28335			0.641676	+	0.766976i	$0.278239\pi$
$252$		0			0
$253$	668.852			0.166207
$254$	−3234.21	+	1867.27i	−0.798947	+	0.461272i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−2269.28	−	3930.51i	−0.550793	−	0.954001i	−0.998218	−	0.0596795i	$-0.980992\pi$
$257$	−2269.28	−	3930.51i	−0.550793	−	0.954001i	0.447425	−	0.894322i	$-0.352341\pi$
$258$		0			0
$259$	23.9586	+	338.511i	0.00574794	+	0.0812127i
$260$			1313.28i			0.313254i
$261$		0			0
$262$	3794.32	+	2190.65i	0.894710	+	0.516561i
$263$	2581.50	+	1490.43i	0.605254	+	0.349444i	0.771106	−	0.636707i	$-0.219704\pi$
$263$	2581.50	+	1490.43i	0.605254	+	0.349444i	−0.165852	+	0.986151i	$0.553037\pi$
$264$		0			0
$265$			527.674i			0.122320i
$266$	1017.29	−	687.424i	0.234489	−	0.158454i
$267$		0			0
$268$	−373.251	−	646.489i	−0.0850743	−	0.147353i
$269$	−2710.50	+	4694.73i	−0.614358	+	1.06410i	0.376139	+	0.926563i	$0.377251\pi$
$269$	−2710.50	+	4694.73i	−0.614358	+	1.06410i	−0.990497	+	0.137536i	$0.956082\pi$
$270$		0			0
$271$	4424.12	−	2554.27i	0.991683	−	0.572548i	0.0859058	−	0.996303i	$-0.472622\pi$
$271$	4424.12	−	2554.27i	0.991683	−	0.572548i	0.905777	+	0.423755i	$0.139288\pi$
$272$	1626.49			0.362574
$273$		0			0
$274$	−3592.34			−0.792048
$275$	−1116.81	+	644.792i	−0.244896	+	0.141391i
$276$		0			0
$277$	−2134.85	+	3697.67i	−0.463071	+	0.802062i	−0.999112	−	0.0421295i	$-0.986586\pi$
$277$	−2134.85	+	3697.67i	−0.463071	+	0.802062i	0.536041	+	0.844192i	$0.319919\pi$
$278$	−269.664	−	467.072i	−0.0581777	−	0.100767i
$279$		0			0
$280$	−1230.09	+	87.0616i	−0.262544	+	0.0185819i
$281$			480.737i			0.102058i	0.998697	+	0.0510291i	$0.0162501\pi$
$281$			480.737i			0.102058i	−0.998697	+	0.0510291i	$0.983750\pi$
$282$		0			0
$283$	−2695.76	−	1556.40i	−0.566241	−	0.326920i	0.189405	−	0.981899i	$-0.439344\pi$
$283$	−2695.76	−	1556.40i	−0.566241	−	0.326920i	−0.755647	+	0.654979i	$0.772677\pi$
$284$	4049.28	+	2337.85i	0.846059	+	0.488472i
$285$		0			0
$286$		−	1825.72i		−	0.377473i
$287$	−2060.92	−	1003.02i	−0.423875	−	0.206294i
$288$		0			0
$289$	−2710.41	−	4694.57i	−0.551681	−	0.955540i
$290$	−1479.76	+	2563.02i	−0.299636	+	0.518985i
$291$		0			0
$292$	−2671.69	+	1542.50i	−0.535441	+	0.309137i
$293$	179.301			0.0357504			0.0178752	−	0.999840i	$-0.494310\pi$
$293$	179.301			0.0357504			0.0178752	+	0.999840i	$0.494310\pi$
$294$		0			0
$295$	3593.97			0.709319
$296$	126.950	−	73.2945i	0.0249284	−	0.0143924i
$297$		0			0
$298$	−2310.47	+	4001.85i	−0.449134	+	0.777923i
$299$	570.053	+	987.360i	0.110258	+	0.190972i
$300$		0			0
$301$	−4033.10	−	1962.85i	−0.772305	−	0.375871i
$302$		−	1344.05i		−	0.256098i
$303$		0			0
$304$	−459.296	−	265.175i	−0.0866528	−	0.0500290i
$305$	6237.75	+	3601.37i	1.17106	+	0.676110i
$306$		0			0
$307$		−	3188.08i		−	0.592681i	−0.955082	−	0.296341i	$-0.904234\pi$
$307$		−	3188.08i		−	0.592681i	0.955082	−	0.296341i	$-0.0957664\pi$
$308$	1710.08	−	121.033i	0.316367	−	0.0223913i
$309$		0			0
$310$	1300.39	+	2252.34i	0.238248	+	0.412658i
$311$	−1864.80	+	3229.93i	−0.340010	+	0.588914i	−0.984434	−	0.175754i	$-0.943764\pi$
$311$	−1864.80	+	3229.93i	−0.340010	+	0.588914i	0.644424	+	0.764668i	$0.277097\pi$
$312$		0			0
$313$	−3895.16	+	2248.87i	−0.703410	+	0.406114i	−0.808616	−	0.588337i	$-0.799783\pi$
$313$	−3895.16	+	2248.87i	−0.703410	+	0.406114i	0.105206	+	0.994450i	$0.466450\pi$
$314$	−2185.24			−0.392740
$315$		0			0
$316$	2588.86			0.460869
$317$	−8301.79	+	4793.04i	−1.47090	+	0.849224i	−0.999466	−	0.0326778i	$-0.989596\pi$
$317$	−8301.79	+	4793.04i	−1.47090	+	0.849224i	−0.471433	+	0.881902i	$0.656263\pi$
$318$		0			0
$319$	2057.17	−	3563.12i	0.361064	−	0.625381i
$320$	266.340	+	461.314i	0.0465277	+	0.0805883i
$321$		0			0
$322$	−887.030	+	599.401i	−0.153516	+	0.103737i
$323$			3369.56i			0.580456i
$324$		0			0
$325$	−1903.69	−	1099.09i	−0.324916	−	0.187590i
$326$	322.211	+	186.028i	0.0547411	+	0.0316048i
$327$		0			0
$328$			990.068i			0.166669i
$329$	−83.0923	−	1174.01i	−0.0139241	−	0.196733i
$330$		0			0
$331$	5457.29	+	9452.31i	0.906224	+	1.56963i	0.819266	+	0.573413i	$0.194381\pi$
$331$	5457.29	+	9452.31i	0.906224	+	1.56963i	0.0869572	+	0.996212i	$0.472286\pi$
$332$	463.426	−	802.677i	0.0766078	−	0.132689i
$333$		0			0
$334$	−2166.41	+	1250.77i	−0.354911	+	0.204908i
$335$	−1553.31			−0.253332
$336$		0			0
$337$	−2637.13			−0.426271			−0.213136	−	0.977023i	$-0.568368\pi$
$337$	−2637.13			−0.426271			−0.213136	+	0.977023i	$0.568368\pi$
$338$	−1110.18	+	640.961i	−0.178656	+	0.103147i
$339$		0			0
$340$	1692.18	−	2930.94i	0.269916	−	0.467508i
$341$	−1807.80	−	3131.21i	−0.287091	−	0.497256i
$342$		0			0
$343$	−1336.50	−	6210.26i	−0.210392	−	0.977617i
$344$			1937.50i			0.303672i
$345$		0			0
$346$	6787.98	+	3919.04i	1.05469	+	0.608928i
$347$	1782.87	+	1029.34i	0.275819	+	0.159244i	0.631529	−	0.775352i	$-0.282428\pi$
$347$	1782.87	+	1029.34i	0.275819	+	0.159244i	−0.355710	+	0.934596i	$0.615761\pi$
$348$		0			0
$349$		−	320.118i		−	0.0490989i	−0.999699	−	0.0245494i	$-0.992185\pi$
$349$		−	320.118i		−	0.0490989i	0.999699	−	0.0245494i	$-0.00781512\pi$
$350$	903.276	−	1855.97i	0.137949	−	0.283445i
$351$		0			0
$352$	−370.267	−	641.321i	−0.0560662	−	0.0971095i
$353$	−4150.23	+	7188.42i	−0.625764	+	1.08386i	0.362629	+	0.931934i	$0.381879\pi$
$353$	−4150.23	+	7188.42i	−0.625764	+	1.08386i	−0.988393	+	0.151921i	$0.951454\pi$
$354$		0			0
$355$	8425.67	−	4864.56i	1.25968	−	0.727279i
$356$	1036.85			0.154362
$357$		0			0
$358$	3874.15			0.571942
$359$	−9676.82	+	5586.92i	−1.42263	+	0.821354i	−0.996523	−	0.0833196i	$-0.973448\pi$
$359$	−9676.82	+	5586.92i	−1.42263	+	0.821354i	−0.426105	+	0.904674i	$0.640114\pi$
$360$		0			0
$361$	−2880.14	+	4988.55i	−0.419907	+	0.727300i
$362$	−822.430	−	1424.49i	−0.119409	−	0.206822i
$363$		0			0
$364$	1636.15	+	2421.27i	0.235598	+	0.348652i
$365$			6419.21i			0.920540i
$366$		0			0
$367$	−1963.55	−	1133.66i	−0.279282	−	0.161244i	0.353816	−	0.935315i	$-0.384884\pi$
$367$	−1963.55	−	1133.66i	−0.279282	−	0.161244i	−0.633098	+	0.774071i	$0.718217\pi$
$368$	400.484	+	231.220i	0.0567301	+	0.0327531i
$369$		0			0
$370$		−	305.020i		−	0.0428573i
$371$	657.404	+	972.866i	0.0919965	+	0.136142i
$372$		0			0
$373$	−4268.99	−	7394.10i	−0.592600	−	1.02641i	−0.993881	−	0.110458i	$-0.964768\pi$
$373$	−4268.99	−	7394.10i	−0.592600	−	1.02641i	0.401281	−	0.915955i	$-0.368565\pi$
$374$	−2352.48	+	4074.61i	−0.325251	+	0.563350i
$375$		0			0
$376$	−440.282	+	254.197i	−0.0603878	+	0.0348649i
$377$	7013.18			0.958083
$378$		0			0
$379$	10475.3			1.41973			0.709866	−	0.704336i	$-0.248755\pi$
$379$	10475.3			1.41973			0.709866	+	0.704336i	$0.248755\pi$
$380$	−955.695	+	551.771i	−0.129016	+	0.0744875i
$381$		0			0
$382$	−1836.83	+	3181.49i	−0.246023	+	0.426124i
$383$	−5289.05	−	9160.91i	−0.705634	−	1.22219i	−0.966462	−	0.256809i	$-0.917329\pi$
$383$	−5289.05	−	9160.91i	−0.705634	−	1.22219i	0.260828	−	0.965385i	$-0.416004\pi$
$384$		0			0
$385$	1561.05	−	3207.51i	0.206645	−	0.424596i
$386$			7239.34i			0.954592i
$387$		0			0
$388$	4703.58	+	2715.61i	0.615433	+	0.355321i
$389$	9695.01	+	5597.42i	1.26364	+	0.729564i	0.973777	−	0.227504i	$-0.0730565\pi$
$389$	9695.01	+	5597.42i	1.26364	+	0.729564i	0.289864	+	0.957068i	$0.406390\pi$
$390$		0			0
$391$		−	2938.09i		−	0.380014i
$392$	−2159.44	+	1693.03i	−0.278235	+	0.218140i
$393$		0			0
$394$	−670.900	−	1162.03i	−0.0857854	−	0.148585i
$395$	2693.42	−	4665.14i	0.343091	−	0.594250i
$396$		0			0
$397$	−966.902	+	558.241i	−0.122235	+	0.0705726i	−0.559871	−	0.828580i	$-0.689149\pi$
$397$	−966.902	+	558.241i	−0.122235	+	0.0705726i	0.437636	+	0.899152i	$0.355816\pi$
$398$	−1636.01			−0.206044
$399$		0			0
$400$	−891.610			−0.111451
$401$	−6854.16	+	3957.25i	−0.853568	+	0.492807i	−0.861853	−	0.507158i	$-0.830696\pi$
$401$	−6854.16	+	3957.25i	−0.853568	+	0.492807i	0.00828539	+	0.999966i	$0.497363\pi$
$402$		0			0
$403$	3081.53	−	5337.37i	0.380898	−	0.659735i
$404$	−2491.96	−	4316.20i	−0.306880	−	0.531532i
$405$		0			0
$406$	464.928	+	6568.96i	0.0568324	+	0.802985i
$407$			424.040i			0.0516434i
$408$		0			0
$409$	−9250.83	−	5340.97i	−1.11840	−	0.645706i	−0.177405	−	0.984138i	$-0.556770\pi$
$409$	−9250.83	−	5340.97i	−1.11840	−	0.645706i	−0.940991	+	0.338431i	$0.890104\pi$
$410$	1784.11	+	1030.06i	0.214905	+	0.124075i
$411$		0			0
$412$		−	1295.98i		−	0.154971i
$413$	6626.16	−	4477.55i	0.789472	−	0.533477i
$414$		0			0
$415$	−964.288	−	1670.19i	−0.114060	−	0.197558i
$416$	631.147	−	1093.18i	0.0743858	−	0.128840i
$417$		0			0
$418$	1328.61	−	767.074i	0.155465	−	0.0897580i
$419$	−522.112			−0.0608755			−0.0304377	−	0.999537i	$-0.509690\pi$
$419$	−522.112			−0.0608755			−0.0304377	+	0.999537i	$0.509690\pi$
$420$		0			0
$421$	−6758.66			−0.782415			−0.391208	−	0.920302i	$-0.627943\pi$
$421$	−6758.66			−0.782415			−0.391208	+	0.920302i	$0.627943\pi$
$422$	1222.30	−	705.692i	0.140996	−	0.0814042i
$423$		0			0
$424$	253.594	−	439.238i	0.0290463	−	0.0503097i
$425$	2832.40	+	4905.87i	0.323275	+	0.559928i
$426$		0			0
$427$	15987.2	−	1131.52i	1.81189	−	0.128239i
$428$			6915.87i			0.781055i
$429$		0			0
$430$	3491.40	+	2015.76i	0.391559	+	0.226067i
$431$	9598.90	+	5541.93i	1.07277	+	0.619362i	0.928936	−	0.370240i	$-0.120724\pi$
$431$	9598.90	+	5541.93i	1.07277	+	0.619362i	0.143831	+	0.989602i	$0.454058\pi$
$432$		0			0
$433$		−	10214.3i		−	1.13364i	−0.823842	−	0.566820i	$-0.808173\pi$
$433$		−	10214.3i		−	1.13364i	0.823842	−	0.566820i	$-0.191827\pi$
$434$	5203.58	+	2532.51i	0.575530	+	0.280102i
$435$		0			0
$436$	3295.47	+	5707.93i	0.361983	+	0.626973i
$437$	−479.013	+	829.674i	−0.0524354	+	0.0908209i
$438$		0			0
$439$	−5474.52	+	3160.71i	−0.595181	+	0.343628i	−0.767143	−	0.641476i	$-0.778323\pi$
$439$	−5474.52	+	3160.71i	−0.595181	+	0.343628i	0.171963	+	0.985103i	$0.444989\pi$
$440$	−1540.89			−0.166952
$441$		0			0
$442$	−8019.93			−0.863053
$443$	−3186.24	+	1839.58i	−0.341722	+	0.197294i	−0.661033	−	0.750356i	$-0.729882\pi$
$443$	−3186.24	+	1839.58i	−0.341722	+	0.197294i	0.319311	+	0.947650i	$0.396549\pi$
$444$		0			0
$445$	1078.73	−	1868.42i	0.114914	−	0.199037i
$446$	−6345.19	−	10990.2i	−0.673663	−	1.16682i
$447$		0			0
$448$	1065.78	+	518.699i	0.112396	+	0.0547014i
$449$		−	16168.6i		−	1.69943i	−0.527242	−	0.849715i	$-0.676774\pi$
$449$		−	16168.6i		−	1.69943i	0.527242	−	0.849715i	$-0.323226\pi$
$450$		0			0
$451$	−2480.28	−	1431.99i	−0.258962	−	0.149512i
$452$	−3652.07	−	2108.52i	−0.380042	−	0.219417i
$453$		0			0
$454$			3500.05i			0.361818i
$455$	6065.39	−	429.286i	0.624945	−	0.0442313i
$456$		0			0
$457$	246.502	+	426.953i	0.0252316	+	0.0437025i	0.878366	−	0.477990i	$-0.158634\pi$
$457$	246.502	+	426.953i	0.0252316	+	0.0437025i	−0.853134	+	0.521692i	$0.825301\pi$
$458$	4058.10	−	7028.83i	0.414023	−	0.717109i
$459$		0			0
$460$	833.320	−	481.117i	0.0844646	−	0.0487657i
$461$	−9098.09			−0.919176			−0.459588	−	0.888132i	$-0.652003\pi$
$461$	−9098.09			−0.919176			−0.459588	+	0.888132i	$0.652003\pi$
$462$		0			0
$463$	9203.93			0.923851			0.461925	−	0.886919i	$-0.347159\pi$
$463$	9203.93			0.923851			0.461925	+	0.886919i	$0.347159\pi$
$464$	2463.52	−	1422.31i	0.246478	−	0.142304i
$465$		0			0
$466$	5877.32	−	10179.8i	0.584253	−	1.01196i
$467$	−7942.44	−	13756.7i	−0.787008	−	1.36314i	−0.927792	−	0.373097i	$-0.878296\pi$
$467$	−7942.44	−	13756.7i	−0.787008	−	1.36314i	0.140785	−	0.990040i	$-0.455037\pi$
$468$		0			0
$469$	−2863.81	+	1935.19i	−0.281958	+	0.190530i
$470$			1057.86i			0.103820i
$471$		0			0
$472$	−2991.64	−	1727.22i	−0.291740	−	0.168436i
$473$	−4853.76	−	2802.32i	−0.471831	−	0.272412i
$474$		0			0
$475$		−	1847.13i		−	0.178425i
$476$	−531.669	−	7511.95i	−0.0511953	−	0.723339i
$477$		0			0
$478$	−5691.65	−	9858.22i	−0.544623	−	0.943315i
$479$	−1082.19	+	1874.41i	−0.103229	+	0.178797i	−0.913013	−	0.407930i	$-0.866251\pi$
$479$	−1082.19	+	1874.41i	−0.103229	+	0.178797i	0.809785	+	0.586727i	$0.199584\pi$
$480$		0			0
$481$	−625.968	+	361.403i	−0.0593382	+	0.0342590i
$482$	8289.84			0.783385
$483$		0			0
$484$	−3181.85			−0.298821
$485$	9787.12	−	5650.60i	0.916310	−	0.529032i
$486$		0			0
$487$	−9822.99	+	17013.9i	−0.914009	+	1.58311i	−0.105662	+	0.994402i	$0.533696\pi$
$487$	−9822.99	+	17013.9i	−0.914009	+	1.58311i	−0.808346	+	0.588707i	$0.799637\pi$
$488$	−3461.55	−	5995.59i	−0.321101	−	0.556163i
$489$		0			0
$490$	804.190	+	5652.75i	0.0741421	+	0.521153i
$491$			11831.2i			1.08744i	0.839266	+	0.543721i	$0.182985\pi$
$491$			11831.2i			1.08744i	−0.839266	+	0.543721i	$0.817015\pi$
$492$		0			0
$493$	−15651.9	−	9036.60i	−1.42987	−	0.825534i
$494$	2264.71	+	1307.53i	0.206264	+	0.119086i
$495$		0			0
$496$		−	2499.80i		−	0.226300i
$497$	9473.77	−	19465.9i	0.855043	−	1.75687i
$498$		0			0
$499$	3858.68	+	6683.42i	0.346168	+	0.599581i	0.985565	−	0.169295i	$-0.0541492\pi$
$499$	3858.68	+	6683.42i	0.346168	+	0.599581i	−0.639397	+	0.768877i	$0.720816\pi$
$500$	−3008.40	+	5210.71i	−0.269080	+	0.466060i
$501$		0			0
$502$	−8839.28	+	5103.36i	−0.785890	+	0.453734i
$503$	17416.4			1.54386			0.771928	−	0.635710i	$-0.219293\pi$
$503$	17416.4			1.54386			0.771928	+	0.635710i	$0.219293\pi$
$504$		0			0
$505$	−10370.4			−0.913819
$506$	−1158.48	+	668.852i	−0.101780	+	0.0587630i
$507$		0			0
$508$	3734.55	−	6468.42i	0.326169	−	0.564941i
$509$	−3499.63	−	6061.54i	−0.304751	−	0.527845i	0.672455	−	0.740138i	$-0.265240\pi$
$509$	−3499.63	−	6061.54i	−0.304751	−	0.527845i	−0.977206	+	0.212294i	$0.931907\pi$
$510$		0			0
$511$	7997.39	+	11835.0i	0.692336	+	1.02456i
$512$		−	512.000i		−	0.0441942i
$513$		0			0
$514$	7861.01	+	4538.56i	0.674581	+	0.389469i
$515$	−2335.36	−	1348.32i	−0.199822	−	0.115367i
$516$		0			0
$517$		−	1470.64i		−	0.125103i
$518$	−380.009	−	562.360i	−0.0322329	−	0.0477002i
$519$		0			0
$520$	−1313.28	−	2274.66i	−0.110752	−	0.191828i
$521$	7707.54	−	13349.9i	0.648126	−	1.12259i	−0.335444	−	0.942060i	$-0.608886\pi$
$521$	7707.54	−	13349.9i	0.648126	−	1.12259i	0.983570	−	0.180527i	$-0.0577803\pi$
$522$		0			0
$523$	1871.65	−	1080.60i	0.156485	−	0.0903465i	−0.419713	−	0.907657i	$-0.637869\pi$
$523$	1871.65	−	1080.60i	0.156485	−	0.0903465i	0.576198	+	0.817310i	$0.304536\pi$
$524$	−8762.61			−0.730528
$525$		0			0
$526$	−5961.71			−0.494188
$527$	−13754.6	+	7941.21i	−1.13692	+	0.656403i
$528$		0			0
$529$	−5665.82	+	9813.50i	−0.465671	+	0.806567i
$530$	−527.674	−	913.959i	−0.0432466	−	0.0749054i
$531$		0			0
$532$	−1074.58	+	2207.95i	−0.0875730	+	0.179937i
$533$		−	4881.86i		−	0.396729i
$534$		0			0
$535$	12462.5	+	7195.21i	1.00710	+	0.581450i
$536$	1292.98	+	746.502i	0.104194	+	0.0601566i
$537$		0			0
$538$		−	10842.0i		−	0.868833i
$539$	−1117.99	−	7858.47i	−0.0893417	−	0.627993i
$540$		0			0
$541$	11313.2	+	19595.0i	0.899060	+	1.55722i	0.828698	+	0.559696i	$0.189082\pi$
$541$	11313.2	+	19595.0i	0.899060	+	1.55722i	0.0703617	+	0.997522i	$0.477585\pi$
$542$	−5108.53	+	8848.23i	−0.404853	+	0.701226i
$543$		0			0
$544$	−2817.16	+	1626.49i	−0.222031	+	0.128189i
$545$	13714.3			1.07790
$546$		0			0
$547$	8918.09			0.697093			0.348547	−	0.937291i	$-0.386675\pi$
$547$	8918.09			0.697093			0.348547	+	0.937291i	$0.386675\pi$
$548$	6222.11	−	3592.34i	0.485028	−	0.280031i
$549$		0			0
$550$	1289.58	−	2233.63i	0.0999783	−	0.173168i
$551$	2946.57	+	5103.61i	0.227819	+	0.394594i
$552$		0			0
$553$	−846.250	−	11956.7i	−0.0650745	−	0.919438i
$554$		−	8539.39i		−	0.654881i
$555$		0			0
$556$	934.145	+	539.329i	0.0712528	+	0.0411378i
$557$	−11159.0	−	6442.63i	−0.848870	−	0.490095i	0.0113997	−	0.999935i	$-0.496371\pi$
$557$	−11159.0	−	6442.63i	−0.848870	−	0.490095i	−0.860269	+	0.509840i	$0.829705\pi$
$558$		0			0
$559$		−	9553.51i		−	0.722845i
$560$	2043.52	−	1380.89i	0.154205	−	0.104202i
$561$		0			0
$562$	−480.737	−	832.660i	−0.0360830	−	0.0624976i
$563$	8851.98	−	15332.1i	0.662640	−	1.14773i	−0.317279	−	0.948332i	$-0.602769\pi$
$563$	8851.98	−	15332.1i	0.662640	−	1.14773i	0.979919	−	0.199394i	$-0.0638975\pi$
$564$		0			0
$565$	−7599.15	+	4387.37i	−0.565839	+	0.326687i
$566$	6225.59			0.462334
$567$		0			0
$568$	−9351.41			−0.690804
$569$	−19264.2	+	11122.2i	−1.41933	+	0.819449i	−0.996240	−	0.0866403i	$-0.972387\pi$
$569$	−19264.2	+	11122.2i	−1.41933	+	0.819449i	−0.423087	+	0.906089i	$0.639054\pi$
$570$		0			0
$571$	−9048.83	+	15673.0i	−0.663191	+	1.14868i	0.316582	+	0.948565i	$0.397465\pi$
$571$	−9048.83	+	15673.0i	−0.663191	+	1.14868i	−0.979772	+	0.200115i	$0.935869\pi$
$572$	1825.72	+	3162.25i	0.133457	+	0.231154i
$573$		0			0
$574$	4572.64	−	323.635i	0.332506	−	0.0235336i
$575$			1610.61i			0.116812i
$576$		0			0
$577$	17496.4	+	10101.5i	1.26236	+	0.728826i	0.973531	−	0.228554i	$-0.0733997\pi$
$577$	17496.4	+	10101.5i	1.26236	+	0.728826i	0.288832	+	0.957380i	$0.406733\pi$
$578$	9389.13	+	5420.82i	0.675669	+	0.390098i
$579$		0			0
$580$		−	5919.04i		−	0.423749i
$581$	−3858.66	−	1877.96i	−0.275532	−	0.134098i
$582$		0			0
$583$	733.575	+	1270.59i	0.0521125	+	0.0902615i
$584$	3085.00	−	5343.38i	0.218593	−	0.378614i
$585$		0			0
$586$	−310.558	+	179.301i	−0.0218926	+	0.0126397i
$587$	−19836.9			−1.39482			−0.697408	−	0.716675i	$-0.745663\pi$
$587$	−19836.9			−1.39482			−0.697408	+	0.716675i	$0.745663\pi$
$588$		0			0
$589$	5178.79			0.362289
$590$	−6224.94	+	3593.97i	−0.434367	+	0.250782i
$591$		0			0
$592$	−146.589	+	253.900i	−0.0101770	+	0.0176270i
$593$	−127.658	−	221.110i	−0.00884026	−	0.0153118i	0.861571	−	0.507636i	$-0.169481\pi$
$593$	−127.658	−	221.110i	−0.00884026	−	0.0153118i	−0.870412	+	0.492325i	$0.836147\pi$
$594$		0			0
$595$	−14089.7	−	6857.29i	−0.970795	−	0.472473i
$596$		−	9241.88i		−	0.635172i
$597$		0			0
$598$	−1974.72	−	1140.11i	−0.135037	−	0.0779638i
$599$	1582.54	+	913.681i	0.107948	+	0.0623238i	0.553002	−	0.833180i	$-0.313482\pi$
$599$	1582.54	+	913.681i	0.107948	+	0.0623238i	−0.445054	+	0.895504i	$0.646816\pi$
$600$		0			0
$601$			9233.47i			0.626690i	0.949639	+	0.313345i	$0.101450\pi$
$601$			9233.47i			0.626690i	−0.949639	+	0.313345i	$0.898550\pi$
$602$	8948.38	−	633.334i	0.605829	−	0.0428784i
$603$		0			0
$604$	1344.05	+	2327.97i	0.0905442	+	0.156827i
$605$	−3310.37	+	5733.72i	−0.222455	+	0.385304i
$606$		0			0
$607$	−3718.73	+	2147.01i	−0.248663	+	0.143566i	−0.619152	−	0.785271i	$-0.712524\pi$
$607$	−3718.73	+	2147.01i	−0.248663	+	0.143566i	0.370489	+	0.928837i	$0.379190\pi$
$608$	1060.70			0.0707517
$609$		0			0
$610$	−14405.5			−0.956165
$611$	2170.96	−	1253.40i	0.143744	−	0.0829905i
$612$		0			0
$613$	−3825.38	+	6625.75i	−0.252048	+	0.436561i	−0.964090	−	0.265577i	$-0.914438\pi$
$613$	−3825.38	+	6625.75i	−0.252048	+	0.436561i	0.712041	+	0.702138i	$0.247771\pi$
$614$	3188.08	+	5521.91i	0.209545	+	0.362942i
$615$		0			0
$616$	−2840.92	+	1919.72i	−0.185818	+	0.125564i
$617$			1771.89i			0.115614i	0.998328	+	0.0578069i	$0.0184108\pi$
$617$			1771.89i			0.115614i	−0.998328	+	0.0578069i	$0.981589\pi$
$618$		0			0
$619$	8545.25	+	4933.60i	0.554867	+	0.320353i	0.751083	−	0.660208i	$-0.229532\pi$
$619$	8545.25	+	4933.60i	0.554867	+	0.320353i	−0.196216	+	0.980561i	$0.562865\pi$
$620$	−4504.67	−	2600.77i	−0.291793	−	0.168467i
$621$		0			0
$622$		−	7459.20i		−	0.480847i
$623$	−338.928	−	4788.71i	−0.0217959	−	0.307955i
$624$		0			0
$625$	2776.98	+	4809.87i	0.177727	+	0.307831i
$626$	4497.74	−	7790.31i	0.287166	−	0.497386i
$627$		0			0
$628$	3784.95	−	2185.24i	0.240503	−	0.138855i
$629$	1862.70			0.118077
$630$		0			0
$631$	−8657.42			−0.546191			−0.273095	−	0.961987i	$-0.588047\pi$
$631$	−8657.42			−0.546191			−0.273095	+	0.961987i	$0.588047\pi$
$632$	−4484.03	+	2588.86i	−0.282223	+	0.162942i
$633$		0			0
$634$	9586.08	−	16603.6i	0.600492	−	1.04008i
$635$	−7770.77	−	13459.4i	−0.485628	−	0.841132i
$636$		0			0
$637$	10647.8	−	8348.04i	0.662297	−	0.519249i
$638$			8228.67i			0.510621i
$639$		0			0
$640$	−922.629	−	532.680i	−0.0569845	−	0.0329000i
$641$	2344.82	+	1353.78i	0.144485	+	0.0834185i	0.570500	−	0.821298i	$-0.306749\pi$
$641$	2344.82	+	1353.78i	0.144485	+	0.0834185i	−0.426015	+	0.904716i	$0.640083\pi$
$642$		0			0
$643$			4164.00i			0.255384i	0.991814	+	0.127692i	$0.0407569\pi$
$643$			4164.00i			0.255384i	−0.991814	+	0.127692i	$0.959243\pi$
$644$	936.980	−	1925.22i	0.0573326	−	0.117802i
$645$		0			0
$646$	−3369.56	−	5836.24i	−0.205222	−	0.355455i
$647$	8381.49	−	14517.2i	0.509289	−	0.882115i	−0.490653	−	0.871355i	$-0.663242\pi$
$647$	8381.49	−	14517.2i	0.509289	−	0.882115i	0.999942	−	0.0107600i	$-0.00342508\pi$
$648$		0			0
$649$	8653.94	−	4996.35i	0.523416	−	0.302194i
$650$	4396.38			0.265293
$651$		0			0
$652$	−744.114			−0.0446959
$653$	10249.1	−	5917.34i	0.614211	−	0.354615i	−0.160401	−	0.987052i	$-0.551279\pi$
$653$	10249.1	−	5917.34i	0.614211	−	0.354615i	0.774612	+	0.632437i	$0.217945\pi$
$654$		0			0
$655$	−9116.54	+	15790.3i	−0.543836	+	0.941952i
$656$	−990.068	−	1714.85i	−0.0589263	−	0.102063i
$657$		0			0
$658$	1317.93	+	1950.35i	0.0780825	+	0.115551i
$659$		−	3368.19i		−	0.199099i	−0.995033	−	0.0995493i	$-0.968260\pi$
$659$		−	3368.19i		−	0.199099i	0.995033	−	0.0995493i	$-0.0317401\pi$
$660$		0			0
$661$	−23010.5	−	13285.1i	−1.35401	−	0.781740i	−0.365205	−	0.930927i	$-0.619001\pi$
$661$	−23010.5	−	13285.1i	−1.35401	−	0.781740i	−0.988809	+	0.149187i	$0.952334\pi$
$662$	−18904.6	−	10914.6i	−1.10989	−	0.640797i
$663$		0			0
$664$			1853.70i			0.108340i
$665$	2860.76	+	4233.53i	0.166820	+	0.246871i
$666$		0			0
$667$	−2569.27	−	4450.10i	−0.149149	−	0.258334i
$668$	2501.55	−	4332.81i	0.144892	−	0.250960i
$669$		0			0
$670$	2690.41	−	1553.31i	0.155133	−	0.0895663i
$671$	20026.5			1.15219
$672$		0			0
$673$	−9144.44			−0.523763			−0.261881	−	0.965100i	$-0.584343\pi$
$673$	−9144.44			−0.523763			−0.261881	+	0.965100i	$0.584343\pi$
$674$	4567.64	−	2637.13i	0.261037	−	0.150710i
$675$		0			0
$676$	1281.92	−	2220.35i	0.0729359	−	0.126329i
$677$	−833.748	−	1444.09i	−0.0473317	−	0.0819809i	0.841389	−	0.540430i	$-0.181738\pi$
$677$	−833.748	−	1444.09i	−0.0473317	−	0.0819809i	−0.888721	+	0.458449i	$0.848405\pi$
$678$		0			0
$679$	11004.6	−	22611.2i	0.621969	−	1.27797i
$680$			6768.72i			0.381719i
$681$		0			0
$682$	6262.41	+	3615.61i	0.351613	+	0.203004i
$683$	7775.82	+	4489.37i	0.435627	+	0.251509i	0.701741	−	0.712432i	$-0.252406\pi$
$683$	7775.82	+	4489.37i	0.435627	+	0.251509i	−0.266114	+	0.963942i	$0.585740\pi$
$684$		0			0
$685$		−	14949.7i		−	0.833869i
$686$	8525.15	+	9419.99i	0.474478	+	0.524281i
$687$		0			0
$688$	−1937.50	−	3355.85i	−0.107364	−	0.185960i
$689$	−1250.43	+	2165.81i	−0.0691403	+	0.119754i
$690$		0			0
$691$	23895.7	−	13796.2i	1.31554	−	0.759527i	0.332531	−	0.943092i	$-0.392097\pi$
$691$	23895.7	−	13796.2i	1.31554	−	0.759527i	0.983007	+	0.183566i	$0.0587640\pi$
$692$	−15676.2			−0.861154
$693$		0			0
$694$	−4117.36			−0.225206
$695$	1943.75	−	1122.23i	0.106087	−	0.0612495i
$696$		0			0
$697$	−6290.36	+	10895.2i	−0.341843	+	0.592089i
$698$	320.118	+	554.460i	0.0173591	+	0.0300668i
$699$		0			0
$700$	291.451	+	4117.91i	0.0157369	+	0.222346i
$701$			26713.1i			1.43929i	0.694344	+	0.719644i	$0.255695\pi$
$701$			26713.1i			1.43929i	−0.694344	+	0.719644i	$0.744305\pi$
$702$		0			0
$703$	−525.998	−	303.685i	−0.0282196	−	0.0162926i
$704$	1282.64	+	740.534i	0.0686668	+	0.0396448i
$705$		0			0
$706$		−	16600.9i		−	0.884964i
$707$	−19119.8	+	12920.0i	−1.01708	+	0.687281i
$708$		0			0
$709$	9177.34	+	15895.6i	0.486124	+	0.841992i	0.999873	−	0.0159488i	$-0.00507688\pi$
$709$	9177.34	+	15895.6i	0.486124	+	0.841992i	−0.513748	+	0.857941i	$0.671744\pi$
$710$	−9729.12	+	16851.3i	−0.514264	+	0.890731i
$711$		0			0
$712$	−1795.88	+	1036.85i	−0.0945273	+	0.0545754i
$713$	−4515.66			−0.237185
$714$		0			0
$715$	7597.87			0.397404
$716$	−6710.23	+	3874.15i	−0.350242	+	0.202212i
$717$		0			0
$718$	11173.8	−	19353.6i	0.580785	−	1.00595i
$719$	−7775.70	−	13467.9i	−0.403317	−	0.698565i	0.590807	−	0.806813i	$-0.298809\pi$
$719$	−7775.70	−	13467.9i	−0.403317	−	0.698565i	−0.994124	+	0.108248i	$0.965476\pi$
$720$		0			0
$721$	−5985.49	+	423.631i	−0.309169	+	0.0218819i
$722$		−	11520.6i		−	0.593838i
$723$		0			0
$724$	2848.98	+	1644.86i	0.146245	+	0.0844348i
$725$	8580.06	+	4953.70i	0.439525	+	0.253760i
$726$		0			0
$727$			27003.3i			1.37757i	0.724964	+	0.688787i	$0.241856\pi$
$727$			27003.3i			1.37757i	−0.724964	+	0.688787i	$0.758144\pi$
$728$	−5255.17	−	2557.62i	−0.267540	−	0.130208i
$729$		0			0
$730$	−6419.21	−	11118.4i	−0.325460	−	0.563713i
$731$	−12309.9	+	21321.3i	−0.622841	+	1.07879i
$732$		0			0
$733$	−13000.2	+	7505.67i	−0.655079	+	0.378210i	−0.790400	−	0.612592i	$-0.790127\pi$
$733$	−13000.2	+	7505.67i	−0.655079	+	0.378210i	0.135320	+	0.990802i	$0.456794\pi$
$734$	4534.62			0.228033
$735$		0			0
$736$	−924.879			−0.0463199
$737$	−3740.21	+	2159.41i	−0.186937	+	0.107928i
$738$		0			0
$739$	7159.79	−	12401.1i	0.356397	−	0.617298i	−0.630959	−	0.775816i	$-0.717338\pi$
$739$	7159.79	−	12401.1i	0.356397	−	0.617298i	0.987356	+	0.158519i	$0.0506718\pi$
$740$	305.020	+	528.309i	0.0151524	+	0.0262447i
$741$		0			0
$742$	−2111.52	−	1027.65i	−0.104470	−	0.0508440i
$743$			27470.1i			1.35637i	0.734892	+	0.678184i	$0.237233\pi$
$743$			27470.1i			1.35637i	−0.734892	+	0.678184i	$0.762767\pi$
$744$		0			0
$745$	−16654.0	−	9615.17i	−0.818998	−	0.472849i
$746$	14788.2	+	8537.97i	0.725783	+	0.419031i
$747$		0			0
$748$		−	9409.91i		−	0.459974i
$749$	31941.0	−	2260.67i	1.55821	−	0.110285i
$750$		0			0
$751$	18562.1	+	32150.5i	0.901919	+	1.56217i	0.825001	+	0.565131i	$0.191174\pi$
$751$	18562.1	+	32150.5i	0.901919	+	1.56217i	0.0769175	+	0.997037i	$0.475492\pi$
$752$	508.394	−	880.564i	0.0246532	−	0.0427006i
$753$		0			0
$754$	−12147.2	+	7013.18i	−0.586703	+	0.338733i
$755$	5593.35			0.269620
$756$		0			0
$757$	30289.1			1.45426			0.727132	−	0.686498i	$-0.240853\pi$
$757$	30289.1			1.45426			0.727132	+	0.686498i	$0.240853\pi$
$758$	−18143.7	+	10475.3i	−0.869405	+	0.501951i
$759$		0			0
$760$	1103.54	−	1911.39i	0.0526706	−	0.0912282i
$761$	18626.4	+	32261.9i	0.887263	+	1.53678i	0.843098	+	0.537760i	$0.180729\pi$
$761$	18626.4	+	32261.9i	0.887263	+	1.53678i	0.0441643	+	0.999024i	$0.485937\pi$
$762$		0			0
$763$	25284.9	−	17086.0i	1.19971	−	0.810688i
$764$		−	7347.34i		−	0.347928i
$765$		0			0
$766$	18321.8	+	10578.1i	0.864222	+	0.498959i
$767$	14751.3	+	8516.64i	0.694442	+	0.400936i
$768$		0			0
$769$			24044.8i			1.12754i	0.825932	+	0.563770i	$0.190650\pi$
$769$			24044.8i			1.12754i	−0.825932	+	0.563770i	$0.809350\pi$
$770$	503.688	+	7116.61i	0.0235736	+	0.333071i
$771$		0			0
$772$	−7239.34	−	12538.9i	−0.337499	−	0.584566i
$773$	−14130.8	+	24475.2i	−0.657501	+	1.13882i	0.323760	+	0.946139i	$0.395053\pi$
$773$	−14130.8	+	24475.2i	−0.657501	+	1.13882i	−0.981261	+	0.192685i	$0.938280\pi$
$774$		0			0
$775$	7540.00	−	4353.22i	0.349477	−	0.201771i
$776$	−10862.5			−0.502499
$777$		0			0
$778$	−22389.7			−1.03176
$779$	3552.61	−	2051.10i	0.163396	−	0.0943368i
$780$		0			0
$781$	13525.5	−	23426.8i	0.619692	−	1.07334i
$782$	2938.09	+	5088.92i	0.134355	+	0.232710i
$783$		0			0
$784$	2047.23	−	5091.85i	0.0932596	−	0.231954i
$785$		−	9094.03i		−	0.413477i
$786$		0			0
$787$	−8810.30	−	5086.63i	−0.399051	−	0.230392i	0.287023	−	0.957924i	$-0.407334\pi$
$787$	−8810.30	−	5086.63i	−0.399051	−	0.230392i	−0.686074	+	0.727531i	$0.740668\pi$
$788$	2324.06	+	1341.80i	0.105065	+	0.0606594i
$789$		0			0
$790$			10773.7i			0.485203i
$791$	−8544.44	+	17556.4i	−0.384078	+	0.789168i
$792$		0			0
$793$	17068.3	+	29563.2i	0.764331	+	1.32386i
$794$	1116.48	−	1933.80i	0.0499024	−	0.0864334i
$795$		0			0
$796$	2833.65	−	1636.01i	0.126176	−	0.0728476i
$797$	−32606.8			−1.44917			−0.724587	−	0.689184i	$-0.757969\pi$
$797$	−32606.8			−1.44917			−0.724587	+	0.689184i	$0.757969\pi$
$798$		0			0
$799$	−6460.12			−0.286036
$800$	1544.31	−	891.610i	0.0682497	−	0.0394040i
$801$		0			0
$802$	7914.51	−	13708.3i	0.348468	−	0.603563i
$803$	8924.02	+	15456.9i	0.392182	+	0.679278i
$804$		0			0
$805$	−2494.44	−	3691.43i	−0.109214	−	0.161622i
$806$			12326.1i			0.538671i
$807$		0			0
$808$	8632.40	+	4983.92i	0.375850	+	0.216997i
$809$	−22305.4	−	12878.0i	−0.969365	−	0.559663i	−0.0703222	−	0.997524i	$-0.522403\pi$
$809$	−22305.4	−	12878.0i	−0.969365	−	0.559663i	−0.899042	+	0.437861i	$0.855736\pi$
$810$		0			0
$811$			3614.74i			0.156511i	0.996933	+	0.0782556i	$0.0249350\pi$
$811$			3614.74i			0.156511i	−0.996933	+	0.0782556i	$0.975065\pi$
$812$	−7374.24	−	10912.8i	−0.318701	−	0.471633i
$813$		0			0
$814$	−424.040	−	734.458i	−0.0182587	−	0.0316250i
$815$	−774.169	+	1340.90i	−0.0332736	+	0.0576315i
$816$		0			0
$817$	6952.25	−	4013.89i	0.297709	−	0.171883i
$818$	21363.9			0.913167
$819$		0			0
$820$	−4120.23			−0.175469
$821$	16618.7	−	9594.81i	0.706451	−	0.407870i	−0.103295	−	0.994651i	$-0.532938\pi$
$821$	16618.7	−	9594.81i	0.706451	−	0.407870i	0.809746	+	0.586781i	$0.199605\pi$
$822$		0			0
$823$	−12693.6	+	21985.9i	−0.537630	+	0.931203i	0.461401	+	0.887192i	$0.347347\pi$
$823$	−12693.6	+	21985.9i	−0.537630	+	0.931203i	−0.999031	+	0.0440111i	$0.985986\pi$
$824$	1295.98	+	2244.70i	0.0547907	+	0.0949002i
$825$		0			0
$826$	−6999.28	+	14381.5i	−0.294838	+	0.605807i
$827$			2224.78i			0.0935468i	0.998906	+	0.0467734i	$0.0148939\pi$
$827$			2224.78i			0.0935468i	−0.998906	+	0.0467734i	$0.985106\pi$
$828$		0			0
$829$	3840.87	+	2217.53i	0.160916	+	0.0929047i	0.578295	−	0.815828i	$-0.303718\pi$
$829$	3840.87	+	2217.53i	0.160916	+	0.0929047i	−0.417380	+	0.908732i	$0.637051\pi$
$830$	3340.39	+	1928.58i	0.139695	+	0.0806528i
$831$		0			0
$832$			2524.59i			0.105197i
$833$	−34520.2	+	4911.03i	−1.43584	+	0.204270i
$834$		0			0
$835$	−5205.18	−	9015.63i	−0.215728	−	0.373651i
$836$	−1534.15	+	2657.22i	−0.0634685	+	0.109931i
$837$		0			0
$838$	904.324	−	522.112i	0.0372785	−	0.0215227i
$839$	−33407.6			−1.37468			−0.687340	−	0.726336i	$-0.741222\pi$
$839$	−33407.6			−1.37468			−0.687340	+	0.726336i	$0.741222\pi$
$840$		0			0
$841$	−7219.91			−0.296031
$842$	11706.3	−	6758.66i	0.479129	−	0.276626i
$843$		0			0
$844$	−1411.38	+	2444.59i	−0.0575615	+	0.0996994i
$845$	−2667.40	−	4620.07i	−0.108593	−	0.188089i
$846$		0			0
$847$	1040.09	+	14695.4i	0.0421934	+	0.596151i
$848$			1014.38i			0.0410777i
$849$		0			0
$850$	−9811.74	−	5664.81i	−0.395929	−	0.228590i
$851$	458.645	+	264.799i	0.0184749	+	0.0106665i
$852$		0			0
$853$		−	36851.9i		−	1.47923i	−0.673029	−	0.739616i	$-0.735007\pi$
$853$		−	36851.9i		−	1.47923i	0.673029	−	0.739616i	$-0.264993\pi$
$854$	−26559.2	+	17947.1i	−1.06421	+	0.719129i
$855$		0			0
$856$	−6915.87	−	11978.6i	−0.276145	−	0.478296i
$857$	645.110	−	1117.36i	0.0257136	−	0.0445372i	−0.852882	−	0.522103i	$-0.825148\pi$
$857$	645.110	−	1117.36i	0.0257136	−	0.0445372i	0.878596	+	0.477566i	$0.158481\pi$
$858$		0			0
$859$	−10713.8	+	6185.59i	−0.425551	+	0.245692i	−0.697450	−	0.716634i	$-0.745682\pi$
$859$	−10713.8	+	6185.59i	−0.425551	+	0.245692i	0.271898	+	0.962326i	$0.412349\pi$
$860$	−8063.04			−0.319706
$861$		0			0
$862$	−22167.7			−0.875911
$863$	−4672.03	+	2697.40i	−0.184285	+	0.106397i	−0.589304	−	0.807911i	$-0.700598\pi$
$863$	−4672.03	+	2697.40i	−0.184285	+	0.106397i	0.405020	+	0.914308i	$0.367265\pi$
$864$		0			0
$865$	−16309.3	+	28248.6i	−0.641080	+	1.11038i
$866$	10214.3	+	17691.6i	0.400802	+	0.694210i
$867$		0			0
$868$	−11545.4	+	817.140i	−0.451470	+	0.0319534i
$869$		−	14977.6i		−	0.584674i
$870$		0			0
$871$	−6375.46	−	3680.87i	−0.248019	−	0.143194i
$872$	−11415.9	−	6590.95i	−0.443337	−	0.255961i
$873$		0			0
$874$		−	1916.05i		−	0.0741549i
$875$	25049.1	+	12191.1i	0.967788	+	0.471009i
$876$		0			0
$877$	−11228.9	−	19449.0i	−0.432352	−	0.748856i	0.564723	−	0.825280i	$-0.308983\pi$
$877$	−11228.9	−	19449.0i	−0.432352	−	0.748856i	−0.997075	+	0.0764243i	$0.975650\pi$
$878$	6321.43	−	10949.0i	0.242982	−	0.420856i
$879$		0			0
$880$	2668.90	−	1540.89i	0.102237	−	0.0590265i
$881$	−16245.3			−0.621247			−0.310623	−	0.950533i	$-0.600538\pi$
$881$	−16245.3			−0.621247			−0.310623	+	0.950533i	$0.600538\pi$
$882$		0			0
$883$	−15876.5			−0.605081			−0.302540	−	0.953137i	$-0.597835\pi$
$883$	−15876.5			−0.605081			−0.302540	+	0.953137i	$0.597835\pi$
$884$	13890.9	−	8019.93i	0.528510	−	0.305135i
$885$		0			0
$886$	3679.16	−	6372.49i	0.139508	−	0.241634i
$887$	−10331.9	−	17895.3i	−0.391105	−	0.677413i	0.601491	−	0.798880i	$-0.294574\pi$
$887$	−10331.9	−	17895.3i	−0.391105	−	0.677413i	−0.992596	+	0.121466i	$0.961240\pi$
$888$		0			0
$889$	−31095.2	−	15133.6i	−1.17312	−	0.570940i
$890$			4314.92i			0.162513i
$891$		0			0
$892$	21980.4	+	12690.4i	0.825065	+	0.476351i
$893$	1824.24	+	1053.23i	0.0683606	+	0.0394680i
$894$		0			0
$895$			16122.5i			0.602141i
$896$	−2364.68	+	167.363i	−0.0881678	+	0.00624020i
$897$		0			0
$898$	16168.6	+	28004.9i	0.600839	+	1.04068i
$899$	−13888.7	+	24055.9i	−0.515254	+	0.892446i
$900$		0			0
$901$	5581.37	−	3222.41i	0.206373	−	0.119150i
$902$	5727.96			0.211441
$903$		0			0
$904$	8434.09			0.310303
$905$	5928.11	−	3422.60i	0.217743	−	0.125714i
$906$		0			0
$907$	12054.3	−	20878.7i	0.441298	−	0.764351i	−0.556488	−	0.830856i	$-0.687851\pi$
$907$	12054.3	−	20878.7i	0.441298	−	0.764351i	0.997786	+	0.0665047i	$0.0211847\pi$
$908$	−3500.05	−	6062.26i	−0.127922	−	0.221567i
$909$		0			0
$910$	−10076.3	+	6808.93i	−0.367061	+	0.248037i
$911$		−	47902.4i		−	1.74213i	−0.491171	−	0.871063i	$-0.663431\pi$
$911$		−	47902.4i		−	1.74213i	0.491171	−	0.871063i	$-0.336569\pi$
$912$		0			0
$913$	−4643.82	−	2681.11i	−0.168333	−	0.0971872i
$914$	−853.907	−	493.003i	−0.0309023	−	0.0178415i
$915$		0			0
$916$			16232.4i			0.585517i
$917$	2864.34	+	40470.2i	0.103150	+	1.45741i
$918$		0			0
$919$	−19752.6	−	34212.6i	−0.709009	−	1.22804i	−0.965225	−	0.261420i	$-0.915809\pi$
$919$	−19752.6	−	34212.6i	−0.709009	−	1.22804i	0.256216	−	0.966619i	$-0.417524\pi$
$920$	−962.235	+	1666.64i	−0.0344825	+	0.0597255i
$921$		0			0
$922$	15758.3	−	9098.09i	0.562878	−	0.324978i
$923$	46110.3			1.64435
$924$		0			0
$925$	−1021.10			−0.0362956
$926$	−15941.7	+	9203.93i	−0.565741	+	0.326631i
$927$		0			0
$928$	−2844.62	+	4927.03i	−0.100624	+	0.174286i
$929$	23541.7	+	40775.4i	0.831407	+	1.44004i	0.896923	+	0.442188i	$0.145797\pi$
$929$	23541.7	+	40775.4i	0.831407	+	1.44004i	−0.0655155	+	0.997852i	$0.520869\pi$
$930$		0			0
$931$	10548.7	+	4241.21i	0.371342	+	0.149302i
$932$			23509.3i			0.826258i
$933$		0			0
$934$	27513.4	+	15884.9i	0.963883	+	0.556498i
$935$	−16956.7	−	9789.98i	−0.593096	−	0.342424i
$936$		0			0
$937$			25357.7i			0.884099i	0.896991	+	0.442050i	$0.145748\pi$
$937$			25357.7i			0.884099i	−0.896991	+	0.442050i	$0.854252\pi$
$938$	3025.08	−	6215.65i	0.105301	−	0.216363i
$939$		0			0
$940$	−1057.86	−	1832.26i	−0.0367058	−	0.0635763i
$941$	8535.03	−	14783.1i	0.295679	−	0.512131i	−0.679464	−	0.733709i	$-0.737787\pi$
$941$	8535.03	−	14783.1i	0.295679	−	0.512131i	0.975143	+	0.221578i	$0.0711208\pi$
$942$		0			0
$943$	−3097.71	+	1788.46i	−0.106973	+	0.0617607i
$944$	6908.89			0.238205
$945$		0			0
$946$	11209.3			0.385248
$947$	33746.1	−	19483.3i	1.15797	−	0.668556i	0.207156	−	0.978308i	$-0.433579\pi$
$947$	33746.1	−	19483.3i	1.15797	−	0.668556i	0.950817	+	0.309752i	$0.100246\pi$
$948$		0			0
$949$	−15211.6	+	26347.3i	−0.520327	+	0.901233i
$950$	1847.13	+	3199.32i	0.0630829	+	0.109263i
$951$		0			0
$952$	8432.82	+	12479.4i	0.287090	+	0.424853i
$953$			22611.7i			0.768589i	0.923211	+	0.384295i	$0.125555\pi$
$953$			22611.7i			0.768589i	−0.923211	+	0.384295i	$0.874445\pi$
$954$		0			0
$955$	−13240.0	−	7644.10i	−0.448623	−	0.259013i
$956$	19716.4	+	11383.3i	0.667025	+	0.385107i
$957$		0			0
$958$		−	4328.76i		−	0.145987i
$959$	−18625.1	−	27562.6i	−0.627151	−	0.928095i
$960$		0			0
$961$	−2690.37	−	4659.86i	−0.0903083	−	0.156419i
$962$	722.806	−	1251.94i	0.0242247	−	0.0419585i
$963$		0			0
$964$	−14358.4	+	8289.84i	−0.479724	+	0.276969i
$965$	−30127.0			−1.00500
$966$		0			0
$967$	8946.41			0.297515			0.148758	−	0.988874i	$-0.452473\pi$
$967$	8946.41			0.297515			0.148758	+	0.988874i	$0.452473\pi$
$968$	5511.12	−	3181.85i	0.182990	−	0.105649i
$969$		0			0
$970$	−11301.2	+	19574.2i	−0.374082	+	0.647929i
$971$	29157.3	+	50501.9i	0.963648	+	1.66909i	0.713205	+	0.700955i	$0.247243\pi$
$971$	29157.3	+	50501.9i	0.963648	+	1.66909i	0.250443	+	0.968131i	$0.419424\pi$
$972$		0			0
$973$	2185.54	−	4490.66i	0.0720095	−	0.147959i
$974$		−	39292.0i		−	1.29260i
$975$		0			0
$976$	11991.2	+	6923.11i	0.393267	+	0.227053i
$977$	−2795.29	−	1613.86i	−0.0915346	−	0.0528475i	0.453534	−	0.891239i	$-0.350163\pi$
$977$	−2795.29	−	1613.86i	−0.0915346	−	0.0528475i	−0.545069	+	0.838391i	$0.683496\pi$
$978$		0			0
$979$		−	5998.62i		−	0.195829i
$980$	−7045.64	−	8986.65i	−0.229658	−	0.292927i
$981$		0			0
$982$	−11831.2	−	20492.2i	−0.384469	−	0.665920i
$983$	−4506.86	+	7806.10i	−0.146232	+	0.253282i	−0.929832	−	0.367984i	$-0.880048\pi$
$983$	−4506.86	+	7806.10i	−0.146232	+	0.253282i	0.783600	+	0.621266i	$0.213381\pi$
$984$		0			0
$985$	4835.87	−	2791.99i	0.156430	−	0.0903149i
$986$	36146.4			1.16748
$987$		0			0
$988$	−5230.13			−0.168414
$989$	−6062.03	+	3499.91i	−0.194905	+	0.112529i
$990$		0			0
$991$	−14685.7	+	25436.4i	−0.470744	+	0.815353i	−0.999440	−	0.0334585i	$-0.989348\pi$
$991$	−14685.7	+	25436.4i	−0.470744	+	0.815353i	0.528696	+	0.848811i	$0.322681\pi$
$992$	2499.80	+	4329.79i	0.0800090	+	0.138580i
$993$		0			0
$994$	3056.81	+	43189.6i	0.0975412	+	1.37816i
$995$		−	6808.34i		−	0.216924i
$996$		0			0
$997$	−16427.1	−	9484.18i	−0.521816	−	0.301271i	0.215861	−	0.976424i	$-0.430744\pi$
$997$	−16427.1	−	9484.18i	−0.521816	−	0.301271i	−0.737677	+	0.675153i	$0.764077\pi$
$998$	−13366.8	−	7717.35i	−0.423968	−	0.244778i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.k.a.269.1	yes	12
3.2	odd	2	inner	378.4.k.a.269.6	yes	12
7.5	odd	6	inner	378.4.k.a.215.6	yes	12
21.5	even	6	inner	378.4.k.a.215.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.k.a.215.1	✓	12	21.5	even	6	inner
378.4.k.a.215.6	yes	12	7.5	odd	6	inner
378.4.k.a.269.1	yes	12	1.1	even	1	trivial
378.4.k.a.269.6	yes	12	3.2	odd	2	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-4.16156 - 7.20804i) q^{5} +(-16.6528 - 8.10467i) q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-4.16156 - 7.20804i) q^{5} +(-16.6528 - 8.10467i) q^{7} +8.00000i q^{8} +(14.4161 + 8.32312i) q^{10} +(-20.0413 - 11.5708i) q^{11} -39.4467i q^{13} +(36.9481 - 2.61505i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(-50.8277 + 88.0362i) q^{17} +(28.7060 - 16.5734i) q^{19} -33.2925 q^{20} +46.2834 q^{22} +(-25.0303 + 14.4512i) q^{23} +(27.8628 - 48.2598i) q^{25} +(39.4467 + 68.3236i) q^{26} +(-61.3809 + 41.4775i) q^{28} +177.789i q^{29} +(135.306 + 78.1189i) q^{31} +(27.7128 + 16.0000i) q^{32} -203.311i q^{34} +(10.8827 + 153.762i) q^{35} +(-9.16181 - 15.8687i) q^{37} +(-33.1469 + 57.4120i) q^{38} +(57.6643 - 33.2925i) q^{40} +123.758 q^{41} +242.188 q^{43} +(-80.1652 + 46.2834i) q^{44} +(28.9025 - 50.0605i) q^{46} +(31.7746 + 55.0352i) q^{47} +(211.629 + 269.930i) q^{49} +111.451i q^{50} +(-136.647 - 78.8933i) q^{52} +(-54.9048 - 31.6993i) q^{53} +192.611i q^{55} +(64.8374 - 133.222i) q^{56} +(-177.789 - 307.939i) q^{58} +(-215.903 + 373.955i) q^{59} +(-749.448 + 432.694i) q^{61} -312.476 q^{62} -64.0000 q^{64} +(-284.333 + 164.160i) q^{65} +(93.3127 - 161.622i) q^{67} +(203.311 + 352.145i) q^{68} +(-172.611 - 255.440i) q^{70} +1168.93i q^{71} +(-667.923 - 385.625i) q^{73} +(31.7375 + 18.3236i) q^{74} -132.587i q^{76} +(239.965 + 355.114i) q^{77} +(323.607 + 560.504i) q^{79} +(-66.5850 + 115.329i) q^{80} +(-214.356 + 123.758i) q^{82} +231.713 q^{83} +846.090 q^{85} +(-419.482 + 242.188i) q^{86} +(92.5667 - 160.330i) q^{88} +(129.606 + 224.485i) q^{89} +(-319.702 + 656.896i) q^{91} +115.610i q^{92} +(-110.070 - 63.5492i) q^{94} +(-238.924 - 137.943i) q^{95} +1357.81i q^{97} +(-636.482 - 255.904i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 24 q^{4} - 42 q^{7}+O(q^{10}) \) 12 * q + 24 * q^4 - 42 * q^7	\( 12 q + 24 q^{4} - 42 q^{7} - 48 q^{10} - 96 q^{16} - 66 q^{19} + 240 q^{22} + 366 q^{25} - 168 q^{28} + 630 q^{31} + 348 q^{37} - 192 q^{40} - 252 q^{43} + 600 q^{46} - 1218 q^{49} + 192 q^{52} - 144 q^{58} + 114 q^{61} - 768 q^{64} + 108 q^{67} - 1344 q^{70} - 978 q^{73} - 744 q^{79} + 240 q^{82} + 7248 q^{85} + 480 q^{88} - 5544 q^{91} - 1920 q^{94}+O(q^{100}) \) 12 * q + 24 * q^4 - 42 * q^7 - 48 * q^10 - 96 * q^16 - 66 * q^19 + 240 * q^22 + 366 * q^25 - 168 * q^28 + 630 * q^31 + 348 * q^37 - 192 * q^40 - 252 * q^43 + 600 * q^46 - 1218 * q^49 + 192 * q^52 - 144 * q^58 + 114 * q^61 - 768 * q^64 + 108 * q^67 - 1344 * q^70 - 978 * q^73 - 744 * q^79 + 240 * q^82 + 7248 * q^85 + 480 * q^88 - 5544 * q^91 - 1920 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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