LMFDB - Embedded newform 168.4.u.a.17.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,4,Mod(17,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

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

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

chi := DirichletCharacter("168.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	168.u (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:	$9.91232088096$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.20
Character	$\chi$	$=$	168.17
Dual form			168.4.u.a.89.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.72903 + 3.61861i) q^{3} +(6.11124 + 10.5850i) q^{5} +(4.39164 - 17.9920i) q^{7} +(0.811307 + 26.9878i) q^{9} +O(q^{10})$	\(q+(3.72903 + 3.61861i) q^{3} +(6.11124 + 10.5850i) q^{5} +(4.39164 - 17.9920i) q^{7} +(0.811307 + 26.9878i) q^{9} +(47.3827 + 27.3564i) q^{11} -27.0247i q^{13} +(-15.5139 + 61.5859i) q^{15} +(-20.3052 + 35.1697i) q^{17} +(-48.4879 + 27.9945i) q^{19} +(81.4827 - 51.2012i) q^{21} +(-93.3394 + 53.8895i) q^{23} +(-12.1945 + 21.1215i) q^{25} +(-94.6330 + 103.574i) q^{27} -38.2431i q^{29} +(257.992 + 148.952i) q^{31} +(77.6993 + 273.473i) q^{33} +(217.284 - 63.4683i) q^{35} +(-142.286 - 246.447i) q^{37} +(97.7918 - 100.776i) q^{39} -28.3958 q^{41} +212.817 q^{43} +(-280.707 + 173.517i) q^{45} +(125.209 + 216.869i) q^{47} +(-304.427 - 158.029i) q^{49} +(-202.984 + 57.6721i) q^{51} +(-294.863 - 170.239i) q^{53} +668.727i q^{55} +(-282.114 - 71.0665i) q^{57} +(451.101 - 781.329i) q^{59} +(499.851 - 288.589i) q^{61} +(489.129 + 103.924i) q^{63} +(286.056 - 165.154i) q^{65} +(-184.464 + 319.500i) q^{67} +(-543.070 - 136.803i) q^{69} -1184.28i q^{71} +(-407.399 - 235.212i) q^{73} +(-121.904 + 34.6355i) q^{75} +(700.286 - 732.373i) q^{77} +(-20.0802 - 34.7799i) q^{79} +(-727.684 + 43.7908i) q^{81} -855.263 q^{83} -496.361 q^{85} +(138.387 - 142.610i) q^{87} +(-322.110 - 557.910i) q^{89} +(-486.229 - 118.683i) q^{91} +(423.061 + 1489.02i) q^{93} +(-592.642 - 342.162i) q^{95} -748.863i q^{97} +(-699.848 + 1300.95i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 12 q^{7} + 14 q^{9}+O(q^{10}) $ 48 * q + 12 * q^7 + 14 * q^9	$ 48 q + 12 q^{7} + 14 q^{9} - 88 q^{15} - 270 q^{19} + 50 q^{21} - 438 q^{25} + 216 q^{31} - 372 q^{33} + 66 q^{37} + 242 q^{39} + 900 q^{43} - 294 q^{45} + 60 q^{49} - 138 q^{51} + 1384 q^{57} + 108 q^{61} + 1096 q^{63} + 6 q^{67} - 1206 q^{73} - 594 q^{75} - 588 q^{79} - 54 q^{81} - 240 q^{85} - 3522 q^{87} + 234 q^{91} - 608 q^{93} + 1988 q^{99}+O(q^{100}) $ 48 * q + 12 * q^7 + 14 * q^9 - 88 * q^15 - 270 * q^19 + 50 * q^21 - 438 * q^25 + 216 * q^31 - 372 * q^33 + 66 * q^37 + 242 * q^39 + 900 * q^43 - 294 * q^45 + 60 * q^49 - 138 * q^51 + 1384 * q^57 + 108 * q^61 + 1096 * q^63 + 6 * q^67 - 1206 * q^73 - 594 * q^75 - 588 * q^79 - 54 * q^81 - 240 * q^85 - 3522 * q^87 + 234 * q^91 - 608 * q^93 + 1988 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$-1$	$1$

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
$3$	3.72903	+	3.61861i	0.717652	+	0.696402i
$3$	3.72903	+	3.61861i	0.717652	+	0.696402i
$4$		0			0
$5$	6.11124	+	10.5850i	0.546606	+	0.946749i	0.998504	+	0.0546796i	$0.0174138\pi$
$5$	6.11124	+	10.5850i	0.546606	+	0.946749i	−0.451898	+	0.892070i	$0.649253\pi$
$6$		0			0
$7$	4.39164	−	17.9920i	0.237126	−	0.971479i
$7$	4.39164	−	17.9920i	0.237126	−	0.971479i
$8$		0			0
$9$	0.811307	+	26.9878i	0.0300484	+	0.999548i
$10$		0			0
$11$	47.3827	+	27.3564i	1.29877	+	0.749843i	0.980191	−	0.198054i	$-0.0634623\pi$
$11$	47.3827	+	27.3564i	1.29877	+	0.749843i	0.318575	+	0.947898i	$0.396796\pi$
$12$		0			0
$13$		−	27.0247i		−	0.576561i	−0.957546	−	0.288281i	$-0.906916\pi$
$13$		−	27.0247i		−	0.576561i	0.957546	−	0.288281i	$-0.0930836\pi$
$14$		0			0
$15$	−15.5139	+	61.5859i	−0.267045	+	1.06009i
$16$		0			0
$17$	−20.3052	+	35.1697i	−0.289691	+	0.501759i	−0.973736	−	0.227680i	$-0.926886\pi$
$17$	−20.3052	+	35.1697i	−0.289691	+	0.501759i	0.684045	+	0.729440i	$0.260219\pi$
$18$		0			0
$19$	−48.4879	+	27.9945i	−0.585467	+	0.338020i	−0.763303	−	0.646040i	$-0.776424\pi$
$19$	−48.4879	+	27.9945i	−0.585467	+	0.338020i	0.177836	+	0.984060i	$0.443090\pi$
$20$		0			0
$21$	81.4827	−	51.2012i	0.846714	−	0.532049i
$22$		0			0
$23$	−93.3394	+	53.8895i	−0.846200	+	0.488554i	−0.859367	−	0.511359i	$-0.829142\pi$
$23$	−93.3394	+	53.8895i	−0.846200	+	0.488554i	0.0131667	+	0.999913i	$0.495809\pi$
$24$		0			0
$25$	−12.1945	+	21.1215i	−0.0975561	+	0.168972i
$26$		0			0
$27$	−94.6330	+	103.574i	−0.674523	+	0.738254i
$28$		0			0
$29$		−	38.2431i		−	0.244882i	−0.992476	−	0.122441i	$-0.960928\pi$
$29$		−	38.2431i		−	0.244882i	0.992476	−	0.122441i	$-0.0390722\pi$
$30$		0			0
$31$	257.992	+	148.952i	1.49473	+	0.862985i	0.999982	−	0.00604814i	$-0.00192519\pi$
$31$	257.992	+	148.952i	1.49473	+	0.862985i	0.494753	+	0.869034i	$0.335259\pi$
$32$		0			0
$33$	77.6993	+	273.473i	0.409870	+	1.44259i
$34$		0			0
$35$	217.284	−	63.4683i	1.04936	−	0.306517i
$36$		0			0
$37$	−142.286	−	246.447i	−0.632209	−	1.09502i	−0.987099	−	0.160110i	$-0.948815\pi$
$37$	−142.286	−	246.447i	−0.632209	−	1.09502i	0.354890	−	0.934908i	$-0.384518\pi$
$38$		0			0
$39$	97.7918	−	100.776i	0.401518	−	0.413770i
$40$		0			0
$41$	−28.3958			−0.108163			−0.0540815	−	0.998537i	$-0.517223\pi$
$41$	−28.3958			−0.108163			−0.0540815	+	0.998537i	$0.517223\pi$
$42$		0			0
$43$	212.817			0.754752			0.377376	−	0.926060i	$-0.376827\pi$
$43$	212.817			0.754752			0.377376	+	0.926060i	$0.376827\pi$
$44$		0			0
$45$	−280.707	+	173.517i	−0.929897	+	0.574807i
$46$		0			0
$47$	125.209	+	216.869i	0.388588	+	0.673055i	0.992260	−	0.124178i	$-0.0396295\pi$
$47$	125.209	+	216.869i	0.388588	+	0.673055i	−0.603672	+	0.797233i	$0.706296\pi$
$48$		0			0
$49$	−304.427	−	158.029i	−0.887542	−	0.460726i
$50$		0			0
$51$	−202.984	+	57.6721i	−0.557323	+	0.158347i
$52$		0			0
$53$	−294.863	−	170.239i	−0.764198	−	0.441210i	0.0666029	−	0.997780i	$-0.478784\pi$
$53$	−294.863	−	170.239i	−0.764198	−	0.441210i	−0.830801	+	0.556570i	$0.812117\pi$
$54$		0			0
$55$			668.727i			1.63947i
$56$		0			0
$57$	−282.114	−	71.0665i	−0.655559	−	0.165140i
$58$		0			0
$59$	451.101	−	781.329i	0.995395	−	1.72407i	0.414681	−	0.909967i	$-0.363893\pi$
$59$	451.101	−	781.329i	0.995395	−	1.72407i	0.580714	−	0.814108i	$-0.302774\pi$
$60$		0			0
$61$	499.851	−	288.589i	1.04917	−	0.605739i	0.126753	−	0.991934i	$-0.459544\pi$
$61$	499.851	−	288.589i	1.04917	−	0.605739i	0.922417	+	0.386196i	$0.126211\pi$
$62$		0			0
$63$	489.129	+	103.924i	0.978165	+	0.207828i
$64$		0			0
$65$	286.056	−	165.154i	0.545859	−	0.315152i
$66$		0			0
$67$	−184.464	+	319.500i	−0.336355	+	0.582585i	−0.983744	−	0.179575i	$-0.942528\pi$
$67$	−184.464	+	319.500i	−0.336355	+	0.582585i	0.647389	+	0.762160i	$0.275861\pi$
$68$		0			0
$69$	−543.070	−	136.803i	−0.947507	−	0.238684i
$70$		0			0
$71$		−	1184.28i		−	1.97955i	−0.142648	−	0.989773i	$-0.545562\pi$
$71$		−	1184.28i		−	1.97955i	0.142648	−	0.989773i	$-0.454438\pi$
$72$		0			0
$73$	−407.399	−	235.212i	−0.653184	−	0.377116i	0.136491	−	0.990641i	$-0.456417\pi$
$73$	−407.399	−	235.212i	−0.653184	−	0.377116i	−0.789675	+	0.613526i	$0.789751\pi$
$74$		0			0
$75$	−121.904	+	34.6355i	−0.187684	+	0.0533249i
$76$		0			0
$77$	700.286	−	732.373i	1.03643	−	1.08392i
$78$		0			0
$79$	−20.0802	−	34.7799i	−0.0285974	−	0.0495321i	0.851373	−	0.524562i	$-0.175771\pi$
$79$	−20.0802	−	34.7799i	−0.0285974	−	0.0495321i	−0.879970	+	0.475029i	$0.842437\pi$
$80$		0			0
$81$	−727.684	+	43.7908i	−0.998194	+	0.0600697i
$82$		0			0
$83$	−855.263			−1.13105			−0.565526	−	0.824731i	$-0.691327\pi$
$83$	−855.263			−1.13105			−0.565526	+	0.824731i	$0.691327\pi$
$84$		0			0
$85$	−496.361			−0.633387
$86$		0			0
$87$	138.387	−	142.610i	0.170536	−	0.175740i
$88$		0			0
$89$	−322.110	−	557.910i	−0.383636	−	0.664476i	0.607943	−	0.793980i	$-0.291995\pi$
$89$	−322.110	−	557.910i	−0.383636	−	0.664476i	−0.991579	+	0.129504i	$0.958661\pi$
$90$		0			0
$91$	−486.229	−	118.683i	−0.560117	−	0.136718i
$92$		0			0
$93$	423.061	+	1489.02i	0.471714	+	1.66026i
$94$		0			0
$95$	−592.642	−	342.162i	−0.640040	−	0.369527i
$96$		0			0
$97$		−	748.863i		−	0.783872i	−0.919993	−	0.391936i	$-0.871806\pi$
$97$		−	748.863i		−	0.783872i	0.919993	−	0.391936i	$-0.128194\pi$
$98$		0			0
$99$	−699.848	+	1300.95i	−0.710479	+	1.32071i
$100$		0			0
$101$	639.339	−	1107.37i	0.629867	−	1.09096i	−0.357711	−	0.933832i	$-0.616443\pi$
$101$	639.339	−	1107.37i	0.629867	−	1.09096i	0.987578	−	0.157130i	$-0.0502241\pi$
$102$		0			0
$103$	−48.6627	+	28.0954i	−0.0465523	+	0.0268770i	−0.523096	−	0.852274i	$-0.675223\pi$
$103$	−48.6627	+	28.0954i	−0.0465523	+	0.0268770i	0.476543	+	0.879151i	$0.341890\pi$
$104$		0			0
$105$	1039.92	+	549.590i	0.966535	+	0.510805i
$106$		0			0
$107$	−96.0313	+	55.4437i	−0.0867635	+	0.0500930i	−0.542754	−	0.839892i	$-0.682618\pi$
$107$	−96.0313	+	55.4437i	−0.0867635	+	0.0500930i	0.455991	+	0.889985i	$0.349285\pi$
$108$		0			0
$109$	486.042	−	841.850i	0.427105	−	0.739767i	−0.569510	−	0.821985i	$-0.692867\pi$
$109$	486.042	−	841.850i	0.427105	−	0.739767i	0.996614	+	0.0822175i	$0.0262002\pi$
$110$		0			0
$111$	361.207	−	1433.89i	0.308867	−	1.22611i
$112$		0			0
$113$			1949.29i			1.62277i	0.584510	+	0.811387i	$0.301287\pi$
$113$			1949.29i			1.62277i	−0.584510	+	0.811387i	$0.698713\pi$
$114$		0			0
$115$	−1140.84	−	658.664i	−0.925076	−	0.534093i
$116$		0			0
$117$	729.337	−	21.9253i	0.576301	−	0.0173248i
$118$		0			0
$119$	543.602	+	519.785i	0.418755	+	0.400409i
$120$		0			0
$121$	831.249	+	1439.76i	0.624529	+	1.08172i
$122$		0			0
$123$	−105.889	−	102.754i	−0.0776234	−	0.0753250i
$124$		0			0
$125$	1229.72			0.879913
$126$		0			0
$127$	2659.56			1.85825			0.929124	−	0.369768i	$-0.120563\pi$
$127$	2659.56			1.85825			0.929124	+	0.369768i	$0.120563\pi$
$128$		0			0
$129$	793.601	+	770.103i	0.541649	+	0.525611i
$130$		0			0
$131$	999.107	+	1730.50i	0.666354	+	1.15416i	0.978916	+	0.204262i	$0.0654793\pi$
$131$	999.107	+	1730.50i	0.666354	+	1.15416i	−0.312562	+	0.949897i	$0.601187\pi$
$132$		0			0
$133$	290.737	+	995.337i	0.189549	+	0.648922i
$134$		0			0
$135$	−1674.65	−	368.722i	−1.06764	−	0.235071i
$136$		0			0
$137$	−1334.65	−	770.561i	−0.832314	−	0.480537i	0.0223305	−	0.999751i	$-0.492891\pi$
$137$	−1334.65	−	770.561i	−0.832314	−	0.480537i	−0.854644	+	0.519214i	$0.826225\pi$
$138$		0			0
$139$			2926.53i			1.78579i	0.450261	+	0.892897i	$0.351331\pi$
$139$			2926.53i			1.78579i	−0.450261	+	0.892897i	$0.648669\pi$
$140$		0			0
$141$	−317.855	+	1261.79i	−0.189846	+	0.753633i
$142$		0			0
$143$	739.299	−	1280.50i	0.432331	−	0.748818i
$144$		0			0
$145$	404.803	−	233.713i	0.231842	−	0.133854i
$146$		0			0
$147$	−563.372	−	1690.90i	−0.316096	−	0.948727i
$148$		0			0
$149$	−2078.26	+	1199.89i	−1.14267	+	0.659721i	−0.947090	−	0.320967i	$-0.895992\pi$
$149$	−2078.26	+	1199.89i	−1.14267	+	0.659721i	−0.195580	+	0.980688i	$0.562659\pi$
$150$		0			0
$151$	89.6992	−	155.363i	0.0483418	−	0.0837305i	−0.840842	−	0.541281i	$-0.817940\pi$
$151$	89.6992	−	155.363i	0.0483418	−	0.0837305i	0.889184	+	0.457550i	$0.151273\pi$
$152$		0			0
$153$	−965.627	−	519.461i	−0.510237	−	0.274483i
$154$		0			0
$155$			3641.12i			1.88685i
$156$		0			0
$157$	−1609.18	−	929.062i	−0.818004	−	0.472275i	0.0317234	−	0.999497i	$-0.489900\pi$
$157$	−1609.18	−	929.062i	−0.818004	−	0.472275i	−0.849728	+	0.527222i	$0.823234\pi$
$158$		0			0
$159$	−483.522	−	1701.82i	−0.241169	−	0.848824i
$160$		0			0
$161$	559.670	+	1916.03i	0.273964	+	0.937915i
$162$		0			0
$163$	−1512.28	−	2619.35i	−0.726693	−	1.25867i	−0.958274	−	0.285853i	$-0.907723\pi$
$163$	−1512.28	−	2619.35i	−0.726693	−	1.25867i	0.231581	−	0.972816i	$-0.425610\pi$
$164$		0			0
$165$	−2419.86	+	2493.70i	−1.14173	+	1.17657i
$166$		0			0
$167$	1811.37			0.839329			0.419664	−	0.907679i	$-0.362148\pi$
$167$	1811.37			0.839329			0.419664	+	0.907679i	$0.362148\pi$
$168$		0			0
$169$	1466.67			0.667577
$170$		0			0
$171$	−794.848	−	1285.87i	−0.355459	−	0.575046i
$172$		0			0
$173$	554.616	+	960.623i	0.243738	+	0.422167i	0.961776	−	0.273837i	$-0.0882930\pi$
$173$	554.616	+	960.623i	0.243738	+	0.422167i	−0.718038	+	0.696004i	$0.754960\pi$
$174$		0			0
$175$	326.465	+	312.162i	0.141020	+	0.134841i
$176$		0			0
$177$	4509.49	−	1281.24i	1.91500	−	0.544090i
$178$		0			0
$179$	−3272.96	−	1889.64i	−1.36666	−	0.789043i	−0.376162	−	0.926554i	$-0.622756\pi$
$179$	−3272.96	−	1889.64i	−1.36666	−	0.789043i	−0.990500	+	0.137512i	$0.956090\pi$
$180$		0			0
$181$		−	1180.13i		−	0.484634i	−0.970197	−	0.242317i	$-0.922093\pi$
$181$		−	1180.13i		−	0.484634i	0.970197	−	0.242317i	$-0.0779074\pi$
$182$		0			0
$183$	2908.25	+	732.610i	1.17478	+	0.295935i
$184$		0			0
$185$	1739.09	−	3012.19i	0.691138	−	1.19709i
$186$		0			0
$187$	−1924.24	+	1110.96i	−0.752481	+	0.434445i
$188$		0			0
$189$	1447.92	+	2157.50i	0.557251	+	0.830344i
$190$		0			0
$191$	−2714.53	+	1567.24i	−1.02836	+	0.593724i	−0.916514	−	0.400002i	$-0.869010\pi$
$191$	−2714.53	+	1567.24i	−1.02836	+	0.593724i	−0.111846	+	0.993726i	$0.535676\pi$
$192$		0			0
$193$	−406.923	+	704.811i	−0.151767	+	0.262867i	−0.931877	−	0.362775i	$-0.881830\pi$
$193$	−406.923	+	704.811i	−0.151767	+	0.262867i	0.780110	+	0.625642i	$0.215163\pi$
$194$		0			0
$195$	1664.34	+	419.259i	0.611209	+	0.153968i
$196$		0			0
$197$			2665.95i			0.964168i	0.876125	+	0.482084i	$0.160120\pi$
$197$			2665.95i			0.964168i	−0.876125	+	0.482084i	$0.839880\pi$
$198$		0			0
$199$	1849.89	+	1068.03i	0.658970	+	0.380457i	0.791885	−	0.610671i	$-0.209100\pi$
$199$	1849.89	+	1068.03i	0.658970	+	0.380457i	−0.132914	+	0.991128i	$0.542433\pi$
$200$		0			0
$201$	−1844.02	+	523.924i	−0.647099	+	0.183854i
$202$		0			0
$203$	−688.072	−	167.950i	−0.237898	−	0.0580679i
$204$		0			0
$205$	−173.534	−	300.569i	−0.0591226	−	0.102403i
$206$		0			0
$207$	−1530.09	−	2475.30i	−0.513760	−	0.831138i
$208$		0			0
$209$	−3063.32			−1.01385
$210$		0			0
$211$	−1542.44			−0.503250			−0.251625	−	0.967825i	$-0.580965\pi$
$211$	−1542.44			−0.503250			−0.251625	+	0.967825i	$0.580965\pi$
$212$		0			0
$213$	4285.44	−	4416.20i	1.37856	−	1.42063i
$214$		0			0
$215$	1300.58	+	2252.67i	0.412552	+	0.714560i
$216$		0			0
$217$	3812.96	−	3987.66i	1.19281	−	1.24747i
$218$		0			0
$219$	−668.061	−	2351.33i	−0.206134	−	0.725516i
$220$		0			0
$221$	950.450	+	548.743i	0.289295	+	0.167025i
$222$		0			0
$223$		−	1581.82i		−	0.475008i	−0.971387	−	0.237504i	$-0.923671\pi$
$223$		−	1581.82i		−	0.475008i	0.971387	−	0.237504i	$-0.0763292\pi$
$224$		0			0
$225$	−579.917	−	311.967i	−0.171827	−	0.0924347i
$226$		0			0
$227$	1283.03	−	2222.27i	0.375143	−	0.649767i	−0.615206	−	0.788367i	$-0.710927\pi$
$227$	1283.03	−	2222.27i	0.375143	−	0.649767i	0.990348	+	0.138600i	$0.0442603\pi$
$228$		0			0
$229$	−1499.61	+	865.802i	−0.432739	+	0.249842i	−0.700513	−	0.713640i	$-0.747045\pi$
$229$	−1499.61	+	865.802i	−0.432739	+	0.249842i	0.267774	+	0.963482i	$0.413712\pi$
$230$		0			0
$231$	5261.56	−	196.976i	1.49864	−	0.0561043i
$232$		0			0
$233$	−2637.07	+	1522.51i	−0.741460	+	0.428082i	−0.822600	−	0.568621i	$-0.807477\pi$
$233$	−2637.07	+	1522.51i	−0.741460	+	0.428082i	0.0811399	+	0.996703i	$0.474144\pi$
$234$		0			0
$235$	−1530.37	+	2650.67i	−0.424809	+	0.735791i
$236$		0			0
$237$	50.9753	−	202.357i	0.0139713	−	0.0554621i
$238$		0			0
$239$			660.825i			0.178850i	0.995994	+	0.0894251i	$0.0285030\pi$
$239$			660.825i			0.178850i	−0.995994	+	0.0894251i	$0.971497\pi$
$240$		0			0
$241$	419.293	+	242.079i	0.112071	+	0.0647041i	0.554988	−	0.831859i	$-0.312723\pi$
$241$	419.293	+	242.079i	0.112071	+	0.0647041i	−0.442917	+	0.896563i	$0.646056\pi$
$242$		0			0
$243$	−2872.01	−	2469.91i	−0.758189	−	0.652035i
$244$		0			0
$245$	−187.694	−	4188.11i	−0.0489441	−	1.09212i
$246$		0			0
$247$	756.542	+	1310.37i	0.194889	+	0.337558i
$248$		0			0
$249$	−3189.30	−	3094.86i	−0.811701	−	0.787667i
$250$		0			0
$251$	2610.69			0.656514			0.328257	−	0.944588i	$-0.393539\pi$
$251$	2610.69			0.656514			0.328257	+	0.944588i	$0.393539\pi$
$252$		0			0
$253$	−5896.90			−1.46536
$254$		0			0
$255$	−1850.94	−	1796.14i	−0.454551	−	0.441092i
$256$		0			0
$257$	−729.496	−	1263.52i	−0.177061	−	0.306679i	0.763812	−	0.645439i	$-0.223326\pi$
$257$	−729.496	−	1263.52i	−0.177061	−	0.306679i	−0.940873	+	0.338760i	$0.889992\pi$
$258$		0			0
$259$	−5058.96	+	1477.72i	−1.21370	+	0.354520i
$260$		0			0
$261$	1032.10	−	31.0269i	0.244771	−	0.00735831i
$262$		0			0
$263$	−4410.38	−	2546.33i	−1.03405	−	0.597010i	−0.115909	−	0.993260i	$-0.536978\pi$
$263$	−4410.38	−	2546.33i	−1.03405	−	0.597010i	−0.918143	+	0.396249i	$0.870312\pi$
$264$		0			0
$265$		−	4161.49i		−	0.964672i
$266$		0			0
$267$	817.704	−	3246.05i	0.187426	−	0.744027i
$268$		0			0
$269$	487.624	−	844.590i	0.110524	−	0.191433i	−0.805458	−	0.592653i	$-0.798080\pi$
$269$	487.624	−	844.590i	0.110524	−	0.191433i	0.915982	+	0.401220i	$0.131414\pi$
$270$		0			0
$271$	3190.71	−	1842.16i	0.715209	−	0.412926i	−0.0977775	−	0.995208i	$-0.531173\pi$
$271$	3190.71	−	1842.16i	0.715209	−	0.412926i	0.812987	+	0.582282i	$0.197840\pi$
$272$		0			0
$273$	−1383.70	−	2202.04i	−0.306759	−	0.488182i
$274$		0			0
$275$	−1155.62	+	667.197i	−0.253405	+	0.146304i
$276$		0			0
$277$	−1650.60	+	2858.93i	−0.358033	+	0.620132i	−0.987632	−	0.156789i	$-0.949886\pi$
$277$	−1650.60	+	2858.93i	−0.358033	+	0.620132i	0.629599	+	0.776920i	$0.283219\pi$
$278$		0			0
$279$	−3810.57	+	7083.49i	−0.817681	+	1.51999i
$280$		0			0
$281$			444.621i			0.0943910i	0.998886	+	0.0471955i	$0.0150284\pi$
$281$			444.621i			0.0943910i	−0.998886	+	0.0471955i	$0.984972\pi$
$282$		0			0
$283$	1569.80	+	906.327i	0.329736	+	0.190373i	0.655724	−	0.755001i	$-0.272364\pi$
$283$	1569.80	+	906.327i	0.329736	+	0.190373i	−0.325988	+	0.945374i	$0.605697\pi$
$284$		0			0
$285$	−971.828	−	3420.47i	−0.201986	−	0.710917i
$286$		0			0
$287$	−124.704	+	510.899i	−0.0256483	+	0.105078i
$288$		0			0
$289$	1631.89	+	2826.52i	0.332158	+	0.575315i
$290$		0			0
$291$	2709.85	−	2792.53i	0.545890	−	0.562547i
$292$		0			0
$293$	−693.371			−0.138250			−0.0691248	−	0.997608i	$-0.522021\pi$
$293$	−693.371			−0.138250			−0.0691248	+	0.997608i	$0.522021\pi$
$294$		0			0
$295$	11027.1			2.17636
$296$		0			0
$297$	−7317.39	+	2318.80i	−1.42962	+	0.453032i
$298$		0			0
$299$	1456.35	+	2522.47i	0.281681	+	0.487886i
$300$		0			0
$301$	934.615	−	3829.02i	0.178971	−	0.733225i
$302$		0			0
$303$	6391.25	−	1815.89i	1.21177	−	0.344290i
$304$		0			0
$305$	6109.42	+	3527.28i	1.14697	+	0.662201i
$306$		0			0
$307$			5574.91i			1.03641i	0.855257	+	0.518203i	$0.173399\pi$
$307$			5574.91i			1.03641i	−0.855257	+	0.518203i	$0.826601\pi$
$308$		0			0
$309$	−283.131	−	71.3228i	−0.0521255	−	0.0131308i
$310$		0			0
$311$	484.794	−	839.688i	0.0883927	−	0.153101i	−0.818439	−	0.574593i	$-0.805160\pi$
$311$	484.794	−	839.688i	0.0883927	−	0.153101i	0.906832	+	0.421492i	$0.138494\pi$
$312$		0			0
$313$	−1698.97	+	980.901i	−0.306810	+	0.177137i	−0.645498	−	0.763762i	$-0.723350\pi$
$313$	−1698.97	+	980.901i	−0.306810	+	0.177137i	0.338688	+	0.940899i	$0.390017\pi$
$314$		0			0
$315$	1889.15	+	5812.52i	0.337910	+	1.03968i
$316$		0			0
$317$	−8969.01	+	5178.26i	−1.58912	+	0.917476i	−0.595662	+	0.803235i	$0.703110\pi$
$317$	−8969.01	+	5178.26i	−1.58912	+	0.917476i	−0.993453	+	0.114241i	$0.963556\pi$
$318$		0			0
$319$	1046.20	−	1812.06i	0.183623	−	0.318044i
$320$		0			0
$321$	−558.733	−	140.749i	−0.0971509	−	0.0244730i
$322$		0			0
$323$		−	2273.74i		−	0.391685i
$324$		0			0
$325$	570.802	+	329.553i	0.0974228	+	0.0562471i
$326$		0			0
$327$	4858.79	−	1380.48i	0.821688	−	0.233459i
$328$		0			0
$329$	4451.79	−	1300.36i	0.746003	−	0.217907i
$330$		0			0
$331$	1990.22	+	3447.16i	0.330491	+	0.572427i	0.982608	−	0.185691i	$-0.0594524\pi$
$331$	1990.22	+	3447.16i	0.330491	+	0.572427i	−0.652117	+	0.758118i	$0.726119\pi$
$332$		0			0
$333$	6535.63	−	4039.94i	1.07553	−	0.664827i
$334$		0			0
$335$	−4509.21			−0.735415
$336$		0			0
$337$	−8369.88			−1.35293			−0.676463	−	0.736476i	$-0.736488\pi$
$337$	−8369.88			−1.35293			−0.676463	+	0.736476i	$0.736488\pi$
$338$		0			0
$339$	−7053.71	+	7268.94i	−1.13010	+	1.16459i
$340$		0			0
$341$	8149.58	+	14115.5i	1.29421	+	2.24163i
$342$		0			0
$343$	−4180.20	+	4783.26i	−0.658045	+	0.752979i
$344$		0			0
$345$	−1870.77	−	6584.43i	−0.291939	−	1.02752i
$346$		0			0
$347$	4876.37	+	2815.37i	0.754401	+	0.435554i	0.827282	−	0.561787i	$-0.189886\pi$
$347$	4876.37	+	2815.37i	0.754401	+	0.435554i	−0.0728808	+	0.997341i	$0.523219\pi$
$348$		0			0
$349$		−	10150.9i		−	1.55691i	−0.627698	−	0.778457i	$-0.716003\pi$
$349$		−	10150.9i		−	1.55691i	0.627698	−	0.778457i	$-0.283997\pi$
$350$		0			0
$351$	2799.06	+	2557.43i	0.425648	+	0.388904i
$352$		0			0
$353$	−1284.89	+	2225.50i	−0.193733	+	0.335556i	−0.946485	−	0.322749i	$-0.895393\pi$
$353$	−1284.89	+	2225.50i	−0.193733	+	0.335556i	0.752751	+	0.658305i	$0.228726\pi$
$354$		0			0
$355$	12535.5	−	7237.40i	1.87413	−	1.08203i
$356$		0			0
$357$	146.205	+	3905.38i	0.0216751	+	0.578976i
$358$		0			0
$359$	1588.86	−	917.327i	0.233584	−	0.134860i	−0.378640	−	0.925544i	$-0.623608\pi$
$359$	1588.86	−	917.327i	0.233584	−	0.134860i	0.612224	+	0.790684i	$0.290275\pi$
$360$		0			0
$361$	−1862.12	+	3225.28i	−0.271485	+	0.470226i
$362$		0			0
$363$	−2110.20	+	8376.89i	−0.305115	+	1.21122i
$364$		0			0
$365$		−	5749.74i		−	0.824535i
$366$		0			0
$367$	−6793.91	−	3922.47i	−0.966320	−	0.557905i	−0.0682076	−	0.997671i	$-0.521728\pi$
$367$	−6793.91	−	3922.47i	−0.966320	−	0.557905i	−0.898112	+	0.439766i	$0.855061\pi$
$368$		0			0
$369$	−23.0378	−	766.342i	−0.00325013	−	0.108114i
$370$		0			0
$371$	−4357.88	+	4557.55i	−0.609837	+	0.637780i
$372$		0			0
$373$	2429.11	+	4207.33i	0.337197	+	0.584042i	0.983904	−	0.178696i	$-0.0571880\pi$
$373$	2429.11	+	4207.33i	0.337197	+	0.584042i	−0.646708	+	0.762738i	$0.723855\pi$
$374$		0			0
$375$	4585.64	+	4449.86i	0.631471	+	0.612773i
$376$		0			0
$377$	−1033.51			−0.141189
$378$		0			0
$379$	−10589.6			−1.43523			−0.717616	−	0.696439i	$-0.754767\pi$
$379$	−10589.6			−1.43523			−0.717616	+	0.696439i	$0.754767\pi$
$380$		0			0
$381$	9917.57	+	9623.90i	1.33358	+	1.29409i
$382$		0			0
$383$	1948.54	+	3374.96i	0.259962	+	0.450268i	0.966231	−	0.257676i	$-0.0829566\pi$
$383$	1948.54	+	3374.96i	0.259962	+	0.450268i	−0.706269	+	0.707943i	$0.749623\pi$
$384$		0			0
$385$	12031.8	+	2936.80i	1.59272	+	0.388762i
$386$		0			0
$387$	172.660	+	5743.47i	0.0226791	+	0.754411i
$388$		0			0
$389$	−7782.20	−	4493.06i	−1.01433	−	0.585622i	−0.101872	−	0.994798i	$-0.532483\pi$
$389$	−7782.20	−	4493.06i	−1.01433	−	0.585622i	−0.912456	+	0.409175i	$0.865816\pi$
$390$		0			0
$391$		−	4376.96i		−	0.566118i
$392$		0			0
$393$	−2536.32	+	10068.5i	−0.325548	+	1.29233i
$394$		0			0
$395$	245.429	−	425.096i	0.0312630	−	0.0541491i
$396$		0			0
$397$	−2777.60	+	1603.65i	−0.351143	+	0.202733i	−0.665189	−	0.746675i	$-0.731649\pi$
$397$	−2777.60	+	1603.65i	−0.351143	+	0.202733i	0.314045	+	0.949408i	$0.398316\pi$
$398$		0			0
$399$	−2517.57	+	4763.70i	−0.315880	+	0.597703i
$400$		0			0
$401$	7566.81	−	4368.70i	0.942315	−	0.544046i	0.0516300	−	0.998666i	$-0.483558\pi$
$401$	7566.81	−	4368.70i	0.942315	−	0.544046i	0.890685	+	0.454620i	$0.150225\pi$
$402$		0			0
$403$	4025.38	−	6972.16i	0.497564	−	0.861806i
$404$		0			0
$405$	−4910.57	−	7434.90i	−0.602490	−	0.912205i
$406$		0			0
$407$		−	15569.8i		−	1.89623i
$408$		0			0
$409$	10056.2	+	5805.94i	1.21576	+	0.701920i	0.964009	−	0.265871i	$-0.0856596\pi$
$409$	10056.2	+	5805.94i	1.21576	+	0.701920i	0.251753	+	0.967792i	$0.418993\pi$
$410$		0			0
$411$	−2188.59	−	7703.03i	−0.262665	−	0.924483i
$412$		0			0
$413$	−12076.6	−	11547.5i	−1.43887	−	1.37583i
$414$		0			0
$415$	−5226.72	−	9052.94i	−0.618240	−	1.07082i
$416$		0			0
$417$	−10590.0	+	10913.1i	−1.24363	+	1.28158i
$418$		0			0
$419$	10111.6			1.17896			0.589482	−	0.807782i	$-0.299332\pi$
$419$	10111.6			1.17896			0.589482	+	0.807782i	$0.299332\pi$
$420$		0			0
$421$	−8005.12			−0.926711			−0.463356	−	0.886172i	$-0.653355\pi$
$421$	−8005.12			−0.926711			−0.463356	+	0.886172i	$0.653355\pi$
$422$		0			0
$423$	−5751.23	+	3555.07i	−0.661074	+	0.408637i
$424$		0			0
$425$	−495.225	−	857.755i	−0.0565222	−	0.0978994i
$426$		0			0
$427$	−2997.14	−	10260.7i	−0.339677	−	1.16288i
$428$		0			0
$429$	7390.51	−	2099.80i	0.831742	−	0.236315i
$430$		0			0
$431$	13312.3	+	7685.84i	1.48777	+	0.858965i	0.999903	−	0.0139531i	$-0.00444154\pi$
$431$	13312.3	+	7685.84i	1.48777	+	0.858965i	0.487868	+	0.872918i	$0.337775\pi$
$432$		0			0
$433$			15508.4i			1.72121i	0.509272	+	0.860606i	$0.329915\pi$
$433$			15508.4i			1.72121i	−0.509272	+	0.860606i	$0.670085\pi$
$434$		0			0
$435$	2355.24	+	593.301i	0.259598	+	0.0653946i
$436$		0			0
$437$	3017.22	−	5225.97i	0.330282	−	0.572065i
$438$		0			0
$439$	−13707.7	+	7914.13i	−1.49028	+	0.860412i	−0.999938	−	0.0111193i	$-0.996461\pi$
$439$	−13707.7	+	7914.13i	−1.49028	+	0.860412i	−0.490339	+	0.871532i	$0.663127\pi$
$440$		0			0
$441$	4017.87	−	8344.03i	0.433849	−	0.900986i
$442$		0			0
$443$	−5655.01	+	3264.92i	−0.606495	+	0.350160i	−0.771593	−	0.636117i	$-0.780540\pi$
$443$	−5655.01	+	3264.92i	−0.606495	+	0.350160i	0.165097	+	0.986277i	$0.447206\pi$
$444$		0			0
$445$	3936.98	−	6819.05i	0.419395	−	0.726413i
$446$		0			0
$447$	−12091.8	−	3046.02i	−1.27947	−	0.322308i
$448$		0			0
$449$		−	12934.7i		−	1.35952i	−0.733434	−	0.679761i	$-0.762083\pi$
$449$		−	12934.7i		−	1.35952i	0.733434	−	0.679761i	$-0.237917\pi$
$450$		0			0
$451$	−1345.47	−	776.809i	−0.140479	−	0.0811053i
$452$		0			0
$453$	896.691	−	254.769i	0.0930027	−	0.0264240i
$454$		0			0
$455$	−1715.21	−	5872.02i	−0.176726	−	0.605021i
$456$		0			0
$457$	4806.68	+	8325.42i	0.492007	+	0.852181i	0.999958	−	0.00920512i	$-0.00293012\pi$
$457$	4806.68	+	8325.42i	0.492007	+	0.852181i	−0.507951	+	0.861386i	$0.669597\pi$
$458$		0			0
$459$	−1721.13	−	5431.31i	−0.175022	−	0.552314i
$460$		0			0
$461$	11205.4			1.13207			0.566037	−	0.824380i	$-0.308476\pi$
$461$	11205.4			1.13207			0.566037	+	0.824380i	$0.308476\pi$
$462$		0			0
$463$	4300.85			0.431701			0.215850	−	0.976426i	$-0.430748\pi$
$463$	4300.85			0.431701			0.215850	+	0.976426i	$0.430748\pi$
$464$		0			0
$465$	−13175.8	+	13577.9i	−1.31401	+	1.35410i
$466$		0			0
$467$	−7282.65	−	12613.9i	−0.721630	−	1.24990i	−0.960346	−	0.278810i	$-0.910060\pi$
$467$	−7282.65	−	12613.9i	−0.721630	−	1.24990i	0.238717	−	0.971089i	$-0.423273\pi$
$468$		0			0
$469$	4938.37	+	4722.01i	0.486210	+	0.464908i
$470$		0			0
$471$	−2638.77	−	9287.50i	−0.258149	−	0.908589i
$472$		0			0
$473$	10083.9	+	5821.92i	0.980246	+	0.565945i
$474$		0			0
$475$		−	1365.52i		−	0.131904i
$476$		0			0
$477$	4355.15	−	8095.81i	0.418048	−	0.777111i
$478$		0			0
$479$	−937.775	+	1624.27i	−0.0894531	+	0.154937i	−0.907280	−	0.420527i	$-0.861845\pi$
$479$	−937.775	+	1624.27i	−0.0894531	+	0.154937i	0.817827	+	0.575464i	$0.195179\pi$
$480$		0			0
$481$	−6660.15	+	3845.24i	−0.631345	+	0.364507i
$482$		0			0
$483$	−4846.34	+	9170.15i	−0.456555	+	0.863885i
$484$		0			0
$485$	7926.70	−	4576.48i	0.742130	−	0.428469i
$486$		0			0
$487$	−615.525	+	1066.12i	−0.0572733	+	0.0992002i	−0.893240	−	0.449579i	$-0.851574\pi$
$487$	−615.525	+	1066.12i	−0.0572733	+	0.0992002i	0.835967	+	0.548779i	$0.184907\pi$
$488$		0			0
$489$	3839.06	−	15240.0i	0.355027	−	1.40936i
$490$		0			0
$491$		−	6644.33i		−	0.610702i	−0.952240	−	0.305351i	$-0.901226\pi$
$491$		−	6644.33i		−	0.610702i	0.952240	−	0.305351i	$-0.0987737\pi$
$492$		0			0
$493$	1345.00	+	776.536i	0.122872	+	0.0709400i
$494$		0			0
$495$	−18047.5	+	542.543i	−1.63873	+	0.0492636i
$496$		0			0
$497$	−21307.6	−	5200.91i	−1.92309	−	0.469402i
$498$		0			0
$499$	−7276.93	−	12604.0i	−0.652826	−	1.13073i	−0.982434	−	0.186609i	$-0.940250\pi$
$499$	−7276.93	−	12604.0i	−0.652826	−	1.13073i	0.329609	−	0.944118i	$-0.393083\pi$
$500$		0			0
$501$	6754.64	+	6554.64i	0.602346	+	0.584510i
$502$		0			0
$503$	6799.96			0.602774			0.301387	−	0.953502i	$-0.402550\pi$
$503$	6799.96			0.602774			0.301387	+	0.953502i	$0.402550\pi$
$504$		0			0
$505$	15628.6			1.37716
$506$		0			0
$507$	5469.24	+	5307.30i	0.479088	+	0.464902i
$508$		0			0
$509$	−1924.80	−	3333.85i	−0.167614	−	0.290315i	0.769967	−	0.638084i	$-0.220273\pi$
$509$	−1924.80	−	3333.85i	−0.167614	−	0.290315i	−0.937580	+	0.347769i	$0.886939\pi$
$510$		0			0
$511$	−6021.09	+	6296.97i	−0.521247	+	0.545130i
$512$		0			0
$513$	1689.05	−	7671.29i	0.145367	−	0.660226i
$514$		0			0
$515$	−594.779	−	343.396i	−0.0508915	−	0.0293822i
$516$		0			0
$517$			13701.1i			1.16552i
$518$		0			0
$519$	−1407.94	+	5589.13i	−0.119079	+	0.472708i
$520$		0			0
$521$	5682.32	−	9842.06i	0.477825	−	0.827617i	−0.521852	−	0.853036i	$-0.674759\pi$
$521$	5682.32	−	9842.06i	0.477825	−	0.827617i	0.999677	+	0.0254190i	$0.00809199\pi$
$522$		0			0
$523$	10037.3	−	5795.05i	0.839200	−	0.484512i	−0.0177924	−	0.999842i	$-0.505664\pi$
$523$	10037.3	−	5795.05i	0.839200	−	0.484512i	0.856992	+	0.515330i	$0.172330\pi$
$524$		0			0
$525$	87.8050	+	2345.41i	0.00729929	+	0.194976i
$526$		0			0
$527$	−10477.2	+	6049.01i	−0.866022	+	0.499998i
$528$		0			0
$529$	−275.341	+	476.904i	−0.0226301	+	0.0391965i
$530$		0			0
$531$	21452.3	+	11540.3i	1.75321	+	0.943140i
$532$		0			0
$533$			767.389i			0.0623626i
$534$		0			0
$535$	−1173.74	−	677.660i	−0.0948509	−	0.0547622i
$536$		0			0
$537$	−5367.07	−	18890.1i	−0.431297	−	1.51800i
$538$		0			0
$539$	−10101.5	−	15815.9i	−0.807238	−	1.26389i
$540$		0			0
$541$	−2756.37	−	4774.17i	−0.219049	−	0.379404i	0.735468	−	0.677559i	$-0.236962\pi$
$541$	−2756.37	−	4774.17i	−0.219049	−	0.379404i	−0.954518	+	0.298155i	$0.903629\pi$
$542$		0			0
$543$	4270.45	−	4400.75i	0.337500	−	0.347798i
$544$		0			0
$545$	11881.3			0.933832
$546$		0			0
$547$	19797.4			1.54749			0.773746	−	0.633497i	$-0.218381\pi$
$547$	19797.4			1.54749			0.773746	+	0.633497i	$0.218381\pi$
$548$		0			0
$549$	8193.92	+	13255.8i	0.636991	+	1.03050i
$550$		0			0
$551$	1070.60	+	1854.33i	0.0827749	+	0.143370i
$552$		0			0
$553$	−713.945	+	208.543i	−0.0549006	+	0.0160364i
$554$		0			0
$555$	17385.1	−	4939.46i	1.32965	−	0.377781i
$556$		0			0
$557$	4626.61	+	2671.18i	0.351950	+	0.203198i	0.665544	−	0.746359i	$-0.268200\pi$
$557$	4626.61	+	2671.18i	0.351950	+	0.203198i	−0.313594	+	0.949557i	$0.601533\pi$
$558$		0			0
$559$		−	5751.32i		−	0.435161i
$560$		0			0
$561$	−11195.7	−	2820.27i	−0.842568	−	0.212249i
$562$		0			0
$563$	5595.50	−	9691.69i	0.418867	−	0.725499i	−0.576959	−	0.816773i	$-0.695761\pi$
$563$	5595.50	−	9691.69i	0.418867	−	0.725499i	0.995826	+	0.0912740i	$0.0290939\pi$
$564$		0			0
$565$	−20633.2	+	11912.6i	−1.53636	+	0.887018i
$566$		0			0
$567$	−2407.83	+	13284.8i	−0.178341	+	0.983969i
$568$		0			0
$569$	−13960.7	+	8060.21i	−1.02858	+	0.593852i	−0.916578	−	0.399856i	$-0.869060\pi$
$569$	−13960.7	+	8060.21i	−1.02858	+	0.593852i	−0.112004	+	0.993708i	$0.535727\pi$
$570$		0			0
$571$	4187.14	−	7252.34i	0.306876	−	0.531525i	−0.670801	−	0.741637i	$-0.734050\pi$
$571$	4187.14	−	7252.34i	0.306876	−	0.531525i	0.977677	+	0.210112i	$0.0673828\pi$
$572$		0			0
$573$	−15793.8	−	3978.57i	−1.15148	−	0.290065i
$574$		0			0
$575$		−	2628.63i		−	0.190646i
$576$		0			0
$577$	−10723.1	−	6190.98i	−0.773671	−	0.446679i	0.0605115	−	0.998167i	$-0.480727\pi$
$577$	−10723.1	−	6190.98i	−0.773671	−	0.446679i	−0.834183	+	0.551488i	$0.814060\pi$
$578$		0			0
$579$	−4067.86	+	1155.76i	−0.291977	+	0.0829567i
$580$		0			0
$581$	−3756.00	+	15387.9i	−0.268202	+	1.09879i
$582$		0			0
$583$	−9314.26	−	16132.8i	−0.661677	−	1.14606i
$584$		0			0
$585$	4689.23	+	7586.02i	0.331412	+	0.536143i
$586$		0			0
$587$	−12886.6			−0.906112			−0.453056	−	0.891482i	$-0.649666\pi$
$587$	−12886.6			−0.906112			−0.453056	+	0.891482i	$0.649666\pi$
$588$		0			0
$589$	−16679.3			−1.16682
$590$		0			0
$591$	−9647.03	+	9941.40i	−0.671448	+	0.691937i
$592$		0			0
$593$	1510.79	+	2616.76i	0.104622	+	0.181210i	0.913584	−	0.406651i	$-0.133304\pi$
$593$	1510.79	+	2616.76i	0.104622	+	0.181210i	−0.808962	+	0.587861i	$0.799970\pi$
$594$		0			0
$595$	−2179.84	+	8930.54i	−0.150193	+	0.615322i
$596$		0			0
$597$	3033.49	+	10676.8i	0.207960	+	0.731944i
$598$		0			0
$599$	7770.56	+	4486.34i	0.530044	+	0.306021i	0.741035	−	0.671467i	$-0.234335\pi$
$599$	7770.56	+	4486.34i	0.530044	+	0.306021i	−0.210990	+	0.977488i	$0.567669\pi$
$600$		0			0
$601$		−	14238.3i		−	0.966374i	−0.875517	−	0.483187i	$-0.839479\pi$
$601$		−	14238.3i		−	0.966374i	0.875517	−	0.483187i	$-0.160521\pi$
$602$		0			0
$603$	−8772.27	−	4719.05i	−0.592429	−	0.318698i
$604$		0			0
$605$	−10159.9	+	17597.5i	−0.682743	+	1.18255i
$606$		0			0
$607$	−3552.68	+	2051.14i	−0.237560	+	0.137155i	−0.614055	−	0.789264i	$-0.710463\pi$
$607$	−3552.68	+	2051.14i	−0.237560	+	0.137155i	0.376495	+	0.926419i	$0.377129\pi$
$608$		0			0
$609$	−1958.10	−	3116.16i	−0.130289	−	0.207345i
$610$		0			0
$611$	5860.81	−	3383.74i	0.388057	−	0.224045i
$612$		0			0
$613$	3234.01	−	5601.47i	0.213084	−	0.369072i	−0.739594	−	0.673053i	$-0.764983\pi$
$613$	3234.01	−	5601.47i	0.213084	−	0.369072i	0.952678	+	0.303981i	$0.0983159\pi$
$614$		0			0
$615$	440.531	−	1748.78i	0.0288844	−	0.114663i
$616$		0			0
$617$			6608.54i			0.431199i	0.976482	+	0.215600i	$0.0691706\pi$
$617$			6608.54i			0.431199i	−0.976482	+	0.215600i	$0.930829\pi$
$618$		0			0
$619$	18325.1	+	10580.0i	1.18990	+	0.686990i	0.958284	−	0.285817i	$-0.0922651\pi$
$619$	18325.1	+	10580.0i	1.18990	+	0.686990i	0.231617	+	0.972807i	$0.425598\pi$
$620$		0			0
$621$	3251.43	−	14767.3i	0.210105	−	0.954251i
$622$		0			0
$623$	−11452.5	+	3345.27i	−0.736495	+	0.215129i
$624$		0			0
$625$	9039.40	+	15656.7i	0.578522	+	1.00203i
$626$		0			0
$627$	−11423.2	−	11085.0i	−0.727589	−	0.706045i
$628$		0			0
$629$	11556.6			0.732580
$630$		0			0
$631$	−7741.19			−0.488387			−0.244193	−	0.969727i	$-0.578523\pi$
$631$	−7741.19			−0.488387			−0.244193	+	0.969727i	$0.578523\pi$
$632$		0			0
$633$	−5751.79	−	5581.48i	−0.361158	−	0.350464i
$634$		0			0
$635$	16253.2	+	28151.4i	1.01573	+	1.75930i
$636$		0			0
$637$	−4270.68	+	8227.04i	−0.265637	+	0.511723i
$638$		0			0
$639$	31961.0	−	960.813i	1.97865	−	0.0594823i
$640$		0			0
$641$	12037.2	+	6949.67i	0.741716	+	0.428230i	0.822693	−	0.568486i	$-0.192471\pi$
$641$	12037.2	+	6949.67i	0.741716	+	0.428230i	−0.0809769	+	0.996716i	$0.525804\pi$
$642$		0			0
$643$			3191.05i			0.195712i	0.995201	+	0.0978560i	$0.0311985\pi$
$643$			3191.05i			0.195712i	−0.995201	+	0.0978560i	$0.968802\pi$
$644$		0			0
$645$	−3301.63	+	13106.5i	−0.201553	+	0.800107i
$646$		0			0
$647$	−14331.5	+	24822.9i	−0.870835	+	1.50833i	−0.00970044	+	0.999953i	$0.503088\pi$
$647$	−14331.5	+	24822.9i	−0.870835	+	1.50833i	−0.861134	+	0.508377i	$0.830246\pi$
$648$		0			0
$649$	42748.8	−	24681.0i	2.58557	−	1.49278i
$650$		0			0
$651$	28648.4	−	1072.51i	1.72476	−	0.0645698i
$652$		0			0
$653$	12927.5	−	7463.67i	0.774717	−	0.447283i	−0.0598375	−	0.998208i	$-0.519058\pi$
$653$	12927.5	−	7463.67i	0.774717	−	0.447283i	0.834555	+	0.550925i	$0.185725\pi$
$654$		0			0
$655$	−12211.6	+	21151.0i	−0.728466	+	1.26174i
$656$		0			0
$657$	6017.32	−	11185.6i	0.357318	−	0.664220i
$658$		0			0
$659$			20808.5i			1.23002i	0.788518	+	0.615012i	$0.210849\pi$
$659$			20808.5i			1.23002i	−0.788518	+	0.615012i	$0.789151\pi$
$660$		0			0
$661$	−12905.7	−	7451.11i	−0.759415	−	0.438448i	0.0696706	−	0.997570i	$-0.477805\pi$
$661$	−12905.7	−	7451.11i	−0.759415	−	0.438448i	−0.829086	+	0.559122i	$0.811139\pi$
$662$		0			0
$663$	1558.57	+	5485.59i	0.0912969	+	0.321331i
$664$		0			0
$665$	−8758.86	+	9160.19i	−0.510758	+	0.534161i
$666$		0			0
$667$	2060.90	+	3569.59i	0.119638	+	0.207219i
$668$		0			0
$669$	5724.00	−	5898.66i	0.330796	−	0.340890i
$670$		0			0
$671$	31579.1			1.81684
$672$		0			0
$673$	20032.6			1.14740			0.573699	−	0.819066i	$-0.305508\pi$
$673$	20032.6			1.14740			0.573699	+	0.819066i	$0.305508\pi$
$674$		0			0
$675$	−1033.64	−	3261.83i	−0.0589404	−	0.185997i
$676$		0			0
$677$	−2839.67	−	4918.45i	−0.161207	−	0.279219i	0.774095	−	0.633070i	$-0.218205\pi$
$677$	−2839.67	−	4918.45i	−0.161207	−	0.279219i	−0.935302	+	0.353851i	$0.884872\pi$
$678$		0			0
$679$	−13473.6	−	3288.74i	−0.761515	−	0.185876i
$680$		0			0
$681$	12826.0	−	3644.12i	0.721721	−	0.205056i
$682$		0			0
$683$	14522.3	+	8384.43i	0.813585	+	0.469724i	0.848199	−	0.529677i	$-0.177687\pi$
$683$	14522.3	+	8384.43i	0.813585	+	0.469724i	−0.0346143	+	0.999401i	$0.511020\pi$
$684$		0			0
$685$		−	18836.3i		−	1.05066i
$686$		0			0
$687$	−8725.10	−	2197.92i	−0.484546	−	0.122061i
$688$		0			0
$689$	−4600.66	+	7968.57i	−0.254385	+	0.440607i
$690$		0			0
$691$	8348.57	−	4820.05i	0.459616	−	0.265359i	−0.252267	−	0.967658i	$-0.581176\pi$
$691$	8348.57	−	4820.05i	0.459616	−	0.265359i	0.711883	+	0.702298i	$0.247843\pi$
$692$		0			0
$693$	20333.3	+	18305.0i	1.11457	+	1.00339i
$694$		0			0
$695$	−30977.3	+	17884.8i	−1.69070	+	0.976126i
$696$		0			0
$697$	576.584	−	998.674i	0.0313338	−	0.0542718i
$698$		0			0
$699$	−15343.1	−	3865.04i	−0.830227	−	0.209140i
$700$		0			0
$701$			4109.58i			0.221422i	0.993853	+	0.110711i	$0.0353127\pi$
$701$			4109.58i			0.221422i	−0.993853	+	0.110711i	$0.964687\pi$
$702$		0			0
$703$	13798.3	+	7966.46i	0.740275	+	0.427398i
$704$		0			0
$705$	−15298.5	+	4346.64i	−0.817272	+	0.232204i
$706$		0			0
$707$	−17116.1	−	16366.2i	−0.910489	−	0.870598i
$708$		0			0
$709$	−4208.59	−	7289.49i	−0.222929	−	0.386125i	0.732767	−	0.680480i	$-0.238229\pi$
$709$	−4208.59	−	7289.49i	−0.222929	−	0.386125i	−0.955696	+	0.294355i	$0.904895\pi$
$710$		0			0
$711$	922.341	−	570.137i	0.0486505	−	0.0300728i
$712$		0			0
$713$	−32107.8			−1.68646
$714$		0			0
$715$	18072.1			0.945258
$716$		0			0
$717$	−2391.27	+	2464.23i	−0.124552	+	0.128352i
$718$		0			0
$719$	5394.98	+	9344.38i	0.279831	+	0.484682i	0.971343	−	0.237684i	$-0.0763881\pi$
$719$	5394.98	+	9344.38i	0.279831	+	0.484682i	−0.691511	+	0.722366i	$0.743055\pi$
$720$		0			0
$721$	291.785	+	998.927i	0.0150716	+	0.0515978i
$722$		0			0
$723$	687.566	+	2419.98i	0.0353677	+	0.124481i
$724$		0			0
$725$	807.753	+	466.357i	0.0413782	+	0.0238897i
$726$		0			0
$727$			7138.00i			0.364146i	0.983285	+	0.182073i	$0.0582807\pi$
$727$			7138.00i			0.364146i	−0.983285	+	0.182073i	$0.941719\pi$
$728$		0			0
$729$	−1772.19	−	19603.1i	−0.0900368	−	0.995938i
$730$		0			0
$731$	−4321.30	+	7484.72i	−0.218645	+	0.378704i
$732$		0			0
$733$	−5673.59	+	3275.65i	−0.285892	+	0.165060i	−0.636088	−	0.771617i	$-0.719448\pi$
$733$	−5673.59	+	3275.65i	−0.285892	+	0.165060i	0.350196	+	0.936677i	$0.386115\pi$
$734$		0			0
$735$	14455.2	−	16296.8i	0.725427	−	0.817844i
$736$		0			0
$737$	−17480.8	+	10092.5i	−0.873694	+	0.504428i
$738$		0			0
$739$	−11962.1	+	20718.9i	−0.595443	+	1.03134i	0.398041	+	0.917368i	$0.369690\pi$
$739$	−11962.1	+	20718.9i	−0.595443	+	1.03134i	−0.993484	+	0.113970i	$0.963643\pi$
$740$		0			0
$741$	−1920.55	+	7624.03i	−0.0952134	+	0.377970i
$742$		0			0
$743$		−	33097.2i		−	1.63421i	−0.576489	−	0.817105i	$-0.695578\pi$
$743$		−	33097.2i		−	1.63421i	0.576489	−	0.817105i	$-0.304422\pi$
$744$		0			0
$745$	−25401.5	−	14665.6i	−1.24918	−	0.721215i
$746$		0			0
$747$	−693.881	−	23081.7i	−0.0339863	−	1.13054i
$748$		0			0
$749$	575.811	+	1971.29i	0.0280904	+	0.0961673i
$750$		0			0
$751$	8613.02	+	14918.2i	0.418500	+	0.724863i	0.995789	−	0.0916766i	$-0.0292226\pi$
$751$	8613.02	+	14918.2i	0.418500	+	0.724863i	−0.577289	+	0.816540i	$0.695889\pi$
$752$		0			0
$753$	9735.33	+	9447.06i	0.471149	+	0.457198i
$754$		0			0
$755$	2192.69			0.105696
$756$		0			0
$757$	−34314.7			−1.64754			−0.823770	−	0.566924i	$-0.808133\pi$
$757$	−34314.7			−1.64754			−0.823770	+	0.566924i	$0.808133\pi$
$758$		0			0
$759$	−21989.7	−	21338.6i	−1.05161	−	1.02048i
$760$		0			0
$761$	14662.6	+	25396.4i	0.698448	+	1.20975i	0.969004	+	0.247043i	$0.0794590\pi$
$761$	14662.6	+	25396.4i	0.698448	+	1.20975i	−0.270556	+	0.962704i	$0.587208\pi$
$762$		0			0
$763$	−13012.1	−	12442.0i	−0.617391	−	0.590341i
$764$		0			0
$765$	−402.701	−	13395.7i	−0.0190323	−	0.633101i
$766$		0			0
$767$	−21115.2	−	12190.8i	−0.994035	−	0.573906i
$768$		0			0
$769$			41511.3i			1.94660i	0.229539	+	0.973299i	$0.426278\pi$
$769$			41511.3i			1.94660i	−0.229539	+	0.973299i	$0.573722\pi$
$770$		0			0
$771$	1851.89	−	7351.48i	0.0865035	−	0.343394i
$772$		0			0
$773$	12697.7	−	21993.1i	0.590821	−	1.02333i	−0.403301	−	0.915068i	$-0.632137\pi$
$773$	12697.7	−	21993.1i	0.590821	−	1.02333i	0.994122	−	0.108265i	$-0.0345295\pi$
$774$		0			0
$775$	−6292.18	+	3632.79i	−0.291641	+	0.168379i
$776$		0			0
$777$	−24212.3	−	12796.0i	−1.11790	−	0.590801i
$778$		0			0
$779$	1376.85	−	794.927i	0.0633259	−	0.0365612i
$780$		0			0
$781$	32397.6	−	56114.3i	1.48435	−	2.57097i
$782$		0			0
$783$	3961.00	+	3619.06i	0.180785	+	0.165178i
$784$		0			0
$785$		−	22710.9i		−	1.03259i
$786$		0			0
$787$	33979.6	+	19618.1i	1.53906	+	0.888578i	0.998894	+	0.0470205i	$0.0149726\pi$
$787$	33979.6	+	19618.1i	1.53906	+	0.888578i	0.540168	+	0.841557i	$0.318361\pi$
$788$		0			0
$789$	−7232.24	−	25454.8i	−0.326330	−	1.14856i
$790$		0			0
$791$	35071.6	+	8560.55i	1.57649	+	0.384802i
$792$		0			0
$793$	−7799.03	−	13508.3i	−0.349246	−	0.604911i
$794$		0			0
$795$	15058.8	−	15518.3i	0.671800	−	0.692299i
$796$		0			0
$797$	−28857.2			−1.28253			−0.641264	−	0.767321i	$-0.721590\pi$
$797$	−28857.2			−1.28253			−0.641264	+	0.767321i	$0.721590\pi$
$798$		0			0
$799$	−10169.6			−0.450282
$800$		0			0
$801$	14795.4	−	9145.67i	0.652649	−	0.403429i
$802$		0			0
$803$	−12869.1	−	22289.9i	−0.565555	−	0.979571i
$804$		0			0
$805$	−16860.8	+	17633.4i	−0.738220	+	0.772045i
$806$		0			0
$807$	4874.61	−	1384.98i	0.212632	−	0.0604133i
$808$		0			0
$809$	−29443.3	−	16999.1i	−1.27957	−	0.738759i	−0.302799	−	0.953055i	$-0.597921\pi$
$809$	−29443.3	−	16999.1i	−1.27957	−	0.738759i	−0.976769	+	0.214296i	$0.931254\pi$
$810$		0			0
$811$			1300.27i			0.0562990i	0.999604	+	0.0281495i	$0.00896145\pi$
$811$			1300.27i			0.0562990i	−0.999604	+	0.0281495i	$0.991039\pi$
$812$		0			0
$813$	18564.3	+	4676.48i	0.800834	+	0.201736i
$814$		0			0
$815$	18483.8	−	32014.9i	0.794429	−	1.37599i
$816$		0			0
$817$	−10319.1	+	5957.71i	−0.441882	+	0.255121i
$818$		0			0
$819$	2808.50	−	13218.5i	0.119825	−	0.563972i
$820$		0			0
$821$	−10717.9	+	6187.98i	−0.455611	+	0.263047i	−0.710197	−	0.704003i	$-0.751394\pi$
$821$	−10717.9	+	6187.98i	−0.455611	+	0.263047i	0.254586	+	0.967050i	$0.418061\pi$
$822$		0			0
$823$	−2282.79	+	3953.90i	−0.0966864	+	0.167466i	−0.910311	−	0.413925i	$-0.864158\pi$
$823$	−2282.79	+	3953.90i	−0.0966864	+	0.167466i	0.813625	+	0.581390i	$0.197491\pi$
$824$		0			0
$825$	−6723.66	−	1693.74i	−0.283743	−	0.0714769i
$826$		0			0
$827$			13907.2i			0.584765i	0.956301	+	0.292383i	$0.0944481\pi$
$827$			13907.2i			0.584765i	−0.956301	+	0.292383i	$0.905552\pi$
$828$		0			0
$829$	−26.4048	−	15.2448i	−0.00110624	−	0.000638690i	0.499447	−	0.866345i	$-0.333537\pi$
$829$	−26.4048	−	15.2448i	−0.00110624	−	0.000638690i	−0.500553	+	0.865706i	$0.666870\pi$
$830$		0			0
$831$	−16500.5	+	4688.14i	−0.688804	+	0.195704i
$832$		0			0
$833$	11739.3	−	7497.80i	0.488286	−	0.311865i
$834$		0			0
$835$	11069.7	+	19173.3i	0.458782	+	0.794634i
$836$		0			0
$837$	−39842.1	+	12625.6i	−1.64534	+	0.521390i
$838$		0			0
$839$	−18157.8			−0.747173			−0.373587	−	0.927595i	$-0.621872\pi$
$839$	−18157.8			−0.747173			−0.373587	+	0.927595i	$0.621872\pi$
$840$		0			0
$841$	22926.5			0.940033
$842$		0			0
$843$	−1608.91	+	1658.00i	−0.0657341	+	0.0677399i
$844$		0			0
$845$	8963.15	+	15524.6i	0.364902	+	0.632028i
$846$		0			0
$847$	29554.8	−	8632.94i	1.19896	−	0.350214i
$848$		0			0
$849$	2574.20	+	9060.23i	0.104059	+	0.366250i
$850$		0			0
$851$	26561.8	+	15335.5i	1.06995	+	0.617736i
$852$		0			0
$853$			13489.4i			0.541464i	0.962655	+	0.270732i	$0.0872657\pi$
$853$			13489.4i			0.541464i	−0.962655	+	0.270732i	$0.912734\pi$
$854$		0			0
$855$	8753.39	−	16271.7i	0.350128	−	0.650855i
$856$		0			0
$857$	1188.83	−	2059.12i	0.0473859	−	0.0820748i	−0.841360	−	0.540476i	$-0.818244\pi$
$857$	1188.83	−	2059.12i	0.0473859	−	0.0820748i	0.888746	+	0.458401i	$0.151578\pi$
$858$		0			0
$859$	−16353.2	+	9441.53i	−0.649551	+	0.375019i	−0.788284	−	0.615311i	$-0.789030\pi$
$859$	−16353.2	+	9441.53i	−0.649551	+	0.375019i	0.138733	+	0.990330i	$0.455697\pi$
$860$		0			0
$861$	−2313.77	+	1453.90i	−0.0915832	+	0.0575480i
$862$		0			0
$863$	−10869.9	+	6275.72i	−0.428754	+	0.247541i	−0.698816	−	0.715302i	$-0.746289\pi$
$863$	−10869.9	+	6275.72i	−0.428754	+	0.247541i	0.270062	+	0.962843i	$0.412956\pi$
$864$		0			0
$865$	−6778.78	+	11741.2i	−0.266457	+	0.461518i
$866$		0			0
$867$	−4142.71	+	16445.4i	−0.162277	+	0.644192i
$868$		0			0
$869$		−	2197.29i		−	0.0857742i
$870$		0			0
$871$	8634.39	+	4985.07i	0.335896	+	0.193930i
$872$		0			0
$873$	20210.2	−	607.558i	0.783518	−	0.0235541i
$874$		0			0
$875$	5400.46	−	22125.1i	0.208650	−	0.854817i
$876$		0			0
$877$	−20663.8	−	35790.8i	−0.795630	−	1.37807i	−0.922439	−	0.386144i	$-0.873807\pi$
$877$	−20663.8	−	35790.8i	−0.795630	−	1.37807i	0.126809	−	0.991927i	$-0.459526\pi$
$878$		0			0
$879$	−2585.60	−	2509.04i	−0.0992151	−	0.0962773i
$880$		0			0
$881$	31002.4			1.18558			0.592791	−	0.805357i	$-0.298026\pi$
$881$	31002.4			1.18558			0.592791	+	0.805357i	$0.298026\pi$
$882$		0			0
$883$	−42592.8			−1.62329			−0.811643	−	0.584154i	$-0.801426\pi$
$883$	−42592.8			−1.62329			−0.811643	+	0.584154i	$0.801426\pi$
$884$		0			0
$885$	41120.5	+	39902.9i	1.56187	+	1.51562i
$886$		0			0
$887$	3129.31	+	5420.12i	0.118458	+	0.205175i	0.919157	−	0.393892i	$-0.128872\pi$
$887$	3129.31	+	5420.12i	0.118458	+	0.205175i	−0.800699	+	0.599067i	$0.795538\pi$
$888$		0			0
$889$	11679.8	−	47850.9i	0.440639	−	1.80525i
$890$		0			0
$891$	−35677.6	−	17831.9i	−1.34146	−	0.670473i
$892$		0			0
$893$	−12142.3	−	7010.34i	−0.455011	−	0.262701i
$894$		0			0
$895$		−	46192.3i		−	1.72518i
$896$		0			0
$897$	−3697.07	+	14676.3i	−0.137616	+	0.546296i
$898$		0			0
$899$	5696.39	−	9866.43i	0.211329	−	0.366033i
$900$		0			0
$901$	11974.5	−	6913.49i	0.442762	−	0.255629i
$902$		0			0
$903$	17340.9	−	10896.5i	0.639059	−	0.401565i
$904$		0			0
$905$	12491.7	−	7212.08i	0.458827	−	0.264904i
$906$		0			0
$907$	−6082.91	+	10535.9i	−0.222690	+	0.385710i	−0.955624	−	0.294589i	$-0.904817\pi$
$907$	−6082.91	+	10535.9i	−0.222690	+	0.385710i	0.732934	+	0.680300i	$0.238150\pi$
$908$		0			0
$909$	30404.1	+	16355.9i	1.10940	+	0.596801i
$910$		0			0
$911$			30486.2i			1.10873i	0.832273	+	0.554366i	$0.187039\pi$
$911$			30486.2i			1.10873i	−0.832273	+	0.554366i	$0.812961\pi$
$912$		0			0
$913$	−40524.7	−	23396.9i	−1.46897	−	0.848111i
$914$		0			0
$915$	10018.4	+	35260.9i	0.361964	+	1.27398i
$916$		0			0
$917$	35523.0	−	10376.2i	1.27925	−	0.373668i
$918$		0			0
$919$	−7103.12	−	12303.0i	−0.254962	−	0.441608i	0.709923	−	0.704279i	$-0.248730\pi$
$919$	−7103.12	−	12303.0i	−0.254962	−	0.441608i	−0.964885	+	0.262672i	$0.915396\pi$
$920$		0			0
$921$	−20173.4	+	20789.0i	−0.721756	+	0.743779i
$922$		0			0
$923$	−32004.7			−1.14133
$924$		0			0
$925$	6940.45			0.246703
$926$		0			0
$927$	−797.715	−	1290.51i	−0.0282636	−	0.0457236i
$928$		0			0
$929$	19290.5	+	33412.2i	0.681272	+	1.18000i	0.974593	+	0.223984i	$0.0719064\pi$
$929$	19290.5	+	33412.2i	0.681272	+	1.18000i	−0.293321	+	0.956014i	$0.594760\pi$
$930$		0			0
$931$	19185.0	−	859.791i	0.675362	−	0.0302669i
$932$		0			0
$933$	4846.31	−	1376.94i	0.170055	−	0.0483161i
$934$		0			0
$935$	−23518.9	−	13578.7i	−0.822622	−	0.474941i
$936$		0			0
$937$			20630.2i			0.719272i	0.933093	+	0.359636i	$0.117099\pi$
$937$			20630.2i			0.719272i	−0.933093	+	0.359636i	$0.882901\pi$
$938$		0			0
$939$	−9885.01	−	2490.11i	−0.343541	−	0.0865405i
$940$		0			0
$941$	8820.50	−	15277.5i	0.305569	−	0.529260i	−0.671819	−	0.740715i	$-0.734487\pi$
$941$	8820.50	−	15277.5i	0.305569	−	0.529260i	0.977388	+	0.211455i	$0.0678202\pi$
$942$		0			0
$943$	2650.45	−	1530.24i	0.0915276	−	0.0528435i
$944$		0			0
$945$	−13988.5	+	28511.2i	−0.481531	+	0.981448i
$946$		0			0
$947$	11311.5	−	6530.68i	0.388145	−	0.224096i	−0.293211	−	0.956048i	$-0.594724\pi$
$947$	11311.5	−	6530.68i	0.388145	−	0.224096i	0.681356	+	0.731952i	$0.261391\pi$
$948$		0			0
$949$	−6356.52	+	11009.8i	−0.217430	+	0.376600i
$950$		0			0
$951$	−52183.8	−	13145.5i	−1.77936	−	0.448235i
$952$		0			0
$953$		−	12172.7i		−	0.413757i	−0.978367	−	0.206879i	$-0.933669\pi$
$953$		−	12172.7i		−	0.413757i	0.978367	−	0.206879i	$-0.0663306\pi$
$954$		0			0
$955$	−33178.3	−	19155.5i	−1.12422	−	0.649066i
$956$		0			0
$957$	10458.5	−	2971.46i	0.353264	−	0.100370i
$958$		0			0
$959$	−19725.3	+	20629.1i	−0.664194	+	0.694627i
$960$		0			0
$961$	29477.8	+	51057.1i	0.989488	+	1.71384i
$962$		0			0
$963$	−1574.22	−	2546.69i	−0.0526774	−	0.0852191i
$964$		0			0
$965$	−9947.21			−0.331826
$966$		0			0
$967$	−3592.90			−0.119483			−0.0597413	−	0.998214i	$-0.519028\pi$
$967$	−3592.90			−0.119483			−0.0597413	+	0.998214i	$0.519028\pi$
$968$		0			0
$969$	8227.78	−	8478.84i	0.272770	−	0.281093i
$970$		0			0
$971$	8552.13	+	14812.7i	0.282648	+	0.489560i	0.972036	−	0.234832i	$-0.0754539\pi$
$971$	8552.13	+	14812.7i	0.282648	+	0.489560i	−0.689388	+	0.724392i	$0.742121\pi$
$972$		0			0
$973$	52654.3	+	12852.3i	1.73486	+	0.423458i
$974$		0			0
$975$	936.014	+	3294.42i	0.0307451	+	0.108211i
$976$		0			0
$977$	2706.99	+	1562.88i	0.0886431	+	0.0511781i	0.543666	−	0.839301i	$-0.317036\pi$
$977$	2706.99	+	1562.88i	0.0886431	+	0.0511781i	−0.455023	+	0.890480i	$0.650369\pi$
$978$		0			0
$979$		−	35247.1i		−	1.15067i
$980$		0			0
$981$	23114.0	+	12434.2i	0.752267	+	0.404683i
$982$		0			0
$983$	13385.5	−	23184.4i	0.434314	−	0.752254i	−0.562925	−	0.826508i	$-0.690324\pi$
$983$	13385.5	−	23184.4i	0.434314	−	0.752254i	0.997239	+	0.0742535i	$0.0236574\pi$
$984$		0			0
$985$	−28219.0	+	16292.3i	−0.912825	+	0.527020i
$986$		0			0
$987$	21306.3	+	11260.2i	0.687121	+	0.363137i
$988$		0			0
$989$	−19864.2	+	11468.6i	−0.638671	+	0.368737i
$990$		0			0
$991$	−18555.4	+	32138.9i	−0.594785	+	1.03020i	0.398792	+	0.917041i	$0.369429\pi$
$991$	−18555.4	+	32138.9i	−0.594785	+	1.03020i	−0.993577	+	0.113156i	$0.963904\pi$
$992$		0			0
$993$	−5052.35	+	20056.4i	−0.161462	+	0.640958i
$994$		0			0
$995$			26108.0i			0.831839i
$996$		0			0
$997$	39407.8	+	22752.1i	1.25181	+	0.722734i	0.971469	−	0.237165i	$-0.0762183\pi$
$997$	39407.8	+	22752.1i	1.25181	+	0.722734i	0.280344	+	0.959900i	$0.409552\pi$
$998$		0			0
$999$	38990.5	+	8584.85i	1.23484	+	0.271884i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.4.u.a.17.20	✓	48
3.2	odd	2	inner	168.4.u.a.17.22	yes	48
4.3	odd	2		336.4.bc.f.17.5		48
7.5	odd	6	inner	168.4.u.a.89.22	yes	48
12.11	even	2		336.4.bc.f.17.3		48
21.5	even	6	inner	168.4.u.a.89.20	yes	48
28.19	even	6		336.4.bc.f.257.3		48
84.47	odd	6		336.4.bc.f.257.5		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.u.a.17.20	✓	48	1.1	even	1	trivial
168.4.u.a.17.22	yes	48	3.2	odd	2	inner
168.4.u.a.89.20	yes	48	21.5	even	6	inner
168.4.u.a.89.22	yes	48	7.5	odd	6	inner
336.4.bc.f.17.3		48	12.11	even	2
336.4.bc.f.17.5		48	4.3	odd	2
336.4.bc.f.257.3		48	28.19	even	6
336.4.bc.f.257.5		48	84.47	odd	6

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:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	168.u (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(3.72903 + 3.61861i) q^{3} +(6.11124 + 10.5850i) q^{5} +(4.39164 - 17.9920i) q^{7} +(0.811307 + 26.9878i) q^{9} +O(q^{10})\)	\(q+(3.72903 + 3.61861i) q^{3} +(6.11124 + 10.5850i) q^{5} +(4.39164 - 17.9920i) q^{7} +(0.811307 + 26.9878i) q^{9} +(47.3827 + 27.3564i) q^{11} -27.0247i q^{13} +(-15.5139 + 61.5859i) q^{15} +(-20.3052 + 35.1697i) q^{17} +(-48.4879 + 27.9945i) q^{19} +(81.4827 - 51.2012i) q^{21} +(-93.3394 + 53.8895i) q^{23} +(-12.1945 + 21.1215i) q^{25} +(-94.6330 + 103.574i) q^{27} -38.2431i q^{29} +(257.992 + 148.952i) q^{31} +(77.6993 + 273.473i) q^{33} +(217.284 - 63.4683i) q^{35} +(-142.286 - 246.447i) q^{37} +(97.7918 - 100.776i) q^{39} -28.3958 q^{41} +212.817 q^{43} +(-280.707 + 173.517i) q^{45} +(125.209 + 216.869i) q^{47} +(-304.427 - 158.029i) q^{49} +(-202.984 + 57.6721i) q^{51} +(-294.863 - 170.239i) q^{53} +668.727i q^{55} +(-282.114 - 71.0665i) q^{57} +(451.101 - 781.329i) q^{59} +(499.851 - 288.589i) q^{61} +(489.129 + 103.924i) q^{63} +(286.056 - 165.154i) q^{65} +(-184.464 + 319.500i) q^{67} +(-543.070 - 136.803i) q^{69} -1184.28i q^{71} +(-407.399 - 235.212i) q^{73} +(-121.904 + 34.6355i) q^{75} +(700.286 - 732.373i) q^{77} +(-20.0802 - 34.7799i) q^{79} +(-727.684 + 43.7908i) q^{81} -855.263 q^{83} -496.361 q^{85} +(138.387 - 142.610i) q^{87} +(-322.110 - 557.910i) q^{89} +(-486.229 - 118.683i) q^{91} +(423.061 + 1489.02i) q^{93} +(-592.642 - 342.162i) q^{95} -748.863i q^{97} +(-699.848 + 1300.95i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 12 q^{7} + 14 q^{9}+O(q^{10}) \) 48 * q + 12 * q^7 + 14 * q^9	\( 48 q + 12 q^{7} + 14 q^{9} - 88 q^{15} - 270 q^{19} + 50 q^{21} - 438 q^{25} + 216 q^{31} - 372 q^{33} + 66 q^{37} + 242 q^{39} + 900 q^{43} - 294 q^{45} + 60 q^{49} - 138 q^{51} + 1384 q^{57} + 108 q^{61} + 1096 q^{63} + 6 q^{67} - 1206 q^{73} - 594 q^{75} - 588 q^{79} - 54 q^{81} - 240 q^{85} - 3522 q^{87} + 234 q^{91} - 608 q^{93} + 1988 q^{99}+O(q^{100}) \) 48 * q + 12 * q^7 + 14 * q^9 - 88 * q^15 - 270 * q^19 + 50 * q^21 - 438 * q^25 + 216 * q^31 - 372 * q^33 + 66 * q^37 + 242 * q^39 + 900 * q^43 - 294 * q^45 + 60 * q^49 - 138 * q^51 + 1384 * q^57 + 108 * q^61 + 1096 * q^63 + 6 * q^67 - 1206 * q^73 - 594 * q^75 - 588 * q^79 - 54 * q^81 - 240 * q^85 - 3522 * q^87 + 234 * q^91 - 608 * q^93 + 1988 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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