LMFDB - Embedded newform 315.2.bb.b.89.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,2,Mod(89,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("315.89");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.bb (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:	$2.51528766367$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			89.8
Character	$\chi$	$=$	315.89
Dual form			315.2.bb.b.269.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.659204 - 1.14177i) q^{2} +(0.130901 + 0.226727i) q^{4} +(0.729935 - 2.11357i) q^{5} +(-2.12635 - 1.57437i) q^{7} +2.98198 q^{8} +O(q^{10})$	\(q+(0.659204 - 1.14177i) q^{2} +(0.130901 + 0.226727i) q^{4} +(0.729935 - 2.11357i) q^{5} +(-2.12635 - 1.57437i) q^{7} +2.98198 q^{8} +(-1.93205 - 2.22670i) q^{10} +(2.08688 - 1.20486i) q^{11} -1.69332 q^{13} +(-3.19927 + 1.38998i) q^{14} +(1.70393 - 2.95129i) q^{16} +(-0.831785 + 0.480231i) q^{17} +(3.56633 + 2.05902i) q^{19} +(0.574754 - 0.111173i) q^{20} -3.17699i q^{22} +(2.88570 - 4.99818i) q^{23} +(-3.93439 - 3.08554i) q^{25} +(-1.11625 + 1.93339i) q^{26} +(0.0786111 - 0.688188i) q^{28} +5.56553i q^{29} +(-7.58148 + 4.37717i) q^{31} +(0.735506 + 1.27393i) q^{32} +1.26628i q^{34} +(-4.87964 + 3.34501i) q^{35} +(3.44079 + 1.98654i) q^{37} +(4.70187 - 2.71463i) q^{38} +(2.17665 - 6.30263i) q^{40} +1.87474 q^{41} +10.2706i q^{43} +(0.546349 + 0.315435i) q^{44} +(-3.80453 - 6.58963i) q^{46} +(-8.75337 - 5.05376i) q^{47} +(2.04272 + 6.69532i) q^{49} +(-6.11656 + 2.45818i) q^{50} +(-0.221658 - 0.383923i) q^{52} +(6.35430 + 11.0060i) q^{53} +(-1.02327 - 5.29024i) q^{55} +(-6.34072 - 4.69473i) q^{56} +(6.35458 + 3.66882i) q^{58} +(-3.38686 - 5.86621i) q^{59} +(1.98485 + 1.14595i) q^{61} +11.5418i q^{62} +8.75510 q^{64} +(-1.23602 + 3.57896i) q^{65} +(-4.15470 + 2.39872i) q^{67} +(-0.217763 - 0.125726i) q^{68} +(0.602567 + 7.77649i) q^{70} -2.24602i q^{71} +(7.34937 + 12.7295i) q^{73} +(4.53637 - 2.61907i) q^{74} +1.07811i q^{76} +(-6.33432 - 0.723564i) q^{77} +(0.892703 - 1.54621i) q^{79} +(-4.99401 - 5.75563i) q^{80} +(1.23583 - 2.14052i) q^{82} -9.62086i q^{83} +(0.407855 + 2.10858i) q^{85} +(11.7267 + 6.77039i) q^{86} +(6.22302 - 3.59286i) q^{88} +(-0.220850 + 0.382523i) q^{89} +(3.60060 + 2.66592i) q^{91} +1.51096 q^{92} +(-11.5405 + 6.66292i) q^{94} +(6.95508 - 6.03475i) q^{95} +12.4926 q^{97} +(8.99111 + 2.08125i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{4}+O(q^{10}) $ 24 * q - 24 * q^4	$ 24 q - 24 q^{4} - 12 q^{10} - 36 q^{19} + 12 q^{25} - 60 q^{31} + 96 q^{40} - 24 q^{46} + 36 q^{49} + 48 q^{61} + 48 q^{64} - 48 q^{70} - 60 q^{79} - 72 q^{85} + 60 q^{91} + 48 q^{94}+O(q^{100}) $ 24 * q - 24 * q^4 - 12 * q^10 - 36 * q^19 + 12 * q^25 - 60 * q^31 + 96 * q^40 - 24 * q^46 + 36 * q^49 + 48 * q^61 + 48 * q^64 - 48 * q^70 - 60 * q^79 - 72 * q^85 + 60 * q^91 + 48 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$-1$

Coefficient data

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

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

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

$2$	0.659204	−	1.14177i	0.466127	−	0.807356i	−0.533124	−	0.846037i	$-0.678982\pi$
$2$	0.659204	−	1.14177i	0.466127	−	0.807356i	0.999252	+	0.0386807i	$0.0123155\pi$
$3$		0			0
$3$		0			0
$4$	0.130901	+	0.226727i	0.0654506	+	0.113364i
$5$	0.729935	−	2.11357i	0.326437	−	0.945219i
$5$	0.729935	−	2.11357i	0.326437	−	0.945219i
$6$		0			0
$7$	−2.12635	−	1.57437i	−0.803685	−	0.595056i
$7$	−2.12635	−	1.57437i	−0.803685	−	0.595056i
$8$	2.98198			1.05429
$9$		0			0
$10$	−1.93205	−	2.22670i	−0.610967	−	0.704143i
$11$	2.08688	−	1.20486i	0.629217	−	0.363279i	−0.151232	−	0.988498i	$-0.548324\pi$
$11$	2.08688	−	1.20486i	0.629217	−	0.363279i	0.780449	+	0.625220i	$0.214991\pi$
$12$		0			0
$13$	−1.69332			−0.469643			−0.234822	−	0.972038i	$-0.575451\pi$
$13$	−1.69332			−0.469643			−0.234822	+	0.972038i	$0.575451\pi$
$14$	−3.19927	+	1.38998i	−0.855041	+	0.371488i
$15$		0			0
$16$	1.70393	−	2.95129i	0.425982	−	0.737822i
$17$	−0.831785	+	0.480231i	−0.201737	+	0.116473i	−0.597466	−	0.801895i	$-0.703826\pi$
$17$	−0.831785	+	0.480231i	−0.201737	+	0.116473i	0.395728	+	0.918368i	$0.370492\pi$
$18$		0			0
$19$	3.56633	+	2.05902i	0.818172	+	0.472372i	0.849786	−	0.527129i	$-0.176731\pi$
$19$	3.56633	+	2.05902i	0.818172	+	0.472372i	−0.0316139	+	0.999500i	$0.510065\pi$
$20$	0.574754	−	0.111173i	0.128519	−	0.0248590i
$21$		0			0
$22$		−	3.17699i		−	0.677336i
$23$	2.88570	−	4.99818i	0.601710	−	1.04219i	−0.390853	−	0.920453i	$-0.627820\pi$
$23$	2.88570	−	4.99818i	0.601710	−	1.04219i	0.992562	−	0.121738i	$-0.0388469\pi$
$24$		0			0
$25$	−3.93439	−	3.08554i	−0.786878	−	0.617109i
$26$	−1.11625	+	1.93339i	−0.218914	+	0.379170i
$27$		0			0
$28$	0.0786111	−	0.688188i	0.0148561	−	0.130055i
$29$			5.56553i			1.03349i	0.856138	+	0.516747i	$0.172857\pi$
$29$			5.56553i			1.03349i	−0.856138	+	0.516747i	$0.827143\pi$
$30$		0			0
$31$	−7.58148	+	4.37717i	−1.36167	+	0.786163i	−0.989847	−	0.142139i	$-0.954602\pi$
$31$	−7.58148	+	4.37717i	−1.36167	+	0.786163i	−0.371827	+	0.928302i	$0.621269\pi$
$32$	0.735506	+	1.27393i	0.130020	+	0.225202i
$33$		0			0
$34$			1.26628i			0.217165i
$35$	−4.87964	+	3.34501i	−0.824810	+	0.565410i
$36$		0			0
$37$	3.44079	+	1.98654i	0.565663	+	0.326586i	0.755415	−	0.655246i	$-0.227435\pi$
$37$	3.44079	+	1.98654i	0.565663	+	0.326586i	−0.189752	+	0.981832i	$0.560769\pi$
$38$	4.70187	−	2.71463i	0.762744	−	0.440371i
$39$		0			0
$40$	2.17665	−	6.30263i	0.344158	−	0.996533i
$41$	1.87474			0.292784			0.146392	−	0.989227i	$-0.453234\pi$
$41$	1.87474			0.292784			0.146392	+	0.989227i	$0.453234\pi$
$42$		0			0
$43$			10.2706i			1.56625i	0.621867	+	0.783123i	$0.286375\pi$
$43$			10.2706i			1.56625i	−0.621867	+	0.783123i	$0.713625\pi$
$44$	0.546349	+	0.315435i	0.0823652	+	0.0475536i
$45$		0			0
$46$	−3.80453	−	6.58963i	−0.560947	−	0.971588i
$47$	−8.75337	−	5.05376i	−1.27681	−	0.737167i	−0.300550	−	0.953766i	$-0.597170\pi$
$47$	−8.75337	−	5.05376i	−1.27681	−	0.737167i	−0.976261	+	0.216599i	$0.930504\pi$
$48$		0			0
$49$	2.04272	+	6.69532i	0.291818	+	0.956474i
$50$	−6.11656	+	2.45818i	−0.865012	+	0.347640i
$51$		0			0
$52$	−0.221658	−	0.383923i	−0.0307384	−	0.0532405i
$53$	6.35430	+	11.0060i	0.872830	+	1.51179i	0.859057	+	0.511880i	$0.171051\pi$
$53$	6.35430	+	11.0060i	0.872830	+	1.51179i	0.0137732	+	0.999905i	$0.495616\pi$
$54$		0			0
$55$	−1.02327	−	5.29024i	−0.137978	−	0.713335i
$56$	−6.34072	−	4.69473i	−0.847315	−	0.627360i
$57$		0			0
$58$	6.35458	+	3.66882i	0.834398	+	0.481740i
$59$	−3.38686	−	5.86621i	−0.440931	−	0.763715i	0.556828	−	0.830628i	$-0.312018\pi$
$59$	−3.38686	−	5.86621i	−0.440931	−	0.763715i	−0.997759	+	0.0669127i	$0.978685\pi$
$60$		0			0
$61$	1.98485	+	1.14595i	0.254134	+	0.146724i	0.621656	−	0.783291i	$-0.286460\pi$
$61$	1.98485	+	1.14595i	0.254134	+	0.146724i	−0.367522	+	0.930015i	$0.619794\pi$
$62$			11.5418i			1.46581i
$63$		0			0
$64$	8.75510			1.09439
$65$	−1.23602	+	3.57896i	−0.153309	+	0.443916i
$66$		0			0
$67$	−4.15470	+	2.39872i	−0.507577	+	0.293050i	−0.731837	−	0.681480i	$-0.761337\pi$
$67$	−4.15470	+	2.39872i	−0.507577	+	0.293050i	0.224260	+	0.974529i	$0.428004\pi$
$68$	−0.217763	−	0.125726i	−0.0264077	−	0.0152465i
$69$		0			0
$70$	0.602567	+	7.77649i	0.0720205	+	0.929469i
$71$		−	2.24602i		−	0.266554i	−0.991079	−	0.133277i	$-0.957450\pi$
$71$		−	2.24602i		−	0.266554i	0.991079	−	0.133277i	$-0.0425500\pi$
$72$		0			0
$73$	7.34937	+	12.7295i	0.860178	+	1.48987i	0.871756	+	0.489940i	$0.162981\pi$
$73$	7.34937	+	12.7295i	0.860178	+	1.48987i	−0.0115780	+	0.999933i	$0.503685\pi$
$74$	4.53637	−	2.61907i	0.527342	−	0.304461i
$75$		0			0
$76$			1.07811i			0.123668i
$77$	−6.33432	−	0.723564i	−0.721863	−	0.0824577i
$78$		0			0
$79$	0.892703	−	1.54621i	0.100437	−	0.173962i	−0.811428	−	0.584453i	$-0.801309\pi$
$79$	0.892703	−	1.54621i	0.100437	−	0.173962i	0.911865	+	0.410491i	$0.134643\pi$
$80$	−4.99401	−	5.75563i	−0.558347	−	0.643499i
$81$		0			0
$82$	1.23583	−	2.14052i	0.136475	−	0.236381i
$83$		−	9.62086i		−	1.05603i	−0.849236	−	0.528013i	$-0.822937\pi$
$83$		−	9.62086i		−	1.05603i	0.849236	−	0.528013i	$-0.177063\pi$
$84$		0			0
$85$	0.407855	+	2.10858i	0.0442381	+	0.228707i
$86$	11.7267	+	6.77039i	1.26452	+	0.730070i
$87$		0			0
$88$	6.22302	−	3.59286i	0.663376	−	0.383000i
$89$	−0.220850	+	0.382523i	−0.0234100	+	0.0405474i	−0.877493	−	0.479589i	$-0.840786\pi$
$89$	−0.220850	+	0.382523i	−0.0234100	+	0.0405474i	0.854083	+	0.520137i	$0.174119\pi$
$90$		0			0
$91$	3.60060	+	2.66592i	0.377445	+	0.279464i
$92$	1.51096			0.157529
$93$		0			0
$94$	−11.5405	+	6.66292i	−1.19031	+	0.687227i
$95$	6.95508	−	6.03475i	0.713576	−	0.619152i
$96$		0			0
$97$	12.4926			1.26843			0.634215	−	0.773157i	$-0.281323\pi$
$97$	12.4926			1.26843			0.634215	+	0.773157i	$0.281323\pi$
$98$	8.99111	+	2.08125i	0.908239	+	0.210238i
$99$		0			0
$100$	0.184561	−	1.29594i	0.0184561	−	0.129594i
$101$	−4.74466	−	8.21799i	−0.472111	−	0.817720i	0.527380	−	0.849630i	$-0.323175\pi$
$101$	−4.74466	−	8.21799i	−0.472111	−	0.817720i	−0.999491	+	0.0319094i	$0.989841\pi$
$102$		0			0
$103$	−4.04133	+	6.99979i	−0.398204	+	0.689710i	−0.993504	−	0.113793i	$-0.963700\pi$
$103$	−4.04133	+	6.99979i	−0.398204	+	0.689710i	0.595300	+	0.803503i	$0.297033\pi$
$104$	−5.04945			−0.495139
$105$		0			0
$106$	16.7551			1.62740
$107$	−2.71312	+	4.69926i	−0.262287	+	0.454294i	−0.966849	−	0.255348i	$-0.917810\pi$
$107$	−2.71312	+	4.69926i	−0.262287	+	0.454294i	0.704562	+	0.709642i	$0.251143\pi$
$108$		0			0
$109$	−8.68208	−	15.0378i	−0.831592	−	1.44036i	−0.896775	−	0.442487i	$-0.854096\pi$
$109$	−8.68208	−	15.0378i	−0.831592	−	1.44036i	0.0651830	−	0.997873i	$-0.479237\pi$
$110$	−6.71480	−	2.31900i	−0.640231	−	0.221108i
$111$		0			0
$112$	−8.26956	+	3.59286i	−0.781400	+	0.339493i
$113$	5.92400			0.557283			0.278642	−	0.960395i	$-0.410116\pi$
$113$	5.92400			0.557283			0.278642	+	0.960395i	$0.410116\pi$
$114$		0			0
$115$	−8.45764	−	9.74748i	−0.788679	−	0.908957i
$116$	−1.26186	+	0.728535i	−0.117161	+	0.0676427i
$117$		0			0
$118$	−8.93051			−0.822121
$119$	2.52473	+	0.288397i	0.231441	+	0.0264373i
$120$		0			0
$121$	−2.59663	+	4.49750i	−0.236057	+	0.408863i
$122$	2.61684	−	1.51083i	0.236917	−	0.136784i
$123$		0			0
$124$	−1.98485	−	1.14595i	−0.178245	−	0.102910i
$125$	−9.39337	+	6.06338i	−0.840169	+	0.542325i
$126$		0			0
$127$		−	5.01325i		−	0.444854i	−0.974949	−	0.222427i	$-0.928602\pi$
$127$		−	5.01325i		−	0.444854i	0.974949	−	0.222427i	$-0.0713979\pi$
$128$	4.30038	−	7.44848i	0.380104	−	0.658359i
$129$		0			0
$130$	3.27158	+	3.77052i	0.286937	+	0.330696i
$131$	−8.52739	+	14.7699i	−0.745042	+	1.29045i	0.205134	+	0.978734i	$0.434237\pi$
$131$	−8.52739	+	14.7699i	−0.745042	+	1.29045i	−0.950175	+	0.311716i	$0.899096\pi$
$132$		0			0
$133$	−4.34160	−	9.99291i	−0.376464	−	0.866495i
$134$			6.32497i			0.546394i
$135$		0			0
$136$	−2.48036	+	1.43204i	−0.212689	+	0.122796i
$137$	−4.95959	−	8.59026i	−0.423726	−	0.733915i	0.572574	−	0.819853i	$-0.305945\pi$
$137$	−4.95959	−	8.59026i	−0.423726	−	0.733915i	−0.996301	+	0.0859376i	$0.972611\pi$
$138$		0			0
$139$		−	15.1104i		−	1.28164i	−0.767689	−	0.640822i	$-0.778593\pi$
$139$		−	15.1104i		−	1.28164i	0.767689	−	0.640822i	$-0.221407\pi$
$140$	−1.39716	−	0.668483i	−0.118081	−	0.0564971i
$141$		0			0
$142$	−2.56445	−	1.48059i	−0.215204	−	0.124248i
$143$	−3.53376	+	2.04022i	−0.295508	+	0.170611i
$144$		0			0
$145$	11.7632	+	4.06248i	0.976878	+	0.337370i
$146$	19.3789			1.60381
$147$		0			0
$148$			1.04016i			0.0855008i
$149$	2.02420	+	1.16867i	0.165829	+	0.0957413i	0.580618	−	0.814176i	$-0.302811\pi$
$149$	2.02420	+	1.16867i	0.165829	+	0.0957413i	−0.414789	+	0.909918i	$0.636144\pi$
$150$		0			0
$151$	−5.01515	−	8.68650i	−0.408127	−	0.706897i	0.586553	−	0.809911i	$-0.300485\pi$
$151$	−5.01515	−	8.68650i	−0.408127	−	0.706897i	−0.994680	+	0.103014i	$0.967151\pi$
$152$	10.6347	+	6.13995i	0.862588	+	0.498016i
$153$		0			0
$154$	−5.00175	+	6.75539i	−0.403053	+	0.544365i
$155$	3.71748	+	19.2191i	0.298595	+	1.54371i
$156$		0			0
$157$	−10.3765	−	17.9727i	−0.828137	−	1.43437i	−0.899498	−	0.436924i	$-0.856068\pi$
$157$	−10.3765	−	17.9727i	−0.828137	−	1.43437i	0.0713615	−	0.997451i	$-0.477266\pi$
$158$	−1.17695	−	2.03853i	−0.0936328	−	0.162177i
$159$		0			0
$160$	3.22943	−	0.624657i	0.255308	−	0.0493835i
$161$	−14.0050	+	6.08471i	−1.10375	+	0.479543i
$162$		0			0
$163$	−13.5273	−	7.81001i	−1.05954	−	0.611727i	−0.134237	−	0.990949i	$-0.542858\pi$
$163$	−13.5273	−	7.81001i	−1.05954	−	0.611727i	−0.925306	+	0.379222i	$0.876192\pi$
$164$	0.245405	+	0.425054i	0.0191629	+	0.0331911i
$165$		0			0
$166$	−10.9848	−	6.34211i	−0.852590	−	0.492243i
$167$		−	14.7858i		−	1.14416i	−0.820198	−	0.572080i	$-0.806137\pi$
$167$		−	14.7858i		−	1.14416i	0.820198	−	0.572080i	$-0.193863\pi$
$168$		0			0
$169$	−10.1327			−0.779435
$170$	2.67638	+	0.924302i	0.205269	+	0.0708908i
$171$		0			0
$172$	−2.32862	+	1.34443i	−0.177555	+	0.102512i
$173$	5.94398	+	3.43176i	0.451912	+	0.260912i	0.708637	−	0.705573i	$-0.249310\pi$
$173$	5.94398	+	3.43176i	0.451912	+	0.260912i	−0.256725	+	0.966484i	$0.582644\pi$
$174$		0			0
$175$	3.50810	+	12.7551i	0.265188	+	0.964197i
$176$		−	8.21197i		−	0.619000i
$177$		0			0
$178$	0.291170	+	0.504321i	0.0218241	+	0.0378005i
$179$	10.1393	−	5.85393i	0.757847	−	0.437543i	−0.0706751	−	0.997499i	$-0.522515\pi$
$179$	10.1393	−	5.85393i	0.757847	−	0.437543i	0.828522	+	0.559956i	$0.189182\pi$
$180$		0			0
$181$			17.3498i			1.28960i	0.764351	+	0.644800i	$0.223059\pi$
$181$			17.3498i			1.28960i	−0.764351	+	0.644800i	$0.776941\pi$
$182$	5.41740	−	2.35369i	0.401565	−	0.174467i
$183$		0			0
$184$	8.60508	−	14.9044i	0.634375	−	1.09877i
$185$	6.71026	−	5.82232i	0.493348	−	0.428066i
$186$		0			0
$187$	−1.15722	+	2.00437i	−0.0846244	+	0.146574i
$188$		−	2.64617i		−	0.192992i
$189$		0			0
$190$	−2.30550	−	11.9193i	−0.167259	−	0.864714i
$191$	−16.3677	−	9.44988i	−1.18432	−	0.683770i	−0.227313	−	0.973822i	$-0.572994\pi$
$191$	−16.3677	−	9.44988i	−1.18432	−	0.683770i	−0.957011	+	0.290052i	$0.906327\pi$
$192$		0			0
$193$	20.1976	−	11.6611i	1.45385	−	0.839382i	0.455155	−	0.890412i	$-0.349584\pi$
$193$	20.1976	−	11.6611i	1.45385	−	0.839382i	0.998697	+	0.0510306i	$0.0162506\pi$
$194$	8.23516	−	14.2637i	0.591250	−	1.02407i
$195$		0			0
$196$	−1.25062	+	1.33957i	−0.0893298	+	0.0956833i
$197$	−0.842929			−0.0600562			−0.0300281	−	0.999549i	$-0.509560\pi$
$197$	−0.842929			−0.0600562			−0.0300281	+	0.999549i	$0.509560\pi$
$198$		0			0
$199$	−0.739952	+	0.427211i	−0.0524538	+	0.0302842i	−0.525998	−	0.850486i	$-0.676308\pi$
$199$	−0.739952	+	0.427211i	−0.0524538	+	0.0302842i	0.473544	+	0.880770i	$0.342975\pi$
$200$	−11.7323	−	9.20102i	−0.829596	−	0.650610i
$201$		0			0
$202$	−12.5108			−0.880255
$203$	8.76220	−	11.8343i	0.614986	−	0.830603i
$204$		0			0
$205$	1.36844	−	3.96239i	0.0955757	−	0.276745i
$206$	5.32812	+	9.22857i	0.371228	+	0.642985i
$207$		0			0
$208$	−2.88530	+	4.99749i	−0.200060	+	0.346513i
$209$	9.92331			0.686410
$210$		0			0
$211$	16.1933			1.11479			0.557395	−	0.830247i	$-0.311801\pi$
$211$	16.1933			1.11479			0.557395	+	0.830247i	$0.311801\pi$
$212$	−1.66357	+	2.88139i	−0.114254	+	0.197894i
$213$		0			0
$214$	3.57699	+	6.19553i	0.244518	+	0.423518i
$215$	21.7076	+	7.49684i	1.48045	+	0.511281i
$216$		0			0
$217$	23.0122	+	2.62866i	1.56217	+	0.178445i
$218$	−22.8930			−1.55051
$219$		0			0
$220$	1.06549	−	0.924502i	0.0718356	−	0.0623299i
$221$	1.40848	−	0.813187i	0.0947447	−	0.0547009i
$222$		0			0
$223$	−10.0027			−0.669832			−0.334916	−	0.942248i	$-0.608708\pi$
$223$	−10.0027			−0.669832			−0.334916	+	0.942248i	$0.608708\pi$
$224$	0.441699	−	3.86679i	0.0295123	−	0.258361i
$225$		0			0
$226$	3.90512	−	6.76387i	0.259765	−	0.449926i
$227$	4.05150	−	2.33913i	0.268908	−	0.155254i	−0.359484	−	0.933151i	$-0.617047\pi$
$227$	4.05150	−	2.33913i	0.268908	−	0.155254i	0.628391	+	0.777898i	$0.283714\pi$
$228$		0			0
$229$	12.3460	+	7.12797i	0.815848	+	0.471030i	0.848982	−	0.528421i	$-0.177216\pi$
$229$	12.3460	+	7.12797i	0.815848	+	0.471030i	−0.0331349	+	0.999451i	$0.510549\pi$
$230$	−16.7047	+	3.23114i	−1.10148	+	0.213055i
$231$		0			0
$232$			16.5963i			1.08960i
$233$	−11.1524	+	19.3166i	−0.730621	+	1.26547i	0.225997	+	0.974128i	$0.427436\pi$
$233$	−11.1524	+	19.3166i	−0.730621	+	1.26547i	−0.956618	+	0.291345i	$0.905897\pi$
$234$		0			0
$235$	−17.0709	+	14.8120i	−1.11358	+	0.966227i
$236$	0.886687	−	1.53579i	0.0577184	−	0.0999712i
$237$		0			0
$238$	1.99359	−	2.69255i	0.129225	−	0.174532i
$239$		−	6.39734i		−	0.413810i	−0.978361	−	0.206905i	$-0.933661\pi$
$239$		−	6.39734i		−	0.413810i	0.978361	−	0.206905i	$-0.0663391\pi$
$240$		0			0
$241$	−6.20841	+	3.58443i	−0.399919	+	0.230893i	−0.686449	−	0.727178i	$-0.740832\pi$
$241$	−6.20841	+	3.58443i	−0.399919	+	0.230893i	0.286530	+	0.958071i	$0.407498\pi$
$242$	3.42342	+	5.92953i	0.220066	+	0.381165i
$243$		0			0
$244$			0.600026i			0.0384127i
$245$	15.6421	+	0.569702i	0.999337	+	0.0363969i
$246$		0			0
$247$	−6.03895	−	3.48659i	−0.384249	−	0.221846i
$248$	−22.6078	+	13.0526i	−1.43560	+	0.828842i
$249$		0			0
$250$	0.730861	+	14.7221i	0.0462237	+	0.931108i
$251$	16.3470			1.03181			0.515906	−	0.856645i	$-0.327455\pi$
$251$	16.3470			1.03181			0.515906	+	0.856645i	$0.327455\pi$
$252$		0			0
$253$		−	13.9074i		−	0.874353i
$254$	−5.72400	−	3.30475i	−0.359156	−	0.207359i
$255$		0			0
$256$	3.08545	+	5.34415i	0.192840	+	0.334009i
$257$	10.5984	+	6.11896i	0.661107	+	0.381690i	0.792699	−	0.609614i	$-0.208675\pi$
$257$	10.5984	+	6.11896i	0.661107	+	0.381690i	−0.131592	+	0.991304i	$0.542009\pi$
$258$		0			0
$259$	−4.18878	−	9.64116i	−0.260278	−	0.599073i
$260$	−0.973245	+	0.188252i	−0.0603581	+	0.0116749i
$261$		0			0
$262$	11.2426	+	19.4727i	0.694569	+	1.20303i
$263$	4.71070	+	8.15918i	0.290474	+	0.503117i	0.973922	−	0.226883i	$-0.0728535\pi$
$263$	4.71070	+	8.15918i	0.290474	+	0.503117i	−0.683448	+	0.730000i	$0.739520\pi$
$264$		0			0
$265$	27.9001	−	5.39664i	1.71389	−	0.331513i
$266$	−14.2716	−	1.63024i	−0.875051	−	0.0999563i
$267$		0			0
$268$	−1.08771	−	0.627989i	−0.0664424	−	0.0383605i
$269$	−3.81600	−	6.60950i	−0.232666	−	0.402988i	0.725926	−	0.687773i	$-0.241411\pi$
$269$	−3.81600	−	6.60950i	−0.232666	−	0.402988i	−0.958592	+	0.284784i	$0.908078\pi$
$270$		0			0
$271$	−4.71908	−	2.72456i	−0.286664	−	0.165505i	0.349773	−	0.936835i	$-0.386259\pi$
$271$	−4.71908	−	2.72456i	−0.286664	−	0.165505i	−0.636436	+	0.771329i	$0.719592\pi$
$272$			3.27312i			0.198462i
$273$		0			0
$274$	−13.0775			−0.790041
$275$	−11.9282	−	1.69876i	−0.719299	−	0.102439i
$276$		0			0
$277$	14.7711	−	8.52809i	0.887508	−	0.512403i	0.0143815	−	0.999897i	$-0.495422\pi$
$277$	14.7711	−	8.52809i	0.887508	−	0.512403i	0.873127	+	0.487494i	$0.162089\pi$
$278$	−17.2526	−	9.96081i	−1.03474	−	0.597410i
$279$		0			0
$280$	−14.5510	+	9.97474i	−0.869587	+	0.596105i
$281$		−	25.3828i		−	1.51421i	−0.653292	−	0.757106i	$-0.726613\pi$
$281$		−	25.3828i		−	1.51421i	0.653292	−	0.757106i	$-0.273387\pi$
$282$		0			0
$283$	4.78999	+	8.29651i	0.284735	+	0.493176i	0.972545	−	0.232715i	$-0.0747609\pi$
$283$	4.78999	+	8.29651i	0.284735	+	0.493176i	−0.687810	+	0.725891i	$0.741428\pi$
$284$	0.509235	−	0.294007i	0.0302175	−	0.0174461i
$285$		0			0
$286$			5.37967i			0.318107i
$287$	−3.98634	−	2.95153i	−0.235306	−	0.174223i
$288$		0			0
$289$	−8.03876	+	13.9235i	−0.472868	+	0.819031i
$290$	12.3928	−	10.7529i	0.727728	−	0.631431i
$291$		0			0
$292$	−1.92408	+	3.33261i	−0.112598	+	0.195026i
$293$			1.44713i			0.0845421i	0.999106	+	0.0422710i	$0.0134593\pi$
$293$			1.44713i			0.0845421i	−0.999106	+	0.0422710i	$0.986541\pi$
$294$		0			0
$295$	−14.8709	+	2.87642i	−0.865815	+	0.167472i
$296$	10.2604	+	5.92382i	0.596372	+	0.344315i
$297$		0			0
$298$	2.66872	−	1.54079i	0.154595	−	0.0892553i
$299$	−4.88642	+	8.46353i	−0.282589	+	0.489459i
$300$		0			0
$301$	16.1697	−	21.8388i	0.932004	−	1.25877i
$302$	−13.2240			−0.760957
$303$		0			0
$304$	12.1535	−	7.01684i	0.697053	−	0.402444i
$305$	3.87087	−	3.35865i	0.221645	−	0.192316i
$306$		0			0
$307$	7.69871			0.439388			0.219694	−	0.975569i	$-0.429494\pi$
$307$	7.69871			0.439388			0.219694	+	0.975569i	$0.429494\pi$
$308$	−0.665118	−	1.53088i	−0.0378986	−	0.0872299i
$309$		0			0
$310$	24.3944	+	8.42475i	1.38551	+	0.478494i
$311$	16.5875	+	28.7304i	0.940592	+	1.62915i	0.764345	+	0.644807i	$0.223062\pi$
$311$	16.5875	+	28.7304i	0.940592	+	1.62915i	0.176247	+	0.984346i	$0.443604\pi$
$312$		0			0
$313$	−4.22823	+	7.32351i	−0.238994	+	0.413950i	−0.960426	−	0.278536i	$-0.910151\pi$
$313$	−4.22823	+	7.32351i	−0.238994	+	0.413950i	0.721432	+	0.692485i	$0.243484\pi$
$314$	−27.3610			−1.54407
$315$		0			0
$316$	0.467423			0.0262946
$317$	−2.97083	+	5.14563i	−0.166859	+	0.289008i	−0.937314	−	0.348487i	$-0.886696\pi$
$317$	−2.97083	+	5.14563i	−0.166859	+	0.289008i	0.770455	+	0.637494i	$0.220029\pi$
$318$		0			0
$319$	6.70568	+	11.6146i	0.375446	+	0.650292i
$320$	6.39066	−	18.5046i	0.357249	−	1.03444i
$321$		0			0
$322$	−2.28476	+	20.0016i	−0.127325	+	1.11464i
$323$	−3.95522			−0.220074
$324$		0			0
$325$	6.66219	+	5.22482i	0.369552	+	0.289821i
$326$	−17.8345	+	10.2968i	−0.987764	+	0.570286i
$327$		0			0
$328$	5.59042			0.308679
$329$	10.6562	+	24.5271i	0.587498	+	1.35222i
$330$		0			0
$331$	−13.2551	+	22.9585i	−0.728566	+	1.26191i	0.228923	+	0.973445i	$0.426480\pi$
$331$	−13.2551	+	22.9585i	−0.728566	+	1.26191i	−0.957489	+	0.288469i	$0.906854\pi$
$332$	2.18131	−	1.25938i	0.119715	−	0.0691175i
$333$		0			0
$334$	−16.8820	−	9.74685i	−0.923744	−	0.533324i
$335$	2.03720	+	10.5322i	0.111304	+	0.575434i
$336$		0			0
$337$			3.42106i			0.186357i	0.995649	+	0.0931784i	$0.0297027\pi$
$337$			3.42106i			0.186357i	−0.995649	+	0.0931784i	$0.970297\pi$
$338$	−6.67948	+	11.5692i	−0.363316	+	0.629282i
$339$		0			0
$340$	−0.424683	+	0.368487i	−0.0230317	+	0.0199840i
$341$	−10.5477	+	18.2692i	−0.571192	+	0.989334i
$342$		0			0
$343$	6.19736	−	17.4526i	0.334626	−	0.942351i
$344$			30.6266i			1.65127i
$345$		0			0
$346$	7.83658	−	4.52445i	0.421297	−	0.243236i
$347$	−6.69946	−	11.6038i	−0.359646	−	0.622925i	0.628256	−	0.778007i	$-0.283769\pi$
$347$	−6.69946	−	11.6038i	−0.359646	−	0.622925i	−0.987902	+	0.155082i	$0.950436\pi$
$348$		0			0
$349$			15.2584i			0.816762i	0.912812	+	0.408381i	$0.133906\pi$
$349$			15.2584i			0.816762i	−0.912812	+	0.408381i	$0.866094\pi$
$350$	16.8760	+	4.40277i	0.902062	+	0.235338i
$351$		0			0
$352$	3.06982	+	1.77236i	0.163622	+	0.0944672i
$353$	−31.9752	+	18.4609i	−1.70187	+	0.982574i	−0.757997	+	0.652258i	$0.773822\pi$
$353$	−31.9752	+	18.4609i	−1.70187	+	0.982574i	−0.943871	+	0.330316i	$0.892845\pi$
$354$		0			0
$355$	−4.74714	−	1.63945i	−0.251952	−	0.0870130i
$356$	−0.115638			−0.00612880
$357$		0			0
$358$		−	15.4357i		−	0.815804i
$359$	−7.71108	−	4.45200i	−0.406975	−	0.234967i	0.282514	−	0.959263i	$-0.408832\pi$
$359$	−7.71108	−	4.45200i	−0.406975	−	0.234967i	−0.689489	+	0.724296i	$0.742165\pi$
$360$		0			0
$361$	−1.02087	−	1.76820i	−0.0537301	−	0.0930633i
$362$	19.8095	+	11.4370i	1.04117	+	0.601118i
$363$		0			0
$364$	−0.133114	+	1.16533i	−0.00697707	+	0.0610797i
$365$	32.2693	−	6.24174i	1.68905	−	0.326708i
$366$		0			0
$367$	−7.22690	−	12.5174i	−0.377241	−	0.653401i	0.613419	−	0.789758i	$-0.289794\pi$
$367$	−7.22690	−	12.5174i	−0.377241	−	0.653401i	−0.990660	+	0.136357i	$0.956461\pi$
$368$	−9.83404	−	17.0331i	−0.512635	−	0.887910i
$369$		0			0
$370$	−2.22435	−	11.4997i	−0.115638	−	0.597841i
$371$	3.81600	−	33.4065i	0.198117	−	1.73438i
$372$		0			0
$373$	−18.2917	−	10.5607i	−0.947107	−	0.546812i	−0.0549259	−	0.998490i	$-0.517492\pi$
$373$	−18.2917	−	10.5607i	−0.947107	−	0.546812i	−0.892181	+	0.451678i	$0.850826\pi$
$374$	1.52569	+	2.64257i	0.0788915	+	0.136644i
$375$		0			0
$376$	−26.1024	−	15.0702i	−1.34613	−	0.777186i
$377$		−	9.42425i		−	0.485374i
$378$		0			0
$379$	−0.0133979			−0.000688205			−0.000344103	−	1.00000i	$-0.500110\pi$
$379$	−0.0133979			−0.000688205			−0.000344103		1.00000i	$0.500110\pi$
$380$	2.27867	+	0.786952i	0.116893	+	0.0403698i
$381$		0			0
$382$	−21.5793	+	12.4588i	−1.10409	+	0.637447i
$383$	−14.9151	−	8.61124i	−0.762127	−	0.440014i	0.0679321	−	0.997690i	$-0.478360\pi$
$383$	−14.9151	−	8.61124i	−0.762127	−	0.440014i	−0.830059	+	0.557676i	$0.811693\pi$
$384$		0			0
$385$	−6.15295	+	12.8599i	−0.313583	+	0.655401i
$386$		−	30.7481i		−	1.56504i
$387$		0			0
$388$	1.63529	+	2.83241i	0.0830194	+	0.143794i
$389$	−22.3077	+	12.8793i	−1.13104	+	0.653008i	−0.944197	−	0.329381i	$-0.893160\pi$
$389$	−22.3077	+	12.8793i	−1.13104	+	0.653008i	−0.186846	+	0.982389i	$0.559827\pi$
$390$		0			0
$391$			5.54321i			0.280332i
$392$	6.09135	+	19.9653i	0.307660	+	1.00840i
$393$		0			0
$394$	−0.555662	+	0.962434i	−0.0279938	+	0.0484867i
$395$	−2.61641	−	3.01543i	−0.131646	−	0.151723i
$396$		0			0
$397$	7.62398	−	13.2051i	0.382637	−	0.662746i	−0.608802	−	0.793322i	$-0.708349\pi$
$397$	7.62398	−	13.2051i	0.382637	−	0.662746i	0.991438	+	0.130576i	$0.0416828\pi$
$398$			1.12648i			0.0564652i
$399$		0			0
$400$	−15.8102	+	6.35398i	−0.790512	+	0.317699i
$401$	11.3640	+	6.56104i	0.567494	+	0.327643i	0.756148	−	0.654401i	$-0.227079\pi$
$401$	11.3640	+	6.56104i	0.567494	+	0.327643i	−0.188654	+	0.982044i	$0.560412\pi$
$402$		0			0
$403$	12.8379	−	7.41196i	0.639501	−	0.369216i
$404$	1.24216	−	2.15149i	0.0617999	−	0.107041i
$405$		0			0
$406$	−7.73598	−	17.8057i	−0.383930	−	0.883680i
$407$	9.57401			0.474566
$408$		0			0
$409$	28.4692	−	16.4367i	1.40771	−	0.812744i	0.412546	−	0.910937i	$-0.364640\pi$
$409$	28.4692	−	16.4367i	1.40771	−	0.812744i	0.995167	+	0.0981933i	$0.0313063\pi$
$410$	−3.62208	−	4.17447i	−0.178882	−	0.206162i
$411$		0			0
$412$	−2.11606			−0.104251
$413$	−2.03394	+	17.8058i	−0.100084	+	0.876165i
$414$		0			0
$415$	−20.3344	−	7.02260i	−0.998176	−	0.344726i
$416$	−1.24545	−	2.15718i	−0.0610632	−	0.105765i
$417$		0			0
$418$	6.54148	−	11.3302i	0.319954	−	0.554177i
$419$	−9.33114			−0.455856			−0.227928	−	0.973678i	$-0.573195\pi$
$419$	−9.33114			−0.455856			−0.227928	+	0.973678i	$0.573195\pi$
$420$		0			0
$421$	−13.0720			−0.637093			−0.318546	−	0.947907i	$-0.603195\pi$
$421$	−13.0720			−0.637093			−0.318546	+	0.947907i	$0.603195\pi$
$422$	10.6747	−	18.4890i	0.519634	−	0.900033i
$423$		0			0
$424$	18.9484	+	32.8195i	0.920214	+	1.59386i
$425$	4.75434	+	0.677092i	0.230619	+	0.0328438i
$426$		0			0
$427$	−2.41633	−	5.56158i	−0.116934	−	0.269144i
$428$	−1.42060			−0.0686673
$429$		0			0
$430$	22.8694	−	19.8432i	1.10286	−	0.956925i
$431$	−5.18251	+	2.99212i	−0.249633	+	0.144125i	−0.619596	−	0.784921i	$-0.712704\pi$
$431$	−5.18251	+	2.99212i	−0.249633	+	0.144125i	0.369963	+	0.929046i	$0.379370\pi$
$432$		0			0
$433$	−19.4335			−0.933916			−0.466958	−	0.884280i	$-0.654650\pi$
$433$	−19.4335			−0.933916			−0.466958	+	0.884280i	$0.654650\pi$
$434$	18.1710	−	24.5419i	0.872237	−	1.17805i
$435$		0			0
$436$	2.27299	−	3.93693i	0.108856	−	0.188545i
$437$	20.5827	−	11.8834i	0.984603	−	0.568461i
$438$		0			0
$439$	24.1175	+	13.9242i	1.15107	+	0.664568i	0.949147	−	0.314834i	$-0.101949\pi$
$439$	24.1175	+	13.9242i	1.15107	+	0.664568i	0.201919	+	0.979402i	$0.435282\pi$
$440$	−3.05138	−	15.7754i	−0.145469	−	0.752061i
$441$		0			0
$442$		−	2.14422i		−	0.101990i
$443$	11.7154	−	20.2916i	0.556614	−	0.964084i	−0.441162	−	0.897428i	$-0.645434\pi$
$443$	11.7154	−	20.2916i	0.556614	−	0.964084i	0.997776	−	0.0666564i	$-0.0212331\pi$
$444$		0			0
$445$	0.647285	+	0.745999i	0.0306842	+	0.0353638i
$446$	−6.59383	+	11.4208i	−0.312227	+	0.540793i
$447$		0			0
$448$	−18.6164	−	13.7838i	−0.879543	−	0.651222i
$449$		−	15.7868i		−	0.745025i	−0.928027	−	0.372513i	$-0.878496\pi$
$449$		−	15.7868i		−	0.745025i	0.928027	−	0.372513i	$-0.121504\pi$
$450$		0			0
$451$	3.91234	−	2.25879i	0.184225	−	0.106362i
$452$	0.775458	+	1.34313i	0.0364745	+	0.0631757i
$453$		0			0
$454$		−	6.16786i		−	0.289472i
$455$	8.26281	−	5.66418i	0.387367	−	0.265541i
$456$		0			0
$457$	23.1358	+	13.3575i	1.08225	+	0.624836i	0.931502	−	0.363737i	$-0.118499\pi$
$457$	23.1358	+	13.3575i	1.08225	+	0.624836i	0.150746	+	0.988573i	$0.451833\pi$
$458$	16.2771	−	9.39757i	0.760578	−	0.439120i
$459$		0			0
$460$	1.10291	−	3.19354i	0.0514233	−	0.148899i
$461$	33.1750			1.54512			0.772558	−	0.634944i	$-0.218977\pi$
$461$	33.1750			1.54512			0.772558	+	0.634944i	$0.218977\pi$
$462$		0			0
$463$			9.01458i			0.418943i	0.977815	+	0.209472i	$0.0671744\pi$
$463$			9.01458i			0.418943i	−0.977815	+	0.209472i	$0.932826\pi$
$464$	16.4255	+	9.48327i	0.762535	+	0.440250i
$465$		0			0
$466$	14.7035	+	25.4672i	0.681125	+	1.17974i
$467$	−28.1192	−	16.2346i	−1.30120	−	0.751248i	−0.320590	−	0.947218i	$-0.603881\pi$
$467$	−28.1192	−	16.2346i	−1.30120	−	0.751248i	−0.980610	+	0.195970i	$0.937214\pi$
$468$		0			0
$469$	12.6108	+	1.44052i	0.582313	+	0.0665171i
$470$	5.65874	+	29.2552i	0.261018	+	1.34944i
$471$		0			0
$472$	−10.0995	−	17.4929i	−0.464869	−	0.805176i
$473$	12.3746	+	21.4334i	0.568984	+	0.985509i
$474$		0			0
$475$	−7.67812	−	19.1050i	−0.352297	−	0.876600i
$476$	0.265102	+	0.610176i	0.0121509	+	0.0279674i
$477$		0			0
$478$	−7.30432	−	4.21715i	−0.334092	−	0.192888i
$479$	2.08303	+	3.60791i	0.0951759	+	0.164849i	0.909682	−	0.415305i	$-0.136325\pi$
$479$	2.08303	+	3.60791i	0.0951759	+	0.164849i	−0.814506	+	0.580155i	$0.802992\pi$
$480$		0			0
$481$	−5.82638	−	3.36386i	−0.265660	−	0.153379i
$482$			9.45148i			0.430503i
$483$		0			0
$484$	−1.35961			−0.0618003
$485$	9.11877	−	26.4040i	0.414062	−	1.19894i
$486$		0			0
$487$	−17.8689	+	10.3166i	−0.809719	+	0.467491i	−0.846858	−	0.531819i	$-0.821509\pi$
$487$	−17.8689	+	10.3166i	−0.809719	+	0.467491i	0.0371394	+	0.999310i	$0.488175\pi$
$488$	5.91877	+	3.41720i	0.267930	+	0.154690i
$489$		0			0
$490$	10.9618	−	17.4842i	0.495204	−	0.789856i
$491$		−	33.6376i		−	1.51805i	−0.651064	−	0.759023i	$-0.725677\pi$
$491$		−	33.6376i		−	1.51805i	0.651064	−	0.759023i	$-0.274323\pi$
$492$		0			0
$493$	−2.67274	−	4.62933i	−0.120374	−	0.208494i
$494$	−7.96179	+	4.59674i	−0.358218	+	0.206817i
$495$		0			0
$496$			29.8335i			1.33956i
$497$	−3.53607	+	4.77583i	−0.158614	+	0.214225i
$498$		0			0
$499$	−13.4181	+	23.2408i	−0.600675	+	1.04040i	0.392044	+	0.919946i	$0.371768\pi$
$499$	−13.4181	+	23.2408i	−0.600675	+	1.04040i	−0.992719	+	0.120453i	$0.961565\pi$
$500$	−2.60434	−	1.33603i	−0.116469	−	0.0597492i
$501$		0			0
$502$	10.7760	−	18.6646i	0.480956	−	0.833040i
$503$		−	22.8321i		−	1.01803i	−0.860757	−	0.509017i	$-0.830009\pi$
$503$		−	22.8321i		−	1.01803i	0.860757	−	0.509017i	$-0.169991\pi$
$504$		0			0
$505$	−20.8326	+	4.02959i	−0.927039	+	0.179314i
$506$	−15.8791	−	9.16783i	−0.705914	−	0.407560i
$507$		0			0
$508$	1.13664	−	0.656240i	0.0504303	−	0.0291159i
$509$	7.67782	−	13.2984i	0.340313	−	0.589440i	−0.644177	−	0.764876i	$-0.722800\pi$
$509$	7.67782	−	13.2984i	0.340313	−	0.589440i	0.984491	+	0.175436i	$0.0561335\pi$
$510$		0			0
$511$	4.41358	−	38.6379i	0.195245	−	1.70924i
$512$	25.3373			1.11976
$513$		0			0
$514$	13.9729	−	8.06729i	0.616320	−	0.355833i
$515$	11.8447	+	13.6510i	0.521938	+	0.601537i
$516$		0			0
$517$	−24.3563			−1.07119
$518$	−13.7693	−	1.57285i	−0.604988	−	0.0691072i
$519$		0			0
$520$	−3.68577	+	10.6724i	−0.161632	+	0.468015i
$521$	9.31915	+	16.1412i	0.408279	+	0.707160i	0.994697	−	0.102848i	$-0.0327957\pi$
$521$	9.31915	+	16.1412i	0.408279	+	0.707160i	−0.586418	+	0.810009i	$0.699462\pi$
$522$		0			0
$523$	4.59801	−	7.96399i	0.201057	−	0.348241i	−0.747812	−	0.663910i	$-0.768896\pi$
$523$	4.59801	−	7.96399i	0.201057	−	0.348241i	0.948869	+	0.315669i	$0.102229\pi$
$524$	−4.46498			−0.195054
$525$		0			0
$526$	12.4213			0.541592
$527$	4.20411	−	7.28172i	0.183134	−	0.317197i
$528$		0			0
$529$	−5.15451	−	8.92787i	−0.224109	−	0.388168i
$530$	12.2301	−	35.4131i	0.531243	−	1.53825i
$531$		0			0
$532$	1.69735	−	2.29244i	0.0735893	−	0.0993900i
$533$	−3.17453			−0.137504
$534$		0			0
$535$	7.95183	+	9.16453i	0.343787	+	0.396217i
$536$	−12.3892	+	7.15291i	−0.535132	+	0.308959i
$537$		0			0
$538$	−10.0621			−0.433807
$539$	12.3298	+	11.5111i	0.531083	+	0.495819i
$540$		0			0
$541$	4.49603	−	7.78736i	0.193300	−	0.334805i	−0.753042	−	0.657972i	$-0.771414\pi$
$541$	4.49603	−	7.78736i	0.193300	−	0.334805i	0.946342	+	0.323168i	$0.104748\pi$
$542$	−6.22167	+	3.59208i	−0.267244	+	0.154293i
$543$		0			0
$544$	−1.22357	−	0.706426i	−0.0524599	−	0.0302878i
$545$	−38.1209	+	7.37359i	−1.63292	+	0.315850i
$546$		0			0
$547$		−	12.8673i		−	0.550165i	−0.961421	−	0.275082i	$-0.911295\pi$
$547$		−	12.8673i		−	0.550165i	0.961421	−	0.275082i	$-0.0887051\pi$
$548$	1.29843	−	2.24895i	0.0554662	−	0.0960703i
$549$		0			0
$550$	−9.80274	+	12.4995i	−0.417990	+	0.532981i
$551$	−11.4595	+	19.8485i	−0.488193	+	0.845575i
$552$		0			0
$553$	−4.33250	+	1.88233i	−0.184237	+	0.0800449i
$554$		−	22.4870i		−	0.955380i
$555$		0			0
$556$	3.42594	−	1.97796i	0.145292	−	0.0838844i
$557$	−3.41227	−	5.91023i	−0.144583	−	0.250424i	0.784635	−	0.619959i	$-0.212851\pi$
$557$	−3.41227	−	5.91023i	−0.144583	−	0.250424i	−0.929217	+	0.369534i	$0.879517\pi$
$558$		0			0
$559$		−	17.3914i		−	0.735577i
$560$	1.55753	+	20.1009i	0.0658177	+	0.849418i
$561$		0			0
$562$	−28.9815	−	16.7324i	−1.22251	−	0.705816i
$563$	7.52241	−	4.34307i	0.317032	−	0.183038i	−0.333037	−	0.942914i	$-0.608073\pi$
$563$	7.52241	−	4.34307i	0.317032	−	0.183038i	0.650069	+	0.759875i	$0.274740\pi$
$564$		0			0
$565$	4.32414	−	12.5208i	0.181918	−	0.526755i
$566$	12.6303			0.530892
$567$		0			0
$568$		−	6.69759i		−	0.281025i
$569$	−28.5427	−	16.4792i	−1.19657	−	0.690842i	−0.236784	−	0.971562i	$-0.576093\pi$
$569$	−28.5427	−	16.4792i	−1.19657	−	0.690842i	−0.959790	+	0.280720i	$0.909427\pi$
$570$		0			0
$571$	−4.38204	−	7.58991i	−0.183383	−	0.317628i	0.759648	−	0.650335i	$-0.225371\pi$
$571$	−4.38204	−	7.58991i	−0.183383	−	0.317628i	−0.943030	+	0.332707i	$0.892038\pi$
$572$	−0.925146	−	0.534133i	−0.0386823	−	0.0223332i
$573$		0			0
$574$	−5.99779	+	2.60585i	−0.250343	+	0.108766i
$575$	−26.7755	+	10.7608i	−1.11662	+	0.448757i
$576$		0			0
$577$	17.7788	+	30.7939i	0.740143	+	1.28197i	0.952430	+	0.304758i	$0.0985756\pi$
$577$	17.7788	+	30.7939i	0.740143	+	1.28197i	−0.212287	+	0.977207i	$0.568091\pi$
$578$	10.5984	+	18.3569i	0.440833	+	0.763546i
$579$		0			0
$580$	0.618737	+	3.19881i	0.0256916	+	0.132824i
$581$	−15.1468	+	20.4573i	−0.628394	+	0.848712i
$582$		0			0
$583$	26.5213	+	15.3121i	1.09840	+	0.634161i
$584$	21.9156	+	37.9590i	0.906876	+	1.57075i
$585$		0			0
$586$	1.65229	+	0.953952i	0.0682556	+	0.0394074i
$587$			31.4362i			1.29751i	0.760997	+	0.648755i	$0.224710\pi$
$587$			31.4362i			1.29751i	−0.760997	+	0.648755i	$0.775290\pi$
$588$		0			0
$589$	−36.0507			−1.48544
$590$	−6.51870	+	18.8753i	−0.268370	+	0.777084i
$591$		0			0
$592$	11.7257	−	6.76985i	0.481924	−	0.278239i
$593$	−11.2487	−	6.49446i	−0.461930	−	0.266695i	0.250926	−	0.968006i	$-0.419265\pi$
$593$	−11.2487	−	6.49446i	−0.461930	−	0.266695i	−0.712855	+	0.701311i	$0.752598\pi$
$594$		0			0
$595$	2.45243	−	5.12568i	0.100540	−	0.210133i
$596$			0.611922i			0.0250653i
$597$		0			0
$598$	6.44229	+	11.1584i	0.263445	+	0.456300i
$599$	−12.7099	+	7.33804i	−0.519311	+	0.299824i	−0.736653	−	0.676271i	$-0.763595\pi$
$599$	−12.7099	+	7.33804i	−0.519311	+	0.299824i	0.217342	+	0.976096i	$0.430261\pi$
$600$		0			0
$601$			9.55020i			0.389561i	0.980847	+	0.194780i	$0.0623994\pi$
$601$			9.55020i			0.389561i	−0.980847	+	0.194780i	$0.937601\pi$
$602$	−14.2759	−	32.8583i	−0.581842	−	1.33920i
$603$		0			0
$604$	1.31298	−	2.27414i	0.0534243	−	0.0925336i
$605$	7.61042	+	8.77105i	0.309408	+	0.356594i
$606$		0			0
$607$	−13.6253	+	23.5998i	−0.553035	+	0.957885i	0.445018	+	0.895522i	$0.353197\pi$
$607$	−13.6253	+	23.5998i	−0.553035	+	0.957885i	−0.998053	+	0.0623639i	$0.980136\pi$
$608$			6.05769i			0.245672i
$609$		0			0
$610$	−1.28313	−	6.63369i	−0.0519526	−	0.268590i
$611$	14.8223	+	8.55765i	0.599646	+	0.346206i
$612$		0			0
$613$	−41.9818	+	24.2382i	−1.69563	+	0.978971i	−0.745816	+	0.666152i	$0.767940\pi$
$613$	−41.9818	+	24.2382i	−1.69563	+	0.978971i	−0.949813	+	0.312819i	$0.898727\pi$
$614$	5.07501	−	8.79018i	0.204811	−	0.354743i
$615$		0			0
$616$	−18.8888	−	2.15765i	−0.761051	−	0.0869342i
$617$	−8.42587			−0.339213			−0.169606	−	0.985512i	$-0.554250\pi$
$617$	−8.42587			−0.339213			−0.169606	+	0.985512i	$0.554250\pi$
$618$		0			0
$619$	−14.9893	+	8.65410i	−0.602472	+	0.347837i	−0.770014	−	0.638028i	$-0.779751\pi$
$619$	−14.9893	+	8.65410i	−0.602472	+	0.347837i	0.167541	+	0.985865i	$0.446417\pi$
$620$	−3.87087	+	3.35865i	−0.155458	+	0.134887i
$621$		0			0
$622$	43.7382			1.75374
$623$	1.07184	−	0.465678i	0.0429422	−	0.0186570i
$624$		0			0
$625$	5.95884	+	24.2795i	0.238354	+	0.971178i
$626$	5.57453	+	9.65537i	0.222803	+	0.385906i
$627$		0			0
$628$	2.71660	−	4.70528i	0.108404	−	0.187761i
$629$	−3.81600			−0.152154
$630$		0			0
$631$	16.3945			0.652653			0.326327	−	0.945257i	$-0.394189\pi$
$631$	16.3945			0.652653			0.326327	+	0.945257i	$0.394189\pi$
$632$	2.66202	−	4.61075i	0.105889	−	0.183406i
$633$		0			0
$634$	3.91677	+	6.78404i	0.155555	+	0.269429i
$635$	−10.5959	−	3.65935i	−0.420484	−	0.145217i
$636$		0			0
$637$	−3.45899	−	11.3373i	−0.137050	−	0.449202i
$638$	17.6816			0.700023
$639$		0			0
$640$	−12.6039	−	14.5261i	−0.498214	−	0.574194i
$641$	−12.9885	+	7.49893i	−0.513016	+	0.296190i	−0.734072	−	0.679071i	$-0.762383\pi$
$641$	−12.9885	+	7.49893i	−0.513016	+	0.296190i	0.221057	+	0.975261i	$0.429049\pi$
$642$		0			0
$643$	26.5806			1.04824			0.524119	−	0.851645i	$-0.324395\pi$
$643$	26.5806			1.04824			0.524119	+	0.851645i	$0.324395\pi$
$644$	−3.21284	−	2.37882i	−0.126604	−	0.0937385i
$645$		0			0
$646$	−2.60730	+	4.51597i	−0.102583	+	0.177678i
$647$	−20.0021	+	11.5482i	−0.786364	+	0.454007i	−0.838681	−	0.544623i	$-0.816673\pi$
$647$	−20.0021	+	11.5482i	−0.786364	+	0.454007i	0.0523172	+	0.998631i	$0.483339\pi$
$648$		0			0
$649$	−14.1359	−	8.16137i	−0.554883	−	0.320362i
$650$	10.3573	−	4.16250i	0.406247	−	0.163267i
$651$		0			0
$652$		−	4.08936i		−	0.160152i
$653$	10.5784	−	18.3223i	0.413964	−	0.717007i	−0.581355	−	0.813650i	$-0.697477\pi$
$653$	10.5784	−	18.3223i	0.413964	−	0.717007i	0.995319	+	0.0966433i	$0.0308106\pi$
$654$		0			0
$655$	24.9928	+	28.8043i	0.976549	+	1.12548i
$656$	3.19441	−	5.53289i	0.124721	−	0.216023i
$657$		0			0
$658$	35.0290	+	4.00134i	1.36557	+	0.155988i
$659$			34.2144i			1.33281i	0.745592	+	0.666403i	$0.232167\pi$
$659$			34.2144i			1.33281i	−0.745592	+	0.666403i	$0.767833\pi$
$660$		0			0
$661$	0.188317	−	0.108725i	0.00732467	−	0.00422890i	−0.496333	−	0.868132i	$-0.665321\pi$
$661$	0.188317	−	0.108725i	0.00732467	−	0.00422890i	0.503658	+	0.863903i	$0.331987\pi$
$662$	17.4756	+	30.2687i	0.679209	+	1.17643i
$663$		0			0
$664$		−	28.6892i		−	1.11336i
$665$	−24.2898	+	1.88212i	−0.941920	+	0.0729853i
$666$		0			0
$667$	27.8175	+	16.0604i	1.07710	+	0.621863i
$668$	3.35234	−	1.93548i	0.129706	−	0.0748859i
$669$		0			0
$670$	13.3683	+	4.61682i	0.516462	+	0.178363i
$671$	5.52284			0.213207
$672$		0			0
$673$			29.8280i			1.14979i	0.818229	+	0.574893i	$0.194956\pi$
$673$			29.8280i			1.14979i	−0.818229	+	0.574893i	$0.805044\pi$
$674$	3.90607	+	2.25517i	0.150456	+	0.0868660i
$675$		0			0
$676$	−1.32638	−	2.29735i	−0.0510145	−	0.0883596i
$677$	−36.4734	−	21.0579i	−1.40179	−	0.809321i	−0.407209	−	0.913335i	$-0.633498\pi$
$677$	−36.4734	−	21.0579i	−1.40179	−	0.809321i	−0.994576	+	0.104014i	$0.966831\pi$
$678$		0			0
$679$	−26.5636	−	19.6679i	−1.01942	−	0.754786i
$680$	1.21621	+	6.28772i	0.0466397	+	0.241123i
$681$		0			0
$682$	13.9062	+	24.0863i	0.532497	+	0.922311i
$683$	−11.6590	−	20.1941i	−0.446121	−	0.772704i	0.552009	−	0.833838i	$-0.313862\pi$
$683$	−11.6590	−	20.1941i	−0.446121	−	0.772704i	−0.998130	+	0.0611342i	$0.980528\pi$
$684$		0			0
$685$	−21.7763	+	4.21212i	−0.832030	+	0.160937i
$686$	−15.8416	−	18.5808i	−0.604835	−	0.709418i
$687$		0			0
$688$	30.3114	+	17.5003i	1.15561	+	0.667192i
$689$	−10.7599	−	18.6367i	−0.409919	−	0.710000i
$690$		0			0
$691$	4.66296	+	2.69216i	0.177387	+	0.102415i	0.586065	−	0.810264i	$-0.300676\pi$
$691$	4.66296	+	2.69216i	0.177387	+	0.102415i	−0.408677	+	0.912679i	$0.634010\pi$
$692$			1.79688i			0.0683073i
$693$		0			0
$694$	−17.6652			−0.670563
$695$	−31.9369	−	11.0296i	−1.21144	−	0.418376i
$696$		0			0
$697$	−1.55938	+	0.900306i	−0.0590656	+	0.0341015i
$698$	17.4216	+	10.0584i	0.659418	+	0.380715i
$699$		0			0
$700$	−2.43272	+	2.46504i	−0.0919482	+	0.0931699i
$701$		−	21.9593i		−	0.829391i	−0.909960	−	0.414696i	$-0.863888\pi$
$701$		−	21.9593i		−	0.829391i	0.909960	−	0.414696i	$-0.136112\pi$
$702$		0			0
$703$	8.18066	+	14.1693i	0.308540	+	0.534406i
$704$	18.2708	−	10.5487i	0.688607	−	0.397568i
$705$		0			0
$706$			48.6779i			1.83202i
$707$	−2.84935	+	24.9442i	−0.107161	+	0.938121i
$708$		0			0
$709$	13.6505	−	23.6434i	0.512657	−	0.887948i	−0.487236	−	0.873271i	$-0.661995\pi$
$709$	13.6505	−	23.6434i	0.512657	−	0.887948i	0.999892	−	0.0146769i	$-0.00467198\pi$
$710$	−5.00121	+	4.33943i	−0.187692	+	0.162856i
$711$		0			0
$712$	−0.658569	+	1.14067i	−0.0246809	+	0.0427486i
$713$			50.5248i			1.89217i
$714$		0			0
$715$	1.73273	+	8.95808i	0.0648006	+	0.335013i
$716$	2.65449	+	1.53257i	0.0992031	+	0.0572749i
$717$		0			0
$718$	−10.1663	+	5.86954i	−0.379405	+	0.219049i
$719$	16.3001	−	28.2327i	0.607893	−	1.05290i	−0.383694	−	0.923460i	$-0.625348\pi$
$719$	16.3001	−	28.2327i	0.607893	−	1.05290i	0.991587	−	0.129441i	$-0.0413183\pi$
$720$		0			0
$721$	19.6135	−	8.52145i	0.730446	−	0.317355i
$722$	−2.69185			−0.100180
$723$		0			0
$724$	−3.93367	+	2.27111i	−0.146194	+	0.0844050i
$725$	17.1727	−	21.8970i	0.637778	−	0.813233i
$726$		0			0
$727$	4.18319			0.155146			0.0775729	−	0.996987i	$-0.475283\pi$
$727$	4.18319			0.155146			0.0775729	+	0.996987i	$0.475283\pi$
$728$	10.7369	+	7.94970i	0.397936	+	0.294636i
$729$		0			0
$730$	14.1454	−	40.9588i	0.523543	−	1.51595i
$731$	−4.93224	−	8.54290i	−0.182426	−	0.315970i
$732$		0			0
$733$	0.324735	−	0.562457i	0.0119944	−	0.0207748i	−0.859966	−	0.510351i	$-0.829515\pi$
$733$	0.324735	−	0.562457i	0.0119944	−	0.0207748i	0.871960	+	0.489577i	$0.162849\pi$
$734$	−19.0560			−0.703370
$735$		0			0
$736$	8.48979			0.312938
$737$	−5.78023	+	10.0116i	−0.212917	+	0.368784i
$738$		0			0
$739$	−11.5360	−	19.9810i	−0.424360	−	0.735012i	0.572001	−	0.820253i	$-0.306167\pi$
$739$	−11.5360	−	19.9810i	−0.424360	−	0.735012i	−0.996360	+	0.0852408i	$0.972834\pi$
$740$	2.19846	+	0.759251i	0.0808170	+	0.0279106i
$741$		0			0
$742$	−35.6272	−	26.3787i	−1.30792	−	0.968393i
$743$	18.5599			0.680897			0.340448	−	0.940263i	$-0.389421\pi$
$743$	18.5599			0.680897			0.340448	+	0.940263i	$0.389421\pi$
$744$		0			0
$745$	3.94761	−	3.42524i	0.144629	−	0.125491i
$746$	−24.1159	+	13.9233i	−0.882945	+	0.509768i
$747$		0			0
$748$	−0.605926			−0.0221549
$749$	13.1674	−	5.72081i	0.481126	−	0.209034i
$750$		0			0
$751$	−11.0882	+	19.2053i	−0.404613	+	0.700811i	−0.994276	−	0.106838i	$-0.965927\pi$
$751$	−11.0882	+	19.2053i	−0.404613	+	0.700811i	0.589663	+	0.807649i	$0.299261\pi$
$752$	−29.8302	+	17.2225i	−1.08780	+	0.628040i
$753$		0			0
$754$	−10.7604	−	6.21250i	−0.391869	−	0.226246i
$755$	−22.0203	+	4.25931i	−0.801400	+	0.155012i
$756$		0			0
$757$		−	22.4056i		−	0.814347i	−0.913351	−	0.407173i	$-0.866515\pi$
$757$		−	22.4056i		−	0.814347i	0.913351	−	0.407173i	$-0.133485\pi$
$758$	−0.00883196	+	0.0152974i	−0.000320791	+	0.000555627i
$759$		0			0
$760$	20.7399	−	17.9955i	0.752315	−	0.652764i
$761$	7.01527	−	12.1508i	0.254303	−	0.440466i	−0.710403	−	0.703795i	$-0.751487\pi$
$761$	7.01527	−	12.1508i	0.254303	−	0.440466i	0.964706	+	0.263329i	$0.0848205\pi$
$762$		0			0
$763$	−5.21392	+	45.6444i	−0.188757	+	1.65244i
$764$		−	4.94800i		−	0.179012i
$765$		0			0
$766$	−19.6642	+	11.3531i	−0.710496	+	0.410205i
$767$	5.73505	+	9.93339i	0.207081	+	0.358674i
$768$		0			0
$769$			40.4788i			1.45970i	0.683606	+	0.729851i	$0.260411\pi$
$769$			40.4788i			1.45970i	−0.683606	+	0.729851i	$0.739589\pi$
$770$	10.6271	+	15.5026i	0.382973	+	0.558674i
$771$		0			0
$772$	5.28777	+	3.05289i	0.190311	+	0.109876i
$773$	27.1171	−	15.6561i	0.975335	−	0.563110i	0.0744762	−	0.997223i	$-0.476272\pi$
$773$	27.1171	−	15.6561i	0.975335	−	0.563110i	0.900858	+	0.434113i	$0.142938\pi$
$774$		0			0
$775$	43.3344	+	6.17150i	1.55662	+	0.221687i
$776$	37.2526			1.33729
$777$		0			0
$778$			33.9604i			1.21754i
$779$	6.68592	+	3.86012i	0.239548	+	0.138303i
$780$		0			0
$781$	−2.70614	−	4.68717i	−0.0968333	−	0.167720i
$782$	6.32909	+	3.65410i	0.226328	+	0.130670i
$783$		0			0
$784$	23.2405	+	5.37967i	0.830017	+	0.192131i
$785$	−45.5607	+	8.81267i	−1.62613	+	0.314538i
$786$		0			0
$787$	−12.1781	−	21.0932i	−0.434104	−	0.751890i	0.563118	−	0.826376i	$-0.309602\pi$
$787$	−12.1781	−	21.0932i	−0.434104	−	0.751890i	−0.997222	+	0.0744866i	$0.976268\pi$
$788$	−0.110340	−	0.191115i	−0.00393071	−	0.00680819i
$789$		0			0
$790$	−5.16768	+	0.999568i	−0.183858	+	0.0355630i
$791$	−12.5965	−	9.32656i	−0.447880	−	0.331614i
$792$		0			0
$793$	−3.36099	−	1.94047i	−0.119352	−	0.0689081i
$794$	−10.0515	−	17.4097i	−0.356715	−	0.617848i
$795$		0			0
$796$	−0.193721	−	0.111845i	−0.00686626	−	0.00396424i
$797$		−	42.7862i		−	1.51557i	−0.652507	−	0.757783i	$-0.726283\pi$
$797$		−	42.7862i		−	1.51557i	0.652507	−	0.757783i	$-0.273717\pi$
$798$		0			0
$799$	9.70789			0.343441
$800$	1.03701	−	7.28159i	0.0366639	−	0.257443i
$801$		0			0
$802$	14.9824	−	8.65012i	0.529049	−	0.305446i
$803$	30.6744	+	17.7099i	1.08248	+	0.624969i
$804$		0			0
$805$	2.63777	+	34.0420i	0.0929691	+	1.19982i
$806$		−	19.5440i		−	0.688407i
$807$		0			0
$808$	−14.1485	−	24.5058i	−0.497741	−	0.862113i
$809$	33.0092	−	19.0579i	1.16054	−	0.670038i	0.209107	−	0.977893i	$-0.432944\pi$
$809$	33.0092	−	19.0579i	1.16054	−	0.670038i	0.951433	+	0.307854i	$0.0996110\pi$
$810$		0			0
$811$		−	43.0019i		−	1.51000i	−0.655725	−	0.755000i	$-0.727637\pi$
$811$		−	43.0019i		−	1.51000i	0.655725	−	0.755000i	$-0.272363\pi$
$812$	3.83014	+	0.437513i	0.134411	+	0.0153537i
$813$		0			0
$814$	6.31122	−	10.9314i	0.221208	−	0.383144i
$815$	−26.3811	+	22.8902i	−0.924090	+	0.801810i
$816$		0			0
$817$	−21.1473	+	36.6282i	−0.739850	+	1.28146i
$818$		−	43.3406i		−	1.51537i
$819$		0			0
$820$	1.07751	−	0.208420i	0.0376284	−	0.00727833i
$821$	−6.74916	−	3.89663i	−0.235547	−	0.135993i	0.377581	−	0.925976i	$-0.376756\pi$
$821$	−6.74916	−	3.89663i	−0.235547	−	0.135993i	−0.613129	+	0.789983i	$0.710089\pi$
$822$		0			0
$823$	19.9581	−	11.5228i	0.695697	−	0.401661i	−0.110046	−	0.993927i	$-0.535100\pi$
$823$	19.9581	−	11.5228i	0.695697	−	0.401661i	0.805743	+	0.592265i	$0.201766\pi$
$824$	−12.0512	+	20.8732i	−0.419822	+	0.727153i
$825$		0			0
$826$	18.9894	+	14.0599i	0.660726	+	0.489207i
$827$	0.555383			0.0193126			0.00965628	−	0.999953i	$-0.496926\pi$
$827$	0.555383			0.0193126			0.00965628	+	0.999953i	$0.496926\pi$
$828$		0			0
$829$	20.8718	−	12.0504i	0.724909	−	0.418526i	−0.0916480	−	0.995791i	$-0.529213\pi$
$829$	20.8718	−	12.0504i	0.724909	−	0.418526i	0.816557	+	0.577265i	$0.195880\pi$
$830$	−21.4227	+	18.5880i	−0.743594	+	0.645198i
$831$		0			0
$832$	−14.8252			−0.513972
$833$	−4.91441	−	4.58808i	−0.170274	−	0.158968i
$834$		0			0
$835$	−31.2509	−	10.7927i	−1.08148	−	0.373496i
$836$	1.29897	+	2.24989i	0.0449259	+	0.0778140i
$837$		0			0
$838$	−6.15112	+	10.6541i	−0.212487	+	0.368038i
$839$	45.5904			1.57395			0.786977	−	0.616982i	$-0.211645\pi$
$839$	45.5904			1.57395			0.786977	+	0.616982i	$0.211645\pi$
$840$		0			0
$841$	−1.97516			−0.0681090
$842$	−8.61714	+	14.9253i	−0.296966	+	0.514361i
$843$		0			0
$844$	2.11972	+	3.67146i	0.0729636	+	0.126377i
$845$	−7.39618	+	21.4161i	−0.254436	+	0.736737i
$846$		0			0
$847$	12.6021	−	5.47519i	0.433012	−	0.188130i
$848$	43.3090			1.48724
$849$		0			0
$850$	3.90716	−	4.98204i	0.134015	−	0.170883i
$851$	19.8582	−	11.4651i	0.680730	−	0.393019i
$852$		0			0
$853$	−0.440934			−0.0150973			−0.00754864	−	0.999972i	$-0.502403\pi$
$853$	−0.440934			−0.0150973			−0.00754864	+	0.999972i	$0.502403\pi$
$854$	−7.94292	−	0.907313i	−0.271801	−	0.0310476i
$855$		0			0
$856$	−8.09045	+	14.0131i	−0.276526	+	0.478957i
$857$	7.23358	−	4.17631i	0.247094	−	0.142660i	−0.371339	−	0.928497i	$-0.621101\pi$
$857$	7.23358	−	4.17631i	0.247094	−	0.142660i	0.618433	+	0.785838i	$0.287768\pi$
$858$		0			0
$859$	2.09709	+	1.21076i	0.0715518	+	0.0413104i	0.535349	−	0.844631i	$-0.320180\pi$
$859$	2.09709	+	1.21076i	0.0715518	+	0.0413104i	−0.463797	+	0.885941i	$0.653513\pi$
$860$	1.14181	+	5.90305i	0.0389353	+	0.201292i
$861$		0			0
$862$			7.88967i			0.268723i
$863$	7.63275	−	13.2203i	0.259822	−	0.450025i	−0.706372	−	0.707841i	$-0.749669\pi$
$863$	7.63275	−	13.2203i	0.259822	−	0.450025i	0.966194	+	0.257816i	$0.0830028\pi$
$864$		0			0
$865$	11.5920	−	10.0581i	0.394139	−	0.341985i
$866$	−12.8107	+	22.1887i	−0.435324	+	0.754003i
$867$		0			0
$868$	2.41633	+	5.56158i	0.0820155	+	0.188772i
$869$		−	4.30233i		−	0.145946i
$870$		0			0
$871$	7.03525	−	4.06180i	0.238380	−	0.137629i
$872$	−25.8897	−	44.8424i	−0.876738	−	1.51855i
$873$		0			0
$874$		−	31.3344i		−	1.05990i
$875$	29.5196	+	1.89578i	0.997944	+	0.0640891i
$876$		0			0
$877$	−29.9865	−	17.3127i	−1.01257	−	0.584609i	−0.100629	−	0.994924i	$-0.532086\pi$
$877$	−29.9865	−	17.3127i	−1.01257	−	0.584609i	−0.911944	+	0.410315i	$0.865419\pi$
$878$	31.7967	−	18.3578i	1.07309	−	0.619547i
$879$		0			0
$880$	−17.3566	−	5.99420i	−0.585091	−	0.202065i
$881$	−18.2805			−0.615887			−0.307943	−	0.951405i	$-0.599641\pi$
$881$	−18.2805			−0.615887			−0.307943	+	0.951405i	$0.599641\pi$
$882$		0			0
$883$		−	15.7048i		−	0.528508i	−0.964453	−	0.264254i	$-0.914874\pi$
$883$		−	15.7048i		−	0.528508i	0.964453	−	0.264254i	$-0.0851257\pi$
$884$	0.368743	+	0.212894i	0.0124022	+	0.00716040i
$885$		0			0
$886$	−15.4456	−	26.7526i	−0.518906	−	0.898772i
$887$	27.4423	+	15.8438i	0.921422	+	0.531983i	0.884088	−	0.467320i	$-0.154780\pi$
$887$	27.4423	+	15.8438i	0.921422	+	0.531983i	0.0373336	+	0.999303i	$0.488114\pi$
$888$		0			0
$889$	−7.89270	+	10.6599i	−0.264713	+	0.357522i
$890$	1.27845	−	0.247287i	0.0428539	−	0.00828909i
$891$		0			0
$892$	−1.30937	−	2.26789i	−0.0438409	−	0.0759346i
$893$	−20.8116	−	36.0467i	−0.696433	−	1.20626i
$894$		0			0
$895$	−4.97168	−	25.7032i	−0.166185	−	0.859162i
$896$	−20.8708	+	9.06769i	−0.697244	+	0.302930i
$897$		0			0
$898$	−18.0250	−	10.4067i	−0.601501	−	0.347277i
$899$	−24.3613	−	42.1950i	−0.812494	−	1.40728i
$900$		0			0
$901$	−10.5708	−	6.10306i	−0.352165	−	0.203322i
$902$		−	5.95601i		−	0.198314i
$903$		0			0
$904$	17.6652			0.587537
$905$	36.6701	+	12.6642i	1.21895	+	0.420973i
$906$		0			0
$907$	2.78222	−	1.60632i	0.0923822	−	0.0533369i	−0.453097	−	0.891461i	$-0.649681\pi$
$907$	2.78222	−	1.60632i	0.0923822	−	0.0533369i	0.545479	+	0.838124i	$0.316348\pi$
$908$	1.06069	+	0.612391i	0.0352003	+	0.0203229i
$909$		0			0
$910$	−1.02034	−	13.1681i	−0.0338240	−	0.436519i
$911$			43.0454i			1.42616i	0.701084	+	0.713079i	$0.252700\pi$
$911$			43.0454i			1.42616i	−0.701084	+	0.713079i	$0.747300\pi$
$912$		0			0
$913$	−11.5918	−	20.0775i	−0.383632	−	0.664470i
$914$	30.5024	−	17.6106i	1.00893	−	0.582506i
$915$		0			0
$916$			3.73224i			0.123317i
$917$	41.3855	−	17.9807i	1.36667	−	0.593773i
$918$		0			0
$919$	−17.9287	+	31.0535i	−0.591414	+	1.02436i	0.402628	+	0.915364i	$0.368097\pi$
$919$	−17.9287	+	31.0535i	−0.591414	+	1.02436i	−0.994042	+	0.108996i	$0.965237\pi$
$920$	−25.2205	−	29.0668i	−0.831495	−	0.958303i
$921$		0			0
$922$	21.8691	−	37.8784i	0.720221	−	1.24746i
$923$			3.80324i			0.125185i
$924$		0			0
$925$	−7.40786	−	18.4325i	−0.243569	−	0.606058i
$926$	10.2926	+	5.94245i	0.338236	+	0.195281i
$927$		0			0
$928$	−7.09012	+	4.09348i	−0.232745	+	0.134375i
$929$	−3.44568	+	5.96810i	−0.113049	+	0.195807i	−0.916998	−	0.398891i	$-0.869395\pi$
$929$	−3.44568	+	5.96810i	−0.113049	+	0.195807i	0.803949	+	0.594698i	$0.202728\pi$
$930$		0			0
$931$	−6.50078	+	28.0837i	−0.213054	+	0.920406i
$932$	−5.83947			−0.191278
$933$		0			0
$934$	−37.0725	+	21.4038i	−1.21305	+	0.700354i
$935$	3.39168	+	3.90893i	0.110920	+	0.127836i
$936$		0			0
$937$	−13.8533			−0.452566			−0.226283	−	0.974062i	$-0.572657\pi$
$937$	−13.8533			−0.452566			−0.226283	+	0.974062i	$0.572657\pi$
$938$	9.95784	−	13.4491i	0.325135	−	0.439129i
$939$		0			0
$940$	−5.59288	−	1.93153i	−0.182420	−	0.0629997i
$941$	15.2003	+	26.3277i	0.495516	+	0.858259i	0.999987	−	0.00517013i	$-0.00164571\pi$
$941$	15.2003	+	26.3277i	0.495516	+	0.858259i	−0.504471	+	0.863429i	$0.668312\pi$
$942$		0			0
$943$	5.40992	−	9.37026i	0.176171	−	0.305138i
$944$	−23.0838			−0.751315
$945$		0			0
$946$	32.6295			1.06088
$947$	−13.3878	+	23.1883i	−0.435044	+	0.753519i	−0.997299	−	0.0734452i	$-0.976601\pi$
$947$	−13.3878	+	23.1883i	−0.435044	+	0.753519i	0.562255	+	0.826964i	$0.309934\pi$
$948$		0			0
$949$	−12.4449	−	21.5551i	−0.403977	−	0.699709i
$950$	−26.8751	−	3.82743i	−0.871943	−	0.124178i
$951$		0			0
$952$	7.52867	+	0.859993i	0.244006	+	0.0278725i
$953$	−23.9462			−0.775693			−0.387846	−	0.921724i	$-0.626781\pi$
$953$	−23.9462			−0.775693			−0.387846	+	0.921724i	$0.626781\pi$
$954$		0			0
$955$	−31.9204	+	27.6965i	−1.03292	+	0.896238i
$956$	1.45045	−	0.837419i	0.0469110	−	0.0270841i
$957$		0			0
$958$	5.49255			0.177456
$959$	−2.97842	+	26.0741i	−0.0961783	+	0.841977i
$960$		0			0
$961$	22.8192	−	39.5240i	0.736104	−	1.27497i
$962$	−7.68154	+	4.43494i	−0.247663	+	0.142988i
$963$		0			0
$964$	−1.62538	−	0.938412i	−0.0523499	−	0.0302242i
$965$	−9.90362	−	51.2008i	−0.318809	−	1.64821i
$966$		0			0
$967$		−	12.8640i		−	0.413678i	−0.978375	−	0.206839i	$-0.933682\pi$
$967$		−	12.8640i		−	0.413678i	0.978375	−	0.206839i	$-0.0663177\pi$
$968$	−7.74309	+	13.4114i	−0.248872	+	0.431060i
$969$		0			0
$970$	−24.1363	−	27.8172i	−0.774969	−	0.893156i
$971$	−17.4878	+	30.2897i	−0.561209	+	0.972043i	0.436182	+	0.899859i	$0.356330\pi$
$971$	−17.4878	+	30.2897i	−0.561209	+	0.972043i	−0.997391	+	0.0721846i	$0.977003\pi$
$972$		0			0
$973$	−23.7893	+	32.1299i	−0.762650	+	1.03004i
$974$			27.2031i			0.871642i
$975$		0			0
$976$	6.76408	−	3.90524i	0.216513	−	0.125004i
$977$	4.41664	+	7.64984i	0.141301	+	0.244740i	0.927987	−	0.372613i	$-0.121538\pi$
$977$	4.41664	+	7.64984i	0.141301	+	0.244740i	−0.786686	+	0.617353i	$0.788205\pi$
$978$		0			0
$979$			1.06437i			0.0340174i
$980$	1.91840	+	3.62107i	0.0612811	+	0.115671i
$981$		0			0
$982$	−38.4066	−	22.1741i	−1.22560	−	0.707603i
$983$	44.4437	−	25.6596i	1.41753	−	0.818414i	0.421453	−	0.906850i	$-0.361520\pi$
$983$	44.4437	−	25.6596i	1.41753	−	0.818414i	0.996082	+	0.0884361i	$0.0281869\pi$
$984$		0			0
$985$	−0.615283	+	1.78159i	−0.0196046	+	0.0567663i
$986$	−7.04753			−0.224439
$987$		0			0
$988$		−	1.82559i		−	0.0580798i
$989$	51.3341	+	29.6377i	1.63233	+	0.942425i
$990$		0			0
$991$	9.04272	+	15.6625i	0.287252	+	0.497534i	0.973153	−	0.230160i	$-0.0739251\pi$
$991$	9.04272	+	15.6625i	0.287252	+	0.497534i	−0.685901	+	0.727695i	$0.740592\pi$
$992$	−11.1524	−	6.43887i	−0.354091	−	0.204434i
$993$		0			0
$994$	3.12193	+	7.18564i	0.0990216	+	0.227915i
$995$	0.362826	+	1.87578i	0.0115024	+	0.0594662i
$996$		0			0
$997$	−16.6125	−	28.7737i	−0.526124	−	0.911273i	−0.999537	−	0.0304328i	$-0.990311\pi$
$997$	−16.6125	−	28.7737i	−0.526124	−	0.911273i	0.473413	−	0.880841i	$-0.343022\pi$
$998$	17.6905	+	30.6408i	0.559982	+	0.969917i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bb.b.89.8	yes	24
3.2	odd	2	inner	315.2.bb.b.89.5	✓	24
5.2	odd	4		1575.2.bk.i.26.7		24
5.3	odd	4		1575.2.bk.i.26.5		24
5.4	even	2	inner	315.2.bb.b.89.6	yes	24
7.2	even	3		2205.2.g.b.2204.10		24
7.3	odd	6	inner	315.2.bb.b.269.7	yes	24
7.5	odd	6		2205.2.g.b.2204.9		24
15.2	even	4		1575.2.bk.i.26.6		24
15.8	even	4		1575.2.bk.i.26.8		24
15.14	odd	2	inner	315.2.bb.b.89.7	yes	24
21.2	odd	6		2205.2.g.b.2204.16		24
21.5	even	6		2205.2.g.b.2204.15		24
21.17	even	6	inner	315.2.bb.b.269.6	yes	24
35.3	even	12		1575.2.bk.i.1151.8		24
35.9	even	6		2205.2.g.b.2204.14		24
35.17	even	12		1575.2.bk.i.1151.6		24
35.19	odd	6		2205.2.g.b.2204.13		24
35.24	odd	6	inner	315.2.bb.b.269.5	yes	24
105.17	odd	12		1575.2.bk.i.1151.7		24
105.38	odd	12		1575.2.bk.i.1151.5		24
105.44	odd	6		2205.2.g.b.2204.12		24
105.59	even	6	inner	315.2.bb.b.269.8	yes	24
105.89	even	6		2205.2.g.b.2204.11		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bb.b.89.5	✓	24	3.2	odd	2	inner
315.2.bb.b.89.6	yes	24	5.4	even	2	inner
315.2.bb.b.89.7	yes	24	15.14	odd	2	inner
315.2.bb.b.89.8	yes	24	1.1	even	1	trivial
315.2.bb.b.269.5	yes	24	35.24	odd	6	inner
315.2.bb.b.269.6	yes	24	21.17	even	6	inner
315.2.bb.b.269.7	yes	24	7.3	odd	6	inner
315.2.bb.b.269.8	yes	24	105.59	even	6	inner
1575.2.bk.i.26.5		24	5.3	odd	4
1575.2.bk.i.26.6		24	15.2	even	4
1575.2.bk.i.26.7		24	5.2	odd	4
1575.2.bk.i.26.8		24	15.8	even	4
1575.2.bk.i.1151.5		24	105.38	odd	12
1575.2.bk.i.1151.6		24	35.17	even	12
1575.2.bk.i.1151.7		24	105.17	odd	12
1575.2.bk.i.1151.8		24	35.3	even	12
2205.2.g.b.2204.9		24	7.5	odd	6
2205.2.g.b.2204.10		24	7.2	even	3
2205.2.g.b.2204.11		24	105.89	even	6
2205.2.g.b.2204.12		24	105.44	odd	6
2205.2.g.b.2204.13		24	35.19	odd	6
2205.2.g.b.2204.14		24	35.9	even	6
2205.2.g.b.2204.15		24	21.5	even	6
2205.2.g.b.2204.16		24	21.2	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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bb (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.659204 - 1.14177i) q^{2} +(0.130901 + 0.226727i) q^{4} +(0.729935 - 2.11357i) q^{5} +(-2.12635 - 1.57437i) q^{7} +2.98198 q^{8} +O(q^{10})\)	\(q+(0.659204 - 1.14177i) q^{2} +(0.130901 + 0.226727i) q^{4} +(0.729935 - 2.11357i) q^{5} +(-2.12635 - 1.57437i) q^{7} +2.98198 q^{8} +(-1.93205 - 2.22670i) q^{10} +(2.08688 - 1.20486i) q^{11} -1.69332 q^{13} +(-3.19927 + 1.38998i) q^{14} +(1.70393 - 2.95129i) q^{16} +(-0.831785 + 0.480231i) q^{17} +(3.56633 + 2.05902i) q^{19} +(0.574754 - 0.111173i) q^{20} -3.17699i q^{22} +(2.88570 - 4.99818i) q^{23} +(-3.93439 - 3.08554i) q^{25} +(-1.11625 + 1.93339i) q^{26} +(0.0786111 - 0.688188i) q^{28} +5.56553i q^{29} +(-7.58148 + 4.37717i) q^{31} +(0.735506 + 1.27393i) q^{32} +1.26628i q^{34} +(-4.87964 + 3.34501i) q^{35} +(3.44079 + 1.98654i) q^{37} +(4.70187 - 2.71463i) q^{38} +(2.17665 - 6.30263i) q^{40} +1.87474 q^{41} +10.2706i q^{43} +(0.546349 + 0.315435i) q^{44} +(-3.80453 - 6.58963i) q^{46} +(-8.75337 - 5.05376i) q^{47} +(2.04272 + 6.69532i) q^{49} +(-6.11656 + 2.45818i) q^{50} +(-0.221658 - 0.383923i) q^{52} +(6.35430 + 11.0060i) q^{53} +(-1.02327 - 5.29024i) q^{55} +(-6.34072 - 4.69473i) q^{56} +(6.35458 + 3.66882i) q^{58} +(-3.38686 - 5.86621i) q^{59} +(1.98485 + 1.14595i) q^{61} +11.5418i q^{62} +8.75510 q^{64} +(-1.23602 + 3.57896i) q^{65} +(-4.15470 + 2.39872i) q^{67} +(-0.217763 - 0.125726i) q^{68} +(0.602567 + 7.77649i) q^{70} -2.24602i q^{71} +(7.34937 + 12.7295i) q^{73} +(4.53637 - 2.61907i) q^{74} +1.07811i q^{76} +(-6.33432 - 0.723564i) q^{77} +(0.892703 - 1.54621i) q^{79} +(-4.99401 - 5.75563i) q^{80} +(1.23583 - 2.14052i) q^{82} -9.62086i q^{83} +(0.407855 + 2.10858i) q^{85} +(11.7267 + 6.77039i) q^{86} +(6.22302 - 3.59286i) q^{88} +(-0.220850 + 0.382523i) q^{89} +(3.60060 + 2.66592i) q^{91} +1.51096 q^{92} +(-11.5405 + 6.66292i) q^{94} +(6.95508 - 6.03475i) q^{95} +12.4926 q^{97} +(8.99111 + 2.08125i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{4}+O(q^{10}) \) 24 * q - 24 * q^4	\( 24 q - 24 q^{4} - 12 q^{10} - 36 q^{19} + 12 q^{25} - 60 q^{31} + 96 q^{40} - 24 q^{46} + 36 q^{49} + 48 q^{61} + 48 q^{64} - 48 q^{70} - 60 q^{79} - 72 q^{85} + 60 q^{91} + 48 q^{94}+O(q^{100}) \) 24 * q - 24 * q^4 - 12 * q^10 - 36 * q^19 + 12 * q^25 - 60 * q^31 + 96 * q^40 - 24 * q^46 + 36 * q^49 + 48 * q^61 + 48 * q^64 - 48 * q^70 - 60 * q^79 - 72 * q^85 + 60 * q^91 + 48 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more