LMFDB - Embedded newform 75.4.e.c.32.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,4,Mod(32,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.32"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	75.e (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,6,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$4.42514325043$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.28356903014400.8
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 209x^{4} + 1600 $ x^8 + 209*x^4 + 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			32.3
Root			$1.18766 + 1.18766i$ of defining polynomial
Character	$\chi$	$=$	75.32
Dual form			75.4.e.c.68.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.18766 - 1.18766i) q^{2} +(-5.11173 - 0.932827i) q^{3} +5.17891i q^{4} +(-7.17891 + 4.96314i) q^{6} +(13.3578 + 13.3578i) q^{7} +(15.6521 + 15.6521i) q^{8} +(25.2597 + 9.53673i) q^{9} +28.7164i q^{11} +(4.83102 - 26.4732i) q^{12} +(-14.1789 + 14.1789i) q^{13} +31.7292 q^{14} -4.25236 q^{16} +(18.5587 - 18.5587i) q^{17} +(41.3264 - 18.6736i) q^{18} +49.0735i q^{19} +(-55.8211 - 80.7421i) q^{21} +(34.1055 + 34.1055i) q^{22} +(-37.7738 - 37.7738i) q^{23} +(-65.4088 - 94.6102i) q^{24} +33.6796i q^{26} +(-120.225 - 72.3121i) q^{27} +(-69.1789 + 69.1789i) q^{28} -125.854 q^{29} +247.367 q^{31} +(-130.267 + 130.267i) q^{32} +(26.7874 - 146.791i) q^{33} -44.0829i q^{34} +(-49.3898 + 130.818i) q^{36} +(127.463 + 127.463i) q^{37} +(58.2828 + 58.2828i) q^{38} +(85.7053 - 59.2524i) q^{39} -390.328i q^{41} +(-162.191 - 29.5978i) q^{42} +(39.3993 - 39.3993i) q^{43} -148.720 q^{44} -89.7251 q^{46} +(124.560 - 124.560i) q^{47} +(21.7369 + 3.96671i) q^{48} +13.8625i q^{49} +(-112.179 + 77.5549i) q^{51} +(-73.4313 - 73.4313i) q^{52} +(-160.441 - 160.441i) q^{53} +(-228.669 + 56.9040i) q^{54} +418.156i q^{56} +(45.7770 - 250.850i) q^{57} +(-149.473 + 149.473i) q^{58} +729.423 q^{59} +2.00000 q^{61} +(293.789 - 293.789i) q^{62} +(210.024 + 464.804i) q^{63} +275.409i q^{64} +(-142.524 - 206.153i) q^{66} +(329.987 + 329.987i) q^{67} +(96.1136 + 96.1136i) q^{68} +(157.853 + 228.326i) q^{69} +171.760i q^{71} +(246.097 + 544.637i) q^{72} +(279.927 - 279.927i) q^{73} +302.767 q^{74} -254.147 q^{76} +(-383.589 + 383.589i) q^{77} +(31.4172 - 172.161i) q^{78} +48.0189i q^{79} +(547.102 + 481.789i) q^{81} +(-463.578 - 463.578i) q^{82} +(144.451 + 144.451i) q^{83} +(418.156 - 289.092i) q^{84} -93.5862i q^{86} +(643.334 + 117.400i) q^{87} +(-449.473 + 449.473i) q^{88} -1417.21 q^{89} -378.799 q^{91} +(195.627 - 195.627i) q^{92} +(-1264.48 - 230.751i) q^{93} -295.872i q^{94} +(787.409 - 544.375i) q^{96} +(-908.111 - 908.111i) q^{97} +(16.4640 + 16.4640i) q^{98} +(-273.861 + 725.367i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{3} - 12 q^{6} + 16 q^{7} - 132 q^{12} - 68 q^{13} + 284 q^{16} + 240 q^{18} - 492 q^{21} + 500 q^{22} - 702 q^{27} - 508 q^{28} + 616 q^{31} + 240 q^{33} - 804 q^{36} + 1156 q^{37} - 540 q^{42}+ \cdots - 1904 q^{97}+O(q^{100}) $ 8 * q + 6 * q^3 - 12 * q^6 + 16 * q^7 - 132 * q^12 - 68 * q^13 + 284 * q^16 + 240 * q^18 - 492 * q^21 + 500 * q^22 - 702 * q^27 - 508 * q^28 + 616 * q^31 + 240 * q^33 - 804 * q^36 + 1156 * q^37 - 540 * q^42 - 548 * q^43 + 736 * q^46 + 1116 * q^48 - 852 * q^51 - 224 * q^52 - 684 * q^57 - 60 * q^58 + 16 * q^61 - 1428 * q^63 + 2040 * q^66 - 404 * q^67 + 1800 * q^72 + 2512 * q^73 - 1488 * q^76 + 360 * q^78 + 288 * q^81 - 2800 * q^82 + 1680 * q^87 - 2460 * q^88 - 1304 * q^91 - 3408 * q^93 + 4164 * q^96 - 1904 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$	1.18766	−	1.18766i	0.419903	−	0.419903i	−0.465267	−	0.885170i	$-0.654042\pi$
$2$	1.18766	−	1.18766i	0.419903	−	0.419903i	0.885170	+	0.465267i	$0.154042\pi$
$3$	−5.11173	−	0.932827i	−0.983754	−	0.179523i
$3$	−5.11173	−	0.932827i	−0.983754	−	0.179523i
$4$			5.17891i			0.647364i
$5$		0			0
$5$		0			0
$6$	−7.17891	+	4.96314i	−0.488463	+	0.337699i
$7$	13.3578	+	13.3578i	0.721254	+	0.721254i	0.968861	−	0.247606i	$-0.0796440\pi$
$7$	13.3578	+	13.3578i	0.721254	+	0.721254i	−0.247606	+	0.968861i	$0.579644\pi$
$8$	15.6521	+	15.6521i	0.691732	+	0.691732i
$9$	25.2597	+	9.53673i	0.935543	+	0.353212i
$10$		0			0
$11$			28.7164i			0.787121i	0.919299	+	0.393560i	$0.128757\pi$
$11$			28.7164i			0.787121i	−0.919299	+	0.393560i	$0.871243\pi$
$12$	4.83102	−	26.4732i	0.116216	−	0.636846i
$13$	−14.1789	+	14.1789i	−0.302502	+	0.302502i	−0.841992	−	0.539490i	$-0.818617\pi$
$13$	−14.1789	+	14.1789i	−0.302502	+	0.302502i	0.539490	+	0.841992i	$0.318617\pi$
$14$	31.7292			0.605713
$15$		0			0
$16$	−4.25236			−0.0664431
$17$	18.5587	−	18.5587i	0.264773	−	0.264773i	−0.562217	−	0.826990i	$-0.690051\pi$
$17$	18.5587	−	18.5587i	0.264773	−	0.264773i	0.826990	+	0.562217i	$0.190051\pi$
$18$	41.3264	−	18.6736i	0.541152	−	0.244522i
$19$			49.0735i			0.592538i	0.955105	+	0.296269i	$0.0957425\pi$
$19$			49.0735i			0.592538i	−0.955105	+	0.296269i	$0.904258\pi$
$20$		0			0
$21$	−55.8211	−	80.7421i	−0.580055	−	0.839018i
$22$	34.1055	+	34.1055i	0.330514	+	0.330514i
$23$	−37.7738	−	37.7738i	−0.342451	−	0.342451i	0.514837	−	0.857288i	$-0.327852\pi$
$23$	−37.7738	−	37.7738i	−0.342451	−	0.342451i	−0.857288	+	0.514837i	$0.827852\pi$
$24$	−65.4088	−	94.6102i	−0.556313	−	0.804676i
$25$		0			0
$26$			33.6796i			0.254042i
$27$	−120.225	−	72.3121i	−0.856935	−	0.515425i
$28$	−69.1789	+	69.1789i	−0.466914	+	0.466914i
$29$	−125.854			−0.805882			−0.402941	−	0.915226i	$-0.632012\pi$
$29$	−125.854			−0.805882			−0.402941	+	0.915226i	$0.632012\pi$
$30$		0			0
$31$	247.367			1.43318			0.716588	−	0.697496i	$-0.245703\pi$
$31$	247.367			1.43318			0.716588	+	0.697496i	$0.245703\pi$
$32$	−130.267	+	130.267i	−0.719632	+	0.719632i
$33$	26.7874	−	146.791i	0.141306	−	0.774333i
$34$		−	44.0829i		−	0.222357i
$35$		0			0
$36$	−49.3898	+	130.818i	−0.228657	+	0.605637i
$37$	127.463	+	127.463i	0.566347	+	0.566347i	0.931103	−	0.364756i	$-0.118848\pi$
$37$	127.463	+	127.463i	0.566347	+	0.566347i	−0.364756	+	0.931103i	$0.618848\pi$
$38$	58.2828	+	58.2828i	0.248808	+	0.248808i
$39$	85.7053	−	59.2524i	0.351893	−	0.243281i
$40$		0			0
$41$		−	390.328i		−	1.48680i	−0.668845	−	0.743402i	$-0.733211\pi$
$41$		−	390.328i		−	1.48680i	0.668845	−	0.743402i	$-0.266789\pi$
$42$	−162.191	−	29.5978i	−0.595873	−	0.108739i
$43$	39.3993	−	39.3993i	0.139729	−	0.139729i	−0.633783	−	0.773511i	$-0.718499\pi$
$43$	39.3993	−	39.3993i	0.139729	−	0.139729i	0.773511	+	0.633783i	$0.218499\pi$
$44$	−148.720			−0.509553
$45$		0			0
$46$	−89.7251			−0.287592
$47$	124.560	−	124.560i	0.386575	−	0.386575i	−0.486889	−	0.873464i	$-0.661868\pi$
$47$	124.560	−	124.560i	0.386575	−	0.386575i	0.873464	+	0.486889i	$0.161868\pi$
$48$	21.7369	+	3.96671i	0.0653637	+	0.0119280i
$49$			13.8625i			0.0404155i
$50$		0			0
$51$	−112.179	+	77.5549i	−0.308004	+	0.212938i
$52$	−73.4313	−	73.4313i	−0.195829	−	0.195829i
$53$	−160.441	−	160.441i	−0.415816	−	0.415816i	0.467943	−	0.883759i	$-0.344995\pi$
$53$	−160.441	−	160.441i	−0.415816	−	0.415816i	−0.883759	+	0.467943i	$0.844995\pi$
$54$	−228.669	+	56.9040i	−0.576257	+	0.143401i
$55$		0			0
$56$			418.156i			0.997830i
$57$	45.7770	−	250.850i	0.106374	−	0.582912i
$58$	−149.473	+	149.473i	−0.338392	+	0.338392i
$59$	729.423			1.60954			0.804769	−	0.593588i	$-0.202289\pi$
$59$	729.423			1.60954			0.804769	+	0.593588i	$0.202289\pi$
$60$		0			0
$61$	2.00000			0.00419793			0.00209897	−	0.999998i	$-0.499332\pi$
$61$	2.00000			0.00419793			0.00209897	+	0.999998i	$0.499332\pi$
$62$	293.789	−	293.789i	0.601795	−	0.601795i
$63$	210.024	+	464.804i	0.420009	+	0.929520i
$64$			275.409i			0.537908i
$65$		0			0
$66$	−142.524	−	206.153i	−0.265810	−	0.384479i
$67$	329.987	+	329.987i	0.601706	+	0.601706i	0.940765	−	0.339059i	$-0.110109\pi$
$67$	329.987	+	329.987i	0.601706	+	0.601706i	−0.339059	+	0.940765i	$0.610109\pi$
$68$	96.1136	+	96.1136i	0.171404	+	0.171404i
$69$	157.853	+	228.326i	0.275410	+	0.398365i
$70$		0			0
$71$			171.760i			0.287100i	0.989643	+	0.143550i	$0.0458518\pi$
$71$			171.760i			0.287100i	−0.989643	+	0.143550i	$0.954148\pi$
$72$	246.097	+	544.637i	0.402817	+	0.891474i
$73$	279.927	−	279.927i	0.448807	−	0.448807i	−0.446151	−	0.894958i	$-0.647205\pi$
$73$	279.927	−	279.927i	0.448807	−	0.448807i	0.894958	+	0.446151i	$0.147205\pi$
$74$	302.767			0.475621
$75$		0			0
$76$	−254.147			−0.383587
$77$	−383.589	+	383.589i	−0.567714	+	0.567714i
$78$	31.4172	−	172.161i	0.0456064	−	0.249915i
$79$			48.0189i			0.0683866i	0.999415	+	0.0341933i	$0.0108862\pi$
$79$			48.0189i			0.0683866i	−0.999415	+	0.0341933i	$0.989114\pi$
$80$		0			0
$81$	547.102	+	481.789i	0.750483	+	0.660890i
$82$	−463.578	−	463.578i	−0.624313	−	0.624313i
$83$	144.451	+	144.451i	0.191031	+	0.191031i	0.796141	−	0.605111i	$-0.206871\pi$
$83$	144.451	+	144.451i	0.191031	+	0.191031i	−0.605111	+	0.796141i	$0.706871\pi$
$84$	418.156	−	289.092i	0.543150	−	0.375507i
$85$		0			0
$86$		−	93.5862i		−	0.117345i
$87$	643.334	+	117.400i	0.792789	+	0.144674i
$88$	−449.473	+	449.473i	−0.544477	+	0.544477i
$89$	−1417.21			−1.68790			−0.843952	−	0.536419i	$-0.819777\pi$
$89$	−1417.21			−1.68790			−0.843952	+	0.536419i	$0.819777\pi$
$90$		0			0
$91$	−378.799			−0.436361
$92$	195.627	−	195.627i	0.221690	−	0.221690i
$93$	−1264.48	−	230.751i	−1.40989	−	0.257288i
$94$		−	295.872i		−	0.324647i
$95$		0			0
$96$	787.409	−	544.375i	0.837131	−	0.578751i
$97$	−908.111	−	908.111i	−0.950564	−	0.950564i	0.0482702	−	0.998834i	$-0.484629\pi$
$97$	−908.111	−	908.111i	−0.950564	−	0.950564i	−0.998834	+	0.0482702i	$0.984629\pi$
$98$	16.4640	+	16.4640i	0.0169706	+	0.0169706i
$99$	−273.861	+	725.367i	−0.278020	+	0.736385i
$100$		0			0
$101$		−	337.668i		−	0.332665i	−0.986070	−	0.166333i	$-0.946807\pi$
$101$		−	337.668i		−	0.332665i	0.986070	−	0.166333i	$-0.0531926\pi$
$102$	−41.1217	+	225.340i	−0.0399182	+	0.218745i
$103$	933.505	−	933.505i	0.893019	−	0.893019i	−0.101787	−	0.994806i	$-0.532456\pi$
$103$	933.505	−	933.505i	0.893019	−	0.893019i	0.994806	+	0.101787i	$0.0324561\pi$
$104$	−443.860			−0.418500
$105$		0			0
$106$	−381.100			−0.349205
$107$	−596.188	+	596.188i	−0.538651	+	0.538651i	−0.923133	−	0.384481i	$-0.874380\pi$
$107$	−596.188	+	596.188i	−0.538651	+	0.538651i	0.384481	+	0.923133i	$0.374380\pi$
$108$	374.498	−	622.632i	0.333667	−	0.554748i
$109$		−	2074.60i		−	1.82303i	−0.411264	−	0.911516i	$-0.634913\pi$
$109$		−	2074.60i		−	1.82303i	0.411264	−	0.911516i	$-0.365087\pi$
$110$		0			0
$111$	−532.657	−	770.460i	−0.455474	−	0.658818i
$112$	−56.8022	−	56.8022i	−0.0479224	−	0.0479224i
$113$	271.193	+	271.193i	0.225767	+	0.225767i	0.810922	−	0.585154i	$-0.198966\pi$
$113$	271.193	+	271.193i	0.225767	+	0.225767i	−0.585154	+	0.810922i	$0.698966\pi$
$114$	−243.558	−	352.294i	−0.200099	−	0.289433i
$115$		0			0
$116$		−	651.788i		−	0.521698i
$117$	−493.375	+	222.934i	−0.389851	+	0.176156i
$118$	866.309	−	866.309i	0.675849	−	0.675849i
$119$	495.806			0.381937
$120$		0			0
$121$	506.367			0.380441
$122$	2.37533	−	2.37533i	0.00176272	−	0.00176272i
$123$	−364.108	+	1995.25i	−0.266915	+	1.46265i
$124$			1281.09i			0.927786i
$125$		0			0
$126$	801.469	+	302.593i	0.566671	+	0.213945i
$127$	−105.588	−	105.588i	−0.0737747	−	0.0737747i	0.669257	−	0.743031i	$-0.266613\pi$
$127$	−105.588	−	105.588i	−0.0737747	−	0.0737747i	−0.743031	+	0.669257i	$0.766613\pi$
$128$	−715.045	−	715.045i	−0.493763	−	0.493763i
$129$	−238.151	+	164.646i	−0.162543	+	0.112374i
$130$		0			0
$131$			1979.28i			1.32008i	0.751231	+	0.660039i	$0.229460\pi$
$131$			1979.28i			1.32008i	−0.751231	+	0.660039i	$0.770540\pi$
$132$	760.216	+	138.730i	0.501275	+	0.0914763i
$133$	−655.514	+	655.514i	−0.427371	+	0.427371i
$134$	783.827			0.505316
$135$		0			0
$136$	580.964			0.366304
$137$	507.451	−	507.451i	0.316456	−	0.316456i	−0.530948	−	0.847404i	$-0.678164\pi$
$137$	507.451	−	507.451i	0.316456	−	0.316456i	0.847404	+	0.530948i	$0.178164\pi$
$138$	458.651	+	83.6979i	0.282920	+	0.0516293i
$139$			68.4333i			0.0417585i	0.999782	+	0.0208793i	$0.00664656\pi$
$139$			68.4333i			0.0417585i	−0.999782	+	0.0208793i	$0.993353\pi$
$140$		0			0
$141$	−752.913	+	520.527i	−0.449693	+	0.310895i
$142$	203.993	+	203.993i	0.120554	+	0.120554i
$143$	−407.167	−	407.167i	−0.238105	−	0.238105i
$144$	−107.413	−	40.5536i	−0.0621604	−	0.0234685i
$145$		0			0
$146$		−	664.917i		−	0.376911i
$147$	12.9313	−	70.8616i	0.00725550	−	0.0397590i
$148$	−660.121	+	660.121i	−0.366632	+	0.366632i
$149$	−363.356			−0.199780			−0.0998902	−	0.994998i	$-0.531849\pi$
$149$	−363.356			−0.199780			−0.0998902	+	0.994998i	$0.531849\pi$
$150$		0			0
$151$	−2083.14			−1.12267			−0.561337	−	0.827588i	$-0.689713\pi$
$151$	−2083.14			−1.12267			−0.561337	+	0.827588i	$0.689713\pi$
$152$	−768.103	+	768.103i	−0.409878	+	0.409878i
$153$	645.774	−	291.797i	0.341227	−	0.154185i
$154$			911.149i			0.476769i
$155$		0			0
$156$	306.863	+	443.860i	0.157491	+	0.227803i
$157$	1208.52	+	1208.52i	0.614333	+	0.614333i	0.944072	−	0.329739i	$-0.106961\pi$
$157$	1208.52	+	1208.52i	0.614333	+	0.614333i	−0.329739	+	0.944072i	$0.606961\pi$
$158$	57.0303	+	57.0303i	0.0287157	+	0.0287157i
$159$	670.468	+	969.795i	0.334412	+	0.483709i
$160$		0			0
$161$		−	1009.15i		−	0.493989i
$162$	1221.98	−	77.5696i	0.592639	−	0.0376200i
$163$	−626.062	+	626.062i	−0.300840	+	0.300840i	−0.841343	−	0.540502i	$-0.818234\pi$
$163$	−626.062	+	626.062i	−0.300840	+	0.300840i	0.540502	+	0.841343i	$0.318234\pi$
$164$	2021.47			0.962502
$165$		0			0
$166$	343.119			0.160429
$167$	3009.65	−	3009.65i	1.39457	−	1.39457i	0.579848	−	0.814724i	$-0.303112\pi$
$167$	3009.65	−	3009.65i	1.39457	−	1.39457i	0.814724	−	0.579848i	$-0.196888\pi$
$168$	390.067	−	2137.50i	0.179133	−	0.981619i
$169$			1794.92i			0.816985i
$170$		0			0
$171$	−468.000	+	1239.58i	−0.209292	+	0.554345i
$172$	204.045	+	204.045i	0.0904552	+	0.0904552i
$173$	1839.23	+	1839.23i	0.808288	+	0.808288i	0.984375	−	0.176086i	$-0.0563438\pi$
$173$	1839.23	+	1839.23i	0.808288	+	0.808288i	−0.176086	+	0.984375i	$0.556344\pi$
$174$	903.497	−	624.633i	0.393643	−	0.272145i
$175$		0			0
$176$		−	122.113i		−	0.0522987i
$177$	−3728.61	−	680.425i	−1.58339	−	0.288948i
$178$	−1683.16	+	1683.16i	−0.708755	+	0.708755i
$179$	821.582			0.343061			0.171530	−	0.985179i	$-0.445129\pi$
$179$	821.582			0.343061			0.171530	+	0.985179i	$0.445129\pi$
$180$		0			0
$181$	2314.20			0.950350			0.475175	−	0.879891i	$-0.342385\pi$
$181$	2314.20			0.950350			0.475175	+	0.879891i	$0.342385\pi$
$182$	−449.885	+	449.885i	−0.183229	+	0.183229i
$183$	−10.2235	−	1.86565i	−0.00412973	−	0.000753623i
$184$		−	1182.48i		−	0.473769i
$185$		0			0
$186$	−1775.83	+	1227.72i	−0.700053	+	0.483982i
$187$	532.938	+	532.938i	0.208408	+	0.208408i
$188$	645.087	+	645.087i	0.250254	+	0.250254i
$189$	−640.007	−	2571.87i	−0.246316	−	0.989820i
$190$		0			0
$191$			931.167i			0.352758i	0.984322	+	0.176379i	$0.0564385\pi$
$191$			931.167i			0.352758i	−0.984322	+	0.176379i	$0.943562\pi$
$192$	256.909	−	1407.82i	0.0965666	−	0.529169i
$193$	2623.28	−	2623.28i	0.978382	−	0.978382i	−0.0213891	−	0.999771i	$-0.506809\pi$
$193$	2623.28	−	2623.28i	0.978382	−	0.978382i	0.999771	+	0.0213891i	$0.00680888\pi$
$194$	−2157.06			−0.798289
$195$		0			0
$196$	−71.7928			−0.0261636
$197$	−2995.05	+	2995.05i	−1.08319	+	1.08319i	−0.0869796	+	0.996210i	$0.527721\pi$
$197$	−2995.05	+	2995.05i	−1.08319	+	1.08319i	−0.996210	+	0.0869796i	$0.972279\pi$
$198$	536.238	+	1186.75i	0.192469	+	0.425952i
$199$		−	109.458i		−	0.0389912i	−0.999810	−	0.0194956i	$-0.993794\pi$
$199$		−	109.458i		−	0.0389912i	0.999810	−	0.0194956i	$-0.00620603\pi$
$200$		0			0
$201$	−1378.98	−	1994.63i	−0.483911	−	0.699951i
$202$	−401.036	−	401.036i	−0.139687	−	0.139687i
$203$	−1681.14	−	1681.14i	−0.581246	−	0.581246i
$204$	−401.650	−	580.964i	−0.137849	−	0.199390i
$205$		0			0
$206$		−	2217.38i		−	0.749962i
$207$	−593.915	−	1314.39i	−0.199420	−	0.441336i
$208$	60.2938	−	60.2938i	0.0200991	−	0.0200991i
$209$	−1409.21			−0.466399
$210$		0			0
$211$	2714.94			0.885801			0.442901	−	0.896571i	$-0.353949\pi$
$211$	2714.94			0.885801			0.442901	+	0.896571i	$0.353949\pi$
$212$	830.909	−	830.909i	0.269184	−	0.269184i
$213$	160.222	−	877.989i	0.0515409	−	0.282436i
$214$			1416.14i			0.452362i
$215$		0			0
$216$	−749.932	−	3013.61i	−0.236233	−	0.949305i
$217$	3304.29	+	3304.29i	1.03368	+	1.03368i
$218$	−2463.93	−	2463.93i	−0.765496	−	0.765496i
$219$	−1692.03	+	1169.79i	−0.522087	+	0.360945i
$220$		0			0
$221$			526.283i			0.160188i
$222$	−1547.67	−	282.429i	−0.467894	−	0.0853847i
$223$	−2830.49	+	2830.49i	−0.849971	+	0.849971i	−0.990129	−	0.140158i	$-0.955239\pi$
$223$	−2830.49	+	2830.49i	−0.849971	+	0.849971i	0.140158	+	0.990129i	$0.455239\pi$
$224$	−3480.17			−1.03808
$225$		0			0
$226$	644.173			0.189601
$227$	−1398.96	+	1398.96i	−0.409042	+	0.409042i	−0.881404	−	0.472362i	$-0.843401\pi$
$227$	−1398.96	+	1398.96i	−0.409042	+	0.409042i	0.472362	+	0.881404i	$0.343401\pi$
$228$	1299.13	+	237.075i	0.377356	+	0.0688626i
$229$		−	3930.38i		−	1.13418i	−0.823656	−	0.567089i	$-0.808069\pi$
$229$		−	3930.38i		−	1.13418i	0.823656	−	0.567089i	$-0.191931\pi$
$230$		0			0
$231$	2318.63	−	1602.98i	0.660408	−	0.456573i
$232$	−1969.89	−	1969.89i	−0.557454	−	0.557454i
$233$	−1980.71	−	1980.71i	−0.556912	−	0.556912i	0.371515	−	0.928427i	$-0.378838\pi$
$233$	−1980.71	−	1980.71i	−0.556912	−	0.556912i	−0.928427	+	0.371515i	$0.878838\pi$
$234$	−321.193	+	850.735i	−0.0897309	+	0.237668i
$235$		0			0
$236$			3777.61i			1.04196i
$237$	44.7933	−	245.460i	0.0122769	−	0.0672756i
$238$	588.851	−	588.851i	0.160376	−	0.160376i
$239$	2976.20			0.805500			0.402750	−	0.915310i	$-0.368054\pi$
$239$	2976.20			0.805500			0.402750	+	0.915310i	$0.368054\pi$
$240$		0			0
$241$	1835.45			0.490587			0.245294	−	0.969449i	$-0.421116\pi$
$241$	1835.45			0.490587			0.245294	+	0.969449i	$0.421116\pi$
$242$	601.394	−	601.394i	0.159748	−	0.159748i
$243$	−2347.21	−	2973.13i	−0.619645	−	0.784882i
$244$			10.3578i			0.00271759i
$245$		0			0
$246$	1937.25	+	2802.13i	0.502092	+	0.726248i
$247$	−695.808	−	695.808i	−0.179244	−	0.179244i
$248$	3871.82	+	3871.82i	0.991374	+	0.991374i
$249$	−603.648	−	873.143i	−0.153633	−	0.222222i
$250$		0			0
$251$		−	1542.14i		−	0.387805i	−0.981021	−	0.193902i	$-0.937885\pi$
$251$		−	1542.14i		−	0.387805i	0.981021	−	0.193902i	$-0.0621145\pi$
$252$	−2407.18	+	1087.70i	−0.601738	+	0.271898i
$253$	1084.73	−	1084.73i	0.269550	−	0.269550i
$254$	−250.805			−0.0619564
$255$		0			0
$256$	−3901.74			−0.952572
$257$	−428.853	+	428.853i	−0.104090	+	0.104090i	−0.757234	−	0.653144i	$-0.773450\pi$
$257$	−428.853	+	428.853i	−0.104090	+	0.104090i	0.653144	+	0.757234i	$0.273450\pi$
$258$	−87.2997	+	478.388i	−0.0210660	+	0.115438i
$259$			3405.26i			0.816960i
$260$		0			0
$261$	−3179.04	−	1200.24i	−0.753937	−	0.284647i
$262$	2350.72	+	2350.72i	0.554304	+	0.554304i
$263$	5256.99	+	5256.99i	1.23255	+	1.23255i	0.962982	+	0.269565i	$0.0868798\pi$
$263$	5256.99	+	5256.99i	1.23255	+	1.23255i	0.269565	+	0.962982i	$0.413120\pi$
$264$	2716.87	−	1878.31i	0.633377	−	0.437885i
$265$		0			0
$266$			1557.06i			0.358908i
$267$	7244.38	+	1322.01i	1.66048	+	0.303017i
$268$	−1708.97	+	1708.97i	−0.389523	+	0.389523i
$269$	1930.34			0.437528			0.218764	−	0.975778i	$-0.429797\pi$
$269$	1930.34			0.437528			0.218764	+	0.975778i	$0.429797\pi$
$270$		0			0
$271$	−3261.67			−0.731116			−0.365558	−	0.930789i	$-0.619122\pi$
$271$	−3261.67			−0.731116			−0.365558	+	0.930789i	$0.619122\pi$
$272$	−78.9180	+	78.9180i	−0.0175923	+	0.0175923i
$273$	1936.32	+	353.353i	0.429272	+	0.0783367i
$274$		−	1205.36i		−	0.265762i
$275$		0			0
$276$	−1182.48	+	817.507i	−0.257887	+	0.178290i
$277$	−4865.57	−	4865.57i	−1.05539	−	1.05539i	−0.998373	−	0.0570194i	$-0.981840\pi$
$277$	−4865.57	−	4865.57i	−1.05539	−	1.05539i	−0.0570194	−	0.998373i	$-0.518160\pi$
$278$	81.2758	+	81.2758i	0.0175345	+	0.0175345i
$279$	6248.41	+	2359.07i	1.34080	+	0.506215i
$280$		0			0
$281$		−	3981.96i		−	0.845351i	−0.906281	−	0.422676i	$-0.861091\pi$
$281$		−	3981.96i		−	0.845351i	0.906281	−	0.422676i	$-0.138909\pi$
$282$	−275.997	+	1512.42i	−0.0582815	+	0.319373i
$283$	4092.66	−	4092.66i	0.859658	−	0.859658i	−0.131640	−	0.991298i	$-0.542024\pi$
$283$	4092.66	−	4092.66i	0.859658	−	0.859658i	0.991298	+	0.131640i	$0.0420242\pi$
$284$	−889.527			−0.185858
$285$		0			0
$286$	−967.156			−0.199962
$287$	5213.93	−	5213.93i	1.07236	−	1.07236i
$288$	−4532.83	+	2048.19i	−0.927429	+	0.419064i
$289$			4224.15i			0.859791i
$290$		0			0
$291$	3794.91	+	5489.13i	0.764473	+	1.10577i
$292$	1449.71	+	1449.71i	0.290541	+	0.290541i
$293$	−4515.35	−	4515.35i	−0.900305	−	0.900305i	0.0951570	−	0.995462i	$-0.469665\pi$
$293$	−4515.35	−	4515.35i	−0.900305	−	0.900305i	−0.995462	+	0.0951570i	$0.969665\pi$
$294$	−68.8017	−	99.5179i	−0.0136483	−	0.0197415i
$295$		0			0
$296$			3990.14i			0.783521i
$297$	2076.54	−	3452.42i	0.405701	−	0.674511i
$298$	−431.545	+	431.545i	−0.0838883	+	0.0838883i
$299$	1071.18			0.207184
$300$		0			0
$301$	1052.58			0.201560
$302$	−2474.07	+	2474.07i	−0.471413	+	0.471413i
$303$	−314.986	+	1726.07i	−0.0597210	+	0.327261i
$304$		−	208.678i		−	0.0393701i
$305$		0			0
$306$	420.406	−	1113.52i	0.0785393	−	0.208025i
$307$	−1831.07	−	1831.07i	−0.340406	−	0.340406i	0.516114	−	0.856520i	$-0.327378\pi$
$307$	−1831.07	−	1831.07i	−0.340406	−	0.340406i	−0.856520	+	0.516114i	$0.827378\pi$
$308$	−1986.57	−	1986.57i	−0.367517	−	0.367517i
$309$	−5642.63	+	3901.03i	−1.03883	+	0.718194i
$310$		0			0
$311$		−	4010.21i		−	0.731184i	−0.930775	−	0.365592i	$-0.880866\pi$
$311$		−	4010.21i		−	0.731184i	0.930775	−	0.365592i	$-0.119134\pi$
$312$	2268.89	+	414.044i	0.411701	+	0.0751303i
$313$	−6072.69	+	6072.69i	−1.09664	+	1.09664i	−0.101841	+	0.994801i	$0.532473\pi$
$313$	−6072.69	+	6072.69i	−1.09664	+	1.09664i	−0.994801	+	0.101841i	$0.967527\pi$
$314$	2870.63			0.515920
$315$		0			0
$316$	−248.685			−0.0442710
$317$	3112.10	−	3112.10i	0.551397	−	0.551397i	−0.375447	−	0.926844i	$-0.622511\pi$
$317$	3112.10	−	3112.10i	0.551397	−	0.551397i	0.926844	+	0.375447i	$0.122511\pi$
$318$	1948.08	+	355.500i	0.343531	+	0.0626901i
$319$		−	3614.09i		−	0.634326i
$320$		0			0
$321$	3603.70	−	2491.42i	0.626600	−	0.433200i
$322$	−1198.53	−	1198.53i	−0.207427	−	0.207427i
$323$	910.737	+	910.737i	0.156888	+	0.156888i
$324$	−2495.14	+	2833.39i	−0.427836	+	0.485835i
$325$		0			0
$326$			1487.10i			0.252647i
$327$	−1935.24	+	10604.8i	−0.327275	+	1.79342i
$328$	6109.45	−	6109.45i	1.02847	−	1.02847i
$329$	3327.71			0.557637
$330$		0			0
$331$	−9589.47			−1.59240			−0.796201	−	0.605033i	$-0.793160\pi$
$331$	−9589.47			−1.59240			−0.796201	+	0.605033i	$0.793160\pi$
$332$	−748.099	+	748.099i	−0.123666	+	0.123666i
$333$	2004.10	+	4435.26i	0.329801	+	0.729883i
$334$		−	7148.90i		−	1.17117i
$335$		0			0
$336$	237.371	+	343.345i	0.0385407	+	0.0557470i
$337$	2561.34	+	2561.34i	0.414021	+	0.414021i	0.883137	−	0.469115i	$-0.155427\pi$
$337$	2561.34	+	2561.34i	0.414021	+	0.414021i	−0.469115	+	0.883137i	$0.655427\pi$
$338$	2131.76	+	2131.76i	0.343054	+	0.343054i
$339$	−1133.29	−	1639.24i	−0.181569	−	0.262630i
$340$		0			0
$341$			7103.50i			1.12808i
$342$	916.377	+	2028.03i	0.144889	+	0.320653i
$343$	4396.56	−	4396.56i	0.692104	−	0.692104i
$344$	1233.36			0.193310
$345$		0			0
$346$	4368.77			0.678805
$347$	−8177.44	+	8177.44i	−1.26509	+	1.26509i	−0.316503	+	0.948592i	$0.602509\pi$
$347$	−8177.44	+	8177.44i	−1.26509	+	1.26509i	−0.948592	+	0.316503i	$0.897491\pi$
$348$	−608.005	+	3331.77i	−0.0936566	+	0.513223i
$349$			2766.04i			0.424249i	0.977243	+	0.212124i	$0.0680382\pi$
$349$			2766.04i			0.424249i	−0.977243	+	0.212124i	$0.931962\pi$
$350$		0			0
$351$	2729.96	−	679.347i	0.415141	−	0.103307i
$352$	−3740.81	−	3740.81i	−0.566437	−	0.566437i
$353$	−6338.53	−	6338.53i	−0.955711	−	0.955711i	0.0433491	−	0.999060i	$-0.486197\pi$
$353$	−6338.53	−	6338.53i	−0.955711	−	0.955711i	−0.999060	+	0.0433491i	$0.986197\pi$
$354$	−5236.46	+	3620.23i	−0.786199	+	0.543539i
$355$		0			0
$356$		−	7339.58i		−	1.09269i
$357$	−2534.43	−	462.501i	−0.375732	−	0.0685663i
$358$	975.763	−	975.763i	0.144052	−	0.144052i
$359$	−8827.47			−1.29776			−0.648880	−	0.760890i	$-0.724762\pi$
$359$	−8827.47			−1.29776			−0.648880	+	0.760890i	$0.724762\pi$
$360$		0			0
$361$	4450.80			0.648899
$362$	2748.50	−	2748.50i	0.399055	−	0.399055i
$363$	−2588.42	−	472.353i	−0.374261	−	0.0682978i
$364$		−	1961.76i		−	0.282484i
$365$		0			0
$366$	−14.3578	+	9.92628i	−0.00205053	+	0.00141764i
$367$	−6191.27	−	6191.27i	−0.880605	−	0.880605i	0.112991	−	0.993596i	$-0.463957\pi$
$367$	−6191.27	−	6191.27i	−0.880605	−	0.880605i	−0.993596	+	0.112991i	$0.963957\pi$
$368$	160.628	+	160.628i	0.0227535	+	0.0227535i
$369$	3722.45	−	9859.55i	0.525157	−	1.39097i
$370$		0			0
$371$		−	4286.28i		−	0.599818i
$372$	1195.04	−	6548.60i	0.166559	−	0.912713i
$373$	−3584.46	+	3584.46i	−0.497577	+	0.497577i	−0.910683	−	0.413106i	$-0.864444\pi$
$373$	−3584.46	+	3584.46i	−0.497577	+	0.497577i	0.413106	+	0.910683i	$0.364444\pi$
$374$	1265.90			0.175022
$375$		0			0
$376$	3899.27			0.534812
$377$	1784.48	−	1784.48i	0.243781	−	0.243781i
$378$	−3814.63	−	2294.40i	−0.519057	−	0.312200i
$379$		−	7110.48i		−	0.963695i	−0.876255	−	0.481848i	$-0.839966\pi$
$379$		−	7110.48i		−	0.963695i	0.876255	−	0.481848i	$-0.160034\pi$
$380$		0			0
$381$	441.241	+	638.231i	0.0593319	+	0.0858203i
$382$	1105.91	+	1105.91i	0.148124	+	0.148124i
$383$	1695.77	+	1695.77i	0.226240	+	0.226240i	0.811120	−	0.584880i	$-0.198858\pi$
$383$	1695.77	+	1695.77i	0.226240	+	0.226240i	−0.584880	+	0.811120i	$0.698858\pi$
$384$	2988.11	+	4322.14i	0.397100	+	0.574383i
$385$		0			0
$386$		−	6231.15i		−	0.821650i
$387$	1370.95	−	619.472i	0.180076	−	0.0813683i
$388$	4703.02	−	4703.02i	0.615361	−	0.615361i
$389$	−12362.2			−1.61128			−0.805640	−	0.592405i	$-0.798178\pi$
$389$	−12362.2			−1.61128			−0.805640	+	0.592405i	$0.798178\pi$
$390$		0			0
$391$	−1402.06			−0.181343
$392$	−216.978	+	216.978i	−0.0279567	+	0.0279567i
$393$	1846.32	−	10117.5i	0.236984	−	1.29863i
$394$			7114.22i			0.909668i
$395$		0			0
$396$	−3756.61	−	1418.30i	−0.476709	−	0.179980i
$397$	1340.12	+	1340.12i	0.169417	+	0.169417i	0.786723	−	0.617306i	$-0.211776\pi$
$397$	1340.12	+	1340.12i	0.169417	+	0.169417i	−0.617306	+	0.786723i	$0.711776\pi$
$398$	−129.999	−	129.999i	−0.0163725	−	0.0163725i
$399$	3962.30	−	2739.33i	0.497150	−	0.343705i
$400$		0			0
$401$			2281.30i			0.284096i	0.989860	+	0.142048i	$0.0453688\pi$
$401$			2281.30i			0.284096i	−0.989860	+	0.142048i	$0.954631\pi$
$402$	−4006.72	−	731.175i	−0.497107	−	0.0907156i
$403$	−3507.40	+	3507.40i	−0.433538	+	0.433538i
$404$	1748.75			0.215356
$405$		0			0
$406$	−3993.26			−0.488133
$407$	−3660.29	+	3660.29i	−0.445783	+	0.445783i
$408$	−2969.74	−	541.939i	−0.360352	−	0.0657597i
$409$			4614.82i			0.557917i	0.960303	+	0.278959i	$0.0899892\pi$
$409$			4614.82i			0.557917i	−0.960303	+	0.278959i	$0.910011\pi$
$410$		0			0
$411$	−3067.32	+	2120.59i	−0.368126	+	0.254504i
$412$	4834.54	+	4834.54i	0.578108	+	0.578108i
$413$	9743.49	+	9743.49i	1.16089	+	1.16089i
$414$	−2266.43	−	855.683i	−0.269055	−	0.101581i
$415$		0			0
$416$		−	3694.10i		−	0.435380i
$417$	63.8364	−	349.813i	0.00749660	−	0.0410801i
$418$	−1673.67	+	1673.67i	−0.195842	+	0.195842i
$419$	−2142.28			−0.249779			−0.124889	−	0.992171i	$-0.539858\pi$
$419$	−2142.28			−0.249779			−0.124889	+	0.992171i	$0.539858\pi$
$420$		0			0
$421$	−8889.30			−1.02907			−0.514534	−	0.857470i	$-0.672035\pi$
$421$	−8889.30			−1.02907			−0.514534	+	0.857470i	$0.672035\pi$
$422$	3224.43	−	3224.43i	0.371950	−	0.371950i
$423$	4334.26	−	1958.46i	0.498200	−	0.225115i
$424$		−	5022.48i		−	0.575267i
$425$		0			0
$426$	−852.466	−	1233.05i	−0.0969534	−	0.140238i
$427$	26.7156	+	26.7156i	0.00302778	+	0.00302778i
$428$	−3087.60	−	3087.60i	−0.348703	−	0.348703i
$429$	1701.52	+	2461.15i	0.191492	+	0.276982i
$430$		0			0
$431$			15707.3i			1.75543i	0.479179	+	0.877717i	$0.340935\pi$
$431$			15707.3i			1.75543i	−0.479179	+	0.877717i	$0.659065\pi$
$432$	511.238	+	307.497i	0.0569374	+	0.0342464i
$433$	−5430.81	+	5430.81i	−0.602744	+	0.602744i	−0.941040	−	0.338296i	$-0.890149\pi$
$433$	−5430.81	+	5430.81i	−0.602744	+	0.602744i	0.338296	+	0.941040i	$0.390149\pi$
$434$	7848.76			0.868094
$435$		0			0
$436$	10744.2			1.18016
$437$	1853.69	−	1853.69i	0.202915	−	0.202915i
$438$	−620.253	+	3398.88i	−0.0676640	+	0.370787i
$439$			8221.92i			0.893874i	0.894565	+	0.446937i	$0.147485\pi$
$439$			8221.92i			0.893874i	−0.894565	+	0.446937i	$0.852515\pi$
$440$		0			0
$441$	−132.203	+	350.163i	−0.0142753	+	0.0378105i
$442$	625.047	+	625.047i	0.0672635	+	0.0672635i
$443$	1960.53	+	1960.53i	0.210265	+	0.210265i	0.804380	−	0.594115i	$-0.202498\pi$
$443$	1960.53	+	1960.53i	0.210265	+	0.210265i	−0.594115	+	0.804380i	$0.702498\pi$
$444$	3990.14	−	2758.58i	0.426495	−	0.294857i
$445$		0			0
$446$			6723.34i			0.713810i
$447$	1857.38	+	338.948i	0.196535	+	0.0358651i
$448$	−3678.86	+	3678.86i	−0.387968	+	0.387968i
$449$	17849.2			1.87607			0.938036	−	0.346537i	$-0.112643\pi$
$449$	17849.2			1.87607			0.938036	+	0.346537i	$0.112643\pi$
$450$		0			0
$451$	11208.8			1.17029
$452$	−1404.49	+	1404.49i	−0.146154	+	0.146154i
$453$	10648.5	+	1943.21i	1.10443	+	0.201545i
$454$			3323.00i			0.343516i
$455$		0			0
$456$	4642.85	−	3209.83i	0.476801	−	0.329636i
$457$	6346.80	+	6346.80i	0.649652	+	0.649652i	0.952909	−	0.303257i	$-0.0980740\pi$
$457$	6346.80	+	6346.80i	0.649652	+	0.649652i	−0.303257	+	0.952909i	$0.598074\pi$
$458$	−4667.97	−	4667.97i	−0.476245	−	0.476245i
$459$	−3573.22	+	889.192i	−0.363363	+	0.0904225i
$460$		0			0
$461$			8848.20i			0.893930i	0.894552	+	0.446965i	$0.147495\pi$
$461$			8848.20i			0.893930i	−0.894552	+	0.446965i	$0.852505\pi$
$462$	849.944	−	4657.55i	0.0855908	−	0.469024i
$463$	−1329.43	+	1329.43i	−0.133443	+	0.133443i	−0.770673	−	0.637230i	$-0.780080\pi$
$463$	−1329.43	+	1329.43i	−0.133443	+	0.133443i	0.637230	+	0.770673i	$0.280080\pi$
$464$	535.178			0.0535453
$465$		0			0
$466$	−4704.83			−0.467697
$467$	1479.96	−	1479.96i	0.146647	−	0.146647i	−0.629971	−	0.776618i	$-0.716933\pi$
$467$	1479.96	−	1479.96i	0.146647	−	0.146647i	0.776618	+	0.629971i	$0.216933\pi$
$468$	−1154.56	−	2555.14i	−0.114037	−	0.252375i
$469$			8815.81i			0.867966i
$470$		0			0
$471$	−5050.29	−	7304.96i	−0.494066	−	0.714639i
$472$	11417.0	+	11417.0i	1.11337	+	1.11337i
$473$	1131.41	+	1131.41i	0.109983	+	0.109983i
$474$	−238.324	−	344.723i	−0.0230941	−	0.0334043i
$475$		0			0
$476$			2567.73i			0.247252i
$477$	−2522.60	−	5582.76i	−0.242143	−	0.535885i
$478$	3534.73	−	3534.73i	0.338231	−	0.338231i
$479$	−5039.60			−0.480720			−0.240360	−	0.970684i	$-0.577266\pi$
$479$	−5039.60			−0.480720			−0.240360	+	0.970684i	$0.577266\pi$
$480$		0			0
$481$	−3614.58			−0.342642
$482$	2179.89	−	2179.89i	0.205999	−	0.205999i
$483$	−941.362	+	5158.51i	−0.0886821	+	0.485963i
$484$			2622.43i			0.246284i
$485$		0			0
$486$	−6318.78	−	743.377i	−0.589765	−	0.0693833i
$487$	−1292.93	−	1292.93i	−0.120305	−	0.120305i	0.644391	−	0.764696i	$-0.277111\pi$
$487$	−1292.93	−	1292.93i	−0.120305	−	0.120305i	−0.764696	+	0.644391i	$0.777111\pi$
$488$	31.3042	+	31.3042i	0.00290384	+	0.00290384i
$489$	3784.27	−	2616.26i	0.349961	−	0.241945i
$490$		0			0
$491$		−	13865.7i		−	1.27444i	−0.770681	−	0.637221i	$-0.780084\pi$
$491$		−	13865.7i		−	1.27444i	0.770681	−	0.637221i	$-0.219916\pi$
$492$	−10333.2	−	1885.68i	−0.946865	−	0.172791i
$493$	−2335.69	+	2335.69i	−0.213375	+	0.213375i
$494$	−1652.77			−0.150530
$495$		0			0
$496$	−1051.89			−0.0952247
$497$	−2294.33	+	2294.33i	−0.207072	+	0.207072i
$498$	−1753.93	−	320.070i	−0.157822	−	0.0288006i
$499$		−	10884.3i		−	0.976453i	−0.872717	−	0.488226i	$-0.837644\pi$
$499$		−	10884.3i		−	0.976453i	0.872717	−	0.488226i	$-0.162356\pi$
$500$		0			0
$501$	−18192.0	+	12577.0i	−1.62227	+	1.12156i
$502$	−1831.54	−	1831.54i	−0.162840	−	0.162840i
$503$	−7880.86	−	7880.86i	−0.698589	−	0.698589i	0.265517	−	0.964106i	$-0.414457\pi$
$503$	−7880.86	−	7880.86i	−0.698589	−	0.698589i	−0.964106	+	0.265517i	$0.914457\pi$
$504$	−3987.84	+	10562.5i	−0.352445	+	0.933513i
$505$		0			0
$506$		−	2576.58i		−	0.226370i
$507$	1674.35	−	9175.14i	0.146667	−	0.803713i
$508$	546.829	−	546.829i	0.0477590	−	0.0477590i
$509$	−1788.46			−0.155741			−0.0778704	−	0.996963i	$-0.524812\pi$
$509$	−1788.46			−0.155741			−0.0778704	+	0.996963i	$0.524812\pi$
$510$		0			0
$511$	7478.42			0.647408
$512$	1086.41	−	1086.41i	0.0937754	−	0.0937754i
$513$	3548.60	−	5899.84i	0.305409	−	0.507766i
$514$			1018.67i			0.0874153i
$515$		0			0
$516$	−852.686	−	1233.36i	−0.0727469	−	0.105224i
$517$	3576.93	+	3576.93i	0.304281	+	0.304281i
$518$	4044.31	+	4044.31i	0.343044	+	0.343044i
$519$	−7685.96	−	11117.3i	−0.650051	−	0.940263i
$520$		0			0
$521$		−	18251.6i		−	1.53478i	−0.641183	−	0.767388i	$-0.721556\pi$
$521$		−	18251.6i		−	1.53478i	0.641183	−	0.767388i	$-0.278444\pi$
$522$	−5201.11	+	2350.15i	−0.436104	+	0.197056i
$523$	2125.69	−	2125.69i	0.177725	−	0.177725i	−0.612639	−	0.790363i	$-0.709892\pi$
$523$	2125.69	−	2125.69i	0.177725	−	0.177725i	0.790363	+	0.612639i	$0.209892\pi$
$524$	−10250.5			−0.854570
$525$		0			0
$526$	12487.1			1.03510
$527$	4590.80	−	4590.80i	0.379466	−	0.379466i
$528$	−113.910	+	624.207i	−0.00938880	+	0.0514491i
$529$		−	9313.29i		−	0.765455i
$530$		0			0
$531$	18425.0	+	6956.30i	1.50579	+	0.568508i
$532$	−3394.85	−	3394.85i	−0.276664	−	0.276664i
$533$	5534.42	+	5534.42i	0.449761	+	0.449761i
$534$	10174.0	−	7033.79i	0.824478	−	0.570003i
$535$		0			0
$536$			10330.0i			0.832439i
$537$	−4199.71	−	766.393i	−0.337488	−	0.0615872i
$538$	2292.60	−	2292.60i	0.183719	−	0.183719i
$539$	−398.082			−0.0318119
$540$		0			0
$541$	−2214.16			−0.175960			−0.0879798	−	0.996122i	$-0.528041\pi$
$541$	−2214.16			−0.175960			−0.0879798	+	0.996122i	$0.528041\pi$
$542$	−3873.77	+	3873.77i	−0.306998	+	0.306998i
$543$	−11829.6	−	2158.75i	−0.934911	−	0.170609i
$544$			4835.17i			0.381078i
$545$		0			0
$546$	2719.36	−	1880.03i	0.213146	−	0.147359i
$547$	−12385.1	−	12385.1i	−0.968098	−	0.968098i	0.0314090	−	0.999507i	$-0.490001\pi$
$547$	−12385.1	−	12385.1i	−0.968098	−	0.968098i	−0.999507	+	0.0314090i	$0.990001\pi$
$548$	2628.04	+	2628.04i	0.204862	+	0.204862i
$549$	50.5193	+	19.0735i	0.00392735	+	0.00148276i
$550$		0			0
$551$		−	6176.11i		−	0.477516i
$552$	−1103.05	+	6044.52i	−0.0850522	+	0.466072i
$553$	−641.427	+	641.427i	−0.0493242	+	0.0493242i
$554$	−11557.3			−0.886324
$555$		0			0
$556$	−354.410			−0.0270329
$557$	−8716.96	+	8716.96i	−0.663105	+	0.663105i	−0.956111	−	0.293006i	$-0.905344\pi$
$557$	−8716.96	+	8716.96i	−0.663105	+	0.663105i	0.293006	+	0.956111i	$0.405344\pi$
$558$	10222.8	−	4619.23i	0.775566	−	0.350444i
$559$			1117.28i			0.0845363i
$560$		0			0
$561$	−2227.10	−	3221.38i	−0.167608	−	0.242436i
$562$	−4729.23	−	4729.23i	−0.354965	−	0.354965i
$563$	−11697.0	−	11697.0i	−0.875615	−	0.875615i	0.117462	−	0.993077i	$-0.462524\pi$
$563$	−11697.0	−	11697.0i	−0.875615	−	0.875615i	−0.993077	+	0.117462i	$0.962524\pi$
$564$	−2695.76	−	3899.27i	−0.201262	−	0.291115i
$565$		0			0
$566$		−	9721.40i		−	0.721945i
$567$	872.435	+	13743.7i	0.0646188	+	1.01796i
$568$	−2688.40	+	2688.40i	−0.198596	+	0.198596i
$569$	11517.5			0.848576			0.424288	−	0.905527i	$-0.360524\pi$
$569$	11517.5			0.848576			0.424288	+	0.905527i	$0.360524\pi$
$570$		0			0
$571$	−11093.9			−0.813072			−0.406536	−	0.913635i	$-0.633263\pi$
$571$	−11093.9			−0.813072			−0.406536	+	0.913635i	$0.633263\pi$
$572$	2108.68	−	2108.68i	0.154141	−	0.154141i
$573$	868.617	−	4759.88i	0.0633281	−	0.347027i
$574$		−	12384.8i		−	0.900577i
$575$		0			0
$576$	−2626.50	+	6956.73i	−0.189995	+	0.503236i
$577$	14119.4	+	14119.4i	1.01871	+	1.01871i	0.999822	+	0.0188920i	$0.00601387\pi$
$577$	14119.4	+	14119.4i	1.01871	+	1.01871i	0.0188920	+	0.999822i	$0.493986\pi$
$578$	5016.87	+	5016.87i	0.361028	+	0.361028i
$579$	−15856.6	+	10962.4i	−1.13813	+	0.786846i
$580$		0			0
$581$			3859.10i			0.275564i
$582$	11026.3	+	2012.16i	0.785320	+	0.143311i
$583$	4607.29	−	4607.29i	0.327297	−	0.327297i
$584$	8762.89			0.620909
$585$		0			0
$586$	−10725.4			−0.756081
$587$	−8524.33	+	8524.33i	−0.599381	+	0.599381i	−0.940148	−	0.340767i	$-0.889313\pi$
$587$	−8524.33	+	8524.33i	−0.599381	+	0.599381i	0.340767	+	0.940148i	$0.389313\pi$
$588$	366.986	+	66.9702i	0.0257385	+	0.00469695i
$589$			12139.2i			0.849211i
$590$		0			0
$591$	18103.8	−	12516.0i	1.26005	−	0.871135i
$592$	−542.020	−	542.020i	−0.0376298	−	0.0376298i
$593$	4580.53	+	4580.53i	0.317200	+	0.317200i	0.847691	−	0.530491i	$-0.177992\pi$
$593$	4580.53	+	4580.53i	0.317200	+	0.317200i	−0.530491	+	0.847691i	$0.677992\pi$
$594$	−1634.08	−	6566.55i	−0.112874	−	0.453584i
$595$		0			0
$596$		−	1881.79i		−	0.129331i
$597$	−102.105	+	559.518i	−0.00699979	+	0.0383577i
$598$	1272.20	−	1272.20i	0.0869971	−	0.0869971i
$599$	−10195.8			−0.695477			−0.347738	−	0.937592i	$-0.613050\pi$
$599$	−10195.8			−0.695477			−0.347738	+	0.937592i	$0.613050\pi$
$600$		0			0
$601$	18915.8			1.28385			0.641923	−	0.766769i	$-0.278137\pi$
$601$	18915.8			1.28385			0.641923	+	0.766769i	$0.278137\pi$
$602$	1250.11	−	1250.11i	0.0846355	−	0.0846355i
$603$	5188.36	+	11482.4i	0.350392	+	0.775452i
$604$		−	10788.4i		−	0.726778i
$605$		0			0
$606$	1675.89	+	2424.09i	0.112341	+	0.162495i
$607$	−12758.2	−	12758.2i	−0.853113	−	0.853113i	0.137402	−	0.990515i	$-0.456125\pi$
$607$	−12758.2	−	12758.2i	−0.853113	−	0.853113i	−0.990515	+	0.137402i	$0.956125\pi$
$608$	−6392.67	−	6392.67i	−0.426409	−	0.426409i
$609$	7025.33	+	10161.8i	0.467456	+	0.676149i
$610$		0			0
$611$			3532.26i			0.233879i
$612$	1511.19	+	3344.41i	0.0998140	+	0.220898i
$613$	−7340.02	+	7340.02i	−0.483622	+	0.483622i	−0.906286	−	0.422664i	$-0.861095\pi$
$613$	−7340.02	+	7340.02i	−0.483622	+	0.483622i	0.422664	+	0.906286i	$0.361095\pi$
$614$	−4349.39			−0.285875
$615$		0			0
$616$	−12007.9			−0.785412
$617$	17118.9	−	17118.9i	1.11698	−	1.11698i	0.124803	−	0.992182i	$-0.460170\pi$
$617$	17118.9	−	17118.9i	1.11698	−	1.11698i	0.992182	−	0.124803i	$-0.0398299\pi$
$618$	−2068.43	+	11334.7i	−0.134635	+	0.737778i
$619$			3176.16i			0.206237i	0.994669	+	0.103118i	$0.0328820\pi$
$619$			3176.16i			0.206237i	−0.994669	+	0.103118i	$0.967118\pi$
$620$		0			0
$621$	1809.84	+	7272.84i	0.116950	+	0.469966i
$622$	−4762.79	−	4762.79i	−0.307026	−	0.307026i
$623$	−18930.8	−	18930.8i	−1.21741	−	1.21741i
$624$	−364.450	+	251.962i	−0.0233809	+	0.0161644i
$625$		0			0
$626$			14424.6i			0.920965i
$627$	7203.53	+	1314.55i	0.458822	+	0.0837291i
$628$	−6258.80	+	6258.80i	−0.397697	+	0.397697i
$629$	4731.09			0.299906
$630$		0			0
$631$	−11467.2			−0.723459			−0.361729	−	0.932283i	$-0.617814\pi$
$631$	−11467.2			−0.723459			−0.361729	+	0.932283i	$0.617814\pi$
$632$	−751.597	+	751.597i	−0.0473052	+	0.0473052i
$633$	−13878.0	−	2532.57i	−0.871410	−	0.159021i
$634$		−	7392.26i		−	0.463066i
$635$		0			0
$636$	−5022.48	+	3472.29i	−0.313136	+	0.216486i
$637$	−196.556	−	196.556i	−0.0122258	−	0.0122258i
$638$	−4292.32	−	4292.32i	−0.266355	−	0.266355i
$639$	−1638.02	+	4338.59i	−0.101407	+	0.268595i
$640$		0			0
$641$			12383.2i			0.763035i	0.924362	+	0.381518i	$0.124598\pi$
$641$			12383.2i			0.763035i	−0.924362	+	0.381518i	$0.875402\pi$
$642$	1321.02	−	7238.95i	0.0812092	−	0.445013i
$643$	11142.9	−	11142.9i	0.683413	−	0.683413i	−0.277354	−	0.960768i	$-0.589458\pi$
$643$	11142.9	−	11142.9i	0.683413	−	0.683413i	0.960768	+	0.277354i	$0.0894576\pi$
$644$	5226.29			0.319790
$645$		0			0
$646$	2163.30			0.131755
$647$	2391.21	−	2391.21i	0.145299	−	0.145299i	−0.630716	−	0.776014i	$-0.717239\pi$
$647$	2391.21	−	2391.21i	0.145299	−	0.145299i	0.776014	+	0.630716i	$0.217239\pi$
$648$	1022.28	+	16104.3i	0.0619738	+	0.976292i
$649$			20946.4i			1.26690i
$650$		0			0
$651$	−13808.3	−	19973.0i	−0.831322	−	1.20246i
$652$	−3242.32	−	3242.32i	−0.194753	−	0.194753i
$653$	16623.3	+	16623.3i	0.996201	+	0.996201i	0.999993	−	0.00379136i	$-0.00120683\pi$
$653$	16623.3	+	16623.3i	0.996201	+	0.996201i	−0.00379136	+	0.999993i	$0.501207\pi$
$654$	10296.5	+	14893.4i	0.615636	+	0.890484i
$655$		0			0
$656$			1659.81i			0.0987878i
$657$	9740.43	−	4401.27i	0.578403	−	0.261354i
$658$	3952.20	−	3952.20i	0.234153	−	0.234153i
$659$	20089.8			1.18754			0.593768	−	0.804636i	$-0.297640\pi$
$659$	20089.8			1.18754			0.593768	+	0.804636i	$0.297640\pi$
$660$		0			0
$661$	541.434			0.0318598			0.0159299	−	0.999873i	$-0.494929\pi$
$661$	541.434			0.0318598			0.0159299	+	0.999873i	$0.494929\pi$
$662$	−11389.1	+	11389.1i	−0.668654	+	0.668654i
$663$	490.931	−	2690.22i	0.0287574	−	0.157586i
$664$			4521.93i			0.264284i
$665$		0			0
$666$	7647.80	+	2887.41i	0.444964	+	0.167995i
$667$	4753.99	+	4753.99i	0.275975	+	0.275975i
$668$	15586.7	+	15586.7i	0.902796	+	0.902796i
$669$	17109.1	−	11828.3i	0.988751	−	0.683573i
$670$		0			0
$671$			57.4328i			0.00330428i
$672$	17789.7	+	3246.40i	1.02121	+	0.186358i
$673$	24314.3	−	24314.3i	1.39264	−	1.39264i	0.573288	−	0.819354i	$-0.305668\pi$
$673$	24314.3	−	24314.3i	1.39264	−	1.39264i	0.819354	−	0.573288i	$-0.194332\pi$
$674$	6084.03			0.347697
$675$		0			0
$676$	−9295.71			−0.528887
$677$	14662.4	−	14662.4i	0.832380	−	0.832380i	−0.155462	−	0.987842i	$-0.549687\pi$
$677$	14662.4	−	14662.4i	0.832380	−	0.832380i	0.987842	+	0.155462i	$0.0496867\pi$
$678$	−3292.84	−	600.902i	−0.186520	−	0.0340376i
$679$		−	24260.8i		−	1.37120i
$680$		0			0
$681$	8456.13	−	5846.14i	0.475829	−	0.328964i
$682$	8436.57	+	8436.57i	0.473685	+	0.473685i
$683$	15981.2	+	15981.2i	0.895320	+	0.895320i	0.995018	−	0.0996976i	$-0.0317875\pi$
$683$	15981.2	+	15981.2i	0.895320	+	0.895320i	−0.0996976	+	0.995018i	$0.531788\pi$
$684$	−6419.67	−	2423.73i	−0.358863	−	0.135488i
$685$		0			0
$686$		−	10443.3i		−	0.581233i
$687$	−3666.36	+	20091.1i	−0.203611	+	1.11575i
$688$	−167.540	+	167.540i	−0.00928400	+	0.00928400i
$689$	4549.75			0.251570
$690$		0			0
$691$	−16714.9			−0.920209			−0.460105	−	0.887865i	$-0.652188\pi$
$691$	−16714.9			−0.920209			−0.460105	+	0.887865i	$0.652188\pi$
$692$	−9525.19	+	9525.19i	−0.523256	+	0.523256i
$693$	−13347.5	+	6031.14i	−0.731645	+	0.330598i
$694$			19424.1i			1.06243i
$695$		0			0
$696$	8231.98	+	11907.1i	0.448322	+	0.648474i
$697$	−7243.96	−	7243.96i	−0.393665	−	0.393665i
$698$	3285.13	+	3285.13i	0.178143	+	0.178143i
$699$	8277.19	+	11972.5i	0.447886	+	0.647842i
$700$		0			0
$701$			6990.49i			0.376643i	0.982107	+	0.188322i	$0.0603048\pi$
$701$			6990.49i			0.376643i	−0.982107	+	0.188322i	$0.939695\pi$
$702$	2435.44	−	4049.11i	0.130940	−	0.217698i
$703$	−6255.06	+	6255.06i	−0.335582	+	0.335582i
$704$	−7908.75			−0.423398
$705$		0			0
$706$	−15056.1			−0.802611
$707$	4510.51	−	4510.51i	0.239936	−	0.239936i
$708$	3523.86	−	19310.2i	0.187055	−	1.02503i
$709$		−	28175.0i		−	1.49243i	−0.665705	−	0.746215i	$-0.731869\pi$
$709$		−	28175.0i		−	1.49243i	0.665705	−	0.746215i	$-0.268131\pi$
$710$		0			0
$711$	−457.943	+	1212.94i	−0.0241550	+	0.0639787i
$712$	−22182.3	−	22182.3i	−1.16758	−	1.16758i
$713$	−9343.99	−	9343.99i	−0.490793	−	0.490793i
$714$	−3559.35	+	2460.75i	−0.186562	+	0.128980i
$715$		0			0
$716$			4254.90i			0.222085i
$717$	−15213.6	−	2776.28i	−0.792414	−	0.144605i
$718$	−10484.1	+	10484.1i	−0.544933	+	0.544933i
$719$	−20143.8			−1.04484			−0.522418	−	0.852690i	$-0.674970\pi$
$719$	−20143.8			−1.04484			−0.522418	+	0.852690i	$0.674970\pi$
$720$		0			0
$721$	24939.2			1.28819
$722$	5286.05	−	5286.05i	0.272474	−	0.272474i
$723$	−9382.32	−	1712.15i	−0.482617	−	0.0880715i
$724$			11985.0i			0.615222i
$725$		0			0
$726$	−3635.16	+	2513.17i	−0.185831	+	0.128475i
$727$	9805.90	+	9805.90i	0.500249	+	0.500249i	0.911515	−	0.411267i	$-0.134913\pi$
$727$	9805.90	+	9805.90i	0.500249	+	0.500249i	−0.411267	+	0.911515i	$0.634913\pi$
$728$	−5929.00	−	5929.00i	−0.301845	−	0.301845i
$729$	9224.92	+	17387.4i	0.468674	+	0.883371i
$730$		0			0
$731$		−	1462.39i		−	0.0739926i
$732$	9.66205	−	52.9464i	0.000487868	−	0.00267344i
$733$	−16533.1	+	16533.1i	−0.833100	+	0.833100i	−0.987940	−	0.154840i	$-0.950514\pi$
$733$	−16533.1	+	16533.1i	−0.833100	+	0.833100i	0.154840	+	0.987940i	$0.450514\pi$
$734$	−14706.3			−0.739536
$735$		0			0
$736$	9841.37			0.492877
$737$	−9476.04	+	9476.04i	−0.473615	+	0.473615i
$738$	−7288.81	−	16130.8i	−0.363557	−	0.804586i
$739$		−	15250.1i		−	0.759114i	−0.925168	−	0.379557i	$-0.876076\pi$
$739$		−	15250.1i		−	0.759114i	0.925168	−	0.379557i	$-0.123924\pi$
$740$		0			0
$741$	2907.72	+	4205.85i	0.144153	+	0.208510i
$742$	−5090.66	−	5090.66i	−0.251865	−	0.251865i
$743$	5438.49	+	5438.49i	0.268531	+	0.268531i	0.828508	−	0.559977i	$-0.189190\pi$
$743$	5438.49	+	5438.49i	0.268531	+	0.268531i	−0.559977	+	0.828508i	$0.689190\pi$
$744$	−16180.0	−	23403.5i	−0.797294	−	1.15324i
$745$		0			0
$746$			8514.27i			0.417868i
$747$	2271.20	+	5026.38i	0.111243	+	0.246192i
$748$	−2760.04	+	2760.04i	−0.134916	+	0.134916i
$749$	−15927.5			−0.777009
$750$		0			0
$751$	2087.82			0.101446			0.0507228	−	0.998713i	$-0.483848\pi$
$751$	2087.82			0.101446			0.0507228	+	0.998713i	$0.483848\pi$
$752$	−529.676	+	529.676i	−0.0256852	+	0.0256852i
$753$	−1438.55	+	7883.01i	−0.0696197	+	0.381505i
$754$		−	4238.72i		−	0.204728i
$755$		0			0
$756$	13319.5	−	3314.54i	0.640774	−	0.159456i
$757$	10258.6	+	10258.6i	0.492546	+	0.492546i	0.909107	−	0.416562i	$-0.136765\pi$
$757$	10258.6	+	10258.6i	0.492546	+	0.492546i	−0.416562	+	0.909107i	$0.636765\pi$
$758$	−8444.86	−	8444.86i	−0.404658	−	0.404658i
$759$	−6556.70	+	4532.98i	−0.313561	+	0.216781i
$760$		0			0
$761$		−	26879.5i		−	1.28040i	−0.768210	−	0.640198i	$-0.778852\pi$
$761$		−	26879.5i		−	1.28040i	0.768210	−	0.640198i	$-0.221148\pi$
$762$	1282.05	+	233.958i	0.0609498	+	0.0111226i
$763$	27712.1	−	27712.1i	1.31487	−	1.31487i
$764$	−4822.43			−0.228363
$765$		0			0
$766$	4028.02			0.189998
$767$	−10342.4	+	10342.4i	−0.486888	+	0.486888i
$768$	19944.6	+	3639.64i	0.937097	+	0.171008i
$769$			25180.9i			1.18082i	0.807105	+	0.590408i	$0.201033\pi$
$769$			25180.9i			1.18082i	−0.807105	+	0.590408i	$0.798967\pi$
$770$		0			0
$771$	2592.23	−	1792.14i	0.121085	−	0.0837124i
$772$	13585.7	+	13585.7i	0.633369	+	0.633369i
$773$	−18871.2	−	18871.2i	−0.878072	−	0.878072i	0.115263	−	0.993335i	$-0.463229\pi$
$773$	−18871.2	−	18871.2i	−0.878072	−	0.878072i	−0.993335	+	0.115263i	$0.963229\pi$
$774$	892.506	−	2363.96i	0.0414476	−	0.109781i
$775$		0			0
$776$		−	28427.7i		−	1.31507i
$777$	3176.52	−	17406.8i	0.146663	−	0.803688i
$778$	−14682.1	+	14682.1i	−0.676581	+	0.676581i
$779$	19154.7			0.880988
$780$		0			0
$781$	−4932.32			−0.225982
$782$	−1665.18	+	1665.18i	−0.0761465	+	0.0761465i
$783$	15130.8	+	9100.79i	0.690588	+	0.415371i
$784$		−	58.9485i		−	0.00268533i
$785$		0			0
$786$	−9823.43	−	14209.0i	−0.445789	−	0.644809i
$787$	−14716.0	−	14716.0i	−0.666542	−	0.666542i	0.290372	−	0.956914i	$-0.406221\pi$
$787$	−14716.0	−	14716.0i	−0.666542	−	0.666542i	−0.956914	+	0.290372i	$0.906221\pi$
$788$	−15511.1	−	15511.1i	−0.701217	−	0.701217i
$789$	−21968.5	−	31776.2i	−0.991253	−	1.43379i
$790$		0			0
$791$			7245.10i			0.325672i
$792$	−15640.0	+	7067.03i	−0.701697	+	0.317066i
$793$	−28.3578	+	28.3578i	−0.00126988	+	0.00126988i
$794$	3183.22			0.142277
$795$		0			0
$796$	566.871			0.0252415
$797$	−17463.9	+	17463.9i	−0.776163	+	0.776163i	−0.979176	−	0.203013i	$-0.934927\pi$
$797$	−17463.9	+	17463.9i	−0.776163	+	0.776163i	0.203013	+	0.979176i	$0.434927\pi$
$798$	1452.47	−	7959.28i	0.0644321	−	0.353077i
$799$		−	4623.35i		−	0.204709i
$800$		0			0
$801$	−35798.1	−	13515.5i	−1.57911	−	0.596188i
$802$	2709.42	+	2709.42i	0.119293	+	0.119293i
$803$	8038.49	+	8038.49i	0.353265	+	0.353265i
$804$	10330.0	−	7141.64i	0.453122	−	0.313266i
$805$		0			0
$806$			8331.22i			0.364088i
$807$	−9867.40	−	1800.68i	−0.430420	−	0.0785462i
$808$	5285.22	−	5285.22i	0.230115	−	0.230115i
$809$	−24097.9			−1.04727			−0.523633	−	0.851944i	$-0.675424\pi$
$809$	−24097.9			−1.04727			−0.523633	+	0.851944i	$0.675424\pi$
$810$		0			0
$811$	25302.7			1.09556			0.547780	−	0.836622i	$-0.315473\pi$
$811$	25302.7			1.09556			0.547780	+	0.836622i	$0.315473\pi$
$812$	8706.47	−	8706.47i	0.376277	−	0.376277i
$813$	16672.8	+	3042.57i	0.719238	+	0.131252i
$814$			8694.39i			0.374371i
$815$		0			0
$816$	477.025	−	329.791i	0.0204647	−	0.0141483i
$817$	1933.46	+	1933.46i	0.0827945	+	0.0827945i
$818$	5480.85	+	5480.85i	0.234271	+	0.234271i
$819$	−9568.33	−	3612.50i	−0.408235	−	0.154128i
$820$		0			0
$821$			27925.3i			1.18709i	0.804802	+	0.593544i	$0.202272\pi$
$821$			27925.3i			1.18709i	−0.804802	+	0.593544i	$0.797728\pi$
$822$	−1124.39	+	6161.50i	−0.0477102	+	0.261444i
$823$	−997.907	+	997.907i	−0.0422659	+	0.0422659i	−0.727924	−	0.685658i	$-0.759515\pi$
$823$	−997.907	+	997.907i	−0.0422659	+	0.0422659i	0.685658	+	0.727924i	$0.259515\pi$
$824$	29222.6			1.23546
$825$		0			0
$826$	23144.0			0.974918
$827$	18683.3	−	18683.3i	0.785590	−	0.785590i	−0.195178	−	0.980768i	$-0.562529\pi$
$827$	18683.3	−	18683.3i	0.785590	−	0.785590i	0.980768	+	0.195178i	$0.0625285\pi$
$828$	6807.11	−	3075.83i	0.285705	−	0.129097i
$829$		−	21146.9i		−	0.885962i	−0.896531	−	0.442981i	$-0.853921\pi$
$829$		−	21146.9i		−	0.885962i	0.896531	−	0.442981i	$-0.146079\pi$
$830$		0			0
$831$	20332.8	+	29410.2i	0.848780	+	1.22771i
$832$	−3904.99	−	3904.99i	−0.162718	−	0.162718i
$833$	257.270	+	257.270i	0.0107009	+	0.0107009i
$834$	−339.644	−	491.276i	−0.0141018	−	0.0203975i
$835$		0			0
$836$		−	7298.19i		−	0.301930i
$837$	−29739.6	−	17887.6i	−1.22814	−	0.738695i
$838$	−2544.31	+	2544.31i	−0.104883	+	0.104883i
$839$	30903.4			1.27164			0.635820	−	0.771838i	$-0.280662\pi$
$839$	30903.4			1.27164			0.635820	+	0.771838i	$0.280662\pi$
$840$		0			0
$841$	−8549.68			−0.350555
$842$	−10557.5	+	10557.5i	−0.432109	+	0.432109i
$843$	−3714.48	+	20354.7i	−0.151760	+	0.831618i
$844$			14060.4i			0.573435i
$845$		0			0
$846$	2821.65	−	7473.63i	0.114669	−	0.303722i
$847$	6763.96	+	6763.96i	0.274395	+	0.274395i
$848$	682.252	+	682.252i	0.0276281	+	0.0276281i
$849$	−24738.3	+	17102.8i	−1.00002	+	0.691364i
$850$		0			0
$851$		−	9629.53i		−	0.387892i
$852$	4547.02	+	829.774i	0.182839	+	0.0333657i
$853$	−181.224	+	181.224i	−0.00727432	+	0.00727432i	−0.710735	−	0.703460i	$-0.751637\pi$
$853$	−181.224	+	181.224i	−0.00727432	+	0.00727432i	0.703460	+	0.710735i	$0.251637\pi$
$854$	63.4584			0.00254274
$855$		0			0
$856$	−18663.2			−0.745205
$857$	13852.9	−	13852.9i	0.552167	−	0.552167i	−0.374899	−	0.927066i	$-0.622323\pi$
$857$	13852.9	−	13852.9i	0.552167	−	0.552167i	0.927066	+	0.374899i	$0.122323\pi$
$858$	4943.85	+	902.189i	0.196713	+	0.0358977i
$859$			8910.47i			0.353925i	0.984218	+	0.176962i	$0.0566271\pi$
$859$			8910.47i			0.353925i	−0.984218	+	0.176962i	$0.943373\pi$
$860$		0			0
$861$	−31515.9	+	21788.5i	−1.24746	+	0.862428i
$862$	18655.0	+	18655.0i	0.737112	+	0.737112i
$863$	6487.75	+	6487.75i	0.255905	+	0.255905i	0.823386	−	0.567481i	$-0.192082\pi$
$863$	6487.75	+	6487.75i	0.255905	+	0.255905i	−0.567481	+	0.823386i	$0.692082\pi$
$864$	25081.2	−	6241.43i	0.987594	−	0.245761i
$865$		0			0
$866$			12900.0i			0.506187i
$867$	3940.40	−	21592.8i	0.154352	−	0.845823i
$868$	−17112.6	+	17112.6i	−0.669170	+	0.669170i
$869$	−1378.93			−0.0538285
$870$		0			0
$871$	−9357.71			−0.364034
$872$	32471.9	−	32471.9i	1.26105	−	1.26105i
$873$	−14278.2	−	31599.0i	−0.553543	−	1.22504i
$874$		−	4403.12i		−	0.170409i
$875$		0			0
$876$	−6058.22	−	8762.89i	−0.233662	−	0.337980i
$877$	20231.6	+	20231.6i	0.778987	+	0.778987i	0.979659	−	0.200672i	$-0.0643124\pi$
$877$	20231.6	+	20231.6i	0.778987	+	0.778987i	−0.200672	+	0.979659i	$0.564312\pi$
$878$	9764.88	+	9764.88i	0.375340	+	0.375340i
$879$	18869.2	+	27293.3i	0.724054	+	1.04730i
$880$		0			0
$881$		−	33209.0i		−	1.26996i	−0.772527	−	0.634982i	$-0.781007\pi$
$881$		−	33209.0i		−	1.26996i	0.772527	−	0.634982i	$-0.218993\pi$
$882$	258.863	+	572.889i	0.00988251	+	0.0218709i
$883$	−26984.3	+	26984.3i	−1.02842	+	1.02842i	−0.0288331	+	0.999584i	$0.509179\pi$
$883$	−26984.3	+	26984.3i	−1.02842	+	1.02842i	−0.999584	+	0.0288331i	$0.990821\pi$
$884$	−2725.57			−0.103700
$885$		0			0
$886$	4656.90			0.176582
$887$	−24008.9	+	24008.9i	−0.908838	+	0.908838i	−0.996179	−	0.0873406i	$-0.972163\pi$
$887$	−24008.9	+	24008.9i	−0.908838	+	0.908838i	0.0873406	+	0.996179i	$0.472163\pi$
$888$	3722.11	−	20396.5i	0.140660	−	0.770792i
$889$		−	2820.84i		−	0.106421i
$890$		0			0
$891$	−13835.3	+	15710.8i	−0.520200	+	0.590720i
$892$	−14658.8	−	14658.8i	−0.550240	−	0.550240i
$893$	6112.61	+	6112.61i	0.229060	+	0.229060i
$894$	2608.50	−	1803.39i	0.0975853	−	0.0674656i
$895$		0			0
$896$		−	19102.9i		−	0.712258i
$897$	−5475.59	−	999.226i	−0.203818	−	0.0371942i
$898$	21198.9	−	21198.9i	0.787768	−	0.787768i
$899$	−31132.2			−1.15497
$900$		0			0
$901$	−5955.13			−0.220193
$902$	13312.3	−	13312.3i	0.491409	−	0.491409i
$903$	−5380.49	−	981.871i	−0.198285	−	0.0361845i
$904$			8489.50i			0.312341i
$905$		0			0
$906$	14954.7	−	10338.9i	0.548384	−	0.379125i
$907$	23026.7	+	23026.7i	0.842989	+	0.842989i	0.989246	−	0.146258i	$-0.0467229\pi$
$907$	23026.7	+	23026.7i	0.842989	+	0.842989i	−0.146258	+	0.989246i	$0.546723\pi$
$908$	−7245.11	−	7245.11i	−0.264799	−	0.264799i
$909$	3220.25	−	8529.38i	0.117501	−	0.311223i
$910$		0			0
$911$			33422.1i			1.21550i	0.794127	+	0.607752i	$0.207929\pi$
$911$			33422.1i			1.21550i	−0.794127	+	0.607752i	$0.792071\pi$
$912$	−194.660	+	1066.71i	−0.00706781	+	0.0387304i
$913$	−4148.12	+	4148.12i	−0.150364	+	0.150364i
$914$	15075.7			0.545581
$915$		0			0
$916$	20355.1			0.734226
$917$	−26438.8	+	26438.8i	−0.952112	+	0.952112i
$918$	−3187.73	+	5299.85i	−0.114609	+	0.190546i
$919$			42542.2i			1.52703i	0.645792	+	0.763513i	$0.276527\pi$
$919$			42542.2i			1.52703i	−0.645792	+	0.763513i	$0.723473\pi$
$920$		0			0
$921$	7651.86	+	11068.0i	0.273765	+	0.395986i
$922$	10508.7	+	10508.7i	0.375363	+	0.375363i
$923$	−2435.36	−	2435.36i	−0.0868482	−	0.0868482i
$924$	8301.70	+	12007.9i	0.295569	+	0.427524i
$925$		0			0
$926$			3157.84i			0.112066i
$927$	32482.6	−	14677.4i	1.15088	−	0.520033i
$928$	16394.7	−	16394.7i	0.579938	−	0.579938i
$929$	5721.21			0.202053			0.101026	−	0.994884i	$-0.467787\pi$
$929$	5721.21			0.202053			0.101026	+	0.994884i	$0.467787\pi$
$930$		0			0
$931$	−680.282			−0.0239477
$932$	10257.9	−	10257.9i	0.360524	−	0.360524i
$933$	−3740.83	+	20499.1i	−0.131264	+	0.719305i
$934$		−	3515.38i		−	0.123155i
$935$		0			0
$936$	−11211.8	−	4232.97i	−0.391525	−	0.147819i
$937$	−1344.01	−	1344.01i	−0.0468589	−	0.0468589i	0.683289	−	0.730148i	$-0.260549\pi$
$937$	−1344.01	−	1344.01i	−0.0468589	−	0.0468589i	−0.730148	+	0.683289i	$0.760549\pi$
$938$	10470.2	+	10470.2i	0.364461	+	0.364461i
$939$	36706.8	−	25377.2i	1.27570	−	0.881953i
$940$		0			0
$941$			9625.77i			0.333466i	0.986002	+	0.166733i	$0.0533217\pi$
$941$			9625.77i			0.333466i	−0.986002	+	0.166733i	$0.946678\pi$
$942$	−14673.9	−	2677.80i	−0.507538	−	0.0926192i
$943$	−14744.1	+	14744.1i	−0.509157	+	0.509157i
$944$	−3101.77			−0.106943
$945$		0			0
$946$	2687.46			0.0923645
$947$	2234.24	−	2234.24i	0.0766664	−	0.0766664i	−0.667734	−	0.744400i	$-0.732736\pi$
$947$	2234.24	−	2234.24i	0.0766664	−	0.0766664i	0.744400	+	0.667734i	$0.232736\pi$
$948$	1271.21	+	231.980i	0.0435518	+	0.00794765i
$949$			7938.11i			0.271530i
$950$		0			0
$951$	−18811.3	+	13005.2i	−0.641427	+	0.443451i
$952$	7760.41	+	7760.41i	0.264198	+	0.264198i
$953$	−6457.14	−	6457.14i	−0.219483	−	0.219483i	0.588798	−	0.808281i	$-0.299602\pi$
$953$	−6457.14	−	6457.14i	−0.219483	−	0.219483i	−0.808281	+	0.588798i	$0.799602\pi$
$954$	−9626.45	−	3634.44i	−0.326696	−	0.123343i
$955$		0			0
$956$			15413.5i			0.521451i
$957$	−3371.32	+	18474.3i	−0.113876	+	0.624021i
$958$	−5985.35	+	5985.35i	−0.201856	+	0.201856i
$959$	13556.9			0.456491
$960$		0			0
$961$	31399.6			1.05399
$962$	−4292.91	+	4292.91i	−0.143876	+	0.143876i
$963$	−20745.2	+	9373.84i	−0.694190	+	0.313674i
$964$			9505.61i			0.317588i
$965$		0			0
$966$	5008.55	+	7244.59i	0.166819	+	0.241295i
$967$	−13166.9	−	13166.9i	−0.437869	−	0.437869i	0.453425	−	0.891294i	$-0.350202\pi$
$967$	−13166.9	−	13166.9i	−0.437869	−	0.437869i	−0.891294	+	0.453425i	$0.850202\pi$
$968$	7925.72	+	7925.72i	0.263163	+	0.263163i
$969$	−3805.89	−	5505.01i	−0.126174	−	0.182504i
$970$		0			0
$971$			21504.3i			0.710716i	0.934730	+	0.355358i	$0.115641\pi$
$971$			21504.3i			0.710716i	−0.934730	+	0.355358i	$0.884359\pi$
$972$	15397.6	−	12156.0i	0.508104	−	0.401136i
$973$	−914.119	+	914.119i	−0.0301185	+	0.0301185i
$974$	−3071.14			−0.101032
$975$		0			0
$976$	−8.50472			−0.000278924
$977$	8996.19	−	8996.19i	0.294589	−	0.294589i	−0.544301	−	0.838890i	$-0.683205\pi$
$977$	8996.19	−	8996.19i	0.294589	−	0.294589i	0.838890	+	0.544301i	$0.183205\pi$
$978$	1387.21	−	7601.68i	0.0453559	−	0.248543i
$979$		−	40697.1i		−	1.32858i
$980$		0			0
$981$	19784.9	−	52403.7i	0.643917	−	1.70553i
$982$	−16467.8	−	16467.8i	−0.535142	−	0.535142i
$983$	16337.3	+	16337.3i	0.530090	+	0.530090i	0.920599	−	0.390509i	$-0.127701\pi$
$983$	16337.3	+	16337.3i	0.530090	+	0.530090i	−0.390509	+	0.920599i	$0.627701\pi$
$984$	−36929.0	+	25530.8i	−1.19639	+	0.827128i
$985$		0			0
$986$			5548.02i			0.179194i
$987$	−17010.4	−	3104.18i	−0.548578	−	0.100108i
$988$	3603.53	−	3603.53i	0.116036	−	0.116036i
$989$	−2976.52			−0.0957004
$990$		0			0
$991$	−18296.9			−0.586500			−0.293250	−	0.956036i	$-0.594737\pi$
$991$	−18296.9			−0.586500			−0.293250	+	0.956036i	$0.594737\pi$
$992$	−32223.9	+	32223.9i	−1.03136	+	1.03136i
$993$	49018.8	+	8945.31i	1.56653	+	0.285872i
$994$			5449.79i			0.173900i
$995$		0			0
$996$	4521.93	−	3126.24i	0.143858	−	0.0994564i
$997$	−20406.0	−	20406.0i	−0.648210	−	0.648210i	0.304350	−	0.952560i	$-0.401561\pi$
$997$	−20406.0	−	20406.0i	−0.648210	−	0.648210i	−0.952560	+	0.304350i	$0.901561\pi$
$998$	−12926.9	−	12926.9i	−0.410015	−	0.410015i
$999$	−6107.09	−	24541.4i	−0.193413	−	0.777232i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.e.c.32.3		8
3.2	odd	2	inner	75.4.e.c.32.2		8
5.2	odd	4		15.4.e.a.8.3	yes	8
5.3	odd	4	inner	75.4.e.c.68.2		8
5.4	even	2		15.4.e.a.2.2	✓	8
15.2	even	4		15.4.e.a.8.2	yes	8
15.8	even	4	inner	75.4.e.c.68.3		8
15.14	odd	2		15.4.e.a.2.3	yes	8
20.7	even	4		240.4.v.c.113.2		8
20.19	odd	2		240.4.v.c.17.1		8
60.47	odd	4		240.4.v.c.113.1		8
60.59	even	2		240.4.v.c.17.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.e.a.2.2	✓	8	5.4	even	2
15.4.e.a.2.3	yes	8	15.14	odd	2
15.4.e.a.8.2	yes	8	15.2	even	4
15.4.e.a.8.3	yes	8	5.2	odd	4
75.4.e.c.32.2		8	3.2	odd	2	inner
75.4.e.c.32.3		8	1.1	even	1	trivial
75.4.e.c.68.2		8	5.3	odd	4	inner
75.4.e.c.68.3		8	15.8	even	4	inner
240.4.v.c.17.1		8	20.19	odd	2
240.4.v.c.17.2		8	60.59	even	2
240.4.v.c.113.1		8	60.47	odd	4
240.4.v.c.113.2		8	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.18766 - 1.18766i) q^{2} +(-5.11173 - 0.932827i) q^{3} +5.17891i q^{4} +(-7.17891 + 4.96314i) q^{6} +(13.3578 + 13.3578i) q^{7} +(15.6521 + 15.6521i) q^{8} +(25.2597 + 9.53673i) q^{9} +28.7164i q^{11} +(4.83102 - 26.4732i) q^{12} +(-14.1789 + 14.1789i) q^{13} +31.7292 q^{14} -4.25236 q^{16} +(18.5587 - 18.5587i) q^{17} +(41.3264 - 18.6736i) q^{18} +49.0735i q^{19} +(-55.8211 - 80.7421i) q^{21} +(34.1055 + 34.1055i) q^{22} +(-37.7738 - 37.7738i) q^{23} +(-65.4088 - 94.6102i) q^{24} +33.6796i q^{26} +(-120.225 - 72.3121i) q^{27} +(-69.1789 + 69.1789i) q^{28} -125.854 q^{29} +247.367 q^{31} +(-130.267 + 130.267i) q^{32} +(26.7874 - 146.791i) q^{33} -44.0829i q^{34} +(-49.3898 + 130.818i) q^{36} +(127.463 + 127.463i) q^{37} +(58.2828 + 58.2828i) q^{38} +(85.7053 - 59.2524i) q^{39} -390.328i q^{41} +(-162.191 - 29.5978i) q^{42} +(39.3993 - 39.3993i) q^{43} -148.720 q^{44} -89.7251 q^{46} +(124.560 - 124.560i) q^{47} +(21.7369 + 3.96671i) q^{48} +13.8625i q^{49} +(-112.179 + 77.5549i) q^{51} +(-73.4313 - 73.4313i) q^{52} +(-160.441 - 160.441i) q^{53} +(-228.669 + 56.9040i) q^{54} +418.156i q^{56} +(45.7770 - 250.850i) q^{57} +(-149.473 + 149.473i) q^{58} +729.423 q^{59} +2.00000 q^{61} +(293.789 - 293.789i) q^{62} +(210.024 + 464.804i) q^{63} +275.409i q^{64} +(-142.524 - 206.153i) q^{66} +(329.987 + 329.987i) q^{67} +(96.1136 + 96.1136i) q^{68} +(157.853 + 228.326i) q^{69} +171.760i q^{71} +(246.097 + 544.637i) q^{72} +(279.927 - 279.927i) q^{73} +302.767 q^{74} -254.147 q^{76} +(-383.589 + 383.589i) q^{77} +(31.4172 - 172.161i) q^{78} +48.0189i q^{79} +(547.102 + 481.789i) q^{81} +(-463.578 - 463.578i) q^{82} +(144.451 + 144.451i) q^{83} +(418.156 - 289.092i) q^{84} -93.5862i q^{86} +(643.334 + 117.400i) q^{87} +(-449.473 + 449.473i) q^{88} -1417.21 q^{89} -378.799 q^{91} +(195.627 - 195.627i) q^{92} +(-1264.48 - 230.751i) q^{93} -295.872i q^{94} +(787.409 - 544.375i) q^{96} +(-908.111 - 908.111i) q^{97} +(16.4640 + 16.4640i) q^{98} +(-273.861 + 725.367i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{3} - 12 q^{6} + 16 q^{7} - 132 q^{12} - 68 q^{13} + 284 q^{16} + 240 q^{18} - 492 q^{21} + 500 q^{22} - 702 q^{27} - 508 q^{28} + 616 q^{31} + 240 q^{33} - 804 q^{36} + 1156 q^{37} - 540 q^{42}+ \cdots - 1904 q^{97}+O(q^{100}) \) 8 * q + 6 * q^3 - 12 * q^6 + 16 * q^7 - 132 * q^12 - 68 * q^13 + 284 * q^16 + 240 * q^18 - 492 * q^21 + 500 * q^22 - 702 * q^27 - 508 * q^28 + 616 * q^31 + 240 * q^33 - 804 * q^36 + 1156 * q^37 - 540 * q^42 - 548 * q^43 + 736 * q^46 + 1116 * q^48 - 852 * q^51 - 224 * q^52 - 684 * q^57 - 60 * q^58 + 16 * q^61 - 1428 * q^63 + 2040 * q^66 - 404 * q^67 + 1800 * q^72 + 2512 * q^73 - 1488 * q^76 + 360 * q^78 + 288 * q^81 - 2800 * q^82 + 1680 * q^87 - 2460 * q^88 - 1304 * q^91 - 3408 * q^93 + 4164 * q^96 - 1904 * q^97

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