LMFDB - Embedded newform 75.4.e.c.32.2

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			32.2
Root			$-1.18766 - 1.18766i$ of defining polynomial
Character	$\chi$	$=$	75.32
Dual form			75.4.e.c.68.2

$q$-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} +(0.932827 + 5.11173i) 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} +(-26.4732 + 4.83102i) q^{12} +(-14.1789 + 14.1789i) q^{13} -31.7292 q^{14} -4.25236 q^{16} +(-18.5587 + 18.5587i) q^{17} +(18.6736 - 41.3264i) 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} +(-72.3121 - 120.225i) q^{27} +(-69.1789 + 69.1789i) q^{28} +125.854 q^{29} +247.367 q^{31} +(130.267 - 130.267i) q^{32} +(146.791 - 26.7874i) 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} +(-29.5978 - 162.191i) q^{42} +(39.3993 - 39.3993i) q^{43} +148.720 q^{44} -89.7251 q^{46} +(-124.560 + 124.560i) q^{47} +(-3.96671 - 21.7369i) 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} +(-250.850 + 45.7770i) q^{57} +(-149.473 + 149.473i) q^{58} -729.423 q^{59} +2.00000 q^{61} +(-293.789 + 293.789i) q^{62} +(-464.804 - 210.024i) 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} +(544.637 + 246.097i) q^{72} +(279.927 - 279.927i) q^{73} -302.767 q^{74} -254.147 q^{76} +(383.589 - 383.589i) q^{77} +(172.161 - 31.4172i) 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} +(117.400 + 643.334i) q^{87} +(-449.473 + 449.473i) q^{88} +1417.21 q^{89} -378.799 q^{91} +(-195.627 + 195.627i) q^{92} +(230.751 + 1264.48i) 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	−0.885170	−	0.465267i	$-0.845958\pi$
$2$	−1.18766	+	1.18766i	−0.419903	+	0.419903i	0.465267	+	0.885170i	$0.345958\pi$
$3$	0.932827	+	5.11173i	0.179523	+	0.983754i
$3$	0.932827	+	5.11173i	0.179523	+	0.983754i
$4$			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	$-0.871243\pi$
$11$		−	28.7164i		−	0.787121i	0.919299	−	0.393560i	$-0.128757\pi$
$12$	−26.4732	+	4.83102i	−0.636846	+	0.116216i
$13$	−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	−0.826990	−	0.562217i	$-0.809949\pi$
$17$	−18.5587	+	18.5587i	−0.264773	+	0.264773i	0.562217	+	0.826990i	$0.309949\pi$
$18$	18.6736	−	41.3264i	0.244522	−	0.541152i
$19$			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	0.857288	−	0.514837i	$-0.172148\pi$
$23$	37.7738	+	37.7738i	0.342451	+	0.342451i	−0.514837	+	0.857288i	$0.672148\pi$
$24$	65.4088	−	94.6102i	0.556313	−	0.804676i
$25$		0			0
$26$		−	33.6796i		−	0.254042i
$27$	−72.3121	−	120.225i	−0.515425	−	0.856935i
$28$	−69.1789	+	69.1789i	−0.466914	+	0.466914i
$29$	125.854			0.805882			0.402941	−	0.915226i	$-0.367988\pi$
$29$	125.854			0.805882			0.402941	+	0.915226i	$0.367988\pi$
$30$		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
$33$	146.791	−	26.7874i	0.774333	−	0.141306i
$34$		−	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	$0.266789\pi$
$41$			390.328i			1.48680i	−0.668845	+	0.743402i	$0.733211\pi$
$42$	−29.5978	−	162.191i	−0.108739	−	0.595873i
$43$	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	−0.873464	−	0.486889i	$-0.838132\pi$
$47$	−124.560	+	124.560i	−0.386575	+	0.386575i	0.486889	+	0.873464i	$0.338132\pi$
$48$	−3.96671	−	21.7369i	−0.0119280	−	0.0653637i
$49$			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	0.883759	−	0.467943i	$-0.155005\pi$
$53$	160.441	+	160.441i	0.415816	+	0.415816i	−0.467943	+	0.883759i	$0.655005\pi$
$54$	228.669	+	56.9040i	0.576257	+	0.143401i
$55$		0			0
$56$		−	418.156i		−	0.997830i
$57$	−250.850	+	45.7770i	−0.582912	+	0.106374i
$58$	−149.473	+	149.473i	−0.338392	+	0.338392i
$59$	−729.423			−1.60954			−0.804769	−	0.593588i	$-0.797711\pi$
$59$	−729.423			−1.60954			−0.804769	+	0.593588i	$0.797711\pi$
$60$		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
$63$	−464.804	−	210.024i	−0.929520	−	0.420009i
$64$			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	$-0.954148\pi$
$71$		−	171.760i		−	0.287100i	0.989643	−	0.143550i	$-0.0458518\pi$
$72$	544.637	+	246.097i	0.891474	+	0.402817i
$73$	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
$78$	172.161	−	31.4172i	0.249915	−	0.0456064i
$79$			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	0.605111	−	0.796141i	$-0.293129\pi$
$83$	−144.451	−	144.451i	−0.191031	−	0.191031i	−0.796141	+	0.605111i	$0.793129\pi$
$84$	−418.156	−	289.092i	−0.543150	−	0.375507i
$85$		0			0
$86$			93.5862i			0.117345i
$87$	117.400	+	643.334i	0.144674	+	0.792789i
$88$	−449.473	+	449.473i	−0.544477	+	0.544477i
$89$	1417.21			1.68790			0.843952	−	0.536419i	$-0.180223\pi$
$89$	1417.21			1.68790			0.843952	+	0.536419i	$0.180223\pi$
$90$		0			0
$91$	−378.799			−0.436361
$92$	−195.627	+	195.627i	−0.221690	+	0.221690i
$93$	230.751	+	1264.48i	0.257288	+	1.40989i
$94$		−	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	$0.0531926\pi$
$101$			337.668i			0.332665i	−0.986070	+	0.166333i	$0.946807\pi$
$102$	225.340	−	41.1217i	0.218745	−	0.0399182i
$103$	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	−0.384481	−	0.923133i	$-0.625620\pi$
$107$	596.188	−	596.188i	0.538651	−	0.538651i	0.923133	+	0.384481i	$0.125620\pi$
$108$	622.632	−	374.498i	0.554748	−	0.333667i
$109$		−	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	0.585154	−	0.810922i	$-0.301034\pi$
$113$	−271.193	−	271.193i	−0.225767	−	0.225767i	−0.810922	+	0.585154i	$0.801034\pi$
$114$	243.558	−	352.294i	0.200099	−	0.289433i
$115$		0			0
$116$			651.788i			0.521698i
$117$	222.934	−	493.375i	0.176156	−	0.389851i
$118$	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
$123$	−1995.25	+	364.108i	−1.46265	+	0.266915i
$124$			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	$-0.770540\pi$
$131$		−	1979.28i		−	1.32008i	0.751231	−	0.660039i	$-0.229460\pi$
$132$	138.730	+	760.216i	0.0914763	+	0.501275i
$133$	−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	−0.847404	−	0.530948i	$-0.821836\pi$
$137$	−507.451	+	507.451i	−0.316456	+	0.316456i	0.530948	+	0.847404i	$0.321836\pi$
$138$	−83.6979	−	458.651i	−0.0516293	−	0.282920i
$139$			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
$147$	−70.8616	+	12.9313i	−0.0397590	+	0.00725550i
$148$	−660.121	+	660.121i	−0.366632	+	0.366632i
$149$	363.356			0.199780			0.0998902	−	0.994998i	$-0.468151\pi$
$149$	363.356			0.199780			0.0998902	+	0.994998i	$0.468151\pi$
$150$		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
$153$	291.797	−	645.774i	0.154185	−	0.341227i
$154$			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
$162$	−77.5696	+	1221.98i	−0.0376200	+	0.592639i
$163$	−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	$0.696888\pi$
$167$	−3009.65	+	3009.65i	−1.39457	+	1.39457i	−0.814724	+	0.579848i	$0.803112\pi$
$168$	2137.50	−	390.067i	0.981619	−	0.179133i
$169$			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	0.176086	−	0.984375i	$-0.443656\pi$
$173$	−1839.23	−	1839.23i	−0.808288	−	0.808288i	−0.984375	+	0.176086i	$0.943656\pi$
$174$	−903.497	−	624.633i	−0.393643	−	0.272145i
$175$		0			0
$176$			122.113i			0.0522987i
$177$	−680.425	−	3728.61i	−0.288948	−	1.58339i
$178$	−1683.16	+	1683.16i	−0.708755	+	0.708755i
$179$	−821.582			−0.343061			−0.171530	−	0.985179i	$-0.554871\pi$
$179$	−821.582			−0.343061			−0.171530	+	0.985179i	$0.554871\pi$
$180$		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
$183$	1.86565	+	10.2235i	0.000753623	+	0.00412973i
$184$		−	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	$-0.943562\pi$
$191$		−	931.167i		−	0.352758i	0.984322	−	0.176379i	$-0.0564385\pi$
$192$	−1407.82	+	256.909i	−0.529169	+	0.0965666i
$193$	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	$-0.472279\pi$
$197$	2995.05	−	2995.05i	1.08319	−	1.08319i	0.996210	−	0.0869796i	$-0.0277215\pi$
$198$	−1186.75	−	536.238i	−0.425952	−	0.192469i
$199$		−	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
$207$	−1314.39	−	593.915i	−0.441336	−	0.199420i
$208$	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
$213$	877.989	−	160.222i	0.282436	−	0.0515409i
$214$			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
$222$	−282.429	−	1547.67i	−0.0853847	−	0.467894i
$223$	−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	−0.472362	−	0.881404i	$-0.656599\pi$
$227$	1398.96	−	1398.96i	0.409042	−	0.409042i	0.881404	+	0.472362i	$0.156599\pi$
$228$	−237.075	−	1299.13i	−0.0688626	−	0.377356i
$229$		−	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	0.928427	−	0.371515i	$-0.121162\pi$
$233$	1980.71	+	1980.71i	0.556912	+	0.556912i	−0.371515	+	0.928427i	$0.621162\pi$
$234$	321.193	+	850.735i	0.0897309	+	0.237668i
$235$		0			0
$236$		−	3777.61i		−	1.04196i
$237$	−245.460	+	44.7933i	−0.0672756	+	0.0122769i
$238$	588.851	−	588.851i	0.160376	−	0.160376i
$239$	−2976.20			−0.805500			−0.402750	−	0.915310i	$-0.631946\pi$
$239$	−2976.20			−0.805500			−0.402750	+	0.915310i	$0.631946\pi$
$240$		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
$243$	2973.13	+	2347.21i	0.784882	+	0.619645i
$244$			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	$0.0621145\pi$
$251$			1542.14i			0.387805i	−0.981021	+	0.193902i	$0.937885\pi$
$252$	1087.70	−	2407.18i	0.271898	−	0.601738i
$253$	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	−0.653144	−	0.757234i	$-0.726550\pi$
$257$	428.853	−	428.853i	0.104090	−	0.104090i	0.757234	+	0.653144i	$0.226550\pi$
$258$	−478.388	+	87.2997i	−0.115438	+	0.0210660i
$259$			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	$-0.913120\pi$
$263$	−5256.99	−	5256.99i	−1.23255	−	1.23255i	−0.269565	−	0.962982i	$-0.586880\pi$
$264$	−2716.87	−	1878.31i	−0.633377	−	0.437885i
$265$		0			0
$266$		−	1557.06i		−	0.358908i
$267$	1322.01	+	7244.38i	0.303017	+	1.66048i
$268$	−1708.97	+	1708.97i	−0.389523	+	0.389523i
$269$	−1930.34			−0.437528			−0.218764	−	0.975778i	$-0.570203\pi$
$269$	−1930.34			−0.437528			−0.218764	+	0.975778i	$0.570203\pi$
$270$		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
$273$	−353.353	−	1936.32i	−0.0783367	−	0.429272i
$274$		−	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	$0.138909\pi$
$281$			3981.96i			0.845351i	−0.906281	+	0.422676i	$0.861091\pi$
$282$	1512.42	−	275.997i	0.319373	−	0.0582815i
$283$	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
$288$	−2048.19	+	4532.83i	−0.419064	+	0.927429i
$289$			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	0.995462	−	0.0951570i	$-0.0303353\pi$
$293$	4515.35	+	4515.35i	0.900305	+	0.900305i	−0.0951570	+	0.995462i	$0.530335\pi$
$294$	68.8017	−	99.5179i	0.0136483	−	0.0197415i
$295$		0			0
$296$		−	3990.14i		−	0.783521i
$297$	−3452.42	+	2076.54i	−0.674511	+	0.405701i
$298$	−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
$303$	−1726.07	+	314.986i	−0.327261	+	0.0597210i
$304$		−	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	$0.119134\pi$
$311$			4010.21i			0.731184i	−0.930775	+	0.365592i	$0.880866\pi$
$312$	414.044	+	2268.89i	0.0751303	+	0.411701i
$313$	−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	−0.926844	−	0.375447i	$-0.877489\pi$
$317$	−3112.10	+	3112.10i	−0.551397	+	0.551397i	0.375447	+	0.926844i	$0.377489\pi$
$318$	−355.500	−	1948.08i	−0.0626901	−	0.343531i
$319$		−	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
$327$	10604.8	−	1935.24i	1.79342	−	0.327275i
$328$	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
$333$	−4435.26	−	2004.10i	−0.729883	−	0.329801i
$334$		−	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
$342$	2028.03	+	916.377i	0.320653	+	0.144889i
$343$	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	$-0.397491\pi$
$347$	8177.44	−	8177.44i	1.26509	−	1.26509i	0.948592	−	0.316503i	$-0.102509\pi$
$348$	−3331.77	+	608.005i	−0.513223	+	0.0936566i
$349$			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	0.999060	−	0.0433491i	$-0.0138028\pi$
$353$	6338.53	+	6338.53i	0.955711	+	0.955711i	−0.0433491	+	0.999060i	$0.513803\pi$
$354$	5236.46	+	3620.23i	0.786199	+	0.543539i
$355$		0			0
$356$			7339.58i			1.09269i
$357$	−462.501	−	2534.43i	−0.0685663	−	0.375732i
$358$	975.763	−	975.763i	0.144052	−	0.144052i
$359$	8827.47			1.29776			0.648880	−	0.760890i	$-0.275238\pi$
$359$	8827.47			1.29776			0.648880	+	0.760890i	$0.275238\pi$
$360$		0			0
$361$	4450.80			0.648899
$362$	−2748.50	+	2748.50i	−0.399055	+	0.399055i
$363$	472.353	+	2588.42i	0.0682978	+	0.374261i
$364$		−	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
$372$	−6548.60	+	1195.04i	−0.912713	+	0.166559i
$373$	−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
$378$	2294.40	+	3814.63i	0.312200	+	0.519057i
$379$		−	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	0.584880	−	0.811120i	$-0.301142\pi$
$383$	−1695.77	−	1695.77i	−0.226240	−	0.226240i	−0.811120	+	0.584880i	$0.801142\pi$
$384$	−2988.11	+	4322.14i	−0.397100	+	0.574383i
$385$		0			0
$386$			6231.15i			0.821650i
$387$	−619.472	+	1370.95i	−0.0813683	+	0.180076i
$388$	4703.02	−	4703.02i	0.615361	−	0.615361i
$389$	12362.2			1.61128			0.805640	−	0.592405i	$-0.201822\pi$
$389$	12362.2			1.61128			0.805640	+	0.592405i	$0.201822\pi$
$390$		0			0
$391$	−1402.06			−0.181343
$392$	216.978	−	216.978i	0.0279567	−	0.0279567i
$393$	10117.5	−	1846.32i	1.29863	−	0.236984i
$394$			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	$-0.954631\pi$
$401$		−	2281.30i		−	0.284096i	0.989860	−	0.142048i	$-0.0453688\pi$
$402$	−731.175	−	4006.72i	−0.0907156	−	0.497107i
$403$	−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
$408$	541.939	+	2969.74i	0.0657597	+	0.360352i
$409$			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
$417$	−349.813	+	63.8364i	−0.0410801	+	0.00749660i
$418$	−1673.67	+	1673.67i	−0.195842	+	0.195842i
$419$	2142.28			0.249779			0.124889	−	0.992171i	$-0.460142\pi$
$419$	2142.28			0.249779			0.124889	+	0.992171i	$0.460142\pi$
$420$		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
$423$	1958.46	−	4334.26i	0.225115	−	0.498200i
$424$		−	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	$-0.659065\pi$
$431$		−	15707.3i		−	1.75543i	0.479179	−	0.877717i	$-0.340935\pi$
$432$	307.497	+	511.238i	0.0342464	+	0.0569374i
$433$	−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
$438$	−3398.88	+	620.253i	−0.370787	+	0.0676640i
$439$			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	0.594115	−	0.804380i	$-0.297502\pi$
$443$	−1960.53	−	1960.53i	−0.210265	−	0.210265i	−0.804380	+	0.594115i	$0.797502\pi$
$444$	−3990.14	−	2758.58i	−0.426495	−	0.294857i
$445$		0			0
$446$		−	6723.34i		−	0.713810i
$447$	338.948	+	1857.38i	0.0358651	+	0.196535i
$448$	−3678.86	+	3678.86i	−0.387968	+	0.387968i
$449$	−17849.2			−1.87607			−0.938036	−	0.346537i	$-0.887357\pi$
$449$	−17849.2			−1.87607			−0.938036	+	0.346537i	$0.887357\pi$
$450$		0			0
$451$	11208.8			1.17029
$452$	1404.49	−	1404.49i	0.146154	−	0.146154i
$453$	−1943.21	−	10648.5i	−0.201545	−	1.10443i
$454$			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	$-0.852505\pi$
$461$		−	8848.20i		−	0.893930i	0.894552	−	0.446965i	$-0.147495\pi$
$462$	−4657.55	+	849.944i	−0.469024	+	0.0855908i
$463$	−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	−0.776618	−	0.629971i	$-0.783067\pi$
$467$	−1479.96	+	1479.96i	−0.146647	+	0.146647i	0.629971	+	0.776618i	$0.283067\pi$
$468$	2555.14	+	1154.56i	0.252375	+	0.114037i
$469$			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
$477$	−5582.76	−	2522.60i	−0.535885	−	0.242143i
$478$	3534.73	−	3534.73i	0.338231	−	0.338231i
$479$	5039.60			0.480720			0.240360	−	0.970684i	$-0.422734\pi$
$479$	5039.60			0.480720			0.240360	+	0.970684i	$0.422734\pi$
$480$		0			0
$481$	−3614.58			−0.342642
$482$	−2179.89	+	2179.89i	−0.205999	+	0.205999i
$483$	−5158.51	+	941.362i	−0.485963	+	0.0886821i
$484$			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	$0.219916\pi$
$491$			13865.7i			1.27444i	−0.770681	+	0.637221i	$0.780084\pi$
$492$	−1885.68	−	10333.2i	−0.172791	−	0.946865i
$493$	−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
$498$	320.070	+	1753.93i	0.0288006	+	0.157822i
$499$		−	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	0.964106	−	0.265517i	$-0.0855427\pi$
$503$	7880.86	+	7880.86i	0.698589	+	0.698589i	−0.265517	+	0.964106i	$0.585543\pi$
$504$	3987.84	+	10562.5i	0.352445	+	0.933513i
$505$		0			0
$506$			2576.58i			0.226370i
$507$	−9175.14	+	1674.35i	−0.803713	+	0.146667i
$508$	546.829	−	546.829i	0.0477590	−	0.0477590i
$509$	1788.46			0.155741			0.0778704	−	0.996963i	$-0.475188\pi$
$509$	1788.46			0.155741			0.0778704	+	0.996963i	$0.475188\pi$
$510$		0			0
$511$	7478.42			0.647408
$512$	−1086.41	+	1086.41i	−0.0937754	+	0.0937754i
$513$	5899.84	−	3548.60i	0.507766	−	0.305409i
$514$			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	$0.278444\pi$
$521$			18251.6i			1.53478i	−0.641183	+	0.767388i	$0.721556\pi$
$522$	2350.15	−	5201.11i	0.197056	−	0.436104i
$523$	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
$528$	−624.207	+	113.910i	−0.0514491	+	0.00938880i
$529$		−	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
$537$	−766.393	−	4199.71i	−0.0615872	−	0.337488i
$538$	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
$543$	2158.75	+	11829.6i	0.170609	+	0.934911i
$544$			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
$552$	6044.52	−	1103.05i	0.466072	−	0.0850522i
$553$	−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	−0.293006	−	0.956111i	$-0.594656\pi$
$557$	8716.96	−	8716.96i	0.663105	−	0.663105i	0.956111	+	0.293006i	$0.0946555\pi$
$558$	4619.23	−	10222.8i	0.350444	−	0.775566i
$559$			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	0.993077	−	0.117462i	$-0.0374759\pi$
$563$	11697.0	+	11697.0i	0.875615	+	0.875615i	−0.117462	+	0.993077i	$0.537476\pi$
$564$	2695.76	−	3899.27i	0.201262	−	0.291115i
$565$		0			0
$566$			9721.40i			0.721945i
$567$	13743.7	+	872.435i	1.01796	+	0.0646188i
$568$	−2688.40	+	2688.40i	−0.198596	+	0.198596i
$569$	−11517.5			−0.848576			−0.424288	−	0.905527i	$-0.639476\pi$
$569$	−11517.5			−0.848576			−0.424288	+	0.905527i	$0.639476\pi$
$570$		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
$573$	4759.88	−	868.617i	0.347027	−	0.0633281i
$574$		−	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
$582$	2012.16	+	11026.3i	0.143311	+	0.785320i
$583$	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	−0.340767	−	0.940148i	$-0.610687\pi$
$587$	8524.33	−	8524.33i	0.599381	−	0.599381i	0.940148	+	0.340767i	$0.110687\pi$
$588$	−66.9702	−	366.986i	−0.00469695	−	0.0257385i
$589$			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	0.530491	−	0.847691i	$-0.322008\pi$
$593$	−4580.53	−	4580.53i	−0.317200	−	0.317200i	−0.847691	+	0.530491i	$0.822008\pi$
$594$	1634.08	−	6566.55i	0.112874	−	0.453584i
$595$		0			0
$596$			1881.79i			0.129331i
$597$	559.518	−	102.105i	0.0383577	−	0.00699979i
$598$	1272.20	−	1272.20i	0.0869971	−	0.0869971i
$599$	10195.8			0.695477			0.347738	−	0.937592i	$-0.386950\pi$
$599$	10195.8			0.695477			0.347738	+	0.937592i	$0.386950\pi$
$600$		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
$603$	−11482.4	−	5188.36i	−0.775452	−	0.350392i
$604$		−	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
$612$	3344.41	+	1511.19i	0.220898	+	0.0998140i
$613$	−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	$0.539830\pi$
$617$	−17118.9	+	17118.9i	−1.11698	+	1.11698i	−0.992182	+	0.124803i	$0.960170\pi$
$618$	−11334.7	+	2068.43i	−0.737778	+	0.134635i
$619$			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
$627$	1314.55	+	7203.53i	0.0837291	+	0.458822i
$628$	−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
$633$	2532.57	+	13878.0i	0.159021	+	0.871410i
$634$		−	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	$-0.875402\pi$
$641$		−	12383.2i		−	0.763035i	0.924362	−	0.381518i	$-0.124598\pi$
$642$	−7238.95	+	1321.02i	−0.445013	+	0.0812092i
$643$	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	−0.776014	−	0.630716i	$-0.782761\pi$
$647$	−2391.21	+	2391.21i	−0.145299	+	0.145299i	0.630716	+	0.776014i	$0.282761\pi$
$648$	−16104.3	−	1022.28i	−0.976292	−	0.0619738i
$649$			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	0.00379136	−	0.999993i	$-0.498793\pi$
$653$	−16623.3	−	16623.3i	−0.996201	−	0.996201i	−0.999993	+	0.00379136i	$0.998793\pi$
$654$	−10296.5	+	14893.4i	−0.615636	+	0.890484i
$655$		0			0
$656$		−	1659.81i		−	0.0987878i
$657$	−4401.27	+	9740.43i	−0.261354	+	0.578403i
$658$	3952.20	−	3952.20i	0.234153	−	0.234153i
$659$	−20089.8			−1.18754			−0.593768	−	0.804636i	$-0.702360\pi$
$659$	−20089.8			−1.18754			−0.593768	+	0.804636i	$0.702360\pi$
$660$		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
$663$	2690.22	−	490.931i	0.157586	−	0.0287574i
$664$			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
$672$	3246.40	+	17789.7i	0.186358	+	1.02121i
$673$	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	−0.987842	−	0.155462i	$-0.950313\pi$
$677$	−14662.4	+	14662.4i	−0.832380	+	0.832380i	0.155462	+	0.987842i	$0.450313\pi$
$678$	600.902	+	3292.84i	0.0340376	+	0.186520i
$679$		−	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	0.0996976	−	0.995018i	$-0.468212\pi$
$683$	−15981.2	−	15981.2i	−0.895320	−	0.895320i	−0.995018	+	0.0996976i	$0.968212\pi$
$684$	6419.67	−	2423.73i	0.358863	−	0.135488i
$685$		0			0
$686$			10443.3i			0.581233i
$687$	20091.1	−	3666.36i	1.11575	−	0.203611i
$688$	−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
$693$	−6031.14	+	13347.5i	−0.330598	+	0.731645i
$694$			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	$-0.939695\pi$
$701$		−	6990.49i		−	0.376643i	0.982107	−	0.188322i	$-0.0603048\pi$
$702$	−4049.11	+	2435.44i	−0.217698	+	0.130940i
$703$	−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
$708$	19310.2	−	3523.86i	1.02503	−	0.187055i
$709$		−	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
$717$	−2776.28	−	15213.6i	−0.144605	−	0.792414i
$718$	−10484.1	+	10484.1i	−0.544933	+	0.544933i
$719$	20143.8			1.04484			0.522418	−	0.852690i	$-0.325030\pi$
$719$	20143.8			1.04484			0.522418	+	0.852690i	$0.325030\pi$
$720$		0			0
$721$	24939.2			1.28819
$722$	−5286.05	+	5286.05i	−0.272474	+	0.272474i
$723$	1712.15	+	9382.32i	0.0880715	+	0.482617i
$724$			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
$732$	−52.9464	+	9.66205i	−0.00267344	+	0.000487868i
$733$	−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
$738$	16130.8	+	7288.81i	0.804586	+	0.363557i
$739$		−	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	0.559977	−	0.828508i	$-0.310810\pi$
$743$	−5438.49	−	5438.49i	−0.268531	−	0.268531i	−0.828508	+	0.559977i	$0.810810\pi$
$744$	16180.0	−	23403.5i	0.797294	−	1.15324i
$745$		0			0
$746$		−	8514.27i		−	0.417868i
$747$	5026.38	+	2271.20i	0.246192	+	0.111243i
$748$	−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
$753$	−7883.01	+	1438.55i	−0.381505	+	0.0696197i
$754$		−	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	$0.221148\pi$
$761$			26879.5i			1.28040i	−0.768210	+	0.640198i	$0.778852\pi$
$762$	233.958	+	1282.05i	0.0111226	+	0.0609498i
$763$	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
$768$	−3639.64	−	19944.6i	−0.171008	−	0.937097i
$769$			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	0.993335	−	0.115263i	$-0.0367711\pi$
$773$	18871.2	+	18871.2i	0.878072	+	0.878072i	−0.115263	+	0.993335i	$0.536771\pi$
$774$	−892.506	−	2363.96i	−0.0414476	−	0.109781i
$775$		0			0
$776$			28427.7i			1.31507i
$777$	−17406.8	+	3176.52i	−0.803688	+	0.146663i
$778$	−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
$783$	−9100.79	−	15130.8i	−0.415371	−	0.690588i
$784$		−	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
$792$	7067.03	−	15640.0i	0.317066	−	0.701697i
$793$	−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	−0.203013	−	0.979176i	$-0.565073\pi$
$797$	17463.9	−	17463.9i	0.776163	−	0.776163i	0.979176	+	0.203013i	$0.0650735\pi$
$798$	7959.28	−	1452.47i	0.353077	−	0.0644321i
$799$		−	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
$807$	−1800.68	−	9867.40i	−0.0785462	−	0.430420i
$808$	5285.22	−	5285.22i	0.230115	−	0.230115i
$809$	24097.9			1.04727			0.523633	−	0.851944i	$-0.324576\pi$
$809$	24097.9			1.04727			0.523633	+	0.851944i	$0.324576\pi$
$810$		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
$813$	−3042.57	−	16672.8i	−0.131252	−	0.719238i
$814$			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	$-0.797728\pi$
$821$		−	27925.3i		−	1.18709i	0.804802	−	0.593544i	$-0.202272\pi$
$822$	6161.50	−	1124.39i	0.261444	−	0.0477102i
$823$	−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	−0.980768	−	0.195178i	$-0.937471\pi$
$827$	−18683.3	+	18683.3i	−0.785590	+	0.785590i	0.195178	+	0.980768i	$0.437471\pi$
$828$	3075.83	−	6807.11i	0.129097	−	0.285705i
$829$		−	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
$837$	−17887.6	−	29739.6i	−0.738695	−	1.22814i
$838$	−2544.31	+	2544.31i	−0.104883	+	0.104883i
$839$	−30903.4			−1.27164			−0.635820	−	0.771838i	$-0.719338\pi$
$839$	−30903.4			−1.27164			−0.635820	+	0.771838i	$0.719338\pi$
$840$		0			0
$841$	−8549.68			−0.350555
$842$	10557.5	−	10557.5i	0.432109	−	0.432109i
$843$	−20354.7	+	3714.48i	−0.831618	+	0.151760i
$844$			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
$852$	829.774	+	4547.02i	0.0333657	+	0.182839i
$853$	−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	−0.927066	−	0.374899i	$-0.877677\pi$
$857$	−13852.9	+	13852.9i	−0.552167	+	0.552167i	0.374899	+	0.927066i	$0.377677\pi$
$858$	−902.189	−	4943.85i	−0.0358977	−	0.196713i
$859$			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	0.567481	−	0.823386i	$-0.307918\pi$
$863$	−6487.75	−	6487.75i	−0.255905	−	0.255905i	−0.823386	+	0.567481i	$0.807918\pi$
$864$	−25081.2	−	6241.43i	−0.987594	−	0.245761i
$865$		0			0
$866$		−	12900.0i		−	0.506187i
$867$	−21592.8	+	3940.40i	−0.845823	+	0.154352i
$868$	−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
$873$	31599.0	+	14278.2i	1.22504	+	0.553543i
$874$		−	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	$0.218993\pi$
$881$			33209.0i			1.26996i	−0.772527	+	0.634982i	$0.781007\pi$
$882$	572.889	+	258.863i	0.0218709	+	0.00988251i
$883$	−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	−0.0873406	−	0.996179i	$-0.527837\pi$
$887$	24008.9	−	24008.9i	0.908838	−	0.908838i	0.996179	+	0.0873406i	$0.0278369\pi$
$888$	20396.5	−	3722.11i	0.770792	−	0.140660i
$889$		−	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
$897$	−999.226	−	5475.59i	−0.0371942	−	0.203818i
$898$	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
$903$	981.871	+	5380.49i	0.0361845	+	0.198285i
$904$			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	$-0.792071\pi$
$911$		−	33422.1i		−	1.21550i	0.794127	−	0.607752i	$-0.207929\pi$
$912$	1066.71	−	194.660i	0.0387304	−	0.00706781i
$913$	−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
$918$	−5299.85	+	3187.73i	−0.190546	+	0.114609i
$919$			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
$927$	−14677.4	+	32482.6i	−0.520033	+	1.15088i
$928$	16394.7	−	16394.7i	0.579938	−	0.579938i
$929$	−5721.21			−0.202053			−0.101026	−	0.994884i	$-0.532213\pi$
$929$	−5721.21			−0.202053			−0.101026	+	0.994884i	$0.532213\pi$
$930$		0			0
$931$	−680.282			−0.0239477
$932$	−10257.9	+	10257.9i	−0.360524	+	0.360524i
$933$	−20499.1	+	3740.83i	−0.719305	+	0.131264i
$934$		−	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	$-0.946678\pi$
$941$		−	9625.77i		−	0.333466i	0.986002	−	0.166733i	$-0.0533217\pi$
$942$	−2677.80	−	14673.9i	−0.0926192	−	0.507538i
$943$	−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	−0.744400	−	0.667734i	$-0.767264\pi$
$947$	−2234.24	+	2234.24i	−0.0766664	+	0.0766664i	0.667734	+	0.744400i	$0.267264\pi$
$948$	−231.980	−	1271.21i	−0.00794765	−	0.0435518i
$949$			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	0.808281	−	0.588798i	$-0.200398\pi$
$953$	6457.14	+	6457.14i	0.219483	+	0.219483i	−0.588798	+	0.808281i	$0.700398\pi$
$954$	9626.45	−	3634.44i	0.326696	−	0.123343i
$955$		0			0
$956$		−	15413.5i		−	0.521451i
$957$	18474.3	−	3371.32i	0.624021	−	0.113876i
$958$	−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
$963$	−9373.84	+	20745.2i	−0.313674	+	0.694190i
$964$			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	$-0.884359\pi$
$971$		−	21504.3i		−	0.710716i	0.934730	−	0.355358i	$-0.115641\pi$
$972$	−12156.0	+	15397.6i	−0.401136	+	0.508104i
$973$	−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	−0.838890	−	0.544301i	$-0.816795\pi$
$977$	−8996.19	+	8996.19i	−0.294589	+	0.294589i	0.544301	+	0.838890i	$0.316795\pi$
$978$	7601.68	−	1387.21i	0.248543	−	0.0453559i
$979$		−	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	0.390509	−	0.920599i	$-0.372299\pi$
$983$	−16337.3	−	16337.3i	−0.530090	−	0.530090i	−0.920599	+	0.390509i	$0.872299\pi$
$984$	36929.0	+	25530.8i	1.19639	+	0.827128i
$985$		0			0
$986$		−	5548.02i		−	0.179194i
$987$	−3104.18	−	17010.4i	−0.100108	−	0.548578i
$988$	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
$993$	−8945.31	−	49018.8i	−0.285872	−	1.56653i
$994$			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.32.2		8
3.2	odd	2	inner	75.4.e.c.32.3		8
5.2	odd	4		15.4.e.a.8.2	yes	8
5.3	odd	4	inner	75.4.e.c.68.3		8
5.4	even	2		15.4.e.a.2.3	yes	8
15.2	even	4		15.4.e.a.8.3	yes	8
15.8	even	4	inner	75.4.e.c.68.2		8
15.14	odd	2		15.4.e.a.2.2	✓	8
20.7	even	4		240.4.v.c.113.1		8
20.19	odd	2		240.4.v.c.17.2		8
60.47	odd	4		240.4.v.c.113.2		8
60.59	even	2		240.4.v.c.17.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.e.a.2.2	✓	8	15.14	odd	2
15.4.e.a.2.3	yes	8	5.4	even	2
15.4.e.a.8.2	yes	8	5.2	odd	4
15.4.e.a.8.3	yes	8	15.2	even	4
75.4.e.c.32.2		8	1.1	even	1	trivial
75.4.e.c.32.3		8	3.2	odd	2	inner
75.4.e.c.68.2		8	15.8	even	4	inner
75.4.e.c.68.3		8	5.3	odd	4	inner
240.4.v.c.17.1		8	60.59	even	2
240.4.v.c.17.2		8	20.19	odd	2
240.4.v.c.113.1		8	20.7	even	4
240.4.v.c.113.2		8	60.47	odd	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} +(0.932827 + 5.11173i) 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} +(-26.4732 + 4.83102i) q^{12} +(-14.1789 + 14.1789i) q^{13} -31.7292 q^{14} -4.25236 q^{16} +(-18.5587 + 18.5587i) q^{17} +(18.6736 - 41.3264i) 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} +(-72.3121 - 120.225i) q^{27} +(-69.1789 + 69.1789i) q^{28} +125.854 q^{29} +247.367 q^{31} +(130.267 - 130.267i) q^{32} +(146.791 - 26.7874i) 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} +(-29.5978 - 162.191i) q^{42} +(39.3993 - 39.3993i) q^{43} +148.720 q^{44} -89.7251 q^{46} +(-124.560 + 124.560i) q^{47} +(-3.96671 - 21.7369i) 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} +(-250.850 + 45.7770i) q^{57} +(-149.473 + 149.473i) q^{58} -729.423 q^{59} +2.00000 q^{61} +(-293.789 + 293.789i) q^{62} +(-464.804 - 210.024i) 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} +(544.637 + 246.097i) q^{72} +(279.927 - 279.927i) q^{73} -302.767 q^{74} -254.147 q^{76} +(383.589 - 383.589i) q^{77} +(172.161 - 31.4172i) 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} +(117.400 + 643.334i) q^{87} +(-449.473 + 449.473i) q^{88} +1417.21 q^{89} -378.799 q^{91} +(-195.627 + 195.627i) q^{92} +(230.751 + 1264.48i) 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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