LMFDB - Embedded newform 45.5.g.e.37.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,5,Mod(28,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45.28"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	45.g (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$4.65164833877$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 60x^{5} + 1973x^{4} - 3300x^{3} + 1800x^{2} + 31560x + 276676 $ x^8 - 60x^5 + 1973x^4 - 3300x^3 + 1800x^2 + 31560*x + 276676
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			37.1
Root			$3.80336 - 3.80336i$ of defining polynomial
Character	$\chi$	$=$	45.37
Dual form			45.5.g.e.28.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-3.80336 - 3.80336i) q^{2} +12.9311i q^{4} +(-6.56915 + 24.1215i) q^{5} +(16.6149 + 16.6149i) q^{7} +(-11.6720 + 11.6720i) q^{8} +(116.728 - 66.7579i) q^{10} +215.278 q^{11} +(-29.9347 + 29.9347i) q^{13} -126.385i q^{14} +295.684 q^{16} +(3.97694 + 3.97694i) q^{17} +604.349i q^{19} +(-311.919 - 84.9467i) q^{20} +(-818.782 - 818.782i) q^{22} +(-376.379 + 376.379i) q^{23} +(-538.692 - 316.916i) q^{25} +227.705 q^{26} +(-214.850 + 214.850i) q^{28} +624.166i q^{29} +263.888 q^{31} +(-937.842 - 937.842i) q^{32} -30.2515i q^{34} +(-509.922 + 291.630i) q^{35} +(979.613 + 979.613i) q^{37} +(2298.56 - 2298.56i) q^{38} +(-204.870 - 358.220i) q^{40} +1579.25 q^{41} +(-30.8409 + 30.8409i) q^{43} +2783.80i q^{44} +2863.01 q^{46} +(-1785.16 - 1785.16i) q^{47} -1848.89i q^{49} +(843.498 + 3254.19i) q^{50} +(-387.089 - 387.089i) q^{52} +(-707.912 + 707.912i) q^{53} +(-1414.20 + 5192.84i) q^{55} -387.857 q^{56} +(2373.93 - 2373.93i) q^{58} -1366.84i q^{59} -2983.71 q^{61} +(-1003.66 - 1003.66i) q^{62} +2402.96i q^{64} +(-525.423 - 918.714i) q^{65} +(-986.003 - 986.003i) q^{67} +(-51.4265 + 51.4265i) q^{68} +(3048.59 + 830.242i) q^{70} -68.0686 q^{71} +(2476.90 - 2476.90i) q^{73} -7451.65i q^{74} -7814.93 q^{76} +(3576.83 + 3576.83i) q^{77} -6047.25i q^{79} +(-1942.39 + 7132.33i) q^{80} +(-6006.45 - 6006.45i) q^{82} +(5158.95 - 5158.95i) q^{83} +(-122.055 + 69.8047i) q^{85} +234.598 q^{86} +(-2512.72 + 2512.72i) q^{88} +7028.50i q^{89} -994.722 q^{91} +(-4867.01 - 4867.01i) q^{92} +13579.2i q^{94} +(-14577.8 - 3970.06i) q^{95} +(10869.3 + 10869.3i) q^{97} +(-7032.00 + 7032.00i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 84 q^{5} + 20 q^{7} - 180 q^{8} + 104 q^{10} + 288 q^{11} - 340 q^{13} + 620 q^{16} - 900 q^{17} - 564 q^{20} - 1100 q^{22} + 1560 q^{23} - 1204 q^{25} + 3024 q^{26} + 3580 q^{28} - 512 q^{31} - 4980 q^{32}+ \cdots - 46440 q^{98}+O(q^{100}) $ 8 * q + 84 * q^5 + 20 * q^7 - 180 * q^8 + 104 * q^10 + 288 * q^11 - 340 * q^13 + 620 * q^16 - 900 * q^17 - 564 * q^20 - 1100 * q^22 + 1560 * q^23 - 1204 * q^25 + 3024 * q^26 + 3580 * q^28 - 512 * q^31 - 4980 * q^32 - 6600 * q^35 - 3820 * q^37 + 7680 * q^38 - 2952 * q^40 + 2712 * q^41 - 1240 * q^43 + 13528 * q^46 - 4800 * q^47 - 3744 * q^50 - 1240 * q^52 - 1020 * q^53 - 3644 * q^55 + 30720 * q^56 + 2340 * q^58 - 4760 * q^61 - 28680 * q^62 + 1212 * q^65 - 8920 * q^67 + 1920 * q^68 + 7380 * q^70 - 7536 * q^71 + 11600 * q^73 + 4344 * q^76 + 360 * q^77 - 10644 * q^80 - 27200 * q^82 + 32400 * q^83 - 15628 * q^85 - 14592 * q^86 - 14340 * q^88 + 16528 * q^91 + 31800 * q^92 - 18864 * q^95 + 58640 * q^97 - 46440 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−3.80336	−	3.80336i	−0.950841	−	0.950841i	0.0480062	−	0.998847i	$-0.484713\pi$
$2$	−3.80336	−	3.80336i	−0.950841	−	0.950841i	−0.998847	+	0.0480062i	$0.984713\pi$
$3$		0			0
$3$		0			0
$4$			12.9311i			0.808197i
$5$	−6.56915	+	24.1215i	−0.262766	+	0.964860i
$5$	−6.56915	+	24.1215i	−0.262766	+	0.964860i
$6$		0			0
$7$	16.6149	+	16.6149i	0.339079	+	0.339079i	0.856021	−	0.516941i	$-0.172929\pi$
$7$	16.6149	+	16.6149i	0.339079	+	0.339079i	−0.516941	+	0.856021i	$0.672929\pi$
$8$	−11.6720	+	11.6720i	−0.182374	+	0.182374i
$9$		0			0
$10$	116.728	−	66.7579i	1.16728	−	0.667579i
$11$	215.278			1.77916			0.889580	−	0.456779i	$-0.150997\pi$
$11$	215.278			1.77916			0.889580	+	0.456779i	$0.150997\pi$
$12$		0			0
$13$	−29.9347	+	29.9347i	−0.177128	+	0.177128i	−0.790103	−	0.612975i	$-0.789973\pi$
$13$	−29.9347	+	29.9347i	−0.177128	+	0.177128i	0.612975	+	0.790103i	$0.289973\pi$
$14$		−	126.385i		−	0.644821i
$15$		0			0
$16$	295.684			1.15501
$17$	3.97694	+	3.97694i	0.0137611	+	0.0137611i	0.713954	−	0.700193i	$-0.246903\pi$
$17$	3.97694	+	3.97694i	0.0137611	+	0.0137611i	−0.700193	+	0.713954i	$0.746903\pi$
$18$		0			0
$19$			604.349i			1.67410i	0.547128	+	0.837049i	$0.315721\pi$
$19$			604.349i			1.67410i	−0.547128	+	0.837049i	$0.684279\pi$
$20$	−311.919	−	84.9467i	−0.779796	−	0.212367i
$21$		0			0
$22$	−818.782	−	818.782i	−1.69170	−	1.69170i
$23$	−376.379	+	376.379i	−0.711491	+	0.711491i	−0.966847	−	0.255356i	$-0.917807\pi$
$23$	−376.379	+	376.379i	−0.711491	+	0.711491i	0.255356	+	0.966847i	$0.417807\pi$
$24$		0			0
$25$	−538.692	−	316.916i	−0.861908	−	0.507065i
$26$	227.705			0.336841
$27$		0			0
$28$	−214.850	+	214.850i	−0.274043	+	0.274043i
$29$			624.166i			0.742171i	0.928599	+	0.371086i	$0.121014\pi$
$29$			624.166i			0.742171i	−0.928599	+	0.371086i	$0.878986\pi$
$30$		0			0
$31$	263.888			0.274597			0.137299	−	0.990530i	$-0.456158\pi$
$31$	263.888			0.274597			0.137299	+	0.990530i	$0.456158\pi$
$32$	−937.842	−	937.842i	−0.915861	−	0.915861i
$33$		0			0
$34$		−	30.2515i		−	0.0261691i
$35$	−509.922	+	291.630i	−0.416263	+	0.238065i
$36$		0			0
$37$	979.613	+	979.613i	0.715568	+	0.715568i	0.967694	−	0.252126i	$-0.0811298\pi$
$37$	979.613	+	979.613i	0.715568	+	0.715568i	−0.252126	+	0.967694i	$0.581130\pi$
$38$	2298.56	−	2298.56i	1.59180	−	1.59180i
$39$		0			0
$40$	−204.870	−	358.220i	−0.128044	−	0.223887i
$41$	1579.25			0.939468			0.469734	−	0.882808i	$-0.344350\pi$
$41$	1579.25			0.939468			0.469734	+	0.882808i	$0.344350\pi$
$42$		0			0
$43$	−30.8409	+	30.8409i	−0.0166797	+	0.0166797i	−0.715397	−	0.698718i	$-0.753754\pi$
$43$	−30.8409	+	30.8409i	−0.0166797	+	0.0166797i	0.698718	+	0.715397i	$0.253754\pi$
$44$			2783.80i			1.43791i
$45$		0			0
$46$	2863.01			1.35303
$47$	−1785.16	−	1785.16i	−0.808129	−	0.808129i	0.176221	−	0.984351i	$-0.443613\pi$
$47$	−1785.16	−	1785.16i	−0.808129	−	0.808129i	−0.984351	+	0.176221i	$0.943613\pi$
$48$		0			0
$49$		−	1848.89i		−	0.770050i
$50$	843.498	+	3254.19i	0.337399	+	1.30168i
$51$		0			0
$52$	−387.089	−	387.089i	−0.143154	−	0.143154i
$53$	−707.912	+	707.912i	−0.252016	+	0.252016i	−0.821797	−	0.569781i	$-0.807028\pi$
$53$	−707.912	+	707.912i	−0.252016	+	0.252016i	0.569781	+	0.821797i	$0.307028\pi$
$54$		0			0
$55$	−1414.20	+	5192.84i	−0.467503	+	1.71664i
$56$	−387.857			−0.123679
$57$		0			0
$58$	2373.93	−	2373.93i	0.705687	−	0.705687i
$59$		−	1366.84i		−	0.392657i	−0.980538	−	0.196329i	$-0.937098\pi$
$59$		−	1366.84i		−	0.392657i	0.980538	−	0.196329i	$-0.0629020\pi$
$60$		0			0
$61$	−2983.71			−0.801856			−0.400928	−	0.916110i	$-0.631312\pi$
$61$	−2983.71			−0.801856			−0.400928	+	0.916110i	$0.631312\pi$
$62$	−1003.66	−	1003.66i	−0.261098	−	0.261098i
$63$		0			0
$64$			2402.96i			0.586661i
$65$	−525.423	−	918.714i	−0.124361	−	0.217447i
$66$		0			0
$67$	−986.003	−	986.003i	−0.219649	−	0.219649i	0.588702	−	0.808350i	$-0.299639\pi$
$67$	−986.003	−	986.003i	−0.219649	−	0.219649i	−0.808350	+	0.588702i	$0.799639\pi$
$68$	−51.4265	+	51.4265i	−0.0111216	+	0.0111216i
$69$		0			0
$70$	3048.59	+	830.242i	0.622162	+	0.169437i
$71$	−68.0686			−0.0135030			−0.00675150	−	0.999977i	$-0.502149\pi$
$71$	−68.0686			−0.0135030			−0.00675150	+	0.999977i	$0.502149\pi$
$72$		0			0
$73$	2476.90	−	2476.90i	0.464797	−	0.464797i	−0.435427	−	0.900224i	$-0.643403\pi$
$73$	2476.90	−	2476.90i	0.464797	−	0.464797i	0.900224	+	0.435427i	$0.143403\pi$
$74$		−	7451.65i		−	1.36078i
$75$		0			0
$76$	−7814.93			−1.35300
$77$	3576.83	+	3576.83i	0.603277	+	0.603277i
$78$		0			0
$79$		−	6047.25i		−	0.968955i	−0.874804	−	0.484478i	$-0.839010\pi$
$79$		−	6047.25i		−	0.968955i	0.874804	−	0.484478i	$-0.160990\pi$
$80$	−1942.39	+	7132.33i	−0.303499	+	1.11443i
$81$		0			0
$82$	−6006.45	−	6006.45i	−0.893285	−	0.893285i
$83$	5158.95	−	5158.95i	0.748868	−	0.748868i	−0.225398	−	0.974267i	$-0.572368\pi$
$83$	5158.95	−	5158.95i	0.748868	−	0.748868i	0.974267	+	0.225398i	$0.0723684\pi$
$84$		0			0
$85$	−122.055	+	69.8047i	−0.0168934	+	0.00966154i
$86$	234.598			0.0317196
$87$		0			0
$88$	−2512.72	+	2512.72i	−0.324473	+	0.324473i
$89$			7028.50i			0.887325i	0.896194	+	0.443662i	$0.146321\pi$
$89$			7028.50i			0.887325i	−0.896194	+	0.443662i	$0.853679\pi$
$90$		0			0
$91$	−994.722			−0.120121
$92$	−4867.01	−	4867.01i	−0.575025	−	0.575025i
$93$		0			0
$94$			13579.2i			1.53681i
$95$	−14577.8	−	3970.06i	−1.61527	−	0.439896i
$96$		0			0
$97$	10869.3	+	10869.3i	1.15520	+	1.15520i	0.985495	+	0.169703i	$0.0542810\pi$
$97$	10869.3	+	10869.3i	1.15520	+	1.15520i	0.169703	+	0.985495i	$0.445719\pi$
$98$	−7032.00	+	7032.00i	−0.732195	+	0.732195i
$99$		0			0
$100$	4098.08	−	6965.91i	0.409808	−	0.696591i
$101$	4043.94			0.396426			0.198213	−	0.980159i	$-0.436486\pi$
$101$	4043.94			0.396426			0.198213	+	0.980159i	$0.436486\pi$
$102$		0			0
$103$	−119.716	+	119.716i	−0.0112844	+	0.0112844i	−0.712726	−	0.701442i	$-0.752540\pi$
$103$	−119.716	+	119.716i	−0.0112844	+	0.0112844i	0.701442	+	0.712726i	$0.252540\pi$
$104$		−	698.792i		−	0.0646073i
$105$		0			0
$106$	5384.89			0.479254
$107$	8330.41	+	8330.41i	0.727610	+	0.727610i	0.970143	−	0.242533i	$-0.0779783\pi$
$107$	8330.41	+	8330.41i	0.727610	+	0.727610i	−0.242533	+	0.970143i	$0.577978\pi$
$108$		0			0
$109$		−	8380.72i		−	0.705389i	−0.935739	−	0.352694i	$-0.885266\pi$
$109$		−	8380.72i		−	0.705389i	0.935739	−	0.352694i	$-0.114734\pi$
$110$	25128.9	−	14371.5i	2.07677	−	1.18773i
$111$		0			0
$112$	4912.75	+	4912.75i	0.391642	+	0.391642i
$113$	−968.067	+	968.067i	−0.0758139	+	0.0758139i	−0.743997	−	0.668183i	$-0.767072\pi$
$113$	−968.067	+	968.067i	−0.0758139	+	0.0758139i	0.668183	+	0.743997i	$0.267072\pi$
$114$		0			0
$115$	−6606.33	−	11551.3i	−0.499533	−	0.873445i
$116$	−8071.19			−0.599821
$117$		0			0
$118$	−5198.59	+	5198.59i	−0.373355	+	0.373355i
$119$			132.153i			0.00933218i
$120$		0			0
$121$	31703.8			2.16541
$122$	11348.1	+	11348.1i	0.762437	+	0.762437i
$123$		0			0
$124$			3412.37i			0.221929i
$125$	11183.2	−	10912.2i	0.715727	−	0.698381i
$126$		0			0
$127$	−12023.3	−	12023.3i	−0.745446	−	0.745446i	0.228174	−	0.973620i	$-0.426724\pi$
$127$	−12023.3	−	12023.3i	−0.745446	−	0.745446i	−0.973620	+	0.228174i	$0.926724\pi$
$128$	−5866.12	+	5866.12i	−0.358039	+	0.358039i
$129$		0			0
$130$	−1495.83	+	5492.58i	−0.0885105	+	0.325005i
$131$	−4866.47			−0.283577			−0.141789	−	0.989897i	$-0.545285\pi$
$131$	−4866.47			−0.283577			−0.141789	+	0.989897i	$0.545285\pi$
$132$		0			0
$133$	−10041.2	+	10041.2i	−0.567652	+	0.567652i
$134$			7500.26i			0.417702i
$135$		0			0
$136$	−92.8374			−0.00501933
$137$	−10167.9	−	10167.9i	−0.541736	−	0.541736i	0.382301	−	0.924038i	$-0.375132\pi$
$137$	−10167.9	−	10167.9i	−0.541736	−	0.541736i	−0.924038	+	0.382301i	$0.875132\pi$
$138$		0			0
$139$		−	4195.51i		−	0.217148i	−0.994088	−	0.108574i	$-0.965372\pi$
$139$		−	4195.51i		−	0.217148i	0.994088	−	0.108574i	$-0.0346284\pi$
$140$	−3771.11	−	6593.87i	−0.192404	−	0.336422i
$141$		0			0
$142$	258.890	+	258.890i	0.0128392	+	0.0128392i
$143$	−6444.29	+	6444.29i	−0.315139	+	0.315139i
$144$		0			0
$145$	−15055.8	−	4100.24i	−0.716091	−	0.195018i
$146$	−18841.1			−0.883895
$147$		0			0
$148$	−12667.5	+	12667.5i	−0.578320	+	0.578320i
$149$		−	24880.4i		−	1.12069i	−0.828259	−	0.560345i	$-0.810669\pi$
$149$		−	24880.4i		−	1.12069i	0.828259	−	0.560345i	$-0.189331\pi$
$150$		0			0
$151$	−37791.8			−1.65746			−0.828732	−	0.559645i	$-0.810937\pi$
$151$	−37791.8			−1.65746			−0.828732	+	0.559645i	$0.810937\pi$
$152$	−7053.94	−	7053.94i	−0.305312	−	0.305312i
$153$		0			0
$154$		−	27208.0i		−	1.14724i
$155$	−1733.52	+	6365.37i	−0.0721548	+	0.264948i
$156$		0			0
$157$	18627.8	+	18627.8i	0.755723	+	0.755723i	0.975541	−	0.219818i	$-0.0705463\pi$
$157$	18627.8	+	18627.8i	0.755723	+	0.755723i	−0.219818	+	0.975541i	$0.570546\pi$
$158$	−22999.9	+	22999.9i	−0.921322	+	0.921322i
$159$		0			0
$160$	28783.0	−	16461.3i	1.12433	−	0.643020i
$161$	−12507.0			−0.482504
$162$		0			0
$163$	−280.411	+	280.411i	−0.0105541	+	0.0105541i	−0.712364	−	0.701810i	$-0.752376\pi$
$163$	−280.411	+	280.411i	−0.0105541	+	0.0105541i	0.701810	+	0.712364i	$0.252376\pi$
$164$			20421.5i			0.759275i
$165$		0			0
$166$	−39242.8			−1.42411
$167$	25855.1	+	25855.1i	0.927073	+	0.927073i	0.997516	−	0.0704428i	$-0.0224412\pi$
$167$	25855.1	+	25855.1i	0.927073	+	0.927073i	−0.0704428	+	0.997516i	$0.522441\pi$
$168$		0			0
$169$			26768.8i			0.937251i
$170$	729.712	+	198.727i	0.0252495	+	0.00687636i
$171$		0			0
$172$	−398.808	−	398.808i	−0.0134805	−	0.0134805i
$173$	31991.2	−	31991.2i	1.06890	−	1.06890i	0.0714600	−	0.997443i	$-0.477234\pi$
$173$	31991.2	−	31991.2i	1.06890	−	1.06890i	0.997443	−	0.0714600i	$-0.0227658\pi$
$174$		0			0
$175$	−3684.80	−	14215.8i	−0.120320	−	0.464190i
$176$	63654.3			2.05496
$177$		0			0
$178$	26731.9	−	26731.9i	0.843705	−	0.843705i
$179$		−	17987.8i		−	0.561400i	−0.959796	−	0.280700i	$-0.909433\pi$
$179$		−	17987.8i		−	0.561400i	0.959796	−	0.280700i	$-0.0905666\pi$
$180$		0			0
$181$	23072.1			0.704255			0.352127	−	0.935952i	$-0.385458\pi$
$181$	23072.1			0.704255			0.352127	+	0.935952i	$0.385458\pi$
$182$	3783.29	+	3783.29i	0.114216	+	0.114216i
$183$		0			0
$184$		−	8786.16i		−	0.259515i
$185$	−30065.0	+	17194.5i	−0.878450	+	0.502396i
$186$		0			0
$187$	856.150	+	856.150i	0.0244831	+	0.0244831i
$188$	23084.1	−	23084.1i	0.653128	−	0.653128i
$189$		0			0
$190$	40345.1	+	70544.3i	1.11759	+	1.95414i
$191$	−51892.0			−1.42244			−0.711220	−	0.702970i	$-0.751857\pi$
$191$	−51892.0			−1.42244			−0.711220	+	0.702970i	$0.751857\pi$
$192$		0			0
$193$	50542.0	−	50542.0i	1.35687	−	1.35687i	0.479118	−	0.877751i	$-0.340957\pi$
$193$	50542.0	−	50542.0i	1.35687	−	1.35687i	0.877751	−	0.479118i	$-0.159043\pi$
$194$		−	82679.5i		−	2.19682i
$195$		0			0
$196$	23908.3			0.622352
$197$	22327.7	+	22327.7i	0.575323	+	0.575323i	0.933611	−	0.358288i	$-0.116640\pi$
$197$	22327.7	+	22327.7i	0.575323	+	0.575323i	−0.358288	+	0.933611i	$0.616640\pi$
$198$		0			0
$199$			57027.8i			1.44006i	0.693943	+	0.720030i	$0.255872\pi$
$199$			57027.8i			1.44006i	−0.693943	+	0.720030i	$0.744128\pi$
$200$	9986.62	−	2588.57i	0.249665	−	0.0647143i
$201$		0			0
$202$	−15380.6	−	15380.6i	−0.376938	−	0.376938i
$203$	−10370.5	+	10370.5i	−0.251655	+	0.251655i
$204$		0			0
$205$	−10374.3	+	38093.8i	−0.246860	+	0.906455i
$206$	910.647			0.0214593
$207$		0			0
$208$	−8851.19	+	8851.19i	−0.204586	+	0.204586i
$209$			130103.i			2.97849i
$210$		0			0
$211$	−50695.3			−1.13868			−0.569341	−	0.822101i	$-0.692802\pi$
$211$	−50695.3			−1.13868			−0.569341	+	0.822101i	$0.692802\pi$
$212$	−9154.12	−	9154.12i	−0.203678	−	0.203678i
$213$		0			0
$214$		−	63367.1i		−	1.38368i
$215$	−541.329	−	946.526i	−0.0117107	−	0.0204765i
$216$		0			0
$217$	4384.47	+	4384.47i	0.0931102	+	0.0931102i
$218$	−31874.9	+	31874.9i	−0.670712	+	0.670712i
$219$		0			0
$220$	−67149.3	−	18287.2i	−1.38738	−	0.377835i
$221$	−238.097			−0.00487494
$222$		0			0
$223$	−18991.4	+	18991.4i	−0.381898	+	0.381898i	−0.871786	−	0.489888i	$-0.837038\pi$
$223$	−18991.4	+	18991.4i	−0.381898	+	0.381898i	0.489888	+	0.871786i	$0.337038\pi$
$224$		−	31164.3i		−	0.621099i
$225$		0			0
$226$	7363.82			0.144174
$227$	−24016.1	−	24016.1i	−0.466069	−	0.466069i	0.434570	−	0.900638i	$-0.356900\pi$
$227$	−24016.1	−	24016.1i	−0.466069	−	0.466069i	−0.900638	+	0.434570i	$0.856900\pi$
$228$		0			0
$229$		−	4953.50i		−	0.0944586i	−0.998884	−	0.0472293i	$-0.984961\pi$
$229$		−	4953.50i		−	0.0944586i	0.998884	−	0.0472293i	$-0.0150391\pi$
$230$	−18807.6	+	69060.1i	−0.355530	+	1.30548i
$231$		0			0
$232$	−7285.24	−	7285.24i	−0.135353	−	0.135353i
$233$	60878.5	−	60878.5i	1.12138	−	1.12138i	0.129844	−	0.991534i	$-0.458552\pi$
$233$	60878.5	−	60878.5i	1.12138	−	1.12138i	0.991534	−	0.129844i	$-0.0414475\pi$
$234$		0			0
$235$	54787.6	−	31333.7i	0.992080	−	0.567382i
$236$	17674.8			0.317345
$237$		0			0
$238$	502.626	−	502.626i	0.00887342	−	0.00887342i
$239$		−	40609.0i		−	0.710929i	−0.934690	−	0.355464i	$-0.884323\pi$
$239$		−	40609.0i		−	0.710929i	0.934690	−	0.355464i	$-0.115677\pi$
$240$		0			0
$241$	24443.6			0.420854			0.210427	−	0.977610i	$-0.432515\pi$
$241$	24443.6			0.420854			0.210427	+	0.977610i	$0.432515\pi$
$242$	−120581.	−	120581.i	−2.05896	−	2.05896i
$243$		0			0
$244$		−	38582.7i		−	0.648057i
$245$	44598.0	+	12145.6i	0.742990	+	0.202343i
$246$		0			0
$247$	−18091.0	−	18091.0i	−0.296530	−	0.296530i
$248$	−3080.09	+	3080.09i	−0.0500795	+	0.0500795i
$249$		0			0
$250$	−84036.9	−	1030.83i	−1.34459	−	0.0164932i
$251$	71951.0			1.14206			0.571030	−	0.820929i	$-0.306544\pi$
$251$	71951.0			1.14206			0.571030	+	0.820929i	$0.306544\pi$
$252$		0			0
$253$	−81026.3	+	81026.3i	−1.26586	+	1.26586i
$254$			91457.9i			1.41760i
$255$		0			0
$256$	83069.4			1.26754
$257$	−33507.3	−	33507.3i	−0.507310	−	0.507310i	0.406390	−	0.913700i	$-0.366787\pi$
$257$	−33507.3	−	33507.3i	−0.507310	−	0.507310i	−0.913700	+	0.406390i	$0.866787\pi$
$258$		0			0
$259$			32552.3i			0.485269i
$260$	11880.0	−	6794.32i	0.175740	−	0.100508i
$261$		0			0
$262$	18508.9	+	18508.9i	0.269637	+	0.269637i
$263$	18051.0	−	18051.0i	0.260969	−	0.260969i	−0.564479	−	0.825448i	$-0.690923\pi$
$263$	18051.0	−	18051.0i	0.260969	−	0.260969i	0.825448	+	0.564479i	$0.190923\pi$
$264$		0			0
$265$	−12425.5	−	21726.3i	−0.176939	−	0.309381i
$266$	76380.6			1.07949
$267$		0			0
$268$	12750.2	−	12750.2i	0.177519	−	0.177519i
$269$			90057.7i			1.24456i	0.782794	+	0.622281i	$0.213794\pi$
$269$			90057.7i			1.24456i	−0.782794	+	0.622281i	$0.786206\pi$
$270$		0			0
$271$	28019.1			0.381518			0.190759	−	0.981637i	$-0.438905\pi$
$271$	28019.1			0.381518			0.190759	+	0.981637i	$0.438905\pi$
$272$	1175.92	+	1175.92i	0.0158942	+	0.0158942i
$273$		0			0
$274$			77344.1i			1.03021i
$275$	−115969.	−	68225.1i	−1.53347	−	0.902150i
$276$		0			0
$277$	−77139.7	−	77139.7i	−1.00535	−	1.00535i	−0.999986	−	0.00536651i	$-0.998292\pi$
$277$	−77139.7	−	77139.7i	−1.00535	−	1.00535i	−0.00536651	−	0.999986i	$-0.501708\pi$
$278$	−15957.1	+	15957.1i	−0.206473	+	0.206473i
$279$		0			0
$280$	2547.89	−	9355.68i	0.0324986	−	0.119333i
$281$	−59704.3			−0.756125			−0.378062	−	0.925780i	$-0.623409\pi$
$281$	−59704.3			−0.756125			−0.378062	+	0.925780i	$0.623409\pi$
$282$		0			0
$283$	71516.3	−	71516.3i	0.892960	−	0.892960i	−0.101841	−	0.994801i	$-0.532473\pi$
$283$	71516.3	−	71516.3i	0.892960	−	0.892960i	0.994801	+	0.101841i	$0.0324733\pi$
$284$		−	880.205i		−	0.0109131i
$285$		0			0
$286$	49019.9			0.599295
$287$	26239.0	+	26239.0i	0.318554	+	0.318554i
$288$		0			0
$289$		−	83489.4i		−	0.999621i
$290$	41668.0	+	72857.5i	0.495458	+	0.866319i
$291$		0			0
$292$	32029.2	+	32029.2i	0.375647	+	0.375647i
$293$	−2048.52	+	2048.52i	−0.0238619	+	0.0238619i	−0.718937	−	0.695075i	$-0.755371\pi$
$293$	−2048.52	+	2048.52i	−0.0238619	+	0.0238619i	0.695075	+	0.718937i	$0.255371\pi$
$294$		0			0
$295$	32970.2	+	8978.99i	0.378859	+	0.103177i
$296$	−22868.0			−0.261003
$297$		0			0
$298$	−94629.4	+	94629.4i	−1.06560	+	1.06560i
$299$		−	22533.5i		−	0.252050i
$300$		0			0
$301$	−1024.83			−0.0113115
$302$	143736.	+	143736.i	1.57598	+	1.57598i
$303$		0			0
$304$			178696.i			1.93361i
$305$	19600.4	−	71971.4i	0.210701	−	0.773678i
$306$		0			0
$307$	−28977.5	−	28977.5i	−0.307457	−	0.307457i	0.536465	−	0.843922i	$-0.319759\pi$
$307$	−28977.5	−	28977.5i	−0.307457	−	0.307457i	−0.843922	+	0.536465i	$0.819759\pi$
$308$	−46252.5	+	46252.5i	−0.487566	+	0.487566i
$309$		0			0
$310$	30803.0	−	17616.6i	0.320531	−	0.183315i
$311$	20134.0			0.208166			0.104083	−	0.994569i	$-0.466809\pi$
$311$	20134.0			0.208166			0.104083	+	0.994569i	$0.466809\pi$
$312$		0			0
$313$	115839.	−	115839.i	1.18240	−	1.18240i	0.203283	−	0.979120i	$-0.434839\pi$
$313$	115839.	−	115839.i	1.18240	−	1.18240i	0.979120	−	0.203283i	$-0.0651611\pi$
$314$		−	141697.i		−	1.43715i
$315$		0			0
$316$	78197.9			0.783106
$317$	−56563.1	−	56563.1i	−0.562879	−	0.562879i	0.367245	−	0.930124i	$-0.380301\pi$
$317$	−56563.1	−	56563.1i	−0.562879	−	0.562879i	−0.930124	+	0.367245i	$0.880301\pi$
$318$		0			0
$319$			134370.i			1.32044i
$320$	−57963.1	−	15785.4i	−0.566046	−	0.154155i
$321$		0			0
$322$	47568.6	+	47568.6i	0.458785	+	0.458785i
$323$	−2403.46	+	2403.46i	−0.0230373	+	0.0230373i
$324$		0			0
$325$	25612.3	−	6638.82i	0.242484	−	0.0628527i
$326$	2133.01			0.0200705
$327$		0			0
$328$	−18432.9	+	18432.9i	−0.171335	+	0.171335i
$329$		−	59320.4i		−	0.548040i
$330$		0			0
$331$	−76249.8			−0.695958			−0.347979	−	0.937502i	$-0.613132\pi$
$331$	−76249.8			−0.695958			−0.347979	+	0.937502i	$0.613132\pi$
$332$	66711.2	+	66711.2i	0.605233	+	0.605233i
$333$		0			0
$334$		−	196673.i		−	1.76300i
$335$	30261.1	−	17306.7i	0.269646	−	0.154214i
$336$		0			0
$337$	88861.9	+	88861.9i	0.782448	+	0.782448i	0.980243	−	0.197795i	$-0.0633780\pi$
$337$	88861.9	+	88861.9i	0.782448	+	0.782448i	−0.197795	+	0.980243i	$0.563378\pi$
$338$	101812.	−	101812.i	0.891177	−	0.891177i
$339$		0			0
$340$	−902.654	−	1578.31i	−0.00780843	−	0.0136532i
$341$	56809.4			0.488552
$342$		0			0
$343$	70611.5	−	70611.5i	0.600188	−	0.600188i
$344$		−	719.946i		−	0.00608392i
$345$		0			0
$346$	−243348.			−2.03271
$347$	41523.5	+	41523.5i	0.344854	+	0.344854i	0.858189	−	0.513334i	$-0.171590\pi$
$347$	41523.5	+	41523.5i	0.344854	+	0.344854i	−0.513334	+	0.858189i	$0.671590\pi$
$348$		0			0
$349$		−	94995.3i		−	0.779922i	−0.920831	−	0.389961i	$-0.872489\pi$
$349$		−	94995.3i		−	0.779922i	0.920831	−	0.389961i	$-0.127511\pi$
$350$	−40053.3	+	68082.6i	−0.326966	+	0.555776i
$351$		0			0
$352$	−201897.	−	201897.i	−1.62946	−	1.62946i
$353$	−54491.3	+	54491.3i	−0.437298	+	0.437298i	−0.891102	−	0.453804i	$-0.850067\pi$
$353$	−54491.3	+	54491.3i	−0.437298	+	0.437298i	0.453804	+	0.891102i	$0.350067\pi$
$354$		0			0
$355$	447.153	−	1641.92i	0.00354813	−	0.0130285i
$356$	−90886.6			−0.717133
$357$		0			0
$358$	−68414.2	+	68414.2i	−0.533802	+	0.533802i
$359$			60457.0i			0.469092i	0.972105	+	0.234546i	$0.0753603\pi$
$359$			60457.0i			0.469092i	−0.972105	+	0.234546i	$0.924640\pi$
$360$		0			0
$361$	−234917.			−1.80260
$362$	−87751.6	−	87751.6i	−0.669634	−	0.669634i
$363$		0			0
$364$		−	12862.9i		−	0.0970814i
$365$	43475.4	+	76017.7i	0.326331	+	0.570596i
$366$		0			0
$367$	84336.6	+	84336.6i	0.626158	+	0.626158i	0.947099	−	0.320941i	$-0.103999\pi$
$367$	84336.6	+	84336.6i	0.626158	+	0.626158i	−0.320941	+	0.947099i	$0.603999\pi$
$368$	−111289.	+	111289.i	−0.821783	+	0.821783i
$369$		0			0
$370$	179745.	+	48951.0i	1.31296	+	0.357568i
$371$	−23523.8			−0.170907
$372$		0			0
$373$	−67604.9	+	67604.9i	−0.485915	+	0.485915i	−0.907015	−	0.421099i	$-0.861644\pi$
$373$	−67604.9	+	67604.9i	−0.485915	+	0.485915i	0.421099	+	0.907015i	$0.361644\pi$
$374$		−	6512.50i		−	0.0465591i
$375$		0			0
$376$	41672.6			0.294764
$377$	−18684.2	−	18684.2i	−0.131459	−	0.131459i
$378$		0			0
$379$		−	128651.i		−	0.895645i	−0.894123	−	0.447822i	$-0.852200\pi$
$379$		−	128651.i		−	0.895645i	0.894123	−	0.447822i	$-0.147800\pi$
$380$	51337.5	−	188508.i	0.355523	−	1.30546i
$381$		0			0
$382$	197364.	+	197364.i	1.35251	+	1.35251i
$383$	−147476.	+	147476.i	−1.00536	+	1.00536i	−0.00537910	+	0.999986i	$0.501712\pi$
$383$	−147476.	+	147476.i	−1.00536	+	1.00536i	−0.999986	+	0.00537910i	$0.998288\pi$
$384$		0			0
$385$	−109775.	+	62781.7i	−0.740598	+	0.423557i
$386$	−384459.			−2.58033
$387$		0			0
$388$	−140552.	+	140552.i	−0.933628	+	0.933628i
$389$		−	175663.i		−	1.16086i	−0.814309	−	0.580432i	$-0.802884\pi$
$389$		−	175663.i		−	1.16086i	0.814309	−	0.580432i	$-0.197116\pi$
$390$		0			0
$391$	−2993.68			−0.0195817
$392$	21580.2	+	21580.2i	0.140437	+	0.140437i
$393$		0			0
$394$		−	169841.i		−	1.09408i
$395$	145869.	+	39725.3i	0.934906	+	0.254609i
$396$		0			0
$397$	59199.8	+	59199.8i	0.375612	+	0.375612i	0.869516	−	0.493904i	$-0.164431\pi$
$397$	59199.8	+	59199.8i	0.375612	+	0.375612i	−0.493904	+	0.869516i	$0.664431\pi$
$398$	216897.	−	216897.i	1.36927	−	1.36927i
$399$		0			0
$400$	−159283.	−	93706.8i	−0.995516	−	0.585667i
$401$	−81929.2			−0.509507			−0.254753	−	0.967006i	$-0.581994\pi$
$401$	−81929.2			−0.509507			−0.254753	+	0.967006i	$0.581994\pi$
$402$		0			0
$403$	−7899.39	+	7899.39i	−0.0486389	+	0.0486389i
$404$			52292.8i			0.320390i
$405$		0			0
$406$	78885.2			0.478568
$407$	210890.	+	210890.i	1.27311	+	1.27311i
$408$		0			0
$409$			107251.i			0.641145i	0.947224	+	0.320573i	$0.103875\pi$
$409$			107251.i			0.641145i	−0.947224	+	0.320573i	$0.896125\pi$
$410$	184342.	−	105427.i	1.09662	−	0.627170i
$411$		0			0
$412$	−1548.07	−	1548.07i	−0.00912000	−	0.00912000i
$413$	22709.9	−	22709.9i	0.133142	−	0.133142i
$414$		0			0
$415$	90551.7	+	158332.i	0.525775	+	0.919330i
$416$	56147.9			0.324449
$417$		0			0
$418$	494830.	−	494830.i	2.83207	−	2.83207i
$419$			25226.4i			0.143690i	0.997416	+	0.0718451i	$0.0228887\pi$
$419$			25226.4i			0.143690i	−0.997416	+	0.0718451i	$0.977111\pi$
$420$		0			0
$421$	165855.			0.935762			0.467881	−	0.883792i	$-0.345018\pi$
$421$	165855.			0.935762			0.467881	+	0.883792i	$0.345018\pi$
$422$	192813.	+	192813.i	1.08271	+	1.08271i
$423$		0			0
$424$		−	16525.4i		−	0.0919224i
$425$	−881.994	−	3402.70i	−0.00488301	−	0.0188385i
$426$		0			0
$427$	−49573.9	−	49573.9i	−0.271893	−	0.271893i
$428$	−107722.	+	107722.i	−0.588052	+	0.588052i
$429$		0			0
$430$	−1541.11	+	5658.85i	−0.00833483	+	0.0306049i
$431$	181899.			0.979210			0.489605	−	0.871944i	$-0.337141\pi$
$431$	181899.			0.979210			0.489605	+	0.871944i	$0.337141\pi$
$432$		0			0
$433$	−178342.	+	178342.i	−0.951214	+	0.951214i	−0.998864	−	0.0476501i	$-0.984827\pi$
$433$	−178342.	+	178342.i	−0.951214	+	0.951214i	0.0476501	+	0.998864i	$0.484827\pi$
$434$		−	33351.5i		−	0.177066i
$435$		0			0
$436$	108372.			0.570093
$437$	−227464.	−	227464.i	−1.19111	−	1.19111i
$438$		0			0
$439$			24924.4i			0.129329i	0.997907	+	0.0646645i	$0.0205977\pi$
$439$			24924.4i			0.129329i	−0.997907	+	0.0646645i	$0.979402\pi$
$440$	−44104.1	−	77117.0i	−0.227811	−	0.398332i
$441$		0			0
$442$	905.569	+	905.569i	0.00463529	+	0.00463529i
$443$	75101.4	−	75101.4i	0.382684	−	0.382684i	−0.489384	−	0.872068i	$-0.662778\pi$
$443$	75101.4	−	75101.4i	0.382684	−	0.382684i	0.872068	+	0.489384i	$0.162778\pi$
$444$		0			0
$445$	−169538.	−	46171.3i	−0.856144	−	0.233159i
$446$	144462.			0.726248
$447$		0			0
$448$	−39925.0	+	39925.0i	−0.198925	+	0.198925i
$449$			226867.i			1.12533i	0.826687	+	0.562663i	$0.190223\pi$
$449$			226867.i			1.12533i	−0.826687	+	0.562663i	$0.809777\pi$
$450$		0			0
$451$	339978.			1.67147
$452$	−12518.2	−	12518.2i	−0.0612725	−	0.0612725i
$453$		0			0
$454$			182684.i			0.886314i
$455$	6534.48	−	23994.2i	0.0315637	−	0.115900i
$456$		0			0
$457$	−3272.88	−	3272.88i	−0.0156710	−	0.0156710i	0.699228	−	0.714899i	$-0.253527\pi$
$457$	−3272.88	−	3272.88i	−0.0156710	−	0.0156710i	−0.714899	+	0.699228i	$0.753527\pi$
$458$	−18840.0	+	18840.0i	−0.0898151	+	0.0898151i
$459$		0			0
$460$	149372.	−	85427.4i	0.705915	−	0.403721i
$461$	−94982.0			−0.446930			−0.223465	−	0.974712i	$-0.571737\pi$
$461$	−94982.0			−0.446930			−0.223465	+	0.974712i	$0.571737\pi$
$462$		0			0
$463$	−216080.	+	216080.i	−1.00798	+	1.00798i	−0.00801234	+	0.999968i	$0.502550\pi$
$463$	−216080.	+	216080.i	−1.00798	+	1.00798i	−0.999968	+	0.00801234i	$0.997450\pi$
$464$			184556.i			0.857219i
$465$		0			0
$466$	−463086.			−2.13250
$467$	−7732.59	−	7732.59i	−0.0354561	−	0.0354561i	0.689156	−	0.724613i	$-0.257981\pi$
$467$	−7732.59	−	7732.59i	−0.0354561	−	0.0354561i	−0.724613	+	0.689156i	$0.757981\pi$
$468$		0			0
$469$		−	32764.7i		−	0.148957i
$470$	−327551.	−	89203.9i	−1.48280	−	0.403820i
$471$		0			0
$472$	15953.7	+	15953.7i	0.0716106	+	0.0716106i
$473$	−6639.37	+	6639.37i	−0.0296759	+	0.0296759i
$474$		0			0
$475$	191528.	−	325558.i	0.848876	−	1.44292i
$476$	−1708.89			−0.00754224
$477$		0			0
$478$	−154451.	+	154451.i	−0.675980	+	0.675980i
$479$		−	320749.i		−	1.39796i	−0.715142	−	0.698980i	$-0.753638\pi$
$479$		−	320749.i		−	1.39796i	0.715142	−	0.698980i	$-0.246362\pi$
$480$		0			0
$481$	−58648.8			−0.253495
$482$	−92967.9	−	92967.9i	−0.400165	−	0.400165i
$483$		0			0
$484$			409967.i			1.75008i
$485$	−333585.	+	190781.i	−1.41815	+	0.811057i
$486$		0			0
$487$	5262.88	+	5262.88i	0.0221904	+	0.0221904i	0.718115	−	0.695925i	$-0.245005\pi$
$487$	5262.88	+	5262.88i	0.0221904	+	0.0221904i	−0.695925	+	0.718115i	$0.745005\pi$
$488$	34825.7	−	34825.7i	0.146238	−	0.146238i
$489$		0			0
$490$	−123428.	−	215817.i	−0.514070	−	0.898862i
$491$	−191704.			−0.795183			−0.397592	−	0.917562i	$-0.630154\pi$
$491$	−191704.			−0.795183			−0.397592	+	0.917562i	$0.630154\pi$
$492$		0			0
$493$	−2482.27	+	2482.27i	−0.0102131	+	0.0102131i
$494$			137613.i			0.563905i
$495$		0			0
$496$	78027.4			0.317164
$497$	−1130.95	−	1130.95i	−0.00457859	−	0.00457859i
$498$		0			0
$499$		−	261730.i		−	1.05112i	−0.850756	−	0.525561i	$-0.823855\pi$
$499$		−	261730.i		−	1.05112i	0.850756	−	0.525561i	$-0.176145\pi$
$500$	141107.	+	144612.i	0.564429	+	0.578448i
$501$		0			0
$502$	−273656.	−	273656.i	−1.08592	−	1.08592i
$503$	−206538.	+	206538.i	−0.816328	+	0.816328i	−0.985574	−	0.169245i	$-0.945867\pi$
$503$	−206538.	+	206538.i	−0.816328	+	0.816328i	0.169245	+	0.985574i	$0.445867\pi$
$504$		0			0
$505$	−26565.3	+	97545.8i	−0.104167	+	0.382495i
$506$	616345.			2.40726
$507$		0			0
$508$	155475.	−	155475.i	0.602467	−	0.602467i
$509$		−	44082.3i		−	0.170149i	−0.996375	−	0.0850743i	$-0.972887\pi$
$509$		−	44082.3i		−	0.170149i	0.996375	−	0.0850743i	$-0.0271128\pi$
$510$		0			0
$511$	82306.9			0.315206
$512$	−222085.	−	222085.i	−0.847188	−	0.847188i
$513$		0			0
$514$			254881.i			0.964741i
$515$	−2101.30	−	3674.16i	−0.00792269	−	0.0138530i
$516$		0			0
$517$	−384306.	−	384306.i	−1.43779	−	1.43779i
$518$	123808.	−	123808.i	0.461414	−	0.461414i
$519$		0			0
$520$	16855.9	+	4590.47i	0.0623369	+	0.0169766i
$521$	507981.			1.87142			0.935712	−	0.352764i	$-0.114758\pi$
$521$	507981.			1.87142			0.935712	+	0.352764i	$0.114758\pi$
$522$		0			0
$523$	−119838.	+	119838.i	−0.438119	+	0.438119i	−0.891378	−	0.453260i	$-0.850261\pi$
$523$	−119838.	+	119838.i	−0.438119	+	0.438119i	0.453260	+	0.891378i	$0.350261\pi$
$524$		−	62929.0i		−	0.229186i
$525$		0			0
$526$	−137309.			−0.496280
$527$	1049.47	+	1049.47i	0.00377875	+	0.00377875i
$528$		0			0
$529$		−	3481.11i		−	0.0124396i
$530$	−35374.2	+	129892.i	−0.125932	+	0.462413i
$531$		0			0
$532$	−129844.	−	129844.i	−0.458775	−	0.458775i
$533$	−47274.2	+	47274.2i	−0.166406	+	0.166406i
$534$		0			0
$535$	−255666.	+	146218.i	−0.893233	+	0.510850i
$536$	23017.2			0.0801166
$537$		0			0
$538$	342522.	−	342522.i	1.18338	−	1.18338i
$539$		−	398026.i		−	1.37004i
$540$		0			0
$541$	509491.			1.74077			0.870387	−	0.492368i	$-0.163869\pi$
$541$	509491.			1.74077			0.870387	+	0.492368i	$0.163869\pi$
$542$	−106567.	−	106567.i	−0.362763	−	0.362763i
$543$		0			0
$544$		−	7459.49i		−	0.0252064i
$545$	202156.	+	55054.3i	0.680601	+	0.185352i
$546$		0			0
$547$	376658.	+	376658.i	1.25885	+	1.25885i	0.951644	+	0.307204i	$0.0993933\pi$
$547$	376658.	+	376658.i	1.25885	+	1.25885i	0.307204	+	0.951644i	$0.400607\pi$
$548$	131482.	−	131482.i	0.437830	−	0.437830i
$549$		0			0
$550$	181587.	+	700557.i	0.600288	+	2.31589i
$551$	−377214.			−1.24247
$552$		0			0
$553$	100474.	−	100474.i	0.328553	−	0.328553i
$554$			586780.i			1.91186i
$555$		0			0
$556$	54252.8			0.175498
$557$	76108.6	+	76108.6i	0.245315	+	0.245315i	0.819045	−	0.573730i	$-0.194504\pi$
$557$	76108.6	+	76108.6i	0.245315	+	0.245315i	−0.573730	+	0.819045i	$0.694504\pi$
$558$		0			0
$559$		−	1846.42i		−	0.00590891i
$560$	−150776.	+	86230.3i	−0.480789	+	0.274969i
$561$		0			0
$562$	227077.	+	227077.i	0.718954	+	0.718954i
$563$	242876.	−	242876.i	0.766244	−	0.766244i	−0.211199	−	0.977443i	$-0.567737\pi$
$563$	242876.	−	242876.i	0.766244	−	0.766244i	0.977443	+	0.211199i	$0.0677368\pi$
$564$		0			0
$565$	−16991.8	−	29710.6i	−0.0532284	−	0.0930711i
$566$	−544005.			−1.69813
$567$		0			0
$568$	794.494	−	794.494i	0.00246260	−	0.00246260i
$569$		−	285859.i		−	0.882934i	−0.897278	−	0.441467i	$-0.854458\pi$
$569$		−	285859.i		−	0.882934i	0.897278	−	0.441467i	$-0.145542\pi$
$570$		0			0
$571$	−28753.2			−0.0881890			−0.0440945	−	0.999027i	$-0.514040\pi$
$571$	−28753.2			−0.0881890			−0.0440945	+	0.999027i	$0.514040\pi$
$572$	−83332.0	−	83332.0i	−0.254695	−	0.254695i
$573$		0			0
$574$		−	199593.i		−	0.605789i
$575$	322033.	−	83472.2i	0.974012	−	0.252468i
$576$		0			0
$577$	−429650.	−	429650.i	−1.29052	−	1.29052i	−0.934467	−	0.356048i	$-0.884124\pi$
$577$	−429650.	−	429650.i	−1.29052	−	1.29052i	−0.356048	−	0.934467i	$-0.615876\pi$
$578$	−317540.	+	317540.i	−0.950481	+	0.950481i
$579$		0			0
$580$	53020.9	−	194689.i	0.157613	−	0.578743i
$581$	171431.			0.507852
$582$		0			0
$583$	−152398.	+	152398.i	−0.448376	+	0.448376i
$584$			57820.6i			0.169534i
$585$		0			0
$586$	15582.6			0.0453778
$587$	116615.	+	116615.i	0.338436	+	0.338436i	0.855778	−	0.517343i	$-0.173079\pi$
$587$	116615.	+	116615.i	0.338436	+	0.338436i	−0.517343	+	0.855778i	$0.673079\pi$
$588$		0			0
$589$			159480.i			0.459703i
$590$	−91247.4	−	159548.i	−0.262130	−	0.458340i
$591$		0			0
$592$	289656.	+	289656.i	0.826492	+	0.826492i
$593$	80973.9	−	80973.9i	0.230269	−	0.230269i	−0.582536	−	0.812805i	$-0.697939\pi$
$593$	80973.9	−	80973.9i	0.230269	−	0.230269i	0.812805	+	0.582536i	$0.197939\pi$
$594$		0			0
$595$	−3187.73	−	868.133i	−0.00900424	−	0.00245218i
$596$	321733.			0.905738
$597$		0			0
$598$	−85703.3	+	85703.3i	−0.239660	+	0.239660i
$599$			683966.i			1.90626i	0.302569	+	0.953128i	$0.402156\pi$
$599$			683966.i			1.90626i	−0.302569	+	0.953128i	$0.597844\pi$
$600$		0			0
$601$	−211089.			−0.584408			−0.292204	−	0.956356i	$-0.594389\pi$
$601$	−211089.			−0.584408			−0.292204	+	0.956356i	$0.594389\pi$
$602$	3897.82	+	3897.82i	0.0107555	+	0.0107555i
$603$		0			0
$604$		−	488692.i		−	1.33956i
$605$	−208267.	+	764743.i	−0.568997	+	2.08932i
$606$		0			0
$607$	502579.	+	502579.i	1.36404	+	1.36404i	0.868699	+	0.495340i	$0.164956\pi$
$607$	502579.	+	502579.i	1.36404	+	1.36404i	0.495340	+	0.868699i	$0.335044\pi$
$608$	566784.	−	566784.i	1.53324	−	1.53324i
$609$		0			0
$610$	−348281.	+	199186.i	−0.935987	+	0.535302i
$611$	106876.			0.286285
$612$		0			0
$613$	402276.	−	402276.i	1.07054	−	1.07054i	0.0732238	−	0.997316i	$-0.476671\pi$
$613$	402276.	−	402276.i	1.07054	−	1.07054i	0.997316	−	0.0732238i	$-0.0233287\pi$
$614$			220424.i			0.584685i
$615$		0			0
$616$	−83497.2			−0.220044
$617$	−439415.	−	439415.i	−1.15426	−	1.15426i	−0.985689	−	0.168574i	$-0.946084\pi$
$617$	−439415.	−	439415.i	−1.15426	−	1.15426i	−0.168574	−	0.985689i	$-0.553916\pi$
$618$		0			0
$619$		−	555813.i		−	1.45060i	−0.688434	−	0.725299i	$-0.741701\pi$
$619$		−	555813.i		−	1.45060i	0.688434	−	0.725299i	$-0.258299\pi$
$620$	−82311.5	−	22416.4i	−0.214130	−	0.0583153i
$621$		0			0
$622$	−76576.9	−	76576.9i	−0.197932	−	0.197932i
$623$	−116778.	+	116778.i	−0.300874	+	0.300874i
$624$		0			0
$625$	189754.	+	341440.i	0.485771	+	0.874086i
$626$	−881154.			−2.24855
$627$		0			0
$628$	−240879.	+	240879.i	−0.610773	+	0.610773i
$629$			7791.73i			0.0196939i
$630$		0			0
$631$	−448846.			−1.12730			−0.563648	−	0.826015i	$-0.690603\pi$
$631$	−448846.			−1.12730			−0.563648	+	0.826015i	$0.690603\pi$
$632$	70583.2	+	70583.2i	0.176713	+	0.176713i
$633$		0			0
$634$			430260.i			1.07042i
$635$	369003.	−	211037.i	0.915128	−	0.523373i
$636$		0			0
$637$	55345.9	+	55345.9i	0.136398	+	0.136398i
$638$	511056.	−	511056.i	1.25553	−	1.25553i
$639$		0			0
$640$	−102964.	−	180035.i	−0.251377	−	0.439538i
$641$	579776.			1.41106			0.705528	−	0.708682i	$-0.250710\pi$
$641$	579776.			1.41106			0.705528	+	0.708682i	$0.250710\pi$
$642$		0			0
$643$	485382.	−	485382.i	1.17398	−	1.17398i	0.192731	−	0.981252i	$-0.438266\pi$
$643$	485382.	−	485382.i	1.17398	−	1.17398i	0.981252	−	0.192731i	$-0.0617344\pi$
$644$		−	161730.i		−	0.389958i
$645$		0			0
$646$	18282.5			0.0438097
$647$	−227180.	−	227180.i	−0.542703	−	0.542703i	0.381618	−	0.924320i	$-0.375367\pi$
$647$	−227180.	−	227180.i	−0.542703	−	0.542703i	−0.924320	+	0.381618i	$0.875367\pi$
$648$		0			0
$649$		−	294251.i		−	0.698601i
$650$	−122663.	−	72163.2i	−0.290326	−	0.170800i
$651$		0			0
$652$	−3626.04	−	3626.04i	−0.00852977	−	0.00852977i
$653$	−51532.9	+	51532.9i	−0.120853	+	0.120853i	−0.764947	−	0.644093i	$-0.777235\pi$
$653$	−51532.9	+	51532.9i	−0.120853	+	0.120853i	0.644093	+	0.764947i	$0.277235\pi$
$654$		0			0
$655$	31968.6	−	117386.i	0.0745145	−	0.273612i
$656$	466958.			1.08510
$657$		0			0
$658$	−225617.	+	225617.i	−0.521099	+	0.521099i
$659$		−	447720.i		−	1.03095i	−0.856906	−	0.515473i	$-0.827616\pi$
$659$		−	447720.i		−	1.03095i	0.856906	−	0.515473i	$-0.172384\pi$
$660$		0			0
$661$	−99640.8			−0.228052			−0.114026	−	0.993478i	$-0.536375\pi$
$661$	−99640.8			−0.228052			−0.114026	+	0.993478i	$0.536375\pi$
$662$	290006.	+	290006.i	0.661745	+	0.661745i
$663$		0			0
$664$			120430.i			0.273149i
$665$	−176246.	−	308171.i	−0.398545	−	0.696864i
$666$		0			0
$667$	−234923.	−	234923.i	−0.528048	−	0.528048i
$668$	−334337.	+	334337.i	−0.749257	+	0.749257i
$669$		0			0
$670$	−180917.	−	49270.3i	−0.403024	−	0.109758i
$671$	−642327.			−1.42663
$672$		0			0
$673$	−170652.	+	170652.i	−0.376774	+	0.376774i	−0.869937	−	0.493163i	$-0.835841\pi$
$673$	−170652.	+	170652.i	−0.376774	+	0.376774i	0.493163	+	0.869937i	$0.335841\pi$
$674$		−	675948.i		−	1.48797i
$675$		0			0
$676$	−346152.			−0.757483
$677$	178335.	+	178335.i	0.389099	+	0.389099i	0.874366	−	0.485267i	$-0.161278\pi$
$677$	178335.	+	178335.i	0.389099	+	0.389099i	−0.485267	+	0.874366i	$0.661278\pi$
$678$		0			0
$679$			361183.i			0.783408i
$680$	609.863	−	2239.38i	0.00131891	−	0.00484294i
$681$		0			0
$682$	−216067.	−	216067.i	−0.464536	−	0.464536i
$683$	−621359.	+	621359.i	−1.33199	+	1.33199i	−0.428402	+	0.903588i	$0.640923\pi$
$683$	−621359.	+	621359.i	−1.33199	+	1.33199i	−0.903588	+	0.428402i	$0.859077\pi$
$684$		0			0
$685$	312058.	−	178470.i	0.665049	−	0.380350i
$686$	−537122.			−1.14137
$687$		0			0
$688$	−9119.14	+	9119.14i	−0.0192654	+	0.0192654i
$689$		−	42382.2i		−	0.0892782i
$690$		0			0
$691$	−142980.			−0.299446			−0.149723	−	0.988728i	$-0.547838\pi$
$691$	−142980.			−0.299446			−0.149723	+	0.988728i	$0.547838\pi$
$692$	413683.	+	413683.i	0.863884	+	0.863884i
$693$		0			0
$694$		−	315858.i		−	0.655803i
$695$	101202.	+	27561.0i	0.209517	+	0.0570591i
$696$		0			0
$697$	6280.57	+	6280.57i	0.0129281	+	0.0129281i
$698$	−361301.	+	361301.i	−0.741582	+	0.741582i
$699$		0			0
$700$	183827.	−	47648.7i	0.375157	−	0.0972422i
$701$	−248096.			−0.504876			−0.252438	−	0.967613i	$-0.581232\pi$
$701$	−248096.			−0.504876			−0.252438	+	0.967613i	$0.581232\pi$
$702$		0			0
$703$	−592028.	+	592028.i	−1.19793	+	1.19793i
$704$			517306.i			1.04376i
$705$		0			0
$706$	414500.			0.831602
$707$	67189.6	+	67189.6i	0.134420	+	0.134420i
$708$		0			0
$709$			669146.i			1.33115i	0.746329	+	0.665577i	$0.231814\pi$
$709$			669146.i			1.33115i	−0.746329	+	0.665577i	$0.768186\pi$
$710$	−7945.49	+	4544.12i	−0.0157617	+	0.00901432i
$711$		0			0
$712$	−82036.4	−	82036.4i	−0.161825	−	0.161825i
$713$	−99321.8	+	99321.8i	−0.195373	+	0.195373i
$714$		0			0
$715$	−113112.	−	197779.i	−0.221257	−	0.386873i
$716$	232603.			0.453722
$717$		0			0
$718$	229940.	−	229940.i	0.446031	−	0.446031i
$719$		−	244719.i		−	0.473380i	−0.971585	−	0.236690i	$-0.923937\pi$
$719$		−	244719.i		−	0.473380i	0.971585	−	0.236690i	$-0.0760626\pi$
$720$		0			0
$721$	−3978.14			−0.00765260
$722$	893475.	+	893475.i	1.71399	+	1.71399i
$723$		0			0
$724$			298349.i			0.569177i
$725$	197808.	−	336234.i	0.376329	−	0.639683i
$726$		0			0
$727$	−515647.	−	515647.i	−0.975626	−	0.975626i	0.0240837	−	0.999710i	$-0.492333\pi$
$727$	−515647.	−	515647.i	−0.975626	−	0.975626i	−0.999710	+	0.0240837i	$0.992333\pi$
$728$	11610.4	−	11610.4i	0.0219070	−	0.0219070i
$729$		0			0
$730$	123770.	−	454476.i	0.232258	−	0.852835i
$731$	−245.305			−0.000459062
$732$		0			0
$733$	98009.3	−	98009.3i	0.182414	−	0.182414i	−0.609993	−	0.792407i	$-0.708828\pi$
$733$	98009.3	−	98009.3i	0.182414	−	0.182414i	0.792407	+	0.609993i	$0.208828\pi$
$734$		−	641526.i		−	1.19075i
$735$		0			0
$736$	705967.			1.30325
$737$	−212265.	−	212265.i	−0.390790	−	0.390790i
$738$		0			0
$739$		−	31323.0i		−	0.0573554i	−0.999589	−	0.0286777i	$-0.990870\pi$
$739$		−	31323.0i		−	0.0573554i	0.999589	−	0.0286777i	$-0.00912965\pi$
$740$	−222345.	−	388774.i	−0.406035	−	0.709961i
$741$		0			0
$742$	89469.4	+	89469.4i	0.162505	+	0.162505i
$743$	−46804.2	+	46804.2i	−0.0847828	+	0.0847828i	−0.748226	−	0.663444i	$-0.769094\pi$
$743$	−46804.2	+	46804.2i	−0.0847828	+	0.0847828i	0.663444	+	0.748226i	$0.269094\pi$
$744$		0			0
$745$	600153.	+	163443.i	1.08131	+	0.294479i
$746$	514252.			0.924056
$747$		0			0
$748$	−11071.0	+	11071.0i	−0.0197872	+	0.0197872i
$749$			276818.i			0.493435i
$750$		0			0
$751$	−186.753			−0.000331121			−0.000165561	−	1.00000i	$-0.500053\pi$
$751$	−186.753			−0.000331121			−0.000165561		1.00000i	$0.500053\pi$
$752$	−527842.	−	527842.i	−0.933401	−	0.933401i
$753$		0			0
$754$			142126.i			0.249994i
$755$	248260.	−	911596.i	0.435525	−	1.59922i
$756$		0			0
$757$	596186.	+	596186.i	1.04037	+	1.04037i	0.999150	+	0.0412248i	$0.0131260\pi$
$757$	596186.	+	596186.i	1.04037	+	1.04037i	0.0412248	+	0.999150i	$0.486874\pi$
$758$	−489308.	+	489308.i	−0.851616	+	0.851616i
$759$		0			0
$760$	216490.	−	123813.i	0.374809	−	0.214358i
$761$	1.04441e6			1.80344			0.901722	−	0.432315i	$-0.142303\pi$
$761$	1.04441e6			1.80344			0.901722	+	0.432315i	$0.142303\pi$
$762$		0			0
$763$	139245.	−	139245.i	0.239183	−	0.239183i
$764$		−	671023.i		−	1.14961i
$765$		0			0
$766$	1.12181e6			1.91188
$767$	40915.9	+	40915.9i	0.0695507	+	0.0695507i
$768$		0			0
$769$			577158.i			0.975982i	0.872849	+	0.487991i	$0.162270\pi$
$769$			577158.i			0.975982i	−0.872849	+	0.487991i	$0.837730\pi$
$770$	656296.	+	178733.i	1.10693	+	0.301456i
$771$		0			0
$772$	653566.	+	653566.i	1.09662	+	1.09662i
$773$	414104.	−	414104.i	0.693027	−	0.693027i	−0.269870	−	0.962897i	$-0.586981\pi$
$773$	414104.	−	414104.i	0.693027	−	0.693027i	0.962897	+	0.269870i	$0.0869807\pi$
$774$		0			0
$775$	−142154.	−	83630.2i	−0.236677	−	0.139239i
$776$	−253731.			−0.421357
$777$		0			0
$778$	−668111.	+	668111.i	−1.10380	+	1.10380i
$779$			954417.i			1.57276i
$780$		0			0
$781$	−14653.7			−0.0240240
$782$	11386.0	+	11386.0i	0.0186191	+	0.0186191i
$783$		0			0
$784$		−	546687.i		−	0.889419i
$785$	−571700.	+	326962.i	−0.927745	+	0.530588i
$786$		0			0
$787$	−337250.	−	337250.i	−0.544505	−	0.544505i	0.380341	−	0.924846i	$-0.375807\pi$
$787$	−337250.	−	337250.i	−0.544505	−	0.544505i	−0.924846	+	0.380341i	$0.875807\pi$
$788$	−288723.	+	288723.i	−0.464974	+	0.464974i
$789$		0			0
$790$	−403702.	−	705881.i	−0.646854	−	1.13104i
$791$	−32168.7			−0.0514138
$792$		0			0
$793$	89316.2	−	89316.2i	0.142031	−	0.142031i
$794$		−	450317.i		−	0.714294i
$795$		0			0
$796$	−737435.			−1.16385
$797$	365276.	+	365276.i	0.575048	+	0.575048i	0.933535	−	0.358487i	$-0.116707\pi$
$797$	365276.	+	365276.i	0.575048	+	0.575048i	−0.358487	+	0.933535i	$0.616707\pi$
$798$		0			0
$799$		−	14198.9i		−	0.0222414i
$800$	207992.	+	802425.i	0.324987	+	1.25379i
$801$		0			0
$802$	311606.	+	311606.i	0.484460	+	0.484460i
$803$	533223.	−	533223.i	0.826948	−	0.826948i
$804$		0			0
$805$	82160.3	−	301687.i	0.126786	−	0.465549i
$806$	60088.5			0.0924957
$807$		0			0
$808$	−47200.7	+	47200.7i	−0.0722979	+	0.0722979i
$809$			1.04908e6i			1.60292i	0.598051	+	0.801458i	$0.295942\pi$
$809$			1.04908e6i			1.60292i	−0.598051	+	0.801458i	$0.704058\pi$
$810$		0			0
$811$	−882064.			−1.34109			−0.670546	−	0.741868i	$-0.733940\pi$
$811$	−882064.			−1.34109			−0.670546	+	0.741868i	$0.733940\pi$
$812$	−134102.	−	134102.i	−0.203387	−	0.203387i
$813$		0			0
$814$		−	1.60418e6i		−	2.42105i
$815$	−4921.87	−	8606.00i	−0.00740995	−	0.0129565i
$816$		0			0
$817$	−18638.6	−	18638.6i	−0.0279235	−	0.0279235i
$818$	407916.	−	407916.i	0.609627	−	0.609627i
$819$		0			0
$820$	−492596.	−	134152.i	−0.732594	−	0.199512i
$821$	−1.11452e6			−1.65348			−0.826742	−	0.562581i	$-0.809809\pi$
$821$	−1.11452e6			−1.65348			−0.826742	+	0.562581i	$0.809809\pi$
$822$		0			0
$823$	−258025.	+	258025.i	−0.380945	+	0.380945i	−0.871443	−	0.490497i	$-0.836815\pi$
$823$	−258025.	+	258025.i	−0.380945	+	0.380945i	0.490497	+	0.871443i	$0.336815\pi$
$824$		−	2794.64i		−	0.00411596i
$825$		0			0
$826$	−172748.			−0.253194
$827$	55607.8	+	55607.8i	0.0813063	+	0.0813063i	0.746590	−	0.665284i	$-0.231690\pi$
$827$	55607.8	+	55607.8i	0.0813063	+	0.0813063i	−0.665284	+	0.746590i	$0.731690\pi$
$828$		0			0
$829$			1.03445e6i			1.50522i	0.658468	+	0.752609i	$0.271205\pi$
$829$			1.03445e6i			1.50522i	−0.658468	+	0.752609i	$0.728795\pi$
$830$	257792.	−	946594.i	0.374208	−	1.37407i
$831$		0			0
$832$	−71931.9	−	71931.9i	−0.103914	−	0.103914i
$833$	7352.94	−	7352.94i	0.0105967	−	0.0105967i
$834$		0			0
$835$	−793511.	+	453818.i	−1.13810	+	0.650892i
$836$	−1.68239e6			−2.40720
$837$		0			0
$838$	95945.1	−	95945.1i	0.136626	−	0.136626i
$839$			426954.i			0.606537i	0.952905	+	0.303268i	$0.0980778\pi$
$839$			426954.i			0.606537i	−0.952905	+	0.303268i	$0.901922\pi$
$840$		0			0
$841$	317698.			0.449182
$842$	−630808.	−	630808.i	−0.889761	−	0.889761i
$843$		0			0
$844$		−	655548.i		−	0.920279i
$845$	−645704.	−	175849.i	−0.904316	−	0.246278i
$846$		0			0
$847$	526755.	+	526755.i	0.734247	+	0.734247i
$848$	−209318.	+	209318.i	−0.291082	+	0.291082i
$849$		0			0
$850$	−9587.18	+	16296.3i	−0.0132695	+	0.0225554i
$851$	−737411.			−1.01824
$852$		0			0
$853$	−63890.0	+	63890.0i	−0.0878081	+	0.0878081i	−0.749647	−	0.661838i	$-0.769776\pi$
$853$	−63890.0	+	63890.0i	−0.0878081	+	0.0878081i	0.661838	+	0.749647i	$0.269776\pi$
$854$			377095.i			0.517053i
$855$		0			0
$856$	−194464.			−0.265395
$857$	104328.	+	104328.i	0.142049	+	0.142049i	0.774555	−	0.632506i	$-0.217974\pi$
$857$	104328.	+	104328.i	0.142049	+	0.142049i	−0.632506	+	0.774555i	$0.717974\pi$
$858$		0			0
$859$		−	1.25585e6i		−	1.70196i	−0.525196	−	0.850982i	$-0.676008\pi$
$859$		−	1.25585e6i		−	1.70196i	0.525196	−	0.850982i	$-0.323992\pi$
$860$	12239.7	−	7000.01i	0.0165490	−	0.00946458i
$861$		0			0
$862$	−691828.	−	691828.i	−0.931073	−	0.931073i
$863$	−618705.	+	618705.i	−0.830734	+	0.830734i	−0.987617	−	0.156883i	$-0.949855\pi$
$863$	−618705.	+	618705.i	−0.830734	+	0.830734i	0.156883	+	0.987617i	$0.449855\pi$
$864$		0			0
$865$	561520.	+	981831.i	0.750470	+	1.31221i
$866$	1.35660e6			1.80891
$867$		0			0
$868$	−56696.2	+	56696.2i	−0.0752514	+	0.0752514i
$869$		−	1.30184e6i		−	1.72393i
$870$		0			0
$871$	59031.3			0.0778119
$872$	97819.4	+	97819.4i	0.128645	+	0.128645i
$873$		0			0
$874$			1.73026e6i			2.26510i
$875$	367113.	+	4503.14i	0.479495	+	0.00588165i
$876$		0			0
$877$	−446207.	−	446207.i	−0.580146	−	0.580146i	0.354797	−	0.934943i	$-0.384550\pi$
$877$	−446207.	−	446207.i	−0.580146	−	0.580146i	−0.934943	+	0.354797i	$0.884550\pi$
$878$	94796.6	−	94796.6i	0.122971	−	0.122971i
$879$		0			0
$880$	−418155.	+	1.53544e6i	−0.539973	+	1.98274i
$881$	226795.			0.292201			0.146101	−	0.989270i	$-0.453328\pi$
$881$	226795.			0.292201			0.146101	+	0.989270i	$0.453328\pi$
$882$		0			0
$883$	235051.	−	235051.i	0.301467	−	0.301467i	−0.540121	−	0.841588i	$-0.681621\pi$
$883$	235051.	−	235051.i	0.301467	−	0.301467i	0.841588	+	0.540121i	$0.181621\pi$
$884$		−	3078.87i		−	0.00393991i
$885$		0			0
$886$	−571276.			−0.727744
$887$	789601.	+	789601.i	1.00360	+	1.00360i	0.999993	+	0.00360590i	$0.00114779\pi$
$887$	789601.	+	789601.i	1.00360	+	1.00360i	0.00360590	+	0.999993i	$0.498852\pi$
$888$		0			0
$889$		−	399531.i		−	0.505531i
$890$	469208.	+	820421.i	0.592360	+	1.03575i
$891$		0			0
$892$	−245581.	−	245581.i	−0.308649	−	0.308649i
$893$	1.07886e6	−	1.07886e6i	1.35289	−	1.35289i
$894$		0			0
$895$	433893.	+	118165.i	0.541672	+	0.147517i
$896$	−194930.			−0.242808
$897$		0			0
$898$	862857.	−	862857.i	1.07001	−	1.07001i
$899$			164710.i			0.203798i
$900$		0			0
$901$	−5630.65			−0.00693600
$902$	−1.29306e6	−	1.29306e6i	−1.58930	−	1.58930i
$903$		0			0
$904$		−	22598.5i		−	0.0276530i
$905$	−151564.	+	556533.i	−0.185054	+	0.679507i
$906$		0			0
$907$	−36080.2	−	36080.2i	−0.0438585	−	0.0438585i	0.684837	−	0.728696i	$-0.259873\pi$
$907$	−36080.2	−	36080.2i	−0.0438585	−	0.0438585i	−0.728696	+	0.684837i	$0.759873\pi$
$908$	310555.	−	310555.i	0.376675	−	0.376675i
$909$		0			0
$910$	−116112.	+	66405.6i	−0.140214	+	0.0801903i
$911$	−216456.			−0.260815			−0.130407	−	0.991460i	$-0.541629\pi$
$911$	−216456.			−0.260815			−0.130407	+	0.991460i	$0.541629\pi$
$912$		0			0
$913$	1.11061e6	−	1.11061e6i	1.33236	−	1.33236i
$914$			24895.9i			0.0298013i
$915$		0			0
$916$	64054.5			0.0763411
$917$	−80855.8	−	80855.8i	−0.0961552	−	0.0961552i
$918$		0			0
$919$		−	358123.i		−	0.424034i	−0.977266	−	0.212017i	$-0.931997\pi$
$919$		−	358123.i		−	0.424034i	0.977266	−	0.212017i	$-0.0680032\pi$
$920$	211935.	+	57717.6i	0.250396	+	0.0681919i
$921$		0			0
$922$	361251.	+	361251.i	0.424959	+	0.424959i
$923$	2037.61	−	2037.61i	0.00239176	−	0.00239176i
$924$		0			0
$925$	−217256.	−	838165.i	−0.253914	−	0.979594i
$926$	1.64366e6			1.91686
$927$		0			0
$928$	585369.	−	585369.i	0.679726	−	0.679726i
$929$		−	12991.3i		−	0.0150530i	−0.999972	−	0.00752649i	$-0.997604\pi$
$929$		−	12991.3i		−	0.0150530i	0.999972	−	0.00752649i	$-0.00239578\pi$
$930$		0			0
$931$	1.11738e6			1.28914
$932$	787229.	+	787229.i	0.906294	+	0.906294i
$933$		0			0
$934$			58819.7i			0.0674262i
$935$	−26275.8	+	15027.4i	−0.0300561	+	0.0171894i
$936$		0			0
$937$	253354.	+	253354.i	0.288568	+	0.288568i	0.836514	−	0.547946i	$-0.184590\pi$
$937$	253354.	+	253354.i	0.288568	+	0.288568i	−0.547946	+	0.836514i	$0.684590\pi$
$938$	−124616.	+	124616.i	−0.141634	+	0.141634i
$939$		0			0
$940$	405181.	+	708467.i	0.458557	+	0.801796i
$941$	1.24657e6			1.40779			0.703893	−	0.710306i	$-0.251443\pi$
$941$	1.24657e6			1.40779			0.703893	+	0.710306i	$0.251443\pi$
$942$		0			0
$943$	−594395.	+	594395.i	−0.668424	+	0.668424i
$944$		−	404153.i		−	0.453525i
$945$		0			0
$946$	50503.9			0.0564342
$947$	−550159.	−	550159.i	−0.613462	−	0.613462i	0.330384	−	0.943847i	$-0.392822\pi$
$947$	−550159.	−	550159.i	−0.613462	−	0.613462i	−0.943847	+	0.330384i	$0.892822\pi$
$948$		0			0
$949$			148290.i			0.164657i
$950$	−1.96667e6	+	509768.i	−2.17913	+	0.564839i
$951$		0			0
$952$	−1542.48	−	1542.48i	−0.00170195	−	0.00170195i
$953$	−457502.	+	457502.i	−0.503741	+	0.503741i	−0.912598	−	0.408858i	$-0.865927\pi$
$953$	−457502.	+	457502.i	−0.503741	+	0.503741i	0.408858	+	0.912598i	$0.365927\pi$
$954$		0			0
$955$	340887.	−	1.25171e6i	0.373769	−	1.37245i
$956$	525121.			0.574570
$957$		0			0
$958$	−1.21993e6	+	1.21993e6i	−1.32924	+	1.32924i
$959$		−	337875.i		−	0.367383i
$960$		0			0
$961$	−853884.			−0.924596
$962$	223063.	+	223063.i	0.241033	+	0.241033i
$963$		0			0
$964$			316084.i			0.340133i
$965$	887130.	+	1.55117e6i	0.952648	+	1.66573i
$966$		0			0
$967$	72229.7	+	72229.7i	0.0772437	+	0.0772437i	0.744673	−	0.667429i	$-0.232605\pi$
$967$	72229.7	+	72229.7i	0.0772437	+	0.0772437i	−0.667429	+	0.744673i	$0.732605\pi$
$968$	−370045.	+	370045.i	−0.394916	+	0.394916i
$969$		0			0
$970$	1.99435e6	+	543134.i	2.11962	+	0.577250i
$971$	−864636.			−0.917054			−0.458527	−	0.888680i	$-0.651623\pi$
$971$	−864636.			−0.917054			−0.458527	+	0.888680i	$0.651623\pi$
$972$		0			0
$973$	69708.0	−	69708.0i	0.0736303	−	0.0736303i
$974$		−	40033.3i		−	0.0421992i
$975$		0			0
$976$	−882233.			−0.926155
$977$	−838896.	−	838896.i	−0.878859	−	0.878859i	0.114558	−	0.993417i	$-0.463455\pi$
$977$	−838896.	−	838896.i	−0.878859	−	0.878859i	−0.993417	+	0.114558i	$0.963455\pi$
$978$		0			0
$979$			1.51308e6i			1.57869i
$980$	−157057.	+	576703.i	−0.163533	+	0.600482i
$981$		0			0
$982$	729119.	+	729119.i	0.756093	+	0.756093i
$983$	1.04189e6	−	1.04189e6i	1.07823	−	1.07823i	0.0815669	−	0.996668i	$-0.474008\pi$
$983$	1.04189e6	−	1.04189e6i	1.07823	−	1.07823i	0.996668	−	0.0815669i	$-0.0259924\pi$
$984$		0			0
$985$	−685251.	+	391903.i	−0.706281	+	0.403930i
$986$	18882.0			0.0194220
$987$		0			0
$988$	233937.	−	233937.i	0.239654	−	0.239654i
$989$		−	23215.7i		−	0.0237350i
$990$		0			0
$991$	−1.02008e6			−1.03869			−0.519347	−	0.854564i	$-0.673825\pi$
$991$	−1.02008e6			−1.03869			−0.519347	+	0.854564i	$0.673825\pi$
$992$	−247485.	−	247485.i	−0.251493	−	0.251493i
$993$		0			0
$994$			8602.85i			0.00870702i
$995$	−1.37560e6	−	374624.i	−1.38945	−	0.378399i
$996$		0			0
$997$	−398909.	−	398909.i	−0.401313	−	0.401313i	0.477382	−	0.878696i	$-0.341586\pi$
$997$	−398909.	−	398909.i	−0.401313	−	0.401313i	−0.878696	+	0.477382i	$0.841586\pi$
$998$	−995456.	+	995456.i	−0.999450	+	0.999450i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.5.g.e.37.1		8
3.2	odd	2		15.5.f.a.7.4	✓	8
5.2	odd	4		225.5.g.m.118.4		8
5.3	odd	4	inner	45.5.g.e.28.1		8
5.4	even	2		225.5.g.m.82.4		8
12.11	even	2		240.5.bg.c.97.4		8
15.2	even	4		75.5.f.e.43.1		8
15.8	even	4		15.5.f.a.13.4	yes	8
15.14	odd	2		75.5.f.e.7.1		8
60.23	odd	4		240.5.bg.c.193.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.5.f.a.7.4	✓	8	3.2	odd	2
15.5.f.a.13.4	yes	8	15.8	even	4
45.5.g.e.28.1		8	5.3	odd	4	inner
45.5.g.e.37.1		8	1.1	even	1	trivial
75.5.f.e.7.1		8	15.14	odd	2
75.5.f.e.43.1		8	15.2	even	4
225.5.g.m.82.4		8	5.4	even	2
225.5.g.m.118.4		8	5.2	odd	4
240.5.bg.c.97.4		8	12.11	even	2
240.5.bg.c.193.4		8	60.23	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	45.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.80336 - 3.80336i) q^{2} +12.9311i q^{4} +(-6.56915 + 24.1215i) q^{5} +(16.6149 + 16.6149i) q^{7} +(-11.6720 + 11.6720i) q^{8} +(116.728 - 66.7579i) q^{10} +215.278 q^{11} +(-29.9347 + 29.9347i) q^{13} -126.385i q^{14} +295.684 q^{16} +(3.97694 + 3.97694i) q^{17} +604.349i q^{19} +(-311.919 - 84.9467i) q^{20} +(-818.782 - 818.782i) q^{22} +(-376.379 + 376.379i) q^{23} +(-538.692 - 316.916i) q^{25} +227.705 q^{26} +(-214.850 + 214.850i) q^{28} +624.166i q^{29} +263.888 q^{31} +(-937.842 - 937.842i) q^{32} -30.2515i q^{34} +(-509.922 + 291.630i) q^{35} +(979.613 + 979.613i) q^{37} +(2298.56 - 2298.56i) q^{38} +(-204.870 - 358.220i) q^{40} +1579.25 q^{41} +(-30.8409 + 30.8409i) q^{43} +2783.80i q^{44} +2863.01 q^{46} +(-1785.16 - 1785.16i) q^{47} -1848.89i q^{49} +(843.498 + 3254.19i) q^{50} +(-387.089 - 387.089i) q^{52} +(-707.912 + 707.912i) q^{53} +(-1414.20 + 5192.84i) q^{55} -387.857 q^{56} +(2373.93 - 2373.93i) q^{58} -1366.84i q^{59} -2983.71 q^{61} +(-1003.66 - 1003.66i) q^{62} +2402.96i q^{64} +(-525.423 - 918.714i) q^{65} +(-986.003 - 986.003i) q^{67} +(-51.4265 + 51.4265i) q^{68} +(3048.59 + 830.242i) q^{70} -68.0686 q^{71} +(2476.90 - 2476.90i) q^{73} -7451.65i q^{74} -7814.93 q^{76} +(3576.83 + 3576.83i) q^{77} -6047.25i q^{79} +(-1942.39 + 7132.33i) q^{80} +(-6006.45 - 6006.45i) q^{82} +(5158.95 - 5158.95i) q^{83} +(-122.055 + 69.8047i) q^{85} +234.598 q^{86} +(-2512.72 + 2512.72i) q^{88} +7028.50i q^{89} -994.722 q^{91} +(-4867.01 - 4867.01i) q^{92} +13579.2i q^{94} +(-14577.8 - 3970.06i) q^{95} +(10869.3 + 10869.3i) q^{97} +(-7032.00 + 7032.00i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 84 q^{5} + 20 q^{7} - 180 q^{8} + 104 q^{10} + 288 q^{11} - 340 q^{13} + 620 q^{16} - 900 q^{17} - 564 q^{20} - 1100 q^{22} + 1560 q^{23} - 1204 q^{25} + 3024 q^{26} + 3580 q^{28} - 512 q^{31} - 4980 q^{32}+ \cdots - 46440 q^{98}+O(q^{100}) \) 8 * q + 84 * q^5 + 20 * q^7 - 180 * q^8 + 104 * q^10 + 288 * q^11 - 340 * q^13 + 620 * q^16 - 900 * q^17 - 564 * q^20 - 1100 * q^22 + 1560 * q^23 - 1204 * q^25 + 3024 * q^26 + 3580 * q^28 - 512 * q^31 - 4980 * q^32 - 6600 * q^35 - 3820 * q^37 + 7680 * q^38 - 2952 * q^40 + 2712 * q^41 - 1240 * q^43 + 13528 * q^46 - 4800 * q^47 - 3744 * q^50 - 1240 * q^52 - 1020 * q^53 - 3644 * q^55 + 30720 * q^56 + 2340 * q^58 - 4760 * q^61 - 28680 * q^62 + 1212 * q^65 - 8920 * q^67 + 1920 * q^68 + 7380 * q^70 - 7536 * q^71 + 11600 * q^73 + 4344 * q^76 + 360 * q^77 - 10644 * q^80 - 27200 * q^82 + 32400 * q^83 - 15628 * q^85 - 14592 * q^86 - 14340 * q^88 + 16528 * q^91 + 31800 * q^92 - 18864 * q^95 + 58640 * q^97 - 46440 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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