LMFDB - Embedded newform 45.5.g.e.28.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,5,Mod(28,45)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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			28.2
Root			$3.30519 + 3.30519i$ of defining polynomial
Character	$\chi$	$=$	45.28
Dual form			45.5.g.e.37.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.30519 + 3.30519i) q^{2} -5.84858i q^{4} +(16.2403 + 19.0066i) q^{5} +(-33.1649 + 33.1649i) q^{7} +(-33.5524 - 33.5524i) q^{8} +O(q^{10})$	\(q+(-3.30519 + 3.30519i) q^{2} -5.84858i q^{4} +(16.2403 + 19.0066i) q^{5} +(-33.1649 + 33.1649i) q^{7} +(-33.5524 - 33.5524i) q^{8} +(-116.498 - 9.14312i) q^{10} -55.3118 q^{11} +(-161.918 - 161.918i) q^{13} -219.233i q^{14} +315.371 q^{16} +(-278.961 + 278.961i) q^{17} +179.871i q^{19} +(111.162 - 94.9829i) q^{20} +(182.816 - 182.816i) q^{22} +(398.080 + 398.080i) q^{23} +(-97.5033 + 617.348i) q^{25} +1070.34 q^{26} +(193.968 + 193.968i) q^{28} -547.472i q^{29} +1538.11 q^{31} +(-505.525 + 505.525i) q^{32} -1844.04i q^{34} +(-1168.96 - 91.7437i) q^{35} +(-1660.74 + 1660.74i) q^{37} +(-594.509 - 594.509i) q^{38} +(92.8156 - 1182.62i) q^{40} +307.601 q^{41} +(104.254 + 104.254i) q^{43} +323.496i q^{44} -2631.46 q^{46} +(346.159 - 346.159i) q^{47} +201.180i q^{49} +(-1718.19 - 2362.72i) q^{50} +(-946.991 + 946.991i) q^{52} +(2025.02 + 2025.02i) q^{53} +(-898.282 - 1051.29i) q^{55} +2225.52 q^{56} +(1809.50 + 1809.50i) q^{58} +2854.49i q^{59} -1057.37 q^{61} +(-5083.73 + 5083.73i) q^{62} +1704.23i q^{64} +(447.912 - 5707.12i) q^{65} +(3756.53 - 3756.53i) q^{67} +(1631.53 + 1631.53i) q^{68} +(4166.87 - 3560.41i) q^{70} -1429.24 q^{71} +(813.690 + 813.690i) q^{73} -10978.2i q^{74} +1051.99 q^{76} +(1834.41 - 1834.41i) q^{77} +4854.69i q^{79} +(5121.74 + 5994.14i) q^{80} +(-1016.68 + 1016.68i) q^{82} +(1316.57 + 1316.57i) q^{83} +(-9832.53 - 771.687i) q^{85} -689.160 q^{86} +(1855.84 + 1855.84i) q^{88} +5185.49i q^{89} +10740.0 q^{91} +(2328.20 - 2328.20i) q^{92} +2288.24i q^{94} +(-3418.74 + 2921.17i) q^{95} +(3495.32 - 3495.32i) q^{97} +(-664.940 - 664.940i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 84 q^{5} + 20 q^{7} - 180 q^{8}+O(q^{10}) $ 8 * q + 84 * q^5 + 20 * q^7 - 180 * q^8	$ 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}+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

Display 100 coefficients 10 coefficients

Character values

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

$n$	$11$	$37$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−3.30519	+	3.30519i	−0.826298	+	0.826298i	−0.987003	−	0.160705i	$-0.948623\pi$
$2$	−3.30519	+	3.30519i	−0.826298	+	0.826298i	0.160705	+	0.987003i	$0.448623\pi$
$3$		0			0
$3$		0			0
$4$		−	5.84858i		−	0.365537i
$5$	16.2403	+	19.0066i	0.649613	+	0.760265i
$5$	16.2403	+	19.0066i	0.649613	+	0.760265i
$6$		0			0
$7$	−33.1649	+	33.1649i	−0.676834	+	0.676834i	−0.959283	−	0.282448i	$-0.908853\pi$
$7$	−33.1649	+	33.1649i	−0.676834	+	0.676834i	0.282448	+	0.959283i	$0.408853\pi$
$8$	−33.5524	−	33.5524i	−0.524256	−	0.524256i
$9$		0			0
$10$	−116.498	−	9.14312i	−1.16498	−	0.0914312i
$11$	−55.3118			−0.457122			−0.228561	−	0.973530i	$-0.573402\pi$
$11$	−55.3118			−0.457122			−0.228561	+	0.973530i	$0.573402\pi$
$12$		0			0
$13$	−161.918	−	161.918i	−0.958095	−	0.958095i	0.0410619	−	0.999157i	$-0.486926\pi$
$13$	−161.918	−	161.918i	−0.958095	−	0.958095i	−0.999157	+	0.0410619i	$0.986926\pi$
$14$		−	219.233i		−	1.11853i
$15$		0			0
$16$	315.371			1.23192
$17$	−278.961	+	278.961i	−0.965264	+	0.965264i	−0.999417	−	0.0341531i	$-0.989127\pi$
$17$	−278.961	+	278.961i	−0.965264	+	0.965264i	0.0341531	+	0.999417i	$0.489127\pi$
$18$		0			0
$19$			179.871i			0.498258i	0.968470	+	0.249129i	$0.0801443\pi$
$19$			179.871i			0.498258i	−0.968470	+	0.249129i	$0.919856\pi$
$20$	111.162	−	94.9829i	0.277905	−	0.237457i
$21$		0			0
$22$	182.816	−	182.816i	0.377719	−	0.377719i
$23$	398.080	+	398.080i	0.752514	+	0.752514i	0.974948	−	0.222434i	$-0.0714001\pi$
$23$	398.080	+	398.080i	0.752514	+	0.752514i	−0.222434	+	0.974948i	$0.571400\pi$
$24$		0			0
$25$	−97.5033	+	617.348i	−0.156005	+	0.987756i
$26$	1070.34			1.58334
$27$		0			0
$28$	193.968	+	193.968i	0.247408	+	0.247408i
$29$		−	547.472i		−	0.650977i	−0.945546	−	0.325488i	$-0.894471\pi$
$29$		−	547.472i		−	0.650977i	0.945546	−	0.325488i	$-0.105529\pi$
$30$		0			0
$31$	1538.11			1.60053			0.800263	−	0.599649i	$-0.204693\pi$
$31$	1538.11			1.60053			0.800263	+	0.599649i	$0.204693\pi$
$32$	−505.525	+	505.525i	−0.493677	+	0.493677i
$33$		0			0
$34$		−	1844.04i		−	1.59519i
$35$	−1168.96	−	91.7437i	−0.954254	−	0.0748928i
$36$		0			0
$37$	−1660.74	+	1660.74i	−1.21311	+	1.21311i	−0.243109	+	0.969999i	$0.578167\pi$
$37$	−1660.74	+	1660.74i	−1.21311	+	1.21311i	−0.969999	+	0.243109i	$0.921833\pi$
$38$	−594.509	−	594.509i	−0.411710	−	0.411710i
$39$		0			0
$40$	92.8156	−	1182.62i	0.0580097	−	0.739137i
$41$	307.601			0.182987			0.0914933	−	0.995806i	$-0.470836\pi$
$41$	307.601			0.182987			0.0914933	+	0.995806i	$0.470836\pi$
$42$		0			0
$43$	104.254	+	104.254i	0.0563841	+	0.0563841i	0.734737	−	0.678353i	$-0.237306\pi$
$43$	104.254	+	104.254i	0.0563841	+	0.0563841i	−0.678353	+	0.734737i	$0.737306\pi$
$44$			323.496i			0.167095i
$45$		0			0
$46$	−2631.46			−1.24360
$47$	346.159	−	346.159i	0.156704	−	0.156704i	−0.624400	−	0.781104i	$-0.714657\pi$
$47$	346.159	−	346.159i	0.156704	−	0.156704i	0.781104	+	0.624400i	$0.214657\pi$
$48$		0			0
$49$			201.180i			0.0837903i
$50$	−1718.19	−	2362.72i	−0.687274	−	0.945088i
$51$		0			0
$52$	−946.991	+	946.991i	−0.350219	+	0.350219i
$53$	2025.02	+	2025.02i	0.720906	+	0.720906i	0.968790	−	0.247884i	$-0.0797352\pi$
$53$	2025.02	+	2025.02i	0.720906	+	0.720906i	−0.247884	+	0.968790i	$0.579735\pi$
$54$		0			0
$55$	−898.282	−	1051.29i	−0.296953	−	0.347534i
$56$	2225.52			0.709669
$57$		0			0
$58$	1809.50	+	1809.50i	0.537901	+	0.537901i
$59$			2854.49i			0.820020i	0.912081	+	0.410010i	$0.134475\pi$
$59$			2854.49i			0.820020i	−0.912081	+	0.410010i	$0.865525\pi$
$60$		0			0
$61$	−1057.37			−0.284163			−0.142082	−	0.989855i	$-0.545380\pi$
$61$	−1057.37			−0.284163			−0.142082	+	0.989855i	$0.545380\pi$
$62$	−5083.73	+	5083.73i	−1.32251	+	1.32251i
$63$		0			0
$64$			1704.23i			0.416071i
$65$	447.912	−	5707.12i	0.106015	−	1.35080i
$66$		0			0
$67$	3756.53	−	3756.53i	0.836830	−	0.836830i	−0.151611	−	0.988440i	$-0.548446\pi$
$67$	3756.53	−	3756.53i	0.836830	−	0.836830i	0.988440	+	0.151611i	$0.0484460\pi$
$68$	1631.53	+	1631.53i	0.352839	+	0.352839i
$69$		0			0
$70$	4166.87	−	3560.41i	0.850382	−	0.726614i
$71$	−1429.24			−0.283523			−0.141761	−	0.989901i	$-0.545277\pi$
$71$	−1429.24			−0.283523			−0.141761	+	0.989901i	$0.545277\pi$
$72$		0			0
$73$	813.690	+	813.690i	0.152691	+	0.152691i	0.779319	−	0.626628i	$-0.215565\pi$
$73$	813.690	+	813.690i	0.152691	+	0.152691i	−0.626628	+	0.779319i	$0.715565\pi$
$74$		−	10978.2i		−	2.00478i
$75$		0			0
$76$	1051.99			0.182132
$77$	1834.41	−	1834.41i	0.309396	−	0.309396i
$78$		0			0
$79$			4854.69i			0.777871i	0.921265	+	0.388936i	$0.127157\pi$
$79$			4854.69i			0.777871i	−0.921265	+	0.388936i	$0.872843\pi$
$80$	5121.74	+	5994.14i	0.800271	+	0.936585i
$81$		0			0
$82$	−1016.68	+	1016.68i	−0.151201	+	0.151201i
$83$	1316.57	+	1316.57i	0.191111	+	0.191111i	0.796176	−	0.605065i	$-0.206853\pi$
$83$	1316.57	+	1316.57i	0.191111	+	0.191111i	−0.605065	+	0.796176i	$0.706853\pi$
$84$		0			0
$85$	−9832.53	−	771.687i	−1.36090	−	0.106808i
$86$	−689.160			−0.0931801
$87$		0			0
$88$	1855.84	+	1855.84i	0.239649	+	0.239649i
$89$			5185.49i			0.654651i	0.944912	+	0.327326i	$0.106147\pi$
$89$			5185.49i			0.654651i	−0.944912	+	0.327326i	$0.893853\pi$
$90$		0			0
$91$	10740.0			1.29694
$92$	2328.20	−	2328.20i	0.275071	−	0.275071i
$93$		0			0
$94$			2288.24i			0.258968i
$95$	−3418.74	+	2921.17i	−0.378808	+	0.323675i
$96$		0			0
$97$	3495.32	−	3495.32i	0.371487	−	0.371487i	−0.496532	−	0.868018i	$-0.665393\pi$
$97$	3495.32	−	3495.32i	0.371487	−	0.371487i	0.868018	+	0.496532i	$0.165393\pi$
$98$	−664.940	−	664.940i	−0.0692357	−	0.0692357i
$99$		0			0
$100$	3610.61	+	570.256i	0.361061	+	0.0570256i
$101$	−7708.86			−0.755697			−0.377848	−	0.925867i	$-0.623336\pi$
$101$	−7708.86			−0.755697			−0.377848	+	0.925867i	$0.623336\pi$
$102$		0			0
$103$	−5098.51	−	5098.51i	−0.480583	−	0.480583i	0.424735	−	0.905318i	$-0.360367\pi$
$103$	−5098.51	−	5098.51i	−0.480583	−	0.480583i	−0.905318	+	0.424735i	$0.860367\pi$
$104$			10865.5i			1.00457i
$105$		0			0
$106$	−13386.2			−1.19137
$107$	2041.19	−	2041.19i	0.178285	−	0.178285i	−0.612323	−	0.790608i	$-0.709765\pi$
$107$	2041.19	−	2041.19i	0.178285	−	0.178285i	0.790608	+	0.612323i	$0.209765\pi$
$108$		0			0
$109$		−	11999.0i		−	1.00993i	−0.863139	−	0.504966i	$-0.831505\pi$
$109$		−	11999.0i		−	1.00993i	0.863139	−	0.504966i	$-0.168495\pi$
$110$	6443.71	+	505.722i	0.532538	+	0.0417952i
$111$		0			0
$112$	−10459.3	+	10459.3i	−0.833806	+	0.833806i
$113$	−8953.64	−	8953.64i	−0.701202	−	0.701202i	0.263467	−	0.964668i	$-0.415134\pi$
$113$	−8953.64	−	8953.64i	−0.701202	−	0.701202i	−0.964668	+	0.263467i	$0.915134\pi$
$114$		0			0
$115$	−1101.20	+	14031.1i	−0.0832669	+	1.06095i
$116$	−3201.93			−0.237956
$117$		0			0
$118$	−9434.64	−	9434.64i	−0.677581	−	0.677581i
$119$		−	18503.4i		−	1.30665i
$120$		0			0
$121$	−11581.6			−0.791039
$122$	3494.82	−	3494.82i	0.234804	−	0.234804i
$123$		0			0
$124$		−	8995.74i		−	0.585051i
$125$	−13317.2	+	8172.72i	−0.852299	+	0.523054i
$126$		0			0
$127$	6051.36	−	6051.36i	0.375185	−	0.375185i	−0.494177	−	0.869361i	$-0.664530\pi$
$127$	6051.36	−	6051.36i	0.375185	−	0.375185i	0.869361	+	0.494177i	$0.164530\pi$
$128$	−13721.2	−	13721.2i	−0.837476	−	0.837476i
$129$		0			0
$130$	17382.7	+	20343.5i	1.02856	+	1.20376i
$131$	22015.7			1.28289			0.641446	−	0.767168i	$-0.278335\pi$
$131$	22015.7			1.28289			0.641446	+	0.767168i	$0.278335\pi$
$132$		0			0
$133$	−5965.41	−	5965.41i	−0.337238	−	0.337238i
$134$			24832.1i			1.38294i
$135$		0			0
$136$	18719.6			1.01209
$137$	11424.0	−	11424.0i	0.608661	−	0.608661i	−0.333935	−	0.942596i	$-0.608377\pi$
$137$	11424.0	−	11424.0i	0.608661	−	0.608661i	0.942596	+	0.333935i	$0.108377\pi$
$138$		0			0
$139$		−	432.165i		−	0.0223676i	−0.999937	−	0.0111838i	$-0.996440\pi$
$139$		−	432.165i		−	0.0223676i	0.999937	−	0.0111838i	$-0.00356000\pi$
$140$	−536.571	+	6836.77i	−0.0273761	+	0.348815i
$141$		0			0
$142$	4723.90	−	4723.90i	0.234274	−	0.234274i
$143$	8955.98	+	8955.98i	0.437967	+	0.437967i
$144$		0			0
$145$	10405.6	−	8891.12i	0.494915	−	0.422883i
$146$	−5378.81			−0.252337
$147$		0			0
$148$	9713.00	+	9713.00i	0.443435	+	0.443435i
$149$			17370.9i			0.782438i	0.920298	+	0.391219i	$0.127947\pi$
$149$			17370.9i			0.782438i	−0.920298	+	0.391219i	$0.872053\pi$
$150$		0			0
$151$	20493.6			0.898805			0.449402	−	0.893329i	$-0.351637\pi$
$151$	20493.6			0.898805			0.449402	+	0.893329i	$0.351637\pi$
$152$	6035.10	−	6035.10i	0.261215	−	0.261215i
$153$		0			0
$154$			12126.2i			0.511307i
$155$	24979.3	+	29234.2i	1.03972	+	1.21682i
$156$		0			0
$157$	2340.31	−	2340.31i	0.0949456	−	0.0949456i	−0.658039	−	0.752984i	$-0.728614\pi$
$157$	2340.31	−	2340.31i	0.0949456	−	0.0949456i	0.752984	+	0.658039i	$0.228614\pi$
$158$	−16045.7	−	16045.7i	−0.642753	−	0.642753i
$159$		0			0
$160$	−17818.2	−	1398.43i	−0.696024	−	0.0546261i
$161$	−26404.6			−1.01866
$162$		0			0
$163$	15303.1	+	15303.1i	0.575977	+	0.575977i	0.933792	−	0.357815i	$-0.116478\pi$
$163$	15303.1	+	15303.1i	0.575977	+	0.575977i	−0.357815	+	0.933792i	$0.616478\pi$
$164$		−	1799.03i		−	0.0668883i
$165$		0			0
$166$	−8703.01			−0.315830
$167$	23457.8	−	23457.8i	0.841113	−	0.841113i	−0.147891	−	0.989004i	$-0.547248\pi$
$167$	23457.8	−	23457.8i	0.841113	−	0.841113i	0.989004	+	0.147891i	$0.0472485\pi$
$168$		0			0
$169$			23873.9i			0.835891i
$170$	35049.0	−	29947.8i	1.21277	−	1.03626i
$171$		0			0
$172$	609.739	−	609.739i	0.0206104	−	0.0206104i
$173$	20512.3	+	20512.3i	0.685366	+	0.685366i	0.961204	−	0.275838i	$-0.0889554\pi$
$173$	20512.3	+	20512.3i	0.685366	+	0.685366i	−0.275838	+	0.961204i	$0.588955\pi$
$174$		0			0
$175$	−17240.6	−	23708.0i	−0.562958	−	0.774137i
$176$	−17443.8			−0.563138
$177$		0			0
$178$	−17139.0	−	17139.0i	−0.540937	−	0.540937i
$179$		−	52460.9i		−	1.63731i	−0.574288	−	0.818653i	$-0.694721\pi$
$179$		−	52460.9i		−	1.63731i	0.574288	−	0.818653i	$-0.305279\pi$
$180$		0			0
$181$	−60242.8			−1.83886			−0.919428	−	0.393258i	$-0.871348\pi$
$181$	−60242.8			−1.83886			−0.919428	+	0.393258i	$0.871348\pi$
$182$	−35497.7	+	35497.7i	−1.07166	+	1.07166i
$183$		0			0
$184$		−	26713.1i		−	0.789020i
$185$	−58536.2	−	4594.10i	−1.71033	−	0.134232i
$186$		0			0
$187$	15429.8	−	15429.8i	0.441244	−	0.441244i
$188$	−2024.54	−	2024.54i	−0.0572810	−	0.0572810i
$189$		0			0
$190$	1644.58	−	20954.6i	0.0455563	−	0.580460i
$191$	−44243.4			−1.21278			−0.606390	−	0.795167i	$-0.707383\pi$
$191$	−44243.4			−1.21278			−0.606390	+	0.795167i	$0.707383\pi$
$192$		0			0
$193$	18449.5	+	18449.5i	0.495303	+	0.495303i	0.909972	−	0.414669i	$-0.136103\pi$
$193$	18449.5	+	18449.5i	0.495303	+	0.495303i	−0.414669	+	0.909972i	$0.636103\pi$
$194$			23105.4i			0.613917i
$195$		0			0
$196$	1176.62			0.0306284
$197$	−18454.6	+	18454.6i	−0.475524	+	0.475524i	−0.903697	−	0.428173i	$-0.859157\pi$
$197$	−18454.6	+	18454.6i	−0.475524	+	0.475524i	0.428173	+	0.903697i	$0.359157\pi$
$198$		0			0
$199$			69350.3i			1.75123i	0.483013	+	0.875613i	$0.339542\pi$
$199$			69350.3i			1.75123i	−0.483013	+	0.875613i	$0.660458\pi$
$200$	23984.9	−	17442.0i	0.599624	−	0.436050i
$201$		0			0
$202$	25479.3	−	25479.3i	0.624431	−	0.624431i
$203$	18156.8	+	18156.8i	0.440604	+	0.440604i
$204$		0			0
$205$	4995.53	+	5846.45i	0.118871	+	0.139118i
$206$	33703.1			0.794210
$207$		0			0
$208$	−51064.3	−	51064.3i	−1.18030	−	1.18030i
$209$		−	9949.00i		−	0.227765i
$210$		0			0
$211$	−19458.0			−0.437052			−0.218526	−	0.975831i	$-0.570125\pi$
$211$	−19458.0			−0.437052			−0.218526	+	0.975831i	$0.570125\pi$
$212$	11843.5	−	11843.5i	0.263517	−	0.263517i
$213$		0			0
$214$			13493.0i			0.294633i
$215$	−288.397	+	3674.64i	−0.00623899	+	0.0794947i
$216$		0			0
$217$	−51011.1	+	51011.1i	−1.08329	+	1.08329i
$218$	39659.0	+	39659.0i	0.834505	+	0.834505i
$219$		0			0
$220$	−6148.56	+	5253.68i	−0.127036	+	0.108547i
$221$	90337.7			1.84963
$222$		0			0
$223$	52732.0	+	52732.0i	1.06039	+	1.06039i	0.998056	+	0.0623311i	$0.0198535\pi$
$223$	52732.0	+	52732.0i	1.06039	+	1.06039i	0.0623311	+	0.998056i	$0.480147\pi$
$224$		−	33531.4i		−	0.668275i
$225$		0			0
$226$	59187.0			1.15880
$227$	−61436.6	+	61436.6i	−1.19227	+	1.19227i	−0.215844	+	0.976428i	$0.569250\pi$
$227$	−61436.6	+	61436.6i	−1.19227	+	1.19227i	−0.976428	+	0.215844i	$0.930750\pi$
$228$		0			0
$229$		−	98741.9i		−	1.88292i	−0.337132	−	0.941458i	$-0.609457\pi$
$229$		−	98741.9i		−	1.88292i	0.337132	−	0.941458i	$-0.390543\pi$
$230$	−42735.8	−	50015.2i	−0.807860	−	0.945467i
$231$		0			0
$232$	−18369.0	+	18369.0i	−0.341278	+	0.341278i
$233$	−26861.4	−	26861.4i	−0.494785	−	0.494785i	0.415025	−	0.909810i	$-0.363773\pi$
$233$	−26861.4	−	26861.4i	−0.494785	−	0.494785i	−0.909810	+	0.415025i	$0.863773\pi$
$234$		0			0
$235$	12201.1	+	957.576i	0.220934	+	0.0173395i
$236$	16694.7			0.299747
$237$		0			0
$238$	61157.4	+	61157.4i	1.07968	+	1.07968i
$239$			64860.8i			1.13550i	0.823202	+	0.567749i	$0.192186\pi$
$239$			64860.8i			1.13550i	−0.823202	+	0.567749i	$0.807814\pi$
$240$		0			0
$241$	11766.6			0.202590			0.101295	−	0.994856i	$-0.467701\pi$
$241$	11766.6			0.202590			0.101295	+	0.994856i	$0.467701\pi$
$242$	38279.4	−	38279.4i	0.653634	−	0.653634i
$243$		0			0
$244$			6184.13i			0.103872i
$245$	−3823.76	+	3267.24i	−0.0637028	+	0.0544313i
$246$		0			0
$247$	29124.4	−	29124.4i	0.477378	−	0.477378i
$248$	−51607.1	−	51607.1i	−0.839085	−	0.839085i
$249$		0			0
$250$	17003.4	−	71028.2i	0.272055	−	1.13645i
$251$	−14659.6			−0.232688			−0.116344	−	0.993209i	$-0.537118\pi$
$251$	−14659.6			−0.232688			−0.116344	+	0.993209i	$0.537118\pi$
$252$		0			0
$253$	−22018.5	−	22018.5i	−0.343991	−	0.343991i
$254$			40001.8i			0.620029i
$255$		0			0
$256$	63434.7			0.967937
$257$	47816.3	−	47816.3i	0.723951	−	0.723951i	−0.245456	−	0.969408i	$-0.578938\pi$
$257$	47816.3	−	47816.3i	0.723951	−	0.723951i	0.969408	+	0.245456i	$0.0789378\pi$
$258$		0			0
$259$		−	110157.i		−	1.64215i
$260$	−33378.6	−	2619.65i	−0.493766	−	0.0387522i
$261$		0			0
$262$	−72766.2	+	72766.2i	−1.06005	+	1.06005i
$263$	45872.6	+	45872.6i	0.663197	+	0.663197i	0.956132	−	0.292936i	$-0.0946322\pi$
$263$	45872.6	+	45872.6i	0.663197	+	0.663197i	−0.292936	+	0.956132i	$0.594632\pi$
$264$		0			0
$265$	−5601.80	+	71375.9i	−0.0797694	+	1.01639i
$266$	39433.6			0.557319
$267$		0			0
$268$	−21970.4	−	21970.4i	−0.305892	−	0.305892i
$269$		−	80295.1i		−	1.10965i	−0.831968	−	0.554823i	$-0.812786\pi$
$269$		−	80295.1i		−	1.10965i	0.831968	−	0.554823i	$-0.187214\pi$
$270$		0			0
$271$	40047.7			0.545304			0.272652	−	0.962113i	$-0.412099\pi$
$271$	40047.7			0.545304			0.272652	+	0.962113i	$0.412099\pi$
$272$	−87976.4	+	87976.4i	−1.18913	+	1.18913i
$273$		0			0
$274$			75516.8i			1.00587i
$275$	5393.08	−	34146.6i	0.0713135	−	0.451525i
$276$		0			0
$277$	−49870.0	+	49870.0i	−0.649950	+	0.649950i	−0.952981	−	0.303031i	$-0.902001\pi$
$277$	−49870.0	+	49870.0i	−0.649950	+	0.649950i	0.303031	+	0.952981i	$0.402001\pi$
$278$	1428.39	+	1428.39i	0.0184823	+	0.0184823i
$279$		0			0
$280$	36143.2	+	42299.6i	0.461010	+	0.539536i
$281$	12842.2			0.162639			0.0813196	−	0.996688i	$-0.474087\pi$
$281$	12842.2			0.162639			0.0813196	+	0.996688i	$0.474087\pi$
$282$		0			0
$283$	−35925.4	−	35925.4i	−0.448568	−	0.448568i	0.446310	−	0.894878i	$-0.352738\pi$
$283$	−35925.4	−	35925.4i	−0.448568	−	0.448568i	−0.894878	+	0.446310i	$0.852738\pi$
$284$			8359.02i			0.103638i
$285$		0			0
$286$	−59202.4			−0.723782
$287$	−10201.5	+	10201.5i	−0.123852	+	0.123852i
$288$		0			0
$289$		−	72117.7i		−	0.863467i
$290$	−5005.60	+	63779.3i	−0.0595196	+	0.758375i
$291$		0			0
$292$	4758.94	−	4758.94i	0.0558141	−	0.0558141i
$293$	50366.0	+	50366.0i	0.586681	+	0.586681i	0.936731	−	0.350050i	$-0.113835\pi$
$293$	50366.0	+	50366.0i	0.586681	+	0.586681i	−0.350050	+	0.936731i	$0.613835\pi$
$294$		0			0
$295$	−54254.2	+	46357.9i	−0.623432	+	0.532696i
$296$	111444.			1.27196
$297$		0			0
$298$	−57414.2	−	57414.2i	−0.646527	−	0.646527i
$299$		−	128913.i		−	1.44196i
$300$		0			0
$301$	−6915.16			−0.0763254
$302$	−67735.4	+	67735.4i	−0.742680	+	0.742680i
$303$		0			0
$304$			56726.2i			0.613814i
$305$	−17172.1	−	20097.1i	−0.184596	−	0.216039i
$306$		0			0
$307$	75581.6	−	75581.6i	0.801936	−	0.801936i	−0.181462	−	0.983398i	$-0.558083\pi$
$307$	75581.6	−	75581.6i	0.801936	−	0.801936i	0.983398	+	0.181462i	$0.0580830\pi$
$308$	−10728.7	−	10728.7i	−0.113096	−	0.113096i
$309$		0			0
$310$	−179186.	−	14063.1i	−1.86458	−	0.146338i
$311$	−107241.			−1.10877			−0.554384	−	0.832261i	$-0.687046\pi$
$311$	−107241.			−1.10877			−0.554384	+	0.832261i	$0.687046\pi$
$312$		0			0
$313$	24848.0	+	24848.0i	0.253631	+	0.253631i	0.822458	−	0.568826i	$-0.192602\pi$
$313$	24848.0	+	24848.0i	0.253631	+	0.253631i	−0.568826	+	0.822458i	$0.692602\pi$
$314$			15470.4i			0.156907i
$315$		0			0
$316$	28393.1			0.284340
$317$	7619.16	−	7619.16i	0.0758209	−	0.0758209i	−0.668179	−	0.744000i	$-0.732926\pi$
$317$	7619.16	−	7619.16i	0.0758209	−	0.0758209i	0.744000	+	0.668179i	$0.232926\pi$
$318$		0			0
$319$			30281.6i			0.297576i
$320$	−32391.6	+	27677.2i	−0.316324	+	0.270286i
$321$		0			0
$322$	87272.1	−	87272.1i	0.841713	−	0.841713i
$323$	−50177.1	−	50177.1i	−0.480950	−	0.480950i
$324$		0			0
$325$	115747.	−	84172.2i	1.09583	−	0.796896i
$326$	−101160.			−0.951857
$327$		0			0
$328$	−10320.7	−	10320.7i	−0.0959318	−	0.0959318i
$329$			22960.7i			0.212125i
$330$		0			0
$331$	28518.2			0.260295			0.130148	−	0.991495i	$-0.458455\pi$
$331$	28518.2			0.260295			0.130148	+	0.991495i	$0.458455\pi$
$332$	7700.05	−	7700.05i	0.0698582	−	0.0698582i
$333$		0			0
$334$			155065.i			1.39002i
$335$	132406.	+	10391.6i	1.17983	+	0.0925965i
$336$		0			0
$337$	−22519.8	+	22519.8i	−0.198292	+	0.198292i	−0.799267	−	0.600976i	$-0.794779\pi$
$337$	−22519.8	+	22519.8i	−0.198292	+	0.198292i	0.600976	+	0.799267i	$0.294779\pi$
$338$	−78907.8	−	78907.8i	−0.690695	−	0.690695i
$339$		0			0
$340$	−4513.28	+	57506.4i	−0.0390422	+	0.497460i
$341$	−85075.4			−0.731636
$342$		0			0
$343$	−86301.0	−	86301.0i	−0.733547	−	0.733547i
$344$		−	6995.95i		−	0.0591194i
$345$		0			0
$346$	−135594.			−1.13263
$347$	123818.	−	123818.i	1.02831	−	1.02831i	0.0287203	−	0.999587i	$-0.490857\pi$
$347$	123818.	−	123818.i	1.02831	−	1.02831i	0.999587	−	0.0287203i	$-0.00914323\pi$
$348$		0			0
$349$		−	53357.0i		−	0.438067i	−0.975717	−	0.219034i	$-0.929710\pi$
$349$		−	53357.0i		−	0.438067i	0.975717	−	0.219034i	$-0.0702905\pi$
$350$	135343.	+	21375.9i	1.10484	+	0.174497i
$351$		0			0
$352$	27961.5	−	27961.5i	0.225671	−	0.225671i
$353$	48279.2	+	48279.2i	0.387445	+	0.387445i	0.873775	−	0.486330i	$-0.161665\pi$
$353$	48279.2	+	48279.2i	0.387445	+	0.387445i	−0.486330	+	0.873775i	$0.661665\pi$
$354$		0			0
$355$	−23211.3	−	27165.0i	−0.184180	−	0.215552i
$356$	30327.8			0.239299
$357$		0			0
$358$	173393.	+	173393.i	1.35290	+	1.35290i
$359$			91012.9i			0.706178i	0.935590	+	0.353089i	$0.114869\pi$
$359$			91012.9i			0.706178i	−0.935590	+	0.353089i	$0.885131\pi$
$360$		0			0
$361$	97967.4			0.751739
$362$	199114.	−	199114.i	1.51944	−	1.51944i
$363$		0			0
$364$		−	62813.7i		−	0.474080i
$365$	−2250.90	+	28680.1i	−0.0168955	+	0.215276i
$366$		0			0
$367$	−94731.0	+	94731.0i	−0.703331	+	0.703331i	−0.965124	−	0.261793i	$-0.915686\pi$
$367$	−94731.0	+	94731.0i	−0.703331	+	0.703331i	0.261793	+	0.965124i	$0.415686\pi$
$368$	125543.	+	125543.i	0.927037	+	0.927037i
$369$		0			0
$370$	208658.	−	178289.i	1.52416	−	1.30233i
$371$	−134319.			−0.975868
$372$		0			0
$373$	174691.	+	174691.i	1.25560	+	1.25560i	0.953172	+	0.302430i	$0.0977978\pi$
$373$	174691.	+	174691.i	1.25560	+	1.25560i	0.302430	+	0.953172i	$0.402202\pi$
$374$			101997.i			0.729197i
$375$		0			0
$376$	−23228.9			−0.164306
$377$	−88645.5	+	88645.5i	−0.623697	+	0.623697i
$378$		0			0
$379$			216082.i			1.50432i	0.658981	+	0.752159i	$0.270988\pi$
$379$			216082.i			1.50432i	−0.658981	+	0.752159i	$0.729012\pi$
$380$	17084.7	+	19994.8i	0.118315	+	0.138468i
$381$		0			0
$382$	146233.	−	146233.i	1.00212	−	1.00212i
$383$	6943.25	+	6943.25i	0.0473331	+	0.0473331i	0.730377	−	0.683044i	$-0.239344\pi$
$383$	6943.25	+	6943.25i	0.0473331	+	0.0473331i	−0.683044	+	0.730377i	$0.739344\pi$
$384$		0			0
$385$	64657.4	+	5074.51i	0.436211	+	0.0342352i
$386$	−121959.			−0.818536
$387$		0			0
$388$	−20442.7	−	20442.7i	−0.135792	−	0.135792i
$389$			171993.i			1.13661i	0.822817	+	0.568306i	$0.192401\pi$
$389$			171993.i			1.13661i	−0.822817	+	0.568306i	$0.807599\pi$
$390$		0			0
$391$	−222098.			−1.45275
$392$	6750.08	−	6750.08i	0.0439275	−	0.0439275i
$393$		0			0
$394$		−	121992.i		−	0.785849i
$395$	−92271.3	+	78841.8i	−0.591388	+	0.505315i
$396$		0			0
$397$	47564.4	−	47564.4i	0.301787	−	0.301787i	−0.539925	−	0.841713i	$-0.681548\pi$
$397$	47564.4	−	47564.4i	0.301787	−	0.301787i	0.841713	+	0.539925i	$0.181548\pi$
$398$	−229216.	−	229216.i	−1.44703	−	1.44703i
$399$		0			0
$400$	−30749.7	+	194694.i	−0.192186	+	1.21684i
$401$	193768.			1.20502			0.602508	−	0.798113i	$-0.294168\pi$
$401$	193768.			1.20502			0.602508	+	0.798113i	$0.294168\pi$
$402$		0			0
$403$	−249047.	−	249047.i	−1.53346	−	1.53346i
$404$			45085.9i			0.276235i
$405$		0			0
$406$	−120024.			−0.728140
$407$	91858.8	−	91858.8i	0.554539	−	0.554539i
$408$		0			0
$409$			265.258i			0.00158570i	1.00000		0.000792852i	$0.000252373\pi$
$409$			265.258i			0.00158570i	−1.00000		0.000792852i	$0.999748\pi$
$410$	−35834.8	−	2812.43i	−0.213176	−	0.0167307i
$411$		0			0
$412$	−29819.0	+	29819.0i	−0.175671	+	0.175671i
$413$	−94668.8	−	94668.8i	−0.555018	−	0.555018i
$414$		0			0
$415$	−3642.00	+	46405.0i	−0.0211468	+	0.269444i
$416$	163707.			0.945978
$417$		0			0
$418$	32883.3	+	32883.3i	0.188202	+	0.188202i
$419$			30926.0i			0.176155i	0.996114	+	0.0880775i	$0.0280723\pi$
$419$			30926.0i			0.176155i	−0.996114	+	0.0880775i	$0.971928\pi$
$420$		0			0
$421$	−62470.0			−0.352458			−0.176229	−	0.984349i	$-0.556390\pi$
$421$	−62470.0			−0.352458			−0.176229	+	0.984349i	$0.556390\pi$
$422$	64312.3	−	64312.3i	0.361135	−	0.361135i
$423$		0			0
$424$		−	135889.i		−	0.755878i
$425$	−145016.	−	199416.i	−0.802859	−	1.10403i
$426$		0			0
$427$	35067.6	−	35067.6i	0.192332	−	0.192332i
$428$	−11938.1	−	11938.1i	−0.0651698	−	0.0651698i
$429$		0			0
$430$	−11192.2	−	13098.6i	−0.0605310	−	0.0708416i
$431$	−69270.8			−0.372903			−0.186451	−	0.982464i	$-0.559699\pi$
$431$	−69270.8			−0.372903			−0.186451	+	0.982464i	$0.559699\pi$
$432$		0			0
$433$	−110961.	−	110961.i	−0.591829	−	0.591829i	0.346296	−	0.938125i	$-0.387439\pi$
$433$	−110961.	−	110961.i	−0.591829	−	0.591829i	−0.938125	+	0.346296i	$0.887439\pi$
$434$		−	337203.i		−	1.79024i
$435$		0			0
$436$	−70177.2			−0.369167
$437$	−71603.1	+	71603.1i	−0.374946	+	0.374946i
$438$		0			0
$439$			167998.i			0.871716i	0.900016	+	0.435858i	$0.143555\pi$
$439$			167998.i			0.871716i	−0.900016	+	0.435858i	$0.856445\pi$
$440$	−5133.80	+	65412.8i	−0.0265176	+	0.337876i
$441$		0			0
$442$	−298583.	+	298583.i	−1.52834	+	1.52834i
$443$	−66278.3	−	66278.3i	−0.337726	−	0.337726i	0.517785	−	0.855511i	$-0.326757\pi$
$443$	−66278.3	−	66278.3i	−0.337726	−	0.337726i	−0.855511	+	0.517785i	$0.826757\pi$
$444$		0			0
$445$	−98558.7	+	84214.1i	−0.497708	+	0.425270i
$446$	−348579.			−1.75239
$447$		0			0
$448$	−56520.6	−	56520.6i	−0.281611	−	0.281611i
$449$			212689.i			1.05500i	0.849556	+	0.527499i	$0.176870\pi$
$449$			212689.i			1.05500i	−0.849556	+	0.527499i	$0.823130\pi$
$450$		0			0
$451$	−17013.9			−0.0836473
$452$	−52366.1	+	52366.1i	−0.256315	+	0.256315i
$453$		0			0
$454$		−	406119.i		−	1.97034i
$455$	174421.	+	204131.i	0.842511	+	0.986020i
$456$		0			0
$457$	188980.	−	188980.i	0.904864	−	0.904864i	−0.0909883	−	0.995852i	$-0.529003\pi$
$457$	188980.	−	188980.i	0.904864	−	0.904864i	0.995852	+	0.0909883i	$0.0290026\pi$
$458$	326361.	+	326361.i	1.55585	+	1.55585i
$459$		0			0
$460$	82062.1	+	6440.49i	0.387817	+	0.0304371i
$461$	203463.			0.957379			0.478690	−	0.877984i	$-0.341112\pi$
$461$	203463.			0.957379			0.478690	+	0.877984i	$0.341112\pi$
$462$		0			0
$463$	280217.	+	280217.i	1.30717	+	1.30717i	0.923448	+	0.383724i	$0.125359\pi$
$463$	280217.	+	280217.i	1.30717	+	1.30717i	0.383724	+	0.923448i	$0.374641\pi$
$464$		−	172657.i		−	0.801951i
$465$		0			0
$466$	177564.			0.817679
$467$	−25143.8	+	25143.8i	−0.115291	+	0.115291i	−0.762399	−	0.647107i	$-0.775979\pi$
$467$	−25143.8	+	25143.8i	−0.115291	+	0.115291i	0.647107	+	0.762399i	$0.275979\pi$
$468$		0			0
$469$			249170.i			1.13279i
$470$	−43491.8	+	37161.8i	−0.196885	+	0.168229i
$471$		0			0
$472$	95774.9	−	95774.9i	0.429900	−	0.429900i
$473$	−5766.49	−	5766.49i	−0.0257744	−	0.0257744i
$474$		0			0
$475$	−111043.	−	17538.0i	−0.492158	−	0.0777309i
$476$	−108219.			−0.477627
$477$		0			0
$478$	−214377.	−	214377.i	−0.938260	−	0.938260i
$479$		−	17282.5i		−	0.0753246i	−0.999291	−	0.0376623i	$-0.988009\pi$
$479$		−	17282.5i		−	0.0753246i	0.999291	−	0.0376623i	$-0.0119911\pi$
$480$		0			0
$481$	537809.			2.32454
$482$	−38891.0	+	38891.0i	−0.167400	+	0.167400i
$483$		0			0
$484$			67736.0i			0.289154i
$485$	123199.	+	9669.06i	0.523751	+	0.0411056i
$486$		0			0
$487$	−250914.	+	250914.i	−1.05795	+	1.05795i	−0.0597410	+	0.998214i	$0.519027\pi$
$487$	−250914.	+	250914.i	−1.05795	+	1.05795i	−0.998214	+	0.0597410i	$0.980973\pi$
$488$	35477.4	+	35477.4i	0.148974	+	0.148974i
$489$		0			0
$490$	1839.42	−	23437.1i	0.00766104	−	0.0976139i
$491$	175178.			0.726634			0.363317	−	0.931666i	$-0.381644\pi$
$491$	175178.			0.726634			0.363317	+	0.931666i	$0.381644\pi$
$492$		0			0
$493$	152723.	+	152723.i	0.628364	+	0.628364i
$494$			192523.i			0.788914i
$495$		0			0
$496$	485075.			1.97172
$497$	47400.5	−	47400.5i	0.191898	−	0.191898i
$498$		0			0
$499$		−	219732.i		−	0.882454i	−0.897396	−	0.441227i	$-0.854543\pi$
$499$		−	219732.i		−	0.882454i	0.897396	−	0.441227i	$-0.145457\pi$
$500$	47798.9	+	77886.6i	0.191195	+	0.311547i
$501$		0			0
$502$	48452.8	−	48452.8i	0.192270	−	0.192270i
$503$	−298149.	−	298149.i	−1.17841	−	1.17841i	−0.980148	−	0.198266i	$-0.936469\pi$
$503$	−298149.	−	298149.i	−1.17841	−	1.17841i	−0.198266	−	0.980148i	$-0.563531\pi$
$504$		0			0
$505$	−125194.	−	146519.i	−0.490911	−	0.574530i
$506$	145551.			0.568478
$507$		0			0
$508$	−35391.9	−	35391.9i	−0.137144	−	0.137144i
$509$		−	476403.i		−	1.83882i	−0.393303	−	0.919409i	$-0.628668\pi$
$509$		−	476403.i		−	1.83882i	0.393303	−	0.919409i	$-0.371332\pi$
$510$		0			0
$511$	−53971.9			−0.206693
$512$	9875.23	−	9875.23i	0.0376710	−	0.0376710i
$513$		0			0
$514$			316084.i			1.19640i
$515$	14103.9	−	179707.i	0.0531773	−	0.677564i
$516$		0			0
$517$	−19146.7	+	19146.7i	−0.0716329	+	0.0716329i
$518$	364089.	+	364089.i	1.35690	+	1.35690i
$519$		0			0
$520$	−206516.	+	176459.i	−0.763742	+	0.652584i
$521$	366162.			1.34896			0.674479	−	0.738294i	$-0.264368\pi$
$521$	366162.			1.34896			0.674479	+	0.738294i	$0.264368\pi$
$522$		0			0
$523$	−236853.	−	236853.i	−0.865916	−	0.865916i	0.126101	−	0.992017i	$-0.459754\pi$
$523$	−236853.	−	236853.i	−0.865916	−	0.865916i	−0.992017	+	0.126101i	$0.959754\pi$
$524$		−	128761.i		−	0.468944i
$525$		0			0
$526$	−303236.			−1.09600
$527$	−429072.	+	429072.i	−1.54493	+	1.54493i
$528$		0			0
$529$			37094.3i			0.132555i
$530$	−217396.	−	254426.i	−0.773927	−	0.905754i
$531$		0			0
$532$	−34889.2	+	34889.2i	−0.123273	+	0.123273i
$533$	−49806.1	−	49806.1i	−0.175319	−	0.175319i
$534$		0			0
$535$	71945.6	+	5646.52i	0.251360	+	0.0197275i
$536$	−252081.			−0.877426
$537$		0			0
$538$	265391.	+	265391.i	0.916898	+	0.916898i
$539$		−	11127.7i		−	0.0383024i
$540$		0			0
$541$	70441.7			0.240678			0.120339	−	0.992733i	$-0.461602\pi$
$541$	70441.7			0.240678			0.120339	+	0.992733i	$0.461602\pi$
$542$	−132365.	+	132365.i	−0.450584	+	0.450584i
$543$		0			0
$544$		−	282044.i		−	0.953056i
$545$	228061.	−	194868.i	0.767816	−	0.656065i
$546$		0			0
$547$	347520.	−	347520.i	1.16146	−	1.16146i	0.177308	−	0.984155i	$-0.443261\pi$
$547$	347520.	−	347520.i	1.16146	−	1.16146i	0.984155	−	0.177308i	$-0.0567387\pi$
$548$	−66814.0	−	66814.0i	−0.222488	−	0.222488i
$549$		0			0
$550$	95035.9	+	130686.i	0.314168	+	0.432021i
$551$	98474.3			0.324354
$552$		0			0
$553$	−161005.	−	161005.i	−0.526490	−	0.526490i
$554$		−	329660.i		−	1.07410i
$555$		0			0
$556$	−2527.56			−0.00817619
$557$	−254820.	+	254820.i	−0.821340	+	0.821340i	−0.986300	−	0.164960i	$-0.947251\pi$
$557$	−254820.	+	254820.i	−0.821340	+	0.821340i	0.164960	+	0.986300i	$0.447251\pi$
$558$		0			0
$559$		−	33761.3i		−	0.108043i
$560$	−368657.	−	28933.3i	−1.17556	−	0.0922619i
$561$		0			0
$562$	−42445.8	+	42445.8i	−0.134389	+	0.134389i
$563$	−31884.1	−	31884.1i	−0.100591	−	0.100591i	0.655021	−	0.755611i	$-0.272660\pi$
$563$	−31884.1	−	31884.1i	−0.100591	−	0.100591i	−0.755611	+	0.655021i	$0.772660\pi$
$564$		0			0
$565$	24768.4	−	315589.i	0.0775891	−	0.988609i
$566$	237480.			0.741302
$567$		0			0
$568$	47954.3	+	47954.3i	0.148638	+	0.148638i
$569$			609312.i			1.88198i	0.338431	+	0.940991i	$0.390104\pi$
$569$			609312.i			1.88198i	−0.338431	+	0.940991i	$0.609896\pi$
$570$		0			0
$571$	−233321.			−0.715618			−0.357809	−	0.933795i	$-0.616476\pi$
$571$	−233321.			−0.715618			−0.357809	+	0.933795i	$0.616476\pi$
$572$	52379.8	−	52379.8i	0.160093	−	0.160093i
$573$		0			0
$574$		−	67436.1i		−	0.204677i
$575$	−284568.	+	206940.i	−0.860697	+	0.625904i
$576$		0			0
$577$	−324793.	+	324793.i	−0.975562	+	0.975562i	−0.999708	−	0.0241466i	$-0.992313\pi$
$577$	−324793.	+	324793.i	−0.975562	+	0.975562i	0.0241466	+	0.999708i	$0.492313\pi$
$578$	238363.	+	238363.i	0.713481	+	0.713481i
$579$		0			0
$580$	−52000.5	−	60857.9i	−0.154579	−	0.180909i
$581$	−87327.6			−0.258702
$582$		0			0
$583$	−112008.	−	112008.i	−0.329542	−	0.329542i
$584$		−	54602.5i		−	0.160098i
$585$		0			0
$586$	−332939.			−0.969547
$587$	369440.	−	369440.i	1.07218	−	1.07218i	0.0749961	−	0.997184i	$-0.476106\pi$
$587$	369440.	−	369440.i	1.07218	−	1.07218i	0.997184	−	0.0749961i	$-0.0238944\pi$
$588$		0			0
$589$			276661.i			0.797475i
$590$	26098.9	−	332542.i	0.0749754	−	0.955306i
$591$		0			0
$592$	−523751.	+	523751.i	−1.49445	+	1.49445i
$593$	263575.	+	263575.i	0.749541	+	0.749541i	0.974393	−	0.224852i	$-0.0721898\pi$
$593$	263575.	+	263575.i	0.749541	+	0.749541i	−0.224852	+	0.974393i	$0.572190\pi$
$594$		0			0
$595$	351688.	−	300502.i	0.993398	−	0.848815i
$596$	101595.			0.286010
$597$		0			0
$598$	426081.	+	426081.i	1.19149	+	1.19149i
$599$			319604.i			0.890756i	0.895343	+	0.445378i	$0.146931\pi$
$599$			319604.i			0.890756i	−0.895343	+	0.445378i	$0.853069\pi$
$600$		0			0
$601$	−558724.			−1.54685			−0.773425	−	0.633887i	$-0.781458\pi$
$601$	−558724.			−1.54685			−0.773425	+	0.633887i	$0.781458\pi$
$602$	22855.9	−	22855.9i	0.0630675	−	0.0630675i
$603$		0			0
$604$		−	119859.i		−	0.328546i
$605$	−188089.	−	220127.i	−0.513870	−	0.601399i
$606$		0			0
$607$	31979.5	−	31979.5i	0.0867950	−	0.0867950i	−0.662376	−	0.749171i	$-0.730452\pi$
$607$	31979.5	−	31979.5i	0.0867950	−	0.0867950i	0.749171	+	0.662376i	$0.230452\pi$
$608$	−90929.4	−	90929.4i	−0.245978	−	0.245978i
$609$		0			0
$610$	123182.	+	9667.68i	0.331045	+	0.0259814i
$611$	−112099.			−0.300275
$612$		0			0
$613$	181360.	+	181360.i	0.482636	+	0.482636i	0.905973	−	0.423336i	$-0.139141\pi$
$613$	181360.	+	181360.i	0.482636	+	0.482636i	−0.423336	+	0.905973i	$0.639141\pi$
$614$			499624.i			1.32528i
$615$		0			0
$616$	−123098.			−0.324405
$617$	−58514.8	+	58514.8i	−0.153708	+	0.153708i	−0.779772	−	0.626064i	$-0.784665\pi$
$617$	−58514.8	+	58514.8i	−0.153708	+	0.153708i	0.626064	+	0.779772i	$0.284665\pi$
$618$		0			0
$619$		−	193773.i		−	0.505722i	−0.967503	−	0.252861i	$-0.918628\pi$
$619$		−	193773.i		−	0.505722i	0.967503	−	0.252861i	$-0.0813716\pi$
$620$	170979.	−	146094.i	0.444794	−	0.380057i
$621$		0			0
$622$	354452.	−	354452.i	0.916172	−	0.916172i
$623$	−171976.	−	171976.i	−0.443090	−	0.443090i
$624$		0			0
$625$	−371611.	−	120387.i	−0.951325	−	0.308190i
$626$	−164255.			−0.419150
$627$		0			0
$628$	−13687.5	−	13687.5i	−0.0347061	−	0.0347061i
$629$		−	926566.i		−	2.34194i
$630$		0			0
$631$	276166.			0.693604			0.346802	−	0.937938i	$-0.387268\pi$
$631$	276166.			0.693604			0.346802	+	0.937938i	$0.387268\pi$
$632$	162886.	−	162886.i	0.407803	−	0.407803i
$633$		0			0
$634$			50365.6i			0.125301i
$635$	213292.	+	16739.8i	0.528965	+	0.0415148i
$636$		0			0
$637$	32574.7	−	32574.7i	0.0802790	−	0.0802790i
$638$	−100087.	−	100087.i	−0.245886	−	0.245886i
$639$		0			0
$640$	37956.8	−	483631.i	0.0926680	−	1.18074i
$641$	−556595.			−1.35464			−0.677319	−	0.735689i	$-0.736858\pi$
$641$	−556595.			−1.35464			−0.677319	+	0.735689i	$0.736858\pi$
$642$		0			0
$643$	288950.	+	288950.i	0.698878	+	0.698878i	0.964169	−	0.265291i	$-0.0854679\pi$
$643$	288950.	+	288950.i	0.698878	+	0.698878i	−0.265291	+	0.964169i	$0.585468\pi$
$644$			154429.i			0.372356i
$645$		0			0
$646$	331690.			0.794817
$647$	435693.	−	435693.i	1.04081	−	1.04081i	0.0416796	−	0.999131i	$-0.486729\pi$
$647$	435693.	−	435693.i	1.04081	−	1.04081i	0.999131	−	0.0416796i	$-0.0132709\pi$
$648$		0			0
$649$		−	157887.i		−	0.374849i
$650$	−104362.	+	660772.i	−0.247010	+	1.56396i
$651$		0			0
$652$	89501.7	−	89501.7i	0.210541	−	0.210541i
$653$	96648.2	+	96648.2i	0.226656	+	0.226656i	0.811294	−	0.584638i	$-0.198763\pi$
$653$	96648.2	+	96648.2i	0.226656	+	0.226656i	−0.584638	+	0.811294i	$0.698763\pi$
$654$		0			0
$655$	357543.	+	418444.i	0.833384	+	0.975338i
$656$	97008.4			0.225425
$657$		0			0
$658$	−75889.4	−	75889.4i	−0.175279	−	0.175279i
$659$			693374.i			1.59660i	0.602259	+	0.798301i	$0.294267\pi$
$659$			693374.i			1.59660i	−0.602259	+	0.798301i	$0.705733\pi$
$660$		0			0
$661$	102052.			0.233572			0.116786	−	0.993157i	$-0.462741\pi$
$661$	102052.			0.233572			0.116786	+	0.993157i	$0.462741\pi$
$662$	−94258.1	+	94258.1i	−0.215081	+	0.215081i
$663$		0			0
$664$		−	88347.9i		−	0.200383i
$665$	16502.0	−	210262.i	0.0373159	−	0.475465i
$666$		0			0
$667$	217937.	−	217937.i	0.489869	−	0.489869i
$668$	−137195.	−	137195.i	−0.307457	−	0.307457i
$669$		0			0
$670$	−471974.	+	403281.i	−1.05140	+	0.898377i
$671$	58485.2			0.129897
$672$		0			0
$673$	−222583.	−	222583.i	−0.491431	−	0.491431i	0.417326	−	0.908757i	$-0.362967\pi$
$673$	−222583.	−	222583.i	−0.491431	−	0.491431i	−0.908757	+	0.417326i	$0.862967\pi$
$674$		−	148865.i		−	0.327696i
$675$		0			0
$676$	139628.			0.305549
$677$	−423786.	+	423786.i	−0.924633	+	0.924633i	−0.997352	−	0.0727193i	$-0.976832\pi$
$677$	−423786.	+	423786.i	−0.924633	+	0.924633i	0.0727193	+	0.997352i	$0.476832\pi$
$678$		0			0
$679$			231844.i			0.502870i
$680$	304013.	+	355797.i	0.657467	+	0.769457i
$681$		0			0
$682$	281191.	−	281191.i	0.604550	−	0.604550i
$683$	−215197.	−	215197.i	−0.461313	−	0.461313i	0.437773	−	0.899086i	$-0.355767\pi$
$683$	−215197.	−	215197.i	−0.461313	−	0.461313i	−0.899086	+	0.437773i	$0.855767\pi$
$684$		0			0
$685$	402660.	+	31602.0i	0.858138	+	0.0673493i
$686$	570483.			1.21226
$687$		0			0
$688$	32878.8	+	32878.8i	0.0694607	+	0.0694607i
$689$		−	655776.i		−	1.38139i
$690$		0			0
$691$	322148.			0.674683			0.337341	−	0.941382i	$-0.390472\pi$
$691$	322148.			0.674683			0.337341	+	0.941382i	$0.390472\pi$
$692$	119968.	−	119968.i	0.250526	−	0.250526i
$693$		0			0
$694$			818481.i			1.69938i
$695$	8214.00	−	7018.51i	0.0170053	−	0.0145303i
$696$		0			0
$697$	−85808.6	+	85808.6i	−0.176630	+	0.176630i
$698$	176355.	+	176355.i	0.361974	+	0.361974i
$699$		0			0
$700$	−138658.	+	100833.i	−0.282975	+	0.205782i
$701$	324568.			0.660495			0.330248	−	0.943894i	$-0.392868\pi$
$701$	324568.			0.660495			0.330248	+	0.943894i	$0.392868\pi$
$702$		0			0
$703$	−298720.	−	298720.i	−0.604441	−	0.604441i
$704$		−	94264.0i		−	0.190196i
$705$		0			0
$706$	−319144.			−0.640291
$707$	255664.	−	255664.i	0.511482	−	0.511482i
$708$		0			0
$709$			248026.i			0.493406i	0.969091	+	0.246703i	$0.0793472\pi$
$709$			248026.i			0.493406i	−0.969091	+	0.246703i	$0.920653\pi$
$710$	166503.	+	13067.7i	0.330298	+	0.0259228i
$711$		0			0
$712$	173986.	−	173986.i	0.343205	−	0.343205i
$713$	612289.	+	612289.i	1.20442	+	1.20442i
$714$		0			0
$715$	−24774.8	+	315671.i	−0.0484617	+	0.617479i
$716$	−306822.			−0.598495
$717$		0			0
$718$	−300815.	−	300815.i	−0.583514	−	0.583514i
$719$			406907.i			0.787113i	0.919300	+	0.393557i	$0.128756\pi$
$719$			406907.i			0.787113i	−0.919300	+	0.393557i	$0.871244\pi$
$720$		0			0
$721$	338183.			0.650550
$722$	−323801.	+	323801.i	−0.621160	+	0.621160i
$723$		0			0
$724$			352335.i			0.672169i
$725$	337980.	+	53380.3i	0.643006	+	0.101556i
$726$		0			0
$727$	−465457.	+	465457.i	−0.880664	+	0.880664i	−0.993602	−	0.112938i	$-0.963974\pi$
$727$	−465457.	+	465457.i	−0.880664	+	0.880664i	0.112938	+	0.993602i	$0.463974\pi$
$728$	−360352.	−	360352.i	−0.679930	−	0.679930i
$729$		0			0
$730$	−87353.6	−	102233.i	−0.163921	−	0.191843i
$731$	−58165.7			−0.108851
$732$		0			0
$733$	−596415.	−	596415.i	−1.11004	−	1.11004i	−0.993144	−	0.116901i	$-0.962704\pi$
$733$	−596415.	−	596415.i	−1.11004	−	1.11004i	−0.116901	−	0.993144i	$-0.537296\pi$
$734$		−	626208.i		−	1.16232i
$735$		0			0
$736$	−402479.			−0.742997
$737$	−207780.	+	207780.i	−0.382534	+	0.382534i
$738$		0			0
$739$		−	672211.i		−	1.23088i	−0.788182	−	0.615442i	$-0.788978\pi$
$739$		−	672211.i		−	1.23088i	0.788182	−	0.615442i	$-0.211022\pi$
$740$	−26869.0	+	342354.i	−0.0490668	+	0.625190i
$741$		0			0
$742$	443951.	−	443951.i	0.806358	−	0.806358i
$743$	−251951.	−	251951.i	−0.456393	−	0.456393i	0.441076	−	0.897470i	$-0.354597\pi$
$743$	−251951.	−	251951.i	−0.456393	−	0.456393i	−0.897470	+	0.441076i	$0.854597\pi$
$744$		0			0
$745$	−330162.	+	282109.i	−0.594860	+	0.508282i
$746$	−1.15477e6			−2.07500
$747$		0			0
$748$	−90242.8	−	90242.8i	−0.161291	−	0.161291i
$749$			135392.i			0.241339i
$750$		0			0
$751$	−37245.9			−0.0660387			−0.0330194	−	0.999455i	$-0.510512\pi$
$751$	−37245.9			−0.0660387			−0.0330194	+	0.999455i	$0.510512\pi$
$752$	109169.	−	109169.i	0.193047	−	0.193047i
$753$		0			0
$754$		−	585981.i		−	1.03072i
$755$	332824.	+	389515.i	0.583875	+	0.683330i
$756$		0			0
$757$	−133738.	+	133738.i	−0.233380	+	0.233380i	−0.814102	−	0.580722i	$-0.802770\pi$
$757$	−133738.	+	133738.i	−0.233380	+	0.233380i	0.580722	+	0.814102i	$0.302770\pi$
$758$	−714192.	−	714192.i	−1.24302	−	1.24302i
$759$		0			0
$760$	212719.	+	16694.8i	0.368281	+	0.0289038i
$761$	−969173.			−1.67352			−0.836762	−	0.547567i	$-0.815554\pi$
$761$	−969173.			−1.67352			−0.836762	+	0.547567i	$0.815554\pi$
$762$		0			0
$763$	397946.	+	397946.i	0.683557	+	0.683557i
$764$			258761.i			0.443315i
$765$		0			0
$766$	−45897.5			−0.0782225
$767$	462193.	−	462193.i	0.785657	−	0.785657i
$768$		0			0
$769$		−	572412.i		−	0.967956i	−0.875080	−	0.483978i	$-0.839191\pi$
$769$		−	572412.i		−	0.967956i	0.875080	−	0.483978i	$-0.160809\pi$
$770$	−230477.	+	196933.i	−0.388729	+	0.332152i
$771$		0			0
$772$	107904.	−	107904.i	0.181051	−	0.181051i
$773$	−345426.	−	345426.i	−0.578090	−	0.578090i	0.356287	−	0.934377i	$-0.384043\pi$
$773$	−345426.	−	345426.i	−0.578090	−	0.578090i	−0.934377	+	0.356287i	$0.884043\pi$
$774$		0			0
$775$	−149970.	+	949546.i	−0.249691	+	1.58093i
$776$	−234552.			−0.389508
$777$		0			0
$778$	−568471.	−	568471.i	−0.939180	−	0.939180i
$779$			55328.5i			0.0911746i
$780$		0			0
$781$	79053.7			0.129605
$782$	734076.	−	734076.i	1.20040	−	1.20040i
$783$		0			0
$784$			63446.5i			0.103223i
$785$	82488.9	+	6473.99i	0.133862	+	0.0105059i
$786$		0			0
$787$	−172088.	+	172088.i	−0.277843	+	0.277843i	−0.832248	−	0.554404i	$-0.812946\pi$
$787$	−172088.	+	172088.i	−0.277843	+	0.277843i	0.554404	+	0.832248i	$0.312946\pi$
$788$	107933.	+	107933.i	0.173821	+	0.173821i
$789$		0			0
$790$	44387.0	−	565562.i	0.0711217	−	0.906204i
$791$	593893.			0.949195
$792$		0			0
$793$	171208.	+	171208.i	0.272256	+	0.272256i
$794$			314419.i			0.498733i
$795$		0			0
$796$	405601.			0.640137
$797$	−296648.	+	296648.i	−0.467009	+	0.467009i	−0.900944	−	0.433935i	$-0.857125\pi$
$797$	−296648.	+	296648.i	−0.467009	+	0.467009i	0.433935	+	0.900944i	$0.357125\pi$
$798$		0			0
$799$			193130.i			0.302521i
$800$	−262794.	−	361375.i	−0.410616	−	0.564648i
$801$		0			0
$802$	−640440.	+	640440.i	−0.995703	+	0.995703i
$803$	−45006.7	−	45006.7i	−0.0697985	−	0.0697985i
$804$		0			0
$805$	−428819.	−	501861.i	−0.661732	−	0.774448i
$806$	1.64630e6			2.53418
$807$		0			0
$808$	258651.	+	258651.i	0.396178	+	0.396178i
$809$			286471.i			0.437707i	0.975758	+	0.218854i	$0.0702317\pi$
$809$			286471.i			0.437707i	−0.975758	+	0.218854i	$0.929768\pi$
$810$		0			0
$811$	−672778.			−1.02289			−0.511447	−	0.859315i	$-0.670890\pi$
$811$	−672778.			−1.02289			−0.511447	+	0.859315i	$0.670890\pi$
$812$	106192.	−	106192.i	0.161057	−	0.161057i
$813$		0			0
$814$			607222.i			0.916428i
$815$	−42332.9	+	539389.i	−0.0637328	+	0.812058i
$816$		0			0
$817$	−18752.3	+	18752.3i	−0.0280938	+	0.0280938i
$818$	−876.729	−	876.729i	−0.00131026	−	0.00131026i
$819$		0			0
$820$	34193.4	−	29216.8i	0.0508528	−	0.0434515i
$821$	270459.			0.401251			0.200625	−	0.979668i	$-0.435703\pi$
$821$	270459.			0.401251			0.200625	+	0.979668i	$0.435703\pi$
$822$		0			0
$823$	445888.	+	445888.i	0.658304	+	0.658304i	0.954979	−	0.296675i	$-0.0958777\pi$
$823$	445888.	+	445888.i	0.658304	+	0.658304i	−0.296675	+	0.954979i	$0.595878\pi$
$824$			342134.i			0.503897i
$825$		0			0
$826$	625797.			0.917220
$827$	505149.	−	505149.i	0.738599	−	0.738599i	−0.233708	−	0.972307i	$-0.575086\pi$
$827$	505149.	−	505149.i	0.738599	−	0.738599i	0.972307	+	0.233708i	$0.0750860\pi$
$828$		0			0
$829$		−	176611.i		−	0.256986i	−0.991710	−	0.128493i	$-0.958986\pi$
$829$		−	176611.i		−	0.256986i	0.991710	−	0.128493i	$-0.0410139\pi$
$830$	−141340.	−	165415.i	−0.205167	−	0.240114i
$831$		0			0
$832$	275945.	−	275945.i	0.398636	−	0.398636i
$833$	−56121.5	−	56121.5i	−0.0808797	−	0.0808797i
$834$		0			0
$835$	826816.	+	64891.1i	1.18587	+	0.0930705i
$836$	−58187.6			−0.0832564
$837$		0			0
$838$	−102216.	−	102216.i	−0.145557	−	0.145557i
$839$			153954.i			0.218709i	0.994003	+	0.109354i	$0.0348783\pi$
$839$			153954.i			0.218709i	−0.994003	+	0.109354i	$0.965122\pi$
$840$		0			0
$841$	407556.			0.576229
$842$	206475.	−	206475.i	0.291235	−	0.291235i
$843$		0			0
$844$			113802.i			0.159758i
$845$	−453762.	+	387720.i	−0.635498	+	0.543006i
$846$		0			0
$847$	384103.	−	384103.i	0.535403	−	0.535403i
$848$	638635.	+	638635.i	0.888098	+	0.888098i
$849$		0			0
$850$	1.13841e6	+	179800.i	1.57566	+	0.248858i
$851$	−1.32222e6			−1.82576
$852$		0			0
$853$	601023.	+	601023.i	0.826025	+	0.826025i	0.986964	−	0.160940i	$-0.0514524\pi$
$853$	601023.	+	601023.i	0.826025	+	0.826025i	−0.160940	+	0.986964i	$0.551452\pi$
$854$			231811.i			0.317846i
$855$		0			0
$856$	−136973.			−0.186934
$857$	−479752.	+	479752.i	−0.653214	+	0.653214i	−0.953766	−	0.300552i	$-0.902829\pi$
$857$	−479752.	+	479752.i	−0.653214	+	0.653214i	0.300552	+	0.953766i	$0.402829\pi$
$858$		0			0
$859$		−	925363.i		−	1.25408i	−0.778987	−	0.627041i	$-0.784266\pi$
$859$		−	925363.i		−	1.25408i	0.778987	−	0.627041i	$-0.215734\pi$
$860$	21491.5	+	1686.72i	0.0290582	+	0.00228058i
$861$		0			0
$862$	228953.	−	228953.i	0.308129	−	0.308129i
$863$	882185.	+	882185.i	1.18451	+	1.18451i	0.978564	+	0.205944i	$0.0660265\pi$
$863$	882185.	+	882185.i	1.18451	+	1.18451i	0.205944	+	0.978564i	$0.433974\pi$
$864$		0			0
$865$	−56743.0	+	722997.i	−0.0758368	+	0.966282i
$866$	733498.			0.978054
$867$		0			0
$868$	298343.	+	298343.i	0.395983	+	0.395983i
$869$		−	268522.i		−	0.355582i
$870$		0			0
$871$	−1.21650e6			−1.60352
$872$	−402595.	+	402595.i	−0.529463	+	0.529463i
$873$		0			0
$874$		−	473324.i		−	0.619635i
$875$	170615.	−	712710.i	0.222844	−	0.930887i
$876$		0			0
$877$	−272599.	+	272599.i	−0.354426	+	0.354426i	−0.861753	−	0.507328i	$-0.830633\pi$
$877$	−272599.	+	272599.i	−0.354426	+	0.354426i	0.507328	+	0.861753i	$0.330633\pi$
$878$	−555265.	−	555265.i	−0.720297	−	0.720297i
$879$		0			0
$880$	−283292.	−	331547.i	−0.365822	−	0.428134i
$881$	557598.			0.718405			0.359202	−	0.933260i	$-0.383049\pi$
$881$	557598.			0.718405			0.359202	+	0.933260i	$0.383049\pi$
$882$		0			0
$883$	180236.	+	180236.i	0.231164	+	0.231164i	0.813179	−	0.582014i	$-0.197735\pi$
$883$	180236.	+	180236.i	0.231164	+	0.231164i	−0.582014	+	0.813179i	$0.697735\pi$
$884$		−	528347.i		−	0.676107i
$885$		0			0
$886$	438125.			0.558124
$887$	−170589.	+	170589.i	−0.216822	+	0.216822i	−0.807158	−	0.590336i	$-0.798995\pi$
$887$	−170589.	+	170589.i	−0.216822	+	0.216822i	0.590336	+	0.807158i	$0.298995\pi$
$888$		0			0
$889$			401385.i			0.507876i
$890$	47411.6	−	604099.i	0.0598555	−	0.762655i
$891$		0			0
$892$	308407.	−	308407.i	0.387610	−	0.387610i
$893$	62264.0	+	62264.0i	0.0780790	+	0.0780790i
$894$		0			0
$895$	997105.	−	851983.i	1.24479	−	1.06362i
$896$	910124.			1.13366
$897$		0			0
$898$	−702977.	−	702977.i	−0.871743	−	0.871743i
$899$		−	842069.i		−	1.04191i
$900$		0			0
$901$	−1.12981e6			−1.39173
$902$	56234.3	−	56234.3i	0.0691176	−	0.0691176i
$903$		0			0
$904$			600832.i			0.735218i
$905$	−978362.	−	1.14501e6i	−1.19455	−	1.39802i
$906$		0			0
$907$	399726.	−	399726.i	0.485901	−	0.485901i	−0.421109	−	0.907010i	$-0.638359\pi$
$907$	399726.	−	399726.i	0.485901	−	0.485901i	0.907010	+	0.421109i	$0.138359\pi$
$908$	359317.	+	359317.i	0.435819	+	0.435819i
$909$		0			0
$910$	−1.25119e6	−	98196.9i	−1.51091	−	0.118581i
$911$	−1.58194e6			−1.90614			−0.953068	−	0.302757i	$-0.902093\pi$
$911$	−1.58194e6			−1.90614			−0.953068	+	0.302757i	$0.902093\pi$
$912$		0			0
$913$	−72821.7	−	72821.7i	−0.0873613	−	0.0873613i
$914$			1.24923e6i			1.49537i
$915$		0			0
$916$	−577501.			−0.688274
$917$	−730149.	+	730149.i	−0.868306	+	0.868306i
$918$		0			0
$919$			1.60006e6i			1.89455i	0.320420	+	0.947276i	$0.396176\pi$
$919$			1.60006e6i			1.89455i	−0.320420	+	0.947276i	$0.603824\pi$
$920$	507725.	−	433829.i	0.599864	−	0.512558i
$921$		0			0
$922$	−672485.	+	672485.i	−0.791081	+	0.791081i
$923$	231419.	+	231419.i	0.271641	+	0.271641i
$924$		0			0
$925$	−863329.	−	1.18718e6i	−1.00900	−	1.38751i
$926$	−1.85234e6			−2.16023
$927$		0			0
$928$	276761.	+	276761.i	0.321372	+	0.321372i
$929$		−	1.50251e6i		−	1.74095i	−0.492209	−	0.870477i	$-0.663811\pi$
$929$		−	1.50251e6i		−	1.74095i	0.492209	−	0.870477i	$-0.336189\pi$
$930$		0			0
$931$	−36186.6			−0.0417492
$932$	−157101.	+	157101.i	−0.180862	+	0.180862i
$933$		0			0
$934$		−	166210.i		−	0.190530i
$935$	543855.	+	42683.4i	0.622100	+	0.0488243i
$936$		0			0
$937$	45116.6	−	45116.6i	0.0513874	−	0.0513874i	−0.680946	−	0.732334i	$-0.738431\pi$
$937$	45116.6	−	45116.6i	0.0513874	−	0.0513874i	0.732334	+	0.680946i	$0.238431\pi$
$938$	−823554.	−	823554.i	−0.936022	−	0.936022i
$939$		0			0
$940$	5600.47	−	71358.9i	0.00633824	−	0.0807593i
$941$	1.62275e6			1.83262			0.916312	−	0.400466i	$-0.131152\pi$
$941$	1.62275e6			1.83262			0.916312	+	0.400466i	$0.131152\pi$
$942$		0			0
$943$	122450.	+	122450.i	0.137700	+	0.137700i
$944$			900224.i			1.01020i
$945$		0			0
$946$	38118.7			0.0425947
$947$	868195.	−	868195.i	0.968094	−	0.968094i	−0.0314128	−	0.999506i	$-0.510001\pi$
$947$	868195.	−	868195.i	0.968094	−	0.968094i	0.999506	+	0.0314128i	$0.0100007\pi$
$948$		0			0
$949$		−	263502.i		−	0.292585i
$950$	424985.	−	309052.i	0.470898	−	0.342440i
$951$		0			0
$952$	−620834.	+	620834.i	−0.685017	+	0.685017i
$953$	221674.	+	221674.i	0.244078	+	0.244078i	0.818535	−	0.574457i	$-0.194787\pi$
$953$	221674.	+	221674.i	0.244078	+	0.244078i	−0.574457	+	0.818535i	$0.694787\pi$
$954$		0			0
$955$	−718528.	−	840918.i	−0.787838	−	0.922034i
$956$	379344.			0.415066
$957$		0			0
$958$	57122.1	+	57122.1i	0.0622406	+	0.0622406i
$959$			757749.i			0.823926i
$960$		0			0
$961$	1.44225e6			1.56168
$962$	−1.77756e6	+	1.77756e6i	−1.92077	+	1.92077i
$963$		0			0
$964$		−	68818.1i		−	0.0740540i
$965$	−51036.8	+	650290.i	−0.0548061	+	0.698317i
$966$		0			0
$967$	310704.	−	310704.i	0.332272	−	0.332272i	−0.521176	−	0.853449i	$-0.674507\pi$
$967$	310704.	−	310704.i	0.332272	−	0.332272i	0.853449	+	0.521176i	$0.174507\pi$
$968$	388590.	+	388590.i	0.414707	+	0.414707i
$969$		0			0
$970$	−439155.	+	375239.i	−0.466740	+	0.398809i
$971$	993419.			1.05364			0.526822	−	0.849976i	$-0.323383\pi$
$971$	993419.			1.05364			0.526822	+	0.849976i	$0.323383\pi$
$972$		0			0
$973$	14332.7	+	14332.7i	0.0151392	+	0.0151392i
$974$		−	1.65864e6i		−	1.74837i
$975$		0			0
$976$	−333465.			−0.350067
$977$	575998.	−	575998.i	0.603437	−	0.603437i	−0.337786	−	0.941223i	$-0.609678\pi$
$977$	575998.	−	575998.i	0.603437	−	0.603437i	0.941223	+	0.337786i	$0.109678\pi$
$978$		0			0
$979$		−	286819.i		−	0.299256i
$980$	19108.7	+	22363.6i	0.0198966	+	0.0232857i
$981$		0			0
$982$	−578996.	+	578996.i	−0.600416	+	0.600416i
$983$	164785.	+	164785.i	0.170533	+	0.170533i	0.787214	−	0.616680i	$-0.211523\pi$
$983$	164785.	+	164785.i	0.170533	+	0.170533i	−0.616680	+	0.787214i	$0.711523\pi$
$984$		0			0
$985$	−650468.	−	51050.8i	−0.670430	−	0.0526175i
$986$	−1.00956e6			−1.03843
$987$		0			0
$988$	−170336.	−	170336.i	−0.174499	−	0.174499i
$989$			83003.0i			0.0848596i
$990$		0			0
$991$	1.06880e6			1.08830			0.544148	−	0.838989i	$-0.316853\pi$
$991$	1.06880e6			1.08830			0.544148	+	0.838989i	$0.316853\pi$
$992$	−777551.	+	777551.i	−0.790143	+	0.790143i
$993$		0			0
$994$			313335.i			0.317130i
$995$	−1.31811e6	+	1.12627e6i	−1.33140	+	1.13762i
$996$		0			0
$997$	205698.	−	205698.i	0.206938	−	0.206938i	−0.596027	−	0.802965i	$-0.703255\pi$
$997$	205698.	−	205698.i	0.206938	−	0.206938i	0.802965	+	0.596027i	$0.203255\pi$
$998$	726256.	+	726256.i	0.729170	+	0.729170i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.5.g.e.28.2		8
3.2	odd	2		15.5.f.a.13.3	yes	8
5.2	odd	4	inner	45.5.g.e.37.2		8
5.3	odd	4		225.5.g.m.82.3		8
5.4	even	2		225.5.g.m.118.3		8
12.11	even	2		240.5.bg.c.193.1		8
15.2	even	4		15.5.f.a.7.3	✓	8
15.8	even	4		75.5.f.e.7.2		8
15.14	odd	2		75.5.f.e.43.2		8
60.47	odd	4		240.5.bg.c.97.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.5.f.a.7.3	✓	8	15.2	even	4
15.5.f.a.13.3	yes	8	3.2	odd	2
45.5.g.e.28.2		8	1.1	even	1	trivial
45.5.g.e.37.2		8	5.2	odd	4	inner
75.5.f.e.7.2		8	15.8	even	4
75.5.f.e.43.2		8	15.14	odd	2
225.5.g.m.82.3		8	5.3	odd	4
225.5.g.m.118.3		8	5.4	even	2
240.5.bg.c.97.1		8	60.47	odd	4
240.5.bg.c.193.1		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-3.30519 + 3.30519i) q^{2} -5.84858i q^{4} +(16.2403 + 19.0066i) q^{5} +(-33.1649 + 33.1649i) q^{7} +(-33.5524 - 33.5524i) q^{8} +O(q^{10})\)	\(q+(-3.30519 + 3.30519i) q^{2} -5.84858i q^{4} +(16.2403 + 19.0066i) q^{5} +(-33.1649 + 33.1649i) q^{7} +(-33.5524 - 33.5524i) q^{8} +(-116.498 - 9.14312i) q^{10} -55.3118 q^{11} +(-161.918 - 161.918i) q^{13} -219.233i q^{14} +315.371 q^{16} +(-278.961 + 278.961i) q^{17} +179.871i q^{19} +(111.162 - 94.9829i) q^{20} +(182.816 - 182.816i) q^{22} +(398.080 + 398.080i) q^{23} +(-97.5033 + 617.348i) q^{25} +1070.34 q^{26} +(193.968 + 193.968i) q^{28} -547.472i q^{29} +1538.11 q^{31} +(-505.525 + 505.525i) q^{32} -1844.04i q^{34} +(-1168.96 - 91.7437i) q^{35} +(-1660.74 + 1660.74i) q^{37} +(-594.509 - 594.509i) q^{38} +(92.8156 - 1182.62i) q^{40} +307.601 q^{41} +(104.254 + 104.254i) q^{43} +323.496i q^{44} -2631.46 q^{46} +(346.159 - 346.159i) q^{47} +201.180i q^{49} +(-1718.19 - 2362.72i) q^{50} +(-946.991 + 946.991i) q^{52} +(2025.02 + 2025.02i) q^{53} +(-898.282 - 1051.29i) q^{55} +2225.52 q^{56} +(1809.50 + 1809.50i) q^{58} +2854.49i q^{59} -1057.37 q^{61} +(-5083.73 + 5083.73i) q^{62} +1704.23i q^{64} +(447.912 - 5707.12i) q^{65} +(3756.53 - 3756.53i) q^{67} +(1631.53 + 1631.53i) q^{68} +(4166.87 - 3560.41i) q^{70} -1429.24 q^{71} +(813.690 + 813.690i) q^{73} -10978.2i q^{74} +1051.99 q^{76} +(1834.41 - 1834.41i) q^{77} +4854.69i q^{79} +(5121.74 + 5994.14i) q^{80} +(-1016.68 + 1016.68i) q^{82} +(1316.57 + 1316.57i) q^{83} +(-9832.53 - 771.687i) q^{85} -689.160 q^{86} +(1855.84 + 1855.84i) q^{88} +5185.49i q^{89} +10740.0 q^{91} +(2328.20 - 2328.20i) q^{92} +2288.24i q^{94} +(-3418.74 + 2921.17i) q^{95} +(3495.32 - 3495.32i) q^{97} +(-664.940 - 664.940i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 84 q^{5} + 20 q^{7} - 180 q^{8}+O(q^{10}) \) 8 * q + 84 * q^5 + 20 * q^7 - 180 * q^8	\( 8 q + 84 q^{5} + 20 q^{7} - 180 q^{8} + 104 q^{10} + 288 q^{11} - 340 q^{13} + 620 q^{16} - 900 q^{17} - 564 q^{20} - 1100 q^{22} + 1560 q^{23} - 1204 q^{25} + 3024 q^{26} + 3580 q^{28} - 512 q^{31} - 4980 q^{32} - 6600 q^{35} - 3820 q^{37} + 7680 q^{38} - 2952 q^{40} + 2712 q^{41} - 1240 q^{43} + 13528 q^{46} - 4800 q^{47} - 3744 q^{50} - 1240 q^{52} - 1020 q^{53} - 3644 q^{55} + 30720 q^{56} + 2340 q^{58} - 4760 q^{61} - 28680 q^{62} + 1212 q^{65} - 8920 q^{67} + 1920 q^{68} + 7380 q^{70} - 7536 q^{71} + 11600 q^{73} + 4344 q^{76} + 360 q^{77} - 10644 q^{80} - 27200 q^{82} + 32400 q^{83} - 15628 q^{85} - 14592 q^{86} - 14340 q^{88} + 16528 q^{91} + 31800 q^{92} - 18864 q^{95} + 58640 q^{97} - 46440 q^{98}+O(q^{100}) \) 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

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