LMFDB - Embedded newform 15.5.d.c.14.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [15,5,Mod(14,15)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("15.14"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	15.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0] 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:	$1.55054944626$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{10}, \sqrt{-26})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 81 $ x^4 + 8*x^2 + 81
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			14.3
Root			$1.58114 - 2.54951i$ of defining polynomial
Character	$\chi$	$=$	15.14
Dual form			15.5.d.c.14.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.16228 q^{2} +(4.74342 - 7.64853i) q^{3} -6.00000 q^{4} +(6.32456 + 24.1868i) q^{5} +(15.0000 - 24.1868i) q^{6} +15.2971i q^{7} -69.5701 q^{8} +(-36.0000 - 72.5603i) q^{9} +(20.0000 + 76.4853i) q^{10} +96.7471i q^{11} +(-28.4605 + 45.8912i) q^{12} -244.753i q^{13} +48.3735i q^{14} +(214.993 + 66.3544i) q^{15} -124.000 q^{16} +278.280 q^{17} +(-113.842 - 229.456i) q^{18} +308.000 q^{19} +(-37.9473 - 145.121i) q^{20} +(117.000 + 72.5603i) q^{21} +305.941i q^{22} -414.258 q^{23} +(-330.000 + 532.109i) q^{24} +(-545.000 + 305.941i) q^{25} -773.977i q^{26} +(-725.743 - 68.8368i) q^{27} -91.7824i q^{28} -193.494i q^{29} +(679.868 + 209.831i) q^{30} +32.0000 q^{31} +720.999 q^{32} +(739.973 + 458.912i) q^{33} +880.000 q^{34} +(-369.986 + 96.7471i) q^{35} +(216.000 + 435.362i) q^{36} -1284.95i q^{37} +973.982 q^{38} +(-1872.00 - 1160.97i) q^{39} +(-440.000 - 1682.68i) q^{40} +2080.06i q^{41} +(369.986 + 229.456i) q^{42} +2585.20i q^{43} -580.483i q^{44} +(1527.32 - 1329.64i) q^{45} -1310.00 q^{46} -2444.44 q^{47} +(-588.184 + 948.418i) q^{48} +2167.00 q^{49} +(-1723.44 + 967.471i) q^{50} +(1320.00 - 2128.44i) q^{51} +1468.52i q^{52} +1416.70 q^{53} +(-2295.00 - 217.681i) q^{54} +(-2340.00 + 611.882i) q^{55} -1064.22i q^{56} +(1460.97 - 2355.75i) q^{57} -611.882i q^{58} +3966.63i q^{59} +(-1289.96 - 398.126i) q^{60} -928.000 q^{61} +101.193 q^{62} +(1109.96 - 550.694i) q^{63} +4264.00 q^{64} +(5919.78 - 1547.95i) q^{65} +(2340.00 + 1451.21i) q^{66} -2585.20i q^{67} -1669.68 q^{68} +(-1965.00 + 3168.47i) q^{69} +(-1170.00 + 305.941i) q^{70} -4643.86i q^{71} +(2504.52 + 5048.03i) q^{72} -4221.99i q^{73} -4063.38i q^{74} +(-245.162 + 5619.65i) q^{75} -1848.00 q^{76} -1479.95 q^{77} +(-5919.78 - 3671.29i) q^{78} +8.00000 q^{79} +(-784.245 - 2999.16i) q^{80} +(-3969.00 + 5224.34i) q^{81} +6577.74i q^{82} -4436.68 q^{83} +(-702.000 - 435.362i) q^{84} +(1760.00 + 6730.71i) q^{85} +8175.13i q^{86} +(-1479.95 - 917.824i) q^{87} -6730.71i q^{88} -9287.72i q^{89} +(4829.80 - 4204.68i) q^{90} +3744.00 q^{91} +2485.55 q^{92} +(151.789 - 244.753i) q^{93} -7730.00 q^{94} +(1947.96 + 7449.53i) q^{95} +(3420.00 - 5514.58i) q^{96} -2508.72i q^{97} +6852.66 q^{98} +(7020.00 - 3482.90i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 24 q^{4} + 60 q^{6} - 144 q^{9} + 80 q^{10} + 120 q^{15} - 496 q^{16} + 1232 q^{19} + 468 q^{21} - 1320 q^{24} - 2180 q^{25} + 2340 q^{30} + 128 q^{31} + 3520 q^{34} + 864 q^{36} - 7488 q^{39} - 1760 q^{40}+ \cdots + 28080 q^{99}+O(q^{100}) $ 4 * q - 24 * q^4 + 60 * q^6 - 144 * q^9 + 80 * q^10 + 120 * q^15 - 496 * q^16 + 1232 * q^19 + 468 * q^21 - 1320 * q^24 - 2180 * q^25 + 2340 * q^30 + 128 * q^31 + 3520 * q^34 + 864 * q^36 - 7488 * q^39 - 1760 * q^40 + 7020 * q^45 - 5240 * q^46 + 8668 * q^49 + 5280 * q^51 - 9180 * q^54 - 9360 * q^55 - 720 * q^60 - 3712 * q^61 + 17056 * q^64 + 9360 * q^66 - 7860 * q^69 - 4680 * q^70 + 9360 * q^75 - 7392 * q^76 + 32 * q^79 - 15876 * q^81 - 2808 * q^84 + 7040 * q^85 - 2880 * q^90 + 14976 * q^91 - 30920 * q^94 + 13680 * q^96 + 28080 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$.

$n$	$7$	$11$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	3.16228			0.790569			0.395285	−	0.918559i	$-0.370646\pi$
$2$	3.16228			0.790569			0.395285	+	0.918559i	$0.370646\pi$
$3$	4.74342	−	7.64853i	0.527046	−	0.849837i
$3$	4.74342	−	7.64853i	0.527046	−	0.849837i
$4$	−6.00000			−0.375000
$5$	6.32456	+	24.1868i	0.252982	+	0.967471i
$5$	6.32456	+	24.1868i	0.252982	+	0.967471i
$6$	15.0000	−	24.1868i	0.416667	−	0.671855i
$7$			15.2971i			0.312185i	0.987742	+	0.156092i	$0.0498898\pi$
$7$			15.2971i			0.312185i	−0.987742	+	0.156092i	$0.950110\pi$
$8$	−69.5701			−1.08703
$9$	−36.0000	−	72.5603i	−0.444444	−	0.895806i
$10$	20.0000	+	76.4853i	0.200000	+	0.764853i
$11$			96.7471i			0.799563i	0.916611	+	0.399781i	$0.130914\pi$
$11$			96.7471i			0.799563i	−0.916611	+	0.399781i	$0.869086\pi$
$12$	−28.4605	+	45.8912i	−0.197642	+	0.318689i
$13$		−	244.753i		−	1.44824i	−0.689673	−	0.724121i	$-0.742246\pi$
$13$		−	244.753i		−	1.44824i	0.689673	−	0.724121i	$-0.257754\pi$
$14$			48.3735i			0.246804i
$15$	214.993	+	66.3544i	0.955526	+	0.294908i
$16$	−124.000			−0.484375
$17$	278.280			0.962908			0.481454	−	0.876471i	$-0.340109\pi$
$17$	278.280			0.962908			0.481454	+	0.876471i	$0.340109\pi$
$18$	−113.842	−	229.456i	−0.351364	−	0.708197i
$19$	308.000			0.853186			0.426593	−	0.904444i	$-0.359714\pi$
$19$	308.000			0.853186			0.426593	+	0.904444i	$0.359714\pi$
$20$	−37.9473	−	145.121i	−0.0948683	−	0.362802i
$21$	117.000	+	72.5603i	0.265306	+	0.164536i
$22$			305.941i			0.632110i
$23$	−414.258			−0.783097			−0.391549	−	0.920157i	$-0.628060\pi$
$23$	−414.258			−0.783097			−0.391549	+	0.920157i	$0.628060\pi$
$24$	−330.000	+	532.109i	−0.572917	+	0.923800i
$25$	−545.000	+	305.941i	−0.872000	+	0.489506i
$26$		−	773.977i		−	1.14494i
$27$	−725.743	−	68.8368i	−0.995532	−	0.0944263i
$28$		−	91.7824i		−	0.117069i
$29$		−	193.494i		−	0.230076i	−0.993361	−	0.115038i	$-0.963301\pi$
$29$		−	193.494i		−	0.230076i	0.993361	−	0.115038i	$-0.0366990\pi$
$30$	679.868	+	209.831i	0.755409	+	0.233146i
$31$	32.0000			0.0332986			0.0166493	−	0.999861i	$-0.494700\pi$
$31$	32.0000			0.0332986			0.0166493	+	0.999861i	$0.494700\pi$
$32$	720.999			0.704101
$33$	739.973	+	458.912i	0.679498	+	0.421407i
$34$	880.000			0.761246
$35$	−369.986	+	96.7471i	−0.302030	+	0.0789772i
$36$	216.000	+	435.362i	0.166667	+	0.335927i
$37$		−	1284.95i		−	0.938607i	−0.883037	−	0.469303i	$-0.844505\pi$
$37$		−	1284.95i		−	0.938607i	0.883037	−	0.469303i	$-0.155495\pi$
$38$	973.982			0.674502
$39$	−1872.00	−	1160.97i	−1.23077	−	0.763291i
$40$	−440.000	−	1682.68i	−0.275000	−	1.05167i
$41$			2080.06i			1.23740i	0.785629	+	0.618698i	$0.212340\pi$
$41$			2080.06i			1.23740i	−0.785629	+	0.618698i	$0.787660\pi$
$42$	369.986	+	229.456i	0.209743	+	0.130077i
$43$			2585.20i			1.39816i	0.715042	+	0.699081i	$0.246407\pi$
$43$			2585.20i			1.39816i	−0.715042	+	0.699081i	$0.753593\pi$
$44$		−	580.483i		−	0.299836i
$45$	1527.32	−	1329.64i	0.754230	−	0.656610i
$46$	−1310.00			−0.619093
$47$	−2444.44			−1.10658			−0.553291	−	0.832988i	$-0.686628\pi$
$47$	−2444.44			−1.10658			−0.553291	+	0.832988i	$0.686628\pi$
$48$	−588.184	+	948.418i	−0.255288	+	0.411640i
$49$	2167.00			0.902541
$50$	−1723.44	+	967.471i	−0.689377	+	0.386988i
$51$	1320.00	−	2128.44i	0.507497	−	0.818315i
$52$			1468.52i			0.543091i
$53$	1416.70			0.504343			0.252172	−	0.967683i	$-0.418855\pi$
$53$	1416.70			0.504343			0.252172	+	0.967683i	$0.418855\pi$
$54$	−2295.00	−	217.681i	−0.787037	−	0.0746505i
$55$	−2340.00	+	611.882i	−0.773554	+	0.202275i
$56$		−	1064.22i		−	0.339355i
$57$	1460.97	−	2355.75i	0.449668	−	0.725068i
$58$		−	611.882i		−	0.181891i
$59$			3966.63i			1.13951i	0.821815	+	0.569754i	$0.192962\pi$
$59$			3966.63i			1.13951i	−0.821815	+	0.569754i	$0.807038\pi$
$60$	−1289.96	−	398.126i	−0.358322	−	0.110591i
$61$	−928.000			−0.249395			−0.124698	−	0.992195i	$-0.539796\pi$
$61$	−928.000			−0.249395			−0.124698	+	0.992195i	$0.539796\pi$
$62$	101.193			0.0263249
$63$	1109.96	−	550.694i	0.279657	−	0.138749i
$64$	4264.00			1.04102
$65$	5919.78	−	1547.95i	1.40113	−	0.366380i
$66$	2340.00	+	1451.21i	0.537190	+	0.333151i
$67$		−	2585.20i		−	0.575897i	−0.957646	−	0.287949i	$-0.907027\pi$
$67$		−	2585.20i		−	0.575897i	0.957646	−	0.287949i	$-0.0929732\pi$
$68$	−1669.68			−0.361091
$69$	−1965.00	+	3168.47i	−0.412728	+	0.665505i
$70$	−1170.00	+	305.941i	−0.238776	+	0.0624370i
$71$		−	4643.86i		−	0.921218i	−0.887603	−	0.460609i	$-0.847631\pi$
$71$		−	4643.86i		−	0.921218i	0.887603	−	0.460609i	$-0.152369\pi$
$72$	2504.52	+	5048.03i	0.483126	+	0.973771i
$73$		−	4221.99i		−	0.792266i	−0.918193	−	0.396133i	$-0.870352\pi$
$73$		−	4221.99i		−	0.792266i	0.918193	−	0.396133i	$-0.129648\pi$
$74$		−	4063.38i		−	0.742034i
$75$	−245.162	+	5619.65i	−0.0435844	+	0.999050i
$76$	−1848.00			−0.319945
$77$	−1479.95			−0.249611
$78$	−5919.78	−	3671.29i	−0.973009	−	0.603434i
$79$	8.00000			0.00128185			0.000640923	−	1.00000i	$-0.499796\pi$
$79$	8.00000			0.00128185			0.000640923		1.00000i	$0.499796\pi$
$80$	−784.245	−	2999.16i	−0.122538	−	0.468619i
$81$	−3969.00	+	5224.34i	−0.604938	+	0.796272i
$82$			6577.74i			0.978247i
$83$	−4436.68			−0.644023			−0.322012	−	0.946736i	$-0.604359\pi$
$83$	−4436.68			−0.644023			−0.322012	+	0.946736i	$0.604359\pi$
$84$	−702.000	−	435.362i	−0.0994898	−	0.0617010i
$85$	1760.00	+	6730.71i	0.243599	+	0.931586i
$86$			8175.13i			1.10534i
$87$	−1479.95	−	917.824i	−0.195527	−	0.121261i
$88$		−	6730.71i		−	0.869151i
$89$		−	9287.72i		−	1.17254i	−0.810114	−	0.586272i	$-0.800595\pi$
$89$		−	9287.72i		−	1.17254i	0.810114	−	0.586272i	$-0.199405\pi$
$90$	4829.80	−	4204.68i	0.596271	−	0.519096i
$91$	3744.00			0.452119
$92$	2485.55			0.293661
$93$	151.789	−	244.753i	0.0175499	−	0.0282984i
$94$	−7730.00			−0.874830
$95$	1947.96	+	7449.53i	0.215841	+	0.825432i
$96$	3420.00	−	5514.58i	0.371094	−	0.598371i
$97$		−	2508.72i		−	0.266630i	−0.991074	−	0.133315i	$-0.957438\pi$
$97$		−	2508.72i		−	0.266630i	0.991074	−	0.133315i	$-0.0425621\pi$
$98$	6852.66			0.713521
$99$	7020.00	−	3482.90i	0.716253	−	0.355361i
$100$	3270.00	−	1835.65i	0.327000	−	0.183565i
$101$		−	4837.35i		−	0.474204i	−0.971485	−	0.237102i	$-0.923802\pi$
$101$		−	4837.35i		−	0.474204i	0.971485	−	0.237102i	$-0.0761976\pi$
$102$	4174.21	−	6730.71i	0.401212	−	0.646934i
$103$			5491.64i			0.517640i	0.965926	+	0.258820i	$0.0833337\pi$
$103$			5491.64i			0.517640i	−0.965926	+	0.258820i	$0.916666\pi$
$104$			17027.5i			1.57429i
$105$	−1015.03	+	3288.76i	−0.0920659	+	0.298301i
$106$	4480.00			0.398718
$107$	2925.11			0.255490			0.127745	−	0.991807i	$-0.459226\pi$
$107$	2925.11			0.255490			0.127745	+	0.991807i	$0.459226\pi$
$108$	4354.46	+	413.021i	0.373324	+	0.0354099i
$109$	−6352.00			−0.534635			−0.267318	−	0.963608i	$-0.586137\pi$
$109$	−6352.00			−0.534635			−0.267318	+	0.963608i	$0.586137\pi$
$110$	−7399.73	+	1934.94i	−0.611548	+	0.159913i
$111$	−9828.00	−	6095.07i	−0.797663	−	0.494689i
$112$		−	1896.84i		−	0.151215i
$113$	−8867.03			−0.694418			−0.347209	−	0.937788i	$-0.612871\pi$
$113$	−8867.03			−0.694418			−0.347209	+	0.937788i	$0.612871\pi$
$114$	4620.00	−	7449.53i	0.355494	−	0.573217i
$115$	−2620.00	−	10019.6i	−0.198110	−	0.757624i
$116$			1160.97i			0.0862786i
$117$	−17759.4	+	8811.11i	−1.29734	+	0.643663i
$118$			12543.6i			0.900861i
$119$			4256.87i			0.300605i
$120$	−14957.1	−	4616.28i	−1.03869	−	0.320575i
$121$	5281.00			0.360699
$122$	−2934.59			−0.197164
$123$	15909.4	+	9866.60i	1.05158	+	0.652165i
$124$	−192.000			−0.0124870
$125$	−10846.6	−	11246.8i	−0.694183	−	0.719798i
$126$	3510.00	−	1741.45i	0.221088	−	0.109691i
$127$		−	3808.97i		−	0.236156i	−0.993004	−	0.118078i	$-0.962327\pi$
$127$		−	3808.97i		−	0.236156i	0.993004	−	0.118078i	$-0.0376734\pi$
$128$	1947.96			0.118894
$129$	19773.0	+	12262.7i	1.18821	+	0.736896i
$130$	18720.0	−	4895.06i	1.10769	−	0.289648i
$131$			8030.01i			0.467922i	0.972246	+	0.233961i	$0.0751688\pi$
$131$			8030.01i			0.467922i	−0.972246	+	0.233961i	$0.924831\pi$
$132$	−4439.84	−	2753.47i	−0.254812	−	0.158027i
$133$			4711.49i			0.266352i
$134$		−	8175.13i		−	0.455287i
$135$	−2925.06	−	17988.7i	−0.160497	−	0.987036i
$136$	−19360.0			−1.04671
$137$	21263.2			1.13289			0.566443	−	0.824101i	$-0.308319\pi$
$137$	21263.2			1.13289			0.566443	+	0.824101i	$0.308319\pi$
$138$	−6213.88	+	10019.6i	−0.326290	+	0.526128i
$139$	−19372.0			−1.00264			−0.501320	−	0.865262i	$-0.667152\pi$
$139$	−19372.0			−1.00264			−0.501320	+	0.865262i	$0.667152\pi$
$140$	2219.92	−	580.483i	0.113261	−	0.0296165i
$141$	−11595.0	+	18696.4i	−0.583220	+	0.940414i
$142$		−	14685.2i		−	0.728287i
$143$	23679.1			1.15796
$144$	4464.00	+	8997.48i	0.215278	+	0.433906i
$145$	4680.00	−	1223.76i	0.222592	−	0.0582052i
$146$		−	13351.1i		−	0.626342i
$147$	10279.0	−	16574.4i	0.475681	−	0.767012i
$148$			7709.72i			0.351978i
$149$		−	16205.1i		−	0.729928i	−0.931022	−	0.364964i	$-0.881081\pi$
$149$		−	16205.1i		−	0.729928i	0.931022	−	0.364964i	$-0.118919\pi$
$150$	−775.270	+	17770.9i	−0.0344565	+	0.789818i
$151$	33272.0			1.45923			0.729617	−	0.683856i	$-0.239698\pi$
$151$	33272.0			1.45923			0.729617	+	0.683856i	$0.239698\pi$
$152$	−21427.6			−0.927441
$153$	−10018.1	−	20192.1i	−0.427959	−	0.862579i
$154$	−4680.00			−0.197335
$155$	202.386	+	773.977i	0.00842397	+	0.0322155i
$156$	11232.0	+	6965.79i	0.461538	+	0.286234i
$157$			29309.2i			1.18906i	0.804073	+	0.594530i	$0.202662\pi$
$157$			29309.2i			1.18906i	−0.804073	+	0.594530i	$0.797338\pi$
$158$	25.2982			0.00101339
$159$	6720.00	−	10835.7i	0.265812	−	0.428609i
$160$	4560.00	+	17438.6i	0.178125	+	0.681197i
$161$		−	6336.93i		−	0.244471i
$162$	−12551.1	+	16520.8i	−0.478246	+	0.629509i
$163$		−	12252.9i		−	0.461174i	−0.973052	−	0.230587i	$-0.925935\pi$
$163$		−	12252.9i		−	0.461174i	0.973052	−	0.230587i	$-0.0740647\pi$
$164$		−	12480.4i		−	0.464023i
$165$	−6419.59	+	20800.0i	−0.235798	+	0.764003i
$166$	−14030.0			−0.509145
$167$	−2520.34			−0.0903702			−0.0451851	−	0.998979i	$-0.514388\pi$
$167$	−2520.34			−0.0903702			−0.0451851	+	0.998979i	$0.514388\pi$
$168$	−8139.70	−	5048.03i	−0.288396	−	0.178856i
$169$	−31343.0			−1.09741
$170$	5565.61	+	21284.4i	0.192582	+	0.736483i
$171$	−11088.0	−	22348.6i	−0.379194	−	0.764289i
$172$		−	15511.2i		−	0.524311i
$173$	−49318.9			−1.64786			−0.823931	−	0.566690i	$-0.808224\pi$
$173$	−49318.9			−1.64786			−0.823931	+	0.566690i	$0.808224\pi$
$174$	−4680.00	−	2902.41i	−0.154578	−	0.0958651i
$175$	−4680.00	−	8336.90i	−0.152816	−	0.272225i
$176$		−	11996.6i		−	0.387288i
$177$	30338.9	+	18815.4i	0.968396	+	0.600574i
$178$		−	29370.4i		−	0.926977i
$179$			38021.6i			1.18665i	0.804961	+	0.593327i	$0.202186\pi$
$179$			38021.6i			1.18665i	−0.804961	+	0.593327i	$0.797814\pi$
$180$	−9163.90	+	7977.81i	−0.282836	+	0.246229i
$181$	20342.0			0.620921			0.310461	−	0.950586i	$-0.399517\pi$
$181$	20342.0			0.620921			0.310461	+	0.950586i	$0.399517\pi$
$182$	11839.6			0.357432
$183$	−4401.89	+	7097.84i	−0.131443	+	0.211945i
$184$	28820.0			0.851252
$185$	31078.9	−	8126.76i	0.908075	−	0.237451i
$186$	480.000	−	773.977i	0.0138744	−	0.0223719i
$187$			26922.8i			0.769905i
$188$	14666.6			0.414968
$189$	1053.00	−	11101.7i	0.0294785	−	0.310790i
$190$	6160.00	+	23557.5i	0.170637	+	0.652562i
$191$			55532.8i			1.52224i	0.648611	+	0.761120i	$0.275350\pi$
$191$			55532.8i			1.52224i	−0.648611	+	0.761120i	$0.724650\pi$
$192$	20225.9	−	32613.3i	0.548663	−	0.884693i
$193$		−	50725.0i		−	1.36178i	−0.732384	−	0.680891i	$-0.761593\pi$
$193$		−	50725.0i		−	1.36178i	0.732384	−	0.680891i	$-0.238407\pi$
$194$		−	7933.26i		−	0.210789i
$195$	16240.4	−	52620.2i	0.427099	−	1.38383i
$196$	−13002.0			−0.338453
$197$	−28865.3			−0.743778			−0.371889	−	0.928277i	$-0.621290\pi$
$197$	−28865.3			−0.743778			−0.371889	+	0.928277i	$0.621290\pi$
$198$	22199.2	−	11013.9i	0.566248	−	0.280938i
$199$	34328.0			0.866847			0.433423	−	0.901190i	$-0.357305\pi$
$199$	34328.0			0.866847			0.433423	+	0.901190i	$0.357305\pi$
$200$	37915.7	−	21284.4i	0.947893	−	0.532109i
$201$	−19773.0	−	12262.7i	−0.489419	−	0.303525i
$202$		−	15297.1i		−	0.374891i
$203$	2959.89			0.0718263
$204$	−7920.00	+	12770.6i	−0.190311	+	0.306868i
$205$	−50310.0	+	13155.5i	−1.19714	+	0.313039i
$206$			17366.1i			0.409230i
$207$	14913.3	+	30058.7i	0.348043	+	0.701503i
$208$			30349.4i			0.701492i
$209$			29798.1i			0.682175i
$210$	−3209.80	+	10400.0i	−0.0727845	+	0.235827i
$211$	−33268.0			−0.747243			−0.373621	−	0.927581i	$-0.621884\pi$
$211$	−33268.0			−0.747243			−0.373621	+	0.927581i	$0.621884\pi$
$212$	−8500.20			−0.189129
$213$	−35518.7	−	22027.8i	−0.782885	−	0.485525i
$214$	9250.00			0.201983
$215$	−62527.7	+	16350.3i	−1.35268	+	0.353710i
$216$	50490.0	+	4788.98i	1.08218	+	0.102644i
$217$			489.506i			0.0103953i
$218$	−20086.8			−0.422666
$219$	−32292.0	−	20026.6i	−0.673297	−	0.417561i
$220$	14040.0	−	3671.29i	0.290083	−	0.0758532i
$221$		−	68110.0i		−	1.39452i
$222$	−31078.9	−	19274.3i	−0.630608	−	0.391086i
$223$			76011.1i			1.52851i	0.644916	+	0.764253i	$0.276892\pi$
$223$			76011.1i			1.52851i	−0.644916	+	0.764253i	$0.723108\pi$
$224$			11029.2i			0.219810i
$225$	41819.2	+	28531.5i	0.826058	+	0.563585i
$226$	−28040.0			−0.548986
$227$	−16788.5			−0.325807			−0.162904	−	0.986642i	$-0.552086\pi$
$227$	−16788.5			−0.325807			−0.162904	+	0.986642i	$0.552086\pi$
$228$	−8765.83	+	14134.5i	−0.168626	+	0.271901i
$229$	40298.0			0.768445			0.384222	−	0.923241i	$-0.374470\pi$
$229$	40298.0			0.768445			0.384222	+	0.923241i	$0.374470\pi$
$230$	−8285.17	−	31684.7i	−0.156619	−	0.598954i
$231$	−7020.00	+	11319.4i	−0.131557	+	0.212129i
$232$			13461.4i			0.250101i
$233$	87253.6			1.60721			0.803603	−	0.595166i	$-0.202914\pi$
$233$	87253.6			1.60721			0.803603	+	0.595166i	$0.202914\pi$
$234$	−56160.0	+	27863.2i	−1.02564	+	0.508860i
$235$	−15460.0	−	59123.1i	−0.279946	−	1.07059i
$236$		−	23799.8i		−	0.427316i
$237$	37.9473	−	61.1882i	0.000675592	−	0.00108936i
$238$			13461.4i			0.237649i
$239$		−	88620.3i		−	1.55145i	−0.631072	−	0.775725i	$-0.717385\pi$
$239$		−	88620.3i		−	1.55145i	0.631072	−	0.775725i	$-0.282615\pi$
$240$	−26659.2	−	8227.94i	−0.462833	−	0.142846i
$241$	72152.0			1.24227			0.621133	−	0.783706i	$-0.286673\pi$
$241$	72152.0			1.24227			0.621133	+	0.783706i	$0.286673\pi$
$242$	16700.0			0.285158
$243$	21131.9	+	55138.2i	0.357871	+	0.933771i
$244$	5568.00			0.0935232
$245$	13705.3	+	52412.7i	0.228327	+	0.873182i
$246$	50310.0	+	31200.9i	0.831350	+	0.515582i
$247$		−	75383.9i		−	1.23562i
$248$	−2226.24			−0.0361967
$249$	−21045.0	+	33934.0i	−0.339430	+	0.547314i
$250$	−34300.0	−	35565.7i	−0.548800	−	0.569051i
$251$			15963.3i			0.253381i	0.991942	+	0.126691i	$0.0404355\pi$
$251$			15963.3i			0.253381i	−0.991942	+	0.126691i	$0.959564\pi$
$252$	−6659.76	+	3304.16i	−0.104871	+	0.0520308i
$253$		−	40078.3i		−	0.626135i
$254$		−	12045.0i		−	0.186698i
$255$	59828.4	+	18465.1i	0.920083	+	0.283970i
$256$	−62064.0			−0.947021
$257$	84938.8			1.28600			0.642998	−	0.765868i	$-0.277690\pi$
$257$	84938.8			1.28600			0.642998	+	0.765868i	$0.277690\pi$
$258$	62527.7	+	38778.0i	0.939362	+	0.582568i
$259$	19656.0			0.293019
$260$	−35518.7	+	9287.72i	−0.525425	+	0.137392i
$261$	−14040.0	+	6965.79i	−0.206104	+	0.102256i
$262$			25393.1i			0.369925i
$263$	2849.21			0.0411920			0.0205960	−	0.999788i	$-0.493444\pi$
$263$	2849.21			0.0411920			0.0205960	+	0.999788i	$0.493444\pi$
$264$	−51480.0	−	31926.5i	−0.738636	−	0.458083i
$265$	8960.00	+	34265.4i	0.127590	+	0.487938i
$266$			14899.1i			0.210569i
$267$	−71037.4	−	44055.5i	−0.996471	−	0.617985i
$268$			15511.2i			0.215961i
$269$		−	3918.26i		−	0.0541487i	−0.999633	−	0.0270744i	$-0.991381\pi$
$269$		−	3918.26i		−	0.0541487i	0.999633	−	0.0270744i	$-0.00861909\pi$
$270$	−9249.85	−	56885.4i	−0.126884	−	0.780321i
$271$	−67048.0			−0.912951			−0.456475	−	0.889736i	$-0.650888\pi$
$271$	−67048.0			−0.912951			−0.456475	+	0.889736i	$0.650888\pi$
$272$	−34506.8			−0.466409
$273$	17759.4	−	28636.1i	0.238288	−	0.384228i
$274$	67240.0			0.895626
$275$	−29598.9	−	52727.2i	−0.391391	−	0.697219i
$276$	11790.0	−	19010.8i	0.154773	−	0.249564i
$277$			10340.8i			0.134771i	0.997727	+	0.0673853i	$0.0214657\pi$
$277$			10340.8i			0.134771i	−0.997727	+	0.0673853i	$0.978534\pi$
$278$	−61259.6			−0.792656
$279$	−1152.00	−	2321.93i	−0.0147994	−	0.0298291i
$280$	25740.0	−	6730.71i	0.328316	−	0.0858508i
$281$			22300.2i			0.282420i	0.989980	+	0.141210i	$0.0450994\pi$
$281$			22300.2i			0.282420i	−0.989980	+	0.141210i	$0.954901\pi$
$282$	−36666.6	+	59123.1i	−0.461076	+	0.743463i
$283$			117466.i			1.46669i	0.679854	+	0.733347i	$0.262043\pi$
$283$			117466.i			1.46669i	−0.679854	+	0.733347i	$0.737957\pi$
$284$			27863.2i			0.345457i
$285$	66217.9	+	20437.2i	0.815241	+	0.251612i
$286$	74880.0			0.915448
$287$	−31818.8			−0.386296
$288$	−25956.0	−	52315.9i	−0.312934	−	0.630738i
$289$	−6081.00			−0.0728080
$290$	14799.5	−	3869.88i	0.175975	−	0.0460153i
$291$	−19188.0	−	11899.9i	−0.226592	−	0.140526i
$292$			25331.9i			0.297100i
$293$	−57363.7			−0.668193			−0.334097	−	0.942539i	$-0.608431\pi$
$293$	−57363.7			−0.668193			−0.334097	+	0.942539i	$0.608431\pi$
$294$	32505.0	−	52412.7i	0.376059	−	0.606376i
$295$	−95940.0	+	25087.2i	−1.10244	+	0.288276i
$296$			89394.3i			1.02030i
$297$	6659.76	−	70213.5i	0.0754997	−	0.795990i
$298$		−	51245.1i		−	0.577059i
$299$			101391.i			1.13411i
$300$	1470.97	−	33717.9i	0.0163441	−	0.374644i
$301$	−39546.0			−0.436485
$302$	105215.			1.15363
$303$	−36998.6	−	22945.6i	−0.402996	−	0.249927i
$304$	−38192.0			−0.413262
$305$	−5869.19	−	22445.3i	−0.0630926	−	0.241283i
$306$	−31680.0	−	63853.1i	−0.338331	−	0.681929i
$307$			10417.3i			0.110530i	0.998472	+	0.0552648i	$0.0176003\pi$
$307$			10417.3i			0.110530i	−0.998472	+	0.0552648i	$0.982400\pi$
$308$	8879.68			0.0936043
$309$	42003.0	+	26049.2i	0.439910	+	0.272820i
$310$	640.000	+	2447.53i	0.00665973	+	0.0254686i
$311$		−	101971.i		−	1.05428i	−0.849777	−	0.527142i	$-0.823264\pi$
$311$		−	101971.i		−	1.05428i	0.849777	−	0.527142i	$-0.176736\pi$
$312$	130235.	+	80768.5i	1.33789	+	0.829722i
$313$		−	182586.i		−	1.86371i	−0.362832	−	0.931854i	$-0.618190\pi$
$313$		−	182586.i		−	1.86371i	0.362832	−	0.931854i	$-0.381810\pi$
$314$			92683.7i			0.940035i
$315$	20339.5	+	23363.4i	0.204984	+	0.235459i
$316$	−48.0000			−0.000480692
$317$	−33494.8			−0.333319			−0.166659	−	0.986015i	$-0.553298\pi$
$317$	−33494.8			−0.333319			−0.166659	+	0.986015i	$0.553298\pi$
$318$	21250.5	−	34265.4i	0.210143	−	0.338845i
$319$	18720.0			0.183960
$320$	26967.9	+	103132.i	0.263358	+	1.00715i
$321$	13875.0	−	22372.8i	0.134655	−	0.217125i
$322$		−	20039.1i		−	0.193271i
$323$	85710.4			0.821539
$324$	23814.0	−	31346.1i	0.226852	−	0.298602i
$325$	74880.0	+	133390.i	0.708923	+	1.26287i
$326$		−	38747.2i		−	0.364590i
$327$	−30130.2	+	48583.5i	−0.281777	+	0.454352i
$328$		−	144710.i		−	1.34509i
$329$		−	37392.8i		−	0.345458i
$330$	−20300.5	+	65775.3i	−0.186415	+	0.603997i
$331$	204092.			1.86282			0.931408	−	0.363977i	$-0.118581\pi$
$331$	204092.			1.86282			0.931408	+	0.363977i	$0.118581\pi$
$332$	26620.1			0.241509
$333$	−93236.6	+	46258.3i	−0.840810	+	0.417159i
$334$	−7970.00			−0.0714439
$335$	62527.7	−	16350.3i	0.557164	−	0.145692i
$336$	−14508.0	−	8997.48i	−0.128508	−	0.0796971i
$337$			44912.2i			0.395461i	0.980256	+	0.197731i	$0.0633572\pi$
$337$			44912.2i			0.395461i	−0.980256	+	0.197731i	$0.936643\pi$
$338$	−99115.3			−0.867575
$339$	−42060.0	+	67819.7i	−0.365991	+	0.590142i
$340$	−10560.0	−	40384.2i	−0.0913495	−	0.349345i
$341$			3095.91i			0.0266244i
$342$	−35063.3	−	70672.4i	−0.299779	−	0.604224i
$343$			69877.0i			0.593944i
$344$		−	179853.i		−	1.51985i
$345$	−89062.8	−	27487.9i	−0.748269	−	0.230942i
$346$	−155960.			−1.30275
$347$	−135260.			−1.12334			−0.561669	−	0.827362i	$-0.689841\pi$
$347$	−135260.			−1.12334			−0.561669	+	0.827362i	$0.689841\pi$
$348$	8879.68	+	5506.94i	0.0733227	+	0.0454728i
$349$	−228682.			−1.87751			−0.938753	−	0.344592i	$-0.888017\pi$
$349$	−228682.			−1.87751			−0.938753	+	0.344592i	$0.888017\pi$
$350$	−14799.5	−	26363.6i	−0.120812	−	0.215213i
$351$	−16848.0	+	177628.i	−0.136752	+	1.44177i
$352$			69754.6i			0.562973i
$353$	−106923.			−0.858067			−0.429034	−	0.903288i	$-0.641146\pi$
$353$	−106923.			−0.858067			−0.429034	+	0.903288i	$0.641146\pi$
$354$	95940.0	+	59499.5i	0.765585	+	0.474795i
$355$	112320.	−	29370.4i	0.891252	−	0.233052i
$356$			55726.3i			0.439704i
$357$	32558.8	+	20192.1i	0.255465	+	0.158433i
$358$			120235.i			0.938133i
$359$		−	176467.i		−	1.36922i	−0.728909	−	0.684611i	$-0.759972\pi$
$359$		−	176467.i		−	1.36922i	0.728909	−	0.684611i	$-0.240028\pi$
$360$	−106256.	+	92502.9i	−0.819873	+	0.713757i
$361$	−35457.0			−0.272074
$362$	64327.1			0.490881
$363$	25050.0	−	40391.9i	0.190105	−	0.306536i
$364$	−22464.0			−0.169545
$365$	102116.	−	26702.2i	0.766495	−	0.200429i
$366$	−13920.0	+	22445.3i	−0.103915	+	0.167557i
$367$		−	178042.i		−	1.32188i	−0.750440	−	0.660939i	$-0.770158\pi$
$367$		−	178042.i		−	1.32188i	0.750440	−	0.660939i	$-0.229842\pi$
$368$	51368.0			0.379313
$369$	150930.	−	74882.2i	1.10847	−	0.549954i
$370$	98280.0	−	25699.1i	0.717896	−	0.187721i
$371$			21671.3i			0.157448i
$372$	−910.736	+	1468.52i	−0.00658122	+	0.0106119i
$373$			77770.2i			0.558979i	0.960149	+	0.279490i	$0.0901653\pi$
$373$			77770.2i			0.558979i	−0.960149	+	0.279490i	$0.909835\pi$
$374$			85137.4i			0.608664i
$375$	−137472.	+	29612.1i	−0.977578	+	0.210575i
$376$	170060.			1.20289
$377$	−47358.3			−0.333206
$378$	3329.88	−	35106.7i	0.0233048	−	0.245701i
$379$	−12292.0			−0.0855745			−0.0427872	−	0.999084i	$-0.513624\pi$
$379$	−12292.0			−0.0855745			−0.0427872	+	0.999084i	$0.513624\pi$
$380$	−11687.8	−	44697.2i	−0.0809403	−	0.309537i
$381$	−29133.0	−	18067.5i	−0.200694	−	0.124465i
$382$			175610.i			1.20344i
$383$	246111.			1.67777			0.838886	−	0.544308i	$-0.183208\pi$
$383$	246111.			1.67777			0.838886	+	0.544308i	$0.183208\pi$
$384$	9240.00	−	14899.1i	0.0626628	−	0.101041i
$385$	−9360.00	−	35795.1i	−0.0631472	−	0.241492i
$386$		−	160407.i		−	1.07658i
$387$	187583.	−	93067.3i	1.25248	−	0.621406i
$388$			15052.3i			0.0999861i
$389$			99794.6i			0.659490i	0.944070	+	0.329745i	$0.106963\pi$
$389$			99794.6i			0.659490i	−0.944070	+	0.329745i	$0.893037\pi$
$390$	51356.8	−	166400.i	0.337651	−	1.09402i
$391$	−115280.			−0.754051
$392$	−150758.			−0.981091
$393$	61417.8	+	38089.7i	0.397657	+	0.246617i
$394$	−91280.0			−0.588008
$395$	50.5964	+	193.494i	0.000324284	+	0.00124015i
$396$	−42120.0	+	20897.4i	−0.268595	+	0.133260i
$397$		−	143548.i		−	0.910783i	−0.890291	−	0.455391i	$-0.849499\pi$
$397$		−	143548.i		−	0.910783i	0.890291	−	0.455391i	$-0.150501\pi$
$398$	108555.			0.685303
$399$	36036.0	+	22348.6i	0.226355	+	0.140380i
$400$	67580.0	−	37936.7i	0.422375	−	0.237104i
$401$			245738.i			1.52821i	0.645092	+	0.764105i	$0.276819\pi$
$401$			245738.i			1.52821i	−0.645092	+	0.764105i	$0.723181\pi$
$402$	−62527.7	−	38778.0i	−0.386919	−	0.239957i
$403$		−	7832.09i		−	0.0482245i
$404$			29024.1i			0.177826i
$405$	−151462.	−	62955.7i	−0.923409	−	0.383817i
$406$	9360.00			0.0567837
$407$	124315.			0.750475
$408$	−91832.5	+	148076.i	−0.551666	+	0.889535i
$409$	34808.0			0.208081			0.104041	−	0.994573i	$-0.466823\pi$
$409$	34808.0			0.208081			0.104041	+	0.994573i	$0.466823\pi$
$410$	−159094.	+	41601.2i	−0.946426	+	0.247479i
$411$	100860.	−	162632.i	0.597084	−	0.962769i
$412$		−	32949.9i		−	0.194115i
$413$	−60677.8			−0.355737
$414$	47160.0	+	95054.0i	0.275152	+	0.554587i
$415$	−28060.0	−	107309.i	−0.162926	−	0.623074i
$416$		−	176467.i		−	1.01971i
$417$	−91889.5	+	148167.i	−0.528437	+	0.852080i
$418$			94229.9i			0.539307i
$419$			219326.i			1.24928i	0.780911	+	0.624642i	$0.214755\pi$
$419$			219326.i			1.24928i	−0.780911	+	0.624642i	$0.785245\pi$
$420$	6090.16	−	19732.6i	0.0345247	−	0.111863i
$421$	168992.			0.953459			0.476729	−	0.879050i	$-0.341822\pi$
$421$	168992.			0.953459			0.476729	+	0.879050i	$0.341822\pi$
$422$	−105203.			−0.590747
$423$	87999.9	+	177369.i	0.491814	+	0.991284i
$424$	−98560.0			−0.548238
$425$	−151663.	+	85137.4i	−0.839656	+	0.471349i
$426$	−112320.	−	69657.9i	−0.618925	−	0.383841i
$427$		−	14195.7i		−	0.0778574i
$428$	−17550.6			−0.0958088
$429$	112320.	−	181111.i	0.610299	−	0.984077i
$430$	−197730.	+	51704.1i	−1.06939	+	0.279633i
$431$		−	154989.i		−	0.834345i	−0.908827	−	0.417173i	$-0.863021\pi$
$431$		−	154989.i		−	0.834345i	0.908827	−	0.417173i	$-0.136979\pi$
$432$	89992.1	+	8535.76i	0.482211	+	0.0457377i
$433$			153644.i			0.819481i	0.912202	+	0.409740i	$0.134381\pi$
$433$			153644.i			0.819481i	−0.912202	+	0.409740i	$0.865619\pi$
$434$			1547.95i			0.00821823i
$435$	12839.2	−	41599.9i	0.0678514	−	0.219844i
$436$	38112.0			0.200488
$437$	−127592.			−0.668127
$438$	−102116.	−	63329.8i	−0.532288	−	0.330111i
$439$	65168.0			0.338147			0.169073	−	0.985603i	$-0.445922\pi$
$439$	65168.0			0.338147			0.169073	+	0.985603i	$0.445922\pi$
$440$	162794.	−	42568.7i	0.840878	−	0.219880i
$441$	−78012.0	−	157238.i	−0.401129	−	0.808502i
$442$		−	215383.i		−	1.10247i
$443$	−116514.			−0.593706			−0.296853	−	0.954923i	$-0.595937\pi$
$443$	−116514.			−0.593706			−0.296853	+	0.954923i	$0.595937\pi$
$444$	58968.0	+	36570.4i	0.299123	+	0.185508i
$445$	224640.	−	58740.7i	1.13440	−	0.296633i
$446$			240368.i			1.20839i
$447$	−123945.	−	76867.7i	−0.620320	−	0.384706i
$448$			65226.7i			0.324989i
$449$		−	79090.7i		−	0.392313i	−0.980573	−	0.196157i	$-0.937154\pi$
$449$		−	79090.7i		−	0.392313i	0.980573	−	0.196157i	$-0.0628461\pi$
$450$	132244.	+	90224.5i	0.653056	+	0.445553i
$451$	−201240.			−0.989376
$452$	53202.2			0.260407
$453$	157823.	−	254482.i	0.769084	−	1.24011i
$454$	−53090.0			−0.257573
$455$	23679.1	+	90555.3i	0.114378	+	0.437412i
$456$	−101640.	+	163890.i	−0.488804	+	0.788173i
$457$			389402.i			1.86451i	0.361797	+	0.932257i	$0.382164\pi$
$457$			389402.i			1.86451i	−0.361797	+	0.932257i	$0.617836\pi$
$458$	127433.			0.607509
$459$	−201960.	−	19155.9i	−0.958606	−	0.0909238i
$460$	15720.0	+	60117.4i	0.0742911	+	0.284109i
$461$			236643.i			1.11351i	0.830678	+	0.556753i	$0.187953\pi$
$461$			236643.i			1.11351i	−0.830678	+	0.556753i	$0.812047\pi$
$462$	−22199.2	+	35795.1i	−0.104005	+	0.167703i
$463$		−	253793.i		−	1.18391i	−0.805971	−	0.591955i	$-0.798356\pi$
$463$		−	253793.i		−	1.18391i	0.805971	−	0.591955i	$-0.201644\pi$
$464$			23993.3i			0.111443i
$465$	6879.78	+	2123.34i	0.0318177	+	0.00982005i
$466$	275920.			1.27061
$467$	365853.			1.67754			0.838771	−	0.544485i	$-0.183275\pi$
$467$	365853.			1.67754			0.838771	+	0.544485i	$0.183275\pi$
$468$	106556.	−	52866.6i	0.486504	−	0.241374i
$469$	39546.0			0.179786
$470$	−48888.8	−	186964.i	−0.221316	−	0.846373i
$471$	224172.	+	139026.i	1.01051	+	0.626690i
$472$		−	275959.i		−	1.23868i
$473$	−250111.			−1.11792
$474$	120.000	−	193.494i	0.000534102	−	0.000861214i
$475$	−167860.	+	94229.9i	−0.743978	+	0.417639i
$476$		−	25541.2i		−	0.112727i
$477$	−51001.2	−	102796.i	−0.224153	−	0.451794i
$478$		−	280242.i		−	1.22653i
$479$		−	255606.i		−	1.11404i	−0.830500	−	0.557019i	$-0.811945\pi$
$479$		−	255606.i		−	1.11404i	0.830500	−	0.557019i	$-0.188055\pi$
$480$	155010.	+	47841.5i	0.672786	+	0.207645i
$481$	−314496.			−1.35933
$482$	228165.			0.982097
$483$	−48468.2	−	30058.7i	−0.207760	−	0.128848i
$484$	−31686.0			−0.135262
$485$	60677.8	−	15866.5i	0.257956	−	0.0674525i
$486$	66825.0	+	174362.i	0.282922	+	0.738211i
$487$			445925.i			1.88020i	0.340902	+	0.940099i	$0.389267\pi$
$487$			445925.i			1.88020i	−0.340902	+	0.940099i	$0.610733\pi$
$488$	64561.1			0.271101
$489$	−93717.0	−	58120.8i	−0.391923	−	0.243060i
$490$	43340.0	+	165744.i	0.180508	+	0.690311i
$491$		−	229387.i		−	0.951495i	−0.879582	−	0.475747i	$-0.842178\pi$
$491$		−	229387.i		−	0.951495i	0.879582	−	0.475747i	$-0.157822\pi$
$492$	−95456.5	−	59199.6i	−0.394344	−	0.244562i
$493$		−	53845.6i		−	0.221542i
$494$		−	238385.i		−	0.976843i
$495$	128638.	+	147763.i	0.525001	+	0.603054i
$496$	−3968.00			−0.0161290
$497$	71037.4			0.287590
$498$	−66550.1	+	107309.i	−0.268343	+	0.432690i
$499$	157748.			0.633524			0.316762	−	0.948505i	$-0.397404\pi$
$499$	157748.			0.633524			0.316762	+	0.948505i	$0.397404\pi$
$500$	65079.7	+	67481.1i	0.260319	+	0.269924i
$501$	−11955.0	+	19276.9i	−0.0476293	+	0.0767999i
$502$			50480.3i			0.200315i
$503$	−499630.			−1.97475			−0.987377	−	0.158390i	$-0.949370\pi$
$503$	−499630.			−1.97475			−0.987377	+	0.158390i	$0.949370\pi$
$504$	−77220.0	+	38311.8i	−0.303997	+	0.150825i
$505$	117000.	−	30594.1i	0.458779	−	0.119965i
$506$		−	126739.i		−	0.495003i
$507$	−148673.	+	239728.i	−0.578384	+	0.932615i
$508$			22853.8i			0.0885587i
$509$			153441.i			0.592251i	0.955149	+	0.296125i	$0.0956946\pi$
$509$			153441.i			0.592251i	−0.955149	+	0.296125i	$0.904305\pi$
$510$	189194.	+	58391.9i	0.727390	+	0.224498i
$511$	64584.0			0.247334
$512$	−227431.			−0.867580
$513$	−223529.	−	21201.7i	−0.849373	−	0.0805631i
$514$	268600.			1.01667
$515$	−132825.	+	34732.2i	−0.500802	+	0.130954i
$516$	−118638.	−	73576.2i	−0.445579	−	0.276336i
$517$		−	236493.i		−	0.884782i
$518$	62157.7			0.231652
$519$	−233940.	+	377217.i	−0.868500	+	1.40041i
$520$	−411840.	+	107691.i	−1.52308	+	0.398267i
$521$		−	242642.i		−	0.893902i	−0.894558	−	0.446951i	$-0.852510\pi$
$521$		−	242642.i		−	0.893902i	0.894558	−	0.446951i	$-0.147490\pi$
$522$	−44398.4	+	22027.8i	−0.162939	+	0.0808406i
$523$		−	133865.i		−	0.489398i	−0.969599	−	0.244699i	$-0.921311\pi$
$523$		−	133865.i		−	0.489398i	0.969599	−	0.244699i	$-0.0786892\pi$
$524$		−	48180.1i		−	0.175471i
$525$	−85964.2	−	3750.26i	−0.311888	−	0.0136064i
$526$	9010.00			0.0325652
$527$	8904.97			0.0320635
$528$	−91756.6	−	56905.1i	−0.329132	−	0.204119i
$529$	−108231.			−0.386759
$530$	28334.0	+	108357.i	0.100869	+	0.385748i
$531$	287820.	−	142799.i	1.02078	−	0.506448i
$532$		−	28269.0i		−	0.0998819i
$533$	509101.			1.79205
$534$	−224640.	−	139316.i	−0.787779	−	0.488560i
$535$	18500.0	+	70748.9i	0.0646345	+	0.247179i
$536$			179853.i			0.626019i
$537$	290809.	+	180352.i	1.00846	+	0.625422i
$538$		−	12390.6i		−	0.0428083i
$539$			209651.i			0.721638i
$540$	17550.4	+	107932.i	0.0601864	+	0.370139i
$541$	−195478.			−0.667888			−0.333944	−	0.942593i	$-0.608380\pi$
$541$	−195478.			−0.667888			−0.333944	+	0.942593i	$0.608380\pi$
$542$	−212024.			−0.721751
$543$	96490.6	−	155586.i	0.327254	−	0.527682i
$544$	200640.			0.677984
$545$	−40173.6	−	153634.i	−0.135253	−	0.517244i
$546$	56160.0	−	90555.3i	0.188383	−	0.303759i
$547$		−	143165.i		−	0.478479i	−0.970961	−	0.239239i	$-0.923102\pi$
$547$		−	143165.i		−	0.478479i	0.970961	−	0.239239i	$-0.0768981\pi$
$548$	−127579.			−0.424833
$549$	33408.0	+	67336.0i	0.110842	+	0.223410i
$550$	−93600.0	−	166738.i	−0.309421	−	0.551200i
$551$		−	59596.2i		−	0.196298i
$552$	136705.	−	220431.i	0.448649	−	0.723425i
$553$			122.376i			0.000400173i
$554$			32700.5i			0.106545i
$555$	85262.3	−	276256.i	0.276803	−	0.896863i
$556$	116232.			0.375990
$557$	401799.			1.29509			0.647543	−	0.762029i	$-0.275797\pi$
$557$	401799.			1.29509			0.647543	+	0.762029i	$0.275797\pi$
$558$	−3642.94	−	7342.59i	−0.0117000	−	0.0235820i
$559$	632736.			2.02488
$560$	45878.3	−	11996.6i	0.146296	−	0.0382546i
$561$	205920.	+	127706.i	0.654294	+	0.405776i
$562$			70519.4i			0.223273i
$563$	126178.			0.398077			0.199038	−	0.979992i	$-0.436218\pi$
$563$	126178.			0.398077			0.199038	+	0.979992i	$0.436218\pi$
$564$	69570.0	−	112178.i	0.218708	−	0.352655i
$565$	−56080.0	−	214465.i	−0.175675	−	0.671829i
$566$			371460.i			1.15952i
$567$	−79917.1	−	60714.0i	−0.248584	−	0.188853i
$568$			323074.i			1.00139i
$569$			355497.i			1.09802i	0.835815	+	0.549012i	$0.184996\pi$
$569$			355497.i			1.09802i	−0.835815	+	0.549012i	$0.815004\pi$
$570$	209399.	+	64628.0i	0.644504	+	0.198916i
$571$	−454108.			−1.39279			−0.696397	−	0.717657i	$-0.745215\pi$
$571$	−454108.			−1.39279			−0.696397	+	0.717657i	$0.745215\pi$
$572$	−142075.			−0.434235
$573$	424744.	+	263415.i	1.29366	+	0.802291i
$574$	−100620.			−0.305394
$575$	225771.	−	126739.i	0.682861	−	0.383331i
$576$	−153504.	−	309397.i	−0.462674	−	0.932548i
$577$		−	103163.i		−	0.309866i	−0.987925	−	0.154933i	$-0.950484\pi$
$577$		−	103163.i		−	0.309866i	0.987925	−	0.154933i	$-0.0495162\pi$
$578$	−19229.8			−0.0575598
$579$	−387972.	−	240610.i	−1.15729	−	0.717723i
$580$	−28080.0	+	7342.59i	−0.0834721	+	0.0218270i
$581$		−	67868.1i		−	0.201054i
$582$	−60677.8	−	37630.8i	−0.179136	−	0.111096i
$583$			137062.i			0.403254i
$584$			293724.i			0.861220i
$585$	−325432.	−	373815.i	−0.950931	−	1.09231i
$586$	−181400.			−0.528253
$587$	−474661.			−1.37755			−0.688775	−	0.724975i	$-0.741851\pi$
$587$	−474661.			−1.37755			−0.688775	+	0.724975i	$0.741851\pi$
$588$	−61673.9	+	99446.2i	−0.178380	+	0.287630i
$589$	9856.00			0.0284099
$590$	−303389.	+	79332.6i	−0.871557	+	0.227902i
$591$	−136920.	+	220777.i	−0.392005	+	0.632090i
$592$			159334.i			0.454638i
$593$	−173103.			−0.492261			−0.246130	−	0.969237i	$-0.579159\pi$
$593$	−173103.			−0.492261			−0.246130	+	0.969237i	$0.579159\pi$
$594$	21060.0	−	222035.i	0.0596878	−	0.629285i
$595$	−102960.	+	26922.8i	−0.290827	+	0.0760478i
$596$			97230.8i			0.273723i
$597$	162832.	−	262559.i	0.456868	−	0.736678i
$598$			320626.i			0.896596i
$599$		−	587255.i		−	1.63671i	−0.574710	−	0.818357i	$-0.694885\pi$
$599$		−	587255.i		−	1.63671i	0.574710	−	0.818357i	$-0.305115\pi$
$600$	17055.9	−	390960.i	0.0473776	−	1.08600i
$601$	475352.			1.31603			0.658016	−	0.753004i	$-0.271396\pi$
$601$	475352.			1.31603			0.658016	+	0.753004i	$0.271396\pi$
$602$	−125055.			−0.345072
$603$	−187583.	+	93067.3i	−0.515892	+	0.255954i
$604$	−199632.			−0.547213
$605$	33400.0	+	127730.i	0.0912505	+	0.348966i
$606$	−117000.	−	72560.3i	−0.318596	−	0.197585i
$607$			311310.i			0.844921i	0.906381	+	0.422461i	$0.138834\pi$
$607$			311310.i			0.844921i	−0.906381	+	0.422461i	$0.861166\pi$
$608$	222068.			0.600729
$609$	14040.0	−	22638.8i	0.0378558	−	0.0610407i
$610$	−18560.0	−	70978.4i	−0.0498791	−	0.190751i
$611$			598284.i			1.60260i
$612$	60108.6	+	121153.i	0.160485	+	0.323467i
$613$			224316.i			0.596952i	0.954417	+	0.298476i	$0.0964783\pi$
$613$			224316.i			0.596952i	−0.954417	+	0.298476i	$0.903522\pi$
$614$			32942.4i			0.0873813i
$615$	−138021.	+	447199.i	−0.364918	+	1.18236i
$616$	102960.			0.271336
$617$	−200463.			−0.526580			−0.263290	−	0.964717i	$-0.584808\pi$
$617$	−200463.			−0.526580			−0.263290	+	0.964717i	$0.584808\pi$
$618$	132825.	+	82374.7i	0.347779	+	0.215683i
$619$	22508.0			0.0587429			0.0293715	−	0.999569i	$-0.490649\pi$
$619$	22508.0			0.0587429			0.0293715	+	0.999569i	$0.490649\pi$
$620$	−1214.31	−	4643.86i	−0.00315899	−	0.0120808i
$621$	300645.	+	28516.2i	0.779598	+	0.0739450i
$622$		−	322462.i		−	0.833485i
$623$	142075.			0.366050
$624$	232128.	+	143960.i	0.596154	+	0.369719i
$625$	203425.	−	333476.i	0.520768	−	0.853698i
$626$		−	577387.i		−	1.47339i
$627$	227912.	+	141345.i	0.579738	+	0.359538i
$628$		−	175855.i		−	0.445898i
$629$		−	357577.i		−	0.903792i
$630$	64319.2	+	73881.7i	0.162054	+	0.186147i
$631$	−157408.			−0.395338			−0.197669	−	0.980269i	$-0.563337\pi$
$631$	−157408.			−0.395338			−0.197669	+	0.980269i	$0.563337\pi$
$632$	−556.561			−0.00139341
$633$	−157804.	+	254451.i	−0.393832	+	0.635034i
$634$	−105920.			−0.263511
$635$	92126.6	−	24090.0i	0.228475	−	0.0597434i
$636$	−40320.0	+	65014.0i	−0.0996796	+	0.160729i
$637$		−	530380.i		−	1.30710i
$638$	59197.8			0.145434
$639$	−336960.	+	167179.i	−0.825233	+	0.409430i
$640$	12320.0	+	47114.9i	0.0300781	+	0.115027i
$641$		−	297836.i		−	0.724871i	−0.932009	−	0.362436i	$-0.881945\pi$
$641$		−	297836.i		−	0.724871i	0.932009	−	0.362436i	$-0.118055\pi$
$642$	43876.6	−	70748.9i	0.106454	−	0.171652i
$643$		−	130805.i		−	0.316376i	−0.987409	−	0.158188i	$-0.949435\pi$
$643$		−	130805.i		−	0.316376i	0.987409	−	0.158188i	$-0.0505651\pi$
$644$			38021.6i			0.0916767i
$645$	−171540.	+	555801.i	−0.412330	+	1.33598i
$646$	271040.			0.649484
$647$	−117520.			−0.280739			−0.140369	−	0.990099i	$-0.544829\pi$
$647$	−117520.			−0.280739			−0.140369	+	0.990099i	$0.544829\pi$
$648$	276124.	−	363458.i	0.657588	−	0.865574i
$649$	−383760.			−0.911109
$650$	236791.	+	421817.i	0.560453	+	0.998384i
$651$	3744.00	+	2321.93i	0.00883433	+	0.00547882i
$652$			73517.7i			0.172940i
$653$	640829.			1.50285			0.751426	−	0.659818i	$-0.229367\pi$
$653$	640829.			1.50285			0.751426	+	0.659818i	$0.229367\pi$
$654$	−95280.0	+	153634.i	−0.222765	+	0.359197i
$655$	−194220.	+	50786.2i	−0.452701	+	0.118376i
$656$		−	257928.i		−	0.599364i
$657$	−306349.	+	151992.i	−0.709717	+	0.352118i
$658$		−	118246.i		−	0.273109i
$659$		−	121805.i		−	0.280474i	−0.990118	−	0.140237i	$-0.955214\pi$
$659$		−	121805.i		−	0.280474i	0.990118	−	0.140237i	$-0.0447865\pi$
$660$	38517.6	−	124800.i	0.0884242	−	0.286501i
$661$	5072.00			0.0116085			0.00580425	−	0.999983i	$-0.498152\pi$
$661$	5072.00			0.0116085			0.00580425	+	0.999983i	$0.498152\pi$
$662$	645396.			1.47269
$663$	−520941.	−	323074.i	−1.18512	−	0.734979i
$664$	308660.			0.700074
$665$	−113956.	+	29798.1i	−0.257687	+	0.0673822i
$666$	−294840.	+	146282.i	−0.664719	+	0.329793i
$667$			80156.6i			0.180172i
$668$	15122.0			0.0338888
$669$	581373.	+	360552.i	1.29898	+	0.805593i
$670$	197730.	−	51704.1i	0.440477	−	0.115179i
$671$		−	89781.3i		−	0.199407i
$672$	84356.9	+	52315.9i	0.186802	+	0.115850i
$673$			181790.i			0.401366i	0.979656	+	0.200683i	$0.0643161\pi$
$673$			181790.i			0.401366i	−0.979656	+	0.200683i	$0.935684\pi$
$674$			142025.i			0.312640i
$675$	416590.	−	184519.i	0.914326	−	0.404979i
$676$	188058.			0.411527
$677$	−304666.			−0.664733			−0.332367	−	0.943150i	$-0.607847\pi$
$677$	−304666.			−0.664733			−0.332367	+	0.943150i	$0.607847\pi$
$678$	−133005.	+	214465.i	−0.289341	+	0.466548i
$679$	38376.0			0.0832377
$680$	−122443.	−	468256.i	−0.264800	−	1.01266i
$681$	−79635.0	+	128408.i	−0.171716	+	0.276883i
$682$			9790.12i			0.0210484i
$683$	−627816.			−1.34583			−0.672917	−	0.739718i	$-0.734959\pi$
$683$	−627816.			−1.34583			−0.672917	+	0.739718i	$0.734959\pi$
$684$	66528.0	+	134091.i	0.142198	+	0.286608i
$685$	134480.	+	514287.i	0.286600	+	1.09604i
$686$			220970.i			0.469554i
$687$	191150.	−	308220.i	0.405006	−	0.653052i
$688$		−	320565.i		−	0.677235i
$689$		−	346742.i		−	0.730411i
$690$	−281641.	−	86924.3i	−0.591559	−	0.182576i
$691$	−17308.0			−0.0362486			−0.0181243	−	0.999836i	$-0.505769\pi$
$691$	−17308.0			−0.0362486			−0.0181243	+	0.999836i	$0.505769\pi$
$692$	295913.			0.617949
$693$	53278.1	+	107385.i	0.110938	+	0.223603i
$694$	−427730.			−0.888077
$695$	−122519.	−	468546.i	−0.253650	−	0.970025i
$696$	102960.	+	63853.1i	0.212545	+	0.131815i
$697$			578841.i			1.19150i
$698$	−723156.			−1.48430
$699$	413880.	−	667361.i	0.847072	−	1.36586i
$700$	28080.0	+	50021.4i	0.0573061	+	0.102084i
$701$			23944.9i			0.0487278i	0.999703	+	0.0243639i	$0.00775604\pi$
$701$			23944.9i			0.0487278i	−0.999703	+	0.0243639i	$0.992244\pi$
$702$	−53278.1	+	561708.i	−0.108112	+	1.13982i
$703$		−	395765.i		−	0.800806i
$704$			412530.i			0.832357i
$705$	−525538.	−	162199.i	−1.05737	−	0.326340i
$706$	−338120.			−0.678362
$707$	73997.3			0.148039
$708$	−182033.	−	112892.i	−0.363149	−	0.225215i
$709$	−381322.			−0.758577			−0.379288	−	0.925279i	$-0.623831\pi$
$709$	−381322.			−0.758577			−0.379288	+	0.925279i	$0.623831\pi$
$710$	355187.	−	92877.2i	0.704596	−	0.184244i
$711$	−288.000	−	580.483i	−0.000569709	−	0.00114829i
$712$			646148.i			1.27459i
$713$	−13256.3			−0.0260761
$714$	102960.	+	63853.1i	0.201963	+	0.125252i
$715$	149760.	+	572722.i	0.292943	+	1.12029i
$716$		−	228130.i		−	0.444996i
$717$	−677815.	−	420363.i	−1.31848	−	0.817686i
$718$		−	558037.i		−	1.08247i
$719$			927031.i			1.79323i	0.442810	+	0.896616i	$0.353982\pi$
$719$			927031.i			1.79323i	−0.442810	+	0.896616i	$0.646018\pi$
$720$	−189387.	+	164875.i	−0.365330	+	0.318046i
$721$	−84006.0			−0.161599
$722$	−112125.			−0.215094
$723$	342247.	−	551857.i	0.654731	−	1.05572i
$724$	−122052.			−0.232845
$725$	59197.8	+	105454.i	0.112624	+	0.200627i
$726$	79215.0	−	127730.i	0.150291	−	0.242338i
$727$		−	373998.i		−	0.707620i	−0.935317	−	0.353810i	$-0.884886\pi$
$727$		−	373998.i		−	0.707620i	0.935317	−	0.353810i	$-0.115114\pi$
$728$	−260470.			−0.491469
$729$	521964.	+	99915.6i	0.982167	+	0.188009i
$730$	322920.	−	84439.8i	0.605967	−	0.158453i
$731$			719411.i			1.34630i
$732$	26411.3	−	42587.0i	0.0492911	−	0.0794795i
$733$			318240.i			0.592307i	0.955140	+	0.296153i	$0.0957040\pi$
$733$			318240.i			0.592307i	−0.955140	+	0.296153i	$0.904296\pi$
$734$		−	563020.i		−	1.04504i
$735$	465890.	+	143790.i	0.862401	+	0.266167i
$736$	−298680.			−0.551379
$737$	250111.			0.460466
$738$	477283.	−	236798.i	0.876320	−	0.434777i
$739$	409268.			0.749409			0.374705	−	0.927144i	$-0.377744\pi$
$739$	409268.			0.749409			0.374705	+	0.927144i	$0.377744\pi$
$740$	−186473.	+	48760.5i	−0.340528	+	0.0890441i
$741$	−576576.	−	357577.i	−1.05007	−	0.651229i
$742$			68530.8i			0.124474i
$743$	−11400.0			−0.0206504			−0.0103252	−	0.999947i	$-0.503287\pi$
$743$	−11400.0			−0.0206504			−0.0103252	+	0.999947i	$0.503287\pi$
$744$	−10560.0	+	17027.5i	−0.0190773	+	0.0307613i
$745$	391950.	−	102490.i	0.706184	−	0.184659i
$746$			245931.i			0.441912i
$747$	159720.	+	321927.i	0.286233	+	0.576920i
$748$		−	161537.i		−	0.288715i
$749$			44745.5i			0.0797602i
$750$	−434724.	+	93641.8i	−0.772843	+	0.166474i
$751$	347432.			0.616013			0.308007	−	0.951384i	$-0.400338\pi$
$751$	347432.			0.616013			0.308007	+	0.951384i	$0.400338\pi$
$752$	303111.			0.536001
$753$	122096.	+	75720.4i	0.215333	+	0.133544i
$754$	−149760.			−0.263423
$755$	210431.	+	804742.i	0.369160	+	1.41177i
$756$	−6318.00	+	66610.4i	−0.0110544	+	0.116546i
$757$		−	886067.i		−	1.54623i	−0.634265	−	0.773116i	$-0.718697\pi$
$757$		−	886067.i		−	1.54623i	0.634265	−	0.773116i	$-0.281303\pi$
$758$	−38870.7			−0.0676525
$759$	−306540.	−	190108.i	−0.532113	−	0.330002i
$760$	−135520.	−	518264.i	−0.234626	−	0.897272i
$761$		−	261604.i		−	0.451726i	−0.974159	−	0.225863i	$-0.927480\pi$
$761$		−	261604.i		−	0.451726i	0.974159	−	0.225863i	$-0.0725202\pi$
$762$	−92126.6	−	57134.5i	−0.158663	−	0.0983985i
$763$		−	97166.9i		−	0.166905i
$764$		−	333197.i		−	0.570840i
$765$	425022.	−	370012.i	0.726254	−	0.632255i
$766$	778270.			1.32639
$767$	970845.			1.65029
$768$	−294395.	+	474698.i	−0.499124	+	0.804814i
$769$	−832882.			−1.40842			−0.704208	−	0.709994i	$-0.748698\pi$
$769$	−832882.			−1.40842			−0.704208	+	0.709994i	$0.748698\pi$
$770$	−29598.9	−	113194.i	−0.0499223	−	0.190916i
$771$	402900.	−	649657.i	0.677780	−	1.09289i
$772$			304350.i			0.510669i
$773$	38529.2			0.0644809			0.0322404	−	0.999480i	$-0.489736\pi$
$773$	38529.2			0.0644809			0.0322404	+	0.999480i	$0.489736\pi$
$774$	593190.	−	294305.i	0.990175	−	0.491264i
$775$	−17440.0	+	9790.12i	−0.0290364	+	0.0162999i
$776$			174532.i			0.289835i
$777$	93236.6	−	150339.i	0.154435	−	0.249018i
$778$			315578.i			0.521372i
$779$			640659.i			1.05573i
$780$	−97442.6	+	315721.i	−0.160162	+	0.518937i
$781$	449280.			0.736572
$782$	−364547.			−0.596129
$783$	−13319.5	+	140427.i	−0.0217253	+	0.229048i
$784$	−268708.			−0.437168
$785$	−708894.	+	185367.i	−1.15038	+	0.300811i
$786$	194220.	+	120450.i	0.314376	+	0.194967i
$787$		−	627164.i		−	1.01259i	−0.862362	−	0.506293i	$-0.831016\pi$
$787$		−	627164.i		−	1.01259i	0.862362	−	0.506293i	$-0.168984\pi$
$788$	173192.			0.278917
$789$	13515.0	−	21792.3i	0.0217101	−	0.0350065i
$790$	160.000	+	611.882i	0.000256369	+	0.000980424i
$791$		−	135639.i		−	0.216787i
$792$	−488382.	+	242305.i	−0.778591	+	0.386289i
$793$			227131.i			0.361185i
$794$		−	453937.i		−	0.720037i
$795$	304581.	+	94004.3i	0.481913	+	0.148735i
$796$	−205968.			−0.325068
$797$	−1.09025e6			−1.71637			−0.858184	−	0.513343i	$-0.828407\pi$
$797$	−1.09025e6			−1.71637			−0.858184	+	0.513343i	$0.828407\pi$
$798$	113956.	+	70672.4i	0.178950	+	0.110980i
$799$	−680240.			−1.06554
$800$	−392945.	+	220583.i	−0.613976	+	0.344662i
$801$	−673920.	+	334358.i	−1.05037	+	0.521131i
$802$			777091.i			1.20816i
$803$	408465.			0.633467
$804$	118638.	+	73576.2i	0.183532	+	0.113822i
$805$	153270.	−	40078.3i	0.236519	−	0.0618468i
$806$		−	24767.3i		−	0.0381248i
$807$	−29968.9	−	18585.9i	−0.0460176	−	0.0285389i
$808$			336535.i			0.515475i
$809$		−	101004.i		−	0.154327i	−0.997018	−	0.0771634i	$-0.975414\pi$
$809$		−	101004.i		−	0.154327i	0.997018	−	0.0771634i	$-0.0245863\pi$
$810$	−478965.	−	199083.i	−0.730019	−	0.303434i
$811$	−400948.			−0.609602			−0.304801	−	0.952416i	$-0.598590\pi$
$811$	−400948.			−0.609602			−0.304801	+	0.952416i	$0.598590\pi$
$812$	−17759.4			−0.0269349
$813$	−318037.	+	512819.i	−0.481167	+	0.775859i
$814$	393120.			0.593303
$815$	296359.	−	77494.4i	0.446173	−	0.116669i
$816$	−163680.	+	263926.i	−0.245819	+	0.396371i
$817$			796242.i			1.19289i
$818$	110073.			0.164502
$819$	−134784.	−	271666.i	−0.200942	−	0.405011i
$820$	301860.	−	78932.8i	0.448929	−	0.117390i
$821$		−	777895.i		−	1.15408i	−0.816717	−	0.577038i	$-0.804208\pi$
$821$		−	777895.i		−	1.15408i	0.816717	−	0.577038i	$-0.195792\pi$
$822$	318947.	−	514287.i	0.472036	−	0.761136i
$823$		−	218457.i		−	0.322528i	−0.986911	−	0.161264i	$-0.948443\pi$
$823$		−	218457.i		−	0.322528i	0.986911	−	0.161264i	$-0.0515570\pi$
$824$		−	382054.i		−	0.562692i
$825$	−543685.	−	23718.7i	−0.798803	−	0.0348484i
$826$	−191880.			−0.281235
$827$	−460867.			−0.673852			−0.336926	−	0.941531i	$-0.609387\pi$
$827$	−460867.			−0.673852			−0.336926	+	0.941531i	$0.609387\pi$
$828$	−89479.8	−	180352.i	−0.130516	−	0.263064i
$829$	−60352.0			−0.0878178			−0.0439089	−	0.999036i	$-0.513981\pi$
$829$	−60352.0			−0.0878178			−0.0439089	+	0.999036i	$0.513981\pi$
$830$	−88733.5	−	339340.i	−0.128805	−	0.492583i
$831$	79092.0	+	49050.8i	0.114533	+	0.0710303i
$832$		−	1.04363e6i		−	1.50764i
$833$	603034.			0.869064
$834$	−290580.	+	468546.i	−0.417767	+	0.673628i
$835$	−15940.0	−	60958.8i	−0.0228621	−	0.0874306i
$836$		−	178789.i		−	0.255816i
$837$	−23223.8	−	2202.78i	−0.0331499	−	0.00314427i
$838$			693569.i			0.987646i
$839$		−	162148.i		−	0.230350i	−0.993345	−	0.115175i	$-0.963257\pi$
$839$		−	162148.i		−	0.230350i	0.993345	−	0.115175i	$-0.0367429\pi$
$840$	70615.5	−	228800.i	0.100079	−	0.324263i
$841$	669841.			0.947065
$842$	534400.			0.753775
$843$	170564.	+	105779.i	0.240011	+	0.148849i
$844$	199608.			0.280216
$845$	−198231.	−	758086.i	−0.277624	−	1.06171i
$846$	278280.	+	560891.i	0.388813	+	0.783679i
$847$			80783.8i			0.112605i
$848$	−175671.			−0.244291
$849$	898443.	+	557191.i	1.24645	+	0.773016i
$850$	−479600.	+	269228.i	−0.663806	+	0.372634i
$851$			532303.i			0.735020i
$852$	213112.	+	132167.i	0.293582	+	0.182072i
$853$			1.27675e6i			1.75472i	0.479828	+	0.877362i	$0.340699\pi$
$853$			1.27675e6i			1.75472i	−0.479828	+	0.877362i	$0.659301\pi$
$854$		−	44890.7i		−	0.0615517i
$855$	470413.	−	409528.i	0.643498	−	0.560210i
$856$	−203500.			−0.277726
$857$	311105.			0.423589			0.211795	−	0.977314i	$-0.432069\pi$
$857$	311105.			0.423589			0.211795	+	0.977314i	$0.432069\pi$
$858$	355187.	−	572722.i	0.482484	−	0.777981i
$859$	1.14631e6			1.55351			0.776757	−	0.629801i	$-0.216863\pi$
$859$	1.14631e6			1.55351			0.776757	+	0.629801i	$0.216863\pi$
$860$	375166.	−	98101.6i	0.507256	−	0.132641i
$861$	−150930.	+	243367.i	−0.203596	+	0.328289i
$862$		−	490118.i		−	0.659608i
$863$	640839.			0.860453			0.430227	−	0.902721i	$-0.358434\pi$
$863$	640839.			0.860453			0.430227	+	0.902721i	$0.358434\pi$
$864$	−523260.	−	49631.3i	−0.700955	−	0.0664856i
$865$	−311920.	−	1.19286e6i	−0.416880	−	1.59426i
$866$			485864.i			0.647857i
$867$	−28844.7	+	46510.7i	−0.0383732	+	0.0618749i
$868$		−	2937.04i		−	0.00389825i
$869$			773.977i			0.00102492i
$870$	40601.1	−	131551.i	0.0536413	−	0.173802i
$871$	−632736.			−0.834039
$872$	441909.			0.581166
$873$	−182033.	+	90313.8i	−0.238848	+	0.118502i
$874$	−403480.			−0.528201
$875$	172044.	−	165921.i	0.224710	−	0.216713i
$876$	193752.	+	120160.i	0.252486	+	0.156585i
$877$		−	447959.i		−	0.582424i	−0.956659	−	0.291212i	$-0.905942\pi$
$877$		−	447959.i		−	0.582424i	0.956659	−	0.291212i	$-0.0940585\pi$
$878$	206079.			0.267329
$879$	−272100.	+	438748.i	−0.352169	+	0.567855i
$880$	290160.	−	75873.4i	0.374690	−	0.0979770i
$881$		−	72995.7i		−	0.0940471i	−0.998894	−	0.0470235i	$-0.985026\pi$
$881$		−	72995.7i		−	0.0940471i	0.998894	−	0.0470235i	$-0.0149736\pi$
$882$	−246696.	−	497231.i	−0.317120	−	0.639177i
$883$		−	797910.i		−	1.02337i	−0.859173	−	0.511685i	$-0.829022\pi$
$883$		−	797910.i		−	1.02337i	0.859173	−	0.511685i	$-0.170978\pi$
$884$			408660.i			0.522947i
$885$	−263203.	+	852799.i	−0.336051	+	1.08883i
$886$	−368450.			−0.469365
$887$	−1.04890e6			−1.33317			−0.666586	−	0.745428i	$-0.732245\pi$
$887$	−1.04890e6			−1.33317			−0.666586	+	0.745428i	$0.732245\pi$
$888$	683735.	+	424034.i	0.867085	+	0.537744i
$889$	58266.0			0.0737245
$890$	710374.	−	185754.i	0.896824	−	0.234509i
$891$	−505440.	−	383989.i	−0.636670	−	0.483686i
$892$		−	456067.i		−	0.573190i
$893$	−752888.			−0.944120
$894$	−391950.	−	243077.i	−0.490406	−	0.304137i
$895$	−919620.	+	240470.i	−1.14805	+	0.300203i
$896$			29798.1i			0.0371170i
$897$	775492.	+	480940.i	0.963812	+	0.597731i
$898$		−	250107.i		−	0.310151i
$899$		−	6191.81i		−	0.00766123i
$900$	−250915.	−	171189.i	−0.309772	−	0.211344i
$901$	394240.			0.485636
$902$	−636377.			−0.782170
$903$	−187583.	+	302469.i	−0.230048	+	0.370941i
$904$	616880.			0.754856
$905$	128654.	+	492007.i	0.157082	+	0.600723i
$906$	499080.	−	804742.i	0.608014	−	0.980394i
$907$			1.06515e6i			1.29478i	0.762159	+	0.647390i	$0.224139\pi$
$907$			1.06515e6i			1.29478i	−0.762159	+	0.647390i	$0.775861\pi$
$908$	100731.			0.122178
$909$	−351000.	+	174145.i	−0.424795	+	0.210757i
$910$	74880.0	+	286361.i	0.0904239	+	0.345805i
$911$		−	1.25520e6i		−	1.51243i	−0.654324	−	0.756215i	$-0.727047\pi$
$911$		−	1.25520e6i		−	1.51243i	0.654324	−	0.756215i	$-0.272953\pi$
$912$	−181161.	+	292113.i	−0.217808	+	0.351205i
$913$		−	429235.i		−	0.514937i
$914$			1.23140e6i			1.47403i
$915$	−199514.	−	61576.9i	−0.238304	−	0.0735488i
$916$	−241788.			−0.288167
$917$	−122836.			−0.146078
$918$	−638654.	−	60576.4i	−0.757844	−	0.0718816i
$919$	552368.			0.654030			0.327015	−	0.945019i	$-0.393957\pi$
$919$	552368.			0.654030			0.327015	+	0.945019i	$0.393957\pi$
$920$	182274.	+	697063.i	0.215352	+	0.823562i
$921$	79677.0	+	49413.6i	0.0939320	+	0.0582542i
$922$			748332.i			0.880304i
$923$	−1.13660e6			−1.33415
$924$	42120.0	−	67916.5i	0.0493338	−	0.0795483i
$925$	393120.	+	700299.i	0.459454	+	0.818465i
$926$		−	802566.i		−	0.935963i
$927$	398475.	−	197699.i	0.463705	−	0.230062i
$928$		−	139509.i		−	0.161997i
$929$		−	748290.i		−	0.867039i	−0.901144	−	0.433520i	$-0.857271\pi$
$929$		−	748290.i		−	0.867039i	0.901144	−	0.433520i	$-0.142729\pi$
$930$	21755.8	+	6714.59i	0.0251541	+	0.00776343i
$931$	667436.			0.770035
$932$	−523521.			−0.602702
$933$	−779932.	−	483693.i	−0.895969	−	0.555657i
$934$	1.15693e6			1.32621
$935$	−651176.	+	170275.i	−0.744861	+	0.194772i
$936$	1.23552e6	−	612990.i	1.41026	−	0.699683i
$937$			173102.i			0.197161i	0.995129	+	0.0985807i	$0.0314302\pi$
$937$			173102.i			0.197161i	−0.995129	+	0.0985807i	$0.968570\pi$
$938$	125055.			0.142134
$939$	−1.39651e6	−	866080.i	−1.58385	−	0.982261i
$940$	92760.0	+	354739.i	0.104980	+	0.401470i
$941$			1.74861e6i			1.97475i	0.158388	+	0.987377i	$0.449370\pi$
$941$			1.74861e6i			1.97475i	−0.158388	+	0.987377i	$0.550630\pi$
$942$	708894.	+	439637.i	0.798876	+	0.495442i
$943$		−	861683.i		−	0.969001i
$944$		−	491862.i		−	0.551950i
$945$	275175.	−	44744.8i	0.308138	−	0.0501048i
$946$	−790920.			−0.883792
$947$	983427.			1.09658			0.548292	−	0.836287i	$-0.315278\pi$
$947$	983427.			1.09658			0.548292	+	0.836287i	$0.315278\pi$
$948$	−227.684	+	367.129i	−0.000253347	+	0.000408510i
$949$	−1.03334e6			−1.14739
$950$	−530820.	+	297981.i	−0.588166	+	0.330173i
$951$	−158880.	+	256186.i	−0.175674	+	0.283266i
$952$		−	296151.i		−	0.326768i
$953$	894507.			0.984913			0.492457	−	0.870337i	$-0.336099\pi$
$953$	894507.			0.984913			0.492457	+	0.870337i	$0.336099\pi$
$954$	−161280.	−	325070.i	−0.177208	−	0.357175i
$955$	−1.34316e6	+	351220.i	−1.47272	+	0.385100i
$956$			531722.i			0.581793i
$957$	88796.8	−	143180.i	0.0969557	−	0.156336i
$958$		−	808297.i		−	0.880724i
$959$			325264.i			0.353670i
$960$	916731.	+	282935.i	0.994717	+	0.307004i
$961$	−922497.			−0.998891
$962$	−994524.			−1.07464
$963$	−105304.	−	212247.i	−0.113551	−	0.228870i
$964$	−432912.			−0.465849
$965$	1.22688e6	−	320813.i	1.31749	−	0.344507i
$966$	−153270.	−	95054.0i	−0.164249	−	0.101863i
$967$			811983.i			0.868349i	0.900829	+	0.434174i	$0.142960\pi$
$967$			811983.i			0.868349i	−0.900829	+	0.434174i	$0.857040\pi$
$968$	−367400.			−0.392092
$969$	406560.	−	655558.i	0.432989	−	0.698174i
$970$	191880.	−	50174.4i	0.203932	−	0.0533259i
$971$			333487.i			0.353705i	0.984237	+	0.176852i	$0.0565915\pi$
$971$			333487.i			0.353705i	−0.984237	+	0.176852i	$0.943409\pi$
$972$	−126792.	−	330829.i	−0.134202	−	0.350164i
$973$		−	296335.i		−	0.313009i
$974$			1.41014e6i			1.48643i
$975$	1.37543e6	+	60004.1i	1.44687	+	0.0631207i
$976$	115072.			0.120801
$977$	1.32479e6			1.38790			0.693951	−	0.720023i	$-0.255869\pi$
$977$	1.32479e6			1.38790			0.693951	+	0.720023i	$0.255869\pi$
$978$	−296359.	−	183794.i	−0.309842	−	0.192156i
$979$	898560.			0.937522
$980$	−82231.9	−	314476.i	−0.0856225	−	0.327443i
$981$	228672.	+	460903.i	0.237616	+	0.478930i
$982$		−	725387.i		−	0.752223i
$983$	−815276.			−0.843719			−0.421859	−	0.906661i	$-0.638622\pi$
$983$	−815276.			−0.843719			−0.421859	+	0.906661i	$0.638622\pi$
$984$	−1.10682e6	−	686421.i	−1.14311	−	0.708925i
$985$	−182560.	−	698158.i	−0.188163	−	0.719583i
$986$		−	170275.i		−	0.175145i
$987$	−286000.	−	177369.i	−0.293583	−	0.182073i
$988$			452303.i			0.463357i
$989$		−	1.07094e6i		−	1.09490i
$990$	406790.	+	467269.i	0.415050	+	0.476756i
$991$	1.57635e6			1.60511			0.802557	−	0.596575i	$-0.203472\pi$
$991$	1.57635e6			1.60511			0.802557	+	0.596575i	$0.203472\pi$
$992$	23072.0			0.0234456
$993$	968093.	−	1.56100e6i	0.981790	−	1.58309i
$994$	224640.			0.227360
$995$	217109.	+	830284.i	0.219297	+	0.838649i
$996$	126270.	−	203604.i	0.127286	−	0.205243i
$997$		−	128251.i		−	0.129024i	−0.997917	−	0.0645118i	$-0.979451\pi$
$997$		−	128251.i		−	0.129024i	0.997917	−	0.0645118i	$-0.0205490\pi$
$998$	498843.			0.500844
$999$	−88452.0	+	932545.i	−0.0886292	+	0.934413i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.5.d.c.14.3	yes	4
3.2	odd	2	inner	15.5.d.c.14.1	✓	4
4.3	odd	2		240.5.c.c.209.2		4
5.2	odd	4		75.5.c.h.26.3		4
5.3	odd	4		75.5.c.h.26.2		4
5.4	even	2	inner	15.5.d.c.14.2	yes	4
12.11	even	2		240.5.c.c.209.4		4
15.2	even	4		75.5.c.h.26.1		4
15.8	even	4		75.5.c.h.26.4		4
15.14	odd	2	inner	15.5.d.c.14.4	yes	4
20.19	odd	2		240.5.c.c.209.3		4
60.59	even	2		240.5.c.c.209.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.5.d.c.14.1	✓	4	3.2	odd	2	inner
15.5.d.c.14.2	yes	4	5.4	even	2	inner
15.5.d.c.14.3	yes	4	1.1	even	1	trivial
15.5.d.c.14.4	yes	4	15.14	odd	2	inner
75.5.c.h.26.1		4	15.2	even	4
75.5.c.h.26.2		4	5.3	odd	4
75.5.c.h.26.3		4	5.2	odd	4
75.5.c.h.26.4		4	15.8	even	4
240.5.c.c.209.1		4	60.59	even	2
240.5.c.c.209.2		4	4.3	odd	2
240.5.c.c.209.3		4	20.19	odd	2
240.5.c.c.209.4		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	15.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.16228 q^{2} +(4.74342 - 7.64853i) q^{3} -6.00000 q^{4} +(6.32456 + 24.1868i) q^{5} +(15.0000 - 24.1868i) q^{6} +15.2971i q^{7} -69.5701 q^{8} +(-36.0000 - 72.5603i) q^{9} +(20.0000 + 76.4853i) q^{10} +96.7471i q^{11} +(-28.4605 + 45.8912i) q^{12} -244.753i q^{13} +48.3735i q^{14} +(214.993 + 66.3544i) q^{15} -124.000 q^{16} +278.280 q^{17} +(-113.842 - 229.456i) q^{18} +308.000 q^{19} +(-37.9473 - 145.121i) q^{20} +(117.000 + 72.5603i) q^{21} +305.941i q^{22} -414.258 q^{23} +(-330.000 + 532.109i) q^{24} +(-545.000 + 305.941i) q^{25} -773.977i q^{26} +(-725.743 - 68.8368i) q^{27} -91.7824i q^{28} -193.494i q^{29} +(679.868 + 209.831i) q^{30} +32.0000 q^{31} +720.999 q^{32} +(739.973 + 458.912i) q^{33} +880.000 q^{34} +(-369.986 + 96.7471i) q^{35} +(216.000 + 435.362i) q^{36} -1284.95i q^{37} +973.982 q^{38} +(-1872.00 - 1160.97i) q^{39} +(-440.000 - 1682.68i) q^{40} +2080.06i q^{41} +(369.986 + 229.456i) q^{42} +2585.20i q^{43} -580.483i q^{44} +(1527.32 - 1329.64i) q^{45} -1310.00 q^{46} -2444.44 q^{47} +(-588.184 + 948.418i) q^{48} +2167.00 q^{49} +(-1723.44 + 967.471i) q^{50} +(1320.00 - 2128.44i) q^{51} +1468.52i q^{52} +1416.70 q^{53} +(-2295.00 - 217.681i) q^{54} +(-2340.00 + 611.882i) q^{55} -1064.22i q^{56} +(1460.97 - 2355.75i) q^{57} -611.882i q^{58} +3966.63i q^{59} +(-1289.96 - 398.126i) q^{60} -928.000 q^{61} +101.193 q^{62} +(1109.96 - 550.694i) q^{63} +4264.00 q^{64} +(5919.78 - 1547.95i) q^{65} +(2340.00 + 1451.21i) q^{66} -2585.20i q^{67} -1669.68 q^{68} +(-1965.00 + 3168.47i) q^{69} +(-1170.00 + 305.941i) q^{70} -4643.86i q^{71} +(2504.52 + 5048.03i) q^{72} -4221.99i q^{73} -4063.38i q^{74} +(-245.162 + 5619.65i) q^{75} -1848.00 q^{76} -1479.95 q^{77} +(-5919.78 - 3671.29i) q^{78} +8.00000 q^{79} +(-784.245 - 2999.16i) q^{80} +(-3969.00 + 5224.34i) q^{81} +6577.74i q^{82} -4436.68 q^{83} +(-702.000 - 435.362i) q^{84} +(1760.00 + 6730.71i) q^{85} +8175.13i q^{86} +(-1479.95 - 917.824i) q^{87} -6730.71i q^{88} -9287.72i q^{89} +(4829.80 - 4204.68i) q^{90} +3744.00 q^{91} +2485.55 q^{92} +(151.789 - 244.753i) q^{93} -7730.00 q^{94} +(1947.96 + 7449.53i) q^{95} +(3420.00 - 5514.58i) q^{96} -2508.72i q^{97} +6852.66 q^{98} +(7020.00 - 3482.90i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 24 q^{4} + 60 q^{6} - 144 q^{9} + 80 q^{10} + 120 q^{15} - 496 q^{16} + 1232 q^{19} + 468 q^{21} - 1320 q^{24} - 2180 q^{25} + 2340 q^{30} + 128 q^{31} + 3520 q^{34} + 864 q^{36} - 7488 q^{39} - 1760 q^{40}+ \cdots + 28080 q^{99}+O(q^{100}) \) 4 * q - 24 * q^4 + 60 * q^6 - 144 * q^9 + 80 * q^10 + 120 * q^15 - 496 * q^16 + 1232 * q^19 + 468 * q^21 - 1320 * q^24 - 2180 * q^25 + 2340 * q^30 + 128 * q^31 + 3520 * q^34 + 864 * q^36 - 7488 * q^39 - 1760 * q^40 + 7020 * q^45 - 5240 * q^46 + 8668 * q^49 + 5280 * q^51 - 9180 * q^54 - 9360 * q^55 - 720 * q^60 - 3712 * q^61 + 17056 * q^64 + 9360 * q^66 - 7860 * q^69 - 4680 * q^70 + 9360 * q^75 - 7392 * q^76 + 32 * q^79 - 15876 * q^81 - 2808 * q^84 + 7040 * q^85 - 2880 * q^90 + 14976 * q^91 - 30920 * q^94 + 13680 * q^96 + 28080 * q^99

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