LMFDB - Embedded newform 15.5.c.a.11.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [15,5,Mod(11,15)] 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("15.11"); 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, 0])) N = Newforms(chi, 5, names="a")

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

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.55054944626$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 73x^{4} + 1096x^{2} + 180 $ x^6 + 73x^4 + 1096x^2 + 180
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.1
Root			$-7.20990i$ of defining polynomial
Character	$\chi$	$=$	15.11
Dual form			15.5.c.a.11.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-7.20990i q^{2} +(8.77108 + 2.01697i) q^{3} -35.9827 q^{4} +11.1803i q^{5} +(14.5422 - 63.2386i) q^{6} +23.3388 q^{7} +144.073i q^{8} +(72.8637 + 35.3820i) q^{9} +80.6091 q^{10} -33.9875i q^{11} +(-315.607 - 72.5760i) q^{12} -125.965 q^{13} -168.271i q^{14} +(-22.5504 + 98.0636i) q^{15} +463.029 q^{16} +308.095i q^{17} +(255.101 - 525.340i) q^{18} -363.724 q^{19} -402.298i q^{20} +(204.707 + 47.0738i) q^{21} -245.046 q^{22} -102.256i q^{23} +(-290.591 + 1263.68i) q^{24} -125.000 q^{25} +908.197i q^{26} +(567.728 + 457.302i) q^{27} -839.794 q^{28} -1565.10i q^{29} +(707.029 + 162.586i) q^{30} +326.954 q^{31} -1033.23i q^{32} +(68.5518 - 298.107i) q^{33} +2221.33 q^{34} +260.936i q^{35} +(-2621.83 - 1273.14i) q^{36} +72.4778 q^{37} +2622.41i q^{38} +(-1104.85 - 254.068i) q^{39} -1610.78 q^{40} -2149.01i q^{41} +(339.397 - 1475.92i) q^{42} +1501.27 q^{43} +1222.96i q^{44} +(-395.583 + 814.640i) q^{45} -737.258 q^{46} +1958.33i q^{47} +(4061.27 + 933.916i) q^{48} -1856.30 q^{49} +901.237i q^{50} +(-621.418 + 2702.32i) q^{51} +4532.57 q^{52} -2965.17i q^{53} +(3297.10 - 4093.26i) q^{54} +379.992 q^{55} +3362.50i q^{56} +(-3190.25 - 733.621i) q^{57} -11284.2 q^{58} +5045.29i q^{59} +(811.424 - 3528.59i) q^{60} +736.716 q^{61} -2357.30i q^{62} +(1700.55 + 825.775i) q^{63} -40.9924 q^{64} -1408.34i q^{65} +(-2149.32 - 494.252i) q^{66} +4254.63 q^{67} -11086.1i q^{68} +(206.248 - 896.898i) q^{69} +1881.32 q^{70} -1303.84i q^{71} +(-5097.59 + 10497.7i) q^{72} +6768.14 q^{73} -522.557i q^{74} +(-1096.38 - 252.121i) q^{75} +13087.8 q^{76} -793.229i q^{77} +(-1831.81 + 7965.87i) q^{78} -6367.07 q^{79} +5176.82i q^{80} +(4057.23 + 5156.13i) q^{81} -15494.1 q^{82} +6892.17i q^{83} +(-7365.90 - 1693.84i) q^{84} -3444.60 q^{85} -10824.0i q^{86} +(3156.76 - 13727.6i) q^{87} +4896.68 q^{88} +6401.82i q^{89} +(5873.48 + 2852.11i) q^{90} -2939.88 q^{91} +3679.45i q^{92} +(2867.74 + 659.456i) q^{93} +14119.3 q^{94} -4066.56i q^{95} +(2083.99 - 9062.51i) q^{96} -9879.94 q^{97} +13383.7i q^{98} +(1202.55 - 2476.45i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{3} - 50 q^{4} - 2 q^{6} + 76 q^{7} + 118 q^{9} + 50 q^{10} - 452 q^{12} - 424 q^{13} + 50 q^{15} + 802 q^{16} + 1160 q^{18} - 244 q^{19} - 876 q^{21} + 340 q^{22} - 786 q^{24} - 750 q^{25} - 352 q^{27}+ \cdots + 9680 q^{99}+O(q^{100}) $ 6 * q + 8 * q^3 - 50 * q^4 - 2 * q^6 + 76 * q^7 + 118 * q^9 + 50 * q^10 - 452 * q^12 - 424 * q^13 + 50 * q^15 + 802 * q^16 + 1160 * q^18 - 244 * q^19 - 876 * q^21 + 340 * q^22 - 786 * q^24 - 750 * q^25 - 352 * q^27 - 3764 * q^28 + 2200 * q^30 + 3772 * q^31 + 4420 * q^33 + 3124 * q^34 - 7606 * q^36 + 1896 * q^37 - 1336 * q^39 - 4650 * q^40 - 1980 * q^42 - 7384 * q^43 + 1900 * q^45 + 8196 * q^46 + 14668 * q^48 - 1318 * q^49 - 8492 * q^51 + 8976 * q^52 - 278 * q^54 - 1300 * q^55 - 11584 * q^57 - 23740 * q^58 + 5050 * q^60 + 6452 * q^61 + 14796 * q^63 + 3174 * q^64 - 12760 * q^66 + 13816 * q^67 + 5472 * q^69 - 2100 * q^70 - 2040 * q^72 + 596 * q^73 - 1000 * q^75 + 21348 * q^76 - 1400 * q^78 - 16124 * q^79 + 5086 * q^81 - 31240 * q^82 - 14736 * q^84 - 3100 * q^85 - 4900 * q^87 + 15660 * q^88 + 7550 * q^90 - 11632 * q^91 - 8184 * q^93 + 34924 * q^94 + 14354 * q^96 + 9756 * q^97 + 9680 * 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 $

$2$		−	7.20990i		−	1.80247i	−0.433325	−	0.901237i	$-0.642660\pi$
$2$		−	7.20990i		−	1.80247i	0.433325	−	0.901237i	$-0.357340\pi$
$3$	8.77108	+	2.01697i	0.974564	+	0.224108i
$3$	8.77108	+	2.01697i	0.974564	+	0.224108i
$4$	−35.9827			−2.24892
$5$			11.1803i			0.447214i
$5$			11.1803i			0.447214i
$6$	14.5422	−	63.2386i	0.403949	−	1.75663i
$7$	23.3388			0.476303			0.238151	−	0.971228i	$-0.423459\pi$
$7$	23.3388			0.476303			0.238151	+	0.971228i	$0.423459\pi$
$8$			144.073i			2.25114i
$9$	72.8637	+	35.3820i	0.899551	+	0.436815i
$10$	80.6091			0.806091
$11$		−	33.9875i		−	0.280888i	−0.990089	−	0.140444i	$-0.955147\pi$
$11$		−	33.9875i		−	0.280888i	0.990089	−	0.140444i	$-0.0448531\pi$
$12$	−315.607	−	72.5760i	−2.19171	−	0.504000i
$13$	−125.965			−0.745357			−0.372678	−	0.927961i	$-0.621561\pi$
$13$	−125.965			−0.745357			−0.372678	+	0.927961i	$0.621561\pi$
$14$		−	168.271i		−	0.858524i
$15$	−22.5504	+	98.0636i	−0.100224	+	0.435838i
$16$	463.029			1.80871
$17$			308.095i			1.06607i	0.846093	+	0.533036i	$0.178949\pi$
$17$			308.095i			1.06607i	−0.846093	+	0.533036i	$0.821051\pi$
$18$	255.101	−	525.340i	0.787348	−	1.62142i
$19$	−363.724			−1.00755			−0.503773	−	0.863836i	$-0.668055\pi$
$19$	−363.724			−1.00755			−0.503773	+	0.863836i	$0.668055\pi$
$20$		−	402.298i		−	1.00575i
$21$	204.707	+	47.0738i	0.464188	+	0.106743i
$22$	−245.046			−0.506294
$23$		−	102.256i		−	0.193301i	−0.995318	−	0.0966506i	$-0.969187\pi$
$23$		−	102.256i		−	0.193301i	0.995318	−	0.0966506i	$-0.0308129\pi$
$24$	−290.591	+	1263.68i	−0.504498	+	2.19388i
$25$	−125.000			−0.200000
$26$			908.197i			1.34349i
$27$	567.728	+	457.302i	0.778777	+	0.627301i
$28$	−839.794			−1.07117
$29$		−	1565.10i		−	1.86100i	−0.366296	−	0.930498i	$-0.619374\pi$
$29$		−	1565.10i		−	1.86100i	0.366296	−	0.930498i	$-0.380626\pi$
$30$	707.029	+	162.586i	0.785588	+	0.180651i
$31$	326.954			0.340222			0.170111	−	0.985425i	$-0.445587\pi$
$31$	326.954			0.340222			0.170111	+	0.985425i	$0.445587\pi$
$32$		−	1033.23i		−	1.00901i
$33$	68.5518	−	298.107i	0.0629493	−	0.273744i
$34$	2221.33			1.92157
$35$			260.936i			0.213009i
$36$	−2621.83	−	1273.14i	−2.02302	−	0.982360i
$37$	72.4778			0.0529421			0.0264711	−	0.999650i	$-0.491573\pi$
$37$	72.4778			0.0529421			0.0264711	+	0.999650i	$0.491573\pi$
$38$			2622.41i			1.81608i
$39$	−1104.85	−	254.068i	−0.726398	−	0.167040i
$40$	−1610.78			−1.00674
$41$		−	2149.01i		−	1.27841i	−0.769036	−	0.639205i	$-0.779264\pi$
$41$		−	2149.01i		−	1.27841i	0.769036	−	0.639205i	$-0.220736\pi$
$42$	339.397	−	1475.92i	0.192402	−	0.836687i
$43$	1501.27			0.811934			0.405967	−	0.913888i	$-0.366935\pi$
$43$	1501.27			0.811934			0.405967	+	0.913888i	$0.366935\pi$
$44$			1222.96i			0.631695i
$45$	−395.583	+	814.640i	−0.195350	+	0.402292i
$46$	−737.258			−0.348420
$47$			1958.33i			0.886522i	0.896393	+	0.443261i	$0.146178\pi$
$47$			1958.33i			0.886522i	−0.896393	+	0.443261i	$0.853822\pi$
$48$	4061.27	+	933.916i	1.76270	+	0.405346i
$49$	−1856.30			−0.773136
$50$			901.237i			0.360495i
$51$	−621.418	+	2702.32i	−0.238915	+	1.03896i
$52$	4532.57			1.67625
$53$		−	2965.17i		−	1.05560i	−0.849370	−	0.527798i	$-0.823018\pi$
$53$		−	2965.17i		−	1.05560i	0.849370	−	0.527798i	$-0.176982\pi$
$54$	3297.10	−	4093.26i	1.13069	−	1.40373i
$55$	379.992			0.125617
$56$			3362.50i			1.07222i
$57$	−3190.25	−	733.621i	−0.981919	−	0.225799i
$58$	−11284.2			−3.35440
$59$			5045.29i			1.44938i	0.689075	+	0.724690i	$0.258017\pi$
$59$			5045.29i			1.44938i	−0.689075	+	0.724690i	$0.741983\pi$
$60$	811.424	−	3528.59i	0.225396	−	0.980164i
$61$	736.716			0.197989			0.0989944	−	0.995088i	$-0.468437\pi$
$61$	736.716			0.197989			0.0989944	+	0.995088i	$0.468437\pi$
$62$		−	2357.30i		−	0.613242i
$63$	1700.55	+	825.775i	0.428459	+	0.208056i
$64$	−40.9924			−0.0100079
$65$		−	1408.34i		−	0.333334i
$66$	−2149.32	−	494.252i	−0.493416	−	0.113465i
$67$	4254.63			0.947791			0.473895	−	0.880581i	$-0.342847\pi$
$67$	4254.63			0.947791			0.473895	+	0.880581i	$0.342847\pi$
$68$		−	11086.1i		−	2.39751i
$69$	206.248	−	896.898i	0.0433203	−	0.188384i
$70$	1881.32			0.383944
$71$		−	1303.84i		−	0.258647i	−0.991602	−	0.129323i	$-0.958719\pi$
$71$		−	1303.84i		−	0.258647i	0.991602	−	0.129323i	$-0.0412805\pi$
$72$	−5097.59	+	10497.7i	−0.983332	+	2.02502i
$73$	6768.14			1.27006			0.635029	−	0.772488i	$-0.280988\pi$
$73$	6768.14			1.27006			0.635029	+	0.772488i	$0.280988\pi$
$74$		−	522.557i		−	0.0954269i
$75$	−1096.38	−	252.121i	−0.194913	−	0.0448216i
$76$	13087.8			2.26589
$77$		−	793.229i		−	0.133788i
$78$	−1831.81	+	7965.87i	−0.301086	+	1.30931i
$79$	−6367.07			−1.02020			−0.510100	−	0.860115i	$-0.670392\pi$
$79$	−6367.07			−1.02020			−0.510100	+	0.860115i	$0.670392\pi$
$80$			5176.82i			0.808879i
$81$	4057.23	+	5156.13i	0.618385	+	0.785875i
$82$	−15494.1			−2.30430
$83$			6892.17i			1.00046i	0.865893	+	0.500230i	$0.166751\pi$
$83$			6892.17i			1.00046i	−0.865893	+	0.500230i	$0.833249\pi$
$84$	−7365.90	−	1693.84i	−1.04392	−	0.240057i
$85$	−3444.60			−0.476762
$86$		−	10824.0i		−	1.46349i
$87$	3156.76	−	13727.6i	0.417064	−	1.81366i
$88$	4896.68			0.632319
$89$			6401.82i			0.808208i	0.914713	+	0.404104i	$0.132417\pi$
$89$			6401.82i			0.808208i	−0.914713	+	0.404104i	$0.867583\pi$
$90$	5873.48	+	2852.11i	0.725121	+	0.352113i
$91$	−2939.88			−0.355016
$92$			3679.45i			0.434718i
$93$	2867.74	+	659.456i	0.331568	+	0.0762465i
$94$	14119.3			1.59793
$95$		−	4066.56i		−	0.450588i
$96$	2083.99	−	9062.51i	0.226127	−	0.983345i
$97$	−9879.94			−1.05005			−0.525026	−	0.851086i	$-0.675944\pi$
$97$	−9879.94			−1.05005			−0.525026	+	0.851086i	$0.675944\pi$
$98$			13383.7i			1.39356i
$99$	1202.55	−	2476.45i	0.122696	−	0.252674i
$100$	4497.83			0.449783
$101$		−	11119.7i		−	1.09006i	−0.838415	−	0.545032i	$-0.816518\pi$
$101$		−	11119.7i		−	1.09006i	0.838415	−	0.545032i	$-0.183482\pi$
$102$	19483.5	+	4480.36i	1.87269	+	0.430638i
$103$	−12245.5			−1.15426			−0.577128	−	0.816653i	$-0.695827\pi$
$103$	−12245.5			−1.15426			−0.577128	+	0.816653i	$0.695827\pi$
$104$		−	18148.2i		−	1.67790i
$105$	−526.301	+	2288.69i	−0.0477370	+	0.207591i
$106$	−21378.6			−1.90269
$107$		−	708.790i		−	0.0619085i	−0.999521	−	0.0309542i	$-0.990145\pi$
$107$		−	708.790i		−	0.0619085i	0.999521	−	0.0309542i	$-0.00985462\pi$
$108$	−20428.4	−	16455.0i	−1.75140	−	1.41075i
$109$	−908.407			−0.0764588			−0.0382294	−	0.999269i	$-0.512172\pi$
$109$	−908.407			−0.0764588			−0.0382294	+	0.999269i	$0.512172\pi$
$110$		−	2739.70i		−	0.226422i
$111$	635.708	+	146.186i	0.0515955	+	0.0118647i
$112$	10806.6			0.861493
$113$			3441.26i			0.269501i	0.990880	+	0.134751i	$0.0430233\pi$
$113$			3441.26i			0.269501i	−0.990880	+	0.134751i	$0.956977\pi$
$114$	−5289.33	+	23001.4i	−0.406997	+	1.76988i
$115$	1143.26			0.0864469
$116$			56316.4i			4.18523i
$117$	−9178.29	−	4456.91i	−0.670487	−	0.325583i
$118$	36376.0			2.61247
$119$			7190.57i			0.507773i
$120$	−14128.3	−	3248.91i	−0.981133	−	0.225618i
$121$	13485.8			0.921102
$122$		−	5311.65i		−	0.356870i
$123$	4334.48	−	18849.1i	0.286502	−	1.24589i
$124$	−11764.7			−0.765131
$125$		−	1397.54i		−	0.0894427i
$126$	5953.76	−	12260.8i	0.375016	−	0.772286i
$127$	28648.5			1.77621			0.888106	−	0.459638i	$-0.152021\pi$
$127$	28648.5			1.77621			0.888106	+	0.459638i	$0.152021\pi$
$128$		−	16236.1i		−	0.990971i
$129$	13167.7	+	3028.01i	0.791282	+	0.181961i
$130$	−10154.0			−0.600826
$131$		−	400.763i		−	0.0233531i	−0.999932	−	0.0116766i	$-0.996283\pi$
$131$		−	400.763i		−	0.0233531i	0.999932	−	0.0116766i	$-0.00371685\pi$
$132$	−2466.68	+	10726.7i	−0.141568	+	0.615627i
$133$	−8488.90			−0.479897
$134$		−	30675.5i		−	1.70837i
$135$	−5112.80	+	6347.40i	−0.280538	+	0.348280i
$136$	−44388.1			−2.39988
$137$			29277.7i			1.55989i	0.625845	+	0.779947i	$0.284754\pi$
$137$			29277.7i			1.55989i	−0.625845	+	0.779947i	$0.715246\pi$
$138$	−6466.55	−	1487.03i	−0.339558	−	0.0780838i
$139$	1852.04			0.0958565			0.0479282	−	0.998851i	$-0.484738\pi$
$139$	1852.04			0.0958565			0.0479282	+	0.998851i	$0.484738\pi$
$140$		−	9389.18i		−	0.479040i
$141$	−3949.89	+	17176.6i	−0.198677	+	0.863973i
$142$	−9400.54			−0.466204
$143$			4281.25i			0.209362i
$144$	33738.0	+	16382.9i	1.62703	+	0.790071i
$145$	17498.3			0.832263
$146$		−	48797.6i		−	2.28925i
$147$	−16281.7	−	3744.10i	−0.753470	−	0.173266i
$148$	−2607.94			−0.119062
$149$		−	19905.0i		−	0.896582i	−0.893888	−	0.448291i	$-0.852033\pi$
$149$		−	19905.0i		−	0.896582i	0.893888	−	0.448291i	$-0.147967\pi$
$150$	−1817.77	+	7904.83i	−0.0807898	+	0.351326i
$151$	−24648.2			−1.08101			−0.540506	−	0.841340i	$-0.681767\pi$
$151$	−24648.2			−1.08101			−0.540506	+	0.841340i	$0.681767\pi$
$152$		−	52402.8i		−	2.26813i
$153$	−10901.0	+	22448.9i	−0.465676	+	0.958986i
$154$	−5719.10			−0.241149
$155$			3655.45i			0.152152i
$156$	39755.5	+	9142.06i	1.63361	+	0.375660i
$157$	1399.18			0.0567642			0.0283821	−	0.999597i	$-0.490964\pi$
$157$	1399.18			0.0567642			0.0283821	+	0.999597i	$0.490964\pi$
$158$			45905.9i			1.83888i
$159$	5980.66	−	26007.8i	0.236568	−	1.02875i
$160$	11551.8			0.451243
$161$		−	2386.54i		−	0.0920699i
$162$	37175.2	−	29252.2i	1.41652	−	1.11462i
$163$	8023.34			0.301981			0.150991	−	0.988535i	$-0.451754\pi$
$163$	8023.34			0.301981			0.150991	+	0.988535i	$0.451754\pi$
$164$			77327.0i			2.87504i
$165$	3332.94	+	766.432i	0.122422	+	0.0281518i
$166$	49691.8			1.80330
$167$			20948.5i			0.751138i	0.926794	+	0.375569i	$0.122553\pi$
$167$			20948.5i			0.751138i	−0.926794	+	0.375569i	$0.877447\pi$
$168$	−6782.06	+	29492.7i	−0.240294	+	1.04495i
$169$	−12693.7			−0.444443
$170$			24835.2i			0.859351i
$171$	−26502.3	−	12869.3i	−0.906339	−	0.440111i
$172$	−54019.5			−1.82597
$173$		−	20564.6i		−	0.687112i	−0.939132	−	0.343556i	$-0.888368\pi$
$173$		−	20564.6i		−	0.687112i	0.939132	−	0.343556i	$-0.111632\pi$
$174$	−98974.6	−	22759.9i	−3.26908	−	0.751747i
$175$	−2917.36			−0.0952606
$176$		−	15737.2i		−	0.508045i
$177$	−10176.2	+	44252.6i	−0.324817	+	1.41251i
$178$	46156.5			1.45678
$179$		−	32395.6i		−	1.01107i	−0.862807	−	0.505533i	$-0.831296\pi$
$179$		−	32395.6i		−	1.01107i	0.862807	−	0.505533i	$-0.168704\pi$
$180$	14234.1	−	29312.9i	0.439325	−	0.904720i
$181$	18055.4			0.551125			0.275562	−	0.961283i	$-0.411136\pi$
$181$	18055.4			0.551125			0.275562	+	0.961283i	$0.411136\pi$
$182$			21196.3i			0.639907i
$183$	6461.79	+	1485.93i	0.192953	+	0.0443708i
$184$	14732.4			0.435148
$185$			810.326i			0.0236764i
$186$	4754.61	−	20676.1i	0.137432	−	0.597644i
$187$	10471.4			0.299447
$188$		−	70465.8i		−	1.99371i
$189$	13250.1	+	10672.9i	0.370934	+	0.298785i
$190$	−29319.5			−0.812174
$191$			11720.8i			0.321285i	0.987013	+	0.160642i	$0.0513566\pi$
$191$			11720.8i			0.321285i	−0.987013	+	0.160642i	$0.948643\pi$
$192$	−359.548	−	82.6806i	−0.00975336	−	0.00224285i
$193$	−28253.2			−0.758496			−0.379248	−	0.925295i	$-0.623817\pi$
$193$	−28253.2			−0.758496			−0.379248	+	0.925295i	$0.623817\pi$
$194$			71233.4i			1.89269i
$195$	2840.57	−	12352.6i	0.0747027	−	0.324855i
$196$	66794.6			1.73872
$197$		−	53312.1i		−	1.37370i	−0.726797	−	0.686852i	$-0.758992\pi$
$197$		−	53312.1i		−	1.37370i	0.726797	−	0.686852i	$-0.241008\pi$
$198$	−17855.0	−	8670.24i	−0.455438	−	0.221157i
$199$	50464.4			1.27432			0.637161	−	0.770731i	$-0.280109\pi$
$199$	50464.4			1.27432			0.637161	+	0.770731i	$0.280109\pi$
$200$		−	18009.1i		−	0.450228i
$201$	37317.7	+	8581.47i	0.923683	+	0.212407i
$202$	−80172.2			−1.96481
$203$		−	36527.6i		−	0.886398i
$204$	22360.3	−	97236.7i	0.537300	−	2.33652i
$205$	24026.6			0.571722
$206$			88288.9i			2.08052i
$207$	3618.03	−	7450.77i	0.0844368	−	0.173884i
$208$	−58325.6			−1.34813
$209$			12362.1i			0.283008i
$210$	16501.2	+	3794.58i	0.374178	+	0.0860448i
$211$	−67329.0			−1.51230			−0.756148	−	0.654400i	$-0.772921\pi$
$211$	−67329.0			−1.51230			−0.756148	+	0.654400i	$0.772921\pi$
$212$			106695.i			2.37395i
$213$	2629.80	−	11436.1i	0.0579648	−	0.252068i
$214$	−5110.31			−0.111589
$215$			16784.7i			0.363108i
$216$	−65884.9	+	81794.3i	−1.41214	+	1.75314i
$217$	7630.72			0.162049
$218$			6549.52i			0.137815i
$219$	59363.9	+	13651.1i	1.23775	+	0.284630i
$220$	−13673.1			−0.282502
$221$		−	38809.2i		−	0.794604i
$222$	1053.98	−	4583.39i	0.0213859	−	0.0929996i
$223$	−56592.7			−1.13802			−0.569011	−	0.822330i	$-0.692674\pi$
$223$	−56592.7			−1.13802			−0.569011	+	0.822330i	$0.692674\pi$
$224$		−	24114.3i		−	0.480594i
$225$	−9107.96	−	4422.75i	−0.179910	−	0.0873630i
$226$	24811.1			0.485769
$227$		−	19015.9i		−	0.369033i	−0.982829	−	0.184516i	$-0.940928\pi$
$227$		−	19015.9i		−	0.369033i	0.982829	−	0.184516i	$-0.0590719\pi$
$228$	114794.	+	26397.6i	2.20825	+	0.507803i
$229$	−79079.9			−1.50798			−0.753989	−	0.656887i	$-0.771873\pi$
$229$	−79079.9			−1.50798			−0.753989	+	0.656887i	$0.771873\pi$
$230$		−	8242.79i		−	0.155818i
$231$	1599.92	−	6957.47i	0.0299829	−	0.130385i
$232$	225488.			4.18936
$233$			36312.6i			0.668876i	0.942418	+	0.334438i	$0.108546\pi$
$233$			36312.6i			0.668876i	−0.942418	+	0.334438i	$0.891454\pi$
$234$	−32133.9	+	66174.6i	−0.586855	+	1.20854i
$235$	−21894.8			−0.396465
$236$		−	181543.i		−	3.25953i
$237$	−55846.0	−	12842.2i	−0.994250	−	0.228635i
$238$	51843.3			0.915248
$239$			46783.7i			0.819028i	0.912304	+	0.409514i	$0.134302\pi$
$239$			46783.7i			0.819028i	−0.912304	+	0.409514i	$0.865698\pi$
$240$	−10441.5	+	45406.3i	−0.181276	+	0.788304i
$241$	30646.4			0.527649			0.263824	−	0.964571i	$-0.415016\pi$
$241$	30646.4			0.527649			0.263824	+	0.964571i	$0.415016\pi$
$242$		−	97231.6i		−	1.66026i
$243$	25186.5	+	53408.1i	0.426535	+	0.904471i
$244$	−26509.0			−0.445260
$245$		−	20754.0i		−	0.345757i
$246$	−135900.	−	31251.2i	−2.24569	−	0.516412i
$247$	45816.6			0.750981
$248$			47105.2i			0.765888i
$249$	−13901.3	+	60451.8i	−0.224211	+	0.975013i
$250$	−10076.1			−0.161218
$251$			114804.i			1.82226i	0.412120	+	0.911130i	$0.364788\pi$
$251$			114804.i			1.82226i	−0.412120	+	0.911130i	$0.635212\pi$
$252$	−61190.4	−	29713.6i	−0.963568	−	0.467901i
$253$	−3475.44			−0.0542960
$254$		−	206553.i		−	3.20158i
$255$	−30212.9	−	6947.66i	−0.464635	−	0.106846i
$256$	−117716.			−1.79621
$257$		−	89602.7i		−	1.35661i	−0.734780	−	0.678305i	$-0.762715\pi$
$257$		−	89602.7i		−	1.35661i	0.734780	−	0.678305i	$-0.237285\pi$
$258$	21831.6	−	94938.0i	0.327980	−	1.42627i
$259$	1691.55			0.0252165
$260$			50675.6i			0.749640i
$261$	55376.3	−	114039.i	0.812911	−	1.67406i
$262$	−2889.46			−0.0420935
$263$			113892.i			1.64658i	0.567620	+	0.823291i	$0.307864\pi$
$263$			113892.i			1.64658i	−0.567620	+	0.823291i	$0.692136\pi$
$264$	42949.2	+	9876.46i	0.616236	+	0.141708i
$265$	33151.6			0.472077
$266$			61204.1i			0.865003i
$267$	−12912.3	+	56150.9i	−0.181126	+	0.787651i
$268$	−153093.			−2.13150
$269$		−	39232.3i		−	0.542174i	−0.962555	−	0.271087i	$-0.912617\pi$
$269$		−	39232.3i		−	0.542174i	0.962555	−	0.271087i	$-0.0873832\pi$
$270$	45764.1	+	36862.7i	0.627765	+	0.505662i
$271$	52549.5			0.715533			0.357767	−	0.933811i	$-0.383538\pi$
$271$	52549.5			0.715533			0.357767	+	0.933811i	$0.383538\pi$
$272$			142657.i			1.92821i
$273$	−25786.0	−	5929.66i	−0.345986	−	0.0795618i
$274$	211089.			2.81167
$275$			4248.44i			0.0561777i
$276$	−7421.35	+	32272.8i	−0.0974237	+	0.423661i
$277$	−28707.3			−0.374139			−0.187069	−	0.982347i	$-0.559899\pi$
$277$	−28707.3			−0.374139			−0.187069	+	0.982347i	$0.559899\pi$
$278$		−	13353.0i		−	0.172779i
$279$	23823.0	+	11568.3i	0.306047	+	0.148614i
$280$	−37593.8			−0.479513
$281$			83537.2i			1.05796i	0.848636	+	0.528978i	$0.177424\pi$
$281$			83537.2i			1.05796i	−0.848636	+	0.528978i	$0.822576\pi$
$282$	123842.	+	28478.3i	1.55729	+	0.358109i
$283$	126719.			1.58223			0.791115	−	0.611667i	$-0.209501\pi$
$283$	126719.			1.58223			0.791115	+	0.611667i	$0.209501\pi$
$284$			46915.6i			0.581675i
$285$	8202.13	−	35668.1i	0.100980	−	0.439127i
$286$	30867.4			0.377370
$287$		−	50155.3i		−	0.608910i
$288$	36557.6	−	75284.7i	0.440751	−	0.907656i
$289$	−11401.3			−0.136508
$290$		−	126161.i		−	1.50013i
$291$	−86657.7	−	19927.5i	−1.02334	−	0.235325i
$292$	−243536.			−2.85626
$293$		−	69385.0i		−	0.808222i	−0.914710	−	0.404111i	$-0.867581\pi$
$293$		−	69385.0i		−	0.808222i	0.914710	−	0.404111i	$-0.132419\pi$
$294$	−26994.6	+	117390.i	−0.312307	+	1.35811i
$295$	−56408.1			−0.648182
$296$			10442.1i			0.119180i
$297$	15542.6	−	19295.7i	0.176202	−	0.218749i
$298$	−143513.			−1.61607
$299$			12880.7i			0.144078i
$300$	39450.8	+	9072.00i	0.438343	+	0.100800i
$301$	35037.8			0.386727
$302$			177711.i			1.94850i
$303$	22428.2	−	97532.1i	0.244292	−	1.06234i
$304$	−168415.			−1.82236
$305$			8236.74i			0.0885432i
$306$	161854.	+	78595.2i	1.72855	+	0.839369i
$307$	63952.5			0.678549			0.339274	−	0.940687i	$-0.389818\pi$
$307$	63952.5			0.678549			0.339274	+	0.940687i	$0.389818\pi$
$308$			28542.5i			0.300878i
$309$	−107406.	−	24698.8i	−1.12490	−	0.258678i
$310$	26355.4			0.274250
$311$		−	18806.5i		−	0.194441i	−0.995263	−	0.0972203i	$-0.969005\pi$
$311$		−	18806.5i		−	0.194441i	0.995263	−	0.0972203i	$-0.0309951\pi$
$312$	36604.4	−	159179.i	0.376031	−	1.63522i
$313$	62746.1			0.640469			0.320235	−	0.947338i	$-0.396238\pi$
$313$	62746.1			0.640469			0.320235	+	0.947338i	$0.396238\pi$
$314$		−	10088.0i		−	0.102316i
$315$	−9232.45	+	19012.8i	−0.0930456	+	0.191613i
$316$	229104.			2.29434
$317$			93427.4i			0.929728i	0.885382	+	0.464864i	$0.153897\pi$
$317$			93427.4i			0.929728i	−0.885382	+	0.464864i	$0.846103\pi$
$318$	−187513.	−	43120.0i	−1.85429	−	0.426407i
$319$	−53193.8			−0.522732
$320$		−	458.309i		−	0.00447568i
$321$	1429.61	−	6216.86i	0.0138742	−	0.0603338i
$322$	−17206.7			−0.165954
$323$		−	112061.i		−	1.07412i
$324$	−145990.	−	185531.i	−1.39070	−	1.76737i
$325$	15745.7			0.149071
$326$		−	57847.5i		−	0.544314i
$327$	−7967.71	−	1832.23i	−0.0745140	−	0.0171350i
$328$	309614.			2.87788
$329$			45705.1i			0.422253i
$330$	5525.90	−	24030.2i	0.0507429	−	0.220663i
$331$	−71675.0			−0.654202			−0.327101	−	0.944989i	$-0.606072\pi$
$331$	−71675.0			−0.654202			−0.327101	+	0.944989i	$0.606072\pi$
$332$		−	247999.i		−	2.24995i
$333$	5281.00	+	2564.41i	0.0476242	+	0.0231259i
$334$	151036.			1.35391
$335$			47568.3i			0.423865i
$336$	94785.2	+	21796.5i	0.839580	+	0.193067i
$337$	−87920.2			−0.774157			−0.387078	−	0.922047i	$-0.626516\pi$
$337$	−87920.2			−0.774157			−0.387078	+	0.922047i	$0.626516\pi$
$338$			91520.6i			0.801098i
$339$	−6940.92	+	30183.6i	−0.0603973	+	0.262646i
$340$	123946.			1.07220
$341$		−	11112.3i		−	0.0955645i
$342$	−92786.3	+	191079.i	−0.793290	+	1.63365i
$343$	−99360.4			−0.844550
$344$			216292.i			1.82778i
$345$	10027.6	+	2305.92i	0.0842481	+	0.0193734i
$346$	−148269.			−1.23850
$347$		−	5941.38i		−	0.0493433i	−0.999696	−	0.0246717i	$-0.992146\pi$
$347$		−	5941.38i		−	0.0493433i	0.999696	−	0.0246717i	$-0.00785403\pi$
$348$	−113589.	+	493955.i	−0.937942	+	4.07877i
$349$	−65214.3			−0.535417			−0.267709	−	0.963500i	$-0.586266\pi$
$349$	−65214.3			−0.535417			−0.267709	+	0.963500i	$0.586266\pi$
$350$			21033.8i			0.171705i
$351$	−71514.1	−	57604.2i	−0.580467	−	0.467563i
$352$	−35116.8			−0.283419
$353$			36220.5i			0.290673i	0.989382	+	0.145337i	$0.0464265\pi$
$353$			36220.5i			0.290673i	−0.989382	+	0.145337i	$0.953573\pi$
$354$	319057.	+	73369.4i	2.54602	+	0.585475i
$355$	14577.4			0.115670
$356$		−	230354.i		−	1.81759i
$357$	−14503.2	+	63069.1i	−0.113796	+	0.494857i
$358$	−233569.			−1.82242
$359$		−	152925.i		−	1.18656i	−0.804997	−	0.593279i	$-0.797833\pi$
$359$		−	152925.i		−	1.18656i	0.804997	−	0.593279i	$-0.202167\pi$
$360$	−117368.	−	56992.8i	−0.905615	−	0.439759i
$361$	1974.27			0.0151493
$362$		−	130178.i		−	0.993389i
$363$	118285.	+	27200.6i	0.897673	+	0.206426i
$364$	105785.			0.798400
$365$			75670.1i			0.567987i
$366$	10713.4	−	46588.9i	0.0799773	−	0.347792i
$367$	143778.			1.06748			0.533740	−	0.845649i	$-0.320786\pi$
$367$	143778.			1.06748			0.533740	+	0.845649i	$0.320786\pi$
$368$		−	47347.6i		−	0.349625i
$369$	76036.2	−	156584.i	0.558429	−	1.15000i
$370$	5842.37			0.0426762
$371$		−	69203.7i		−	0.502784i
$372$	−103189.	−	23729.0i	−0.745670	−	0.171472i
$373$	102685.			0.738056			0.369028	−	0.929418i	$-0.379691\pi$
$373$	102685.			0.738056			0.369028	+	0.929418i	$0.379691\pi$
$374$		−	75497.5i		−	0.539746i
$375$	2818.80	−	12258.0i	0.0200448	−	0.0871677i
$376$	−282142.			−1.99568
$377$			197148.i			1.38711i
$378$	76950.6	−	95532.1i	0.538553	−	0.668599i
$379$	−148649.			−1.03486			−0.517432	−	0.855724i	$-0.673112\pi$
$379$	−148649.			−1.03486			−0.517432	+	0.855724i	$0.673112\pi$
$380$			146326.i			1.01334i
$381$	251279.	+	57783.3i	1.73103	+	0.398063i
$382$	84505.7			0.579108
$383$		−	88390.0i		−	0.602568i	−0.953535	−	0.301284i	$-0.902585\pi$
$383$		−	88390.0i		−	0.602568i	0.953535	−	0.301284i	$-0.0974152\pi$
$384$	32747.7	−	142408.i	0.222084	−	0.965765i
$385$	8868.57			0.0598318
$386$			203703.i			1.36717i
$387$	109388.	+	53117.8i	0.730376	+	0.354665i
$388$	355506.			2.36148
$389$			107542.i			0.710687i	0.934736	+	0.355344i	$0.115636\pi$
$389$			107542.i			0.710687i	−0.934736	+	0.355344i	$0.884364\pi$
$390$	−89061.1	−	20480.2i	−0.585543	−	0.134650i
$391$	31504.6			0.206073
$392$		−	267442.i		−	1.74044i
$393$	808.328	−	3515.13i	0.00523362	−	0.0227591i
$394$	−384375.			−2.47607
$395$		−	71186.0i		−	0.456247i
$396$	−43270.8	+	89109.4i	−0.275934	+	0.568242i
$397$	−131243.			−0.832710			−0.416355	−	0.909202i	$-0.636693\pi$
$397$	−131243.			−0.832710			−0.416355	+	0.909202i	$0.636693\pi$
$398$		−	363844.i		−	2.29693i
$399$	−74456.8	−	17121.9i	−0.467691	−	0.107549i
$400$	−57878.6			−0.361742
$401$		−	21480.0i		−	0.133581i	−0.997767	−	0.0667907i	$-0.978724\pi$
$401$		−	21480.0i		−	0.133581i	0.997767	−	0.0667907i	$-0.0212760\pi$
$402$	61871.6	−	269057.i	0.382859	−	1.66492i
$403$	−41184.8			−0.253587
$404$			400118.i			2.45146i
$405$	−57647.3	+	45361.2i	−0.351454	+	0.276550i
$406$	−263360.			−1.59771
$407$		−	2463.34i		−	0.0148708i
$408$	−389332.	−	89529.5i	−2.33883	−	0.537831i
$409$	118822.			0.710313			0.355156	−	0.934807i	$-0.384428\pi$
$409$	118822.			0.710313			0.355156	+	0.934807i	$0.384428\pi$
$410$		−	173230.i		−	1.03052i
$411$	−59052.2	+	256797.i	−0.349585	+	1.52022i
$412$	440626.			2.59583
$413$			117751.i			0.690344i
$414$	−53719.3	−	26085.7i	−0.313422	−	0.152195i
$415$	−77056.8			−0.447419
$416$			130151.i			0.752073i
$417$	16244.4	+	3735.52i	0.0934183	+	0.0214822i
$418$	89129.3			0.510115
$419$		−	95945.4i		−	0.546507i	−0.961942	−	0.273254i	$-0.911900\pi$
$419$		−	95945.4i		−	0.546507i	0.961942	−	0.273254i	$-0.0880998\pi$
$420$	18937.7	−	82353.2i	0.107357	−	0.466855i
$421$	73818.2			0.416485			0.208242	−	0.978077i	$-0.433226\pi$
$421$	73818.2			0.416485			0.208242	+	0.978077i	$0.433226\pi$
$422$			485435.i			2.72588i
$423$	−69289.6	+	142691.i	−0.387246	+	0.797472i
$424$	427201.			2.37630
$425$		−	38511.8i		−	0.213214i
$426$	−82452.9	−	18960.6i	−0.454346	−	0.104480i
$427$	17194.1			0.0943026
$428$			25504.2i			0.139227i
$429$	−8635.15	+	37551.1i	−0.0469197	+	0.204037i
$430$	121016.			0.654493
$431$		−	107733.i		−	0.579955i	−0.957033	−	0.289977i	$-0.906352\pi$
$431$		−	107733.i		−	0.579955i	0.957033	−	0.289977i	$-0.0936478\pi$
$432$	262875.	+	211744.i	1.40858	+	1.13460i
$433$	−114738.			−0.611974			−0.305987	−	0.952036i	$-0.598986\pi$
$433$	−114738.			−0.611974			−0.305987	+	0.952036i	$0.598986\pi$
$434$		−	55016.7i		−	0.292089i
$435$	153479.	+	35293.6i	0.811094	+	0.186517i
$436$	32686.9			0.171949
$437$			37193.1i			0.194760i
$438$	98423.4	−	428008.i	0.513039	−	2.23102i
$439$	257591.			1.33660			0.668300	−	0.743892i	$-0.267022\pi$
$439$	257591.			1.33660			0.668300	+	0.743892i	$0.267022\pi$
$440$			54746.5i			0.282782i
$441$	−135257.	−	65679.6i	−0.695475	−	0.337717i
$442$	−279811.			−1.43225
$443$		−	35360.6i		−	0.180182i	−0.995934	−	0.0900911i	$-0.971284\pi$
$443$		−	35360.6i		−	0.180182i	0.995934	−	0.0900911i	$-0.0287158\pi$
$444$	−22874.5	−	5260.14i	−0.116034	−	0.0266828i
$445$	−71574.5			−0.361442
$446$			408028.i			2.05126i
$447$	40147.8	−	174588.i	0.200931	−	0.873777i
$448$	−956.716			−0.00476680
$449$		−	8230.74i		−	0.0408269i	−0.999792	−	0.0204134i	$-0.993502\pi$
$449$		−	8230.74i		−	0.0408269i	0.999792	−	0.0204134i	$-0.00649825\pi$
$450$	−31887.6	+	65667.5i	−0.157470	+	0.324284i
$451$	−73039.4			−0.359091
$452$		−	123826.i		−	0.606086i
$453$	−216191.	−	49714.6i	−1.05352	−	0.242263i
$454$	−137103.			−0.665172
$455$		−	32868.9i		−	0.158768i
$456$	105695.	−	459629.i	0.508305	−	2.21044i
$457$	−180660.			−0.865027			−0.432514	−	0.901627i	$-0.642373\pi$
$457$	−180660.			−0.865027			−0.432514	+	0.901627i	$0.642373\pi$
$458$			570158.i			2.71809i
$459$	−140892.	+	174914.i	−0.668748	+	0.830232i
$460$	−41137.5			−0.194412
$461$			34875.9i			0.164106i	0.996628	+	0.0820529i	$0.0261476\pi$
$461$			34875.9i			0.164106i	−0.996628	+	0.0820529i	$0.973852\pi$
$462$	−50162.7	−	11535.3i	−0.235016	−	0.0540435i
$463$	−206020.			−0.961054			−0.480527	−	0.876980i	$-0.659555\pi$
$463$	−206020.			−0.961054			−0.480527	+	0.876980i	$0.659555\pi$
$464$		−	724686.i		−	3.36600i
$465$	−7372.94	+	32062.3i	−0.0340985	+	0.148282i
$466$	261810.			1.20563
$467$		−	274766.i		−	1.25988i	−0.776644	−	0.629940i	$-0.783079\pi$
$467$		−	274766.i		−	1.25988i	0.776644	−	0.629940i	$-0.216921\pi$
$468$	330259.	+	160371.i	1.50787	+	0.732209i
$469$	99298.2			0.451436
$470$			157859.i			0.714618i
$471$	12272.3	+	2822.11i	0.0553204	+	0.0127213i
$472$	−726890.			−3.26276
$473$		−	51024.3i		−	0.228063i
$474$	−92590.9	+	402644.i	−0.412108	+	1.79211i
$475$	45465.5			0.201509
$476$		−	258736.i		−	1.14194i
$477$	104914.	−	216053.i	0.461101	−	0.949564i
$478$	337306.			1.47628
$479$			160960.i			0.701533i	0.936463	+	0.350766i	$0.114079\pi$
$479$			160960.i			0.701533i	−0.936463	+	0.350766i	$0.885921\pi$
$480$	101322.	+	23299.7i	0.439765	+	0.101127i
$481$	−9129.68			−0.0394608
$482$		−	220957.i		−	0.951074i
$483$	4813.59	−	20932.6i	0.0206336	−	0.0897280i
$484$	−485257.			−2.07148
$485$		−	110461.i		−	0.469597i
$486$	385067.	−	181592.i	1.63029	−	0.768819i
$487$	−69827.5			−0.294421			−0.147210	−	0.989105i	$-0.547029\pi$
$487$	−69827.5			−0.294421			−0.147210	+	0.989105i	$0.547029\pi$
$488$			106141.i			0.445700i
$489$	70373.4	+	16182.9i	0.294300	+	0.0676764i
$490$	−149635.			−0.623218
$491$		−	430893.i		−	1.78734i	−0.448727	−	0.893669i	$-0.648122\pi$
$491$		−	430893.i		−	1.78734i	0.448727	−	0.893669i	$-0.351878\pi$
$492$	−155966.	+	678241.i	−0.644318	+	2.80191i
$493$	482198.			1.98396
$494$		−	330333.i		−	1.35363i
$495$	27687.6	+	13444.9i	0.112999	+	0.0548715i
$496$	151389.			0.615362
$497$		−	30430.1i		−	0.123194i
$498$	435851.	+	100227.i	1.75744	+	0.404135i
$499$	282872.			1.13603			0.568013	−	0.823019i	$-0.307712\pi$
$499$	282872.			1.13603			0.568013	+	0.823019i	$0.307712\pi$
$500$			50287.3i			0.201149i
$501$	−42252.5	+	183741.i	−0.168336	+	0.732032i
$502$	827726.			3.28458
$503$			240734.i			0.951483i	0.879585	+	0.475742i	$0.157820\pi$
$503$			240734.i			0.951483i	−0.879585	+	0.475742i	$0.842180\pi$
$504$	−118972.	+	245004.i	−0.468364	+	0.964521i
$505$	124322.			0.487491
$506$			25057.5i			0.0978673i
$507$	−111338.	−	25602.9i	−0.433138	−	0.0996032i
$508$	−1.03085e6			−3.99455
$509$			225427.i			0.870102i	0.900406	+	0.435051i	$0.143270\pi$
$509$			225427.i			0.870102i	−0.900406	+	0.435051i	$0.856730\pi$
$510$	−50092.0	+	217832.i	−0.192587	+	0.837493i
$511$	157961.			0.604933
$512$			588946.i			2.24665i
$513$	−206497.	−	166332.i	−0.784654	−	0.632035i
$514$	−646027.			−2.44526
$515$		−	136909.i		−	0.516199i
$516$	−473810.	−	108956.i	−1.77953	−	0.409215i
$517$	66558.6			0.249014
$518$		−	12195.9i		−	0.0454521i
$519$	41478.2	−	180374.i	0.153987	−	0.669635i
$520$	202903.			0.750381
$521$		−	154960.i		−	0.570880i	−0.958397	−	0.285440i	$-0.907860\pi$
$521$		−	154960.i		−	0.570880i	0.958397	−	0.285440i	$-0.0921397\pi$
$522$	−822208.	−	399258.i	−3.01745	−	1.46525i
$523$	−128163.			−0.468553			−0.234277	−	0.972170i	$-0.575272\pi$
$523$	−128163.			−0.468553			−0.234277	+	0.972170i	$0.575272\pi$
$524$			14420.5i			0.0525193i
$525$	−25588.4	−	5884.22i	−0.0928376	−	0.0213486i
$526$	821153.			2.96792
$527$			100733.i			0.362701i
$528$	31741.5	−	138032.i	0.113857	−	0.495123i
$529$	269385.			0.962635
$530$		−	239020.i		−	0.850907i
$531$	−178513.	+	367618.i	−0.633111	+	1.30379i
$532$	305453.			1.07925
$533$			270700.i			0.952872i
$534$	404842.	+	93096.3i	1.41972	+	0.326475i
$535$	7924.52			0.0276863
$536$			612978.i			2.13361i
$537$	65340.9	−	284144.i	0.226588	−	0.985349i
$538$	−282861.			−0.977256
$539$			63090.9i			0.217165i
$540$	183972.	−	228396.i	0.630905	−	0.783252i
$541$	−499660.			−1.70718			−0.853591	−	0.520943i	$-0.825580\pi$
$541$	−499660.			−1.70718			−0.853591	+	0.520943i	$0.825580\pi$
$542$		−	378876.i		−	1.28973i
$543$	158365.	+	36417.2i	0.537107	+	0.123511i
$544$	318332.			1.07568
$545$		−	10156.3i		−	0.0341934i
$546$	−42752.3	+	185914.i	−0.143408	+	0.623630i
$547$	−244301.			−0.816490			−0.408245	−	0.912872i	$-0.633859\pi$
$547$	−244301.			−0.816490			−0.408245	+	0.912872i	$0.633859\pi$
$548$		−	1.05349e6i		−	3.50807i
$549$	53679.8	+	26066.5i	0.178101	+	0.0864845i
$550$	30630.8			0.101259
$551$			569264.i			1.87504i
$552$	129219.	+	29714.8i	0.424080	+	0.0975201i
$553$	−148600.			−0.485924
$554$			206977.i			0.674376i
$555$	−1634.40	+	7107.43i	−0.00530608	+	0.0230742i
$556$	−66641.4			−0.215573
$557$			247875.i			0.798954i	0.916743	+	0.399477i	$0.130808\pi$
$557$			247875.i			0.798954i	−0.916743	+	0.399477i	$0.869192\pi$
$558$	83406.1	−	171762.i	0.267873	−	0.551643i
$559$	−189107.			−0.605181
$560$			120821.i			0.385271i
$561$	91845.2	+	21120.4i	0.291830	+	0.0671085i
$562$	602295.			1.90694
$563$		−	265392.i		−	0.837279i	−0.908152	−	0.418640i	$-0.862507\pi$
$563$		−	265392.i		−	0.837279i	0.908152	−	0.418640i	$-0.137493\pi$
$564$	142127.	−	618061.i	0.446807	−	1.94300i
$565$	−38474.5			−0.120525
$566$		−	913633.i		−	2.85193i
$567$	94690.9	+	120338.i	0.294539	+	0.374315i
$568$	187848.			0.582250
$569$		−	551671.i		−	1.70394i	−0.523587	−	0.851972i	$-0.675407\pi$
$569$		−	551671.i		−	1.70394i	0.523587	−	0.851972i	$-0.324593\pi$
$570$	−257164.	−	59136.6i	−0.791516	−	0.182015i
$571$	325456.			0.998205			0.499102	−	0.866543i	$-0.333663\pi$
$571$	325456.			0.998205			0.499102	+	0.866543i	$0.333663\pi$
$572$		−	154051.i		−	0.470838i
$573$	−23640.5	+	102804.i	−0.0720024	+	0.313113i
$574$	−361615.			−1.09755
$575$			12782.0i			0.0386602i
$576$	−2986.86	−	1450.40i	−0.00900264	−	0.00437161i
$577$	−262408.			−0.788181			−0.394091	−	0.919072i	$-0.628940\pi$
$577$	−262408.			−0.788181			−0.394091	+	0.919072i	$0.628940\pi$
$578$			82202.3i			0.246053i
$579$	−247811.	−	56985.9i	−0.739203	−	0.169985i
$580$	−629636.			−1.87169
$581$			160855.i			0.476522i
$582$	−143676.	+	624793.i	−0.424167	+	1.84455i
$583$	−100779.			−0.296505
$584$			975106.i			2.85908i
$585$	49829.7	−	102616.i	0.145605	−	0.299851i
$586$	−500259.			−1.45680
$587$			427538.i			1.24079i	0.784290	+	0.620395i	$0.213028\pi$
$587$			427538.i			1.24079i	−0.784290	+	0.620395i	$0.786972\pi$
$588$	585860.	+	134723.i	1.69449	+	0.389660i
$589$	−118921.			−0.342790
$590$			406696.i			1.16833i
$591$	107529.	−	467605.i	0.307858	−	1.33876i
$592$	33559.3			0.0957568
$593$		−	63018.6i		−	0.179209i	−0.995977	−	0.0896043i	$-0.971440\pi$
$593$		−	63018.6i		−	0.179209i	0.995977	−	0.0896043i	$-0.0285603\pi$
$594$	−139120.	−	112060.i	−0.394290	−	0.317599i
$595$	−80393.0			−0.227083
$596$			716235.i			2.01634i
$597$	442628.	+	101785.i	1.24191	+	0.285586i
$598$	92868.9			0.259698
$599$			634561.i			1.76856i	0.466959	+	0.884279i	$0.345350\pi$
$599$			634561.i			1.76856i	−0.466959	+	0.884279i	$0.654650\pi$
$600$	36323.9	−	157959.i	0.100900	−	0.438776i
$601$	−474806.			−1.31452			−0.657260	−	0.753664i	$-0.728285\pi$
$601$	−474806.			−1.31452			−0.657260	+	0.753664i	$0.728285\pi$
$602$		−	252619.i		−	0.697065i
$603$	310008.	+	150538.i	0.852587	+	0.414009i
$604$	886906.			2.43111
$605$			150776.i			0.411929i
$606$	−703197.	−	161705.i	−1.91484	−	0.440330i
$607$	−163861.			−0.444731			−0.222365	−	0.974963i	$-0.571378\pi$
$607$	−163861.			−0.444731			−0.222365	+	0.974963i	$0.571378\pi$
$608$			375809.i			1.01662i
$609$	73675.1	−	320386.i	0.198649	−	0.863852i
$610$	59386.0			0.159597
$611$		−	246681.i		−	0.660775i
$612$	392247.	−	807771.i	1.04727	−	2.15668i
$613$	328099.			0.873139			0.436570	−	0.899670i	$-0.356193\pi$
$613$	328099.			0.873139			0.436570	+	0.899670i	$0.356193\pi$
$614$		−	461091.i		−	1.22307i
$615$	210739.	+	48461.0i	0.557180	+	0.128127i
$616$	114283.			0.301175
$617$		−	405778.i		−	1.06590i	−0.846146	−	0.532952i	$-0.821083\pi$
$617$		−	405778.i		−	1.06590i	0.846146	−	0.532952i	$-0.178917\pi$
$618$	−178076.	+	774389.i	−0.466261	+	2.02760i
$619$	184587.			0.481747			0.240874	−	0.970557i	$-0.422566\pi$
$619$	184587.			0.481747			0.240874	+	0.970557i	$0.422566\pi$
$620$		−	131533.i		−	0.342177i
$621$	46762.0	−	58053.8i	0.121258	−	0.150538i
$622$	−135593.			−0.350474
$623$			149411.i			0.384952i
$624$	−511579.	−	117641.i	−1.31384	−	0.302127i
$625$	15625.0			0.0400000
$626$		−	452393.i		−	1.15443i
$627$	−24933.9	+	108429.i	−0.0634243	+	0.275810i
$628$	−50346.2			−0.127658
$629$			22330.0i			0.0564401i
$630$	137080.	+	66565.0i	0.345377	+	0.167712i
$631$	712031.			1.78830			0.894149	−	0.447769i	$-0.147781\pi$
$631$	712031.			1.78830			0.894149	+	0.447769i	$0.147781\pi$
$632$		−	917322.i		−	2.29661i
$633$	−590548.	−	135801.i	−1.47383	−	0.338918i
$634$	673602.			1.67581
$635$			320300.i			0.794346i
$636$	−215200.	+	935828.i	−0.532021	+	2.31357i
$637$	233829.			0.576262
$638$			383522.i			0.942212i
$639$	46132.4	−	95002.4i	0.112981	−	0.232666i
$640$	181525.			0.443176
$641$		−	388033.i		−	0.944393i	−0.881493	−	0.472196i	$-0.843461\pi$
$641$		−	388033.i		−	0.944393i	0.881493	−	0.472196i	$-0.156539\pi$
$642$	−44822.9	−	10307.3i	−0.108750	−	0.0250079i
$643$	−276131.			−0.667872			−0.333936	−	0.942596i	$-0.608377\pi$
$643$	−276131.			−0.667872			−0.333936	+	0.942596i	$0.608377\pi$
$644$			85874.2i			0.207057i
$645$	−33854.2	+	147220.i	−0.0813753	+	0.353872i
$646$	−807952.			−1.93607
$647$			603278.i			1.44115i	0.693378	+	0.720574i	$0.256122\pi$
$647$			603278.i			1.44115i	−0.693378	+	0.720574i	$0.743878\pi$
$648$	−742859.	+	584536.i	−1.76912	+	1.39207i
$649$	171477.			0.407114
$650$		−	113525.i		−	0.268697i
$651$	66929.6	+	15390.9i	0.157927	+	0.0363164i
$652$	−288701.			−0.679131
$653$		−	625614.i		−	1.46717i	−0.679598	−	0.733584i	$-0.737846\pi$
$653$		−	625614.i		−	1.46717i	0.679598	−	0.733584i	$-0.262154\pi$
$654$	−13210.2	+	57446.4i	−0.0308854	+	0.134310i
$655$	4480.67			0.0104438
$656$		−	995053.i		−	2.31227i
$657$	493152.	+	239471.i	1.14248	+	0.554781i
$658$	329529.			0.761100
$659$		−	706201.i		−	1.62614i	−0.582167	−	0.813069i	$-0.697795\pi$
$659$		−	706201.i		−	1.62614i	0.582167	−	0.813069i	$-0.302205\pi$
$660$	−119928.	−	27578.3i	−0.275317	−	0.0633110i
$661$	−296423.			−0.678437			−0.339219	−	0.940708i	$-0.610163\pi$
$661$	−296423.			−0.678437			−0.339219	+	0.940708i	$0.610163\pi$
$662$			516769.i			1.17918i
$663$	78277.1	−	340399.i	0.178077	−	0.774392i
$664$	−992975.			−2.25218
$665$		−	94908.8i		−	0.214617i
$666$	18489.1	−	38075.4i	0.0416839	−	0.0858414i
$667$	−160041.			−0.359733
$668$		−	753782.i		−	1.68925i
$669$	−496379.	−	114146.i	−1.10908	−	0.255040i
$670$	342962.			0.764006
$671$		−	25039.1i		−	0.0556127i
$672$	48637.9	−	211508.i	0.107705	−	0.468370i
$673$	52622.9			0.116184			0.0580918	−	0.998311i	$-0.481498\pi$
$673$	52622.9			0.116184			0.0580918	+	0.998311i	$0.481498\pi$
$674$			633896.i			1.39540i
$675$	−70966.0	−	57162.8i	−0.155755	−	0.125460i
$676$	456754.			0.999515
$677$		−	341720.i		−	0.745578i	−0.927916	−	0.372789i	$-0.878401\pi$
$677$		−	341720.i		−	0.745578i	0.927916	−	0.372789i	$-0.121599\pi$
$678$	217621.	+	50043.4i	0.473413	+	0.108865i
$679$	−230586.			−0.500143
$680$		−	496274.i		−	1.07326i
$681$	38354.5	−	166790.i	0.0827032	−	0.359646i
$682$	−80118.8			−0.172253
$683$			247237.i			0.529996i	0.964249	+	0.264998i	$0.0853714\pi$
$683$			247237.i			0.529996i	−0.964249	+	0.264998i	$0.914629\pi$
$684$	953622.	+	463071.i	2.03828	+	0.989774i
$685$	−327334.			−0.697606
$686$			716379.i			1.52228i
$687$	−693616.	−	159502.i	−1.46962	−	0.337950i
$688$	695130.			1.46855
$689$			373509.i			0.786796i
$690$	16625.5	−	72298.2i	0.0349201	−	0.151855i
$691$	471510.			0.987496			0.493748	−	0.869605i	$-0.335627\pi$
$691$	471510.			0.987496			0.493748	+	0.869605i	$0.335627\pi$
$692$			739968.i			1.54526i
$693$	28066.0	−	57797.6i	0.0584406	−	0.120349i
$694$	−42836.8			−0.0889401
$695$			20706.5i			0.0428683i
$696$	1.97778e6	+	454803.i	4.08280	+	0.938869i
$697$	662097.			1.36288
$698$			470189.i			0.965076i
$699$	−73241.5	+	318501.i	−0.149900	+	0.651863i
$700$	104974.			0.214233
$701$			530831.i			1.08024i	0.841588	+	0.540120i	$0.181621\pi$
$701$			530831.i			1.08024i	−0.841588	+	0.540120i	$0.818379\pi$
$702$	−415321.	+	515609.i	−0.842771	+	1.04628i
$703$	−26361.9			−0.0533416
$704$			1393.23i			0.00281111i
$705$	−192041.	−	44161.1i	−0.386380	−	0.0888508i
$706$	261146.			0.523931
$707$		−	259522.i		−	0.519200i
$708$	366167.	−	1.59233e6i	0.730487	−	3.17662i
$709$	−938943.			−1.86787			−0.933936	−	0.357441i	$-0.883649\pi$
$709$	−938943.			−1.86787			−0.933936	+	0.357441i	$0.883649\pi$
$710$		−	105101.i		−	0.208493i
$711$	−463928.	−	225280.i	−0.917722	−	0.445639i
$712$	−922329.			−1.81939
$713$		−	33433.1i		−	0.0657653i
$714$	454722.	+	104566.i	0.891968	+	0.205114i
$715$	−47865.8			−0.0936296
$716$			1.16568e6i			2.27380i
$717$	−94361.3	+	410343.i	−0.183551	+	0.798195i
$718$	−1.10257e6			−2.13874
$719$		−	918433.i		−	1.77660i	−0.459263	−	0.888300i	$-0.651886\pi$
$719$		−	918433.i		−	1.77660i	0.459263	−	0.888300i	$-0.348114\pi$
$720$	−183166.	+	377202.i	−0.353330	+	0.727628i
$721$	−285796.			−0.549776
$722$		−	14234.3i		−	0.0273062i
$723$	268802.	+	61812.8i	0.514228	+	0.118250i
$724$	−649681.			−1.23943
$725$			195637.i			0.372199i
$726$	196113.	−	852826.i	0.372078	−	1.61803i
$727$	−314965.			−0.595928			−0.297964	−	0.954577i	$-0.596308\pi$
$727$	−314965.			−0.595928			−0.297964	+	0.954577i	$0.596308\pi$
$728$		−	423558.i		−	0.799190i
$729$	113190.	+	519247.i	0.212987	+	0.977055i
$730$	545574.			1.02378
$731$			462532.i			0.865580i
$732$	−232513.	−	53467.9i	−0.433935	−	0.0997863i
$733$	958567.			1.78408			0.892040	−	0.451956i	$-0.149274\pi$
$733$	958567.			1.78408			0.892040	+	0.451956i	$0.149274\pi$
$734$		−	1.03662e6i		−	1.92411i
$735$	41860.3	−	182035.i	0.0774868	−	0.336962i
$736$	−105654.			−0.195043
$737$		−	144604.i		−	0.266224i
$738$	−1.12896e6	−	548213.i	−2.07284	−	1.00655i
$739$	197682.			0.361976			0.180988	−	0.983485i	$-0.442071\pi$
$739$	197682.			0.361976			0.180988	+	0.983485i	$0.442071\pi$
$740$		−	29157.7i		−	0.0532463i
$741$	401861.	+	92410.8i	0.731880	+	0.168301i
$742$	−498952.			−0.906255
$743$		−	81590.4i		−	0.147796i	−0.997266	−	0.0738978i	$-0.976456\pi$
$743$		−	81590.4i		−	0.147796i	0.997266	−	0.0738978i	$-0.0235439\pi$
$744$	−95009.7	+	413163.i	−0.171641	+	0.746407i
$745$	222545.			0.400964
$746$		−	740349.i		−	1.33033i
$747$	−243859.	+	502189.i	−0.437016	+	0.899965i
$748$	−376788.			−0.673431
$749$		−	16542.3i		−	0.0294872i
$750$	−88378.6	−	20323.3i	−0.157118	−	0.0361303i
$751$	132505.			0.234938			0.117469	−	0.993077i	$-0.462522\pi$
$751$	132505.			0.234938			0.117469	+	0.993077i	$0.462522\pi$
$752$			906762.i			1.60346i
$753$	−231557.	+	1.00696e6i	−0.408383	+	1.77591i
$754$	1.42142e6			2.50022
$755$		−	275575.i		−	0.483443i
$756$	−476775.	−	384040.i	−0.834199	−	0.671943i
$757$	997153.			1.74008			0.870042	−	0.492978i	$-0.164092\pi$
$757$	997153.			1.74008			0.870042	+	0.492978i	$0.164092\pi$
$758$			1.07174e6i			1.86532i
$759$	−30483.3	−	7009.85i	−0.0529150	−	0.0121682i
$760$	585881.			1.01434
$761$			316363.i			0.546281i	0.961974	+	0.273140i	$0.0880623\pi$
$761$			316363.i			0.546281i	−0.961974	+	0.273140i	$0.911938\pi$
$762$	416612.	−	1.81169e6i	0.717499	−	3.12014i
$763$	−21201.2			−0.0364175
$764$		−	421745.i		−	0.722542i
$765$	−250986.	−	121877.i	−0.428872	−	0.208257i
$766$	−637283.			−1.08611
$767$		−	635532.i		−	1.08031i
$768$	−1.03250e6	−	237430.i	−1.75052	−	0.402544i
$769$	584632.			0.988621			0.494311	−	0.869285i	$-0.335421\pi$
$769$	584632.			0.988621			0.494311	+	0.869285i	$0.335421\pi$
$770$		−	63941.5i		−	0.107845i
$771$	180726.	−	785913.i	0.304027	−	1.32210i
$772$	1.01663e6			1.70579
$773$			662736.i			1.10913i	0.832141	+	0.554564i	$0.187115\pi$
$773$			662736.i			1.10913i	−0.832141	+	0.554564i	$0.812885\pi$
$774$	382974.	−	788675.i	0.639275	−	1.31649i
$775$	−40869.2			−0.0680444
$776$		−	1.42343e6i		−	2.36381i
$777$	14836.7	+	3411.80i	0.0245751	+	0.00565121i
$778$	775367.			1.28100
$779$			781646.i			1.28806i
$780$	−102211.	+	444480.i	−0.168000	+	0.730572i
$781$	−44314.2			−0.0726509
$782$		−	227145.i		−	0.371441i
$783$	715723.	−	888551.i	1.16740	−	1.44930i
$784$	−859520.			−1.39838
$785$			15643.3i			0.0253857i
$786$	−25343.7	−	5827.96i	−0.0410228	−	0.00943348i
$787$	−773806.			−1.24935			−0.624673	−	0.780886i	$-0.714768\pi$
$787$	−773806.			−1.24935			−0.624673	+	0.780886i	$0.714768\pi$
$788$			1.91831e6i			3.08935i
$789$	−229718.	+	998959.i	−0.369012	+	1.60470i
$790$	−513244.			−0.822374
$791$			80315.0i			0.128364i
$792$	356790.	+	173254.i	0.568804	+	0.276207i
$793$	−92800.7			−0.147572
$794$			946246.i			1.50094i
$795$	290776.	+	66865.9i	0.460070	+	0.105796i
$796$	−1.81584e6			−2.86584
$797$		−	481401.i		−	0.757863i	−0.925425	−	0.378932i	$-0.876292\pi$
$797$		−	481401.i		−	0.757863i	0.925425	−	0.378932i	$-0.123708\pi$
$798$	−123447.	+	536826.i	−0.193854	+	0.843001i
$799$	−603350.			−0.945096
$800$			129153.i			0.201802i
$801$	−226509.	+	466460.i	−0.353038	+	0.727025i
$802$	−154869.			−0.240777
$803$		−	230032.i		−	0.356745i
$804$	−1.34279e6	−	308784.i	−2.07729	−	0.477686i
$805$	26682.4			0.0411749
$806$			296938.i			0.457084i
$807$	79130.4	−	344109.i	0.121506	−	0.528384i
$808$	1.60205e6			2.45389
$809$		−	391432.i		−	0.598081i	−0.954240	−	0.299040i	$-0.903333\pi$
$809$		−	391432.i		−	0.598081i	0.954240	−	0.299040i	$-0.0966665\pi$
$810$	327049.	+	415631.i	0.498475	+	0.633487i
$811$	1.15536e6			1.75661			0.878305	−	0.478101i	$-0.158675\pi$
$811$	1.15536e6			1.75661			0.878305	+	0.478101i	$0.158675\pi$
$812$			1.31436e6i			1.99343i
$813$	460916.	+	105991.i	0.697333	+	0.160357i
$814$	−17760.4			−0.0268043
$815$			89703.7i			0.135050i
$816$	−287735.	+	1.25125e6i	−0.432127	+	1.87917i
$817$	−546047.			−0.818061
$818$		−	856694.i		−	1.28032i
$819$	−214211.	−	104019.i	−0.319355	−	0.155076i
$820$	−864542.			−1.28576
$821$		−	299979.i		−	0.445045i	−0.974928	−	0.222522i	$-0.928571\pi$
$821$		−	299979.i		−	0.445045i	0.974928	−	0.222522i	$-0.0714291\pi$
$822$	1.85148e6	+	425760.i	2.74015	+	0.630118i
$823$	32530.4			0.0480275			0.0240138	−	0.999712i	$-0.492355\pi$
$823$	32530.4			0.0480275			0.0240138	+	0.999712i	$0.492355\pi$
$824$		−	1.76425e6i		−	2.59839i
$825$	−8568.97	+	37263.4i	−0.0125899	+	0.0547488i
$826$	848975.			1.24433
$827$		−	764819.i		−	1.11827i	−0.829076	−	0.559136i	$-0.811133\pi$
$827$		−	764819.i		−	1.11827i	0.829076	−	0.559136i	$-0.188867\pi$
$828$	−130186.	+	268098.i	−0.189891	+	0.391051i
$829$	−137003.			−0.199352			−0.0996758	−	0.995020i	$-0.531781\pi$
$829$	−137003.			−0.199352			−0.0996758	+	0.995020i	$0.531781\pi$
$830$			555572.i			0.806462i
$831$	−251794.	−	57901.8i	−0.364622	−	0.0838475i
$832$	5163.63			0.00745947
$833$		−	571916.i		−	0.824218i
$834$	26932.7	−	117121.i	0.0387211	−	0.168384i
$835$	−234211.			−0.335919
$836$		−	444820.i		−	0.636461i
$837$	185621.	+	149517.i	0.264957	+	0.213422i
$838$	−691757.			−0.985066
$839$			1.12228e6i			1.59433i	0.603759	+	0.797167i	$0.293669\pi$
$839$			1.12228e6i			1.59433i	−0.603759	+	0.797167i	$0.706331\pi$
$840$	−329739.	−	75825.7i	−0.467317	−	0.107463i
$841$	−1.74225e6			−2.46331
$842$		−	532222.i		−	0.750704i
$843$	−168492.	+	732711.i	−0.237096	+	1.03105i
$844$	2.42267e6			3.40103
$845$		−	141920.i		−	0.198761i
$846$	1.02879e6	+	499571.i	1.43742	+	0.698001i
$847$	314744.			0.438723
$848$		−	1.37296e6i		−	1.90927i
$849$	1.11146e6	+	255589.i	1.54199	+	0.354590i
$850$	−277666.			−0.384313
$851$		−	7411.31i		−	0.0102338i
$852$	−94627.3	+	411500.i	−0.130358	+	0.566879i
$853$	43550.6			0.0598544			0.0299272	−	0.999552i	$-0.490472\pi$
$853$	43550.6			0.0598544			0.0299272	+	0.999552i	$0.490472\pi$
$854$		−	123968.i		−	0.169978i
$855$	143883.	−	296304.i	0.196824	−	0.405327i
$856$	102118.			0.139365
$857$		−	300075.i		−	0.408572i	−0.978911	−	0.204286i	$-0.934513\pi$
$857$		−	300075.i		−	0.408572i	0.978911	−	0.204286i	$-0.0654872\pi$
$858$	270740.	+	62258.6i	0.367771	+	0.0845716i
$859$	−498647.			−0.675782			−0.337891	−	0.941185i	$-0.609713\pi$
$859$	−498647.			−0.675782			−0.337891	+	0.941185i	$0.609713\pi$
$860$		−	603957.i		−	0.816599i
$861$	101162.	−	439916.i	0.136462	−	0.593422i
$862$	−776744.			−1.04535
$863$			13487.9i			0.0181102i	0.999959	+	0.00905512i	$0.00288237\pi$
$863$			13487.9i			0.0181102i	−0.999959	+	0.00905512i	$0.997118\pi$
$864$	472497.	−	586592.i	0.632953	−	0.785794i
$865$	229919.			0.307286
$866$			827252.i			1.10307i
$867$	−100002.	−	22996.1i	−0.133036	−	0.0305926i
$868$	−274573.			−0.364434
$869$			216401.i			0.286562i
$870$	254463.	−	1.10657e6i	0.336192	−	1.46198i
$871$	−535936.			−0.706443
$872$		−	130877.i		−	0.172119i
$873$	−719888.	−	349572.i	−0.944575	−	0.458678i
$874$	268158.			0.351050
$875$		−	32617.0i		−	0.0426018i
$876$	−2.13607e6	−	491205.i	−2.78360	−	0.640109i
$877$	−944732.			−1.22831			−0.614157	−	0.789184i	$-0.710504\pi$
$877$	−944732.			−1.22831			−0.614157	+	0.789184i	$0.710504\pi$
$878$		−	1.85720e6i		−	2.40919i
$879$	139948.	−	608582.i	0.181129	−	0.787664i
$880$	175947.			0.227205
$881$			1.28495e6i			1.65552i	0.561082	+	0.827760i	$0.310385\pi$
$881$			1.28495e6i			1.65552i	−0.561082	+	0.827760i	$0.689615\pi$
$882$	−473543.	+	975187.i	−0.608727	+	1.25358i
$883$	−1.22382e6			−1.56963			−0.784815	−	0.619731i	$-0.787242\pi$
$883$	−1.22382e6			−1.56963			−0.784815	+	0.619731i	$0.787242\pi$
$884$			1.39646e6i			1.78700i
$885$	−494760.	−	113773.i	−0.631695	−	0.145263i
$886$	−254946.			−0.324774
$887$			755435.i			0.960173i	0.877221	+	0.480087i	$0.159395\pi$
$887$			755435.i			0.960173i	−0.877221	+	0.480087i	$0.840605\pi$
$888$	−21061.4	+	91588.4i	−0.0267092	+	0.116149i
$889$	668624.			0.846015
$890$			516045.i			0.651490i
$891$	175244.	−	137895.i	0.220743	−	0.173697i
$892$	2.03636e6			2.55932
$893$		−	712291.i		−	0.893212i
$894$	−1.25877e6	−	289462.i	−1.57496	−	0.362173i
$895$	362193.			0.452162
$896$		−	378931.i		−	0.472002i
$897$	−25980.1	+	112978.i	−0.0322891	+	0.140414i
$898$	−59342.8			−0.0735894
$899$		−	511714.i		−	0.633152i
$900$	327729.	+	159142.i	0.404603	+	0.196472i
$901$	913553.			1.12534
$902$			526607.i			0.647252i
$903$	307319.	+	70670.2i	0.376890	+	0.0866685i
$904$	−495793.			−0.606685
$905$			201866.i			0.246471i
$906$	−358437.	+	1.55871e6i	−0.436674	+	1.89894i
$907$	1.28653e6			1.56389			0.781946	−	0.623346i	$-0.214227\pi$
$907$	1.28653e6			1.56389			0.781946	+	0.623346i	$0.214227\pi$
$908$			684242.i			0.829924i
$909$	393439.	−	810225.i	0.476156	−	0.980568i
$910$	−236982.			−0.286175
$911$		−	501962.i		−	0.604831i	−0.953176	−	0.302416i	$-0.902207\pi$
$911$		−	501962.i		−	0.604831i	0.953176	−	0.302416i	$-0.0977931\pi$
$912$	−1.47718e6	−	339688.i	−1.77600	−	0.408404i
$913$	234248.			0.281018
$914$			1.30254e6i			1.55919i
$915$	−16613.3	+	72245.1i	−0.0198432	+	0.0862911i
$916$	2.84551e6			3.39132
$917$		−	9353.35i		−	0.0111232i
$918$	1.26111e6	+	1.01582e6i	1.49647	+	1.20540i
$919$	−252142.			−0.298548			−0.149274	−	0.988796i	$-0.547694\pi$
$919$	−252142.			−0.298548			−0.149274	+	0.988796i	$0.547694\pi$
$920$			164713.i			0.194604i
$921$	560933.	+	128990.i	0.661289	+	0.152068i
$922$	251452.			0.295797
$923$			164238.i			0.192784i
$924$	−57569.4	+	250348.i	−0.0674291	+	0.293225i
$925$	−9059.72			−0.0105884
$926$			1.48539e6i			1.73228i
$927$	−892253.	−	433271.i	−1.03831	−	0.504197i
$928$	−1.61710e6			−1.87776
$929$		−	1.28900e6i		−	1.49356i	−0.665074	−	0.746778i	$-0.731600\pi$
$929$		−	1.28900e6i		−	1.49356i	0.665074	−	0.746778i	$-0.268400\pi$
$930$	231166.	+	53158.2i	0.267274	+	0.0614616i
$931$	675181.			0.778970
$932$		−	1.30662e6i		−	1.50425i
$933$	37932.1	−	164953.i	0.0435757	−	0.189495i
$934$	−1.98104e6			−2.27090
$935$			117073.i			0.133917i
$936$	642120.	−	1.32234e6i	0.732933	−	1.50936i
$937$	596815.			0.679768			0.339884	−	0.940467i	$-0.389612\pi$
$937$	596815.			0.679768			0.339884	+	0.940467i	$0.389612\pi$
$938$		−	715930.i		−	0.813701i
$939$	550351.	+	126557.i	0.624178	+	0.143534i
$940$	787832.			0.891616
$941$		−	1.55891e6i		−	1.76052i	−0.474493	−	0.880259i	$-0.657369\pi$
$941$		−	1.55891e6i		−	1.76052i	0.474493	−	0.880259i	$-0.342631\pi$
$942$	20347.1	−	88482.2i	0.0229298	−	0.0997136i
$943$	−219749.			−0.247118
$944$			2.33612e6i			2.62150i
$945$	−119327.	+	148141.i	−0.133621	+	0.165887i
$946$	−367880.			−0.411078
$947$			226143.i			0.252164i	0.992020	+	0.126082i	$0.0402402\pi$
$947$			226143.i			0.252164i	−0.992020	+	0.126082i	$0.959760\pi$
$948$	2.00949e6	+	462096.i	2.23599	+	0.514180i
$949$	−852551.			−0.946647
$950$		−	327802.i		−	0.363215i
$951$	−188440.	+	819459.i	−0.208359	+	0.906080i
$952$	−1.03597e6			−1.14307
$953$			1.71407e6i			1.88731i	0.330927	+	0.943656i	$0.392639\pi$
$953$			1.71407e6i			1.88731i	−0.330927	+	0.943656i	$0.607361\pi$
$954$	−1.55772e6	−	756418.i	−1.71156	−	0.831122i
$955$	−131042.			−0.143683
$956$		−	1.68340e6i		−	1.84192i
$957$	−466567.	−	107290.i	−0.509436	−	0.117148i
$958$	1.16051e6			1.26449
$959$			683307.i			0.742982i
$960$	924.397	−	4019.87i	0.00100303	−	0.00436184i
$961$	−816622.			−0.884249
$962$			65824.1i			0.0711271i
$963$	25078.4	−	51645.1i	0.0270426	−	0.0556899i
$964$	−1.10274e6			−1.18664
$965$		−	315880.i		−	0.339210i
$966$	−150922.	−	34705.5i	−0.161733	−	0.0371915i
$967$	1.48328e6			1.58625			0.793123	−	0.609061i	$-0.208454\pi$
$967$	1.48328e6			1.58625			0.793123	+	0.609061i	$0.208454\pi$
$968$			1.94295e6i			2.07353i
$969$	226025.	−	982900.i	0.240718	−	1.04680i
$970$	−796413.			−0.846438
$971$		−	25773.5i		−	0.0273360i	−0.999907	−	0.0136680i	$-0.995649\pi$
$971$		−	25773.5i		−	0.0273360i	0.999907	−	0.0136680i	$-0.00435080\pi$
$972$	−906277.	−	1.92177e6i	−0.959242	−	2.03408i
$973$	43224.5			0.0456567
$974$			503449.i			0.530686i
$975$	138106.	+	31758.5i	0.145280	+	0.0334081i
$976$	341121.			0.358104
$977$			1.63957e6i			1.71767i	0.512249	+	0.858837i	$0.328813\pi$
$977$			1.63957e6i			1.71767i	−0.512249	+	0.858837i	$0.671187\pi$
$978$	116677.	−	507385.i	0.121985	−	0.530469i
$979$	217582.			0.227016
$980$			746786.i			0.777578i
$981$	−66189.9	−	32141.3i	−0.0687786	−	0.0333984i
$982$	−3.10670e6			−3.22163
$983$			1.02737e6i			1.06322i	0.846991	+	0.531608i	$0.178412\pi$
$983$			1.02737e6i			1.06322i	−0.846991	+	0.531608i	$0.821588\pi$
$984$	2.71565e6	+	624482.i	2.80468	+	0.644955i
$985$	596047.			0.614339
$986$		−	3.47660e6i		−	3.57603i
$987$	−92185.8	+	400883.i	−0.0946302	+	0.411513i
$988$	−1.64860e6			−1.68889
$989$		−	153514.i		−	0.156948i
$990$	96936.2	−	199625.i	0.0989044	−	0.203678i
$991$	101778.			0.103635			0.0518174	−	0.998657i	$-0.483499\pi$
$991$	101778.			0.103635			0.0518174	+	0.998657i	$0.483499\pi$
$992$		−	337817.i		−	0.343288i
$993$	−628667.	−	144566.i	−0.637562	−	0.146612i
$994$	−219398.			−0.222054
$995$			564210.i			0.569894i
$996$	500206.	−	2.17521e6i	0.504232	−	2.19272i
$997$	−466189.			−0.468999			−0.234499	−	0.972116i	$-0.575345\pi$
$997$	−466189.			−0.468999			−0.234499	+	0.972116i	$0.575345\pi$
$998$		−	2.03948e6i		−	2.04766i
$999$	41147.7	+	33144.3i	0.0412301	+	0.0332106i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.5.c.a.11.1	✓	6
3.2	odd	2	inner	15.5.c.a.11.6	yes	6
4.3	odd	2		240.5.l.d.161.1		6
5.2	odd	4		75.5.d.d.74.11		12
5.3	odd	4		75.5.d.d.74.2		12
5.4	even	2		75.5.c.i.26.6		6
12.11	even	2		240.5.l.d.161.2		6
15.2	even	4		75.5.d.d.74.1		12
15.8	even	4		75.5.d.d.74.12		12
15.14	odd	2		75.5.c.i.26.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.5.c.a.11.1	✓	6	1.1	even	1	trivial
15.5.c.a.11.6	yes	6	3.2	odd	2	inner
75.5.c.i.26.1		6	15.14	odd	2
75.5.c.i.26.6		6	5.4	even	2
75.5.d.d.74.1		12	15.2	even	4
75.5.d.d.74.2		12	5.3	odd	4
75.5.d.d.74.11		12	5.2	odd	4
75.5.d.d.74.12		12	15.8	even	4
240.5.l.d.161.1		6	4.3	odd	2
240.5.l.d.161.2		6	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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.20990i q^{2} +(8.77108 + 2.01697i) q^{3} -35.9827 q^{4} +11.1803i q^{5} +(14.5422 - 63.2386i) q^{6} +23.3388 q^{7} +144.073i q^{8} +(72.8637 + 35.3820i) q^{9} +80.6091 q^{10} -33.9875i q^{11} +(-315.607 - 72.5760i) q^{12} -125.965 q^{13} -168.271i q^{14} +(-22.5504 + 98.0636i) q^{15} +463.029 q^{16} +308.095i q^{17} +(255.101 - 525.340i) q^{18} -363.724 q^{19} -402.298i q^{20} +(204.707 + 47.0738i) q^{21} -245.046 q^{22} -102.256i q^{23} +(-290.591 + 1263.68i) q^{24} -125.000 q^{25} +908.197i q^{26} +(567.728 + 457.302i) q^{27} -839.794 q^{28} -1565.10i q^{29} +(707.029 + 162.586i) q^{30} +326.954 q^{31} -1033.23i q^{32} +(68.5518 - 298.107i) q^{33} +2221.33 q^{34} +260.936i q^{35} +(-2621.83 - 1273.14i) q^{36} +72.4778 q^{37} +2622.41i q^{38} +(-1104.85 - 254.068i) q^{39} -1610.78 q^{40} -2149.01i q^{41} +(339.397 - 1475.92i) q^{42} +1501.27 q^{43} +1222.96i q^{44} +(-395.583 + 814.640i) q^{45} -737.258 q^{46} +1958.33i q^{47} +(4061.27 + 933.916i) q^{48} -1856.30 q^{49} +901.237i q^{50} +(-621.418 + 2702.32i) q^{51} +4532.57 q^{52} -2965.17i q^{53} +(3297.10 - 4093.26i) q^{54} +379.992 q^{55} +3362.50i q^{56} +(-3190.25 - 733.621i) q^{57} -11284.2 q^{58} +5045.29i q^{59} +(811.424 - 3528.59i) q^{60} +736.716 q^{61} -2357.30i q^{62} +(1700.55 + 825.775i) q^{63} -40.9924 q^{64} -1408.34i q^{65} +(-2149.32 - 494.252i) q^{66} +4254.63 q^{67} -11086.1i q^{68} +(206.248 - 896.898i) q^{69} +1881.32 q^{70} -1303.84i q^{71} +(-5097.59 + 10497.7i) q^{72} +6768.14 q^{73} -522.557i q^{74} +(-1096.38 - 252.121i) q^{75} +13087.8 q^{76} -793.229i q^{77} +(-1831.81 + 7965.87i) q^{78} -6367.07 q^{79} +5176.82i q^{80} +(4057.23 + 5156.13i) q^{81} -15494.1 q^{82} +6892.17i q^{83} +(-7365.90 - 1693.84i) q^{84} -3444.60 q^{85} -10824.0i q^{86} +(3156.76 - 13727.6i) q^{87} +4896.68 q^{88} +6401.82i q^{89} +(5873.48 + 2852.11i) q^{90} -2939.88 q^{91} +3679.45i q^{92} +(2867.74 + 659.456i) q^{93} +14119.3 q^{94} -4066.56i q^{95} +(2083.99 - 9062.51i) q^{96} -9879.94 q^{97} +13383.7i q^{98} +(1202.55 - 2476.45i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{3} - 50 q^{4} - 2 q^{6} + 76 q^{7} + 118 q^{9} + 50 q^{10} - 452 q^{12} - 424 q^{13} + 50 q^{15} + 802 q^{16} + 1160 q^{18} - 244 q^{19} - 876 q^{21} + 340 q^{22} - 786 q^{24} - 750 q^{25} - 352 q^{27}+ \cdots + 9680 q^{99}+O(q^{100}) \) 6 * q + 8 * q^3 - 50 * q^4 - 2 * q^6 + 76 * q^7 + 118 * q^9 + 50 * q^10 - 452 * q^12 - 424 * q^13 + 50 * q^15 + 802 * q^16 + 1160 * q^18 - 244 * q^19 - 876 * q^21 + 340 * q^22 - 786 * q^24 - 750 * q^25 - 352 * q^27 - 3764 * q^28 + 2200 * q^30 + 3772 * q^31 + 4420 * q^33 + 3124 * q^34 - 7606 * q^36 + 1896 * q^37 - 1336 * q^39 - 4650 * q^40 - 1980 * q^42 - 7384 * q^43 + 1900 * q^45 + 8196 * q^46 + 14668 * q^48 - 1318 * q^49 - 8492 * q^51 + 8976 * q^52 - 278 * q^54 - 1300 * q^55 - 11584 * q^57 - 23740 * q^58 + 5050 * q^60 + 6452 * q^61 + 14796 * q^63 + 3174 * q^64 - 12760 * q^66 + 13816 * q^67 + 5472 * q^69 - 2100 * q^70 - 2040 * q^72 + 596 * q^73 - 1000 * q^75 + 21348 * q^76 - 1400 * q^78 - 16124 * q^79 + 5086 * q^81 - 31240 * q^82 - 14736 * q^84 - 3100 * q^85 - 4900 * q^87 + 15660 * q^88 + 7550 * q^90 - 11632 * q^91 - 8184 * q^93 + 34924 * q^94 + 14354 * q^96 + 9756 * q^97 + 9680 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more