LMFDB - Embedded newform 42.5.b.a.29.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [42,5,Mod(29,42)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(42, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("42.29");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	42.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.34153844952$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 44x^{6} + 646x^{4} - 3060x^{2} + 6561 $ x^8 - 44x^6 + 646x^4 - 3060*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{3}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			29.7
Root			$-1.88752 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	42.29
Dual form			42.5.b.a.29.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.82843i q^{2} +(6.59792 - 6.12106i) q^{3} -8.00000 q^{4} -47.9925i q^{5} +(17.3130 + 18.6617i) q^{6} -18.5203 q^{7} -22.6274i q^{8} +(6.06518 - 80.7726i) q^{9} +135.743 q^{10} +123.237i q^{11} +(-52.7834 + 48.9685i) q^{12} +241.627 q^{13} -52.3832i q^{14} +(-293.765 - 316.651i) q^{15} +64.0000 q^{16} +64.4425i q^{17} +(228.459 + 17.1549i) q^{18} +125.357 q^{19} +383.940i q^{20} +(-122.195 + 113.364i) q^{21} -348.566 q^{22} +287.613i q^{23} +(-138.504 - 149.294i) q^{24} -1678.28 q^{25} +683.426i q^{26} +(-454.397 - 570.057i) q^{27} +148.162 q^{28} +94.2775i q^{29} +(895.624 - 830.893i) q^{30} +429.829 q^{31} +181.019i q^{32} +(754.339 + 813.106i) q^{33} -182.271 q^{34} +888.834i q^{35} +(-48.5215 + 646.181i) q^{36} +2118.52 q^{37} +354.563i q^{38} +(1594.24 - 1479.02i) q^{39} -1085.95 q^{40} -1220.23i q^{41} +(-320.641 - 345.620i) q^{42} +976.206 q^{43} -985.893i q^{44} +(-3876.48 - 291.083i) q^{45} -813.493 q^{46} +526.553i q^{47} +(422.267 - 391.748i) q^{48} +343.000 q^{49} -4746.90i q^{50} +(394.457 + 425.187i) q^{51} -1933.02 q^{52} +1567.07i q^{53} +(1612.36 - 1285.23i) q^{54} +5914.43 q^{55} +419.066i q^{56} +(827.096 - 767.318i) q^{57} -266.657 q^{58} +5826.08i q^{59} +(2350.12 + 2533.21i) q^{60} -4955.04 q^{61} +1215.74i q^{62} +(-112.329 + 1495.93i) q^{63} -512.000 q^{64} -11596.3i q^{65} +(-2299.81 + 2133.59i) q^{66} -1321.11 q^{67} -515.540i q^{68} +(1760.50 + 1897.65i) q^{69} -2514.00 q^{70} +3717.89i q^{71} +(-1827.68 - 137.239i) q^{72} +2023.01 q^{73} +5992.07i q^{74} +(-11073.2 + 10272.9i) q^{75} -1002.86 q^{76} -2282.37i q^{77} +(4183.29 + 4509.19i) q^{78} +4711.86 q^{79} -3071.52i q^{80} +(-6487.43 - 979.801i) q^{81} +3451.32 q^{82} -4420.87i q^{83} +(977.562 - 906.909i) q^{84} +3092.76 q^{85} +2761.13i q^{86} +(577.078 + 622.036i) q^{87} +2788.53 q^{88} -8482.72i q^{89} +(823.308 - 10964.3i) q^{90} -4475.00 q^{91} -2300.91i q^{92} +(2835.98 - 2631.01i) q^{93} -1489.32 q^{94} -6016.20i q^{95} +(1108.03 + 1194.35i) q^{96} -16341.1 q^{97} +970.151i q^{98} +(9954.14 + 747.452i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{3} - 64 q^{4} - 32 q^{6} + 152 q^{9} + 256 q^{10} - 96 q^{12} - 520 q^{13} - 616 q^{15} + 512 q^{16} + 768 q^{18} + 1336 q^{19} - 196 q^{21} - 1024 q^{22} + 256 q^{24} - 3672 q^{25} + 36 q^{27}+ \cdots + 10976 q^{99}+O(q^{100}) $ 8 * q + 12 * q^3 - 64 * q^4 - 32 * q^6 + 152 * q^9 + 256 * q^10 - 96 * q^12 - 520 * q^13 - 616 * q^15 + 512 * q^16 + 768 * q^18 + 1336 * q^19 - 196 * q^21 - 1024 * q^22 + 256 * q^24 - 3672 * q^25 + 36 * q^27 - 1216 * q^30 + 4880 * q^31 + 1432 * q^33 + 1280 * q^34 - 1216 * q^36 - 800 * q^37 + 5672 * q^39 - 2048 * q^40 - 2816 * q^43 - 10712 * q^45 + 896 * q^46 + 768 * q^48 + 2744 * q^49 - 5768 * q^51 + 4160 * q^52 + 3616 * q^54 - 2320 * q^55 + 12864 * q^57 - 5504 * q^58 + 4928 * q^60 + 2696 * q^61 + 8232 * q^63 - 4096 * q^64 - 15680 * q^66 - 6960 * q^67 - 2864 * q^69 - 3136 * q^70 - 6144 * q^72 + 27648 * q^73 - 15052 * q^75 - 10688 * q^76 + 15808 * q^78 - 1024 * q^79 + 17528 * q^81 + 18048 * q^82 + 1568 * q^84 - 19504 * q^85 - 13960 * q^87 + 8192 * q^88 + 7360 * q^90 - 23128 * q^91 - 30016 * q^93 + 19584 * q^94 - 2048 * q^96 - 27408 * q^97 + 10976 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/42\mathbb{Z}\right)^\times$.

$n$	$29$	$31$
$\chi(n)$	$-1$	$1$

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 $

$2$			2.82843i			0.707107i
$2$			2.82843i			0.707107i
$3$	6.59792	−	6.12106i	0.733103	−	0.680118i
$3$	6.59792	−	6.12106i	0.733103	−	0.680118i
$4$	−8.00000			−0.500000
$5$		−	47.9925i		−	1.91970i	−0.280513	−	0.959850i	$-0.590505\pi$
$5$		−	47.9925i		−	1.91970i	0.280513	−	0.959850i	$-0.409495\pi$
$6$	17.3130	+	18.6617i	0.480916	+	0.518382i
$7$	−18.5203			−0.377964
$7$	−18.5203			−0.377964
$8$		−	22.6274i		−	0.353553i
$9$	6.06518	−	80.7726i	0.0748788	−	0.997193i
$10$	135.743			1.35743
$11$			123.237i			1.01848i	0.860623	+	0.509242i	$0.170074\pi$
$11$			123.237i			1.01848i	−0.860623	+	0.509242i	$0.829926\pi$
$12$	−52.7834	+	48.9685i	−0.366551	+	0.340059i
$13$	241.627			1.42975			0.714874	−	0.699253i	$-0.246484\pi$
$13$	241.627			1.42975			0.714874	+	0.699253i	$0.246484\pi$
$14$		−	52.3832i		−	0.267261i
$15$	−293.765	−	316.651i	−1.30562	−	1.40734i
$16$	64.0000			0.250000
$17$			64.4425i			0.222984i	0.993765	+	0.111492i	$0.0355630\pi$
$17$			64.4425i			0.222984i	−0.993765	+	0.111492i	$0.964437\pi$
$18$	228.459	+	17.1549i	0.705122	+	0.0529473i
$19$	125.357			0.347249			0.173625	−	0.984812i	$-0.444452\pi$
$19$	125.357			0.347249			0.173625	+	0.984812i	$0.444452\pi$
$20$			383.940i			0.959850i
$21$	−122.195	+	113.364i	−0.277087	+	0.257060i
$22$	−348.566			−0.720177
$23$			287.613i			0.543693i	0.962341	+	0.271846i	$0.0876342\pi$
$23$			287.613i			0.543693i	−0.962341	+	0.271846i	$0.912366\pi$
$24$	−138.504	−	149.294i	−0.240458	−	0.259191i
$25$	−1678.28			−2.68525
$26$			683.426i			1.01098i
$27$	−454.397	−	570.057i	−0.623315	−	0.781971i
$28$	148.162			0.188982
$29$			94.2775i			0.112102i	0.998428	+	0.0560508i	$0.0178509\pi$
$29$			94.2775i			0.112102i	−0.998428	+	0.0560508i	$0.982149\pi$
$30$	895.624	−	830.893i	0.995138	−	0.923215i
$31$	429.829			0.447273			0.223637	−	0.974673i	$-0.428207\pi$
$31$	429.829			0.447273			0.223637	+	0.974673i	$0.428207\pi$
$32$			181.019i			0.176777i
$33$	754.339	+	813.106i	0.692690	+	0.746653i
$34$	−182.271			−0.157674
$35$			888.834i			0.725579i
$36$	−48.5215	+	646.181i	−0.0374394	+	0.498596i
$37$	2118.52			1.54749			0.773746	−	0.633496i	$-0.218381\pi$
$37$	2118.52			1.54749			0.773746	+	0.633496i	$0.218381\pi$
$38$			354.563i			0.245542i
$39$	1594.24	−	1479.02i	1.04815	−	0.972398i
$40$	−1085.95			−0.678717
$41$		−	1220.23i		−	0.725893i	−0.931810	−	0.362946i	$-0.881771\pi$
$41$		−	1220.23i		−	0.725893i	0.931810	−	0.362946i	$-0.118229\pi$
$42$	−320.641	−	345.620i	−0.181769	−	0.195930i
$43$	976.206			0.527964			0.263982	−	0.964528i	$-0.414964\pi$
$43$	976.206			0.527964			0.263982	+	0.964528i	$0.414964\pi$
$44$		−	985.893i		−	0.509242i
$45$	−3876.48	−	291.083i	−1.91431	−	0.143745i
$46$	−813.493			−0.384449
$47$			526.553i			0.238367i	0.992872	+	0.119183i	$0.0380277\pi$
$47$			526.553i			0.238367i	−0.992872	+	0.119183i	$0.961972\pi$
$48$	422.267	−	391.748i	0.183276	−	0.170030i
$49$	343.000			0.142857
$50$		−	4746.90i		−	1.89876i
$51$	394.457	+	425.187i	0.151656	+	0.163470i
$52$	−1933.02			−0.714874
$53$			1567.07i			0.557875i	0.960309	+	0.278937i	$0.0899822\pi$
$53$			1567.07i			0.557875i	−0.960309	+	0.278937i	$0.910018\pi$
$54$	1612.36	−	1285.23i	0.552937	−	0.440750i
$55$	5914.43			1.95518
$56$			419.066i			0.133631i
$57$	827.096	−	767.318i	0.254569	−	0.236171i
$58$	−266.657			−0.0792678
$59$			5826.08i			1.67368i	0.547448	+	0.836840i	$0.315599\pi$
$59$			5826.08i			1.67368i	−0.547448	+	0.836840i	$0.684401\pi$
$60$	2350.12	+	2533.21i	0.652811	+	0.703669i
$61$	−4955.04			−1.33164			−0.665821	−	0.746112i	$-0.731918\pi$
$61$	−4955.04			−1.33164			−0.665821	+	0.746112i	$0.731918\pi$
$62$			1215.74i			0.316270i
$63$	−112.329	+	1495.93i	−0.0283015	+	0.376903i
$64$	−512.000			−0.125000
$65$		−	11596.3i		−	2.74469i
$66$	−2299.81	+	2133.59i	−0.527964	+	0.489806i
$67$	−1321.11			−0.294300			−0.147150	−	0.989114i	$-0.547010\pi$
$67$	−1321.11			−0.294300			−0.147150	+	0.989114i	$0.547010\pi$
$68$		−	515.540i		−	0.111492i
$69$	1760.50	+	1897.65i	0.369775	+	0.398582i
$70$	−2514.00			−0.513062
$71$			3717.89i			0.737531i	0.929522	+	0.368765i	$0.120219\pi$
$71$			3717.89i			0.737531i	−0.929522	+	0.368765i	$0.879781\pi$
$72$	−1827.68	−	137.239i	−0.352561	−	0.0264737i
$73$	2023.01			0.379622			0.189811	−	0.981821i	$-0.439212\pi$
$73$	2023.01			0.379622			0.189811	+	0.981821i	$0.439212\pi$
$74$			5992.07i			1.09424i
$75$	−11073.2	+	10272.9i	−1.96856	+	1.82629i
$76$	−1002.86			−0.173625
$77$		−	2282.37i		−	0.384951i
$78$	4183.29	+	4509.19i	0.687589	+	0.741156i
$79$	4711.86			0.754985			0.377493	−	0.926013i	$-0.376786\pi$
$79$	4711.86			0.754985			0.377493	+	0.926013i	$0.376786\pi$
$80$		−	3071.52i		−	0.479925i
$81$	−6487.43	−	979.801i	−0.988786	−	0.149337i
$82$	3451.32			0.513284
$83$		−	4420.87i		−	0.641729i	−0.947125	−	0.320865i	$-0.896027\pi$
$83$		−	4420.87i		−	0.641729i	0.947125	−	0.320865i	$-0.103973\pi$
$84$	977.562	−	906.909i	0.138543	−	0.128530i
$85$	3092.76			0.428063
$86$			2761.13i			0.373327i
$87$	577.078	+	622.036i	0.0762424	+	0.0821820i
$88$	2788.53			0.360089
$89$		−	8482.72i		−	1.07091i	−0.844562	−	0.535457i	$-0.820139\pi$
$89$		−	8482.72i		−	1.07091i	0.844562	−	0.535457i	$-0.179861\pi$
$90$	823.308	−	10964.3i	0.101643	−	1.35362i
$91$	−4475.00			−0.540394
$92$		−	2300.91i		−	0.271846i
$93$	2835.98	−	2631.01i	0.327897	−	0.304199i
$94$	−1489.32			−0.168551
$95$		−	6016.20i		−	0.666615i
$96$	1108.03	+	1194.35i	0.120229	+	0.129595i
$97$	−16341.1			−1.73675			−0.868375	−	0.495908i	$-0.834835\pi$
$97$	−16341.1			−1.73675			−0.868375	+	0.495908i	$0.834835\pi$
$98$			970.151i			0.101015i
$99$	9954.14	+	747.452i	1.01563	+	0.0762629i
$100$	13426.2			1.34262
$101$			5191.39i			0.508910i	0.967085	+	0.254455i	$0.0818960\pi$
$101$			5191.39i			0.508910i	−0.967085	+	0.254455i	$0.918104\pi$
$102$	−1202.61	+	1115.69i	−0.115591	+	0.107237i
$103$	9498.09			0.895286			0.447643	−	0.894212i	$-0.352264\pi$
$103$	9498.09			0.895286			0.447643	+	0.894212i	$0.352264\pi$
$104$		−	5467.41i		−	0.505492i
$105$	5440.61	+	5864.46i	0.493479	+	0.531924i
$106$	−4432.35			−0.394477
$107$		−	3676.19i		−	0.321093i	−0.987028	−	0.160546i	$-0.948674\pi$
$107$		−	3676.19i		−	0.321093i	0.987028	−	0.160546i	$-0.0513256\pi$
$108$	3635.17	+	4560.45i	0.311657	+	0.390985i
$109$	−11597.2			−0.976113			−0.488056	−	0.872812i	$-0.662294\pi$
$109$	−11597.2			−0.976113			−0.488056	+	0.872812i	$0.662294\pi$
$110$			16728.5i			1.38252i
$111$	13977.8	−	12967.6i	1.13447	−	1.05248i
$112$	−1185.30			−0.0944911
$113$		−	1074.40i		−	0.0841413i	−0.999115	−	0.0420706i	$-0.986605\pi$
$113$		−	1074.40i		−	0.0841413i	0.999115	−	0.0420706i	$-0.0133955\pi$
$114$	2170.30	+	2339.38i	0.166998	+	0.180008i
$115$	13803.3			1.04373
$116$		−	754.220i		−	0.0560508i
$117$	1465.51	−	19516.9i	0.107058	−	1.42573i
$118$	−16478.6			−1.18347
$119$		−	1193.49i		−	0.0842802i
$120$	−7164.99	+	6647.15i	−0.497569	+	0.461607i
$121$	−546.261			−0.0373103
$122$		−	14015.0i		−	0.941613i
$123$	−7469.08	−	8050.96i	−0.493693	−	0.532154i
$124$	−3438.64			−0.223637
$125$			50549.6i			3.23517i
$126$	−4231.13	−	317.714i	−0.266511	−	0.0200122i
$127$	−19449.2			−1.20585			−0.602927	−	0.797797i	$-0.705999\pi$
$127$	−19449.2			−1.20585			−0.602927	+	0.797797i	$0.705999\pi$
$128$		−	1448.15i		−	0.0883883i
$129$	6440.93	−	5975.42i	0.387052	−	0.359078i
$130$	32799.3			1.94079
$131$		−	14919.5i		−	0.869382i	−0.900580	−	0.434691i	$-0.856858\pi$
$131$		−	14919.5i		−	0.869382i	0.900580	−	0.434691i	$-0.143142\pi$
$132$	−6034.71	−	6504.85i	−0.346345	−	0.373327i
$133$	−2321.64			−0.131248
$134$		−	3736.67i		−	0.208102i
$135$	−27358.5	+	21807.6i	−1.50115	+	1.19658i
$136$	1458.17			0.0788369
$137$			19699.7i			1.04959i	0.851229	+	0.524795i	$0.175858\pi$
$137$			19699.7i			1.04959i	−0.851229	+	0.524795i	$0.824142\pi$
$138$	−5367.37	+	4979.44i	−0.281840	+	0.261471i
$139$	22226.3			1.15037			0.575184	−	0.818024i	$-0.304930\pi$
$139$	22226.3			1.15037			0.575184	+	0.818024i	$0.304930\pi$
$140$		−	7110.67i		−	0.362789i
$141$	3223.06	+	3474.15i	0.162118	+	0.174747i
$142$	−10515.8			−0.521513
$143$			29777.4i			1.45618i
$144$	388.172	−	5169.45i	0.0187197	−	0.249298i
$145$	4524.61			0.215202
$146$			5721.92i			0.268433i
$147$	2263.09	−	2099.52i	0.104729	−	0.0971597i
$148$	−16948.1			−0.773746
$149$		−	6791.29i		−	0.305900i	−0.988234	−	0.152950i	$-0.951123\pi$
$149$		−	6791.29i		−	0.305900i	0.988234	−	0.152950i	$-0.0488774\pi$
$150$	−29056.0	−	31319.7i	−1.29138	−	1.39198i
$151$	−8470.50			−0.371497			−0.185748	−	0.982597i	$-0.559471\pi$
$151$	−8470.50			−0.371497			−0.185748	+	0.982597i	$0.559471\pi$
$152$		−	2836.51i		−	0.122771i
$153$	5205.19	+	390.856i	0.222358	+	0.0166968i
$154$	6455.53			0.272201
$155$		−	20628.6i		−	0.858630i
$156$	−12753.9	+	11832.1i	−0.524076	+	0.486199i
$157$	−13638.2			−0.553297			−0.276649	−	0.960971i	$-0.589224\pi$
$157$	−13638.2			−0.553297			−0.276649	+	0.960971i	$0.589224\pi$
$158$			13327.2i			0.533855i
$159$	9592.14	+	10339.4i	0.379421	+	0.408980i
$160$	8687.57			0.339358
$161$		−	5326.67i		−	0.205496i
$162$	2771.30	−	18349.2i	0.105597	−	0.699178i
$163$	−38643.6			−1.45446			−0.727232	−	0.686392i	$-0.759193\pi$
$163$	−38643.6			−1.45446			−0.727232	+	0.686392i	$0.759193\pi$
$164$			9761.81i			0.362946i
$165$	39023.0	−	36202.6i	1.43335	−	1.32976i
$166$	12504.1			0.453771
$167$			33982.6i			1.21850i	0.792980	+	0.609248i	$0.208529\pi$
$167$			33982.6i			1.21850i	−0.792980	+	0.609248i	$0.791471\pi$
$168$	2565.13	+	2764.96i	0.0908846	+	0.0979650i
$169$	29822.8			1.04418
$170$			8747.64i			0.302686i
$171$	760.313	−	10125.4i	0.0260016	−	0.346275i
$172$	−7809.65			−0.263982
$173$			12224.7i			0.408457i	0.978923	+	0.204229i	$0.0654686\pi$
$173$			12224.7i			0.408457i	−0.978923	+	0.204229i	$0.934531\pi$
$174$	−1759.38	+	1632.22i	−0.0581115	+	0.0539115i
$175$	31082.2			1.01493
$176$			7887.14i			0.254621i
$177$	35661.8	+	38440.0i	1.13830	+	1.22698i
$178$	23992.7			0.757251
$179$		−	53278.8i		−	1.66283i	−0.555651	−	0.831416i	$-0.687531\pi$
$179$		−	53278.8i		−	1.66283i	0.555651	−	0.831416i	$-0.312469\pi$
$180$	31011.8	+	2328.67i	0.957156	+	0.0718724i
$181$	19789.7			0.604064			0.302032	−	0.953298i	$-0.402335\pi$
$181$	19789.7			0.604064			0.302032	+	0.953298i	$0.402335\pi$
$182$		−	12657.2i		−	0.382116i
$183$	−32693.0	+	30330.1i	−0.976230	+	0.905673i
$184$	6507.95			0.192224
$185$		−	101673.i		−	2.97072i
$186$	7441.63	+	8021.37i	0.215101	+	0.231858i
$187$	−7941.68			−0.227106
$188$		−	4212.42i		−	0.119183i
$189$	8415.54	+	10557.6i	0.235591	+	0.295557i
$190$	17016.4			0.471368
$191$			22349.2i			0.612626i	0.951931	+	0.306313i	$0.0990954\pi$
$191$			22349.2i			0.612626i	−0.951931	+	0.306313i	$0.900905\pi$
$192$	−3378.14	+	3133.98i	−0.0916378	+	0.0850148i
$193$	−55747.5			−1.49662			−0.748308	−	0.663351i	$-0.769134\pi$
$193$	−55747.5			−1.49662			−0.748308	+	0.663351i	$0.769134\pi$
$194$		−	46219.5i		−	1.22807i
$195$	−70981.7	−	76511.6i	−1.86671	−	2.01214i
$196$	−2744.00			−0.0714286
$197$		−	3307.33i		−	0.0852208i	−0.999092	−	0.0426104i	$-0.986433\pi$
$197$		−	3307.33i		−	0.0852208i	0.999092	−	0.0426104i	$-0.0135674\pi$
$198$	−2114.11	+	28154.6i	−0.0539260	+	0.718155i
$199$	31962.3			0.807107			0.403554	−	0.914956i	$-0.367775\pi$
$199$	31962.3			0.807107			0.403554	+	0.914956i	$0.367775\pi$
$200$			37975.2i			0.949379i
$201$	−8716.61	+	8086.62i	−0.215752	+	0.200159i
$202$	−14683.5			−0.359854
$203$		−	1746.04i		−	0.0423704i
$204$	−3155.65	−	3401.49i	−0.0758279	−	0.0817352i
$205$	−58561.7			−1.39350
$206$			26864.7i			0.633063i
$207$	23231.3	+	1744.43i	0.542166	+	0.0407110i
$208$	15464.2			0.357437
$209$			15448.6i			0.353668i
$210$	−16587.2	+	15388.4i	−0.376127	+	0.348942i
$211$	8820.58			0.198122			0.0990609	−	0.995081i	$-0.468416\pi$
$211$	8820.58			0.198122			0.0990609	+	0.995081i	$0.468416\pi$
$212$		−	12536.6i		−	0.278937i
$213$	22757.5	+	24530.4i	0.501608	+	0.540686i
$214$	10397.8			0.227047
$215$		−	46850.6i		−	1.01353i
$216$	−12898.9	+	10281.8i	−0.276468	+	0.220375i
$217$	−7960.55			−0.169053
$218$		−	32801.8i		−	0.690216i
$219$	13347.6	−	12382.9i	0.278302	−	0.258188i
$220$	−47315.5			−0.977592
$221$			15571.1i			0.318812i
$222$	36677.8	+	39535.2i	0.744214	+	0.802191i
$223$	−52110.8			−1.04790			−0.523948	−	0.851750i	$-0.675541\pi$
$223$	−52110.8			−1.04790			−0.523948	+	0.851750i	$0.675541\pi$
$224$		−	3352.53i		−	0.0668153i
$225$	−10179.1	+	135559.i	−0.201068	+	2.67771i
$226$	3038.86			0.0594969
$227$			60377.1i			1.17171i	0.810415	+	0.585856i	$0.199241\pi$
$227$			60377.1i			1.17171i	−0.810415	+	0.585856i	$0.800759\pi$
$228$	−6616.77	+	6138.55i	−0.127285	+	0.118085i
$229$	3035.47			0.0578835			0.0289418	−	0.999581i	$-0.490786\pi$
$229$	3035.47			0.0578835			0.0289418	+	0.999581i	$0.490786\pi$
$230$			39041.6i			0.738026i
$231$	−13970.6	−	15058.9i	−0.261812	−	0.282208i
$232$	2133.26			0.0396339
$233$			34594.5i			0.637228i	0.947885	+	0.318614i	$0.103217\pi$
$233$			34594.5i			0.637228i	−0.947885	+	0.318614i	$0.896783\pi$
$234$	55202.1	+	4145.10i	1.00815	+	0.0757013i
$235$	25270.6			0.457593
$236$		−	46608.6i		−	0.836840i
$237$	31088.5	−	28841.6i	0.553482	−	0.513479i
$238$	3375.70			0.0595951
$239$		−	78309.8i		−	1.37095i	−0.728098	−	0.685473i	$-0.759595\pi$
$239$		−	78309.8i		−	1.37095i	0.728098	−	0.685473i	$-0.240405\pi$
$240$	−18801.0	−	20265.7i	−0.326406	−	0.351834i
$241$	28362.2			0.488322			0.244161	−	0.969735i	$-0.421487\pi$
$241$	28362.2			0.488322			0.244161	+	0.969735i	$0.421487\pi$
$242$		−	1545.06i		−	0.0263824i
$243$	−48801.0	+	33245.3i	−0.826449	+	0.563012i
$244$	39640.3			0.665821
$245$		−	16461.4i		−	0.274243i
$246$	22771.5	−	21125.7i	0.376290	−	0.349094i
$247$	30289.7			0.496479
$248$		−	9725.93i		−	0.158135i
$249$	−27060.4	−	29168.6i	−0.436452	−	0.470453i
$250$	−142976.			−2.28761
$251$			89655.5i			1.42308i	0.702646	+	0.711540i	$0.252002\pi$
$251$			89655.5i			1.42308i	−0.702646	+	0.711540i	$0.747998\pi$
$252$	898.630	−	11967.4i	0.0141508	−	0.188452i
$253$	−35444.5			−0.553742
$254$		−	55010.7i		−	0.852667i
$255$	20405.8	−	18931.0i	0.313814	−	0.291134i
$256$	4096.00			0.0625000
$257$		−	12337.1i		−	0.186787i	−0.995629	−	0.0933933i	$-0.970229\pi$
$257$		−	12337.1i		−	0.186787i	0.995629	−	0.0933933i	$-0.0297714\pi$
$258$	16901.0	+	18217.7i	0.253906	+	0.273687i
$259$	−39235.5			−0.584897
$260$			92770.5i			1.37234i
$261$	7615.04	+	571.810i	0.111787	+	0.00839404i
$262$	42198.6			0.614746
$263$			39803.9i			0.575458i	0.957712	+	0.287729i	$0.0929003\pi$
$263$			39803.9i			0.575458i	−0.957712	+	0.287729i	$0.907100\pi$
$264$	18398.5	−	17068.7i	0.263982	−	0.244903i
$265$	75207.7			1.07095
$266$		−	6566.60i		−	0.0928063i
$267$	−51923.2	−	55968.3i	−0.728349	−	0.785091i
$268$	10568.9			0.147150
$269$			38209.6i			0.528042i	0.964517	+	0.264021i	$0.0850488\pi$
$269$			38209.6i			0.528042i	−0.964517	+	0.264021i	$0.914951\pi$
$270$	−61681.3	−	77381.4i	−0.846108	−	1.06147i
$271$	134877.			1.83653			0.918267	−	0.395961i	$-0.129588\pi$
$271$	134877.			1.83653			0.918267	+	0.395961i	$0.129588\pi$
$272$			4124.32i			0.0557461i
$273$	−29525.7	+	27391.8i	−0.396164	+	0.367532i
$274$	−55719.3			−0.742172
$275$		−	206826.i		−	2.73488i
$276$	−14084.0	−	15181.2i	−0.184888	−	0.199291i
$277$	134803.			1.75688			0.878438	−	0.477856i	$-0.158586\pi$
$277$	134803.			1.75688			0.878438	+	0.477856i	$0.158586\pi$
$278$			62865.4i			0.813434i
$279$	2606.99	−	34718.4i	0.0334913	−	0.446017i
$280$	20112.0			0.256531
$281$		−	3288.42i		−	0.0416462i	−0.999783	−	0.0208231i	$-0.993371\pi$
$281$		−	3288.42i		−	0.0416462i	0.999783	−	0.0208231i	$-0.00662868\pi$
$282$	−9826.39	+	9116.20i	−0.123565	+	0.114635i
$283$	−129266.			−1.61403			−0.807016	−	0.590529i	$-0.798919\pi$
$283$	−129266.			−1.61403			−0.807016	+	0.590529i	$0.798919\pi$
$284$		−	29743.1i		−	0.368765i
$285$	−36825.5	−	39694.4i	−0.453377	−	0.488697i
$286$	−84223.1			−1.02967
$287$			22598.9i			0.274362i
$288$	14621.4	+	1097.92i	0.176280	+	0.0132368i
$289$	79368.2			0.950278
$290$			12797.5i			0.152171i
$291$	−107817.	+	100025.i	−1.27322	+	1.18119i
$292$	−16184.0			−0.189811
$293$		−	168398.i		−	1.96157i	−0.195104	−	0.980783i	$-0.562504\pi$
$293$		−	168398.i		−	1.96157i	0.195104	−	0.980783i	$-0.437496\pi$
$294$	5938.35	+	6400.98i	0.0687023	+	0.0740545i
$295$	279608.			3.21296
$296$		−	47936.5i		−	0.547121i
$297$	70251.9	−	55998.3i	0.796425	−	0.634836i
$298$	19208.7			0.216304
$299$			69495.3i			0.777344i
$300$	88585.4	−	82182.9i	0.984282	−	0.913143i
$301$	−18079.6			−0.199552
$302$		−	23958.2i		−	0.262688i
$303$	31776.8	+	34252.4i	0.346119	+	0.373083i
$304$	8022.85			0.0868123
$305$			237805.i			2.55635i
$306$	−1105.51	+	14722.5i	−0.0118064	+	0.157231i
$307$	−44393.8			−0.471027			−0.235514	−	0.971871i	$-0.575677\pi$
$307$	−44393.8			−0.471027			−0.235514	+	0.971871i	$0.575677\pi$
$308$			18259.0i			0.192475i
$309$	62667.7	−	58138.4i	0.656336	−	0.608900i
$310$	58346.5			0.607143
$311$			146724.i			1.51699i	0.651681	+	0.758493i	$0.274064\pi$
$311$			146724.i			1.51699i	−0.651681	+	0.758493i	$0.725936\pi$
$312$	−33466.3	−	36073.5i	−0.343795	−	0.370578i
$313$	−54255.1			−0.553799			−0.276899	−	0.960899i	$-0.589307\pi$
$313$	−54255.1			−0.553799			−0.276899	+	0.960899i	$0.589307\pi$
$314$		−	38574.7i		−	0.391240i
$315$	71793.4	+	5390.94i	0.723542	+	0.0543304i
$316$	−37694.9			−0.377493
$317$			130350.i			1.29715i	0.761149	+	0.648577i	$0.224636\pi$
$317$			130350.i			1.29715i	−0.761149	+	0.648577i	$0.775364\pi$
$318$	−29244.3	+	27130.7i	−0.289192	+	0.268291i
$319$	−11618.4			−0.114174
$320$			24572.2i			0.239963i
$321$	−22502.2	−	24255.2i	−0.218381	−	0.235394i
$322$	15066.1			0.145308
$323$			8078.32i			0.0774312i
$324$	51899.4	+	7838.41i	0.494393	+	0.0746686i
$325$	−405519.			−3.83923
$326$		−	109301.i		−	1.02846i
$327$	−76517.4	+	70987.2i	−0.715591	+	0.663872i
$328$	−27610.6			−0.256642
$329$		−	9751.89i		−	0.0900943i
$330$	102396.	+	110374.i	0.940280	+	1.01353i
$331$	−35032.0			−0.319749			−0.159875	−	0.987137i	$-0.551109\pi$
$331$	−35032.0			−0.319749			−0.159875	+	0.987137i	$0.551109\pi$
$332$			35367.0i			0.320865i
$333$	12849.2	−	171118.i	0.115874	−	1.54315i
$334$	−96117.4			−0.861607
$335$			63403.6i			0.564968i
$336$	−7820.50	+	7255.27i	−0.0692717	+	0.0642651i
$337$	−184305.			−1.62285			−0.811423	−	0.584459i	$-0.801307\pi$
$337$	−184305.			−1.62285			−0.811423	+	0.584459i	$0.801307\pi$
$338$			84351.7i			0.738347i
$339$	−6576.47	−	7088.81i	−0.0572260	−	0.0616842i
$340$	−24742.1			−0.214032
$341$			52970.7i			0.455541i
$342$	28639.0	+	2150.49i	0.244853	+	0.0183859i
$343$	−6352.45			−0.0539949
$344$		−	22089.0i		−	0.186664i
$345$	91073.0	−	84490.8i	0.765159	−	0.709858i
$346$	−34576.7			−0.288823
$347$		−	15320.3i		−	0.127235i	−0.997974	−	0.0636176i	$-0.979736\pi$
$347$		−	15320.3i		−	0.127235i	0.997974	−	0.0636176i	$-0.0202638\pi$
$348$	−4616.63	−	4976.29i	−0.0381212	−	0.0410910i
$349$	85296.5			0.700294			0.350147	−	0.936695i	$-0.386132\pi$
$349$	85296.5			0.700294			0.350147	+	0.936695i	$0.386132\pi$
$350$			87913.7i			0.717663i
$351$	−109795.	−	137741.i	−0.891184	−	1.11802i
$352$	−22308.2			−0.180044
$353$		−	120060.i		−	0.963493i	−0.876311	−	0.481746i	$-0.840003\pi$
$353$		−	120060.i		−	0.963493i	0.876311	−	0.481746i	$-0.159997\pi$
$354$	−108725.	+	100867.i	−0.867605	+	0.804899i
$355$	178431.			1.41584
$356$			67861.7i			0.535457i
$357$	−7305.44	−	7874.57i	−0.0573205	−	0.0617860i
$358$	150695.			1.17580
$359$		−	164672.i		−	1.27771i	−0.769328	−	0.638854i	$-0.779409\pi$
$359$		−	164672.i		−	1.27771i	0.769328	−	0.638854i	$-0.220591\pi$
$360$	−6586.46	+	87714.7i	−0.0508215	+	0.676811i
$361$	−114607.			−0.879418
$362$			55973.8i			0.427138i
$363$	−3604.19	+	3343.70i	−0.0273523	+	0.0253754i
$364$	35800.0			0.270197
$365$		−	97089.1i		−	0.728760i
$366$	−85786.5	−	92469.7i	−0.640408	−	0.690299i
$367$	179523.			1.33287			0.666435	−	0.745563i	$-0.267819\pi$
$367$	179523.			1.33287			0.666435	+	0.745563i	$0.267819\pi$
$368$			18407.3i			0.135923i
$369$	−98560.8	−	7400.89i	−0.723855	−	0.0543540i
$370$	287574.			2.10062
$371$		−	29022.6i		−	0.210857i
$372$	−22687.9	+	21048.1i	−0.163949	+	0.152099i
$373$	154860.			1.11307			0.556534	−	0.830825i	$-0.312131\pi$
$373$	154860.			1.11307			0.556534	+	0.830825i	$0.312131\pi$
$374$		−	22462.4i		−	0.160588i
$375$	309417.	+	333522.i	2.20030	+	2.37171i
$376$	11914.5			0.0842755
$377$			22780.0i			0.160277i
$378$	−29861.4	+	23802.7i	−0.208991	+	0.166588i
$379$	−241643.			−1.68227			−0.841135	−	0.540826i	$-0.818112\pi$
$379$	−241643.			−1.68227			−0.841135	+	0.540826i	$0.818112\pi$
$380$			48129.6i			0.333307i
$381$	−128324.	+	119050.i	−0.884014	+	0.820123i
$382$	−63213.1			−0.433192
$383$			95222.8i			0.649147i	0.945860	+	0.324574i	$0.105221\pi$
$383$			95222.8i			0.649147i	−0.945860	+	0.324574i	$0.894779\pi$
$384$	−8864.25	−	9554.81i	−0.0601145	−	0.0647977i
$385$	−109537.			−0.738990
$386$		−	157678.i		−	1.05827i
$387$	5920.87	−	78850.7i	0.0395333	−	0.526482i
$388$	130729.			0.868375
$389$		−	224301.i		−	1.48228i	−0.671349	−	0.741142i	$-0.734285\pi$
$389$		−	224301.i		−	1.48228i	0.671349	−	0.741142i	$-0.265715\pi$
$390$	216407.	−	200767.i	1.42280	−	1.31997i
$391$	−18534.5			−0.121235
$392$		−	7761.20i		−	0.0505076i
$393$	−91323.0	−	98437.5i	−0.591283	−	0.637347i
$394$	9354.55			0.0602602
$395$		−	226134.i		−	1.44935i
$396$	−79633.1	−	5979.62i	−0.507813	−	0.0381314i
$397$	−79675.1			−0.505524			−0.252762	−	0.967528i	$-0.581339\pi$
$397$	−79675.1			−0.505524			−0.252762	+	0.967528i	$0.581339\pi$
$398$			90402.9i			0.570711i
$399$	−15318.0	+	14210.9i	−0.0962182	+	0.0892641i
$400$	−107410.			−0.671312
$401$		−	87504.0i		−	0.544176i	−0.962272	−	0.272088i	$-0.912286\pi$
$401$		−	87504.0i		−	0.544176i	0.962272	−	0.272088i	$-0.0877142\pi$
$402$	−22872.4	−	24654.3i	−0.141534	−	0.152560i
$403$	103859.			0.639488
$404$		−	41531.1i		−	0.254455i
$405$	−47023.1	+	311348.i	−0.286683	+	1.89817i
$406$	4938.56			0.0299604
$407$			261079.i			1.57610i
$408$	9620.88	−	8925.53i	0.0577955	−	0.0536184i
$409$	260159.			1.55522			0.777612	−	0.628745i	$-0.216431\pi$
$409$	260159.			1.55522			0.777612	+	0.628745i	$0.216431\pi$
$410$		−	165637.i		−	0.985351i
$411$	120583.	+	129977.i	0.713845	+	0.769457i
$412$	−75984.7			−0.447643
$413$		−	107900.i		−	0.632591i
$414$	−4933.99	+	65708.0i	−0.0287871	+	0.383369i
$415$	−212169.			−1.23193
$416$			43739.2i			0.252746i
$417$	146647.	−	136048.i	0.843338	−	0.782387i
$418$	−43695.2			−0.250081
$419$		−	222361.i		−	1.26658i	−0.773916	−	0.633288i	$-0.781705\pi$
$419$		−	222361.i		−	1.26658i	0.773916	−	0.633288i	$-0.218295\pi$
$420$	−43524.9	−	46915.7i	−0.246740	−	0.265962i
$421$	274759.			1.55020			0.775100	−	0.631838i	$-0.217699\pi$
$421$	274759.			1.55020			0.775100	+	0.631838i	$0.217699\pi$
$422$			24948.4i			0.140093i
$423$	42531.0	+	3193.64i	0.237698	+	0.0178486i
$424$	35458.8			0.197239
$425$		−	108153.i		−	0.598769i
$426$	−69382.4	+	64367.8i	−0.382323	+	0.354690i
$427$	91768.6			0.503313
$428$			29409.5i			0.160546i
$429$	182269.	+	196469.i	0.990372	+	1.06753i
$430$	132513.			0.716676
$431$			79239.0i			0.426564i	0.976991	+	0.213282i	$0.0684154\pi$
$431$			79239.0i			0.426564i	−0.976991	+	0.213282i	$0.931585\pi$
$432$	−29081.4	−	36483.6i	−0.155829	−	0.195493i
$433$	−56355.8			−0.300582			−0.150291	−	0.988642i	$-0.548021\pi$
$433$	−56355.8			−0.300582			−0.150291	+	0.988642i	$0.548021\pi$
$434$		−	22515.8i		−	0.119539i
$435$	29853.1	−	27695.4i	0.157765	−	0.146363i
$436$	92777.6			0.488056
$437$			36054.4i			0.188797i
$438$	35024.3	+	37752.8i	0.182566	+	0.196789i
$439$	−83094.4			−0.431164			−0.215582	−	0.976486i	$-0.569165\pi$
$439$	−83094.4			−0.431164			−0.215582	+	0.976486i	$0.569165\pi$
$440$		−	133828.i		−	0.691262i
$441$	2080.36	−	27705.0i	0.0106970	−	0.142456i
$442$	−44041.7			−0.225434
$443$			139917.i			0.712955i	0.934304	+	0.356477i	$0.116022\pi$
$443$			139917.i			0.712955i	−0.934304	+	0.356477i	$0.883978\pi$
$444$	−111822.	+	103741.i	−0.567235	+	0.526238i
$445$	−407107.			−2.05584
$446$		−	147392.i		−	0.740974i
$447$	−41569.9	−	44808.4i	−0.208048	−	0.224256i
$448$	9482.37			0.0472456
$449$			100269.i			0.497361i	0.968586	+	0.248681i	$0.0799970\pi$
$449$			100269.i			0.497361i	−0.968586	+	0.248681i	$0.920003\pi$
$450$	−383419.	−	28790.8i	−1.89343	−	0.142177i
$451$	150376.			0.739310
$452$			8595.20i			0.0420706i
$453$	−55887.7	+	51848.4i	−0.272345	+	0.252662i
$454$	−170772.			−0.828525
$455$			214767.i			1.03739i
$456$	−17362.4	−	18715.0i	−0.0834989	−	0.0900039i
$457$	−82991.3			−0.397375			−0.198687	−	0.980063i	$-0.563668\pi$
$457$	−82991.3			−0.397375			−0.198687	+	0.980063i	$0.563668\pi$
$458$			8585.60i			0.0409298i
$459$	36735.9	−	29282.5i	0.174367	−	0.138990i
$460$	−110426.			−0.521863
$461$		−	321734.i		−	1.51389i	−0.653478	−	0.756946i	$-0.726691\pi$
$461$		−	321734.i		−	1.51389i	0.653478	−	0.756946i	$-0.273309\pi$
$462$	42593.1	−	39514.7i	0.199552	−	0.185129i
$463$	−102874.			−0.479894			−0.239947	−	0.970786i	$-0.577130\pi$
$463$	−102874.			−0.479894			−0.239947	+	0.970786i	$0.577130\pi$
$464$			6033.76i			0.0280254i
$465$	−126269.	−	136106.i	−0.583970	−	0.629464i
$466$	−97847.9			−0.450588
$467$		−	87221.5i		−	0.399935i	−0.979803	−	0.199968i	$-0.935916\pi$
$467$		−	87221.5i		−	0.399935i	0.979803	−	0.199968i	$-0.0640837\pi$
$468$	−11724.1	+	156135.i	−0.0535289	+	0.712867i
$469$	24467.4			0.111235
$470$			71476.0i			0.323567i
$471$	−89984.0	+	83480.4i	−0.405624	+	0.376308i
$472$	131829.			0.591735
$473$			120304.i			0.537723i
$474$	81576.4	+	87931.6i	0.363085	+	0.391371i
$475$	−210384.			−0.932451
$476$			9547.93i			0.0421401i
$477$	126576.	+	9504.57i	0.556309	+	0.0417730i
$478$	221494.			0.969406
$479$			131399.i			0.572691i	0.958126	+	0.286345i	$0.0924405\pi$
$479$			131399.i			0.572691i	−0.958126	+	0.286345i	$0.907559\pi$
$480$	57319.9	−	53177.2i	0.248784	−	0.230804i
$481$	511892.			2.21252
$482$			80220.5i			0.345296i
$483$	−32604.9	−	35145.0i	−0.139762	−	0.150650i
$484$	4370.09			0.0186552
$485$			784249.i			3.33404i
$486$	−94031.9	−	138030.i	−0.398110	−	0.584387i
$487$	−115823.			−0.488355			−0.244178	−	0.969731i	$-0.578518\pi$
$487$	−115823.			−0.488355			−0.244178	+	0.969731i	$0.578518\pi$
$488$			112120.i			0.470806i
$489$	−254968.	+	236540.i	−1.06627	+	0.989207i
$490$	46560.0			0.193919
$491$			455525.i			1.88951i	0.327775	+	0.944756i	$0.393701\pi$
$491$			455525.i			1.88951i	−0.327775	+	0.944756i	$0.606299\pi$
$492$	59752.6	+	64407.6i	0.246846	+	0.266077i
$493$	−6075.48			−0.0249969
$494$			85672.2i			0.351064i
$495$	35872.1	−	477724.i	0.146402	−	1.94970i
$496$	27509.1			0.111818
$497$		−	68856.3i		−	0.278760i
$498$	82501.2	−	76538.5i	0.332661	−	0.308618i
$499$	−353733.			−1.42061			−0.710305	−	0.703894i	$-0.751443\pi$
$499$	−353733.			−1.42061			−0.710305	+	0.703894i	$0.751443\pi$
$500$		−	404397.i		−	1.61759i
$501$	208010.	+	224215.i	0.828721	+	0.893282i
$502$	−253584.			−1.00627
$503$		−	424746.i		−	1.67878i	−0.543530	−	0.839390i	$-0.682912\pi$
$503$		−	424746.i		−	1.67878i	0.543530	−	0.839390i	$-0.317088\pi$
$504$	33849.0	+	2541.71i	0.133255	+	0.0100061i
$505$	249148.			0.976954
$506$		−	100252.i		−	0.391555i
$507$	196769.	−	182548.i	0.765492	−	0.710166i
$508$	155594.			0.602927
$509$			49005.7i			0.189152i	0.995518	+	0.0945760i	$0.0301495\pi$
$509$			49005.7i			0.189152i	−0.995518	+	0.0945760i	$0.969850\pi$
$510$	53544.8	+	57716.3i	0.205863	+	0.221900i
$511$	−37466.6			−0.143484
$512$			11585.2i			0.0441942i
$513$	−56961.8	−	71460.6i	−0.216446	−	0.271539i
$514$	34894.5			0.132078
$515$		−	455837.i		−	1.71868i
$516$	−51527.4	+	47803.3i	−0.193526	+	0.179539i
$517$	−64890.6			−0.242773
$518$		−	110975.i		−	0.413584i
$519$	74828.2	+	80657.7i	0.277799	+	0.299441i
$520$	−262395.			−0.970394
$521$		−	158536.i		−	0.584054i	−0.956410	−	0.292027i	$-0.905670\pi$
$521$		−	158536.i		−	0.584054i	0.956410	−	0.292027i	$-0.0943298\pi$
$522$	−1617.32	+	21538.6i	−0.00593548	+	0.0790453i
$523$	137079.			0.501151			0.250575	−	0.968097i	$-0.419380\pi$
$523$	137079.			0.501151			0.250575	+	0.968097i	$0.419380\pi$
$524$			119356.i			0.434691i
$525$	205078.	−	190256.i	0.744047	−	0.690272i
$526$	−112582.			−0.406910
$527$			27699.3i			0.0997349i
$528$	48277.7	+	52038.8i	0.173172	+	0.186663i
$529$	197120.			0.704398
$530$			212719.i			0.757278i
$531$	470588.	+	35336.2i	1.66898	+	0.125323i
$532$	18573.2			0.0656240
$533$		−	294840.i		−	1.03784i
$534$	158302.	−	146861.i	0.555143	−	0.515020i
$535$	−176430.			−0.616402
$536$			29893.4i			0.104051i
$537$	−326123.	−	351529.i	−1.13092	−	1.21903i
$538$	−108073.			−0.373382
$539$			42270.2i			0.145498i
$540$	218868.	−	174461.i	0.750575	−	0.598289i
$541$	−171295.			−0.585263			−0.292632	−	0.956225i	$-0.594531\pi$
$541$	−171295.			−0.585263			−0.292632	+	0.956225i	$0.594531\pi$
$542$			381490.i			1.29863i
$543$	130571.	−	121134.i	0.442841	−	0.410835i
$544$	−11665.3			−0.0394185
$545$			556578.i			1.87384i
$546$	−77475.7	−	83511.4i	−0.259884	−	0.280130i
$547$	−131375.			−0.439075			−0.219537	−	0.975604i	$-0.570455\pi$
$547$	−131375.			−0.439075			−0.219537	+	0.975604i	$0.570455\pi$
$548$		−	157598.i		−	0.524795i
$549$	−30053.2	+	400231.i	−0.0997117	+	1.32790i
$550$	584991.			1.93386
$551$			11818.3i			0.0389272i
$552$	42938.9	−	39835.6i	0.140920	−	0.130735i
$553$	−87264.9			−0.285358
$554$			381281.i			1.24230i
$555$	−622346.	−	670830.i	−2.02044	−	2.17784i
$556$	−177810.			−0.575184
$557$		−	256492.i		−	0.826729i	−0.910566	−	0.413365i	$-0.864353\pi$
$557$		−	256492.i		−	0.826729i	0.910566	−	0.413365i	$-0.135647\pi$
$558$	98198.6	+	7373.69i	0.315382	+	0.0236819i
$559$	235878.			0.754856
$560$			56885.4i			0.181395i
$561$	−52398.6	+	48611.5i	−0.166492	+	0.154459i
$562$	9301.07			0.0294483
$563$		−	167229.i		−	0.527587i	−0.964579	−	0.263794i	$-0.915026\pi$
$563$		−	167229.i		−	0.527587i	0.964579	−	0.263794i	$-0.0849738\pi$
$564$	−25784.5	−	27793.2i	−0.0810589	−	0.0873737i
$565$	−51563.2			−0.161526
$566$		−	365620.i		−	1.14129i
$567$	120149.	+	18146.2i	0.373726	+	0.0564441i
$568$	84126.3			0.260757
$569$			227471.i			0.702591i	0.936265	+	0.351295i	$0.114259\pi$
$569$			227471.i			0.702591i	−0.936265	+	0.351295i	$0.885741\pi$
$570$	112273.	−	104158.i	0.345561	−	0.320586i
$571$	162502.			0.498409			0.249205	−	0.968451i	$-0.419831\pi$
$571$	162502.			0.498409			0.249205	+	0.968451i	$0.419831\pi$
$572$		−	238219.i		−	0.728088i
$573$	136801.	+	147458.i	0.416658	+	0.449118i
$574$	−63919.3			−0.194003
$575$		−	482696.i		−	1.45995i
$576$	−3105.37	+	41355.6i	−0.00935985	+	0.124649i
$577$	159511.			0.479113			0.239557	−	0.970882i	$-0.422998\pi$
$577$	159511.			0.479113			0.239557	+	0.970882i	$0.422998\pi$
$578$			224487.i			0.671948i
$579$	−367817.	+	341234.i	−1.09717	+	1.01788i
$580$	−36196.9			−0.107601
$581$			81875.7i			0.242551i
$582$	−282913.	−	304953.i	−0.835231	−	0.900299i
$583$	−193120.			−0.568187
$584$		−	45775.4i		−	0.134217i
$585$	−936664.	−	70333.7i	−2.73698	−	0.205519i
$586$	476303.			1.38704
$587$			207293.i			0.601601i	0.953687	+	0.300801i	$0.0972539\pi$
$587$			207293.i			0.601601i	−0.953687	+	0.300801i	$0.902746\pi$
$588$	−18104.7	+	16796.2i	−0.0523645	+	0.0485799i
$589$	53882.1			0.155315
$590$			790851.i			2.27191i
$591$	−20244.4	−	21821.5i	−0.0579602	−	0.0624756i
$592$	135585.			0.386873
$593$		−	373791.i		−	1.06297i	−0.847069	−	0.531483i	$-0.821635\pi$
$593$		−	373791.i		−	1.06297i	0.847069	−	0.531483i	$-0.178365\pi$
$594$	158387.	+	198702.i	0.448897	+	0.563158i
$595$	−57278.7			−0.161793
$596$			54330.3i			0.152950i
$597$	210884.	−	195643.i	0.591692	−	0.548928i
$598$	−196562.			−0.549665
$599$			153142.i			0.426815i	0.976963	+	0.213408i	$0.0684563\pi$
$599$			153142.i			0.426815i	−0.976963	+	0.213408i	$0.931544\pi$
$600$	232448.	+	250557.i	0.645690	+	0.695992i
$601$	352527.			0.975987			0.487993	−	0.872847i	$-0.337729\pi$
$601$	352527.			0.975987			0.487993	+	0.872847i	$0.337729\pi$
$602$		−	51136.8i		−	0.141104i
$603$	−8012.79	+	106710.i	−0.0220368	+	0.293474i
$604$	67764.0			0.185748
$605$			26216.4i			0.0716247i
$606$	−96880.4	+	89878.4i	−0.263810	+	0.244743i
$607$	179592.			0.487427			0.243714	−	0.969847i	$-0.421634\pi$
$607$	179592.			0.487427			0.243714	+	0.969847i	$0.421634\pi$
$608$			22692.0i			0.0613856i
$609$	−10687.6	−	11520.3i	−0.0288169	−	0.0310619i
$610$	−672613.			−1.80761
$611$			127230.i			0.340805i
$612$	−41641.5	−	3126.84i	−0.111179	−	0.00834840i
$613$	235777.			0.627452			0.313726	−	0.949513i	$-0.398423\pi$
$613$	235777.			0.627452			0.313726	+	0.949513i	$0.398423\pi$
$614$		−	125565.i		−	0.333067i
$615$	−386386.	+	358460.i	−1.02158	+	0.947742i
$616$	−51644.2			−0.136101
$617$			477167.i			1.25343i	0.779248	+	0.626716i	$0.215601\pi$
$617$			477167.i			1.25343i	−0.779248	+	0.626716i	$0.784399\pi$
$618$	164440.	+	177251.i	0.430557	+	0.464100i
$619$	51174.4			0.133559			0.0667793	−	0.997768i	$-0.478728\pi$
$619$	51174.4			0.133559			0.0667793	+	0.997768i	$0.478728\pi$
$620$			165029.i			0.429315i
$621$	163956.	−	130691.i	0.425152	−	0.338892i
$622$	−414999.			−1.07267
$623$			157102.i			0.404768i
$624$	102031.	−	94657.1i	0.262038	−	0.243099i
$625$	1.37708e6			3.52532
$626$		−	153457.i		−	0.391595i
$627$	94561.7	+	101929.i	0.240536	+	0.259275i
$628$	109106.			0.276649
$629$			136522.i			0.345067i
$630$	−15247.9	+	203062.i	−0.0384174	+	0.511621i
$631$	−570880.			−1.43379			−0.716896	−	0.697180i	$-0.754438\pi$
$631$	−570880.			−1.43379			−0.716896	+	0.697180i	$0.754438\pi$
$632$		−	106617.i		−	0.266928i
$633$	58197.5	−	53991.3i	0.145244	−	0.134746i
$634$	−368685.			−0.917226
$635$			933416.i			2.31488i
$636$	−76737.1	−	82715.3i	−0.189710	−	0.204490i
$637$	82878.2			0.204250
$638$		−	32861.9i		−	0.0807331i
$639$	300304.	+	22549.7i	0.735460	+	0.0552254i
$640$	−69500.6			−0.169679
$641$		−	570798.i		−	1.38921i	−0.719394	−	0.694603i	$-0.755580\pi$
$641$		−	570798.i		−	1.38921i	0.719394	−	0.694603i	$-0.244420\pi$
$642$	68604.1	−	63645.8i	0.166449	−	0.154419i
$643$	−708381.			−1.71335			−0.856673	−	0.515860i	$-0.827473\pi$
$643$	−708381.			−1.71335			−0.856673	+	0.515860i	$0.827473\pi$
$644$			42613.4i			0.102748i
$645$	−286775.	−	309116.i	−0.689322	−	0.743024i
$646$	−22848.9			−0.0547521
$647$		−	61606.2i		−	0.147169i	−0.997289	−	0.0735844i	$-0.976556\pi$
$647$		−	61606.2i		−	0.147169i	0.997289	−	0.0735844i	$-0.0234438\pi$
$648$	−22170.4	+	146794.i	−0.0527987	+	0.349589i
$649$	−717986.			−1.70462
$650$		−	1.14698e6i		−	2.71475i
$651$	−52523.1	+	48727.0i	−0.123933	+	0.114976i
$652$	309149.			0.727232
$653$		−	524113.i		−	1.22913i	−0.788865	−	0.614567i	$-0.789331\pi$
$653$		−	524113.i		−	1.22913i	0.788865	−	0.614567i	$-0.210669\pi$
$654$	−200782.	−	216424.i	−0.469428	−	0.505999i
$655$	−716023.			−1.66895
$656$		−	78094.4i		−	0.181473i
$657$	12269.9	−	163403.i	0.0284256	−	0.378556i
$658$	27582.5			0.0637063
$659$		−	234214.i		−	0.539313i	−0.962957	−	0.269657i	$-0.913090\pi$
$659$		−	234214.i		−	0.539313i	0.962957	−	0.269657i	$-0.0869102\pi$
$660$	−312184.	+	289621.i	−0.716676	+	0.664878i
$661$	−227410.			−0.520484			−0.260242	−	0.965543i	$-0.583802\pi$
$661$	−227410.			−0.520484			−0.260242	+	0.965543i	$0.583802\pi$
$662$		−	99085.6i		−	0.226097i
$663$	95311.6	+	102737.i	0.216830	+	0.233722i
$664$	−100033.			−0.226885
$665$			111422.i			0.251957i
$666$	483995.	+	36343.0i	1.09117	+	0.0819355i
$667$	−27115.5			−0.0609488
$668$		−	271861.i		−	0.609248i
$669$	−343823.	+	318973.i	−0.768215	+	0.712693i
$670$	−179332.			−0.399493
$671$		−	610642.i		−	1.35626i
$672$	−20521.0	−	22119.7i	−0.0454423	−	0.0489825i
$673$	203616.			0.449553			0.224776	−	0.974410i	$-0.427835\pi$
$673$	203616.			0.449553			0.224776	+	0.974410i	$0.427835\pi$
$674$		−	521293.i		−	1.14753i
$675$	762605.	+	956716.i	1.67376	+	2.09979i
$676$	−238583.			−0.522090
$677$			494274.i			1.07843i	0.842169	+	0.539213i	$0.181278\pi$
$677$			494274.i			1.07843i	−0.842169	+	0.539213i	$0.818722\pi$
$678$	20050.2	−	18601.1i	0.0436173	−	0.0404649i
$679$	302641.			0.656430
$680$		−	69981.1i		−	0.151343i
$681$	369572.	+	398364.i	0.796902	+	0.858985i
$682$	−149824.			−0.322116
$683$			43360.8i			0.0929514i	0.998919	+	0.0464757i	$0.0147990\pi$
$683$			43360.8i			0.0929514i	−0.998919	+	0.0464757i	$0.985201\pi$
$684$	−6082.51	+	81003.3i	−0.0130008	+	0.173137i
$685$	945440.			2.01490
$686$		−	17967.4i		−	0.0381802i
$687$	20027.8	−	18580.3i	0.0424345	−	0.0393676i
$688$	62477.2			0.131991
$689$			378647.i			0.797621i
$690$	238976.	+	257593.i	0.501945	+	0.541049i
$691$	429593.			0.899708			0.449854	−	0.893102i	$-0.351476\pi$
$691$	429593.			0.899708			0.449854	+	0.893102i	$0.351476\pi$
$692$		−	97797.7i		−	0.204229i
$693$	−184353.	−	13843.0i	−0.383870	−	0.0288247i
$694$	43332.2			0.0899689
$695$		−	1.06669e6i		−	2.20836i
$696$	14075.1	−	13057.8i	0.0290557	−	0.0269557i
$697$	78634.4			0.161863
$698$			241255.i			0.495182i
$699$	211755.	+	228252.i	0.433390	+	0.467153i
$700$	−248658.			−0.507464
$701$			281406.i			0.572660i	0.958131	+	0.286330i	$0.0924354\pi$
$701$			281406.i			0.572660i	−0.958131	+	0.286330i	$0.907565\pi$
$702$	389591.	−	310546.i	0.790561	−	0.630162i
$703$	265571.			0.537365
$704$		−	63097.1i		−	0.127311i
$705$	166733.	−	154683.i	0.335463	−	0.311217i
$706$	339581.			0.681292
$707$		−	96145.9i		−	0.192350i
$708$	−285294.	−	307520.i	−0.569150	−	0.613489i
$709$	−273533.			−0.544148			−0.272074	−	0.962276i	$-0.587709\pi$
$709$	−273533.			−0.544148			−0.272074	+	0.962276i	$0.587709\pi$
$710$			504679.i			1.00115i
$711$	28578.3	−	380589.i	0.0565324	−	0.752866i
$712$	−191942.			−0.378626
$713$			123625.i			0.243179i
$714$	22272.6	−	20662.9i	0.0436893	−	0.0405317i
$715$	1.42909e6			2.79542
$716$			426230.i			0.831416i
$717$	−479339.	−	516682.i	−0.932405	−	1.00504i
$718$	465763.			0.903475
$719$		−	886466.i		−	1.71476i	−0.514681	−	0.857381i	$-0.672090\pi$
$719$		−	886466.i		−	1.71476i	0.514681	−	0.857381i	$-0.327910\pi$
$720$	−248095.	−	18629.3i	−0.478578	−	0.0359362i
$721$	−175907.			−0.338386
$722$		−	324156.i		−	0.621842i
$723$	187132.	−	173607.i	0.357990	−	0.332117i
$724$	−158318.			−0.302032
$725$		−	158224.i		−	0.301021i
$726$	−9457.40	−	10194.2i	−0.0179431	−	0.0193410i
$727$	541251.			1.02407			0.512036	−	0.858964i	$-0.328892\pi$
$727$	541251.			1.02407			0.512036	+	0.858964i	$0.328892\pi$
$728$			101258.i			0.191058i
$729$	−118489.	+	518064.i	−0.222957	+	0.974828i
$730$	274609.			0.515311
$731$			62909.1i			0.117728i
$732$	261544.	−	242641.i	0.488115	−	0.452837i
$733$	−698311.			−1.29969			−0.649846	−	0.760066i	$-0.725167\pi$
$733$	−698311.			−1.29969			−0.649846	+	0.760066i	$0.725167\pi$
$734$			507768.i			0.942482i
$735$	−100761.	−	108611.i	−0.186518	−	0.201048i
$736$	−52063.6			−0.0961122
$737$		−	162810.i		−	0.299740i
$738$	20932.9	−	278772.i	0.0384341	−	0.511843i
$739$	−584503.			−1.07028			−0.535140	−	0.844763i	$-0.679741\pi$
$739$	−584503.			−1.07028			−0.535140	+	0.844763i	$0.679741\pi$
$740$			813383.i			1.48536i
$741$	199849.	−	185405.i	0.363970	−	0.337665i
$742$	82088.2			0.149098
$743$		−	1.06823e6i		−	1.93503i	−0.252822	−	0.967513i	$-0.581359\pi$
$743$		−	1.06823e6i		−	1.93503i	0.252822	−	0.967513i	$-0.418641\pi$
$744$	−59533.0	−	64170.9i	−0.107550	−	0.115929i
$745$	−325931.			−0.587237
$746$			438011.i			0.787059i
$747$	−357085.	−	26813.4i	−0.639928	−	0.0480519i
$748$	63533.4			0.113553
$749$			68084.0i			0.121362i
$750$	−943344.	+	875164.i	−1.67706	+	1.55585i
$751$	−526620.			−0.933722			−0.466861	−	0.884331i	$-0.654615\pi$
$751$	−526620.			−0.933722			−0.466861	+	0.884331i	$0.654615\pi$
$752$			33699.4i			0.0595917i
$753$	548787.	+	591540.i	0.967862	+	1.04326i
$754$	−64431.7			−0.113333
$755$			406520.i			0.713162i
$756$	−67324.3	−	84460.8i	−0.117795	−	0.147779i
$757$	−412218.			−0.719342			−0.359671	−	0.933079i	$-0.617111\pi$
$757$	−412218.			−0.719342			−0.359671	+	0.933079i	$0.617111\pi$
$758$		−	683469.i		−	1.18954i
$759$	−233860.	+	216958.i	−0.405950	+	0.376610i
$760$	−136131.			−0.235684
$761$			51994.6i			0.0897819i	0.998992	+	0.0448909i	$0.0142940\pi$
$761$			51994.6i			0.0897819i	−0.998992	+	0.0448909i	$0.985706\pi$
$762$	−336724.	−	362956.i	−0.579914	−	0.625092i
$763$	214783.			0.368936
$764$		−	178794.i		−	0.306313i
$765$	18758.1	−	249810.i	0.0320529	−	0.426862i
$766$	−269331.			−0.459017
$767$			1.40774e6i			2.39294i
$768$	27025.1	−	25071.9i	0.0458189	−	0.0425074i
$769$	260902.			0.441189			0.220595	−	0.975366i	$-0.429200\pi$
$769$	260902.			0.441189			0.220595	+	0.975366i	$0.429200\pi$
$770$		−	309817.i		−	0.522545i
$771$	−75516.0	−	81399.0i	−0.127037	−	0.136934i
$772$	445980.			0.748308
$773$		−	323742.i		−	0.541801i	−0.962607	−	0.270900i	$-0.912679\pi$
$773$		−	323742.i		−	0.541801i	0.962607	−	0.270900i	$-0.0873214\pi$
$774$	223023.	+	16746.7i	0.372279	+	0.0279543i
$775$	−721375.			−1.20104
$776$			369756.i			0.614034i
$777$	−258873.	+	240163.i	−0.428789	+	0.397799i
$778$	634418.			1.04813
$779$		−	152964.i		−	0.252066i
$780$	567854.	+	612093.i	0.933356	+	1.00607i
$781$	−458181.			−0.751164
$782$		−	52423.6i		−	0.0857261i
$783$	53743.5	−	42839.4i	0.0876602	−	0.0698746i
$784$	21952.0			0.0357143
$785$			654533.i			1.06217i
$786$	278423.	−	258301.i	0.450672	−	0.418100i
$787$	−642009.			−1.03655			−0.518277	−	0.855213i	$-0.673426\pi$
$787$	−642009.			−1.03655			−0.518277	+	0.855213i	$0.673426\pi$
$788$			26458.7i			0.0426104i
$789$	243642.	+	262623.i	0.391380	+	0.421870i
$790$	639604.			1.02484
$791$			19898.2i			0.0318024i
$792$	16912.9	−	225237.i	0.0269630	−	0.359078i
$793$	−1.19727e6			−1.90391
$794$		−	225355.i		−	0.357459i
$795$	496214.	−	460351.i	0.785118	−	0.728374i
$796$	−255698.			−0.403554
$797$		−	26635.0i		−	0.0419311i	−0.999780	−	0.0209656i	$-0.993326\pi$
$797$		−	26635.0i		−	0.0419311i	0.999780	−	0.0209656i	$-0.00667404\pi$
$798$	−40194.6	−	43325.9i	−0.0631192	−	0.0680365i
$799$	−33932.4			−0.0531521
$800$		−	303801.i		−	0.474690i
$801$	−685171.	−	51449.2i	−1.06791	−	0.0801888i
$802$	247499.			0.384790
$803$			249308.i			0.386639i
$804$	69732.8	−	64693.0i	0.107876	−	0.100079i
$805$	−255640.			−0.394492
$806$			293757.i			0.452186i
$807$	233883.	+	252104.i	0.359131	+	0.387109i
$808$	117468.			0.179927
$809$		−	904872.i		−	1.38258i	−0.722578	−	0.691290i	$-0.757043\pi$
$809$		−	904872.i		−	1.38258i	0.722578	−	0.691290i	$-0.242957\pi$
$810$	−880625.	−	133001.i	−1.34221	−	0.202715i
$811$	−528283.			−0.803203			−0.401601	−	0.915815i	$-0.631546\pi$
$811$	−528283.			−0.803203			−0.401601	+	0.915815i	$0.631546\pi$
$812$			13968.3i			0.0211852i
$813$	889908.	−	825590.i	1.34637	−	1.24906i
$814$	−738442.			−1.11447
$815$			1.85460e6i			2.79213i
$816$	25245.2	+	27211.9i	0.0379139	+	0.0408676i
$817$	122374.			0.183335
$818$			735842.i			1.09971i
$819$	−27141.7	+	361458.i	−0.0404641	+	0.538877i
$820$	468494.			0.696748
$821$			277365.i			0.411495i	0.978605	+	0.205748i	$0.0659626\pi$
$821$			277365.i			0.411495i	−0.978605	+	0.205748i	$0.934037\pi$
$822$	−367632.	+	341061.i	−0.544088	+	0.504764i
$823$	554443.			0.818572			0.409286	−	0.912406i	$-0.365778\pi$
$823$	554443.			0.818572			0.409286	+	0.912406i	$0.365778\pi$
$824$		−	214917.i		−	0.316531i
$825$	−1.26599e6	−	1.36462e6i	−1.86004	−	2.00495i
$826$	305189.			0.447310
$827$			844426.i			1.23467i	0.786701	+	0.617335i	$0.211787\pi$
$827$			844426.i			1.23467i	−0.786701	+	0.617335i	$0.788213\pi$
$828$	−185850.	−	13955.4i	−0.271083	−	0.0203555i
$829$	74784.2			0.108818			0.0544090	−	0.998519i	$-0.482673\pi$
$829$	74784.2			0.108818			0.0544090	+	0.998519i	$0.482673\pi$
$830$		−	600104.i		−	0.871104i
$831$	889422.	−	825140.i	1.28797	−	1.19488i
$832$	−123713.			−0.178719
$833$			22103.8i			0.0318549i
$834$	384803.	+	414781.i	0.553231	+	0.596330i
$835$	1.63091e6			2.33915
$836$		−	123589.i		−	0.176834i
$837$	−195313.	−	245027.i	−0.278792	−	0.349755i
$838$	628933.			0.895605
$839$		−	463620.i		−	0.658625i	−0.944221	−	0.329313i	$-0.893183\pi$
$839$		−	463620.i		−	0.658625i	0.944221	−	0.329313i	$-0.106817\pi$
$840$	132698.	−	123107.i	0.188063	−	0.174471i
$841$	698393.			0.987433
$842$			777136.i			1.09616i
$843$	−20128.7	−	21696.8i	−0.0283243	−	0.0305309i
$844$	−70564.6			−0.0990609
$845$		−	1.43127e6i		−	2.00451i
$846$	−9032.97	+	120296.i	−0.0126209	+	0.168078i
$847$	10116.9			0.0141020
$848$			100293.i			0.139469i
$849$	−852889.	+	791247.i	−1.18325	+	1.09773i
$850$	305902.			0.423394
$851$			609314.i			0.841360i
$852$	−182060.	−	196243.i	−0.250804	−	0.270343i
$853$	301646.			0.414572			0.207286	−	0.978280i	$-0.433537\pi$
$853$	301646.			0.414572			0.207286	+	0.978280i	$0.433537\pi$
$854$			259561.i			0.355896i
$855$	−485944.	−	36489.3i	−0.664743	−	0.0499153i
$856$	−83182.7			−0.113523
$857$		−	656980.i		−	0.894520i	−0.894404	−	0.447260i	$-0.852400\pi$
$857$		−	656980.i		−	0.894520i	0.894404	−	0.447260i	$-0.147600\pi$
$858$	−555697.	+	515535.i	−0.754855	+	0.700299i
$859$	−104640.			−0.141812			−0.0709060	−	0.997483i	$-0.522589\pi$
$859$	−104640.			−0.141812			−0.0709060	+	0.997483i	$0.522589\pi$
$860$			374805.i			0.506767i
$861$	138329.	+	149106.i	0.186598	+	0.201135i
$862$	−224122.			−0.301627
$863$			1.12056e6i			1.50457i	0.658839	+	0.752284i	$0.271048\pi$
$863$			1.12056e6i			1.50457i	−0.658839	+	0.752284i	$0.728952\pi$
$864$	103191.	−	82254.6i	0.138234	−	0.110188i
$865$	586695.			0.784115
$866$		−	159398.i		−	0.212543i
$867$	523665.	−	485818.i	0.696651	−	0.646301i
$868$	63684.4			0.0845267
$869$			580674.i			0.768941i
$870$	78334.5	+	84437.2i	0.103494	+	0.111557i
$871$	−319217.			−0.420775
$872$			262415.i			0.345108i
$873$	−99111.6	+	1.31991e6i	−0.130046	+	1.73187i
$874$	−101977.			−0.133500
$875$		−	936192.i		−	1.22278i
$876$	−106781.	+	99063.6i	−0.139151	+	0.129094i
$877$	−20694.4			−0.0269062			−0.0134531	−	0.999910i	$-0.504282\pi$
$877$	−20694.4			−0.0269062			−0.0134531	+	0.999910i	$0.504282\pi$
$878$		−	235027.i		−	0.304879i
$879$	−1.03078e6	−	1.11108e6i	−1.33410	−	1.43803i
$880$	378524.			0.488796
$881$		−	711224.i		−	0.916336i	−0.888866	−	0.458168i	$-0.848506\pi$
$881$		−	711224.i		−	0.916336i	0.888866	−	0.458168i	$-0.151494\pi$
$882$	78361.6	+	5884.14i	0.100732	+	0.00756390i
$883$	260420.			0.334004			0.167002	−	0.985957i	$-0.446591\pi$
$883$	260420.			0.334004			0.167002	+	0.985957i	$0.446591\pi$
$884$		−	124569.i		−	0.159406i
$885$	1.84483e6	−	1.71150e6i	2.35543	−	2.18519i
$886$	−395744.			−0.504135
$887$			649369.i			0.825362i	0.910876	+	0.412681i	$0.135408\pi$
$887$			649369.i			0.825362i	−0.910876	+	0.412681i	$0.864592\pi$
$888$	−293423.	−	316282.i	−0.372107	−	0.401096i
$889$	360204.			0.455770
$890$		−	1.15147e6i		−	1.45370i
$891$	120747.	−	799488.i	0.152098	−	1.00706i
$892$	416886.			0.523948
$893$			66007.1i			0.0827728i
$894$	126737.	−	117577.i	0.158573	−	0.147112i
$895$	−2.55698e6			−3.19214
$896$			26820.2i			0.0334077i
$897$	425385.	+	458525.i	0.528685	+	0.569873i
$898$	−283602.			−0.351688
$899$			40523.2i			0.0501401i
$900$	81432.6	−	1.08447e6i	0.100534	−	1.33886i
$901$	−100986.			−0.124397
$902$			425329.i			0.522771i
$903$	−119288.	+	110666.i	−0.146292	+	0.135719i
$904$	−24310.9			−0.0297484
$905$		−	949759.i		−	1.15962i
$906$	−146650.	−	158074.i	−0.178659	−	0.192577i
$907$	1.11128e6			1.35086			0.675429	−	0.737425i	$-0.263958\pi$
$907$	1.11128e6			1.35086			0.675429	+	0.737425i	$0.263958\pi$
$908$		−	483017.i		−	0.585856i
$909$	419322.	+	31486.7i	0.507481	+	0.0381066i
$910$	−607452.			−0.733549
$911$			125363.i			0.151054i	0.997144	+	0.0755272i	$0.0240640\pi$
$911$			125363.i			0.151054i	−0.997144	+	0.0755272i	$0.975936\pi$
$912$	52934.1	−	49108.4i	0.0636424	−	0.0590426i
$913$	544813.			0.653591
$914$		−	234735.i		−	0.280986i
$915$	1.45562e6	+	1.56902e6i	1.73862	+	1.87407i
$916$	−24283.8			−0.0289418
$917$			276313.i			0.328596i
$918$	82823.3	+	103905.i	0.0982804	+	0.123296i
$919$	748471.			0.886225			0.443113	−	0.896466i	$-0.353874\pi$
$919$	748471.			0.886225			0.443113	+	0.896466i	$0.353874\pi$
$920$		−	312333.i		−	0.369013i
$921$	−292907.	+	271738.i	−0.345311	+	0.320354i
$922$	910000.			1.07048
$923$			898345.i			1.05448i
$924$	111764.	+	120471.i	0.130906	+	0.141104i
$925$	−3.55547e6			−4.15540
$926$		−	290973.i		−	0.339336i
$927$	57607.6	−	767185.i	0.0670379	−	0.892773i
$928$	−17066.0			−0.0198170
$929$		−	132284.i		−	0.153276i	−0.997059	−	0.0766381i	$-0.975581\pi$
$929$		−	132284.i		−	0.153276i	0.997059	−	0.0766381i	$-0.0244186\pi$
$930$	384966.	−	357142.i	0.445098	−	0.412929i
$931$	42997.5			0.0496071
$932$		−	276756.i		−	0.318614i
$933$	898109.	+	968076.i	1.03173	+	1.11211i
$934$	246700.			0.282797
$935$			381141.i			0.435976i
$936$	−441617.	−	33160.8i	−0.504073	−	0.0378507i
$937$	318721.			0.363020			0.181510	−	0.983389i	$-0.441901\pi$
$937$	318721.			0.363020			0.181510	+	0.983389i	$0.441901\pi$
$938$			69204.2i			0.0786550i
$939$	−357971.	+	332099.i	−0.405991	+	0.376649i
$940$	−202165.			−0.228797
$941$		−	897526.i		−	1.01360i	−0.862063	−	0.506802i	$-0.830828\pi$
$941$		−	897526.i		−	1.01360i	0.862063	−	0.506802i	$-0.169172\pi$
$942$	−236118.	−	254513.i	−0.266090	−	0.286819i
$943$	350953.			0.394663
$944$			372869.i			0.418420i
$945$	506686.	−	403883.i	0.567381	−	0.452264i
$946$	−340272.			−0.380228
$947$			416790.i			0.464748i	0.972626	+	0.232374i	$0.0746494\pi$
$947$			416790.i			0.464748i	−0.972626	+	0.232374i	$0.925351\pi$
$948$	−248708.	+	230733.i	−0.276741	+	0.256740i
$949$	488814.			0.542764
$950$		−	595057.i		−	0.659343i
$951$	797879.	+	860037.i	0.882218	+	0.950947i
$952$	−27005.6			−0.0297975
$953$		−	1.57654e6i		−	1.73587i	−0.496676	−	0.867936i	$-0.665446\pi$
$953$		−	1.57654e6i		−	1.73587i	0.496676	−	0.867936i	$-0.334554\pi$
$954$	−26883.0	+	358012.i	−0.0295380	+	0.393370i
$955$	1.07259e6			1.17606
$956$			626479.i			0.685473i
$957$	−76657.6	+	71117.2i	−0.0837011	+	0.0776517i
$958$	−371652.			−0.404954
$959$		−	364844.i		−	0.396707i
$960$	150408.	+	162125.i	0.163203	+	0.175917i
$961$	−738768.			−0.799947
$962$			1.44785e6i			1.56449i
$963$	−296935.	−	22296.8i	−0.320191	−	0.0240430i
$964$	−226898.			−0.244161
$965$			2.67546e6i			2.87305i
$966$	99405.0	−	92220.6i	0.106526	−	0.0988266i
$967$	−449565.			−0.480772			−0.240386	−	0.970677i	$-0.577274\pi$
$967$	−449565.			−0.480772			−0.240386	+	0.970677i	$0.577274\pi$
$968$			12360.5i			0.0131912i
$969$	49447.9	+	53300.1i	0.0526624	+	0.0567650i
$970$	−2.21819e6			−2.35752
$971$		−	397697.i		−	0.421807i	−0.977507	−	0.210904i	$-0.932359\pi$
$971$		−	397697.i		−	0.421807i	0.977507	−	0.210904i	$-0.0676407\pi$
$972$	390408.	−	265962.i	0.413224	−	0.281506i
$973$	−411636.			−0.434799
$974$		−	327596.i		−	0.345319i
$975$	−2.67558e6	+	2.48221e6i	−2.81455	+	2.61113i
$976$	−317122.			−0.332910
$977$			1.19821e6i			1.25529i	0.778499	+	0.627646i	$0.215982\pi$
$977$			1.19821e6i			1.25529i	−0.778499	+	0.627646i	$0.784018\pi$
$978$	−669036.	−	721158.i	−0.699475	−	0.753967i
$979$	1.04538e6			1.09071
$980$			131691.i			0.137121i
$981$	−70339.1	+	936736.i	−0.0730901	+	0.973372i
$982$	−1.28842e6			−1.33609
$983$		−	1.59614e6i		−	1.65183i	−0.563798	−	0.825913i	$-0.690660\pi$
$983$		−	1.59614e6i		−	1.65183i	0.563798	−	0.825913i	$-0.309340\pi$
$984$	−182172.	+	169006.i	−0.188145	+	0.174547i
$985$	−158727.			−0.163598
$986$		−	17184.0i		−	0.0176755i
$987$	−59691.9	−	64342.2i	−0.0612747	−	0.0660483i
$988$	−242318.			−0.248240
$989$			280770.i			0.287050i
$990$	1.35121e6	+	101462.i	1.37864	+	0.103522i
$991$	1.43265e6			1.45879			0.729396	−	0.684092i	$-0.239801\pi$
$991$	1.43265e6			1.45879			0.729396	+	0.684092i	$0.239801\pi$
$992$			77807.4i			0.0790675i
$993$	−231139.	+	214433.i	−0.234409	+	0.217467i
$994$	194755.			0.197113
$995$		−	1.53395e6i		−	1.54940i
$996$	216483.	+	233349.i	0.218226	+	0.235227i
$997$	−397969.			−0.400368			−0.200184	−	0.979758i	$-0.564154\pi$
$997$	−397969.			−0.400368			−0.200184	+	0.979758i	$0.564154\pi$
$998$		−	1.00051e6i		−	1.00452i
$999$	−962646.	−	1.20767e6i	−0.964575	−	1.21009i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.5.b.a.29.7	yes	8
3.2	odd	2	inner	42.5.b.a.29.3	✓	8
4.3	odd	2		336.5.d.a.113.4		8
7.6	odd	2		294.5.b.d.197.6		8
12.11	even	2		336.5.d.a.113.3		8
21.20	even	2		294.5.b.d.197.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.5.b.a.29.3	✓	8	3.2	odd	2	inner
42.5.b.a.29.7	yes	8	1.1	even	1	trivial
294.5.b.d.197.2		8	21.20	even	2
294.5.b.d.197.6		8	7.6	odd	2
336.5.d.a.113.3		8	12.11	even	2
336.5.d.a.113.4		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	42.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} +(6.59792 - 6.12106i) q^{3} -8.00000 q^{4} -47.9925i q^{5} +(17.3130 + 18.6617i) q^{6} -18.5203 q^{7} -22.6274i q^{8} +(6.06518 - 80.7726i) q^{9} +135.743 q^{10} +123.237i q^{11} +(-52.7834 + 48.9685i) q^{12} +241.627 q^{13} -52.3832i q^{14} +(-293.765 - 316.651i) q^{15} +64.0000 q^{16} +64.4425i q^{17} +(228.459 + 17.1549i) q^{18} +125.357 q^{19} +383.940i q^{20} +(-122.195 + 113.364i) q^{21} -348.566 q^{22} +287.613i q^{23} +(-138.504 - 149.294i) q^{24} -1678.28 q^{25} +683.426i q^{26} +(-454.397 - 570.057i) q^{27} +148.162 q^{28} +94.2775i q^{29} +(895.624 - 830.893i) q^{30} +429.829 q^{31} +181.019i q^{32} +(754.339 + 813.106i) q^{33} -182.271 q^{34} +888.834i q^{35} +(-48.5215 + 646.181i) q^{36} +2118.52 q^{37} +354.563i q^{38} +(1594.24 - 1479.02i) q^{39} -1085.95 q^{40} -1220.23i q^{41} +(-320.641 - 345.620i) q^{42} +976.206 q^{43} -985.893i q^{44} +(-3876.48 - 291.083i) q^{45} -813.493 q^{46} +526.553i q^{47} +(422.267 - 391.748i) q^{48} +343.000 q^{49} -4746.90i q^{50} +(394.457 + 425.187i) q^{51} -1933.02 q^{52} +1567.07i q^{53} +(1612.36 - 1285.23i) q^{54} +5914.43 q^{55} +419.066i q^{56} +(827.096 - 767.318i) q^{57} -266.657 q^{58} +5826.08i q^{59} +(2350.12 + 2533.21i) q^{60} -4955.04 q^{61} +1215.74i q^{62} +(-112.329 + 1495.93i) q^{63} -512.000 q^{64} -11596.3i q^{65} +(-2299.81 + 2133.59i) q^{66} -1321.11 q^{67} -515.540i q^{68} +(1760.50 + 1897.65i) q^{69} -2514.00 q^{70} +3717.89i q^{71} +(-1827.68 - 137.239i) q^{72} +2023.01 q^{73} +5992.07i q^{74} +(-11073.2 + 10272.9i) q^{75} -1002.86 q^{76} -2282.37i q^{77} +(4183.29 + 4509.19i) q^{78} +4711.86 q^{79} -3071.52i q^{80} +(-6487.43 - 979.801i) q^{81} +3451.32 q^{82} -4420.87i q^{83} +(977.562 - 906.909i) q^{84} +3092.76 q^{85} +2761.13i q^{86} +(577.078 + 622.036i) q^{87} +2788.53 q^{88} -8482.72i q^{89} +(823.308 - 10964.3i) q^{90} -4475.00 q^{91} -2300.91i q^{92} +(2835.98 - 2631.01i) q^{93} -1489.32 q^{94} -6016.20i q^{95} +(1108.03 + 1194.35i) q^{96} -16341.1 q^{97} +970.151i q^{98} +(9954.14 + 747.452i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{3} - 64 q^{4} - 32 q^{6} + 152 q^{9} + 256 q^{10} - 96 q^{12} - 520 q^{13} - 616 q^{15} + 512 q^{16} + 768 q^{18} + 1336 q^{19} - 196 q^{21} - 1024 q^{22} + 256 q^{24} - 3672 q^{25} + 36 q^{27}+ \cdots + 10976 q^{99}+O(q^{100}) \) 8 * q + 12 * q^3 - 64 * q^4 - 32 * q^6 + 152 * q^9 + 256 * q^10 - 96 * q^12 - 520 * q^13 - 616 * q^15 + 512 * q^16 + 768 * q^18 + 1336 * q^19 - 196 * q^21 - 1024 * q^22 + 256 * q^24 - 3672 * q^25 + 36 * q^27 - 1216 * q^30 + 4880 * q^31 + 1432 * q^33 + 1280 * q^34 - 1216 * q^36 - 800 * q^37 + 5672 * q^39 - 2048 * q^40 - 2816 * q^43 - 10712 * q^45 + 896 * q^46 + 768 * q^48 + 2744 * q^49 - 5768 * q^51 + 4160 * q^52 + 3616 * q^54 - 2320 * q^55 + 12864 * q^57 - 5504 * q^58 + 4928 * q^60 + 2696 * q^61 + 8232 * q^63 - 4096 * q^64 - 15680 * q^66 - 6960 * q^67 - 2864 * q^69 - 3136 * q^70 - 6144 * q^72 + 27648 * q^73 - 15052 * q^75 - 10688 * q^76 + 15808 * q^78 - 1024 * q^79 + 17528 * q^81 + 18048 * q^82 + 1568 * q^84 - 19504 * q^85 - 13960 * q^87 + 8192 * q^88 + 7360 * q^90 - 23128 * q^91 - 30016 * q^93 + 19584 * q^94 - 2048 * q^96 - 27408 * q^97 + 10976 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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