LMFDB - Embedded newform 38.9.b.a.37.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,9,Mod(37,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("38.37");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	38.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:	$15.4803871823$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 $ x^12 + 46118x^10 + 738386961x^8 + 5214446299656x^6 + 17370647163698184x^4 + 24830681474333400768*x^2 + 9218779084612644462864
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{21} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.10
Root			$23.4825i$ of defining polynomial
Character	$\chi$	$=$	38.37
Dual form			38.9.b.a.37.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.3137i q^{2} +10.7546i q^{3} -128.000 q^{4} -919.278 q^{5} -121.674 q^{6} -343.629 q^{7} -1448.15i q^{8} +6445.34 q^{9} +O(q^{10})$	\(q+11.3137i q^{2} +10.7546i q^{3} -128.000 q^{4} -919.278 q^{5} -121.674 q^{6} -343.629 q^{7} -1448.15i q^{8} +6445.34 q^{9} -10400.4i q^{10} +4190.29 q^{11} -1376.59i q^{12} -16507.2i q^{13} -3887.72i q^{14} -9886.47i q^{15} +16384.0 q^{16} +5020.47 q^{17} +72920.7i q^{18} +(39747.8 - 124112. i) q^{19} +117668. q^{20} -3695.59i q^{21} +47407.7i q^{22} +350263. q^{23} +15574.3 q^{24} +454447. q^{25} +186758. q^{26} +139878. i q^{27} +43984.5 q^{28} -484397. i q^{29} +111853. q^{30} +263837. i q^{31} +185364. i q^{32} +45064.9i q^{33} +56800.1i q^{34} +315891. q^{35} -825003. q^{36} -2.12516e6i q^{37} +(1.40416e6 + 449695. i) q^{38} +177529. q^{39} +1.33126e6i q^{40} -3.03570e6i q^{41} +41810.9 q^{42} -2.35738e6 q^{43} -536357. q^{44} -5.92506e6 q^{45} +3.96277e6i q^{46} +727875. q^{47} +176203. i q^{48} -5.64672e6 q^{49} +5.14148e6i q^{50} +53993.1i q^{51} +2.11293e6i q^{52} -6.39013e6i q^{53} -1.58254e6 q^{54} -3.85204e6 q^{55} +497628. i q^{56} +(1.33477e6 + 427472. i) q^{57} +5.48032e6 q^{58} +1.53982e7i q^{59} +1.26547e6i q^{60} +963914. q^{61} -2.98497e6 q^{62} -2.21481e6 q^{63} -2.09715e6 q^{64} +1.51747e7i q^{65} -509851. q^{66} -3.33818e7i q^{67} -642620. q^{68} +3.76694e6i q^{69} +3.57390e6i q^{70} -2.44350e7i q^{71} -9.33385e6i q^{72} +2.36687e7 q^{73} +2.40434e7 q^{74} +4.88740e6i q^{75} +(-5.08772e6 + 1.58863e7i) q^{76} -1.43991e6 q^{77} +2.00851e6i q^{78} -5.45025e7i q^{79} -1.50615e7 q^{80} +4.07835e7 q^{81} +3.43450e7 q^{82} +4.12252e7 q^{83} +473036. i q^{84} -4.61521e6 q^{85} -2.66707e7i q^{86} +5.20949e6 q^{87} -6.06818e6i q^{88} +7.15445e7i q^{89} -6.70344e7i q^{90} +5.67237e6i q^{91} -4.48337e7 q^{92} -2.83746e6 q^{93} +8.23496e6i q^{94} +(-3.65393e7 + 1.14093e8i) q^{95} -1.99351e6 q^{96} +1.33289e8i q^{97} -6.38853e7i q^{98} +2.70078e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9}+O(q^{10}) $ 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9	$ 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9} - 12546 q^{11} + 196608 q^{16} + 270810 q^{17} + 41512 q^{19} - 71424 q^{20} - 823956 q^{23} - 229376 q^{24} + 865538 q^{25} - 431616 q^{26} + 694016 q^{28} + 71168 q^{30} - 1194378 q^{35} + 1995776 q^{36} + 998784 q^{38} + 5786100 q^{39} - 8383744 q^{42} + 7586646 q^{43} + 1605888 q^{44} + 2226046 q^{45} - 20260530 q^{47} - 19498842 q^{49} + 16933888 q^{54} - 14858554 q^{55} + 14430564 q^{57} - 5506560 q^{58} - 41363266 q^{61} + 32266752 q^{62} + 84235798 q^{63} - 25165824 q^{64} + 14371328 q^{66} - 34663680 q^{68} + 87906498 q^{73} - 2149632 q^{74} - 5313536 q^{76} - 78817962 q^{77} + 9142272 q^{80} - 100904812 q^{81} - 49609728 q^{82} - 55944960 q^{83} + 25440254 q^{85} + 119189604 q^{87} + 105466368 q^{92} + 105500856 q^{93} + 81396774 q^{95} + 29360128 q^{96} - 85554938 q^{99}+O(q^{100}) $ 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9 - 12546 * q^11 + 196608 * q^16 + 270810 * q^17 + 41512 * q^19 - 71424 * q^20 - 823956 * q^23 - 229376 * q^24 + 865538 * q^25 - 431616 * q^26 + 694016 * q^28 + 71168 * q^30 - 1194378 * q^35 + 1995776 * q^36 + 998784 * q^38 + 5786100 * q^39 - 8383744 * q^42 + 7586646 * q^43 + 1605888 * q^44 + 2226046 * q^45 - 20260530 * q^47 - 19498842 * q^49 + 16933888 * q^54 - 14858554 * q^55 + 14430564 * q^57 - 5506560 * q^58 - 41363266 * q^61 + 32266752 * q^62 + 84235798 * q^63 - 25165824 * q^64 + 14371328 * q^66 - 34663680 * q^68 + 87906498 * q^73 - 2149632 * q^74 - 5313536 * q^76 - 78817962 * q^77 + 9142272 * q^80 - 100904812 * q^81 - 49609728 * q^82 - 55944960 * q^83 + 25440254 * q^85 + 119189604 * q^87 + 105466368 * q^92 + 105500856 * q^93 + 81396774 * q^95 + 29360128 * q^96 - 85554938 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$.

$n$	$21$
$\chi(n)$	$-1$

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 $

$2$			11.3137i			0.707107i
$2$			11.3137i			0.707107i
$3$			10.7546i			0.132773i	0.997794	+	0.0663864i	$0.0211470\pi$
$3$			10.7546i			0.132773i	−0.997794	+	0.0663864i	$0.978853\pi$
$4$	−128.000			−0.500000
$5$	−919.278			−1.47085			−0.735423	−	0.677609i	$-0.763016\pi$
$5$	−919.278			−1.47085			−0.735423	+	0.677609i	$0.763016\pi$
$6$	−121.674			−0.0938846
$7$	−343.629			−0.143119			−0.0715596	−	0.997436i	$-0.522798\pi$
$7$	−343.629			−0.143119			−0.0715596	+	0.997436i	$0.522798\pi$
$8$		−	1448.15i		−	0.353553i
$9$	6445.34			0.982371
$10$		−	10400.4i		−	1.04004i
$11$	4190.29			0.286202			0.143101	−	0.989708i	$-0.454293\pi$
$11$	4190.29			0.286202			0.143101	+	0.989708i	$0.454293\pi$
$12$		−	1376.59i		−	0.0663864i
$13$		−	16507.2i		−	0.577964i	−0.957335	−	0.288982i	$-0.906683\pi$
$13$		−	16507.2i		−	0.577964i	0.957335	−	0.288982i	$-0.0933168\pi$
$14$		−	3887.72i		−	0.101201i
$15$		−	9886.47i		−	0.195288i
$16$	16384.0			0.250000
$17$	5020.47			0.0601103			0.0300551	−	0.999548i	$-0.490432\pi$
$17$	5020.47			0.0601103			0.0300551	+	0.999548i	$0.490432\pi$
$18$			72920.7i			0.694641i
$19$	39747.8	−	124112.i	0.304999	−	0.952353i
$19$	39747.8	−	124112.i	0.304999	−	0.952353i
$20$	117668.			0.735423
$21$		−	3695.59i		−	0.0190023i
$22$			47407.7i			0.202376i
$23$	350263.			1.25165			0.625825	−	0.779963i	$-0.284762\pi$
$23$	350263.			1.25165			0.625825	+	0.779963i	$0.284762\pi$
$24$	15574.3			0.0469423
$25$	454447.			1.16339
$26$	186758.			0.408682
$27$			139878.i			0.263205i
$28$	43984.5			0.0715596
$29$		−	484397.i		−	0.684872i	−0.939541	−	0.342436i	$-0.888748\pi$
$29$		−	484397.i		−	0.684872i	0.939541	−	0.342436i	$-0.111252\pi$
$30$	111853.			0.138090
$31$			263837.i			0.285686i	0.989745	+	0.142843i	$0.0456244\pi$
$31$			263837.i			0.285686i	−0.989745	+	0.142843i	$0.954376\pi$
$32$			185364.i			0.176777i
$33$			45064.9i			0.0379999i
$34$			56800.1i			0.0425044i
$35$	315891.			0.210506
$36$	−825003.			−0.491186
$37$		−	2.12516e6i		−	1.13393i	−0.823743	−	0.566963i	$-0.808118\pi$
$37$		−	2.12516e6i		−	1.13393i	0.823743	−	0.566963i	$-0.191882\pi$
$38$	1.40416e6	+	449695.i	0.673415	+	0.215667i
$39$	177529.			0.0767379
$40$			1.33126e6i			0.520022i
$41$		−	3.03570e6i		−	1.07429i	−0.843488	−	0.537147i	$-0.819502\pi$
$41$		−	3.03570e6i		−	1.07429i	0.843488	−	0.537147i	$-0.180498\pi$
$42$	41810.9			0.0134367
$43$	−2.35738e6			−0.689533			−0.344767	−	0.938688i	$-0.612042\pi$
$43$	−2.35738e6			−0.689533			−0.344767	+	0.938688i	$0.612042\pi$
$44$	−536357.			−0.143101
$45$	−5.92506e6			−1.44492
$46$			3.96277e6i			0.885051i
$47$	727875.			0.149164			0.0745822	−	0.997215i	$-0.476238\pi$
$47$	727875.			0.149164			0.0745822	+	0.997215i	$0.476238\pi$
$48$			176203.i			0.0331932i
$49$	−5.64672e6			−0.979517
$50$			5.14148e6i			0.822638i
$51$			53993.1i			0.00798101i
$52$			2.11293e6i			0.288982i
$53$		−	6.39013e6i		−	0.809853i	−0.914349	−	0.404927i	$-0.867297\pi$
$53$		−	6.39013e6i		−	0.809853i	0.914349	−	0.404927i	$-0.132703\pi$
$54$	−1.58254e6			−0.186114
$55$	−3.85204e6			−0.420959
$56$			497628.i			0.0506003i
$57$	1.33477e6	+	427472.i	0.126447	+	0.0404956i
$58$	5.48032e6			0.484277
$59$			1.53982e7i			1.27075i	0.772203	+	0.635376i	$0.219155\pi$
$59$			1.53982e7i			1.27075i	−0.772203	+	0.635376i	$0.780845\pi$
$60$			1.26547e6i			0.0976441i
$61$	963914.			0.0696176			0.0348088	−	0.999394i	$-0.488918\pi$
$61$	963914.			0.0696176			0.0348088	+	0.999394i	$0.488918\pi$
$62$	−2.98497e6			−0.202010
$63$	−2.21481e6			−0.140596
$64$	−2.09715e6			−0.125000
$65$			1.51747e7i			0.850095i
$66$	−509851.			−0.0268700
$67$		−	3.33818e7i		−	1.65657i	−0.560306	−	0.828286i	$-0.689316\pi$
$67$		−	3.33818e7i		−	1.65657i	0.560306	−	0.828286i	$-0.310684\pi$
$68$	−642620.			−0.0300551
$69$			3.76694e6i			0.166185i
$70$			3.57390e6i			0.148850i
$71$		−	2.44350e7i		−	0.961567i	−0.876839	−	0.480783i	$-0.840352\pi$
$71$		−	2.44350e7i		−	0.961567i	0.876839	−	0.480783i	$-0.159648\pi$
$72$		−	9.33385e6i		−	0.347321i
$73$	2.36687e7			0.833455			0.416728	−	0.909031i	$-0.363177\pi$
$73$	2.36687e7			0.833455			0.416728	+	0.909031i	$0.363177\pi$
$74$	2.40434e7			0.801807
$75$			4.88740e6i			0.154466i
$76$	−5.08772e6	+	1.58863e7i	−0.152500	+	0.476176i
$77$	−1.43991e6			−0.0409610
$78$			2.00851e6i			0.0542619i
$79$		−	5.45025e7i		−	1.39929i	−0.714490	−	0.699646i	$-0.753341\pi$
$79$		−	5.45025e7i		−	1.39929i	0.714490	−	0.699646i	$-0.246659\pi$
$80$	−1.50615e7			−0.367711
$81$	4.07835e7			0.947425
$82$	3.43450e7			0.759641
$83$	4.12252e7			0.868662			0.434331	−	0.900753i	$-0.356985\pi$
$83$	4.12252e7			0.868662			0.434331	+	0.900753i	$0.356985\pi$
$84$			473036.i			0.00950117i
$85$	−4.61521e6			−0.0884129
$86$		−	2.66707e7i		−	0.487574i
$87$	5.20949e6			0.0909324
$88$		−	6.06818e6i		−	0.101188i
$89$			7.15445e7i			1.14029i	0.821543	+	0.570146i	$0.193113\pi$
$89$			7.15445e7i			1.14029i	−0.821543	+	0.570146i	$0.806887\pi$
$90$		−	6.70344e7i		−	1.02171i
$91$			5.67237e6i			0.0827178i
$92$	−4.48337e7			−0.625825
$93$	−2.83746e6			−0.0379313
$94$			8.23496e6i			0.105475i
$95$	−3.65393e7	+	1.14093e8i	−0.448607	+	1.40076i
$96$	−1.99351e6			−0.0234711
$97$			1.33289e8i			1.50559i	0.658255	+	0.752795i	$0.271295\pi$
$97$			1.33289e8i			1.50559i	−0.658255	+	0.752795i	$0.728705\pi$
$98$		−	6.38853e7i		−	0.692623i
$99$	2.70078e7			0.281157
$100$	−5.81693e7			−0.581693
$101$	−2.20883e7			−0.212264			−0.106132	−	0.994352i	$-0.533847\pi$
$101$	−2.20883e7			−0.212264			−0.106132	+	0.994352i	$0.533847\pi$
$102$	−610862.			−0.00564343
$103$			1.42603e7i			0.126701i	0.997991	+	0.0633503i	$0.0201786\pi$
$103$			1.42603e7i			0.126701i	−0.997991	+	0.0633503i	$0.979821\pi$
$104$	−2.39050e7			−0.204341
$105$			3.39728e6i			0.0279495i
$106$	7.22961e7			0.572653
$107$		−	5.24572e6i		−	0.0400193i	−0.999800	−	0.0200097i	$-0.993630\pi$
$107$		−	5.24572e6i		−	0.0400193i	0.999800	−	0.0200097i	$-0.00636970\pi$
$108$		−	1.79044e7i		−	0.131603i
$109$		−	1.90488e8i		−	1.34947i	−0.738061	−	0.674734i	$-0.764258\pi$
$109$		−	1.90488e8i		−	1.34947i	0.738061	−	0.674734i	$-0.235742\pi$
$110$		−	4.35809e7i		−	0.297663i
$111$	2.28552e7			0.150555
$112$	−5.63002e6			−0.0357798
$113$			2.04711e8i			1.25553i	0.778403	+	0.627765i	$0.216030\pi$
$113$			2.04711e8i			1.25553i	−0.778403	+	0.627765i	$0.783970\pi$
$114$	−4.83629e6	+	1.51012e7i	−0.0286347	+	0.0894112i
$115$	−3.21989e8			−1.84098
$116$			6.20028e7i			0.342436i
$117$		−	1.06395e8i		−	0.567775i
$118$	−1.74210e8			−0.898557
$119$	−1.72518e6			−0.00860293
$120$	−1.43171e7			−0.0690448
$121$	−1.96800e8			−0.918088
$122$			1.09054e7i			0.0492271i
$123$	3.26477e7			0.142637
$124$		−	3.37711e7i		−	0.142843i
$125$	−5.86705e7			−0.240314
$126$		−	2.50577e7i		−	0.0994165i
$127$			1.24877e7i			0.0480027i	0.999712	+	0.0240014i	$0.00764061\pi$
$127$			1.24877e7i			0.0480027i	−0.999712	+	0.0240014i	$0.992359\pi$
$128$		−	2.37266e7i		−	0.0883883i
$129$		−	2.53526e7i		−	0.0915513i
$130$	−1.71683e8			−0.601108
$131$	1.65537e8			0.562094			0.281047	−	0.959694i	$-0.409318\pi$
$131$	1.65537e8			0.562094			0.281047	+	0.959694i	$0.409318\pi$
$132$		−	5.76830e6i		−	0.0189999i
$133$	−1.36585e7	+	4.26484e7i	−0.0436512	+	0.136300i
$134$	3.77672e8			1.17137
$135$		−	1.28587e8i		−	0.387134i
$136$		−	7.27042e6i		−	0.0212522i
$137$	5.08631e8			1.44384			0.721922	−	0.691974i	$-0.243259\pi$
$137$	5.08631e8			1.44384			0.721922	+	0.691974i	$0.243259\pi$
$138$	−4.26181e7			−0.117511
$139$	1.29060e8			0.345727			0.172864	−	0.984946i	$-0.444698\pi$
$139$	1.29060e8			0.345727			0.172864	+	0.984946i	$0.444698\pi$
$140$	−4.04340e7			−0.105253
$141$			7.82800e6i			0.0198050i
$142$	2.76451e8			0.679930
$143$		−	6.91700e7i		−	0.165415i
$144$	1.05600e8			0.245593
$145$			4.45295e8i			1.00734i
$146$			2.67780e8i			0.589342i
$147$		−	6.07282e7i		−	0.130053i
$148$			2.72021e8i			0.566963i
$149$	7.74775e8			1.57192			0.785961	−	0.618277i	$-0.212169\pi$
$149$	7.74775e8			1.57192			0.785961	+	0.618277i	$0.212169\pi$
$150$	−5.52946e7			−0.109224
$151$		−	7.13683e8i		−	1.37277i	−0.727238	−	0.686385i	$-0.759197\pi$
$151$		−	7.13683e8i		−	1.37277i	0.727238	−	0.686385i	$-0.240803\pi$
$152$	−1.79733e8	−	5.75610e7i	−0.336707	−	0.107833i
$153$	3.23586e7			0.0590506
$154$		−	1.62907e7i		−	0.0289638i
$155$		−	2.42539e8i		−	0.420200i
$156$	−2.27237e7			−0.0383690
$157$	−7.63715e8			−1.25699			−0.628497	−	0.777812i	$-0.716329\pi$
$157$	−7.63715e8			−1.25699			−0.628497	+	0.777812i	$0.716329\pi$
$158$	6.16626e8			0.989449
$159$	6.87233e7			0.107527
$160$		−	1.70401e8i		−	0.260011i
$161$	−1.20361e8			−0.179135
$162$			4.61413e8i			0.669931i
$163$	−5.73887e8			−0.812972			−0.406486	−	0.913657i	$-0.633246\pi$
$163$	−5.73887e8			−0.812972			−0.406486	+	0.913657i	$0.633246\pi$
$164$			3.88570e8i			0.537147i
$165$		−	4.14271e7i		−	0.0558919i
$166$			4.66410e8i			0.614237i
$167$		−	5.55377e8i		−	0.714040i	−0.934097	−	0.357020i	$-0.883793\pi$
$167$		−	5.55377e8i		−	0.714040i	0.934097	−	0.357020i	$-0.116207\pi$
$168$	−5.35179e6			−0.00671834
$169$	5.43242e8			0.665958
$170$		−	5.22151e7i		−	0.0625173i
$171$	2.56188e8	−	7.99941e8i	0.299622	−	0.935564i
$172$	3.01744e8			0.344767
$173$		−	6.56811e8i		−	0.733256i	−0.930368	−	0.366628i	$-0.880512\pi$
$173$		−	6.56811e8i		−	0.733256i	0.930368	−	0.366628i	$-0.119488\pi$
$174$			5.89387e7i			0.0642989i
$175$	−1.56161e8			−0.166503
$176$	6.86537e7			0.0715506
$177$	−1.65601e8			−0.168721
$178$	−8.09434e8			−0.806309
$179$		−	1.17573e9i		−	1.14523i	−0.819823	−	0.572617i	$-0.805928\pi$
$179$		−	1.17573e9i		−	1.14523i	0.819823	−	0.572617i	$-0.194072\pi$
$180$	7.58408e8			0.722458
$181$			6.73071e8i			0.627114i	0.949569	+	0.313557i	$0.101521\pi$
$181$			6.73071e8i			0.627114i	−0.949569	+	0.313557i	$0.898479\pi$
$182$	−6.41755e7			−0.0584903
$183$			1.03665e7i			0.00924332i
$184$		−	5.07235e8i		−	0.442525i
$185$			1.95361e9i			1.66783i
$186$		−	3.21022e7i		−	0.0268215i
$187$	2.10372e7			0.0172037
$188$	−9.31680e7			−0.0745822
$189$		−	4.80662e7i		−	0.0376697i
$190$	−1.29082e9	−	4.13395e8i	−0.990489	−	0.317213i
$191$	−2.49800e9			−1.87698			−0.938488	−	0.345311i	$-0.887773\pi$
$191$	−2.49800e9			−1.87698			−0.938488	+	0.345311i	$0.887773\pi$
$192$		−	2.25540e7i		−	0.0165966i
$193$			2.38380e9i			1.71807i	0.511921	+	0.859033i	$0.328934\pi$
$193$			2.38380e9i			1.71807i	−0.511921	+	0.859033i	$0.671066\pi$
$194$	−1.50799e9			−1.06461
$195$	−1.63198e8			−0.112870
$196$	7.22780e8			0.489758
$197$	−4.08382e8			−0.271145			−0.135573	−	0.990767i	$-0.543287\pi$
$197$	−4.08382e8			−0.271145			−0.135573	+	0.990767i	$0.543287\pi$
$198$			3.05559e8i			0.198808i
$199$	−1.99524e9			−1.27228			−0.636139	−	0.771574i	$-0.719470\pi$
$199$	−1.99524e9			−1.27228			−0.636139	+	0.771574i	$0.719470\pi$
$200$		−	6.58110e8i		−	0.411319i
$201$	3.59008e8			0.219948
$202$		−	2.49900e8i		−	0.150093i
$203$			1.66453e8i			0.0980183i
$204$		−	6.91112e6i		−	0.00399050i
$205$			2.79065e9i			1.58012i
$206$	−1.61337e8			−0.0895909
$207$	2.25756e9			1.22959
$208$		−	2.70454e8i		−	0.144491i
$209$	1.66555e8	−	5.20063e8i	0.0872915	−	0.272565i
$210$	−3.84358e7			−0.0197633
$211$		−	2.21085e9i		−	1.11540i	−0.830044	−	0.557698i	$-0.811685\pi$
$211$		−	2.21085e9i		−	1.11540i	0.830044	−	0.557698i	$-0.188315\pi$
$212$			8.17937e8i			0.404927i
$213$	2.62789e8			0.127670
$214$	5.93485e7			0.0282979
$215$	2.16708e9			1.01420
$216$	2.02565e8			0.0930570
$217$		−	9.06621e7i		−	0.0408871i
$218$	2.15513e9			0.954218
$219$			2.54547e8i			0.110660i
$220$	4.93061e8			0.210480
$221$		−	8.28740e7i		−	0.0347416i
$222$			2.58578e8i			0.106458i
$223$		−	1.00945e9i		−	0.408193i	−0.978951	−	0.204096i	$-0.934574\pi$
$223$		−	1.00945e9i		−	0.408193i	0.978951	−	0.204096i	$-0.0654256\pi$
$224$		−	6.36964e7i		−	0.0253001i
$225$	2.92907e9			1.14288
$226$	−2.31604e9			−0.887794
$227$			1.07230e9i			0.403844i	0.979402	+	0.201922i	$0.0647187\pi$
$227$			1.07230e9i			0.403844i	−0.979402	+	0.201922i	$0.935281\pi$
$228$	−1.70851e8	−	5.47164e7i	−0.0632233	−	0.0202478i
$229$	−2.18741e9			−0.795406			−0.397703	−	0.917514i	$-0.630193\pi$
$229$	−2.18741e9			−0.795406			−0.397703	+	0.917514i	$0.630193\pi$
$230$		−	3.64289e9i		−	1.30177i
$231$		−	1.54856e7i		−	0.00543851i
$232$	−7.01481e8			−0.242139
$233$	−1.60064e9			−0.543087			−0.271544	−	0.962426i	$-0.587534\pi$
$233$	−1.60064e9			−0.543087			−0.271544	+	0.962426i	$0.587534\pi$
$234$	1.20372e9			0.401478
$235$	−6.69119e8			−0.219398
$236$		−	1.97096e9i		−	0.635376i
$237$	5.86153e8			0.185788
$238$		−	1.95182e7i		−	0.00608319i
$239$	2.42499e9			0.743222			0.371611	−	0.928388i	$-0.378805\pi$
$239$	2.42499e9			0.743222			0.371611	+	0.928388i	$0.378805\pi$
$240$		−	1.61980e8i		−	0.0488221i
$241$			1.99574e9i			0.591610i	0.955248	+	0.295805i	$0.0955879\pi$
$241$			1.99574e9i			0.591610i	−0.955248	+	0.295805i	$0.904412\pi$
$242$		−	2.22654e9i		−	0.649186i
$243$			1.35635e9i			0.388997i
$244$	−1.23381e8			−0.0348088
$245$	5.19091e9			1.44072
$246$			3.69367e8i			0.100860i
$247$	−2.04874e9	−	6.56126e8i	−0.550426	−	0.176279i
$248$	3.82077e8			0.101005
$249$			4.43361e8i			0.115335i
$250$		−	6.63781e8i		−	0.169928i
$251$	2.54465e9			0.641110			0.320555	−	0.947230i	$-0.396131\pi$
$251$	2.54465e9			0.641110			0.320555	+	0.947230i	$0.396131\pi$
$252$	2.83495e8			0.0702981
$253$	1.46770e9			0.358225
$254$	−1.41282e8			−0.0339431
$255$		−	4.96347e7i		−	0.0117388i
$256$	2.68435e8			0.0625000
$257$		−	5.40032e9i		−	1.23790i	−0.785429	−	0.618952i	$-0.787558\pi$
$257$		−	5.40032e9i		−	1.23790i	0.785429	−	0.618952i	$-0.212442\pi$
$258$	2.86832e8			0.0647365
$259$			7.30267e8i			0.162287i
$260$		−	1.94237e9i		−	0.425048i
$261$		−	3.12210e9i		−	0.672798i
$262$			1.87283e9i			0.397460i
$263$	4.20312e9			0.878514			0.439257	−	0.898361i	$-0.355242\pi$
$263$	4.20312e9			0.878514			0.439257	+	0.898361i	$0.355242\pi$
$264$	6.52609e7			0.0134350
$265$			5.87431e9i			1.19117i
$266$	−4.82511e8	−	1.54528e8i	−0.0963786	−	0.0308661i
$267$	−7.69432e8			−0.151400
$268$			4.27287e9i			0.828286i
$269$		−	1.65507e7i		−	0.00316087i	−0.999999	−	0.00158044i	$-0.999497\pi$
$269$		−	1.65507e7i		−	0.00316087i	0.999999	−	0.00158044i	$-0.000503069\pi$
$270$	1.45479e9			0.273745
$271$	−8.10231e9			−1.50221			−0.751107	−	0.660180i	$-0.770480\pi$
$271$	−8.10231e9			−1.50221			−0.751107	+	0.660180i	$0.770480\pi$
$272$	8.22554e7			0.0150276
$273$	−6.10040e7			−0.0109827
$274$			5.75450e9i			1.02095i
$275$	1.90427e9			0.332964
$276$		−	4.82168e8i		−	0.0830926i
$277$	−4.75322e9			−0.807363			−0.403682	−	0.914900i	$-0.632270\pi$
$277$	−4.75322e9			−0.807363			−0.403682	+	0.914900i	$0.632270\pi$
$278$			1.46015e9i			0.244466i
$279$			1.70052e9i			0.280650i
$280$		−	4.57459e8i		−	0.0744252i
$281$		−	1.26345e9i		−	0.202643i	−0.994854	−	0.101321i	$-0.967693\pi$
$281$		−	1.26345e9i		−	0.202643i	0.994854	−	0.101321i	$-0.0323070\pi$
$282$	−8.85637e7			−0.0140042
$283$	−6.11988e9			−0.954107			−0.477054	−	0.878874i	$-0.658295\pi$
$283$	−6.11988e9			−0.954107			−0.477054	+	0.878874i	$0.658295\pi$
$284$			3.12768e9i			0.480783i
$285$	−1.22702e9	−	3.92965e8i	−0.185983	−	0.0595628i
$286$	7.82570e8			0.116966
$287$			1.04316e9i			0.153752i
$288$			1.19473e9i			0.173660i
$289$	−6.95055e9			−0.996387
$290$	−5.03794e9			−0.712297
$291$	−1.43347e9			−0.199901
$292$	−3.02959e9			−0.416728
$293$			8.14824e9i			1.10559i	0.833318	+	0.552794i	$0.186438\pi$
$293$			8.14824e9i			1.10559i	−0.833318	+	0.552794i	$0.813562\pi$
$294$	6.87061e8			0.0919615
$295$		−	1.41552e10i		−	1.86908i
$296$	−3.07756e9			−0.400903
$297$			5.86129e8i			0.0753299i
$298$			8.76558e9i			1.11152i
$299$		−	5.78187e9i		−	0.723409i
$300$		−	6.25587e8i		−	0.0772330i
$301$	8.10063e8			0.0986854
$302$	8.07440e9			0.970695
$303$		−	2.37550e8i		−	0.0281829i
$304$	6.51228e8	−	2.03344e9i	0.0762498	−	0.238088i
$305$	−8.86105e8			−0.102397
$306$			3.66096e8i			0.0417551i
$307$		−	7.97886e9i		−	0.898229i	−0.893474	−	0.449115i	$-0.851739\pi$
$307$		−	7.97886e9i		−	0.898229i	0.893474	−	0.449115i	$-0.148261\pi$
$308$	1.84308e8			0.0204805
$309$	−1.53364e8			−0.0168224
$310$	2.74402e9			0.297126
$311$	−1.02790e10			−1.09878			−0.549390	−	0.835566i	$-0.685140\pi$
$311$	−1.02790e10			−1.09878			−0.549390	+	0.835566i	$0.685140\pi$
$312$		−	2.57089e8i		−	0.0271309i
$313$	6.84361e9			0.713030			0.356515	−	0.934290i	$-0.383965\pi$
$313$	6.84361e9			0.713030			0.356515	+	0.934290i	$0.383965\pi$
$314$		−	8.64045e9i		−	0.888828i
$315$	2.03602e9			0.206795
$316$			6.97632e9i			0.699646i
$317$			1.07817e10i			1.06770i	0.845579	+	0.533851i	$0.179256\pi$
$317$			1.07817e10i			1.06770i	−0.845579	+	0.533851i	$0.820744\pi$
$318$			7.77515e8i			0.0760327i
$319$		−	2.02976e9i		−	0.196012i
$320$	1.92787e9			0.183856
$321$	5.64156e7			0.00531348
$322$		−	1.36173e9i		−	0.126668i
$323$	1.99553e8	−	6.23098e8i	0.0183336	−	0.0572462i
$324$	−5.22029e9			−0.473712
$325$		−	7.50167e9i		−	0.672395i
$326$		−	6.49279e9i		−	0.574858i
$327$	2.04863e9			0.179173
$328$	−4.39616e9			−0.379821
$329$	−2.50119e8			−0.0213483
$330$	4.68695e8			0.0395216
$331$		−	4.83111e9i		−	0.402471i	−0.979543	−	0.201235i	$-0.935504\pi$
$331$		−	4.83111e9i		−	0.402471i	0.979543	−	0.201235i	$-0.0644956\pi$
$332$	−5.27683e9			−0.434331
$333$		−	1.36974e10i		−	1.11394i
$334$	6.28338e9			0.504902
$335$			3.06871e10i			2.43656i
$336$		−	6.05486e7i		−	0.00475059i
$337$		−	1.24834e10i		−	0.967860i	−0.875107	−	0.483930i	$-0.839209\pi$
$337$		−	1.24834e10i		−	0.967860i	0.875107	−	0.483930i	$-0.160791\pi$
$338$			6.14608e9i			0.470903i
$339$	−2.20158e9			−0.166700
$340$	5.90747e8			0.0442064
$341$			1.10555e9i			0.0817639i
$342$	9.05030e9	+	2.89844e9i	0.661544	+	0.211865i
$343$	3.92133e9			0.283307
$344$			3.41385e9i			0.243787i
$345$		−	3.46287e9i		−	0.244433i
$346$	7.43096e9			0.518490
$347$	2.24032e10			1.54522			0.772612	−	0.634879i	$-0.218950\pi$
$347$	2.24032e10			1.54522			0.772612	+	0.634879i	$0.218950\pi$
$348$	−6.66815e8			−0.0454662
$349$	2.47555e10			1.66867			0.834335	−	0.551257i	$-0.185852\pi$
$349$	2.47555e10			1.66867			0.834335	+	0.551257i	$0.185852\pi$
$350$		−	1.76676e9i		−	0.117735i
$351$	2.30900e9			0.152123
$352$			7.76728e8i			0.0505939i
$353$	7.58628e9			0.488573			0.244287	−	0.969703i	$-0.421446\pi$
$353$	7.58628e9			0.488573			0.244287	+	0.969703i	$0.421446\pi$
$354$		−	1.87356e9i		−	0.119304i
$355$			2.24626e10i			1.41432i
$356$		−	9.15770e9i		−	0.570146i
$357$		−	1.85536e7i		−	0.00114224i
$358$	1.33018e10			0.809802
$359$	−6.91505e9			−0.416311			−0.208155	−	0.978096i	$-0.566746\pi$
$359$	−6.91505e9			−0.416311			−0.208155	+	0.978096i	$0.566746\pi$
$360$			8.58040e9i			0.510855i
$361$	−1.38238e10	−	9.86632e9i	−0.813951	−	0.580934i
$362$	−7.61493e9			−0.443436
$363$		−	2.11651e9i		−	0.121897i
$364$		−	7.26063e8i		−	0.0413589i
$365$	−2.17581e10			−1.22588
$366$	−1.17284e8			−0.00653602
$367$	3.46443e10			1.90971			0.954855	−	0.297072i	$-0.0960101\pi$
$367$	3.46443e10			1.90971			0.954855	+	0.297072i	$0.0960101\pi$
$368$	5.73871e9			0.312913
$369$		−	1.95661e10i		−	1.05536i
$370$	−2.21026e10			−1.17933
$371$			2.19584e9i			0.115906i
$372$	3.63195e8			0.0189657
$373$			1.32696e10i			0.685521i	0.939423	+	0.342761i	$0.111362\pi$
$373$			1.32696e10i			0.685521i	−0.939423	+	0.342761i	$0.888638\pi$
$374$			2.38009e8i			0.0121648i
$375$		−	6.30978e8i		−	0.0319072i
$376$		−	1.05408e9i		−	0.0527376i
$377$	−7.99605e9			−0.395831
$378$	5.43806e8			0.0266365
$379$			1.37892e10i			0.668318i	0.942517	+	0.334159i	$0.108452\pi$
$379$			1.37892e10i			0.668318i	−0.942517	+	0.334159i	$0.891548\pi$
$380$	4.67703e9	−	1.46039e10i	0.224303	−	0.700382i
$381$	−1.34300e8			−0.00637346
$382$		−	2.82616e10i		−	1.32722i
$383$			2.21901e10i			1.03125i	0.856814	+	0.515626i	$0.172440\pi$
$383$			2.21901e10i			1.03125i	−0.856814	+	0.515626i	$0.827560\pi$
$384$	2.55170e8			0.0117356
$385$	1.32367e9			0.0602474
$386$	−2.69696e10			−1.21486
$387$	−1.51941e10			−0.677378
$388$		−	1.70610e10i		−	0.752795i
$389$	5.26015e9			0.229721			0.114860	−	0.993382i	$-0.463358\pi$
$389$	5.26015e9			0.229721			0.114860	+	0.993382i	$0.463358\pi$
$390$		−	1.84638e9i		−	0.0798108i
$391$	1.75848e9			0.0752370
$392$			8.17732e9i			0.346312i
$393$			1.78028e9i			0.0746308i
$394$		−	4.62032e9i		−	0.191729i
$395$			5.01030e10i			2.05814i
$396$	−3.45700e9			−0.140578
$397$	−8.36911e9			−0.336913			−0.168456	−	0.985709i	$-0.553878\pi$
$397$	−8.36911e9			−0.336913			−0.168456	+	0.985709i	$0.553878\pi$
$398$		−	2.25735e10i		−	0.899637i
$399$	−4.58666e8	−	1.46892e8i	−0.0180969	−	0.00579570i
$400$	7.44567e9			0.290846
$401$		−	3.79985e10i		−	1.46957i	−0.678302	−	0.734783i	$-0.737284\pi$
$401$		−	3.79985e10i		−	1.46957i	0.678302	−	0.734783i	$-0.262716\pi$
$402$			4.06171e9i			0.155527i
$403$	4.35522e9			0.165116
$404$	2.82730e9			0.106132
$405$	−3.74914e10			−1.39352
$406$	−1.88320e9			−0.0693094
$407$		−	8.90503e9i		−	0.324532i
$408$	7.81904e7			0.00282171
$409$		−	4.18057e9i		−	0.149397i	−0.997206	−	0.0746985i	$-0.976201\pi$
$409$		−	4.18057e9i		−	0.149397i	0.997206	−	0.0746985i	$-0.0237994\pi$
$410$	−3.15726e10			−1.11731
$411$			5.47012e9i			0.191703i
$412$		−	1.82532e9i		−	0.0633503i
$413$		−	5.29126e9i		−	0.181869i
$414$			2.55414e10i			0.869448i
$415$	−3.78975e10			−1.27767
$416$	3.05984e9			0.102171
$417$			1.38799e9i			0.0459032i
$418$	5.88384e9	+	1.88435e9i	0.192733	+	0.0617244i
$419$	6.81207e9			0.221016			0.110508	−	0.993875i	$-0.464752\pi$
$419$	6.81207e9			0.221016			0.110508	+	0.993875i	$0.464752\pi$
$420$		−	4.34852e8i		−	0.0139748i
$421$			4.75587e9i			0.151392i	0.997131	+	0.0756958i	$0.0241178\pi$
$421$			4.75587e9i			0.151392i	−0.997131	+	0.0756958i	$0.975882\pi$
$422$	2.50129e10			0.788704
$423$	4.69140e9			0.146535
$424$	−9.25390e9			−0.286326
$425$	2.28154e9			0.0699314
$426$			2.97312e9i			0.0902763i
$427$	−3.31229e8			−0.00996362
$428$			6.71452e8i			0.0200097i
$429$	7.43896e8			0.0219626
$430$			2.45178e10i			0.717145i
$431$			5.49100e10i			1.59126i	0.605780	+	0.795632i	$0.292861\pi$
$431$			5.49100e10i			1.59126i	−0.605780	+	0.795632i	$0.707139\pi$
$432$			2.29176e9i			0.0658013i
$433$			5.67002e10i			1.61300i	0.591237	+	0.806498i	$0.298640\pi$
$433$			5.67002e10i			1.61300i	−0.591237	+	0.806498i	$0.701360\pi$
$434$	1.02572e9			0.0289116
$435$	−4.78897e9			−0.133747
$436$			2.43825e10i			0.674734i
$437$	1.39222e10	−	4.34717e10i	0.381752	−	1.19201i
$438$	−2.87987e9			−0.0782486
$439$		−	6.67486e10i		−	1.79715i	−0.438822	−	0.898574i	$-0.644604\pi$
$439$		−	6.67486e10i		−	1.79715i	0.438822	−	0.898574i	$-0.355396\pi$
$440$			5.57835e9i			0.148832i
$441$	−3.63950e10			−0.962249
$442$	9.37613e8			0.0245660
$443$	−4.33866e9			−0.112653			−0.0563263	−	0.998412i	$-0.517939\pi$
$443$	−4.33866e9			−0.112653			−0.0563263	+	0.998412i	$0.517939\pi$
$444$	−2.92547e9			−0.0752773
$445$		−	6.57693e10i		−	1.67719i
$446$	1.14206e10			0.288636
$447$			8.33240e9i			0.208708i
$448$	7.20643e8			0.0178899
$449$			7.17821e10i			1.76616i	0.469219	+	0.883082i	$0.344536\pi$
$449$			7.17821e10i			1.76616i	−0.469219	+	0.883082i	$0.655464\pi$
$450$			3.31386e10i			0.808136i
$451$		−	1.27205e10i		−	0.307466i
$452$		−	2.62030e10i		−	0.627765i
$453$	7.67538e9			0.182267
$454$	−1.21317e10			−0.285561
$455$		−	5.21448e9i		−	0.121665i
$456$	6.19045e8	−	1.93295e9i	0.0143174	−	0.0447056i
$457$	8.06293e10			1.84854			0.924268	−	0.381743i	$-0.124676\pi$
$457$	8.06293e10			1.84854			0.924268	+	0.381743i	$0.124676\pi$
$458$		−	2.47478e10i		−	0.562437i
$459$			7.02253e8i			0.0158213i
$460$	4.12146e10			0.920492
$461$	−5.53201e9			−0.122484			−0.0612420	−	0.998123i	$-0.519506\pi$
$461$	−5.53201e9			−0.122484			−0.0612420	+	0.998123i	$0.519506\pi$
$462$	1.75200e8			0.00384561
$463$	4.32453e10			0.941054			0.470527	−	0.882386i	$-0.344064\pi$
$463$	4.32453e10			0.941054			0.470527	+	0.882386i	$0.344064\pi$
$464$		−	7.93636e9i		−	0.171218i
$465$	2.60841e9			0.0557911
$466$		−	1.81092e10i		−	0.384021i
$467$	2.49413e10			0.524386			0.262193	−	0.965015i	$-0.415554\pi$
$467$	2.49413e10			0.524386			0.262193	+	0.965015i	$0.415554\pi$
$468$			1.36185e10i			0.283888i
$469$			1.14710e10i			0.237087i
$470$		−	7.57022e9i		−	0.155138i
$471$		−	8.21345e9i		−	0.166895i
$472$	2.22989e10			0.449279
$473$	−9.87809e9			−0.197346
$474$			6.63156e9i			0.131372i
$475$	1.80633e10	−	5.64022e10i	0.354832	−	1.10795i
$476$	2.20823e8			0.00430147
$477$		−	4.11866e10i		−	0.795577i
$478$			2.74357e10i			0.525538i
$479$	5.76944e10			1.09595			0.547977	−	0.836494i	$-0.315398\pi$
$479$	5.76944e10			1.09595			0.547977	+	0.836494i	$0.315398\pi$
$480$	1.83259e9			0.0345224
$481$	−3.50805e10			−0.655369
$482$	−2.25792e10			−0.418332
$483$		−	1.29443e9i		−	0.0237843i
$484$	2.51904e10			0.459044
$485$		−	1.22529e11i		−	2.21449i
$486$	−1.53453e10			−0.275063
$487$			3.94004e10i			0.700462i	0.936663	+	0.350231i	$0.113897\pi$
$487$			3.94004e10i			0.700462i	−0.936663	+	0.350231i	$0.886103\pi$
$488$		−	1.39590e9i		−	0.0246135i
$489$		−	6.17192e9i		−	0.107941i
$490$			5.87284e10i			1.01874i
$491$	−1.08173e11			−1.86120			−0.930599	−	0.366041i	$-0.880713\pi$
$491$	−1.08173e11			−1.86120			−0.930599	+	0.366041i	$0.880713\pi$
$492$	−4.17891e9			−0.0713186
$493$		−	2.43190e9i		−	0.0411678i
$494$	7.42322e9	−	2.31788e10i	0.124648	−	0.389210i
$495$	−2.48277e10			−0.413538
$496$			4.32270e9i			0.0714215i
$497$			8.39659e9i			0.137619i
$498$	−5.01606e9			−0.0815539
$499$	8.97074e10			1.44686			0.723430	−	0.690398i	$-0.242565\pi$
$499$	8.97074e10			1.44686			0.723430	+	0.690398i	$0.242565\pi$
$500$	7.50982e9			0.120157
$501$	5.97286e9			0.0948051
$502$			2.87894e10i			0.453333i
$503$	−7.42237e10			−1.15950			−0.579750	−	0.814795i	$-0.696850\pi$
$503$	−7.42237e10			−1.15950			−0.579750	+	0.814795i	$0.696850\pi$
$504$			3.20738e9i			0.0497083i
$505$	2.03053e10			0.312207
$506$			1.66052e10i			0.253303i
$507$			5.84235e9i			0.0884211i
$508$		−	1.59842e9i		−	0.0240014i
$509$		−	3.30960e10i		−	0.493066i	−0.969134	−	0.246533i	$-0.920709\pi$
$509$		−	3.30960e10i		−	0.493066i	0.969134	−	0.246533i	$-0.0792913\pi$
$510$	5.61553e8			0.00830060
$511$	−8.13324e9			−0.119283
$512$			3.03700e9i			0.0441942i
$513$	1.73605e10	+	5.55984e9i	0.250664	+	0.0802773i
$514$	6.10976e10			0.875330
$515$		−	1.31092e10i		−	0.186357i
$516$			3.24514e9i			0.0457756i
$517$	3.05000e9			0.0426912
$518$	−8.26203e9			−0.114754
$519$	7.06373e9			0.0973565
$520$	2.19754e10			0.300554
$521$			1.15808e11i			1.57177i	0.618372	+	0.785886i	$0.287793\pi$
$521$			1.15808e11i			1.57177i	−0.618372	+	0.785886i	$0.712207\pi$
$522$	3.53225e10			0.475740
$523$		−	1.39644e11i		−	1.86644i	−0.359305	−	0.933220i	$-0.616986\pi$
$523$		−	1.39644e11i		−	1.86644i	0.359305	−	0.933220i	$-0.383014\pi$
$524$	−2.11887e10			−0.281047
$525$		−	1.67945e9i		−	0.0221070i
$526$			4.75529e10i			0.621204i
$527$			1.32458e9i			0.0171726i
$528$			7.38343e8i			0.00949997i
$529$	4.43733e10			0.566629
$530$	−6.64602e10			−0.842283
$531$			9.92463e10i			1.24835i
$532$	1.74829e9	−	5.45899e9i	0.0218256	−	0.0681500i
$533$	−5.01110e10			−0.620904
$534$		−	8.70513e9i		−	0.107056i
$535$			4.82227e9i			0.0588622i
$536$	−4.83420e10			−0.585687
$537$	1.26445e10			0.152056
$538$	1.87250e8			0.00223508
$539$	−2.36614e10			−0.280340
$540$			1.64591e10i			0.193567i
$541$	−1.09728e11			−1.28094			−0.640470	−	0.767983i	$-0.721260\pi$
$541$	−1.09728e11			−1.28094			−0.640470	+	0.767983i	$0.721260\pi$
$542$		−	9.16672e10i		−	1.06223i
$543$	−7.23861e9			−0.0832637
$544$			9.30613e8i			0.0106261i
$545$			1.75112e11i			1.98486i
$546$		−	6.90182e8i		−	0.00776592i
$547$			7.70367e10i			0.860495i	0.902711	+	0.430247i	$0.141574\pi$
$547$			7.70367e10i			0.860495i	−0.902711	+	0.430247i	$0.858426\pi$
$548$	−6.51047e10			−0.721922
$549$	6.21275e9			0.0683903
$550$			2.15443e10i			0.235441i
$551$	−6.01192e10	−	1.92537e10i	−0.652239	−	0.208885i
$552$	5.45511e9			0.0587553
$553$			1.87287e10i			0.200266i
$554$		−	5.37766e10i		−	0.570892i
$555$	−2.10103e10			−0.221442
$556$	−1.65197e10			−0.172864
$557$	−8.33803e10			−0.866248			−0.433124	−	0.901334i	$-0.642589\pi$
$557$	−8.33803e10			−0.866248			−0.433124	+	0.901334i	$0.642589\pi$
$558$	−1.92392e10			−0.198449
$559$			3.89138e10i			0.398525i
$560$	5.17556e9			0.0526265
$561$			2.26247e8i			0.00228418i
$562$	1.42943e10			0.143290
$563$			1.15774e11i			1.15233i	0.817334	+	0.576164i	$0.195451\pi$
$563$			1.15774e11i			1.15233i	−0.817334	+	0.576164i	$0.804549\pi$
$564$		−	1.00198e9i		−	0.00990249i
$565$		−	1.88186e11i		−	1.84669i
$566$		−	6.92386e10i		−	0.674656i
$567$	−1.40144e10			−0.135595
$568$	−3.53857e10			−0.339965
$569$			7.99151e10i			0.762394i	0.924494	+	0.381197i	$0.124488\pi$
$569$			7.99151e10i			0.762394i	−0.924494	+	0.381197i	$0.875512\pi$
$570$	4.44590e9	−	1.38822e10i	0.0421172	−	0.131510i
$571$	−1.62039e11			−1.52432			−0.762159	−	0.647390i	$-0.775861\pi$
$571$	−1.62039e11			−1.52432			−0.762159	+	0.647390i	$0.775861\pi$
$572$			8.85377e9i			0.0827073i
$573$		−	2.68650e10i		−	0.249212i
$574$	−1.18020e10			−0.108719
$575$	1.59176e11			1.45615
$576$	−1.35169e10			−0.122796
$577$	1.78544e11			1.61080			0.805402	−	0.592728i	$-0.201949\pi$
$577$	1.78544e11			1.61080			0.805402	+	0.592728i	$0.201949\pi$
$578$		−	7.86365e10i		−	0.704552i
$579$	−2.56368e10			−0.228112
$580$		−	5.69978e10i		−	0.503670i
$581$	−1.41662e10			−0.124322
$582$		−	1.62178e10i		−	0.141352i
$583$		−	2.67765e10i		−	0.231782i
$584$		−	3.42759e10i		−	0.294671i
$585$			9.78063e10i			0.835109i
$586$	−9.21868e10			−0.781768
$587$	2.06207e11			1.73681			0.868404	−	0.495857i	$-0.165146\pi$
$587$	2.06207e11			1.73681			0.868404	+	0.495857i	$0.165146\pi$
$588$			7.77321e9i			0.0650266i
$589$	3.27452e10	+	1.04869e10i	0.272074	+	0.0871339i
$590$	1.60148e11			1.32164
$591$		−	4.39199e9i		−	0.0360007i
$592$		−	3.48186e10i		−	0.283482i
$593$	−8.30747e10			−0.671815			−0.335908	−	0.941895i	$-0.609043\pi$
$593$	−8.30747e10			−0.671815			−0.335908	+	0.941895i	$0.609043\pi$
$594$	−6.63129e9			−0.0532663
$595$	1.58592e9			0.0126536
$596$	−9.91712e10			−0.785961
$597$		−	2.14580e10i		−	0.168924i
$598$	6.54144e10			0.511527
$599$		−	2.05057e11i		−	1.59282i	−0.604757	−	0.796410i	$-0.706730\pi$
$599$		−	2.05057e11i		−	1.59282i	0.604757	−	0.796410i	$-0.293270\pi$
$600$	7.07771e9			0.0546120
$601$			1.04583e11i			0.801611i	0.916163	+	0.400805i	$0.131270\pi$
$601$			1.04583e11i			0.801611i	−0.916163	+	0.400805i	$0.868730\pi$
$602$			9.16482e9i			0.0697811i
$603$		−	2.15157e11i		−	1.62737i
$604$			9.13515e10i			0.686385i
$605$	1.80914e11			1.35037
$606$	2.68758e9			0.0199283
$607$			1.21178e11i			0.892626i	0.894877	+	0.446313i	$0.147263\pi$
$607$			1.21178e11i			0.892626i	−0.894877	+	0.446313i	$0.852737\pi$
$608$	2.30058e10	+	7.36780e9i	0.168354	+	0.0539167i
$609$	−1.79013e9			−0.0130142
$610$		−	1.00251e10i		−	0.0724054i
$611$		−	1.20152e10i		−	0.0862117i
$612$	−4.14190e9			−0.0295253
$613$	−5.78476e10			−0.409678			−0.204839	−	0.978796i	$-0.565667\pi$
$613$	−5.78476e10			−0.409678			−0.204839	+	0.978796i	$0.565667\pi$
$614$	9.02705e10			0.635144
$615$	−3.00124e10			−0.209797
$616$			2.08521e9i			0.0144819i
$617$	1.37356e11			0.947776			0.473888	−	0.880585i	$-0.342850\pi$
$617$	1.37356e11			0.947776			0.473888	+	0.880585i	$0.342850\pi$
$618$		−	1.73511e9i		−	0.0118952i
$619$	−8.69099e10			−0.591979			−0.295990	−	0.955191i	$-0.595649\pi$
$619$	−8.69099e10			−0.591979			−0.295990	+	0.955191i	$0.595649\pi$
$620$			3.10451e10i			0.210100i
$621$			4.89941e10i			0.329441i
$622$		−	1.16294e11i		−	0.776955i
$623$		−	2.45848e10i		−	0.163198i
$624$	2.90863e9			0.0191845
$625$	−1.23584e11			−0.809920
$626$			7.74266e10i			0.504189i
$627$	5.59307e9	+	1.79123e9i	0.0361893	+	0.0115899i
$628$	9.77556e10			0.628497
$629$		−	1.06693e10i		−	0.0681606i
$630$			2.30350e10i			0.146226i
$631$	1.40125e11			0.883889			0.441945	−	0.897042i	$-0.354289\pi$
$631$	1.40125e11			0.883889			0.441945	+	0.897042i	$0.354289\pi$
$632$	−7.89281e10			−0.494724
$633$	2.37768e10			0.148094
$634$	−1.21981e11			−0.754979
$635$		−	1.14796e10i		−	0.0706046i
$636$	−8.79658e9			−0.0537633
$637$			9.32117e10i			0.566125i
$638$	2.29641e10			0.138601
$639$		−	1.57492e11i		−	0.944616i
$640$			2.18113e10i			0.130006i
$641$			2.18831e11i			1.29621i	0.761549	+	0.648107i	$0.224439\pi$
$641$			2.18831e11i			1.29621i	−0.761549	+	0.648107i	$0.775561\pi$
$642$			6.38270e8i			0.00375720i
$643$	6.72532e10			0.393431			0.196716	−	0.980461i	$-0.436972\pi$
$643$	6.72532e10			0.393431			0.196716	+	0.980461i	$0.436972\pi$
$644$	1.54062e10			0.0895676
$645$			2.33061e10i			0.134658i
$646$	7.04955e9	+	2.25768e9i	0.0404791	+	0.0129638i
$647$	2.70481e9			0.0154355			0.00771773	−	0.999970i	$-0.497543\pi$
$647$	2.70481e9			0.0154355			0.00771773	+	0.999970i	$0.497543\pi$
$648$		−	5.90609e10i		−	0.334965i
$649$			6.45227e10i			0.363692i
$650$	8.48717e10			0.475455
$651$	9.75034e8			0.00542870
$652$	7.34575e10			0.406486
$653$	2.53210e11			1.39260			0.696302	−	0.717749i	$-0.254828\pi$
$653$	2.53210e11			1.39260			0.696302	+	0.717749i	$0.254828\pi$
$654$			2.31776e10i			0.126694i
$655$	−1.52174e11			−0.826753
$656$		−	4.97369e10i		−	0.268574i
$657$	1.52553e11			0.818762
$658$		−	2.82977e9i		−	0.0150955i
$659$		−	1.31513e9i		−	0.00697310i	−0.999994	−	0.00348655i	$-0.998890\pi$
$659$		−	1.31513e9i		−	0.00697310i	0.999994	−	0.00348655i	$-0.00110981\pi$
$660$			5.30267e9i			0.0279460i
$661$			1.05346e11i			0.551836i	0.961181	+	0.275918i	$0.0889819\pi$
$661$			1.05346e11i			0.551836i	−0.961181	+	0.275918i	$0.911018\pi$
$662$	5.46577e10			0.284590
$663$	8.91277e8			0.00461274
$664$		−	5.97005e10i		−	0.307118i
$665$	1.25560e10	−	3.92057e10i	0.0642042	−	0.200476i
$666$	1.54968e11			0.787672
$667$		−	1.69666e11i		−	0.857220i
$668$			7.10883e10i			0.357020i
$669$	1.08562e10			0.0541969
$670$	−3.47185e11			−1.72291
$671$	4.03908e9			0.0199247
$672$	6.85029e8			0.00335917
$673$		−	1.01126e11i		−	0.492948i	−0.969149	−	0.246474i	$-0.920728\pi$
$673$		−	1.01126e11i		−	0.492948i	0.969149	−	0.246474i	$-0.0792719\pi$
$674$	1.41233e11			0.684381
$675$			6.35672e10i			0.306209i
$676$	−6.95350e10			−0.332979
$677$			1.62812e11i			0.775053i	0.921858	+	0.387527i	$0.126671\pi$
$677$			1.62812e11i			0.775053i	−0.921858	+	0.387527i	$0.873329\pi$
$678$		−	2.49081e10i		−	0.117875i
$679$		−	4.58019e10i		−	0.215479i
$680$			6.68353e9i			0.0312587i
$681$	−1.15322e10			−0.0536195
$682$	−1.25079e10			−0.0578158
$683$		−	4.24838e11i		−	1.95227i	−0.217157	−	0.976137i	$-0.569678\pi$
$683$		−	4.24838e11i		−	1.95227i	0.217157	−	0.976137i	$-0.430322\pi$
$684$	−3.27921e10	+	1.02392e11i	−0.149811	+	0.467782i
$685$	−4.67573e11			−2.12367
$686$			4.43648e10i			0.200328i
$687$		−	2.35247e10i		−	0.105608i
$688$	−3.86233e10			−0.172383
$689$	−1.05483e11			−0.468066
$690$	3.91778e10			0.172840
$691$	−4.07739e11			−1.78842			−0.894210	−	0.447647i	$-0.852262\pi$
$691$	−4.07739e11			−1.78842			−0.894210	+	0.447647i	$0.852262\pi$
$692$			8.40718e10i			0.366628i
$693$	−9.28068e9			−0.0402390
$694$			2.53463e11i			1.09264i
$695$	−1.18642e11			−0.508511
$696$		−	7.54415e9i		−	0.0321494i
$697$		−	1.52406e10i		−	0.0645761i
$698$			2.80077e11i			1.17993i
$699$		−	1.72142e10i		−	0.0721072i
$700$	1.99887e10			0.0832514
$701$	−6.60489e10			−0.273523			−0.136761	−	0.990604i	$-0.543669\pi$
$701$	−6.60489e10			−0.273523			−0.136761	+	0.990604i	$0.543669\pi$
$702$			2.61233e10i			0.107567i
$703$	−2.63757e11	−	8.44704e10i	−1.07990	−	0.345847i
$704$	−8.78767e9			−0.0357753
$705$		−	7.19611e9i		−	0.0291301i
$706$			8.58289e10i			0.345474i
$707$	7.59017e9			0.0303790
$708$	2.11969e10			0.0843607
$709$	−8.16942e10			−0.323300			−0.161650	−	0.986848i	$-0.551682\pi$
$709$	−8.16942e10			−0.323300			−0.161650	+	0.986848i	$0.551682\pi$
$710$	−2.54135e11			−1.00007
$711$		−	3.51287e11i		−	1.37462i
$712$	1.03608e11			0.403154
$713$			9.24123e10i			0.357579i
$714$	2.09910e8			0.000807683
$715$			6.35865e10i			0.243299i
$716$			1.50493e11i			0.572617i
$717$			2.60798e10i			0.0986797i
$718$		−	7.82349e10i		−	0.294376i
$719$	−2.01728e11			−0.754834			−0.377417	−	0.926043i	$-0.623188\pi$
$719$	−2.01728e11			−0.754834			−0.377417	+	0.926043i	$0.623188\pi$
$720$	−9.70762e10			−0.361229
$721$		−	4.90025e9i		−	0.0181333i
$722$	1.11625e11	−	1.56398e11i	0.410782	−	0.575550i
$723$	−2.14634e10			−0.0785498
$724$		−	8.61530e10i		−	0.313557i
$725$		−	2.20133e11i		−	0.796770i
$726$	2.39456e10			0.0861943
$727$	1.68993e11			0.604966			0.302483	−	0.953155i	$-0.402184\pi$
$727$	1.68993e11			0.604966			0.302483	+	0.953155i	$0.402184\pi$
$728$	8.21446e9			0.0292451
$729$	2.52994e11			0.895777
$730$		−	2.46165e11i		−	0.866830i
$731$	−1.18351e10			−0.0414480
$732$		−	1.32691e9i		−	0.00462166i
$733$	3.42616e11			1.18684			0.593420	−	0.804893i	$-0.297777\pi$
$733$	3.42616e11			1.18684			0.593420	+	0.804893i	$0.297777\pi$
$734$			3.91955e11i			1.35037i
$735$			5.58261e10i			0.191288i
$736$			6.49261e10i			0.221263i
$737$		−	1.39879e11i		−	0.474115i
$738$	2.21365e11			0.746250
$739$	5.33253e11			1.78795			0.893975	−	0.448117i	$-0.147905\pi$
$739$	5.33253e11			1.78795			0.893975	+	0.448117i	$0.147905\pi$
$740$		−	2.50063e11i		−	0.833915i
$741$	7.05637e9	−	2.20334e10i	0.0234050	−	0.0730815i
$742$	−2.48430e10			−0.0819576
$743$			3.52979e10i			0.115823i	0.998322	+	0.0579113i	$0.0184441\pi$
$743$			3.52979e10i			0.115823i	−0.998322	+	0.0579113i	$0.981556\pi$
$744$			4.10908e9i			0.0134107i
$745$	−7.12234e11			−2.31205
$746$	−1.50128e11			−0.484737
$747$	2.65711e11			0.853349
$748$	−2.69276e9			−0.00860185
$749$			1.80258e9i			0.00572754i
$750$	7.13870e9			0.0225618
$751$			4.77175e11i			1.50009i	0.661386	+	0.750045i	$0.269968\pi$
$751$			4.77175e11i			1.50009i	−0.661386	+	0.750045i	$0.730032\pi$
$752$	1.19255e10			0.0372911
$753$			2.73667e10i			0.0851220i
$754$		−	9.04650e10i		−	0.279895i
$755$			6.56073e11i			2.01913i
$756$			6.15247e9i			0.0188348i
$757$	3.54055e11			1.07817			0.539084	−	0.842252i	$-0.318770\pi$
$757$	3.54055e11			1.07817			0.539084	+	0.842252i	$0.318770\pi$
$758$	−1.56007e11			−0.472572
$759$			1.57846e10i			0.0475626i
$760$	1.65224e11	+	5.29145e10i	0.495245	+	0.158606i
$761$	−3.22849e11			−0.962632			−0.481316	−	0.876547i	$-0.659841\pi$
$761$	−3.22849e11			−0.962632			−0.481316	+	0.876547i	$0.659841\pi$
$762$		−	1.51943e9i		−	0.00450672i
$763$			6.54574e10i			0.193135i
$764$	3.19744e11			0.938488
$765$	−2.97466e10			−0.0868543
$766$	−2.51053e11			−0.729205
$767$	2.54181e11			0.734449
$768$			2.88692e9i			0.00829830i
$769$	8.99891e10			0.257326			0.128663	−	0.991688i	$-0.458931\pi$
$769$	8.99891e10			0.257326			0.128663	+	0.991688i	$0.458931\pi$
$770$			1.49757e10i			0.0426013i
$771$	5.80783e10			0.164360
$772$		−	3.05126e11i		−	0.859033i
$773$		−	3.43312e11i		−	0.961548i	−0.876845	−	0.480774i	$-0.840356\pi$
$773$		−	3.43312e11i		−	0.961548i	0.876845	−	0.480774i	$-0.159644\pi$
$774$		−	1.71901e11i		−	0.478978i
$775$			1.19900e11i			0.332363i
$776$	1.93023e11			0.532306
$777$	−7.85373e9			−0.0215473
$778$			5.95119e10i			0.162437i
$779$	−3.76765e11	−	1.20662e11i	−1.02311	−	0.327659i
$780$	2.08894e10			0.0564348
$781$		−	1.02390e11i		−	0.275203i
$782$			1.98950e10i			0.0532006i
$783$	6.77564e10			0.180262
$784$	−9.25159e10			−0.244879
$785$	7.02067e11			1.84884
$786$	−2.01416e10			−0.0527719
$787$			2.82012e11i			0.735137i	0.929997	+	0.367568i	$0.119810\pi$
$787$			2.82012e11i			0.735137i	−0.929997	+	0.367568i	$0.880190\pi$
$788$	5.22729e10			0.135573
$789$			4.52029e10i			0.116643i
$790$	−5.66851e11			−1.45533
$791$		−	7.03446e10i		−	0.179690i
$792$		−	3.91115e10i		−	0.0994040i
$793$		−	1.59116e10i		−	0.0402365i
$794$		−	9.46857e10i		−	0.238233i
$795$	−6.31758e10			−0.158155
$796$	2.55390e11			0.636139
$797$		−	6.48442e11i		−	1.60708i	−0.595250	−	0.803540i	$-0.702947\pi$
$797$		−	6.48442e11i		−	1.60708i	0.595250	−	0.803540i	$-0.297053\pi$
$798$	1.66189e9	−	5.18921e9i	0.00409818	−	0.0127965i
$799$	3.65427e9			0.00896631
$800$			8.42381e10i			0.205659i
$801$			4.61129e11i			1.12019i
$802$	4.29904e11			1.03914
$803$	9.91785e10			0.238537
$804$	−4.59530e10			−0.109974
$805$	1.10645e11			0.263480
$806$			4.92736e10i			0.116755i
$807$	1.77996e8			0.000419678
$808$			3.19872e10i			0.0750466i
$809$	6.90473e11			1.61195			0.805977	−	0.591947i	$-0.201641\pi$
$809$	6.90473e11			1.61195			0.805977	+	0.591947i	$0.201641\pi$
$810$		−	4.24167e11i		−	0.985364i
$811$		−	1.59384e11i		−	0.368434i	−0.982886	−	0.184217i	$-0.941025\pi$
$811$		−	1.59384e11i		−	0.368434i	0.982886	−	0.184217i	$-0.0589750\pi$
$812$		−	2.13060e10i		−	0.0490092i
$813$		−	8.71371e10i		−	0.199453i
$814$	1.00749e11			0.229479
$815$	5.27561e11			1.19576
$816$			8.84623e8i			0.00199525i
$817$	−9.37005e10	+	2.92578e11i	−0.210307	+	0.656679i
$818$	4.72977e10			0.105640
$819$			3.65603e10i			0.0812596i
$820$		−	3.57204e11i		−	0.790061i
$821$	−7.47845e11			−1.64603			−0.823017	−	0.568016i	$-0.807711\pi$
$821$	−7.47845e11			−1.64603			−0.823017	+	0.568016i	$0.807711\pi$
$822$	−6.18873e10			−0.135555
$823$	7.29538e10			0.159019			0.0795095	−	0.996834i	$-0.474665\pi$
$823$	7.29538e10			0.159019			0.0795095	+	0.996834i	$0.474665\pi$
$824$	2.06511e10			0.0447955
$825$			2.04796e10i			0.0442085i
$826$	5.98637e10			0.128601
$827$			1.71778e11i			0.367236i	0.982998	+	0.183618i	$0.0587809\pi$
$827$			1.71778e11i			0.367236i	−0.982998	+	0.183618i	$0.941219\pi$
$828$	−2.88968e11			−0.614793
$829$		−	8.66395e11i		−	1.83442i	−0.398408	−	0.917208i	$-0.630437\pi$
$829$		−	8.66395e11i		−	1.83442i	0.398408	−	0.917208i	$-0.369563\pi$
$830$		−	4.28761e11i		−	0.903447i
$831$		−	5.11190e10i		−	0.107196i
$832$			3.46182e10i			0.0722455i
$833$	−2.83492e10			−0.0588790
$834$	−1.57033e10			−0.0324585
$835$			5.10546e11i			1.05024i
$836$	−2.13190e10	+	6.65681e10i	−0.0436457	+	0.136283i
$837$	−3.69050e10			−0.0751940
$838$			7.70698e10i			0.156282i
$839$		−	2.46507e11i		−	0.497487i	−0.968569	−	0.248743i	$-0.919982\pi$
$839$		−	2.46507e11i		−	0.497487i	0.968569	−	0.248743i	$-0.0800176\pi$
$840$	4.91979e9			0.00988164
$841$	2.65606e11			0.530951
$842$	−5.38065e10			−0.107050
$843$	1.35878e10			0.0269055
$844$			2.82989e11i			0.557698i
$845$	−4.99391e11			−0.979521
$846$			5.30771e10i			0.103616i
$847$	6.76264e10			0.131396
$848$		−	1.04696e11i		−	0.202463i
$849$		−	6.58169e10i		−	0.126680i
$850$			2.58127e10i			0.0494490i
$851$		−	7.44365e11i		−	1.41928i
$852$	−3.36370e10			−0.0638350
$853$	−6.05188e11			−1.14313			−0.571563	−	0.820558i	$-0.693663\pi$
$853$	−6.05188e11			−1.14313			−0.571563	+	0.820558i	$0.693663\pi$
$854$		−	3.74743e9i		−	0.00704534i
$855$	−2.35508e11	+	7.35368e11i	−0.440698	+	1.37607i
$856$	−7.59661e9			−0.0141490
$857$			8.23654e11i			1.52694i	0.645844	+	0.763469i	$0.276505\pi$
$857$			8.23654e11i			1.52694i	−0.645844	+	0.763469i	$0.723495\pi$
$858$			8.41622e9i			0.0155299i
$859$	−3.99209e11			−0.733209			−0.366605	−	0.930377i	$-0.619480\pi$
$859$	−3.99209e11			−0.733209			−0.366605	+	0.930377i	$0.619480\pi$
$860$	−2.77387e11			−0.507098
$861$	−1.12187e10			−0.0204141
$862$	−6.21236e11			−1.12519
$863$			6.65968e11i			1.20063i	0.799762	+	0.600317i	$0.204959\pi$
$863$			6.65968e11i			1.20063i	−0.799762	+	0.600317i	$0.795041\pi$
$864$	−2.59283e10			−0.0465285
$865$			6.03792e11i			1.07851i
$866$	−6.41490e11			−1.14056
$867$		−	7.47504e10i		−	0.132293i
$868$			1.16047e10i			0.0204436i
$869$		−	2.28381e11i		−	0.400480i
$870$		−	5.41810e10i		−	0.0945737i
$871$	−5.51041e11			−0.957439
$872$	−2.75857e11			−0.477109
$873$			8.59091e11i			1.47905i
$874$	4.91826e11	+	1.57512e11i	0.842880	+	0.269940i
$875$	2.01609e10			0.0343936
$876$		−	3.25820e10i		−	0.0553301i
$877$		−	1.42492e11i		−	0.240875i	−0.992721	−	0.120438i	$-0.961570\pi$
$877$		−	1.42492e11i		−	0.240875i	0.992721	−	0.120438i	$-0.0384297\pi$
$878$	7.55174e11			1.27078
$879$	−8.76310e10			−0.146792
$880$	−6.31118e10			−0.105240
$881$	−6.50983e10			−0.108060			−0.0540302	−	0.998539i	$-0.517207\pi$
$881$	−6.50983e10			−0.108060			−0.0540302	+	0.998539i	$0.517207\pi$
$882$		−	4.11763e11i		−	0.680413i
$883$	3.09709e11			0.509462			0.254731	−	0.967012i	$-0.418013\pi$
$883$	3.09709e11			0.509462			0.254731	+	0.967012i	$0.418013\pi$
$884$			1.06079e10i			0.0173708i
$885$	1.52233e11			0.248163
$886$		−	4.90864e10i		−	0.0796574i
$887$		−	4.50470e11i		−	0.727732i	−0.931451	−	0.363866i	$-0.881457\pi$
$887$		−	4.50470e11i		−	0.727732i	0.931451	−	0.363866i	$-0.118543\pi$
$888$		−	3.30979e10i		−	0.0532291i
$889$		−	4.29112e9i		−	0.00687012i
$890$	7.44095e11			1.18595
$891$	1.70895e11			0.271155
$892$			1.29210e11i			0.204096i
$893$	2.89314e10	−	9.03377e10i	0.0454950	−	0.142057i
$894$	−9.42703e10			−0.147579
$895$			1.08082e12i			1.68446i
$896$			8.15314e9i			0.0126501i
$897$	6.21817e10			0.0960490
$898$	−8.12122e11			−1.24887
$899$	1.27802e11			0.195658
$900$	−3.74921e11			−0.571438
$901$		−	3.20815e10i		−	0.0486805i
$902$	1.43916e11			0.217411
$903$			8.71191e9i			0.0131027i
$904$	2.96453e11			0.443897
$905$		−	6.18739e11i		−	0.922387i
$906$			8.68370e10i			0.128882i
$907$		−	5.45609e11i		−	0.806218i	−0.915152	−	0.403109i	$-0.867929\pi$
$907$		−	5.45609e11i		−	0.806218i	0.915152	−	0.403109i	$-0.132071\pi$
$908$		−	1.37255e11i		−	0.201922i
$909$	−1.42366e11			−0.208522
$910$	5.89951e10			0.0860301
$911$		−	5.13209e11i		−	0.745111i	−0.928010	−	0.372556i	$-0.878482\pi$
$911$		−	5.13209e11i		−	0.745111i	0.928010	−	0.372556i	$-0.121518\pi$
$912$	2.18689e10	+	7.00369e9i	0.0316116	+	0.0101239i
$913$	1.72746e11			0.248613
$914$			9.12216e11i			1.30711i
$915$		−	9.52971e9i		−	0.0135955i
$916$	2.79989e11			0.397703
$917$	−5.68832e10			−0.0804464
$918$	−7.94508e9			−0.0111874
$919$	−1.31908e10			−0.0184930			−0.00924652	−	0.999957i	$-0.502943\pi$
$919$	−1.31908e10			−0.0184930			−0.00924652	+	0.999957i	$0.502943\pi$
$920$			4.66290e11i			0.650886i
$921$	8.58094e10			0.119260
$922$		−	6.25875e10i		−	0.0866092i
$923$	−4.03355e11			−0.555751
$924$			1.98216e9i			0.00271926i
$925$		−	9.65774e11i		−	1.31919i
$926$			4.89264e11i			0.665426i
$927$			9.19123e10i			0.124467i
$928$	8.97896e10			0.121069
$929$	−6.52670e11			−0.876256			−0.438128	−	0.898913i	$-0.644358\pi$
$929$	−6.52670e11			−0.876256			−0.438128	+	0.898913i	$0.644358\pi$
$930$			2.95108e10i			0.0394503i
$931$	−2.24445e11	+	7.00823e11i	−0.298752	+	0.932845i
$932$	2.04882e11			0.271544
$933$		−	1.10547e11i		−	0.145888i
$934$			2.82178e11i			0.370797i
$935$	−1.93390e10			−0.0253040
$936$	−1.54076e11			−0.200739
$937$	−4.27565e11			−0.554681			−0.277341	−	0.960772i	$-0.589453\pi$
$937$	−4.27565e11			−0.554681			−0.277341	+	0.960772i	$0.589453\pi$
$938$	−1.29779e11			−0.167646
$939$			7.36003e10i			0.0946711i
$940$	8.56473e10			0.109699
$941$		−	1.36175e12i		−	1.73676i	−0.495898	−	0.868381i	$-0.665161\pi$
$941$		−	1.36175e12i		−	1.73676i	0.495898	−	0.868381i	$-0.334839\pi$
$942$	9.29246e10			0.118012
$943$		−	1.06329e12i		−	1.34464i
$944$			2.52283e11i			0.317688i
$945$			4.41862e10i			0.0554063i
$946$		−	1.11758e11i		−	0.139545i
$947$	−1.00481e12			−1.24935			−0.624675	−	0.780885i	$-0.714768\pi$
$947$	−1.00481e12			−1.24935			−0.624675	+	0.780885i	$0.714768\pi$
$948$	−7.50276e10			−0.0928940
$949$		−	3.90704e11i		−	0.481707i
$950$	6.38118e11	+	2.04363e11i	0.783441	+	0.250904i
$951$	−1.15953e11			−0.141762
$952$			2.49833e9i			0.00304160i
$953$		−	3.27699e11i		−	0.397286i	−0.980072	−	0.198643i	$-0.936347\pi$
$953$		−	3.27699e11i		−	0.397286i	0.980072	−	0.198643i	$-0.0636534\pi$
$954$	4.65973e11			0.562558
$955$	2.29636e12			2.76074
$956$	−3.10399e11			−0.371611
$957$	2.18293e10			0.0260250
$958$			6.52738e11i			0.774956i
$959$	−1.74780e11			−0.206642
$960$			2.07334e10i			0.0244110i
$961$	7.83281e11			0.918384
$962$		−	3.96891e11i		−	0.463416i
$963$		−	3.38104e10i		−	0.0393138i
$964$		−	2.55455e11i		−	0.295805i
$965$		−	2.19137e12i		−	2.52701i
$966$	1.46448e10			0.0168180
$967$	1.10076e12			1.25888			0.629442	−	0.777048i	$-0.283284\pi$
$967$	1.10076e12			1.25888			0.629442	+	0.777048i	$0.283284\pi$
$968$			2.84997e11i			0.324593i
$969$	6.70117e9	+	2.14611e9i	0.00760073	+	0.00243420i
$970$	1.38626e12			1.56588
$971$			8.23490e11i			0.926363i	0.886263	+	0.463182i	$0.153292\pi$
$971$			8.23490e11i			0.926363i	−0.886263	+	0.463182i	$0.846708\pi$
$972$		−	1.73613e11i		−	0.194499i
$973$	−4.43489e10			−0.0494802
$974$	−4.45764e11			−0.495301
$975$	8.06774e10			0.0892758
$976$	1.57928e10			0.0174044
$977$			1.22250e12i			1.34174i	0.741574	+	0.670871i	$0.234080\pi$
$977$			1.22250e12i			1.34174i	−0.741574	+	0.670871i	$0.765920\pi$
$978$	6.98273e10			0.0763255
$979$			2.99792e11i			0.326354i
$980$	−6.64436e11			−0.720359
$981$		−	1.22776e12i		−	1.32568i
$982$		−	1.22384e12i		−	1.31607i
$983$			1.79480e12i			1.92222i	0.276167	+	0.961110i	$0.410936\pi$
$983$			1.79480e12i			1.92222i	−0.276167	+	0.961110i	$0.589064\pi$
$984$		−	4.72790e10i		−	0.0504299i
$985$	3.75417e11			0.398813
$986$	2.75138e10			0.0291100
$987$		−	2.68993e9i		−	0.00283447i
$988$	2.62238e11	+	8.39841e10i	0.275213	+	0.0881393i
$989$	−8.25702e11			−0.863054
$990$		−	2.80893e11i		−	0.292416i
$991$		−	1.61330e11i		−	0.167271i	−0.996496	−	0.0836355i	$-0.973347\pi$
$991$		−	1.61330e11i		−	0.167271i	0.996496	−	0.0836355i	$-0.0266532\pi$
$992$	−4.89058e10			−0.0505026
$993$	5.19566e10			0.0534372
$994$	−9.49966e10			−0.0973111
$995$	1.83418e12			1.87132
$996$		−	5.67502e10i		−	0.0576673i
$997$	6.29339e11			0.636948			0.318474	−	0.947932i	$-0.396830\pi$
$997$	6.29339e11			0.636948			0.318474	+	0.947932i	$0.396830\pi$
$998$			1.01492e12i			1.02308i
$999$	2.97263e11			0.298455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.b.a.37.10	yes	12
3.2	odd	2		342.9.d.a.37.6		12
4.3	odd	2		304.9.e.e.113.6		12
19.18	odd	2	inner	38.9.b.a.37.3	✓	12
57.56	even	2		342.9.d.a.37.12		12
76.75	even	2		304.9.e.e.113.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.b.a.37.3	✓	12	19.18	odd	2	inner
38.9.b.a.37.10	yes	12	1.1	even	1	trivial
304.9.e.e.113.6		12	4.3	odd	2
304.9.e.e.113.7		12	76.75	even	2
342.9.d.a.37.6		12	3.2	odd	2
342.9.d.a.37.12		12	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	38.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.3137i q^{2} +10.7546i q^{3} -128.000 q^{4} -919.278 q^{5} -121.674 q^{6} -343.629 q^{7} -1448.15i q^{8} +6445.34 q^{9} +O(q^{10})\)	\(q+11.3137i q^{2} +10.7546i q^{3} -128.000 q^{4} -919.278 q^{5} -121.674 q^{6} -343.629 q^{7} -1448.15i q^{8} +6445.34 q^{9} -10400.4i q^{10} +4190.29 q^{11} -1376.59i q^{12} -16507.2i q^{13} -3887.72i q^{14} -9886.47i q^{15} +16384.0 q^{16} +5020.47 q^{17} +72920.7i q^{18} +(39747.8 - 124112. i) q^{19} +117668. q^{20} -3695.59i q^{21} +47407.7i q^{22} +350263. q^{23} +15574.3 q^{24} +454447. q^{25} +186758. q^{26} +139878. i q^{27} +43984.5 q^{28} -484397. i q^{29} +111853. q^{30} +263837. i q^{31} +185364. i q^{32} +45064.9i q^{33} +56800.1i q^{34} +315891. q^{35} -825003. q^{36} -2.12516e6i q^{37} +(1.40416e6 + 449695. i) q^{38} +177529. q^{39} +1.33126e6i q^{40} -3.03570e6i q^{41} +41810.9 q^{42} -2.35738e6 q^{43} -536357. q^{44} -5.92506e6 q^{45} +3.96277e6i q^{46} +727875. q^{47} +176203. i q^{48} -5.64672e6 q^{49} +5.14148e6i q^{50} +53993.1i q^{51} +2.11293e6i q^{52} -6.39013e6i q^{53} -1.58254e6 q^{54} -3.85204e6 q^{55} +497628. i q^{56} +(1.33477e6 + 427472. i) q^{57} +5.48032e6 q^{58} +1.53982e7i q^{59} +1.26547e6i q^{60} +963914. q^{61} -2.98497e6 q^{62} -2.21481e6 q^{63} -2.09715e6 q^{64} +1.51747e7i q^{65} -509851. q^{66} -3.33818e7i q^{67} -642620. q^{68} +3.76694e6i q^{69} +3.57390e6i q^{70} -2.44350e7i q^{71} -9.33385e6i q^{72} +2.36687e7 q^{73} +2.40434e7 q^{74} +4.88740e6i q^{75} +(-5.08772e6 + 1.58863e7i) q^{76} -1.43991e6 q^{77} +2.00851e6i q^{78} -5.45025e7i q^{79} -1.50615e7 q^{80} +4.07835e7 q^{81} +3.43450e7 q^{82} +4.12252e7 q^{83} +473036. i q^{84} -4.61521e6 q^{85} -2.66707e7i q^{86} +5.20949e6 q^{87} -6.06818e6i q^{88} +7.15445e7i q^{89} -6.70344e7i q^{90} +5.67237e6i q^{91} -4.48337e7 q^{92} -2.83746e6 q^{93} +8.23496e6i q^{94} +(-3.65393e7 + 1.14093e8i) q^{95} -1.99351e6 q^{96} +1.33289e8i q^{97} -6.38853e7i q^{98} +2.70078e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9}+O(q^{10}) \) 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9	\( 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9} - 12546 q^{11} + 196608 q^{16} + 270810 q^{17} + 41512 q^{19} - 71424 q^{20} - 823956 q^{23} - 229376 q^{24} + 865538 q^{25} - 431616 q^{26} + 694016 q^{28} + 71168 q^{30} - 1194378 q^{35} + 1995776 q^{36} + 998784 q^{38} + 5786100 q^{39} - 8383744 q^{42} + 7586646 q^{43} + 1605888 q^{44} + 2226046 q^{45} - 20260530 q^{47} - 19498842 q^{49} + 16933888 q^{54} - 14858554 q^{55} + 14430564 q^{57} - 5506560 q^{58} - 41363266 q^{61} + 32266752 q^{62} + 84235798 q^{63} - 25165824 q^{64} + 14371328 q^{66} - 34663680 q^{68} + 87906498 q^{73} - 2149632 q^{74} - 5313536 q^{76} - 78817962 q^{77} + 9142272 q^{80} - 100904812 q^{81} - 49609728 q^{82} - 55944960 q^{83} + 25440254 q^{85} + 119189604 q^{87} + 105466368 q^{92} + 105500856 q^{93} + 81396774 q^{95} + 29360128 q^{96} - 85554938 q^{99}+O(q^{100}) \) 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9 - 12546 * q^11 + 196608 * q^16 + 270810 * q^17 + 41512 * q^19 - 71424 * q^20 - 823956 * q^23 - 229376 * q^24 + 865538 * q^25 - 431616 * q^26 + 694016 * q^28 + 71168 * q^30 - 1194378 * q^35 + 1995776 * q^36 + 998784 * q^38 + 5786100 * q^39 - 8383744 * q^42 + 7586646 * q^43 + 1605888 * q^44 + 2226046 * q^45 - 20260530 * q^47 - 19498842 * q^49 + 16933888 * q^54 - 14858554 * q^55 + 14430564 * q^57 - 5506560 * q^58 - 41363266 * q^61 + 32266752 * q^62 + 84235798 * q^63 - 25165824 * q^64 + 14371328 * q^66 - 34663680 * q^68 + 87906498 * q^73 - 2149632 * q^74 - 5313536 * q^76 - 78817962 * q^77 + 9142272 * q^80 - 100904812 * q^81 - 49609728 * q^82 - 55944960 * q^83 + 25440254 * q^85 + 119189604 * q^87 + 105466368 * q^92 + 105500856 * q^93 + 81396774 * q^95 + 29360128 * q^96 - 85554938 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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