LMFDB - Embedded newform 108.6.b.c.107.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,6,Mod(107,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("108.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	108.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:	$17.3214525398$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 94 x^{18} + 5872 x^{16} - 207192 x^{14} + 5271952 x^{12} - 76648960 x^{10} + 792478720 x^{8} + \cdots + 41943040000 $ x^20 - 94x^18 + 5872x^16 - 207192x^14 + 5271952x^12 - 76648960x^10 + 792478720x^8 - 4371873792x^6 + 17152147456x^4 - 32033996800*x^2 + 41943040000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{50}\cdot 3^{40} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			107.2
Root			$3.14287 + 1.81454i$ of defining polynomial
Character	$\chi$	$=$	108.107
Dual form			108.6.b.c.107.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.64474 + 0.370087i) q^{2} +(31.7261 - 4.17809i) q^{4} +38.9183i q^{5} -132.403i q^{7} +(-177.539 + 35.3256i) q^{8} +O(q^{10})$	\(q+(-5.64474 + 0.370087i) q^{2} +(31.7261 - 4.17809i) q^{4} +38.9183i q^{5} -132.403i q^{7} +(-177.539 + 35.3256i) q^{8} +(-14.4032 - 219.683i) q^{10} +190.441 q^{11} -416.631 q^{13} +(49.0008 + 747.382i) q^{14} +(989.087 - 265.109i) q^{16} +1145.55i q^{17} -768.976i q^{19} +(162.604 + 1234.72i) q^{20} +(-1074.99 + 70.4798i) q^{22} -2231.96 q^{23} +1610.37 q^{25} +(2351.77 - 154.190i) q^{26} +(-553.193 - 4200.64i) q^{28} -5919.93i q^{29} -10009.3i q^{31} +(-5485.02 + 1862.52i) q^{32} +(-423.953 - 6466.32i) q^{34} +5152.91 q^{35} +4317.45 q^{37} +(284.588 + 4340.67i) q^{38} +(-1374.81 - 6909.51i) q^{40} -18511.5i q^{41} -5568.60i q^{43} +(6041.95 - 795.680i) q^{44} +(12598.8 - 826.019i) q^{46} +12618.3 q^{47} -723.662 q^{49} +(-9090.10 + 595.977i) q^{50} +(-13218.1 + 1740.72i) q^{52} -30907.9i q^{53} +7411.64i q^{55} +(4677.23 + 23506.8i) q^{56} +(2190.89 + 33416.4i) q^{58} -39249.3 q^{59} -45204.0 q^{61} +(3704.31 + 56499.8i) q^{62} +(30272.2 - 12543.4i) q^{64} -16214.6i q^{65} +24713.4i q^{67} +(4786.20 + 36343.8i) q^{68} +(-29086.8 + 1907.03i) q^{70} +18755.2 q^{71} +81273.7 q^{73} +(-24370.9 + 1597.83i) q^{74} +(-3212.85 - 24396.6i) q^{76} -25215.0i q^{77} +33373.3i q^{79} +(10317.6 + 38493.6i) q^{80} +(6850.85 + 104492. i) q^{82} +47528.7 q^{83} -44582.8 q^{85} +(2060.87 + 31433.3i) q^{86} +(-33810.7 + 6727.45i) q^{88} -65440.5i q^{89} +55163.4i q^{91} +(-70811.3 + 9325.32i) q^{92} +(-71226.9 + 4669.86i) q^{94} +29927.2 q^{95} -29170.2 q^{97} +(4084.88 - 267.818i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{4}+O(q^{10}) $ 20 * q + 20 * q^4	$ 20 q + 20 q^{4} + 184 q^{10} - 116 q^{13} - 4168 q^{16} + 696 q^{22} - 15228 q^{25} - 4764 q^{28} - 16520 q^{34} - 6452 q^{37} + 1504 q^{40} - 9336 q^{46} - 44464 q^{49} + 8236 q^{52} - 58736 q^{58} + 84604 q^{61} - 6496 q^{64} + 138696 q^{70} + 85420 q^{73} + 89172 q^{76} + 221200 q^{82} + 180320 q^{85} - 85824 q^{88} - 60936 q^{94} - 219908 q^{97}+O(q^{100}) $ 20 * q + 20 * q^4 + 184 * q^10 - 116 * q^13 - 4168 * q^16 + 696 * q^22 - 15228 * q^25 - 4764 * q^28 - 16520 * q^34 - 6452 * q^37 + 1504 * q^40 - 9336 * q^46 - 44464 * q^49 + 8236 * q^52 - 58736 * q^58 + 84604 * q^61 - 6496 * q^64 + 138696 * q^70 + 85420 * q^73 + 89172 * q^76 + 221200 * q^82 + 180320 * q^85 - 85824 * q^88 - 60936 * q^94 - 219908 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−5.64474	+	0.370087i	−0.997858	+	0.0654228i
$2$	−5.64474	+	0.370087i	−0.997858	+	0.0654228i
$3$		0			0
$3$		0			0
$4$	31.7261	−	4.17809i	0.991440	−	0.130565i
$5$			38.9183i			0.696191i	0.937459	+	0.348096i	$0.113172\pi$
$5$			38.9183i			0.696191i	−0.937459	+	0.348096i	$0.886828\pi$
$6$		0			0
$7$		−	132.403i		−	1.02130i	−0.859788	−	0.510651i	$-0.829404\pi$
$7$		−	132.403i		−	1.02130i	0.859788	−	0.510651i	$-0.170596\pi$
$8$	−177.539	+	35.3256i	−0.980774	+	0.195148i
$9$		0			0
$10$	−14.4032	−	219.683i	−0.0455468	−	0.694700i
$11$	190.441			0.474547			0.237273	−	0.971443i	$-0.423746\pi$
$11$	190.441			0.474547			0.237273	+	0.971443i	$0.423746\pi$
$12$		0			0
$13$	−416.631			−0.683744			−0.341872	−	0.939746i	$-0.611061\pi$
$13$	−416.631			−0.683744			−0.341872	+	0.939746i	$0.611061\pi$
$14$	49.0008	+	747.382i	0.0668164	+	1.01911i
$15$		0			0
$16$	989.087	−	265.109i	0.965905	−	0.258895i
$17$			1145.55i			0.961371i	0.876893	+	0.480686i	$0.159612\pi$
$17$			1145.55i			0.961371i	−0.876893	+	0.480686i	$0.840388\pi$
$18$		0			0
$19$		−	768.976i		−	0.488685i	−0.969689	−	0.244342i	$-0.921428\pi$
$19$		−	768.976i		−	0.488685i	0.969689	−	0.244342i	$-0.0785721\pi$
$20$	162.604	+	1234.72i	0.0908984	+	0.690232i
$21$		0			0
$22$	−1074.99	+	70.4798i	−0.473530	+	0.0310462i
$23$	−2231.96			−0.879765			−0.439882	−	0.898055i	$-0.644980\pi$
$23$	−2231.96			−0.879765			−0.439882	+	0.898055i	$0.644980\pi$
$24$		0			0
$25$	1610.37			0.515318
$26$	2351.77	−	154.190i	0.682279	−	0.0447325i
$27$		0			0
$28$	−553.193	−	4200.64i	−0.133347	−	1.01256i
$29$		−	5919.93i		−	1.30714i	−0.756867	−	0.653569i	$-0.773271\pi$
$29$		−	5919.93i		−	1.30714i	0.756867	−	0.653569i	$-0.226729\pi$
$30$		0			0
$31$		−	10009.3i		−	1.87068i	−0.353751	−	0.935340i	$-0.615094\pi$
$31$		−	10009.3i		−	1.87068i	0.353751	−	0.935340i	$-0.384906\pi$
$32$	−5485.02	+	1862.52i	−0.946898	+	0.321533i
$33$		0			0
$34$	−423.953	−	6466.32i	−0.0628956	−	0.959312i
$35$	5152.91			0.711021
$36$		0			0
$37$	4317.45			0.518470			0.259235	−	0.965814i	$-0.416530\pi$
$37$	4317.45			0.518470			0.259235	+	0.965814i	$0.416530\pi$
$38$	284.588	+	4340.67i	0.0319711	+	0.487638i
$39$		0			0
$40$	−1374.81	−	6909.51i	−0.135861	−	0.682806i
$41$		−	18511.5i		−	1.71981i	−0.510453	−	0.859906i	$-0.670522\pi$
$41$		−	18511.5i		−	1.71981i	0.510453	−	0.859906i	$-0.329478\pi$
$42$		0			0
$43$		−	5568.60i		−	0.459277i	−0.973276	−	0.229638i	$-0.926246\pi$
$43$		−	5568.60i		−	0.459277i	0.973276	−	0.229638i	$-0.0737544\pi$
$44$	6041.95	−	795.680i	0.470484	−	0.0619593i
$45$		0			0
$46$	12598.8	−	826.019i	0.877880	−	0.0575567i
$47$	12618.3			0.833212			0.416606	−	0.909087i	$-0.363219\pi$
$47$	12618.3			0.833212			0.416606	+	0.909087i	$0.363219\pi$
$48$		0			0
$49$	−723.662			−0.0430572
$50$	−9090.10	+	595.977i	−0.514214	+	0.0337135i
$51$		0			0
$52$	−13218.1	+	1740.72i	−0.677891	+	0.0892732i
$53$		−	30907.9i		−	1.51140i	−0.654918	−	0.755700i	$-0.727297\pi$
$53$		−	30907.9i		−	1.51140i	0.654918	−	0.755700i	$-0.272703\pi$
$54$		0			0
$55$			7411.64i			0.330375i
$56$	4677.23	+	23506.8i	0.199305	+	1.00167i
$57$		0			0
$58$	2190.89	+	33416.4i	0.0855167	+	1.30434i
$59$	−39249.3			−1.46792			−0.733959	−	0.679194i	$-0.762329\pi$
$59$	−39249.3			−1.46792			−0.733959	+	0.679194i	$0.762329\pi$
$60$		0			0
$61$	−45204.0			−1.55544			−0.777719	−	0.628612i	$-0.783623\pi$
$61$	−45204.0			−1.55544			−0.777719	+	0.628612i	$0.783623\pi$
$62$	3704.31	+	56499.8i	0.122385	+	1.86667i
$63$		0			0
$64$	30272.2	−	12543.4i	0.923834	−	0.382793i
$65$		−	16214.6i		−	0.476017i
$66$		0			0
$67$			24713.4i			0.672582i	0.941758	+	0.336291i	$0.109173\pi$
$67$			24713.4i			0.672582i	−0.941758	+	0.336291i	$0.890827\pi$
$68$	4786.20	+	36343.8i	0.125522	+	0.953142i
$69$		0			0
$70$	−29086.8	+	1907.03i	−0.709498	+	0.0465170i
$71$	18755.2			0.441546			0.220773	−	0.975325i	$-0.429142\pi$
$71$	18755.2			0.441546			0.220773	+	0.975325i	$0.429142\pi$
$72$		0			0
$73$	81273.7			1.78502			0.892510	−	0.451029i	$-0.148943\pi$
$73$	81273.7			1.78502			0.892510	+	0.451029i	$0.148943\pi$
$74$	−24370.9	+	1597.83i	−0.517359	+	0.0339197i
$75$		0			0
$76$	−3212.85	−	24396.6i	−0.0638053	−	0.484502i
$77$		−	25215.0i		−	0.484655i
$78$		0			0
$79$			33373.3i			0.601633i	0.953682	+	0.300817i	$0.0972592\pi$
$79$			33373.3i			0.601633i	−0.953682	+	0.300817i	$0.902741\pi$
$80$	10317.6	+	38493.6i	0.180241	+	0.672455i
$81$		0			0
$82$	6850.85	+	104492.i	0.112515	+	1.71613i
$83$	47528.7			0.757288			0.378644	−	0.925542i	$-0.376391\pi$
$83$	47528.7			0.757288			0.378644	+	0.925542i	$0.376391\pi$
$84$		0			0
$85$	−44582.8			−0.669298
$86$	2060.87	+	31433.3i	0.0300472	+	0.458293i
$87$		0			0
$88$	−33810.7	+	6727.45i	−0.465423	+	0.0926070i
$89$		−	65440.5i		−	0.875733i	−0.899040	−	0.437866i	$-0.855734\pi$
$89$		−	65440.5i		−	0.875733i	0.899040	−	0.437866i	$-0.144266\pi$
$90$		0			0
$91$			55163.4i			0.698309i
$92$	−70811.3	+	9325.32i	−0.872234	+	0.114867i
$93$		0			0
$94$	−71226.9	+	4669.86i	−0.831427	+	0.0545111i
$95$	29927.2			0.340218
$96$		0			0
$97$	−29170.2			−0.314782			−0.157391	−	0.987536i	$-0.550308\pi$
$97$	−29170.2			−0.314782			−0.157391	+	0.987536i	$0.550308\pi$
$98$	4084.88	−	267.818i	0.0429649	−	0.00281692i
$99$		0			0
$100$	51090.6	−	6728.26i	0.510906	−	0.0672826i
$101$		−	94523.4i		−	0.922011i	−0.887397	−	0.461005i	$-0.847489\pi$
$101$		−	94523.4i		−	0.922011i	0.887397	−	0.461005i	$-0.152511\pi$
$102$		0			0
$103$			69615.0i			0.646561i	0.946303	+	0.323281i	$0.104786\pi$
$103$			69615.0i			0.646561i	−0.946303	+	0.323281i	$0.895214\pi$
$104$	73968.3	−	14717.8i	0.670598	−	0.133432i
$105$		0			0
$106$	11438.6	+	174467.i	0.0988800	+	1.50816i
$107$	−174764.			−1.47568			−0.737840	−	0.674976i	$-0.764154\pi$
$107$	−174764.			−1.47568			−0.737840	+	0.674976i	$0.764154\pi$
$108$		0			0
$109$	90100.9			0.726378			0.363189	−	0.931715i	$-0.381688\pi$
$109$	90100.9			0.726378			0.363189	+	0.931715i	$0.381688\pi$
$110$	−2742.95	−	41836.7i	−0.0216141	−	0.329667i
$111$		0			0
$112$	−35101.3	−	130959.i	−0.264410	−	0.986481i
$113$		−	53319.6i		−	0.392817i	−0.980522	−	0.196409i	$-0.937072\pi$
$113$		−	53319.6i		−	0.392817i	0.980522	−	0.196409i	$-0.0629279\pi$
$114$		0			0
$115$		−	86864.0i		−	0.612485i
$116$	−24734.0	−	187816.i	−0.170667	−	1.29595i
$117$		0			0
$118$	221552.	−	14525.7i	1.46477	−	0.0960353i
$119$	151675.			0.981850
$120$		0			0
$121$	−124783.			−0.774806
$122$	255165.	−	16729.4i	1.55211	−	0.101761i
$123$		0			0
$124$	−41819.7	−	317556.i	−0.244246	−	1.85467i
$125$			184292.i			1.05495i
$126$		0			0
$127$		−	226128.i		−	1.24407i	−0.782988	−	0.622036i	$-0.786306\pi$
$127$		−	226128.i		−	1.24407i	0.782988	−	0.622036i	$-0.213694\pi$
$128$	−166236.	+	82007.3i	−0.896812	+	0.442412i
$129$		0			0
$130$	6000.81	+	91527.0i	0.0311423	+	0.474997i
$131$	−52645.4			−0.268029			−0.134015	−	0.990979i	$-0.542787\pi$
$131$	−52645.4			−0.268029			−0.134015	+	0.990979i	$0.542787\pi$
$132$		0			0
$133$	−101815.			−0.499095
$134$	−9146.11	−	139501.i	−0.0440022	−	0.671141i
$135$		0			0
$136$	−40467.2	−	203380.i	−0.187610	−	0.942888i
$137$			288620.i			1.31379i	0.753984	+	0.656893i	$0.228130\pi$
$137$			288620.i			1.31379i	−0.753984	+	0.656893i	$0.771870\pi$
$138$		0			0
$139$		−	241199.i		−	1.05886i	−0.848354	−	0.529430i	$-0.822406\pi$
$139$		−	241199.i		−	1.05886i	0.848354	−	0.529430i	$-0.177594\pi$
$140$	163482.	−	21529.3i	0.704935	−	0.0928347i
$141$		0			0
$142$	−105868.	+	6941.07i	−0.440600	+	0.0288872i
$143$	−79343.7			−0.324468
$144$		0			0
$145$	230393.			0.910018
$146$	−458768.	+	30078.4i	−1.78119	+	0.116781i
$147$		0			0
$148$	136976.	−	18038.7i	0.514031	−	0.0676941i
$149$			404813.i			1.49379i	0.664943	+	0.746894i	$0.268456\pi$
$149$			404813.i			1.49379i	−0.664943	+	0.746894i	$0.731544\pi$
$150$		0			0
$151$		−	107181.i		−	0.382540i	−0.981537	−	0.191270i	$-0.938739\pi$
$151$		−	107181.i		−	0.382540i	0.981537	−	0.191270i	$-0.0612605\pi$
$152$	27164.6	+	136523.i	0.0953660	+	0.479289i
$153$		0			0
$154$	9331.77	+	142332.i	0.0317075	+	0.483617i
$155$	389545.			1.30235
$156$		0			0
$157$	289913.			0.938682			0.469341	−	0.883017i	$-0.344492\pi$
$157$	289913.			0.938682			0.469341	+	0.883017i	$0.344492\pi$
$158$	−12351.0	−	188384.i	−0.0393605	−	0.600344i
$159$		0			0
$160$	−72485.9	−	213468.i	−0.223848	−	0.659222i
$161$			295519.i			0.898505i
$162$		0			0
$163$			383691.i			1.13113i	0.824703	+	0.565566i	$0.191342\pi$
$163$			383691.i			1.13113i	−0.824703	+	0.565566i	$0.808658\pi$
$164$	−77342.5	−	587296.i	−0.224548	−	1.70509i
$165$		0			0
$166$	−268287.	+	17589.8i	−0.755665	+	0.0495439i
$167$	−552670.			−1.53347			−0.766735	−	0.641964i	$-0.778120\pi$
$167$	−552670.			−1.53347			−0.766735	+	0.641964i	$0.778120\pi$
$168$		0			0
$169$	−197711.			−0.532494
$170$	251658.	−	16499.5i	0.667864	−	0.0437874i
$171$		0			0
$172$	−23266.1	−	176670.i	−0.0599656	−	0.455345i
$173$			388771.i			0.987594i	0.869577	+	0.493797i	$0.164391\pi$
$173$			388771.i			0.987594i	−0.869577	+	0.493797i	$0.835609\pi$
$174$		0			0
$175$		−	213218.i		−	0.526295i
$176$	188363.	−	50487.6i	0.458367	−	0.122858i
$177$		0			0
$178$	24218.7	+	369394.i	0.0572929	+	0.873857i
$179$	−300195.			−0.700278			−0.350139	−	0.936698i	$-0.613866\pi$
$179$	−300195.			−0.700278			−0.350139	+	0.936698i	$0.613866\pi$
$180$		0			0
$181$	−197901.			−0.449005			−0.224503	−	0.974474i	$-0.572076\pi$
$181$	−197901.			−0.449005			−0.224503	+	0.974474i	$0.572076\pi$
$182$	−20415.3	−	311383.i	−0.0456853	−	0.696813i
$183$		0			0
$184$	396260.	−	78845.3i	0.862850	−	0.171685i
$185$			168028.i			0.360954i
$186$		0			0
$187$			218159.i			0.456215i
$188$	400328.	−	52720.3i	0.826080	−	0.108789i
$189$		0			0
$190$	−168931.	+	11075.7i	−0.339489	+	0.0222580i
$191$	911538.			1.80797			0.903986	−	0.427563i	$-0.140628\pi$
$191$	911538.			1.80797			0.903986	+	0.427563i	$0.140628\pi$
$192$		0			0
$193$	−148579.			−0.287120			−0.143560	−	0.989642i	$-0.545855\pi$
$193$	−148579.			−0.287120			−0.143560	+	0.989642i	$0.545855\pi$
$194$	164658.	−	10795.5i	0.314108	−	0.0205939i
$195$		0			0
$196$	−22959.0	+	3023.52i	−0.0426886	+	0.00562177i
$197$		−	537867.i		−	0.987437i	−0.869622	−	0.493718i	$-0.835637\pi$
$197$		−	537867.i		−	0.987437i	0.869622	−	0.493718i	$-0.164363\pi$
$198$		0			0
$199$		−	199552.i		−	0.357211i	−0.983921	−	0.178605i	$-0.942841\pi$
$199$		−	199552.i		−	0.357211i	0.983921	−	0.178605i	$-0.0571585\pi$
$200$	−285903.	+	56887.2i	−0.505410	+	0.100563i
$201$		0			0
$202$	34981.9	+	533560.i	0.0603205	+	0.920035i
$203$	−783819.			−1.33498
$204$		0			0
$205$	720434.			1.19732
$206$	−25763.6	−	392958.i	−0.0422998	−	0.645176i
$207$		0			0
$208$	−412085.	+	110453.i	−0.660432	+	0.177018i
$209$		−	146445.i		−	0.231904i
$210$		0			0
$211$			778163.i			1.20327i	0.798770	+	0.601637i	$0.205485\pi$
$211$			778163.i			1.20327i	−0.798770	+	0.601637i	$0.794515\pi$
$212$	−129136.	−	980585.i	−0.197336	−	1.49846i
$213$		0			0
$214$	986496.	−	64677.9i	1.47252	−	0.0965431i
$215$	216720.			0.319745
$216$		0			0
$217$	−1.32526e6			−1.91053
$218$	−508596.	+	33345.2i	−0.724822	+	0.0475217i
$219$		0			0
$220$	30966.5	+	235142.i	0.0431355	+	0.327547i
$221$		−	477271.i		−	0.657332i
$222$		0			0
$223$			681625.i			0.917875i	0.888469	+	0.458937i	$0.151770\pi$
$223$			681625.i			0.917875i	−0.888469	+	0.458937i	$0.848230\pi$
$224$	246604.	+	726236.i	0.328382	+	0.967069i
$225$		0			0
$226$	19732.9	+	300975.i	0.0256992	+	0.391976i
$227$	1.28670e6			1.65734			0.828671	−	0.559736i	$-0.189098\pi$
$227$	1.28670e6			1.65734			0.828671	+	0.559736i	$0.189098\pi$
$228$		0			0
$229$	−74657.5			−0.0940773			−0.0470386	−	0.998893i	$-0.514978\pi$
$229$	−74657.5			−0.0940773			−0.0470386	+	0.998893i	$0.514978\pi$
$230$	32147.3	+	490324.i	0.0400705	+	0.611172i
$231$		0			0
$232$	209125.	+	1.05102e6i	0.255086	+	1.28201i
$233$			638465.i			0.770455i	0.922822	+	0.385227i	$0.125877\pi$
$233$			638465.i			0.770455i	−0.922822	+	0.385227i	$0.874123\pi$
$234$		0			0
$235$			491082.i			0.580075i
$236$	−1.24523e6	+	163987.i	−1.45535	+	0.191659i
$237$		0			0
$238$	−856162.	+	56132.8i	−0.979747	+	0.0642354i
$239$	−1.43424e6			−1.62415			−0.812077	−	0.583550i	$-0.801663\pi$
$239$	−1.43424e6			−1.62415			−0.812077	+	0.583550i	$0.801663\pi$
$240$		0			0
$241$	989623.			1.09756			0.548778	−	0.835968i	$-0.315093\pi$
$241$	989623.			1.09756			0.548778	+	0.835968i	$0.315093\pi$
$242$	704368.	−	46180.7i	0.773146	−	0.0506899i
$243$		0			0
$244$	−1.43415e6	+	188866.i	−1.54212	+	0.203086i
$245$		−	28163.7i		−	0.0299760i
$246$		0			0
$247$			320380.i			0.334135i
$248$	353585.	+	1.77704e6i	0.365060	+	1.83471i
$249$		0			0
$250$	−68204.2	−	1.04028e6i	−0.0690178	−	1.05269i
$251$	272737.			0.273250			0.136625	−	0.990623i	$-0.456375\pi$
$251$	272737.			0.273250			0.136625	+	0.990623i	$0.456375\pi$
$252$		0			0
$253$	−425057.			−0.417489
$254$	83687.2	+	1.27643e6i	0.0813907	+	1.24141i
$255$		0			0
$256$	908011.	−	524431.i	0.865947	−	0.500137i
$257$		−	1.22534e6i		−	1.15725i	−0.815595	−	0.578623i	$-0.803590\pi$
$257$		−	1.22534e6i		−	1.15725i	0.815595	−	0.578623i	$-0.196410\pi$
$258$		0			0
$259$		−	571645.i		−	0.529514i
$260$	−67745.9	−	514425.i	−0.0621513	−	0.471942i
$261$		0			0
$262$	297169.	−	19483.4i	0.267455	−	0.0175352i
$263$	202375.			0.180413			0.0902065	−	0.995923i	$-0.471247\pi$
$263$	202375.			0.180413			0.0902065	+	0.995923i	$0.471247\pi$
$264$		0			0
$265$	1.20288e6			1.05222
$266$	574719.	−	37680.5i	0.498025	−	0.0326522i
$267$		0			0
$268$	103255.	+	784059.i	0.0878159	+	0.666824i
$269$		−	546051.i		−	0.460100i	−0.973179	−	0.230050i	$-0.926111\pi$
$269$		−	546051.i		−	0.460100i	0.973179	−	0.230050i	$-0.0738890\pi$
$270$		0			0
$271$		−	1.95161e6i		−	1.61425i	−0.590383	−	0.807123i	$-0.701023\pi$
$271$		−	1.95161e6i		−	1.61425i	0.590383	−	0.807123i	$-0.298977\pi$
$272$	303695.	+	1.13305e6i	0.248894	+	0.928594i
$273$		0			0
$274$	−106814.	−	1.62918e6i	−0.0859515	−	1.31097i
$275$	306680.			0.244542
$276$		0			0
$277$	−128661.			−0.100751			−0.0503754	−	0.998730i	$-0.516042\pi$
$277$	−128661.			−0.100751			−0.0503754	+	0.998730i	$0.516042\pi$
$278$	89264.7	+	1.36150e6i	0.0692736	+	1.05659i
$279$		0			0
$280$	−914843.	+	182030.i	−0.697351	+	0.138755i
$281$		−	1.55530e6i		−	1.17503i	−0.809213	−	0.587515i	$-0.800106\pi$
$281$		−	1.55530e6i		−	1.17503i	0.809213	−	0.587515i	$-0.199894\pi$
$282$		0			0
$283$		−	520162.i		−	0.386076i	−0.981191	−	0.193038i	$-0.938166\pi$
$283$		−	520162.i		−	0.386076i	0.981191	−	0.193038i	$-0.0618341\pi$
$284$	595030.	−	78361.0i	0.437767	−	0.0576506i
$285$		0			0
$286$	447874.	−	29364.1i	0.323773	−	0.0212276i
$287$	−2.45098e6			−1.75645
$288$		0			0
$289$	107576.			0.0757652
$290$	−1.30051e6	+	85265.7i	−0.908069	+	0.0595360i
$291$		0			0
$292$	2.57850e6	−	339569.i	1.76974	−	0.233062i
$293$			398515.i			0.271191i	0.990764	+	0.135596i	$0.0432948\pi$
$293$			398515.i			0.271191i	−0.990764	+	0.135596i	$0.956705\pi$
$294$		0			0
$295$		−	1.52751e6i		−	1.02195i
$296$	−766516.	+	152517.i	−0.508501	+	0.101178i
$297$		0			0
$298$	−149816.	−	2.28506e6i	−0.0977278	−	1.49059i
$299$	929904.			0.601534
$300$		0			0
$301$	−737301.			−0.469060
$302$	39666.4	+	605010.i	0.0250268	+	0.381720i
$303$		0			0
$304$	−203862.	−	760585.i	−0.126518	−	0.472023i
$305$		−	1.75926e6i		−	1.08288i
$306$		0			0
$307$			1.24181e6i			0.751987i	0.926622	+	0.375994i	$0.122699\pi$
$307$			1.24181e6i			0.751987i	−0.926622	+	0.375994i	$0.877301\pi$
$308$	−105351.	−	799974.i	−0.0632791	−	0.480506i
$309$		0			0
$310$	−2.19888e6	+	144165.i	−1.29956	+	0.0852034i
$311$	2.77430e6			1.62649			0.813246	−	0.581921i	$-0.197699\pi$
$311$	2.77430e6			1.62649			0.813246	+	0.581921i	$0.197699\pi$
$312$		0			0
$313$	−1.10249e6			−0.636080			−0.318040	−	0.948077i	$-0.603025\pi$
$313$	−1.10249e6			−0.636080			−0.318040	+	0.948077i	$0.603025\pi$
$314$	−1.63648e6	+	107293.i	−0.936671	+	0.0614112i
$315$		0			0
$316$	139437.	+	1.05880e6i	0.0785524	+	0.596483i
$317$		−	2.09886e6i		−	1.17310i	−0.809914	−	0.586549i	$-0.800486\pi$
$317$		−	2.09886e6i		−	1.17310i	0.809914	−	0.586549i	$-0.199514\pi$
$318$		0			0
$319$		−	1.12740e6i		−	0.620298i
$320$	488166.	+	1.17814e6i	0.266497	+	0.643165i
$321$		0			0
$322$	−109368.	−	1.66813e6i	−0.0587827	−	0.896580i
$323$	880900.			0.469808
$324$		0			0
$325$	−670930.			−0.352345
$326$	−141999.	−	2.16584e6i	−0.0740018	−	1.12871i
$327$		0			0
$328$	653929.	+	3.28651e6i	0.335618	+	1.68675i
$329$		−	1.67070e6i		−	0.850961i
$330$		0			0
$331$			1.27129e6i			0.637786i	0.947791	+	0.318893i	$0.103311\pi$
$331$			1.27129e6i			0.637786i	−0.947791	+	0.318893i	$0.896689\pi$
$332$	1.50790e6	−	198579.i	0.750805	−	0.0988755i
$333$		0			0
$334$	3.11968e6	−	204536.i	1.53018	−	0.100324i
$335$	−961802.			−0.468246
$336$		0			0
$337$	1.39426e6			0.668760			0.334380	−	0.942438i	$-0.391473\pi$
$337$	1.39426e6			0.668760			0.334380	+	0.942438i	$0.391473\pi$
$338$	1.11603e6	−	73170.4i	0.531353	−	0.0348372i
$339$		0			0
$340$	−1.41444e6	+	186271.i	−0.663569	+	0.0873871i
$341$		−	1.90618e6i		−	0.887724i
$342$		0			0
$343$		−	2.12949e6i		−	0.977327i
$344$	196714.	+	988643.i	0.0896271	+	0.450447i
$345$		0			0
$346$	−143879.	−	2.19451e6i	−0.0646111	−	0.985478i
$347$	−762430.			−0.339920			−0.169960	−	0.985451i	$-0.554364\pi$
$347$	−762430.			−0.339920			−0.169960	+	0.985451i	$0.554364\pi$
$348$		0			0
$349$	356038.			0.156471			0.0782353	−	0.996935i	$-0.475071\pi$
$349$	356038.			0.156471			0.0782353	+	0.996935i	$0.475071\pi$
$350$	78909.3	+	1.20356e6i	0.0344317	+	0.525167i
$351$		0			0
$352$	−1.04457e6	+	354700.i	−0.449347	+	0.152582i
$353$			3.82852e6i			1.63529i	0.575725	+	0.817643i	$0.304720\pi$
$353$			3.82852e6i			1.63529i	−0.575725	+	0.817643i	$0.695280\pi$
$354$		0			0
$355$			729921.i			0.307401i
$356$	−273416.	−	2.07617e6i	−0.114340	−	0.868236i
$357$		0			0
$358$	1.69452e6	−	111098.i	0.698778	−	0.0458141i
$359$	976058.			0.399705			0.199852	−	0.979826i	$-0.435954\pi$
$359$	976058.			0.399705			0.199852	+	0.979826i	$0.435954\pi$
$360$		0			0
$361$	1.88477e6			0.761187
$362$	1.11710e6	−	73240.6i	0.448043	−	0.0293752i
$363$		0			0
$364$	230478.	+	1.75012e6i	0.0911749	+	0.692331i
$365$			3.16303e6i			1.24271i
$366$		0			0
$367$		−	378329.i		−	0.146624i	−0.997309	−	0.0733119i	$-0.976643\pi$
$367$		−	378329.i		−	0.146624i	0.997309	−	0.0733119i	$-0.0233569\pi$
$368$	−2.20760e6	+	591712.i	−0.849770	+	0.227767i
$369$		0			0
$370$	−62184.9	−	948472.i	−0.0236146	−	0.360181i
$371$	−4.09230e6			−1.54359
$372$		0			0
$373$	−2.27236e6			−0.845678			−0.422839	−	0.906205i	$-0.638966\pi$
$373$	−2.27236e6			−0.845678			−0.422839	+	0.906205i	$0.638966\pi$
$374$	−80738.0	−	1.23145e6i	−0.0298469	−	0.455238i
$375$		0			0
$376$	−2.24024e6	+	445749.i	−0.817193	+	0.162600i
$377$			2.46643e6i			0.893748i
$378$		0			0
$379$			5.13906e6i			1.83775i	0.394551	+	0.918874i	$0.370900\pi$
$379$			5.13906e6i			1.83775i	−0.394551	+	0.918874i	$0.629100\pi$
$380$	949473.	−	125039.i	0.337306	−	0.0444207i
$381$		0			0
$382$	−5.14539e6	+	337349.i	−1.80410	+	0.118283i
$383$	70733.0			0.0246391			0.0123196	−	0.999924i	$-0.496078\pi$
$383$	70733.0			0.0246391			0.0123196	+	0.999924i	$0.496078\pi$
$384$		0			0
$385$	981326.			0.337413
$386$	838686.	−	54987.0i	0.286504	−	0.0187842i
$387$		0			0
$388$	−925456.	+	121876.i	−0.312088	+	0.0410996i
$389$			1.49179e6i			0.499843i	0.968266	+	0.249922i	$0.0804049\pi$
$389$			1.49179e6i			0.499843i	−0.968266	+	0.249922i	$0.919595\pi$
$390$		0			0
$391$		−	2.55682e6i		−	0.845781i
$392$	128478.	−	25563.8i	0.0422294	−	0.00840254i
$393$		0			0
$394$	199058.	+	3.03612e6i	0.0646009	+	0.985321i
$395$	−1.29883e6			−0.418852
$396$		0			0
$397$	2.91290e6			0.927575			0.463787	−	0.885947i	$-0.346490\pi$
$397$	2.91290e6			0.927575			0.463787	+	0.885947i	$0.346490\pi$
$398$	73851.8	+	1.12642e6i	0.0233697	+	0.356445i
$399$		0			0
$400$	1.59279e6	−	426923.i	0.497748	−	0.133413i
$401$			1.18881e6i			0.369191i	0.982815	+	0.184595i	$0.0590975\pi$
$401$			1.18881e6i			0.369191i	−0.982815	+	0.184595i	$0.940903\pi$
$402$		0			0
$403$			4.17019e6i			1.27907i
$404$	−394927.	−	2.99886e6i	−0.120383	−	0.914118i
$405$		0			0
$406$	4.42445e6	−	290081.i	1.33212	−	0.0873383i
$407$	822220.			0.246038
$408$		0			0
$409$	−3.31777e6			−0.980705			−0.490352	−	0.871524i	$-0.663132\pi$
$409$	−3.31777e6			−0.980705			−0.490352	+	0.871524i	$0.663132\pi$
$410$	−4.06666e6	+	266623.i	−1.19475	+	0.0783319i
$411$		0			0
$412$	290858.	+	2.20861e6i	0.0844184	+	0.641026i
$413$			5.19674e6i			1.49919i
$414$		0			0
$415$			1.84974e6i			0.527217i
$416$	2.28523e6	−	775983.i	0.647436	−	0.219846i
$417$		0			0
$418$	54197.3	+	826641.i	0.0151718	+	0.231407i
$419$	4.24309e6			1.18072			0.590361	−	0.807140i	$-0.298985\pi$
$419$	4.24309e6			1.18072			0.590361	+	0.807140i	$0.298985\pi$
$420$		0			0
$421$	−3.70532e6			−1.01887			−0.509437	−	0.860508i	$-0.670146\pi$
$421$	−3.70532e6			−1.01887			−0.509437	+	0.860508i	$0.670146\pi$
$422$	−287988.	−	4.39252e6i	−0.0787215	−	1.20070i
$423$		0			0
$424$	1.09184e6	+	5.48735e6i	0.294947	+	1.48234i
$425$			1.84475e6i			0.495412i
$426$		0			0
$427$			5.98517e6i			1.58857i
$428$	−5.54457e6	+	730179.i	−1.46305	+	0.192673i
$429$		0			0
$430$	−1.22333e6	+	80205.4i	−0.319060	+	0.0209186i
$431$	4.36762e6			1.13253			0.566267	−	0.824222i	$-0.308387\pi$
$431$	4.36762e6			1.13253			0.566267	+	0.824222i	$0.308387\pi$
$432$		0			0
$433$	2.08282e6			0.533865			0.266932	−	0.963715i	$-0.413990\pi$
$433$	2.08282e6			0.533865			0.266932	+	0.963715i	$0.413990\pi$
$434$	7.48077e6	−	490464.i	1.90643	−	0.124992i
$435$		0			0
$436$	2.85855e6	−	376450.i	0.720160	−	0.0948398i
$437$			1.71632e6i			0.429928i
$438$		0			0
$439$		−	2.60275e6i		−	0.644571i	−0.946642	−	0.322286i	$-0.895549\pi$
$439$		−	2.60275e6i		−	0.644571i	0.946642	−	0.322286i	$-0.104451\pi$
$440$	−261821.	−	1.31585e6i	−0.0644722	−	0.324023i
$441$		0			0
$442$	176632.	+	2.69407e6i	0.0430045	+	0.655924i
$443$	−4.93001e6			−1.19354			−0.596772	−	0.802411i	$-0.703551\pi$
$443$	−4.93001e6			−1.19354			−0.596772	+	0.802411i	$0.703551\pi$
$444$		0			0
$445$	2.54683e6			0.609678
$446$	−252261.	−	3.84759e6i	−0.0600499	−	0.915908i
$447$		0			0
$448$	−1.66078e6	−	4.00814e6i	−0.390947	−	0.943514i
$449$			3.42135e6i			0.800906i	0.916317	+	0.400453i	$0.131147\pi$
$449$			3.42135e6i			0.800906i	−0.916317	+	0.400453i	$0.868853\pi$
$450$		0			0
$451$		−	3.52534e6i		−	0.816131i
$452$	−222774.	−	1.69162e6i	−0.0512883	−	0.389455i
$453$		0			0
$454$	−7.26307e6	+	476190.i	−1.65379	+	0.108428i
$455$	−2.14687e6			−0.486157
$456$		0			0
$457$	−7.87833e6			−1.76459			−0.882295	−	0.470697i	$-0.844002\pi$
$457$	−7.87833e6			−1.76459			−0.882295	+	0.470697i	$0.844002\pi$
$458$	421422.	−	27629.8i	0.0938757	−	0.00615480i
$459$		0			0
$460$	−362925.	−	2.75585e6i	−0.0799692	−	0.607241i
$461$		−	3.24234e6i		−	0.710569i	−0.934758	−	0.355284i	$-0.884384\pi$
$461$		−	3.24234e6i		−	0.710569i	0.934758	−	0.355284i	$-0.115616\pi$
$462$		0			0
$463$		−	3.95642e6i		−	0.857728i	−0.903369	−	0.428864i	$-0.858914\pi$
$463$		−	3.95642e6i		−	0.857728i	0.903369	−	0.428864i	$-0.141086\pi$
$464$	−1.56942e6	−	5.85533e6i	−0.338412	−	1.26257i
$465$		0			0
$466$	−236288.	−	3.60396e6i	−0.0504053	−	0.768804i
$467$	3.80560e6			0.807478			0.403739	−	0.914874i	$-0.367710\pi$
$467$	3.80560e6			0.807478			0.403739	+	0.914874i	$0.367710\pi$
$468$		0			0
$469$	3.27214e6			0.686909
$470$	−181743.	−	2.77203e6i	−0.0379501	−	0.578832i
$471$		0			0
$472$	6.96828e6	−	1.38650e6i	1.43969	−	0.286462i
$473$		−	1.06049e6i		−	0.217948i
$474$		0			0
$475$		−	1.23833e6i		−	0.251828i
$476$	4.81204e6	−	633710.i	0.973445	−	0.128196i
$477$		0			0
$478$	8.09591e6	−	530794.i	1.62067	−	0.106257i
$479$	−6.76059e6			−1.34631			−0.673156	−	0.739500i	$-0.735062\pi$
$479$	−6.76059e6			−1.34631			−0.673156	+	0.739500i	$0.735062\pi$
$480$		0			0
$481$	−1.79879e6			−0.354501
$482$	−5.58616e6	+	366247.i	−1.09521	+	0.0718052i
$483$		0			0
$484$	−3.95888e6	+	521355.i	−0.768173	+	0.101163i
$485$		−	1.13525e6i		−	0.219149i
$486$		0			0
$487$		−	3.94101e6i		−	0.752982i	−0.926420	−	0.376491i	$-0.877131\pi$
$487$		−	3.94101e6i		−	0.752982i	0.926420	−	0.376491i	$-0.122869\pi$
$488$	8.02548e6	−	1.59686e6i	1.52553	−	0.303541i
$489$		0			0
$490$	10423.0	+	158977.i	0.00196112	+	0.0299118i
$491$	−2.38312e6			−0.446111			−0.223055	−	0.974806i	$-0.571603\pi$
$491$	−2.38312e6			−0.446111			−0.223055	+	0.974806i	$0.571603\pi$
$492$		0			0
$493$	6.78157e6			1.25665
$494$	−118568.	−	1.80846e6i	−0.0218601	−	0.333420i
$495$		0			0
$496$	−2.65355e6	−	9.90007e6i	−0.484310	−	1.80690i
$497$		−	2.48326e6i		−	0.450952i
$498$		0			0
$499$			5.97843e6i			1.07482i	0.843321	+	0.537410i	$0.180597\pi$
$499$			5.97843e6i			1.07482i	−0.843321	+	0.537410i	$0.819403\pi$
$500$	769990.	+	5.84687e6i	0.137740	+	1.04592i
$501$		0			0
$502$	−1.53953e6	+	100936.i	−0.272664	+	0.0178767i
$503$	2.75877e6			0.486179			0.243090	−	0.970004i	$-0.421839\pi$
$503$	2.75877e6			0.486179			0.243090	+	0.970004i	$0.421839\pi$
$504$		0			0
$505$	3.67869e6			0.641896
$506$	2.39933e6	−	157308.i	0.416595	−	0.0273133i
$507$		0			0
$508$	−944785.	−	7.17417e6i	−0.162433	−	1.23342i
$509$		−	198012.i		−	0.0338764i	−0.999857	−	0.0169382i	$-0.994608\pi$
$509$		−	198012.i		−	0.0338764i	0.999857	−	0.0169382i	$-0.00539186\pi$
$510$		0			0
$511$		−	1.07609e7i		−	1.82304i
$512$	−4.93140e6	+	3.29632e6i	−0.831371	+	0.555718i
$513$		0			0
$514$	453484.	+	6.91675e6i	0.0757102	+	1.15477i
$515$	−2.70929e6			−0.450130
$516$		0			0
$517$	2.40304e6			0.395398
$518$	211559.	+	3.22679e6i	0.0346423	+	0.528379i
$519$		0			0
$520$	572790.	+	2.87872e6i	0.0928939	+	0.466865i
$521$		−	499429.i		−	0.0806082i	−0.999187	−	0.0403041i	$-0.987167\pi$
$521$		−	499429.i		−	0.0806082i	0.999187	−	0.0403041i	$-0.0128327\pi$
$522$		0			0
$523$		−	2.10979e6i		−	0.337276i	−0.985678	−	0.168638i	$-0.946063\pi$
$523$		−	2.10979e6i		−	0.337276i	0.985678	−	0.168638i	$-0.0539369\pi$
$524$	−1.67023e6	+	219957.i	−0.265735	+	0.0349953i
$525$		0			0
$526$	−1.14235e6	+	74896.5i	−0.180027	+	0.0118031i
$527$	1.14661e7			1.79842
$528$		0			0
$529$	−1.45470e6			−0.226014
$530$	−6.78994e6	+	445171.i	−1.04997	+	0.0688394i
$531$		0			0
$532$	−3.23019e6	+	425392.i	−0.494822	+	0.0651644i
$533$			7.71245e6i			1.17591i
$534$		0			0
$535$		−	6.80151e6i		−	1.02736i
$536$	−873016.	−	4.38759e6i	−0.131253	−	0.659651i
$537$		0			0
$538$	202086.	+	3.08231e6i	0.0301010	+	0.459114i
$539$	−137815.			−0.0204326
$540$		0			0
$541$	−7.08611e6			−1.04091			−0.520457	−	0.853888i	$-0.674238\pi$
$541$	−7.08611e6			−1.04091			−0.520457	+	0.853888i	$0.674238\pi$
$542$	722266.	+	1.10163e7i	0.105609	+	1.61079i
$543$		0			0
$544$	−2.13360e6	−	6.28336e6i	−0.309112	−	0.910321i
$545$			3.50657e6i			0.505698i
$546$		0			0
$547$			1.14218e7i			1.63218i	0.577925	+	0.816090i	$0.303863\pi$
$547$			1.14218e7i			1.63218i	−0.577925	+	0.816090i	$0.696137\pi$
$548$	1.20588e6	+	9.15677e6i	0.171535	+	1.30254i
$549$		0			0
$550$	−1.73113e6	+	113498.i	−0.244018	+	0.0159986i
$551$	−4.55229e6			−0.638779
$552$		0			0
$553$	4.41874e6			0.614449
$554$	726259.	−	47615.9i	0.100535	−	0.00659140i
$555$		0			0
$556$	−1.00775e6	−	7.65230e6i	−0.138250	−	1.04980i
$557$		−	1.11633e7i		−	1.52459i	−0.647227	−	0.762297i	$-0.724071\pi$
$557$		−	1.11633e7i		−	1.52459i	0.647227	−	0.762297i	$-0.275929\pi$
$558$		0			0
$559$			2.32005e6i			0.314028i
$560$	5.09668e6	−	1.36608e6i	0.686779	−	0.184080i
$561$		0			0
$562$	575598.	+	8.77927e6i	0.0768738	+	1.17251i
$563$	6.63847e6			0.882668			0.441334	−	0.897343i	$-0.354505\pi$
$563$	6.63847e6			0.882668			0.441334	+	0.897343i	$0.354505\pi$
$564$		0			0
$565$	2.07511e6			0.273476
$566$	192506.	+	2.93618e6i	0.0252582	+	0.385249i
$567$		0			0
$568$	−3.32978e6	+	662540.i	−0.433057	+	0.0861670i
$569$		−	3.66619e6i		−	0.474716i	−0.971422	−	0.237358i	$-0.923719\pi$
$569$		−	3.66619e6i		−	0.474716i	0.971422	−	0.237358i	$-0.0762815\pi$
$570$		0			0
$571$		−	4.47869e6i		−	0.574858i	−0.957802	−	0.287429i	$-0.907199\pi$
$571$		−	4.47869e6i		−	0.574858i	0.957802	−	0.287429i	$-0.0928006\pi$
$572$	−2.51726e6	+	331505.i	−0.321691	+	0.0423643i
$573$		0			0
$574$	1.38351e7	−	907076.i	1.75268	−	0.114912i
$575$	−3.59427e6			−0.453358
$576$		0			0
$577$	−8.10380e6			−1.01333			−0.506663	−	0.862144i	$-0.669121\pi$
$577$	−8.10380e6			−1.01333			−0.506663	+	0.862144i	$0.669121\pi$
$578$	−607236.	+	39812.4i	−0.0756029	+	0.00495677i
$579$		0			0
$580$	7.30948e6	−	962604.i	0.902228	−	0.118817i
$581$		−	6.29296e6i		−	0.773419i
$582$		0			0
$583$		−	5.88612e6i		−	0.717229i
$584$	−1.44293e7	+	2.87104e6i	−1.75070	+	0.348343i
$585$		0			0
$586$	−147485.	−	2.24951e6i	−0.0177421	−	0.270610i
$587$	−6.56387e6			−0.786258			−0.393129	−	0.919483i	$-0.628607\pi$
$587$	−6.56387e6			−0.786258			−0.393129	+	0.919483i	$0.628607\pi$
$588$		0			0
$589$	−7.69691e6			−0.914173
$590$	565313.	+	8.62241e6i	0.0668589	+	1.01976i
$591$		0			0
$592$	4.27034e6	−	1.14459e6i	0.500793	−	0.134229i
$593$		−	8.37601e6i		−	0.978139i	−0.872245	−	0.489069i	$-0.837336\pi$
$593$		−	8.37601e6i		−	0.978139i	0.872245	−	0.489069i	$-0.162664\pi$
$594$		0			0
$595$			5.90291e6i			0.683556i
$596$	1.69135e6	+	1.28431e7i	0.195037	+	1.48100i
$597$		0			0
$598$	−5.24906e6	+	344146.i	−0.600245	+	0.0393540i
$599$	3.96714e6			0.451763			0.225882	−	0.974155i	$-0.427474\pi$
$599$	3.96714e6			0.451763			0.225882	+	0.974155i	$0.427474\pi$
$600$		0			0
$601$	1.31467e7			1.48467			0.742335	−	0.670028i	$-0.233718\pi$
$601$	1.31467e7			1.48467			0.742335	+	0.670028i	$0.233718\pi$
$602$	4.16187e6	−	272866.i	0.468055	−	0.0306872i
$603$		0			0
$604$	−447813.	−	3.40044e6i	−0.0499464	−	0.379265i
$605$		−	4.85635e6i		−	0.539413i
$606$		0			0
$607$		−	2.24278e6i		−	0.247067i	−0.992340	−	0.123533i	$-0.960577\pi$
$607$		−	2.24278e6i		−	0.247067i	0.992340	−	0.123533i	$-0.0394226\pi$
$608$	1.43223e6	+	4.21785e6i	0.157128	+	0.462735i
$609$		0			0
$610$	651081.	+	9.93058e6i	0.0708452	+	1.08056i
$611$	−5.25717e6			−0.569704
$612$		0			0
$613$	−3.46298e6			−0.372220			−0.186110	−	0.982529i	$-0.559588\pi$
$613$	−3.46298e6			−0.372220			−0.186110	+	0.982529i	$0.559588\pi$
$614$	−459579.	−	7.00971e6i	−0.0491971	−	0.750376i
$615$		0			0
$616$	890737.	+	4.47665e6i	0.0945797	+	0.475337i
$617$		−	9.38415e6i		−	0.992389i	−0.868211	−	0.496195i	$-0.834730\pi$
$617$		−	9.38415e6i		−	0.992389i	0.868211	−	0.496195i	$-0.165270\pi$
$618$		0			0
$619$			1.62470e7i			1.70430i	0.523294	+	0.852152i	$0.324703\pi$
$619$			1.62470e7i			1.70430i	−0.523294	+	0.852152i	$0.675297\pi$
$620$	1.23587e7	−	1.62755e6i	1.29120	−	0.170042i
$621$		0			0
$622$	−1.56602e7	+	1.02673e6i	−1.62301	+	0.106410i
$623$	−8.66455e6			−0.894387
$624$		0			0
$625$	−2.13994e6			−0.219130
$626$	6.22324e6	−	408016.i	0.634718	−	0.0416142i
$627$		0			0
$628$	9.19780e6	−	1.21128e6i	0.930646	−	0.122559i
$629$			4.94585e6i			0.498442i
$630$		0			0
$631$		−	5.28301e6i		−	0.528212i	−0.964494	−	0.264106i	$-0.914923\pi$
$631$		−	5.28301e6i		−	0.528212i	0.964494	−	0.264106i	$-0.0850768\pi$
$632$	−1.17893e6	−	5.92507e6i	−0.117408	−	0.590066i
$633$		0			0
$634$	776760.	+	1.18475e7i	0.0767474	+	1.17059i
$635$	8.80053e6			0.866113
$636$		0			0
$637$	301500.			0.0294401
$638$	417235.	+	6.36386e6i	0.0405816	+	0.618969i
$639$		0			0
$640$	−3.19158e6	−	6.46963e6i	−0.308004	−	0.624352i
$641$		−	1.82753e7i		−	1.75679i	−0.477940	−	0.878393i	$-0.658616\pi$
$641$		−	1.82753e7i		−	1.75679i	0.477940	−	0.878393i	$-0.341384\pi$
$642$		0			0
$643$			1.43208e7i			1.36596i	0.730435	+	0.682982i	$0.239317\pi$
$643$			1.43208e7i			1.36596i	−0.730435	+	0.682982i	$0.760683\pi$
$644$	1.23470e6	+	9.37565e6i	0.117314	+	0.890814i
$645$		0			0
$646$	−4.97244e6	+	326010.i	−0.468801	+	0.0307361i
$647$	4.52939e6			0.425382			0.212691	−	0.977120i	$-0.431777\pi$
$647$	4.52939e6			0.425382			0.212691	+	0.977120i	$0.431777\pi$
$648$		0			0
$649$	−7.47467e6			−0.696595
$650$	3.78722e6	−	248303.i	0.351591	−	0.0230514i
$651$		0			0
$652$	1.60310e6	+	1.21730e7i	0.147686	+	1.12145i
$653$		−	5.33309e6i		−	0.489436i	−0.969594	−	0.244718i	$-0.921305\pi$
$653$		−	5.33309e6i		−	0.489436i	0.969594	−	0.244718i	$-0.0786954\pi$
$654$		0			0
$655$		−	2.04887e6i		−	0.186600i
$656$	−4.90755e6	−	1.83094e7i	−0.445251	−	1.66118i
$657$		0			0
$658$	618306.	+	9.43068e6i	0.0556722	+	0.849138i
$659$	1.52786e7			1.37048			0.685238	−	0.728319i	$-0.259698\pi$
$659$	1.52786e7			1.37048			0.685238	+	0.728319i	$0.259698\pi$
$660$		0			0
$661$	1.14110e7			1.01582			0.507912	−	0.861409i	$-0.330418\pi$
$661$	1.14110e7			1.01582			0.507912	+	0.861409i	$0.330418\pi$
$662$	−470488.	−	7.17610e6i	−0.0417257	−	0.636419i
$663$		0			0
$664$	−8.43820e6	+	1.67898e6i	−0.742728	+	0.147783i
$665$		−	3.96247e6i		−	0.347465i
$666$		0			0
$667$			1.32130e7i			1.14997i
$668$	−1.75341e7	+	2.30911e6i	−1.52034	+	0.200218i
$669$		0			0
$670$	5.42912e6	−	355951.i	0.467243	−	0.0306339i
$671$	−8.60870e6			−0.738128
$672$		0			0
$673$	−6.26029e6			−0.532791			−0.266395	−	0.963864i	$-0.585833\pi$
$673$	−6.26029e6			−0.532791			−0.266395	+	0.963864i	$0.585833\pi$
$674$	−7.87025e6	+	515999.i	−0.667327	+	0.0437521i
$675$		0			0
$676$	−6.27260e6	+	826055.i	−0.527936	+	0.0695252i
$677$		−	1.15594e7i		−	0.969311i	−0.874705	−	0.484655i	$-0.838945\pi$
$677$		−	1.15594e7i		−	0.969311i	0.874705	−	0.484655i	$-0.161055\pi$
$678$		0			0
$679$			3.86224e6i			0.321488i
$680$	7.91518e6	−	1.57491e6i	0.656430	−	0.130612i
$681$		0			0
$682$	705453.	+	1.07599e7i	0.0580774	+	0.885823i
$683$	3.36657e6			0.276145			0.138072	−	0.990422i	$-0.455909\pi$
$683$	3.36657e6			0.276145			0.138072	+	0.990422i	$0.455909\pi$
$684$		0			0
$685$	−1.12326e7			−0.914646
$686$	788097.	+	1.20204e7i	0.0639395	+	0.975234i
$687$		0			0
$688$	−1.47628e6	−	5.50783e6i	−0.118905	−	0.443618i
$689$			1.28772e7i			1.03341i
$690$		0			0
$691$			297628.i			0.0237126i	0.999930	+	0.0118563i	$0.00377406\pi$
$691$			297628.i			0.0237126i	−0.999930	+	0.0118563i	$0.996226\pi$
$692$	1.62432e6	+	1.23342e7i	0.128945	+	0.979139i
$693$		0			0
$694$	4.30372e6	−	282166.i	0.339192	−	0.0222385i
$695$	9.38705e6			0.737169
$696$		0			0
$697$	2.12058e7			1.65338
$698$	−2.00974e6	+	131765.i	−0.156135	+	0.0102367i
$699$		0			0
$700$	−890845.	−	6.76458e6i	−0.0687158	−	0.521790i
$701$			1.45408e7i			1.11762i	0.829297	+	0.558808i	$0.188741\pi$
$701$			1.45408e7i			1.11762i	−0.829297	+	0.558808i	$0.811259\pi$
$702$		0			0
$703$		−	3.32002e6i		−	0.253368i
$704$	5.76507e6	−	2.38877e6i	0.438402	−	0.181653i
$705$		0			0
$706$	−1.41689e6	−	2.16110e7i	−0.106985	−	1.63178i
$707$	−1.25152e7			−0.941651
$708$		0			0
$709$	1.52103e7			1.13638			0.568188	−	0.822899i	$-0.307645\pi$
$709$	1.52103e7			1.13638			0.568188	+	0.822899i	$0.307645\pi$
$710$	−270134.	−	4.12021e6i	−0.0201110	−	0.306742i
$711$		0			0
$712$	2.31173e6	+	1.16182e7i	0.170898	+	0.858896i
$713$			2.23403e7i			1.64576i
$714$		0			0
$715$		−	3.08792e6i		−	0.225892i
$716$	−9.52400e6	+	1.25424e6i	−0.694283	+	0.0914320i
$717$		0			0
$718$	−5.50959e6	+	361227.i	−0.398849	+	0.0261498i
$719$	−1.04620e7			−0.754729			−0.377364	−	0.926065i	$-0.623170\pi$
$719$	−1.04620e7			−0.754729			−0.377364	+	0.926065i	$0.623170\pi$
$720$		0			0
$721$	9.21726e6			0.660334
$722$	−1.06391e7	+	697531.i	−0.759556	+	0.0497990i
$723$		0			0
$724$	−6.27862e6	+	826847.i	−0.445161	+	0.0586245i
$725$		−	9.53327e6i		−	0.673592i
$726$		0			0
$727$		−	8.49478e6i		−	0.596096i	−0.954551	−	0.298048i	$-0.903664\pi$
$727$		−	8.49478e6i		−	0.596096i	0.954551	−	0.298048i	$-0.0963355\pi$
$728$	−1.94868e6	−	9.79366e6i	−0.136274	−	0.684883i
$729$		0			0
$730$	−1.17060e6	−	1.78545e7i	−0.0813019	−	1.24005i
$731$	6.37910e6			0.441536
$732$		0			0
$733$	−2.47868e7			−1.70397			−0.851984	−	0.523568i	$-0.824600\pi$
$733$	−2.47868e7			−1.70397			−0.851984	+	0.523568i	$0.824600\pi$
$734$	140015.	+	2.13557e6i	0.00959254	+	0.146310i
$735$		0			0
$736$	1.22423e7	−	4.15706e6i	0.833048	−	0.282873i
$737$			4.70644e6i			0.319171i
$738$		0			0
$739$		−	7.22886e6i		−	0.486921i	−0.969911	−	0.243460i	$-0.921717\pi$
$739$		−	7.22886e6i		−	0.486921i	0.969911	−	0.243460i	$-0.0782826\pi$
$740$	702035.	+	5.33086e6i	0.0471281	+	0.357864i
$741$		0			0
$742$	2.31000e7	−	1.51451e6i	1.54029	−	0.100986i
$743$	−1.87060e7			−1.24311			−0.621555	−	0.783370i	$-0.713499\pi$
$743$	−1.87060e7			−1.24311			−0.621555	+	0.783370i	$0.713499\pi$
$744$		0			0
$745$	−1.57546e7			−1.03996
$746$	1.28269e7	−	840972.i	0.843866	−	0.0553266i
$747$		0			0
$748$	911490.	+	6.92134e6i	0.0595659	+	0.452310i
$749$			2.31393e7i			1.50711i
$750$		0			0
$751$		−	2.63625e7i		−	1.70564i	−0.522206	−	0.852819i	$-0.674891\pi$
$751$		−	2.63625e7i		−	1.70564i	0.522206	−	0.852819i	$-0.325109\pi$
$752$	1.24806e7	−	3.34522e6i	0.804804	−	0.215715i
$753$		0			0
$754$	−912794.	−	1.39223e7i	−0.0584715	−	0.891834i
$755$	4.17131e6			0.266321
$756$		0			0
$757$	−1.33984e7			−0.849793			−0.424897	−	0.905242i	$-0.639690\pi$
$757$	−1.33984e7			−0.849793			−0.424897	+	0.905242i	$0.639690\pi$
$758$	−1.90190e6	−	2.90087e7i	−0.120231	−	1.83381i
$759$		0			0
$760$	−5.31325e6	+	1.05720e6i	−0.333677	+	0.0663930i
$761$			2.01529e7i			1.26147i	0.775998	+	0.630735i	$0.217247\pi$
$761$			2.01529e7i			1.26147i	−0.775998	+	0.630735i	$0.782753\pi$
$762$		0			0
$763$		−	1.19297e7i		−	0.741851i
$764$	2.89195e7	−	3.80849e6i	1.79249	−	0.236058i
$765$		0			0
$766$	−399269.	+	26177.4i	−0.0245863	+	0.00161196i
$767$	1.63525e7			1.00368
$768$		0			0
$769$	9.59359e6			0.585013			0.292506	−	0.956264i	$-0.405511\pi$
$769$	9.59359e6			0.585013			0.292506	+	0.956264i	$0.405511\pi$
$770$	−5.53933e6	+	363176.i	−0.336690	+	0.0220745i
$771$		0			0
$772$	−4.71381e6	+	620774.i	−0.284662	+	0.0374878i
$773$			6.90703e6i			0.415760i	0.978154	+	0.207880i	$0.0666564\pi$
$773$			6.90703e6i			0.415760i	−0.978154	+	0.207880i	$0.933344\pi$
$774$		0			0
$775$		−	1.61186e7i		−	0.963994i
$776$	5.17885e6	−	1.03046e6i	0.308730	−	0.0614292i
$777$		0			0
$778$	−552093.	−	8.42076e6i	−0.0327011	−	0.498772i
$779$	−1.42349e7			−0.840446
$780$		0			0
$781$	3.57176e6			0.209534
$782$	946245.	+	1.44326e7i	0.0553333	+	0.843969i
$783$		0			0
$784$	−715765.	+	191849.i	−0.0415892	+	0.0111473i
$785$			1.12829e7i			0.653502i
$786$		0			0
$787$			2.50408e7i			1.44116i	0.693373	+	0.720579i	$0.256124\pi$
$787$			2.50408e7i			1.44116i	−0.693373	+	0.720579i	$0.743876\pi$
$788$	−2.24726e6	−	1.70644e7i	−0.128925	−	0.978984i
$789$		0			0
$790$	7.33156e6	−	480681.i	0.417954	−	0.0274024i
$791$	−7.05969e6			−0.401185
$792$		0			0
$793$	1.88334e7			1.06352
$794$	−1.64425e7	+	1.07803e6i	−0.925588	+	0.0606845i
$795$		0			0
$796$	−833748.	−	6.33102e6i	−0.0466393	−	0.354153i
$797$			5.81142e6i			0.324068i	0.986785	+	0.162034i	$0.0518055\pi$
$797$			5.81142e6i			0.324068i	−0.986785	+	0.162034i	$0.948195\pi$
$798$		0			0
$799$			1.44549e7i			0.801026i
$800$	−8.83290e6	+	2.99934e6i	−0.487954	+	0.165692i
$801$		0			0
$802$	−439963.	−	6.71051e6i	−0.0241535	−	0.368400i
$803$	1.54778e7			0.847075
$804$		0			0
$805$	−1.15011e7			−0.625532
$806$	−1.54333e6	−	2.35396e7i	−0.0836801	−	1.27633i
$807$		0			0
$808$	3.33910e6	+	1.67816e7i	0.179929	+	0.904284i
$809$		−	7.88107e6i		−	0.423364i	−0.977339	−	0.211682i	$-0.932106\pi$
$809$		−	7.88107e6i		−	0.423364i	0.977339	−	0.211682i	$-0.0678941\pi$
$810$		0			0
$811$		−	2.02506e7i		−	1.08115i	−0.841296	−	0.540575i	$-0.818207\pi$
$811$		−	2.02506e7i		−	1.08115i	0.841296	−	0.540575i	$-0.181793\pi$
$812$	−2.48675e7	+	3.27487e6i	−1.32355	+	0.174302i
$813$		0			0
$814$	−4.64121e6	+	304293.i	−0.245511	+	0.0160965i
$815$	−1.49326e7			−0.787484
$816$		0			0
$817$	−4.28212e6			−0.224442
$818$	1.87279e7	−	1.22787e6i	0.978604	−	0.0641605i
$819$		0			0
$820$	2.28565e7	−	3.01004e6i	1.18707	−	0.156328i
$821$			1.09505e7i			0.566992i	0.958973	+	0.283496i	$0.0914943\pi$
$821$			1.09505e7i			0.566992i	−0.958973	+	0.283496i	$0.908506\pi$
$822$		0			0
$823$			1.49437e7i			0.769060i	0.923113	+	0.384530i	$0.125636\pi$
$823$			1.49437e7i			0.769060i	−0.923113	+	0.384530i	$0.874364\pi$
$824$	−2.45919e6	−	1.23594e7i	−0.126175	−	0.634130i
$825$		0			0
$826$	−1.92325e6	−	2.93342e7i	−0.0980810	−	1.49597i
$827$	9.00457e6			0.457825			0.228912	−	0.973447i	$-0.426483\pi$
$827$	9.00457e6			0.457825			0.228912	+	0.973447i	$0.426483\pi$
$828$		0			0
$829$	2.68460e7			1.35673			0.678365	−	0.734725i	$-0.262689\pi$
$829$	2.68460e7			1.35673			0.678365	+	0.734725i	$0.262689\pi$
$830$	−684564.	−	1.04413e7i	−0.0344920	−	0.526088i
$831$		0			0
$832$	−1.26124e7	+	5.22595e6i	−0.631666	+	0.261732i
$833$		−	828990.i		−	0.0413939i
$834$		0			0
$835$		−	2.15090e7i		−	1.06759i
$836$	−611859.	−	4.64611e6i	−0.0302786	−	0.229919i
$837$		0			0
$838$	−2.39511e7	+	1.57031e6i	−1.17819	+	0.0772461i
$839$	1.52985e7			0.750316			0.375158	−	0.926961i	$-0.377588\pi$
$839$	1.52985e7			0.750316			0.375158	+	0.926961i	$0.377588\pi$
$840$		0			0
$841$	−1.45344e7			−0.708611
$842$	2.09156e7	−	1.37129e6i	1.01669	−	0.0666576i
$843$		0			0
$844$	3.25123e6	+	2.46880e7i	0.157106	+	1.19297i
$845$		−	7.69458e6i		−	0.370718i
$846$		0			0
$847$			1.65217e7i			0.791310i
$848$	−8.19394e6	−	3.05706e7i	−0.391294	−	1.45987i
$849$		0			0
$850$	−682720.	−	1.04132e7i	−0.0324112	−	0.494350i
$851$	−9.63638e6			−0.456131
$852$		0			0
$853$	1.45321e7			0.683841			0.341920	−	0.939729i	$-0.388923\pi$
$853$	1.45321e7			0.683841			0.341920	+	0.939729i	$0.388923\pi$
$854$	−2.21503e6	−	3.37847e7i	−0.103929	−	1.58517i
$855$		0			0
$856$	3.10274e7	−	6.17364e6i	1.44731	−	0.287976i
$857$		−	7.09825e6i		−	0.330141i	−0.986282	−	0.165070i	$-0.947215\pi$
$857$		−	7.09825e6i		−	0.330141i	0.986282	−	0.165070i	$-0.0527851\pi$
$858$		0			0
$859$		−	2.43203e7i		−	1.12457i	−0.826944	−	0.562284i	$-0.809923\pi$
$859$		−	2.43203e7i		−	1.12457i	0.826944	−	0.562284i	$-0.190077\pi$
$860$	6.87568e6	−	905476.i	0.317007	−	0.0417475i
$861$		0			0
$862$	−2.46541e7	+	1.61640e6i	−1.13011	+	0.0740936i
$863$	1.22346e7			0.559194			0.279597	−	0.960117i	$-0.409799\pi$
$863$	1.22346e7			0.559194			0.279597	+	0.960117i	$0.409799\pi$
$864$		0			0
$865$	−1.51303e7			−0.687554
$866$	−1.17570e7	+	770824.i	−0.532721	+	0.0349269i
$867$		0			0
$868$	−4.20455e7	+	5.53707e6i	−1.89417	+	0.249449i
$869$			6.35565e6i			0.285503i
$870$		0			0
$871$		−	1.02964e7i		−	0.459874i
$872$	−1.59964e7	+	3.18287e6i	−0.712413	+	0.141752i
$873$		0			0
$874$	−635189.	−	9.68819e6i	−0.0281271	−	0.429007i
$875$	2.44009e7			1.07742
$876$		0			0
$877$	6.34255e6			0.278461			0.139231	−	0.990260i	$-0.455537\pi$
$877$	6.34255e6			0.278461			0.139231	+	0.990260i	$0.455537\pi$
$878$	963244.	+	1.46918e7i	0.0421697	+	0.643190i
$879$		0			0
$880$	1.96489e6	+	7.33075e6i	0.0855325	+	0.319111i
$881$			2.37666e7i			1.03164i	0.856697	+	0.515820i	$0.172513\pi$
$881$			2.37666e7i			1.03164i	−0.856697	+	0.515820i	$0.827487\pi$
$882$		0			0
$883$			4.03790e7i			1.74282i	0.490552	+	0.871412i	$0.336795\pi$
$883$			4.03790e7i			1.74282i	−0.490552	+	0.871412i	$0.663205\pi$
$884$	−1.99408e6	−	1.51419e7i	−0.0858247	−	0.651705i
$885$		0			0
$886$	2.78286e7	−	1.82453e6i	1.19099	−	0.0780850i
$887$	−9.71146e6			−0.414453			−0.207227	−	0.978293i	$-0.566444\pi$
$887$	−9.71146e6			−0.414453			−0.207227	+	0.978293i	$0.566444\pi$
$888$		0			0
$889$	−2.99402e7			−1.27057
$890$	−1.43762e7	+	942550.i	−0.608371	+	0.0398868i
$891$		0			0
$892$	2.84789e6	+	2.16253e7i	0.119843	+	0.910017i
$893$		−	9.70316e6i		−	0.407178i
$894$		0			0
$895$		−	1.16831e7i		−	0.487527i
$896$	1.08580e7	+	2.20103e7i	0.451837	+	0.915915i
$897$		0			0
$898$	−1.26620e6	−	1.93126e7i	−0.0523975	−	0.799190i
$899$	−5.92543e7			−2.44524
$900$		0			0
$901$	3.54064e7			1.45302
$902$	1.30468e6	+	1.98996e7i	0.0533936	+	0.814382i
$903$		0			0
$904$	1.88355e6	+	9.46631e6i	0.0766576	+	0.385265i
$905$		−	7.70196e6i		−	0.312593i
$906$		0			0
$907$			3.22603e7i			1.30212i	0.759027	+	0.651060i	$0.225675\pi$
$907$			3.22603e7i			1.30212i	−0.759027	+	0.651060i	$0.774325\pi$
$908$	4.08219e7	−	5.37594e6i	1.64315	−	0.216391i
$909$		0			0
$910$	1.21185e7	−	794527.i	0.485115	−	0.0318057i
$911$	3.72296e7			1.48625			0.743126	−	0.669151i	$-0.233342\pi$
$911$	3.72296e7			1.48625			0.743126	+	0.669151i	$0.233342\pi$
$912$		0			0
$913$	9.05142e6			0.359368
$914$	4.44711e7	−	2.91567e6i	1.76081	−	0.115444i
$915$		0			0
$916$	−2.36859e6	+	311926.i	−0.0932720	+	0.0122832i
$917$			6.97043e6i			0.273739i
$918$		0			0
$919$		−	1.63109e7i		−	0.637073i	−0.947911	−	0.318537i	$-0.896809\pi$
$919$		−	1.63109e7i		−	0.637073i	0.947911	−	0.318537i	$-0.103191\pi$
$920$	3.06852e6	+	1.54217e7i	0.119525	+	0.600709i
$921$		0			0
$922$	1.19995e6	+	1.83021e7i	0.0464874	+	0.709046i
$923$	−7.81402e6			−0.301905
$924$		0			0
$925$	6.95269e6			0.267177
$926$	1.46422e6	+	2.23329e7i	0.0561150	+	0.855890i
$927$		0			0
$928$	1.10260e7	+	3.24709e7i	0.420288	+	1.23773i
$929$		−	1.30936e7i		−	0.497760i	−0.968534	−	0.248880i	$-0.919938\pi$
$929$		−	1.30936e7i		−	0.497760i	0.968534	−	0.248880i	$-0.0800624\pi$
$930$		0			0
$931$			556479.i			0.0210414i
$932$	2.66756e6	+	2.02560e7i	0.100595	+	0.763859i
$933$		0			0
$934$	−2.14816e7	+	1.40840e6i	−0.805748	+	0.0528275i
$935$	−8.49039e6			−0.317613
$936$		0			0
$937$	1.74601e7			0.649678			0.324839	−	0.945769i	$-0.394690\pi$
$937$	1.74601e7			0.649678			0.324839	+	0.945769i	$0.394690\pi$
$938$	−1.84703e7	+	1.21098e6i	−0.685438	+	0.0449395i
$939$		0			0
$940$	2.05178e6	+	1.55801e7i	0.0757377	+	0.575109i
$941$			1.83646e7i			0.676094i	0.941129	+	0.338047i	$0.109766\pi$
$941$			1.83646e7i			0.676094i	−0.941129	+	0.338047i	$0.890234\pi$
$942$		0			0
$943$			4.13168e7i			1.51303i
$944$	−3.88210e7	+	1.04053e7i	−1.41787	+	0.380037i
$945$		0			0
$946$	392474.	+	5.98618e6i	0.0142588	+	0.217481i
$947$	−4.05438e7			−1.46909			−0.734547	−	0.678558i	$-0.762605\pi$
$947$	−4.05438e7			−1.46909			−0.734547	+	0.678558i	$0.762605\pi$
$948$		0			0
$949$	−3.38612e7			−1.22050
$950$	458292.	+	6.99007e6i	0.0164753	+	0.251288i
$951$		0			0
$952$	−2.69281e7	+	5.35800e6i	−0.962973	+	0.191606i
$953$		−	1.80656e7i		−	0.644347i	−0.946681	−	0.322174i	$-0.895587\pi$
$953$		−	1.80656e7i		−	0.644347i	0.946681	−	0.322174i	$-0.104413\pi$
$954$		0			0
$955$			3.54755e7i			1.25869i
$956$	−4.55028e7	+	5.99238e6i	−1.61025	+	0.212058i
$957$		0			0
$958$	3.81617e7	−	2.50201e6i	1.34343	−	0.0880795i
$959$	3.82142e7			1.34177
$960$		0			0
$961$	−7.15569e7			−2.49944
$962$	1.01537e7	−	665708.i	0.353741	−	0.0231924i
$963$		0			0
$964$	3.13968e7	−	4.13473e6i	1.08816	−	0.143303i
$965$		−	5.78242e6i		−	0.199890i
$966$		0			0
$967$			3.63809e7i			1.25114i	0.780167	+	0.625571i	$0.215134\pi$
$967$			3.63809e7i			1.25114i	−0.780167	+	0.625571i	$0.784866\pi$
$968$	2.21539e7	−	4.40804e6i	0.759909	−	0.151202i
$969$		0			0
$970$	420143.	+	6.40821e6i	0.0143373	+	0.218679i
$971$	8.19775e6			0.279027			0.139514	−	0.990220i	$-0.455446\pi$
$971$	8.19775e6			0.279027			0.139514	+	0.990220i	$0.455446\pi$
$972$		0			0
$973$	−3.19356e7			−1.08142
$974$	1.45852e6	+	2.22459e7i	0.0492622	+	0.751369i
$975$		0			0
$976$	−4.47107e7	+	1.19840e7i	−1.50241	+	0.402695i
$977$		−	3.29491e7i		−	1.10435i	−0.833728	−	0.552176i	$-0.813798\pi$
$977$		−	3.29491e7i		−	1.10435i	0.833728	−	0.552176i	$-0.186202\pi$
$978$		0			0
$979$		−	1.24626e7i		−	0.415576i
$980$	−117670.	−	893523.i	−0.00391383	−	0.0297194i
$981$		0			0
$982$	1.34521e7	−	881964.i	0.445155	−	0.0291858i
$983$	−2.15949e6			−0.0712798			−0.0356399	−	0.999365i	$-0.511347\pi$
$983$	−2.15949e6			−0.0712798			−0.0356399	+	0.999365i	$0.511347\pi$
$984$		0			0
$985$	2.09329e7			0.687445
$986$	−3.82802e7	+	2.50977e6i	−1.25395	+	0.0822133i
$987$		0			0
$988$	1.33857e6	+	1.01644e7i	0.0436265	+	0.331275i
$989$			1.24289e7i			0.404056i
$990$		0			0
$991$			7.72802e6i			0.249968i	0.992159	+	0.124984i	$0.0398879\pi$
$991$			7.72802e6i			0.249968i	−0.992159	+	0.124984i	$0.960112\pi$
$992$	1.86425e7	+	5.49012e7i	0.601485	+	1.77134i
$993$		0			0
$994$	919021.	+	1.40173e7i	0.0295025	+	0.449986i
$995$	7.76624e6			0.248687
$996$		0			0
$997$	7.86897e6			0.250715			0.125357	−	0.992112i	$-0.459992\pi$
$997$	7.86897e6			0.250715			0.125357	+	0.992112i	$0.459992\pi$
$998$	−2.21254e6	−	3.37466e7i	−0.0703177	−	1.07252i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.b.c.107.2	yes	20
3.2	odd	2	inner	108.6.b.c.107.19	yes	20
4.3	odd	2	inner	108.6.b.c.107.20	yes	20
12.11	even	2	inner	108.6.b.c.107.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.b.c.107.1	✓	20	12.11	even	2	inner
108.6.b.c.107.2	yes	20	1.1	even	1	trivial
108.6.b.c.107.19	yes	20	3.2	odd	2	inner
108.6.b.c.107.20	yes	20	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	108.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.64474 + 0.370087i) q^{2} +(31.7261 - 4.17809i) q^{4} +38.9183i q^{5} -132.403i q^{7} +(-177.539 + 35.3256i) q^{8} +O(q^{10})\)	\(q+(-5.64474 + 0.370087i) q^{2} +(31.7261 - 4.17809i) q^{4} +38.9183i q^{5} -132.403i q^{7} +(-177.539 + 35.3256i) q^{8} +(-14.4032 - 219.683i) q^{10} +190.441 q^{11} -416.631 q^{13} +(49.0008 + 747.382i) q^{14} +(989.087 - 265.109i) q^{16} +1145.55i q^{17} -768.976i q^{19} +(162.604 + 1234.72i) q^{20} +(-1074.99 + 70.4798i) q^{22} -2231.96 q^{23} +1610.37 q^{25} +(2351.77 - 154.190i) q^{26} +(-553.193 - 4200.64i) q^{28} -5919.93i q^{29} -10009.3i q^{31} +(-5485.02 + 1862.52i) q^{32} +(-423.953 - 6466.32i) q^{34} +5152.91 q^{35} +4317.45 q^{37} +(284.588 + 4340.67i) q^{38} +(-1374.81 - 6909.51i) q^{40} -18511.5i q^{41} -5568.60i q^{43} +(6041.95 - 795.680i) q^{44} +(12598.8 - 826.019i) q^{46} +12618.3 q^{47} -723.662 q^{49} +(-9090.10 + 595.977i) q^{50} +(-13218.1 + 1740.72i) q^{52} -30907.9i q^{53} +7411.64i q^{55} +(4677.23 + 23506.8i) q^{56} +(2190.89 + 33416.4i) q^{58} -39249.3 q^{59} -45204.0 q^{61} +(3704.31 + 56499.8i) q^{62} +(30272.2 - 12543.4i) q^{64} -16214.6i q^{65} +24713.4i q^{67} +(4786.20 + 36343.8i) q^{68} +(-29086.8 + 1907.03i) q^{70} +18755.2 q^{71} +81273.7 q^{73} +(-24370.9 + 1597.83i) q^{74} +(-3212.85 - 24396.6i) q^{76} -25215.0i q^{77} +33373.3i q^{79} +(10317.6 + 38493.6i) q^{80} +(6850.85 + 104492. i) q^{82} +47528.7 q^{83} -44582.8 q^{85} +(2060.87 + 31433.3i) q^{86} +(-33810.7 + 6727.45i) q^{88} -65440.5i q^{89} +55163.4i q^{91} +(-70811.3 + 9325.32i) q^{92} +(-71226.9 + 4669.86i) q^{94} +29927.2 q^{95} -29170.2 q^{97} +(4084.88 - 267.818i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{4}+O(q^{10}) \) 20 * q + 20 * q^4	\( 20 q + 20 q^{4} + 184 q^{10} - 116 q^{13} - 4168 q^{16} + 696 q^{22} - 15228 q^{25} - 4764 q^{28} - 16520 q^{34} - 6452 q^{37} + 1504 q^{40} - 9336 q^{46} - 44464 q^{49} + 8236 q^{52} - 58736 q^{58} + 84604 q^{61} - 6496 q^{64} + 138696 q^{70} + 85420 q^{73} + 89172 q^{76} + 221200 q^{82} + 180320 q^{85} - 85824 q^{88} - 60936 q^{94} - 219908 q^{97}+O(q^{100}) \) 20 * q + 20 * q^4 + 184 * q^10 - 116 * q^13 - 4168 * q^16 + 696 * q^22 - 15228 * q^25 - 4764 * q^28 - 16520 * q^34 - 6452 * q^37 + 1504 * q^40 - 9336 * q^46 - 44464 * q^49 + 8236 * q^52 - 58736 * q^58 + 84604 * q^61 - 6496 * q^64 + 138696 * q^70 + 85420 * q^73 + 89172 * q^76 + 221200 * q^82 + 180320 * q^85 - 85824 * q^88 - 60936 * q^94 - 219908 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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