LMFDB - Embedded newform 690.2.e.b.551.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,2,Mod(551,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("690.551");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	690.e (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:	$5.50967773947$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 12 x^{13} + 15 x^{12} - 4 x^{11} + 45 x^{10} - 66 x^{9} - 32 x^{8} + \cdots + 6561 $ x^16 - 2x^15 + 3x^14 - 12x^13 + 15x^12 - 4x^11 + 45x^10 - 66x^9 - 32x^8 - 198x^7 + 405x^6 - 108x^5 + 1215x^4 - 2916x^3 + 2187x^2 - 4374*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			551.9
Root			$-0.251873 - 1.71364i$ of defining polynomial
Character	$\chi$	$=$	690.551
Dual form			690.2.e.b.551.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +(-1.71364 + 0.251873i) q^{3} -1.00000 q^{4} +1.00000 q^{5} +(-0.251873 - 1.71364i) q^{6} -0.194278i q^{7} -1.00000i q^{8} +(2.87312 - 0.863238i) q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +(-1.71364 + 0.251873i) q^{3} -1.00000 q^{4} +1.00000 q^{5} +(-0.251873 - 1.71364i) q^{6} -0.194278i q^{7} -1.00000i q^{8} +(2.87312 - 0.863238i) q^{9} +1.00000i q^{10} -3.67639 q^{11} +(1.71364 - 0.251873i) q^{12} +3.85351 q^{13} +0.194278 q^{14} +(-1.71364 + 0.251873i) q^{15} +1.00000 q^{16} +3.71391 q^{17} +(0.863238 + 2.87312i) q^{18} -0.273524i q^{19} -1.00000 q^{20} +(0.0489334 + 0.332923i) q^{21} -3.67639i q^{22} +(4.59371 + 1.37760i) q^{23} +(0.251873 + 1.71364i) q^{24} +1.00000 q^{25} +3.85351i q^{26} +(-4.70607 + 2.20294i) q^{27} +0.194278i q^{28} +9.11317i q^{29} +(-0.251873 - 1.71364i) q^{30} +2.24052 q^{31} +1.00000i q^{32} +(6.30001 - 0.925983i) q^{33} +3.71391i q^{34} -0.194278i q^{35} +(-2.87312 + 0.863238i) q^{36} -11.7876i q^{37} +0.273524 q^{38} +(-6.60352 + 0.970593i) q^{39} -1.00000i q^{40} +9.65442i q^{41} +(-0.332923 + 0.0489334i) q^{42} +10.4535i q^{43} +3.67639 q^{44} +(2.87312 - 0.863238i) q^{45} +(-1.37760 + 4.59371i) q^{46} +4.64640i q^{47} +(-1.71364 + 0.251873i) q^{48} +6.96226 q^{49} +1.00000i q^{50} +(-6.36430 + 0.935432i) q^{51} -3.85351 q^{52} -4.09935 q^{53} +(-2.20294 - 4.70607i) q^{54} -3.67639 q^{55} -0.194278 q^{56} +(0.0688932 + 0.468721i) q^{57} -9.11317 q^{58} -2.49823i q^{59} +(1.71364 - 0.251873i) q^{60} +13.9867i q^{61} +2.24052i q^{62} +(-0.167708 - 0.558185i) q^{63} -1.00000 q^{64} +3.85351 q^{65} +(0.925983 + 6.30001i) q^{66} -10.9816i q^{67} -3.71391 q^{68} +(-8.21895 - 1.20368i) q^{69} +0.194278 q^{70} +9.72869i q^{71} +(-0.863238 - 2.87312i) q^{72} +11.0817 q^{73} +11.7876 q^{74} +(-1.71364 + 0.251873i) q^{75} +0.273524i q^{76} +0.714243i q^{77} +(-0.970593 - 6.60352i) q^{78} -6.43875i q^{79} +1.00000 q^{80} +(7.50964 - 4.96037i) q^{81} -9.65442 q^{82} +9.42725 q^{83} +(-0.0489334 - 0.332923i) q^{84} +3.71391 q^{85} -10.4535 q^{86} +(-2.29536 - 15.6167i) q^{87} +3.67639i q^{88} -9.92316 q^{89} +(0.863238 + 2.87312i) q^{90} -0.748652i q^{91} +(-4.59371 - 1.37760i) q^{92} +(-3.83945 + 0.564326i) q^{93} -4.64640 q^{94} -0.273524i q^{95} +(-0.251873 - 1.71364i) q^{96} -8.91482i q^{97} +6.96226i q^{98} +(-10.5627 + 3.17360i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} + 16 q^{5} + 2 q^{6} + 2 q^{9}+O(q^{10}) $ 16 * q - 16 * q^4 + 16 * q^5 + 2 * q^6 + 2 * q^9	$ 16 q - 16 q^{4} + 16 q^{5} + 2 q^{6} + 2 q^{9} + 12 q^{11} + 12 q^{14} + 16 q^{16} + 8 q^{18} - 16 q^{20} + 4 q^{21} - 4 q^{23} - 2 q^{24} + 16 q^{25} + 24 q^{27} + 2 q^{30} + 4 q^{31} + 28 q^{33} - 2 q^{36} + 16 q^{38} - 8 q^{39} - 12 q^{44} + 2 q^{45} - 4 q^{46} - 4 q^{49} + 2 q^{51} + 8 q^{53} - 26 q^{54} + 12 q^{55} - 12 q^{56} - 28 q^{57} - 8 q^{58} - 16 q^{64} - 10 q^{66} - 22 q^{69} + 12 q^{70} - 8 q^{72} - 16 q^{73} + 24 q^{74} - 12 q^{78} + 16 q^{80} + 22 q^{81} - 16 q^{82} + 40 q^{83} - 4 q^{84} + 40 q^{86} + 20 q^{87} - 80 q^{89} + 8 q^{90} + 4 q^{92} - 4 q^{93} - 24 q^{94} + 2 q^{96} + 8 q^{99}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^5 + 2 * q^6 + 2 * q^9 + 12 * q^11 + 12 * q^14 + 16 * q^16 + 8 * q^18 - 16 * q^20 + 4 * q^21 - 4 * q^23 - 2 * q^24 + 16 * q^25 + 24 * q^27 + 2 * q^30 + 4 * q^31 + 28 * q^33 - 2 * q^36 + 16 * q^38 - 8 * q^39 - 12 * q^44 + 2 * q^45 - 4 * q^46 - 4 * q^49 + 2 * q^51 + 8 * q^53 - 26 * q^54 + 12 * q^55 - 12 * q^56 - 28 * q^57 - 8 * q^58 - 16 * q^64 - 10 * q^66 - 22 * q^69 + 12 * q^70 - 8 * q^72 - 16 * q^73 + 24 * q^74 - 12 * q^78 + 16 * q^80 + 22 * q^81 - 16 * q^82 + 40 * q^83 - 4 * q^84 + 40 * q^86 + 20 * q^87 - 80 * q^89 + 8 * q^90 + 4 * q^92 - 4 * q^93 - 24 * q^94 + 2 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$511$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	−1.71364	+	0.251873i	−0.989370	+	0.145419i
$3$	−1.71364	+	0.251873i	−0.989370	+	0.145419i
$4$	−1.00000			−0.500000
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$	−0.251873	−	1.71364i	−0.102827	−	0.699590i
$7$		−	0.194278i		−	0.0734302i	−0.999326	−	0.0367151i	$-0.988311\pi$
$7$		−	0.194278i		−	0.0734302i	0.999326	−	0.0367151i	$-0.0116894\pi$
$8$		−	1.00000i		−	0.353553i
$9$	2.87312	−	0.863238i	0.957707	−	0.287746i
$10$			1.00000i			0.316228i
$11$	−3.67639			−1.10847			−0.554237	−	0.832359i	$-0.686990\pi$
$11$	−3.67639			−1.10847			−0.554237	+	0.832359i	$0.686990\pi$
$12$	1.71364	−	0.251873i	0.494685	−	0.0727094i
$13$	3.85351			1.06877			0.534385	−	0.845241i	$-0.320543\pi$
$13$	3.85351			1.06877			0.534385	+	0.845241i	$0.320543\pi$
$14$	0.194278			0.0519230
$15$	−1.71364	+	0.251873i	−0.442460	+	0.0650333i
$16$	1.00000			0.250000
$17$	3.71391			0.900754			0.450377	−	0.892838i	$-0.351290\pi$
$17$	3.71391			0.900754			0.450377	+	0.892838i	$0.351290\pi$
$18$	0.863238	+	2.87312i	0.203467	+	0.677201i
$19$		−	0.273524i		−	0.0627507i	−0.999508	−	0.0313753i	$-0.990011\pi$
$19$		−	0.273524i		−	0.0627507i	0.999508	−	0.0313753i	$-0.00998872\pi$
$20$	−1.00000			−0.223607
$21$	0.0489334	+	0.332923i	0.0106781	+	0.0726497i
$22$		−	3.67639i		−	0.783810i
$23$	4.59371	+	1.37760i	0.957856	+	0.287250i
$23$	4.59371	+	1.37760i	0.957856	+	0.287250i
$24$	0.251873	+	1.71364i	0.0514133	+	0.349795i
$25$	1.00000			0.200000
$26$			3.85351i			0.755735i
$27$	−4.70607	+	2.20294i	−0.905683	+	0.423956i
$28$			0.194278i			0.0367151i
$29$			9.11317i			1.69227i	0.532967	+	0.846136i	$0.321077\pi$
$29$			9.11317i			1.69227i	−0.532967	+	0.846136i	$0.678923\pi$
$30$	−0.251873	−	1.71364i	−0.0459855	−	0.312866i
$31$	2.24052			0.402410			0.201205	−	0.979549i	$-0.435514\pi$
$31$	2.24052			0.402410			0.201205	+	0.979549i	$0.435514\pi$
$32$			1.00000i			0.176777i
$33$	6.30001	−	0.925983i	1.09669	−	0.161193i
$34$			3.71391i			0.636930i
$35$		−	0.194278i		−	0.0328390i
$36$	−2.87312	+	0.863238i	−0.478853	+	0.143873i
$37$		−	11.7876i		−	1.93787i	−0.247321	−	0.968934i	$-0.579550\pi$
$37$		−	11.7876i		−	1.93787i	0.247321	−	0.968934i	$-0.420450\pi$
$38$	0.273524			0.0443714
$39$	−6.60352	+	0.970593i	−1.05741	+	0.155419i
$40$		−	1.00000i		−	0.158114i
$41$			9.65442i			1.50777i	0.657008	+	0.753884i	$0.271822\pi$
$41$			9.65442i			1.50777i	−0.657008	+	0.753884i	$0.728178\pi$
$42$	−0.332923	+	0.0489334i	−0.0513711	+	0.00755058i
$43$			10.4535i			1.59415i	0.603882	+	0.797074i	$0.293620\pi$
$43$			10.4535i			1.59415i	−0.603882	+	0.797074i	$0.706380\pi$
$44$	3.67639			0.554237
$45$	2.87312	−	0.863238i	0.428299	−	0.128684i
$46$	−1.37760	+	4.59371i	−0.203116	+	0.677306i
$47$			4.64640i			0.677747i	0.940832	+	0.338874i	$0.110046\pi$
$47$			4.64640i			0.677747i	−0.940832	+	0.338874i	$0.889954\pi$
$48$	−1.71364	+	0.251873i	−0.247343	+	0.0363547i
$49$	6.96226			0.994608
$50$			1.00000i			0.141421i
$51$	−6.36430	+	0.935432i	−0.891180	+	0.130987i
$52$	−3.85351			−0.534385
$53$	−4.09935			−0.563089			−0.281545	−	0.959548i	$-0.590847\pi$
$53$	−4.09935			−0.563089			−0.281545	+	0.959548i	$0.590847\pi$
$54$	−2.20294	−	4.70607i	−0.299782	−	0.640414i
$55$	−3.67639			−0.495725
$56$	−0.194278			−0.0259615
$57$	0.0688932	+	0.468721i	0.00912513	+	0.0620837i
$58$	−9.11317			−1.19662
$59$		−	2.49823i		−	0.325242i	−0.986689	−	0.162621i	$-0.948005\pi$
$59$		−	2.49823i		−	0.325242i	0.986689	−	0.162621i	$-0.0519947\pi$
$60$	1.71364	−	0.251873i	0.221230	−	0.0325166i
$61$			13.9867i			1.79082i	0.445245	+	0.895409i	$0.353117\pi$
$61$			13.9867i			1.79082i	−0.445245	+	0.895409i	$0.646883\pi$
$62$			2.24052i			0.284547i
$63$	−0.167708	−	0.558185i	−0.0211293	−	0.0703246i
$64$	−1.00000			−0.125000
$65$	3.85351			0.477968
$66$	0.925983	+	6.30001i	0.113981	+	0.775478i
$67$		−	10.9816i		−	1.34161i	−0.741632	−	0.670807i	$-0.765948\pi$
$67$		−	10.9816i		−	1.34161i	0.741632	−	0.670807i	$-0.234052\pi$
$68$	−3.71391			−0.450377
$69$	−8.21895	−	1.20368i	−0.989445	−	0.144906i
$70$	0.194278			0.0232207
$71$			9.72869i			1.15458i	0.816538	+	0.577291i	$0.195890\pi$
$71$			9.72869i			1.15458i	−0.816538	+	0.577291i	$0.804110\pi$
$72$	−0.863238	−	2.87312i	−0.101734	−	0.338600i
$73$	11.0817			1.29702			0.648508	−	0.761208i	$-0.275393\pi$
$73$	11.0817			1.29702			0.648508	+	0.761208i	$0.275393\pi$
$74$	11.7876			1.37028
$75$	−1.71364	+	0.251873i	−0.197874	+	0.0290838i
$76$			0.273524i			0.0313753i
$77$			0.714243i			0.0813955i
$78$	−0.970593	−	6.60352i	−0.109898	−	0.747701i
$79$		−	6.43875i		−	0.724416i	−0.932097	−	0.362208i	$-0.882023\pi$
$79$		−	6.43875i		−	0.724416i	0.932097	−	0.362208i	$-0.117977\pi$
$80$	1.00000			0.111803
$81$	7.50964	−	4.96037i	0.834404	−	0.551153i
$82$	−9.65442			−1.06615
$83$	9.42725			1.03477			0.517387	−	0.855751i	$-0.326905\pi$
$83$	9.42725			1.03477			0.517387	+	0.855751i	$0.326905\pi$
$84$	−0.0489334	−	0.332923i	−0.00533907	−	0.0363248i
$85$	3.71391			0.402830
$86$	−10.4535			−1.12723
$87$	−2.29536	−	15.6167i	−0.246088	−	1.67428i
$88$			3.67639i			0.391905i
$89$	−9.92316			−1.05185			−0.525926	−	0.850530i	$-0.676281\pi$
$89$	−9.92316			−1.05185			−0.525926	+	0.850530i	$0.676281\pi$
$90$	0.863238	+	2.87312i	0.0909933	+	0.302853i
$91$		−	0.748652i		−	0.0784800i
$92$	−4.59371	−	1.37760i	−0.478928	−	0.143625i
$93$	−3.83945	+	0.564326i	−0.398132	+	0.0585179i
$94$	−4.64640			−0.479240
$95$		−	0.273524i		−	0.0280630i
$96$	−0.251873	−	1.71364i	−0.0257067	−	0.174898i
$97$		−	8.91482i		−	0.905163i	−0.891723	−	0.452582i	$-0.850503\pi$
$97$		−	8.91482i		−	0.905163i	0.891723	−	0.452582i	$-0.149497\pi$
$98$			6.96226i			0.703294i
$99$	−10.5627	+	3.17360i	−1.06159	+	0.318959i
$100$	−1.00000			−0.100000
$101$		−	8.39174i		−	0.835009i	−0.908675	−	0.417505i	$-0.862905\pi$
$101$		−	8.39174i		−	0.835009i	0.908675	−	0.417505i	$-0.137095\pi$
$102$	−0.935432	−	6.36430i	−0.0926215	−	0.630159i
$103$		−	3.35240i		−	0.330322i	−0.986267	−	0.165161i	$-0.947186\pi$
$103$		−	3.35240i		−	0.330322i	0.986267	−	0.165161i	$-0.0528144\pi$
$104$		−	3.85351i		−	0.377867i
$105$	0.0489334	+	0.332923i	0.00477541	+	0.0324899i
$106$		−	4.09935i		−	0.398164i
$107$	17.6874			1.70991			0.854954	−	0.518704i	$-0.173585\pi$
$107$	17.6874			1.70991			0.854954	+	0.518704i	$0.173585\pi$
$108$	4.70607	−	2.20294i	0.452841	−	0.211978i
$109$		−	6.90800i		−	0.661667i	−0.943689	−	0.330833i	$-0.892670\pi$
$109$		−	6.90800i		−	0.661667i	0.943689	−	0.330833i	$-0.107330\pi$
$110$		−	3.67639i		−	0.350530i
$111$	2.96897	+	20.1997i	0.281802	+	1.91727i
$112$		−	0.194278i		−	0.0183576i
$113$	3.33657			0.313878			0.156939	−	0.987608i	$-0.449837\pi$
$113$	3.33657			0.313878			0.156939	+	0.987608i	$0.449837\pi$
$114$	−0.468721	+	0.0688932i	−0.0438998	+	0.00645244i
$115$	4.59371	+	1.37760i	0.428366	+	0.128462i
$116$		−	9.11317i		−	0.846136i
$117$	11.0716	−	3.32649i	1.02357	−	0.307534i
$118$	2.49823			0.229980
$119$		−	0.721531i		−	0.0661426i
$120$	0.251873	+	1.71364i	0.0229927	+	0.156433i
$121$	2.51586			0.228715
$122$	−13.9867			−1.26630
$123$	−2.43169	−	16.5442i	−0.219258	−	1.49174i
$124$	−2.24052			−0.201205
$125$	1.00000			0.0894427
$126$	0.558185	−	0.167708i	0.0497270	−	0.0149406i
$127$	13.8315			1.22734			0.613672	−	0.789561i	$-0.289692\pi$
$127$	13.8315			1.22734			0.613672	+	0.789561i	$0.289692\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−2.63296	−	17.9136i	−0.231819	−	1.57720i
$130$			3.85351i			0.337975i
$131$		−	11.1092i		−	0.970613i	−0.874344	−	0.485306i	$-0.838708\pi$
$131$		−	11.1092i		−	0.970613i	0.874344	−	0.485306i	$-0.161292\pi$
$132$	−6.30001	+	0.925983i	−0.548346	+	0.0805965i
$133$	−0.0531397			−0.00460780
$134$	10.9816			0.948665
$135$	−4.70607	+	2.20294i	−0.405034	+	0.189599i
$136$		−	3.71391i		−	0.318465i
$137$	−6.61230			−0.564927			−0.282463	−	0.959278i	$-0.591152\pi$
$137$	−6.61230			−0.564927			−0.282463	+	0.959278i	$0.591152\pi$
$138$	1.20368	−	8.21895i	0.102464	−	0.699644i
$139$	−2.36109			−0.200265			−0.100133	−	0.994974i	$-0.531927\pi$
$139$	−2.36109			−0.200265			−0.100133	+	0.994974i	$0.531927\pi$
$140$			0.194278i			0.0164195i
$141$	−1.17030	−	7.96226i	−0.0985572	−	0.670543i
$142$	−9.72869			−0.816413
$143$	−14.1670			−1.18470
$144$	2.87312	−	0.863238i	0.239427	−	0.0719365i
$145$			9.11317i			0.756807i
$146$			11.0817i			0.917128i
$147$	−11.9308	+	1.75360i	−0.984036	+	0.144635i
$148$			11.7876i			0.968934i
$149$	−9.03818			−0.740436			−0.370218	−	0.928945i	$-0.620717\pi$
$149$	−9.03818			−0.740436			−0.370218	+	0.928945i	$0.620717\pi$
$150$	−0.251873	−	1.71364i	−0.0205653	−	0.139918i
$151$	7.21414			0.587078			0.293539	−	0.955947i	$-0.405167\pi$
$151$	7.21414			0.587078			0.293539	+	0.955947i	$0.405167\pi$
$152$	−0.273524			−0.0221857
$153$	10.6705	−	3.20598i	0.862659	−	0.259189i
$154$	−0.714243			−0.0575553
$155$	2.24052			0.179963
$156$	6.60352	−	0.970593i	0.528705	−	0.0777096i
$157$			0.699510i			0.0558270i	0.999610	+	0.0279135i	$0.00888629\pi$
$157$			0.699510i			0.0558270i	−0.999610	+	0.0279135i	$0.991114\pi$
$158$	6.43875			0.512239
$159$	7.02481	−	1.03252i	0.557104	−	0.0818838i
$160$			1.00000i			0.0790569i
$161$	0.267638	−	0.892458i	0.0210928	−	0.0703356i
$162$	4.96037	+	7.50964i	0.389724	+	0.590013i
$163$	−13.2190			−1.03539			−0.517694	−	0.855566i	$-0.673210\pi$
$163$	−13.2190			−1.03539			−0.517694	+	0.855566i	$0.673210\pi$
$164$		−	9.65442i		−	0.753884i
$165$	6.30001	−	0.925983i	0.490455	−	0.0720877i
$166$			9.42725i			0.731696i
$167$		−	10.7904i		−	0.834986i	−0.908680	−	0.417493i	$-0.862909\pi$
$167$		−	10.7904i		−	0.834986i	0.908680	−	0.417493i	$-0.137091\pi$
$168$	0.332923	−	0.0489334i	0.0256855	−	0.00377529i
$169$	1.84950			0.142269
$170$			3.71391i			0.284844i
$171$	−0.236116	−	0.785867i	−0.0180563	−	0.0600968i
$172$		−	10.4535i		−	0.797074i
$173$			3.90535i			0.296918i	0.988919	+	0.148459i	$0.0474314\pi$
$173$			3.90535i			0.296918i	−0.988919	+	0.148459i	$0.952569\pi$
$174$	15.6167	−	2.29536i	1.18390	−	0.174011i
$175$		−	0.194278i		−	0.0146860i
$176$	−3.67639			−0.277119
$177$	0.629235	+	4.28106i	0.0472962	+	0.321784i
$178$		−	9.92316i		−	0.743772i
$179$			5.90397i			0.441284i	0.975355	+	0.220642i	$0.0708152\pi$
$179$			5.90397i			0.441284i	−0.975355	+	0.220642i	$0.929185\pi$
$180$	−2.87312	+	0.863238i	−0.214150	+	0.0643420i
$181$		−	3.65493i		−	0.271669i	−0.990732	−	0.135834i	$-0.956628\pi$
$181$		−	3.65493i		−	0.271669i	0.990732	−	0.135834i	$-0.0433715\pi$
$182$	0.748652			0.0554938
$183$	−3.52288	−	23.9682i	−0.260419	−	1.77178i
$184$	1.37760	−	4.59371i	0.101558	−	0.338653i
$185$		−	11.7876i		−	0.866641i
$186$	−0.564326	−	3.83945i	−0.0413784	−	0.281522i
$187$	−13.6538			−0.998463
$188$		−	4.64640i		−	0.338874i
$189$	0.427983	+	0.914286i	0.0311312	+	0.0665045i
$190$	0.273524			0.0198435
$191$	−12.5903			−0.911005			−0.455503	−	0.890234i	$-0.650540\pi$
$191$	−12.5903			−0.911005			−0.455503	+	0.890234i	$0.650540\pi$
$192$	1.71364	−	0.251873i	0.123671	−	0.0181773i
$193$	23.4908			1.69090			0.845452	−	0.534051i	$-0.179331\pi$
$193$	23.4908			1.69090			0.845452	+	0.534051i	$0.179331\pi$
$194$	8.91482			0.640047
$195$	−6.60352	+	0.970593i	−0.472888	+	0.0695056i
$196$	−6.96226			−0.497304
$197$			15.4325i			1.09952i	0.835322	+	0.549761i	$0.185281\pi$
$197$			15.4325i			1.09952i	−0.835322	+	0.549761i	$0.814719\pi$
$198$	−3.17360	−	10.5627i	−0.225538	−	0.750660i
$199$			21.1429i			1.49878i	0.662127	+	0.749391i	$0.269654\pi$
$199$			21.1429i			1.49878i	−0.662127	+	0.749391i	$0.730346\pi$
$200$		−	1.00000i		−	0.0707107i
$201$	2.76596	+	18.8185i	0.195096	+	1.32735i
$202$	8.39174			0.590441
$203$	1.77049			0.124264
$204$	6.36430	−	0.935432i	0.445590	−	0.0654933i
$205$			9.65442i			0.674294i
$206$	3.35240			0.233573
$207$	14.3875	−	0.00745181i	1.00000	−	0.000517937i
$208$	3.85351			0.267193
$209$			1.00558i			0.0695575i
$210$	−0.332923	+	0.0489334i	−0.0229739	+	0.00337672i
$211$	−0.599814			−0.0412929			−0.0206464	−	0.999787i	$-0.506572\pi$
$211$	−0.599814			−0.0412929			−0.0206464	+	0.999787i	$0.506572\pi$
$212$	4.09935			0.281545
$213$	−2.45039	−	16.6715i	−0.167898	−	1.14231i
$214$			17.6874i			1.20909i
$215$			10.4535i			0.712925i
$216$	2.20294	+	4.70607i	0.149891	+	0.320207i
$217$		−	0.435285i		−	0.0295490i
$218$	6.90800			0.467869
$219$	−18.9900	+	2.79118i	−1.28323	+	0.188610i
$220$	3.67639			0.247862
$221$	14.3116			0.962699
$222$	−20.1997	+	2.96897i	−1.35571	+	0.199264i
$223$	−10.6634			−0.714072			−0.357036	−	0.934091i	$-0.616213\pi$
$223$	−10.6634			−0.714072			−0.357036	+	0.934091i	$0.616213\pi$
$224$	0.194278			0.0129808
$225$	2.87312	−	0.863238i	0.191541	−	0.0575492i
$226$			3.33657i			0.221945i
$227$	−5.57422			−0.369974			−0.184987	−	0.982741i	$-0.559224\pi$
$227$	−5.57422			−0.369974			−0.184987	+	0.982741i	$0.559224\pi$
$228$	−0.0688932	−	0.468721i	−0.00456256	−	0.0310418i
$229$		−	14.9060i		−	0.985015i	−0.870308	−	0.492508i	$-0.836080\pi$
$229$		−	14.9060i		−	0.985015i	0.870308	−	0.492508i	$-0.163920\pi$
$230$	−1.37760	+	4.59371i	−0.0908364	+	0.302901i
$231$	−0.179898	−	1.22395i	−0.0118364	−	0.0805303i
$232$	9.11317			0.598309
$233$			15.3425i			1.00512i	0.864543	+	0.502559i	$0.167608\pi$
$233$			15.3425i			1.00512i	−0.864543	+	0.502559i	$0.832392\pi$
$234$	3.32649	+	11.0716i	0.217460	+	0.723772i
$235$			4.64640i			0.303098i
$236$			2.49823i			0.162621i
$237$	1.62174	+	11.0337i	0.105344	+	0.716715i
$238$	0.721531			0.0467699
$239$		−	11.6220i		−	0.751763i	−0.926668	−	0.375881i	$-0.877340\pi$
$239$		−	11.6220i		−	0.751763i	0.926668	−	0.375881i	$-0.122660\pi$
$240$	−1.71364	+	0.251873i	−0.110615	+	0.0162583i
$241$		−	6.44721i		−	0.415301i	−0.978203	−	0.207651i	$-0.933418\pi$
$241$		−	6.44721i		−	0.415301i	0.978203	−	0.207651i	$-0.0665817\pi$
$242$			2.51586i			0.161726i
$243$	−11.6194	+	10.3918i	−0.745387	+	0.666632i
$244$		−	13.9867i		−	0.895409i
$245$	6.96226			0.444802
$246$	16.5442	−	2.43169i	1.05482	−	0.155039i
$247$		−	1.05403i		−	0.0670661i
$248$		−	2.24052i		−	0.142273i
$249$	−16.1549	+	2.37447i	−1.02378	+	0.150476i
$250$			1.00000i			0.0632456i
$251$	0.366988			0.0231641			0.0115820	−	0.999933i	$-0.496313\pi$
$251$	0.366988			0.0231641			0.0115820	+	0.999933i	$0.496313\pi$
$252$	0.167708	+	0.558185i	0.0105646	+	0.0351623i
$253$	−16.8883	−	5.06461i	−1.06176	−	0.318409i
$254$			13.8315i			0.867863i
$255$	−6.36430	+	0.935432i	−0.398548	+	0.0585790i
$256$	1.00000			0.0625000
$257$		−	20.9857i		−	1.30905i	−0.756040	−	0.654525i	$-0.772869\pi$
$257$		−	20.9857i		−	1.30905i	0.756040	−	0.654525i	$-0.227131\pi$
$258$	17.9136	−	2.63296i	1.11525	−	0.163921i
$259$	−2.29007			−0.142298
$260$	−3.85351			−0.238984
$261$	7.86683	+	26.1832i	0.486945	+	1.62070i
$262$	11.1092			0.686327
$263$	−28.4952			−1.75709			−0.878544	−	0.477661i	$-0.841485\pi$
$263$	−28.4952			−1.75709			−0.878544	+	0.477661i	$0.841485\pi$
$264$	−0.925983	−	6.30001i	−0.0569903	−	0.387739i
$265$	−4.09935			−0.251821
$266$		−	0.0531397i		−	0.00325821i
$267$	17.0047	−	2.49937i	1.04067	−	0.152959i
$268$			10.9816i			0.670807i
$269$		−	5.95349i		−	0.362991i	−0.983392	−	0.181495i	$-0.941906\pi$
$269$		−	5.95349i		−	0.362991i	0.983392	−	0.181495i	$-0.0580938\pi$
$270$	−2.20294	−	4.70607i	−0.134067	−	0.286402i
$271$	−10.8969			−0.661937			−0.330969	−	0.943642i	$-0.607375\pi$
$271$	−10.8969			−0.661937			−0.330969	+	0.943642i	$0.607375\pi$
$272$	3.71391			0.225189
$273$	0.188565	+	1.28292i	0.0114125	+	0.0776458i
$274$		−	6.61230i		−	0.399463i
$275$	−3.67639			−0.221695
$276$	8.21895	+	1.20368i	0.494723	+	0.0724532i
$277$	−12.5927			−0.756623			−0.378312	−	0.925678i	$-0.623495\pi$
$277$	−12.5927			−0.756623			−0.378312	+	0.925678i	$0.623495\pi$
$278$		−	2.36109i		−	0.141609i
$279$	6.43729	−	1.93410i	0.385390	−	0.115792i
$280$	−0.194278			−0.0116103
$281$	10.5936			0.631961			0.315980	−	0.948766i	$-0.397667\pi$
$281$	10.5936			0.631961			0.315980	+	0.948766i	$0.397667\pi$
$282$	7.96226	−	1.17030i	0.474145	−	0.0696904i
$283$			31.7364i			1.88653i	0.332039	+	0.943266i	$0.392263\pi$
$283$			31.7364i			1.88653i	−0.332039	+	0.943266i	$0.607737\pi$
$284$		−	9.72869i		−	0.577291i
$285$	0.0688932	+	0.468721i	0.00408088	+	0.0277647i
$286$		−	14.1670i		−	0.837712i
$287$	1.87564			0.110716
$288$	0.863238	+	2.87312i	0.0508668	+	0.169300i
$289$	−3.20690			−0.188641
$290$	−9.11317			−0.535144
$291$	2.24540	+	15.2768i	0.131628	+	0.895542i
$292$	−11.0817			−0.648508
$293$	−6.77543			−0.395825			−0.197912	−	0.980220i	$-0.563416\pi$
$293$	−6.77543			−0.395825			−0.197912	+	0.980220i	$0.563416\pi$
$294$	−1.75360	−	11.9308i	−0.102272	−	0.695818i
$295$		−	2.49823i		−	0.145452i
$296$	−11.7876			−0.685139
$297$	17.3013	−	8.09887i	1.00393	−	0.469944i
$298$		−	9.03818i		−	0.523567i
$299$	17.7019	+	5.30860i	1.02373	+	0.307004i
$300$	1.71364	−	0.251873i	0.0989370	−	0.0145419i
$301$	2.03089			0.117059
$302$			7.21414i			0.415127i
$303$	2.11365	+	14.3804i	0.121426	+	0.826133i
$304$		−	0.273524i		−	0.0156877i
$305$			13.9867i			0.800878i
$306$	3.20598	+	10.6705i	0.183274	+	0.609992i
$307$	−13.5532			−0.773522			−0.386761	−	0.922180i	$-0.626406\pi$
$307$	−13.5532			−0.773522			−0.386761	+	0.922180i	$0.626406\pi$
$308$		−	0.714243i		−	0.0406978i
$309$	0.844379	+	5.74481i	0.0480350	+	0.326811i
$310$			2.24052i			0.127253i
$311$		−	3.63405i		−	0.206068i	−0.994678	−	0.103034i	$-0.967145\pi$
$311$		−	3.63405i		−	0.206068i	0.994678	−	0.103034i	$-0.0328550\pi$
$312$	0.970593	+	6.60352i	0.0549490	+	0.373851i
$313$		−	20.2986i		−	1.14734i	−0.819085	−	0.573672i	$-0.805519\pi$
$313$		−	20.2986i		−	1.14734i	0.819085	−	0.573672i	$-0.194481\pi$
$314$	−0.699510			−0.0394756
$315$	−0.167708	−	0.558185i	−0.00944929	−	0.0314501i
$316$			6.43875i			0.362208i
$317$		−	22.0942i		−	1.24093i	−0.784233	−	0.620467i	$-0.786943\pi$
$317$		−	22.0942i		−	1.24093i	0.784233	−	0.620467i	$-0.213057\pi$
$318$	1.03252	+	7.02481i	0.0579006	+	0.393932i
$319$		−	33.5036i		−	1.87584i
$320$	−1.00000			−0.0559017
$321$	−30.3099	+	4.45498i	−1.69173	+	0.248653i
$322$	0.892458	+	0.267638i	0.0497348	+	0.0149149i
$323$		−	1.01584i		−	0.0565230i
$324$	−7.50964	+	4.96037i	−0.417202	+	0.275576i
$325$	3.85351			0.213754
$326$		−	13.2190i		−	0.732130i
$327$	1.73994	+	11.8378i	0.0962188	+	0.654633i
$328$	9.65442			0.533076
$329$	0.902694			0.0497671
$330$	0.925983	+	6.30001i	0.0509737	+	0.346804i
$331$	−27.8332			−1.52985			−0.764925	−	0.644120i	$-0.777224\pi$
$331$	−27.8332			−1.52985			−0.764925	+	0.644120i	$0.777224\pi$
$332$	−9.42725			−0.517387
$333$	−10.1755	−	33.8672i	−0.557614	−	1.85591i
$334$	10.7904			0.590424
$335$		−	10.9816i		−	0.599988i
$336$	0.0489334	+	0.332923i	0.00266953	+	0.0181624i
$337$			27.7015i			1.50900i	0.656302	+	0.754498i	$0.272120\pi$
$337$			27.7015i			1.50900i	−0.656302	+	0.754498i	$0.727880\pi$
$338$			1.84950i			0.100600i
$339$	−5.71767	+	0.840390i	−0.310541	+	0.0456437i
$340$	−3.71391			−0.201415
$341$	−8.23704			−0.446061
$342$	0.785867	−	0.236116i	0.0424948	−	0.0127677i
$343$		−	2.71256i		−	0.146465i
$344$	10.4535			0.563616
$345$	−8.21895	−	1.20368i	−0.442493	−	0.0648041i
$346$	−3.90535			−0.209953
$347$		−	25.4305i		−	1.36518i	−0.730801	−	0.682590i	$-0.760853\pi$
$347$		−	25.4305i		−	1.36518i	0.730801	−	0.682590i	$-0.239147\pi$
$348$	2.29536	+	15.6167i	0.123044	+	0.837142i
$349$	−25.3679			−1.35791			−0.678955	−	0.734180i	$-0.737567\pi$
$349$	−25.3679			−1.35791			−0.678955	+	0.734180i	$0.737567\pi$
$350$	0.194278			0.0103846
$351$	−18.1348	+	8.48904i	−0.967967	+	0.453111i
$352$		−	3.67639i		−	0.195952i
$353$		−	6.58726i		−	0.350605i	−0.984515	−	0.175302i	$-0.943910\pi$
$353$		−	6.58726i		−	0.350605i	0.984515	−	0.175302i	$-0.0560903\pi$
$354$	−4.28106	+	0.629235i	−0.227536	+	0.0334435i
$355$			9.72869i			0.516345i
$356$	9.92316			0.525926
$357$	0.181734	+	1.23644i	0.00961838	+	0.0654395i
$358$	−5.90397			−0.312035
$359$	34.1462			1.80217			0.901083	−	0.433647i	$-0.142774\pi$
$359$	34.1462			1.80217			0.901083	+	0.433647i	$0.142774\pi$
$360$	−0.863238	−	2.87312i	−0.0454966	−	0.151427i
$361$	18.9252			0.996062
$362$	3.65493			0.192099
$363$	−4.31128	+	0.633677i	−0.226284	+	0.0332594i
$364$			0.748652i			0.0392400i
$365$	11.0817			0.580043
$366$	23.9682	−	3.52288i	1.25284	−	0.184144i
$367$		−	2.78712i		−	0.145487i	−0.997351	−	0.0727433i	$-0.976825\pi$
$367$		−	2.78712i		−	0.145487i	0.997351	−	0.0727433i	$-0.0231754\pi$
$368$	4.59371	+	1.37760i	0.239464	+	0.0718125i
$369$	8.33407	+	27.7383i	0.433854	+	1.44400i
$370$	11.7876			0.612807
$371$			0.796415i			0.0413478i
$372$	3.83945	−	0.564326i	0.199066	−	0.0292590i
$373$			36.8414i			1.90757i	0.300485	+	0.953787i	$0.402852\pi$
$373$			36.8414i			1.90757i	−0.300485	+	0.953787i	$0.597148\pi$
$374$		−	13.6538i		−	0.706020i
$375$	−1.71364	+	0.251873i	−0.0884920	+	0.0130067i
$376$	4.64640			0.239620
$377$			35.1176i			1.80865i
$378$	−0.914286	+	0.427983i	−0.0470258	+	0.0220131i
$379$		−	12.8044i		−	0.657717i	−0.944379	−	0.328859i	$-0.893336\pi$
$379$		−	12.8044i		−	0.657717i	0.944379	−	0.328859i	$-0.106664\pi$
$380$			0.273524i			0.0140315i
$381$	−23.7021	+	3.48377i	−1.21430	+	0.178479i
$382$		−	12.5903i		−	0.644178i
$383$	−14.2092			−0.726055			−0.363028	−	0.931778i	$-0.618257\pi$
$383$	−14.2092			−0.726055			−0.363028	+	0.931778i	$0.618257\pi$
$384$	0.251873	+	1.71364i	0.0128533	+	0.0874488i
$385$			0.714243i			0.0364012i
$386$			23.4908i			1.19565i
$387$	9.02388	+	30.0342i	0.458710	+	1.52673i
$388$			8.91482i			0.452582i
$389$	−7.98762			−0.404988			−0.202494	−	0.979283i	$-0.564905\pi$
$389$	−7.98762			−0.404988			−0.202494	+	0.979283i	$0.564905\pi$
$390$	−0.970593	−	6.60352i	−0.0491479	−	0.334382i
$391$	17.0606	+	5.11629i	0.862793	+	0.258742i
$392$		−	6.96226i		−	0.351647i
$393$	2.79810	+	19.0371i	0.141145	+	0.960295i
$394$	−15.4325			−0.777479
$395$		−	6.43875i		−	0.323968i
$396$	10.5627	−	3.17360i	0.530797	−	0.159479i
$397$	6.37102			0.319752			0.159876	−	0.987137i	$-0.448891\pi$
$397$	6.37102			0.319752			0.159876	+	0.987137i	$0.448891\pi$
$398$	−21.1429			−1.05980
$399$	0.0910623	−	0.0133844i	0.00455882	−	0.000670061i
$400$	1.00000			0.0500000
$401$	−4.08280			−0.203885			−0.101943	−	0.994790i	$-0.532506\pi$
$401$	−4.08280			−0.203885			−0.101943	+	0.994790i	$0.532506\pi$
$402$	−18.8185	+	2.76596i	−0.938580	+	0.137954i
$403$	8.63386			0.430083
$404$			8.39174i			0.417505i
$405$	7.50964	−	4.96037i	0.373157	−	0.246483i
$406$			1.77049i			0.0878679i
$407$			43.3358i			2.14808i
$408$	0.935432	+	6.36430i	0.0463108	+	0.315080i
$409$	−18.0794			−0.893971			−0.446985	−	0.894541i	$-0.647502\pi$
$409$	−18.0794			−0.893971			−0.446985	+	0.894541i	$0.647502\pi$
$410$	−9.65442			−0.476798
$411$	11.3311	−	1.66546i	0.558922	−	0.0821509i
$412$			3.35240i			0.165161i
$413$	−0.485351			−0.0238826
$414$	0.00745181	+	14.3875i	0.000366237	+	0.707107i
$415$	9.42725			0.462765
$416$			3.85351i			0.188934i
$417$	4.04606	−	0.594694i	0.198136	−	0.0291223i
$418$	−1.00558			−0.0491846
$419$	−25.7595			−1.25843			−0.629217	−	0.777229i	$-0.716624\pi$
$419$	−25.7595			−1.25843			−0.629217	+	0.777229i	$0.716624\pi$
$420$	−0.0489334	−	0.332923i	−0.00238770	−	0.0162450i
$421$			13.8562i			0.675308i	0.941270	+	0.337654i	$0.109633\pi$
$421$			13.8562i			0.675308i	−0.941270	+	0.337654i	$0.890367\pi$
$422$		−	0.599814i		−	0.0291985i
$423$	4.01095	+	13.3497i	0.195019	+	0.649083i
$424$			4.09935i			0.199082i
$425$	3.71391			0.180151
$426$	16.6715	−	2.45039i	0.807735	−	0.118722i
$427$	2.71732			0.131500
$428$	−17.6874			−0.854954
$429$	24.2771	−	3.56828i	1.17211	−	0.172278i
$430$	−10.4535			−0.504114
$431$	23.1260			1.11394			0.556970	−	0.830533i	$-0.311964\pi$
$431$	23.1260			1.11394			0.556970	+	0.830533i	$0.311964\pi$
$432$	−4.70607	+	2.20294i	−0.226421	+	0.105989i
$433$			15.8236i			0.760432i	0.924898	+	0.380216i	$0.124150\pi$
$433$			15.8236i			0.760432i	−0.924898	+	0.380216i	$0.875850\pi$
$434$	0.435285			0.0208943
$435$	−2.29536	−	15.6167i	−0.110054	−	0.748763i
$436$			6.90800i			0.330833i
$437$	0.376807	−	1.25649i	0.0180251	−	0.0601061i
$438$	−2.79118	−	18.9900i	−0.133368	−	0.907379i
$439$	31.7678			1.51619			0.758096	−	0.652143i	$-0.226130\pi$
$439$	31.7678			1.51619			0.758096	+	0.652143i	$0.226130\pi$
$440$			3.67639i			0.175265i
$441$	20.0034	−	6.01008i	0.952543	−	0.286194i
$442$			14.3116i			0.680731i
$443$			11.4636i			0.544652i	0.962205	+	0.272326i	$0.0877930\pi$
$443$			11.4636i			0.544652i	−0.962205	+	0.272326i	$0.912207\pi$
$444$	−2.96897	−	20.1997i	−0.140901	−	0.958634i
$445$	−9.92316			−0.470403
$446$		−	10.6634i		−	0.504925i
$447$	15.4882	−	2.27647i	0.732565	−	0.107673i
$448$			0.194278i			0.00917878i
$449$			6.54091i			0.308685i	0.988017	+	0.154342i	$0.0493259\pi$
$449$			6.54091i			0.308685i	−0.988017	+	0.154342i	$0.950674\pi$
$450$	0.863238	+	2.87312i	0.0406934	+	0.135440i
$451$		−	35.4935i		−	1.67132i
$452$	−3.33657			−0.156939
$453$	−12.3624	+	1.81704i	−0.580838	+	0.0853722i
$454$		−	5.57422i		−	0.261611i
$455$		−	0.748652i		−	0.0350973i
$456$	0.468721	−	0.0688932i	0.0219499	−	0.00322622i
$457$		−	30.6183i		−	1.43226i	−0.697966	−	0.716131i	$-0.745911\pi$
$457$		−	30.6183i		−	1.43226i	0.697966	−	0.716131i	$-0.254089\pi$
$458$	14.9060			0.696511
$459$	−17.4779	+	8.18151i	−0.815798	+	0.381880i
$460$	−4.59371	−	1.37760i	−0.214183	−	0.0642310i
$461$		−	22.9570i		−	1.06921i	−0.845101	−	0.534606i	$-0.820460\pi$
$461$		−	22.9570i		−	1.06921i	0.845101	−	0.534606i	$-0.179540\pi$
$462$	1.22395	−	0.179898i	0.0569435	−	0.00836963i
$463$	22.2784			1.03536			0.517682	−	0.855573i	$-0.326795\pi$
$463$	22.2784			1.03536			0.517682	+	0.855573i	$0.326795\pi$
$464$			9.11317i			0.423068i
$465$	−3.83945	+	0.564326i	−0.178050	+	0.0261700i
$466$	−15.3425			−0.710726
$467$	21.8911			1.01300			0.506500	−	0.862240i	$-0.330939\pi$
$467$	21.8911			1.01300			0.506500	+	0.862240i	$0.330939\pi$
$468$	−11.0716	+	3.32649i	−0.511784	+	0.153767i
$469$	−2.13348			−0.0985151
$470$	−4.64640			−0.214322
$471$	−0.176187	−	1.19871i	−0.00811829	−	0.0552335i
$472$	−2.49823			−0.114990
$473$		−	38.4313i		−	1.76707i
$474$	−11.0337	+	1.62174i	−0.506794	+	0.0744892i
$475$		−	0.273524i		−	0.0125501i
$476$			0.721531i			0.0330713i
$477$	−11.7779	+	3.53872i	−0.539275	+	0.162027i
$478$	11.6220			0.531576
$479$	−5.71132			−0.260957			−0.130478	−	0.991451i	$-0.541651\pi$
$479$	−5.71132			−0.260957			−0.130478	+	0.991451i	$0.541651\pi$
$480$	−0.251873	−	1.71364i	−0.0114964	−	0.0782166i
$481$		−	45.4235i		−	2.07113i
$482$	6.44721			0.293662
$483$	−0.233849	+	1.59676i	−0.0106405	+	0.0726552i
$484$	−2.51586			−0.114357
$485$		−	8.91482i		−	0.404801i
$486$	−10.3918	−	11.6194i	−0.471380	−	0.527068i
$487$	22.2185			1.00682			0.503409	−	0.864048i	$-0.332079\pi$
$487$	22.2185			1.00682			0.503409	+	0.864048i	$0.332079\pi$
$488$	13.9867			0.633150
$489$	22.6525	−	3.32949i	1.02438	−	0.150565i
$490$			6.96226i			0.314523i
$491$			9.94225i			0.448688i	0.974510	+	0.224344i	$0.0720239\pi$
$491$			9.94225i			0.448688i	−0.974510	+	0.224344i	$0.927976\pi$
$492$	2.43169	+	16.5442i	0.109629	+	0.745870i
$493$			33.8454i			1.52432i
$494$	1.05403			0.0474229
$495$	−10.5627	+	3.17360i	−0.474759	+	0.142643i
$496$	2.24052			0.100602
$497$	1.89007			0.0847813
$498$	−2.37447	−	16.1549i	−0.106402	−	0.723919i
$499$	5.95037			0.266375			0.133187	−	0.991091i	$-0.457479\pi$
$499$	5.95037			0.266375			0.133187	+	0.991091i	$0.457479\pi$
$500$	−1.00000			−0.0447214
$501$	2.71781	+	18.4908i	0.121423	+	0.826110i
$502$			0.366988i			0.0163795i
$503$	−27.9653			−1.24691			−0.623455	−	0.781859i	$-0.714272\pi$
$503$	−27.9653			−1.24691			−0.623455	+	0.781859i	$0.714272\pi$
$504$	−0.558185	+	0.167708i	−0.0248635	+	0.00747032i
$505$		−	8.39174i		−	0.373427i
$506$	5.06461	−	16.8883i	0.225149	−	0.750776i
$507$	−3.16938	+	0.465839i	−0.140757	+	0.0206886i
$508$	−13.8315			−0.613672
$509$		−	3.52122i		−	0.156075i	−0.996950	−	0.0780376i	$-0.975135\pi$
$509$		−	3.52122i		−	0.156075i	0.996950	−	0.0780376i	$-0.0248654\pi$
$510$	−0.935432	−	6.36430i	−0.0414216	−	0.281816i
$511$		−	2.15293i		−	0.0952402i
$512$			1.00000i			0.0441942i
$513$	0.602557	+	1.28722i	0.0266035	+	0.0568322i
$514$	20.9857			0.925638
$515$		−	3.35240i		−	0.147724i
$516$	2.63296	+	17.9136i	0.115910	+	0.788601i
$517$		−	17.0820i		−	0.751265i
$518$		−	2.29007i		−	0.100620i
$519$	−0.983651	−	6.69236i	−0.0431775	−	0.293762i
$520$		−	3.85351i		−	0.168987i
$521$	25.1110			1.10013			0.550067	−	0.835121i	$-0.314602\pi$
$521$	25.1110			1.10013			0.550067	+	0.835121i	$0.314602\pi$
$522$	−26.1832	+	7.86683i	−1.14601	+	0.344322i
$523$		−	23.9422i		−	1.04692i	−0.852050	−	0.523460i	$-0.824641\pi$
$523$		−	23.9422i		−	1.04692i	0.852050	−	0.523460i	$-0.175359\pi$
$524$			11.1092i			0.485306i
$525$	0.0489334	+	0.332923i	0.00213563	+	0.0145299i
$526$		−	28.4952i		−	1.24245i
$527$	8.32109			0.362472
$528$	6.30001	−	0.925983i	0.274173	−	0.0402982i
$529$	19.2044	+	12.6566i	0.834975	+	0.550288i
$530$		−	4.09935i		−	0.178065i
$531$	−2.15656	−	7.17771i	−0.0935869	−	0.311486i
$532$	0.0531397			0.00230390
$533$			37.2034i			1.61146i
$534$	2.49937	+	17.0047i	0.108158	+	0.735866i
$535$	17.6874			0.764694
$536$	−10.9816			−0.474332
$537$	−1.48705	−	10.1173i	−0.0641709	−	0.436593i
$538$	5.95349			0.256673
$539$	−25.5960			−1.10250
$540$	4.70607	−	2.20294i	0.202517	−	0.0947994i
$541$	−3.04910			−0.131091			−0.0655455	−	0.997850i	$-0.520879\pi$
$541$	−3.04910			−0.131091			−0.0655455	+	0.997850i	$0.520879\pi$
$542$		−	10.8969i		−	0.468060i
$543$	0.920578	+	6.26324i	0.0395058	+	0.268781i
$544$			3.71391i			0.159232i
$545$		−	6.90800i		−	0.295906i
$546$	−1.28292	+	0.188565i	−0.0549039	+	0.00806984i
$547$	23.7201			1.01420			0.507099	−	0.861888i	$-0.330718\pi$
$547$	23.7201			1.01420			0.507099	+	0.861888i	$0.330718\pi$
$548$	6.61230			0.282463
$549$	12.0739	+	40.1856i	0.515301	+	1.71508i
$550$		−	3.67639i		−	0.156762i
$551$	2.49267			0.106191
$552$	−1.20368	+	8.21895i	−0.0512321	+	0.349822i
$553$	−1.25091			−0.0531940
$554$		−	12.5927i		−	0.535013i
$555$	2.96897	+	20.1997i	0.126026	+	0.857428i
$556$	2.36109			0.100133
$557$	−3.97743			−0.168529			−0.0842645	−	0.996443i	$-0.526854\pi$
$557$	−3.97743			−0.168529			−0.0842645	+	0.996443i	$0.526854\pi$
$558$	1.93410	+	6.43729i	0.0818772	+	0.272512i
$559$			40.2827i			1.70378i
$560$		−	0.194278i		−	0.00820975i
$561$	23.3976	−	3.43901i	0.987849	−	0.145195i
$562$			10.5936i			0.446864i
$563$	−19.2899			−0.812971			−0.406486	−	0.913657i	$-0.633246\pi$
$563$	−19.2899			−0.812971			−0.406486	+	0.913657i	$0.633246\pi$
$564$	1.17030	+	7.96226i	0.0492786	+	0.335271i
$565$	3.33657			0.140370
$566$	−31.7364			−1.33398
$567$	−0.963692	−	1.45896i	−0.0404713	−	0.0612705i
$568$	9.72869			0.408207
$569$	−10.8881			−0.456454			−0.228227	−	0.973608i	$-0.573293\pi$
$569$	−10.8881			−0.456454			−0.228227	+	0.973608i	$0.573293\pi$
$570$	−0.468721	+	0.0688932i	−0.0196326	+	0.00288562i
$571$			4.04576i			0.169310i	0.996410	+	0.0846549i	$0.0269788\pi$
$571$			4.04576i			0.169310i	−0.996410	+	0.0846549i	$0.973021\pi$
$572$	14.1670			0.592352
$573$	21.5753	−	3.17116i	0.901321	−	0.132477i
$574$			1.87564i			0.0782879i
$575$	4.59371	+	1.37760i	0.191571	+	0.0574500i
$576$	−2.87312	+	0.863238i	−0.119713	+	0.0359683i
$577$	−35.0273			−1.45821			−0.729103	−	0.684404i	$-0.760062\pi$
$577$	−35.0273			−1.45821			−0.729103	+	0.684404i	$0.760062\pi$
$578$		−	3.20690i		−	0.133390i
$579$	−40.2548	+	5.91669i	−1.67293	+	0.245889i
$580$		−	9.11317i		−	0.378404i
$581$		−	1.83151i		−	0.0759838i
$582$	−15.2768	+	2.24540i	−0.633244	+	0.0930749i
$583$	15.0708			0.624170
$584$		−	11.0817i		−	0.458564i
$585$	11.0716	−	3.32649i	0.457754	−	0.137534i
$586$		−	6.77543i		−	0.279890i
$587$		−	39.2631i		−	1.62056i	−0.586043	−	0.810280i	$-0.699315\pi$
$587$		−	39.2631i		−	1.62056i	0.586043	−	0.810280i	$-0.300685\pi$
$588$	11.9308	−	1.75360i	0.492018	−	0.0723173i
$589$		−	0.612836i		−	0.0252515i
$590$	2.49823			0.102850
$591$	−3.88703	−	26.4458i	−0.159891	−	1.08783i
$592$		−	11.7876i		−	0.484467i
$593$		−	20.0039i		−	0.821461i	−0.911757	−	0.410730i	$-0.865274\pi$
$593$		−	20.0039i		−	0.821461i	0.911757	−	0.410730i	$-0.134726\pi$
$594$	8.09887	+	17.3013i	0.332301	+	0.709883i
$595$		−	0.721531i		−	0.0295799i
$596$	9.03818			0.370218
$597$	−5.32533	−	36.2314i	−0.217951	−	1.48285i
$598$	−5.30860	+	17.7019i	−0.217085	+	0.723885i
$599$			9.77816i			0.399525i	0.979844	+	0.199762i	$0.0640170\pi$
$599$			9.77816i			0.399525i	−0.979844	+	0.199762i	$0.935983\pi$
$600$	0.251873	+	1.71364i	0.0102827	+	0.0699590i
$601$	−33.3197			−1.35914			−0.679569	−	0.733612i	$-0.737833\pi$
$601$	−33.3197			−1.35914			−0.679569	+	0.733612i	$0.737833\pi$
$602$			2.03089i			0.0827730i
$603$	−9.47972	−	31.5514i	−0.386044	−	1.28487i
$604$	−7.21414			−0.293539
$605$	2.51586			0.102284
$606$	−14.3804	+	2.11365i	−0.584164	+	0.0858612i
$607$	−6.68091			−0.271170			−0.135585	−	0.990766i	$-0.543291\pi$
$607$	−6.68091			−0.271170			−0.135585	+	0.990766i	$0.543291\pi$
$608$	0.273524			0.0110929
$609$	−3.03398	+	0.445938i	−0.122943	+	0.0180703i
$610$	−13.9867			−0.566306
$611$			17.9049i			0.724356i
$612$	−10.6705	+	3.20598i	−0.431329	+	0.129594i
$613$			2.66775i			0.107749i	0.998548	+	0.0538746i	$0.0171571\pi$
$613$			2.66775i			0.107749i	−0.998548	+	0.0538746i	$0.982843\pi$
$614$		−	13.5532i		−	0.546962i
$615$	−2.43169	−	16.5442i	−0.0980550	−	0.667127i
$616$	0.714243			0.0287777
$617$	−39.4541			−1.58836			−0.794182	−	0.607680i	$-0.792100\pi$
$617$	−39.4541			−1.58836			−0.794182	+	0.607680i	$0.792100\pi$
$618$	−5.74481	+	0.844379i	−0.231090	+	0.0339659i
$619$			2.35902i			0.0948171i	0.998876	+	0.0474086i	$0.0150963\pi$
$619$			2.35902i			0.0948171i	−0.998876	+	0.0474086i	$0.984904\pi$
$620$	−2.24052			−0.0899815
$621$	−24.6531	+	3.63659i	−0.989295	+	0.145931i
$622$	3.63405			0.145712
$623$			1.92785i			0.0772378i
$624$	−6.60352	+	0.970593i	−0.264352	+	0.0388548i
$625$	1.00000			0.0400000
$626$	20.2986			0.811294
$627$	−0.253279	−	1.72320i	−0.0101150	−	0.0688181i
$628$		−	0.699510i		−	0.0279135i
$629$		−	43.7780i		−	1.74554i
$630$	0.558185	−	0.167708i	0.0222386	−	0.00668166i
$631$		−	0.472138i		−	0.0187955i	−0.999956	−	0.00939776i	$-0.997009\pi$
$631$		−	0.472138i		−	0.0187955i	0.999956	−	0.00939776i	$-0.00299144\pi$
$632$	−6.43875			−0.256120
$633$	1.02786	−	0.151077i	0.0408540	−	0.00600476i
$634$	22.0942			0.877472
$635$	13.8315			0.548884
$636$	−7.02481	+	1.03252i	−0.278552	+	0.0409419i
$637$	26.8291			1.06301
$638$	33.5036			1.32642
$639$	8.39817	+	27.9517i	0.332227	+	1.10575i
$640$		−	1.00000i		−	0.0395285i
$641$	17.5371			0.692675			0.346337	−	0.938110i	$-0.387425\pi$
$641$	17.5371			0.692675			0.346337	+	0.938110i	$0.387425\pi$
$642$	−4.45498	−	30.3099i	−0.175824	−	1.19623i
$643$		−	3.71442i		−	0.146482i	−0.997314	−	0.0732412i	$-0.976666\pi$
$643$		−	3.71442i		−	0.146482i	0.997314	−	0.0732412i	$-0.0233343\pi$
$644$	−0.267638	+	0.892458i	−0.0105464	+	0.0351678i
$645$	−2.63296	−	17.9136i	−0.103673	−	0.705346i
$646$	1.01584			0.0399678
$647$		−	37.9848i		−	1.49334i	−0.665197	−	0.746668i	$-0.731653\pi$
$647$		−	37.9848i		−	1.49334i	0.665197	−	0.746668i	$-0.268347\pi$
$648$	−4.96037	−	7.50964i	−0.194862	−	0.295007i
$649$			9.18446i			0.360522i
$650$			3.85351i			0.151147i
$651$	0.109636	+	0.745921i	0.00429699	+	0.0292349i
$652$	13.2190			0.517694
$653$		−	28.7963i		−	1.12689i	−0.826155	−	0.563443i	$-0.809476\pi$
$653$		−	28.7963i		−	1.12689i	0.826155	−	0.563443i	$-0.190524\pi$
$654$	−11.8378	+	1.73994i	−0.462896	+	0.0680369i
$655$		−	11.1092i		−	0.434071i
$656$			9.65442i			0.376942i
$657$	31.8391	−	9.56615i	1.24216	−	0.373211i
$658$			0.902694i			0.0351907i
$659$	6.49917			0.253172			0.126586	−	0.991956i	$-0.459598\pi$
$659$	6.49917			0.253172			0.126586	+	0.991956i	$0.459598\pi$
$660$	−6.30001	+	0.925983i	−0.245228	+	0.0360438i
$661$		−	39.1114i		−	1.52126i	−0.649187	−	0.760629i	$-0.724891\pi$
$661$		−	39.1114i		−	1.52126i	0.649187	−	0.760629i	$-0.275109\pi$
$662$		−	27.8332i		−	1.08177i
$663$	−24.5248	+	3.60469i	−0.952466	+	0.139995i
$664$		−	9.42725i		−	0.365848i
$665$	−0.0531397			−0.00206067
$666$	33.8672	−	10.1755i	1.31233	−	0.394292i
$667$	−12.5543	+	41.8633i	−0.486105	+	1.62095i
$668$			10.7904i			0.417493i
$669$	18.2732	−	2.68581i	0.706481	−	0.103839i
$670$	10.9816			0.424256
$671$		−	51.4207i		−	1.98507i
$672$	−0.332923	+	0.0489334i	−0.0128428	+	0.00188765i
$673$	−13.0249			−0.502071			−0.251036	−	0.967978i	$-0.580771\pi$
$673$	−13.0249			−0.502071			−0.251036	+	0.967978i	$0.580771\pi$
$674$	−27.7015			−1.06702
$675$	−4.70607	+	2.20294i	−0.181137	+	0.0847912i
$676$	−1.84950			−0.0711347
$677$	−26.1392			−1.00461			−0.502305	−	0.864691i	$-0.667515\pi$
$677$	−26.1392			−1.00461			−0.502305	+	0.864691i	$0.667515\pi$
$678$	−0.840390	−	5.71767i	−0.0322750	−	0.219586i
$679$	−1.73196			−0.0664664
$680$		−	3.71391i		−	0.142422i
$681$	9.55221	−	1.40399i	0.366041	−	0.0538012i
$682$		−	8.23704i		−	0.315413i
$683$		−	15.5123i		−	0.593560i	−0.954946	−	0.296780i	$-0.904087\pi$
$683$		−	15.5123i		−	0.593560i	0.954946	−	0.296780i	$-0.0959128\pi$
$684$	0.236116	+	0.785867i	0.00902813	+	0.0300484i
$685$	−6.61230			−0.252643
$686$	2.71256			0.103566
$687$	3.75441	+	25.5435i	0.143240	+	0.974545i
$688$			10.4535i			0.398537i
$689$	−15.7969			−0.601813
$690$	1.20368	−	8.21895i	0.0458234	−	0.312890i
$691$	31.1673			1.18566			0.592830	−	0.805327i	$-0.298010\pi$
$691$	31.1673			1.18566			0.592830	+	0.805327i	$0.298010\pi$
$692$		−	3.90535i		−	0.148459i
$693$	0.616562	+	2.05211i	0.0234212	+	0.0779530i
$694$	25.4305			0.965329
$695$	−2.36109			−0.0895612
$696$	−15.6167	+	2.29536i	−0.591949	+	0.0870053i
$697$			35.8556i			1.35813i
$698$		−	25.3679i		−	0.960188i
$699$	−3.86435	−	26.2915i	−0.146163	−	0.994434i
$700$			0.194278i			0.00734302i
$701$	−39.7108			−1.49986			−0.749929	−	0.661519i	$-0.769912\pi$
$701$	−39.7108			−1.49986			−0.749929	+	0.661519i	$0.769912\pi$
$702$	−8.48904	−	18.1348i	−0.320398	−	0.684456i
$703$	−3.22419			−0.121603
$704$	3.67639			0.138559
$705$	−1.17030	−	7.96226i	−0.0440761	−	0.299876i
$706$	6.58726			0.247915
$707$	−1.63033			−0.0613149
$708$	−0.629235	−	4.28106i	−0.0236481	−	0.160892i
$709$			29.9453i			1.12462i	0.826926	+	0.562311i	$0.190087\pi$
$709$			29.9453i			1.12462i	−0.826926	+	0.562311i	$0.809913\pi$
$710$	−9.72869			−0.365111
$711$	−5.55817	−	18.4993i	−0.208448	−	0.693778i
$712$			9.92316i			0.371886i
$713$	10.2923	+	3.08655i	0.385450	+	0.115592i
$714$	−1.23644	+	0.181734i	−0.0462727	+	0.00680122i
$715$	−14.1670			−0.529816
$716$		−	5.90397i		−	0.220642i
$717$	2.92726	+	19.9159i	0.109320	+	0.743772i
$718$			34.1462i			1.27432i
$719$		−	19.0401i		−	0.710077i	−0.934852	−	0.355038i	$-0.884468\pi$
$719$		−	19.0401i		−	0.710077i	0.934852	−	0.355038i	$-0.115532\pi$
$720$	2.87312	−	0.863238i	0.107075	−	0.0321710i
$721$	−0.651299			−0.0242556
$722$			18.9252i			0.704322i
$723$	1.62388	+	11.0482i	0.0603926	+	0.410886i
$724$			3.65493i			0.135834i
$725$			9.11317i			0.338454i
$726$	−0.633677	−	4.31128i	−0.0235180	−	0.160007i
$727$		−	28.0357i		−	1.03979i	−0.854231	−	0.519893i	$-0.825972\pi$
$727$		−	28.0357i		−	1.03979i	0.854231	−	0.519893i	$-0.174028\pi$
$728$	−0.748652			−0.0277469
$729$	17.2941	−	20.7344i	0.640523	−	0.767939i
$730$			11.0817i			0.410152i
$731$			38.8234i			1.43594i
$732$	3.52288	+	23.9682i	0.130209	+	0.885891i
$733$		−	33.0849i		−	1.22202i	−0.791623	−	0.611010i	$-0.790763\pi$
$733$		−	33.0849i		−	1.22202i	0.791623	−	0.611010i	$-0.209237\pi$
$734$	2.78712			0.102875
$735$	−11.9308	+	1.75360i	−0.440074	+	0.0646826i
$736$	−1.37760	+	4.59371i	−0.0507791	+	0.169327i
$737$			40.3726i			1.48714i
$738$	−27.7383	+	8.33407i	−1.02106	+	0.306781i
$739$	0.332871			0.0122449			0.00612243	−	0.999981i	$-0.498051\pi$
$739$	0.332871			0.0122449			0.00612243	+	0.999981i	$0.498051\pi$
$740$			11.7876i			0.433320i
$741$	0.265480	+	1.80622i	0.00975266	+	0.0663532i
$742$	−0.796415			−0.0292373
$743$	16.2658			0.596734			0.298367	−	0.954451i	$-0.403558\pi$
$743$	16.2658			0.596734			0.298367	+	0.954451i	$0.403558\pi$
$744$	0.564326	+	3.83945i	0.0206892	+	0.140761i
$745$	−9.03818			−0.331133
$746$	−36.8414			−1.34886
$747$	27.0856	−	8.13796i	0.991011	−	0.297752i
$748$	13.6538			0.499231
$749$		−	3.43628i		−	0.125559i
$750$	−0.251873	−	1.71364i	−0.00919709	−	0.0625733i
$751$		−	40.0221i		−	1.46043i	−0.683218	−	0.730214i	$-0.739420\pi$
$751$		−	40.0221i		−	1.46043i	0.683218	−	0.730214i	$-0.260580\pi$
$752$			4.64640i			0.169437i
$753$	−0.628885	+	0.0924342i	−0.0229178	+	0.00336849i
$754$	−35.1176			−1.27891
$755$	7.21414			0.262549
$756$	−0.427983	−	0.914286i	−0.0155656	−	0.0332523i
$757$			20.5788i			0.747950i	0.927439	+	0.373975i	$0.122005\pi$
$757$			20.5788i			0.747950i	−0.927439	+	0.373975i	$0.877995\pi$
$758$	12.8044			0.465076
$759$	30.2161	+	4.42521i	1.09677	+	0.160625i
$760$	−0.273524			−0.00992176
$761$		−	5.93470i		−	0.215133i	−0.994198	−	0.107566i	$-0.965694\pi$
$761$		−	5.93470i		−	0.215133i	0.994198	−	0.107566i	$-0.0343058\pi$
$762$	−3.48377	−	23.7021i	−0.126204	−	0.858637i
$763$	−1.34207			−0.0485864
$764$	12.5903			0.455503
$765$	10.6705	−	3.20598i	0.385793	−	0.115913i
$766$		−	14.2092i		−	0.513399i
$767$		−	9.62693i		−	0.347608i
$768$	−1.71364	+	0.251873i	−0.0618356	+	0.00908867i
$769$			9.97467i			0.359696i	0.983694	+	0.179848i	$0.0575606\pi$
$769$			9.97467i			0.359696i	−0.983694	+	0.179848i	$0.942439\pi$
$770$	−0.714243			−0.0257395
$771$	5.28572	+	35.9619i	0.190361	+	1.29514i
$772$	−23.4908			−0.845452
$773$	44.7513			1.60959			0.804796	−	0.593552i	$-0.202275\pi$
$773$	44.7513			1.60959			0.804796	+	0.593552i	$0.202275\pi$
$774$	−30.0342	+	9.02388i	−1.07956	+	0.324357i
$775$	2.24052			0.0804819
$776$	−8.91482			−0.320024
$777$	3.92436	−	0.576806i	0.140785	−	0.0206928i
$778$		−	7.98762i		−	0.286370i
$779$	2.64072			0.0946135
$780$	6.60352	−	0.970593i	0.236444	−	0.0347528i
$781$		−	35.7665i		−	1.27983i
$782$	−5.11629	+	17.0606i	−0.182958	+	0.610087i
$783$	−20.0758	−	42.8872i	−0.717449	−	1.53266i
$784$	6.96226			0.248652
$785$			0.699510i			0.0249666i
$786$	−19.0371	+	2.79810i	−0.679031	+	0.0998048i
$787$			23.8924i			0.851671i	0.904801	+	0.425835i	$0.140020\pi$
$787$			23.8924i			0.851671i	−0.904801	+	0.425835i	$0.859980\pi$
$788$		−	15.4325i		−	0.549761i
$789$	48.8305	−	7.17716i	1.73841	−	0.255514i
$790$	6.43875			0.229080
$791$		−	0.648222i		−	0.0230481i
$792$	3.17360	+	10.5627i	0.112769	+	0.375330i
$793$			53.8979i			1.91397i
$794$			6.37102i			0.226099i
$795$	7.02481	−	1.03252i	0.249144	−	0.0366195i
$796$		−	21.1429i		−	0.749391i
$797$	16.9611			0.600793			0.300396	−	0.953814i	$-0.402881\pi$
$797$	16.9611			0.600793			0.300396	+	0.953814i	$0.402881\pi$
$798$	0.0133844	+	0.0910623i	0.000473804	+	0.00322357i
$799$			17.2563i			0.610484i
$800$			1.00000i			0.0353553i
$801$	−28.5104	+	8.56605i	−1.00737	+	0.302666i
$802$		−	4.08280i		−	0.144169i
$803$	−40.7407			−1.43771
$804$	−2.76596	−	18.8185i	−0.0975480	−	0.663677i
$805$	0.267638	−	0.892458i	0.00943300	−	0.0314550i
$806$			8.63386i			0.304115i
$807$	1.49952	+	10.2021i	0.0527857	+	0.359132i
$808$	−8.39174			−0.295220
$809$			18.7429i			0.658965i	0.944162	+	0.329483i	$0.106874\pi$
$809$			18.7429i			0.658965i	−0.944162	+	0.329483i	$0.893126\pi$
$810$	4.96037	+	7.50964i	0.174290	+	0.263862i
$811$	−38.4375			−1.34972			−0.674861	−	0.737944i	$-0.735797\pi$
$811$	−38.4375			−1.34972			−0.674861	+	0.737944i	$0.735797\pi$
$812$	−1.77049			−0.0621320
$813$	18.6733	−	2.74462i	0.654901	−	0.0962581i
$814$	−43.3358			−1.51892
$815$	−13.2190			−0.463040
$816$	−6.36430	+	0.935432i	−0.222795	+	0.0327467i
$817$	2.85929			0.100034
$818$		−	18.0794i		−	0.632133i
$819$	−0.646265	−	2.15097i	−0.0225823	−	0.0751609i
$820$		−	9.65442i		−	0.337147i
$821$			32.7980i			1.14466i	0.820023	+	0.572330i	$0.193960\pi$
$821$			32.7980i			1.14466i	−0.820023	+	0.572330i	$0.806040\pi$
$822$	1.66546	+	11.3311i	0.0580895	+	0.395217i
$823$	36.3523			1.26716			0.633581	−	0.773676i	$-0.281584\pi$
$823$	36.3523			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$824$	−3.35240			−0.116786
$825$	6.30001	−	0.925983i	0.219338	−	0.0322386i
$826$		−	0.485351i		−	0.0168875i
$827$	−1.85363			−0.0644572			−0.0322286	−	0.999481i	$-0.510260\pi$
$827$	−1.85363			−0.0644572			−0.0322286	+	0.999481i	$0.510260\pi$
$828$	−14.3875	+	0.00745181i	−0.500000	+	0.000258968i
$829$	−23.4117			−0.813124			−0.406562	−	0.913623i	$-0.633272\pi$
$829$	−23.4117			−0.813124			−0.406562	+	0.913623i	$0.633272\pi$
$830$			9.42725i			0.327225i
$831$	21.5794	−	3.17176i	0.748580	−	0.110027i
$832$	−3.85351			−0.133596
$833$	25.8572			0.895898
$834$	0.594694	+	4.04606i	0.0205926	+	0.140103i
$835$		−	10.7904i		−	0.373417i
$836$		−	1.00558i		−	0.0347788i
$837$	−10.5440	+	4.93573i	−0.364456	+	0.170604i
$838$		−	25.7595i		−	0.889848i
$839$	19.0303			0.656998			0.328499	−	0.944504i	$-0.393457\pi$
$839$	19.0303			0.656998			0.328499	+	0.944504i	$0.393457\pi$
$840$	0.332923	−	0.0489334i	0.0114869	−	0.00168836i
$841$	−54.0498			−1.86379
$842$	−13.8562			−0.477515
$843$	−18.1536	+	2.66824i	−0.625243	+	0.0918990i
$844$	0.599814			0.0206464
$845$	1.84950			0.0636248
$846$	−13.3497	+	4.01095i	−0.458971	+	0.137899i
$847$		−	0.488777i		−	0.0167946i
$848$	−4.09935			−0.140772
$849$	−7.99352	−	54.3847i	−0.274337	−	1.86648i
$850$			3.71391i			0.127386i
$851$	16.2386	−	54.1488i	0.556652	−	1.85620i
$852$	2.45039	+	16.6715i	0.0839490	+	0.571155i
$853$	−8.98937			−0.307790			−0.153895	−	0.988087i	$-0.549182\pi$
$853$	−8.98937			−0.307790			−0.153895	+	0.988087i	$0.549182\pi$
$854$			2.71732i			0.0929847i
$855$	−0.236116	−	0.785867i	−0.00807501	−	0.0268761i
$856$		−	17.6874i		−	0.604544i
$857$		−	35.0879i		−	1.19858i	−0.800532	−	0.599290i	$-0.795450\pi$
$857$		−	35.0879i		−	1.19858i	0.800532	−	0.599290i	$-0.204550\pi$
$858$	3.56828	+	24.2771i	0.121819	+	0.828807i
$859$	−20.3531			−0.694439			−0.347220	−	0.937784i	$-0.612874\pi$
$859$	−20.3531			−0.694439			−0.347220	+	0.937784i	$0.612874\pi$
$860$		−	10.4535i		−	0.356462i
$861$	−3.21418	+	0.472424i	−0.109539	+	0.0161002i
$862$			23.1260i			0.787674i
$863$			55.7757i			1.89863i	0.314333	+	0.949313i	$0.398219\pi$
$863$			55.7757i			1.89863i	−0.314333	+	0.949313i	$0.601781\pi$
$864$	−2.20294	−	4.70607i	−0.0749455	−	0.160104i
$865$			3.90535i			0.132786i
$866$	−15.8236			−0.537707
$867$	5.49548	−	0.807732i	0.186636	−	0.0274320i
$868$			0.435285i			0.0147745i
$869$			23.6714i			0.802996i
$870$	15.6167	−	2.29536i	0.529455	−	0.0778199i
$871$		−	42.3176i		−	1.43388i
$872$	−6.90800			−0.233935
$873$	−7.69562	−	25.6134i	−0.260457	−	0.866881i
$874$	1.25649	+	0.376807i	0.0425014	+	0.0127457i
$875$		−	0.194278i		−	0.00656780i
$876$	18.9900	−	2.79118i	0.641614	−	0.0943052i
$877$	26.1377			0.882608			0.441304	−	0.897358i	$-0.354516\pi$
$877$	26.1377			0.882608			0.441304	+	0.897358i	$0.354516\pi$
$878$			31.7678i			1.07211i
$879$	11.6106	−	1.70655i	0.391617	−	0.0575604i
$880$	−3.67639			−0.123931
$881$	15.6301			0.526590			0.263295	−	0.964715i	$-0.415191\pi$
$881$	15.6301			0.526590			0.263295	+	0.964715i	$0.415191\pi$
$882$	6.01008	+	20.0034i	0.202370	+	0.673549i
$883$	26.2855			0.884577			0.442289	−	0.896873i	$-0.354167\pi$
$883$	26.2855			0.884577			0.442289	+	0.896873i	$0.354167\pi$
$884$	−14.3116			−0.481350
$885$	0.629235	+	4.28106i	0.0211515	+	0.143906i
$886$	−11.4636			−0.385127
$887$			21.1883i			0.711435i	0.934594	+	0.355717i	$0.115763\pi$
$887$			21.1883i			0.711435i	−0.934594	+	0.355717i	$0.884237\pi$
$888$	20.1997	−	2.96897i	0.677857	−	0.0996321i
$889$		−	2.68715i		−	0.0901241i
$890$		−	9.92316i		−	0.332625i
$891$	−27.6084	+	18.2363i	−0.924916	+	0.610938i
$892$	10.6634			0.357036
$893$	1.27090			0.0425291
$894$	2.27647	+	15.4882i	0.0761365	+	0.518002i
$895$			5.90397i			0.197348i
$896$	−0.194278			−0.00649038
$897$	−31.6718	−	4.63840i	−1.05749	−	0.154872i
$898$	−6.54091			−0.218273
$899$			20.4183i			0.680987i
$900$	−2.87312	+	0.863238i	−0.0957707	+	0.0287746i
$901$	−15.2246			−0.507205
$902$	35.4935			1.18180
$903$	−3.48022	+	0.511526i	−0.115814	+	0.0170225i
$904$		−	3.33657i		−	0.110973i
$905$		−	3.65493i		−	0.121494i
$906$	−1.81704	−	12.3624i	−0.0603672	−	0.410714i
$907$			30.7874i			1.02228i	0.859498	+	0.511140i	$0.170777\pi$
$907$			30.7874i			1.02228i	−0.859498	+	0.511140i	$0.829223\pi$
$908$	5.57422			0.184987
$909$	−7.24407	−	24.1105i	−0.240271	−	0.799694i
$910$	0.748652			0.0248176
$911$	43.6844			1.44733			0.723664	−	0.690152i	$-0.242456\pi$
$911$	43.6844			1.44733			0.723664	+	0.690152i	$0.242456\pi$
$912$	0.0688932	+	0.468721i	0.00228128	+	0.0155209i
$913$	−34.6583			−1.14702
$914$	30.6183			1.01276
$915$	−3.52288	−	23.9682i	−0.116463	−	0.792365i
$916$			14.9060i			0.492508i
$917$	−2.15827			−0.0712723
$918$	−8.18151	−	17.4779i	−0.270030	−	0.576856i
$919$		−	53.3411i		−	1.75956i	−0.475381	−	0.879780i	$-0.657690\pi$
$919$		−	53.3411i		−	1.75956i	0.475381	−	0.879780i	$-0.342310\pi$
$920$	1.37760	−	4.59371i	0.0454182	−	0.151450i
$921$	23.2253	−	3.41368i	0.765299	−	0.112485i
$922$	22.9570			0.756047
$923$			37.4895i			1.23398i
$924$	0.179898	+	1.22395i	0.00591822	+	0.0402652i
$925$		−	11.7876i		−	0.387573i
$926$			22.2784i			0.732114i
$927$	−2.89392	−	9.63185i	−0.0950488	−	0.316352i
$928$	−9.11317			−0.299154
$929$			40.8893i			1.34153i	0.741668	+	0.670767i	$0.234035\pi$
$929$			40.8893i			1.34153i	−0.741668	+	0.670767i	$0.765965\pi$
$930$	−0.564326	−	3.83945i	−0.0185050	−	0.125900i
$931$		−	1.90434i		−	0.0624123i
$932$		−	15.3425i		−	0.502559i
$933$	0.915317	+	6.22745i	0.0299661	+	0.203877i
$934$			21.8911i			0.716299i
$935$	−13.6538			−0.446526
$936$	−3.32649	−	11.0716i	−0.108730	−	0.361886i
$937$			12.8925i			0.421181i	0.977574	+	0.210590i	$0.0675386\pi$
$937$			12.8925i			0.421181i	−0.977574	+	0.210590i	$0.932461\pi$
$938$		−	2.13348i		−	0.0696607i
$939$	5.11266	+	34.7844i	0.166845	+	1.13515i
$940$		−	4.64640i		−	0.151549i
$941$	−28.2649			−0.921410			−0.460705	−	0.887553i	$-0.652403\pi$
$941$	−28.2649			−0.921410			−0.460705	+	0.887553i	$0.652403\pi$
$942$	1.19871	−	0.176187i	0.0390560	−	0.00574050i
$943$	−13.3000	+	44.3497i	−0.433106	+	1.44422i
$944$		−	2.49823i		−	0.0813104i
$945$	0.427983	+	0.914286i	0.0139223	+	0.0297417i
$946$	38.4313			1.24951
$947$			53.6576i			1.74364i	0.489829	+	0.871818i	$0.337059\pi$
$947$			53.6576i			1.74364i	−0.489829	+	0.871818i	$0.662941\pi$
$948$	−1.62174	−	11.0337i	−0.0526718	−	0.358358i
$949$	42.7034			1.38621
$950$	0.273524			0.00887429
$951$	5.56492	+	37.8615i	0.180455	+	1.22774i
$952$	−0.721531			−0.0233849
$953$	47.7407			1.54647			0.773236	−	0.634118i	$-0.218637\pi$
$953$	47.7407			1.54647			0.773236	+	0.634118i	$0.218637\pi$
$954$	−3.53872	−	11.7779i	−0.114570	−	0.381325i
$955$	−12.5903			−0.407414
$956$			11.6220i			0.375881i
$957$	8.43864	+	57.4130i	0.272782	+	1.85590i
$958$		−	5.71132i		−	0.184524i
$959$			1.28462i			0.0414827i
$960$	1.71364	−	0.251873i	0.0553075	−	0.00812916i
$961$	−25.9801			−0.838066
$962$	45.4235			1.46451
$963$	50.8181	−	15.2685i	1.63759	−	0.492019i
$964$			6.44721i			0.207651i
$965$	23.4908			0.756196
$966$	−1.59676	−	0.233849i	−0.0513750	−	0.00752398i
$967$	−7.74638			−0.249107			−0.124553	−	0.992213i	$-0.539750\pi$
$967$	−7.74638			−0.249107			−0.124553	+	0.992213i	$0.539750\pi$
$968$		−	2.51586i		−	0.0808629i
$969$	0.255863	+	1.74079i	0.00821950	+	0.0559221i
$970$	8.91482			0.286238
$971$	−54.7216			−1.75610			−0.878051	−	0.478568i	$-0.841156\pi$
$971$	−54.7216			−1.75610			−0.878051	+	0.478568i	$0.841156\pi$
$972$	11.6194	−	10.3918i	0.372693	−	0.333316i
$973$			0.458708i			0.0147055i
$974$			22.2185i			0.711928i
$975$	−6.60352	+	0.970593i	−0.211482	+	0.0310838i
$976$			13.9867i			0.447704i
$977$	−30.6187			−0.979578			−0.489789	−	0.871841i	$-0.662926\pi$
$977$	−30.6187			−0.979578			−0.489789	+	0.871841i	$0.662926\pi$
$978$	3.32949	+	22.6525i	0.106465	+	0.724348i
$979$	36.4814			1.16595
$980$	−6.96226			−0.222401
$981$	−5.96325	−	19.8475i	−0.190392	−	0.633683i
$982$	−9.94225			−0.317270
$983$	−18.1788			−0.579812			−0.289906	−	0.957055i	$-0.593624\pi$
$983$	−18.1788			−0.579812			−0.289906	+	0.957055i	$0.593624\pi$
$984$	−16.5442	+	2.43169i	−0.527410	+	0.0775193i
$985$			15.4325i			0.491721i
$986$	−33.8454			−1.07786
$987$	−1.54689	+	0.227364i	−0.0492381	+	0.00723708i
$988$			1.05403i			0.0335330i
$989$	−14.4008	+	48.0205i	−0.457919	+	1.52696i
$990$	−3.17360	−	10.5627i	−0.100864	−	0.335705i
$991$	28.0460			0.890912			0.445456	−	0.895304i	$-0.353042\pi$
$991$	28.0460			0.890912			0.445456	+	0.895304i	$0.353042\pi$
$992$			2.24052i			0.0711367i
$993$	47.6960	−	7.01042i	1.51359	−	0.222469i
$994$			1.89007i			0.0599494i
$995$			21.1429i			0.670276i
$996$	16.1549	−	2.37447i	0.511888	−	0.0752379i
$997$	−34.7005			−1.09898			−0.549488	−	0.835502i	$-0.685177\pi$
$997$	−34.7005			−1.09898			−0.549488	+	0.835502i	$0.685177\pi$
$998$			5.95037i			0.188356i
$999$	25.9673	+	55.4732i	0.821570	+	1.75509i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.2.e.b.551.9	yes	16
3.2	odd	2		690.2.e.a.551.1	✓	16
23.22	odd	2		690.2.e.a.551.9	yes	16
69.68	even	2	inner	690.2.e.b.551.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.e.a.551.1	✓	16	3.2	odd	2
690.2.e.a.551.9	yes	16	23.22	odd	2
690.2.e.b.551.1	yes	16	69.68	even	2	inner
690.2.e.b.551.9	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	690.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +(-1.71364 + 0.251873i) q^{3} -1.00000 q^{4} +1.00000 q^{5} +(-0.251873 - 1.71364i) q^{6} -0.194278i q^{7} -1.00000i q^{8} +(2.87312 - 0.863238i) q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +(-1.71364 + 0.251873i) q^{3} -1.00000 q^{4} +1.00000 q^{5} +(-0.251873 - 1.71364i) q^{6} -0.194278i q^{7} -1.00000i q^{8} +(2.87312 - 0.863238i) q^{9} +1.00000i q^{10} -3.67639 q^{11} +(1.71364 - 0.251873i) q^{12} +3.85351 q^{13} +0.194278 q^{14} +(-1.71364 + 0.251873i) q^{15} +1.00000 q^{16} +3.71391 q^{17} +(0.863238 + 2.87312i) q^{18} -0.273524i q^{19} -1.00000 q^{20} +(0.0489334 + 0.332923i) q^{21} -3.67639i q^{22} +(4.59371 + 1.37760i) q^{23} +(0.251873 + 1.71364i) q^{24} +1.00000 q^{25} +3.85351i q^{26} +(-4.70607 + 2.20294i) q^{27} +0.194278i q^{28} +9.11317i q^{29} +(-0.251873 - 1.71364i) q^{30} +2.24052 q^{31} +1.00000i q^{32} +(6.30001 - 0.925983i) q^{33} +3.71391i q^{34} -0.194278i q^{35} +(-2.87312 + 0.863238i) q^{36} -11.7876i q^{37} +0.273524 q^{38} +(-6.60352 + 0.970593i) q^{39} -1.00000i q^{40} +9.65442i q^{41} +(-0.332923 + 0.0489334i) q^{42} +10.4535i q^{43} +3.67639 q^{44} +(2.87312 - 0.863238i) q^{45} +(-1.37760 + 4.59371i) q^{46} +4.64640i q^{47} +(-1.71364 + 0.251873i) q^{48} +6.96226 q^{49} +1.00000i q^{50} +(-6.36430 + 0.935432i) q^{51} -3.85351 q^{52} -4.09935 q^{53} +(-2.20294 - 4.70607i) q^{54} -3.67639 q^{55} -0.194278 q^{56} +(0.0688932 + 0.468721i) q^{57} -9.11317 q^{58} -2.49823i q^{59} +(1.71364 - 0.251873i) q^{60} +13.9867i q^{61} +2.24052i q^{62} +(-0.167708 - 0.558185i) q^{63} -1.00000 q^{64} +3.85351 q^{65} +(0.925983 + 6.30001i) q^{66} -10.9816i q^{67} -3.71391 q^{68} +(-8.21895 - 1.20368i) q^{69} +0.194278 q^{70} +9.72869i q^{71} +(-0.863238 - 2.87312i) q^{72} +11.0817 q^{73} +11.7876 q^{74} +(-1.71364 + 0.251873i) q^{75} +0.273524i q^{76} +0.714243i q^{77} +(-0.970593 - 6.60352i) q^{78} -6.43875i q^{79} +1.00000 q^{80} +(7.50964 - 4.96037i) q^{81} -9.65442 q^{82} +9.42725 q^{83} +(-0.0489334 - 0.332923i) q^{84} +3.71391 q^{85} -10.4535 q^{86} +(-2.29536 - 15.6167i) q^{87} +3.67639i q^{88} -9.92316 q^{89} +(0.863238 + 2.87312i) q^{90} -0.748652i q^{91} +(-4.59371 - 1.37760i) q^{92} +(-3.83945 + 0.564326i) q^{93} -4.64640 q^{94} -0.273524i q^{95} +(-0.251873 - 1.71364i) q^{96} -8.91482i q^{97} +6.96226i q^{98} +(-10.5627 + 3.17360i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{5} + 2 q^{6} + 2 q^{9}+O(q^{10}) \) 16 * q - 16 * q^4 + 16 * q^5 + 2 * q^6 + 2 * q^9	\( 16 q - 16 q^{4} + 16 q^{5} + 2 q^{6} + 2 q^{9} + 12 q^{11} + 12 q^{14} + 16 q^{16} + 8 q^{18} - 16 q^{20} + 4 q^{21} - 4 q^{23} - 2 q^{24} + 16 q^{25} + 24 q^{27} + 2 q^{30} + 4 q^{31} + 28 q^{33} - 2 q^{36} + 16 q^{38} - 8 q^{39} - 12 q^{44} + 2 q^{45} - 4 q^{46} - 4 q^{49} + 2 q^{51} + 8 q^{53} - 26 q^{54} + 12 q^{55} - 12 q^{56} - 28 q^{57} - 8 q^{58} - 16 q^{64} - 10 q^{66} - 22 q^{69} + 12 q^{70} - 8 q^{72} - 16 q^{73} + 24 q^{74} - 12 q^{78} + 16 q^{80} + 22 q^{81} - 16 q^{82} + 40 q^{83} - 4 q^{84} + 40 q^{86} + 20 q^{87} - 80 q^{89} + 8 q^{90} + 4 q^{92} - 4 q^{93} - 24 q^{94} + 2 q^{96} + 8 q^{99}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^5 + 2 * q^6 + 2 * q^9 + 12 * q^11 + 12 * q^14 + 16 * q^16 + 8 * q^18 - 16 * q^20 + 4 * q^21 - 4 * q^23 - 2 * q^24 + 16 * q^25 + 24 * q^27 + 2 * q^30 + 4 * q^31 + 28 * q^33 - 2 * q^36 + 16 * q^38 - 8 * q^39 - 12 * q^44 + 2 * q^45 - 4 * q^46 - 4 * q^49 + 2 * q^51 + 8 * q^53 - 26 * q^54 + 12 * q^55 - 12 * q^56 - 28 * q^57 - 8 * q^58 - 16 * q^64 - 10 * q^66 - 22 * q^69 + 12 * q^70 - 8 * q^72 - 16 * q^73 + 24 * q^74 - 12 * q^78 + 16 * q^80 + 22 * q^81 - 16 * q^82 + 40 * q^83 - 4 * q^84 + 40 * q^86 + 20 * q^87 - 80 * q^89 + 8 * q^90 + 4 * q^92 - 4 * q^93 - 24 * q^94 + 2 * q^96 + 8 * q^99

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