LMFDB - Embedded newform 690.2.e.b.551.10

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			551.10
Root			$-1.14349 - 1.30094i$ of defining polynomial
Character	$\chi$	$=$	690.551
Dual form			690.2.e.b.551.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +(-1.30094 + 1.14349i) q^{3} -1.00000 q^{4} +1.00000 q^{5} +(-1.14349 - 1.30094i) q^{6} +1.02070i q^{7} -1.00000i q^{8} +(0.384868 - 2.97521i) q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +(-1.30094 + 1.14349i) q^{3} -1.00000 q^{4} +1.00000 q^{5} +(-1.14349 - 1.30094i) q^{6} +1.02070i q^{7} -1.00000i q^{8} +(0.384868 - 2.97521i) q^{9} +1.00000i q^{10} -0.336117 q^{11} +(1.30094 - 1.14349i) q^{12} -4.92792 q^{13} -1.02070 q^{14} +(-1.30094 + 1.14349i) q^{15} +1.00000 q^{16} -7.57164 q^{17} +(2.97521 + 0.384868i) q^{18} +3.95042i q^{19} -1.00000 q^{20} +(-1.16716 - 1.32787i) q^{21} -0.336117i q^{22} +(-2.61089 - 4.02284i) q^{23} +(1.14349 + 1.30094i) q^{24} +1.00000 q^{25} -4.92792i q^{26} +(2.90143 + 4.31065i) q^{27} -1.02070i q^{28} -2.20183i q^{29} +(-1.14349 - 1.30094i) q^{30} -1.27271 q^{31} +1.00000i q^{32} +(0.437267 - 0.384346i) q^{33} -7.57164i q^{34} +1.02070i q^{35} +(-0.384868 + 2.97521i) q^{36} -0.378822i q^{37} -3.95042 q^{38} +(6.41091 - 5.63502i) q^{39} -1.00000i q^{40} -7.82260i q^{41} +(1.32787 - 1.16716i) q^{42} -10.0534i q^{43} +0.336117 q^{44} +(0.384868 - 2.97521i) q^{45} +(4.02284 - 2.61089i) q^{46} +5.34859i q^{47} +(-1.30094 + 1.14349i) q^{48} +5.95817 q^{49} +1.00000i q^{50} +(9.85021 - 8.65808i) q^{51} +4.92792 q^{52} -13.2494 q^{53} +(-4.31065 + 2.90143i) q^{54} -0.336117 q^{55} +1.02070 q^{56} +(-4.51726 - 5.13924i) q^{57} +2.20183 q^{58} +2.53151i q^{59} +(1.30094 - 1.14349i) q^{60} +12.9663i q^{61} -1.27271i q^{62} +(3.03680 + 0.392835i) q^{63} -1.00000 q^{64} -4.92792 q^{65} +(0.384346 + 0.437267i) q^{66} +3.67460i q^{67} +7.57164 q^{68} +(7.99667 + 2.24794i) q^{69} -1.02070 q^{70} -10.8426i q^{71} +(-2.97521 - 0.384868i) q^{72} -7.22073 q^{73} +0.378822 q^{74} +(-1.30094 + 1.14349i) q^{75} -3.95042i q^{76} -0.343075i q^{77} +(5.63502 + 6.41091i) q^{78} +4.78248i q^{79} +1.00000 q^{80} +(-8.70375 - 2.29012i) q^{81} +7.82260 q^{82} +16.0711 q^{83} +(1.16716 + 1.32787i) q^{84} -7.57164 q^{85} +10.0534 q^{86} +(2.51776 + 2.86443i) q^{87} +0.336117i q^{88} -14.1373 q^{89} +(2.97521 + 0.384868i) q^{90} -5.02994i q^{91} +(2.61089 + 4.02284i) q^{92} +(1.65571 - 1.45533i) q^{93} -5.34859 q^{94} +3.95042i q^{95} +(-1.14349 - 1.30094i) q^{96} +11.6476i q^{97} +5.95817i q^{98} +(-0.129361 + 1.00002i) 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
$3$	−1.30094	+	1.14349i	−0.751096	+	0.660193i
$3$	−1.30094	+	1.14349i	−0.751096	+	0.660193i
$4$	−1.00000			−0.500000
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$	−1.14349	−	1.30094i	−0.466827	−	0.531105i
$7$			1.02070i			0.385789i	0.981220	+	0.192894i	$0.0617875\pi$
$7$			1.02070i			0.385789i	−0.981220	+	0.192894i	$0.938213\pi$
$8$		−	1.00000i		−	0.353553i
$9$	0.384868	−	2.97521i	0.128289	−	0.991737i
$10$			1.00000i			0.316228i
$11$	−0.336117			−0.101343			−0.0506716	−	0.998715i	$-0.516136\pi$
$11$	−0.336117			−0.101343			−0.0506716	+	0.998715i	$0.516136\pi$
$12$	1.30094	−	1.14349i	0.375548	−	0.330097i
$13$	−4.92792			−1.36676			−0.683380	−	0.730063i	$-0.739491\pi$
$13$	−4.92792			−1.36676			−0.683380	+	0.730063i	$0.739491\pi$
$14$	−1.02070			−0.272794
$15$	−1.30094	+	1.14349i	−0.335900	+	0.295247i
$16$	1.00000			0.250000
$17$	−7.57164			−1.83639			−0.918196	−	0.396127i	$-0.870354\pi$
$17$	−7.57164			−1.83639			−0.918196	+	0.396127i	$0.870354\pi$
$18$	2.97521	+	0.384868i	0.701264	+	0.0907141i
$19$			3.95042i			0.906289i	0.891437	+	0.453144i	$0.149698\pi$
$19$			3.95042i			0.906289i	−0.891437	+	0.453144i	$0.850302\pi$
$20$	−1.00000			−0.223607
$21$	−1.16716	−	1.32787i	−0.254695	−	0.289764i
$22$		−	0.336117i		−	0.0716604i
$23$	−2.61089	−	4.02284i	−0.544408	−	0.838821i
$23$	−2.61089	−	4.02284i	−0.544408	−	0.838821i
$24$	1.14349	+	1.30094i	0.233414	+	0.265552i
$25$	1.00000			0.200000
$26$		−	4.92792i		−	0.966445i
$27$	2.90143	+	4.31065i	0.558381	+	0.829585i
$28$		−	1.02070i		−	0.192894i
$29$		−	2.20183i		−	0.408869i	−0.978880	−	0.204434i	$-0.934464\pi$
$29$		−	2.20183i		−	0.408869i	0.978880	−	0.204434i	$-0.0655355\pi$
$30$	−1.14349	−	1.30094i	−0.208772	−	0.237517i
$31$	−1.27271			−0.228586			−0.114293	−	0.993447i	$-0.536460\pi$
$31$	−1.27271			−0.228586			−0.114293	+	0.993447i	$0.536460\pi$
$32$			1.00000i			0.176777i
$33$	0.437267	−	0.384346i	0.0761184	−	0.0669061i
$34$		−	7.57164i		−	1.29853i
$35$			1.02070i			0.172530i
$36$	−0.384868	+	2.97521i	−0.0641446	+	0.495868i
$37$		−	0.378822i		−	0.0622779i	−0.999515	−	0.0311390i	$-0.990087\pi$
$37$		−	0.378822i		−	0.0622779i	0.999515	−	0.0311390i	$-0.00991345\pi$
$38$	−3.95042			−0.640843
$39$	6.41091	−	5.63502i	1.02657	−	0.902326i
$40$		−	1.00000i		−	0.158114i
$41$		−	7.82260i		−	1.22169i	−0.791752	−	0.610843i	$-0.790831\pi$
$41$		−	7.82260i		−	1.22169i	0.791752	−	0.610843i	$-0.209169\pi$
$42$	1.32787	−	1.16716i	0.204894	−	0.180097i
$43$		−	10.0534i		−	1.53313i	−0.642168	−	0.766564i	$-0.721965\pi$
$43$		−	10.0534i		−	1.53313i	0.642168	−	0.766564i	$-0.278035\pi$
$44$	0.336117			0.0506716
$45$	0.384868	−	2.97521i	0.0573727	−	0.443518i
$46$	4.02284	−	2.61089i	0.593136	−	0.384954i
$47$			5.34859i			0.780172i	0.920778	+	0.390086i	$0.127555\pi$
$47$			5.34859i			0.780172i	−0.920778	+	0.390086i	$0.872445\pi$
$48$	−1.30094	+	1.14349i	−0.187774	+	0.165048i
$49$	5.95817			0.851167
$50$			1.00000i			0.141421i
$51$	9.85021	−	8.65808i	1.37931	−	1.21237i
$52$	4.92792			0.683380
$53$	−13.2494			−1.81995			−0.909975	−	0.414664i	$-0.863899\pi$
$53$	−13.2494			−1.81995			−0.909975	+	0.414664i	$0.863899\pi$
$54$	−4.31065	+	2.90143i	−0.586605	+	0.394835i
$55$	−0.336117			−0.0453220
$56$	1.02070			0.136397
$57$	−4.51726	−	5.13924i	−0.598326	−	0.680709i
$58$	2.20183			0.289114
$59$			2.53151i			0.329574i	0.986329	+	0.164787i	$0.0526937\pi$
$59$			2.53151i			0.329574i	−0.986329	+	0.164787i	$0.947306\pi$
$60$	1.30094	−	1.14349i	0.167950	−	0.147624i
$61$			12.9663i			1.66016i	0.557644	+	0.830080i	$0.311705\pi$
$61$			12.9663i			1.66016i	−0.557644	+	0.830080i	$0.688295\pi$
$62$		−	1.27271i		−	0.161634i
$63$	3.03680	+	0.392835i	0.382601	+	0.0494925i
$64$	−1.00000			−0.125000
$65$	−4.92792			−0.611234
$66$	0.384346	+	0.437267i	0.0473097	+	0.0538238i
$67$			3.67460i			0.448924i	0.974483	+	0.224462i	$0.0720625\pi$
$67$			3.67460i			0.448924i	−0.974483	+	0.224462i	$0.927938\pi$
$68$	7.57164			0.918196
$69$	7.99667	+	2.24794i	0.962686	+	0.270620i
$70$	−1.02070			−0.121997
$71$		−	10.8426i		−	1.28677i	−0.765541	−	0.643387i	$-0.777529\pi$
$71$		−	10.8426i		−	1.28677i	0.765541	−	0.643387i	$-0.222471\pi$
$72$	−2.97521	−	0.384868i	−0.350632	−	0.0453571i
$73$	−7.22073			−0.845123			−0.422561	−	0.906334i	$-0.638869\pi$
$73$	−7.22073			−0.845123			−0.422561	+	0.906334i	$0.638869\pi$
$74$	0.378822			0.0440372
$75$	−1.30094	+	1.14349i	−0.150219	+	0.132039i
$76$		−	3.95042i		−	0.453144i
$77$		−	0.343075i		−	0.0390970i
$78$	5.63502	+	6.41091i	0.638041	+	0.725893i
$79$			4.78248i			0.538070i	0.963130	+	0.269035i	$0.0867048\pi$
$79$			4.78248i			0.538070i	−0.963130	+	0.269035i	$0.913295\pi$
$80$	1.00000			0.111803
$81$	−8.70375	−	2.29012i	−0.967084	−	0.254458i
$82$	7.82260			0.863862
$83$	16.0711			1.76403			0.882017	−	0.471218i	$-0.156186\pi$
$83$	16.0711			1.76403			0.882017	+	0.471218i	$0.156186\pi$
$84$	1.16716	+	1.32787i	0.127348	+	0.144882i
$85$	−7.57164			−0.821259
$86$	10.0534			1.08408
$87$	2.51776	+	2.86443i	0.269933	+	0.307100i
$88$			0.336117i			0.0358302i
$89$	−14.1373			−1.49855			−0.749274	−	0.662260i	$-0.769597\pi$
$89$	−14.1373			−1.49855			−0.749274	+	0.662260i	$0.769597\pi$
$90$	2.97521	+	0.384868i	0.313615	+	0.0405686i
$91$		−	5.02994i		−	0.527281i
$92$	2.61089	+	4.02284i	0.272204	+	0.419410i
$93$	1.65571	−	1.45533i	0.171690	−	0.150911i
$94$	−5.34859			−0.551665
$95$			3.95042i			0.405305i
$96$	−1.14349	−	1.30094i	−0.116707	−	0.132776i
$97$			11.6476i			1.18263i	0.806441	+	0.591315i	$0.201391\pi$
$97$			11.6476i			1.18263i	−0.806441	+	0.591315i	$0.798609\pi$
$98$			5.95817i			0.601866i
$99$	−0.129361	+	1.00002i	−0.0130012	+	0.100506i
$100$	−1.00000			−0.100000
$101$		−	12.2651i		−	1.22042i	−0.792239	−	0.610211i	$-0.791085\pi$
$101$		−	12.2651i		−	1.22042i	0.792239	−	0.610211i	$-0.208915\pi$
$102$	8.65808	+	9.85021i	0.857278	+	0.975316i
$103$			6.63946i			0.654206i	0.944989	+	0.327103i	$0.106072\pi$
$103$			6.63946i			0.654206i	−0.944989	+	0.327103i	$0.893928\pi$
$104$			4.92792i			0.483223i
$105$	−1.16716	−	1.32787i	−0.113903	−	0.129587i
$106$		−	13.2494i		−	1.28690i
$107$	−8.84979			−0.855541			−0.427771	−	0.903887i	$-0.640701\pi$
$107$	−8.84979			−0.855541			−0.427771	+	0.903887i	$0.640701\pi$
$108$	−2.90143	−	4.31065i	−0.279190	−	0.414792i
$109$			13.4015i			1.28363i	0.766858	+	0.641817i	$0.221819\pi$
$109$			13.4015i			1.28363i	−0.766858	+	0.641817i	$0.778181\pi$
$110$		−	0.336117i		−	0.0320475i
$111$	0.433179	+	0.492823i	0.0411155	+	0.0467767i
$112$			1.02070i			0.0964472i
$113$	−1.71841			−0.161655			−0.0808274	−	0.996728i	$-0.525756\pi$
$113$	−1.71841			−0.161655			−0.0808274	+	0.996728i	$0.525756\pi$
$114$	5.13924	−	4.51726i	0.481334	−	0.423080i
$115$	−2.61089	−	4.02284i	−0.243467	−	0.375132i
$116$			2.20183i			0.204434i
$117$	−1.89660	+	14.6616i	−0.175341	+	1.35547i
$118$	−2.53151			−0.233044
$119$		−	7.72838i		−	0.708460i
$120$	1.14349	+	1.30094i	0.104386	+	0.118759i
$121$	−10.8870			−0.989730
$122$	−12.9663			−1.17391
$123$	8.94506	+	10.1767i	0.806549	+	0.917603i
$124$	1.27271			0.114293
$125$	1.00000			0.0894427
$126$	−0.392835	+	3.03680i	−0.0349965	+	0.270540i
$127$	17.6630			1.56733			0.783667	−	0.621181i	$-0.213347\pi$
$127$	17.6630			1.56733			0.783667	+	0.621181i	$0.213347\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	11.4959	+	13.0788i	1.01216	+	1.15153i
$130$		−	4.92792i		−	0.432207i
$131$			12.4875i			1.09104i	0.838098	+	0.545519i	$0.183667\pi$
$131$			12.4875i			1.09104i	−0.838098	+	0.545519i	$0.816333\pi$
$132$	−0.437267	+	0.384346i	−0.0380592	+	0.0334530i
$133$	−4.03220			−0.349636
$134$	−3.67460			−0.317437
$135$	2.90143	+	4.31065i	0.249715	+	0.371002i
$136$			7.57164i			0.649263i
$137$	0.245755			0.0209962			0.0104981	−	0.999945i	$-0.496658\pi$
$137$	0.245755			0.0209962			0.0104981	+	0.999945i	$0.496658\pi$
$138$	−2.24794	+	7.99667i	−0.191357	+	0.680722i
$139$	−5.22537			−0.443210			−0.221605	−	0.975137i	$-0.571130\pi$
$139$	−5.22537			−0.443210			−0.221605	+	0.975137i	$0.571130\pi$
$140$		−	1.02070i		−	0.0862650i
$141$	−6.11605	−	6.95817i	−0.515064	−	0.585984i
$142$	10.8426			0.909887
$143$	1.65636			0.138512
$144$	0.384868	−	2.97521i	0.0320723	−	0.247934i
$145$		−	2.20183i		−	0.182852i
$146$		−	7.22073i		−	0.597592i
$147$	−7.75119	+	6.81310i	−0.639308	+	0.561935i
$148$			0.378822i			0.0311390i
$149$	3.43726			0.281591			0.140796	−	0.990039i	$-0.455034\pi$
$149$	3.43726			0.281591			0.140796	+	0.990039i	$0.455034\pi$
$150$	−1.14349	−	1.30094i	−0.0933655	−	0.106221i
$151$	−19.8169			−1.61268			−0.806339	−	0.591454i	$-0.798554\pi$
$151$	−19.8169			−1.61268			−0.806339	+	0.591454i	$0.798554\pi$
$152$	3.95042			0.320421
$153$	−2.91408	+	22.5272i	−0.235589	+	1.82122i
$154$	0.343075			0.0276458
$155$	−1.27271			−0.102227
$156$	−6.41091	+	5.63502i	−0.513284	+	0.451163i
$157$			12.8500i			1.02554i	0.858525	+	0.512772i	$0.171381\pi$
$157$			12.8500i			1.02554i	−0.858525	+	0.512772i	$0.828619\pi$
$158$	−4.78248			−0.380473
$159$	17.2367	−	15.1506i	1.36696	−	1.20152i
$160$			1.00000i			0.0790569i
$161$	4.10612	−	2.66494i	0.323608	−	0.210026i
$162$	2.29012	−	8.70375i	0.179929	−	0.683831i
$163$	−7.48989			−0.586654			−0.293327	−	0.956012i	$-0.594762\pi$
$163$	−7.48989			−0.586654			−0.293327	+	0.956012i	$0.594762\pi$
$164$			7.82260i			0.610843i
$165$	0.437267	−	0.384346i	0.0340412	−	0.0299213i
$166$			16.0711i			1.24736i
$167$			10.1932i			0.788776i	0.918944	+	0.394388i	$0.129043\pi$
$167$			10.1932i			0.788776i	−0.918944	+	0.394388i	$0.870957\pi$
$168$	−1.32787	+	1.16716i	−0.102447	+	0.0900484i
$169$	11.2844			0.868033
$170$		−	7.57164i		−	0.580718i
$171$	11.7533	+	1.52039i	0.898800	+	0.116267i
$172$			10.0534i			0.766564i
$173$		−	4.41012i		−	0.335295i	−0.985847	−	0.167648i	$-0.946383\pi$
$173$		−	4.41012i		−	0.335295i	0.985847	−	0.167648i	$-0.0536171\pi$
$174$	−2.86443	+	2.51776i	−0.217152	+	0.190871i
$175$			1.02070i			0.0771578i
$176$	−0.336117			−0.0253358
$177$	−2.89475	−	3.29333i	−0.217583	−	0.247542i
$178$		−	14.1373i		−	1.05963i
$179$			7.99808i			0.597804i	0.954284	+	0.298902i	$0.0966204\pi$
$179$			7.99808i			0.597804i	−0.954284	+	0.298902i	$0.903380\pi$
$180$	−0.384868	+	2.97521i	−0.0286863	+	0.221759i
$181$		−	14.7941i		−	1.09963i	−0.835285	−	0.549817i	$-0.814698\pi$
$181$		−	14.7941i		−	1.09963i	0.835285	−	0.549817i	$-0.185302\pi$
$182$	5.02994			0.372844
$183$	−14.8268	−	16.8683i	−1.09603	−	1.24694i
$184$	−4.02284	+	2.61089i	−0.296568	+	0.192477i
$185$		−	0.378822i		−	0.0278515i
$186$	1.45533	+	1.65571i	0.106710	+	0.121403i
$187$	2.54496			0.186106
$188$		−	5.34859i		−	0.390086i
$189$	−4.39989	+	2.96150i	−0.320045	+	0.215417i
$190$	−3.95042			−0.286594
$191$	−13.0642			−0.945294			−0.472647	−	0.881252i	$-0.656701\pi$
$191$	−13.0642			−0.945294			−0.472647	+	0.881252i	$0.656701\pi$
$192$	1.30094	−	1.14349i	0.0938869	−	0.0825242i
$193$	−1.34388			−0.0967343			−0.0483672	−	0.998830i	$-0.515402\pi$
$193$	−1.34388			−0.0967343			−0.0483672	+	0.998830i	$0.515402\pi$
$194$	−11.6476			−0.836245
$195$	6.41091	−	5.63502i	0.459095	−	0.403532i
$196$	−5.95817			−0.425583
$197$		−	6.30397i		−	0.449139i	−0.974458	−	0.224570i	$-0.927902\pi$
$197$		−	6.30397i		−	0.449139i	0.974458	−	0.224570i	$-0.0720976\pi$
$198$	−1.00002	−	0.129361i	−0.0710683	−	0.00919325i
$199$			7.67517i			0.544078i	0.962286	+	0.272039i	$0.0876980\pi$
$199$			7.67517i			0.544078i	−0.962286	+	0.272039i	$0.912302\pi$
$200$		−	1.00000i		−	0.0707107i
$201$	−4.20186	−	4.78042i	−0.296377	−	0.337185i
$202$	12.2651			0.862969
$203$	2.24741			0.157737
$204$	−9.85021	+	8.65808i	−0.689653	+	0.606187i
$205$		−	7.82260i		−	0.546354i
$206$	−6.63946			−0.462593
$207$	−12.9737	+	6.21968i	−0.901731	+	0.432297i
$208$	−4.92792			−0.341690
$209$		−	1.32780i		−	0.0918461i
$210$	1.32787	−	1.16716i	0.0916315	−	0.0805417i
$211$	17.7592			1.22260			0.611298	−	0.791400i	$-0.290648\pi$
$211$	17.7592			1.22260			0.611298	+	0.791400i	$0.290648\pi$
$212$	13.2494			0.909975
$213$	12.3983	+	14.1055i	0.849520	+	0.966490i
$214$		−	8.84979i		−	0.604959i
$215$		−	10.0534i		−	0.685635i
$216$	4.31065	−	2.90143i	0.293303	−	0.197417i
$217$		−	1.29906i		−	0.0881858i
$218$	−13.4015			−0.907667
$219$	9.39371	−	8.25682i	0.634768	−	0.557945i
$220$	0.336117			0.0226610
$221$	37.3124			2.50991
$222$	−0.492823	+	0.433179i	−0.0330761	+	0.0290730i
$223$	−2.68511			−0.179808			−0.0899042	−	0.995950i	$-0.528656\pi$
$223$	−2.68511			−0.179808			−0.0899042	+	0.995950i	$0.528656\pi$
$224$	−1.02070			−0.0681985
$225$	0.384868	−	2.97521i	0.0256578	−	0.198347i
$226$		−	1.71841i		−	0.114307i
$227$	4.16668			0.276552			0.138276	−	0.990394i	$-0.455844\pi$
$227$	4.16668			0.276552			0.138276	+	0.990394i	$0.455844\pi$
$228$	4.51726	+	5.13924i	0.299163	+	0.340355i
$229$		−	9.82184i		−	0.649045i	−0.945878	−	0.324523i	$-0.894796\pi$
$229$		−	9.82184i		−	0.649045i	0.945878	−	0.324523i	$-0.105204\pi$
$230$	4.02284	−	2.61089i	0.265258	−	0.172157i
$231$	0.392303	+	0.446319i	0.0258116	+	0.0293656i
$232$	−2.20183			−0.144557
$233$			20.4118i			1.33722i	0.743613	+	0.668610i	$0.233110\pi$
$233$			20.4118i			1.33722i	−0.743613	+	0.668610i	$0.766890\pi$
$234$	−14.6616	−	1.89660i	−0.958459	−	0.123984i
$235$			5.34859i			0.348903i
$236$		−	2.53151i		−	0.164787i
$237$	−5.46871	−	6.22169i	−0.355231	−	0.404142i
$238$	7.72838			0.500957
$239$			8.99553i			0.581873i	0.956742	+	0.290936i	$0.0939668\pi$
$239$			8.99553i			0.581873i	−0.956742	+	0.290936i	$0.906033\pi$
$240$	−1.30094	+	1.14349i	−0.0839750	+	0.0738119i
$241$		−	9.97182i		−	0.642341i	−0.947021	−	0.321171i	$-0.895924\pi$
$241$		−	9.97182i		−	0.642341i	0.947021	−	0.321171i	$-0.104076\pi$
$242$		−	10.8870i		−	0.699844i
$243$	13.9418	−	6.97334i	0.894364	−	0.447340i
$244$		−	12.9663i		−	0.830080i
$245$	5.95817			0.380653
$246$	−10.1767	+	8.94506i	−0.648843	+	0.570316i
$247$		−	19.4674i		−	1.23868i
$248$			1.27271i			0.0808172i
$249$	−20.9075	+	18.3771i	−1.32496	+	1.16460i
$250$			1.00000i			0.0632456i
$251$	13.0633			0.824550			0.412275	−	0.911059i	$-0.364734\pi$
$251$	13.0633			0.824550			0.412275	+	0.911059i	$0.364734\pi$
$252$	−3.03680	−	0.392835i	−0.191301	−	0.0247463i
$253$	0.877564	+	1.35215i	0.0551720	+	0.0850087i
$254$			17.6630i			1.10827i
$255$	9.85021	−	8.65808i	0.616844	−	0.542190i
$256$	1.00000			0.0625000
$257$		−	4.77735i		−	0.298003i	−0.988837	−	0.149001i	$-0.952394\pi$
$257$		−	4.77735i		−	0.298003i	0.988837	−	0.149001i	$-0.0476059\pi$
$258$	−13.0788	+	11.4959i	−0.814251	+	0.715706i
$259$	0.386664			0.0240261
$260$	4.92792			0.305617
$261$	−6.55090	−	0.847412i	−0.405490	−	0.0524535i
$262$	−12.4875			−0.771480
$263$	3.18728			0.196536			0.0982680	−	0.995160i	$-0.468670\pi$
$263$	3.18728			0.196536			0.0982680	+	0.995160i	$0.468670\pi$
$264$	−0.384346	−	0.437267i	−0.0236549	−	0.0269119i
$265$	−13.2494			−0.813906
$266$		−	4.03220i		−	0.247230i
$267$	18.3917	−	16.1658i	1.12555	−	0.989331i
$268$		−	3.67460i		−	0.224462i
$269$		−	6.77166i		−	0.412876i	−0.978460	−	0.206438i	$-0.933813\pi$
$269$		−	6.77166i		−	0.412876i	0.978460	−	0.206438i	$-0.0661871\pi$
$270$	−4.31065	+	2.90143i	−0.262338	+	0.176575i
$271$	−6.43609			−0.390965			−0.195482	−	0.980707i	$-0.562627\pi$
$271$	−6.43609			−0.390965			−0.195482	+	0.980707i	$0.562627\pi$
$272$	−7.57164			−0.459098
$273$	5.75168	+	6.54363i	0.348107	+	0.396038i
$274$			0.245755i			0.0148466i
$275$	−0.336117			−0.0202686
$276$	−7.99667	−	2.24794i	−0.481343	−	0.135310i
$277$	4.47192			0.268692			0.134346	−	0.990934i	$-0.457107\pi$
$277$	4.47192			0.268692			0.134346	+	0.990934i	$0.457107\pi$
$278$		−	5.22537i		−	0.313397i
$279$	−0.489825	+	3.78658i	−0.0293251	+	0.226697i
$280$	1.02070			0.0609986
$281$	−30.8476			−1.84021			−0.920106	−	0.391669i	$-0.871898\pi$
$281$	−30.8476			−1.84021			−0.920106	+	0.391669i	$0.871898\pi$
$282$	6.95817	−	6.11605i	0.414353	−	0.364205i
$283$			12.4633i			0.740868i	0.928859	+	0.370434i	$0.120791\pi$
$283$			12.4633i			0.740868i	−0.928859	+	0.370434i	$0.879209\pi$
$284$			10.8426i			0.643387i
$285$	−4.51726	−	5.13924i	−0.267579	−	0.304423i
$286$			1.65636i			0.0979426i
$287$	7.98454			0.471313
$288$	2.97521	+	0.384868i	0.175316	+	0.0226785i
$289$	40.3297			2.37233
$290$	2.20183			0.129296
$291$	−13.3188	−	15.1527i	−0.780764	−	0.888268i
$292$	7.22073			0.422561
$293$	12.9662			0.757491			0.378745	−	0.925501i	$-0.376356\pi$
$293$	12.9662			0.757491			0.378745	+	0.925501i	$0.376356\pi$
$294$	−6.81310	−	7.75119i	−0.397348	−	0.452059i
$295$			2.53151i			0.147390i
$296$	−0.378822			−0.0220186
$297$	−0.975221	−	1.44888i	−0.0565880	−	0.0840727i
$298$			3.43726i			0.199115i
$299$	12.8663	+	19.8243i	0.744075	+	1.14647i
$300$	1.30094	−	1.14349i	0.0751096	−	0.0660193i
$301$	10.2615			0.591464
$302$		−	19.8169i		−	1.14034i
$303$	14.0250	+	15.9561i	0.805715	+	0.916654i
$304$			3.95042i			0.226572i
$305$			12.9663i			0.742446i
$306$	−22.5272	−	2.91408i	−1.28780	−	0.166587i
$307$	−24.4391			−1.39481			−0.697406	−	0.716676i	$-0.745663\pi$
$307$	−24.4391			−1.39481			−0.697406	+	0.716676i	$0.745663\pi$
$308$			0.343075i			0.0195485i
$309$	−7.59215	−	8.63752i	−0.431902	−	0.491371i
$310$		−	1.27271i		−	0.0722851i
$311$			20.4152i			1.15764i	0.815455	+	0.578820i	$0.196487\pi$
$311$			20.4152i			1.15764i	−0.815455	+	0.578820i	$0.803513\pi$
$312$	−5.63502	−	6.41091i	−0.319020	−	0.362946i
$313$		−	27.4387i		−	1.55093i	−0.631392	−	0.775464i	$-0.717516\pi$
$313$		−	27.4387i		−	1.55093i	0.631392	−	0.775464i	$-0.282484\pi$
$314$	−12.8500			−0.725169
$315$	3.03680	+	0.392835i	0.171104	+	0.0221337i
$316$		−	4.78248i		−	0.269035i
$317$			2.34160i			0.131517i	0.997836	+	0.0657587i	$0.0209468\pi$
$317$			2.34160i			0.131517i	−0.997836	+	0.0657587i	$0.979053\pi$
$318$	15.1506	+	17.2367i	0.849602	+	0.966584i
$319$			0.740072i			0.0414361i
$320$	−1.00000			−0.0559017
$321$	11.5130	−	10.1196i	0.642593	−	0.564823i
$322$	2.66494	+	4.10612i	0.148511	+	0.228825i
$323$		−	29.9112i		−	1.66430i
$324$	8.70375	+	2.29012i	0.483542	+	0.127229i
$325$	−4.92792			−0.273352
$326$		−	7.48989i		−	0.414827i
$327$	−15.3245	−	17.4345i	−0.847447	−	0.964132i
$328$	−7.82260			−0.431931
$329$	−5.45931			−0.300982
$330$	0.384346	+	0.437267i	0.0211576	+	0.0240707i
$331$	27.4881			1.51088			0.755442	−	0.655216i	$-0.227422\pi$
$331$	27.4881			1.51088			0.755442	+	0.655216i	$0.227422\pi$
$332$	−16.0711			−0.882017
$333$	−1.12708	−	0.145796i	−0.0617633	−	0.00798959i
$334$	−10.1932			−0.557749
$335$			3.67460i			0.200765i
$336$	−1.16716	−	1.32787i	−0.0636738	−	0.0724411i
$337$			22.1863i			1.20857i	0.796770	+	0.604283i	$0.206540\pi$
$337$			22.1863i			1.20857i	−0.796770	+	0.604283i	$0.793460\pi$
$338$			11.2844i			0.613792i
$339$	2.23555	−	1.96499i	0.121418	−	0.106723i
$340$	7.57164			0.410630
$341$	0.427780			0.0231656
$342$	−1.52039	+	11.7533i	−0.0822132	+	0.635547i
$343$			13.2264i			0.714160i
$344$	−10.0534			−0.542042
$345$	7.99667	+	2.24794i	0.430526	+	0.121025i
$346$	4.41012			0.237090
$347$			24.4010i			1.30992i	0.755665	+	0.654958i	$0.227314\pi$
$347$			24.4010i			1.30992i	−0.755665	+	0.654958i	$0.772686\pi$
$348$	−2.51776	−	2.86443i	−0.134966	−	0.153550i
$349$	−11.9864			−0.641618			−0.320809	−	0.947144i	$-0.603955\pi$
$349$	−11.9864			−0.641618			−0.320809	+	0.947144i	$0.603955\pi$
$350$	−1.02070			−0.0545588
$351$	−14.2980	−	21.2425i	−0.763172	−	1.13384i
$352$		−	0.336117i		−	0.0179151i
$353$		−	22.4575i		−	1.19529i	−0.801761	−	0.597645i	$-0.796103\pi$
$353$		−	22.4575i		−	1.19529i	0.801761	−	0.597645i	$-0.203897\pi$
$354$	3.29333	−	2.89475i	0.175039	−	0.153854i
$355$		−	10.8426i		−	0.575463i
$356$	14.1373			0.749274
$357$	8.83732	+	10.0541i	0.467720	+	0.532121i
$358$	−7.99808			−0.422712
$359$	10.4831			0.553277			0.276638	−	0.960974i	$-0.410780\pi$
$359$	10.4831			0.553277			0.276638	+	0.960974i	$0.410780\pi$
$360$	−2.97521	−	0.384868i	−0.156807	−	0.0202843i
$361$	3.39418			0.178641
$362$	14.7941			0.777558
$363$	14.1633	−	12.4492i	0.743382	−	0.653413i
$364$			5.02994i			0.263640i
$365$	−7.22073			−0.377950
$366$	16.8683	−	14.8268i	0.881719	−	0.775008i
$367$		−	22.8694i		−	1.19377i	−0.802326	−	0.596886i	$-0.796404\pi$
$367$		−	22.8694i		−	1.19377i	0.802326	−	0.596886i	$-0.203596\pi$
$368$	−2.61089	−	4.02284i	−0.136102	−	0.209705i
$369$	−23.2739	−	3.01067i	−1.21159	−	0.156729i
$370$	0.378822			0.0196940
$371$		−	13.5237i		−	0.702116i
$372$	−1.65571	+	1.45533i	−0.0858448	+	0.0754554i
$373$		−	5.47431i		−	0.283449i	−0.989906	−	0.141725i	$-0.954735\pi$
$373$		−	5.47431i		−	0.283449i	0.989906	−	0.141725i	$-0.0452647\pi$
$374$			2.54496i			0.131597i
$375$	−1.30094	+	1.14349i	−0.0671800	+	0.0590495i
$376$	5.34859			0.275832
$377$			10.8504i			0.558826i
$378$	−2.96150	−	4.39989i	−0.152323	−	0.226306i
$379$		−	30.3390i		−	1.55841i	−0.626768	−	0.779206i	$-0.715623\pi$
$379$		−	30.3390i		−	1.55841i	0.626768	−	0.779206i	$-0.284377\pi$
$380$		−	3.95042i		−	0.202652i
$381$	−22.9784	+	20.1974i	−1.17722	+	1.03474i
$382$		−	13.0642i		−	0.668424i
$383$	14.8764			0.760149			0.380075	−	0.924956i	$-0.375898\pi$
$383$	14.8764			0.760149			0.380075	+	0.924956i	$0.375898\pi$
$384$	1.14349	+	1.30094i	0.0583534	+	0.0663881i
$385$		−	0.343075i		−	0.0174847i
$386$		−	1.34388i		−	0.0684015i
$387$	−29.9110	−	3.86922i	−1.52046	−	0.196684i
$388$		−	11.6476i		−	0.591315i
$389$	12.5719			0.637422			0.318711	−	0.947852i	$-0.396750\pi$
$389$	12.5719			0.637422			0.318711	+	0.947852i	$0.396750\pi$
$390$	5.63502	+	6.41091i	0.285341	+	0.324629i
$391$	19.7687	+	30.4595i	0.999746	+	1.54040i
$392$		−	5.95817i		−	0.300933i
$393$	−14.2793	−	16.2454i	−0.720296	−	0.819474i
$394$	6.30397			0.317589
$395$			4.78248i			0.240632i
$396$	0.129361	−	1.00002i	0.00650061	−	0.0502528i
$397$	24.1720			1.21316			0.606579	−	0.795023i	$-0.292542\pi$
$397$	24.1720			1.21316			0.606579	+	0.795023i	$0.292542\pi$
$398$	−7.67517			−0.384721
$399$	5.24563	−	4.61078i	0.262610	−	0.230827i
$400$	1.00000			0.0500000
$401$	−21.4707			−1.07220			−0.536098	−	0.844155i	$-0.680102\pi$
$401$	−21.4707			−1.07220			−0.536098	+	0.844155i	$0.680102\pi$
$402$	4.78042	−	4.20186i	0.238426	−	0.209570i
$403$	6.27182			0.312422
$404$			12.2651i			0.610211i
$405$	−8.70375	−	2.29012i	−0.432493	−	0.113797i
$406$			2.24741i			0.111537i
$407$			0.127329i			0.00631144i
$408$	−8.65808	−	9.85021i	−0.428639	−	0.487658i
$409$	−13.5579			−0.670397			−0.335198	−	0.942148i	$-0.608803\pi$
$409$	−13.5579			−0.670397			−0.335198	+	0.942148i	$0.608803\pi$
$410$	7.82260			0.386331
$411$	−0.319711	+	0.281018i	−0.0157702	+	0.0138616i
$412$		−	6.63946i		−	0.327103i
$413$	−2.58391			−0.127146
$414$	−6.21968	−	12.9737i	−0.305680	−	0.637620i
$415$	16.0711			0.788900
$416$		−	4.92792i		−	0.241611i
$417$	6.79787	−	5.97515i	0.332893	−	0.292604i
$418$	1.32780			0.0649450
$419$	3.19459			0.156066			0.0780330	−	0.996951i	$-0.475136\pi$
$419$	3.19459			0.156066			0.0780330	+	0.996951i	$0.475136\pi$
$420$	1.16716	+	1.32787i	0.0569516	+	0.0647933i
$421$			23.8342i			1.16161i	0.814043	+	0.580805i	$0.197262\pi$
$421$			23.8342i			1.16161i	−0.814043	+	0.580805i	$0.802738\pi$
$422$			17.7592i			0.864506i
$423$	15.9132	+	2.05850i	0.773725	+	0.100088i
$424$			13.2494i			0.643449i
$425$	−7.57164			−0.367278
$426$	−14.1055	+	12.3983i	−0.683412	+	0.600701i
$427$	−13.2347			−0.640471
$428$	8.84979			0.427771
$429$	−2.15482	+	1.89403i	−0.104036	+	0.0914445i
$430$	10.0534			0.484818
$431$	−14.0920			−0.678787			−0.339394	−	0.940644i	$-0.610222\pi$
$431$	−14.0920			−0.678787			−0.339394	+	0.940644i	$0.610222\pi$
$432$	2.90143	+	4.31065i	0.139595	+	0.207396i
$433$		−	24.3424i		−	1.16982i	−0.811098	−	0.584910i	$-0.801130\pi$
$433$		−	24.3424i		−	1.16982i	0.811098	−	0.584910i	$-0.198870\pi$
$434$	1.29906			0.0623568
$435$	2.51776	+	2.86443i	0.120718	+	0.137339i
$436$		−	13.4015i		−	0.641817i
$437$	15.8919	−	10.3141i	0.760214	−	0.493391i
$438$	8.25682	+	9.39371i	0.394526	+	0.448849i
$439$	−1.40497			−0.0670555			−0.0335277	−	0.999438i	$-0.510674\pi$
$439$	−1.40497			−0.0670555			−0.0335277	+	0.999438i	$0.510674\pi$
$440$			0.336117i			0.0160238i
$441$	2.29311	−	17.7268i	0.109196	−	0.844134i
$442$			37.3124i			1.77477i
$443$		−	39.5844i		−	1.88071i	−0.340195	−	0.940355i	$-0.610493\pi$
$443$		−	39.5844i		−	1.88071i	0.340195	−	0.940355i	$-0.389507\pi$
$444$	−0.433179	−	0.492823i	−0.0205577	−	0.0233883i
$445$	−14.1373			−0.670171
$446$		−	2.68511i		−	0.127144i
$447$	−4.47165	+	3.93047i	−0.211502	+	0.185905i
$448$		−	1.02070i		−	0.0482236i
$449$		−	29.0379i		−	1.37038i	−0.728364	−	0.685191i	$-0.759719\pi$
$449$		−	29.0379i		−	1.37038i	0.728364	−	0.685191i	$-0.240281\pi$
$450$	2.97521	+	0.384868i	0.140253	+	0.0181428i
$451$			2.62931i			0.123809i
$452$	1.71841			0.0808274
$453$	25.7805	−	22.6604i	1.21128	−	1.06468i
$454$			4.16668i			0.195552i
$455$		−	5.02994i		−	0.235807i
$456$	−5.13924	+	4.51726i	−0.240667	+	0.211540i
$457$		−	35.5772i		−	1.66423i	−0.554603	−	0.832115i	$-0.687130\pi$
$457$		−	35.5772i		−	1.66423i	0.554603	−	0.832115i	$-0.312870\pi$
$458$	9.82184			0.458944
$459$	−21.9686	−	32.6387i	−1.02541	−	1.52344i
$460$	2.61089	+	4.02284i	0.121733	+	0.187566i
$461$			29.4329i			1.37083i	0.728153	+	0.685415i	$0.240379\pi$
$461$			29.4329i			1.37083i	−0.728153	+	0.685415i	$0.759621\pi$
$462$	−0.446319	+	0.392303i	−0.0207646	+	0.0182516i
$463$	−12.4050			−0.576508			−0.288254	−	0.957554i	$-0.593075\pi$
$463$	−12.4050			−0.576508			−0.288254	+	0.957554i	$0.593075\pi$
$464$		−	2.20183i		−	0.102217i
$465$	1.65571	−	1.45533i	0.0767819	−	0.0674893i
$466$	−20.4118			−0.945557
$467$	−37.4350			−1.73229			−0.866143	−	0.499796i	$-0.833408\pi$
$467$	−37.4350			−1.73229			−0.866143	+	0.499796i	$0.833408\pi$
$468$	1.89660	−	14.6616i	0.0876703	−	0.677733i
$469$	−3.75067			−0.173190
$470$	−5.34859			−0.246712
$471$	−14.6939	−	16.7171i	−0.677057	−	0.770281i
$472$	2.53151			0.116522
$473$			3.37912i			0.155372i
$474$	6.22169	−	5.46871i	0.285772	−	0.251186i
$475$			3.95042i			0.181258i
$476$			7.72838i			0.354230i
$477$	−5.09927	+	39.4198i	−0.233480	+	1.80491i
$478$	−8.99553			−0.411446
$479$	−18.9516			−0.865920			−0.432960	−	0.901413i	$-0.642531\pi$
$479$	−18.9516			−0.865920			−0.432960	+	0.901413i	$0.642531\pi$
$480$	−1.14349	−	1.30094i	−0.0521929	−	0.0593793i
$481$			1.86681i			0.0851190i
$482$	9.97182			0.454204
$483$	−2.29448	+	8.16221i	−0.104402	+	0.371394i
$484$	10.8870			0.494865
$485$			11.6476i			0.528888i
$486$	6.97334	+	13.9418i	0.316317	+	0.632411i
$487$	−41.8360			−1.89577			−0.947885	−	0.318612i	$-0.896783\pi$
$487$	−41.8360			−1.89577			−0.947885	+	0.318612i	$0.896783\pi$
$488$	12.9663			0.586955
$489$	9.74387	−	8.56461i	0.440633	−	0.387305i
$490$			5.95817i			0.269163i
$491$			35.6738i			1.60994i	0.593318	+	0.804968i	$0.297818\pi$
$491$			35.6738i			1.60994i	−0.593318	+	0.804968i	$0.702182\pi$
$492$	−8.94506	−	10.1767i	−0.403274	−	0.458801i
$493$			16.6714i			0.750844i
$494$	19.4674			0.875878
$495$	−0.129361	+	1.00002i	−0.00581432	+	0.0449475i
$496$	−1.27271			−0.0571464
$497$	11.0670			0.496423
$498$	−18.3771	−	20.9075i	−0.823499	−	0.936887i
$499$	3.39015			0.151764			0.0758821	−	0.997117i	$-0.475823\pi$
$499$	3.39015			0.151764			0.0758821	+	0.997117i	$0.475823\pi$
$500$	−1.00000			−0.0447214
$501$	−11.6558	−	13.2607i	−0.520745	−	0.592446i
$502$			13.0633i			0.583045i
$503$	2.86791			0.127874			0.0639369	−	0.997954i	$-0.479634\pi$
$503$	2.86791			0.127874			0.0639369	+	0.997954i	$0.479634\pi$
$504$	0.392835	−	3.03680i	0.0174983	−	0.135270i
$505$		−	12.2651i		−	0.545790i
$506$	−1.35215	+	0.877564i	−0.0601102	+	0.0390125i
$507$	−14.6803	+	12.9036i	−0.651976	+	0.573070i
$508$	−17.6630			−0.783667
$509$		−	4.03161i		−	0.178698i	−0.996000	−	0.0893491i	$-0.971521\pi$
$509$		−	4.03161i		−	0.178698i	0.996000	−	0.0893491i	$-0.0284787\pi$
$510$	8.65808	+	9.85021i	0.383386	+	0.436175i
$511$		−	7.37021i		−	0.326039i
$512$			1.00000i			0.0441942i
$513$	−17.0289	+	11.4619i	−0.751843	+	0.506054i
$514$	4.77735			0.210720
$515$			6.63946i			0.292570i
$516$	−11.4959	−	13.0788i	−0.506080	−	0.575763i
$517$		−	1.79775i		−	0.0790650i
$518$			0.386664i			0.0169890i
$519$	5.04292	+	5.73728i	0.221360	+	0.251839i
$520$			4.92792i			0.216104i
$521$	14.2955			0.626296			0.313148	−	0.949704i	$-0.398616\pi$
$521$	14.2955			0.626296			0.313148	+	0.949704i	$0.398616\pi$
$522$	0.847412	−	6.55090i	0.0370902	−	0.286725i
$523$			8.12221i			0.355159i	0.984106	+	0.177580i	$0.0568267\pi$
$523$			8.12221i			0.355159i	−0.984106	+	0.177580i	$0.943173\pi$
$524$		−	12.4875i		−	0.545519i
$525$	−1.16716	−	1.32787i	−0.0509391	−	0.0579529i
$526$			3.18728i			0.138972i
$527$	9.63650			0.419773
$528$	0.437267	−	0.384346i	0.0190296	−	0.0167265i
$529$	−9.36653	+	21.0064i	−0.407241	+	0.913321i
$530$		−	13.2494i		−	0.575518i
$531$	7.53177	+	0.974295i	0.326851	+	0.0422808i
$532$	4.03220			0.174818
$533$			38.5492i			1.66975i
$534$	16.1658	+	18.3917i	0.699563	+	0.795886i
$535$	−8.84979			−0.382610
$536$	3.67460			0.158719
$537$	−9.14571	−	10.4050i	−0.394667	−	0.449008i
$538$	6.77166			0.291947
$539$	−2.00264			−0.0862599
$540$	−2.90143	−	4.31065i	−0.124858	−	0.185501i
$541$	−1.95561			−0.0840783			−0.0420391	−	0.999116i	$-0.513385\pi$
$541$	−1.95561			−0.0840783			−0.0420391	+	0.999116i	$0.513385\pi$
$542$		−	6.43609i		−	0.276454i
$543$	16.9168	+	19.2461i	0.725971	+	0.825930i
$544$		−	7.57164i		−	0.324631i
$545$			13.4015i			0.574059i
$546$	−6.54363	+	5.75168i	−0.280041	+	0.246149i
$547$	33.2009			1.41957			0.709784	−	0.704420i	$-0.248793\pi$
$547$	33.2009			1.41957			0.709784	+	0.704420i	$0.248793\pi$
$548$	−0.245755			−0.0104981
$549$	38.5774	+	4.99029i	1.64644	+	0.212981i
$550$		−	0.336117i		−	0.0143321i
$551$	8.69814			0.370553
$552$	2.24794	−	7.99667i	0.0956787	−	0.340361i
$553$	−4.88148			−0.207582
$554$			4.47192i			0.189994i
$555$	0.433179	+	0.492823i	0.0183874	+	0.0209192i
$556$	5.22537			0.221605
$557$	−5.11839			−0.216873			−0.108437	−	0.994103i	$-0.534584\pi$
$557$	−5.11839			−0.216873			−0.108437	+	0.994103i	$0.534584\pi$
$558$	−3.78658	−	0.489825i	−0.160299	−	0.0207359i
$559$			49.5423i			2.09542i
$560$			1.02070i			0.0431325i
$561$	−3.31082	+	2.91013i	−0.139783	+	0.122866i
$562$		−	30.8476i		−	1.30123i
$563$	34.6454			1.46013			0.730065	−	0.683378i	$-0.239490\pi$
$563$	34.6454			1.46013			0.730065	+	0.683378i	$0.239490\pi$
$564$	6.11605	+	6.95817i	0.257532	+	0.292992i
$565$	−1.71841			−0.0722943
$566$	−12.4633			−0.523873
$567$	2.33753	−	8.88393i	0.0981671	−	0.373090i
$568$	−10.8426			−0.454943
$569$	2.79385			0.117124			0.0585621	−	0.998284i	$-0.481348\pi$
$569$	2.79385			0.117124			0.0585621	+	0.998284i	$0.481348\pi$
$570$	5.13924	−	4.51726i	0.215259	−	0.189207i
$571$			18.8564i			0.789118i	0.918871	+	0.394559i	$0.129103\pi$
$571$			18.8564i			0.789118i	−0.918871	+	0.394559i	$0.870897\pi$
$572$	−1.65636			−0.0692559
$573$	16.9957	−	14.9388i	0.710006	−	0.624077i
$574$			7.98454i			0.333268i
$575$	−2.61089	−	4.02284i	−0.108882	−	0.167764i
$576$	−0.384868	+	2.97521i	−0.0160361	+	0.123967i
$577$	12.8591			0.535331			0.267665	−	0.963512i	$-0.413748\pi$
$577$	12.8591			0.535331			0.267665	+	0.963512i	$0.413748\pi$
$578$			40.3297i			1.67749i
$579$	1.74830	−	1.53671i	0.0726567	−	0.0638634i
$580$			2.20183i			0.0914259i
$581$			16.4038i			0.680545i
$582$	15.1527	−	13.3188i	0.628100	−	0.552084i
$583$	4.45336			0.184439
$584$			7.22073i			0.298796i
$585$	−1.89660	+	14.6616i	−0.0784147	+	0.606183i
$586$			12.9662i			0.535627i
$587$			4.49967i			0.185721i	0.995679	+	0.0928607i	$0.0296011\pi$
$587$			4.49967i			0.185721i	−0.995679	+	0.0928607i	$0.970399\pi$
$588$	7.75119	−	6.81310i	0.319654	−	0.280967i
$589$		−	5.02774i		−	0.207165i
$590$	−2.53151			−0.104221
$591$	7.20851	+	8.20106i	0.296519	+	0.337346i
$592$		−	0.378822i		−	0.0155695i
$593$			6.11300i			0.251031i	0.992092	+	0.125515i	$0.0400584\pi$
$593$			6.11300i			0.251031i	−0.992092	+	0.125515i	$0.959942\pi$
$594$	1.44888	−	0.975221i	0.0594484	−	0.0400138i
$595$		−	7.72838i		−	0.316833i
$596$	−3.43726			−0.140796
$597$	−8.77647	−	9.98490i	−0.359197	−	0.408655i
$598$	−19.8243	+	12.8663i	−0.810674	+	0.526140i
$599$		−	30.8437i		−	1.26024i	−0.776498	−	0.630120i	$-0.783006\pi$
$599$		−	30.8437i		−	1.26024i	0.776498	−	0.630120i	$-0.216994\pi$
$600$	1.14349	+	1.30094i	0.0466827	+	0.0531105i
$601$	20.6559			0.842574			0.421287	−	0.906927i	$-0.361579\pi$
$601$	20.6559			0.842574			0.421287	+	0.906927i	$0.361579\pi$
$602$			10.2615i			0.418228i
$603$	10.9327	+	1.41423i	0.445214	+	0.0575921i
$604$	19.8169			0.806339
$605$	−10.8870			−0.442621
$606$	−15.9561	+	14.0250i	−0.648172	+	0.569727i
$607$	−37.9955			−1.54219			−0.771095	−	0.636720i	$-0.780291\pi$
$607$	−37.9955			−1.54219			−0.771095	+	0.636720i	$0.780291\pi$
$608$	−3.95042			−0.160211
$609$	−2.92373	+	2.56989i	−0.118476	+	0.104137i
$610$	−12.9663			−0.524989
$611$		−	26.3574i		−	1.06631i
$612$	2.91408	−	22.5272i	0.117795	−	0.910609i
$613$			38.5259i			1.55604i	0.628237	+	0.778022i	$0.283777\pi$
$613$			38.5259i			1.55604i	−0.628237	+	0.778022i	$0.716223\pi$
$614$		−	24.4391i		−	0.986281i
$615$	8.94506	+	10.1767i	0.360700	+	0.410364i
$616$	−0.343075			−0.0138229
$617$	18.7768			0.755925			0.377963	−	0.925821i	$-0.376625\pi$
$617$	18.7768			0.755925			0.377963	+	0.925821i	$0.376625\pi$
$618$	8.63752	−	7.59215i	0.347452	−	0.305401i
$619$			10.9690i			0.440881i	0.975400	+	0.220441i	$0.0707496\pi$
$619$			10.9690i			0.440881i	−0.975400	+	0.220441i	$0.929250\pi$
$620$	1.27271			0.0511133
$621$	9.76575	−	22.9266i	0.391886	−	0.920014i
$622$	−20.4152			−0.818575
$623$		−	14.4299i		−	0.578123i
$624$	6.41091	−	5.63502i	0.256642	−	0.225582i
$625$	1.00000			0.0400000
$626$	27.4387			1.09667
$627$	1.51833	+	1.72739i	0.0606362	+	0.0689852i
$628$		−	12.8500i		−	0.512772i
$629$			2.86830i			0.114367i
$630$	−0.392835	+	3.03680i	−0.0156509	+	0.120989i
$631$			22.4999i			0.895709i	0.894107	+	0.447854i	$0.147812\pi$
$631$			22.4999i			0.895709i	−0.894107	+	0.447854i	$0.852188\pi$
$632$	4.78248			0.190237
$633$	−23.1036	+	20.3075i	−0.918287	+	0.807150i
$634$	−2.34160			−0.0929969
$635$	17.6630			0.700933
$636$	−17.2367	+	15.1506i	−0.683478	+	0.600759i
$637$	−29.3614			−1.16334
$638$	−0.740072			−0.0292997
$639$	−32.2589	−	4.17295i	−1.27614	−	0.165079i
$640$		−	1.00000i		−	0.0395285i
$641$	10.2836			0.406178			0.203089	−	0.979160i	$-0.434902\pi$
$641$	10.2836			0.406178			0.203089	+	0.979160i	$0.434902\pi$
$642$	10.1196	+	11.5130i	0.399390	+	0.454382i
$643$		−	37.6763i		−	1.48581i	−0.669399	−	0.742903i	$-0.733448\pi$
$643$		−	37.6763i		−	1.48581i	0.669399	−	0.742903i	$-0.266552\pi$
$644$	−4.10612	+	2.66494i	−0.161804	+	0.105013i
$645$	11.4959	+	13.0788i	0.452652	+	0.514978i
$646$	29.9112			1.17684
$647$		−	27.6384i		−	1.08658i	−0.839546	−	0.543289i	$-0.817179\pi$
$647$		−	27.6384i		−	1.08658i	0.839546	−	0.543289i	$-0.182821\pi$
$648$	−2.29012	+	8.70375i	−0.0899646	+	0.341916i
$649$		−	0.850883i		−	0.0334001i
$650$		−	4.92792i		−	0.193289i
$651$	1.48546	+	1.68999i	0.0582197	+	0.0662360i
$652$	7.48989			0.293327
$653$			23.6302i			0.924720i	0.886692	+	0.462360i	$0.152997\pi$
$653$			23.6302i			0.924720i	−0.886692	+	0.462360i	$0.847003\pi$
$654$	17.4345	−	15.3245i	0.681744	−	0.599236i
$655$			12.4875i			0.487927i
$656$		−	7.82260i		−	0.305421i
$657$	−2.77903	+	21.4832i	−0.108420	+	0.838139i
$658$		−	5.45931i		−	0.212826i
$659$	4.24256			0.165267			0.0826333	−	0.996580i	$-0.473667\pi$
$659$	4.24256			0.165267			0.0826333	+	0.996580i	$0.473667\pi$
$660$	−0.437267	+	0.384346i	−0.0170206	+	0.0149607i
$661$		−	1.60933i		−	0.0625956i	−0.999510	−	0.0312978i	$-0.990036\pi$
$661$		−	1.60933i		−	0.0625956i	0.999510	−	0.0312978i	$-0.00996402\pi$
$662$			27.4881i			1.06836i
$663$	−48.5411	+	42.6664i	−1.88518	+	1.65702i
$664$		−	16.0711i		−	0.623680i
$665$	−4.03220			−0.156362
$666$	0.145796	−	1.12708i	0.00564949	−	0.0436733i
$667$	−8.85760	+	5.74872i	−0.342968	+	0.222591i
$668$		−	10.1932i		−	0.394388i
$669$	3.49316	−	3.07039i	0.135053	−	0.118708i
$670$	−3.67460			−0.141962
$671$		−	4.35818i		−	0.168246i
$672$	1.32787	−	1.16716i	0.0512236	−	0.0450242i
$673$	−25.6047			−0.986988			−0.493494	−	0.869749i	$-0.664280\pi$
$673$	−25.6047			−0.986988			−0.493494	+	0.869749i	$0.664280\pi$
$674$	−22.1863			−0.854586
$675$	2.90143	+	4.31065i	0.111676	+	0.165917i
$676$	−11.2844			−0.434016
$677$	−19.9065			−0.765071			−0.382535	−	0.923941i	$-0.624949\pi$
$677$	−19.9065			−0.765071			−0.382535	+	0.923941i	$0.624949\pi$
$678$	1.96499	+	2.23555i	0.0754649	+	0.0858557i
$679$	−11.8887			−0.456245
$680$			7.57164i			0.290359i
$681$	−5.42058	+	4.76455i	−0.207717	+	0.182578i
$682$			0.427780i			0.0163805i
$683$		−	14.5455i		−	0.556569i	−0.960499	−	0.278284i	$-0.910234\pi$
$683$		−	14.5455i		−	0.556569i	0.960499	−	0.278284i	$-0.0897657\pi$
$684$	−11.7533	−	1.52039i	−0.449400	−	0.0581335i
$685$	0.245755			0.00938980
$686$	−13.2264			−0.504987
$687$	11.2312	+	12.7776i	0.428495	+	0.487495i
$688$		−	10.0534i		−	0.383282i
$689$	65.2922			2.48743
$690$	−2.24794	+	7.99667i	−0.0855776	+	0.304428i
$691$	12.9697			0.493391			0.246696	−	0.969093i	$-0.420655\pi$
$691$	12.9697			0.493391			0.246696	+	0.969093i	$0.420655\pi$
$692$			4.41012i			0.167648i
$693$	−1.02072	−	0.132038i	−0.0387740	−	0.00501573i
$694$	−24.4010			−0.926250
$695$	−5.22537			−0.198209
$696$	2.86443	−	2.51776i	0.108576	−	0.0954356i
$697$			59.2299i			2.24349i
$698$		−	11.9864i		−	0.453693i
$699$	−23.3406	−	26.5544i	−0.882824	−	1.00438i
$700$		−	1.02070i		−	0.0385789i
$701$	−18.3901			−0.694586			−0.347293	−	0.937757i	$-0.612899\pi$
$701$	−18.3901			−0.694586			−0.347293	+	0.937757i	$0.612899\pi$
$702$	21.2425	−	14.2980i	0.801748	−	0.539644i
$703$	1.49651			0.0564418
$704$	0.336117			0.0126679
$705$	−6.11605	−	6.95817i	−0.230344	−	0.262060i
$706$	22.4575			0.845197
$707$	12.5190			0.470825
$708$	2.89475	+	3.29333i	0.108791	+	0.123771i
$709$		−	52.2675i		−	1.96295i	−0.191596	−	0.981474i	$-0.561366\pi$
$709$		−	52.2675i		−	1.96295i	0.191596	−	0.981474i	$-0.438634\pi$
$710$	10.8426			0.406914
$711$	14.2289	+	1.84062i	0.533624	+	0.0690286i
$712$			14.1373i			0.529817i
$713$	3.32290	+	5.11992i	0.124444	+	0.191742i
$714$	−10.0541	+	8.83732i	−0.376266	+	0.330728i
$715$	1.65636			0.0619443
$716$		−	7.99808i		−	0.298902i
$717$	−10.2863	−	11.7026i	−0.384149	−	0.437042i
$718$			10.4831i			0.391226i
$719$			10.1289i			0.377743i	0.982002	+	0.188872i	$0.0604830\pi$
$719$			10.1289i			0.377743i	−0.982002	+	0.188872i	$0.939517\pi$
$720$	0.384868	−	2.97521i	0.0143432	−	0.110880i
$721$	−6.77691			−0.252385
$722$			3.39418i			0.126318i
$723$	11.4027	+	12.9727i	0.424070	+	0.482460i
$724$			14.7941i			0.549817i
$725$		−	2.20183i		−	0.0817738i
$726$	12.4492	+	14.1633i	0.462033	+	0.525650i
$727$			50.8542i			1.88608i	0.332680	+	0.943040i	$0.392047\pi$
$727$			50.8542i			1.88608i	−0.332680	+	0.943040i	$0.607953\pi$
$728$	−5.02994			−0.186422
$729$	−10.1634	+	25.0141i	−0.376422	+	0.926448i
$730$		−	7.22073i		−	0.267251i
$731$			76.1206i			2.81542i
$732$	14.8268	+	16.8683i	0.548013	+	0.623469i
$733$		−	6.58068i		−	0.243063i	−0.992588	−	0.121531i	$-0.961219\pi$
$733$		−	6.58068i		−	0.243063i	0.992588	−	0.121531i	$-0.0387805\pi$
$734$	22.8694			0.844124
$735$	−7.75119	+	6.81310i	−0.285907	+	0.251305i
$736$	4.02284	−	2.61089i	0.148284	−	0.0962386i
$737$		−	1.23510i		−	0.0454954i
$738$	3.01067	−	23.2739i	0.110824	−	0.856724i
$739$	−12.4255			−0.457080			−0.228540	−	0.973535i	$-0.573395\pi$
$739$	−12.4255			−0.457080			−0.228540	+	0.973535i	$0.573395\pi$
$740$			0.378822i			0.0139258i
$741$	22.2607	+	25.3258i	0.817768	+	0.930366i
$742$	13.5237			0.496471
$743$	−17.8783			−0.655890			−0.327945	−	0.944697i	$-0.606356\pi$
$743$	−17.8783			−0.655890			−0.327945	+	0.944697i	$0.606356\pi$
$744$	−1.45533	−	1.65571i	−0.0533550	−	0.0607015i
$745$	3.43726			0.125931
$746$	5.47431			0.200429
$747$	6.18525	−	47.8149i	0.226306	−	1.74946i
$748$	−2.54496			−0.0930528
$749$		−	9.03299i		−	0.330058i
$750$	−1.14349	−	1.30094i	−0.0417543	−	0.0475035i
$751$			14.4641i			0.527802i	0.964550	+	0.263901i	$0.0850093\pi$
$751$			14.4641i			0.527802i	−0.964550	+	0.263901i	$0.914991\pi$
$752$			5.34859i			0.195043i
$753$	−16.9946	+	14.9378i	−0.619316	+	0.544363i
$754$	−10.8504			−0.395149
$755$	−19.8169			−0.721211
$756$	4.39989	−	2.96150i	0.160022	−	0.107709i
$757$		−	44.8565i		−	1.63034i	−0.579224	−	0.815169i	$-0.696644\pi$
$757$		−	44.8565i		−	1.63034i	0.579224	−	0.815169i	$-0.303356\pi$
$758$	30.3390			1.10196
$759$	−2.68782	−	0.755571i	−0.0975616	−	0.0274255i
$760$	3.95042			0.143297
$761$			8.43284i			0.305690i	0.988250	+	0.152845i	$0.0488435\pi$
$761$			8.43284i			0.305690i	−0.988250	+	0.152845i	$0.951156\pi$
$762$	−20.1974	−	22.9784i	−0.731674	−	0.832419i
$763$	−13.6790			−0.495212
$764$	13.0642			0.472647
$765$	−2.91408	+	22.5272i	−0.105359	+	0.814473i
$766$			14.8764i			0.537507i
$767$		−	12.4751i		−	0.450449i
$768$	−1.30094	+	1.14349i	−0.0469435	+	0.0412621i
$769$			12.1869i			0.439472i	0.975559	+	0.219736i	$0.0705196\pi$
$769$			12.1869i			0.439472i	−0.975559	+	0.219736i	$0.929480\pi$
$770$	0.343075			0.0123636
$771$	5.46284	+	6.21502i	0.196739	+	0.223829i
$772$	1.34388			0.0483672
$773$	−10.4195			−0.374764			−0.187382	−	0.982287i	$-0.560000\pi$
$773$	−10.4195			−0.374764			−0.187382	+	0.982287i	$0.560000\pi$
$774$	3.86922	−	29.9110i	0.139076	−	1.07513i
$775$	−1.27271			−0.0457171
$776$	11.6476			0.418123
$777$	−0.503025	+	0.442146i	−0.0180459	+	0.0158619i
$778$			12.5719i			0.450726i
$779$	30.9026			1.10720
$780$	−6.41091	+	5.63502i	−0.229547	+	0.201766i
$781$			3.64437i			0.130406i
$782$	−30.4595	+	19.7687i	−1.08923	+	0.706927i
$783$	9.49130	−	6.38845i	0.339191	−	0.228305i
$784$	5.95817			0.212792
$785$			12.8500i			0.458637i
$786$	16.2454	−	14.2793i	0.579456	−	0.509326i
$787$		−	27.8222i		−	0.991756i	−0.868392	−	0.495878i	$-0.834846\pi$
$787$		−	27.8222i		−	0.991756i	0.868392	−	0.495878i	$-0.165154\pi$
$788$			6.30397i			0.224570i
$789$	−4.14645	+	3.64462i	−0.147617	+	0.129752i
$790$	−4.78248			−0.170153
$791$		−	1.75399i		−	0.0623647i
$792$	1.00002	+	0.129361i	0.0355341	+	0.00459663i
$793$		−	63.8967i		−	2.26904i
$794$			24.1720i			0.857832i
$795$	17.2367	−	15.1506i	0.611321	−	0.537335i
$796$		−	7.67517i		−	0.272039i
$797$	43.0453			1.52474			0.762372	−	0.647139i	$-0.224035\pi$
$797$	43.0453			1.52474			0.762372	+	0.647139i	$0.224035\pi$
$798$	4.61078	+	5.24563i	0.163220	+	0.185693i
$799$		−	40.4976i		−	1.43270i
$800$			1.00000i			0.0353553i
$801$	−5.44098	+	42.0614i	−0.192247	+	1.48617i
$802$		−	21.4707i		−	0.758158i
$803$	2.42701			0.0856474
$804$	4.20186	+	4.78042i	0.148188	+	0.168592i
$805$	4.10612	−	2.66494i	0.144722	−	0.0939267i
$806$			6.27182i			0.220915i
$807$	7.74332	+	8.80950i	0.272578	+	0.310109i
$808$	−12.2651			−0.431485
$809$		−	31.2888i		−	1.10006i	−0.835146	−	0.550028i	$-0.814617\pi$
$809$		−	31.2888i		−	1.10006i	0.835146	−	0.550028i	$-0.185383\pi$
$810$	2.29012	−	8.70375i	0.0804667	−	0.305819i
$811$	7.67811			0.269615			0.134807	−	0.990872i	$-0.456958\pi$
$811$	7.67811			0.269615			0.134807	+	0.990872i	$0.456958\pi$
$812$	−2.24741			−0.0788685
$813$	8.37294	−	7.35960i	0.293652	−	0.258112i
$814$	−0.127329			−0.00446286
$815$	−7.48989			−0.262359
$816$	9.85021	−	8.65808i	0.344826	−	0.303093i
$817$	39.7151			1.38946
$818$		−	13.5579i		−	0.474042i
$819$	−14.9651	−	1.93586i	−0.522924	−	0.0676444i
$820$			7.82260i			0.273177i
$821$		−	20.6930i		−	0.722191i	−0.932529	−	0.361095i	$-0.882403\pi$
$821$		−	20.6930i		−	0.722191i	0.932529	−	0.361095i	$-0.117597\pi$
$822$	−0.281018	−	0.319711i	−0.00980162	−	0.0111512i
$823$	26.1496			0.911519			0.455760	−	0.890103i	$-0.349368\pi$
$823$	26.1496			0.911519			0.455760	+	0.890103i	$0.349368\pi$
$824$	6.63946			0.231297
$825$	0.437267	−	0.384346i	0.0152237	−	0.0133812i
$826$		−	2.58391i		−	0.0899059i
$827$	35.5900			1.23759			0.618793	−	0.785554i	$-0.287622\pi$
$827$	35.5900			1.23759			0.618793	+	0.785554i	$0.287622\pi$
$828$	12.9737	−	6.21968i	0.450866	−	0.216149i
$829$	−27.5242			−0.955954			−0.477977	−	0.878372i	$-0.658630\pi$
$829$	−27.5242			−0.955954			−0.477977	+	0.878372i	$0.658630\pi$
$830$			16.0711i			0.557836i
$831$	−5.81768	+	5.11359i	−0.201813	+	0.177389i
$832$	4.92792			0.170845
$833$	−45.1131			−1.56308
$834$	5.97515	+	6.79787i	0.206902	+	0.235391i
$835$			10.1932i			0.352751i
$836$			1.32780i			0.0459231i
$837$	−3.69268	−	5.48621i	−0.127638	−	0.189631i
$838$			3.19459i			0.110355i
$839$	−30.4186			−1.05017			−0.525083	−	0.851051i	$-0.675966\pi$
$839$	−30.4186			−1.05017			−0.525083	+	0.851051i	$0.675966\pi$
$840$	−1.32787	+	1.16716i	−0.0458158	+	0.0402709i
$841$	24.1520			0.832826
$842$	−23.8342			−0.821382
$843$	40.1307	−	35.2739i	1.38218	−	1.21490i
$844$	−17.7592			−0.611298
$845$	11.2844			0.388196
$846$	−2.05850	+	15.9132i	−0.0707726	+	0.547106i
$847$		−	11.1124i		−	0.381827i
$848$	−13.2494			−0.454987
$849$	−14.2517	−	16.2140i	−0.489116	−	0.556463i
$850$		−	7.57164i		−	0.259705i
$851$	−1.52394	+	0.989061i	−0.0522400	+	0.0339046i
$852$	−12.3983	−	14.1055i	−0.424760	−	0.483245i
$853$	−17.8266			−0.610370			−0.305185	−	0.952293i	$-0.598718\pi$
$853$	−17.8266			−0.610370			−0.305185	+	0.952293i	$0.598718\pi$
$854$		−	13.2347i		−	0.452882i
$855$	11.7533	+	1.52039i	0.401956	+	0.0519962i
$856$			8.84979i			0.302479i
$857$		−	29.7593i		−	1.01656i	−0.861192	−	0.508279i	$-0.830282\pi$
$857$		−	29.7593i		−	1.01656i	0.861192	−	0.508279i	$-0.169718\pi$
$858$	−1.89403	−	2.15482i	−0.0646610	−	0.0735642i
$859$	22.5038			0.767819			0.383910	−	0.923371i	$-0.374577\pi$
$859$	22.5038			0.767819			0.383910	+	0.923371i	$0.374577\pi$
$860$			10.0534i			0.342818i
$861$	−10.3874	+	9.13023i	−0.354001	+	0.311158i
$862$		−	14.0920i		−	0.479975i
$863$			9.58024i			0.326115i	0.986617	+	0.163058i	$0.0521356\pi$
$863$			9.58024i			0.326115i	−0.986617	+	0.163058i	$0.947864\pi$
$864$	−4.31065	+	2.90143i	−0.146651	+	0.0987087i
$865$		−	4.41012i		−	0.149949i
$866$	24.3424			0.827188
$867$	−52.4663	+	46.1165i	−1.78185	+	1.56620i
$868$			1.29906i			0.0440929i
$869$		−	1.60747i		−	0.0545297i
$870$	−2.86443	+	2.51776i	−0.0971134	+	0.0853602i
$871$		−	18.1082i		−	0.613571i
$872$	13.4015			0.453833
$873$	34.6539	+	4.48276i	1.17286	+	0.151719i
$874$	10.3141	+	15.8919i	0.348880	+	0.537552i
$875$			1.02070i			0.0345060i
$876$	−9.39371	+	8.25682i	−0.317384	+	0.278972i
$877$	13.9728			0.471827			0.235914	−	0.971774i	$-0.424192\pi$
$877$	13.9728			0.471827			0.235914	+	0.971774i	$0.424192\pi$
$878$		−	1.40497i		−	0.0474154i
$879$	−16.8681	+	14.8266i	−0.568948	+	0.500090i
$880$	−0.336117			−0.0113305
$881$	0.000653309	0		2.20105e−5	0		1.10053e−5	−	1.00000i	$-0.499996\pi$
$881$	0.000653309	0		2.20105e−5	0		1.10053e−5		1.00000i	$0.499996\pi$
$882$	17.7268	+	2.29311i	0.596893	+	0.0772129i
$883$	28.3523			0.954131			0.477066	−	0.878868i	$-0.341700\pi$
$883$	28.3523			0.954131			0.477066	+	0.878868i	$0.341700\pi$
$884$	−37.3124			−1.25495
$885$	−2.89475	−	3.29333i	−0.0973060	−	0.110704i
$886$	39.5844			1.32986
$887$		−	7.93986i		−	0.266594i	−0.991076	−	0.133297i	$-0.957444\pi$
$887$		−	7.93986i		−	0.266594i	0.991076	−	0.133297i	$-0.0425565\pi$
$888$	0.492823	−	0.433179i	0.0165381	−	0.0145365i
$889$			18.0286i			0.604660i
$890$		−	14.1373i		−	0.473882i
$891$	2.92548	+	0.769750i	0.0980073	+	0.0257876i
$892$	2.68511			0.0899042
$893$	−21.1292			−0.707061
$894$	−3.93047	−	4.47165i	−0.131454	−	0.149554i
$895$			7.99808i			0.267346i
$896$	1.02070			0.0340992
$897$	−39.4070	−	11.0777i	−1.31576	−	0.369873i
$898$	29.0379			0.969006
$899$			2.80229i			0.0934616i
$900$	−0.384868	+	2.97521i	−0.0128289	+	0.0991737i
$901$	100.320			3.34214
$902$	−2.62931			−0.0875465
$903$	−13.3496	+	11.7339i	−0.444246	+	0.390480i
$904$			1.71841i			0.0571536i
$905$		−	14.7941i		−	0.491771i
$906$	22.6604	+	25.7805i	0.752842	+	0.856501i
$907$			18.1858i			0.603849i	0.953332	+	0.301924i	$0.0976290\pi$
$907$			18.1858i			0.603849i	−0.953332	+	0.301924i	$0.902371\pi$
$908$	−4.16668			−0.138276
$909$	−36.4912	−	4.72044i	−1.21034	−	0.156567i
$910$	5.02994			0.166741
$911$	−18.7950			−0.622707			−0.311354	−	0.950294i	$-0.600782\pi$
$911$	−18.7950			−0.622707			−0.311354	+	0.950294i	$0.600782\pi$
$912$	−4.51726	−	5.13924i	−0.149581	−	0.170177i
$913$	−5.40177			−0.178773
$914$	35.5772			1.17679
$915$	−14.8268	−	16.8683i	−0.490158	−	0.557648i
$916$			9.82184i			0.324523i
$917$	−12.7460			−0.420910
$918$	32.6387	−	21.9686i	1.07724	−	0.725071i
$919$		−	2.72538i		−	0.0899019i	−0.998989	−	0.0449510i	$-0.985687\pi$
$919$		−	2.72538i		−	0.0899019i	0.998989	−	0.0449510i	$-0.0143132\pi$
$920$	−4.02284	+	2.61089i	−0.132629	+	0.0860784i
$921$	31.7937	−	27.9458i	1.04764	−	0.920846i
$922$	−29.4329			−0.969323
$923$			53.4313i			1.75871i
$924$	−0.392303	−	0.446319i	−0.0129058	−	0.0146828i
$925$		−	0.378822i		−	0.0124556i
$926$		−	12.4050i		−	0.407652i
$927$	19.7538	+	2.55531i	0.648800	+	0.0839275i
$928$	2.20183			0.0722785
$929$		−	27.8642i		−	0.914196i	−0.889416	−	0.457098i	$-0.848889\pi$
$929$		−	27.8642i		−	0.914196i	0.889416	−	0.457098i	$-0.151111\pi$
$930$	1.45533	+	1.65571i	0.0477222	+	0.0542930i
$931$			23.5373i			0.771403i
$932$		−	20.4118i		−	0.668610i
$933$	−23.3446	−	26.5589i	−0.764267	−	0.869499i
$934$		−	37.4350i		−	1.22491i
$935$	2.54496			0.0832290
$936$	14.6616	+	1.89660i	0.479230	+	0.0619922i
$937$		−	58.4892i		−	1.91076i	−0.295379	−	0.955380i	$-0.595446\pi$
$937$		−	58.4892i		−	1.91076i	0.295379	−	0.955380i	$-0.404554\pi$
$938$		−	3.75067i		−	0.122464i
$939$	31.3759	+	35.6960i	1.02391	+	1.16490i
$940$		−	5.34859i		−	0.174452i
$941$	−48.4064			−1.57800			−0.789001	−	0.614392i	$-0.789402\pi$
$941$	−48.4064			−1.57800			−0.789001	+	0.614392i	$0.789402\pi$
$942$	16.7171	−	14.6939i	0.544671	−	0.478752i
$943$	−31.4691	+	20.4239i	−1.02478	+	0.665095i
$944$			2.53151i			0.0823936i
$945$	−4.39989	+	2.96150i	−0.143128	+	0.0963374i
$946$	−3.37912			−0.109865
$947$		−	35.0021i		−	1.13742i	−0.822539	−	0.568708i	$-0.807443\pi$
$947$		−	35.0021i		−	1.13742i	0.822539	−	0.568708i	$-0.192557\pi$
$948$	5.46871	+	6.22169i	0.177615	+	0.202071i
$949$	35.5832			1.15508
$950$	−3.95042			−0.128169
$951$	−2.67759	−	3.04627i	−0.0868270	−	0.0987822i
$952$	−7.72838			−0.250478
$953$	−54.6907			−1.77160			−0.885802	−	0.464063i	$-0.846391\pi$
$953$	−54.6907			−1.77160			−0.885802	+	0.464063i	$0.846391\pi$
$954$	−39.4198	−	5.09927i	−1.27626	−	0.165095i
$955$	−13.0642			−0.422748
$956$		−	8.99553i		−	0.290936i
$957$	−0.846263	−	0.962786i	−0.0273558	−	0.0311224i
$958$		−	18.9516i		−	0.612298i
$959$			0.250842i			0.00810011i
$960$	1.30094	−	1.14349i	0.0419875	−	0.0369059i
$961$	−29.3802			−0.947749
$962$	−1.86681			−0.0601882
$963$	−3.40599	+	26.3300i	−0.109757	+	0.848472i
$964$			9.97182i			0.321171i
$965$	−1.34388			−0.0432609
$966$	−8.16221	−	2.29448i	−0.262615	−	0.0738236i
$967$	−41.7162			−1.34150			−0.670751	−	0.741683i	$-0.734028\pi$
$967$	−41.7162			−1.34150			−0.670751	+	0.741683i	$0.734028\pi$
$968$			10.8870i			0.349922i
$969$	34.2031	+	38.9125i	1.09876	+	1.25005i
$970$	−11.6476			−0.373980
$971$	−33.7924			−1.08445			−0.542226	−	0.840233i	$-0.682418\pi$
$971$	−33.7924			−1.08445			−0.542226	+	0.840233i	$0.682418\pi$
$972$	−13.9418	+	6.97334i	−0.447182	+	0.223670i
$973$		−	5.33354i		−	0.170985i
$974$		−	41.8360i		−	1.34051i
$975$	6.41091	−	5.63502i	0.205313	−	0.180465i
$976$			12.9663i			0.415040i
$977$	−36.3672			−1.16349			−0.581746	−	0.813371i	$-0.697630\pi$
$977$	−36.3672			−1.16349			−0.581746	+	0.813371i	$0.697630\pi$
$978$	8.56461	+	9.74387i	0.273866	+	0.311574i
$979$	4.75178			0.151868
$980$	−5.95817			−0.190327
$981$	39.8724	+	5.15782i	1.27303	+	0.164676i
$982$	−35.6738			−1.13840
$983$	−21.7604			−0.694050			−0.347025	−	0.937856i	$-0.612808\pi$
$983$	−21.7604			−0.694050			−0.347025	+	0.937856i	$0.612808\pi$
$984$	10.1767	−	8.94506i	0.324421	−	0.285158i
$985$		−	6.30397i		−	0.200861i
$986$	−16.6714			−0.530927
$987$	7.10221	−	6.24266i	0.226066	−	0.198706i
$988$			19.4674i			0.619340i
$989$	−40.4432	+	26.2483i	−1.28602	+	0.834646i
$990$	−1.00002	−	0.129361i	−0.0317827	−	0.00411135i
$991$	−39.6331			−1.25899			−0.629493	−	0.777006i	$-0.716738\pi$
$991$	−39.6331			−1.25899			−0.629493	+	0.777006i	$0.716738\pi$
$992$		−	1.27271i		−	0.0404086i
$993$	−35.7603	+	31.4323i	−1.13482	+	0.997475i
$994$			11.0670i			0.351024i
$995$			7.67517i			0.243319i
$996$	20.9075	−	18.3771i	0.662479	−	0.582302i
$997$	51.0995			1.61834			0.809169	−	0.587576i	$-0.199918\pi$
$997$	51.0995			1.61834			0.809169	+	0.587576i	$0.199918\pi$
$998$			3.39015i			0.107313i
$999$	1.63297	−	1.09913i	0.0516648	−	0.0347748i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.2.e.b.551.10	yes	16
3.2	odd	2		690.2.e.a.551.2	✓	16
23.22	odd	2		690.2.e.a.551.10	yes	16
69.68	even	2	inner	690.2.e.b.551.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.e.a.551.2	✓	16	3.2	odd	2
690.2.e.a.551.10	yes	16	23.22	odd	2
690.2.e.b.551.2	yes	16	69.68	even	2	inner
690.2.e.b.551.10	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +(-1.30094 + 1.14349i) q^{3} -1.00000 q^{4} +1.00000 q^{5} +(-1.14349 - 1.30094i) q^{6} +1.02070i q^{7} -1.00000i q^{8} +(0.384868 - 2.97521i) q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +(-1.30094 + 1.14349i) q^{3} -1.00000 q^{4} +1.00000 q^{5} +(-1.14349 - 1.30094i) q^{6} +1.02070i q^{7} -1.00000i q^{8} +(0.384868 - 2.97521i) q^{9} +1.00000i q^{10} -0.336117 q^{11} +(1.30094 - 1.14349i) q^{12} -4.92792 q^{13} -1.02070 q^{14} +(-1.30094 + 1.14349i) q^{15} +1.00000 q^{16} -7.57164 q^{17} +(2.97521 + 0.384868i) q^{18} +3.95042i q^{19} -1.00000 q^{20} +(-1.16716 - 1.32787i) q^{21} -0.336117i q^{22} +(-2.61089 - 4.02284i) q^{23} +(1.14349 + 1.30094i) q^{24} +1.00000 q^{25} -4.92792i q^{26} +(2.90143 + 4.31065i) q^{27} -1.02070i q^{28} -2.20183i q^{29} +(-1.14349 - 1.30094i) q^{30} -1.27271 q^{31} +1.00000i q^{32} +(0.437267 - 0.384346i) q^{33} -7.57164i q^{34} +1.02070i q^{35} +(-0.384868 + 2.97521i) q^{36} -0.378822i q^{37} -3.95042 q^{38} +(6.41091 - 5.63502i) q^{39} -1.00000i q^{40} -7.82260i q^{41} +(1.32787 - 1.16716i) q^{42} -10.0534i q^{43} +0.336117 q^{44} +(0.384868 - 2.97521i) q^{45} +(4.02284 - 2.61089i) q^{46} +5.34859i q^{47} +(-1.30094 + 1.14349i) q^{48} +5.95817 q^{49} +1.00000i q^{50} +(9.85021 - 8.65808i) q^{51} +4.92792 q^{52} -13.2494 q^{53} +(-4.31065 + 2.90143i) q^{54} -0.336117 q^{55} +1.02070 q^{56} +(-4.51726 - 5.13924i) q^{57} +2.20183 q^{58} +2.53151i q^{59} +(1.30094 - 1.14349i) q^{60} +12.9663i q^{61} -1.27271i q^{62} +(3.03680 + 0.392835i) q^{63} -1.00000 q^{64} -4.92792 q^{65} +(0.384346 + 0.437267i) q^{66} +3.67460i q^{67} +7.57164 q^{68} +(7.99667 + 2.24794i) q^{69} -1.02070 q^{70} -10.8426i q^{71} +(-2.97521 - 0.384868i) q^{72} -7.22073 q^{73} +0.378822 q^{74} +(-1.30094 + 1.14349i) q^{75} -3.95042i q^{76} -0.343075i q^{77} +(5.63502 + 6.41091i) q^{78} +4.78248i q^{79} +1.00000 q^{80} +(-8.70375 - 2.29012i) q^{81} +7.82260 q^{82} +16.0711 q^{83} +(1.16716 + 1.32787i) q^{84} -7.57164 q^{85} +10.0534 q^{86} +(2.51776 + 2.86443i) q^{87} +0.336117i q^{88} -14.1373 q^{89} +(2.97521 + 0.384868i) q^{90} -5.02994i q^{91} +(2.61089 + 4.02284i) q^{92} +(1.65571 - 1.45533i) q^{93} -5.34859 q^{94} +3.95042i q^{95} +(-1.14349 - 1.30094i) q^{96} +11.6476i q^{97} +5.95817i q^{98} +(-0.129361 + 1.00002i) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more