LMFDB - Embedded newform 690.2.d.d.139.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,2,Mod(139,690)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("690.139");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	690.d (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:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			139.2
Root			$0.432320 - 0.432320i$ of defining polynomial
Character	$\chi$	$=$	690.139
Dual form			690.2.d.d.139.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(0.432320 - 2.19388i) q^{5} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(0.432320 - 2.19388i) q^{5} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +(-2.19388 - 0.432320i) q^{10} -0.864641 q^{11} +1.00000i q^{12} -5.52311i q^{13} +2.00000 q^{14} +(-2.19388 - 0.432320i) q^{15} +1.00000 q^{16} -3.52311i q^{17} +1.00000i q^{18} -8.11704 q^{19} +(-0.432320 + 2.19388i) q^{20} +2.00000 q^{21} +0.864641i q^{22} -1.00000i q^{23} +1.00000 q^{24} +(-4.62620 - 1.89692i) q^{25} -5.52311 q^{26} +1.00000i q^{27} -2.00000i q^{28} +2.00000 q^{29} +(-0.432320 + 2.19388i) q^{30} -3.25240 q^{31} -1.00000i q^{32} +0.864641i q^{33} -3.52311 q^{34} +(4.38776 + 0.864641i) q^{35} +1.00000 q^{36} +5.13536i q^{37} +8.11704i q^{38} -5.52311 q^{39} +(2.19388 + 0.432320i) q^{40} +3.52311 q^{41} -2.00000i q^{42} -1.13536i q^{43} +0.864641 q^{44} +(-0.432320 + 2.19388i) q^{45} -1.00000 q^{46} +5.52311i q^{47} -1.00000i q^{48} +3.00000 q^{49} +(-1.89692 + 4.62620i) q^{50} -3.52311 q^{51} +5.52311i q^{52} -1.34153i q^{53} +1.00000 q^{54} +(-0.373802 + 1.89692i) q^{55} -2.00000 q^{56} +8.11704i q^{57} -2.00000i q^{58} +10.9817 q^{59} +(2.19388 + 0.432320i) q^{60} -5.91087 q^{61} +3.25240i q^{62} -2.00000i q^{63} -1.00000 q^{64} +(-12.1170 - 2.38776i) q^{65} +0.864641 q^{66} -5.91087i q^{67} +3.52311i q^{68} -1.00000 q^{69} +(0.864641 - 4.38776i) q^{70} -5.79383 q^{71} -1.00000i q^{72} -15.2524i q^{73} +5.13536 q^{74} +(-1.89692 + 4.62620i) q^{75} +8.11704 q^{76} -1.72928i q^{77} +5.52311i q^{78} +3.79383 q^{79} +(0.432320 - 2.19388i) q^{80} +1.00000 q^{81} -3.52311i q^{82} -8.32320i q^{83} -2.00000 q^{84} +(-7.72928 - 1.52311i) q^{85} -1.13536 q^{86} -2.00000i q^{87} -0.864641i q^{88} -4.77551 q^{89} +(2.19388 + 0.432320i) q^{90} +11.0462 q^{91} +1.00000i q^{92} +3.25240i q^{93} +5.52311 q^{94} +(-3.50916 + 17.8078i) q^{95} -1.00000 q^{96} -1.04623i q^{97} -3.00000i q^{98} +0.864641 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 6 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^4 - 6 * q^6 - 6 * q^9	$ 6 q - 6 q^{4} - 6 q^{6} - 6 q^{9} + 2 q^{10} + 12 q^{14} + 2 q^{15} + 6 q^{16} - 8 q^{19} + 12 q^{21} + 6 q^{24} - 10 q^{25} - 8 q^{26} + 12 q^{29} + 16 q^{31} + 4 q^{34} - 4 q^{35} + 6 q^{36} - 8 q^{39} - 2 q^{40} - 4 q^{41} - 6 q^{46} + 18 q^{49} - 4 q^{50} + 4 q^{51} + 6 q^{54} - 20 q^{55} - 12 q^{56} + 20 q^{59} - 2 q^{60} + 20 q^{61} - 6 q^{64} - 32 q^{65} - 6 q^{69} - 20 q^{71} + 36 q^{74} - 4 q^{75} + 8 q^{76} + 8 q^{79} + 6 q^{81} - 12 q^{84} - 36 q^{85} - 12 q^{86} + 32 q^{89} - 2 q^{90} + 16 q^{91} + 8 q^{94} - 44 q^{95} - 6 q^{96}+O(q^{100}) $ 6 * q - 6 * q^4 - 6 * q^6 - 6 * q^9 + 2 * q^10 + 12 * q^14 + 2 * q^15 + 6 * q^16 - 8 * q^19 + 12 * q^21 + 6 * q^24 - 10 * q^25 - 8 * q^26 + 12 * q^29 + 16 * q^31 + 4 * q^34 - 4 * q^35 + 6 * q^36 - 8 * q^39 - 2 * q^40 - 4 * q^41 - 6 * q^46 + 18 * q^49 - 4 * q^50 + 4 * q^51 + 6 * q^54 - 20 * q^55 - 12 * q^56 + 20 * q^59 - 2 * q^60 + 20 * q^61 - 6 * q^64 - 32 * q^65 - 6 * q^69 - 20 * q^71 + 36 * q^74 - 4 * q^75 + 8 * q^76 + 8 * q^79 + 6 * q^81 - 12 * q^84 - 36 * q^85 - 12 * q^86 + 32 * q^89 - 2 * q^90 + 16 * q^91 + 8 * q^94 - 44 * q^95 - 6 * q^96

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−1.00000			−0.500000
$5$	0.432320	−	2.19388i	0.193340	−	0.981132i
$5$	0.432320	−	2.19388i	0.193340	−	0.981132i
$6$	−1.00000			−0.408248
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$			1.00000i			0.353553i
$9$	−1.00000			−0.333333
$10$	−2.19388	−	0.432320i	−0.693765	−	0.136712i
$11$	−0.864641			−0.260699			−0.130350	−	0.991468i	$-0.541610\pi$
$11$	−0.864641			−0.260699			−0.130350	+	0.991468i	$0.541610\pi$
$12$			1.00000i			0.288675i
$13$		−	5.52311i		−	1.53184i	−0.642938	−	0.765918i	$-0.722285\pi$
$13$		−	5.52311i		−	1.53184i	0.642938	−	0.765918i	$-0.277715\pi$
$14$	2.00000			0.534522
$15$	−2.19388	−	0.432320i	−0.566457	−	0.111625i
$16$	1.00000			0.250000
$17$		−	3.52311i		−	0.854481i	−0.904138	−	0.427240i	$-0.859486\pi$
$17$		−	3.52311i		−	0.854481i	0.904138	−	0.427240i	$-0.140514\pi$
$18$			1.00000i			0.235702i
$19$	−8.11704			−1.86218			−0.931088	−	0.364795i	$-0.881139\pi$
$19$	−8.11704			−1.86218			−0.931088	+	0.364795i	$0.881139\pi$
$20$	−0.432320	+	2.19388i	−0.0966698	+	0.490566i
$21$	2.00000			0.436436
$22$			0.864641i			0.184342i
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$	1.00000			0.204124
$25$	−4.62620	−	1.89692i	−0.925240	−	0.379383i
$26$	−5.52311			−1.08317
$27$			1.00000i			0.192450i
$28$		−	2.00000i		−	0.377964i
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$	−0.432320	+	2.19388i	−0.0789306	+	0.400545i
$31$	−3.25240			−0.584148			−0.292074	−	0.956396i	$-0.594345\pi$
$31$	−3.25240			−0.584148			−0.292074	+	0.956396i	$0.594345\pi$
$32$		−	1.00000i		−	0.176777i
$33$			0.864641i			0.150515i
$34$	−3.52311			−0.604209
$35$	4.38776	+	0.864641i	0.741666	+	0.146151i
$36$	1.00000			0.166667
$37$			5.13536i			0.844248i	0.906538	+	0.422124i	$0.138715\pi$
$37$			5.13536i			0.844248i	−0.906538	+	0.422124i	$0.861285\pi$
$38$			8.11704i			1.31676i
$39$	−5.52311			−0.884406
$40$	2.19388	+	0.432320i	0.346883	+	0.0683559i
$41$	3.52311			0.550218			0.275109	−	0.961413i	$-0.411286\pi$
$41$	3.52311			0.550218			0.275109	+	0.961413i	$0.411286\pi$
$42$		−	2.00000i		−	0.308607i
$43$		−	1.13536i		−	0.173141i	−0.996246	−	0.0865703i	$-0.972409\pi$
$43$		−	1.13536i		−	0.173141i	0.996246	−	0.0865703i	$-0.0275907\pi$
$44$	0.864641			0.130350
$45$	−0.432320	+	2.19388i	−0.0644465	+	0.327044i
$46$	−1.00000			−0.147442
$47$			5.52311i			0.805629i	0.915282	+	0.402815i	$0.131968\pi$
$47$			5.52311i			0.805629i	−0.915282	+	0.402815i	$0.868032\pi$
$48$		−	1.00000i		−	0.144338i
$49$	3.00000			0.428571
$50$	−1.89692	+	4.62620i	−0.268264	+	0.654243i
$51$	−3.52311			−0.493335
$52$			5.52311i			0.765918i
$53$		−	1.34153i		−	0.184273i	−0.995746	−	0.0921364i	$-0.970630\pi$
$53$		−	1.34153i		−	0.184273i	0.995746	−	0.0921364i	$-0.0293696\pi$
$54$	1.00000			0.136083
$55$	−0.373802	+	1.89692i	−0.0504034	+	0.255780i
$56$	−2.00000			−0.267261
$57$			8.11704i			1.07513i
$58$		−	2.00000i		−	0.262613i
$59$	10.9817			1.42969			0.714846	−	0.699282i	$-0.246497\pi$
$59$	10.9817			1.42969			0.714846	+	0.699282i	$0.246497\pi$
$60$	2.19388	+	0.432320i	0.283228	+	0.0558123i
$61$	−5.91087			−0.756809			−0.378405	−	0.925640i	$-0.623527\pi$
$61$	−5.91087			−0.756809			−0.378405	+	0.925640i	$0.623527\pi$
$62$			3.25240i			0.413055i
$63$		−	2.00000i		−	0.251976i
$64$	−1.00000			−0.125000
$65$	−12.1170	−	2.38776i	−1.50293	−	0.296165i
$66$	0.864641			0.106430
$67$		−	5.91087i		−	0.722128i	−0.932541	−	0.361064i	$-0.882414\pi$
$67$		−	5.91087i		−	0.722128i	0.932541	−	0.361064i	$-0.117586\pi$
$68$			3.52311i			0.427240i
$69$	−1.00000			−0.120386
$70$	0.864641	−	4.38776i	0.103344	−	0.524437i
$71$	−5.79383			−0.687601			−0.343801	−	0.939043i	$-0.611714\pi$
$71$	−5.79383			−0.687601			−0.343801	+	0.939043i	$0.611714\pi$
$72$		−	1.00000i		−	0.117851i
$73$		−	15.2524i		−	1.78516i	−0.450891	−	0.892579i	$-0.648894\pi$
$73$		−	15.2524i		−	1.78516i	0.450891	−	0.892579i	$-0.351106\pi$
$74$	5.13536			0.596973
$75$	−1.89692	+	4.62620i	−0.219037	+	0.534187i
$76$	8.11704			0.931088
$77$		−	1.72928i		−	0.197070i
$78$			5.52311i			0.625370i
$79$	3.79383			0.426840			0.213420	−	0.976961i	$-0.431540\pi$
$79$	3.79383			0.426840			0.213420	+	0.976961i	$0.431540\pi$
$80$	0.432320	−	2.19388i	0.0483349	−	0.245283i
$81$	1.00000			0.111111
$82$		−	3.52311i		−	0.389063i
$83$		−	8.32320i		−	0.913590i	−0.889572	−	0.456795i	$-0.848997\pi$
$83$		−	8.32320i		−	0.913590i	0.889572	−	0.456795i	$-0.151003\pi$
$84$	−2.00000			−0.218218
$85$	−7.72928	−	1.52311i	−0.838358	−	0.165205i
$86$	−1.13536			−0.122429
$87$		−	2.00000i		−	0.214423i
$88$		−	0.864641i		−	0.0921710i
$89$	−4.77551			−0.506203			−0.253102	−	0.967440i	$-0.581451\pi$
$89$	−4.77551			−0.506203			−0.253102	+	0.967440i	$0.581451\pi$
$90$	2.19388	+	0.432320i	0.231255	+	0.0455706i
$91$	11.0462			1.15796
$92$			1.00000i			0.104257i
$93$			3.25240i			0.337258i
$94$	5.52311			0.569666
$95$	−3.50916	+	17.8078i	−0.360032	+	1.82704i
$96$	−1.00000			−0.102062
$97$		−	1.04623i		−	0.106228i	−0.998588	−	0.0531142i	$-0.983085\pi$
$97$		−	1.04623i		−	0.106228i	0.998588	−	0.0531142i	$-0.0169147\pi$
$98$		−	3.00000i		−	0.303046i
$99$	0.864641			0.0868997
$100$	4.62620	+	1.89692i	0.462620	+	0.189692i
$101$	9.04623			0.900133			0.450067	−	0.892995i	$-0.351400\pi$
$101$	9.04623			0.900133			0.450067	+	0.892995i	$0.351400\pi$
$102$			3.52311i			0.348840i
$103$		−	11.3169i		−	1.11509i	−0.830146	−	0.557546i	$-0.811743\pi$
$103$		−	11.3169i		−	1.11509i	0.830146	−	0.557546i	$-0.188257\pi$
$104$	5.52311			0.541586
$105$	0.864641	−	4.38776i	0.0843803	−	0.428201i
$106$	−1.34153			−0.130301
$107$			5.64015i			0.545254i	0.962120	+	0.272627i	$0.0878925\pi$
$107$			5.64015i			0.545254i	−0.962120	+	0.272627i	$0.912108\pi$
$108$		−	1.00000i		−	0.0962250i
$109$	12.1816			1.16678			0.583392	−	0.812191i	$-0.301725\pi$
$109$	12.1816			1.16678			0.583392	+	0.812191i	$0.301725\pi$
$110$	1.89692	+	0.373802i	0.180864	+	0.0356406i
$111$	5.13536			0.487427
$112$			2.00000i			0.188982i
$113$		−	5.79383i		−	0.545038i	−0.962150	−	0.272519i	$-0.912143\pi$
$113$		−	5.79383i		−	0.545038i	0.962150	−	0.272519i	$-0.0878567\pi$
$114$	8.11704			0.760230
$115$	−2.19388	−	0.432320i	−0.204580	−	0.0403141i
$116$	−2.00000			−0.185695
$117$			5.52311i			0.510612i
$118$		−	10.9817i		−	1.01095i
$119$	7.04623			0.645927
$120$	0.432320	−	2.19388i	0.0394653	−	0.200273i
$121$	−10.2524			−0.932036
$122$			5.91087i			0.535145i
$123$		−	3.52311i		−	0.317669i
$124$	3.25240			0.292074
$125$	−6.16160	+	9.32924i	−0.551110	+	0.834432i
$126$	−2.00000			−0.178174
$127$			0.747604i			0.0663391i	0.999450	+	0.0331696i	$0.0105601\pi$
$127$			0.747604i			0.0663391i	−0.999450	+	0.0331696i	$0.989440\pi$
$128$			1.00000i			0.0883883i
$129$	−1.13536			−0.0999628
$130$	−2.38776	+	12.1170i	−0.209420	+	1.06273i
$131$	19.7572			1.72619			0.863097	−	0.505039i	$-0.168522\pi$
$131$	19.7572			1.72619			0.863097	+	0.505039i	$0.168522\pi$
$132$		−	0.864641i		−	0.0752573i
$133$		−	16.2341i		−	1.40767i
$134$	−5.91087			−0.510621
$135$	2.19388	+	0.432320i	0.188819	+	0.0372082i
$136$	3.52311			0.302105
$137$			0.476886i			0.0407431i	0.999792	+	0.0203715i	$0.00648491\pi$
$137$			0.476886i			0.0407431i	−0.999792	+	0.0203715i	$0.993515\pi$
$138$			1.00000i			0.0851257i
$139$	3.45856			0.293352			0.146676	−	0.989185i	$-0.453143\pi$
$139$	3.45856			0.293352			0.146676	+	0.989185i	$0.453143\pi$
$140$	−4.38776	−	0.864641i	−0.370833	−	0.0730755i
$141$	5.52311			0.465130
$142$			5.79383i			0.486208i
$143$			4.77551i			0.399348i
$144$	−1.00000			−0.0833333
$145$	0.864641	−	4.38776i	0.0718045	−	0.364383i
$146$	−15.2524			−1.26230
$147$		−	3.00000i		−	0.247436i
$148$		−	5.13536i		−	0.422124i
$149$	8.86464			0.726220			0.363110	−	0.931746i	$-0.381715\pi$
$149$	8.86464			0.726220			0.363110	+	0.931746i	$0.381715\pi$
$150$	4.62620	+	1.89692i	0.377727	+	0.154883i
$151$	−6.71096			−0.546130			−0.273065	−	0.961996i	$-0.588037\pi$
$151$	−6.71096			−0.546130			−0.273065	+	0.961996i	$0.588037\pi$
$152$		−	8.11704i		−	0.658379i
$153$			3.52311i			0.284827i
$154$	−1.72928			−0.139349
$155$	−1.40608	+	7.13536i	−0.112939	+	0.573126i
$156$	5.52311			0.442203
$157$		−	14.6864i		−	1.17210i	−0.810275	−	0.586050i	$-0.800682\pi$
$157$		−	14.6864i		−	1.17210i	0.810275	−	0.586050i	$-0.199318\pi$
$158$		−	3.79383i		−	0.301821i
$159$	−1.34153			−0.106390
$160$	−2.19388	−	0.432320i	−0.173441	−	0.0341779i
$161$	2.00000			0.157622
$162$		−	1.00000i		−	0.0785674i
$163$			23.2803i			1.82345i	0.410797	+	0.911727i	$0.365251\pi$
$163$			23.2803i			1.82345i	−0.410797	+	0.911727i	$0.634749\pi$
$164$	−3.52311			−0.275109
$165$	1.89692	+	0.373802i	0.147675	+	0.0291004i
$166$	−8.32320			−0.646006
$167$		−	5.93545i		−	0.459299i	−0.973273	−	0.229649i	$-0.926242\pi$
$167$		−	5.93545i		−	0.459299i	0.973273	−	0.229649i	$-0.0737580\pi$
$168$			2.00000i			0.154303i
$169$	−17.5048			−1.34652
$170$	−1.52311	+	7.72928i	−0.116818	+	0.592809i
$171$	8.11704			0.620725
$172$			1.13536i			0.0865703i
$173$		−	25.8217i		−	1.96319i	−0.190973	−	0.981595i	$-0.561164\pi$
$173$		−	25.8217i		−	1.96319i	0.190973	−	0.981595i	$-0.438836\pi$
$174$	−2.00000			−0.151620
$175$	3.79383	−	9.25240i	0.286787	−	0.699415i
$176$	−0.864641			−0.0651748
$177$		−	10.9817i		−	0.825433i
$178$			4.77551i			0.357940i
$179$	23.3449			1.74488			0.872438	−	0.488725i	$-0.162538\pi$
$179$	23.3449			1.74488			0.872438	+	0.488725i	$0.162538\pi$
$180$	0.432320	−	2.19388i	0.0322233	−	0.163522i
$181$	13.3694			0.993742			0.496871	−	0.867824i	$-0.334482\pi$
$181$	13.3694			0.993742			0.496871	+	0.867824i	$0.334482\pi$
$182$		−	11.0462i		−	0.818801i
$183$			5.91087i			0.436944i
$184$	1.00000			0.0737210
$185$	11.2663	+	2.22012i	0.828318	+	0.163227i
$186$	3.25240			0.238477
$187$			3.04623i			0.222762i
$188$		−	5.52311i		−	0.402815i
$189$	−2.00000			−0.145479
$190$	17.8078	+	3.50916i	1.29191	+	0.254581i
$191$	12.2341			0.885227			0.442613	−	0.896713i	$-0.354051\pi$
$191$	12.2341			0.885227			0.442613	+	0.896713i	$0.354051\pi$
$192$			1.00000i			0.0721688i
$193$		−	7.45856i		−	0.536879i	−0.963297	−	0.268440i	$-0.913492\pi$
$193$		−	7.45856i		−	0.536879i	0.963297	−	0.268440i	$-0.0865080\pi$
$194$	−1.04623			−0.0751148
$195$	−2.38776	+	12.1170i	−0.170991	+	0.867719i
$196$	−3.00000			−0.214286
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		−	0.864641i		−	0.0614474i
$199$	−22.8401			−1.61909			−0.809545	−	0.587059i	$-0.800286\pi$
$199$	−22.8401			−1.61909			−0.809545	+	0.587059i	$0.800286\pi$
$200$	1.89692	−	4.62620i	0.134132	−	0.327122i
$201$	−5.91087			−0.416921
$202$		−	9.04623i		−	0.636490i
$203$			4.00000i			0.280745i
$204$	3.52311			0.246667
$205$	1.52311	−	7.72928i	0.106379	−	0.539836i
$206$	−11.3169			−0.788489
$207$			1.00000i			0.0695048i
$208$		−	5.52311i		−	0.382959i
$209$	7.01832			0.485467
$210$	−4.38776	−	0.864641i	−0.302784	−	0.0596659i
$211$	24.2341			1.66834			0.834171	−	0.551506i	$-0.185946\pi$
$211$	24.2341			1.66834			0.834171	+	0.551506i	$0.185946\pi$
$212$			1.34153i			0.0921364i
$213$			5.79383i			0.396987i
$214$	5.64015			0.385553
$215$	−2.49084	−	0.490839i	−0.169874	−	0.0334749i
$216$	−1.00000			−0.0680414
$217$		−	6.50479i		−	0.441574i
$218$		−	12.1816i		−	0.825041i
$219$	−15.2524			−1.03066
$220$	0.373802	−	1.89692i	0.0252017	−	0.127890i
$221$	−19.4586			−1.30892
$222$		−	5.13536i		−	0.344663i
$223$		−	13.9634i		−	0.935055i	−0.883978	−	0.467528i	$-0.845145\pi$
$223$		−	13.9634i		−	0.935055i	0.883978	−	0.467528i	$-0.154855\pi$
$224$	2.00000			0.133631
$225$	4.62620	+	1.89692i	0.308413	+	0.126461i
$226$	−5.79383			−0.385400
$227$		−	4.45231i		−	0.295510i	−0.989024	−	0.147755i	$-0.952795\pi$
$227$		−	4.45231i		−	0.295510i	0.989024	−	0.147755i	$-0.0472047\pi$
$228$		−	8.11704i		−	0.537564i
$229$	4.05249			0.267796			0.133898	−	0.990995i	$-0.457251\pi$
$229$	4.05249			0.267796			0.133898	+	0.990995i	$0.457251\pi$
$230$	−0.432320	+	2.19388i	−0.0285064	+	0.144660i
$231$	−1.72928			−0.113778
$232$			2.00000i			0.131306i
$233$			24.5327i			1.60719i	0.595176	+	0.803595i	$0.297082\pi$
$233$			24.5327i			1.60719i	−0.595176	+	0.803595i	$0.702918\pi$
$234$	5.52311			0.361057
$235$	12.1170	+	2.38776i	0.790428	+	0.155760i
$236$	−10.9817			−0.714846
$237$		−	3.79383i		−	0.246436i
$238$		−	7.04623i		−	0.456739i
$239$	−10.7755			−0.697010			−0.348505	−	0.937307i	$-0.613311\pi$
$239$	−10.7755			−0.697010			−0.348505	+	0.937307i	$0.613311\pi$
$240$	−2.19388	−	0.432320i	−0.141614	−	0.0279062i
$241$	−10.7755			−0.694112			−0.347056	−	0.937844i	$-0.612819\pi$
$241$	−10.7755			−0.694112			−0.347056	+	0.937844i	$0.612819\pi$
$242$			10.2524i			0.659049i
$243$		−	1.00000i		−	0.0641500i
$244$	5.91087			0.378405
$245$	1.29696	−	6.58163i	0.0828598	−	0.420485i
$246$	−3.52311			−0.224626
$247$			44.8313i			2.85255i
$248$		−	3.25240i		−	0.206527i
$249$	−8.32320			−0.527462
$250$	9.32924	+	6.16160i	0.590033	+	0.389694i
$251$	−3.91087			−0.246852			−0.123426	−	0.992354i	$-0.539388\pi$
$251$	−3.91087			−0.246852			−0.123426	+	0.992354i	$0.539388\pi$
$252$			2.00000i			0.125988i
$253$			0.864641i			0.0543595i
$254$	0.747604			0.0469088
$255$	−1.52311	+	7.72928i	−0.0953811	+	0.484026i
$256$	1.00000			0.0625000
$257$			28.0925i			1.75236i	0.481985	+	0.876180i	$0.339916\pi$
$257$			28.0925i			1.75236i	−0.481985	+	0.876180i	$0.660084\pi$
$258$			1.13536i			0.0706844i
$259$	−10.2707			−0.638191
$260$	12.1170	+	2.38776i	0.751467	+	0.148082i
$261$	−2.00000			−0.123797
$262$		−	19.7572i		−	1.22060i
$263$		−	29.5510i		−	1.82219i	−0.412192	−	0.911097i	$-0.635237\pi$
$263$		−	29.5510i		−	1.82219i	0.412192	−	0.911097i	$-0.364763\pi$
$264$	−0.864641			−0.0532150
$265$	−2.94315	−	0.579969i	−0.180796	−	0.0356272i
$266$	−16.2341			−0.995375
$267$			4.77551i			0.292256i
$268$			5.91087i			0.361064i
$269$	17.2803			1.05360			0.526799	−	0.849990i	$-0.323392\pi$
$269$	17.2803			1.05360			0.526799	+	0.849990i	$0.323392\pi$
$270$	0.432320	−	2.19388i	0.0263102	−	0.133515i
$271$	9.96336			0.605231			0.302615	−	0.953113i	$-0.402140\pi$
$271$	9.96336			0.605231			0.302615	+	0.953113i	$0.402140\pi$
$272$		−	3.52311i		−	0.213620i
$273$		−	11.0462i		−	0.668548i
$274$	0.476886			0.0288097
$275$	4.00000	+	1.64015i	0.241209	+	0.0989049i
$276$	1.00000			0.0601929
$277$			27.0741i			1.62673i	0.581756	+	0.813364i	$0.302366\pi$
$277$			27.0741i			1.62673i	−0.581756	+	0.813364i	$0.697634\pi$
$278$		−	3.45856i		−	0.207431i
$279$	3.25240			0.194716
$280$	−0.864641	+	4.38776i	−0.0516722	+	0.262219i
$281$	−27.2803			−1.62741			−0.813703	−	0.581281i	$-0.802552\pi$
$281$	−27.2803			−1.62741			−0.813703	+	0.581281i	$0.802552\pi$
$282$		−	5.52311i		−	0.328897i
$283$			30.9205i			1.83803i	0.394222	+	0.919015i	$0.371014\pi$
$283$			30.9205i			1.83803i	−0.394222	+	0.919015i	$0.628986\pi$
$284$	5.79383			0.343801
$285$	17.8078	+	3.50916i	1.05484	+	0.207865i
$286$	4.77551			0.282382
$287$			7.04623i			0.415926i
$288$			1.00000i			0.0589256i
$289$	4.58767			0.269863
$290$	−4.38776	−	0.864641i	−0.257658	−	0.0507735i
$291$	−1.04623			−0.0613310
$292$			15.2524i			0.892579i
$293$		−	11.0708i		−	0.646764i	−0.946269	−	0.323382i	$-0.895180\pi$
$293$		−	11.0708i		−	0.646764i	0.946269	−	0.323382i	$-0.104820\pi$
$294$	−3.00000			−0.174964
$295$	4.74760	−	24.0925i	0.276416	−	1.40272i
$296$	−5.13536			−0.298487
$297$		−	0.864641i		−	0.0501716i
$298$		−	8.86464i		−	0.513515i
$299$	−5.52311			−0.319410
$300$	1.89692	−	4.62620i	0.109519	−	0.267094i
$301$	2.27072			0.130882
$302$			6.71096i			0.386172i
$303$		−	9.04623i		−	0.519692i
$304$	−8.11704			−0.465544
$305$	−2.55539	+	12.9677i	−0.146321	+	0.742530i
$306$	3.52311			0.201403
$307$			4.23407i			0.241651i	0.992674	+	0.120826i	$0.0385542\pi$
$307$			4.23407i			0.241651i	−0.992674	+	0.120826i	$0.961446\pi$
$308$			1.72928i			0.0985350i
$309$	−11.3169			−0.643799
$310$	7.13536	+	1.40608i	0.405261	+	0.0798598i
$311$	−13.2803			−0.753057			−0.376528	−	0.926405i	$-0.622882\pi$
$311$	−13.2803			−0.753057			−0.376528	+	0.926405i	$0.622882\pi$
$312$		−	5.52311i		−	0.312685i
$313$			31.9634i			1.80668i	0.428930	+	0.903338i	$0.358891\pi$
$313$			31.9634i			1.80668i	−0.428930	+	0.903338i	$0.641109\pi$
$314$	−14.6864			−0.828800
$315$	−4.38776	−	0.864641i	−0.247222	−	0.0487170i
$316$	−3.79383			−0.213420
$317$		−	13.2803i		−	0.745896i	−0.927852	−	0.372948i	$-0.878347\pi$
$317$		−	13.2803i		−	0.745896i	0.927852	−	0.372948i	$-0.121653\pi$
$318$			1.34153i			0.0752291i
$319$	−1.72928			−0.0968212
$320$	−0.432320	+	2.19388i	−0.0241674	+	0.122641i
$321$	5.64015			0.314803
$322$		−	2.00000i		−	0.111456i
$323$			28.5972i			1.59119i
$324$	−1.00000			−0.0555556
$325$	−10.4769	+	25.5510i	−0.581153	+	1.41732i
$326$	23.2803			1.28938
$327$		−	12.1816i		−	0.673643i
$328$			3.52311i			0.194531i
$329$	−11.0462			−0.608998
$330$	0.373802	−	1.89692i	0.0205771	−	0.104422i
$331$	−11.8217			−0.649782			−0.324891	−	0.945752i	$-0.605328\pi$
$331$	−11.8217			−0.649782			−0.324891	+	0.945752i	$0.605328\pi$
$332$			8.32320i			0.456795i
$333$		−	5.13536i		−	0.281416i
$334$	−5.93545			−0.324773
$335$	−12.9677	−	2.55539i	−0.708502	−	0.139616i
$336$	2.00000			0.109109
$337$			13.2803i			0.723424i	0.932290	+	0.361712i	$0.117808\pi$
$337$			13.2803i			0.723424i	−0.932290	+	0.361712i	$0.882192\pi$
$338$			17.5048i			0.952135i
$339$	−5.79383			−0.314678
$340$	7.72928	+	1.52311i	0.419179	+	0.0826025i
$341$	2.81215			0.152287
$342$		−	8.11704i		−	0.438919i
$343$			20.0000i			1.07990i
$344$	1.13536			0.0612145
$345$	−0.432320	+	2.19388i	−0.0232754	+	0.118114i
$346$	−25.8217			−1.38819
$347$		−	24.3632i		−	1.30788i	−0.756545	−	0.653942i	$-0.773114\pi$
$347$		−	24.3632i		−	1.30788i	0.756545	−	0.653942i	$-0.226886\pi$
$348$			2.00000i			0.107211i
$349$	−3.49521			−0.187094			−0.0935471	−	0.995615i	$-0.529821\pi$
$349$	−3.49521			−0.187094			−0.0935471	+	0.995615i	$0.529821\pi$
$350$	−9.25240	−	3.79383i	−0.494561	−	0.202789i
$351$	5.52311			0.294802
$352$			0.864641i			0.0460855i
$353$		−	31.5789i		−	1.68078i	−0.541985	−	0.840388i	$-0.682327\pi$
$353$		−	31.5789i		−	1.68078i	0.541985	−	0.840388i	$-0.317673\pi$
$354$	−10.9817			−0.583670
$355$	−2.50479	+	12.7110i	−0.132941	+	0.674628i
$356$	4.77551			0.253102
$357$		−	7.04623i		−	0.372926i
$358$		−	23.3449i		−	1.23381i
$359$	21.1878			1.11825			0.559126	−	0.829083i	$-0.311137\pi$
$359$	21.1878			1.11825			0.559126	+	0.829083i	$0.311137\pi$
$360$	−2.19388	−	0.432320i	−0.115628	−	0.0227853i
$361$	46.8863			2.46770
$362$		−	13.3694i		−	0.702682i
$363$			10.2524i			0.538111i
$364$	−11.0462			−0.578980
$365$	−33.4619	−	6.59392i	−1.75148	−	0.345142i
$366$	5.91087			0.308966
$367$			13.0462i			0.681008i	0.940243	+	0.340504i	$0.110598\pi$
$367$			13.0462i			0.681008i	−0.940243	+	0.340504i	$0.889402\pi$
$368$		−	1.00000i		−	0.0521286i
$369$	−3.52311			−0.183406
$370$	2.22012	−	11.2663i	0.115419	−	0.585710i
$371$	2.68305			0.139297
$372$		−	3.25240i		−	0.168629i
$373$		−	18.8646i		−	0.976774i	−0.872627	−	0.488387i	$-0.837585\pi$
$373$		−	18.8646i		−	0.976774i	0.872627	−	0.488387i	$-0.162415\pi$
$374$	3.04623			0.157517
$375$	9.32924	+	6.16160i	0.481760	+	0.318184i
$376$	−5.52311			−0.284833
$377$		−	11.0462i		−	0.568910i
$378$			2.00000i			0.102869i
$379$	−26.7509			−1.37410			−0.687052	−	0.726609i	$-0.741095\pi$
$379$	−26.7509			−1.37410			−0.687052	+	0.726609i	$0.741095\pi$
$380$	3.50916	−	17.8078i	0.180016	−	0.913520i
$381$	0.747604			0.0383009
$382$		−	12.2341i		−	0.625950i
$383$			22.5048i			1.14994i	0.818174	+	0.574971i	$0.194987\pi$
$383$			22.5048i			1.14994i	−0.818174	+	0.574971i	$0.805013\pi$
$384$	1.00000			0.0510310
$385$	−3.79383	−	0.747604i	−0.193352	−	0.0381014i
$386$	−7.45856			−0.379631
$387$			1.13536i			0.0577135i
$388$			1.04623i			0.0531142i
$389$	7.67680			0.389229			0.194614	−	0.980880i	$-0.437654\pi$
$389$	7.67680			0.389229			0.194614	+	0.980880i	$0.437654\pi$
$390$	12.1170	+	2.38776i	0.613570	+	0.120909i
$391$	−3.52311			−0.178172
$392$			3.00000i			0.151523i
$393$		−	19.7572i		−	0.996618i
$394$	2.00000			0.100759
$395$	1.64015	−	8.32320i	0.0825250	−	0.418786i
$396$	−0.864641			−0.0434498
$397$		−	31.7938i		−	1.59569i	−0.602865	−	0.797843i	$-0.705974\pi$
$397$		−	31.7938i		−	1.59569i	0.602865	−	0.797843i	$-0.294026\pi$
$398$			22.8401i			1.14487i
$399$	−16.2341			−0.812720
$400$	−4.62620	−	1.89692i	−0.231310	−	0.0948458i
$401$	−18.0925			−0.903494			−0.451747	−	0.892146i	$-0.649199\pi$
$401$	−18.0925			−0.903494			−0.451747	+	0.892146i	$0.649199\pi$
$402$			5.91087i			0.294807i
$403$			17.9634i			0.894818i
$404$	−9.04623			−0.450067
$405$	0.432320	−	2.19388i	0.0214822	−	0.109015i
$406$	4.00000			0.198517
$407$		−	4.44024i		−	0.220095i
$408$		−	3.52311i		−	0.174420i
$409$	−2.00000			−0.0988936			−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000			−0.0988936			−0.0494468	+	0.998777i	$0.515746\pi$
$410$	−7.72928	−	1.52311i	−0.381722	−	0.0752213i
$411$	0.476886			0.0235230
$412$			11.3169i			0.557546i
$413$			21.9634i			1.08075i
$414$	1.00000			0.0491473
$415$	−18.2601	−	3.59829i	−0.896353	−	0.176633i
$416$	−5.52311			−0.270793
$417$		−	3.45856i		−	0.169367i
$418$		−	7.01832i		−	0.343277i
$419$	−20.4523			−0.999161			−0.499580	−	0.866268i	$-0.666512\pi$
$419$	−20.4523			−0.999161			−0.499580	+	0.866268i	$0.666512\pi$
$420$	−0.864641	+	4.38776i	−0.0421902	+	0.214101i
$421$	−12.9571			−0.631490			−0.315745	−	0.948844i	$-0.602254\pi$
$421$	−12.9571			−0.631490			−0.315745	+	0.948844i	$0.602254\pi$
$422$		−	24.2341i		−	1.17970i
$423$		−	5.52311i		−	0.268543i
$424$	1.34153			0.0651503
$425$	−6.68305	+	16.2986i	−0.324176	+	0.790599i
$426$	5.79383			0.280712
$427$		−	11.8217i		−	0.572094i
$428$		−	5.64015i		−	0.272627i
$429$	4.77551			0.230564
$430$	−0.490839	+	2.49084i	−0.0236704	+	0.120119i
$431$	33.3728			1.60751			0.803755	−	0.594961i	$-0.202832\pi$
$431$	33.3728			1.60751			0.803755	+	0.594961i	$0.202832\pi$
$432$			1.00000i			0.0481125i
$433$		−	19.0096i		−	0.913542i	−0.889584	−	0.456771i	$-0.849006\pi$
$433$		−	19.0096i		−	0.913542i	0.889584	−	0.456771i	$-0.150994\pi$
$434$	−6.50479			−0.312240
$435$	−4.38776	−	0.864641i	−0.210377	−	0.0414564i
$436$	−12.1816			−0.583392
$437$			8.11704i			0.388291i
$438$			15.2524i			0.728788i
$439$	−32.5972			−1.55578			−0.777891	−	0.628399i	$-0.783710\pi$
$439$	−32.5972			−1.55578			−0.777891	+	0.628399i	$0.783710\pi$
$440$	−1.89692	−	0.373802i	−0.0904319	−	0.0178203i
$441$	−3.00000			−0.142857
$442$			19.4586i			0.925549i
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	−5.13536			−0.243713
$445$	−2.06455	+	10.4769i	−0.0978691	+	0.496652i
$446$	−13.9634			−0.661184
$447$		−	8.86464i		−	0.419283i
$448$		−	2.00000i		−	0.0944911i
$449$	−39.5789			−1.86785			−0.933923	−	0.357475i	$-0.883638\pi$
$449$	−39.5789			−1.86785			−0.933923	+	0.357475i	$0.883638\pi$
$450$	1.89692	−	4.62620i	0.0894215	−	0.218081i
$451$	−3.04623			−0.143441
$452$			5.79383i			0.272519i
$453$			6.71096i			0.315308i
$454$	−4.45231			−0.208957
$455$	4.77551	−	24.2341i	0.223879	−	1.13611i
$456$	−8.11704			−0.380115
$457$			24.5048i			1.14629i	0.819455	+	0.573143i	$0.194276\pi$
$457$			24.5048i			1.14629i	−0.819455	+	0.573143i	$0.805724\pi$
$458$		−	4.05249i		−	0.189360i
$459$	3.52311			0.164445
$460$	2.19388	+	0.432320i	0.102290	+	0.0201570i
$461$	18.2341			0.849245			0.424623	−	0.905370i	$-0.360407\pi$
$461$	18.2341			0.849245			0.424623	+	0.905370i	$0.360407\pi$
$462$			1.72928i			0.0804535i
$463$			4.20617i			0.195477i	0.995212	+	0.0977386i	$0.0311609\pi$
$463$			4.20617i			0.195477i	−0.995212	+	0.0977386i	$0.968839\pi$
$464$	2.00000			0.0928477
$465$	7.13536	+	1.40608i	0.330894	+	0.0652053i
$466$	24.5327			1.13646
$467$			19.1912i			0.888062i	0.896012	+	0.444031i	$0.146452\pi$
$467$			19.1912i			0.888062i	−0.896012	+	0.444031i	$0.853548\pi$
$468$		−	5.52311i		−	0.255306i
$469$	11.8217			0.545877
$470$	2.38776	−	12.1170i	0.110139	−	0.558917i
$471$	−14.6864			−0.676713
$472$			10.9817i			0.505473i
$473$			0.981678i			0.0451376i
$474$	−3.79383			−0.174257
$475$	37.5510	+	15.3973i	1.72296	+	0.706478i
$476$	−7.04623			−0.322963
$477$			1.34153i			0.0614243i
$478$			10.7755i			0.492860i
$479$	25.6801			1.17335			0.586677	−	0.809821i	$-0.300436\pi$
$479$	25.6801			1.17335			0.586677	+	0.809821i	$0.300436\pi$
$480$	−0.432320	+	2.19388i	−0.0197326	+	0.100136i
$481$	28.3632			1.29325
$482$			10.7755i			0.490811i
$483$		−	2.00000i		−	0.0910032i
$484$	10.2524			0.466018
$485$	−2.29530	−	0.452306i	−0.104224	−	0.0205382i
$486$	−1.00000			−0.0453609
$487$			8.74760i			0.396392i	0.980162	+	0.198196i	$0.0635082\pi$
$487$			8.74760i			0.396392i	−0.980162	+	0.198196i	$0.936492\pi$
$488$		−	5.91087i		−	0.267572i
$489$	23.2803			1.05277
$490$	−6.58163	−	1.29696i	−0.297328	−	0.0585907i
$491$	23.8863			1.07797			0.538987	−	0.842314i	$-0.318807\pi$
$491$	23.8863			1.07797			0.538987	+	0.842314i	$0.318807\pi$
$492$			3.52311i			0.158834i
$493$		−	7.04623i		−	0.317346i
$494$	44.8313			2.01706
$495$	0.373802	−	1.89692i	0.0168011	−	0.0852600i
$496$	−3.25240			−0.146037
$497$		−	11.5877i		−	0.519778i
$498$			8.32320i			0.372972i
$499$	−29.7293			−1.33087			−0.665433	−	0.746458i	$-0.731753\pi$
$499$	−29.7293			−1.33087			−0.665433	+	0.746458i	$0.731753\pi$
$500$	6.16160	−	9.32924i	0.275555	−	0.417216i
$501$	−5.93545			−0.265176
$502$			3.91087i			0.174551i
$503$		−	0.775511i		−	0.0345783i	−0.999851	−	0.0172892i	$-0.994496\pi$
$503$		−	0.775511i		−	0.0345783i	0.999851	−	0.0172892i	$-0.00550358\pi$
$504$	2.00000			0.0890871
$505$	3.91087	−	19.8463i	0.174031	−	0.883150i
$506$	0.864641			0.0384380
$507$			17.5048i			0.777415i
$508$		−	0.747604i		−	0.0331696i
$509$	26.0000			1.15243			0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000			1.15243			0.576215	+	0.817298i	$0.304529\pi$
$510$	7.72928	+	1.52311i	0.342258	+	0.0674446i
$511$	30.5048			1.34945
$512$		−	1.00000i		−	0.0441942i
$513$		−	8.11704i		−	0.358376i
$514$	28.0925			1.23911
$515$	−24.8280	−	4.89255i	−1.09405	−	0.215591i
$516$	1.13536			0.0499814
$517$		−	4.77551i		−	0.210027i
$518$			10.2707i			0.451269i
$519$	−25.8217			−1.13345
$520$	2.38776	−	12.1170i	0.104710	−	0.531367i
$521$	13.4219			0.588025			0.294012	−	0.955802i	$-0.405009\pi$
$521$	13.4219			0.588025			0.294012	+	0.955802i	$0.405009\pi$
$522$			2.00000i			0.0875376i
$523$			41.9667i			1.83507i	0.397649	+	0.917537i	$0.369826\pi$
$523$			41.9667i			1.83507i	−0.397649	+	0.917537i	$0.630174\pi$
$524$	−19.7572			−0.863097
$525$	−9.25240	−	3.79383i	−0.403808	−	0.165576i
$526$	−29.5510			−1.28849
$527$			11.4586i			0.499143i
$528$			0.864641i			0.0376287i
$529$	−1.00000			−0.0434783
$530$	−0.579969	+	2.94315i	−0.0251923	+	0.127842i
$531$	−10.9817			−0.476564
$532$			16.2341i			0.703836i
$533$		−	19.4586i		−	0.842844i
$534$	4.77551			0.206657
$535$	12.3738	+	2.43835i	0.534966	+	0.105419i
$536$	5.91087			0.255311
$537$		−	23.3449i		−	1.00740i
$538$		−	17.2803i		−	0.745007i
$539$	−2.59392			−0.111728
$540$	−2.19388	−	0.432320i	−0.0944095	−	0.0186041i
$541$	−1.58767			−0.0682591			−0.0341295	−	0.999417i	$-0.510866\pi$
$541$	−1.58767			−0.0682591			−0.0341295	+	0.999417i	$0.510866\pi$
$542$		−	9.96336i		−	0.427963i
$543$		−	13.3694i		−	0.573737i
$544$	−3.52311			−0.151052
$545$	5.26635	−	26.7249i	0.225586	−	1.14477i
$546$	−11.0462			−0.472735
$547$		−	20.6464i		−	0.882777i	−0.897316	−	0.441388i	$-0.854486\pi$
$547$		−	20.6464i		−	0.882777i	0.897316	−	0.441388i	$-0.145514\pi$
$548$		−	0.476886i		−	0.0203715i
$549$	5.91087			0.252270
$550$	1.64015	−	4.00000i	0.0699363	−	0.170561i
$551$	−16.2341			−0.691595
$552$		−	1.00000i		−	0.0425628i
$553$			7.58767i			0.322660i
$554$	27.0741			1.15027
$555$	2.22012	−	11.2663i	0.0942389	−	0.478230i
$556$	−3.45856			−0.146676
$557$			9.47063i			0.401283i	0.979665	+	0.200642i	$0.0643027\pi$
$557$			9.47063i			0.401283i	−0.979665	+	0.200642i	$0.935697\pi$
$558$		−	3.25240i		−	0.137685i
$559$	−6.27072			−0.265223
$560$	4.38776	+	0.864641i	0.185417	+	0.0365377i
$561$	3.04623			0.128612
$562$			27.2803i			1.15075i
$563$		−	1.51105i		−	0.0636832i	−0.999493	−	0.0318416i	$-0.989863\pi$
$563$		−	1.51105i		−	0.0636832i	0.999493	−	0.0318416i	$-0.0101372\pi$
$564$	−5.52311			−0.232565
$565$	−12.7110	−	2.50479i	−0.534754	−	0.105377i
$566$	30.9205			1.29968
$567$			2.00000i			0.0839921i
$568$		−	5.79383i		−	0.243104i
$569$	20.0000			0.838444			0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000			0.838444			0.419222	+	0.907884i	$0.362303\pi$
$570$	3.50916	−	17.8078i	0.146983	−	0.745886i
$571$	−35.1633			−1.47154			−0.735768	−	0.677233i	$-0.763179\pi$
$571$	−35.1633			−1.47154			−0.735768	+	0.677233i	$0.763179\pi$
$572$		−	4.77551i		−	0.199674i
$573$		−	12.2341i		−	0.511086i
$574$	7.04623			0.294104
$575$	−1.89692	+	4.62620i	−0.0791069	+	0.192926i
$576$	1.00000			0.0416667
$577$		−	22.7110i		−	0.945470i	−0.881205	−	0.472735i	$-0.843267\pi$
$577$		−	22.7110i		−	0.945470i	0.881205	−	0.472735i	$-0.156733\pi$
$578$		−	4.58767i		−	0.190822i
$579$	−7.45856			−0.309967
$580$	−0.864641	+	4.38776i	−0.0359023	+	0.182192i
$581$	16.6464			0.690609
$582$			1.04623i			0.0433676i
$583$			1.15994i			0.0480398i
$584$	15.2524			0.631149
$585$	12.1170	+	2.38776i	0.500978	+	0.0987215i
$586$	−11.0708			−0.457331
$587$			3.76593i			0.155436i	0.996975	+	0.0777182i	$0.0247634\pi$
$587$			3.76593i			0.155436i	−0.996975	+	0.0777182i	$0.975237\pi$
$588$			3.00000i			0.123718i
$589$	26.3998			1.08779
$590$	−24.0925	−	4.74760i	−0.991871	−	0.195456i
$591$	2.00000			0.0822690
$592$			5.13536i			0.211062i
$593$		−	1.04623i		−	0.0429635i	−0.999769	−	0.0214817i	$-0.993162\pi$
$593$		−	1.04623i		−	0.0429635i	0.999769	−	0.0214817i	$-0.00683837\pi$
$594$	−0.864641			−0.0354766
$595$	3.04623	−	15.4586i	0.124883	−	0.633739i
$596$	−8.86464			−0.363110
$597$			22.8401i			0.934781i
$598$			5.52311i			0.225857i
$599$	21.2803			0.869490			0.434745	−	0.900554i	$-0.356839\pi$
$599$	21.2803			0.869490			0.434745	+	0.900554i	$0.356839\pi$
$600$	−4.62620	−	1.89692i	−0.188864	−	0.0774413i
$601$	30.8034			1.25650			0.628249	−	0.778012i	$-0.283772\pi$
$601$	30.8034			1.25650			0.628249	+	0.778012i	$0.283772\pi$
$602$		−	2.27072i		−	0.0925476i
$603$			5.91087i			0.240709i
$604$	6.71096			0.273065
$605$	−4.43232	+	22.4925i	−0.180199	+	0.914450i
$606$	−9.04623			−0.367478
$607$		−	22.0925i		−	0.896705i	−0.893857	−	0.448353i	$-0.852011\pi$
$607$		−	22.0925i		−	0.896705i	0.893857	−	0.448353i	$-0.147989\pi$
$608$			8.11704i			0.329189i
$609$	4.00000			0.162088
$610$	12.9677	+	2.55539i	0.525048	+	0.103465i
$611$	30.5048			1.23409
$612$		−	3.52311i		−	0.142413i
$613$			41.4985i			1.67611i	0.545586	+	0.838055i	$0.316307\pi$
$613$			41.4985i			1.67611i	−0.545586	+	0.838055i	$0.683693\pi$
$614$	4.23407			0.170873
$615$	−7.72928	−	1.52311i	−0.311675	−	0.0614179i
$616$	1.72928			0.0696747
$617$		−	30.9325i		−	1.24530i	−0.782502	−	0.622648i	$-0.786057\pi$
$617$		−	30.9325i		−	1.24530i	0.782502	−	0.622648i	$-0.213943\pi$
$618$			11.3169i			0.455234i
$619$	−14.5660			−0.585458			−0.292729	−	0.956196i	$-0.594563\pi$
$619$	−14.5660			−0.585458			−0.292729	+	0.956196i	$0.594563\pi$
$620$	1.40608	−	7.13536i	0.0564694	−	0.286563i
$621$	1.00000			0.0401286
$622$			13.2803i			0.532492i
$623$		−	9.55102i		−	0.382654i
$624$	−5.52311			−0.221102
$625$	17.8034	+	17.5510i	0.712137	+	0.702041i
$626$	31.9634			1.27751
$627$		−	7.01832i		−	0.280285i
$628$			14.6864i			0.586050i
$629$	18.0925			0.721394
$630$	−0.864641	+	4.38776i	−0.0344481	+	0.174812i
$631$	−38.4036			−1.52882			−0.764412	−	0.644729i	$-0.776970\pi$
$631$	−38.4036			−1.52882			−0.764412	+	0.644729i	$0.776970\pi$
$632$			3.79383i			0.150911i
$633$		−	24.2341i		−	0.963218i
$634$	−13.2803			−0.527428
$635$	1.64015	+	0.323204i	0.0650874	+	0.0128260i
$636$	1.34153			0.0531950
$637$		−	16.5693i		−	0.656501i
$638$			1.72928i			0.0684629i
$639$	5.79383			0.229200
$640$	2.19388	+	0.432320i	0.0867206	+	0.0170890i
$641$	−31.9267			−1.26103			−0.630515	−	0.776177i	$-0.717156\pi$
$641$	−31.9267			−1.26103			−0.630515	+	0.776177i	$0.717156\pi$
$642$		−	5.64015i		−	0.222599i
$643$			4.72302i			0.186258i	0.995654	+	0.0931289i	$0.0296869\pi$
$643$			4.72302i			0.186258i	−0.995654	+	0.0931289i	$0.970313\pi$
$644$	−2.00000			−0.0788110
$645$	−0.490839	+	2.49084i	−0.0193268	+	0.0980767i
$646$	28.5972			1.12514
$647$		−	43.4586i		−	1.70853i	−0.519836	−	0.854266i	$-0.674007\pi$
$647$		−	43.4586i		−	1.70853i	0.519836	−	0.854266i	$-0.325993\pi$
$648$			1.00000i			0.0392837i
$649$	−9.49521			−0.372720
$650$	25.5510	+	10.4769i	1.00219	+	0.410937i
$651$	−6.50479			−0.254943
$652$		−	23.2803i		−	0.911727i
$653$		−	2.36318i		−	0.0924782i	−0.998930	−	0.0462391i	$-0.985276\pi$
$653$		−	2.36318i		−	0.0924782i	0.998930	−	0.0462391i	$-0.0147236\pi$
$654$	−12.1816			−0.476338
$655$	8.54144	−	43.3449i	0.333742	−	1.69362i
$656$	3.52311			0.137555
$657$			15.2524i			0.595053i
$658$			11.0462i			0.430627i
$659$	22.4157			0.873190			0.436595	−	0.899658i	$-0.356184\pi$
$659$	22.4157			0.873190			0.436595	+	0.899658i	$0.356184\pi$
$660$	−1.89692	−	0.373802i	−0.0738374	−	0.0145502i
$661$	−14.9937			−0.583189			−0.291594	−	0.956542i	$-0.594186\pi$
$661$	−14.9937			−0.583189			−0.291594	+	0.956542i	$0.594186\pi$
$662$			11.8217i			0.459465i
$663$			19.4586i			0.755708i
$664$	8.32320			0.323003
$665$	−35.6156	−	7.01832i	−1.38111	−	0.272159i
$666$	−5.13536			−0.198991
$667$		−	2.00000i		−	0.0774403i
$668$			5.93545i			0.229649i
$669$	−13.9634			−0.539855
$670$	−2.55539	+	12.9677i	−0.0987233	+	0.500987i
$671$	5.11078			0.197299
$672$		−	2.00000i		−	0.0771517i
$673$			49.2158i			1.89713i	0.316586	+	0.948564i	$0.397464\pi$
$673$			49.2158i			1.89713i	−0.316586	+	0.948564i	$0.602536\pi$
$674$	13.2803			0.511538
$675$	1.89692	−	4.62620i	0.0730123	−	0.178062i
$676$	17.5048			0.673261
$677$		−	5.34153i		−	0.205292i	−0.994718	−	0.102646i	$-0.967269\pi$
$677$		−	5.34153i		−	0.205292i	0.994718	−	0.102646i	$-0.0327308\pi$
$678$			5.79383i			0.222511i
$679$	2.09246			0.0803011
$680$	1.52311	−	7.72928i	0.0584088	−	0.296404i
$681$	−4.45231			−0.170613
$682$		−	2.81215i		−	0.107683i
$683$		−	27.1512i		−	1.03891i	−0.854497	−	0.519456i	$-0.826135\pi$
$683$		−	27.1512i		−	1.03891i	0.854497	−	0.519456i	$-0.173865\pi$
$684$	−8.11704			−0.310363
$685$	1.04623	+	0.206167i	0.0399743	+	0.00787725i
$686$	20.0000			0.763604
$687$		−	4.05249i		−	0.154612i
$688$		−	1.13536i		−	0.0432852i
$689$	−7.40940			−0.282276
$690$	2.19388	+	0.432320i	0.0835195	+	0.0164582i
$691$	12.1050			0.460495			0.230247	−	0.973132i	$-0.426046\pi$
$691$	12.1050			0.460495			0.230247	+	0.973132i	$0.426046\pi$
$692$			25.8217i			0.981595i
$693$			1.72928i			0.0656900i
$694$	−24.3632			−0.924814
$695$	1.49521	−	7.58767i	0.0567165	−	0.287817i
$696$	2.00000			0.0758098
$697$		−	12.4123i		−	0.470151i
$698$			3.49521i			0.132296i
$699$	24.5327			0.927912
$700$	−3.79383	+	9.25240i	−0.143393	+	0.349708i
$701$	2.59392			0.0979711			0.0489856	−	0.998799i	$-0.484401\pi$
$701$	2.59392			0.0979711			0.0489856	+	0.998799i	$0.484401\pi$
$702$		−	5.52311i		−	0.208457i
$703$		−	41.6839i		−	1.57214i
$704$	0.864641			0.0325874
$705$	2.38776	−	12.1170i	0.0899281	−	0.456354i
$706$	−31.5789			−1.18849
$707$			18.0925i			0.680437i
$708$			10.9817i			0.412717i
$709$	−4.00333			−0.150348			−0.0751741	−	0.997170i	$-0.523951\pi$
$709$	−4.00333			−0.150348			−0.0751741	+	0.997170i	$0.523951\pi$
$710$	12.7110	+	2.50479i	0.477034	+	0.0940032i
$711$	−3.79383			−0.142280
$712$		−	4.77551i		−	0.178970i
$713$			3.25240i			0.121803i
$714$	−7.04623			−0.263698
$715$	10.4769	+	2.06455i	0.391813	+	0.0772098i
$716$	−23.3449			−0.872438
$717$			10.7755i			0.402419i
$718$		−	21.1878i		−	0.790723i
$719$	−20.2707			−0.755970			−0.377985	−	0.925812i	$-0.623383\pi$
$719$	−20.2707			−0.755970			−0.377985	+	0.925812i	$0.623383\pi$
$720$	−0.432320	+	2.19388i	−0.0161116	+	0.0817610i
$721$	22.6339			0.842930
$722$		−	46.8863i		−	1.74493i
$723$			10.7755i			0.400746i
$724$	−13.3694			−0.496871
$725$	−9.25240	−	3.79383i	−0.343625	−	0.140899i
$726$	10.2524			0.380502
$727$		−	32.6339i		−	1.21032i	−0.796102	−	0.605162i	$-0.793108\pi$
$727$		−	32.6339i		−	1.21032i	0.796102	−	0.605162i	$-0.206892\pi$
$728$			11.0462i			0.409400i
$729$	−1.00000			−0.0370370
$730$	−6.59392	+	33.4619i	−0.244052	+	1.23848i
$731$	−4.00000			−0.147945
$732$		−	5.91087i		−	0.218472i
$733$			1.13536i			0.0419354i	0.999780	+	0.0209677i	$0.00667472\pi$
$733$			1.13536i			0.0419354i	−0.999780	+	0.0209677i	$0.993325\pi$
$734$	13.0462			0.481545
$735$	−6.58163	−	1.29696i	−0.242767	−	0.0478391i
$736$	−1.00000			−0.0368605
$737$			5.11078i			0.188258i
$738$			3.52311i			0.129688i
$739$	29.4219			1.08230			0.541151	−	0.840925i	$-0.317989\pi$
$739$	29.4219			1.08230			0.541151	+	0.840925i	$0.317989\pi$
$740$	−11.2663	−	2.22012i	−0.414159	−	0.0816133i
$741$	44.8313			1.64692
$742$		−	2.68305i		−	0.0984980i
$743$			27.4586i			1.00736i	0.863891	+	0.503678i	$0.168020\pi$
$743$			27.4586i			1.00736i	−0.863891	+	0.503678i	$0.831980\pi$
$744$	−3.25240			−0.119239
$745$	3.83237	−	19.4479i	0.140407	−	0.712517i
$746$	−18.8646			−0.690684
$747$			8.32320i			0.304530i
$748$		−	3.04623i		−	0.111381i
$749$	−11.2803			−0.412173
$750$	6.16160	−	9.32924i	0.224990	−	0.340656i
$751$	4.02791			0.146980			0.0734902	−	0.997296i	$-0.476586\pi$
$751$	4.02791			0.146980			0.0734902	+	0.997296i	$0.476586\pi$
$752$			5.52311i			0.201407i
$753$			3.91087i			0.142520i
$754$	−11.0462			−0.402280
$755$	−2.90129	+	14.7230i	−0.105589	+	0.535826i
$756$	2.00000			0.0727393
$757$		−	31.5110i		−	1.14529i	−0.819804	−	0.572644i	$-0.805918\pi$
$757$		−	31.5110i		−	1.14529i	0.819804	−	0.572644i	$-0.194082\pi$
$758$			26.7509i			0.971638i
$759$	0.864641			0.0313845
$760$	−17.8078	−	3.50916i	−0.645956	−	0.127291i
$761$	−19.9634			−0.723671			−0.361836	−	0.932242i	$-0.617850\pi$
$761$	−19.9634			−0.723671			−0.361836	+	0.932242i	$0.617850\pi$
$762$		−	0.747604i		−	0.0270828i
$763$			24.3632i			0.882006i
$764$	−12.2341			−0.442613
$765$	7.72928	+	1.52311i	0.279453	+	0.0550683i
$766$	22.5048			0.813131
$767$		−	60.6531i		−	2.19006i
$768$		−	1.00000i		−	0.0360844i
$769$	46.4190			1.67391			0.836956	−	0.547271i	$-0.184333\pi$
$769$	46.4190			1.67391			0.836956	+	0.547271i	$0.184333\pi$
$770$	−0.747604	+	3.79383i	−0.0269418	+	0.136720i
$771$	28.0925			1.01173
$772$			7.45856i			0.268440i
$773$			29.0342i			1.04429i	0.852858	+	0.522143i	$0.174867\pi$
$773$			29.0342i			1.04429i	−0.852858	+	0.522143i	$0.825133\pi$
$774$	1.13536			0.0408096
$775$	15.0462	+	6.16952i	0.540476	+	0.221616i
$776$	1.04623			0.0375574
$777$			10.2707i			0.368460i
$778$		−	7.67680i		−	0.275226i
$779$	−28.5972			−1.02460
$780$	2.38776	−	12.1170i	0.0854954	−	0.433860i
$781$	5.00958			0.179257
$782$			3.52311i			0.125986i
$783$			2.00000i			0.0714742i
$784$	3.00000			0.107143
$785$	−32.2201	−	6.34922i	−1.14999	−	0.226613i
$786$	−19.7572			−0.704716
$787$			35.7693i			1.27504i	0.770435	+	0.637518i	$0.220039\pi$
$787$			35.7693i			1.27504i	−0.770435	+	0.637518i	$0.779961\pi$
$788$		−	2.00000i		−	0.0712470i
$789$	−29.5510			−1.05204
$790$	−8.32320	−	1.64015i	−0.296126	−	0.0583540i
$791$	11.5877			0.412010
$792$			0.864641i			0.0307237i
$793$			32.6464i			1.15931i
$794$	−31.7938			−1.12832
$795$	−0.579969	+	2.94315i	−0.0205694	+	0.104383i
$796$	22.8401			0.809545
$797$		−	22.8367i		−	0.808919i	−0.914556	−	0.404459i	$-0.867460\pi$
$797$		−	22.8367i		−	0.808919i	0.914556	−	0.404459i	$-0.132540\pi$
$798$			16.2341i			0.574680i
$799$	19.4586			0.688394
$800$	−1.89692	+	4.62620i	−0.0670661	+	0.163561i
$801$	4.77551			0.168734
$802$			18.0925i			0.638867i
$803$			13.1878i			0.465389i
$804$	5.91087			0.208460
$805$	0.864641	−	4.38776i	0.0304746	−	0.154648i
$806$	17.9634			0.632732
$807$		−	17.2803i		−	0.608295i
$808$			9.04623i			0.318245i
$809$	36.9171			1.29794			0.648969	−	0.760815i	$-0.275201\pi$
$809$	36.9171			1.29794			0.648969	+	0.760815i	$0.275201\pi$
$810$	−2.19388	−	0.432320i	−0.0770850	−	0.0151902i
$811$	51.8217			1.81971			0.909854	−	0.414929i	$-0.136194\pi$
$811$	51.8217			1.81971			0.909854	+	0.414929i	$0.136194\pi$
$812$		−	4.00000i		−	0.140372i
$813$		−	9.96336i		−	0.349430i
$814$	−4.44024			−0.155630
$815$	51.0741	+	10.0646i	1.78905	+	0.352546i
$816$	−3.52311			−0.123334
$817$			9.21575i			0.322418i
$818$			2.00000i			0.0699284i
$819$	−11.0462			−0.385986
$820$	−1.52311	+	7.72928i	−0.0531895	+	0.269918i
$821$	−17.8709			−0.623699			−0.311849	−	0.950132i	$-0.600948\pi$
$821$	−17.8709			−0.623699			−0.311849	+	0.950132i	$0.600948\pi$
$822$		−	0.476886i		−	0.0166333i
$823$		−	9.00958i		−	0.314054i	−0.987594	−	0.157027i	$-0.949809\pi$
$823$		−	9.00958i		−	0.314054i	0.987594	−	0.157027i	$-0.0501910\pi$
$824$	11.3169			0.394245
$825$	1.64015	−	4.00000i	0.0571027	−	0.139262i
$826$	21.9634			0.764203
$827$			42.1083i			1.46425i	0.681171	+	0.732125i	$0.261471\pi$
$827$			42.1083i			1.46425i	−0.681171	+	0.732125i	$0.738529\pi$
$828$		−	1.00000i		−	0.0347524i
$829$	35.5510			1.23474			0.617369	−	0.786674i	$-0.288199\pi$
$829$	35.5510			1.23474			0.617369	+	0.786674i	$0.288199\pi$
$830$	−3.59829	+	18.2601i	−0.124899	+	0.633817i
$831$	27.0741			0.939191
$832$			5.52311i			0.191480i
$833$		−	10.5693i		−	0.366206i
$834$	−3.45856			−0.119760
$835$	−13.0216	−	2.56602i	−0.450633	−	0.0888006i
$836$	−7.01832			−0.242734
$837$		−	3.25240i		−	0.112419i
$838$			20.4523i			0.706513i
$839$	25.6801			0.886576			0.443288	−	0.896379i	$-0.353812\pi$
$839$	25.6801			0.886576			0.443288	+	0.896379i	$0.353812\pi$
$840$	4.38776	+	0.864641i	0.151392	+	0.0298329i
$841$	−25.0000			−0.862069
$842$			12.9571i			0.446531i
$843$			27.2803i			0.939584i
$844$	−24.2341			−0.834171
$845$	−7.56768	+	38.4034i	−0.260336	+	1.32112i
$846$	−5.52311			−0.189889
$847$		−	20.5048i		−	0.704553i
$848$		−	1.34153i		−	0.0460682i
$849$	30.9205			1.06119
$850$	16.2986	+	6.68305i	0.559038	+	0.229227i
$851$	5.13536			0.176038
$852$		−	5.79383i		−	0.198493i
$853$			11.6647i			0.399393i	0.979858	+	0.199696i	$0.0639956\pi$
$853$			11.6647i			0.399393i	−0.979858	+	0.199696i	$0.936004\pi$
$854$	−11.8217			−0.404532
$855$	3.50916	−	17.8078i	0.120011	−	0.609013i
$856$	−5.64015			−0.192776
$857$			41.5423i			1.41906i	0.704677	+	0.709529i	$0.251092\pi$
$857$			41.5423i			1.41906i	−0.704677	+	0.709529i	$0.748908\pi$
$858$		−	4.77551i		−	0.163033i
$859$	−21.5510			−0.735311			−0.367656	−	0.929962i	$-0.619839\pi$
$859$	−21.5510			−0.735311			−0.367656	+	0.929962i	$0.619839\pi$
$860$	2.49084	+	0.490839i	0.0849369	+	0.0167375i
$861$	7.04623			0.240135
$862$		−	33.3728i		−	1.13668i
$863$			6.91713i			0.235462i	0.993046	+	0.117731i	$0.0375620\pi$
$863$			6.91713i			0.235462i	−0.993046	+	0.117731i	$0.962438\pi$
$864$	1.00000			0.0340207
$865$	−56.6497	−	11.1633i	−1.92615	−	0.379562i
$866$	−19.0096			−0.645972
$867$		−	4.58767i		−	0.155805i
$868$			6.50479i			0.220787i
$869$	−3.28030			−0.111277
$870$	−0.864641	+	4.38776i	−0.0293141	+	0.148759i
$871$	−32.6464			−1.10618
$872$			12.1816i			0.412521i
$873$			1.04623i			0.0354095i
$874$	8.11704			0.274563
$875$	−18.6585	−	12.3232i	−0.630772	−	0.416600i
$876$	15.2524			0.515331
$877$			2.53270i			0.0855232i	0.999085	+	0.0427616i	$0.0136156\pi$
$877$			2.53270i			0.0855232i	−0.999085	+	0.0427616i	$0.986384\pi$
$878$			32.5972i			1.10010i
$879$	−11.0708			−0.373409
$880$	−0.373802	+	1.89692i	−0.0126009	+	0.0639450i
$881$	−10.8680			−0.366151			−0.183076	−	0.983099i	$-0.558605\pi$
$881$	−10.8680			−0.366151			−0.183076	+	0.983099i	$0.558605\pi$
$882$			3.00000i			0.101015i
$883$		−	40.4190i		−	1.36021i	−0.733116	−	0.680104i	$-0.761935\pi$
$883$		−	40.4190i		−	1.36021i	0.733116	−	0.680104i	$-0.238065\pi$
$884$	19.4586			0.654462
$885$	−24.0925	−	4.74760i	−0.809859	−	0.159589i
$886$	−4.00000			−0.134383
$887$		−	7.48647i		−	0.251371i	−0.992070	−	0.125686i	$-0.959887\pi$
$887$		−	7.48647i		−	0.251371i	0.992070	−	0.125686i	$-0.0401130\pi$
$888$			5.13536i			0.172331i
$889$	−1.49521			−0.0501477
$890$	10.4769	+	2.06455i	0.351186	+	0.0692039i
$891$	−0.864641			−0.0289666
$892$			13.9634i			0.467528i
$893$		−	44.8313i		−	1.50022i
$894$	−8.86464			−0.296478
$895$	10.0925	−	51.2158i	0.337354	−	1.71195i
$896$	−2.00000			−0.0668153
$897$			5.52311i			0.184411i
$898$			39.5789i			1.32077i
$899$	−6.50479			−0.216947
$900$	−4.62620	−	1.89692i	−0.154207	−	0.0632305i
$901$	−4.72635			−0.157458
$902$			3.04623i			0.101428i
$903$		−	2.27072i		−	0.0755648i
$904$	5.79383			0.192700
$905$	5.77988	−	29.3309i	0.192130	−	0.974992i
$906$	6.71096			0.222957
$907$		−	22.6864i		−	0.753289i	−0.926358	−	0.376644i	$-0.877078\pi$
$907$		−	22.6864i		−	0.753289i	0.926358	−	0.376644i	$-0.122922\pi$
$908$			4.45231i			0.147755i
$909$	−9.04623			−0.300044
$910$	−24.2341	−	4.77551i	−0.803352	−	0.158307i
$911$	−35.8217			−1.18683			−0.593414	−	0.804898i	$-0.702220\pi$
$911$	−35.8217			−1.18683			−0.593414	+	0.804898i	$0.702220\pi$
$912$			8.11704i			0.268782i
$913$			7.19658i			0.238172i
$914$	24.5048			0.810546
$915$	12.9677	+	2.55539i	0.428700	+	0.0844786i
$916$	−4.05249			−0.133898
$917$			39.5144i			1.30488i
$918$		−	3.52311i		−	0.116280i
$919$	13.2890			0.438365			0.219182	−	0.975684i	$-0.429661\pi$
$919$	13.2890			0.438365			0.219182	+	0.975684i	$0.429661\pi$
$920$	0.432320	−	2.19388i	0.0142532	−	0.0723300i
$921$	4.23407			0.139517
$922$		−	18.2341i		−	0.600507i
$923$			32.0000i			1.05329i
$924$	1.72928			0.0568892
$925$	9.74135	−	23.7572i	0.320293	−	0.781132i
$926$	4.20617			0.138223
$927$			11.3169i			0.371697i
$928$		−	2.00000i		−	0.0656532i
$929$	−41.1299			−1.34943			−0.674715	−	0.738078i	$-0.735733\pi$
$929$	−41.1299			−1.34943			−0.674715	+	0.738078i	$0.735733\pi$
$930$	1.40608	−	7.13536i	0.0461071	−	0.233978i
$931$	−24.3511			−0.798075
$932$		−	24.5327i		−	0.803595i
$933$			13.2803i			0.434778i
$934$	19.1912			0.627954
$935$	6.68305	+	1.31695i	0.218559	+	0.0430688i
$936$	−5.52311			−0.180529
$937$			22.7755i			0.744043i	0.928224	+	0.372022i	$0.121335\pi$
$937$			22.7755i			0.744043i	−0.928224	+	0.372022i	$0.878665\pi$
$938$		−	11.8217i		−	0.385993i
$939$	31.9634			1.04308
$940$	−12.1170	−	2.38776i	−0.395214	−	0.0778800i
$941$	−54.7788			−1.78574			−0.892870	−	0.450315i	$-0.851312\pi$
$941$	−54.7788			−1.78574			−0.892870	+	0.450315i	$0.851312\pi$
$942$			14.6864i			0.478508i
$943$		−	3.52311i		−	0.114728i
$944$	10.9817			0.357423
$945$	−0.864641	+	4.38776i	−0.0281268	+	0.142734i
$946$	0.981678			0.0319171
$947$		−	11.4094i		−	0.370756i	−0.982667	−	0.185378i	$-0.940649\pi$
$947$		−	11.4094i		−	0.370756i	0.982667	−	0.185378i	$-0.0593509\pi$
$948$			3.79383i			0.123218i
$949$	−84.2407			−2.73457
$950$	15.3973	−	37.5510i	0.499556	−	1.21832i
$951$	−13.2803			−0.430643
$952$			7.04623i			0.228370i
$953$			41.6714i			1.34987i	0.737878	+	0.674934i	$0.235828\pi$
$953$			41.6714i			1.34987i	−0.737878	+	0.674934i	$0.764172\pi$
$954$	1.34153			0.0434335
$955$	5.28904	−	26.8401i	0.171149	−	0.868524i
$956$	10.7755			0.348505
$957$			1.72928i			0.0558997i
$958$		−	25.6801i		−	0.829687i
$959$	−0.953771			−0.0307989
$960$	2.19388	+	0.432320i	0.0708071	+	0.0139531i
$961$	−20.4219			−0.658772
$962$		−	28.3632i		−	0.914465i
$963$		−	5.64015i		−	0.181751i
$964$	10.7755			0.347056
$965$	−16.3632	−	3.22449i	−0.526749	−	0.103800i
$966$	−2.00000			−0.0643489
$967$		−	3.17533i		−	0.102112i	−0.998696	−	0.0510559i	$-0.983741\pi$
$967$		−	3.17533i		−	0.102112i	0.998696	−	0.0510559i	$-0.0162587\pi$
$968$		−	10.2524i		−	0.329524i
$969$	28.5972			0.918676
$970$	−0.452306	+	2.29530i	−0.0145227	+	0.0736976i
$971$	−35.7326			−1.14671			−0.573357	−	0.819306i	$-0.694359\pi$
$971$	−35.7326			−1.14671			−0.573357	+	0.819306i	$0.694359\pi$
$972$			1.00000i			0.0320750i
$973$			6.91713i			0.221753i
$974$	8.74760			0.280291
$975$	25.5510	+	10.4769i	0.818288	+	0.335529i
$976$	−5.91087			−0.189202
$977$		−	2.02791i		−	0.0648785i	−0.999474	−	0.0324392i	$-0.989672\pi$
$977$		−	2.02791i		−	0.0648785i	0.999474	−	0.0324392i	$-0.0103275\pi$
$978$		−	23.2803i		−	0.744422i
$979$	4.12910			0.131967
$980$	−1.29696	+	6.58163i	−0.0414299	+	0.210243i
$981$	−12.1816			−0.388928
$982$		−	23.8863i		−	0.762242i
$983$		−	1.67347i		−	0.0533754i	−0.999644	−	0.0266877i	$-0.991504\pi$
$983$		−	1.67347i		−	0.0533754i	0.999644	−	0.0266877i	$-0.00849596\pi$
$984$	3.52311			0.112313
$985$	4.38776	+	0.864641i	0.139806	+	0.0275497i
$986$	−7.04623			−0.224398
$987$			11.0462i			0.351605i
$988$		−	44.8313i		−	1.42627i
$989$	−1.13536			−0.0361023
$990$	−1.89692	−	0.373802i	−0.0602880	−	0.0118802i
$991$	−55.8496			−1.77412			−0.887061	−	0.461652i	$-0.847257\pi$
$991$	−55.8496			−1.77412			−0.887061	+	0.461652i	$0.847257\pi$
$992$			3.25240i			0.103264i
$993$			11.8217i			0.375152i
$994$	−11.5877			−0.367538
$995$	−9.87423	+	50.1083i	−0.313034	+	1.58854i
$996$	8.32320			0.263731
$997$		−	44.2062i		−	1.40002i	−0.714131	−	0.700012i	$-0.753178\pi$
$997$		−	44.2062i		−	1.40002i	0.714131	−	0.700012i	$-0.246822\pi$
$998$			29.7293i			0.941064i
$999$	−5.13536			−0.162476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.2.d.d.139.2	✓	6
3.2	odd	2		2070.2.d.d.829.5		6
5.2	odd	4		3450.2.a.br.1.2		3
5.3	odd	4		3450.2.a.bq.1.2		3
5.4	even	2	inner	690.2.d.d.139.5	yes	6
15.14	odd	2		2070.2.d.d.829.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.d.d.139.2	✓	6	1.1	even	1	trivial
690.2.d.d.139.5	yes	6	5.4	even	2	inner
2070.2.d.d.829.2		6	15.14	odd	2
2070.2.d.d.829.5		6	3.2	odd	2
3450.2.a.bq.1.2		3	5.3	odd	4
3450.2.a.br.1.2		3	5.2	odd	4

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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(0.432320 - 2.19388i) q^{5} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(0.432320 - 2.19388i) q^{5} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +(-2.19388 - 0.432320i) q^{10} -0.864641 q^{11} +1.00000i q^{12} -5.52311i q^{13} +2.00000 q^{14} +(-2.19388 - 0.432320i) q^{15} +1.00000 q^{16} -3.52311i q^{17} +1.00000i q^{18} -8.11704 q^{19} +(-0.432320 + 2.19388i) q^{20} +2.00000 q^{21} +0.864641i q^{22} -1.00000i q^{23} +1.00000 q^{24} +(-4.62620 - 1.89692i) q^{25} -5.52311 q^{26} +1.00000i q^{27} -2.00000i q^{28} +2.00000 q^{29} +(-0.432320 + 2.19388i) q^{30} -3.25240 q^{31} -1.00000i q^{32} +0.864641i q^{33} -3.52311 q^{34} +(4.38776 + 0.864641i) q^{35} +1.00000 q^{36} +5.13536i q^{37} +8.11704i q^{38} -5.52311 q^{39} +(2.19388 + 0.432320i) q^{40} +3.52311 q^{41} -2.00000i q^{42} -1.13536i q^{43} +0.864641 q^{44} +(-0.432320 + 2.19388i) q^{45} -1.00000 q^{46} +5.52311i q^{47} -1.00000i q^{48} +3.00000 q^{49} +(-1.89692 + 4.62620i) q^{50} -3.52311 q^{51} +5.52311i q^{52} -1.34153i q^{53} +1.00000 q^{54} +(-0.373802 + 1.89692i) q^{55} -2.00000 q^{56} +8.11704i q^{57} -2.00000i q^{58} +10.9817 q^{59} +(2.19388 + 0.432320i) q^{60} -5.91087 q^{61} +3.25240i q^{62} -2.00000i q^{63} -1.00000 q^{64} +(-12.1170 - 2.38776i) q^{65} +0.864641 q^{66} -5.91087i q^{67} +3.52311i q^{68} -1.00000 q^{69} +(0.864641 - 4.38776i) q^{70} -5.79383 q^{71} -1.00000i q^{72} -15.2524i q^{73} +5.13536 q^{74} +(-1.89692 + 4.62620i) q^{75} +8.11704 q^{76} -1.72928i q^{77} +5.52311i q^{78} +3.79383 q^{79} +(0.432320 - 2.19388i) q^{80} +1.00000 q^{81} -3.52311i q^{82} -8.32320i q^{83} -2.00000 q^{84} +(-7.72928 - 1.52311i) q^{85} -1.13536 q^{86} -2.00000i q^{87} -0.864641i q^{88} -4.77551 q^{89} +(2.19388 + 0.432320i) q^{90} +11.0462 q^{91} +1.00000i q^{92} +3.25240i q^{93} +5.52311 q^{94} +(-3.50916 + 17.8078i) q^{95} -1.00000 q^{96} -1.04623i q^{97} -3.00000i q^{98} +0.864641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 6 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^4 - 6 * q^6 - 6 * q^9	\( 6 q - 6 q^{4} - 6 q^{6} - 6 q^{9} + 2 q^{10} + 12 q^{14} + 2 q^{15} + 6 q^{16} - 8 q^{19} + 12 q^{21} + 6 q^{24} - 10 q^{25} - 8 q^{26} + 12 q^{29} + 16 q^{31} + 4 q^{34} - 4 q^{35} + 6 q^{36} - 8 q^{39} - 2 q^{40} - 4 q^{41} - 6 q^{46} + 18 q^{49} - 4 q^{50} + 4 q^{51} + 6 q^{54} - 20 q^{55} - 12 q^{56} + 20 q^{59} - 2 q^{60} + 20 q^{61} - 6 q^{64} - 32 q^{65} - 6 q^{69} - 20 q^{71} + 36 q^{74} - 4 q^{75} + 8 q^{76} + 8 q^{79} + 6 q^{81} - 12 q^{84} - 36 q^{85} - 12 q^{86} + 32 q^{89} - 2 q^{90} + 16 q^{91} + 8 q^{94} - 44 q^{95} - 6 q^{96}+O(q^{100}) \) 6 * q - 6 * q^4 - 6 * q^6 - 6 * q^9 + 2 * q^10 + 12 * q^14 + 2 * q^15 + 6 * q^16 - 8 * q^19 + 12 * q^21 + 6 * q^24 - 10 * q^25 - 8 * q^26 + 12 * q^29 + 16 * q^31 + 4 * q^34 - 4 * q^35 + 6 * q^36 - 8 * q^39 - 2 * q^40 - 4 * q^41 - 6 * q^46 + 18 * q^49 - 4 * q^50 + 4 * q^51 + 6 * q^54 - 20 * q^55 - 12 * q^56 + 20 * q^59 - 2 * q^60 + 20 * q^61 - 6 * q^64 - 32 * q^65 - 6 * q^69 - 20 * q^71 + 36 * q^74 - 4 * q^75 + 8 * q^76 + 8 * q^79 + 6 * q^81 - 12 * q^84 - 36 * q^85 - 12 * q^86 + 32 * q^89 - 2 * q^90 + 16 * q^91 + 8 * q^94 - 44 * q^95 - 6 * q^96

Introduction

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