LMFDB - Embedded newform 165.2.c.a.34.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,2,Mod(34,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, 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("165.34");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 165 = 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	165.c (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:	$1.31753163335$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			34.6
Root			$0.403032 + 0.403032i$ of defining polynomial
Character	$\chi$	$=$	165.34
Dual form			165.2.c.a.34.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.67513i q^{2} +1.00000i q^{3} -5.15633 q^{4} +(-1.48119 - 1.67513i) q^{5} -2.67513 q^{6} +2.80606i q^{7} -8.44358i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.67513i q^{2} +1.00000i q^{3} -5.15633 q^{4} +(-1.48119 - 1.67513i) q^{5} -2.67513 q^{6} +2.80606i q^{7} -8.44358i q^{8} -1.00000 q^{9} +(4.48119 - 3.96239i) q^{10} -1.00000 q^{11} -5.15633i q^{12} +5.11871i q^{13} -7.50659 q^{14} +(1.67513 - 1.48119i) q^{15} +12.2750 q^{16} +4.54420i q^{17} -2.67513i q^{18} +4.57452 q^{19} +(7.63752 + 8.63752i) q^{20} -2.80606 q^{21} -2.67513i q^{22} -4.00000i q^{23} +8.44358 q^{24} +(-0.612127 + 4.96239i) q^{25} -13.6932 q^{26} -1.00000i q^{27} -14.4690i q^{28} +2.38787 q^{29} +(3.96239 + 4.48119i) q^{30} -0.962389 q^{31} +15.9502i q^{32} -1.00000i q^{33} -12.1563 q^{34} +(4.70052 - 4.15633i) q^{35} +5.15633 q^{36} +1.61213i q^{37} +12.2374i q^{38} -5.11871 q^{39} +(-14.1441 + 12.5066i) q^{40} -2.38787 q^{41} -7.50659i q^{42} -2.80606i q^{43} +5.15633 q^{44} +(1.48119 + 1.67513i) q^{45} +10.7005 q^{46} -4.31265i q^{47} +12.2750i q^{48} -0.873992 q^{49} +(-13.2750 - 1.63752i) q^{50} -4.54420 q^{51} -26.3938i q^{52} +6.57452i q^{53} +2.67513 q^{54} +(1.48119 + 1.67513i) q^{55} +23.6932 q^{56} +4.57452i q^{57} +6.38787i q^{58} +13.2750 q^{59} +(-8.63752 + 7.63752i) q^{60} +7.92478 q^{61} -2.57452i q^{62} -2.80606i q^{63} -18.1187 q^{64} +(8.57452 - 7.58181i) q^{65} +2.67513 q^{66} +10.7005i q^{67} -23.4314i q^{68} +4.00000 q^{69} +(11.1187 + 12.5745i) q^{70} -7.35026 q^{71} +8.44358i q^{72} -6.41819i q^{73} -4.31265 q^{74} +(-4.96239 - 0.612127i) q^{75} -23.5877 q^{76} -2.80606i q^{77} -13.6932i q^{78} -1.35026 q^{79} +(-18.1817 - 20.5623i) q^{80} +1.00000 q^{81} -6.38787i q^{82} +0.806063i q^{83} +14.4690 q^{84} +(7.61213 - 6.73084i) q^{85} +7.50659 q^{86} +2.38787i q^{87} +8.44358i q^{88} +2.96239 q^{89} +(-4.48119 + 3.96239i) q^{90} -14.3634 q^{91} +20.6253i q^{92} -0.962389i q^{93} +11.5369 q^{94} +(-6.77575 - 7.66291i) q^{95} -15.9502 q^{96} -9.92478i q^{97} -2.33804i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 10 * q^4 + 2 * q^5 - 6 * q^6 - 6 * q^9	$ 6 q - 10 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{9} + 16 q^{10} - 6 q^{11} - 4 q^{14} + 10 q^{16} + 4 q^{19} + 14 q^{20} - 16 q^{21} + 18 q^{24} - 2 q^{25} - 16 q^{26} + 16 q^{29} + 2 q^{30} + 16 q^{31} - 52 q^{34} - 12 q^{35} + 10 q^{36} + 12 q^{39} - 12 q^{40} - 16 q^{41} + 10 q^{44} - 2 q^{45} + 24 q^{46} - 22 q^{49} - 16 q^{50} - 8 q^{51} + 6 q^{54} - 2 q^{55} + 76 q^{56} + 16 q^{59} - 20 q^{60} + 4 q^{61} - 66 q^{64} + 28 q^{65} + 6 q^{66} + 24 q^{69} + 24 q^{70} - 24 q^{71} + 16 q^{74} - 8 q^{75} - 36 q^{76} + 12 q^{79} - 58 q^{80} + 6 q^{81} + 24 q^{84} + 44 q^{85} + 4 q^{86} - 4 q^{89} - 16 q^{90} + 16 q^{91} + 24 q^{94} - 44 q^{95} - 22 q^{96} + 6 q^{99}+O(q^{100}) $ 6 * q - 10 * q^4 + 2 * q^5 - 6 * q^6 - 6 * q^9 + 16 * q^10 - 6 * q^11 - 4 * q^14 + 10 * q^16 + 4 * q^19 + 14 * q^20 - 16 * q^21 + 18 * q^24 - 2 * q^25 - 16 * q^26 + 16 * q^29 + 2 * q^30 + 16 * q^31 - 52 * q^34 - 12 * q^35 + 10 * q^36 + 12 * q^39 - 12 * q^40 - 16 * q^41 + 10 * q^44 - 2 * q^45 + 24 * q^46 - 22 * q^49 - 16 * q^50 - 8 * q^51 + 6 * q^54 - 2 * q^55 + 76 * q^56 + 16 * q^59 - 20 * q^60 + 4 * q^61 - 66 * q^64 + 28 * q^65 + 6 * q^66 + 24 * q^69 + 24 * q^70 - 24 * q^71 + 16 * q^74 - 8 * q^75 - 36 * q^76 + 12 * q^79 - 58 * q^80 + 6 * q^81 + 24 * q^84 + 44 * q^85 + 4 * q^86 - 4 * q^89 - 16 * q^90 + 16 * q^91 + 24 * q^94 - 44 * q^95 - 22 * q^96 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/165\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$67$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$			2.67513i			1.89160i	0.324745	+	0.945802i	$0.394721\pi$
$2$			2.67513i			1.89160i	−0.324745	+	0.945802i	$0.605279\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	−5.15633			−2.57816
$5$	−1.48119	−	1.67513i	−0.662410	−	0.749141i
$5$	−1.48119	−	1.67513i	−0.662410	−	0.749141i
$6$	−2.67513			−1.09212
$7$			2.80606i			1.06059i	0.847812	+	0.530296i	$0.177919\pi$
$7$			2.80606i			1.06059i	−0.847812	+	0.530296i	$0.822081\pi$
$8$		−	8.44358i		−	2.98526i
$9$	−1.00000			−0.333333
$10$	4.48119	−	3.96239i	1.41708	−	1.25302i
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		−	5.15633i		−	1.48850i
$13$			5.11871i			1.41968i	0.704365	+	0.709838i	$0.251232\pi$
$13$			5.11871i			1.41968i	−0.704365	+	0.709838i	$0.748768\pi$
$14$	−7.50659			−2.00622
$15$	1.67513	−	1.48119i	0.432517	−	0.382443i
$16$	12.2750			3.06876
$17$			4.54420i			1.10213i	0.834462	+	0.551065i	$0.185778\pi$
$17$			4.54420i			1.10213i	−0.834462	+	0.551065i	$0.814222\pi$
$18$		−	2.67513i		−	0.630534i
$19$	4.57452			1.04947			0.524733	−	0.851267i	$-0.324165\pi$
$19$	4.57452			1.04947			0.524733	+	0.851267i	$0.324165\pi$
$20$	7.63752	+	8.63752i	1.70780	+	1.93141i
$21$	−2.80606			−0.612333
$22$		−	2.67513i		−	0.570340i
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	8.44358			1.72354
$25$	−0.612127	+	4.96239i	−0.122425	+	0.992478i
$26$	−13.6932			−2.68546
$27$		−	1.00000i		−	0.192450i
$28$		−	14.4690i		−	2.73438i
$29$	2.38787			0.443417			0.221708	−	0.975113i	$-0.428837\pi$
$29$	2.38787			0.443417			0.221708	+	0.975113i	$0.428837\pi$
$30$	3.96239	+	4.48119i	0.723430	+	0.818150i
$31$	−0.962389			−0.172850			−0.0864250	−	0.996258i	$-0.527544\pi$
$31$	−0.962389			−0.172850			−0.0864250	+	0.996258i	$0.527544\pi$
$32$			15.9502i			2.81962i
$33$		−	1.00000i		−	0.174078i
$34$	−12.1563			−2.08479
$35$	4.70052	−	4.15633i	0.794533	−	0.702547i
$36$	5.15633			0.859388
$37$			1.61213i			0.265032i	0.991181	+	0.132516i	$0.0423056\pi$
$37$			1.61213i			0.265032i	−0.991181	+	0.132516i	$0.957694\pi$
$38$			12.2374i			1.98517i
$39$	−5.11871			−0.819650
$40$	−14.1441	+	12.5066i	−2.23638	+	1.97747i
$41$	−2.38787			−0.372923			−0.186462	−	0.982462i	$-0.559702\pi$
$41$	−2.38787			−0.372923			−0.186462	+	0.982462i	$0.559702\pi$
$42$		−	7.50659i		−	1.15829i
$43$		−	2.80606i		−	0.427921i	−0.976842	−	0.213960i	$-0.931364\pi$
$43$		−	2.80606i		−	0.427921i	0.976842	−	0.213960i	$-0.0686363\pi$
$44$	5.15633			0.777345
$45$	1.48119	+	1.67513i	0.220803	+	0.249714i
$46$	10.7005			1.57771
$47$		−	4.31265i		−	0.629065i	−0.949247	−	0.314532i	$-0.898152\pi$
$47$		−	4.31265i		−	0.629065i	0.949247	−	0.314532i	$-0.101848\pi$
$48$			12.2750i			1.77175i
$49$	−0.873992			−0.124856
$50$	−13.2750	−	1.63752i	−1.87737	−	0.231580i
$51$	−4.54420			−0.636315
$52$		−	26.3938i		−	3.66015i
$53$			6.57452i			0.903079i	0.892251	+	0.451540i	$0.149125\pi$
$53$			6.57452i			0.903079i	−0.892251	+	0.451540i	$0.850875\pi$
$54$	2.67513			0.364039
$55$	1.48119	+	1.67513i	0.199724	+	0.225875i
$56$	23.6932			3.16614
$57$			4.57452i			0.605909i
$58$			6.38787i			0.838769i
$59$	13.2750			1.72826			0.864131	−	0.503266i	$-0.167868\pi$
$59$	13.2750			1.72826			0.864131	+	0.503266i	$0.167868\pi$
$60$	−8.63752	+	7.63752i	−1.11510	+	0.986000i
$61$	7.92478			1.01466			0.507332	−	0.861751i	$-0.330632\pi$
$61$	7.92478			1.01466			0.507332	+	0.861751i	$0.330632\pi$
$62$		−	2.57452i		−	0.326964i
$63$		−	2.80606i		−	0.353531i
$64$	−18.1187			−2.26484
$65$	8.57452	−	7.58181i	1.06354	−	0.940408i
$66$	2.67513			0.329286
$67$			10.7005i			1.30728i	0.756807	+	0.653639i	$0.226758\pi$
$67$			10.7005i			1.30728i	−0.756807	+	0.653639i	$0.773242\pi$
$68$		−	23.4314i		−	2.84147i
$69$	4.00000			0.481543
$70$	11.1187	+	12.5745i	1.32894	+	1.50294i
$71$	−7.35026			−0.872316			−0.436158	−	0.899870i	$-0.643661\pi$
$71$	−7.35026			−0.872316			−0.436158	+	0.899870i	$0.643661\pi$
$72$			8.44358i			0.995086i
$73$		−	6.41819i		−	0.751192i	−0.926783	−	0.375596i	$-0.877438\pi$
$73$		−	6.41819i		−	0.751192i	0.926783	−	0.375596i	$-0.122562\pi$
$74$	−4.31265			−0.501335
$75$	−4.96239	−	0.612127i	−0.573007	−	0.0706823i
$76$	−23.5877			−2.70569
$77$		−	2.80606i		−	0.319781i
$78$		−	13.6932i		−	1.55045i
$79$	−1.35026			−0.151916			−0.0759582	−	0.997111i	$-0.524202\pi$
$79$	−1.35026			−0.151916			−0.0759582	+	0.997111i	$0.524202\pi$
$80$	−18.1817	−	20.5623i	−2.03278	−	2.29893i
$81$	1.00000			0.111111
$82$		−	6.38787i		−	0.705423i
$83$			0.806063i			0.0884770i	0.999021	+	0.0442385i	$0.0140861\pi$
$83$			0.806063i			0.0884770i	−0.999021	+	0.0442385i	$0.985914\pi$
$84$	14.4690			1.57869
$85$	7.61213	−	6.73084i	0.825651	−	0.730062i
$86$	7.50659			0.809456
$87$			2.38787i			0.256007i
$88$			8.44358i			0.900089i
$89$	2.96239			0.314013			0.157006	−	0.987598i	$-0.449816\pi$
$89$	2.96239			0.314013			0.157006	+	0.987598i	$0.449816\pi$
$90$	−4.48119	+	3.96239i	−0.472359	+	0.417672i
$91$	−14.3634			−1.50570
$92$			20.6253i			2.15034i
$93$		−	0.962389i		−	0.0997950i
$94$	11.5369			1.18994
$95$	−6.77575	−	7.66291i	−0.695177	−	0.786198i
$96$	−15.9502			−1.62791
$97$		−	9.92478i		−	1.00771i	−0.863789	−	0.503854i	$-0.831915\pi$
$97$		−	9.92478i		−	1.00771i	0.863789	−	0.503854i	$-0.168085\pi$
$98$		−	2.33804i		−	0.236178i
$99$	1.00000			0.100504
$100$	3.15633	−	25.5877i	0.315633	−	2.55877i
$101$	−13.6121			−1.35446			−0.677229	−	0.735773i	$-0.736819\pi$
$101$	−13.6121			−1.35446			−0.677229	+	0.735773i	$0.736819\pi$
$102$		−	12.1563i		−	1.20366i
$103$		−	16.3127i		−	1.60733i	−0.595080	−	0.803667i	$-0.702880\pi$
$103$		−	16.3127i		−	1.60733i	0.595080	−	0.803667i	$-0.297120\pi$
$104$	43.2203			4.23810
$105$	4.15633	+	4.70052i	0.405616	+	0.458724i
$106$	−17.5877			−1.70827
$107$		−	9.43136i		−	0.911764i	−0.890040	−	0.455882i	$-0.849324\pi$
$107$		−	9.43136i		−	0.911764i	0.890040	−	0.455882i	$-0.150676\pi$
$108$			5.15633i			0.496168i
$109$	15.4010			1.47515			0.737576	−	0.675264i	$-0.235970\pi$
$109$	15.4010			1.47515			0.737576	+	0.675264i	$0.235970\pi$
$110$	−4.48119	+	3.96239i	−0.427265	+	0.377799i
$111$	−1.61213			−0.153016
$112$			34.4445i			3.25470i
$113$			13.7381i			1.29238i	0.763179	+	0.646188i	$0.223638\pi$
$113$			13.7381i			1.29238i	−0.763179	+	0.646188i	$0.776362\pi$
$114$	−12.2374			−1.14614
$115$	−6.70052	+	5.92478i	−0.624827	+	0.552488i
$116$	−12.3127			−1.14320
$117$		−	5.11871i		−	0.473225i
$118$			35.5125i			3.26919i
$119$	−12.7513			−1.16891
$120$	−12.5066	−	14.1441i	−1.14169	−	1.29117i
$121$	1.00000			0.0909091
$122$			21.1998i			1.91934i
$123$		−	2.38787i		−	0.215307i
$124$	4.96239			0.445636
$125$	9.21933	−	6.32487i	0.824602	−	0.565713i
$126$	7.50659			0.668740
$127$			12.4182i			1.10194i	0.834526	+	0.550968i	$0.185741\pi$
$127$			12.4182i			1.10194i	−0.834526	+	0.550968i	$0.814259\pi$
$128$		−	16.5696i		−	1.46456i
$129$	2.80606			0.247060
$130$	20.2823	+	22.9380i	1.77888	+	2.01179i
$131$	5.92478			0.517650			0.258825	−	0.965924i	$-0.416665\pi$
$131$	5.92478			0.517650			0.258825	+	0.965924i	$0.416665\pi$
$132$			5.15633i			0.448800i
$133$			12.8364i			1.11306i
$134$	−28.6253			−2.47285
$135$	−1.67513	+	1.48119i	−0.144172	+	0.127481i
$136$	38.3693			3.29014
$137$			9.79877i			0.837165i	0.908179	+	0.418583i	$0.137473\pi$
$137$			9.79877i			0.837165i	−0.908179	+	0.418583i	$0.862527\pi$
$138$			10.7005i			0.910889i
$139$	2.12601			0.180326			0.0901628	−	0.995927i	$-0.471261\pi$
$139$	2.12601			0.180326			0.0901628	+	0.995927i	$0.471261\pi$
$140$	−24.2374	+	21.4314i	−2.04844	+	1.81128i
$141$	4.31265			0.363191
$142$		−	19.6629i		−	1.65007i
$143$		−	5.11871i		−	0.428048i
$144$	−12.2750			−1.02292
$145$	−3.53690	−	4.00000i	−0.293724	−	0.332182i
$146$	17.1695			1.42096
$147$		−	0.873992i		−	0.0720856i
$148$		−	8.31265i		−	0.683296i
$149$	−8.31265			−0.680999			−0.340499	−	0.940245i	$-0.610596\pi$
$149$	−8.31265			−0.680999			−0.340499	+	0.940245i	$0.610596\pi$
$150$	1.63752	−	13.2750i	0.133703	−	1.08390i
$151$	−15.2750			−1.24307			−0.621533	−	0.783388i	$-0.713490\pi$
$151$	−15.2750			−1.24307			−0.621533	+	0.783388i	$0.713490\pi$
$152$		−	38.6253i		−	3.13293i
$153$		−	4.54420i		−	0.367377i
$154$	7.50659			0.604898
$155$	1.42548	+	1.61213i	0.114498	+	0.129489i
$156$	26.3938			2.11319
$157$			7.01317i			0.559712i	0.960042	+	0.279856i	$0.0902868\pi$
$157$			7.01317i			0.559712i	−0.960042	+	0.279856i	$0.909713\pi$
$158$		−	3.61213i		−	0.287365i
$159$	−6.57452			−0.521393
$160$	26.7186	−	23.6253i	2.11229	−	1.86774i
$161$	11.2243			0.884595
$162$			2.67513i			0.210178i
$163$		−	6.76116i		−	0.529575i	−0.964307	−	0.264787i	$-0.914698\pi$
$163$		−	6.76116i		−	0.529575i	0.964307	−	0.264787i	$-0.0853018\pi$
$164$	12.3127			0.961456
$165$	−1.67513	+	1.48119i	−0.130409	+	0.115311i
$166$	−2.15633			−0.167363
$167$		−	11.8192i		−	0.914600i	−0.889312	−	0.457300i	$-0.848817\pi$
$167$		−	11.8192i		−	0.914600i	0.889312	−	0.457300i	$-0.151183\pi$
$168$			23.6932i			1.82797i
$169$	−13.2012			−1.01548
$170$	18.0059	+	20.3634i	1.38099	+	1.56180i
$171$	−4.57452			−0.349822
$172$			14.4690i			1.10325i
$173$		−	6.99271i		−	0.531646i	−0.964022	−	0.265823i	$-0.914356\pi$
$173$		−	6.99271i		−	0.531646i	0.964022	−	0.265823i	$-0.0856436\pi$
$174$	−6.38787			−0.484263
$175$	−13.9248	−	1.71767i	−1.05261	−	0.129843i
$176$	−12.2750			−0.925266
$177$			13.2750i			0.997813i
$178$			7.92478i			0.593987i
$179$	2.70052			0.201847			0.100923	−	0.994894i	$-0.467820\pi$
$179$	2.70052			0.201847			0.100923	+	0.994894i	$0.467820\pi$
$180$	−7.63752	−	8.63752i	−0.569267	−	0.643803i
$181$	−21.1998			−1.57577			−0.787885	−	0.615822i	$-0.788824\pi$
$181$	−21.1998			−1.57577			−0.787885	+	0.615822i	$0.788824\pi$
$182$		−	38.4241i		−	2.84818i
$183$			7.92478i			0.585816i
$184$	−33.7743			−2.48988
$185$	2.70052	−	2.38787i	0.198546	−	0.175560i
$186$	2.57452			0.188773
$187$		−	4.54420i		−	0.332305i
$188$			22.2374i			1.62183i
$189$	2.80606			0.204111
$190$	20.4993	−	18.1260i	1.48717	−	1.31500i
$191$	13.1490			0.951430			0.475715	−	0.879599i	$-0.342189\pi$
$191$	13.1490			0.951430			0.475715	+	0.879599i	$0.342189\pi$
$192$		−	18.1187i		−	1.30761i
$193$		−	5.89446i		−	0.424293i	−0.977238	−	0.212146i	$-0.931955\pi$
$193$		−	5.89446i		−	0.424293i	0.977238	−	0.212146i	$-0.0680453\pi$
$194$	26.5501			1.90618
$195$	7.58181	+	8.57452i	0.542945	+	0.614034i
$196$	4.50659			0.321899
$197$			20.3938i			1.45299i	0.687169	+	0.726497i	$0.258853\pi$
$197$			20.3938i			1.45299i	−0.687169	+	0.726497i	$0.741147\pi$
$198$			2.67513i			0.190113i
$199$	−17.4010			−1.23353			−0.616764	−	0.787148i	$-0.711557\pi$
$199$	−17.4010			−1.23353			−0.616764	+	0.787148i	$0.711557\pi$
$200$	41.9003	+	5.16854i	2.96280	+	0.365471i
$201$	−10.7005			−0.754757
$202$		−	36.4142i		−	2.56210i
$203$			6.70052i			0.470285i
$204$	23.4314			1.64052
$205$	3.53690	+	4.00000i	0.247028	+	0.279372i
$206$	43.6385			3.04044
$207$			4.00000i			0.278019i
$208$			62.8324i			4.35664i
$209$	−4.57452			−0.316426
$210$	−12.5745	+	11.1187i	−0.867724	+	0.767264i
$211$	18.1260			1.24785			0.623923	−	0.781486i	$-0.285538\pi$
$211$	18.1260			1.24785			0.623923	+	0.781486i	$0.285538\pi$
$212$		−	33.9003i		−	2.32828i
$213$		−	7.35026i		−	0.503632i
$214$	25.2301			1.72470
$215$	−4.70052	+	4.15633i	−0.320573	+	0.283459i
$216$	−8.44358			−0.574513
$217$		−	2.70052i		−	0.183323i
$218$			41.1998i			2.79040i
$219$	6.41819			0.433701
$220$	−7.63752	−	8.63752i	−0.514921	−	0.582341i
$221$	−23.2605			−1.56467
$222$		−	4.31265i		−	0.289446i
$223$		−	23.6385i		−	1.58295i	−0.611202	−	0.791475i	$-0.709314\pi$
$223$		−	23.6385i		−	1.58295i	0.611202	−	0.791475i	$-0.290686\pi$
$224$	−44.7572			−2.99047
$225$	0.612127	−	4.96239i	0.0408085	−	0.330826i
$226$	−36.7513			−2.44466
$227$		−	1.26916i		−	0.0842371i	−0.999113	−	0.0421185i	$-0.986589\pi$
$227$		−	1.26916i		−	0.0842371i	0.999113	−	0.0421185i	$-0.0134107\pi$
$228$		−	23.5877i		−	1.56213i
$229$	−16.1768			−1.06899			−0.534496	−	0.845171i	$-0.679499\pi$
$229$	−16.1768			−1.06899			−0.534496	+	0.845171i	$0.679499\pi$
$230$	−15.8496	−	17.9248i	−1.04509	−	1.18192i
$231$	2.80606			0.184625
$232$		−	20.1622i		−	1.32371i
$233$			18.4690i			1.20994i	0.796247	+	0.604971i	$0.206815\pi$
$233$			18.4690i			1.20994i	−0.796247	+	0.604971i	$0.793185\pi$
$234$	13.6932			0.895154
$235$	−7.22425	+	6.38787i	−0.471258	+	0.416699i
$236$	−68.4504			−4.45574
$237$		−	1.35026i		−	0.0877089i
$238$		−	34.1114i		−	2.21111i
$239$	−19.3258			−1.25008			−0.625042	−	0.780591i	$-0.714918\pi$
$239$	−19.3258			−1.25008			−0.625042	+	0.780591i	$0.714918\pi$
$240$	20.5623	−	18.1817i	1.32729	−	1.17362i
$241$	28.5501			1.83907			0.919536	−	0.393006i	$-0.128565\pi$
$241$	28.5501			1.83907			0.919536	+	0.393006i	$0.128565\pi$
$242$			2.67513i			0.171964i
$243$			1.00000i			0.0641500i
$244$	−40.8627			−2.61597
$245$	1.29455	+	1.46405i	0.0827059	+	0.0935348i
$246$	6.38787			0.407276
$247$			23.4156i			1.48990i
$248$			8.12601i			0.516002i
$249$	−0.806063			−0.0510822
$250$	16.9199	+	24.6629i	1.07011	+	1.55982i
$251$	23.1998			1.46436			0.732180	−	0.681112i	$-0.238503\pi$
$251$	23.1998			1.46436			0.732180	+	0.681112i	$0.238503\pi$
$252$			14.4690i			0.911460i
$253$			4.00000i			0.251478i
$254$	−33.2203			−2.08443
$255$	6.73084	+	7.61213i	0.421502	+	0.476690i
$256$	8.08840			0.505525
$257$		−	10.8872i		−	0.679123i	−0.940584	−	0.339561i	$-0.889721\pi$
$257$		−	10.8872i		−	0.679123i	0.940584	−	0.339561i	$-0.110279\pi$
$258$			7.50659i			0.467340i
$259$	−4.52373			−0.281091
$260$	−44.2130	+	39.0943i	−2.74197	+	2.42452i
$261$	−2.38787			−0.147806
$262$			15.8496i			0.979189i
$263$		−	9.11871i		−	0.562284i	−0.959666	−	0.281142i	$-0.909287\pi$
$263$		−	9.11871i		−	0.562284i	0.959666	−	0.281142i	$-0.0907132\pi$
$264$	−8.44358			−0.519667
$265$	11.0132	−	9.73813i	0.676534	−	0.598209i
$266$	−34.3390			−2.10546
$267$			2.96239i			0.181295i
$268$		−	55.1754i		−	3.37037i
$269$	11.2995			0.688941			0.344471	−	0.938797i	$-0.388058\pi$
$269$	11.2995			0.688941			0.344471	+	0.938797i	$0.388058\pi$
$270$	−3.96239	−	4.48119i	−0.241143	−	0.272717i
$271$	25.1998			1.53078			0.765390	−	0.643567i	$-0.222546\pi$
$271$	25.1998			1.53078			0.765390	+	0.643567i	$0.222546\pi$
$272$			55.7802i			3.38217i
$273$		−	14.3634i		−	0.869315i
$274$	−26.2130			−1.58358
$275$	0.612127	−	4.96239i	0.0369126	−	0.299243i
$276$	−20.6253			−1.24150
$277$		−	2.41819i		−	0.145295i	−0.997358	−	0.0726475i	$-0.976855\pi$
$277$		−	2.41819i		−	0.145295i	0.997358	−	0.0726475i	$-0.0231448\pi$
$278$			5.68735i			0.341105i
$279$	0.962389			0.0576167
$280$	−35.0943	−	39.6893i	−2.09728	−	2.37189i
$281$	30.4894			1.81885			0.909424	−	0.415870i	$-0.136523\pi$
$281$	30.4894			1.81885			0.909424	+	0.415870i	$0.136523\pi$
$282$			11.5369i			0.687013i
$283$		−	8.35756i		−	0.496805i	−0.968657	−	0.248403i	$-0.920094\pi$
$283$		−	8.35756i		−	0.496805i	0.968657	−	0.248403i	$-0.0799056\pi$
$284$	37.9003			2.24897
$285$	7.66291	−	6.77575i	0.453912	−	0.401361i
$286$	13.6932			0.809698
$287$		−	6.70052i		−	0.395519i
$288$		−	15.9502i		−	0.939873i
$289$	−3.64974			−0.214690
$290$	10.7005	−	9.46168i	0.628356	−	0.555609i
$291$	9.92478			0.581801
$292$			33.0943i			1.93670i
$293$			23.0943i			1.34918i	0.738192	+	0.674591i	$0.235680\pi$
$293$			23.0943i			1.34918i	−0.738192	+	0.674591i	$0.764320\pi$
$294$	2.33804			0.136357
$295$	−19.6629	−	22.2374i	−1.14482	−	1.29471i
$296$	13.6121			0.791189
$297$			1.00000i			0.0580259i
$298$		−	22.2374i		−	1.28818i
$299$	20.4749			1.18409
$300$	25.5877	+	3.15633i	1.47731	+	0.182231i
$301$	7.87399			0.453849
$302$		−	40.8627i		−	2.35139i
$303$		−	13.6121i		−	0.781996i
$304$	56.1524			3.22056
$305$	−11.7381	−	13.2750i	−0.672124	−	0.760127i
$306$	12.1563			0.694931
$307$			2.65562i			0.151564i	0.997124	+	0.0757821i	$0.0241453\pi$
$307$			2.65562i			0.151564i	−0.997124	+	0.0757821i	$0.975855\pi$
$308$			14.4690i			0.824446i
$309$	16.3127			0.927994
$310$	−4.31265	+	3.81336i	−0.244942	+	0.216584i
$311$	−15.9756			−0.905891			−0.452946	−	0.891538i	$-0.649627\pi$
$311$	−15.9756			−0.905891			−0.452946	+	0.891538i	$0.649627\pi$
$312$			43.2203i			2.44687i
$313$			0.0606343i			0.00342726i	0.999999	+	0.00171363i	$0.000545465\pi$
$313$			0.0606343i			0.00342726i	−0.999999	+	0.00171363i	$0.999455\pi$
$314$	−18.7612			−1.05875
$315$	−4.70052	+	4.15633i	−0.264844	+	0.234182i
$316$	6.96239			0.391665
$317$			3.81336i			0.214180i	0.994249	+	0.107090i	$0.0341532\pi$
$317$			3.81336i			0.214180i	−0.994249	+	0.107090i	$0.965847\pi$
$318$		−	17.5877i		−	0.986269i
$319$	−2.38787			−0.133695
$320$	26.8373	+	30.3512i	1.50025	+	1.69668i
$321$	9.43136			0.526407
$322$			30.0263i			1.67330i
$323$			20.7875i			1.15665i
$324$	−5.15633			−0.286463
$325$	−25.4010	−	3.13330i	−1.40900	−	0.173804i
$326$	18.0870			1.00175
$327$			15.4010i			0.851680i
$328$			20.1622i			1.11327i
$329$	12.1016			0.667181
$330$	−3.96239	−	4.48119i	−0.218122	−	0.246682i
$331$	24.2882			1.33500			0.667500	−	0.744609i	$-0.267364\pi$
$331$	24.2882			1.33500			0.667500	+	0.744609i	$0.267364\pi$
$332$		−	4.15633i		−	0.228108i
$333$		−	1.61213i		−	0.0883440i
$334$	31.6180			1.73006
$335$	17.9248	−	15.8496i	0.979335	−	0.865954i
$336$	−34.4445			−1.87910
$337$		−	22.5804i		−	1.23003i	−0.788514	−	0.615016i	$-0.789149\pi$
$337$		−	22.5804i		−	1.23003i	0.788514	−	0.615016i	$-0.210851\pi$
$338$		−	35.3150i		−	1.92088i
$339$	−13.7381			−0.746153
$340$	−39.2506	+	34.7064i	−2.12866	+	1.88222i
$341$	0.962389			0.0521163
$342$		−	12.2374i		−	0.661724i
$343$			17.1900i			0.928171i
$344$	−23.6932			−1.27745
$345$	−5.92478	−	6.70052i	−0.318979	−	0.360744i
$346$	18.7064			1.00566
$347$			20.1925i			1.08399i	0.840381	+	0.541996i	$0.182331\pi$
$347$			20.1925i			1.08399i	−0.840381	+	0.541996i	$0.817669\pi$
$348$		−	12.3127i		−	0.660027i
$349$	27.2506			1.45869			0.729346	−	0.684145i	$-0.239825\pi$
$349$	27.2506			1.45869			0.729346	+	0.684145i	$0.239825\pi$
$350$	4.59498	−	37.2506i	0.245612	−	1.99113i
$351$	5.11871			0.273217
$352$		−	15.9502i		−	0.850147i
$353$		−	5.02302i		−	0.267349i	−0.991025	−	0.133674i	$-0.957322\pi$
$353$		−	5.02302i		−	0.267349i	0.991025	−	0.133674i	$-0.0426776\pi$
$354$	−35.5125			−1.88747
$355$	10.8872	+	12.3127i	0.577831	+	0.653488i
$356$	−15.2750			−0.809575
$357$		−	12.7513i		−	0.674871i
$358$			7.22425i			0.381814i
$359$	18.1768			0.959334			0.479667	−	0.877450i	$-0.340757\pi$
$359$	18.1768			0.959334			0.479667	+	0.877450i	$0.340757\pi$
$360$	14.1441	−	12.5066i	0.745460	−	0.659155i
$361$	1.92619			0.101379
$362$		−	56.7123i		−	2.98073i
$363$			1.00000i			0.0524864i
$364$	74.0625			3.88193
$365$	−10.7513	+	9.50659i	−0.562749	+	0.497598i
$366$	−21.1998			−1.10813
$367$		−	8.56467i		−	0.447072i	−0.974696	−	0.223536i	$-0.928240\pi$
$367$		−	8.56467i		−	0.447072i	0.974696	−	0.223536i	$-0.0717600\pi$
$368$		−	49.1002i		−	2.55952i
$369$	2.38787			0.124308
$370$	6.38787	+	7.22425i	0.332090	+	0.375571i
$371$	−18.4485			−0.957799
$372$			4.96239i			0.257288i
$373$		−	7.81924i		−	0.404865i	−0.979296	−	0.202432i	$-0.935115\pi$
$373$		−	7.81924i		−	0.404865i	0.979296	−	0.202432i	$-0.0648846\pi$
$374$	12.1563			0.628589
$375$	6.32487	+	9.21933i	0.326615	+	0.476084i
$376$	−36.4142			−1.87792
$377$			12.2228i			0.629508i
$378$			7.50659i			0.386097i
$379$	−1.14903			−0.0590218			−0.0295109	−	0.999564i	$-0.509395\pi$
$379$	−1.14903			−0.0590218			−0.0295109	+	0.999564i	$0.509395\pi$
$380$	34.9380	+	39.5125i	1.79228	+	2.02695i
$381$	−12.4182			−0.636203
$382$			35.1754i			1.79973i
$383$		−	34.0870i		−	1.74176i	−0.491493	−	0.870882i	$-0.663549\pi$
$383$		−	34.0870i		−	1.74176i	0.491493	−	0.870882i	$-0.336451\pi$
$384$	16.5696			0.845563
$385$	−4.70052	+	4.15633i	−0.239561	+	0.211826i
$386$	15.7685			0.802593
$387$			2.80606i			0.142640i
$388$			51.1754i			2.59804i
$389$	−23.7743			−1.20541			−0.602703	−	0.797965i	$-0.705910\pi$
$389$	−23.7743			−1.20541			−0.602703	+	0.797965i	$0.705910\pi$
$390$	−22.9380	+	20.2823i	−1.16151	+	1.02704i
$391$	18.1768			0.919240
$392$			7.37962i			0.372727i
$393$			5.92478i			0.298865i
$394$	−54.5560			−2.74849
$395$	2.00000	+	2.26187i	0.100631	+	0.113807i
$396$	−5.15633			−0.259115
$397$		−	4.15045i		−	0.208305i	−0.994561	−	0.104152i	$-0.966787\pi$
$397$		−	4.15045i		−	0.208305i	0.994561	−	0.104152i	$-0.0332130\pi$
$398$		−	46.5501i		−	2.33334i
$399$	−12.8364			−0.642623
$400$	−7.51388	+	60.9135i	−0.375694	+	3.04568i
$401$	−33.9149			−1.69363			−0.846815	−	0.531887i	$-0.821483\pi$
$401$	−33.9149			−1.69363			−0.846815	+	0.531887i	$0.821483\pi$
$402$		−	28.6253i		−	1.42770i
$403$		−	4.92619i		−	0.245391i
$404$	70.1886			3.49201
$405$	−1.48119	−	1.67513i	−0.0736011	−	0.0832379i
$406$	−17.9248			−0.889592
$407$		−	1.61213i		−	0.0799102i
$408$			38.3693i			1.89956i
$409$	−8.55008			−0.422774			−0.211387	−	0.977402i	$-0.567798\pi$
$409$	−8.55008			−0.422774			−0.211387	+	0.977402i	$0.567798\pi$
$410$	−10.7005	+	9.46168i	−0.528461	+	0.467279i
$411$	−9.79877			−0.483338
$412$			84.1133i			4.14397i
$413$			37.2506i			1.83298i
$414$	−10.7005			−0.525902
$415$	1.35026	−	1.19394i	0.0662817	−	0.0586080i
$416$	−81.6444			−4.00294
$417$			2.12601i			0.104111i
$418$		−	12.2374i		−	0.598552i
$419$	−13.2995			−0.649722			−0.324861	−	0.945762i	$-0.605318\pi$
$419$	−13.2995			−0.649722			−0.324861	+	0.945762i	$0.605318\pi$
$420$	−21.4314	−	24.2374i	−1.04574	−	1.18267i
$421$	33.3014			1.62301			0.811505	−	0.584345i	$-0.198649\pi$
$421$	33.3014			1.62301			0.811505	+	0.584345i	$0.198649\pi$
$422$			48.4894i			2.36043i
$423$			4.31265i			0.209688i
$424$	55.5125			2.69592
$425$	−22.5501	−	2.78163i	−1.09384	−	0.134929i
$426$	19.6629			0.952671
$427$			22.2374i			1.07614i
$428$			48.6312i			2.35068i
$429$	5.11871			0.247134
$430$	−11.1187	−	12.5745i	−0.536192	−	0.606397i
$431$	2.07522			0.0999600			0.0499800	−	0.998750i	$-0.484084\pi$
$431$	2.07522			0.0999600			0.0499800	+	0.998750i	$0.484084\pi$
$432$		−	12.2750i		−	0.590583i
$433$			37.4010i			1.79738i	0.438585	+	0.898690i	$0.355480\pi$
$433$			37.4010i			1.79738i	−0.438585	+	0.898690i	$0.644520\pi$
$434$	7.22425			0.346775
$435$	4.00000	−	3.53690i	0.191785	−	0.169582i
$436$	−79.4128			−3.80318
$437$		−	18.2981i		−	0.875315i
$438$			17.1695i			0.820390i
$439$	6.02444			0.287531			0.143765	−	0.989612i	$-0.454079\pi$
$439$	6.02444			0.287531			0.143765	+	0.989612i	$0.454079\pi$
$440$	14.1441	−	12.5066i	0.674294	−	0.596228i
$441$	0.873992			0.0416187
$442$		−	62.2247i		−	2.95973i
$443$			10.1359i			0.481569i	0.970579	+	0.240785i	$0.0774047\pi$
$443$			10.1359i			0.481569i	−0.970579	+	0.240785i	$0.922595\pi$
$444$	8.31265			0.394501
$445$	−4.38787	−	4.96239i	−0.208005	−	0.235240i
$446$	63.2360			2.99431
$447$		−	8.31265i		−	0.393175i
$448$		−	50.8423i		−	2.40207i
$449$	11.4861			0.542063			0.271032	−	0.962570i	$-0.412635\pi$
$449$	11.4861			0.542063			0.271032	+	0.962570i	$0.412635\pi$
$450$	13.2750	+	1.63752i	0.625791	+	0.0771934i
$451$	2.38787			0.112441
$452$		−	70.8383i		−	3.33195i
$453$		−	15.2750i		−	0.717684i
$454$	3.39517			0.159343
$455$	21.2750	+	24.0606i	0.997389	+	1.12798i
$456$	38.6253			1.80880
$457$		−	15.0435i		−	0.703705i	−0.936055	−	0.351852i	$-0.885552\pi$
$457$		−	15.0435i		−	0.703705i	0.936055	−	0.351852i	$-0.114448\pi$
$458$		−	43.2750i		−	2.02211i
$459$	4.54420			0.212105
$460$	34.5501	−	30.5501i	1.61091	−	1.42440i
$461$	−26.2374			−1.22200			−0.610999	−	0.791631i	$-0.709232\pi$
$461$	−26.2374			−1.22200			−0.610999	+	0.791631i	$0.709232\pi$
$462$			7.50659i			0.349238i
$463$			5.46168i			0.253826i	0.991914	+	0.126913i	$0.0405069\pi$
$463$			5.46168i			0.253826i	−0.991914	+	0.126913i	$0.959493\pi$
$464$	29.3112			1.36074
$465$	−1.61213	+	1.42548i	−0.0747606	+	0.0661053i
$466$	−49.4069			−2.28873
$467$		−	2.70052i		−	0.124965i	−0.998046	−	0.0624827i	$-0.980098\pi$
$467$		−	2.70052i		−	0.124965i	0.998046	−	0.0624827i	$-0.0199018\pi$
$468$			26.3938i			1.22005i
$469$	−30.0263			−1.38649
$470$	−17.0884	−	19.3258i	−0.788229	−	0.891434i
$471$	−7.01317			−0.323150
$472$		−	112.089i		−	5.15931i
$473$			2.80606i			0.129023i
$474$	3.61213			0.165910
$475$	−2.80018	+	22.7005i	−0.128481	+	1.04157i
$476$	65.7499			3.01364
$477$		−	6.57452i		−	0.301026i
$478$		−	51.6991i		−	2.36466i
$479$	−16.7757			−0.766503			−0.383252	−	0.923644i	$-0.625196\pi$
$479$	−16.7757			−0.766503			−0.383252	+	0.923644i	$0.625196\pi$
$480$	23.6253	+	26.7186i	1.07834	+	1.21953i
$481$	−8.25202			−0.376260
$482$			76.3752i			3.47879i
$483$			11.2243i			0.510721i
$484$	−5.15633			−0.234378
$485$	−16.6253	+	14.7005i	−0.754916	+	0.667516i
$486$	−2.67513			−0.121346
$487$		−	34.3996i		−	1.55880i	−0.626529	−	0.779398i	$-0.715525\pi$
$487$		−	34.3996i		−	1.55880i	0.626529	−	0.779398i	$-0.284475\pi$
$488$		−	66.9135i		−	3.02903i
$489$	6.76116			0.305750
$490$	−3.91653	+	3.46310i	−0.176931	+	0.156447i
$491$	14.9525			0.674799			0.337399	−	0.941362i	$-0.390453\pi$
$491$	14.9525			0.674799			0.337399	+	0.941362i	$0.390453\pi$
$492$			12.3127i			0.555097i
$493$			10.8510i			0.488703i
$494$	−62.6399			−2.81830
$495$	−1.48119	−	1.67513i	−0.0665747	−	0.0752915i
$496$	−11.8134			−0.530435
$497$		−	20.6253i		−	0.925171i
$498$		−	2.15633i		−	0.0966272i
$499$	2.85097			0.127627			0.0638135	−	0.997962i	$-0.479674\pi$
$499$	2.85097			0.127627			0.0638135	+	0.997962i	$0.479674\pi$
$500$	−47.5379	+	32.6131i	−2.12596	+	1.45850i
$501$	11.8192			0.528045
$502$			62.0625i			2.76999i
$503$		−	1.48024i		−	0.0660006i	−0.999455	−	0.0330003i	$-0.989494\pi$
$503$		−	1.48024i		−	0.0660006i	0.999455	−	0.0330003i	$-0.0105062\pi$
$504$	−23.6932			−1.05538
$505$	20.1622	+	22.8021i	0.897206	+	1.01468i
$506$	−10.7005			−0.475696
$507$		−	13.2012i		−	0.586287i
$508$		−	64.0322i		−	2.84097i
$509$	23.2995			1.03273			0.516366	−	0.856368i	$-0.327285\pi$
$509$	23.2995			1.03273			0.516366	+	0.856368i	$0.327285\pi$
$510$	−20.3634	+	18.0059i	−0.901708	+	0.797314i
$511$	18.0098			0.796709
$512$		−	11.5017i		−	0.508306i
$513$		−	4.57452i		−	0.201970i
$514$	29.1246			1.28463
$515$	−27.3258	+	24.1622i	−1.20412	+	1.06471i
$516$	−14.4690			−0.636961
$517$			4.31265i			0.189670i
$518$		−	12.1016i		−	0.531712i
$519$	6.99271			0.306946
$520$	−64.0176	−	72.3996i	−2.80736	−	3.17493i
$521$	−6.81194			−0.298437			−0.149218	−	0.988804i	$-0.547676\pi$
$521$	−6.81194			−0.298437			−0.149218	+	0.988804i	$0.547676\pi$
$522$		−	6.38787i		−	0.279590i
$523$			16.0567i			0.702109i	0.936355	+	0.351054i	$0.114177\pi$
$523$			16.0567i			0.702109i	−0.936355	+	0.351054i	$0.885823\pi$
$524$	−30.5501			−1.33459
$525$	1.71767	−	13.9248i	0.0749651	−	0.607727i
$526$	24.3938			1.06362
$527$		−	4.37328i		−	0.190503i
$528$		−	12.2750i		−	0.534203i
$529$	7.00000			0.304348
$530$	26.0508	+	29.4617i	1.13157	+	1.27973i
$531$	−13.2750			−0.576088
$532$		−	66.1886i		−	2.86964i
$533$		−	12.2228i		−	0.529430i
$534$	−7.92478			−0.342939
$535$	−15.7988	+	13.9697i	−0.683040	+	0.603962i
$536$	90.3508			3.90256
$537$			2.70052i			0.116536i
$538$			30.2276i			1.30320i
$539$	0.873992			0.0376455
$540$	8.63752	−	7.63752i	0.371700	−	0.328667i
$541$	6.62530			0.284844			0.142422	−	0.989806i	$-0.454511\pi$
$541$	6.62530			0.284844			0.142422	+	0.989806i	$0.454511\pi$
$542$			67.4128i			2.89563i
$543$		−	21.1998i		−	0.909771i
$544$	−72.4807			−3.10759
$545$	−22.8119	−	25.7988i	−0.977156	−	1.10510i
$546$	38.4241			1.64440
$547$		−	5.75860i		−	0.246220i	−0.992393	−	0.123110i	$-0.960713\pi$
$547$		−	5.75860i		−	0.246220i	0.992393	−	0.123110i	$-0.0392868\pi$
$548$		−	50.5256i		−	2.15835i
$549$	−7.92478			−0.338221
$550$	13.2750	+	1.63752i	0.566050	+	0.0698241i
$551$	10.9234			0.465351
$552$		−	33.7743i		−	1.43753i
$553$		−	3.78892i		−	0.161121i
$554$	6.46898			0.274840
$555$	2.38787	+	2.70052i	0.101360	+	0.114631i
$556$	−10.9624			−0.464909
$557$			19.8700i			0.841920i	0.907079	+	0.420960i	$0.138307\pi$
$557$			19.8700i			0.841920i	−0.907079	+	0.420960i	$0.861693\pi$
$558$			2.57452i			0.108988i
$559$	14.3634			0.607509
$560$	57.6991	−	51.0191i	2.43823	−	2.15595i
$561$	4.54420			0.191856
$562$			81.5633i			3.44054i
$563$		−	5.83383i		−	0.245866i	−0.992415	−	0.122933i	$-0.960770\pi$
$563$		−	5.83383i		−	0.245866i	0.992415	−	0.122933i	$-0.0392301\pi$
$564$	−22.2374			−0.936365
$565$	23.0132	−	20.3488i	0.968171	−	0.856082i
$566$	22.3576			0.939758
$567$			2.80606i			0.117844i
$568$			62.0625i			2.60409i
$569$	−46.7123			−1.95828			−0.979140	−	0.203185i	$-0.934871\pi$
$569$	−46.7123			−1.95828			−0.979140	+	0.203185i	$0.934871\pi$
$570$	18.1260	+	20.4993i	0.759215	+	0.858621i
$571$	−24.4241			−1.02212			−0.511058	−	0.859546i	$-0.670746\pi$
$571$	−24.4241			−1.02212			−0.511058	+	0.859546i	$0.670746\pi$
$572$			26.3938i			1.10358i
$573$			13.1490i			0.549309i
$574$	17.9248			0.748166
$575$	19.8496	+	2.44851i	0.827784	+	0.102110i
$576$	18.1187			0.754946
$577$			16.5647i			0.689596i	0.938677	+	0.344798i	$0.112053\pi$
$577$			16.5647i			0.689596i	−0.938677	+	0.344798i	$0.887947\pi$
$578$		−	9.76353i		−	0.406109i
$579$	5.89446			0.244965
$580$	18.2374	+	20.6253i	0.757268	+	0.856419i
$581$	−2.26187			−0.0938380
$582$			26.5501i			1.10054i
$583$		−	6.57452i		−	0.272289i
$584$	−54.1925			−2.24250
$585$	−8.57452	+	7.58181i	−0.354513	+	0.313469i
$586$	−61.7802			−2.55212
$587$			27.3258i			1.12786i	0.825823	+	0.563929i	$0.190711\pi$
$587$			27.3258i			1.12786i	−0.825823	+	0.563929i	$0.809289\pi$
$588$			4.50659i			0.185849i
$589$	−4.40246			−0.181400
$590$	59.4880	−	52.6009i	2.44908	−	2.16554i
$591$	−20.3938			−0.838887
$592$			19.7889i			0.813320i
$593$			12.6048i			0.517618i	0.965928	+	0.258809i	$0.0833301\pi$
$593$			12.6048i			0.517618i	−0.965928	+	0.258809i	$0.916670\pi$
$594$	−2.67513			−0.109762
$595$	18.8872	+	21.3601i	0.774298	+	0.875679i
$596$	42.8627			1.75573
$597$		−	17.4010i		−	0.712177i
$598$			54.7729i			2.23983i
$599$	−10.5990			−0.433061			−0.216531	−	0.976276i	$-0.569474\pi$
$599$	−10.5990			−0.433061			−0.216531	+	0.976276i	$0.569474\pi$
$600$	−5.16854	+	41.9003i	−0.211005	+	1.71057i
$601$	−22.4749			−0.916768			−0.458384	−	0.888754i	$-0.651572\pi$
$601$	−22.4749			−0.916768			−0.458384	+	0.888754i	$0.651572\pi$
$602$			21.0640i			0.858503i
$603$		−	10.7005i		−	0.435759i
$604$	78.7631			3.20482
$605$	−1.48119	−	1.67513i	−0.0602191	−	0.0681038i
$606$	36.4142			1.47923
$607$			33.5183i			1.36047i	0.732995	+	0.680234i	$0.238122\pi$
$607$			33.5183i			1.36047i	−0.732995	+	0.680234i	$0.761878\pi$
$608$			72.9643i			2.95909i
$609$	−6.70052			−0.271519
$610$	35.5125	−	31.4010i	1.43786	−	1.27139i
$611$	22.0752			0.893068
$612$			23.4314i			0.947157i
$613$		−	11.5672i		−	0.467196i	−0.972333	−	0.233598i	$-0.924950\pi$
$613$		−	11.5672i		−	0.467196i	0.972333	−	0.233598i	$-0.0750499\pi$
$614$	−7.10413			−0.286699
$615$	−4.00000	+	3.53690i	−0.161296	+	0.142622i
$616$	−23.6932			−0.954627
$617$		−	33.3357i		−	1.34204i	−0.741438	−	0.671022i	$-0.765856\pi$
$617$		−	33.3357i		−	1.34204i	0.741438	−	0.671022i	$-0.234144\pi$
$618$			43.6385i			1.75540i
$619$	−4.43866			−0.178405			−0.0892024	−	0.996014i	$-0.528432\pi$
$619$	−4.43866			−0.178405			−0.0892024	+	0.996014i	$0.528432\pi$
$620$	−7.35026	−	8.31265i	−0.295194	−	0.333844i
$621$	−4.00000			−0.160514
$622$		−	42.7367i		−	1.71359i
$623$			8.31265i			0.333039i
$624$	−62.8324			−2.51531
$625$	−24.2506	−	6.07522i	−0.970024	−	0.243009i
$626$	−0.162205			−0.00648301
$627$		−	4.57452i		−	0.182689i
$628$		−	36.1622i		−	1.44303i
$629$	−7.32582			−0.292100
$630$	−11.1187	−	12.5745i	−0.442980	−	0.500981i
$631$	−39.8496			−1.58639			−0.793193	−	0.608971i	$-0.791583\pi$
$631$	−39.8496			−1.58639			−0.793193	+	0.608971i	$0.791583\pi$
$632$			11.4010i			0.453509i
$633$			18.1260i			0.720444i
$634$	−10.2012			−0.405143
$635$	20.8021	−	18.3938i	0.825506	−	0.729934i
$636$	33.9003			1.34424
$637$		−	4.47371i		−	0.177255i
$638$		−	6.38787i		−	0.252898i
$639$	7.35026			0.290772
$640$	−27.7562	+	24.5428i	−1.09716	+	0.970139i
$641$	−3.88858			−0.153590			−0.0767948	−	0.997047i	$-0.524469\pi$
$641$	−3.88858			−0.153590			−0.0767948	+	0.997047i	$0.524469\pi$
$642$			25.2301i			0.995754i
$643$		−	40.9380i		−	1.61444i	−0.590254	−	0.807218i	$-0.700972\pi$
$643$		−	40.9380i		−	1.61444i	0.590254	−	0.807218i	$-0.299028\pi$
$644$	−57.8759			−2.28063
$645$	−4.15633	−	4.70052i	−0.163655	−	0.185083i
$646$	−55.6093			−2.18792
$647$			1.76257i			0.0692939i	0.999400	+	0.0346469i	$0.0110307\pi$
$647$			1.76257i			0.0692939i	−0.999400	+	0.0346469i	$0.988969\pi$
$648$		−	8.44358i		−	0.331695i
$649$	−13.2750			−0.521091
$650$	8.38199	−	67.9511i	0.328769	−	2.66526i
$651$	2.70052			0.105842
$652$			34.8627i			1.36533i
$653$			7.03761i			0.275403i	0.990474	+	0.137702i	$0.0439715\pi$
$653$			7.03761i			0.275403i	−0.990474	+	0.137702i	$0.956029\pi$
$654$	−41.1998			−1.61104
$655$	−8.77575	−	9.92478i	−0.342897	−	0.387793i
$656$	−29.3112			−1.14441
$657$			6.41819i			0.250397i
$658$			32.3733i			1.26204i
$659$	12.6253			0.491812			0.245906	−	0.969294i	$-0.420915\pi$
$659$	12.6253			0.491812			0.245906	+	0.969294i	$0.420915\pi$
$660$	8.63752	−	7.63752i	0.336215	−	0.297290i
$661$	13.2243			0.514364			0.257182	−	0.966363i	$-0.417206\pi$
$661$	13.2243			0.514364			0.257182	+	0.966363i	$0.417206\pi$
$662$			64.9741i			2.52529i
$663$		−	23.2605i		−	0.903361i
$664$	6.80606			0.264126
$665$	21.5026	−	19.0132i	0.833836	−	0.737299i
$666$	4.31265			0.167112
$667$		−	9.55149i		−	0.369835i
$668$			60.9438i			2.35799i
$669$	23.6385			0.913916
$670$	42.3996	+	47.9511i	1.63804	+	1.85251i
$671$	−7.92478			−0.305933
$672$		−	44.7572i		−	1.72655i
$673$		−	18.2677i		−	0.704170i	−0.935968	−	0.352085i	$-0.885473\pi$
$673$		−	18.2677i		−	0.704170i	0.935968	−	0.352085i	$-0.114527\pi$
$674$	60.4055			2.32673
$675$	4.96239	+	0.612127i	0.191002	+	0.0235608i
$676$	68.0698			2.61807
$677$		−	33.0191i		−	1.26903i	−0.772912	−	0.634513i	$-0.781201\pi$
$677$		−	33.0191i		−	1.26903i	0.772912	−	0.634513i	$-0.218799\pi$
$678$		−	36.7513i		−	1.41143i
$679$	27.8496			1.06877
$680$	−56.8324	−	64.2736i	−2.17942	−	2.46478i
$681$	1.26916			0.0486343
$682$			2.57452i			0.0985833i
$683$		−	30.8627i		−	1.18093i	−0.807063	−	0.590465i	$-0.798944\pi$
$683$		−	30.8627i		−	1.18093i	0.807063	−	0.590465i	$-0.201056\pi$
$684$	23.5877			0.901898
$685$	16.4142	−	14.5139i	0.627155	−	0.554547i
$686$	−45.9854			−1.75573
$687$		−	16.1768i		−	0.617183i
$688$		−	34.4445i		−	1.31319i
$689$	−33.6531			−1.28208
$690$	17.9248	−	15.8496i	0.682385	−	0.603382i
$691$	27.6991			1.05372			0.526862	−	0.849951i	$-0.323369\pi$
$691$	27.6991			1.05372			0.526862	+	0.849951i	$0.323369\pi$
$692$			36.0567i			1.37067i
$693$			2.80606i			0.106594i
$694$	−54.0176			−2.05048
$695$	−3.14903	−	3.56134i	−0.119450	−	0.135089i
$696$	20.1622			0.764246
$697$		−	10.8510i		−	0.411010i
$698$			72.8989i			2.75927i
$699$	−18.4690			−0.698561
$700$	71.8007	+	8.85685i	2.71381	+	0.334757i
$701$	−38.9643			−1.47166			−0.735831	−	0.677166i	$-0.763208\pi$
$701$	−38.9643			−1.47166			−0.735831	+	0.677166i	$0.763208\pi$
$702$			13.6932i			0.516818i
$703$			7.37470i			0.278142i
$704$	18.1187			0.682875
$705$	−6.38787	−	7.22425i	−0.240581	−	0.272081i
$706$	13.4372			0.505717
$707$		−	38.1965i		−	1.43653i
$708$		−	68.4504i		−	2.57252i
$709$	4.32250			0.162335			0.0811674	−	0.996700i	$-0.474135\pi$
$709$	4.32250			0.162335			0.0811674	+	0.996700i	$0.474135\pi$
$710$	−32.9380	+	29.1246i	−1.23614	+	1.09303i
$711$	1.35026			0.0506388
$712$		−	25.0132i		−	0.937408i
$713$			3.84955i			0.144167i
$714$	34.1114			1.27659
$715$	−8.57452	+	7.58181i	−0.320669	+	0.283544i
$716$	−13.9248			−0.520393
$717$		−	19.3258i		−	0.721736i
$718$			48.6253i			1.81468i
$719$	38.3996			1.43206			0.716032	−	0.698067i	$-0.245956\pi$
$719$	38.3996			1.43206			0.716032	+	0.698067i	$0.245956\pi$
$720$	18.1817	+	20.5623i	0.677593	+	0.766312i
$721$	45.7743			1.70473
$722$			5.15282i			0.191768i
$723$			28.5501i			1.06179i
$724$	109.313			4.06259
$725$	−1.46168	+	11.8496i	−0.0542855	+	0.440081i
$726$	−2.67513			−0.0992834
$727$		−	21.6728i		−	0.803798i	−0.915684	−	0.401899i	$-0.868350\pi$
$727$		−	21.6728i		−	0.803798i	0.915684	−	0.401899i	$-0.131650\pi$
$728$			121.279i			4.49489i
$729$	−1.00000			−0.0370370
$730$	−25.4314	−	28.7612i	−0.941257	−	1.06450i
$731$	12.7513			0.471624
$732$		−	40.8627i		−	1.51033i
$733$		−	25.0698i		−	0.925976i	−0.886365	−	0.462988i	$-0.846777\pi$
$733$		−	25.0698i		−	0.925976i	0.886365	−	0.462988i	$-0.153223\pi$
$734$	22.9116			0.845683
$735$	−1.46405	+	1.29455i	−0.0540023	+	0.0477503i
$736$	63.8007			2.35172
$737$		−	10.7005i		−	0.394159i
$738$			6.38787i			0.235141i
$739$	18.9018			0.695312			0.347656	−	0.937622i	$-0.386978\pi$
$739$	18.9018			0.695312			0.347656	+	0.937622i	$0.386978\pi$
$740$	−13.9248	+	12.3127i	−0.511885	+	0.452622i
$741$	−23.4156			−0.860195
$742$		−	49.3522i		−	1.81178i
$743$		−	43.6688i		−	1.60205i	−0.598629	−	0.801026i	$-0.704288\pi$
$743$		−	43.6688i		−	1.60205i	0.598629	−	0.801026i	$-0.295712\pi$
$744$	−8.12601			−0.297914
$745$	12.3127	+	13.9248i	0.451101	+	0.510164i
$746$	20.9175			0.765843
$747$		−	0.806063i		−	0.0294923i
$748$			23.4314i			0.856736i
$749$	26.4650			0.967010
$750$	−24.6629	+	16.9199i	−0.900562	+	0.617826i
$751$	−47.0541			−1.71703			−0.858514	−	0.512789i	$-0.828612\pi$
$751$	−47.0541			−1.71703			−0.858514	+	0.512789i	$0.828612\pi$
$752$		−	52.9380i		−	1.93045i
$753$			23.1998i			0.845448i
$754$	−32.6977			−1.19078
$755$	22.6253	+	25.5877i	0.823419	+	0.931231i
$756$	−14.4690			−0.526232
$757$		−	26.9116i		−	0.978119i	−0.872250	−	0.489059i	$-0.837340\pi$
$757$		−	26.9116i		−	0.978119i	0.872250	−	0.489059i	$-0.162660\pi$
$758$		−	3.07381i		−	0.111646i
$759$	−4.00000			−0.145191
$760$	−64.7024	+	57.2116i	−2.34700	+	2.07528i
$761$	16.6859			0.604865			0.302432	−	0.953171i	$-0.402201\pi$
$761$	16.6859			0.604865			0.302432	+	0.953171i	$0.402201\pi$
$762$		−	33.2203i		−	1.20344i
$763$			43.2163i			1.56454i
$764$	−67.8007			−2.45294
$765$	−7.61213	+	6.73084i	−0.275217	+	0.243354i
$766$	91.1871			3.29473
$767$			67.9511i			2.45357i
$768$			8.08840i			0.291865i
$769$	−23.2995			−0.840201			−0.420100	−	0.907478i	$-0.638005\pi$
$769$	−23.2995			−0.840201			−0.420100	+	0.907478i	$0.638005\pi$
$770$	−11.1187	−	12.5745i	−0.400691	−	0.453154i
$771$	10.8872			0.392092
$772$			30.3938i			1.09390i
$773$			1.63656i			0.0588631i	0.999567	+	0.0294316i	$0.00936971\pi$
$773$			1.63656i			0.0588631i	−0.999567	+	0.0294316i	$0.990630\pi$
$774$	−7.50659			−0.269819
$775$	0.589104	−	4.77575i	0.0211612	−	0.171550i
$776$	−83.8007			−3.00827
$777$		−	4.52373i		−	0.162288i
$778$		−	63.5994i		−	2.28015i
$779$	−10.9234			−0.391370
$780$	−39.0943	−	44.2130i	−1.39980	−	1.58308i
$781$	7.35

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:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	165.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.67513i q^{2} +1.00000i q^{3} -5.15633 q^{4} +(-1.48119 - 1.67513i) q^{5} -2.67513 q^{6} +2.80606i q^{7} -8.44358i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.67513i q^{2} +1.00000i q^{3} -5.15633 q^{4} +(-1.48119 - 1.67513i) q^{5} -2.67513 q^{6} +2.80606i q^{7} -8.44358i q^{8} -1.00000 q^{9} +(4.48119 - 3.96239i) q^{10} -1.00000 q^{11} -5.15633i q^{12} +5.11871i q^{13} -7.50659 q^{14} +(1.67513 - 1.48119i) q^{15} +12.2750 q^{16} +4.54420i q^{17} -2.67513i q^{18} +4.57452 q^{19} +(7.63752 + 8.63752i) q^{20} -2.80606 q^{21} -2.67513i q^{22} -4.00000i q^{23} +8.44358 q^{24} +(-0.612127 + 4.96239i) q^{25} -13.6932 q^{26} -1.00000i q^{27} -14.4690i q^{28} +2.38787 q^{29} +(3.96239 + 4.48119i) q^{30} -0.962389 q^{31} +15.9502i q^{32} -1.00000i q^{33} -12.1563 q^{34} +(4.70052 - 4.15633i) q^{35} +5.15633 q^{36} +1.61213i q^{37} +12.2374i q^{38} -5.11871 q^{39} +(-14.1441 + 12.5066i) q^{40} -2.38787 q^{41} -7.50659i q^{42} -2.80606i q^{43} +5.15633 q^{44} +(1.48119 + 1.67513i) q^{45} +10.7005 q^{46} -4.31265i q^{47} +12.2750i q^{48} -0.873992 q^{49} +(-13.2750 - 1.63752i) q^{50} -4.54420 q^{51} -26.3938i q^{52} +6.57452i q^{53} +2.67513 q^{54} +(1.48119 + 1.67513i) q^{55} +23.6932 q^{56} +4.57452i q^{57} +6.38787i q^{58} +13.2750 q^{59} +(-8.63752 + 7.63752i) q^{60} +7.92478 q^{61} -2.57452i q^{62} -2.80606i q^{63} -18.1187 q^{64} +(8.57452 - 7.58181i) q^{65} +2.67513 q^{66} +10.7005i q^{67} -23.4314i q^{68} +4.00000 q^{69} +(11.1187 + 12.5745i) q^{70} -7.35026 q^{71} +8.44358i q^{72} -6.41819i q^{73} -4.31265 q^{74} +(-4.96239 - 0.612127i) q^{75} -23.5877 q^{76} -2.80606i q^{77} -13.6932i q^{78} -1.35026 q^{79} +(-18.1817 - 20.5623i) q^{80} +1.00000 q^{81} -6.38787i q^{82} +0.806063i q^{83} +14.4690 q^{84} +(7.61213 - 6.73084i) q^{85} +7.50659 q^{86} +2.38787i q^{87} +8.44358i q^{88} +2.96239 q^{89} +(-4.48119 + 3.96239i) q^{90} -14.3634 q^{91} +20.6253i q^{92} -0.962389i q^{93} +11.5369 q^{94} +(-6.77575 - 7.66291i) q^{95} -15.9502 q^{96} -9.92478i q^{97} -2.33804i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 10 * q^4 + 2 * q^5 - 6 * q^6 - 6 * q^9	\( 6 q - 10 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{9} + 16 q^{10} - 6 q^{11} - 4 q^{14} + 10 q^{16} + 4 q^{19} + 14 q^{20} - 16 q^{21} + 18 q^{24} - 2 q^{25} - 16 q^{26} + 16 q^{29} + 2 q^{30} + 16 q^{31} - 52 q^{34} - 12 q^{35} + 10 q^{36} + 12 q^{39} - 12 q^{40} - 16 q^{41} + 10 q^{44} - 2 q^{45} + 24 q^{46} - 22 q^{49} - 16 q^{50} - 8 q^{51} + 6 q^{54} - 2 q^{55} + 76 q^{56} + 16 q^{59} - 20 q^{60} + 4 q^{61} - 66 q^{64} + 28 q^{65} + 6 q^{66} + 24 q^{69} + 24 q^{70} - 24 q^{71} + 16 q^{74} - 8 q^{75} - 36 q^{76} + 12 q^{79} - 58 q^{80} + 6 q^{81} + 24 q^{84} + 44 q^{85} + 4 q^{86} - 4 q^{89} - 16 q^{90} + 16 q^{91} + 24 q^{94} - 44 q^{95} - 22 q^{96} + 6 q^{99}+O(q^{100}) \) 6 * q - 10 * q^4 + 2 * q^5 - 6 * q^6 - 6 * q^9 + 16 * q^10 - 6 * q^11 - 4 * q^14 + 10 * q^16 + 4 * q^19 + 14 * q^20 - 16 * q^21 + 18 * q^24 - 2 * q^25 - 16 * q^26 + 16 * q^29 + 2 * q^30 + 16 * q^31 - 52 * q^34 - 12 * q^35 + 10 * q^36 + 12 * q^39 - 12 * q^40 - 16 * q^41 + 10 * q^44 - 2 * q^45 + 24 * q^46 - 22 * q^49 - 16 * q^50 - 8 * q^51 + 6 * q^54 - 2 * q^55 + 76 * q^56 + 16 * q^59 - 20 * q^60 + 4 * q^61 - 66 * q^64 + 28 * q^65 + 6 * q^66 + 24 * q^69 + 24 * q^70 - 24 * q^71 + 16 * q^74 - 8 * q^75 - 36 * q^76 + 12 * q^79 - 58 * q^80 + 6 * q^81 + 24 * q^84 + 44 * q^85 + 4 * q^86 - 4 * q^89 - 16 * q^90 + 16 * q^91 + 24 * q^94 - 44 * q^95 - 22 * q^96 + 6 * q^99

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