LMFDB - Embedded newform 1690.2.b.b.339.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1690,2,Mod(339,1690)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1690, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1690.339"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-6,0,0,0,0,2,2,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.4947179416$
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:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			339.1
Root			$1.45161 + 1.45161i$ of defining polynomial
Character	$\chi$	$=$	1690.339
Dual form			1690.2.b.b.339.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -2.21432i q^{3} -1.00000 q^{4} +(-2.21432 + 0.311108i) q^{5} -2.21432 q^{6} +3.90321i q^{7} +1.00000i q^{8} -1.90321 q^{9} +(0.311108 + 2.21432i) q^{10} -1.06668 q^{11} +2.21432i q^{12} +3.90321 q^{14} +(0.688892 + 4.90321i) q^{15} +1.00000 q^{16} +1.31111i q^{17} +1.90321i q^{18} -6.59210 q^{19} +(2.21432 - 0.311108i) q^{20} +8.64296 q^{21} +1.06668i q^{22} -2.14764i q^{23} +2.21432 q^{24} +(4.80642 - 1.37778i) q^{25} -2.42864i q^{27} -3.90321i q^{28} +9.05086 q^{29} +(4.90321 - 0.688892i) q^{30} +6.92396 q^{31} -1.00000i q^{32} +2.36196i q^{33} +1.31111 q^{34} +(-1.21432 - 8.64296i) q^{35} +1.90321 q^{36} -5.02074i q^{37} +6.59210i q^{38} +(-0.311108 - 2.21432i) q^{40} +5.95407 q^{41} -8.64296i q^{42} -2.00000i q^{43} +1.06668 q^{44} +(4.21432 - 0.592104i) q^{45} -2.14764 q^{46} +0.0967881i q^{47} -2.21432i q^{48} -8.23506 q^{49} +(-1.37778 - 4.80642i) q^{50} +2.90321 q^{51} +1.49532i q^{53} -2.42864 q^{54} +(2.36196 - 0.331851i) q^{55} -3.90321 q^{56} +14.5970i q^{57} -9.05086i q^{58} +7.18421 q^{59} +(-0.688892 - 4.90321i) q^{60} +7.88739 q^{61} -6.92396i q^{62} -7.42864i q^{63} -1.00000 q^{64} +2.36196 q^{66} +8.42864i q^{67} -1.31111i q^{68} -4.75557 q^{69} +(-8.64296 + 1.21432i) q^{70} +15.4795 q^{71} -1.90321i q^{72} -15.3526i q^{73} -5.02074 q^{74} +(-3.05086 - 10.6430i) q^{75} +6.59210 q^{76} -4.16346i q^{77} -4.30174 q^{79} +(-2.21432 + 0.311108i) q^{80} -11.0874 q^{81} -5.95407i q^{82} +9.69381i q^{83} -8.64296 q^{84} +(-0.407896 - 2.90321i) q^{85} -2.00000 q^{86} -20.0415i q^{87} -1.06668i q^{88} +4.52543 q^{89} +(-0.592104 - 4.21432i) q^{90} +2.14764i q^{92} -15.3319i q^{93} +0.0967881 q^{94} +(14.5970 - 2.05086i) q^{95} -2.21432 q^{96} +4.26025i q^{97} +8.23506i q^{98} +2.03011 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} + 2 q^{9} + 2 q^{10} - 6 q^{11} + 10 q^{14} + 4 q^{15} + 6 q^{16} - 26 q^{19} + 12 q^{21} + 2 q^{25} + 28 q^{29} + 16 q^{30} - 12 q^{31} + 8 q^{34} + 6 q^{35} - 2 q^{36} - 2 q^{40} - 4 q^{41}+ \cdots + 26 q^{99}+O(q^{100}) $ 6 * q - 6 * q^4 + 2 * q^9 + 2 * q^10 - 6 * q^11 + 10 * q^14 + 4 * q^15 + 6 * q^16 - 26 * q^19 + 12 * q^21 + 2 * q^25 + 28 * q^29 + 16 * q^30 - 12 * q^31 + 8 * q^34 + 6 * q^35 - 2 * q^36 - 2 * q^40 - 4 * q^41 + 6 * q^44 + 12 * q^45 + 4 * q^49 - 8 * q^50 + 4 * q^51 + 12 * q^54 - 12 * q^55 - 10 * q^56 + 16 * q^59 - 4 * q^60 + 8 * q^61 - 6 * q^64 - 12 * q^66 - 28 * q^69 - 12 * q^70 + 40 * q^71 + 10 * q^74 + 8 * q^75 + 26 * q^76 + 28 * q^79 - 26 * q^81 - 12 * q^84 - 16 * q^85 - 12 * q^86 + 14 * q^89 + 10 * q^90 + 14 * q^94 + 8 * q^95 + 26 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		−	2.21432i		−	1.27844i	−0.769025	−	0.639219i	$-0.779258\pi$
$3$		−	2.21432i		−	1.27844i	0.769025	−	0.639219i	$-0.220742\pi$
$4$	−1.00000			−0.500000
$5$	−2.21432	+	0.311108i	−0.990274	+	0.139132i
$5$	−2.21432	+	0.311108i	−0.990274	+	0.139132i
$6$	−2.21432			−0.903992
$7$			3.90321i			1.47528i	0.675197	+	0.737638i	$0.264059\pi$
$7$			3.90321i			1.47528i	−0.675197	+	0.737638i	$0.735941\pi$
$8$			1.00000i			0.353553i
$9$	−1.90321			−0.634404
$10$	0.311108	+	2.21432i	0.0983809	+	0.700229i
$11$	−1.06668			−0.321615			−0.160808	−	0.986986i	$-0.551410\pi$
$11$	−1.06668			−0.321615			−0.160808	+	0.986986i	$0.551410\pi$
$12$			2.21432i			0.639219i
$13$		0			0
$13$		0			0
$14$	3.90321			1.04318
$15$	0.688892	+	4.90321i	0.177871	+	1.26600i
$16$	1.00000			0.250000
$17$			1.31111i			0.317990i	0.987279	+	0.158995i	$0.0508254\pi$
$17$			1.31111i			0.317990i	−0.987279	+	0.158995i	$0.949175\pi$
$18$			1.90321i			0.448591i
$19$	−6.59210			−1.51233			−0.756166	−	0.654380i	$-0.772930\pi$
$19$	−6.59210			−1.51233			−0.756166	+	0.654380i	$0.772930\pi$
$20$	2.21432	−	0.311108i	0.495137	−	0.0695658i
$21$	8.64296			1.88605
$22$			1.06668i			0.227416i
$23$		−	2.14764i		−	0.447815i	−0.974610	−	0.223907i	$-0.928119\pi$
$23$		−	2.14764i		−	0.447815i	0.974610	−	0.223907i	$-0.0718813\pi$
$24$	2.21432			0.451996
$25$	4.80642	−	1.37778i	0.961285	−	0.275557i
$26$		0			0
$27$		−	2.42864i		−	0.467392i
$28$		−	3.90321i		−	0.737638i
$29$	9.05086			1.68070			0.840351	−	0.542043i	$-0.182349\pi$
$29$	9.05086			1.68070			0.840351	+	0.542043i	$0.182349\pi$
$30$	4.90321	−	0.688892i	0.895200	−	0.125774i
$31$	6.92396			1.24358			0.621790	−	0.783184i	$-0.286406\pi$
$31$	6.92396			1.24358			0.621790	+	0.783184i	$0.286406\pi$
$32$		−	1.00000i		−	0.176777i
$33$			2.36196i			0.411165i
$34$	1.31111			0.224853
$35$	−1.21432	−	8.64296i	−0.205257	−	1.46093i
$36$	1.90321			0.317202
$37$		−	5.02074i		−	0.825405i	−0.910866	−	0.412703i	$-0.864585\pi$
$37$		−	5.02074i		−	0.825405i	0.910866	−	0.412703i	$-0.135415\pi$
$38$			6.59210i			1.06938i
$39$		0			0
$40$	−0.311108	−	2.21432i	−0.0491905	−	0.350115i
$41$	5.95407			0.929869			0.464935	−	0.885345i	$-0.346078\pi$
$41$	5.95407			0.929869			0.464935	+	0.885345i	$0.346078\pi$
$42$		−	8.64296i		−	1.33364i
$43$		−	2.00000i		−	0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$		−	2.00000i		−	0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	1.06668			0.160808
$45$	4.21432	−	0.592104i	0.628234	−	0.0882657i
$46$	−2.14764			−0.316653
$47$			0.0967881i			0.0141180i	0.999975	+	0.00705900i	$0.00224697\pi$
$47$			0.0967881i			0.0141180i	−0.999975	+	0.00705900i	$0.997753\pi$
$48$		−	2.21432i		−	0.319610i
$49$	−8.23506			−1.17644
$50$	−1.37778	−	4.80642i	−0.194848	−	0.679731i
$51$	2.90321			0.406531
$52$		0			0
$53$			1.49532i			0.205397i	0.994713	+	0.102699i	$0.0327478\pi$
$53$			1.49532i			0.205397i	−0.994713	+	0.102699i	$0.967252\pi$
$54$	−2.42864			−0.330496
$55$	2.36196	−	0.331851i	0.318487	−	0.0447468i
$56$	−3.90321			−0.521589
$57$			14.5970i			1.93342i
$58$		−	9.05086i		−	1.18844i
$59$	7.18421			0.935304			0.467652	−	0.883913i	$-0.345100\pi$
$59$	7.18421			0.935304			0.467652	+	0.883913i	$0.345100\pi$
$60$	−0.688892	−	4.90321i	−0.0889356	−	0.633002i
$61$	7.88739			1.00988			0.504938	−	0.863155i	$-0.331515\pi$
$61$	7.88739			1.00988			0.504938	+	0.863155i	$0.331515\pi$
$62$		−	6.92396i		−	0.879343i
$63$		−	7.42864i		−	0.935921i
$64$	−1.00000			−0.125000
$65$		0			0
$66$	2.36196			0.290738
$67$			8.42864i			1.02972i	0.857274	+	0.514861i	$0.172157\pi$
$67$			8.42864i			1.02972i	−0.857274	+	0.514861i	$0.827843\pi$
$68$		−	1.31111i		−	0.158995i
$69$	−4.75557			−0.572503
$70$	−8.64296	+	1.21432i	−1.03303	+	0.145139i
$71$	15.4795			1.83708			0.918539	−	0.395330i	$-0.129370\pi$
$71$	15.4795			1.83708			0.918539	+	0.395330i	$0.129370\pi$
$72$		−	1.90321i		−	0.224296i
$73$		−	15.3526i		−	1.79689i	−0.439091	−	0.898443i	$-0.644699\pi$
$73$		−	15.3526i		−	1.79689i	0.439091	−	0.898443i	$-0.355301\pi$
$74$	−5.02074			−0.583650
$75$	−3.05086	−	10.6430i	−0.352282	−	1.22894i
$76$	6.59210			0.756166
$77$		−	4.16346i		−	0.474471i
$78$		0			0
$79$	−4.30174			−0.483984			−0.241992	−	0.970278i	$-0.577801\pi$
$79$	−4.30174			−0.483984			−0.241992	+	0.970278i	$0.577801\pi$
$80$	−2.21432	+	0.311108i	−0.247568	+	0.0347829i
$81$	−11.0874			−1.23194
$82$		−	5.95407i		−	0.657517i
$83$			9.69381i			1.06403i	0.846734	+	0.532017i	$0.178566\pi$
$83$			9.69381i			1.06403i	−0.846734	+	0.532017i	$0.821434\pi$
$84$	−8.64296			−0.943024
$85$	−0.407896	−	2.90321i	−0.0442425	−	0.314898i
$86$	−2.00000			−0.215666
$87$		−	20.0415i		−	2.14867i
$88$		−	1.06668i		−	0.113708i
$89$	4.52543			0.479694			0.239847	−	0.970811i	$-0.422903\pi$
$89$	4.52543			0.479694			0.239847	+	0.970811i	$0.422903\pi$
$90$	−0.592104	−	4.21432i	−0.0624133	−	0.444228i
$91$		0			0
$92$			2.14764i			0.223907i
$93$		−	15.3319i		−	1.58984i
$94$	0.0967881			0.00998293
$95$	14.5970	−	2.05086i	1.49762	−	0.210413i
$96$	−2.21432			−0.225998
$97$			4.26025i			0.432563i	0.976331	+	0.216282i	$0.0693929\pi$
$97$			4.26025i			0.432563i	−0.976331	+	0.216282i	$0.930607\pi$
$98$			8.23506i			0.831867i
$99$	2.03011			0.204034
$100$	−4.80642	+	1.37778i	−0.480642	+	0.137778i
$101$	1.45875			0.145151			0.0725756	−	0.997363i	$-0.476878\pi$
$101$	1.45875			0.145151			0.0725756	+	0.997363i	$0.476878\pi$
$102$		−	2.90321i		−	0.287461i
$103$			10.2444i			1.00941i	0.863291	+	0.504707i	$0.168399\pi$
$103$			10.2444i			1.00941i	−0.863291	+	0.504707i	$0.831601\pi$
$104$		0			0
$105$	−19.1383	+	2.68889i	−1.86770	+	0.262409i
$106$	1.49532			0.145238
$107$		−	2.21432i		−	0.214066i	−0.994255	−	0.107033i	$-0.965865\pi$
$107$		−	2.21432i		−	0.214066i	0.994255	−	0.107033i	$-0.0341351\pi$
$108$			2.42864i			0.233696i
$109$	0.133353			0.0127729			0.00638645	−	0.999980i	$-0.497967\pi$
$109$	0.133353			0.0127729			0.00638645	+	0.999980i	$0.497967\pi$
$110$	−0.331851	−	2.36196i	−0.0316408	−	0.225204i
$111$	−11.1175			−1.05523
$112$			3.90321i			0.368819i
$113$			14.0000i			1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$			14.0000i			1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	14.5970			1.36714
$115$	0.668149	+	4.75557i	0.0623052	+	0.443459i
$116$	−9.05086			−0.840351
$117$		0			0
$118$		−	7.18421i		−	0.661360i
$119$	−5.11753			−0.469123
$120$	−4.90321	+	0.688892i	−0.447600	+	0.0628870i
$121$	−9.86220			−0.896564
$122$		−	7.88739i		−	0.714091i
$123$		−	13.1842i		−	1.18878i
$124$	−6.92396			−0.621790
$125$	−10.2143	+	4.54617i	−0.913597	+	0.406622i
$126$	−7.42864			−0.661796
$127$			9.99063i			0.886525i	0.896392	+	0.443263i	$0.146179\pi$
$127$			9.99063i			0.886525i	−0.896392	+	0.443263i	$0.853821\pi$
$128$			1.00000i			0.0883883i
$129$	−4.42864			−0.389920
$130$		0			0
$131$	9.25581			0.808684			0.404342	−	0.914608i	$-0.367501\pi$
$131$	9.25581			0.808684			0.404342	+	0.914608i	$0.367501\pi$
$132$		−	2.36196i		−	0.205582i
$133$		−	25.7304i		−	2.23111i
$134$	8.42864			0.728124
$135$	0.755569	+	5.37778i	0.0650290	+	0.462846i
$136$	−1.31111			−0.112427
$137$			14.7906i			1.26365i	0.775113	+	0.631823i	$0.217693\pi$
$137$			14.7906i			1.26365i	−0.775113	+	0.631823i	$0.782307\pi$
$138$			4.75557i			0.404821i
$139$	12.6938			1.07668			0.538338	−	0.842729i	$-0.319053\pi$
$139$	12.6938			1.07668			0.538338	+	0.842729i	$0.319053\pi$
$140$	1.21432	+	8.64296i	0.102629	+	0.730463i
$141$	0.214320			0.0180490
$142$		−	15.4795i		−	1.29901i
$143$		0			0
$144$	−1.90321			−0.158601
$145$	−20.0415	+	2.81579i	−1.66435	+	0.233839i
$146$	−15.3526			−1.27059
$147$			18.2351i			1.50400i
$148$			5.02074i			0.412703i
$149$	7.28592			0.596886			0.298443	−	0.954428i	$-0.403533\pi$
$149$	7.28592			0.596886			0.298443	+	0.954428i	$0.403533\pi$
$150$	−10.6430	+	3.05086i	−0.868994	+	0.249101i
$151$	−20.1082			−1.63638			−0.818190	−	0.574949i	$-0.805022\pi$
$151$	−20.1082			−1.63638			−0.818190	+	0.574949i	$0.805022\pi$
$152$		−	6.59210i		−	0.534690i
$153$		−	2.49532i		−	0.201734i
$154$	−4.16346			−0.335502
$155$	−15.3319	+	2.15410i	−1.23148	+	0.173021i
$156$		0			0
$157$		−	1.98418i		−	0.158355i	−0.996861	−	0.0791773i	$-0.974771\pi$
$157$		−	1.98418i		−	0.158355i	0.996861	−	0.0791773i	$-0.0252293\pi$
$158$			4.30174i			0.342228i
$159$	3.31111			0.262588
$160$	0.311108	+	2.21432i	0.0245952	+	0.175057i
$161$	8.38271			0.660650
$162$			11.0874i			0.871110i
$163$		−	4.70318i		−	0.368382i	−0.982891	−	0.184191i	$-0.941034\pi$
$163$		−	4.70318i		−	0.368382i	0.982891	−	0.184191i	$-0.0589664\pi$
$164$	−5.95407			−0.464935
$165$	−0.734825	−	5.23014i	−0.0572061	−	0.407166i
$166$	9.69381			0.752386
$167$			9.69535i			0.750248i	0.926975	+	0.375124i	$0.122400\pi$
$167$			9.69535i			0.750248i	−0.926975	+	0.375124i	$0.877600\pi$
$168$			8.64296i			0.666819i
$169$		0			0
$170$	−2.90321	+	0.407896i	−0.222666	+	0.0312842i
$171$	12.5462			0.959430
$172$			2.00000i			0.152499i
$173$		−	24.0716i		−	1.83013i	−0.403307	−	0.915065i	$-0.632139\pi$
$173$		−	24.0716i		−	1.83013i	0.403307	−	0.915065i	$-0.367861\pi$
$174$	−20.0415			−1.51934
$175$	5.37778	+	18.7605i	0.406522	+	1.41816i
$176$	−1.06668			−0.0804038
$177$		−	15.9081i		−	1.19573i
$178$		−	4.52543i		−	0.339195i
$179$	−3.05086			−0.228032			−0.114016	−	0.993479i	$-0.536371\pi$
$179$	−3.05086			−0.228032			−0.114016	+	0.993479i	$0.536371\pi$
$180$	−4.21432	+	0.592104i	−0.314117	+	0.0441328i
$181$	10.6430			0.791085			0.395542	−	0.918448i	$-0.370557\pi$
$181$	10.6430			0.791085			0.395542	+	0.918448i	$0.370557\pi$
$182$		0			0
$183$		−	17.4652i		−	1.29107i
$184$	2.14764			0.158326
$185$	1.56199	+	11.1175i	0.114840	+	0.817377i
$186$	−15.3319			−1.12419
$187$		−	1.39853i		−	0.102270i
$188$		−	0.0967881i		−	0.00705900i
$189$	9.47949			0.689532
$190$	−2.05086	−	14.5970i	−0.148785	−	1.05898i
$191$	14.7304			1.06585			0.532926	−	0.846162i	$-0.321092\pi$
$191$	14.7304			1.06585			0.532926	+	0.846162i	$0.321092\pi$
$192$			2.21432i			0.159805i
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	4.26025			0.305868
$195$		0			0
$196$	8.23506			0.588219
$197$			10.4128i			0.741883i	0.928656	+	0.370941i	$0.120965\pi$
$197$			10.4128i			0.741883i	−0.928656	+	0.370941i	$0.879035\pi$
$198$		−	2.03011i		−	0.144274i
$199$	4.36842			0.309669			0.154834	−	0.987940i	$-0.450516\pi$
$199$	4.36842			0.309669			0.154834	+	0.987940i	$0.450516\pi$
$200$	1.37778	+	4.80642i	0.0974241	+	0.339865i
$201$	18.6637			1.31644
$202$		−	1.45875i		−	0.102637i
$203$			35.3274i			2.47950i
$204$	−2.90321			−0.203265
$205$	−13.1842	+	1.85236i	−0.920825	+	0.129374i
$206$	10.2444			0.713763
$207$			4.08742i			0.284095i
$208$		0			0
$209$	7.03164			0.486389
$210$	2.68889	+	19.1383i	0.185551	+	1.32067i
$211$	5.49532			0.378313			0.189157	−	0.981947i	$-0.439425\pi$
$211$	5.49532			0.378313			0.189157	+	0.981947i	$0.439425\pi$
$212$		−	1.49532i		−	0.102699i
$213$		−	34.2766i		−	2.34859i
$214$	−2.21432			−0.151368
$215$	0.622216	+	4.42864i	0.0424348	+	0.302031i
$216$	2.42864			0.165248
$217$			27.0257i			1.83462i
$218$		−	0.133353i		−	0.00903181i
$219$	−33.9956			−2.29721
$220$	−2.36196	+	0.331851i	−0.159244	+	0.0223734i
$221$		0			0
$222$			11.1175i			0.746160i
$223$		−	20.0968i		−	1.34578i	−0.739742	−	0.672890i	$-0.765053\pi$
$223$		−	20.0968i		−	1.34578i	0.739742	−	0.672890i	$-0.234947\pi$
$224$	3.90321			0.260794
$225$	−9.14764	+	2.62222i	−0.609843	+	0.174814i
$226$	14.0000			0.931266
$227$		−	19.3590i		−	1.28491i	−0.766325	−	0.642453i	$-0.777917\pi$
$227$		−	19.3590i		−	1.28491i	0.766325	−	0.642453i	$-0.222083\pi$
$228$		−	14.5970i		−	0.966712i
$229$	15.7255			1.03917			0.519584	−	0.854420i	$-0.326087\pi$
$229$	15.7255			1.03917			0.519584	+	0.854420i	$0.326087\pi$
$230$	4.75557	−	0.668149i	0.313573	−	0.0440564i
$231$	−9.21924			−0.606582
$232$			9.05086i			0.594218i
$233$			14.8825i			0.974983i	0.873128	+	0.487491i	$0.162088\pi$
$233$			14.8825i			0.974983i	−0.873128	+	0.487491i	$0.837912\pi$
$234$		0			0
$235$	−0.0301115	−	0.214320i	−0.00196426	−	0.0139807i
$236$	−7.18421			−0.467652
$237$			9.52543i			0.618743i
$238$			5.11753i			0.331720i
$239$	4.42219			0.286047			0.143024	−	0.989719i	$-0.454317\pi$
$239$	4.42219			0.286047			0.143024	+	0.989719i	$0.454317\pi$
$240$	0.688892	+	4.90321i	0.0444678	+	0.316501i
$241$	11.2810			0.726673			0.363336	−	0.931658i	$-0.381638\pi$
$241$	11.2810			0.726673			0.363336	+	0.931658i	$0.381638\pi$
$242$			9.86220i			0.633966i
$243$			17.2652i			1.10756i
$244$	−7.88739			−0.504938
$245$	18.2351	−	2.56199i	1.16500	−	0.163680i
$246$	−13.1842			−0.840594
$247$		0			0
$248$			6.92396i			0.439672i
$249$	21.4652			1.36030
$250$	4.54617	+	10.2143i	0.287525	+	0.646010i
$251$	2.65233			0.167413			0.0837067	−	0.996490i	$-0.473324\pi$
$251$	2.65233			0.167413			0.0837067	+	0.996490i	$0.473324\pi$
$252$			7.42864i			0.467960i
$253$			2.29084i			0.144024i
$254$	9.99063			0.626868
$255$	−6.42864	+	0.903212i	−0.402577	+	0.0565613i
$256$	1.00000			0.0625000
$257$			4.10171i			0.255858i	0.991783	+	0.127929i	$0.0408329\pi$
$257$			4.10171i			0.255858i	−0.991783	+	0.127929i	$0.959167\pi$
$258$			4.42864i			0.275715i
$259$	19.5970			1.21770
$260$		0			0
$261$	−17.2257			−1.06624
$262$		−	9.25581i		−	0.571826i
$263$			10.8020i			0.666079i	0.942913	+	0.333039i	$0.108074\pi$
$263$			10.8020i			0.666079i	−0.942913	+	0.333039i	$0.891926\pi$
$264$	−2.36196			−0.145369
$265$	−0.465205	−	3.31111i	−0.0285773	−	0.203400i
$266$	−25.7304			−1.57763
$267$		−	10.0207i		−	0.613260i
$268$		−	8.42864i		−	0.514861i
$269$	18.5827			1.13301			0.566505	−	0.824059i	$-0.308295\pi$
$269$	18.5827			1.13301			0.566505	+	0.824059i	$0.308295\pi$
$270$	5.37778	−	0.755569i	0.327282	−	0.0459824i
$271$	−19.7748			−1.20123			−0.600616	−	0.799537i	$-0.705078\pi$
$271$	−19.7748			−1.20123			−0.600616	+	0.799537i	$0.705078\pi$
$272$			1.31111i			0.0794976i
$273$		0			0
$274$	14.7906			0.893533
$275$	−5.12690	+	1.46965i	−0.309164	+	0.0886232i
$276$	4.75557			0.286252
$277$			0.204952i			0.0123144i	0.999981	+	0.00615718i	$0.00195990\pi$
$277$			0.204952i			0.0123144i	−0.999981	+	0.00615718i	$0.998040\pi$
$278$		−	12.6938i		−	0.761324i
$279$	−13.1778			−0.788932
$280$	8.64296	−	1.21432i	0.516516	−	0.0725695i
$281$	20.4558			1.22029			0.610146	−	0.792289i	$-0.291111\pi$
$281$	20.4558			1.22029			0.610146	+	0.792289i	$0.291111\pi$
$282$		−	0.214320i		−	0.0127626i
$283$		−	1.43801i		−	0.0854807i	−0.999086	−	0.0427403i	$-0.986391\pi$
$283$		−	1.43801i		−	0.0854807i	0.999086	−	0.0427403i	$-0.0136088\pi$
$284$	−15.4795			−0.918539
$285$	−4.54125	−	32.3225i	−0.269000	−	1.91462i
$286$		0			0
$287$			23.2400i			1.37181i
$288$			1.90321i			0.112148i
$289$	15.2810			0.898882
$290$	2.81579	+	20.0415i	0.165349	+	1.17688i
$291$	9.43356			0.553005
$292$			15.3526i			0.898443i
$293$		−	20.9146i		−	1.22184i	−0.791691	−	0.610922i	$-0.790799\pi$
$293$		−	20.9146i		−	1.22184i	0.791691	−	0.610922i	$-0.209201\pi$
$294$	18.2351			1.06349
$295$	−15.9081	+	2.23506i	−0.926207	+	0.130130i
$296$	5.02074			0.291825
$297$			2.59057i			0.150320i
$298$		−	7.28592i		−	0.422062i
$299$		0			0
$300$	3.05086	+	10.6430i	0.176141	+	0.614472i
$301$	7.80642			0.449955
$302$			20.1082i			1.15709i
$303$		−	3.23014i		−	0.185567i
$304$	−6.59210			−0.378083
$305$	−17.4652	+	2.45383i	−1.00005	+	0.140506i
$306$	−2.49532			−0.142648
$307$		−	12.5303i		−	0.715145i	−0.933885	−	0.357572i	$-0.883605\pi$
$307$		−	12.5303i		−	0.715145i	0.933885	−	0.357572i	$-0.116395\pi$
$308$			4.16346i			0.237235i
$309$	22.6844			1.29047
$310$	2.15410	+	15.3319i	0.122344	+	0.870791i
$311$	−9.78123			−0.554643			−0.277321	−	0.960777i	$-0.589447\pi$
$311$	−9.78123			−0.554643			−0.277321	+	0.960777i	$0.589447\pi$
$312$		0			0
$313$		−	17.2128i		−	0.972924i	−0.873702	−	0.486462i	$-0.838287\pi$
$313$		−	17.2128i		−	0.972924i	0.873702	−	0.486462i	$-0.161713\pi$
$314$	−1.98418			−0.111974
$315$	2.31111	+	16.4494i	0.130216	+	0.926818i
$316$	4.30174			0.241992
$317$		−	13.7447i		−	0.771978i	−0.922503	−	0.385989i	$-0.873860\pi$
$317$		−	13.7447i		−	0.771978i	0.922503	−	0.385989i	$-0.126140\pi$
$318$		−	3.31111i		−	0.185678i
$319$	−9.65433			−0.540539
$320$	2.21432	−	0.311108i	0.123784	−	0.0173915i
$321$	−4.90321			−0.273671
$322$		−	8.38271i		−	0.467150i
$323$		−	8.64296i		−	0.480907i
$324$	11.0874			0.615968
$325$		0			0
$326$	−4.70318			−0.260485
$327$		−	0.295286i		−	0.0163294i
$328$			5.95407i			0.328758i
$329$	−0.377784			−0.0208279
$330$	−5.23014	+	0.734825i	−0.287910	+	0.0404508i
$331$	−19.3876			−1.06564			−0.532820	−	0.846228i	$-0.678868\pi$
$331$	−19.3876			−1.06564			−0.532820	+	0.846228i	$0.678868\pi$
$332$		−	9.69381i		−	0.532017i
$333$			9.55554i			0.523640i
$334$	9.69535			0.530506
$335$	−2.62222	−	18.6637i	−0.143267	−	1.01971i
$336$	8.64296			0.471512
$337$			0.555539i			0.0302621i	0.999886	+	0.0151311i	$0.00481655\pi$
$337$			0.555539i			0.0302621i	−0.999886	+	0.0151311i	$0.995183\pi$
$338$		0			0
$339$	31.0005			1.68371
$340$	0.407896	+	2.90321i	0.0221213	+	0.157449i
$341$	−7.38562			−0.399954
$342$		−	12.5462i		−	0.678419i
$343$		−	4.82071i		−	0.260294i
$344$	2.00000			0.107833
$345$	10.5303	−	1.47949i	0.566935	−	0.0796533i
$346$	−24.0716			−1.29410
$347$			6.68445i			0.358840i	0.983773	+	0.179420i	$0.0574221\pi$
$347$			6.68445i			0.358840i	−0.983773	+	0.179420i	$0.942578\pi$
$348$			20.0415i			1.07434i
$349$	10.5827			0.566481			0.283240	−	0.959049i	$-0.408591\pi$
$349$	10.5827			0.566481			0.283240	+	0.959049i	$0.408591\pi$
$350$	18.7605	−	5.37778i	1.00279	−	0.287455i
$351$		0			0
$352$			1.06668i			0.0568541i
$353$		−	9.20940i		−	0.490167i	−0.969502	−	0.245083i	$-0.921185\pi$
$353$		−	9.20940i		−	0.490167i	0.969502	−	0.245083i	$-0.0788153\pi$
$354$	−15.9081			−0.845508
$355$	−34.2766	+	4.81579i	−1.81921	+	0.255596i
$356$	−4.52543			−0.239847
$357$			11.3319i			0.599745i
$358$			3.05086i			0.161243i
$359$	−4.19358			−0.221328			−0.110664	−	0.993858i	$-0.535298\pi$
$359$	−4.19358			−0.221328			−0.110664	+	0.993858i	$0.535298\pi$
$360$	0.592104	+	4.21432i	0.0312066	+	0.222114i
$361$	24.4558			1.28715
$362$		−	10.6430i		−	0.559382i
$363$			21.8381i			1.14620i
$364$		0			0
$365$	4.77631	+	33.9956i	0.250004	+	1.77941i
$366$	−17.4652			−0.912921
$367$			13.9684i			0.729142i	0.931176	+	0.364571i	$0.118784\pi$
$367$			13.9684i			0.729142i	−0.931176	+	0.364571i	$0.881216\pi$
$368$		−	2.14764i		−	0.111954i
$369$	−11.3319			−0.589913
$370$	11.1175	−	1.56199i	0.577973	−	0.0812041i
$371$	−5.83654			−0.303018
$372$			15.3319i			0.794919i
$373$		−	11.8479i		−	0.613462i	−0.951796	−	0.306731i	$-0.900765\pi$
$373$		−	11.8479i		−	0.613462i	0.951796	−	0.306731i	$-0.0992351\pi$
$374$	−1.39853			−0.0723162
$375$	10.0667	+	22.6178i	0.519841	+	1.16798i
$376$	−0.0967881			−0.00499146
$377$		0			0
$378$		−	9.47949i		−	0.487573i
$379$	0.606394			0.0311484			0.0155742	−	0.999879i	$-0.495042\pi$
$379$	0.606394			0.0311484			0.0155742	+	0.999879i	$0.495042\pi$
$380$	−14.5970	+	2.05086i	−0.748812	+	0.105207i
$381$	22.1225			1.13337
$382$		−	14.7304i		−	0.753672i
$383$		−	34.7797i		−	1.77716i	−0.458722	−	0.888580i	$-0.651693\pi$
$383$		−	34.7797i		−	1.77716i	0.458722	−	0.888580i	$-0.348307\pi$
$384$	2.21432			0.112999
$385$	1.29529	+	9.21924i	0.0660139	+	0.469856i
$386$	−14.0000			−0.712581
$387$			3.80642i			0.193491i
$388$		−	4.26025i		−	0.216282i
$389$	18.6844			0.947339			0.473670	−	0.880703i	$-0.342929\pi$
$389$	18.6844			0.947339			0.473670	+	0.880703i	$0.342929\pi$
$390$		0			0
$391$	2.81579			0.142401
$392$		−	8.23506i		−	0.415934i
$393$		−	20.4953i		−	1.03385i
$394$	10.4128			0.524590
$395$	9.52543	−	1.33830i	0.479276	−	0.0673374i
$396$	−2.03011			−0.102017
$397$		−	22.9748i		−	1.15307i	−0.817072	−	0.576536i	$-0.804404\pi$
$397$		−	22.9748i		−	1.15307i	0.817072	−	0.576536i	$-0.195596\pi$
$398$		−	4.36842i		−	0.218969i
$399$	−56.9753			−2.85233
$400$	4.80642	−	1.37778i	0.240321	−	0.0688892i
$401$	4.37334			0.218394			0.109197	−	0.994020i	$-0.465172\pi$
$401$	4.37334			0.218394			0.109197	+	0.994020i	$0.465172\pi$
$402$		−	18.6637i		−	0.930861i
$403$		0			0
$404$	−1.45875			−0.0725756
$405$	24.5511	−	3.44938i	1.21995	−	0.171401i
$406$	35.3274			1.75327
$407$			5.35551i			0.265463i
$408$			2.90321i			0.143730i
$409$	−3.76986			−0.186408			−0.0932038	−	0.995647i	$-0.529711\pi$
$409$	−3.76986			−0.186408			−0.0932038	+	0.995647i	$0.529711\pi$
$410$	1.85236	+	13.1842i	0.0914814	+	0.651122i
$411$	32.7511			1.61549
$412$		−	10.2444i		−	0.504707i
$413$			28.0415i			1.37983i
$414$	4.08742			0.200886
$415$	−3.01582	−	21.4652i	−0.148041	−	1.05369i
$416$		0			0
$417$		−	28.1082i		−	1.37646i
$418$		−	7.03164i		−	0.343929i
$419$	−7.45230			−0.364069			−0.182034	−	0.983292i	$-0.558268\pi$
$419$	−7.45230			−0.364069			−0.182034	+	0.983292i	$0.558268\pi$
$420$	19.1383	−	2.68889i	0.933852	−	0.131205i
$421$	−27.6751			−1.34880			−0.674400	−	0.738366i	$-0.735598\pi$
$421$	−27.6751			−1.34880			−0.674400	+	0.738366i	$0.735598\pi$
$422$		−	5.49532i		−	0.267508i
$423$		−	0.184208i		−	0.00895651i
$424$	−1.49532			−0.0726190
$425$	1.80642	+	6.30174i	0.0876244	+	0.305679i
$426$	−34.2766			−1.66070
$427$			30.7862i			1.48985i
$428$			2.21432i			0.107033i
$429$		0			0
$430$	4.42864	−	0.622216i	0.213568	−	0.0300059i
$431$	−19.6894			−0.948404			−0.474202	−	0.880416i	$-0.657263\pi$
$431$	−19.6894			−0.948404			−0.474202	+	0.880416i	$0.657263\pi$
$432$		−	2.42864i		−	0.116848i
$433$			13.7462i			0.660600i	0.943876	+	0.330300i	$0.107150\pi$
$433$			13.7462i			0.660600i	−0.943876	+	0.330300i	$0.892850\pi$
$434$	27.0257			1.29727
$435$	6.23506	+	44.3783i	0.298948	+	2.12777i
$436$	−0.133353			−0.00638645
$437$			14.1575i			0.677244i
$438$			33.9956i			1.62437i
$439$	−41.8227			−1.99609			−0.998045	−	0.0625026i	$-0.980092\pi$
$439$	−41.8227			−1.99609			−0.998045	+	0.0625026i	$0.980092\pi$
$440$	0.331851	+	2.36196i	0.0158204	+	0.112602i
$441$	15.6731			0.746337
$442$		0			0
$443$			23.6829i			1.12521i	0.826726	+	0.562605i	$0.190201\pi$
$443$			23.6829i			1.12521i	−0.826726	+	0.562605i	$0.809799\pi$
$444$	11.1175			0.527615
$445$	−10.0207	+	1.40790i	−0.475029	+	0.0667407i
$446$	−20.0968			−0.951610
$447$		−	16.1334i		−	0.763081i
$448$		−	3.90321i		−	0.184409i
$449$	7.05578			0.332983			0.166491	−	0.986043i	$-0.446756\pi$
$449$	7.05578			0.332983			0.166491	+	0.986043i	$0.446756\pi$
$450$	2.62222	+	9.14764i	0.123612	+	0.431224i
$451$	−6.35106			−0.299060
$452$		−	14.0000i		−	0.658505i
$453$			44.5259i			2.09201i
$454$	−19.3590			−0.908565
$455$		0			0
$456$	−14.5970			−0.683568
$457$			35.6795i			1.66902i	0.550995	+	0.834509i	$0.314249\pi$
$457$			35.6795i			1.66902i	−0.550995	+	0.834509i	$0.685751\pi$
$458$		−	15.7255i		−	0.734802i
$459$	3.18421			0.148626
$460$	−0.668149	−	4.75557i	−0.0311526	−	0.221730i
$461$	14.6015			0.680058			0.340029	−	0.940415i	$-0.389563\pi$
$461$	14.6015			0.680058			0.340029	+	0.940415i	$0.389563\pi$
$462$			9.21924i			0.428918i
$463$			30.7511i			1.42913i	0.699571	+	0.714563i	$0.253374\pi$
$463$			30.7511i			1.42913i	−0.699571	+	0.714563i	$0.746626\pi$
$464$	9.05086			0.420175
$465$	4.76986	+	33.9496i	0.221197	+	1.57438i
$466$	14.8825			0.689417
$467$		−	20.5620i		−	0.951496i	−0.879582	−	0.475748i	$-0.842178\pi$
$467$		−	20.5620i		−	0.951496i	0.879582	−	0.475748i	$-0.157822\pi$
$468$		0			0
$469$	−32.8988			−1.51912
$470$	−0.214320	+	0.0301115i	−0.00988583	+	0.00138894i
$471$	−4.39361			−0.202447
$472$			7.18421i			0.330680i
$473$			2.13335i			0.0980917i
$474$	9.52543			0.437517
$475$	−31.6844	+	9.08250i	−1.45378	+	0.416734i
$476$	5.11753			0.234562
$477$		−	2.84590i		−	0.130305i
$478$		−	4.42219i		−	0.202266i
$479$	11.9333			0.545247			0.272624	−	0.962121i	$-0.412109\pi$
$479$	11.9333			0.545247			0.272624	+	0.962121i	$0.412109\pi$
$480$	4.90321	−	0.688892i	0.223800	−	0.0314435i
$481$		0			0
$482$		−	11.2810i		−	0.513835i
$483$		−	18.5620i		−	0.844600i
$484$	9.86220			0.448282
$485$	−1.32540	−	9.43356i	−0.0601832	−	0.428356i
$486$	17.2652			0.783164
$487$			27.9719i			1.26753i	0.773527	+	0.633764i	$0.218491\pi$
$487$			27.9719i			1.26753i	−0.773527	+	0.633764i	$0.781509\pi$
$488$			7.88739i			0.357045i
$489$	−10.4143			−0.470953
$490$	−2.56199	−	18.2351i	−0.115739	−	0.823776i
$491$	−20.7812			−0.937844			−0.468922	−	0.883240i	$-0.655357\pi$
$491$	−20.7812			−0.937844			−0.468922	+	0.883240i	$0.655357\pi$
$492$			13.1842i			0.594390i
$493$			11.8666i			0.534447i
$494$		0			0
$495$	−4.49532	+	0.631584i	−0.202049	+	0.0283876i
$496$	6.92396			0.310895
$497$			60.4197i			2.71020i
$498$		−	21.4652i		−	0.961879i
$499$	34.2908			1.53507			0.767534	−	0.641008i	$-0.221483\pi$
$499$	34.2908			1.53507			0.767534	+	0.641008i	$0.221483\pi$
$500$	10.2143	−	4.54617i	0.456798	−	0.203311i
$501$	21.4686			0.959146
$502$		−	2.65233i		−	0.118379i
$503$			20.2257i			0.901819i	0.892570	+	0.450910i	$0.148900\pi$
$503$			20.2257i			0.901819i	−0.892570	+	0.450910i	$0.851100\pi$
$504$	7.42864			0.330898
$505$	−3.23014	+	0.453829i	−0.143739	+	0.0201951i
$506$	2.29084			0.101840
$507$		0			0
$508$		−	9.99063i		−	0.443263i
$509$	−27.3778			−1.21350			−0.606749	−	0.794893i	$-0.707527\pi$
$509$	−27.3778			−1.21350			−0.606749	+	0.794893i	$0.707527\pi$
$510$	0.903212	+	6.42864i	0.0399949	+	0.284665i
$511$	59.9244			2.65090
$512$		−	1.00000i		−	0.0441942i
$513$			16.0098i			0.706852i
$514$	4.10171			0.180919
$515$	−3.18712	−	22.6844i	−0.140441	−	0.999596i
$516$	4.42864			0.194960
$517$		−	0.103242i		−	0.00454056i
$518$		−	19.5970i		−	0.861044i
$519$	−53.3022			−2.33971
$520$		0			0
$521$	−31.2034			−1.36705			−0.683523	−	0.729929i	$-0.739553\pi$
$521$	−31.2034			−1.36705			−0.683523	+	0.729929i	$0.739553\pi$
$522$			17.2257i			0.753948i
$523$			9.84791i			0.430619i	0.976546	+	0.215310i	$0.0690760\pi$
$523$			9.84791i			0.430619i	−0.976546	+	0.215310i	$0.930924\pi$
$524$	−9.25581			−0.404342
$525$	41.5417	−	11.9081i	1.81303	−	0.519714i
$526$	10.8020			0.470989
$527$			9.07805i			0.395446i
$528$			2.36196i			0.102791i
$529$	18.3876			0.799462
$530$	−3.31111	+	0.465205i	−0.143825	+	0.0202072i
$531$	−13.6731			−0.593361
$532$			25.7304i			1.11555i
$533$		0			0
$534$	−10.0207			−0.433640
$535$	0.688892	+	4.90321i	0.0297834	+	0.211984i
$536$	−8.42864			−0.364062
$537$			6.75557i			0.291524i
$538$		−	18.5827i		−	0.801159i
$539$	8.78415			0.378360
$540$	−0.755569	−	5.37778i	−0.0325145	−	0.231423i
$541$	−41.6149			−1.78916			−0.894581	−	0.446905i	$-0.852526\pi$
$541$	−41.6149			−1.78916			−0.894581	+	0.446905i	$0.852526\pi$
$542$			19.7748i			0.849400i
$543$		−	23.5669i		−	1.01135i
$544$	1.31111			0.0562133
$545$	−0.295286	+	0.0414872i	−0.0126487	+	0.00177712i
$546$		0			0
$547$			6.77430i			0.289648i	0.989457	+	0.144824i	$0.0462617\pi$
$547$			6.77430i			0.289648i	−0.989457	+	0.144824i	$0.953738\pi$
$548$		−	14.7906i		−	0.631823i
$549$	−15.0114			−0.640670
$550$	1.46965	+	5.12690i	0.0626661	+	0.218612i
$551$	−59.6642			−2.54178
$552$		−	4.75557i		−	0.202410i
$553$		−	16.7906i		−	0.714009i
$554$	0.204952			0.00870757
$555$	24.6178	−	3.45875i	1.04497	−	0.146816i
$556$	−12.6938			−0.538338
$557$		−	25.4449i		−	1.07814i	−0.842262	−	0.539068i	$-0.818776\pi$
$557$		−	25.4449i		−	1.07814i	0.842262	−	0.539068i	$-0.181224\pi$
$558$			13.1778i			0.557859i
$559$		0			0
$560$	−1.21432	−	8.64296i	−0.0513144	−	0.365232i
$561$	−3.09679			−0.130746
$562$		−	20.4558i		−	0.862877i
$563$		−	13.1842i		−	0.555648i	−0.960632	−	0.277824i	$-0.910387\pi$
$563$		−	13.1842i		−	0.555648i	0.960632	−	0.277824i	$-0.0896132\pi$
$564$	−0.214320			−0.00902449
$565$	−4.35551	−	31.0005i	−0.183238	−	1.30420i
$566$	−1.43801			−0.0604440
$567$		−	43.2766i		−	1.81744i
$568$			15.4795i			0.649505i
$569$	24.4608			1.02545			0.512724	−	0.858553i	$-0.328636\pi$
$569$	24.4608			1.02545			0.512724	+	0.858553i	$0.328636\pi$
$570$	−32.3225	+	4.54125i	−1.35384	+	0.190212i
$571$	24.3526			1.01912			0.509562	−	0.860434i	$-0.329807\pi$
$571$	24.3526			1.01912			0.509562	+	0.860434i	$0.329807\pi$
$572$		0			0
$573$		−	32.6178i		−	1.36263i
$574$	23.2400			0.970018
$575$	−2.95899	−	10.3225i	−0.123398	−	0.430477i
$576$	1.90321			0.0793005
$577$			42.0479i			1.75048i	0.483690	+	0.875239i	$0.339296\pi$
$577$			42.0479i			1.75048i	−0.483690	+	0.875239i	$0.660704\pi$
$578$		−	15.2810i		−	0.635606i
$579$	−31.0005			−1.28834
$580$	20.0415	−	2.81579i	0.832177	−	0.116919i
$581$	−37.8370			−1.56974
$582$		−	9.43356i		−	0.391034i
$583$		−	1.59502i		−	0.0660589i
$584$	15.3526			0.635295
$585$		0			0
$586$	−20.9146			−0.863974
$587$			7.34614i			0.303208i	0.988441	+	0.151604i	$0.0484438\pi$
$587$			7.34614i			0.303208i	−0.988441	+	0.151604i	$0.951556\pi$
$588$		−	18.2351i		−	0.752001i
$589$	−45.6434			−1.88071
$590$	2.23506	+	15.9081i	0.0920161	+	0.654928i
$591$	23.0573			0.948451
$592$		−	5.02074i		−	0.206351i
$593$			4.19358i			0.172210i	0.996286	+	0.0861048i	$0.0274420\pi$
$593$			4.19358i			0.172210i	−0.996286	+	0.0861048i	$0.972558\pi$
$594$	2.59057			0.106292
$595$	11.3319	−	1.59210i	0.464561	−	0.0652699i
$596$	−7.28592			−0.298443
$597$		−	9.67307i		−	0.395892i
$598$		0			0
$599$	−9.33630			−0.381471			−0.190735	−	0.981641i	$-0.561087\pi$
$599$	−9.33630			−0.381471			−0.190735	+	0.981641i	$0.561087\pi$
$600$	10.6430	−	3.05086i	0.434497	−	0.124551i
$601$	−2.97142			−0.121207			−0.0606034	−	0.998162i	$-0.519302\pi$
$601$	−2.97142			−0.121207			−0.0606034	+	0.998162i	$0.519302\pi$
$602$		−	7.80642i		−	0.318166i
$603$		−	16.0415i		−	0.653260i
$604$	20.1082			0.818190
$605$	21.8381	−	3.06821i	0.887844	−	0.124740i
$606$	−3.23014			−0.131216
$607$		−	17.1432i		−	0.695821i	−0.937528	−	0.347910i	$-0.886891\pi$
$607$		−	17.1432i		−	0.695821i	0.937528	−	0.347910i	$-0.113109\pi$
$608$			6.59210i			0.267345i
$609$	78.2262			3.16988
$610$	2.45383	+	17.4652i	0.0993526	+	0.707145i
$611$		0			0
$612$			2.49532i			0.100867i
$613$		−	29.2844i		−	1.18279i	−0.806384	−	0.591393i	$-0.798578\pi$
$613$		−	29.2844i		−	1.18279i	0.806384	−	0.591393i	$-0.201422\pi$
$614$	−12.5303			−0.505684
$615$	4.10171	+	29.1941i	0.165397	+	1.17722i
$616$	4.16346			0.167751
$617$			13.2159i			0.532050i	0.963966	+	0.266025i	$0.0857104\pi$
$617$			13.2159i			0.532050i	−0.963966	+	0.266025i	$0.914290\pi$
$618$		−	22.6844i		−	0.912502i
$619$	−18.9333			−0.760995			−0.380497	−	0.924782i	$-0.624247\pi$
$619$	−18.9333			−0.760995			−0.380497	+	0.924782i	$0.624247\pi$
$620$	15.3319	−	2.15410i	0.615742	−	0.0865106i
$621$	−5.21585			−0.209305
$622$			9.78123i			0.392192i
$623$			17.6637i			0.707681i
$624$		0			0
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$	−17.2128			−0.687961
$627$		−	15.5703i		−	0.621818i
$628$			1.98418i			0.0791773i
$629$	6.58274			0.262471
$630$	16.4494	−	2.31111i	0.655359	−	0.0920767i
$631$	−2.42864			−0.0966826			−0.0483413	−	0.998831i	$-0.515394\pi$
$631$	−2.42864			−0.0966826			−0.0483413	+	0.998831i	$0.515394\pi$
$632$		−	4.30174i		−	0.171114i
$633$		−	12.1684i		−	0.483650i
$634$	−13.7447			−0.545871
$635$	−3.10816	−	22.1225i	−0.123344	−	0.877903i
$636$	−3.31111			−0.131294
$637$		0			0
$638$			9.65433i			0.382219i
$639$	−29.4608			−1.16545
$640$	−0.311108	−	2.21432i	−0.0122976	−	0.0875287i
$641$	15.0874			0.595917			0.297959	−	0.954579i	$-0.403694\pi$
$641$	15.0874			0.595917			0.297959	+	0.954579i	$0.403694\pi$
$642$			4.90321i			0.193514i
$643$		−	6.70318i		−	0.264348i	−0.991227	−	0.132174i	$-0.957804\pi$
$643$		−	6.70318i		−	0.264348i	0.991227	−	0.132174i	$-0.0421957\pi$
$644$	−8.38271			−0.330325
$645$	9.80642	−	1.37778i	0.386128	−	0.0542502i
$646$	−8.64296			−0.340053
$647$			22.3733i			0.879587i	0.898099	+	0.439793i	$0.144948\pi$
$647$			22.3733i			0.879587i	−0.898099	+	0.439793i	$0.855052\pi$
$648$		−	11.0874i		−	0.435555i
$649$	−7.66323			−0.300808
$650$		0			0
$651$	59.8435			2.34545
$652$			4.70318i			0.184191i
$653$		−	16.4128i		−	0.642283i	−0.947031	−	0.321142i	$-0.895933\pi$
$653$		−	16.4128i		−	0.642283i	0.947031	−	0.321142i	$-0.104067\pi$
$654$	−0.295286			−0.0115466
$655$	−20.4953	+	2.87955i	−0.800818	+	0.112513i
$656$	5.95407			0.232467
$657$			29.2192i			1.13995i
$658$			0.377784i			0.0147276i
$659$	−8.66815			−0.337663			−0.168832	−	0.985645i	$-0.553999\pi$
$659$	−8.66815			−0.337663			−0.168832	+	0.985645i	$0.553999\pi$
$660$	0.734825	+	5.23014i	0.0286030	+	0.203583i
$661$	−17.4302			−0.677955			−0.338978	−	0.940794i	$-0.610081\pi$
$661$	−17.4302			−0.677955			−0.338978	+	0.940794i	$0.610081\pi$
$662$			19.3876i			0.753522i
$663$		0			0
$664$	−9.69381			−0.376193
$665$	8.00492	+	56.9753i	0.310418	+	2.20941i
$666$	9.55554			0.370270
$667$		−	19.4380i		−	0.752643i
$668$		−	9.69535i		−	0.375124i
$669$	−44.5007			−1.72050
$670$	−18.6637	+	2.62222i	−0.721042	+	0.101305i
$671$	−8.41329			−0.324792
$672$		−	8.64296i		−	0.333409i
$673$		−	40.4830i		−	1.56051i	−0.625464	−	0.780253i	$-0.715090\pi$
$673$		−	40.4830i		−	1.56051i	0.625464	−	0.780253i	$-0.284910\pi$
$674$	0.555539			0.0213986
$675$	−3.34614	−	11.6731i	−0.128793	−	0.449297i
$676$		0			0
$677$			40.0228i			1.53820i	0.639129	+	0.769100i	$0.279295\pi$
$677$			40.0228i			1.53820i	−0.639129	+	0.769100i	$0.720705\pi$
$678$		−	31.0005i		−	1.19057i
$679$	−16.6287			−0.638150
$680$	2.90321	−	0.407896i	0.111333	−	0.0156421i
$681$	−42.8671			−1.64267
$682$			7.38562i			0.282810i
$683$			43.3087i			1.65716i	0.559871	+	0.828580i	$0.310851\pi$
$683$			43.3087i			1.65716i	−0.559871	+	0.828580i	$0.689149\pi$
$684$	−12.5462			−0.479715
$685$	−4.60147	−	32.7511i	−0.175813	−	1.25136i
$686$	−4.82071			−0.184056
$687$		−	34.8212i		−	1.32851i
$688$		−	2.00000i		−	0.0762493i
$689$		0			0
$690$	−1.47949	−	10.5303i	−0.0563234	−	0.400884i
$691$	−28.3481			−1.07841			−0.539207	−	0.842173i	$-0.681276\pi$
$691$	−28.3481			−1.07841			−0.539207	+	0.842173i	$0.681276\pi$
$692$			24.0716i			0.915065i
$693$			7.92396i			0.301006i
$694$	6.68445			0.253738
$695$	−28.1082	+	3.94914i	−1.06620	+	0.149800i
$696$	20.0415			0.759671
$697$			7.80642i			0.295689i
$698$		−	10.5827i		−	0.400562i
$699$	32.9545			1.24646
$700$	−5.37778	−	18.7605i	−0.203261	−	0.709080i
$701$	12.8080			0.483750			0.241875	−	0.970307i	$-0.422238\pi$
$701$	12.8080			0.483750			0.241875	+	0.970307i	$0.422238\pi$
$702$		0			0
$703$			33.0973i			1.24829i
$704$	1.06668			0.0402019
$705$	−0.474572	+	0.0666765i	−0.0178734	+	0.00251118i
$706$	−9.20940			−0.346600
$707$			5.69381i			0.214138i
$708$			15.9081i			0.597864i
$709$	2.52251			0.0947350			0.0473675	−	0.998878i	$-0.484917\pi$
$709$	2.52251			0.0947350			0.0473675	+	0.998878i	$0.484917\pi$
$710$	4.81579	+	34.2766i	0.180733	+	1.28638i
$711$	8.18712			0.307041
$712$			4.52543i			0.169598i
$713$		−	14.8702i		−	0.556893i
$714$	11.3319			0.424084
$715$		0			0
$716$	3.05086			0.114016
$717$		−	9.79213i		−	0.365694i
$718$			4.19358i			0.156503i
$719$	−17.0223			−0.634824			−0.317412	−	0.948288i	$-0.602814\pi$
$719$	−17.0223			−0.634824			−0.317412	+	0.948288i	$0.602814\pi$
$720$	4.21432	−	0.592104i	0.157058	−	0.0220664i
$721$	−39.9862			−1.48916
$722$		−	24.4558i		−	0.910152i
$723$		−	24.9797i		−	0.929006i
$724$	−10.6430			−0.395542
$725$	43.5022	−	12.4701i	1.61563	−	0.463129i
$726$	21.8381			0.810487
$727$		−	30.5353i		−	1.13249i	−0.824237	−	0.566245i	$-0.808395\pi$
$727$		−	30.5353i		−	1.13249i	0.824237	−	0.566245i	$-0.191605\pi$
$728$		0			0
$729$	4.96836			0.184013
$730$	33.9956	−	4.77631i	1.25823	−	0.176779i
$731$	2.62222			0.0969861
$732$			17.4652i			0.645533i
$733$			24.4499i			0.903076i	0.892252	+	0.451538i	$0.149124\pi$
$733$			24.4499i			0.903076i	−0.892252	+	0.451538i	$0.850876\pi$
$734$	13.9684			0.515581
$735$	−5.67307	−	40.3783i	−0.209254	−	1.48937i
$736$	−2.14764			−0.0791632
$737$		−	8.99063i		−	0.331174i
$738$			11.3319i			0.417131i
$739$	−10.7955			−0.397120			−0.198560	−	0.980089i	$-0.563626\pi$
$739$	−10.7955			−0.397120			−0.198560	+	0.980089i	$0.563626\pi$
$740$	−1.56199	−	11.1175i	−0.0574200	−	0.408689i
$741$		0			0
$742$			5.83654i			0.214266i
$743$			29.9353i			1.09822i	0.835750	+	0.549110i	$0.185033\pi$
$743$			29.9353i			1.09822i	−0.835750	+	0.549110i	$0.814967\pi$
$744$	15.3319			0.562093
$745$	−16.1334	+	2.26671i	−0.591080	+	0.0830457i
$746$	−11.8479			−0.433783
$747$		−	18.4494i		−	0.675028i
$748$			1.39853i			0.0511352i
$749$	8.64296			0.315807
$750$	22.6178	−	10.0667i	0.825884	−	0.367583i
$751$	26.9175			0.982234			0.491117	−	0.871094i	$-0.336589\pi$
$751$	26.9175			0.982234			0.491117	+	0.871094i	$0.336589\pi$
$752$			0.0967881i			0.00352950i
$753$		−	5.87310i		−	0.214028i
$754$		0			0
$755$	44.5259	−	6.25581i	1.62046	−	0.227672i
$756$	−9.47949			−0.344766
$757$		−	35.9066i		−	1.30505i	−0.757768	−	0.652524i	$-0.773710\pi$
$757$		−	35.9066i		−	1.30505i	0.757768	−	0.652524i	$-0.226290\pi$
$758$		−	0.606394i		−	0.0220252i
$759$	5.07265			0.184126
$760$	2.05086	+	14.5970i	0.0743923	+	0.529490i
$761$	−22.5397			−0.817064			−0.408532	−	0.912744i	$-0.633959\pi$
$761$	−22.5397			−0.817064			−0.408532	+	0.912744i	$0.633959\pi$
$762$		−	22.1225i		−	0.801412i
$763$			0.520505i			0.0188436i
$764$	−14.7304			−0.532926
$765$	0.776312	+	5.52543i	0.0280676	+	0.199772i
$766$	−34.7797			−1.25664
$767$		0			0
$768$		−	2.21432i		−	0.0799024i
$769$	2.47457			0.0892354			0.0446177	−	0.999004i	$-0.485793\pi$
$769$	2.47457			0.0892354			0.0446177	+	0.999004i	$0.485793\pi$
$770$	9.21924	−	1.29529i	0.332238	−	0.0466789i
$771$	9.08250			0.327098
$772$			14.0000i			0.503871i
$773$		−	44.2005i		−	1.58978i	−0.606752	−	0.794891i	$-0.707528\pi$
$773$		−	44.2005i		−	1.58978i	0.606752	−	0.794891i	$-0.292472\pi$
$774$	3.80642			0.136819
$775$	33.2795	−	9.53972i	1.19543	−	0.342677i
$776$	−4.26025			−0.152934
$777$		−	43.3941i		−	1.55675i
$778$		−	18.6844i		−	0.669870i
$779$	−39.2498			−1.40627
$780$		0			0
$781$	−16.5116			−0.590832
$782$		−	2.81579i		−	0.100693i
$783$		−	21.9813i		−	0.785546i
$784$	−8.23506			−0.294109
$785$	0.617293	+	4.39361i	0.0220321	+	0.156815i
$786$	−20.4953			−0.731044
$787$		−	34.0306i		−	1.21306i	−0.795061	−	0.606530i	$-0.792561\pi$
$787$		−	34.0306i		−	1.21306i	0.795061	−	0.606530i	$-0.207439\pi$
$788$		−	10.4128i		−	0.370941i
$789$	23.9190			0.851540
$790$	−1.33830	−	9.52543i	−0.0476148	−	0.338900i
$791$	−54.6450			−1.94295
$792$			2.03011i			0.0721369i
$793$		0			0
$794$	−22.9748			−0.815346
$795$	−7.33185	+	1.03011i	−0.260034	+	0.0365343i
$796$	−4.36842			−0.154834
$797$			8.13780i			0.288256i	0.989559	+	0.144128i	$0.0460376\pi$
$797$			8.13780i			0.288256i	−0.989559	+	0.144128i	$0.953962\pi$
$798$			56.9753i			2.01690i
$799$	−0.126900			−0.00448939
$800$	−1.37778	−	4.80642i	−0.0487120	−	0.169933i
$801$	−8.61285			−0.304320
$802$		−	4.37334i		−	0.154428i
$803$			16.3763i			0.577905i
$804$	−18.6637			−0.658218
$805$	−18.5620	+	2.60793i	−0.654224	+	0.0919173i
$806$		0			0
$807$		−	41.1481i		−	1.44848i
$808$			1.45875i			0.0513187i
$809$	7.85236			0.276074			0.138037	−	0.990427i	$-0.455921\pi$
$809$	7.85236			0.276074			0.138037	+	0.990427i	$0.455921\pi$
$810$	−3.44938	−	24.5511i	−0.121199	−	0.862637i
$811$	−9.33477			−0.327788			−0.163894	−	0.986478i	$-0.552405\pi$
$811$	−9.33477			−0.327788			−0.163894	+	0.986478i	$0.552405\pi$
$812$		−	35.3274i		−	1.23975i
$813$			43.7877i			1.53570i
$814$	5.35551			0.187711
$815$	1.46320	+	10.4143i	0.0512535	+	0.364799i
$816$	2.90321			0.101633
$817$			13.1842i			0.461257i
$818$			3.76986i			0.131810i
$819$		0			0
$820$	13.1842	−	1.85236i	0.460413	−	0.0646871i
$821$	47.5319			1.65887			0.829437	−	0.558600i	$-0.188661\pi$
$821$	47.5319			1.65887			0.829437	+	0.558600i	$0.188661\pi$
$822$		−	32.7511i		−	1.14233i
$823$			1.38715i			0.0483531i	0.999708	+	0.0241765i	$0.00769638\pi$
$823$			1.38715i			0.0483531i	−0.999708	+	0.0241765i	$0.992304\pi$
$824$	−10.2444			−0.356882
$825$	3.25428	+	11.3526i	0.113299	+	0.395247i
$826$	28.0415			0.975688
$827$			31.5131i			1.09582i	0.836537	+	0.547910i	$0.184576\pi$
$827$			31.5131i			1.09582i	−0.836537	+	0.547910i	$0.815424\pi$
$828$		−	4.08742i		−	0.142048i
$829$	28.8256			1.00116			0.500578	−	0.865692i	$-0.333121\pi$
$829$	28.8256			1.00116			0.500578	+	0.865692i	$0.333121\pi$
$830$	−21.4652	+	3.01582i	−0.745068	+	0.104681i
$831$	0.453829			0.0157431
$832$		0			0
$833$		−	10.7971i		−	0.374096i
$834$	−28.1082			−0.973306
$835$	−3.01630	−	21.4686i	−0.104383	−	0.742951i
$836$	−7.03164			−0.243194
$837$		−	16.8158i		−	0.581239i
$838$			7.45230i			0.257435i
$839$	8.43509			0.291212			0.145606	−	0.989343i	$-0.453487\pi$
$839$	8.43509			0.291212			0.145606	+	0.989343i	$0.453487\pi$
$840$	−2.68889	−	19.1383i	−0.0927756	−	0.660333i
$841$	52.9180			1.82476
$842$			27.6751i			0.953746i
$843$		−	45.2958i		−	1.56007i
$844$	−5.49532			−0.189157
$845$		0			0
$846$	−0.184208			−0.00633321
$847$		−	38.4943i		−	1.32268i
$848$			1.49532i			0.0513494i
$849$	−3.18421			−0.109282
$850$	6.30174	−	1.80642i	0.216148	−	0.0619598i
$851$	−10.7828			−0.369628
$852$			34.2766i			1.17430i
$853$		−	40.0656i		−	1.37182i	−0.727686	−	0.685910i	$-0.759404\pi$
$853$		−	40.0656i		−	1.37182i	0.727686	−	0.685910i	$-0.240596\pi$
$854$	30.7862			1.05348
$855$	−27.7812	+	3.90321i	−0.950098	+	0.133487i
$856$	2.21432			0.0756839
$857$			32.0479i			1.09474i	0.836892	+	0.547368i	$0.184370\pi$
$857$			32.0479i			1.09474i	−0.836892	+	0.547368i	$0.815630\pi$
$858$		0			0
$859$	26.6894			0.910630			0.455315	−	0.890331i	$-0.349527\pi$
$859$	26.6894			0.910630			0.455315	+	0.890331i	$0.349527\pi$
$860$	−0.622216	−	4.42864i	−0.0212174	−	0.151015i
$861$	51.4608			1.75378
$862$			19.6894i			0.670623i
$863$			30.6593i			1.04365i	0.853052	+	0.521827i	$0.174749\pi$
$863$			30.6593i			1.04365i	−0.853052	+	0.521827i	$0.825251\pi$
$864$	−2.42864			−0.0826240
$865$	7.48886	+	53.3022i	0.254629	+	1.81233i
$866$	13.7462			0.467115
$867$		−	33.8370i		−	1.14917i
$868$		−	27.0257i		−	0.917311i
$869$	4.58857			0.155656
$870$	44.3783	−	6.23506i	1.50456	−	0.211388i
$871$		0			0
$872$			0.133353i			0.00451591i
$873$		−	8.10816i		−	0.274420i
$874$	14.1575			0.478884
$875$	−17.7447	−	39.8687i	−0.599879	−	1.34781i
$876$	33.9956			1.14860
$877$		−	20.9304i		−	0.706770i	−0.935478	−	0.353385i	$-0.885031\pi$
$877$		−	20.9304i		−	0.706770i	0.935478	−	0.353385i	$-0.114969\pi$
$878$			41.8227i			1.41145i
$879$	−46.3116			−1.56205
$880$	2.36196	−	0.331851i	0.0796218	−	0.0111867i
$881$	21.4701			0.723347			0.361673	−	0.932305i	$-0.382206\pi$
$881$	21.4701			0.723347			0.361673	+	0.932305i	$0.382206\pi$
$882$		−	15.6731i		−	0.527740i
$883$			7.73530i			0.260314i	0.991493	+	0.130157i	$0.0415481\pi$
$883$			7.73530i			0.260314i	−0.991493	+	0.130157i	$0.958452\pi$
$884$		0			0
$885$	4.94914	+	35.2257i	0.166364	+	1.18410i
$886$	23.6829			0.795643
$887$		−	48.4465i		−	1.62667i	−0.581793	−	0.813337i	$-0.697649\pi$
$887$		−	48.4465i		−	1.62667i	0.581793	−	0.813337i	$-0.302351\pi$
$888$		−	11.1175i		−	0.373080i
$889$	−38.9956			−1.30787
$890$	1.40790	+	10.0207i	0.0471928	+	0.335896i
$891$	11.8267			0.396209
$892$			20.0968i			0.672890i
$893$		−	0.638037i		−	0.0213511i
$894$	−16.1334			−0.539580
$895$	6.75557	−	0.949145i	0.225814	−	0.0317264i
$896$	−3.90321			−0.130397
$897$		0			0
$898$		−	7.05578i		−	0.235454i
$899$	62.6677			2.09009
$900$	9.14764	−	2.62222i	0.304921	−	0.0874072i
$901$	−1.96052			−0.0653144
$902$			6.35106i			0.211467i
$903$		−	17.2859i		−	0.575239i
$904$	−14.0000			−0.465633
$905$	−23.5669	+	3.31111i	−0.783391	+	0.110065i
$906$	44.5259			1.47927
$907$			12.0109i			0.398815i	0.979917	+	0.199408i	$0.0639018\pi$
$907$			12.0109i			0.398815i	−0.979917	+	0.199408i	$0.936098\pi$
$908$			19.3590i			0.642453i
$909$	−2.77631			−0.0920845
$910$		0			0
$911$	−8.10171			−0.268422			−0.134211	−	0.990953i	$-0.542850\pi$
$911$	−8.10171			−0.268422			−0.134211	+	0.990953i	$0.542850\pi$
$912$			14.5970i			0.483356i
$913$		−	10.3402i		−	0.342209i
$914$	35.6795			1.18017
$915$	5.43356	+	38.6735i	0.179628	+	1.27851i
$916$	−15.7255			−0.519584
$917$			36.1274i			1.19303i
$918$		−	3.18421i		−	0.105095i
$919$	−16.5936			−0.547374			−0.273687	−	0.961819i	$-0.588243\pi$
$919$	−16.5936			−0.547374			−0.273687	+	0.961819i	$0.588243\pi$
$920$	−4.75557	+	0.668149i	−0.156786	+	0.0220282i
$921$	−27.7462			−0.914268
$922$		−	14.6015i		−	0.480874i
$923$		0			0
$924$	9.21924			0.303291
$925$	−6.91750	−	24.1318i	−0.227446	−	0.793449i
$926$	30.7511			1.01054
$927$		−	19.4973i		−	0.640376i
$928$		−	9.05086i		−	0.297109i
$929$	57.1209			1.87408			0.937038	−	0.349227i	$-0.113556\pi$
$929$	57.1209			1.87408			0.937038	+	0.349227i	$0.113556\pi$
$930$	33.9496	−	4.76986i	1.11325	−	0.156410i
$931$	54.2864			1.77916
$932$		−	14.8825i		−	0.487491i
$933$			21.6588i			0.709077i
$934$	−20.5620			−0.672809
$935$	0.435093	+	3.09679i	0.0142291	+	0.101276i
$936$		0			0
$937$		−	11.1842i		−	0.365372i	−0.983171	−	0.182686i	$-0.941521\pi$
$937$		−	11.1842i		−	0.365372i	0.983171	−	0.182686i	$-0.0584792\pi$
$938$			32.8988i			1.07418i
$939$	−38.1146			−1.24382
$940$	0.0301115	+	0.214320i	0.000982130	+	0.00699034i
$941$	7.68598			0.250556			0.125278	−	0.992122i	$-0.460018\pi$
$941$	7.68598			0.250556			0.125278	+	0.992122i	$0.460018\pi$
$942$			4.39361i			0.143151i
$943$		−	12.7872i		−	0.416409i
$944$	7.18421			0.233826
$945$	−20.9906	+	2.94914i	−0.682825	+	0.0959357i
$946$	2.13335			0.0693613
$947$		−	20.6746i		−	0.671834i	−0.941892	−	0.335917i	$-0.890954\pi$
$947$		−	20.6746i		−	0.671834i	0.941892	−	0.335917i	$-0.109046\pi$
$948$		−	9.52543i		−	0.309372i
$949$		0			0
$950$	9.08250	+	31.6844i	0.294675	+	1.02798i
$951$	−30.4351			−0.986926
$952$		−	5.11753i		−	0.165860i
$953$			30.9753i			1.00339i	0.865045	+	0.501694i	$0.167290\pi$
$953$			30.9753i			1.00339i	−0.865045	+	0.501694i	$0.832710\pi$
$954$	−2.84590			−0.0921395
$955$	−32.6178	+	4.58274i	−1.05549	+	0.148294i
$956$	−4.42219			−0.143024
$957$			21.3778i			0.691046i
$958$		−	11.9333i		−	0.385548i
$959$	−57.7309			−1.86423
$960$	−0.688892	−	4.90321i	−0.0222339	−	0.158250i
$961$	16.9412			0.546489
$962$		0			0
$963$			4.21432i			0.135805i
$964$	−11.2810			−0.363336
$965$	4.35551	+	31.0005i	0.140209	+	0.997941i
$966$	−18.5620			−0.597222
$967$			0.529873i			0.0170396i	0.999964	+	0.00851979i	$0.00271197\pi$
$967$			0.529873i			0.0170396i	−0.999964	+	0.00851979i	$0.997288\pi$
$968$		−	9.86220i		−	0.316983i
$969$	−19.1383			−0.614810
$970$	−9.43356	+	1.32540i	−0.302893	+	0.0425560i
$971$	−34.3240			−1.10151			−0.550755	−	0.834667i	$-0.685660\pi$
$971$	−34.3240			−1.10151			−0.550755	+	0.834667i	$0.685660\pi$
$972$		−	17.2652i		−	0.553781i
$973$			49.5466i			1.58839i
$974$	27.9719			0.896277
$975$		0			0
$976$	7.88739			0.252469
$977$		−	19.9684i		−	0.638844i	−0.947613	−	0.319422i	$-0.896511\pi$
$977$		−	19.9684i		−	0.638844i	0.947613	−	0.319422i	$-0.103489\pi$
$978$			10.4143i			0.333014i
$979$	−4.82717			−0.154277
$980$	−18.2351	+	2.56199i	−0.582498	+	0.0818399i
$981$	−0.253799			−0.00810318
$982$			20.7812i			0.663156i
$983$		−	20.8524i		−	0.665087i	−0.943088	−	0.332543i	$-0.892093\pi$
$983$		−	20.8524i		−	0.665087i	0.943088	−	0.332543i	$-0.107907\pi$
$984$	13.1842			0.420297
$985$	−3.23951	−	23.0573i	−0.103219	−	0.734667i
$986$	11.8666			0.377911
$987$			0.836535i			0.0266272i
$988$		0			0
$989$	−4.29529			−0.136582
$990$	0.631584	+	4.49532i	0.0200730	+	0.142871i
$991$	−2.95899			−0.0939954			−0.0469977	−	0.998895i	$-0.514965\pi$
$991$	−2.95899			−0.0939954			−0.0469977	+	0.998895i	$0.514965\pi$
$992$		−	6.92396i		−	0.219836i
$993$			42.9304i			1.36236i
$994$	60.4197			1.91640
$995$	−9.67307	+	1.35905i	−0.306657	+	0.0430847i
$996$	−21.4652			−0.680151
$997$		−	21.0479i		−	0.666595i	−0.942822	−	0.333297i	$-0.891839\pi$
$997$		−	21.0479i		−	0.666595i	0.942822	−	0.333297i	$-0.108161\pi$
$998$		−	34.2908i		−	1.08546i
$999$	−12.1936			−0.385788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.b.b.339.1		6
5.2	odd	4		8450.2.a.ca.1.1		3
5.3	odd	4		8450.2.a.bt.1.3		3
5.4	even	2	inner	1690.2.b.b.339.6		6
13.4	even	6		130.2.n.a.29.4	yes	12
13.5	odd	4		1690.2.c.b.1689.1		6
13.8	odd	4		1690.2.c.c.1689.1		6
13.10	even	6		130.2.n.a.9.3	✓	12
13.12	even	2		1690.2.b.c.339.4		6
39.17	odd	6		1170.2.bp.h.289.1		12
39.23	odd	6		1170.2.bp.h.919.4		12
52.23	odd	6		1040.2.dh.b.529.1		12
52.43	odd	6		1040.2.dh.b.289.6		12
65.4	even	6		130.2.n.a.29.3	yes	12
65.12	odd	4		8450.2.a.bu.1.1		3
65.17	odd	12		650.2.e.k.601.3		6
65.23	odd	12		650.2.e.j.451.1		6
65.34	odd	4		1690.2.c.b.1689.6		6
65.38	odd	4		8450.2.a.cb.1.3		3
65.43	odd	12		650.2.e.j.601.1		6
65.44	odd	4		1690.2.c.c.1689.6		6
65.49	even	6		130.2.n.a.9.4	yes	12
65.62	odd	12		650.2.e.k.451.3		6
65.64	even	2		1690.2.b.c.339.3		6
195.134	odd	6		1170.2.bp.h.289.4		12
195.179	odd	6		1170.2.bp.h.919.1		12
260.179	odd	6		1040.2.dh.b.529.6		12
260.199	odd	6		1040.2.dh.b.289.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.n.a.9.3	✓	12	13.10	even	6
130.2.n.a.9.4	yes	12	65.49	even	6
130.2.n.a.29.3	yes	12	65.4	even	6
130.2.n.a.29.4	yes	12	13.4	even	6
650.2.e.j.451.1		6	65.23	odd	12
650.2.e.j.601.1		6	65.43	odd	12
650.2.e.k.451.3		6	65.62	odd	12
650.2.e.k.601.3		6	65.17	odd	12
1040.2.dh.b.289.1		12	260.199	odd	6
1040.2.dh.b.289.6		12	52.43	odd	6
1040.2.dh.b.529.1		12	52.23	odd	6
1040.2.dh.b.529.6		12	260.179	odd	6
1170.2.bp.h.289.1		12	39.17	odd	6
1170.2.bp.h.289.4		12	195.134	odd	6
1170.2.bp.h.919.1		12	195.179	odd	6
1170.2.bp.h.919.4		12	39.23	odd	6
1690.2.b.b.339.1		6	1.1	even	1	trivial
1690.2.b.b.339.6		6	5.4	even	2	inner
1690.2.b.c.339.3		6	65.64	even	2
1690.2.b.c.339.4		6	13.12	even	2
1690.2.c.b.1689.1		6	13.5	odd	4
1690.2.c.b.1689.6		6	65.34	odd	4
1690.2.c.c.1689.1		6	13.8	odd	4
1690.2.c.c.1689.6		6	65.44	odd	4
8450.2.a.bt.1.3		3	5.3	odd	4
8450.2.a.bu.1.1		3	65.12	odd	4
8450.2.a.ca.1.1		3	5.2	odd	4
8450.2.a.cb.1.3		3	65.38	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -2.21432i q^{3} -1.00000 q^{4} +(-2.21432 + 0.311108i) q^{5} -2.21432 q^{6} +3.90321i q^{7} +1.00000i q^{8} -1.90321 q^{9} +(0.311108 + 2.21432i) q^{10} -1.06668 q^{11} +2.21432i q^{12} +3.90321 q^{14} +(0.688892 + 4.90321i) q^{15} +1.00000 q^{16} +1.31111i q^{17} +1.90321i q^{18} -6.59210 q^{19} +(2.21432 - 0.311108i) q^{20} +8.64296 q^{21} +1.06668i q^{22} -2.14764i q^{23} +2.21432 q^{24} +(4.80642 - 1.37778i) q^{25} -2.42864i q^{27} -3.90321i q^{28} +9.05086 q^{29} +(4.90321 - 0.688892i) q^{30} +6.92396 q^{31} -1.00000i q^{32} +2.36196i q^{33} +1.31111 q^{34} +(-1.21432 - 8.64296i) q^{35} +1.90321 q^{36} -5.02074i q^{37} +6.59210i q^{38} +(-0.311108 - 2.21432i) q^{40} +5.95407 q^{41} -8.64296i q^{42} -2.00000i q^{43} +1.06668 q^{44} +(4.21432 - 0.592104i) q^{45} -2.14764 q^{46} +0.0967881i q^{47} -2.21432i q^{48} -8.23506 q^{49} +(-1.37778 - 4.80642i) q^{50} +2.90321 q^{51} +1.49532i q^{53} -2.42864 q^{54} +(2.36196 - 0.331851i) q^{55} -3.90321 q^{56} +14.5970i q^{57} -9.05086i q^{58} +7.18421 q^{59} +(-0.688892 - 4.90321i) q^{60} +7.88739 q^{61} -6.92396i q^{62} -7.42864i q^{63} -1.00000 q^{64} +2.36196 q^{66} +8.42864i q^{67} -1.31111i q^{68} -4.75557 q^{69} +(-8.64296 + 1.21432i) q^{70} +15.4795 q^{71} -1.90321i q^{72} -15.3526i q^{73} -5.02074 q^{74} +(-3.05086 - 10.6430i) q^{75} +6.59210 q^{76} -4.16346i q^{77} -4.30174 q^{79} +(-2.21432 + 0.311108i) q^{80} -11.0874 q^{81} -5.95407i q^{82} +9.69381i q^{83} -8.64296 q^{84} +(-0.407896 - 2.90321i) q^{85} -2.00000 q^{86} -20.0415i q^{87} -1.06668i q^{88} +4.52543 q^{89} +(-0.592104 - 4.21432i) q^{90} +2.14764i q^{92} -15.3319i q^{93} +0.0967881 q^{94} +(14.5970 - 2.05086i) q^{95} -2.21432 q^{96} +4.26025i q^{97} +8.23506i q^{98} +2.03011 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} + 2 q^{9} + 2 q^{10} - 6 q^{11} + 10 q^{14} + 4 q^{15} + 6 q^{16} - 26 q^{19} + 12 q^{21} + 2 q^{25} + 28 q^{29} + 16 q^{30} - 12 q^{31} + 8 q^{34} + 6 q^{35} - 2 q^{36} - 2 q^{40} - 4 q^{41}+ \cdots + 26 q^{99}+O(q^{100}) \) 6 * q - 6 * q^4 + 2 * q^9 + 2 * q^10 - 6 * q^11 + 10 * q^14 + 4 * q^15 + 6 * q^16 - 26 * q^19 + 12 * q^21 + 2 * q^25 + 28 * q^29 + 16 * q^30 - 12 * q^31 + 8 * q^34 + 6 * q^35 - 2 * q^36 - 2 * q^40 - 4 * q^41 + 6 * q^44 + 12 * q^45 + 4 * q^49 - 8 * q^50 + 4 * q^51 + 12 * q^54 - 12 * q^55 - 10 * q^56 + 16 * q^59 - 4 * q^60 + 8 * q^61 - 6 * q^64 - 12 * q^66 - 28 * q^69 - 12 * q^70 + 40 * q^71 + 10 * q^74 + 8 * q^75 + 26 * q^76 + 28 * q^79 - 26 * q^81 - 12 * q^84 - 16 * q^85 - 12 * q^86 + 14 * q^89 + 10 * q^90 + 14 * q^94 + 8 * q^95 + 26 * q^99

Introduction

L-functions

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