LMFDB - Embedded newform 169.2.c.a.146.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,2,Mod(22,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("169.22"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	169.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$1.34947179416$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			146.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	169.146
Dual form			169.2.c.a.22.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.866025 - 1.50000i) q^{2} +(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{4} -1.73205 q^{5} +(-1.73205 + 3.00000i) q^{6} -1.73205 q^{8} +(-0.500000 + 0.866025i) q^{9} +(1.50000 + 2.59808i) q^{10} +2.00000 q^{12} +(1.73205 + 3.00000i) q^{15} +(2.50000 + 4.33013i) q^{16} +(-1.50000 + 2.59808i) q^{17} +1.73205 q^{18} +(1.73205 - 3.00000i) q^{19} +(0.866025 - 1.50000i) q^{20} +(-3.00000 - 5.19615i) q^{23} +(1.73205 + 3.00000i) q^{24} -2.00000 q^{25} -4.00000 q^{27} +(-1.50000 - 2.59808i) q^{29} +(3.00000 - 5.19615i) q^{30} +3.46410 q^{31} +(2.59808 - 4.50000i) q^{32} +5.19615 q^{34} +(-0.500000 - 0.866025i) q^{36} +(-4.33013 - 7.50000i) q^{37} -6.00000 q^{38} +3.00000 q^{40} +(-2.59808 - 4.50000i) q^{41} +(4.00000 - 6.92820i) q^{43} +(0.866025 - 1.50000i) q^{45} +(-5.19615 + 9.00000i) q^{46} +3.46410 q^{47} +(5.00000 - 8.66025i) q^{48} +(3.50000 + 6.06218i) q^{49} +(1.73205 + 3.00000i) q^{50} +6.00000 q^{51} -3.00000 q^{53} +(3.46410 + 6.00000i) q^{54} -6.92820 q^{57} +(-2.59808 + 4.50000i) q^{58} +(3.46410 - 6.00000i) q^{59} -3.46410 q^{60} +(-0.500000 + 0.866025i) q^{61} +(-3.00000 - 5.19615i) q^{62} +1.00000 q^{64} +(1.73205 + 3.00000i) q^{67} +(-1.50000 - 2.59808i) q^{68} +(-6.00000 + 10.3923i) q^{69} +(-1.73205 + 3.00000i) q^{71} +(0.866025 - 1.50000i) q^{72} -1.73205 q^{73} +(-7.50000 + 12.9904i) q^{74} +(2.00000 + 3.46410i) q^{75} +(1.73205 + 3.00000i) q^{76} +4.00000 q^{79} +(-4.33013 - 7.50000i) q^{80} +(5.50000 + 9.52628i) q^{81} +(-4.50000 + 7.79423i) q^{82} +13.8564 q^{83} +(2.59808 - 4.50000i) q^{85} -13.8564 q^{86} +(-3.00000 + 5.19615i) q^{87} +(3.46410 + 6.00000i) q^{89} -3.00000 q^{90} +6.00000 q^{92} +(-3.46410 - 6.00000i) q^{93} +(-3.00000 - 5.19615i) q^{94} +(-3.00000 + 5.19615i) q^{95} -10.3923 q^{96} +(-3.46410 + 6.00000i) q^{97} +(6.06218 - 10.5000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 2 q^{4} - 2 q^{9} + 6 q^{10} + 8 q^{12} + 10 q^{16} - 6 q^{17} - 12 q^{23} - 8 q^{25} - 16 q^{27} - 6 q^{29} + 12 q^{30} - 2 q^{36} - 24 q^{38} + 12 q^{40} + 16 q^{43} + 20 q^{48} + 14 q^{49}+ \cdots - 12 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 - 2 * q^4 - 2 * q^9 + 6 * q^10 + 8 * q^12 + 10 * q^16 - 6 * q^17 - 12 * q^23 - 8 * q^25 - 16 * q^27 - 6 * q^29 + 12 * q^30 - 2 * q^36 - 24 * q^38 + 12 * q^40 + 16 * q^43 + 20 * q^48 + 14 * q^49 + 24 * q^51 - 12 * q^53 - 2 * q^61 - 12 * q^62 + 4 * q^64 - 6 * q^68 - 24 * q^69 - 30 * q^74 + 8 * q^75 + 16 * q^79 + 22 * q^81 - 18 * q^82 - 12 * q^87 - 12 * q^90 + 24 * q^92 - 12 * q^94 - 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−0.866025	−	1.50000i	−0.612372	−	1.06066i	−0.990839	−	0.135045i	$-0.956882\pi$
$2$	−0.866025	−	1.50000i	−0.612372	−	1.06066i	0.378467	−	0.925615i	$-0.376451\pi$
$3$	−1.00000	−	1.73205i	−0.577350	−	1.00000i	−0.995782	−	0.0917517i	$-0.970753\pi$
$3$	−1.00000	−	1.73205i	−0.577350	−	1.00000i	0.418432	−	0.908248i	$-0.362580\pi$
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	−1.73205			−0.774597			−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205			−0.774597			−0.387298	+	0.921954i	$0.626592\pi$
$6$	−1.73205	+	3.00000i	−0.707107	+	1.22474i
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$	−1.73205			−0.612372
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$	1.50000	+	2.59808i	0.474342	+	0.821584i
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$	2.00000			0.577350
$13$		0			0
$13$		0			0
$14$		0			0
$15$	1.73205	+	3.00000i	0.447214	+	0.774597i
$16$	2.50000	+	4.33013i	0.625000	+	1.08253i
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	−0.988583	−	0.150675i	$-0.951855\pi$
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	0.624780	+	0.780801i	$0.285189\pi$
$18$	1.73205			0.408248
$19$	1.73205	−	3.00000i	0.397360	−	0.688247i	−0.596040	−	0.802955i	$-0.703260\pi$
$19$	1.73205	−	3.00000i	0.397360	−	0.688247i	0.993399	+	0.114708i	$0.0365932\pi$
$20$	0.866025	−	1.50000i	0.193649	−	0.335410i
$21$		0			0
$22$		0			0
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$	1.73205	+	3.00000i	0.353553	+	0.612372i
$25$	−2.00000			−0.400000
$26$		0			0
$27$	−4.00000			−0.769800
$28$		0			0
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	0.692480	−	0.721437i	$-0.256518\pi$
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	−0.971023	+	0.238987i	$0.923185\pi$
$30$	3.00000	−	5.19615i	0.547723	−	0.948683i
$31$	3.46410			0.622171			0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410			0.622171			0.311086	+	0.950382i	$0.399307\pi$
$32$	2.59808	−	4.50000i	0.459279	−	0.795495i
$33$		0			0
$34$	5.19615			0.891133
$35$		0			0
$36$	−0.500000	−	0.866025i	−0.0833333	−	0.144338i
$37$	−4.33013	−	7.50000i	−0.711868	−	1.23299i	−0.964155	−	0.265340i	$-0.914516\pi$
$37$	−4.33013	−	7.50000i	−0.711868	−	1.23299i	0.252286	−	0.967653i	$-0.418817\pi$
$38$	−6.00000			−0.973329
$39$		0			0
$40$	3.00000			0.474342
$41$	−2.59808	−	4.50000i	−0.405751	−	0.702782i	0.588657	−	0.808383i	$-0.299657\pi$
$41$	−2.59808	−	4.50000i	−0.405751	−	0.702782i	−0.994409	+	0.105601i	$0.966323\pi$
$42$		0			0
$43$	4.00000	−	6.92820i	0.609994	−	1.05654i	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	4.00000	−	6.92820i	0.609994	−	1.05654i	0.991241	−	0.132068i	$-0.0421616\pi$
$44$		0			0
$45$	0.866025	−	1.50000i	0.129099	−	0.223607i
$46$	−5.19615	+	9.00000i	−0.766131	+	1.32698i
$47$	3.46410			0.505291			0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410			0.505291			0.252646	+	0.967559i	$0.418699\pi$
$48$	5.00000	−	8.66025i	0.721688	−	1.25000i
$49$	3.50000	+	6.06218i	0.500000	+	0.866025i
$50$	1.73205	+	3.00000i	0.244949	+	0.424264i
$51$	6.00000			0.840168
$52$		0			0
$53$	−3.00000			−0.412082			−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000			−0.412082			−0.206041	+	0.978543i	$0.566058\pi$
$54$	3.46410	+	6.00000i	0.471405	+	0.816497i
$55$		0			0
$56$		0			0
$57$	−6.92820			−0.917663
$58$	−2.59808	+	4.50000i	−0.341144	+	0.590879i
$59$	3.46410	−	6.00000i	0.450988	−	0.781133i	−0.547460	−	0.836832i	$-0.684405\pi$
$59$	3.46410	−	6.00000i	0.450988	−	0.781133i	0.998448	+	0.0556984i	$0.0177385\pi$
$60$	−3.46410			−0.447214
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	0.832240	+	0.554416i	$0.187058\pi$
$62$	−3.00000	−	5.19615i	−0.381000	−	0.659912i
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	1.73205	+	3.00000i	0.211604	+	0.366508i	0.952217	−	0.305424i	$-0.0987981\pi$
$67$	1.73205	+	3.00000i	0.211604	+	0.366508i	−0.740613	+	0.671932i	$0.765465\pi$
$68$	−1.50000	−	2.59808i	−0.181902	−	0.315063i
$69$	−6.00000	+	10.3923i	−0.722315	+	1.25109i
$70$		0			0
$71$	−1.73205	+	3.00000i	−0.205557	+	0.356034i	−0.950310	−	0.311305i	$-0.899234\pi$
$71$	−1.73205	+	3.00000i	−0.205557	+	0.356034i	0.744753	+	0.667340i	$0.232567\pi$
$72$	0.866025	−	1.50000i	0.102062	−	0.176777i
$73$	−1.73205			−0.202721			−0.101361	−	0.994850i	$-0.532320\pi$
$73$	−1.73205			−0.202721			−0.101361	+	0.994850i	$0.532320\pi$
$74$	−7.50000	+	12.9904i	−0.871857	+	1.51010i
$75$	2.00000	+	3.46410i	0.230940	+	0.400000i
$76$	1.73205	+	3.00000i	0.198680	+	0.344124i
$77$		0			0
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$	−4.33013	−	7.50000i	−0.484123	−	0.838525i
$81$	5.50000	+	9.52628i	0.611111	+	1.05848i
$82$	−4.50000	+	7.79423i	−0.496942	+	0.860729i
$83$	13.8564			1.52094			0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564			1.52094			0.760469	+	0.649374i	$0.224969\pi$
$84$		0			0
$85$	2.59808	−	4.50000i	0.281801	−	0.488094i
$86$	−13.8564			−1.49417
$87$	−3.00000	+	5.19615i	−0.321634	+	0.557086i
$88$		0			0
$89$	3.46410	+	6.00000i	0.367194	+	0.635999i	0.989126	−	0.147073i	$-0.0469852\pi$
$89$	3.46410	+	6.00000i	0.367194	+	0.635999i	−0.621932	+	0.783072i	$0.713652\pi$
$90$	−3.00000			−0.316228
$91$		0			0
$92$	6.00000			0.625543
$93$	−3.46410	−	6.00000i	−0.359211	−	0.622171i
$94$	−3.00000	−	5.19615i	−0.309426	−	0.535942i
$95$	−3.00000	+	5.19615i	−0.307794	+	0.533114i
$96$	−10.3923			−1.06066
$97$	−3.46410	+	6.00000i	−0.351726	+	0.609208i	−0.986552	−	0.163448i	$-0.947739\pi$
$97$	−3.46410	+	6.00000i	−0.351726	+	0.609208i	0.634826	+	0.772655i	$0.281072\pi$
$98$	6.06218	−	10.5000i	0.612372	−	1.06066i
$99$		0			0
$100$	1.00000	−	1.73205i	0.100000	−	0.173205i
$101$	−1.50000	−	2.59808i	−0.149256	−	0.258518i	0.781697	−	0.623658i	$-0.214354\pi$
$101$	−1.50000	−	2.59808i	−0.149256	−	0.258518i	−0.930953	+	0.365140i	$0.881021\pi$
$102$	−5.19615	−	9.00000i	−0.514496	−	0.891133i
$103$	10.0000			0.985329			0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000			0.985329			0.492665	+	0.870219i	$0.336023\pi$
$104$		0			0
$105$		0			0
$106$	2.59808	+	4.50000i	0.252347	+	0.437079i
$107$	−3.00000	−	5.19615i	−0.290021	−	0.502331i	0.683793	−	0.729676i	$-0.260329\pi$
$107$	−3.00000	−	5.19615i	−0.290021	−	0.502331i	−0.973814	+	0.227345i	$0.926996\pi$
$108$	2.00000	−	3.46410i	0.192450	−	0.333333i
$109$	−13.8564			−1.32720			−0.663602	−	0.748086i	$-0.730973\pi$
$109$	−13.8564			−1.32720			−0.663602	+	0.748086i	$0.730973\pi$
$110$		0			0
$111$	−8.66025	+	15.0000i	−0.821995	+	1.42374i
$112$		0			0
$113$	7.50000	−	12.9904i	0.705541	−	1.22203i	−0.260955	−	0.965351i	$-0.584038\pi$
$113$	7.50000	−	12.9904i	0.705541	−	1.22203i	0.966496	−	0.256681i	$-0.0826291\pi$
$114$	6.00000	+	10.3923i	0.561951	+	0.973329i
$115$	5.19615	+	9.00000i	0.484544	+	0.839254i
$116$	3.00000			0.278543
$117$		0			0
$118$	−12.0000			−1.10469
$119$		0			0
$120$	−3.00000	−	5.19615i	−0.273861	−	0.474342i
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$	1.73205			0.156813
$123$	−5.19615	+	9.00000i	−0.468521	+	0.811503i
$124$	−1.73205	+	3.00000i	−0.155543	+	0.269408i
$125$	12.1244			1.08444
$126$		0			0
$127$	−1.00000	−	1.73205i	−0.0887357	−	0.153695i	0.818241	−	0.574875i	$-0.194949\pi$
$127$	−1.00000	−	1.73205i	−0.0887357	−	0.153695i	−0.906977	+	0.421180i	$0.861616\pi$
$128$	−6.06218	−	10.5000i	−0.535826	−	0.928078i
$129$	−16.0000			−1.40872
$130$		0			0
$131$	18.0000			1.57267			0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000			1.57267			0.786334	+	0.617802i	$0.211977\pi$
$132$		0			0
$133$		0			0
$134$	3.00000	−	5.19615i	0.259161	−	0.448879i
$135$	6.92820			0.596285
$136$	2.59808	−	4.50000i	0.222783	−	0.385872i
$137$	−7.79423	+	13.5000i	−0.665906	+	1.15338i	0.313133	+	0.949709i	$0.398621\pi$
$137$	−7.79423	+	13.5000i	−0.665906	+	1.15338i	−0.979039	+	0.203674i	$0.934712\pi$
$138$	20.7846			1.76930
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	−0.768655	−	0.639664i	$-0.779074\pi$
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	0.938293	+	0.345843i	$0.112407\pi$
$140$		0			0
$141$	−3.46410	−	6.00000i	−0.291730	−	0.505291i
$142$	6.00000			0.503509
$143$		0			0
$144$	−5.00000			−0.416667
$145$	2.59808	+	4.50000i	0.215758	+	0.373705i
$146$	1.50000	+	2.59808i	0.124141	+	0.215018i
$147$	7.00000	−	12.1244i	0.577350	−	1.00000i
$148$	8.66025			0.711868
$149$	9.52628	−	16.5000i	0.780423	−	1.35173i	−0.151272	−	0.988492i	$-0.548337\pi$
$149$	9.52628	−	16.5000i	0.780423	−	1.35173i	0.931695	−	0.363241i	$-0.118330\pi$
$150$	3.46410	−	6.00000i	0.282843	−	0.489898i
$151$	−17.3205			−1.40952			−0.704761	−	0.709444i	$-0.748946\pi$
$151$	−17.3205			−1.40952			−0.704761	+	0.709444i	$0.748946\pi$
$152$	−3.00000	+	5.19615i	−0.243332	+	0.421464i
$153$	−1.50000	−	2.59808i	−0.121268	−	0.210042i
$154$		0			0
$155$	−6.00000			−0.481932
$156$		0			0
$157$	−13.0000			−1.03751			−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000			−1.03751			−0.518756	+	0.854922i	$0.673605\pi$
$158$	−3.46410	−	6.00000i	−0.275589	−	0.477334i
$159$	3.00000	+	5.19615i	0.237915	+	0.412082i
$160$	−4.50000	+	7.79423i	−0.355756	+	0.616188i
$161$		0			0
$162$	9.52628	−	16.5000i	0.748455	−	1.29636i
$163$	10.3923	−	18.0000i	0.813988	−	1.40987i	−0.0960641	−	0.995375i	$-0.530625\pi$
$163$	10.3923	−	18.0000i	0.813988	−	1.40987i	0.910052	−	0.414494i	$-0.136041\pi$
$164$	5.19615			0.405751
$165$		0			0
$166$	−12.0000	−	20.7846i	−0.931381	−	1.61320i
$167$	6.92820	+	12.0000i	0.536120	+	0.928588i	0.999108	+	0.0422232i	$0.0134441\pi$
$167$	6.92820	+	12.0000i	0.536120	+	0.928588i	−0.462988	+	0.886365i	$0.653223\pi$
$168$		0			0
$169$		0			0
$170$	−9.00000			−0.690268
$171$	1.73205	+	3.00000i	0.132453	+	0.229416i
$172$	4.00000	+	6.92820i	0.304997	+	0.528271i
$173$	3.00000	−	5.19615i	0.228086	−	0.395056i	−0.729155	−	0.684349i	$-0.760087\pi$
$173$	3.00000	−	5.19615i	0.228086	−	0.395056i	0.957241	+	0.289292i	$0.0934200\pi$
$174$	10.3923			0.787839
$175$		0			0
$176$		0			0
$177$	−13.8564			−1.04151
$178$	6.00000	−	10.3923i	0.449719	−	0.778936i
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$	0.866025	+	1.50000i	0.0645497	+	0.111803i
$181$	−11.0000			−0.817624			−0.408812	−	0.912619i	$-0.634057\pi$
$181$	−11.0000			−0.817624			−0.408812	+	0.912619i	$0.634057\pi$
$182$		0			0
$183$	2.00000			0.147844
$184$	5.19615	+	9.00000i	0.383065	+	0.663489i
$185$	7.50000	+	12.9904i	0.551411	+	0.955072i
$186$	−6.00000	+	10.3923i	−0.439941	+	0.762001i
$187$		0			0
$188$	−1.73205	+	3.00000i	−0.126323	+	0.218797i
$189$		0			0
$190$	10.3923			0.753937
$191$	−9.00000	+	15.5885i	−0.651217	+	1.12794i	0.331611	+	0.943416i	$0.392408\pi$
$191$	−9.00000	+	15.5885i	−0.651217	+	1.12794i	−0.982828	+	0.184525i	$0.940925\pi$
$192$	−1.00000	−	1.73205i	−0.0721688	−	0.125000i
$193$	2.59808	+	4.50000i	0.187014	+	0.323917i	0.944253	−	0.329220i	$-0.106786\pi$
$193$	2.59808	+	4.50000i	0.187014	+	0.323917i	−0.757240	+	0.653137i	$0.773452\pi$
$194$	12.0000			0.861550
$195$		0			0
$196$	−7.00000			−0.500000
$197$	6.92820	+	12.0000i	0.493614	+	0.854965i	0.999973	−	0.00735824i	$-0.00234222\pi$
$197$	6.92820	+	12.0000i	0.493614	+	0.854965i	−0.506359	+	0.862323i	$0.669009\pi$
$198$		0			0
$199$	−1.00000	+	1.73205i	−0.0708881	+	0.122782i	−0.899291	−	0.437351i	$-0.855917\pi$
$199$	−1.00000	+	1.73205i	−0.0708881	+	0.122782i	0.828403	+	0.560133i	$0.189250\pi$
$200$	3.46410			0.244949
$201$	3.46410	−	6.00000i	0.244339	−	0.423207i
$202$	−2.59808	+	4.50000i	−0.182800	+	0.316619i
$203$		0			0
$204$	−3.00000	+	5.19615i	−0.210042	+	0.363803i
$205$	4.50000	+	7.79423i	0.314294	+	0.544373i
$206$	−8.66025	−	15.0000i	−0.603388	−	1.04510i
$207$	6.00000			0.417029
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−5.00000	−	8.66025i	−0.344214	−	0.596196i	0.640996	−	0.767544i	$-0.278521\pi$
$211$	−5.00000	−	8.66025i	−0.344214	−	0.596196i	−0.985211	+	0.171347i	$0.945188\pi$
$212$	1.50000	−	2.59808i	0.103020	−	0.178437i
$213$	6.92820			0.474713
$214$	−5.19615	+	9.00000i	−0.355202	+	0.615227i
$215$	−6.92820	+	12.0000i	−0.472500	+	0.818393i
$216$	6.92820			0.471405
$217$		0			0
$218$	12.0000	+	20.7846i	0.812743	+	1.40771i
$219$	1.73205	+	3.00000i	0.117041	+	0.202721i
$220$		0			0
$221$		0			0
$222$	30.0000			2.01347
$223$	−5.19615	−	9.00000i	−0.347960	−	0.602685i	0.637927	−	0.770097i	$-0.279792\pi$
$223$	−5.19615	−	9.00000i	−0.347960	−	0.602685i	−0.985887	+	0.167412i	$0.946459\pi$
$224$		0			0
$225$	1.00000	−	1.73205i	0.0666667	−	0.115470i
$226$	−25.9808			−1.72821
$227$	−12.1244	+	21.0000i	−0.804722	+	1.39382i	0.111757	+	0.993736i	$0.464352\pi$
$227$	−12.1244	+	21.0000i	−0.804722	+	1.39382i	−0.916479	+	0.400083i	$0.868981\pi$
$228$	3.46410	−	6.00000i	0.229416	−	0.397360i
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$	9.00000	−	15.5885i	0.593442	−	1.02787i
$231$		0			0
$232$	2.59808	+	4.50000i	0.170572	+	0.295439i
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$	−6.00000			−0.391397
$236$	3.46410	+	6.00000i	0.225494	+	0.390567i
$237$	−4.00000	−	6.92820i	−0.259828	−	0.450035i
$238$		0			0
$239$	−20.7846			−1.34444			−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846			−1.34444			−0.672222	+	0.740349i	$0.734660\pi$
$240$	−8.66025	+	15.0000i	−0.559017	+	0.968246i
$241$	−0.866025	+	1.50000i	−0.0557856	+	0.0966235i	−0.892570	−	0.450910i	$-0.851100\pi$
$241$	−0.866025	+	1.50000i	−0.0557856	+	0.0966235i	0.836784	+	0.547533i	$0.184433\pi$
$242$	−19.0526			−1.22474
$243$	5.00000	−	8.66025i	0.320750	−	0.555556i
$244$	−0.500000	−	0.866025i	−0.0320092	−	0.0554416i
$245$	−6.06218	−	10.5000i	−0.387298	−	0.670820i
$246$	18.0000			1.14764
$247$		0			0
$248$	−6.00000			−0.381000
$249$	−13.8564	−	24.0000i	−0.878114	−	1.52094i
$250$	−10.5000	−	18.1865i	−0.664078	−	1.15022i
$251$	−9.00000	+	15.5885i	−0.568075	+	0.983935i	0.428681	+	0.903456i	$0.358978\pi$
$251$	−9.00000	+	15.5885i	−0.568075	+	0.983935i	−0.996756	+	0.0804789i	$0.974355\pi$
$252$		0			0
$253$		0			0
$254$	−1.73205	+	3.00000i	−0.108679	+	0.188237i
$255$	−10.3923			−0.650791
$256$	−9.50000	+	16.4545i	−0.593750	+	1.02841i
$257$	1.50000	+	2.59808i	0.0935674	+	0.162064i	0.909010	−	0.416775i	$-0.136840\pi$
$257$	1.50000	+	2.59808i	0.0935674	+	0.162064i	−0.815442	+	0.578838i	$0.803506\pi$
$258$	13.8564	+	24.0000i	0.862662	+	1.49417i
$259$		0			0
$260$		0			0
$261$	3.00000			0.185695
$262$	−15.5885	−	27.0000i	−0.963058	−	1.66807i
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	0.619586	−	0.784929i	$-0.287301\pi$
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	−0.989561	+	0.144112i	$0.953967\pi$
$264$		0			0
$265$	5.19615			0.319197
$266$		0			0
$267$	6.92820	−	12.0000i	0.423999	−	0.734388i
$268$	−3.46410			−0.211604
$269$	3.00000	−	5.19615i	0.182913	−	0.316815i	−0.759958	−	0.649972i	$-0.774781\pi$
$269$	3.00000	−	5.19615i	0.182913	−	0.316815i	0.942871	+	0.333157i	$0.108114\pi$
$270$	−6.00000	−	10.3923i	−0.365148	−	0.632456i
$271$	−10.3923	−	18.0000i	−0.631288	−	1.09342i	−0.987289	−	0.158937i	$-0.949193\pi$
$271$	−10.3923	−	18.0000i	−0.631288	−	1.09342i	0.356001	−	0.934485i	$-0.384140\pi$
$272$	−15.0000			−0.909509
$273$		0			0
$274$	27.0000			1.63113
$275$		0			0
$276$	−6.00000	−	10.3923i	−0.361158	−	0.625543i
$277$	−3.50000	+	6.06218i	−0.210295	+	0.364241i	−0.951807	−	0.306699i	$-0.900776\pi$
$277$	−3.50000	+	6.06218i	−0.210295	+	0.364241i	0.741512	+	0.670940i	$0.234109\pi$
$278$	−6.92820			−0.415526
$279$	−1.73205	+	3.00000i	−0.103695	+	0.179605i
$280$		0			0
$281$	22.5167			1.34323			0.671616	−	0.740900i	$-0.265601\pi$
$281$	22.5167			1.34323			0.671616	+	0.740900i	$0.265601\pi$
$282$	−6.00000	+	10.3923i	−0.357295	+	0.618853i
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	0.919327	−	0.393494i	$-0.128734\pi$
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	−0.800439	+	0.599414i	$0.795400\pi$
$284$	−1.73205	−	3.00000i	−0.102778	−	0.178017i
$285$	12.0000			0.710819
$286$		0			0
$287$		0			0
$288$	2.59808	+	4.50000i	0.153093	+	0.265165i
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$	4.50000	−	7.79423i	0.264249	−	0.457693i
$291$	13.8564			0.812277
$292$	0.866025	−	1.50000i	0.0506803	−	0.0877809i
$293$	2.59808	−	4.50000i	0.151781	−	0.262893i	−0.780101	−	0.625653i	$-0.784832\pi$
$293$	2.59808	−	4.50000i	0.151781	−	0.262893i	0.931882	+	0.362761i	$0.118166\pi$
$294$	−24.2487			−1.41421
$295$	−6.00000	+	10.3923i	−0.349334	+	0.605063i
$296$	7.50000	+	12.9904i	0.435929	+	0.755051i
$297$		0			0
$298$	−33.0000			−1.91164
$299$		0			0
$300$	−4.00000			−0.230940
$301$		0			0
$302$	15.0000	+	25.9808i	0.863153	+	1.49502i
$303$	−3.00000	+	5.19615i	−0.172345	+	0.298511i
$304$	17.3205			0.993399
$305$	0.866025	−	1.50000i	0.0495885	−	0.0858898i
$306$	−2.59808	+	4.50000i	−0.148522	+	0.257248i
$307$	17.3205			0.988534			0.494267	−	0.869310i	$-0.335437\pi$
$307$	17.3205			0.988534			0.494267	+	0.869310i	$0.335437\pi$
$308$		0			0
$309$	−10.0000	−	17.3205i	−0.568880	−	0.985329i
$310$	5.19615	+	9.00000i	0.295122	+	0.511166i
$311$	30.0000			1.70114			0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000			1.70114			0.850572	+	0.525859i	$0.176256\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$	11.2583	+	19.5000i	0.635344	+	1.10045i
$315$		0			0
$316$	−2.00000	+	3.46410i	−0.112509	+	0.194871i
$317$	−5.19615			−0.291845			−0.145922	−	0.989296i	$-0.546615\pi$
$317$	−5.19615			−0.291845			−0.145922	+	0.989296i	$0.546615\pi$
$318$	5.19615	−	9.00000i	0.291386	−	0.504695i
$319$		0			0
$320$	−1.73205			−0.0968246
$321$	−6.00000	+	10.3923i	−0.334887	+	0.580042i
$322$		0			0
$323$	5.19615	+	9.00000i	0.289122	+	0.500773i
$324$	−11.0000			−0.611111
$325$		0			0
$326$	−36.0000			−1.99386
$327$	13.8564	+	24.0000i	0.766261	+	1.32720i
$328$	4.50000	+	7.79423i	0.248471	+	0.430364i
$329$		0			0
$330$		0			0
$331$	13.8564	−	24.0000i	0.761617	−	1.31916i	−0.180400	−	0.983593i	$-0.557739\pi$
$331$	13.8564	−	24.0000i	0.761617	−	1.31916i	0.942017	−	0.335566i	$-0.108928\pi$
$332$	−6.92820	+	12.0000i	−0.380235	+	0.658586i
$333$	8.66025			0.474579
$334$	12.0000	−	20.7846i	0.656611	−	1.13728i
$335$	−3.00000	−	5.19615i	−0.163908	−	0.283896i
$336$		0			0
$337$	23.0000			1.25289			0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000			1.25289			0.626445	+	0.779466i	$0.284509\pi$
$338$		0			0
$339$	−30.0000			−1.62938
$340$	2.59808	+	4.50000i	0.140900	+	0.244047i
$341$		0			0
$342$	3.00000	−	5.19615i	0.162221	−	0.280976i
$343$		0			0
$344$	−6.92820	+	12.0000i	−0.373544	+	0.646997i
$345$	10.3923	−	18.0000i	0.559503	−	0.969087i
$346$	−10.3923			−0.558694
$347$	15.0000	−	25.9808i	0.805242	−	1.39472i	−0.110885	−	0.993833i	$-0.535369\pi$
$347$	15.0000	−	25.9808i	0.805242	−	1.39472i	0.916127	−	0.400887i	$-0.131298\pi$
$348$	−3.00000	−	5.19615i	−0.160817	−	0.278543i
$349$	−6.92820	−	12.0000i	−0.370858	−	0.642345i	0.618840	−	0.785517i	$-0.287603\pi$
$349$	−6.92820	−	12.0000i	−0.370858	−	0.642345i	−0.989698	+	0.143172i	$0.954270\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	16.4545	+	28.5000i	0.875784	+	1.51690i	0.855926	+	0.517099i	$0.172988\pi$
$353$	16.4545	+	28.5000i	0.875784	+	1.51690i	0.0198582	+	0.999803i	$0.493679\pi$
$354$	12.0000	+	20.7846i	0.637793	+	1.10469i
$355$	3.00000	−	5.19615i	0.159223	−	0.275783i
$356$	−6.92820			−0.367194
$357$		0			0
$358$		0			0
$359$	6.92820			0.365657			0.182828	−	0.983145i	$-0.441475\pi$
$359$	6.92820			0.365657			0.182828	+	0.983145i	$0.441475\pi$
$360$	−1.50000	+	2.59808i	−0.0790569	+	0.136931i
$361$	3.50000	+	6.06218i	0.184211	+	0.319062i
$362$	9.52628	+	16.5000i	0.500690	+	0.867221i
$363$	−22.0000			−1.15470
$364$		0			0
$365$	3.00000			0.157027
$366$	−1.73205	−	3.00000i	−0.0905357	−	0.156813i
$367$	11.0000	+	19.0526i	0.574195	+	0.994535i	0.996129	+	0.0879086i	$0.0280183\pi$
$367$	11.0000	+	19.0526i	0.574195	+	0.994535i	−0.421933	+	0.906627i	$0.638648\pi$
$368$	15.0000	−	25.9808i	0.781929	−	1.35434i
$369$	5.19615			0.270501
$370$	12.9904	−	22.5000i	0.675338	−	1.16972i
$371$		0			0
$372$	6.92820			0.359211
$373$	−9.50000	+	16.4545i	−0.491891	+	0.851981i	−0.999956	−	0.00933789i	$-0.997028\pi$
$373$	−9.50000	+	16.4545i	−0.491891	+	0.851981i	0.508065	+	0.861319i	$0.330361\pi$
$374$		0			0
$375$	−12.1244	−	21.0000i	−0.626099	−	1.08444i
$376$	−6.00000			−0.309426
$377$		0			0
$378$		0			0
$379$	−12.1244	−	21.0000i	−0.622786	−	1.07870i	−0.988964	−	0.148153i	$-0.952667\pi$
$379$	−12.1244	−	21.0000i	−0.622786	−	1.07870i	0.366178	−	0.930545i	$-0.380666\pi$
$380$	−3.00000	−	5.19615i	−0.153897	−	0.266557i
$381$	−2.00000	+	3.46410i	−0.102463	+	0.177471i
$382$	31.1769			1.59515
$383$	10.3923	−	18.0000i	0.531022	−	0.919757i	−0.468323	−	0.883558i	$-0.655141\pi$
$383$	10.3923	−	18.0000i	0.531022	−	0.919757i	0.999345	−	0.0361995i	$-0.0115252\pi$
$384$	−12.1244	+	21.0000i	−0.618718	+	1.07165i
$385$		0			0
$386$	4.50000	−	7.79423i	0.229044	−	0.396716i
$387$	4.00000	+	6.92820i	0.203331	+	0.352180i
$388$	−3.46410	−	6.00000i	−0.175863	−	0.304604i
$389$	9.00000			0.456318			0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000			0.456318			0.228159	+	0.973624i	$0.426729\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$	−6.06218	−	10.5000i	−0.306186	−	0.530330i
$393$	−18.0000	−	31.1769i	−0.907980	−	1.57267i
$394$	12.0000	−	20.7846i	0.604551	−	1.04711i
$395$	−6.92820			−0.348596
$396$		0			0
$397$	−6.92820	+	12.0000i	−0.347717	+	0.602263i	−0.985843	−	0.167668i	$-0.946376\pi$
$397$	−6.92820	+	12.0000i	−0.347717	+	0.602263i	0.638127	+	0.769931i	$0.279710\pi$
$398$	3.46410			0.173640
$399$		0			0
$400$	−5.00000	−	8.66025i	−0.250000	−	0.433013i
$401$	0.866025	+	1.50000i	0.0432472	+	0.0749064i	0.886839	−	0.462079i	$-0.152896\pi$
$401$	0.866025	+	1.50000i	0.0432472	+	0.0749064i	−0.843592	+	0.536985i	$0.819563\pi$
$402$	−12.0000			−0.598506
$403$		0			0
$404$	3.00000			0.149256
$405$	−9.52628	−	16.5000i	−0.473365	−	0.819892i
$406$		0			0
$407$		0			0
$408$	−10.3923			−0.514496
$409$	−7.79423	+	13.5000i	−0.385400	+	0.667532i	−0.991825	−	0.127609i	$-0.959270\pi$
$409$	−7.79423	+	13.5000i	−0.385400	+	0.667532i	0.606425	+	0.795141i	$0.292603\pi$
$410$	7.79423	−	13.5000i	0.384930	−	0.666717i
$411$	31.1769			1.53784
$412$	−5.00000	+	8.66025i	−0.246332	+	0.426660i
$413$		0			0
$414$	−5.19615	−	9.00000i	−0.255377	−	0.442326i
$415$	−24.0000			−1.17811
$416$		0			0
$417$	−8.00000			−0.391762
$418$		0			0
$419$	−9.00000	−	15.5885i	−0.439679	−	0.761546i	0.557986	−	0.829851i	$-0.311574\pi$
$419$	−9.00000	−	15.5885i	−0.439679	−	0.761546i	−0.997665	+	0.0683046i	$0.978241\pi$
$420$		0			0
$421$	15.5885			0.759735			0.379867	−	0.925041i	$-0.375970\pi$
$421$	15.5885			0.759735			0.379867	+	0.925041i	$0.375970\pi$
$422$	−8.66025	+	15.0000i	−0.421575	+	0.730189i
$423$	−1.73205	+	3.00000i	−0.0842152	+	0.145865i
$424$	5.19615			0.252347
$425$	3.00000	−	5.19615i	0.145521	−	0.252050i
$426$	−6.00000	−	10.3923i	−0.290701	−	0.503509i
$427$		0			0
$428$	6.00000			0.290021
$429$		0			0
$430$	24.0000			1.15738
$431$	−3.46410	−	6.00000i	−0.166860	−	0.289010i	0.770454	−	0.637495i	$-0.220029\pi$
$431$	−3.46410	−	6.00000i	−0.166860	−	0.289010i	−0.937314	+	0.348485i	$0.886696\pi$
$432$	−10.0000	−	17.3205i	−0.481125	−	0.833333i
$433$	−8.50000	+	14.7224i	−0.408484	+	0.707515i	−0.994720	−	0.102625i	$-0.967276\pi$
$433$	−8.50000	+	14.7224i	−0.408484	+	0.707515i	0.586236	+	0.810140i	$0.300609\pi$
$434$		0			0
$435$	5.19615	−	9.00000i	0.249136	−	0.431517i
$436$	6.92820	−	12.0000i	0.331801	−	0.574696i
$437$	−20.7846			−0.994263
$438$	3.00000	−	5.19615i	0.143346	−	0.248282i
$439$	−14.0000	−	24.2487i	−0.668184	−	1.15733i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−14.0000	−	24.2487i	−0.668184	−	1.15733i	0.310228	−	0.950662i	$-0.399595\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		0			0
$443$	−12.0000			−0.570137			−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000			−0.570137			−0.285069	+	0.958507i	$0.592016\pi$
$444$	−8.66025	−	15.0000i	−0.410997	−	0.711868i
$445$	−6.00000	−	10.3923i	−0.284427	−	0.492642i
$446$	−9.00000	+	15.5885i	−0.426162	+	0.738135i
$447$	−38.1051			−1.80231
$448$		0			0
$449$	−3.46410	+	6.00000i	−0.163481	+	0.283158i	−0.936115	−	0.351694i	$-0.885606\pi$
$449$	−3.46410	+	6.00000i	−0.163481	+	0.283158i	0.772634	+	0.634852i	$0.218939\pi$
$450$	−3.46410			−0.163299
$451$		0			0
$452$	7.50000	+	12.9904i	0.352770	+	0.611016i
$453$	17.3205	+	30.0000i	0.813788	+	1.40952i
$454$	42.0000			1.97116
$455$		0			0
$456$	12.0000			0.561951
$457$	0.866025	+	1.50000i	0.0405110	+	0.0701670i	0.885570	−	0.464506i	$-0.153768\pi$
$457$	0.866025	+	1.50000i	0.0405110	+	0.0701670i	−0.845059	+	0.534673i	$0.820435\pi$
$458$		0			0
$459$	6.00000	−	10.3923i	0.280056	−	0.485071i
$460$	−10.3923			−0.484544
$461$	−11.2583	+	19.5000i	−0.524353	+	0.908206i	0.475245	+	0.879853i	$0.342359\pi$
$461$	−11.2583	+	19.5000i	−0.524353	+	0.908206i	−0.999598	+	0.0283522i	$0.990974\pi$
$462$		0			0
$463$	−13.8564			−0.643962			−0.321981	−	0.946746i	$-0.604349\pi$
$463$	−13.8564			−0.643962			−0.321981	+	0.946746i	$0.604349\pi$
$464$	7.50000	−	12.9904i	0.348179	−	0.603063i
$465$	6.00000	+	10.3923i	0.278243	+	0.481932i
$466$	5.19615	+	9.00000i	0.240707	+	0.416917i
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$		0			0
$470$	5.19615	+	9.00000i	0.239681	+	0.415139i
$471$	13.0000	+	22.5167i	0.599008	+	1.03751i
$472$	−6.00000	+	10.3923i	−0.276172	+	0.478345i
$473$		0			0
$474$	−6.92820	+	12.0000i	−0.318223	+	0.551178i
$475$	−3.46410	+	6.00000i	−0.158944	+	0.275299i
$476$		0			0
$477$	1.50000	−	2.59808i	0.0686803	−	0.118958i
$478$	18.0000	+	31.1769i	0.823301	+	1.42600i
$479$	−12.1244	−	21.0000i	−0.553976	−	0.959514i	−0.997982	−	0.0634909i	$-0.979777\pi$
$479$	−12.1244	−	21.0000i	−0.553976	−	0.959514i	0.444006	−	0.896024i	$-0.353557\pi$
$480$	18.0000			0.821584
$481$		0			0
$482$	3.00000			0.136646
$483$		0			0
$484$	5.50000	+	9.52628i	0.250000	+	0.433013i
$485$	6.00000	−	10.3923i	0.272446	−	0.471890i
$486$	−17.3205			−0.785674
$487$	3.46410	−	6.00000i	0.156973	−	0.271886i	−0.776802	−	0.629744i	$-0.783160\pi$
$487$	3.46410	−	6.00000i	0.156973	−	0.271886i	0.933776	+	0.357858i	$0.116493\pi$
$488$	0.866025	−	1.50000i	0.0392031	−	0.0679018i
$489$	−41.5692			−1.87983
$490$	−10.5000	+	18.1865i	−0.474342	+	0.821584i
$491$	6.00000	+	10.3923i	0.270776	+	0.468998i	0.969061	−	0.246822i	$-0.0793863\pi$
$491$	6.00000	+	10.3923i	0.270776	+	0.468998i	−0.698285	+	0.715820i	$0.746053\pi$
$492$	−5.19615	−	9.00000i	−0.234261	−	0.405751i
$493$	9.00000			0.405340
$494$		0			0
$495$		0			0
$496$	8.66025	+	15.0000i	0.388857	+	0.673520i
$497$		0			0
$498$	−24.0000	+	41.5692i	−1.07547	+	1.86276i
$499$	31.1769			1.39567			0.697835	−	0.716258i	$-0.254147\pi$
$499$	31.1769			1.39567			0.697835	+	0.716258i	$0.254147\pi$
$500$	−6.06218	+	10.5000i	−0.271109	+	0.469574i
$501$	13.8564	−	24.0000i	0.619059	−	1.07224i
$502$	31.1769			1.39149
$503$	−18.0000	+	31.1769i	−0.802580	+	1.39011i	0.115332	+	0.993327i	$0.463207\pi$
$503$	−18.0000	+	31.1769i	−0.802580	+	1.39011i	−0.917912	+	0.396783i	$0.870127\pi$
$504$		0			0
$505$	2.59808	+	4.50000i	0.115613	+	0.200247i
$506$		0			0
$507$		0			0
$508$	2.00000			0.0887357
$509$	9.52628	+	16.5000i	0.422245	+	0.731350i	0.996159	−	0.0875661i	$-0.0279089\pi$
$509$	9.52628	+	16.5000i	0.422245	+	0.731350i	−0.573914	+	0.818916i	$0.694576\pi$
$510$	9.00000	+	15.5885i	0.398527	+	0.690268i
$511$		0			0
$512$	8.66025			0.382733
$513$	−6.92820	+	12.0000i	−0.305888	+	0.529813i
$514$	2.59808	−	4.50000i	0.114596	−	0.198486i
$515$	−17.3205			−0.763233
$516$	8.00000	−	13.8564i	0.352180	−	0.609994i
$517$		0			0
$518$		0			0
$519$	−12.0000			−0.526742
$520$		0			0
$521$	9.00000			0.394297			0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000			0.394297			0.197149	+	0.980374i	$0.436832\pi$
$522$	−2.59808	−	4.50000i	−0.113715	−	0.196960i
$523$	8.00000	+	13.8564i	0.349816	+	0.605898i	0.986216	−	0.165460i	$-0.0529109\pi$
$523$	8.00000	+	13.8564i	0.349816	+	0.605898i	−0.636401	+	0.771358i	$0.719578\pi$
$524$	−9.00000	+	15.5885i	−0.393167	+	0.680985i
$525$		0			0
$526$	−10.3923	+	18.0000i	−0.453126	+	0.784837i
$527$	−5.19615	+	9.00000i	−0.226348	+	0.392046i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$	−4.50000	−	7.79423i	−0.195468	−	0.338560i
$531$	3.46410	+	6.00000i	0.150329	+	0.260378i
$532$		0			0
$533$		0			0
$534$	−24.0000			−1.03858
$535$	5.19615	+	9.00000i	0.224649	+	0.389104i
$536$	−3.00000	−	5.19615i	−0.129580	−	0.224440i
$537$		0			0
$538$	−10.3923			−0.448044
$539$		0			0
$540$	−3.46410	+	6.00000i	−0.149071	+	0.258199i
$541$	−29.4449			−1.26593			−0.632967	−	0.774179i	$-0.718163\pi$
$541$	−29.4449			−1.26593			−0.632967	+	0.774179i	$0.718163\pi$
$542$	−18.0000	+	31.1769i	−0.773166	+	1.33916i
$543$	11.0000	+	19.0526i	0.472055	+	0.817624i
$544$	7.79423	+	13.5000i	0.334175	+	0.578808i
$545$	24.0000			1.02805
$546$		0			0
$547$	−22.0000			−0.940652			−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000			−0.940652			−0.470326	+	0.882493i	$0.655864\pi$
$548$	−7.79423	−	13.5000i	−0.332953	−	0.576691i
$549$	−0.500000	−	0.866025i	−0.0213395	−	0.0369611i
$550$		0			0
$551$	−10.3923			−0.442727
$552$	10.3923	−	18.0000i	0.442326	−	0.766131i
$553$		0			0
$554$	12.1244			0.515115
$555$	15.0000	−	25.9808i	0.636715	−	1.10282i
$556$	2.00000	+	3.46410i	0.0848189	+	0.146911i
$557$	7.79423	+	13.5000i	0.330252	+	0.572013i	0.982561	−	0.185940i	$-0.0595329\pi$
$557$	7.79423	+	13.5000i	0.330252	+	0.572013i	−0.652309	+	0.757953i	$0.726200\pi$
$558$	6.00000			0.254000
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−19.5000	−	33.7750i	−0.822558	−	1.42471i
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$	6.92820			0.291730
$565$	−12.9904	+	22.5000i	−0.546509	+	0.946582i
$566$	3.46410	−	6.00000i	0.145607	−	0.252199i
$567$		0			0
$568$	3.00000	−	5.19615i	0.125877	−	0.218026i
$569$	−21.0000	−	36.3731i	−0.880366	−	1.52484i	−0.850935	−	0.525271i	$-0.823964\pi$
$569$	−21.0000	−	36.3731i	−0.880366	−	1.52484i	−0.0294311	−	0.999567i	$-0.509370\pi$
$570$	−10.3923	−	18.0000i	−0.435286	−	0.753937i
$571$	−40.0000			−1.67395			−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000			−1.67395			−0.836974	+	0.547243i	$0.815677\pi$
$572$		0			0
$573$	36.0000			1.50392
$574$		0			0
$575$	6.00000	+	10.3923i	0.250217	+	0.433389i
$576$	−0.500000	+	0.866025i	−0.0208333	+	0.0360844i
$577$	−19.0526			−0.793168			−0.396584	−	0.917998i	$-0.629805\pi$
$577$	−19.0526			−0.793168			−0.396584	+	0.917998i	$0.629805\pi$
$578$	6.92820	−	12.0000i	0.288175	−	0.499134i
$579$	5.19615	−	9.00000i	0.215945	−	0.374027i
$580$	−5.19615			−0.215758
$581$		0			0
$582$	−12.0000	−	20.7846i	−0.497416	−	0.861550i
$583$		0			0
$584$	3.00000			0.124141
$585$		0			0
$586$	−9.00000			−0.371787
$587$	−10.3923	−	18.0000i	−0.428936	−	0.742940i	0.567843	−	0.823137i	$-0.307778\pi$
$587$	−10.3923	−	18.0000i	−0.428936	−	0.742940i	−0.996779	+	0.0801976i	$0.974445\pi$
$588$	7.00000	+	12.1244i	0.288675	+	0.500000i
$589$	6.00000	−	10.3923i	0.247226	−	0.428207i
$590$	20.7846			0.855689
$591$	13.8564	−	24.0000i	0.569976	−	0.987228i
$592$	21.6506	−	37.5000i	0.889836	−	1.54124i
$593$	25.9808			1.06690			0.533451	−	0.845831i	$-0.320895\pi$
$593$	25.9808			1.06690			0.533451	+	0.845831i	$0.320895\pi$
$594$		0			0
$595$		0			0
$596$	9.52628	+	16.5000i	0.390212	+	0.675866i
$597$	4.00000			0.163709
$598$		0			0
$599$	30.0000			1.22577			0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000			1.22577			0.612883	+	0.790173i	$0.290010\pi$
$600$	−3.46410	−	6.00000i	−0.141421	−	0.244949i
$601$	−12.5000	−	21.6506i	−0.509886	−	0.883148i	−0.999934	−	0.0114528i	$-0.996354\pi$
$601$	−12.5000	−	21.6506i	−0.509886	−	0.883148i	0.490049	−	0.871695i	$-0.336979\pi$
$602$		0			0
$603$	−3.46410			−0.141069
$604$	8.66025	−	15.0000i	0.352381	−	0.610341i
$605$	−9.52628	+	16.5000i	−0.387298	+	0.670820i
$606$	10.3923			0.422159
$607$	17.0000	−	29.4449i	0.690009	−	1.19513i	−0.281826	−	0.959466i	$-0.590940\pi$
$607$	17.0000	−	29.4449i	0.690009	−	1.19513i	0.971834	−	0.235665i	$-0.0757267\pi$
$608$	−9.00000	−	15.5885i	−0.364998	−	0.632195i
$609$		0			0
$610$	−3.00000			−0.121466
$611$		0			0
$612$	3.00000			0.121268
$613$	−6.06218	−	10.5000i	−0.244849	−	0.424091i	0.717240	−	0.696826i	$-0.245405\pi$
$613$	−6.06218	−	10.5000i	−0.244849	−	0.424091i	−0.962089	+	0.272735i	$0.912072\pi$
$614$	−15.0000	−	25.9808i	−0.605351	−	1.04850i
$615$	9.00000	−	15.5885i	0.362915	−	0.628587i
$616$		0			0
$617$	11.2583	−	19.5000i	0.453243	−	0.785040i	−0.545342	−	0.838214i	$-0.683600\pi$
$617$	11.2583	−	19.5000i	0.453243	−	0.785040i	0.998585	+	0.0531732i	$0.0169335\pi$
$618$	−17.3205	+	30.0000i	−0.696733	+	1.20678i
$619$	20.7846			0.835404			0.417702	−	0.908584i	$-0.362836\pi$
$619$	20.7846			0.835404			0.417702	+	0.908584i	$0.362836\pi$
$620$	3.00000	−	5.19615i	0.120483	−	0.208683i
$621$	12.0000	+	20.7846i	0.481543	+	0.834058i
$622$	−25.9808	−	45.0000i	−1.04173	−	1.80434i
$623$		0			0
$624$		0			0
$625$	−11.0000			−0.440000
$626$	−8.66025	−	15.0000i	−0.346133	−	0.599521i
$627$		0			0
$628$	6.50000	−	11.2583i	0.259378	−	0.449256i
$629$	25.9808			1.03592
$630$		0			0
$631$	−24.2487	+	42.0000i	−0.965326	+	1.67199i	−0.256589	+	0.966521i	$0.582599\pi$
$631$	−24.2487	+	42.0000i	−0.965326	+	1.67199i	−0.708737	+	0.705473i	$0.750735\pi$
$632$	−6.92820			−0.275589
$633$	−10.0000	+	17.3205i	−0.397464	+	0.688428i
$634$	4.50000	+	7.79423i	0.178718	+	0.309548i
$635$	1.73205	+	3.00000i	0.0687343	+	0.119051i
$636$	−6.00000			−0.237915
$637$		0			0
$638$		0			0
$639$	−1.73205	−	3.00000i	−0.0685189	−	0.118678i
$640$	10.5000	+	18.1865i	0.415049	+	0.718886i
$641$	16.5000	−	28.5788i	0.651711	−	1.12880i	−0.330997	−	0.943632i	$-0.607385\pi$
$641$	16.5000	−	28.5788i	0.651711	−	1.12880i	0.982708	−	0.185164i	$-0.0592817\pi$
$642$	20.7846			0.820303
$643$	−6.92820	+	12.0000i	−0.273222	+	0.473234i	−0.969685	−	0.244359i	$-0.921423\pi$
$643$	−6.92820	+	12.0000i	−0.273222	+	0.473234i	0.696463	+	0.717592i	$0.254756\pi$
$644$		0			0
$645$	27.7128			1.09119
$646$	9.00000	−	15.5885i	0.354100	−	0.613320i
$647$	9.00000	+	15.5885i	0.353827	+	0.612845i	0.986916	−	0.161233i	$-0.0515470\pi$
$647$	9.00000	+	15.5885i	0.353827	+	0.612845i	−0.633090	+	0.774078i	$0.718214\pi$
$648$	−9.52628	−	16.5000i	−0.374228	−	0.648181i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	10.3923	+	18.0000i	0.406994	+	0.704934i
$653$	15.0000	+	25.9808i	0.586995	+	1.01671i	0.994623	+	0.103558i	$0.0330227\pi$
$653$	15.0000	+	25.9808i	0.586995	+	1.01671i	−0.407628	+	0.913148i	$0.633644\pi$
$654$	24.0000	−	41.5692i	0.938474	−	1.62549i
$655$	−31.1769			−1.21818
$656$	12.9904	−	22.5000i	0.507189	−	0.878477i
$657$	0.866025	−	1.50000i	0.0337869	−	0.0585206i
$658$		0			0
$659$	6.00000	−	10.3923i	0.233727	−	0.404827i	−0.725175	−	0.688565i	$-0.758241\pi$
$659$	6.00000	−	10.3923i	0.233727	−	0.404827i	0.958902	+	0.283738i	$0.0915745\pi$
$660$		0			0
$661$	23.3827	+	40.5000i	0.909481	+	1.57527i	0.814787	+	0.579761i	$0.196854\pi$
$661$	23.3827	+	40.5000i	0.909481	+	1.57527i	0.0946945	+	0.995506i	$0.469813\pi$
$662$	−48.0000			−1.86557
$663$		0			0
$664$	−24.0000			−0.931381
$665$		0			0
$666$	−7.50000	−	12.9904i	−0.290619	−	0.503367i
$667$	−9.00000	+	15.5885i	−0.348481	+	0.603587i
$668$	−13.8564			−0.536120
$669$	−10.3923	+	18.0000i	−0.401790	+	0.695920i
$670$	−5.19615	+	9.00000i	−0.200745	+	0.347700i
$671$		0			0
$672$		0			0
$673$	−9.50000	−	16.4545i	−0.366198	−	0.634274i	0.622770	−	0.782405i	$-0.286007\pi$
$673$	−9.50000	−	16.4545i	−0.366198	−	0.634274i	−0.988968	+	0.148132i	$0.952674\pi$
$674$	−19.9186	−	34.5000i	−0.767235	−	1.32889i
$675$	8.00000			0.307920
$676$		0			0
$677$	−6.00000			−0.230599			−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000			−0.230599			−0.115299	+	0.993331i	$0.536783\pi$
$678$	25.9808	+	45.0000i	0.997785	+	1.72821i
$679$		0			0
$680$	−4.50000	+	7.79423i	−0.172567	+	0.298895i
$681$	48.4974			1.85843
$682$		0			0
$683$	12.1244	−	21.0000i	0.463926	−	0.803543i	−0.535227	−	0.844708i	$-0.679774\pi$
$683$	12.1244	−	21.0000i	0.463926	−	0.803543i	0.999152	+	0.0411658i	$0.0131072\pi$
$684$	−3.46410			−0.132453
$685$	13.5000	−	23.3827i	0.515808	−	0.893407i
$686$		0			0
$687$		0			0
$688$	40.0000			1.52499
$689$		0			0
$690$	−36.0000			−1.37050
$691$	6.92820	+	12.0000i	0.263561	+	0.456502i	0.967186	−	0.254071i	$-0.0817696\pi$
$691$	6.92820	+	12.0000i	0.263561	+	0.456502i	−0.703624	+	0.710572i	$0.748436\pi$
$692$	3.00000	+	5.19615i	0.114043	+	0.197528i
$693$		0			0
$694$	−51.9615			−1.97243
$695$	−3.46410	+	6.00000i	−0.131401	+	0.227593i
$696$	5.19615	−	9.00000i	0.196960	−	0.341144i
$697$	15.5885			0.590455
$698$	−12.0000	+	20.7846i	−0.454207	+	0.786709i
$699$	6.00000	+	10.3923i	0.226941	+	0.393073i
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	−30.0000			−1.13147
$704$		0			0
$705$	6.00000	+	10.3923i	0.225973	+	0.391397i
$706$	28.5000	−	49.3634i	1.07261	−	1.85782i
$707$		0			0
$708$	6.92820	−	12.0000i	0.260378	−	0.450988i
$709$	−2.59808	+	4.50000i	−0.0975728	+	0.169001i	−0.910679	−	0.413114i	$-0.864441\pi$
$709$	−2.59808	+	4.50000i	−0.0975728	+	0.169001i	0.813107	+	0.582115i	$0.197775\pi$
$710$	−10.3923			−0.390016
$711$	−2.00000	+	3.46410i	−0.0750059	+	0.129914i
$712$	−6.00000	−	10.3923i	−0.224860	−	0.389468i
$713$	−10.3923	−	18.0000i	−0.389195	−	0.674105i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	20.7846	+	36.0000i	0.776215	+	1.34444i
$718$	−6.00000	−	10.3923i	−0.223918	−	0.387837i
$719$	−24.0000	+	41.5692i	−0.895049	+	1.55027i	−0.0613050	+	0.998119i	$0.519526\pi$
$719$	−24.0000	+	41.5692i	−0.895049	+	1.55027i	−0.833744	+	0.552151i	$0.813807\pi$
$720$	8.66025			0.322749
$721$		0			0
$722$	6.06218	−	10.5000i	0.225611	−	0.390770i
$723$	3.46410			0.128831
$724$	5.50000	−	9.52628i	0.204406	−	0.354041i
$725$	3.00000	+	5.19615i	0.111417	+	0.192980i
$726$	19.0526	+	33.0000i	0.707107	+	1.22474i
$727$	32.0000			1.18681			0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000			1.18681			0.593407	+	0.804902i	$0.297782\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$	−2.59808	−	4.50000i	−0.0961591	−	0.166552i
$731$	12.0000	+	20.7846i	0.443836	+	0.768747i
$732$	−1.00000	+	1.73205i	−0.0369611	+	0.0640184i
$733$	12.1244			0.447823			0.223912	−	0.974609i	$-0.428117\pi$
$733$	12.1244			0.447823			0.223912	+	0.974609i	$0.428117\pi$
$734$	19.0526	−	33.0000i	0.703243	−	1.21805i
$735$	−12.1244	+	21.0000i	−0.447214	+	0.774597i
$736$	−31.1769			−1.14920
$737$		0			0
$738$	−4.50000	−	7.79423i	−0.165647	−	0.286910i
$739$	10.3923	+	18.0000i	0.382287	+	0.662141i	0.991389	−	0.130951i	$-0.0418032\pi$
$739$	10.3923	+	18.0000i	0.382287	+	0.662141i	−0.609102	+	0.793092i	$0.708470\pi$
$740$	−15.0000			−0.551411
$741$		0			0
$742$		0			0
$743$	−17.3205	−	30.0000i	−0.635428	−	1.10059i	−0.986424	−	0.164216i	$-0.947490\pi$
$743$	−17.3205	−	30.0000i	−0.635428	−	1.10059i	0.350997	−	0.936377i	$-0.385843\pi$
$744$	6.00000	+	10.3923i	0.219971	+	0.381000i
$745$	−16.5000	+	28.5788i	−0.604513	+	1.04705i
$746$	32.9090			1.20488
$747$	−6.92820	+	12.0000i	−0.253490	+	0.439057i
$748$		0			0
$749$		0			0
$750$	−21.0000	+	36.3731i	−0.766812	+	1.32816i
$751$	−8.00000	−	13.8564i	−0.291924	−	0.505627i	0.682341	−	0.731034i	$-0.260962\pi$
$751$	−8.00000	−	13.8564i	−0.291924	−	0.505627i	−0.974265	+	0.225407i	$0.927629\pi$
$752$	8.66025	+	15.0000i	0.315807	+	0.546994i
$753$	36.0000			1.31191
$754$		0			0
$755$	30.0000			1.09181
$756$		0			0
$757$	13.0000	+	22.5167i	0.472493	+	0.818382i	0.999505	−	0.0314762i	$-0.0100208\pi$
$757$	13.0000	+	22.5167i	0.472493	+	0.818382i	−0.527011	+	0.849858i	$0.676688\pi$
$758$	−21.0000	+	36.3731i	−0.762754	+	1.32113i
$759$		0			0
$760$	5.19615	−	9.00000i	0.188484	−	0.326464i
$761$	17.3205	−	30.0000i	0.627868	−	1.08750i	−0.360111	−	0.932910i	$-0.617261\pi$
$761$	17.3205	−	30.0000i	0.627868	−	1.08750i	0.987979	−	0.154590i	$-0.0494055\pi$
$762$	6.92820			0.250982
$763$		0			0
$764$	−9.00000	−	15.5885i	−0.325609	−	0.563971i
$765$	2.59808	+	4.50000i	0.0939336	+	0.162698i
$766$	−36.0000			−1.30073
$767$		0			0
$768$	38.0000			1.37121
$769$	3.46410	+	6.00000i	0.124919	+	0.216366i	0.921701	−	0.387901i	$-0.126800\pi$
$769$	3.46410	+	6.00000i	0.124919	+	0.216366i	−0.796782	+	0.604266i	$0.793466\pi$
$770$		0			0
$771$	3.00000	−	5.19615i	0.108042	−	0.187135i
$772$	−5.19615			−0.187014
$773$	17.3205	−	30.0000i	0.622975	−	1.07903i	−0.365953	−	0.930633i	$-0.619257\pi$
$773$	17.3205	−	30.0000i	0.622975	−	1.07903i	0.988929	−	0.148392i	$-0.0474097\pi$
$774$	6.92820	−	12.0000i	0.249029	−	0.431331i
$775$	−6.92820			−0.248868
$776$	6.00000	−	10.3923i	0.215387	−	0.373062i
$777$		0			0
$778$	−7.79423	−	13.5000i	−0.279437	−	0.483998i
$779$	−18.0000			−0.644917
$780$		0			0
$781$		0			0
$782$	−15.5885	−	27.0000i	−0.557442	−	0.965518i
$783$	6.00000	+	10.3923i	0.214423	+	0.371391i
$784$	−17.5000	+	30.3109i	−0.625000	+	1.08253i
$785$	22.5167			0.803654
$786$	−31.1769	+	54.0000i	−1.11204	+	1.92612i
$787$	19.0526	−	33.0000i	0.679150	−	1.17632i	−0.296087	−	0.955161i	$-0.595682\pi$
$787$	19.0526	−	33.0000i	0.679150	−	1.17632i	0.975237	−	0.221162i	$-0.0709849\pi$
$788$	−13.8564			−0.493614
$789$	−12.0000	+	20.7846i	−0.427211	+	0.739952i
$790$	6.00000	+	10.3923i	0.213470	+	0.369742i
$791$		0			0
$792$		0			0
$793$		0			0
$794$	24.0000			0.851728
$795$	−5.19615	−	9.00000i	−0.184289	−	0.319197i
$796$	−1.00000	−	1.73205i	−0.0354441	−	0.0613909i
$797$	21.0000	−	36.3731i	0.743858	−	1.28840i	−0.206868	−	0.978369i	$-0.566327\pi$
$797$	21.0000	−	36.3731i	0.743858	−	1.28840i	0.950726	−	0.310031i	$-0.100340\pi$
$798$		0			0
$799$	−5.19615	+	9.00000i	−0.183827	+	0.318397i
$800$	−5.19615	+	9.00000i	−0.183712	+	0.318198i
$801$	−6.92820			−0.244796
$802$	1.50000	−	2.59808i	0.0529668	−	0.0917413i
$803$		0			0
$804$	3.46410	+	6.00000i	0.122169	+	0.211604i
$805$		0			0
$806$		0			0
$807$	−12.0000			−0.422420
$808$	2.59808	+	4.50000i	0.0914000	+	0.158309i
$809$	−16.5000	−	28.5788i	−0.580109	−	1.00478i	−0.995466	−	0.0951198i	$-0.969677\pi$
$809$	−16.5000	−	28.5788i	−0.580109	−	1.00478i	0.415357	−	0.909659i	$-0.363657\pi$
$810$	−16.5000	+	28.5788i	−0.579751	+	1.00416i
$811$	−38.1051			−1.33805			−0.669026	−	0.743239i	$-0.733288\pi$
$811$	−38.1051			−1.33805			−0.669026	+	0.743239i	$0.733288\pi$
$812$		0			0
$813$	−20.7846	+	36.0000i	−0.728948	+	1.26258i
$814$		0			0
$815$	−18.0000	+	31.1769i	−0.630512	+	1.09208i
$816$	15.0000	+	25.9808i	0.525105	+	0.909509i
$817$	−13.8564	−	24.0000i	−0.484774	−	0.839654i
$818$	27.0000			0.944033
$819$		0			0
$820$	−9.00000			−0.314294
$821$	−20.7846	−	36.0000i	−0.725388	−	1.25641i	−0.958814	−	0.284034i	$-0.908327\pi$
$821$	−20.7846	−	36.0000i	−0.725388	−	1.25641i	0.233426	−	0.972375i	$-0.425006\pi$
$822$	−27.0000	−	46.7654i	−0.941733	−	1.63113i
$823$	−2.00000	+	3.46410i	−0.0697156	+	0.120751i	−0.898776	−	0.438408i	$-0.855543\pi$
$823$	−2.00000	+	3.46410i	−0.0697156	+	0.120751i	0.829060	+	0.559159i	$0.188876\pi$
$824$	−17.3205			−0.603388
$825$		0			0
$826$		0			0
$827$	−20.7846			−0.722752			−0.361376	−	0.932420i	$-0.617693\pi$
$827$	−20.7846			−0.722752			−0.361376	+	0.932420i	$0.617693\pi$
$828$	−3.00000	+	5.19615i	−0.104257	+	0.180579i
$829$	12.5000	+	21.6506i	0.434143	+	0.751958i	0.997225	−	0.0744432i	$-0.0237179\pi$
$829$	12.5000	+	21.6506i	0.434143	+	0.751958i	−0.563082	+	0.826401i	$0.690385\pi$
$830$	20.7846	+	36.0000i	0.721444	+	1.24958i
$831$	14.0000			0.485655
$832$		0			0
$833$	−21.0000			−0.727607
$834$	6.92820	+	12.0000i	0.239904	+	0.415526i
$835$	−12.0000	−	20.7846i	−0.415277	−	0.719281i
$836$		0			0
$837$	−13.8564			−0.478947
$838$	−15.5885	+	27.0000i	−0.538494	+	0.932700i
$839$	22.5167	−	39.0000i	0.777361	−	1.34643i	−0.156096	−	0.987742i	$-0.549891\pi$
$839$	22.5167	−	39.0000i	0.777361	−	1.34643i	0.933458	−	0.358688i	$-0.116776\pi$
$840$		0			0
$841$	10.0000	−	17.3205i	0.344828	−	0.597259i
$842$	−13.5000	−	23.3827i	−0.465241	−	0.805821i
$843$	−22.5167	−	39.0000i	−0.775515	−	1.34323i
$844$	10.0000			0.344214
$845$		0			0
$846$	6.00000			0.206284
$847$		0			0
$848$	−7.50000	−	12.9904i	−0.257551	−	0.446092i
$849$	4.00000	−	6.92820i	0.137280	−	0.237775i
$850$	−10.3923			−0.356453
$851$	−25.9808	+	45.0000i	−0.890609	+	1.54258i
$852$	−3.46410	+	6.00000i	−0.118678	+	0.205557i
$853$	−25.9808			−0.889564			−0.444782	−	0.895639i	$-0.646719\pi$
$853$	−25.9808			−0.889564			−0.444782	+	0.895639i	$0.646719\pi$
$854$		0			0
$855$	−3.00000	−	5.19615i	−0.102598	−	0.177705i
$856$	5.19615	+	9.00000i	0.177601	+	0.307614i
$857$	−3.00000			−0.102478			−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000			−0.102478			−0.0512390	+	0.998686i	$0.516317\pi$
$858$		0			0
$859$	−14.0000			−0.477674			−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000			−0.477674			−0.238837	+	0.971060i	$0.576766\pi$
$860$	−6.92820	−	12.0000i	−0.236250	−	0.409197i
$861$		0			0
$862$	−6.00000	+	10.3923i	−0.204361	+	0.353963i
$863$	−27.7128			−0.943355			−0.471678	−	0.881771i	$-0.656351\pi$
$863$	−27.7128			−0.943355			−0.471678	+	0.881771i	$0.656351\pi$
$864$	−10.3923	+	18.0000i	−0.353553	+	0.612372i
$865$	−5.19615	+	9.00000i	−0.176674	+	0.306009i
$866$	29.4449			1.00058
$867$	8.00000	−	13.8564i	0.271694	−	0.470588i
$868$		0			0
$869$		0			0
$870$	−18.0000			−0.610257
$871$		0			0
$872$	24.0000			0.812743
$873$	−3.46410	−	6.00000i	−0.117242	−	0.203069i
$874$	18.0000	+	31.1769i	0.608859	+	1.05457i
$875$		0			0
$876$	−3.46410			−0.117041
$877$	−6.06218	+	10.5000i	−0.204705	+	0.354560i	−0.950039	−	0.312132i	$-0.898957\pi$
$877$	−6.06218	+	10.5000i	−0.204705	+	0.354560i	0.745334	+	0.666692i	$0.232290\pi$
$878$	−24.2487	+	42.0000i	−0.818354	+	1.41743i
$879$	−10.3923			−0.350524
$880$		0			0
$881$	13.5000	+	23.3827i	0.454827	+	0.787783i	0.998678	−	0.0513987i	$-0.0163679\pi$
$881$	13.5000	+	23.3827i	0.454827	+	0.787783i	−0.543852	+	0.839181i	$0.683035\pi$
$882$	6.06218	+	10.5000i	0.204124	+	0.353553i
$883$	−10.0000			−0.336527			−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000			−0.336527			−0.168263	+	0.985742i	$0.553816\pi$
$884$		0			0
$885$	24.0000			0.806751
$886$	10.3923	+	18.0000i	0.349136	+	0.604722i
$887$	−18.0000	−	31.1769i	−0.604381	−	1.04682i	−0.992149	−	0.125061i	$-0.960087\pi$
$887$	−18.0000	−	31.1769i	−0.604381	−	1.04682i	0.387768	−	0.921757i	$-0.373246\pi$
$888$	15.0000	−	25.9808i	0.503367	−	0.871857i
$889$		0			0
$890$	−10.3923	+	18.0000i	−0.348351	+	0.603361i
$891$		0			0
$892$	10.3923			0.347960
$893$	6.00000	−	10.3923i	0.200782	−	0.347765i
$894$	33.0000	+	57.1577i	1.10369	+	1.91164i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	12.0000			0.400445
$899$	−5.19615	−	9.00000i	−0.173301	−	0.300167i
$900$	1.00000	+	1.73205i	0.0333333	+	0.0577350i
$901$	4.50000	−	7.79423i	0.149917	−	0.259663i
$902$		0			0
$903$		0			0
$904$	−12.9904	+	22.5000i	−0.432054	+	0.748339i
$905$	19.0526			0.633328
$906$	30.0000	−	51.9615i	0.996683	−	1.72631i
$907$	14.0000	+	24.2487i	0.464862	+	0.805165i	0.999195	−	0.0401089i	$-0.0127705\pi$
$907$	14.0000	+	24.2487i	0.464862	+	0.805165i	−0.534333	+	0.845274i	$0.679437\pi$
$908$	−12.1244	−	21.0000i	−0.402361	−	0.696909i
$909$	3.00000			0.0995037
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−17.3205	−	30.0000i	−0.573539	−	0.993399i
$913$		0			0
$914$	1.50000	−	2.59808i	0.0496156	−	0.0859367i
$915$	−3.46410			−0.114520
$916$		0			0
$917$		0			0
$918$	−20.7846			−0.685994
$919$	−11.0000	+	19.0526i	−0.362857	+	0.628486i	−0.988430	−	0.151680i	$-0.951532\pi$
$919$	−11.0000	+	19.0526i	−0.362857	+	0.628486i	0.625573	+	0.780165i	$0.284865\pi$
$920$	−9.00000	−	15.5885i	−0.296721	−	0.513936i
$921$	−17.3205	−	30.0000i	−0.570730	−	0.988534i
$922$	39.0000			1.28440
$923$		0			0
$924$		0			0
$925$	8.66025	+	15.0000i	0.284747	+	0.493197i
$926$	12.0000	+	20.7846i	0.394344	+	0.683025i
$927$	−5.00000	+	8.66025i	−0.164222	+	0.284440i
$928$	−15.5885			−0.511716
$929$	23.3827	−	40.5000i	0.767161	−	1.32876i	−0.171935	−	0.985108i	$-0.555002\pi$
$929$	23.3827	−	40.5000i	0.767161	−	1.32876i	0.939096	−	0.343654i	$-0.111665\pi$
$930$	10.3923	−	18.0000i	0.340777	−	0.590243i
$931$	24.2487			0.794719
$932$	3.00000	−	5.19615i	0.0982683	−	0.170206i
$933$	−30.0000	−	51.9615i	−0.982156	−	1.70114i
$934$	10.3923	+	18.0000i	0.340047	+	0.588978i
$935$		0			0
$936$		0			0
$937$	7.00000			0.228680			0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000			0.228680			0.114340	+	0.993442i	$0.463525\pi$
$938$		0			0
$939$	−10.0000	−	17.3205i	−0.326338	−	0.565233i
$940$	3.00000	−	5.19615i	0.0978492	−	0.169480i
$941$	20.7846			0.677559			0.338779	−	0.940866i	$-0.389986\pi$
$941$	20.7846			0.677559			0.338779	+	0.940866i	$0.389986\pi$
$942$	22.5167	−	39.0000i	0.733632	−	1.27069i
$943$	−15.5885	+	27.0000i	−0.507630	+	0.879241i
$944$	34.6410			1.12747
$945$		0			0
$946$		0			0
$947$	8.66025	+	15.0000i	0.281420	+	0.487435i	0.971735	−	0.236075i	$-0.0758611\pi$
$947$	8.66025	+	15.0000i	0.281420	+	0.487435i	−0.690314	+	0.723510i	$0.742528\pi$
$948$	8.00000			0.259828
$949$		0			0
$950$	12.0000			0.389331
$951$	5.19615	+	9.00000i	0.168497	+	0.291845i
$952$		0			0
$953$	−3.00000	+	5.19615i	−0.0971795	+	0.168320i	−0.910516	−	0.413473i	$-0.864315\pi$
$953$	−3.00000	+	5.19615i	−0.0971795	+	0.168320i	0.813337	+	0.581793i	$0.197649\pi$
$954$	−5.19615			−0.168232
$955$	15.5885	−	27.0000i	0.504431	−	0.873699i
$956$	10.3923	−	18.0000i	0.336111	−	0.582162i
$957$		0			0
$958$	−21.0000	+	36.3731i	−0.678479	+	1.17516i
$959$		0			0
$960$	1.73205	+	3.00000i	0.0559017	+	0.0968246i
$961$	−19.0000			−0.612903
$962$		0			0
$963$	6.00000			0.193347
$964$	−0.866025	−	1.50000i	−0.0278928	−	0.0483117i
$965$	−4.50000	−	7.79423i	−0.144860	−	0.250905i
$966$		0			0
$967$	58.8897			1.89377			0.946883	−	0.321578i	$-0.104213\pi$
$967$	58.8897			1.89377			0.946883	+	0.321578i	$0.104213\pi$
$968$	−9.52628	+	16.5000i	−0.306186	+	0.530330i
$969$	10.3923	−	18.0000i	0.333849	−	0.578243i
$970$	−20.7846			−0.667354
$971$	−3.00000	+	5.19615i	−0.0962746	+	0.166752i	−0.910140	−	0.414301i	$-0.864026\pi$
$971$	−3.00000	+	5.19615i	−0.0962746	+	0.166752i	0.813865	+	0.581054i	$0.197359\pi$
$972$	5.00000	+	8.66025i	0.160375	+	0.277778i
$973$		0			0
$974$	−12.0000			−0.384505
$975$		0			0
$976$	−5.00000			−0.160046
$977$	21.6506	+	37.5000i	0.692665	+	1.19973i	0.970961	+	0.239236i	$0.0768970\pi$
$977$	21.6506	+	37.5000i	0.692665	+	1.19973i	−0.278296	+	0.960495i	$0.589770\pi$
$978$	36.0000	+	62.3538i	1.15115	+	1.99386i
$979$		0			0
$980$	12.1244			0.387298
$981$	6.92820	−	12.0000i	0.221201	−	0.383131i
$982$	10.3923	−	18.0000i	0.331632	−	0.574403i
$983$	−51.9615			−1.65732			−0.828658	−	0.559756i	$-0.810895\pi$
$983$	−51.9615			−1.65732			−0.828658	+	0.559756i	$0.810895\pi$
$984$	9.00000	−	15.5885i	0.286910	−	0.496942i
$985$	−12.0000	−	20.7846i	−0.382352	−	0.662253i
$986$	−7.79423	−	13.5000i	−0.248219	−	0.429928i
$987$		0			0
$988$		0			0
$989$	−48.0000			−1.52631
$990$		0			0
$991$	−1.00000	−	1.73205i	−0.0317660	−	0.0550204i	0.849705	−	0.527258i	$-0.176780\pi$
$991$	−1.00000	−	1.73205i	−0.0317660	−	0.0550204i	−0.881471	+	0.472237i	$0.843446\pi$
$992$	9.00000	−	15.5885i	0.285750	−	0.494934i
$993$	−55.4256			−1.75888
$994$		0			0
$995$	1.73205	−	3.00000i	0.0549097	−	0.0951064i
$996$	27.7128			0.878114
$997$	−8.50000	+	14.7224i	−0.269198	+	0.466264i	−0.968655	−	0.248410i	$-0.920092\pi$
$997$	−8.50000	+	14.7224i	−0.269198	+	0.466264i	0.699457	+	0.714675i	$0.253425\pi$
$998$	−27.0000	−	46.7654i	−0.854670	−	1.48033i
$999$	17.3205	+	30.0000i	0.547997	+	0.949158i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.c.a.146.1		4
13.2	odd	12		169.2.b.a.168.1		2
13.3	even	3		169.2.a.a.1.2		2
13.4	even	6	inner	169.2.c.a.22.2		4
13.5	odd	4		13.2.e.a.10.1	yes	2
13.6	odd	12		169.2.e.a.147.1		2
13.7	odd	12		13.2.e.a.4.1	✓	2
13.8	odd	4		169.2.e.a.23.1		2
13.9	even	3	inner	169.2.c.a.22.1		4
13.10	even	6		169.2.a.a.1.1		2
13.11	odd	12		169.2.b.a.168.2		2
13.12	even	2	inner	169.2.c.a.146.2		4
39.2	even	12		1521.2.b.a.1351.2		2
39.5	even	4		117.2.q.c.10.1		2
39.11	even	12		1521.2.b.a.1351.1		2
39.20	even	12		117.2.q.c.82.1		2
39.23	odd	6		1521.2.a.k.1.2		2
39.29	odd	6		1521.2.a.k.1.1		2
52.3	odd	6		2704.2.a.o.1.1		2
52.7	even	12		208.2.w.b.17.1		2
52.11	even	12		2704.2.f.b.337.1		2
52.15	even	12		2704.2.f.b.337.2		2
52.23	odd	6		2704.2.a.o.1.2		2
52.31	even	4		208.2.w.b.49.1		2
65.7	even	12		325.2.m.a.199.2		4
65.18	even	4		325.2.m.a.49.2		4
65.29	even	6		4225.2.a.v.1.1		2
65.33	even	12		325.2.m.a.199.1		4
65.44	odd	4		325.2.n.a.101.1		2
65.49	even	6		4225.2.a.v.1.2		2
65.57	even	4		325.2.m.a.49.1		4
65.59	odd	12		325.2.n.a.251.1		2
91.5	even	12		637.2.u.b.361.1		2
91.18	odd	12		637.2.k.a.569.1		2
91.20	even	12		637.2.q.a.589.1		2
91.31	even	12		637.2.k.c.569.1		2
91.33	even	12		637.2.k.c.459.1		2
91.44	odd	12		637.2.u.c.361.1		2
91.46	odd	12		637.2.u.c.30.1		2
91.55	odd	6		8281.2.a.q.1.2		2
91.59	even	12		637.2.u.b.30.1		2
91.62	odd	6		8281.2.a.q.1.1		2
91.72	odd	12		637.2.k.a.459.1		2
91.83	even	4		637.2.q.a.491.1		2
104.5	odd	4		832.2.w.d.257.1		2
104.59	even	12		832.2.w.a.641.1		2
104.83	even	4		832.2.w.a.257.1		2
104.85	odd	12		832.2.w.d.641.1		2
156.59	odd	12		1872.2.by.d.433.1		2
156.83	odd	4		1872.2.by.d.1297.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.2.e.a.4.1	✓	2	13.7	odd	12
13.2.e.a.10.1	yes	2	13.5	odd	4
117.2.q.c.10.1		2	39.5	even	4
117.2.q.c.82.1		2	39.20	even	12
169.2.a.a.1.1		2	13.10	even	6
169.2.a.a.1.2		2	13.3	even	3
169.2.b.a.168.1		2	13.2	odd	12
169.2.b.a.168.2		2	13.11	odd	12
169.2.c.a.22.1		4	13.9	even	3	inner
169.2.c.a.22.2		4	13.4	even	6	inner
169.2.c.a.146.1		4	1.1	even	1	trivial
169.2.c.a.146.2		4	13.12	even	2	inner
169.2.e.a.23.1		2	13.8	odd	4
169.2.e.a.147.1		2	13.6	odd	12
208.2.w.b.17.1		2	52.7	even	12
208.2.w.b.49.1		2	52.31	even	4
325.2.m.a.49.1		4	65.57	even	4
325.2.m.a.49.2		4	65.18	even	4
325.2.m.a.199.1		4	65.33	even	12
325.2.m.a.199.2		4	65.7	even	12
325.2.n.a.101.1		2	65.44	odd	4
325.2.n.a.251.1		2	65.59	odd	12
637.2.k.a.459.1		2	91.72	odd	12
637.2.k.a.569.1		2	91.18	odd	12
637.2.k.c.459.1		2	91.33	even	12
637.2.k.c.569.1		2	91.31	even	12
637.2.q.a.491.1		2	91.83	even	4
637.2.q.a.589.1		2	91.20	even	12
637.2.u.b.30.1		2	91.59	even	12
637.2.u.b.361.1		2	91.5	even	12
637.2.u.c.30.1		2	91.46	odd	12
637.2.u.c.361.1		2	91.44	odd	12
832.2.w.a.257.1		2	104.83	even	4
832.2.w.a.641.1		2	104.59	even	12
832.2.w.d.257.1		2	104.5	odd	4
832.2.w.d.641.1		2	104.85	odd	12
1521.2.a.k.1.1		2	39.29	odd	6
1521.2.a.k.1.2		2	39.23	odd	6
1521.2.b.a.1351.1		2	39.11	even	12
1521.2.b.a.1351.2		2	39.2	even	12
1872.2.by.d.433.1		2	156.59	odd	12
1872.2.by.d.1297.1		2	156.83	odd	4
2704.2.a.o.1.1		2	52.3	odd	6
2704.2.a.o.1.2		2	52.23	odd	6
2704.2.f.b.337.1		2	52.11	even	12
2704.2.f.b.337.2		2	52.15	even	12
4225.2.a.v.1.1		2	65.29	even	6
4225.2.a.v.1.2		2	65.49	even	6
8281.2.a.q.1.1		2	91.62	odd	6
8281.2.a.q.1.2		2	91.55	odd	6

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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 1.50000i) q^{2} +(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{4} -1.73205 q^{5} +(-1.73205 + 3.00000i) q^{6} -1.73205 q^{8} +(-0.500000 + 0.866025i) q^{9} +(1.50000 + 2.59808i) q^{10} +2.00000 q^{12} +(1.73205 + 3.00000i) q^{15} +(2.50000 + 4.33013i) q^{16} +(-1.50000 + 2.59808i) q^{17} +1.73205 q^{18} +(1.73205 - 3.00000i) q^{19} +(0.866025 - 1.50000i) q^{20} +(-3.00000 - 5.19615i) q^{23} +(1.73205 + 3.00000i) q^{24} -2.00000 q^{25} -4.00000 q^{27} +(-1.50000 - 2.59808i) q^{29} +(3.00000 - 5.19615i) q^{30} +3.46410 q^{31} +(2.59808 - 4.50000i) q^{32} +5.19615 q^{34} +(-0.500000 - 0.866025i) q^{36} +(-4.33013 - 7.50000i) q^{37} -6.00000 q^{38} +3.00000 q^{40} +(-2.59808 - 4.50000i) q^{41} +(4.00000 - 6.92820i) q^{43} +(0.866025 - 1.50000i) q^{45} +(-5.19615 + 9.00000i) q^{46} +3.46410 q^{47} +(5.00000 - 8.66025i) q^{48} +(3.50000 + 6.06218i) q^{49} +(1.73205 + 3.00000i) q^{50} +6.00000 q^{51} -3.00000 q^{53} +(3.46410 + 6.00000i) q^{54} -6.92820 q^{57} +(-2.59808 + 4.50000i) q^{58} +(3.46410 - 6.00000i) q^{59} -3.46410 q^{60} +(-0.500000 + 0.866025i) q^{61} +(-3.00000 - 5.19615i) q^{62} +1.00000 q^{64} +(1.73205 + 3.00000i) q^{67} +(-1.50000 - 2.59808i) q^{68} +(-6.00000 + 10.3923i) q^{69} +(-1.73205 + 3.00000i) q^{71} +(0.866025 - 1.50000i) q^{72} -1.73205 q^{73} +(-7.50000 + 12.9904i) q^{74} +(2.00000 + 3.46410i) q^{75} +(1.73205 + 3.00000i) q^{76} +4.00000 q^{79} +(-4.33013 - 7.50000i) q^{80} +(5.50000 + 9.52628i) q^{81} +(-4.50000 + 7.79423i) q^{82} +13.8564 q^{83} +(2.59808 - 4.50000i) q^{85} -13.8564 q^{86} +(-3.00000 + 5.19615i) q^{87} +(3.46410 + 6.00000i) q^{89} -3.00000 q^{90} +6.00000 q^{92} +(-3.46410 - 6.00000i) q^{93} +(-3.00000 - 5.19615i) q^{94} +(-3.00000 + 5.19615i) q^{95} -10.3923 q^{96} +(-3.46410 + 6.00000i) q^{97} +(6.06218 - 10.5000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 2 q^{4} - 2 q^{9} + 6 q^{10} + 8 q^{12} + 10 q^{16} - 6 q^{17} - 12 q^{23} - 8 q^{25} - 16 q^{27} - 6 q^{29} + 12 q^{30} - 2 q^{36} - 24 q^{38} + 12 q^{40} + 16 q^{43} + 20 q^{48} + 14 q^{49}+ \cdots - 12 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 - 2 * q^4 - 2 * q^9 + 6 * q^10 + 8 * q^12 + 10 * q^16 - 6 * q^17 - 12 * q^23 - 8 * q^25 - 16 * q^27 - 6 * q^29 + 12 * q^30 - 2 * q^36 - 24 * q^38 + 12 * q^40 + 16 * q^43 + 20 * q^48 + 14 * q^49 + 24 * q^51 - 12 * q^53 - 2 * q^61 - 12 * q^62 + 4 * q^64 - 6 * q^68 - 24 * q^69 - 30 * q^74 + 8 * q^75 + 16 * q^79 + 22 * q^81 - 18 * q^82 - 12 * q^87 - 12 * q^90 + 24 * q^92 - 12 * q^94 - 12 * q^95

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