LMFDB - Embedded newform 1344.2.q.a.193.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,2,Mod(193,1344)] 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("1344.193"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,-4,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$10.7318940317$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			193.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1344.193
Dual form			1344.2.q.a.961.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 + 0.866025i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(3.00000 - 5.19615i) q^{11} -5.00000 q^{13} +4.00000 q^{15} +(-1.00000 + 1.73205i) q^{17} +(0.500000 + 0.866025i) q^{19} +(2.00000 - 1.73205i) q^{21} +(3.00000 + 5.19615i) q^{23} +(-5.50000 + 9.52628i) q^{25} +1.00000 q^{27} +(1.50000 - 2.59808i) q^{31} +(3.00000 + 5.19615i) q^{33} +(2.00000 + 10.3923i) q^{35} +(1.50000 + 2.59808i) q^{37} +(2.50000 - 4.33013i) q^{39} -6.00000 q^{41} -5.00000 q^{43} +(-2.00000 + 3.46410i) q^{45} +(2.00000 + 3.46410i) q^{47} +(5.50000 + 4.33013i) q^{49} +(-1.00000 - 1.73205i) q^{51} +(-3.00000 + 5.19615i) q^{53} -24.0000 q^{55} -1.00000 q^{57} +(-3.00000 + 5.19615i) q^{59} +(-1.00000 - 1.73205i) q^{61} +(0.500000 + 2.59808i) q^{63} +(10.0000 + 17.3205i) q^{65} +(3.50000 - 6.06218i) q^{67} -6.00000 q^{69} +16.0000 q^{71} +(1.50000 - 2.59808i) q^{73} +(-5.50000 - 9.52628i) q^{75} +(-12.0000 + 10.3923i) q^{77} +(-5.50000 - 9.52628i) q^{79} +(-0.500000 + 0.866025i) q^{81} -12.0000 q^{83} +8.00000 q^{85} +(-2.00000 - 3.46410i) q^{89} +(12.5000 + 4.33013i) q^{91} +(1.50000 + 2.59808i) q^{93} +(2.00000 - 3.46410i) q^{95} -6.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 4 q^{5} - 5 q^{7} - q^{9} + 6 q^{11} - 10 q^{13} + 8 q^{15} - 2 q^{17} + q^{19} + 4 q^{21} + 6 q^{23} - 11 q^{25} + 2 q^{27} + 3 q^{31} + 6 q^{33} + 4 q^{35} + 3 q^{37} + 5 q^{39} - 12 q^{41}+ \cdots - 12 q^{99}+O(q^{100}) $ 2 * q - q^3 - 4 * q^5 - 5 * q^7 - q^9 + 6 * q^11 - 10 * q^13 + 8 * q^15 - 2 * q^17 + q^19 + 4 * q^21 + 6 * q^23 - 11 * q^25 + 2 * q^27 + 3 * q^31 + 6 * q^33 + 4 * q^35 + 3 * q^37 + 5 * q^39 - 12 * q^41 - 10 * q^43 - 4 * q^45 + 4 * q^47 + 11 * q^49 - 2 * q^51 - 6 * q^53 - 48 * q^55 - 2 * q^57 - 6 * q^59 - 2 * q^61 + q^63 + 20 * q^65 + 7 * q^67 - 12 * q^69 + 32 * q^71 + 3 * q^73 - 11 * q^75 - 24 * q^77 - 11 * q^79 - q^81 - 24 * q^83 + 16 * q^85 - 4 * q^89 + 25 * q^91 + 3 * q^93 + 4 * q^95 - 12 * q^97 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$		0			0
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.0599153	−	0.998203i	$-0.519083\pi$
$6$		0			0
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	3.00000	−	5.19615i	0.904534	−	1.56670i	0.0829925	−	0.996550i	$-0.473552\pi$
$11$	3.00000	−	5.19615i	0.904534	−	1.56670i	0.821541	−	0.570149i	$-0.193114\pi$
$12$		0			0
$13$	−5.00000			−1.38675			−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000			−1.38675			−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$	4.00000			1.03280
$16$		0			0
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	0.718900	+	0.695113i	$0.244646\pi$
$18$		0			0
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	0.917663	−	0.397360i	$-0.130073\pi$
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$	2.00000	−	1.73205i	0.436436	−	0.377964i
$22$		0			0
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	1.50000	−	2.59808i	0.269408	−	0.466628i	−0.699301	−	0.714827i	$-0.746505\pi$
$31$	1.50000	−	2.59808i	0.269408	−	0.466628i	0.968709	+	0.248199i	$0.0798387\pi$
$32$		0			0
$33$	3.00000	+	5.19615i	0.522233	+	0.904534i
$34$		0			0
$35$	2.00000	+	10.3923i	0.338062	+	1.75662i
$36$		0			0
$37$	1.50000	+	2.59808i	0.246598	+	0.427121i	0.962580	−	0.270998i	$-0.0873538\pi$
$37$	1.50000	+	2.59808i	0.246598	+	0.427121i	−0.715981	+	0.698119i	$0.754020\pi$
$38$		0			0
$39$	2.50000	−	4.33013i	0.400320	−	0.693375i
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	−5.00000			−0.762493			−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000			−0.762493			−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$	−2.00000	+	3.46410i	−0.298142	+	0.516398i
$46$		0			0
$47$	2.00000	+	3.46410i	0.291730	+	0.505291i	0.974219	−	0.225605i	$-0.0724358\pi$
$47$	2.00000	+	3.46410i	0.291730	+	0.505291i	−0.682489	+	0.730896i	$0.739102\pi$
$48$		0			0
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$		0			0
$51$	−1.00000	−	1.73205i	−0.140028	−	0.242536i
$52$		0			0
$53$	−3.00000	+	5.19615i	−0.412082	+	0.713746i	−0.995117	−	0.0987002i	$-0.968532\pi$
$53$	−3.00000	+	5.19615i	−0.412082	+	0.713746i	0.583036	+	0.812447i	$0.301865\pi$
$54$		0			0
$55$	−24.0000			−3.23616
$56$		0			0
$57$	−1.00000			−0.132453
$58$		0			0
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	−0.992524	−	0.122047i	$-0.961054\pi$
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	0.601958	+	0.798528i	$0.294388\pi$
$60$		0			0
$61$	−1.00000	−	1.73205i	−0.128037	−	0.221766i	0.794879	−	0.606768i	$-0.207534\pi$
$61$	−1.00000	−	1.73205i	−0.128037	−	0.221766i	−0.922916	+	0.385002i	$0.874201\pi$
$62$		0			0
$63$	0.500000	+	2.59808i	0.0629941	+	0.327327i
$64$		0			0
$65$	10.0000	+	17.3205i	1.24035	+	2.14834i
$66$		0			0
$67$	3.50000	−	6.06218i	0.427593	−	0.740613i	−0.569066	−	0.822292i	$-0.692695\pi$
$67$	3.50000	−	6.06218i	0.427593	−	0.740613i	0.996659	+	0.0816792i	$0.0260283\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$	16.0000			1.89885			0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000			1.89885			0.949425	+	0.313993i	$0.101667\pi$
$72$		0			0
$73$	1.50000	−	2.59808i	0.175562	−	0.304082i	−0.764794	−	0.644275i	$-0.777159\pi$
$73$	1.50000	−	2.59808i	0.175562	−	0.304082i	0.940356	+	0.340193i	$0.110493\pi$
$74$		0			0
$75$	−5.50000	−	9.52628i	−0.635085	−	1.10000i
$76$		0			0
$77$	−12.0000	+	10.3923i	−1.36753	+	1.18431i
$78$		0			0
$79$	−5.50000	−	9.52628i	−0.618798	−	1.07179i	−0.989705	−	0.143120i	$-0.954286\pi$
$79$	−5.50000	−	9.52628i	−0.618798	−	1.07179i	0.370907	−	0.928670i	$-0.379047\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$	8.00000			0.867722
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.00000	−	3.46410i	−0.212000	−	0.367194i	0.740341	−	0.672232i	$-0.234664\pi$
$89$	−2.00000	−	3.46410i	−0.212000	−	0.367194i	−0.952340	+	0.305038i	$0.901331\pi$
$90$		0			0
$91$	12.5000	+	4.33013i	1.31036	+	0.453921i
$92$		0			0
$93$	1.50000	+	2.59808i	0.155543	+	0.269408i
$94$		0			0
$95$	2.00000	−	3.46410i	0.205196	−	0.355409i
$96$		0			0
$97$	−6.00000			−0.609208			−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000			−0.609208			−0.304604	+	0.952479i	$0.598524\pi$
$98$		0			0
$99$	−6.00000			−0.603023
$100$		0			0
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	−0.811976	−	0.583691i	$-0.801608\pi$
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	0.911479	+	0.411346i	$0.134941\pi$
$102$		0			0
$103$	5.50000	+	9.52628i	0.541931	+	0.938652i	0.998793	+	0.0491146i	$0.0156400\pi$
$103$	5.50000	+	9.52628i	0.541931	+	0.938652i	−0.456862	+	0.889538i	$0.651027\pi$
$104$		0			0
$105$	−10.0000	−	3.46410i	−0.975900	−	0.338062i
$106$		0			0
$107$	5.00000	+	8.66025i	0.483368	+	0.837218i	0.999818	−	0.0190994i	$-0.00607989\pi$
$107$	5.00000	+	8.66025i	0.483368	+	0.837218i	−0.516449	+	0.856318i	$0.672747\pi$
$108$		0			0
$109$	−7.50000	+	12.9904i	−0.718370	+	1.24425i	0.243276	+	0.969957i	$0.421778\pi$
$109$	−7.50000	+	12.9904i	−0.718370	+	1.24425i	−0.961645	+	0.274296i	$0.911555\pi$
$110$		0			0
$111$	−3.00000			−0.284747
$112$		0			0
$113$	−16.0000			−1.50515			−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000			−1.50515			−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$	12.0000	−	20.7846i	1.11901	−	1.93817i
$116$		0			0
$117$	2.50000	+	4.33013i	0.231125	+	0.400320i
$118$		0			0
$119$	4.00000	−	3.46410i	0.366679	−	0.317554i
$120$		0			0
$121$	−12.5000	−	21.6506i	−1.13636	−	1.96824i
$122$		0			0
$123$	3.00000	−	5.19615i	0.270501	−	0.468521i
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$	−7.00000			−0.621150			−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000			−0.621150			−0.310575	+	0.950549i	$0.600522\pi$
$128$		0			0
$129$	2.50000	−	4.33013i	0.220113	−	0.381246i
$130$		0			0
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	0.966803	−	0.255524i	$-0.0822479\pi$
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	−0.704692	+	0.709514i	$0.748915\pi$
$132$		0			0
$133$	−0.500000	−	2.59808i	−0.0433555	−	0.225282i
$134$		0			0
$135$	−2.00000	−	3.46410i	−0.172133	−	0.298142i
$136$		0			0
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	−0.487278	−	0.873247i	$-0.662010\pi$
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	0.999893	−	0.0146279i	$-0.00465636\pi$
$138$		0			0
$139$	−5.00000			−0.424094			−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000			−0.424094			−0.212047	+	0.977259i	$0.568013\pi$
$140$		0			0
$141$	−4.00000			−0.336861
$142$		0			0
$143$	−15.0000	+	25.9808i	−1.25436	+	2.17262i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−6.50000	+	2.59808i	−0.536111	+	0.214286i
$148$		0			0
$149$	−2.00000	−	3.46410i	−0.163846	−	0.283790i	0.772399	−	0.635138i	$-0.219057\pi$
$149$	−2.00000	−	3.46410i	−0.163846	−	0.283790i	−0.936245	+	0.351348i	$0.885723\pi$
$150$		0			0
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	−0.656101	−	0.754673i	$-0.727796\pi$
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	0.981617	+	0.190864i	$0.0611289\pi$
$152$		0			0
$153$	2.00000			0.161690
$154$		0			0
$155$	−12.0000			−0.963863
$156$		0			0
$157$	5.00000	−	8.66025i	0.399043	−	0.691164i	−0.594565	−	0.804048i	$-0.702676\pi$
$157$	5.00000	−	8.66025i	0.399043	−	0.691164i	0.993608	+	0.112884i	$0.0360089\pi$
$158$		0			0
$159$	−3.00000	−	5.19615i	−0.237915	−	0.412082i
$160$		0			0
$161$	−3.00000	−	15.5885i	−0.236433	−	1.22854i
$162$		0			0
$163$	10.0000	+	17.3205i	0.783260	+	1.35665i	0.930033	+	0.367477i	$0.119778\pi$
$163$	10.0000	+	17.3205i	0.783260	+	1.35665i	−0.146772	+	0.989170i	$0.546888\pi$
$164$		0			0
$165$	12.0000	−	20.7846i	0.934199	−	1.61808i
$166$		0			0
$167$	−8.00000			−0.619059			−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000			−0.619059			−0.309529	+	0.950890i	$0.600171\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$	0.500000	−	0.866025i	0.0382360	−	0.0662266i
$172$		0			0
$173$	11.0000	+	19.0526i	0.836315	+	1.44854i	0.892956	+	0.450145i	$0.148628\pi$
$173$	11.0000	+	19.0526i	0.836315	+	1.44854i	−0.0566411	+	0.998395i	$0.518039\pi$
$174$		0			0
$175$	22.0000	−	19.0526i	1.66304	−	1.44024i
$176$		0			0
$177$	−3.00000	−	5.19615i	−0.225494	−	0.390567i
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−25.0000			−1.85824			−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000			−1.85824			−0.929118	+	0.369784i	$0.879432\pi$
$182$		0			0
$183$	2.00000			0.147844
$184$		0			0
$185$	6.00000	−	10.3923i	0.441129	−	0.764057i
$186$		0			0
$187$	6.00000	+	10.3923i	0.438763	+	0.759961i
$188$		0			0
$189$	−2.50000	−	0.866025i	−0.181848	−	0.0629941i
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	0.500000	−	0.866025i	0.0359908	−	0.0623379i	−0.847469	−	0.530845i	$-0.821875\pi$
$193$	0.500000	−	0.866025i	0.0359908	−	0.0623379i	0.883460	+	0.468507i	$0.155208\pi$
$194$		0			0
$195$	−20.0000			−1.43223
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$	−10.0000	+	17.3205i	−0.708881	+	1.22782i	0.256391	+	0.966573i	$0.417466\pi$
$199$	−10.0000	+	17.3205i	−0.708881	+	1.22782i	−0.965272	+	0.261245i	$0.915867\pi$
$200$		0			0
$201$	3.50000	+	6.06218i	0.246871	+	0.427593i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	12.0000	+	20.7846i	0.838116	+	1.45166i
$206$		0			0
$207$	3.00000	−	5.19615i	0.208514	−	0.361158i
$208$		0			0
$209$	6.00000			0.415029
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$	−8.00000	+	13.8564i	−0.548151	+	0.949425i
$214$		0			0
$215$	10.0000	+	17.3205i	0.681994	+	1.18125i
$216$		0			0
$217$	−6.00000	+	5.19615i	−0.407307	+	0.352738i
$218$		0			0
$219$	1.50000	+	2.59808i	0.101361	+	0.175562i
$220$		0			0
$221$	5.00000	−	8.66025i	0.336336	−	0.582552i
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		0			0
$225$	11.0000			0.733333
$226$		0			0
$227$	11.0000	−	19.0526i	0.730096	−	1.26456i	−0.226746	−	0.973954i	$-0.572809\pi$
$227$	11.0000	−	19.0526i	0.730096	−	1.26456i	0.956842	−	0.290609i	$-0.0938578\pi$
$228$		0			0
$229$	−5.50000	−	9.52628i	−0.363450	−	0.629514i	0.625076	−	0.780564i	$-0.285068\pi$
$229$	−5.50000	−	9.52628i	−0.363450	−	0.629514i	−0.988526	+	0.151050i	$0.951735\pi$
$230$		0			0
$231$	−3.00000	−	15.5885i	−0.197386	−	1.02565i
$232$		0			0
$233$	4.00000	+	6.92820i	0.262049	+	0.453882i	0.966786	−	0.255586i	$-0.0822686\pi$
$233$	4.00000	+	6.92820i	0.262049	+	0.453882i	−0.704737	+	0.709468i	$0.748935\pi$
$234$		0			0
$235$	8.00000	−	13.8564i	0.521862	−	0.903892i
$236$		0			0
$237$	11.0000			0.714527
$238$		0			0
$239$	2.00000			0.129369			0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000			0.129369			0.0646846	+	0.997906i	$0.479396\pi$
$240$		0			0
$241$	−5.00000	+	8.66025i	−0.322078	+	0.557856i	−0.980917	−	0.194429i	$-0.937715\pi$
$241$	−5.00000	+	8.66025i	−0.322078	+	0.557856i	0.658838	+	0.752285i	$0.271048\pi$
$242$		0			0
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	4.00000	−	27.7128i	0.255551	−	1.77051i
$246$		0			0
$247$	−2.50000	−	4.33013i	−0.159071	−	0.275519i
$248$		0			0
$249$	6.00000	−	10.3923i	0.380235	−	0.658586i
$250$		0			0
$251$	14.0000			0.883672			0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000			0.883672			0.441836	+	0.897096i	$0.354327\pi$
$252$		0			0
$253$	36.0000			2.26330
$254$		0			0
$255$	−4.00000	+	6.92820i	−0.250490	+	0.433861i
$256$		0			0
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	0.990217	−	0.139533i	$-0.0445601\pi$
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	−0.615948	+	0.787787i	$0.711227\pi$
$258$		0			0
$259$	−1.50000	−	7.79423i	−0.0932055	−	0.484310i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	24.0000			1.47431
$266$		0			0
$267$	4.00000			0.244796
$268$		0			0
$269$	−1.00000	+	1.73205i	−0.0609711	+	0.105605i	−0.894900	−	0.446267i	$-0.852753\pi$
$269$	−1.00000	+	1.73205i	−0.0609711	+	0.105605i	0.833929	+	0.551872i	$0.186086\pi$
$270$		0			0
$271$	−8.00000	−	13.8564i	−0.485965	−	0.841717i	0.513905	−	0.857847i	$-0.328199\pi$
$271$	−8.00000	−	13.8564i	−0.485965	−	0.841717i	−0.999870	+	0.0161307i	$0.994865\pi$
$272$		0			0
$273$	−10.0000	+	8.66025i	−0.605228	+	0.524142i
$274$		0			0
$275$	33.0000	+	57.1577i	1.98997	+	3.44674i
$276$		0			0
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	−0.850613	−	0.525792i	$-0.823769\pi$
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	0.880656	+	0.473757i	$0.157103\pi$
$278$		0			0
$279$	−3.00000			−0.179605
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	−5.50000	+	9.52628i	−0.326941	+	0.566279i	−0.981903	−	0.189383i	$-0.939351\pi$
$283$	−5.50000	+	9.52628i	−0.326941	+	0.566279i	0.654962	+	0.755662i	$0.272685\pi$
$284$		0			0
$285$	2.00000	+	3.46410i	0.118470	+	0.205196i
$286$		0			0
$287$	15.0000	+	5.19615i	0.885422	+	0.306719i
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$	3.00000	−	5.19615i	0.175863	−	0.304604i
$292$		0			0
$293$	12.0000			0.701047			0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000			0.701047			0.350524	+	0.936554i	$0.386004\pi$
$294$		0			0
$295$	24.0000			1.39733
$296$		0			0
$297$	3.00000	−	5.19615i	0.174078	−	0.301511i
$298$		0			0
$299$	−15.0000	−	25.9808i	−0.867472	−	1.50251i
$300$		0			0
$301$	12.5000	+	4.33013i	0.720488	+	0.249584i
$302$		0			0
$303$	1.00000	+	1.73205i	0.0574485	+	0.0995037i
$304$		0			0
$305$	−4.00000	+	6.92820i	−0.229039	+	0.396708i
$306$		0			0
$307$	−11.0000			−0.627803			−0.313902	−	0.949456i	$-0.601636\pi$
$307$	−11.0000			−0.627803			−0.313902	+	0.949456i	$0.601636\pi$
$308$		0			0
$309$	−11.0000			−0.625768
$310$		0			0
$311$	1.00000	−	1.73205i	0.0567048	−	0.0982156i	−0.836280	−	0.548303i	$-0.815274\pi$
$311$	1.00000	−	1.73205i	0.0567048	−	0.0982156i	0.892984	+	0.450088i	$0.148607\pi$
$312$		0			0
$313$	−15.5000	−	26.8468i	−0.876112	−	1.51747i	−0.855574	−	0.517681i	$-0.826795\pi$
$313$	−15.5000	−	26.8468i	−0.876112	−	1.51747i	−0.0205381	−	0.999789i	$-0.506538\pi$
$314$		0			0
$315$	8.00000	−	6.92820i	0.450749	−	0.390360i
$316$		0			0
$317$	−10.0000	−	17.3205i	−0.561656	−	0.972817i	−0.997352	−	0.0727229i	$-0.976831\pi$
$317$	−10.0000	−	17.3205i	−0.561656	−	0.972817i	0.435696	−	0.900094i	$-0.356502\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−10.0000			−0.558146
$322$		0			0
$323$	−2.00000			−0.111283
$324$		0			0
$325$	27.5000	−	47.6314i	1.52543	−	2.64211i
$326$		0			0
$327$	−7.50000	−	12.9904i	−0.414751	−	0.718370i
$328$		0			0
$329$	−2.00000	−	10.3923i	−0.110264	−	0.572946i
$330$		0			0
$331$	−2.50000	−	4.33013i	−0.137412	−	0.238005i	0.789104	−	0.614260i	$-0.210545\pi$
$331$	−2.50000	−	4.33013i	−0.137412	−	0.238005i	−0.926516	+	0.376254i	$0.877212\pi$
$332$		0			0
$333$	1.50000	−	2.59808i	0.0821995	−	0.142374i
$334$		0			0
$335$	−28.0000			−1.52980
$336$		0			0
$337$	1.00000			0.0544735			0.0272367	−	0.999629i	$-0.491329\pi$
$337$	1.00000			0.0544735			0.0272367	+	0.999629i	$0.491329\pi$
$338$		0			0
$339$	8.00000	−	13.8564i	0.434500	−	0.752577i
$340$		0			0
$341$	−9.00000	−	15.5885i	−0.487377	−	0.844162i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$	12.0000	+	20.7846i	0.646058	+	1.11901i
$346$		0			0
$347$	−11.0000	+	19.0526i	−0.590511	+	1.02279i	0.403653	+	0.914912i	$0.367740\pi$
$347$	−11.0000	+	19.0526i	−0.590511	+	1.02279i	−0.994164	+	0.107883i	$0.965593\pi$
$348$		0			0
$349$	−26.0000			−1.39175			−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000			−1.39175			−0.695874	+	0.718164i	$0.744983\pi$
$350$		0			0
$351$	−5.00000			−0.266880
$352$		0			0
$353$	2.00000	−	3.46410i	0.106449	−	0.184376i	−0.807880	−	0.589347i	$-0.799385\pi$
$353$	2.00000	−	3.46410i	0.106449	−	0.184376i	0.914329	+	0.404971i	$0.132718\pi$
$354$		0			0
$355$	−32.0000	−	55.4256i	−1.69838	−	2.94169i
$356$		0			0
$357$	1.00000	+	5.19615i	0.0529256	+	0.275010i
$358$		0			0
$359$	12.0000	+	20.7846i	0.633336	+	1.09697i	0.986865	+	0.161546i	$0.0516481\pi$
$359$	12.0000	+	20.7846i	0.633336	+	1.09697i	−0.353529	+	0.935423i	$0.615019\pi$
$360$		0			0
$361$	9.00000	−	15.5885i	0.473684	−	0.820445i
$362$		0			0
$363$	25.0000			1.31216
$364$		0			0
$365$	−12.0000			−0.628109
$366$		0			0
$367$	−13.5000	+	23.3827i	−0.704694	+	1.22057i	0.262108	+	0.965039i	$0.415582\pi$
$367$	−13.5000	+	23.3827i	−0.704694	+	1.22057i	−0.966802	+	0.255528i	$0.917751\pi$
$368$		0			0
$369$	3.00000	+	5.19615i	0.156174	+	0.270501i
$370$		0			0
$371$	12.0000	−	10.3923i	0.623009	−	0.539542i
$372$		0			0
$373$	−14.5000	−	25.1147i	−0.750782	−	1.30039i	−0.947444	−	0.319921i	$-0.896344\pi$
$373$	−14.5000	−	25.1147i	−0.750782	−	1.30039i	0.196663	−	0.980471i	$-0.436990\pi$
$374$		0			0
$375$	−12.0000	+	20.7846i	−0.619677	+	1.07331i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−27.0000			−1.38690			−0.693448	−	0.720506i	$-0.743909\pi$
$379$	−27.0000			−1.38690			−0.693448	+	0.720506i	$0.743909\pi$
$380$		0			0
$381$	3.50000	−	6.06218i	0.179310	−	0.310575i
$382$		0			0
$383$	−13.0000	−	22.5167i	−0.664269	−	1.15055i	−0.979483	−	0.201527i	$-0.935410\pi$
$383$	−13.0000	−	22.5167i	−0.664269	−	1.15055i	0.315214	−	0.949021i	$-0.397924\pi$
$384$		0			0
$385$	60.0000	+	20.7846i	3.05788	+	1.05928i
$386$		0			0
$387$	2.50000	+	4.33013i	0.127082	+	0.220113i
$388$		0			0
$389$	−2.00000	+	3.46410i	−0.101404	+	0.175637i	−0.912263	−	0.409604i	$-0.865667\pi$
$389$	−2.00000	+	3.46410i	−0.101404	+	0.175637i	0.810859	+	0.585241i	$0.199000\pi$
$390$		0			0
$391$	−12.0000			−0.606866
$392$		0			0
$393$	−6.00000			−0.302660
$394$		0			0
$395$	−22.0000	+	38.1051i	−1.10694	+	1.91728i
$396$		0			0
$397$	−10.5000	−	18.1865i	−0.526980	−	0.912756i	−0.999506	−	0.0314391i	$-0.989991\pi$
$397$	−10.5000	−	18.1865i	−0.526980	−	0.912756i	0.472526	−	0.881317i	$-0.343342\pi$
$398$		0			0
$399$	2.50000	+	0.866025i	0.125157	+	0.0433555i
$400$		0			0
$401$	1.00000	+	1.73205i	0.0499376	+	0.0864945i	0.889914	−	0.456129i	$-0.150764\pi$
$401$	1.00000	+	1.73205i	0.0499376	+	0.0864945i	−0.839976	+	0.542623i	$0.817431\pi$
$402$		0			0
$403$	−7.50000	+	12.9904i	−0.373602	+	0.647097i
$404$		0			0
$405$	4.00000			0.198762
$406$		0			0
$407$	18.0000			0.892227
$408$		0			0
$409$	−6.50000	+	11.2583i	−0.321404	+	0.556689i	−0.980778	−	0.195127i	$-0.937488\pi$
$409$	−6.50000	+	11.2583i	−0.321404	+	0.556689i	0.659374	+	0.751815i	$0.270822\pi$
$410$		0			0
$411$	6.00000	+	10.3923i	0.295958	+	0.512615i
$412$		0			0
$413$	12.0000	−	10.3923i	0.590481	−	0.511372i
$414$		0			0
$415$	24.0000	+	41.5692i	1.17811	+	2.04055i
$416$		0			0
$417$	2.50000	−	4.33013i	0.122426	−	0.212047i
$418$		0			0
$419$	26.0000			1.27018			0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000			1.27018			0.635092	+	0.772437i	$0.280962\pi$
$420$		0			0
$421$	−17.0000			−0.828529			−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000			−0.828529			−0.414265	+	0.910156i	$0.635961\pi$
$422$		0			0
$423$	2.00000	−	3.46410i	0.0972433	−	0.168430i
$424$		0			0
$425$	−11.0000	−	19.0526i	−0.533578	−	0.924185i
$426$		0			0
$427$	1.00000	+	5.19615i	0.0483934	+	0.251459i
$428$		0			0
$429$	−15.0000	−	25.9808i	−0.724207	−	1.25436i
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$		0			0
$433$	−25.0000			−1.20142			−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000			−1.20142			−0.600712	+	0.799466i	$0.705116\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.00000	+	5.19615i	−0.143509	+	0.248566i
$438$		0			0
$439$	−12.0000	−	20.7846i	−0.572729	−	0.991995i	−0.996284	−	0.0861252i	$-0.972552\pi$
$439$	−12.0000	−	20.7846i	−0.572729	−	0.991995i	0.423556	−	0.905870i	$-0.360782\pi$
$440$		0			0
$441$	1.00000	−	6.92820i	0.0476190	−	0.329914i
$442$		0			0
$443$	−13.0000	−	22.5167i	−0.617649	−	1.06980i	−0.989914	−	0.141672i	$-0.954752\pi$
$443$	−13.0000	−	22.5167i	−0.617649	−	1.06980i	0.372265	−	0.928126i	$-0.378581\pi$
$444$		0			0
$445$	−8.00000	+	13.8564i	−0.379236	+	0.656857i
$446$		0			0
$447$	4.00000			0.189194
$448$		0			0
$449$	36.0000			1.69895			0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000			1.69895			0.849473	+	0.527633i	$0.176920\pi$
$450$		0			0
$451$	−18.0000	+	31.1769i	−0.847587	+	1.46806i
$452$		0			0
$453$	4.00000	+	6.92820i	0.187936	+	0.325515i
$454$		0			0
$455$	−10.0000	−	51.9615i	−0.468807	−	2.43599i
$456$		0			0
$457$	−12.5000	−	21.6506i	−0.584725	−	1.01277i	−0.994910	−	0.100771i	$-0.967869\pi$
$457$	−12.5000	−	21.6506i	−0.584725	−	1.01277i	0.410184	−	0.912003i	$-0.365464\pi$
$458$		0			0
$459$	−1.00000	+	1.73205i	−0.0466760	+	0.0808452i
$460$		0			0
$461$	4.00000			0.186299			0.0931493	−	0.995652i	$-0.470307\pi$
$461$	4.00000			0.186299			0.0931493	+	0.995652i	$0.470307\pi$
$462$		0			0
$463$	−5.00000			−0.232370			−0.116185	−	0.993228i	$-0.537067\pi$
$463$	−5.00000			−0.232370			−0.116185	+	0.993228i	$0.537067\pi$
$464$		0			0
$465$	6.00000	−	10.3923i	0.278243	−	0.481932i
$466$		0			0
$467$	4.00000	+	6.92820i	0.185098	+	0.320599i	0.943610	−	0.331061i	$-0.107406\pi$
$467$	4.00000	+	6.92820i	0.185098	+	0.320599i	−0.758512	+	0.651660i	$0.774073\pi$
$468$		0			0
$469$	−14.0000	+	12.1244i	−0.646460	+	0.559851i
$470$		0			0
$471$	5.00000	+	8.66025i	0.230388	+	0.399043i
$472$		0			0
$473$	−15.0000	+	25.9808i	−0.689701	+	1.19460i
$474$		0			0
$475$	−11.0000			−0.504715
$476$		0			0
$477$	6.00000			0.274721
$478$		0			0
$479$	−1.00000	+	1.73205i	−0.0456912	+	0.0791394i	−0.887967	−	0.459908i	$-0.847882\pi$
$479$	−1.00000	+	1.73205i	−0.0456912	+	0.0791394i	0.842275	+	0.539048i	$0.181216\pi$
$480$		0			0
$481$	−7.50000	−	12.9904i	−0.341971	−	0.592310i
$482$		0			0
$483$	15.0000	+	5.19615i	0.682524	+	0.236433i
$484$		0			0
$485$	12.0000	+	20.7846i	0.544892	+	0.943781i
$486$		0			0
$487$	0.500000	−	0.866025i	0.0226572	−	0.0392434i	−0.854475	−	0.519493i	$-0.826121\pi$
$487$	0.500000	−	0.866025i	0.0226572	−	0.0392434i	0.877132	+	0.480250i	$0.159454\pi$
$488$		0			0
$489$	−20.0000			−0.904431
$490$		0			0
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	12.0000	+	20.7846i	0.539360	+	0.934199i
$496$		0			0
$497$	−40.0000	−	13.8564i	−1.79425	−	0.621545i
$498$		0			0
$499$	14.5000	+	25.1147i	0.649109	+	1.12429i	0.983336	+	0.181797i	$0.0581915\pi$
$499$	14.5000	+	25.1147i	0.649109	+	1.12429i	−0.334227	+	0.942493i	$0.608475\pi$
$500$		0			0
$501$	4.00000	−	6.92820i	0.178707	−	0.309529i
$502$		0			0
$503$	18.0000			0.802580			0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000			0.802580			0.401290	+	0.915951i	$0.368562\pi$
$504$		0			0
$505$	−8.00000			−0.355995
$506$		0			0
$507$	−6.00000	+	10.3923i	−0.266469	+	0.461538i
$508$		0			0
$509$	−12.0000	−	20.7846i	−0.531891	−	0.921262i	−0.999307	−	0.0372243i	$-0.988148\pi$
$509$	−12.0000	−	20.7846i	−0.531891	−	0.921262i	0.467416	−	0.884037i	$-0.345185\pi$
$510$		0			0
$511$	−6.00000	+	5.19615i	−0.265424	+	0.229864i
$512$		0			0
$513$	0.500000	+	0.866025i	0.0220755	+	0.0382360i
$514$		0			0
$515$	22.0000	−	38.1051i	0.969436	−	1.67911i
$516$		0			0
$517$	24.0000			1.05552
$518$		0			0
$519$	−22.0000			−0.965693
$520$		0			0
$521$	−12.0000	+	20.7846i	−0.525730	+	0.910590i	0.473821	+	0.880621i	$0.342874\pi$
$521$	−12.0000	+	20.7846i	−0.525730	+	0.910590i	−0.999551	+	0.0299693i	$0.990459\pi$
$522$		0			0
$523$	−8.50000	−	14.7224i	−0.371679	−	0.643767i	0.618145	−	0.786064i	$-0.287884\pi$
$523$	−8.50000	−	14.7224i	−0.371679	−	0.643767i	−0.989824	+	0.142297i	$0.954551\pi$
$524$		0			0
$525$	5.50000	+	28.5788i	0.240040	+	1.24728i
$526$		0			0
$527$	3.00000	+	5.19615i	0.130682	+	0.226348i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$	6.00000			0.260378
$532$		0			0
$533$	30.0000			1.29944
$534$		0			0
$535$	20.0000	−	34.6410i	0.864675	−	1.49766i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	39.0000	−	15.5885i	1.67985	−	0.671442i
$540$		0			0
$541$	18.5000	+	32.0429i	0.795377	+	1.37763i	0.922599	+	0.385759i	$0.126061\pi$
$541$	18.5000	+	32.0429i	0.795377	+	1.37763i	−0.127222	+	0.991874i	$0.540606\pi$
$542$		0			0
$543$	12.5000	−	21.6506i	0.536426	−	0.929118i
$544$		0			0
$545$	60.0000			2.57012
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$		0			0
$549$	−1.00000	+	1.73205i	−0.0426790	+	0.0739221i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	5.50000	+	28.5788i	0.233884	+	1.21530i
$554$		0			0
$555$	6.00000	+	10.3923i	0.254686	+	0.441129i
$556$		0			0
$557$	−3.00000	+	5.19615i	−0.127114	+	0.220168i	−0.922557	−	0.385860i	$-0.873905\pi$
$557$	−3.00000	+	5.19615i	−0.127114	+	0.220168i	0.795443	+	0.606028i	$0.207238\pi$
$558$		0			0
$559$	25.0000			1.05739
$560$		0			0
$561$	−12.0000			−0.506640
$562$		0			0
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	−0.494238	−	0.869326i	$-0.664553\pi$
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	0.999978	−	0.00664037i	$-0.00211371\pi$
$564$		0			0
$565$	32.0000	+	55.4256i	1.34625	+	2.33177i
$566$		0			0
$567$	2.00000	−	1.73205i	0.0839921	−	0.0727393i
$568$		0			0
$569$	−9.00000	−	15.5885i	−0.377300	−	0.653502i	0.613369	−	0.789797i	$-0.289814\pi$
$569$	−9.00000	−	15.5885i	−0.377300	−	0.653502i	−0.990668	+	0.136295i	$0.956481\pi$
$570$		0			0
$571$	14.5000	−	25.1147i	0.606806	−	1.05102i	−0.384957	−	0.922934i	$-0.625784\pi$
$571$	14.5000	−	25.1147i	0.606806	−	1.05102i	0.991763	−	0.128085i	$-0.0408829\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−66.0000			−2.75239
$576$		0			0
$577$	−11.5000	+	19.9186i	−0.478751	+	0.829222i	−0.999703	−	0.0243645i	$-0.992244\pi$
$577$	−11.5000	+	19.9186i	−0.478751	+	0.829222i	0.520952	+	0.853586i	$0.325577\pi$
$578$		0			0
$579$	0.500000	+	0.866025i	0.0207793	+	0.0359908i
$580$		0			0
$581$	30.0000	+	10.3923i	1.24461	+	0.431145i
$582$		0			0
$583$	18.0000	+	31.1769i	0.745484	+	1.29122i
$584$		0			0
$585$	10.0000	−	17.3205i	0.413449	−	0.716115i
$586$		0			0
$587$	−40.0000			−1.65098			−0.825488	−	0.564419i	$-0.809100\pi$
$587$	−40.0000			−1.65098			−0.825488	+	0.564419i	$0.809100\pi$
$588$		0			0
$589$	3.00000			0.123613
$590$		0			0
$591$	9.00000	−	15.5885i	0.370211	−	0.641223i
$592$		0			0
$593$	13.0000	+	22.5167i	0.533846	+	0.924648i	0.999218	+	0.0395334i	$0.0125871\pi$
$593$	13.0000	+	22.5167i	0.533846	+	0.924648i	−0.465372	+	0.885115i	$0.654080\pi$
$594$		0			0
$595$	−20.0000	−	6.92820i	−0.819920	−	0.284029i
$596$		0			0
$597$	−10.0000	−	17.3205i	−0.409273	−	0.708881i
$598$		0			0
$599$	−10.0000	+	17.3205i	−0.408589	+	0.707697i	−0.994732	−	0.102511i	$-0.967312\pi$
$599$	−10.0000	+	17.3205i	−0.408589	+	0.707697i	0.586143	+	0.810208i	$0.300646\pi$
$600$		0			0
$601$	−21.0000			−0.856608			−0.428304	−	0.903635i	$-0.640889\pi$
$601$	−21.0000			−0.856608			−0.428304	+	0.903635i	$0.640889\pi$
$602$		0			0
$603$	−7.00000			−0.285062
$604$		0			0
$605$	−50.0000	+	86.6025i	−2.03279	+	3.52089i
$606$		0			0
$607$	2.50000	+	4.33013i	0.101472	+	0.175754i	0.912291	−	0.409542i	$-0.134311\pi$
$607$	2.50000	+	4.33013i	0.101472	+	0.175754i	−0.810819	+	0.585296i	$0.800978\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10.0000	−	17.3205i	−0.404557	−	0.700713i
$612$		0			0
$613$	−23.0000	+	39.8372i	−0.928961	+	1.60901i	−0.143898	+	0.989593i	$0.545964\pi$
$613$	−23.0000	+	39.8372i	−0.928961	+	1.60901i	−0.785063	+	0.619416i	$0.787370\pi$
$614$		0			0
$615$	−24.0000			−0.967773
$616$		0			0
$617$	−14.0000			−0.563619			−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000			−0.563619			−0.281809	+	0.959470i	$0.590935\pi$
$618$		0			0
$619$	−12.5000	+	21.6506i	−0.502417	+	0.870212i	0.497579	+	0.867419i	$0.334223\pi$
$619$	−12.5000	+	21.6506i	−0.502417	+	0.870212i	−0.999996	+	0.00279365i	$0.999111\pi$
$620$		0			0
$621$	3.00000	+	5.19615i	0.120386	+	0.208514i
$622$		0			0
$623$	2.00000	+	10.3923i	0.0801283	+	0.416359i
$624$		0			0
$625$	−20.5000	−	35.5070i	−0.820000	−	1.42028i
$626$		0			0
$627$	−3.00000	+	5.19615i	−0.119808	+	0.207514i
$628$		0			0
$629$	−6.00000			−0.239236
$630$		0			0
$631$	28.0000			1.11466			0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000			1.11466			0.557331	+	0.830290i	$0.311825\pi$
$632$		0			0
$633$	6.00000	−	10.3923i	0.238479	−	0.413057i
$634$		0			0
$635$	14.0000	+	24.2487i	0.555573	+	0.962281i
$636$		0			0
$637$	−27.5000	−	21.6506i	−1.08959	−	0.857829i
$638$		0			0
$639$	−8.00000	−	13.8564i	−0.316475	−	0.548151i
$640$		0			0
$641$	−19.0000	+	32.9090i	−0.750455	+	1.29983i	0.197148	+	0.980374i	$0.436832\pi$
$641$	−19.0000	+	32.9090i	−0.750455	+	1.29983i	−0.947602	+	0.319452i	$0.896501\pi$
$642$		0			0
$643$	23.0000			0.907031			0.453516	−	0.891248i	$-0.350170\pi$
$643$	23.0000			0.907031			0.453516	+	0.891248i	$0.350170\pi$
$644$		0			0
$645$	−20.0000			−0.787499
$646$		0			0
$647$	−5.00000	+	8.66025i	−0.196570	+	0.340470i	−0.947414	−	0.320010i	$-0.896314\pi$
$647$	−5.00000	+	8.66025i	−0.196570	+	0.340470i	0.750844	+	0.660480i	$0.229647\pi$
$648$		0			0
$649$	18.0000	+	31.1769i	0.706562	+	1.22380i
$650$		0			0
$651$	−1.50000	−	7.79423i	−0.0587896	−	0.305480i
$652$		0			0
$653$	17.0000	+	29.4449i	0.665261	+	1.15227i	0.979214	+	0.202828i	$0.0650132\pi$
$653$	17.0000	+	29.4449i	0.665261	+	1.15227i	−0.313953	+	0.949439i	$0.601653\pi$
$654$		0			0
$655$	12.0000	−	20.7846i	0.468879	−	0.812122i
$656$		0			0
$657$	−3.00000			−0.117041
$658$		0			0
$659$	−6.00000			−0.233727			−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000			−0.233727			−0.116863	+	0.993148i	$0.537284\pi$
$660$		0			0
$661$	1.50000	−	2.59808i	0.0583432	−	0.101053i	−0.835379	−	0.549675i	$-0.814752\pi$
$661$	1.50000	−	2.59808i	0.0583432	−	0.101053i	0.893722	+	0.448622i	$0.148085\pi$
$662$		0			0
$663$	5.00000	+	8.66025i	0.194184	+	0.336336i
$664$		0			0
$665$	−8.00000	+	6.92820i	−0.310227	+	0.268664i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−8.00000	+	13.8564i	−0.309298	+	0.535720i
$670$		0			0
$671$	−12.0000			−0.463255
$672$		0			0
$673$	−29.0000			−1.11787			−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000			−1.11787			−0.558934	+	0.829212i	$0.688789\pi$
$674$		0			0
$675$	−5.50000	+	9.52628i	−0.211695	+	0.366667i
$676$		0			0
$677$	−6.00000	−	10.3923i	−0.230599	−	0.399409i	0.727386	−	0.686229i	$-0.240735\pi$
$677$	−6.00000	−	10.3923i	−0.230599	−	0.399409i	−0.957984	+	0.286820i	$0.907402\pi$
$678$		0			0
$679$	15.0000	+	5.19615i	0.575647	+	0.199410i
$680$		0			0
$681$	11.0000	+	19.0526i	0.421521	+	0.730096i
$682$		0			0
$683$	9.00000	−	15.5885i	0.344375	−	0.596476i	−0.640865	−	0.767654i	$-0.721424\pi$
$683$	9.00000	−	15.5885i	0.344375	−	0.596476i	0.985240	+	0.171178i	$0.0547574\pi$
$684$		0			0
$685$	−48.0000			−1.83399
$686$		0			0
$687$	11.0000			0.419676
$688$		0			0
$689$	15.0000	−	25.9808i	0.571454	−	0.989788i
$690$		0			0
$691$	11.5000	+	19.9186i	0.437481	+	0.757739i	0.997494	−	0.0707446i	$-0.0225375\pi$
$691$	11.5000	+	19.9186i	0.437481	+	0.757739i	−0.560014	+	0.828483i	$0.689204\pi$
$692$		0			0
$693$	15.0000	+	5.19615i	0.569803	+	0.197386i
$694$		0			0
$695$	10.0000	+	17.3205i	0.379322	+	0.657004i
$696$		0			0
$697$	6.00000	−	10.3923i	0.227266	−	0.393637i
$698$		0			0
$699$	−8.00000			−0.302588
$700$		0			0
$701$	−22.0000			−0.830929			−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000			−0.830929			−0.415464	+	0.909610i	$0.636381\pi$
$702$		0			0
$703$	−1.50000	+	2.59808i	−0.0565736	+	0.0979883i
$704$		0			0
$705$	8.00000	+	13.8564i	0.301297	+	0.521862i
$706$		0			0
$707$	−4.00000	+	3.46410i	−0.150435	+	0.130281i
$708$		0			0
$709$	−3.00000	−	5.19615i	−0.112667	−	0.195146i	0.804178	−	0.594389i	$-0.202606\pi$
$709$	−3.00000	−	5.19615i	−0.112667	−	0.195146i	−0.916845	+	0.399244i	$0.869273\pi$
$710$		0			0
$711$	−5.50000	+	9.52628i	−0.206266	+	0.357263i
$712$		0			0
$713$	18.0000			0.674105
$714$		0			0
$715$	120.000			4.48775
$716$		0			0
$717$	−1.00000	+	1.73205i	−0.0373457	+	0.0646846i
$718$		0			0
$719$	−15.0000	−	25.9808i	−0.559406	−	0.968919i	−0.997546	−	0.0700124i	$-0.977696\pi$
$719$	−15.0000	−	25.9808i	−0.559406	−	0.968919i	0.438141	−	0.898906i	$-0.355637\pi$
$720$		0			0
$721$	−5.50000	−	28.5788i	−0.204831	−	1.06433i
$722$		0			0
$723$	−5.00000	−	8.66025i	−0.185952	−	0.322078i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−41.0000			−1.52061			−0.760303	−	0.649569i	$-0.774949\pi$
$727$	−41.0000			−1.52061			−0.760303	+	0.649569i	$0.774949\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	5.00000	−	8.66025i	0.184932	−	0.320311i
$732$		0			0
$733$	10.5000	+	18.1865i	0.387826	+	0.671735i	0.992157	−	0.124999i	$-0.0398927\pi$
$733$	10.5000	+	18.1865i	0.387826	+	0.671735i	−0.604331	+	0.796734i	$0.706559\pi$
$734$		0			0
$735$	22.0000	+	17.3205i	0.811482	+	0.638877i
$736$		0			0
$737$	−21.0000	−	36.3731i	−0.773545	−	1.33982i
$738$		0			0
$739$	−25.5000	+	44.1673i	−0.938033	+	1.62472i	−0.168898	+	0.985634i	$0.554021\pi$
$739$	−25.5000	+	44.1673i	−0.938033	+	1.62472i	−0.769135	+	0.639087i	$0.779313\pi$
$740$		0			0
$741$	5.00000			0.183680
$742$		0			0
$743$	−18.0000			−0.660356			−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000			−0.660356			−0.330178	+	0.943919i	$0.607109\pi$
$744$		0			0
$745$	−8.00000	+	13.8564i	−0.293097	+	0.507659i
$746$		0			0
$747$	6.00000	+	10.3923i	0.219529	+	0.380235i
$748$		0			0
$749$	−5.00000	−	25.9808i	−0.182696	−	0.949316i
$750$		0			0
$751$	3.50000	+	6.06218i	0.127717	+	0.221212i	0.922792	−	0.385299i	$-0.125902\pi$
$751$	3.50000	+	6.06218i	0.127717	+	0.221212i	−0.795075	+	0.606511i	$0.792568\pi$
$752$		0			0
$753$	−7.00000	+	12.1244i	−0.255094	+	0.441836i
$754$		0			0
$755$	−32.0000			−1.16460
$756$		0			0
$757$	26.0000			0.944986			0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000			0.944986			0.472493	+	0.881334i	$0.343354\pi$
$758$		0			0
$759$	−18.0000	+	31.1769i	−0.653359	+	1.13165i
$760$		0			0
$761$	−25.0000	−	43.3013i	−0.906249	−	1.56967i	−0.819231	−	0.573463i	$-0.805600\pi$
$761$	−25.0000	−	43.3013i	−0.906249	−	1.56967i	−0.0870179	−	0.996207i	$-0.527734\pi$
$762$		0			0
$763$	30.0000	−	25.9808i	1.08607	−	0.940567i
$764$		0			0
$765$	−4.00000	−	6.92820i	−0.144620	−	0.250490i
$766$		0			0
$767$	15.0000	−	25.9808i	0.541619	−	0.938111i
$768$		0			0
$769$	31.0000			1.11789			0.558944	−	0.829205i	$-0.311207\pi$
$769$	31.0000			1.11789			0.558944	+	0.829205i	$0.311207\pi$
$770$		0			0
$771$	−12.0000			−0.432169
$772$		0			0
$773$	−11.0000	+	19.0526i	−0.395643	+	0.685273i	−0.993183	−	0.116566i	$-0.962811\pi$
$773$	−11.0000	+	19.0526i	−0.395643	+	0.685273i	0.597540	+	0.801839i	$0.296145\pi$
$774$		0			0
$775$	16.5000	+	28.5788i	0.592697	+	1.02658i
$776$		0			0
$777$	7.50000	+	2.59808i	0.269061	+	0.0932055i
$778$		0			0
$779$	−3.00000	−	5.19615i	−0.107486	−	0.186171i
$780$		0			0
$781$	48.0000	−	83.1384i	1.71758	−	2.97493i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−40.0000			−1.42766
$786$		0			0
$787$	16.0000	−	27.7128i	0.570338	−	0.987855i	−0.426193	−	0.904632i	$-0.640145\pi$
$787$	16.0000	−	27.7128i	0.570338	−	0.987855i	0.996531	−	0.0832226i	$-0.0265213\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	40.0000	+	13.8564i	1.42224	+	0.492677i
$792$		0			0
$793$	5.00000	+	8.66025i	0.177555	+	0.307535i
$794$		0			0
$795$	−12.0000	+	20.7846i	−0.425596	+	0.737154i
$796$		0			0
$797$	46.0000			1.62940			0.814702	−	0.579880i	$-0.196901\pi$
$797$	46.0000			1.62940			0.814702	+	0.579880i	$0.196901\pi$
$798$		0			0
$799$	−8.00000			−0.283020
$800$		0			0
$801$	−2.00000	+	3.46410i	−0.0706665	+	0.122398i
$802$		0			0
$803$	−9.00000	−	15.5885i	−0.317603	−	0.550105i
$804$		0			0
$805$	−48.0000	+	41.5692i	−1.69178	+	1.46512i
$806$		0			0
$807$	−1.00000	−	1.73205i	−0.0352017	−	0.0609711i
$808$		0			0
$809$	−4.00000	+	6.92820i	−0.140633	+	0.243583i	−0.927735	−	0.373240i	$-0.878247\pi$
$809$	−4.00000	+	6.92820i	−0.140633	+	0.243583i	0.787102	+	0.616822i	$0.211580\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$		0			0
$813$	16.0000			0.561144
$814$		0			0
$815$	40.0000	−	69.2820i	1.40114	−	2.42684i
$816$		0			0
$817$	−2.50000	−	4.33013i	−0.0874639	−	0.151492i
$818$		0			0
$819$	−2.50000	−	12.9904i	−0.0873571	−	0.453921i
$820$		0			0
$821$	2.00000	+	3.46410i	0.0698005	+	0.120898i	0.898813	−	0.438331i	$-0.144430\pi$
$821$	2.00000	+	3.46410i	0.0698005	+	0.120898i	−0.829013	+	0.559229i	$0.811097\pi$
$822$		0			0
$823$	12.0000	−	20.7846i	0.418294	−	0.724506i	−0.577474	−	0.816409i	$-0.695962\pi$
$823$	12.0000	−	20.7846i	0.418294	−	0.724506i	0.995768	+	0.0919029i	$0.0292950\pi$
$824$		0			0
$825$	−66.0000			−2.29783
$826$		0			0
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$	2.50000	−	4.33013i	0.0868286	−	0.150392i	−0.819340	−	0.573307i	$-0.805660\pi$
$829$	2.50000	−	4.33013i	0.0868286	−	0.150392i	0.906169	+	0.422916i	$0.138993\pi$
$830$		0			0
$831$	0.500000	+	0.866025i	0.0173448	+	0.0300421i
$832$		0			0
$833$	−13.0000	+	5.19615i	−0.450423	+	0.180036i
$834$		0			0
$835$	16.0000	+	27.7128i	0.553703	+	0.959041i
$836$		0			0
$837$	1.50000	−	2.59808i	0.0518476	−	0.0898027i
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	−3.00000	+	5.19615i	−0.103325	+	0.178965i
$844$		0			0
$845$	−24.0000	−	41.5692i	−0.825625	−	1.43002i
$846$		0			0
$847$	12.5000	+	64.9519i	0.429505	+	2.23177i
$848$		0			0
$849$	−5.50000	−	9.52628i	−0.188760	−	0.326941i
$850$		0			0
$851$	−9.00000	+	15.5885i	−0.308516	+	0.534365i
$852$		0			0
$853$	33.0000			1.12990			0.564949	−	0.825126i	$-0.308896\pi$
$853$	33.0000			1.12990			0.564949	+	0.825126i	$0.308896\pi$
$854$		0			0
$855$	−4.00000			−0.136797
$856$		0			0
$857$	−12.0000	+	20.7846i	−0.409912	+	0.709989i	−0.994880	−	0.101068i	$-0.967774\pi$
$857$	−12.0000	+	20.7846i	−0.409912	+	0.709989i	0.584967	+	0.811057i	$0.301107\pi$
$858$		0			0
$859$	26.0000	+	45.0333i	0.887109	+	1.53652i	0.843278	+	0.537478i	$0.180623\pi$
$859$	26.0000	+	45.0333i	0.887109	+	1.53652i	0.0438309	+	0.999039i	$0.486044\pi$
$860$		0			0
$861$	−12.0000	+	10.3923i	−0.408959	+	0.354169i
$862$		0			0
$863$	−27.0000	−	46.7654i	−0.919091	−	1.59191i	−0.800799	−	0.598933i	$-0.795592\pi$
$863$	−27.0000	−	46.7654i	−0.919091	−	1.59191i	−0.118291	−	0.992979i	$-0.537742\pi$
$864$		0			0
$865$	44.0000	−	76.2102i	1.49604	−	2.59123i
$866$		0			0
$867$	−13.0000			−0.441503
$868$		0			0
$869$	−66.0000			−2.23890
$870$		0			0
$871$	−17.5000	+	30.3109i	−0.592965	+	1.02705i
$872$		0			0
$873$	3.00000	+	5.19615i	0.101535	+	0.175863i
$874$		0			0
$875$	−60.0000	−	20.7846i	−2.02837	−	0.702648i
$876$		0			0
$877$	1.00000	+	1.73205i	0.0337676	+	0.0584872i	0.882415	−	0.470471i	$-0.155916\pi$
$877$	1.00000	+	1.73205i	0.0337676	+	0.0584872i	−0.848648	+	0.528958i	$0.822583\pi$
$878$		0			0
$879$	−6.00000	+	10.3923i	−0.202375	+	0.350524i
$880$		0			0
$881$	−42.0000			−1.41502			−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000			−1.41502			−0.707508	+	0.706705i	$0.750181\pi$
$882$		0			0
$883$	−7.00000			−0.235569			−0.117784	−	0.993039i	$-0.537579\pi$
$883$	−7.00000			−0.235569			−0.117784	+	0.993039i	$0.537579\pi$
$884$		0			0
$885$	−12.0000	+	20.7846i	−0.403376	+	0.698667i
$886$		0			0
$887$	−17.0000	−	29.4449i	−0.570804	−	0.988662i	−0.996484	−	0.0837878i	$-0.973298\pi$
$887$	−17.0000	−	29.4449i	−0.570804	−	0.988662i	0.425679	−	0.904874i	$-0.360035\pi$
$888$		0			0
$889$	17.5000	+	6.06218i	0.586931	+	0.203319i
$890$		0			0
$891$	3.00000	+	5.19615i	0.100504	+	0.174078i
$892$		0			0
$893$	−2.00000	+	3.46410i	−0.0669274	+	0.115922i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	30.0000			1.00167
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−6.00000	−	10.3923i	−0.199889	−	0.346218i
$902$		0			0
$903$	−10.0000	+	8.66025i	−0.332779	+	0.288195i
$904$		0			0
$905$	50.0000	+	86.6025i	1.66206	+	2.87877i
$906$		0			0
$907$	21.5000	−	37.2391i	0.713896	−	1.23650i	−0.249488	−	0.968378i	$-0.580262\pi$
$907$	21.5000	−	37.2391i	0.713896	−	1.23650i	0.963384	−	0.268126i	$-0.0864043\pi$
$908$		0			0
$909$	−2.00000			−0.0663358
$910$		0			0
$911$	−26.0000			−0.861418			−0.430709	−	0.902491i	$-0.641737\pi$
$911$	−26.0000			−0.861418			−0.430709	+	0.902491i	$0.641737\pi$
$912$		0			0
$913$	−36.0000	+	62.3538i	−1.19143	+	2.06361i
$914$		0			0
$915$	−4.00000	−	6.92820i	−0.132236	−	0.229039i
$916$		0			0
$917$	−3.00000	−	15.5885i	−0.0990687	−	0.514776i
$918$		0			0
$919$	−21.5000	−	37.2391i	−0.709220	−	1.22840i	−0.965147	−	0.261708i	$-0.915714\pi$
$919$	−21.5000	−	37.2391i	−0.709220	−	1.22840i	0.255927	−	0.966696i	$-0.417619\pi$
$920$		0			0
$921$	5.50000	−	9.52628i	0.181231	−	0.313902i
$922$		0			0
$923$	−80.0000			−2.63323
$924$		0			0
$925$	−33.0000			−1.08503
$926$		0			0
$927$	5.50000	−	9.52628i	0.180644	−	0.312884i
$928$		0			0
$929$	18.0000	+	31.1769i	0.590561	+	1.02288i	0.994157	+	0.107944i	$0.0344268\pi$
$929$	18.0000	+	31.1769i	0.590561	+	1.02288i	−0.403596	+	0.914937i	$0.632240\pi$
$930$		0			0
$931$	−1.00000	+	6.92820i	−0.0327737	+	0.227063i
$932$		0			0
$933$	1.00000	+	1.73205i	0.0327385	+	0.0567048i
$934$		0			0
$935$	24.0000	−	41.5692i	0.784884	−	1.35946i
$936$		0			0
$937$	51.0000			1.66610			0.833049	−	0.553200i	$-0.186593\pi$
$937$	51.0000			1.66610			0.833049	+	0.553200i	$0.186593\pi$
$938$		0			0
$939$	31.0000			1.01165
$940$		0			0
$941$	27.0000	−	46.7654i	0.880175	−	1.52451i	0.0290288	−	0.999579i	$-0.490759\pi$
$941$	27.0000	−	46.7654i	0.880175	−	1.52451i	0.851146	−	0.524929i	$-0.175908\pi$
$942$		0			0
$943$	−18.0000	−	31.1769i	−0.586161	−	1.01526i
$944$		0			0
$945$	2.00000	+	10.3923i	0.0650600	+	0.338062i
$946$		0			0
$947$	−15.0000	−	25.9808i	−0.487435	−	0.844261i	0.512461	−	0.858710i	$-0.328734\pi$
$947$	−15.0000	−	25.9808i	−0.487435	−	0.844261i	−0.999896	+	0.0144491i	$0.995401\pi$
$948$		0			0
$949$	−7.50000	+	12.9904i	−0.243460	+	0.421686i
$950$		0			0
$951$	20.0000			0.648544
$952$		0			0
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−24.0000	+	20.7846i	−0.775000	+	0.671170i
$960$		0			0
$961$	11.0000	+	19.0526i	0.354839	+	0.614599i
$962$		0			0
$963$	5.00000	−	8.66025i	0.161123	−	0.279073i
$964$		0			0
$965$	−4.00000			−0.128765
$966$		0			0
$967$	−53.0000			−1.70437			−0.852183	−	0.523245i	$-0.824721\pi$
$967$	−53.0000			−1.70437			−0.852183	+	0.523245i	$0.824721\pi$
$968$		0			0
$969$	1.00000	−	1.73205i	0.0321246	−	0.0556415i
$970$		0			0
$971$	−13.0000	−	22.5167i	−0.417190	−	0.722594i	0.578466	−	0.815707i	$-0.303652\pi$
$971$	−13.0000	−	22.5167i	−0.417190	−	0.722594i	−0.995656	+	0.0931127i	$0.970318\pi$
$972$		0			0
$973$	12.5000	+	4.33013i	0.400732	+	0.138817i
$974$		0			0
$975$	27.5000	+	47.6314i	0.880705	+	1.52543i
$976$		0			0
$977$	−15.0000	+	25.9808i	−0.479893	+	0.831198i	−0.999734	−	0.0230645i	$-0.992658\pi$
$977$	−15.0000	+	25.9808i	−0.479893	+	0.831198i	0.519841	+	0.854263i	$0.325991\pi$
$978$		0			0
$979$	−24.0000			−0.767043
$980$		0			0
$981$	15.0000			0.478913
$982$		0			0
$983$	−13.0000	+	22.5167i	−0.414636	+	0.718170i	−0.995390	−	0.0959088i	$-0.969424\pi$
$983$	−13.0000	+	22.5167i	−0.414636	+	0.718170i	0.580755	+	0.814079i	$0.302758\pi$
$984$		0			0
$985$	36.0000	+	62.3538i	1.14706	+	1.98676i
$986$		0			0
$987$	10.0000	+	3.46410i	0.318304	+	0.110264i
$988$		0			0
$989$	−15.0000	−	25.9808i	−0.476972	−	0.826140i
$990$		0			0
$991$	−12.5000	+	21.6506i	−0.397076	+	0.687755i	−0.993364	−	0.115015i	$-0.963308\pi$
$991$	−12.5000	+	21.6506i	−0.397076	+	0.687755i	0.596288	+	0.802771i	$0.296642\pi$
$992$		0			0
$993$	5.00000			0.158670
$994$		0			0
$995$	80.0000			2.53617
$996$		0			0
$997$	−2.50000	+	4.33013i	−0.0791758	+	0.137136i	−0.902895	−	0.429862i	$-0.858562\pi$
$997$	−2.50000	+	4.33013i	−0.0791758	+	0.137136i	0.823719	+	0.566999i	$0.191896\pi$
$998$		0			0
$999$	1.50000	+	2.59808i	0.0474579	+	0.0821995i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.q.a.193.1		2
4.3	odd	2		1344.2.q.l.193.1		2
7.2	even	3	inner	1344.2.q.a.961.1		2
7.3	odd	6		9408.2.a.a.1.1		1
7.4	even	3		9408.2.a.dd.1.1		1
8.3	odd	2		672.2.q.e.193.1	✓	2
8.5	even	2		672.2.q.j.193.1	yes	2
24.5	odd	2		2016.2.s.a.865.1		2
24.11	even	2		2016.2.s.b.865.1		2
28.3	even	6		9408.2.a.bs.1.1		1
28.11	odd	6		9408.2.a.bp.1.1		1
28.23	odd	6		1344.2.q.l.961.1		2
56.3	even	6		4704.2.a.p.1.1		1
56.11	odd	6		4704.2.a.r.1.1		1
56.37	even	6		672.2.q.j.289.1	yes	2
56.45	odd	6		4704.2.a.bh.1.1		1
56.51	odd	6		672.2.q.e.289.1	yes	2
56.53	even	6		4704.2.a.a.1.1		1
168.107	even	6		2016.2.s.b.289.1		2
168.149	odd	6		2016.2.s.a.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.q.e.193.1	✓	2	8.3	odd	2
672.2.q.e.289.1	yes	2	56.51	odd	6
672.2.q.j.193.1	yes	2	8.5	even	2
672.2.q.j.289.1	yes	2	56.37	even	6
1344.2.q.a.193.1		2	1.1	even	1	trivial
1344.2.q.a.961.1		2	7.2	even	3	inner
1344.2.q.l.193.1		2	4.3	odd	2
1344.2.q.l.961.1		2	28.23	odd	6
2016.2.s.a.289.1		2	168.149	odd	6
2016.2.s.a.865.1		2	24.5	odd	2
2016.2.s.b.289.1		2	168.107	even	6
2016.2.s.b.865.1		2	24.11	even	2
4704.2.a.a.1.1		1	56.53	even	6
4704.2.a.p.1.1		1	56.3	even	6
4704.2.a.r.1.1		1	56.11	odd	6
4704.2.a.bh.1.1		1	56.45	odd	6
9408.2.a.a.1.1		1	7.3	odd	6
9408.2.a.bp.1.1		1	28.11	odd	6
9408.2.a.bs.1.1		1	28.3	even	6
9408.2.a.dd.1.1		1	7.4	even	3

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(3.00000 - 5.19615i) q^{11} -5.00000 q^{13} +4.00000 q^{15} +(-1.00000 + 1.73205i) q^{17} +(0.500000 + 0.866025i) q^{19} +(2.00000 - 1.73205i) q^{21} +(3.00000 + 5.19615i) q^{23} +(-5.50000 + 9.52628i) q^{25} +1.00000 q^{27} +(1.50000 - 2.59808i) q^{31} +(3.00000 + 5.19615i) q^{33} +(2.00000 + 10.3923i) q^{35} +(1.50000 + 2.59808i) q^{37} +(2.50000 - 4.33013i) q^{39} -6.00000 q^{41} -5.00000 q^{43} +(-2.00000 + 3.46410i) q^{45} +(2.00000 + 3.46410i) q^{47} +(5.50000 + 4.33013i) q^{49} +(-1.00000 - 1.73205i) q^{51} +(-3.00000 + 5.19615i) q^{53} -24.0000 q^{55} -1.00000 q^{57} +(-3.00000 + 5.19615i) q^{59} +(-1.00000 - 1.73205i) q^{61} +(0.500000 + 2.59808i) q^{63} +(10.0000 + 17.3205i) q^{65} +(3.50000 - 6.06218i) q^{67} -6.00000 q^{69} +16.0000 q^{71} +(1.50000 - 2.59808i) q^{73} +(-5.50000 - 9.52628i) q^{75} +(-12.0000 + 10.3923i) q^{77} +(-5.50000 - 9.52628i) q^{79} +(-0.500000 + 0.866025i) q^{81} -12.0000 q^{83} +8.00000 q^{85} +(-2.00000 - 3.46410i) q^{89} +(12.5000 + 4.33013i) q^{91} +(1.50000 + 2.59808i) q^{93} +(2.00000 - 3.46410i) q^{95} -6.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 4 q^{5} - 5 q^{7} - q^{9} + 6 q^{11} - 10 q^{13} + 8 q^{15} - 2 q^{17} + q^{19} + 4 q^{21} + 6 q^{23} - 11 q^{25} + 2 q^{27} + 3 q^{31} + 6 q^{33} + 4 q^{35} + 3 q^{37} + 5 q^{39} - 12 q^{41}+ \cdots - 12 q^{99}+O(q^{100}) \) 2 * q - q^3 - 4 * q^5 - 5 * q^7 - q^9 + 6 * q^11 - 10 * q^13 + 8 * q^15 - 2 * q^17 + q^19 + 4 * q^21 + 6 * q^23 - 11 * q^25 + 2 * q^27 + 3 * q^31 + 6 * q^33 + 4 * q^35 + 3 * q^37 + 5 * q^39 - 12 * q^41 - 10 * q^43 - 4 * q^45 + 4 * q^47 + 11 * q^49 - 2 * q^51 - 6 * q^53 - 48 * q^55 - 2 * q^57 - 6 * q^59 - 2 * q^61 + q^63 + 20 * q^65 + 7 * q^67 - 12 * q^69 + 32 * q^71 + 3 * q^73 - 11 * q^75 - 24 * q^77 - 11 * q^79 - q^81 - 24 * q^83 + 16 * q^85 - 4 * q^89 + 25 * q^91 + 3 * q^93 + 4 * q^95 - 12 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

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