LMFDB - Embedded newform 1344.2.q.o.193.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,2,Mod(193,1344)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$10.7318940317$
Analytic rank:	$0$
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 168)
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.o.961.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(1.50000 - 2.59808i) q^{11} -4.00000 q^{13} -1.00000 q^{15} +(-2.00000 - 3.46410i) q^{19} +(2.50000 + 0.866025i) q^{21} +(-4.00000 - 6.92820i) q^{23} +(2.00000 - 3.46410i) q^{25} -1.00000 q^{27} +3.00000 q^{29} +(2.50000 - 4.33013i) q^{31} +(-1.50000 - 2.59808i) q^{33} +(2.00000 - 1.73205i) q^{35} +(4.00000 + 6.92820i) q^{37} +(-2.00000 + 3.46410i) q^{39} +8.00000 q^{41} -6.00000 q^{43} +(-0.500000 + 0.866025i) q^{45} +(-5.00000 - 8.66025i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(4.50000 - 7.79423i) q^{53} -3.00000 q^{55} -4.00000 q^{57} +(-2.50000 + 4.33013i) q^{59} +(-5.00000 - 8.66025i) q^{61} +(2.00000 - 1.73205i) q^{63} +(2.00000 + 3.46410i) q^{65} +(3.00000 - 5.19615i) q^{67} -8.00000 q^{69} +10.0000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(-2.00000 - 3.46410i) q^{75} +(7.50000 + 2.59808i) q^{77} +(-5.50000 - 9.52628i) q^{79} +(-0.500000 + 0.866025i) q^{81} -7.00000 q^{83} +(1.50000 - 2.59808i) q^{87} +(9.00000 + 15.5885i) q^{89} +(-2.00000 - 10.3923i) q^{91} +(-2.50000 - 4.33013i) q^{93} +(-2.00000 + 3.46410i) q^{95} -17.0000 q^{97} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - q^{5} + q^{7} - q^{9}+O(q^{10}) $ 2 * q + q^3 - q^5 + q^7 - q^9	$ 2 q + q^{3} - q^{5} + q^{7} - q^{9} + 3 q^{11} - 8 q^{13} - 2 q^{15} - 4 q^{19} + 5 q^{21} - 8 q^{23} + 4 q^{25} - 2 q^{27} + 6 q^{29} + 5 q^{31} - 3 q^{33} + 4 q^{35} + 8 q^{37} - 4 q^{39} + 16 q^{41} - 12 q^{43} - q^{45} - 10 q^{47} - 13 q^{49} + 9 q^{53} - 6 q^{55} - 8 q^{57} - 5 q^{59} - 10 q^{61} + 4 q^{63} + 4 q^{65} + 6 q^{67} - 16 q^{69} + 20 q^{71} - 2 q^{73} - 4 q^{75} + 15 q^{77} - 11 q^{79} - q^{81} - 14 q^{83} + 3 q^{87} + 18 q^{89} - 4 q^{91} - 5 q^{93} - 4 q^{95} - 34 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q + q^3 - q^5 + q^7 - q^9 + 3 * q^11 - 8 * q^13 - 2 * q^15 - 4 * q^19 + 5 * q^21 - 8 * q^23 + 4 * q^25 - 2 * q^27 + 6 * q^29 + 5 * q^31 - 3 * q^33 + 4 * q^35 + 8 * q^37 - 4 * q^39 + 16 * q^41 - 12 * q^43 - q^45 - 10 * q^47 - 13 * q^49 + 9 * q^53 - 6 * q^55 - 8 * q^57 - 5 * q^59 - 10 * q^61 + 4 * q^63 + 4 * q^65 + 6 * q^67 - 16 * q^69 + 20 * q^71 - 2 * q^73 - 4 * q^75 + 15 * q^77 - 11 * q^79 - q^81 - 14 * q^83 + 3 * q^87 + 18 * q^89 - 4 * q^91 - 5 * q^93 - 4 * q^95 - 34 * q^97 - 6 * q^99

Display 100 coefficients 10 coefficients

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
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	0.732294	−	0.680989i	$-0.238450\pi$
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	−0.955901	+	0.293691i	$0.905116\pi$
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	0.998526	+	0.0542666i	$0.0172821\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	0.540068	−	0.841621i	$-0.318398\pi$
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	−0.998899	+	0.0469020i	$0.985065\pi$
$20$		0			0
$21$	2.50000	+	0.866025i	0.545545	+	0.188982i
$22$		0			0
$23$	−4.00000	−	6.92820i	−0.834058	−	1.44463i	−0.894795	−	0.446476i	$-0.852679\pi$
$23$	−4.00000	−	6.92820i	−0.834058	−	1.44463i	0.0607377	−	0.998154i	$-0.480655\pi$
$24$		0			0
$25$	2.00000	−	3.46410i	0.400000	−	0.692820i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	3.00000			0.557086			0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000			0.557086			0.278543	+	0.960424i	$0.410149\pi$
$30$		0			0
$31$	2.50000	−	4.33013i	0.449013	−	0.777714i	−0.549309	−	0.835619i	$-0.685109\pi$
$31$	2.50000	−	4.33013i	0.449013	−	0.777714i	0.998322	+	0.0579057i	$0.0184423\pi$
$32$		0			0
$33$	−1.50000	−	2.59808i	−0.261116	−	0.452267i
$34$		0			0
$35$	2.00000	−	1.73205i	0.338062	−	0.292770i
$36$		0			0
$37$	4.00000	+	6.92820i	0.657596	+	1.13899i	0.981236	+	0.192809i	$0.0617599\pi$
$37$	4.00000	+	6.92820i	0.657596	+	1.13899i	−0.323640	+	0.946180i	$0.604907\pi$
$38$		0			0
$39$	−2.00000	+	3.46410i	−0.320256	+	0.554700i
$40$		0			0
$41$	8.00000			1.24939			0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000			1.24939			0.624695	+	0.780869i	$0.285223\pi$
$42$		0			0
$43$	−6.00000			−0.914991			−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000			−0.914991			−0.457496	+	0.889212i	$0.651253\pi$
$44$		0			0
$45$	−0.500000	+	0.866025i	−0.0745356	+	0.129099i
$46$		0			0
$47$	−5.00000	−	8.66025i	−0.729325	−	1.26323i	−0.957169	−	0.289530i	$-0.906501\pi$
$47$	−5.00000	−	8.66025i	−0.729325	−	1.26323i	0.227844	−	0.973698i	$-0.426832\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.50000	−	7.79423i	0.618123	−	1.07062i	−0.371706	−	0.928351i	$-0.621227\pi$
$53$	4.50000	−	7.79423i	0.618123	−	1.07062i	0.989828	−	0.142269i	$-0.0454398\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$		0			0
$57$	−4.00000			−0.529813
$58$		0			0
$59$	−2.50000	+	4.33013i	−0.325472	+	0.563735i	−0.981608	−	0.190909i	$-0.938857\pi$
$59$	−2.50000	+	4.33013i	−0.325472	+	0.563735i	0.656136	+	0.754643i	$0.272190\pi$
$60$		0			0
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	−0.985391	−	0.170305i	$-0.945525\pi$
$61$	−5.00000	−	8.66025i	−0.640184	−	1.10883i	0.345207	−	0.938527i	$-0.387809\pi$
$62$		0			0
$63$	2.00000	−	1.73205i	0.251976	−	0.218218i
$64$		0			0
$65$	2.00000	+	3.46410i	0.248069	+	0.429669i
$66$		0			0
$67$	3.00000	−	5.19615i	0.366508	−	0.634811i	−0.622509	−	0.782613i	$-0.713886\pi$
$67$	3.00000	−	5.19615i	0.366508	−	0.634811i	0.989017	+	0.147802i	$0.0472198\pi$
$68$		0			0
$69$	−8.00000			−0.963087
$70$		0			0
$71$	10.0000			1.18678			0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000			1.18678			0.593391	+	0.804914i	$0.297789\pi$
$72$		0			0
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	−0.918594	−	0.395203i	$-0.870674\pi$
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	0.801553	+	0.597924i	$0.204008\pi$
$74$		0			0
$75$	−2.00000	−	3.46410i	−0.230940	−	0.400000i
$76$		0			0
$77$	7.50000	+	2.59808i	0.854704	+	0.296078i
$78$		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
$83$	−7.00000			−0.768350			−0.384175	−	0.923260i	$-0.625514\pi$
$83$	−7.00000			−0.768350			−0.384175	+	0.923260i	$0.625514\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	1.50000	−	2.59808i	0.160817	−	0.278543i
$88$		0			0
$89$	9.00000	+	15.5885i	0.953998	+	1.65237i	0.736644	+	0.676280i	$0.236409\pi$
$89$	9.00000	+	15.5885i	0.953998	+	1.65237i	0.217354	+	0.976093i	$0.430258\pi$
$90$		0			0
$91$	−2.00000	−	10.3923i	−0.209657	−	1.08941i
$92$		0			0
$93$	−2.50000	−	4.33013i	−0.259238	−	0.449013i
$94$		0			0
$95$	−2.00000	+	3.46410i	−0.205196	+	0.355409i
$96$		0			0
$97$	−17.0000			−1.72609			−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000			−1.72609			−0.863044	+	0.505128i	$0.831445\pi$
$98$		0			0
$99$	−3.00000			−0.301511
$100$		0			0
$101$	−1.00000	+	1.73205i	−0.0995037	+	0.172345i	−0.911479	−	0.411346i	$-0.865059\pi$
$101$	−1.00000	+	1.73205i	−0.0995037	+	0.172345i	0.811976	+	0.583691i	$0.198392\pi$
$102$		0			0
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$	−0.500000	−	2.59808i	−0.0487950	−	0.253546i
$106$		0			0
$107$	−5.50000	−	9.52628i	−0.531705	−	0.920940i	−0.999315	−	0.0370053i	$-0.988218\pi$
$107$	−5.50000	−	9.52628i	−0.531705	−	0.920940i	0.467610	−	0.883935i	$-0.345115\pi$
$108$		0			0
$109$	−5.00000	+	8.66025i	−0.478913	+	0.829502i	−0.999708	−	0.0241802i	$-0.992302\pi$
$109$	−5.00000	+	8.66025i	−0.478913	+	0.829502i	0.520794	+	0.853682i	$0.325636\pi$
$110$		0			0
$111$	8.00000			0.759326
$112$		0			0
$113$	−8.00000			−0.752577			−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000			−0.752577			−0.376288	+	0.926503i	$0.622800\pi$
$114$		0			0
$115$	−4.00000	+	6.92820i	−0.373002	+	0.646058i
$116$		0			0
$117$	2.00000	+	3.46410i	0.184900	+	0.320256i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$	4.00000	−	6.92820i	0.360668	−	0.624695i
$124$		0			0
$125$	−9.00000			−0.804984
$126$		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
$129$	−3.00000	+	5.19615i	−0.264135	+	0.457496i
$130$		0			0
$131$	7.50000	+	12.9904i	0.655278	+	1.13497i	0.981824	+	0.189794i	$0.0607819\pi$
$131$	7.50000	+	12.9904i	0.655278	+	1.13497i	−0.326546	+	0.945181i	$0.605885\pi$
$132$		0			0
$133$	8.00000	−	6.92820i	0.693688	−	0.600751i
$134$		0			0
$135$	0.500000	+	0.866025i	0.0430331	+	0.0745356i
$136$		0			0
$137$	7.00000	−	12.1244i	0.598050	−	1.03585i	−0.395058	−	0.918656i	$-0.629276\pi$
$137$	7.00000	−	12.1244i	0.598050	−	1.03585i	0.993109	−	0.117198i	$-0.0373911\pi$
$138$		0			0
$139$	22.0000			1.86602			0.933008	−	0.359856i	$-0.117174\pi$
$139$	22.0000			1.86602			0.933008	+	0.359856i	$0.117174\pi$
$140$		0			0
$141$	−10.0000			−0.842152
$142$		0			0
$143$	−6.00000	+	10.3923i	−0.501745	+	0.869048i
$144$		0			0
$145$	−1.50000	−	2.59808i	−0.124568	−	0.215758i
$146$		0			0
$147$	−1.00000	+	6.92820i	−0.0824786	+	0.571429i
$148$		0			0
$149$	9.00000	+	15.5885i	0.737309	+	1.27706i	0.953703	+	0.300750i	$0.0972370\pi$
$149$	9.00000	+	15.5885i	0.737309	+	1.27706i	−0.216394	+	0.976306i	$0.569430\pi$
$150$		0			0
$151$	2.50000	−	4.33013i	0.203447	−	0.352381i	−0.746190	−	0.665733i	$-0.768119\pi$
$151$	2.50000	−	4.33013i	0.203447	−	0.352381i	0.949637	+	0.313353i	$0.101452\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.00000			−0.401610
$156$		0			0
$157$	−2.00000	+	3.46410i	−0.159617	+	0.276465i	−0.934731	−	0.355357i	$-0.884359\pi$
$157$	−2.00000	+	3.46410i	−0.159617	+	0.276465i	0.775113	+	0.631822i	$0.217693\pi$
$158$		0			0
$159$	−4.50000	−	7.79423i	−0.356873	−	0.618123i
$160$		0			0
$161$	16.0000	−	13.8564i	1.26098	−	1.09204i
$162$		0			0
$163$	4.00000	+	6.92820i	0.313304	+	0.542659i	0.979076	−	0.203497i	$-0.0652307\pi$
$163$	4.00000	+	6.92820i	0.313304	+	0.542659i	−0.665771	+	0.746156i	$0.731897\pi$
$164$		0			0
$165$	−1.50000	+	2.59808i	−0.116775	+	0.202260i
$166$		0			0
$167$	−2.00000			−0.154765			−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000			−0.154765			−0.0773823	+	0.997001i	$0.524656\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	−2.00000	+	3.46410i	−0.152944	+	0.264906i
$172$		0			0
$173$	1.00000	+	1.73205i	0.0760286	+	0.131685i	0.901533	−	0.432710i	$-0.142443\pi$
$173$	1.00000	+	1.73205i	0.0760286	+	0.131685i	−0.825505	+	0.564396i	$0.809109\pi$
$174$		0			0
$175$	10.0000	+	3.46410i	0.755929	+	0.261861i
$176$		0			0
$177$	2.50000	+	4.33013i	0.187912	+	0.325472i
$178$		0			0
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	−0.549825	−	0.835280i	$-0.685306\pi$
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	0.998286	+	0.0585225i	$0.0186389\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$	−10.0000			−0.739221
$184$		0			0
$185$	4.00000	−	6.92820i	0.294086	−	0.509372i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.500000	−	2.59808i	−0.0363696	−	0.188982i
$190$		0			0
$191$	8.00000	+	13.8564i	0.578860	+	1.00261i	0.995610	+	0.0935936i	$0.0298354\pi$
$191$	8.00000	+	13.8564i	0.578860	+	1.00261i	−0.416751	+	0.909021i	$0.636831\pi$
$192$		0			0
$193$	13.5000	−	23.3827i	0.971751	−	1.68312i	0.281487	−	0.959565i	$-0.409172\pi$
$193$	13.5000	−	23.3827i	0.971751	−	1.68312i	0.690264	−	0.723558i	$-0.257494\pi$
$194$		0			0
$195$	4.00000			0.286446
$196$		0			0
$197$	26.0000			1.85242			0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000			1.85242			0.926212	+	0.377004i	$0.123046\pi$
$198$		0			0
$199$	−6.00000	+	10.3923i	−0.425329	+	0.736691i	−0.996451	−	0.0841740i	$-0.973175\pi$
$199$	−6.00000	+	10.3923i	−0.425329	+	0.736691i	0.571122	+	0.820865i	$0.306508\pi$
$200$		0			0
$201$	−3.00000	−	5.19615i	−0.211604	−	0.366508i
$202$		0			0
$203$	1.50000	+	7.79423i	0.105279	+	0.547048i
$204$		0			0
$205$	−4.00000	−	6.92820i	−0.279372	−	0.483887i
$206$		0			0
$207$	−4.00000	+	6.92820i	−0.278019	+	0.481543i
$208$		0			0
$209$	−12.0000			−0.830057
$210$		0			0
$211$	−6.00000			−0.413057			−0.206529	−	0.978441i	$-0.566217\pi$
$211$	−6.00000			−0.413057			−0.206529	+	0.978441i	$0.566217\pi$
$212$		0			0
$213$	5.00000	−	8.66025i	0.342594	−	0.593391i
$214$		0			0
$215$	3.00000	+	5.19615i	0.204598	+	0.354375i
$216$		0			0
$217$	12.5000	+	4.33013i	0.848555	+	0.293948i
$218$		0			0
$219$	1.00000	+	1.73205i	0.0675737	+	0.117041i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.00000			0.0669650			0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000			0.0669650			0.0334825	+	0.999439i	$0.489340\pi$
$224$		0			0
$225$	−4.00000			−0.266667
$226$		0			0
$227$	10.5000	−	18.1865i	0.696909	−	1.20708i	−0.272623	−	0.962121i	$-0.587891\pi$
$227$	10.5000	−	18.1865i	0.696909	−	1.20708i	0.969533	−	0.244962i	$-0.0787754\pi$
$228$		0			0
$229$	−6.00000	−	10.3923i	−0.396491	−	0.686743i	0.596799	−	0.802391i	$-0.296439\pi$
$229$	−6.00000	−	10.3923i	−0.396491	−	0.686743i	−0.993290	+	0.115648i	$0.963106\pi$
$230$		0			0
$231$	6.00000	−	5.19615i	0.394771	−	0.341882i
$232$		0			0
$233$	12.0000	+	20.7846i	0.786146	+	1.36165i	0.928312	+	0.371802i	$0.121260\pi$
$233$	12.0000	+	20.7846i	0.786146	+	1.36165i	−0.142166	+	0.989843i	$0.545407\pi$
$234$		0			0
$235$	−5.00000	+	8.66025i	−0.326164	+	0.564933i
$236$		0			0
$237$	−11.0000			−0.714527
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−11.5000	+	19.9186i	−0.740780	+	1.28307i	0.211360	+	0.977408i	$0.432211\pi$
$241$	−11.5000	+	19.9186i	−0.740780	+	1.28307i	−0.952141	+	0.305661i	$0.901123\pi$
$242$		0			0
$243$	0.500000	+	0.866025i	0.0320750	+	0.0555556i
$244$		0			0
$245$	5.50000	+	4.33013i	0.351382	+	0.276642i
$246$		0			0
$247$	8.00000	+	13.8564i	0.509028	+	0.881662i
$248$		0			0
$249$	−3.50000	+	6.06218i	−0.221803	+	0.384175i
$250$		0			0
$251$	−11.0000			−0.694314			−0.347157	−	0.937807i	$-0.612853\pi$
$251$	−11.0000			−0.694314			−0.347157	+	0.937807i	$0.612853\pi$
$252$		0			0
$253$	−24.0000			−1.50887
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.00000	−	12.1244i	−0.436648	−	0.756297i	0.560781	−	0.827964i	$-0.310501\pi$
$257$	−7.00000	−	12.1244i	−0.436648	−	0.756297i	−0.997429	+	0.0716680i	$0.977168\pi$
$258$		0			0
$259$	−16.0000	+	13.8564i	−0.994192	+	0.860995i
$260$		0			0
$261$	−1.50000	−	2.59808i	−0.0928477	−	0.160817i
$262$		0			0
$263$	7.00000	−	12.1244i	0.431638	−	0.747620i	−0.565376	−	0.824833i	$-0.691269\pi$
$263$	7.00000	−	12.1244i	0.431638	−	0.747620i	0.997015	+	0.0772134i	$0.0246023\pi$
$264$		0			0
$265$	−9.00000			−0.552866
$266$		0			0
$267$	18.0000			1.10158
$268$		0			0
$269$	0.500000	−	0.866025i	0.0304855	−	0.0528025i	−0.850380	−	0.526169i	$-0.823628\pi$
$269$	0.500000	−	0.866025i	0.0304855	−	0.0528025i	0.880866	+	0.473366i	$0.156961\pi$
$270$		0			0
$271$	8.50000	+	14.7224i	0.516338	+	0.894324i	0.999820	+	0.0189696i	$0.00603859\pi$
$271$	8.50000	+	14.7224i	0.516338	+	0.894324i	−0.483482	+	0.875354i	$0.660628\pi$
$272$		0			0
$273$	−10.0000	−	3.46410i	−0.605228	−	0.209657i
$274$		0			0
$275$	−6.00000	−	10.3923i	−0.361814	−	0.626680i
$276$		0			0
$277$	4.00000	−	6.92820i	0.240337	−	0.416275i	−0.720473	−	0.693482i	$-0.756075\pi$
$277$	4.00000	−	6.92820i	0.240337	−	0.416275i	0.960810	+	0.277207i	$0.0894088\pi$
$278$		0			0
$279$	−5.00000			−0.299342
$280$		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
$283$	−9.00000	+	15.5885i	−0.534994	+	0.926638i	0.464169	+	0.885747i	$0.346353\pi$
$283$	−9.00000	+	15.5885i	−0.534994	+	0.926638i	−0.999164	+	0.0408910i	$0.986980\pi$
$284$		0			0
$285$	2.00000	+	3.46410i	0.118470	+	0.205196i
$286$		0			0
$287$	4.00000	+	20.7846i	0.236113	+	1.22688i
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$	−8.50000	+	14.7224i	−0.498279	+	0.863044i
$292$		0			0
$293$	19.0000			1.10999			0.554996	−	0.831853i	$-0.312720\pi$
$293$	19.0000			1.10999			0.554996	+	0.831853i	$0.312720\pi$
$294$		0			0
$295$	5.00000			0.291111
$296$		0			0
$297$	−1.50000	+	2.59808i	−0.0870388	+	0.150756i
$298$		0			0
$299$	16.0000	+	27.7128i	0.925304	+	1.60267i
$300$		0			0
$301$	−3.00000	−	15.5885i	−0.172917	−	0.898504i
$302$		0			0
$303$	1.00000	+	1.73205i	0.0574485	+	0.0995037i
$304$		0			0
$305$	−5.00000	+	8.66025i	−0.286299	+	0.495885i
$306$		0			0
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−4.50000	−	7.79423i	−0.254355	−	0.440556i	0.710365	−	0.703833i	$-0.248530\pi$
$313$	−4.50000	−	7.79423i	−0.254355	−	0.440556i	−0.964720	+	0.263278i	$0.915197\pi$
$314$		0			0
$315$	−2.50000	−	0.866025i	−0.140859	−	0.0487950i
$316$		0			0
$317$	6.50000	+	11.2583i	0.365076	+	0.632331i	0.988788	−	0.149323i	$-0.0477095\pi$
$317$	6.50000	+	11.2583i	0.365076	+	0.632331i	−0.623712	+	0.781654i	$0.714376\pi$
$318$		0			0
$319$	4.50000	−	7.79423i	0.251952	−	0.436393i
$320$		0			0
$321$	−11.0000			−0.613960
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−8.00000	+	13.8564i	−0.443760	+	0.768615i
$326$		0			0
$327$	5.00000	+	8.66025i	0.276501	+	0.478913i
$328$		0			0
$329$	20.0000	−	17.3205i	1.10264	−	0.954911i
$330$		0			0
$331$	2.00000	+	3.46410i	0.109930	+	0.190404i	0.915742	−	0.401768i	$-0.131604\pi$
$331$	2.00000	+	3.46410i	0.109930	+	0.190404i	−0.805812	+	0.592172i	$0.798271\pi$
$332$		0			0
$333$	4.00000	−	6.92820i	0.219199	−	0.379663i
$334$		0			0
$335$	−6.00000			−0.327815
$336$		0			0
$337$	−15.0000			−0.817102			−0.408551	−	0.912735i	$-0.633966\pi$
$337$	−15.0000			−0.817102			−0.408551	+	0.912735i	$0.633966\pi$
$338$		0			0
$339$	−4.00000	+	6.92820i	−0.217250	+	0.376288i
$340$		0			0
$341$	−7.50000	−	12.9904i	−0.406148	−	0.703469i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$	4.00000	+	6.92820i	0.215353	+	0.373002i
$346$		0			0
$347$	−10.0000	+	17.3205i	−0.536828	+	0.929814i	0.462244	+	0.886753i	$0.347044\pi$
$347$	−10.0000	+	17.3205i	−0.536828	+	0.929814i	−0.999072	+	0.0430610i	$0.986289\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$	4.00000			0.213504
$352$		0			0
$353$	−4.00000	+	6.92820i	−0.212899	+	0.368751i	−0.952620	−	0.304162i	$-0.901624\pi$
$353$	−4.00000	+	6.92820i	−0.212899	+	0.368751i	0.739722	+	0.672913i	$0.234957\pi$
$354$		0			0
$355$	−5.00000	−	8.66025i	−0.265372	−	0.459639i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.00000	−	5.19615i	−0.158334	−	0.274242i	0.775934	−	0.630814i	$-0.217279\pi$
$359$	−3.00000	−	5.19615i	−0.158334	−	0.274242i	−0.934268	+	0.356572i	$0.883946\pi$
$360$		0			0
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$		0			0
$363$	2.00000			0.104973
$364$		0			0
$365$	2.00000			0.104685
$366$		0			0
$367$	3.50000	−	6.06218i	0.182699	−	0.316443i	−0.760100	−	0.649806i	$-0.774850\pi$
$367$	3.50000	−	6.06218i	0.182699	−	0.316443i	0.942799	+	0.333363i	$0.108183\pi$
$368$		0			0
$369$	−4.00000	−	6.92820i	−0.208232	−	0.360668i
$370$		0			0
$371$	22.5000	+	7.79423i	1.16814	+	0.404656i
$372$		0			0
$373$	8.00000	+	13.8564i	0.414224	+	0.717458i	0.995347	−	0.0963587i	$-0.0307196\pi$
$373$	8.00000	+	13.8564i	0.414224	+	0.717458i	−0.581122	+	0.813816i	$0.697386\pi$
$374$		0			0
$375$	−4.50000	+	7.79423i	−0.232379	+	0.402492i
$376$		0			0
$377$	−12.0000			−0.618031
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$	−3.50000	+	6.06218i	−0.179310	+	0.310575i
$382$		0			0
$383$	−1.00000	−	1.73205i	−0.0510976	−	0.0885037i	0.839345	−	0.543599i	$-0.182939\pi$
$383$	−1.00000	−	1.73205i	−0.0510976	−	0.0885037i	−0.890443	+	0.455095i	$0.849605\pi$
$384$		0			0
$385$	−1.50000	−	7.79423i	−0.0764471	−	0.397231i
$386$		0			0
$387$	3.00000	+	5.19615i	0.152499	+	0.264135i
$388$		0			0
$389$	13.0000	−	22.5167i	0.659126	−	1.14164i	−0.321716	−	0.946836i	$-0.604260\pi$
$389$	13.0000	−	22.5167i	0.659126	−	1.14164i	0.980842	−	0.194804i	$-0.0624070\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	15.0000			0.756650
$394$		0			0
$395$	−5.50000	+	9.52628i	−0.276735	+	0.479319i
$396$		0			0
$397$	14.0000	+	24.2487i	0.702640	+	1.21701i	0.967537	+	0.252731i	$0.0813288\pi$
$397$	14.0000	+	24.2487i	0.702640	+	1.21701i	−0.264897	+	0.964277i	$0.585338\pi$
$398$		0			0
$399$	−2.00000	−	10.3923i	−0.100125	−	0.520266i
$400$		0			0
$401$	4.00000	+	6.92820i	0.199750	+	0.345978i	0.948447	−	0.316934i	$-0.102654\pi$
$401$	4.00000	+	6.92820i	0.199750	+	0.345978i	−0.748697	+	0.662912i	$0.769320\pi$
$402$		0			0
$403$	−10.0000	+	17.3205i	−0.498135	+	0.862796i
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$	−3.50000	+	6.06218i	−0.173064	+	0.299755i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−3.50000	+	6.06218i	−0.173064	+	0.299755i	0.766426	+	0.642333i	$0.222033\pi$
$410$		0			0
$411$	−7.00000	−	12.1244i	−0.345285	−	0.598050i
$412$		0			0
$413$	−12.5000	−	4.33013i	−0.615085	−	0.213072i
$414$		0			0
$415$	3.50000	+	6.06218i	0.171808	+	0.297581i
$416$		0			0
$417$	11.0000	−	19.0526i	0.538672	−	0.933008i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		0			0
$423$	−5.00000	+	8.66025i	−0.243108	+	0.421076i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	20.0000	−	17.3205i	0.967868	−	0.838198i
$428$		0			0
$429$	6.00000	+	10.3923i	0.289683	+	0.501745i
$430$		0			0
$431$	6.00000	−	10.3923i	0.289010	−	0.500580i	−0.684564	−	0.728953i	$-0.740007\pi$
$431$	6.00000	−	10.3923i	0.289010	−	0.500580i	0.973574	+	0.228373i	$0.0733406\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$	−3.00000			−0.143839
$436$		0			0
$437$	−16.0000	+	27.7128i	−0.765384	+	1.32568i
$438$		0			0
$439$	4.50000	+	7.79423i	0.214773	+	0.371998i	0.953202	−	0.302333i	$-0.0977654\pi$
$439$	4.50000	+	7.79423i	0.214773	+	0.371998i	−0.738429	+	0.674331i	$0.764432\pi$
$440$		0			0
$441$	5.50000	+	4.33013i	0.261905	+	0.206197i
$442$		0			0
$443$	−0.500000	−	0.866025i	−0.0237557	−	0.0411461i	0.853903	−	0.520432i	$-0.174229\pi$
$443$	−0.500000	−	0.866025i	−0.0237557	−	0.0411461i	−0.877659	+	0.479286i	$0.840896\pi$
$444$		0			0
$445$	9.00000	−	15.5885i	0.426641	−	0.738964i
$446$		0			0
$447$	18.0000			0.851371
$448$		0			0
$449$	4.00000			0.188772			0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000			0.188772			0.0943858	+	0.995536i	$0.469911\pi$
$450$		0			0
$451$	12.0000	−	20.7846i	0.565058	−	0.978709i
$452$		0			0
$453$	−2.50000	−	4.33013i	−0.117460	−	0.203447i
$454$		0			0
$455$	−8.00000	+	6.92820i	−0.375046	+	0.324799i
$456$		0			0
$457$	8.50000	+	14.7224i	0.397613	+	0.688686i	0.993431	−	0.114433i	$-0.0365053\pi$
$457$	8.50000	+	14.7224i	0.397613	+	0.688686i	−0.595818	+	0.803120i	$0.703172\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−30.0000			−1.39724			−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000			−1.39724			−0.698620	+	0.715493i	$0.746202\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$	−2.50000	+	4.33013i	−0.115935	+	0.200805i
$466$		0			0
$467$	−14.0000	−	24.2487i	−0.647843	−	1.12210i	−0.983637	−	0.180161i	$-0.942338\pi$
$467$	−14.0000	−	24.2487i	−0.647843	−	1.12210i	0.335794	−	0.941935i	$-0.390995\pi$
$468$		0			0
$469$	15.0000	+	5.19615i	0.692636	+	0.239936i
$470$		0			0
$471$	2.00000	+	3.46410i	0.0921551	+	0.159617i
$472$		0			0
$473$	−9.00000	+	15.5885i	−0.413820	+	0.716758i
$474$		0			0
$475$	−16.0000			−0.734130
$476$		0			0
$477$	−9.00000			−0.412082
$478$		0			0
$479$	−3.00000	+	5.19615i	−0.137073	+	0.237418i	−0.926388	−	0.376571i	$-0.877103\pi$
$479$	−3.00000	+	5.19615i	−0.137073	+	0.237418i	0.789314	+	0.613990i	$0.210436\pi$
$480$		0			0
$481$	−16.0000	−	27.7128i	−0.729537	−	1.26360i
$482$		0			0
$483$	−4.00000	−	20.7846i	−0.182006	−	0.945732i
$484$		0			0
$485$	8.50000	+	14.7224i	0.385965	+	0.668511i
$486$		0			0
$487$	−18.5000	+	32.0429i	−0.838315	+	1.45200i	0.0529875	+	0.998595i	$0.483126\pi$
$487$	−18.5000	+	32.0429i	−0.838315	+	1.45200i	−0.891303	+	0.453409i	$0.850208\pi$
$488$		0			0
$489$	8.00000			0.361773
$490$		0			0
$491$	9.00000			0.406164			0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000			0.406164			0.203082	+	0.979162i	$0.434904\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	1.50000	+	2.59808i	0.0674200	+	0.116775i
$496$		0			0
$497$	5.00000	+	25.9808i	0.224281	+	1.16540i
$498$		0			0
$499$	5.00000	+	8.66025i	0.223831	+	0.387686i	0.955968	−	0.293471i	$-0.0948104\pi$
$499$	5.00000	+	8.66025i	0.223831	+	0.387686i	−0.732137	+	0.681157i	$0.761477\pi$
$500$		0			0
$501$	−1.00000	+	1.73205i	−0.0446767	+	0.0773823i
$502$		0			0
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$	2.00000			0.0889988
$506$		0			0
$507$	1.50000	−	2.59808i	0.0666173	−	0.115385i
$508$		0			0
$509$	−19.5000	−	33.7750i	−0.864322	−	1.49705i	−0.867719	−	0.497056i	$-0.834414\pi$
$509$	−19.5000	−	33.7750i	−0.864322	−	1.49705i	0.00339621	−	0.999994i	$-0.498919\pi$
$510$		0			0
$511$	−5.00000	−	1.73205i	−0.221187	−	0.0766214i
$512$		0			0
$513$	2.00000	+	3.46410i	0.0883022	+	0.152944i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−30.0000			−1.31940
$518$		0			0
$519$	2.00000			0.0877903
$520$		0			0
$521$	9.00000	−	15.5885i	0.394297	−	0.682943i	−0.598714	−	0.800963i	$-0.704321\pi$
$521$	9.00000	−	15.5885i	0.394297	−	0.682943i	0.993011	+	0.118020i	$0.0376547\pi$
$522$		0			0
$523$	−6.00000	−	10.3923i	−0.262362	−	0.454424i	0.704507	−	0.709697i	$-0.251168\pi$
$523$	−6.00000	−	10.3923i	−0.262362	−	0.454424i	−0.966869	+	0.255273i	$0.917835\pi$
$524$		0			0
$525$	8.00000	−	6.92820i	0.349149	−	0.302372i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−20.5000	+	35.5070i	−0.891304	+	1.54378i
$530$		0			0
$531$	5.00000			0.216982
$532$		0			0
$533$	−32.0000			−1.38607
$534$		0			0
$535$	−5.50000	+	9.52628i	−0.237786	+	0.411857i
$536$		0			0
$537$	−6.00000	−	10.3923i	−0.258919	−	0.448461i
$538$		0			0
$539$	−3.00000	+	20.7846i	−0.129219	+	0.895257i
$540$		0			0
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	0.956504	+	0.291718i	$0.0942267\pi$
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	−0.225617	+	0.974216i	$0.572440\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.0000			0.428353
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$		0			0
$549$	−5.00000	+	8.66025i	−0.213395	+	0.369611i
$550$		0			0
$551$	−6.00000	−	10.3923i	−0.255609	−	0.442727i
$552$		0			0
$553$	22.0000	−	19.0526i	0.935535	−	0.810197i
$554$		0			0
$555$	−4.00000	−	6.92820i	−0.169791	−	0.294086i
$556$		0			0
$557$	7.50000	−	12.9904i	0.317785	−	0.550420i	−0.662240	−	0.749291i	$-0.730394\pi$
$557$	7.50000	−	12.9904i	0.317785	−	0.550420i	0.980026	+	0.198871i	$0.0637276\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.5000	−	33.7750i	0.821827	−	1.42345i	−0.0824933	−	0.996592i	$-0.526288\pi$
$563$	19.5000	−	33.7750i	0.821827	−	1.42345i	0.904320	−	0.426855i	$-0.140378\pi$
$564$		0			0
$565$	4.00000	+	6.92820i	0.168281	+	0.291472i
$566$		0			0
$567$	−2.50000	−	0.866025i	−0.104990	−	0.0363696i
$568$		0			0
$569$	−6.00000	−	10.3923i	−0.251533	−	0.435668i	0.712415	−	0.701758i	$-0.247601\pi$
$569$	−6.00000	−	10.3923i	−0.251533	−	0.435668i	−0.963948	+	0.266090i	$0.914268\pi$
$570$		0			0
$571$	11.0000	−	19.0526i	0.460336	−	0.797325i	−0.538642	−	0.842535i	$-0.681062\pi$
$571$	11.0000	−	19.0526i	0.460336	−	0.797325i	0.998978	+	0.0452101i	$0.0143957\pi$
$572$		0			0
$573$	16.0000			0.668410
$574$		0			0
$575$	−32.0000			−1.33449
$576$		0			0
$577$	16.5000	−	28.5788i	0.686904	−	1.18975i	−0.285930	−	0.958250i	$-0.592303\pi$
$577$	16.5000	−	28.5788i	0.686904	−	1.18975i	0.972834	−	0.231502i	$-0.0743641\pi$
$578$		0			0
$579$	−13.5000	−	23.3827i	−0.561041	−	0.971751i
$580$		0			0
$581$	−3.50000	−	18.1865i	−0.145204	−	0.754505i
$582$		0			0
$583$	−13.5000	−	23.3827i	−0.559113	−	0.968412i
$584$		0			0
$585$	2.00000	−	3.46410i	0.0826898	−	0.143223i
$586$		0			0
$587$	−45.0000			−1.85735			−0.928674	−	0.370896i	$-0.879051\pi$
$587$	−45.0000			−1.85735			−0.928674	+	0.370896i	$0.879051\pi$
$588$		0			0
$589$	−20.0000			−0.824086
$590$		0			0
$591$	13.0000	−	22.5167i	0.534749	−	0.926212i
$592$		0			0
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	6.00000	+	10.3923i	0.245564	+	0.425329i
$598$		0			0
$599$	21.0000	−	36.3731i	0.858037	−	1.48616i	−0.0157622	−	0.999876i	$-0.505017\pi$
$599$	21.0000	−	36.3731i	0.858037	−	1.48616i	0.873799	−	0.486287i	$-0.161649\pi$
$600$		0			0
$601$	−13.0000			−0.530281			−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000			−0.530281			−0.265141	+	0.964210i	$0.585418\pi$
$602$		0			0
$603$	−6.00000			−0.244339
$604$		0			0
$605$	1.00000	−	1.73205i	0.0406558	−	0.0704179i
$606$		0			0
$607$	−6.50000	−	11.2583i	−0.263827	−	0.456962i	0.703429	−	0.710766i	$-0.251651\pi$
$607$	−6.50000	−	11.2583i	−0.263827	−	0.456962i	−0.967256	+	0.253804i	$0.918318\pi$
$608$		0			0
$609$	7.50000	+	2.59808i	0.303915	+	0.105279i
$610$		0			0
$611$	20.0000	+	34.6410i	0.809113	+	1.40143i
$612$		0			0
$613$	4.00000	−	6.92820i	0.161558	−	0.279827i	−0.773869	−	0.633345i	$-0.781681\pi$
$613$	4.00000	−	6.92820i	0.161558	−	0.279827i	0.935428	+	0.353518i	$0.115015\pi$
$614$		0			0
$615$	−8.00000			−0.322591
$616$		0			0
$617$	−30.0000			−1.20775			−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000			−1.20775			−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	3.00000	−	5.19615i	0.120580	−	0.208851i	−0.799416	−	0.600777i	$-0.794858\pi$
$619$	3.00000	−	5.19615i	0.120580	−	0.208851i	0.919997	+	0.391926i	$0.128191\pi$
$620$		0			0
$621$	4.00000	+	6.92820i	0.160514	+	0.278019i
$622$		0			0
$623$	−36.0000	+	31.1769i	−1.44231	+	1.24908i
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$		0			0
$627$	−6.00000	+	10.3923i	−0.239617	+	0.415029i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−11.0000			−0.437903			−0.218952	−	0.975736i	$-0.570264\pi$
$631$	−11.0000			−0.437903			−0.218952	+	0.975736i	$0.570264\pi$
$632$		0			0
$633$	−3.00000	+	5.19615i	−0.119239	+	0.206529i
$634$		0			0
$635$	3.50000	+	6.06218i	0.138893	+	0.240570i
$636$		0			0
$637$	26.0000	−	10.3923i	1.03016	−	0.411758i
$638$		0			0
$639$	−5.00000	−	8.66025i	−0.197797	−	0.342594i
$640$		0			0
$641$	11.0000	−	19.0526i	0.434474	−	0.752531i	−0.562779	−	0.826608i	$-0.690268\pi$
$641$	11.0000	−	19.0526i	0.434474	−	0.752531i	0.997253	+	0.0740768i	$0.0236010\pi$
$642$		0			0
$643$	−46.0000			−1.81406			−0.907031	−	0.421063i	$-0.861657\pi$
$643$	−46.0000			−1.81406			−0.907031	+	0.421063i	$0.861657\pi$
$644$		0			0
$645$	6.00000			0.236250
$646$		0			0
$647$	25.0000	−	43.3013i	0.982851	−	1.70235i	0.331731	−	0.943374i	$-0.392367\pi$
$647$	25.0000	−	43.3013i	0.982851	−	1.70235i	0.651120	−	0.758975i	$-0.274299\pi$
$648$		0			0
$649$	7.50000	+	12.9904i	0.294401	+	0.509917i
$650$		0			0
$651$	10.0000	−	8.66025i	0.391931	−	0.339422i
$652$		0			0
$653$	7.50000	+	12.9904i	0.293498	+	0.508353i	0.974634	−	0.223803i	$-0.0718474\pi$
$653$	7.50000	+	12.9904i	0.293498	+	0.508353i	−0.681137	+	0.732156i	$0.738514\pi$
$654$		0			0
$655$	7.50000	−	12.9904i	0.293049	−	0.507576i
$656$		0			0
$657$	2.00000			0.0780274
$658$		0			0
$659$	−24.0000			−0.934907			−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000			−0.934907			−0.467454	+	0.884018i	$0.654829\pi$
$660$		0			0
$661$	19.0000	−	32.9090i	0.739014	−	1.28001i	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	19.0000	−	32.9090i	0.739014	−	1.28001i	0.952940	−	0.303160i	$-0.0980418\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−10.0000	−	3.46410i	−0.387783	−	0.134332i
$666$		0			0
$667$	−12.0000	−	20.7846i	−0.464642	−	0.804783i
$668$		0			0
$669$	0.500000	−	0.866025i	0.0193311	−	0.0334825i
$670$		0			0
$671$	−30.0000			−1.15814
$672$		0			0
$673$	29.0000			1.11787			0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000			1.11787			0.558934	+	0.829212i	$0.311211\pi$
$674$		0			0
$675$	−2.00000	+	3.46410i	−0.0769800	+	0.133333i
$676$		0			0
$677$	9.50000	+	16.4545i	0.365115	+	0.632397i	0.988795	−	0.149283i	$-0.0476964\pi$
$677$	9.50000	+	16.4545i	0.365115	+	0.632397i	−0.623680	+	0.781680i	$0.714363\pi$
$678$		0			0
$679$	−8.50000	−	44.1673i	−0.326200	−	1.69499i
$680$		0			0
$681$	−10.5000	−	18.1865i	−0.402361	−	0.696909i
$682$		0			0
$683$	8.50000	−	14.7224i	0.325243	−	0.563338i	−0.656318	−	0.754484i	$-0.727887\pi$
$683$	8.50000	−	14.7224i	0.325243	−	0.563338i	0.981562	+	0.191146i	$0.0612204\pi$
$684$		0			0
$685$	−14.0000			−0.534913
$686$		0			0
$687$	−12.0000			−0.457829
$688$		0			0
$689$	−18.0000	+	31.1769i	−0.685745	+	1.18775i
$690$		0			0
$691$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$691$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$692$		0			0
$693$	−1.50000	−	7.79423i	−0.0569803	−	0.296078i
$694$		0			0
$695$	−11.0000	−	19.0526i	−0.417254	−	0.722705i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	24.0000			0.907763
$700$		0			0
$701$	−5.00000			−0.188847			−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000			−0.188847			−0.0944237	+	0.995532i	$0.530101\pi$
$702$		0			0
$703$	16.0000	−	27.7128i	0.603451	−	1.04521i
$704$		0			0
$705$	5.00000	+	8.66025i	0.188311	+	0.326164i
$706$		0			0
$707$	−5.00000	−	1.73205i	−0.188044	−	0.0651405i
$708$		0			0
$709$	3.00000	+	5.19615i	0.112667	+	0.195146i	0.916845	−	0.399244i	$-0.130727\pi$
$709$	3.00000	+	5.19615i	0.112667	+	0.195146i	−0.804178	+	0.594389i	$0.797394\pi$
$710$		0			0
$711$	−5.50000	+	9.52628i	−0.206266	+	0.357263i
$712$		0			0
$713$	−40.0000			−1.49801
$714$		0			0
$715$	12.0000			0.448775
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17.0000	+	29.4449i	0.633993	+	1.09811i	0.986728	+	0.162385i	$0.0519185\pi$
$719$	17.0000	+	29.4449i	0.633993	+	1.09811i	−0.352735	+	0.935723i	$0.614748\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	11.5000	+	19.9186i	0.427690	+	0.740780i
$724$		0			0
$725$	6.00000	−	10.3923i	0.222834	−	0.385961i
$726$		0			0
$727$	47.0000			1.74313			0.871567	−	0.490277i	$-0.163104\pi$
$727$	47.0000			1.74313			0.871567	+	0.490277i	$0.163104\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		0			0
$732$		0			0
$733$	7.00000	+	12.1244i	0.258551	+	0.447823i	0.965854	−	0.259087i	$-0.0834217\pi$
$733$	7.00000	+	12.1244i	0.258551	+	0.447823i	−0.707303	+	0.706910i	$0.750088\pi$
$734$		0			0
$735$	6.50000	−	2.59808i	0.239756	−	0.0958315i
$736$		0			0
$737$	−9.00000	−	15.5885i	−0.331519	−	0.574208i
$738$		0			0
$739$	5.00000	−	8.66025i	0.183928	−	0.318573i	−0.759287	−	0.650756i	$-0.774452\pi$
$739$	5.00000	−	8.66025i	0.183928	−	0.318573i	0.943215	+	0.332184i	$0.107785\pi$
$740$		0			0
$741$	16.0000			0.587775
$742$		0			0
$743$	34.0000			1.24734			0.623670	−	0.781688i	$-0.285641\pi$
$743$	34.0000			1.24734			0.623670	+	0.781688i	$0.285641\pi$
$744$		0			0
$745$	9.00000	−	15.5885i	0.329734	−	0.571117i
$746$		0			0
$747$	3.50000	+	6.06218i	0.128058	+	0.221803i
$748$		0			0
$749$	22.0000	−	19.0526i	0.803863	−	0.696165i
$750$		0			0
$751$	13.5000	+	23.3827i	0.492622	+	0.853246i	0.999964	−	0.00849853i	$-0.00270520\pi$
$751$	13.5000	+	23.3827i	0.492622	+	0.853246i	−0.507342	+	0.861745i	$0.669372\pi$
$752$		0			0
$753$	−5.50000	+	9.52628i	−0.200431	+	0.347157i
$754$		0			0
$755$	−5.00000			−0.181969
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$		0			0
$759$	−12.0000	+	20.7846i	−0.435572	+	0.754434i
$760$		0			0
$761$	6.00000	+	10.3923i	0.217500	+	0.376721i	0.954043	−	0.299670i	$-0.0968765\pi$
$761$	6.00000	+	10.3923i	0.217500	+	0.376721i	−0.736543	+	0.676391i	$0.763543\pi$
$762$		0			0
$763$	−25.0000	−	8.66025i	−0.905061	−	0.313522i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.0000	−	17.3205i	0.361079	−	0.625407i
$768$		0			0
$769$	37.0000			1.33425			0.667127	−	0.744944i	$-0.267524\pi$
$769$	37.0000			1.33425			0.667127	+	0.744944i	$0.267524\pi$
$770$		0			0
$771$	−14.0000			−0.504198
$772$		0			0
$773$	−13.0000	+	22.5167i	−0.467578	+	0.809868i	−0.999314	−	0.0370420i	$-0.988206\pi$
$773$	−13.0000	+	22.5167i	−0.467578	+	0.809868i	0.531736	+	0.846910i	$0.321540\pi$
$774$		0			0
$775$	−10.0000	−	17.3205i	−0.359211	−	0.622171i
$776$		0			0
$777$	4.00000	+	20.7846i	0.143499	+	0.745644i
$778$		0			0
$779$	−16.0000	−	27.7128i	−0.573259	−	0.992915i
$780$		0			0
$781$	15.0000	−	25.9808i	0.536742	−	0.929665i
$782$		0			0
$783$	−3.00000			−0.107211
$784$		0			0
$785$	4.00000			0.142766
$786$		0			0
$787$	−11.0000	+	19.0526i	−0.392108	+	0.679150i	−0.992727	−	0.120384i	$-0.961587\pi$
$787$	−11.0000	+	19.0526i	−0.392108	+	0.679150i	0.600620	+	0.799535i	$0.294921\pi$
$788$		0			0
$789$	−7.00000	−	12.1244i	−0.249207	−	0.431638i
$790$		0			0
$791$	−4.00000	−	20.7846i	−0.142224	−	0.739016i
$792$		0			0
$793$	20.0000	+	34.6410i	0.710221	+	1.23014i
$794$		0			0
$795$	−4.50000	+	7.79423i	−0.159599	+	0.276433i
$796$		0			0
$797$	−3.00000			−0.106265			−0.0531327	−	0.998587i	$-0.516921\pi$
$797$	−3.00000			−0.106265			−0.0531327	+	0.998587i	$0.516921\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	9.00000	−	15.5885i	0.317999	−	0.550791i
$802$		0			0
$803$	3.00000	+	5.19615i	0.105868	+	0.183368i
$804$		0			0
$805$	−20.0000	−	6.92820i	−0.704907	−	0.244187i
$806$		0			0
$807$	−0.500000	−	0.866025i	−0.0176008	−	0.0304855i
$808$		0			0
$809$	8.00000	−	13.8564i	0.281265	−	0.487165i	−0.690432	−	0.723398i	$-0.742579\pi$
$809$	8.00000	−	13.8564i	0.281265	−	0.487165i	0.971697	+	0.236232i	$0.0759127\pi$
$810$		0			0
$811$	−2.00000			−0.0702295			−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000			−0.0702295			−0.0351147	+	0.999383i	$0.511180\pi$
$812$		0			0
$813$	17.0000			0.596216
$814$		0			0
$815$	4.00000	−	6.92820i	0.140114	−	0.242684i
$816$		0			0
$817$	12.0000	+	20.7846i	0.419827	+	0.727161i
$818$		0			0
$819$	−8.00000	+	6.92820i	−0.279543	+	0.242091i
$820$		0			0
$821$	−19.5000	−	33.7750i	−0.680555	−	1.17876i	−0.974812	−	0.223029i	$-0.928406\pi$
$821$	−19.5000	−	33.7750i	−0.680555	−	1.17876i	0.294257	−	0.955726i	$-0.404928\pi$
$822$		0			0
$823$	−16.0000	+	27.7128i	−0.557725	+	0.966008i	0.439961	+	0.898017i	$0.354992\pi$
$823$	−16.0000	+	27.7128i	−0.557725	+	0.966008i	−0.997686	+	0.0679910i	$0.978341\pi$
$824$		0			0
$825$	−12.0000			−0.417786
$826$		0			0
$827$	41.0000			1.42571			0.712855	−	0.701312i	$-0.247402\pi$
$827$	41.0000			1.42571			0.712855	+	0.701312i	$0.247402\pi$
$828$		0			0
$829$	−2.00000	+	3.46410i	−0.0694629	+	0.120313i	−0.898665	−	0.438636i	$-0.855462\pi$
$829$	−2.00000	+	3.46410i	−0.0694629	+	0.120313i	0.829202	+	0.558949i	$0.188795\pi$
$830$		0			0
$831$	−4.00000	−	6.92820i	−0.138758	−	0.240337i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	1.00000	+	1.73205i	0.0346064	+	0.0599401i
$836$		0			0
$837$	−2.50000	+	4.33013i	−0.0864126	+	0.149671i
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−20.0000			−0.689655
$842$		0			0
$843$	3.00000	−	5.19615i	0.103325	−	0.178965i
$844$		0			0
$845$	−1.50000	−	2.59808i	−0.0516016	−	0.0893765i
$846$		0			0
$847$	−4.00000	+	3.46410i	−0.137442	+	0.119028i
$848$		0			0
$849$	9.00000	+	15.5885i	0.308879	+	0.534994i
$850$		0			0
$851$	32.0000	−	55.4256i	1.09695	−	1.89997i
$852$		0			0
$853$	−14.0000			−0.479351			−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000			−0.479351			−0.239675	+	0.970853i	$0.577041\pi$
$854$		0			0
$855$	4.00000			0.136797
$856$		0			0
$857$	−7.00000	+	12.1244i	−0.239115	+	0.414160i	−0.960461	−	0.278416i	$-0.910191\pi$
$857$	−7.00000	+	12.1244i	−0.239115	+	0.414160i	0.721345	+	0.692576i	$0.243524\pi$
$858$		0			0
$859$	−7.00000	−	12.1244i	−0.238837	−	0.413678i	0.721544	−	0.692369i	$-0.243433\pi$
$859$	−7.00000	−	12.1244i	−0.238837	−	0.413678i	−0.960381	+	0.278691i	$0.910099\pi$
$860$		0			0
$861$	20.0000	+	6.92820i	0.681598	+	0.236113i
$862$		0			0
$863$	−11.0000	−	19.0526i	−0.374444	−	0.648557i	0.615799	−	0.787903i	$-0.288833\pi$
$863$	−11.0000	−	19.0526i	−0.374444	−	0.648557i	−0.990244	+	0.139346i	$0.955500\pi$
$864$		0			0
$865$	1.00000	−	1.73205i	0.0340010	−	0.0588915i
$866$		0			0
$867$	17.0000			0.577350
$868$		0			0
$869$	−33.0000			−1.11945
$870$		0			0
$871$	−12.0000	+	20.7846i	−0.406604	+	0.704260i
$872$		0			0
$873$	8.50000	+	14.7224i	0.287681	+	0.498279i
$874$		0			0
$875$	−4.50000	−	23.3827i	−0.152128	−	0.790479i
$876$		0			0
$877$	−16.0000	−	27.7128i	−0.540282	−	0.935795i	−0.998888	−	0.0471555i	$-0.984984\pi$
$877$	−16.0000	−	27.7128i	−0.540282	−	0.935795i	0.458606	−	0.888640i	$-0.348349\pi$
$878$		0			0
$879$	9.50000	−	16.4545i	0.320427	−	0.554996i
$880$		0			0
$881$	−54.0000			−1.81931			−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000			−1.81931			−0.909653	+	0.415369i	$0.863653\pi$
$882$		0			0
$883$	−40.0000			−1.34611			−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000			−1.34611			−0.673054	+	0.739594i	$0.735018\pi$
$884$		0			0
$885$	2.50000	−	4.33013i	0.0840366	−	0.145556i
$886$		0			0
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	0.915722	+	0.401813i	$0.131620\pi$
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	−0.109881	+	0.993945i	$0.535047\pi$
$888$		0			0
$889$	−3.50000	−	18.1865i	−0.117386	−	0.609957i
$890$		0			0
$891$	1.50000	+	2.59808i	0.0502519	+	0.0870388i
$892$		0			0
$893$	−20.0000	+	34.6410i	−0.669274	+	1.15922i
$894$		0			0
$895$	−12.0000			−0.401116
$896$		0			0
$897$	32.0000			1.06845
$898$		0			0
$899$	7.50000	−	12.9904i	0.250139	−	0.433253i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−15.0000	−	5.19615i	−0.499169	−	0.172917i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−8.00000	+	13.8564i	−0.265636	+	0.460094i	−0.967730	−	0.251990i	$-0.918915\pi$
$907$	−8.00000	+	13.8564i	−0.265636	+	0.460094i	0.702094	+	0.712084i	$0.252248\pi$
$908$		0			0
$909$	2.00000			0.0663358
$910$		0			0
$911$	6.00000			0.198789			0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000			0.198789			0.0993944	+	0.995048i	$0.468309\pi$
$912$		0			0
$913$	−10.5000	+	18.1865i	−0.347499	+	0.601886i
$914$		0			0
$915$	5.00000	+	8.66025i	0.165295	+	0.286299i
$916$		0			0
$917$	−30.0000	+	25.9808i	−0.990687	+	0.857960i
$918$		0			0
$919$	8.00000	+	13.8564i	0.263896	+	0.457081i	0.967274	−	0.253735i	$-0.0816592\pi$
$919$	8.00000	+	13.8564i	0.263896	+	0.457081i	−0.703378	+	0.710816i	$0.748326\pi$
$920$		0			0
$921$	8.00000	−	13.8564i	0.263609	−	0.456584i
$922$		0			0
$923$	−40.0000			−1.31662
$924$		0			0
$925$	32.0000			1.05215
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−21.0000	−	36.3731i	−0.688988	−	1.19336i	−0.972166	−	0.234294i	$-0.924722\pi$
$929$	−21.0000	−	36.3731i	−0.688988	−	1.19336i	0.283178	−	0.959067i	$-0.408611\pi$
$930$		0			0
$931$	22.0000	+	17.3205i	0.721021	+	0.567657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	19.0000			0.620703			0.310351	−	0.950622i	$-0.399553\pi$
$937$	19.0000			0.620703			0.310351	+	0.950622i	$0.399553\pi$
$938$		0			0
$939$	−9.00000			−0.293704
$940$		0			0
$941$	−6.50000	+	11.2583i	−0.211894	+	0.367011i	−0.952307	−	0.305141i	$-0.901296\pi$
$941$	−6.50000	+	11.2583i	−0.211894	+	0.367011i	0.740413	+	0.672152i	$0.234630\pi$
$942$		0			0
$943$	−32.0000	−	55.4256i	−1.04206	−	1.80491i
$944$		0			0
$945$	−2.00000	+	1.73205i	−0.0650600	+	0.0563436i
$946$		0			0
$947$	−24.0000	−	41.5692i	−0.779895	−	1.35082i	−0.932002	−	0.362454i	$-0.881939\pi$
$947$	−24.0000	−	41.5692i	−0.779895	−	1.35082i	0.152106	−	0.988364i	$-0.451394\pi$
$948$		0			0
$949$	4.00000	−	6.92820i	0.129845	−	0.224899i
$950$		0			0
$951$	13.0000			0.421554
$952$		0			0
$953$	−38.0000			−1.23094			−0.615470	−	0.788160i	$-0.711034\pi$
$953$	−38.0000			−1.23094			−0.615470	+	0.788160i	$0.711034\pi$
$954$		0			0
$955$	8.00000	−	13.8564i	0.258874	−	0.448383i
$956$		0			0
$957$	−4.50000	−	7.79423i	−0.145464	−	0.251952i
$958$		0			0
$959$	35.0000	+	12.1244i	1.13021	+	0.391516i
$960$		0			0
$961$	3.00000	+	5.19615i	0.0967742	+	0.167618i
$962$		0			0
$963$	−5.50000	+	9.52628i	−0.177235	+	0.306980i
$964$		0			0
$965$	−27.0000			−0.869161
$966$		0			0
$967$	3.00000			0.0964735			0.0482367	−	0.998836i	$-0.484640\pi$
$967$	3.00000			0.0964735			0.0482367	+	0.998836i	$0.484640\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	12.5000	+	21.6506i	0.401144	+	0.694802i	0.993864	−	0.110607i	$-0.0352793\pi$
$971$	12.5000	+	21.6506i	0.401144	+	0.694802i	−0.592720	+	0.805408i	$0.701946\pi$
$972$		0			0
$973$	11.0000	+	57.1577i	0.352644	+	1.83239i
$974$		0			0
$975$	8.00000	+	13.8564i	0.256205	+	0.443760i
$976$		0			0
$977$	11.0000	−	19.0526i	0.351921	−	0.609545i	−0.634665	−	0.772787i	$-0.718862\pi$
$977$	11.0000	−	19.0526i	0.351921	−	0.609545i	0.986586	+	0.163242i	$0.0521952\pi$
$978$		0			0
$979$	54.0000			1.72585
$980$		0			0
$981$	10.0000			0.319275
$982$		0			0
$983$	−22.0000	+	38.1051i	−0.701691	+	1.21536i	0.266181	+	0.963923i	$0.414238\pi$
$983$	−22.0000	+	38.1051i	−0.701691	+	1.21536i	−0.967872	+	0.251442i	$0.919095\pi$
$984$		0			0
$985$	−13.0000	−	22.5167i	−0.414214	−	0.717440i
$986$		0			0
$987$	−5.00000	−	25.9808i	−0.159152	−	0.826977i
$988$		0			0
$989$	24.0000	+	41.5692i	0.763156	+	1.32182i
$990$		0			0
$991$	−7.50000	+	12.9904i	−0.238245	+	0.412653i	−0.960211	−	0.279276i	$-0.909906\pi$
$991$	−7.50000	+	12.9904i	−0.238245	+	0.412653i	0.721966	+	0.691929i	$0.243239\pi$
$992$		0			0
$993$	4.00000			0.126936
$994$		0			0
$995$	12.0000			0.380426
$996$		0			0
$997$	−13.0000	+	22.5167i	−0.411714	+	0.713110i	−0.995077	−	0.0991016i	$-0.968403\pi$
$997$	−13.0000	+	22.5167i	−0.411714	+	0.713110i	0.583363	+	0.812211i	$0.301736\pi$
$998$		0			0
$999$	−4.00000	−	6.92820i	−0.126554	−	0.219199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.q.o.193.1		2
4.3	odd	2		1344.2.q.d.193.1		2
7.2	even	3	inner	1344.2.q.o.961.1		2
7.3	odd	6		9408.2.a.cf.1.1		1
7.4	even	3		9408.2.a.ba.1.1		1
8.3	odd	2		336.2.q.e.193.1		2
8.5	even	2		168.2.q.a.25.1	✓	2
24.5	odd	2		504.2.s.d.361.1		2
24.11	even	2		1008.2.s.f.865.1		2
28.3	even	6		9408.2.a.p.1.1		1
28.11	odd	6		9408.2.a.cq.1.1		1
28.23	odd	6		1344.2.q.d.961.1		2
56.3	even	6		2352.2.a.u.1.1		1
56.5	odd	6		1176.2.q.g.961.1		2
56.11	odd	6		2352.2.a.g.1.1		1
56.13	odd	2		1176.2.q.g.361.1		2
56.19	even	6		2352.2.q.f.961.1		2
56.27	even	2		2352.2.q.f.1537.1		2
56.37	even	6		168.2.q.a.121.1	yes	2
56.45	odd	6		1176.2.a.c.1.1		1
56.51	odd	6		336.2.q.e.289.1		2
56.53	even	6		1176.2.a.g.1.1		1
168.5	even	6		3528.2.s.p.3313.1		2
168.11	even	6		7056.2.a.bk.1.1		1
168.53	odd	6		3528.2.a.q.1.1		1
168.59	odd	6		7056.2.a.t.1.1		1
168.101	even	6		3528.2.a.i.1.1		1
168.107	even	6		1008.2.s.f.289.1		2
168.125	even	2		3528.2.s.p.361.1		2
168.149	odd	6		504.2.s.d.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.q.a.25.1	✓	2	8.5	even	2
168.2.q.a.121.1	yes	2	56.37	even	6
336.2.q.e.193.1		2	8.3	odd	2
336.2.q.e.289.1		2	56.51	odd	6
504.2.s.d.289.1		2	168.149	odd	6
504.2.s.d.361.1		2	24.5	odd	2
1008.2.s.f.289.1		2	168.107	even	6
1008.2.s.f.865.1		2	24.11	even	2
1176.2.a.c.1.1		1	56.45	odd	6
1176.2.a.g.1.1		1	56.53	even	6
1176.2.q.g.361.1		2	56.13	odd	2
1176.2.q.g.961.1		2	56.5	odd	6
1344.2.q.d.193.1		2	4.3	odd	2
1344.2.q.d.961.1		2	28.23	odd	6
1344.2.q.o.193.1		2	1.1	even	1	trivial
1344.2.q.o.961.1		2	7.2	even	3	inner
2352.2.a.g.1.1		1	56.11	odd	6
2352.2.a.u.1.1		1	56.3	even	6
2352.2.q.f.961.1		2	56.19	even	6
2352.2.q.f.1537.1		2	56.27	even	2
3528.2.a.i.1.1		1	168.101	even	6
3528.2.a.q.1.1		1	168.53	odd	6
3528.2.s.p.361.1		2	168.125	even	2
3528.2.s.p.3313.1		2	168.5	even	6
7056.2.a.t.1.1		1	168.59	odd	6
7056.2.a.bk.1.1		1	168.11	even	6
9408.2.a.p.1.1		1	28.3	even	6
9408.2.a.ba.1.1		1	7.4	even	3
9408.2.a.cf.1.1		1	7.3	odd	6
9408.2.a.cq.1.1		1	28.11	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(1.50000 - 2.59808i) q^{11} -4.00000 q^{13} -1.00000 q^{15} +(-2.00000 - 3.46410i) q^{19} +(2.50000 + 0.866025i) q^{21} +(-4.00000 - 6.92820i) q^{23} +(2.00000 - 3.46410i) q^{25} -1.00000 q^{27} +3.00000 q^{29} +(2.50000 - 4.33013i) q^{31} +(-1.50000 - 2.59808i) q^{33} +(2.00000 - 1.73205i) q^{35} +(4.00000 + 6.92820i) q^{37} +(-2.00000 + 3.46410i) q^{39} +8.00000 q^{41} -6.00000 q^{43} +(-0.500000 + 0.866025i) q^{45} +(-5.00000 - 8.66025i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(4.50000 - 7.79423i) q^{53} -3.00000 q^{55} -4.00000 q^{57} +(-2.50000 + 4.33013i) q^{59} +(-5.00000 - 8.66025i) q^{61} +(2.00000 - 1.73205i) q^{63} +(2.00000 + 3.46410i) q^{65} +(3.00000 - 5.19615i) q^{67} -8.00000 q^{69} +10.0000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(-2.00000 - 3.46410i) q^{75} +(7.50000 + 2.59808i) q^{77} +(-5.50000 - 9.52628i) q^{79} +(-0.500000 + 0.866025i) q^{81} -7.00000 q^{83} +(1.50000 - 2.59808i) q^{87} +(9.00000 + 15.5885i) q^{89} +(-2.00000 - 10.3923i) q^{91} +(-2.50000 - 4.33013i) q^{93} +(-2.00000 + 3.46410i) q^{95} -17.0000 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - q^{5} + q^{7} - q^{9}+O(q^{10}) \) 2 * q + q^3 - q^5 + q^7 - q^9	\( 2 q + q^{3} - q^{5} + q^{7} - q^{9} + 3 q^{11} - 8 q^{13} - 2 q^{15} - 4 q^{19} + 5 q^{21} - 8 q^{23} + 4 q^{25} - 2 q^{27} + 6 q^{29} + 5 q^{31} - 3 q^{33} + 4 q^{35} + 8 q^{37} - 4 q^{39} + 16 q^{41} - 12 q^{43} - q^{45} - 10 q^{47} - 13 q^{49} + 9 q^{53} - 6 q^{55} - 8 q^{57} - 5 q^{59} - 10 q^{61} + 4 q^{63} + 4 q^{65} + 6 q^{67} - 16 q^{69} + 20 q^{71} - 2 q^{73} - 4 q^{75} + 15 q^{77} - 11 q^{79} - q^{81} - 14 q^{83} + 3 q^{87} + 18 q^{89} - 4 q^{91} - 5 q^{93} - 4 q^{95} - 34 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q + q^3 - q^5 + q^7 - q^9 + 3 * q^11 - 8 * q^13 - 2 * q^15 - 4 * q^19 + 5 * q^21 - 8 * q^23 + 4 * q^25 - 2 * q^27 + 6 * q^29 + 5 * q^31 - 3 * q^33 + 4 * q^35 + 8 * q^37 - 4 * q^39 + 16 * q^41 - 12 * q^43 - q^45 - 10 * q^47 - 13 * q^49 + 9 * q^53 - 6 * q^55 - 8 * q^57 - 5 * q^59 - 10 * q^61 + 4 * q^63 + 4 * q^65 + 6 * q^67 - 16 * q^69 + 20 * q^71 - 2 * q^73 - 4 * q^75 + 15 * q^77 - 11 * q^79 - q^81 - 14 * q^83 + 3 * q^87 + 18 * q^89 - 4 * q^91 - 5 * q^93 - 4 * q^95 - 34 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

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