LMFDB - Embedded newform 1176.2.c.c.589.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,2,Mod(589,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1176.589");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1176.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.39040727770$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.386672896.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 $ x^8 - x^6 - 2x^5 + 2x^4 - 4x^3 - 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			589.6
Root			$-0.835949 - 1.14070i$ of defining polynomial
Character	$\chi$	$=$	1176.589
Dual form			1176.2.c.c.589.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.835949 + 1.14070i) q^{2} -1.00000i q^{3} +(-0.602380 + 1.90713i) q^{4} -0.467138i q^{5} +(1.14070 - 0.835949i) q^{6} +(-2.67901 + 0.907128i) q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+(0.835949 + 1.14070i) q^{2} -1.00000i q^{3} +(-0.602380 + 1.90713i) q^{4} -0.467138i q^{5} +(1.14070 - 0.835949i) q^{6} +(-2.67901 + 0.907128i) q^{8} -1.00000 q^{9} +(0.532862 - 0.390503i) q^{10} -4.87666i q^{11} +(1.90713 + 0.602380i) q^{12} -4.56279i q^{13} -0.467138 q^{15} +(-3.27428 - 2.29763i) q^{16} -6.09565 q^{17} +(-0.835949 - 1.14070i) q^{18} +1.34379i q^{19} +(0.890891 + 0.281394i) q^{20} +(5.56279 - 4.07663i) q^{22} -4.09565 q^{23} +(0.907128 + 2.67901i) q^{24} +4.78178 q^{25} +(5.20476 - 3.81426i) q^{26} +1.00000i q^{27} -7.78178i q^{29} +(-0.390503 - 0.532862i) q^{30} -4.40952 q^{31} +(-0.116226 - 5.65566i) q^{32} -4.87666 q^{33} +(-5.09565 - 6.95329i) q^{34} +(0.602380 - 1.90713i) q^{36} +4.40952i q^{37} +(-1.53286 + 1.12334i) q^{38} -4.56279 q^{39} +(0.423754 + 1.25147i) q^{40} +6.09565 q^{41} -4.15327i q^{43} +(9.30041 + 2.93760i) q^{44} +0.467138i q^{45} +(-3.42375 - 4.67190i) q^{46} +6.68759 q^{47} +(-2.29763 + 3.27428i) q^{48} +(3.99732 + 5.45457i) q^{50} +6.09565i q^{51} +(8.70182 + 2.74853i) q^{52} +1.34379i q^{53} +(-1.14070 + 0.835949i) q^{54} -2.27807 q^{55} +1.34379 q^{57} +(8.87666 - 6.50517i) q^{58} +4.00000i q^{59} +(0.281394 - 0.890891i) q^{60} -5.49706i q^{61} +(-3.68613 - 5.02993i) q^{62} +(6.35424 - 4.86042i) q^{64} -2.13145 q^{65} +(-4.07663 - 5.56279i) q^{66} -5.90658i q^{67} +(3.67190 - 11.6252i) q^{68} +4.09565i q^{69} -4.72339 q^{71} +(2.67901 - 0.907128i) q^{72} +12.0599 q^{73} +(-5.02993 + 3.68613i) q^{74} -4.78178i q^{75} +(-2.56279 - 0.809475i) q^{76} +(-3.81426 - 5.20476i) q^{78} -16.1913 q^{79} +(-1.07331 + 1.52954i) q^{80} +1.00000 q^{81} +(5.09565 + 6.95329i) q^{82} -13.7533i q^{83} +2.84751i q^{85} +(4.73762 - 3.47192i) q^{86} -7.78178 q^{87} +(4.42375 + 13.0646i) q^{88} -7.96420 q^{89} +(-0.532862 + 0.390503i) q^{90} +(2.46714 - 7.81093i) q^{92} +4.40952i q^{93} +(5.59048 + 7.62851i) q^{94} +0.627737 q^{95} +(-5.65566 + 0.116226i) q^{96} +12.8789 q^{97} +4.87666i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{4} - 2 q^{6} - 6 q^{8} - 8 q^{9}+O(q^{10}) $ 8 * q + 2 * q^4 - 2 * q^6 - 6 * q^8 - 8 * q^9	$ 8 q + 2 q^{4} - 2 q^{6} - 6 q^{8} - 8 q^{9} + 4 q^{10} + 4 q^{12} - 4 q^{15} - 6 q^{16} - 4 q^{17} - 24 q^{20} + 12 q^{23} - 4 q^{24} - 24 q^{25} + 28 q^{26} - 12 q^{30} - 8 q^{31} - 30 q^{32} - 12 q^{33} + 4 q^{34} - 2 q^{36} - 12 q^{38} + 8 q^{39} - 28 q^{40} + 4 q^{41} + 16 q^{44} + 4 q^{46} - 16 q^{48} - 20 q^{50} + 12 q^{52} + 2 q^{54} + 8 q^{55} - 16 q^{57} + 44 q^{58} - 20 q^{60} - 12 q^{62} + 26 q^{64} - 16 q^{65} - 24 q^{66} + 16 q^{68} - 28 q^{71} + 6 q^{72} + 8 q^{73} + 4 q^{74} + 24 q^{76} - 8 q^{78} - 40 q^{79} + 4 q^{80} + 8 q^{81} - 4 q^{82} + 24 q^{86} + 4 q^{88} - 20 q^{89} - 4 q^{90} + 20 q^{92} + 72 q^{94} + 40 q^{95} - 12 q^{96} - 40 q^{97}+O(q^{100}) $ 8 * q + 2 * q^4 - 2 * q^6 - 6 * q^8 - 8 * q^9 + 4 * q^10 + 4 * q^12 - 4 * q^15 - 6 * q^16 - 4 * q^17 - 24 * q^20 + 12 * q^23 - 4 * q^24 - 24 * q^25 + 28 * q^26 - 12 * q^30 - 8 * q^31 - 30 * q^32 - 12 * q^33 + 4 * q^34 - 2 * q^36 - 12 * q^38 + 8 * q^39 - 28 * q^40 + 4 * q^41 + 16 * q^44 + 4 * q^46 - 16 * q^48 - 20 * q^50 + 12 * q^52 + 2 * q^54 + 8 * q^55 - 16 * q^57 + 44 * q^58 - 20 * q^60 - 12 * q^62 + 26 * q^64 - 16 * q^65 - 24 * q^66 + 16 * q^68 - 28 * q^71 + 6 * q^72 + 8 * q^73 + 4 * q^74 + 24 * q^76 - 8 * q^78 - 40 * q^79 + 4 * q^80 + 8 * q^81 - 4 * q^82 + 24 * q^86 + 4 * q^88 - 20 * q^89 - 4 * q^90 + 20 * q^92 + 72 * q^94 + 40 * q^95 - 12 * q^96 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0.835949	+	1.14070i	0.591105	+	0.806595i
$2$	0.835949	+	1.14070i	0.591105	+	0.806595i
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−0.602380	+	1.90713i	−0.301190	+	0.953564i
$5$		−	0.467138i		−	0.208910i	−0.994530	−	0.104455i	$-0.966690\pi$
$5$		−	0.467138i		−	0.208910i	0.994530	−	0.104455i	$-0.0333099\pi$
$6$	1.14070	−	0.835949i	0.465688	−	0.341275i
$7$		0			0
$7$		0			0
$8$	−2.67901	+	0.907128i	−0.947175	+	0.320718i
$9$	−1.00000			−0.333333
$10$	0.532862	−	0.390503i	0.168506	−	0.123488i
$11$		−	4.87666i		−	1.47037i	−0.677868	−	0.735184i	$-0.737096\pi$
$11$		−	4.87666i		−	1.47037i	0.677868	−	0.735184i	$-0.262904\pi$
$12$	1.90713	+	0.602380i	0.550541	+	0.173892i
$13$		−	4.56279i		−	1.26549i	−0.774360	−	0.632745i	$-0.781928\pi$
$13$		−	4.56279i		−	1.26549i	0.774360	−	0.632745i	$-0.218072\pi$
$14$		0			0
$15$	−0.467138			−0.120614
$16$	−3.27428	−	2.29763i	−0.818569	−	0.574408i
$17$	−6.09565			−1.47841			−0.739206	−	0.673479i	$-0.764799\pi$
$17$	−6.09565			−1.47841			−0.739206	+	0.673479i	$0.764799\pi$
$18$	−0.835949	−	1.14070i	−0.197035	−	0.268865i
$19$			1.34379i			0.308288i	0.988048	+	0.154144i	$0.0492619\pi$
$19$			1.34379i			0.308288i	−0.988048	+	0.154144i	$0.950738\pi$
$20$	0.890891	+	0.281394i	0.199209	+	0.0629217i
$21$		0			0
$22$	5.56279	−	4.07663i	1.18599	−	0.869141i
$23$	−4.09565			−0.854002			−0.427001	−	0.904251i	$-0.640430\pi$
$23$	−4.09565			−0.854002			−0.427001	+	0.904251i	$0.640430\pi$
$24$	0.907128	+	2.67901i	0.185167	+	0.546852i
$25$	4.78178			0.956357
$26$	5.20476	−	3.81426i	1.02074	−	0.748037i
$27$			1.00000i			0.192450i
$28$		0			0
$29$		−	7.78178i		−	1.44504i	−0.691350	−	0.722520i	$-0.742984\pi$
$29$		−	7.78178i		−	1.44504i	0.691350	−	0.722520i	$-0.257016\pi$
$30$	−0.390503	−	0.532862i	−0.0712958	−	0.0972869i
$31$	−4.40952			−0.791973			−0.395987	−	0.918256i	$-0.629597\pi$
$31$	−4.40952			−0.791973			−0.395987	+	0.918256i	$0.629597\pi$
$32$	−0.116226	−	5.65566i	−0.0205460	−	0.999789i
$33$	−4.87666			−0.848917
$34$	−5.09565	−	6.95329i	−0.873897	−	1.19248i
$35$		0			0
$36$	0.602380	−	1.90713i	0.100397	−	0.317855i
$37$			4.40952i			0.724921i	0.931999	+	0.362460i	$0.118063\pi$
$37$			4.40952i			0.724921i	−0.931999	+	0.362460i	$0.881937\pi$
$38$	−1.53286	+	1.12334i	−0.248663	+	0.182230i
$39$	−4.56279			−0.730631
$40$	0.423754	+	1.25147i	0.0670013	+	0.197874i
$41$	6.09565			0.951981			0.475990	−	0.879450i	$-0.342090\pi$
$41$	6.09565			0.951981			0.475990	+	0.879450i	$0.342090\pi$
$42$		0			0
$43$		−	4.15327i		−	0.633368i	−0.948531	−	0.316684i	$-0.897431\pi$
$43$		−	4.15327i		−	0.633368i	0.948531	−	0.316684i	$-0.102569\pi$
$44$	9.30041	+	2.93760i	1.40209	+	0.442860i
$45$			0.467138i			0.0696368i
$46$	−3.42375	−	4.67190i	−0.504805	−	0.688834i
$47$	6.68759			0.975485			0.487743	−	0.872988i	$-0.337820\pi$
$47$	6.68759			0.975485			0.487743	+	0.872988i	$0.337820\pi$
$48$	−2.29763	+	3.27428i	−0.331635	+	0.472601i
$49$		0			0
$50$	3.99732	+	5.45457i	0.565307	+	0.771392i
$51$			6.09565i			0.853562i
$52$	8.70182	+	2.74853i	1.20673	+	0.381153i
$53$			1.34379i			0.184584i	0.995732	+	0.0922922i	$0.0294194\pi$
$53$			1.34379i			0.184584i	−0.995732	+	0.0922922i	$0.970581\pi$
$54$	−1.14070	+	0.835949i	−0.155229	+	0.113758i
$55$	−2.27807			−0.307175
$56$		0			0
$57$	1.34379			0.177990
$58$	8.87666	−	6.50517i	1.16556	−	0.854171i
$59$			4.00000i			0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$			4.00000i			0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0.281394	−	0.890891i	0.0363278	−	0.115014i
$61$		−	5.49706i		−	0.703827i	−0.936033	−	0.351913i	$-0.885531\pi$
$61$		−	5.49706i		−	0.703827i	0.936033	−	0.351913i	$-0.114469\pi$
$62$	−3.68613	−	5.02993i	−0.468139	−	0.638801i
$63$		0			0
$64$	6.35424	−	4.86042i	0.794280	−	0.607552i
$65$	−2.13145			−0.264374
$66$	−4.07663	−	5.56279i	−0.501799	−	0.684732i
$67$		−	5.90658i		−	0.721604i	−0.932642	−	0.360802i	$-0.882503\pi$
$67$		−	5.90658i		−	0.721604i	0.932642	−	0.360802i	$-0.117497\pi$
$68$	3.67190	−	11.6252i	0.445283	−	1.40976i
$69$			4.09565i			0.493058i
$70$		0			0
$71$	−4.72339			−0.560563			−0.280281	−	0.959918i	$-0.590428\pi$
$71$	−4.72339			−0.560563			−0.280281	+	0.959918i	$0.590428\pi$
$72$	2.67901	−	0.907128i	0.315725	−	0.106906i
$73$	12.0599			1.41150			0.705749	−	0.708462i	$-0.250610\pi$
$73$	12.0599			1.41150			0.705749	+	0.708462i	$0.250610\pi$
$74$	−5.02993	+	3.68613i	−0.584717	+	0.428504i
$75$		−	4.78178i		−	0.552153i
$76$	−2.56279	−	0.809475i	−0.293972	−	0.0928531i
$77$		0			0
$78$	−3.81426	−	5.20476i	−0.431880	−	0.589323i
$79$	−16.1913			−1.82166			−0.910832	−	0.412778i	$-0.864559\pi$
$79$	−16.1913			−1.82166			−0.910832	+	0.412778i	$0.864559\pi$
$80$	−1.07331	+	1.52954i	−0.120000	+	0.171008i
$81$	1.00000			0.111111
$82$	5.09565	+	6.95329i	0.562721	+	0.767863i
$83$		−	13.7533i		−	1.50962i	−0.655942	−	0.754811i	$-0.727728\pi$
$83$		−	13.7533i		−	1.50962i	0.655942	−	0.754811i	$-0.272272\pi$
$84$		0			0
$85$			2.84751i			0.308856i
$86$	4.73762	−	3.47192i	0.510871	−	0.374387i
$87$	−7.78178			−0.834295
$88$	4.42375	+	13.0646i	0.471574	+	1.39269i
$89$	−7.96420			−0.844204			−0.422102	−	0.906548i	$-0.638708\pi$
$89$	−7.96420			−0.844204			−0.422102	+	0.906548i	$0.638708\pi$
$90$	−0.532862	+	0.390503i	−0.0561686	+	0.0411626i
$91$		0			0
$92$	2.46714	−	7.81093i	0.257217	−	0.814346i
$93$			4.40952i			0.457246i
$94$	5.59048	+	7.62851i	0.576614	+	0.786821i
$95$	0.627737			0.0644044
$96$	−5.65566	+	0.116226i	−0.577228	+	0.0118623i
$97$	12.8789			1.30765			0.653827	−	0.756644i	$-0.273163\pi$
$97$	12.8789			1.30765			0.653827	+	0.756644i	$0.273163\pi$
$98$		0			0
$99$			4.87666i			0.490122i
$100$	−2.88045	+	9.11947i	−0.288045	+	0.911947i
$101$			2.22045i			0.220943i	0.993879	+	0.110472i	$0.0352361\pi$
$101$			2.22045i			0.220943i	−0.993879	+	0.110472i	$0.964764\pi$
$102$	−6.95329	+	5.09565i	−0.688478	+	0.504545i
$103$	−11.0971			−1.09343			−0.546715	−	0.837319i	$-0.684122\pi$
$103$	−11.0971			−1.09343			−0.546715	+	0.837319i	$0.684122\pi$
$104$	4.13903	+	12.2238i	0.405866	+	1.19864i
$105$		0			0
$106$	−1.53286	+	1.12334i	−0.148885	+	0.109109i
$107$		−	3.12334i		−	0.301945i	−0.988538	−	0.150972i	$-0.951760\pi$
$107$		−	3.12334i		−	0.301945i	0.988538	−	0.150972i	$-0.0482405\pi$
$108$	−1.90713	−	0.602380i	−0.183514	−	0.0579640i
$109$			10.6876i			1.02369i	0.859079	+	0.511843i	$0.171037\pi$
$109$			10.6876i			1.02369i	−0.859079	+	0.511843i	$0.828963\pi$
$110$	−1.90435	−	2.59859i	−0.181573	−	0.247766i
$111$	4.40952			0.418533
$112$		0			0
$113$	−12.0599			−1.13450			−0.567248	−	0.823547i	$-0.691992\pi$
$113$	−12.0599			−1.13450			−0.567248	+	0.823547i	$0.691992\pi$
$114$	1.12334	+	1.53286i	0.105211	+	0.143566i
$115$			1.91323i			0.178410i
$116$	14.8409	+	4.68759i	1.37794	+	0.435232i
$117$			4.56279i			0.421830i
$118$	−4.56279	+	3.34379i	−0.420039	+	0.307821i
$119$		0			0
$120$	1.25147	−	0.423754i	0.114243	−	0.0386832i
$121$	−12.7818			−1.16198
$122$	6.27048	−	4.59526i	0.567703	−	0.416036i
$123$		−	6.09565i		−	0.549626i
$124$	2.65621	−	8.40952i	0.238534	−	0.755197i
$125$		−	4.56944i		−	0.408703i
$126$		0			0
$127$	18.8789			1.67523			0.837615	−	0.546261i	$-0.183949\pi$
$127$	18.8789			1.67523			0.837615	+	0.546261i	$0.183949\pi$
$128$	10.8561	+	3.18520i	0.959551	+	0.281534i
$129$	−4.15327			−0.365675
$130$	−1.78178	−	2.43134i	−0.156273	−	0.213243i
$131$		−	4.93428i		−	0.431110i	−0.976492	−	0.215555i	$-0.930844\pi$
$131$		−	4.93428i		−	0.431110i	0.976492	−	0.215555i	$-0.0691560\pi$
$132$	2.93760	−	9.30041i	0.255685	−	0.809497i
$133$		0			0
$134$	6.73762	−	4.93760i	0.582042	−	0.426544i
$135$	0.467138			0.0402048
$136$	16.3303	−	5.52954i	1.40031	−	0.474154i
$137$	19.0103			1.62416			0.812082	−	0.583544i	$-0.198334\pi$
$137$	19.0103			1.62416			0.812082	+	0.583544i	$0.198334\pi$
$138$	−4.67190	+	3.42375i	−0.397698	+	0.291449i
$139$			10.4380i			0.885339i	0.896685	+	0.442669i	$0.145968\pi$
$139$			10.4380i			0.885339i	−0.896685	+	0.442669i	$0.854032\pi$
$140$		0			0
$141$		−	6.68759i		−	0.563197i
$142$	−3.94851	−	5.38795i	−0.331352	−	0.452147i
$143$	−22.2512			−1.86073
$144$	3.27428	+	2.29763i	0.272856	+	0.191469i
$145$	−3.63516			−0.301884
$146$	10.0814	+	13.7566i	0.834344	+	1.13851i
$147$		0			0
$148$	−8.40952	−	2.65621i	−0.691258	−	0.218339i
$149$		−	6.65621i		−	0.545298i	−0.962114	−	0.272649i	$-0.912100\pi$
$149$		−	6.65621i		−	0.545298i	0.962114	−	0.272649i	$-0.0878997\pi$
$150$	5.45457	−	3.99732i	0.445363	−	0.326380i
$151$	−2.68759			−0.218713			−0.109356	−	0.994003i	$-0.534879\pi$
$151$	−2.68759			−0.218713			−0.109356	+	0.994003i	$0.534879\pi$
$152$	−1.21899	−	3.60004i	−0.0988735	−	0.292002i
$153$	6.09565			0.492804
$154$		0			0
$155$			2.05985i			0.165451i
$156$	2.74853	−	8.70182i	0.220059	−	0.696703i
$157$			23.1351i			1.84639i	0.384338	+	0.923193i	$0.374430\pi$
$157$			23.1351i			1.84639i	−0.384338	+	0.923193i	$0.625570\pi$
$158$	−13.5351	−	18.4694i	−1.07679	−	1.46934i
$159$	1.34379			0.106570
$160$	−2.64197	+	0.0542935i	−0.208866	+	0.00429228i
$161$		0			0
$162$	0.835949	+	1.14070i	0.0656783	+	0.0896216i
$163$			12.0380i			0.942891i	0.881895	+	0.471446i	$0.156268\pi$
$163$			12.0380i			0.942891i	−0.881895	+	0.471446i	$0.843732\pi$
$164$	−3.67190	+	11.6252i	−0.286727	+	0.907775i
$165$			2.27807i			0.177347i
$166$	15.6884	−	11.4971i	1.21765	−	0.892345i
$167$	−9.01034			−0.697241			−0.348621	−	0.937264i	$-0.613350\pi$
$167$	−9.01034			−0.697241			−0.348621	+	0.937264i	$0.613350\pi$
$168$		0			0
$169$	−7.81904			−0.601465
$170$	−3.24814	+	2.38037i	−0.249121	+	0.182566i
$171$		−	1.34379i		−	0.102763i
$172$	7.92082	+	2.50185i	0.603957	+	0.190764i
$173$			9.59271i			0.729321i	0.931141	+	0.364660i	$0.118815\pi$
$173$			9.59271i			0.729321i	−0.931141	+	0.364660i	$0.881185\pi$
$174$	−6.50517	−	8.87666i	−0.493156	−	0.672938i
$175$		0			0
$176$	−11.2048	+	15.9675i	−0.844591	+	1.20360i
$177$	4.00000			0.300658
$178$	−6.65766	−	9.08474i	−0.499013	−	0.680930i
$179$			7.69278i			0.574985i	0.957783	+	0.287493i	$0.0928217\pi$
$179$			7.69278i			0.574985i	−0.957783	+	0.287493i	$0.907178\pi$
$180$	−0.890891	−	0.281394i	−0.0664031	−	0.0209739i
$181$			15.2504i			1.13355i	0.823872	+	0.566776i	$0.191809\pi$
$181$			15.2504i			1.13355i	−0.823872	+	0.566776i	$0.808191\pi$
$182$		0			0
$183$	−5.49706			−0.406355
$184$	10.9723	−	3.71528i	0.808889	−	0.273894i
$185$	2.05985			0.151443
$186$	−5.02993	+	3.68613i	−0.368812	+	0.270280i
$187$			29.7264i			2.17381i
$188$	−4.02847	+	12.7541i	−0.293806	+	0.930188i
$189$		0			0
$190$	0.524756	+	0.716058i	0.0380698	+	0.0519483i
$191$	−4.72339			−0.341772			−0.170886	−	0.985291i	$-0.554663\pi$
$191$	−4.72339			−0.341772			−0.170886	+	0.985291i	$0.554663\pi$
$192$	−4.86042	−	6.35424i	−0.350771	−	0.458578i
$193$	22.3379			1.60792			0.803959	−	0.594684i	$-0.202723\pi$
$193$	22.3379			1.60792			0.803959	+	0.594684i	$0.202723\pi$
$194$	10.7661	+	14.6909i	0.772960	+	1.05475i
$195$			2.13145i			0.152636i
$196$		0			0
$197$		−	1.53510i		−	0.109371i	−0.998504	−	0.0546855i	$-0.982584\pi$
$197$		−	1.53510i		−	0.109371i	0.998504	−	0.0546855i	$-0.0174156\pi$
$198$	−5.56279	+	4.07663i	−0.395330	+	0.289714i
$199$	14.6876			1.04118			0.520588	−	0.853808i	$-0.325713\pi$
$199$	14.6876			1.04118			0.520588	+	0.853808i	$0.325713\pi$
$200$	−12.8105	+	4.33769i	−0.905837	+	0.306721i
$201$	−5.90658			−0.416618
$202$	−2.53286	+	1.85618i	−0.178212	+	0.130601i
$203$		0			0
$204$	−11.6252	−	3.67190i	−0.813926	−	0.257084i
$205$		−	2.84751i		−	0.198879i
$206$	−9.27661	−	12.6584i	−0.646332	−	0.881955i
$207$	4.09565			0.284667
$208$	−10.4836	+	14.9398i	−0.726907	+	1.03589i
$209$	6.55322			0.453296
$210$		0			0
$211$			22.8409i			1.57243i	0.617953	+	0.786215i	$0.287962\pi$
$211$			22.8409i			1.57243i	−0.617953	+	0.786215i	$0.712038\pi$
$212$	−2.56279	−	0.809475i	−0.176013	−	0.0555949i
$213$			4.72339i			0.323641i
$214$	3.56279	−	2.61095i	0.243547	−	0.178481i
$215$	−1.94015			−0.132317
$216$	−0.907128	−	2.67901i	−0.0617223	−	0.182284i
$217$		0			0
$218$	−12.1913	+	8.93428i	−0.825699	+	0.605105i
$219$		−	12.0599i		−	0.814929i
$220$	1.37226	−	4.34457i	0.0925180	−	0.292911i
$221$			27.8132i			1.87092i
$222$	3.68613	+	5.02993i	0.247397	+	0.337587i
$223$	−21.4321			−1.43520			−0.717600	−	0.696455i	$-0.754760\pi$
$223$	−21.4321			−1.43520			−0.717600	+	0.696455i	$0.754760\pi$
$224$		0			0
$225$	−4.78178			−0.318786
$226$	−10.0814	−	13.7566i	−0.670606	−	0.915078i
$227$		−	29.3169i		−	1.94583i	−0.231164	−	0.972915i	$-0.574253\pi$
$227$		−	29.3169i		−	1.94583i	0.231164	−	0.972915i	$-0.425747\pi$
$228$	−0.809475	+	2.56279i	−0.0536088	+	0.169725i
$229$		−	22.5074i		−	1.48733i	−0.668552	−	0.743666i	$-0.733086\pi$
$229$		−	22.5074i		−	1.48733i	0.668552	−	0.743666i	$-0.266914\pi$
$230$	−2.18242	+	1.59936i	−0.143904	+	0.105459i
$231$		0			0
$232$	7.05908	+	20.8475i	0.463451	+	1.36871i
$233$	−0.687589			−0.0450454			−0.0225227	−	0.999746i	$-0.507170\pi$
$233$	−0.687589			−0.0450454			−0.0225227	+	0.999746i	$0.507170\pi$
$234$	−5.20476	+	3.81426i	−0.340246	+	0.249346i
$235$		−	3.12402i		−	0.203789i
$236$	−7.62851	−	2.40952i	−0.496574	−	0.156846i
$237$			16.1913i			1.05174i
$238$		0			0
$239$	−9.59936			−0.620931			−0.310466	−	0.950585i	$-0.600485\pi$
$239$	−9.59936			−0.620931			−0.310466	+	0.950585i	$0.600485\pi$
$240$	1.52954	+	1.07331i	0.0987312	+	0.0692819i
$241$	−0.496287			−0.0319687			−0.0159843	−	0.999872i	$-0.505088\pi$
$241$	−0.496287			−0.0319687			−0.0159843	+	0.999872i	$0.505088\pi$
$242$	−10.6849	−	14.5801i	−0.686852	−	0.937247i
$243$		−	1.00000i		−	0.0641500i
$244$	10.4836	+	3.31132i	0.671144	+	0.211986i
$245$		0			0
$246$	6.95329	−	5.09565i	0.443326	−	0.324887i
$247$	6.13145			0.390135
$248$	11.8132	−	4.00000i	0.750137	−	0.254000i
$249$	−13.7533			−0.871581
$250$	5.21234	−	3.81982i	0.329658	−	0.241586i
$251$		−	22.1359i		−	1.39721i	−0.715509	−	0.698603i	$-0.753805\pi$
$251$		−	22.1359i		−	1.39721i	0.715509	−	0.698603i	$-0.246195\pi$
$252$		0			0
$253$			19.9731i			1.25570i
$254$	15.7818	+	21.5351i	0.990237	+	1.35123i
$255$	2.84751			0.178318
$256$	5.44178	+	15.0462i	0.340111	+	0.940385i
$257$	−2.28695			−0.142656			−0.0713281	−	0.997453i	$-0.522724\pi$
$257$	−2.28695			−0.142656			−0.0713281	+	0.997453i	$0.522724\pi$
$258$	−3.47192	−	4.73762i	−0.216152	−	0.294951i
$259$		0			0
$260$	1.28394	−	4.06495i	0.0796267	−	0.252097i
$261$			7.78178i			0.481680i
$262$	5.62851	−	4.12480i	0.347731	−	0.254831i
$263$	−30.1555			−1.85947			−0.929734	−	0.368232i	$-0.879963\pi$
$263$	−30.1555			−1.85947			−0.929734	+	0.368232i	$0.879963\pi$
$264$	13.0646	−	4.42375i	0.804073	−	0.272263i
$265$	0.627737			0.0385616
$266$		0			0
$267$			7.96420i			0.487401i
$268$	11.2646	+	3.55801i	0.688096	+	0.217340i
$269$		−	9.47748i		−	0.577852i	−0.957351	−	0.288926i	$-0.906702\pi$
$269$		−	9.47748i		−	0.577852i	0.957351	−	0.288926i	$-0.0932982\pi$
$270$	0.390503	+	0.532862i	0.0237653	+	0.0324290i
$271$	13.2855			0.807036			0.403518	−	0.914972i	$-0.367787\pi$
$271$	13.2855			0.807036			0.403518	+	0.914972i	$0.367787\pi$
$272$	19.9589	+	14.0056i	1.21018	+	0.849212i
$273$		0			0
$274$	15.8917	+	21.6850i	0.960051	+	1.31004i
$275$		−	23.3191i		−	1.40620i
$276$	−7.81093	−	2.46714i	−0.470163	−	0.148504i
$277$		−	26.8475i		−	1.61311i	−0.591159	−	0.806555i	$-0.701329\pi$
$277$		−	26.8475i		−	1.61311i	0.591159	−	0.806555i	$-0.298671\pi$
$278$	−11.9066	+	8.72562i	−0.714109	+	0.523328i
$279$	4.40952			0.263991
$280$		0			0
$281$	11.8686			0.708018			0.354009	−	0.935242i	$-0.384818\pi$
$281$	11.8686			0.708018			0.354009	+	0.935242i	$0.384818\pi$
$282$	7.62851	−	5.59048i	0.454271	−	0.332908i
$283$		−	2.46937i		−	0.146789i	−0.997303	−	0.0733944i	$-0.976617\pi$
$283$		−	2.46937i		−	0.146789i	0.997303	−	0.0733944i	$-0.0233832\pi$
$284$	2.84527	−	9.00811i	0.168836	−	0.534533i
$285$		−	0.627737i		−	0.0371839i
$286$	−18.6008	−	25.3818i	−1.09989	−	1.50086i
$287$		0			0
$288$	0.116226	+	5.65566i	0.00684868	+	0.333263i
$289$	20.1570			1.18570
$290$	−3.03881	−	4.14662i	−0.178445	−	0.243498i
$291$		−	12.8789i		−	0.754974i
$292$	−7.26461	+	22.9997i	−0.425129	+	1.34595i
$293$		−	22.7899i		−	1.33140i	−0.746220	−	0.665700i	$-0.768133\pi$
$293$		−	22.7899i		−	1.33140i	0.746220	−	0.665700i	$-0.231867\pi$
$294$		0			0
$295$	1.86855			0.108791
$296$	−4.00000	−	11.8132i	−0.232495	−	0.686626i
$297$	4.87666			0.282972
$298$	7.59271	−	5.56425i	0.439834	−	0.322328i
$299$			18.6876i			1.08073i
$300$	9.11947	+	2.88045i	0.526513	+	0.166303i
$301$		0			0
$302$	−2.24669	−	3.06572i	−0.129282	−	0.176413i
$303$	2.22045			0.127562
$304$	3.08754	−	4.39996i	0.177083	−	0.252355i
$305$	−2.56788			−0.147037
$306$	5.09565	+	6.95329i	0.291299	+	0.397493i
$307$		−	9.34379i		−	0.533279i	−0.963796	−	0.266639i	$-0.914087\pi$
$307$		−	9.34379i		−	0.533279i	0.963796	−	0.266639i	$-0.0859132\pi$
$308$		0			0
$309$			11.0971i			0.631292i
$310$	−2.34967	+	1.72193i	−0.133452	+	0.0977991i
$311$	−6.68759			−0.379218			−0.189609	−	0.981860i	$-0.560722\pi$
$311$	−6.68759			−0.379218			−0.189609	+	0.981860i	$0.560722\pi$
$312$	12.2238	−	4.13903i	0.692035	−	0.234327i
$313$	15.6381			0.883916			0.441958	−	0.897036i	$-0.354284\pi$
$313$	15.6381			0.883916			0.441958	+	0.897036i	$0.354284\pi$
$314$	−26.3902	+	19.3398i	−1.48928	+	1.09141i
$315$		0			0
$316$	9.75331	−	30.8789i	0.548667	−	1.73707i
$317$			9.34379i			0.524800i	0.964959	+	0.262400i	$0.0845139\pi$
$317$			9.34379i			0.524800i	−0.964959	+	0.262400i	$0.915486\pi$
$318$	1.12334	+	1.53286i	0.0629940	+	0.0859587i
$319$	−37.9491			−2.12474
$320$	−2.27048	−	2.96830i	−0.126924	−	0.165933i
$321$	−3.12334			−0.174328
$322$		0			0
$323$		−	8.19130i		−	0.455776i
$324$	−0.602380	+	1.90713i	−0.0334655	+	0.105952i
$325$		−	21.8183i		−	1.21026i
$326$	−13.7317	+	10.0632i	−0.760531	+	0.557348i
$327$	10.6876			0.591025
$328$	−16.3303	+	5.52954i	−0.901692	+	0.305318i
$329$		0			0
$330$	−2.59859	+	1.90435i	−0.143048	+	0.104831i
$331$		−	5.40874i		−	0.297291i	−0.988891	−	0.148646i	$-0.952509\pi$
$331$		−	5.40874i		−	0.297291i	0.988891	−	0.148646i	$-0.0474914\pi$
$332$	26.2293	+	8.28472i	1.43952	+	0.454683i
$333$		−	4.40952i		−	0.241640i
$334$	−7.53218	−	10.2781i	−0.412143	−	0.562391i
$335$	−2.75919			−0.150750
$336$		0			0
$337$	−6.55614			−0.357136			−0.178568	−	0.983928i	$-0.557146\pi$
$337$	−6.55614			−0.357136			−0.178568	+	0.983928i	$0.557146\pi$
$338$	−6.53631	−	8.91915i	−0.355529	−	0.485138i
$339$			12.0599i			0.655001i
$340$	−5.43056	−	1.71528i	−0.294514	−	0.0930242i
$341$			21.5037i			1.16449i
$342$	1.53286	−	1.12334i	0.0828877	−	0.0607434i
$343$		0			0
$344$	3.76755	+	11.1267i	0.203133	+	0.599910i
$345$	1.91323			0.103005
$346$	−10.9424	+	8.01902i	−0.588266	+	0.431105i
$347$		−	24.9964i		−	1.34187i	−0.741514	−	0.670937i	$-0.765892\pi$
$347$		−	24.9964i		−	1.34187i	0.741514	−	0.670937i	$-0.234108\pi$
$348$	4.68759	−	14.8409i	0.251281	−	0.795553i
$349$			27.1921i			1.45556i	0.685811	+	0.727779i	$0.259447\pi$
$349$			27.1921i			1.45556i	−0.685811	+	0.727779i	$0.740553\pi$
$350$		0			0
$351$	4.56279			0.243544
$352$	−27.5807	+	0.566794i	−1.47006	+	0.0302102i
$353$	30.2870			1.61201			0.806006	−	0.591907i	$-0.201625\pi$
$353$	30.2870			1.61201			0.806006	+	0.591907i	$0.201625\pi$
$354$	3.34379	+	4.56279i	0.177721	+	0.242509i
$355$			2.20647i			0.117107i
$356$	4.79747	−	15.1888i	0.254266	−	0.805002i
$357$		0			0
$358$	−8.77513	+	6.43077i	−0.463780	+	0.339877i
$359$	6.59194			0.347909			0.173955	−	0.984754i	$-0.444345\pi$
$359$	6.59194			0.347909			0.173955	+	0.984754i	$0.444345\pi$
$360$	−0.423754	−	1.25147i	−0.0223338	−	0.0659582i
$361$	17.1942			0.904959
$362$	−17.3961	+	12.7485i	−0.914317	+	0.670048i
$363$			12.7818i			0.670870i
$364$		0			0
$365$		−	5.63361i		−	0.294877i
$366$	−4.59526	−	6.27048i	−0.240198	−	0.327763i
$367$	6.68759			0.349089			0.174545	−	0.984649i	$-0.444155\pi$
$367$	6.68759			0.349089			0.174545	+	0.984649i	$0.444155\pi$
$368$	13.4103	+	9.41030i	0.699060	+	0.490546i
$369$	−6.09565			−0.317327
$370$	1.72193	+	2.34967i	0.0895189	+	0.122153i
$371$		0			0
$372$	−8.40952	−	2.65621i	−0.436013	−	0.137718i
$373$		−	17.1256i		−	0.886729i	−0.896341	−	0.443364i	$-0.853785\pi$
$373$		−	17.1256i		−	0.886729i	0.896341	−	0.443364i	$-0.146215\pi$
$374$	−33.9088	+	24.8497i	−1.75338	+	1.28495i
$375$	−4.56944			−0.235965
$376$	−17.9161	+	6.06650i	−0.923955	+	0.312856i
$377$	−35.5066			−1.82868
$378$		0			0
$379$		−	16.7094i		−	0.858305i	−0.903232	−	0.429152i	$-0.858812\pi$
$379$		−	16.7094i		−	0.858305i	0.903232	−	0.429152i	$-0.141188\pi$
$380$	−0.378136	+	1.19717i	−0.0193980	+	0.0614138i
$381$		−	18.8789i		−	0.967195i
$382$	−3.94851	−	5.38795i	−0.202023	−	0.275672i
$383$	9.94015			0.507918			0.253959	−	0.967215i	$-0.418267\pi$
$383$	9.94015			0.507918			0.253959	+	0.967215i	$0.418267\pi$
$384$	3.18520	−	10.8561i	0.162544	−	0.553997i
$385$		0			0
$386$	18.6734	+	25.4808i	0.950449	+	1.29694i
$387$			4.15327i			0.211123i
$388$	−7.75798	+	24.5617i	−0.393852	+	1.24693i
$389$			19.2884i			0.977961i	0.872295	+	0.488981i	$0.162631\pi$
$389$			19.2884i			0.977961i	−0.872295	+	0.488981i	$0.837369\pi$
$390$	−2.43134	+	1.78178i	−0.123116	+	0.0902241i
$391$	24.9657			1.26257
$392$		0			0
$393$	−4.93428			−0.248901
$394$	1.75108	−	1.28326i	0.0882181	−	0.0646498i
$395$			7.56357i			0.380564i
$396$	−9.30041	−	2.93760i	−0.467363	−	0.147620i
$397$		−	23.4417i		−	1.17650i	−0.808678	−	0.588252i	$-0.799816\pi$
$397$		−	23.4417i		−	1.17650i	0.808678	−	0.588252i	$-0.200184\pi$
$398$	12.2781	+	16.7541i	0.615444	+	0.839807i
$399$		0			0
$400$	−15.6569	−	10.9868i	−0.782844	−	0.549339i
$401$	−16.0029			−0.799147			−0.399574	−	0.916701i	$-0.630842\pi$
$401$	−16.0029			−0.799147			−0.399574	+	0.916701i	$0.630842\pi$
$402$	−4.93760	−	6.73762i	−0.246265	−	0.336042i
$403$			20.1197i			1.00223i
$404$	−4.23469	−	1.33756i	−0.210683	−	0.0665459i
$405$		−	0.467138i		−	0.0232123i
$406$		0			0
$407$	21.5037			1.06590
$408$	−5.52954	−	16.3303i	−0.273753	−	0.808472i
$409$	−15.5665			−0.769713			−0.384856	−	0.922976i	$-0.625749\pi$
$409$	−15.5665			−0.769713			−0.384856	+	0.922976i	$0.625749\pi$
$410$	3.24814	−	2.38037i	0.160414	−	0.117558i
$411$		−	19.0103i		−	0.937711i
$412$	6.68467	−	21.1636i	0.329330	−	1.04266i
$413$		0			0
$414$	3.42375	+	4.67190i	0.168268	+	0.229611i
$415$	−6.42469			−0.315376
$416$	−25.8056	+	0.530314i	−1.26522	+	0.0260008i
$417$	10.4380			0.511150
$418$	5.47816	+	7.47524i	0.267946	+	0.365626i
$419$		−	1.31241i		−	0.0641155i	−0.999486	−	0.0320577i	$-0.989794\pi$
$419$		−	1.31241i		−	0.0641155i	0.999486	−	0.0320577i	$-0.0102060\pi$
$420$		0			0
$421$		−	3.66655i		−	0.178697i	−0.996000	−	0.0893483i	$-0.971522\pi$
$421$		−	3.66655i		−	0.178697i	0.996000	−	0.0893483i	$-0.0284784\pi$
$422$	−26.0545	+	19.0938i	−1.26831	+	0.929471i
$423$	−6.68759			−0.325162
$424$	−1.21899	−	3.60004i	−0.0591996	−	0.174834i
$425$	−29.1481			−1.41389
$426$	−5.38795	+	3.94851i	−0.261047	+	0.191306i
$427$		0			0
$428$	5.95662	+	1.88144i	0.287924	+	0.0909428i
$429$			22.2512i			1.07430i
$430$	−1.62186	−	2.21312i	−0.0782132	−	0.106726i
$431$	20.0957			0.967973			0.483987	−	0.875075i	$-0.339188\pi$
$431$	20.0957			0.967973			0.483987	+	0.875075i	$0.339188\pi$
$432$	2.29763	−	3.27428i	0.110545	−	0.157534i
$433$	−8.68759			−0.417499			−0.208749	−	0.977969i	$-0.566939\pi$
$433$	−8.68759			−0.417499			−0.208749	+	0.977969i	$0.566939\pi$
$434$		0			0
$435$			3.63516i			0.174293i
$436$	−20.3826	−	6.43799i	−0.976150	−	0.308324i
$437$		−	5.50371i		−	0.263278i
$438$	13.7566	−	10.0814i	0.657318	−	0.481709i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	6.10298	−	2.06650i	0.290948	−	0.0985166i
$441$		0			0
$442$	−31.7264	+	23.2504i	−1.50907	+	1.10591i
$443$		−	4.57012i		−	0.217133i	−0.994089	−	0.108566i	$-0.965374\pi$
$443$		−	4.57012i		−	0.217133i	0.994089	−	0.108566i	$-0.0346260\pi$
$444$	−2.65621	+	8.40952i	−0.126058	+	0.399098i
$445$			3.72038i			0.176363i
$446$	−17.9161	−	24.4476i	−0.848354	−	1.15763i
$447$	−6.65621			−0.314828
$448$		0			0
$449$	−3.94015			−0.185947			−0.0929735	−	0.995669i	$-0.529637\pi$
$449$	−3.94015			−0.185947			−0.0929735	+	0.995669i	$0.529637\pi$
$450$	−3.99732	−	5.45457i	−0.188436	−	0.257131i
$451$		−	29.7264i		−	1.39976i
$452$	7.26461	−	22.9997i	0.341699	−	1.08181i
$453$			2.68759i			0.126274i
$454$	33.4417	−	24.5074i	1.56950	−	1.15019i
$455$		0			0
$456$	−3.60004	+	1.21899i	−0.168588	+	0.0570846i
$457$	40.0207			1.87209			0.936044	−	0.351882i	$-0.114458\pi$
$457$	40.0207			1.87209			0.936044	+	0.351882i	$0.114458\pi$
$458$	25.6741	−	18.8150i	1.19967	−	0.879169i
$459$		−	6.09565i		−	0.284521i
$460$	−3.64878	−	1.15249i	−0.170125	−	0.0537352i
$461$			34.6031i			1.61162i	0.592171	+	0.805812i	$0.298271\pi$
$461$			34.6031i			1.61162i	−0.592171	+	0.805812i	$0.701729\pi$
$462$		0			0
$463$	0.191302			0.00889055			0.00444528	−	0.999990i	$-0.498585\pi$
$463$	0.191302			0.00889055			0.00444528	+	0.999990i	$0.498585\pi$
$464$	−17.8797	+	25.4797i	−0.830043	+	1.18287i
$465$	2.05985			0.0955234
$466$	−0.574789	−	0.784331i	−0.0266266	−	0.0363334i
$467$		−	4.49784i		−	0.208135i	−0.994570	−	0.104068i	$-0.966814\pi$
$467$		−	4.49784i		−	0.208135i	0.994570	−	0.104068i	$-0.0331858\pi$
$468$	−8.70182	−	2.74853i	−0.402242	−	0.127051i
$469$		0			0
$470$	3.56357	−	2.61152i	0.164375	−	0.120461i
$471$	23.1351			1.06601
$472$	−3.62851	−	10.7161i	−0.167016	−	0.493247i
$473$	−20.2541			−0.931283
$474$	−18.4694	+	13.5351i	−0.848326	+	0.621688i
$475$			6.42573i			0.294833i
$476$		0			0
$477$		−	1.34379i		−	0.0615281i
$478$	−8.02457	−	10.9500i	−0.367036	−	0.500840i
$479$	22.2512			1.01668			0.508341	−	0.861156i	$-0.330259\pi$
$479$	22.2512			1.01668			0.508341	+	0.861156i	$0.330259\pi$
$480$	0.0542935	+	2.64197i	0.00247815	+	0.120589i
$481$	20.1197			0.917380
$482$	−0.414870	−	0.566113i	−0.0188968	−	0.0257857i
$483$		0			0
$484$	7.69949	−	24.3765i	0.349977	−	1.10802i
$485$		−	6.01621i		−	0.273182i
$486$	1.14070	−	0.835949i	0.0517431	−	0.0379194i
$487$	−1.86855			−0.0846721			−0.0423360	−	0.999103i	$-0.513480\pi$
$487$	−1.86855			−0.0846721			−0.0423360	+	0.999103i	$0.513480\pi$
$488$	4.98654	+	14.7267i	0.225730	+	0.666647i
$489$	12.0380			0.544379
$490$		0			0
$491$		−	17.6374i		−	0.795965i	−0.917393	−	0.397982i	$-0.869711\pi$
$491$		−	17.6374i		−	0.795965i	0.917393	−	0.397982i	$-0.130289\pi$
$492$	11.6252	+	3.67190i	0.524104	+	0.165542i
$493$			47.4350i			2.13637i
$494$	5.12558	+	6.99413i	0.230611	+	0.314681i
$495$	2.27807			0.102392
$496$	14.4380	+	10.1314i	0.648285	+	0.454916i
$497$		0			0
$498$	−11.4971	−	15.6884i	−0.515196	−	0.703012i
$499$			21.5854i			0.966295i	0.875539	+	0.483147i	$0.160506\pi$
$499$			21.5854i			0.966295i	−0.875539	+	0.483147i	$0.839494\pi$
$500$	8.71450	+	2.75254i	0.389724	+	0.123097i
$501$			9.01034i			0.402552i
$502$	25.2504	−	18.5045i	1.12698	−	0.825896i
$503$	−5.24081			−0.233676			−0.116838	−	0.993151i	$-0.537276\pi$
$503$	−5.24081			−0.233676			−0.116838	+	0.993151i	$0.537276\pi$
$504$		0			0
$505$	1.03726			0.0461573
$506$	−22.7832	+	16.6965i	−1.01284	+	0.742249i
$507$			7.81904i			0.347256i
$508$	−11.3723	+	36.0045i	−0.504563	+	1.59744i
$509$		−	18.3357i		−	0.812715i	−0.913714	−	0.406358i	$-0.866799\pi$
$509$		−	18.3357i		−	0.812715i	0.913714	−	0.406358i	$-0.133201\pi$
$510$	2.38037	+	3.24814i	0.105405	+	0.143830i
$511$		0			0
$512$	−12.6141	+	18.7852i	−0.557468	+	0.830198i
$513$	−1.34379			−0.0593300
$514$	−1.91178	−	2.60872i	−0.0843248	−	0.115066i
$515$			5.18388i			0.228429i
$516$	2.50185	−	7.92082i	0.110138	−	0.348695i
$517$		−	32.6131i		−	1.43432i
$518$		0			0
$519$	9.59271			0.421073
$520$	5.71018	−	1.93350i	0.250408	−	0.0847895i
$521$	−6.78033			−0.297051			−0.148526	−	0.988909i	$-0.547453\pi$
$521$	−6.78033			−0.297051			−0.148526	+	0.988909i	$0.547453\pi$
$522$	−8.87666	+	6.50517i	−0.388521	+	0.284724i
$523$		−	23.5066i		−	1.02787i	−0.857828	−	0.513937i	$-0.828187\pi$
$523$		−	23.5066i		−	1.02787i	0.857828	−	0.513937i	$-0.171813\pi$
$524$	9.41030	+	2.97231i	0.411091	+	0.129846i
$525$		0			0
$526$	−25.2085	−	34.3983i	−1.09914	−	1.49984i
$527$	26.8789			1.17086
$528$	15.9675	+	11.2048i	0.694897	+	0.487625i
$529$	−6.22564			−0.270680
$530$	0.524756	+	0.716058i	0.0227939	+	0.0311036i
$531$		−	4.00000i		−	0.173585i
$532$		0			0
$533$		−	27.8132i		−	1.20472i
$534$	−9.08474	+	6.65766i	−0.393135	+	0.288105i
$535$	−1.45903			−0.0630794
$536$	5.35803	+	15.8238i	0.231432	+	0.683485i
$537$	7.69278			0.331968
$538$	10.8109	−	7.92268i	0.466092	−	0.341571i
$539$		0			0
$540$	−0.281394	+	0.890891i	−0.0121093	+	0.0383379i
$541$		−	5.21526i		−	0.224222i	−0.993696	−	0.112111i	$-0.964239\pi$
$541$		−	5.21526i		−	0.224222i	0.993696	−	0.112111i	$-0.0357611\pi$
$542$	11.1060	+	15.1547i	0.477043	+	0.650951i
$543$	15.2504			0.654456
$544$	0.708473	+	34.4749i	0.0303755	+	1.47810i
$545$	4.99257			0.213858
$546$		0			0
$547$		−	27.2951i		−	1.16705i	−0.812094	−	0.583526i	$-0.801673\pi$
$547$		−	27.2951i		−	1.16705i	0.812094	−	0.583526i	$-0.198327\pi$
$548$	−11.4514	+	36.2552i	−0.489182	+	1.54874i
$549$			5.49706i			0.234609i
$550$	26.6000	−	19.4936i	1.13423	−	0.831209i
$551$	10.4571			0.445488
$552$	−3.71528	−	10.9723i	−0.158133	−	0.467012i
$553$		0			0
$554$	30.6249	−	22.4431i	1.30113	−	0.953518i
$555$		−	2.05985i		−	0.0874359i
$556$	−19.9066	−	6.28763i	−0.844227	−	0.266655i
$557$			4.78766i			0.202859i	0.994843	+	0.101430i	$0.0323417\pi$
$557$			4.78766i			0.202859i	−0.994843	+	0.101430i	$0.967658\pi$
$558$	3.68613	+	5.02993i	0.156046	+	0.212934i
$559$	−18.9505			−0.801520
$560$		0			0
$561$	29.7264			1.25505
$562$	9.92150	+	13.5384i	0.418513	+	0.571084i
$563$			18.7445i			0.789988i	0.918684	+	0.394994i	$0.129253\pi$
$563$			18.7445i			0.789988i	−0.918684	+	0.394994i	$0.870747\pi$
$564$	12.7541	+	4.02847i	0.537044	+	0.169629i
$565$			5.63361i			0.237008i
$566$	2.81680	−	2.06427i	0.118399	−	0.0867676i
$567$		0			0
$568$	12.6540	−	4.28472i	0.530951	−	0.179783i
$569$	27.6950			1.16104			0.580518	−	0.814248i	$-0.302850\pi$
$569$	27.6950			1.16104			0.580518	+	0.814248i	$0.302850\pi$
$570$	0.716058	−	0.524756i	0.0299924	−	0.0219796i
$571$			17.4132i			0.728720i	0.931258	+	0.364360i	$0.118712\pi$
$571$			17.4132i			0.728720i	−0.931258	+	0.364360i	$0.881288\pi$
$572$	13.4036	−	42.4358i	0.560435	−	1.77433i
$573$			4.72339i			0.197322i
$574$		0			0
$575$	−19.5845			−0.816731
$576$	−6.35424	+	4.86042i	−0.264760	+	0.202517i
$577$	6.81904			0.283880			0.141940	−	0.989875i	$-0.454666\pi$
$577$	6.81904			0.283880			0.141940	+	0.989875i	$0.454666\pi$
$578$	16.8502	+	22.9930i	0.700875	+	0.956382i
$579$		−	22.3379i		−	0.928332i
$580$	2.18975	−	6.93272i	0.0909244	−	0.287866i
$581$		0			0
$582$	14.6909	−	10.7661i	0.608958	−	0.446269i
$583$	6.55322			0.271407
$584$	−32.3085	+	10.9398i	−1.33694	+	0.452694i
$585$	2.13145			0.0881246
$586$	25.9964	−	19.0512i	1.07390	−	0.786997i
$587$		−	9.81025i		−	0.404912i	−0.979291	−	0.202456i	$-0.935108\pi$
$587$		−	9.81025i		−	0.404912i	0.979291	−	0.202456i	$-0.0648924\pi$
$588$		0			0
$589$		−	5.92549i		−	0.244155i
$590$	1.56201	+	2.13145i	0.0643070	+	0.0877504i
$591$	−1.53510			−0.0631454
$592$	10.1314	−	14.4380i	0.416400	−	0.593398i
$593$	39.0913			1.60529			0.802644	−	0.596458i	$-0.203426\pi$
$593$	39.0913			1.60529			0.802644	+	0.596458i	$0.203426\pi$
$594$	4.07663	+	5.56279i	0.167266	+	0.228244i
$595$		0			0
$596$	12.6942	+	4.00956i	0.519976	+	0.164238i
$597$		−	14.6876i		−	0.601123i
$598$	−21.3169	+	15.6219i	−0.871712	+	0.638826i
$599$	22.0270			0.899998			0.449999	−	0.893029i	$-0.351424\pi$
$599$	22.0270			0.899998			0.449999	+	0.893029i	$0.351424\pi$
$600$	4.33769	+	12.8105i	0.177085	+	0.522985i
$601$	20.2512			0.826062			0.413031	−	0.910717i	$-0.364470\pi$
$601$	20.2512			0.826062			0.413031	+	0.910717i	$0.364470\pi$
$602$		0			0
$603$			5.90658i			0.240535i
$604$	1.61895	−	5.12558i	0.0658741	−	0.208557i
$605$			5.97085i			0.242750i
$606$	1.85618	+	2.53286i	0.0754023	+	0.102891i
$607$	−44.4454			−1.80398			−0.901991	−	0.431755i	$-0.857895\pi$
$607$	−44.4454			−1.80398			−0.901991	+	0.431755i	$0.857895\pi$
$608$	7.60004	−	0.156184i	0.308223	−	0.00633409i
$609$		0			0
$610$	−2.14662	−	2.92918i	−0.0869141	−	0.118599i
$611$		−	30.5141i		−	1.23447i
$612$	−3.67190	+	11.6252i	−0.148428	+	0.469920i
$613$		−	47.6263i		−	1.92361i	−0.273737	−	0.961805i	$-0.588260\pi$
$613$		−	47.6263i		−	1.92361i	0.273737	−	0.961805i	$-0.411740\pi$
$614$	10.6584	−	7.81093i	0.430140	−	0.315224i
$615$	−2.84751			−0.114823
$616$		0			0
$617$	−3.01034			−0.121192			−0.0605959	−	0.998162i	$-0.519300\pi$
$617$	−3.01034			−0.121192			−0.0605959	+	0.998162i	$0.519300\pi$
$618$	−12.6584	+	9.27661i	−0.509197	+	0.373160i
$619$		−	18.5141i		−	0.744143i	−0.928204	−	0.372071i	$-0.878648\pi$
$619$		−	18.5141i		−	0.744143i	0.928204	−	0.372071i	$-0.121352\pi$
$620$	−3.92840	−	1.24081i	−0.157768	−	0.0498323i
$621$		−	4.09565i		−	0.164353i
$622$	−5.59048	−	7.62851i	−0.224158	−	0.305876i
$623$		0			0
$624$	14.9398	+	10.4836i	0.598072	+	0.419680i
$625$	21.7744			0.870974
$626$	13.0726	+	17.8383i	0.522487	+	0.712962i
$627$		−	6.55322i		−	0.261711i
$628$	−44.1217	−	13.9361i	−1.76065	−	0.556113i
$629$		−	26.8789i		−	1.07173i
$630$		0			0
$631$	10.2658			0.408676			0.204338	−	0.978900i	$-0.434496\pi$
$631$	10.2658			0.408676			0.204338	+	0.978900i	$0.434496\pi$
$632$	43.3767	−	14.6876i	1.72543	−	0.584241i
$633$	22.8409			0.907843
$634$	−10.6584	+	7.81093i	−0.423301	+	0.310212i
$635$		−	8.81904i		−	0.349973i
$636$	−0.809475	+	2.56279i	−0.0320978	+	0.101621i
$637$		0			0
$638$	−31.7235	−	43.2884i	−1.25594	−	1.71380i
$639$	4.72339			0.186854
$640$	1.48793	−	5.07128i	0.0588154	−	0.200460i
$641$	−16.9358			−0.668925			−0.334462	−	0.942409i	$-0.608555\pi$
$641$	−16.9358			−0.668925			−0.334462	+	0.942409i	$0.608555\pi$
$642$	−2.61095	−	3.56279i	−0.103046	−	0.140612i
$643$			19.5189i			0.769750i	0.922969	+	0.384875i	$0.125755\pi$
$643$			19.5189i			0.769750i	−0.922969	+	0.384875i	$0.874245\pi$
$644$		0			0
$645$			1.94015i			0.0763933i
$646$	9.34379	−	6.84751i	0.367627	−	0.269412i
$647$	23.3723			0.918858			0.459429	−	0.888214i	$-0.348054\pi$
$647$	23.3723			0.918858			0.459429	+	0.888214i	$0.348054\pi$
$648$	−2.67901	+	0.907128i	−0.105242	+	0.0356354i
$649$	19.5066			0.765702
$650$	24.8880	−	18.2389i	0.976189	−	0.715390i
$651$		0			0
$652$	−22.9581	−	7.25147i	−0.899108	−	0.283989i
$653$			1.45903i			0.0570963i	0.999592	+	0.0285481i	$0.00908839\pi$
$653$			1.45903i			0.0570963i	−0.999592	+	0.0285481i	$0.990912\pi$
$654$	8.93428	+	12.1913i	0.349358	+	0.476718i
$655$	−2.30499			−0.0900632
$656$	−19.9589	−	14.0056i	−0.779262	−	0.546825i
$657$	−12.0599			−0.470500
$658$		0			0
$659$			10.8929i			0.424326i	0.977234	+	0.212163i	$0.0680508\pi$
$659$			10.8929i			0.424326i	−0.977234	+	0.212163i	$0.931949\pi$
$660$	−4.34457	−	1.37226i	−0.169112	−	0.0534153i
$661$		−	8.69715i		−	0.338280i	−0.985592	−	0.169140i	$-0.945901\pi$
$661$		−	8.69715i		−	0.338280i	0.985592	−	0.169140i	$-0.0540990\pi$
$662$	6.16974	−	4.52143i	0.239794	−	0.175730i
$663$	27.8132			1.08017
$664$	12.4760	+	36.8453i	0.484164	+	1.42988i
$665$		0			0
$666$	5.02993	−	3.68613i	0.194906	−	0.142835i
$667$			31.8715i			1.23407i
$668$	5.42765	−	17.1839i	0.210002	−	0.664864i
$669$			21.4321i			0.828613i
$670$	−2.30654	−	3.14740i	−0.0891093	−	0.121595i
$671$	−26.8073			−1.03488
$672$		0			0
$673$	18.0447			0.695571			0.347786	−	0.937574i	$-0.386934\pi$
$673$	18.0447			0.695571			0.347786	+	0.937574i	$0.386934\pi$
$674$	−5.48060	−	7.47857i	−0.211105	−	0.288064i
$675$			4.78178i			0.184051i
$676$	4.71003	−	14.9119i	0.181155	−	0.573535i
$677$			28.7781i			1.10603i	0.833170	+	0.553017i	$0.186524\pi$
$677$			28.7781i			1.10603i	−0.833170	+	0.553017i	$0.813476\pi$
$678$	−13.7566	+	10.0814i	−0.528321	+	0.387174i
$679$		0			0
$680$	−2.58305	−	7.62851i	−0.0990556	−	0.292540i
$681$	−29.3169			−1.12343
$682$	−24.5292	+	17.9760i	−0.939273	+	0.688337i
$683$			34.4431i			1.31793i	0.752174	+	0.658965i	$0.229005\pi$
$683$			34.4431i			1.31793i	−0.752174	+	0.658965i	$0.770995\pi$
$684$	2.56279	+	0.809475i	0.0979907	+	0.0309510i
$685$		−	8.88044i		−	0.339304i
$686$		0			0
$687$	−22.5074			−0.858711
$688$	−9.54268	+	13.5990i	−0.363811	+	0.518455i
$689$	6.13145			0.233590
$690$	1.59936	+	2.18242i	0.0608867	+	0.0830833i
$691$			41.5651i			1.58121i	0.612325	+	0.790606i	$0.290234\pi$
$691$			41.5651i			1.58121i	−0.612325	+	0.790606i	$0.709766\pi$
$692$	−18.2945	−	5.77846i	−0.695454	−	0.219664i
$693$		0			0
$694$	28.5133	−	20.8957i	1.08235	−	0.793189i
$695$	4.87598			0.184956
$696$	20.8475	−	7.05908i	0.790223	−	0.267574i
$697$	−37.1570			−1.40742
$698$	−31.0179	+	22.7312i	−1.17405	+	0.860388i
$699$			0.687589i			0.0260070i
$700$		0			0
$701$		−	46.7804i		−	1.76687i	−0.468553	−	0.883435i	$-0.655225\pi$
$701$		−	46.7804i		−	1.76687i	0.468553	−	0.883435i	$-0.344775\pi$
$702$	3.81426	+	5.20476i	0.143960	+	0.196441i
$703$	−5.92549			−0.223484
$704$	−23.7026	−	30.9874i	−0.893325	−	1.16788i
$705$	−3.12402			−0.117658
$706$	25.3183	+	34.5482i	0.952868	+	1.30024i
$707$		0			0
$708$	−2.40952	+	7.62851i	−0.0905553	+	0.286697i
$709$			8.43643i			0.316837i	0.987372	+	0.158418i	$0.0506395\pi$
$709$			8.43643i			0.316837i	−0.987372	+	0.158418i	$0.949360\pi$
$710$	−2.51692	+	1.84450i	−0.0944582	+	0.0692227i
$711$	16.1913			0.607221
$712$	21.3362	−	7.22455i	0.799608	−	0.270752i
$713$	18.0599			0.676347
$714$		0			0
$715$			10.3943i			0.388727i
$716$	−14.6711	−	4.63398i	−0.548286	−	0.173180i
$717$			9.59936i			0.358495i
$718$	5.51052	+	7.51940i	0.205651	+	0.280622i
$719$	38.4425			1.43366			0.716831	−	0.697247i	$-0.245592\pi$
$719$	38.4425			1.43366			0.716831	+	0.697247i	$0.245592\pi$
$720$	1.07331	−	1.52954i	0.0399999	−	0.0570025i
$721$		0			0
$722$	14.3735	+	19.6134i	0.534926	+	0.729935i
$723$			0.496287i			0.0184571i
$724$	−29.0844	−	9.18652i	−1.08091	−	0.341414i
$725$		−	37.2108i		−	1.38197i
$726$	−14.5801	+	10.6849i	−0.541120	+	0.396554i
$727$	17.5484			0.650834			0.325417	−	0.945571i	$-0.394495\pi$
$727$	17.5484			0.650834			0.325417	+	0.945571i	$0.394495\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$	6.42624	−	4.70941i	0.237846	−	0.174303i
$731$			25.3169i			0.936379i
$732$	3.31132	−	10.4836i	0.122390	−	0.387485i
$733$		−	38.8272i		−	1.43412i	−0.697013	−	0.717058i	$-0.745488\pi$
$733$		−	38.8272i		−	1.43412i	0.697013	−	0.717058i	$-0.254512\pi$
$734$	5.59048	+	7.62851i	0.206348	+	0.281574i
$735$		0			0
$736$	0.476021	+	23.1636i	0.0175464	+	0.853822i
$737$	−28.8044			−1.06102
$738$	−5.09565	−	6.95329i	−0.187574	−	0.255954i
$739$			13.9066i			0.511562i	0.966735	+	0.255781i	$0.0823326\pi$
$739$			13.9066i			0.511562i	−0.966735	+	0.255781i	$0.917667\pi$
$740$	−1.24081	+	3.92840i	−0.0456132	+	0.144411i
$741$		−	6.13145i		−	0.225244i
$742$		0			0
$743$	40.2300			1.47590			0.737948	−	0.674858i	$-0.235795\pi$
$743$	40.2300			1.47590			0.737948	+	0.674858i	$0.235795\pi$
$744$	−4.00000	−	11.8132i	−0.146647	−	0.433092i
$745$	−3.10936			−0.113918
$746$	19.5351	−	14.3161i	0.715231	−	0.524150i
$747$			13.7533i			0.503207i
$748$	−56.6921	−	17.9066i	−2.07287	−	0.654730i
$749$		0			0
$750$	−3.81982	−	5.21234i	−0.139480	−	0.190328i
$751$	16.9957			0.620181			0.310091	−	0.950707i	$-0.399641\pi$
$751$	16.9957			0.620181			0.310091	+	0.950707i	$0.399641\pi$
$752$	−21.8970	−	15.3656i	−0.798502	−	0.560326i
$753$	−22.1359			−0.806678
$754$	−29.6817	−	40.5023i	−1.08094	−	1.47501i
$755$			1.25547i			0.0456914i
$756$		0			0
$757$			9.43212i			0.342816i	0.985200	+	0.171408i	$0.0548316\pi$
$757$			9.43212i			0.342816i	−0.985200	+	0.171408i	$0.945168\pi$
$758$	19.0604	−	13.9682i	0.692304	−	0.507348i
$759$	19.9731			0.724977
$760$	−1.68172	+	0.569438i	−0.0610023	+	0.0206557i
$761$	9.41098			0.341148			0.170574	−	0.985345i	$-0.445438\pi$
$761$	9.41098			0.341148			0.170574	+	0.985345i	$0.445438\pi$
$762$	21.5351	−	15.7818i	0.780134	−	0.571714i
$763$		0			0
$764$	2.84527	−	9.00811i	0.102938	−	0.325902i
$765$		−	2.84751i		−	0.102952i
$766$	8.30945	+	11.3387i	0.300233	+	0.409684i
$767$	18.2512			0.659011
$768$	15.0462	−	5.44178i	0.542932	−	0.196363i
$769$	−27.6950			−0.998708			−0.499354	−	0.866398i	$-0.666429\pi$
$769$	−27.6950			−0.998708			−0.499354	+	0.866398i	$0.666429\pi$
$770$		0			0
$771$			2.28695i			0.0823626i
$772$	−13.4559	+	42.6013i	−0.484289	+	1.53325i
$773$		−	33.8993i		−	1.21927i	−0.792682	−	0.609636i	$-0.791316\pi$
$773$		−	33.8993i		−	1.21927i	0.792682	−	0.609636i	$-0.208684\pi$
$774$	−4.73762	+	3.47192i	−0.170290	+	0.124796i
$775$	−21.0854			−0.757409
$776$	−34.5027	+	11.6828i	−1.23858	+	0.419388i
$777$		0			0
$778$	−22.0022	+	16.1241i	−0.788818	+	0.578078i
$779$			8.19130i			0.293484i
$780$	−4.06495	−	1.28394i	−0.145548	−	0.0459725i
$781$			23.0343i			0.824234i
$782$	20.8700	+	28.4783i	0.746310	+	1.01838i
$783$	7.78178			0.278098
$784$		0			0
$785$	10.8073			0.385729
$786$	−4.12480	−	5.62851i	−0.147127	−	0.200762i
$787$			31.2001i			1.11216i	0.831128	+	0.556082i	$0.187696\pi$
$787$			31.2001i			1.11216i	−0.831128	+	0.556082i	$0.812304\pi$
$788$	2.92763	+	0.924711i	0.104292	+	0.0329415i
$789$			30.1555i			1.07356i
$790$	−8.62774	+	6.32275i	−0.306961	+	0.224953i
$791$		0			0
$792$	−4.42375	−	13.0646i	−0.157191	−	0.464232i
$793$	−25.0819			−0.890686
$794$	26.7399	−	19.5960i	0.948962	−	0.695437i
$795$		−	0.627737i		−	0.0222635i
$796$	−8.84751	+	28.0111i	−0.313592	+	0.992828i
$797$		−	32.2848i		−	1.14359i	−0.820398	−	0.571793i	$-0.806248\pi$
$797$		−	32.2848i		−	1.14359i	0.820398	−	0.571793i	$-0.193752\pi$
$798$		0			0
$799$	−40.7652			−1.44217
$800$	−0.555767	−	27.0441i	−0.0196493	−	0.956155i
$801$	7.96420			0.281401
$802$	−13.3776	−	18.2545i	−0.472380	−	0.644588i
$803$		−	58.8118i		−	2.07542i
$804$	3.55801	−	11.2646i	0.125481	−	0.397272i
$805$		0			0
$806$	−22.9505	+	16.8190i	−0.808396	+	0.592425i
$807$	−9.47748			−0.333623
$808$	−2.01423	−	5.94862i	−0.0708605	−	0.209272i
$809$	8.49629			0.298714			0.149357	−	0.988783i	$-0.452280\pi$
$809$	8.49629			0.298714			0.149357	+	0.988783i	$0.452280\pi$
$810$	0.532862	−	0.390503i	0.0187229	−	0.0137209i
$811$			12.3826i			0.434812i	0.976081	+	0.217406i	$0.0697596\pi$
$811$			12.3826i			0.434812i	−0.976081	+	0.217406i	$0.930240\pi$
$812$		0			0
$813$		−	13.2855i		−	0.465943i
$814$	17.9760	+	24.5292i	0.630058	+	0.859749i
$815$	5.62342			0.196980
$816$	14.0056	−	19.9589i	0.490293	−	0.698700i
$817$	5.58114			0.195259
$818$	−13.0128	−	17.7566i	−0.454981	−	0.620846i
$819$		0			0
$820$	5.43056	+	1.71528i	0.189643	+	0.0599002i
$821$		−	21.8999i		−	0.764313i	−0.924098	−	0.382156i	$-0.875182\pi$
$821$		−	21.8999i		−	0.764313i	0.924098	−	0.382156i	$-0.124818\pi$
$822$	21.6850	−	15.8917i	0.756353	−	0.554286i
$823$	−1.88321			−0.0656446			−0.0328223	−	0.999461i	$-0.510450\pi$
$823$	−1.88321			−0.0656446			−0.0328223	+	0.999461i	$0.510450\pi$
$824$	29.7293	−	10.0665i	1.03567	−	0.350683i
$825$	−23.3191			−0.811867
$826$		0			0
$827$			32.8051i			1.14074i	0.821387	+	0.570372i	$0.193201\pi$
$827$			32.8051i			1.14074i	−0.821387	+	0.570372i	$0.806799\pi$
$828$	−2.46714	+	7.81093i	−0.0857390	+	0.271449i
$829$		−	9.63143i		−	0.334513i	−0.985913	−	0.167257i	$-0.946509\pi$
$829$		−	9.63143i		−	0.334513i	0.985913	−	0.167257i	$-0.0534909\pi$
$830$	−5.37071	−	7.32862i	−0.186420	−	0.254380i
$831$	−26.8475			−0.931330
$832$	−22.1771	−	28.9930i	−0.768851	−	1.00515i
$833$		0			0
$834$	8.72562	+	11.9066i	0.302144	+	0.412291i
$835$			4.20907i			0.145661i
$836$	−3.94753	+	12.4978i	−0.136528	+	0.432247i
$837$		−	4.40952i		−	0.152415i
$838$	1.49706	−	1.09711i	0.0517152	−	0.0378990i
$839$	48.0481			1.65880			0.829402	−	0.558652i	$-0.188681\pi$
$839$	48.0481			1.65880			0.829402	+	0.558652i	$0.188681\pi$
$840$		0			0
$841$	−31.5561			−1.08814
$842$	4.18242	−	3.06504i	0.144136	−	0.105628i
$843$		−	11.8686i		−	0.408775i
$844$	−43.5604	−	13.7589i	−1.49941	−	0.473600i
$845$			3.65257i			0.125652i
$846$	−5.59048	−	7.62851i	−0.192205	−	0.262274i
$847$		0			0
$848$	3.08754	−	4.39996i	0.106027	−	0.151095i
$849$	−2.46937			−0.0847486
$850$	−24.3663	−	33.2491i	−0.835757	−	1.14044i
$851$		−	18.0599i		−	0.619084i
$852$	−9.00811	−	2.84527i	−0.308613	−	0.0974775i
$853$		−	1.44013i		−	0.0493090i	−0.999696	−	0.0246545i	$-0.992151\pi$
$853$		−	1.44013i		−	0.0493090i	0.999696	−	0.0246545i	$-0.00784856\pi$
$854$		0			0
$855$	−0.627737			−0.0214681
$856$	2.83327	+	8.36748i	0.0968393	+	0.285995i
$857$	−34.9775			−1.19481			−0.597404	−	0.801941i	$-0.703801\pi$
$857$	−34.9775			−1.19481			−0.597404	+	0.801941i	$0.703801\pi$
$858$	−25.3818	+	18.6008i	−0.866521	+	0.635022i
$859$		−	6.45024i		−	0.220079i	−0.993927	−	0.110040i	$-0.964902\pi$
$859$		−	6.45024i		−	0.220079i	0.993927	−	0.110040i	$-0.0350978\pi$
$860$	1.16871	−	3.70011i	0.0398525	−	0.126173i
$861$		0			0
$862$	16.7989	+	22.9231i	0.572174	+	0.780762i
$863$	−34.4037			−1.17112			−0.585559	−	0.810630i	$-0.699125\pi$
$863$	−34.4037			−1.17112			−0.585559	+	0.810630i	$0.699125\pi$
$864$	5.65566	−	0.116226i	0.192409	−	0.00395409i
$865$	4.48112			0.152363
$866$	−7.26238	−	9.90991i	−0.246786	−	0.336752i
$867$		−	20.1570i		−	0.684566i
$868$		0			0
$869$			78.9594i			2.67852i
$870$	−4.14662	+	3.03881i	−0.140584	+	0.103025i
$871$	−26.9505			−0.913182
$872$	−9.69501	−	28.6322i	−0.328315	−	0.969609i
$873$	−12.8789			−0.435884
$874$	6.27807	−	4.60082i	0.212359	−	0.155625i
$875$		0			0
$876$	22.9997	+	7.26461i	0.777087	+	0.245448i
$877$		−	40.3826i		−	1.36362i	−0.731528	−	0.681812i	$-0.761192\pi$
$877$		−	40.3826i		−	1.36362i	0.731528	−	0.681812i	$-0.238808\pi$
$878$		0			0
$879$	−22.7899			−0.768684
$880$	7.45903	+	5.23416i	0.251444	+	0.176444i
$881$	−15.8926			−0.535435			−0.267718	−	0.963497i	$-0.586269\pi$
$881$	−15.8926			−0.535435			−0.267718	+	0.963497i	$0.586269\pi$
$882$		0			0
$883$		−	25.3113i		−	0.851792i	−0.904772	−	0.425896i	$-0.859959\pi$
$883$		−	25.3113i		−	0.851792i	0.904772	−	0.425896i	$-0.140041\pi$
$884$	−53.0433	−	16.7541i	−1.78404	−	0.563501i
$885$		−	1.86855i		−	0.0628106i
$886$	5.21312	−	3.82038i	0.175138	−	0.128348i
$887$	−19.6352			−0.659284			−0.329642	−	0.944106i	$-0.606928\pi$
$887$	−19.6352			−0.659284			−0.329642	+	0.944106i	$0.606928\pi$
$888$	−11.8132	+	4.00000i	−0.396424	+	0.134231i
$889$		0			0
$890$	−4.24382	+	3.11004i	−0.142253	+	0.104249i
$891$		−	4.87666i		−	0.163374i
$892$	12.9103	−	40.8738i	0.432268	−	1.36856i
$893$			8.98674i			0.300730i
$894$	−5.56425	−	7.59271i	−0.186096	−	0.253938i
$895$	3.59359			0.120120
$896$		0			0
$897$	18.6876			0.623960
$898$	−3.29376	−	4.49452i	−0.109914	−	0.149984i
$899$			34.3139i			1.14443i
$900$	2.88045	−	9.11947i	0.0960150	−	0.303982i
$901$		−	8.19130i		−	0.272892i
$902$	33.9088	−	24.8497i	1.12904	−	0.827406i
$903$		0			0
$904$	32.3085	−	10.9398i	1.07457	−	0.363853i
$905$	7.12402			0.236811
$906$	−3.06572	+	2.24669i	−0.101852	+	0.0746411i
$907$		−	4.84241i		−	0.160790i	−0.996763	−	0.0803948i	$-0.974382\pi$
$907$		−	4.84241i		−	0.160790i	0.996763	−	0.0803948i	$-0.0256181\pi$
$908$	55.9111	+	17.6599i	1.85547	+	0.586064i
$909$		−	2.22045i		−	0.0736477i
$910$		0			0
$911$	−18.9176			−0.626768			−0.313384	−	0.949626i	$-0.601463\pi$
$911$	−18.9176			−0.626768			−0.313384	+	0.949626i	$0.601463\pi$
$912$	−4.39996	−	3.08754i	−0.145697	−	0.102239i
$913$	−67.0702			−2.21970
$914$	33.4552	+	45.6515i	1.10660	+	1.51002i
$915$			2.56788i			0.0848917i
$916$	42.9245	+	13.5580i	1.41827	+	0.447969i
$917$		0			0
$918$	6.95329	−	5.09565i	0.229493	−	0.168182i
$919$	−9.11228			−0.300586			−0.150293	−	0.988641i	$-0.548022\pi$
$919$	−9.11228			−0.300586			−0.150293	+	0.988641i	$0.548022\pi$
$920$	−1.73555	−	5.12558i	−0.0572193	−	0.168985i
$921$	−9.34379			−0.307888
$922$	−39.4716	+	28.9264i	−1.29993	+	0.952639i
$923$			21.5518i			0.709387i
$924$		0			0
$925$			21.0854i			0.693282i
$926$	0.159919	+	0.218217i	0.00525525	+	0.00717107i
$927$	11.0971			0.364477
$928$	−44.0111	+	0.904445i	−1.44474	+	0.0296899i
$929$	−32.7294			−1.07382			−0.536909	−	0.843640i	$-0.680408\pi$
$929$	−32.7294			−1.07382			−0.536909	+	0.843640i	$0.680408\pi$
$930$	1.72193	+	2.34967i	0.0564643	+	0.0770486i
$931$		0			0
$932$	0.414190	−	1.31132i	0.0135672	−	0.0429537i
$933$			6.68759i			0.218942i
$934$	5.13067	−	3.75996i	0.167881	−	0.123030i
$935$	13.8863			0.454131
$936$	−4.13903	−	12.2238i	−0.135289	−	0.399547i
$937$	21.5066			0.702591			0.351295	−	0.936265i	$-0.385741\pi$
$937$	21.5066			0.702591			0.351295	+	0.936265i	$0.385741\pi$
$938$		0			0
$939$		−	15.6381i		−	0.510329i
$940$	5.95791	+	1.88185i	0.194326	+	0.0613791i
$941$			1.85115i			0.0603456i	0.999545	+	0.0301728i	$0.00960577\pi$
$941$			1.85115i			0.0603456i	−0.999545	+	0.0301728i	$0.990394\pi$
$942$	19.3398	+	26.3902i	0.630124	+	0.859839i
$943$	−24.9657			−0.812994
$944$	9.19053	−	13.0971i	0.299126	−	0.426275i
$945$		0			0
$946$	−16.9314	−	23.1038i	−0.550486	−	0.751168i
$947$			45.3353i			1.47320i	0.676329	+	0.736600i	$0.263570\pi$
$947$			45.3353i			1.47320i	−0.676329	+	0.736600i	$0.736430\pi$
$948$	−30.8789	−	9.75331i	−1.00290	−	0.316773i
$949$		−	55.0266i		−	1.78624i
$950$	−7.32981	+	5.37158i	−0.237811	+	0.174277i
$951$	9.34379			0.302993
$952$		0			0
$953$	40.5768			1.31441			0.657206	−	0.753711i	$-0.271738\pi$
$953$	40.5768			1.31441			0.657206	+	0.753711i	$0.271738\pi$
$954$	1.53286	−	1.12334i	0.0496283	−	0.0363696i
$955$			2.20647i			0.0713997i
$956$	5.78246	−	18.3072i	0.187018	−	0.592098i
$957$			37.9491i			1.22672i
$958$	18.6008	+	25.3818i	0.600965	+	0.820050i
$959$		0			0
$960$	−2.96830	+	2.27048i	−0.0958015	+	0.0732796i
$961$	−11.5561			−0.372779
$962$	16.8190	+	22.9505i	0.542268	+	0.739953i
$963$			3.12334i			0.100648i
$964$	0.298953	−	0.946483i	0.00962864	−	0.0304842i
$965$		−	10.4349i		−	0.335911i
$966$		0			0
$967$	15.8715			0.510392			0.255196	−	0.966889i	$-0.417860\pi$
$967$	15.8715			0.510392			0.255196	+	0.966889i	$0.417860\pi$
$968$	34.2426	−	11.5947i	1.10060	−	0.372668i
$969$	−8.19130			−0.263143
$970$	6.86268	−	5.02925i	0.220347	−	0.161479i
$971$		−	14.6876i		−	0.471347i	−0.971832	−	0.235674i	$-0.924270\pi$
$971$		−	14.6876i		−	0.471347i	0.971832	−	0.235674i	$-0.0757296\pi$
$972$	1.90713	+	0.602380i	0.0611712	+	0.0193213i
$973$		0			0
$974$	−1.56201	−	2.13145i	−0.0500501	−	0.0682961i
$975$	−21.8183			−0.698744
$976$	−12.6302	+	17.9989i	−0.404284	+	0.576131i
$977$	−21.2437			−0.679647			−0.339824	−	0.940489i	$-0.610367\pi$
$977$	−21.2437			−0.679647			−0.339824	+	0.940489i	$0.610367\pi$
$978$	10.0632	+	13.7317i	0.321785	+	0.439093i
$979$			38.8387i			1.24129i
$980$		0			0
$981$		−	10.6876i		−	0.341228i
$982$	20.1189	−	14.7440i	0.642021	−	0.470499i
$983$	−11.8265			−0.377206			−0.188603	−	0.982053i	$-0.560396\pi$
$983$	−11.8265			−0.377206			−0.188603	+	0.982053i	$0.560396\pi$
$984$	5.52954	+	16.3303i	0.176275	+	0.520592i
$985$	−0.717101			−0.0228487
$986$	−54.1090	+	39.6532i	−1.72318	+	1.26282i
$987$		0			0
$988$	−3.69346	+	11.6935i	−0.117505	+	0.372019i
$989$			17.0103i			0.540897i
$990$	1.90435	+	2.59859i	0.0605242	+	0.0825885i
$991$	28.9387			0.919269			0.459635	−	0.888108i	$-0.347980\pi$
$991$	28.9387			0.919269			0.459635	+	0.888108i	$0.347980\pi$
$992$	0.512500	+	24.9387i	0.0162719	+	0.791806i
$993$	−5.40874			−0.171641
$994$		0			0
$995$		−	6.86112i		−	0.217512i
$996$	8.28472	−	26.2293i	0.262511	−	0.831108i
$997$		−	56.0548i		−	1.77527i	−0.460546	−	0.887636i	$-0.652346\pi$
$997$		−	56.0548i		−	1.77527i	0.460546	−	0.887636i	$-0.347654\pi$
$998$	−24.6224	+	18.0443i	−0.779408	+	0.571181i
$999$	−4.40952			−0.139511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.2.c.c.589.6		8
4.3	odd	2		4704.2.c.c.2353.7		8
7.6	odd	2		168.2.c.b.85.6	yes	8
8.3	odd	2		4704.2.c.c.2353.2		8
8.5	even	2	inner	1176.2.c.c.589.5		8
21.20	even	2		504.2.c.f.253.3		8
28.27	even	2		672.2.c.b.337.2		8
56.13	odd	2		168.2.c.b.85.5	✓	8
56.27	even	2		672.2.c.b.337.7		8
84.83	odd	2		2016.2.c.e.1009.4		8
112.13	odd	4		5376.2.a.bm.1.2		4
112.27	even	4		5376.2.a.bl.1.3		4
112.69	odd	4		5376.2.a.bp.1.3		4
112.83	even	4		5376.2.a.bq.1.2		4
168.83	odd	2		2016.2.c.e.1009.5		8
168.125	even	2		504.2.c.f.253.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.c.b.85.5	✓	8	56.13	odd	2
168.2.c.b.85.6	yes	8	7.6	odd	2
504.2.c.f.253.3		8	21.20	even	2
504.2.c.f.253.4		8	168.125	even	2
672.2.c.b.337.2		8	28.27	even	2
672.2.c.b.337.7		8	56.27	even	2
1176.2.c.c.589.5		8	8.5	even	2	inner
1176.2.c.c.589.6		8	1.1	even	1	trivial
2016.2.c.e.1009.4		8	84.83	odd	2
2016.2.c.e.1009.5		8	168.83	odd	2
4704.2.c.c.2353.2		8	8.3	odd	2
4704.2.c.c.2353.7		8	4.3	odd	2
5376.2.a.bl.1.3		4	112.27	even	4
5376.2.a.bm.1.2		4	112.13	odd	4
5376.2.a.bp.1.3		4	112.69	odd	4
5376.2.a.bq.1.2		4	112.83	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.835949 + 1.14070i) q^{2} -1.00000i q^{3} +(-0.602380 + 1.90713i) q^{4} -0.467138i q^{5} +(1.14070 - 0.835949i) q^{6} +(-2.67901 + 0.907128i) q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+(0.835949 + 1.14070i) q^{2} -1.00000i q^{3} +(-0.602380 + 1.90713i) q^{4} -0.467138i q^{5} +(1.14070 - 0.835949i) q^{6} +(-2.67901 + 0.907128i) q^{8} -1.00000 q^{9} +(0.532862 - 0.390503i) q^{10} -4.87666i q^{11} +(1.90713 + 0.602380i) q^{12} -4.56279i q^{13} -0.467138 q^{15} +(-3.27428 - 2.29763i) q^{16} -6.09565 q^{17} +(-0.835949 - 1.14070i) q^{18} +1.34379i q^{19} +(0.890891 + 0.281394i) q^{20} +(5.56279 - 4.07663i) q^{22} -4.09565 q^{23} +(0.907128 + 2.67901i) q^{24} +4.78178 q^{25} +(5.20476 - 3.81426i) q^{26} +1.00000i q^{27} -7.78178i q^{29} +(-0.390503 - 0.532862i) q^{30} -4.40952 q^{31} +(-0.116226 - 5.65566i) q^{32} -4.87666 q^{33} +(-5.09565 - 6.95329i) q^{34} +(0.602380 - 1.90713i) q^{36} +4.40952i q^{37} +(-1.53286 + 1.12334i) q^{38} -4.56279 q^{39} +(0.423754 + 1.25147i) q^{40} +6.09565 q^{41} -4.15327i q^{43} +(9.30041 + 2.93760i) q^{44} +0.467138i q^{45} +(-3.42375 - 4.67190i) q^{46} +6.68759 q^{47} +(-2.29763 + 3.27428i) q^{48} +(3.99732 + 5.45457i) q^{50} +6.09565i q^{51} +(8.70182 + 2.74853i) q^{52} +1.34379i q^{53} +(-1.14070 + 0.835949i) q^{54} -2.27807 q^{55} +1.34379 q^{57} +(8.87666 - 6.50517i) q^{58} +4.00000i q^{59} +(0.281394 - 0.890891i) q^{60} -5.49706i q^{61} +(-3.68613 - 5.02993i) q^{62} +(6.35424 - 4.86042i) q^{64} -2.13145 q^{65} +(-4.07663 - 5.56279i) q^{66} -5.90658i q^{67} +(3.67190 - 11.6252i) q^{68} +4.09565i q^{69} -4.72339 q^{71} +(2.67901 - 0.907128i) q^{72} +12.0599 q^{73} +(-5.02993 + 3.68613i) q^{74} -4.78178i q^{75} +(-2.56279 - 0.809475i) q^{76} +(-3.81426 - 5.20476i) q^{78} -16.1913 q^{79} +(-1.07331 + 1.52954i) q^{80} +1.00000 q^{81} +(5.09565 + 6.95329i) q^{82} -13.7533i q^{83} +2.84751i q^{85} +(4.73762 - 3.47192i) q^{86} -7.78178 q^{87} +(4.42375 + 13.0646i) q^{88} -7.96420 q^{89} +(-0.532862 + 0.390503i) q^{90} +(2.46714 - 7.81093i) q^{92} +4.40952i q^{93} +(5.59048 + 7.62851i) q^{94} +0.627737 q^{95} +(-5.65566 + 0.116226i) q^{96} +12.8789 q^{97} +4.87666i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{4} - 2 q^{6} - 6 q^{8} - 8 q^{9}+O(q^{10}) \) 8 * q + 2 * q^4 - 2 * q^6 - 6 * q^8 - 8 * q^9	\( 8 q + 2 q^{4} - 2 q^{6} - 6 q^{8} - 8 q^{9} + 4 q^{10} + 4 q^{12} - 4 q^{15} - 6 q^{16} - 4 q^{17} - 24 q^{20} + 12 q^{23} - 4 q^{24} - 24 q^{25} + 28 q^{26} - 12 q^{30} - 8 q^{31} - 30 q^{32} - 12 q^{33} + 4 q^{34} - 2 q^{36} - 12 q^{38} + 8 q^{39} - 28 q^{40} + 4 q^{41} + 16 q^{44} + 4 q^{46} - 16 q^{48} - 20 q^{50} + 12 q^{52} + 2 q^{54} + 8 q^{55} - 16 q^{57} + 44 q^{58} - 20 q^{60} - 12 q^{62} + 26 q^{64} - 16 q^{65} - 24 q^{66} + 16 q^{68} - 28 q^{71} + 6 q^{72} + 8 q^{73} + 4 q^{74} + 24 q^{76} - 8 q^{78} - 40 q^{79} + 4 q^{80} + 8 q^{81} - 4 q^{82} + 24 q^{86} + 4 q^{88} - 20 q^{89} - 4 q^{90} + 20 q^{92} + 72 q^{94} + 40 q^{95} - 12 q^{96} - 40 q^{97}+O(q^{100}) \) 8 * q + 2 * q^4 - 2 * q^6 - 6 * q^8 - 8 * q^9 + 4 * q^10 + 4 * q^12 - 4 * q^15 - 6 * q^16 - 4 * q^17 - 24 * q^20 + 12 * q^23 - 4 * q^24 - 24 * q^25 + 28 * q^26 - 12 * q^30 - 8 * q^31 - 30 * q^32 - 12 * q^33 + 4 * q^34 - 2 * q^36 - 12 * q^38 + 8 * q^39 - 28 * q^40 + 4 * q^41 + 16 * q^44 + 4 * q^46 - 16 * q^48 - 20 * q^50 + 12 * q^52 + 2 * q^54 + 8 * q^55 - 16 * q^57 + 44 * q^58 - 20 * q^60 - 12 * q^62 + 26 * q^64 - 16 * q^65 - 24 * q^66 + 16 * q^68 - 28 * q^71 + 6 * q^72 + 8 * q^73 + 4 * q^74 + 24 * q^76 - 8 * q^78 - 40 * q^79 + 4 * q^80 + 8 * q^81 - 4 * q^82 + 24 * q^86 + 4 * q^88 - 20 * q^89 - 4 * q^90 + 20 * q^92 + 72 * q^94 + 40 * q^95 - 12 * q^96 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more