LMFDB - Embedded newform 1045.2.b.a.419.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(419,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1045.419");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.b (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:	$8.34436701122$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			419.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1045.419
Dual form			1045.2.b.a.419.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214i q^{2} +2.00000i q^{3} +1.82843 q^{4} +(-1.00000 - 2.00000i) q^{5} -0.828427 q^{6} +0.828427i q^{7} +1.58579i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+0.414214i q^{2} +2.00000i q^{3} +1.82843 q^{4} +(-1.00000 - 2.00000i) q^{5} -0.828427 q^{6} +0.828427i q^{7} +1.58579i q^{8} -1.00000 q^{9} +(0.828427 - 0.414214i) q^{10} -1.00000 q^{11} +3.65685i q^{12} +6.82843i q^{13} -0.343146 q^{14} +(4.00000 - 2.00000i) q^{15} +3.00000 q^{16} -6.82843i q^{17} -0.414214i q^{18} +1.00000 q^{19} +(-1.82843 - 3.65685i) q^{20} -1.65685 q^{21} -0.414214i q^{22} +7.65685i q^{23} -3.17157 q^{24} +(-3.00000 + 4.00000i) q^{25} -2.82843 q^{26} +4.00000i q^{27} +1.51472i q^{28} +4.82843 q^{29} +(0.828427 + 1.65685i) q^{30} -6.82843 q^{31} +4.41421i q^{32} -2.00000i q^{33} +2.82843 q^{34} +(1.65685 - 0.828427i) q^{35} -1.82843 q^{36} +8.48528i q^{37} +0.414214i q^{38} -13.6569 q^{39} +(3.17157 - 1.58579i) q^{40} +6.48528 q^{41} -0.686292i q^{42} -0.828427i q^{43} -1.82843 q^{44} +(1.00000 + 2.00000i) q^{45} -3.17157 q^{46} -11.6569i q^{47} +6.00000i q^{48} +6.31371 q^{49} +(-1.65685 - 1.24264i) q^{50} +13.6569 q^{51} +12.4853i q^{52} +10.8284i q^{53} -1.65685 q^{54} +(1.00000 + 2.00000i) q^{55} -1.31371 q^{56} +2.00000i q^{57} +2.00000i q^{58} +2.82843 q^{59} +(7.31371 - 3.65685i) q^{60} -2.00000 q^{61} -2.82843i q^{62} -0.828427i q^{63} +4.17157 q^{64} +(13.6569 - 6.82843i) q^{65} +0.828427 q^{66} -6.00000i q^{67} -12.4853i q^{68} -15.3137 q^{69} +(0.343146 + 0.686292i) q^{70} -14.8284 q^{71} -1.58579i q^{72} +1.17157i q^{73} -3.51472 q^{74} +(-8.00000 - 6.00000i) q^{75} +1.82843 q^{76} -0.828427i q^{77} -5.65685i q^{78} -5.65685 q^{79} +(-3.00000 - 6.00000i) q^{80} -11.0000 q^{81} +2.68629i q^{82} -6.48528i q^{83} -3.02944 q^{84} +(-13.6569 + 6.82843i) q^{85} +0.343146 q^{86} +9.65685i q^{87} -1.58579i q^{88} +4.34315 q^{89} +(-0.828427 + 0.414214i) q^{90} -5.65685 q^{91} +14.0000i q^{92} -13.6569i q^{93} +4.82843 q^{94} +(-1.00000 - 2.00000i) q^{95} -8.82843 q^{96} -1.17157i q^{97} +2.61522i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{5} + 8 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^5 + 8 * q^6 - 4 * q^9	$ 4 q - 4 q^{4} - 4 q^{5} + 8 q^{6} - 4 q^{9} - 8 q^{10} - 4 q^{11} - 24 q^{14} + 16 q^{15} + 12 q^{16} + 4 q^{19} + 4 q^{20} + 16 q^{21} - 24 q^{24} - 12 q^{25} + 8 q^{29} - 8 q^{30} - 16 q^{31} - 16 q^{35} + 4 q^{36} - 32 q^{39} + 24 q^{40} - 8 q^{41} + 4 q^{44} + 4 q^{45} - 24 q^{46} - 20 q^{49} + 16 q^{50} + 32 q^{51} + 16 q^{54} + 4 q^{55} + 40 q^{56} - 16 q^{60} - 8 q^{61} + 28 q^{64} + 32 q^{65} - 8 q^{66} - 16 q^{69} + 24 q^{70} - 48 q^{71} - 48 q^{74} - 32 q^{75} - 4 q^{76} - 12 q^{80} - 44 q^{81} - 80 q^{84} - 32 q^{85} + 24 q^{86} + 40 q^{89} + 8 q^{90} + 8 q^{94} - 4 q^{95} - 24 q^{96} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^5 + 8 * q^6 - 4 * q^9 - 8 * q^10 - 4 * q^11 - 24 * q^14 + 16 * q^15 + 12 * q^16 + 4 * q^19 + 4 * q^20 + 16 * q^21 - 24 * q^24 - 12 * q^25 + 8 * q^29 - 8 * q^30 - 16 * q^31 - 16 * q^35 + 4 * q^36 - 32 * q^39 + 24 * q^40 - 8 * q^41 + 4 * q^44 + 4 * q^45 - 24 * q^46 - 20 * q^49 + 16 * q^50 + 32 * q^51 + 16 * q^54 + 4 * q^55 + 40 * q^56 - 16 * q^60 - 8 * q^61 + 28 * q^64 + 32 * q^65 - 8 * q^66 - 16 * q^69 + 24 * q^70 - 48 * q^71 - 48 * q^74 - 32 * q^75 - 4 * q^76 - 12 * q^80 - 44 * q^81 - 80 * q^84 - 32 * q^85 + 24 * q^86 + 40 * q^89 + 8 * q^90 + 8 * q^94 - 4 * q^95 - 24 * q^96 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$.

$n$	$496$	$761$	$837$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$			0.414214i			0.292893i	0.989219	+	0.146447i	$0.0467837\pi$
$2$			0.414214i			0.292893i	−0.989219	+	0.146447i	$0.953216\pi$
$3$			2.00000i			1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$			2.00000i			1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	1.82843			0.914214
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$	−0.828427			−0.338204
$7$			0.828427i			0.313116i	0.987669	+	0.156558i	$0.0500398\pi$
$7$			0.828427i			0.313116i	−0.987669	+	0.156558i	$0.949960\pi$
$8$			1.58579i			0.560660i
$9$	−1.00000			−0.333333
$10$	0.828427	−	0.414214i	0.261972	−	0.130986i
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$			3.65685i			1.05564i
$13$			6.82843i			1.89386i	0.321433	+	0.946932i	$0.395836\pi$
$13$			6.82843i			1.89386i	−0.321433	+	0.946932i	$0.604164\pi$
$14$	−0.343146			−0.0917096
$15$	4.00000	−	2.00000i	1.03280	−	0.516398i
$16$	3.00000			0.750000
$17$		−	6.82843i		−	1.65614i	−0.560627	−	0.828068i	$-0.689440\pi$
$17$		−	6.82843i		−	1.65614i	0.560627	−	0.828068i	$-0.310560\pi$
$18$		−	0.414214i		−	0.0976311i
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$	−1.82843	−	3.65685i	−0.408849	−	0.817697i
$21$	−1.65685			−0.361555
$22$		−	0.414214i		−	0.0883106i
$23$			7.65685i			1.59656i	0.602284	+	0.798282i	$0.294258\pi$
$23$			7.65685i			1.59656i	−0.602284	+	0.798282i	$0.705742\pi$
$24$	−3.17157			−0.647395
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$	−2.82843			−0.554700
$27$			4.00000i			0.769800i
$28$			1.51472i			0.286255i
$29$	4.82843			0.896616			0.448308	−	0.893879i	$-0.352027\pi$
$29$	4.82843			0.896616			0.448308	+	0.893879i	$0.352027\pi$
$30$	0.828427	+	1.65685i	0.151249	+	0.302499i
$31$	−6.82843			−1.22642			−0.613211	−	0.789919i	$-0.710122\pi$
$31$	−6.82843			−1.22642			−0.613211	+	0.789919i	$0.710122\pi$
$32$			4.41421i			0.780330i
$33$		−	2.00000i		−	0.348155i
$34$	2.82843			0.485071
$35$	1.65685	−	0.828427i	0.280059	−	0.140030i
$36$	−1.82843			−0.304738
$37$			8.48528i			1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$			8.48528i			1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$			0.414214i			0.0671943i
$39$	−13.6569			−2.18685
$40$	3.17157	−	1.58579i	0.501470	−	0.250735i
$41$	6.48528			1.01283			0.506415	−	0.862290i	$-0.330970\pi$
$41$	6.48528			1.01283			0.506415	+	0.862290i	$0.330970\pi$
$42$		−	0.686292i		−	0.105897i
$43$		−	0.828427i		−	0.126334i	−0.998003	−	0.0631670i	$-0.979880\pi$
$43$		−	0.828427i		−	0.126334i	0.998003	−	0.0631670i	$-0.0201201\pi$
$44$	−1.82843			−0.275646
$45$	1.00000	+	2.00000i	0.149071	+	0.298142i
$46$	−3.17157			−0.467623
$47$		−	11.6569i		−	1.70033i	−0.526519	−	0.850163i	$-0.676503\pi$
$47$		−	11.6569i		−	1.70033i	0.526519	−	0.850163i	$-0.323497\pi$
$48$			6.00000i			0.866025i
$49$	6.31371			0.901958
$50$	−1.65685	−	1.24264i	−0.234315	−	0.175736i
$51$	13.6569			1.91234
$52$			12.4853i			1.73140i
$53$			10.8284i			1.48740i	0.668514	+	0.743699i	$0.266931\pi$
$53$			10.8284i			1.48740i	−0.668514	+	0.743699i	$0.733069\pi$
$54$	−1.65685			−0.225469
$55$	1.00000	+	2.00000i	0.134840	+	0.269680i
$56$	−1.31371			−0.175552
$57$			2.00000i			0.264906i
$58$			2.00000i			0.262613i
$59$	2.82843			0.368230			0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843			0.368230			0.184115	+	0.982905i	$0.441058\pi$
$60$	7.31371	−	3.65685i	0.944196	−	0.472098i
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		−	2.82843i		−	0.359211i
$63$		−	0.828427i		−	0.104372i
$64$	4.17157			0.521447
$65$	13.6569	−	6.82843i	1.69392	−	0.846962i
$66$	0.828427			0.101972
$67$		−	6.00000i		−	0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$		−	6.00000i		−	0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$		−	12.4853i		−	1.51406i
$69$	−15.3137			−1.84355
$70$	0.343146	+	0.686292i	0.0410138	+	0.0820275i
$71$	−14.8284			−1.75981			−0.879905	−	0.475149i	$-0.842394\pi$
$71$	−14.8284			−1.75981			−0.879905	+	0.475149i	$0.842394\pi$
$72$		−	1.58579i		−	0.186887i
$73$			1.17157i			0.137122i	0.997647	+	0.0685611i	$0.0218408\pi$
$73$			1.17157i			0.137122i	−0.997647	+	0.0685611i	$0.978159\pi$
$74$	−3.51472			−0.408578
$75$	−8.00000	−	6.00000i	−0.923760	−	0.692820i
$76$	1.82843			0.209735
$77$		−	0.828427i		−	0.0944080i
$78$		−	5.65685i		−	0.640513i
$79$	−5.65685			−0.636446			−0.318223	−	0.948016i	$-0.603086\pi$
$79$	−5.65685			−0.636446			−0.318223	+	0.948016i	$0.603086\pi$
$80$	−3.00000	−	6.00000i	−0.335410	−	0.670820i
$81$	−11.0000			−1.22222
$82$			2.68629i			0.296651i
$83$		−	6.48528i		−	0.711852i	−0.934514	−	0.355926i	$-0.884165\pi$
$83$		−	6.48528i		−	0.711852i	0.934514	−	0.355926i	$-0.115835\pi$
$84$	−3.02944			−0.330539
$85$	−13.6569	+	6.82843i	−1.48129	+	0.740647i
$86$	0.343146			0.0370024
$87$			9.65685i			1.03532i
$88$		−	1.58579i		−	0.169045i
$89$	4.34315			0.460373			0.230186	−	0.973147i	$-0.426066\pi$
$89$	4.34315			0.460373			0.230186	+	0.973147i	$0.426066\pi$
$90$	−0.828427	+	0.414214i	−0.0873239	+	0.0436619i
$91$	−5.65685			−0.592999
$92$			14.0000i			1.45960i
$93$		−	13.6569i		−	1.41615i
$94$	4.82843			0.498014
$95$	−1.00000	−	2.00000i	−0.102598	−	0.205196i
$96$	−8.82843			−0.901048
$97$		−	1.17157i		−	0.118955i	−0.998230	−	0.0594776i	$-0.981057\pi$
$97$		−	1.17157i		−	0.118955i	0.998230	−	0.0594776i	$-0.0189435\pi$
$98$			2.61522i			0.264177i
$99$	1.00000			0.100504
$100$	−5.48528	+	7.31371i	−0.548528	+	0.731371i
$101$	9.31371			0.926749			0.463374	−	0.886163i	$-0.346639\pi$
$101$	9.31371			0.926749			0.463374	+	0.886163i	$0.346639\pi$
$102$			5.65685i			0.560112i
$103$		−	4.34315i		−	0.427943i	−0.976840	−	0.213971i	$-0.931360\pi$
$103$		−	4.34315i		−	0.427943i	0.976840	−	0.213971i	$-0.0686399\pi$
$104$	−10.8284			−1.06181
$105$	1.65685	+	3.31371i	0.161692	+	0.323385i
$106$	−4.48528			−0.435649
$107$		−	18.9706i		−	1.83395i	−0.398941	−	0.916977i	$-0.630622\pi$
$107$		−	18.9706i		−	1.83395i	0.398941	−	0.916977i	$-0.369378\pi$
$108$			7.31371i			0.703762i
$109$	−0.828427			−0.0793489			−0.0396745	−	0.999213i	$-0.512632\pi$
$109$	−0.828427			−0.0793489			−0.0396745	+	0.999213i	$0.512632\pi$
$110$	−0.828427	+	0.414214i	−0.0789874	+	0.0394937i
$111$	−16.9706			−1.61077
$112$			2.48528i			0.234837i
$113$			10.1421i			0.954092i	0.878878	+	0.477046i	$0.158292\pi$
$113$			10.1421i			0.954092i	−0.878878	+	0.477046i	$0.841708\pi$
$114$	−0.828427			−0.0775893
$115$	15.3137	−	7.65685i	1.42801	−	0.714005i
$116$	8.82843			0.819699
$117$		−	6.82843i		−	0.631288i
$118$			1.17157i			0.107852i
$119$	5.65685			0.518563
$120$	3.17157	+	6.34315i	0.289524	+	0.579047i
$121$	1.00000			0.0909091
$122$		−	0.828427i		−	0.0750023i
$123$			12.9706i			1.16952i
$124$	−12.4853			−1.12121
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$	0.343146			0.0305699
$127$		−	14.0000i		−	1.24230i	−0.783692	−	0.621150i	$-0.786666\pi$
$127$		−	14.0000i		−	1.24230i	0.783692	−	0.621150i	$-0.213334\pi$
$128$			10.5563i			0.933058i
$129$	1.65685			0.145878
$130$	2.82843	+	5.65685i	0.248069	+	0.496139i
$131$	7.31371			0.639002			0.319501	−	0.947586i	$-0.396485\pi$
$131$	7.31371			0.639002			0.319501	+	0.947586i	$0.396485\pi$
$132$		−	3.65685i		−	0.318288i
$133$			0.828427i			0.0718337i
$134$	2.48528			0.214696
$135$	8.00000	−	4.00000i	0.688530	−	0.344265i
$136$	10.8284			0.928530
$137$		−	8.00000i		−	0.683486i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$		−	8.00000i		−	0.683486i	0.939793	−	0.341743i	$-0.111017\pi$
$138$		−	6.34315i		−	0.539964i
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$	3.02944	−	1.51472i	0.256034	−	0.128017i
$141$	23.3137			1.96337
$142$		−	6.14214i		−	0.515437i
$143$		−	6.82843i		−	0.571022i
$144$	−3.00000			−0.250000
$145$	−4.82843	−	9.65685i	−0.400979	−	0.801958i
$146$	−0.485281			−0.0401622
$147$			12.6274i			1.04149i
$148$			15.5147i			1.27530i
$149$	8.34315			0.683497			0.341749	−	0.939791i	$-0.388981\pi$
$149$	8.34315			0.683497			0.341749	+	0.939791i	$0.388981\pi$
$150$	2.48528	−	3.31371i	0.202922	−	0.270563i
$151$	10.3431			0.841713			0.420857	−	0.907127i	$-0.361730\pi$
$151$	10.3431			0.841713			0.420857	+	0.907127i	$0.361730\pi$
$152$			1.58579i			0.128624i
$153$			6.82843i			0.552046i
$154$	0.343146			0.0276515
$155$	6.82843	+	13.6569i	0.548472	+	1.09694i
$156$	−24.9706			−1.99925
$157$		−	12.0000i		−	0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$		−	12.0000i		−	0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$		−	2.34315i		−	0.186411i
$159$	−21.6569			−1.71750
$160$	8.82843	−	4.41421i	0.697948	−	0.348974i
$161$	−6.34315			−0.499910
$162$		−	4.55635i		−	0.357981i
$163$		−	2.00000i		−	0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$		−	2.00000i		−	0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	11.8579			0.925944
$165$	−4.00000	+	2.00000i	−0.311400	+	0.155700i
$166$	2.68629			0.208497
$167$		−	8.34315i		−	0.645612i	−0.946465	−	0.322806i	$-0.895374\pi$
$167$		−	8.34315i		−	0.645612i	0.946465	−	0.322806i	$-0.104626\pi$
$168$		−	2.62742i		−	0.202710i
$169$	−33.6274			−2.58672
$170$	−2.82843	−	5.65685i	−0.216930	−	0.433861i
$171$	−1.00000			−0.0764719
$172$		−	1.51472i		−	0.115496i
$173$			12.4853i			0.949238i	0.880191	+	0.474619i	$0.157414\pi$
$173$			12.4853i			0.949238i	−0.880191	+	0.474619i	$0.842586\pi$
$174$	−4.00000			−0.303239
$175$	−3.31371	−	2.48528i	−0.250493	−	0.187870i
$176$	−3.00000			−0.226134
$177$			5.65685i			0.425195i
$178$			1.79899i			0.134840i
$179$	−16.4853			−1.23217			−0.616084	−	0.787681i	$-0.711282\pi$
$179$	−16.4853			−1.23217			−0.616084	+	0.787681i	$0.711282\pi$
$180$	1.82843	+	3.65685i	0.136283	+	0.272566i
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		−	2.34315i		−	0.173686i
$183$		−	4.00000i		−	0.295689i
$184$	−12.1421			−0.895130
$185$	16.9706	−	8.48528i	1.24770	−	0.623850i
$186$	5.65685			0.414781
$187$			6.82843i			0.499344i
$188$		−	21.3137i		−	1.55446i
$189$	−3.31371			−0.241037
$190$	0.828427	−	0.414214i	0.0601004	−	0.0300502i
$191$	16.9706			1.22795			0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706			1.22795			0.613973	+	0.789327i	$0.289570\pi$
$192$			8.34315i			0.602115i
$193$		−	14.1421i		−	1.01797i	−0.860774	−	0.508987i	$-0.830020\pi$
$193$		−	14.1421i		−	1.01797i	0.860774	−	0.508987i	$-0.169980\pi$
$194$	0.485281			0.0348412
$195$	13.6569	+	27.3137i	0.977988	+	1.95598i
$196$	11.5442			0.824583
$197$			6.14214i			0.437609i	0.975769	+	0.218805i	$0.0702157\pi$
$197$			6.14214i			0.437609i	−0.975769	+	0.218805i	$0.929784\pi$
$198$			0.414214i			0.0294369i
$199$	5.65685			0.401004			0.200502	−	0.979693i	$-0.435743\pi$
$199$	5.65685			0.401004			0.200502	+	0.979693i	$0.435743\pi$
$200$	−6.34315	−	4.75736i	−0.448528	−	0.336396i
$201$	12.0000			0.846415
$202$			3.85786i			0.271438i
$203$			4.00000i			0.280745i
$204$	24.9706			1.74829
$205$	−6.48528	−	12.9706i	−0.452952	−	0.905903i
$206$	1.79899			0.125342
$207$		−	7.65685i		−	0.532188i
$208$			20.4853i			1.42040i
$209$	−1.00000			−0.0691714
$210$	−1.37258	+	0.686292i	−0.0947172	+	0.0473586i
$211$	15.3137			1.05424			0.527120	−	0.849791i	$-0.323272\pi$
$211$	15.3137			1.05424			0.527120	+	0.849791i	$0.323272\pi$
$212$			19.7990i			1.35980i
$213$		−	29.6569i		−	2.03205i
$214$	7.85786			0.537153
$215$	−1.65685	+	0.828427i	−0.112997	+	0.0564983i
$216$	−6.34315			−0.431596
$217$		−	5.65685i		−	0.384012i
$218$		−	0.343146i		−	0.0232408i
$219$	−2.34315			−0.158335
$220$	1.82843	+	3.65685i	0.123273	+	0.246545i
$221$	46.6274			3.13650
$222$		−	7.02944i		−	0.471785i
$223$			11.6569i			0.780601i	0.920688	+	0.390300i	$0.127629\pi$
$223$			11.6569i			0.780601i	−0.920688	+	0.390300i	$0.872371\pi$
$224$	−3.65685			−0.244334
$225$	3.00000	−	4.00000i	0.200000	−	0.266667i
$226$	−4.20101			−0.279447
$227$		−	12.3431i		−	0.819243i	−0.912255	−	0.409622i	$-0.865661\pi$
$227$		−	12.3431i		−	0.819243i	0.912255	−	0.409622i	$-0.134339\pi$
$228$			3.65685i			0.242181i
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$	3.17157	+	6.34315i	0.209127	+	0.418255i
$231$	1.65685			0.109013
$232$			7.65685i			0.502697i
$233$		−	4.48528i		−	0.293841i	−0.989148	−	0.146920i	$-0.953064\pi$
$233$		−	4.48528i		−	0.293841i	0.989148	−	0.146920i	$-0.0469361\pi$
$234$	2.82843			0.184900
$235$	−23.3137	+	11.6569i	−1.52082	+	0.760409i
$236$	5.17157			0.336641
$237$		−	11.3137i		−	0.734904i
$238$			2.34315i			0.151884i
$239$	−4.68629			−0.303131			−0.151565	−	0.988447i	$-0.548431\pi$
$239$	−4.68629			−0.303131			−0.151565	+	0.988447i	$0.548431\pi$
$240$	12.0000	−	6.00000i	0.774597	−	0.387298i
$241$	6.48528			0.417754			0.208877	−	0.977942i	$-0.433019\pi$
$241$	6.48528			0.417754			0.208877	+	0.977942i	$0.433019\pi$
$242$			0.414214i			0.0266267i
$243$		−	10.0000i		−	0.641500i
$244$	−3.65685			−0.234106
$245$	−6.31371	−	12.6274i	−0.403368	−	0.806736i
$246$	−5.37258			−0.342543
$247$			6.82843i			0.434482i
$248$		−	10.8284i		−	0.687606i
$249$	12.9706			0.821976
$250$	−0.828427	+	4.55635i	−0.0523943	+	0.288169i
$251$	−4.00000			−0.252478			−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000			−0.252478			−0.126239	+	0.992000i	$0.540291\pi$
$252$		−	1.51472i		−	0.0954183i
$253$		−	7.65685i		−	0.481382i
$254$	5.79899			0.363861
$255$	−13.6569	−	27.3137i	−0.855225	−	1.71045i
$256$	3.97056			0.248160
$257$			18.1421i			1.13168i	0.824517	+	0.565838i	$0.191447\pi$
$257$			18.1421i			1.13168i	−0.824517	+	0.565838i	$0.808553\pi$
$258$			0.686292i			0.0427266i
$259$	−7.02944			−0.436788
$260$	24.9706	−	12.4853i	1.54861	−	0.774304i
$261$	−4.82843			−0.298872
$262$			3.02944i			0.187159i
$263$		−	18.4853i		−	1.13985i	−0.821696	−	0.569926i	$-0.806972\pi$
$263$		−	18.4853i		−	1.13985i	0.821696	−	0.569926i	$-0.193028\pi$
$264$	3.17157			0.195197
$265$	21.6569	−	10.8284i	1.33037	−	0.665185i
$266$	−0.343146			−0.0210396
$267$			8.68629i			0.531592i
$268$		−	10.9706i		−	0.670134i
$269$	22.9706			1.40054			0.700270	−	0.713878i	$-0.253063\pi$
$269$	22.9706			1.40054			0.700270	+	0.713878i	$0.253063\pi$
$270$	1.65685	+	3.31371i	0.100833	+	0.201666i
$271$	11.3137			0.687259			0.343629	−	0.939105i	$-0.388344\pi$
$271$	11.3137			0.687259			0.343629	+	0.939105i	$0.388344\pi$
$272$		−	20.4853i		−	1.24210i
$273$		−	11.3137i		−	0.684737i
$274$	3.31371			0.200188
$275$	3.00000	−	4.00000i	0.180907	−	0.241209i
$276$	−28.0000			−1.68540
$277$			18.8284i			1.13129i	0.824649	+	0.565645i	$0.191373\pi$
$277$			18.8284i			1.13129i	−0.824649	+	0.565645i	$0.808627\pi$
$278$		−	1.65685i		−	0.0993715i
$279$	6.82843			0.408807
$280$	1.31371	+	2.62742i	0.0785091	+	0.157018i
$281$	−19.4558			−1.16064			−0.580319	−	0.814389i	$-0.697072\pi$
$281$	−19.4558			−1.16064			−0.580319	+	0.814389i	$0.697072\pi$
$282$			9.65685i			0.575057i
$283$		−	6.48528i		−	0.385510i	−0.981247	−	0.192755i	$-0.938258\pi$
$283$		−	6.48528i		−	0.385510i	0.981247	−	0.192755i	$-0.0617423\pi$
$284$	−27.1127			−1.60884
$285$	4.00000	−	2.00000i	0.236940	−	0.118470i
$286$	2.82843			0.167248
$287$			5.37258i			0.317134i
$288$		−	4.41421i		−	0.260110i
$289$	−29.6274			−1.74279
$290$	4.00000	−	2.00000i	0.234888	−	0.117444i
$291$	2.34315			0.137358
$292$			2.14214i			0.125359i
$293$		−	26.1421i		−	1.52724i	−0.645666	−	0.763620i	$-0.723420\pi$
$293$		−	26.1421i		−	1.52724i	0.645666	−	0.763620i	$-0.276580\pi$
$294$	−5.23045			−0.305046
$295$	−2.82843	−	5.65685i	−0.164677	−	0.329355i
$296$	−13.4558			−0.782105
$297$		−	4.00000i		−	0.232104i
$298$			3.45584i			0.200192i
$299$	−52.2843			−3.02368
$300$	−14.6274	−	10.9706i	−0.844514	−	0.633386i
$301$	0.686292			0.0395572
$302$			4.28427i			0.246532i
$303$			18.6274i			1.07012i
$304$	3.00000			0.172062
$305$	2.00000	+	4.00000i	0.114520	+	0.229039i
$306$	−2.82843			−0.161690
$307$		−	26.9706i		−	1.53929i	−0.638471	−	0.769646i	$-0.720433\pi$
$307$		−	26.9706i		−	1.53929i	0.638471	−	0.769646i	$-0.279567\pi$
$308$		−	1.51472i		−	0.0863091i
$309$	8.68629			0.494146
$310$	−5.65685	+	2.82843i	−0.321288	+	0.160644i
$311$	0.970563			0.0550356			0.0275178	−	0.999621i	$-0.491240\pi$
$311$	0.970563			0.0550356			0.0275178	+	0.999621i	$0.491240\pi$
$312$		−	21.6569i		−	1.22608i
$313$			16.9706i			0.959233i	0.877478	+	0.479616i	$0.159224\pi$
$313$			16.9706i			0.959233i	−0.877478	+	0.479616i	$0.840776\pi$
$314$	4.97056			0.280505
$315$	−1.65685	+	0.828427i	−0.0933532	+	0.0466766i
$316$	−10.3431			−0.581847
$317$			6.14214i			0.344977i	0.985012	+	0.172488i	$0.0551807\pi$
$317$			6.14214i			0.344977i	−0.985012	+	0.172488i	$0.944819\pi$
$318$		−	8.97056i		−	0.503044i
$319$	−4.82843			−0.270340
$320$	−4.17157	−	8.34315i	−0.233198	−	0.466396i
$321$	37.9411			2.11767
$322$		−	2.62742i		−	0.146420i
$323$		−	6.82843i		−	0.379944i
$324$	−20.1127			−1.11737
$325$	−27.3137	−	20.4853i	−1.51509	−	1.13632i
$326$	0.828427			0.0458823
$327$		−	1.65685i		−	0.0916242i
$328$			10.2843i			0.567854i
$329$	9.65685			0.532400
$330$	−0.828427	−	1.65685i	−0.0456034	−	0.0912068i
$331$	−6.14214			−0.337602			−0.168801	−	0.985650i	$-0.553990\pi$
$331$	−6.14214			−0.337602			−0.168801	+	0.985650i	$0.553990\pi$
$332$		−	11.8579i		−	0.650785i
$333$		−	8.48528i		−	0.464991i
$334$	3.45584			0.189095
$335$	−12.0000	+	6.00000i	−0.655630	+	0.327815i
$336$	−4.97056			−0.271166
$337$		−	2.82843i		−	0.154074i	−0.997028	−	0.0770371i	$-0.975454\pi$
$337$		−	2.82843i		−	0.154074i	0.997028	−	0.0770371i	$-0.0245460\pi$
$338$		−	13.9289i		−	0.757634i
$339$	−20.2843			−1.10169
$340$	−24.9706	+	12.4853i	−1.35422	+	0.677109i
$341$	6.82843			0.369780
$342$		−	0.414214i		−	0.0223981i
$343$			11.0294i			0.595534i
$344$	1.31371			0.0708304
$345$	15.3137	+	30.6274i	0.824462	+	1.64892i
$346$	−5.17157			−0.278025
$347$		−	24.8284i		−	1.33286i	−0.745568	−	0.666430i	$-0.767822\pi$
$347$		−	24.8284i		−	1.33286i	0.745568	−	0.666430i	$-0.232178\pi$
$348$			17.6569i			0.946507i
$349$	27.6569			1.48044			0.740219	−	0.672366i	$-0.234722\pi$
$349$	27.6569			1.48044			0.740219	+	0.672366i	$0.234722\pi$
$350$	1.02944	−	1.37258i	0.0550257	−	0.0733676i
$351$	−27.3137			−1.45790
$352$		−	4.41421i		−	0.235278i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$	−2.34315			−0.124537
$355$	14.8284	+	29.6569i	0.787011	+	1.57402i
$356$	7.94113			0.420879
$357$			11.3137i			0.598785i
$358$		−	6.82843i		−	0.360894i
$359$	−16.9706			−0.895672			−0.447836	−	0.894116i	$-0.647805\pi$
$359$	−16.9706			−0.895672			−0.447836	+	0.894116i	$0.647805\pi$
$360$	−3.17157	+	1.58579i	−0.167157	+	0.0835783i
$361$	1.00000			0.0526316
$362$			5.79899i			0.304788i
$363$			2.00000i			0.104973i
$364$	−10.3431			−0.542128
$365$	2.34315	−	1.17157i	0.122646	−	0.0613229i
$366$	1.65685			0.0866052
$367$			20.3431i			1.06190i	0.847402	+	0.530952i	$0.178165\pi$
$367$			20.3431i			1.06190i	−0.847402	+	0.530952i	$0.821835\pi$
$368$			22.9706i			1.19742i
$369$	−6.48528			−0.337610
$370$	3.51472	+	7.02944i	0.182722	+	0.365443i
$371$	−8.97056			−0.465728
$372$		−	24.9706i		−	1.29466i
$373$		−	17.1716i		−	0.889110i	−0.895751	−	0.444555i	$-0.853362\pi$
$373$		−	17.1716i		−	0.889110i	0.895751	−	0.444555i	$-0.146638\pi$
$374$	−2.82843			−0.146254
$375$	−4.00000	+	22.0000i	−0.206559	+	1.13608i
$376$	18.4853			0.953306
$377$			32.9706i			1.69807i
$378$		−	1.37258i		−	0.0705981i
$379$	−10.8284			−0.556219			−0.278109	−	0.960549i	$-0.589708\pi$
$379$	−10.8284			−0.556219			−0.278109	+	0.960549i	$0.589708\pi$
$380$	−1.82843	−	3.65685i	−0.0937963	−	0.187593i
$381$	28.0000			1.43448
$382$			7.02944i			0.359657i
$383$		−	26.9706i		−	1.37813i	−0.724699	−	0.689066i	$-0.758021\pi$
$383$		−	26.9706i		−	1.37813i	0.724699	−	0.689066i	$-0.241979\pi$
$384$	−21.1127			−1.07740
$385$	−1.65685	+	0.828427i	−0.0844411	+	0.0422206i
$386$	5.85786			0.298157
$387$			0.828427i			0.0421113i
$388$		−	2.14214i		−	0.108750i
$389$	−13.3137			−0.675032			−0.337516	−	0.941320i	$-0.609587\pi$
$389$	−13.3137			−0.675032			−0.337516	+	0.941320i	$0.609587\pi$
$390$	−11.3137	+	5.65685i	−0.572892	+	0.286446i
$391$	52.2843			2.64413
$392$			10.0122i			0.505692i
$393$			14.6274i			0.737856i
$394$	−2.54416			−0.128173
$395$	5.65685	+	11.3137i	0.284627	+	0.569254i
$396$	1.82843			0.0918819
$397$		−	7.31371i		−	0.367065i	−0.983014	−	0.183532i	$-0.941247\pi$
$397$		−	7.31371i		−	0.367065i	0.983014	−	0.183532i	$-0.0587532\pi$
$398$			2.34315i			0.117451i
$399$	−1.65685			−0.0829465
$400$	−9.00000	+	12.0000i	−0.450000	+	0.600000i
$401$	5.31371			0.265354			0.132677	−	0.991159i	$-0.457643\pi$
$401$	5.31371			0.265354			0.132677	+	0.991159i	$0.457643\pi$
$402$			4.97056i			0.247909i
$403$		−	46.6274i		−	2.32268i
$404$	17.0294			0.847246
$405$	11.0000	+	22.0000i	0.546594	+	1.09319i
$406$	−1.65685			−0.0822283
$407$		−	8.48528i		−	0.420600i
$408$			21.6569i			1.07217i
$409$	24.8284			1.22769			0.613843	−	0.789428i	$-0.289623\pi$
$409$	24.8284			1.22769			0.613843	+	0.789428i	$0.289623\pi$
$410$	5.37258	−	2.68629i	0.265333	−	0.132666i
$411$	16.0000			0.789222
$412$		−	7.94113i		−	0.391231i
$413$			2.34315i			0.115299i
$414$	3.17157			0.155874
$415$	−12.9706	+	6.48528i	−0.636700	+	0.318350i
$416$	−30.1421			−1.47784
$417$		−	8.00000i		−	0.391762i
$418$		−	0.414214i		−	0.0202598i
$419$	20.9706			1.02448			0.512240	−	0.858843i	$-0.328816\pi$
$419$	20.9706			1.02448			0.512240	+	0.858843i	$0.328816\pi$
$420$	3.02944	+	6.05887i	0.147821	+	0.295643i
$421$	−26.9706			−1.31446			−0.657232	−	0.753688i	$-0.728273\pi$
$421$	−26.9706			−1.31446			−0.657232	+	0.753688i	$0.728273\pi$
$422$			6.34315i			0.308780i
$423$			11.6569i			0.566776i
$424$	−17.1716			−0.833925
$425$	27.3137	+	20.4853i	1.32491	+	0.993682i
$426$	12.2843			0.595175
$427$		−	1.65685i		−	0.0801808i
$428$		−	34.6863i		−	1.67663i
$429$	13.6569			0.659359
$430$	−0.343146	−	0.686292i	−0.0165480	−	0.0330959i
$431$	6.62742			0.319231			0.159616	−	0.987179i	$-0.448974\pi$
$431$	6.62742			0.319231			0.159616	+	0.987179i	$0.448974\pi$
$432$			12.0000i			0.577350i
$433$			1.17157i			0.0563022i	0.999604	+	0.0281511i	$0.00896196\pi$
$433$			1.17157i			0.0563022i	−0.999604	+	0.0281511i	$0.991038\pi$
$434$	2.34315			0.112475
$435$	19.3137	−	9.65685i	0.926021	−	0.463011i
$436$	−1.51472			−0.0725419
$437$			7.65685i			0.366277i
$438$		−	0.970563i		−	0.0463753i
$439$	−18.3431			−0.875471			−0.437735	−	0.899104i	$-0.644219\pi$
$439$	−18.3431			−0.875471			−0.437735	+	0.899104i	$0.644219\pi$
$440$	−3.17157	+	1.58579i	−0.151199	+	0.0755994i
$441$	−6.31371			−0.300653
$442$			19.3137i			0.918659i
$443$			9.31371i			0.442508i	0.975216	+	0.221254i	$0.0710149\pi$
$443$			9.31371i			0.442508i	−0.975216	+	0.221254i	$0.928985\pi$
$444$	−31.0294			−1.47259
$445$	−4.34315	−	8.68629i	−0.205885	−	0.411770i
$446$	−4.82843			−0.228633
$447$			16.6863i			0.789235i
$448$			3.45584i			0.163273i
$449$	2.97056			0.140190			0.0700948	−	0.997540i	$-0.477670\pi$
$449$	2.97056			0.140190			0.0700948	+	0.997540i	$0.477670\pi$
$450$	1.65685	+	1.24264i	0.0781049	+	0.0585786i
$451$	−6.48528			−0.305380
$452$			18.5442i			0.872244i
$453$			20.6863i			0.971927i
$454$	5.11270			0.239951
$455$	5.65685	+	11.3137i	0.265197	+	0.530395i
$456$	−3.17157			−0.148523
$457$			15.7990i			0.739046i	0.929222	+	0.369523i	$0.120479\pi$
$457$			15.7990i			0.739046i	−0.929222	+	0.369523i	$0.879521\pi$
$458$			9.11270i			0.425808i
$459$	27.3137			1.27489
$460$	28.0000	−	14.0000i	1.30551	−	0.652753i
$461$	−22.2843			−1.03788			−0.518941	−	0.854810i	$-0.673674\pi$
$461$	−22.2843			−1.03788			−0.518941	+	0.854810i	$0.673674\pi$
$462$			0.686292i			0.0319292i
$463$			30.2843i			1.40743i	0.710483	+	0.703715i	$0.248477\pi$
$463$			30.2843i			1.40743i	−0.710483	+	0.703715i	$0.751523\pi$
$464$	14.4853			0.672462
$465$	−27.3137	+	13.6569i	−1.26664	+	0.633321i
$466$	1.85786			0.0860639
$467$		−	4.34315i		−	0.200977i	−0.994938	−	0.100488i	$-0.967959\pi$
$467$		−	4.34315i		−	0.200977i	0.994938	−	0.100488i	$-0.0320405\pi$
$468$		−	12.4853i		−	0.577132i
$469$	4.97056			0.229519
$470$	−4.82843	−	9.65685i	−0.222719	−	0.445437i
$471$	24.0000			1.10586
$472$			4.48528i			0.206452i
$473$			0.828427i			0.0380911i
$474$	4.68629			0.215248
$475$	−3.00000	+	4.00000i	−0.137649	+	0.183533i
$476$	10.3431			0.474077
$477$		−	10.8284i		−	0.495800i
$478$		−	1.94113i		−	0.0887850i
$479$	3.31371			0.151407			0.0757036	−	0.997130i	$-0.475880\pi$
$479$	3.31371			0.151407			0.0757036	+	0.997130i	$0.475880\pi$
$480$	8.82843	+	17.6569i	0.402961	+	0.805921i
$481$	−57.9411			−2.64189
$482$			2.68629i			0.122357i
$483$		−	12.6863i		−	0.577246i
$484$	1.82843			0.0831103
$485$	−2.34315	+	1.17157i	−0.106397	+	0.0531984i
$486$	4.14214			0.187891
$487$		−	15.6569i		−	0.709480i	−0.934965	−	0.354740i	$-0.884569\pi$
$487$		−	15.6569i		−	0.709480i	0.934965	−	0.354740i	$-0.115431\pi$
$488$		−	3.17157i		−	0.143570i
$489$	4.00000			0.180886
$490$	5.23045	−	2.61522i	0.236288	−	0.118144i
$491$	−36.9706			−1.66846			−0.834229	−	0.551418i	$-0.814087\pi$
$491$	−36.9706			−1.66846			−0.834229	+	0.551418i	$0.814087\pi$
$492$			23.7157i			1.06919i
$493$		−	32.9706i		−	1.48492i
$494$	−2.82843			−0.127257
$495$	−1.00000	−	2.00000i	−0.0449467	−	0.0898933i
$496$	−20.4853			−0.919816
$497$		−	12.2843i		−	0.551025i
$498$			5.37258i			0.240751i
$499$	−0.686292			−0.0307226			−0.0153613	−	0.999882i	$-0.504890\pi$
$499$	−0.686292			−0.0307226			−0.0153613	+	0.999882i	$0.504890\pi$
$500$	20.1127	+	3.65685i	0.899467	+	0.163539i
$501$	16.6863			0.745489
$502$		−	1.65685i		−	0.0739490i
$503$			41.7990i			1.86372i	0.362812	+	0.931862i	$0.381817\pi$
$503$			41.7990i			1.86372i	−0.362812	+	0.931862i	$0.618183\pi$
$504$	1.31371			0.0585172
$505$	−9.31371	−	18.6274i	−0.414455	−	0.828909i
$506$	3.17157			0.140994
$507$		−	67.2548i		−	2.98689i
$508$		−	25.5980i		−	1.13573i
$509$	−31.6569			−1.40317			−0.701583	−	0.712588i	$-0.747523\pi$
$509$	−31.6569			−1.40317			−0.701583	+	0.712588i	$0.747523\pi$
$510$	11.3137	−	5.65685i	0.500979	−	0.250490i
$511$	−0.970563			−0.0429352
$512$			22.7574i			1.00574i
$513$			4.00000i			0.176604i
$514$	−7.51472			−0.331460
$515$	−8.68629	+	4.34315i	−0.382764	+	0.191382i
$516$	3.02944			0.133364
$517$			11.6569i			0.512668i
$518$		−	2.91169i		−	0.127932i
$519$	−24.9706			−1.09609
$520$	10.8284	+	21.6569i	0.474858	+	0.949716i
$521$	−14.0000			−0.613351			−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000			−0.613351			−0.306676	+	0.951814i	$0.599217\pi$
$522$		−	2.00000i		−	0.0875376i
$523$			6.97056i			0.304801i	0.988319	+	0.152401i	$0.0487004\pi$
$523$			6.97056i			0.304801i	−0.988319	+	0.152401i	$0.951300\pi$
$524$	13.3726			0.584184
$525$	4.97056	−	6.62742i	0.216933	−	0.289244i
$526$	7.65685			0.333855
$527$			46.6274i			2.03112i
$528$		−	6.00000i		−	0.261116i
$529$	−35.6274			−1.54902
$530$	4.48528	+	8.97056i	0.194828	+	0.389656i
$531$	−2.82843			−0.122743
$532$			1.51472i			0.0656714i
$533$			44.2843i			1.91816i
$534$	−3.59798			−0.155700
$535$	−37.9411	+	18.9706i	−1.64034	+	0.820169i
$536$	9.51472			0.410973
$537$		−	32.9706i		−	1.42278i
$538$			9.51472i			0.410209i
$539$	−6.31371			−0.271951
$540$	14.6274	−	7.31371i	0.629464	−	0.314732i
$541$	−12.3431			−0.530673			−0.265337	−	0.964156i	$-0.585483\pi$
$541$	−12.3431			−0.530673			−0.265337	+	0.964156i	$0.585483\pi$
$542$			4.68629i			0.201293i
$543$			28.0000i			1.20160i
$544$	30.1421			1.29233
$545$	0.828427	+	1.65685i	0.0354859	+	0.0709718i
$546$	4.68629			0.200555
$547$			34.2843i			1.46589i	0.680288	+	0.732945i	$0.261855\pi$
$547$			34.2843i			1.46589i	−0.680288	+	0.732945i	$0.738145\pi$
$548$		−	14.6274i		−	0.624852i
$549$	2.00000			0.0853579
$550$	1.65685	+	1.24264i	0.0706485	+	0.0529864i
$551$	4.82843			0.205698
$552$		−	24.2843i		−	1.03361i
$553$		−	4.68629i		−	0.199281i
$554$	−7.79899			−0.331347
$555$	16.9706	+	33.9411i	0.720360	+	1.44072i
$556$	−7.31371			−0.310170
$557$		−	23.1127i		−	0.979316i	−0.871914	−	0.489658i	$-0.837122\pi$
$557$		−	23.1127i		−	0.979316i	0.871914	−	0.489658i	$-0.162878\pi$
$558$			2.82843i			0.119737i
$559$	5.65685			0.239259
$560$	4.97056	−	2.48528i	0.210045	−	0.105022i
$561$	−13.6569			−0.576593
$562$		−	8.05887i		−	0.339943i
$563$		−	18.9706i		−	0.799514i	−0.916621	−	0.399757i	$-0.869095\pi$
$563$		−	18.9706i		−	0.799514i	0.916621	−	0.399757i	$-0.130905\pi$
$564$	42.6274			1.79494
$565$	20.2843	−	10.1421i	0.853366	−	0.426683i
$566$	2.68629			0.112913
$567$		−	9.11270i		−	0.382697i
$568$		−	23.5147i		−	0.986656i
$569$	32.8284			1.37624			0.688120	−	0.725597i	$-0.258436\pi$
$569$	32.8284			1.37624			0.688120	+	0.725597i	$0.258436\pi$
$570$	0.828427	+	1.65685i	0.0346990	+	0.0693980i
$571$	−24.2843			−1.01627			−0.508133	−	0.861279i	$-0.669664\pi$
$571$	−24.2843			−1.01627			−0.508133	+	0.861279i	$0.669664\pi$
$572$		−	12.4853i		−	0.522036i
$573$			33.9411i			1.41791i
$574$	−2.22540			−0.0928863
$575$	−30.6274	−	22.9706i	−1.27725	−	0.957939i
$576$	−4.17157			−0.173816
$577$		−	32.0000i		−	1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$		−	32.0000i		−	1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$		−	12.2721i		−	0.510451i
$579$	28.2843			1.17545
$580$	−8.82843	−	17.6569i	−0.366580	−	0.733161i
$581$	5.37258			0.222892
$582$			0.970563i			0.0402311i
$583$		−	10.8284i		−	0.448468i
$584$	−1.85786			−0.0768790
$585$	−13.6569	+	6.82843i	−0.564641	+	0.282321i
$586$	10.8284			0.447318
$587$		−	22.2843i		−	0.919770i	−0.887978	−	0.459885i	$-0.847891\pi$
$587$		−	22.2843i		−	0.919770i	0.887978	−	0.459885i	$-0.152109\pi$
$588$			23.0883i			0.952146i
$589$	−6.82843			−0.281360
$590$	2.34315	−	1.17157i	0.0964658	−	0.0482329i
$591$	−12.2843			−0.505307
$592$			25.4558i			1.04623i
$593$		−	5.45584i		−	0.224045i	−0.993706	−	0.112022i	$-0.964267\pi$
$593$		−	5.45584i		−	0.224045i	0.993706	−	0.112022i	$-0.0357328\pi$
$594$	1.65685			0.0679816
$595$	−5.65685	−	11.3137i	−0.231908	−	0.463817i
$596$	15.2548			0.624862
$597$			11.3137i			0.463039i
$598$		−	21.6569i		−	0.885615i
$599$	−30.8284			−1.25962			−0.629808	−	0.776751i	$-0.716866\pi$
$599$	−30.8284			−1.25962			−0.629808	+	0.776751i	$0.716866\pi$
$600$	9.51472	−	12.6863i	0.388437	−	0.517916i
$601$	−25.1127			−1.02437			−0.512184	−	0.858876i	$-0.671163\pi$
$601$	−25.1127			−1.02437			−0.512184	+	0.858876i	$0.671163\pi$
$602$			0.284271i			0.0115860i
$603$			6.00000i			0.244339i
$604$	18.9117			0.769506
$605$	−1.00000	−	2.00000i	−0.0406558	−	0.0813116i
$606$	−7.71573			−0.313430
$607$			28.3431i			1.15041i	0.818008	+	0.575206i	$0.195078\pi$
$607$			28.3431i			1.15041i	−0.818008	+	0.575206i	$0.804922\pi$
$608$			4.41421i			0.179020i
$609$	−8.00000			−0.324176
$610$	−1.65685	+	0.828427i	−0.0670841	+	0.0335420i
$611$	79.5980			3.22019
$612$			12.4853i			0.504688i
$613$		−	13.1716i		−	0.531995i	−0.963974	−	0.265997i	$-0.914299\pi$
$613$		−	13.1716i		−	0.531995i	0.963974	−	0.265997i	$-0.0857013\pi$
$614$	11.1716			0.450848
$615$	25.9411	−	12.9706i	1.04605	−	0.523024i
$616$	1.31371			0.0529308
$617$		−	21.6569i		−	0.871872i	−0.899978	−	0.435936i	$-0.856417\pi$
$617$		−	21.6569i		−	0.871872i	0.899978	−	0.435936i	$-0.143583\pi$
$618$			3.59798i			0.144732i
$619$	26.6274			1.07025			0.535123	−	0.844774i	$-0.320265\pi$
$619$	26.6274			1.07025			0.535123	+	0.844774i	$0.320265\pi$
$620$	12.4853	+	24.9706i	0.501421	+	1.00284i
$621$	−30.6274			−1.22904
$622$			0.402020i			0.0161195i
$623$			3.59798i			0.144150i
$624$	−40.9706			−1.64014
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−7.02944			−0.280953
$627$		−	2.00000i		−	0.0798723i
$628$		−	21.9411i		−	0.875546i
$629$	57.9411			2.31026
$630$	−0.343146	−	0.686292i	−0.0136713	−	0.0273425i
$631$	32.9706			1.31254			0.656269	−	0.754527i	$-0.272134\pi$
$631$	32.9706			1.31254			0.656269	+	0.754527i	$0.272134\pi$
$632$		−	8.97056i		−	0.356830i
$633$			30.6274i			1.21733i
$634$	−2.54416			−0.101041
$635$	−28.0000	+	14.0000i	−1.11115	+	0.555573i
$636$	−39.5980			−1.57016
$637$			43.1127i			1.70819i
$638$		−	2.00000i		−	0.0791808i
$639$	14.8284			0.586604
$640$	21.1127	−	10.5563i	0.834553	−	0.417276i
$641$	7.65685			0.302428			0.151214	−	0.988501i	$-0.451682\pi$
$641$	7.65685			0.302428			0.151214	+	0.988501i	$0.451682\pi$
$642$			15.7157i			0.620250i
$643$			47.9411i			1.89061i	0.326183	+	0.945307i	$0.394238\pi$
$643$			47.9411i			1.89061i	−0.326183	+	0.945307i	$0.605762\pi$
$644$	−11.5980			−0.457024
$645$	−1.65685	−	3.31371i	−0.0652386	−	0.130477i
$646$	2.82843			0.111283
$647$		−	2.68629i		−	0.105609i	−0.998605	−	0.0528045i	$-0.983184\pi$
$647$		−	2.68629i		−	0.105609i	0.998605	−	0.0528045i	$-0.0168160\pi$
$648$		−	17.4437i		−	0.685251i
$649$	−2.82843			−0.111025
$650$	8.48528	−	11.3137i	0.332820	−	0.443760i
$651$	11.3137			0.443419
$652$		−	3.65685i		−	0.143213i
$653$		−	34.6274i		−	1.35508i	−0.735488	−	0.677538i	$-0.763047\pi$
$653$		−	34.6274i		−	1.35508i	0.735488	−	0.677538i	$-0.236953\pi$
$654$	0.686292			0.0268361
$655$	−7.31371	−	14.6274i	−0.285770	−	0.571540i
$656$	19.4558			0.759623
$657$		−	1.17157i		−	0.0457074i
$658$			4.00000i			0.155936i
$659$	−26.6274			−1.03726			−0.518628	−	0.855000i	$-0.673557\pi$
$659$	−26.6274			−1.03726			−0.518628	+	0.855000i	$0.673557\pi$
$660$	−7.31371	+	3.65685i	−0.284686	+	0.142343i
$661$	−5.31371			−0.206679			−0.103340	−	0.994646i	$-0.532953\pi$
$661$	−5.31371			−0.206679			−0.103340	+	0.994646i	$0.532953\pi$
$662$		−	2.54416i		−	0.0988814i
$663$			93.2548i			3.62172i
$664$	10.2843			0.399107
$665$	1.65685	−	0.828427i	0.0642501	−	0.0321250i
$666$	3.51472			0.136193
$667$			36.9706i			1.43151i
$668$		−	15.2548i		−	0.590227i
$669$	−23.3137			−0.901360
$670$	−2.48528	−	4.97056i	−0.0960148	−	0.192030i
$671$	2.00000			0.0772091
$672$		−	7.31371i		−	0.282132i
$673$			22.1421i			0.853517i	0.904366	+	0.426758i	$0.140344\pi$
$673$			22.1421i			0.853517i	−0.904366	+	0.426758i	$0.859656\pi$
$674$	1.17157			0.0451273
$675$	−16.0000	−	12.0000i	−0.615840	−	0.461880i
$676$	−61.4853			−2.36482
$677$		−	19.5147i		−	0.750012i	−0.927022	−	0.375006i	$-0.877641\pi$
$677$		−	19.5147i		−	0.750012i	0.927022	−	0.375006i	$-0.122359\pi$
$678$		−	8.40202i		−	0.322678i
$679$	0.970563			0.0372468
$680$	−10.8284	−	21.6569i	−0.415251	−	0.830502i
$681$	24.6863			0.945981
$682$			2.82843i			0.108306i
$683$			16.6274i			0.636230i	0.948052	+	0.318115i	$0.103050\pi$
$683$			16.6274i			0.636230i	−0.948052	+	0.318115i	$0.896950\pi$
$684$	−1.82843			−0.0699117
$685$	−16.0000	+	8.00000i	−0.611329	+	0.305664i
$686$	−4.56854			−0.174428
$687$			44.0000i			1.67870i
$688$		−	2.48528i		−	0.0947505i
$689$	−73.9411			−2.81693
$690$	−12.6863	+	6.34315i	−0.482959	+	0.241479i
$691$	−28.0000			−1.06517			−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000			−1.06517			−0.532585	+	0.846376i	$0.678779\pi$
$692$			22.8284i			0.867807i
$693$			0.828427i			0.0314693i
$694$	10.2843			0.390386
$695$	4.00000	+	8.00000i	0.151729	+	0.303457i
$696$	−15.3137			−0.580465
$697$		−	44.2843i		−	1.67739i
$698$			11.4558i			0.433610i
$699$	8.97056			0.339298
$700$	−6.05887	−	4.54416i	−0.229004	−	0.171753i
$701$	34.2843			1.29490			0.647450	−	0.762108i	$-0.275836\pi$
$701$	34.2843			1.29490			0.647450	+	0.762108i	$0.275836\pi$
$702$		−	11.3137i		−	0.427008i
$703$			8.48528i			0.320028i
$704$	−4.17157			−0.157222
$705$	−23.3137	−	46.6274i	−0.878045	−	1.75609i
$706$		0			0
$707$			7.71573i			0.290180i
$708$			10.3431i			0.388719i
$709$	−6.68629			−0.251109			−0.125554	−	0.992087i	$-0.540071\pi$
$709$	−6.68629			−0.251109			−0.125554	+	0.992087i	$0.540071\pi$
$710$	−12.2843	+	6.14214i	−0.461020	+	0.230510i
$711$	5.65685			0.212149
$712$			6.88730i			0.258113i
$713$		−	52.2843i		−	1.95806i
$714$	−4.68629			−0.175380
$715$	−13.6569	+	6.82843i	−0.510737	+	0.255369i
$716$	−30.1421			−1.12646
$717$		−	9.37258i		−	0.350026i
$718$		−	7.02944i		−	0.262336i
$719$	−14.6274			−0.545511			−0.272755	−	0.962083i	$-0.587935\pi$
$719$	−14.6274			−0.545511			−0.272755	+	0.962083i	$0.587935\pi$
$720$	3.00000	+	6.00000i	0.111803	+	0.223607i
$721$	3.59798			0.133996
$722$			0.414214i			0.0154154i
$723$			12.9706i			0.482380i
$724$	25.5980			0.951341
$725$	−14.4853	+	19.3137i	−0.537970	+	0.717293i
$726$	−0.828427			−0.0307458
$727$			13.3137i			0.493778i	0.969044	+	0.246889i	$0.0794083\pi$
$727$			13.3137i			0.493778i	−0.969044	+	0.246889i	$0.920592\pi$
$728$		−	8.97056i		−	0.332471i
$729$	−13.0000			−0.481481
$730$	0.485281	+	0.970563i	0.0179611	+	0.0359221i
$731$	−5.65685			−0.209226
$732$		−	7.31371i		−	0.270322i
$733$		−	8.48528i		−	0.313411i	−0.987645	−	0.156706i	$-0.949913\pi$
$733$		−	8.48528i		−	0.313411i	0.987645	−	0.156706i	$-0.0500874\pi$
$734$	−8.42641			−0.311024
$735$	25.2548	−	12.6274i	0.931539	−	0.465769i
$736$	−33.7990			−1.24585
$737$			6.00000i			0.221013i
$738$		−	2.68629i		−	0.0988838i
$739$	44.0000			1.61857			0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000			1.61857			0.809283	+	0.587419i	$0.199856\pi$
$740$	31.0294	−	15.5147i	1.14066	−	0.570332i
$741$	−13.6569			−0.501697
$742$		−	3.71573i		−	0.136409i
$743$			10.9706i			0.402471i	0.979543	+	0.201235i	$0.0644956\pi$
$743$			10.9706i			0.402471i	−0.979543	+	0.201235i	$0.935504\pi$
$744$	21.6569			0.793979
$745$	−8.34315	−	16.6863i	−0.305669	−	0.611338i
$746$	7.11270			0.260414
$747$			6.48528i			0.237284i
$748$			12.4853i			0.456507i
$749$	15.7157			0.574240
$750$	−9.11270	−	1.65685i	−0.332749	−	0.0604998i
$751$	36.4853			1.33137			0.665683	−	0.746234i	$-0.268140\pi$
$751$	36.4853			1.33137			0.665683	+	0.746234i	$0.268140\pi$
$752$		−	34.9706i		−	1.27525i
$753$		−	8.00000i		−	0.291536i
$754$	−13.6569			−0.497353
$755$	−10.3431	−	20.6863i	−0.376426	−	0.752851i
$756$	−6.05887			−0.220359
$757$		−	20.0000i		−	0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$		−	20.0000i		−	0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$		−	4.48528i		−	0.162913i
$759$	15.3137			0.555852
$760$	3.17157	−	1.58579i	0.115045	−	0.0575225i
$761$	−47.9411			−1.73786			−0.868932	−	0.494931i	$-0.835193\pi$
$761$	−47.9411			−1.73786			−0.868932	+	0.494931i	$0.835193\pi$
$762$			11.5980i			0.420150i
$763$		−	0.686292i		−	0.0248454i
$764$	31.0294			1.12261
$765$	13.6569	−	6.82843i	0.493765	−	0.246882i
$766$	11.1716			0.403645
$767$			19.3137i			0.697378i
$768$			7.94113i			0.286551i
$769$	−8.34315			−0.300862			−0.150431	−	0.988621i	$-0.548066\pi$
$769$	−8.34315			−0.300862			−0.150431	+	0.988621i	$0.548066\pi$
$770$	−0.343146	−	0.686292i	−0.0123661	−	0.0247322i
$771$	−36.2843			−1.30675
$772$		−	25.8579i		−	0.930645i
$773$		−	5.17157i		−	0.186009i	−0.995666	−	0.0930043i	$-0.970353\pi$
$773$		−	5.17157i		−	0.186009i	0.995666	−	0.0930043i	$-0.0296470\pi$
$774$	−0.343146			−0.0123341
$775$	20.4853	−	27.3137i	0.735853	−	0.981137i
$776$	1.85786			0.0666934
$777$		−	14.0589i		−	0.504359i
$778$		−	5.51472i		−	0.197712i
$779$	6.48528			0.232359
$780$	24.9706	+	49.9411i	0.894090	+	1.78818i
$781$	14.8284			0.530603
$782$			21.6569i			0.774448i
$783$			19.3137i			0.690216i
$784$	18.9411			0.676469
$785$	−24.0000	+	12.0000i	−0.856597	+	0.428298i
$786$	−6.05887			−0.216113
$787$		−	2.97056i		−	0.105889i	−0.998597	−	0.0529446i	$-0.983139\pi$
$787$		−	2.97056i		−	0.105889i	0.998597	−	0.0529446i	$-0.0168607\pi$
$788$			11.2304i			0.400068i
$789$	36.9706			1.31619
$790$	−4.68629	+	2.34315i	−0.166731	+	0.0833654i
$791$	−8.40202			−0.298741
$792$			1.58579i			0.0563485i
$793$		−	13.6569i		−	0.484969i
$794$	3.02944			0.107511
$795$	21.6569	+	43.3137i	0.768089	+	1.53618i
$796$	10.3431			0.366603
$797$			10.8284i			0.383563i	0.981438	+	0.191781i	$0.0614264\pi$
$797$			10.8284i			0.383563i	−0.981438	+	0.191781i	$0.938574\pi$
$798$		−	0.686292i		−	0.0242945i
$799$	−79.5980			−2.81597
$800$	−17.6569	−	13.2426i	−0.624264	−	0.468198i
$801$	−4.34315			−0.153458
$802$			2.20101i			0.0777204i
$803$		−	1.17157i		−	0.0413439i
$804$	21.9411			0.773804
$805$	6.34315	+	12.6863i	0.223567	+	0.447133i
$806$	19.3137			0.680296
$807$			45.9411i			1.61720i
$808$			14.7696i			0.519591i
$809$	53.3137			1.87441			0.937205	−	0.348779i	$-0.113404\pi$
$809$	53.3137			1.87441			0.937205	+	0.348779i	$0.113404\pi$
$810$	−9.11270	+	4.55635i	−0.320188	+	0.160094i
$811$	−16.2843			−0.571818			−0.285909	−	0.958257i	$-0.592296\pi$
$811$	−16.2843			−0.571818			−0.285909	+	0.958257i	$0.592296\pi$
$812$			7.31371i			0.256661i
$813$			22.6274i			0.793578i
$814$	3.51472			0.123191
$815$	−4.00000	+	2.00000i	−0.140114	+	0.0700569i
$816$	40.9706			1.43426
$817$		−	0.828427i		−	0.0289830i
$818$			10.2843i			0.359581i
$819$	5.65685			0.197666
$820$	−11.8579	−	23.7157i	−0.414095	−	0.828189i
$821$	−51.9411			−1.81276			−0.906379	−	0.422466i	$-0.861165\pi$
$821$	−51.9411			−1.81276			−0.906379	+	0.422466i	$0.861165\pi$
$822$			6.62742i			0.231158i
$823$		−	9.31371i		−	0.324655i	−0.986737	−	0.162328i	$-0.948100\pi$
$823$		−	9.31371i		−	0.324655i	0.986737	−	0.162328i	$-0.0519002\pi$
$824$	6.88730			0.239931
$825$	8.00000	+	6.00000i	0.278524	+	0.208893i
$826$	−0.970563			−0.0337702
$827$		−	14.6863i		−	0.510692i	−0.966850	−	0.255346i	$-0.917811\pi$
$827$		−	14.6863i		−	0.510692i	0.966850	−	0.255346i	$-0.0821894\pi$
$828$		−	14.0000i		−	0.486534i
$829$	−52.9117			−1.83770			−0.918849	−	0.394608i	$-0.870880\pi$
$829$	−52.9117			−1.83770			−0.918849	+	0.394608i	$0.870880\pi$
$830$	−2.68629	−	5.37258i	−0.0932425	−	0.186485i
$831$	−37.6569			−1.30630
$832$			28.4853i			0.987549i
$833$		−	43.1127i		−	1.49377i
$834$	3.31371			0.114744
$835$	−16.6863	+	8.34315i	−0.577453	+	0.288726i
$836$	−1.82843			−0.0632375
$837$		−	27.3137i		−	0.944100i
$838$			8.68629i			0.300063i
$839$	−7.79899			−0.269251			−0.134626	−	0.990897i	$-0.542983\pi$
$839$	−7.79899			−0.269251			−0.134626	+	0.990897i	$0.542983\pi$
$840$	−5.25483	+	2.62742i	−0.181309	+	0.0906545i
$841$	−5.68629			−0.196079
$842$		−	11.1716i		−	0.384998i
$843$		−	38.9117i		−	1.34019i
$844$	28.0000			0.963800
$845$	33.6274	+	67.2548i	1.15682	+	2.31364i
$846$	−4.82843			−0.166005
$847$			0.828427i			0.0284651i
$848$			32.4853i			1.11555i
$849$	12.9706			0.445149
$850$	−8.48528	+	11.3137i	−0.291043	+	0.388057i
$851$	−64.9706			−2.22716
$852$		−	54.2254i		−	1.85773i
$853$			56.4853i			1.93402i	0.254740	+	0.967010i	$0.418010\pi$
$853$			56.4853i			1.93402i	−0.254740	+	0.967010i	$0.581990\pi$
$854$	0.686292			0.0234844
$855$	1.00000	+	2.00000i	0.0341993	+	0.0683986i
$856$	30.0833			1.02822
$857$		−	48.0833i		−	1.64249i	−0.570574	−	0.821246i	$-0.693279\pi$
$857$		−	48.0833i		−	1.64249i	0.570574	−	0.821246i	$-0.306721\pi$
$858$			5.65685i			0.193122i
$859$	47.3137			1.61432			0.807161	−	0.590331i	$-0.201003\pi$
$859$	47.3137			1.61432			0.807161	+	0.590331i	$0.201003\pi$
$860$	−3.02944	+	1.51472i	−0.103303	+	0.0516515i
$861$	−10.7452			−0.366194
$862$			2.74517i			0.0935007i
$863$		−	42.9706i		−	1.46273i	−0.681984	−	0.731367i	$-0.738882\pi$
$863$		−	42.9706i		−	1.46273i	0.681984	−	0.731367i	$-0.261118\pi$
$864$	−17.6569			−0.600698
$865$	24.9706	−	12.4853i	0.849025	−	0.424512i
$866$	−0.485281			−0.0164905
$867$		−	59.2548i		−	2.01240i
$868$		−	10.3431i		−	0.351069i
$869$	5.65685			0.191896
$870$	4.00000	+	8.00000i	0.135613	+	0.271225i
$871$	40.9706			1.38823
$872$		−	1.31371i		−	0.0444878i
$873$			1.17157i			0.0396517i
$874$	−3.17157			−0.107280
$875$	−1.65685	+	9.11270i	−0.0560119	+	0.308065i
$876$	−4.28427			−0.144752
$877$			2.14214i			0.0723348i	0.999346	+	0.0361674i	$0.0115149\pi$
$877$			2.14214i			0.0723348i	−0.999346	+	0.0361674i	$0.988485\pi$
$878$		−	7.59798i		−	0.256419i
$879$	52.2843			1.76350
$880$	3.00000	+	6.00000i	0.101130	+	0.202260i
$881$	−31.9411			−1.07612			−0.538062	−	0.842905i	$-0.680843\pi$
$881$	−31.9411			−1.07612			−0.538062	+	0.842905i	$0.680843\pi$
$882$		−	2.61522i		−	0.0880592i
$883$			33.3137i			1.12110i	0.828122	+	0.560548i	$0.189409\pi$
$883$			33.3137i			1.12110i	−0.828122	+	0.560548i	$0.810591\pi$
$884$	85.2548			2.86743
$885$	11.3137	−	5.65685i	0.380306	−	0.190153i
$886$	−3.85786			−0.129607
$887$		−	15.9411i		−	0.535251i	−0.963523	−	0.267625i	$-0.913761\pi$
$887$		−	15.9411i		−	0.535251i	0.963523	−	0.267625i	$-0.0862389\pi$
$888$		−	26.9117i		−	0.903097i
$889$	11.5980			0.388984
$890$	3.59798	−	1.79899i	0.120605	−	0.0603023i
$891$	11.0000			0.368514
$892$			21.3137i			0.713636i
$893$		−	11.6569i		−	0.390082i
$894$	−6.91169			−0.231161
$895$	16.4853	+	32.9706i	0.551042	+	1.10208i
$896$	−8.74517			−0.292155
$897$		−	104.569i		−	3.49144i
$898$			1.23045i			0.0410606i
$899$	−32.9706			−1.09963
$900$	5.48528	−	7.31371i	0.182843	−	0.243790i
$901$	73.9411			2.46334
$902$		−	2.68629i		−	0.0894437i
$903$			1.37258i			0.0456767i
$904$	−16.0833			−0.534921
$905$	−14.0000	−	28.0000i	−0.465376	−	0.930751i
$906$	−8.56854			−0.284671
$907$		−	28.6274i		−	0.950558i	−0.879835	−	0.475279i	$-0.842347\pi$
$907$		−	28.6274i		−	0.950558i	0.879835	−	0.475279i	$-0.157653\pi$
$908$		−	22.5685i		−	0.748963i
$909$	−9.31371			−0.308916
$910$	−4.68629	+	2.34315i	−0.155349	+	0.0776745i
$911$	25.1716			0.833971			0.416986	−	0.908913i	$-0.363087\pi$
$911$	25.1716			0.833971			0.416986	+	0.908913i	$0.363087\pi$
$912$			6.00000i			0.198680i
$913$			6.48528i			0.214631i
$914$	−6.54416			−0.216461
$915$	−8.00000	+	4.00000i	−0.264472	+	0.132236i
$916$	40.2254			1.32908
$917$			6.05887i			0.200082i
$918$			11.3137i			0.373408i
$919$	48.9706			1.61539			0.807695	−	0.589601i	$-0.200715\pi$
$919$	48.9706			1.61539			0.807695	+	0.589601i	$0.200715\pi$
$920$	12.1421	+	24.2843i	0.400314	+	0.800629i
$921$	53.9411			1.77742
$922$		−	9.23045i		−	0.303989i
$923$		−	101.255i		−	3.33284i
$924$	3.02944			0.0996612
$925$	−33.9411	−	25.4558i	−1.11598	−	0.836983i
$926$	−12.5442			−0.412227
$927$			4.34315i			0.142648i
$928$			21.3137i			0.699657i
$929$	−55.9411			−1.83537			−0.917684	−	0.397310i	$-0.869944\pi$
$929$	−55.9411			−1.83537			−0.917684	+	0.397310i	$0.869944\pi$
$930$	−5.65685	−	11.3137i	−0.185496	−	0.370991i
$931$	6.31371			0.206923
$932$		−	8.20101i		−	0.268633i
$933$			1.94113i			0.0635496i
$934$	1.79899			0.0588647
$935$	13.6569	−	6.82843i	0.446627	−	0.223313i
$936$	10.8284			0.353938
$937$			19.1127i			0.624385i	0.950019	+	0.312192i	$0.101063\pi$
$937$			19.1127i			0.624385i	−0.950019	+	0.312192i	$0.898937\pi$
$938$			2.05887i			0.0672246i
$939$	−33.9411			−1.10763
$940$	−42.6274	+	21.3137i	−1.39035	+	0.695177i
$941$	−23.8579			−0.777744			−0.388872	−	0.921292i	$-0.627135\pi$
$941$	−23.8579			−0.777744			−0.388872	+	0.921292i	$0.627135\pi$
$942$			9.94113i			0.323899i
$943$			49.6569i			1.61705i
$944$	8.48528			0.276172
$945$	3.31371	+	6.62742i	0.107795	+	0.215590i
$946$	−0.343146			−0.0111566
$947$			58.2843i			1.89398i	0.321257	+	0.946992i	$0.395895\pi$
$947$			58.2843i			1.89398i	−0.321257	+	0.946992i	$0.604105\pi$
$948$		−	20.6863i		−	0.671860i
$949$	−8.00000			−0.259691
$950$	−1.65685	−	1.24264i	−0.0537555	−	0.0403166i
$951$	−12.2843			−0.398345
$952$			8.97056i			0.290738i
$953$			39.1127i			1.26698i	0.773749	+	0.633492i	$0.218379\pi$
$953$			39.1127i			1.26698i	−0.773749	+	0.633492i	$0.781621\pi$
$954$	4.48528			0.145216
$955$	−16.9706	−	33.9411i	−0.549155	−	1.09831i
$956$	−8.56854			−0.277126
$957$		−	9.65685i		−	0.312162i
$958$			1.37258i			0.0443461i
$959$	6.62742			0.214010
$960$	16.6863	−	8.34315i	0.538548	−	0.269274i
$961$	15.6274			0.504110
$962$		−	24.0000i		−	0.773791i
$963$			18.9706i			0.611318i
$964$	11.8579			0.381916
$965$	−28.2843	+	14.1421i	−0.910503	+	0.455251i
$966$	5.25483			0.169072
$967$			12.1421i			0.390465i	0.980757	+	0.195232i	$0.0625461\pi$
$967$			12.1421i			0.390465i	−0.980757	+	0.195232i	$0.937454\pi$
$968$			1.58579i			0.0509691i
$969$	13.6569			0.438721
$970$	−0.485281	−	0.970563i	−0.0155814	−	0.0311629i
$971$	−15.1127			−0.484990			−0.242495	−	0.970153i	$-0.577966\pi$
$971$	−15.1127			−0.484990			−0.242495	+	0.970153i	$0.577966\pi$
$972$		−	18.2843i		−	0.586468i
$973$		−	3.31371i		−	0.106233i
$974$	6.48528			0.207802
$975$	40.9706	−	54.6274i	1.31211	−	1.74948i
$976$	−6.00000			−0.192055
$977$			7.79899i			0.249512i	0.992187	+	0.124756i	$0.0398148\pi$
$977$			7.79899i			0.249512i	−0.992187	+	0.124756i	$0.960185\pi$
$978$			1.65685i			0.0529804i
$979$	−4.34315			−0.138808
$980$	−11.5442	−	23.0883i	−0.368765	−	0.737529i
$981$	0.828427			0.0264496
$982$		−	15.3137i		−	0.488680i
$983$			8.34315i			0.266105i	0.991109	+	0.133053i	$0.0424779\pi$
$983$			8.34315i			0.266105i	−0.991109	+	0.133053i	$0.957522\pi$
$984$	−20.5685			−0.655701
$985$	12.2843	−	6.14214i	0.391409	−	0.195705i
$986$	13.6569			0.434923
$987$			19.3137i			0.614762i
$988$			12.4853i			0.397210i
$989$	6.34315			0.201700
$990$	0.828427	−	0.414214i	0.0263291	−	0.0131646i
$991$	22.8284			0.725169			0.362584	−	0.931951i	$-0.381894\pi$
$991$	22.8284			0.725169			0.362584	+	0.931951i	$0.381894\pi$
$992$		−	30.1421i		−	0.957014i
$993$		−	12.2843i		−	0.389830i
$994$	5.08831			0.161391
$995$	−5.65685	−	11.3137i	−0.179334	−	0.358669i
$996$	23.7157			0.751462
$997$			16.4853i			0.522094i	0.965326	+	0.261047i	$0.0840678\pi$
$997$			16.4853i			0.522094i	−0.965326	+	0.261047i	$0.915932\pi$
$998$		−	0.284271i		−	0.00899845i
$999$	−33.9411			−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.b.a.419.3	yes	4
5.2	odd	4		5225.2.a.g.1.1		2
5.3	odd	4		5225.2.a.d.1.2		2
5.4	even	2	inner	1045.2.b.a.419.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.a.419.2	✓	4	5.4	even	2	inner
1045.2.b.a.419.3	yes	4	1.1	even	1	trivial
5225.2.a.d.1.2		2	5.3	odd	4
5225.2.a.g.1.1		2	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.414214i q^{2} +2.00000i q^{3} +1.82843 q^{4} +(-1.00000 - 2.00000i) q^{5} -0.828427 q^{6} +0.828427i q^{7} +1.58579i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+0.414214i q^{2} +2.00000i q^{3} +1.82843 q^{4} +(-1.00000 - 2.00000i) q^{5} -0.828427 q^{6} +0.828427i q^{7} +1.58579i q^{8} -1.00000 q^{9} +(0.828427 - 0.414214i) q^{10} -1.00000 q^{11} +3.65685i q^{12} +6.82843i q^{13} -0.343146 q^{14} +(4.00000 - 2.00000i) q^{15} +3.00000 q^{16} -6.82843i q^{17} -0.414214i q^{18} +1.00000 q^{19} +(-1.82843 - 3.65685i) q^{20} -1.65685 q^{21} -0.414214i q^{22} +7.65685i q^{23} -3.17157 q^{24} +(-3.00000 + 4.00000i) q^{25} -2.82843 q^{26} +4.00000i q^{27} +1.51472i q^{28} +4.82843 q^{29} +(0.828427 + 1.65685i) q^{30} -6.82843 q^{31} +4.41421i q^{32} -2.00000i q^{33} +2.82843 q^{34} +(1.65685 - 0.828427i) q^{35} -1.82843 q^{36} +8.48528i q^{37} +0.414214i q^{38} -13.6569 q^{39} +(3.17157 - 1.58579i) q^{40} +6.48528 q^{41} -0.686292i q^{42} -0.828427i q^{43} -1.82843 q^{44} +(1.00000 + 2.00000i) q^{45} -3.17157 q^{46} -11.6569i q^{47} +6.00000i q^{48} +6.31371 q^{49} +(-1.65685 - 1.24264i) q^{50} +13.6569 q^{51} +12.4853i q^{52} +10.8284i q^{53} -1.65685 q^{54} +(1.00000 + 2.00000i) q^{55} -1.31371 q^{56} +2.00000i q^{57} +2.00000i q^{58} +2.82843 q^{59} +(7.31371 - 3.65685i) q^{60} -2.00000 q^{61} -2.82843i q^{62} -0.828427i q^{63} +4.17157 q^{64} +(13.6569 - 6.82843i) q^{65} +0.828427 q^{66} -6.00000i q^{67} -12.4853i q^{68} -15.3137 q^{69} +(0.343146 + 0.686292i) q^{70} -14.8284 q^{71} -1.58579i q^{72} +1.17157i q^{73} -3.51472 q^{74} +(-8.00000 - 6.00000i) q^{75} +1.82843 q^{76} -0.828427i q^{77} -5.65685i q^{78} -5.65685 q^{79} +(-3.00000 - 6.00000i) q^{80} -11.0000 q^{81} +2.68629i q^{82} -6.48528i q^{83} -3.02944 q^{84} +(-13.6569 + 6.82843i) q^{85} +0.343146 q^{86} +9.65685i q^{87} -1.58579i q^{88} +4.34315 q^{89} +(-0.828427 + 0.414214i) q^{90} -5.65685 q^{91} +14.0000i q^{92} -13.6569i q^{93} +4.82843 q^{94} +(-1.00000 - 2.00000i) q^{95} -8.82843 q^{96} -1.17157i q^{97} +2.61522i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{5} + 8 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^5 + 8 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} - 4 q^{5} + 8 q^{6} - 4 q^{9} - 8 q^{10} - 4 q^{11} - 24 q^{14} + 16 q^{15} + 12 q^{16} + 4 q^{19} + 4 q^{20} + 16 q^{21} - 24 q^{24} - 12 q^{25} + 8 q^{29} - 8 q^{30} - 16 q^{31} - 16 q^{35} + 4 q^{36} - 32 q^{39} + 24 q^{40} - 8 q^{41} + 4 q^{44} + 4 q^{45} - 24 q^{46} - 20 q^{49} + 16 q^{50} + 32 q^{51} + 16 q^{54} + 4 q^{55} + 40 q^{56} - 16 q^{60} - 8 q^{61} + 28 q^{64} + 32 q^{65} - 8 q^{66} - 16 q^{69} + 24 q^{70} - 48 q^{71} - 48 q^{74} - 32 q^{75} - 4 q^{76} - 12 q^{80} - 44 q^{81} - 80 q^{84} - 32 q^{85} + 24 q^{86} + 40 q^{89} + 8 q^{90} + 8 q^{94} - 4 q^{95} - 24 q^{96} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^5 + 8 * q^6 - 4 * q^9 - 8 * q^10 - 4 * q^11 - 24 * q^14 + 16 * q^15 + 12 * q^16 + 4 * q^19 + 4 * q^20 + 16 * q^21 - 24 * q^24 - 12 * q^25 + 8 * q^29 - 8 * q^30 - 16 * q^31 - 16 * q^35 + 4 * q^36 - 32 * q^39 + 24 * q^40 - 8 * q^41 + 4 * q^44 + 4 * q^45 - 24 * q^46 - 20 * q^49 + 16 * q^50 + 32 * q^51 + 16 * q^54 + 4 * q^55 + 40 * q^56 - 16 * q^60 - 8 * q^61 + 28 * q^64 + 32 * q^65 - 8 * q^66 - 16 * q^69 + 24 * q^70 - 48 * q^71 - 48 * q^74 - 32 * q^75 - 4 * q^76 - 12 * q^80 - 44 * q^81 - 80 * q^84 - 32 * q^85 + 24 * q^86 + 40 * q^89 + 8 * q^90 + 8 * q^94 - 4 * q^95 - 24 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

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Database

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