LMFDB - Embedded newform 44.2.c.a.43.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [44,2,Mod(43,44)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(44, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("44.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 44 = 2^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	44.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:	$0.351341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			43.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	44.43
Dual form			44.2.c.a.43.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 1.22474i) q^{2} -1.73205i q^{3} +(-1.00000 + 1.73205i) q^{4} -1.00000 q^{5} +(-2.12132 + 1.22474i) q^{6} +2.82843 q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+(-0.707107 - 1.22474i) q^{2} -1.73205i q^{3} +(-1.00000 + 1.73205i) q^{4} -1.00000 q^{5} +(-2.12132 + 1.22474i) q^{6} +2.82843 q^{7} +2.82843 q^{8} +(0.707107 + 1.22474i) q^{10} +(-2.82843 + 1.73205i) q^{11} +(3.00000 + 1.73205i) q^{12} +4.89898i q^{13} +(-2.00000 - 3.46410i) q^{14} +1.73205i q^{15} +(-2.00000 - 3.46410i) q^{16} -4.89898i q^{17} -2.82843 q^{19} +(1.00000 - 1.73205i) q^{20} -4.89898i q^{21} +(4.12132 + 2.23936i) q^{22} +5.19615i q^{23} -4.89898i q^{24} -4.00000 q^{25} +(6.00000 - 3.46410i) q^{26} -5.19615i q^{27} +(-2.82843 + 4.89898i) q^{28} +(2.12132 - 1.22474i) q^{30} -1.73205i q^{31} +(-2.82843 + 4.89898i) q^{32} +(3.00000 + 4.89898i) q^{33} +(-6.00000 + 3.46410i) q^{34} -2.82843 q^{35} +3.00000 q^{37} +(2.00000 + 3.46410i) q^{38} +8.48528 q^{39} -2.82843 q^{40} +4.89898i q^{41} +(-6.00000 + 3.46410i) q^{42} +5.65685 q^{43} +(-0.171573 - 6.63103i) q^{44} +(6.36396 - 3.67423i) q^{46} -3.46410i q^{47} +(-6.00000 + 3.46410i) q^{48} +1.00000 q^{49} +(2.82843 + 4.89898i) q^{50} -8.48528 q^{51} +(-8.48528 - 4.89898i) q^{52} -2.00000 q^{53} +(-6.36396 + 3.67423i) q^{54} +(2.82843 - 1.73205i) q^{55} +8.00000 q^{56} +4.89898i q^{57} -1.73205i q^{59} +(-3.00000 - 1.73205i) q^{60} -9.79796i q^{61} +(-2.12132 + 1.22474i) q^{62} +8.00000 q^{64} -4.89898i q^{65} +(3.87868 - 7.13834i) q^{66} -8.66025i q^{67} +(8.48528 + 4.89898i) q^{68} +9.00000 q^{69} +(2.00000 + 3.46410i) q^{70} +12.1244i q^{71} -4.89898i q^{73} +(-2.12132 - 3.67423i) q^{74} +6.92820i q^{75} +(2.82843 - 4.89898i) q^{76} +(-8.00000 + 4.89898i) q^{77} +(-6.00000 - 10.3923i) q^{78} -11.3137 q^{79} +(2.00000 + 3.46410i) q^{80} -9.00000 q^{81} +(6.00000 - 3.46410i) q^{82} +(8.48528 + 4.89898i) q^{84} +4.89898i q^{85} +(-4.00000 - 6.92820i) q^{86} +(-8.00000 + 4.89898i) q^{88} -1.00000 q^{89} +13.8564i q^{91} +(-9.00000 - 5.19615i) q^{92} -3.00000 q^{93} +(-4.24264 + 2.44949i) q^{94} +2.82843 q^{95} +(8.48528 + 4.89898i) q^{96} +7.00000 q^{97} +(-0.707107 - 1.22474i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{5}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^5	$ 4 q - 4 q^{4} - 4 q^{5} + 12 q^{12} - 8 q^{14} - 8 q^{16} + 4 q^{20} + 8 q^{22} - 16 q^{25} + 24 q^{26} + 12 q^{33} - 24 q^{34} + 12 q^{37} + 8 q^{38} - 24 q^{42} - 12 q^{44} - 24 q^{48} + 4 q^{49} - 8 q^{53} + 32 q^{56} - 12 q^{60} + 32 q^{64} + 24 q^{66} + 36 q^{69} + 8 q^{70} - 32 q^{77} - 24 q^{78} + 8 q^{80} - 36 q^{81} + 24 q^{82} - 16 q^{86} - 32 q^{88} - 4 q^{89} - 36 q^{92} - 12 q^{93} + 28 q^{97}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^5 + 12 * q^12 - 8 * q^14 - 8 * q^16 + 4 * q^20 + 8 * q^22 - 16 * q^25 + 24 * q^26 + 12 * q^33 - 24 * q^34 + 12 * q^37 + 8 * q^38 - 24 * q^42 - 12 * q^44 - 24 * q^48 + 4 * q^49 - 8 * q^53 + 32 * q^56 - 12 * q^60 + 32 * q^64 + 24 * q^66 + 36 * q^69 + 8 * q^70 - 32 * q^77 - 24 * q^78 + 8 * q^80 - 36 * q^81 + 24 * q^82 - 16 * q^86 - 32 * q^88 - 4 * q^89 - 36 * q^92 - 12 * q^93 + 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/44\mathbb{Z}\right)^\times$.

$n$	$13$	$23$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−0.707107	−	1.22474i	−0.500000	−	0.866025i
$2$	−0.707107	−	1.22474i	−0.500000	−	0.866025i
$3$		−	1.73205i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$3$		−	1.73205i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$4$	−1.00000	+	1.73205i	−0.500000	+	0.866025i
$5$	−1.00000			−0.447214			−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000			−0.447214			−0.223607	+	0.974679i	$0.571783\pi$
$6$	−2.12132	+	1.22474i	−0.866025	+	0.500000i
$7$	2.82843			1.06904			0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843			1.06904			0.534522	+	0.845154i	$0.320491\pi$
$8$	2.82843			1.00000
$9$		0			0
$10$	0.707107	+	1.22474i	0.223607	+	0.387298i
$11$	−2.82843	+	1.73205i	−0.852803	+	0.522233i
$11$	−2.82843	+	1.73205i	−0.852803	+	0.522233i
$12$	3.00000	+	1.73205i	0.866025	+	0.500000i
$13$			4.89898i			1.35873i	0.733799	+	0.679366i	$0.237745\pi$
$13$			4.89898i			1.35873i	−0.733799	+	0.679366i	$0.762255\pi$
$14$	−2.00000	−	3.46410i	−0.534522	−	0.925820i
$15$			1.73205i			0.447214i
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$		−	4.89898i		−	1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$		−	4.89898i		−	1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$		0			0
$19$	−2.82843			−0.648886			−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843			−0.648886			−0.324443	+	0.945905i	$0.605177\pi$
$20$	1.00000	−	1.73205i	0.223607	−	0.387298i
$21$		−	4.89898i		−	1.06904i
$22$	4.12132	+	2.23936i	0.878668	+	0.477432i
$23$			5.19615i			1.08347i	0.840548	+	0.541736i	$0.182233\pi$
$23$			5.19615i			1.08347i	−0.840548	+	0.541736i	$0.817767\pi$
$24$		−	4.89898i		−	1.00000i
$25$	−4.00000			−0.800000
$26$	6.00000	−	3.46410i	1.17670	−	0.679366i
$27$		−	5.19615i		−	1.00000i
$28$	−2.82843	+	4.89898i	−0.534522	+	0.925820i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	2.12132	−	1.22474i	0.387298	−	0.223607i
$31$		−	1.73205i		−	0.311086i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$		−	1.73205i		−	0.311086i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$	−2.82843	+	4.89898i	−0.500000	+	0.866025i
$33$	3.00000	+	4.89898i	0.522233	+	0.852803i
$34$	−6.00000	+	3.46410i	−1.02899	+	0.594089i
$35$	−2.82843			−0.478091
$36$		0			0
$37$	3.00000			0.493197			0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000			0.493197			0.246598	+	0.969118i	$0.420687\pi$
$38$	2.00000	+	3.46410i	0.324443	+	0.561951i
$39$	8.48528			1.35873
$40$	−2.82843			−0.447214
$41$			4.89898i			0.765092i	0.923936	+	0.382546i	$0.124953\pi$
$41$			4.89898i			0.765092i	−0.923936	+	0.382546i	$0.875047\pi$
$42$	−6.00000	+	3.46410i	−0.925820	+	0.534522i
$43$	5.65685			0.862662			0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685			0.862662			0.431331	+	0.902194i	$0.358044\pi$
$44$	−0.171573	−	6.63103i	−0.0258656	−	0.999665i
$45$		0			0
$46$	6.36396	−	3.67423i	0.938315	−	0.541736i
$47$		−	3.46410i		−	0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$		−	3.46410i		−	0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$	−6.00000	+	3.46410i	−0.866025	+	0.500000i
$49$	1.00000			0.142857
$50$	2.82843	+	4.89898i	0.400000	+	0.692820i
$51$	−8.48528			−1.18818
$52$	−8.48528	−	4.89898i	−1.17670	−	0.679366i
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$	−6.36396	+	3.67423i	−0.866025	+	0.500000i
$55$	2.82843	−	1.73205i	0.381385	−	0.233550i
$56$	8.00000			1.06904
$57$			4.89898i			0.648886i
$58$		0			0
$59$		−	1.73205i		−	0.225494i	−0.993624	−	0.112747i	$-0.964035\pi$
$59$		−	1.73205i		−	0.225494i	0.993624	−	0.112747i	$-0.0359649\pi$
$60$	−3.00000	−	1.73205i	−0.387298	−	0.223607i
$61$		−	9.79796i		−	1.25450i	−0.778818	−	0.627250i	$-0.784180\pi$
$61$		−	9.79796i		−	1.25450i	0.778818	−	0.627250i	$-0.215820\pi$
$62$	−2.12132	+	1.22474i	−0.269408	+	0.155543i
$63$		0			0
$64$	8.00000			1.00000
$65$		−	4.89898i		−	0.607644i
$66$	3.87868	−	7.13834i	0.477432	−	0.878668i
$67$		−	8.66025i		−	1.05802i	−0.848616	−	0.529009i	$-0.822564\pi$
$67$		−	8.66025i		−	1.05802i	0.848616	−	0.529009i	$-0.177436\pi$
$68$	8.48528	+	4.89898i	1.02899	+	0.594089i
$69$	9.00000			1.08347
$70$	2.00000	+	3.46410i	0.239046	+	0.414039i
$71$			12.1244i			1.43890i	0.694546	+	0.719448i	$0.255605\pi$
$71$			12.1244i			1.43890i	−0.694546	+	0.719448i	$0.744395\pi$
$72$		0			0
$73$		−	4.89898i		−	0.573382i	−0.958023	−	0.286691i	$-0.907445\pi$
$73$		−	4.89898i		−	0.573382i	0.958023	−	0.286691i	$-0.0925553\pi$
$74$	−2.12132	−	3.67423i	−0.246598	−	0.427121i
$75$			6.92820i			0.800000i
$76$	2.82843	−	4.89898i	0.324443	−	0.561951i
$77$	−8.00000	+	4.89898i	−0.911685	+	0.558291i
$78$	−6.00000	−	10.3923i	−0.679366	−	1.17670i
$79$	−11.3137			−1.27289			−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137			−1.27289			−0.636446	+	0.771321i	$0.719596\pi$
$80$	2.00000	+	3.46410i	0.223607	+	0.387298i
$81$	−9.00000			−1.00000
$82$	6.00000	−	3.46410i	0.662589	−	0.382546i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	8.48528	+	4.89898i	0.925820	+	0.534522i
$85$			4.89898i			0.531369i
$86$	−4.00000	−	6.92820i	−0.431331	−	0.747087i
$87$		0			0
$88$	−8.00000	+	4.89898i	−0.852803	+	0.522233i
$89$	−1.00000			−0.106000			−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000			−0.106000			−0.0529999	+	0.998595i	$0.516878\pi$
$90$		0			0
$91$			13.8564i			1.45255i
$92$	−9.00000	−	5.19615i	−0.938315	−	0.541736i
$93$	−3.00000			−0.311086
$94$	−4.24264	+	2.44949i	−0.437595	+	0.252646i
$95$	2.82843			0.290191
$96$	8.48528	+	4.89898i	0.866025	+	0.500000i
$97$	7.00000			0.710742			0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000			0.710742			0.355371	+	0.934725i	$0.384354\pi$
$98$	−0.707107	−	1.22474i	−0.0714286	−	0.123718i
$99$		0			0
$100$	4.00000	−	6.92820i	0.400000	−	0.692820i
$101$		−	4.89898i		−	0.487467i	−0.969842	−	0.243733i	$-0.921628\pi$
$101$		−	4.89898i		−	0.487467i	0.969842	−	0.243733i	$-0.0783722\pi$
$102$	6.00000	+	10.3923i	0.594089	+	1.02899i
$103$		−	3.46410i		−	0.341328i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$		−	3.46410i		−	0.341328i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$			13.8564i			1.35873i
$105$			4.89898i			0.478091i
$106$	1.41421	+	2.44949i	0.137361	+	0.237915i
$107$	8.48528			0.820303			0.410152	−	0.912017i	$-0.365476\pi$
$107$	8.48528			0.820303			0.410152	+	0.912017i	$0.365476\pi$
$108$	9.00000	+	5.19615i	0.866025	+	0.500000i
$109$			9.79796i			0.938474i	0.883072	+	0.469237i	$0.155471\pi$
$109$			9.79796i			0.938474i	−0.883072	+	0.469237i	$0.844529\pi$
$110$	−4.12132	−	2.23936i	−0.392952	−	0.213514i
$111$		−	5.19615i		−	0.493197i
$112$	−5.65685	−	9.79796i	−0.534522	−	0.925820i
$113$	−5.00000			−0.470360			−0.235180	−	0.971952i	$-0.575568\pi$
$113$	−5.00000			−0.470360			−0.235180	+	0.971952i	$0.575568\pi$
$114$	6.00000	−	3.46410i	0.561951	−	0.324443i
$115$		−	5.19615i		−	0.484544i
$116$		0			0
$117$		0			0
$118$	−2.12132	+	1.22474i	−0.195283	+	0.112747i
$119$		−	13.8564i		−	1.27021i
$120$			4.89898i			0.447214i
$121$	5.00000	−	9.79796i	0.454545	−	0.890724i
$122$	−12.0000	+	6.92820i	−1.08643	+	0.627250i
$123$	8.48528			0.765092
$124$	3.00000	+	1.73205i	0.269408	+	0.155543i
$125$	9.00000			0.804984
$126$		0			0
$127$	−14.1421			−1.25491			−0.627456	−	0.778652i	$-0.715904\pi$
$127$	−14.1421			−1.25491			−0.627456	+	0.778652i	$0.715904\pi$
$128$	−5.65685	−	9.79796i	−0.500000	−	0.866025i
$129$		−	9.79796i		−	0.862662i
$130$	−6.00000	+	3.46410i	−0.526235	+	0.303822i
$131$	−2.82843			−0.247121			−0.123560	−	0.992337i	$-0.539431\pi$
$131$	−2.82843			−0.247121			−0.123560	+	0.992337i	$0.539431\pi$
$132$	−11.4853	+	0.297173i	−0.999665	+	0.0258656i
$133$	−8.00000			−0.693688
$134$	−10.6066	+	6.12372i	−0.916271	+	0.529009i
$135$			5.19615i			0.447214i
$136$		−	13.8564i		−	1.18818i
$137$	19.0000			1.62328			0.811640	−	0.584158i	$-0.198575\pi$
$137$	19.0000			1.62328			0.811640	+	0.584158i	$0.198575\pi$
$138$	−6.36396	−	11.0227i	−0.541736	−	0.938315i
$139$	19.7990			1.67933			0.839664	−	0.543106i	$-0.182752\pi$
$139$	19.7990			1.67933			0.839664	+	0.543106i	$0.182752\pi$
$140$	2.82843	−	4.89898i	0.239046	−	0.414039i
$141$	−6.00000			−0.505291
$142$	14.8492	−	8.57321i	1.24612	−	0.719448i
$143$	−8.48528	−	13.8564i	−0.709575	−	1.15873i
$144$		0			0
$145$		0			0
$146$	−6.00000	+	3.46410i	−0.496564	+	0.286691i
$147$		−	1.73205i		−	0.142857i
$148$	−3.00000	+	5.19615i	−0.246598	+	0.427121i
$149$			14.6969i			1.20402i	0.798489	+	0.602010i	$0.205633\pi$
$149$			14.6969i			1.20402i	−0.798489	+	0.602010i	$0.794367\pi$
$150$	8.48528	−	4.89898i	0.692820	−	0.400000i
$151$	−5.65685			−0.460348			−0.230174	−	0.973149i	$-0.573930\pi$
$151$	−5.65685			−0.460348			−0.230174	+	0.973149i	$0.573930\pi$
$152$	−8.00000			−0.648886
$153$		0			0
$154$	11.6569	+	6.33386i	0.939336	+	0.510397i
$155$			1.73205i			0.139122i
$156$	−8.48528	+	14.6969i	−0.679366	+	1.17670i
$157$	11.0000			0.877896			0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000			0.877896			0.438948	+	0.898513i	$0.355351\pi$
$158$	8.00000	+	13.8564i	0.636446	+	1.10236i
$159$			3.46410i			0.274721i
$160$	2.82843	−	4.89898i	0.223607	−	0.387298i
$161$			14.6969i			1.15828i
$162$	6.36396	+	11.0227i	0.500000	+	0.866025i
$163$			17.3205i			1.35665i	0.734763	+	0.678323i	$0.237293\pi$
$163$			17.3205i			1.35665i	−0.734763	+	0.678323i	$0.762707\pi$
$164$	−8.48528	−	4.89898i	−0.662589	−	0.382546i
$165$	−3.00000	−	4.89898i	−0.233550	−	0.381385i
$166$		0			0
$167$	19.7990			1.53209			0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990			1.53209			0.766046	+	0.642786i	$0.222221\pi$
$168$		−	13.8564i		−	1.06904i
$169$	−11.0000			−0.846154
$170$	6.00000	−	3.46410i	0.460179	−	0.265684i
$171$		0			0
$172$	−5.65685	+	9.79796i	−0.431331	+	0.747087i
$173$		−	9.79796i		−	0.744925i	−0.928047	−	0.372463i	$-0.878514\pi$
$173$		−	9.79796i		−	0.744925i	0.928047	−	0.372463i	$-0.121486\pi$
$174$		0			0
$175$	−11.3137			−0.855236
$176$	11.6569	+	6.33386i	0.878668	+	0.477432i
$177$	−3.00000			−0.225494
$178$	0.707107	+	1.22474i	0.0529999	+	0.0917985i
$179$		−	8.66025i		−	0.647298i	−0.946177	−	0.323649i	$-0.895090\pi$
$179$		−	8.66025i		−	0.647298i	0.946177	−	0.323649i	$-0.104910\pi$
$180$		0			0
$181$	−21.0000			−1.56092			−0.780459	−	0.625207i	$-0.785014\pi$
$181$	−21.0000			−1.56092			−0.780459	+	0.625207i	$0.785014\pi$
$182$	16.9706	−	9.79796i	1.25794	−	0.726273i
$183$	−16.9706			−1.25450
$184$			14.6969i			1.08347i
$185$	−3.00000			−0.220564
$186$	2.12132	+	3.67423i	0.155543	+	0.269408i
$187$	8.48528	+	13.8564i	0.620505	+	1.01328i
$188$	6.00000	+	3.46410i	0.437595	+	0.252646i
$189$		−	14.6969i		−	1.06904i
$190$	−2.00000	−	3.46410i	−0.145095	−	0.251312i
$191$			5.19615i			0.375980i	0.982171	+	0.187990i	$0.0601973\pi$
$191$			5.19615i			0.375980i	−0.982171	+	0.187990i	$0.939803\pi$
$192$		−	13.8564i		−	1.00000i
$193$			9.79796i			0.705273i	0.935760	+	0.352636i	$0.114715\pi$
$193$			9.79796i			0.705273i	−0.935760	+	0.352636i	$0.885285\pi$
$194$	−4.94975	−	8.57321i	−0.355371	−	0.615521i
$195$	−8.48528			−0.607644
$196$	−1.00000	+	1.73205i	−0.0714286	+	0.123718i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$			10.3923i			0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$			10.3923i			0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$	−11.3137			−0.800000
$201$	−15.0000			−1.05802
$202$	−6.00000	+	3.46410i	−0.422159	+	0.243733i
$203$		0			0
$204$	8.48528	−	14.6969i	0.594089	−	1.02899i
$205$		−	4.89898i		−	0.342160i
$206$	−4.24264	+	2.44949i	−0.295599	+	0.170664i
$207$		0			0
$208$	16.9706	−	9.79796i	1.17670	−	0.679366i
$209$	8.00000	−	4.89898i	0.553372	−	0.338869i
$210$	6.00000	−	3.46410i	0.414039	−	0.239046i
$211$	5.65685			0.389434			0.194717	−	0.980859i	$-0.437621\pi$
$211$	5.65685			0.389434			0.194717	+	0.980859i	$0.437621\pi$
$212$	2.00000	−	3.46410i	0.137361	−	0.237915i
$213$	21.0000			1.43890
$214$	−6.00000	−	10.3923i	−0.410152	−	0.710403i
$215$	−5.65685			−0.385794
$216$		−	14.6969i		−	1.00000i
$217$		−	4.89898i		−	0.332564i
$218$	12.0000	−	6.92820i	0.812743	−	0.469237i
$219$	−8.48528			−0.573382
$220$	0.171573	+	6.63103i	0.0115674	+	0.447064i
$221$	24.0000			1.61441
$222$	−6.36396	+	3.67423i	−0.427121	+	0.246598i
$223$		−	22.5167i		−	1.50783i	−0.656974	−	0.753914i	$-0.728164\pi$
$223$		−	22.5167i		−	1.50783i	0.656974	−	0.753914i	$-0.271836\pi$
$224$	−8.00000	+	13.8564i	−0.534522	+	0.925820i
$225$		0			0
$226$	3.53553	+	6.12372i	0.235180	+	0.407344i
$227$	−8.48528			−0.563188			−0.281594	−	0.959534i	$-0.590863\pi$
$227$	−8.48528			−0.563188			−0.281594	+	0.959534i	$0.590863\pi$
$228$	−8.48528	−	4.89898i	−0.561951	−	0.324443i
$229$	−1.00000			−0.0660819			−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000			−0.0660819			−0.0330409	+	0.999454i	$0.510519\pi$
$230$	−6.36396	+	3.67423i	−0.419627	+	0.242272i
$231$	8.48528	+	13.8564i	0.558291	+	0.911685i
$232$		0			0
$233$		−	19.5959i		−	1.28377i	−0.766800	−	0.641886i	$-0.778152\pi$
$233$		−	19.5959i		−	1.28377i	0.766800	−	0.641886i	$-0.221848\pi$
$234$		0			0
$235$			3.46410i			0.225973i
$236$	3.00000	+	1.73205i	0.195283	+	0.112747i
$237$			19.5959i			1.27289i
$238$	−16.9706	+	9.79796i	−1.10004	+	0.635107i
$239$	5.65685			0.365911			0.182956	−	0.983121i	$-0.441433\pi$
$239$	5.65685			0.365911			0.182956	+	0.983121i	$0.441433\pi$
$240$	6.00000	−	3.46410i	0.387298	−	0.223607i
$241$			29.3939i			1.89343i	0.322078	+	0.946713i	$0.395619\pi$
$241$			29.3939i			1.89343i	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−15.5355	+	0.804479i	−0.998662	+	0.0517139i
$243$		0			0
$244$	16.9706	+	9.79796i	1.08643	+	0.627250i
$245$	−1.00000			−0.0638877
$246$	−6.00000	−	10.3923i	−0.382546	−	0.662589i
$247$		−	13.8564i		−	0.881662i
$248$		−	4.89898i		−	0.311086i
$249$		0			0
$250$	−6.36396	−	11.0227i	−0.402492	−	0.697137i
$251$		−	22.5167i		−	1.42124i	−0.703577	−	0.710620i	$-0.748415\pi$
$251$		−	22.5167i		−	1.42124i	0.703577	−	0.710620i	$-0.251585\pi$
$252$		0			0
$253$	−9.00000	−	14.6969i	−0.565825	−	0.923989i
$254$	10.0000	+	17.3205i	0.627456	+	1.08679i
$255$	8.48528			0.531369
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	−22.0000			−1.37232			−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000			−1.37232			−0.686161	+	0.727450i	$0.740706\pi$
$258$	−12.0000	+	6.92820i	−0.747087	+	0.431331i
$259$	8.48528			0.527250
$260$	8.48528	+	4.89898i	0.526235	+	0.303822i
$261$		0			0
$262$	2.00000	+	3.46410i	0.123560	+	0.214013i
$263$	14.1421			0.872041			0.436021	−	0.899937i	$-0.356387\pi$
$263$	14.1421			0.872041			0.436021	+	0.899937i	$0.356387\pi$
$264$	8.48528	+	13.8564i	0.522233	+	0.852803i
$265$	2.00000			0.122859
$266$	5.65685	+	9.79796i	0.346844	+	0.600751i
$267$			1.73205i			0.106000i
$268$	15.0000	+	8.66025i	0.916271	+	0.529009i
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$	6.36396	−	3.67423i	0.387298	−	0.223607i
$271$	−2.82843			−0.171815			−0.0859074	−	0.996303i	$-0.527379\pi$
$271$	−2.82843			−0.171815			−0.0859074	+	0.996303i	$0.527379\pi$
$272$	−16.9706	+	9.79796i	−1.02899	+	0.594089i
$273$	24.0000			1.45255
$274$	−13.4350	−	23.2702i	−0.811640	−	1.40580i
$275$	11.3137	−	6.92820i	0.682242	−	0.417786i
$276$	−9.00000	+	15.5885i	−0.541736	+	0.938315i
$277$			9.79796i			0.588702i	0.955697	+	0.294351i	$0.0951035\pi$
$277$			9.79796i			0.588702i	−0.955697	+	0.294351i	$0.904896\pi$
$278$	−14.0000	−	24.2487i	−0.839664	−	1.45434i
$279$		0			0
$280$	−8.00000			−0.478091
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	4.24264	+	7.34847i	0.252646	+	0.437595i
$283$	11.3137			0.672530			0.336265	−	0.941767i	$-0.390836\pi$
$283$	11.3137			0.672530			0.336265	+	0.941767i	$0.390836\pi$
$284$	−21.0000	−	12.1244i	−1.24612	−	0.719448i
$285$		−	4.89898i		−	0.290191i
$286$	−10.9706	+	20.1903i	−0.648703	+	1.19388i
$287$			13.8564i			0.817918i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$		−	12.1244i		−	0.710742i
$292$	8.48528	+	4.89898i	0.496564	+	0.286691i
$293$			4.89898i			0.286201i	0.989708	+	0.143101i	$0.0457073\pi$
$293$			4.89898i			0.286201i	−0.989708	+	0.143101i	$0.954293\pi$
$294$	−2.12132	+	1.22474i	−0.123718	+	0.0714286i
$295$			1.73205i			0.100844i
$296$	8.48528			0.493197
$297$	9.00000	+	14.6969i	0.522233	+	0.852803i
$298$	18.0000	−	10.3923i	1.04271	−	0.602010i
$299$	−25.4558			−1.47215
$300$	−12.0000	−	6.92820i	−0.692820	−	0.400000i
$301$	16.0000			0.922225
$302$	4.00000	+	6.92820i	0.230174	+	0.398673i
$303$	−8.48528			−0.487467
$304$	5.65685	+	9.79796i	0.324443	+	0.561951i
$305$			9.79796i			0.561029i
$306$		0			0
$307$	−19.7990			−1.12999			−0.564994	−	0.825095i	$-0.691122\pi$
$307$	−19.7990			−1.12999			−0.564994	+	0.825095i	$0.691122\pi$
$308$	−0.485281	−	18.7554i	−0.0276515	−	1.06869i
$309$	−6.00000			−0.341328
$310$	2.12132	−	1.22474i	0.120483	−	0.0695608i
$311$			10.3923i			0.589294i	0.955606	+	0.294647i	$0.0952020\pi$
$311$			10.3923i			0.589294i	−0.955606	+	0.294647i	$0.904798\pi$
$312$	24.0000			1.35873
$313$	27.0000			1.52613			0.763065	−	0.646322i	$-0.223694\pi$
$313$	27.0000			1.52613			0.763065	+	0.646322i	$0.223694\pi$
$314$	−7.77817	−	13.4722i	−0.438948	−	0.760280i
$315$		0			0
$316$	11.3137	−	19.5959i	0.636446	−	1.10236i
$317$	−25.0000			−1.40414			−0.702070	−	0.712108i	$-0.747741\pi$
$317$	−25.0000			−1.40414			−0.702070	+	0.712108i	$0.747741\pi$
$318$	4.24264	−	2.44949i	0.237915	−	0.137361i
$319$		0			0
$320$	−8.00000			−0.447214
$321$		−	14.6969i		−	0.820303i
$322$	18.0000	−	10.3923i	1.00310	−	0.579141i
$323$			13.8564i			0.770991i
$324$	9.00000	−	15.5885i	0.500000	−	0.866025i
$325$		−	19.5959i		−	1.08699i
$326$	21.2132	−	12.2474i	1.17489	−	0.678323i
$327$	16.9706			0.938474
$328$			13.8564i			0.765092i
$329$		−	9.79796i		−	0.540179i
$330$	−3.87868	+	7.13834i	−0.213514	+	0.392952i
$331$			5.19615i			0.285606i	0.989751	+	0.142803i	$0.0456116\pi$
$331$			5.19615i			0.285606i	−0.989751	+	0.142803i	$0.954388\pi$
$332$		0			0
$333$		0			0
$334$	−14.0000	−	24.2487i	−0.766046	−	1.32683i
$335$			8.66025i			0.473160i
$336$	−16.9706	+	9.79796i	−0.925820	+	0.534522i
$337$		−	34.2929i		−	1.86805i	−0.357206	−	0.934025i	$-0.616271\pi$
$337$		−	34.2929i		−	1.86805i	0.357206	−	0.934025i	$-0.383729\pi$
$338$	7.77817	+	13.4722i	0.423077	+	0.732791i
$339$			8.66025i			0.470360i
$340$	−8.48528	−	4.89898i	−0.460179	−	0.265684i
$341$	3.00000	+	4.89898i	0.162459	+	0.265295i
$342$		0			0
$343$	−16.9706			−0.916324
$344$	16.0000			0.862662
$345$	−9.00000			−0.484544
$346$	−12.0000	+	6.92820i	−0.645124	+	0.372463i
$347$	−25.4558			−1.36654			−0.683271	−	0.730165i	$-0.739443\pi$
$347$	−25.4558			−1.36654			−0.683271	+	0.730165i	$0.739443\pi$
$348$		0			0
$349$			4.89898i			0.262236i	0.991367	+	0.131118i	$0.0418567\pi$
$349$			4.89898i			0.262236i	−0.991367	+	0.131118i	$0.958143\pi$
$350$	8.00000	+	13.8564i	0.427618	+	0.740656i
$351$	25.4558			1.35873
$352$	−0.485281	−	18.7554i	−0.0258656	−	0.999665i
$353$	−17.0000			−0.904819			−0.452409	−	0.891810i	$-0.649435\pi$
$353$	−17.0000			−0.904819			−0.452409	+	0.891810i	$0.649435\pi$
$354$	2.12132	+	3.67423i	0.112747	+	0.195283i
$355$		−	12.1244i		−	0.643494i
$356$	1.00000	−	1.73205i	0.0529999	−	0.0917985i
$357$	−24.0000			−1.27021
$358$	−10.6066	+	6.12372i	−0.560576	+	0.323649i
$359$	8.48528			0.447836			0.223918	−	0.974608i	$-0.428115\pi$
$359$	8.48528			0.447836			0.223918	+	0.974608i	$0.428115\pi$
$360$		0			0
$361$	−11.0000			−0.578947
$362$	14.8492	+	25.7196i	0.780459	+	1.35179i
$363$	−16.9706	−	8.66025i	−0.890724	−	0.454545i
$364$	−24.0000	−	13.8564i	−1.25794	−	0.726273i
$365$			4.89898i			0.256424i
$366$	12.0000	+	20.7846i	0.627250	+	1.08643i
$367$		−	1.73205i		−	0.0904123i	−0.998978	−	0.0452062i	$-0.985606\pi$
$367$		−	1.73205i		−	0.0904123i	0.998978	−	0.0452062i	$-0.0143945\pi$
$368$	18.0000	−	10.3923i	0.938315	−	0.541736i
$369$		0			0
$370$	2.12132	+	3.67423i	0.110282	+	0.191014i
$371$	−5.65685			−0.293689
$372$	3.00000	−	5.19615i	0.155543	−	0.269408i
$373$			14.6969i			0.760979i	0.924785	+	0.380489i	$0.124244\pi$
$373$			14.6969i			0.760979i	−0.924785	+	0.380489i	$0.875756\pi$
$374$	10.9706	−	20.1903i	0.567274	−	1.04401i
$375$		−	15.5885i		−	0.804984i
$376$		−	9.79796i		−	0.505291i
$377$		0			0
$378$	−18.0000	+	10.3923i	−0.925820	+	0.534522i
$379$			12.1244i			0.622786i	0.950281	+	0.311393i	$0.100796\pi$
$379$			12.1244i			0.622786i	−0.950281	+	0.311393i	$0.899204\pi$
$380$	−2.82843	+	4.89898i	−0.145095	+	0.251312i
$381$			24.4949i			1.25491i
$382$	6.36396	−	3.67423i	0.325609	−	0.187990i
$383$			32.9090i			1.68157i	0.541370	+	0.840785i	$0.317906\pi$
$383$			32.9090i			1.68157i	−0.541370	+	0.840785i	$0.682094\pi$
$384$	−16.9706	+	9.79796i	−0.866025	+	0.500000i
$385$	8.00000	−	4.89898i	0.407718	−	0.249675i
$386$	12.0000	−	6.92820i	0.610784	−	0.352636i
$387$		0			0
$388$	−7.00000	+	12.1244i	−0.355371	+	0.615521i
$389$	19.0000			0.963338			0.481669	−	0.876353i	$-0.340031\pi$
$389$	19.0000			0.963338			0.481669	+	0.876353i	$0.340031\pi$
$390$	6.00000	+	10.3923i	0.303822	+	0.526235i
$391$	25.4558			1.28736
$392$	2.82843			0.142857
$393$			4.89898i			0.247121i
$394$		0			0
$395$	11.3137			0.569254
$396$		0			0
$397$	6.00000			0.301131			0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000			0.301131			0.150566	+	0.988600i	$0.451890\pi$
$398$	12.7279	−	7.34847i	0.637993	−	0.368345i
$399$			13.8564i			0.693688i
$400$	8.00000	+	13.8564i	0.400000	+	0.692820i
$401$	10.0000			0.499376			0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000			0.499376			0.249688	+	0.968326i	$0.419672\pi$
$402$	10.6066	+	18.3712i	0.529009	+	0.916271i
$403$	8.48528			0.422682
$404$	8.48528	+	4.89898i	0.422159	+	0.243733i
$405$	9.00000			0.447214
$406$		0			0
$407$	−8.48528	+	5.19615i	−0.420600	+	0.257564i
$408$	−24.0000			−1.18818
$409$			29.3939i			1.45343i	0.686937	+	0.726717i	$0.258955\pi$
$409$			29.3939i			1.45343i	−0.686937	+	0.726717i	$0.741045\pi$
$410$	−6.00000	+	3.46410i	−0.296319	+	0.171080i
$411$		−	32.9090i		−	1.62328i
$412$	6.00000	+	3.46410i	0.295599	+	0.170664i
$413$		−	4.89898i		−	0.241063i
$414$		0			0
$415$		0			0
$416$	−24.0000	−	13.8564i	−1.17670	−	0.679366i
$417$		−	34.2929i		−	1.67933i
$418$	−11.6569	−	6.33386i	−0.570155	−	0.309799i
$419$		−	10.3923i		−	0.507697i	−0.967244	−	0.253849i	$-0.918303\pi$
$419$		−	10.3923i		−	0.507697i	0.967244	−	0.253849i	$-0.0816965\pi$
$420$	−8.48528	−	4.89898i	−0.414039	−	0.239046i
$421$	30.0000			1.46211			0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000			1.46211			0.731055	+	0.682318i	$0.239028\pi$
$422$	−4.00000	−	6.92820i	−0.194717	−	0.337260i
$423$		0			0
$424$	−5.65685			−0.274721
$425$			19.5959i			0.950542i
$426$	−14.8492	−	25.7196i	−0.719448	−	1.24612i
$427$		−	27.7128i		−	1.34112i
$428$	−8.48528	+	14.6969i	−0.410152	+	0.710403i
$429$	−24.0000	+	14.6969i	−1.15873	+	0.709575i
$430$	4.00000	+	6.92820i	0.192897	+	0.334108i
$431$	14.1421			0.681203			0.340601	−	0.940208i	$-0.389369\pi$
$431$	14.1421			0.681203			0.340601	+	0.940208i	$0.389369\pi$
$432$	−18.0000	+	10.3923i	−0.866025	+	0.500000i
$433$	−21.0000			−1.00920			−0.504598	−	0.863355i	$-0.668359\pi$
$433$	−21.0000			−1.00920			−0.504598	+	0.863355i	$0.668359\pi$
$434$	−6.00000	+	3.46410i	−0.288009	+	0.166282i
$435$		0			0
$436$	−16.9706	−	9.79796i	−0.812743	−	0.469237i
$437$		−	14.6969i		−	0.703050i
$438$	6.00000	+	10.3923i	0.286691	+	0.496564i
$439$	−22.6274			−1.07995			−0.539974	−	0.841682i	$-0.681566\pi$
$439$	−22.6274			−1.07995			−0.539974	+	0.841682i	$0.681566\pi$
$440$	8.00000	−	4.89898i	0.381385	−	0.233550i
$441$		0			0
$442$	−16.9706	−	29.3939i	−0.807207	−	1.39812i
$443$			5.19615i			0.246877i	0.992352	+	0.123438i	$0.0393921\pi$
$443$			5.19615i			0.246877i	−0.992352	+	0.123438i	$0.960608\pi$
$444$	9.00000	+	5.19615i	0.427121	+	0.246598i
$445$	1.00000			0.0474045
$446$	−27.5772	+	15.9217i	−1.30582	+	0.753914i
$447$	25.4558			1.20402
$448$	22.6274			1.06904
$449$	23.0000			1.08544			0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000			1.08544			0.542719	+	0.839915i	$0.317395\pi$
$450$		0			0
$451$	−8.48528	−	13.8564i	−0.399556	−	0.652473i
$452$	5.00000	−	8.66025i	0.235180	−	0.407344i
$453$			9.79796i			0.460348i
$454$	6.00000	+	10.3923i	0.281594	+	0.487735i
$455$		−	13.8564i		−	0.649598i
$456$			13.8564i			0.648886i
$457$		−	19.5959i		−	0.916658i	−0.888783	−	0.458329i	$-0.848448\pi$
$457$		−	19.5959i		−	0.916658i	0.888783	−	0.458329i	$-0.151552\pi$
$458$	0.707107	+	1.22474i	0.0330409	+	0.0572286i
$459$	−25.4558			−1.18818
$460$	9.00000	+	5.19615i	0.419627	+	0.242272i
$461$		−	29.3939i		−	1.36901i	−0.729008	−	0.684505i	$-0.760019\pi$
$461$		−	29.3939i		−	1.36901i	0.729008	−	0.684505i	$-0.239981\pi$
$462$	10.9706	−	20.1903i	0.510397	−	0.939336i
$463$		−	36.3731i		−	1.69040i	−0.534450	−	0.845200i	$-0.679481\pi$
$463$		−	36.3731i		−	1.69040i	0.534450	−	0.845200i	$-0.320519\pi$
$464$		0			0
$465$	3.00000			0.139122
$466$	−24.0000	+	13.8564i	−1.11178	+	0.641886i
$467$			32.9090i			1.52285i	0.648256	+	0.761423i	$0.275499\pi$
$467$			32.9090i			1.52285i	−0.648256	+	0.761423i	$0.724501\pi$
$468$		0			0
$469$		−	24.4949i		−	1.13107i
$470$	4.24264	−	2.44949i	0.195698	−	0.112987i
$471$		−	19.0526i		−	0.877896i
$472$		−	4.89898i		−	0.225494i
$473$	−16.0000	+	9.79796i	−0.735681	+	0.450511i
$474$	24.0000	−	13.8564i	1.10236	−	0.636446i
$475$	11.3137			0.519109
$476$	24.0000	+	13.8564i	1.10004	+	0.635107i
$477$		0			0
$478$	−4.00000	−	6.92820i	−0.182956	−	0.316889i
$479$	31.1127			1.42158			0.710788	−	0.703407i	$-0.248339\pi$
$479$	31.1127			1.42158			0.710788	+	0.703407i	$0.248339\pi$
$480$	−8.48528	−	4.89898i	−0.387298	−	0.223607i
$481$			14.6969i			0.670123i
$482$	36.0000	−	20.7846i	1.63976	−	0.946713i
$483$	25.4558			1.15828
$484$	11.9706	+	18.4582i	0.544116	+	0.839010i
$485$	−7.00000			−0.317854
$486$		0			0
$487$			39.8372i			1.80519i	0.430486	+	0.902597i	$0.358342\pi$
$487$			39.8372i			1.80519i	−0.430486	+	0.902597i	$0.641658\pi$
$488$		−	27.7128i		−	1.25450i
$489$	30.0000			1.35665
$490$	0.707107	+	1.22474i	0.0319438	+	0.0553283i
$491$	−33.9411			−1.53174			−0.765871	−	0.642995i	$-0.777692\pi$
$491$	−33.9411			−1.53174			−0.765871	+	0.642995i	$0.777692\pi$
$492$	−8.48528	+	14.6969i	−0.382546	+	0.662589i
$493$		0			0
$494$	−16.9706	+	9.79796i	−0.763542	+	0.440831i
$495$		0			0
$496$	−6.00000	+	3.46410i	−0.269408	+	0.155543i
$497$			34.2929i			1.53824i
$498$		0			0
$499$			17.3205i			0.775372i	0.921791	+	0.387686i	$0.126726\pi$
$499$			17.3205i			0.775372i	−0.921791	+	0.387686i	$0.873274\pi$
$500$	−9.00000	+	15.5885i	−0.402492	+	0.697137i
$501$		−	34.2929i		−	1.53209i
$502$	−27.5772	+	15.9217i	−1.23083	+	0.710620i
$503$	−39.5980			−1.76559			−0.882793	−	0.469762i	$-0.844340\pi$
$503$	−39.5980			−1.76559			−0.882793	+	0.469762i	$0.844340\pi$
$504$		0			0
$505$			4.89898i			0.218002i
$506$	−11.6360	+	21.4150i	−0.517285	+	0.952013i
$507$			19.0526i			0.846154i
$508$	14.1421	−	24.4949i	0.627456	−	1.08679i
$509$	−1.00000			−0.0443242			−0.0221621	−	0.999754i	$-0.507055\pi$
$509$	−1.00000			−0.0443242			−0.0221621	+	0.999754i	$0.507055\pi$
$510$	−6.00000	−	10.3923i	−0.265684	−	0.460179i
$511$		−	13.8564i		−	0.612971i
$512$	22.6274			1.00000
$513$			14.6969i			0.648886i
$514$	15.5563	+	26.9444i	0.686161	+	1.18847i
$515$			3.46410i			0.152647i
$516$	16.9706	+	9.79796i	0.747087	+	0.431331i
$517$	6.00000	+	9.79796i	0.263880	+	0.430914i
$518$	−6.00000	−	10.3923i	−0.263625	−	0.456612i
$519$	−16.9706			−0.744925
$520$		−	13.8564i		−	0.607644i
$521$	−17.0000			−0.744784			−0.372392	−	0.928076i	$-0.621462\pi$
$521$	−17.0000			−0.744784			−0.372392	+	0.928076i	$0.621462\pi$
$522$		0			0
$523$	36.7696			1.60782			0.803910	−	0.594751i	$-0.202749\pi$
$523$	36.7696			1.60782			0.803910	+	0.594751i	$0.202749\pi$
$524$	2.82843	−	4.89898i	0.123560	−	0.214013i
$525$			19.5959i			0.855236i
$526$	−10.0000	−	17.3205i	−0.436021	−	0.755210i
$527$	−8.48528			−0.369625
$528$	10.9706	−	20.1903i	0.477432	−	0.878668i
$529$	−4.00000			−0.173913
$530$	−1.41421	−	2.44949i	−0.0614295	−	0.106399i
$531$		0			0
$532$	8.00000	−	13.8564i	0.346844	−	0.600751i
$533$	−24.0000			−1.03956
$534$	2.12132	−	1.22474i	0.0917985	−	0.0529999i
$535$	−8.48528			−0.366851
$536$		−	24.4949i		−	1.05802i
$537$	−15.0000			−0.647298
$538$	7.07107	+	12.2474i	0.304855	+	0.528025i
$539$	−2.82843	+	1.73205i	−0.121829	+	0.0746047i
$540$	−9.00000	−	5.19615i	−0.387298	−	0.223607i
$541$			14.6969i			0.631871i	0.948781	+	0.315935i	$0.102318\pi$
$541$			14.6969i			0.631871i	−0.948781	+	0.315935i	$0.897682\pi$
$542$	2.00000	+	3.46410i	0.0859074	+	0.148796i
$543$			36.3731i			1.56092i
$544$	24.0000	+	13.8564i	1.02899	+	0.594089i
$545$		−	9.79796i		−	0.419698i
$546$	−16.9706	−	29.3939i	−0.726273	−	1.25794i
$547$	5.65685			0.241870			0.120935	−	0.992660i	$-0.461411\pi$
$547$	5.65685			0.241870			0.120935	+	0.992660i	$0.461411\pi$
$548$	−19.0000	+	32.9090i	−0.811640	+	1.40580i
$549$		0			0
$550$	−16.4853	−	8.95743i	−0.702935	−	0.381946i
$551$		0			0
$552$	25.4558			1.08347
$553$	−32.0000			−1.36078
$554$	12.0000	−	6.92820i	0.509831	−	0.294351i
$555$			5.19615i			0.220564i
$556$	−19.7990	+	34.2929i	−0.839664	+	1.45434i
$557$		−	24.4949i		−	1.03788i	−0.854810	−	0.518941i	$-0.826326\pi$
$557$		−	24.4949i		−	1.03788i	0.854810	−	0.518941i	$-0.173674\pi$
$558$		0			0
$559$			27.7128i			1.17213i
$560$	5.65685	+	9.79796i	0.239046	+	0.414039i
$561$	24.0000	−	14.6969i	1.01328	−	0.620505i
$562$		0			0
$563$	11.3137			0.476816			0.238408	−	0.971165i	$-0.423374\pi$
$563$	11.3137			0.476816			0.238408	+	0.971165i	$0.423374\pi$
$564$	6.00000	−	10.3923i	0.252646	−	0.437595i
$565$	5.00000			0.210352
$566$	−8.00000	−	13.8564i	−0.336265	−	0.582428i
$567$	−25.4558			−1.06904
$568$			34.2929i			1.43890i
$569$		−	19.5959i		−	0.821504i	−0.911747	−	0.410752i	$-0.865266\pi$
$569$		−	19.5959i		−	0.821504i	0.911747	−	0.410752i	$-0.134734\pi$
$570$	−6.00000	+	3.46410i	−0.251312	+	0.145095i
$571$	−5.65685			−0.236732			−0.118366	−	0.992970i	$-0.537766\pi$
$571$	−5.65685			−0.236732			−0.118366	+	0.992970i	$0.537766\pi$
$572$	32.4853	−	0.840532i	1.35828	−	0.0351444i
$573$	9.00000			0.375980
$574$	16.9706	−	9.79796i	0.708338	−	0.408959i
$575$		−	20.7846i		−	0.866778i
$576$		0			0
$577$	−17.0000			−0.707719			−0.353860	−	0.935299i	$-0.615131\pi$
$577$	−17.0000			−0.707719			−0.353860	+	0.935299i	$0.615131\pi$
$578$	4.94975	+	8.57321i	0.205882	+	0.356599i
$579$	16.9706			0.705273
$580$		0			0
$581$		0			0
$582$	−14.8492	+	8.57321i	−0.615521	+	0.355371i
$583$	5.65685	−	3.46410i	0.234283	−	0.143468i
$584$		−	13.8564i		−	0.573382i
$585$		0			0
$586$	6.00000	−	3.46410i	0.247858	−	0.143101i
$587$		−	10.3923i		−	0.428936i	−0.976731	−	0.214468i	$-0.931198\pi$
$587$		−	10.3923i		−	0.428936i	0.976731	−	0.214468i	$-0.0688018\pi$
$588$	3.00000	+	1.73205i	0.123718	+	0.0714286i
$589$			4.89898i			0.201859i
$590$	2.12132	−	1.22474i	0.0873334	−	0.0504219i
$591$		0			0
$592$	−6.00000	−	10.3923i	−0.246598	−	0.427121i
$593$		−	9.79796i		−	0.402354i	−0.979555	−	0.201177i	$-0.935523\pi$
$593$		−	9.79796i		−	0.402354i	0.979555	−	0.201177i	$-0.0644766\pi$
$594$	11.6360	−	21.4150i	0.477432	−	0.878668i
$595$			13.8564i			0.568057i
$596$	−25.4558	−	14.6969i	−1.04271	−	0.602010i
$597$	18.0000			0.736691
$598$	18.0000	+	31.1769i	0.736075	+	1.27492i
$599$		−	45.0333i		−	1.84001i	−0.391905	−	0.920006i	$-0.628184\pi$
$599$		−	45.0333i		−	1.84001i	0.391905	−	0.920006i	$-0.371816\pi$
$600$			19.5959i			0.800000i
$601$			4.89898i			0.199834i	0.994996	+	0.0999168i	$0.0318577\pi$
$601$			4.89898i			0.199834i	−0.994996	+	0.0999168i	$0.968142\pi$
$602$	−11.3137	−	19.5959i	−0.461112	−	0.798670i
$603$		0			0
$604$	5.65685	−	9.79796i	0.230174	−	0.398673i
$605$	−5.00000	+	9.79796i	−0.203279	+	0.398344i
$606$	6.00000	+	10.3923i	0.243733	+	0.422159i
$607$	−22.6274			−0.918419			−0.459209	−	0.888328i	$-0.651867\pi$
$607$	−22.6274			−0.918419			−0.459209	+	0.888328i	$0.651867\pi$
$608$	8.00000	−	13.8564i	0.324443	−	0.561951i
$609$		0			0
$610$	12.0000	−	6.92820i	0.485866	−	0.280515i
$611$	16.9706			0.686555
$612$		0			0
$613$			9.79796i			0.395736i	0.980229	+	0.197868i	$0.0634017\pi$
$613$			9.79796i			0.395736i	−0.980229	+	0.197868i	$0.936598\pi$
$614$	14.0000	+	24.2487i	0.564994	+	0.978598i
$615$	−8.48528			−0.342160
$616$	−22.6274	+	13.8564i	−0.911685	+	0.558291i
$617$	2.00000			0.0805170			0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000			0.0805170			0.0402585	+	0.999189i	$0.487182\pi$
$618$	4.24264	+	7.34847i	0.170664	+	0.295599i
$619$		−	1.73205i		−	0.0696170i	−0.999394	−	0.0348085i	$-0.988918\pi$
$619$		−	1.73205i		−	0.0696170i	0.999394	−	0.0348085i	$-0.0110821\pi$
$620$	−3.00000	−	1.73205i	−0.120483	−	0.0695608i
$621$	27.0000			1.08347
$622$	12.7279	−	7.34847i	0.510343	−	0.294647i
$623$	−2.82843			−0.113319
$624$	−16.9706	−	29.3939i	−0.679366	−	1.17670i
$625$	11.0000			0.440000
$626$	−19.0919	−	33.0681i	−0.763065	−	1.32167i
$627$	−8.48528	−	13.8564i	−0.338869	−	0.553372i
$628$	−11.0000	+	19.0526i	−0.438948	+	0.760280i
$629$		−	14.6969i		−	0.586005i
$630$		0			0
$631$		−	36.3731i		−	1.44799i	−0.689806	−	0.723994i	$-0.742304\pi$
$631$		−	36.3731i		−	1.44799i	0.689806	−	0.723994i	$-0.257696\pi$
$632$	−32.0000			−1.27289
$633$		−	9.79796i		−	0.389434i
$634$	17.6777	+	30.6186i	0.702070	+	1.21602i
$635$	14.1421			0.561214
$636$	−6.00000	−	3.46410i	−0.237915	−	0.137361i
$637$			4.89898i			0.194105i
$638$		0			0
$639$		0			0
$640$	5.65685	+	9.79796i	0.223607	+	0.387298i
$641$	19.0000			0.750455			0.375227	−	0.926933i	$-0.377565\pi$
$641$	19.0000			0.750455			0.375227	+	0.926933i	$0.377565\pi$
$642$	−18.0000	+	10.3923i	−0.710403	+	0.410152i
$643$			25.9808i			1.02458i	0.858812	+	0.512291i	$0.171203\pi$
$643$			25.9808i			1.02458i	−0.858812	+	0.512291i	$0.828797\pi$
$644$	−25.4558	−	14.6969i	−1.00310	−	0.579141i
$645$			9.79796i			0.385794i
$646$	16.9706	−	9.79796i	0.667698	−	0.385496i
$647$		−	1.73205i		−	0.0680939i	−0.999420	−	0.0340470i	$-0.989160\pi$
$647$		−	1.73205i		−	0.0680939i	0.999420	−	0.0340470i	$-0.0108396\pi$
$648$	−25.4558			−1.00000
$649$	3.00000	+	4.89898i	0.117760	+	0.192302i
$650$	−24.0000	+	13.8564i	−0.941357	+	0.543493i
$651$	−8.48528			−0.332564
$652$	−30.0000	−	17.3205i	−1.17489	−	0.678323i
$653$	31.0000			1.21312			0.606562	−	0.795036i	$-0.292548\pi$
$653$	31.0000			1.21312			0.606562	+	0.795036i	$0.292548\pi$
$654$	−12.0000	−	20.7846i	−0.469237	−	0.812743i
$655$	2.82843			0.110516
$656$	16.9706	−	9.79796i	0.662589	−	0.382546i
$657$		0			0
$658$	−12.0000	+	6.92820i	−0.467809	+	0.270089i
$659$	45.2548			1.76288			0.881439	−	0.472298i	$-0.156575\pi$
$659$	45.2548			1.76288			0.881439	+	0.472298i	$0.156575\pi$
$660$	11.4853	−	0.297173i	0.447064	−	0.0115674i
$661$	−17.0000			−0.661223			−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000			−0.661223			−0.330612	+	0.943767i	$0.607255\pi$
$662$	6.36396	−	3.67423i	0.247342	−	0.142803i
$663$		−	41.5692i		−	1.61441i
$664$		0			0
$665$	8.00000			0.310227
$666$		0			0
$667$		0			0
$668$	−19.7990	+	34.2929i	−0.766046	+	1.32683i
$669$	−39.0000			−1.50783
$670$	10.6066	−	6.12372i	0.409769	−	0.236580i
$671$	16.9706	+	27.7128i	0.655141	+	1.06984i
$672$	24.0000	+	13.8564i	0.925820	+	0.534522i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	−42.0000	+	24.2487i	−1.61778	+	0.934025i
$675$			20.7846i			0.800000i
$676$	11.0000	−	19.0526i	0.423077	−	0.732791i
$677$			24.4949i			0.941415i	0.882289	+	0.470708i	$0.156001\pi$
$677$			24.4949i			0.941415i	−0.882289	+	0.470708i	$0.843999\pi$
$678$	10.6066	−	6.12372i	0.407344	−	0.235180i
$679$	19.7990			0.759815
$680$			13.8564i			0.531369i
$681$			14.6969i			0.563188i
$682$	3.87868	−	7.13834i	0.148522	−	0.273341i
$683$			3.46410i			0.132550i	0.997801	+	0.0662751i	$0.0211115\pi$
$683$			3.46410i			0.132550i	−0.997801	+	0.0662751i	$0.978889\pi$
$684$		0			0
$685$	−19.0000			−0.725953
$686$	12.0000	+	20.7846i	0.458162	+	0.793560i
$687$			1.73205i			0.0660819i
$688$	−11.3137	−	19.5959i	−0.431331	−	0.747087i
$689$		−	9.79796i		−	0.373273i
$690$	6.36396	+	11.0227i	0.242272	+	0.419627i
$691$		−	15.5885i		−	0.593013i	−0.955031	−	0.296506i	$-0.904178\pi$
$691$		−	15.5885i		−	0.593013i	0.955031	−	0.296506i	$-0.0958216\pi$
$692$	16.9706	+	9.79796i	0.645124	+	0.372463i
$693$		0			0
$694$	18.0000	+	31.1769i	0.683271	+	1.18346i
$695$	−19.7990			−0.751018
$696$		0			0
$697$	24.0000			0.909065
$698$	6.00000	−	3.46410i	0.227103	−	0.131118i
$699$	−33.9411			−1.28377
$700$	11.3137	−	19.5959i	0.427618	−	0.740656i
$701$			34.2929i			1.29522i	0.761971	+	0.647612i	$0.224232\pi$
$701$			34.2929i			1.29522i	−0.761971	+	0.647612i	$0.775768\pi$
$702$	−18.0000	−	31.1769i	−0.679366	−	1.17670i
$703$	−8.48528			−0.320028
$704$	−22.6274	+	13.8564i	−0.852803	+	0.522233i
$705$	6.00000			0.225973
$706$	12.0208	+	20.8207i	0.452409	+	0.783596i
$707$		−	13.8564i		−	0.521124i
$708$	3.00000	−	5.19615i	0.112747	−	0.195283i
$709$	−25.0000			−0.938895			−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000			−0.938895			−0.469447	+	0.882960i	$0.655547\pi$
$710$	−14.8492	+	8.57321i	−0.557282	+	0.321747i
$711$		0			0
$712$	−2.82843			−0.106000
$713$	9.00000			0.337053
$714$	16.9706	+	29.3939i	0.635107	+	1.10004i
$715$	8.48528	+	13.8564i	0.317332	+	0.518200i
$716$	15.0000	+	8.66025i	0.560576	+	0.323649i
$717$		−	9.79796i		−	0.365911i
$718$	−6.00000	−	10.3923i	−0.223918	−	0.387837i
$719$		−	15.5885i		−	0.581351i	−0.956822	−	0.290676i	$-0.906120\pi$
$719$		−	15.5885i		−	0.581351i	0.956822	−	0.290676i	$-0.0938801\pi$
$720$		0			0
$721$		−	9.79796i		−	0.364895i
$722$	7.77817	+	13.4722i	0.289474	+	0.501383i
$723$	50.9117			1.89343
$724$	21.0000	−	36.3731i	0.780459	−	1.35179i
$725$		0			0
$726$	1.39340	+	26.9083i	0.0517139	+	0.998662i
$727$			25.9808i			0.963573i	0.876289	+	0.481787i	$0.160012\pi$
$727$			25.9808i			0.963573i	−0.876289	+	0.481787i	$0.839988\pi$
$728$			39.1918i			1.45255i
$729$	−27.0000			−1.00000
$730$	6.00000	−	3.46410i	0.222070	−	0.128212i
$731$		−	27.7128i		−	1.02500i
$732$	16.9706	−	29.3939i	0.627250	−	1.08643i
$733$		−	9.79796i		−	0.361896i	−0.983493	−	0.180948i	$-0.942083\pi$
$733$		−	9.79796i		−	0.361896i	0.983493	−	0.180948i	$-0.0579166\pi$
$734$	−2.12132	+	1.22474i	−0.0782994	+	0.0452062i
$735$			1.73205i			0.0638877i
$736$	−25.4558	−	14.6969i	−0.938315	−	0.541736i
$737$	15.0000	+	24.4949i	0.552532	+	0.902281i
$738$		0			0
$739$	−5.65685			−0.208091			−0.104045	−	0.994573i	$-0.533179\pi$
$739$	−5.65685			−0.208091			−0.104045	+	0.994573i	$0.533179\pi$
$740$	3.00000	−	5.19615i	0.110282	−	0.191014i
$741$	−24.0000			−0.881662
$742$	4.00000	+	6.92820i	0.146845	+	0.254342i
$743$	−33.9411			−1.24518			−0.622590	−	0.782549i	$-0.713919\pi$
$743$	−33.9411			−1.24518			−0.622590	+	0.782549i	$0.713919\pi$
$744$	−8.48528			−0.311086
$745$		−	14.6969i		−	0.538454i
$746$	18.0000	−	10.3923i	0.659027	−	0.380489i
$747$		0			0
$748$	−32.4853	+	0.840532i	−1.18778	+	0.0307329i
$749$	24.0000			0.876941
$750$	−19.0919	+	11.0227i	−0.697137	+	0.402492i
$751$			5.19615i			0.189610i	0.995496	+	0.0948051i	$0.0302228\pi$
$751$			5.19615i			0.189610i	−0.995496	+	0.0948051i	$0.969777\pi$
$752$	−12.0000	+	6.92820i	−0.437595	+	0.252646i
$753$	−39.0000			−1.42124
$754$		0			0
$755$	5.65685			0.205874
$756$	25.4558	+	14.6969i	0.925820	+	0.534522i
$757$	30.0000			1.09037			0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000			1.09037			0.545184	+	0.838316i	$0.316460\pi$
$758$	14.8492	−	8.57321i	0.539349	−	0.311393i
$759$	−25.4558	+	15.5885i	−0.923989	+	0.565825i
$760$	8.00000			0.290191
$761$		−	44.0908i		−	1.59829i	−0.601138	−	0.799145i	$-0.705286\pi$
$761$		−	44.0908i		−	1.59829i	0.601138	−	0.799145i	$-0.294714\pi$
$762$	30.0000	−	17.3205i	1.08679	−	0.627456i
$763$			27.7128i			1.00327i
$764$	−9.00000	−	5.19615i	−0.325609	−	0.187990i
$765$		0			0
$766$	40.3051	−	23.2702i	1.45628	−	0.840785i
$767$	8.48528			0.306386
$768$	24.0000	+	13.8564i	0.866025	+	0.500000i
$769$			4.89898i			0.176662i	0.996091	+	0.0883309i	$0.0281533\pi$
$769$			4.89898i			0.176662i	−0.996091	+	0.0883309i	$0.971847\pi$
$770$	−11.6569	−	6.33386i	−0.420084	−	0.228256i
$771$			38.1051i			1.37232i
$772$	−16.9706	−	9.79796i	−0.610784	−	0.352636i
$773$	−10.0000			−0.359675			−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000			−0.359675			−0.179838	+	0.983696i	$0.557557\pi$
$774$		0			0
$775$			6.92820i			0.248868i
$776$	19.7990			0.710742
$777$		−	14.6969i		−	0.527250i
$778$	−13.4350	−	23.2702i	−0.481669	−	0.834275i
$779$		−	13.8564i		−	0.496457i
$780$	8.48528	−	14.6969i	0.303822	−	0.526235i
$781$	−21.0000	−	34.2929i	−0.751439	−	1.22709i
$782$	−18.0000	−	31.1769i	−0.643679	−	1.11488i
$783$		0			0
$784$	−2.00000	−	3.46410i	−0.0714286	−	0.123718i
$785$	−11.0000			−0.392607
$786$	6.00000	−	3.46410i	0.214013	−	0.123560i
$787$	−28.2843			−1.00823			−0.504113	−	0.863638i	$-0.668180\pi$
$787$	−28.2843			−1.00823			−0.504113	+	0.863638i	$0.668180\pi$
$788$		0			0
$789$		−	24.4949i		−	0.872041i
$790$	−8.00000	−	13.8564i	−0.284627	−	0.492989i
$791$	−14.1421			−0.502836
$792$		0			0
$793$	48.0000			1.70453
$794$	−4.24264	−	7.34847i	−0.150566	−	0.260787i
$795$		−	3.46410i		−	0.122859i
$796$	−18.0000	−	10.3923i	−0.637993	−	0.368345i
$797$	7.00000			0.247953			0.123976	−	0.992285i	$-0.460435\pi$
$797$	7.00000			0.247953			0.123976	+	0.992285i	$0.460435\pi$
$798$	16.9706	−	9.79796i	0.600751	−	0.346844i
$799$	−16.9706			−0.600375
$800$	11.3137	−	19.5959i	0.400000	−	0.692820i
$801$		0			0
$802$	−7.07107	−	12.2474i	−0.249688	−	0.432472i
$803$	8.48528	+	13.8564i	0.299439	+	0.488982i
$804$	15.0000	−	25.9808i	0.529009	−	0.916271i
$805$		−	14.6969i		−	0.517999i
$806$	−6.00000	−	10.3923i	−0.211341	−	0.366053i
$807$			17.3205i			0.609711i
$808$		−	13.8564i		−	0.487467i
$809$			44.0908i			1.55015i	0.631869	+	0.775075i	$0.282288\pi$
$809$			44.0908i			1.55015i	−0.631869	+	0.775075i	$0.717712\pi$
$810$	−6.36396	−	11.0227i	−0.223607	−	0.387298i
$811$	−39.5980			−1.39047			−0.695237	−	0.718781i	$-0.744700\pi$
$811$	−39.5980			−1.39047			−0.695237	+	0.718781i	$0.744700\pi$
$812$		0			0
$813$			4.89898i			0.171815i
$814$	12.3640	+	6.71807i	0.433357	+	0.235468i
$815$		−	17.3205i		−	0.606711i
$816$	16.9706	+	29.3939i	0.594089	+	1.02899i
$817$	−16.0000			−0.559769
$818$	36.0000	−	20.7846i	1.25871	−	0.726717i
$819$		0			0
$820$	8.48528	+	4.89898i	0.296319	+	0.171080i
$821$			48.9898i			1.70976i	0.518829	+	0.854878i	$0.326368\pi$
$821$			48.9898i			1.70976i	−0.518829	+	0.854878i	$0.673632\pi$
$822$	−40.3051	+	23.2702i	−1.40580	+	0.811640i
$823$		−	8.66025i		−	0.301877i	−0.988543	−	0.150939i	$-0.951770\pi$
$823$		−	8.66025i		−	0.301877i	0.988543	−	0.150939i	$-0.0482296\pi$
$824$		−	9.79796i		−	0.341328i
$825$	−12.0000	−	19.5959i	−0.417786	−	0.682242i
$826$	−6.00000	+	3.46410i	−0.208767	+	0.120532i
$827$	25.4558			0.885186			0.442593	−	0.896723i	$-0.354059\pi$
$827$	25.4558			0.885186			0.442593	+	0.896723i	$0.354059\pi$
$828$		0			0
$829$	−17.0000			−0.590434			−0.295217	−	0.955430i	$-0.595392\pi$
$829$	−17.0000			−0.590434			−0.295217	+	0.955430i	$0.595392\pi$
$830$		0			0
$831$	16.9706			0.588702
$832$			39.1918i			1.35873i
$833$		−	4.89898i		−	0.169740i
$834$	−42.0000	+	24.2487i	−1.45434	+	0.839664i
$835$	−19.7990			−0.685172
$836$	0.485281	+	18.7554i	0.0167838	+	0.648669i
$837$	−9.00000			−0.311086
$838$	−12.7279	+	7.34847i	−0.439679	+	0.253849i
$839$			25.9808i			0.896956i	0.893794	+	0.448478i	$0.148034\pi$
$839$			25.9808i			0.896956i	−0.893794	+	0.448478i	$0.851966\pi$
$840$			13.8564i			0.478091i
$841$	29.0000			1.00000
$842$	−21.2132	−	36.7423i	−0.731055	−	1.26622i
$843$		0			0
$844$	−5.65685	+	9.79796i	−0.194717	+	0.337260i
$845$	11.0000			0.378412
$846$		0			0
$847$	14.1421	−	27.7128i	0.485930	−	0.952224i
$848$	4.00000	+	6.92820i	0.137361	+	0.237915i
$849$		−	19.5959i		−	0.672530i
$850$	24.0000	−	13.8564i	0.823193	−	0.475271i
$851$			15.5885i			0.534365i
$852$	−21.0000	+	36.3731i	−0.719448	+	1.24612i
$853$			39.1918i			1.34190i	0.741501	+	0.670951i	$0.234114\pi$
$853$			39.1918i			1.34190i	−0.741501	+	0.670951i	$0.765886\pi$
$854$	−33.9411	+	19.5959i	−1.16144	+	0.670559i
$855$		0			0
$856$	24.0000			0.820303
$857$			29.3939i			1.00408i	0.864846	+	0.502038i	$0.167416\pi$
$857$			29.3939i			1.00408i	−0.864846	+	0.502038i	$0.832584\pi$
$858$	34.9706	+	19.0016i	1.19388	+	0.648703i
$859$		−	15.5885i		−	0.531871i	−0.963991	−	0.265936i	$-0.914319\pi$
$859$		−	15.5885i		−	0.531871i	0.963991	−	0.265936i	$-0.0856809\pi$
$860$	5.65685	−	9.79796i	0.192897	−	0.334108i
$861$	24.0000			0.817918
$862$	−10.0000	−	17.3205i	−0.340601	−	0.589939i
$863$		−	31.1769i		−	1.06127i	−0.847599	−	0.530637i	$-0.821953\pi$
$863$		−	31.1769i		−	1.06127i	0.847599	−	0.530637i	$-0.178047\pi$
$864$	25.4558	+	14.6969i	0.866025	+	0.500000i
$865$			9.79796i			0.333141i
$866$	14.8492	+	25.7196i	0.504598	+	0.873989i
$867$			12.1244i			0.411765i
$868$	8.48528	+	4.89898i	0.288009	+	0.166282i
$869$	32.0000	−	19.5959i	1.08553	−	0.664746i
$870$		0			0
$871$	42.4264			1.43756
$872$			27.7128i			0.938474i
$873$		0			0
$874$	−18.0000	+	10.3923i	−0.608859	+	0.351525i
$875$	25.4558			0.860565
$876$	8.48528	−	14.6969i	0.286691	−	0.496564i
$877$		−	53.8888i		−	1.81969i	−0.414943	−	0.909847i	$-0.636199\pi$
$877$		−	53.8888i		−	1.81969i	0.414943	−	0.909847i	$-0.363801\pi$
$878$	16.0000	+	27.7128i	0.539974	+	0.935262i
$879$	8.48528			0.286201
$880$	−11.6569	−	6.33386i	−0.392952	−	0.213514i
$881$	−49.0000			−1.65085			−0.825426	−	0.564510i	$-0.809065\pi$
$881$	−49.0000			−1.65085			−0.825426	+	0.564510i	$0.809065\pi$
$882$		0			0
$883$		−	24.2487i		−	0.816034i	−0.912974	−	0.408017i	$-0.866220\pi$
$883$		−	24.2487i		−	0.816034i	0.912974	−	0.408017i	$-0.133780\pi$
$884$	−24.0000	+	41.5692i	−0.807207	+	1.39812i
$885$	3.00000			0.100844
$886$	6.36396	−	3.67423i	0.213801	−	0.123438i
$887$	16.9706			0.569816			0.284908	−	0.958555i	$-0.408037\pi$
$887$	16.9706			0.569816			0.284908	+	0.958555i	$0.408037\pi$
$888$		−	14.6969i		−	0.493197i
$889$	−40.0000			−1.34156
$890$	−0.707107	−	1.22474i	−0.0237023	−	0.0410535i
$891$	25.4558	−	15.5885i	0.852803	−	0.522233i
$892$	39.0000	+	22.5167i	1.30582	+	0.753914i
$893$			9.79796i			0.327876i
$894$	−18.0000	−	31.1769i	−0.602010	−	1.04271i
$895$			8.66025i			0.289480i
$896$	−16.0000	−	27.7128i	−0.534522	−	0.925820i
$897$			44.0908i			1.47215i
$898$	−16.2635	−	28.1691i	−0.542719	−	0.940016i
$899$		0			0
$900$		0			0
$901$			9.79796i			0.326417i
$902$	−10.9706	+	20.1903i	−0.365280	+	0.672262i
$903$		−	27.7128i		−	0.922225i
$904$	−14.1421			−0.470360
$905$	21.0000			0.698064
$906$	12.0000	−	6.92820i	0.398673	−	0.230174i
$907$			3.46410i			0.115024i	0.998345	+	0.0575118i	$0.0183167\pi$
$907$			3.46410i			0.115024i	−0.998345	+	0.0575118i	$0.981683\pi$
$908$	8.48528	−	14.6969i	0.281594	−	0.487735i
$909$		0			0
$910$	−16.9706	+	9.79796i	−0.562569	+	0.324799i
$911$			24.2487i			0.803396i	0.915772	+	0.401698i	$0.131580\pi$
$911$			24.2487i			0.803396i	−0.915772	+	0.401698i	$0.868420\pi$
$912$	16.9706	−	9.79796i	0.561951	−	0.324443i
$913$		0			0
$914$	−24.0000	+	13.8564i	−0.793849	+	0.458329i
$915$	16.9706			0.561029
$916$	1.00000	−	1.73205i	0.0330409	−	0.0572286i
$917$	−8.00000			−0.264183
$918$	18.0000	+	31.1769i	0.594089	+	1.02899i
$919$	−11.3137			−0.373205			−0.186602	−	0.982436i	$-0.559748\pi$
$919$	−11.3137			−0.373205			−0.186602	+	0.982436i	$0.559748\pi$
$920$		−	14.6969i		−	0.484544i
$921$			34.2929i			1.12999i
$922$	−36.0000	+	20.7846i	−1.18560	+	0.684505i
$923$	−59.3970			−1.95508
$924$	−32.4853	+	0.840532i	−1.06869	+	0.0276515i
$925$	−12.0000			−0.394558
$926$	−44.5477	+	25.7196i	−1.46393	+	0.845200i
$927$		0			0
$928$		0			0
$929$	−22.0000			−0.721797			−0.360898	−	0.932605i	$-0.617530\pi$
$929$	−22.0000			−0.721797			−0.360898	+	0.932605i	$0.617530\pi$
$930$	−2.12132	−	3.67423i	−0.0695608	−	0.120483i
$931$	−2.82843			−0.0926980
$932$	33.9411	+	19.5959i	1.11178	+	0.641886i
$933$	18.0000			0.589294
$934$	40.3051	−	23.2702i	1.31882	−	0.761423i
$935$	−8.48528	−	13.8564i	−0.277498	−	0.453153i
$936$		0			0
$937$			4.89898i			0.160043i	0.996793	+	0.0800213i	$0.0254988\pi$
$937$			4.89898i			0.160043i	−0.996793	+	0.0800213i	$0.974501\pi$
$938$	−30.0000	+	17.3205i	−0.979535	+	0.565535i
$939$		−	46.7654i		−	1.52613i
$940$	−6.00000	−	3.46410i	−0.195698	−	0.112987i
$941$		−	39.1918i		−	1.27762i	−0.769366	−	0.638809i	$-0.779428\pi$
$941$		−	39.1918i		−	1.27762i	0.769366	−	0.638809i	$-0.220572\pi$
$942$	−23.3345	+	13.4722i	−0.760280	+	0.438948i
$943$	−25.4558			−0.828956
$944$	−6.00000	+	3.46410i	−0.195283	+	0.112747i
$945$			14.6969i			0.478091i
$946$	23.3137	+	12.6677i	0.757994	+	0.411863i
$947$		−	43.3013i		−	1.40710i	−0.710645	−	0.703551i	$-0.751597\pi$
$947$		−	43.3013i		−	1.40710i	0.710645	−	0.703551i	$-0.248403\pi$
$948$	−33.9411	−	19.5959i	−1.10236	−	0.636446i
$949$	24.0000			0.779073
$950$	−8.00000	−	13.8564i	−0.259554	−	0.449561i
$951$			43.3013i			1.40414i
$952$		−	39.1918i		−	1.27021i
$953$		−	34.2929i		−	1.11085i	−0.831565	−	0.555427i	$-0.812555\pi$
$953$		−	34.2929i		−	1.11085i	0.831565	−	0.555427i	$-0.187445\pi$
$954$		0			0
$955$		−	5.19615i		−	0.168144i
$956$	−5.65685	+	9.79796i	−0.182956	+	0.316889i
$957$		0			0
$958$	−22.0000	−	38.1051i	−0.710788	−	1.23112i
$959$	53.7401			1.73536
$960$			13.8564i			0.447214i
$961$	28.0000			0.903226
$962$	18.0000	−	10.3923i	0.580343	−	0.335061i
$963$		0			0
$964$	−50.9117	−	29.3939i	−1.63976	−	0.946713i
$965$		−	9.79796i		−	0.315407i
$966$	−18.0000	−	31.1769i	−0.579141	−	1.00310i
$967$	45.2548			1.45530			0.727649	−	0.685950i	$-0.240613\pi$
$967$	45.2548			1.45530			0.727649	+	0.685950i	$0.240613\pi$
$968$	14.1421	−	27.7128i	0.454545	−	0.890724i
$969$	24.0000			0.770991
$970$	4.94975	+	8.57321i	0.158927	+	0.275269i
$971$			32.9090i			1.05610i	0.849214	+	0.528049i	$0.177076\pi$
$971$			32.9090i			1.05610i	−0.849214	+	0.528049i	$0.822924\pi$
$972$		0			0
$973$	56.0000			1.79528
$974$	48.7904	−	28.1691i	1.56334	−	0.902597i
$975$	−33.9411			−1.08699
$976$	−33.9411	+	19.5959i	−1.08643	+	0.627250i
$977$	19.0000			0.607864			0.303932	−	0.952694i	$-0.401700\pi$
$977$	19.0000			0.607864			0.303932	+	0.952694i	$0.401700\pi$
$978$	−21.2132	−	36.7423i	−0.678323	−	1.17489i
$979$	2.82843	−	1.73205i	0.0903969	−	0.0553566i
$980$	1.00000	−	1.73205i	0.0319438	−	0.0553283i
$981$		0			0
$982$	24.0000	+	41.5692i	0.765871	+	1.32653i
$983$		−	1.73205i		−	0.0552438i	−0.999618	−	0.0276219i	$-0.991207\pi$
$983$		−	1.73205i		−	0.0552438i	0.999618	−	0.0276219i	$-0.00879345\pi$
$984$	24.0000			0.765092
$985$		0			0
$986$		0			0
$987$	−16.9706			−0.540179
$988$	24.0000	+	13.8564i	0.763542	+	0.440831i
$989$			29.3939i			0.934671i
$990$		0			0
$991$			51.9615i			1.65061i	0.564686	+	0.825306i	$0.308997\pi$
$991$			51.9615i			1.65061i	−0.564686	+	0.825306i	$0.691003\pi$
$992$	8.48528	+	4.89898i	0.269408	+	0.155543i
$993$	9.00000			0.285606
$994$	42.0000	−	24.2487i	1.33216	−	0.769122i
$995$		−	10.3923i		−	0.329458i
$996$		0			0
$997$		−	4.89898i		−	0.155152i	−0.996986	−	0.0775761i	$-0.975282\pi$
$997$		−	4.89898i		−	0.155152i	0.996986	−	0.0775761i	$-0.0247181\pi$
$998$	21.2132	−	12.2474i	0.671492	−	0.387686i
$999$		−	15.5885i		−	0.493197i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	44.2.c.a.43.1	✓	4
3.2	odd	2		396.2.h.b.307.4		4
4.3	odd	2	inner	44.2.c.a.43.3	yes	4
8.3	odd	2		704.2.e.b.703.1		4
8.5	even	2		704.2.e.b.703.4		4
11.2	odd	10		484.2.g.g.403.3		16
11.3	even	5		484.2.g.g.475.1		16
11.4	even	5		484.2.g.g.215.3		16
11.5	even	5		484.2.g.g.239.4		16
11.6	odd	10		484.2.g.g.239.1		16
11.7	odd	10		484.2.g.g.215.2		16
11.8	odd	10		484.2.g.g.475.4		16
11.9	even	5		484.2.g.g.403.2		16
11.10	odd	2	inner	44.2.c.a.43.4	yes	4
12.11	even	2		396.2.h.b.307.2		4
16.3	odd	4		2816.2.g.b.1407.4		8
16.5	even	4		2816.2.g.b.1407.1		8
16.11	odd	4		2816.2.g.b.1407.6		8
16.13	even	4		2816.2.g.b.1407.7		8
33.32	even	2		396.2.h.b.307.1		4
44.3	odd	10		484.2.g.g.475.2		16
44.7	even	10		484.2.g.g.215.1		16
44.15	odd	10		484.2.g.g.215.4		16
44.19	even	10		484.2.g.g.475.3		16
44.27	odd	10		484.2.g.g.239.3		16
44.31	odd	10		484.2.g.g.403.1		16
44.35	even	10		484.2.g.g.403.4		16
44.39	even	10		484.2.g.g.239.2		16
44.43	even	2	inner	44.2.c.a.43.2	yes	4
88.21	odd	2		704.2.e.b.703.3		4
88.43	even	2		704.2.e.b.703.2		4
132.131	odd	2		396.2.h.b.307.3		4
176.21	odd	4		2816.2.g.b.1407.2		8
176.43	even	4		2816.2.g.b.1407.5		8
176.109	odd	4		2816.2.g.b.1407.8		8
176.131	even	4		2816.2.g.b.1407.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.2.c.a.43.1	✓	4	1.1	even	1	trivial
44.2.c.a.43.2	yes	4	44.43	even	2	inner
44.2.c.a.43.3	yes	4	4.3	odd	2	inner
44.2.c.a.43.4	yes	4	11.10	odd	2	inner
396.2.h.b.307.1		4	33.32	even	2
396.2.h.b.307.2		4	12.11	even	2
396.2.h.b.307.3		4	132.131	odd	2
396.2.h.b.307.4		4	3.2	odd	2
484.2.g.g.215.1		16	44.7	even	10
484.2.g.g.215.2		16	11.7	odd	10
484.2.g.g.215.3		16	11.4	even	5
484.2.g.g.215.4		16	44.15	odd	10
484.2.g.g.239.1		16	11.6	odd	10
484.2.g.g.239.2		16	44.39	even	10
484.2.g.g.239.3		16	44.27	odd	10
484.2.g.g.239.4		16	11.5	even	5
484.2.g.g.403.1		16	44.31	odd	10
484.2.g.g.403.2		16	11.9	even	5
484.2.g.g.403.3		16	11.2	odd	10
484.2.g.g.403.4		16	44.35	even	10
484.2.g.g.475.1		16	11.3	even	5
484.2.g.g.475.2		16	44.3	odd	10
484.2.g.g.475.3		16	44.19	even	10
484.2.g.g.475.4		16	11.8	odd	10
704.2.e.b.703.1		4	8.3	odd	2
704.2.e.b.703.2		4	88.43	even	2
704.2.e.b.703.3		4	88.21	odd	2
704.2.e.b.703.4		4	8.5	even	2
2816.2.g.b.1407.1		8	16.5	even	4
2816.2.g.b.1407.2		8	176.21	odd	4
2816.2.g.b.1407.3		8	176.131	even	4
2816.2.g.b.1407.4		8	16.3	odd	4
2816.2.g.b.1407.5		8	176.43	even	4
2816.2.g.b.1407.6		8	16.11	odd	4
2816.2.g.b.1407.7		8	16.13	even	4
2816.2.g.b.1407.8		8	176.109	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:	\( N \)	\(=\)	\( 44 = 2^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	44.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} -1.73205i q^{3} +(-1.00000 + 1.73205i) q^{4} -1.00000 q^{5} +(-2.12132 + 1.22474i) q^{6} +2.82843 q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+(-0.707107 - 1.22474i) q^{2} -1.73205i q^{3} +(-1.00000 + 1.73205i) q^{4} -1.00000 q^{5} +(-2.12132 + 1.22474i) q^{6} +2.82843 q^{7} +2.82843 q^{8} +(0.707107 + 1.22474i) q^{10} +(-2.82843 + 1.73205i) q^{11} +(3.00000 + 1.73205i) q^{12} +4.89898i q^{13} +(-2.00000 - 3.46410i) q^{14} +1.73205i q^{15} +(-2.00000 - 3.46410i) q^{16} -4.89898i q^{17} -2.82843 q^{19} +(1.00000 - 1.73205i) q^{20} -4.89898i q^{21} +(4.12132 + 2.23936i) q^{22} +5.19615i q^{23} -4.89898i q^{24} -4.00000 q^{25} +(6.00000 - 3.46410i) q^{26} -5.19615i q^{27} +(-2.82843 + 4.89898i) q^{28} +(2.12132 - 1.22474i) q^{30} -1.73205i q^{31} +(-2.82843 + 4.89898i) q^{32} +(3.00000 + 4.89898i) q^{33} +(-6.00000 + 3.46410i) q^{34} -2.82843 q^{35} +3.00000 q^{37} +(2.00000 + 3.46410i) q^{38} +8.48528 q^{39} -2.82843 q^{40} +4.89898i q^{41} +(-6.00000 + 3.46410i) q^{42} +5.65685 q^{43} +(-0.171573 - 6.63103i) q^{44} +(6.36396 - 3.67423i) q^{46} -3.46410i q^{47} +(-6.00000 + 3.46410i) q^{48} +1.00000 q^{49} +(2.82843 + 4.89898i) q^{50} -8.48528 q^{51} +(-8.48528 - 4.89898i) q^{52} -2.00000 q^{53} +(-6.36396 + 3.67423i) q^{54} +(2.82843 - 1.73205i) q^{55} +8.00000 q^{56} +4.89898i q^{57} -1.73205i q^{59} +(-3.00000 - 1.73205i) q^{60} -9.79796i q^{61} +(-2.12132 + 1.22474i) q^{62} +8.00000 q^{64} -4.89898i q^{65} +(3.87868 - 7.13834i) q^{66} -8.66025i q^{67} +(8.48528 + 4.89898i) q^{68} +9.00000 q^{69} +(2.00000 + 3.46410i) q^{70} +12.1244i q^{71} -4.89898i q^{73} +(-2.12132 - 3.67423i) q^{74} +6.92820i q^{75} +(2.82843 - 4.89898i) q^{76} +(-8.00000 + 4.89898i) q^{77} +(-6.00000 - 10.3923i) q^{78} -11.3137 q^{79} +(2.00000 + 3.46410i) q^{80} -9.00000 q^{81} +(6.00000 - 3.46410i) q^{82} +(8.48528 + 4.89898i) q^{84} +4.89898i q^{85} +(-4.00000 - 6.92820i) q^{86} +(-8.00000 + 4.89898i) q^{88} -1.00000 q^{89} +13.8564i q^{91} +(-9.00000 - 5.19615i) q^{92} -3.00000 q^{93} +(-4.24264 + 2.44949i) q^{94} +2.82843 q^{95} +(8.48528 + 4.89898i) q^{96} +7.00000 q^{97} +(-0.707107 - 1.22474i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{5}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^5	\( 4 q - 4 q^{4} - 4 q^{5} + 12 q^{12} - 8 q^{14} - 8 q^{16} + 4 q^{20} + 8 q^{22} - 16 q^{25} + 24 q^{26} + 12 q^{33} - 24 q^{34} + 12 q^{37} + 8 q^{38} - 24 q^{42} - 12 q^{44} - 24 q^{48} + 4 q^{49} - 8 q^{53} + 32 q^{56} - 12 q^{60} + 32 q^{64} + 24 q^{66} + 36 q^{69} + 8 q^{70} - 32 q^{77} - 24 q^{78} + 8 q^{80} - 36 q^{81} + 24 q^{82} - 16 q^{86} - 32 q^{88} - 4 q^{89} - 36 q^{92} - 12 q^{93} + 28 q^{97}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^5 + 12 * q^12 - 8 * q^14 - 8 * q^16 + 4 * q^20 + 8 * q^22 - 16 * q^25 + 24 * q^26 + 12 * q^33 - 24 * q^34 + 12 * q^37 + 8 * q^38 - 24 * q^42 - 12 * q^44 - 24 * q^48 + 4 * q^49 - 8 * q^53 + 32 * q^56 - 12 * q^60 + 32 * q^64 + 24 * q^66 + 36 * q^69 + 8 * q^70 - 32 * q^77 - 24 * q^78 + 8 * q^80 - 36 * q^81 + 24 * q^82 - 16 * q^86 - 32 * q^88 - 4 * q^89 - 36 * q^92 - 12 * q^93 + 28 * q^97

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