LMFDB - Embedded newform 45.2.f.a.8.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,2,Mod(8,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("45.8");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	45.f (of order $4$, degree $2$, 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:	$0.359326809096$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
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:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			8.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	45.8
Dual form			45.2.f.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(2.12132 - 0.707107i) q^{5} +(-2.00000 + 2.00000i) q^{7} +(-2.12132 + 2.12132i) q^{8} +O(q^{10})$	\(q+(-0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(2.12132 - 0.707107i) q^{5} +(-2.00000 + 2.00000i) q^{7} +(-2.12132 + 2.12132i) q^{8} +(-2.00000 - 1.00000i) q^{10} +2.82843i q^{11} +(1.00000 + 1.00000i) q^{13} +2.82843 q^{14} +1.00000 q^{16} +(-2.82843 - 2.82843i) q^{17} +(-0.707107 - 2.12132i) q^{20} +(2.00000 - 2.00000i) q^{22} +(2.82843 - 2.82843i) q^{23} +(4.00000 - 3.00000i) q^{25} -1.41421i q^{26} +(2.00000 + 2.00000i) q^{28} -4.24264 q^{29} -4.00000 q^{31} +(3.53553 + 3.53553i) q^{32} +4.00000i q^{34} +(-2.82843 + 5.65685i) q^{35} +(1.00000 - 1.00000i) q^{37} +(-3.00000 + 6.00000i) q^{40} -1.41421i q^{41} +(-8.00000 - 8.00000i) q^{43} +2.82843 q^{44} -4.00000 q^{46} +(5.65685 + 5.65685i) q^{47} -1.00000i q^{49} +(-4.94975 - 0.707107i) q^{50} +(1.00000 - 1.00000i) q^{52} +(2.82843 - 2.82843i) q^{53} +(2.00000 + 6.00000i) q^{55} -8.48528i q^{56} +(3.00000 + 3.00000i) q^{58} -8.48528 q^{59} +8.00000 q^{61} +(2.82843 + 2.82843i) q^{62} -7.00000i q^{64} +(2.82843 + 1.41421i) q^{65} +(4.00000 - 4.00000i) q^{67} +(-2.82843 + 2.82843i) q^{68} +(6.00000 - 2.00000i) q^{70} -5.65685i q^{71} +(1.00000 + 1.00000i) q^{73} -1.41421 q^{74} +(-5.65685 - 5.65685i) q^{77} +12.0000i q^{79} +(2.12132 - 0.707107i) q^{80} +(-1.00000 + 1.00000i) q^{82} +(2.82843 - 2.82843i) q^{83} +(-8.00000 - 4.00000i) q^{85} +11.3137i q^{86} +(-6.00000 - 6.00000i) q^{88} +12.7279 q^{89} -4.00000 q^{91} +(-2.82843 - 2.82843i) q^{92} -8.00000i q^{94} +(-11.0000 + 11.0000i) q^{97} +(-0.707107 + 0.707107i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7}+O(q^{10}) $ 4 * q - 8 * q^7	$ 4 q - 8 q^{7} - 8 q^{10} + 4 q^{13} + 4 q^{16} + 8 q^{22} + 16 q^{25} + 8 q^{28} - 16 q^{31} + 4 q^{37} - 12 q^{40} - 32 q^{43} - 16 q^{46} + 4 q^{52} + 8 q^{55} + 12 q^{58} + 32 q^{61} + 16 q^{67} + 24 q^{70} + 4 q^{73} - 4 q^{82} - 32 q^{85} - 24 q^{88} - 16 q^{91} - 44 q^{97}+O(q^{100}) $ 4 * q - 8 * q^7 - 8 * q^10 + 4 * q^13 + 4 * q^16 + 8 * q^22 + 16 * q^25 + 8 * q^28 - 16 * q^31 + 4 * q^37 - 12 * q^40 - 32 * q^43 - 16 * q^46 + 4 * q^52 + 8 * q^55 + 12 * q^58 + 32 * q^61 + 16 * q^67 + 24 * q^70 + 4 * q^73 - 4 * q^82 - 32 * q^85 - 24 * q^88 - 16 * q^91 - 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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	−	0.707107i	−0.500000	−	0.500000i	0.411438	−	0.911438i	$-0.365027\pi$
$2$	−0.707107	−	0.707107i	−0.500000	−	0.500000i	−0.911438	+	0.411438i	$0.865027\pi$
$3$		0			0
$3$		0			0
$4$		−	1.00000i		−	0.500000i
$5$	2.12132	−	0.707107i	0.948683	−	0.316228i
$5$	2.12132	−	0.707107i	0.948683	−	0.316228i
$6$		0			0
$7$	−2.00000	+	2.00000i	−0.755929	+	0.755929i	−0.975579	−	0.219650i	$-0.929509\pi$
$7$	−2.00000	+	2.00000i	−0.755929	+	0.755929i	0.219650	+	0.975579i	$0.429509\pi$
$8$	−2.12132	+	2.12132i	−0.750000	+	0.750000i
$9$		0			0
$10$	−2.00000	−	1.00000i	−0.632456	−	0.316228i
$11$			2.82843i			0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$			2.82843i			0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$	1.00000	+	1.00000i	0.277350	+	0.277350i	0.832050	−	0.554700i	$-0.187167\pi$
$13$	1.00000	+	1.00000i	0.277350	+	0.277350i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	2.82843			0.755929
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.82843	−	2.82843i	−0.685994	−	0.685994i	0.275350	−	0.961344i	$-0.411206\pi$
$17$	−2.82843	−	2.82843i	−0.685994	−	0.685994i	−0.961344	+	0.275350i	$0.911206\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−0.707107	−	2.12132i	−0.158114	−	0.474342i
$21$		0			0
$22$	2.00000	−	2.00000i	0.426401	−	0.426401i
$23$	2.82843	−	2.82843i	0.589768	−	0.589768i	−0.347801	−	0.937568i	$-0.613071\pi$
$23$	2.82843	−	2.82843i	0.589768	−	0.589768i	0.937568	+	0.347801i	$0.113071\pi$
$24$		0			0
$25$	4.00000	−	3.00000i	0.800000	−	0.600000i
$26$		−	1.41421i		−	0.277350i
$27$		0			0
$28$	2.00000	+	2.00000i	0.377964	+	0.377964i
$29$	−4.24264			−0.787839			−0.393919	−	0.919145i	$-0.628881\pi$
$29$	−4.24264			−0.787839			−0.393919	+	0.919145i	$0.628881\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$	3.53553	+	3.53553i	0.625000	+	0.625000i
$33$		0			0
$34$			4.00000i			0.685994i
$35$	−2.82843	+	5.65685i	−0.478091	+	0.956183i
$36$		0			0
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	−0.620113	−	0.784512i	$-0.712913\pi$
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	0.784512	+	0.620113i	$0.212913\pi$
$38$		0			0
$39$		0			0
$40$	−3.00000	+	6.00000i	−0.474342	+	0.948683i
$41$		−	1.41421i		−	0.220863i	−0.993884	−	0.110432i	$-0.964777\pi$
$41$		−	1.41421i		−	0.220863i	0.993884	−	0.110432i	$-0.0352233\pi$
$42$		0			0
$43$	−8.00000	−	8.00000i	−1.21999	−	1.21999i	−0.967635	−	0.252353i	$-0.918795\pi$
$43$	−8.00000	−	8.00000i	−1.21999	−	1.21999i	−0.252353	−	0.967635i	$-0.581205\pi$
$44$	2.82843			0.426401
$45$		0			0
$46$	−4.00000			−0.589768
$47$	5.65685	+	5.65685i	0.825137	+	0.825137i	0.986840	−	0.161703i	$-0.0516985\pi$
$47$	5.65685	+	5.65685i	0.825137	+	0.825137i	−0.161703	+	0.986840i	$0.551699\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$	−4.94975	−	0.707107i	−0.700000	−	0.100000i
$51$		0			0
$52$	1.00000	−	1.00000i	0.138675	−	0.138675i
$53$	2.82843	−	2.82843i	0.388514	−	0.388514i	−0.485643	−	0.874157i	$-0.661414\pi$
$53$	2.82843	−	2.82843i	0.388514	−	0.388514i	0.874157	+	0.485643i	$0.161414\pi$
$54$		0			0
$55$	2.00000	+	6.00000i	0.269680	+	0.809040i
$56$		−	8.48528i		−	1.13389i
$57$		0			0
$58$	3.00000	+	3.00000i	0.393919	+	0.393919i
$59$	−8.48528			−1.10469			−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528			−1.10469			−0.552345	+	0.833616i	$0.686267\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$	2.82843	+	2.82843i	0.359211	+	0.359211i
$63$		0			0
$64$		−	7.00000i		−	0.875000i
$65$	2.82843	+	1.41421i	0.350823	+	0.175412i
$66$		0			0
$67$	4.00000	−	4.00000i	0.488678	−	0.488678i	−0.419211	−	0.907889i	$-0.637693\pi$
$67$	4.00000	−	4.00000i	0.488678	−	0.488678i	0.907889	+	0.419211i	$0.137693\pi$
$68$	−2.82843	+	2.82843i	−0.342997	+	0.342997i
$69$		0			0
$70$	6.00000	−	2.00000i	0.717137	−	0.239046i
$71$		−	5.65685i		−	0.671345i	−0.941979	−	0.335673i	$-0.891036\pi$
$71$		−	5.65685i		−	0.671345i	0.941979	−	0.335673i	$-0.108964\pi$
$72$		0			0
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	0.763202	−	0.646160i	$-0.223626\pi$
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	−0.646160	+	0.763202i	$0.723626\pi$
$74$	−1.41421			−0.164399
$75$		0			0
$76$		0			0
$77$	−5.65685	−	5.65685i	−0.644658	−	0.644658i
$78$		0			0
$79$			12.0000i			1.35011i	0.737769	+	0.675053i	$0.235879\pi$
$79$			12.0000i			1.35011i	−0.737769	+	0.675053i	$0.764121\pi$
$80$	2.12132	−	0.707107i	0.237171	−	0.0790569i
$81$		0			0
$82$	−1.00000	+	1.00000i	−0.110432	+	0.110432i
$83$	2.82843	−	2.82843i	0.310460	−	0.310460i	−0.534628	−	0.845088i	$-0.679548\pi$
$83$	2.82843	−	2.82843i	0.310460	−	0.310460i	0.845088	+	0.534628i	$0.179548\pi$
$84$		0			0
$85$	−8.00000	−	4.00000i	−0.867722	−	0.433861i
$86$			11.3137i			1.21999i
$87$		0			0
$88$	−6.00000	−	6.00000i	−0.639602	−	0.639602i
$89$	12.7279			1.34916			0.674579	−	0.738203i	$-0.264325\pi$
$89$	12.7279			1.34916			0.674579	+	0.738203i	$0.264325\pi$
$90$		0			0
$91$	−4.00000			−0.419314
$92$	−2.82843	−	2.82843i	−0.294884	−	0.294884i
$93$		0			0
$94$		−	8.00000i		−	0.825137i
$95$		0			0
$96$		0			0
$97$	−11.0000	+	11.0000i	−1.11688	+	1.11688i	−0.124684	+	0.992196i	$0.539792\pi$
$97$	−11.0000	+	11.0000i	−1.11688	+	1.11688i	−0.992196	+	0.124684i	$0.960208\pi$
$98$	−0.707107	+	0.707107i	−0.0714286	+	0.0714286i
$99$		0			0
$100$	−3.00000	−	4.00000i	−0.300000	−	0.400000i
$101$			15.5563i			1.54791i	0.633238	+	0.773957i	$0.281726\pi$
$101$			15.5563i			1.54791i	−0.633238	+	0.773957i	$0.718274\pi$
$102$		0			0
$103$	10.0000	+	10.0000i	0.985329	+	0.985329i	0.999894	−	0.0145647i	$-0.00463624\pi$
$103$	10.0000	+	10.0000i	0.985329	+	0.985329i	−0.0145647	+	0.999894i	$0.504636\pi$
$104$	−4.24264			−0.416025
$105$		0			0
$106$	−4.00000			−0.388514
$107$	−2.82843	−	2.82843i	−0.273434	−	0.273434i	0.557047	−	0.830481i	$-0.311934\pi$
$107$	−2.82843	−	2.82843i	−0.273434	−	0.273434i	−0.830481	+	0.557047i	$0.811934\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	2.82843	−	5.65685i	0.269680	−	0.539360i
$111$		0			0
$112$	−2.00000	+	2.00000i	−0.188982	+	0.188982i
$113$	−9.89949	+	9.89949i	−0.931266	+	0.931266i	−0.997785	−	0.0665190i	$-0.978811\pi$
$113$	−9.89949	+	9.89949i	−0.931266	+	0.931266i	0.0665190	+	0.997785i	$0.478811\pi$
$114$		0			0
$115$	4.00000	−	8.00000i	0.373002	−	0.746004i
$116$			4.24264i			0.393919i
$117$		0			0
$118$	6.00000	+	6.00000i	0.552345	+	0.552345i
$119$	11.3137			1.03713
$120$		0			0
$121$	3.00000			0.272727
$122$	−5.65685	−	5.65685i	−0.512148	−	0.512148i
$123$		0			0
$124$			4.00000i			0.359211i
$125$	6.36396	−	9.19239i	0.569210	−	0.822192i
$126$		0			0
$127$	10.0000	−	10.0000i	0.887357	−	0.887357i	−0.106912	−	0.994268i	$-0.534096\pi$
$127$	10.0000	−	10.0000i	0.887357	−	0.887357i	0.994268	+	0.106912i	$0.0340963\pi$
$128$	2.12132	−	2.12132i	0.187500	−	0.187500i
$129$		0			0
$130$	−1.00000	−	3.00000i	−0.0877058	−	0.263117i
$131$		−	14.1421i		−	1.23560i	−0.786334	−	0.617802i	$-0.788023\pi$
$131$		−	14.1421i		−	1.23560i	0.786334	−	0.617802i	$-0.211977\pi$
$132$		0			0
$133$		0			0
$134$	−5.65685			−0.488678
$135$		0			0
$136$	12.0000			1.02899
$137$	−7.07107	−	7.07107i	−0.604122	−	0.604122i	0.337282	−	0.941404i	$-0.390493\pi$
$137$	−7.07107	−	7.07107i	−0.604122	−	0.604122i	−0.941404	+	0.337282i	$0.890493\pi$
$138$		0			0
$139$		−	12.0000i		−	1.01783i	−0.860818	−	0.508913i	$-0.830047\pi$
$139$		−	12.0000i		−	1.01783i	0.860818	−	0.508913i	$-0.169953\pi$
$140$	5.65685	+	2.82843i	0.478091	+	0.239046i
$141$		0			0
$142$	−4.00000	+	4.00000i	−0.335673	+	0.335673i
$143$	−2.82843	+	2.82843i	−0.236525	+	0.236525i
$144$		0			0
$145$	−9.00000	+	3.00000i	−0.747409	+	0.249136i
$146$		−	1.41421i		−	0.117041i
$147$		0			0
$148$	−1.00000	−	1.00000i	−0.0821995	−	0.0821995i
$149$	4.24264			0.347571			0.173785	−	0.984784i	$-0.444400\pi$
$149$	4.24264			0.347571			0.173785	+	0.984784i	$0.444400\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$		0			0
$154$			8.00000i			0.644658i
$155$	−8.48528	+	2.82843i	−0.681554	+	0.227185i
$156$		0			0
$157$	−5.00000	+	5.00000i	−0.399043	+	0.399043i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−5.00000	+	5.00000i	−0.399043	+	0.399043i	0.478852	+	0.877896i	$0.341053\pi$
$158$	8.48528	−	8.48528i	0.675053	−	0.675053i
$159$		0			0
$160$	10.0000	+	5.00000i	0.790569	+	0.395285i
$161$			11.3137i			0.891645i
$162$		0			0
$163$	−8.00000	−	8.00000i	−0.626608	−	0.626608i	0.320605	−	0.947213i	$-0.396114\pi$
$163$	−8.00000	−	8.00000i	−0.626608	−	0.626608i	−0.947213	+	0.320605i	$0.896114\pi$
$164$	−1.41421			−0.110432
$165$		0			0
$166$	−4.00000			−0.310460
$167$	14.1421	+	14.1421i	1.09435	+	1.09435i	0.995058	+	0.0992931i	$0.0316581\pi$
$167$	14.1421	+	14.1421i	1.09435	+	1.09435i	0.0992931	+	0.995058i	$0.468342\pi$
$168$		0			0
$169$		−	11.0000i		−	0.846154i
$170$	2.82843	+	8.48528i	0.216930	+	0.650791i
$171$		0			0
$172$	−8.00000	+	8.00000i	−0.609994	+	0.609994i
$173$	−9.89949	+	9.89949i	−0.752645	+	0.752645i	−0.974972	−	0.222327i	$-0.928635\pi$
$173$	−9.89949	+	9.89949i	−0.752645	+	0.752645i	0.222327	+	0.974972i	$0.428635\pi$
$174$		0			0
$175$	−2.00000	+	14.0000i	−0.151186	+	1.05830i
$176$			2.82843i			0.213201i
$177$		0			0
$178$	−9.00000	−	9.00000i	−0.674579	−	0.674579i
$179$	−25.4558			−1.90266			−0.951330	−	0.308175i	$-0.900282\pi$
$179$	−25.4558			−1.90266			−0.951330	+	0.308175i	$0.900282\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$	2.82843	+	2.82843i	0.209657	+	0.209657i
$183$		0			0
$184$			12.0000i			0.884652i
$185$	1.41421	−	2.82843i	0.103975	−	0.207950i
$186$		0			0
$187$	8.00000	−	8.00000i	0.585018	−	0.585018i
$188$	5.65685	−	5.65685i	0.412568	−	0.412568i
$189$		0			0
$190$		0			0
$191$		−	22.6274i		−	1.63726i	−0.574320	−	0.818631i	$-0.694733\pi$
$191$		−	22.6274i		−	1.63726i	0.574320	−	0.818631i	$-0.305267\pi$
$192$		0			0
$193$	1.00000	+	1.00000i	0.0719816	+	0.0719816i	0.742181	−	0.670199i	$-0.233791\pi$
$193$	1.00000	+	1.00000i	0.0719816	+	0.0719816i	−0.670199	+	0.742181i	$0.733791\pi$
$194$	15.5563			1.11688
$195$		0			0
$196$	−1.00000			−0.0714286
$197$	9.89949	+	9.89949i	0.705310	+	0.705310i	0.965545	−	0.260235i	$-0.0838002\pi$
$197$	9.89949	+	9.89949i	0.705310	+	0.705310i	−0.260235	+	0.965545i	$0.583800\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−2.12132	+	14.8492i	−0.150000	+	1.05000i
$201$		0			0
$202$	11.0000	−	11.0000i	0.773957	−	0.773957i
$203$	8.48528	−	8.48528i	0.595550	−	0.595550i
$204$		0			0
$205$	−1.00000	−	3.00000i	−0.0698430	−	0.209529i
$206$		−	14.1421i		−	0.985329i
$207$		0			0
$208$	1.00000	+	1.00000i	0.0693375	+	0.0693375i
$209$		0			0
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$	−2.82843	−	2.82843i	−0.194257	−	0.194257i
$213$		0			0
$214$			4.00000i			0.273434i
$215$	−22.6274	−	11.3137i	−1.54318	−	0.771589i
$216$		0			0
$217$	8.00000	−	8.00000i	0.543075	−	0.543075i
$218$		0			0
$219$		0			0
$220$	6.00000	−	2.00000i	0.404520	−	0.134840i
$221$		−	5.65685i		−	0.380521i
$222$		0			0
$223$	10.0000	+	10.0000i	0.669650	+	0.669650i	0.957635	−	0.287985i	$-0.0929854\pi$
$223$	10.0000	+	10.0000i	0.669650	+	0.669650i	−0.287985	+	0.957635i	$0.592985\pi$
$224$	−14.1421			−0.944911
$225$		0			0
$226$	14.0000			0.931266
$227$	5.65685	+	5.65685i	0.375459	+	0.375459i	0.869461	−	0.494002i	$-0.164466\pi$
$227$	5.65685	+	5.65685i	0.375459	+	0.375459i	−0.494002	+	0.869461i	$0.664466\pi$
$228$		0			0
$229$			6.00000i			0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$			6.00000i			0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$	−8.48528	+	2.82843i	−0.559503	+	0.186501i
$231$		0			0
$232$	9.00000	−	9.00000i	0.590879	−	0.590879i
$233$	2.82843	−	2.82843i	0.185296	−	0.185296i	−0.608363	−	0.793659i	$-0.708173\pi$
$233$	2.82843	−	2.82843i	0.185296	−	0.185296i	0.793659	+	0.608363i	$0.208173\pi$
$234$		0			0
$235$	16.0000	+	8.00000i	1.04372	+	0.521862i
$236$			8.48528i			0.552345i
$237$		0			0
$238$	−8.00000	−	8.00000i	−0.518563	−	0.518563i
$239$	16.9706			1.09773			0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706			1.09773			0.548867	+	0.835910i	$0.315059\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$	−2.12132	−	2.12132i	−0.136364	−	0.136364i
$243$		0			0
$244$		−	8.00000i		−	0.512148i
$245$	−0.707107	−	2.12132i	−0.0451754	−	0.135526i
$246$		0			0
$247$		0			0
$248$	8.48528	−	8.48528i	0.538816	−	0.538816i
$249$		0			0
$250$	−11.0000	+	2.00000i	−0.695701	+	0.126491i
$251$			19.7990i			1.24970i	0.780744	+	0.624851i	$0.214840\pi$
$251$			19.7990i			1.24970i	−0.780744	+	0.624851i	$0.785160\pi$
$252$		0			0
$253$	8.00000	+	8.00000i	0.502956	+	0.502956i
$254$	−14.1421			−0.887357
$255$		0			0
$256$	−17.0000			−1.06250
$257$	1.41421	+	1.41421i	0.0882162	+	0.0882162i	0.749838	−	0.661622i	$-0.230131\pi$
$257$	1.41421	+	1.41421i	0.0882162	+	0.0882162i	−0.661622	+	0.749838i	$0.730131\pi$
$258$		0			0
$259$			4.00000i			0.248548i
$260$	1.41421	−	2.82843i	0.0877058	−	0.175412i
$261$		0			0
$262$	−10.0000	+	10.0000i	−0.617802	+	0.617802i
$263$	2.82843	−	2.82843i	0.174408	−	0.174408i	−0.614505	−	0.788913i	$-0.710644\pi$
$263$	2.82843	−	2.82843i	0.174408	−	0.174408i	0.788913	+	0.614505i	$0.210644\pi$
$264$		0			0
$265$	4.00000	−	8.00000i	0.245718	−	0.491436i
$266$		0			0
$267$		0			0
$268$	−4.00000	−	4.00000i	−0.244339	−	0.244339i
$269$	12.7279			0.776035			0.388018	−	0.921652i	$-0.373160\pi$
$269$	12.7279			0.776035			0.388018	+	0.921652i	$0.373160\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$	−2.82843	−	2.82843i	−0.171499	−	0.171499i
$273$		0			0
$274$			10.0000i			0.604122i
$275$	8.48528	+	11.3137i	0.511682	+	0.682242i
$276$		0			0
$277$	−11.0000	+	11.0000i	−0.660926	+	0.660926i	−0.955598	−	0.294672i	$-0.904789\pi$
$277$	−11.0000	+	11.0000i	−0.660926	+	0.660926i	0.294672	+	0.955598i	$0.404789\pi$
$278$	−8.48528	+	8.48528i	−0.508913	+	0.508913i
$279$		0			0
$280$	−6.00000	−	18.0000i	−0.358569	−	1.07571i
$281$		−	9.89949i		−	0.590554i	−0.955412	−	0.295277i	$-0.904588\pi$
$281$		−	9.89949i		−	0.590554i	0.955412	−	0.295277i	$-0.0954120\pi$
$282$		0			0
$283$	−8.00000	−	8.00000i	−0.475551	−	0.475551i	0.428155	−	0.903705i	$-0.359164\pi$
$283$	−8.00000	−	8.00000i	−0.475551	−	0.475551i	−0.903705	+	0.428155i	$0.859164\pi$
$284$	−5.65685			−0.335673
$285$		0			0
$286$	4.00000			0.236525
$287$	2.82843	+	2.82843i	0.166957	+	0.166957i
$288$		0			0
$289$		−	1.00000i		−	0.0588235i
$290$	8.48528	+	4.24264i	0.498273	+	0.249136i
$291$		0			0
$292$	1.00000	−	1.00000i	0.0585206	−	0.0585206i
$293$	−9.89949	+	9.89949i	−0.578335	+	0.578335i	−0.934444	−	0.356110i	$-0.884103\pi$
$293$	−9.89949	+	9.89949i	−0.578335	+	0.578335i	0.356110	+	0.934444i	$0.384103\pi$
$294$		0			0
$295$	−18.0000	+	6.00000i	−1.04800	+	0.349334i
$296$			4.24264i			0.246598i
$297$		0			0
$298$	−3.00000	−	3.00000i	−0.173785	−	0.173785i
$299$	5.65685			0.327144
$300$		0			0
$301$	32.0000			1.84445
$302$	−5.65685	−	5.65685i	−0.325515	−	0.325515i
$303$		0			0
$304$		0			0
$305$	16.9706	−	5.65685i	0.971732	−	0.323911i
$306$		0			0
$307$	−8.00000	+	8.00000i	−0.456584	+	0.456584i	−0.897532	−	0.440948i	$-0.854642\pi$
$307$	−8.00000	+	8.00000i	−0.456584	+	0.456584i	0.440948	+	0.897532i	$0.354642\pi$
$308$	−5.65685	+	5.65685i	−0.322329	+	0.322329i
$309$		0			0
$310$	8.00000	+	4.00000i	0.454369	+	0.227185i
$311$			11.3137i			0.641542i	0.947157	+	0.320771i	$0.103942\pi$
$311$			11.3137i			0.641542i	−0.947157	+	0.320771i	$0.896058\pi$
$312$		0			0
$313$	19.0000	+	19.0000i	1.07394	+	1.07394i	0.997038	+	0.0769051i	$0.0245038\pi$
$313$	19.0000	+	19.0000i	1.07394	+	1.07394i	0.0769051	+	0.997038i	$0.475496\pi$
$314$	7.07107			0.399043
$315$		0			0
$316$	12.0000			0.675053
$317$	−19.7990	−	19.7990i	−1.11202	−	1.11202i	−0.992877	−	0.119145i	$-0.961985\pi$
$317$	−19.7990	−	19.7990i	−1.11202	−	1.11202i	−0.119145	−	0.992877i	$-0.538015\pi$
$318$		0			0
$319$		−	12.0000i		−	0.671871i
$320$	−4.94975	−	14.8492i	−0.276699	−	0.830098i
$321$		0			0
$322$	8.00000	−	8.00000i	0.445823	−	0.445823i
$323$		0			0
$324$		0			0
$325$	7.00000	+	1.00000i	0.388290	+	0.0554700i
$326$			11.3137i			0.626608i
$327$		0			0
$328$	3.00000	+	3.00000i	0.165647	+	0.165647i
$329$	−22.6274			−1.24749
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$	−2.82843	−	2.82843i	−0.155230	−	0.155230i
$333$		0			0
$334$		−	20.0000i		−	1.09435i
$335$	5.65685	−	11.3137i	0.309067	−	0.618134i
$336$		0			0
$337$	−5.00000	+	5.00000i	−0.272367	+	0.272367i	−0.830053	−	0.557685i	$-0.811690\pi$
$337$	−5.00000	+	5.00000i	−0.272367	+	0.272367i	0.557685	+	0.830053i	$0.311690\pi$
$338$	−7.77817	+	7.77817i	−0.423077	+	0.423077i
$339$		0			0
$340$	−4.00000	+	8.00000i	−0.216930	+	0.433861i
$341$		−	11.3137i		−	0.612672i
$342$		0			0
$343$	−12.0000	−	12.0000i	−0.647939	−	0.647939i
$344$	33.9411			1.82998
$345$		0			0
$346$	14.0000			0.752645
$347$	−11.3137	−	11.3137i	−0.607352	−	0.607352i	0.334901	−	0.942253i	$-0.391297\pi$
$347$	−11.3137	−	11.3137i	−0.607352	−	0.607352i	−0.942253	+	0.334901i	$0.891297\pi$
$348$		0			0
$349$		−	24.0000i		−	1.28469i	−0.766415	−	0.642345i	$-0.777962\pi$
$349$		−	24.0000i		−	1.28469i	0.766415	−	0.642345i	$-0.222038\pi$
$350$	11.3137	−	8.48528i	0.604743	−	0.453557i
$351$		0			0
$352$	−10.0000	+	10.0000i	−0.533002	+	0.533002i
$353$	2.82843	−	2.82843i	0.150542	−	0.150542i	−0.627818	−	0.778360i	$-0.716052\pi$
$353$	2.82843	−	2.82843i	0.150542	−	0.150542i	0.778360	+	0.627818i	$0.216052\pi$
$354$		0			0
$355$	−4.00000	−	12.0000i	−0.212298	−	0.636894i
$356$		−	12.7279i		−	0.674579i
$357$		0			0
$358$	18.0000	+	18.0000i	0.951330	+	0.951330i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$	11.3137	+	11.3137i	0.594635	+	0.594635i
$363$		0			0
$364$			4.00000i			0.209657i
$365$	2.82843	+	1.41421i	0.148047	+	0.0740233i
$366$		0			0
$367$	−2.00000	+	2.00000i	−0.104399	+	0.104399i	−0.757377	−	0.652978i	$-0.773519\pi$
$367$	−2.00000	+	2.00000i	−0.104399	+	0.104399i	0.652978	+	0.757377i	$0.273519\pi$
$368$	2.82843	−	2.82843i	0.147442	−	0.147442i
$369$		0			0
$370$	−3.00000	+	1.00000i	−0.155963	+	0.0519875i
$371$			11.3137i			0.587378i
$372$		0			0
$373$	−17.0000	−	17.0000i	−0.880227	−	0.880227i	0.113331	−	0.993557i	$-0.463848\pi$
$373$	−17.0000	−	17.0000i	−0.880227	−	0.880227i	−0.993557	+	0.113331i	$0.963848\pi$
$374$	−11.3137			−0.585018
$375$		0			0
$376$	−24.0000			−1.23771
$377$	−4.24264	−	4.24264i	−0.218507	−	0.218507i
$378$		0			0
$379$			36.0000i			1.84920i	0.380945	+	0.924598i	$0.375599\pi$
$379$			36.0000i			1.84920i	−0.380945	+	0.924598i	$0.624401\pi$
$380$		0			0
$381$		0			0
$382$	−16.0000	+	16.0000i	−0.818631	+	0.818631i
$383$	−22.6274	+	22.6274i	−1.15621	+	1.15621i	−0.170923	+	0.985284i	$0.554675\pi$
$383$	−22.6274	+	22.6274i	−1.15621	+	1.15621i	−0.985284	+	0.170923i	$0.945325\pi$
$384$		0			0
$385$	−16.0000	−	8.00000i	−0.815436	−	0.407718i
$386$		−	1.41421i		−	0.0719816i
$387$		0			0
$388$	11.0000	+	11.0000i	0.558440	+	0.558440i
$389$	−4.24264			−0.215110			−0.107555	−	0.994199i	$-0.534302\pi$
$389$	−4.24264			−0.215110			−0.107555	+	0.994199i	$0.534302\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$	2.12132	+	2.12132i	0.107143	+	0.107143i
$393$		0			0
$394$		−	14.0000i		−	0.705310i
$395$	8.48528	+	25.4558i	0.426941	+	1.28082i
$396$		0			0
$397$	19.0000	−	19.0000i	0.953583	−	0.953583i	−0.0453868	−	0.998969i	$-0.514452\pi$
$397$	19.0000	−	19.0000i	0.953583	−	0.953583i	0.998969	+	0.0453868i	$0.0144520\pi$
$398$		0			0
$399$		0			0
$400$	4.00000	−	3.00000i	0.200000	−	0.150000i
$401$			24.0416i			1.20058i	0.799782	+	0.600291i	$0.204949\pi$
$401$			24.0416i			1.20058i	−0.799782	+	0.600291i	$0.795051\pi$
$402$		0			0
$403$	−4.00000	−	4.00000i	−0.199254	−	0.199254i
$404$	15.5563			0.773957
$405$		0			0
$406$	−12.0000			−0.595550
$407$	2.82843	+	2.82843i	0.140200	+	0.140200i
$408$		0			0
$409$			24.0000i			1.18672i	0.804936	+	0.593362i	$0.202200\pi$
$409$			24.0000i			1.18672i	−0.804936	+	0.593362i	$0.797800\pi$
$410$	−1.41421	+	2.82843i	−0.0698430	+	0.139686i
$411$		0			0
$412$	10.0000	−	10.0000i	0.492665	−	0.492665i
$413$	16.9706	−	16.9706i	0.835067	−	0.835067i
$414$		0			0
$415$	4.00000	−	8.00000i	0.196352	−	0.392705i
$416$			7.07107i			0.346688i
$417$		0			0
$418$		0			0
$419$	−8.48528			−0.414533			−0.207267	−	0.978285i	$-0.566457\pi$
$419$	−8.48528			−0.414533			−0.207267	+	0.978285i	$0.566457\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$	2.82843	+	2.82843i	0.137686	+	0.137686i
$423$		0			0
$424$			12.0000i			0.582772i
$425$	−19.7990	−	2.82843i	−0.960392	−	0.137199i
$426$		0			0
$427$	−16.0000	+	16.0000i	−0.774294	+	0.774294i
$428$	−2.82843	+	2.82843i	−0.136717	+	0.136717i
$429$		0			0
$430$	8.00000	+	24.0000i	0.385794	+	1.15738i
$431$		−	5.65685i		−	0.272481i	−0.990676	−	0.136241i	$-0.956498\pi$
$431$		−	5.65685i		−	0.272481i	0.990676	−	0.136241i	$-0.0435020\pi$
$432$		0			0
$433$	−17.0000	−	17.0000i	−0.816968	−	0.816968i	0.168700	−	0.985668i	$-0.446043\pi$
$433$	−17.0000	−	17.0000i	−0.816968	−	0.816968i	−0.985668	+	0.168700i	$0.946043\pi$
$434$	−11.3137			−0.543075
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	24.0000i		−	1.14546i	−0.819745	−	0.572729i	$-0.805885\pi$
$439$		−	24.0000i		−	1.14546i	0.819745	−	0.572729i	$-0.194115\pi$
$440$	−16.9706	−	8.48528i	−0.809040	−	0.404520i
$441$		0			0
$442$	−4.00000	+	4.00000i	−0.190261	+	0.190261i
$443$	28.2843	−	28.2843i	1.34383	−	1.34383i	0.451612	−	0.892215i	$-0.350849\pi$
$443$	28.2843	−	28.2843i	1.34383	−	1.34383i	0.892215	−	0.451612i	$-0.149151\pi$
$444$		0			0
$445$	27.0000	−	9.00000i	1.27992	−	0.426641i
$446$		−	14.1421i		−	0.669650i
$447$		0			0
$448$	14.0000	+	14.0000i	0.661438	+	0.661438i
$449$	12.7279			0.600668			0.300334	−	0.953834i	$-0.402902\pi$
$449$	12.7279			0.600668			0.300334	+	0.953834i	$0.402902\pi$
$450$		0			0
$451$	4.00000			0.188353
$452$	9.89949	+	9.89949i	0.465633	+	0.465633i
$453$		0			0
$454$		−	8.00000i		−	0.375459i
$455$	−8.48528	+	2.82843i	−0.397796	+	0.132599i
$456$		0			0
$457$	25.0000	−	25.0000i	1.16945	−	1.16945i	0.187112	−	0.982339i	$-0.440087\pi$
$457$	25.0000	−	25.0000i	1.16945	−	1.16945i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	4.24264	−	4.24264i	0.198246	−	0.198246i
$459$		0			0
$460$	−8.00000	−	4.00000i	−0.373002	−	0.186501i
$461$		−	9.89949i		−	0.461065i	−0.973065	−	0.230533i	$-0.925953\pi$
$461$		−	9.89949i		−	0.461065i	0.973065	−	0.230533i	$-0.0740469\pi$
$462$		0			0
$463$	10.0000	+	10.0000i	0.464739	+	0.464739i	0.900205	−	0.435466i	$-0.143416\pi$
$463$	10.0000	+	10.0000i	0.464739	+	0.464739i	−0.435466	+	0.900205i	$0.643416\pi$
$464$	−4.24264			−0.196960
$465$		0			0
$466$	−4.00000			−0.185296
$467$	−2.82843	−	2.82843i	−0.130884	−	0.130884i	0.638630	−	0.769514i	$-0.279501\pi$
$467$	−2.82843	−	2.82843i	−0.130884	−	0.130884i	−0.769514	+	0.638630i	$0.779501\pi$
$468$		0			0
$469$			16.0000i			0.738811i
$470$	−5.65685	−	16.9706i	−0.260931	−	0.782794i
$471$		0			0
$472$	18.0000	−	18.0000i	0.828517	−	0.828517i
$473$	22.6274	−	22.6274i	1.04041	−	1.04041i
$474$		0			0
$475$		0			0
$476$		−	11.3137i		−	0.518563i
$477$		0			0
$478$	−12.0000	−	12.0000i	−0.548867	−	0.548867i
$479$	−16.9706			−0.775405			−0.387702	−	0.921785i	$-0.626731\pi$
$479$	−16.9706			−0.775405			−0.387702	+	0.921785i	$0.626731\pi$
$480$		0			0
$481$	2.00000			0.0911922
$482$	7.07107	+	7.07107i	0.322078	+	0.322078i
$483$		0			0
$484$		−	3.00000i		−	0.136364i
$485$	−15.5563	+	31.1127i	−0.706377	+	1.41275i
$486$		0			0
$487$	10.0000	−	10.0000i	0.453143	−	0.453143i	−0.443253	−	0.896396i	$-0.646176\pi$
$487$	10.0000	−	10.0000i	0.453143	−	0.453143i	0.896396	+	0.443253i	$0.146176\pi$
$488$	−16.9706	+	16.9706i	−0.768221	+	0.768221i
$489$		0			0
$490$	−1.00000	+	2.00000i	−0.0451754	+	0.0903508i
$491$		−	14.1421i		−	0.638226i	−0.947717	−	0.319113i	$-0.896615\pi$
$491$		−	14.1421i		−	0.638226i	0.947717	−	0.319113i	$-0.103385\pi$
$492$		0			0
$493$	12.0000	+	12.0000i	0.540453	+	0.540453i
$494$		0			0
$495$		0			0
$496$	−4.00000			−0.179605
$497$	11.3137	+	11.3137i	0.507489	+	0.507489i
$498$		0			0
$499$			24.0000i			1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$			24.0000i			1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$	−9.19239	−	6.36396i	−0.411096	−	0.284605i
$501$		0			0
$502$	14.0000	−	14.0000i	0.624851	−	0.624851i
$503$	−22.6274	+	22.6274i	−1.00891	+	1.00891i	−0.00894668	+	0.999960i	$0.502848\pi$
$503$	−22.6274	+	22.6274i	−1.00891	+	1.00891i	−0.999960	+	0.00894668i	$0.997152\pi$
$504$		0			0
$505$	11.0000	+	33.0000i	0.489494	+	1.46848i
$506$		−	11.3137i		−	0.502956i
$507$		0			0
$508$	−10.0000	−	10.0000i	−0.443678	−	0.443678i
$509$	4.24264			0.188052			0.0940259	−	0.995570i	$-0.470026\pi$
$509$	4.24264			0.188052			0.0940259	+	0.995570i	$0.470026\pi$
$510$		0			0
$511$	−4.00000			−0.176950
$512$	7.77817	+	7.77817i	0.343750	+	0.343750i
$513$		0			0
$514$		−	2.00000i		−	0.0882162i
$515$	28.2843	+	14.1421i	1.24635	+	0.623177i
$516$		0			0
$517$	−16.0000	+	16.0000i	−0.703679	+	0.703679i
$518$	2.82843	−	2.82843i	0.124274	−	0.124274i
$519$		0			0
$520$	−9.00000	+	3.00000i	−0.394676	+	0.131559i
$521$		−	18.3848i		−	0.805452i	−0.915321	−	0.402726i	$-0.868063\pi$
$521$		−	18.3848i		−	0.805452i	0.915321	−	0.402726i	$-0.131937\pi$
$522$		0			0
$523$	−8.00000	−	8.00000i	−0.349816	−	0.349816i	0.510225	−	0.860041i	$-0.329562\pi$
$523$	−8.00000	−	8.00000i	−0.349816	−	0.349816i	−0.860041	+	0.510225i	$0.829562\pi$
$524$	−14.1421			−0.617802
$525$		0			0
$526$	−4.00000			−0.174408
$527$	11.3137	+	11.3137i	0.492833	+	0.492833i
$528$		0			0
$529$			7.00000i			0.304348i
$530$	−8.48528	+	2.82843i	−0.368577	+	0.122859i
$531$		0			0
$532$		0			0
$533$	1.41421	−	1.41421i	0.0612564	−	0.0612564i
$534$		0			0
$535$	−8.00000	−	4.00000i	−0.345870	−	0.172935i
$536$			16.9706i			0.733017i
$537$		0			0
$538$	−9.00000	−	9.00000i	−0.388018	−	0.388018i
$539$	2.82843			0.121829
$540$		0			0
$541$	−16.0000			−0.687894			−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000			−0.687894			−0.343947	+	0.938989i	$0.611764\pi$
$542$	11.3137	+	11.3137i	0.485965	+	0.485965i
$543$		0			0
$544$		−	20.0000i		−	0.857493i
$545$		0			0
$546$		0			0
$547$	−20.0000	+	20.0000i	−0.855138	+	0.855138i	−0.990761	−	0.135622i	$-0.956697\pi$
$547$	−20.0000	+	20.0000i	−0.855138	+	0.855138i	0.135622	+	0.990761i	$0.456697\pi$
$548$	−7.07107	+	7.07107i	−0.302061	+	0.302061i
$549$		0			0
$550$	2.00000	−	14.0000i	0.0852803	−	0.596962i
$551$		0			0
$552$		0			0
$553$	−24.0000	−	24.0000i	−1.02058	−	1.02058i
$554$	15.5563			0.660926
$555$		0			0
$556$	−12.0000			−0.508913
$557$	−2.82843	−	2.82843i	−0.119844	−	0.119844i	0.644641	−	0.764485i	$-0.277007\pi$
$557$	−2.82843	−	2.82843i	−0.119844	−	0.119844i	−0.764485	+	0.644641i	$0.777007\pi$
$558$		0			0
$559$		−	16.0000i		−	0.676728i
$560$	−2.82843	+	5.65685i	−0.119523	+	0.239046i
$561$		0			0
$562$	−7.00000	+	7.00000i	−0.295277	+	0.295277i
$563$	28.2843	−	28.2843i	1.19204	−	1.19204i	0.215546	−	0.976494i	$-0.430847\pi$
$563$	28.2843	−	28.2843i	1.19204	−	1.19204i	0.976494	−	0.215546i	$-0.0691532\pi$
$564$		0			0
$565$	−14.0000	+	28.0000i	−0.588984	+	1.17797i
$566$			11.3137i			0.475551i
$567$		0			0
$568$	12.0000	+	12.0000i	0.503509	+	0.503509i
$569$	−29.6985			−1.24503			−0.622513	−	0.782610i	$-0.713888\pi$
$569$	−29.6985			−1.24503			−0.622513	+	0.782610i	$0.713888\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$	2.82843	+	2.82843i	0.118262	+	0.118262i
$573$		0			0
$574$		−	4.00000i		−	0.166957i
$575$	2.82843	−	19.7990i	0.117954	−	0.825675i
$576$		0			0
$577$	−17.0000	+	17.0000i	−0.707719	+	0.707719i	−0.966055	−	0.258336i	$-0.916826\pi$
$577$	−17.0000	+	17.0000i	−0.707719	+	0.707719i	0.258336	+	0.966055i	$0.416826\pi$
$578$	−0.707107	+	0.707107i	−0.0294118	+	0.0294118i
$579$		0			0
$580$	3.00000	+	9.00000i	0.124568	+	0.373705i
$581$			11.3137i			0.469372i
$582$		0			0
$583$	8.00000	+	8.00000i	0.331326	+	0.331326i
$584$	−4.24264			−0.175562
$585$		0			0
$586$	14.0000			0.578335
$587$	−19.7990	−	19.7990i	−0.817192	−	0.817192i	0.168508	−	0.985700i	$-0.446105\pi$
$587$	−19.7990	−	19.7990i	−0.817192	−	0.817192i	−0.985700	+	0.168508i	$0.946105\pi$
$588$		0			0
$589$		0			0
$590$	16.9706	+	8.48528i	0.698667	+	0.349334i
$591$		0			0
$592$	1.00000	−	1.00000i	0.0410997	−	0.0410997i
$593$	−9.89949	+	9.89949i	−0.406524	+	0.406524i	−0.880524	−	0.474001i	$-0.842809\pi$
$593$	−9.89949	+	9.89949i	−0.406524	+	0.406524i	0.474001	+	0.880524i	$0.342809\pi$
$594$		0			0
$595$	24.0000	−	8.00000i	0.983904	−	0.327968i
$596$		−	4.24264i		−	0.173785i
$597$		0			0
$598$	−4.00000	−	4.00000i	−0.163572	−	0.163572i
$599$	16.9706			0.693398			0.346699	−	0.937976i	$-0.387302\pi$
$599$	16.9706			0.693398			0.346699	+	0.937976i	$0.387302\pi$
$600$		0			0
$601$	8.00000			0.326327			0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000			0.326327			0.163163	+	0.986599i	$0.447830\pi$
$602$	−22.6274	−	22.6274i	−0.922225	−	0.922225i
$603$		0			0
$604$		−	8.00000i		−	0.325515i
$605$	6.36396	−	2.12132i	0.258732	−	0.0862439i
$606$		0			0
$607$	22.0000	−	22.0000i	0.892952	−	0.892952i	−0.101848	−	0.994800i	$-0.532475\pi$
$607$	22.0000	−	22.0000i	0.892952	−	0.892952i	0.994800	+	0.101848i	$0.0324754\pi$
$608$		0			0
$609$		0			0
$610$	−16.0000	−	8.00000i	−0.647821	−	0.323911i
$611$			11.3137i			0.457704i
$612$		0			0
$613$	1.00000	+	1.00000i	0.0403896	+	0.0403896i	0.727013	−	0.686624i	$-0.240908\pi$
$613$	1.00000	+	1.00000i	0.0403896	+	0.0403896i	−0.686624	+	0.727013i	$0.740908\pi$
$614$	11.3137			0.456584
$615$		0			0
$616$	24.0000			0.966988
$617$	14.1421	+	14.1421i	0.569341	+	0.569341i	0.931944	−	0.362603i	$-0.118112\pi$
$617$	14.1421	+	14.1421i	0.569341	+	0.569341i	−0.362603	+	0.931944i	$0.618112\pi$
$618$		0			0
$619$			12.0000i			0.482321i	0.970485	+	0.241160i	$0.0775280\pi$
$619$			12.0000i			0.482321i	−0.970485	+	0.241160i	$0.922472\pi$
$620$	2.82843	+	8.48528i	0.113592	+	0.340777i
$621$		0			0
$622$	8.00000	−	8.00000i	0.320771	−	0.320771i
$623$	−25.4558	+	25.4558i	−1.01987	+	1.01987i
$624$		0			0
$625$	7.00000	−	24.0000i	0.280000	−	0.960000i
$626$		−	26.8701i		−	1.07394i
$627$		0			0
$628$	5.00000	+	5.00000i	0.199522	+	0.199522i
$629$	−5.65685			−0.225554
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$	−25.4558	−	25.4558i	−1.01258	−	1.01258i
$633$		0			0
$634$			28.0000i			1.11202i
$635$	14.1421	−	28.2843i	0.561214	−	1.12243i
$636$		0			0
$637$	1.00000	−	1.00000i	0.0396214	−	0.0396214i
$638$	−8.48528	+	8.48528i	−0.335936	+	0.335936i
$639$		0			0
$640$	3.00000	−	6.00000i	0.118585	−	0.237171i
$641$			15.5563i			0.614439i	0.951639	+	0.307219i	$0.0993986\pi$
$641$			15.5563i			0.614439i	−0.951639	+	0.307219i	$0.900601\pi$
$642$		0			0
$643$	28.0000	+	28.0000i	1.10421	+	1.10421i	0.993897	+	0.110316i	$0.0351862\pi$
$643$	28.0000	+	28.0000i	1.10421	+	1.10421i	0.110316	+	0.993897i	$0.464814\pi$
$644$	11.3137			0.445823
$645$		0			0
$646$		0			0
$647$	−28.2843	−	28.2843i	−1.11197	−	1.11197i	−0.992884	−	0.119085i	$-0.962004\pi$
$647$	−28.2843	−	28.2843i	−1.11197	−	1.11197i	−0.119085	−	0.992884i	$-0.537996\pi$
$648$		0			0
$649$		−	24.0000i		−	0.942082i
$650$	−4.24264	−	5.65685i	−0.166410	−	0.221880i
$651$		0			0
$652$	−8.00000	+	8.00000i	−0.313304	+	0.313304i
$653$	2.82843	−	2.82843i	0.110685	−	0.110685i	−0.649595	−	0.760280i	$-0.725062\pi$
$653$	2.82843	−	2.82843i	0.110685	−	0.110685i	0.760280	+	0.649595i	$0.225062\pi$
$654$		0			0
$655$	−10.0000	−	30.0000i	−0.390732	−	1.17220i
$656$		−	1.41421i		−	0.0552158i
$657$		0			0
$658$	16.0000	+	16.0000i	0.623745	+	0.623745i
$659$	8.48528			0.330540			0.165270	−	0.986248i	$-0.447151\pi$
$659$	8.48528			0.330540			0.165270	+	0.986248i	$0.447151\pi$
$660$		0			0
$661$	−40.0000			−1.55582			−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000			−1.55582			−0.777910	+	0.628376i	$0.783720\pi$
$662$	−5.65685	−	5.65685i	−0.219860	−	0.219860i
$663$		0			0
$664$			12.0000i			0.465690i
$665$		0			0
$666$		0			0
$667$	−12.0000	+	12.0000i	−0.464642	+	0.464642i
$668$	14.1421	−	14.1421i	0.547176	−	0.547176i
$669$		0			0
$670$	−12.0000	+	4.00000i	−0.463600	+	0.154533i
$671$			22.6274i			0.873522i
$672$		0			0
$673$	1.00000	+	1.00000i	0.0385472	+	0.0385472i	0.726118	−	0.687570i	$-0.241323\pi$
$673$	1.00000	+	1.00000i	0.0385472	+	0.0385472i	−0.687570	+	0.726118i	$0.741323\pi$
$674$	7.07107			0.272367
$675$		0			0
$676$	−11.0000			−0.423077
$677$	31.1127	+	31.1127i	1.19576	+	1.19576i	0.975425	+	0.220334i	$0.0707146\pi$
$677$	31.1127	+	31.1127i	1.19576	+	1.19576i	0.220334	+	0.975425i	$0.429285\pi$
$678$		0			0
$679$		−	44.0000i		−	1.68857i
$680$	25.4558	−	8.48528i	0.976187	−	0.325396i
$681$		0			0
$682$	−8.00000	+	8.00000i	−0.306336	+	0.306336i
$683$	2.82843	−	2.82843i	0.108227	−	0.108227i	−0.650920	−	0.759147i	$-0.725617\pi$
$683$	2.82843	−	2.82843i	0.108227	−	0.108227i	0.759147	+	0.650920i	$0.225617\pi$
$684$		0			0
$685$	−20.0000	−	10.0000i	−0.764161	−	0.382080i
$686$			16.9706i			0.647939i
$687$		0			0
$688$	−8.00000	−	8.00000i	−0.304997	−	0.304997i
$689$	5.65685			0.215509
$690$		0			0
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$	9.89949	+	9.89949i	0.376322	+	0.376322i
$693$		0			0
$694$			16.0000i			0.607352i
$695$	−8.48528	−	25.4558i	−0.321865	−	0.965595i
$696$		0			0
$697$	−4.00000	+	4.00000i	−0.151511	+	0.151511i
$698$	−16.9706	+	16.9706i	−0.642345	+	0.642345i
$699$		0			0
$700$	14.0000	+	2.00000i	0.529150	+	0.0755929i
$701$			7.07107i			0.267071i	0.991044	+	0.133535i	$0.0426329\pi$
$701$			7.07107i			0.267071i	−0.991044	+	0.133535i	$0.957367\pi$
$702$		0			0
$703$		0			0
$704$	19.7990			0.746203
$705$		0			0
$706$	−4.00000			−0.150542
$707$	−31.1127	−	31.1127i	−1.17011	−	1.17011i
$708$		0			0
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$	−5.65685	+	11.3137i	−0.212298	+	0.424596i
$711$		0			0
$712$	−27.0000	+	27.0000i	−1.01187	+	1.01187i
$713$	−11.3137	+	11.3137i	−0.423702	+	0.423702i
$714$		0			0
$715$	−4.00000	+	8.00000i	−0.149592	+	0.299183i
$716$			25.4558i			0.951330i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−40.0000			−1.48968
$722$	−13.4350	−	13.4350i	−0.500000	−	0.500000i
$723$		0			0
$724$			16.0000i			0.594635i
$725$	−16.9706	+	12.7279i	−0.630271	+	0.472703i
$726$		0			0
$727$	−2.00000	+	2.00000i	−0.0741759	+	0.0741759i	−0.743221	−	0.669046i	$-0.766703\pi$
$727$	−2.00000	+	2.00000i	−0.0741759	+	0.0741759i	0.669046	+	0.743221i	$0.266703\pi$
$728$	8.48528	−	8.48528i	0.314485	−	0.314485i
$729$		0			0
$730$	−1.00000	−	3.00000i	−0.0370117	−	0.111035i
$731$			45.2548i			1.67381i
$732$		0			0
$733$	1.00000	+	1.00000i	0.0369358	+	0.0369358i	0.725333	−	0.688398i	$-0.241686\pi$
$733$	1.00000	+	1.00000i	0.0369358	+	0.0369358i	−0.688398	+	0.725333i	$0.741686\pi$
$734$	2.82843			0.104399
$735$		0			0
$736$	20.0000			0.737210
$737$	11.3137	+	11.3137i	0.416746	+	0.416746i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	−2.82843	−	1.41421i	−0.103975	−	0.0519875i
$741$		0			0
$742$	8.00000	−	8.00000i	0.293689	−	0.293689i
$743$	−22.6274	+	22.6274i	−0.830119	+	0.830119i	−0.987533	−	0.157413i	$-0.949684\pi$
$743$	−22.6274	+	22.6274i	−0.830119	+	0.830119i	0.157413	+	0.987533i	$0.449684\pi$
$744$		0			0
$745$	9.00000	−	3.00000i	0.329734	−	0.109911i
$746$			24.0416i			0.880227i
$747$		0			0
$748$	−8.00000	−	8.00000i	−0.292509	−	0.292509i
$749$	11.3137			0.413394
$750$		0			0
$751$	−4.00000			−0.145962			−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000			−0.145962			−0.0729810	+	0.997333i	$0.523251\pi$
$752$	5.65685	+	5.65685i	0.206284	+	0.206284i
$753$		0			0
$754$			6.00000i			0.218507i
$755$	16.9706	−	5.65685i	0.617622	−	0.205874i
$756$		0			0
$757$	19.0000	−	19.0000i	0.690567	−	0.690567i	−0.271790	−	0.962357i	$-0.587616\pi$
$757$	19.0000	−	19.0000i	0.690567	−	0.690567i	0.962357	+	0.271790i	$0.0876156\pi$
$758$	25.4558	−	25.4558i	0.924598	−	0.924598i
$759$		0			0
$760$		0			0
$761$		−	52.3259i		−	1.89681i	−0.317058	−	0.948406i	$-0.602695\pi$
$761$		−	52.3259i		−	1.89681i	0.317058	−	0.948406i	$-0.397305\pi$
$762$		0			0
$763$		0			0
$764$	−22.6274			−0.818631
$765$		0			0
$766$	32.0000			1.15621
$767$	−8.48528	−	8.48528i	−0.306386	−	0.306386i
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$	5.65685	+	16.9706i	0.203859	+	0.611577i
$771$		0			0
$772$	1.00000	−	1.00000i	0.0359908	−	0.0359908i
$773$	2.82843	−	2.82843i	0.101731	−	0.101731i	−0.654409	−	0.756141i	$-0.727083\pi$
$773$	2.82843	−	2.82843i	0.101731	−	0.101731i	0.756141	+	0.654409i	$0.227083\pi$
$774$		0			0
$775$	−16.0000	+	12.0000i	−0.574737	+	0.431053i
$776$		−	46.6690i		−	1.67532i
$777$		0			0
$778$	3.00000	+	3.00000i	0.107555	+	0.107555i
$779$		0			0
$780$		0			0
$781$	16.0000			0.572525
$782$	11.3137	+	11.3137i	0.404577	+	0.404577i
$783$		0			0
$784$		−	1.00000i		−	0.0357143i
$785$	−7.07107	+	14.1421i	−0.252377	+	0.504754i
$786$		0			0
$787$	4.00000	−	4.00000i	0.142585	−	0.142585i	−0.632211	−	0.774796i	$-0.717853\pi$
$787$	4.00000	−	4.00000i	0.142585	−	0.142585i	0.774796	+	0.632211i	$0.217853\pi$
$788$	9.89949	−	9.89949i	0.352655	−	0.352655i
$789$		0			0
$790$	12.0000	−	24.0000i	0.426941	−	0.853882i
$791$		−	39.5980i		−	1.40794i
$792$		0			0
$793$	8.00000	+	8.00000i	0.284088	+	0.284088i
$794$	−26.8701			−0.953583
$795$		0			0
$796$		0			0
$797$	1.41421	+	1.41421i	0.0500940	+	0.0500940i	0.731710	−	0.681616i	$-0.238723\pi$
$797$	1.41421	+	1.41421i	0.0500940	+	0.0500940i	−0.681616	+	0.731710i	$0.738723\pi$
$798$		0			0
$799$		−	32.0000i		−	1.13208i
$800$	24.7487	+	3.53553i	0.875000	+	0.125000i
$801$		0			0
$802$	17.0000	−	17.0000i	0.600291	−	0.600291i
$803$	−2.82843	+	2.82843i	−0.0998130	+	0.0998130i
$804$		0			0
$805$	8.00000	+	24.0000i	0.281963	+	0.845889i
$806$			5.65685i			0.199254i
$807$		0			0
$808$	−33.0000	−	33.0000i	−1.16094	−	1.16094i
$809$	−12.7279			−0.447490			−0.223745	−	0.974648i	$-0.571828\pi$
$809$	−12.7279			−0.447490			−0.223745	+	0.974648i	$0.571828\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$	−8.48528	−	8.48528i	−0.297775	−	0.297775i
$813$		0			0
$814$		−	4.00000i		−	0.140200i
$815$	−22.6274	−	11.3137i	−0.792604	−	0.396302i
$816$		0			0
$817$		0			0
$818$	16.9706	−	16.9706i	0.593362	−	0.593362i
$819$		0			0
$820$	−3.00000	+	1.00000i	−0.104765	+	0.0349215i
$821$		−	35.3553i		−	1.23391i	−0.786998	−	0.616955i	$-0.788366\pi$
$821$		−	35.3553i		−	1.23391i	0.786998	−	0.616955i	$-0.211634\pi$
$822$		0			0
$823$	10.0000	+	10.0000i	0.348578	+	0.348578i	0.859580	−	0.511002i	$-0.170725\pi$
$823$	10.0000	+	10.0000i	0.348578	+	0.348578i	−0.511002	+	0.859580i	$0.670725\pi$
$824$	−42.4264			−1.47799
$825$		0			0
$826$	−24.0000			−0.835067
$827$	22.6274	+	22.6274i	0.786832	+	0.786832i	0.980974	−	0.194141i	$-0.0621920\pi$
$827$	22.6274	+	22.6274i	0.786832	+	0.786832i	−0.194141	+	0.980974i	$0.562192\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$	−8.48528	+	2.82843i	−0.294528	+	0.0981761i
$831$		0			0
$832$	7.00000	−	7.00000i	0.242681	−	0.242681i
$833$	−2.82843	+	2.82843i	−0.0979992	+	0.0979992i
$834$		0			0
$835$	40.0000	+	20.0000i	1.38426	+	0.692129i
$836$		0			0
$837$		0			0
$838$	6.00000	+	6.00000i	0.207267	+	0.207267i
$839$	33.9411			1.17178			0.585889	−	0.810391i	$-0.300745\pi$
$839$	33.9411			1.17178			0.585889	+	0.810391i	$0.300745\pi$
$840$		0			0
$841$	−11.0000			−0.379310
$842$	7.07107	+	7.07107i	0.243685	+	0.243685i
$843$		0			0
$844$			4.00000i			0.137686i
$845$	−7.77817	−	23.3345i	−0.267577	−	0.802732i
$846$		0			0
$847$	−6.00000	+	6.00000i	−0.206162	+	0.206162i
$848$	2.82843	−	2.82843i	0.0971286	−	0.0971286i
$849$		0			0
$850$	12.0000	+	16.0000i	0.411597	+	0.548795i
$851$		−	5.65685i		−	0.193914i
$852$		0			0
$853$	−17.0000	−	17.0000i	−0.582069	−	0.582069i	0.353402	−	0.935471i	$-0.385025\pi$
$853$	−17.0000	−	17.0000i	−0.582069	−	0.582069i	−0.935471	+	0.353402i	$0.885025\pi$
$854$	22.6274			0.774294
$855$		0			0
$856$	12.0000			0.410152
$857$	31.1127	+	31.1127i	1.06279	+	1.06279i	0.997892	+	0.0648976i	$0.0206721\pi$
$857$	31.1127	+	31.1127i	1.06279	+	1.06279i	0.0648976	+	0.997892i	$0.479328\pi$
$858$		0			0
$859$		−	12.0000i		−	0.409435i	−0.978821	−	0.204717i	$-0.934372\pi$
$859$		−	12.0000i		−	0.409435i	0.978821	−	0.204717i	$-0.0656275\pi$
$860$	−11.3137	+	22.6274i	−0.385794	+	0.771589i
$861$		0			0
$862$	−4.00000	+	4.00000i	−0.136241	+	0.136241i
$863$	2.82843	−	2.82843i	0.0962808	−	0.0962808i	−0.657326	−	0.753607i	$-0.728312\pi$
$863$	2.82843	−	2.82843i	0.0962808	−	0.0962808i	0.753607	+	0.657326i	$0.228312\pi$
$864$		0			0
$865$	−14.0000	+	28.0000i	−0.476014	+	0.952029i
$866$			24.0416i			0.816968i
$867$		0			0
$868$	−8.00000	−	8.00000i	−0.271538	−	0.271538i
$869$	−33.9411			−1.15137
$870$		0			0
$871$	8.00000			0.271070
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5.65685	+	31.1127i	0.191237	+	1.05180i
$876$		0			0
$877$	13.0000	−	13.0000i	0.438979	−	0.438979i	−0.452689	−	0.891668i	$-0.649535\pi$
$877$	13.0000	−	13.0000i	0.438979	−	0.438979i	0.891668	+	0.452689i	$0.149535\pi$
$878$	−16.9706	+	16.9706i	−0.572729	+	0.572729i
$879$		0			0
$880$	2.00000	+	6.00000i	0.0674200	+	0.202260i
$881$			7.07107i			0.238230i	0.992880	+	0.119115i	$0.0380058\pi$
$881$			7.07107i			0.238230i	−0.992880	+	0.119115i	$0.961994\pi$
$882$		0			0
$883$	−8.00000	−	8.00000i	−0.269221	−	0.269221i	0.559565	−	0.828786i	$-0.310968\pi$
$883$	−8.00000	−	8.00000i	−0.269221	−	0.269221i	−0.828786	+	0.559565i	$0.810968\pi$
$884$	−5.65685			−0.190261
$885$		0			0
$886$	−40.0000			−1.34383
$887$	14.1421	+	14.1421i	0.474846	+	0.474846i	0.903479	−	0.428632i	$-0.141004\pi$
$887$	14.1421	+	14.1421i	0.474846	+	0.474846i	−0.428632	+	0.903479i	$0.641004\pi$
$888$		0			0
$889$			40.0000i			1.34156i
$890$	−25.4558	−	12.7279i	−0.853282	−	0.426641i
$891$		0			0
$892$	10.0000	−	10.0000i	0.334825	−	0.334825i
$893$		0			0
$894$		0			0
$895$	−54.0000	+	18.0000i	−1.80502	+	0.601674i
$896$			8.48528i			0.283473i
$897$		0			0
$898$	−9.00000	−	9.00000i	−0.300334	−	0.300334i
$899$	16.9706			0.566000
$900$		0			0
$901$	−16.0000			−0.533037
$902$	−2.82843	−	2.82843i	−0.0941763	−	0.0941763i
$903$		0			0
$904$		−	42.0000i		−	1.39690i
$905$	−33.9411	+	11.3137i	−1.12824	+	0.376080i
$906$		0			0
$907$	−20.0000	+	20.0000i	−0.664089	+	0.664089i	−0.956341	−	0.292252i	$-0.905595\pi$
$907$	−20.0000	+	20.0000i	−0.664089	+	0.664089i	0.292252	+	0.956341i	$0.405595\pi$
$908$	5.65685	−	5.65685i	0.187729	−	0.187729i
$909$		0			0
$910$	8.00000	+	4.00000i	0.265197	+	0.132599i
$911$		−	22.6274i		−	0.749680i	−0.927090	−	0.374840i	$-0.877698\pi$
$911$		−	22.6274i		−	0.749680i	0.927090	−	0.374840i	$-0.122302\pi$
$912$		0			0
$913$	8.00000	+	8.00000i	0.264761	+	0.264761i
$914$	−35.3553			−1.16945
$915$		0			0
$916$	6.00000			0.198246
$917$	28.2843	+	28.2843i	0.934029	+	0.934029i
$918$		0			0
$919$			36.0000i			1.18753i	0.804638	+	0.593765i	$0.202359\pi$
$919$			36.0000i			1.18753i	−0.804638	+	0.593765i	$0.797641\pi$
$920$	8.48528	+	25.4558i	0.279751	+	0.839254i
$921$		0			0
$922$	−7.00000	+	7.00000i	−0.230533	+	0.230533i
$923$	5.65685	−	5.65685i	0.186198	−	0.186198i
$924$		0			0
$925$	1.00000	−	7.00000i	0.0328798	−	0.230159i
$926$		−	14.1421i		−	0.464739i
$927$		0			0
$928$	−15.0000	−	15.0000i	−0.492399	−	0.492399i
$929$	46.6690			1.53116			0.765581	−	0.643340i	$-0.222452\pi$
$929$	46.6690			1.53116			0.765581	+	0.643340i	$0.222452\pi$
$930$		0			0
$931$		0			0
$932$	−2.82843	−	2.82843i	−0.0926482	−	0.0926482i
$933$		0			0
$934$			4.00000i			0.130884i
$935$	11.3137	−	22.6274i	0.369998	−	0.739996i
$936$		0			0
$937$	37.0000	−	37.0000i	1.20874	−	1.20874i	0.237301	−	0.971436i	$-0.423737\pi$
$937$	37.0000	−	37.0000i	1.20874	−	1.20874i	0.971436	−	0.237301i	$-0.0762628\pi$
$938$	11.3137	−	11.3137i	0.369406	−	0.369406i
$939$		0			0
$940$	8.00000	−	16.0000i	0.260931	−	0.521862i
$941$		−	1.41421i		−	0.0461020i	−0.999734	−	0.0230510i	$-0.992662\pi$
$941$		−	1.41421i		−	0.0461020i	0.999734	−	0.0230510i	$-0.00733802\pi$
$942$		0			0
$943$	−4.00000	−	4.00000i	−0.130258	−	0.130258i
$944$	−8.48528			−0.276172
$945$		0			0
$946$	−32.0000			−1.04041
$947$	5.65685	+	5.65685i	0.183823	+	0.183823i	0.793019	−	0.609196i	$-0.208508\pi$
$947$	5.65685	+	5.65685i	0.183823	+	0.183823i	−0.609196	+	0.793019i	$0.708508\pi$
$948$		0			0
$949$			2.00000i			0.0649227i
$950$		0			0
$951$		0			0
$952$	−24.0000	+	24.0000i	−0.777844	+	0.777844i
$953$	15.5563	−	15.5563i	0.503920	−	0.503920i	−0.408734	−	0.912654i	$-0.634030\pi$
$953$	15.5563	−	15.5563i	0.503920	−	0.503920i	0.912654	+	0.408734i	$0.134030\pi$
$954$		0			0
$955$	−16.0000	−	48.0000i	−0.517748	−	1.55324i
$956$		−	16.9706i		−	0.548867i
$957$		0			0
$958$	12.0000	+	12.0000i	0.387702	+	0.387702i
$959$	28.2843			0.913347
$960$		0			0
$961$	−15.0000			−0.483871
$962$	−1.41421	−	1.41421i	−0.0455961	−	0.0455961i
$963$		0			0
$964$			10.0000i			0.322078i
$965$	2.82843	+	1.41421i	0.0910503	+	0.0455251i
$966$		0			0
$967$	22.0000	−	22.0000i	0.707472	−	0.707472i	−0.258531	−	0.966003i	$-0.583238\pi$
$967$	22.0000	−	22.0000i	0.707472	−	0.707472i	0.966003	+	0.258531i	$0.0832383\pi$
$968$	−6.36396	+	6.36396i	−0.204545	+	0.204545i
$969$		0			0
$970$	33.0000	−	11.0000i	1.05957	−	0.353189i
$971$			19.7990i			0.635380i	0.948195	+	0.317690i	$0.102907\pi$
$971$			19.7990i			0.635380i	−0.948195	+	0.317690i	$0.897093\pi$
$972$		0			0
$973$	24.0000	+	24.0000i	0.769405	+	0.769405i
$974$	−14.1421			−0.453143
$975$		0			0
$976$	8.00000			0.256074
$977$	−36.7696	−	36.7696i	−1.17636	−	1.17636i	−0.980664	−	0.195698i	$-0.937303\pi$
$977$	−36.7696	−	36.7696i	−1.17636	−	1.17636i	−0.195698	−	0.980664i	$-0.562697\pi$
$978$		0			0
$979$			36.0000i			1.15056i
$980$	−2.12132	+	0.707107i	−0.0677631	+	0.0225877i
$981$		0			0
$982$	−10.0000	+	10.0000i	−0.319113	+	0.319113i
$983$	−22.6274	+	22.6274i	−0.721703	+	0.721703i	−0.968952	−	0.247249i	$-0.920473\pi$
$983$	−22.6274	+	22.6274i	−0.721703	+	0.721703i	0.247249	+	0.968952i	$0.420473\pi$
$984$		0			0
$985$	28.0000	+	14.0000i	0.892154	+	0.446077i
$986$		−	16.9706i		−	0.540453i
$987$		0			0
$988$		0			0
$989$	−45.2548			−1.43902
$990$		0			0
$991$	−16.0000			−0.508257			−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000			−0.508257			−0.254128	+	0.967170i	$0.581789\pi$
$992$	−14.1421	−	14.1421i	−0.449013	−	0.449013i
$993$		0			0
$994$		−	16.0000i		−	0.507489i
$995$		0			0
$996$		0			0
$997$	25.0000	−	25.0000i	0.791758	−	0.791758i	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	25.0000	−	25.0000i	0.791758	−	0.791758i	0.981780	+	0.190022i	$0.0608559\pi$
$998$	16.9706	−	16.9706i	0.537194	−	0.537194i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.2.f.a.8.1	✓	4
3.2	odd	2	inner	45.2.f.a.8.2	yes	4
4.3	odd	2		720.2.w.d.593.2		4
5.2	odd	4	inner	45.2.f.a.17.2	yes	4
5.3	odd	4		225.2.f.a.107.1		4
5.4	even	2		225.2.f.a.143.2		4
8.3	odd	2		2880.2.w.k.2753.1		4
8.5	even	2		2880.2.w.b.2753.1		4
9.2	odd	6		405.2.m.a.53.2		8
9.4	even	3		405.2.m.a.188.2		8
9.5	odd	6		405.2.m.a.188.1		8
9.7	even	3		405.2.m.a.53.1		8
12.11	even	2		720.2.w.d.593.1		4
15.2	even	4	inner	45.2.f.a.17.1	yes	4
15.8	even	4		225.2.f.a.107.2		4
15.14	odd	2		225.2.f.a.143.1		4
20.3	even	4		3600.2.w.b.1457.1		4
20.7	even	4		720.2.w.d.17.1		4
20.19	odd	2		3600.2.w.b.593.1		4
24.5	odd	2		2880.2.w.b.2753.2		4
24.11	even	2		2880.2.w.k.2753.2		4
40.27	even	4		2880.2.w.k.2177.2		4
40.37	odd	4		2880.2.w.b.2177.2		4
45.2	even	12		405.2.m.a.377.2		8
45.7	odd	12		405.2.m.a.377.1		8
45.22	odd	12		405.2.m.a.107.2		8
45.32	even	12		405.2.m.a.107.1		8
60.23	odd	4		3600.2.w.b.1457.2		4
60.47	odd	4		720.2.w.d.17.2		4
60.59	even	2		3600.2.w.b.593.2		4
120.77	even	4		2880.2.w.b.2177.1		4
120.107	odd	4		2880.2.w.k.2177.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.f.a.8.1	✓	4	1.1	even	1	trivial
45.2.f.a.8.2	yes	4	3.2	odd	2	inner
45.2.f.a.17.1	yes	4	15.2	even	4	inner
45.2.f.a.17.2	yes	4	5.2	odd	4	inner
225.2.f.a.107.1		4	5.3	odd	4
225.2.f.a.107.2		4	15.8	even	4
225.2.f.a.143.1		4	15.14	odd	2
225.2.f.a.143.2		4	5.4	even	2
405.2.m.a.53.1		8	9.7	even	3
405.2.m.a.53.2		8	9.2	odd	6
405.2.m.a.107.1		8	45.32	even	12
405.2.m.a.107.2		8	45.22	odd	12
405.2.m.a.188.1		8	9.5	odd	6
405.2.m.a.188.2		8	9.4	even	3
405.2.m.a.377.1		8	45.7	odd	12
405.2.m.a.377.2		8	45.2	even	12
720.2.w.d.17.1		4	20.7	even	4
720.2.w.d.17.2		4	60.47	odd	4
720.2.w.d.593.1		4	12.11	even	2
720.2.w.d.593.2		4	4.3	odd	2
2880.2.w.b.2177.1		4	120.77	even	4
2880.2.w.b.2177.2		4	40.37	odd	4
2880.2.w.b.2753.1		4	8.5	even	2
2880.2.w.b.2753.2		4	24.5	odd	2
2880.2.w.k.2177.1		4	120.107	odd	4
2880.2.w.k.2177.2		4	40.27	even	4
2880.2.w.k.2753.1		4	8.3	odd	2
2880.2.w.k.2753.2		4	24.11	even	2
3600.2.w.b.593.1		4	20.19	odd	2
3600.2.w.b.593.2		4	60.59	even	2
3600.2.w.b.1457.1		4	20.3	even	4
3600.2.w.b.1457.2		4	60.23	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	45.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(2.12132 - 0.707107i) q^{5} +(-2.00000 + 2.00000i) q^{7} +(-2.12132 + 2.12132i) q^{8} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(2.12132 - 0.707107i) q^{5} +(-2.00000 + 2.00000i) q^{7} +(-2.12132 + 2.12132i) q^{8} +(-2.00000 - 1.00000i) q^{10} +2.82843i q^{11} +(1.00000 + 1.00000i) q^{13} +2.82843 q^{14} +1.00000 q^{16} +(-2.82843 - 2.82843i) q^{17} +(-0.707107 - 2.12132i) q^{20} +(2.00000 - 2.00000i) q^{22} +(2.82843 - 2.82843i) q^{23} +(4.00000 - 3.00000i) q^{25} -1.41421i q^{26} +(2.00000 + 2.00000i) q^{28} -4.24264 q^{29} -4.00000 q^{31} +(3.53553 + 3.53553i) q^{32} +4.00000i q^{34} +(-2.82843 + 5.65685i) q^{35} +(1.00000 - 1.00000i) q^{37} +(-3.00000 + 6.00000i) q^{40} -1.41421i q^{41} +(-8.00000 - 8.00000i) q^{43} +2.82843 q^{44} -4.00000 q^{46} +(5.65685 + 5.65685i) q^{47} -1.00000i q^{49} +(-4.94975 - 0.707107i) q^{50} +(1.00000 - 1.00000i) q^{52} +(2.82843 - 2.82843i) q^{53} +(2.00000 + 6.00000i) q^{55} -8.48528i q^{56} +(3.00000 + 3.00000i) q^{58} -8.48528 q^{59} +8.00000 q^{61} +(2.82843 + 2.82843i) q^{62} -7.00000i q^{64} +(2.82843 + 1.41421i) q^{65} +(4.00000 - 4.00000i) q^{67} +(-2.82843 + 2.82843i) q^{68} +(6.00000 - 2.00000i) q^{70} -5.65685i q^{71} +(1.00000 + 1.00000i) q^{73} -1.41421 q^{74} +(-5.65685 - 5.65685i) q^{77} +12.0000i q^{79} +(2.12132 - 0.707107i) q^{80} +(-1.00000 + 1.00000i) q^{82} +(2.82843 - 2.82843i) q^{83} +(-8.00000 - 4.00000i) q^{85} +11.3137i q^{86} +(-6.00000 - 6.00000i) q^{88} +12.7279 q^{89} -4.00000 q^{91} +(-2.82843 - 2.82843i) q^{92} -8.00000i q^{94} +(-11.0000 + 11.0000i) q^{97} +(-0.707107 + 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7}+O(q^{10}) \) 4 * q - 8 * q^7	\( 4 q - 8 q^{7} - 8 q^{10} + 4 q^{13} + 4 q^{16} + 8 q^{22} + 16 q^{25} + 8 q^{28} - 16 q^{31} + 4 q^{37} - 12 q^{40} - 32 q^{43} - 16 q^{46} + 4 q^{52} + 8 q^{55} + 12 q^{58} + 32 q^{61} + 16 q^{67} + 24 q^{70} + 4 q^{73} - 4 q^{82} - 32 q^{85} - 24 q^{88} - 16 q^{91} - 44 q^{97}+O(q^{100}) \) 4 * q - 8 * q^7 - 8 * q^10 + 4 * q^13 + 4 * q^16 + 8 * q^22 + 16 * q^25 + 8 * q^28 - 16 * q^31 + 4 * q^37 - 12 * q^40 - 32 * q^43 - 16 * q^46 + 4 * q^52 + 8 * q^55 + 12 * q^58 + 32 * q^61 + 16 * q^67 + 24 * q^70 + 4 * q^73 - 4 * q^82 - 32 * q^85 - 24 * q^88 - 16 * q^91 - 44 * q^97

Introduction

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