LMFDB - Embedded newform 45.2.f.a.17.1

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			17.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	45.17
Dual form			45.2.f.a.8.1

$q$-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 $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

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

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	−0.911438	−	0.411438i	$-0.865027\pi$
$2$	−0.707107	+	0.707107i	−0.500000	+	0.500000i	0.411438	+	0.911438i	$0.365027\pi$
$3$		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	0.219650	−	0.975579i	$-0.429509\pi$
$7$	−2.00000	−	2.00000i	−0.755929	−	0.755929i	−0.975579	+	0.219650i	$0.929509\pi$
$8$	−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	$-0.859781\pi$
$11$		−	2.82843i		−	0.852803i	0.904534	−	0.426401i	$-0.140219\pi$
$12$		0			0
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	0.832050	+	0.554700i	$0.187167\pi$
$14$	2.82843			0.755929
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.82843	+	2.82843i	−0.685994	+	0.685994i	−0.961344	−	0.275350i	$-0.911206\pi$
$17$	−2.82843	+	2.82843i	−0.685994	+	0.685994i	0.275350	+	0.961344i	$0.411206\pi$
$18$		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	0.937568	−	0.347801i	$-0.113071\pi$
$23$	2.82843	+	2.82843i	0.589768	+	0.589768i	−0.347801	+	0.937568i	$0.613071\pi$
$24$		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	0.784512	−	0.620113i	$-0.212913\pi$
$37$	1.00000	+	1.00000i	0.164399	+	0.164399i	−0.620113	+	0.784512i	$0.712913\pi$
$38$		0			0
$39$		0			0
$40$	−3.00000	−	6.00000i	−0.474342	−	0.948683i
$41$			1.41421i			0.220863i	0.993884	+	0.110432i	$0.0352233\pi$
$41$			1.41421i			0.220863i	−0.993884	+	0.110432i	$0.964777\pi$
$42$		0			0
$43$	−8.00000	+	8.00000i	−1.21999	+	1.21999i	−0.252353	+	0.967635i	$0.581205\pi$
$43$	−8.00000	+	8.00000i	−1.21999	+	1.21999i	−0.967635	+	0.252353i	$0.918795\pi$
$44$	2.82843			0.426401
$45$		0			0
$46$	−4.00000			−0.589768
$47$	5.65685	−	5.65685i	0.825137	−	0.825137i	−0.161703	−	0.986840i	$-0.551699\pi$
$47$	5.65685	−	5.65685i	0.825137	−	0.825137i	0.986840	+	0.161703i	$0.0516985\pi$
$48$		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	0.874157	−	0.485643i	$-0.161414\pi$
$53$	2.82843	+	2.82843i	0.388514	+	0.388514i	−0.485643	+	0.874157i	$0.661414\pi$
$54$		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	0.907889	−	0.419211i	$-0.137693\pi$
$67$	4.00000	+	4.00000i	0.488678	+	0.488678i	−0.419211	+	0.907889i	$0.637693\pi$
$68$	−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	$0.108964\pi$
$71$			5.65685i			0.671345i	−0.941979	+	0.335673i	$0.891036\pi$
$72$		0			0
$73$	1.00000	−	1.00000i	0.117041	−	0.117041i	−0.646160	−	0.763202i	$-0.723626\pi$
$73$	1.00000	−	1.00000i	0.117041	−	0.117041i	0.763202	+	0.646160i	$0.223626\pi$
$74$	−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	$-0.764121\pi$
$79$		−	12.0000i		−	1.35011i	0.737769	−	0.675053i	$-0.235879\pi$
$80$	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	0.845088	−	0.534628i	$-0.179548\pi$
$83$	2.82843	+	2.82843i	0.310460	+	0.310460i	−0.534628	+	0.845088i	$0.679548\pi$
$84$		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	−0.992196	−	0.124684i	$-0.960208\pi$
$97$	−11.0000	−	11.0000i	−1.11688	−	1.11688i	−0.124684	−	0.992196i	$-0.539792\pi$
$98$	−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	$-0.718274\pi$
$101$		−	15.5563i		−	1.54791i	0.633238	−	0.773957i	$-0.281726\pi$
$102$		0			0
$103$	10.0000	−	10.0000i	0.985329	−	0.985329i	−0.0145647	−	0.999894i	$-0.504636\pi$
$103$	10.0000	−	10.0000i	0.985329	−	0.985329i	0.999894	+	0.0145647i	$0.00463624\pi$
$104$	−4.24264			−0.416025
$105$		0			0
$106$	−4.00000			−0.388514
$107$	−2.82843	+	2.82843i	−0.273434	+	0.273434i	−0.830481	−	0.557047i	$-0.811934\pi$
$107$	−2.82843	+	2.82843i	−0.273434	+	0.273434i	0.557047	+	0.830481i	$0.311934\pi$
$108$		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	0.0665190	−	0.997785i	$-0.478811\pi$
$113$	−9.89949	−	9.89949i	−0.931266	−	0.931266i	−0.997785	+	0.0665190i	$0.978811\pi$
$114$		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	0.994268	−	0.106912i	$-0.0340963\pi$
$127$	10.0000	+	10.0000i	0.887357	+	0.887357i	−0.106912	+	0.994268i	$0.534096\pi$
$128$	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	$0.211977\pi$
$131$			14.1421i			1.23560i	−0.786334	+	0.617802i	$0.788023\pi$
$132$		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	−0.941404	−	0.337282i	$-0.890493\pi$
$137$	−7.07107	+	7.07107i	−0.604122	+	0.604122i	0.337282	+	0.941404i	$0.390493\pi$
$138$		0			0
$139$			12.0000i			1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$			12.0000i			1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$	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	0.478852	−	0.877896i	$-0.341053\pi$
$157$	−5.00000	−	5.00000i	−0.399043	−	0.399043i	−0.877896	+	0.478852i	$0.841053\pi$
$158$	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	−0.947213	−	0.320605i	$-0.896114\pi$
$163$	−8.00000	+	8.00000i	−0.626608	+	0.626608i	0.320605	+	0.947213i	$0.396114\pi$
$164$	−1.41421			−0.110432
$165$		0			0
$166$	−4.00000			−0.310460
$167$	14.1421	−	14.1421i	1.09435	−	1.09435i	0.0992931	−	0.995058i	$-0.468342\pi$
$167$	14.1421	−	14.1421i	1.09435	−	1.09435i	0.995058	−	0.0992931i	$-0.0316581\pi$
$168$		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	0.222327	−	0.974972i	$-0.428635\pi$
$173$	−9.89949	−	9.89949i	−0.752645	−	0.752645i	−0.974972	+	0.222327i	$0.928635\pi$
$174$		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	$0.305267\pi$
$191$			22.6274i			1.63726i	−0.574320	+	0.818631i	$0.694733\pi$
$192$		0			0
$193$	1.00000	−	1.00000i	0.0719816	−	0.0719816i	−0.670199	−	0.742181i	$-0.733791\pi$
$193$	1.00000	−	1.00000i	0.0719816	−	0.0719816i	0.742181	+	0.670199i	$0.233791\pi$
$194$	15.5563			1.11688
$195$		0			0
$196$	−1.00000			−0.0714286
$197$	9.89949	−	9.89949i	0.705310	−	0.705310i	−0.260235	−	0.965545i	$-0.583800\pi$
$197$	9.89949	−	9.89949i	0.705310	−	0.705310i	0.965545	+	0.260235i	$0.0838002\pi$
$198$		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	−0.287985	−	0.957635i	$-0.592985\pi$
$223$	10.0000	−	10.0000i	0.669650	−	0.669650i	0.957635	+	0.287985i	$0.0929854\pi$
$224$	−14.1421			−0.944911
$225$		0			0
$226$	14.0000			0.931266
$227$	5.65685	−	5.65685i	0.375459	−	0.375459i	−0.494002	−	0.869461i	$-0.664466\pi$
$227$	5.65685	−	5.65685i	0.375459	−	0.375459i	0.869461	+	0.494002i	$0.164466\pi$
$228$		0			0
$229$		−	6.00000i		−	0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$		−	6.00000i		−	0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$	−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	0.793659	−	0.608363i	$-0.208173\pi$
$233$	2.82843	+	2.82843i	0.185296	+	0.185296i	−0.608363	+	0.793659i	$0.708173\pi$
$234$		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	$-0.785160\pi$
$251$		−	19.7990i		−	1.24970i	0.780744	−	0.624851i	$-0.214840\pi$
$252$		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	−0.661622	−	0.749838i	$-0.730131\pi$
$257$	1.41421	−	1.41421i	0.0882162	−	0.0882162i	0.749838	+	0.661622i	$0.230131\pi$
$258$		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	0.788913	−	0.614505i	$-0.210644\pi$
$263$	2.82843	+	2.82843i	0.174408	+	0.174408i	−0.614505	+	0.788913i	$0.710644\pi$
$264$		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	0.294672	−	0.955598i	$-0.404789\pi$
$277$	−11.0000	−	11.0000i	−0.660926	−	0.660926i	−0.955598	+	0.294672i	$0.904789\pi$
$278$	−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	$0.0954120\pi$
$281$			9.89949i			0.590554i	−0.955412	+	0.295277i	$0.904588\pi$
$282$		0			0
$283$	−8.00000	+	8.00000i	−0.475551	+	0.475551i	−0.903705	−	0.428155i	$-0.859164\pi$
$283$	−8.00000	+	8.00000i	−0.475551	+	0.475551i	0.428155	+	0.903705i	$0.359164\pi$
$284$	−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	0.356110	−	0.934444i	$-0.384103\pi$
$293$	−9.89949	−	9.89949i	−0.578335	−	0.578335i	−0.934444	+	0.356110i	$0.884103\pi$
$294$		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	0.440948	−	0.897532i	$-0.354642\pi$
$307$	−8.00000	−	8.00000i	−0.456584	−	0.456584i	−0.897532	+	0.440948i	$0.854642\pi$
$308$	−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	$-0.896058\pi$
$311$		−	11.3137i		−	0.641542i	0.947157	−	0.320771i	$-0.103942\pi$
$312$		0			0
$313$	19.0000	−	19.0000i	1.07394	−	1.07394i	0.0769051	−	0.997038i	$-0.475496\pi$
$313$	19.0000	−	19.0000i	1.07394	−	1.07394i	0.997038	−	0.0769051i	$-0.0245038\pi$
$314$	7.07107			0.399043
$315$		0			0
$316$	12.0000			0.675053
$317$	−19.7990	+	19.7990i	−1.11202	+	1.11202i	−0.119145	+	0.992877i	$0.538015\pi$
$317$	−19.7990	+	19.7990i	−1.11202	+	1.11202i	−0.992877	+	0.119145i	$0.961985\pi$
$318$		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	0.557685	−	0.830053i	$-0.311690\pi$
$337$	−5.00000	−	5.00000i	−0.272367	−	0.272367i	−0.830053	+	0.557685i	$0.811690\pi$
$338$	−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	−0.942253	−	0.334901i	$-0.891297\pi$
$347$	−11.3137	+	11.3137i	−0.607352	+	0.607352i	0.334901	+	0.942253i	$0.391297\pi$
$348$		0			0
$349$			24.0000i			1.28469i	0.766415	+	0.642345i	$0.222038\pi$
$349$			24.0000i			1.28469i	−0.766415	+	0.642345i	$0.777962\pi$
$350$	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	0.778360	−	0.627818i	$-0.216052\pi$
$353$	2.82843	+	2.82843i	0.150542	+	0.150542i	−0.627818	+	0.778360i	$0.716052\pi$
$354$		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	0.652978	−	0.757377i	$-0.273519\pi$
$367$	−2.00000	−	2.00000i	−0.104399	−	0.104399i	−0.757377	+	0.652978i	$0.773519\pi$
$368$	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	−0.993557	−	0.113331i	$-0.963848\pi$
$373$	−17.0000	+	17.0000i	−0.880227	+	0.880227i	0.113331	+	0.993557i	$0.463848\pi$
$374$	−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	$-0.624401\pi$
$379$		−	36.0000i		−	1.84920i	0.380945	−	0.924598i	$-0.375599\pi$
$380$		0			0
$381$		0			0
$382$	−16.0000	−	16.0000i	−0.818631	−	0.818631i
$383$	−22.6274	−	22.6274i	−1.15621	−	1.15621i	−0.985284	−	0.170923i	$-0.945325\pi$
$383$	−22.6274	−	22.6274i	−1.15621	−	1.15621i	−0.170923	−	0.985284i	$-0.554675\pi$
$384$		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	0.998969	−	0.0453868i	$-0.0144520\pi$
$397$	19.0000	+	19.0000i	0.953583	+	0.953583i	−0.0453868	+	0.998969i	$0.514452\pi$
$398$		0			0
$399$		0			0
$400$	4.00000	+	3.00000i	0.200000	+	0.150000i
$401$		−	24.0416i		−	1.20058i	−0.799782	−	0.600291i	$-0.795051\pi$
$401$		−	24.0416i		−	1.20058i	0.799782	−	0.600291i	$-0.204949\pi$
$402$		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	$-0.797800\pi$
$409$		−	24.0000i		−	1.18672i	0.804936	−	0.593362i	$-0.202200\pi$
$410$	−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	$0.0435020\pi$
$431$			5.65685i			0.272481i	−0.990676	+	0.136241i	$0.956498\pi$
$432$		0			0
$433$	−17.0000	+	17.0000i	−0.816968	+	0.816968i	−0.985668	−	0.168700i	$-0.946043\pi$
$433$	−17.0000	+	17.0000i	−0.816968	+	0.816968i	0.168700	+	0.985668i	$0.446043\pi$
$434$	−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	$0.194115\pi$
$439$			24.0000i			1.14546i	−0.819745	+	0.572729i	$0.805885\pi$
$440$	−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	0.892215	+	0.451612i	$0.149151\pi$
$443$	28.2843	+	28.2843i	1.34383	+	1.34383i	0.451612	+	0.892215i	$0.350849\pi$
$444$		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	0.982339	+	0.187112i	$0.0599128\pi$
$457$	25.0000	+	25.0000i	1.16945	+	1.16945i	0.187112	+	0.982339i	$0.440087\pi$
$458$	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	$0.0740469\pi$
$461$			9.89949i			0.461065i	−0.973065	+	0.230533i	$0.925953\pi$
$462$		0			0
$463$	10.0000	−	10.0000i	0.464739	−	0.464739i	−0.435466	−	0.900205i	$-0.643416\pi$
$463$	10.0000	−	10.0000i	0.464739	−	0.464739i	0.900205	+	0.435466i	$0.143416\pi$
$464$	−4.24264			−0.196960
$465$		0			0
$466$	−4.00000			−0.185296
$467$	−2.82843	+	2.82843i	−0.130884	+	0.130884i	−0.769514	−	0.638630i	$-0.779501\pi$
$467$	−2.82843	+	2.82843i	−0.130884	+	0.130884i	0.638630	+	0.769514i	$0.279501\pi$
$468$		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	0.896396	−	0.443253i	$-0.146176\pi$
$487$	10.0000	+	10.0000i	0.453143	+	0.453143i	−0.443253	+	0.896396i	$0.646176\pi$
$488$	−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	$0.103385\pi$
$491$			14.1421i			0.638226i	−0.947717	+	0.319113i	$0.896615\pi$
$492$		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	$-0.819484\pi$
$499$		−	24.0000i		−	1.07439i	0.843459	−	0.537194i	$-0.180516\pi$
$500$	−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	−0.999960	−	0.00894668i	$-0.997152\pi$
$503$	−22.6274	−	22.6274i	−1.00891	−	1.00891i	−0.00894668	−	0.999960i	$-0.502848\pi$
$504$		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	$0.131937\pi$
$521$			18.3848i			0.805452i	−0.915321	+	0.402726i	$0.868063\pi$
$522$		0			0
$523$	−8.00000	+	8.00000i	−0.349816	+	0.349816i	−0.860041	−	0.510225i	$-0.829562\pi$
$523$	−8.00000	+	8.00000i	−0.349816	+	0.349816i	0.510225	+	0.860041i	$0.329562\pi$
$524$	−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	0.135622	−	0.990761i	$-0.456697\pi$
$547$	−20.0000	−	20.0000i	−0.855138	−	0.855138i	−0.990761	+	0.135622i	$0.956697\pi$
$548$	−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	−0.764485	−	0.644641i	$-0.777007\pi$
$557$	−2.82843	+	2.82843i	−0.119844	+	0.119844i	0.644641	+	0.764485i	$0.277007\pi$
$558$		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	0.976494	+	0.215546i	$0.0691532\pi$
$563$	28.2843	+	28.2843i	1.19204	+	1.19204i	0.215546	+	0.976494i	$0.430847\pi$
$564$		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	0.258336	−	0.966055i	$-0.416826\pi$
$577$	−17.0000	−	17.0000i	−0.707719	−	0.707719i	−0.966055	+	0.258336i	$0.916826\pi$
$578$	−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	−0.985700	−	0.168508i	$-0.946105\pi$
$587$	−19.7990	+	19.7990i	−0.817192	+	0.817192i	0.168508	+	0.985700i	$0.446105\pi$
$588$		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	0.474001	−	0.880524i	$-0.342809\pi$
$593$	−9.89949	−	9.89949i	−0.406524	−	0.406524i	−0.880524	+	0.474001i	$0.842809\pi$
$594$		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	0.994800	−	0.101848i	$-0.0324754\pi$
$607$	22.0000	+	22.0000i	0.892952	+	0.892952i	−0.101848	+	0.994800i	$0.532475\pi$
$608$		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	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	1.00000	−	1.00000i	0.0403896	−	0.0403896i	0.727013	+	0.686624i	$0.240908\pi$
$614$	11.3137			0.456584
$615$		0			0
$616$	24.0000			0.966988
$617$	14.1421	−	14.1421i	0.569341	−	0.569341i	−0.362603	−	0.931944i	$-0.618112\pi$
$617$	14.1421	−	14.1421i	0.569341	−	0.569341i	0.931944	+	0.362603i	$0.118112\pi$
$618$		0			0
$619$		−	12.0000i		−	0.482321i	−0.970485	−	0.241160i	$-0.922472\pi$
$619$		−	12.0000i		−	0.482321i	0.970485	−	0.241160i	$-0.0775280\pi$
$620$	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	$-0.900601\pi$
$641$		−	15.5563i		−	0.614439i	0.951639	−	0.307219i	$-0.0993986\pi$
$642$		0			0
$643$	28.0000	−	28.0000i	1.10421	−	1.10421i	0.110316	−	0.993897i	$-0.464814\pi$
$643$	28.0000	−	28.0000i	1.10421	−	1.10421i	0.993897	−	0.110316i	$-0.0351862\pi$
$644$	11.3137			0.445823
$645$		0			0
$646$		0			0
$647$	−28.2843	+	28.2843i	−1.11197	+	1.11197i	−0.119085	+	0.992884i	$0.537996\pi$
$647$	−28.2843	+	28.2843i	−1.11197	+	1.11197i	−0.992884	+	0.119085i	$0.962004\pi$
$648$		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	0.760280	−	0.649595i	$-0.225062\pi$
$653$	2.82843	+	2.82843i	0.110685	+	0.110685i	−0.649595	+	0.760280i	$0.725062\pi$
$654$		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	−0.687570	−	0.726118i	$-0.741323\pi$
$673$	1.00000	−	1.00000i	0.0385472	−	0.0385472i	0.726118	+	0.687570i	$0.241323\pi$
$674$	7.07107			0.272367
$675$		0			0
$676$	−11.0000			−0.423077
$677$	31.1127	−	31.1127i	1.19576	−	1.19576i	0.220334	−	0.975425i	$-0.429285\pi$
$677$	31.1127	−	31.1127i	1.19576	−	1.19576i	0.975425	−	0.220334i	$-0.0707146\pi$
$678$		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	0.759147	−	0.650920i	$-0.225617\pi$
$683$	2.82843	+	2.82843i	0.108227	+	0.108227i	−0.650920	+	0.759147i	$0.725617\pi$
$684$		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	$-0.957367\pi$
$701$		−	7.07107i		−	0.267071i	0.991044	−	0.133535i	$-0.0426329\pi$
$702$		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	$0.0359394\pi$
$709$			6.00000i			0.225335i	−0.993633	+	0.112667i	$0.964061\pi$
$710$	−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	0.669046	−	0.743221i	$-0.266703\pi$
$727$	−2.00000	−	2.00000i	−0.0741759	−	0.0741759i	−0.743221	+	0.669046i	$0.766703\pi$
$728$	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	−0.688398	−	0.725333i	$-0.741686\pi$
$733$	1.00000	−	1.00000i	0.0369358	−	0.0369358i	0.725333	+	0.688398i	$0.241686\pi$
$734$	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	0.157413	−	0.987533i	$-0.449684\pi$
$743$	−22.6274	−	22.6274i	−0.830119	−	0.830119i	−0.987533	+	0.157413i	$0.949684\pi$
$744$		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	0.962357	−	0.271790i	$-0.0876156\pi$
$757$	19.0000	+	19.0000i	0.690567	+	0.690567i	−0.271790	+	0.962357i	$0.587616\pi$
$758$	25.4558	+	25.4558i	0.924598	+	0.924598i
$759$		0			0
$760$		0			0
$761$			52.3259i			1.89681i	0.317058	+	0.948406i	$0.397305\pi$
$761$			52.3259i			1.89681i	−0.317058	+	0.948406i	$0.602695\pi$
$762$		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	$-0.857550\pi$
$769$		−	24.0000i		−	0.865462i	0.901523	−	0.432731i	$-0.142450\pi$
$770$	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	0.756141	−	0.654409i	$-0.227083\pi$
$773$	2.82843	+	2.82843i	0.101731	+	0.101731i	−0.654409	+	0.756141i	$0.727083\pi$
$774$		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

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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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