LMFDB - Embedded newform 120.2.w.a.77.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [120,2,Mod(53,120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(120, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 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("120.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 120 = 2^{3} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	120.w (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:	$0.958204824255$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			77.1
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	120.77
Dual form			120.2.w.a.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.00000i) q^{2} +(-1.22474 + 1.22474i) q^{3} +2.00000i q^{4} +(-0.224745 - 2.22474i) q^{5} +2.44949 q^{6} +(-3.44949 - 3.44949i) q^{7} +(2.00000 - 2.00000i) q^{8} -3.00000i q^{9} +O(q^{10})$	\(q+(-1.00000 - 1.00000i) q^{2} +(-1.22474 + 1.22474i) q^{3} +2.00000i q^{4} +(-0.224745 - 2.22474i) q^{5} +2.44949 q^{6} +(-3.44949 - 3.44949i) q^{7} +(2.00000 - 2.00000i) q^{8} -3.00000i q^{9} +(-2.00000 + 2.44949i) q^{10} -1.55051 q^{11} +(-2.44949 - 2.44949i) q^{12} +6.89898i q^{14} +(3.00000 + 2.44949i) q^{15} -4.00000 q^{16} +(-3.00000 + 3.00000i) q^{18} +(4.44949 - 0.449490i) q^{20} +8.44949 q^{21} +(1.55051 + 1.55051i) q^{22} +4.89898i q^{24} +(-4.89898 + 1.00000i) q^{25} +(3.67423 + 3.67423i) q^{27} +(6.89898 - 6.89898i) q^{28} -5.34847i q^{29} +(-0.550510 - 5.44949i) q^{30} +4.89898 q^{31} +(4.00000 + 4.00000i) q^{32} +(1.89898 - 1.89898i) q^{33} +(-6.89898 + 8.44949i) q^{35} +6.00000 q^{36} +(-4.89898 - 4.00000i) q^{40} +(-8.44949 - 8.44949i) q^{42} -3.10102i q^{44} +(-6.67423 + 0.674235i) q^{45} +(4.89898 - 4.89898i) q^{48} +16.7980i q^{49} +(5.89898 + 3.89898i) q^{50} +(-2.44949 + 2.44949i) q^{53} -7.34847i q^{54} +(0.348469 + 3.44949i) q^{55} -13.7980 q^{56} +(-5.34847 + 5.34847i) q^{58} -15.3485i q^{59} +(-4.89898 + 6.00000i) q^{60} +(-4.89898 - 4.89898i) q^{62} +(-10.3485 + 10.3485i) q^{63} -8.00000i q^{64} -3.79796 q^{66} +(15.3485 - 1.55051i) q^{70} +(-6.00000 - 6.00000i) q^{72} +(11.8990 - 11.8990i) q^{73} +(4.77526 - 7.22474i) q^{75} +(5.34847 + 5.34847i) q^{77} -14.6969i q^{79} +(0.898979 + 8.89898i) q^{80} -9.00000 q^{81} +(4.00000 - 4.00000i) q^{83} +16.8990i q^{84} +(6.55051 + 6.55051i) q^{87} +(-3.10102 + 3.10102i) q^{88} +(7.34847 + 6.00000i) q^{90} +(-6.00000 + 6.00000i) q^{93} -9.79796 q^{96} +(8.79796 + 8.79796i) q^{97} +(16.7980 - 16.7980i) q^{98} +4.65153i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{5} - 4 q^{7} + 8 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^5 - 4 * q^7 + 8 * q^8	$ 4 q - 4 q^{2} + 4 q^{5} - 4 q^{7} + 8 q^{8} - 8 q^{10} - 16 q^{11} + 12 q^{15} - 16 q^{16} - 12 q^{18} + 8 q^{20} + 24 q^{21} + 16 q^{22} + 8 q^{28} - 12 q^{30} + 16 q^{32} - 12 q^{33} - 8 q^{35} + 24 q^{36} - 24 q^{42} - 12 q^{45} + 4 q^{50} - 28 q^{55} - 16 q^{56} + 8 q^{58} - 12 q^{63} + 24 q^{66} + 32 q^{70} - 24 q^{72} + 28 q^{73} + 24 q^{75} - 8 q^{77} - 16 q^{80} - 36 q^{81} + 16 q^{83} + 36 q^{87} - 32 q^{88} - 24 q^{93} - 4 q^{97} + 28 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^5 - 4 * q^7 + 8 * q^8 - 8 * q^10 - 16 * q^11 + 12 * q^15 - 16 * q^16 - 12 * q^18 + 8 * q^20 + 24 * q^21 + 16 * q^22 + 8 * q^28 - 12 * q^30 + 16 * q^32 - 12 * q^33 - 8 * q^35 + 24 * q^36 - 24 * q^42 - 12 * q^45 + 4 * q^50 - 28 * q^55 - 16 * q^56 + 8 * q^58 - 12 * q^63 + 24 * q^66 + 32 * q^70 - 24 * q^72 + 28 * q^73 + 24 * q^75 - 8 * q^77 - 16 * q^80 - 36 * q^81 + 16 * q^83 + 36 * q^87 - 32 * q^88 - 24 * q^93 - 4 * q^97 + 28 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$.

$n$	$31$	$41$	$61$	$97$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$3$	−1.22474	+	1.22474i	−0.707107	+	0.707107i
$3$	−1.22474	+	1.22474i	−0.707107	+	0.707107i
$4$			2.00000i			1.00000i
$5$	−0.224745	−	2.22474i	−0.100509	−	0.994936i
$5$	−0.224745	−	2.22474i	−0.100509	−	0.994936i
$6$	2.44949			1.00000
$7$	−3.44949	−	3.44949i	−1.30378	−	1.30378i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	−3.44949	−	3.44949i	−1.30378	−	1.30378i	−0.377964	−	0.925820i	$-0.623376\pi$
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$		−	3.00000i		−	1.00000i
$10$	−2.00000	+	2.44949i	−0.632456	+	0.774597i
$11$	−1.55051			−0.467496			−0.233748	−	0.972297i	$-0.575099\pi$
$11$	−1.55051			−0.467496			−0.233748	+	0.972297i	$0.575099\pi$
$12$	−2.44949	−	2.44949i	−0.707107	−	0.707107i
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$			6.89898i			1.84383i
$15$	3.00000	+	2.44949i	0.774597	+	0.632456i
$16$	−4.00000			−1.00000
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$	−3.00000	+	3.00000i	−0.707107	+	0.707107i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	4.44949	−	0.449490i	0.994936	−	0.100509i
$21$	8.44949			1.84383
$22$	1.55051	+	1.55051i	0.330570	+	0.330570i
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$			4.89898i			1.00000i
$25$	−4.89898	+	1.00000i	−0.979796	+	0.200000i
$26$		0			0
$27$	3.67423	+	3.67423i	0.707107	+	0.707107i
$28$	6.89898	−	6.89898i	1.30378	−	1.30378i
$29$		−	5.34847i		−	0.993186i	−0.867984	−	0.496593i	$-0.834584\pi$
$29$		−	5.34847i		−	0.993186i	0.867984	−	0.496593i	$-0.165416\pi$
$30$	−0.550510	−	5.44949i	−0.100509	−	0.994936i
$31$	4.89898			0.879883			0.439941	−	0.898027i	$-0.354999\pi$
$31$	4.89898			0.879883			0.439941	+	0.898027i	$0.354999\pi$
$32$	4.00000	+	4.00000i	0.707107	+	0.707107i
$33$	1.89898	−	1.89898i	0.330570	−	0.330570i
$34$		0			0
$35$	−6.89898	+	8.44949i	−1.16614	+	1.42822i
$36$	6.00000			1.00000
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$	−4.89898	−	4.00000i	−0.774597	−	0.632456i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−8.44949	−	8.44949i	−1.30378	−	1.30378i
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		−	3.10102i		−	0.467496i
$45$	−6.67423	+	0.674235i	−0.994936	+	0.100509i
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	4.89898	−	4.89898i	0.707107	−	0.707107i
$49$			16.7980i			2.39971i
$50$	5.89898	+	3.89898i	0.834242	+	0.551399i
$51$		0			0
$52$		0			0
$53$	−2.44949	+	2.44949i	−0.336463	+	0.336463i	−0.855034	−	0.518571i	$-0.826464\pi$
$53$	−2.44949	+	2.44949i	−0.336463	+	0.336463i	0.518571	+	0.855034i	$0.326464\pi$
$54$		−	7.34847i		−	1.00000i
$55$	0.348469	+	3.44949i	0.0469876	+	0.465129i
$56$	−13.7980			−1.84383
$57$		0			0
$58$	−5.34847	+	5.34847i	−0.702288	+	0.702288i
$59$		−	15.3485i		−	1.99820i	−0.0424110	−	0.999100i	$-0.513504\pi$
$59$		−	15.3485i		−	1.99820i	0.0424110	−	0.999100i	$-0.486496\pi$
$60$	−4.89898	+	6.00000i	−0.632456	+	0.774597i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	−4.89898	−	4.89898i	−0.622171	−	0.622171i
$63$	−10.3485	+	10.3485i	−1.30378	+	1.30378i
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$	−3.79796			−0.467496
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$		0			0
$70$	15.3485	−	1.55051i	1.83449	−	0.185321i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−6.00000	−	6.00000i	−0.707107	−	0.707107i
$73$	11.8990	−	11.8990i	1.39267	−	1.39267i	0.573382	−	0.819288i	$-0.305631\pi$
$73$	11.8990	−	11.8990i	1.39267	−	1.39267i	0.819288	−	0.573382i	$-0.194369\pi$
$74$		0			0
$75$	4.77526	−	7.22474i	0.551399	−	0.834242i
$76$		0			0
$77$	5.34847	+	5.34847i	0.609515	+	0.609515i
$78$		0			0
$79$		−	14.6969i		−	1.65353i	−0.562544	−	0.826767i	$-0.690177\pi$
$79$		−	14.6969i		−	1.65353i	0.562544	−	0.826767i	$-0.309823\pi$
$80$	0.898979	+	8.89898i	0.100509	+	0.994936i
$81$	−9.00000			−1.00000
$82$		0			0
$83$	4.00000	−	4.00000i	0.439057	−	0.439057i	−0.452638	−	0.891695i	$-0.649517\pi$
$83$	4.00000	−	4.00000i	0.439057	−	0.439057i	0.891695	+	0.452638i	$0.149517\pi$
$84$			16.8990i			1.84383i
$85$		0			0
$86$		0			0
$87$	6.55051	+	6.55051i	0.702288	+	0.702288i
$88$	−3.10102	+	3.10102i	−0.330570	+	0.330570i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	7.34847	+	6.00000i	0.774597	+	0.632456i
$91$		0			0
$92$		0			0
$93$	−6.00000	+	6.00000i	−0.622171	+	0.622171i
$94$		0			0
$95$		0			0
$96$	−9.79796			−1.00000
$97$	8.79796	+	8.79796i	0.893297	+	0.893297i	0.994832	−	0.101535i	$-0.0323753\pi$
$97$	8.79796	+	8.79796i	0.893297	+	0.893297i	−0.101535	+	0.994832i	$0.532375\pi$
$98$	16.7980	−	16.7980i	1.69685	−	1.69685i
$99$			4.65153i			0.467496i
$100$	−2.00000	−	9.79796i	−0.200000	−	0.979796i
$101$	−11.5505			−1.14932			−0.574659	−	0.818393i	$-0.694865\pi$
$101$	−11.5505			−1.14932			−0.574659	+	0.818393i	$0.694865\pi$
$102$		0			0
$103$	−0.348469	+	0.348469i	−0.0343357	+	0.0343357i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$	−0.348469	+	0.348469i	−0.0343357	+	0.0343357i	0.689730	+	0.724066i	$0.257729\pi$
$104$		0			0
$105$	−1.89898	−	18.7980i	−0.185321	−	1.83449i
$106$	4.89898			0.475831
$107$	8.00000	+	8.00000i	0.773389	+	0.773389i	0.978697	−	0.205308i	$-0.0658197\pi$
$107$	8.00000	+	8.00000i	0.773389	+	0.773389i	−0.205308	+	0.978697i	$0.565820\pi$
$108$	−7.34847	+	7.34847i	−0.707107	+	0.707107i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	3.10102	−	3.79796i	0.295671	−	0.362121i
$111$		0			0
$112$	13.7980	+	13.7980i	1.30378	+	1.30378i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$	10.6969			0.993186
$117$		0			0
$118$	−15.3485	+	15.3485i	−1.41294	+	1.41294i
$119$		0			0
$120$	10.8990	−	1.10102i	0.994936	−	0.100509i
$121$	−8.59592			−0.781447
$122$		0			0
$123$		0			0
$124$			9.79796i			0.879883i
$125$	3.32577	+	10.6742i	0.297465	+	0.954733i
$126$	20.6969			1.84383
$127$	−13.4495	−	13.4495i	−1.19345	−	1.19345i	−0.976092	−	0.217357i	$-0.930256\pi$
$127$	−13.4495	−	13.4495i	−1.19345	−	1.19345i	−0.217357	−	0.976092i	$-0.569744\pi$
$128$	−8.00000	+	8.00000i	−0.707107	+	0.707107i
$129$		0			0
$130$		0			0
$131$	18.4495			1.61194			0.805970	−	0.591957i	$-0.201644\pi$
$131$	18.4495			1.61194			0.805970	+	0.591957i	$0.201644\pi$
$132$	3.79796	+	3.79796i	0.330570	+	0.330570i
$133$		0			0
$134$		0			0
$135$	7.34847	−	9.00000i	0.632456	−	0.774597i
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	−16.8990	−	13.7980i	−1.42822	−	1.16614i
$141$		0			0
$142$		0			0
$143$		0			0
$144$			12.0000i			1.00000i
$145$	−11.8990	+	1.20204i	−0.988156	+	0.0998241i
$146$	−23.7980			−1.96953
$147$	−20.5732	−	20.5732i	−1.69685	−	1.69685i
$148$		0			0
$149$			19.1464i			1.56854i	0.620422	+	0.784268i	$0.286961\pi$
$149$			19.1464i			1.56854i	−0.620422	+	0.784268i	$0.713039\pi$
$150$	−12.0000	+	2.44949i	−0.979796	+	0.200000i
$151$	2.00000			0.162758			0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000			0.162758			0.0813788	+	0.996683i	$0.474068\pi$
$152$		0			0
$153$		0			0
$154$		−	10.6969i		−	0.861984i
$155$	−1.10102	−	10.8990i	−0.0884361	−	0.875427i
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$	−14.6969	+	14.6969i	−1.16923	+	1.16923i
$159$		−	6.00000i		−	0.475831i
$160$	8.00000	−	9.79796i	0.632456	−	0.774597i
$161$		0			0
$162$	9.00000	+	9.00000i	0.707107	+	0.707107i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$	−4.65153	−	3.79796i	−0.362121	−	0.295671i
$166$	−8.00000			−0.620920
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$	16.8990	−	16.8990i	1.30378	−	1.30378i
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.0000	−	14.0000i	1.06440	−	1.06440i	0.0666220	−	0.997778i	$-0.478778\pi$
$173$	14.0000	−	14.0000i	1.06440	−	1.06440i	0.997778	−	0.0666220i	$-0.0212222\pi$
$174$		−	13.1010i		−	0.993186i
$175$	20.3485	+	13.4495i	1.53820	+	1.01669i
$176$	6.20204			0.467496
$177$	18.7980	+	18.7980i	1.41294	+	1.41294i
$178$		0			0
$179$			9.14643i			0.683636i	0.939766	+	0.341818i	$0.111043\pi$
$179$			9.14643i			0.683636i	−0.939766	+	0.341818i	$0.888957\pi$
$180$	−1.34847	−	13.3485i	−0.100509	−	0.994936i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$	12.0000			0.879883
$187$		0			0
$188$		0			0
$189$		−	25.3485i		−	1.84383i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	9.79796	+	9.79796i	0.707107	+	0.707107i
$193$	−8.10102	+	8.10102i	−0.583124	+	0.583124i	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−8.10102	+	8.10102i	−0.583124	+	0.583124i	0.352636	+	0.935760i	$0.385285\pi$
$194$		−	17.5959i		−	1.26331i
$195$		0			0
$196$	−33.5959			−2.39971
$197$	−17.1464	−	17.1464i	−1.22163	−	1.22163i	−0.967051	−	0.254581i	$-0.918062\pi$
$197$	−17.1464	−	17.1464i	−1.22163	−	1.22163i	−0.254581	−	0.967051i	$-0.581938\pi$
$198$	4.65153	−	4.65153i	0.330570	−	0.330570i
$199$			14.0000i			0.992434i	0.868199	+	0.496217i	$0.165278\pi$
$199$			14.0000i			0.992434i	−0.868199	+	0.496217i	$0.834722\pi$
$200$	−7.79796	+	11.7980i	−0.551399	+	0.834242i
$201$		0			0
$202$	11.5505	+	11.5505i	0.812691	+	0.812691i
$203$	−18.4495	+	18.4495i	−1.29490	+	1.29490i
$204$		0			0
$205$		0			0
$206$	0.696938			0.0485580
$207$		0			0
$208$		0			0
$209$		0			0
$210$	−16.8990	+	20.6969i	−1.16614	+	1.42822i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−4.89898	−	4.89898i	−0.336463	−	0.336463i
$213$		0			0
$214$		−	16.0000i		−	1.09374i
$215$		0			0
$216$	14.6969			1.00000
$217$	−16.8990	−	16.8990i	−1.14718	−	1.14718i
$218$		0			0
$219$			29.1464i			1.96953i
$220$	−6.89898	+	0.696938i	−0.465129	+	0.0469876i
$221$		0			0
$222$		0			0
$223$	−20.3485	+	20.3485i	−1.36263	+	1.36263i	−0.492090	+	0.870544i	$0.663767\pi$
$223$	−20.3485	+	20.3485i	−1.36263	+	1.36263i	−0.870544	+	0.492090i	$0.836233\pi$
$224$		−	27.5959i		−	1.84383i
$225$	3.00000	+	14.6969i	0.200000	+	0.979796i
$226$		0			0
$227$	7.34847	+	7.34847i	0.487735	+	0.487735i	0.907591	−	0.419856i	$-0.137919\pi$
$227$	7.34847	+	7.34847i	0.487735	+	0.487735i	−0.419856	+	0.907591i	$0.637919\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$	−13.1010			−0.861984
$232$	−10.6969	−	10.6969i	−0.702288	−	0.702288i
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$	30.6969			1.99820
$237$	18.0000	+	18.0000i	1.16923	+	1.16923i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−12.0000	−	9.79796i	−0.774597	−	0.632456i
$241$	29.3939			1.89343			0.946713	−	0.322078i	$-0.104381\pi$
$241$	29.3939			1.89343			0.946713	+	0.322078i	$0.104381\pi$
$242$	8.59592	+	8.59592i	0.552567	+	0.552567i
$243$	11.0227	−	11.0227i	0.707107	−	0.707107i
$244$		0			0
$245$	37.3712	−	3.77526i	2.38756	−	0.241192i
$246$		0			0
$247$		0			0
$248$	9.79796	−	9.79796i	0.622171	−	0.622171i
$249$			9.79796i			0.620920i
$250$	7.34847	−	14.0000i	0.464758	−	0.885438i
$251$	−26.0454			−1.64397			−0.821986	−	0.569508i	$-0.807134\pi$
$251$	−26.0454			−1.64397			−0.821986	+	0.569508i	$0.807134\pi$
$252$	−20.6969	−	20.6969i	−1.30378	−	1.30378i
$253$		0			0
$254$			26.8990i			1.68779i
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−16.0454			−0.993186
$262$	−18.4495	−	18.4495i	−1.13981	−	1.13981i
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		−	7.59592i		−	0.467496i
$265$	6.00000	+	4.89898i	0.368577	+	0.300942i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			14.6515i			0.893320i	0.894704	+	0.446660i	$0.147387\pi$
$269$			14.6515i			0.893320i	−0.894704	+	0.446660i	$0.852613\pi$
$270$	−16.3485	+	1.65153i	−0.994936	+	0.100509i
$271$	22.0000			1.33640			0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000			1.33640			0.668202	+	0.743980i	$0.267064\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.59592	−	1.55051i	0.458051	−	0.0934993i
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		0			0
$279$		−	14.6969i		−	0.879883i
$280$	3.10102	+	30.6969i	0.185321	+	1.83449i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	12.0000	−	12.0000i	0.707107	−	0.707107i
$289$			17.0000i			1.00000i
$290$	13.1010	+	10.6969i	0.769318	+	0.628146i
$291$	−21.5505			−1.26331
$292$	23.7980	+	23.7980i	1.39267	+	1.39267i
$293$	22.0454	−	22.0454i	1.28791	−	1.28791i	0.351850	−	0.936056i	$-0.385553\pi$
$293$	22.0454	−	22.0454i	1.28791	−	1.28791i	0.936056	−	0.351850i	$-0.114447\pi$
$294$			41.1464i			2.39971i
$295$	−34.1464	+	3.44949i	−1.98808	+	0.200837i
$296$		0			0
$297$	−5.69694	−	5.69694i	−0.330570	−	0.330570i
$298$	19.1464	−	19.1464i	1.10912	−	1.10912i
$299$		0			0
$300$	14.4495	+	9.55051i	0.834242	+	0.551399i
$301$		0			0
$302$	−2.00000	−	2.00000i	−0.115087	−	0.115087i
$303$	14.1464	−	14.1464i	0.812691	−	0.812691i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$	−10.6969	+	10.6969i	−0.609515	+	0.609515i
$309$		−	0.853572i		−	0.0485580i
$310$	−9.79796	+	12.0000i	−0.556487	+	0.681554i
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	21.8990	−	21.8990i	1.23780	−	1.23780i	0.276907	−	0.960897i	$-0.410691\pi$
$313$	21.8990	−	21.8990i	1.23780	−	1.23780i	0.960897	−	0.276907i	$-0.0893093\pi$
$314$		0			0
$315$	25.3485	+	20.6969i	1.42822	+	1.16614i
$316$	29.3939			1.65353
$317$	−22.0000	−	22.0000i	−1.23564	−	1.23564i	−0.961764	−	0.273879i	$-0.911693\pi$
$317$	−22.0000	−	22.0000i	−1.23564	−	1.23564i	−0.273879	−	0.961764i	$-0.588307\pi$
$318$	−6.00000	+	6.00000i	−0.336463	+	0.336463i
$319$			8.29286i			0.464311i
$320$	−17.7980	+	1.79796i	−0.994936	+	0.100509i
$321$	−19.5959			−1.09374
$322$		0			0
$323$		0			0
$324$		−	18.0000i		−	1.00000i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	0.853572	+	8.44949i	0.0469876	+	0.465129i
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	8.00000	+	8.00000i	0.439057	+	0.439057i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−33.7980			−1.84383
$337$	−25.6969	−	25.6969i	−1.39980	−	1.39980i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	−25.6969	−	25.6969i	−1.39980	−	1.39980i	−0.599208	−	0.800593i	$-0.704518\pi$
$338$	−13.0000	+	13.0000i	−0.707107	+	0.707107i
$339$		0			0
$340$		0			0
$341$	−7.59592			−0.411342
$342$		0			0
$343$	33.7980	−	33.7980i	1.82492	−	1.82492i
$344$		0			0
$345$		0			0
$346$	−28.0000			−1.50529
$347$	−17.1464	−	17.1464i	−0.920468	−	0.920468i	0.0765939	−	0.997062i	$-0.475596\pi$
$347$	−17.1464	−	17.1464i	−0.920468	−	0.920468i	−0.997062	+	0.0765939i	$0.975596\pi$
$348$	−13.1010	+	13.1010i	−0.702288	+	0.702288i
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	−6.89898	−	33.7980i	−0.368766	−	1.80658i
$351$		0			0
$352$	−6.20204	−	6.20204i	−0.330570	−	0.330570i
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		−	37.5959i		−	1.99820i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	9.14643	−	9.14643i	0.483404	−	0.483404i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−12.0000	+	14.6969i	−0.632456	+	0.774597i
$361$	−19.0000			−1.00000
$362$		0			0
$363$	10.5278	−	10.5278i	0.552567	−	0.552567i
$364$		0			0
$365$	−29.1464	−	23.7980i	−1.52559	−	1.24564i
$366$		0			0
$367$	16.5505	+	16.5505i	0.863930	+	0.863930i	0.991792	−	0.127862i	$-0.0408116\pi$
$367$	16.5505	+	16.5505i	0.863930	+	0.863930i	−0.127862	+	0.991792i	$0.540812\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	16.8990			0.877351
$372$	−12.0000	−	12.0000i	−0.622171	−	0.622171i
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$	−17.1464	−	9.00000i	−0.885438	−	0.464758i
$376$		0			0
$377$		0			0
$378$	−25.3485	+	25.3485i	−1.30378	+	1.30378i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	32.9444			1.68779
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		−	19.5959i		−	1.00000i
$385$	10.6969	−	13.1010i	0.545166	−	0.667690i
$386$	16.2020			0.824662
$387$		0			0
$388$	−17.5959	+	17.5959i	−0.893297	+	0.893297i
$389$			39.1464i			1.98480i	0.123043	+	0.992401i	$0.460735\pi$
$389$			39.1464i			1.98480i	−0.123043	+	0.992401i	$0.539265\pi$
$390$		0			0
$391$		0			0
$392$	33.5959	+	33.5959i	1.69685	+	1.69685i
$393$	−22.5959	+	22.5959i	−1.13981	+	1.13981i
$394$			34.2929i			1.72765i
$395$	−32.6969	+	3.30306i	−1.64516	+	0.166195i
$396$	−9.30306			−0.467496
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	14.0000	−	14.0000i	0.701757	−	0.701757i
$399$		0			0
$400$	19.5959	−	4.00000i	0.979796	−	0.200000i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		−	23.1010i		−	1.14932i
$405$	2.02270	+	20.0227i	0.100509	+	0.994936i
$406$	36.8990			1.83127
$407$		0			0
$408$		0			0
$409$		−	39.1918i		−	1.93791i	−0.247234	−	0.968956i	$-0.579522\pi$
$409$		−	39.1918i		−	1.93791i	0.247234	−	0.968956i	$-0.420478\pi$
$410$		0			0
$411$		0			0
$412$	−0.696938	−	0.696938i	−0.0343357	−	0.0343357i
$413$	−52.9444	+	52.9444i	−2.60522	+	2.60522i
$414$		0			0
$415$	−9.79796	−	8.00000i	−0.480963	−	0.392705i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	35.3485i		−	1.72689i	−0.504447	−	0.863443i	$-0.668303\pi$
$419$		−	35.3485i		−	1.72689i	0.504447	−	0.863443i	$-0.331697\pi$
$420$	37.5959	−	3.79796i	1.83449	−	0.185321i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$			9.79796i			0.475831i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−16.0000	+	16.0000i	−0.773389	+	0.773389i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−14.6969	−	14.6969i	−0.707107	−	0.707107i
$433$	−12.5959	+	12.5959i	−0.605321	+	0.605321i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	−12.5959	+	12.5959i	−0.605321	+	0.605321i	0.336399	+	0.941720i	$0.390791\pi$
$434$			33.7980i			1.62235i
$435$	13.1010	−	16.0454i	0.628146	−	0.769318i
$436$		0			0
$437$		0			0
$438$	29.1464	−	29.1464i	1.39267	−	1.39267i
$439$			34.0000i			1.62273i	0.584539	+	0.811366i	$0.301275\pi$
$439$			34.0000i			1.62273i	−0.584539	+	0.811366i	$0.698725\pi$
$440$	7.59592	+	6.20204i	0.362121	+	0.295671i
$441$	50.3939			2.39971
$442$		0			0
$443$	22.0454	−	22.0454i	1.04741	−	1.04741i	0.0485901	−	0.998819i	$-0.484527\pi$
$443$	22.0454	−	22.0454i	1.04741	−	1.04741i	0.998819	−	0.0485901i	$-0.0154728\pi$
$444$		0			0
$445$		0			0
$446$	40.6969			1.92706
$447$	−23.4495	−	23.4495i	−1.10912	−	1.10912i
$448$	−27.5959	+	27.5959i	−1.30378	+	1.30378i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	11.6969	−	17.6969i	0.551399	−	0.834242i
$451$		0			0
$452$		0			0
$453$	−2.44949	+	2.44949i	−0.115087	+	0.115087i
$454$		−	14.6969i		−	0.689761i
$455$		0			0
$456$		0			0
$457$	28.7980	+	28.7980i	1.34711	+	1.34711i	0.888783	+	0.458329i	$0.151552\pi$
$457$	28.7980	+	28.7980i	1.34711	+	1.34711i	0.458329	+	0.888783i	$0.348448\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12.9444			0.602880			0.301440	−	0.953485i	$-0.402533\pi$
$461$	12.9444			0.602880			0.301440	+	0.953485i	$0.402533\pi$
$462$	13.1010	+	13.1010i	0.609515	+	0.609515i
$463$	4.14643	−	4.14643i	0.192701	−	0.192701i	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	4.14643	−	4.14643i	0.192701	−	0.192701i	0.796862	+	0.604161i	$0.206492\pi$
$464$			21.3939i			0.993186i
$465$	14.6969	+	12.0000i	0.681554	+	0.556487i
$466$		0			0
$467$	28.0000	+	28.0000i	1.29569	+	1.29569i	0.931215	+	0.364471i	$0.118750\pi$
$467$	28.0000	+	28.0000i	1.29569	+	1.29569i	0.364471	+	0.931215i	$0.381250\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−30.6969	−	30.6969i	−1.41294	−	1.41294i
$473$		0			0
$474$		−	36.0000i		−	1.65353i
$475$		0			0
$476$		0			0
$477$	7.34847	+	7.34847i	0.336463	+	0.336463i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	2.20204	+	21.7980i	0.100509	+	0.994936i
$481$		0			0
$482$	−29.3939	−	29.3939i	−1.33885	−	1.33885i
$483$		0			0
$484$		−	17.1918i		−	0.781447i
$485$	17.5959	−	21.5505i	0.798989	−	0.978558i
$486$	−22.0454			−1.00000
$487$	21.0454	+	21.0454i	0.953658	+	0.953658i	0.998973	−	0.0453143i	$-0.0144289\pi$
$487$	21.0454	+	21.0454i	0.953658	+	0.953658i	−0.0453143	+	0.998973i	$0.514429\pi$
$488$		0			0
$489$		0			0
$490$	−41.1464	−	33.5959i	−1.85881	−	1.51771i
$491$	42.9444			1.93805			0.969027	−	0.246957i	$-0.0794305\pi$
$491$	42.9444			1.93805			0.969027	+	0.246957i	$0.0794305\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	10.3485	−	1.04541i	0.465129	−	0.0469876i
$496$	−19.5959			−0.879883
$497$		0			0
$498$	9.79796	−	9.79796i	0.439057	−	0.439057i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−21.3485	+	6.65153i	−0.954733	+	0.297465i
$501$		0			0
$502$	26.0454	+	26.0454i	1.16246	+	1.16246i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$			41.3939i			1.84383i
$505$	2.59592	+	25.6969i	0.115517	+	1.14350i
$506$		0			0
$507$	15.9217	+	15.9217i	0.707107	+	0.707107i
$508$	26.8990	−	26.8990i	1.19345	−	1.19345i
$509$		−	29.8434i		−	1.32278i	−0.750040	−	0.661392i	$-0.769966\pi$
$509$		−	29.8434i		−	1.32278i	0.750040	−	0.661392i	$-0.230034\pi$
$510$		0			0
$511$	−82.0908			−3.63148
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$		0			0
$514$		0			0
$515$	0.853572	+	0.696938i	0.0376129	+	0.0307108i
$516$		0			0
$517$		0			0
$518$		0			0
$519$			34.2929i			1.50529i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	16.0454	+	16.0454i	0.702288	+	0.702288i
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$			36.8990i			1.61194i
$525$	−41.3939	+	8.44949i	−1.80658	+	0.368766i
$526$		0			0
$527$		0			0
$528$	−7.59592	+	7.59592i	−0.330570	+	0.330570i
$529$		−	23.0000i		−	1.00000i
$530$	−1.10102	−	10.8990i	−0.0478253	−	0.473421i
$531$	−46.0454			−1.99820
$532$		0			0
$533$		0			0
$534$		0			0
$535$	16.0000	−	19.5959i	0.691740	−	0.847205i
$536$		0			0
$537$	−11.2020	−	11.2020i	−0.483404	−	0.483404i
$538$	14.6515	−	14.6515i	0.631672	−	0.631672i
$539$		−	26.0454i		−	1.12186i
$540$	18.0000	+	14.6969i	0.774597	+	0.632456i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	−22.0000	−	22.0000i	−0.944981	−	0.944981i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$		0			0
$550$	−9.14643	−	6.04541i	−0.390005	−	0.257777i
$551$		0			0
$552$		0			0
$553$	−50.6969	+	50.6969i	−2.15585	+	2.15585i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	31.8434	+	31.8434i	1.34925	+	1.34925i	0.886480	+	0.462767i	$0.153143\pi$
$557$	31.8434	+	31.8434i	1.34925	+	1.34925i	0.462767	+	0.886480i	$0.346857\pi$
$558$	−14.6969	+	14.6969i	−0.622171	+	0.622171i
$559$		0			0
$560$	27.5959	−	33.7980i	1.16614	−	1.42822i
$561$		0			0
$562$		0			0
$563$	−26.9444	+	26.9444i	−1.13557	+	1.13557i	−0.146336	+	0.989235i	$0.546748\pi$
$563$	−26.9444	+	26.9444i	−1.13557	+	1.13557i	−0.989235	+	0.146336i	$0.953252\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	31.0454	+	31.0454i	1.30378	+	1.30378i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−24.0000			−1.00000
$577$	4.30306	+	4.30306i	0.179139	+	0.179139i	0.790980	−	0.611842i	$-0.209571\pi$
$577$	4.30306	+	4.30306i	0.179139	+	0.179139i	−0.611842	+	0.790980i	$0.709571\pi$
$578$	17.0000	−	17.0000i	0.707107	−	0.707107i
$579$		−	19.8434i		−	0.824662i
$580$	−2.40408	−	23.7980i	−0.0998241	−	0.988156i
$581$	−27.5959			−1.14487
$582$	21.5505	+	21.5505i	0.893297	+	0.893297i
$583$	3.79796	−	3.79796i	0.157295	−	0.157295i
$584$		−	47.5959i		−	1.96953i
$585$		0			0
$586$	−44.0908			−1.82137
$587$	−32.0000	−	32.0000i	−1.32078	−	1.32078i	−0.913144	−	0.407638i	$-0.866353\pi$
$587$	−32.0000	−	32.0000i	−1.32078	−	1.32078i	−0.407638	−	0.913144i	$-0.633647\pi$
$588$	41.1464	−	41.1464i	1.69685	−	1.69685i
$589$		0			0
$590$	37.5959	+	30.6969i	1.54780	+	1.26377i
$591$	42.0000			1.72765
$592$		0			0
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$			11.3939i			0.467496i
$595$		0			0
$596$	−38.2929			−1.56854
$597$	−17.1464	−	17.1464i	−0.701757	−	0.701757i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−4.89898	−	24.0000i	−0.200000	−	0.979796i
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$		0			0
$604$			4.00000i			0.162758i
$605$	1.93189	+	19.1237i	0.0785424	+	0.777490i
$606$	−28.2929			−1.14932
$607$	11.0454	+	11.0454i	0.448319	+	0.448319i	0.894795	−	0.446476i	$-0.147321\pi$
$607$	11.0454	+	11.0454i	0.448319	+	0.448319i	−0.446476	+	0.894795i	$0.647321\pi$
$608$		0			0
$609$		−	45.1918i		−	1.83127i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$	21.3939			0.861984
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$	−0.853572	+	0.853572i	−0.0343357	+	0.0343357i
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$	21.7980	−	2.20204i	0.875427	−	0.0884361i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	23.0000	−	9.79796i	0.920000	−	0.391918i
$626$	−43.7980			−1.75052
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−4.65153	−	46.0454i	−0.185321	−	1.83449i
$631$	4.89898			0.195025			0.0975126	−	0.995234i	$-0.468911\pi$
$631$	4.89898			0.195025			0.0975126	+	0.995234i	$0.468911\pi$
$632$	−29.3939	−	29.3939i	−1.16923	−	1.16923i
$633$		0			0
$634$			44.0000i			1.74746i
$635$	−26.8990	+	32.9444i	−1.06745	+	1.30736i
$636$	12.0000			0.475831
$637$		0			0
$638$	8.29286	−	8.29286i	0.328317	−	0.328317i
$639$		0			0
$640$	19.5959	+	16.0000i	0.774597	+	0.632456i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	19.5959	+	19.5959i	0.773389	+	0.773389i
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	−18.0000	+	18.0000i	−0.707107	+	0.707107i
$649$			23.7980i			0.934152i
$650$		0			0
$651$	41.3939			1.62235
$652$		0			0
$653$	34.0000	−	34.0000i	1.33052	−	1.33052i	0.425622	−	0.904901i	$-0.360055\pi$
$653$	34.0000	−	34.0000i	1.33052	−	1.33052i	0.904901	−	0.425622i	$-0.139945\pi$
$654$		0			0
$655$	−4.14643	−	41.0454i	−0.162014	−	1.60378i
$656$		0			0
$657$	−35.6969	−	35.6969i	−1.39267	−	1.39267i
$658$		0			0
$659$			49.1464i			1.91447i	0.289307	+	0.957237i	$0.406575\pi$
$659$			49.1464i			1.91447i	−0.289307	+	0.957237i	$0.593425\pi$
$660$	7.59592	−	9.30306i	0.295671	−	0.362121i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		−	16.0000i		−	0.620920i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	49.8434i		−	1.92706i
$670$		0			0
$671$		0			0
$672$	33.7980	+	33.7980i	1.30378	+	1.30378i
$673$	−2.59592	+	2.59592i	−0.100065	+	0.100065i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	−2.59592	+	2.59592i	−0.100065	+	0.100065i	0.655302	+	0.755367i	$0.272541\pi$
$674$			51.3939i			1.97962i
$675$	−21.6742	−	14.3258i	−0.834242	−	0.551399i
$676$	26.0000			1.00000
$677$	−2.00000	−	2.00000i	−0.0768662	−	0.0768662i	0.667628	−	0.744495i	$-0.267310\pi$
$677$	−2.00000	−	2.00000i	−0.0768662	−	0.0768662i	−0.744495	+	0.667628i	$0.767310\pi$
$678$		0			0
$679$		−	60.6969i		−	2.32933i
$680$		0			0
$681$	−18.0000			−0.689761
$682$	7.59592	+	7.59592i	0.290863	+	0.290863i
$683$	4.00000	−	4.00000i	0.153056	−	0.153056i	−0.626426	−	0.779481i	$-0.715483\pi$
$683$	4.00000	−	4.00000i	0.153056	−	0.153056i	0.779481	+	0.626426i	$0.215483\pi$
$684$		0			0
$685$		0			0
$686$	−67.5959			−2.58082
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	28.0000	+	28.0000i	1.06440	+	1.06440i
$693$	16.0454	−	16.0454i	0.609515	−	0.609515i
$694$			34.2929i			1.30174i
$695$		0			0
$696$	26.2020			0.993186
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−26.8990	+	40.6969i	−1.01669	+	1.53820i
$701$	52.9444			1.99968			0.999841	−	0.0178345i	$-0.00567720\pi$
$701$	52.9444			1.99968			0.999841	+	0.0178345i	$0.00567720\pi$
$702$		0			0
$703$		0			0
$704$			12.4041i			0.467496i
$705$		0			0
$706$		0			0
$707$	39.8434	+	39.8434i	1.49846	+	1.49846i
$708$	−37.5959	+	37.5959i	−1.41294	+	1.41294i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$	−44.0908			−1.65353
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−18.2929			−0.683636
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	26.6969	−	2.69694i	0.994936	−	0.100509i
$721$	2.40408			0.0895327
$722$	19.0000	+	19.0000i	0.707107	+	0.707107i
$723$	−36.0000	+	36.0000i	−1.33885	+	1.33885i
$724$		0			0
$725$	5.34847	+	26.2020i	0.198637	+	0.973119i
$726$	−21.0556			−0.781447
$727$	−27.9444	−	27.9444i	−1.03640	−	1.03640i	−0.999312	−	0.0370879i	$-0.988192\pi$
$727$	−27.9444	−	27.9444i	−1.03640	−	1.03640i	−0.0370879	−	0.999312i	$-0.511808\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$	5.34847	+	52.9444i	0.197956	+	1.95956i
$731$		0			0
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		−	33.1010i		−	1.22178i
$735$	−41.1464	+	50.3939i	−1.51771	+	1.85881i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$	−16.8990	−	16.8990i	−0.620381	−	0.620381i
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$			24.0000i			0.879883i
$745$	42.5959	−	4.30306i	1.56059	−	0.157652i
$746$		0			0
$747$	−12.0000	−	12.0000i	−0.439057	−	0.439057i
$748$		0			0
$749$		−	55.1918i		−	2.01667i
$750$	8.14643	+	26.1464i	0.297465	+	0.954733i
$751$	53.8888			1.96643			0.983215	−	0.182453i	$-0.0584036\pi$
$751$	53.8888			1.96643			0.983215	+	0.182453i	$0.0584036\pi$
$752$		0			0
$753$	31.8990	−	31.8990i	1.16246	−	1.16246i
$754$		0			0
$755$	−0.449490	−	4.44949i	−0.0163586	−	0.161934i
$756$	50.6969			1.84383
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	−32.9444	−	32.9444i	−1.19345	−	1.19345i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−19.5959	+	19.5959i	−0.707107	+	0.707107i
$769$		−	26.0000i		−	0.937584i	−0.883309	−	0.468792i	$-0.844689\pi$
$769$		−	26.0000i		−	0.937584i	0.883309	−	0.468792i	$-0.155311\pi$
$770$	−23.7980	+	2.40408i	−0.857619	+	0.0866371i
$771$		0			0
$772$	−16.2020	−	16.2020i	−0.583124	−	0.583124i
$773$	14.0000	−	14.0000i	0.503545	−	0.503545i	−0.408993	−	0.912538i	$-0.634120\pi$
$773$	14.0000	−	14.0000i	0.503545	−	0.503545i	0.912538	+	0.408993i	$0.134120\pi$
$774$		0			0
$775$	−24.0000	+	4.89898i	−0.862105	+	0.175977i
$776$	35.1918			1.26331
$777$		0			0
$778$	39.1464	−	39.1464i	1.40347	−	1.40347i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	19.6515	−	19.6515i	0.702288	−	0.702288i
$784$		−	67.1918i		−	2.39971i
$785$		0			0
$786$	45.1918			1.61194
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	34.2929	−	34.2929i	1.22163	−	1.22163i
$789$		0			0
$790$	36.0000	+	29.3939i	1.28082	+	1.04579i
$791$

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:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	120.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +(-1.22474 + 1.22474i) q^{3} +2.00000i q^{4} +(-0.224745 - 2.22474i) q^{5} +2.44949 q^{6} +(-3.44949 - 3.44949i) q^{7} +(2.00000 - 2.00000i) q^{8} -3.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 - 1.00000i) q^{2} +(-1.22474 + 1.22474i) q^{3} +2.00000i q^{4} +(-0.224745 - 2.22474i) q^{5} +2.44949 q^{6} +(-3.44949 - 3.44949i) q^{7} +(2.00000 - 2.00000i) q^{8} -3.00000i q^{9} +(-2.00000 + 2.44949i) q^{10} -1.55051 q^{11} +(-2.44949 - 2.44949i) q^{12} +6.89898i q^{14} +(3.00000 + 2.44949i) q^{15} -4.00000 q^{16} +(-3.00000 + 3.00000i) q^{18} +(4.44949 - 0.449490i) q^{20} +8.44949 q^{21} +(1.55051 + 1.55051i) q^{22} +4.89898i q^{24} +(-4.89898 + 1.00000i) q^{25} +(3.67423 + 3.67423i) q^{27} +(6.89898 - 6.89898i) q^{28} -5.34847i q^{29} +(-0.550510 - 5.44949i) q^{30} +4.89898 q^{31} +(4.00000 + 4.00000i) q^{32} +(1.89898 - 1.89898i) q^{33} +(-6.89898 + 8.44949i) q^{35} +6.00000 q^{36} +(-4.89898 - 4.00000i) q^{40} +(-8.44949 - 8.44949i) q^{42} -3.10102i q^{44} +(-6.67423 + 0.674235i) q^{45} +(4.89898 - 4.89898i) q^{48} +16.7980i q^{49} +(5.89898 + 3.89898i) q^{50} +(-2.44949 + 2.44949i) q^{53} -7.34847i q^{54} +(0.348469 + 3.44949i) q^{55} -13.7980 q^{56} +(-5.34847 + 5.34847i) q^{58} -15.3485i q^{59} +(-4.89898 + 6.00000i) q^{60} +(-4.89898 - 4.89898i) q^{62} +(-10.3485 + 10.3485i) q^{63} -8.00000i q^{64} -3.79796 q^{66} +(15.3485 - 1.55051i) q^{70} +(-6.00000 - 6.00000i) q^{72} +(11.8990 - 11.8990i) q^{73} +(4.77526 - 7.22474i) q^{75} +(5.34847 + 5.34847i) q^{77} -14.6969i q^{79} +(0.898979 + 8.89898i) q^{80} -9.00000 q^{81} +(4.00000 - 4.00000i) q^{83} +16.8990i q^{84} +(6.55051 + 6.55051i) q^{87} +(-3.10102 + 3.10102i) q^{88} +(7.34847 + 6.00000i) q^{90} +(-6.00000 + 6.00000i) q^{93} -9.79796 q^{96} +(8.79796 + 8.79796i) q^{97} +(16.7980 - 16.7980i) q^{98} +4.65153i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{5} - 4 q^{7} + 8 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^5 - 4 * q^7 + 8 * q^8	\( 4 q - 4 q^{2} + 4 q^{5} - 4 q^{7} + 8 q^{8} - 8 q^{10} - 16 q^{11} + 12 q^{15} - 16 q^{16} - 12 q^{18} + 8 q^{20} + 24 q^{21} + 16 q^{22} + 8 q^{28} - 12 q^{30} + 16 q^{32} - 12 q^{33} - 8 q^{35} + 24 q^{36} - 24 q^{42} - 12 q^{45} + 4 q^{50} - 28 q^{55} - 16 q^{56} + 8 q^{58} - 12 q^{63} + 24 q^{66} + 32 q^{70} - 24 q^{72} + 28 q^{73} + 24 q^{75} - 8 q^{77} - 16 q^{80} - 36 q^{81} + 16 q^{83} + 36 q^{87} - 32 q^{88} - 24 q^{93} - 4 q^{97} + 28 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^5 - 4 * q^7 + 8 * q^8 - 8 * q^10 - 16 * q^11 + 12 * q^15 - 16 * q^16 - 12 * q^18 + 8 * q^20 + 24 * q^21 + 16 * q^22 + 8 * q^28 - 12 * q^30 + 16 * q^32 - 12 * q^33 - 8 * q^35 + 24 * q^36 - 24 * q^42 - 12 * q^45 + 4 * q^50 - 28 * q^55 - 16 * q^56 + 8 * q^58 - 12 * q^63 + 24 * q^66 + 32 * q^70 - 24 * q^72 + 28 * q^73 + 24 * q^75 - 8 * q^77 - 16 * q^80 - 36 * q^81 + 16 * q^83 + 36 * q^87 - 32 * q^88 - 24 * q^93 - 4 * q^97 + 28 * q^98

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