LMFDB - Embedded newform 60.2.h.b.59.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,2,Mod(59,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("60.59");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	60.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.479102412128$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			59.1
Root			$-0.309017 + 0.535233i$ of defining polynomial
Character	$\chi$	$=$	60.59
Dual form			60.2.h.b.59.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.11803 - 0.866025i) q^{2} +1.73205i q^{3} +(0.500000 + 1.93649i) q^{4} +2.23607 q^{5} +(1.50000 - 1.93649i) q^{6} +(1.11803 - 2.59808i) q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+(-1.11803 - 0.866025i) q^{2} +1.73205i q^{3} +(0.500000 + 1.93649i) q^{4} +2.23607 q^{5} +(1.50000 - 1.93649i) q^{6} +(1.11803 - 2.59808i) q^{8} -3.00000 q^{9} +(-2.50000 - 1.93649i) q^{10} +(-3.35410 + 0.866025i) q^{12} +3.87298i q^{15} +(-3.50000 + 1.93649i) q^{16} -4.47214 q^{17} +(3.35410 + 2.59808i) q^{18} -7.74597i q^{19} +(1.11803 + 4.33013i) q^{20} -3.46410i q^{23} +(4.50000 + 1.93649i) q^{24} +5.00000 q^{25} -5.19615i q^{27} +(3.35410 - 4.33013i) q^{30} +7.74597i q^{31} +(5.59017 + 0.866025i) q^{32} +(5.00000 + 3.87298i) q^{34} +(-1.50000 - 5.80948i) q^{36} +(-6.70820 + 8.66025i) q^{38} +(2.50000 - 5.80948i) q^{40} -6.70820 q^{45} +(-3.00000 + 3.87298i) q^{46} +10.3923i q^{47} +(-3.35410 - 6.06218i) q^{48} -7.00000 q^{49} +(-5.59017 - 4.33013i) q^{50} -7.74597i q^{51} +4.47214 q^{53} +(-4.50000 + 5.80948i) q^{54} +13.4164 q^{57} +(-7.50000 + 1.93649i) q^{60} -2.00000 q^{61} +(6.70820 - 8.66025i) q^{62} +(-5.50000 - 5.80948i) q^{64} +(-2.23607 - 8.66025i) q^{68} +6.00000 q^{69} +(-3.35410 + 7.79423i) q^{72} +8.66025i q^{75} +(15.0000 - 3.87298i) q^{76} +7.74597i q^{79} +(-7.82624 + 4.33013i) q^{80} +9.00000 q^{81} +3.46410i q^{83} -10.0000 q^{85} +(7.50000 + 5.80948i) q^{90} +(6.70820 - 1.73205i) q^{92} -13.4164 q^{93} +(9.00000 - 11.6190i) q^{94} -17.3205i q^{95} +(-1.50000 + 9.68246i) q^{96} +(7.82624 + 6.06218i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 + 6 * q^6 - 12 * q^9	$ 4 q + 2 q^{4} + 6 q^{6} - 12 q^{9} - 10 q^{10} - 14 q^{16} + 18 q^{24} + 20 q^{25} + 20 q^{34} - 6 q^{36} + 10 q^{40} - 12 q^{46} - 28 q^{49} - 18 q^{54} - 30 q^{60} - 8 q^{61} - 22 q^{64} + 24 q^{69} + 60 q^{76} + 36 q^{81} - 40 q^{85} + 30 q^{90} + 36 q^{94} - 6 q^{96}+O(q^{100}) $ 4 * q + 2 * q^4 + 6 * q^6 - 12 * q^9 - 10 * q^10 - 14 * q^16 + 18 * q^24 + 20 * q^25 + 20 * q^34 - 6 * q^36 + 10 * q^40 - 12 * q^46 - 28 * q^49 - 18 * q^54 - 30 * q^60 - 8 * q^61 - 22 * q^64 + 24 * q^69 + 60 * q^76 + 36 * q^81 - 40 * q^85 + 30 * q^90 + 36 * q^94 - 6 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.11803	−	0.866025i	−0.790569	−	0.612372i
$2$	−1.11803	−	0.866025i	−0.790569	−	0.612372i
$3$			1.73205i			1.00000i
$3$			1.73205i			1.00000i
$4$	0.500000	+	1.93649i	0.250000	+	0.968246i
$5$	2.23607			1.00000
$5$	2.23607			1.00000
$6$	1.50000	−	1.93649i	0.612372	−	0.790569i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	1.11803	−	2.59808i	0.395285	−	0.918559i
$9$	−3.00000			−1.00000
$10$	−2.50000	−	1.93649i	−0.790569	−	0.612372i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−3.35410	+	0.866025i	−0.968246	+	0.250000i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$			3.87298i			1.00000i
$16$	−3.50000	+	1.93649i	−0.875000	+	0.484123i
$17$	−4.47214			−1.08465			−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214			−1.08465			−0.542326	+	0.840168i	$0.682456\pi$
$18$	3.35410	+	2.59808i	0.790569	+	0.612372i
$19$		−	7.74597i		−	1.77705i	−0.458831	−	0.888523i	$-0.651732\pi$
$19$		−	7.74597i		−	1.77705i	0.458831	−	0.888523i	$-0.348268\pi$
$20$	1.11803	+	4.33013i	0.250000	+	0.968246i
$21$		0			0
$22$		0			0
$23$		−	3.46410i		−	0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$		−	3.46410i		−	0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$	4.50000	+	1.93649i	0.918559	+	0.395285i
$25$	5.00000			1.00000
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	3.35410	−	4.33013i	0.612372	−	0.790569i
$31$			7.74597i			1.39122i	0.718421	+	0.695608i	$0.244865\pi$
$31$			7.74597i			1.39122i	−0.718421	+	0.695608i	$0.755135\pi$
$32$	5.59017	+	0.866025i	0.988212	+	0.153093i
$33$		0			0
$34$	5.00000	+	3.87298i	0.857493	+	0.664211i
$35$		0			0
$36$	−1.50000	−	5.80948i	−0.250000	−	0.968246i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−6.70820	+	8.66025i	−1.08821	+	1.40488i
$39$		0			0
$40$	2.50000	−	5.80948i	0.395285	−	0.918559i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	−6.70820			−1.00000
$46$	−3.00000	+	3.87298i	−0.442326	+	0.571040i
$47$			10.3923i			1.51587i	0.652328	+	0.757937i	$0.273792\pi$
$47$			10.3923i			1.51587i	−0.652328	+	0.757937i	$0.726208\pi$
$48$	−3.35410	−	6.06218i	−0.484123	−	0.875000i
$49$	−7.00000			−1.00000
$50$	−5.59017	−	4.33013i	−0.790569	−	0.612372i
$51$		−	7.74597i		−	1.08465i
$52$		0			0
$53$	4.47214			0.614295			0.307148	−	0.951662i	$-0.400625\pi$
$53$	4.47214			0.614295			0.307148	+	0.951662i	$0.400625\pi$
$54$	−4.50000	+	5.80948i	−0.612372	+	0.790569i
$55$		0			0
$56$		0			0
$57$	13.4164			1.77705
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−7.50000	+	1.93649i	−0.968246	+	0.250000i
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$	6.70820	−	8.66025i	0.851943	−	1.09985i
$63$		0			0
$64$	−5.50000	−	5.80948i	−0.687500	−	0.726184i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−2.23607	−	8.66025i	−0.271163	−	1.05021i
$69$	6.00000			0.722315
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−3.35410	+	7.79423i	−0.395285	+	0.918559i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$			8.66025i			1.00000i
$76$	15.0000	−	3.87298i	1.72062	−	0.444262i
$77$		0			0
$78$		0			0
$79$			7.74597i			0.871489i	0.900070	+	0.435745i	$0.143515\pi$
$79$			7.74597i			0.871489i	−0.900070	+	0.435745i	$0.856485\pi$
$80$	−7.82624	+	4.33013i	−0.875000	+	0.484123i
$81$	9.00000			1.00000
$82$		0			0
$83$			3.46410i			0.380235i	0.981761	+	0.190117i	$0.0608868\pi$
$83$			3.46410i			0.380235i	−0.981761	+	0.190117i	$0.939113\pi$
$84$		0			0
$85$	−10.0000			−1.08465
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	7.50000	+	5.80948i	0.790569	+	0.612372i
$91$		0			0
$92$	6.70820	−	1.73205i	0.699379	−	0.180579i
$93$	−13.4164			−1.39122
$94$	9.00000	−	11.6190i	0.928279	−	1.19840i
$95$		−	17.3205i		−	1.77705i
$96$	−1.50000	+	9.68246i	−0.153093	+	0.988212i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	7.82624	+	6.06218i	0.790569	+	0.612372i
$99$		0			0
$100$	2.50000	+	9.68246i	0.250000	+	0.968246i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−6.70820	+	8.66025i	−0.664211	+	0.857493i
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	−5.00000	−	3.87298i	−0.485643	−	0.376177i
$107$		−	10.3923i		−	1.00466i	−0.864675	−	0.502331i	$-0.832476\pi$
$107$		−	10.3923i		−	1.00466i	0.864675	−	0.502331i	$-0.167524\pi$
$108$	10.0623	−	2.59808i	0.968246	−	0.250000i
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.47214			−0.420703			−0.210352	−	0.977626i	$-0.567461\pi$
$113$	−4.47214			−0.420703			−0.210352	+	0.977626i	$0.567461\pi$
$114$	−15.0000	−	11.6190i	−1.40488	−	1.08821i
$115$		−	7.74597i		−	0.722315i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	10.0623	+	4.33013i	0.918559	+	0.395285i
$121$	−11.0000			−1.00000
$122$	2.23607	+	1.73205i	0.202444	+	0.156813i
$123$		0			0
$124$	−15.0000	+	3.87298i	−1.34704	+	0.347804i
$125$	11.1803			1.00000
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	1.11803	+	11.2583i	0.0988212	+	0.995105i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		−	11.6190i		−	1.00000i
$136$	−5.00000	+	11.6190i	−0.428746	+	0.996317i
$137$	−22.3607			−1.91040			−0.955201	−	0.295958i	$-0.904361\pi$
$137$	−22.3607			−1.91040			−0.955201	+	0.295958i	$0.904361\pi$
$138$	−6.70820	−	5.19615i	−0.571040	−	0.442326i
$139$			23.2379i			1.97101i	0.169638	+	0.985506i	$0.445740\pi$
$139$			23.2379i			1.97101i	−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$	−18.0000			−1.51587
$142$		0			0
$143$		0			0
$144$	10.5000	−	5.80948i	0.875000	−	0.484123i
$145$		0			0
$146$		0			0
$147$		−	12.1244i		−	1.00000i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	7.50000	−	9.68246i	0.612372	−	0.790569i
$151$		−	23.2379i		−	1.89107i	−0.325515	−	0.945537i	$-0.605538\pi$
$151$		−	23.2379i		−	1.89107i	0.325515	−	0.945537i	$-0.394462\pi$
$152$	−20.1246	−	8.66025i	−1.63232	−	0.702439i
$153$	13.4164			1.08465
$154$		0			0
$155$			17.3205i			1.39122i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	6.70820	−	8.66025i	0.533676	−	0.688973i
$159$			7.74597i			0.614295i
$160$	12.5000	+	1.93649i	0.988212	+	0.153093i
$161$		0			0
$162$	−10.0623	−	7.79423i	−0.790569	−	0.612372i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	3.00000	−	3.87298i	0.232845	−	0.300602i
$167$			24.2487i			1.87642i	0.346064	+	0.938211i	$0.387518\pi$
$167$			24.2487i			1.87642i	−0.346064	+	0.938211i	$0.612482\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$	11.1803	+	8.66025i	0.857493	+	0.664211i
$171$			23.2379i			1.77705i
$172$		0			0
$173$	22.3607			1.70005			0.850026	−	0.526742i	$-0.176586\pi$
$173$	22.3607			1.70005			0.850026	+	0.526742i	$0.176586\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−3.35410	−	12.9904i	−0.250000	−	0.968246i
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$		−	3.46410i		−	0.256074i
$184$	−9.00000	−	3.87298i	−0.663489	−	0.285520i
$185$		0			0
$186$	15.0000	+	11.6190i	1.09985	+	0.851943i
$187$		0			0
$188$	−20.1246	+	5.19615i	−1.46774	+	0.378968i
$189$		0			0
$190$	−15.0000	+	19.3649i	−1.08821	+	1.40488i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	10.0623	−	9.52628i	0.726184	−	0.687500i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−3.50000	−	13.5554i	−0.250000	−	0.968246i
$197$	4.47214			0.318626			0.159313	−	0.987228i	$-0.449072\pi$
$197$	4.47214			0.318626			0.159313	+	0.987228i	$0.449072\pi$
$198$		0			0
$199$		−	23.2379i		−	1.64729i	−0.567105	−	0.823646i	$-0.691937\pi$
$199$		−	23.2379i		−	1.64729i	0.567105	−	0.823646i	$-0.308063\pi$
$200$	5.59017	−	12.9904i	0.395285	−	0.918559i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	15.0000	−	3.87298i	1.05021	−	0.271163i
$205$		0			0
$206$		0			0
$207$			10.3923i			0.722315i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	7.74597i		−	0.533254i	−0.963800	−	0.266627i	$-0.914091\pi$
$211$		−	7.74597i		−	0.533254i	0.963800	−	0.266627i	$-0.0859092\pi$
$212$	2.23607	+	8.66025i	0.153574	+	0.594789i
$213$		0			0
$214$	−9.00000	+	11.6190i	−0.615227	+	0.794255i
$215$		0			0
$216$	−13.5000	−	5.80948i	−0.918559	−	0.395285i
$217$		0			0
$218$	−15.6525	−	12.1244i	−1.06012	−	0.821165i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−15.0000			−1.00000
$226$	5.00000	+	3.87298i	0.332595	+	0.257627i
$227$		−	24.2487i		−	1.60944i	−0.593652	−	0.804722i	$-0.702314\pi$
$227$		−	24.2487i		−	1.60944i	0.593652	−	0.804722i	$-0.297686\pi$
$228$	6.70820	+	25.9808i	0.444262	+	1.72062i
$229$	−26.0000			−1.71813			−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000			−1.71813			−0.859064	+	0.511868i	$0.828954\pi$
$230$	−6.70820	+	8.66025i	−0.442326	+	0.571040i
$231$		0			0
$232$		0			0
$233$	−22.3607			−1.46490			−0.732448	−	0.680823i	$-0.761622\pi$
$233$	−22.3607			−1.46490			−0.732448	+	0.680823i	$0.761622\pi$
$234$		0			0
$235$			23.2379i			1.51587i
$236$		0			0
$237$	−13.4164			−0.871489
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−7.50000	−	13.5554i	−0.484123	−	0.875000i
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$	12.2984	+	9.52628i	0.790569	+	0.612372i
$243$			15.5885i			1.00000i
$244$	−1.00000	−	3.87298i	−0.0640184	−	0.247942i
$245$	−15.6525			−1.00000
$246$		0			0
$247$		0			0
$248$	20.1246	+	8.66025i	1.27791	+	0.549927i
$249$	−6.00000			−0.380235
$250$	−12.5000	−	9.68246i	−0.790569	−	0.612372i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		−	17.3205i		−	1.08465i
$256$	8.50000	−	13.5554i	0.531250	−	0.847215i
$257$	31.3050			1.95275			0.976375	−	0.216085i	$-0.0693287\pi$
$257$	31.3050			1.95275			0.976375	+	0.216085i	$0.0693287\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	31.1769i		−	1.92245i	−0.275764	−	0.961225i	$-0.588931\pi$
$263$		−	31.1769i		−	1.92245i	0.275764	−	0.961225i	$-0.411069\pi$
$264$		0			0
$265$	10.0000			0.614295
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	−10.0623	+	12.9904i	−0.612372	+	0.790569i
$271$			7.74597i			0.470534i	0.971931	+	0.235267i	$0.0755965\pi$
$271$			7.74597i			0.470534i	−0.971931	+	0.235267i	$0.924404\pi$
$272$	15.6525	−	8.66025i	0.949071	−	0.525105i
$273$		0			0
$274$	25.0000	+	19.3649i	1.51031	+	1.16988i
$275$		0			0
$276$	3.00000	+	11.6190i	0.180579	+	0.699379i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	20.1246	−	25.9808i	1.20699	−	1.55822i
$279$		−	23.2379i		−	1.39122i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	20.1246	+	15.5885i	1.19840	+	0.928279i
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$	30.0000			1.77705
$286$		0			0
$287$		0			0
$288$	−16.7705	−	2.59808i	−0.988212	−	0.153093i
$289$	3.00000			0.176471
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−31.3050			−1.82885			−0.914427	−	0.404750i	$-0.867359\pi$
$293$	−31.3050			−1.82885			−0.914427	+	0.404750i	$0.867359\pi$
$294$	−10.5000	+	13.5554i	−0.612372	+	0.790569i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−16.7705	+	4.33013i	−0.968246	+	0.250000i
$301$		0			0
$302$	−20.1246	+	25.9808i	−1.15804	+	1.49502i
$303$		0			0
$304$	15.0000	+	27.1109i	0.860309	+	1.55492i
$305$	−4.47214			−0.256074
$306$	−15.0000	−	11.6190i	−0.857493	−	0.664211i
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$	15.0000	−	19.3649i	0.851943	−	1.09985i
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$	−15.0000	+	3.87298i	−0.843816	+	0.217872i
$317$	22.3607			1.25590			0.627950	−	0.778253i	$-0.283894\pi$
$317$	22.3607			1.25590			0.627950	+	0.778253i	$0.283894\pi$
$318$	6.70820	−	8.66025i	0.376177	−	0.485643i
$319$		0			0
$320$	−12.2984	−	12.9904i	−0.687500	−	0.726184i
$321$	18.0000			1.00466
$322$		0			0
$323$			34.6410i			1.92748i
$324$	4.50000	+	17.4284i	0.250000	+	0.968246i
$325$		0			0
$326$		0			0
$327$			24.2487i			1.34096i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			23.2379i			1.27727i	0.769510	+	0.638635i	$0.220501\pi$
$331$			23.2379i			1.27727i	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−6.70820	+	1.73205i	−0.368161	+	0.0950586i
$333$		0			0
$334$	21.0000	−	27.1109i	1.14907	−	1.48344i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−14.5344	−	11.2583i	−0.790569	−	0.612372i
$339$		−	7.74597i		−	0.420703i
$340$	−5.00000	−	19.3649i	−0.271163	−	1.05021i
$341$		0			0
$342$	20.1246	−	25.9808i	1.08821	−	1.40488i
$343$		0			0
$344$		0			0
$345$	13.4164			0.722315
$346$	−25.0000	−	19.3649i	−1.34401	−	1.04106i
$347$		−	10.3923i		−	0.557888i	−0.960307	−	0.278944i	$-0.910016\pi$
$347$		−	10.3923i		−	0.557888i	0.960307	−	0.278944i	$-0.0899844\pi$
$348$		0			0
$349$	−34.0000			−1.81998			−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000			−1.81998			−0.909989	+	0.414632i	$0.863910\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	31.3050			1.66619			0.833097	−	0.553127i	$-0.186565\pi$
$353$	31.3050			1.66619			0.833097	+	0.553127i	$0.186565\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−7.50000	+	17.4284i	−0.395285	+	0.918559i
$361$	−41.0000			−2.15789
$362$	−24.5967	−	19.0526i	−1.29278	−	1.00138i
$363$		−	19.0526i		−	1.00000i
$364$		0			0
$365$		0			0
$366$	−3.00000	+	3.87298i	−0.156813	+	0.202444i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	6.70820	+	12.1244i	0.349689	+	0.632026i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−6.70820	−	25.9808i	−0.347804	−	1.34704i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$			19.3649i			1.00000i
$376$	27.0000	+	11.6190i	1.39242	+	0.599202i
$377$		0			0
$378$		0			0
$379$		−	38.7298i		−	1.98942i	−0.102733	−	0.994709i	$-0.532759\pi$
$379$		−	38.7298i		−	1.98942i	0.102733	−	0.994709i	$-0.467241\pi$
$380$	33.5410	−	8.66025i	1.72062	−	0.444262i
$381$		0			0
$382$		0			0
$383$			38.1051i			1.94708i	0.228515	+	0.973540i	$0.426613\pi$
$383$			38.1051i			1.94708i	−0.228515	+	0.973540i	$0.573387\pi$
$384$	−19.5000	+	1.93649i	−0.995105	+	0.0988212i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$			15.4919i			0.783461i
$392$	−7.82624	+	18.1865i	−0.395285	+	0.918559i
$393$		0			0
$394$	−5.00000	−	3.87298i	−0.251896	−	0.195118i
$395$			17.3205i			0.871489i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−20.1246	+	25.9808i	−1.00876	+	1.30230i
$399$		0			0
$400$	−17.5000	+	9.68246i	−0.875000	+	0.484123i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	20.1246			1.00000
$406$		0			0
$407$		0			0
$408$	−20.1246	−	8.66025i	−0.996317	−	0.428746i
$409$	26.0000			1.28562			0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000			1.28562			0.642809	+	0.766027i	$0.277769\pi$
$410$		0			0
$411$		−	38.7298i		−	1.91040i
$412$		0			0
$413$		0			0
$414$	9.00000	−	11.6190i	0.442326	−	0.571040i
$415$			7.74597i			0.380235i
$416$		0			0
$417$	−40.2492			−1.97101
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	38.0000			1.85201			0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000			1.85201			0.926003	+	0.377515i	$0.123221\pi$
$422$	−6.70820	+	8.66025i	−0.326550	+	0.421575i
$423$		−	31.1769i		−	1.51587i
$424$	5.00000	−	11.6190i	0.242821	−	0.564266i
$425$	−22.3607			−1.08465
$426$		0			0
$427$		0			0
$428$	20.1246	−	5.19615i	0.972760	−	0.251166i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	10.0623	+	18.1865i	0.484123	+	0.875000i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	7.00000	+	27.1109i	0.335239	+	1.29838i
$437$	−26.8328			−1.28359
$438$		0			0
$439$			38.7298i			1.84847i	0.381819	+	0.924237i	$0.375298\pi$
$439$			38.7298i			1.84847i	−0.381819	+	0.924237i	$0.624702\pi$
$440$		0			0
$441$	21.0000			1.00000
$442$		0			0
$443$		−	38.1051i		−	1.81043i	−0.424955	−	0.905214i	$-0.639710\pi$
$443$		−	38.1051i		−	1.81043i	0.424955	−	0.905214i	$-0.360290\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	16.7705	+	12.9904i	0.790569	+	0.612372i
$451$		0			0
$452$	−2.23607	−	8.66025i	−0.105176	−	0.407344i
$453$	40.2492			1.89107
$454$	−21.0000	+	27.1109i	−0.985579	+	1.27238i
$455$		0			0
$456$	15.0000	−	34.8569i	0.702439	−	1.63232i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	29.0689	+	22.5167i	1.35830	+	1.05213i
$459$			23.2379i			1.08465i
$460$	15.0000	−	3.87298i	0.699379	−	0.180579i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$	−30.0000			−1.39122
$466$	25.0000	+	19.3649i	1.15810	+	0.897062i
$467$		−	24.2487i		−	1.12210i	−0.827783	−	0.561048i	$-0.810398\pi$
$467$		−	24.2487i		−	1.12210i	0.827783	−	0.561048i	$-0.189602\pi$
$468$		0			0
$469$		0			0
$470$	20.1246	−	25.9808i	0.928279	−	1.19840i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	15.0000	+	11.6190i	0.688973	+	0.533676i
$475$		−	38.7298i		−	1.77705i
$476$		0			0
$477$	−13.4164			−0.614295
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	−3.35410	+	21.6506i	−0.153093	+	0.988212i
$481$		0			0
$482$	−2.23607	−	1.73205i	−0.101850	−	0.0788928i
$483$		0			0
$484$	−5.50000	−	21.3014i	−0.250000	−	0.968246i
$485$		0			0
$486$	13.5000	−	17.4284i	0.612372	−	0.790569i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	−2.23607	+	5.19615i	−0.101222	+	0.235219i
$489$		0			0
$490$	17.5000	+	13.5554i	0.790569	+	0.612372i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−15.0000	−	27.1109i	−0.673520	−	1.21731i
$497$		0			0
$498$	6.70820	+	5.19615i	0.300602	+	0.232845i
$499$		−	7.74597i		−	0.346757i	−0.984855	−	0.173379i	$-0.944532\pi$
$499$		−	7.74597i		−	0.346757i	0.984855	−	0.173379i	$-0.0554684\pi$
$500$	5.59017	+	21.6506i	0.250000	+	0.968246i
$501$	−42.0000			−1.87642
$502$		0			0
$503$		−	3.46410i		−	0.154457i	−0.997013	−	0.0772283i	$-0.975393\pi$
$503$		−	3.46410i		−	0.154457i	0.997013	−	0.0772283i	$-0.0246070\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			22.5167i			1.00000i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$	−15.0000	+	19.3649i	−0.664211	+	0.857493i
$511$		0			0
$512$	−21.2426	+	7.79423i	−0.938801	+	0.344459i
$513$	−40.2492			−1.77705
$514$	−35.0000	−	27.1109i	−1.54378	−	1.19581i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			38.7298i			1.70005i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	−27.0000	+	34.8569i	−1.17726	+	1.51983i
$527$		−	34.6410i		−	1.50899i
$528$		0			0
$529$	11.0000			0.478261
$530$	−11.1803	−	8.66025i	−0.485643	−	0.376177i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	23.2379i		−	1.00466i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	22.5000	−	5.80948i	0.968246	−	0.250000i
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$	6.70820	−	8.66025i	0.288142	−	0.371990i
$543$			38.1051i			1.63525i
$544$	−25.0000	−	3.87298i	−1.07187	−	0.166053i
$545$	31.3050			1.34096
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	−11.1803	−	43.3013i	−0.477600	−	1.84974i
$549$	6.00000			0.256074
$550$		0			0
$551$		0			0
$552$	6.70820	−	15.5885i	0.285520	−	0.663489i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−45.0000	+	11.6190i	−1.90843	+	0.492753i
$557$	22.3607			0.947452			0.473726	−	0.880672i	$-0.342909\pi$
$557$	22.3607			0.947452			0.473726	+	0.880672i	$0.342909\pi$
$558$	−20.1246	+	25.9808i	−0.851943	+	1.09985i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$			31.1769i			1.31395i	0.753912	+	0.656975i	$0.228164\pi$
$563$			31.1769i			1.31395i	−0.753912	+	0.656975i	$0.771836\pi$
$564$	−9.00000	−	34.8569i	−0.378968	−	1.46774i
$565$	−10.0000			−0.420703
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$	−33.5410	−	25.9808i	−1.40488	−	1.08821i
$571$		−	38.7298i		−	1.62079i	−0.585882	−	0.810397i	$-0.699252\pi$
$571$		−	38.7298i		−	1.62079i	0.585882	−	0.810397i	$-0.300748\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	17.3205i		−	0.722315i
$576$	16.5000	+	17.4284i	0.687500	+	0.726184i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−3.35410	−	2.59808i	−0.139512	−	0.108066i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	35.0000	+	27.1109i	1.44584	+	1.11994i
$587$			45.0333i			1.85872i	0.369170	+	0.929362i	$0.379642\pi$
$587$			45.0333i			1.85872i	−0.369170	+	0.929362i	$0.620358\pi$
$588$	23.4787	−	6.06218i	0.968246	−	0.250000i
$589$	60.0000			2.47226
$590$		0			0
$591$			7.74597i			0.318626i
$592$		0			0
$593$	−4.47214			−0.183649			−0.0918243	−	0.995775i	$-0.529270\pi$
$593$	−4.47214			−0.183649			−0.0918243	+	0.995775i	$0.529270\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	40.2492			1.64729
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	22.5000	+	9.68246i	0.918559	+	0.395285i
$601$	−38.0000			−1.55005			−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000			−1.55005			−0.775026	+	0.631929i	$0.782263\pi$
$602$		0			0
$603$		0			0
$604$	45.0000	−	11.6190i	1.83102	−	0.472768i
$605$	−24.5967			−1.00000
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	6.70820	−	43.3013i	0.272054	−	1.75610i
$609$		0			0
$610$	5.00000	+	3.87298i	0.202444	+	0.156813i
$611$		0			0
$612$	6.70820	+	25.9808i	0.271163	+	1.05021i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	49.1935			1.98046			0.990228	−	0.139459i	$-0.0445365\pi$
$617$	49.1935			1.98046			0.990228	+	0.139459i	$0.0445365\pi$
$618$		0			0
$619$			23.2379i			0.934010i	0.884255	+	0.467005i	$0.154667\pi$
$619$			23.2379i			0.934010i	−0.884255	+	0.467005i	$0.845333\pi$
$620$	−33.5410	+	8.66025i	−1.34704	+	0.347804i
$621$	−18.0000			−0.722315
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			38.7298i			1.54181i	0.636950	+	0.770905i	$0.280196\pi$
$631$			38.7298i			1.54181i	−0.636950	+	0.770905i	$0.719804\pi$
$632$	20.1246	+	8.66025i	0.800514	+	0.344486i
$633$	13.4164			0.533254
$634$	−25.0000	−	19.3649i	−0.992877	−	0.769079i
$635$		0			0
$636$	−15.0000	+	3.87298i	−0.594789	+	0.153574i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	2.50000	+	25.1744i	0.0988212	+	0.995105i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	−20.1246	−	15.5885i	−0.794255	−	0.615227i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$	30.0000	−	38.7298i	1.18033	−	1.52380i
$647$			24.2487i			0.953315i	0.879089	+	0.476658i	$0.158152\pi$
$647$			24.2487i			0.953315i	−0.879089	+	0.476658i	$0.841848\pi$
$648$	10.0623	−	23.3827i	0.395285	−	0.918559i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−49.1935			−1.92509			−0.962545	−	0.271122i	$-0.912605\pi$
$653$	−49.1935			−1.92509			−0.962545	+	0.271122i	$0.912605\pi$
$654$	21.0000	−	27.1109i	0.821165	−	1.06012i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	22.0000			0.855701			0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000			0.855701			0.427850	+	0.903850i	$0.359271\pi$
$662$	20.1246	−	25.9808i	0.782165	−	1.00977i
$663$		0			0
$664$	9.00000	+	3.87298i	0.349268	+	0.150301i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−46.9574	+	12.1244i	−1.81684	+	0.469105i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		−	25.9808i		−	1.00000i
$676$	6.50000	+	25.1744i	0.250000	+	0.968246i
$677$	−31.3050			−1.20315			−0.601574	−	0.798817i	$-0.705459\pi$
$677$	−31.3050			−1.20315			−0.601574	+	0.798817i	$0.705459\pi$
$678$	−6.70820	+	8.66025i	−0.257627	+	0.332595i
$679$		0			0
$680$	−11.1803	+	25.9808i	−0.428746	+	0.996317i
$681$	42.0000			1.60944
$682$		0			0
$683$		−	38.1051i		−	1.45805i	−0.684486	−	0.729026i	$-0.739973\pi$
$683$		−	38.1051i		−	1.45805i	0.684486	−	0.729026i	$-0.260027\pi$
$684$	−45.0000	+	11.6190i	−1.72062	+	0.444262i
$685$	−50.0000			−1.91040
$686$		0			0
$687$		−	45.0333i		−	1.71813i
$688$		0			0
$689$		0			0
$690$	−15.0000	−	11.6190i	−0.571040	−	0.442326i
$691$		−	7.74597i		−	0.294670i	−0.989087	−	0.147335i	$-0.952930\pi$
$691$		−	7.74597i		−	0.294670i	0.989087	−	0.147335i	$-0.0470696\pi$
$692$	11.1803	+	43.3013i	0.425013	+	1.64607i
$693$		0			0
$694$	−9.00000	+	11.6190i	−0.341635	+	0.441049i
$695$			51.9615i			1.97101i
$696$		0			0
$697$		0			0
$698$	38.0132	+	29.4449i	1.43882	+	1.11450i
$699$		−	38.7298i		−	1.46490i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−40.2492			−1.51587
$706$	−35.0000	−	27.1109i	−1.31724	−	1.02033i
$707$		0			0
$708$		0			0
$709$	−26.0000			−0.976450			−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000			−0.976450			−0.488225	+	0.872718i	$0.662356\pi$
$710$		0			0
$711$		−	23.2379i		−	0.871489i
$712$		0			0
$713$	26.8328			1.00490
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	23.4787	−	12.9904i	0.875000	−	0.484123i
$721$		0			0
$722$	45.8394	+	35.5070i	1.70597	+	1.32144i
$723$			3.46410i			0.128831i
$724$	11.0000	+	42.6028i	0.408812	+	1.58332i
$725$		0			0
$726$	−16.5000	+	21.3014i	−0.612372	+	0.790569i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$	6.70820	−	1.73205i	0.247942	−	0.0640184i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		−	27.1109i		−	1.00000i
$736$	3.00000	−	19.3649i	0.110581	−	0.713800i
$737$		0			0
$738$		0			0
$739$			54.2218i			1.99458i	0.0735712	+	0.997290i	$0.476560\pi$
$739$			54.2218i			1.99458i	−0.0735712	+	0.997290i	$0.523440\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	31.1769i		−	1.14377i	−0.820334	−	0.571885i	$-0.806212\pi$
$743$		−	31.1769i		−	1.14377i	0.820334	−	0.571885i	$-0.193788\pi$
$744$	−15.0000	+	34.8569i	−0.549927	+	1.27791i
$745$		0			0
$746$		0			0
$747$		−	10.3923i		−	0.380235i
$748$		0			0
$749$		0			0
$750$	16.7705	−	21.6506i	0.612372	−	0.790569i
$751$		−	54.2218i		−	1.97858i	−0.145962	−	0.989290i	$-0.546628\pi$
$751$		−	54.2218i		−	1.97858i	0.145962	−	0.989290i	$-0.453372\pi$
$752$	−20.1246	−	36.3731i	−0.733869	−	1.32639i
$753$		0			0
$754$		0			0
$755$		−	51.9615i		−	1.89107i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−33.5410	+	43.3013i	−1.21826	+	1.57277i
$759$		0			0
$760$	−45.0000	−	19.3649i	−1.63232	−	0.702439i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	30.0000			1.08465
$766$	33.0000	−	42.6028i	1.19234	−	1.53930i
$767$		0			0
$768$	23.4787	+	14.7224i	0.847215	+	0.531250i
$769$	−46.0000			−1.65880			−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000			−1.65880			−0.829401	+	0.558653i	$0.811318\pi$
$770$		0			0
$771$			54.2218i			1.95275i
$772$		0			0
$773$	4.47214			0.160852			0.0804258	−	0.996761i	$-0.474372\pi$
$773$	4.47214			0.160852			0.0804258	+	0.996761i	$0.474372\pi$
$774$		0			0
$775$			38.7298i			1.39122i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	13.4164	−	17.3205i	0.479770	−	0.619380i
$783$		0			0
$784$	24.5000	−	13.5554i	0.875000	−	0.484123i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	2.23607	+	8.66025i	0.0796566	+	0.308509i
$789$	54.0000			1.92245
$790$	15.0000	−	19.3649i	0.533676	−	0.688973i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$			17.3205i			0.614295i
$796$	45.0000	−	11.6190i	1.59498	−	0.411823i
$797$	−49.1935			−1.74252			−0.871262	−	0.490819i	$-0.836698\pi$
$797$	−49.1935			−1.74252			−0.871262	+	0.490819i	$0.836698\pi$
$798$		0			0
$799$		−	46.4758i		−	1.64420i
$800$	27.9508	+	4.33013i	0.988212	+	0.153093i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−22.5000	−	17.4284i	−0.790569	−	0.612372i
$811$			23.2379i			0.815993i	0.912983	+	0.407997i	$0.133772\pi$
$811$			23.2379i			0.815993i	−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$	−13.4164			−0.470534
$814$		0			0
$815$		0			0
$816$	15.0000	+	27.1109i	0.525105	+	0.949071i
$817$		0			0
$818$	−29.0689	−	22.5167i	−1.01637	−	0.787277i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−33.5410	+	43.3013i	−1.16988	+	1.51031i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			45.0333i			1.56596i	0.622046	+	0.782981i	$0.286302\pi$
$827$			45.0333i			1.56596i	−0.622046	+	0.782981i	$0.713698\pi$
$828$	−20.1246	+	5.19615i	−0.699379	+	0.180579i
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$	6.70820	−	8.66025i	0.232845	−	0.300602i
$831$		0			0
$832$		0			0
$833$	31.3050			1.08465
$834$	45.0000	+	34.8569i	1.55822	+	1.20699i
$835$			54.2218i			1.87642i
$836$		0			0
$837$	40.2492			1.39122
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$	−42.4853	−	32.9090i	−1.46414	−	1.13412i
$843$		0			0
$844$	15.0000	−	3.87298i	0.516321	−	0.133314i
$845$	29.0689			1.00000
$846$	−27.0000	+	34.8569i	−0.928279	+	1.19840i
$847$		0			0
$848$	−15.6525	+	8.66025i	−0.537508	+	0.297394i
$849$		0			0
$850$	25.0000	+	19.3649i	0.857493	+	0.664211i
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$			51.9615i			1.77705i
$856$	−27.0000	−	11.6190i	−0.922841	−	0.397128i
$857$	−58.1378			−1.98595			−0.992974	−	0.118331i	$-0.962245\pi$
$857$	−58.1378			−1.98595			−0.992974	+	0.118331i	$0.962245\pi$
$858$		0			0
$859$		−	38.7298i		−	1.32144i	−0.750630	−	0.660722i	$-0.770250\pi$
$859$		−	38.7298i		−	1.32144i	0.750630	−	0.660722i	$-0.229750\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			38.1051i			1.29711i	0.761166	+	0.648557i	$0.224627\pi$
$863$			38.1051i			1.29711i	−0.761166	+	0.648557i	$0.775373\pi$
$864$	4.50000	−	29.0474i	0.153093	−	0.988212i
$865$	50.0000			1.70005
$866$		0			0
$867$			5.19615i			0.176471i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	15.6525	−	36.3731i	0.530060	−	1.23175i
$873$		0			0
$874$	30.0000	+	23.2379i	1.01477	+	0.786034i
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$	33.5410	−	43.3013i	1.13195	−	1.46135i
$879$		−	54.2218i		−	1.82885i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−23.4787	−	18.1865i	−0.790569	−	0.612372i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	−33.0000	+	42.6028i	−1.10866	+	1.43127i
$887$		−	58.8897i		−	1.97732i	−0.150160	−	0.988662i	$-0.547979\pi$
$887$		−	58.8897i		−	1.97732i	0.150160	−	0.988662i	$-0.452021\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	80.4984			2.69378
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−7.50000	−	29.0474i	−0.250000	−	0.968246i
$901$	−20.0000			−0.666297
$902$		0			0
$903$		0			0
$904$	−5.00000	+	11.6190i	−0.166298	+	0.386441i
$905$	49.1935			1.63525
$906$	−45.0000	−	34.8569i	−1.49502	−	1.15804i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$	46.9574	−	12.1244i	1.55834	−	0.402361i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−46.9574	+	25.9808i	−1.55492	+	0.860309i
$913$		0			0
$914$		0			0
$915$		−	7.74597i		−	0.256074i
$916$	−13.0000	−	50.3488i	−0.429532	−	1.66357i
$917$		0			0
$918$	20.1246	−	25.9808i	0.664211	−	0.857493i
$919$		−	23.2379i		−	0.766548i	−0.923635	−	0.383274i	$-0.874797\pi$
$919$		−	23.2379i		−	0.766548i	0.923635	−	0.383274i	$-0.125203\pi$
$920$	−20.1246	−	8.66025i	−0.663489	−	0.285520i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$	33.5410	+	25.9808i	1.09985	+	0.851943i
$931$			54.2218i			1.77705i
$932$	−11.1803	−	43.3013i	−0.366224	−	1.41838i
$933$		0			0
$934$	−21.0000	+	27.1109i	−0.687141	+	0.887095i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$	−45.0000	+	11.6190i	−1.46774	+	0.378968i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			58.8897i			1.91366i	0.290650	+	0.956830i	$0.406129\pi$
$947$			58.8897i			1.91366i	−0.290650	+	0.956830i	$0.593871\pi$
$948$	−6.70820	−	25.9808i	−0.217872	−	0.843816i
$949$		0			0
$950$	−33.5410	+	43.3013i	−1.08821	+	1.40488i
$951$			38.7298i			1.25590i
$952$		0			0
$953$	−58.1378			−1.88327			−0.941634	−	0.336640i	$-0.890710\pi$
$953$	−58.1378			−1.88327			−0.941634	+	0.336640i	$0.890710\pi$
$954$	15.0000	+	11.6190i	0.485643	+	0.376177i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	22.5000	−	21.3014i	0.726184	−	0.687500i
$961$	−29.0000			−0.935484
$962$		0			0
$963$			31.1769i			1.00466i
$964$	1.00000	+	3.87298i	0.0322078	+	0.124740i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−12.2984	+	28.5788i	−0.395285	+	0.918559i
$969$	−60.0000			−1.92748
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−30.1869	+	7.79423i	−0.968246	+	0.250000i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	7.00000	−	3.87298i	0.224065	−	0.123971i
$977$	−4.47214			−0.143076			−0.0715382	−	0.997438i	$-0.522791\pi$
$977$	−4.47214			−0.143076			−0.0715382	+	0.997438i	$0.522791\pi$
$978$		0			0
$979$		0			0
$980$	−7.82624	−	30.3109i	−0.250000	−	0.968246i
$981$	−42.0000			−1.34096
$982$		0			0
$983$		−	3.46410i		−	0.110488i	−0.998473	−	0.0552438i	$-0.982406\pi$
$983$		−	3.46410i		−	0.110488i	0.998473	−	0.0552438i	$-0.0175936\pi$
$984$		0			0
$985$	10.0000			0.318626
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	54.2218i		−	1.72241i	−0.508257	−	0.861206i	$-0.669710\pi$
$991$		−	54.2218i		−	1.72241i	0.508257	−	0.861206i	$-0.330290\pi$
$992$	−6.70820	+	43.3013i	−0.212986	+	1.37482i
$993$	−40.2492			−1.27727
$994$		0			0
$995$		−	51.9615i		−	1.64729i
$996$	−3.00000	−	11.6190i	−0.0950586	−	0.368161i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	−6.70820	+	8.66025i	−0.212344	+	0.274136i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.2.h.b.59.1	✓	4
3.2	odd	2	inner	60.2.h.b.59.4	yes	4
4.3	odd	2	inner	60.2.h.b.59.2	yes	4
5.2	odd	4		300.2.e.a.251.3		4
5.3	odd	4		300.2.e.a.251.2		4
5.4	even	2	inner	60.2.h.b.59.4	yes	4
8.3	odd	2		960.2.o.a.959.3		4
8.5	even	2		960.2.o.a.959.1		4
12.11	even	2	inner	60.2.h.b.59.3	yes	4
15.2	even	4		300.2.e.a.251.2		4
15.8	even	4		300.2.e.a.251.3		4
15.14	odd	2	CM	60.2.h.b.59.1	✓	4
20.3	even	4		300.2.e.a.251.4		4
20.7	even	4		300.2.e.a.251.1		4
20.19	odd	2	inner	60.2.h.b.59.3	yes	4
24.5	odd	2		960.2.o.a.959.4		4
24.11	even	2		960.2.o.a.959.2		4
40.19	odd	2		960.2.o.a.959.2		4
40.29	even	2		960.2.o.a.959.4		4
60.23	odd	4		300.2.e.a.251.1		4
60.47	odd	4		300.2.e.a.251.4		4
60.59	even	2	inner	60.2.h.b.59.2	yes	4
120.29	odd	2		960.2.o.a.959.1		4
120.59	even	2		960.2.o.a.959.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.2.h.b.59.1	✓	4	1.1	even	1	trivial
60.2.h.b.59.1	✓	4	15.14	odd	2	CM
60.2.h.b.59.2	yes	4	4.3	odd	2	inner
60.2.h.b.59.2	yes	4	60.59	even	2	inner
60.2.h.b.59.3	yes	4	12.11	even	2	inner
60.2.h.b.59.3	yes	4	20.19	odd	2	inner
60.2.h.b.59.4	yes	4	3.2	odd	2	inner
60.2.h.b.59.4	yes	4	5.4	even	2	inner
300.2.e.a.251.1		4	20.7	even	4
300.2.e.a.251.1		4	60.23	odd	4
300.2.e.a.251.2		4	5.3	odd	4
300.2.e.a.251.2		4	15.2	even	4
300.2.e.a.251.3		4	5.2	odd	4
300.2.e.a.251.3		4	15.8	even	4
300.2.e.a.251.4		4	20.3	even	4
300.2.e.a.251.4		4	60.47	odd	4
960.2.o.a.959.1		4	8.5	even	2
960.2.o.a.959.1		4	120.29	odd	2
960.2.o.a.959.2		4	24.11	even	2
960.2.o.a.959.2		4	40.19	odd	2
960.2.o.a.959.3		4	8.3	odd	2
960.2.o.a.959.3		4	120.59	even	2
960.2.o.a.959.4		4	24.5	odd	2
960.2.o.a.959.4		4	40.29	even	2

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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	60.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.11803 - 0.866025i) q^{2} +1.73205i q^{3} +(0.500000 + 1.93649i) q^{4} +2.23607 q^{5} +(1.50000 - 1.93649i) q^{6} +(1.11803 - 2.59808i) q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+(-1.11803 - 0.866025i) q^{2} +1.73205i q^{3} +(0.500000 + 1.93649i) q^{4} +2.23607 q^{5} +(1.50000 - 1.93649i) q^{6} +(1.11803 - 2.59808i) q^{8} -3.00000 q^{9} +(-2.50000 - 1.93649i) q^{10} +(-3.35410 + 0.866025i) q^{12} +3.87298i q^{15} +(-3.50000 + 1.93649i) q^{16} -4.47214 q^{17} +(3.35410 + 2.59808i) q^{18} -7.74597i q^{19} +(1.11803 + 4.33013i) q^{20} -3.46410i q^{23} +(4.50000 + 1.93649i) q^{24} +5.00000 q^{25} -5.19615i q^{27} +(3.35410 - 4.33013i) q^{30} +7.74597i q^{31} +(5.59017 + 0.866025i) q^{32} +(5.00000 + 3.87298i) q^{34} +(-1.50000 - 5.80948i) q^{36} +(-6.70820 + 8.66025i) q^{38} +(2.50000 - 5.80948i) q^{40} -6.70820 q^{45} +(-3.00000 + 3.87298i) q^{46} +10.3923i q^{47} +(-3.35410 - 6.06218i) q^{48} -7.00000 q^{49} +(-5.59017 - 4.33013i) q^{50} -7.74597i q^{51} +4.47214 q^{53} +(-4.50000 + 5.80948i) q^{54} +13.4164 q^{57} +(-7.50000 + 1.93649i) q^{60} -2.00000 q^{61} +(6.70820 - 8.66025i) q^{62} +(-5.50000 - 5.80948i) q^{64} +(-2.23607 - 8.66025i) q^{68} +6.00000 q^{69} +(-3.35410 + 7.79423i) q^{72} +8.66025i q^{75} +(15.0000 - 3.87298i) q^{76} +7.74597i q^{79} +(-7.82624 + 4.33013i) q^{80} +9.00000 q^{81} +3.46410i q^{83} -10.0000 q^{85} +(7.50000 + 5.80948i) q^{90} +(6.70820 - 1.73205i) q^{92} -13.4164 q^{93} +(9.00000 - 11.6190i) q^{94} -17.3205i q^{95} +(-1.50000 + 9.68246i) q^{96} +(7.82624 + 6.06218i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 + 6 * q^6 - 12 * q^9	\( 4 q + 2 q^{4} + 6 q^{6} - 12 q^{9} - 10 q^{10} - 14 q^{16} + 18 q^{24} + 20 q^{25} + 20 q^{34} - 6 q^{36} + 10 q^{40} - 12 q^{46} - 28 q^{49} - 18 q^{54} - 30 q^{60} - 8 q^{61} - 22 q^{64} + 24 q^{69} + 60 q^{76} + 36 q^{81} - 40 q^{85} + 30 q^{90} + 36 q^{94} - 6 q^{96}+O(q^{100}) \) 4 * q + 2 * q^4 + 6 * q^6 - 12 * q^9 - 10 * q^10 - 14 * q^16 + 18 * q^24 + 20 * q^25 + 20 * q^34 - 6 * q^36 + 10 * q^40 - 12 * q^46 - 28 * q^49 - 18 * q^54 - 30 * q^60 - 8 * q^61 - 22 * q^64 + 24 * q^69 + 60 * q^76 + 36 * q^81 - 40 * q^85 + 30 * q^90 + 36 * q^94 - 6 * q^96

Introduction

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