Embedded newform 114.4.l.b.59.6

• Label: 114.4.l.b.59.6
• Level: $114$
• Weight: $4$
• Character: 114.59
• Analytic conductor: $6.726$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	114.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.72621774065\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			59.6
Character	\(\chi\)	\(=\)	114.59
Dual form			114.4.l.b.29.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.347296 - 1.96962i) q^{2} +(0.999139 - 5.09919i) q^{3} +(-3.75877 - 1.36808i) q^{4} +(2.49532 + 6.85584i) q^{5} +(-9.69644 - 3.73885i) q^{6} +(-13.0267 - 22.5629i) q^{7} +(-4.00000 + 6.92820i) q^{8} +(-25.0034 - 10.1896i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 9 q^{3} + 12 q^{6} - 240 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{18}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.347296	−	1.96962i	0.122788	−	0.696364i
							\(3\)	0.999139	−	5.09919i	0.192284	−	0.981339i
\(5\)	2.49532	+	6.85584i	0.223188	+	0.613205i								0.999861	−	0.0167000i	\(-0.00531602\pi\)
							−0.776672	+	0.629905i	\(0.783094\pi\)
\(7\)	−13.0267	−	22.5629i	−0.703376	−	1.21828i	−0.967275	−	0.253732i	\(-0.918342\pi\)
							0.263899	−	0.964550i	\(-0.414992\pi\)
\(11\)	1.59600	+	0.921452i	0.0437466	+	0.0252571i	0.521714	−	0.853121i	\(-0.325293\pi\)
							−0.477967	+	0.878378i	\(0.658626\pi\)
\(13\)	−33.5080	−	39.9333i	−0.714881	−	0.851962i	0.279242	−	0.960221i	\(-0.409917\pi\)
							−0.994123	+	0.108259i	\(0.965472\pi\)
\(17\)	−9.49454	−	1.67414i	−0.135457	−	0.0238847i	0.105509	−	0.994418i	\(-0.466353\pi\)
							−0.240965	+	0.970534i	\(0.577464\pi\)
\(19\)	27.9917	+	77.9453i	0.337986	+	0.941151i
							\(23\)	−26.1337	+	71.8017i	−0.236924	+	0.650943i	0.763065	+	0.646321i	\(0.223693\pi\)
−0.999989	+	0.00462191i	\(0.998529\pi\)
\(29\)	−48.6924	−	276.148i	−0.311791	−	1.76826i	−0.589675	−	0.807641i	\(-0.700744\pi\)
							0.277884	−	0.960615i	\(-0.410367\pi\)
\(31\)	199.067	−	114.931i	1.15334	−	0.665879i	0.203638	−	0.979046i	\(-0.434723\pi\)
							0.949698	+	0.313167i	\(0.101390\pi\)
\(37\)		−	150.045i		−	0.666682i	−0.942806	−	0.333341i	\(-0.891824\pi\)
							0.942806	−	0.333341i	\(-0.108176\pi\)
\(41\)	−170.197	−	142.812i	−0.648298	−	0.543987i	0.258256	−	0.966077i	\(-0.416852\pi\)
							−0.906554	+	0.422090i	\(0.861297\pi\)
\(43\)	−71.9175	+	26.1758i	−0.255054	+	0.0928321i	−0.466383	−	0.884583i	\(-0.654443\pi\)
							0.211329	+	0.977415i	\(0.432221\pi\)
\(47\)	499.385	−	88.0551i	1.54985	−	0.273280i	0.667764	−	0.744373i	\(-0.267251\pi\)
							0.882084	+	0.471093i	\(0.156140\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.l.b.59.6	yes	60
3.2	odd	2		114.4.l.a.59.9	yes	60
19.10	odd	18		114.4.l.a.29.9	✓	60
57.29	even	18	inner	114.4.l.b.29.6	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.l.a.29.9	✓	60	19.10	odd	18
114.4.l.a.59.9	yes	60	3.2	odd	2
114.4.l.b.29.6	yes	60	57.29	even	18	inner
114.4.l.b.59.6	yes	60	1.1	even	1	trivial