Embedded newform 57.3.k.b.13.2

• Label: 57.3.k.b.13.2
• Level: $57$
• Weight: $3$
• Character: 57.13
• Analytic conductor: $1.553$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	57.k (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.55313750685\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			13.2
Character	\(\chi\)	\(=\)	57.13
Dual form			57.3.k.b.22.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.01678 - 1.21175i) q^{2} +(-0.592396 + 1.62760i) q^{3} +(0.260091 - 1.47505i) q^{4} +(-0.649199 - 3.68179i) q^{5} +(2.57458 - 0.937072i) q^{6} +(6.08998 - 10.5482i) q^{7} +(-7.53148 + 4.34830i) q^{8} +(-2.29813 - 1.92836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.01678	−	1.21175i	−0.508391	−	0.605877i	0.449404	−	0.893329i	\(-0.351636\pi\)
							−0.957795	+	0.287452i	\(0.907192\pi\)
\(3\)	−0.592396	+	1.62760i	−0.197465	+	0.542532i
							\(5\)	−0.649199	−	3.68179i	−0.129840	−	0.736358i	−0.978315	−	0.207123i	\(-0.933590\pi\)
0.848475	−	0.529235i	\(-0.177521\pi\)
\(7\)	6.08998	−	10.5482i	0.869997	−	1.50688i	0.00799842	−	0.999968i	\(-0.497454\pi\)
							0.861998	−	0.506911i	\(-0.169213\pi\)
\(11\)	−1.24952	−	2.16423i	−0.113592	−	0.196748i	0.803624	−	0.595138i	\(-0.202902\pi\)
							−0.917216	+	0.398390i	\(0.869569\pi\)
\(13\)	3.75968	+	10.3296i	0.289206	+	0.794588i	0.996178	+	0.0873460i	\(0.0278386\pi\)
							−0.706972	+	0.707242i	\(0.749939\pi\)
\(17\)	9.80142	−	8.22437i	0.576554	−	0.483787i	−0.307259	−	0.951626i	\(-0.599412\pi\)
							0.883814	+	0.467839i	\(0.154967\pi\)
\(19\)	7.95994	+	17.2522i	0.418944	+	0.908012i
							\(23\)	−5.56864	+	31.5813i	−0.242115	+	1.37310i	0.584985	+	0.811044i	\(0.301100\pi\)
−0.827099	+	0.562056i	\(0.810011\pi\)
\(29\)	12.6273	−	15.0486i	0.435423	−	0.518917i	−0.503056	−	0.864254i	\(-0.667791\pi\)
							0.938479	+	0.345337i	\(0.112235\pi\)
\(31\)	−36.5955	−	21.1284i	−1.18050	−	0.681562i	−0.224370	−	0.974504i	\(-0.572032\pi\)
							−0.956130	+	0.292942i	\(0.905366\pi\)
\(37\)			9.59066i			0.259207i	0.991566	+	0.129604i	\(0.0413705\pi\)
							−0.991566	+	0.129604i	\(0.958630\pi\)
\(41\)	22.3264	−	61.3412i	0.544546	−	1.49613i	−0.296431	−	0.955054i	\(-0.595796\pi\)
							0.840976	−	0.541072i	\(-0.181981\pi\)
\(43\)	5.64680	+	32.0246i	0.131321	+	0.744758i	0.977351	+	0.211623i	\(0.0678750\pi\)
							−0.846030	+	0.533135i	\(0.821014\pi\)
\(47\)	53.8584	+	45.1926i	1.14592	+	0.961544i	0.999616	−	0.0276954i	\(-0.00881685\pi\)
							0.146307	+	0.989239i	\(0.453261\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.b.13.2	✓	24
3.2	odd	2		171.3.ba.d.127.3		24
19.3	odd	18	inner	57.3.k.b.22.2	yes	24
57.41	even	18		171.3.ba.d.136.3		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.b.13.2	✓	24	1.1	even	1	trivial
57.3.k.b.22.2	yes	24	19.3	odd	18	inner
171.3.ba.d.127.3		24	3.2	odd	2
171.3.ba.d.136.3		24	57.41	even	18