Embedded newform 57.3.k.b.10.2

• Label: 57.3.k.b.10.2
• Level: $57$
• Weight: $3$
• Character: 57.10
• 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			10.2
Character	\(\chi\)	\(=\)	57.10
Dual form			57.3.k.b.40.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68828 + 0.297689i) q^{2} +(-1.11334 - 1.32683i) q^{3} +(-0.997107 + 0.362917i) q^{4} +(8.34658 + 3.03791i) q^{5} +(2.27461 + 1.90863i) q^{6} +(5.55292 - 9.61794i) q^{7} +(7.51394 - 4.33818i) q^{8} +(-0.520945 + 2.95442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{17}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68828	+	0.297689i	−0.844139	+	0.148844i	−0.578961	−	0.815355i	\(-0.696542\pi\)
							−0.265178	+	0.964200i	\(0.585431\pi\)
\(3\)	−1.11334	−	1.32683i	−0.371114	−	0.442276i
							\(5\)	8.34658	+	3.03791i	1.66932	+	0.607581i	0.991785	−	0.127913i	\(-0.0408280\pi\)
0.677530	+	0.735495i	\(0.263050\pi\)
\(7\)	5.55292	−	9.61794i	0.793275	−	1.37399i	−0.130654	−	0.991428i	\(-0.541708\pi\)
							0.923929	−	0.382564i	\(-0.124959\pi\)
\(11\)	4.39522	+	7.61274i	0.399565	+	0.692068i	0.993672	−	0.112319i	\(-0.0358277\pi\)
							−0.594107	+	0.804386i	\(0.702494\pi\)
\(13\)	−3.70161	+	4.41141i	−0.284739	+	0.339339i	−0.889388	−	0.457153i	\(-0.848869\pi\)
							0.604649	+	0.796492i	\(0.293314\pi\)
\(17\)	0.256446	+	1.45438i	0.0150850	+	0.0855515i	0.991421	−	0.130709i	\(-0.0417252\pi\)
							−0.976336	+	0.216260i	\(0.930614\pi\)
\(19\)	−6.57170	−	17.8273i	−0.345879	−	0.938279i
							\(23\)	9.77800	−	3.55890i	0.425130	−	0.154735i	−0.120590	−	0.992702i	\(-0.538478\pi\)
0.545720	+	0.837968i	\(0.316256\pi\)
\(29\)	−43.0983	−	7.59939i	−1.48615	−	0.262048i	−0.629116	−	0.777311i	\(-0.716583\pi\)
							−0.857032	+	0.515263i	\(0.827694\pi\)
\(31\)	7.87913	+	4.54902i	0.254166	+	0.146743i	0.621670	−	0.783279i	\(-0.286454\pi\)
							−0.367505	+	0.930022i	\(0.619788\pi\)
\(37\)			14.2554i			0.385281i	0.981269	+	0.192640i	\(0.0617051\pi\)
							−0.981269	+	0.192640i	\(0.938295\pi\)
\(41\)	10.3704	+	12.3590i	0.252936	+	0.301438i	0.877539	−	0.479505i	\(-0.159184\pi\)
							−0.624603	+	0.780942i	\(0.714739\pi\)
\(43\)	−56.1217	−	20.4266i	−1.30516	−	0.475038i	−0.406484	−	0.913658i	\(-0.633245\pi\)
							−0.898673	+	0.438620i	\(0.855467\pi\)
\(47\)	−12.9389	+	73.3799i	−0.275295	+	1.56127i	0.462729	+	0.886500i	\(0.346870\pi\)
							−0.738024	+	0.674775i	\(0.764241\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.b.10.2	✓	24
3.2	odd	2		171.3.ba.d.10.3		24
19.2	odd	18	inner	57.3.k.b.40.2	yes	24
57.2	even	18		171.3.ba.d.154.3		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.b.10.2	✓	24	1.1	even	1	trivial
57.3.k.b.40.2	yes	24	19.2	odd	18	inner
171.3.ba.d.10.3		24	3.2	odd	2
171.3.ba.d.154.3		24	57.2	even	18