Embedded newform 57.3.k.b.10.4

• Label: 57.3.k.b.10.4
• Level: $57$
• Weight: $3$
• Character: 57.10
• Analytic conductor: $1.553$
• Analytic rank: $0$
• Dimension: $24$
• 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			10.4
Character	\(\chi\)	\(=\)	57.10
Dual form			57.3.k.b.40.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.87872 - 0.507596i) q^{2} +(-1.11334 - 1.32683i) q^{3} +(4.27061 - 1.55438i) q^{4} +(1.04130 + 0.379004i) q^{5} +(-3.87849 - 3.25444i) q^{6} +(-1.13703 + 1.96939i) q^{7} +(1.37889 - 0.796102i) q^{8} +(-0.520945 + 2.95442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.87872	−	0.507596i	1.43936	−	0.253798i	0.601146	−	0.799139i	\(-0.294711\pi\)
							0.838214	+	0.545341i	\(0.183600\pi\)
\(3\)	−1.11334	−	1.32683i	−0.371114	−	0.442276i
							\(5\)	1.04130	+	0.379004i	0.208261	+	0.0758007i	0.444044	−	0.896005i	\(-0.353543\pi\)
−0.235783	+	0.971806i	\(0.575766\pi\)
\(7\)	−1.13703	+	1.96939i	−0.162433	+	0.281341i	−0.935741	−	0.352689i	\(-0.885267\pi\)
							0.773308	+	0.634031i	\(0.218601\pi\)
\(11\)	−0.705093	−	1.22126i	−0.0640994	−	0.111023i	0.832195	−	0.554483i	\(-0.187084\pi\)
							−0.896294	+	0.443460i	\(0.853751\pi\)
\(13\)	−6.19810	+	7.38661i	−0.476777	+	0.568201i	−0.949803	−	0.312848i	\(-0.898717\pi\)
							0.473026	+	0.881048i	\(0.343162\pi\)
\(17\)	0.749300	+	4.24949i	0.0440765	+	0.249970i	0.998883	−	0.0472589i	\(-0.0150486\pi\)
							−0.954806	+	0.297229i	\(0.903937\pi\)
\(19\)	0.262905	−	18.9982i	0.0138371	−	0.999904i
							\(23\)	34.8335	−	12.6784i	1.51450	−	0.551233i	0.554732	−	0.832029i	\(-0.312821\pi\)
0.959767	+	0.280796i	\(0.0905985\pi\)
\(29\)	41.2012	+	7.26488i	1.42073	+	0.250513i	0.830633	−	0.556820i	\(-0.187979\pi\)
							0.590096	+	0.807333i	\(0.299090\pi\)
\(31\)	−44.1867	−	25.5112i	−1.42538	−	0.822942i	−0.428625	−	0.903482i	\(-0.641002\pi\)
							−0.996751	+	0.0805407i	\(0.974335\pi\)
\(37\)			1.97816i			0.0534638i	0.999643	+	0.0267319i	\(0.00851005\pi\)
							−0.999643	+	0.0267319i	\(0.991490\pi\)
\(41\)	0.341304	+	0.406751i	0.00832450	+	0.00992075i	0.770191	−	0.637814i	\(-0.220161\pi\)
							−0.761866	+	0.647735i	\(0.775717\pi\)
\(43\)	39.8336	+	14.4983i	0.926364	+	0.337169i	0.760767	−	0.649025i	\(-0.224823\pi\)
							0.165597	+	0.986194i	\(0.447045\pi\)
\(47\)	11.2771	−	63.9557i	0.239939	−	1.36076i	−0.592020	−	0.805923i	\(-0.701670\pi\)
							0.831959	−	0.554837i	\(-0.187219\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.b.10.4	✓	24
3.2	odd	2		171.3.ba.d.10.1		24
19.2	odd	18	inner	57.3.k.b.40.4	yes	24
57.2	even	18		171.3.ba.d.154.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.b.10.4	✓	24	1.1	even	1	trivial
57.3.k.b.40.4	yes	24	19.2	odd	18	inner
171.3.ba.d.10.1		24	3.2	odd	2
171.3.ba.d.154.1		24	57.2	even	18