Embedded newform 57.3.k.b.40.4

• Label: 57.3.k.b.40.4
• Level: $57$
• Weight: $3$
• Character: 57.40
• 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			40.4
Character	\(\chi\)	\(=\)	57.40
Dual form			57.3.k.b.10.4

$q$-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 \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.b.10.4	✓	24	19.10	odd	18	inner
57.3.k.b.40.4	yes	24	1.1	even	1	trivial
171.3.ba.d.10.1		24	57.29	even	18
171.3.ba.d.154.1		24	3.2	odd	2