Embedded newform 57.3.k.b.22.4

• Label: 57.3.k.b.22.4
• Level: $57$
• Weight: $3$
• Character: 57.22
• 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			22.4
Character	\(\chi\)	\(=\)	57.22
Dual form			57.3.k.b.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.13460 - 2.54392i) q^{2} +(-0.592396 - 1.62760i) q^{3} +(-1.22041 - 6.92131i) q^{4} +(-0.870886 + 4.93904i) q^{5} +(-5.40501 - 1.96726i) q^{6} +(2.46536 + 4.27012i) q^{7} +(-8.70859 - 5.02791i) q^{8} +(-2.29813 + 1.92836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.13460	−	2.54392i	1.06730	−	1.27196i	0.106623	−	0.994299i	\(-0.465996\pi\)
							0.960679	−	0.277662i	\(-0.0895595\pi\)
\(3\)	−0.592396	−	1.62760i	−0.197465	−	0.542532i
							\(5\)	−0.870886	+	4.93904i	−0.174177	+	0.987808i	0.764912	+	0.644135i	\(0.222782\pi\)
−0.939089	+	0.343673i	\(0.888329\pi\)
\(7\)	2.46536	+	4.27012i	0.352194	+	0.610018i	0.986634	−	0.162955i	\(-0.0521024\pi\)
							−0.634440	+	0.772972i	\(0.718769\pi\)
\(11\)	7.89760	−	13.6790i	0.717964	−	1.24355i	−0.243842	−	0.969815i	\(-0.578408\pi\)
							0.961805	−	0.273735i	\(-0.0882590\pi\)
\(13\)	−4.52977	+	12.4455i	−0.348444	+	0.957342i	0.634416	+	0.772992i	\(0.281240\pi\)
							−0.982860	+	0.184351i	\(0.940982\pi\)
\(17\)	−18.3243	−	15.3759i	−1.07790	−	0.904464i	−0.0821539	−	0.996620i	\(-0.526180\pi\)
							−0.995745	+	0.0921554i	\(0.970624\pi\)
\(19\)	−11.5559	+	15.0818i	−0.608207	+	0.793778i
							\(23\)	4.49694	+	25.5034i	0.195519	+	1.10884i	0.911678	+	0.410906i	\(0.134788\pi\)
−0.716159	+	0.697938i	\(0.754101\pi\)
\(29\)	−9.60989	−	11.4526i	−0.331376	−	0.394918i	0.574470	−	0.818525i	\(-0.305208\pi\)
							−0.905846	+	0.423607i	\(0.860764\pi\)
\(31\)	−32.4354	+	18.7266i	−1.04630	+	0.604084i	−0.921613	−	0.388111i	\(-0.873128\pi\)
							−0.124692	+	0.992195i	\(0.539794\pi\)
\(37\)			22.7831i			0.615760i	0.951425	+	0.307880i	\(0.0996195\pi\)
							−0.951425	+	0.307880i	\(0.900381\pi\)
\(41\)	−20.3362	−	55.8732i	−0.496005	−	1.36276i	−0.895105	−	0.445855i	\(-0.852900\pi\)
							0.399101	−	0.916907i	\(-0.369322\pi\)
\(43\)	9.77397	−	55.4310i	0.227302	−	1.28909i	−0.630934	−	0.775836i	\(-0.717328\pi\)
							0.858236	−	0.513256i	\(-0.171561\pi\)
\(47\)	58.7306	−	49.2808i	1.24959	−	1.04853i	0.252875	−	0.967499i	\(-0.418624\pi\)
							0.996712	−	0.0810288i	\(-0.0258206\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.b.22.4	yes	24
3.2	odd	2		171.3.ba.d.136.1		24
19.13	odd	18	inner	57.3.k.b.13.4	✓	24
57.32	even	18		171.3.ba.d.127.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.b.13.4	✓	24	19.13	odd	18	inner
57.3.k.b.22.4	yes	24	1.1	even	1	trivial
171.3.ba.d.127.1		24	57.32	even	18
171.3.ba.d.136.1		24	3.2	odd	2