Embedded newform 57.4.i.a.4.4

• Label: 57.4.i.a.4.4
• Level: $57$
• Weight: $4$
• Character: 57.4
• Analytic conductor: $3.363$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	57.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.36310887033\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.4
Character	\(\chi\)	\(=\)	57.4
Dual form			57.4.i.a.43.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.04549 + 1.47244i) q^{2} +(0.520945 + 2.95442i) q^{3} +(8.06959 + 6.77119i) q^{4} +(1.37796 - 1.15625i) q^{5} +(-2.24273 + 12.7192i) q^{6} +(0.604600 - 1.04720i) q^{7} +(5.45480 + 9.44800i) q^{8} +(-8.45723 + 3.07818i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 3 q^{2} + 9 q^{4} - 12 q^{5} - 9 q^{6} + 36 q^{7} + 57 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.04549	+	1.47244i	1.43030	+	0.520586i	0.937017	−	0.349284i	\(-0.113575\pi\)
							0.493281	+	0.869870i	\(0.335797\pi\)
\(3\)	0.520945	+	2.95442i	0.100256	+	0.568579i
							\(5\)	1.37796	−	1.15625i	0.123249	−	0.103418i	−0.579080	−	0.815271i	\(-0.696588\pi\)
0.702329	+	0.711853i	\(0.252144\pi\)
\(7\)	0.604600	−	1.04720i	0.0326453	−	0.0565434i	−0.849241	−	0.528005i	\(-0.822940\pi\)
							0.881886	+	0.471462i	\(0.156273\pi\)
\(11\)	−1.72922	−	2.99509i	−0.0473980	−	0.0820958i	0.841353	−	0.540486i	\(-0.181760\pi\)
							−0.888751	+	0.458390i	\(0.848426\pi\)
\(13\)	3.37724	−	19.1533i	0.0720521	−	0.408628i	−0.927355	−	0.374184i	\(-0.877923\pi\)
							0.999407	−	0.0344438i	\(-0.0109660\pi\)
\(17\)	15.7527	+	5.73350i	0.224740	+	0.0817987i	0.451936	−	0.892050i	\(-0.350734\pi\)
							−0.227196	+	0.973849i	\(0.572956\pi\)
\(19\)	−27.6768	−	78.0576i	−0.334184	−	0.942508i
							\(23\)	2.34578	+	1.96834i	0.0212664	+	0.0178447i	0.653359	−	0.757048i	\(-0.273359\pi\)
−0.632092	+	0.774893i	\(0.717804\pi\)
\(29\)	116.445	−	42.3826i	0.745632	−	0.271388i	0.0588653	−	0.998266i	\(-0.481252\pi\)
							0.686766	+	0.726878i	\(0.259030\pi\)
\(31\)	−118.107	+	204.567i	−0.684278	+	1.18520i	0.289385	+	0.957213i	\(0.406549\pi\)
							−0.973663	+	0.227992i	\(0.926784\pi\)
\(37\)	−234.992			−1.04412			−0.522059	−	0.852909i	\(-0.674836\pi\)
							−0.522059	+	0.852909i	\(0.674836\pi\)
\(41\)	−7.52042	−	42.6504i	−0.0286462	−	0.162460i	0.967129	−	0.254287i	\(-0.0818407\pi\)
							−0.995775	+	0.0918263i	\(0.970730\pi\)
\(43\)	−283.973	+	238.282i	−1.00710	+	0.845060i	−0.987953	−	0.154756i	\(-0.950541\pi\)
							−0.0191511	+	0.999817i	\(0.506096\pi\)
\(47\)	49.2800	−	17.9364i	0.152941	−	0.0556660i	−0.264415	−	0.964409i	\(-0.585179\pi\)
							0.417356	+	0.908743i	\(0.362957\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.4.i.a.4.4	✓	24
3.2	odd	2		171.4.u.a.118.1		24
19.5	even	9	inner	57.4.i.a.43.4	yes	24
19.9	even	9		1083.4.a.p.1.2		12
19.10	odd	18		1083.4.a.o.1.11		12
57.5	odd	18		171.4.u.a.100.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.4.i.a.4.4	✓	24	1.1	even	1	trivial
57.4.i.a.43.4	yes	24	19.5	even	9	inner
171.4.u.a.100.1		24	57.5	odd	18
171.4.u.a.118.1		24	3.2	odd	2
1083.4.a.o.1.11		12	19.10	odd	18
1083.4.a.p.1.2		12	19.9	even	9