Embedded newform 57.3.k.b.13.3

• Label: 57.3.k.b.13.3
• Level: $57$
• Weight: $3$
• Character: 57.13
• 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			13.3
Character	\(\chi\)	\(=\)	57.13
Dual form			57.3.k.b.22.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.924561 + 1.10185i) q^{2} +(-0.592396 + 1.62760i) q^{3} +(0.335335 - 1.90178i) q^{4} +(1.45131 + 8.23078i) q^{5} +(-2.34107 + 0.852080i) q^{6} +(0.258464 - 0.447673i) q^{7} +(7.38814 - 4.26554i) q^{8} +(-2.29813 - 1.92836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(20\)	\(40\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.924561	+	1.10185i	0.462280	+	0.550924i	0.945944	−	0.324330i	\(-0.105139\pi\)
							−0.483664	+	0.875254i	\(0.660694\pi\)
\(3\)	−0.592396	+	1.62760i	−0.197465	+	0.542532i
							\(5\)	1.45131	+	8.23078i	0.290262	+	1.64616i	0.685861	+	0.727733i	\(0.259426\pi\)
−0.395599	+	0.918423i	\(0.629463\pi\)
\(7\)	0.258464	−	0.447673i	0.0369234	−	0.0639533i	−0.846973	−	0.531636i	\(-0.821578\pi\)
							0.883897	+	0.467682i	\(0.154911\pi\)
\(11\)	−4.89878	−	8.48493i	−0.445343	−	0.771357i	0.552733	−	0.833359i	\(-0.313585\pi\)
							−0.998076	+	0.0620013i	\(0.980252\pi\)
\(13\)	−1.34686	−	3.70046i	−0.103604	−	0.284651i	0.877050	−	0.480400i	\(-0.159508\pi\)
							−0.980654	+	0.195749i	\(0.937286\pi\)
\(17\)	16.1349	−	13.5388i	0.949113	−	0.796400i	−0.0300348	−	0.999549i	\(-0.509562\pi\)
							0.979148	+	0.203148i	\(0.0651174\pi\)
\(19\)	−2.93139	−	18.7725i	−0.154284	−	0.988027i
							\(23\)	−3.85583	+	21.8675i	−0.167645	+	0.950761i	0.778651	+	0.627458i	\(0.215904\pi\)
−0.946295	+	0.323303i	\(0.895207\pi\)
\(29\)	−9.08449	+	10.8265i	−0.313258	+	0.373327i	−0.899583	−	0.436749i	\(-0.856130\pi\)
							0.586325	+	0.810076i	\(0.300574\pi\)
\(31\)	−4.45134	−	2.56998i	−0.143592	−	0.0829026i	0.426483	−	0.904496i	\(-0.359752\pi\)
							−0.570075	+	0.821593i	\(0.693086\pi\)
\(37\)			54.7798i			1.48053i	0.672313	+	0.740267i	\(0.265301\pi\)
							−0.672313	+	0.740267i	\(0.734699\pi\)
\(41\)	−19.5194	+	53.6292i	−0.476083	+	1.30803i	0.436708	+	0.899603i	\(0.356144\pi\)
							−0.912792	+	0.408425i	\(0.866078\pi\)
\(43\)	−13.8268	−	78.4154i	−0.321552	−	1.82361i	−0.532871	−	0.846197i	\(-0.678887\pi\)
							0.211318	−	0.977417i	\(-0.432224\pi\)
\(47\)	34.7736	+	29.1786i	0.739865	+	0.620820i	0.932801	−	0.360391i	\(-0.117357\pi\)
							−0.192936	+	0.981211i	\(0.561801\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.b.13.3	✓	24
3.2	odd	2		171.3.ba.d.127.2		24
19.3	odd	18	inner	57.3.k.b.22.3	yes	24
57.41	even	18		171.3.ba.d.136.2		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.b.13.3	✓	24	1.1	even	1	trivial
57.3.k.b.22.3	yes	24	19.3	odd	18	inner
171.3.ba.d.127.2		24	3.2	odd	2
171.3.ba.d.136.2		24	57.41	even	18