Embedded newform 57.3.k.b.13.1

• Label: 57.3.k.b.13.1
• 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			13.1
Character	\(\chi\)	\(=\)	57.13
Dual form			57.3.k.b.22.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.42904 - 2.89482i) q^{2} +(-0.592396 + 1.62760i) q^{3} +(-1.78514 + 10.1240i) q^{4} +(0.487525 + 2.76489i) q^{5} +(6.15055 - 2.23862i) q^{6} +(-3.07210 + 5.32104i) q^{7} +(20.5529 - 11.8662i) q^{8} +(-2.29813 - 1.92836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.42904	−	2.89482i	−1.21452	−	1.44741i	−0.858408	−	0.512967i	\(-0.828546\pi\)
							−0.356113	−	0.934443i	\(-0.615898\pi\)
\(3\)	−0.592396	+	1.62760i	−0.197465	+	0.542532i
							\(5\)	0.487525	+	2.76489i	0.0975049	+	0.552978i	0.993951	+	0.109825i	\(0.0350290\pi\)
−0.896446	+	0.443153i	\(0.853860\pi\)
\(7\)	−3.07210	+	5.32104i	−0.438872	+	0.760149i	−0.997603	−	0.0692005i	\(-0.977955\pi\)
							0.558731	+	0.829349i	\(0.311288\pi\)
\(11\)	−0.536297	−	0.928893i	−0.0487543	−	0.0844449i	0.840618	−	0.541628i	\(-0.182192\pi\)
							−0.889373	+	0.457183i	\(0.848858\pi\)
\(13\)	7.47451	+	20.5361i	0.574962	+	1.57970i	0.796561	+	0.604558i	\(0.206650\pi\)
							−0.221599	+	0.975138i	\(0.571127\pi\)
\(17\)	−25.6656	+	21.5360i	−1.50974	+	1.26682i	−0.645478	+	0.763778i	\(0.723342\pi\)
							−0.864261	+	0.503043i	\(0.832214\pi\)
\(19\)	6.42772	−	17.8797i	0.338301	−	0.941038i
							\(23\)	−0.258286	+	1.46481i	−0.0112298	+	0.0636876i	−0.989908	−	0.141714i	\(-0.954739\pi\)
0.978678	+	0.205401i	\(0.0658499\pi\)
\(29\)	−1.42531	+	1.69862i	−0.0491486	+	0.0585730i	−0.790059	−	0.613031i	\(-0.789950\pi\)
							0.740910	+	0.671604i	\(0.234394\pi\)
\(31\)	−3.13721	−	1.81127i	−0.101200	−	0.0584279i	0.448546	−	0.893760i	\(-0.351942\pi\)
							−0.549746	+	0.835332i	\(0.685275\pi\)
\(37\)		−	5.05759i		−	0.136692i	−0.997662	−	0.0683458i	\(-0.978228\pi\)
							0.997662	−	0.0683458i	\(-0.0217721\pi\)
\(41\)	9.14465	−	25.1247i	0.223040	−	0.612798i	−0.776816	−	0.629727i	\(-0.783167\pi\)
							0.999857	+	0.0169290i	\(0.00538894\pi\)
\(43\)	−6.27939	−	35.6122i	−0.146032	−	0.828190i	−0.966533	−	0.256544i	\(-0.917416\pi\)
							0.820500	−	0.571646i	\(-0.193695\pi\)
\(47\)	21.1311	+	17.7311i	0.449597	+	0.377257i	0.839286	−	0.543690i	\(-0.182973\pi\)
							−0.389689	+	0.920946i	\(0.627418\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.3.k.b.13.1	✓	24
3.2	odd	2		171.3.ba.d.127.4		24
19.3	odd	18	inner	57.3.k.b.22.1	yes	24
57.41	even	18		171.3.ba.d.136.4		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.k.b.13.1	✓	24	1.1	even	1	trivial
57.3.k.b.22.1	yes	24	19.3	odd	18	inner
171.3.ba.d.127.4		24	3.2	odd	2
171.3.ba.d.136.4		24	57.41	even	18