Embedded newform 57.2.j.b.2.4

• Label: 57.2.j.b.2.4
• Level: $57$
• Weight: $2$
• Character: 57.2
• Analytic conductor: $0.455$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	57.j (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.455147291521\)
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			2.4
Character	\(\chi\)	\(=\)	57.2
Dual form			57.2.j.b.29.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.448036 - 2.54094i) q^{2} +(0.504201 + 1.65704i) q^{3} +(-4.37626 - 1.59283i) q^{4} +(-0.533860 - 1.46677i) q^{5} +(4.43634 - 0.538732i) q^{6} +(1.49687 + 2.59266i) q^{7} +(-3.42787 + 5.93724i) q^{8} +(-2.49156 + 1.67096i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 9 q^{3} - 18 q^{4} - 9 q^{6} - 6 q^{7} + 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.448036	−	2.54094i	0.316810	−	1.79672i	−0.245078	−	0.969503i	\(-0.578814\pi\)
							0.561888	−	0.827213i	\(-0.310075\pi\)
\(3\)	0.504201	+	1.65704i	0.291101	+	0.956692i
							\(5\)	−0.533860	−	1.46677i	−0.238750	−	0.655959i	−0.999972	−	0.00746821i	\(-0.997623\pi\)
0.761223	−	0.648491i	\(-0.224599\pi\)
\(7\)	1.49687	+	2.59266i	0.565765	+	0.979934i	0.996978	+	0.0776837i	\(0.0247524\pi\)
							−0.431213	+	0.902250i	\(0.641914\pi\)
\(11\)	1.05900	+	0.611412i	0.319299	+	0.184348i	0.651080	−	0.759009i	\(-0.274316\pi\)
							−0.331781	+	0.943356i	\(0.607649\pi\)
\(13\)	0.203530	+	0.242558i	0.0564491	+	0.0672734i	0.793530	−	0.608531i	\(-0.208241\pi\)
							−0.737081	+	0.675804i	\(0.763796\pi\)
\(17\)	−5.00468	−	0.882460i	−1.21381	−	0.214028i	−0.470152	−	0.882585i	\(-0.655801\pi\)
							−0.743660	+	0.668558i	\(0.766912\pi\)
\(19\)	−0.461268	−	4.33442i	−0.105822	−	0.994385i
							\(23\)	0.211619	−	0.581418i	0.0441256	−	0.121234i	−0.915673	−	0.401925i	\(-0.868341\pi\)
0.959798	+	0.280691i	\(0.0905635\pi\)
\(29\)	0.204862	+	1.16183i	0.0380420	+	0.215747i	0.997903	−	0.0647299i	\(-0.0206186\pi\)
							−0.959861	+	0.280477i	\(0.909507\pi\)
\(31\)	−4.50103	+	2.59867i	−0.808409	+	0.466735i	−0.846403	−	0.532543i	\(-0.821237\pi\)
							0.0379939	+	0.999278i	\(0.487903\pi\)
\(37\)			3.64847i			0.599805i	0.953970	+	0.299902i	\(0.0969541\pi\)
							−0.953970	+	0.299902i	\(0.903046\pi\)
\(41\)	0.118123	+	0.0991172i	0.0184477	+	0.0154795i	0.651965	−	0.758249i	\(-0.273945\pi\)
							−0.633517	+	0.773729i	\(0.718389\pi\)
\(43\)	8.99072	−	3.27235i	1.37107	−	0.499029i	0.451613	−	0.892214i	\(-0.350849\pi\)
							0.919460	+	0.393185i	\(0.128627\pi\)
\(47\)	8.03246	−	1.41634i	1.17165	−	0.206594i	0.446244	−	0.894911i	\(-0.352761\pi\)
							0.725410	+	0.688317i	\(0.241650\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.2.j.b.2.4	yes	24
3.2	odd	2	inner	57.2.j.b.2.1	✓	24
4.3	odd	2		912.2.cc.e.401.2		24
12.11	even	2		912.2.cc.e.401.4		24
19.3	odd	18		1083.2.d.d.1082.2		24
19.10	odd	18	inner	57.2.j.b.29.1	yes	24
19.16	even	9		1083.2.d.d.1082.24		24
57.29	even	18	inner	57.2.j.b.29.4	yes	24
57.35	odd	18		1083.2.d.d.1082.1		24
57.41	even	18		1083.2.d.d.1082.23		24
76.67	even	18		912.2.cc.e.257.4		24
228.143	odd	18		912.2.cc.e.257.2		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.j.b.2.1	✓	24	3.2	odd	2	inner
57.2.j.b.2.4	yes	24	1.1	even	1	trivial
57.2.j.b.29.1	yes	24	19.10	odd	18	inner
57.2.j.b.29.4	yes	24	57.29	even	18	inner
912.2.cc.e.257.2		24	228.143	odd	18
912.2.cc.e.257.4		24	76.67	even	18
912.2.cc.e.401.2		24	4.3	odd	2
912.2.cc.e.401.4		24	12.11	even	2
1083.2.d.d.1082.1		24	57.35	odd	18
1083.2.d.d.1082.2		24	19.3	odd	18
1083.2.d.d.1082.23		24	57.41	even	18
1083.2.d.d.1082.24		24	19.16	even	9