Embedded newform 23.13.d.a.5.11

• Label: 23.13.d.a.5.11
• Level: $23$
• Weight: $13$
• Character: 23.5
• Analytic conductor: $21.022$
• Analytic rank: $0$
• Dimension: $230$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.0218577974\)
Analytic rank:	\(0\)
Dimension:	\(230\)
Relative dimension:	\(23\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.11
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.4298 - 4.53059i) q^{2} +(-161.338 + 186.194i) q^{3} +(-3228.22 - 2074.65i) q^{4} +(3924.46 + 564.252i) q^{5} +(3332.97 - 2141.97i) q^{6} +(124602. - 56904.0i) q^{7} +(83546.0 + 96417.2i) q^{8} +(66993.7 + 465951. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.4298	−	4.53059i	−0.241090	−	0.0707904i	0.158955	−	0.987286i	\(-0.449187\pi\)
							−0.400045	+	0.916495i	\(0.631006\pi\)
\(3\)	−161.338	+	186.194i	−0.221314	+	0.255410i	−0.855539	−	0.517739i	\(-0.826774\pi\)
							0.634225	+	0.773149i	\(0.281319\pi\)
\(5\)	3924.46	+	564.252i	0.251165	+	0.0361121i	0.266748	−	0.963766i	\(-0.414051\pi\)
							−0.0155820	+	0.999879i	\(0.504960\pi\)
\(7\)	124602.	−	56904.0i	1.05910	−	0.483676i	0.191788	−	0.981436i	\(-0.438571\pi\)
							0.867315	+	0.497761i	\(0.165844\pi\)
\(11\)	−410023.	−	1.39641e6i	−0.231447	−	0.788237i	−0.990536	−	0.137250i	\(-0.956174\pi\)
							0.759089	−	0.650987i	\(-0.225645\pi\)
\(13\)	−1.21672e6	+	2.66424e6i	−0.252075	+	0.551967i	−0.992792	−	0.119850i	\(-0.961759\pi\)
							0.740717	+	0.671817i	\(0.234486\pi\)
\(17\)	−1.62088e7	−	2.52214e7i	−0.671518	−	1.04490i	−0.995116	−	0.0987085i	\(-0.968529\pi\)
							0.323598	−	0.946195i	\(-0.395107\pi\)
\(19\)	2.08224e7	−	3.24003e7i	0.442598	−	0.688696i	−0.546250	−	0.837622i	\(-0.683945\pi\)
							0.988848	+	0.148926i	\(0.0475816\pi\)
\(23\)	−1.46199e8	+	2.32487e7i	−0.987591	+	0.157048i
							\(29\)	−3.72776e8	+	2.39569e8i	−0.626701	+	0.402756i	−0.815087	−	0.579339i	\(-0.803311\pi\)
0.188386	+	0.982095i	\(0.439674\pi\)
\(31\)	−3.35346e8	−	3.87010e8i	−0.377853	−	0.436066i	0.534689	−	0.845049i	\(-0.320429\pi\)
							−0.912542	+	0.408983i	\(0.865883\pi\)
\(37\)	−4.73794e9	+	6.81213e8i	−1.84663	+	0.265505i	−0.974600	−	0.223952i	\(-0.928104\pi\)
							−0.872027	+	0.489457i	\(0.837195\pi\)
\(41\)	1.94388e8	−	1.35200e9i	0.0409229	−	0.284625i	−0.959076	−	0.283149i	\(-0.908621\pi\)
							0.999999	−	0.00147617i	\(-0.000469879\pi\)
\(43\)	1.73971e9	+	1.50747e9i	0.275211	+	0.238472i	0.781520	−	0.623881i	\(-0.214445\pi\)
							−0.506309	+	0.862352i	\(0.668990\pi\)
\(47\)	1.72518e10			1.60047			0.800236	−	0.599685i	\(-0.204707\pi\)
							0.800236	+	0.599685i	\(0.204707\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.11	✓	230
23.14	odd	22	inner	23.13.d.a.14.11	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.11	✓	230	1.1	even	1	trivial
23.13.d.a.14.11	yes	230	23.14	odd	22	inner