Embedded newform 23.13.d.a.5.9

• Label: 23.13.d.a.5.9
• 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			5.9
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-35.7422 - 10.4949i) q^{2} +(347.101 - 400.576i) q^{3} +(-2278.41 - 1464.25i) q^{4} +(28946.6 + 4161.89i) q^{5} +(-16610.1 + 10674.7i) q^{6} +(-39526.9 + 18051.3i) q^{7} +(165987. + 191560. i) q^{8} +(35650.0 + 247951. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−35.7422	−	10.4949i	−0.558472	−	0.163982i	−0.00969798	−	0.999953i	\(-0.503087\pi\)
							−0.548774	+	0.835971i	\(0.684905\pi\)
\(3\)	347.101	−	400.576i	0.476133	−	0.549487i	−0.465974	−	0.884798i	\(-0.654296\pi\)
							0.942107	+	0.335312i	\(0.108842\pi\)
\(5\)	28946.6	+	4161.89i	1.85258	+	0.266361i	0.976455	−	0.215721i	\(-0.0692101\pi\)
							0.876126	+	0.482082i	\(0.160119\pi\)
\(7\)	−39526.9	+	18051.3i	−0.335973	+	0.153434i	−0.576256	−	0.817269i	\(-0.695487\pi\)
							0.240283	+	0.970703i	\(0.422760\pi\)
\(11\)	85735.5	+	291988.i	0.0483954	+	0.164820i	0.980145	−	0.198282i	\(-0.0635361\pi\)
							−0.931750	+	0.363101i	\(0.881718\pi\)
\(13\)	−1.43367e6	+	3.13929e6i	−0.297021	+	0.650386i	−0.998028	−	0.0627686i	\(-0.980007\pi\)
							0.701007	+	0.713155i	\(0.252734\pi\)
\(17\)	223827.	+	348281.i	0.00927297	+	0.0144290i	0.845859	−	0.533406i	\(-0.179088\pi\)
							−0.836586	+	0.547835i	\(0.815452\pi\)
\(19\)	−3.11133e7	+	4.84132e7i	−0.661339	+	1.02906i	0.334886	+	0.942259i	\(0.391302\pi\)
							−0.996225	+	0.0868051i	\(0.972334\pi\)
\(23\)	1.48030e8	−	1.26592e6i	0.999963	−	0.00855145i
							\(29\)	4.08855e8	−	2.62755e8i	0.687356	−	0.441737i	−0.149789	−	0.988718i	\(-0.547859\pi\)
0.837145	+	0.546981i	\(0.184223\pi\)
\(31\)	6.01425e8	+	6.94081e8i	0.677659	+	0.782060i	0.985554	−	0.169360i	\(-0.0541701\pi\)
							−0.307895	+	0.951420i	\(0.599625\pi\)
\(37\)	−1.31397e8	+	1.88920e7i	−0.0512124	+	0.00736323i	−0.167873	−	0.985809i	\(-0.553690\pi\)
							0.116661	+	0.993172i	\(0.462781\pi\)
\(41\)	6.52757e8	−	4.54003e9i	0.137420	−	0.955774i	−0.798106	−	0.602516i	\(-0.794165\pi\)
							0.935526	−	0.353258i	\(-0.114926\pi\)
\(43\)	−3.43965e9	−	2.98047e9i	−0.544131	−	0.471492i	0.338890	−	0.940826i	\(-0.389949\pi\)
							−0.883021	+	0.469334i	\(0.844494\pi\)
\(47\)	−1.00774e10			−0.934895			−0.467448	−	0.884021i	\(-0.654826\pi\)
							−0.467448	+	0.884021i	\(0.654826\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.9	✓	230
23.14	odd	22	inner	23.13.d.a.14.9	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.9	✓	230	1.1	even	1	trivial
23.13.d.a.14.9	yes	230	23.14	odd	22	inner