Embedded newform 23.13.d.a.5.14

• Label: 23.13.d.a.5.14
• 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			5.14
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.1031 + 4.72830i) q^{2} +(-479.913 + 553.849i) q^{3} +(-3208.82 - 2062.18i) q^{4} +(-13434.2 - 1931.55i) q^{5} +(-10346.9 + 6649.52i) q^{6} +(-82122.5 + 37504.1i) q^{7} +(-86938.4 - 100332. i) q^{8} +(-800.303 - 5566.23i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	16.1031	+	4.72830i	0.251611	+	0.0738797i	0.405106	−	0.914270i	\(-0.367235\pi\)
							−0.153495	+	0.988149i	\(0.549053\pi\)
\(3\)	−479.913	+	553.849i	−0.658316	+	0.759738i	−0.982501	−	0.186256i	\(-0.940365\pi\)
							0.324185	+	0.945994i	\(0.394910\pi\)
\(5\)	−13434.2	−	1931.55i	−0.859792	−	0.123619i	−0.301704	−	0.953402i	\(-0.597556\pi\)
							−0.558088	+	0.829782i	\(0.688465\pi\)
\(7\)	−82122.5	+	37504.1i	−0.698030	+	0.318780i	−0.732635	−	0.680621i	\(-0.761710\pi\)
							0.0346052	+	0.999401i	\(0.488983\pi\)
\(11\)	−418247.	−	1.42442e6i	−0.236089	−	0.804047i	−0.989253	−	0.146214i	\(-0.953291\pi\)
							0.753164	−	0.657833i	\(-0.228527\pi\)
\(13\)	49455.1	−	108292.i	0.0102459	−	0.0224355i	−0.904440	−	0.426601i	\(-0.859711\pi\)
							0.914686	+	0.404166i	\(0.132438\pi\)
\(17\)	8.81487e6	+	1.37162e7i	0.365193	+	0.568251i	0.974417	−	0.224747i	\(-0.0721555\pi\)
							−0.609224	+	0.792998i	\(0.708519\pi\)
\(19\)	−6.20796e6	+	9.65978e6i	−0.131955	+	0.205327i	−0.900943	−	0.433937i	\(-0.857124\pi\)
							0.768988	+	0.639263i	\(0.220761\pi\)
\(23\)	1.46109e8	+	2.38046e7i	0.986987	+	0.160803i
							\(29\)	6.31757e8	−	4.06006e8i	1.06209	−	0.682565i	0.111739	−	0.993738i	\(-0.464358\pi\)
0.950353	+	0.311173i	\(0.100722\pi\)
\(31\)	4.40023e8	+	5.07813e8i	0.495798	+	0.572182i	0.947406	−	0.320035i	\(-0.103695\pi\)
							−0.451607	+	0.892217i	\(0.649149\pi\)
\(37\)	1.55924e9	−	2.24184e8i	0.607718	−	0.0873766i	0.168416	−	0.985716i	\(-0.446135\pi\)
							0.439302	+	0.898339i	\(0.355226\pi\)
\(41\)	6.45317e8	−	4.48828e9i	0.135853	−	0.944880i	−0.801870	−	0.597498i	\(-0.796162\pi\)
							0.937724	−	0.347382i	\(-0.112929\pi\)
\(43\)	−6.73814e9	−	5.83863e9i	−1.06593	−	0.923635i	−0.0686741	−	0.997639i	\(-0.521877\pi\)
							−0.997258	+	0.0740037i	\(0.976422\pi\)
\(47\)	−6.19348e8			−0.0574576			−0.0287288	−	0.999587i	\(-0.509146\pi\)
							−0.0287288	+	0.999587i	\(0.509146\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.14	✓	230
23.14	odd	22	inner	23.13.d.a.14.14	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.14	✓	230	1.1	even	1	trivial
23.13.d.a.14.14	yes	230	23.14	odd	22	inner