Embedded newform 23.13.d.a.5.2

• Label: 23.13.d.a.5.2
• 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			5.2
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-107.972 - 31.7035i) q^{2} +(-294.207 + 339.533i) q^{3} +(7207.11 + 4631.73i) q^{4} +(15513.5 + 2230.51i) q^{5} +(42530.6 - 27332.7i) q^{6} +(176390. - 80554.6i) q^{7} +(-329484. - 380244. i) q^{8} +(46907.1 + 326246. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−107.972	−	31.7035i	−1.68707	−	0.495367i	−0.709271	−	0.704935i	\(-0.750976\pi\)
							−0.977794	+	0.209568i	\(0.932794\pi\)
\(3\)	−294.207	+	339.533i	−0.403576	+	0.465752i	−0.920764	−	0.390120i	\(-0.872433\pi\)
							0.517188	+	0.855872i	\(0.326979\pi\)
\(5\)	15513.5	+	2230.51i	0.992865	+	0.142753i	0.619554	−	0.784954i	\(-0.287313\pi\)
							0.373311	+	0.927706i	\(0.378222\pi\)
\(7\)	176390.	−	80554.6i	1.49929	−	0.684703i	0.514347	−	0.857582i	\(-0.328034\pi\)
							0.984943	+	0.172879i	\(0.0553070\pi\)
\(11\)	571650.	+	1.94686e6i	0.322681	+	1.09895i	0.947918	+	0.318516i	\(0.103184\pi\)
							−0.625236	+	0.780436i	\(0.714997\pi\)
\(13\)	1.80032e6	−	3.94215e6i	0.372984	−	0.816721i	−0.626326	−	0.779562i	\(-0.715442\pi\)
							0.999309	−	0.0371590i	\(-0.0118308\pi\)
\(17\)	−5.47696e6	−	8.52231e6i	−0.226906	−	0.353073i	0.709069	−	0.705139i	\(-0.249115\pi\)
							−0.935975	+	0.352067i	\(0.885479\pi\)
\(19\)	6.09594e6	−	9.48547e6i	0.129574	−	0.201622i	−0.770404	−	0.637556i	\(-0.779946\pi\)
							0.899979	+	0.435934i	\(0.143582\pi\)
\(23\)	1.47981e8	+	4.03875e6i	0.999628	+	0.0272822i
							\(29\)	7.05520e8	−	4.53410e8i	1.18610	−	0.762260i	0.209602	−	0.977787i	\(-0.432783\pi\)
0.976498	+	0.215527i	\(0.0691468\pi\)
\(31\)	4.31112e8	+	4.97530e8i	0.485758	+	0.560594i	0.944727	−	0.327858i	\(-0.106327\pi\)
							−0.458969	+	0.888452i	\(0.651781\pi\)
\(37\)	−3.19676e9	+	4.59625e8i	−1.24595	+	0.179140i	−0.733567	−	0.679617i	\(-0.762146\pi\)
							−0.512382	+	0.858758i	\(0.671237\pi\)
\(41\)	−7.74391e8	+	5.38601e9i	−0.163026	+	1.13387i	0.729863	+	0.683594i	\(0.239584\pi\)
							−0.892889	+	0.450277i	\(0.851325\pi\)
\(43\)	−6.68455e9	−	5.79219e9i	−1.05745	−	0.916289i	−0.0608100	−	0.998149i	\(-0.519368\pi\)
							−0.996644	+	0.0818604i	\(0.973914\pi\)
\(47\)	1.32643e10			1.23055			0.615274	−	0.788313i	\(-0.289045\pi\)
							0.615274	+	0.788313i	\(0.289045\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.2	✓	230
23.14	odd	22	inner	23.13.d.a.14.2	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.2	✓	230	1.1	even	1	trivial
23.13.d.a.14.2	yes	230	23.14	odd	22	inner