Embedded newform 23.15.d.a.5.2

• Label: 23.15.d.a.5.2
• Level: $23$
• Weight: $15$
• Character: 23.5
• Analytic conductor: $28.596$
• Analytic rank: $0$
• Dimension: $270$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.5956626749\)
Analytic rank:	\(0\)
Dimension:	\(270\)
Relative dimension:	\(27\) 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			23.15.d.a.14.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-215.886 - 63.3897i) q^{2} +(-2125.96 + 2453.49i) q^{3} +(28805.2 + 18512.0i) q^{4} +(7903.43 + 1136.34i) q^{5} +(614491. - 394909. i) q^{6} +(96970.1 - 44284.8i) q^{7} +(-2.63109e6 - 3.03644e6i) q^{8} +(-819216. - 5.69777e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−215.886	−	63.3897i	−1.68661	−	0.495232i	−0.708918	−	0.705291i	\(-0.750816\pi\)
							−0.977689	+	0.210059i	\(0.932634\pi\)
\(3\)	−2125.96	+	2453.49i	−0.972090	+	1.12185i	0.0204321	+	0.999791i	\(0.493496\pi\)
							−0.992523	+	0.122061i	\(0.961050\pi\)
\(5\)	7903.43	+	1136.34i	0.101164	+	0.0145452i	0.192711	−	0.981256i	\(-0.438272\pi\)
							−0.0915470	+	0.995801i	\(0.529181\pi\)
\(7\)	96970.1	−	44284.8i	0.117747	−	0.0537735i	−0.355670	−	0.934612i	\(-0.615747\pi\)
							0.473418	+	0.880838i	\(0.343020\pi\)
\(11\)	1.97736e6	+	6.73427e6i	0.101470	+	0.345575i	0.994542	−	0.104339i	\(-0.0332728\pi\)
							−0.893072	+	0.449914i	\(0.851455\pi\)
\(13\)	−8.80947e6	+	1.92901e7i	−0.140393	+	0.307419i	−0.966748	−	0.255732i	\(-0.917684\pi\)
							0.826354	+	0.563150i	\(0.190411\pi\)
\(17\)	4.50102e7	+	7.00373e7i	0.109690	+	0.170682i	0.891759	−	0.452511i	\(-0.149472\pi\)
							−0.782068	+	0.623193i	\(0.785835\pi\)
\(19\)	−8.77241e8	+	1.36501e9i	−0.981395	+	1.52708i	−0.137550	+	0.990495i	\(0.543923\pi\)
							−0.843846	+	0.536586i	\(0.819714\pi\)
\(23\)	−1.46104e9	−	3.07542e9i	−0.429110	−	0.903252i
							\(29\)	−6.79635e9	+	4.36775e9i	−0.393994	+	0.253205i	−0.722606	−	0.691261i	\(-0.757056\pi\)
0.328612	+	0.944465i	\(0.393419\pi\)
\(31\)	−2.49711e10	−	2.88182e10i	−0.907625	−	1.04746i	−0.998667	−	0.0516081i	\(-0.983565\pi\)
							0.0910424	−	0.995847i	\(-0.470980\pi\)
\(37\)	1.26654e11	−	1.82101e10i	1.33415	−	0.191822i	0.561936	−	0.827181i	\(-0.310057\pi\)
							0.772217	+	0.635358i	\(0.219148\pi\)
\(41\)	−2.04537e10	+	1.42259e11i	−0.105023	+	0.730453i	0.867465	+	0.497499i	\(0.165748\pi\)
							−0.972488	+	0.232954i	\(0.925161\pi\)
\(43\)	−6.97253e10	−	6.04174e10i	−0.256514	−	0.222271i	0.517104	−	0.855922i	\(-0.327010\pi\)
							−0.773619	+	0.633651i	\(0.781555\pi\)
\(47\)	4.22159e11			0.833280			0.416640	−	0.909072i	\(-0.363208\pi\)
							0.416640	+	0.909072i	\(0.363208\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.d.a.5.2	✓	270
23.14	odd	22	inner	23.15.d.a.14.2	yes	270

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.d.a.5.2	✓	270	1.1	even	1	trivial
23.15.d.a.14.2	yes	270	23.14	odd	22	inner