Embedded newform 23.5.d.a.7.2

• Label: 23.5.d.a.7.2
• Level: $23$
• Weight: $5$
• Character: 23.7
• Analytic conductor: $2.378$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.37750915093\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(7\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			7.2
Character	\(\chi\)	\(=\)	23.7
Dual form			23.5.d.a.10.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.88218 + 4.48028i) q^{2} +(14.3217 - 9.20399i) q^{3} +(-2.72451 - 18.9494i) q^{4} +(14.9307 - 6.81860i) q^{5} +(-14.3630 + 99.8968i) q^{6} +(18.1967 + 61.9723i) q^{7} +(15.6809 + 10.0775i) q^{8} +(86.7488 - 189.953i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 3 q^{2} + q^{3} - 35 q^{4} - 11 q^{5} - 110 q^{6} - 11 q^{7} + 146 q^{8} - 182 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{19}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.88218	+	4.48028i	−0.970546	+	1.12007i	0.0221896	+	0.999754i	\(0.492936\pi\)
							−0.992736	+	0.120316i	\(0.961609\pi\)
\(3\)	14.3217	−	9.20399i	1.59130	−	1.02267i	0.620062	−	0.784553i	\(-0.287108\pi\)
							0.971237	−	0.238113i	\(-0.0765288\pi\)
\(5\)	14.9307	−	6.81860i	0.597226	−	0.272744i	−0.0937799	−	0.995593i	\(-0.529895\pi\)
							0.691006	+	0.722849i	\(0.257168\pi\)
\(7\)	18.1967	+	61.9723i	0.371362	+	1.26474i	0.907300	+	0.420484i	\(0.138140\pi\)
							−0.535938	+	0.844257i	\(0.680042\pi\)
\(11\)	−52.0751	+	45.1234i	−0.430373	+	0.372920i	−0.842941	−	0.538007i	\(-0.819178\pi\)
							0.412568	+	0.910927i	\(0.364632\pi\)
\(13\)	49.7599	+	14.6108i	0.294437	+	0.0864546i	0.425615	−	0.904904i	\(-0.360058\pi\)
							−0.131178	+	0.991359i	\(0.541876\pi\)
\(17\)	−389.304	−	55.9734i	−1.34707	−	0.193680i	−0.569250	−	0.822165i	\(-0.692766\pi\)
							−0.777822	+	0.628485i	\(0.783675\pi\)
\(19\)	−73.4690	+	10.5632i	−0.203515	+	0.0292611i	−0.243318	−	0.969946i	\(-0.578236\pi\)
							0.0398032	+	0.999208i	\(0.487327\pi\)
\(23\)	−524.969	−	65.1797i	−0.992380	−	0.123213i
							\(29\)	18.0568	−	125.587i	0.0214706	−	0.149331i	−0.976266	−	0.216575i	\(-0.930511\pi\)
0.997737	+	0.0672438i	\(0.0214205\pi\)
\(31\)	−173.933	−	111.780i	−0.180992	−	0.116317i	0.447008	−	0.894530i	\(-0.352490\pi\)
							−0.628000	+	0.778214i	\(0.716126\pi\)
\(37\)	244.461	+	111.641i	0.178569	+	0.0815496i	0.502693	−	0.864465i	\(-0.332342\pi\)
							−0.324125	+	0.946014i	\(0.605070\pi\)
\(41\)	−200.652	−	439.366i	−0.119364	−	0.261372i	0.840513	−	0.541791i	\(-0.182254\pi\)
							−0.959878	+	0.280420i	\(0.909526\pi\)
\(43\)	1152.69	+	1793.62i	0.623411	+	0.970047i	0.999063	+	0.0432858i	\(0.0137826\pi\)
							−0.375651	+	0.926761i	\(0.622581\pi\)
\(47\)	2755.21			1.24727			0.623634	−	0.781717i	\(-0.285656\pi\)
							0.623634	+	0.781717i	\(0.285656\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.5.d.a.7.2	✓	70
23.10	odd	22	inner	23.5.d.a.10.2	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.5.d.a.7.2	✓	70	1.1	even	1	trivial
23.5.d.a.10.2	yes	70	23.10	odd	22	inner