Embedded newform 23.15.d.a.5.13

• Label: 23.15.d.a.5.13
• 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			5.13
Character	\(\chi\)	\(=\)	23.5
Dual form			23.15.d.a.14.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-18.9656 - 5.56881i) q^{2} +(1300.36 - 1500.69i) q^{3} +(-13454.4 - 8646.63i) q^{4} +(40364.7 + 5803.57i) q^{5} +(-33019.1 + 21220.1i) q^{6} +(-455018. + 207800. i) q^{7} +(419097. + 483664. i) q^{8} +(119540. + 831420. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.9656	−	5.56881i	−0.148169	−	0.0435063i	0.206807	−	0.978382i	\(-0.433693\pi\)
							−0.354976	+	0.934875i	\(0.615511\pi\)
\(3\)	1300.36	−	1500.69i	0.594584	−	0.686187i	−0.376091	−	0.926583i	\(-0.622732\pi\)
							0.970675	+	0.240396i	\(0.0772773\pi\)
\(5\)	40364.7	+	5803.57i	0.516668	+	0.0742857i	0.395716	−	0.918373i	\(-0.370496\pi\)
							0.120952	+	0.992658i	\(0.461405\pi\)
\(7\)	−455018.	+	207800.i	−0.552513	+	0.252324i	−0.672050	−	0.740505i	\(-0.734586\pi\)
							0.119537	+	0.992830i	\(0.461859\pi\)
\(11\)	7.81903e6	+	2.66292e7i	0.401240	+	1.36650i	0.874265	+	0.485448i	\(0.161344\pi\)
							−0.473026	+	0.881049i	\(0.656838\pi\)
\(13\)	7.07287e6	−	1.54874e7i	0.112718	−	0.246818i	−0.844863	−	0.534983i	\(-0.820318\pi\)
							0.957581	+	0.288165i	\(0.0930453\pi\)
\(17\)	8.17997e7	+	1.27283e8i	0.199347	+	0.310190i	0.926507	−	0.376279i	\(-0.122796\pi\)
							−0.727160	+	0.686468i	\(0.759160\pi\)
\(19\)	5.75494e8	−	8.95486e8i	0.643821	−	1.00181i	−0.353962	−	0.935260i	\(-0.615166\pi\)
							0.997783	−	0.0665459i	\(-0.0211979\pi\)
\(23\)	−3.40482e9	−	3.28138e6i	−1.00000	−	0.000963744i
							\(29\)	1.98916e10	−	1.27835e10i	1.15314	−	0.741079i	0.182880	−	0.983135i	\(-0.441458\pi\)
0.970262	+	0.242056i	\(0.0778217\pi\)
\(31\)	2.99789e10	+	3.45975e10i	1.08964	+	1.25752i	0.964132	+	0.265423i	\(0.0855116\pi\)
							0.125511	+	0.992092i	\(0.459943\pi\)
\(37\)	1.32569e11	−	1.90606e10i	1.39647	−	0.200782i	0.597362	−	0.801972i	\(-0.296216\pi\)
							0.799106	+	0.601190i	\(0.205307\pi\)
\(41\)	−1.60220e10	+	1.11435e11i	−0.0822676	+	0.572184i	0.906441	+	0.422333i	\(0.138789\pi\)
							−0.988709	+	0.149851i	\(0.952121\pi\)
\(43\)	1.00380e11	+	8.69799e10i	0.369291	+	0.319993i	0.819660	−	0.572850i	\(-0.194162\pi\)
							−0.450369	+	0.892842i	\(0.648708\pi\)
\(47\)	7.46253e11			1.47299			0.736497	−	0.676441i	\(-0.236479\pi\)
							0.736497	+	0.676441i	\(0.236479\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.d.a.5.13	✓	270
23.14	odd	22	inner	23.15.d.a.14.13	yes	270

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.d.a.5.13	✓	270	1.1	even	1	trivial
23.15.d.a.14.13	yes	270	23.14	odd	22	inner