Embedded newform 23.15.d.a.5.9

• Label: 23.15.d.a.5.9
• 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			5.9
Character	\(\chi\)	\(=\)	23.5
Dual form			23.15.d.a.14.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-101.219 - 29.7205i) q^{2} +(-1577.18 + 1820.17i) q^{3} +(-4421.18 - 2841.32i) q^{4} +(-68025.4 - 9780.58i) q^{5} +(213737. - 137360. i) q^{6} +(169076. - 77214.4i) q^{7} +(1.49491e6 + 1.72522e6i) q^{8} +(-144812. - 1.00719e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−101.219	−	29.7205i	−0.790771	−	0.232191i	−0.138685	−	0.990337i	\(-0.544288\pi\)
							−0.652086	+	0.758145i	\(0.726106\pi\)
\(3\)	−1577.18	+	1820.17i	−0.721163	+	0.832267i	−0.991446	−	0.130514i	\(-0.958337\pi\)
							0.270283	+	0.962781i	\(0.412883\pi\)
\(5\)	−68025.4	−	9780.58i	−0.870725	−	0.125191i	−0.307553	−	0.951531i	\(-0.599510\pi\)
							−0.563172	+	0.826340i	\(0.690419\pi\)
\(7\)	169076.	−	77214.4i	0.205303	−	0.0937588i	−0.310110	−	0.950701i	\(-0.600366\pi\)
							0.515413	+	0.856942i	\(0.327639\pi\)
\(11\)	5.54321e6	+	1.88784e7i	0.284454	+	0.968763i	0.970479	+	0.241185i	\(0.0775359\pi\)
							−0.686025	+	0.727578i	\(0.740646\pi\)
\(13\)	−3.66754e7	+	8.03079e7i	−0.584482	+	1.27984i	0.354238	+	0.935155i	\(0.384740\pi\)
							−0.938720	+	0.344681i	\(0.887987\pi\)
\(17\)	3.72422e8	+	5.79501e8i	0.907598	+	1.41225i	0.911068	+	0.412256i	\(0.135259\pi\)
							−0.00347039	+	0.999994i	\(0.501105\pi\)
\(19\)	4.20370e8	−	6.54109e8i	0.470280	−	0.731770i	−0.522379	−	0.852713i	\(-0.674955\pi\)
							0.992659	+	0.120943i	\(0.0385918\pi\)
\(23\)	3.23447e9	−	1.06352e9i	0.949965	−	0.312356i
							\(29\)	−2.44727e10	+	1.57276e10i	−1.41872	+	0.911754i	−0.418723	+	0.908114i	\(0.637522\pi\)
−0.999993	+	0.00363995i	\(0.998841\pi\)
\(31\)	3.22152e10	+	3.71784e10i	1.17093	+	1.35132i	0.924049	+	0.382274i	\(0.124859\pi\)
							0.246877	+	0.969047i	\(0.420596\pi\)
\(37\)	−1.51559e11	+	2.17909e10i	−1.59651	+	0.229543i	−0.882402	−	0.470496i	\(-0.844075\pi\)
							−0.714104	+	0.700039i	\(0.753166\pi\)
\(41\)	1.49213e10	−	1.03780e11i	0.0766162	−	0.532878i	−0.914979	−	0.403501i	\(-0.867793\pi\)
							0.991596	−	0.129377i	\(-0.0412977\pi\)
\(43\)	8.00608e10	+	6.93730e10i	0.294537	+	0.255218i	0.789576	−	0.613653i	\(-0.210301\pi\)
							−0.495038	+	0.868871i	\(0.664846\pi\)
\(47\)	6.48038e11			1.27913			0.639566	−	0.768736i	\(-0.279114\pi\)
							0.639566	+	0.768736i	\(0.279114\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.d.a.5.9	✓	270
23.14	odd	22	inner	23.15.d.a.14.9	yes	270

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.d.a.5.9	✓	270	1.1	even	1	trivial
23.15.d.a.14.9	yes	270	23.14	odd	22	inner