Embedded newform 23.15.d.a.5.10

• Label: 23.15.d.a.5.10
• 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			5.10
Character	\(\chi\)	\(=\)	23.5
Dual form			23.15.d.a.14.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-76.7679 - 22.5411i) q^{2} +(184.251 - 212.637i) q^{3} +(-8397.89 - 5397.00i) q^{4} +(-65621.6 - 9434.97i) q^{5} +(-18937.6 + 12170.5i) q^{6} +(-1.11550e6 + 509430. i) q^{7} +(1.38147e6 + 1.59430e6i) q^{8} +(669421. + 4.65593e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−76.7679	−	22.5411i	−0.599749	−	0.176102i	−0.0322549	−	0.999480i	\(-0.510269\pi\)
							−0.567494	+	0.823377i	\(0.692087\pi\)
\(3\)	184.251	−	212.637i	0.0842482	−	0.0972276i	−0.712060	−	0.702118i	\(-0.752238\pi\)
							0.796309	+	0.604891i	\(0.206783\pi\)
\(5\)	−65621.6	−	9434.97i	−0.839957	−	0.120768i	−0.291109	−	0.956690i	\(-0.594024\pi\)
							−0.548848	+	0.835922i	\(0.684933\pi\)
\(7\)	−1.11550e6	+	509430.i	−1.35451	+	0.618584i	−0.954578	−	0.297961i	\(-0.903693\pi\)
							−0.399931	+	0.916545i	\(0.630966\pi\)
\(11\)	−9.19379e6	−	3.13112e7i	−0.471787	−	1.60676i	−0.760497	−	0.649342i	\(-0.775045\pi\)
							0.288710	−	0.957417i	\(-0.406774\pi\)
\(13\)	−4.08151e7	+	8.93725e7i	−0.650455	+	1.42430i	0.240701	+	0.970599i	\(0.422623\pi\)
							−0.891156	+	0.453698i	\(0.850105\pi\)
\(17\)	−1.07279e8	−	1.66930e8i	−0.261441	−	0.406810i	0.685572	−	0.728005i	\(-0.259552\pi\)
							−0.947013	+	0.321194i	\(0.895916\pi\)
\(19\)	−5.62417e8	+	8.75139e8i	−0.629193	+	0.979043i	0.369565	+	0.929205i	\(0.379507\pi\)
							−0.998757	+	0.0498378i	\(0.984130\pi\)
\(23\)	−2.18229e9	−	2.61351e9i	−0.640941	−	0.767590i
							\(29\)	1.47103e10	−	9.45371e9i	0.852775	−	0.548045i	−0.0396644	−	0.999213i	\(-0.512629\pi\)
0.892439	+	0.451168i	\(0.148993\pi\)
\(31\)	−1.12015e10	−	1.29273e10i	−0.407142	−	0.469867i	0.514735	−	0.857349i	\(-0.327890\pi\)
							−0.921877	+	0.387483i	\(0.873345\pi\)
\(37\)	1.21771e11	−	1.75080e10i	1.28272	−	0.184427i	0.532960	−	0.846140i	\(-0.321079\pi\)
							0.749757	+	0.661713i	\(0.230170\pi\)
\(41\)	2.11330e10	−	1.46983e11i	0.108511	−	0.754710i	−0.860813	−	0.508922i	\(-0.830044\pi\)
							0.969324	−	0.245788i	\(-0.0790468\pi\)
\(43\)	1.59179e11	+	1.37929e11i	0.585606	+	0.507430i	0.896517	−	0.443009i	\(-0.146089\pi\)
							−0.310912	+	0.950439i	\(0.600634\pi\)
\(47\)	−4.23555e11			−0.836035			−0.418018	−	0.908439i	\(-0.637275\pi\)
							−0.418018	+	0.908439i	\(0.637275\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.d.a.5.10	✓	270
23.14	odd	22	inner	23.15.d.a.14.10	yes	270

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.d.a.5.10	✓	270	1.1	even	1	trivial
23.15.d.a.14.10	yes	270	23.14	odd	22	inner