Embedded newform 23.13.d.a.5.15

• Label: 23.13.d.a.5.15
• Level: $23$
• Weight: $13$
• Character: 23.5
• Analytic conductor: $21.022$
• Analytic rank: $0$
• Dimension: $230$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	23.d (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.0218577974\)
Analytic rank:	\(0\)
Dimension:	\(230\)
Relative dimension:	\(23\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.15
Character	\(\chi\)	\(=\)	23.5
Dual form			23.13.d.a.14.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(35.2606 + 10.3534i) q^{2} +(563.490 - 650.302i) q^{3} +(-2309.66 - 1484.33i) q^{4} +(2550.09 + 366.648i) q^{5} +(26601.9 - 17096.0i) q^{6} +(-174986. + 79913.5i) q^{7} +(-164645. - 190010. i) q^{8} +(-29740.0 - 206846. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	35.2606	+	10.3534i	0.550947	+	0.161773i	0.545345	−	0.838212i	\(-0.316399\pi\)
							0.00560191	+	0.999984i	\(0.498217\pi\)
\(3\)	563.490	−	650.302i	0.772963	−	0.892047i	−0.223617	−	0.974677i	\(-0.571786\pi\)
							0.996580	+	0.0826299i	\(0.0263319\pi\)
\(5\)	2550.09	+	366.648i	0.163206	+	0.0234655i	0.223433	−	0.974719i	\(-0.428273\pi\)
							−0.0602274	+	0.998185i	\(0.519183\pi\)
\(7\)	−174986.	+	79913.5i	−1.48736	+	0.679253i	−0.982893	−	0.184179i	\(-0.941037\pi\)
							−0.504465	+	0.863432i	\(0.668310\pi\)
\(11\)	−59133.6	−	201391.i	−0.0333794	−	0.113680i	0.941118	−	0.338078i	\(-0.109777\pi\)
							−0.974498	+	0.224398i	\(0.927958\pi\)
\(13\)	−2.68503e6	+	5.87940e6i	−0.556275	+	1.21807i	0.397515	+	0.917596i	\(0.369873\pi\)
							−0.953789	+	0.300476i	\(0.902854\pi\)
\(17\)	1.68064e7	+	2.61512e7i	0.696274	+	1.08342i	0.991764	+	0.128077i	\(0.0408806\pi\)
							−0.295490	+	0.955346i	\(0.595483\pi\)
\(19\)	2.68851e7	−	4.18340e7i	0.571466	−	0.889218i	−0.428431	−	0.903574i	\(-0.640934\pi\)
							0.999897	+	0.0143565i	\(0.00456998\pi\)
\(23\)	−1.04504e8	−	1.04850e8i	−0.705938	−	0.708274i
							\(29\)	−7.07517e8	+	4.54694e8i	−1.18946	+	0.764418i	−0.977101	−	0.212774i	\(-0.931750\pi\)
−0.212356	+	0.977192i	\(0.568114\pi\)
\(31\)	−1.03280e9	−	1.19192e9i	−1.16372	−	1.34300i	−0.928621	−	0.371029i	\(-0.879005\pi\)
							−0.235096	−	0.971972i	\(-0.575540\pi\)
\(37\)	−2.02276e9	+	2.90828e8i	−0.788376	+	0.113351i	−0.524731	−	0.851268i	\(-0.675834\pi\)
							−0.263645	+	0.964620i	\(0.584925\pi\)
\(41\)	2.25447e8	−	1.56802e9i	0.0474615	−	0.330102i	−0.952233	−	0.305373i	\(-0.901219\pi\)
							0.999694	−	0.0247287i	\(-0.00787218\pi\)
\(43\)	−6.98736e9	−	6.05458e9i	−1.10536	−	0.957796i	−0.106025	−	0.994364i	\(-0.533812\pi\)
							−0.999331	+	0.0365672i	\(0.988358\pi\)
\(47\)	2.50183e9			0.232098			0.116049	−	0.993244i	\(-0.462977\pi\)
							0.116049	+	0.993244i	\(0.462977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.d.a.5.15	✓	230
23.14	odd	22	inner	23.13.d.a.14.15	yes	230

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.d.a.5.15	✓	230	1.1	even	1	trivial
23.13.d.a.14.15	yes	230	23.14	odd	22	inner