Embedded newform 138.5.h.a.7.3

• Label: 138.5.h.a.7.3
• Level: $138$
• Weight: $5$
• Character: 138.7
• Analytic conductor: $14.265$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	138.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2650549056\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(16\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			7.3
Character	\(\chi\)	\(=\)	138.7
Dual form			138.5.h.a.79.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85223 + 2.13758i) q^{2} +(-4.37128 + 2.80925i) q^{3} +(-1.13852 - 7.91857i) q^{4} +(0.0789724 - 0.0360655i) q^{5} +(2.09159 - 14.5473i) q^{6} +(-7.37685 - 25.1232i) q^{7} +(19.0354 + 12.2333i) q^{8} +(11.2162 - 24.5601i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 128 q^{4} - 432 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85223	+	2.13758i	−0.463056	+	0.534396i
							\(3\)	−4.37128	+	2.80925i	−0.485698	+	0.312139i
\(5\)	0.0789724	−	0.0360655i	0.00315890	−	0.00144262i								−0.413835	−	0.910352i	\(-0.635811\pi\)
							0.416994	+	0.908909i	\(0.363084\pi\)
\(7\)	−7.37685	−	25.1232i	−0.150548	−	0.512719i	0.849338	−	0.527850i	\(-0.177002\pi\)
							−0.999886	+	0.0151308i	\(0.995184\pi\)
\(11\)	−9.96646	+	8.63599i	−0.0823675	+	0.0713718i	−0.695071	−	0.718941i	\(-0.744627\pi\)
							0.612704	+	0.790313i	\(0.290082\pi\)
\(13\)	276.806	+	81.2777i	1.63791	+	0.480933i	0.965750	−	0.259475i	\(-0.0835495\pi\)
							0.672158	+	0.740408i	\(0.265368\pi\)
\(17\)	−342.965	−	49.3110i	−1.18673	−	0.170626i	−0.479457	−	0.877565i	\(-0.659166\pi\)
							−0.707274	+	0.706939i	\(0.750075\pi\)
\(19\)	189.818	−	27.2917i	0.525812	−	0.0756003i	0.125703	−	0.992068i	\(-0.459881\pi\)
							0.400109	+	0.916468i	\(0.368972\pi\)
\(23\)	446.830	+	283.168i	0.844669	+	0.535289i
							\(29\)	−222.923	+	1550.46i	−0.265069	+	1.84359i	0.228073	+	0.973644i	\(0.426758\pi\)
−0.493141	+	0.869949i	\(0.664151\pi\)
\(31\)	−94.6870	−	60.8517i	−0.0985297	−	0.0633212i	0.490448	−	0.871470i	\(-0.336833\pi\)
							−0.588978	+	0.808149i	\(0.700469\pi\)
\(37\)	−1756.72	−	802.265i	−1.28321	−	0.586023i	−0.347133	−	0.937816i	\(-0.612845\pi\)
							−0.936078	+	0.351793i	\(0.885572\pi\)
\(41\)	1142.32	+	2501.32i	0.679546	+	1.48800i	0.863124	+	0.504992i	\(0.168505\pi\)
							−0.183578	+	0.983005i	\(0.558768\pi\)
\(43\)	1507.05	+	2345.02i	0.815064	+	1.26826i	0.960328	+	0.278872i	\(0.0899605\pi\)
							−0.145264	+	0.989393i	\(0.546403\pi\)
\(47\)	−2359.97			−1.06834			−0.534172	−	0.845376i	\(-0.679377\pi\)
							−0.534172	+	0.845376i	\(0.679377\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.h.a.7.3	✓	160
23.10	odd	22	inner	138.5.h.a.79.3	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.h.a.7.3	✓	160	1.1	even	1	trivial
138.5.h.a.79.3	yes	160	23.10	odd	22	inner