Embedded newform 138.5.h.a.7.2

• Label: 138.5.h.a.7.2
• 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			7.2
Character	\(\chi\)	\(=\)	138.7
Dual form			138.5.h.a.79.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85223 + 2.13758i) q^{2} +(-4.37128 + 2.80925i) q^{3} +(-1.13852 - 7.91857i) q^{4} +(-16.9332 + 7.73311i) q^{5} +(2.09159 - 14.5473i) q^{6} +(26.3243 + 89.6523i) q^{7} +(19.0354 + 12.2333i) q^{8} +(11.2162 - 24.5601i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 128 q^{4} - 432 q^{9}+O(q^{10}) \)

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
\(5\)	−16.9332	+	7.73311i	−0.677326	+	0.309324i								−0.724214	−	0.689576i	\(-0.757797\pi\)
							0.0468875	+	0.998900i	\(0.485070\pi\)
\(7\)	26.3243	+	89.6523i	0.537230	+	1.82964i	0.557981	+	0.829854i	\(0.311576\pi\)
							−0.0207510	+	0.999785i	\(0.506606\pi\)
\(11\)	−67.9548	+	58.8832i	−0.561610	+	0.486638i	−0.888784	−	0.458326i	\(-0.848449\pi\)
							0.327174	+	0.944964i	\(0.393904\pi\)
\(13\)	170.243	+	49.9877i	1.00735	+	0.295785i	0.743470	−	0.668769i	\(-0.233179\pi\)
							0.263883	+	0.964555i	\(0.414997\pi\)
\(17\)	21.1927	+	3.04705i	0.0733311	+	0.0105434i	0.178883	−	0.983870i	\(-0.442752\pi\)
							−0.105552	+	0.994414i	\(0.533661\pi\)
\(19\)	−642.589	+	92.3903i	−1.78002	+	0.255929i	−0.952291	−	0.305193i	\(-0.901279\pi\)
							−0.827733	+	0.561122i	\(0.810370\pi\)
\(23\)	−72.1215	−	524.061i	−0.136336	−	0.990663i
							\(29\)	171.172	−	1190.53i	0.203534	−	1.41561i	−0.590155	−	0.807290i	\(-0.700933\pi\)
0.793689	−	0.608323i	\(-0.208158\pi\)
\(31\)	−46.7271	−	30.0297i	−0.0486234	−	0.0312484i	0.516103	−	0.856526i	\(-0.327382\pi\)
							−0.564727	+	0.825278i	\(0.691018\pi\)
\(37\)	2263.40	+	1033.66i	1.65333	+	0.755048i	0.999998	−	0.00202691i	\(-0.000645185\pi\)
							0.653328	+	0.757075i	\(0.273372\pi\)
\(41\)	−898.592	−	1967.64i	−0.534558	−	1.17052i	−0.963628	−	0.267247i	\(-0.913886\pi\)
							0.429070	−	0.903271i	\(-0.358841\pi\)
\(43\)	−1538.33	−	2393.68i	−0.831978	−	1.29458i	−0.953317	−	0.301972i	\(-0.902355\pi\)
							0.121338	−	0.992611i	\(-0.461281\pi\)
\(47\)	−3027.27			−1.37043			−0.685213	−	0.728343i	\(-0.740291\pi\)
							−0.685213	+	0.728343i	\(0.740291\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.h.a.7.2	✓	160
23.10	odd	22	inner	138.5.h.a.79.2	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.h.a.7.2	✓	160	1.1	even	1	trivial
138.5.h.a.79.2	yes	160	23.10	odd	22	inner