Embedded newform 138.5.h.a.79.13

• Label: 138.5.h.a.79.13
• Level: $138$
• Weight: $5$
• Character: 138.79
• 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			79.13
Character	\(\chi\)	\(=\)	138.79
Dual form			138.5.h.a.7.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.85223 + 2.13758i) q^{2} +(4.37128 + 2.80925i) q^{3} +(-1.13852 + 7.91857i) q^{4} +(-26.7504 - 12.2165i) q^{5} +(2.09159 + 14.5473i) q^{6} +(18.8860 - 64.3200i) q^{7} +(-19.0354 + 12.2333i) q^{8} +(11.2162 + 24.5601i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 128 q^{4} - 432 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{22}\right)\)

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
\(5\)	−26.7504	−	12.2165i	−1.07002	−	0.488660i								−0.199042	−	0.979991i	\(-0.563783\pi\)
							−0.870973	+	0.491331i	\(0.836510\pi\)
\(7\)	18.8860	−	64.3200i	0.385430	−	1.31265i	−0.507187	−	0.861836i	\(-0.669315\pi\)
							0.892617	−	0.450817i	\(-0.148867\pi\)
\(11\)	−171.194	−	148.340i	−1.41483	−	1.22595i	−0.937852	−	0.347035i	\(-0.887188\pi\)
							−0.476973	−	0.878918i	\(-0.658266\pi\)
\(13\)	267.146	−	78.4411i	1.58074	−	0.464148i	0.630638	−	0.776077i	\(-0.282793\pi\)
							0.950106	+	0.311929i	\(0.100975\pi\)
\(17\)	−82.5631	+	11.8708i	−0.285685	+	0.0410754i	−0.283668	−	0.958923i	\(-0.591551\pi\)
							−0.00201756	+	0.999998i	\(0.500642\pi\)
\(19\)	−18.4786	−	2.65683i	−0.0511874	−	0.00735963i	0.116674	−	0.993170i	\(-0.462777\pi\)
							−0.167861	+	0.985811i	\(0.553686\pi\)
\(23\)	121.909	−	514.761i	0.230452	−	0.973084i
							\(29\)	112.790	+	784.469i	0.134114	+	0.932781i	0.940113	+	0.340864i	\(0.110720\pi\)
−0.805999	+	0.591917i	\(0.798371\pi\)
\(31\)	−451.355	+	290.068i	−0.469672	+	0.301840i	−0.753985	−	0.656892i	\(-0.771871\pi\)
							0.284313	+	0.958731i	\(0.408234\pi\)
\(37\)	−767.998	+	350.733i	−0.560992	+	0.256197i	−0.675668	−	0.737207i	\(-0.736144\pi\)
							0.114675	+	0.993403i	\(0.463417\pi\)
\(41\)	1122.64	−	2458.24i	0.667842	−	1.46237i	−0.207189	−	0.978301i	\(-0.566431\pi\)
							0.875030	−	0.484068i	\(-0.160841\pi\)
\(43\)	907.131	−	1411.52i	0.490606	−	0.763398i	−0.504372	−	0.863486i	\(-0.668276\pi\)
							0.994979	+	0.100088i	\(0.0319124\pi\)
\(47\)	−2286.05			−1.03488			−0.517439	−	0.855720i	\(-0.673115\pi\)
							−0.517439	+	0.855720i	\(0.673115\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.h.a.79.13	yes	160
23.7	odd	22	inner	138.5.h.a.7.13	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.h.a.7.13	✓	160	23.7	odd	22	inner
138.5.h.a.79.13	yes	160	1.1	even	1	trivial