Embedded newform 138.4.e.c.73.2

• Label: 138.4.e.c.73.2
• Level: $138$
• Weight: $4$
• Character: 138.73
• Analytic conductor: $8.142$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	138.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.14226358079\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(3\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			73.2
Character	\(\chi\)	\(=\)	138.73
Dual form			138.4.e.c.121.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.830830 - 1.81926i) q^{2} +(-2.87848 + 0.845198i) q^{3} +(-2.61944 + 3.02300i) q^{4} +(-3.38842 - 2.17761i) q^{5} +(3.92916 + 4.53450i) q^{6} +(-3.30636 - 22.9962i) q^{7} +(7.67594 + 2.25386i) q^{8} +(7.57128 - 4.86577i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.830830	−	1.81926i	−0.293743	−	0.643207i
							\(3\)	−2.87848	+	0.845198i	−0.553964	+	0.162658i
\(5\)	−3.38842	−	2.17761i	−0.303070	−	0.194771i								0.380263	−	0.924879i	\(-0.375834\pi\)
							−0.683332	+	0.730107i	\(0.739470\pi\)
\(7\)	−3.30636	−	22.9962i	−0.178526	−	1.24168i	−0.860175	−	0.509999i	\(-0.829646\pi\)
							0.681648	−	0.731680i	\(-0.261263\pi\)
\(11\)	−8.80431	+	19.2788i	−0.241327	+	0.528433i	−0.991077	−	0.133288i	\(-0.957446\pi\)
							0.749750	+	0.661721i	\(0.230174\pi\)
\(13\)	−9.66445	+	67.2178i	−0.206187	+	1.43407i	0.579263	+	0.815141i	\(0.303340\pi\)
							−0.785450	+	0.618925i	\(0.787569\pi\)
\(17\)	45.5628	+	52.5822i	0.650035	+	0.750181i	0.981116	−	0.193422i	\(-0.0619586\pi\)
							−0.331081	+	0.943602i	\(0.607413\pi\)
\(19\)	−100.762	+	116.286i	−1.21665	+	1.40409i	−0.328525	+	0.944495i	\(0.606551\pi\)
							−0.888128	+	0.459597i	\(0.847994\pi\)
\(23\)	109.359	+	14.4111i	0.991429	+	0.130649i
							\(29\)	140.429	+	162.064i	0.899207	+	1.03774i	0.999086	+	0.0427401i	\(0.0136088\pi\)
−0.0998797	+	0.995000i	\(0.531846\pi\)
\(31\)	−48.1267	−	14.1313i	−0.278833	−	0.0818727i	0.139326	−	0.990247i	\(-0.455506\pi\)
							−0.418159	+	0.908374i	\(0.637324\pi\)
\(37\)	−288.384	+	185.333i	−1.28135	+	0.823474i	−0.991054	−	0.133463i	\(-0.957390\pi\)
							−0.290297	+	0.956937i	\(0.593754\pi\)
\(41\)	−130.130	−	83.6296i	−0.495681	−	0.318555i	0.268806	−	0.963194i	\(-0.413371\pi\)
							−0.764487	+	0.644640i	\(0.777007\pi\)
\(43\)	327.668	−	96.2119i	1.16207	−	0.341213i	0.356831	−	0.934169i	\(-0.383857\pi\)
							0.805235	+	0.592955i	\(0.202039\pi\)
\(47\)	−257.120			−0.797974			−0.398987	−	0.916957i	\(-0.630638\pi\)
							−0.398987	+	0.916957i	\(0.630638\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.c.73.2	✓	30
23.6	even	11	inner	138.4.e.c.121.2	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.c.73.2	✓	30	1.1	even	1	trivial
138.4.e.c.121.2	yes	30	23.6	even	11	inner