Embedded newform 138.4.e.b.49.3

• Label: 138.4.e.b.49.3
• Level: $138$
• Weight: $4$
• Character: 138.49
• 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			49.3
Character	\(\chi\)	\(=\)	138.49
Dual form			138.4.e.b.31.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.284630 - 1.97964i) q^{2} +(-1.24625 + 2.72890i) q^{3} +(-3.83797 + 1.12693i) q^{4} +(11.2596 - 12.9943i) q^{5} +(5.75696 + 1.69040i) q^{6} +(-30.1361 + 19.3673i) q^{7} +(3.32332 + 7.27706i) q^{8} +(-5.89375 - 6.80175i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{8}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.284630	−	1.97964i	−0.100632	−	0.699909i
							\(3\)	−1.24625	+	2.72890i	−0.239840	+	0.525176i
\(5\)	11.2596	−	12.9943i	1.00709	−	1.16225i								0.0203749	−	0.999792i	\(-0.493514\pi\)
							0.986716	−	0.162453i	\(-0.0519405\pi\)
\(7\)	−30.1361	+	19.3673i	−1.62720	+	1.04574i	−0.676119	+	0.736793i	\(0.736339\pi\)
							−0.951080	+	0.308944i	\(0.900024\pi\)
\(11\)	7.73571	−	53.8030i	0.212037	−	1.47475i	−0.554305	−	0.832314i	\(-0.687016\pi\)
							0.766341	−	0.642434i	\(-0.222075\pi\)
\(13\)	−5.24834	−	3.37290i	−0.111971	−	0.0719596i	0.483457	−	0.875368i	\(-0.339381\pi\)
							−0.595428	+	0.803409i	\(0.703017\pi\)
\(17\)	−80.6104	−	23.6693i	−1.15005	−	0.337686i	−0.349493	−	0.936939i	\(-0.613646\pi\)
							−0.800559	+	0.599253i	\(0.795464\pi\)
\(19\)	−97.2776	+	28.5633i	−1.17458	+	0.344888i	−0.810082	−	0.586317i	\(-0.800577\pi\)
							−0.364497	+	0.931204i	\(0.618759\pi\)
\(23\)	−74.7264	+	81.1355i	−0.677458	+	0.735562i
							\(29\)	−202.030	−	59.3213i	−1.29366	−	0.379851i	−0.438738	−	0.898615i	\(-0.644574\pi\)
−0.854918	+	0.518764i	\(0.826392\pi\)
\(31\)	−27.0163	−	59.1574i	−0.156525	−	0.342742i	0.815081	−	0.579347i	\(-0.196692\pi\)
							−0.971606	+	0.236605i	\(0.923965\pi\)
\(37\)	103.772	+	119.759i	0.461080	+	0.532115i	0.937909	−	0.346881i	\(-0.112759\pi\)
							−0.476829	+	0.878996i	\(0.658214\pi\)
\(41\)	335.933	−	387.688i	1.27961	−	1.47675i	0.478677	−	0.877991i	\(-0.341117\pi\)
							0.800932	−	0.598756i	\(-0.204338\pi\)
\(43\)	−26.9381	+	58.9862i	−0.0955353	+	0.209193i	−0.951366	−	0.308063i	\(-0.900319\pi\)
							0.855831	+	0.517256i	\(0.173047\pi\)
\(47\)	160.908			0.499380			0.249690	−	0.968326i	\(-0.419671\pi\)
							0.249690	+	0.968326i	\(0.419671\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.b.49.3	yes	30
23.8	even	11	inner	138.4.e.b.31.3	✓	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.b.31.3	✓	30	23.8	even	11	inner
138.4.e.b.49.3	yes	30	1.1	even	1	trivial