Embedded newform 49.6.e.a.15.10

• Label: 49.6.e.a.15.10
• Level: $49$
• Weight: $6$
• Character: 49.15
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $138$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85880717084\)
Analytic rank:	\(0\)
Dimension:	\(138\)
Relative dimension:	\(23\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			15.10
Character	\(\chi\)	\(=\)	49.15
Dual form			49.6.e.a.36.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.417387 + 1.82869i) q^{2} +(-8.97479 + 11.2540i) q^{3} +(25.6611 + 12.3577i) q^{4} +(41.8164 - 52.4361i) q^{5} +(-16.8342 - 21.1094i) q^{6} +(124.202 - 37.1611i) q^{7} +(-70.7328 + 88.6962i) q^{8} +(7.96626 + 34.9025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.417387	+	1.82869i	−0.0737843	+	0.323270i	−0.998328	−	0.0578064i	\(-0.981589\pi\)
							0.924544	+	0.381076i	\(0.124447\pi\)
\(3\)	−8.97479	+	11.2540i	−0.575733	+	0.721946i	−0.981378	−	0.192084i	\(-0.938475\pi\)
							0.405646	+	0.914030i	\(0.367047\pi\)
\(5\)	41.8164	−	52.4361i	0.748035	−	0.938006i	−0.251519	−	0.967852i	\(-0.580930\pi\)
							0.999555	+	0.0298460i	\(0.00950167\pi\)
\(7\)	124.202	−	37.1611i	0.958037	−	0.286644i
							\(11\)	−23.8365	+	104.435i	−0.0593966	+	0.260234i	−0.995904	−	0.0904135i	\(-0.971181\pi\)
0.936508	+	0.350647i	\(0.114038\pi\)
\(13\)	−166.799	+	730.796i	−0.273738	+	1.19933i	0.631823	+	0.775112i	\(0.282307\pi\)
							−0.905562	+	0.424214i	\(0.860550\pi\)
\(17\)	1520.60	−	732.282i	1.27612	−	0.614548i	0.331731	−	0.943374i	\(-0.392367\pi\)
							0.944391	+	0.328826i	\(0.106653\pi\)
\(19\)	−1963.92			−1.24807			−0.624037	−	0.781395i	\(-0.714509\pi\)
							−0.624037	+	0.781395i	\(0.714509\pi\)
\(23\)	1370.87	+	660.175i	0.540351	+	0.260219i	0.684091	−	0.729396i	\(-0.260199\pi\)
							−0.143741	+	0.989615i	\(0.545913\pi\)
\(29\)	−2351.66	+	1132.50i	−0.519254	+	0.250060i	−0.675106	−	0.737721i	\(-0.735902\pi\)
							0.155852	+	0.987780i	\(0.450188\pi\)
\(31\)	7644.59			1.42873			0.714364	−	0.699774i	\(-0.246716\pi\)
							0.714364	+	0.699774i	\(0.246716\pi\)
\(37\)	−1603.22	+	772.071i	−0.192526	+	0.0927156i	−0.527663	−	0.849454i	\(-0.676932\pi\)
							0.335137	+	0.942169i	\(0.391217\pi\)
\(41\)	11120.5	−	13944.6i	1.03315	−	1.29553i	0.0787815	−	0.996892i	\(-0.474897\pi\)
							0.954367	−	0.298636i	\(-0.0965315\pi\)
\(43\)	−12215.4	−	15317.6i	−1.00748	−	1.26334i	−0.964451	−	0.264263i	\(-0.914871\pi\)
							−0.0430271	−	0.999074i	\(-0.513700\pi\)
\(47\)	−1139.01	+	4990.31i	−0.0752110	+	0.329521i	−0.998511	−	0.0545598i	\(-0.982624\pi\)
							0.923299	+	0.384081i	\(0.125482\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.e.a.15.10	✓	138
49.36	even	7	inner	49.6.e.a.36.10	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.e.a.15.10	✓	138	1.1	even	1	trivial
49.6.e.a.36.10	yes	138	49.36	even	7	inner