Embedded newform 49.4.g.a.16.4

• Label: 49.4.g.a.16.4
• Level: $49$
• Weight: $4$
• Character: 49.16
• Analytic conductor: $2.891$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89109359028\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(13\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			16.4
Character	\(\chi\)	\(=\)	49.16
Dual form			49.4.g.a.46.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.05418 - 2.68600i) q^{2} +(-0.556902 - 7.43134i) q^{3} +(-0.238907 + 0.221674i) q^{4} +(6.01178 - 4.09876i) q^{5} +(-19.3735 + 9.32980i) q^{6} +(4.14079 + 18.0514i) q^{7} +(-19.9504 - 9.60763i) q^{8} +(-28.2163 + 4.25292i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{10}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.05418	−	2.68600i	−0.372708	−	0.949646i	−0.987022	−	0.160584i	\(-0.948662\pi\)
							0.614314	−	0.789062i	\(-0.289433\pi\)
\(3\)	−0.556902	−	7.43134i	−0.107176	−	1.43016i	−0.748623	−	0.662995i	\(-0.769285\pi\)
							0.641448	−	0.767167i	\(-0.278334\pi\)
\(5\)	6.01178	−	4.09876i	0.537710	−	0.366604i	−0.263808	−	0.964575i	\(-0.584979\pi\)
							0.801518	+	0.597971i	\(0.204026\pi\)
\(7\)	4.14079	+	18.0514i	0.223582	+	0.974685i
							\(11\)	21.5006	+	3.24070i	0.589335	+	0.0888280i	0.436938	−	0.899491i	\(-0.356063\pi\)
0.152397	+	0.988319i	\(0.451301\pi\)
\(13\)	−10.5804	+	13.2674i	−0.225729	+	0.283055i	−0.881780	−	0.471662i	\(-0.843654\pi\)
							0.656051	+	0.754717i	\(0.272226\pi\)
\(17\)	39.0217	−	12.0366i	0.556715	−	0.171724i	−0.00361499	−	0.999993i	\(-0.501151\pi\)
							0.560330	+	0.828270i	\(0.310674\pi\)
\(19\)	46.3519	−	80.2839i	0.559677	−	0.969389i	−0.437846	−	0.899050i	\(-0.644258\pi\)
							0.997523	−	0.0703392i	\(-0.0224082\pi\)
\(23\)	4.74621	+	1.46401i	0.0430284	+	0.0132725i	0.316195	−	0.948694i	\(-0.397595\pi\)
							−0.273166	+	0.961967i	\(0.588071\pi\)
\(29\)	49.3191	−	216.081i	0.315804	−	1.38363i	−0.529031	−	0.848602i	\(-0.677444\pi\)
							0.844835	−	0.535026i	\(-0.179698\pi\)
\(31\)	130.489	+	226.013i	0.756014	+	1.30945i	0.944869	+	0.327450i	\(0.106189\pi\)
							−0.188855	+	0.982005i	\(0.560477\pi\)
\(37\)	−311.010	−	288.575i	−1.38188	−	1.28220i	−0.918153	−	0.396225i	\(-0.870320\pi\)
							−0.463731	−	0.885976i	\(-0.653490\pi\)
\(41\)	129.390	+	62.3110i	0.492862	+	0.237350i	0.663769	−	0.747937i	\(-0.268956\pi\)
							−0.170908	+	0.985287i	\(0.554670\pi\)
\(43\)	−260.030	+	125.224i	−0.922192	+	0.444104i	−0.833853	−	0.551986i	\(-0.813870\pi\)
							−0.0883385	+	0.996091i	\(0.528156\pi\)
\(47\)	196.019	+	499.448i	0.608346	+	1.55004i	0.819798	+	0.572652i	\(0.194086\pi\)
							−0.211452	+	0.977388i	\(0.567819\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.16.4	✓	156
49.12	odd	42		2401.4.a.f.1.22		78
49.37	even	21		2401.4.a.g.1.22		78
49.46	even	21	inner	49.4.g.a.46.4	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.16.4	✓	156	1.1	even	1	trivial
49.4.g.a.46.4	yes	156	49.46	even	21	inner
2401.4.a.f.1.22		78	49.12	odd	42
2401.4.a.g.1.22		78	49.37	even	21