Embedded newform 49.4.e.a.8.4

• Label: 49.4.e.a.8.4
• Level: $49$
• Weight: $4$
• Character: 49.8
• Analytic conductor: $2.891$
• Analytic rank: $0$
• Dimension: $78$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89109359028\)
Analytic rank:	\(0\)
Dimension:	\(78\)
Relative dimension:	\(13\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			8.4
Character	\(\chi\)	\(=\)	49.8
Dual form			49.4.e.a.43.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.61233 - 2.02180i) q^{2} +(-3.25484 + 1.56745i) q^{3} +(0.292112 - 1.27982i) q^{4} +(-3.53374 + 1.70176i) q^{5} +(8.41693 + 4.05338i) q^{6} +(-10.4442 + 15.2944i) q^{7} +(-21.6976 + 10.4490i) q^{8} +(-8.69714 + 10.9059i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.61233	−	2.02180i	−0.570044	−	0.714813i	0.410335	−	0.911935i	\(-0.365412\pi\)
							−0.980379	+	0.197122i	\(0.936840\pi\)
\(3\)	−3.25484	+	1.56745i	−0.626394	+	0.301656i	−0.720021	−	0.693952i	\(-0.755868\pi\)
							0.0936273	+	0.995607i	\(0.470154\pi\)
\(5\)	−3.53374	+	1.70176i	−0.316068	+	0.152210i	−0.585191	−	0.810896i	\(-0.698980\pi\)
							0.269123	+	0.963106i	\(0.413266\pi\)
\(7\)	−10.4442	+	15.2944i	−0.563936	+	0.825819i
							\(11\)	32.0120	+	40.1418i	0.877453	+	1.10029i	0.994244	+	0.107135i	\(0.0341677\pi\)
−0.116791	+	0.993156i	\(0.537261\pi\)
\(13\)	−49.9374	−	62.6195i	−1.06540	−	1.33596i	−0.938974	−	0.343987i	\(-0.888222\pi\)
							−0.126421	−	0.991977i	\(-0.540349\pi\)
\(17\)	−2.87731	−	12.6063i	−0.0410500	−	0.179852i	0.950247	−	0.311498i	\(-0.100831\pi\)
							−0.991297	+	0.131646i	\(0.957974\pi\)
\(19\)	−124.338			−1.50133			−0.750663	−	0.660685i	\(-0.770266\pi\)
							−0.750663	+	0.660685i	\(0.770266\pi\)
\(23\)	38.3461	−	168.005i	0.347640	−	1.52311i	−0.434882	−	0.900487i	\(-0.643210\pi\)
							0.782522	−	0.622623i	\(-0.213933\pi\)
\(29\)	33.7103	+	147.695i	0.215857	+	0.945731i	0.960503	+	0.278271i	\(0.0897613\pi\)
							−0.744646	+	0.667460i	\(0.767382\pi\)
\(31\)	122.844			0.711726			0.355863	−	0.934538i	\(-0.384187\pi\)
							0.355863	+	0.934538i	\(0.384187\pi\)
\(37\)	22.3426	+	97.8895i	0.0992731	+	0.434944i	1.00000		0.000557817i	\(0.000177559\pi\)
							−0.900727	+	0.434386i	\(0.856965\pi\)
\(41\)	−270.080	+	130.064i	−1.02877	+	0.495428i	−0.870604	−	0.491985i	\(-0.836272\pi\)
							−0.158164	+	0.987413i	\(0.550557\pi\)
\(43\)	−325.158	−	156.588i	−1.15317	−	0.555336i	−0.243183	−	0.969980i	\(-0.578192\pi\)
							−0.909984	+	0.414644i	\(0.863906\pi\)
\(47\)	204.920	+	256.961i	0.635971	+	0.797482i	0.990493	−	0.137566i	\(-0.0439278\pi\)
							−0.354522	+	0.935048i	\(0.615356\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.8.4	✓	78
49.22	even	7		2401.4.a.d.1.10		39
49.27	odd	14		2401.4.a.c.1.10		39
49.43	even	7	inner	49.4.e.a.43.4	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.8.4	✓	78	1.1	even	1	trivial
49.4.e.a.43.4	yes	78	49.43	even	7	inner
2401.4.a.c.1.10		39	49.27	odd	14
2401.4.a.d.1.10		39	49.22	even	7