Embedded newform 49.4.e.a.8.3

• Label: 49.4.e.a.8.3
• 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			8.3
Character	\(\chi\)	\(=\)	49.8
Dual form			49.4.e.a.43.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.45388 - 3.07706i) q^{2} +(2.79026 - 1.34372i) q^{3} +(-1.66664 + 7.30203i) q^{4} +(16.4856 - 7.93907i) q^{5} +(-10.9817 - 5.28849i) q^{6} +(-6.70370 - 17.2644i) q^{7} +(-1.80909 + 0.871210i) q^{8} +(-10.8543 + 13.6108i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.45388	−	3.07706i	−0.867576	−	1.08791i	−0.995371	−	0.0961052i	\(-0.969361\pi\)
							0.127795	−	0.991801i	\(-0.459210\pi\)
\(3\)	2.79026	−	1.34372i	0.536986	−	0.258599i	−0.145677	−	0.989332i	\(-0.546536\pi\)
							0.682663	+	0.730733i	\(0.260822\pi\)
\(5\)	16.4856	−	7.93907i	1.47452	−	0.710092i	0.487866	−	0.872919i	\(-0.337776\pi\)
							0.986655	+	0.162827i	\(0.0520612\pi\)
\(7\)	−6.70370	−	17.2644i	−0.361966	−	0.932191i
							\(11\)	6.93723	+	8.69901i	0.190150	+	0.238441i	0.867763	−	0.496978i	\(-0.165557\pi\)
−0.677613	+	0.735419i	\(0.736986\pi\)
\(13\)	−20.0311	−	25.1182i	−0.427356	−	0.535887i	0.520806	−	0.853675i	\(-0.325631\pi\)
							−0.948162	+	0.317788i	\(0.897060\pi\)
\(17\)	−15.1891	−	66.5479i	−0.216700	−	0.949425i	−0.959897	−	0.280353i	\(-0.909549\pi\)
							0.743197	−	0.669073i	\(-0.233309\pi\)
\(19\)	113.492			1.37036			0.685182	−	0.728372i	\(-0.259722\pi\)
							0.685182	+	0.728372i	\(0.259722\pi\)
\(23\)	−33.6662	+	147.501i	−0.305213	+	1.33722i	0.556930	+	0.830559i	\(0.311979\pi\)
							−0.862143	+	0.506665i	\(0.830878\pi\)
\(29\)	40.8875	+	179.140i	0.261814	+	1.14708i	0.919282	+	0.393600i	\(0.128771\pi\)
							−0.657467	+	0.753483i	\(0.728372\pi\)
\(31\)	167.796			0.972162			0.486081	−	0.873914i	\(-0.338426\pi\)
							0.486081	+	0.873914i	\(0.338426\pi\)
\(37\)	−56.8583	−	249.112i	−0.252634	−	1.10686i	−0.928937	−	0.370237i	\(-0.879276\pi\)
							0.676304	−	0.736623i	\(-0.263581\pi\)
\(41\)	214.875	−	103.478i	0.818482	−	0.394160i	0.0226994	−	0.999742i	\(-0.492774\pi\)
							0.795783	+	0.605582i	\(0.207060\pi\)
\(43\)	81.3480	+	39.1751i	0.288499	+	0.138934i	0.572536	−	0.819880i	\(-0.305960\pi\)
							−0.284037	+	0.958813i	\(0.591674\pi\)
\(47\)	220.566	+	276.581i	0.684529	+	0.858373i	0.995763	−	0.0919610i	\(-0.0293135\pi\)
							−0.311233	+	0.950334i	\(0.600742\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.8.3	✓	78
49.22	even	7		2401.4.a.d.1.7		39
49.27	odd	14		2401.4.a.c.1.7		39
49.43	even	7	inner	49.4.e.a.43.3	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.8.3	✓	78	1.1	even	1	trivial
49.4.e.a.43.3	yes	78	49.43	even	7	inner
2401.4.a.c.1.7		39	49.27	odd	14
2401.4.a.d.1.7		39	49.22	even	7