Embedded newform 49.4.e.a.8.11

• Label: 49.4.e.a.8.11
• 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			8.11
Character	\(\chi\)	\(=\)	49.8
Dual form			49.4.e.a.43.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.59695 + 3.25648i) q^{2} +(2.96773 - 1.42918i) q^{3} +(-2.08030 + 9.11440i) q^{4} +(9.52410 - 4.58657i) q^{5} +(12.3612 + 5.95282i) q^{6} +(-16.3066 + 8.78031i) q^{7} +(-5.06162 + 2.43755i) q^{8} +(-10.0694 + 12.6266i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.59695	+	3.25648i	0.918161	+	1.15134i	0.988104	+	0.153787i	\(0.0491469\pi\)
							−0.0699428	+	0.997551i	\(0.522282\pi\)
\(3\)	2.96773	−	1.42918i	0.571140	−	0.275047i	−0.125942	−	0.992038i	\(-0.540195\pi\)
							0.697082	+	0.716991i	\(0.254481\pi\)
\(5\)	9.52410	−	4.58657i	0.851862	−	0.410235i	0.0435933	−	0.999049i	\(-0.486119\pi\)
							0.808268	+	0.588814i	\(0.200405\pi\)
\(7\)	−16.3066	+	8.78031i	−0.880475	+	0.474092i
							\(11\)	−34.1739	−	42.8527i	−0.936710	−	1.17460i	−0.984437	−	0.175736i	\(-0.943769\pi\)
0.0477277	−	0.998860i	\(-0.484802\pi\)
\(13\)	−18.7277	−	23.4838i	−0.399549	−	0.501019i	0.540837	−	0.841127i	\(-0.318108\pi\)
							−0.940386	+	0.340109i	\(0.889536\pi\)
\(17\)	9.96373	+	43.6539i	0.142150	+	0.622802i	0.994933	+	0.100536i	\(0.0320559\pi\)
							−0.852783	+	0.522266i	\(0.825087\pi\)
\(19\)	71.5587			0.864036			0.432018	−	0.901865i	\(-0.357802\pi\)
							0.432018	+	0.901865i	\(0.357802\pi\)
\(23\)	37.1988	−	162.978i	0.337238	−	1.47754i	−0.467546	−	0.883969i	\(-0.654862\pi\)
							0.804784	−	0.593568i	\(-0.202281\pi\)
\(29\)	−4.18669	−	18.3431i	−0.0268086	−	0.117456i	0.959753	−	0.280845i	\(-0.0906146\pi\)
							−0.986562	+	0.163389i	\(0.947757\pi\)
\(31\)	−114.344			−0.662477			−0.331238	−	0.943547i	\(-0.607466\pi\)
							−0.331238	+	0.943547i	\(0.607466\pi\)
\(37\)	18.8099	+	82.4114i	0.0835763	+	0.366172i	0.999371	−	0.0354760i	\(-0.0112947\pi\)
							−0.915794	+	0.401648i	\(0.868438\pi\)
\(41\)	−55.7401	+	26.8430i	−0.212320	+	0.102248i	−0.537021	−	0.843569i	\(-0.680451\pi\)
							0.324701	+	0.945817i	\(0.394736\pi\)
\(43\)	−148.440	−	71.4850i	−0.526440	−	0.253520i	0.151734	−	0.988421i	\(-0.451514\pi\)
							−0.678174	+	0.734901i	\(0.737228\pi\)
\(47\)	337.800	+	423.588i	1.04837	+	1.31461i	0.947516	+	0.319709i	\(0.103585\pi\)
							0.100851	+	0.994902i	\(0.467843\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.8.11	✓	78
49.22	even	7		2401.4.a.d.1.33		39
49.27	odd	14		2401.4.a.c.1.33		39
49.43	even	7	inner	49.4.e.a.43.11	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.8.11	✓	78	1.1	even	1	trivial
49.4.e.a.43.11	yes	78	49.43	even	7	inner
2401.4.a.c.1.33		39	49.27	odd	14
2401.4.a.d.1.33		39	49.22	even	7