Embedded newform 49.4.e.a.43.8

• Label: 49.4.e.a.43.8
• Level: $49$
• Weight: $4$
• Character: 49.43
• 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			43.8
Character	\(\chi\)	\(=\)	49.43
Dual form			49.4.e.a.8.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.521845 - 0.654373i) q^{2} +(5.78023 + 2.78361i) q^{3} +(1.62429 + 7.11646i) q^{4} +(-1.28950 - 0.620991i) q^{5} +(4.83790 - 2.32981i) q^{6} +(-17.3557 + 6.46381i) q^{7} +(11.5371 + 5.55599i) q^{8} +(8.82835 + 11.0704i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.521845	−	0.654373i	0.184500	−	0.231356i	−0.680976	−	0.732305i	\(-0.738444\pi\)
							0.865476	+	0.500950i	\(0.167016\pi\)
\(3\)	5.78023	+	2.78361i	1.11241	+	0.535707i	0.897538	−	0.440937i	\(-0.145354\pi\)
							0.214868	+	0.976643i	\(0.431068\pi\)
\(5\)	−1.28950	−	0.620991i	−0.115336	−	0.0555431i	0.375326	−	0.926893i	\(-0.377531\pi\)
							−0.490662	+	0.871350i	\(0.663245\pi\)
\(7\)	−17.3557	+	6.46381i	−0.937118	+	0.349013i
							\(11\)	33.8930	−	42.5005i	0.929012	−	1.16494i	−0.0570176	−	0.998373i	\(-0.518159\pi\)
0.986029	−	0.166571i	\(-0.0532694\pi\)
\(13\)	41.8811	−	52.5173i	0.893518	−	1.12044i	−0.0985995	−	0.995127i	\(-0.531436\pi\)
							0.992118	−	0.125309i	\(-0.0399923\pi\)
\(17\)	−4.21013	+	18.4458i	−0.0600651	+	0.263163i	−0.996040	−	0.0889037i	\(-0.971664\pi\)
							0.935975	+	0.352066i	\(0.114521\pi\)
\(19\)	−58.6213			−0.707823			−0.353912	−	0.935279i	\(-0.615149\pi\)
							−0.353912	+	0.935279i	\(0.615149\pi\)
\(23\)	−4.27322	−	18.7222i	−0.0387403	−	0.169732i	0.951857	−	0.306542i	\(-0.0991722\pi\)
							−0.990597	+	0.136810i	\(0.956315\pi\)
\(29\)	−50.8862	+	222.947i	−0.325839	+	1.42759i	0.501142	+	0.865365i	\(0.332913\pi\)
							−0.826981	+	0.562229i	\(0.809944\pi\)
\(31\)	165.221			0.957244			0.478622	−	0.878021i	\(-0.341136\pi\)
							0.478622	+	0.878021i	\(0.341136\pi\)
\(37\)	−53.1486	+	232.859i	−0.236151	+	1.03464i	0.708280	+	0.705932i	\(0.249472\pi\)
							−0.944430	+	0.328711i	\(0.893386\pi\)
\(41\)	−174.801	−	84.1798i	−0.665838	−	0.320651i	0.0702570	−	0.997529i	\(-0.477618\pi\)
							−0.736095	+	0.676878i	\(0.763332\pi\)
\(43\)	−206.174	+	99.2879i	−0.731190	+	0.352123i	−0.762152	−	0.647398i	\(-0.775857\pi\)
							0.0309621	+	0.999521i	\(0.490143\pi\)
\(47\)	153.504	−	192.488i	0.476402	−	0.597389i	−0.484324	−	0.874889i	\(-0.660934\pi\)
							0.960726	+	0.277500i	\(0.0895059\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.43.8	yes	78
49.8	even	7	inner	49.4.e.a.8.8	✓	78
49.20	odd	14		2401.4.a.c.1.23		39
49.29	even	7		2401.4.a.d.1.23		39

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.8.8	✓	78	49.8	even	7	inner
49.4.e.a.43.8	yes	78	1.1	even	1	trivial
2401.4.a.c.1.23		39	49.20	odd	14
2401.4.a.d.1.23		39	49.29	even	7