Embedded newform 49.4.e.a.15.1

• Label: 49.4.e.a.15.1
• Level: $49$
• Weight: $4$
• Character: 49.15
• 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			15.1
Character	\(\chi\)	\(=\)	49.15
Dual form			49.4.e.a.36.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16701 + 5.11301i) q^{2} +(-1.24065 + 1.55573i) q^{3} +(-17.5732 - 8.46282i) q^{4} +(0.849213 - 1.06488i) q^{5} +(-6.50660 - 8.15902i) q^{6} +(-18.2375 + 3.22369i) q^{7} +(37.6195 - 47.1733i) q^{8} +(5.12699 + 22.4628i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16701	+	5.11301i	−0.412601	+	1.80772i	0.159103	+	0.987262i	\(0.449140\pi\)
							−0.571703	+	0.820460i	\(0.693717\pi\)
\(3\)	−1.24065	+	1.55573i	−0.238763	+	0.299400i	−0.886748	−	0.462254i	\(-0.847041\pi\)
							0.647984	+	0.761654i	\(0.275612\pi\)
\(5\)	0.849213	−	1.06488i	0.0759559	−	0.0952457i	−0.742402	−	0.669955i	\(-0.766313\pi\)
							0.818358	+	0.574709i	\(0.194885\pi\)
\(7\)	−18.2375	+	3.22369i	−0.984735	+	0.174063i
							\(11\)	4.10883	−	18.0020i	0.112623	−	0.493436i	−0.886882	−	0.461996i	\(-0.847134\pi\)
0.999506	−	0.0314400i	\(-0.0100093\pi\)
\(13\)	−4.72905	+	20.7193i	−0.100892	+	0.442039i	0.899098	+	0.437747i	\(0.144223\pi\)
							−0.999991	+	0.00429192i	\(0.998634\pi\)
\(17\)	−113.591	+	54.7028i	−1.62059	+	0.780433i	−0.999993	−	0.00369931i	\(-0.998822\pi\)
							−0.620593	+	0.784133i	\(0.713108\pi\)
\(19\)	−50.2875			−0.607197			−0.303598	−	0.952800i	\(-0.598188\pi\)
							−0.303598	+	0.952800i	\(0.598188\pi\)
\(23\)	74.0563	+	35.6637i	0.671383	+	0.323321i	0.738334	−	0.674435i	\(-0.235613\pi\)
							−0.0669509	+	0.997756i	\(0.521327\pi\)
\(29\)	−52.6355	+	25.3479i	−0.337040	+	0.162310i	−0.594746	−	0.803913i	\(-0.702748\pi\)
							0.257706	+	0.966223i	\(0.417033\pi\)
\(31\)	319.654			1.85199			0.925993	−	0.377541i	\(-0.123230\pi\)
							0.925993	+	0.377541i	\(0.123230\pi\)
\(37\)	−240.081	+	115.617i	−1.06673	+	0.513712i	−0.883052	−	0.469274i	\(-0.844516\pi\)
							−0.183681	+	0.982986i	\(0.558801\pi\)
\(41\)	153.419	−	192.382i	0.584392	−	0.732805i	−0.398463	−	0.917184i	\(-0.630456\pi\)
							0.982855	+	0.184380i	\(0.0590277\pi\)
\(43\)	225.042	+	282.193i	0.798105	+	1.00079i	0.999772	+	0.0213378i	\(0.00679255\pi\)
							−0.201667	+	0.979454i	\(0.564636\pi\)
\(47\)	−31.0669	+	136.113i	−0.0964164	+	0.422428i	−0.999982	−	0.00602220i	\(-0.998083\pi\)
							0.903565	+	0.428450i	\(0.140940\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.15.1	✓	78
49.6	odd	14		2401.4.a.c.1.39		39
49.36	even	7	inner	49.4.e.a.36.1	yes	78
49.43	even	7		2401.4.a.d.1.39		39

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.1	✓	78	1.1	even	1	trivial
49.4.e.a.36.1	yes	78	49.36	even	7	inner
2401.4.a.c.1.39		39	49.6	odd	14
2401.4.a.d.1.39		39	49.43	even	7