Embedded newform 49.4.e.a.36.1

• Label: 49.4.e.a.36.1
• Level: $49$
• Weight: $4$
• Character: 49.36
• 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			36.1
Character	\(\chi\)	\(=\)	49.36
Dual form			49.4.e.a.15.1

$q$-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 \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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	−0.571703	−	0.820460i	\(-0.693717\pi\)
							0.159103	−	0.987262i	\(-0.449140\pi\)
\(3\)	−1.24065	−	1.55573i	−0.238763	−	0.299400i	0.647984	−	0.761654i	\(-0.275612\pi\)
							−0.886748	+	0.462254i	\(0.847041\pi\)
\(5\)	0.849213	+	1.06488i	0.0759559	+	0.0952457i	0.818358	−	0.574709i	\(-0.194885\pi\)
							−0.742402	+	0.669955i	\(0.766313\pi\)
\(7\)	−18.2375	−	3.22369i	−0.984735	−	0.174063i
							\(11\)	4.10883	+	18.0020i	0.112623	+	0.493436i	0.999506	+	0.0314400i	\(0.0100093\pi\)
−0.886882	+	0.461996i	\(0.847134\pi\)
\(13\)	−4.72905	−	20.7193i	−0.100892	−	0.442039i	−0.999991	−	0.00429192i	\(-0.998634\pi\)
							0.899098	−	0.437747i	\(-0.144223\pi\)
\(17\)	−113.591	−	54.7028i	−1.62059	−	0.780433i	−0.620593	−	0.784133i	\(-0.713108\pi\)
							−0.999993	+	0.00369931i	\(0.998822\pi\)
\(19\)	−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	−0.0669509	−	0.997756i	\(-0.521327\pi\)
							0.738334	+	0.674435i	\(0.235613\pi\)
\(29\)	−52.6355	−	25.3479i	−0.337040	−	0.162310i	0.257706	−	0.966223i	\(-0.417033\pi\)
							−0.594746	+	0.803913i	\(0.702748\pi\)
\(31\)	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	−0.183681	−	0.982986i	\(-0.558801\pi\)
							−0.883052	+	0.469274i	\(0.844516\pi\)
\(41\)	153.419	+	192.382i	0.584392	+	0.732805i	0.982855	−	0.184380i	\(-0.0590277\pi\)
							−0.398463	+	0.917184i	\(0.630456\pi\)
\(43\)	225.042	−	282.193i	0.798105	−	1.00079i	−0.201667	−	0.979454i	\(-0.564636\pi\)
							0.999772	−	0.0213378i	\(-0.00679255\pi\)
\(47\)	−31.0669	−	136.113i	−0.0964164	−	0.422428i	0.903565	−	0.428450i	\(-0.140940\pi\)
							−0.999982	+	0.00602220i	\(0.998083\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.1	✓	78	49.15	even	7	inner
49.4.e.a.36.1	yes	78	1.1	even	1	trivial
2401.4.a.c.1.39		39	49.41	odd	14
2401.4.a.d.1.39		39	49.8	even	7