Embedded newform 49.6.g.a.2.17

• Label: 49.6.g.a.2.17
• Level: $49$
• Weight: $6$
• Character: 49.2
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85880717084\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			2.17
Character	\(\chi\)	\(=\)	49.2
Dual form			49.6.g.a.25.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.34627 + 1.64911i) q^{2} +(-3.67157 + 9.35501i) q^{3} +(-0.576558 - 0.393090i) q^{4} +(96.5461 - 14.5520i) q^{5} +(-35.0566 + 43.9596i) q^{6} +(41.8489 + 122.702i) q^{7} +(-114.061 - 143.027i) q^{8} +(104.096 + 96.5868i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.34627	+	1.64911i	0.945096	+	0.291524i	0.728740	−	0.684791i	\(-0.240106\pi\)
							0.216357	+	0.976314i	\(0.430583\pi\)
\(3\)	−3.67157	+	9.35501i	−0.235531	+	0.600124i	−0.998952	−	0.0457778i	\(-0.985423\pi\)
							0.763420	+	0.645902i	\(0.223519\pi\)
\(5\)	96.5461	−	14.5520i	1.72707	−	0.260314i	0.790921	−	0.611918i	\(-0.209602\pi\)
							0.936149	+	0.351604i	\(0.114364\pi\)
\(7\)	41.8489	+	122.702i	0.322804	+	0.946466i
							\(11\)	−160.772	+	149.175i	−0.400616	+	0.371718i	−0.854595	−	0.519295i	\(-0.826195\pi\)
0.453979	+	0.891012i	\(0.350004\pi\)
\(13\)	241.656	+	1058.76i	0.396587	+	1.73756i	0.640679	+	0.767809i	\(0.278653\pi\)
							−0.244092	+	0.969752i	\(0.578490\pi\)
\(17\)	−132.666	−	1770.31i	−0.111337	−	1.48568i	−0.720952	−	0.692985i	\(-0.756295\pi\)
							0.609615	−	0.792697i	\(-0.291324\pi\)
\(19\)	548.533	−	950.088i	0.348593	−	0.603781i	−0.637406	−	0.770528i	\(-0.719993\pi\)
							0.986000	+	0.166746i	\(0.0533261\pi\)
\(23\)	98.4796	−	1314.12i	0.0388174	−	0.517983i	−0.943576	−	0.331157i	\(-0.892561\pi\)
							0.982393	−	0.186826i	\(-0.0598199\pi\)
\(29\)	−1279.27	−	616.066i	−0.282467	−	0.136029i	0.287286	−	0.957845i	\(-0.407247\pi\)
							−0.569753	+	0.821816i	\(0.692961\pi\)
\(31\)	−2960.80	−	5128.26i	−0.553357	−	0.958443i	−0.998029	−	0.0627492i	\(-0.980013\pi\)
							0.444672	−	0.895693i	\(-0.353320\pi\)
\(37\)	7079.90	−	4827.00i	0.850204	−	0.579659i	−0.0579390	−	0.998320i	\(-0.518453\pi\)
							0.908143	+	0.418661i	\(0.137501\pi\)
\(41\)	−2833.77	−	3553.44i	−0.263272	−	0.330133i	0.632572	−	0.774502i	\(-0.281999\pi\)
							−0.895844	+	0.444369i	\(0.853428\pi\)
\(43\)	−3444.24	+	4318.94i	−0.284068	+	0.356210i	−0.903309	−	0.428992i	\(-0.858869\pi\)
							0.619241	+	0.785201i	\(0.287440\pi\)
\(47\)	−25024.4	−	7718.99i	−1.65241	−	0.509701i	−0.678112	−	0.734958i	\(-0.737202\pi\)
							−0.974299	+	0.225257i	\(0.927678\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.2.17	✓	264
49.25	even	21	inner	49.6.g.a.25.17	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.2.17	✓	264	1.1	even	1	trivial
49.6.g.a.25.17	yes	264	49.25	even	21	inner