Embedded newform 49.6.g.a.2.11

• Label: 49.6.g.a.2.11
• 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			2.11
Character	\(\chi\)	\(=\)	49.2
Dual form			49.6.g.a.25.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11459 - 0.343805i) q^{2} +(1.70006 - 4.33168i) q^{3} +(-25.3155 - 17.2598i) q^{4} +(-34.5506 + 5.20767i) q^{5} +(-3.38412 + 4.24356i) q^{6} +(61.0553 + 114.365i) q^{7} +(45.5542 + 57.1232i) q^{8} +(162.258 + 150.554i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11459	−	0.343805i	−0.197033	−	0.0607767i	0.194668	−	0.980869i	\(-0.437637\pi\)
							−0.391701	+	0.920092i	\(0.628113\pi\)
\(3\)	1.70006	−	4.33168i	0.109059	−	0.277878i	−0.865917	−	0.500187i	\(-0.833264\pi\)
							0.974976	+	0.222310i	\(0.0713597\pi\)
\(5\)	−34.5506	+	5.20767i	−0.618060	+	0.0931576i	−0.450607	−	0.892722i	\(-0.648792\pi\)
							−0.167453	+	0.985880i	\(0.553554\pi\)
\(7\)	61.0553	+	114.365i	0.470954	+	0.882158i
							\(11\)	−26.0271	+	24.1496i	−0.0648551	+	0.0601767i	−0.711934	−	0.702246i	\(-0.752181\pi\)
0.647079	+	0.762423i	\(0.275990\pi\)
\(13\)	43.5494	+	190.802i	0.0714699	+	0.313130i	0.998009	−	0.0630677i	\(-0.0200884\pi\)
							−0.926539	+	0.376198i	\(0.877231\pi\)
\(17\)	43.0935	+	575.043i	0.0361651	+	0.482589i	0.985653	+	0.168785i	\(0.0539845\pi\)
							−0.949488	+	0.313804i	\(0.898396\pi\)
\(19\)	−534.755	+	926.222i	−0.339837	+	0.588615i	−0.984402	−	0.175935i	\(-0.943705\pi\)
							0.644565	+	0.764550i	\(0.277039\pi\)
\(23\)	−140.050	+	1868.84i	−0.0552032	+	0.736635i	0.899090	+	0.437764i	\(0.144229\pi\)
							−0.954293	+	0.298872i	\(0.903390\pi\)
\(29\)	−1556.28	−	749.465i	−0.343631	−	0.165484i	0.254106	−	0.967176i	\(-0.418219\pi\)
							−0.597737	+	0.801692i	\(0.703933\pi\)
\(31\)	−864.764	−	1497.82i	−0.161619	−	0.279933i	0.773830	−	0.633393i	\(-0.218338\pi\)
							−0.935450	+	0.353460i	\(0.885005\pi\)
\(37\)	−2637.51	+	1798.22i	−0.316730	+	0.215943i	−0.711241	−	0.702948i	\(-0.751866\pi\)
							0.394511	+	0.918891i	\(0.370914\pi\)
\(41\)	2252.98	+	2825.14i	0.209313	+	0.262471i	0.875395	−	0.483408i	\(-0.160601\pi\)
							−0.666082	+	0.745879i	\(0.732030\pi\)
\(43\)	2976.43	−	3732.32i	0.245485	−	0.307828i	−0.643789	−	0.765203i	\(-0.722639\pi\)
							0.889274	+	0.457375i	\(0.151210\pi\)
\(47\)	2369.66	+	730.945i	0.156474	+	0.0482658i	0.372003	−	0.928232i	\(-0.378671\pi\)
							−0.215529	+	0.976498i	\(0.569147\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.2.11	✓	264
49.25	even	21	inner	49.6.g.a.25.11	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.2.11	✓	264	1.1	even	1	trivial
49.6.g.a.25.11	yes	264	49.25	even	21	inner