Embedded newform 49.6.g.a.2.14

• Label: 49.6.g.a.2.14
• 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			2.14
Character	\(\chi\)	\(=\)	49.2
Dual form			49.6.g.a.25.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.73931 + 1.15342i) q^{2} +(0.0993195 - 0.253062i) q^{3} +(-13.7876 - 9.40021i) q^{4} +(41.9031 - 6.31588i) q^{5} +(0.663274 - 0.831720i) q^{6} +(64.1502 - 112.658i) q^{7} +(-118.788 - 148.955i) q^{8} +(178.077 + 165.232i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.73931	+	1.15342i	0.661023	+	0.203899i	0.607074	−	0.794645i	\(-0.292343\pi\)
							0.0539488	+	0.998544i	\(0.482819\pi\)
\(3\)	0.0993195	−	0.253062i	0.00637135	−	0.0162339i	−0.927653	−	0.373444i	\(-0.878177\pi\)
							0.934024	+	0.357210i	\(0.116272\pi\)
\(5\)	41.9031	−	6.31588i	0.749586	−	0.112982i	0.236871	−	0.971541i	\(-0.423878\pi\)
							0.512715	+	0.858559i	\(0.328640\pi\)
\(7\)	64.1502	−	112.658i	0.494826	−	0.868992i
							\(11\)	391.800	−	363.537i	0.976298	−	0.905872i	−0.0192794	−	0.999814i	\(-0.506137\pi\)
0.995578	+	0.0939416i	\(0.0299467\pi\)
\(13\)	−122.837	−	538.183i	−0.201591	−	0.883226i	−0.969969	−	0.243230i	\(-0.921793\pi\)
							0.768378	−	0.639996i	\(-0.221064\pi\)
\(17\)	95.5982	+	1275.67i	0.0802282	+	1.07057i	0.881473	+	0.472235i	\(0.156552\pi\)
							−0.801245	+	0.598337i	\(0.795828\pi\)
\(19\)	−8.21888	+	14.2355i	−0.00522310	+	0.00904668i	−0.868625	−	0.495470i	\(-0.834996\pi\)
							0.863402	+	0.504517i	\(0.168329\pi\)
\(23\)	113.036	−	1508.36i	0.0445550	−	0.594545i	−0.929586	−	0.368605i	\(-0.879836\pi\)
							0.974141	−	0.225940i	\(-0.0725453\pi\)
\(29\)	429.722	+	206.943i	0.0948840	+	0.0456937i	0.480724	−	0.876872i	\(-0.340374\pi\)
							−0.385840	+	0.922566i	\(0.626088\pi\)
\(31\)	626.100	+	1084.44i	0.117014	+	0.202675i	0.918583	−	0.395228i	\(-0.129334\pi\)
							−0.801569	+	0.597903i	\(0.796001\pi\)
\(37\)	−10672.5	+	7276.40i	−1.28163	+	0.873800i	−0.996410	−	0.0846622i	\(-0.973019\pi\)
							−0.285220	+	0.958462i	\(0.592067\pi\)
\(41\)	299.854	+	376.006i	0.0278581	+	0.0349329i	0.795565	−	0.605868i	\(-0.207174\pi\)
							−0.767707	+	0.640801i	\(0.778603\pi\)
\(43\)	−10855.8	+	13612.8i	−0.895348	+	1.12273i	0.0965040	+	0.995333i	\(0.469234\pi\)
							−0.991852	+	0.127398i	\(0.959337\pi\)
\(47\)	26997.0	+	8327.47i	1.78267	+	0.549880i	0.997582	−	0.0695048i	\(-0.0221419\pi\)
							0.785087	+	0.619385i	\(0.212618\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.2.14	✓	264
49.25	even	21	inner	49.6.g.a.25.14	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.2.14	✓	264	1.1	even	1	trivial
49.6.g.a.25.14	yes	264	49.25	even	21	inner