Embedded newform 49.6.g.a.2.12

• Label: 49.6.g.a.2.12
• 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			2.12
Character	\(\chi\)	\(=\)	49.2
Dual form			49.6.g.a.25.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.768968 - 0.237195i) q^{2} +(-8.72763 + 22.2376i) q^{3} +(-25.9046 - 17.6615i) q^{4} +(28.5592 - 4.30461i) q^{5} +(11.9859 - 15.0299i) q^{6} +(-107.947 + 71.7951i) q^{7} +(31.7861 + 39.8585i) q^{8} +(-240.209 - 222.882i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.768968	−	0.237195i	−0.135936	−	0.0419306i	0.226041	−	0.974118i	\(-0.427422\pi\)
							−0.361976	+	0.932187i	\(0.617898\pi\)
\(3\)	−8.72763	+	22.2376i	−0.559878	+	1.42654i	0.318199	+	0.948024i	\(0.396922\pi\)
							−0.878076	+	0.478521i	\(0.841173\pi\)
\(5\)	28.5592	−	4.30461i	0.510882	−	0.0770031i	0.111458	−	0.993769i	\(-0.464448\pi\)
							0.399425	+	0.916766i	\(0.369210\pi\)
\(7\)	−107.947	+	71.7951i	−0.832653	+	0.553796i
							\(11\)	432.017	−	400.853i	1.07651	−	0.998857i	0.0765131	−	0.997069i	\(-0.475621\pi\)
0.999998	−	0.00178816i	\(-0.000569191\pi\)
\(13\)	−206.502	−	904.746i	−0.338896	−	1.48480i	−0.801370	−	0.598169i	\(-0.795895\pi\)
							0.462474	−	0.886633i	\(-0.346962\pi\)
\(17\)	−156.851	−	2093.03i	−0.131633	−	1.75652i	−0.538124	−	0.842865i	\(-0.680867\pi\)
							0.406491	−	0.913655i	\(-0.366752\pi\)
\(19\)	−953.349	+	1651.25i	−0.605854	+	1.04937i	0.386062	+	0.922473i	\(0.373835\pi\)
							−0.991916	+	0.126897i	\(0.959498\pi\)
\(23\)	−24.4696	+	326.523i	−0.00964509	+	0.128705i	−0.999948	−	0.0102219i	\(-0.996746\pi\)
							0.990303	+	0.138927i	\(0.0443653\pi\)
\(29\)	3836.61	+	1847.61i	0.847134	+	0.407958i	0.806513	−	0.591216i	\(-0.201352\pi\)
							0.0406211	+	0.999175i	\(0.487066\pi\)
\(31\)	−3561.03	−	6167.89i	−0.665536	−	1.15274i	−0.979140	−	0.203189i	\(-0.934869\pi\)
							0.313603	−	0.949554i	\(-0.398464\pi\)
\(37\)	−5543.25	+	3779.32i	−0.665671	+	0.453847i	−0.848428	−	0.529311i	\(-0.822450\pi\)
							0.182756	+	0.983158i	\(0.441498\pi\)
\(41\)	−8496.33	−	10654.1i	−0.789353	−	0.989818i	−0.999925	−	0.0122394i	\(-0.996104\pi\)
							0.210572	−	0.977578i	\(-0.432467\pi\)
\(43\)	−2372.92	+	2975.55i	−0.195710	+	0.245412i	−0.869997	−	0.493057i	\(-0.835880\pi\)
							0.674287	+	0.738469i	\(0.264451\pi\)
\(47\)	−14695.8	−	4533.05i	−0.970395	−	0.299327i	−0.231266	−	0.972891i	\(-0.574287\pi\)
							−0.739129	+	0.673563i	\(0.764763\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.2.12	✓	264
49.25	even	21	inner	49.6.g.a.25.12	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.2.12	✓	264	1.1	even	1	trivial
49.6.g.a.25.12	yes	264	49.25	even	21	inner