Embedded newform 49.6.e.a.8.20

• Label: 49.6.e.a.8.20
• Level: $49$
• Weight: $6$
• Character: 49.8
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $138$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85880717084\)
Analytic rank:	\(0\)
Dimension:	\(138\)
Relative dimension:	\(23\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			8.20
Character	\(\chi\)	\(=\)	49.8
Dual form			49.6.e.a.43.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.98712 + 6.25365i) q^{2} +(7.50846 - 3.61589i) q^{3} +(-7.11609 + 31.1776i) q^{4} +(29.9404 - 14.4185i) q^{5} +(60.0580 + 28.9224i) q^{6} +(75.7713 + 105.194i) q^{7} +(0.148300 - 0.0714174i) q^{8} +(-108.206 + 135.686i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.98712	+	6.25365i	0.881606	+	1.10550i	0.993730	+	0.111805i	\(0.0356631\pi\)
							−0.112124	+	0.993694i	\(0.535765\pi\)
\(3\)	7.50846	−	3.61589i	0.481668	−	0.231959i	−0.177261	−	0.984164i	\(-0.556724\pi\)
							0.658930	+	0.752205i	\(0.271010\pi\)
\(5\)	29.9404	−	14.4185i	0.535590	−	0.257927i	−0.146480	−	0.989214i	\(-0.546794\pi\)
							0.682070	+	0.731287i	\(0.261080\pi\)
\(7\)	75.7713	+	105.194i	0.584467	+	0.811418i
							\(11\)	295.204	+	370.174i	0.735598	+	0.922410i	0.999107	−	0.0422464i	\(-0.0134515\pi\)
−0.263509	+	0.964657i	\(0.584880\pi\)
\(13\)	−480.854	−	602.972i	−0.789142	−	0.989553i	−0.999928	−	0.0120205i	\(-0.996174\pi\)
							0.210786	−	0.977532i	\(-0.432398\pi\)
\(17\)	−363.471	−	1592.47i	−0.305034	−	1.33644i	−0.862423	−	0.506188i	\(-0.831054\pi\)
							0.557389	−	0.830251i	\(-0.311803\pi\)
\(19\)	−821.238			−0.521897			−0.260949	−	0.965353i	\(-0.584035\pi\)
							−0.260949	+	0.965353i	\(0.584035\pi\)
\(23\)	171.373	−	750.835i	0.0675497	−	0.295955i	−0.929857	−	0.367920i	\(-0.880070\pi\)
							0.997407	+	0.0719655i	\(0.0229272\pi\)
\(29\)	−1896.55	−	8309.33i	−0.418764	−	1.83473i	−0.539427	−	0.842032i	\(-0.681359\pi\)
							0.120663	−	0.992693i	\(-0.461498\pi\)
\(31\)	7841.12			1.46546			0.732730	−	0.680519i	\(-0.238246\pi\)
							0.732730	+	0.680519i	\(0.238246\pi\)
\(37\)	−1622.52	−	7108.71i	−0.194843	−	0.853664i	−0.973948	−	0.226770i	\(-0.927183\pi\)
							0.779105	−	0.626893i	\(-0.215674\pi\)
\(41\)	2256.45	−	1086.65i	0.209636	−	0.100955i	−0.326120	−	0.945329i	\(-0.605741\pi\)
							0.535755	+	0.844373i	\(0.320027\pi\)
\(43\)	−10274.0	−	4947.72i	−0.847365	−	0.408069i	−0.0407661	−	0.999169i	\(-0.512980\pi\)
							−0.806599	+	0.591099i	\(0.798694\pi\)
\(47\)	−14623.5	−	18337.3i	−0.965620	−	1.21085i	−0.977503	−	0.210919i	\(-0.932354\pi\)
							0.0118839	−	0.999929i	\(-0.496217\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.e.a.8.20	✓	138
49.43	even	7	inner	49.6.e.a.43.20	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.e.a.8.20	✓	138	1.1	even	1	trivial
49.6.e.a.43.20	yes	138	49.43	even	7	inner