Embedded newform 49.4.g.a.11.6

• Label: 49.4.g.a.11.6
• Level: $49$
• Weight: $4$
• Character: 49.11
• Analytic conductor: $2.891$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89109359028\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(13\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			11.6
Character	\(\chi\)	\(=\)	49.11
Dual form			49.4.g.a.9.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16645 + 1.08230i) q^{2} +(8.79637 - 1.32584i) q^{3} +(-0.408626 + 5.45273i) q^{4} +(6.71029 - 17.0976i) q^{5} +(-8.82553 + 11.0669i) q^{6} +(-9.75859 + 15.7407i) q^{7} +(-13.3617 - 16.7551i) q^{8} +(49.8178 - 15.3668i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16645	+	1.08230i	−0.412401	+	0.382652i	−0.858910	−	0.512127i	\(-0.828858\pi\)
							0.446509	+	0.894779i	\(0.352667\pi\)
\(3\)	8.79637	−	1.32584i	1.69286	−	0.255158i	0.769378	−	0.638794i	\(-0.220566\pi\)
							0.923484	+	0.383636i	\(0.125328\pi\)
\(5\)	6.71029	−	17.0976i	0.600187	−	1.52925i	−0.230824	−	0.972996i	\(-0.574142\pi\)
							0.831011	−	0.556256i	\(-0.187763\pi\)
\(7\)	−9.75859	+	15.7407i	−0.526914	+	0.849919i
							\(11\)	10.2100	+	3.14935i	0.279856	+	0.0863242i	0.431506	−	0.902110i	\(-0.357982\pi\)
−0.151650	+	0.988434i	\(0.548459\pi\)
\(13\)	11.0102	+	48.2390i	0.234899	+	1.02916i	0.945514	+	0.325580i	\(0.105560\pi\)
							−0.710615	+	0.703581i	\(0.751583\pi\)
\(17\)	−60.2371	+	41.0690i	−0.859391	+	0.585923i	−0.910843	−	0.412753i	\(-0.864567\pi\)
							0.0514522	+	0.998675i	\(0.483615\pi\)
\(19\)	−47.6119	−	82.4662i	−0.574890	−	0.995739i	−0.996054	−	0.0887545i	\(-0.971711\pi\)
							0.421163	−	0.906985i	\(-0.361622\pi\)
\(23\)	−83.3246	−	56.8097i	−0.755408	−	0.515028i	0.123311	−	0.992368i	\(-0.460649\pi\)
							−0.878719	+	0.477340i	\(0.841601\pi\)
\(29\)	8.24463	+	3.97040i	0.0527927	+	0.0254236i	0.460094	−	0.887870i	\(-0.347816\pi\)
							−0.407301	+	0.913294i	\(0.633530\pi\)
\(31\)	−49.3758	+	85.5214i	−0.286070	+	0.495487i	−0.972868	−	0.231361i	\(-0.925682\pi\)
							0.686798	+	0.726848i	\(0.259016\pi\)
\(37\)	1.29081	+	17.2246i	0.00573533	+	0.0765327i	0.999337	−	0.0364087i	\(-0.0115918\pi\)
							−0.993602	+	0.112941i	\(0.963973\pi\)
\(41\)	48.9837	+	61.4236i	0.186585	+	0.233970i	0.866322	−	0.499486i	\(-0.166478\pi\)
							−0.679737	+	0.733456i	\(0.737906\pi\)
\(43\)	176.799	−	221.699i	0.627015	−	0.786252i	−0.362297	−	0.932063i	\(-0.618008\pi\)
							0.989313	+	0.145810i	\(0.0465790\pi\)
\(47\)	44.0165	−	40.8413i	0.136606	−	0.126751i	−0.608911	−	0.793239i	\(-0.708393\pi\)
							0.745517	+	0.666487i	\(0.232203\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.11.6	yes	156
49.3	odd	42		2401.4.a.f.1.48		78
49.9	even	21	inner	49.4.g.a.9.6	✓	156
49.46	even	21		2401.4.a.g.1.48		78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.9.6	✓	156	49.9	even	21	inner
49.4.g.a.11.6	yes	156	1.1	even	1	trivial
2401.4.a.f.1.48		78	49.3	odd	42
2401.4.a.g.1.48		78	49.46	even	21