Embedded newform 49.4.g.a.16.5

• Label: 49.4.g.a.16.5
• Level: $49$
• Weight: $4$
• Character: 49.16
• 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			16.5
Character	\(\chi\)	\(=\)	49.16
Dual form			49.4.g.a.46.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03329 - 2.63279i) q^{2} +(0.108753 + 1.45121i) q^{3} +(0.000535829 - 0.000497176i) q^{4} +(8.19228 - 5.58540i) q^{5} +(3.70835 - 1.78585i) q^{6} +(-11.6777 - 14.3747i) q^{7} +(-20.3875 - 9.81812i) q^{8} +(24.6043 - 3.70850i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{10}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03329	−	2.63279i	−0.365324	−	0.930831i	−0.988835	−	0.149016i	\(-0.952389\pi\)
							0.623510	−	0.781815i	\(-0.285706\pi\)
\(3\)	0.108753	+	1.45121i	0.0209295	+	0.279285i	0.997843	+	0.0656487i	\(0.0209117\pi\)
							−0.976913	+	0.213636i	\(0.931469\pi\)
\(5\)	8.19228	−	5.58540i	0.732740	−	0.499574i	−0.138542	−	0.990357i	\(-0.544242\pi\)
							0.871282	+	0.490783i	\(0.163289\pi\)
\(7\)	−11.6777	−	14.3747i	−0.630535	−	0.776161i
							\(11\)	−28.8060	−	4.34181i	−0.789577	−	0.119010i	−0.258140	−	0.966107i	\(-0.583110\pi\)
−0.531437	+	0.847098i	\(0.678348\pi\)
\(13\)	1.24064	−	1.55571i	0.0264686	−	0.0331906i	−0.768421	−	0.639945i	\(-0.778957\pi\)
							0.794890	+	0.606754i	\(0.207529\pi\)
\(17\)	124.586	−	38.4296i	1.77744	−	0.548267i	0.780407	−	0.625272i	\(-0.215012\pi\)
							0.997031	+	0.0770048i	\(0.0245357\pi\)
\(19\)	−32.4196	+	56.1524i	−0.391451	+	0.678013i	−0.992641	−	0.121093i	\(-0.961360\pi\)
							0.601190	+	0.799106i	\(0.294693\pi\)
\(23\)	175.231	+	54.0517i	1.58862	+	0.490024i	0.958168	−	0.286205i	\(-0.0923938\pi\)
							0.630451	+	0.776229i	\(0.282870\pi\)
\(29\)	−60.6947	+	265.921i	−0.388646	+	1.70277i	0.280681	+	0.959801i	\(0.409440\pi\)
							−0.669327	+	0.742968i	\(0.733418\pi\)
\(31\)	−105.697	−	183.073i	−0.612381	−	1.06068i	−0.990838	−	0.135056i	\(-0.956878\pi\)
							0.378457	−	0.925619i	\(-0.376455\pi\)
\(37\)	−1.14244	−	1.06002i	−0.00507609	−	0.00470992i	0.677631	−	0.735402i	\(-0.263007\pi\)
							−0.682707	+	0.730693i	\(0.739197\pi\)
\(41\)	169.603	+	81.6763i	0.646036	+	0.311114i	0.728057	−	0.685516i	\(-0.240423\pi\)
							−0.0820217	+	0.996631i	\(0.526138\pi\)
\(43\)	77.2080	−	37.1814i	0.273816	−	0.131863i	−0.291937	−	0.956438i	\(-0.594300\pi\)
							0.565753	+	0.824575i	\(0.308585\pi\)
\(47\)	−56.3541	−	143.588i	−0.174896	−	0.445627i	0.816366	−	0.577534i	\(-0.195985\pi\)
							−0.991262	+	0.131908i	\(0.957890\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.16.5	✓	156
49.12	odd	42		2401.4.a.f.1.23		78
49.37	even	21		2401.4.a.g.1.23		78
49.46	even	21	inner	49.4.g.a.46.5	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.16.5	✓	156	1.1	even	1	trivial
49.4.g.a.46.5	yes	156	49.46	even	21	inner
2401.4.a.f.1.23		78	49.12	odd	42
2401.4.a.g.1.23		78	49.37	even	21