Embedded newform 49.4.g.a.11.3

• Label: 49.4.g.a.11.3
• 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			11.3
Character	\(\chi\)	\(=\)	49.11
Dual form			49.4.g.a.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.81651 + 2.61334i) q^{2} +(3.09679 - 0.466766i) q^{3} +(0.505348 - 6.74340i) q^{4} +(-3.96357 + 10.0990i) q^{5} +(-7.50233 + 9.40762i) q^{6} +(11.1259 + 14.8059i) q^{7} +(-2.96496 - 3.71794i) q^{8} +(-16.4282 + 5.06744i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.81651	+	2.61334i	−0.995788	+	0.923956i	−0.997105	−	0.0760429i	\(-0.975771\pi\)
							0.00131657	+	0.999999i	\(0.499581\pi\)
\(3\)	3.09679	−	0.466766i	0.595977	−	0.0898291i	0.155875	−	0.987777i	\(-0.450180\pi\)
							0.440102	+	0.897948i	\(0.354942\pi\)
\(5\)	−3.96357	+	10.0990i	−0.354512	+	0.903283i	0.636713	+	0.771101i	\(0.280294\pi\)
							−0.991225	+	0.132182i	\(0.957802\pi\)
\(7\)	11.1259	+	14.8059i	0.600740	+	0.799444i
							\(11\)	−54.5080	−	16.8135i	−1.49407	−	0.460860i	−0.563234	−	0.826297i	\(-0.690443\pi\)
−0.930836	+	0.365438i	\(0.880919\pi\)
\(13\)	14.0625	+	61.6117i	0.300018	+	1.31446i	0.870100	+	0.492875i	\(0.164054\pi\)
							−0.570083	+	0.821587i	\(0.693089\pi\)
\(17\)	54.5946	−	37.2220i	0.778890	−	0.531038i	−0.107374	−	0.994219i	\(-0.534244\pi\)
							0.886264	+	0.463180i	\(0.153292\pi\)
\(19\)	20.4413	+	35.4054i	0.246819	+	0.427503i	0.962642	−	0.270779i	\(-0.0872813\pi\)
							−0.715822	+	0.698283i	\(0.753948\pi\)
\(23\)	43.9204	+	29.9444i	0.398176	+	0.271472i	0.745810	−	0.666159i	\(-0.232063\pi\)
							−0.347634	+	0.937630i	\(0.613015\pi\)
\(29\)	216.488	+	104.255i	1.38623	+	0.667574i	0.970319	−	0.241830i	\(-0.0777475\pi\)
							0.415914	+	0.909404i	\(0.363462\pi\)
\(31\)	96.9682	−	167.954i	0.561806	−	0.973077i	−0.435533	−	0.900173i	\(-0.643440\pi\)
							0.997339	−	0.0729043i	\(-0.0232268\pi\)
\(37\)	−13.6892	−	182.669i	−0.0608239	−	0.811638i	−0.941305	−	0.337557i	\(-0.890399\pi\)
							0.880481	−	0.474081i	\(-0.157220\pi\)
\(41\)	−71.1605	−	89.2324i	−0.271059	−	0.339897i	0.627608	−	0.778530i	\(-0.284034\pi\)
							−0.898666	+	0.438633i	\(0.855463\pi\)
\(43\)	64.4015	−	80.7569i	0.228398	−	0.286403i	−0.654406	−	0.756143i	\(-0.727081\pi\)
							0.882804	+	0.469741i	\(0.155653\pi\)
\(47\)	157.175	−	145.837i	0.487794	−	0.452607i	−0.397524	−	0.917592i	\(-0.630131\pi\)
							0.885319	+	0.464985i	\(0.153940\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.11.3	yes	156
49.3	odd	42		2401.4.a.f.1.64		78
49.9	even	21	inner	49.4.g.a.9.3	✓	156
49.46	even	21		2401.4.a.g.1.64		78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.9.3	✓	156	49.9	even	21	inner
49.4.g.a.11.3	yes	156	1.1	even	1	trivial
2401.4.a.f.1.64		78	49.3	odd	42
2401.4.a.g.1.64		78	49.46	even	21