Embedded newform 49.4.g.a.11.5

• Label: 49.4.g.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	49.11
Dual form			49.4.g.a.9.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38163 + 1.28196i) q^{2} +(-8.55569 + 1.28956i) q^{3} +(-0.332376 + 4.43525i) q^{4} +(5.71607 - 14.5643i) q^{5} +(10.1676 - 12.7497i) q^{6} +(17.7236 - 5.37350i) q^{7} +(-14.6276 - 18.3425i) q^{8} +(45.7364 - 14.1078i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38163	+	1.28196i	−0.488478	+	0.453242i	−0.885549	−	0.464547i	\(-0.846217\pi\)
							0.397070	+	0.917788i	\(0.370027\pi\)
\(3\)	−8.55569	+	1.28956i	−1.64654	+	0.248176i	−0.905829	−	0.423644i	\(-0.860751\pi\)
							−0.740714	+	0.671820i	\(0.765513\pi\)
\(5\)	5.71607	−	14.5643i	0.511261	−	1.30267i	−0.409417	−	0.912347i	\(-0.634268\pi\)
							0.920678	−	0.390324i	\(-0.127637\pi\)
\(7\)	17.7236	−	5.37350i	0.956984	−	0.290142i
							\(11\)	−52.4248	−	16.1709i	−1.43697	−	0.443246i	−0.524156	−	0.851622i	\(-0.675619\pi\)
−0.912814	+	0.408376i	\(0.866095\pi\)
\(13\)	−0.201194	−	0.881490i	−0.00429240	−	0.0188062i	0.972736	−	0.231914i	\(-0.0744987\pi\)
							−0.977029	+	0.213107i	\(0.931642\pi\)
\(17\)	84.4030	−	57.5450i	1.20416	−	0.820983i	0.216301	−	0.976327i	\(-0.430601\pi\)
							0.987860	+	0.155344i	\(0.0496485\pi\)
\(19\)	−55.2064	−	95.6202i	−0.666590	−	1.15457i	−0.978852	−	0.204571i	\(-0.934420\pi\)
							0.312262	−	0.949996i	\(-0.398913\pi\)
\(23\)	−19.2729	−	13.1401i	−0.174725	−	0.119126i	0.472800	−	0.881170i	\(-0.343243\pi\)
							−0.647525	+	0.762044i	\(0.724196\pi\)
\(29\)	−133.865	−	64.4659i	−0.857175	−	0.412794i	−0.0469391	−	0.998898i	\(-0.514947\pi\)
							−0.810236	+	0.586104i	\(0.800661\pi\)
\(31\)	−63.3788	+	109.775i	−0.367199	+	0.636007i	−0.989126	−	0.147068i	\(-0.953017\pi\)
							0.621928	+	0.783075i	\(0.286350\pi\)
\(37\)	6.86817	+	91.6493i	0.0305168	+	0.407218i	0.991454	+	0.130453i	\(0.0416432\pi\)
							−0.960938	+	0.276765i	\(0.910738\pi\)
\(41\)	−123.068	−	154.323i	−0.468782	−	0.587834i	0.490090	−	0.871672i	\(-0.336964\pi\)
							−0.958872	+	0.283838i	\(0.908392\pi\)
\(43\)	−16.5160	+	20.7104i	−0.0585735	+	0.0734488i	−0.810255	−	0.586077i	\(-0.800672\pi\)
							0.751682	+	0.659526i	\(0.229243\pi\)
\(47\)	112.263	−	104.165i	0.348409	−	0.323276i	−0.486483	−	0.873690i	\(-0.661721\pi\)
							0.834892	+	0.550414i	\(0.185530\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.11.5	yes	156
49.3	odd	42		2401.4.a.f.1.51		78
49.9	even	21	inner	49.4.g.a.9.5	✓	156
49.46	even	21		2401.4.a.g.1.51		78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.9.5	✓	156	49.9	even	21	inner
49.4.g.a.11.5	yes	156	1.1	even	1	trivial
2401.4.a.f.1.51		78	49.3	odd	42
2401.4.a.g.1.51		78	49.46	even	21