Embedded newform 49.4.g.a.2.9

• Label: 49.4.g.a.2.9
• Level: $49$
• Weight: $4$
• Character: 49.2
• 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			2.9
Character	\(\chi\)	\(=\)	49.2
Dual form			49.4.g.a.25.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.871702 + 0.268884i) q^{2} +(-2.65215 + 6.75756i) q^{3} +(-5.92235 - 4.03779i) q^{4} +(-5.10865 + 0.770006i) q^{5} +(-4.12888 + 5.17746i) q^{6} +(-7.37832 + 16.9871i) q^{7} +(-8.62695 - 10.8178i) q^{8} +(-18.8383 - 17.4794i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.871702	+	0.268884i	0.308193	+	0.0950650i	0.444994	−	0.895534i	\(-0.353206\pi\)
							−0.136801	+	0.990599i	\(0.543682\pi\)
\(3\)	−2.65215	+	6.75756i	−0.510406	+	1.30049i	0.410909	+	0.911676i	\(0.365211\pi\)
							−0.921315	+	0.388817i	\(0.872884\pi\)
\(5\)	−5.10865	+	0.770006i	−0.456932	+	0.0688714i	−0.373475	−	0.927640i	\(-0.621834\pi\)
							−0.0834563	+	0.996511i	\(0.526596\pi\)
\(7\)	−7.37832	+	16.9871i	−0.398392	+	0.917215i
							\(11\)	0.907615	−	0.842144i	0.0248778	−	0.0230833i	−0.667636	−	0.744488i	\(-0.732694\pi\)
0.692514	+	0.721405i	\(0.256503\pi\)
\(13\)	10.4050	+	45.5873i	0.221987	+	0.972589i	0.955980	+	0.293432i	\(0.0947973\pi\)
							−0.733993	+	0.679157i	\(0.762346\pi\)
\(17\)	9.49190	+	126.661i	0.135419	+	1.80704i	0.489687	+	0.871898i	\(0.337111\pi\)
							−0.354268	+	0.935144i	\(0.615270\pi\)
\(19\)	74.5439	−	129.114i	0.900081	−	1.55899i	0.0726932	−	0.997354i	\(-0.476841\pi\)
							0.827388	−	0.561631i	\(-0.189826\pi\)
\(23\)	5.92266	−	79.0324i	0.0536939	−	0.716495i	−0.903784	−	0.427989i	\(-0.859222\pi\)
							0.957478	−	0.288507i	\(-0.0931587\pi\)
\(29\)	57.8111	+	27.8404i	0.370181	+	0.178270i	0.609720	−	0.792617i	\(-0.291282\pi\)
							−0.239539	+	0.970887i	\(0.576996\pi\)
\(31\)	−15.0671	−	26.0969i	−0.0872944	−	0.151198i	0.819072	−	0.573690i	\(-0.194489\pi\)
							−0.906367	+	0.422492i	\(0.861155\pi\)
\(37\)	−95.1743	+	64.8887i	−0.422880	+	0.288315i	−0.756002	−	0.654569i	\(-0.772850\pi\)
							0.333122	+	0.942884i	\(0.391898\pi\)
\(41\)	−18.1258	−	22.7290i	−0.0690433	−	0.0865776i	0.746110	−	0.665822i	\(-0.231919\pi\)
							−0.815154	+	0.579245i	\(0.803348\pi\)
\(43\)	−197.391	+	247.521i	−0.700043	+	0.877827i	−0.997027	−	0.0770559i	\(-0.975448\pi\)
							0.296983	+	0.954883i	\(0.404019\pi\)
\(47\)	187.529	+	57.8451i	0.581999	+	0.179523i	0.571755	−	0.820425i	\(-0.306263\pi\)
							0.0102442	+	0.999948i	\(0.496739\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.2.9	✓	156
49.5	odd	42		2401.4.a.f.1.45		78
49.25	even	21	inner	49.4.g.a.25.9	yes	156
49.44	even	21		2401.4.a.g.1.45		78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.2.9	✓	156	1.1	even	1	trivial
49.4.g.a.25.9	yes	156	49.25	even	21	inner
2401.4.a.f.1.45		78	49.5	odd	42
2401.4.a.g.1.45		78	49.44	even	21