Embedded newform 49.4.e.a.15.4

• Label: 49.4.e.a.15.4
• Level: $49$
• Weight: $4$
• Character: 49.15
• Analytic conductor: $2.891$
• Analytic rank: $0$
• Dimension: $78$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89109359028\)
Analytic rank:	\(0\)
Dimension:	\(78\)
Relative dimension:	\(13\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			15.4
Character	\(\chi\)	\(=\)	49.15
Dual form			49.4.e.a.36.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.764068 + 3.34760i) q^{2} +(-5.99643 + 7.51929i) q^{3} +(-3.41489 - 1.64452i) q^{4} +(0.968529 - 1.21450i) q^{5} +(-20.5899 - 25.8189i) q^{6} +(18.5012 + 0.839908i) q^{7} +(-9.01256 + 11.3014i) q^{8} +(-14.5744 - 63.8547i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.764068	+	3.34760i	−0.270139	+	1.18356i	0.639710	+	0.768616i	\(0.279054\pi\)
							−0.909849	+	0.414940i	\(0.863803\pi\)
\(3\)	−5.99643	+	7.51929i	−1.15401	+	1.44709i	−0.280789	+	0.959769i	\(0.590596\pi\)
							−0.873225	+	0.487318i	\(0.837975\pi\)
\(5\)	0.968529	−	1.21450i	0.0866279	−	0.108628i	−0.736629	−	0.676297i	\(-0.763583\pi\)
							0.823257	+	0.567670i	\(0.192155\pi\)
\(7\)	18.5012	+	0.839908i	0.998971	+	0.0453508i
							\(11\)	3.27050	−	14.3290i	0.0896449	−	0.392760i	−0.910122	−	0.414340i	\(-0.864012\pi\)
0.999767	+	0.0215799i	\(0.00686962\pi\)
\(13\)	−15.5577	+	68.1626i	−0.331917	+	1.45422i	0.483497	+	0.875346i	\(0.339366\pi\)
							−0.815414	+	0.578878i	\(0.803491\pi\)
\(17\)	−63.7286	+	30.6901i	−0.909203	+	0.437849i	−0.829204	−	0.558946i	\(-0.811206\pi\)
							−0.0799990	+	0.996795i	\(0.525492\pi\)
\(19\)	87.5128			1.05667			0.528337	−	0.849035i	\(-0.322816\pi\)
							0.528337	+	0.849035i	\(0.322816\pi\)
\(23\)	−67.8425	−	32.6712i	−0.615049	−	0.296192i	0.100302	−	0.994957i	\(-0.468019\pi\)
							−0.715351	+	0.698765i	\(0.753733\pi\)
\(29\)	85.8713	−	41.3534i	0.549858	−	0.264798i	−0.138260	−	0.990396i	\(-0.544151\pi\)
							0.688119	+	0.725598i	\(0.258437\pi\)
\(31\)	24.8262			0.143836			0.0719181	−	0.997411i	\(-0.477088\pi\)
							0.0719181	+	0.997411i	\(0.477088\pi\)
\(37\)	118.552	−	57.0918i	0.526754	−	0.253671i	−0.151554	−	0.988449i	\(-0.548428\pi\)
							0.678308	+	0.734778i	\(0.262714\pi\)
\(41\)	−269.658	+	338.140i	−1.02716	+	1.28802i	−0.0702803	+	0.997527i	\(0.522389\pi\)
							−0.956878	+	0.290489i	\(0.906182\pi\)
\(43\)	−5.18157	−	6.49749i	−0.0183763	−	0.0230432i	0.772558	−	0.634944i	\(-0.218977\pi\)
							−0.790935	+	0.611901i	\(0.790405\pi\)
\(47\)	0.791524	−	3.46789i	0.00245650	−	0.0107626i	−0.973685	−	0.227899i	\(-0.926814\pi\)
							0.976141	+	0.217136i	\(0.0696716\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.15.4	✓	78
49.6	odd	14		2401.4.a.c.1.30		39
49.36	even	7	inner	49.4.e.a.36.4	yes	78
49.43	even	7		2401.4.a.d.1.30		39

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.4	✓	78	1.1	even	1	trivial
49.4.e.a.36.4	yes	78	49.36	even	7	inner
2401.4.a.c.1.30		39	49.6	odd	14
2401.4.a.d.1.30		39	49.43	even	7