Embedded newform 49.4.e.a.15.3

• Label: 49.4.e.a.15.3
• 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			15.3
Character	\(\chi\)	\(=\)	49.15
Dual form			49.4.e.a.36.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.846844 + 3.71026i) q^{2} +(3.04811 - 3.82222i) q^{3} +(-5.84116 - 2.81296i) q^{4} +(-12.1071 + 15.1818i) q^{5} +(11.6002 + 14.5461i) q^{6} +(13.1709 + 13.0203i) q^{7} +(-3.59906 + 4.51308i) q^{8} +(0.689741 + 3.02195i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.846844	+	3.71026i	−0.299404	+	1.31178i	0.571613	+	0.820524i	\(0.306318\pi\)
							−0.871017	+	0.491253i	\(0.836539\pi\)
\(3\)	3.04811	−	3.82222i	0.586610	−	0.735586i	−0.396615	−	0.917985i	\(-0.629815\pi\)
							0.983224	+	0.182400i	\(0.0583865\pi\)
\(5\)	−12.1071	+	15.1818i	−1.08289	+	1.35790i	−0.153781	+	0.988105i	\(0.549145\pi\)
							−0.929112	+	0.369799i	\(0.879426\pi\)
\(7\)	13.1709	+	13.0203i	0.711162	+	0.703028i
							\(11\)	3.84932	−	16.8650i	0.105510	−	0.462271i	−0.894378	−	0.447312i	\(-0.852381\pi\)
0.999888	−	0.0149587i	\(-0.00476167\pi\)
\(13\)	6.63306	−	29.0614i	0.141514	−	0.620013i	−0.853570	−	0.520978i	\(-0.825567\pi\)
							0.995084	−	0.0990349i	\(-0.0315756\pi\)
\(17\)	30.1062	−	14.4984i	0.429519	−	0.206845i	−0.206618	−	0.978422i	\(-0.566246\pi\)
							0.636137	+	0.771576i	\(0.280531\pi\)
\(19\)	122.424			1.47821			0.739104	−	0.673592i	\(-0.235249\pi\)
							0.739104	+	0.673592i	\(0.235249\pi\)
\(23\)	35.7422	+	17.2126i	0.324033	+	0.156046i	0.588828	−	0.808259i	\(-0.299590\pi\)
							−0.264794	+	0.964305i	\(0.585304\pi\)
\(29\)	−257.246	+	123.883i	−1.64722	+	0.793260i	−0.647714	+	0.761884i	\(0.724275\pi\)
							−0.999508	+	0.0313759i	\(0.990011\pi\)
\(31\)	61.6857			0.357389			0.178695	−	0.983905i	\(-0.442813\pi\)
							0.178695	+	0.983905i	\(0.442813\pi\)
\(37\)	34.5744	−	16.6502i	0.153622	−	0.0739802i	−0.355493	−	0.934679i	\(-0.615687\pi\)
							0.509115	+	0.860699i	\(0.329973\pi\)
\(41\)	312.646	−	392.045i	1.19090	−	1.49335i	0.363634	−	0.931542i	\(-0.381536\pi\)
							0.827270	−	0.561804i	\(-0.189893\pi\)
\(43\)	−51.8869	−	65.0641i	−0.184016	−	0.230748i	0.681264	−	0.732038i	\(-0.261431\pi\)
							−0.865280	+	0.501289i	\(0.832859\pi\)
\(47\)	−5.83314	+	25.5567i	−0.0181032	+	0.0793154i	−0.983173	−	0.182679i	\(-0.941523\pi\)
							0.965069	+	0.261994i	\(0.0843802\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.15.3	✓	78
49.6	odd	14		2401.4.a.c.1.31		39
49.36	even	7	inner	49.4.e.a.36.3	yes	78
49.43	even	7		2401.4.a.d.1.31		39

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.3	✓	78	1.1	even	1	trivial
49.4.e.a.36.3	yes	78	49.36	even	7	inner
2401.4.a.c.1.31		39	49.6	odd	14
2401.4.a.d.1.31		39	49.43	even	7