Embedded newform 49.4.e.a.36.2

• Label: 49.4.e.a.36.2
• Level: $49$
• Weight: $4$
• Character: 49.36
• 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			36.2
Character	\(\chi\)	\(=\)	49.36
Dual form			49.4.e.a.15.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.892344 - 3.90961i) q^{2} +(5.02886 + 6.30599i) q^{3} +(-7.28105 + 3.50637i) q^{4} +(8.76329 + 10.9888i) q^{5} +(20.1665 - 25.2880i) q^{6} +(3.97541 + 18.0886i) q^{7} +(0.203423 + 0.255085i) q^{8} +(-8.46803 + 37.1008i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.892344	−	3.90961i	−0.315491	−	1.38226i	−0.845369	−	0.534183i	\(-0.820619\pi\)
							0.529878	−	0.848074i	\(-0.322238\pi\)
\(3\)	5.02886	+	6.30599i	0.967805	+	1.21359i	0.976915	+	0.213630i	\(0.0685287\pi\)
							−0.00911002	+	0.999959i	\(0.502900\pi\)
\(5\)	8.76329	+	10.9888i	0.783813	+	0.982870i	0.999979	+	0.00651323i	\(0.00207324\pi\)
							−0.216166	+	0.976357i	\(0.569355\pi\)
\(7\)	3.97541	+	18.0886i	0.214652	+	0.976691i
							\(11\)	−11.9449	−	52.3342i	−0.327412	−	1.43449i	−0.824044	−	0.566525i	\(-0.808287\pi\)
0.496632	−	0.867961i	\(-0.334570\pi\)
\(13\)	−14.3712	−	62.9644i	−0.306604	−	1.34332i	−0.859954	−	0.510372i	\(-0.829508\pi\)
							0.553350	−	0.832949i	\(-0.313349\pi\)
\(17\)	47.5604	+	22.9039i	0.678535	+	0.326765i	0.741214	−	0.671268i	\(-0.234250\pi\)
							−0.0626790	+	0.998034i	\(0.519964\pi\)
\(19\)	−11.4048			−0.137707			−0.0688535	−	0.997627i	\(-0.521934\pi\)
							−0.0688535	+	0.997627i	\(0.521934\pi\)
\(23\)	89.1380	−	42.9266i	0.808111	−	0.389166i	0.0162506	−	0.999868i	\(-0.494827\pi\)
							0.791860	+	0.610702i	\(0.209113\pi\)
\(29\)	−38.3802	−	18.4829i	−0.245759	−	0.118351i	0.306950	−	0.951726i	\(-0.400692\pi\)
							−0.552709	+	0.833374i	\(0.686406\pi\)
\(31\)	−67.2545			−0.389654			−0.194827	−	0.980838i	\(-0.562414\pi\)
							−0.194827	+	0.980838i	\(0.562414\pi\)
\(37\)	−386.104	−	185.938i	−1.71554	−	0.826162i	−0.990509	−	0.137449i	\(-0.956110\pi\)
							−0.725034	−	0.688713i	\(-0.758176\pi\)
\(41\)	28.5253	+	35.7695i	0.108656	+	0.136250i	0.833186	−	0.552993i	\(-0.186514\pi\)
							−0.724530	+	0.689244i	\(0.757943\pi\)
\(43\)	122.569	−	153.697i	0.434688	−	0.545082i	−0.515446	−	0.856922i	\(-0.672374\pi\)
							0.950135	+	0.311840i	\(0.100945\pi\)
\(47\)	126.548	+	554.444i	0.392744	+	1.72072i	0.654917	+	0.755701i	\(0.272704\pi\)
							−0.262173	+	0.965021i	\(0.584439\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.36.2	yes	78
49.8	even	7		2401.4.a.d.1.32		39
49.15	even	7	inner	49.4.e.a.15.2	✓	78
49.41	odd	14		2401.4.a.c.1.32		39

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.2	✓	78	49.15	even	7	inner
49.4.e.a.36.2	yes	78	1.1	even	1	trivial
2401.4.a.c.1.32		39	49.41	odd	14
2401.4.a.d.1.32		39	49.8	even	7