Embedded newform 49.6.e.a.36.14

• Label: 49.6.e.a.36.14
• Level: $49$
• Weight: $6$
• Character: 49.36
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $138$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85880717084\)
Analytic rank:	\(0\)
Dimension:	\(138\)
Relative dimension:	\(23\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			36.14
Character	\(\chi\)	\(=\)	49.36
Dual form			49.6.e.a.15.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.349896 + 1.53300i) q^{2} +(-18.7448 - 23.5052i) q^{3} +(26.6034 - 12.8115i) q^{4} +(-56.1895 - 70.4594i) q^{5} +(29.4746 - 36.9600i) q^{6} +(13.7252 + 128.913i) q^{7} +(60.3208 + 75.6399i) q^{8} +(-147.055 + 644.291i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.349896	+	1.53300i	0.0618535	+	0.270998i	0.996393	−	0.0848623i	\(-0.0270450\pi\)
							−0.934539	+	0.355860i	\(0.884188\pi\)
\(3\)	−18.7448	−	23.5052i	−1.20248	−	1.50786i	−0.808244	−	0.588848i	\(-0.799582\pi\)
							−0.394233	−	0.919010i	\(-0.628990\pi\)
\(5\)	−56.1895	−	70.4594i	−1.00515	−	1.26042i	−0.965282	−	0.261210i	\(-0.915878\pi\)
							−0.0398657	−	0.999205i	\(-0.512693\pi\)
\(7\)	13.7252	+	128.913i	0.105870	+	0.994380i
							\(11\)	−88.9749	−	389.825i	−0.221710	−	0.971376i	−0.956190	−	0.292746i	\(-0.905431\pi\)
0.734480	−	0.678630i	\(-0.237426\pi\)
\(13\)	26.5719	+	116.419i	0.0436078	+	0.191058i	0.992040	−	0.125921i	\(-0.0401886\pi\)
							−0.948432	+	0.316979i	\(0.897331\pi\)
\(17\)	10.2166	+	4.92007i	0.00857403	+	0.00412904i	0.438166	−	0.898894i	\(-0.355628\pi\)
							−0.429592	+	0.903023i	\(0.641343\pi\)
\(19\)	−513.784			−0.326510			−0.163255	−	0.986584i	\(-0.552199\pi\)
							−0.163255	+	0.986584i	\(0.552199\pi\)
\(23\)	−2183.48	+	1051.51i	−0.860654	+	0.414469i	−0.811521	−	0.584323i	\(-0.801360\pi\)
							−0.0491332	+	0.998792i	\(0.515646\pi\)
\(29\)	−4930.30	−	2374.31i	−1.08863	−	0.524255i	−0.198561	−	0.980088i	\(-0.563627\pi\)
							−0.890065	+	0.455834i	\(0.849341\pi\)
\(31\)	−730.365			−0.136501			−0.0682505	−	0.997668i	\(-0.521742\pi\)
							−0.0682505	+	0.997668i	\(0.521742\pi\)
\(37\)	−1661.34	−	800.057i	−0.199505	−	0.0960764i	0.331464	−	0.943468i	\(-0.392457\pi\)
							−0.530969	+	0.847392i	\(0.678172\pi\)
\(41\)	−6718.10	−	8424.23i	−0.624147	−	0.782655i	0.364775	−	0.931096i	\(-0.381146\pi\)
							−0.988921	+	0.148441i	\(0.952575\pi\)
\(43\)	2189.72	−	2745.83i	0.180600	−	0.226466i	−0.683288	−	0.730149i	\(-0.739451\pi\)
							0.863888	+	0.503683i	\(0.168022\pi\)
\(47\)	−2300.75	−	10080.2i	−0.151923	−	0.665619i	−0.992325	−	0.123656i	\(-0.960538\pi\)
							0.840402	−	0.541964i	\(-0.182319\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.e.a.36.14	yes	138
49.15	even	7	inner	49.6.e.a.15.14	✓	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.e.a.15.14	✓	138	49.15	even	7	inner
49.6.e.a.36.14	yes	138	1.1	even	1	trivial