Embedded newform 61.6.g.a.3.11

• Label: 61.6.g.a.3.11
• Level: $61$
• Weight: $6$
• Character: 61.3
• Analytic conductor: $9.783$
• Analytic rank: $0$
• Dimension: $104$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	61.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.78341300859\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			3.11
Character	\(\chi\)	\(=\)	61.3
Dual form			61.6.g.a.41.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.95760 - 0.960982i) q^{2} +(2.65010 - 8.15618i) q^{3} +(-18.0646 - 13.1247i) q^{4} +(60.4383 + 43.9110i) q^{5} +(-15.6759 + 21.5760i) q^{6} +(11.3775 - 3.69676i) q^{7} +(99.3080 + 136.686i) q^{8} +(137.091 + 99.6024i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 5 q^{2} + 15 q^{3} + 465 q^{4} - 122 q^{5} + 445 q^{6} + 230 q^{7} - 5 q^{8} - 2841 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/61\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.95760	−	0.960982i	−0.522835	−	0.169879i	0.0356968	−	0.999363i	\(-0.488635\pi\)
							−0.558531	+	0.829483i	\(0.688635\pi\)
\(3\)	2.65010	−	8.15618i	0.170004	−	0.523219i	−0.829366	−	0.558706i	\(-0.811298\pi\)
							0.999370	+	0.0354868i	\(0.0112982\pi\)
\(5\)	60.4383	+	43.9110i	1.08115	+	0.785504i	0.977884	−	0.209150i	\(-0.0670696\pi\)
							0.103270	+	0.994653i	\(0.467070\pi\)
\(7\)	11.3775	−	3.69676i	0.0877608	−	0.0285152i	−0.264807	−	0.964301i	\(-0.585308\pi\)
							0.352568	+	0.935786i	\(0.385308\pi\)
\(11\)		−	576.962i		−	1.43769i	−0.695170	−	0.718845i	\(-0.744671\pi\)
							0.695170	−	0.718845i	\(-0.255329\pi\)
\(13\)	−183.922			−0.301838			−0.150919	−	0.988546i	\(-0.548223\pi\)
							−0.150919	+	0.988546i	\(0.548223\pi\)
\(17\)	788.476	−	1085.24i	0.661707	−	0.910762i	−0.337829	−	0.941207i	\(-0.609693\pi\)
							0.999536	+	0.0304455i	\(0.00969259\pi\)
\(19\)	347.047	−	1068.10i	0.220549	−	0.678779i	−0.778164	−	0.628061i	\(-0.783849\pi\)
							0.998713	−	0.0507183i	\(-0.0161511\pi\)
\(23\)	1260.36	−	1734.73i	0.496791	−	0.683775i	−0.484831	−	0.874608i	\(-0.661119\pi\)
							0.981622	+	0.190833i	\(0.0611189\pi\)
\(29\)			375.268i			0.0828602i	0.999141	+	0.0414301i	\(0.0131914\pi\)
							−0.999141	+	0.0414301i	\(0.986809\pi\)
\(31\)	1734.57	−	563.595i	0.324180	−	0.105333i	−0.142406	−	0.989808i	\(-0.545484\pi\)
							0.466586	+	0.884476i	\(0.345484\pi\)
\(37\)	−2401.32	+	780.236i	−0.288367	+	0.0936962i	−0.449629	−	0.893216i	\(-0.648444\pi\)
							0.161262	+	0.986912i	\(0.448444\pi\)
\(41\)	2900.60	+	8927.12i	0.269481	+	0.829376i	0.990627	+	0.136594i	\(0.0436155\pi\)
							−0.721146	+	0.692783i	\(0.756384\pi\)
\(43\)	5480.88	+	7543.78i	0.452042	+	0.622182i	0.972835	−	0.231501i	\(-0.0743636\pi\)
							−0.520793	+	0.853683i	\(0.674364\pi\)
\(47\)	−4754.69			−0.313962			−0.156981	−	0.987602i	\(-0.550176\pi\)
							−0.156981	+	0.987602i	\(0.550176\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.6.g.a.3.11	✓	104
61.41	even	10	inner	61.6.g.a.41.11	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.6.g.a.3.11	✓	104	1.1	even	1	trivial
61.6.g.a.41.11	yes	104	61.41	even	10	inner