Embedded newform 61.6.g.a.3.15

• Label: 61.6.g.a.3.15
• Level: $61$
• Weight: $6$
• Character: 61.3
• Analytic conductor: $9.783$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• 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			3.15
Character	\(\chi\)	\(=\)	61.3
Dual form			61.6.g.a.41.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.632478 + 0.205504i) q^{2} +(8.01843 - 24.6782i) q^{3} +(-25.5307 - 18.5492i) q^{4} +(-4.43129 - 3.21952i) q^{5} +(10.1430 - 13.9606i) q^{6} +(128.232 - 41.6652i) q^{7} +(-24.8443 - 34.1952i) q^{8} +(-348.127 - 252.929i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 5 q^{2} + 15 q^{3} + 465 q^{4} - 122 q^{5} + 445 q^{6} + 230 q^{7} - 5 q^{8} - 2841 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.632478	+	0.205504i	0.111807	+	0.0363284i	0.364386	−	0.931248i	\(-0.381279\pi\)
							−0.252579	+	0.967576i	\(0.581279\pi\)
\(3\)	8.01843	−	24.6782i	0.514383	−	1.58311i	−0.270020	−	0.962855i	\(-0.587030\pi\)
							0.784402	−	0.620252i	\(-0.212970\pi\)
\(5\)	−4.43129	−	3.21952i	−0.0792693	−	0.0575925i	0.547445	−	0.836842i	\(-0.315601\pi\)
							−0.626714	+	0.779249i	\(0.715601\pi\)
\(7\)	128.232	−	41.6652i	0.989128	−	0.321387i	0.230615	−	0.973045i	\(-0.425926\pi\)
							0.758513	+	0.651658i	\(0.225926\pi\)
\(11\)			82.9904i			0.206798i	0.994640	+	0.103399i	\(0.0329718\pi\)
							−0.994640	+	0.103399i	\(0.967028\pi\)
\(13\)	−526.185			−0.863536			−0.431768	−	0.901985i	\(-0.642110\pi\)
							−0.431768	+	0.901985i	\(0.642110\pi\)
\(17\)	−347.717	+	478.592i	−0.291813	+	0.401646i	−0.929602	−	0.368565i	\(-0.879849\pi\)
							0.637789	+	0.770211i	\(0.279849\pi\)
\(19\)	144.764	−	445.539i	0.0919979	−	0.283140i	−0.894462	−	0.447144i	\(-0.852441\pi\)
							0.986460	+	0.164004i	\(0.0524410\pi\)
\(23\)	−812.982	+	1118.97i	−0.320451	+	0.441063i	−0.938605	−	0.344994i	\(-0.887881\pi\)
							0.618154	+	0.786057i	\(0.287881\pi\)
\(29\)		−	5340.16i		−	1.17912i	−0.807723	−	0.589561i	\(-0.799301\pi\)
							0.807723	−	0.589561i	\(-0.200699\pi\)
\(31\)	7314.52	−	2376.63i	1.36704	−	0.444179i	0.468654	−	0.883382i	\(-0.344739\pi\)
							0.898388	+	0.439203i	\(0.144739\pi\)
\(37\)	7607.74	−	2471.90i	0.913590	−	0.296843i	0.185755	−	0.982596i	\(-0.440527\pi\)
							0.727835	+	0.685753i	\(0.240527\pi\)
\(41\)	351.123	+	1080.65i	0.0326212	+	0.100398i	0.966041	−	0.258387i	\(-0.0831911\pi\)
							−0.933420	+	0.358785i	\(0.883191\pi\)
\(43\)	−2713.16	−	3734.34i	−0.223771	−	0.307994i	0.682339	−	0.731035i	\(-0.260963\pi\)
							−0.906111	+	0.423041i	\(0.860963\pi\)
\(47\)	17515.3			1.15657			0.578287	−	0.815833i	\(-0.303721\pi\)
							0.578287	+	0.815833i	\(0.303721\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.6.g.a.3.15	✓	104
61.41	even	10	inner	61.6.g.a.41.15	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.6.g.a.3.15	✓	104	1.1	even	1	trivial
61.6.g.a.41.15	yes	104	61.41	even	10	inner