Embedded newform 61.7.j.a.8.8

• Label: 61.7.j.a.8.8
• Level: $61$
• Weight: $7$
• Character: 61.8
• Analytic conductor: $14.033$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	61.j (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.0332991008\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			8.8
Character	\(\chi\)	\(=\)	61.8
Dual form			61.7.j.a.23.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.70749 - 1.37913i) q^{2} +(11.9365 + 16.4292i) q^{3} +(13.0507 + 4.24042i) q^{4} +(-207.954 - 67.5685i) q^{5} +(-81.2790 - 159.519i) q^{6} +(-31.8422 - 201.044i) q^{7} +(394.938 + 201.231i) q^{8} +(97.8350 - 301.105i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 16 q^{2} - 10 q^{3} - 10 q^{4} - 230 q^{5} + 1384 q^{6} + 248 q^{7} + 632 q^{8} + 13602 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.70749	−	1.37913i	−1.08844	−	0.172391i	−0.413673	−	0.910426i	\(-0.635754\pi\)
							−0.674763	+	0.738034i	\(0.735754\pi\)
\(3\)	11.9365	+	16.4292i	0.442093	+	0.608489i	0.970676	−	0.240393i	\(-0.0772762\pi\)
							−0.528582	+	0.848882i	\(0.677276\pi\)
\(5\)	−207.954	−	67.5685i	−1.66363	−	0.540548i	−0.682005	−	0.731347i	\(-0.738892\pi\)
							−0.981629	+	0.190799i	\(0.938892\pi\)
\(7\)	−31.8422	−	201.044i	−0.0928345	−	0.586134i	−0.989625	−	0.143678i	\(-0.954107\pi\)
							0.896790	−	0.442456i	\(-0.145893\pi\)
\(11\)	200.760	−	200.760i	0.150834	−	0.150834i	−0.627657	−	0.778490i	\(-0.715986\pi\)
							0.778490	+	0.627657i	\(0.215986\pi\)
\(13\)	−3041.73			−1.38449			−0.692246	−	0.721662i	\(-0.743379\pi\)
							−0.692246	+	0.721662i	\(0.743379\pi\)
\(17\)	951.788	+	1867.99i	0.193729	+	0.380214i	0.967354	−	0.253430i	\(-0.0815587\pi\)
							−0.773625	+	0.633643i	\(0.781559\pi\)
\(19\)	5988.53	+	8242.50i	0.873090	+	1.20171i	0.978287	+	0.207254i	\(0.0664528\pi\)
							−0.105197	+	0.994451i	\(0.533547\pi\)
\(23\)	−4380.47	+	2231.96i	−0.360029	+	0.183444i	−0.624643	−	0.780911i	\(-0.714755\pi\)
							0.264614	+	0.964354i	\(0.414755\pi\)
\(29\)	−25596.0	+	25596.0i	−1.04949	+	1.04949i	−0.0507817	+	0.998710i	\(0.516171\pi\)
							−0.998710	+	0.0507817i	\(0.983829\pi\)
\(31\)	27997.0	−	4434.29i	0.939780	−	0.148847i	0.332293	−	0.943176i	\(-0.392178\pi\)
							0.607487	+	0.794330i	\(0.292178\pi\)
\(37\)	79632.0	−	12612.5i	1.57211	−	0.248997i	0.691334	−	0.722535i	\(-0.257023\pi\)
							0.880773	+	0.473538i	\(0.157023\pi\)
\(41\)	−57022.0	+	78484.1i	−0.827354	+	1.13875i	0.161056	+	0.986945i	\(0.448510\pi\)
							−0.988410	+	0.151810i	\(0.951490\pi\)
\(43\)	−4463.64	−	2274.34i	−0.0561414	−	0.0286055i	0.425694	−	0.904867i	\(-0.360030\pi\)
							−0.481836	+	0.876261i	\(0.660030\pi\)
\(47\)	75951.8			0.731551			0.365775	−	0.930703i	\(-0.380804\pi\)
							0.365775	+	0.930703i	\(0.380804\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.7.j.a.8.8	✓	240
61.23	odd	20	inner	61.7.j.a.23.8	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.7.j.a.8.8	✓	240	1.1	even	1	trivial
61.7.j.a.23.8	yes	240	61.23	odd	20	inner