Embedded newform 61.7.j.a.23.8

• Label: 61.7.j.a.23.8
• Level: $61$
• Weight: $7$
• Character: 61.23
• Analytic conductor: $14.033$
• Analytic rank: $0$
• Dimension: $240$
• 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			23.8
Character	\(\chi\)	\(=\)	61.23
Dual form			61.7.j.a.8.8

$q$-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 \(\left(\mathbb{Z}/61\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{19}{20}\right)\)

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	−0.674763	−	0.738034i	\(-0.735754\pi\)
							−0.413673	+	0.910426i	\(0.635754\pi\)
\(3\)	11.9365	−	16.4292i	0.442093	−	0.608489i	−0.528582	−	0.848882i	\(-0.677276\pi\)
							0.970676	+	0.240393i	\(0.0772762\pi\)
\(5\)	−207.954	+	67.5685i	−1.66363	+	0.540548i	−0.981629	−	0.190799i	\(-0.938892\pi\)
							−0.682005	+	0.731347i	\(0.738892\pi\)
\(7\)	−31.8422	+	201.044i	−0.0928345	+	0.586134i	0.896790	+	0.442456i	\(0.145893\pi\)
							−0.989625	+	0.143678i	\(0.954107\pi\)
\(11\)	200.760	+	200.760i	0.150834	+	0.150834i	0.778490	−	0.627657i	\(-0.215986\pi\)
							−0.627657	+	0.778490i	\(0.715986\pi\)
\(13\)	−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	−0.773625	−	0.633643i	\(-0.781559\pi\)
							0.967354	+	0.253430i	\(0.0815587\pi\)
\(19\)	5988.53	−	8242.50i	0.873090	−	1.20171i	−0.105197	−	0.994451i	\(-0.533547\pi\)
							0.978287	−	0.207254i	\(-0.0664528\pi\)
\(23\)	−4380.47	−	2231.96i	−0.360029	−	0.183444i	0.264614	−	0.964354i	\(-0.414755\pi\)
							−0.624643	+	0.780911i	\(0.714755\pi\)
\(29\)	−25596.0	−	25596.0i	−1.04949	−	1.04949i	−0.998710	−	0.0507817i	\(-0.983829\pi\)
							−0.0507817	−	0.998710i	\(-0.516171\pi\)
\(31\)	27997.0	+	4434.29i	0.939780	+	0.148847i	0.607487	−	0.794330i	\(-0.292178\pi\)
							0.332293	+	0.943176i	\(0.392178\pi\)
\(37\)	79632.0	+	12612.5i	1.57211	+	0.248997i	0.880773	−	0.473538i	\(-0.157023\pi\)
							0.691334	+	0.722535i	\(0.257023\pi\)
\(41\)	−57022.0	−	78484.1i	−0.827354	−	1.13875i	−0.988410	−	0.151810i	\(-0.951490\pi\)
							0.161056	−	0.986945i	\(-0.448510\pi\)
\(43\)	−4463.64	+	2274.34i	−0.0561414	+	0.0286055i	−0.481836	−	0.876261i	\(-0.660030\pi\)
							0.425694	+	0.904867i	\(0.360030\pi\)
\(47\)	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.23.8	yes	240
61.8	odd	20	inner	61.7.j.a.8.8	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.7.j.a.8.8	✓	240	61.8	odd	20	inner
61.7.j.a.23.8	yes	240	1.1	even	1	trivial