Embedded newform 61.7.j.a.8.11

• Label: 61.7.j.a.8.11
• 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			8.11
Character	\(\chi\)	\(=\)	61.8
Dual form			61.7.j.a.23.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.80061 - 0.918727i) q^{2} +(-14.6267 - 20.1319i) q^{3} +(-28.0646 - 9.11873i) q^{4} +(-3.89918 - 1.26692i) q^{5} +(66.3479 + 130.215i) q^{6} +(92.8806 + 586.425i) q^{7} +(489.314 + 249.318i) q^{8} +(33.9202 - 104.396i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 16 q^{2} - 10 q^{3} - 10 q^{4} - 230 q^{5} + 1384 q^{6} + 248 q^{7} + 632 q^{8} + 13602 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.80061	−	0.918727i	−0.725077	−	0.114841i	−0.217023	−	0.976166i	\(-0.569635\pi\)
							−0.508053	+	0.861326i	\(0.669635\pi\)
\(3\)	−14.6267	−	20.1319i	−0.541728	−	0.745625i	0.447133	−	0.894468i	\(-0.352445\pi\)
							−0.988861	+	0.148843i	\(0.952445\pi\)
\(5\)	−3.89918	−	1.26692i	−0.0311934	−	0.0101354i	0.293379	−	0.955996i	\(-0.405220\pi\)
							−0.324572	+	0.945861i	\(0.605220\pi\)
\(7\)	92.8806	+	586.425i	0.270789	+	1.70969i	0.630132	+	0.776488i	\(0.283001\pi\)
							−0.359343	+	0.933205i	\(0.616999\pi\)
\(11\)	−14.7331	+	14.7331i	−0.0110692	+	0.0110692i	−0.712620	−	0.701551i	\(-0.752491\pi\)
							0.701551	+	0.712620i	\(0.252491\pi\)
\(13\)	−1081.53			−0.492275			−0.246137	−	0.969235i	\(-0.579161\pi\)
							−0.246137	+	0.969235i	\(0.579161\pi\)
\(17\)	−2536.95	−	4979.04i	−0.516374	−	1.01344i	−0.991076	−	0.133295i	\(-0.957444\pi\)
							0.474702	−	0.880146i	\(-0.342556\pi\)
\(19\)	3620.44	+	4983.11i	0.527839	+	0.726507i	0.986799	−	0.161949i	\(-0.0517781\pi\)
							−0.458961	+	0.888457i	\(0.651778\pi\)
\(23\)	8996.89	−	4584.15i	0.739450	−	0.376769i	−0.0433350	−	0.999061i	\(-0.513798\pi\)
							0.782785	+	0.622292i	\(0.213798\pi\)
\(29\)	12748.0	−	12748.0i	0.522695	−	0.522695i	−0.395689	−	0.918384i	\(-0.629494\pi\)
							0.918384	+	0.395689i	\(0.129494\pi\)
\(31\)	40651.9	−	6438.63i	1.36457	−	0.216127i	0.569174	−	0.822217i	\(-0.307263\pi\)
							0.795396	+	0.606090i	\(0.207263\pi\)
\(37\)	59072.0	−	9356.09i	1.16621	−	0.184710i	0.456842	−	0.889548i	\(-0.348980\pi\)
							0.709368	+	0.704838i	\(0.248980\pi\)
\(41\)	−813.044	+	1119.06i	−0.0117968	+	0.0162368i	−0.814875	−	0.579637i	\(-0.803194\pi\)
							0.803078	+	0.595874i	\(0.203194\pi\)
\(43\)	−62170.2	−	31677.3i	−0.781946	−	0.398421i	0.0169761	−	0.999856i	\(-0.494596\pi\)
							−0.798922	+	0.601434i	\(0.794596\pi\)
\(47\)	110022.			1.05971			0.529855	−	0.848088i	\(-0.322246\pi\)
							0.529855	+	0.848088i	\(0.322246\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.7.j.a.8.11	✓	240
61.23	odd	20	inner	61.7.j.a.23.11	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.7.j.a.8.11	✓	240	1.1	even	1	trivial
61.7.j.a.23.11	yes	240	61.23	odd	20	inner