Embedded newform 61.7.j.a.23.11

• Label: 61.7.j.a.23.11
• 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			23.11
Character	\(\chi\)	\(=\)	61.23
Dual form			61.7.j.a.8.11

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.80061	+	0.918727i	−0.725077	+	0.114841i	−0.508053	−	0.861326i	\(-0.669635\pi\)
							−0.217023	+	0.976166i	\(0.569635\pi\)
\(3\)	−14.6267	+	20.1319i	−0.541728	+	0.745625i	−0.988861	−	0.148843i	\(-0.952445\pi\)
							0.447133	+	0.894468i	\(0.352445\pi\)
\(5\)	−3.89918	+	1.26692i	−0.0311934	+	0.0101354i	−0.324572	−	0.945861i	\(-0.605220\pi\)
							0.293379	+	0.955996i	\(0.405220\pi\)
\(7\)	92.8806	−	586.425i	0.270789	−	1.70969i	−0.359343	−	0.933205i	\(-0.616999\pi\)
							0.630132	−	0.776488i	\(-0.283001\pi\)
\(11\)	−14.7331	−	14.7331i	−0.0110692	−	0.0110692i	0.701551	−	0.712620i	\(-0.252491\pi\)
							−0.712620	+	0.701551i	\(0.752491\pi\)
\(13\)	−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	0.474702	+	0.880146i	\(0.342556\pi\)
							−0.991076	+	0.133295i	\(0.957444\pi\)
\(19\)	3620.44	−	4983.11i	0.527839	−	0.726507i	−0.458961	−	0.888457i	\(-0.651778\pi\)
							0.986799	+	0.161949i	\(0.0517781\pi\)
\(23\)	8996.89	+	4584.15i	0.739450	+	0.376769i	0.782785	−	0.622292i	\(-0.213798\pi\)
							−0.0433350	+	0.999061i	\(0.513798\pi\)
\(29\)	12748.0	+	12748.0i	0.522695	+	0.522695i	0.918384	−	0.395689i	\(-0.129494\pi\)
							−0.395689	+	0.918384i	\(0.629494\pi\)
\(31\)	40651.9	+	6438.63i	1.36457	+	0.216127i	0.795396	−	0.606090i	\(-0.207263\pi\)
							0.569174	+	0.822217i	\(0.307263\pi\)
\(37\)	59072.0	+	9356.09i	1.16621	+	0.184710i	0.709368	−	0.704838i	\(-0.248980\pi\)
							0.456842	+	0.889548i	\(0.348980\pi\)
\(41\)	−813.044	−	1119.06i	−0.0117968	−	0.0162368i	0.803078	−	0.595874i	\(-0.203194\pi\)
							−0.814875	+	0.579637i	\(0.803194\pi\)
\(43\)	−62170.2	+	31677.3i	−0.781946	+	0.398421i	−0.798922	−	0.601434i	\(-0.794596\pi\)
							0.0169761	+	0.999856i	\(0.494596\pi\)
\(47\)	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.23.11	yes	240
61.8	odd	20	inner	61.7.j.a.8.11	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.7.j.a.8.11	✓	240	61.8	odd	20	inner
61.7.j.a.23.11	yes	240	1.1	even	1	trivial