Embedded newform 240.4.bf.a.53.70

• Label: 240.4.bf.a.53.70
• Level: $240$
• Weight: $4$
• Character: 240.53
• Analytic conductor: $14.160$
• Analytic rank: $0$
• Dimension: $280$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	240.bf (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.1604584014\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(140\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			53.70
Character	\(\chi\)	\(=\)	240.53
Dual form			240.4.bf.a.77.70

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0459492 - 2.82805i) q^{2} +(0.422093 - 5.17898i) q^{3} +(-7.99578 + 0.259893i) q^{4} +(-3.69195 + 10.5532i) q^{5} +(-14.6658 - 0.955732i) q^{6} +(-12.0982 - 12.0982i) q^{7} +(1.10239 + 22.6005i) q^{8} +(-26.6437 - 4.37202i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 4 q^{3} - 12 q^{4} - 4 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(97\)	\(161\)	\(181\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0459492	−	2.82805i	−0.0162455	−	0.999868i
							\(3\)	0.422093	−	5.17898i	0.0812318	−	0.996695i
\(5\)	−3.69195	+	10.5532i	−0.330218	+	0.943905i
							\(7\)	−12.0982	−	12.0982i	−0.653241	−	0.653241i	0.300531	−	0.953772i	\(-0.402836\pi\)
−0.953772	+	0.300531i	\(0.902836\pi\)
\(11\)	−19.8066	+	19.8066i	−0.542903	+	0.542903i	−0.924379	−	0.381476i	\(-0.875416\pi\)
							0.381476	+	0.924379i	\(0.375416\pi\)
\(13\)	59.2447			1.26396			0.631982	−	0.774983i	\(-0.282242\pi\)
							0.631982	+	0.774983i	\(0.282242\pi\)
\(17\)	−31.5199	−	31.5199i	−0.449689	−	0.449689i	0.445562	−	0.895251i	\(-0.353004\pi\)
							−0.895251	+	0.445562i	\(0.853004\pi\)
\(19\)	46.7903	+	46.7903i	0.564970	+	0.564970i	0.930715	−	0.365745i	\(-0.119186\pi\)
							−0.365745	+	0.930715i	\(0.619186\pi\)
\(23\)	139.684	+	139.684i	1.26636	+	1.26636i	0.947956	+	0.318401i	\(0.103146\pi\)
							0.318401	+	0.947956i	\(0.396854\pi\)
\(29\)	−171.453	+	171.453i	−1.09786	+	1.09786i	−0.103200	+	0.994661i	\(0.532908\pi\)
							−0.994661	+	0.103200i	\(0.967092\pi\)
\(31\)	4.54261			0.0263186			0.0131593	−	0.999913i	\(-0.495811\pi\)
							0.0131593	+	0.999913i	\(0.495811\pi\)
\(37\)	50.6108			0.224875			0.112437	−	0.993659i	\(-0.464134\pi\)
							0.112437	+	0.993659i	\(0.464134\pi\)
\(41\)	−147.772			−0.562882			−0.281441	−	0.959578i	\(-0.590812\pi\)
							−0.281441	+	0.959578i	\(0.590812\pi\)
\(43\)			37.2795i			0.132211i	0.997813	+	0.0661055i	\(0.0210574\pi\)
							−0.997813	+	0.0661055i	\(0.978943\pi\)
\(47\)	164.558	+	164.558i	0.510706	+	0.510706i	0.914743	−	0.404037i	\(-0.132393\pi\)
							−0.404037	+	0.914743i	\(0.632393\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.4.bf.a.53.70	yes	280
3.2	odd	2	inner	240.4.bf.a.53.71	yes	280
5.2	odd	4		240.4.bb.a.197.140	yes	280
15.2	even	4		240.4.bb.a.197.1	yes	280
16.13	even	4		240.4.bb.a.173.1	✓	280
48.29	odd	4		240.4.bb.a.173.140	yes	280
80.77	odd	4	inner	240.4.bf.a.77.71	yes	280
240.77	even	4	inner	240.4.bf.a.77.70	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.4.bb.a.173.1	✓	280	16.13	even	4
240.4.bb.a.173.140	yes	280	48.29	odd	4
240.4.bb.a.197.1	yes	280	15.2	even	4
240.4.bb.a.197.140	yes	280	5.2	odd	4
240.4.bf.a.53.70	yes	280	1.1	even	1	trivial
240.4.bf.a.53.71	yes	280	3.2	odd	2	inner
240.4.bf.a.77.70	yes	280	240.77	even	4	inner
240.4.bf.a.77.71	yes	280	80.77	odd	4	inner