Embedded newform 240.4.bb.a.173.1

• Label: 240.4.bb.a.173.1
• Level: $240$
• Weight: $4$
• Character: 240.173
• Analytic conductor: $14.160$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	240.bb (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			173.1
Character	\(\chi\)	\(=\)	240.173
Dual form			240.4.bb.a.197.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.82805 - 0.0459492i) q^{2} +(5.17898 + 0.422093i) q^{3} +(7.99578 + 0.259893i) q^{4} +(10.5532 + 3.69195i) q^{5} +(-14.6270 - 1.43167i) q^{6} +(12.0982 + 12.0982i) q^{7} +(-22.6005 - 1.10239i) q^{8} +(26.6437 + 4.37202i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + 12 q^{4} - 4 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.82805	−	0.0459492i	−0.999868	−	0.0162455i
							\(3\)	5.17898	+	0.422093i	0.996695	+	0.0812318i
\(5\)	10.5532	+	3.69195i	0.943905	+	0.330218i
							\(7\)	12.0982	+	12.0982i	0.653241	+	0.653241i	0.953772	−	0.300531i	\(-0.0971638\pi\)
−0.300531	+	0.953772i	\(0.597164\pi\)
\(11\)	19.8066	+	19.8066i	0.542903	+	0.542903i	0.924379	−	0.381476i	\(-0.124584\pi\)
							−0.381476	+	0.924379i	\(0.624584\pi\)
\(13\)			59.2447i			1.26396i	0.774983	+	0.631982i	\(0.217758\pi\)
							−0.774983	+	0.631982i	\(0.782242\pi\)
\(17\)	−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	\(-0.880814\pi\)
							0.365745	+	0.930715i	\(0.380814\pi\)
\(23\)	−139.684	−	139.684i	−1.26636	−	1.26636i	−0.947956	−	0.318401i	\(-0.896854\pi\)
							−0.318401	−	0.947956i	\(-0.603146\pi\)
\(29\)	−171.453	−	171.453i	−1.09786	−	1.09786i	−0.994661	−	0.103200i	\(-0.967092\pi\)
							−0.103200	−	0.994661i	\(-0.532908\pi\)
\(31\)	4.54261			0.0263186			0.0131593	−	0.999913i	\(-0.495811\pi\)
							0.0131593	+	0.999913i	\(0.495811\pi\)
\(37\)		−	50.6108i		−	0.224875i	−0.993659	−	0.112437i	\(-0.964134\pi\)
							0.993659	−	0.112437i	\(-0.0358657\pi\)
\(41\)	147.772			0.562882			0.281441	−	0.959578i	\(-0.409188\pi\)
							0.281441	+	0.959578i	\(0.409188\pi\)
\(43\)	37.2795			0.132211			0.0661055	−	0.997813i	\(-0.478943\pi\)
							0.0661055	+	0.997813i	\(0.478943\pi\)
\(47\)	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.bb.a.173.1	✓	280
3.2	odd	2	inner	240.4.bb.a.173.140	yes	280
5.2	odd	4		240.4.bf.a.77.71	yes	280
15.2	even	4		240.4.bf.a.77.70	yes	280
16.5	even	4		240.4.bf.a.53.70	yes	280
48.5	odd	4		240.4.bf.a.53.71	yes	280
80.37	odd	4	inner	240.4.bb.a.197.140	yes	280
240.197	even	4	inner	240.4.bb.a.197.1	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.4.bb.a.173.1	✓	280	1.1	even	1	trivial
240.4.bb.a.173.140	yes	280	3.2	odd	2	inner
240.4.bb.a.197.1	yes	280	240.197	even	4	inner
240.4.bb.a.197.140	yes	280	80.37	odd	4	inner
240.4.bf.a.53.70	yes	280	16.5	even	4
240.4.bf.a.53.71	yes	280	48.5	odd	4
240.4.bf.a.77.70	yes	280	15.2	even	4
240.4.bf.a.77.71	yes	280	5.2	odd	4