Embedded newform 240.4.y.a.163.15

• Label: 240.4.y.a.163.15
• Level: $240$
• Weight: $4$
• Character: 240.163
• Analytic conductor: $14.160$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(14.1604584014\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			163.15
Character	\(\chi\)	\(=\)	240.163
Dual form			240.4.y.a.187.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.756265 - 2.72545i) q^{2} -3.00000 q^{3} +(-6.85613 + 4.12232i) q^{4} +(-1.38253 - 11.0945i) q^{5} +(2.26880 + 8.17634i) q^{6} +(18.9545 - 18.9545i) q^{7} +(16.4202 + 15.5684i) q^{8} +9.00000 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 216 q^{3} + 2 q^{4} - 42 q^{8} + 648 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.756265	−	2.72545i	−0.267380	−	0.963591i
							\(3\)	−3.00000			−0.577350
\(5\)	−1.38253	−	11.0945i	−0.123657	−	0.992325i
							\(7\)	18.9545	−	18.9545i	1.02345	−	1.02345i	0.0237283	−	0.999718i	\(-0.492446\pi\)
0.999718	−	0.0237283i	\(-0.00755368\pi\)
\(11\)	−17.0048	−	17.0048i	−0.466103	−	0.466103i	0.434546	−	0.900649i	\(-0.356909\pi\)
							−0.900649	+	0.434546i	\(0.856909\pi\)
\(13\)		−	38.3421i		−	0.818014i	−0.912531	−	0.409007i	\(-0.865875\pi\)
							0.912531	−	0.409007i	\(-0.134125\pi\)
\(17\)	68.7675	−	68.7675i	0.981092	−	0.981092i	−0.0187329	−	0.999825i	\(-0.505963\pi\)
							0.999825	+	0.0187329i	\(0.00596320\pi\)
\(19\)	48.9687	+	48.9687i	0.591274	+	0.591274i	0.937975	−	0.346702i	\(-0.112698\pi\)
							−0.346702	+	0.937975i	\(0.612698\pi\)
\(23\)	−95.9316	−	95.9316i	−0.869701	−	0.869701i	0.122738	−	0.992439i	\(-0.460833\pi\)
							−0.992439	+	0.122738i	\(0.960833\pi\)
\(29\)	−201.677	+	201.677i	−1.29140	+	1.29140i	−0.357471	+	0.933924i	\(0.616361\pi\)
							−0.933924	+	0.357471i	\(0.883639\pi\)
\(31\)			71.1904i			0.412457i	0.978504	+	0.206229i	\(0.0661191\pi\)
							−0.978504	+	0.206229i	\(0.933881\pi\)
\(37\)			197.077i			0.875657i	0.899058	+	0.437829i	\(0.144252\pi\)
							−0.899058	+	0.437829i	\(0.855748\pi\)
\(41\)		−	466.331i		−	1.77631i	−0.459546	−	0.888154i	\(-0.651988\pi\)
							0.459546	−	0.888154i	\(-0.348012\pi\)
\(43\)			387.879i			1.37560i	0.725899	+	0.687802i	\(0.241424\pi\)
							−0.725899	+	0.687802i	\(0.758576\pi\)
\(47\)	177.994	+	177.994i	0.552405	+	0.552405i	0.927134	−	0.374729i	\(-0.122264\pi\)
							−0.374729	+	0.927134i	\(0.622264\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.4.y.a.163.15	✓	72
5.2	odd	4		240.4.bc.b.67.33	yes	72
16.11	odd	4		240.4.bc.b.43.33	yes	72
80.27	even	4	inner	240.4.y.a.187.15	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.4.y.a.163.15	✓	72	1.1	even	1	trivial
240.4.y.a.187.15	yes	72	80.27	even	4	inner
240.4.bc.b.43.33	yes	72	16.11	odd	4
240.4.bc.b.67.33	yes	72	5.2	odd	4