Embedded newform 240.4.y.a.163.19

• Label: 240.4.y.a.163.19
• 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			163.19
Character	\(\chi\)	\(=\)	240.163
Dual form			240.4.y.a.187.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.401137 - 2.79984i) q^{2} -3.00000 q^{3} +(-7.67818 - 2.24624i) q^{4} +(-0.685918 + 11.1593i) q^{5} +(-1.20341 + 8.39951i) q^{6} +(12.6294 - 12.6294i) q^{7} +(-9.36910 + 20.5966i) q^{8} +9.00000 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 216 q^{3} + 2 q^{4} - 42 q^{8} + 648 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.401137	−	2.79984i	0.141823	−	0.989892i
							\(3\)	−3.00000			−0.577350
\(5\)	−0.685918	+	11.1593i	−0.0613504	+	0.998116i
							\(7\)	12.6294	−	12.6294i	0.681926	−	0.681926i	−0.278508	−	0.960434i	\(-0.589840\pi\)
0.960434	+	0.278508i	\(0.0898399\pi\)
\(11\)	28.6557	+	28.6557i	0.785455	+	0.785455i	0.980745	−	0.195290i	\(-0.0625650\pi\)
							−0.195290	+	0.980745i	\(0.562565\pi\)
\(13\)		−	89.4523i		−	1.90843i	−0.299119	−	0.954216i	\(-0.596693\pi\)
							0.299119	−	0.954216i	\(-0.403307\pi\)
\(17\)	−17.7327	+	17.7327i	−0.252989	+	0.252989i	−0.822195	−	0.569206i	\(-0.807251\pi\)
							0.569206	+	0.822195i	\(0.307251\pi\)
\(19\)	−54.1460	−	54.1460i	−0.653786	−	0.653786i	0.300116	−	0.953903i	\(-0.402974\pi\)
							−0.953903	+	0.300116i	\(0.902974\pi\)
\(23\)	−48.4975	−	48.4975i	−0.439671	−	0.439671i	0.452230	−	0.891901i	\(-0.350628\pi\)
							−0.891901	+	0.452230i	\(0.850628\pi\)
\(29\)	169.242	−	169.242i	1.08371	−	1.08371i	0.0875479	−	0.996160i	\(-0.472097\pi\)
							0.996160	−	0.0875479i	\(-0.0279031\pi\)
\(31\)			94.5056i			0.547539i	0.961795	+	0.273769i	\(0.0882705\pi\)
							−0.961795	+	0.273769i	\(0.911729\pi\)
\(37\)		−	292.559i		−	1.29990i	−0.759977	−	0.649951i	\(-0.774790\pi\)
							0.759977	−	0.649951i	\(-0.225210\pi\)
\(41\)		−	298.228i		−	1.13598i	−0.823034	−	0.567992i	\(-0.807720\pi\)
							0.823034	−	0.567992i	\(-0.192280\pi\)
\(43\)			97.4144i			0.345478i	0.984968	+	0.172739i	\(0.0552617\pi\)
							−0.984968	+	0.172739i	\(0.944738\pi\)
\(47\)	−305.146	−	305.146i	−0.947024	−	0.947024i	0.0516414	−	0.998666i	\(-0.483555\pi\)
							−0.998666	+	0.0516414i	\(0.983555\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.4.y.a.163.19	✓	72
5.2	odd	4		240.4.bc.b.67.36	yes	72
16.11	odd	4		240.4.bc.b.43.36	yes	72
80.27	even	4	inner	240.4.y.a.187.19	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.4.y.a.163.19	✓	72	1.1	even	1	trivial
240.4.y.a.187.19	yes	72	80.27	even	4	inner
240.4.bc.b.43.36	yes	72	16.11	odd	4
240.4.bc.b.67.36	yes	72	5.2	odd	4