Embedded newform 240.2.t.b.59.13

• Label: 240.2.t.b.59.13
• Level: $240$
• Weight: $2$
• Character: 240.59
• Analytic conductor: $1.916$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.91640964851\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			59.13
Character	\(\chi\)	\(=\)	240.59
Dual form			240.2.t.b.179.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.743883 - 1.20276i) q^{2} +(-1.02525 + 1.39602i) q^{3} +(-0.893276 + 1.78943i) q^{4} +(-0.920160 - 2.03797i) q^{5} +(2.44174 + 0.194661i) q^{6} +0.440232i q^{7} +(2.81675 - 0.256728i) q^{8} +(-0.897721 - 2.86253i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 8 q^{4} - 4 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.743883	−	1.20276i	−0.526005	−	0.850482i
							\(3\)	−1.02525	+	1.39602i	−0.591929	+	0.805990i
\(5\)	−0.920160	−	2.03797i	−0.411508	−	0.911406i
							\(7\)			0.440232i			0.166392i	0.996533	+	0.0831961i	\(0.0265128\pi\)
−0.996533	+	0.0831961i	\(0.973487\pi\)
\(11\)	3.34512	+	3.34512i	1.00859	+	1.00859i	0.999963	+	0.00862901i	\(0.00274673\pi\)
							0.00862901	+	0.999963i	\(0.497253\pi\)
\(13\)	3.79299	+	3.79299i	1.05199	+	1.05199i	0.998572	+	0.0534141i	\(0.0170103\pi\)
							0.0534141	+	0.998572i	\(0.482990\pi\)
\(17\)	3.44344			0.835156			0.417578	−	0.908641i	\(-0.362879\pi\)
							0.417578	+	0.908641i	\(0.362879\pi\)
\(19\)	3.74228	+	3.74228i	0.858538	+	0.858538i	0.991166	−	0.132628i	\(-0.0423416\pi\)
							−0.132628	+	0.991166i	\(0.542342\pi\)
\(23\)	0.0184980			0.00385710			0.00192855	−	0.999998i	\(-0.499386\pi\)
							0.00192855	+	0.999998i	\(0.499386\pi\)
\(29\)	−4.40505	−	4.40505i	−0.817998	−	0.817998i	0.167820	−	0.985818i	\(-0.446327\pi\)
							−0.985818	+	0.167820i	\(0.946327\pi\)
\(31\)		−	3.65248i		−	0.656004i	−0.944677	−	0.328002i	\(-0.893625\pi\)
							0.944677	−	0.328002i	\(-0.106375\pi\)
\(37\)	2.54315	−	2.54315i	0.418091	−	0.418091i	−0.466454	−	0.884545i	\(-0.654469\pi\)
							0.884545	+	0.466454i	\(0.154469\pi\)
\(41\)	−7.93346			−1.23900			−0.619499	−	0.784998i	\(-0.712664\pi\)
							−0.619499	+	0.784998i	\(0.712664\pi\)
\(43\)	4.73534	+	4.73534i	0.722133	+	0.722133i	0.969039	−	0.246907i	\(-0.0794141\pi\)
							−0.246907	+	0.969039i	\(0.579414\pi\)
\(47\)		−	3.28433i		−	0.479069i	−0.970888	−	0.239535i	\(-0.923005\pi\)
							0.970888	−	0.239535i	\(-0.0769948\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.2.t.b.59.13	✓	80
3.2	odd	2	inner	240.2.t.b.59.27	yes	80
4.3	odd	2		960.2.t.b.719.29		80
5.4	even	2	inner	240.2.t.b.59.28	yes	80
12.11	even	2		960.2.t.b.719.32		80
15.14	odd	2	inner	240.2.t.b.59.14	yes	80
16.3	odd	4	inner	240.2.t.b.179.14	yes	80
16.13	even	4		960.2.t.b.239.9		80
20.19	odd	2		960.2.t.b.719.12		80
48.29	odd	4		960.2.t.b.239.12		80
48.35	even	4	inner	240.2.t.b.179.28	yes	80
60.59	even	2		960.2.t.b.719.9		80
80.19	odd	4	inner	240.2.t.b.179.27	yes	80
80.29	even	4		960.2.t.b.239.32		80
240.29	odd	4		960.2.t.b.239.29		80
240.179	even	4	inner	240.2.t.b.179.13	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.t.b.59.13	✓	80	1.1	even	1	trivial
240.2.t.b.59.14	yes	80	15.14	odd	2	inner
240.2.t.b.59.27	yes	80	3.2	odd	2	inner
240.2.t.b.59.28	yes	80	5.4	even	2	inner
240.2.t.b.179.13	yes	80	240.179	even	4	inner
240.2.t.b.179.14	yes	80	16.3	odd	4	inner
240.2.t.b.179.27	yes	80	80.19	odd	4	inner
240.2.t.b.179.28	yes	80	48.35	even	4	inner
960.2.t.b.239.9		80	16.13	even	4
960.2.t.b.239.12		80	48.29	odd	4
960.2.t.b.239.29		80	240.29	odd	4
960.2.t.b.239.32		80	80.29	even	4
960.2.t.b.719.9		80	60.59	even	2
960.2.t.b.719.12		80	20.19	odd	2
960.2.t.b.719.29		80	4.3	odd	2
960.2.t.b.719.32		80	12.11	even	2