Embedded newform 240.2.t.b.59.7

• Label: 240.2.t.b.59.7
• 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			59.7
Character	\(\chi\)	\(=\)	240.59
Dual form			240.2.t.b.179.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22048 - 0.714442i) q^{2} +(0.591463 + 1.62793i) q^{3} +(0.979146 + 1.74392i) q^{4} +(2.22111 - 0.258204i) q^{5} +(0.441196 - 2.40943i) q^{6} +4.18600i q^{7} +(0.0509042 - 2.82797i) q^{8} +(-2.30034 + 1.92573i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 8 q^{4} - 4 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22048	−	0.714442i	−0.863010	−	0.505187i
							\(3\)	0.591463	+	1.62793i	0.341481	+	0.939889i
\(5\)	2.22111	−	0.258204i	0.993311	−	0.115472i
							\(7\)			4.18600i			1.58216i	0.611713	+	0.791080i	\(0.290481\pi\)
−0.611713	+	0.791080i	\(0.709519\pi\)
\(11\)	−0.975442	−	0.975442i	−0.294107	−	0.294107i	0.544593	−	0.838700i	\(-0.316684\pi\)
							−0.838700	+	0.544593i	\(0.816684\pi\)
\(13\)	−2.97668	−	2.97668i	−0.825582	−	0.825582i	0.161320	−	0.986902i	\(-0.448425\pi\)
							−0.986902	+	0.161320i	\(0.948425\pi\)
\(17\)	−0.313777			−0.0761021			−0.0380510	−	0.999276i	\(-0.512115\pi\)
							−0.0380510	+	0.999276i	\(0.512115\pi\)
\(19\)	4.54172	+	4.54172i	1.04194	+	1.04194i	0.999081	+	0.0428606i	\(0.0136471\pi\)
							0.0428606	+	0.999081i	\(0.486353\pi\)
\(23\)	−0.374730			−0.0781366			−0.0390683	−	0.999237i	\(-0.512439\pi\)
							−0.0390683	+	0.999237i	\(0.512439\pi\)
\(29\)	4.71030	+	4.71030i	0.874681	+	0.874681i	0.992978	−	0.118297i	\(-0.0377435\pi\)
							−0.118297	+	0.992978i	\(0.537743\pi\)
\(31\)		−	5.53854i		−	0.994751i	−0.867535	−	0.497375i	\(-0.834297\pi\)
							0.867535	−	0.497375i	\(-0.165703\pi\)
\(37\)	4.61988	−	4.61988i	0.759503	−	0.759503i	−0.216729	−	0.976232i	\(-0.569539\pi\)
							0.976232	+	0.216729i	\(0.0695387\pi\)
\(41\)	−3.91134			−0.610848			−0.305424	−	0.952216i	\(-0.598798\pi\)
							−0.305424	+	0.952216i	\(0.598798\pi\)
\(43\)	6.73708	+	6.73708i	1.02740	+	1.02740i	0.999614	+	0.0277811i	\(0.00884412\pi\)
							0.0277811	+	0.999614i	\(0.491156\pi\)
\(47\)		−	6.23699i		−	0.909759i	−0.890553	−	0.454880i	\(-0.849682\pi\)
							0.890553	−	0.454880i	\(-0.150318\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.2.t.b.59.7	✓	80
3.2	odd	2	inner	240.2.t.b.59.33	yes	80
4.3	odd	2		960.2.t.b.719.15		80
5.4	even	2	inner	240.2.t.b.59.34	yes	80
12.11	even	2		960.2.t.b.719.35		80
15.14	odd	2	inner	240.2.t.b.59.8	yes	80
16.3	odd	4	inner	240.2.t.b.179.8	yes	80
16.13	even	4		960.2.t.b.239.6		80
20.19	odd	2		960.2.t.b.719.26		80
48.29	odd	4		960.2.t.b.239.26		80
48.35	even	4	inner	240.2.t.b.179.34	yes	80
60.59	even	2		960.2.t.b.719.6		80
80.19	odd	4	inner	240.2.t.b.179.33	yes	80
80.29	even	4		960.2.t.b.239.35		80
240.29	odd	4		960.2.t.b.239.15		80
240.179	even	4	inner	240.2.t.b.179.7	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.t.b.59.7	✓	80	1.1	even	1	trivial
240.2.t.b.59.8	yes	80	15.14	odd	2	inner
240.2.t.b.59.33	yes	80	3.2	odd	2	inner
240.2.t.b.59.34	yes	80	5.4	even	2	inner
240.2.t.b.179.7	yes	80	240.179	even	4	inner
240.2.t.b.179.8	yes	80	16.3	odd	4	inner
240.2.t.b.179.33	yes	80	80.19	odd	4	inner
240.2.t.b.179.34	yes	80	48.35	even	4	inner
960.2.t.b.239.6		80	16.13	even	4
960.2.t.b.239.15		80	240.29	odd	4
960.2.t.b.239.26		80	48.29	odd	4
960.2.t.b.239.35		80	80.29	even	4
960.2.t.b.719.6		80	60.59	even	2
960.2.t.b.719.15		80	4.3	odd	2
960.2.t.b.719.26		80	20.19	odd	2
960.2.t.b.719.35		80	12.11	even	2