Embedded newform 240.2.bb.a.173.3

• Label: 240.2.bb.a.173.3
• Level: $240$
• Weight: $2$
• Character: 240.173
• Analytic conductor: $1.916$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.91640964851\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			173.3
Character	\(\chi\)	\(=\)	240.173
Dual form			240.2.bb.a.197.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37825 + 0.316900i) q^{2} +(1.69498 + 0.356415i) q^{3} +(1.79915 - 0.873536i) q^{4} +(1.08277 - 1.95643i) q^{5} +(-2.44906 + 0.0459120i) q^{6} +(2.05875 + 2.05875i) q^{7} +(-2.20285 + 1.77410i) q^{8} +(2.74594 + 1.20823i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 4 q^{4} - 4 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37825	+	0.316900i	−0.974570	+	0.224082i
							\(3\)	1.69498	+	0.356415i	0.978599	+	0.205776i
\(5\)	1.08277	−	1.95643i	0.484230	−	0.874941i
							\(7\)	2.05875	+	2.05875i	0.778136	+	0.778136i	0.979514	−	0.201378i	\(-0.0645418\pi\)
−0.201378	+	0.979514i	\(0.564542\pi\)
\(11\)	−3.62397	−	3.62397i	−1.09267	−	1.09267i	−0.995243	−	0.0974259i	\(-0.968939\pi\)
							−0.0974259	−	0.995243i	\(-0.531061\pi\)
\(13\)		−	3.27409i		−	0.908068i	−0.890984	−	0.454034i	\(-0.849984\pi\)
							0.890984	−	0.454034i	\(-0.150016\pi\)
\(17\)	1.90342	+	1.90342i	0.461646	+	0.461646i	0.899195	−	0.437548i	\(-0.144153\pi\)
							−0.437548	+	0.899195i	\(0.644153\pi\)
\(19\)	−4.35033	+	4.35033i	−0.998034	+	0.998034i	−0.999998	−	0.00196364i	\(-0.999375\pi\)
							0.00196364	+	0.999998i	\(0.499375\pi\)
\(23\)	2.84115	+	2.84115i	0.592421	+	0.592421i	0.938285	−	0.345864i	\(-0.112414\pi\)
							−0.345864	+	0.938285i	\(0.612414\pi\)
\(29\)	1.64493	+	1.64493i	0.305455	+	0.305455i	0.843144	−	0.537688i	\(-0.180702\pi\)
							−0.537688	+	0.843144i	\(0.680702\pi\)
\(31\)	2.97963			0.535158			0.267579	−	0.963536i	\(-0.413776\pi\)
							0.267579	+	0.963536i	\(0.413776\pi\)
\(37\)			0.174228i			0.0286429i	0.999897	+	0.0143215i	\(0.00455882\pi\)
							−0.999897	+	0.0143215i	\(0.995441\pi\)
\(41\)	−9.82542			−1.53447			−0.767237	−	0.641364i	\(-0.778369\pi\)
							−0.767237	+	0.641364i	\(0.778369\pi\)
\(43\)	−3.04783			−0.464789			−0.232395	−	0.972622i	\(-0.574656\pi\)
							−0.232395	+	0.972622i	\(0.574656\pi\)
\(47\)	5.17716	+	5.17716i	0.755166	+	0.755166i	0.975438	−	0.220272i	\(-0.0706946\pi\)
							−0.220272	+	0.975438i	\(0.570695\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.2.bb.a.173.3	✓	88
3.2	odd	2	inner	240.2.bb.a.173.42	yes	88
4.3	odd	2		960.2.bb.a.593.3		88
5.2	odd	4		240.2.bf.a.77.20	yes	88
12.11	even	2		960.2.bb.a.593.42		88
15.2	even	4		240.2.bf.a.77.25	yes	88
16.5	even	4		240.2.bf.a.53.25	yes	88
16.11	odd	4		960.2.bf.a.113.20		88
20.7	even	4		960.2.bf.a.17.20		88
48.5	odd	4		240.2.bf.a.53.20	yes	88
48.11	even	4		960.2.bf.a.113.19		88
60.47	odd	4		960.2.bf.a.17.19		88
80.27	even	4		960.2.bb.a.497.42		88
80.37	odd	4	inner	240.2.bb.a.197.42	yes	88
240.107	odd	4		960.2.bb.a.497.3		88
240.197	even	4	inner	240.2.bb.a.197.3	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.bb.a.173.3	✓	88	1.1	even	1	trivial
240.2.bb.a.173.42	yes	88	3.2	odd	2	inner
240.2.bb.a.197.3	yes	88	240.197	even	4	inner
240.2.bb.a.197.42	yes	88	80.37	odd	4	inner
240.2.bf.a.53.20	yes	88	48.5	odd	4
240.2.bf.a.53.25	yes	88	16.5	even	4
240.2.bf.a.77.20	yes	88	5.2	odd	4
240.2.bf.a.77.25	yes	88	15.2	even	4
960.2.bb.a.497.3		88	240.107	odd	4
960.2.bb.a.497.42		88	80.27	even	4
960.2.bb.a.593.3		88	4.3	odd	2
960.2.bb.a.593.42		88	12.11	even	2
960.2.bf.a.17.19		88	60.47	odd	4
960.2.bf.a.17.20		88	20.7	even	4
960.2.bf.a.113.19		88	48.11	even	4
960.2.bf.a.113.20		88	16.11	odd	4