Newform orbit 240.2.bl.a

• Label: 240.2.bl.a
• Level: $240$
• Weight: $2$
• Character orbit: 240.bl
• Analytic conductor: $1.916$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.91640964851\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
109.1	−1.40976	−	0.112110i	−0.707107	+	0.707107i	1.97486	+	0.316097i	2.18678	+	0.466917i	1.07613	−	0.917579i	1.00010			−2.74865	−	0.667024i		−	1.00000i	−3.03049	−	0.903402i
109.2	−1.39033	−	0.258835i	0.707107	−	0.707107i	1.86601	+	0.719729i	0.404088	−	2.19925i	−1.16613	+	0.800085i	1.81567			−2.40807	−	1.48364i		−	1.00000i	−1.13106	+	2.95308i
109.3	−1.36038	+	0.386472i	0.707107	−	0.707107i	1.70128	−	1.05150i	−1.98421	+	1.03097i	−0.688658	+	1.23521i	−3.91927			−1.90801	+	2.08794i		−	1.00000i	2.30085	−	2.16936i
109.4	−1.22294	−	0.710230i	−0.707107	+	0.707107i	0.991146	+	1.73713i	−2.15195	−	0.607542i	1.36696	−	0.362538i	2.25286			0.0216568	−	2.82834i		−	1.00000i	2.20020	+	2.27137i
109.5	−1.20386	+	0.742113i	−0.707107	+	0.707107i	0.898535	−	1.78679i	−2.06370	−	0.860885i	0.326501	−	1.37601i	0.707398			0.244298	+	2.81786i		−	1.00000i	3.12328	−	0.495122i
109.6	−1.06224	−	0.933621i	0.707107	−	0.707107i	0.256702	+	1.98346i	−1.86022	+	1.24079i	−1.41129	+	0.0909462i	1.58988			1.57912	−	2.34657i		−	1.00000i	3.13443	+	0.418728i
109.7	−0.903247	+	1.08818i	−0.707107	+	0.707107i	−0.368289	−	1.96580i	0.770325	+	2.09919i	−0.130770	−	1.40815i	−3.05002			2.47181	+	1.37484i		−	1.00000i	−2.98010	−	1.05783i
109.8	−0.750333	−	1.19875i	−0.707107	+	0.707107i	−0.874002	+	1.79892i	1.07735	−	1.95942i	1.37821	+	0.317079i	−1.22137			2.81225	−	0.302081i		−	1.00000i	−3.15722	+	0.178735i
109.9	−0.550383	+	1.30272i	0.707107	−	0.707107i	−1.39416	−	1.43399i	0.162008	+	2.23019i	0.531983	+	1.31034i	2.93661			2.63541	−	1.02695i		−	1.00000i	−2.99448	−	1.01641i
109.10	−0.456856	+	1.33839i	0.707107	−	0.707107i	−1.58257	−	1.22290i	−1.50085	−	1.65754i	0.623338	+	1.26943i	−2.58977			2.35972	−	1.55940i		−	1.00000i	2.90411	−	1.25146i
109.11	−0.382275	+	1.36157i	−0.707107	+	0.707107i	−1.70773	−	1.04099i	1.38805	−	1.75308i	−0.692464	−	1.23308i	4.66030			2.07020	−	1.92725i		−	1.00000i	1.85633	+	2.56009i
109.12	−0.345118	−	1.37146i	0.707107	−	0.707107i	−1.76179	+	0.946629i	2.16437	+	0.561697i	−1.21380	−	0.725731i	4.51614			1.90629	+	2.08952i		−	1.00000i	0.0233792	−	3.16219i
109.13	0.345118	+	1.37146i	−0.707107	+	0.707107i	−1.76179	+	0.946629i	−0.561697	−	2.16437i	−1.21380	−	0.725731i	−4.51614			−1.90629	−	2.08952i		−	1.00000i	2.77449	−	1.51731i
109.14	0.382275	−	1.36157i	0.707107	−	0.707107i	−1.70773	−	1.04099i	1.75308	−	1.38805i	−0.692464	−	1.23308i	−4.66030			−2.07020	+	1.92725i		−	1.00000i	−1.21977	−	2.91756i
109.15	0.456856	−	1.33839i	−0.707107	+	0.707107i	−1.58257	−	1.22290i	1.65754	+	1.50085i	0.623338	+	1.26943i	2.58977			−2.35972	+	1.55940i		−	1.00000i	2.76598	−	1.53277i
109.16	0.550383	−	1.30272i	−0.707107	+	0.707107i	−1.39416	−	1.43399i	−2.23019	−	0.162008i	0.531983	+	1.31034i	−2.93661			−2.63541	+	1.02695i		−	1.00000i	−1.43851	+	2.81615i
109.17	0.750333	+	1.19875i	0.707107	−	0.707107i	−0.874002	+	1.79892i	1.95942	−	1.07735i	1.37821	+	0.317079i	1.22137			−2.81225	+	0.302081i		−	1.00000i	2.76169	+	1.54047i
109.18	0.903247	−	1.08818i	0.707107	−	0.707107i	−0.368289	−	1.96580i	−2.09919	−	0.770325i	−0.130770	−	1.40815i	3.05002			−2.47181	−	1.37484i		−	1.00000i	−2.73434	+	1.58851i
109.19	1.06224	+	0.933621i	−0.707107	+	0.707107i	0.256702	+	1.98346i	−1.24079	+	1.86022i	−1.41129	+	0.0909462i	−1.58988			−1.57912	+	2.34657i		−	1.00000i	−3.05476	+	0.817572i
109.20	1.20386	−	0.742113i	0.707107	−	0.707107i	0.898535	−	1.78679i	0.860885	+	2.06370i	0.326501	−	1.37601i	−0.707398			−0.244298	−	2.81786i		−	1.00000i	2.56788	+	1.84553i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
16.e	even	4	1	inner
80.q	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	240.2.bl.a	✓	48
3.b	odd	2	1		720.2.bm.h		48
4.b	odd	2	1		960.2.bl.a		48
5.b	even	2	1	inner	240.2.bl.a	✓	48
8.b	even	2	1		1920.2.bl.a		48
8.d	odd	2	1		1920.2.bl.b		48
15.d	odd	2	1		720.2.bm.h		48
16.e	even	4	1	inner	240.2.bl.a	✓	48
16.e	even	4	1		1920.2.bl.a		48
16.f	odd	4	1		960.2.bl.a		48
16.f	odd	4	1		1920.2.bl.b		48
20.d	odd	2	1		960.2.bl.a		48
40.e	odd	2	1		1920.2.bl.b		48
40.f	even	2	1		1920.2.bl.a		48
48.i	odd	4	1		720.2.bm.h		48
80.k	odd	4	1		960.2.bl.a		48
80.k	odd	4	1		1920.2.bl.b		48
80.q	even	4	1	inner	240.2.bl.a	✓	48
80.q	even	4	1		1920.2.bl.a		48
240.bm	odd	4	1		720.2.bm.h		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
240.2.bl.a	✓	48	1.a	even	1	1	trivial
240.2.bl.a	✓	48	5.b	even	2	1	inner
240.2.bl.a	✓	48	16.e	even	4	1	inner
240.2.bl.a	✓	48	80.q	even	4	1	inner
720.2.bm.h		48	3.b	odd	2	1
720.2.bm.h		48	15.d	odd	2	1
720.2.bm.h		48	48.i	odd	4	1
720.2.bm.h		48	240.bm	odd	4	1
960.2.bl.a		48	4.b	odd	2	1
960.2.bl.a		48	16.f	odd	4	1
960.2.bl.a		48	20.d	odd	2	1
960.2.bl.a		48	80.k	odd	4	1
1920.2.bl.a		48	8.b	even	2	1
1920.2.bl.a		48	16.e	even	4	1
1920.2.bl.a		48	40.f	even	2	1
1920.2.bl.a		48	80.q	even	4	1
1920.2.bl.b		48	8.d	odd	2	1
1920.2.bl.b		48	16.f	odd	4	1
1920.2.bl.b		48	40.e	odd	2	1
1920.2.bl.b		48	80.k	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(240, [\chi])\).