Newform orbit 240.2.bc.d

• Label: 240.2.bc.d
• Level: $240$
• Weight: $2$
• Character orbit: 240.bc
• Analytic conductor: $1.916$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.91640964851\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(i)\)
Coefficient field:	6.0.399424.1
Defining polynomial:	\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b1 * q^2 - b3 * q^3 + (-b3 + b2) * q^4 + (-b3 + 2) * q^5 - b4 * q^6 + (-b5 + b4 + b2 + b1) * q^7 + (-b4 - b3 - b2 + b1 - 2) * q^8 - q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - 2 q^{4} + 12 q^{5} - 2 q^{7} - 8 q^{8} - 6 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{5} + 3\nu^{3} + 2\nu - 8 ) / 4 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 4 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{5} - \nu^{3} + 2\nu^{2} + 4 ) / 2 \)
\(\beta_{5}\)	\(=\)	\( -\nu^{5} + \nu^{4} - 2\nu^{3} + 3\nu^{2} - 2\nu + 5 \)

Character values

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

\(n\)	\(31\)	\(97\)	\(161\)	\(181\)
\(\chi(n)\)	\(-1\)	\(\beta_{3}\)	\(1\)	\(-\beta_{3}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

43.1

1.40680	−	0.144584i
0.264658	+	1.38923i
−0.671462	−	1.24464i
1.40680	+	0.144584i
0.264658	−	1.38923i
−0.671462	+	1.24464i

−1.40680

+

0.144584i

1.00000i

1.95819

−

0.406803i

2.00000

+

1.00000i

−0.144584

−

1.40680i

2.10278

−

2.10278i

−2.69597

+

0.855416i

−1.00000

−2.95819

−

1.11763i

43.2

−0.264658

−

1.38923i

1.00000i

−1.85991

+

0.735342i

2.00000

+

1.00000i

1.38923

−

0.264658i

−3.24914

+

3.24914i

1.51380

+

2.38923i

−1.00000

0.859912

−

3.04312i

43.3

0.671462

+

1.24464i

1.00000i

−1.09828

+

1.67146i

2.00000

+

1.00000i

−1.24464

+

0.671462i

0.146365

−

0.146365i

−2.81783

−

0.244644i

−1.00000

0.0982788

+

3.16075i

67.1

−1.40680

−

0.144584i

−

1.00000i

1.95819

+

0.406803i

2.00000

−

1.00000i

−0.144584

+

1.40680i

2.10278

+

2.10278i

−2.69597

−

0.855416i

−1.00000

−2.95819

+

1.11763i

67.2

−0.264658

+

1.38923i

−

1.00000i

−1.85991

−

0.735342i

2.00000

−

1.00000i

1.38923

+

0.264658i

−3.24914

−

3.24914i

1.51380

−

2.38923i

−1.00000

0.859912

+

3.04312i

67.3

0.671462

−

1.24464i

−

1.00000i

−1.09828

−

1.67146i

2.00000

−

1.00000i

−1.24464

−

0.671462i

0.146365

+

0.146365i

−2.81783

+

0.244644i

−1.00000

0.0982788

−

3.16075i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
80.j	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.bc.d	yes	6
3.b	odd	2	1		720.2.bd.e		6
4.b	odd	2	1		960.2.bc.d		6
5.c	odd	4	1		240.2.y.d	✓	6
8.b	even	2	1		1920.2.bc.g		6
8.d	odd	2	1		1920.2.bc.h		6
15.e	even	4	1		720.2.z.e		6
16.e	even	4	1		960.2.y.d		6
16.e	even	4	1		1920.2.y.h		6
16.f	odd	4	1		240.2.y.d	✓	6
16.f	odd	4	1		1920.2.y.g		6
20.e	even	4	1		960.2.y.d		6
40.i	odd	4	1		1920.2.y.g		6
40.k	even	4	1		1920.2.y.h		6
48.k	even	4	1		720.2.z.e		6
80.i	odd	4	1		1920.2.bc.h		6
80.j	even	4	1	inner	240.2.bc.d	yes	6
80.s	even	4	1		1920.2.bc.g		6
80.t	odd	4	1		960.2.bc.d		6
240.bd	odd	4	1		720.2.bd.e		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
240.2.y.d	✓	6	5.c	odd	4	1
240.2.y.d	✓	6	16.f	odd	4	1
240.2.bc.d	yes	6	1.a	even	1	1	trivial
240.2.bc.d	yes	6	80.j	even	4	1	inner
720.2.z.e		6	15.e	even	4	1
720.2.z.e		6	48.k	even	4	1
720.2.bd.e		6	3.b	odd	2	1
720.2.bd.e		6	240.bd	odd	4	1
960.2.y.d		6	16.e	even	4	1
960.2.y.d		6	20.e	even	4	1
960.2.bc.d		6	4.b	odd	2	1
960.2.bc.d		6	80.t	odd	4	1
1920.2.y.g		6	16.f	odd	4	1
1920.2.y.g		6	40.i	odd	4	1
1920.2.y.h		6	16.e	even	4	1
1920.2.y.h		6	40.k	even	4	1
1920.2.bc.g		6	8.b	even	2	1
1920.2.bc.g		6	80.s	even	4	1
1920.2.bc.h		6	8.d	odd	2	1
1920.2.bc.h		6	80.i	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{6} + 2T_{7}^{5} + 2T_{7}^{4} - 32T_{7}^{3} + 196T_{7}^{2} - 56T_{7} + 8 \)

\( T_{11}^{6} + 2T_{11}^{5} + 2T_{11}^{4} - 32T_{11}^{3} + 196T_{11}^{2} - 56T_{11} + 8 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^6 + 2*T^5 + 3*T^4 + 6*T^3 + 6*T^2 + 8*T + 8
$3$	\( (T^{2} + 1)^{3} \)
$5$	\( (T^{2} - 4 T + 5)^{3} \)
$7$	T^6 + 2*T^5 + 2*T^4 - 32*T^3 + 196*T^2 - 56*T + 8
$11$	T^6 + 2*T^5 + 2*T^4 - 32*T^3 + 196*T^2 - 56*T + 8