Newform orbit 240.2.y.b

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

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.91640964851\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - i - 1) q^{2} + q^{3} + 2 i q^{4} + ( - 2 i - 1) q^{5} + ( - i - 1) q^{6} + (3 i + 3) q^{7} + ( - 2 i + 2) q^{8} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} - 2 q^{5} - 2 q^{6} + 6 q^{7} + 4 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(97\)	\(161\)	\(181\)
\(\chi(n)\)	\(-1\)	\(i\)	\(1\)	\(i\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

163.1

	−	1.00000i
		1.00000i

−1.00000

+

1.00000i

1.00000

−

2.00000i

−1.00000

+

2.00000i

−1.00000

+

1.00000i

3.00000

−

3.00000i

2.00000

+

2.00000i

1.00000

−1.00000

−

3.00000i

187.1

−1.00000

−

1.00000i

1.00000

2.00000i

−1.00000

−

2.00000i

−1.00000

−

1.00000i

3.00000

+

3.00000i

2.00000

−

2.00000i

1.00000

−1.00000

+

3.00000i

Inner twists

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

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
240.2.y.b	✓	2	1.a	even	1	1	trivial
240.2.y.b	✓	2	80.s	even	4	1	inner
240.2.bc.a	yes	2	5.c	odd	4	1
240.2.bc.a	yes	2	16.f	odd	4	1
720.2.z.c		2	3.b	odd	2	1
720.2.z.c		2	240.z	odd	4	1
720.2.bd.d		2	15.e	even	4	1
720.2.bd.d		2	48.k	even	4	1
960.2.y.a		2	4.b	odd	2	1
960.2.y.a		2	80.i	odd	4	1
960.2.bc.a		2	16.e	even	4	1
960.2.bc.a		2	20.e	even	4	1
1920.2.y.c		2	8.b	even	2	1
1920.2.y.c		2	80.j	even	4	1
1920.2.y.e		2	8.d	odd	2	1
1920.2.y.e		2	80.t	odd	4	1
1920.2.bc.b		2	16.e	even	4	1
1920.2.bc.b		2	40.k	even	4	1
1920.2.bc.e		2	16.f	odd	4	1
1920.2.bc.e		2	40.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 \(S_{2}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{2} - 6T_{7} + 18 \)

\( T_{11}^{2} - 2T_{11} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T + 2 \)
$3$	\( (T - 1)^{2} \)
$5$	\( T^{2} + 2T + 5 \)
$7$	\( T^{2} - 6T + 18 \)
$11$	\( T^{2} - 2T + 2 \)