Newform orbit 240.1.bm.a

• Label: 240.1.bm.a
• Level: $240$
• Weight: $1$
• Character orbit: 240.bm
• Analytic conductor: $0.120$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{4}$
• CM discriminant: -15
• Inner twists: $8$

Newspace parameters

Level:	$N$	$=$	$240 = 2^{4} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	240.bm (of order $4$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.119775603032$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
Defining polynomial:	$x^{4} + 1$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.92160.2

$q$-expansion

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	q - z * q^2 + z * q^3 + z^2 * q^4 + z^3 * q^5 - z^2 * q^6 - z^3 * q^8 + z^2 * q^9 + q^10 + z^3 * q^12 - q^15 - q^16 + (-z^3 + z) * q^17 - z^3 * q^18 + (-z^2 - 1) * q^19 - z * q^20 + (-z^3 - z) * q^23 + q^24 - z^2 * q^25 + z^3 * q^27 + z * q^30 + z * q^32 + (-z^2 - 1) * q^34 - q^36 + (z^3 + z) * q^38 + z^2 * q^40 - z * q^45 + (z^2 - 1) * q^46 + (z^3 - z) * q^47 - z * q^48 - q^49 + z^3 * q^50 + (z^2 + 1) * q^51 + q^54 + (-z^3 - z) * q^57 - z^2 * q^60 + (z^2 + 1) * q^61 - z^2 * q^64 + (z^3 + z) * q^68 + (-z^2 + 1) * q^69 + z * q^72 - z^3 * q^75 + (-z^2 + 1) * q^76 + 2 * q^79 - z^3 * q^80 - q^81 + (z^2 - 1) * q^85 + z^2 * q^90 + (-z^3 + z) * q^92 + (z^2 + 1) * q^94 + (-z^3 + z) * q^95 + z^2 * q^96 + z * q^98
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 4 q^{10} - 4 q^{15} - 4 q^{16} - 4 q^{19} + 4 q^{24} - 4 q^{34} - 4 q^{36} - 4 q^{46} - 4 q^{49} + 4 q^{51} + 4 q^{54} + 4 q^{61} + 4 q^{69} + 4 q^{76} + 8 q^{79} - 4 q^{81} - 4 q^{85} + 4 q^{94}+O(q^{100})$

Character values

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

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$1$	$-1$	$-1$	$-\zeta_{8}^{2}$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

29.1

0.707107	+	0.707107i
−0.707107	−	0.707107i
0.707107	−	0.707107i
−0.707107	+	0.707107i

−0.707107

−

0.707107i

0.707107

+

0.707107i

1.00000i

−0.707107

+

0.707107i

−

1.00000i

0

0.707107

−

0.707107i

1.00000i

1.00000

29.2

0.707107

+

0.707107i

−0.707107

−

0.707107i

1.00000i

0.707107

−

0.707107i

−

1.00000i

0

−0.707107

+

0.707107i

1.00000i

1.00000

149.1

−0.707107

+

0.707107i

0.707107

−

0.707107i

−

1.00000i

−0.707107

−

0.707107i

1.00000i

0

0.707107

+

0.707107i

−

1.00000i

1.00000

149.2

0.707107

−

0.707107i

−0.707107

+

0.707107i

−

1.00000i

0.707107

+

0.707107i

1.00000i

0

−0.707107

−

0.707107i

−

1.00000i

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15})$
3.b	odd	2	1	inner
5.b	even	2	1	inner
16.e	even	4	1	inner
48.i	odd	4	1	inner
80.q	even	4	1	inner
240.bm	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	240.1.bm.a	✓	4
3.b	odd	2	1	inner	240.1.bm.a	✓	4
4.b	odd	2	1		960.1.bm.a		4
5.b	even	2	1	inner	240.1.bm.a	✓	4
5.c	odd	4	2		1200.1.r.a		4
8.b	even	2	1		1920.1.bm.a		4
8.d	odd	2	1		1920.1.bm.b		4
12.b	even	2	1		960.1.bm.a		4
15.d	odd	2	1	CM	240.1.bm.a	✓	4
15.e	even	4	2		1200.1.r.a		4
16.e	even	4	1	inner	240.1.bm.a	✓	4
16.e	even	4	1		1920.1.bm.a		4
16.f	odd	4	1		960.1.bm.a		4
16.f	odd	4	1		1920.1.bm.b		4
20.d	odd	2	1		960.1.bm.a		4
24.f	even	2	1		1920.1.bm.b		4
24.h	odd	2	1		1920.1.bm.a		4
40.e	odd	2	1		1920.1.bm.b		4
40.f	even	2	1		1920.1.bm.a		4
48.i	odd	4	1	inner	240.1.bm.a	✓	4
48.i	odd	4	1		1920.1.bm.a		4
48.k	even	4	1		960.1.bm.a		4
48.k	even	4	1		1920.1.bm.b		4
60.h	even	2	1		960.1.bm.a		4
80.i	odd	4	1		1200.1.r.a		4
80.k	odd	4	1		960.1.bm.a		4
80.k	odd	4	1		1920.1.bm.b		4
80.q	even	4	1	inner	240.1.bm.a	✓	4
80.q	even	4	1		1920.1.bm.a		4
80.t	odd	4	1		1200.1.r.a		4
120.i	odd	2	1		1920.1.bm.a		4
120.m	even	2	1		1920.1.bm.b		4
240.t	even	4	1		960.1.bm.a		4
240.t	even	4	1		1920.1.bm.b		4
240.bb	even	4	1		1200.1.r.a		4
240.bf	even	4	1		1200.1.r.a		4
240.bm	odd	4	1	inner	240.1.bm.a	✓	4
240.bm	odd	4	1		1920.1.bm.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
240.1.bm.a	✓	4	1.a	even	1	1	trivial
240.1.bm.a	✓	4	3.b	odd	2	1	inner
240.1.bm.a	✓	4	5.b	even	2	1	inner
240.1.bm.a	✓	4	15.d	odd	2	1	CM
240.1.bm.a	✓	4	16.e	even	4	1	inner
240.1.bm.a	✓	4	48.i	odd	4	1	inner
240.1.bm.a	✓	4	80.q	even	4	1	inner
240.1.bm.a	✓	4	240.bm	odd	4	1	inner
960.1.bm.a		4	4.b	odd	2	1
960.1.bm.a		4	12.b	even	2	1
960.1.bm.a		4	16.f	odd	4	1
960.1.bm.a		4	20.d	odd	2	1
960.1.bm.a		4	48.k	even	4	1
960.1.bm.a		4	60.h	even	2	1
960.1.bm.a		4	80.k	odd	4	1
960.1.bm.a		4	240.t	even	4	1
1200.1.r.a		4	5.c	odd	4	2
1200.1.r.a		4	15.e	even	4	2
1200.1.r.a		4	80.i	odd	4	1
1200.1.r.a		4	80.t	odd	4	1
1200.1.r.a		4	240.bb	even	4	1
1200.1.r.a		4	240.bf	even	4	1
1920.1.bm.a		4	8.b	even	2	1
1920.1.bm.a		4	16.e	even	4	1
1920.1.bm.a		4	24.h	odd	2	1
1920.1.bm.a		4	40.f	even	2	1
1920.1.bm.a		4	48.i	odd	4	1
1920.1.bm.a		4	80.q	even	4	1
1920.1.bm.a		4	120.i	odd	2	1
1920.1.bm.a		4	240.bm	odd	4	1
1920.1.bm.b		4	8.d	odd	2	1
1920.1.bm.b		4	16.f	odd	4	1
1920.1.bm.b		4	24.f	even	2	1
1920.1.bm.b		4	40.e	odd	2	1
1920.1.bm.b		4	48.k	even	4	1
1920.1.bm.b		4	80.k	odd	4	1
1920.1.bm.b		4	120.m	even	2	1
1920.1.bm.b		4	240.t	even	4	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(240, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4} + 1$
$3$	$T^{4} + 1$
$5$	$T^{4} + 1$
$7$	$T^{4}$
$11$	$T^{4}$