Newform orbit 300.1.l.a

• Label: 300.1.l.a
• Level: $300$
• Weight: $1$
• Character orbit: 300.l
• Analytic conductor: $0.150$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{2}$
• CM/RM discs: -15, -20, 12
• Inner twists: $16$

Newspace parameters

Level:	$N$	$=$	$300 = 2^{2} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	300.l (of order $4$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.149719503790$
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:	$D_{2}$
Projective field:	Galois closure of $\Q(\sqrt{3}, \sqrt{-5})$
Artin image:	$\OD_{16}:C_2$
Artin field:	Galois closure of 16.0.10251562500000000.1

$q$-expansion

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

$f(q)$	$=$	q - z * q^2 - z^3 * q^3 + z^2 * q^4 - q^6 - z^3 * q^8 - z^2 * q^9 + z * q^12 - q^16 + z^3 * q^18 + 2*z^3 * q^23 - z^2 * q^24 - z * q^27 + z * q^32 + q^36 + 2 * q^46 + 2*z * q^47 + z^3 * q^48 + z^2 * q^49 + z^2 * q^54 - 2 * q^61 - z^2 * q^64 + 2*z^2 * q^69 - z * q^72 - q^81 - 2*z^3 * q^83 - 2*z * q^92 - 2*z^2 * q^94 + q^96 - z^3 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{6} - 4 q^{16} + 4 q^{36} + 8 q^{46} - 8 q^{61} - 4 q^{81} + 4 q^{96}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$-1$	$-1$	$\zeta_{8}^{2}$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

107.1

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

−1.00000

0

0.707107

−

0.707107i

−

1.00000i

0

107.2

0.707107

+

0.707107i

−0.707107

+

0.707107i

1.00000i

0

−1.00000

0

−0.707107

+

0.707107i

−

1.00000i

0

143.1

−0.707107

+

0.707107i

0.707107

+

0.707107i

−

1.00000i

0

−1.00000

0

0.707107

+

0.707107i

1.00000i

0

143.2

0.707107

−

0.707107i

−0.707107

−

0.707107i

−

1.00000i

0

−1.00000

0

−0.707107

−

0.707107i

1.00000i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
12.b	even	2	1	RM by $\Q(\sqrt{3})$
15.d	odd	2	1	CM by $\Q(\sqrt{-15})$
20.d	odd	2	1	CM by $\Q(\sqrt{-5})$
3.b	odd	2	1	inner
4.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
15.e	even	4	2	inner
20.e	even	4	2	inner
60.h	even	2	1	inner
60.l	odd	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	300.1.l.a	✓	4
3.b	odd	2	1	inner	300.1.l.a	✓	4
4.b	odd	2	1	inner	300.1.l.a	✓	4
5.b	even	2	1	inner	300.1.l.a	✓	4
5.c	odd	4	2	inner	300.1.l.a	✓	4
12.b	even	2	1	RM	300.1.l.a	✓	4
15.d	odd	2	1	CM	300.1.l.a	✓	4
15.e	even	4	2	inner	300.1.l.a	✓	4
20.d	odd	2	1	CM	300.1.l.a	✓	4
20.e	even	4	2	inner	300.1.l.a	✓	4
60.h	even	2	1	inner	300.1.l.a	✓	4
60.l	odd	4	2	inner	300.1.l.a	✓	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
300.1.l.a	✓	4	1.a	even	1	1	trivial
300.1.l.a	✓	4	3.b	odd	2	1	inner
300.1.l.a	✓	4	4.b	odd	2	1	inner
300.1.l.a	✓	4	5.b	even	2	1	inner
300.1.l.a	✓	4	5.c	odd	4	2	inner
300.1.l.a	✓	4	12.b	even	2	1	RM
300.1.l.a	✓	4	15.d	odd	2	1	CM
300.1.l.a	✓	4	15.e	even	4	2	inner
300.1.l.a	✓	4	20.d	odd	2	1	CM
300.1.l.a	✓	4	20.e	even	4	2	inner
300.1.l.a	✓	4	60.h	even	2	1	inner
300.1.l.a	✓	4	60.l	odd	4	2	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(300, [\chi])$.

Hecke characteristic polynomials

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