Newform orbit 1225.1.q.a

• Label: 1225.1.q.a
• Level: $1225$
• Weight: $1$
• Character orbit: 1225.q
• Analytic conductor: $0.611$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{2}$
• CM/RM discs: -7, -35, 5
• Inner twists: $16$

Newspace parameters

Level:	$N$	$=$	$1225 = 5^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1225.q (of order $12$, degree $4$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.611354640475$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
Defining polynomial:	$x^{4} - x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(\sqrt{5}, \sqrt{-7})$
Artin image:	$C_3\times \OD_{16}$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{24} - \cdots)$

$q$-expansion

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

$f(q)$	$=$	q + z * q^4 - z^5 * q^9 + 2*z^4 * q^11 + z^2 * q^16 - 2*z^3 * q^29 + q^36 + 2*z^5 * q^44 + z^3 * q^64 - 2 * q^71 - 2*z^5 * q^79 - z^4 * q^81 + 2*z^3 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{11} + 2 q^{16} + 4 q^{36} - 8 q^{71} + 2 q^{81}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$-\zeta_{12}^{2}$	$\zeta_{12}^{3}$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

18.1

−0.866025	−	0.500000i
0.866025	+	0.500000i
0.866025	−	0.500000i
−0.866025	+	0.500000i

0

0

−0.866025

−

0.500000i

0

0

0

0

−0.866025

+

0.500000i

0

557.1

0

0

0.866025

+

0.500000i

0

0

0

0

0.866025

−

0.500000i

0

618.1

0

0

0.866025

−

0.500000i

0

0

0

0

0.866025

+

0.500000i

0

1157.1

0

0

−0.866025

+

0.500000i

0

0

0

0

−0.866025

−

0.500000i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	RM by $\Q(\sqrt{5})$
7.b	odd	2	1	CM by $\Q(\sqrt{-7})$
35.c	odd	2	1	CM by $\Q(\sqrt{-35})$
5.c	odd	4	2	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
35.f	even	4	2	inner
35.i	odd	6	1	inner
35.j	even	6	1	inner
35.k	even	12	2	inner
35.l	odd	12	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1225.1.q.a		4
5.b	even	2	1	RM	1225.1.q.a		4
5.c	odd	4	2	inner	1225.1.q.a		4
7.b	odd	2	1	CM	1225.1.q.a		4
7.c	even	3	1		1225.1.g.a	✓	2
7.c	even	3	1	inner	1225.1.q.a		4
7.d	odd	6	1		1225.1.g.a	✓	2
7.d	odd	6	1	inner	1225.1.q.a		4
35.c	odd	2	1	CM	1225.1.q.a		4
35.f	even	4	2	inner	1225.1.q.a		4
35.i	odd	6	1		1225.1.g.a	✓	2
35.i	odd	6	1	inner	1225.1.q.a		4
35.j	even	6	1		1225.1.g.a	✓	2
35.j	even	6	1	inner	1225.1.q.a		4
35.k	even	12	2		1225.1.g.a	✓	2
35.k	even	12	2	inner	1225.1.q.a		4
35.l	odd	12	2		1225.1.g.a	✓	2
35.l	odd	12	2	inner	1225.1.q.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1225.1.g.a	✓	2	7.c	even	3	1
1225.1.g.a	✓	2	7.d	odd	6	1
1225.1.g.a	✓	2	35.i	odd	6	1
1225.1.g.a	✓	2	35.j	even	6	1
1225.1.g.a	✓	2	35.k	even	12	2
1225.1.g.a	✓	2	35.l	odd	12	2
1225.1.q.a		4	1.a	even	1	1	trivial
1225.1.q.a		4	5.b	even	2	1	RM
1225.1.q.a		4	5.c	odd	4	2	inner
1225.1.q.a		4	7.b	odd	2	1	CM
1225.1.q.a		4	7.c	even	3	1	inner
1225.1.q.a		4	7.d	odd	6	1	inner
1225.1.q.a		4	35.c	odd	2	1	CM
1225.1.q.a		4	35.f	even	4	2	inner
1225.1.q.a		4	35.i	odd	6	1	inner
1225.1.q.a		4	35.j	even	6	1	inner
1225.1.q.a		4	35.k	even	12	2	inner
1225.1.q.a		4	35.l	odd	12	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}$ acting on $S_{1}^{\mathrm{new}}(1225, [\chi])$.

Hecke characteristic polynomials

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