Properties

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 $16$

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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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{4} -\zeta_{12}^{5} q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{4} -\zeta_{12}^{5} q^{9} + 2 \zeta_{12}^{4} q^{11} + \zeta_{12}^{2} q^{16} -2 \zeta_{12}^{3} q^{29} + q^{36} + 2 \zeta_{12}^{5} q^{44} + \zeta_{12}^{3} q^{64} -2 q^{71} -2 \zeta_{12}^{5} q^{79} -\zeta_{12}^{4} q^{81} + 2 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{11} + 2q^{16} + 4q^{36} - 8q^{71} + 2q^{81} + O(q^{100}) \)

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
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 \(S_{1}^{\mathrm{new}}(1225, [\chi])\).