Properties

Label 2025.1.p.a
Level $2025$
Weight $1$
Character orbit 2025.p
Analytic conductor $1.011$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -3, -15, 5
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2025,1,Mod(757,2025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2025, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2025.757");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2025.p (of order \(12\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.01060665058\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 225)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{5})\)
Artin image: $C_3\times \OD_{16}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{12}^{5} q^{4} - \zeta_{12}^{4} q^{16} - 2 \zeta_{12}^{3} q^{19} + 2 \zeta_{12}^{2} q^{31} - \zeta_{12}^{5} q^{49} + 2 \zeta_{12}^{4} q^{61} - \zeta_{12}^{3} q^{64} - 2 \zeta_{12}^{2} q^{76} + \cdots - 2 \zeta_{12} q^{79} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{16} + 4 q^{31} - 4 q^{61} - 4 q^{76}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(1702\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-\zeta_{12}^{3}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
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 0
1243.1 0 0 −0.866025 0.500000i 0 0 0 0 0 0
1432.1 0 0 −0.866025 + 0.500000i 0 0 0 0 0 0
1918.1 0 0 0.866025 0.500000i 0 0 0 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 RM by \(\Q(\sqrt{5}) \)
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
5.c odd 4 2 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.e even 4 2 inner
45.h odd 6 1 inner
45.j even 6 1 inner
45.k odd 12 2 inner
45.l even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2025.1.p.a 4
3.b odd 2 1 CM 2025.1.p.a 4
5.b even 2 1 RM 2025.1.p.a 4
5.c odd 4 2 inner 2025.1.p.a 4
9.c even 3 1 225.1.g.a 2
9.c even 3 1 inner 2025.1.p.a 4
9.d odd 6 1 225.1.g.a 2
9.d odd 6 1 inner 2025.1.p.a 4
15.d odd 2 1 CM 2025.1.p.a 4
15.e even 4 2 inner 2025.1.p.a 4
36.f odd 6 1 3600.1.bh.a 2
36.h even 6 1 3600.1.bh.a 2
45.h odd 6 1 225.1.g.a 2
45.h odd 6 1 inner 2025.1.p.a 4
45.j even 6 1 225.1.g.a 2
45.j even 6 1 inner 2025.1.p.a 4
45.k odd 12 2 225.1.g.a 2
45.k odd 12 2 inner 2025.1.p.a 4
45.l even 12 2 225.1.g.a 2
45.l even 12 2 inner 2025.1.p.a 4
180.n even 6 1 3600.1.bh.a 2
180.p odd 6 1 3600.1.bh.a 2
180.v odd 12 2 3600.1.bh.a 2
180.x even 12 2 3600.1.bh.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.1.g.a 2 9.c even 3 1
225.1.g.a 2 9.d odd 6 1
225.1.g.a 2 45.h odd 6 1
225.1.g.a 2 45.j even 6 1
225.1.g.a 2 45.k odd 12 2
225.1.g.a 2 45.l even 12 2
2025.1.p.a 4 1.a even 1 1 trivial
2025.1.p.a 4 3.b odd 2 1 CM
2025.1.p.a 4 5.b even 2 1 RM
2025.1.p.a 4 5.c odd 4 2 inner
2025.1.p.a 4 9.c even 3 1 inner
2025.1.p.a 4 9.d odd 6 1 inner
2025.1.p.a 4 15.d odd 2 1 CM
2025.1.p.a 4 15.e even 4 2 inner
2025.1.p.a 4 45.h odd 6 1 inner
2025.1.p.a 4 45.j even 6 1 inner
2025.1.p.a 4 45.k odd 12 2 inner
2025.1.p.a 4 45.l even 12 2 inner
3600.1.bh.a 2 36.f odd 6 1
3600.1.bh.a 2 36.h even 6 1
3600.1.bh.a 2 180.n even 6 1
3600.1.bh.a 2 180.p odd 6 1
3600.1.bh.a 2 180.v odd 12 2
3600.1.bh.a 2 180.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2025, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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