Properties

Label 225.1.g.a
Level $225$
Weight $1$
Character orbit 225.g
Analytic conductor $0.112$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -3, -15, 5
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,1,Mod(82,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.82");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 225.g (of order \(4\), degree \(2\), 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: \(0.112289627842\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{5})\)
Artin image: $\OD_{16}$
Artin field: Galois closure of 8.4.56953125.1

$q$-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.

\(f(q)\) \(=\) \( q + i q^{4}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{4} - q^{16} - 2 i q^{19} - 2 q^{31} + i q^{49} + 2 q^{61} - i q^{64} + 2 q^{76} + 2 i q^{79} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{16} - 4 q^{31} + 4 q^{61} + 4 q^{76}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
1.00000i
1.00000i
0 0 1.00000i 0 0 0 0 0 0
118.1 0 0 1.00000i 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
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.1.g.a 2
3.b odd 2 1 CM 225.1.g.a 2
4.b odd 2 1 3600.1.bh.a 2
5.b even 2 1 RM 225.1.g.a 2
5.c odd 4 2 inner 225.1.g.a 2
9.c even 3 2 2025.1.p.a 4
9.d odd 6 2 2025.1.p.a 4
12.b even 2 1 3600.1.bh.a 2
15.d odd 2 1 CM 225.1.g.a 2
15.e even 4 2 inner 225.1.g.a 2
20.d odd 2 1 3600.1.bh.a 2
20.e even 4 2 3600.1.bh.a 2
45.h odd 6 2 2025.1.p.a 4
45.j even 6 2 2025.1.p.a 4
45.k odd 12 4 2025.1.p.a 4
45.l even 12 4 2025.1.p.a 4
60.h even 2 1 3600.1.bh.a 2
60.l odd 4 2 3600.1.bh.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.1.g.a 2 1.a even 1 1 trivial
225.1.g.a 2 3.b odd 2 1 CM
225.1.g.a 2 5.b even 2 1 RM
225.1.g.a 2 5.c odd 4 2 inner
225.1.g.a 2 15.d odd 2 1 CM
225.1.g.a 2 15.e even 4 2 inner
2025.1.p.a 4 9.c even 3 2
2025.1.p.a 4 9.d odd 6 2
2025.1.p.a 4 45.h odd 6 2
2025.1.p.a 4 45.j even 6 2
2025.1.p.a 4 45.k odd 12 4
2025.1.p.a 4 45.l even 12 4
3600.1.bh.a 2 4.b odd 2 1
3600.1.bh.a 2 12.b even 2 1
3600.1.bh.a 2 20.d odd 2 1
3600.1.bh.a 2 20.e even 4 2
3600.1.bh.a 2 60.h even 2 1
3600.1.bh.a 2 60.l odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(225, [\chi])\).

Hecke characteristic polynomials

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