Properties

Label 225.4.a.d
Level $225$
Weight $4$
Character orbit 225.a
Self dual yes
Analytic conductor $13.275$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,4,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

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: yes
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 8 q^{4} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{4} - 20 q^{7} + 70 q^{13} + 64 q^{16} + 56 q^{19} + 160 q^{28} + 308 q^{31} - 110 q^{37} + 520 q^{43} + 57 q^{49} - 560 q^{52} + 182 q^{61} - 512 q^{64} + 880 q^{67} - 1190 q^{73} - 448 q^{76} + 884 q^{79} - 1400 q^{91} + 1330 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 −8.00000 0 0 −20.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.a.d 1
3.b odd 2 1 CM 225.4.a.d 1
5.b even 2 1 9.4.a.a 1
5.c odd 4 2 225.4.b.g 2
15.d odd 2 1 9.4.a.a 1
15.e even 4 2 225.4.b.g 2
20.d odd 2 1 144.4.a.d 1
35.c odd 2 1 441.4.a.f 1
35.i odd 6 2 441.4.e.j 2
35.j even 6 2 441.4.e.i 2
40.e odd 2 1 576.4.a.l 1
40.f even 2 1 576.4.a.m 1
45.h odd 6 2 81.4.c.b 2
45.j even 6 2 81.4.c.b 2
55.d odd 2 1 1089.4.a.g 1
60.h even 2 1 144.4.a.d 1
65.d even 2 1 1521.4.a.g 1
105.g even 2 1 441.4.a.f 1
105.o odd 6 2 441.4.e.i 2
105.p even 6 2 441.4.e.j 2
120.i odd 2 1 576.4.a.m 1
120.m even 2 1 576.4.a.l 1
165.d even 2 1 1089.4.a.g 1
195.e odd 2 1 1521.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 5.b even 2 1
9.4.a.a 1 15.d odd 2 1
81.4.c.b 2 45.h odd 6 2
81.4.c.b 2 45.j even 6 2
144.4.a.d 1 20.d odd 2 1
144.4.a.d 1 60.h even 2 1
225.4.a.d 1 1.a even 1 1 trivial
225.4.a.d 1 3.b odd 2 1 CM
225.4.b.g 2 5.c odd 4 2
225.4.b.g 2 15.e even 4 2
441.4.a.f 1 35.c odd 2 1
441.4.a.f 1 105.g even 2 1
441.4.e.i 2 35.j even 6 2
441.4.e.i 2 105.o odd 6 2
441.4.e.j 2 35.i odd 6 2
441.4.e.j 2 105.p even 6 2
576.4.a.l 1 40.e odd 2 1
576.4.a.l 1 120.m even 2 1
576.4.a.m 1 40.f even 2 1
576.4.a.m 1 120.i odd 2 1
1089.4.a.g 1 55.d odd 2 1
1089.4.a.g 1 165.d even 2 1
1521.4.a.g 1 65.d even 2 1
1521.4.a.g 1 195.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 20 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 70 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 56 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 308 \) Copy content Toggle raw display
$37$ \( T + 110 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 520 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 182 \) Copy content Toggle raw display
$67$ \( T - 880 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 1190 \) Copy content Toggle raw display
$79$ \( T - 884 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 1330 \) Copy content Toggle raw display
show more
show less