Properties

Label 1225.2.a.f
Level $1225$
Weight $2$
Character orbit 1225.a
Self dual yes
Analytic conductor $9.782$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1225,2,Mod(1,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.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: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{3} - q^{4} - q^{6} - 3 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} - q^{4} - q^{6} - 3 q^{8} - 2 q^{9} + q^{12} + 2 q^{13} - q^{16} + 2 q^{17} - 2 q^{18} + 6 q^{19} - 3 q^{23} + 3 q^{24} + 2 q^{26} + 5 q^{27} + 7 q^{29} + 2 q^{31} + 5 q^{32} + 2 q^{34} + 2 q^{36} - 8 q^{37} + 6 q^{38} - 2 q^{39} + 5 q^{41} + 7 q^{43} - 3 q^{46} + q^{48} - 2 q^{51} - 2 q^{52} + 6 q^{53} + 5 q^{54} - 6 q^{57} + 7 q^{58} + 10 q^{59} + 7 q^{61} + 2 q^{62} + 7 q^{64} - 5 q^{67} - 2 q^{68} + 3 q^{69} - 2 q^{71} + 6 q^{72} - 6 q^{73} - 8 q^{74} - 6 q^{76} - 2 q^{78} - 2 q^{79} + q^{81} + 5 q^{82} - 11 q^{83} + 7 q^{86} - 7 q^{87} + 9 q^{89} + 3 q^{92} - 2 q^{93} - 5 q^{96} - 16 q^{97}+O(q^{100}) \) 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
1.00000 −1.00000 −1.00000 0 −1.00000 0 −3.00000 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.2.a.f 1
5.b even 2 1 1225.2.a.d 1
5.c odd 4 2 245.2.b.c 2
7.b odd 2 1 1225.2.a.g 1
7.c even 3 2 175.2.e.a 2
15.e even 4 2 2205.2.d.d 2
35.c odd 2 1 1225.2.a.b 1
35.f even 4 2 245.2.b.b 2
35.j even 6 2 175.2.e.b 2
35.k even 12 4 245.2.j.c 4
35.l odd 12 4 35.2.j.a 4
105.k odd 4 2 2205.2.d.e 2
105.x even 12 4 315.2.bf.a 4
140.w even 12 4 560.2.bw.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.j.a 4 35.l odd 12 4
175.2.e.a 2 7.c even 3 2
175.2.e.b 2 35.j even 6 2
245.2.b.b 2 35.f even 4 2
245.2.b.c 2 5.c odd 4 2
245.2.j.c 4 35.k even 12 4
315.2.bf.a 4 105.x even 12 4
560.2.bw.b 4 140.w even 12 4
1225.2.a.b 1 35.c odd 2 1
1225.2.a.d 1 5.b even 2 1
1225.2.a.f 1 1.a even 1 1 trivial
1225.2.a.g 1 7.b odd 2 1
2205.2.d.d 2 15.e even 4 2
2205.2.d.e 2 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T - 6 \) Copy content Toggle raw display
$23$ \( T + 3 \) Copy content Toggle raw display
$29$ \( T - 7 \) Copy content Toggle raw display
$31$ \( T - 2 \) Copy content Toggle raw display
$37$ \( T + 8 \) Copy content Toggle raw display
$41$ \( T - 5 \) Copy content Toggle raw display
$43$ \( T - 7 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 10 \) Copy content Toggle raw display
$61$ \( T - 7 \) Copy content Toggle raw display
$67$ \( T + 5 \) Copy content Toggle raw display
$71$ \( T + 2 \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T + 2 \) Copy content Toggle raw display
$83$ \( T + 11 \) Copy content Toggle raw display
$89$ \( T - 9 \) Copy content Toggle raw display
$97$ \( T + 16 \) Copy content Toggle raw display
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