Properties

Label 9025.2.a.m
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
Dimension $2$
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] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, 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("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.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: \(72.0649878242\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (2 \beta - 1) q^{3} + 3 \beta q^{4} + ( - 3 \beta - 1) q^{6} + (2 \beta + 1) q^{7} + ( - 4 \beta - 1) q^{8} + 2 q^{9} + (4 \beta - 1) q^{11} + (3 \beta + 6) q^{12} + 2 q^{13}+ \cdots + (8 \beta - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9} + 2 q^{11} + 15 q^{12} + 4 q^{13} - 11 q^{14} + 13 q^{16} + 3 q^{17} - 6 q^{18} + 10 q^{21} - 13 q^{22} - 2 q^{23} - 20 q^{24} - 6 q^{26}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−2.61803 2.23607 4.85410 0 −5.85410 4.23607 −7.47214 2.00000 0
1.2 −0.381966 −2.23607 −1.85410 0 0.854102 −0.236068 1.47214 2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(19\) \( -1 \)

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 9025.2.a.m yes 2
5.b even 2 1 9025.2.a.t yes 2
19.b odd 2 1 9025.2.a.u yes 2
95.d odd 2 1 9025.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9025.2.a.l 2 95.d odd 2 1
9025.2.a.m yes 2 1.a even 1 1 trivial
9025.2.a.t yes 2 5.b even 2 1
9025.2.a.u yes 2 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 5 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 19 \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 5 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 19 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$37$ \( T^{2} + 15T + 45 \) Copy content Toggle raw display
$41$ \( T^{2} + 15T + 55 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 101 \) Copy content Toggle raw display
$47$ \( T^{2} - 13T + 41 \) Copy content Toggle raw display
$53$ \( T^{2} - 15T + 55 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$61$ \( T^{2} - 9T - 81 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
$71$ \( T^{2} + 18T + 61 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 29 \) Copy content Toggle raw display
$79$ \( (T - 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 9T - 11 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 17T + 71 \) Copy content Toggle raw display
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