Properties

Label 9025.2.a.br
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $6$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.41289040.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 8x^{4} + 21x^{3} + 18x^{2} - 25x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 475)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_{3} + \beta_1 - 1) q^{3} + (\beta_{4} - \beta_{3}) q^{4} + ( - \beta_{5} + \beta_{4} - \beta_1) q^{6} + \beta_{2} q^{7} + (\beta_{3} - 1) q^{8} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots + 1) q^{9}+ \cdots + ( - 2 \beta_{5} + 6 \beta_{4} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{6} - 2 q^{7} - 6 q^{8} + 7 q^{9} - q^{11} - 7 q^{12} - 5 q^{13} - 6 q^{14} - 6 q^{16} + 3 q^{17} - 7 q^{18} + 3 q^{21} - 9 q^{22} + 6 q^{23} + 11 q^{24} + 19 q^{26}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 8x^{4} + 21x^{3} + 18x^{2} - 25x - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - \nu^{4} - 10\nu^{3} + 6\nu^{2} + 20\nu - 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{5} - 2\nu^{4} - 15\nu^{3} + 7\nu^{2} + 15\nu - 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} + 3\nu^{4} - 20\nu^{3} - 28\nu^{2} + 25\nu + 20 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 2\beta_{3} + \beta_{2} + 6\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - 2\beta_{3} + 8\beta_{2} + 11\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 10\beta_{4} - 17\beta_{3} + 12\beta_{2} + 45\beta _1 + 37 \) 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
−1.41269
2.95188
−2.24415
1.02983
−0.181608
2.85674
−2.17009 −2.95188 2.70928 0 6.40583 −0.591620 −1.53919 5.71358 0
1.2 −2.17009 1.41269 2.70928 0 −3.06566 1.76171 −1.53919 −1.00431 0
1.3 −0.311108 −1.02983 −1.90321 0 0.320390 3.28038 1.21432 −1.93944 0
1.4 −0.311108 2.24415 −1.90321 0 −0.698174 −3.96928 1.21432 2.03623 0
1.5 1.48119 −2.85674 0.193937 0 −4.23138 −3.78541 −2.67513 5.16096 0
1.6 1.48119 0.181608 0.193937 0 0.268996 1.30422 −2.67513 −2.96702 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.br 6
5.b even 2 1 9025.2.a.bz 6
19.b odd 2 1 9025.2.a.by 6
19.c even 3 2 475.2.e.h yes 12
95.d odd 2 1 9025.2.a.bs 6
95.i even 6 2 475.2.e.f 12
95.m odd 12 4 475.2.j.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.e.f 12 95.i even 6 2
475.2.e.h yes 12 19.c even 3 2
475.2.j.d 24 95.m odd 12 4
9025.2.a.br 6 1.a even 1 1 trivial
9025.2.a.bs 6 95.d odd 2 1
9025.2.a.by 6 19.b odd 2 1
9025.2.a.bz 6 5.b even 2 1

Hecke kernels

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

\( T_{2}^{3} + T_{2}^{2} - 3T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{6} + 3T_{3}^{5} - 8T_{3}^{4} - 21T_{3}^{3} + 18T_{3}^{2} + 25T_{3} - 5 \) Copy content Toggle raw display
\( T_{7}^{6} + 2T_{7}^{5} - 21T_{7}^{4} - 20T_{7}^{3} + 123T_{7}^{2} - 38T_{7} - 67 \) Copy content Toggle raw display
\( T_{11}^{6} + T_{11}^{5} - 46T_{11}^{4} + 9T_{11}^{3} + 522T_{11}^{2} - 871T_{11} + 247 \) Copy content Toggle raw display
\( T_{29}^{6} - 3T_{29}^{5} - 108T_{29}^{4} + 97T_{29}^{3} + 3140T_{29}^{2} + 2693T_{29} - 12781 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} + T^{2} - 3 T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{6} + 3 T^{5} + \cdots - 5 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} + \cdots - 67 \) Copy content Toggle raw display
$11$ \( T^{6} + T^{5} + \cdots + 247 \) Copy content Toggle raw display
$13$ \( T^{6} + 5 T^{5} + \cdots - 317 \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} + \cdots + 1073 \) Copy content Toggle raw display
$29$ \( T^{6} - 3 T^{5} + \cdots - 12781 \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{5} + \cdots - 631 \) Copy content Toggle raw display
$37$ \( T^{6} - 6 T^{5} + \cdots - 64 \) Copy content Toggle raw display
$41$ \( T^{6} - 11 T^{5} + \cdots + 1319 \) Copy content Toggle raw display
$43$ \( T^{6} + 13 T^{5} + \cdots - 911 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + \cdots + 23179 \) Copy content Toggle raw display
$53$ \( T^{6} + 18 T^{5} + \cdots + 341861 \) Copy content Toggle raw display
$59$ \( T^{6} - 4 T^{5} + \cdots + 11593 \) Copy content Toggle raw display
$61$ \( T^{6} - 25 T^{5} + \cdots - 17491 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + \cdots - 12080 \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} + \cdots - 11657 \) Copy content Toggle raw display
$73$ \( T^{6} + T^{5} + \cdots - 68555 \) Copy content Toggle raw display
$79$ \( T^{6} - 3 T^{5} + \cdots + 326351 \) Copy content Toggle raw display
$83$ \( T^{6} - 23 T^{5} + \cdots + 378053 \) Copy content Toggle raw display
$89$ \( T^{6} - 12 T^{5} + \cdots - 349609 \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{5} + \cdots - 159631 \) Copy content Toggle raw display
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