Properties

Label 4725.2.a.bk
Level $4725$
Weight $2$
Character orbit 4725.a
Self dual yes
Analytic conductor $37.729$
Analytic rank $1$
Dimension $3$
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] = [4725,2,Mod(1,4725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4725, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4725.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4725 = 3^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4725.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: \(37.7293149551\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{2} + \beta_1 + 1) q^{4} + q^{7} + (\beta_{2} - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{2} + \beta_1 + 1) q^{4} + q^{7} + (\beta_{2} - 3) q^{8} + \beta_1 q^{11} + ( - \beta_{2} - \beta_1 - 1) q^{13} + \beta_{2} q^{14} + ( - 2 \beta_{2} - \beta_1 + 1) q^{16} + (\beta_{2} + 2 \beta_1 - 3) q^{17} + ( - \beta_{2} - 3 \beta_1 + 2) q^{19} + (\beta_{2} + \beta_1) q^{22} + (\beta_{2} - \beta_1) q^{23} + ( - \beta_{2} - 2 \beta_1 - 3) q^{26} + ( - \beta_{2} + \beta_1 + 1) q^{28} + (\beta_{2} - \beta_1) q^{29} + ( - 3 \beta_{2} - \beta_1 - 1) q^{31} - 3 \beta_1 q^{32} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{34} + (\beta_{2} - 3 \beta_1 + 2) q^{37} + ( - 4 \beta_1 - 3) q^{38} + ( - 2 \beta_{2} + 5 \beta_1) q^{41} + ( - \beta_1 - 7) q^{43} + 3 q^{44} + ( - 2 \beta_{2} + 3) q^{46} + ( - 3 \beta_{2} - \beta_1 - 3) q^{47} + q^{49} + ( - 2 \beta_{2} - \beta_1 - 1) q^{52} + (2 \beta_{2} - \beta_1 - 6) q^{53} + (\beta_{2} - 3) q^{56} + ( - 2 \beta_{2} + 3) q^{58} + (4 \beta_{2} - 3 \beta_1) q^{59} + ( - \beta_{2} + 4 \beta_1 - 4) q^{61} + (\beta_{2} - 4 \beta_1 - 9) q^{62} + (\beta_{2} - \beta_1 - 2) q^{64} + (6 \beta_{2} + 5) q^{67} + (6 \beta_{2} - 3 \beta_1) q^{68} + (5 \beta_1 - 3) q^{71} + ( - 2 \beta_{2} + 4 \beta_1 - 1) q^{73} + ( - 2 \beta_{2} - 2 \beta_1 + 3) q^{74} + ( - 5 \beta_{2} + 2 \beta_1 - 4) q^{76} + \beta_1 q^{77} + ( - 2 \beta_{2} + 4 \beta_1 - 1) q^{79} + (7 \beta_{2} + 3 \beta_1 - 6) q^{82} + ( - 4 \beta_{2} - \beta_1 - 3) q^{83} + ( - 8 \beta_{2} - \beta_1) q^{86} + (\beta_{2} - 2 \beta_1) q^{88} + ( - 2 \beta_{2} - 4 \beta_1) q^{89} + ( - \beta_{2} - \beta_1 - 1) q^{91} + (3 \beta_{2} - 6) q^{92} + ( - \beta_{2} - 4 \beta_1 - 9) q^{94} + (4 \beta_1 + 2) q^{97} + \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{4} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4 q^{4} + 3 q^{7} - 9 q^{8} + q^{11} - 4 q^{13} + 2 q^{16} - 7 q^{17} + 3 q^{19} + q^{22} - q^{23} - 11 q^{26} + 4 q^{28} - q^{29} - 4 q^{31} - 3 q^{32} + 12 q^{34} + 3 q^{37} - 13 q^{38} + 5 q^{41} - 22 q^{43} + 9 q^{44} + 9 q^{46} - 10 q^{47} + 3 q^{49} - 4 q^{52} - 19 q^{53} - 9 q^{56} + 9 q^{58} - 3 q^{59} - 8 q^{61} - 31 q^{62} - 7 q^{64} + 15 q^{67} - 3 q^{68} - 4 q^{71} + q^{73} + 7 q^{74} - 10 q^{76} + q^{77} + q^{79} - 15 q^{82} - 10 q^{83} - q^{86} - 2 q^{88} - 4 q^{89} - 4 q^{91} - 18 q^{92} - 31 q^{94} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
0.713538
−1.91223
2.19869
−2.49086 0 4.20440 0 0 1.00000 −5.49086 0 0
1.2 0.656620 0 −1.56885 0 0 1.00000 −2.34338 0 0
1.3 1.83424 0 1.36445 0 0 1.00000 −1.16576 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 4725.2.a.bk yes 3
3.b odd 2 1 4725.2.a.bl yes 3
5.b even 2 1 4725.2.a.bj yes 3
15.d odd 2 1 4725.2.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4725.2.a.bi 3 15.d odd 2 1
4725.2.a.bj yes 3 5.b even 2 1
4725.2.a.bk yes 3 1.a even 1 1 trivial
4725.2.a.bl yes 3 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4725))\):

\( T_{2}^{3} - 5T_{2} + 3 \) Copy content Toggle raw display
\( T_{11}^{3} - T_{11}^{2} - 4T_{11} + 3 \) Copy content Toggle raw display
\( T_{13}^{3} + 4T_{13}^{2} - 5T_{13} + 1 \) Copy content Toggle raw display
\( T_{37}^{3} - 3T_{37}^{2} - 38T_{37} - 61 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 5T + 3 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - T^{2} - 4T + 3 \) Copy content Toggle raw display
$13$ \( T^{3} + 4 T^{2} - 5 T + 1 \) Copy content Toggle raw display
$17$ \( T^{3} + 7 T^{2} - 8 T - 81 \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} - 44 T + 107 \) Copy content Toggle raw display
$23$ \( T^{3} + T^{2} - 8T - 3 \) Copy content Toggle raw display
$29$ \( T^{3} + T^{2} - 8T - 3 \) Copy content Toggle raw display
$31$ \( T^{3} + 4 T^{2} - 47 T - 53 \) Copy content Toggle raw display
$37$ \( T^{3} - 3 T^{2} - 38 T - 61 \) Copy content Toggle raw display
$41$ \( T^{3} - 5 T^{2} - 110 T + 681 \) Copy content Toggle raw display
$43$ \( T^{3} + 22 T^{2} + 157 T + 361 \) Copy content Toggle raw display
$47$ \( T^{3} + 10 T^{2} - 19 T - 123 \) Copy content Toggle raw display
$53$ \( T^{3} + 19 T^{2} + 98 T + 147 \) Copy content Toggle raw display
$59$ \( T^{3} + 3 T^{2} - 104 T + 75 \) Copy content Toggle raw display
$61$ \( T^{3} + 8 T^{2} - 49 T + 49 \) Copy content Toggle raw display
$67$ \( T^{3} - 15 T^{2} - 105 T + 1423 \) Copy content Toggle raw display
$71$ \( T^{3} + 4 T^{2} - 103 T + 57 \) Copy content Toggle raw display
$73$ \( T^{3} - T^{2} - 81 T + 281 \) Copy content Toggle raw display
$79$ \( T^{3} - T^{2} - 81 T + 281 \) Copy content Toggle raw display
$83$ \( T^{3} + 10 T^{2} - 55 T - 291 \) Copy content Toggle raw display
$89$ \( T^{3} + 4 T^{2} - 92 T + 168 \) Copy content Toggle raw display
$97$ \( T^{3} - 10 T^{2} - 36 T + 296 \) Copy content Toggle raw display
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