Properties

Label 4725.2.a.bw
Level $4725$
Weight $2$
Character orbit 4725.a
Self dual yes
Analytic conductor $37.729$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 1) q^{4} + q^{7} + 2 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 1) q^{4} + q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{2} + \beta_1) q^{11} + (\beta_{3} + 2) q^{13} + \beta_1 q^{14} + ( - 3 \beta_{2} + 2 \beta_1) q^{17} - 2 \beta_{3} q^{19} + 2 q^{22} + (\beta_{2} - \beta_1) q^{23} + (2 \beta_{2} + 3 \beta_1) q^{26} + (\beta_{3} + 1) q^{28} + (4 \beta_{2} + \beta_1) q^{29} + 3 \beta_{3} q^{31} - 4 \beta_{2} q^{32} + ( - \beta_{3} + 3) q^{34} + 3 q^{37} + ( - 4 \beta_{2} - 2 \beta_1) q^{38} + (3 \beta_{2} + 2 \beta_1) q^{41} + (2 \beta_{3} + 1) q^{43} + 2 \beta_{2} q^{44} - 2 q^{46} + (5 \beta_{2} - \beta_1) q^{47} + q^{49} + (3 \beta_{3} + 7) q^{52} + ( - 3 \beta_{2} - \beta_1) q^{53} + 2 \beta_{2} q^{56} + (5 \beta_{3} + 7) q^{58} + ( - 5 \beta_{2} + \beta_1) q^{59} + (3 \beta_{3} + 3) q^{61} + (6 \beta_{2} + 3 \beta_1) q^{62} + ( - 4 \beta_{3} - 4) q^{64} + ( - \beta_{3} + 4) q^{67} + (4 \beta_{2} - 2 \beta_1) q^{68} + ( - 8 \beta_{2} + 4 \beta_1) q^{71} + ( - 2 \beta_{3} + 1) q^{73} + 3 \beta_1 q^{74} + ( - 2 \beta_{3} - 10) q^{76} + ( - \beta_{2} + \beta_1) q^{77} + ( - 2 \beta_{3} + 3) q^{79} + (5 \beta_{3} + 9) q^{82} + (7 \beta_{2} - 5 \beta_1) q^{83} + (4 \beta_{2} + 3 \beta_1) q^{86} + (2 \beta_{3} - 2) q^{88} + ( - \beta_{2} - 3 \beta_1) q^{89} + (\beta_{3} + 2) q^{91} - 2 \beta_{2} q^{92} + (4 \beta_{3} + 2) q^{94} + ( - 2 \beta_{3} + 6) q^{97} + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 4 q^{7} + 8 q^{13} + 8 q^{22} + 4 q^{28} + 12 q^{34} + 12 q^{37} + 4 q^{43} - 8 q^{46} + 4 q^{49} + 28 q^{52} + 28 q^{58} + 12 q^{61} - 16 q^{64} + 16 q^{67} + 4 q^{73} - 40 q^{76} + 12 q^{79} + 36 q^{82} - 8 q^{88} + 8 q^{91} + 8 q^{94} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 4\beta_1 \) 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
−2.28825
−0.874032
0.874032
2.28825
−2.28825 0 3.23607 0 0 1.00000 −2.82843 0 0
1.2 −0.874032 0 −1.23607 0 0 1.00000 2.82843 0 0
1.3 0.874032 0 −1.23607 0 0 1.00000 −2.82843 0 0
1.4 2.28825 0 3.23607 0 0 1.00000 2.82843 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4725.2.a.bw yes 4
3.b odd 2 1 inner 4725.2.a.bw yes 4
5.b even 2 1 4725.2.a.bt 4
15.d odd 2 1 4725.2.a.bt 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4725.2.a.bt 4 5.b even 2 1
4725.2.a.bt 4 15.d odd 2 1
4725.2.a.bw yes 4 1.a even 1 1 trivial
4725.2.a.bw yes 4 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4725))\):

\( T_{2}^{4} - 6T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 6T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} - 1 \) Copy content Toggle raw display
\( T_{37} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 6T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 6T^{2} + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 36T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 6T^{2} + 4 \) Copy content Toggle raw display
$29$ \( T^{4} - 86T^{2} + 1444 \) Copy content Toggle raw display
$31$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$37$ \( (T - 3)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 84T^{2} + 484 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 19)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 86T^{2} + 1444 \) Copy content Toggle raw display
$53$ \( T^{4} - 54T^{2} + 484 \) Copy content Toggle raw display
$59$ \( T^{4} - 86T^{2} + 1444 \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T - 36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T + 11)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 224T^{2} + 1024 \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T - 19)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 6 T - 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 206T^{2} + 484 \) Copy content Toggle raw display
$89$ \( T^{4} - 70T^{2} + 100 \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
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