Properties

Label 4851.2.a.bu
Level $4851$
Weight $2$
Character orbit 4851.a
Self dual yes
Analytic conductor $38.735$
Analytic rank $0$
Dimension $4$
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] = [4851,2,Mod(1,4851)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4851, 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("4851.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4851.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: \(38.7354300205\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{8} + (\beta_{3} - 2) q^{10} + q^{11} - \beta_{3} q^{13} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 3) q^{16} + ( - 2 \beta_{2} - \beta_1) q^{17} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{19} + (\beta_{2} + \beta_1) q^{20} - \beta_1 q^{22} + (2 \beta_{3} + 2 \beta_1 - 1) q^{23} + ( - 2 \beta_{3} - 2 \beta_{2} - 1) q^{25} + (\beta_{2} - \beta_1) q^{26} + (\beta_{3} + \beta_1 - 2) q^{29} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{31} + ( - 4 \beta_{2} - 5 \beta_1 - 6) q^{32} + (2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 5) q^{34} + ( - \beta_{2} - \beta_1 - 1) q^{37} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{38}+ \cdots + ( - 2 \beta_{3} + \beta_{2} + \cdots + 11) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 4 q^{4} + 4 q^{5} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 4 q^{4} + 4 q^{5} - 12 q^{8} - 10 q^{10} + 4 q^{11} + 2 q^{13} + 12 q^{16} + 2 q^{17} - 2 q^{22} - 4 q^{23} + 4 q^{25} - 4 q^{26} - 8 q^{29} + 12 q^{31} - 26 q^{32} + 16 q^{34} - 4 q^{37} + 8 q^{38} + 6 q^{40} + 2 q^{41} + 18 q^{43} + 4 q^{44} - 14 q^{46} + 12 q^{47} - 2 q^{50} + 6 q^{52} + 12 q^{53} + 4 q^{55} - 4 q^{58} + 12 q^{59} - 2 q^{61} - 26 q^{62} + 56 q^{64} + 4 q^{65} + 28 q^{67} - 48 q^{68} - 12 q^{71} - 6 q^{73} + 16 q^{74} + 18 q^{76} + 2 q^{79} + 16 q^{80} + 12 q^{82} - 12 q^{83} + 18 q^{85} - 36 q^{86} - 12 q^{88} + 8 q^{89} + 16 q^{92} - 20 q^{94} - 34 q^{95} + 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \) 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
2.78165
1.28734
−0.552409
−1.51658
−2.78165 0 5.73760 0.825711 0 0 −10.3967 0 −2.29684
1.2 −1.28734 0 −0.342766 3.91744 0 0 3.01593 0 −5.04306
1.3 0.552409 0 −1.69484 1.59002 0 0 −2.04107 0 0.878345
1.4 1.51658 0 0.300014 −2.33317 0 0 −2.57816 0 −3.53844
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( -1 \)
\(11\) \( -1 \)

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 4851.2.a.bu 4
3.b odd 2 1 1617.2.a.x 4
7.b odd 2 1 4851.2.a.bt 4
7.d odd 6 2 693.2.i.i 8
21.c even 2 1 1617.2.a.z 4
21.g even 6 2 231.2.i.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.e 8 21.g even 6 2
693.2.i.i 8 7.d odd 6 2
1617.2.a.x 4 3.b odd 2 1
1617.2.a.z 4 21.c even 2 1
4851.2.a.bt 4 7.b odd 2 1
4851.2.a.bu 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4851))\):

\( T_{2}^{4} + 2T_{2}^{3} - 4T_{2}^{2} - 4T_{2} + 3 \) Copy content Toggle raw display
\( T_{5}^{4} - 4T_{5}^{3} - 4T_{5}^{2} + 20T_{5} - 12 \) Copy content Toggle raw display
\( T_{13}^{4} - 2T_{13}^{3} - 8T_{13}^{2} + 16T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{4} - 2T_{17}^{3} - 40T_{17}^{2} + 124T_{17} + 15 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + \cdots - 12 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + \cdots + 15 \) Copy content Toggle raw display
$19$ \( T^{4} - 40 T^{2} + \cdots - 89 \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + \cdots + 465 \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$31$ \( T^{4} - 12 T^{3} + \cdots - 1396 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} + \cdots + 60 \) Copy content Toggle raw display
$43$ \( T^{4} - 18 T^{3} + \cdots - 1385 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + \cdots + 6252 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + \cdots + 249 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + \cdots + 20 \) Copy content Toggle raw display
$67$ \( T^{4} - 28 T^{3} + \cdots - 1460 \) Copy content Toggle raw display
$71$ \( T^{4} + 12 T^{3} + \cdots - 699 \) Copy content Toggle raw display
$73$ \( T^{4} + 6 T^{3} + \cdots + 17156 \) Copy content Toggle raw display
$79$ \( T^{4} - 2 T^{3} + \cdots + 388 \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} + \cdots + 2592 \) Copy content Toggle raw display
$89$ \( T^{4} - 8 T^{3} + \cdots + 12048 \) Copy content Toggle raw display
$97$ \( T^{4} - 44 T^{3} + \cdots + 8501 \) Copy content Toggle raw display
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