Properties

Label 2100.4.a.ba
Level $2100$
Weight $4$
Character orbit 2100.a
Self dual yes
Analytic conductor $123.904$
Analytic rank $0$
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] = [2100,4,Mod(1,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2100.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: \(123.904011012\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.698565.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 205x + 190 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + 3 q^{3} + 7 q^{7} + 9 q^{9} + (\beta_{2} - \beta_1 + 2) q^{11} + (\beta_1 + 16) q^{13} + ( - \beta_1 + 10) q^{17} + ( - 3 \beta_{2} + 35) q^{19} + 21 q^{21} + ( - 2 \beta_{2} - 4 \beta_1 + 11) q^{23}+ \cdots + (9 \beta_{2} - 9 \beta_1 + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 9 q^{3} + 21 q^{7} + 27 q^{9} + 8 q^{11} + 47 q^{13} + 31 q^{17} + 102 q^{19} + 63 q^{21} + 35 q^{23} + 81 q^{27} + 123 q^{29} + 143 q^{31} + 24 q^{33} + 56 q^{37} + 141 q^{39} + 67 q^{41} - 53 q^{43}+ \cdots + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 205x + 190 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{2} + 17\nu + 129 ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{2} - 8\nu + 279 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + 2\beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -17\beta_{2} - 8\beta _1 + 825 ) / 6 \) 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
14.3570
0.926522
−14.2835
0 3.00000 0 0 0 7.00000 0 9.00000 0
1.2 0 3.00000 0 0 0 7.00000 0 9.00000 0
1.3 0 3.00000 0 0 0 7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)
\(7\) \( -1 \)

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 2100.4.a.ba yes 3
5.b even 2 1 2100.4.a.w 3
5.c odd 4 2 2100.4.k.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.4.a.w 3 5.b even 2 1
2100.4.a.ba yes 3 1.a even 1 1 trivial
2100.4.k.q 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2100))\):

\( T_{11}^{3} - 8T_{11}^{2} - 2835T_{11} + 51894 \) Copy content Toggle raw display
\( T_{13}^{3} - 47T_{13}^{2} - 790T_{13} + 42836 \) Copy content Toggle raw display
\( T_{17}^{3} - 31T_{17}^{2} - 1206T_{17} - 8100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T - 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 8 T^{2} + \cdots + 51894 \) Copy content Toggle raw display
$13$ \( T^{3} - 47 T^{2} + \cdots + 42836 \) Copy content Toggle raw display
$17$ \( T^{3} - 31 T^{2} + \cdots - 8100 \) Copy content Toggle raw display
$19$ \( T^{3} - 102 T^{2} + \cdots + 471136 \) Copy content Toggle raw display
$23$ \( T^{3} - 35 T^{2} + \cdots - 394497 \) Copy content Toggle raw display
$29$ \( T^{3} - 123 T^{2} + \cdots + 2352159 \) Copy content Toggle raw display
$31$ \( T^{3} - 143 T^{2} + \cdots + 81980 \) Copy content Toggle raw display
$37$ \( T^{3} - 56 T^{2} + \cdots - 5181034 \) Copy content Toggle raw display
$41$ \( T^{3} - 67 T^{2} + \cdots + 5080320 \) Copy content Toggle raw display
$43$ \( T^{3} + 53 T^{2} + \cdots - 1224417 \) Copy content Toggle raw display
$47$ \( T^{3} + 34 T^{2} + \cdots - 1357344 \) Copy content Toggle raw display
$53$ \( T^{3} + 27 T^{2} + \cdots - 1116828 \) Copy content Toggle raw display
$59$ \( T^{3} - 41 T^{2} + \cdots + 6921180 \) Copy content Toggle raw display
$61$ \( T^{3} - 617 T^{2} + \cdots + 84831440 \) Copy content Toggle raw display
$67$ \( T^{3} - 928 T^{2} + \cdots + 4368084 \) Copy content Toggle raw display
$71$ \( T^{3} - 754 T^{2} + \cdots + 664039404 \) Copy content Toggle raw display
$73$ \( T^{3} - 92 T^{2} + \cdots + 338232920 \) Copy content Toggle raw display
$79$ \( T^{3} - 430 T^{2} + \cdots + 11601594 \) Copy content Toggle raw display
$83$ \( T^{3} + 897 T^{2} + \cdots - 57279636 \) Copy content Toggle raw display
$89$ \( T^{3} + 212 T^{2} + \cdots + 28448280 \) Copy content Toggle raw display
$97$ \( T^{3} - 2018 T^{2} + \cdots + 220691240 \) Copy content Toggle raw display
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