Properties

Label 1911.2.a.s
Level $1911$
Weight $2$
Character orbit 1911.a
Self dual yes
Analytic conductor $15.259$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1911,2,Mod(1,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, 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("1911.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.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: \(15.2594118263\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 273)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{2} + 1) q^{5} - \beta_1 q^{6} + (\beta_{3} + 2 \beta_1) q^{8} + q^{9} + (\beta_{3} + 3 \beta_1) q^{10} + ( - \beta_{3} - \beta_1) q^{11} + ( - \beta_{2} - 2) q^{12}+ \cdots + ( - \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 4 q^{3} + 7 q^{4} + 3 q^{5} - q^{6} + 3 q^{8} + 4 q^{9} + 4 q^{10} - 2 q^{11} - 7 q^{12} - 4 q^{13} - 3 q^{15} + 9 q^{16} + 2 q^{17} + q^{18} - 7 q^{19} + 32 q^{20} - 8 q^{22} + 3 q^{23}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + 2\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + \beta_{2} + 2\beta _1 + 1 \) 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
−0.787711
1.52616
−2.10710
2.36865
−2.37951 −1.00000 3.66208 2.66208 2.37951 0 −3.95493 1.00000 −6.33445
1.2 −0.670843 −1.00000 −1.54997 −2.54997 0.670843 0 2.38147 1.00000 1.71063
1.3 1.43986 −1.00000 0.0731828 −0.926817 −1.43986 0 −2.77434 1.00000 −1.33448
1.4 2.61050 −1.00000 4.81471 3.81471 −2.61050 0 7.34780 1.00000 9.95830
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( -1 \)
\(13\) \( +1 \)

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 1911.2.a.s 4
3.b odd 2 1 5733.2.a.bf 4
7.b odd 2 1 273.2.a.e 4
21.c even 2 1 819.2.a.k 4
28.d even 2 1 4368.2.a.br 4
35.c odd 2 1 6825.2.a.bg 4
91.b odd 2 1 3549.2.a.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.e 4 7.b odd 2 1
819.2.a.k 4 21.c even 2 1
1911.2.a.s 4 1.a even 1 1 trivial
3549.2.a.w 4 91.b odd 2 1
4368.2.a.br 4 28.d even 2 1
5733.2.a.bf 4 3.b odd 2 1
6825.2.a.bg 4 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1911))\):

\( T_{2}^{4} - T_{2}^{3} - 7T_{2}^{2} + 5T_{2} + 6 \) Copy content Toggle raw display
\( T_{5}^{4} - 3T_{5}^{3} - 10T_{5}^{2} + 20T_{5} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 7 T^{2} + \cdots + 6 \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + \cdots + 24 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 96 \) Copy content Toggle raw display
$13$ \( (T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + \cdots + 96 \) Copy content Toggle raw display
$19$ \( T^{4} + 7 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{4} - 3 T^{3} + \cdots - 288 \) Copy content Toggle raw display
$29$ \( T^{4} - T^{3} + \cdots + 72 \) Copy content Toggle raw display
$31$ \( T^{4} + 3 T^{3} + \cdots + 3968 \) Copy content Toggle raw display
$37$ \( T^{4} - 10 T^{3} + \cdots - 128 \) Copy content Toggle raw display
$41$ \( T^{4} - 16 T^{3} + \cdots - 1392 \) Copy content Toggle raw display
$43$ \( T^{4} - 3 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 5 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$53$ \( T^{4} - 5 T^{3} + \cdots - 24 \) Copy content Toggle raw display
$59$ \( T^{4} - 20 T^{3} + \cdots - 1536 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + \cdots + 496 \) Copy content Toggle raw display
$67$ \( T^{4} + 22 T^{3} + \cdots - 15488 \) Copy content Toggle raw display
$71$ \( T^{4} - 232 T^{2} + \cdots + 10176 \) Copy content Toggle raw display
$73$ \( T^{4} - 13 T^{3} + \cdots - 11672 \) Copy content Toggle raw display
$79$ \( T^{4} - 11 T^{3} + \cdots - 3456 \) Copy content Toggle raw display
$83$ \( T^{4} + T^{3} + \cdots - 48 \) Copy content Toggle raw display
$89$ \( T^{4} - 5 T^{3} + \cdots - 1704 \) Copy content Toggle raw display
$97$ \( T^{4} - 17 T^{3} + \cdots - 1528 \) Copy content Toggle raw display
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