Properties

Label 2156.2.a.l
Level $2156$
Weight $2$
Character orbit 2156.a
Self dual yes
Analytic conductor $17.216$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2156,2,Mod(1,2156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2156, 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("2156.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2156.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: \(17.2157466758\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} + 2 \beta_1) q^{5} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} + 2 \beta_1) q^{5} + (\beta_{2} - 1) q^{9} - q^{11} + ( - 2 \beta_{3} + 2 \beta_1) q^{13} + (2 \beta_{2} + 3) q^{15} + 2 \beta_{3} q^{17} + (2 \beta_{3} + 4 \beta_1) q^{19} - 3 \beta_{2} q^{23} + (3 \beta_{2} + 1) q^{25} + (\beta_{3} - 2 \beta_1) q^{27} + ( - 4 \beta_{2} + 2) q^{29} + ( - 4 \beta_{3} + 3 \beta_1) q^{31} - \beta_1 q^{33} + ( - 4 \beta_{2} + 5) q^{37} + (2 \beta_{2} + 6) q^{39} - 2 \beta_1 q^{41} + 4 q^{43} + ( - \beta_{3} + \beta_1) q^{45} + (7 \beta_{3} + 5 \beta_1) q^{47} - 2 q^{51} + ( - 4 \beta_{2} - 4) q^{53} + ( - \beta_{3} - 2 \beta_1) q^{55} + (4 \beta_{2} + 6) q^{57} + ( - \beta_{3} - 4 \beta_1) q^{59} + (2 \beta_{3} + 8 \beta_1) q^{61} + (6 \beta_{2} + 6) q^{65} + (\beta_{2} + 8) q^{67} + ( - 3 \beta_{3} - 6 \beta_1) q^{69} + ( - 3 \beta_{2} + 4) q^{71} - 6 \beta_1 q^{73} + (3 \beta_{3} + 7 \beta_1) q^{75} + (2 \beta_{2} + 4) q^{79} + ( - 5 \beta_{2} - 2) q^{81} + ( - 4 \beta_{3} + 4 \beta_1) q^{83} - 2 \beta_{2} q^{85} + ( - 4 \beta_{3} - 6 \beta_1) q^{87} + (5 \beta_{3} + 2 \beta_1) q^{89} + (3 \beta_{2} + 10) q^{93} + (6 \beta_{2} + 12) q^{95} + ( - 9 \beta_{3} - 10 \beta_1) q^{97} + ( - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 4 q^{11} + 12 q^{15} + 4 q^{25} + 8 q^{29} + 20 q^{37} + 24 q^{39} + 16 q^{43} - 8 q^{51} - 16 q^{53} + 24 q^{57} + 24 q^{65} + 32 q^{67} + 16 q^{71} + 16 q^{79} - 8 q^{81} + 40 q^{93} + 48 q^{95} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) 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
−1.93185
−0.517638
0.517638
1.93185
0 −1.93185 0 −3.34607 0 0 0 0.732051 0
1.2 0 −0.517638 0 0.896575 0 0 0 −2.73205 0
1.3 0 0.517638 0 −0.896575 0 0 0 −2.73205 0
1.4 0 1.93185 0 3.34607 0 0 0 0.732051 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(11\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2156.2.a.l 4
4.b odd 2 1 8624.2.a.cs 4
7.b odd 2 1 inner 2156.2.a.l 4
7.c even 3 2 2156.2.i.o 8
7.d odd 6 2 2156.2.i.o 8
28.d even 2 1 8624.2.a.cs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2156.2.a.l 4 1.a even 1 1 trivial
2156.2.a.l 4 7.b odd 2 1 inner
2156.2.i.o 8 7.c even 3 2
2156.2.i.o 8 7.d odd 6 2
8624.2.a.cs 4 4.b odd 2 1
8624.2.a.cs 4 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2156))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 12T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 16T^{2} + 16 \) Copy content Toggle raw display
$19$ \( T^{4} - 48T^{2} + 144 \) Copy content Toggle raw display
$23$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 148T^{2} + 5329 \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T - 23)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 16T^{2} + 16 \) Copy content Toggle raw display
$43$ \( (T - 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 156T^{2} + 4356 \) Copy content Toggle raw display
$53$ \( (T^{2} + 8 T - 32)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 52T^{2} + 1 \) Copy content Toggle raw display
$61$ \( T^{4} - 208T^{2} + 16 \) Copy content Toggle raw display
$67$ \( (T^{2} - 16 T + 61)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 8 T - 11)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 144T^{2} + 1296 \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 76T^{2} + 121 \) Copy content Toggle raw display
$97$ \( T^{4} - 364 T^{2} + 32041 \) Copy content Toggle raw display
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