Properties

Label 2156.2.a.j
Level $2156$
Weight $2$
Character orbit 2156.a
Self dual yes
Analytic conductor $17.216$
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] = [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: \(3\)
Coefficient field: 3.3.1016.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 308)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{2} + \beta_1 + 1) q^{9} + q^{11} + ( - \beta_{2} - 4) q^{13} + (\beta_{2} + 3 \beta_1 + 2) q^{15} + (\beta_{2} - 2 \beta_1) q^{17} + 2 \beta_1 q^{19} + (\beta_{2} - \beta_1 - 2) q^{23} + ( - \beta_{2} + 3 \beta_1 + 5) q^{25} + (\beta_{2} + \beta_1 + 2) q^{27} + ( - 2 \beta_{2} + 2) q^{29} + ( - 3 \beta_1 + 4) q^{31} + \beta_1 q^{33} + (\beta_{2} + \beta_1 + 2) q^{37} + ( - 6 \beta_1 + 2) q^{39} + ( - \beta_{2} + 2 \beta_1) q^{41} + (2 \beta_1 - 2) q^{43} + (4 \beta_1 + 10) q^{45} + ( - \beta_{2} - 2 \beta_1 + 2) q^{47} + ( - 2 \beta_{2} - 10) q^{51} + (2 \beta_{2} + 2 \beta_1) q^{53} + (\beta_{2} + \beta_1) q^{55} + (2 \beta_{2} + 2 \beta_1 + 8) q^{57} + ( - \beta_1 + 8) q^{59} + ( - 3 \beta_{2} - 2 \beta_1 - 4) q^{61} + ( - 2 \beta_{2} - 4 \beta_1 - 8) q^{65} + ( - \beta_{2} + \beta_1 - 2) q^{67} + ( - \beta_{2} - \beta_1 - 6) q^{69} + (3 \beta_{2} + \beta_1 - 2) q^{71} + ( - \beta_{2} + 2 \beta_1 + 4) q^{73} + (3 \beta_{2} + 6 \beta_1 + 14) q^{75} + ( - 2 \beta_{2} - 4 \beta_1 + 6) q^{79} + ( - 2 \beta_{2} + 2 \beta_1 - 1) q^{81} + (2 \beta_{2} - 4 \beta_1) q^{83} + ( - 4 \beta_{2} - 6 \beta_1 + 4) q^{85} + ( - 2 \beta_1 + 4) q^{87} + ( - \beta_{2} - 5 \beta_1 + 8) q^{89} + ( - 3 \beta_{2} + \beta_1 - 12) q^{93} + (2 \beta_{2} + 6 \beta_1 + 4) q^{95} + ( - \beta_{2} - 3 \beta_1 - 4) q^{97} + (\beta_{2} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} + q^{5} + 4 q^{9} + 3 q^{11} - 12 q^{13} + 9 q^{15} - 2 q^{17} + 2 q^{19} - 7 q^{23} + 18 q^{25} + 7 q^{27} + 6 q^{29} + 9 q^{31} + q^{33} + 7 q^{37} + 2 q^{41} - 4 q^{43} + 34 q^{45} + 4 q^{47} - 30 q^{51} + 2 q^{53} + q^{55} + 26 q^{57} + 23 q^{59} - 14 q^{61} - 28 q^{65} - 5 q^{67} - 19 q^{69} - 5 q^{71} + 14 q^{73} + 48 q^{75} + 14 q^{79} - q^{81} - 4 q^{83} + 6 q^{85} + 10 q^{87} + 19 q^{89} - 35 q^{93} + 18 q^{95} - 15 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) 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
−2.17741
0.321637
2.85577
0 −2.17741 0 0.741113 0 0 0 1.74111 0
1.2 0 0.321637 0 −3.89655 0 0 0 −2.89655 0
1.3 0 2.85577 0 4.15544 0 0 0 5.15544 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

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 2156.2.a.j 3
4.b odd 2 1 8624.2.a.cj 3
7.b odd 2 1 308.2.a.c 3
7.c even 3 2 2156.2.i.k 6
7.d odd 6 2 2156.2.i.m 6
21.c even 2 1 2772.2.a.s 3
28.d even 2 1 1232.2.a.r 3
35.c odd 2 1 7700.2.a.y 3
35.f even 4 2 7700.2.e.p 6
56.e even 2 1 4928.2.a.bx 3
56.h odd 2 1 4928.2.a.ca 3
77.b even 2 1 3388.2.a.o 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.a.c 3 7.b odd 2 1
1232.2.a.r 3 28.d even 2 1
2156.2.a.j 3 1.a even 1 1 trivial
2156.2.i.k 6 7.c even 3 2
2156.2.i.m 6 7.d odd 6 2
2772.2.a.s 3 21.c even 2 1
3388.2.a.o 3 77.b even 2 1
4928.2.a.bx 3 56.e even 2 1
4928.2.a.ca 3 56.h odd 2 1
7700.2.a.y 3 35.c odd 2 1
7700.2.e.p 6 35.f even 4 2
8624.2.a.cj 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - T_{3}^{2} - 6T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2156))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 6T + 2 \) Copy content Toggle raw display
$5$ \( T^{3} - T^{2} + \cdots + 12 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 12 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} + \cdots - 156 \) Copy content Toggle raw display
$19$ \( T^{3} - 2 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{3} + 7 T^{2} + \cdots - 72 \) Copy content Toggle raw display
$29$ \( T^{3} - 6 T^{2} + \cdots - 24 \) Copy content Toggle raw display
$31$ \( T^{3} - 9 T^{2} + \cdots + 146 \) Copy content Toggle raw display
$37$ \( T^{3} - 7T^{2} + 32 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} + \cdots + 156 \) Copy content Toggle raw display
$43$ \( T^{3} + 4 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$47$ \( T^{3} - 4 T^{2} + \cdots + 96 \) Copy content Toggle raw display
$53$ \( T^{3} - 2 T^{2} + \cdots + 96 \) Copy content Toggle raw display
$59$ \( T^{3} - 23 T^{2} + \cdots - 402 \) Copy content Toggle raw display
$61$ \( T^{3} + 14 T^{2} + \cdots - 916 \) Copy content Toggle raw display
$67$ \( T^{3} + 5 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$71$ \( T^{3} + 5 T^{2} + \cdots + 312 \) Copy content Toggle raw display
$73$ \( T^{3} - 14 T^{2} + \cdots + 244 \) Copy content Toggle raw display
$79$ \( T^{3} - 14 T^{2} + \cdots + 936 \) Copy content Toggle raw display
$83$ \( T^{3} + 4 T^{2} + \cdots - 1248 \) Copy content Toggle raw display
$89$ \( T^{3} - 19 T^{2} + \cdots + 1284 \) Copy content Toggle raw display
$97$ \( T^{3} + 15 T^{2} + \cdots + 4 \) Copy content Toggle raw display
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