Properties

Label 2156.2.i.g
Level $2156$
Weight $2$
Character orbit 2156.i
Analytic conductor $17.216$
Analytic rank $0$
Dimension $4$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2156,2,Mod(177,2156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2156, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2156.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2156.i (of order \(3\), degree \(2\), not minimal)

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: no
Analytic conductor: \(17.2157466758\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1079\) \(1277\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 −0.707107 1.22474i 0 0 0 0 0 0.500000 0.866025i 0
177.2 0 0.707107 + 1.22474i 0 0 0 0 0 0.500000 0.866025i 0
1145.1 0 −0.707107 + 1.22474i 0 0 0 0 0 0.500000 + 0.866025i 0
1145.2 0 0.707107 1.22474i 0 0 0 0 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2156, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 50T^{2} + 2500 \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$23$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 98T^{2} + 9604 \) Copy content Toggle raw display
$53$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$61$ \( T^{4} + 162 T^{2} + 26244 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 128 T^{2} + 16384 \) Copy content Toggle raw display
$97$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
show more
show less