Properties

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

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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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + (3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + (3 \beta_{2} + 3) q^{9} + \beta_{2} q^{11} + 3 \beta_{3} q^{13} + ( - \beta_{3} - \beta_1) q^{17} - 2 \beta_1 q^{19} - 3 \beta_{2} q^{25} - 4 q^{29} + ( - 2 \beta_{3} - 2 \beta_1) q^{31} + 3 \beta_{3} q^{41} - 8 q^{43} + ( - 3 \beta_{3} - 3 \beta_1) q^{45} - 2 \beta_1 q^{47} - 6 \beta_{2} q^{53} - \beta_{3} q^{55} + ( - 8 \beta_{3} - 8 \beta_1) q^{59} - 9 \beta_1 q^{61} + (6 \beta_{2} + 6) q^{65} - 4 \beta_{2} q^{67} - 12 q^{71} + (9 \beta_{3} + 9 \beta_1) q^{73} + (12 \beta_{2} + 12) q^{79} + 9 \beta_{2} q^{81} - 2 \beta_{3} q^{83} - 2 q^{85} - 5 \beta_1 q^{89} + 4 \beta_{2} q^{95} + 5 \beta_{3} q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{9} - 2 q^{11} + 6 q^{25} - 16 q^{29} - 32 q^{43} + 12 q^{53} + 12 q^{65} + 8 q^{67} - 48 q^{71} + 24 q^{79} - 18 q^{81} - 8 q^{85} - 8 q^{95} - 12 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times\).

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

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 0 −0.707107 + 1.22474i 0 0 0 1.50000 2.59808i 0
177.2 0 0 0 0.707107 1.22474i 0 0 0 1.50000 2.59808i 0
1145.1 0 0 0 −0.707107 1.22474i 0 0 0 1.50000 + 2.59808i 0
1145.2 0 0 0 0.707107 + 1.22474i 0 0 0 1.50000 + 2.59808i 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.h 4
7.b odd 2 1 inner 2156.2.i.h 4
7.c even 3 1 2156.2.a.d 2
7.c even 3 1 inner 2156.2.i.h 4
7.d odd 6 1 2156.2.a.d 2
7.d odd 6 1 inner 2156.2.i.h 4
28.f even 6 1 8624.2.a.bm 2
28.g odd 6 1 8624.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2156.2.a.d 2 7.c even 3 1
2156.2.a.d 2 7.d odd 6 1
2156.2.i.h 4 1.a even 1 1 trivial
2156.2.i.h 4 7.b odd 2 1 inner
2156.2.i.h 4 7.c even 3 1 inner
2156.2.i.h 4 7.d odd 6 1 inner
8624.2.a.bm 2 28.f even 6 1
8624.2.a.bm 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 \(S_{2}^{\mathrm{new}}(2156, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T + 4)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$43$ \( (T + 8)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$53$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 128 T^{2} + 16384 \) Copy content Toggle raw display
$61$ \( T^{4} + 162 T^{2} + 26244 \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T + 12)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 162 T^{2} + 26244 \) Copy content Toggle raw display
$79$ \( (T^{2} - 12 T + 144)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 50T^{2} + 2500 \) Copy content Toggle raw display
$97$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
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