Properties

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

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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: \( 3 \)
Twist minimal: no (minimal twist has level 308)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (2 \beta_1 - 2) q^{5} + (3 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (2 \beta_1 - 2) q^{5} + (3 \beta_1 - 3) q^{9} + \beta_1 q^{11} + ( - \beta_{3} + 2) q^{13} + 2 \beta_{3} q^{15} + (\beta_{2} - 2 \beta_1) q^{17} + (2 \beta_{3} - 2 \beta_{2}) q^{19} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{23} + \beta_1 q^{25} + (2 \beta_{3} - 2) q^{29} + ( - \beta_{2} - 4 \beta_1) q^{31} + (\beta_{3} - \beta_{2}) q^{33} + (4 \beta_1 - 4) q^{37} + ( - 2 \beta_{2} + 6 \beta_1) q^{39} + ( - 3 \beta_{3} - 2) q^{41} + (2 \beta_{3} - 2) q^{43} - 6 \beta_1 q^{45} + (\beta_{3} - \beta_{2} - 4 \beta_1 + 4) q^{47} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 6) q^{51}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 6 q^{9} + 2 q^{11} + 8 q^{13} - 4 q^{17} - 8 q^{23} + 2 q^{25} - 8 q^{29} - 8 q^{31} - 8 q^{37} + 12 q^{39} - 8 q^{41} - 8 q^{43} - 12 q^{45} + 8 q^{47} + 12 q^{51} - 8 q^{55} - 48 q^{57} + 4 q^{61} - 8 q^{65} - 48 q^{69} - 32 q^{71} - 20 q^{73} + 12 q^{79} + 18 q^{81} - 48 q^{83} + 16 q^{85} - 24 q^{87} - 12 q^{89} - 12 q^{93} + 40 q^{97} - 12 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 4\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
0 −1.22474 2.12132i 0 −1.00000 + 1.73205i 0 0 0 −1.50000 + 2.59808i 0
177.2 0 1.22474 + 2.12132i 0 −1.00000 + 1.73205i 0 0 0 −1.50000 + 2.59808i 0
1145.1 0 −1.22474 + 2.12132i 0 −1.00000 1.73205i 0 0 0 −1.50000 2.59808i 0
1145.2 0 1.22474 2.12132i 0 −1.00000 1.73205i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2156.2.i.e 4
7.b odd 2 1 2156.2.i.i 4
7.c even 3 1 308.2.a.b 2
7.c even 3 1 inner 2156.2.i.e 4
7.d odd 6 1 2156.2.a.c 2
7.d odd 6 1 2156.2.i.i 4
21.h odd 6 1 2772.2.a.n 2
28.f even 6 1 8624.2.a.bj 2
28.g odd 6 1 1232.2.a.n 2
35.j even 6 1 7700.2.a.s 2
35.l odd 12 2 7700.2.e.j 4
56.k odd 6 1 4928.2.a.bq 2
56.p even 6 1 4928.2.a.bp 2
77.h odd 6 1 3388.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.a.b 2 7.c even 3 1
1232.2.a.n 2 28.g odd 6 1
2156.2.a.c 2 7.d odd 6 1
2156.2.i.e 4 1.a even 1 1 trivial
2156.2.i.e 4 7.c even 3 1 inner
2156.2.i.i 4 7.b odd 2 1
2156.2.i.i 4 7.d odd 6 1
2772.2.a.n 2 21.h odd 6 1
3388.2.a.h 2 77.h odd 6 1
4928.2.a.bp 2 56.p even 6 1
4928.2.a.bq 2 56.k odd 6 1
7700.2.a.s 2 35.j even 6 1
7700.2.e.j 4 35.l odd 12 2
8624.2.a.bj 2 28.f even 6 1

Hecke kernels

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

\( T_{3}^{4} + 6T_{3}^{2} + 36 \) Copy content Toggle raw display
\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 20)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} + \cdots + 100 \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 50)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 20)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + \cdots + 100 \) Copy content Toggle raw display
$53$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$59$ \( T^{4} + 54T^{2} + 2916 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + \cdots + 2500 \) Copy content Toggle raw display
$67$ \( T^{4} + 216 T^{2} + 46656 \) Copy content Toggle raw display
$71$ \( (T^{2} + 16 T + 40)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 20 T^{3} + \cdots + 8836 \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$83$ \( (T^{2} + 24 T + 120)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 20 T + 76)^{2} \) Copy content Toggle raw display
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