Properties

Label 2156.1.bk.a
Level $2156$
Weight $1$
Character orbit 2156.bk
Analytic conductor $1.076$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -7
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2156,1,Mod(471,2156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2156, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 10, 18]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2156.471");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2156.bk (of order \(30\), degree \(8\), 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: \(1.07598416724\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.527027889439744.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{30} q^{2} + \zeta_{30}^{2} q^{4} + \zeta_{30}^{3} q^{8} + \zeta_{30}^{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{30} q^{2} + \zeta_{30}^{2} q^{4} + \zeta_{30}^{3} q^{8} + \zeta_{30}^{4} q^{9} + \zeta_{30}^{11} q^{11} + \zeta_{30}^{4} q^{16} + \zeta_{30}^{5} q^{18} + \zeta_{30}^{12} q^{22} + (\zeta_{30}^{13} - \zeta_{30}^{7}) q^{23} - \zeta_{30}^{2} q^{25} + ( - \zeta_{30}^{12} + \zeta_{30}^{9}) q^{29} + \zeta_{30}^{5} q^{32} + \zeta_{30}^{6} q^{36} + ( - \zeta_{30}^{10} + \zeta_{30}) q^{37} + ( - \zeta_{30}^{9} - \zeta_{30}^{6}) q^{43} + \zeta_{30}^{13} q^{44} + (\zeta_{30}^{14} - \zeta_{30}^{8}) q^{46} - \zeta_{30}^{3} q^{50} + (\zeta_{30}^{8} - \zeta_{30}^{5}) q^{53} + ( - \zeta_{30}^{13} + \zeta_{30}^{10}) q^{58} + \zeta_{30}^{6} q^{64} + (\zeta_{30}^{14} + \zeta_{30}^{11}) q^{67} + ( - \zeta_{30}^{12} + 1) q^{71} + \zeta_{30}^{7} q^{72} + ( - \zeta_{30}^{11} + \zeta_{30}^{2}) q^{74} + ( - \zeta_{30}^{13} - \zeta_{30}^{10}) q^{79} + \zeta_{30}^{8} q^{81} + ( - \zeta_{30}^{10} - \zeta_{30}^{7}) q^{86} + \zeta_{30}^{14} q^{88} + ( - \zeta_{30}^{9} - 1) q^{92} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + q^{4} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} + q^{4} + 2 q^{8} + q^{9} - q^{11} + q^{16} + 4 q^{18} - 2 q^{22} - q^{25} + 4 q^{29} + 4 q^{32} - 2 q^{36} + 3 q^{37} - q^{44} - 2 q^{50} - 3 q^{53} - 3 q^{58} - 2 q^{64} + 10 q^{71} - q^{72} + 2 q^{74} + 5 q^{79} + q^{81} + 5 q^{86} + q^{88} - 10 q^{92} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(981\) \(1079\) \(1277\)
\(\chi(n)\) \(-\zeta_{30}^{3}\) \(-1\) \(\zeta_{30}^{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
471.1
−0.978148 0.207912i
0.913545 + 0.406737i
0.669131 + 0.743145i
0.913545 0.406737i
0.669131 0.743145i
−0.978148 + 0.207912i
−0.104528 + 0.994522i
−0.104528 0.994522i
0.978148 + 0.207912i 0 0.913545 + 0.406737i 0 0 0 0.809017 + 0.587785i 0.669131 + 0.743145i 0
851.1 −0.913545 0.406737i 0 0.669131 + 0.743145i 0 0 0 −0.309017 0.951057i −0.104528 + 0.994522i 0
863.1 −0.669131 0.743145i 0 −0.104528 + 0.994522i 0 0 0 0.809017 0.587785i −0.978148 0.207912i 0
1059.1 −0.913545 + 0.406737i 0 0.669131 0.743145i 0 0 0 −0.309017 + 0.951057i −0.104528 0.994522i 0
1439.1 −0.669131 + 0.743145i 0 −0.104528 0.994522i 0 0 0 0.809017 + 0.587785i −0.978148 + 0.207912i 0
1831.1 0.978148 0.207912i 0 0.913545 0.406737i 0 0 0 0.809017 0.587785i 0.669131 0.743145i 0
2027.1 0.104528 0.994522i 0 −0.978148 0.207912i 0 0 0 −0.309017 + 0.951057i 0.913545 + 0.406737i 0
2039.1 0.104528 + 0.994522i 0 −0.978148 + 0.207912i 0 0 0 −0.309017 0.951057i 0.913545 0.406737i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 471.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
7.c even 3 1 inner
7.d odd 6 1 inner
44.h odd 10 1 inner
308.t even 10 1 inner
308.bb odd 30 1 inner
308.be even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2156.1.bk.a 8
4.b odd 2 1 2156.1.bk.b 8
7.b odd 2 1 CM 2156.1.bk.a 8
7.c even 3 1 2156.1.v.b yes 4
7.c even 3 1 inner 2156.1.bk.a 8
7.d odd 6 1 2156.1.v.b yes 4
7.d odd 6 1 inner 2156.1.bk.a 8
11.c even 5 1 2156.1.bk.b 8
28.d even 2 1 2156.1.bk.b 8
28.f even 6 1 2156.1.v.a 4
28.f even 6 1 2156.1.bk.b 8
28.g odd 6 1 2156.1.v.a 4
28.g odd 6 1 2156.1.bk.b 8
44.h odd 10 1 inner 2156.1.bk.a 8
77.j odd 10 1 2156.1.bk.b 8
77.m even 15 1 2156.1.v.a 4
77.m even 15 1 2156.1.bk.b 8
77.p odd 30 1 2156.1.v.a 4
77.p odd 30 1 2156.1.bk.b 8
308.t even 10 1 inner 2156.1.bk.a 8
308.bb odd 30 1 2156.1.v.b yes 4
308.bb odd 30 1 inner 2156.1.bk.a 8
308.be even 30 1 2156.1.v.b yes 4
308.be even 30 1 inner 2156.1.bk.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2156.1.v.a 4 28.f even 6 1
2156.1.v.a 4 28.g odd 6 1
2156.1.v.a 4 77.m even 15 1
2156.1.v.a 4 77.p odd 30 1
2156.1.v.b yes 4 7.c even 3 1
2156.1.v.b yes 4 7.d odd 6 1
2156.1.v.b yes 4 308.bb odd 30 1
2156.1.v.b yes 4 308.be even 30 1
2156.1.bk.a 8 1.a even 1 1 trivial
2156.1.bk.a 8 7.b odd 2 1 CM
2156.1.bk.a 8 7.c even 3 1 inner
2156.1.bk.a 8 7.d odd 6 1 inner
2156.1.bk.a 8 44.h odd 10 1 inner
2156.1.bk.a 8 308.t even 10 1 inner
2156.1.bk.a 8 308.bb odd 30 1 inner
2156.1.bk.a 8 308.be even 30 1 inner
2156.1.bk.b 8 4.b odd 2 1
2156.1.bk.b 8 11.c even 5 1
2156.1.bk.b 8 28.d even 2 1
2156.1.bk.b 8 28.f even 6 1
2156.1.bk.b 8 28.g odd 6 1
2156.1.bk.b 8 77.j odd 10 1
2156.1.bk.b 8 77.m even 15 1
2156.1.bk.b 8 77.p odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{71}^{4} - 5T_{71}^{3} + 10T_{71}^{2} - 10T_{71} + 5 \) acting on \(S_{1}^{\mathrm{new}}(2156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} - 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$71$ \( (T^{4} - 5 T^{3} + 10 T^{2} + \cdots + 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} - 5 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
show more
show less