Properties

Label 56.2.i.b
Level $56$
Weight $2$
Character orbit 56.i
Analytic conductor $0.447$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,2,Mod(9,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 56.i (of order \(3\), degree \(2\), 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: \(0.447162251319\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{3} + ( - \zeta_{6} + 1) q^{5} + (2 \zeta_{6} - 3) q^{7} + ( - 2 \zeta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{3} + ( - \zeta_{6} + 1) q^{5} + (2 \zeta_{6} - 3) q^{7} + ( - 2 \zeta_{6} + 2) q^{9} - 3 \zeta_{6} q^{11} - 6 q^{13} + q^{15} + 5 \zeta_{6} q^{17} + (\zeta_{6} - 1) q^{19} + ( - \zeta_{6} - 2) q^{21} + ( - 7 \zeta_{6} + 7) q^{23} + 4 \zeta_{6} q^{25} + 5 q^{27} + 2 q^{29} + 5 \zeta_{6} q^{31} + ( - 3 \zeta_{6} + 3) q^{33} + (3 \zeta_{6} - 1) q^{35} + (3 \zeta_{6} - 3) q^{37} - 6 \zeta_{6} q^{39} - 2 q^{41} - 4 q^{43} - 2 \zeta_{6} q^{45} + (5 \zeta_{6} - 5) q^{47} + ( - 8 \zeta_{6} + 5) q^{49} + (5 \zeta_{6} - 5) q^{51} + \zeta_{6} q^{53} - 3 q^{55} - q^{57} - 15 \zeta_{6} q^{59} + ( - 5 \zeta_{6} + 5) q^{61} + (6 \zeta_{6} - 2) q^{63} + (6 \zeta_{6} - 6) q^{65} + 9 \zeta_{6} q^{67} + 7 q^{69} - 7 \zeta_{6} q^{73} + (4 \zeta_{6} - 4) q^{75} + (3 \zeta_{6} + 6) q^{77} + (\zeta_{6} - 1) q^{79} - \zeta_{6} q^{81} + 12 q^{83} + 5 q^{85} + 2 \zeta_{6} q^{87} + (7 \zeta_{6} - 7) q^{89} + ( - 12 \zeta_{6} + 18) q^{91} + (5 \zeta_{6} - 5) q^{93} + \zeta_{6} q^{95} - 2 q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9} - 3 q^{11} - 12 q^{13} + 2 q^{15} + 5 q^{17} - q^{19} - 5 q^{21} + 7 q^{23} + 4 q^{25} + 10 q^{27} + 4 q^{29} + 5 q^{31} + 3 q^{33} + q^{35} - 3 q^{37} - 6 q^{39} - 4 q^{41} - 8 q^{43} - 2 q^{45} - 5 q^{47} + 2 q^{49} - 5 q^{51} + q^{53} - 6 q^{55} - 2 q^{57} - 15 q^{59} + 5 q^{61} + 2 q^{63} - 6 q^{65} + 9 q^{67} + 14 q^{69} - 7 q^{73} - 4 q^{75} + 15 q^{77} - q^{79} - q^{81} + 24 q^{83} + 10 q^{85} + 2 q^{87} - 7 q^{89} + 24 q^{91} - 5 q^{93} + q^{95} - 4 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 −2.00000 + 1.73205i 0 1.00000 1.73205i 0
25.1 0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 −2.00000 1.73205i 0 1.00000 + 1.73205i 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 56.2.i.b 2
3.b odd 2 1 504.2.s.c 2
4.b odd 2 1 112.2.i.a 2
5.b even 2 1 1400.2.q.d 2
5.c odd 4 2 1400.2.bh.a 4
7.b odd 2 1 392.2.i.b 2
7.c even 3 1 inner 56.2.i.b 2
7.c even 3 1 392.2.a.c 1
7.d odd 6 1 392.2.a.e 1
7.d odd 6 1 392.2.i.b 2
8.b even 2 1 448.2.i.b 2
8.d odd 2 1 448.2.i.d 2
12.b even 2 1 1008.2.s.g 2
21.c even 2 1 3528.2.s.q 2
21.g even 6 1 3528.2.a.j 1
21.g even 6 1 3528.2.s.q 2
21.h odd 6 1 504.2.s.c 2
21.h odd 6 1 3528.2.a.p 1
28.d even 2 1 784.2.i.h 2
28.f even 6 1 784.2.a.c 1
28.f even 6 1 784.2.i.h 2
28.g odd 6 1 112.2.i.a 2
28.g odd 6 1 784.2.a.h 1
35.i odd 6 1 9800.2.a.s 1
35.j even 6 1 1400.2.q.d 2
35.j even 6 1 9800.2.a.be 1
35.l odd 12 2 1400.2.bh.a 4
56.j odd 6 1 3136.2.a.i 1
56.k odd 6 1 448.2.i.d 2
56.k odd 6 1 3136.2.a.j 1
56.m even 6 1 3136.2.a.t 1
56.p even 6 1 448.2.i.b 2
56.p even 6 1 3136.2.a.u 1
84.j odd 6 1 7056.2.a.u 1
84.n even 6 1 1008.2.s.g 2
84.n even 6 1 7056.2.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.b 2 1.a even 1 1 trivial
56.2.i.b 2 7.c even 3 1 inner
112.2.i.a 2 4.b odd 2 1
112.2.i.a 2 28.g odd 6 1
392.2.a.c 1 7.c even 3 1
392.2.a.e 1 7.d odd 6 1
392.2.i.b 2 7.b odd 2 1
392.2.i.b 2 7.d odd 6 1
448.2.i.b 2 8.b even 2 1
448.2.i.b 2 56.p even 6 1
448.2.i.d 2 8.d odd 2 1
448.2.i.d 2 56.k odd 6 1
504.2.s.c 2 3.b odd 2 1
504.2.s.c 2 21.h odd 6 1
784.2.a.c 1 28.f even 6 1
784.2.a.h 1 28.g odd 6 1
784.2.i.h 2 28.d even 2 1
784.2.i.h 2 28.f even 6 1
1008.2.s.g 2 12.b even 2 1
1008.2.s.g 2 84.n even 6 1
1400.2.q.d 2 5.b even 2 1
1400.2.q.d 2 35.j even 6 1
1400.2.bh.a 4 5.c odd 4 2
1400.2.bh.a 4 35.l odd 12 2
3136.2.a.i 1 56.j odd 6 1
3136.2.a.j 1 56.k odd 6 1
3136.2.a.t 1 56.m even 6 1
3136.2.a.u 1 56.p even 6 1
3528.2.a.j 1 21.g even 6 1
3528.2.a.p 1 21.h odd 6 1
3528.2.s.q 2 21.c even 2 1
3528.2.s.q 2 21.g even 6 1
7056.2.a.u 1 84.j odd 6 1
7056.2.a.bj 1 84.n even 6 1
9800.2.a.s 1 35.i odd 6 1
9800.2.a.be 1 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(56, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$53$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$61$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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