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Properties

Label 2793.1.cf.b
Level $2793$
Weight $1$
Character orbit 2793.cf
Analytic conductor $1.394$
Analytic rank $0$
Dimension $6$
Projective image $D_{18}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2793,1,Mod(146,2793)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2793, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2793.146");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2793.cf (of order \(18\), degree \(6\), 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: \(1.39388858028\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 399)
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{18}^{2} q^{3} + \zeta_{18} q^{4} + \zeta_{18}^{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{18}^{2} q^{3} + \zeta_{18} q^{4} + \zeta_{18}^{4} q^{9} + \zeta_{18}^{3} q^{12} + (\zeta_{18}^{3} - \zeta_{18}^{2}) q^{13} + \zeta_{18}^{2} q^{16} + \zeta_{18}^{6} q^{19} + \zeta_{18}^{7} q^{25} + \zeta_{18}^{6} q^{27} + ( - \zeta_{18}^{4} - \zeta_{18}^{2}) q^{31} + \zeta_{18}^{5} q^{36} + ( - \zeta_{18}^{7} - \zeta_{18}^{2}) q^{37} + (\zeta_{18}^{5} - \zeta_{18}^{4}) q^{39} + ( - \zeta_{18}^{6} + \zeta_{18}) q^{43} + \zeta_{18}^{4} q^{48} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{52} + \zeta_{18}^{8} q^{57} + ( - \zeta_{18}^{2} + 1) q^{61} + \zeta_{18}^{3} q^{64} + ( - \zeta_{18}^{8} + 1) q^{67} + ( - \zeta_{18}^{7} - \zeta_{18}^{6}) q^{73} - q^{75} + \zeta_{18}^{7} q^{76} + (\zeta_{18}^{3} - \zeta_{18}) q^{79} + \zeta_{18}^{8} q^{81} + ( - \zeta_{18}^{6} - \zeta_{18}^{4}) q^{93} + \zeta_{18}^{5} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{12} + 3 q^{13} - 3 q^{19} - 3 q^{27} + 3 q^{43} - 3 q^{52} + 6 q^{61} + 3 q^{64} + 6 q^{67} + 3 q^{73} - 6 q^{75} + 3 q^{79} + 3 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2793\mathbb{Z}\right)^\times\).

\(n\) \(932\) \(2110\) \(2206\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{18}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
−0.173648 + 0.984808i
−0.173648 − 0.984808i
0.939693 + 0.342020i
−0.766044 − 0.642788i
0.939693 − 0.342020i
−0.766044 + 0.642788i
0 −0.939693 − 0.342020i −0.173648 + 0.984808i 0 0 0 0 0.766044 + 0.642788i 0
440.1 0 −0.939693 + 0.342020i −0.173648 − 0.984808i 0 0 0 0 0.766044 − 0.642788i 0
1028.1 0 0.766044 + 0.642788i 0.939693 + 0.342020i 0 0 0 0 0.173648 + 0.984808i 0
1763.1 0 0.173648 + 0.984808i −0.766044 − 0.642788i 0 0 0 0 −0.939693 + 0.342020i 0
1910.1 0 0.766044 − 0.642788i 0.939693 − 0.342020i 0 0 0 0 0.173648 − 0.984808i 0
2351.1 0 0.173648 − 0.984808i −0.766044 + 0.642788i 0 0 0 0 −0.939693 − 0.342020i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
133.ba even 18 1 inner
399.bx odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2793.1.cf.b 6
3.b odd 2 1 CM 2793.1.cf.b 6
7.b odd 2 1 2793.1.cf.a 6
7.c even 3 1 399.1.bs.a ✓ 6
7.c even 3 1 2793.1.cg.a 6
7.d odd 6 1 399.1.by.a yes 6
7.d odd 6 1 2793.1.ca.a 6
19.f odd 18 1 2793.1.cf.a 6
21.c even 2 1 2793.1.cf.a 6
21.g even 6 1 399.1.by.a yes 6
21.g even 6 1 2793.1.ca.a 6
21.h odd 6 1 399.1.bs.a ✓ 6
21.h odd 6 1 2793.1.cg.a 6
57.j even 18 1 2793.1.cf.a 6
133.ba even 18 1 inner 2793.1.cf.b 6
133.bb even 18 1 399.1.bs.a ✓ 6
133.bd odd 18 1 2793.1.ca.a 6
133.be odd 18 1 399.1.by.a yes 6
133.bf even 18 1 2793.1.cg.a 6
399.br even 18 1 399.1.by.a yes 6
399.bs odd 18 1 2793.1.cg.a 6
399.bv even 18 1 2793.1.ca.a 6
399.bx odd 18 1 inner 2793.1.cf.b 6
399.by odd 18 1 399.1.bs.a ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.1.bs.a ✓ 6 7.c even 3 1
399.1.bs.a ✓ 6 21.h odd 6 1
399.1.bs.a ✓ 6 133.bb even 18 1
399.1.bs.a ✓ 6 399.by odd 18 1
399.1.by.a yes 6 7.d odd 6 1
399.1.by.a yes 6 21.g even 6 1
399.1.by.a yes 6 133.be odd 18 1
399.1.by.a yes 6 399.br even 18 1
2793.1.ca.a 6 7.d odd 6 1
2793.1.ca.a 6 21.g even 6 1
2793.1.ca.a 6 133.bd odd 18 1
2793.1.ca.a 6 399.bv even 18 1
2793.1.cf.a 6 7.b odd 2 1
2793.1.cf.a 6 19.f odd 18 1
2793.1.cf.a 6 21.c even 2 1
2793.1.cf.a 6 57.j even 18 1
2793.1.cf.b 6 1.a even 1 1 trivial
2793.1.cf.b 6 3.b odd 2 1 CM
2793.1.cf.b 6 133.ba even 18 1 inner
2793.1.cf.b 6 399.bx odd 18 1 inner
2793.1.cg.a 6 7.c even 3 1
2793.1.cg.a 6 21.h odd 6 1
2793.1.cg.a 6 133.bf even 18 1
2793.1.cg.a 6 399.bs odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{6} - 3T_{13}^{5} + 6T_{13}^{4} - 8T_{13}^{3} + 3T_{13}^{2} + 3T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2793, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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