Properties

Label 399.1.q.a
Level $399$
Weight $1$
Character orbit 399.q
Analytic conductor $0.199$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
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] = [399,1,Mod(293,399)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(399, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("399.293");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 399 = 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 399.q (of order \(6\), 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: \(0.199126940041\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
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
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.7643717613.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{3} - \zeta_{6}^{2} q^{4} - \zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{3} - \zeta_{6}^{2} q^{4} - \zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} - q^{12} + \zeta_{6}^{2} q^{13} - \zeta_{6} q^{16} + q^{19} + \zeta_{6}^{2} q^{21} - \zeta_{6}^{2} q^{25} + q^{27} - q^{28} + q^{31} + \zeta_{6} q^{36} + (\zeta_{6}^{2} + \zeta_{6}) q^{37} + q^{39} - \zeta_{6} q^{43} + \zeta_{6}^{2} q^{48} + \zeta_{6}^{2} q^{49} + \zeta_{6} q^{52} - \zeta_{6} q^{57} + (\zeta_{6} + 1) q^{61} + q^{63} - q^{64} + (\zeta_{6} + 1) q^{67} + (\zeta_{6}^{2} - 1) q^{73} - q^{75} - \zeta_{6}^{2} q^{76} + (\zeta_{6}^{2} - 1) q^{79} - \zeta_{6} q^{81} + \zeta_{6} q^{84} + q^{91} - \zeta_{6} q^{93} - 2 \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{4} - q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + q^{4} - q^{7} - q^{9} - 2 q^{12} - q^{13} - q^{16} + 2 q^{19} - q^{21} + q^{25} + 2 q^{27} - 2 q^{28} + 2 q^{31} + q^{36} + 2 q^{39} - q^{43} - q^{48} - q^{49} + q^{52} - q^{57} + 3 q^{61} + 2 q^{63} - 2 q^{64} + 3 q^{67} - 3 q^{73} - 2 q^{75} + q^{76} - 3 q^{79} - q^{81} + q^{84} + 2 q^{91} - q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/399\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(134\) \(211\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{6}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0
335.1 0 −0.500000 0.866025i 0.500000 0.866025i 0 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
133.p even 6 1 inner
399.q odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 399.1.q.a 2
3.b odd 2 1 CM 399.1.q.a 2
7.b odd 2 1 399.1.q.b yes 2
7.c even 3 1 2793.1.r.b 2
7.c even 3 1 2793.1.bn.a 2
7.d odd 6 1 2793.1.r.a 2
7.d odd 6 1 2793.1.bn.b 2
19.d odd 6 1 399.1.q.b yes 2
21.c even 2 1 399.1.q.b yes 2
21.g even 6 1 2793.1.r.a 2
21.g even 6 1 2793.1.bn.b 2
21.h odd 6 1 2793.1.r.b 2
21.h odd 6 1 2793.1.bn.a 2
57.f even 6 1 399.1.q.b yes 2
133.i even 6 1 2793.1.r.b 2
133.j odd 6 1 2793.1.r.a 2
133.n odd 6 1 2793.1.bn.b 2
133.p even 6 1 inner 399.1.q.a 2
133.s even 6 1 2793.1.bn.a 2
399.q odd 6 1 inner 399.1.q.a 2
399.r odd 6 1 2793.1.bn.a 2
399.x even 6 1 2793.1.bn.b 2
399.bm even 6 1 2793.1.r.a 2
399.bn odd 6 1 2793.1.r.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.1.q.a 2 1.a even 1 1 trivial
399.1.q.a 2 3.b odd 2 1 CM
399.1.q.a 2 133.p even 6 1 inner
399.1.q.a 2 399.q odd 6 1 inner
399.1.q.b yes 2 7.b odd 2 1
399.1.q.b yes 2 19.d odd 6 1
399.1.q.b yes 2 21.c even 2 1
399.1.q.b yes 2 57.f even 6 1
2793.1.r.a 2 7.d odd 6 1
2793.1.r.a 2 21.g even 6 1
2793.1.r.a 2 133.j odd 6 1
2793.1.r.a 2 399.bm even 6 1
2793.1.r.b 2 7.c even 3 1
2793.1.r.b 2 21.h odd 6 1
2793.1.r.b 2 133.i even 6 1
2793.1.r.b 2 399.bn odd 6 1
2793.1.bn.a 2 7.c even 3 1
2793.1.bn.a 2 21.h odd 6 1
2793.1.bn.a 2 133.s even 6 1
2793.1.bn.a 2 399.r odd 6 1
2793.1.bn.b 2 7.d odd 6 1
2793.1.bn.b 2 21.g even 6 1
2793.1.bn.b 2 133.n odd 6 1
2793.1.bn.b 2 399.x even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} + T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(399, [\chi])\). 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 3 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$79$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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