Properties

Label 1407.1.g.i
Level $1407$
Weight $1$
Character orbit 1407.g
Analytic conductor $0.702$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
RM discriminant 469
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1407,1,Mod(1406,1407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1407, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1407.1406");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1407 = 3 \cdot 7 \cdot 67 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1407.g (of order \(2\), degree \(1\), 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: \(0.702184472775\)
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 6.0.5938947.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{3} - q^{4} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{5} + q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{3} - q^{4} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{5} + q^{7} - \zeta_{6} q^{9} + \zeta_{6}^{2} q^{12} + q^{13} + ( - \zeta_{6} - 1) q^{15} + q^{16} + (\zeta_{6}^{2} + \zeta_{6}) q^{20} - \zeta_{6}^{2} q^{21} + (\zeta_{6}^{2} + \zeta_{6}) q^{23} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{25} - q^{27} - q^{28} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{29} - q^{31} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{35} + \zeta_{6} q^{36} - q^{37} - \zeta_{6}^{2} q^{39} + (\zeta_{6}^{2} + \zeta_{6}) q^{41} + (\zeta_{6}^{2} - 1) q^{45} - \zeta_{6}^{2} q^{48} + q^{49} - q^{52} + (\zeta_{6} + 1) q^{60} + q^{61} - \zeta_{6} q^{63} - q^{64} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{65} - q^{67} + (\zeta_{6} + 1) q^{69} + (\zeta_{6}^{2} + \zeta_{6}) q^{71} + (\zeta_{6}^{2} + \zeta_{6} - 1) q^{75} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{80} + \zeta_{6}^{2} q^{81} + \zeta_{6}^{2} q^{84} + ( - \zeta_{6} - 1) q^{87} + q^{91} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{92} + \zeta_{6}^{2} q^{93} + q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{4} + 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 2 q^{4} + 2 q^{7} - q^{9} - q^{12} + 2 q^{13} - 3 q^{15} + 2 q^{16} + q^{21} - 4 q^{25} - 2 q^{27} - 2 q^{28} - 2 q^{31} + q^{36} - 2 q^{37} + q^{39} - 3 q^{45} + q^{48} + 2 q^{49} - 2 q^{52} + 3 q^{60} + 4 q^{61} - q^{63} - 2 q^{64} - 2 q^{67} + 3 q^{69} - 2 q^{75} - q^{81} - q^{84} - 3 q^{87} + 2 q^{91} - q^{93} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1407\mathbb{Z}\right)^\times\).

\(n\) \(337\) \(470\) \(1207\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1406.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i −1.00000 1.73205i 0 1.00000 0 −0.500000 0.866025i 0
1406.2 0 0.500000 + 0.866025i −1.00000 1.73205i 0 1.00000 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
469.c even 2 1 RM by \(\Q(\sqrt{469}) \)
3.b odd 2 1 inner
1407.g odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1407.1.g.i yes 2
3.b odd 2 1 inner 1407.1.g.i yes 2
7.b odd 2 1 1407.1.g.h 2
21.c even 2 1 1407.1.g.h 2
67.b odd 2 1 1407.1.g.h 2
201.d even 2 1 1407.1.g.h 2
469.c even 2 1 RM 1407.1.g.i yes 2
1407.g odd 2 1 inner 1407.1.g.i yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1407.1.g.h 2 7.b odd 2 1
1407.1.g.h 2 21.c even 2 1
1407.1.g.h 2 67.b odd 2 1
1407.1.g.h 2 201.d even 2 1
1407.1.g.i yes 2 1.a even 1 1 trivial
1407.1.g.i yes 2 3.b odd 2 1 inner
1407.1.g.i yes 2 469.c even 2 1 RM
1407.1.g.i yes 2 1407.g odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1407, [\chi])\):

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