Properties

Label 1407.1.g.b
Level $1407$
Weight $1$
Character orbit 1407.g
Self dual yes
Analytic conductor $0.702$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -1407
Inner twists $2$

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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: yes
Analytic conductor: \(0.702184472775\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1407.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.1407.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} - q^{6} + q^{7} + q^{8} + q^{9} - q^{11} - q^{13} - q^{14} - q^{16} + 2 q^{17} - q^{18} + q^{21} + q^{22} + q^{24} + q^{25} + q^{26} + q^{27} - q^{31} - q^{33} - 2 q^{34} - q^{37} - q^{39} - q^{42} - q^{47} - q^{48} + q^{49} - q^{50} + 2 q^{51} + 2 q^{53} - q^{54} + q^{56} - q^{59} - q^{61} + q^{62} + q^{63} + q^{64} + q^{66} + q^{67} + q^{72} + q^{74} + q^{75} - q^{77} + q^{78} + q^{81} + 2 q^{83} - q^{88} - q^{89} - q^{91} - q^{93} + q^{94} - q^{97} - q^{98} - q^{99}+O(q^{100}) \) 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
−1.00000 1.00000 0 0 −1.00000 1.00000 1.00000 1.00000 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
1407.g odd 2 1 CM by \(\Q(\sqrt{-1407}) \)

Twists

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

Hecke kernels

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

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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