Properties

Label 3240.1.p.b
Level $3240$
Weight $1$
Character orbit 3240.p
Self dual yes
Analytic conductor $1.617$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -40
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3240,1,Mod(1459,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3240.1459");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3240.p (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: \(1.61697064093\)
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 3.1.3240.1
Artin image: $D_6$
Artin field: Galois closure of 6.2.52488000.1
Stark unit: Root of $x^{6} - 1086x^{5} - 68565x^{4} - 1318700x^{3} - 68565x^{2} - 1086x + 1$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} + q^{10} + 2 q^{11} + q^{13} - q^{14} + q^{16} - q^{19} - q^{20} - 2 q^{22} + q^{23} + q^{25} - q^{26} + q^{28} - q^{32} - q^{35} - 2 q^{37} + q^{38} + q^{40} - q^{41} + 2 q^{44} - q^{46} + q^{47} - q^{50} + q^{52} + q^{53} - 2 q^{55} - q^{56} - q^{59} + q^{64} - q^{65} + q^{70} + 2 q^{74} - q^{76} + 2 q^{77} - q^{80} + q^{82} - 2 q^{88} + 2 q^{89} + q^{91} + q^{92} - q^{94} + q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1459.1
0
−1.00000 0 1.00000 −1.00000 0 1.00000 −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3240.1.p.b yes 1
3.b odd 2 1 3240.1.p.d yes 1
5.b even 2 1 3240.1.p.c yes 1
8.d odd 2 1 3240.1.p.c yes 1
9.c even 3 2 3240.1.z.h 2
9.d odd 6 2 3240.1.z.b 2
15.d odd 2 1 3240.1.p.a 1
24.f even 2 1 3240.1.p.a 1
40.e odd 2 1 CM 3240.1.p.b yes 1
45.h odd 6 2 3240.1.z.i 2
45.j even 6 2 3240.1.z.c 2
72.l even 6 2 3240.1.z.i 2
72.p odd 6 2 3240.1.z.c 2
120.m even 2 1 3240.1.p.d yes 1
360.z odd 6 2 3240.1.z.h 2
360.bd even 6 2 3240.1.z.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.1.p.a 1 15.d odd 2 1
3240.1.p.a 1 24.f even 2 1
3240.1.p.b yes 1 1.a even 1 1 trivial
3240.1.p.b yes 1 40.e odd 2 1 CM
3240.1.p.c yes 1 5.b even 2 1
3240.1.p.c yes 1 8.d odd 2 1
3240.1.p.d yes 1 3.b odd 2 1
3240.1.p.d yes 1 120.m even 2 1
3240.1.z.b 2 9.d odd 6 2
3240.1.z.b 2 360.bd even 6 2
3240.1.z.c 2 45.j even 6 2
3240.1.z.c 2 72.p odd 6 2
3240.1.z.h 2 9.c even 3 2
3240.1.z.h 2 360.z odd 6 2
3240.1.z.i 2 45.h odd 6 2
3240.1.z.i 2 72.l even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3240, [\chi])\):

\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{23} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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