Properties

Label 3840.1.c.b
Level $3840$
Weight $1$
Character orbit 3840.c
Self dual yes
Analytic conductor $1.916$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -15, -24, 40
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3840,1,Mod(3329,3840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("3840.3329");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3840.c (of order \(2\), degree \(1\), not 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: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-6}, \sqrt{10})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.92160.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{3} + q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} + q^{9} - q^{15} + q^{25} - q^{27} + 2 q^{31} + q^{45} + q^{49} - 2 q^{53} - q^{75} - 2 q^{79} + q^{81} + 2 q^{83} - 2 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(0\) \(1\) \(1\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3329.1
0
0 −1.00000 0 1.00000 0 0 0 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
40.f even 2 1 RM by \(\Q(\sqrt{10}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3840.1.c.b 1
3.b odd 2 1 3840.1.c.c 1
4.b odd 2 1 3840.1.c.d 1
5.b even 2 1 3840.1.c.c 1
8.b even 2 1 3840.1.c.c 1
8.d odd 2 1 3840.1.c.a 1
12.b even 2 1 3840.1.c.a 1
15.d odd 2 1 CM 3840.1.c.b 1
16.e even 4 2 480.1.i.a 2
16.f odd 4 2 120.1.i.a 2
20.d odd 2 1 3840.1.c.a 1
24.f even 2 1 3840.1.c.d 1
24.h odd 2 1 CM 3840.1.c.b 1
40.e odd 2 1 3840.1.c.d 1
40.f even 2 1 RM 3840.1.c.b 1
48.i odd 4 2 480.1.i.a 2
48.k even 4 2 120.1.i.a 2
60.h even 2 1 3840.1.c.d 1
80.i odd 4 2 2400.1.n.b 1
80.j even 4 2 600.1.n.a 1
80.k odd 4 2 120.1.i.a 2
80.q even 4 2 480.1.i.a 2
80.s even 4 2 600.1.n.b 1
80.t odd 4 2 2400.1.n.a 1
120.i odd 2 1 3840.1.c.c 1
120.m even 2 1 3840.1.c.a 1
144.u even 12 4 3240.1.bh.h 4
144.v odd 12 4 3240.1.bh.h 4
240.t even 4 2 120.1.i.a 2
240.z odd 4 2 600.1.n.a 1
240.bb even 4 2 2400.1.n.a 1
240.bd odd 4 2 600.1.n.b 1
240.bf even 4 2 2400.1.n.b 1
240.bm odd 4 2 480.1.i.a 2
720.cz odd 12 4 3240.1.bh.h 4
720.da even 12 4 3240.1.bh.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.1.i.a 2 16.f odd 4 2
120.1.i.a 2 48.k even 4 2
120.1.i.a 2 80.k odd 4 2
120.1.i.a 2 240.t even 4 2
480.1.i.a 2 16.e even 4 2
480.1.i.a 2 48.i odd 4 2
480.1.i.a 2 80.q even 4 2
480.1.i.a 2 240.bm odd 4 2
600.1.n.a 1 80.j even 4 2
600.1.n.a 1 240.z odd 4 2
600.1.n.b 1 80.s even 4 2
600.1.n.b 1 240.bd odd 4 2
2400.1.n.a 1 80.t odd 4 2
2400.1.n.a 1 240.bb even 4 2
2400.1.n.b 1 80.i odd 4 2
2400.1.n.b 1 240.bf even 4 2
3240.1.bh.h 4 144.u even 12 4
3240.1.bh.h 4 144.v odd 12 4
3240.1.bh.h 4 720.cz odd 12 4
3240.1.bh.h 4 720.da even 12 4
3840.1.c.a 1 8.d odd 2 1
3840.1.c.a 1 12.b even 2 1
3840.1.c.a 1 20.d odd 2 1
3840.1.c.a 1 120.m even 2 1
3840.1.c.b 1 1.a even 1 1 trivial
3840.1.c.b 1 15.d odd 2 1 CM
3840.1.c.b 1 24.h odd 2 1 CM
3840.1.c.b 1 40.f even 2 1 RM
3840.1.c.c 1 3.b odd 2 1
3840.1.c.c 1 5.b even 2 1
3840.1.c.c 1 8.b even 2 1
3840.1.c.c 1 120.i odd 2 1
3840.1.c.d 1 4.b odd 2 1
3840.1.c.d 1 24.f even 2 1
3840.1.c.d 1 40.e odd 2 1
3840.1.c.d 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3840, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display
\( T_{31} - 2 \) Copy content Toggle raw display
\( T_{53} + 2 \) Copy content Toggle raw display
\( T_{83} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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