Properties

Label 3840.1.c.h
Level $3840$
Weight $1$
Character orbit 3840.c
Analytic conductor $1.916$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -20, -24, 120
Inner twists $8$

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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: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1920)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-5}, \sqrt{-6})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.33973862400.4

$q$-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.

\(f(q)\) \(=\) \( q + q^{3} - i q^{5} - 2 i q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - i q^{5} - 2 i q^{7} + q^{9} - i q^{15} - 2 i q^{21} - q^{25} + q^{27} + 2 i q^{29} - 2 q^{35} - i q^{45} - 3 q^{49} - 2 i q^{63} - q^{75} + q^{81} + 2 q^{83} + 2 i q^{87} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{9} - 2 q^{25} + 2 q^{27} - 4 q^{35} - 6 q^{49} - 2 q^{75} + 2 q^{81} + 4 q^{83}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3329.1
1.00000i
1.00000i
0 1.00000 0 1.00000i 0 2.00000i 0 1.00000 0
3329.2 0 1.00000 0 1.00000i 0 2.00000i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
120.m even 2 1 RM by \(\Q(\sqrt{30}) \)
8.d odd 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3840.1.c.h 2
3.b odd 2 1 3840.1.c.e 2
4.b odd 2 1 3840.1.c.e 2
5.b even 2 1 3840.1.c.e 2
8.b even 2 1 3840.1.c.e 2
8.d odd 2 1 inner 3840.1.c.h 2
12.b even 2 1 inner 3840.1.c.h 2
15.d odd 2 1 inner 3840.1.c.h 2
16.e even 4 1 1920.1.i.e 2
16.e even 4 1 1920.1.i.f yes 2
16.f odd 4 1 1920.1.i.e 2
16.f odd 4 1 1920.1.i.f yes 2
20.d odd 2 1 CM 3840.1.c.h 2
24.f even 2 1 3840.1.c.e 2
24.h odd 2 1 CM 3840.1.c.h 2
40.e odd 2 1 3840.1.c.e 2
40.f even 2 1 inner 3840.1.c.h 2
48.i odd 4 1 1920.1.i.e 2
48.i odd 4 1 1920.1.i.f yes 2
48.k even 4 1 1920.1.i.e 2
48.k even 4 1 1920.1.i.f yes 2
60.h even 2 1 3840.1.c.e 2
80.k odd 4 1 1920.1.i.e 2
80.k odd 4 1 1920.1.i.f yes 2
80.q even 4 1 1920.1.i.e 2
80.q even 4 1 1920.1.i.f yes 2
120.i odd 2 1 3840.1.c.e 2
120.m even 2 1 RM 3840.1.c.h 2
240.t even 4 1 1920.1.i.e 2
240.t even 4 1 1920.1.i.f yes 2
240.bm odd 4 1 1920.1.i.e 2
240.bm odd 4 1 1920.1.i.f yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1920.1.i.e 2 16.e even 4 1
1920.1.i.e 2 16.f odd 4 1
1920.1.i.e 2 48.i odd 4 1
1920.1.i.e 2 48.k even 4 1
1920.1.i.e 2 80.k odd 4 1
1920.1.i.e 2 80.q even 4 1
1920.1.i.e 2 240.t even 4 1
1920.1.i.e 2 240.bm odd 4 1
1920.1.i.f yes 2 16.e even 4 1
1920.1.i.f yes 2 16.f odd 4 1
1920.1.i.f yes 2 48.i odd 4 1
1920.1.i.f yes 2 48.k even 4 1
1920.1.i.f yes 2 80.k odd 4 1
1920.1.i.f yes 2 80.q even 4 1
1920.1.i.f yes 2 240.t even 4 1
1920.1.i.f yes 2 240.bm odd 4 1
3840.1.c.e 2 3.b odd 2 1
3840.1.c.e 2 4.b odd 2 1
3840.1.c.e 2 5.b even 2 1
3840.1.c.e 2 8.b even 2 1
3840.1.c.e 2 24.f even 2 1
3840.1.c.e 2 40.e odd 2 1
3840.1.c.e 2 60.h even 2 1
3840.1.c.e 2 120.i odd 2 1
3840.1.c.h 2 1.a even 1 1 trivial
3840.1.c.h 2 8.d odd 2 1 inner
3840.1.c.h 2 12.b even 2 1 inner
3840.1.c.h 2 15.d odd 2 1 inner
3840.1.c.h 2 20.d odd 2 1 CM
3840.1.c.h 2 24.h odd 2 1 CM
3840.1.c.h 2 40.f even 2 1 inner
3840.1.c.h 2 120.m even 2 1 RM

Hecke kernels

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

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display
\( T_{53} \) Copy content Toggle raw display
\( T_{83} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) 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
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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