Properties

Label 3840.2.d.e
Level $3840$
Weight $2$
Character orbit 3840.d
Analytic conductor $30.663$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3840,2,Mod(2689,3840)]
 
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, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3840.2689");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.d (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: \(30.6625543762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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: \( 2 \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + (\beta - 1) q^{5} + \beta q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + (\beta - 1) q^{5} + \beta q^{7} + q^{9} - 3 \beta q^{11} - 2 q^{13} + ( - \beta + 1) q^{15} - 3 \beta q^{17} + 2 \beta q^{19} - \beta q^{21} + 4 \beta q^{23} + ( - 2 \beta - 3) q^{25} - q^{27} + 8 q^{31} + 3 \beta q^{33} + ( - \beta - 4) q^{35} + 2 q^{37} + 2 q^{39} + 6 q^{41} + 4 q^{43} + (\beta - 1) q^{45} + 2 \beta q^{47} + 3 q^{49} + 3 \beta q^{51} - 6 q^{53} + (3 \beta + 12) q^{55} - 2 \beta q^{57} + 3 \beta q^{59} - 3 \beta q^{61} + \beta q^{63} + ( - 2 \beta + 2) q^{65} - 4 \beta q^{69} - 4 q^{71} + 6 \beta q^{73} + (2 \beta + 3) q^{75} + 12 q^{77} - 8 q^{79} + q^{81} - 12 q^{83} + (3 \beta + 12) q^{85} + 14 q^{89} - 2 \beta q^{91} - 8 q^{93} + ( - 2 \beta - 8) q^{95} + 4 \beta q^{97} - 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{13} + 2 q^{15} - 6 q^{25} - 2 q^{27} + 16 q^{31} - 8 q^{35} + 4 q^{37} + 4 q^{39} + 12 q^{41} + 8 q^{43} - 2 q^{45} + 6 q^{49} - 12 q^{53} + 24 q^{55} + 4 q^{65} - 8 q^{71} + 6 q^{75} + 24 q^{77} - 16 q^{79} + 2 q^{81} - 24 q^{83} + 24 q^{85} + 28 q^{89} - 16 q^{93} - 16 q^{95}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2689.1
1.00000i
1.00000i
0 −1.00000 0 −1.00000 2.00000i 0 2.00000i 0 1.00000 0
2689.2 0 −1.00000 0 −1.00000 + 2.00000i 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
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 3840.2.d.e 2
4.b odd 2 1 3840.2.d.u 2
5.b even 2 1 3840.2.d.bc 2
8.b even 2 1 3840.2.d.bc 2
8.d odd 2 1 3840.2.d.k 2
16.e even 4 1 480.2.f.b yes 2
16.e even 4 1 960.2.f.g 2
16.f odd 4 1 480.2.f.a 2
16.f odd 4 1 960.2.f.j 2
20.d odd 2 1 3840.2.d.k 2
40.e odd 2 1 3840.2.d.u 2
40.f even 2 1 inner 3840.2.d.e 2
48.i odd 4 1 1440.2.f.e 2
48.i odd 4 1 2880.2.f.g 2
48.k even 4 1 1440.2.f.g 2
48.k even 4 1 2880.2.f.a 2
80.i odd 4 1 2400.2.a.bf 1
80.i odd 4 1 4800.2.a.bt 1
80.j even 4 1 2400.2.a.be 1
80.j even 4 1 4800.2.a.bu 1
80.k odd 4 1 480.2.f.a 2
80.k odd 4 1 960.2.f.j 2
80.q even 4 1 480.2.f.b yes 2
80.q even 4 1 960.2.f.g 2
80.s even 4 1 2400.2.a.c 1
80.s even 4 1 4800.2.a.ba 1
80.t odd 4 1 2400.2.a.d 1
80.t odd 4 1 4800.2.a.z 1
240.t even 4 1 1440.2.f.g 2
240.t even 4 1 2880.2.f.a 2
240.z odd 4 1 7200.2.a.p 1
240.bb even 4 1 7200.2.a.bl 1
240.bd odd 4 1 7200.2.a.br 1
240.bf even 4 1 7200.2.a.j 1
240.bm odd 4 1 1440.2.f.e 2
240.bm odd 4 1 2880.2.f.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.a 2 16.f odd 4 1
480.2.f.a 2 80.k odd 4 1
480.2.f.b yes 2 16.e even 4 1
480.2.f.b yes 2 80.q even 4 1
960.2.f.g 2 16.e even 4 1
960.2.f.g 2 80.q even 4 1
960.2.f.j 2 16.f odd 4 1
960.2.f.j 2 80.k odd 4 1
1440.2.f.e 2 48.i odd 4 1
1440.2.f.e 2 240.bm odd 4 1
1440.2.f.g 2 48.k even 4 1
1440.2.f.g 2 240.t even 4 1
2400.2.a.c 1 80.s even 4 1
2400.2.a.d 1 80.t odd 4 1
2400.2.a.be 1 80.j even 4 1
2400.2.a.bf 1 80.i odd 4 1
2880.2.f.a 2 48.k even 4 1
2880.2.f.a 2 240.t even 4 1
2880.2.f.g 2 48.i odd 4 1
2880.2.f.g 2 240.bm odd 4 1
3840.2.d.e 2 1.a even 1 1 trivial
3840.2.d.e 2 40.f even 2 1 inner
3840.2.d.k 2 8.d odd 2 1
3840.2.d.k 2 20.d odd 2 1
3840.2.d.u 2 4.b odd 2 1
3840.2.d.u 2 40.e odd 2 1
3840.2.d.bc 2 5.b even 2 1
3840.2.d.bc 2 8.b even 2 1
4800.2.a.z 1 80.t odd 4 1
4800.2.a.ba 1 80.s even 4 1
4800.2.a.bt 1 80.i odd 4 1
4800.2.a.bu 1 80.j even 4 1
7200.2.a.j 1 240.bf even 4 1
7200.2.a.p 1 240.z odd 4 1
7200.2.a.bl 1 240.bb even 4 1
7200.2.a.br 1 240.bd odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3840, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 36 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{31} - 8 \) Copy content Toggle raw display
\( T_{37} - 2 \) Copy content Toggle raw display
\( T_{43} - 4 \) 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
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 36 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 64 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 16 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 36 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 144 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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