Properties

Label 3840.2.k.a
Level $3840$
Weight $2$
Character orbit 3840.k
Analytic conductor $30.663$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3840,2,Mod(1921,3840)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3840.1921");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.k (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{3} - i q^{5} - 4 q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{3} - i q^{5} - 4 q^{7} - q^{9} + 6 i q^{13} - q^{15} - 2 q^{17} - 4 i q^{19} + 4 i q^{21} + 8 q^{23} - q^{25} + i q^{27} + 6 i q^{29} + 4 i q^{35} - 6 i q^{37} + 6 q^{39} - 10 q^{41} - 4 i q^{43} + i q^{45} + 8 q^{47} + 9 q^{49} + 2 i q^{51} + 10 i q^{53} - 4 q^{57} - 6 i q^{61} + 4 q^{63} + 6 q^{65} + 4 i q^{67} - 8 i q^{69} + 14 q^{73} + i q^{75} + 16 q^{79} + q^{81} - 12 i q^{83} + 2 i q^{85} + 6 q^{87} - 2 q^{89} - 24 i q^{91} - 4 q^{95} + 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{7} - 2 q^{9} - 2 q^{15} - 4 q^{17} + 16 q^{23} - 2 q^{25} + 12 q^{39} - 20 q^{41} + 16 q^{47} + 18 q^{49} - 8 q^{57} + 8 q^{63} + 12 q^{65} + 28 q^{73} + 32 q^{79} + 2 q^{81} + 12 q^{87} - 4 q^{89} - 8 q^{95} + 4 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1921.1
1.00000i
1.00000i
0 1.00000i 0 1.00000i 0 −4.00000 0 −1.00000 0
1921.2 0 1.00000i 0 1.00000i 0 −4.00000 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
8.b 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.k.a 2
4.b odd 2 1 3840.2.k.z 2
8.b even 2 1 inner 3840.2.k.a 2
8.d odd 2 1 3840.2.k.z 2
16.e even 4 1 120.2.a.a 1
16.e even 4 1 960.2.a.g 1
16.f odd 4 1 240.2.a.a 1
16.f odd 4 1 960.2.a.n 1
48.i odd 4 1 360.2.a.e 1
48.i odd 4 1 2880.2.a.r 1
48.k even 4 1 720.2.a.f 1
48.k even 4 1 2880.2.a.b 1
80.i odd 4 1 600.2.f.c 2
80.i odd 4 1 4800.2.f.u 2
80.j even 4 1 1200.2.f.f 2
80.j even 4 1 4800.2.f.n 2
80.k odd 4 1 1200.2.a.r 1
80.k odd 4 1 4800.2.a.bh 1
80.q even 4 1 600.2.a.a 1
80.q even 4 1 4800.2.a.bl 1
80.s even 4 1 1200.2.f.f 2
80.s even 4 1 4800.2.f.n 2
80.t odd 4 1 600.2.f.c 2
80.t odd 4 1 4800.2.f.u 2
112.l odd 4 1 5880.2.a.p 1
144.w odd 12 2 3240.2.q.a 2
144.x even 12 2 3240.2.q.m 2
240.t even 4 1 3600.2.a.bo 1
240.z odd 4 1 3600.2.f.l 2
240.bb even 4 1 1800.2.f.g 2
240.bd odd 4 1 3600.2.f.l 2
240.bf even 4 1 1800.2.f.g 2
240.bm odd 4 1 1800.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.a 1 16.e even 4 1
240.2.a.a 1 16.f odd 4 1
360.2.a.e 1 48.i odd 4 1
600.2.a.a 1 80.q even 4 1
600.2.f.c 2 80.i odd 4 1
600.2.f.c 2 80.t odd 4 1
720.2.a.f 1 48.k even 4 1
960.2.a.g 1 16.e even 4 1
960.2.a.n 1 16.f odd 4 1
1200.2.a.r 1 80.k odd 4 1
1200.2.f.f 2 80.j even 4 1
1200.2.f.f 2 80.s even 4 1
1800.2.a.c 1 240.bm odd 4 1
1800.2.f.g 2 240.bb even 4 1
1800.2.f.g 2 240.bf even 4 1
2880.2.a.b 1 48.k even 4 1
2880.2.a.r 1 48.i odd 4 1
3240.2.q.a 2 144.w odd 12 2
3240.2.q.m 2 144.x even 12 2
3600.2.a.bo 1 240.t even 4 1
3600.2.f.l 2 240.z odd 4 1
3600.2.f.l 2 240.bd odd 4 1
3840.2.k.a 2 1.a even 1 1 trivial
3840.2.k.a 2 8.b even 2 1 inner
3840.2.k.z 2 4.b odd 2 1
3840.2.k.z 2 8.d odd 2 1
4800.2.a.bh 1 80.k odd 4 1
4800.2.a.bl 1 80.q even 4 1
4800.2.f.n 2 80.j even 4 1
4800.2.f.n 2 80.s even 4 1
4800.2.f.u 2 80.i odd 4 1
4800.2.f.u 2 80.t odd 4 1
5880.2.a.p 1 112.l odd 4 1

Hecke kernels

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

\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} + 36 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display
\( T_{23} - 8 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display
\( T_{47} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T - 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 100 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 36 \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 14)^{2} \) Copy content Toggle raw display
$79$ \( (T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
show more
show less