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Properties

Label 1530.2.r.d
Level $1530$
Weight $2$
Character orbit 1530.r
Analytic conductor $12.217$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1530,2,Mod(917,1530)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1530, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1530.917");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1530.r (of order \(4\), degree \(2\), 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: \(12.2171115093\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{13} - 40 q^{16} - 8 q^{25} + 32 q^{43} - 8 q^{52} - 96 q^{55} - 64 q^{67} - 8 q^{70} + 16 q^{76} + 72 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
917.1 −0.707107 + 0.707107i 0 − 1.00000i −2.23194 − 0.135831i 0 −3.43960 + 3.43960i 0.707107 + 0.707107i 0 1.67427 − 1.48217i
917.2 −0.707107 + 0.707107i 0 − 1.00000i 2.23194 + 0.135831i 0 3.43960 − 3.43960i 0.707107 + 0.707107i 0 −1.67427 + 1.48217i
917.3 −0.707107 + 0.707107i 0 − 1.00000i −0.449991 + 2.19032i 0 −1.73799 + 1.73799i 0.707107 + 0.707107i 0 −1.23060 − 1.86698i
917.4 −0.707107 + 0.707107i 0 − 1.00000i 0.449991 − 2.19032i 0 1.73799 − 1.73799i 0.707107 + 0.707107i 0 1.23060 + 1.86698i
917.5 −0.707107 + 0.707107i 0 − 1.00000i −1.41348 + 1.73265i 0 3.48299 − 3.48299i 0.707107 + 0.707107i 0 −0.225690 − 2.22465i
917.6 −0.707107 + 0.707107i 0 − 1.00000i 1.41348 − 1.73265i 0 −3.48299 + 3.48299i 0.707107 + 0.707107i 0 0.225690 + 2.22465i
917.7 −0.707107 + 0.707107i 0 − 1.00000i −2.19353 − 0.434058i 0 1.26606 − 1.26606i 0.707107 + 0.707107i 0 1.85799 − 1.24414i
917.8 −0.707107 + 0.707107i 0 − 1.00000i 2.19353 + 0.434058i 0 −1.26606 + 1.26606i 0.707107 + 0.707107i 0 −1.85799 + 1.24414i
917.9 −0.707107 + 0.707107i 0 − 1.00000i −0.0802825 − 2.23463i 0 0.643778 − 0.643778i 0.707107 + 0.707107i 0 1.63689 + 1.52335i
917.10 −0.707107 + 0.707107i 0 − 1.00000i 0.0802825 + 2.23463i 0 −0.643778 + 0.643778i 0.707107 + 0.707107i 0 −1.63689 − 1.52335i
917.11 0.707107 − 0.707107i 0 − 1.00000i −0.449991 + 2.19032i 0 1.73799 − 1.73799i −0.707107 − 0.707107i 0 1.23060 + 1.86698i
917.12 0.707107 − 0.707107i 0 − 1.00000i 0.449991 − 2.19032i 0 −1.73799 + 1.73799i −0.707107 − 0.707107i 0 −1.23060 − 1.86698i
917.13 0.707107 − 0.707107i 0 − 1.00000i −2.19353 − 0.434058i 0 −1.26606 + 1.26606i −0.707107 − 0.707107i 0 −1.85799 + 1.24414i
917.14 0.707107 − 0.707107i 0 − 1.00000i 2.19353 + 0.434058i 0 1.26606 − 1.26606i −0.707107 − 0.707107i 0 1.85799 − 1.24414i
917.15 0.707107 − 0.707107i 0 − 1.00000i −0.0802825 − 2.23463i 0 −0.643778 + 0.643778i −0.707107 − 0.707107i 0 −1.63689 − 1.52335i
917.16 0.707107 − 0.707107i 0 − 1.00000i 0.0802825 + 2.23463i 0 0.643778 − 0.643778i −0.707107 − 0.707107i 0 1.63689 + 1.52335i
917.17 0.707107 − 0.707107i 0 − 1.00000i −2.23194 − 0.135831i 0 3.43960 − 3.43960i −0.707107 − 0.707107i 0 −1.67427 + 1.48217i
917.18 0.707107 − 0.707107i 0 − 1.00000i 2.23194 + 0.135831i 0 −3.43960 + 3.43960i −0.707107 − 0.707107i 0 1.67427 − 1.48217i
917.19 0.707107 − 0.707107i 0 − 1.00000i −1.41348 + 1.73265i 0 −3.48299 + 3.48299i −0.707107 − 0.707107i 0 0.225690 + 2.22465i
917.20 0.707107 − 0.707107i 0 − 1.00000i 1.41348 − 1.73265i 0 3.48299 − 3.48299i −0.707107 − 0.707107i 0 −0.225690 − 2.22465i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 917.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner
17.b even 2 1 inner
51.c odd 2 1 inner
85.g odd 4 1 inner
255.o even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1530.2.r.d ✓ 40
3.b odd 2 1 inner 1530.2.r.d ✓ 40
5.c odd 4 1 inner 1530.2.r.d ✓ 40
15.e even 4 1 inner 1530.2.r.d ✓ 40
17.b even 2 1 inner 1530.2.r.d ✓ 40
51.c odd 2 1 inner 1530.2.r.d ✓ 40
85.g odd 4 1 inner 1530.2.r.d ✓ 40
255.o even 4 1 inner 1530.2.r.d ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1530.2.r.d ✓ 40 1.a even 1 1 trivial
1530.2.r.d ✓ 40 3.b odd 2 1 inner
1530.2.r.d ✓ 40 5.c odd 4 1 inner
1530.2.r.d ✓ 40 15.e even 4 1 inner
1530.2.r.d ✓ 40 17.b even 2 1 inner
1530.2.r.d ✓ 40 51.c odd 2 1 inner
1530.2.r.d ✓ 40 85.g odd 4 1 inner
1530.2.r.d ✓ 40 255.o even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1530, [\chi])\):

\( T_{7}^{20} + 1196T_{7}^{16} + 384496T_{7}^{12} + 16109888T_{7}^{8} + 134505472T_{7}^{4} + 84934656 \) Copy content Toggle raw display
\( T_{11}^{10} - 72T_{11}^{8} + 1568T_{11}^{6} - 10624T_{11}^{4} + 23552T_{11}^{2} - 16384 \) Copy content Toggle raw display