↑

Properties

Label 504.2.t.d
Level $504$
Weight $2$
Character orbit 504.t
Analytic conductor $4.024$
Analytic rank $0$
Dimension $22$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(193,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.t (of order \(3\), 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{3} - 6 q^{5} + 7 q^{7} - 8 q^{9} + 6 q^{11} - 3 q^{13} - q^{15} + 7 q^{17} - q^{19} - 15 q^{21} - 4 q^{23} + 20 q^{25} - 4 q^{27} + 9 q^{29} - 4 q^{31} - 31 q^{33} + 14 q^{35} + 2 q^{37} + 8 q^{39}+ \cdots - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1 0 −1.61774 − 0.618811i 0 −1.83657 0 2.45061 + 0.997255i 0 2.23415 + 2.00215i 0
193.2 0 −1.40598 − 1.01154i 0 3.84095 0 0.676469 − 2.55781i 0 0.953553 + 2.84442i 0
193.3 0 −1.21966 + 1.22980i 0 −0.481387 0 2.53326 − 0.763277i 0 −0.0248369 − 2.99990i 0
193.4 0 −0.849966 − 1.50916i 0 −1.58188 0 −1.80922 + 1.93047i 0 −1.55512 + 2.56547i 0
193.5 0 0.134843 + 1.72679i 0 −3.43592 0 −1.83889 − 1.90223i 0 −2.96363 + 0.465691i 0
193.6 0 0.371921 + 1.69165i 0 1.68316 0 0.960133 + 2.46539i 0 −2.72335 + 1.25832i 0
193.7 0 0.444471 − 1.67405i 0 −3.52959 0 1.16715 − 2.37440i 0 −2.60489 − 1.48813i 0
193.8 0 0.577666 − 1.63288i 0 1.85591 0 −2.60465 − 0.464545i 0 −2.33261 − 1.88652i 0
193.9 0 1.34414 − 1.09237i 0 2.66802 0 1.94471 + 1.79391i 0 0.613444 − 2.93661i 0
193.10 0 1.51940 + 0.831519i 0 −2.52290 0 −1.07705 + 2.41660i 0 1.61715 + 2.52682i 0
193.11 0 1.70090 + 0.327002i 0 0.340200 0 1.09748 − 2.40739i 0 2.78614 + 1.11240i 0
457.1 0 −1.61774 + 0.618811i 0 −1.83657 0 2.45061 − 0.997255i 0 2.23415 − 2.00215i 0
457.2 0 −1.40598 + 1.01154i 0 3.84095 0 0.676469 + 2.55781i 0 0.953553 − 2.84442i 0
457.3 0 −1.21966 − 1.22980i 0 −0.481387 0 2.53326 + 0.763277i 0 −0.0248369 + 2.99990i 0
457.4 0 −0.849966 + 1.50916i 0 −1.58188 0 −1.80922 − 1.93047i 0 −1.55512 − 2.56547i 0
457.5 0 0.134843 − 1.72679i 0 −3.43592 0 −1.83889 + 1.90223i 0 −2.96363 − 0.465691i 0
457.6 0 0.371921 − 1.69165i 0 1.68316 0 0.960133 − 2.46539i 0 −2.72335 − 1.25832i 0
457.7 0 0.444471 + 1.67405i 0 −3.52959 0 1.16715 + 2.37440i 0 −2.60489 + 1.48813i 0
457.8 0 0.577666 + 1.63288i 0 1.85591 0 −2.60465 + 0.464545i 0 −2.33261 + 1.88652i 0
457.9 0 1.34414 + 1.09237i 0 2.66802 0 1.94471 − 1.79391i 0 0.613444 + 2.93661i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.t.d yes 22
3.b odd 2 1 1512.2.t.d 22
4.b odd 2 1 1008.2.t.k 22
7.c even 3 1 504.2.q.d ✓ 22
9.c even 3 1 504.2.q.d ✓ 22
9.d odd 6 1 1512.2.q.c 22
12.b even 2 1 3024.2.t.l 22
21.h odd 6 1 1512.2.q.c 22
28.g odd 6 1 1008.2.q.k 22
36.f odd 6 1 1008.2.q.k 22
36.h even 6 1 3024.2.q.k 22
63.g even 3 1 inner 504.2.t.d yes 22
63.n odd 6 1 1512.2.t.d 22
84.n even 6 1 3024.2.q.k 22
252.o even 6 1 3024.2.t.l 22
252.bl odd 6 1 1008.2.t.k 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.d ✓ 22 7.c even 3 1
504.2.q.d ✓ 22 9.c even 3 1
504.2.t.d yes 22 1.a even 1 1 trivial
504.2.t.d yes 22 63.g even 3 1 inner
1008.2.q.k 22 28.g odd 6 1
1008.2.q.k 22 36.f odd 6 1
1008.2.t.k 22 4.b odd 2 1
1008.2.t.k 22 252.bl odd 6 1
1512.2.q.c 22 9.d odd 6 1
1512.2.q.c 22 21.h odd 6 1
1512.2.t.d 22 3.b odd 2 1
1512.2.t.d 22 63.n odd 6 1
3024.2.q.k 22 36.h even 6 1
3024.2.q.k 22 84.n even 6 1
3024.2.t.l 22 12.b even 2 1
3024.2.t.l 22 252.o even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{11} + 3 T_{5}^{10} - 28 T_{5}^{9} - 85 T_{5}^{8} + 249 T_{5}^{7} + 766 T_{5}^{6} - 841 T_{5}^{5} + \cdots - 466 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display