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Properties

Label 504.2.q.c
Level $504$
Weight $2$
Character orbit 504.q
Analytic conductor $4.024$
Analytic rank $0$
Dimension $22$
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] = [504,2,Mod(25,504)]
 
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, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.q (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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 2 q^{3} + q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 2 q^{3} + q^{5} + 5 q^{7} + 6 q^{9} + 3 q^{11} + 7 q^{13} - q^{15} - q^{17} + 13 q^{19} - 22 q^{25} - 2 q^{27} - 7 q^{29} - 12 q^{31} - 3 q^{33} + 2 q^{35} + 6 q^{37} - 4 q^{39} + 4 q^{41} + 2 q^{43} - 3 q^{45} - 34 q^{47} - 25 q^{49} + 53 q^{51} + q^{53} + 2 q^{55} - 21 q^{57} + 42 q^{59} - 62 q^{61} - 22 q^{63} + 6 q^{65} + 52 q^{67} - 40 q^{69} - 32 q^{71} + 17 q^{73} + 53 q^{75} - q^{77} + 32 q^{79} - 6 q^{81} - 36 q^{83} + 28 q^{85} - 5 q^{87} - 2 q^{89} + 15 q^{91} - 11 q^{93} + 48 q^{95} + 19 q^{97} + 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −1.71769 + 0.222607i 0 0.234085 − 0.405446i 0 0.212345 + 2.63722i 0 2.90089 − 0.764737i 0
25.2 0 −1.69989 − 0.332219i 0 −1.59750 + 2.76695i 0 −1.66645 − 2.05498i 0 2.77926 + 1.12947i 0
25.3 0 −1.46869 − 0.918117i 0 1.89970 − 3.29038i 0 −0.841809 − 2.50826i 0 1.31412 + 2.69686i 0
25.4 0 −0.914508 + 1.47094i 0 0.891774 − 1.54460i 0 −2.54386 − 0.727153i 0 −1.32735 − 2.69038i 0
25.5 0 −0.704143 − 1.58246i 0 −1.05220 + 1.82246i 0 2.58382 + 0.569079i 0 −2.00837 + 2.22856i 0
25.6 0 −0.633073 + 1.61221i 0 −1.70368 + 2.95086i 0 −0.410295 + 2.61374i 0 −2.19844 − 2.04129i 0
25.7 0 0.748111 + 1.56216i 0 2.11148 − 3.65719i 0 2.19338 + 1.47956i 0 −1.88066 + 2.33733i 0
25.8 0 0.987504 + 1.42297i 0 −1.38590 + 2.40045i 0 1.74026 − 1.99286i 0 −1.04967 + 2.81037i 0
25.9 0 1.04182 − 1.38370i 0 1.33425 − 2.31099i 0 2.54743 − 0.714566i 0 −0.829236 − 2.88312i 0
25.10 0 1.64138 − 0.553060i 0 −0.263002 + 0.455533i 0 −0.333150 − 2.62469i 0 2.38825 − 1.81556i 0
25.11 0 1.71918 + 0.210723i 0 0.0309846 − 0.0536670i 0 −0.981674 + 2.45689i 0 2.91119 + 0.724543i 0
121.1 0 −1.71769 − 0.222607i 0 0.234085 + 0.405446i 0 0.212345 − 2.63722i 0 2.90089 + 0.764737i 0
121.2 0 −1.69989 + 0.332219i 0 −1.59750 − 2.76695i 0 −1.66645 + 2.05498i 0 2.77926 − 1.12947i 0
121.3 0 −1.46869 + 0.918117i 0 1.89970 + 3.29038i 0 −0.841809 + 2.50826i 0 1.31412 − 2.69686i 0
121.4 0 −0.914508 − 1.47094i 0 0.891774 + 1.54460i 0 −2.54386 + 0.727153i 0 −1.32735 + 2.69038i 0
121.5 0 −0.704143 + 1.58246i 0 −1.05220 − 1.82246i 0 2.58382 − 0.569079i 0 −2.00837 − 2.22856i 0
121.6 0 −0.633073 − 1.61221i 0 −1.70368 − 2.95086i 0 −0.410295 − 2.61374i 0 −2.19844 + 2.04129i 0
121.7 0 0.748111 − 1.56216i 0 2.11148 + 3.65719i 0 2.19338 − 1.47956i 0 −1.88066 − 2.33733i 0
121.8 0 0.987504 − 1.42297i 0 −1.38590 − 2.40045i 0 1.74026 + 1.99286i 0 −1.04967 − 2.81037i 0
121.9 0 1.04182 + 1.38370i 0 1.33425 + 2.31099i 0 2.54743 + 0.714566i 0 −0.829236 + 2.88312i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.11
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{22} - T_{5}^{21} + 39 T_{5}^{20} - 4 T_{5}^{19} + 952 T_{5}^{18} + 120 T_{5}^{17} + 14145 T_{5}^{16} + \cdots + 5476 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display