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Properties

Label 504.2.bu.a
Level $504$
Weight $2$
Character orbit 504.bu
Analytic conductor $4.024$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,2,Mod(41,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("504.41"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.bu (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{9} + 8 q^{15} - 4 q^{21} + 12 q^{23} - 24 q^{25} - 36 q^{29} + 32 q^{39} + 12 q^{43} + 6 q^{49} + 24 q^{51} + 28 q^{57} - 14 q^{63} + 36 q^{65} - 60 q^{77} - 12 q^{79} - 36 q^{81} - 12 q^{91}+ \cdots + 44 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 0 −1.73104 − 0.0591464i 0 1.11364 − 1.92889i 0 −2.58429 + 0.566975i 0 2.99300 + 0.204770i 0
41.2 0 −1.69695 − 0.346907i 0 −1.79123 + 3.10250i 0 1.83349 − 1.90743i 0 2.75931 + 1.17737i 0
41.3 0 −1.53463 + 0.803066i 0 0.0977451 − 0.169300i 0 −2.48762 − 0.900962i 0 1.71017 − 2.46482i 0
41.4 0 −1.50601 − 0.855530i 0 0.422480 − 0.731757i 0 0.327684 + 2.62538i 0 1.53614 + 2.57688i 0
41.5 0 −1.31185 − 1.13095i 0 0.965651 − 1.67256i 0 2.53170 − 0.768427i 0 0.441899 + 2.96728i 0
41.6 0 −1.29861 + 1.14613i 0 1.79302 − 3.10561i 0 2.25543 − 1.38312i 0 0.372767 − 2.97675i 0
41.7 0 −1.27553 + 1.17176i 0 −0.00869840 + 0.0150661i 0 2.50492 + 0.851694i 0 0.253935 − 2.98923i 0
41.8 0 −1.01807 − 1.40126i 0 −1.60290 + 2.77631i 0 −1.96276 − 1.77414i 0 −0.927062 + 2.85317i 0
41.9 0 −0.703080 + 1.58293i 0 −1.16173 + 2.01217i 0 −1.25073 − 2.33145i 0 −2.01136 − 2.22586i 0
41.10 0 −0.610344 + 1.62095i 0 −1.91834 + 3.32266i 0 −0.283334 + 2.63054i 0 −2.25496 − 1.97867i 0
41.11 0 −0.233051 − 1.71630i 0 −0.0868503 + 0.150429i 0 −2.60056 + 0.486915i 0 −2.89137 + 0.799970i 0
41.12 0 −0.0936255 − 1.72952i 0 −1.26858 + 2.19724i 0 1.98582 + 1.74829i 0 −2.98247 + 0.323854i 0
41.13 0 0.0936255 + 1.72952i 0 1.26858 − 2.19724i 0 0.521158 + 2.59391i 0 −2.98247 + 0.323854i 0
41.14 0 0.233051 + 1.71630i 0 0.0868503 − 0.150429i 0 1.72196 − 2.00869i 0 −2.89137 + 0.799970i 0
41.15 0 0.610344 − 1.62095i 0 1.91834 − 3.32266i 0 2.41978 + 1.06989i 0 −2.25496 − 1.97867i 0
41.16 0 0.703080 − 1.58293i 0 1.16173 − 2.01217i 0 −1.39373 − 2.24889i 0 −2.01136 − 2.22586i 0
41.17 0 1.01807 + 1.40126i 0 1.60290 − 2.77631i 0 −0.555071 − 2.58687i 0 −0.927062 + 2.85317i 0
41.18 0 1.27553 − 1.17176i 0 0.00869840 − 0.0150661i 0 −0.514871 + 2.59517i 0 0.253935 − 2.98923i 0
41.19 0 1.29861 − 1.14613i 0 −1.79302 + 3.10561i 0 −2.32553 + 1.26170i 0 0.372767 − 2.97675i 0
41.20 0 1.31185 + 1.13095i 0 −0.965651 + 1.67256i 0 −1.93133 + 1.80831i 0 0.441899 + 2.96728i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.bu.a ✓ 48
3.b odd 2 1 1512.2.bu.a 48
4.b odd 2 1 1008.2.cc.d 48
7.b odd 2 1 inner 504.2.bu.a ✓ 48
9.c even 3 1 1512.2.bu.a 48
9.c even 3 1 4536.2.k.a 48
9.d odd 6 1 inner 504.2.bu.a ✓ 48
9.d odd 6 1 4536.2.k.a 48
12.b even 2 1 3024.2.cc.d 48
21.c even 2 1 1512.2.bu.a 48
28.d even 2 1 1008.2.cc.d 48
36.f odd 6 1 3024.2.cc.d 48
36.h even 6 1 1008.2.cc.d 48
63.l odd 6 1 1512.2.bu.a 48
63.l odd 6 1 4536.2.k.a 48
63.o even 6 1 inner 504.2.bu.a ✓ 48
63.o even 6 1 4536.2.k.a 48
84.h odd 2 1 3024.2.cc.d 48
252.s odd 6 1 1008.2.cc.d 48
252.bi even 6 1 3024.2.cc.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bu.a ✓ 48 1.a even 1 1 trivial
504.2.bu.a ✓ 48 7.b odd 2 1 inner
504.2.bu.a ✓ 48 9.d odd 6 1 inner
504.2.bu.a ✓ 48 63.o even 6 1 inner
1008.2.cc.d 48 4.b odd 2 1
1008.2.cc.d 48 28.d even 2 1
1008.2.cc.d 48 36.h even 6 1
1008.2.cc.d 48 252.s odd 6 1
1512.2.bu.a 48 3.b odd 2 1
1512.2.bu.a 48 9.c even 3 1
1512.2.bu.a 48 21.c even 2 1
1512.2.bu.a 48 63.l odd 6 1
3024.2.cc.d 48 12.b even 2 1
3024.2.cc.d 48 36.f odd 6 1
3024.2.cc.d 48 84.h odd 2 1
3024.2.cc.d 48 252.bi even 6 1
4536.2.k.a 48 9.c even 3 1
4536.2.k.a 48 9.d odd 6 1
4536.2.k.a 48 63.l odd 6 1
4536.2.k.a 48 63.o even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(504, [\chi])\).