# Properties

 Label 504.2.bu.a Level 504 Weight 2 Character orbit 504.bu Analytic conductor 4.024 Analytic rank 0 Dimension 48 CM no Inner twists 4

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 4q^{9} + 8q^{15} - 4q^{21} + 12q^{23} - 24q^{25} - 36q^{29} + 32q^{39} + 12q^{43} + 6q^{49} + 24q^{51} + 28q^{57} - 14q^{63} + 36q^{65} - 60q^{77} - 12q^{79} - 36q^{81} - 12q^{91} + 16q^{93} - 108q^{95} + 44q^{99} + O(q^{100})$$

## 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.

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 209.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database