# Properties

 Label 504.2.cj.e Level 504 Weight 2 Character orbit 504.cj Analytic conductor 4.024 Analytic rank 0 Dimension 32 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.cj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 168) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 2q^{2} - 2q^{4} + 16q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 2q^{2} - 2q^{4} + 16q^{8} + 6q^{10} - 22q^{14} - 10q^{16} + 40q^{20} - 12q^{22} + 8q^{23} + 16q^{25} - 6q^{26} - 26q^{28} - 24q^{31} + 8q^{32} - 24q^{34} + 26q^{38} - 6q^{40} - 20q^{44} + 16q^{46} + 24q^{47} + 8q^{49} - 52q^{50} + 44q^{52} - 64q^{55} - 40q^{56} + 34q^{58} - 100q^{62} - 20q^{64} - 16q^{68} + 38q^{70} + 80q^{71} + 8q^{73} - 10q^{74} - 32q^{76} + 8q^{79} + 56q^{80} + 22q^{86} + 50q^{88} - 64q^{92} - 48q^{94} - 24q^{95} - 48q^{97} + 64q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −1.36256 0.378724i 0 1.71314 + 1.03207i 0.586448 + 0.338586i 0 2.23683 + 1.41301i −1.94338 2.05506i 0 −0.670840 0.683446i
37.2 −1.30853 + 0.536416i 0 1.42451 1.40384i 3.09843 + 1.78888i 0 0.993295 + 2.45222i −1.11098 + 2.60110i 0 −5.01398 0.678758i
37.3 −1.10325 0.884778i 0 0.434335 + 1.95227i −0.0402223 0.0232224i 0 −1.97032 + 1.76574i 1.24814 2.53814i 0 0.0238287 + 0.0612079i
37.4 −1.06853 + 0.926418i 0 0.283500 1.97980i −1.23074 0.710569i 0 1.39545 2.24783i 1.53120 + 2.37811i 0 1.97336 0.380919i
37.5 −0.902242 + 1.08902i 0 −0.371918 1.96512i −3.08781 1.78275i 0 −2.38336 + 1.14873i 2.47560 + 1.36799i 0 4.72740 1.75421i
37.6 −0.867144 1.11717i 0 −0.496121 + 1.93749i −2.93503 1.69454i 0 −1.85242 1.88906i 2.59471 1.12583i 0 0.652012 + 4.74833i
37.7 −0.491996 + 1.32587i 0 −1.51588 1.30465i 3.08781 + 1.78275i 0 −2.38336 + 1.14873i 2.47560 1.36799i 0 −3.88289 + 3.21694i
37.8 −0.268038 + 1.38858i 0 −1.85631 0.744384i 1.23074 + 0.710569i 0 1.39545 2.24783i 1.53120 2.37811i 0 −1.31657 + 1.51852i
37.9 −0.267238 1.38873i 0 −1.85717 + 0.742246i −1.56250 0.902108i 0 2.63683 0.217074i 1.52709 + 2.38076i 0 −0.835229 + 2.41097i
37.10 0.189716 + 1.40143i 0 −1.92802 + 0.531748i −3.09843 1.78888i 0 0.993295 + 2.45222i −1.11098 2.60110i 0 1.91917 4.68162i
37.11 0.446345 1.34193i 0 −1.60155 1.19793i 1.98722 + 1.14732i 0 −1.05630 2.42574i −2.32238 + 1.61448i 0 2.42662 2.15461i
37.12 0.938973 1.05751i 0 −0.236659 1.98595i −1.98722 1.14732i 0 −1.05630 2.42574i −2.32238 1.61448i 0 −3.07926 + 1.02420i
37.13 1.00926 + 0.990649i 0 0.0372299 + 1.99965i −0.586448 0.338586i 0 2.23683 + 1.41301i −1.94338 + 2.05506i 0 −0.256462 0.922687i
37.14 1.31787 + 0.513056i 0 1.47355 + 1.35228i 0.0402223 + 0.0232224i 0 −1.97032 + 1.76574i 1.24814 + 2.53814i 0 0.0410933 + 0.0512403i
37.15 1.33630 0.462932i 0 1.57139 1.23723i 1.56250 + 0.902108i 0 2.63683 0.217074i 1.52709 2.38076i 0 2.50558 + 0.482155i
37.16 1.40107 + 0.192386i 0 1.92598 + 0.539091i 2.93503 + 1.69454i 0 −1.85242 1.88906i 2.59471 + 1.12583i 0 3.78617 + 2.93882i
109.1 −1.36256 + 0.378724i 0 1.71314 1.03207i 0.586448 0.338586i 0 2.23683 1.41301i −1.94338 + 2.05506i 0 −0.670840 + 0.683446i
109.2 −1.30853 0.536416i 0 1.42451 + 1.40384i 3.09843 1.78888i 0 0.993295 2.45222i −1.11098 2.60110i 0 −5.01398 + 0.678758i
109.3 −1.10325 + 0.884778i 0 0.434335 1.95227i −0.0402223 + 0.0232224i 0 −1.97032 1.76574i 1.24814 + 2.53814i 0 0.0238287 0.0612079i
109.4 −1.06853 0.926418i 0 0.283500 + 1.97980i −1.23074 + 0.710569i 0 1.39545 + 2.24783i 1.53120 2.37811i 0 1.97336 + 0.380919i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.16 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.c even 3 1 inner
8.b even 2 1 inner
56.p 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.cj.e 32
3.b odd 2 1 168.2.bc.a 32
4.b odd 2 1 2016.2.cr.e 32
7.c even 3 1 inner 504.2.cj.e 32
8.b even 2 1 inner 504.2.cj.e 32
8.d odd 2 1 2016.2.cr.e 32
12.b even 2 1 672.2.bk.a 32
21.g even 6 1 1176.2.c.f 16
21.h odd 6 1 168.2.bc.a 32
21.h odd 6 1 1176.2.c.e 16
24.f even 2 1 672.2.bk.a 32
24.h odd 2 1 168.2.bc.a 32
28.g odd 6 1 2016.2.cr.e 32
56.k odd 6 1 2016.2.cr.e 32
56.p even 6 1 inner 504.2.cj.e 32
84.j odd 6 1 4704.2.c.f 16
84.n even 6 1 672.2.bk.a 32
84.n even 6 1 4704.2.c.e 16
168.s odd 6 1 168.2.bc.a 32
168.s odd 6 1 1176.2.c.e 16
168.v even 6 1 672.2.bk.a 32
168.v even 6 1 4704.2.c.e 16
168.ba even 6 1 1176.2.c.f 16
168.be odd 6 1 4704.2.c.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.bc.a 32 3.b odd 2 1
168.2.bc.a 32 21.h odd 6 1
168.2.bc.a 32 24.h odd 2 1
168.2.bc.a 32 168.s odd 6 1
504.2.cj.e 32 1.a even 1 1 trivial
504.2.cj.e 32 7.c even 3 1 inner
504.2.cj.e 32 8.b even 2 1 inner
504.2.cj.e 32 56.p even 6 1 inner
672.2.bk.a 32 12.b even 2 1
672.2.bk.a 32 24.f even 2 1
672.2.bk.a 32 84.n even 6 1
672.2.bk.a 32 168.v even 6 1
1176.2.c.e 16 21.h odd 6 1
1176.2.c.e 16 168.s odd 6 1
1176.2.c.f 16 21.g even 6 1
1176.2.c.f 16 168.ba even 6 1
2016.2.cr.e 32 4.b odd 2 1
2016.2.cr.e 32 8.d odd 2 1
2016.2.cr.e 32 28.g odd 6 1
2016.2.cr.e 32 56.k odd 6 1
4704.2.c.e 16 84.n even 6 1
4704.2.c.e 16 168.v even 6 1
4704.2.c.f 16 84.j odd 6 1
4704.2.c.f 16 168.be odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{32} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database