Properties

 Label 504.2.bk.c Level 504 Weight 2 Character orbit 504.bk 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.bk (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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 2q^{2} - 2q^{4} - 16q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 2q^{2} - 2q^{4} - 16q^{8} - 18q^{10} - 8q^{11} + 10q^{14} + 6q^{16} - 20q^{22} - 16q^{25} + 30q^{26} - 14q^{28} + 12q^{32} + 24q^{35} + 18q^{38} - 30q^{40} - 16q^{43} - 24q^{44} + 8q^{46} + 8q^{49} - 76q^{50} + 36q^{52} - 16q^{56} - 6q^{58} + 96q^{59} + 76q^{64} - 32q^{67} - 96q^{68} + 6q^{70} - 24q^{73} + 34q^{74} - 36q^{80} - 36q^{82} - 50q^{86} - 14q^{88} + 56q^{91} + 128q^{92} + 36q^{94} - 60q^{98} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.41405 0.0213058i 0 1.99909 + 0.0602550i 1.14053 1.97545i 0 1.95181 + 1.78618i −2.82554 0.127796i 0 −1.65485 + 2.76909i
19.2 −1.34646 0.432485i 0 1.62591 + 1.16465i 0.155280 0.268953i 0 −2.58581 0.560001i −1.68554 2.27133i 0 −0.325396 + 0.294978i
19.3 −1.34597 + 0.434022i 0 1.62325 1.16836i −1.44142 + 2.49662i 0 −2.63862 + 0.194181i −1.67775 + 2.27710i 0 0.856518 3.98597i
19.4 −1.09919 + 0.889821i 0 0.416438 1.95616i 0.225540 0.390646i 0 0.458196 + 2.60577i 1.28289 + 2.52075i 0 0.0996941 + 0.630084i
19.5 −0.647235 1.25741i 0 −1.16217 + 1.62768i 2.08776 3.61611i 0 2.39694 1.12013i 2.79887 + 0.407838i 0 −5.89822 0.284706i
19.6 −0.582416 1.28872i 0 −1.32158 + 1.50114i 0.128707 0.222928i 0 −0.623918 + 2.57113i 2.70425 + 0.828860i 0 −0.362252 0.0360307i
19.7 −0.221012 + 1.39684i 0 −1.90231 0.617436i −0.225540 + 0.390646i 0 −0.458196 2.60577i 1.28289 2.52075i 0 −0.495822 0.401380i
19.8 0.297109 + 1.38265i 0 −1.82345 + 0.821596i 1.44142 2.49662i 0 2.63862 0.194181i −1.67775 2.27710i 0 3.88021 + 1.25122i
19.9 0.321935 1.37708i 0 −1.79272 0.886663i −1.25150 + 2.16767i 0 1.36321 2.26752i −1.79815 + 2.18327i 0 2.58215 + 2.42127i
19.10 0.647418 1.25732i 0 −1.16170 1.62802i 1.61398 2.79550i 0 −1.82725 1.91341i −2.79905 + 0.406616i 0 −2.46991 3.83914i
19.11 0.725478 + 1.21395i 0 −0.947364 + 1.76139i −1.14053 + 1.97545i 0 −1.95181 1.78618i −2.82554 + 0.127796i 0 −3.22553 + 0.0485996i
19.12 0.765161 1.18934i 0 −0.829058 1.82007i −1.61398 + 2.79550i 0 1.82725 + 1.91341i −2.79905 0.406616i 0 2.08984 + 4.05858i
19.13 1.03162 0.967346i 0 0.128485 1.99587i 1.25150 2.16767i 0 −1.36321 + 2.26752i −1.79815 2.18327i 0 −0.805806 3.44685i
19.14 1.04777 + 0.949827i 0 0.195657 + 1.99041i −0.155280 + 0.268953i 0 2.58581 + 0.560001i −1.68554 + 2.27133i 0 −0.418156 + 0.134312i
19.15 1.40727 0.139971i 0 1.96082 0.393955i −0.128707 + 0.222928i 0 0.623918 2.57113i 2.70425 0.828860i 0 −0.149923 + 0.331735i
19.16 1.41257 0.0681843i 0 1.99070 0.192630i −2.08776 + 3.61611i 0 −2.39694 + 1.12013i 2.79887 0.407838i 0 −2.70255 + 5.25036i
451.1 −1.41405 + 0.0213058i 0 1.99909 0.0602550i 1.14053 + 1.97545i 0 1.95181 1.78618i −2.82554 + 0.127796i 0 −1.65485 2.76909i
451.2 −1.34646 + 0.432485i 0 1.62591 1.16465i 0.155280 + 0.268953i 0 −2.58581 + 0.560001i −1.68554 + 2.27133i 0 −0.325396 0.294978i
451.3 −1.34597 0.434022i 0 1.62325 + 1.16836i −1.44142 2.49662i 0 −2.63862 0.194181i −1.67775 2.27710i 0 0.856518 + 3.98597i
451.4 −1.09919 0.889821i 0 0.416438 + 1.95616i 0.225540 + 0.390646i 0 0.458196 2.60577i 1.28289 2.52075i 0 0.0996941 0.630084i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 451.16 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.d odd 6 1 inner
8.d odd 2 1 inner
56.m 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.bk.c 32
3.b odd 2 1 168.2.t.a 32
4.b odd 2 1 2016.2.bs.c 32
7.d odd 6 1 inner 504.2.bk.c 32
8.b even 2 1 2016.2.bs.c 32
8.d odd 2 1 inner 504.2.bk.c 32
12.b even 2 1 672.2.bb.a 32
21.g even 6 1 168.2.t.a 32
21.g even 6 1 1176.2.p.a 32
21.h odd 6 1 1176.2.p.a 32
24.f even 2 1 168.2.t.a 32
24.h odd 2 1 672.2.bb.a 32
28.f even 6 1 2016.2.bs.c 32
56.j odd 6 1 2016.2.bs.c 32
56.m even 6 1 inner 504.2.bk.c 32
84.j odd 6 1 672.2.bb.a 32
84.j odd 6 1 4704.2.p.a 32
84.n even 6 1 4704.2.p.a 32
168.s odd 6 1 4704.2.p.a 32
168.v even 6 1 1176.2.p.a 32
168.ba even 6 1 672.2.bb.a 32
168.ba even 6 1 4704.2.p.a 32
168.be odd 6 1 168.2.t.a 32
168.be odd 6 1 1176.2.p.a 32

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