# Properties

 Label 504.4.bl.a Level $504$ Weight $4$ Character orbit 504.bl Analytic conductor $29.737$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 504.bl (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.7369626429$$ 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 dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 24q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 24q^{7} + 540q^{19} - 924q^{25} + 648q^{31} - 132q^{37} - 792q^{43} + 672q^{49} - 12q^{67} + 2412q^{73} + 1680q^{79} + 480q^{85} + 1404q^{91} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 0 0 −10.9253 18.9231i 0 −12.1367 13.9893i 0 0 0
17.2 0 0 0 −9.74197 16.8736i 0 −18.4446 + 1.67197i 0 0 0
17.3 0 0 0 −8.48442 14.6954i 0 3.52711 18.1813i 0 0 0
17.4 0 0 0 −7.89184 13.6691i 0 10.6185 + 15.1739i 0 0 0
17.5 0 0 0 −7.29543 12.6361i 0 18.4933 + 0.998739i 0 0 0
17.6 0 0 0 −5.74088 9.94350i 0 17.2542 + 6.73007i 0 0 0
17.7 0 0 0 −5.00025 8.66069i 0 1.56940 18.4536i 0 0 0
17.8 0 0 0 −3.34783 5.79860i 0 −12.7404 + 13.4418i 0 0 0
17.9 0 0 0 −3.30930 5.73188i 0 −4.31556 + 18.0104i 0 0 0
17.10 0 0 0 −2.08296 3.60779i 0 −18.2790 2.97950i 0 0 0
17.11 0 0 0 −1.56246 2.70625i 0 17.1949 6.88015i 0 0 0
17.12 0 0 0 −1.35769 2.35159i 0 −8.74105 16.3277i 0 0 0
17.13 0 0 0 1.35769 + 2.35159i 0 −8.74105 16.3277i 0 0 0
17.14 0 0 0 1.56246 + 2.70625i 0 17.1949 6.88015i 0 0 0
17.15 0 0 0 2.08296 + 3.60779i 0 −18.2790 2.97950i 0 0 0
17.16 0 0 0 3.30930 + 5.73188i 0 −4.31556 + 18.0104i 0 0 0
17.17 0 0 0 3.34783 + 5.79860i 0 −12.7404 + 13.4418i 0 0 0
17.18 0 0 0 5.00025 + 8.66069i 0 1.56940 18.4536i 0 0 0
17.19 0 0 0 5.74088 + 9.94350i 0 17.2542 + 6.73007i 0 0 0
17.20 0 0 0 7.29543 + 12.6361i 0 18.4933 + 0.998739i 0 0 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.4.bl.a 48
3.b odd 2 1 inner 504.4.bl.a 48
4.b odd 2 1 1008.4.bt.d 48
7.d odd 6 1 inner 504.4.bl.a 48
12.b even 2 1 1008.4.bt.d 48
21.g even 6 1 inner 504.4.bl.a 48
28.f even 6 1 1008.4.bt.d 48
84.j odd 6 1 1008.4.bt.d 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.4.bl.a 48 1.a even 1 1 trivial
504.4.bl.a 48 3.b odd 2 1 inner
504.4.bl.a 48 7.d odd 6 1 inner
504.4.bl.a 48 21.g even 6 1 inner
1008.4.bt.d 48 4.b odd 2 1
1008.4.bt.d 48 12.b even 2 1
1008.4.bt.d 48 28.f even 6 1
1008.4.bt.d 48 84.j odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(504, [\chi])$$.