# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.bs (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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 4q^{9} + 8q^{15} + 8q^{21} - 12q^{23} - 24q^{25} - 18q^{27} + 18q^{29} - 10q^{39} + 6q^{41} - 6q^{43} + 6q^{45} + 36q^{47} + 6q^{49} - 12q^{51} + 12q^{53} + 4q^{57} + 46q^{63} - 54q^{75} - 36q^{77} - 12q^{79} - 24q^{87} + 18q^{89} + 6q^{91} + 16q^{93} - 64q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1 0 −1.70597 + 0.299447i 0 1.76701 + 3.06054i 0 1.34480 + 2.27849i 0 2.82066 1.02170i 0
257.2 0 −1.69760 + 0.343722i 0 −0.0262870 0.0455305i 0 −2.10354 + 1.60471i 0 2.76371 1.16701i 0
257.3 0 −1.68581 0.397560i 0 −0.311798 0.540051i 0 2.62086 + 0.362103i 0 2.68389 + 1.34042i 0
257.4 0 −1.54049 + 0.791772i 0 −0.905867 1.56901i 0 −2.60735 + 0.449161i 0 1.74619 2.43943i 0
257.5 0 −1.41991 0.991901i 0 −1.11047 1.92339i 0 0.362456 2.62081i 0 1.03226 + 2.81681i 0
257.6 0 −1.32026 1.12112i 0 1.38468 + 2.39834i 0 −1.42373 2.23002i 0 0.486198 + 2.96034i 0
257.7 0 −1.14780 1.29713i 0 −0.527910 0.914367i 0 0.781227 + 2.52778i 0 −0.365108 + 2.97770i 0
257.8 0 −1.09564 + 1.34148i 0 −0.271038 0.469451i 0 1.60655 2.10214i 0 −0.599164 2.93956i 0
257.9 0 −0.559017 + 1.63936i 0 −1.82284 3.15725i 0 1.57353 + 2.12697i 0 −2.37500 1.83286i 0
257.10 0 −0.445548 + 1.67376i 0 −0.101589 0.175958i 0 −1.37377 2.26114i 0 −2.60297 1.49148i 0
257.11 0 −0.419673 1.68044i 0 1.02449 + 1.77447i 0 −2.64365 + 0.105420i 0 −2.64775 + 1.41047i 0
257.12 0 0.0250939 1.73187i 0 −1.42382 2.46612i 0 1.38343 2.25524i 0 −2.99874 0.0869188i 0
257.13 0 0.0709339 + 1.73060i 0 2.04942 + 3.54970i 0 −2.21443 1.44785i 0 −2.98994 + 0.245516i 0
257.14 0 0.589575 1.62862i 0 1.11415 + 1.92977i 0 −0.553477 + 2.58721i 0 −2.30480 1.92039i 0
257.15 0 0.690387 + 1.58851i 0 1.10442 + 1.91291i 0 0.234181 + 2.63537i 0 −2.04673 + 2.19337i 0
257.16 0 0.859678 1.50365i 0 1.29668 + 2.24592i 0 1.66807 2.05367i 0 −1.52191 2.58530i 0
257.17 0 0.872943 + 1.49598i 0 −2.09905 3.63567i 0 −2.61186 + 0.422135i 0 −1.47594 + 2.61182i 0
257.18 0 0.957290 1.44347i 0 −1.80389 3.12442i 0 2.05506 + 1.66636i 0 −1.16719 2.76363i 0
257.19 0 1.10617 + 1.33281i 0 −0.103514 0.179292i 0 2.64517 0.0556151i 0 −0.552770 + 2.94863i 0
257.20 0 1.29769 1.14717i 0 −0.643917 1.11530i 0 −2.63392 0.249885i 0 0.368015 2.97734i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i 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.bs.a 48
3.b odd 2 1 1512.2.bs.a 48
4.b odd 2 1 1008.2.ca.e 48
7.d odd 6 1 504.2.cx.a yes 48
9.c even 3 1 1512.2.cx.a 48
9.d odd 6 1 504.2.cx.a yes 48
12.b even 2 1 3024.2.ca.e 48
21.g even 6 1 1512.2.cx.a 48
28.f even 6 1 1008.2.df.e 48
36.f odd 6 1 3024.2.df.e 48
36.h even 6 1 1008.2.df.e 48
63.i even 6 1 inner 504.2.bs.a 48
63.t odd 6 1 1512.2.bs.a 48
84.j odd 6 1 3024.2.df.e 48
252.r odd 6 1 1008.2.ca.e 48
252.bj even 6 1 3024.2.ca.e 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bs.a 48 1.a even 1 1 trivial
504.2.bs.a 48 63.i even 6 1 inner
504.2.cx.a yes 48 7.d odd 6 1
504.2.cx.a yes 48 9.d odd 6 1
1008.2.ca.e 48 4.b odd 2 1
1008.2.ca.e 48 252.r odd 6 1
1008.2.df.e 48 28.f even 6 1
1008.2.df.e 48 36.h even 6 1
1512.2.bs.a 48 3.b odd 2 1
1512.2.bs.a 48 63.t odd 6 1
1512.2.cx.a 48 9.c even 3 1
1512.2.cx.a 48 21.g even 6 1
3024.2.ca.e 48 12.b even 2 1
3024.2.ca.e 48 252.bj even 6 1
3024.2.df.e 48 36.f odd 6 1
3024.2.df.e 48 84.j odd 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