# Properties

 Label 504.2.bm.c Level 504 Weight 2 Character orbit 504.bm Analytic conductor 4.024 Analytic rank 0 Dimension 48 CM no Inner twists 8

# Related objects

## Newspace parameters

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 8q^{10} - 28q^{16} - 32q^{19} + 32q^{22} + 4q^{28} + 112q^{34} - 36q^{40} - 160q^{43} + 40q^{46} + 56q^{49} - 36q^{52} + 12q^{58} - 24q^{64} + 92q^{70} + 16q^{73} - 120q^{76} + 20q^{82} - 100q^{88} - 32q^{91} - 20q^{94} + 240q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −1.41266 + 0.0663565i 0 1.99119 0.187478i 1.02787 1.78033i 0 −1.24680 + 2.33356i −2.80043 + 0.396971i 0 −1.33390 + 2.58320i
107.2 −1.33462 0.467740i 0 1.56244 + 1.24851i −0.316953 + 0.548978i 0 2.06297 1.65655i −1.50129 2.39711i 0 0.679792 0.584428i
107.3 −1.31291 0.525613i 0 1.44746 + 1.38016i −1.51483 + 2.62375i 0 1.48957 + 2.18659i −1.17496 2.57283i 0 3.36791 2.64854i
107.4 −1.23551 + 0.688125i 0 1.05297 1.70037i 0.317795 0.550436i 0 −2.11900 1.58425i −0.130884 + 2.82540i 0 −0.0138692 + 0.898752i
107.5 −1.11165 0.874207i 0 0.471526 + 1.94362i 1.51483 2.62375i 0 −1.48957 2.18659i 1.17496 2.57283i 0 −3.97766 + 1.59242i
107.6 −1.07239 0.921948i 0 0.300025 + 1.97737i 0.316953 0.548978i 0 −2.06297 + 1.65655i 1.50129 2.39711i 0 −0.846025 + 0.296503i
107.7 −1.00991 + 0.989990i 0 0.0398381 1.99960i 0.635051 1.09994i 0 2.64362 + 0.106220i 1.93936 + 2.05886i 0 0.447586 + 1.73954i
107.8 −0.830529 + 1.14465i 0 −0.620442 1.90133i −1.88256 + 3.26069i 0 2.23426 1.41706i 2.69165 + 0.868922i 0 −2.16882 4.86297i
107.9 −0.648862 1.25657i 0 −1.15796 + 1.63069i −1.02787 + 1.78033i 0 1.24680 2.33356i 2.80043 + 0.396971i 0 2.90407 + 0.136412i
107.10 −0.576030 + 1.29158i 0 −1.33638 1.48798i −1.88256 + 3.26069i 0 −2.23426 + 1.41706i 2.69165 0.868922i 0 −3.12704 4.30974i
107.11 −0.352402 + 1.36960i 0 −1.75163 0.965301i 0.635051 1.09994i 0 −2.64362 0.106220i 1.93936 2.05886i 0 1.28269 + 1.25739i
107.12 −0.0218210 1.41405i 0 −1.99905 + 0.0617118i −0.317795 + 0.550436i 0 2.11900 + 1.58425i 0.130884 + 2.82540i 0 0.785276 + 0.437365i
107.13 0.0218210 + 1.41405i 0 −1.99905 + 0.0617118i 0.317795 0.550436i 0 2.11900 + 1.58425i −0.130884 2.82540i 0 0.785276 + 0.437365i
107.14 0.352402 1.36960i 0 −1.75163 0.965301i −0.635051 + 1.09994i 0 −2.64362 0.106220i −1.93936 + 2.05886i 0 1.28269 + 1.25739i
107.15 0.576030 1.29158i 0 −1.33638 1.48798i 1.88256 3.26069i 0 −2.23426 + 1.41706i −2.69165 + 0.868922i 0 −3.12704 4.30974i
107.16 0.648862 + 1.25657i 0 −1.15796 + 1.63069i 1.02787 1.78033i 0 1.24680 2.33356i −2.80043 0.396971i 0 2.90407 + 0.136412i
107.17 0.830529 1.14465i 0 −0.620442 1.90133i 1.88256 3.26069i 0 2.23426 1.41706i −2.69165 0.868922i 0 −2.16882 4.86297i
107.18 1.00991 0.989990i 0 0.0398381 1.99960i −0.635051 + 1.09994i 0 2.64362 + 0.106220i −1.93936 2.05886i 0 0.447586 + 1.73954i
107.19 1.07239 + 0.921948i 0 0.300025 + 1.97737i −0.316953 + 0.548978i 0 −2.06297 + 1.65655i −1.50129 + 2.39711i 0 −0.846025 + 0.296503i
107.20 1.11165 + 0.874207i 0 0.471526 + 1.94362i −1.51483 + 2.62375i 0 −1.48957 2.18659i −1.17496 + 2.57283i 0 −3.97766 + 1.59242i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 179.24 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.c even 3 1 inner
8.d odd 2 1 inner
21.h odd 6 1 inner
24.f even 2 1 inner
56.k odd 6 1 inner
168.v 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.bm.c 48
3.b odd 2 1 inner 504.2.bm.c 48
4.b odd 2 1 2016.2.bu.c 48
7.c even 3 1 inner 504.2.bm.c 48
8.b even 2 1 2016.2.bu.c 48
8.d odd 2 1 inner 504.2.bm.c 48
12.b even 2 1 2016.2.bu.c 48
21.h odd 6 1 inner 504.2.bm.c 48
24.f even 2 1 inner 504.2.bm.c 48
24.h odd 2 1 2016.2.bu.c 48
28.g odd 6 1 2016.2.bu.c 48
56.k odd 6 1 inner 504.2.bm.c 48
56.p even 6 1 2016.2.bu.c 48
84.n even 6 1 2016.2.bu.c 48
168.s odd 6 1 2016.2.bu.c 48
168.v even 6 1 inner 504.2.bm.c 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bm.c 48 1.a even 1 1 trivial
504.2.bm.c 48 3.b odd 2 1 inner
504.2.bm.c 48 7.c even 3 1 inner
504.2.bm.c 48 8.d odd 2 1 inner
504.2.bm.c 48 21.h odd 6 1 inner
504.2.bm.c 48 24.f even 2 1 inner
504.2.bm.c 48 56.k odd 6 1 inner
504.2.bm.c 48 168.v even 6 1 inner
2016.2.bu.c 48 4.b odd 2 1
2016.2.bu.c 48 8.b even 2 1
2016.2.bu.c 48 12.b even 2 1
2016.2.bu.c 48 24.h odd 2 1
2016.2.bu.c 48 28.g odd 6 1
2016.2.bu.c 48 56.p even 6 1
2016.2.bu.c 48 84.n even 6 1
2016.2.bu.c 48 168.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{24} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database