# Properties

 Label 504.2.j.a Level 504 Weight 2 Character orbit 504.j Analytic conductor 4.024 Analytic rank 0 Dimension 24 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.j (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$24$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 8q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 8q^{4} + 24q^{10} + 12q^{16} + 32q^{19} + 12q^{22} + 24q^{25} + 4q^{28} - 8q^{40} - 64q^{43} - 12q^{46} - 24q^{49} - 16q^{52} - 12q^{58} + 16q^{64} + 16q^{67} + 24q^{70} + 8q^{76} + 24q^{82} - 84q^{88} - 72q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
323.1 −1.40979 0.111820i 0 1.97499 + 0.315285i −0.472391 0 1.00000i −2.74906 0.665328i 0 0.665969 + 0.0528227i
323.2 −1.40979 + 0.111820i 0 1.97499 0.315285i −0.472391 0 1.00000i −2.74906 + 0.665328i 0 0.665969 0.0528227i
323.3 −1.25516 0.651592i 0 1.15086 + 1.63571i −1.82464 0 1.00000i −0.378695 2.80296i 0 2.29021 + 1.18892i
323.4 −1.25516 + 0.651592i 0 1.15086 1.63571i −1.82464 0 1.00000i −0.378695 + 2.80296i 0 2.29021 1.18892i
323.5 −0.937511 1.05881i 0 −0.242146 + 1.98529i −4.34020 0 1.00000i 2.32905 1.60484i 0 4.06898 + 4.59543i
323.6 −0.937511 + 1.05881i 0 −0.242146 1.98529i −4.34020 0 1.00000i 2.32905 + 1.60484i 0 4.06898 4.59543i
323.7 −0.619645 1.27124i 0 −1.23208 + 1.57543i −0.753720 0 1.00000i 2.76619 + 0.590057i 0 0.467039 + 0.958156i
323.8 −0.619645 + 1.27124i 0 −1.23208 1.57543i −0.753720 0 1.00000i 2.76619 0.590057i 0 0.467039 0.958156i
323.9 −0.386593 1.36035i 0 −1.70109 + 1.05180i 3.11999 0 1.00000i 2.08845 + 1.90746i 0 −1.20617 4.24428i
323.10 −0.386593 + 1.36035i 0 −1.70109 1.05180i 3.11999 0 1.00000i 2.08845 1.90746i 0 −1.20617 + 4.24428i
323.11 −0.157273 1.40544i 0 −1.95053 + 0.442076i 1.81873 0 1.00000i 0.928077 + 2.67183i 0 −0.286037 2.55612i
323.12 −0.157273 + 1.40544i 0 −1.95053 0.442076i 1.81873 0 1.00000i 0.928077 2.67183i 0 −0.286037 + 2.55612i
323.13 0.157273 1.40544i 0 −1.95053 0.442076i −1.81873 0 1.00000i −0.928077 + 2.67183i 0 −0.286037 + 2.55612i
323.14 0.157273 + 1.40544i 0 −1.95053 + 0.442076i −1.81873 0 1.00000i −0.928077 2.67183i 0 −0.286037 2.55612i
323.15 0.386593 1.36035i 0 −1.70109 1.05180i −3.11999 0 1.00000i −2.08845 + 1.90746i 0 −1.20617 + 4.24428i
323.16 0.386593 + 1.36035i 0 −1.70109 + 1.05180i −3.11999 0 1.00000i −2.08845 1.90746i 0 −1.20617 4.24428i
323.17 0.619645 1.27124i 0 −1.23208 1.57543i 0.753720 0 1.00000i −2.76619 + 0.590057i 0 0.467039 0.958156i
323.18 0.619645 + 1.27124i 0 −1.23208 + 1.57543i 0.753720 0 1.00000i −2.76619 0.590057i 0 0.467039 + 0.958156i
323.19 0.937511 1.05881i 0 −0.242146 1.98529i 4.34020 0 1.00000i −2.32905 1.60484i 0 4.06898 4.59543i
323.20 0.937511 + 1.05881i 0 −0.242146 + 1.98529i 4.34020 0 1.00000i −2.32905 + 1.60484i 0 4.06898 + 4.59543i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.24 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
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.j.a 24
3.b odd 2 1 inner 504.2.j.a 24
4.b odd 2 1 2016.2.j.a 24
8.b even 2 1 2016.2.j.a 24
8.d odd 2 1 inner 504.2.j.a 24
12.b even 2 1 2016.2.j.a 24
24.f even 2 1 inner 504.2.j.a 24
24.h odd 2 1 2016.2.j.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.j.a 24 1.a even 1 1 trivial
504.2.j.a 24 3.b odd 2 1 inner
504.2.j.a 24 8.d odd 2 1 inner
504.2.j.a 24 24.f even 2 1 inner
2016.2.j.a 24 4.b odd 2 1
2016.2.j.a 24 8.b even 2 1
2016.2.j.a 24 12.b even 2 1
2016.2.j.a 24 24.h odd 2 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