# Properties

 Label 504.1.cu.a Level $504$ Weight $1$ Character orbit 504.cu Analytic conductor $0.252$ Analytic rank $0$ Dimension $4$ Projective image $S_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 504.cu (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.251528766367$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$S_{4}$$ Projective field Galois closure of 4.2.21168.2

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{3} ) q^{5} + \beta_{2} q^{7} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{3} ) q^{5} + \beta_{2} q^{7} -\beta_{1} q^{11} - q^{13} + \beta_{2} q^{19} + ( 1 - \beta_{2} ) q^{25} + ( 1 - \beta_{2} ) q^{31} + \beta_{1} q^{35} -\beta_{2} q^{37} + \beta_{3} q^{41} - q^{43} + ( -\beta_{1} + \beta_{3} ) q^{47} + ( -1 + \beta_{2} ) q^{49} -2 q^{55} + ( -\beta_{1} + \beta_{3} ) q^{65} + ( 1 - \beta_{2} ) q^{67} -\beta_{3} q^{71} + ( -1 + \beta_{2} ) q^{73} -\beta_{3} q^{77} -\beta_{2} q^{79} + \beta_{3} q^{83} -\beta_{2} q^{91} + \beta_{1} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{7} + O(q^{10})$$ $$4q + 2q^{7} - 4q^{13} + 2q^{19} + 2q^{25} + 2q^{31} - 2q^{37} - 4q^{43} - 2q^{49} - 8q^{55} + 2q^{67} - 2q^{73} - 2q^{79} - 2q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$$.

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
233.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
0 0 0 −1.22474 0.707107i 0 0.500000 0.866025i 0 0 0
233.2 0 0 0 1.22474 + 0.707107i 0 0.500000 0.866025i 0 0 0
305.1 0 0 0 −1.22474 + 0.707107i 0 0.500000 + 0.866025i 0 0 0
305.2 0 0 0 1.22474 0.707107i 0 0.500000 + 0.866025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.1.cu.a 4
3.b odd 2 1 inner 504.1.cu.a 4
4.b odd 2 1 1008.1.dc.a 4
7.b odd 2 1 3528.1.cu.a 4
7.c even 3 1 inner 504.1.cu.a 4
7.c even 3 1 3528.1.d.a 2
7.d odd 6 1 3528.1.d.b 2
7.d odd 6 1 3528.1.cu.a 4
12.b even 2 1 1008.1.dc.a 4
21.c even 2 1 3528.1.cu.a 4
21.g even 6 1 3528.1.d.b 2
21.g even 6 1 3528.1.cu.a 4
21.h odd 6 1 inner 504.1.cu.a 4
21.h odd 6 1 3528.1.d.a 2
28.g odd 6 1 1008.1.dc.a 4
84.n even 6 1 1008.1.dc.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.cu.a 4 1.a even 1 1 trivial
504.1.cu.a 4 3.b odd 2 1 inner
504.1.cu.a 4 7.c even 3 1 inner
504.1.cu.a 4 21.h odd 6 1 inner
1008.1.dc.a 4 4.b odd 2 1
1008.1.dc.a 4 12.b even 2 1
1008.1.dc.a 4 28.g odd 6 1
1008.1.dc.a 4 84.n even 6 1
3528.1.d.a 2 7.c even 3 1
3528.1.d.a 2 21.h odd 6 1
3528.1.d.b 2 7.d odd 6 1
3528.1.d.b 2 21.g even 6 1
3528.1.cu.a 4 7.b odd 2 1
3528.1.cu.a 4 7.d odd 6 1
3528.1.cu.a 4 21.c even 2 1
3528.1.cu.a 4 21.g even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(504, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$4 - 2 T^{2} + T^{4}$$
$7$ $$( 1 - T + T^{2} )^{2}$$
$11$ $$4 - 2 T^{2} + T^{4}$$
$13$ $$( 1 + T )^{4}$$
$17$ $$T^{4}$$
$19$ $$( 1 - T + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$( 1 - T + T^{2} )^{2}$$
$37$ $$( 1 + T + T^{2} )^{2}$$
$41$ $$( 2 + T^{2} )^{2}$$
$43$ $$( 1 + T )^{4}$$
$47$ $$4 - 2 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( 1 - T + T^{2} )^{2}$$
$71$ $$( 2 + T^{2} )^{2}$$
$73$ $$( 1 + T + T^{2} )^{2}$$
$79$ $$( 1 + T + T^{2} )^{2}$$
$83$ $$( 2 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$