# Properties

 Label 504.2.s.f Level $504$ Weight $2$ Character orbit 504.s Analytic conductor $4.024$ Analytic rank $0$ Dimension $2$ 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.s (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{5} + ( -2 - \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{5} + ( -2 - \zeta_{6} ) q^{7} -5 \zeta_{6} q^{11} + 2 q^{13} -6 \zeta_{6} q^{17} + ( -2 + 2 \zeta_{6} ) q^{19} + ( 6 - 6 \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} -3 q^{29} -5 \zeta_{6} q^{31} + ( -3 + 2 \zeta_{6} ) q^{35} + ( 2 - 2 \zeta_{6} ) q^{37} + 8 q^{41} -4 q^{43} + ( -4 + 4 \zeta_{6} ) q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} -9 \zeta_{6} q^{53} -5 q^{55} + 3 \zeta_{6} q^{59} + ( 12 - 12 \zeta_{6} ) q^{61} + ( 2 - 2 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} -8 q^{71} + 14 \zeta_{6} q^{73} + ( -5 + 15 \zeta_{6} ) q^{77} + ( -1 + \zeta_{6} ) q^{79} + 17 q^{83} -6 q^{85} + ( -18 + 18 \zeta_{6} ) q^{89} + ( -4 - 2 \zeta_{6} ) q^{91} + 2 \zeta_{6} q^{95} + 3 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{5} - 5q^{7} + O(q^{10})$$ $$2q + q^{5} - 5q^{7} - 5q^{11} + 4q^{13} - 6q^{17} - 2q^{19} + 6q^{23} + 4q^{25} - 6q^{29} - 5q^{31} - 4q^{35} + 2q^{37} + 16q^{41} - 8q^{43} - 4q^{47} + 11q^{49} - 9q^{53} - 10q^{55} + 3q^{59} + 12q^{61} + 2q^{65} - 2q^{67} - 16q^{71} + 14q^{73} + 5q^{77} - q^{79} + 34q^{83} - 12q^{85} - 18q^{89} - 10q^{91} + 2q^{95} + 6q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0.500000 0.866025i 0 −2.50000 0.866025i 0 0 0
361.1 0 0 0 0.500000 + 0.866025i 0 −2.50000 + 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.s.f yes 2
3.b odd 2 1 504.2.s.b 2
4.b odd 2 1 1008.2.s.l 2
7.b odd 2 1 3528.2.s.l 2
7.c even 3 1 inner 504.2.s.f yes 2
7.c even 3 1 3528.2.a.l 1
7.d odd 6 1 3528.2.a.s 1
7.d odd 6 1 3528.2.s.l 2
12.b even 2 1 1008.2.s.h 2
21.c even 2 1 3528.2.s.r 2
21.g even 6 1 3528.2.a.h 1
21.g even 6 1 3528.2.s.r 2
21.h odd 6 1 504.2.s.b 2
21.h odd 6 1 3528.2.a.o 1
28.f even 6 1 7056.2.a.bh 1
28.g odd 6 1 1008.2.s.l 2
28.g odd 6 1 7056.2.a.r 1
84.j odd 6 1 7056.2.a.v 1
84.n even 6 1 1008.2.s.h 2
84.n even 6 1 7056.2.a.bm 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.s.b 2 3.b odd 2 1
504.2.s.b 2 21.h odd 6 1
504.2.s.f yes 2 1.a even 1 1 trivial
504.2.s.f yes 2 7.c even 3 1 inner
1008.2.s.h 2 12.b even 2 1
1008.2.s.h 2 84.n even 6 1
1008.2.s.l 2 4.b odd 2 1
1008.2.s.l 2 28.g odd 6 1
3528.2.a.h 1 21.g even 6 1
3528.2.a.l 1 7.c even 3 1
3528.2.a.o 1 21.h odd 6 1
3528.2.a.s 1 7.d odd 6 1
3528.2.s.l 2 7.b odd 2 1
3528.2.s.l 2 7.d odd 6 1
3528.2.s.r 2 21.c even 2 1
3528.2.s.r 2 21.g even 6 1
7056.2.a.r 1 28.g odd 6 1
7056.2.a.v 1 84.j odd 6 1
7056.2.a.bh 1 28.f even 6 1
7056.2.a.bm 1 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$:

 $$T_{5}^{2} - T_{5} + 1$$ $$T_{11}^{2} + 5 T_{11} + 25$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4}$$
$7$ $$1 + 5 T + 7 T^{2}$$
$11$ $$1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 2 T + 13 T^{2} )^{2}$$
$17$ $$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4}$$
$23$ $$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 3 T + 29 T^{2} )^{2}$$
$31$ $$1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4}$$
$37$ $$1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4}$$
$41$ $$( 1 - 8 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{2}$$
$47$ $$1 + 4 T - 31 T^{2} + 188 T^{3} + 2209 T^{4}$$
$53$ $$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$
$59$ $$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$
$61$ $$1 - 12 T + 83 T^{2} - 732 T^{3} + 3721 T^{4}$$
$67$ $$1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 8 T + 71 T^{2} )^{2}$$
$73$ $$1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4}$$
$79$ $$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 17 T + 83 T^{2} )^{2}$$
$89$ $$1 + 18 T + 235 T^{2} + 1602 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 3 T + 97 T^{2} )^{2}$$