# Properties

 Label 504.2.q.a Level 504 Weight 2 Character orbit 504.q 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.q (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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} + ( -1 + \zeta_{6} ) q^{13} + ( -2 + \zeta_{6} ) q^{15} -3 \zeta_{6} q^{17} + ( -5 + 5 \zeta_{6} ) q^{19} + ( -1 - 4 \zeta_{6} ) q^{21} -\zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} -9 \zeta_{6} q^{29} + 4 q^{31} + ( -3 - 3 \zeta_{6} ) q^{33} + ( -2 - \zeta_{6} ) q^{35} + ( -5 + 5 \zeta_{6} ) q^{37} + ( 1 + \zeta_{6} ) q^{39} + ( -7 + 7 \zeta_{6} ) q^{41} -3 \zeta_{6} q^{43} + 3 \zeta_{6} q^{45} + 8 q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -6 + 3 \zeta_{6} ) q^{51} -9 \zeta_{6} q^{53} -3 q^{55} + ( 5 + 5 \zeta_{6} ) q^{57} -4 q^{59} + 2 q^{61} + ( -9 + 6 \zeta_{6} ) q^{63} + q^{65} + 12 q^{67} + ( -2 + \zeta_{6} ) q^{69} + 8 q^{71} + 13 \zeta_{6} q^{73} + ( -4 - 4 \zeta_{6} ) q^{75} + ( 3 - 9 \zeta_{6} ) q^{77} + 8 q^{79} + 9 q^{81} + 13 \zeta_{6} q^{83} + ( -3 + 3 \zeta_{6} ) q^{85} + ( -18 + 9 \zeta_{6} ) q^{87} + ( 9 - 9 \zeta_{6} ) q^{89} + ( -1 + 3 \zeta_{6} ) q^{91} + ( 4 - 8 \zeta_{6} ) q^{93} + 5 q^{95} + 17 \zeta_{6} q^{97} + ( -9 + 9 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{5} + 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q - q^{5} + 4q^{7} - 6q^{9} + 3q^{11} - q^{13} - 3q^{15} - 3q^{17} - 5q^{19} - 6q^{21} - q^{23} + 4q^{25} - 9q^{29} + 8q^{31} - 9q^{33} - 5q^{35} - 5q^{37} + 3q^{39} - 7q^{41} - 3q^{43} + 3q^{45} + 16q^{47} + 2q^{49} - 9q^{51} - 9q^{53} - 6q^{55} + 15q^{57} - 8q^{59} + 4q^{61} - 12q^{63} + 2q^{65} + 24q^{67} - 3q^{69} + 16q^{71} + 13q^{73} - 12q^{75} - 3q^{77} + 16q^{79} + 18q^{81} + 13q^{83} - 3q^{85} - 27q^{87} + 9q^{89} + q^{91} + 10q^{95} + 17q^{97} - 9q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 0 −0.500000 + 0.866025i 0 2.00000 + 1.73205i 0 −3.00000 0
121.1 0 1.73205i 0 −0.500000 0.866025i 0 2.00000 1.73205i 0 −3.00000 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
63.h 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.q.a 2
3.b odd 2 1 1512.2.q.b 2
4.b odd 2 1 1008.2.q.b 2
7.c even 3 1 504.2.t.a yes 2
9.c even 3 1 504.2.t.a yes 2
9.d odd 6 1 1512.2.t.a 2
12.b even 2 1 3024.2.q.d 2
21.h odd 6 1 1512.2.t.a 2
28.g odd 6 1 1008.2.t.e 2
36.f odd 6 1 1008.2.t.e 2
36.h even 6 1 3024.2.t.c 2
63.h even 3 1 inner 504.2.q.a 2
63.j odd 6 1 1512.2.q.b 2
84.n even 6 1 3024.2.t.c 2
252.u odd 6 1 1008.2.q.b 2
252.bb even 6 1 3024.2.q.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.a 2 1.a even 1 1 trivial
504.2.q.a 2 63.h even 3 1 inner
504.2.t.a yes 2 7.c even 3 1
504.2.t.a yes 2 9.c even 3 1
1008.2.q.b 2 4.b odd 2 1
1008.2.q.b 2 252.u odd 6 1
1008.2.t.e 2 28.g odd 6 1
1008.2.t.e 2 36.f odd 6 1
1512.2.q.b 2 3.b odd 2 1
1512.2.q.b 2 63.j odd 6 1
1512.2.t.a 2 9.d odd 6 1
1512.2.t.a 2 21.h odd 6 1
3024.2.q.d 2 12.b even 2 1
3024.2.q.d 2 252.bb even 6 1
3024.2.t.c 2 36.h even 6 1
3024.2.t.c 2 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T^{2}$$
$5$ $$1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$13$ $$1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4}$$
$17$ $$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$
$19$ $$1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4}$$
$23$ $$1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4}$$
$29$ $$1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}$$
$37$ $$1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4}$$
$41$ $$1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4}$$
$43$ $$1 + 3 T - 34 T^{2} + 129 T^{3} + 1849 T^{4}$$
$47$ $$( 1 - 8 T + 47 T^{2} )^{2}$$
$53$ $$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$
$59$ $$( 1 + 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 2 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 12 T + 67 T^{2} )^{2}$$
$71$ $$( 1 - 8 T + 71 T^{2} )^{2}$$
$73$ $$1 - 13 T + 96 T^{2} - 949 T^{3} + 5329 T^{4}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}$$
$83$ $$1 - 13 T + 86 T^{2} - 1079 T^{3} + 6889 T^{4}$$
$89$ $$1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4}$$
$97$ $$1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4}$$