# Properties

 Label 504.2.r.a Level $504$ Weight $2$ Character orbit 504.r 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.r (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.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} -\zeta_{6} q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} -\zeta_{6} q^{7} -3 q^{9} + 6 \zeta_{6} q^{11} + ( -6 + 6 \zeta_{6} ) q^{13} + ( 1 + \zeta_{6} ) q^{15} -2 q^{17} + 7 q^{19} + ( 2 - \zeta_{6} ) q^{21} + ( 1 - \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} -2 \zeta_{6} q^{29} + ( -10 + 10 \zeta_{6} ) q^{31} + ( -12 + 6 \zeta_{6} ) q^{33} - q^{35} -6 q^{37} + ( -6 - 6 \zeta_{6} ) q^{39} + ( 8 - 8 \zeta_{6} ) q^{41} + 10 \zeta_{6} q^{43} + ( -3 + 3 \zeta_{6} ) q^{45} -8 \zeta_{6} q^{47} + ( -1 + \zeta_{6} ) q^{49} + ( 2 - 4 \zeta_{6} ) q^{51} + 2 q^{53} + 6 q^{55} + ( -7 + 14 \zeta_{6} ) q^{57} -7 \zeta_{6} q^{61} + 3 \zeta_{6} q^{63} + 6 \zeta_{6} q^{65} + ( 12 - 12 \zeta_{6} ) q^{67} + ( 1 + \zeta_{6} ) q^{69} + 15 q^{71} -2 q^{73} + ( -8 + 4 \zeta_{6} ) q^{75} + ( 6 - 6 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + 9 q^{81} -12 \zeta_{6} q^{83} + ( -2 + 2 \zeta_{6} ) q^{85} + ( 4 - 2 \zeta_{6} ) q^{87} + 4 q^{89} + 6 q^{91} + ( -10 - 10 \zeta_{6} ) q^{93} + ( 7 - 7 \zeta_{6} ) q^{95} + 2 \zeta_{6} q^{97} -18 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{5} - q^{7} - 6q^{9} + O(q^{10})$$ $$2q + q^{5} - q^{7} - 6q^{9} + 6q^{11} - 6q^{13} + 3q^{15} - 4q^{17} + 14q^{19} + 3q^{21} + q^{23} + 4q^{25} - 2q^{29} - 10q^{31} - 18q^{33} - 2q^{35} - 12q^{37} - 18q^{39} + 8q^{41} + 10q^{43} - 3q^{45} - 8q^{47} - q^{49} + 4q^{53} + 12q^{55} - 7q^{61} + 3q^{63} + 6q^{65} + 12q^{67} + 3q^{69} + 30q^{71} - 4q^{73} - 12q^{75} + 6q^{77} - q^{79} + 18q^{81} - 12q^{83} - 2q^{85} + 6q^{87} + 8q^{89} + 12q^{91} - 30q^{93} + 7q^{95} + 2q^{97} - 18q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 −3.00000 0
337.1 0 1.73205i 0 0.500000 0.866025i 0 −0.500000 0.866025i 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
9.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.r.a 2
3.b odd 2 1 1512.2.r.a 2
4.b odd 2 1 1008.2.r.c 2
9.c even 3 1 inner 504.2.r.a 2
9.c even 3 1 4536.2.a.d 1
9.d odd 6 1 1512.2.r.a 2
9.d odd 6 1 4536.2.a.g 1
12.b even 2 1 3024.2.r.b 2
36.f odd 6 1 1008.2.r.c 2
36.f odd 6 1 9072.2.a.i 1
36.h even 6 1 3024.2.r.b 2
36.h even 6 1 9072.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.a 2 1.a even 1 1 trivial
504.2.r.a 2 9.c even 3 1 inner
1008.2.r.c 2 4.b odd 2 1
1008.2.r.c 2 36.f odd 6 1
1512.2.r.a 2 3.b odd 2 1
1512.2.r.a 2 9.d odd 6 1
3024.2.r.b 2 12.b even 2 1
3024.2.r.b 2 36.h even 6 1
4536.2.a.d 1 9.c even 3 1
4536.2.a.g 1 9.d odd 6 1
9072.2.a.i 1 36.f odd 6 1
9072.2.a.n 1 36.h 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
$3$ $$1 + 3 T^{2}$$
$5$ $$1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4}$$
$7$ $$1 + T + T^{2}$$
$11$ $$1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4}$$
$13$ $$1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 2 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 7 T + 19 T^{2} )^{2}$$
$23$ $$1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4}$$
$29$ $$1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4}$$
$31$ $$1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 6 T + 37 T^{2} )^{2}$$
$41$ $$1 - 8 T + 23 T^{2} - 328 T^{3} + 1681 T^{4}$$
$43$ $$1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4}$$
$47$ $$1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 2 T + 53 T^{2} )^{2}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4}$$
$67$ $$1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 15 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 2 T + 73 T^{2} )^{2}$$
$79$ $$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$
$83$ $$1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 4 T + 89 T^{2} )^{2}$$
$97$ $$1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4}$$