# Properties

 Label 504.2.r.b Level 504 Weight 2 Character orbit 504.r Analytic conductor 4.024 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Learn more about

## 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: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{5} -\zeta_{6} q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( 2 - \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{5} -\zeta_{6} q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} + 3 \zeta_{6} q^{11} + ( 6 - 6 \zeta_{6} ) q^{13} + ( -2 + 4 \zeta_{6} ) q^{15} + 7 q^{17} + q^{19} + ( -1 - \zeta_{6} ) q^{21} + ( 4 - 4 \zeta_{6} ) q^{23} + \zeta_{6} q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + 4 \zeta_{6} q^{29} + ( -10 + 10 \zeta_{6} ) q^{31} + ( 3 + 3 \zeta_{6} ) q^{33} + 2 q^{35} -6 q^{37} + ( 6 - 12 \zeta_{6} ) q^{39} + ( -7 + 7 \zeta_{6} ) q^{41} -11 \zeta_{6} q^{43} + 6 \zeta_{6} q^{45} -8 \zeta_{6} q^{47} + ( -1 + \zeta_{6} ) q^{49} + ( 14 - 7 \zeta_{6} ) q^{51} -4 q^{53} -6 q^{55} + ( 2 - \zeta_{6} ) q^{57} + ( -9 + 9 \zeta_{6} ) q^{59} -4 \zeta_{6} q^{61} -3 q^{63} + 12 \zeta_{6} q^{65} + ( -9 + 9 \zeta_{6} ) q^{67} + ( 4 - 8 \zeta_{6} ) q^{69} + 13 q^{73} + ( 1 + \zeta_{6} ) q^{75} + ( 3 - 3 \zeta_{6} ) q^{77} -10 \zeta_{6} q^{79} -9 \zeta_{6} q^{81} + ( -14 + 14 \zeta_{6} ) q^{85} + ( 4 + 4 \zeta_{6} ) q^{87} + 10 q^{89} -6 q^{91} + ( -10 + 20 \zeta_{6} ) q^{93} + ( -2 + 2 \zeta_{6} ) q^{95} -7 \zeta_{6} q^{97} + 9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - 2q^{5} - q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} - 2q^{5} - q^{7} + 3q^{9} + 3q^{11} + 6q^{13} + 14q^{17} + 2q^{19} - 3q^{21} + 4q^{23} + q^{25} + 4q^{29} - 10q^{31} + 9q^{33} + 4q^{35} - 12q^{37} - 7q^{41} - 11q^{43} + 6q^{45} - 8q^{47} - q^{49} + 21q^{51} - 8q^{53} - 12q^{55} + 3q^{57} - 9q^{59} - 4q^{61} - 6q^{63} + 12q^{65} - 9q^{67} + 26q^{73} + 3q^{75} + 3q^{77} - 10q^{79} - 9q^{81} - 14q^{85} + 12q^{87} + 20q^{89} - 12q^{91} - 2q^{95} - 7q^{97} + 18q^{99} + O(q^{100})$$

## 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 1.50000 + 0.866025i 0 −1.00000 1.73205i 0 −0.500000 + 0.866025i 0 1.50000 + 2.59808i 0
337.1 0 1.50000 0.866025i 0 −1.00000 + 1.73205i 0 −0.500000 0.866025i 0 1.50000 2.59808i 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.b 2
3.b odd 2 1 1512.2.r.b 2
4.b odd 2 1 1008.2.r.b 2
9.c even 3 1 inner 504.2.r.b 2
9.c even 3 1 4536.2.a.h 1
9.d odd 6 1 1512.2.r.b 2
9.d odd 6 1 4536.2.a.c 1
12.b even 2 1 3024.2.r.d 2
36.f odd 6 1 1008.2.r.b 2
36.f odd 6 1 9072.2.a.s 1
36.h even 6 1 3024.2.r.d 2
36.h even 6 1 9072.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.b 2 1.a even 1 1 trivial
504.2.r.b 2 9.c even 3 1 inner
1008.2.r.b 2 4.b odd 2 1
1008.2.r.b 2 36.f odd 6 1
1512.2.r.b 2 3.b odd 2 1
1512.2.r.b 2 9.d odd 6 1
3024.2.r.d 2 12.b even 2 1
3024.2.r.d 2 36.h even 6 1
4536.2.a.c 1 9.d odd 6 1
4536.2.a.h 1 9.c even 3 1
9072.2.a.d 1 36.h even 6 1
9072.2.a.s 1 36.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2 T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 3 T + 3 T^{2}$$
$5$ $$1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4}$$
$7$ $$1 + T + T^{2}$$
$11$ $$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$13$ $$1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4}$$
$17$ $$( 1 - 7 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - T + 19 T^{2} )^{2}$$
$23$ $$1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4}$$
$29$ $$1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4}$$
$31$ $$1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 6 T + 37 T^{2} )^{2}$$
$41$ $$1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4}$$
$43$ $$1 + 11 T + 78 T^{2} + 473 T^{3} + 1849 T^{4}$$
$47$ $$1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 4 T + 53 T^{2} )^{2}$$
$59$ $$1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4}$$
$61$ $$1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4}$$
$67$ $$1 + 9 T + 14 T^{2} + 603 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 - 13 T + 73 T^{2} )^{2}$$
$79$ $$1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4}$$
$83$ $$1 - 83 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 10 T + 89 T^{2} )^{2}$$
$97$ $$1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4}$$
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