# Properties

 Label 504.2.cj.b Level 504 Weight 2 Character orbit 504.cj Analytic conductor 4.024 Analytic rank 0 Dimension 8 CM discriminant -24 Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + 2 \zeta_{24}^{4} q^{4} + ( -3 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + 2 \zeta_{24}^{4} q^{4} + ( -3 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} + ( -3 \zeta_{24} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{10} + ( -2 \zeta_{24} - 3 \zeta_{24}^{2} + 3 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{11} + ( \zeta_{24} - 4 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{14} + ( -4 + 4 \zeta_{24}^{4} ) q^{16} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{20} + ( -4 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{22} + ( 6 \zeta_{24} + 6 \zeta_{24}^{4} + 6 \zeta_{24}^{7} ) q^{25} + ( 2 - 4 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{28} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 9 \zeta_{24}^{6} ) q^{29} + ( 3 \zeta_{24} - 5 \zeta_{24}^{4} + 3 \zeta_{24}^{7} ) q^{31} + ( -4 \zeta_{24} + 4 \zeta_{24}^{7} ) q^{32} + ( 2 \zeta_{24} + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{35} + ( 4 + 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{40} + ( -6 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{44} + ( 5 + 4 \zeta_{24} + 2 \zeta_{24}^{3} - 5 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{49} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 12 \zeta_{24}^{6} ) q^{50} + ( -5 \zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} + 5 \zeta_{24}^{7} ) q^{53} + ( 13 + 9 \zeta_{24} + 9 \zeta_{24}^{3} - 9 \zeta_{24}^{5} ) q^{55} + ( 2 \zeta_{24} - 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{56} + ( -2 + 9 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 9 \zeta_{24}^{5} - 9 \zeta_{24}^{7} ) q^{58} + ( 4 \zeta_{24} - 9 \zeta_{24}^{2} + 9 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{59} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{62} -8 q^{64} + ( 4 + 4 \zeta_{24} - \zeta_{24}^{3} + 8 \zeta_{24}^{4} + \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{70} -14 \zeta_{24}^{4} q^{73} + ( 6 \zeta_{24} + 7 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 8 \zeta_{24}^{6} - 5 \zeta_{24}^{7} ) q^{77} + ( -5 - 9 \zeta_{24}^{3} + 5 \zeta_{24}^{4} + 9 \zeta_{24}^{5} + 9 \zeta_{24}^{7} ) q^{79} + ( 4 \zeta_{24} + 12 \zeta_{24}^{2} - 12 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{80} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 15 \zeta_{24}^{6} ) q^{83} + ( -6 \zeta_{24} - 8 \zeta_{24}^{4} - 6 \zeta_{24}^{7} ) q^{88} + ( -1 - 12 \zeta_{24} - 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} ) q^{97} + ( 5 \zeta_{24} + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} - 5 \zeta_{24}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{4} - 4q^{7} + O(q^{10})$$ $$8q + 8q^{4} - 4q^{7} - 8q^{10} - 16q^{16} - 32q^{22} + 24q^{25} + 8q^{28} - 20q^{31} + 16q^{40} + 20q^{49} + 104q^{55} - 8q^{58} - 64q^{64} + 64q^{70} - 56q^{73} - 20q^{79} - 32q^{88} - 8q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-1 + \zeta_{24}$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.965926 + 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i
−1.22474 0.707107i 0 1.00000 + 1.73205i −1.37333 0.792893i 0 1.62132 2.09077i 2.82843i 0 1.12132 + 1.94218i
37.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 3.82282 + 2.20711i 0 −2.62132 + 0.358719i 2.82843i 0 −3.12132 5.40629i
37.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −3.82282 2.20711i 0 −2.62132 + 0.358719i 2.82843i 0 −3.12132 5.40629i
37.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.37333 + 0.792893i 0 1.62132 2.09077i 2.82843i 0 1.12132 + 1.94218i
109.1 −1.22474 + 0.707107i 0 1.00000 1.73205i −1.37333 + 0.792893i 0 1.62132 + 2.09077i 2.82843i 0 1.12132 1.94218i
109.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 3.82282 2.20711i 0 −2.62132 0.358719i 2.82843i 0 −3.12132 + 5.40629i
109.3 1.22474 0.707107i 0 1.00000 1.73205i −3.82282 + 2.20711i 0 −2.62132 0.358719i 2.82843i 0 −3.12132 + 5.40629i
109.4 1.22474 0.707107i 0 1.00000 1.73205i 1.37333 0.792893i 0 1.62132 + 2.09077i 2.82843i 0 1.12132 1.94218i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
7.c even 3 1 inner
8.b even 2 1 inner
21.h odd 6 1 inner
56.p even 6 1 inner
168.s odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.cj.b 8
3.b odd 2 1 inner 504.2.cj.b 8
4.b odd 2 1 2016.2.cr.b 8
7.c even 3 1 inner 504.2.cj.b 8
8.b even 2 1 inner 504.2.cj.b 8
8.d odd 2 1 2016.2.cr.b 8
12.b even 2 1 2016.2.cr.b 8
21.h odd 6 1 inner 504.2.cj.b 8
24.f even 2 1 2016.2.cr.b 8
24.h odd 2 1 CM 504.2.cj.b 8
28.g odd 6 1 2016.2.cr.b 8
56.k odd 6 1 2016.2.cr.b 8
56.p even 6 1 inner 504.2.cj.b 8
84.n even 6 1 2016.2.cr.b 8
168.s odd 6 1 inner 504.2.cj.b 8
168.v even 6 1 2016.2.cr.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cj.b 8 1.a even 1 1 trivial
504.2.cj.b 8 3.b odd 2 1 inner
504.2.cj.b 8 7.c even 3 1 inner
504.2.cj.b 8 8.b even 2 1 inner
504.2.cj.b 8 21.h odd 6 1 inner
504.2.cj.b 8 24.h odd 2 1 CM
504.2.cj.b 8 56.p even 6 1 inner
504.2.cj.b 8 168.s odd 6 1 inner
2016.2.cr.b 8 4.b odd 2 1
2016.2.cr.b 8 8.d odd 2 1
2016.2.cr.b 8 12.b even 2 1
2016.2.cr.b 8 24.f even 2 1
2016.2.cr.b 8 28.g odd 6 1
2016.2.cr.b 8 56.k odd 6 1
2016.2.cr.b 8 84.n even 6 1
2016.2.cr.b 8 168.v even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 22 T_{5}^{6} + 435 T_{5}^{4} - 1078 T_{5}^{2} + 2401$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} + 4 T^{4} )^{2}$$
$3$ 
$5$ $$( 1 - 2 T^{2} + 25 T^{4} )^{2}( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} )$$
$7$ $$( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 10 T^{2} + 121 T^{4} )^{2}( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} )$$
$13$ $$( 1 - 13 T^{2} )^{8}$$
$17$ $$( 1 - 17 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 + 19 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 23 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 + 50 T^{2} + 1659 T^{4} + 42050 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 10 T + 31 T^{2} )^{4}( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 37 T^{2} + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 41 T^{2} )^{8}$$
$43$ $$( 1 - 43 T^{2} )^{8}$$
$47$ $$( 1 - 47 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 + 94 T^{2} + 2809 T^{4} )^{2}( 1 - 94 T^{2} + 6027 T^{4} - 264046 T^{6} + 7890481 T^{8} )$$
$59$ $$( 1 + 10 T^{2} + 3481 T^{4} )^{2}( 1 - 10 T^{2} - 3381 T^{4} - 34810 T^{6} + 12117361 T^{8} )$$
$61$ $$( 1 + 61 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 67 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 + 71 T^{2} )^{8}$$
$73$ $$( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} )^{4}$$
$79$ $$( 1 + 10 T + 79 T^{2} )^{4}( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 134 T^{2} + 11067 T^{4} + 923126 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 89 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} )^{4}$$