# Properties

 Label 504.2.bf.a Level 504 Weight 2 Character orbit 504.bf Analytic conductor 4.024 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{12}^{3} ) q^{2} + ( 1 - 2 \zeta_{12}^{2} ) q^{3} + 2 \zeta_{12}^{3} q^{4} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{5} + ( 1 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{6} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( -2 + 2 \zeta_{12}^{3} ) q^{8} -3 q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{12}^{3} ) q^{2} + ( 1 - 2 \zeta_{12}^{2} ) q^{3} + 2 \zeta_{12}^{3} q^{4} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{5} + ( 1 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{6} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( -2 + 2 \zeta_{12}^{3} ) q^{8} -3 q^{9} + ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{10} + 5 \zeta_{12}^{2} q^{11} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{12} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{13} + ( -1 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{14} + 3 \zeta_{12} q^{15} -4 q^{16} + ( 1 + \zeta_{12}^{2} ) q^{17} + ( -3 - 3 \zeta_{12}^{3} ) q^{18} + ( -2 + \zeta_{12}^{2} ) q^{19} + ( -2 - 2 \zeta_{12}^{2} ) q^{20} + ( 4 \zeta_{12} + \zeta_{12}^{3} ) q^{21} + ( -5 \zeta_{12} + 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{22} -7 \zeta_{12} q^{23} + ( -2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{24} + 2 \zeta_{12}^{2} q^{25} + ( 2 - \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{26} + ( -3 + 6 \zeta_{12}^{2} ) q^{27} + ( -2 - 4 \zeta_{12}^{2} ) q^{28} + 7 \zeta_{12} q^{29} + ( -3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{30} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{31} + ( -4 - 4 \zeta_{12}^{3} ) q^{32} + ( 10 - 5 \zeta_{12}^{2} ) q^{33} + ( 1 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{34} + ( 1 - 5 \zeta_{12}^{2} ) q^{35} -6 \zeta_{12}^{3} q^{36} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{37} + ( -2 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{38} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{39} + ( -2 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{40} + ( 14 - 7 \zeta_{12}^{2} ) q^{41} + ( -5 + 4 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{42} + ( -9 + 9 \zeta_{12}^{2} ) q^{43} + ( -10 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{44} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{45} + ( 7 - 7 \zeta_{12} - 7 \zeta_{12}^{2} ) q^{46} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{47} + ( -4 + 8 \zeta_{12}^{2} ) q^{48} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{50} + ( 3 - 3 \zeta_{12}^{2} ) q^{51} + ( 4 - 2 \zeta_{12}^{2} ) q^{52} + 11 \zeta_{12} q^{53} + ( -3 - 6 \zeta_{12} + 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{54} + ( -10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{55} + ( -2 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{56} + 3 \zeta_{12}^{2} q^{57} + ( -7 + 7 \zeta_{12} + 7 \zeta_{12}^{2} ) q^{58} + ( 4 - 8 \zeta_{12}^{2} ) q^{59} + ( -6 + 6 \zeta_{12}^{2} ) q^{60} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{61} + ( -4 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{62} + ( 6 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{63} -8 \zeta_{12}^{3} q^{64} + 3 q^{65} + ( 10 + 5 \zeta_{12} - 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{66} -8 q^{67} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{68} + ( -7 \zeta_{12} + 14 \zeta_{12}^{3} ) q^{69} + ( 1 + 5 \zeta_{12} - 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{70} -10 \zeta_{12}^{3} q^{71} + ( 6 - 6 \zeta_{12}^{3} ) q^{72} + ( -5 - 5 \zeta_{12}^{2} ) q^{73} + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{74} + ( 4 - 2 \zeta_{12}^{2} ) q^{75} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{76} + ( -15 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{77} + ( -3 \zeta_{12} - 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{78} -6 \zeta_{12}^{3} q^{79} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{80} + 9 q^{81} + ( 14 + 7 \zeta_{12} - 7 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{82} + ( 9 + 9 \zeta_{12}^{2} ) q^{83} + ( -10 + 8 \zeta_{12}^{2} ) q^{84} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{85} + ( -9 - 9 \zeta_{12} + 9 \zeta_{12}^{2} ) q^{86} + ( 7 \zeta_{12} - 14 \zeta_{12}^{3} ) q^{87} + ( -10 \zeta_{12} - 10 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{88} + ( 10 - 5 \zeta_{12}^{2} ) q^{89} + ( 3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{90} + ( 4 + \zeta_{12}^{2} ) q^{91} + ( 14 - 14 \zeta_{12}^{2} ) q^{92} -12 \zeta_{12}^{3} q^{93} + ( 2 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{94} -3 \zeta_{12}^{3} q^{95} + ( -4 - 8 \zeta_{12} + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{96} + ( 1 + \zeta_{12}^{2} ) q^{97} + ( 3 + 8 \zeta_{12} - 8 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{98} -15 \zeta_{12}^{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} - 8q^{8} - 12q^{9} + O(q^{10})$$ $$4q + 4q^{2} - 8q^{8} - 12q^{9} - 6q^{10} + 10q^{11} - 8q^{14} - 16q^{16} + 6q^{17} - 12q^{18} - 6q^{19} - 12q^{20} + 10q^{22} + 4q^{25} + 6q^{26} - 16q^{28} - 6q^{30} - 16q^{32} + 30q^{33} + 6q^{34} - 6q^{35} - 6q^{38} - 12q^{40} + 42q^{41} - 12q^{42} - 18q^{43} + 14q^{46} - 4q^{49} + 4q^{50} + 6q^{51} + 12q^{52} - 16q^{56} + 6q^{57} - 14q^{58} - 12q^{60} + 12q^{65} + 30q^{66} - 32q^{67} - 6q^{70} + 24q^{72} - 30q^{73} - 2q^{74} + 12q^{75} - 6q^{78} + 36q^{81} + 42q^{82} + 54q^{83} - 24q^{84} - 18q^{86} - 20q^{88} + 30q^{89} + 18q^{90} + 18q^{91} + 28q^{92} + 6q^{97} - 4q^{98} - 30q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$-1$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
115.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
1.00000 1.00000i 1.73205i 2.00000i 0.866025 1.50000i −1.73205 1.73205i 1.73205 2.00000i −2.00000 2.00000i −3.00000 −0.633975 2.36603i
115.2 1.00000 + 1.00000i 1.73205i 2.00000i −0.866025 + 1.50000i 1.73205 1.73205i −1.73205 + 2.00000i −2.00000 + 2.00000i −3.00000 −2.36603 + 0.633975i
355.1 1.00000 1.00000i 1.73205i 2.00000i −0.866025 1.50000i 1.73205 + 1.73205i −1.73205 2.00000i −2.00000 2.00000i −3.00000 −2.36603 0.633975i
355.2 1.00000 + 1.00000i 1.73205i 2.00000i 0.866025 + 1.50000i −1.73205 + 1.73205i 1.73205 + 2.00000i −2.00000 + 2.00000i −3.00000 −0.633975 + 2.36603i
 $$n$$: 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
8.d odd 2 1 inner
63.t odd 6 1 inner
504.bf even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.bf.a 4
7.d odd 6 1 504.2.cz.a yes 4
8.d odd 2 1 inner 504.2.bf.a 4
9.c even 3 1 504.2.cz.a yes 4
56.m even 6 1 504.2.cz.a yes 4
63.t odd 6 1 inner 504.2.bf.a 4
72.p odd 6 1 504.2.cz.a yes 4
504.bf even 6 1 inner 504.2.bf.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bf.a 4 1.a even 1 1 trivial
504.2.bf.a 4 8.d odd 2 1 inner
504.2.bf.a 4 63.t odd 6 1 inner
504.2.bf.a 4 504.bf even 6 1 inner
504.2.cz.a yes 4 7.d odd 6 1
504.2.cz.a yes 4 9.c even 3 1
504.2.cz.a yes 4 56.m even 6 1
504.2.cz.a yes 4 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 3 T_{5}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} )^{2}$$
$3$ $$( 1 + 3 T^{2} )^{2}$$
$5$ $$1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8}$$
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 22 T^{2} + 169 T^{4} )( 1 - T^{2} + 169 T^{4} )$$
$17$ $$( 1 - 3 T + 20 T^{2} - 51 T^{3} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 3 T^{2} - 520 T^{4} - 1587 T^{6} + 279841 T^{8}$$
$29$ $$1 + 9 T^{2} - 760 T^{4} + 7569 T^{6} + 707281 T^{8}$$
$31$ $$( 1 + 14 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 26 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} )$$
$41$ $$( 1 - 21 T + 188 T^{2} - 861 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 9 T + 38 T^{2} + 387 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 82 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$1 - 15 T^{2} - 2584 T^{4} - 42135 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 - 70 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 110 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 8 T + 67 T^{2} )^{4}$$
$71$ $$( 1 - 42 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 15 T + 148 T^{2} + 1095 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 122 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 27 T + 326 T^{2} - 2241 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 15 T + 164 T^{2} - 1335 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 3 T + 100 T^{2} - 291 T^{3} + 9409 T^{4} )^{2}$$