# Properties

 Label 504.2.bm.a Level 504 Weight 2 Character orbit 504.bm Analytic conductor 4.024 Analytic rank 0 Dimension 8 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.bm (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: $$2^{2}$$ 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{2} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{4} + ( 1 + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{7} + ( -2 + 2 \zeta_{24}^{6} ) q^{8} +O(q^{10})$$ $$q + ( -1 - \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{2} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{4} + ( 1 + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{7} + ( -2 + 2 \zeta_{24}^{6} ) q^{8} + ( -\zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{10} + ( \zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{11} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{13} + ( -1 - 2 \zeta_{24} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{14} + ( 4 - 4 \zeta_{24}^{4} ) q^{16} + ( -3 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{17} + ( 4 + \zeta_{24}^{3} - 4 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{19} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{20} + ( -3 - 2 \zeta_{24} + 2 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{22} + ( -2 - \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{23} + ( -4 \zeta_{24} - 4 \zeta_{24}^{4} - 4 \zeta_{24}^{7} ) q^{25} + ( 4 - 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{26} + ( 2 + 4 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{28} -3 q^{29} + ( -\zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{31} + ( -4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{32} + ( -2 + 6 \zeta_{24} - 6 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{34} + ( 3 \zeta_{24} + 8 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 5 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{35} + ( 2 \zeta_{24}^{2} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{37} + ( -4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{38} + ( -2 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 8 \zeta_{24}^{5} ) q^{40} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{41} + ( 10 - \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{43} + ( 6 + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{44} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{46} + ( 4 + \zeta_{24}^{3} - 4 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{47} + ( -5 + 4 \zeta_{24} + 2 \zeta_{24}^{3} + 5 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{49} + ( 4 + 8 \zeta_{24}^{3} + 4 \zeta_{24}^{6} ) q^{50} + ( 2 \zeta_{24} + 8 \zeta_{24}^{4} + 2 \zeta_{24}^{7} ) q^{52} + ( 2 \zeta_{24} + 9 \zeta_{24}^{4} + 2 \zeta_{24}^{7} ) q^{53} + ( 7 \zeta_{24} - 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} - 7 \zeta_{24}^{6} ) q^{55} + ( -2 \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{56} + ( 3 + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} ) q^{58} + ( -5 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{6} + 5 \zeta_{24}^{7} ) q^{59} + ( -2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{61} + ( -3 + 2 \zeta_{24} - 2 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{62} -8 \zeta_{24}^{6} q^{64} + ( 8 \zeta_{24}^{2} + 9 \zeta_{24}^{3} + 9 \zeta_{24}^{5} - 9 \zeta_{24}^{7} ) q^{65} + ( -4 \zeta_{24} + 2 \zeta_{24}^{4} - 4 \zeta_{24}^{7} ) q^{67} + ( 4 - 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{68} + ( -5 - 4 \zeta_{24} - 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 8 \zeta_{24}^{6} ) q^{70} + ( 4 - 7 \zeta_{24} - 7 \zeta_{24}^{3} + 7 \zeta_{24}^{5} ) q^{71} + ( -8 \zeta_{24} - 4 \zeta_{24}^{4} - 8 \zeta_{24}^{7} ) q^{73} + ( 10 \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{74} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 8 \zeta_{24}^{6} ) q^{76} + ( 5 + 6 \zeta_{24} + 4 \zeta_{24}^{3} - \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{77} + ( -\zeta_{24}^{2} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{79} + ( -8 \zeta_{24} - 4 \zeta_{24}^{4} - 8 \zeta_{24}^{7} ) q^{80} + ( -6 + 6 \zeta_{24}^{2} + 6 \zeta_{24}^{4} - 4 \zeta_{24}^{5} ) q^{82} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{83} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 10 \zeta_{24}^{6} ) q^{85} + ( -10 - 10 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 10 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{86} + ( -6 \zeta_{24}^{2} + 6 \zeta_{24}^{4} + 6 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{88} + ( -4 - 3 \zeta_{24} - 8 \zeta_{24}^{3} + 6 \zeta_{24}^{4} + 8 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{91} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{92} + ( -4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{94} + ( -9 \zeta_{24} - 8 \zeta_{24}^{4} - 9 \zeta_{24}^{7} ) q^{95} + ( -13 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{97} + ( 5 \zeta_{24}^{2} - 8 \zeta_{24}^{3} - 5 \zeta_{24}^{4} - 5 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{2} + 4q^{5} - 16q^{8} + O(q^{10})$$ $$8q - 4q^{2} + 4q^{5} - 16q^{8} + 4q^{10} - 8q^{14} + 16q^{16} + 16q^{19} - 24q^{22} - 8q^{23} - 16q^{25} + 16q^{26} + 8q^{28} - 24q^{29} + 16q^{32} - 16q^{34} + 16q^{38} - 8q^{40} + 80q^{43} + 24q^{44} - 8q^{46} + 16q^{47} - 20q^{49} + 32q^{50} + 32q^{52} + 36q^{53} + 8q^{56} + 12q^{58} - 24q^{62} + 8q^{67} + 16q^{68} - 52q^{70} + 32q^{71} - 16q^{73} - 8q^{74} + 36q^{77} - 16q^{80} - 24q^{82} - 40q^{86} + 24q^{88} - 8q^{91} + 16q^{94} - 32q^{95} - 104q^{97} - 20q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-1 + \zeta_{24}^{2}$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i
−1.36603 + 0.366025i 0 1.73205 1.00000i −0.914214 + 1.58346i 0 −0.358719 2.62132i −2.00000 + 2.00000i 0 0.669251 2.49768i
107.2 −1.36603 + 0.366025i 0 1.73205 1.00000i 1.91421 3.31552i 0 2.09077 + 1.62132i −2.00000 + 2.00000i 0 −1.40130 + 5.22973i
107.3 0.366025 + 1.36603i 0 −1.73205 + 1.00000i −0.914214 + 1.58346i 0 0.358719 + 2.62132i −2.00000 2.00000i 0 −2.49768 0.669251i
107.4 0.366025 + 1.36603i 0 −1.73205 + 1.00000i 1.91421 3.31552i 0 −2.09077 1.62132i −2.00000 2.00000i 0 5.22973 + 1.40130i
179.1 −1.36603 0.366025i 0 1.73205 + 1.00000i −0.914214 1.58346i 0 −0.358719 + 2.62132i −2.00000 2.00000i 0 0.669251 + 2.49768i
179.2 −1.36603 0.366025i 0 1.73205 + 1.00000i 1.91421 + 3.31552i 0 2.09077 1.62132i −2.00000 2.00000i 0 −1.40130 5.22973i
179.3 0.366025 1.36603i 0 −1.73205 1.00000i −0.914214 1.58346i 0 0.358719 2.62132i −2.00000 + 2.00000i 0 −2.49768 + 0.669251i
179.4 0.366025 1.36603i 0 −1.73205 1.00000i 1.91421 + 3.31552i 0 −2.09077 + 1.62132i −2.00000 + 2.00000i 0 5.22973 1.40130i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 179.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
24.f even 2 1 inner
168.v 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.bm.a 8
3.b odd 2 1 504.2.bm.b yes 8
4.b odd 2 1 2016.2.bu.b 8
7.c even 3 1 inner 504.2.bm.a 8
8.b even 2 1 2016.2.bu.a 8
8.d odd 2 1 504.2.bm.b yes 8
12.b even 2 1 2016.2.bu.a 8
21.h odd 6 1 504.2.bm.b yes 8
24.f even 2 1 inner 504.2.bm.a 8
24.h odd 2 1 2016.2.bu.b 8
28.g odd 6 1 2016.2.bu.b 8
56.k odd 6 1 504.2.bm.b yes 8
56.p even 6 1 2016.2.bu.a 8
84.n even 6 1 2016.2.bu.a 8
168.s odd 6 1 2016.2.bu.b 8
168.v even 6 1 inner 504.2.bm.a 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 2 T_{5}^{3} + 11 T_{5}^{2} + 14 T_{5} + 49$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$
$3$ 1
$5$ $$( 1 - 2 T + T^{2} + 14 T^{3} - 36 T^{4} + 70 T^{5} + 25 T^{6} - 250 T^{7} + 625 T^{8} )^{2}$$
$7$ $$1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8}$$
$11$ $$1 + 22 T^{2} + 193 T^{4} + 1078 T^{6} + 11476 T^{8} + 130438 T^{10} + 2825713 T^{12} + 38974342 T^{14} + 214358881 T^{16}$$
$13$ $$( 1 - 16 T^{2} + 274 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$1 + 24 T^{2} + 142 T^{4} - 3456 T^{6} - 61629 T^{8} - 998784 T^{10} + 11859982 T^{12} + 579301656 T^{14} + 6975757441 T^{16}$$
$19$ $$( 1 - 8 T + 12 T^{2} - 112 T^{3} + 1127 T^{4} - 2128 T^{5} + 4332 T^{6} - 54872 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 4 T - 32 T^{2} + 8 T^{3} + 1407 T^{4} + 184 T^{5} - 16928 T^{6} + 48668 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 3 T + 29 T^{2} )^{8}$$
$31$ $$1 + 102 T^{2} + 5953 T^{4} + 257958 T^{6} + 8889636 T^{8} + 247897638 T^{10} + 5497720513 T^{12} + 90525375462 T^{14} + 852891037441 T^{16}$$
$37$ $$1 + 40 T^{2} - 738 T^{4} - 16000 T^{6} + 1401683 T^{8} - 21904000 T^{10} - 1383130818 T^{12} + 102629056360 T^{14} + 3512479453921 T^{16}$$
$41$ $$( 1 - 76 T^{2} + 3654 T^{4} - 127756 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 20 T + 184 T^{2} - 860 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$( 1 - 8 T - 44 T^{2} - 112 T^{3} + 6447 T^{4} - 5264 T^{5} - 97196 T^{6} - 830584 T^{7} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 18 T + 145 T^{2} - 1314 T^{3} + 12060 T^{4} - 69642 T^{5} + 407305 T^{6} - 2679786 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$1 + 134 T^{2} + 6705 T^{4} + 574726 T^{6} + 51962804 T^{8} + 2000621206 T^{10} + 81246905505 T^{12} + 5652191507894 T^{14} + 146830437604321 T^{16}$$
$61$ $$1 + 200 T^{2} + 22846 T^{4} + 1942400 T^{6} + 131573875 T^{8} + 7227670400 T^{10} + 316322083486 T^{12} + 10304074872200 T^{14} + 191707312997281 T^{16}$$
$67$ $$( 1 - 4 T - 90 T^{2} + 112 T^{3} + 5675 T^{4} + 7504 T^{5} - 404010 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 8 T + 60 T^{2} - 568 T^{3} + 5041 T^{4} )^{4}$$
$73$ $$( 1 + 8 T + 30 T^{2} - 896 T^{3} - 8845 T^{4} - 65408 T^{5} + 159870 T^{6} + 3112136 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$1 + 214 T^{2} + 22065 T^{4} + 2407286 T^{6} + 238397444 T^{8} + 15023871926 T^{10} + 859433537265 T^{12} + 52020715481494 T^{14} + 1517108809906561 T^{16}$$
$83$ $$( 1 - 230 T^{2} + 26803 T^{4} - 1584470 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 89 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 + 26 T + 355 T^{2} + 2522 T^{3} + 9409 T^{4} )^{4}$$