# Properties

 Label 504.2.s.i Level 504 Weight 2 Character orbit 504.s Analytic conductor 4.024 Analytic rank 0 Dimension 4 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.s (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( -1 - \beta_{1} + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( -1 - \beta_{1} + \beta_{3} ) q^{7} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{11} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -4 + 4 \beta_{2} ) q^{17} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{19} + 4 \beta_{2} q^{23} + ( -9 - \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{25} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + ( 1 - \beta_{2} ) q^{31} + ( 11 - 3 \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{35} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{43} -6 \beta_{2} q^{47} + ( -4 + 3 \beta_{1} - 3 \beta_{3} ) q^{49} + ( 6 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 15 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{55} + ( 2 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{59} -10 \beta_{2} q^{61} + ( -2 + 4 \beta_{1} + 14 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -3 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{67} -2 q^{71} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{73} + ( -5 + 2 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{77} + ( 2 - 4 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{79} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{83} + ( 4 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{85} + ( 2 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 3 - 4 \beta_{1} - 11 \beta_{2} + \beta_{3} ) q^{91} + ( -14 + 2 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} ) q^{95} + ( 13 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{5} - 6q^{7} + O(q^{10})$$ $$4q - q^{5} - 6q^{7} - q^{11} + 10q^{13} - 8q^{17} + 5q^{19} + 8q^{23} - 19q^{25} - 6q^{29} + 2q^{31} + 30q^{35} - 3q^{37} - 12q^{41} - 14q^{43} - 12q^{47} - 10q^{49} + 11q^{53} + 58q^{55} + 5q^{59} - 20q^{61} + 26q^{65} - 7q^{67} - 8q^{71} + q^{73} - 27q^{77} - 8q^{79} - 14q^{83} + 8q^{85} - 6q^{89} - 15q^{91} - 26q^{95} + 50q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 5$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu + 5$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{3} + 4 \beta_{1} + 5$$

## Character values

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

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −1.63746 + 1.52274i 2.13746 − 0.656712i −1.63746 − 1.52274i 2.13746 + 0.656712i
0 0 0 −2.13746 + 3.70219i 0 −1.50000 2.17945i 0 0 0
289.2 0 0 0 1.63746 2.83616i 0 −1.50000 + 2.17945i 0 0 0
361.1 0 0 0 −2.13746 3.70219i 0 −1.50000 + 2.17945i 0 0 0
361.2 0 0 0 1.63746 + 2.83616i 0 −1.50000 2.17945i 0 0 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
7.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.s.i 4
3.b odd 2 1 168.2.q.c 4
4.b odd 2 1 1008.2.s.r 4
7.b odd 2 1 3528.2.s.bk 4
7.c even 3 1 inner 504.2.s.i 4
7.c even 3 1 3528.2.a.bk 2
7.d odd 6 1 3528.2.a.bd 2
7.d odd 6 1 3528.2.s.bk 4
12.b even 2 1 336.2.q.g 4
21.c even 2 1 1176.2.q.l 4
21.g even 6 1 1176.2.a.n 2
21.g even 6 1 1176.2.q.l 4
21.h odd 6 1 168.2.q.c 4
21.h odd 6 1 1176.2.a.k 2
24.f even 2 1 1344.2.q.x 4
24.h odd 2 1 1344.2.q.w 4
28.f even 6 1 7056.2.a.ch 2
28.g odd 6 1 1008.2.s.r 4
28.g odd 6 1 7056.2.a.cu 2
84.h odd 2 1 2352.2.q.bf 4
84.j odd 6 1 2352.2.a.ba 2
84.j odd 6 1 2352.2.q.bf 4
84.n even 6 1 336.2.q.g 4
84.n even 6 1 2352.2.a.bf 2
168.s odd 6 1 1344.2.q.w 4
168.s odd 6 1 9408.2.a.ec 2
168.v even 6 1 1344.2.q.x 4
168.v even 6 1 9408.2.a.dp 2
168.ba even 6 1 9408.2.a.dj 2
168.be odd 6 1 9408.2.a.dw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 3.b odd 2 1
168.2.q.c 4 21.h odd 6 1
336.2.q.g 4 12.b even 2 1
336.2.q.g 4 84.n even 6 1
504.2.s.i 4 1.a even 1 1 trivial
504.2.s.i 4 7.c even 3 1 inner
1008.2.s.r 4 4.b odd 2 1
1008.2.s.r 4 28.g odd 6 1
1176.2.a.k 2 21.h odd 6 1
1176.2.a.n 2 21.g even 6 1
1176.2.q.l 4 21.c even 2 1
1176.2.q.l 4 21.g even 6 1
1344.2.q.w 4 24.h odd 2 1
1344.2.q.w 4 168.s odd 6 1
1344.2.q.x 4 24.f even 2 1
1344.2.q.x 4 168.v even 6 1
2352.2.a.ba 2 84.j odd 6 1
2352.2.a.bf 2 84.n even 6 1
2352.2.q.bf 4 84.h odd 2 1
2352.2.q.bf 4 84.j odd 6 1
3528.2.a.bd 2 7.d odd 6 1
3528.2.a.bk 2 7.c even 3 1
3528.2.s.bk 4 7.b odd 2 1
3528.2.s.bk 4 7.d odd 6 1
7056.2.a.ch 2 28.f even 6 1
7056.2.a.cu 2 28.g odd 6 1
9408.2.a.dj 2 168.ba even 6 1
9408.2.a.dp 2 168.v even 6 1
9408.2.a.dw 2 168.be odd 6 1
9408.2.a.ec 2 168.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$:

 $$T_{5}^{4} + T_{5}^{3} + 15 T_{5}^{2} - 14 T_{5} + 196$$ $$T_{11}^{4} + T_{11}^{3} + 15 T_{11}^{2} - 14 T_{11} + 196$$ $$T_{13}^{2} - 5 T_{13} - 8$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$( 1 + T + 5 T^{2} )^{2}( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} )$$
$7$ $$( 1 + 3 T + 7 T^{2} )^{2}$$
$11$ $$1 + T - 7 T^{2} - 14 T^{3} - 68 T^{4} - 154 T^{5} - 847 T^{6} + 1331 T^{7} + 14641 T^{8}$$
$13$ $$( 1 - 5 T + 18 T^{2} - 65 T^{3} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 - 5 T - 5 T^{2} + 40 T^{3} + 64 T^{4} + 760 T^{5} - 1805 T^{6} - 34295 T^{7} + 130321 T^{8}$$
$23$ $$( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 3 T + 46 T^{2} + 87 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 3 T - 53 T^{2} - 36 T^{3} + 2142 T^{4} - 1332 T^{5} - 72557 T^{6} + 151959 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 + 6 T + 34 T^{2} + 246 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 7 T + 84 T^{2} + 301 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$1 - 11 T - T^{2} - 176 T^{3} + 5662 T^{4} - 9328 T^{5} - 2809 T^{6} - 1637647 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 5 T - 85 T^{2} + 40 T^{3} + 7144 T^{4} + 2360 T^{5} - 295885 T^{6} - 1026895 T^{7} + 12117361 T^{8}$$
$61$ $$( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 7 T - 83 T^{2} - 14 T^{3} + 9652 T^{4} - 938 T^{5} - 372587 T^{6} + 2105341 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 2 T + 71 T^{2} )^{4}$$
$73$ $$1 - T - 131 T^{2} + 14 T^{3} + 12022 T^{4} + 1022 T^{5} - 698099 T^{6} - 389017 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 8 T - 53 T^{2} - 328 T^{3} + 2392 T^{4} - 25912 T^{5} - 330773 T^{6} + 3944312 T^{7} + 38950081 T^{8}$$
$83$ $$( 1 + 7 T + 164 T^{2} + 581 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$1 + 6 T - 94 T^{2} - 288 T^{3} + 5775 T^{4} - 25632 T^{5} - 744574 T^{6} + 4229814 T^{7} + 62742241 T^{8}$$
$97$ $$( 1 - 25 T + 336 T^{2} - 2425 T^{3} + 9409 T^{4} )^{2}$$