# Properties

 Label 504.4.s.e Level $504$ Weight $4$ Character orbit 504.s Analytic conductor $29.737$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 504.s (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.7369626429$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 7 - 7 \zeta_{6} ) q^{5} + ( -14 - 7 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 7 - 7 \zeta_{6} ) q^{5} + ( -14 - 7 \zeta_{6} ) q^{7} + 7 \zeta_{6} q^{11} -52 q^{13} + 72 \zeta_{6} q^{17} + ( -20 + 20 \zeta_{6} ) q^{19} + ( -48 + 48 \zeta_{6} ) q^{23} + 76 \zeta_{6} q^{25} + 243 q^{29} -95 \zeta_{6} q^{31} + ( -147 + 98 \zeta_{6} ) q^{35} + ( -352 + 352 \zeta_{6} ) q^{37} + 296 q^{41} + 158 q^{43} + ( -142 + 142 \zeta_{6} ) q^{47} + ( 147 + 245 \zeta_{6} ) q^{49} -375 \zeta_{6} q^{53} + 49 q^{55} + 279 \zeta_{6} q^{59} + ( -246 + 246 \zeta_{6} ) q^{61} + ( -364 + 364 \zeta_{6} ) q^{65} + 730 \zeta_{6} q^{67} -338 q^{71} + 542 \zeta_{6} q^{73} + ( 49 - 147 \zeta_{6} ) q^{77} + ( 305 - 305 \zeta_{6} ) q^{79} -1123 q^{83} + 504 q^{85} + ( -426 + 426 \zeta_{6} ) q^{89} + ( 728 + 364 \zeta_{6} ) q^{91} + 140 \zeta_{6} q^{95} -369 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 7q^{5} - 35q^{7} + O(q^{10})$$ $$2q + 7q^{5} - 35q^{7} + 7q^{11} - 104q^{13} + 72q^{17} - 20q^{19} - 48q^{23} + 76q^{25} + 486q^{29} - 95q^{31} - 196q^{35} - 352q^{37} + 592q^{41} + 316q^{43} - 142q^{47} + 539q^{49} - 375q^{53} + 98q^{55} + 279q^{59} - 246q^{61} - 364q^{65} + 730q^{67} - 676q^{71} + 542q^{73} - 49q^{77} + 305q^{79} - 2246q^{83} + 1008q^{85} - 426q^{89} + 1820q^{91} + 140q^{95} - 738q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 3.50000 6.06218i 0 −17.5000 6.06218i 0 0 0
361.1 0 0 0 3.50000 + 6.06218i 0 −17.5000 + 6.06218i 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 504.4.s.e 2
3.b odd 2 1 168.4.q.a 2
7.c even 3 1 inner 504.4.s.e 2
12.b even 2 1 336.4.q.a 2
21.g even 6 1 1176.4.a.i 1
21.h odd 6 1 168.4.q.a 2
21.h odd 6 1 1176.4.a.f 1
84.j odd 6 1 2352.4.a.c 1
84.n even 6 1 336.4.q.a 2
84.n even 6 1 2352.4.a.bg 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.a 2 3.b odd 2 1
168.4.q.a 2 21.h odd 6 1
336.4.q.a 2 12.b even 2 1
336.4.q.a 2 84.n even 6 1
504.4.s.e 2 1.a even 1 1 trivial
504.4.s.e 2 7.c even 3 1 inner
1176.4.a.f 1 21.h odd 6 1
1176.4.a.i 1 21.g even 6 1
2352.4.a.c 1 84.j odd 6 1
2352.4.a.bg 1 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 7 T_{5} + 49$$ acting on $$S_{4}^{\mathrm{new}}(504, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$49 - 7 T + T^{2}$$
$7$ $$343 + 35 T + T^{2}$$
$11$ $$49 - 7 T + T^{2}$$
$13$ $$( 52 + T )^{2}$$
$17$ $$5184 - 72 T + T^{2}$$
$19$ $$400 + 20 T + T^{2}$$
$23$ $$2304 + 48 T + T^{2}$$
$29$ $$( -243 + T )^{2}$$
$31$ $$9025 + 95 T + T^{2}$$
$37$ $$123904 + 352 T + T^{2}$$
$41$ $$( -296 + T )^{2}$$
$43$ $$( -158 + T )^{2}$$
$47$ $$20164 + 142 T + T^{2}$$
$53$ $$140625 + 375 T + T^{2}$$
$59$ $$77841 - 279 T + T^{2}$$
$61$ $$60516 + 246 T + T^{2}$$
$67$ $$532900 - 730 T + T^{2}$$
$71$ $$( 338 + T )^{2}$$
$73$ $$293764 - 542 T + T^{2}$$
$79$ $$93025 - 305 T + T^{2}$$
$83$ $$( 1123 + T )^{2}$$
$89$ $$181476 + 426 T + T^{2}$$
$97$ $$( 369 + T )^{2}$$