Properties

 Label 504.2.s.a Level $504$ Weight $2$ Character orbit 504.s Analytic conductor $4.024$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 504.s (of order $$3$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -4 + 4 \zeta_{6} ) q^{5} + ( 3 - \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -4 + 4 \zeta_{6} ) q^{5} + ( 3 - \zeta_{6} ) q^{7} -3 q^{13} + 4 \zeta_{6} q^{17} + ( -7 + 7 \zeta_{6} ) q^{19} + ( -4 + 4 \zeta_{6} ) q^{23} -11 \zeta_{6} q^{25} -8 q^{29} + 5 \zeta_{6} q^{31} + ( -8 + 12 \zeta_{6} ) q^{35} + ( -3 + 3 \zeta_{6} ) q^{37} + 8 q^{41} + 11 q^{43} + ( -4 + 4 \zeta_{6} ) q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{53} -12 \zeta_{6} q^{59} + ( 2 - 2 \zeta_{6} ) q^{61} + ( 12 - 12 \zeta_{6} ) q^{65} + 3 \zeta_{6} q^{67} + 12 q^{71} -\zeta_{6} q^{73} + ( -1 + \zeta_{6} ) q^{79} + 12 q^{83} -16 q^{85} + ( -8 + 8 \zeta_{6} ) q^{89} + ( -9 + 3 \zeta_{6} ) q^{91} -28 \zeta_{6} q^{95} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} + 5q^{7} + O(q^{10})$$ $$2q - 4q^{5} + 5q^{7} - 6q^{13} + 4q^{17} - 7q^{19} - 4q^{23} - 11q^{25} - 16q^{29} + 5q^{31} - 4q^{35} - 3q^{37} + 16q^{41} + 22q^{43} - 4q^{47} + 11q^{49} - 4q^{53} - 12q^{59} + 2q^{61} + 12q^{65} + 3q^{67} + 24q^{71} - q^{73} - q^{79} + 24q^{83} - 32q^{85} - 8q^{89} - 15q^{91} - 28q^{95} - 4q^{97} + O(q^{100})$$

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 −2.00000 + 3.46410i 0 2.50000 0.866025i 0 0 0
361.1 0 0 0 −2.00000 3.46410i 0 2.50000 + 0.866025i 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.a 2
3.b odd 2 1 504.2.s.h yes 2
4.b odd 2 1 1008.2.s.a 2
7.b odd 2 1 3528.2.s.bb 2
7.c even 3 1 inner 504.2.s.a 2
7.c even 3 1 3528.2.a.z 1
7.d odd 6 1 3528.2.a.c 1
7.d odd 6 1 3528.2.s.bb 2
12.b even 2 1 1008.2.s.q 2
21.c even 2 1 3528.2.s.b 2
21.g even 6 1 3528.2.a.ba 1
21.g even 6 1 3528.2.s.b 2
21.h odd 6 1 504.2.s.h yes 2
21.h odd 6 1 3528.2.a.a 1
28.f even 6 1 7056.2.a.d 1
28.g odd 6 1 1008.2.s.a 2
28.g odd 6 1 7056.2.a.cb 1
84.j odd 6 1 7056.2.a.cc 1
84.n even 6 1 1008.2.s.q 2
84.n even 6 1 7056.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.s.a 2 1.a even 1 1 trivial
504.2.s.a 2 7.c even 3 1 inner
504.2.s.h yes 2 3.b odd 2 1
504.2.s.h yes 2 21.h odd 6 1
1008.2.s.a 2 4.b odd 2 1
1008.2.s.a 2 28.g odd 6 1
1008.2.s.q 2 12.b even 2 1
1008.2.s.q 2 84.n even 6 1
3528.2.a.a 1 21.h odd 6 1
3528.2.a.c 1 7.d odd 6 1
3528.2.a.z 1 7.c even 3 1
3528.2.a.ba 1 21.g even 6 1
3528.2.s.b 2 21.c even 2 1
3528.2.s.b 2 21.g even 6 1
3528.2.s.bb 2 7.b odd 2 1
3528.2.s.bb 2 7.d odd 6 1
7056.2.a.b 1 84.n even 6 1
7056.2.a.d 1 28.f even 6 1
7056.2.a.cb 1 28.g odd 6 1
7056.2.a.cc 1 84.j odd 6 1

Hecke kernels

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

 $$T_{5}^{2} + 4 T_{5} + 16$$ $$T_{11}$$ $$T_{13} + 3$$