# Properties

 Label 504.2.a.h Level $504$ Weight $2$ Character orbit 504.a Self dual yes Analytic conductor $4.024$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,2,Mod(1,504)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("504.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 504.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 4 q^{5} + q^{7}+O(q^{10})$$ q + 4 * q^5 + q^7 $$q + 4 q^{5} + q^{7} + 2 q^{17} - 2 q^{19} - 8 q^{23} + 11 q^{25} - 2 q^{29} + 4 q^{31} + 4 q^{35} - 6 q^{37} + 2 q^{41} + 8 q^{43} + 4 q^{47} + q^{49} + 10 q^{53} - 6 q^{59} + 4 q^{61} - 12 q^{67} - 14 q^{73} - 8 q^{79} - 6 q^{83} + 8 q^{85} - 10 q^{89} - 8 q^{95} - 2 q^{97}+O(q^{100})$$ q + 4 * q^5 + q^7 + 2 * q^17 - 2 * q^19 - 8 * q^23 + 11 * q^25 - 2 * q^29 + 4 * q^31 + 4 * q^35 - 6 * q^37 + 2 * q^41 + 8 * q^43 + 4 * q^47 + q^49 + 10 * q^53 - 6 * q^59 + 4 * q^61 - 12 * q^67 - 14 * q^73 - 8 * q^79 - 6 * q^83 + 8 * q^85 - 10 * q^89 - 8 * q^95 - 2 * q^97

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 4.00000 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.a.h 1
3.b odd 2 1 56.2.a.b 1
4.b odd 2 1 1008.2.a.m 1
7.b odd 2 1 3528.2.a.b 1
7.c even 3 2 3528.2.s.a 2
7.d odd 6 2 3528.2.s.ba 2
8.b even 2 1 4032.2.a.d 1
8.d odd 2 1 4032.2.a.a 1
12.b even 2 1 112.2.a.a 1
15.d odd 2 1 1400.2.a.a 1
15.e even 4 2 1400.2.g.b 2
21.c even 2 1 392.2.a.b 1
21.g even 6 2 392.2.i.e 2
21.h odd 6 2 392.2.i.a 2
24.f even 2 1 448.2.a.h 1
24.h odd 2 1 448.2.a.c 1
28.d even 2 1 7056.2.a.c 1
33.d even 2 1 6776.2.a.h 1
39.d odd 2 1 9464.2.a.h 1
48.i odd 4 2 1792.2.b.a 2
48.k even 4 2 1792.2.b.h 2
60.h even 2 1 2800.2.a.bd 1
60.l odd 4 2 2800.2.g.g 2
84.h odd 2 1 784.2.a.i 1
84.j odd 6 2 784.2.i.b 2
84.n even 6 2 784.2.i.j 2
105.g even 2 1 9800.2.a.bj 1
168.e odd 2 1 3136.2.a.c 1
168.i even 2 1 3136.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 3.b odd 2 1
112.2.a.a 1 12.b even 2 1
392.2.a.b 1 21.c even 2 1
392.2.i.a 2 21.h odd 6 2
392.2.i.e 2 21.g even 6 2
448.2.a.c 1 24.h odd 2 1
448.2.a.h 1 24.f even 2 1
504.2.a.h 1 1.a even 1 1 trivial
784.2.a.i 1 84.h odd 2 1
784.2.i.b 2 84.j odd 6 2
784.2.i.j 2 84.n even 6 2
1008.2.a.m 1 4.b odd 2 1
1400.2.a.a 1 15.d odd 2 1
1400.2.g.b 2 15.e even 4 2
1792.2.b.a 2 48.i odd 4 2
1792.2.b.h 2 48.k even 4 2
2800.2.a.bd 1 60.h even 2 1
2800.2.g.g 2 60.l odd 4 2
3136.2.a.c 1 168.e odd 2 1
3136.2.a.w 1 168.i even 2 1
3528.2.a.b 1 7.b odd 2 1
3528.2.s.a 2 7.c even 3 2
3528.2.s.ba 2 7.d odd 6 2
4032.2.a.a 1 8.d odd 2 1
4032.2.a.d 1 8.b even 2 1
6776.2.a.h 1 33.d even 2 1
7056.2.a.c 1 28.d even 2 1
9464.2.a.h 1 39.d odd 2 1
9800.2.a.bj 1 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(504))$$:

 $$T_{5} - 4$$ T5 - 4 $$T_{11}$$ T11 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 4$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T - 2$$
$19$ $$T + 2$$
$23$ $$T + 8$$
$29$ $$T + 2$$
$31$ $$T - 4$$
$37$ $$T + 6$$
$41$ $$T - 2$$
$43$ $$T - 8$$
$47$ $$T - 4$$
$53$ $$T - 10$$
$59$ $$T + 6$$
$61$ $$T - 4$$
$67$ $$T + 12$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T + 8$$
$83$ $$T + 6$$
$89$ $$T + 10$$
$97$ $$T + 2$$