# Properties

 Label 56.2.a.a Level $56$ Weight $2$ Character orbit 56.a Self dual yes Analytic conductor $0.447$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [56,2,Mod(1,56)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(56, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("56.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$56 = 2^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 56.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: $$0.447162251319$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{5} - q^{7} - 3 q^{9}+O(q^{10})$$ q + 2 * q^5 - q^7 - 3 * q^9 $$q + 2 q^{5} - q^{7} - 3 q^{9} - 4 q^{11} + 2 q^{13} - 6 q^{17} + 8 q^{19} - q^{25} + 6 q^{29} + 8 q^{31} - 2 q^{35} - 2 q^{37} + 2 q^{41} - 4 q^{43} - 6 q^{45} - 8 q^{47} + q^{49} + 6 q^{53} - 8 q^{55} - 6 q^{61} + 3 q^{63} + 4 q^{65} - 4 q^{67} - 8 q^{71} + 10 q^{73} + 4 q^{77} + 16 q^{79} + 9 q^{81} + 8 q^{83} - 12 q^{85} - 6 q^{89} - 2 q^{91} + 16 q^{95} - 6 q^{97} + 12 q^{99}+O(q^{100})$$ q + 2 * q^5 - q^7 - 3 * q^9 - 4 * q^11 + 2 * q^13 - 6 * q^17 + 8 * q^19 - q^25 + 6 * q^29 + 8 * q^31 - 2 * q^35 - 2 * q^37 + 2 * q^41 - 4 * q^43 - 6 * q^45 - 8 * q^47 + q^49 + 6 * q^53 - 8 * q^55 - 6 * q^61 + 3 * q^63 + 4 * q^65 - 4 * q^67 - 8 * q^71 + 10 * q^73 + 4 * q^77 + 16 * q^79 + 9 * q^81 + 8 * q^83 - 12 * q^85 - 6 * q^89 - 2 * q^91 + 16 * q^95 - 6 * q^97 + 12 * q^99

## 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 2.00000 0 −1.00000 0 −3.00000 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
$$2$$ $$-1$$
$$7$$ $$+1$$

## 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 56.2.a.a 1
3.b odd 2 1 504.2.a.c 1
4.b odd 2 1 112.2.a.b 1
5.b even 2 1 1400.2.a.g 1
5.c odd 4 2 1400.2.g.g 2
7.b odd 2 1 392.2.a.d 1
7.c even 3 2 392.2.i.c 2
7.d odd 6 2 392.2.i.d 2
8.b even 2 1 448.2.a.d 1
8.d odd 2 1 448.2.a.e 1
11.b odd 2 1 6776.2.a.g 1
12.b even 2 1 1008.2.a.d 1
13.b even 2 1 9464.2.a.c 1
16.e even 4 2 1792.2.b.i 2
16.f odd 4 2 1792.2.b.d 2
20.d odd 2 1 2800.2.a.p 1
20.e even 4 2 2800.2.g.p 2
21.c even 2 1 3528.2.a.x 1
21.g even 6 2 3528.2.s.e 2
21.h odd 6 2 3528.2.s.t 2
24.f even 2 1 4032.2.a.bk 1
24.h odd 2 1 4032.2.a.bb 1
28.d even 2 1 784.2.a.e 1
28.f even 6 2 784.2.i.g 2
28.g odd 6 2 784.2.i.e 2
35.c odd 2 1 9800.2.a.u 1
56.e even 2 1 3136.2.a.p 1
56.h odd 2 1 3136.2.a.q 1
84.h odd 2 1 7056.2.a.bo 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 1.a even 1 1 trivial
112.2.a.b 1 4.b odd 2 1
392.2.a.d 1 7.b odd 2 1
392.2.i.c 2 7.c even 3 2
392.2.i.d 2 7.d odd 6 2
448.2.a.d 1 8.b even 2 1
448.2.a.e 1 8.d odd 2 1
504.2.a.c 1 3.b odd 2 1
784.2.a.e 1 28.d even 2 1
784.2.i.e 2 28.g odd 6 2
784.2.i.g 2 28.f even 6 2
1008.2.a.d 1 12.b even 2 1
1400.2.a.g 1 5.b even 2 1
1400.2.g.g 2 5.c odd 4 2
1792.2.b.d 2 16.f odd 4 2
1792.2.b.i 2 16.e even 4 2
2800.2.a.p 1 20.d odd 2 1
2800.2.g.p 2 20.e even 4 2
3136.2.a.p 1 56.e even 2 1
3136.2.a.q 1 56.h odd 2 1
3528.2.a.x 1 21.c even 2 1
3528.2.s.e 2 21.g even 6 2
3528.2.s.t 2 21.h odd 6 2
4032.2.a.bb 1 24.h odd 2 1
4032.2.a.bk 1 24.f even 2 1
6776.2.a.g 1 11.b odd 2 1
7056.2.a.bo 1 84.h odd 2 1
9464.2.a.c 1 13.b even 2 1
9800.2.a.u 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(56))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 2$$
$7$ $$T + 1$$
$11$ $$T + 4$$
$13$ $$T - 2$$
$17$ $$T + 6$$
$19$ $$T - 8$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T - 8$$
$37$ $$T + 2$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T + 8$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T + 6$$
$67$ $$T + 4$$
$71$ $$T + 8$$
$73$ $$T - 10$$
$79$ $$T - 16$$
$83$ $$T - 8$$
$89$ $$T + 6$$
$97$ $$T + 6$$
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