# Properties

 Label 52.2.a.a Level $52$ Weight $2$ Character orbit 52.a Self dual yes Analytic conductor $0.415$ 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] = [52,2,Mod(1,52)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(52, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("52.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$52 = 2^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 52.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: $$0.415222090511$$ 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10})$$ q + 2 * q^5 - 2 * q^7 - 3 * q^9 $$q + 2 q^{5} - 2 q^{7} - 3 q^{9} - 2 q^{11} - q^{13} + 6 q^{17} - 6 q^{19} + 8 q^{23} - q^{25} + 2 q^{29} + 10 q^{31} - 4 q^{35} - 6 q^{37} - 6 q^{41} + 4 q^{43} - 6 q^{45} - 2 q^{47} - 3 q^{49} + 6 q^{53} - 4 q^{55} - 10 q^{59} - 2 q^{61} + 6 q^{63} - 2 q^{65} + 10 q^{67} + 10 q^{71} + 2 q^{73} + 4 q^{77} - 4 q^{79} + 9 q^{81} - 6 q^{83} + 12 q^{85} - 6 q^{89} + 2 q^{91} - 12 q^{95} + 2 q^{97} + 6 q^{99}+O(q^{100})$$ q + 2 * q^5 - 2 * q^7 - 3 * q^9 - 2 * q^11 - q^13 + 6 * q^17 - 6 * q^19 + 8 * q^23 - q^25 + 2 * q^29 + 10 * q^31 - 4 * q^35 - 6 * q^37 - 6 * q^41 + 4 * q^43 - 6 * q^45 - 2 * q^47 - 3 * q^49 + 6 * q^53 - 4 * q^55 - 10 * q^59 - 2 * q^61 + 6 * q^63 - 2 * q^65 + 10 * q^67 + 10 * q^71 + 2 * q^73 + 4 * q^77 - 4 * q^79 + 9 * q^81 - 6 * q^83 + 12 * q^85 - 6 * q^89 + 2 * q^91 - 12 * q^95 + 2 * q^97 + 6 * 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 −2.00000 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
$$2$$ $$-1$$
$$13$$ $$1$$

## 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 52.2.a.a 1
3.b odd 2 1 468.2.a.b 1
4.b odd 2 1 208.2.a.c 1
5.b even 2 1 1300.2.a.d 1
5.c odd 4 2 1300.2.c.c 2
7.b odd 2 1 2548.2.a.e 1
7.c even 3 2 2548.2.j.e 2
7.d odd 6 2 2548.2.j.f 2
8.b even 2 1 832.2.a.e 1
8.d odd 2 1 832.2.a.f 1
9.c even 3 2 4212.2.i.d 2
9.d odd 6 2 4212.2.i.i 2
11.b odd 2 1 6292.2.a.g 1
12.b even 2 1 1872.2.a.f 1
13.b even 2 1 676.2.a.c 1
13.c even 3 2 676.2.e.c 2
13.d odd 4 2 676.2.d.c 2
13.e even 6 2 676.2.e.b 2
13.f odd 12 4 676.2.h.c 4
16.e even 4 2 3328.2.b.q 2
16.f odd 4 2 3328.2.b.e 2
20.d odd 2 1 5200.2.a.q 1
24.f even 2 1 7488.2.a.bw 1
24.h odd 2 1 7488.2.a.bn 1
39.d odd 2 1 6084.2.a.m 1
39.f even 4 2 6084.2.b.m 2
52.b odd 2 1 2704.2.a.g 1
52.f even 4 2 2704.2.f.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 1.a even 1 1 trivial
208.2.a.c 1 4.b odd 2 1
468.2.a.b 1 3.b odd 2 1
676.2.a.c 1 13.b even 2 1
676.2.d.c 2 13.d odd 4 2
676.2.e.b 2 13.e even 6 2
676.2.e.c 2 13.c even 3 2
676.2.h.c 4 13.f odd 12 4
832.2.a.e 1 8.b even 2 1
832.2.a.f 1 8.d odd 2 1
1300.2.a.d 1 5.b even 2 1
1300.2.c.c 2 5.c odd 4 2
1872.2.a.f 1 12.b even 2 1
2548.2.a.e 1 7.b odd 2 1
2548.2.j.e 2 7.c even 3 2
2548.2.j.f 2 7.d odd 6 2
2704.2.a.g 1 52.b odd 2 1
2704.2.f.f 2 52.f even 4 2
3328.2.b.e 2 16.f odd 4 2
3328.2.b.q 2 16.e even 4 2
4212.2.i.d 2 9.c even 3 2
4212.2.i.i 2 9.d odd 6 2
5200.2.a.q 1 20.d odd 2 1
6084.2.a.m 1 39.d odd 2 1
6084.2.b.m 2 39.f even 4 2
6292.2.a.g 1 11.b odd 2 1
7488.2.a.bn 1 24.h odd 2 1
7488.2.a.bw 1 24.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(52))$$.

## Hecke characteristic polynomials

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