# Properties

 Label 52.2.f.a Level $52$ Weight $2$ Character orbit 52.f Analytic conductor $0.415$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [52,2,Mod(31,52)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(52, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("52.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$52 = 2^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 52.f (of order $$4$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$0.415222090511$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (i + 1) q^{2} + 2 i q^{4} + ( - 3 i - 3) q^{5} + (2 i - 2) q^{8} + 3 q^{9}+O(q^{10})$$ q + (i + 1) * q^2 + 2*i * q^4 + (-3*i - 3) * q^5 + (2*i - 2) * q^8 + 3 * q^9 $$q + (i + 1) q^{2} + 2 i q^{4} + ( - 3 i - 3) q^{5} + (2 i - 2) q^{8} + 3 q^{9} - 6 i q^{10} + (3 i - 2) q^{13} - 4 q^{16} - 2 i q^{17} + (3 i + 3) q^{18} + ( - 6 i + 6) q^{20} + 13 i q^{25} + (i - 5) q^{26} + 4 q^{29} + ( - 4 i - 4) q^{32} + ( - 2 i + 2) q^{34} + 6 i q^{36} + ( - 5 i + 5) q^{37} + 12 q^{40} + ( - i - 1) q^{41} + ( - 9 i - 9) q^{45} - 7 i q^{49} + (13 i - 13) q^{50} + ( - 4 i - 6) q^{52} - 14 q^{53} + (4 i + 4) q^{58} + 10 q^{61} - 8 i q^{64} + ( - 3 i + 15) q^{65} + 4 q^{68} + (6 i - 6) q^{72} + (11 i - 11) q^{73} + 10 q^{74} + (12 i + 12) q^{80} + 9 q^{81} - 2 i q^{82} + (6 i - 6) q^{85} + ( - 3 i + 3) q^{89} - 18 i q^{90} + (5 i + 5) q^{97} + ( - 7 i + 7) q^{98} +O(q^{100})$$ q + (i + 1) * q^2 + 2*i * q^4 + (-3*i - 3) * q^5 + (2*i - 2) * q^8 + 3 * q^9 - 6*i * q^10 + (3*i - 2) * q^13 - 4 * q^16 - 2*i * q^17 + (3*i + 3) * q^18 + (-6*i + 6) * q^20 + 13*i * q^25 + (i - 5) * q^26 + 4 * q^29 + (-4*i - 4) * q^32 + (-2*i + 2) * q^34 + 6*i * q^36 + (-5*i + 5) * q^37 + 12 * q^40 + (-i - 1) * q^41 + (-9*i - 9) * q^45 - 7*i * q^49 + (13*i - 13) * q^50 + (-4*i - 6) * q^52 - 14 * q^53 + (4*i + 4) * q^58 + 10 * q^61 - 8*i * q^64 + (-3*i + 15) * q^65 + 4 * q^68 + (6*i - 6) * q^72 + (11*i - 11) * q^73 + 10 * q^74 + (12*i + 12) * q^80 + 9 * q^81 - 2*i * q^82 + (6*i - 6) * q^85 + (-3*i + 3) * q^89 - 18*i * q^90 + (5*i + 5) * q^97 + (-7*i + 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 6 q^{5} - 4 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 6 * q^5 - 4 * q^8 + 6 * q^9 $$2 q + 2 q^{2} - 6 q^{5} - 4 q^{8} + 6 q^{9} - 4 q^{13} - 8 q^{16} + 6 q^{18} + 12 q^{20} - 10 q^{26} + 8 q^{29} - 8 q^{32} + 4 q^{34} + 10 q^{37} + 24 q^{40} - 2 q^{41} - 18 q^{45} - 26 q^{50} - 12 q^{52} - 28 q^{53} + 8 q^{58} + 20 q^{61} + 30 q^{65} + 8 q^{68} - 12 q^{72} - 22 q^{73} + 20 q^{74} + 24 q^{80} + 18 q^{81} - 12 q^{85} + 6 q^{89} + 10 q^{97} + 14 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 6 * q^5 - 4 * q^8 + 6 * q^9 - 4 * q^13 - 8 * q^16 + 6 * q^18 + 12 * q^20 - 10 * q^26 + 8 * q^29 - 8 * q^32 + 4 * q^34 + 10 * q^37 + 24 * q^40 - 2 * q^41 - 18 * q^45 - 26 * q^50 - 12 * q^52 - 28 * q^53 + 8 * q^58 + 20 * q^61 + 30 * q^65 + 8 * q^68 - 12 * q^72 - 22 * q^73 + 20 * q^74 + 24 * q^80 + 18 * q^81 - 12 * q^85 + 6 * q^89 + 10 * q^97 + 14 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/52\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-1$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 − 1.00000i 1.00000i
1.00000 1.00000i 0 2.00000i −3.00000 + 3.00000i 0 0 −2.00000 2.00000i 3.00000 6.00000i
47.1 1.00000 + 1.00000i 0 2.00000i −3.00000 3.00000i 0 0 −2.00000 + 2.00000i 3.00000 6.00000i
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
13.d odd 4 1 inner
52.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.2.f.a 2
3.b odd 2 1 468.2.n.c 2
4.b odd 2 1 CM 52.2.f.a 2
8.b even 2 1 832.2.k.d 2
8.d odd 2 1 832.2.k.d 2
12.b even 2 1 468.2.n.c 2
13.b even 2 1 676.2.f.b 2
13.c even 3 2 676.2.l.a 4
13.d odd 4 1 inner 52.2.f.a 2
13.d odd 4 1 676.2.f.b 2
13.e even 6 2 676.2.l.g 4
13.f odd 12 2 676.2.l.a 4
13.f odd 12 2 676.2.l.g 4
39.f even 4 1 468.2.n.c 2
52.b odd 2 1 676.2.f.b 2
52.f even 4 1 inner 52.2.f.a 2
52.f even 4 1 676.2.f.b 2
52.i odd 6 2 676.2.l.g 4
52.j odd 6 2 676.2.l.a 4
52.l even 12 2 676.2.l.a 4
52.l even 12 2 676.2.l.g 4
104.j odd 4 1 832.2.k.d 2
104.m even 4 1 832.2.k.d 2
156.l odd 4 1 468.2.n.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.f.a 2 1.a even 1 1 trivial
52.2.f.a 2 4.b odd 2 1 CM
52.2.f.a 2 13.d odd 4 1 inner
52.2.f.a 2 52.f even 4 1 inner
468.2.n.c 2 3.b odd 2 1
468.2.n.c 2 12.b even 2 1
468.2.n.c 2 39.f even 4 1
468.2.n.c 2 156.l odd 4 1
676.2.f.b 2 13.b even 2 1
676.2.f.b 2 13.d odd 4 1
676.2.f.b 2 52.b odd 2 1
676.2.f.b 2 52.f even 4 1
676.2.l.a 4 13.c even 3 2
676.2.l.a 4 13.f odd 12 2
676.2.l.a 4 52.j odd 6 2
676.2.l.a 4 52.l even 12 2
676.2.l.g 4 13.e even 6 2
676.2.l.g 4 13.f odd 12 2
676.2.l.g 4 52.i odd 6 2
676.2.l.g 4 52.l even 12 2
832.2.k.d 2 8.b even 2 1
832.2.k.d 2 8.d odd 2 1
832.2.k.d 2 104.j odd 4 1
832.2.k.d 2 104.m even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(52, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 6T + 18$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4T + 13$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 10T + 50$$
$41$ $$T^{2} + 2T + 2$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$(T + 14)^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 22T + 242$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 6T + 18$$
$97$ $$T^{2} - 10T + 50$$