# Properties

 Label 52.1.j.a Level $52$ Weight $1$ Character orbit 52.j Analytic conductor $0.026$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -4 Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [52,1,Mod(3,52)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(52, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("52.3");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$52 = 2^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 52.j (of order $$6$$, 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: $$0.0259513806569$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.676.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.10816.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{5} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ q - z * q^2 + z^2 * q^4 - q^5 + q^8 + z^2 * q^9 $$q - \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{5} + q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{10} - \zeta_{6} q^{13} - \zeta_{6} q^{16} - \zeta_{6}^{2} q^{17} + q^{18} - \zeta_{6}^{2} q^{20} + \zeta_{6}^{2} q^{26} + \zeta_{6} q^{29} + \zeta_{6}^{2} q^{32} - q^{34} - \zeta_{6} q^{36} + \zeta_{6} q^{37} - q^{40} + \zeta_{6} q^{41} - \zeta_{6}^{2} q^{45} - \zeta_{6} q^{49} + q^{52} - q^{53} - \zeta_{6}^{2} q^{58} - \zeta_{6}^{2} q^{61} + q^{64} + \zeta_{6} q^{65} + \zeta_{6} q^{68} + \zeta_{6}^{2} q^{72} - q^{73} - \zeta_{6}^{2} q^{74} + \zeta_{6} q^{80} - \zeta_{6} q^{81} - \zeta_{6}^{2} q^{82} + \zeta_{6}^{2} q^{85} - \zeta_{6} q^{89} - q^{90} + \zeta_{6}^{2} q^{97} + \zeta_{6}^{2} q^{98} +O(q^{100})$$ q - z * q^2 + z^2 * q^4 - q^5 + q^8 + z^2 * q^9 + z * q^10 - z * q^13 - z * q^16 - z^2 * q^17 + q^18 - z^2 * q^20 + z^2 * q^26 + z * q^29 + z^2 * q^32 - q^34 - z * q^36 + z * q^37 - q^40 + z * q^41 - z^2 * q^45 - z * q^49 + q^52 - q^53 - z^2 * q^58 - z^2 * q^61 + q^64 + z * q^65 + z * q^68 + z^2 * q^72 - q^73 - z^2 * q^74 + z * q^80 - z * q^81 - z^2 * q^82 + z^2 * q^85 - z * q^89 - q^90 + z^2 * q^97 + z^2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 - q^4 - 2 * q^5 + 2 * q^8 - q^9 $$2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{8} - q^{9} + q^{10} - q^{13} - q^{16} + q^{17} + 2 q^{18} + q^{20} - q^{26} + q^{29} - q^{32} - 2 q^{34} - q^{36} + q^{37} - 2 q^{40} + q^{41} + q^{45} - q^{49} + 2 q^{52} - 2 q^{53} + q^{58} + q^{61} + 2 q^{64} + q^{65} + q^{68} - q^{72} - 2 q^{73} + q^{74} + q^{80} - q^{81} + q^{82} - q^{85} - 2 q^{89} - 2 q^{90} - 2 q^{97} - q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 - 2 * q^5 + 2 * q^8 - q^9 + q^10 - q^13 - q^16 + q^17 + 2 * q^18 + q^20 - q^26 + q^29 - q^32 - 2 * q^34 - q^36 + q^37 - 2 * q^40 + q^41 + q^45 - q^49 + 2 * q^52 - 2 * q^53 + q^58 + q^61 + 2 * q^64 + q^65 + q^68 - q^72 - 2 * q^73 + q^74 + q^80 - q^81 + q^82 - q^85 - 2 * q^89 - 2 * q^90 - 2 * q^97 - q^98

## Character values

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

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-1$$ $$\zeta_{6}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 0 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
35.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.00000 0 0 1.00000 −0.500000 0.866025i 0.500000 0.866025i
 $$n$$: 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.c even 3 1 inner
52.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.1.j.a 2
3.b odd 2 1 468.1.br.a 2
4.b odd 2 1 CM 52.1.j.a 2
5.b even 2 1 1300.1.bc.a 2
5.c odd 4 2 1300.1.w.a 4
7.b odd 2 1 2548.1.bn.a 2
7.c even 3 1 2548.1.q.b 2
7.c even 3 1 2548.1.bi.b 2
7.d odd 6 1 2548.1.q.a 2
7.d odd 6 1 2548.1.bi.a 2
8.b even 2 1 832.1.bb.a 2
8.d odd 2 1 832.1.bb.a 2
12.b even 2 1 468.1.br.a 2
13.b even 2 1 676.1.j.a 2
13.c even 3 1 inner 52.1.j.a 2
13.c even 3 1 676.1.c.b 1
13.d odd 4 2 676.1.i.a 4
13.e even 6 1 676.1.c.a 1
13.e even 6 1 676.1.j.a 2
13.f odd 12 2 676.1.b.a 2
13.f odd 12 2 676.1.i.a 4
16.e even 4 2 3328.1.v.b 4
16.f odd 4 2 3328.1.v.b 4
20.d odd 2 1 1300.1.bc.a 2
20.e even 4 2 1300.1.w.a 4
28.d even 2 1 2548.1.bn.a 2
28.f even 6 1 2548.1.q.a 2
28.f even 6 1 2548.1.bi.a 2
28.g odd 6 1 2548.1.q.b 2
28.g odd 6 1 2548.1.bi.b 2
39.i odd 6 1 468.1.br.a 2
52.b odd 2 1 676.1.j.a 2
52.f even 4 2 676.1.i.a 4
52.i odd 6 1 676.1.c.a 1
52.i odd 6 1 676.1.j.a 2
52.j odd 6 1 inner 52.1.j.a 2
52.j odd 6 1 676.1.c.b 1
52.l even 12 2 676.1.b.a 2
52.l even 12 2 676.1.i.a 4
65.n even 6 1 1300.1.bc.a 2
65.q odd 12 2 1300.1.w.a 4
91.g even 3 1 2548.1.bi.b 2
91.h even 3 1 2548.1.q.b 2
91.m odd 6 1 2548.1.bi.a 2
91.n odd 6 1 2548.1.bn.a 2
91.v odd 6 1 2548.1.q.a 2
104.n odd 6 1 832.1.bb.a 2
104.r even 6 1 832.1.bb.a 2
156.p even 6 1 468.1.br.a 2
208.bg odd 12 2 3328.1.v.b 4
208.bj even 12 2 3328.1.v.b 4
260.v odd 6 1 1300.1.bc.a 2
260.bj even 12 2 1300.1.w.a 4
364.q odd 6 1 2548.1.bi.b 2
364.v even 6 1 2548.1.bn.a 2
364.ba even 6 1 2548.1.q.a 2
364.bi odd 6 1 2548.1.q.b 2
364.br even 6 1 2548.1.bi.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 1.a even 1 1 trivial
52.1.j.a 2 4.b odd 2 1 CM
52.1.j.a 2 13.c even 3 1 inner
52.1.j.a 2 52.j odd 6 1 inner
468.1.br.a 2 3.b odd 2 1
468.1.br.a 2 12.b even 2 1
468.1.br.a 2 39.i odd 6 1
468.1.br.a 2 156.p even 6 1
676.1.b.a 2 13.f odd 12 2
676.1.b.a 2 52.l even 12 2
676.1.c.a 1 13.e even 6 1
676.1.c.a 1 52.i odd 6 1
676.1.c.b 1 13.c even 3 1
676.1.c.b 1 52.j odd 6 1
676.1.i.a 4 13.d odd 4 2
676.1.i.a 4 13.f odd 12 2
676.1.i.a 4 52.f even 4 2
676.1.i.a 4 52.l even 12 2
676.1.j.a 2 13.b even 2 1
676.1.j.a 2 13.e even 6 1
676.1.j.a 2 52.b odd 2 1
676.1.j.a 2 52.i odd 6 1
832.1.bb.a 2 8.b even 2 1
832.1.bb.a 2 8.d odd 2 1
832.1.bb.a 2 104.n odd 6 1
832.1.bb.a 2 104.r even 6 1
1300.1.w.a 4 5.c odd 4 2
1300.1.w.a 4 20.e even 4 2
1300.1.w.a 4 65.q odd 12 2
1300.1.w.a 4 260.bj even 12 2
1300.1.bc.a 2 5.b even 2 1
1300.1.bc.a 2 20.d odd 2 1
1300.1.bc.a 2 65.n even 6 1
1300.1.bc.a 2 260.v odd 6 1
2548.1.q.a 2 7.d odd 6 1
2548.1.q.a 2 28.f even 6 1
2548.1.q.a 2 91.v odd 6 1
2548.1.q.a 2 364.ba even 6 1
2548.1.q.b 2 7.c even 3 1
2548.1.q.b 2 28.g odd 6 1
2548.1.q.b 2 91.h even 3 1
2548.1.q.b 2 364.bi odd 6 1
2548.1.bi.a 2 7.d odd 6 1
2548.1.bi.a 2 28.f even 6 1
2548.1.bi.a 2 91.m odd 6 1
2548.1.bi.a 2 364.br even 6 1
2548.1.bi.b 2 7.c even 3 1
2548.1.bi.b 2 28.g odd 6 1
2548.1.bi.b 2 91.g even 3 1
2548.1.bi.b 2 364.q odd 6 1
2548.1.bn.a 2 7.b odd 2 1
2548.1.bn.a 2 28.d even 2 1
2548.1.bn.a 2 91.n odd 6 1
2548.1.bn.a 2 364.v even 6 1
3328.1.v.b 4 16.e even 4 2
3328.1.v.b 4 16.f odd 4 2
3328.1.v.b 4 208.bg odd 12 2
3328.1.v.b 4 208.bj even 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(52, [\chi])$$.

## Hecke characteristic polynomials

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