# Properties

 Label 52.2.e.a Level $52$ Weight $2$ Character orbit 52.e Analytic conductor $0.415$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [52,2,Mod(9,52)]

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([0, 4]))

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

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

chi := DirichletCharacter("52.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$52 = 2^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 52.e (of order $$3$$, 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: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (3 \zeta_{6} - 3) q^{3} + 2 q^{5} - \zeta_{6} q^{7} - 6 \zeta_{6} q^{9} +O(q^{10})$$ q + (3*z - 3) * q^3 + 2 * q^5 - z * q^7 - 6*z * q^9 $$q + (3 \zeta_{6} - 3) q^{3} + 2 q^{5} - \zeta_{6} q^{7} - 6 \zeta_{6} q^{9} + ( - 5 \zeta_{6} + 5) q^{11} + (4 \zeta_{6} - 3) q^{13} + (6 \zeta_{6} - 6) q^{15} - 3 \zeta_{6} q^{17} + 3 \zeta_{6} q^{19} + 3 q^{21} + ( - \zeta_{6} + 1) q^{23} - q^{25} + 9 q^{27} + ( - \zeta_{6} + 1) q^{29} - 8 q^{31} + 15 \zeta_{6} q^{33} - 2 \zeta_{6} q^{35} + (3 \zeta_{6} - 3) q^{37} + ( - 9 \zeta_{6} - 3) q^{39} + (3 \zeta_{6} - 3) q^{41} - \zeta_{6} q^{43} - 12 \zeta_{6} q^{45} + 4 q^{47} + ( - 6 \zeta_{6} + 6) q^{49} + 9 q^{51} - 6 q^{53} + ( - 10 \zeta_{6} + 10) q^{55} - 9 q^{57} - 5 \zeta_{6} q^{59} + 5 \zeta_{6} q^{61} + (6 \zeta_{6} - 6) q^{63} + (8 \zeta_{6} - 6) q^{65} + (7 \zeta_{6} - 7) q^{67} + 3 \zeta_{6} q^{69} + 11 \zeta_{6} q^{71} + 14 q^{73} + ( - 3 \zeta_{6} + 3) q^{75} - 5 q^{77} - 4 q^{79} + (9 \zeta_{6} - 9) q^{81} + 12 q^{83} - 6 \zeta_{6} q^{85} + 3 \zeta_{6} q^{87} + ( - 9 \zeta_{6} + 9) q^{89} + ( - \zeta_{6} + 4) q^{91} + ( - 24 \zeta_{6} + 24) q^{93} + 6 \zeta_{6} q^{95} + \zeta_{6} q^{97} - 30 q^{99} +O(q^{100})$$ q + (3*z - 3) * q^3 + 2 * q^5 - z * q^7 - 6*z * q^9 + (-5*z + 5) * q^11 + (4*z - 3) * q^13 + (6*z - 6) * q^15 - 3*z * q^17 + 3*z * q^19 + 3 * q^21 + (-z + 1) * q^23 - q^25 + 9 * q^27 + (-z + 1) * q^29 - 8 * q^31 + 15*z * q^33 - 2*z * q^35 + (3*z - 3) * q^37 + (-9*z - 3) * q^39 + (3*z - 3) * q^41 - z * q^43 - 12*z * q^45 + 4 * q^47 + (-6*z + 6) * q^49 + 9 * q^51 - 6 * q^53 + (-10*z + 10) * q^55 - 9 * q^57 - 5*z * q^59 + 5*z * q^61 + (6*z - 6) * q^63 + (8*z - 6) * q^65 + (7*z - 7) * q^67 + 3*z * q^69 + 11*z * q^71 + 14 * q^73 + (-3*z + 3) * q^75 - 5 * q^77 - 4 * q^79 + (9*z - 9) * q^81 + 12 * q^83 - 6*z * q^85 + 3*z * q^87 + (-9*z + 9) * q^89 + (-z + 4) * q^91 + (-24*z + 24) * q^93 + 6*z * q^95 + z * q^97 - 30 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + 4 q^{5} - q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + 4 * q^5 - q^7 - 6 * q^9 $$2 q - 3 q^{3} + 4 q^{5} - q^{7} - 6 q^{9} + 5 q^{11} - 2 q^{13} - 6 q^{15} - 3 q^{17} + 3 q^{19} + 6 q^{21} + q^{23} - 2 q^{25} + 18 q^{27} + q^{29} - 16 q^{31} + 15 q^{33} - 2 q^{35} - 3 q^{37} - 15 q^{39} - 3 q^{41} - q^{43} - 12 q^{45} + 8 q^{47} + 6 q^{49} + 18 q^{51} - 12 q^{53} + 10 q^{55} - 18 q^{57} - 5 q^{59} + 5 q^{61} - 6 q^{63} - 4 q^{65} - 7 q^{67} + 3 q^{69} + 11 q^{71} + 28 q^{73} + 3 q^{75} - 10 q^{77} - 8 q^{79} - 9 q^{81} + 24 q^{83} - 6 q^{85} + 3 q^{87} + 9 q^{89} + 7 q^{91} + 24 q^{93} + 6 q^{95} + q^{97} - 60 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + 4 * q^5 - q^7 - 6 * q^9 + 5 * q^11 - 2 * q^13 - 6 * q^15 - 3 * q^17 + 3 * q^19 + 6 * q^21 + q^23 - 2 * q^25 + 18 * q^27 + q^29 - 16 * q^31 + 15 * q^33 - 2 * q^35 - 3 * q^37 - 15 * q^39 - 3 * q^41 - q^43 - 12 * q^45 + 8 * q^47 + 6 * q^49 + 18 * q^51 - 12 * q^53 + 10 * q^55 - 18 * q^57 - 5 * q^59 + 5 * q^61 - 6 * q^63 - 4 * q^65 - 7 * q^67 + 3 * q^69 + 11 * q^71 + 28 * q^73 + 3 * q^75 - 10 * q^77 - 8 * q^79 - 9 * q^81 + 24 * q^83 - 6 * q^85 + 3 * q^87 + 9 * q^89 + 7 * q^91 + 24 * q^93 + 6 * q^95 + q^97 - 60 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 + 2.59808i 0 2.00000 0 −0.500000 0.866025i 0 −3.00000 5.19615i 0
29.1 0 −1.50000 2.59808i 0 2.00000 0 −0.500000 + 0.866025i 0 −3.00000 + 5.19615i 0
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.2.e.a 2
3.b odd 2 1 468.2.l.a 2
4.b odd 2 1 208.2.i.d 2
5.b even 2 1 1300.2.i.f 2
5.c odd 4 2 1300.2.bb.f 4
7.b odd 2 1 2548.2.k.d 2
7.c even 3 1 2548.2.i.a 2
7.c even 3 1 2548.2.l.h 2
7.d odd 6 1 2548.2.i.h 2
7.d odd 6 1 2548.2.l.a 2
8.b even 2 1 832.2.i.j 2
8.d odd 2 1 832.2.i.a 2
12.b even 2 1 1872.2.t.f 2
13.b even 2 1 676.2.e.a 2
13.c even 3 1 inner 52.2.e.a 2
13.c even 3 1 676.2.a.e 1
13.d odd 4 2 676.2.h.b 4
13.e even 6 1 676.2.a.d 1
13.e even 6 1 676.2.e.a 2
13.f odd 12 2 676.2.d.d 2
13.f odd 12 2 676.2.h.b 4
39.h odd 6 1 6084.2.a.k 1
39.i odd 6 1 468.2.l.a 2
39.i odd 6 1 6084.2.a.f 1
39.k even 12 2 6084.2.b.l 2
52.i odd 6 1 2704.2.a.a 1
52.j odd 6 1 208.2.i.d 2
52.j odd 6 1 2704.2.a.b 1
52.l even 12 2 2704.2.f.a 2
65.n even 6 1 1300.2.i.f 2
65.q odd 12 2 1300.2.bb.f 4
91.g even 3 1 2548.2.i.a 2
91.h even 3 1 2548.2.l.h 2
91.m odd 6 1 2548.2.i.h 2
91.n odd 6 1 2548.2.k.d 2
91.v odd 6 1 2548.2.l.a 2
104.n odd 6 1 832.2.i.a 2
104.r even 6 1 832.2.i.j 2
156.p even 6 1 1872.2.t.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.e.a 2 1.a even 1 1 trivial
52.2.e.a 2 13.c even 3 1 inner
208.2.i.d 2 4.b odd 2 1
208.2.i.d 2 52.j odd 6 1
468.2.l.a 2 3.b odd 2 1
468.2.l.a 2 39.i odd 6 1
676.2.a.d 1 13.e even 6 1
676.2.a.e 1 13.c even 3 1
676.2.d.d 2 13.f odd 12 2
676.2.e.a 2 13.b even 2 1
676.2.e.a 2 13.e even 6 1
676.2.h.b 4 13.d odd 4 2
676.2.h.b 4 13.f odd 12 2
832.2.i.a 2 8.d odd 2 1
832.2.i.a 2 104.n odd 6 1
832.2.i.j 2 8.b even 2 1
832.2.i.j 2 104.r even 6 1
1300.2.i.f 2 5.b even 2 1
1300.2.i.f 2 65.n even 6 1
1300.2.bb.f 4 5.c odd 4 2
1300.2.bb.f 4 65.q odd 12 2
1872.2.t.f 2 12.b even 2 1
1872.2.t.f 2 156.p even 6 1
2548.2.i.a 2 7.c even 3 1
2548.2.i.a 2 91.g even 3 1
2548.2.i.h 2 7.d odd 6 1
2548.2.i.h 2 91.m odd 6 1
2548.2.k.d 2 7.b odd 2 1
2548.2.k.d 2 91.n odd 6 1
2548.2.l.a 2 7.d odd 6 1
2548.2.l.a 2 91.v odd 6 1
2548.2.l.h 2 7.c even 3 1
2548.2.l.h 2 91.h even 3 1
2704.2.a.a 1 52.i odd 6 1
2704.2.a.b 1 52.j odd 6 1
2704.2.f.a 2 52.l even 12 2
6084.2.a.f 1 39.i odd 6 1
6084.2.a.k 1 39.h odd 6 1
6084.2.b.l 2 39.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3T_{3} + 9$$ acting on $$S_{2}^{\mathrm{new}}(52, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$(T - 2)^{2}$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$T^{2} + 2T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - 3T + 9$$
$23$ $$T^{2} - T + 1$$
$29$ $$T^{2} - T + 1$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} + T + 1$$
$47$ $$(T - 4)^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 5T + 25$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} + 7T + 49$$
$71$ $$T^{2} - 11T + 121$$
$73$ $$(T - 14)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} - 9T + 81$$
$97$ $$T^{2} - T + 1$$