# Properties

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

# Learn more

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: $$\Z[a_1, \ldots, a_{9}]$$ 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.

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

## 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 1.00000 1.73205i 0 −3.00000 0 2.00000 + 3.46410i 0 −0.500000 0.866025i 0
29.1 0 1.00000 + 1.73205i 0 −3.00000 0 2.00000 3.46410i 0 −0.500000 + 0.866025i 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.b 2
3.b odd 2 1 468.2.l.d 2
4.b odd 2 1 208.2.i.a 2
5.b even 2 1 1300.2.i.b 2
5.c odd 4 2 1300.2.bb.d 4
7.b odd 2 1 2548.2.k.a 2
7.c even 3 1 2548.2.i.g 2
7.c even 3 1 2548.2.l.b 2
7.d odd 6 1 2548.2.i.b 2
7.d odd 6 1 2548.2.l.g 2
8.b even 2 1 832.2.i.c 2
8.d odd 2 1 832.2.i.i 2
12.b even 2 1 1872.2.t.m 2
13.b even 2 1 676.2.e.d 2
13.c even 3 1 inner 52.2.e.b 2
13.c even 3 1 676.2.a.a 1
13.d odd 4 2 676.2.h.d 4
13.e even 6 1 676.2.a.b 1
13.e even 6 1 676.2.e.d 2
13.f odd 12 2 676.2.d.a 2
13.f odd 12 2 676.2.h.d 4
39.h odd 6 1 6084.2.a.c 1
39.i odd 6 1 468.2.l.d 2
39.i odd 6 1 6084.2.a.o 1
39.k even 12 2 6084.2.b.k 2
52.i odd 6 1 2704.2.a.m 1
52.j odd 6 1 208.2.i.a 2
52.j odd 6 1 2704.2.a.l 1
52.l even 12 2 2704.2.f.i 2
65.n even 6 1 1300.2.i.b 2
65.q odd 12 2 1300.2.bb.d 4
91.g even 3 1 2548.2.i.g 2
91.h even 3 1 2548.2.l.b 2
91.m odd 6 1 2548.2.i.b 2
91.n odd 6 1 2548.2.k.a 2
91.v odd 6 1 2548.2.l.g 2
104.n odd 6 1 832.2.i.i 2
104.r even 6 1 832.2.i.c 2
156.p even 6 1 1872.2.t.m 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(52, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$(T + 3)^{2}$$
$7$ $$T^{2} - 4T + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 7T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} + 9T + 81$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} - 7T + 49$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$(T + 6)^{2}$$
$53$ $$(T - 9)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$(T + 1)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$T^{2} + 14T + 196$$
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