# Properties

 Label 52.2.h.a Level $52$ Weight $2$ Character orbit 52.h 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(17,52)]

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, 1]))

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

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

chi := DirichletCharacter("52.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$52 = 2^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 52.h (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: $$0.415222090511$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 2) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z + 1) * q^3 + (z - 2) * q^7 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 2) q^{7} + 2 \zeta_{6} q^{9} + ( - 3 \zeta_{6} - 3) q^{11} + (4 \zeta_{6} - 3) q^{13} - 3 \zeta_{6} q^{17} + ( - 3 \zeta_{6} + 6) q^{19} + (2 \zeta_{6} - 1) q^{21} + (3 \zeta_{6} - 3) q^{23} + 5 q^{25} + 5 q^{27} + ( - 9 \zeta_{6} + 9) q^{29} + (4 \zeta_{6} - 2) q^{31} + (3 \zeta_{6} - 6) q^{33} + ( - 3 \zeta_{6} - 3) q^{37} + (3 \zeta_{6} + 1) q^{39} + ( - 3 \zeta_{6} - 3) q^{41} - 5 \zeta_{6} q^{43} + (12 \zeta_{6} - 6) q^{47} + (4 \zeta_{6} - 4) q^{49} - 3 q^{51} - 6 q^{53} + ( - 6 \zeta_{6} + 3) q^{57} + ( - 3 \zeta_{6} + 6) q^{59} + 5 \zeta_{6} q^{61} + ( - 2 \zeta_{6} - 2) q^{63} + (\zeta_{6} + 1) q^{67} + 3 \zeta_{6} q^{69} + ( - 3 \zeta_{6} + 6) q^{71} + ( - 8 \zeta_{6} + 4) q^{73} + ( - 5 \zeta_{6} + 5) q^{75} + 9 q^{77} + 4 q^{79} + (\zeta_{6} - 1) q^{81} + (12 \zeta_{6} - 6) q^{83} - 9 \zeta_{6} q^{87} + (9 \zeta_{6} + 9) q^{89} + ( - 7 \zeta_{6} + 2) q^{91} + (2 \zeta_{6} + 2) q^{93} + (7 \zeta_{6} - 14) q^{97} + ( - 12 \zeta_{6} + 6) q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + (z - 2) * q^7 + 2*z * q^9 + (-3*z - 3) * q^11 + (4*z - 3) * q^13 - 3*z * q^17 + (-3*z + 6) * q^19 + (2*z - 1) * q^21 + (3*z - 3) * q^23 + 5 * q^25 + 5 * q^27 + (-9*z + 9) * q^29 + (4*z - 2) * q^31 + (3*z - 6) * q^33 + (-3*z - 3) * q^37 + (3*z + 1) * q^39 + (-3*z - 3) * q^41 - 5*z * q^43 + (12*z - 6) * q^47 + (4*z - 4) * q^49 - 3 * q^51 - 6 * q^53 + (-6*z + 3) * q^57 + (-3*z + 6) * q^59 + 5*z * q^61 + (-2*z - 2) * q^63 + (z + 1) * q^67 + 3*z * q^69 + (-3*z + 6) * q^71 + (-8*z + 4) * q^73 + (-5*z + 5) * q^75 + 9 * q^77 + 4 * q^79 + (z - 1) * q^81 + (12*z - 6) * q^83 - 9*z * q^87 + (9*z + 9) * q^89 + (-7*z + 2) * q^91 + (2*z + 2) * q^93 + (7*z - 14) * q^97 + (-12*z + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 3 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 - 3 * q^7 + 2 * q^9 $$2 q + q^{3} - 3 q^{7} + 2 q^{9} - 9 q^{11} - 2 q^{13} - 3 q^{17} + 9 q^{19} - 3 q^{23} + 10 q^{25} + 10 q^{27} + 9 q^{29} - 9 q^{33} - 9 q^{37} + 5 q^{39} - 9 q^{41} - 5 q^{43} - 4 q^{49} - 6 q^{51} - 12 q^{53} + 9 q^{59} + 5 q^{61} - 6 q^{63} + 3 q^{67} + 3 q^{69} + 9 q^{71} + 5 q^{75} + 18 q^{77} + 8 q^{79} - q^{81} - 9 q^{87} + 27 q^{89} - 3 q^{91} + 6 q^{93} - 21 q^{97}+O(q^{100})$$ 2 * q + q^3 - 3 * q^7 + 2 * q^9 - 9 * q^11 - 2 * q^13 - 3 * q^17 + 9 * q^19 - 3 * q^23 + 10 * q^25 + 10 * q^27 + 9 * q^29 - 9 * q^33 - 9 * q^37 + 5 * q^39 - 9 * q^41 - 5 * q^43 - 4 * q^49 - 6 * q^51 - 12 * q^53 + 9 * q^59 + 5 * q^61 - 6 * q^63 + 3 * q^67 + 3 * q^69 + 9 * q^71 + 5 * q^75 + 18 * q^77 + 8 * q^79 - q^81 - 9 * q^87 + 27 * q^89 - 3 * q^91 + 6 * q^93 - 21 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 0 0 −1.50000 + 0.866025i 0 1.00000 + 1.73205i 0
49.1 0 0.500000 + 0.866025i 0 0 0 −1.50000 0.866025i 0 1.00000 1.73205i 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.e even 6 1 inner

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 3T + 3$$
$11$ $$T^{2} + 9T + 27$$
$13$ $$T^{2} + 2T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - 9T + 27$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$T^{2} - 9T + 81$$
$31$ $$T^{2} + 12$$
$37$ $$T^{2} + 9T + 27$$
$41$ $$T^{2} + 9T + 27$$
$43$ $$T^{2} + 5T + 25$$
$47$ $$T^{2} + 108$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} - 9T + 27$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} - 3T + 3$$
$71$ $$T^{2} - 9T + 27$$
$73$ $$T^{2} + 48$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 108$$
$89$ $$T^{2} - 27T + 243$$
$97$ $$T^{2} + 21T + 147$$