# Properties

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

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [52,2,Mod(7,52)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(52, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 11]))

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

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

chi := DirichletCharacter("52.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$52 = 2^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 52.l (of order $$12$$, degree $$4$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{5} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10})$$ q + (z^3 - z^2 - z) * q^2 + 2*z * q^4 + (-2*z^3 - z^2 + z + 2) * q^5 + (-2*z^3 - 2) * q^8 + (3*z^2 - 3) * q^9 $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{5} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + (3 \zeta_{12}^{3} + \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{10} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 3 \zeta_{12}) q^{13} + 4 \zeta_{12}^{2} q^{16} + ( - 4 \zeta_{12}^{2} + \zeta_{12} - 4) q^{17} + ( - 3 \zeta_{12}^{3} + 3) q^{18} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{20} + ( - \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{25} + ( - 5 \zeta_{12}^{2} + \zeta_{12} + 5) q^{26} + (5 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 5 \zeta_{12}) q^{29} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{32} + ( - 5 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12} - 5) q^{34} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{36} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} - 6 \zeta_{12} + 7) q^{37} + (2 \zeta_{12}^{3} - 4 \zeta_{12} - 6) q^{40} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 5 \zeta_{12} - 9) q^{41} + (3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{45} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{49} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} - 6) q^{50} + (4 \zeta_{12}^{3} - 6) q^{52} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 7) q^{53} + ( - 10 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{58} + ( - 12 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 6 \zeta_{12} + 5) q^{61} + 8 \zeta_{12}^{3} q^{64} + (4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + \zeta_{12} - 1) q^{65} + ( - 8 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 8 \zeta_{12}) q^{68} + ( - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{72} + (8 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 5 \zeta_{12} + 8) q^{73} + (14 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 7 \zeta_{12} + 5) q^{74} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{80} - 9 \zeta_{12}^{2} q^{81} + (9 \zeta_{12}^{2} + \zeta_{12} + 9) q^{82} + (11 \zeta_{12}^{3} - \zeta_{12}^{2} - 10 \zeta_{12} - 10) q^{85} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{89} + ( - 9 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{90} + (5 \zeta_{12}^{2} + 5 \zeta_{12} - 5) q^{97} + (7 \zeta_{12}^{2} - 7 \zeta_{12} - 7) q^{98} +O(q^{100})$$ q + (z^3 - z^2 - z) * q^2 + 2*z * q^4 + (-2*z^3 - z^2 + z + 2) * q^5 + (-2*z^3 - 2) * q^8 + (3*z^2 - 3) * q^9 + (3*z^3 + z^2 - 3*z - 2) * q^10 + (3*z^3 + 2*z^2 - 3*z) * q^13 + 4*z^2 * q^16 + (-4*z^2 + z - 4) * q^17 + (-3*z^3 + 3) * q^18 + (-2*z^3 - 2*z^2 + 4*z + 4) * q^20 + (-z^3 - 6*z^2 + 3) * q^25 + (-5*z^2 + z + 5) * q^26 + (5*z^3 + 2*z^2 + 5*z) * q^29 + (-4*z^2 - 4*z + 4) * q^32 + (-5*z^3 + 8*z^2 + 8*z - 5) * q^34 + (6*z^3 - 6*z) * q^36 + (-z^3 - z^2 - 6*z + 7) * q^37 + (2*z^3 - 4*z - 6) * q^40 + (-4*z^3 + 4*z^2 - 5*z - 9) * q^41 + (3*z^3 + 6*z^2 + 3*z - 3) * q^45 + (-7*z^3 + 7*z) * q^49 + (4*z^3 + 4*z^2 + 2*z - 6) * q^50 + (4*z^3 - 6) * q^52 + (2*z^3 - 4*z + 7) * q^53 + (-10*z^3 - 7*z^2 + 3*z - 3) * q^58 + (-12*z^3 - 5*z^2 + 6*z + 5) * q^61 + 8*z^3 * q^64 + (4*z^3 + 8*z^2 + z - 1) * q^65 + (-8*z^3 + 2*z^2 - 8*z) * q^68 + (-6*z^2 + 6*z + 6) * q^72 + (8*z^3 - 5*z^2 - 5*z + 8) * q^73 + (14*z^3 - 5*z^2 - 7*z + 5) * q^74 + (-4*z^3 + 4*z^2 + 8*z + 4) * q^80 - 9*z^2 * q^81 + (9*z^2 + z + 9) * q^82 + (11*z^3 - z^2 - 10*z - 10) * q^85 + (-3*z^3 - 3*z^2 + 3*z) * q^89 + (-9*z^3 - 6*z^2 + 3) * q^90 + (5*z^2 + 5*z - 5) * q^97 + (7*z^2 - 7*z - 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 6 q^{5} - 8 q^{8} - 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 6 * q^5 - 8 * q^8 - 6 * q^9 $$4 q - 2 q^{2} + 6 q^{5} - 8 q^{8} - 6 q^{9} - 6 q^{10} + 4 q^{13} + 8 q^{16} - 24 q^{17} + 12 q^{18} + 12 q^{20} + 10 q^{26} + 4 q^{29} + 8 q^{32} - 4 q^{34} + 26 q^{37} - 24 q^{40} - 28 q^{41} - 16 q^{50} - 24 q^{52} + 28 q^{53} - 26 q^{58} + 10 q^{61} + 12 q^{65} + 4 q^{68} + 12 q^{72} + 22 q^{73} + 10 q^{74} + 24 q^{80} - 18 q^{81} + 54 q^{82} - 42 q^{85} - 6 q^{89} - 10 q^{97} - 14 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 + 6 * q^5 - 8 * q^8 - 6 * q^9 - 6 * q^10 + 4 * q^13 + 8 * q^16 - 24 * q^17 + 12 * q^18 + 12 * q^20 + 10 * q^26 + 4 * q^29 + 8 * q^32 - 4 * q^34 + 26 * q^37 - 24 * q^40 - 28 * q^41 - 16 * q^50 - 24 * q^52 + 28 * q^53 - 26 * q^58 + 10 * q^61 + 12 * q^65 + 4 * q^68 + 12 * q^72 + 22 * q^73 + 10 * q^74 + 24 * q^80 - 18 * q^81 + 54 * q^82 - 42 * q^85 - 6 * q^89 - 10 * q^97 - 14 * q^98

## Character values

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

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
−1.36603 + 0.366025i 0 1.73205 1.00000i 2.36603 + 2.36603i 0 0 −2.00000 + 2.00000i −1.50000 2.59808i −4.09808 2.36603i
11.1 0.366025 1.36603i 0 −1.73205 1.00000i 0.633975 + 0.633975i 0 0 −2.00000 + 2.00000i −1.50000 + 2.59808i 1.09808 0.633975i
15.1 −1.36603 0.366025i 0 1.73205 + 1.00000i 2.36603 2.36603i 0 0 −2.00000 2.00000i −1.50000 + 2.59808i −4.09808 + 2.36603i
19.1 0.366025 + 1.36603i 0 −1.73205 + 1.00000i 0.633975 0.633975i 0 0 −2.00000 2.00000i −1.50000 2.59808i 1.09808 + 0.633975i
 $$n$$: 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.f odd 12 1 inner
52.l even 12 1 inner

## Twists

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

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

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 6 T^{3} + 18 T^{2} - 18 T + 9$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 4 T^{3} + 3 T^{2} - 52 T + 169$$
$17$ $$T^{4} + 24 T^{3} + 239 T^{2} + \cdots + 2209$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} - 4 T^{3} + 87 T^{2} + \cdots + 5041$$
$31$ $$T^{4}$$
$37$ $$T^{4} - 26 T^{3} + 233 T^{2} + \cdots + 3721$$
$41$ $$T^{4} + 28 T^{3} + 365 T^{2} + \cdots + 14641$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} - 14 T + 37)^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4} - 10 T^{3} + 183 T^{2} + \cdots + 6889$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 22 T^{3} + 242 T^{2} + \cdots + 529$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324$$
$97$ $$T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 2500$$
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