# Properties

 Label 1521.2.a.h Level $1521$ Weight $2$ Character orbit 1521.a Self dual yes Analytic conductor $12.145$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## 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: yes Analytic conductor: $$12.1452461474$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{4} + 2 \beta q^{5} - \beta q^{7} +O(q^{10})$$ q - 2 * q^4 + 2*b * q^5 - b * q^7 $$q - 2 q^{4} + 2 \beta q^{5} - \beta q^{7} - 2 \beta q^{11} + 4 q^{16} - 2 \beta q^{19} - 4 \beta q^{20} - 6 q^{23} + 7 q^{25} + 2 \beta q^{28} - 6 q^{29} - \beta q^{31} - 6 q^{35} + 4 \beta q^{41} + q^{43} + 4 \beta q^{44} - 2 \beta q^{47} - 4 q^{49} - 12 q^{53} - 12 q^{55} + 2 \beta q^{59} + q^{61} - 8 q^{64} + 5 \beta q^{67} - 6 \beta q^{71} + \beta q^{73} + 4 \beta q^{76} + 6 q^{77} - 11 q^{79} + 8 \beta q^{80} - 8 \beta q^{83} + 4 \beta q^{89} + 12 q^{92} - 12 q^{95} + 3 \beta q^{97} +O(q^{100})$$ q - 2 * q^4 + 2*b * q^5 - b * q^7 - 2*b * q^11 + 4 * q^16 - 2*b * q^19 - 4*b * q^20 - 6 * q^23 + 7 * q^25 + 2*b * q^28 - 6 * q^29 - b * q^31 - 6 * q^35 + 4*b * q^41 + q^43 + 4*b * q^44 - 2*b * q^47 - 4 * q^49 - 12 * q^53 - 12 * q^55 + 2*b * q^59 + q^61 - 8 * q^64 + 5*b * q^67 - 6*b * q^71 + b * q^73 + 4*b * q^76 + 6 * q^77 - 11 * q^79 + 8*b * q^80 - 8*b * q^83 + 4*b * q^89 + 12 * q^92 - 12 * q^95 + 3*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4}+O(q^{10})$$ 2 * q - 4 * q^4 $$2 q - 4 q^{4} + 8 q^{16} - 12 q^{23} + 14 q^{25} - 12 q^{29} - 12 q^{35} + 2 q^{43} - 8 q^{49} - 24 q^{53} - 24 q^{55} + 2 q^{61} - 16 q^{64} + 12 q^{77} - 22 q^{79} + 24 q^{92} - 24 q^{95}+O(q^{100})$$ 2 * q - 4 * q^4 + 8 * q^16 - 12 * q^23 + 14 * q^25 - 12 * q^29 - 12 * q^35 + 2 * q^43 - 8 * q^49 - 24 * q^53 - 24 * q^55 + 2 * q^61 - 16 * q^64 + 12 * q^77 - 22 * q^79 + 24 * q^92 - 24 * q^95

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
0 0 −2.00000 −3.46410 0 1.73205 0 0 0
1.2 0 0 −2.00000 3.46410 0 −1.73205 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.2.a.h 2
3.b odd 2 1 507.2.a.e 2
12.b even 2 1 8112.2.a.bu 2
13.b even 2 1 inner 1521.2.a.h 2
13.d odd 4 2 1521.2.b.f 2
13.f odd 12 2 117.2.q.a 2
39.d odd 2 1 507.2.a.e 2
39.f even 4 2 507.2.b.c 2
39.h odd 6 2 507.2.e.f 4
39.i odd 6 2 507.2.e.f 4
39.k even 12 2 39.2.j.a 2
39.k even 12 2 507.2.j.b 2
52.l even 12 2 1872.2.by.f 2
156.h even 2 1 8112.2.a.bu 2
156.v odd 12 2 624.2.bv.b 2
195.bc odd 12 2 975.2.w.d 4
195.bh even 12 2 975.2.bc.c 2
195.bn odd 12 2 975.2.w.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 39.k even 12 2
117.2.q.a 2 13.f odd 12 2
507.2.a.e 2 3.b odd 2 1
507.2.a.e 2 39.d odd 2 1
507.2.b.c 2 39.f even 4 2
507.2.e.f 4 39.h odd 6 2
507.2.e.f 4 39.i odd 6 2
507.2.j.b 2 39.k even 12 2
624.2.bv.b 2 156.v odd 12 2
975.2.w.d 4 195.bc odd 12 2
975.2.w.d 4 195.bn odd 12 2
975.2.bc.c 2 195.bh even 12 2
1521.2.a.h 2 1.a even 1 1 trivial
1521.2.a.h 2 13.b even 2 1 inner
1521.2.b.f 2 13.d odd 4 2
1872.2.by.f 2 52.l even 12 2
8112.2.a.bu 2 12.b even 2 1
8112.2.a.bu 2 156.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2}$$ T2 $$T_{5}^{2} - 12$$ T5^2 - 12 $$T_{7}^{2} - 3$$ T7^2 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 12$$
$7$ $$T^{2} - 3$$
$11$ $$T^{2} - 12$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 12$$
$23$ $$(T + 6)^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} - 3$$
$37$ $$T^{2}$$
$41$ $$T^{2} - 48$$
$43$ $$(T - 1)^{2}$$
$47$ $$T^{2} - 12$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} - 12$$
$61$ $$(T - 1)^{2}$$
$67$ $$T^{2} - 75$$
$71$ $$T^{2} - 108$$
$73$ $$T^{2} - 3$$
$79$ $$(T + 11)^{2}$$
$83$ $$T^{2} - 192$$
$89$ $$T^{2} - 48$$
$97$ $$T^{2} - 27$$