# Properties

 Label 1521.4.a.g Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,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, 4, names="a")

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$89.7419051187$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) Fricke sign: $$+1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 8 q^{4} - 20 q^{7}+O(q^{10})$$ q - 8 * q^4 - 20 * q^7 $$q - 8 q^{4} - 20 q^{7} + 64 q^{16} - 56 q^{19} - 125 q^{25} + 160 q^{28} - 308 q^{31} - 110 q^{37} - 520 q^{43} + 57 q^{49} + 182 q^{61} - 512 q^{64} + 880 q^{67} - 1190 q^{73} + 448 q^{76} + 884 q^{79} + 1330 q^{97}+O(q^{100})$$ q - 8 * q^4 - 20 * q^7 + 64 * q^16 - 56 * q^19 - 125 * q^25 + 160 * q^28 - 308 * q^31 - 110 * q^37 - 520 * q^43 + 57 * q^49 + 182 * q^61 - 512 * q^64 + 880 * q^67 - 1190 * q^73 + 448 * q^76 + 884 * q^79 + 1330 * q^97

## 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
 0
0 0 −8.00000 0 0 −20.0000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.4.a.g 1
3.b odd 2 1 CM 1521.4.a.g 1
13.b even 2 1 9.4.a.a 1
39.d odd 2 1 9.4.a.a 1
52.b odd 2 1 144.4.a.d 1
65.d even 2 1 225.4.a.d 1
65.h odd 4 2 225.4.b.g 2
91.b odd 2 1 441.4.a.f 1
91.r even 6 2 441.4.e.i 2
91.s odd 6 2 441.4.e.j 2
104.e even 2 1 576.4.a.m 1
104.h odd 2 1 576.4.a.l 1
117.n odd 6 2 81.4.c.b 2
117.t even 6 2 81.4.c.b 2
143.d odd 2 1 1089.4.a.g 1
156.h even 2 1 144.4.a.d 1
195.e odd 2 1 225.4.a.d 1
195.s even 4 2 225.4.b.g 2
273.g even 2 1 441.4.a.f 1
273.w odd 6 2 441.4.e.i 2
273.ba even 6 2 441.4.e.j 2
312.b odd 2 1 576.4.a.m 1
312.h even 2 1 576.4.a.l 1
429.e even 2 1 1089.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 13.b even 2 1
9.4.a.a 1 39.d odd 2 1
81.4.c.b 2 117.n odd 6 2
81.4.c.b 2 117.t even 6 2
144.4.a.d 1 52.b odd 2 1
144.4.a.d 1 156.h even 2 1
225.4.a.d 1 65.d even 2 1
225.4.a.d 1 195.e odd 2 1
225.4.b.g 2 65.h odd 4 2
225.4.b.g 2 195.s even 4 2
441.4.a.f 1 91.b odd 2 1
441.4.a.f 1 273.g even 2 1
441.4.e.i 2 91.r even 6 2
441.4.e.i 2 273.w odd 6 2
441.4.e.j 2 91.s odd 6 2
441.4.e.j 2 273.ba even 6 2
576.4.a.l 1 104.h odd 2 1
576.4.a.l 1 312.h even 2 1
576.4.a.m 1 104.e even 2 1
576.4.a.m 1 312.b odd 2 1
1089.4.a.g 1 143.d odd 2 1
1089.4.a.g 1 429.e even 2 1
1521.4.a.g 1 1.a even 1 1 trivial
1521.4.a.g 1 3.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{7} + 20$$ T7 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 20$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T + 56$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 308$$
$37$ $$T + 110$$
$41$ $$T$$
$43$ $$T + 520$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 182$$
$67$ $$T - 880$$
$71$ $$T$$
$73$ $$T + 1190$$
$79$ $$T - 884$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T - 1330$$