# Properties

 Label 9025.2.a.f Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $1$ Dimension $1$ CM discriminant -19 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.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: $$72.0649878242$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 361) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{4} - 3 q^{7} - 3 q^{9}+O(q^{10})$$ q - 2 * q^4 - 3 * q^7 - 3 * q^9 $$q - 2 q^{4} - 3 q^{7} - 3 q^{9} - 5 q^{11} + 4 q^{16} + 7 q^{17} + 4 q^{23} + 6 q^{28} + 6 q^{36} + q^{43} + 10 q^{44} - 13 q^{47} + 2 q^{49} + 15 q^{61} + 9 q^{63} - 8 q^{64} - 14 q^{68} + 11 q^{73} + 15 q^{77} + 9 q^{81} + 16 q^{83} - 8 q^{92} + 15 q^{99}+O(q^{100})$$ q - 2 * q^4 - 3 * q^7 - 3 * q^9 - 5 * q^11 + 4 * q^16 + 7 * q^17 + 4 * q^23 + 6 * q^28 + 6 * q^36 + q^43 + 10 * q^44 - 13 * q^47 + 2 * q^49 + 15 * q^61 + 9 * q^63 - 8 * q^64 - 14 * q^68 + 11 * q^73 + 15 * q^77 + 9 * q^81 + 16 * q^83 - 8 * q^92 + 15 * q^99

## 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 −2.00000 0 0 −3.00000 0 −3.00000 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
$$5$$ $$+1$$
$$19$$ $$+1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9025.2.a.f 1
5.b even 2 1 361.2.a.a 1
15.d odd 2 1 3249.2.a.e 1
19.b odd 2 1 CM 9025.2.a.f 1
20.d odd 2 1 5776.2.a.i 1
95.d odd 2 1 361.2.a.a 1
95.h odd 6 2 361.2.c.b 2
95.i even 6 2 361.2.c.b 2
95.o odd 18 6 361.2.e.c 6
95.p even 18 6 361.2.e.c 6
285.b even 2 1 3249.2.a.e 1
380.d even 2 1 5776.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
361.2.a.a 1 5.b even 2 1
361.2.a.a 1 95.d odd 2 1
361.2.c.b 2 95.h odd 6 2
361.2.c.b 2 95.i even 6 2
361.2.e.c 6 95.o odd 18 6
361.2.e.c 6 95.p even 18 6
3249.2.a.e 1 15.d odd 2 1
3249.2.a.e 1 285.b even 2 1
5776.2.a.i 1 20.d odd 2 1
5776.2.a.i 1 380.d even 2 1
9025.2.a.f 1 1.a even 1 1 trivial
9025.2.a.f 1 19.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9025))$$:

 $$T_{2}$$ T2 $$T_{3}$$ T3 $$T_{7} + 3$$ T7 + 3 $$T_{11} + 5$$ T11 + 5 $$T_{29}$$ T29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 3$$
$11$ $$T + 5$$
$13$ $$T$$
$17$ $$T - 7$$
$19$ $$T$$
$23$ $$T - 4$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T - 1$$
$47$ $$T + 13$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 15$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T - 11$$
$79$ $$T$$
$83$ $$T - 16$$
$89$ $$T$$
$97$ $$T$$