# Properties

 Label 9025.2.a.g Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# 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 95) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} - 2 q^{4} + 4 q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^3 - 2 * q^4 + 4 * q^7 + q^9 $$q + 2 q^{3} - 2 q^{4} + 4 q^{7} + q^{9} + 3 q^{11} - 4 q^{12} - 2 q^{13} + 4 q^{16} - 6 q^{17} + 8 q^{21} - 4 q^{27} - 8 q^{28} - 3 q^{29} - 7 q^{31} + 6 q^{33} - 2 q^{36} - 8 q^{37} - 4 q^{39} - 6 q^{41} + 4 q^{43} - 6 q^{44} - 6 q^{47} + 8 q^{48} + 9 q^{49} - 12 q^{51} + 4 q^{52} + 6 q^{53} - 15 q^{59} + 5 q^{61} + 4 q^{63} - 8 q^{64} - 2 q^{67} + 12 q^{68} - 3 q^{71} - 8 q^{73} + 12 q^{77} + 5 q^{79} - 11 q^{81} - 12 q^{83} - 16 q^{84} - 6 q^{87} - 15 q^{89} - 8 q^{91} - 14 q^{93} - 8 q^{97} + 3 q^{99}+O(q^{100})$$ q + 2 * q^3 - 2 * q^4 + 4 * q^7 + q^9 + 3 * q^11 - 4 * q^12 - 2 * q^13 + 4 * q^16 - 6 * q^17 + 8 * q^21 - 4 * q^27 - 8 * q^28 - 3 * q^29 - 7 * q^31 + 6 * q^33 - 2 * q^36 - 8 * q^37 - 4 * q^39 - 6 * q^41 + 4 * q^43 - 6 * q^44 - 6 * q^47 + 8 * q^48 + 9 * q^49 - 12 * q^51 + 4 * q^52 + 6 * q^53 - 15 * q^59 + 5 * q^61 + 4 * q^63 - 8 * q^64 - 2 * q^67 + 12 * q^68 - 3 * q^71 - 8 * q^73 + 12 * q^77 + 5 * q^79 - 11 * q^81 - 12 * q^83 - 16 * q^84 - 6 * q^87 - 15 * q^89 - 8 * q^91 - 14 * q^93 - 8 * q^97 + 3 * 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 2.00000 −2.00000 0 0 4.00000 0 1.00000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9025.2.a.g 1
5.b even 2 1 1805.2.a.a 1
19.b odd 2 1 9025.2.a.e 1
19.c even 3 2 475.2.e.b 2
95.d odd 2 1 1805.2.a.b 1
95.i even 6 2 95.2.e.a 2
95.m odd 12 4 475.2.j.a 4
285.n odd 6 2 855.2.k.b 2
380.p odd 6 2 1520.2.q.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.a 2 95.i even 6 2
475.2.e.b 2 19.c even 3 2
475.2.j.a 4 95.m odd 12 4
855.2.k.b 2 285.n odd 6 2
1520.2.q.c 2 380.p odd 6 2
1805.2.a.a 1 5.b even 2 1
1805.2.a.b 1 95.d odd 2 1
9025.2.a.e 1 19.b odd 2 1
9025.2.a.g 1 1.a even 1 1 trivial

## 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 $$T_{3} - 2$$ T3 - 2 $$T_{7} - 4$$ T7 - 4 $$T_{11} - 3$$ T11 - 3 $$T_{29} + 3$$ T29 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T - 4$$
$11$ $$T - 3$$
$13$ $$T + 2$$
$17$ $$T + 6$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 3$$
$31$ $$T + 7$$
$37$ $$T + 8$$
$41$ $$T + 6$$
$43$ $$T - 4$$
$47$ $$T + 6$$
$53$ $$T - 6$$
$59$ $$T + 15$$
$61$ $$T - 5$$
$67$ $$T + 2$$
$71$ $$T + 3$$
$73$ $$T + 8$$
$79$ $$T - 5$$
$83$ $$T + 12$$
$89$ $$T + 15$$
$97$ $$T + 8$$