# Properties

 Label 9025.2.a.a 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 3 q^{9}+O(q^{10})$$ q - 2 * q^2 + 2 * q^4 - 4 * q^7 - 3 * q^9 $$q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 3 q^{9} - q^{11} + 2 q^{13} + 8 q^{14} - 4 q^{16} - 2 q^{17} + 6 q^{18} + 2 q^{22} + 6 q^{23} - 4 q^{26} - 8 q^{28} - 9 q^{29} + 7 q^{31} + 8 q^{32} + 4 q^{34} - 6 q^{36} - 2 q^{37} - 2 q^{41} + 2 q^{43} - 2 q^{44} - 12 q^{46} + 6 q^{47} + 9 q^{49} + 4 q^{52} - 4 q^{53} + 18 q^{58} - 9 q^{59} - 7 q^{61} - 14 q^{62} + 12 q^{63} - 8 q^{64} + 10 q^{67} - 4 q^{68} - q^{71} + 10 q^{73} + 4 q^{74} + 4 q^{77} - q^{79} + 9 q^{81} + 4 q^{82} - 6 q^{83} - 4 q^{86} + 11 q^{89} - 8 q^{91} + 12 q^{92} - 12 q^{94} + 6 q^{97} - 18 q^{98} + 3 q^{99}+O(q^{100})$$ q - 2 * q^2 + 2 * q^4 - 4 * q^7 - 3 * q^9 - q^11 + 2 * q^13 + 8 * q^14 - 4 * q^16 - 2 * q^17 + 6 * q^18 + 2 * q^22 + 6 * q^23 - 4 * q^26 - 8 * q^28 - 9 * q^29 + 7 * q^31 + 8 * q^32 + 4 * q^34 - 6 * q^36 - 2 * q^37 - 2 * q^41 + 2 * q^43 - 2 * q^44 - 12 * q^46 + 6 * q^47 + 9 * q^49 + 4 * q^52 - 4 * q^53 + 18 * q^58 - 9 * q^59 - 7 * q^61 - 14 * q^62 + 12 * q^63 - 8 * q^64 + 10 * q^67 - 4 * q^68 - q^71 + 10 * q^73 + 4 * q^74 + 4 * q^77 - q^79 + 9 * q^81 + 4 * q^82 - 6 * q^83 - 4 * q^86 + 11 * q^89 - 8 * q^91 + 12 * q^92 - 12 * q^94 + 6 * q^97 - 18 * q^98 + 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
−2.00000 0 2.00000 0 0 −4.00000 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

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.a 1
5.b even 2 1 9025.2.a.j 1
5.c odd 4 2 1805.2.b.a 2
19.b odd 2 1 9025.2.a.i 1
19.d odd 6 2 475.2.e.a 2
95.d odd 2 1 9025.2.a.b 1
95.g even 4 2 1805.2.b.b 2
95.h odd 6 2 475.2.e.c 2
95.l even 12 4 95.2.i.a 4
285.w odd 12 4 855.2.be.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.a 4 95.l even 12 4
475.2.e.a 2 19.d odd 6 2
475.2.e.c 2 95.h odd 6 2
855.2.be.a 4 285.w odd 12 4
1805.2.b.a 2 5.c odd 4 2
1805.2.b.b 2 95.g even 4 2
9025.2.a.a 1 1.a even 1 1 trivial
9025.2.a.b 1 95.d odd 2 1
9025.2.a.i 1 19.b odd 2 1
9025.2.a.j 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9025))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{3}$$ T3 $$T_{7} + 4$$ T7 + 4 $$T_{11} + 1$$ T11 + 1 $$T_{29} + 9$$ T29 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T + 1$$
$13$ $$T - 2$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T - 6$$
$29$ $$T + 9$$
$31$ $$T - 7$$
$37$ $$T + 2$$
$41$ $$T + 2$$
$43$ $$T - 2$$
$47$ $$T - 6$$
$53$ $$T + 4$$
$59$ $$T + 9$$
$61$ $$T + 7$$
$67$ $$T - 10$$
$71$ $$T + 1$$
$73$ $$T - 10$$
$79$ $$T + 1$$
$83$ $$T + 6$$
$89$ $$T - 11$$
$97$ $$T - 6$$