Properties

 Label 2025.2.a.a Level $2025$ Weight $2$ Character orbit 2025.a Self dual yes Analytic conductor $16.170$ 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] = [2025,2,Mod(1,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, 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("2025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.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: $$16.1697064093$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 405) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4}+O(q^{10})$$ q - 2 * q^2 + 2 * q^4 $$q - 2 q^{2} + 2 q^{4} + 5 q^{11} - 4 q^{13} - 4 q^{16} + 4 q^{17} - 5 q^{19} - 10 q^{22} - 6 q^{23} + 8 q^{26} - 5 q^{29} - 9 q^{31} + 8 q^{32} - 8 q^{34} + 10 q^{37} + 10 q^{38} + 7 q^{41} + 2 q^{43} + 10 q^{44} + 12 q^{46} - 2 q^{47} - 7 q^{49} - 8 q^{52} - 8 q^{53} + 10 q^{58} - q^{59} - 2 q^{61} + 18 q^{62} - 8 q^{64} - 6 q^{67} + 8 q^{68} + q^{71} + 8 q^{73} - 20 q^{74} - 10 q^{76} + 12 q^{79} - 14 q^{82} - 6 q^{83} - 4 q^{86} - 9 q^{89} - 12 q^{92} + 4 q^{94} - 14 q^{97} + 14 q^{98}+O(q^{100})$$ q - 2 * q^2 + 2 * q^4 + 5 * q^11 - 4 * q^13 - 4 * q^16 + 4 * q^17 - 5 * q^19 - 10 * q^22 - 6 * q^23 + 8 * q^26 - 5 * q^29 - 9 * q^31 + 8 * q^32 - 8 * q^34 + 10 * q^37 + 10 * q^38 + 7 * q^41 + 2 * q^43 + 10 * q^44 + 12 * q^46 - 2 * q^47 - 7 * q^49 - 8 * q^52 - 8 * q^53 + 10 * q^58 - q^59 - 2 * q^61 + 18 * q^62 - 8 * q^64 - 6 * q^67 + 8 * q^68 + q^71 + 8 * q^73 - 20 * q^74 - 10 * q^76 + 12 * q^79 - 14 * q^82 - 6 * q^83 - 4 * q^86 - 9 * q^89 - 12 * q^92 + 4 * q^94 - 14 * q^97 + 14 * q^98

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 0 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
$$3$$ $$1$$
$$5$$ $$1$$

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 2025.2.a.a 1
3.b odd 2 1 2025.2.a.f 1
5.b even 2 1 405.2.a.f yes 1
5.c odd 4 2 2025.2.b.b 2
15.d odd 2 1 405.2.a.a 1
15.e even 4 2 2025.2.b.a 2
20.d odd 2 1 6480.2.a.r 1
45.h odd 6 2 405.2.e.g 2
45.j even 6 2 405.2.e.a 2
60.h even 2 1 6480.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.a 1 15.d odd 2 1
405.2.a.f yes 1 5.b even 2 1
405.2.e.a 2 45.j even 6 2
405.2.e.g 2 45.h odd 6 2
2025.2.a.a 1 1.a even 1 1 trivial
2025.2.a.f 1 3.b odd 2 1
2025.2.b.a 2 15.e even 4 2
2025.2.b.b 2 5.c odd 4 2
6480.2.a.f 1 60.h even 2 1
6480.2.a.r 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2025))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{7}$$ T7 $$T_{11} - 5$$ T11 - 5

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 5$$
$13$ $$T + 4$$
$17$ $$T - 4$$
$19$ $$T + 5$$
$23$ $$T + 6$$
$29$ $$T + 5$$
$31$ $$T + 9$$
$37$ $$T - 10$$
$41$ $$T - 7$$
$43$ $$T - 2$$
$47$ $$T + 2$$
$53$ $$T + 8$$
$59$ $$T + 1$$
$61$ $$T + 2$$
$67$ $$T + 6$$
$71$ $$T - 1$$
$73$ $$T - 8$$
$79$ $$T - 12$$
$83$ $$T + 6$$
$89$ $$T + 9$$
$97$ $$T + 14$$