# Properties

 Label 2025.2.b.b Level $2025$ Weight $2$ Character orbit 2025.b Analytic conductor $16.170$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(649,2025)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2025.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$16.1697064093$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 405) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 2 q^{4}+O(q^{10})$$ q + b * q^2 - 2 * q^4 $$q + \beta q^{2} - 2 q^{4} + 5 q^{11} - 2 \beta q^{13} - 4 q^{16} - 2 \beta q^{17} + 5 q^{19} + 5 \beta q^{22} - 3 \beta q^{23} + 8 q^{26} + 5 q^{29} - 9 q^{31} - 4 \beta q^{32} + 8 q^{34} - 5 \beta q^{37} + 5 \beta q^{38} + 7 q^{41} + \beta q^{43} - 10 q^{44} + 12 q^{46} + \beta q^{47} + 7 q^{49} + 4 \beta q^{52} - 4 \beta q^{53} + 5 \beta q^{58} + q^{59} - 2 q^{61} - 9 \beta q^{62} + 8 q^{64} + 3 \beta q^{67} + 4 \beta q^{68} + q^{71} + 4 \beta q^{73} + 20 q^{74} - 10 q^{76} - 12 q^{79} + 7 \beta q^{82} - 3 \beta q^{83} - 4 q^{86} + 9 q^{89} + 6 \beta q^{92} - 4 q^{94} + 7 \beta q^{97} + 7 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 2 * q^4 + 5 * q^11 - 2*b * q^13 - 4 * q^16 - 2*b * q^17 + 5 * q^19 + 5*b * q^22 - 3*b * q^23 + 8 * q^26 + 5 * q^29 - 9 * q^31 - 4*b * q^32 + 8 * q^34 - 5*b * q^37 + 5*b * q^38 + 7 * q^41 + b * q^43 - 10 * q^44 + 12 * q^46 + b * q^47 + 7 * q^49 + 4*b * q^52 - 4*b * q^53 + 5*b * q^58 + q^59 - 2 * q^61 - 9*b * q^62 + 8 * q^64 + 3*b * q^67 + 4*b * q^68 + q^71 + 4*b * q^73 + 20 * q^74 - 10 * q^76 - 12 * q^79 + 7*b * q^82 - 3*b * q^83 - 4 * q^86 + 9 * q^89 + 6*b * q^92 - 4 * q^94 + 7*b * q^97 + 7*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4}+O(q^{10})$$ 2 * q - 4 * q^4 $$2 q - 4 q^{4} + 10 q^{11} - 8 q^{16} + 10 q^{19} + 16 q^{26} + 10 q^{29} - 18 q^{31} + 16 q^{34} + 14 q^{41} - 20 q^{44} + 24 q^{46} + 14 q^{49} + 2 q^{59} - 4 q^{61} + 16 q^{64} + 2 q^{71} + 40 q^{74} - 20 q^{76} - 24 q^{79} - 8 q^{86} + 18 q^{89} - 8 q^{94}+O(q^{100})$$ 2 * q - 4 * q^4 + 10 * q^11 - 8 * q^16 + 10 * q^19 + 16 * q^26 + 10 * q^29 - 18 * q^31 + 16 * q^34 + 14 * q^41 - 20 * q^44 + 24 * q^46 + 14 * q^49 + 2 * q^59 - 4 * q^61 + 16 * q^64 + 2 * q^71 + 40 * q^74 - 20 * q^76 - 24 * q^79 - 8 * q^86 + 18 * q^89 - 8 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$1702$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
2.00000i 0 −2.00000 0 0 0 0 0 0
649.2 2.00000i 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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2025.2.b.b 2
3.b odd 2 1 2025.2.b.a 2
5.b even 2 1 inner 2025.2.b.b 2
5.c odd 4 1 405.2.a.f yes 1
5.c odd 4 1 2025.2.a.a 1
15.d odd 2 1 2025.2.b.a 2
15.e even 4 1 405.2.a.a 1
15.e even 4 1 2025.2.a.f 1
20.e even 4 1 6480.2.a.r 1
45.k odd 12 2 405.2.e.a 2
45.l even 12 2 405.2.e.g 2
60.l odd 4 1 6480.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.a 1 15.e even 4 1
405.2.a.f yes 1 5.c odd 4 1
405.2.e.a 2 45.k odd 12 2
405.2.e.g 2 45.l even 12 2
2025.2.a.a 1 5.c odd 4 1
2025.2.a.f 1 15.e even 4 1
2025.2.b.a 2 3.b odd 2 1
2025.2.b.a 2 15.d odd 2 1
2025.2.b.b 2 1.a even 1 1 trivial
2025.2.b.b 2 5.b even 2 1 inner
6480.2.a.f 1 60.l odd 4 1
6480.2.a.r 1 20.e even 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 4$$ T2^2 + 4 $$T_{11} - 5$$ T11 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 16$$
$19$ $$(T - 5)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 5)^{2}$$
$31$ $$(T + 9)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T - 7)^{2}$$
$43$ $$T^{2} + 4$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 64$$
$59$ $$(T - 1)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 36$$
$71$ $$(T - 1)^{2}$$
$73$ $$T^{2} + 64$$
$79$ $$(T + 12)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 9)^{2}$$
$97$ $$T^{2} + 196$$