# Properties

 Label 2025.2.b.f 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} - \beta q^{7} +O(q^{10})$$ q + 2 * q^4 - b * q^7 $$q + 2 q^{4} - \beta q^{7} + 3 q^{11} - 2 \beta q^{13} + 4 q^{16} - 3 \beta q^{17} + q^{19} + 3 \beta q^{23} - 2 \beta q^{28} - 9 q^{29} - q^{31} - 4 \beta q^{37} - 3 q^{41} - 2 \beta q^{43} + 6 q^{44} + 6 \beta q^{47} + 3 q^{49} - 4 \beta q^{52} - 3 \beta q^{53} + 3 q^{59} - 10 q^{61} + 8 q^{64} - 7 \beta q^{67} - 6 \beta q^{68} + 3 q^{71} + \beta q^{73} + 2 q^{76} - 3 \beta q^{77} + 16 q^{79} + 6 \beta q^{83} + 15 q^{89} - 8 q^{91} + 6 \beta q^{92} + 2 \beta q^{97} +O(q^{100})$$ q + 2 * q^4 - b * q^7 + 3 * q^11 - 2*b * q^13 + 4 * q^16 - 3*b * q^17 + q^19 + 3*b * q^23 - 2*b * q^28 - 9 * q^29 - q^31 - 4*b * q^37 - 3 * q^41 - 2*b * q^43 + 6 * q^44 + 6*b * q^47 + 3 * q^49 - 4*b * q^52 - 3*b * q^53 + 3 * q^59 - 10 * q^61 + 8 * q^64 - 7*b * q^67 - 6*b * q^68 + 3 * q^71 + b * q^73 + 2 * q^76 - 3*b * q^77 + 16 * q^79 + 6*b * q^83 + 15 * q^89 - 8 * q^91 + 6*b * q^92 + 2*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4}+O(q^{10})$$ 2 * q + 4 * q^4 $$2 q + 4 q^{4} + 6 q^{11} + 8 q^{16} + 2 q^{19} - 18 q^{29} - 2 q^{31} - 6 q^{41} + 12 q^{44} + 6 q^{49} + 6 q^{59} - 20 q^{61} + 16 q^{64} + 6 q^{71} + 4 q^{76} + 32 q^{79} + 30 q^{89} - 16 q^{91}+O(q^{100})$$ 2 * q + 4 * q^4 + 6 * q^11 + 8 * q^16 + 2 * q^19 - 18 * q^29 - 2 * q^31 - 6 * q^41 + 12 * q^44 + 6 * q^49 + 6 * q^59 - 20 * q^61 + 16 * q^64 + 6 * q^71 + 4 * q^76 + 32 * q^79 + 30 * q^89 - 16 * q^91

## 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
0 0 2.00000 0 0 2.00000i 0 0 0
649.2 0 0 2.00000 0 0 2.00000i 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.f 2
3.b odd 2 1 2025.2.b.e 2
5.b even 2 1 inner 2025.2.b.f 2
5.c odd 4 1 405.2.a.d yes 1
5.c odd 4 1 2025.2.a.d 1
15.d odd 2 1 2025.2.b.e 2
15.e even 4 1 405.2.a.c 1
15.e even 4 1 2025.2.a.c 1
20.e even 4 1 6480.2.a.o 1
45.k odd 12 2 405.2.e.d 2
45.l even 12 2 405.2.e.e 2
60.l odd 4 1 6480.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.c 1 15.e even 4 1
405.2.a.d yes 1 5.c odd 4 1
405.2.e.d 2 45.k odd 12 2
405.2.e.e 2 45.l even 12 2
2025.2.a.c 1 15.e even 4 1
2025.2.a.d 1 5.c odd 4 1
2025.2.b.e 2 3.b odd 2 1
2025.2.b.e 2 15.d odd 2 1
2025.2.b.f 2 1.a even 1 1 trivial
2025.2.b.f 2 5.b even 2 1 inner
6480.2.a.c 1 60.l odd 4 1
6480.2.a.o 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 $$S_{2}^{\mathrm{new}}(2025, [\chi])$$:

 $$T_{2}$$ T2 $$T_{11} - 3$$ T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T + 9)^{2}$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T + 3)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 3)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 196$$
$71$ $$(T - 3)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 16)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} + 16$$