# Properties

 Label 2025.2.b.k Level $2025$ Weight $2$ Character orbit 2025.b Analytic conductor $16.170$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 81) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - q^{4} - 2 \beta_1 q^{7} + \beta_{2} q^{8}+O(q^{10})$$ q + b2 * q^2 - q^4 - 2*b1 * q^7 + b2 * q^8 $$q + \beta_{2} q^{2} - q^{4} - 2 \beta_1 q^{7} + \beta_{2} q^{8} + 2 \beta_{3} q^{11} - \beta_1 q^{13} + 2 \beta_{3} q^{14} - 5 q^{16} + 3 \beta_{2} q^{17} - 2 q^{19} + 6 \beta_1 q^{22} - 2 \beta_{2} q^{23} + \beta_{3} q^{26} + 2 \beta_1 q^{28} + \beta_{3} q^{29} + 8 q^{31} - 3 \beta_{2} q^{32} - 9 q^{34} + 7 \beta_1 q^{37} - 2 \beta_{2} q^{38} + 4 \beta_{3} q^{41} + 2 \beta_1 q^{43} - 2 \beta_{3} q^{44} + 6 q^{46} + 4 \beta_{2} q^{47} + 3 q^{49} + \beta_1 q^{52} + 2 \beta_{3} q^{56} + 3 \beta_1 q^{58} + 8 \beta_{3} q^{59} - 7 q^{61} + 8 \beta_{2} q^{62} - q^{64} + 10 \beta_1 q^{67} - 3 \beta_{2} q^{68} + 6 \beta_{3} q^{71} - 7 \beta_1 q^{73} - 7 \beta_{3} q^{74} + 2 q^{76} - 4 \beta_{2} q^{77} - 2 q^{79} + 12 \beta_1 q^{82} + 8 \beta_{2} q^{83} - 2 \beta_{3} q^{86} + 6 \beta_1 q^{88} - 3 \beta_{3} q^{89} - 2 q^{91} + 2 \beta_{2} q^{92} - 12 q^{94} - 2 \beta_1 q^{97} + 3 \beta_{2} q^{98}+O(q^{100})$$ q + b2 * q^2 - q^4 - 2*b1 * q^7 + b2 * q^8 + 2*b3 * q^11 - b1 * q^13 + 2*b3 * q^14 - 5 * q^16 + 3*b2 * q^17 - 2 * q^19 + 6*b1 * q^22 - 2*b2 * q^23 + b3 * q^26 + 2*b1 * q^28 + b3 * q^29 + 8 * q^31 - 3*b2 * q^32 - 9 * q^34 + 7*b1 * q^37 - 2*b2 * q^38 + 4*b3 * q^41 + 2*b1 * q^43 - 2*b3 * q^44 + 6 * q^46 + 4*b2 * q^47 + 3 * q^49 + b1 * q^52 + 2*b3 * q^56 + 3*b1 * q^58 + 8*b3 * q^59 - 7 * q^61 + 8*b2 * q^62 - q^64 + 10*b1 * q^67 - 3*b2 * q^68 + 6*b3 * q^71 - 7*b1 * q^73 - 7*b3 * q^74 + 2 * q^76 - 4*b2 * q^77 - 2 * q^79 + 12*b1 * q^82 + 8*b2 * q^83 - 2*b3 * q^86 + 6*b1 * q^88 - 3*b3 * q^89 - 2 * q^91 + 2*b2 * q^92 - 12 * q^94 - 2*b1 * q^97 + 3*b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} - 20 q^{16} - 8 q^{19} + 32 q^{31} - 36 q^{34} + 24 q^{46} + 12 q^{49} - 28 q^{61} - 4 q^{64} + 8 q^{76} - 8 q^{79} - 8 q^{91} - 48 q^{94}+O(q^{100})$$ 4 * q - 4 * q^4 - 20 * q^16 - 8 * q^19 + 32 * q^31 - 36 * q^34 + 24 * q^46 + 12 * q^49 - 28 * q^61 - 4 * q^64 + 8 * q^76 - 8 * q^79 - 8 * q^91 - 48 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## 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
 −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
649.2 1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
649.3 1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
649.4 1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 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.k 4
3.b odd 2 1 inner 2025.2.b.k 4
5.b even 2 1 inner 2025.2.b.k 4
5.c odd 4 1 81.2.a.a 2
5.c odd 4 1 2025.2.a.j 2
15.d odd 2 1 inner 2025.2.b.k 4
15.e even 4 1 81.2.a.a 2
15.e even 4 1 2025.2.a.j 2
20.e even 4 1 1296.2.a.o 2
35.f even 4 1 3969.2.a.i 2
40.i odd 4 1 5184.2.a.br 2
40.k even 4 1 5184.2.a.bq 2
45.k odd 12 2 81.2.c.b 4
45.l even 12 2 81.2.c.b 4
55.e even 4 1 9801.2.a.v 2
60.l odd 4 1 1296.2.a.o 2
105.k odd 4 1 3969.2.a.i 2
120.q odd 4 1 5184.2.a.bq 2
120.w even 4 1 5184.2.a.br 2
135.q even 36 6 729.2.e.o 12
135.r odd 36 6 729.2.e.o 12
165.l odd 4 1 9801.2.a.v 2
180.v odd 12 2 1296.2.i.s 4
180.x even 12 2 1296.2.i.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.a.a 2 5.c odd 4 1
81.2.a.a 2 15.e even 4 1
81.2.c.b 4 45.k odd 12 2
81.2.c.b 4 45.l even 12 2
729.2.e.o 12 135.q even 36 6
729.2.e.o 12 135.r odd 36 6
1296.2.a.o 2 20.e even 4 1
1296.2.a.o 2 60.l odd 4 1
1296.2.i.s 4 180.v odd 12 2
1296.2.i.s 4 180.x even 12 2
2025.2.a.j 2 5.c odd 4 1
2025.2.a.j 2 15.e even 4 1
2025.2.b.k 4 1.a even 1 1 trivial
2025.2.b.k 4 3.b odd 2 1 inner
2025.2.b.k 4 5.b even 2 1 inner
2025.2.b.k 4 15.d odd 2 1 inner
3969.2.a.i 2 35.f even 4 1
3969.2.a.i 2 105.k odd 4 1
5184.2.a.bq 2 40.k even 4 1
5184.2.a.bq 2 120.q odd 4 1
5184.2.a.br 2 40.i odd 4 1
5184.2.a.br 2 120.w even 4 1
9801.2.a.v 2 55.e even 4 1
9801.2.a.v 2 165.l odd 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 3$$ T2^2 + 3 $$T_{11}^{2} - 12$$ T11^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 3)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} - 12)^{2}$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$(T^{2} + 27)^{2}$$
$19$ $$(T + 2)^{4}$$
$23$ $$(T^{2} + 12)^{2}$$
$29$ $$(T^{2} - 3)^{2}$$
$31$ $$(T - 8)^{4}$$
$37$ $$(T^{2} + 49)^{2}$$
$41$ $$(T^{2} - 48)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$(T^{2} + 48)^{2}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} - 192)^{2}$$
$61$ $$(T + 7)^{4}$$
$67$ $$(T^{2} + 100)^{2}$$
$71$ $$(T^{2} - 108)^{2}$$
$73$ $$(T^{2} + 49)^{2}$$
$79$ $$(T + 2)^{4}$$
$83$ $$(T^{2} + 192)^{2}$$
$89$ $$(T^{2} - 27)^{2}$$
$97$ $$(T^{2} + 4)^{2}$$