# Properties

 Label 2025.2.a.q Level $2025$ Weight $2$ Character orbit 2025.a Self dual yes Analytic conductor $16.170$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.11661.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 3$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 225) 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)

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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{3} + \beta_{2} + 1) q^{7} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{8}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b2 - b1 + 2) * q^4 + (-b3 + b2 + 1) * q^7 + (b3 - 2*b2 + b1 - 4) * q^8 $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{3} + \beta_{2} + 1) q^{7} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{8} + (\beta_{3} + \beta_{2}) q^{11} + ( - 2 \beta_{2} - \beta_1 - 1) q^{13} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{14} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{16} + ( - \beta_{3} - \beta_{2} - 3) q^{17} + ( - \beta_{2} - 2 \beta_1 + 1) q^{19} + (\beta_{3} + 2 \beta_1 - 2) q^{22} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{23} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1) q^{26} + (2 \beta_{2} - 2 \beta_1 + 3) q^{28} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{29} + (2 \beta_{3} + \beta_1 - 2) q^{31} + ( - 3 \beta_{2} + 2 \beta_1 - 5) q^{32} + ( - \beta_{3} - 5 \beta_1 + 5) q^{34} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 - 3) q^{37} + ( - \beta_{3} - \beta_{2} - 6) q^{38} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{41} + (2 \beta_{3} - 2 \beta_1 + 3) q^{43} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 7) q^{44} + (2 \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{46} + (2 \beta_{3} - \beta_{2} - 6) q^{47} + (\beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{49} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 6) q^{52} + (3 \beta_{2} + \beta_1 - 4) q^{53} + (3 \beta_1 - 9) q^{56} + ( - 2 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 9) q^{58} + ( - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 7) q^{59} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 - 6) q^{61} + (3 \beta_{2} + 3) q^{62} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 6) q^{64} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 5) q^{67} + (2 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 13) q^{68} + (3 \beta_{2} + \beta_1 - 1) q^{71} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{73} + ( - 2 \beta_{3} + 7 \beta_{2} - 4 \beta_1 + 16) q^{74} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 6) q^{76} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{77} + (\beta_{3} + 2 \beta_{2} - 1) q^{79} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{82} + ( - 3 \beta_1 - 6) q^{83} + (5 \beta_1 - 11) q^{86} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 5) q^{88} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 6) q^{89} + ( - 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{91} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 5) q^{92} + ( - \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 5) q^{94} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 5) q^{97} + ( - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 1) q^{98}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b2 - b1 + 2) * q^4 + (-b3 + b2 + 1) * q^7 + (b3 - 2*b2 + b1 - 4) * q^8 + (b3 + b2) * q^11 + (-2*b2 - b1 - 1) * q^13 + (b3 - 2*b2 + b1 - 1) * q^14 + (-2*b3 + 2*b2 - 3*b1 + 4) * q^16 + (-b3 - b2 - 3) * q^17 + (-b2 - 2*b1 + 1) * q^19 + (b3 + 2*b1 - 2) * q^22 + (-b3 + 2*b2 - b1 - 2) * q^23 + (-2*b3 + b2 - 3*b1) * q^26 + (2*b2 - 2*b1 + 3) * q^28 + (b3 - 2*b2 + 2*b1 - 2) * q^29 + (2*b3 + b1 - 2) * q^31 + (-3*b2 + 2*b1 - 5) * q^32 + (-b3 - 5*b1 + 5) * q^34 + (b3 - 2*b2 + 4*b1 - 3) * q^37 + (-b3 - b2 - 6) * q^38 + (b3 - 2*b2 - b1 - 2) * q^41 + (2*b3 - 2*b1 + 3) * q^43 + (-2*b3 + b2 - b1 + 7) * q^44 + (2*b3 - 4*b2 - b1 - 2) * q^46 + (2*b3 - b2 - 6) * q^47 + (b3 - 3*b2 - b1 - 2) * q^49 + (b3 - 2*b2 + b1 - 6) * q^52 + (3*b2 + b1 - 4) * q^53 + (3*b1 - 9) * q^56 + (-2*b3 + 5*b2 - 3*b1 + 9) * q^58 + (-3*b3 + 3*b2 - b1 + 7) * q^59 + (b3 - 3*b2 + 2*b1 - 6) * q^61 + (3*b2 + 3) * q^62 + (b3 + b2 - 2*b1 + 6) * q^64 + (-b3 + b2 + 3*b1 - 5) * q^67 + (2*b3 - 4*b2 + 4*b1 - 13) * q^68 + (3*b2 + b1 - 1) * q^71 + (-2*b3 - b2 - 3*b1 + 1) * q^73 + (-2*b3 + 7*b2 - 4*b1 + 16) * q^74 + (-b3 + 2*b2 - 4*b1 + 6) * q^76 + (2*b3 + 2*b2 - b1 - 2) * q^77 + (b3 + 2*b2 - 1) * q^79 + (-2*b3 + 2*b2 - 3*b1) * q^82 + (-3*b1 - 6) * q^83 + (5*b1 - 11) * q^86 + (-b3 - 4*b2 + 2*b1 - 5) * q^88 + (3*b3 - 3*b2 + 3*b1 - 6) * q^89 + (-3*b3 + 2*b2 + b1 - 3) * q^91 + (-2*b3 + b2 - 2*b1 + 5) * q^92 + (-b3 + 3*b2 - 5*b1 + 5) * q^94 + (-b3 + 2*b2 + 2*b1 + 5) * q^97 + (-3*b3 + 3*b2 - 4*b1 + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 4 q^{4} + q^{7} - 9 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 + 4 * q^4 + q^7 - 9 * q^8 $$4 q - 2 q^{2} + 4 q^{4} + q^{7} - 9 q^{8} - q^{11} - 2 q^{13} + 3 q^{14} + 4 q^{16} - 11 q^{17} + 2 q^{19} - 3 q^{22} - 15 q^{23} - 10 q^{26} + 4 q^{28} + q^{29} - 4 q^{31} - 10 q^{32} + 9 q^{34} + q^{37} - 23 q^{38} - 5 q^{41} + 10 q^{43} + 22 q^{44} - 20 q^{47} - 3 q^{49} - 17 q^{52} - 20 q^{53} - 30 q^{56} + 18 q^{58} + 17 q^{59} - 13 q^{61} + 6 q^{62} + 19 q^{64} - 17 q^{67} - 34 q^{68} - 8 q^{71} - 2 q^{73} + 40 q^{74} + 11 q^{76} - 12 q^{77} - 7 q^{79} - 12 q^{82} - 30 q^{83} - 34 q^{86} - 9 q^{88} - 9 q^{89} - 17 q^{91} + 12 q^{92} + 3 q^{94} + 19 q^{97} - 13 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 + 4 * q^4 + q^7 - 9 * q^8 - q^11 - 2 * q^13 + 3 * q^14 + 4 * q^16 - 11 * q^17 + 2 * q^19 - 3 * q^22 - 15 * q^23 - 10 * q^26 + 4 * q^28 + q^29 - 4 * q^31 - 10 * q^32 + 9 * q^34 + q^37 - 23 * q^38 - 5 * q^41 + 10 * q^43 + 22 * q^44 - 20 * q^47 - 3 * q^49 - 17 * q^52 - 20 * q^53 - 30 * q^56 + 18 * q^58 + 17 * q^59 - 13 * q^61 + 6 * q^62 + 19 * q^64 - 17 * q^67 - 34 * q^68 - 8 * q^71 - 2 * q^73 + 40 * q^74 + 11 * q^76 - 12 * q^77 - 7 * q^79 - 12 * q^82 - 30 * q^83 - 34 * q^86 - 9 * q^88 - 9 * q^89 - 17 * q^91 + 12 * q^92 + 3 * q^94 + 19 * q^97 - 13 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 4x^{2} + 5x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4\nu + 1$$ v^3 - v^2 - 4*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5\beta _1 + 2$$ b3 + b2 + 5*b1 + 2

## 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
 −1.63372 −0.473255 1.47325 2.63372
−2.63372 0 4.93650 0 0 1.79743 −7.73393 0 0
1.2 −1.47325 0 0.170479 0 0 −3.86583 2.69535 0 0
1.3 0.473255 0 −1.77603 0 0 2.56305 −1.78702 0 0
1.4 1.63372 0 0.669052 0 0 0.505348 −2.17440 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.q 4
3.b odd 2 1 2025.2.a.z 4
5.b even 2 1 2025.2.a.y 4
5.c odd 4 2 2025.2.b.n 8
9.c even 3 2 225.2.e.e yes 8
9.d odd 6 2 675.2.e.c 8
15.d odd 2 1 2025.2.a.p 4
15.e even 4 2 2025.2.b.o 8
45.h odd 6 2 675.2.e.e 8
45.j even 6 2 225.2.e.c 8
45.k odd 12 4 225.2.k.c 16
45.l even 12 4 675.2.k.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.e.c 8 45.j even 6 2
225.2.e.e yes 8 9.c even 3 2
225.2.k.c 16 45.k odd 12 4
675.2.e.c 8 9.d odd 6 2
675.2.e.e 8 45.h odd 6 2
675.2.k.c 16 45.l even 12 4
2025.2.a.p 4 15.d odd 2 1
2025.2.a.q 4 1.a even 1 1 trivial
2025.2.a.y 4 5.b even 2 1
2025.2.a.z 4 3.b odd 2 1
2025.2.b.n 8 5.c odd 4 2
2025.2.b.o 8 15.e even 4 2

## Hecke kernels

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

 $$T_{2}^{4} + 2T_{2}^{3} - 4T_{2}^{2} - 5T_{2} + 3$$ T2^4 + 2*T2^3 - 4*T2^2 - 5*T2 + 3 $$T_{7}^{4} - T_{7}^{3} - 12T_{7}^{2} + 24T_{7} - 9$$ T7^4 - T7^3 - 12*T7^2 + 24*T7 - 9 $$T_{11}^{4} + T_{11}^{3} - 25T_{11}^{2} + 41T_{11} - 9$$ T11^4 + T11^3 - 25*T11^2 + 41*T11 - 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} - 4 T^{2} - 5 T + 3$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - T^{3} - 12 T^{2} + 24 T - 9$$
$11$ $$T^{4} + T^{3} - 25 T^{2} + 41 T - 9$$
$13$ $$T^{4} + 2 T^{3} - 30 T^{2} - 5 T + 107$$
$17$ $$T^{4} + 11 T^{3} + 20 T^{2} + \cdots - 303$$
$19$ $$T^{4} - 2 T^{3} - 27 T^{2} + 80 T - 25$$
$23$ $$T^{4} + 15 T^{3} + 57 T^{2} + \cdots - 243$$
$29$ $$T^{4} - T^{3} - 40 T^{2} + 142 T - 129$$
$31$ $$T^{4} + 4 T^{3} - 42 T^{2} - 27 T + 243$$
$37$ $$T^{4} - T^{3} - 99 T^{2} + 503 T - 647$$
$41$ $$T^{4} + 5 T^{3} - 25 T^{2} - 161 T - 207$$
$43$ $$T^{4} - 10 T^{3} - 48 T^{2} + \cdots - 673$$
$47$ $$T^{4} + 20 T^{3} + 107 T^{2} + \cdots - 381$$
$53$ $$T^{4} + 20 T^{3} + 86 T^{2} + \cdots - 471$$
$59$ $$T^{4} - 17 T^{3} + 2 T^{2} + \cdots - 2313$$
$61$ $$T^{4} + 13 T^{3} - 3 T^{2} - 91 T - 1$$
$67$ $$T^{4} + 17 T^{3} + 36 T^{2} + \cdots + 243$$
$71$ $$T^{4} + 8 T^{3} - 40 T^{2} - 263 T + 381$$
$73$ $$T^{4} + 2 T^{3} - 96 T^{2} + 241 T + 113$$
$79$ $$T^{4} + 7 T^{3} - 33 T^{2} - 69 T + 207$$
$83$ $$T^{4} + 30 T^{3} + 288 T^{2} + \cdots + 729$$
$89$ $$T^{4} + 9 T^{3} - 99 T^{2} + \cdots + 2025$$
$97$ $$T^{4} - 19 T^{3} + 81 T^{2} + \cdots - 953$$