# Properties

 Label 2025.2.b.i Level $2025$ Weight $2$ Character orbit 2025.b Analytic conductor $16.170$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - 2) q^{4} + (2 \beta_{2} - \beta_1) q^{7} + 3 \beta_{2} q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 - 2) * q^4 + (2*b2 - b1) * q^7 + 3*b2 * q^8 $$q + \beta_1 q^{2} + (\beta_{3} - 2) q^{4} + (2 \beta_{2} - \beta_1) q^{7} + 3 \beta_{2} q^{8} + (\beta_{3} - 4) q^{11} + ( - 2 \beta_{2} - \beta_1) q^{13} + ( - 3 \beta_{3} + 6) q^{14} + ( - \beta_{3} - 1) q^{16} + \beta_1 q^{17} + ( - 4 \beta_{3} + 2) q^{19} + (3 \beta_{2} - 4 \beta_1) q^{22} + 3 \beta_1 q^{23} + (\beta_{3} + 2) q^{26} + ( - 5 \beta_{2} + 4 \beta_1) q^{28} + ( - 2 \beta_{3} - 4) q^{29} + ( - 4 \beta_{3} + 3) q^{31} + (3 \beta_{2} - \beta_1) q^{32} + (\beta_{3} - 4) q^{34} + (5 \beta_{2} + 3 \beta_1) q^{37} + ( - 12 \beta_{2} + 2 \beta_1) q^{38} + ( - \beta_{3} + 4) q^{41} + ( - 8 \beta_{2} - \beta_1) q^{43} + ( - 5 \beta_{3} + 11) q^{44} + (3 \beta_{3} - 12) q^{46} + (3 \beta_{2} + 4 \beta_1) q^{47} + (5 \beta_{3} - 5) q^{49} - \beta_{2} q^{52} + (6 \beta_{2} + 5 \beta_1) q^{53} + (3 \beta_{3} - 9) q^{56} + ( - 6 \beta_{2} - 4 \beta_1) q^{58} + ( - \beta_{3} - 5) q^{59} + (\beta_{3} + 1) q^{61} + ( - 12 \beta_{2} + 3 \beta_1) q^{62} + ( - 6 \beta_{3} + 5) q^{64} + ( - \beta_{2} + 2 \beta_1) q^{67} + (3 \beta_{2} - 2 \beta_1) q^{68} + (2 \beta_{3} + 7) q^{71} + ( - 5 \beta_{2} + 2 \beta_1) q^{73} + ( - 2 \beta_{3} - 7) q^{74} + (6 \beta_{3} - 16) q^{76} + ( - 9 \beta_{2} + 6 \beta_1) q^{77} - 2 \beta_{3} q^{79} + ( - 3 \beta_{2} + 4 \beta_1) q^{82} - 9 \beta_{2} q^{83} + (7 \beta_{3} - 4) q^{86} + ( - 9 \beta_{2} + 3 \beta_1) q^{88} + 12 q^{89} + \beta_{3} q^{91} + (9 \beta_{2} - 6 \beta_1) q^{92} + (\beta_{3} - 13) q^{94} + 2 \beta_{2} q^{97} + (15 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 - 2) * q^4 + (2*b2 - b1) * q^7 + 3*b2 * q^8 + (b3 - 4) * q^11 + (-2*b2 - b1) * q^13 + (-3*b3 + 6) * q^14 + (-b3 - 1) * q^16 + b1 * q^17 + (-4*b3 + 2) * q^19 + (3*b2 - 4*b1) * q^22 + 3*b1 * q^23 + (b3 + 2) * q^26 + (-5*b2 + 4*b1) * q^28 + (-2*b3 - 4) * q^29 + (-4*b3 + 3) * q^31 + (3*b2 - b1) * q^32 + (b3 - 4) * q^34 + (5*b2 + 3*b1) * q^37 + (-12*b2 + 2*b1) * q^38 + (-b3 + 4) * q^41 + (-8*b2 - b1) * q^43 + (-5*b3 + 11) * q^44 + (3*b3 - 12) * q^46 + (3*b2 + 4*b1) * q^47 + (5*b3 - 5) * q^49 - b2 * q^52 + (6*b2 + 5*b1) * q^53 + (3*b3 - 9) * q^56 + (-6*b2 - 4*b1) * q^58 + (-b3 - 5) * q^59 + (b3 + 1) * q^61 + (-12*b2 + 3*b1) * q^62 + (-6*b3 + 5) * q^64 + (-b2 + 2*b1) * q^67 + (3*b2 - 2*b1) * q^68 + (2*b3 + 7) * q^71 + (-5*b2 + 2*b1) * q^73 + (-2*b3 - 7) * q^74 + (6*b3 - 16) * q^76 + (-9*b2 + 6*b1) * q^77 - 2*b3 * q^79 + (-3*b2 + 4*b1) * q^82 - 9*b2 * q^83 + (7*b3 - 4) * q^86 + (-9*b2 + 3*b1) * q^88 + 12 * q^89 + b3 * q^91 + (9*b2 - 6*b1) * q^92 + (b3 - 13) * q^94 + 2*b2 * q^97 + (15*b2 - 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{4}+O(q^{10})$$ 4 * q - 6 * q^4 $$4 q - 6 q^{4} - 14 q^{11} + 18 q^{14} - 6 q^{16} + 10 q^{26} - 20 q^{29} + 4 q^{31} - 14 q^{34} + 14 q^{41} + 34 q^{44} - 42 q^{46} - 10 q^{49} - 30 q^{56} - 22 q^{59} + 6 q^{61} + 8 q^{64} + 32 q^{71} - 32 q^{74} - 52 q^{76} - 4 q^{79} - 2 q^{86} + 48 q^{89} + 2 q^{91} - 50 q^{94}+O(q^{100})$$ 4 * q - 6 * q^4 - 14 * q^11 + 18 * q^14 - 6 * q^16 + 10 * q^26 - 20 * q^29 + 4 * q^31 - 14 * q^34 + 14 * q^41 + 34 * q^44 - 42 * q^46 - 10 * q^49 - 30 * q^56 - 22 * q^59 + 6 * q^61 + 8 * q^64 + 32 * q^71 - 32 * q^74 - 52 * q^76 - 4 * q^79 - 2 * q^86 + 48 * q^89 + 2 * q^91 - 50 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 3$$ (v^3 + 4*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ b3 - 4 $$\nu^{3}$$ $$=$$ $$3\beta_{2} - 4\beta_1$$ 3*b2 - 4*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
 − 2.30278i − 1.30278i 1.30278i 2.30278i
2.30278i 0 −3.30278 0 0 4.30278i 3.00000i 0 0
649.2 1.30278i 0 0.302776 0 0 0.697224i 3.00000i 0 0
649.3 1.30278i 0 0.302776 0 0 0.697224i 3.00000i 0 0
649.4 2.30278i 0 −3.30278 0 0 4.30278i 3.00000i 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.i 4
3.b odd 2 1 2025.2.b.j 4
5.b even 2 1 inner 2025.2.b.i 4
5.c odd 4 1 2025.2.a.i yes 2
5.c odd 4 1 2025.2.a.k yes 2
15.d odd 2 1 2025.2.b.j 4
15.e even 4 1 2025.2.a.h 2
15.e even 4 1 2025.2.a.l yes 2

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

 $$T_{2}^{4} + 7T_{2}^{2} + 9$$ T2^4 + 7*T2^2 + 9 $$T_{11}^{2} + 7T_{11} + 9$$ T11^2 + 7*T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 9$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 19T^{2} + 9$$
$11$ $$(T^{2} + 7 T + 9)^{2}$$
$13$ $$T^{4} + 11T^{2} + 1$$
$17$ $$T^{4} + 7T^{2} + 9$$
$19$ $$(T^{2} - 52)^{2}$$
$23$ $$T^{4} + 63T^{2} + 729$$
$29$ $$(T^{2} + 10 T + 12)^{2}$$
$31$ $$(T^{2} - 2 T - 51)^{2}$$
$37$ $$T^{4} + 83T^{2} + 289$$
$41$ $$(T^{2} - 7 T + 9)^{2}$$
$43$ $$T^{4} + 119T^{2} + 2809$$
$47$ $$T^{4} + 106T^{2} + 2601$$
$53$ $$T^{4} + 187T^{2} + 4761$$
$59$ $$(T^{2} + 11 T + 27)^{2}$$
$61$ $$(T^{2} - 3 T - 1)^{2}$$
$67$ $$T^{4} + 34T^{2} + 81$$
$71$ $$(T^{2} - 16 T + 51)^{2}$$
$73$ $$T^{4} + 98T^{2} + 529$$
$79$ $$(T^{2} + 2 T - 12)^{2}$$
$83$ $$(T^{2} + 81)^{2}$$
$89$ $$(T - 12)^{4}$$
$97$ $$(T^{2} + 4)^{2}$$