# Properties

 Label 225.3.g.d Level $225$ Weight $3$ Character orbit 225.g Analytic conductor $6.131$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,3,Mod(82,225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("225.82");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 225.g (of order $$4$$, degree $$2$$, 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: $$6.13080594811$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - 4 \beta_{2} q^{4} + \beta_1 q^{7}+O(q^{10})$$ q - 4*b2 * q^4 + b1 * q^7 $$q - 4 \beta_{2} q^{4} + \beta_1 q^{7} - 3 \beta_{3} q^{13} - 16 q^{16} - 37 \beta_{2} q^{19} - 4 \beta_{3} q^{28} + 13 q^{31} + 8 \beta_1 q^{37} + 7 \beta_{3} q^{43} + 26 \beta_{2} q^{49} - 12 \beta_1 q^{52} + 47 q^{61} + 64 \beta_{2} q^{64} - 9 \beta_1 q^{67} + 16 \beta_{3} q^{73} - 148 q^{76} + 142 \beta_{2} q^{79} + 225 q^{91} + 11 \beta_1 q^{97}+O(q^{100})$$ q - 4*b2 * q^4 + b1 * q^7 - 3*b3 * q^13 - 16 * q^16 - 37*b2 * q^19 - 4*b3 * q^28 + 13 * q^31 + 8*b1 * q^37 + 7*b3 * q^43 + 26*b2 * q^49 - 12*b1 * q^52 + 47 * q^61 + 64*b2 * q^64 - 9*b1 * q^67 + 16*b3 * q^73 - 148 * q^76 + 142*b2 * q^79 + 225 * q^91 + 11*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 64 q^{16} + 52 q^{31} + 188 q^{61} - 592 q^{76} + 900 q^{91}+O(q^{100})$$ 4 * q - 64 * q^16 + 52 * q^31 + 188 * q^61 - 592 * q^76 + 900 * q^91

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

 $$\beta_{1}$$ $$=$$ $$5\nu$$ 5*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( 5\nu^{3} ) / 3$$ (5*v^3) / 3
 $$\nu$$ $$=$$ $$( \beta_1 ) / 5$$ (b1) / 5 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{3} ) / 5$$ (3*b3) / 5

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
0 0 4.00000i 0 0 −6.12372 6.12372i 0 0 0
82.2 0 0 4.00000i 0 0 6.12372 + 6.12372i 0 0 0
118.1 0 0 4.00000i 0 0 −6.12372 + 6.12372i 0 0 0
118.2 0 0 4.00000i 0 0 6.12372 6.12372i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.3.g.d 4
3.b odd 2 1 CM 225.3.g.d 4
5.b even 2 1 inner 225.3.g.d 4
5.c odd 4 2 inner 225.3.g.d 4
15.d odd 2 1 inner 225.3.g.d 4
15.e even 4 2 inner 225.3.g.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.3.g.d 4 1.a even 1 1 trivial
225.3.g.d 4 3.b odd 2 1 CM
225.3.g.d 4 5.b even 2 1 inner
225.3.g.d 4 5.c odd 4 2 inner
225.3.g.d 4 15.d odd 2 1 inner
225.3.g.d 4 15.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7}^{4} + 5625$$ T7^4 + 5625

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 5625$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 455625$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 1369)^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$(T - 13)^{4}$$
$37$ $$T^{4} + 23040000$$
$41$ $$T^{4}$$
$43$ $$T^{4} + 13505625$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T - 47)^{4}$$
$67$ $$T^{4} + 36905625$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 368640000$$
$79$ $$(T^{2} + 20164)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 82355625$$