# Properties

 Label 225.3.g.c Level $225$ Weight $3$ Character orbit 225.g Analytic conductor $6.131$ Analytic rank $0$ Dimension $4$ CM discriminant -15 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: $$\Q(i, \sqrt{30})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 225$$ x^4 + 225 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 11 \beta_{2} q^{4} + 7 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 11*b2 * q^4 + 7*b3 * q^8 $$q + \beta_1 q^{2} + 11 \beta_{2} q^{4} + 7 \beta_{3} q^{8} - 61 q^{16} + 8 \beta_1 q^{17} - 22 \beta_{2} q^{19} - 8 \beta_{3} q^{23} - 2 q^{31} - 33 \beta_1 q^{32} + 120 \beta_{2} q^{34} - 22 \beta_{3} q^{38} + 120 q^{46} + 24 \beta_1 q^{47} - 49 \beta_{2} q^{49} - 16 \beta_{3} q^{53} - 118 q^{61} - 2 \beta_1 q^{62} - 251 \beta_{2} q^{64} + 88 \beta_{3} q^{68} + 242 q^{76} - 98 \beta_{2} q^{79} + 16 \beta_{3} q^{83} + 88 \beta_1 q^{92} + 360 \beta_{2} q^{94} - 49 \beta_{3} q^{98}+O(q^{100})$$ q + b1 * q^2 + 11*b2 * q^4 + 7*b3 * q^8 - 61 * q^16 + 8*b1 * q^17 - 22*b2 * q^19 - 8*b3 * q^23 - 2 * q^31 - 33*b1 * q^32 + 120*b2 * q^34 - 22*b3 * q^38 + 120 * q^46 + 24*b1 * q^47 - 49*b2 * q^49 - 16*b3 * q^53 - 118 * q^61 - 2*b1 * q^62 - 251*b2 * q^64 + 88*b3 * q^68 + 242 * q^76 - 98*b2 * q^79 + 16*b3 * q^83 + 88*b1 * q^92 + 360*b2 * q^94 - 49*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 244 q^{16} - 8 q^{31} + 480 q^{46} - 472 q^{61} + 968 q^{76}+O(q^{100})$$ 4 * q - 244 * q^16 - 8 * q^31 + 480 * q^46 - 472 * q^61 + 968 * q^76

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 225$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 15$$ (v^2) / 15 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 15$$ (v^3) / 15
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$15\beta_{2}$$ 15*b2 $$\nu^{3}$$ $$=$$ $$15\beta_{3}$$ 15*b3

## 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
 −2.73861 − 2.73861i 2.73861 + 2.73861i −2.73861 + 2.73861i 2.73861 − 2.73861i
−2.73861 2.73861i 0 11.0000i 0 0 0 19.1703 19.1703i 0 0
82.2 2.73861 + 2.73861i 0 11.0000i 0 0 0 −19.1703 + 19.1703i 0 0
118.1 −2.73861 + 2.73861i 0 11.0000i 0 0 0 19.1703 + 19.1703i 0 0
118.2 2.73861 2.73861i 0 11.0000i 0 0 0 −19.1703 19.1703i 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 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.c 4
3.b odd 2 1 inner 225.3.g.c 4
5.b even 2 1 inner 225.3.g.c 4
5.c odd 4 2 inner 225.3.g.c 4
15.d odd 2 1 CM 225.3.g.c 4
15.e even 4 2 inner 225.3.g.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.3.g.c 4 1.a even 1 1 trivial
225.3.g.c 4 3.b odd 2 1 inner
225.3.g.c 4 5.b even 2 1 inner
225.3.g.c 4 5.c odd 4 2 inner
225.3.g.c 4 15.d odd 2 1 CM
225.3.g.c 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 $$S_{3}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{4} + 225$$ T2^4 + 225 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 225$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 921600$$
$19$ $$(T^{2} + 484)^{2}$$
$23$ $$T^{4} + 921600$$
$29$ $$T^{4}$$
$31$ $$(T + 2)^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4} + 74649600$$
$53$ $$T^{4} + 14745600$$
$59$ $$T^{4}$$
$61$ $$(T + 118)^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} + 9604)^{2}$$
$83$ $$T^{4} + 14745600$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$