# Properties

 Label 725.1.g.a Level $725$ Weight $1$ Character orbit 725.g Analytic conductor $0.362$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ RM discriminant 5 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,1,Mod(476,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("725.476");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 725.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: $$0.361822134159$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.121945.1 Artin image: $C_4^2:C_4$ Artin field: Galois closure of 16.0.145220537353515625.2

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - i q^{4} - i q^{9} +O(q^{10})$$ q - z * q^4 - z * q^9 $$q - i q^{4} - i q^{9} + ( - i - 1) q^{11} - q^{16} + (i + 1) q^{19} - i q^{29} + (i + 1) q^{31} - q^{36} + ( - i + 1) q^{41} + (i - 1) q^{44} - q^{49} + (i + 1) q^{61} + i q^{64} + i q^{71} + ( - i + 1) q^{76} + (i + 1) q^{79} - q^{81} + (i + 1) q^{89} + (i - 1) q^{99} +O(q^{100})$$ q - z * q^4 - z * q^9 + (-z - 1) * q^11 - q^16 + (z + 1) * q^19 - z * q^29 + (z + 1) * q^31 - q^36 + (-z + 1) * q^41 + (z - 1) * q^44 - q^49 + (z + 1) * q^61 + z * q^64 + z * q^71 + (-z + 1) * q^76 + (z + 1) * q^79 - q^81 + (z + 1) * q^89 + (z - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 2 q^{11} - 2 q^{16} + 2 q^{19} + 2 q^{31} - 2 q^{36} + 2 q^{41} - 2 q^{44} - 2 q^{49} + 2 q^{61} + 2 q^{76} + 2 q^{79} - 2 q^{81} + 2 q^{89} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^11 - 2 * q^16 + 2 * q^19 + 2 * q^31 - 2 * q^36 + 2 * q^41 - 2 * q^44 - 2 * q^49 + 2 * q^61 + 2 * q^76 + 2 * q^79 - 2 * q^81 + 2 * q^89 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$$.

 $$n$$ $$176$$ $$552$$ $$\chi(n)$$ $$i$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
476.1
 1.00000i − 1.00000i
0 0 1.00000i 0 0 0 0 1.00000i 0
626.1 0 0 1.00000i 0 0 0 0 1.00000i 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 RM by $$\Q(\sqrt{5})$$
29.c odd 4 1 inner
145.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.1.g.a 2
5.b even 2 1 RM 725.1.g.a 2
5.c odd 4 2 145.1.f.a 2
15.e even 4 2 1305.1.l.a 2
20.e even 4 2 2320.1.bj.a 2
29.c odd 4 1 inner 725.1.g.a 2
145.e even 4 1 145.1.f.a 2
145.f odd 4 1 inner 725.1.g.a 2
145.j even 4 1 145.1.f.a 2
435.i odd 4 1 1305.1.l.a 2
435.t odd 4 1 1305.1.l.a 2
580.i odd 4 1 2320.1.bj.a 2
580.t odd 4 1 2320.1.bj.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.1.f.a 2 5.c odd 4 2
145.1.f.a 2 145.e even 4 1
145.1.f.a 2 145.j even 4 1
725.1.g.a 2 1.a even 1 1 trivial
725.1.g.a 2 5.b even 2 1 RM
725.1.g.a 2 29.c odd 4 1 inner
725.1.g.a 2 145.f odd 4 1 inner
1305.1.l.a 2 15.e even 4 2
1305.1.l.a 2 435.i odd 4 1
1305.1.l.a 2 435.t odd 4 1
2320.1.bj.a 2 20.e even 4 2
2320.1.bj.a 2 580.i odd 4 1
2320.1.bj.a 2 580.t odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(725, [\chi])$$.

## Hecke characteristic polynomials

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