# Properties

 Label 825.1.l.b Level $825$ Weight $1$ Character orbit 825.l Analytic conductor $0.412$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -11 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,1,Mod(32,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

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

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

chi := DirichletCharacter("825.32");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 825.l (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: $$0.411728635422$$ 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 165) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.12375.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.16471125.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{3} + i q^{4} + q^{9}+O(q^{10})$$ q + q^3 + z * q^4 + q^9 $$q + q^{3} + i q^{4} + q^{9} - i q^{11} + i q^{12} - q^{16} + (i - 1) q^{23} + q^{27} - i q^{33} + i q^{36} + (i - 1) q^{37} + q^{44} + ( - i - 1) q^{47} - q^{48} - i q^{49} + ( - i + 1) q^{53} - q^{59} - i q^{64} + ( - i + 1) q^{67} + (i - 1) q^{69} + q^{81} + ( - i - 1) q^{92} + ( - i + 1) q^{97} - i q^{99} +O(q^{100})$$ q + q^3 + z * q^4 + q^9 - z * q^11 + z * q^12 - q^16 + (z - 1) * q^23 + q^27 - z * q^33 + z * q^36 + (z - 1) * q^37 + q^44 + (-z - 1) * q^47 - q^48 - z * q^49 + (-z + 1) * q^53 - q^59 - z * q^64 + (-z + 1) * q^67 + (z - 1) * q^69 + q^81 + (-z - 1) * q^92 + (-z + 1) * q^97 - z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{9} - 2 q^{16} - 2 q^{23} + 2 q^{27} - 2 q^{37} + 2 q^{44} - 2 q^{47} - 2 q^{48} + 2 q^{53} - 4 q^{59} + 2 q^{67} - 2 q^{69} + 2 q^{81} - 2 q^{92} + 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^9 - 2 * q^16 - 2 * q^23 + 2 * q^27 - 2 * q^37 + 2 * q^44 - 2 * q^47 - 2 * q^48 + 2 * q^53 - 4 * q^59 + 2 * q^67 - 2 * q^69 + 2 * q^81 - 2 * q^92 + 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 − 1.00000i 1.00000i
0 1.00000 1.00000i 0 0 0 0 1.00000 0
593.1 0 1.00000 1.00000i 0 0 0 0 1.00000 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
15.e even 4 1 inner
165.l odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.1.l.b 2
3.b odd 2 1 825.1.l.a 2
5.b even 2 1 165.1.l.a 2
5.c odd 4 1 165.1.l.b yes 2
5.c odd 4 1 825.1.l.a 2
11.b odd 2 1 CM 825.1.l.b 2
15.d odd 2 1 165.1.l.b yes 2
15.e even 4 1 165.1.l.a 2
15.e even 4 1 inner 825.1.l.b 2
20.d odd 2 1 2640.1.ch.b 2
20.e even 4 1 2640.1.ch.a 2
33.d even 2 1 825.1.l.a 2
55.d odd 2 1 165.1.l.a 2
55.e even 4 1 165.1.l.b yes 2
55.e even 4 1 825.1.l.a 2
55.h odd 10 4 1815.1.v.b 8
55.j even 10 4 1815.1.v.b 8
55.k odd 20 4 1815.1.v.a 8
55.l even 20 4 1815.1.v.a 8
60.h even 2 1 2640.1.ch.a 2
60.l odd 4 1 2640.1.ch.b 2
165.d even 2 1 165.1.l.b yes 2
165.l odd 4 1 165.1.l.a 2
165.l odd 4 1 inner 825.1.l.b 2
165.o odd 10 4 1815.1.v.a 8
165.r even 10 4 1815.1.v.a 8
165.u odd 20 4 1815.1.v.b 8
165.v even 20 4 1815.1.v.b 8
220.g even 2 1 2640.1.ch.b 2
220.i odd 4 1 2640.1.ch.a 2
660.g odd 2 1 2640.1.ch.a 2
660.q even 4 1 2640.1.ch.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.1.l.a 2 5.b even 2 1
165.1.l.a 2 15.e even 4 1
165.1.l.a 2 55.d odd 2 1
165.1.l.a 2 165.l odd 4 1
165.1.l.b yes 2 5.c odd 4 1
165.1.l.b yes 2 15.d odd 2 1
165.1.l.b yes 2 55.e even 4 1
165.1.l.b yes 2 165.d even 2 1
825.1.l.a 2 3.b odd 2 1
825.1.l.a 2 5.c odd 4 1
825.1.l.a 2 33.d even 2 1
825.1.l.a 2 55.e even 4 1
825.1.l.b 2 1.a even 1 1 trivial
825.1.l.b 2 11.b odd 2 1 CM
825.1.l.b 2 15.e even 4 1 inner
825.1.l.b 2 165.l odd 4 1 inner
1815.1.v.a 8 55.k odd 20 4
1815.1.v.a 8 55.l even 20 4
1815.1.v.a 8 165.o odd 10 4
1815.1.v.a 8 165.r even 10 4
1815.1.v.b 8 55.h odd 10 4
1815.1.v.b 8 55.j even 10 4
1815.1.v.b 8 165.u odd 20 4
1815.1.v.b 8 165.v even 20 4
2640.1.ch.a 2 20.e even 4 1
2640.1.ch.a 2 60.h even 2 1
2640.1.ch.a 2 220.i odd 4 1
2640.1.ch.a 2 660.g odd 2 1
2640.1.ch.b 2 20.d odd 2 1
2640.1.ch.b 2 60.l odd 4 1
2640.1.ch.b 2 220.g even 2 1
2640.1.ch.b 2 660.q even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23}^{2} + 2T_{23} + 2$$ acting on $$S_{1}^{\mathrm{new}}(825, [\chi])$$.

## Hecke characteristic polynomials

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