# Properties

 Label 1045.1.br.a Level $1045$ Weight $1$ Character orbit 1045.br Analytic conductor $0.522$ Analytic rank $0$ Dimension $8$ Projective image $D_{20}$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,1,Mod(18,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([15, 14, 10]))

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

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

chi := DirichletCharacter("1045.18");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1045.br (of order $$20$$, degree $$8$$, 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: $$0.521522938201$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{20})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{20}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{20} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{20}^{3} q^{4} + \zeta_{20}^{4} q^{5} + ( - \zeta_{20} - 1) q^{7} + \zeta_{20}^{9} q^{9} +O(q^{10})$$ q - z^3 * q^4 + z^4 * q^5 + (-z - 1) * q^7 + z^9 * q^9 $$q - \zeta_{20}^{3} q^{4} + \zeta_{20}^{4} q^{5} + ( - \zeta_{20} - 1) q^{7} + \zeta_{20}^{9} q^{9} - \zeta_{20}^{7} q^{11} + \zeta_{20}^{6} q^{16} + (\zeta_{20}^{5} - \zeta_{20}^{2}) q^{17} - \zeta_{20}^{2} q^{19} - \zeta_{20}^{7} q^{20} + ( - \zeta_{20}^{4} - \zeta_{20}) q^{23} + \zeta_{20}^{8} q^{25} + (\zeta_{20}^{4} + \zeta_{20}^{3}) q^{28} + ( - \zeta_{20}^{5} - \zeta_{20}^{4}) q^{35} + \zeta_{20}^{2} q^{36} + (\zeta_{20}^{9} + \zeta_{20}^{6}) q^{43} - q^{44} - \zeta_{20}^{3} q^{45} + ( - \zeta_{20}^{6} + \zeta_{20}^{3}) q^{47} + (\zeta_{20}^{2} + \zeta_{20} + 1) q^{49} + \zeta_{20} q^{55} + (\zeta_{20}^{7} - \zeta_{20}^{5}) q^{61} + ( - \zeta_{20}^{9} + 1) q^{63} - \zeta_{20}^{9} q^{64} + ( - \zeta_{20}^{8} + \zeta_{20}^{5}) q^{68} + ( - \zeta_{20}^{8} - \zeta_{20}^{3}) q^{73} + \zeta_{20}^{5} q^{76} + (\zeta_{20}^{8} + \zeta_{20}^{7}) q^{77} - q^{80} - \zeta_{20}^{8} q^{81} + ( - \zeta_{20}^{9} + \zeta_{20}^{8}) q^{83} + (\zeta_{20}^{9} - \zeta_{20}^{6}) q^{85} + (\zeta_{20}^{7} + \zeta_{20}^{4}) q^{92} - \zeta_{20}^{6} q^{95} + \zeta_{20}^{6} q^{99} +O(q^{100})$$ q - z^3 * q^4 + z^4 * q^5 + (-z - 1) * q^7 + z^9 * q^9 - z^7 * q^11 + z^6 * q^16 + (z^5 - z^2) * q^17 - z^2 * q^19 - z^7 * q^20 + (-z^4 - z) * q^23 + z^8 * q^25 + (z^4 + z^3) * q^28 + (-z^5 - z^4) * q^35 + z^2 * q^36 + (z^9 + z^6) * q^43 - q^44 - z^3 * q^45 + (-z^6 + z^3) * q^47 + (z^2 + z + 1) * q^49 + z * q^55 + (z^7 - z^5) * q^61 + (-z^9 + 1) * q^63 - z^9 * q^64 + (-z^8 + z^5) * q^68 + (-z^8 - z^3) * q^73 + z^5 * q^76 + (z^8 + z^7) * q^77 - q^80 - z^8 * q^81 + (-z^9 + z^8) * q^83 + (z^9 - z^6) * q^85 + (z^7 + z^4) * q^92 - z^6 * q^95 + z^6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{5} - 8 q^{7}+O(q^{10})$$ 8 * q - 2 * q^5 - 8 * q^7 $$8 q - 2 q^{5} - 8 q^{7} + 2 q^{16} - 2 q^{17} - 2 q^{19} + 2 q^{23} - 2 q^{25} - 2 q^{28} + 2 q^{35} + 2 q^{36} + 2 q^{43} - 8 q^{44} - 2 q^{47} + 10 q^{49} + 8 q^{63} + 2 q^{68} + 2 q^{73} - 2 q^{77} - 8 q^{80} + 2 q^{81} - 2 q^{83} - 2 q^{85} - 2 q^{92} - 2 q^{95} + 2 q^{99}+O(q^{100})$$ 8 * q - 2 * q^5 - 8 * q^7 + 2 * q^16 - 2 * q^17 - 2 * q^19 + 2 * q^23 - 2 * q^25 - 2 * q^28 + 2 * q^35 + 2 * q^36 + 2 * q^43 - 8 * q^44 - 2 * q^47 + 10 * q^49 + 8 * q^63 + 2 * q^68 + 2 * q^73 - 2 * q^77 - 8 * q^80 + 2 * q^81 - 2 * q^83 - 2 * q^85 - 2 * q^92 - 2 * q^95 + 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$$.

 $$n$$ $$496$$ $$761$$ $$837$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{20}^{8}$$ $$-\zeta_{20}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 −0.587785 − 0.809017i 0.587785 + 0.809017i 0.951057 + 0.309017i −0.951057 + 0.309017i −0.951057 − 0.309017i 0.951057 − 0.309017i 0.587785 − 0.809017i −0.587785 + 0.809017i
0 0 −0.951057 + 0.309017i −0.809017 0.587785i 0 −0.412215 + 0.809017i 0 0.587785 0.809017i 0
227.1 0 0 0.951057 0.309017i −0.809017 0.587785i 0 −1.58779 0.809017i 0 −0.587785 + 0.809017i 0
303.1 0 0 −0.587785 0.809017i 0.309017 + 0.951057i 0 −1.95106 0.309017i 0 −0.951057 + 0.309017i 0
398.1 0 0 0.587785 0.809017i 0.309017 0.951057i 0 −0.0489435 0.309017i 0 0.951057 + 0.309017i 0
512.1 0 0 0.587785 + 0.809017i 0.309017 + 0.951057i 0 −0.0489435 + 0.309017i 0 0.951057 0.309017i 0
607.1 0 0 −0.587785 + 0.809017i 0.309017 0.951057i 0 −1.95106 + 0.309017i 0 −0.951057 0.309017i 0
778.1 0 0 0.951057 + 0.309017i −0.809017 + 0.587785i 0 −1.58779 + 0.809017i 0 −0.587785 0.809017i 0
987.1 0 0 −0.951057 0.309017i −0.809017 + 0.587785i 0 −0.412215 0.809017i 0 0.587785 + 0.809017i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 18.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
55.l even 20 1 inner
1045.br odd 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.1.br.a 8
5.c odd 4 1 1045.1.br.b yes 8
11.d odd 10 1 1045.1.br.b yes 8
19.b odd 2 1 CM 1045.1.br.a 8
55.l even 20 1 inner 1045.1.br.a 8
95.g even 4 1 1045.1.br.b yes 8
209.k even 10 1 1045.1.br.b yes 8
1045.br odd 20 1 inner 1045.1.br.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.br.a 8 1.a even 1 1 trivial
1045.1.br.a 8 19.b odd 2 1 CM
1045.1.br.a 8 55.l even 20 1 inner
1045.1.br.a 8 1045.br odd 20 1 inner
1045.1.br.b yes 8 5.c odd 4 1
1045.1.br.b yes 8 11.d odd 10 1
1045.1.br.b yes 8 95.g even 4 1
1045.1.br.b yes 8 209.k even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 8T_{7}^{7} + 27T_{7}^{6} + 50T_{7}^{5} + 56T_{7}^{4} + 40T_{7}^{3} + 18T_{7}^{2} + 4T_{7} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1045, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$7$ $$T^{8} + 8 T^{7} + 27 T^{6} + 50 T^{5} + \cdots + 1$$
$11$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$13$ $$T^{8}$$
$17$ $$T^{8} + 2 T^{7} + 7 T^{6} + 10 T^{5} + \cdots + 1$$
$19$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$23$ $$T^{8} - 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8} - 2 T^{7} + 2 T^{6} + 11 T^{4} + \cdots + 1$$
$47$ $$T^{8} + 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 1$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8} - 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 16$$
$79$ $$T^{8}$$
$83$ $$T^{8} + 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 1$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$