# Properties

 Label 1045.1.k.c Level $1045$ Weight $1$ Character orbit 1045.k Analytic conductor $0.522$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ 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(208,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3, 2, 2]))

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

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

chi := DirichletCharacter("1045.208");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1045.k (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: $$0.521522938201$$ 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: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.287375.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - i q^{4} + q^{5} + ( - i + 1) q^{7} - i q^{9} +O(q^{10})$$ q - z * q^4 + q^5 + (-z + 1) * q^7 - z * q^9 $$q - i q^{4} + q^{5} + ( - i + 1) q^{7} - i q^{9} + i q^{11} - q^{16} + (i - 1) q^{17} - q^{19} - i q^{20} + (i - 1) q^{23} + q^{25} + ( - i - 1) q^{28} + ( - i + 1) q^{35} - q^{36} + (i + 1) q^{43} + q^{44} - i q^{45} + (i + 1) q^{47} - i q^{49} + i q^{55} - i q^{61} + ( - i - 1) q^{63} + i q^{64} + (i + 1) q^{68} + (i + 1) q^{73} + i q^{76} + (i + 1) q^{77} - q^{80} - q^{81} + ( - i - 1) q^{83} + (i - 1) q^{85} + (i + 1) q^{92} - q^{95} + q^{99} +O(q^{100})$$ q - z * q^4 + q^5 + (-z + 1) * q^7 - z * q^9 + z * q^11 - q^16 + (z - 1) * q^17 - q^19 - z * q^20 + (z - 1) * q^23 + q^25 + (-z - 1) * q^28 + (-z + 1) * q^35 - q^36 + (z + 1) * q^43 + q^44 - z * q^45 + (z + 1) * q^47 - z * q^49 + z * q^55 - z * q^61 + (-z - 1) * q^63 + z * q^64 + (z + 1) * q^68 + (z + 1) * q^73 + z * q^76 + (z + 1) * q^77 - q^80 - q^81 + (-z - 1) * q^83 + (z - 1) * q^85 + (z + 1) * q^92 - q^95 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + 2 * q^7 $$2 q + 2 q^{5} + 2 q^{7} - 2 q^{16} - 2 q^{17} - 2 q^{19} - 2 q^{23} + 2 q^{25} - 2 q^{28} + 2 q^{35} - 2 q^{36} + 2 q^{43} + 2 q^{44} + 2 q^{47} - 2 q^{63} + 2 q^{68} + 2 q^{73} + 2 q^{77} - 2 q^{80} - 2 q^{81} - 2 q^{83} - 2 q^{85} + 2 q^{92} - 2 q^{95} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^5 + 2 * 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 + 2 * q^44 + 2 * q^47 - 2 * q^63 + 2 * q^68 + 2 * q^73 + 2 * q^77 - 2 * 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 $$\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$$.

 $$n$$ $$496$$ $$761$$ $$837$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
208.1
 − 1.00000i 1.00000i
0 0 1.00000i 1.00000 0 1.00000 + 1.00000i 0 1.00000i 0
417.1 0 0 1.00000i 1.00000 0 1.00000 1.00000i 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
55.e even 4 1 inner
1045.k odd 4 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1045, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7}^{2} - 2T_{7} + 2$$ T7^2 - 2*T7 + 2

## Hecke characteristic polynomials

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