# Properties

 Label 1805.1.h.a Level $1805$ Weight $1$ Character orbit 1805.h Analytic conductor $0.901$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -19, -95, 5 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,1,Mod(69,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("1805.69");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1805.h (of order $$6$$, degree $$2$$, not 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: $$0.900812347803$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{-19})$$ Artin image: $C_3\times D_4$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{4} + \zeta_{6}^{2} q^{5} + \zeta_{6} q^{9}+O(q^{10})$$ q + z * q^4 + z^2 * q^5 + z * q^9 $$q + \zeta_{6} q^{4} + \zeta_{6}^{2} q^{5} + \zeta_{6} q^{9} - 2 q^{11} + \zeta_{6}^{2} q^{16} - q^{20} - \zeta_{6} q^{25} + \zeta_{6}^{2} q^{36} - 2 \zeta_{6} q^{44} - q^{45} + q^{49} - 2 \zeta_{6}^{2} q^{55} + 2 \zeta_{6} q^{61} - q^{64} - \zeta_{6} q^{80} + \zeta_{6}^{2} q^{81} - 2 \zeta_{6} q^{99} +O(q^{100})$$ q + z * q^4 + z^2 * q^5 + z * q^9 - 2 * q^11 + z^2 * q^16 - q^20 - z * q^25 + z^2 * q^36 - 2*z * q^44 - q^45 + q^49 - 2*z^2 * q^55 + 2*z * q^61 - q^64 - z * q^80 + z^2 * q^81 - 2*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{4} - q^{5} + q^{9}+O(q^{10})$$ 2 * q + q^4 - q^5 + q^9 $$2 q + q^{4} - q^{5} + q^{9} - 4 q^{11} - q^{16} - 2 q^{20} - q^{25} - q^{36} - 2 q^{44} - 2 q^{45} + 2 q^{49} + 2 q^{55} + 2 q^{61} - 2 q^{64} - q^{80} - q^{81} - 2 q^{99}+O(q^{100})$$ 2 * q + q^4 - q^5 + q^9 - 4 * q^11 - q^16 - 2 * q^20 - q^25 - q^36 - 2 * q^44 - 2 * q^45 + 2 * q^49 + 2 * q^55 + 2 * q^61 - 2 * q^64 - q^80 - q^81 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times$$.

 $$n$$ $$362$$ $$1446$$ $$\chi(n)$$ $$-1$$ $$\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
69.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0.500000 0.866025i −0.500000 0.866025i 0 0 0 0.500000 0.866025i 0
654.1 0 0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0 0 0.500000 + 0.866025i 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 $$\Q(\sqrt{5})$$
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
19.c even 3 1 inner
19.d odd 6 1 inner
95.h odd 6 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.1.h.a 2
5.b even 2 1 RM 1805.1.h.a 2
19.b odd 2 1 CM 1805.1.h.a 2
19.c even 3 1 95.1.d.a 1
19.c even 3 1 inner 1805.1.h.a 2
19.d odd 6 1 95.1.d.a 1
19.d odd 6 1 inner 1805.1.h.a 2
19.e even 9 6 1805.1.o.a 6
19.f odd 18 6 1805.1.o.a 6
57.f even 6 1 855.1.g.a 1
57.h odd 6 1 855.1.g.a 1
76.f even 6 1 1520.1.m.a 1
76.g odd 6 1 1520.1.m.a 1
95.d odd 2 1 CM 1805.1.h.a 2
95.h odd 6 1 95.1.d.a 1
95.h odd 6 1 inner 1805.1.h.a 2
95.i even 6 1 95.1.d.a 1
95.i even 6 1 inner 1805.1.h.a 2
95.l even 12 2 475.1.c.a 1
95.m odd 12 2 475.1.c.a 1
95.o odd 18 6 1805.1.o.a 6
95.p even 18 6 1805.1.o.a 6
285.n odd 6 1 855.1.g.a 1
285.q even 6 1 855.1.g.a 1
380.p odd 6 1 1520.1.m.a 1
380.s even 6 1 1520.1.m.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.a 1 19.c even 3 1
95.1.d.a 1 19.d odd 6 1
95.1.d.a 1 95.h odd 6 1
95.1.d.a 1 95.i even 6 1
475.1.c.a 1 95.l even 12 2
475.1.c.a 1 95.m odd 12 2
855.1.g.a 1 57.f even 6 1
855.1.g.a 1 57.h odd 6 1
855.1.g.a 1 285.n odd 6 1
855.1.g.a 1 285.q even 6 1
1520.1.m.a 1 76.f even 6 1
1520.1.m.a 1 76.g odd 6 1
1520.1.m.a 1 380.p odd 6 1
1520.1.m.a 1 380.s even 6 1
1805.1.h.a 2 1.a even 1 1 trivial
1805.1.h.a 2 5.b even 2 1 RM
1805.1.h.a 2 19.b odd 2 1 CM
1805.1.h.a 2 19.c even 3 1 inner
1805.1.h.a 2 19.d odd 6 1 inner
1805.1.h.a 2 95.d odd 2 1 CM
1805.1.h.a 2 95.h odd 6 1 inner
1805.1.h.a 2 95.i even 6 1 inner
1805.1.o.a 6 19.e even 9 6
1805.1.o.a 6 19.f odd 18 6
1805.1.o.a 6 95.o odd 18 6
1805.1.o.a 6 95.p even 18 6

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{1}^{\mathrm{new}}(1805, [\chi])$$.

## Hecke characteristic polynomials

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