# Properties

 Label 1805.1.m.a Level $1805$ Weight $1$ Character orbit 1805.m Analytic conductor $0.901$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -19 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,1,Mod(68,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([9, 8]))

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

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

chi := DirichletCharacter("1805.68");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1805.m (of order $$12$$, degree $$4$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - 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 4.2.2375.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{5} q^{4} - \zeta_{12}^{4} q^{5} + ( - \zeta_{12}^{3} + 1) q^{7} + \zeta_{12}^{5} q^{9} +O(q^{10})$$ q - z^5 * q^4 - z^4 * q^5 + (-z^3 + 1) * q^7 + z^5 * q^9 $$q - \zeta_{12}^{5} q^{4} - \zeta_{12}^{4} q^{5} + ( - \zeta_{12}^{3} + 1) q^{7} + \zeta_{12}^{5} q^{9} - \zeta_{12}^{4} q^{16} + (\zeta_{12}^{4} + \zeta_{12}) q^{17} - \zeta_{12}^{3} q^{20} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{23} - \zeta_{12}^{2} q^{25} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{28} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{35} + \zeta_{12}^{4} q^{36} + (\zeta_{12}^{4} - \zeta_{12}) q^{43} + \zeta_{12}^{3} q^{45} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{47} - \zeta_{12}^{3} q^{49} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{63} - \zeta_{12}^{3} q^{64} + (\zeta_{12}^{3} + 1) q^{68} + (\zeta_{12}^{4} - \zeta_{12}) q^{73} - \zeta_{12}^{2} q^{80} - \zeta_{12}^{4} q^{81} + (\zeta_{12}^{3} + 1) q^{83} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{85} + (\zeta_{12}^{4} + \zeta_{12}) q^{92} +O(q^{100})$$ q - z^5 * q^4 - z^4 * q^5 + (-z^3 + 1) * q^7 + z^5 * q^9 - z^4 * q^16 + (z^4 + z) * q^17 - z^3 * q^20 + (z^5 + z^2) * q^23 - z^2 * q^25 + (-z^5 - z^2) * q^28 + (-z^4 - z) * q^35 + z^4 * q^36 + (z^4 - z) * q^43 + z^3 * q^45 + (z^5 - z^2) * q^47 - z^3 * q^49 + (z^5 + z^2) * q^63 - z^3 * q^64 + (z^3 + 1) * q^68 + (z^4 - z) * q^73 - z^2 * q^80 - z^4 * q^81 + (z^3 + 1) * q^83 + (-z^5 + z^2) * q^85 + (z^4 + z) * q^92 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5} + 4 q^{7}+O(q^{10})$$ 4 * q + 2 * q^5 + 4 * q^7 $$4 q + 2 q^{5} + 4 q^{7} + 2 q^{16} - 2 q^{17} + 2 q^{23} - 2 q^{25} - 2 q^{28} + 2 q^{35} - 2 q^{36} - 2 q^{43} - 2 q^{47} + 2 q^{63} + 4 q^{68} - 2 q^{73} - 2 q^{80} + 2 q^{81} + 4 q^{83} + 2 q^{85} - 2 q^{92}+O(q^{100})$$ 4 * q + 2 * q^5 + 4 * q^7 + 2 * q^16 - 2 * q^17 + 2 * q^23 - 2 * q^25 - 2 * q^28 + 2 * q^35 - 2 * q^36 - 2 * q^43 - 2 * q^47 + 2 * q^63 + 4 * q^68 - 2 * q^73 - 2 * q^80 + 2 * q^81 + 4 * q^83 + 2 * q^85 - 2 * q^92

## Character values

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

 $$n$$ $$362$$ $$1446$$ $$\chi(n)$$ $$-\zeta_{12}^{3}$$ $$-\zeta_{12}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
68.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 0.866025 0.500000i 0.500000 0.866025i 0 1.00000 1.00000i 0 −0.866025 + 0.500000i 0
292.1 0 0 0.866025 + 0.500000i 0.500000 + 0.866025i 0 1.00000 + 1.00000i 0 −0.866025 0.500000i 0
653.1 0 0 −0.866025 0.500000i 0.500000 + 0.866025i 0 1.00000 1.00000i 0 0.866025 + 0.500000i 0
1512.1 0 0 −0.866025 + 0.500000i 0.500000 0.866025i 0 1.00000 + 1.00000i 0 0.866025 0.500000i 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 $$\Q(\sqrt{-19})$$
5.c odd 4 1 inner
19.c even 3 1 inner
19.d odd 6 1 inner
95.g even 4 1 inner
95.l even 12 1 inner
95.m odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.1.m.a 4
5.c odd 4 1 inner 1805.1.m.a 4
19.b odd 2 1 CM 1805.1.m.a 4
19.c even 3 1 1805.1.f.a 2
19.c even 3 1 inner 1805.1.m.a 4
19.d odd 6 1 1805.1.f.a 2
19.d odd 6 1 inner 1805.1.m.a 4
19.e even 9 6 1805.1.r.a 12
19.f odd 18 6 1805.1.r.a 12
95.g even 4 1 inner 1805.1.m.a 4
95.l even 12 1 1805.1.f.a 2
95.l even 12 1 inner 1805.1.m.a 4
95.m odd 12 1 1805.1.f.a 2
95.m odd 12 1 inner 1805.1.m.a 4
95.q odd 36 6 1805.1.r.a 12
95.r even 36 6 1805.1.r.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.1.f.a 2 19.c even 3 1
1805.1.f.a 2 19.d odd 6 1
1805.1.f.a 2 95.l even 12 1
1805.1.f.a 2 95.m odd 12 1
1805.1.m.a 4 1.a even 1 1 trivial
1805.1.m.a 4 5.c odd 4 1 inner
1805.1.m.a 4 19.b odd 2 1 CM
1805.1.m.a 4 19.c even 3 1 inner
1805.1.m.a 4 19.d odd 6 1 inner
1805.1.m.a 4 95.g even 4 1 inner
1805.1.m.a 4 95.l even 12 1 inner
1805.1.m.a 4 95.m odd 12 1 inner
1805.1.r.a 12 19.e even 9 6
1805.1.r.a 12 19.f odd 18 6
1805.1.r.a 12 95.q odd 36 6
1805.1.r.a 12 95.r even 36 6

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1805, [\chi])$$.

## Hecke characteristic polynomials

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