# Properties

 Label 405.1.h.a Level $405$ Weight $1$ Character orbit 405.h Analytic conductor $0.202$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -15 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,1,Mod(134,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("405.134");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 405.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: $$0.202121330116$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 135) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.135.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of 12.0.6053445140625.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} - \zeta_{6}^{2} q^{5} - q^{8} +O(q^{10})$$ q - z * q^2 - z^2 * q^5 - q^8 $$q - \zeta_{6} q^{2} - \zeta_{6}^{2} q^{5} - q^{8} - q^{10} + \zeta_{6} q^{16} + q^{17} - q^{19} + \zeta_{6}^{2} q^{23} - \zeta_{6} q^{25} - \zeta_{6}^{2} q^{31} - \zeta_{6} q^{34} + \zeta_{6} q^{38} + \zeta_{6}^{2} q^{40} + q^{46} + \zeta_{6} q^{47} + \zeta_{6}^{2} q^{49} + \zeta_{6}^{2} q^{50} + q^{53} + \zeta_{6} q^{61} - q^{62} + q^{64} + \zeta_{6} q^{79} + q^{80} - \zeta_{6} q^{83} - \zeta_{6}^{2} q^{85} - 2 \zeta_{6}^{2} q^{94} + \zeta_{6}^{2} q^{95} + q^{98} +O(q^{100})$$ q - z * q^2 - z^2 * q^5 - q^8 - q^10 + z * q^16 + q^17 - q^19 + z^2 * q^23 - z * q^25 - z^2 * q^31 - z * q^34 + z * q^38 + z^2 * q^40 + q^46 + z * q^47 + z^2 * q^49 + z^2 * q^50 + q^53 + z * q^61 - q^62 + q^64 + z * q^79 + q^80 - z * q^83 - z^2 * q^85 - 2*z^2 * q^94 + z^2 * q^95 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q - q^2 + q^5 - 2 * q^8 $$2 q - q^{2} + q^{5} - 2 q^{8} - 2 q^{10} + q^{16} + 2 q^{17} - 2 q^{19} - q^{23} - q^{25} + q^{31} - q^{34} + q^{38} - q^{40} + 2 q^{46} + 2 q^{47} - q^{49} - q^{50} + 2 q^{53} + q^{61} - 2 q^{62} + 2 q^{64} + q^{79} + 2 q^{80} - q^{83} + q^{85} + 2 q^{94} - q^{95} + 2 q^{98}+O(q^{100})$$ 2 * q - q^2 + q^5 - 2 * q^8 - 2 * q^10 + q^16 + 2 * q^17 - 2 * q^19 - q^23 - q^25 + q^31 - q^34 + q^38 - q^40 + 2 * q^46 + 2 * q^47 - q^49 - q^50 + 2 * q^53 + q^61 - 2 * q^62 + 2 * q^64 + q^79 + 2 * q^80 - q^83 + q^85 + 2 * q^94 - q^95 + 2 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$$.

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{6}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
134.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i 0 0 0.500000 0.866025i 0 0 −1.00000 0 −1.00000
269.1 −0.500000 + 0.866025i 0 0 0.500000 + 0.866025i 0 0 −1.00000 0 −1.00000
 $$n$$: 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
9.c even 3 1 inner
45.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.1.h.a 2
3.b odd 2 1 405.1.h.b 2
5.b even 2 1 405.1.h.b 2
5.c odd 4 2 2025.1.j.c 4
9.c even 3 1 135.1.d.b yes 1
9.c even 3 1 inner 405.1.h.a 2
9.d odd 6 1 135.1.d.a 1
9.d odd 6 1 405.1.h.b 2
15.d odd 2 1 CM 405.1.h.a 2
15.e even 4 2 2025.1.j.c 4
27.e even 9 6 3645.1.n.e 6
27.f odd 18 6 3645.1.n.d 6
36.f odd 6 1 2160.1.c.a 1
36.h even 6 1 2160.1.c.b 1
45.h odd 6 1 135.1.d.b yes 1
45.h odd 6 1 inner 405.1.h.a 2
45.j even 6 1 135.1.d.a 1
45.j even 6 1 405.1.h.b 2
45.k odd 12 2 675.1.c.c 2
45.k odd 12 2 2025.1.j.c 4
45.l even 12 2 675.1.c.c 2
45.l even 12 2 2025.1.j.c 4
135.n odd 18 6 3645.1.n.e 6
135.p even 18 6 3645.1.n.d 6
180.n even 6 1 2160.1.c.a 1
180.p odd 6 1 2160.1.c.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.1.d.a 1 9.d odd 6 1
135.1.d.a 1 45.j even 6 1
135.1.d.b yes 1 9.c even 3 1
135.1.d.b yes 1 45.h odd 6 1
405.1.h.a 2 1.a even 1 1 trivial
405.1.h.a 2 9.c even 3 1 inner
405.1.h.a 2 15.d odd 2 1 CM
405.1.h.a 2 45.h odd 6 1 inner
405.1.h.b 2 3.b odd 2 1
405.1.h.b 2 5.b even 2 1
405.1.h.b 2 9.d odd 6 1
405.1.h.b 2 45.j even 6 1
675.1.c.c 2 45.k odd 12 2
675.1.c.c 2 45.l even 12 2
2025.1.j.c 4 5.c odd 4 2
2025.1.j.c 4 15.e even 4 2
2025.1.j.c 4 45.k odd 12 2
2025.1.j.c 4 45.l even 12 2
2160.1.c.a 1 36.f odd 6 1
2160.1.c.a 1 180.n even 6 1
2160.1.c.b 1 36.h even 6 1
2160.1.c.b 1 180.p odd 6 1
3645.1.n.d 6 27.f odd 18 6
3645.1.n.d 6 135.p even 18 6
3645.1.n.e 6 27.e even 9 6
3645.1.n.e 6 135.n odd 18 6

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(405, [\chi])$$.

## Hecke characteristic polynomials

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