# Properties

 Label 3645.1.n.e Level $3645$ Weight $1$ Character orbit 3645.n Analytic conductor $1.819$ Analytic rank $0$ Dimension $6$ Projective image $D_{3}$ CM discriminant -15 Inner twists $12$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3645,1,Mod(404,3645)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3645, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([5, 9]))

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

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

chi := DirichletCharacter("3645.404");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3645 = 3^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3645.n (of order $$18$$, degree $$6$$, 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: $$1.81909197105$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 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: $C_{18}\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{54} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{8} q^{2} - \zeta_{18}^{4} q^{5} - \zeta_{18}^{6} q^{8} +O(q^{10})$$ q + z^8 * q^2 - z^4 * q^5 - z^6 * q^8 $$q + \zeta_{18}^{8} q^{2} - \zeta_{18}^{4} q^{5} - \zeta_{18}^{6} q^{8} + \zeta_{18}^{3} q^{10} + \zeta_{18}^{5} q^{16} - \zeta_{18}^{3} q^{17} - \zeta_{18}^{6} q^{19} - \zeta_{18}^{7} q^{23} + \zeta_{18}^{8} q^{25} + \zeta_{18}^{7} q^{31} + \zeta_{18}^{4} q^{32} + \zeta_{18}^{2} q^{34} + \zeta_{18}^{5} q^{38} - \zeta_{18} q^{40} + \zeta_{18}^{6} q^{46} - \zeta_{18}^{2} q^{47} + \zeta_{18}^{4} q^{49} - \zeta_{18}^{7} q^{50} + q^{53} - \zeta_{18}^{2} q^{61} - \zeta_{18}^{6} q^{62} - \zeta_{18}^{3} q^{64} - \zeta_{18}^{8} q^{79} + q^{80} + \zeta_{18}^{8} q^{83} + \zeta_{18}^{7} q^{85} + 2 \zeta_{18} q^{94} - \zeta_{18} q^{95} - \zeta_{18}^{3} q^{98} +O(q^{100})$$ q + z^8 * q^2 - z^4 * q^5 - z^6 * q^8 + z^3 * q^10 + z^5 * q^16 - z^3 * q^17 - z^6 * q^19 - z^7 * q^23 + z^8 * q^25 + z^7 * q^31 + z^4 * q^32 + z^2 * q^34 + z^5 * q^38 - z * q^40 + z^6 * q^46 - z^2 * q^47 + z^4 * q^49 - z^7 * q^50 + q^53 - z^2 * q^61 - z^6 * q^62 - z^3 * q^64 - z^8 * q^79 + q^80 + z^8 * q^83 + z^7 * q^85 + 2*z * q^94 - z * q^95 - z^3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{8}+O(q^{10})$$ 6 * q + 3 * q^8 $$6 q + 3 q^{8} + 3 q^{10} - 3 q^{17} + 3 q^{19} - 3 q^{46} + 6 q^{53} + 3 q^{62} - 3 q^{64} + 6 q^{80} - 3 q^{98}+O(q^{100})$$ 6 * q + 3 * q^8 + 3 * q^10 - 3 * q^17 + 3 * q^19 - 3 * q^46 + 6 * q^53 + 3 * q^62 - 3 * q^64 + 6 * q^80 - 3 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3645\mathbb{Z}\right)^\times$$.

 $$n$$ $$731$$ $$2917$$ $$\chi(n)$$ $$\zeta_{18}^{7}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
404.1
 −0.766044 − 0.642788i −0.173648 − 0.984808i 0.939693 − 0.342020i 0.939693 + 0.342020i −0.173648 + 0.984808i −0.766044 + 0.642788i
0.766044 0.642788i 0 0 0.939693 0.342020i 0 0 0.500000 + 0.866025i 0 0.500000 0.866025i
809.1 0.173648 0.984808i 0 0 −0.766044 + 0.642788i 0 0 0.500000 0.866025i 0 0.500000 + 0.866025i
1619.1 −0.939693 0.342020i 0 0 −0.173648 + 0.984808i 0 0 0.500000 + 0.866025i 0 0.500000 0.866025i
2024.1 −0.939693 + 0.342020i 0 0 −0.173648 0.984808i 0 0 0.500000 0.866025i 0 0.500000 + 0.866025i
2834.1 0.173648 + 0.984808i 0 0 −0.766044 0.642788i 0 0 0.500000 + 0.866025i 0 0.500000 0.866025i
3239.1 0.766044 + 0.642788i 0 0 0.939693 + 0.342020i 0 0 0.500000 0.866025i 0 0.500000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3239.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
9.c even 3 2 inner
27.e even 9 3 inner
45.h odd 6 2 inner
135.n odd 18 3 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.1.d.a 1 27.f odd 18 1
135.1.d.a 1 135.p even 18 1
135.1.d.b yes 1 27.e even 9 1
135.1.d.b yes 1 135.n odd 18 1
405.1.h.a 2 27.e even 9 2
405.1.h.a 2 135.n odd 18 2
405.1.h.b 2 27.f odd 18 2
405.1.h.b 2 135.p even 18 2
675.1.c.c 2 135.q even 36 2
675.1.c.c 2 135.r odd 36 2
2025.1.j.c 4 135.q even 36 4
2025.1.j.c 4 135.r odd 36 4
2160.1.c.a 1 108.j odd 18 1
2160.1.c.a 1 540.bb even 18 1
2160.1.c.b 1 108.l even 18 1
2160.1.c.b 1 540.bf odd 18 1
3645.1.n.d 6 3.b odd 2 1
3645.1.n.d 6 5.b even 2 1
3645.1.n.d 6 9.d odd 6 2
3645.1.n.d 6 27.f odd 18 3
3645.1.n.d 6 45.j even 6 2
3645.1.n.d 6 135.p even 18 3
3645.1.n.e 6 1.a even 1 1 trivial
3645.1.n.e 6 9.c even 3 2 inner
3645.1.n.e 6 15.d odd 2 1 CM
3645.1.n.e 6 27.e even 9 3 inner
3645.1.n.e 6 45.h odd 6 2 inner
3645.1.n.e 6 135.n odd 18 3 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + T_{2}^{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3645, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{3} + 1$$
$3$ $$T^{6}$$
$5$ $$T^{6} - T^{3} + 1$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$T^{6}$$
$17$ $$(T^{2} + T + 1)^{3}$$
$19$ $$(T^{2} - T + 1)^{3}$$
$23$ $$T^{6} + T^{3} + 1$$
$29$ $$T^{6}$$
$31$ $$T^{6} - T^{3} + 1$$
$37$ $$T^{6}$$
$41$ $$T^{6}$$
$43$ $$T^{6}$$
$47$ $$T^{6} - 8T^{3} + 64$$
$53$ $$(T - 1)^{6}$$
$59$ $$T^{6}$$
$61$ $$T^{6} - T^{3} + 1$$
$67$ $$T^{6}$$
$71$ $$T^{6}$$
$73$ $$T^{6}$$
$79$ $$T^{6} - T^{3} + 1$$
$83$ $$T^{6} + T^{3} + 1$$
$89$ $$T^{6}$$
$97$ $$T^{6}$$