# Properties

 Label 1215.1.h.a Level $1215$ Weight $1$ Character orbit 1215.h Analytic conductor $0.606$ Analytic rank $0$ Dimension $6$ Projective image $D_{9}$ CM discriminant -15 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1215,1,Mod(404,1215)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1215, 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("1215.404");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1215 = 3^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1215.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: $$0.606363990349$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.242137805625.3

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{2} + (\zeta_{18}^{8} - \zeta_{18}^{7} - \zeta_{18}^{3}) q^{4} - \zeta_{18}^{3} q^{5} + ( - \zeta_{18}^{7} + \zeta_{18}^{6} - \zeta_{18}^{3} - \zeta_{18}^{2}) q^{8} +O(q^{10})$$ q + (z^8 + z^4) * q^2 + (z^8 - z^7 - z^3) * q^4 - z^3 * q^5 + (-z^7 + z^6 - z^3 - z^2) * q^8 $$q + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{2} + (\zeta_{18}^{8} - \zeta_{18}^{7} - \zeta_{18}^{3}) q^{4} - \zeta_{18}^{3} q^{5} + ( - \zeta_{18}^{7} + \zeta_{18}^{6} - \zeta_{18}^{3} - \zeta_{18}^{2}) q^{8} + ( - \zeta_{18}^{7} + \zeta_{18}^{2}) q^{10} + ( - \zeta_{18}^{7} + \zeta_{18}^{6} - \zeta_{18}^{5} - \zeta_{18}^{2} + \zeta_{18}) q^{16} + (\zeta_{18}^{8} - \zeta_{18}) q^{17} + (\zeta_{18}^{8} - \zeta_{18}) q^{19} + (\zeta_{18}^{6} + \zeta_{18}^{2} - \zeta_{18}) q^{20} + (\zeta_{18}^{8} - \zeta_{18}^{7}) q^{23} + \zeta_{18}^{6} q^{25} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{31} + (\zeta_{18}^{6} - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 1) q^{32} + ( - \zeta_{18}^{7} - \zeta_{18}^{5} - \zeta_{18}^{3} + 1) q^{34} + ( - \zeta_{18}^{7} - \zeta_{18}^{5} - \zeta_{18}^{3} + 1) q^{38} + (\zeta_{18}^{6} - \zeta_{18}^{5} - \zeta_{18} + 1) q^{40} + ( - \zeta_{18}^{7} + \zeta_{18}^{6} - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{46} - \zeta_{18}^{6} q^{47} - \zeta_{18}^{3} q^{49} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{50} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{53} + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{61} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 2) q^{62} + (\zeta_{18}^{8} + \zeta_{18}^{6} - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18} - 1) q^{64} + (\zeta_{18}^{8} - \zeta_{18}^{7} + \zeta_{18}^{6} + \zeta_{18}^{4} + \zeta_{18}^{2} + 1) q^{68} + (\zeta_{18}^{8} - \zeta_{18}^{7} + \zeta_{18}^{6} + \zeta_{18}^{4} + \zeta_{18}^{2} + 1) q^{76} + (\zeta_{18}^{2} - \zeta_{18}) q^{79} + (\zeta_{18}^{8} - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18} + 1) q^{80} + ( - \zeta_{18}^{7} - \zeta_{18}^{5}) q^{83} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{85} + ( - \zeta_{18}^{7} + 2 \zeta_{18}^{6} - \zeta_{18}^{5} + \zeta_{18}^{2} - \zeta_{18}) q^{92} + (\zeta_{18}^{5} + \zeta_{18}) q^{94} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{95} + ( - \zeta_{18}^{7} + \zeta_{18}^{2}) q^{98} +O(q^{100})$$ q + (z^8 + z^4) * q^2 + (z^8 - z^7 - z^3) * q^4 - z^3 * q^5 + (-z^7 + z^6 - z^3 - z^2) * q^8 + (-z^7 + z^2) * q^10 + (-z^7 + z^6 - z^5 - z^2 + z) * q^16 + (z^8 - z) * q^17 + (z^8 - z) * q^19 + (z^6 + z^2 - z) * q^20 + (z^8 - z^7) * q^23 + z^6 * q^25 + (-z^5 - z) * q^31 + (z^6 - z^5 + z^4 + z^2 + z - 1) * q^32 + (-z^7 - z^5 - z^3 + 1) * q^34 + (-z^7 - z^5 - z^3 + 1) * q^38 + (z^6 - z^5 - z + 1) * q^40 + (-z^7 + z^6 - z^3 + z^2) * q^46 - z^6 * q^47 - z^3 * q^49 + (-z^5 - z) * q^50 + (-z^5 + z^4) * q^53 + (z^8 + z^4) * q^61 + (-z^5 + z^4 + 2) * q^62 + (z^8 + z^6 - z^5 + z^4 - z^3 + z - 1) * q^64 + (z^8 - z^7 + z^6 + z^4 + z^2 + 1) * q^68 + (z^8 - z^7 + z^6 + z^4 + z^2 + 1) * q^76 + (z^2 - z) * q^79 + (z^8 - z^5 + z^4 - z + 1) * q^80 + (-z^7 - z^5) * q^83 + (z^4 + z^2) * q^85 + (-z^7 + 2*z^6 - z^5 + z^2 - z) * q^92 + (z^5 + z) * q^94 + (z^4 + z^2) * q^95 + (-z^7 + z^2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{4} - 3 q^{5} - 6 q^{8}+O(q^{10})$$ 6 * q - 3 * q^4 - 3 * q^5 - 6 * q^8 $$6 q - 3 q^{4} - 3 q^{5} - 6 q^{8} - 3 q^{16} - 3 q^{20} - 3 q^{25} + 3 q^{32} + 3 q^{34} + 3 q^{38} + 3 q^{40} - 6 q^{46} + 3 q^{47} - 3 q^{49} + 12 q^{62} + 3 q^{68} + 3 q^{76} + 6 q^{80} - 6 q^{92}+O(q^{100})$$ 6 * q - 3 * q^4 - 3 * q^5 - 6 * q^8 - 3 * q^16 - 3 * q^20 - 3 * q^25 + 3 * q^32 + 3 * q^34 + 3 * q^38 + 3 * q^40 - 6 * q^46 + 3 * q^47 - 3 * q^49 + 12 * q^62 + 3 * q^68 + 3 * q^76 + 6 * q^80 - 6 * q^92

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1215\mathbb{Z}\right)^\times$$.

 $$n$$ $$487$$ $$731$$ $$\chi(n)$$ $$-1$$ $$\zeta_{18}^{3}$$

## 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
 0.939693 − 0.342020i −0.766044 − 0.642788i −0.173648 + 0.984808i 0.939693 + 0.342020i −0.766044 + 0.642788i −0.173648 − 0.984808i
−0.766044 1.32683i 0 −0.673648 + 1.16679i −0.500000 + 0.866025i 0 0 0.532089 0 1.53209
404.2 −0.173648 0.300767i 0 0.439693 0.761570i −0.500000 + 0.866025i 0 0 −0.652704 0 0.347296
404.3 0.939693 + 1.62760i 0 −1.26604 + 2.19285i −0.500000 + 0.866025i 0 0 −2.87939 0 −1.87939
809.1 −0.766044 + 1.32683i 0 −0.673648 1.16679i −0.500000 0.866025i 0 0 0.532089 0 1.53209
809.2 −0.173648 + 0.300767i 0 0.439693 + 0.761570i −0.500000 0.866025i 0 0 −0.652704 0 0.347296
809.3 0.939693 1.62760i 0 −1.26604 2.19285i −0.500000 0.866025i 0 0 −2.87939 0 −1.87939
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 404.3 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 1215.1.h.a 6
3.b odd 2 1 1215.1.h.b 6
5.b even 2 1 1215.1.h.b 6
9.c even 3 1 1215.1.d.b yes 3
9.c even 3 1 inner 1215.1.h.a 6
9.d odd 6 1 1215.1.d.a 3
9.d odd 6 1 1215.1.h.b 6
15.d odd 2 1 CM 1215.1.h.a 6
27.e even 9 2 3645.1.n.b 6
27.e even 9 2 3645.1.n.c 6
27.e even 9 2 3645.1.n.h 6
27.f odd 18 2 3645.1.n.a 6
27.f odd 18 2 3645.1.n.f 6
27.f odd 18 2 3645.1.n.g 6
45.h odd 6 1 1215.1.d.b yes 3
45.h odd 6 1 inner 1215.1.h.a 6
45.j even 6 1 1215.1.d.a 3
45.j even 6 1 1215.1.h.b 6
135.n odd 18 2 3645.1.n.b 6
135.n odd 18 2 3645.1.n.c 6
135.n odd 18 2 3645.1.n.h 6
135.p even 18 2 3645.1.n.a 6
135.p even 18 2 3645.1.n.f 6
135.p even 18 2 3645.1.n.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1215.1.d.a 3 9.d odd 6 1
1215.1.d.a 3 45.j even 6 1
1215.1.d.b yes 3 9.c even 3 1
1215.1.d.b yes 3 45.h odd 6 1
1215.1.h.a 6 1.a even 1 1 trivial
1215.1.h.a 6 9.c even 3 1 inner
1215.1.h.a 6 15.d odd 2 1 CM
1215.1.h.a 6 45.h odd 6 1 inner
1215.1.h.b 6 3.b odd 2 1
1215.1.h.b 6 5.b even 2 1
1215.1.h.b 6 9.d odd 6 1
1215.1.h.b 6 45.j even 6 1
3645.1.n.a 6 27.f odd 18 2
3645.1.n.a 6 135.p even 18 2
3645.1.n.b 6 27.e even 9 2
3645.1.n.b 6 135.n odd 18 2
3645.1.n.c 6 27.e even 9 2
3645.1.n.c 6 135.n odd 18 2
3645.1.n.f 6 27.f odd 18 2
3645.1.n.f 6 135.p even 18 2
3645.1.n.g 6 27.f odd 18 2
3645.1.n.g 6 135.p even 18 2
3645.1.n.h 6 27.e even 9 2
3645.1.n.h 6 135.n odd 18 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 3T_{2}^{4} + 2T_{2}^{3} + 9T_{2}^{2} + 3T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1215, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$3$ $$T^{6}$$
$5$ $$(T^{2} + T + 1)^{3}$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$T^{6}$$
$17$ $$(T^{3} - 3 T + 1)^{2}$$
$19$ $$(T^{3} - 3 T + 1)^{2}$$
$23$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$29$ $$T^{6}$$
$31$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$37$ $$T^{6}$$
$41$ $$T^{6}$$
$43$ $$T^{6}$$
$47$ $$(T^{2} - T + 1)^{3}$$
$53$ $$(T^{3} - 3 T + 1)^{2}$$
$59$ $$T^{6}$$
$61$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$67$ $$T^{6}$$
$71$ $$T^{6}$$
$73$ $$T^{6}$$
$79$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$83$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$89$ $$T^{6}$$
$97$ $$T^{6}$$