# Properties

 Label 1215.1.d.a Level $1215$ Weight $1$ Character orbit 1215.d Self dual yes Analytic conductor $0.606$ Analytic rank $0$ Dimension $3$ Projective image $D_{9}$ CM discriminant -15 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1215,1,Mod(1214,1215)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1215, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("1215.1214");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1215 = 3^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1215.d (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$0.606363990349$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_1 + 1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 1) * q^4 - q^5 + (b1 + 1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_1 + 1) q^{8} - \beta_1 q^{10} + (\beta_1 + 1) q^{16} + (\beta_{2} - \beta_1) q^{17} + ( - \beta_{2} + \beta_1) q^{19} + ( - \beta_{2} - 1) q^{20} - \beta_{2} q^{23} + q^{25} - \beta_1 q^{31} + (\beta_{2} + 1) q^{32} + ( - \beta_{2} + \beta_1 - 1) q^{34} + (\beta_{2} - \beta_1 + 1) q^{38} + ( - \beta_1 - 1) q^{40} + ( - \beta_1 - 1) q^{46} + q^{47} + q^{49} + \beta_1 q^{50} - \beta_{2} q^{53} - \beta_1 q^{61} + ( - \beta_{2} - 2) q^{62} + \beta_1 q^{64} + ( - \beta_1 + 1) q^{68} + (\beta_1 - 1) q^{76} + \beta_{2} q^{79} + ( - \beta_1 - 1) q^{80} + (\beta_{2} - \beta_1) q^{83} + ( - \beta_{2} + \beta_1) q^{85} + ( - \beta_1 - 2) q^{92} + \beta_1 q^{94} + (\beta_{2} - \beta_1) q^{95} + \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 1) * q^4 - q^5 + (b1 + 1) * q^8 - b1 * q^10 + (b1 + 1) * q^16 + (b2 - b1) * q^17 + (-b2 + b1) * q^19 + (-b2 - 1) * q^20 - b2 * q^23 + q^25 - b1 * q^31 + (b2 + 1) * q^32 + (-b2 + b1 - 1) * q^34 + (b2 - b1 + 1) * q^38 + (-b1 - 1) * q^40 + (-b1 - 1) * q^46 + q^47 + q^49 + b1 * q^50 - b2 * q^53 - b1 * q^61 + (-b2 - 2) * q^62 + b1 * q^64 + (-b1 + 1) * q^68 + (b1 - 1) * q^76 + b2 * q^79 + (-b1 - 1) * q^80 + (b2 - b1) * q^83 + (-b2 + b1) * q^85 + (-b1 - 2) * q^92 + b1 * q^94 + (b2 - b1) * q^95 + b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{4} - 3 q^{5} + 3 q^{8}+O(q^{10})$$ 3 * q + 3 * q^4 - 3 * q^5 + 3 * q^8 $$3 q + 3 q^{4} - 3 q^{5} + 3 q^{8} + 3 q^{16} - 3 q^{20} + 3 q^{25} + 3 q^{32} - 3 q^{34} + 3 q^{38} - 3 q^{40} - 3 q^{46} + 3 q^{47} + 3 q^{49} - 6 q^{62} + 3 q^{68} - 3 q^{76} - 3 q^{80} - 6 q^{92}+O(q^{100})$$ 3 * q + 3 * q^4 - 3 * q^5 + 3 * q^8 + 3 * q^16 - 3 * q^20 + 3 * q^25 + 3 * q^32 - 3 * q^34 + 3 * q^38 - 3 * q^40 - 3 * q^46 + 3 * q^47 + 3 * q^49 - 6 * q^62 + 3 * q^68 - 3 * q^76 - 3 * q^80 - 6 * q^92

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## Character values

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

 $$n$$ $$487$$ $$731$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1214.1
 −1.53209 −0.347296 1.87939
−1.53209 0 1.34730 −1.00000 0 0 −0.532089 0 1.53209
1214.2 −0.347296 0 −0.879385 −1.00000 0 0 0.652704 0 0.347296
1214.3 1.87939 0 2.53209 −1.00000 0 0 2.87939 0 −1.87939
 $$n$$: 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 $$\Q(\sqrt{-15})$$

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

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