# Properties

 Label 2960.1.fx.a Level $2960$ Weight $1$ Character orbit 2960.fx Analytic conductor $1.477$ Analytic rank $0$ Dimension $6$ Projective image $D_{18}$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,1,Mod(719,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 0, 9, 2]))

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

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

chi := DirichletCharacter("2960.719");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2960.fx (of order $$18$$, degree $$6$$, 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: $$1.47723243739$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{18}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{18} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{18}^{4} q^{5} + \zeta_{18}^{7} q^{9} +O(q^{10})$$ q - z^4 * q^5 + z^7 * q^9 $$q - \zeta_{18}^{4} q^{5} + \zeta_{18}^{7} q^{9} + (\zeta_{18}^{8} + \zeta_{18}^{5}) q^{13} + ( - \zeta_{18}^{3} - \zeta_{18}^{2}) q^{17} + \zeta_{18}^{8} q^{25} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{29} - \zeta_{18}^{2} q^{37} + (\zeta_{18}^{7} - \zeta_{18}^{6}) q^{41} + \zeta_{18}^{2} q^{45} + \zeta_{18} q^{49} + (\zeta_{18}^{7} - \zeta_{18}) q^{53} + ( - \zeta_{18}^{4} - 1) q^{61} + (\zeta_{18}^{3} + 1) q^{65} + (\zeta_{18}^{6} + \zeta_{18}^{3}) q^{73} - \zeta_{18}^{5} q^{81} + (\zeta_{18}^{7} + \zeta_{18}^{6}) q^{85} + (\zeta_{18}^{8} + 1) q^{89} + ( - \zeta_{18}^{8} + \zeta_{18}^{4}) q^{97} +O(q^{100})$$ q - z^4 * q^5 + z^7 * q^9 + (z^8 + z^5) * q^13 + (-z^3 - z^2) * q^17 + z^8 * q^25 + (-z^5 - z) * q^29 - z^2 * q^37 + (z^7 - z^6) * q^41 + z^2 * q^45 + z * q^49 + (z^7 - z) * q^53 + (-z^4 - 1) * q^61 + (z^3 + 1) * q^65 + (z^6 + z^3) * q^73 - z^5 * q^81 + (z^7 + z^6) * q^85 + (z^8 + 1) * q^89 + (-z^8 + z^4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q - 3 q^{17} + 3 q^{41} - 6 q^{61} + 9 q^{65} - 3 q^{85} + 6 q^{89}+O(q^{100})$$ 6 * q - 3 * q^17 + 3 * q^41 - 6 * q^61 + 9 * q^65 - 3 * q^85 + 6 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2960\mathbb{Z}\right)^\times$$.

 $$n$$ $$741$$ $$1777$$ $$2481$$ $$2591$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{18}^{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
719.1
 0.939693 + 0.342020i −0.766044 − 0.642788i −0.173648 − 0.984808i −0.173648 + 0.984808i 0.939693 − 0.342020i −0.766044 + 0.642788i
0 0 0 −0.173648 0.984808i 0 0 0 −0.766044 + 0.642788i 0
959.1 0 0 0 0.939693 0.342020i 0 0 0 −0.173648 + 0.984808i 0
1119.1 0 0 0 −0.766044 + 0.642788i 0 0 0 0.939693 + 0.342020i 0
1439.1 0 0 0 −0.766044 0.642788i 0 0 0 0.939693 0.342020i 0
2079.1 0 0 0 −0.173648 + 0.984808i 0 0 0 −0.766044 0.642788i 0
2639.1 0 0 0 0.939693 + 0.342020i 0 0 0 −0.173648 0.984808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 719.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
185.x even 18 1 inner
740.bs odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.fx.a 6
4.b odd 2 1 CM 2960.1.fx.a 6
5.b even 2 1 2960.1.fx.b yes 6
20.d odd 2 1 2960.1.fx.b yes 6
37.f even 9 1 2960.1.fx.b yes 6
148.p odd 18 1 2960.1.fx.b yes 6
185.x even 18 1 inner 2960.1.fx.a 6
740.bs odd 18 1 inner 2960.1.fx.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.fx.a 6 1.a even 1 1 trivial
2960.1.fx.a 6 4.b odd 2 1 CM
2960.1.fx.a 6 185.x even 18 1 inner
2960.1.fx.a 6 740.bs odd 18 1 inner
2960.1.fx.b yes 6 5.b even 2 1
2960.1.fx.b yes 6 20.d odd 2 1
2960.1.fx.b yes 6 37.f even 9 1
2960.1.fx.b yes 6 148.p odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{6} - 9T_{13}^{3} + 27$$ acting on $$S_{1}^{\mathrm{new}}(2960, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6} - T^{3} + 1$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$T^{6} - 9T^{3} + 27$$
$17$ $$T^{6} + 3 T^{5} + 6 T^{4} + 6 T^{3} + \cdots + 3$$
$19$ $$T^{6}$$
$23$ $$T^{6}$$
$29$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$31$ $$T^{6}$$
$37$ $$T^{6} - T^{3} + 1$$
$41$ $$T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1$$
$43$ $$T^{6}$$
$47$ $$T^{6}$$
$53$ $$T^{6} + 9T^{3} + 27$$
$59$ $$T^{6}$$
$61$ $$T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1$$
$67$ $$T^{6}$$
$71$ $$T^{6}$$
$73$ $$(T^{2} + 3)^{3}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1$$
$97$ $$T^{6} - 3 T^{4} + 9 T^{2} - 9 T + 3$$