# Properties

 Label 2960.1.cw.a Level $2960$ Weight $1$ Character orbit 2960.cw Analytic conductor $1.477$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ 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(639,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))

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

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

chi := DirichletCharacter("2960.639");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2960.cw (of order $$6$$, degree $$2$$, 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: $$2$$ 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.3748322000.5

## $q$-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.

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

## Character values

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

 $$n$$ $$741$$ $$1777$$ $$2481$$ $$2591$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{6}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
639.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −1.00000 0 0 0 0.500000 0.866025i 0
1839.1 0 0 0 −1.00000 0 0 0 0.500000 + 0.866025i 0
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
185.n even 6 1 inner
740.w odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.cw.a 2
4.b odd 2 1 CM 2960.1.cw.a 2
5.b even 2 1 2960.1.cw.b yes 2
20.d odd 2 1 2960.1.cw.b yes 2
37.c even 3 1 2960.1.cw.b yes 2
148.i odd 6 1 2960.1.cw.b yes 2
185.n even 6 1 inner 2960.1.cw.a 2
740.w odd 6 1 inner 2960.1.cw.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.cw.a 2 1.a even 1 1 trivial
2960.1.cw.a 2 4.b odd 2 1 CM
2960.1.cw.a 2 185.n even 6 1 inner
2960.1.cw.a 2 740.w odd 6 1 inner
2960.1.cw.b yes 2 5.b even 2 1
2960.1.cw.b yes 2 20.d odd 2 1
2960.1.cw.b yes 2 37.c even 3 1
2960.1.cw.b yes 2 148.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{2} + 3T_{17} + 3$$ acting on $$S_{1}^{\mathrm{new}}(2960, [\chi])$$.

## Hecke characteristic polynomials

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