# Properties

 Label 2960.1.fx.b 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

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$1.47723243739$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{18}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{18} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

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

## 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.

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 0.500000 0.866025i 0 0 0 −0.766044 + 0.642788i 0
959.1 0 0 0 0.500000 + 0.866025i 0 0 0 −0.173648 + 0.984808i 0
1119.1 0 0 0 0.500000 0.866025i 0 0 0 0.939693 + 0.342020i 0
1439.1 0 0 0 0.500000 + 0.866025i 0 0 0 0.939693 0.342020i 0
2079.1 0 0 0 0.500000 + 0.866025i 0 0 0 −0.766044 0.642788i 0
2639.1 0 0 0 0.500000 0.866025i 0 0 0 −0.173648 0.984808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2639.1 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.b yes 6
4.b odd 2 1 CM 2960.1.fx.b yes 6
5.b even 2 1 2960.1.fx.a 6
20.d odd 2 1 2960.1.fx.a 6
37.f even 9 1 2960.1.fx.a 6
148.p odd 18 1 2960.1.fx.a 6
185.x even 18 1 inner 2960.1.fx.b yes 6
740.bs odd 18 1 inner 2960.1.fx.b yes 6

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$( 1 - T + T^{2} )^{3}$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$27 + 9 T^{3} + T^{6}$$
$17$ $$3 - 9 T + 9 T^{2} - 6 T^{3} + 6 T^{4} - 3 T^{5} + T^{6}$$
$19$ $$T^{6}$$
$23$ $$T^{6}$$
$29$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$31$ $$T^{6}$$
$37$ $$1 + T^{3} + T^{6}$$
$41$ $$1 + 3 T + 3 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6}$$
$43$ $$T^{6}$$
$47$ $$T^{6}$$
$53$ $$27 - 9 T^{3} + T^{6}$$
$59$ $$T^{6}$$
$61$ $$1 + 3 T + 12 T^{2} + 19 T^{3} + 15 T^{4} + 6 T^{5} + T^{6}$$
$67$ $$T^{6}$$
$71$ $$T^{6}$$
$73$ $$( 3 + T^{2} )^{3}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$1 - 3 T + 12 T^{2} - 19 T^{3} + 15 T^{4} - 6 T^{5} + T^{6}$$
$97$ $$3 + 9 T + 9 T^{2} - 3 T^{4} + T^{6}$$