# Properties

 Label 2960.1.eo.a Level $2960$ Weight $1$ Character orbit 2960.eo Analytic conductor $1.477$ Analytic rank $0$ Dimension $4$ Projective image $D_{12}$ 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.eo (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.47723243739$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{3} q^{5} -\zeta_{12} q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{3} q^{5} -\zeta_{12} q^{9} -2 \zeta_{12}^{2} q^{13} + ( -1 - \zeta_{12}^{2} ) q^{17} - q^{25} + ( -\zeta_{12}^{4} + \zeta_{12}^{5} ) q^{29} -\zeta_{12}^{5} q^{37} + ( -1 + \zeta_{12}^{4} ) q^{41} -\zeta_{12}^{4} q^{45} -\zeta_{12} q^{49} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{53} + ( 1 + \zeta_{12} ) q^{61} -2 \zeta_{12}^{5} q^{65} + ( 1 + \zeta_{12}^{3} ) q^{73} + \zeta_{12}^{2} q^{81} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{85} + ( -\zeta_{12}^{2} + \zeta_{12}^{3} ) q^{89} -\zeta_{12}^{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{13} - 6q^{17} - 4q^{25} + 2q^{29} - 6q^{41} + 2q^{45} + 2q^{53} + 4q^{61} + 4q^{73} + 2q^{81} - 2q^{89} + O(q^{100})$$

## Character values

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

 $$n$$ $$741$$ $$1777$$ $$2481$$ $$2591$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}^{3}$$ $$\zeta_{12}^{5}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1007.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 0 1.00000i 0 0 0 0.866025 0.500000i 0
1583.1 0 0 0 1.00000i 0 0 0 −0.866025 + 0.500000i 0
2687.1 0 0 0 1.00000i 0 0 0 −0.866025 0.500000i 0
2863.1 0 0 0 1.00000i 0 0 0 0.866025 + 0.500000i 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.p even 12 1 inner
740.bj odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.eo.a 4
4.b odd 2 1 CM 2960.1.eo.a 4
5.c odd 4 1 2960.1.et.a yes 4
20.e even 4 1 2960.1.et.a yes 4
37.g odd 12 1 2960.1.et.a yes 4
148.l even 12 1 2960.1.et.a yes 4
185.p even 12 1 inner 2960.1.eo.a 4
740.bj odd 12 1 inner 2960.1.eo.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.eo.a 4 1.a even 1 1 trivial
2960.1.eo.a 4 4.b odd 2 1 CM
2960.1.eo.a 4 185.p even 12 1 inner
2960.1.eo.a 4 740.bj odd 12 1 inner
2960.1.et.a yes 4 5.c odd 4 1
2960.1.et.a yes 4 20.e even 4 1
2960.1.et.a yes 4 37.g odd 12 1
2960.1.et.a yes 4 148.l even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2960, [\chi])$$.

## Hecke characteristic polynomials

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