# Properties

 Label 2960.1.cj.a Level $2960$ Weight $1$ Character orbit 2960.cj Analytic conductor $1.477$ Analytic rank $0$ Dimension $2$ Projective image $S_{4}$ CM/RM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2960.cj (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.47723243739$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 740) Projective image: $$S_{4}$$ Projective field: Galois closure of 4.0.5065300.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{3} + i q^{5} -i q^{7} +O(q^{10})$$ $$q - q^{3} + i q^{5} -i q^{7} + i q^{11} -i q^{15} + i q^{21} - q^{25} + q^{27} + ( -1 + i ) q^{31} -i q^{33} + q^{35} + i q^{37} -i q^{41} + ( -1 + i ) q^{43} + i q^{47} -i q^{53} - q^{55} + ( -1 + i ) q^{61} + q^{71} - q^{73} + q^{75} + q^{77} + ( -1 - i ) q^{79} - q^{81} + i q^{83} + ( -1 + i ) q^{89} + ( 1 - i ) q^{93} + ( -1 + i ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{25} + 2q^{27} - 2q^{31} + 2q^{35} - 2q^{43} - 2q^{55} - 2q^{61} + 2q^{71} - 2q^{73} + 2q^{75} + 2q^{77} - 2q^{79} - 2q^{81} - 2q^{89} + 2q^{93} - 2q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$741$$ $$1777$$ $$2481$$ $$2591$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-i$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2769.1
 − 1.00000i 1.00000i
0 −1.00000 0 1.00000i 0 1.00000i 0 0 0
2929.1 0 −1.00000 0 1.00000i 0 1.00000i 0 0 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
185.j odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.cj.a 2
4.b odd 2 1 740.1.t.b yes 2
5.b even 2 1 2960.1.cj.b 2
20.d odd 2 1 740.1.t.a 2
20.e even 4 1 3700.1.j.a 2
20.e even 4 1 3700.1.j.b 2
37.d odd 4 1 2960.1.cj.b 2
148.g even 4 1 740.1.t.a 2
185.j odd 4 1 inner 2960.1.cj.a 2
740.k even 4 1 740.1.t.b yes 2
740.p odd 4 1 3700.1.j.a 2
740.s odd 4 1 3700.1.j.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.t.a 2 20.d odd 2 1
740.1.t.a 2 148.g even 4 1
740.1.t.b yes 2 4.b odd 2 1
740.1.t.b yes 2 740.k even 4 1
2960.1.cj.a 2 1.a even 1 1 trivial
2960.1.cj.a 2 185.j odd 4 1 inner
2960.1.cj.b 2 5.b even 2 1
2960.1.cj.b 2 37.d odd 4 1
3700.1.j.a 2 20.e even 4 1
3700.1.j.a 2 740.p odd 4 1
3700.1.j.b 2 20.e even 4 1
3700.1.j.b 2 740.s odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

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