# Properties

 Label 2960.2.a.s Level $2960$ Weight $2$ Character orbit 2960.a Self dual yes Analytic conductor $23.636$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2960.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$23.6357189983$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1480) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + q^{5} + \beta_{1} q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + q^{5} + \beta_{1} q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} + ( -2 - \beta_{2} ) q^{11} -\beta_{1} q^{15} + ( -2 - \beta_{1} + \beta_{2} ) q^{19} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{21} -4 q^{23} + q^{25} + ( -6 - 2 \beta_{1} - \beta_{2} ) q^{27} + ( -2 - 2 \beta_{2} ) q^{29} + ( -2 + \beta_{1} + \beta_{2} ) q^{31} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{33} + \beta_{1} q^{35} + q^{37} + ( -2 \beta_{1} - \beta_{2} ) q^{41} + ( -4 - 2 \beta_{2} ) q^{43} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{45} + ( \beta_{1} + 4 \beta_{2} ) q^{47} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{49} + ( 4 + 2 \beta_{1} + 3 \beta_{2} ) q^{53} + ( -2 - \beta_{2} ) q^{55} + ( 6 + 4 \beta_{1} + 2 \beta_{2} ) q^{57} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{59} + ( -2 - 4 \beta_{1} ) q^{61} + ( 6 + 5 \beta_{1} + \beta_{2} ) q^{63} + ( -10 - \beta_{1} - 3 \beta_{2} ) q^{67} + 4 \beta_{1} q^{69} + ( -6 + 2 \beta_{1} - \beta_{2} ) q^{71} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{73} -\beta_{1} q^{75} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{77} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{79} + ( 3 + 4 \beta_{1} - 2 \beta_{2} ) q^{81} + ( -4 + 5 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{87} + ( -2 + 4 \beta_{1} + 4 \beta_{2} ) q^{89} -2 q^{93} + ( -2 - \beta_{1} + \beta_{2} ) q^{95} + 6 q^{97} + ( -4 - 2 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{3} + 3q^{5} + q^{7} + 4q^{9} + O(q^{10})$$ $$3q - q^{3} + 3q^{5} + q^{7} + 4q^{9} - 5q^{11} - q^{15} - 8q^{19} - 13q^{21} - 12q^{23} + 3q^{25} - 19q^{27} - 4q^{29} - 6q^{31} - 3q^{33} + q^{35} + 3q^{37} - q^{41} - 10q^{43} + 4q^{45} - 3q^{47} - 8q^{49} + 11q^{53} - 5q^{55} + 20q^{57} + 4q^{59} - 10q^{61} + 22q^{63} - 28q^{67} + 4q^{69} - 15q^{71} + 5q^{73} - q^{75} + 3q^{77} - 2q^{79} + 15q^{81} - 5q^{83} - 8q^{87} - 6q^{89} - 6q^{93} - 8q^{95} + 18q^{97} - 14q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.12489 −0.363328 −1.76156
0 −3.12489 0 1.00000 0 3.12489 0 6.76491 0
1.2 0 0.363328 0 1.00000 0 −0.363328 0 −2.86799 0
1.3 0 1.76156 0 1.00000 0 −1.76156 0 0.103084 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$
$$37$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.2.a.s 3
4.b odd 2 1 1480.2.a.f 3
20.d odd 2 1 7400.2.a.m 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.a.f 3 4.b odd 2 1
2960.2.a.s 3 1.a even 1 1 trivial
7400.2.a.m 3 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2960))$$:

 $$T_{3}^{3} + T_{3}^{2} - 6 T_{3} + 2$$ $$T_{7}^{3} - T_{7}^{2} - 6 T_{7} - 2$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$2 - 6 T + T^{2} + T^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$-2 - 6 T - T^{2} + T^{3}$$
$11$ $$-8 + 5 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$T^{3}$$
$19$ $$-64 + 2 T + 8 T^{2} + T^{3}$$
$23$ $$( 4 + T )^{3}$$
$29$ $$-32 - 28 T + 4 T^{2} + T^{3}$$
$31$ $$-4 + 2 T + 6 T^{2} + T^{3}$$
$37$ $$( -1 + T )^{3}$$
$41$ $$20 - 24 T + T^{2} + T^{3}$$
$43$ $$-64 + 10 T^{2} + T^{3}$$
$47$ $$134 - 118 T + 3 T^{2} + T^{3}$$
$53$ $$452 - 32 T - 11 T^{2} + T^{3}$$
$59$ $$232 - 62 T - 4 T^{2} + T^{3}$$
$61$ $$-40 - 68 T + 10 T^{2} + T^{3}$$
$67$ $$40 + 194 T + 28 T^{2} + T^{3}$$
$71$ $$-32 + 32 T + 15 T^{2} + T^{3}$$
$73$ $$764 - 120 T - 5 T^{2} + T^{3}$$
$79$ $$212 - 94 T + 2 T^{2} + T^{3}$$
$83$ $$106 - 230 T + 5 T^{2} + T^{3}$$
$89$ $$200 - 148 T + 6 T^{2} + T^{3}$$
$97$ $$( -6 + T )^{3}$$