# Properties

 Label 2960.2.p.b Level $2960$ Weight $2$ Character orbit 2960.p Analytic conductor $23.636$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$23.6357189983$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1480) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{3} + i q^{5} -4 q^{7} + q^{9} +O(q^{10})$$ $$q -2 q^{3} + i q^{5} -4 q^{7} + q^{9} -4 q^{11} -2 i q^{13} -2 i q^{15} + 6 i q^{17} + 8 i q^{19} + 8 q^{21} + 4 i q^{23} - q^{25} + 4 q^{27} + 6 i q^{29} -2 i q^{31} + 8 q^{33} -4 i q^{35} + ( -6 + i ) q^{37} + 4 i q^{39} -2 q^{41} + 4 i q^{43} + i q^{45} -12 q^{47} + 9 q^{49} -12 i q^{51} -12 q^{53} -4 i q^{55} -16 i q^{57} -8 i q^{59} -2 i q^{61} -4 q^{63} + 2 q^{65} + 2 q^{67} -8 i q^{69} + 8 q^{71} + 6 q^{73} + 2 q^{75} + 16 q^{77} -6 i q^{79} -11 q^{81} -14 q^{83} -6 q^{85} -12 i q^{87} -4 i q^{89} + 8 i q^{91} + 4 i q^{93} -8 q^{95} + 2 i q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{3} - 8q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 4q^{3} - 8q^{7} + 2q^{9} - 8q^{11} + 16q^{21} - 2q^{25} + 8q^{27} + 16q^{33} - 12q^{37} - 4q^{41} - 24q^{47} + 18q^{49} - 24q^{53} - 8q^{63} + 4q^{65} + 4q^{67} + 16q^{71} + 12q^{73} + 4q^{75} + 32q^{77} - 22q^{81} - 28q^{83} - 12q^{85} - 16q^{95} - 8q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 − 1.00000i 1.00000i
0 −2.00000 0 1.00000i 0 −4.00000 0 1.00000 0
961.2 0 −2.00000 0 1.00000i 0 −4.00000 0 1.00000 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
37.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.2.p.b 2
4.b odd 2 1 1480.2.p.a 2
37.b even 2 1 inner 2960.2.p.b 2
148.b odd 2 1 1480.2.p.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.p.a 2 4.b odd 2 1
1480.2.p.a 2 148.b odd 2 1
2960.2.p.b 2 1.a even 1 1 trivial
2960.2.p.b 2 37.b even 2 1 inner

## Hecke kernels

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

 $$T_{3} + 2$$ $$T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 2 + T )^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$( 4 + T )^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$64 + T^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$36 + T^{2}$$
$31$ $$4 + T^{2}$$
$37$ $$37 + 12 T + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$( 12 + T )^{2}$$
$53$ $$( 12 + T )^{2}$$
$59$ $$64 + T^{2}$$
$61$ $$4 + T^{2}$$
$67$ $$( -2 + T )^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$36 + T^{2}$$
$83$ $$( 14 + T )^{2}$$
$89$ $$16 + T^{2}$$
$97$ $$4 + T^{2}$$