# Properties

 Label 2960.2.p.c 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 185) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + i q^{5} - q^{7} -2 q^{9} +O(q^{10})$$ $$q - q^{3} + i q^{5} - q^{7} -2 q^{9} + 3 q^{11} -i q^{15} -6 i q^{17} -6 i q^{19} + q^{21} + 6 i q^{23} - q^{25} + 5 q^{27} + 6 i q^{29} + 6 i q^{31} -3 q^{33} -i q^{35} + ( 1 + 6 i ) q^{37} -3 q^{41} + 6 i q^{43} -2 i q^{45} + 3 q^{47} -6 q^{49} + 6 i q^{51} -9 q^{53} + 3 i q^{55} + 6 i q^{57} -12 i q^{59} -6 i q^{61} + 2 q^{63} -4 q^{67} -6 i q^{69} -9 q^{71} + 7 q^{73} + q^{75} -3 q^{77} -12 i q^{79} + q^{81} -15 q^{83} + 6 q^{85} -6 i q^{87} -6 i q^{93} + 6 q^{95} -6 i q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{7} - 4q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{7} - 4q^{9} + 6q^{11} + 2q^{21} - 2q^{25} + 10q^{27} - 6q^{33} + 2q^{37} - 6q^{41} + 6q^{47} - 12q^{49} - 18q^{53} + 4q^{63} - 8q^{67} - 18q^{71} + 14q^{73} + 2q^{75} - 6q^{77} + 2q^{81} - 30q^{83} + 12q^{85} + 12q^{95} - 12q^{99} + O(q^{100})$$

## 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 −1.00000 0 1.00000i 0 −1.00000 0 −2.00000 0
961.2 0 −1.00000 0 1.00000i 0 −1.00000 0 −2.00000 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.c 2
4.b odd 2 1 185.2.c.a 2
12.b even 2 1 1665.2.e.b 2
20.d odd 2 1 925.2.c.a 2
20.e even 4 1 925.2.d.b 2
20.e even 4 1 925.2.d.c 2
37.b even 2 1 inner 2960.2.p.c 2
148.b odd 2 1 185.2.c.a 2
148.g even 4 1 6845.2.a.c 1
148.g even 4 1 6845.2.a.d 1
444.g even 2 1 1665.2.e.b 2
740.g odd 2 1 925.2.c.a 2
740.m even 4 1 925.2.d.b 2
740.m even 4 1 925.2.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.c.a 2 4.b odd 2 1
185.2.c.a 2 148.b odd 2 1
925.2.c.a 2 20.d odd 2 1
925.2.c.a 2 740.g odd 2 1
925.2.d.b 2 20.e even 4 1
925.2.d.b 2 740.m even 4 1
925.2.d.c 2 20.e even 4 1
925.2.d.c 2 740.m even 4 1
1665.2.e.b 2 12.b even 2 1
1665.2.e.b 2 444.g even 2 1
2960.2.p.c 2 1.a even 1 1 trivial
2960.2.p.c 2 37.b even 2 1 inner
6845.2.a.c 1 148.g even 4 1
6845.2.a.d 1 148.g even 4 1

## Hecke kernels

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

 $$T_{3} + 1$$ $$T_{7} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$36 + T^{2}$$
$31$ $$36 + T^{2}$$
$37$ $$37 - 2 T + T^{2}$$
$41$ $$( 3 + T )^{2}$$
$43$ $$36 + T^{2}$$
$47$ $$( -3 + T )^{2}$$
$53$ $$( 9 + T )^{2}$$
$59$ $$144 + T^{2}$$
$61$ $$36 + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$( 9 + T )^{2}$$
$73$ $$( -7 + T )^{2}$$
$79$ $$144 + T^{2}$$
$83$ $$( 15 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$36 + T^{2}$$