# Properties

 Label 960.2.k.a Level $960$ Weight $2$ Character orbit 960.k Analytic conductor $7.666$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 960.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.66563859404$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - i q^{3} - i q^{5} - 4 q^{7} - q^{9} +O(q^{10})$$ q - i * q^3 - i * q^5 - 4 * q^7 - q^9 $$q - i q^{3} - i q^{5} - 4 q^{7} - q^{9} - 4 i q^{11} + 2 i q^{13} - q^{15} + 6 i q^{19} + 4 i q^{21} - 2 q^{23} - q^{25} + i q^{27} + 10 i q^{29} - 8 q^{31} - 4 q^{33} + 4 i q^{35} - 6 i q^{37} + 2 q^{39} - 2 q^{41} + i q^{45} - 10 q^{47} + 9 q^{49} + 6 i q^{53} - 4 q^{55} + 6 q^{57} + 4 i q^{61} + 4 q^{63} + 2 q^{65} - 12 i q^{67} + 2 i q^{69} + 12 q^{71} - 14 q^{73} + i q^{75} + 16 i q^{77} - 8 q^{79} + q^{81} - 4 i q^{83} + 10 q^{87} + 18 q^{89} - 8 i q^{91} + 8 i q^{93} + 6 q^{95} - 6 q^{97} + 4 i q^{99} +O(q^{100})$$ q - i * q^3 - i * q^5 - 4 * q^7 - q^9 - 4*i * q^11 + 2*i * q^13 - q^15 + 6*i * q^19 + 4*i * q^21 - 2 * q^23 - q^25 + i * q^27 + 10*i * q^29 - 8 * q^31 - 4 * q^33 + 4*i * q^35 - 6*i * q^37 + 2 * q^39 - 2 * q^41 + i * q^45 - 10 * q^47 + 9 * q^49 + 6*i * q^53 - 4 * q^55 + 6 * q^57 + 4*i * q^61 + 4 * q^63 + 2 * q^65 - 12*i * q^67 + 2*i * q^69 + 12 * q^71 - 14 * q^73 + i * q^75 + 16*i * q^77 - 8 * q^79 + q^81 - 4*i * q^83 + 10 * q^87 + 18 * q^89 - 8*i * q^91 + 8*i * q^93 + 6 * q^95 - 6 * q^97 + 4*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q - 8 * q^7 - 2 * q^9 $$2 q - 8 q^{7} - 2 q^{9} - 2 q^{15} - 4 q^{23} - 2 q^{25} - 16 q^{31} - 8 q^{33} + 4 q^{39} - 4 q^{41} - 20 q^{47} + 18 q^{49} - 8 q^{55} + 12 q^{57} + 8 q^{63} + 4 q^{65} + 24 q^{71} - 28 q^{73} - 16 q^{79} + 2 q^{81} + 20 q^{87} + 36 q^{89} + 12 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q - 8 * q^7 - 2 * q^9 - 2 * q^15 - 4 * q^23 - 2 * q^25 - 16 * q^31 - 8 * q^33 + 4 * q^39 - 4 * q^41 - 20 * q^47 + 18 * q^49 - 8 * q^55 + 12 * q^57 + 8 * q^63 + 4 * q^65 + 24 * q^71 - 28 * q^73 - 16 * q^79 + 2 * q^81 + 20 * q^87 + 36 * q^89 + 12 * q^95 - 12 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$577$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
481.1
 1.00000i − 1.00000i
0 1.00000i 0 1.00000i 0 −4.00000 0 −1.00000 0
481.2 0 1.00000i 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.k.a 2
3.b odd 2 1 2880.2.k.a 2
4.b odd 2 1 960.2.k.d yes 2
5.b even 2 1 4800.2.k.h 2
5.c odd 4 1 4800.2.d.c 2
5.c odd 4 1 4800.2.d.f 2
8.b even 2 1 inner 960.2.k.a 2
8.d odd 2 1 960.2.k.d yes 2
12.b even 2 1 2880.2.k.d 2
16.e even 4 1 3840.2.a.n 1
16.e even 4 1 3840.2.a.u 1
16.f odd 4 1 3840.2.a.a 1
16.f odd 4 1 3840.2.a.v 1
20.d odd 2 1 4800.2.k.a 2
20.e even 4 1 4800.2.d.b 2
20.e even 4 1 4800.2.d.g 2
24.f even 2 1 2880.2.k.d 2
24.h odd 2 1 2880.2.k.a 2
40.e odd 2 1 4800.2.k.a 2
40.f even 2 1 4800.2.k.h 2
40.i odd 4 1 4800.2.d.c 2
40.i odd 4 1 4800.2.d.f 2
40.k even 4 1 4800.2.d.b 2
40.k even 4 1 4800.2.d.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.k.a 2 1.a even 1 1 trivial
960.2.k.a 2 8.b even 2 1 inner
960.2.k.d yes 2 4.b odd 2 1
960.2.k.d yes 2 8.d odd 2 1
2880.2.k.a 2 3.b odd 2 1
2880.2.k.a 2 24.h odd 2 1
2880.2.k.d 2 12.b even 2 1
2880.2.k.d 2 24.f even 2 1
3840.2.a.a 1 16.f odd 4 1
3840.2.a.n 1 16.e even 4 1
3840.2.a.u 1 16.e even 4 1
3840.2.a.v 1 16.f odd 4 1
4800.2.d.b 2 20.e even 4 1
4800.2.d.b 2 40.k even 4 1
4800.2.d.c 2 5.c odd 4 1
4800.2.d.c 2 40.i odd 4 1
4800.2.d.f 2 5.c odd 4 1
4800.2.d.f 2 40.i odd 4 1
4800.2.d.g 2 20.e even 4 1
4800.2.d.g 2 40.k even 4 1
4800.2.k.a 2 20.d odd 2 1
4800.2.k.a 2 40.e odd 2 1
4800.2.k.h 2 5.b even 2 1
4800.2.k.h 2 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(960, [\chi])$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{23} + 2$$ T23 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 1$$
$7$ $$(T + 4)^{2}$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T + 2)^{2}$$
$29$ $$T^{2} + 100$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2}$$
$47$ $$(T + 10)^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 16$$
$67$ $$T^{2} + 144$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T + 14)^{2}$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 18)^{2}$$
$97$ $$(T + 6)^{2}$$