# Properties

 Label 960.2.o.d Level $960$ Weight $2$ Character orbit 960.o Analytic conductor $7.666$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 960.o (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.66563859404$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}\cdot 3$$ Twist minimal: no (minimal twist has level 240) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{3} + ( \zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{5} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{3} + ( \zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{5} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{9} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{11} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{13} + ( -1 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{15} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{17} + ( 2 - 4 \zeta_{24}^{4} ) q^{19} + 6 \zeta_{24}^{6} q^{23} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{25} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{27} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{29} + ( 2 - 4 \zeta_{24}^{4} ) q^{31} + ( 2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{33} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{37} + ( 4 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{39} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{41} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{43} + ( 4 + 3 \zeta_{24} + 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{45} + 6 \zeta_{24}^{6} q^{47} -7 q^{49} + ( 2 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{51} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{53} + ( 4 + 6 \zeta_{24} + 6 \zeta_{24}^{3} - 8 \zeta_{24}^{4} - 6 \zeta_{24}^{5} ) q^{55} + ( -2 \zeta_{24} + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{57} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{59} + 2 q^{61} + ( 6 \zeta_{24} - 8 \zeta_{24}^{2} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{65} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{67} + ( -6 - 6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{69} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{71} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{73} + ( 4 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + \zeta_{24}^{5} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{75} + ( -6 + 12 \zeta_{24}^{4} ) q^{79} + ( -7 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{81} -6 \zeta_{24}^{6} q^{83} + ( -6 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{85} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{87} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{89} + ( -2 \zeta_{24} + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{93} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 6 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{95} + ( -8 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 16 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{9} + O(q^{10})$$ $$8 q + 8 q^{9} + 8 q^{25} + 32 q^{45} - 56 q^{49} + 16 q^{61} - 48 q^{69} - 56 q^{81} - 48 q^{85} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
959.1
 0.258819 − 0.965926i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 − 0.258819i −0.258819 − 0.965926i 0.965926 + 0.258819i
0 −1.41421 1.00000i 0 −1.73205 1.41421i 0 0 0 1.00000 + 2.82843i 0
959.2 0 −1.41421 1.00000i 0 1.73205 1.41421i 0 0 0 1.00000 + 2.82843i 0
959.3 0 −1.41421 + 1.00000i 0 −1.73205 + 1.41421i 0 0 0 1.00000 2.82843i 0
959.4 0 −1.41421 + 1.00000i 0 1.73205 + 1.41421i 0 0 0 1.00000 2.82843i 0
959.5 0 1.41421 1.00000i 0 −1.73205 + 1.41421i 0 0 0 1.00000 2.82843i 0
959.6 0 1.41421 1.00000i 0 1.73205 + 1.41421i 0 0 0 1.00000 2.82843i 0
959.7 0 1.41421 + 1.00000i 0 −1.73205 1.41421i 0 0 0 1.00000 + 2.82843i 0
959.8 0 1.41421 + 1.00000i 0 1.73205 1.41421i 0 0 0 1.00000 + 2.82843i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 959.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h 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.o.d 8
3.b odd 2 1 inner 960.2.o.d 8
4.b odd 2 1 inner 960.2.o.d 8
5.b even 2 1 inner 960.2.o.d 8
8.b even 2 1 240.2.o.b 8
8.d odd 2 1 240.2.o.b 8
12.b even 2 1 inner 960.2.o.d 8
15.d odd 2 1 inner 960.2.o.d 8
20.d odd 2 1 inner 960.2.o.d 8
24.f even 2 1 240.2.o.b 8
24.h odd 2 1 240.2.o.b 8
40.e odd 2 1 240.2.o.b 8
40.f even 2 1 240.2.o.b 8
40.i odd 4 1 1200.2.h.j 4
40.i odd 4 1 1200.2.h.n 4
40.k even 4 1 1200.2.h.j 4
40.k even 4 1 1200.2.h.n 4
60.h even 2 1 inner 960.2.o.d 8
120.i odd 2 1 240.2.o.b 8
120.m even 2 1 240.2.o.b 8
120.q odd 4 1 1200.2.h.j 4
120.q odd 4 1 1200.2.h.n 4
120.w even 4 1 1200.2.h.j 4
120.w even 4 1 1200.2.h.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.o.b 8 8.b even 2 1
240.2.o.b 8 8.d odd 2 1
240.2.o.b 8 24.f even 2 1
240.2.o.b 8 24.h odd 2 1
240.2.o.b 8 40.e odd 2 1
240.2.o.b 8 40.f even 2 1
240.2.o.b 8 120.i odd 2 1
240.2.o.b 8 120.m even 2 1
960.2.o.d 8 1.a even 1 1 trivial
960.2.o.d 8 3.b odd 2 1 inner
960.2.o.d 8 4.b odd 2 1 inner
960.2.o.d 8 5.b even 2 1 inner
960.2.o.d 8 12.b even 2 1 inner
960.2.o.d 8 15.d odd 2 1 inner
960.2.o.d 8 20.d odd 2 1 inner
960.2.o.d 8 60.h even 2 1 inner
1200.2.h.j 4 40.i odd 4 1
1200.2.h.j 4 40.k even 4 1
1200.2.h.j 4 120.q odd 4 1
1200.2.h.j 4 120.w even 4 1
1200.2.h.n 4 40.i odd 4 1
1200.2.h.n 4 40.k even 4 1
1200.2.h.n 4 120.q odd 4 1
1200.2.h.n 4 120.w even 4 1

## Hecke kernels

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

 $$T_{7}$$ $$T_{11}^{2} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 9 - 2 T^{2} + T^{4} )^{2}$$
$5$ $$( 25 - 2 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( -24 + T^{2} )^{4}$$
$13$ $$( 24 + T^{2} )^{4}$$
$17$ $$( -12 + T^{2} )^{4}$$
$19$ $$( 12 + T^{2} )^{4}$$
$23$ $$( 36 + T^{2} )^{4}$$
$29$ $$( 8 + T^{2} )^{4}$$
$31$ $$( 12 + T^{2} )^{4}$$
$37$ $$( 24 + T^{2} )^{4}$$
$41$ $$( 32 + T^{2} )^{4}$$
$43$ $$( -72 + T^{2} )^{4}$$
$47$ $$( 36 + T^{2} )^{4}$$
$53$ $$( -108 + T^{2} )^{4}$$
$59$ $$( -24 + T^{2} )^{4}$$
$61$ $$( -2 + T )^{8}$$
$67$ $$( -72 + T^{2} )^{4}$$
$71$ $$( -96 + T^{2} )^{4}$$
$73$ $$( 96 + T^{2} )^{4}$$
$79$ $$( 108 + T^{2} )^{4}$$
$83$ $$( 36 + T^{2} )^{4}$$
$89$ $$( 32 + T^{2} )^{4}$$
$97$ $$T^{8}$$