# Properties

 Label 2880.2.h.f Level $2880$ Weight $2$ Character orbit 2880.h Analytic conductor $22.997$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2880 = 2^{6} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2880.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.9969157821$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{8}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 720) 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{6} q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} +O(q^{10})$$ $$q -\zeta_{24}^{6} q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - 4 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{11} + ( 2 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{13} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} + ( 2 \zeta_{24} + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{23} - q^{25} -6 \zeta_{24}^{6} q^{29} + ( 2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{31} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{35} + ( -2 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{37} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{41} + ( 4 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{43} + ( -2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{47} + q^{49} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{53} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{55} + ( \zeta_{24} - 4 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{59} + ( -2 - 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{61} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{65} + ( -4 + 8 \zeta_{24}^{4} ) q^{67} + ( -4 \zeta_{24} - 8 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{71} + ( -2 - 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{73} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{77} + ( 2 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{79} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{83} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 12 \zeta_{24}^{6} ) q^{89} + ( -6 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 12 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{91} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{95} + ( -2 - 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 16q^{13} - 8q^{25} - 16q^{37} + 8q^{49} - 16q^{61} - 16q^{73} - 16q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$641$$ $$901$$ $$2431$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 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 0.258819 − 0.965926i
0 0 0 1.00000i 0 2.44949i 0 0 0
1151.2 0 0 0 1.00000i 0 2.44949i 0 0 0
1151.3 0 0 0 1.00000i 0 2.44949i 0 0 0
1151.4 0 0 0 1.00000i 0 2.44949i 0 0 0
1151.5 0 0 0 1.00000i 0 2.44949i 0 0 0
1151.6 0 0 0 1.00000i 0 2.44949i 0 0 0
1151.7 0 0 0 1.00000i 0 2.44949i 0 0 0
1151.8 0 0 0 1.00000i 0 2.44949i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1151.8 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
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.h.f 8
3.b odd 2 1 inner 2880.2.h.f 8
4.b odd 2 1 inner 2880.2.h.f 8
8.b even 2 1 720.2.h.a 8
8.d odd 2 1 720.2.h.a 8
12.b even 2 1 inner 2880.2.h.f 8
24.f even 2 1 720.2.h.a 8
24.h odd 2 1 720.2.h.a 8
40.e odd 2 1 3600.2.h.j 8
40.f even 2 1 3600.2.h.j 8
40.i odd 4 1 3600.2.o.c 8
40.i odd 4 1 3600.2.o.d 8
40.k even 4 1 3600.2.o.c 8
40.k even 4 1 3600.2.o.d 8
120.i odd 2 1 3600.2.h.j 8
120.m even 2 1 3600.2.h.j 8
120.q odd 4 1 3600.2.o.c 8
120.q odd 4 1 3600.2.o.d 8
120.w even 4 1 3600.2.o.c 8
120.w even 4 1 3600.2.o.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.h.a 8 8.b even 2 1
720.2.h.a 8 8.d odd 2 1
720.2.h.a 8 24.f even 2 1
720.2.h.a 8 24.h odd 2 1
2880.2.h.f 8 1.a even 1 1 trivial
2880.2.h.f 8 3.b odd 2 1 inner
2880.2.h.f 8 4.b odd 2 1 inner
2880.2.h.f 8 12.b even 2 1 inner
3600.2.h.j 8 40.e odd 2 1
3600.2.h.j 8 40.f even 2 1
3600.2.h.j 8 120.i odd 2 1
3600.2.h.j 8 120.m even 2 1
3600.2.o.c 8 40.i odd 4 1
3600.2.o.c 8 40.k even 4 1
3600.2.o.c 8 120.q odd 4 1
3600.2.o.c 8 120.w even 4 1
3600.2.o.d 8 40.i odd 4 1
3600.2.o.d 8 40.k even 4 1
3600.2.o.d 8 120.q odd 4 1
3600.2.o.d 8 120.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 $$S_{2}^{\mathrm{new}}(2880, [\chi])$$:

 $$T_{7}^{2} + 6$$ $$T_{11}^{4} - 36 T_{11}^{2} + 36$$ $$T_{23}^{4} - 72 T_{23}^{2} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$( 6 + T^{2} )^{4}$$
$11$ $$( 36 - 36 T^{2} + T^{4} )^{2}$$
$13$ $$( -14 - 4 T + T^{2} )^{4}$$
$17$ $$T^{8}$$
$19$ $$( 24 + T^{2} )^{4}$$
$23$ $$( 144 - 72 T^{2} + T^{4} )^{2}$$
$29$ $$( 36 + T^{2} )^{4}$$
$31$ $$( 144 + 72 T^{2} + T^{4} )^{2}$$
$37$ $$( -14 + 4 T + T^{2} )^{4}$$
$41$ $$( 18 + T^{2} )^{4}$$
$43$ $$( 576 + 144 T^{2} + T^{4} )^{2}$$
$47$ $$( 576 - 144 T^{2} + T^{4} )^{2}$$
$53$ $$( 72 + T^{2} )^{4}$$
$59$ $$( 36 - 36 T^{2} + T^{4} )^{2}$$
$61$ $$( -68 + 4 T + T^{2} )^{4}$$
$67$ $$( 48 + T^{2} )^{4}$$
$71$ $$( 2304 - 288 T^{2} + T^{4} )^{2}$$
$73$ $$( -68 + 4 T + T^{2} )^{4}$$
$79$ $$( 144 + 72 T^{2} + T^{4} )^{2}$$
$83$ $$( -216 + T^{2} )^{4}$$
$89$ $$( 15876 + 324 T^{2} + T^{4} )^{2}$$
$97$ $$( -68 + 4 T + T^{2} )^{4}$$