# Properties

 Label 2880.2.b.a Level $2880$ Weight $2$ Character orbit 2880.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.9969157821$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{11}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + \beta_{3} q^{7} +O(q^{10})$$ $$q - q^{5} + \beta_{3} q^{7} + ( -\beta_{1} + \beta_{3} ) q^{11} + \beta_{2} q^{13} + ( \beta_{2} - \beta_{4} ) q^{17} + \beta_{5} q^{19} + q^{25} + ( -2 + \beta_{6} ) q^{29} + ( \beta_{1} - 2 \beta_{3} ) q^{31} -\beta_{3} q^{35} + ( -\beta_{2} + 2 \beta_{4} ) q^{37} + ( -2 \beta_{2} + \beta_{4} ) q^{41} + ( \beta_{5} - \beta_{7} ) q^{43} + \beta_{7} q^{47} -3 q^{49} + ( -4 - \beta_{6} ) q^{53} + ( \beta_{1} - \beta_{3} ) q^{55} + ( -5 \beta_{1} - \beta_{3} ) q^{59} + 2 \beta_{4} q^{61} -\beta_{2} q^{65} + ( -\beta_{5} - \beta_{7} ) q^{67} + ( -\beta_{5} - \beta_{7} ) q^{71} -2 q^{73} + ( -10 - \beta_{6} ) q^{77} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{79} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{83} + ( -\beta_{2} + \beta_{4} ) q^{85} + ( 2 \beta_{2} + \beta_{4} ) q^{89} + ( \beta_{5} - 2 \beta_{7} ) q^{91} -\beta_{5} q^{95} + ( 6 + \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{5} + O(q^{10})$$ $$8q - 8q^{5} + 8q^{25} - 16q^{29} - 24q^{49} - 32q^{53} - 16q^{73} - 80q^{77} + 48q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{6} + 16 \nu^{2}$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{7} + 6 \nu^{6} - \nu^{5} - 13 \nu^{3} + 36 \nu^{2} - 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{7} - \nu^{5} - 29 \nu^{3} - 11 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$-2 \nu^{7} - \nu^{5} - 13 \nu^{3} - 5 \nu$$ $$\beta_{5}$$ $$=$$ $$($$$$-4 \nu^{7} + 2 \nu^{5} + 4 \nu^{4} - 26 \nu^{3} + 10 \nu + 14$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$-8 \nu^{7} + 2 \nu^{5} - 58 \nu^{3} + 22 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{7} - 4 \nu^{5} + 4 \nu^{4} + 52 \nu^{3} - 20 \nu + 14$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - 6 \beta_{3}$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} - 3 \beta_{2} + 9 \beta_{1}$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 6 \beta_{3}$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{7} + 2 \beta_{5} - 14$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{7} - 15 \beta_{6} + 11 \beta_{5} - 22 \beta_{4} + 30 \beta_{3}$$$$)/24$$ $$\nu^{6}$$ $$=$$ $$($$$$-4 \beta_{4} + 12 \beta_{2} - 27 \beta_{1}$$$$)/6$$ $$\nu^{7}$$ $$=$$ $$($$$$29 \beta_{7} + 39 \beta_{6} - 29 \beta_{5} - 58 \beta_{4} + 78 \beta_{3}$$$$)/24$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2591.1
 1.14412 + 1.14412i 0.437016 + 0.437016i −0.437016 + 0.437016i −1.14412 + 1.14412i −1.14412 − 1.14412i −0.437016 − 0.437016i 0.437016 − 0.437016i 1.14412 − 1.14412i
0 0 0 −1.00000 0 3.16228i 0 0 0
2591.2 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.3 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.4 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.5 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.6 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.7 0 0 0 −1.00000 0 3.16228i 0 0 0
2591.8 0 0 0 −1.00000 0 3.16228i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2591.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.b.a 8
3.b odd 2 1 2880.2.b.d yes 8
4.b odd 2 1 inner 2880.2.b.a 8
8.b even 2 1 2880.2.b.d yes 8
8.d odd 2 1 2880.2.b.d yes 8
12.b even 2 1 2880.2.b.d yes 8
24.f even 2 1 inner 2880.2.b.a 8
24.h odd 2 1 inner 2880.2.b.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2880.2.b.a 8 1.a even 1 1 trivial
2880.2.b.a 8 4.b odd 2 1 inner
2880.2.b.a 8 24.f even 2 1 inner
2880.2.b.a 8 24.h odd 2 1 inner
2880.2.b.d yes 8 3.b odd 2 1
2880.2.b.d yes 8 8.b even 2 1
2880.2.b.d yes 8 8.d odd 2 1
2880.2.b.d yes 8 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2880, [\chi])$$:

 $$T_{7}^{2} + 10$$ $$T_{29}^{2} + 4 T_{29} - 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 1 + T )^{8}$$
$7$ $$( 10 + T^{2} )^{4}$$
$11$ $$( 36 + 28 T^{2} + T^{4} )^{2}$$
$13$ $$( 324 + 44 T^{2} + T^{4} )^{2}$$
$17$ $$( 144 + 56 T^{2} + T^{4} )^{2}$$
$19$ $$( 144 - 56 T^{2} + T^{4} )^{2}$$
$23$ $$T^{8}$$
$29$ $$( -36 + 4 T + T^{2} )^{4}$$
$31$ $$( 1296 + 88 T^{2} + T^{4} )^{2}$$
$37$ $$( 900 + 140 T^{2} + T^{4} )^{2}$$
$41$ $$( 6084 + 164 T^{2} + T^{4} )^{2}$$
$43$ $$( -72 + T^{2} )^{4}$$
$47$ $$( 144 - 104 T^{2} + T^{4} )^{2}$$
$53$ $$( -24 + 8 T + T^{2} )^{4}$$
$59$ $$( 8100 + 220 T^{2} + T^{4} )^{2}$$
$61$ $$( 72 + T^{2} )^{4}$$
$67$ $$( 5184 - 176 T^{2} + T^{4} )^{2}$$
$71$ $$( 5184 - 176 T^{2} + T^{4} )^{2}$$
$73$ $$( 2 + T )^{8}$$
$79$ $$( 16 + 152 T^{2} + T^{4} )^{2}$$
$83$ $$( 576 + 112 T^{2} + T^{4} )^{2}$$
$89$ $$( 900 + 260 T^{2} + T^{4} )^{2}$$
$97$ $$( -4 - 12 T + T^{2} )^{4}$$