# Properties

 Label 2880.2.bl.a Level $2880$ Weight $2$ Character orbit 2880.bl 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.bl (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.9969157821$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.3317760000.4 Defining polynomial: $$x^{8} + 17 x^{4} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 720) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + ( -1 + \beta_{6} ) q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( -1 + \beta_{6} ) q^{7} + ( \beta_{3} - \beta_{5} ) q^{11} + ( -\beta_{4} + \beta_{6} ) q^{13} + ( \beta_{1} + \beta_{5} ) q^{17} + ( -2 - 2 \beta_{2} - \beta_{4} - \beta_{6} ) q^{19} + ( -\beta_{3} + \beta_{7} ) q^{23} -\beta_{2} q^{25} + 4 \beta_{5} q^{29} -4 \beta_{2} q^{31} + ( -\beta_{1} - \beta_{7} ) q^{35} + ( -1 - \beta_{2} ) q^{37} + ( -4 \beta_{1} + \beta_{3} + 4 \beta_{5} + \beta_{7} ) q^{41} + ( 6 - 6 \beta_{2} ) q^{43} + ( -\beta_{1} + \beta_{5} ) q^{47} + ( 9 - 2 \beta_{6} ) q^{49} + 2 \beta_{1} q^{53} + ( 1 - \beta_{6} ) q^{55} + ( -\beta_{3} - 9 \beta_{5} ) q^{59} + ( -3 + 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{61} + ( \beta_{3} - \beta_{7} ) q^{65} + ( -7 - 7 \beta_{2} + \beta_{4} + \beta_{6} ) q^{67} + ( 3 \beta_{1} + \beta_{3} + 3 \beta_{5} - \beta_{7} ) q^{71} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{73} + ( -2 \beta_{3} + 16 \beta_{5} ) q^{77} + ( -4 \beta_{2} + 2 \beta_{4} ) q^{79} -2 \beta_{1} q^{83} + ( -1 - \beta_{2} ) q^{85} + ( -8 \beta_{1} - \beta_{3} + 8 \beta_{5} - \beta_{7} ) q^{89} + ( 15 - 15 \beta_{2} + \beta_{4} - \beta_{6} ) q^{91} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{95} + ( 2 + 4 \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{7} + O(q^{10})$$ $$8q - 8q^{7} - 16q^{19} - 8q^{37} + 48q^{43} + 72q^{49} + 8q^{55} - 24q^{61} - 56q^{67} - 8q^{85} + 120q^{91} + 16q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 17 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 5 \nu$$$$)/28$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 33 \nu^{2}$$$$)/112$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 61 \nu$$$$)/28$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{4} + 17$$$$)/7$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 13 \nu^{3}$$$$)/448$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - \nu^{2}$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-11 \nu^{7} - 251 \nu^{3}$$$$)/448$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 7 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{7} + 11 \beta_{5}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$7 \beta_{4} - 17$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{3} + 61 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-33 \beta_{6} - 7 \beta_{2}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{7} - 251 \beta_{5}$$$$)/2$$

## Character values

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

 $$n$$ $$577$$ $$641$$ $$901$$ $$2431$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
431.1
 −1.01575 + 1.72286i 1.72286 − 1.01575i 1.01575 − 1.72286i −1.72286 + 1.01575i −1.01575 − 1.72286i 1.72286 + 1.01575i 1.01575 + 1.72286i −1.72286 − 1.01575i
0 0 0 −0.707107 0.707107i 0 −4.87298 0 0 0
431.2 0 0 0 −0.707107 0.707107i 0 2.87298 0 0 0
431.3 0 0 0 0.707107 + 0.707107i 0 −4.87298 0 0 0
431.4 0 0 0 0.707107 + 0.707107i 0 2.87298 0 0 0
1871.1 0 0 0 −0.707107 + 0.707107i 0 −4.87298 0 0 0
1871.2 0 0 0 −0.707107 + 0.707107i 0 2.87298 0 0 0
1871.3 0 0 0 0.707107 0.707107i 0 −4.87298 0 0 0
1871.4 0 0 0 0.707107 0.707107i 0 2.87298 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1871.4 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
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.bl.a 8
3.b odd 2 1 inner 2880.2.bl.a 8
4.b odd 2 1 720.2.bl.a 8
12.b even 2 1 720.2.bl.a 8
16.e even 4 1 720.2.bl.a 8
16.f odd 4 1 inner 2880.2.bl.a 8
48.i odd 4 1 720.2.bl.a 8
48.k even 4 1 inner 2880.2.bl.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.bl.a 8 4.b odd 2 1
720.2.bl.a 8 12.b even 2 1
720.2.bl.a 8 16.e even 4 1
720.2.bl.a 8 48.i odd 4 1
2880.2.bl.a 8 1.a even 1 1 trivial
2880.2.bl.a 8 3.b odd 2 1 inner
2880.2.bl.a 8 16.f odd 4 1 inner
2880.2.bl.a 8 48.k even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 2 T_{7} - 14$$ acting on $$S_{2}^{\mathrm{new}}(2880, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 1 + T^{4} )^{2}$$
$7$ $$( -14 + 2 T + T^{2} )^{4}$$
$11$ $$38416 + 632 T^{4} + T^{8}$$
$13$ $$( 900 + T^{4} )^{2}$$
$17$ $$( 2 + T^{2} )^{4}$$
$19$ $$( 484 - 176 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$23$ $$( 30 + T^{2} )^{4}$$
$29$ $$( 256 + T^{4} )^{2}$$
$31$ $$( 16 + T^{2} )^{4}$$
$37$ $$( 2 + 2 T + T^{2} )^{4}$$
$41$ $$( 4 - 124 T^{2} + T^{4} )^{2}$$
$43$ $$( 72 - 12 T + T^{2} )^{4}$$
$47$ $$( -2 + T^{2} )^{4}$$
$53$ $$( 16 + T^{4} )^{2}$$
$59$ $$18974736 + 28152 T^{4} + T^{8}$$
$61$ $$( 10404 - 1224 T + 72 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$67$ $$( 4624 + 1904 T + 392 T^{2} + 28 T^{3} + T^{4} )^{2}$$
$71$ $$( 144 + 96 T^{2} + T^{4} )^{2}$$
$73$ $$( 3136 + 128 T^{2} + T^{4} )^{2}$$
$79$ $$( 1936 + 152 T^{2} + T^{4} )^{2}$$
$83$ $$( 16 + T^{4} )^{2}$$
$89$ $$( 9604 - 316 T^{2} + T^{4} )^{2}$$
$97$ $$( -236 - 4 T + T^{2} )^{4}$$