# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$22.9969157821$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 960) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{5} + ( -\beta_{4} + \beta_{5} ) q^{7} +O(q^{10})$$ $$q -\beta_{4} q^{5} + ( -\beta_{4} + \beta_{5} ) q^{7} + \beta_{1} q^{11} -\beta_{2} q^{13} + \beta_{1} q^{17} + \beta_{3} q^{19} + \beta_{6} q^{25} + \beta_{7} q^{29} + ( -\beta_{2} + \beta_{4} + \beta_{5} ) q^{31} + ( -5 + \beta_{6} ) q^{35} + ( -\beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{37} -6 q^{41} + ( 2 - \beta_{3} - 2 \beta_{6} ) q^{43} + ( \beta_{4} - \beta_{5} + \beta_{7} ) q^{47} + ( -3 + \beta_{3} + 2 \beta_{6} ) q^{49} + ( -2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{53} + ( 3 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{55} + ( \beta_{1} - \beta_{3} ) q^{59} + ( 3 \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{61} + ( -2 + \beta_{1} + \beta_{3} ) q^{65} + ( -6 - \beta_{3} - 2 \beta_{6} ) q^{67} + ( -2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{71} -3 \beta_{3} q^{73} + ( 6 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{77} + ( \beta_{2} - \beta_{4} - \beta_{5} ) q^{79} + ( -2 - \beta_{3} - 2 \beta_{6} ) q^{83} + ( 3 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{85} + 2 q^{89} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{91} + ( 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{95} + 2 \beta_{1} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 40q^{35} - 48q^{41} + 16q^{43} - 24q^{49} - 16q^{65} - 48q^{67} - 16q^{83} + 16q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{5} + 15 \nu^{3} + 42 \nu$$$$)/10$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 3 \nu^{5} - 5 \nu^{3} - 4 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu$$$$)/10$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{7} - \nu^{6} + 5 \nu^{5} + 5 \nu^{4} + 15 \nu^{3} + 15 \nu^{2} + 30 \nu + 8$$$$)/20$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{7} + \nu^{6} + 5 \nu^{5} - 5 \nu^{4} + 15 \nu^{3} - 15 \nu^{2} + 30 \nu - 8$$$$)/20$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} + 10 \nu^{6} + 5 \nu^{5} + 30 \nu^{4} - 5 \nu^{3} + 10 \nu^{2} + 16 \nu + 60$$$$)/20$$ $$\beta_{7}$$ $$=$$ $$($$$$13 \nu^{6} + 15 \nu^{4} + 45 \nu^{2} + 96$$$$)/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 2 \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - 2 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - \beta_{3} - 6$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{7} + 6 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} - 2$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{5} - 5 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$5 \beta_{7} + 15 \beta_{5} - 15 \beta_{4} - 36$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$20 \beta_{5} + 20 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - 14 \beta_{1}$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −0.228425 + 1.39564i −0.228425 − 1.39564i −1.09445 − 0.895644i −1.09445 + 0.895644i 1.09445 + 0.895644i 1.09445 − 0.895644i 0.228425 − 1.39564i 0.228425 + 1.39564i
0 0 0 −2.18890 0.456850i 0 0.913701i 0 0 0
289.2 0 0 0 −2.18890 + 0.456850i 0 0.913701i 0 0 0
289.3 0 0 0 −0.456850 2.18890i 0 4.37780i 0 0 0
289.4 0 0 0 −0.456850 + 2.18890i 0 4.37780i 0 0 0
289.5 0 0 0 0.456850 2.18890i 0 4.37780i 0 0 0
289.6 0 0 0 0.456850 + 2.18890i 0 4.37780i 0 0 0
289.7 0 0 0 2.18890 0.456850i 0 0.913701i 0 0 0
289.8 0 0 0 2.18890 + 0.456850i 0 0.913701i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
20.d odd 2 1 inner
40.f 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.d.i 8
3.b odd 2 1 960.2.d.f yes 8
4.b odd 2 1 2880.2.d.j 8
5.b even 2 1 2880.2.d.j 8
8.b even 2 1 2880.2.d.j 8
8.d odd 2 1 inner 2880.2.d.i 8
12.b even 2 1 960.2.d.e 8
15.d odd 2 1 960.2.d.e 8
15.e even 4 1 4800.2.k.q 8
15.e even 4 1 4800.2.k.r 8
20.d odd 2 1 inner 2880.2.d.i 8
24.f even 2 1 960.2.d.f yes 8
24.h odd 2 1 960.2.d.e 8
40.e odd 2 1 2880.2.d.j 8
40.f even 2 1 inner 2880.2.d.i 8
48.i odd 4 1 3840.2.f.i 8
48.i odd 4 1 3840.2.f.k 8
48.k even 4 1 3840.2.f.i 8
48.k even 4 1 3840.2.f.k 8
60.h even 2 1 960.2.d.f yes 8
60.l odd 4 1 4800.2.k.q 8
60.l odd 4 1 4800.2.k.r 8
120.i odd 2 1 960.2.d.f yes 8
120.m even 2 1 960.2.d.e 8
120.q odd 4 1 4800.2.k.q 8
120.q odd 4 1 4800.2.k.r 8
120.w even 4 1 4800.2.k.q 8
120.w even 4 1 4800.2.k.r 8
240.t even 4 1 3840.2.f.i 8
240.t even 4 1 3840.2.f.k 8
240.bm odd 4 1 3840.2.f.i 8
240.bm odd 4 1 3840.2.f.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.d.e 8 12.b even 2 1
960.2.d.e 8 15.d odd 2 1
960.2.d.e 8 24.h odd 2 1
960.2.d.e 8 120.m even 2 1
960.2.d.f yes 8 3.b odd 2 1
960.2.d.f yes 8 24.f even 2 1
960.2.d.f yes 8 60.h even 2 1
960.2.d.f yes 8 120.i odd 2 1
2880.2.d.i 8 1.a even 1 1 trivial
2880.2.d.i 8 8.d odd 2 1 inner
2880.2.d.i 8 20.d odd 2 1 inner
2880.2.d.i 8 40.f even 2 1 inner
2880.2.d.j 8 4.b odd 2 1
2880.2.d.j 8 5.b even 2 1
2880.2.d.j 8 8.b even 2 1
2880.2.d.j 8 40.e odd 2 1
3840.2.f.i 8 48.i odd 4 1
3840.2.f.i 8 48.k even 4 1
3840.2.f.i 8 240.t even 4 1
3840.2.f.i 8 240.bm odd 4 1
3840.2.f.k 8 48.i odd 4 1
3840.2.f.k 8 48.k even 4 1
3840.2.f.k 8 240.t even 4 1
3840.2.f.k 8 240.bm odd 4 1
4800.2.k.q 8 15.e even 4 1
4800.2.k.q 8 60.l odd 4 1
4800.2.k.q 8 120.q odd 4 1
4800.2.k.q 8 120.w even 4 1
4800.2.k.r 8 15.e even 4 1
4800.2.k.r 8 60.l odd 4 1
4800.2.k.r 8 120.q odd 4 1
4800.2.k.r 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 $$S_{2}^{\mathrm{new}}(2880, [\chi])$$:

 $$T_{7}^{4} + 20 T_{7}^{2} + 16$$ $$T_{11}^{4} + 44 T_{11}^{2} + 400$$ $$T_{13}^{4} - 20 T_{13}^{2} + 16$$ $$T_{31}^{2} - 28$$ $$T_{41} + 6$$ $$T_{43}^{2} - 4 T_{43} - 80$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$625 - 34 T^{4} + T^{8}$$
$7$ $$( 16 + 20 T^{2} + T^{4} )^{2}$$
$11$ $$( 400 + 44 T^{2} + T^{4} )^{2}$$
$13$ $$( 16 - 20 T^{2} + T^{4} )^{2}$$
$17$ $$( 400 + 44 T^{2} + T^{4} )^{2}$$
$19$ $$( 16 + T^{2} )^{4}$$
$23$ $$T^{8}$$
$29$ $$( 400 + 68 T^{2} + T^{4} )^{2}$$
$31$ $$( -28 + T^{2} )^{4}$$
$37$ $$( 400 - 68 T^{2} + T^{4} )^{2}$$
$41$ $$( 6 + T )^{8}$$
$43$ $$( -80 - 4 T + T^{2} )^{4}$$
$47$ $$( 48 + T^{2} )^{4}$$
$53$ $$( 400 - 68 T^{2} + T^{4} )^{2}$$
$59$ $$( 144 + 60 T^{2} + T^{4} )^{2}$$
$61$ $$( 112 + T^{2} )^{4}$$
$67$ $$( -48 + 12 T + T^{2} )^{4}$$
$71$ $$( -48 + T^{2} )^{4}$$
$73$ $$( 144 + T^{2} )^{4}$$
$79$ $$( -28 + T^{2} )^{4}$$
$83$ $$( -80 + 4 T + T^{2} )^{4}$$
$89$ $$( -2 + T )^{8}$$
$97$ $$( 6400 + 176 T^{2} + T^{4} )^{2}$$