# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2880 = 2^{6} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2880.t (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: 8.0.18939904.2 Defining polynomial: $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 240) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + ( \beta_{2} + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + \beta_{3} q^{5} + ( \beta_{2} + \beta_{3} ) q^{7} + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{11} + ( 1 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{13} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{17} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{7} ) q^{19} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} ) q^{23} -\beta_{4} q^{25} + ( -3 + \beta_{1} - 2 \beta_{2} - 3 \beta_{4} - \beta_{7} ) q^{29} + ( -1 - \beta_{1} + \beta_{5} - \beta_{6} ) q^{31} + ( -1 - \beta_{4} ) q^{35} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{37} + ( -4 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{41} + ( -4 \beta_{3} + 2 \beta_{5} ) q^{43} + ( -4 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} + 3 \beta_{6} ) q^{47} + 5 q^{49} + ( 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{7} ) q^{53} + ( \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{55} + ( 1 + 3 \beta_{1} + 3 \beta_{3} - \beta_{4} + 5 \beta_{5} + 3 \beta_{7} ) q^{59} + ( -2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{6} - \beta_{7} ) q^{61} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{65} + ( -2 \beta_{1} + 2 \beta_{7} ) q^{67} + ( 3 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{71} + ( 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + \beta_{5} + \beta_{6} ) q^{73} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{77} + ( 5 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{79} + ( 4 + 2 \beta_{1} + 4 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} ) q^{83} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{85} + ( -6 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{89} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{7} ) q^{91} + ( -2 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{95} + ( -6 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{11} + 8q^{13} - 8q^{17} - 8q^{19} - 24q^{29} - 8q^{31} - 8q^{35} - 8q^{37} + 40q^{49} + 8q^{59} - 16q^{61} + 8q^{65} + 8q^{77} + 40q^{79} + 32q^{83} + 8q^{85} + 8q^{91} - 16q^{95} - 48q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{6} + 3 \nu^{5} - 11 \nu^{4} + 17 \nu^{3} - 24 \nu^{2} + 16 \nu - 5$$ $$\beta_{2}$$ $$=$$ $$5 \nu^{7} - 17 \nu^{6} + 60 \nu^{5} - 105 \nu^{4} + 155 \nu^{3} - 133 \nu^{2} + 77 \nu - 19$$ $$\beta_{3}$$ $$=$$ $$5 \nu^{7} - 18 \nu^{6} + 63 \nu^{5} - 115 \nu^{4} + 170 \nu^{3} - 152 \nu^{2} + 89 \nu - 23$$ $$\beta_{4}$$ $$=$$ $$8 \nu^{7} - 28 \nu^{6} + 98 \nu^{5} - 175 \nu^{4} + 256 \nu^{3} - 223 \nu^{2} + 126 \nu - 31$$ $$\beta_{5}$$ $$=$$ $$9 \nu^{7} - 31 \nu^{6} + 108 \nu^{5} - 190 \nu^{4} + 275 \nu^{3} - 236 \nu^{2} + 131 \nu - 33$$ $$\beta_{6}$$ $$=$$ $$9 \nu^{7} - 32 \nu^{6} + 111 \nu^{5} - 200 \nu^{4} + 290 \nu^{3} - 253 \nu^{2} + 141 \nu - 33$$ $$\beta_{7}$$ $$=$$ $$10 \nu^{7} - 35 \nu^{6} + 123 \nu^{5} - 220 \nu^{4} + 325 \nu^{3} - 285 \nu^{2} + 168 \nu - 43$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{3} - \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} + 2 \beta_{6} - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} - 5$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{7} - \beta_{6} + 3 \beta_{5} - 6 \beta_{4} + 12 \beta_{3} + 4 \beta_{2} - 2 \beta_{1} + 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$3 \beta_{7} - 10 \beta_{6} + 5 \beta_{5} + 10 \beta_{4} + 6 \beta_{3} - 19 \beta_{2} - 5 \beta_{1} + 26$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$22 \beta_{7} - 9 \beta_{6} - 11 \beta_{5} + 45 \beta_{4} - 48 \beta_{3} - 32 \beta_{2} + 5 \beta_{1} - 6$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$7 \beta_{7} + 33 \beta_{6} - 30 \beta_{5} - 83 \beta_{3} + 64 \beta_{2} + 35 \beta_{1} - 118$$$$)/2$$

## Character values

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

 $$n$$ $$577$$ $$641$$ $$901$$ $$2431$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
721.1
 0.5 + 0.0297061i 0.5 − 1.44392i 0.5 + 2.10607i 0.5 − 0.691860i 0.5 − 0.0297061i 0.5 + 1.44392i 0.5 − 2.10607i 0.5 + 0.691860i
0 0 0 −0.707107 + 0.707107i 0 1.41421i 0 0 0
721.2 0 0 0 −0.707107 + 0.707107i 0 1.41421i 0 0 0
721.3 0 0 0 0.707107 0.707107i 0 1.41421i 0 0 0
721.4 0 0 0 0.707107 0.707107i 0 1.41421i 0 0 0
2161.1 0 0 0 −0.707107 0.707107i 0 1.41421i 0 0 0
2161.2 0 0 0 −0.707107 0.707107i 0 1.41421i 0 0 0
2161.3 0 0 0 0.707107 + 0.707107i 0 1.41421i 0 0 0
2161.4 0 0 0 0.707107 + 0.707107i 0 1.41421i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2161.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e 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.t.b 8
3.b odd 2 1 960.2.s.b 8
4.b odd 2 1 720.2.t.b 8
12.b even 2 1 240.2.s.b 8
16.e even 4 1 inner 2880.2.t.b 8
16.f odd 4 1 720.2.t.b 8
24.f even 2 1 1920.2.s.c 8
24.h odd 2 1 1920.2.s.d 8
48.i odd 4 1 960.2.s.b 8
48.i odd 4 1 1920.2.s.d 8
48.k even 4 1 240.2.s.b 8
48.k even 4 1 1920.2.s.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.s.b 8 12.b even 2 1
240.2.s.b 8 48.k even 4 1
720.2.t.b 8 4.b odd 2 1
720.2.t.b 8 16.f odd 4 1
960.2.s.b 8 3.b odd 2 1
960.2.s.b 8 48.i odd 4 1
1920.2.s.c 8 24.f even 2 1
1920.2.s.c 8 48.k even 4 1
1920.2.s.d 8 24.h odd 2 1
1920.2.s.d 8 48.i odd 4 1
2880.2.t.b 8 1.a even 1 1 trivial
2880.2.t.b 8 16.e even 4 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 1 + T^{4} )^{2}$$
$7$ $$( 2 + T^{2} )^{4}$$
$11$ $$64 + 128 T + 128 T^{2} + 16 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$13$ $$16 + 64 T + 128 T^{2} + 32 T^{3} - 8 T^{4} - 16 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$17$ $$( 8 - 12 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$19$ $$256 - 512 T + 512 T^{2} + 384 T^{3} + 224 T^{4} - 96 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$23$ $$200704 + 73728 T^{2} + 5248 T^{4} + 128 T^{6} + T^{8}$$
$29$ $$135424 - 11776 T + 512 T^{2} + 11392 T^{3} + 7136 T^{4} + 1888 T^{5} + 288 T^{6} + 24 T^{7} + T^{8}$$
$31$ $$( -28 - 88 T - 40 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$37$ $$414736 - 154560 T + 28800 T^{2} + 6368 T^{3} + 1016 T^{4} - 144 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$41$ $$17707264 + 1276672 T^{2} + 31072 T^{4} + 304 T^{6} + T^{8}$$
$43$ $$12544 - 28672 T + 32768 T^{2} - 18432 T^{3} + 5408 T^{4} - 256 T^{5} + T^{8}$$
$47$ $$( -224 - 416 T - 152 T^{2} + T^{4} )^{2}$$
$53$ $$73984 + 33312 T^{4} + T^{8}$$
$59$ $$7529536 - 6453888 T + 2765952 T^{2} - 482944 T^{3} + 43904 T^{4} - 784 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$61$ $$4343056 - 1667200 T + 320000 T^{2} + 27456 T^{3} + 1608 T^{4} - 416 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$67$ $$1183744 + 4224 T^{4} + T^{8}$$
$71$ $$5161984 + 699904 T^{2} + 23168 T^{4} + 272 T^{6} + T^{8}$$
$73$ $$50176 + 27136 T^{2} + 4224 T^{4} + 176 T^{6} + T^{8}$$
$79$ $$( 164 - 264 T + 120 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$83$ $$2166784 - 1130496 T + 294912 T^{2} + 14336 T^{3} + 9344 T^{4} - 3328 T^{5} + 512 T^{6} - 32 T^{7} + T^{8}$$
$89$ $$118026496 + 6544640 T^{2} + 102752 T^{4} + 592 T^{6} + T^{8}$$
$97$ $$( -736 - 32 T + 120 T^{2} + 24 T^{3} + T^{4} )^{2}$$