# Properties

 Label 2880.1.ec.a Level 2880 Weight 1 Character orbit 2880.ec Analytic conductor 1.437 Analytic rank 0 Dimension 16 Projective image $$D_{16}$$ CM discriminant -15 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$2880 = 2^{6} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 2880.ec (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.43730723638$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{16})$$ Coefficient field: $$\Q(\zeta_{32})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{16}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{16} + \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{32}^{7} q^{2} + \zeta_{32}^{14} q^{4} -\zeta_{32} q^{5} + \zeta_{32}^{5} q^{8} +O(q^{10})$$ $$q -\zeta_{32}^{7} q^{2} + \zeta_{32}^{14} q^{4} -\zeta_{32} q^{5} + \zeta_{32}^{5} q^{8} + \zeta_{32}^{8} q^{10} -\zeta_{32}^{12} q^{16} + ( -\zeta_{32}^{11} + \zeta_{32}^{13} ) q^{17} + ( \zeta_{32}^{2} + \zeta_{32}^{12} ) q^{19} -\zeta_{32}^{15} q^{20} + ( \zeta_{32}^{5} - \zeta_{32}^{7} ) q^{23} + \zeta_{32}^{2} q^{25} + ( \zeta_{32}^{6} - \zeta_{32}^{10} ) q^{31} -\zeta_{32}^{3} q^{32} + ( -\zeta_{32}^{2} + \zeta_{32}^{4} ) q^{34} + ( \zeta_{32}^{3} - \zeta_{32}^{9} ) q^{38} -\zeta_{32}^{6} q^{40} + ( -\zeta_{32}^{12} + \zeta_{32}^{14} ) q^{46} + ( -\zeta_{32}^{9} - \zeta_{32}^{15} ) q^{47} + \zeta_{32}^{4} q^{49} -\zeta_{32}^{9} q^{50} + ( -\zeta_{32}^{11} + \zeta_{32}^{15} ) q^{53} + ( -\zeta_{32}^{8} - \zeta_{32}^{14} ) q^{61} + ( -\zeta_{32} - \zeta_{32}^{13} ) q^{62} + \zeta_{32}^{10} q^{64} + ( \zeta_{32}^{9} - \zeta_{32}^{11} ) q^{68} + ( -1 - \zeta_{32}^{10} ) q^{76} + ( 1 - \zeta_{32}^{8} ) q^{79} + \zeta_{32}^{13} q^{80} + ( \zeta_{32} - \zeta_{32}^{13} ) q^{83} + ( \zeta_{32}^{12} - \zeta_{32}^{14} ) q^{85} + ( -\zeta_{32}^{3} + \zeta_{32}^{5} ) q^{92} + ( -1 - \zeta_{32}^{6} ) q^{94} + ( -\zeta_{32}^{3} - \zeta_{32}^{13} ) q^{95} -\zeta_{32}^{11} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 16q^{76} + 16q^{79} - 16q^{94} + O(q^{100})$$

## Character values

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

 $$n$$ $$577$$ $$641$$ $$901$$ $$2431$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{32}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.980785 − 0.195090i −0.980785 + 0.195090i −0.195090 + 0.980785i 0.195090 − 0.980785i −0.831470 − 0.555570i 0.831470 + 0.555570i −0.831470 + 0.555570i 0.831470 − 0.555570i −0.195090 − 0.980785i 0.195090 + 0.980785i 0.980785 + 0.195090i −0.980785 − 0.195090i 0.555570 − 0.831470i −0.555570 + 0.831470i 0.555570 + 0.831470i −0.555570 − 0.831470i
−0.195090 + 0.980785i 0 −0.923880 0.382683i −0.980785 + 0.195090i 0 0 0.555570 0.831470i 0 1.00000i
19.2 0.195090 0.980785i 0 −0.923880 0.382683i 0.980785 0.195090i 0 0 −0.555570 + 0.831470i 0 1.00000i
379.1 −0.980785 + 0.195090i 0 0.923880 0.382683i 0.195090 0.980785i 0 0 −0.831470 + 0.555570i 0 1.00000i
379.2 0.980785 0.195090i 0 0.923880 0.382683i −0.195090 + 0.980785i 0 0 0.831470 0.555570i 0 1.00000i
739.1 −0.555570 0.831470i 0 −0.382683 + 0.923880i 0.831470 + 0.555570i 0 0 0.980785 0.195090i 0 1.00000i
739.2 0.555570 + 0.831470i 0 −0.382683 + 0.923880i −0.831470 0.555570i 0 0 −0.980785 + 0.195090i 0 1.00000i
1099.1 −0.555570 + 0.831470i 0 −0.382683 0.923880i 0.831470 0.555570i 0 0 0.980785 + 0.195090i 0 1.00000i
1099.2 0.555570 0.831470i 0 −0.382683 0.923880i −0.831470 + 0.555570i 0 0 −0.980785 0.195090i 0 1.00000i
1459.1 −0.980785 0.195090i 0 0.923880 + 0.382683i 0.195090 + 0.980785i 0 0 −0.831470 0.555570i 0 1.00000i
1459.2 0.980785 + 0.195090i 0 0.923880 + 0.382683i −0.195090 0.980785i 0 0 0.831470 + 0.555570i 0 1.00000i
1819.1 −0.195090 0.980785i 0 −0.923880 + 0.382683i −0.980785 0.195090i 0 0 0.555570 + 0.831470i 0 1.00000i
1819.2 0.195090 + 0.980785i 0 −0.923880 + 0.382683i 0.980785 + 0.195090i 0 0 −0.555570 0.831470i 0 1.00000i
2179.1 −0.831470 + 0.555570i 0 0.382683 0.923880i −0.555570 + 0.831470i 0 0 0.195090 + 0.980785i 0 1.00000i
2179.2 0.831470 0.555570i 0 0.382683 0.923880i 0.555570 0.831470i 0 0 −0.195090 0.980785i 0 1.00000i
2539.1 −0.831470 0.555570i 0 0.382683 + 0.923880i −0.555570 0.831470i 0 0 0.195090 0.980785i 0 1.00000i
2539.2 0.831470 + 0.555570i 0 0.382683 + 0.923880i 0.555570 + 0.831470i 0 0 −0.195090 + 0.980785i 0 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2539.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
64.j odd 16 1 inner
192.s even 16 1 inner
320.bh odd 16 1 inner
960.cp even 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.1.ec.a 16
3.b odd 2 1 inner 2880.1.ec.a 16
5.b even 2 1 inner 2880.1.ec.a 16
15.d odd 2 1 CM 2880.1.ec.a 16
64.j odd 16 1 inner 2880.1.ec.a 16
192.s even 16 1 inner 2880.1.ec.a 16
320.bh odd 16 1 inner 2880.1.ec.a 16
960.cp even 16 1 inner 2880.1.ec.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2880.1.ec.a 16 1.a even 1 1 trivial
2880.1.ec.a 16 3.b odd 2 1 inner
2880.1.ec.a 16 5.b even 2 1 inner
2880.1.ec.a 16 15.d odd 2 1 CM
2880.1.ec.a 16 64.j odd 16 1 inner
2880.1.ec.a 16 192.s even 16 1 inner
2880.1.ec.a 16 320.bh odd 16 1 inner
2880.1.ec.a 16 960.cp even 16 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2880, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{16}$$
$3$ 
$5$ $$1 + T^{16}$$
$7$ $$( 1 + T^{8} )^{4}$$
$11$ $$( 1 + T^{16} )^{2}$$
$13$ $$( 1 + T^{16} )^{2}$$
$17$ $$( 1 + T^{16} )^{2}$$
$19$ $$( 1 + T^{4} )^{4}( 1 + T^{8} )^{2}$$
$23$ $$( 1 + T^{16} )^{2}$$
$29$ $$( 1 + T^{16} )^{2}$$
$31$ $$( 1 + T^{8} )^{4}$$
$37$ $$( 1 + T^{16} )^{2}$$
$41$ $$( 1 + T^{8} )^{4}$$
$43$ $$( 1 + T^{16} )^{2}$$
$47$ $$( 1 + T^{16} )^{2}$$
$53$ $$( 1 + T^{16} )^{2}$$
$59$ $$( 1 + T^{16} )^{2}$$
$61$ $$( 1 + T^{2} )^{8}( 1 + T^{8} )^{2}$$
$67$ $$( 1 + T^{16} )^{2}$$
$71$ $$( 1 + T^{8} )^{4}$$
$73$ $$( 1 + T^{8} )^{4}$$
$79$ $$( 1 - T )^{16}( 1 + T^{2} )^{8}$$
$83$ $$( 1 + T^{16} )^{2}$$
$89$ $$( 1 + T^{8} )^{4}$$
$97$ $$( 1 + T^{2} )^{16}$$