# Properties

 Label 2880.1.p.b Level 2880 Weight 1 Character orbit 2880.p Analytic conductor 1.437 Analytic rank 0 Dimension 2 Projective image $$D_{2}$$ CM/RM discs -24, -120, 5 Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.43730723638$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{5}, \sqrt{-6})$$ Artin image $D_4:C_2$ Artin field Galois closure of 8.0.19110297600.5

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -i q^{5} +O(q^{10})$$ $$q -i q^{5} + 2 q^{11} - q^{25} -2 i q^{29} + 2 i q^{31} - q^{49} -2 i q^{55} + 2 q^{59} -2 i q^{79} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 4q^{11} - 2q^{25} - 2q^{49} + 4q^{59} + O(q^{100})$$

## Character values

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

 $$n$$ $$577$$ $$641$$ $$901$$ $$2431$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2719.1
 1.00000i − 1.00000i
0 0 0 1.00000i 0 0 0 0 0
2719.2 0 0 0 1.00000i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 RM by $$\Q(\sqrt{5})$$
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
120.i odd 2 1 CM by $$\Q(\sqrt{-30})$$
8.d odd 2 1 inner
12.b even 2 1 inner
40.e odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.1.p.b yes 2
3.b odd 2 1 2880.1.p.a 2
4.b odd 2 1 2880.1.p.a 2
5.b even 2 1 RM 2880.1.p.b yes 2
8.b even 2 1 2880.1.p.a 2
8.d odd 2 1 inner 2880.1.p.b yes 2
12.b even 2 1 inner 2880.1.p.b yes 2
15.d odd 2 1 2880.1.p.a 2
20.d odd 2 1 2880.1.p.a 2
24.f even 2 1 2880.1.p.a 2
24.h odd 2 1 CM 2880.1.p.b yes 2
40.e odd 2 1 inner 2880.1.p.b yes 2
40.f even 2 1 2880.1.p.a 2
60.h even 2 1 inner 2880.1.p.b yes 2
120.i odd 2 1 CM 2880.1.p.b yes 2
120.m even 2 1 2880.1.p.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2880.1.p.a 2 3.b odd 2 1
2880.1.p.a 2 4.b odd 2 1
2880.1.p.a 2 8.b even 2 1
2880.1.p.a 2 15.d odd 2 1
2880.1.p.a 2 20.d odd 2 1
2880.1.p.a 2 24.f even 2 1
2880.1.p.a 2 40.f even 2 1
2880.1.p.a 2 120.m even 2 1
2880.1.p.b yes 2 1.a even 1 1 trivial
2880.1.p.b yes 2 5.b even 2 1 RM
2880.1.p.b yes 2 8.d odd 2 1 inner
2880.1.p.b yes 2 12.b even 2 1 inner
2880.1.p.b yes 2 24.h odd 2 1 CM
2880.1.p.b yes 2 40.e odd 2 1 inner
2880.1.p.b yes 2 60.h even 2 1 inner
2880.1.p.b yes 2 120.i odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} - 2$$ acting on $$S_{1}^{\mathrm{new}}(2880, [\chi])$$.

## Hecke Characteristic Polynomials

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