# Properties

 Label 2880.1.bh.a Level $2880$ Weight $1$ Character orbit 2880.bh Analytic conductor $1.437$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2880 = 2^{6} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2880.bh (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.43730723638$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1440) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.18000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{5}+O(q^{10})$$ q - q^5 $$q - q^{5} + ( - i - 1) q^{13} + (i - 1) q^{17} + q^{25} + i q^{29} + (i - 1) q^{37} - q^{41} + i q^{49} + ( - i - 1) q^{53} + (i + 1) q^{65} + ( - i - 1) q^{73} + ( - i + 1) q^{85} - i q^{89} + (i - 1) q^{97} +O(q^{100})$$ q - q^5 + (-z - 1) * q^13 + (z - 1) * q^17 + q^25 + z * q^29 + (z - 1) * q^37 - q^41 + z * q^49 + (-z - 1) * q^53 + (z + 1) * q^65 + (-z - 1) * q^73 + (-z + 1) * q^85 - z * q^89 + (z - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^5 $$2 q - 2 q^{5} - 2 q^{13} - 2 q^{17} + 2 q^{25} - 2 q^{37} - 4 q^{41} - 2 q^{53} + 2 q^{65} - 2 q^{73} + 2 q^{85} - 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^13 - 2 * q^17 + 2 * q^25 - 2 * q^37 - 4 * q^41 - 2 * q^53 + 2 * q^65 - 2 * q^73 + 2 * q^85 - 2 * q^97

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 − 1.00000i 1.00000i
0 0 0 −1.00000 0 0 0 0 0
1153.1 0 0 0 −1.00000 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.1.bh.a 2
3.b odd 2 1 2880.1.bh.c 2
4.b odd 2 1 CM 2880.1.bh.a 2
5.c odd 4 1 inner 2880.1.bh.a 2
8.b even 2 1 1440.1.bh.c yes 2
8.d odd 2 1 1440.1.bh.c yes 2
12.b even 2 1 2880.1.bh.c 2
15.e even 4 1 2880.1.bh.c 2
20.e even 4 1 inner 2880.1.bh.a 2
24.f even 2 1 1440.1.bh.a 2
24.h odd 2 1 1440.1.bh.a 2
40.i odd 4 1 1440.1.bh.c yes 2
40.k even 4 1 1440.1.bh.c yes 2
60.l odd 4 1 2880.1.bh.c 2
120.q odd 4 1 1440.1.bh.a 2
120.w even 4 1 1440.1.bh.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.1.bh.a 2 24.f even 2 1
1440.1.bh.a 2 24.h odd 2 1
1440.1.bh.a 2 120.q odd 4 1
1440.1.bh.a 2 120.w even 4 1
1440.1.bh.c yes 2 8.b even 2 1
1440.1.bh.c yes 2 8.d odd 2 1
1440.1.bh.c yes 2 40.i odd 4 1
1440.1.bh.c yes 2 40.k even 4 1
2880.1.bh.a 2 1.a even 1 1 trivial
2880.1.bh.a 2 4.b odd 2 1 CM
2880.1.bh.a 2 5.c odd 4 1 inner
2880.1.bh.a 2 20.e even 4 1 inner
2880.1.bh.c 2 3.b odd 2 1
2880.1.bh.c 2 12.b even 2 1
2880.1.bh.c 2 15.e even 4 1
2880.1.bh.c 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2880, [\chi])$$:

 $$T_{13}^{2} + 2T_{13} + 2$$ T13^2 + 2*T13 + 2 $$T_{17}^{2} + 2T_{17} + 2$$ T17^2 + 2*T17 + 2

## Hecke characteristic polynomials

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