# Properties

 Label 2880.1.i.b Level $2880$ Weight $1$ Character orbit 2880.i Analytic conductor $1.437$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -40 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.43730723638$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.5400.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{2} q^{5} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{7}+O(q^{10})$$ q + z^2 * q^5 + (-z^3 - z) * q^7 $$q + \zeta_{8}^{2} q^{5} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{7} + (\zeta_{8}^{3} - \zeta_{8}) q^{11} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{13} + q^{23} - q^{25} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{35} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{37} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{41} - q^{49} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{55} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{59} + (\zeta_{8}^{3} + \zeta_{8}) q^{65} + \zeta_{8}^{2} q^{77} + (\zeta_{8}^{3} + \zeta_{8}) q^{89} - \zeta_{8}^{2} q^{91} +O(q^{100})$$ q + z^2 * q^5 + (-z^3 - z) * q^7 + (z^3 - z) * q^11 + (-z^3 + z) * q^13 + q^23 - q^25 + (-z^3 + z) * q^35 + (-z^3 + z) * q^37 + (-z^3 - z) * q^41 - q^49 + (-z^3 - z) * q^55 + (-z^3 + z) * q^59 + (z^3 + z) * q^65 + z^2 * q^77 + (z^3 + z) * q^89 - z^2 * q^91 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 8 q^{23} - 4 q^{25} - 4 q^{49}+O(q^{100})$$ 4 * q + 8 * q^23 - 4 * q^25 - 4 * q^49

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1889.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 1.00000i 0 1.41421i 0 0 0
1889.2 0 0 0 1.00000i 0 1.41421i 0 0 0
1889.3 0 0 0 1.00000i 0 1.41421i 0 0 0
1889.4 0 0 0 1.00000i 0 1.41421i 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
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
8.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
24.f even 2 1 inner
120.i odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23} - 2$$ acting on $$S_{1}^{\mathrm{new}}(2880, [\chi])$$.

## Hecke characteristic polynomials

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