# Properties

 Label 2880.1.bu.c Level $2880$ Weight $1$ Character orbit 2880.bu Analytic conductor $1.437$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -20 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.43730723638$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 720) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.10497600.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{5} q^{3} - \zeta_{12}^{2} q^{5} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{7} - \zeta_{12}^{4} q^{9} +O(q^{10})$$ q - z^5 * q^3 - z^2 * q^5 + (-z^5 - z^3) * q^7 - z^4 * q^9 $$q - \zeta_{12}^{5} q^{3} - \zeta_{12}^{2} q^{5} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{7} - \zeta_{12}^{4} q^{9} - \zeta_{12} q^{15} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{21} + (\zeta_{12}^{3} + \zeta_{12}) q^{23} + \zeta_{12}^{4} q^{25} - \zeta_{12}^{3} q^{27} + \zeta_{12}^{4} q^{29} + (\zeta_{12}^{5} - \zeta_{12}) q^{35} - \zeta_{12}^{2} q^{41} - q^{45} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{47} + ( - \zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{49} - \zeta_{12}^{4} q^{61} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{63} + (\zeta_{12}^{3} + \zeta_{12}) q^{67} + (\zeta_{12}^{2} + 1) q^{69} + \zeta_{12}^{3} q^{75} - \zeta_{12}^{2} q^{81} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{83} + \zeta_{12}^{3} q^{87} - q^{89} +O(q^{100})$$ q - z^5 * q^3 - z^2 * q^5 + (-z^5 - z^3) * q^7 - z^4 * q^9 - z * q^15 + (-z^4 - z^2) * q^21 + (z^3 + z) * q^23 + z^4 * q^25 - z^3 * q^27 + z^4 * q^29 + (z^5 - z) * q^35 - z^2 * q^41 - q^45 + (z^5 + z^3) * q^47 + (-z^4 - z^2 - 1) * q^49 - z^4 * q^61 + (-z^3 - z) * q^63 + (z^3 + z) * q^67 + (z^2 + 1) * q^69 + z^3 * q^75 - z^2 * q^81 + (z^5 + z^3) * q^83 + z^3 * q^87 - q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5} + 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^5 + 2 * q^9 $$4 q - 2 q^{5} + 2 q^{9} - 2 q^{25} - 2 q^{29} - 2 q^{41} - 4 q^{45} - 4 q^{49} + 2 q^{61} + 6 q^{69} - 2 q^{81} - 4 q^{89}+O(q^{100})$$ 4 * q - 2 * q^5 + 2 * q^9 - 2 * q^25 - 2 * q^29 - 2 * q^41 - 4 * q^45 - 4 * q^49 + 2 * q^61 + 6 * q^69 - 2 * q^81 - 4 * q^89

## Character values

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

 $$n$$ $$577$$ $$641$$ $$901$$ $$2431$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
0 −0.866025 0.500000i 0 −0.500000 + 0.866025i 0 −0.866025 1.50000i 0 0.500000 + 0.866025i 0
319.2 0 0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 0.866025 + 1.50000i 0 0.500000 + 0.866025i 0
2239.1 0 −0.866025 + 0.500000i 0 −0.500000 0.866025i 0 −0.866025 + 1.50000i 0 0.500000 0.866025i 0
2239.2 0 0.866025 0.500000i 0 −0.500000 0.866025i 0 0.866025 1.50000i 0 0.500000 0.866025i 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
45.j even 6 1 inner
180.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.1.bu.c 4
4.b odd 2 1 inner 2880.1.bu.c 4
5.b even 2 1 inner 2880.1.bu.c 4
8.b even 2 1 720.1.bu.a 4
8.d odd 2 1 720.1.bu.a 4
9.c even 3 1 inner 2880.1.bu.c 4
20.d odd 2 1 CM 2880.1.bu.c 4
24.f even 2 1 2160.1.bu.a 4
24.h odd 2 1 2160.1.bu.a 4
36.f odd 6 1 inner 2880.1.bu.c 4
40.e odd 2 1 720.1.bu.a 4
40.f even 2 1 720.1.bu.a 4
40.i odd 4 1 3600.1.cc.a 2
40.i odd 4 1 3600.1.cc.b 2
40.k even 4 1 3600.1.cc.a 2
40.k even 4 1 3600.1.cc.b 2
45.j even 6 1 inner 2880.1.bu.c 4
72.j odd 6 1 2160.1.bu.a 4
72.l even 6 1 2160.1.bu.a 4
72.n even 6 1 720.1.bu.a 4
72.p odd 6 1 720.1.bu.a 4
120.i odd 2 1 2160.1.bu.a 4
120.m even 2 1 2160.1.bu.a 4
180.p odd 6 1 inner 2880.1.bu.c 4
360.z odd 6 1 720.1.bu.a 4
360.bd even 6 1 2160.1.bu.a 4
360.bh odd 6 1 2160.1.bu.a 4
360.bk even 6 1 720.1.bu.a 4
360.bo even 12 1 3600.1.cc.a 2
360.bo even 12 1 3600.1.cc.b 2
360.bu odd 12 1 3600.1.cc.a 2
360.bu odd 12 1 3600.1.cc.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.bu.a 4 8.b even 2 1
720.1.bu.a 4 8.d odd 2 1
720.1.bu.a 4 40.e odd 2 1
720.1.bu.a 4 40.f even 2 1
720.1.bu.a 4 72.n even 6 1
720.1.bu.a 4 72.p odd 6 1
720.1.bu.a 4 360.z odd 6 1
720.1.bu.a 4 360.bk even 6 1
2160.1.bu.a 4 24.f even 2 1
2160.1.bu.a 4 24.h odd 2 1
2160.1.bu.a 4 72.j odd 6 1
2160.1.bu.a 4 72.l even 6 1
2160.1.bu.a 4 120.i odd 2 1
2160.1.bu.a 4 120.m even 2 1
2160.1.bu.a 4 360.bd even 6 1
2160.1.bu.a 4 360.bh odd 6 1
2880.1.bu.c 4 1.a even 1 1 trivial
2880.1.bu.c 4 4.b odd 2 1 inner
2880.1.bu.c 4 5.b even 2 1 inner
2880.1.bu.c 4 9.c even 3 1 inner
2880.1.bu.c 4 20.d odd 2 1 CM
2880.1.bu.c 4 36.f odd 6 1 inner
2880.1.bu.c 4 45.j even 6 1 inner
2880.1.bu.c 4 180.p odd 6 1 inner
3600.1.cc.a 2 40.i odd 4 1
3600.1.cc.a 2 40.k even 4 1
3600.1.cc.a 2 360.bo even 12 1
3600.1.cc.a 2 360.bu odd 12 1
3600.1.cc.b 2 40.i odd 4 1
3600.1.cc.b 2 40.k even 4 1
3600.1.cc.b 2 360.bo even 12 1
3600.1.cc.b 2 360.bu odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 3T_{7}^{2} + 9$$ acting on $$S_{1}^{\mathrm{new}}(2880, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} + 3T^{2} + 9$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 3T^{2} + 9$$
$29$ $$(T^{2} + T + 1)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + T + 1)^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4} + 3T^{2} + 9$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - T + 1)^{2}$$
$67$ $$T^{4} + 3T^{2} + 9$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 3T^{2} + 9$$
$89$ $$(T + 1)^{4}$$
$97$ $$T^{4}$$