# Properties

 Label 1440.1.cd.a Level $1440$ Weight $1$ Character orbit 1440.cd Analytic conductor $0.719$ Analytic rank $0$ Dimension $8$ Projective image $D_{12}$ CM discriminant -20 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1440.cd (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.718653618192$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{24}^{7} q^{3} - \zeta_{24}^{10} q^{5} + ( - \zeta_{24}^{3} - \zeta_{24}) q^{7} - \zeta_{24}^{2} q^{9} +O(q^{10})$$ q - z^7 * q^3 - z^10 * q^5 + (-z^3 - z) * q^7 - z^2 * q^9 $$q - \zeta_{24}^{7} q^{3} - \zeta_{24}^{10} q^{5} + ( - \zeta_{24}^{3} - \zeta_{24}) q^{7} - \zeta_{24}^{2} q^{9} - \zeta_{24}^{5} q^{15} + (\zeta_{24}^{10} + \zeta_{24}^{8}) q^{21} + ( - \zeta_{24}^{11} + \zeta_{24}^{9}) q^{23} - \zeta_{24}^{8} q^{25} + \zeta_{24}^{9} q^{27} + ( - \zeta_{24}^{4} - 1) q^{29} + (\zeta_{24}^{11} - \zeta_{24}) q^{35} - \zeta_{24}^{10} q^{41} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{43} - q^{45} + ( - \zeta_{24}^{9} - \zeta_{24}^{7}) q^{47} + (\zeta_{24}^{6} + \zeta_{24}^{4} + \zeta_{24}^{2}) q^{49} + (\zeta_{24}^{10} + \zeta_{24}^{6}) q^{61} + (\zeta_{24}^{5} + \zeta_{24}^{3}) q^{63} + ( - \zeta_{24}^{5} + \zeta_{24}^{3}) q^{67} + ( - \zeta_{24}^{6} + \zeta_{24}^{4}) q^{69} - \zeta_{24}^{3} q^{75} + \zeta_{24}^{4} q^{81} + ( - \zeta_{24}^{3} + \zeta_{24}) q^{83} + (\zeta_{24}^{11} + \zeta_{24}^{7}) q^{87} + ( - \zeta_{24}^{8} - \zeta_{24}^{4}) q^{89} +O(q^{100})$$ q - z^7 * q^3 - z^10 * q^5 + (-z^3 - z) * q^7 - z^2 * q^9 - z^5 * q^15 + (z^10 + z^8) * q^21 + (-z^11 + z^9) * q^23 - z^8 * q^25 + z^9 * q^27 + (-z^4 - 1) * q^29 + (z^11 - z) * q^35 - z^10 * q^41 + (-z^11 + z^5) * q^43 - q^45 + (-z^9 - z^7) * q^47 + (z^6 + z^4 + z^2) * q^49 + (z^10 + z^6) * q^61 + (z^5 + z^3) * q^63 + (-z^5 + z^3) * q^67 + (-z^6 + z^4) * q^69 - z^3 * q^75 + z^4 * q^81 + (-z^3 + z) * q^83 + (z^11 + z^7) * q^87 + (-z^8 - z^4) * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 4 q^{21} + 4 q^{25} - 12 q^{29} - 8 q^{45} + 4 q^{49} + 4 q^{69} + 4 q^{81}+O(q^{100})$$ 8 * q - 4 * q^21 + 4 * q^25 - 12 * q^29 - 8 * q^45 + 4 * q^49 + 4 * q^69 + 4 * q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$641$$ $$901$$ $$991$$ $$\chi(n)$$ $$-1$$ $$\zeta_{24}^{4}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
929.1
 −0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i
0 −0.965926 0.258819i 0 −0.866025 + 0.500000i 0 −0.448288 0.258819i 0 0.866025 + 0.500000i 0
929.2 0 −0.258819 + 0.965926i 0 0.866025 0.500000i 0 1.67303 + 0.965926i 0 −0.866025 0.500000i 0
929.3 0 0.258819 0.965926i 0 0.866025 0.500000i 0 −1.67303 0.965926i 0 −0.866025 0.500000i 0
929.4 0 0.965926 + 0.258819i 0 −0.866025 + 0.500000i 0 0.448288 + 0.258819i 0 0.866025 + 0.500000i 0
1409.1 0 −0.965926 + 0.258819i 0 −0.866025 0.500000i 0 −0.448288 + 0.258819i 0 0.866025 0.500000i 0
1409.2 0 −0.258819 0.965926i 0 0.866025 + 0.500000i 0 1.67303 0.965926i 0 −0.866025 + 0.500000i 0
1409.3 0 0.258819 + 0.965926i 0 0.866025 + 0.500000i 0 −1.67303 + 0.965926i 0 −0.866025 + 0.500000i 0
1409.4 0 0.965926 0.258819i 0 −0.866025 0.500000i 0 0.448288 0.258819i 0 0.866025 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1409.4 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.d odd 6 1 inner
36.h even 6 1 inner
45.h odd 6 1 inner
180.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.1.cd.a 8
4.b odd 2 1 inner 1440.1.cd.a 8
5.b even 2 1 inner 1440.1.cd.a 8
8.b even 2 1 2880.1.cd.a 8
8.d odd 2 1 2880.1.cd.a 8
9.d odd 6 1 inner 1440.1.cd.a 8
20.d odd 2 1 CM 1440.1.cd.a 8
36.h even 6 1 inner 1440.1.cd.a 8
40.e odd 2 1 2880.1.cd.a 8
40.f even 2 1 2880.1.cd.a 8
45.h odd 6 1 inner 1440.1.cd.a 8
72.j odd 6 1 2880.1.cd.a 8
72.l even 6 1 2880.1.cd.a 8
180.n even 6 1 inner 1440.1.cd.a 8
360.bd even 6 1 2880.1.cd.a 8
360.bh odd 6 1 2880.1.cd.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.1.cd.a 8 1.a even 1 1 trivial
1440.1.cd.a 8 4.b odd 2 1 inner
1440.1.cd.a 8 5.b even 2 1 inner
1440.1.cd.a 8 9.d odd 6 1 inner
1440.1.cd.a 8 20.d odd 2 1 CM
1440.1.cd.a 8 36.h even 6 1 inner
1440.1.cd.a 8 45.h odd 6 1 inner
1440.1.cd.a 8 180.n even 6 1 inner
2880.1.cd.a 8 8.b even 2 1
2880.1.cd.a 8 8.d odd 2 1
2880.1.cd.a 8 40.e odd 2 1
2880.1.cd.a 8 40.f even 2 1
2880.1.cd.a 8 72.j odd 6 1
2880.1.cd.a 8 72.l even 6 1
2880.1.cd.a 8 360.bd even 6 1
2880.1.cd.a 8 360.bh odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1440, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - T^{4} + 1$$
$5$ $$(T^{4} - T^{2} + 1)^{2}$$
$7$ $$T^{8} - 4 T^{6} + 15 T^{4} - 4 T^{2} + \cdots + 1$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1$$
$29$ $$(T^{2} + 3 T + 3)^{4}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$(T^{4} - T^{2} + 1)^{2}$$
$43$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$47$ $$T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$(T^{4} + 3 T^{2} + 9)^{2}$$
$67$ $$T^{8} - 4 T^{6} + 15 T^{4} - 4 T^{2} + \cdots + 1$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1$$
$89$ $$(T^{2} + 3)^{4}$$
$97$ $$T^{8}$$