# Properties

 Label 1440.1.cm.a Level $1440$ Weight $1$ Character orbit 1440.cm Analytic conductor $0.719$ Analytic rank $0$ Dimension $8$ Projective image $D_{8}$ CM discriminant -15 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1440.cm (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.718653618192$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.2.36238786560000.11

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{16}^{3} q^{2} + \zeta_{16}^{6} q^{4} -\zeta_{16}^{5} q^{5} -\zeta_{16} q^{8} +O(q^{10})$$ $$q + \zeta_{16}^{3} q^{2} + \zeta_{16}^{6} q^{4} -\zeta_{16}^{5} q^{5} -\zeta_{16} q^{8} + q^{10} -\zeta_{16}^{4} q^{16} + ( \zeta_{16} - \zeta_{16}^{7} ) q^{17} + ( \zeta_{16}^{2} + \zeta_{16}^{4} ) q^{19} + \zeta_{16}^{3} q^{20} + ( \zeta_{16} - \zeta_{16}^{3} ) q^{23} -\zeta_{16}^{2} q^{25} + ( \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{31} -\zeta_{16}^{7} q^{32} + ( \zeta_{16}^{2} + \zeta_{16}^{4} ) q^{34} + ( \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{38} + \zeta_{16}^{6} q^{40} + ( \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{46} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} ) q^{47} -\zeta_{16}^{4} q^{49} -\zeta_{16}^{5} q^{50} + ( -\zeta_{16}^{3} + \zeta_{16}^{7} ) q^{53} + ( 1 + \zeta_{16}^{6} ) q^{61} + ( -\zeta_{16} + \zeta_{16}^{5} ) q^{62} + \zeta_{16}^{2} q^{64} + ( \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{68} + ( -1 - \zeta_{16}^{2} ) q^{76} -2 q^{79} -\zeta_{16} q^{80} + ( -\zeta_{16} + \zeta_{16}^{5} ) q^{83} + ( -\zeta_{16}^{4} - \zeta_{16}^{6} ) q^{85} + ( \zeta_{16} + \zeta_{16}^{7} ) q^{92} + ( 1 - \zeta_{16}^{6} ) q^{94} + ( \zeta_{16} - \zeta_{16}^{7} ) q^{95} -\zeta_{16}^{7} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 8 q^{10} + 8 q^{61} - 8 q^{76} - 16 q^{79} + 8 q^{94} + O(q^{100})$$

## Character values

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

 $$n$$ $$577$$ $$641$$ $$901$$ $$991$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\zeta_{16}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.923880 + 0.382683i 0.923880 − 0.382683i −0.923880 − 0.382683i 0.923880 + 0.382683i 0.382683 + 0.923880i −0.382683 − 0.923880i 0.382683 − 0.923880i −0.382683 + 0.923880i
−0.382683 + 0.923880i 0 −0.707107 0.707107i −0.382683 0.923880i 0 0 0.923880 0.382683i 0 1.00000
19.2 0.382683 0.923880i 0 −0.707107 0.707107i 0.382683 + 0.923880i 0 0 −0.923880 + 0.382683i 0 1.00000
379.1 −0.382683 0.923880i 0 −0.707107 + 0.707107i −0.382683 + 0.923880i 0 0 0.923880 + 0.382683i 0 1.00000
379.2 0.382683 + 0.923880i 0 −0.707107 + 0.707107i 0.382683 0.923880i 0 0 −0.923880 0.382683i 0 1.00000
739.1 −0.923880 0.382683i 0 0.707107 + 0.707107i −0.923880 + 0.382683i 0 0 −0.382683 0.923880i 0 1.00000
739.2 0.923880 + 0.382683i 0 0.707107 + 0.707107i 0.923880 0.382683i 0 0 0.382683 + 0.923880i 0 1.00000
1099.1 −0.923880 + 0.382683i 0 0.707107 0.707107i −0.923880 0.382683i 0 0 −0.382683 + 0.923880i 0 1.00000
1099.2 0.923880 0.382683i 0 0.707107 0.707107i 0.923880 + 0.382683i 0 0 0.382683 0.923880i 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1099.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner
160.y odd 8 1 inner
480.bs even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.1.cm.a 8
3.b odd 2 1 inner 1440.1.cm.a 8
5.b even 2 1 inner 1440.1.cm.a 8
15.d odd 2 1 CM 1440.1.cm.a 8
32.h odd 8 1 inner 1440.1.cm.a 8
96.o even 8 1 inner 1440.1.cm.a 8
160.y odd 8 1 inner 1440.1.cm.a 8
480.bs even 8 1 inner 1440.1.cm.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.1.cm.a 8 1.a even 1 1 trivial
1440.1.cm.a 8 3.b odd 2 1 inner
1440.1.cm.a 8 5.b even 2 1 inner
1440.1.cm.a 8 15.d odd 2 1 CM
1440.1.cm.a 8 32.h odd 8 1 inner
1440.1.cm.a 8 96.o even 8 1 inner
1440.1.cm.a 8 160.y odd 8 1 inner
1440.1.cm.a 8 480.bs even 8 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{8}$$
$3$ $$T^{8}$$
$5$ $$1 + T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$( 2 - 4 T^{2} + T^{4} )^{2}$$
$19$ $$( 2 + 4 T + 2 T^{2} + T^{4} )^{2}$$
$23$ $$4 + 12 T^{4} + T^{8}$$
$29$ $$T^{8}$$
$31$ $$( 2 + T^{2} )^{4}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$( 2 + 4 T^{2} + T^{4} )^{2}$$
$53$ $$16 + T^{8}$$
$59$ $$T^{8}$$
$61$ $$( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$( 2 + T )^{8}$$
$83$ $$16 + T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$