# Properties

 Label 1440.1.c.a Level $1440$ Weight $1$ Character orbit 1440.c Analytic conductor $0.719$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -4 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.718653618192$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.10800.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8} q^{5} +O(q^{10})$$ $$q -\zeta_{8} q^{5} -2 \zeta_{8}^{2} q^{13} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{17} + \zeta_{8}^{2} q^{25} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{29} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} + q^{49} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{53} -2 q^{61} + 2 \zeta_{8}^{3} q^{65} -2 \zeta_{8}^{2} q^{73} + ( 1 + \zeta_{8}^{2} ) q^{85} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{89} + 2 \zeta_{8}^{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 4q^{49} - 8q^{61} + 4q^{85} + O(q^{100})$$

## Character values

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

 $$n$$ $$577$$ $$641$$ $$901$$ $$991$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 −0.707107 0.707107i 0 0 0 0 0
449.2 0 0 0 −0.707107 + 0.707107i 0 0 0 0 0
449.3 0 0 0 0.707107 0.707107i 0 0 0 0 0
449.4 0 0 0 0.707107 + 0.707107i 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 $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

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