# Properties

 Label 1440.1.bh.a Level 1440 Weight 1 Character orbit 1440.bh Analytic conductor 0.719 Analytic rank 0 Dimension 2 Projective image $$D_{4}$$ CM discriminant -4 Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.718653618192$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{4}$$ Projective field Galois closure of 4.2.18000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{5} +O(q^{10})$$ $$q - q^{5} + ( 1 + i ) q^{13} + ( 1 - i ) q^{17} + q^{25} + 2 i q^{29} + ( 1 - i ) q^{37} + 2 q^{41} + i q^{49} + ( -1 - i ) q^{53} + ( -1 - i ) q^{65} + ( -1 - i ) q^{73} + ( -1 + i ) q^{85} + 2 i q^{89} + ( -1 + i ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} + O(q^{10})$$ $$2q - 2q^{5} + 2q^{13} + 2q^{17} + 2q^{25} + 2q^{37} + 4q^{41} - 2q^{53} - 2q^{65} - 2q^{73} - 2q^{85} - 2q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 − 1.00000i 1.00000i
0 0 0 −1.00000 0 0 0 0 0
1153.1 0 0 0 −1.00000 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 $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1440, [\chi])$$:

 $$T_{13}^{2} - 2 T_{13} + 2$$ $$T_{17}^{2} - 2 T_{17} + 2$$

## Hecke Characteristic Polynomials

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