# Properties

 Label 1440.1.p.b Level $1440$ Weight $1$ Character orbit 1440.p Self dual yes Analytic conductor $0.719$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -15, -40, 24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1440.p (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.718653618192$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 360) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{6}, \sqrt{-10})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.21600.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} + O(q^{10})$$ $$q + q^{5} + 2q^{19} - 2q^{23} + q^{25} + 2q^{47} - q^{49} - 2q^{53} + 2q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 0
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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
24.f even 2 1 RM by $$\Q(\sqrt{6})$$
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.1.p.b 1
3.b odd 2 1 1440.1.p.a 1
4.b odd 2 1 360.1.p.a 1
5.b even 2 1 1440.1.p.a 1
8.b even 2 1 360.1.p.b yes 1
8.d odd 2 1 1440.1.p.a 1
12.b even 2 1 360.1.p.b yes 1
15.d odd 2 1 CM 1440.1.p.b 1
20.d odd 2 1 360.1.p.b yes 1
20.e even 4 2 1800.1.g.c 2
24.f even 2 1 RM 1440.1.p.b 1
24.h odd 2 1 360.1.p.a 1
36.f odd 6 2 3240.1.z.f 2
36.h even 6 2 3240.1.z.d 2
40.e odd 2 1 CM 1440.1.p.b 1
40.f even 2 1 360.1.p.a 1
40.i odd 4 2 1800.1.g.c 2
60.h even 2 1 360.1.p.a 1
60.l odd 4 2 1800.1.g.c 2
72.j odd 6 2 3240.1.z.f 2
72.n even 6 2 3240.1.z.d 2
120.i odd 2 1 360.1.p.b yes 1
120.m even 2 1 1440.1.p.a 1
120.w even 4 2 1800.1.g.c 2
180.n even 6 2 3240.1.z.f 2
180.p odd 6 2 3240.1.z.d 2
360.bh odd 6 2 3240.1.z.d 2
360.bk even 6 2 3240.1.z.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.1.p.a 1 4.b odd 2 1
360.1.p.a 1 24.h odd 2 1
360.1.p.a 1 40.f even 2 1
360.1.p.a 1 60.h even 2 1
360.1.p.b yes 1 8.b even 2 1
360.1.p.b yes 1 12.b even 2 1
360.1.p.b yes 1 20.d odd 2 1
360.1.p.b yes 1 120.i odd 2 1
1440.1.p.a 1 3.b odd 2 1
1440.1.p.a 1 5.b even 2 1
1440.1.p.a 1 8.d odd 2 1
1440.1.p.a 1 120.m even 2 1
1440.1.p.b 1 1.a even 1 1 trivial
1440.1.p.b 1 15.d odd 2 1 CM
1440.1.p.b 1 24.f even 2 1 RM
1440.1.p.b 1 40.e odd 2 1 CM
1800.1.g.c 2 20.e even 4 2
1800.1.g.c 2 40.i odd 4 2
1800.1.g.c 2 60.l odd 4 2
1800.1.g.c 2 120.w even 4 2
3240.1.z.d 2 36.h even 6 2
3240.1.z.d 2 72.n even 6 2
3240.1.z.d 2 180.p odd 6 2
3240.1.z.d 2 360.bh odd 6 2
3240.1.z.f 2 36.f odd 6 2
3240.1.z.f 2 72.j odd 6 2
3240.1.z.f 2 180.n even 6 2
3240.1.z.f 2 360.bk even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23} + 2$$ acting on $$S_{1}^{\mathrm{new}}(1440, [\chi])$$.

## Hecke characteristic polynomials

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