# Properties

 Label 1440.2.f.i Level $1440$ Weight $2$ Character orbit 1440.f Analytic conductor $11.498$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$11.4984578911$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{5} -\beta_{1} q^{7} +O(q^{10})$$ $$q -\beta_{3} q^{5} -\beta_{1} q^{7} + ( -\beta_{1} + 2 \beta_{2} ) q^{23} + 5 q^{25} -6 q^{29} + ( -2 \beta_{1} + \beta_{2} ) q^{35} + 2 \beta_{3} q^{41} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{43} + ( \beta_{1} + 4 \beta_{2} ) q^{47} + ( -7 + 6 \beta_{3} ) q^{49} -6 \beta_{3} q^{61} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{67} + ( 2 \beta_{1} + 5 \beta_{2} ) q^{83} + 6 q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 20q^{25} - 24q^{29} - 28q^{49} + 24q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} + 2 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 6 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 3 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.618034i − 0.618034i − 1.61803i 1.61803i
0 0 0 −2.23607 0 0.763932i 0 0 0
289.2 0 0 0 −2.23607 0 0.763932i 0 0 0
289.3 0 0 0 2.23607 0 5.23607i 0 0 0
289.4 0 0 0 2.23607 0 5.23607i 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.f.i 4
3.b odd 2 1 160.2.c.b 4
4.b odd 2 1 inner 1440.2.f.i 4
5.b even 2 1 inner 1440.2.f.i 4
5.c odd 4 1 7200.2.a.cb 2
5.c odd 4 1 7200.2.a.cr 2
8.b even 2 1 2880.2.f.w 4
8.d odd 2 1 2880.2.f.w 4
12.b even 2 1 160.2.c.b 4
15.d odd 2 1 160.2.c.b 4
15.e even 4 1 800.2.a.j 2
15.e even 4 1 800.2.a.n 2
20.d odd 2 1 CM 1440.2.f.i 4
20.e even 4 1 7200.2.a.cb 2
20.e even 4 1 7200.2.a.cr 2
24.f even 2 1 320.2.c.d 4
24.h odd 2 1 320.2.c.d 4
40.e odd 2 1 2880.2.f.w 4
40.f even 2 1 2880.2.f.w 4
48.i odd 4 1 1280.2.f.g 4
48.i odd 4 1 1280.2.f.h 4
48.k even 4 1 1280.2.f.g 4
48.k even 4 1 1280.2.f.h 4
60.h even 2 1 160.2.c.b 4
60.l odd 4 1 800.2.a.j 2
60.l odd 4 1 800.2.a.n 2
120.i odd 2 1 320.2.c.d 4
120.m even 2 1 320.2.c.d 4
120.q odd 4 1 1600.2.a.z 2
120.q odd 4 1 1600.2.a.bd 2
120.w even 4 1 1600.2.a.z 2
120.w even 4 1 1600.2.a.bd 2
240.t even 4 1 1280.2.f.g 4
240.t even 4 1 1280.2.f.h 4
240.bm odd 4 1 1280.2.f.g 4
240.bm odd 4 1 1280.2.f.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.c.b 4 3.b odd 2 1
160.2.c.b 4 12.b even 2 1
160.2.c.b 4 15.d odd 2 1
160.2.c.b 4 60.h even 2 1
320.2.c.d 4 24.f even 2 1
320.2.c.d 4 24.h odd 2 1
320.2.c.d 4 120.i odd 2 1
320.2.c.d 4 120.m even 2 1
800.2.a.j 2 15.e even 4 1
800.2.a.j 2 60.l odd 4 1
800.2.a.n 2 15.e even 4 1
800.2.a.n 2 60.l odd 4 1
1280.2.f.g 4 48.i odd 4 1
1280.2.f.g 4 48.k even 4 1
1280.2.f.g 4 240.t even 4 1
1280.2.f.g 4 240.bm odd 4 1
1280.2.f.h 4 48.i odd 4 1
1280.2.f.h 4 48.k even 4 1
1280.2.f.h 4 240.t even 4 1
1280.2.f.h 4 240.bm odd 4 1
1440.2.f.i 4 1.a even 1 1 trivial
1440.2.f.i 4 4.b odd 2 1 inner
1440.2.f.i 4 5.b even 2 1 inner
1440.2.f.i 4 20.d odd 2 1 CM
1600.2.a.z 2 120.q odd 4 1
1600.2.a.z 2 120.w even 4 1
1600.2.a.bd 2 120.q odd 4 1
1600.2.a.bd 2 120.w even 4 1
2880.2.f.w 4 8.b even 2 1
2880.2.f.w 4 8.d odd 2 1
2880.2.f.w 4 40.e odd 2 1
2880.2.f.w 4 40.f even 2 1
7200.2.a.cb 2 5.c odd 4 1
7200.2.a.cb 2 20.e even 4 1
7200.2.a.cr 2 5.c odd 4 1
7200.2.a.cr 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1440, [\chi])$$:

 $$T_{7}^{4} + 28 T_{7}^{2} + 16$$ $$T_{11}$$ $$T_{17}$$ $$T_{19}$$ $$T_{29} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$16 + 28 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$1936 + 92 T^{2} + T^{4}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( -20 + T^{2} )^{2}$$
$43$ $$5776 + 172 T^{2} + T^{4}$$
$47$ $$16 + 188 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -180 + T^{2} )^{2}$$
$67$ $$13456 + 268 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$5776 + 332 T^{2} + T^{4}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$T^{4}$$