# Properties

 Label 1440.2.f.h Level $1440$ Weight $2$ Character orbit 1440.f Analytic conductor $11.498$ Analytic rank $0$ Dimension $4$ CM no 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 480) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{5} + \beta_{2} q^{7} +O(q^{10})$$ $$q -\beta_{1} q^{5} + \beta_{2} q^{7} -\beta_{3} q^{11} + 2 \beta_{1} q^{13} -2 \beta_{1} q^{17} + 2 \beta_{2} q^{23} -5 q^{25} + 4 q^{29} -2 \beta_{3} q^{31} -\beta_{3} q^{35} + 2 \beta_{1} q^{37} -10 q^{41} + 2 \beta_{2} q^{43} -4 \beta_{2} q^{47} + 3 q^{49} + 2 \beta_{1} q^{53} -5 \beta_{2} q^{55} -3 \beta_{3} q^{59} + 10 q^{61} + 10 q^{65} -4 \beta_{2} q^{67} -2 \beta_{3} q^{71} -4 \beta_{1} q^{73} + 4 \beta_{1} q^{77} -2 \beta_{3} q^{79} -2 \beta_{2} q^{83} -10 q^{85} + 6 q^{89} + 2 \beta_{3} q^{91} -8 \beta_{1} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 20q^{25} + 16q^{29} - 40q^{41} + 12q^{49} + 40q^{61} + 40q^{65} - 40q^{85} + 24q^{89} + O(q^{100})$$

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

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

## 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 − 1.61803i 1.61803i 0.618034i
0 0 0 2.23607i 0 2.00000i 0 0 0
289.2 0 0 0 2.23607i 0 2.00000i 0 0 0
289.3 0 0 0 2.23607i 0 2.00000i 0 0 0
289.4 0 0 0 2.23607i 0 2.00000i 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 inner
5.b even 2 1 inner
20.d odd 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.h 4
3.b odd 2 1 480.2.f.e 4
4.b odd 2 1 inner 1440.2.f.h 4
5.b even 2 1 inner 1440.2.f.h 4
5.c odd 4 1 7200.2.a.cc 2
5.c odd 4 1 7200.2.a.cq 2
8.b even 2 1 2880.2.f.v 4
8.d odd 2 1 2880.2.f.v 4
12.b even 2 1 480.2.f.e 4
15.d odd 2 1 480.2.f.e 4
15.e even 4 1 2400.2.a.bi 2
15.e even 4 1 2400.2.a.bj 2
20.d odd 2 1 inner 1440.2.f.h 4
20.e even 4 1 7200.2.a.cc 2
20.e even 4 1 7200.2.a.cq 2
24.f even 2 1 960.2.f.k 4
24.h odd 2 1 960.2.f.k 4
40.e odd 2 1 2880.2.f.v 4
40.f even 2 1 2880.2.f.v 4
48.i odd 4 1 3840.2.d.bg 4
48.i odd 4 1 3840.2.d.bh 4
48.k even 4 1 3840.2.d.bg 4
48.k even 4 1 3840.2.d.bh 4
60.h even 2 1 480.2.f.e 4
60.l odd 4 1 2400.2.a.bi 2
60.l odd 4 1 2400.2.a.bj 2
120.i odd 2 1 960.2.f.k 4
120.m even 2 1 960.2.f.k 4
120.q odd 4 1 4800.2.a.cu 2
120.q odd 4 1 4800.2.a.cv 2
120.w even 4 1 4800.2.a.cu 2
120.w even 4 1 4800.2.a.cv 2
240.t even 4 1 3840.2.d.bg 4
240.t even 4 1 3840.2.d.bh 4
240.bm odd 4 1 3840.2.d.bg 4
240.bm odd 4 1 3840.2.d.bh 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.e 4 3.b odd 2 1
480.2.f.e 4 12.b even 2 1
480.2.f.e 4 15.d odd 2 1
480.2.f.e 4 60.h even 2 1
960.2.f.k 4 24.f even 2 1
960.2.f.k 4 24.h odd 2 1
960.2.f.k 4 120.i odd 2 1
960.2.f.k 4 120.m even 2 1
1440.2.f.h 4 1.a even 1 1 trivial
1440.2.f.h 4 4.b odd 2 1 inner
1440.2.f.h 4 5.b even 2 1 inner
1440.2.f.h 4 20.d odd 2 1 inner
2400.2.a.bi 2 15.e even 4 1
2400.2.a.bi 2 60.l odd 4 1
2400.2.a.bj 2 15.e even 4 1
2400.2.a.bj 2 60.l odd 4 1
2880.2.f.v 4 8.b even 2 1
2880.2.f.v 4 8.d odd 2 1
2880.2.f.v 4 40.e odd 2 1
2880.2.f.v 4 40.f even 2 1
3840.2.d.bg 4 48.i odd 4 1
3840.2.d.bg 4 48.k even 4 1
3840.2.d.bg 4 240.t even 4 1
3840.2.d.bg 4 240.bm odd 4 1
3840.2.d.bh 4 48.i odd 4 1
3840.2.d.bh 4 48.k even 4 1
3840.2.d.bh 4 240.t even 4 1
3840.2.d.bh 4 240.bm odd 4 1
4800.2.a.cu 2 120.q odd 4 1
4800.2.a.cu 2 120.w even 4 1
4800.2.a.cv 2 120.q odd 4 1
4800.2.a.cv 2 120.w even 4 1
7200.2.a.cc 2 5.c odd 4 1
7200.2.a.cc 2 20.e even 4 1
7200.2.a.cq 2 5.c odd 4 1
7200.2.a.cq 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 $$S_{2}^{\mathrm{new}}(1440, [\chi])$$:

 $$T_{7}^{2} + 4$$ $$T_{11}^{2} - 20$$ $$T_{17}^{2} + 20$$ $$T_{19}$$ $$T_{29} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$( 4 + T^{2} )^{2}$$
$11$ $$( -20 + T^{2} )^{2}$$
$13$ $$( 20 + T^{2} )^{2}$$
$17$ $$( 20 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$( -4 + T )^{4}$$
$31$ $$( -80 + T^{2} )^{2}$$
$37$ $$( 20 + T^{2} )^{2}$$
$41$ $$( 10 + T )^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$( 64 + T^{2} )^{2}$$
$53$ $$( 20 + T^{2} )^{2}$$
$59$ $$( -180 + T^{2} )^{2}$$
$61$ $$( -10 + T )^{4}$$
$67$ $$( 64 + T^{2} )^{2}$$
$71$ $$( -80 + T^{2} )^{2}$$
$73$ $$( 80 + T^{2} )^{2}$$
$79$ $$( -80 + T^{2} )^{2}$$
$83$ $$( 16 + T^{2} )^{2}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$( 320 + T^{2} )^{2}$$