# Properties

 Label 1440.2.f.g Level $1440$ Weight $2$ Character orbit 1440.f Analytic conductor $11.498$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + i ) q^{5} + 2 i q^{7} +O(q^{10})$$ $$q + ( 2 + i ) q^{5} + 2 i q^{7} + 6 q^{11} + 2 i q^{13} + 6 i q^{17} -4 q^{19} -8 i q^{23} + ( 3 + 4 i ) q^{25} -8 q^{31} + ( -2 + 4 i ) q^{35} + 2 i q^{37} + 6 q^{41} -4 i q^{43} + 4 i q^{47} + 3 q^{49} + 6 i q^{53} + ( 12 + 6 i ) q^{55} -6 q^{59} -6 q^{61} + ( -2 + 4 i ) q^{65} + 4 q^{71} -12 i q^{73} + 12 i q^{77} + 8 q^{79} + 12 i q^{83} + ( -6 + 12 i ) q^{85} + 14 q^{89} -4 q^{91} + ( -8 - 4 i ) q^{95} + 8 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{5} + O(q^{10})$$ $$2q + 4q^{5} + 12q^{11} - 8q^{19} + 6q^{25} - 16q^{31} - 4q^{35} + 12q^{41} + 6q^{49} + 24q^{55} - 12q^{59} - 12q^{61} - 4q^{65} + 8q^{71} + 16q^{79} - 12q^{85} + 28q^{89} - 8q^{91} - 16q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 − 1.00000i 1.00000i
0 0 0 2.00000 1.00000i 0 2.00000i 0 0 0
289.2 0 0 0 2.00000 + 1.00000i 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
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.g 2
3.b odd 2 1 480.2.f.a 2
4.b odd 2 1 1440.2.f.e 2
5.b even 2 1 inner 1440.2.f.g 2
5.c odd 4 1 7200.2.a.p 1
5.c odd 4 1 7200.2.a.br 1
8.b even 2 1 2880.2.f.a 2
8.d odd 2 1 2880.2.f.g 2
12.b even 2 1 480.2.f.b yes 2
15.d odd 2 1 480.2.f.a 2
15.e even 4 1 2400.2.a.c 1
15.e even 4 1 2400.2.a.be 1
20.d odd 2 1 1440.2.f.e 2
20.e even 4 1 7200.2.a.j 1
20.e even 4 1 7200.2.a.bl 1
24.f even 2 1 960.2.f.g 2
24.h odd 2 1 960.2.f.j 2
40.e odd 2 1 2880.2.f.g 2
40.f even 2 1 2880.2.f.a 2
48.i odd 4 1 3840.2.d.k 2
48.i odd 4 1 3840.2.d.u 2
48.k even 4 1 3840.2.d.e 2
48.k even 4 1 3840.2.d.bc 2
60.h even 2 1 480.2.f.b yes 2
60.l odd 4 1 2400.2.a.d 1
60.l odd 4 1 2400.2.a.bf 1
120.i odd 2 1 960.2.f.j 2
120.m even 2 1 960.2.f.g 2
120.q odd 4 1 4800.2.a.z 1
120.q odd 4 1 4800.2.a.bt 1
120.w even 4 1 4800.2.a.ba 1
120.w even 4 1 4800.2.a.bu 1
240.t even 4 1 3840.2.d.e 2
240.t even 4 1 3840.2.d.bc 2
240.bm odd 4 1 3840.2.d.k 2
240.bm odd 4 1 3840.2.d.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.a 2 3.b odd 2 1
480.2.f.a 2 15.d odd 2 1
480.2.f.b yes 2 12.b even 2 1
480.2.f.b yes 2 60.h even 2 1
960.2.f.g 2 24.f even 2 1
960.2.f.g 2 120.m even 2 1
960.2.f.j 2 24.h odd 2 1
960.2.f.j 2 120.i odd 2 1
1440.2.f.e 2 4.b odd 2 1
1440.2.f.e 2 20.d odd 2 1
1440.2.f.g 2 1.a even 1 1 trivial
1440.2.f.g 2 5.b even 2 1 inner
2400.2.a.c 1 15.e even 4 1
2400.2.a.d 1 60.l odd 4 1
2400.2.a.be 1 15.e even 4 1
2400.2.a.bf 1 60.l odd 4 1
2880.2.f.a 2 8.b even 2 1
2880.2.f.a 2 40.f even 2 1
2880.2.f.g 2 8.d odd 2 1
2880.2.f.g 2 40.e odd 2 1
3840.2.d.e 2 48.k even 4 1
3840.2.d.e 2 240.t even 4 1
3840.2.d.k 2 48.i odd 4 1
3840.2.d.k 2 240.bm odd 4 1
3840.2.d.u 2 48.i odd 4 1
3840.2.d.u 2 240.bm odd 4 1
3840.2.d.bc 2 48.k even 4 1
3840.2.d.bc 2 240.t even 4 1
4800.2.a.z 1 120.q odd 4 1
4800.2.a.ba 1 120.w even 4 1
4800.2.a.bt 1 120.q odd 4 1
4800.2.a.bu 1 120.w even 4 1
7200.2.a.j 1 20.e even 4 1
7200.2.a.p 1 5.c odd 4 1
7200.2.a.bl 1 20.e even 4 1
7200.2.a.br 1 5.c odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ $$T_{11} - 6$$ $$T_{17}^{2} + 36$$ $$T_{19} + 4$$ $$T_{29}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 - 4 T + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( -6 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$( 6 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$( -4 + T )^{2}$$
$73$ $$144 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -14 + T )^{2}$$
$97$ $$64 + T^{2}$$