# Properties

 Label 1440.4.a.p Level $1440$ Weight $4$ Character orbit 1440.a Self dual yes Analytic conductor $84.963$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,4,Mod(1,1440)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1440.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1440.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$84.9627504083$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 5 q^{5} + 8 q^{7}+O(q^{10})$$ q + 5 * q^5 + 8 * q^7 $$q + 5 q^{5} + 8 q^{7} + 4 q^{11} - 6 q^{13} + 2 q^{17} + 16 q^{19} - 60 q^{23} + 25 q^{25} + 142 q^{29} + 176 q^{31} + 40 q^{35} - 214 q^{37} + 278 q^{41} + 68 q^{43} + 116 q^{47} - 279 q^{49} + 350 q^{53} + 20 q^{55} + 684 q^{59} - 394 q^{61} - 30 q^{65} - 108 q^{67} - 96 q^{71} - 398 q^{73} + 32 q^{77} - 136 q^{79} + 436 q^{83} + 10 q^{85} + 750 q^{89} - 48 q^{91} + 80 q^{95} + 82 q^{97}+O(q^{100})$$ q + 5 * q^5 + 8 * q^7 + 4 * q^11 - 6 * q^13 + 2 * q^17 + 16 * q^19 - 60 * q^23 + 25 * q^25 + 142 * q^29 + 176 * q^31 + 40 * q^35 - 214 * q^37 + 278 * q^41 + 68 * q^43 + 116 * q^47 - 279 * q^49 + 350 * q^53 + 20 * q^55 + 684 * q^59 - 394 * q^61 - 30 * q^65 - 108 * q^67 - 96 * q^71 - 398 * q^73 + 32 * q^77 - 136 * q^79 + 436 * q^83 + 10 * q^85 + 750 * q^89 - 48 * q^91 + 80 * q^95 + 82 * q^97

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 5.00000 0 8.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.4.a.p 1
3.b odd 2 1 480.4.a.b 1
4.b odd 2 1 1440.4.a.m 1
12.b even 2 1 480.4.a.g yes 1
15.d odd 2 1 2400.4.a.q 1
24.f even 2 1 960.4.a.m 1
24.h odd 2 1 960.4.a.bh 1
60.h even 2 1 2400.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.4.a.b 1 3.b odd 2 1
480.4.a.g yes 1 12.b even 2 1
960.4.a.m 1 24.f even 2 1
960.4.a.bh 1 24.h odd 2 1
1440.4.a.m 1 4.b odd 2 1
1440.4.a.p 1 1.a even 1 1 trivial
2400.4.a.f 1 60.h even 2 1
2400.4.a.q 1 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1440))$$:

 $$T_{7} - 8$$ T7 - 8 $$T_{11} - 4$$ T11 - 4 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T - 8$$
$11$ $$T - 4$$
$13$ $$T + 6$$
$17$ $$T - 2$$
$19$ $$T - 16$$
$23$ $$T + 60$$
$29$ $$T - 142$$
$31$ $$T - 176$$
$37$ $$T + 214$$
$41$ $$T - 278$$
$43$ $$T - 68$$
$47$ $$T - 116$$
$53$ $$T - 350$$
$59$ $$T - 684$$
$61$ $$T + 394$$
$67$ $$T + 108$$
$71$ $$T + 96$$
$73$ $$T + 398$$
$79$ $$T + 136$$
$83$ $$T - 436$$
$89$ $$T - 750$$
$97$ $$T - 82$$