# Properties

 Label 1444.2.a.b Level $1444$ Weight $2$ Character orbit 1444.a Self dual yes Analytic conductor $11.530$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,2,Mod(1,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1444.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1444 = 2^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1444.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: $$11.5303980519$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} - 2 q^{9}+O(q^{10})$$ q - q^3 - q^5 - 2 * q^9 $$q - q^{3} - q^{5} - 2 q^{9} - 4 q^{11} - q^{13} + q^{15} + 3 q^{17} + 5 q^{23} - 4 q^{25} + 5 q^{27} + 7 q^{29} + 4 q^{31} + 4 q^{33} + 10 q^{37} + q^{39} - 5 q^{41} - 5 q^{43} + 2 q^{45} - 7 q^{47} - 7 q^{49} - 3 q^{51} + 11 q^{53} + 4 q^{55} + 3 q^{59} + 11 q^{61} + q^{65} - 3 q^{67} - 5 q^{69} + 11 q^{71} + 15 q^{73} + 4 q^{75} - 13 q^{79} + q^{81} - 3 q^{85} - 7 q^{87} + 3 q^{89} - 4 q^{93} - 5 q^{97} + 8 q^{99}+O(q^{100})$$ q - q^3 - q^5 - 2 * q^9 - 4 * q^11 - q^13 + q^15 + 3 * q^17 + 5 * q^23 - 4 * q^25 + 5 * q^27 + 7 * q^29 + 4 * q^31 + 4 * q^33 + 10 * q^37 + q^39 - 5 * q^41 - 5 * q^43 + 2 * q^45 - 7 * q^47 - 7 * q^49 - 3 * q^51 + 11 * q^53 + 4 * q^55 + 3 * q^59 + 11 * q^61 + q^65 - 3 * q^67 - 5 * q^69 + 11 * q^71 + 15 * q^73 + 4 * q^75 - 13 * q^79 + q^81 - 3 * q^85 - 7 * q^87 + 3 * q^89 - 4 * q^93 - 5 * q^97 + 8 * q^99

## 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 −1.00000 0 −1.00000 0 0 0 −2.00000 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
$$2$$ $$-1$$
$$19$$ $$+1$$

## 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 1444.2.a.b 1
4.b odd 2 1 5776.2.a.k 1
19.b odd 2 1 1444.2.a.c 1
19.c even 3 2 76.2.e.a 2
19.d odd 6 2 1444.2.e.b 2
57.h odd 6 2 684.2.k.b 2
76.d even 2 1 5776.2.a.f 1
76.g odd 6 2 304.2.i.a 2
95.i even 6 2 1900.2.i.a 2
95.m odd 12 4 1900.2.s.a 4
152.k odd 6 2 1216.2.i.g 2
152.p even 6 2 1216.2.i.c 2
228.m even 6 2 2736.2.s.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.e.a 2 19.c even 3 2
304.2.i.a 2 76.g odd 6 2
684.2.k.b 2 57.h odd 6 2
1216.2.i.c 2 152.p even 6 2
1216.2.i.g 2 152.k odd 6 2
1444.2.a.b 1 1.a even 1 1 trivial
1444.2.a.c 1 19.b odd 2 1
1444.2.e.b 2 19.d odd 6 2
1900.2.i.a 2 95.i even 6 2
1900.2.s.a 4 95.m odd 12 4
2736.2.s.g 2 228.m even 6 2
5776.2.a.f 1 76.d even 2 1
5776.2.a.k 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1444))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T + 1$$
$17$ $$T - 3$$
$19$ $$T$$
$23$ $$T - 5$$
$29$ $$T - 7$$
$31$ $$T - 4$$
$37$ $$T - 10$$
$41$ $$T + 5$$
$43$ $$T + 5$$
$47$ $$T + 7$$
$53$ $$T - 11$$
$59$ $$T - 3$$
$61$ $$T - 11$$
$67$ $$T + 3$$
$71$ $$T - 11$$
$73$ $$T - 15$$
$79$ $$T + 13$$
$83$ $$T$$
$89$ $$T - 3$$
$97$ $$T + 5$$
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