# Properties

 Label 1872.4.a.e Level $1872$ Weight $4$ Character orbit 1872.a Self dual yes Analytic conductor $110.452$ Analytic rank $1$ 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] = [1872,4,Mod(1,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, 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("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1872 = 2^{4} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1872.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: $$110.451575531$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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)

 $$f(q)$$ $$=$$ $$q - 6 q^{5} - 20 q^{7}+O(q^{10})$$ q - 6 * q^5 - 20 * q^7 $$q - 6 q^{5} - 20 q^{7} + 24 q^{11} + 13 q^{13} + 30 q^{17} + 16 q^{19} - 72 q^{23} - 89 q^{25} + 282 q^{29} - 164 q^{31} + 120 q^{35} + 110 q^{37} + 126 q^{41} - 164 q^{43} - 204 q^{47} + 57 q^{49} + 738 q^{53} - 144 q^{55} + 120 q^{59} + 614 q^{61} - 78 q^{65} - 848 q^{67} + 132 q^{71} + 218 q^{73} - 480 q^{77} + 1096 q^{79} + 552 q^{83} - 180 q^{85} - 210 q^{89} - 260 q^{91} - 96 q^{95} - 1726 q^{97}+O(q^{100})$$ q - 6 * q^5 - 20 * q^7 + 24 * q^11 + 13 * q^13 + 30 * q^17 + 16 * q^19 - 72 * q^23 - 89 * q^25 + 282 * q^29 - 164 * q^31 + 120 * q^35 + 110 * q^37 + 126 * q^41 - 164 * q^43 - 204 * q^47 + 57 * q^49 + 738 * q^53 - 144 * q^55 + 120 * q^59 + 614 * q^61 - 78 * q^65 - 848 * q^67 + 132 * q^71 + 218 * q^73 - 480 * q^77 + 1096 * q^79 + 552 * q^83 - 180 * q^85 - 210 * q^89 - 260 * q^91 - 96 * q^95 - 1726 * 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 −6.00000 0 −20.0000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$13$$ $$-1$$

## 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 1872.4.a.e 1
3.b odd 2 1 624.4.a.i 1
4.b odd 2 1 234.4.a.b 1
12.b even 2 1 78.4.a.e 1
24.f even 2 1 2496.4.a.k 1
24.h odd 2 1 2496.4.a.b 1
60.h even 2 1 1950.4.a.c 1
156.h even 2 1 1014.4.a.b 1
156.l odd 4 2 1014.4.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.e 1 12.b even 2 1
234.4.a.b 1 4.b odd 2 1
624.4.a.i 1 3.b odd 2 1
1014.4.a.b 1 156.h even 2 1
1014.4.b.c 2 156.l odd 4 2
1872.4.a.e 1 1.a even 1 1 trivial
1950.4.a.c 1 60.h even 2 1
2496.4.a.b 1 24.h odd 2 1
2496.4.a.k 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1872))$$:

 $$T_{5} + 6$$ T5 + 6 $$T_{7} + 20$$ T7 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 6$$
$7$ $$T + 20$$
$11$ $$T - 24$$
$13$ $$T - 13$$
$17$ $$T - 30$$
$19$ $$T - 16$$
$23$ $$T + 72$$
$29$ $$T - 282$$
$31$ $$T + 164$$
$37$ $$T - 110$$
$41$ $$T - 126$$
$43$ $$T + 164$$
$47$ $$T + 204$$
$53$ $$T - 738$$
$59$ $$T - 120$$
$61$ $$T - 614$$
$67$ $$T + 848$$
$71$ $$T - 132$$
$73$ $$T - 218$$
$79$ $$T - 1096$$
$83$ $$T - 552$$
$89$ $$T + 210$$
$97$ $$T + 1726$$