# Properties

 Label 2448.4.a.f Level $2448$ Weight $4$ Character orbit 2448.a Self dual yes Analytic conductor $144.437$ 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] = [2448,4,Mod(1,2448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2448, 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("2448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2448 = 2^{4} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2448.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: $$144.436675694$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) 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)

 $$f(q)$$ $$=$$ $$q - 6 q^{5} + 28 q^{7}+O(q^{10})$$ q - 6 * q^5 + 28 * q^7 $$q - 6 q^{5} + 28 q^{7} - 24 q^{11} - 58 q^{13} - 17 q^{17} - 116 q^{19} - 60 q^{23} - 89 q^{25} - 30 q^{29} + 172 q^{31} - 168 q^{35} - 58 q^{37} + 342 q^{41} + 148 q^{43} + 288 q^{47} + 441 q^{49} - 318 q^{53} + 144 q^{55} + 252 q^{59} + 110 q^{61} + 348 q^{65} + 484 q^{67} - 708 q^{71} + 362 q^{73} - 672 q^{77} + 484 q^{79} + 756 q^{83} + 102 q^{85} + 774 q^{89} - 1624 q^{91} + 696 q^{95} - 382 q^{97}+O(q^{100})$$ q - 6 * q^5 + 28 * q^7 - 24 * q^11 - 58 * q^13 - 17 * q^17 - 116 * q^19 - 60 * q^23 - 89 * q^25 - 30 * q^29 + 172 * q^31 - 168 * q^35 - 58 * q^37 + 342 * q^41 + 148 * q^43 + 288 * q^47 + 441 * q^49 - 318 * q^53 + 144 * q^55 + 252 * q^59 + 110 * q^61 + 348 * q^65 + 484 * q^67 - 708 * q^71 + 362 * q^73 - 672 * q^77 + 484 * q^79 + 756 * q^83 + 102 * q^85 + 774 * q^89 - 1624 * q^91 + 696 * q^95 - 382 * 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 28.0000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$17$$ $$1$$

## 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 2448.4.a.f 1
3.b odd 2 1 272.4.a.d 1
4.b odd 2 1 153.4.a.d 1
12.b even 2 1 17.4.a.a 1
24.f even 2 1 1088.4.a.l 1
24.h odd 2 1 1088.4.a.a 1
60.h even 2 1 425.4.a.d 1
60.l odd 4 2 425.4.b.c 2
84.h odd 2 1 833.4.a.a 1
132.d odd 2 1 2057.4.a.d 1
204.h even 2 1 289.4.a.a 1
204.l even 4 2 289.4.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.a 1 12.b even 2 1
153.4.a.d 1 4.b odd 2 1
272.4.a.d 1 3.b odd 2 1
289.4.a.a 1 204.h even 2 1
289.4.b.a 2 204.l even 4 2
425.4.a.d 1 60.h even 2 1
425.4.b.c 2 60.l odd 4 2
833.4.a.a 1 84.h odd 2 1
1088.4.a.a 1 24.h odd 2 1
1088.4.a.l 1 24.f even 2 1
2057.4.a.d 1 132.d odd 2 1
2448.4.a.f 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2448))$$:

 $$T_{5} + 6$$ T5 + 6 $$T_{7} - 28$$ T7 - 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 6$$
$7$ $$T - 28$$
$11$ $$T + 24$$
$13$ $$T + 58$$
$17$ $$T + 17$$
$19$ $$T + 116$$
$23$ $$T + 60$$
$29$ $$T + 30$$
$31$ $$T - 172$$
$37$ $$T + 58$$
$41$ $$T - 342$$
$43$ $$T - 148$$
$47$ $$T - 288$$
$53$ $$T + 318$$
$59$ $$T - 252$$
$61$ $$T - 110$$
$67$ $$T - 484$$
$71$ $$T + 708$$
$73$ $$T - 362$$
$79$ $$T - 484$$
$83$ $$T - 756$$
$89$ $$T - 774$$
$97$ $$T + 382$$