# Properties

 Label 4800.2.a.bq Level $4800$ Weight $2$ Character orbit 4800.a Self dual yes Analytic conductor $38.328$ 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] = [4800,2,Mod(1,4800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4800 = 2^{6} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4800.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: $$38.3281929702$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 3 q^{7} + q^{9}+O(q^{10})$$ q + q^3 - 3 * q^7 + q^9 $$q + q^{3} - 3 q^{7} + q^{9} - 2 q^{11} - q^{13} + 2 q^{17} + 5 q^{19} - 3 q^{21} + 6 q^{23} + q^{27} - 10 q^{29} - 3 q^{31} - 2 q^{33} - 2 q^{37} - q^{39} - 8 q^{41} - q^{43} + 2 q^{47} + 2 q^{49} + 2 q^{51} + 4 q^{53} + 5 q^{57} + 10 q^{59} - 7 q^{61} - 3 q^{63} + 3 q^{67} + 6 q^{69} - 8 q^{71} - 14 q^{73} + 6 q^{77} + q^{81} - 6 q^{83} - 10 q^{87} + 3 q^{91} - 3 q^{93} + 17 q^{97} - 2 q^{99}+O(q^{100})$$ q + q^3 - 3 * q^7 + q^9 - 2 * q^11 - q^13 + 2 * q^17 + 5 * q^19 - 3 * q^21 + 6 * q^23 + q^27 - 10 * q^29 - 3 * q^31 - 2 * q^33 - 2 * q^37 - q^39 - 8 * q^41 - q^43 + 2 * q^47 + 2 * q^49 + 2 * q^51 + 4 * q^53 + 5 * q^57 + 10 * q^59 - 7 * q^61 - 3 * q^63 + 3 * q^67 + 6 * q^69 - 8 * q^71 - 14 * q^73 + 6 * q^77 + q^81 - 6 * q^83 - 10 * q^87 + 3 * q^91 - 3 * q^93 + 17 * q^97 - 2 * 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 0 0 −3.00000 0 1.00000 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## 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 4800.2.a.bq 1
4.b odd 2 1 4800.2.a.be 1
5.b even 2 1 4800.2.a.bb 1
5.c odd 4 2 4800.2.f.l 2
8.b even 2 1 75.2.a.c yes 1
8.d odd 2 1 1200.2.a.p 1
20.d odd 2 1 4800.2.a.br 1
20.e even 4 2 4800.2.f.y 2
24.f even 2 1 3600.2.a.bk 1
24.h odd 2 1 225.2.a.a 1
40.e odd 2 1 1200.2.a.c 1
40.f even 2 1 75.2.a.a 1
40.i odd 4 2 75.2.b.a 2
40.k even 4 2 1200.2.f.d 2
56.h odd 2 1 3675.2.a.q 1
88.b odd 2 1 9075.2.a.a 1
120.i odd 2 1 225.2.a.e 1
120.m even 2 1 3600.2.a.j 1
120.q odd 4 2 3600.2.f.p 2
120.w even 4 2 225.2.b.a 2
280.c odd 2 1 3675.2.a.b 1
440.o odd 2 1 9075.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 40.f even 2 1
75.2.a.c yes 1 8.b even 2 1
75.2.b.a 2 40.i odd 4 2
225.2.a.a 1 24.h odd 2 1
225.2.a.e 1 120.i odd 2 1
225.2.b.a 2 120.w even 4 2
1200.2.a.c 1 40.e odd 2 1
1200.2.a.p 1 8.d odd 2 1
1200.2.f.d 2 40.k even 4 2
3600.2.a.j 1 120.m even 2 1
3600.2.a.bk 1 24.f even 2 1
3600.2.f.p 2 120.q odd 4 2
3675.2.a.b 1 280.c odd 2 1
3675.2.a.q 1 56.h odd 2 1
4800.2.a.bb 1 5.b even 2 1
4800.2.a.be 1 4.b odd 2 1
4800.2.a.bq 1 1.a even 1 1 trivial
4800.2.a.br 1 20.d odd 2 1
4800.2.f.l 2 5.c odd 4 2
4800.2.f.y 2 20.e even 4 2
9075.2.a.a 1 88.b odd 2 1
9075.2.a.s 1 440.o odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4800))$$:

 $$T_{7} + 3$$ T7 + 3 $$T_{11} + 2$$ T11 + 2 $$T_{13} + 1$$ T13 + 1 $$T_{19} - 5$$ T19 - 5 $$T_{23} - 6$$ T23 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 3$$
$11$ $$T + 2$$
$13$ $$T + 1$$
$17$ $$T - 2$$
$19$ $$T - 5$$
$23$ $$T - 6$$
$29$ $$T + 10$$
$31$ $$T + 3$$
$37$ $$T + 2$$
$41$ $$T + 8$$
$43$ $$T + 1$$
$47$ $$T - 2$$
$53$ $$T - 4$$
$59$ $$T - 10$$
$61$ $$T + 7$$
$67$ $$T - 3$$
$71$ $$T + 8$$
$73$ $$T + 14$$
$79$ $$T$$
$83$ $$T + 6$$
$89$ $$T$$
$97$ $$T - 17$$