# Properties

 Label 2400.2.a.f Level $2400$ Weight $2$ Character orbit 2400.a Self dual yes Analytic conductor $19.164$ 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] = [2400,2,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, 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("2400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2400 = 2^{5} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2400.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: $$19.1640964851$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{7} + q^{9}+O(q^{10})$$ q - q^3 - q^7 + q^9 $$q - q^{3} - q^{7} + q^{9} - q^{13} + 3 q^{19} + q^{21} - 4 q^{23} - q^{27} + 4 q^{29} - 7 q^{31} + 6 q^{37} + q^{39} + 6 q^{41} - 9 q^{43} - 6 q^{47} - 6 q^{49} - 2 q^{53} - 3 q^{57} + 10 q^{59} - q^{61} - q^{63} + 3 q^{67} + 4 q^{69} - 14 q^{71} - 10 q^{73} + 8 q^{79} + q^{81} - 18 q^{83} - 4 q^{87} + q^{91} + 7 q^{93} + 3 q^{97}+O(q^{100})$$ q - q^3 - q^7 + q^9 - q^13 + 3 * q^19 + q^21 - 4 * q^23 - q^27 + 4 * q^29 - 7 * q^31 + 6 * q^37 + q^39 + 6 * q^41 - 9 * q^43 - 6 * q^47 - 6 * q^49 - 2 * q^53 - 3 * q^57 + 10 * q^59 - q^61 - q^63 + 3 * q^67 + 4 * q^69 - 14 * q^71 - 10 * q^73 + 8 * q^79 + q^81 - 18 * q^83 - 4 * q^87 + q^91 + 7 * q^93 + 3 * 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 −1.00000 0 0 0 −1.00000 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 2400.2.a.f 1
3.b odd 2 1 7200.2.a.r 1
4.b odd 2 1 2400.2.a.bb yes 1
5.b even 2 1 2400.2.a.bc yes 1
5.c odd 4 2 2400.2.f.h 2
8.b even 2 1 4800.2.a.bx 1
8.d odd 2 1 4800.2.a.x 1
12.b even 2 1 7200.2.a.bi 1
15.d odd 2 1 7200.2.a.bj 1
15.e even 4 2 7200.2.f.l 2
20.d odd 2 1 2400.2.a.g yes 1
20.e even 4 2 2400.2.f.k 2
40.e odd 2 1 4800.2.a.bw 1
40.f even 2 1 4800.2.a.w 1
40.i odd 4 2 4800.2.f.t 2
40.k even 4 2 4800.2.f.q 2
60.h even 2 1 7200.2.a.s 1
60.l odd 4 2 7200.2.f.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.2.a.f 1 1.a even 1 1 trivial
2400.2.a.g yes 1 20.d odd 2 1
2400.2.a.bb yes 1 4.b odd 2 1
2400.2.a.bc yes 1 5.b even 2 1
2400.2.f.h 2 5.c odd 4 2
2400.2.f.k 2 20.e even 4 2
4800.2.a.w 1 40.f even 2 1
4800.2.a.x 1 8.d odd 2 1
4800.2.a.bw 1 40.e odd 2 1
4800.2.a.bx 1 8.b even 2 1
4800.2.f.q 2 40.k even 4 2
4800.2.f.t 2 40.i odd 4 2
7200.2.a.r 1 3.b odd 2 1
7200.2.a.s 1 60.h even 2 1
7200.2.a.bi 1 12.b even 2 1
7200.2.a.bj 1 15.d odd 2 1
7200.2.f.l 2 15.e even 4 2
7200.2.f.r 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2400))$$:

 $$T_{7} + 1$$ T7 + 1 $$T_{11}$$ T11 $$T_{13} + 1$$ T13 + 1 $$T_{19} - 3$$ T19 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T + 1$$
$17$ $$T$$
$19$ $$T - 3$$
$23$ $$T + 4$$
$29$ $$T - 4$$
$31$ $$T + 7$$
$37$ $$T - 6$$
$41$ $$T - 6$$
$43$ $$T + 9$$
$47$ $$T + 6$$
$53$ $$T + 2$$
$59$ $$T - 10$$
$61$ $$T + 1$$
$67$ $$T - 3$$
$71$ $$T + 14$$
$73$ $$T + 10$$
$79$ $$T - 8$$
$83$ $$T + 18$$
$89$ $$T$$
$97$ $$T - 3$$