# Properties

 Label 2400.4.a.g Level $2400$ Weight $4$ Character orbit 2400.a Self dual yes Analytic conductor $141.605$ 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] = [2400,4,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, 4, names="a")

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

chi := DirichletCharacter("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2400 = 2^{5} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$141.604584014$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 12 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 12 * q^7 + 9 * q^9 $$q - 3 q^{3} + 12 q^{7} + 9 q^{9} - 24 q^{11} - 38 q^{13} + 6 q^{17} + 104 q^{19} - 36 q^{21} - 100 q^{23} - 27 q^{27} + 230 q^{29} - 56 q^{31} + 72 q^{33} - 190 q^{37} + 114 q^{39} + 202 q^{41} + 148 q^{43} - 124 q^{47} - 199 q^{49} - 18 q^{51} - 206 q^{53} - 312 q^{57} - 128 q^{59} + 190 q^{61} + 108 q^{63} + 204 q^{67} + 300 q^{69} - 440 q^{71} - 1210 q^{73} - 288 q^{77} + 816 q^{79} + 81 q^{81} + 1412 q^{83} - 690 q^{87} - 214 q^{89} - 456 q^{91} + 168 q^{93} - 1202 q^{97} - 216 q^{99}+O(q^{100})$$ q - 3 * q^3 + 12 * q^7 + 9 * q^9 - 24 * q^11 - 38 * q^13 + 6 * q^17 + 104 * q^19 - 36 * q^21 - 100 * q^23 - 27 * q^27 + 230 * q^29 - 56 * q^31 + 72 * q^33 - 190 * q^37 + 114 * q^39 + 202 * q^41 + 148 * q^43 - 124 * q^47 - 199 * q^49 - 18 * q^51 - 206 * q^53 - 312 * q^57 - 128 * q^59 + 190 * q^61 + 108 * q^63 + 204 * q^67 + 300 * q^69 - 440 * q^71 - 1210 * q^73 - 288 * q^77 + 816 * q^79 + 81 * q^81 + 1412 * q^83 - 690 * q^87 - 214 * q^89 - 456 * q^91 + 168 * q^93 - 1202 * q^97 - 216 * 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 −3.00000 0 0 0 12.0000 0 9.00000 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 2400.4.a.g 1
4.b odd 2 1 2400.4.a.p 1
5.b even 2 1 480.4.a.j yes 1
15.d odd 2 1 1440.4.a.d 1
20.d odd 2 1 480.4.a.e 1
40.e odd 2 1 960.4.a.y 1
40.f even 2 1 960.4.a.d 1
60.h even 2 1 1440.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.4.a.e 1 20.d odd 2 1
480.4.a.j yes 1 5.b even 2 1
960.4.a.d 1 40.f even 2 1
960.4.a.y 1 40.e odd 2 1
1440.4.a.d 1 15.d odd 2 1
1440.4.a.g 1 60.h even 2 1
2400.4.a.g 1 1.a even 1 1 trivial
2400.4.a.p 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2400))$$:

 $$T_{7} - 12$$ T7 - 12 $$T_{11} + 24$$ T11 + 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T$$
$7$ $$T - 12$$
$11$ $$T + 24$$
$13$ $$T + 38$$
$17$ $$T - 6$$
$19$ $$T - 104$$
$23$ $$T + 100$$
$29$ $$T - 230$$
$31$ $$T + 56$$
$37$ $$T + 190$$
$41$ $$T - 202$$
$43$ $$T - 148$$
$47$ $$T + 124$$
$53$ $$T + 206$$
$59$ $$T + 128$$
$61$ $$T - 190$$
$67$ $$T - 204$$
$71$ $$T + 440$$
$73$ $$T + 1210$$
$79$ $$T - 816$$
$83$ $$T - 1412$$
$89$ $$T + 214$$
$97$ $$T + 1202$$