# Properties

 Label 4600.2.a.f Level $4600$ Weight $2$ Character orbit 4600.a Self dual yes Analytic conductor $36.731$ 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] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.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: $$36.7311849298$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} + 2 q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 + 2 * q^7 - 2 * q^9 $$q - q^{3} + 2 q^{7} - 2 q^{9} - q^{13} + 4 q^{17} - 4 q^{19} - 2 q^{21} - q^{23} + 5 q^{27} - 3 q^{29} - q^{31} + 8 q^{37} + q^{39} - 5 q^{41} + 6 q^{43} - 9 q^{47} - 3 q^{49} - 4 q^{51} - 2 q^{53} + 4 q^{57} - 4 q^{63} - 4 q^{67} + q^{69} + 3 q^{71} - 7 q^{73} + 4 q^{79} + q^{81} - 8 q^{83} + 3 q^{87} - 14 q^{89} - 2 q^{91} + q^{93} + 14 q^{97}+O(q^{100})$$ q - q^3 + 2 * q^7 - 2 * q^9 - q^13 + 4 * q^17 - 4 * q^19 - 2 * q^21 - q^23 + 5 * q^27 - 3 * q^29 - q^31 + 8 * q^37 + q^39 - 5 * q^41 + 6 * q^43 - 9 * q^47 - 3 * q^49 - 4 * q^51 - 2 * q^53 + 4 * q^57 - 4 * q^63 - 4 * q^67 + q^69 + 3 * q^71 - 7 * q^73 + 4 * q^79 + q^81 - 8 * q^83 + 3 * q^87 - 14 * q^89 - 2 * q^91 + q^93 + 14 * 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 2.00000 0 −2.00000 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
$$2$$ $$+1$$
$$5$$ $$+1$$
$$23$$ $$+1$$

## 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 4600.2.a.f 1
4.b odd 2 1 9200.2.a.z 1
5.b even 2 1 920.2.a.d 1
5.c odd 4 2 4600.2.e.g 2
15.d odd 2 1 8280.2.a.o 1
20.d odd 2 1 1840.2.a.b 1
40.e odd 2 1 7360.2.a.u 1
40.f even 2 1 7360.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.d 1 5.b even 2 1
1840.2.a.b 1 20.d odd 2 1
4600.2.a.f 1 1.a even 1 1 trivial
4600.2.e.g 2 5.c odd 4 2
7360.2.a.j 1 40.f even 2 1
7360.2.a.u 1 40.e odd 2 1
8280.2.a.o 1 15.d odd 2 1
9200.2.a.z 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4600))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{7} - 2$$ T7 - 2 $$T_{11}$$ T11

## Hecke characteristic polynomials

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