# Properties

 Label 5808.2.a.bh Level $5808$ Weight $2$ Character orbit 5808.a Self dual yes Analytic conductor $46.377$ 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] = [5808,2,Mod(1,5808)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5808, 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("5808.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5808 = 2^{4} \cdot 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5808.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: $$46.3771134940$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 363) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + 4 q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 + 4 * q^5 - q^7 + q^9 $$q + q^{3} + 4 q^{5} - q^{7} + q^{9} - 2 q^{13} + 4 q^{15} + 4 q^{17} + 3 q^{19} - q^{21} - 2 q^{23} + 11 q^{25} + q^{27} + 6 q^{29} + 5 q^{31} - 4 q^{35} + 3 q^{37} - 2 q^{39} - 2 q^{41} - 12 q^{43} + 4 q^{45} - 2 q^{47} - 6 q^{49} + 4 q^{51} + 6 q^{53} + 3 q^{57} + 10 q^{59} + 3 q^{61} - q^{63} - 8 q^{65} + q^{67} - 2 q^{69} - 11 q^{73} + 11 q^{75} - 11 q^{79} + q^{81} - 6 q^{83} + 16 q^{85} + 6 q^{87} + 12 q^{89} + 2 q^{91} + 5 q^{93} + 12 q^{95} + 5 q^{97}+O(q^{100})$$ q + q^3 + 4 * q^5 - q^7 + q^9 - 2 * q^13 + 4 * q^15 + 4 * q^17 + 3 * q^19 - q^21 - 2 * q^23 + 11 * q^25 + q^27 + 6 * q^29 + 5 * q^31 - 4 * q^35 + 3 * q^37 - 2 * q^39 - 2 * q^41 - 12 * q^43 + 4 * q^45 - 2 * q^47 - 6 * q^49 + 4 * q^51 + 6 * q^53 + 3 * q^57 + 10 * q^59 + 3 * q^61 - q^63 - 8 * q^65 + q^67 - 2 * q^69 - 11 * q^73 + 11 * q^75 - 11 * q^79 + q^81 - 6 * q^83 + 16 * q^85 + 6 * q^87 + 12 * q^89 + 2 * q^91 + 5 * q^93 + 12 * q^95 + 5 * 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 4.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$-1$$

## 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 5808.2.a.bh 1
4.b odd 2 1 363.2.a.a 1
11.b odd 2 1 5808.2.a.bi 1
12.b even 2 1 1089.2.a.k 1
20.d odd 2 1 9075.2.a.t 1
44.c even 2 1 363.2.a.c yes 1
44.g even 10 4 363.2.e.d 4
44.h odd 10 4 363.2.e.i 4
132.d odd 2 1 1089.2.a.a 1
220.g even 2 1 9075.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.a 1 4.b odd 2 1
363.2.a.c yes 1 44.c even 2 1
363.2.e.d 4 44.g even 10 4
363.2.e.i 4 44.h odd 10 4
1089.2.a.a 1 132.d odd 2 1
1089.2.a.k 1 12.b even 2 1
5808.2.a.bh 1 1.a even 1 1 trivial
5808.2.a.bi 1 11.b odd 2 1
9075.2.a.b 1 220.g even 2 1
9075.2.a.t 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5808))$$:

 $$T_{5} - 4$$ T5 - 4 $$T_{7} + 1$$ T7 + 1 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 4$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T - 4$$
$19$ $$T - 3$$
$23$ $$T + 2$$
$29$ $$T - 6$$
$31$ $$T - 5$$
$37$ $$T - 3$$
$41$ $$T + 2$$
$43$ $$T + 12$$
$47$ $$T + 2$$
$53$ $$T - 6$$
$59$ $$T - 10$$
$61$ $$T - 3$$
$67$ $$T - 1$$
$71$ $$T$$
$73$ $$T + 11$$
$79$ $$T + 11$$
$83$ $$T + 6$$
$89$ $$T - 12$$
$97$ $$T - 5$$