# Properties

 Label 5824.2.a.t Level $5824$ Weight $2$ Character orbit 5824.a Self dual yes Analytic conductor $46.505$ 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] = [5824,2,Mod(1,5824)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5824, 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("5824.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5824 = 2^{6} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5824.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: $$46.5048741372$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{5} + q^{7} - 3 q^{9}+O(q^{10})$$ q + 3 * q^5 + q^7 - 3 * q^9 $$q + 3 q^{5} + q^{7} - 3 q^{9} - 6 q^{11} + q^{13} + 4 q^{17} + 5 q^{19} - 3 q^{23} + 4 q^{25} + 5 q^{29} + 3 q^{31} + 3 q^{35} + 4 q^{37} - 6 q^{41} - q^{43} - 9 q^{45} - 7 q^{47} + q^{49} + 9 q^{53} - 18 q^{55} + 8 q^{59} + 10 q^{61} - 3 q^{63} + 3 q^{65} - 6 q^{67} + 8 q^{71} - 13 q^{73} - 6 q^{77} - 3 q^{79} + 9 q^{81} + 15 q^{83} + 12 q^{85} + 3 q^{89} + q^{91} + 15 q^{95} + 7 q^{97} + 18 q^{99}+O(q^{100})$$ q + 3 * q^5 + q^7 - 3 * q^9 - 6 * q^11 + q^13 + 4 * q^17 + 5 * q^19 - 3 * q^23 + 4 * q^25 + 5 * q^29 + 3 * q^31 + 3 * q^35 + 4 * q^37 - 6 * q^41 - q^43 - 9 * q^45 - 7 * q^47 + q^49 + 9 * q^53 - 18 * q^55 + 8 * q^59 + 10 * q^61 - 3 * q^63 + 3 * q^65 - 6 * q^67 + 8 * q^71 - 13 * q^73 - 6 * q^77 - 3 * q^79 + 9 * q^81 + 15 * q^83 + 12 * q^85 + 3 * q^89 + q^91 + 15 * q^95 + 7 * q^97 + 18 * 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 0 0 3.00000 0 1.00000 0 −3.00000 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 5824.2.a.t 1
4.b odd 2 1 5824.2.a.s 1
8.b even 2 1 1456.2.a.g 1
8.d odd 2 1 91.2.a.a 1
24.f even 2 1 819.2.a.f 1
40.e odd 2 1 2275.2.a.h 1
56.e even 2 1 637.2.a.a 1
56.k odd 6 2 637.2.e.e 2
56.m even 6 2 637.2.e.d 2
104.h odd 2 1 1183.2.a.b 1
104.m even 4 2 1183.2.c.b 2
168.e odd 2 1 5733.2.a.l 1
728.b even 2 1 8281.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.a 1 8.d odd 2 1
637.2.a.a 1 56.e even 2 1
637.2.e.d 2 56.m even 6 2
637.2.e.e 2 56.k odd 6 2
819.2.a.f 1 24.f even 2 1
1183.2.a.b 1 104.h odd 2 1
1183.2.c.b 2 104.m even 4 2
1456.2.a.g 1 8.b even 2 1
2275.2.a.h 1 40.e odd 2 1
5733.2.a.l 1 168.e odd 2 1
5824.2.a.s 1 4.b odd 2 1
5824.2.a.t 1 1.a even 1 1 trivial
8281.2.a.l 1 728.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5824))$$:

 $$T_{3}$$ T3 $$T_{5} - 3$$ T5 - 3 $$T_{11} + 6$$ T11 + 6 $$T_{17} - 4$$ T17 - 4 $$T_{19} - 5$$ T19 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 3$$
$7$ $$T - 1$$
$11$ $$T + 6$$
$13$ $$T - 1$$
$17$ $$T - 4$$
$19$ $$T - 5$$
$23$ $$T + 3$$
$29$ $$T - 5$$
$31$ $$T - 3$$
$37$ $$T - 4$$
$41$ $$T + 6$$
$43$ $$T + 1$$
$47$ $$T + 7$$
$53$ $$T - 9$$
$59$ $$T - 8$$
$61$ $$T - 10$$
$67$ $$T + 6$$
$71$ $$T - 8$$
$73$ $$T + 13$$
$79$ $$T + 3$$
$83$ $$T - 15$$
$89$ $$T - 3$$
$97$ $$T - 7$$