# Properties

 Label 2496.4.a.k Level $2496$ Weight $4$ Character orbit 2496.a Self dual yes Analytic conductor $147.269$ 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] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, 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("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2496.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: $$147.268767374$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 6 q^{5} + 20 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 6 * q^5 + 20 * q^7 + 9 * q^9 $$q + 3 q^{3} - 6 q^{5} + 20 q^{7} + 9 q^{9} - 24 q^{11} - 13 q^{13} - 18 q^{15} - 30 q^{17} + 16 q^{19} + 60 q^{21} - 72 q^{23} - 89 q^{25} + 27 q^{27} + 282 q^{29} + 164 q^{31} - 72 q^{33} - 120 q^{35} - 110 q^{37} - 39 q^{39} - 126 q^{41} - 164 q^{43} - 54 q^{45} - 204 q^{47} + 57 q^{49} - 90 q^{51} + 738 q^{53} + 144 q^{55} + 48 q^{57} - 120 q^{59} - 614 q^{61} + 180 q^{63} + 78 q^{65} - 848 q^{67} - 216 q^{69} + 132 q^{71} + 218 q^{73} - 267 q^{75} - 480 q^{77} - 1096 q^{79} + 81 q^{81} - 552 q^{83} + 180 q^{85} + 846 q^{87} + 210 q^{89} - 260 q^{91} + 492 q^{93} - 96 q^{95} - 1726 q^{97} - 216 q^{99}+O(q^{100})$$ q + 3 * q^3 - 6 * q^5 + 20 * q^7 + 9 * q^9 - 24 * q^11 - 13 * q^13 - 18 * q^15 - 30 * q^17 + 16 * q^19 + 60 * q^21 - 72 * q^23 - 89 * q^25 + 27 * q^27 + 282 * q^29 + 164 * q^31 - 72 * q^33 - 120 * q^35 - 110 * q^37 - 39 * q^39 - 126 * q^41 - 164 * q^43 - 54 * q^45 - 204 * q^47 + 57 * q^49 - 90 * q^51 + 738 * q^53 + 144 * q^55 + 48 * q^57 - 120 * q^59 - 614 * q^61 + 180 * q^63 + 78 * q^65 - 848 * q^67 - 216 * q^69 + 132 * q^71 + 218 * q^73 - 267 * q^75 - 480 * q^77 - 1096 * q^79 + 81 * q^81 - 552 * q^83 + 180 * q^85 + 846 * q^87 + 210 * q^89 - 260 * q^91 + 492 * q^93 - 96 * q^95 - 1726 * 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 −6.00000 0 20.0000 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$13$$ $$+1$$

## 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 2496.4.a.k 1
4.b odd 2 1 2496.4.a.b 1
8.b even 2 1 78.4.a.e 1
8.d odd 2 1 624.4.a.i 1
24.f even 2 1 1872.4.a.e 1
24.h odd 2 1 234.4.a.b 1
40.f even 2 1 1950.4.a.c 1
104.e even 2 1 1014.4.a.b 1
104.j odd 4 2 1014.4.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.e 1 8.b even 2 1
234.4.a.b 1 24.h odd 2 1
624.4.a.i 1 8.d odd 2 1
1014.4.a.b 1 104.e even 2 1
1014.4.b.c 2 104.j odd 4 2
1872.4.a.e 1 24.f even 2 1
1950.4.a.c 1 40.f even 2 1
2496.4.a.b 1 4.b odd 2 1
2496.4.a.k 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2496))$$:

 $$T_{5} + 6$$ T5 + 6 $$T_{7} - 20$$ T7 - 20 $$T_{11} + 24$$ T11 + 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 6$$
$7$ $$T - 20$$
$11$ $$T + 24$$
$13$ $$T + 13$$
$17$ $$T + 30$$
$19$ $$T - 16$$
$23$ $$T + 72$$
$29$ $$T - 282$$
$31$ $$T - 164$$
$37$ $$T + 110$$
$41$ $$T + 126$$
$43$ $$T + 164$$
$47$ $$T + 204$$
$53$ $$T - 738$$
$59$ $$T + 120$$
$61$ $$T + 614$$
$67$ $$T + 848$$
$71$ $$T - 132$$
$73$ $$T - 218$$
$79$ $$T + 1096$$
$83$ $$T + 552$$
$89$ $$T - 210$$
$97$ $$T + 1726$$