# Properties

 Label 2496.2.a.h Level $2496$ Weight $2$ Character orbit 2496.a Self dual yes Analytic conductor $19.931$ 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] = [2496,2,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, 2, names="a")

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

chi := DirichletCharacter("2496.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$19.9306603445$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 156) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} + 2 q^{7} + q^{9}+O(q^{10})$$ q - q^3 + 2 * q^7 + q^9 $$q - q^{3} + 2 q^{7} + q^{9} - q^{13} - 6 q^{17} - 2 q^{19} - 2 q^{21} - 5 q^{25} - q^{27} + 6 q^{29} + 2 q^{31} - 2 q^{37} + q^{39} - 12 q^{41} + 4 q^{43} - 3 q^{49} + 6 q^{51} - 6 q^{53} + 2 q^{57} - 12 q^{59} - 2 q^{61} + 2 q^{63} + 10 q^{67} + 12 q^{71} + 14 q^{73} + 5 q^{75} + 8 q^{79} + q^{81} - 12 q^{83} - 6 q^{87} - 2 q^{91} - 2 q^{93} - 10 q^{97}+O(q^{100})$$ q - q^3 + 2 * q^7 + q^9 - q^13 - 6 * q^17 - 2 * q^19 - 2 * q^21 - 5 * q^25 - q^27 + 6 * q^29 + 2 * q^31 - 2 * q^37 + q^39 - 12 * q^41 + 4 * q^43 - 3 * q^49 + 6 * q^51 - 6 * q^53 + 2 * q^57 - 12 * q^59 - 2 * q^61 + 2 * q^63 + 10 * q^67 + 12 * q^71 + 14 * q^73 + 5 * q^75 + 8 * q^79 + q^81 - 12 * q^83 - 6 * q^87 - 2 * q^91 - 2 * q^93 - 10 * 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 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
$$2$$ $$1$$
$$3$$ $$1$$
$$13$$ $$1$$

## 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 2496.2.a.h 1
3.b odd 2 1 7488.2.a.bf 1
4.b odd 2 1 2496.2.a.v 1
8.b even 2 1 156.2.a.b 1
8.d odd 2 1 624.2.a.b 1
12.b even 2 1 7488.2.a.bb 1
24.f even 2 1 1872.2.a.i 1
24.h odd 2 1 468.2.a.c 1
40.f even 2 1 3900.2.a.a 1
40.i odd 4 2 3900.2.h.e 2
56.h odd 2 1 7644.2.a.a 1
72.j odd 6 2 4212.2.i.g 2
72.n even 6 2 4212.2.i.f 2
104.e even 2 1 2028.2.a.e 1
104.h odd 2 1 8112.2.a.i 1
104.j odd 4 2 2028.2.b.d 2
104.r even 6 2 2028.2.i.b 2
104.s even 6 2 2028.2.i.c 2
104.x odd 12 4 2028.2.q.d 4
312.b odd 2 1 6084.2.a.h 1
312.y even 4 2 6084.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.b 1 8.b even 2 1
468.2.a.c 1 24.h odd 2 1
624.2.a.b 1 8.d odd 2 1
1872.2.a.i 1 24.f even 2 1
2028.2.a.e 1 104.e even 2 1
2028.2.b.d 2 104.j odd 4 2
2028.2.i.b 2 104.r even 6 2
2028.2.i.c 2 104.s even 6 2
2028.2.q.d 4 104.x odd 12 4
2496.2.a.h 1 1.a even 1 1 trivial
2496.2.a.v 1 4.b odd 2 1
3900.2.a.a 1 40.f even 2 1
3900.2.h.e 2 40.i odd 4 2
4212.2.i.f 2 72.n even 6 2
4212.2.i.g 2 72.j odd 6 2
6084.2.a.h 1 312.b odd 2 1
6084.2.b.a 2 312.y even 4 2
7488.2.a.bb 1 12.b even 2 1
7488.2.a.bf 1 3.b odd 2 1
7644.2.a.a 1 56.h odd 2 1
8112.2.a.i 1 104.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2496))$$:

 $$T_{5}$$ T5 $$T_{7} - 2$$ T7 - 2 $$T_{11}$$ T11 $$T_{17} + 6$$ T17 + 6 $$T_{19} + 2$$ T19 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T + 1$$
$17$ $$T + 6$$
$19$ $$T + 2$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T - 2$$
$37$ $$T + 2$$
$41$ $$T + 12$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T + 12$$
$61$ $$T + 2$$
$67$ $$T - 10$$
$71$ $$T - 12$$
$73$ $$T - 14$$
$79$ $$T - 8$$
$83$ $$T + 12$$
$89$ $$T$$
$97$ $$T + 10$$