# Properties

 Label 1296.2.a.i Level $1296$ Weight $2$ Character orbit 1296.a Self dual yes Analytic conductor $10.349$ 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] = [1296,2,Mod(1,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1296.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1296.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: $$10.3486121020$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) 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)

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

## 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 1296.2.a.i 1
3.b odd 2 1 1296.2.a.e 1
4.b odd 2 1 648.2.a.c 1
8.b even 2 1 5184.2.a.n 1
8.d odd 2 1 5184.2.a.i 1
9.c even 3 2 432.2.i.a 2
9.d odd 6 2 144.2.i.b 2
12.b even 2 1 648.2.a.a 1
24.f even 2 1 5184.2.a.s 1
24.h odd 2 1 5184.2.a.x 1
36.f odd 6 2 216.2.i.a 2
36.h even 6 2 72.2.i.a 2
72.j odd 6 2 576.2.i.c 2
72.l even 6 2 576.2.i.d 2
72.n even 6 2 1728.2.i.g 2
72.p odd 6 2 1728.2.i.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.a 2 36.h even 6 2
144.2.i.b 2 9.d odd 6 2
216.2.i.a 2 36.f odd 6 2
432.2.i.a 2 9.c even 3 2
576.2.i.c 2 72.j odd 6 2
576.2.i.d 2 72.l even 6 2
648.2.a.a 1 12.b even 2 1
648.2.a.c 1 4.b odd 2 1
1296.2.a.e 1 3.b odd 2 1
1296.2.a.i 1 1.a even 1 1 trivial
1728.2.i.g 2 72.n even 6 2
1728.2.i.h 2 72.p odd 6 2
5184.2.a.i 1 8.d odd 2 1
5184.2.a.n 1 8.b even 2 1
5184.2.a.s 1 24.f even 2 1
5184.2.a.x 1 24.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1296))$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{7} - 3$$ T7 - 3 $$T_{11} - 5$$ T11 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T - 3$$
$11$ $$T - 5$$
$13$ $$T + 5$$
$17$ $$T - 2$$
$19$ $$T - 4$$
$23$ $$T + 1$$
$29$ $$T - 9$$
$31$ $$T - 1$$
$37$ $$T + 6$$
$41$ $$T + 3$$
$43$ $$T + 1$$
$47$ $$T + 3$$
$53$ $$T + 2$$
$59$ $$T - 11$$
$61$ $$T - 7$$
$67$ $$T - 1$$
$71$ $$T - 4$$
$73$ $$T + 2$$
$79$ $$T + 1$$
$83$ $$T - 1$$
$89$ $$T - 18$$
$97$ $$T + 13$$