# Properties

 Label 1296.2.a.k 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 36) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{5} + q^{7}+O(q^{10})$$ q + 3 * q^5 + q^7 $$q + 3 q^{5} + q^{7} - 3 q^{11} - q^{13} + 6 q^{17} + 4 q^{19} + 3 q^{23} + 4 q^{25} + 3 q^{29} - 5 q^{31} + 3 q^{35} + 2 q^{37} + 3 q^{41} + q^{43} + 9 q^{47} - 6 q^{49} - 6 q^{53} - 9 q^{55} + 3 q^{59} - 13 q^{61} - 3 q^{65} + 7 q^{67} + 12 q^{71} - 10 q^{73} - 3 q^{77} - 11 q^{79} + 9 q^{83} + 18 q^{85} + 6 q^{89} - q^{91} + 12 q^{95} + 11 q^{97}+O(q^{100})$$ q + 3 * q^5 + q^7 - 3 * q^11 - q^13 + 6 * q^17 + 4 * q^19 + 3 * q^23 + 4 * q^25 + 3 * q^29 - 5 * q^31 + 3 * q^35 + 2 * q^37 + 3 * q^41 + q^43 + 9 * q^47 - 6 * q^49 - 6 * q^53 - 9 * q^55 + 3 * q^59 - 13 * q^61 - 3 * q^65 + 7 * q^67 + 12 * q^71 - 10 * q^73 - 3 * q^77 - 11 * q^79 + 9 * q^83 + 18 * q^85 + 6 * q^89 - q^91 + 12 * q^95 + 11 * 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 3.00000 0 1.00000 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.k 1
3.b odd 2 1 1296.2.a.b 1
4.b odd 2 1 324.2.a.c 1
8.b even 2 1 5184.2.a.f 1
8.d odd 2 1 5184.2.a.e 1
9.c even 3 2 144.2.i.a 2
9.d odd 6 2 432.2.i.c 2
12.b even 2 1 324.2.a.a 1
20.d odd 2 1 8100.2.a.j 1
20.e even 4 2 8100.2.d.h 2
24.f even 2 1 5184.2.a.ba 1
24.h odd 2 1 5184.2.a.bb 1
36.f odd 6 2 36.2.e.a 2
36.h even 6 2 108.2.e.a 2
60.h even 2 1 8100.2.a.g 1
60.l odd 4 2 8100.2.d.c 2
72.j odd 6 2 1728.2.i.c 2
72.l even 6 2 1728.2.i.d 2
72.n even 6 2 576.2.i.e 2
72.p odd 6 2 576.2.i.f 2
180.n even 6 2 2700.2.i.b 2
180.p odd 6 2 900.2.i.b 2
180.v odd 12 4 2700.2.s.b 4
180.x even 12 4 900.2.s.b 4
252.n even 6 2 1764.2.l.a 2
252.o even 6 2 5292.2.l.a 2
252.r odd 6 2 5292.2.i.a 2
252.s odd 6 2 5292.2.j.a 2
252.u odd 6 2 1764.2.i.a 2
252.bb even 6 2 5292.2.i.c 2
252.bi even 6 2 1764.2.j.b 2
252.bj even 6 2 1764.2.i.c 2
252.bl odd 6 2 1764.2.l.c 2
252.bn odd 6 2 5292.2.l.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 36.f odd 6 2
108.2.e.a 2 36.h even 6 2
144.2.i.a 2 9.c even 3 2
324.2.a.a 1 12.b even 2 1
324.2.a.c 1 4.b odd 2 1
432.2.i.c 2 9.d odd 6 2
576.2.i.e 2 72.n even 6 2
576.2.i.f 2 72.p odd 6 2
900.2.i.b 2 180.p odd 6 2
900.2.s.b 4 180.x even 12 4
1296.2.a.b 1 3.b odd 2 1
1296.2.a.k 1 1.a even 1 1 trivial
1728.2.i.c 2 72.j odd 6 2
1728.2.i.d 2 72.l even 6 2
1764.2.i.a 2 252.u odd 6 2
1764.2.i.c 2 252.bj even 6 2
1764.2.j.b 2 252.bi even 6 2
1764.2.l.a 2 252.n even 6 2
1764.2.l.c 2 252.bl odd 6 2
2700.2.i.b 2 180.n even 6 2
2700.2.s.b 4 180.v odd 12 4
5184.2.a.e 1 8.d odd 2 1
5184.2.a.f 1 8.b even 2 1
5184.2.a.ba 1 24.f even 2 1
5184.2.a.bb 1 24.h odd 2 1
5292.2.i.a 2 252.r odd 6 2
5292.2.i.c 2 252.bb even 6 2
5292.2.j.a 2 252.s odd 6 2
5292.2.l.a 2 252.o even 6 2
5292.2.l.c 2 252.bn odd 6 2
8100.2.a.g 1 60.h even 2 1
8100.2.a.j 1 20.d odd 2 1
8100.2.d.c 2 60.l odd 4 2
8100.2.d.h 2 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1296))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 3$$
$7$ $$T - 1$$
$11$ $$T + 3$$
$13$ $$T + 1$$
$17$ $$T - 6$$
$19$ $$T - 4$$
$23$ $$T - 3$$
$29$ $$T - 3$$
$31$ $$T + 5$$
$37$ $$T - 2$$
$41$ $$T - 3$$
$43$ $$T - 1$$
$47$ $$T - 9$$
$53$ $$T + 6$$
$59$ $$T - 3$$
$61$ $$T + 13$$
$67$ $$T - 7$$
$71$ $$T - 12$$
$73$ $$T + 10$$
$79$ $$T + 11$$
$83$ $$T - 9$$
$89$ $$T - 6$$
$97$ $$T - 11$$