# Properties

 Label 1296.2.a.l 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 162) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{5} + 4 q^{7}+O(q^{10})$$ q + 3 * q^5 + 4 * q^7 $$q + 3 q^{5} + 4 q^{7} - q^{13} + 3 q^{17} + 4 q^{19} + 4 q^{25} - 9 q^{29} + 4 q^{31} + 12 q^{35} - q^{37} - 6 q^{41} - 8 q^{43} - 12 q^{47} + 9 q^{49} + 6 q^{53} - q^{61} - 3 q^{65} + 4 q^{67} - 12 q^{71} + 11 q^{73} + 16 q^{79} - 12 q^{83} + 9 q^{85} + 3 q^{89} - 4 q^{91} + 12 q^{95} + 2 q^{97}+O(q^{100})$$ q + 3 * q^5 + 4 * q^7 - q^13 + 3 * q^17 + 4 * q^19 + 4 * q^25 - 9 * q^29 + 4 * q^31 + 12 * q^35 - q^37 - 6 * q^41 - 8 * q^43 - 12 * q^47 + 9 * q^49 + 6 * q^53 - q^61 - 3 * q^65 + 4 * q^67 - 12 * q^71 + 11 * q^73 + 16 * q^79 - 12 * q^83 + 9 * q^85 + 3 * q^89 - 4 * q^91 + 12 * q^95 + 2 * 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 4.00000 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.l 1
3.b odd 2 1 1296.2.a.c 1
4.b odd 2 1 162.2.a.d yes 1
8.b even 2 1 5184.2.a.h 1
8.d odd 2 1 5184.2.a.c 1
9.c even 3 2 1296.2.i.b 2
9.d odd 6 2 1296.2.i.n 2
12.b even 2 1 162.2.a.a 1
20.d odd 2 1 4050.2.a.r 1
20.e even 4 2 4050.2.c.n 2
24.f even 2 1 5184.2.a.y 1
24.h odd 2 1 5184.2.a.bd 1
28.d even 2 1 7938.2.a.s 1
36.f odd 6 2 162.2.c.a 2
36.h even 6 2 162.2.c.d 2
60.h even 2 1 4050.2.a.bh 1
60.l odd 4 2 4050.2.c.g 2
84.h odd 2 1 7938.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 12.b even 2 1
162.2.a.d yes 1 4.b odd 2 1
162.2.c.a 2 36.f odd 6 2
162.2.c.d 2 36.h even 6 2
1296.2.a.c 1 3.b odd 2 1
1296.2.a.l 1 1.a even 1 1 trivial
1296.2.i.b 2 9.c even 3 2
1296.2.i.n 2 9.d odd 6 2
4050.2.a.r 1 20.d odd 2 1
4050.2.a.bh 1 60.h even 2 1
4050.2.c.g 2 60.l odd 4 2
4050.2.c.n 2 20.e even 4 2
5184.2.a.c 1 8.d odd 2 1
5184.2.a.h 1 8.b even 2 1
5184.2.a.y 1 24.f even 2 1
5184.2.a.bd 1 24.h odd 2 1
7938.2.a.n 1 84.h odd 2 1
7938.2.a.s 1 28.d even 2 1

## Hecke kernels

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

 $$T_{5} - 3$$ T5 - 3 $$T_{7} - 4$$ T7 - 4 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 3$$
$7$ $$T - 4$$
$11$ $$T$$
$13$ $$T + 1$$
$17$ $$T - 3$$
$19$ $$T - 4$$
$23$ $$T$$
$29$ $$T + 9$$
$31$ $$T - 4$$
$37$ $$T + 1$$
$41$ $$T + 6$$
$43$ $$T + 8$$
$47$ $$T + 12$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T + 1$$
$67$ $$T - 4$$
$71$ $$T + 12$$
$73$ $$T - 11$$
$79$ $$T - 16$$
$83$ $$T + 12$$
$89$ $$T - 3$$
$97$ $$T - 2$$