# Properties

 Label 1296.2.a.o Level $1296$ Weight $2$ Character orbit 1296.a Self dual yes Analytic conductor $10.349$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 81) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} - 2 q^{7}+O(q^{10})$$ q + b * q^5 - 2 * q^7 $$q + \beta q^{5} - 2 q^{7} - 2 \beta q^{11} - q^{13} - 3 \beta q^{17} - 2 q^{19} + 2 \beta q^{23} - 2 q^{25} - \beta q^{29} - 8 q^{31} - 2 \beta q^{35} - 7 q^{37} + 4 \beta q^{41} - 2 q^{43} + 4 \beta q^{47} - 3 q^{49} - 6 q^{55} + 8 \beta q^{59} - 7 q^{61} - \beta q^{65} + 10 q^{67} - 6 \beta q^{71} - 7 q^{73} + 4 \beta q^{77} - 2 q^{79} - 8 \beta q^{83} - 9 q^{85} + 3 \beta q^{89} + 2 q^{91} - 2 \beta q^{95} + 2 q^{97} +O(q^{100})$$ q + b * q^5 - 2 * q^7 - 2*b * q^11 - q^13 - 3*b * q^17 - 2 * q^19 + 2*b * q^23 - 2 * q^25 - b * q^29 - 8 * q^31 - 2*b * q^35 - 7 * q^37 + 4*b * q^41 - 2 * q^43 + 4*b * q^47 - 3 * q^49 - 6 * q^55 + 8*b * q^59 - 7 * q^61 - b * q^65 + 10 * q^67 - 6*b * q^71 - 7 * q^73 + 4*b * q^77 - 2 * q^79 - 8*b * q^83 - 9 * q^85 + 3*b * q^89 + 2 * q^91 - 2*b * q^95 + 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7}+O(q^{10})$$ 2 * q - 4 * q^7 $$2 q - 4 q^{7} - 2 q^{13} - 4 q^{19} - 4 q^{25} - 16 q^{31} - 14 q^{37} - 4 q^{43} - 6 q^{49} - 12 q^{55} - 14 q^{61} + 20 q^{67} - 14 q^{73} - 4 q^{79} - 18 q^{85} + 4 q^{91} + 4 q^{97}+O(q^{100})$$ 2 * q - 4 * q^7 - 2 * q^13 - 4 * q^19 - 4 * q^25 - 16 * q^31 - 14 * q^37 - 4 * q^43 - 6 * q^49 - 12 * q^55 - 14 * q^61 + 20 * q^67 - 14 * q^73 - 4 * q^79 - 18 * q^85 + 4 * q^91 + 4 * 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
 −1.73205 1.73205
0 0 0 −1.73205 0 −2.00000 0 0 0
1.2 0 0 0 1.73205 0 −2.00000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.a.o 2
3.b odd 2 1 inner 1296.2.a.o 2
4.b odd 2 1 81.2.a.a 2
8.b even 2 1 5184.2.a.bq 2
8.d odd 2 1 5184.2.a.br 2
9.c even 3 2 1296.2.i.s 4
9.d odd 6 2 1296.2.i.s 4
12.b even 2 1 81.2.a.a 2
20.d odd 2 1 2025.2.a.j 2
20.e even 4 2 2025.2.b.k 4
24.f even 2 1 5184.2.a.br 2
24.h odd 2 1 5184.2.a.bq 2
28.d even 2 1 3969.2.a.i 2
36.f odd 6 2 81.2.c.b 4
36.h even 6 2 81.2.c.b 4
44.c even 2 1 9801.2.a.v 2
60.h even 2 1 2025.2.a.j 2
60.l odd 4 2 2025.2.b.k 4
84.h odd 2 1 3969.2.a.i 2
108.j odd 18 6 729.2.e.o 12
108.l even 18 6 729.2.e.o 12
132.d odd 2 1 9801.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.a.a 2 4.b odd 2 1
81.2.a.a 2 12.b even 2 1
81.2.c.b 4 36.f odd 6 2
81.2.c.b 4 36.h even 6 2
729.2.e.o 12 108.j odd 18 6
729.2.e.o 12 108.l even 18 6
1296.2.a.o 2 1.a even 1 1 trivial
1296.2.a.o 2 3.b odd 2 1 inner
1296.2.i.s 4 9.c even 3 2
1296.2.i.s 4 9.d odd 6 2
2025.2.a.j 2 20.d odd 2 1
2025.2.a.j 2 60.h even 2 1
2025.2.b.k 4 20.e even 4 2
2025.2.b.k 4 60.l odd 4 2
3969.2.a.i 2 28.d even 2 1
3969.2.a.i 2 84.h odd 2 1
5184.2.a.bq 2 8.b even 2 1
5184.2.a.bq 2 24.h odd 2 1
5184.2.a.br 2 8.d odd 2 1
5184.2.a.br 2 24.f even 2 1
9801.2.a.v 2 44.c even 2 1
9801.2.a.v 2 132.d odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 3$$ T5^2 - 3 $$T_{7} + 2$$ T7 + 2 $$T_{11}^{2} - 12$$ T11^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} - 12$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 27$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2} - 12$$
$29$ $$T^{2} - 3$$
$31$ $$(T + 8)^{2}$$
$37$ $$(T + 7)^{2}$$
$41$ $$T^{2} - 48$$
$43$ $$(T + 2)^{2}$$
$47$ $$T^{2} - 48$$
$53$ $$T^{2}$$
$59$ $$T^{2} - 192$$
$61$ $$(T + 7)^{2}$$
$67$ $$(T - 10)^{2}$$
$71$ $$T^{2} - 108$$
$73$ $$(T + 7)^{2}$$
$79$ $$(T + 2)^{2}$$
$83$ $$T^{2} - 192$$
$89$ $$T^{2} - 27$$
$97$ $$(T - 2)^{2}$$