# Properties

 Label 1296.2.a.n Level $1296$ Weight $2$ Character orbit 1296.a Self dual yes Analytic conductor $10.349$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{5} + (\beta - 2) q^{7} +O(q^{10})$$ q - b * q^5 + (b - 2) * q^7 $$q - \beta q^{5} + (\beta - 2) q^{7} - q^{11} + (\beta + 2) q^{13} + (\beta - 3) q^{17} + ( - \beta - 3) q^{19} + ( - \beta - 2) q^{23} + (\beta + 3) q^{25} + (\beta - 2) q^{29} + (\beta - 4) q^{31} + (\beta - 8) q^{35} + ( - 2 \beta + 4) q^{37} + (2 \beta - 7) q^{41} + ( - 2 \beta - 3) q^{43} + (\beta - 2) q^{47} + ( - 3 \beta + 5) q^{49} + ( - 2 \beta - 4) q^{53} + \beta q^{55} - 7 q^{59} - \beta q^{61} + ( - 3 \beta - 8) q^{65} + ( - 2 \beta - 1) q^{67} + 4 q^{71} + (3 \beta - 5) q^{73} + ( - \beta + 2) q^{77} + ( - \beta + 4) q^{79} + ( - \beta - 12) q^{83} + (2 \beta - 8) q^{85} + 6 q^{89} + (\beta + 4) q^{91} + (4 \beta + 8) q^{95} + ( - 2 \beta - 3) q^{97} +O(q^{100})$$ q - b * q^5 + (b - 2) * q^7 - q^11 + (b + 2) * q^13 + (b - 3) * q^17 + (-b - 3) * q^19 + (-b - 2) * q^23 + (b + 3) * q^25 + (b - 2) * q^29 + (b - 4) * q^31 + (b - 8) * q^35 + (-2*b + 4) * q^37 + (2*b - 7) * q^41 + (-2*b - 3) * q^43 + (b - 2) * q^47 + (-3*b + 5) * q^49 + (-2*b - 4) * q^53 + b * q^55 - 7 * q^59 - b * q^61 + (-3*b - 8) * q^65 + (-2*b - 1) * q^67 + 4 * q^71 + (3*b - 5) * q^73 + (-b + 2) * q^77 + (-b + 4) * q^79 + (-b - 12) * q^83 + (2*b - 8) * q^85 + 6 * q^89 + (b + 4) * q^91 + (4*b + 8) * q^95 + (-2*b - 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} - 3 q^{7}+O(q^{10})$$ 2 * q - q^5 - 3 * q^7 $$2 q - q^{5} - 3 q^{7} - 2 q^{11} + 5 q^{13} - 5 q^{17} - 7 q^{19} - 5 q^{23} + 7 q^{25} - 3 q^{29} - 7 q^{31} - 15 q^{35} + 6 q^{37} - 12 q^{41} - 8 q^{43} - 3 q^{47} + 7 q^{49} - 10 q^{53} + q^{55} - 14 q^{59} - q^{61} - 19 q^{65} - 4 q^{67} + 8 q^{71} - 7 q^{73} + 3 q^{77} + 7 q^{79} - 25 q^{83} - 14 q^{85} + 12 q^{89} + 9 q^{91} + 20 q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q - q^5 - 3 * q^7 - 2 * q^11 + 5 * q^13 - 5 * q^17 - 7 * q^19 - 5 * q^23 + 7 * q^25 - 3 * q^29 - 7 * q^31 - 15 * q^35 + 6 * q^37 - 12 * q^41 - 8 * q^43 - 3 * q^47 + 7 * q^49 - 10 * q^53 + q^55 - 14 * q^59 - q^61 - 19 * q^65 - 4 * q^67 + 8 * q^71 - 7 * q^73 + 3 * q^77 + 7 * q^79 - 25 * q^83 - 14 * q^85 + 12 * q^89 + 9 * q^91 + 20 * q^95 - 8 * 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
 3.37228 −2.37228
0 0 0 −3.37228 0 1.37228 0 0 0
1.2 0 0 0 2.37228 0 −4.37228 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.n 2
3.b odd 2 1 1296.2.a.p 2
4.b odd 2 1 648.2.a.f 2
8.b even 2 1 5184.2.a.bs 2
8.d odd 2 1 5184.2.a.bt 2
9.c even 3 2 144.2.i.d 4
9.d odd 6 2 432.2.i.d 4
12.b even 2 1 648.2.a.g 2
24.f even 2 1 5184.2.a.bp 2
24.h odd 2 1 5184.2.a.bo 2
36.f odd 6 2 72.2.i.b 4
36.h even 6 2 216.2.i.b 4
72.j odd 6 2 1728.2.i.j 4
72.l even 6 2 1728.2.i.i 4
72.n even 6 2 576.2.i.l 4
72.p odd 6 2 576.2.i.j 4

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

 $$T_{5}^{2} + T_{5} - 8$$ T5^2 + T5 - 8 $$T_{7}^{2} + 3T_{7} - 6$$ T7^2 + 3*T7 - 6 $$T_{11} + 1$$ T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T - 8$$
$7$ $$T^{2} + 3T - 6$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 5T - 2$$
$17$ $$T^{2} + 5T - 2$$
$19$ $$T^{2} + 7T + 4$$
$23$ $$T^{2} + 5T - 2$$
$29$ $$T^{2} + 3T - 6$$
$31$ $$T^{2} + 7T + 4$$
$37$ $$T^{2} - 6T - 24$$
$41$ $$T^{2} + 12T + 3$$
$43$ $$T^{2} + 8T - 17$$
$47$ $$T^{2} + 3T - 6$$
$53$ $$T^{2} + 10T - 8$$
$59$ $$(T + 7)^{2}$$
$61$ $$T^{2} + T - 8$$
$67$ $$T^{2} + 4T - 29$$
$71$ $$(T - 4)^{2}$$
$73$ $$T^{2} + 7T - 62$$
$79$ $$T^{2} - 7T + 4$$
$83$ $$T^{2} + 25T + 148$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 8T - 17$$