# Properties

 Label 1296.2.i.s Level $1296$ Weight $2$ Character orbit 1296.i Analytic conductor $10.349$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,2,Mod(433,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1296.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1296.i (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$10.3486121020$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 81) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{5} + ( - 2 \beta_1 + 2) q^{7}+O(q^{10})$$ q - b2 * q^5 + (-2*b1 + 2) * q^7 $$q - \beta_{2} q^{5} + ( - 2 \beta_1 + 2) q^{7} + (2 \beta_{3} - 2 \beta_{2}) q^{11} + \beta_1 q^{13} - 3 \beta_{3} q^{17} - 2 q^{19} - 2 \beta_{2} q^{23} + ( - 2 \beta_1 + 2) q^{25} + (\beta_{3} - \beta_{2}) q^{29} + 8 \beta_1 q^{31} - 2 \beta_{3} q^{35} - 7 q^{37} - 4 \beta_{2} q^{41} + ( - 2 \beta_1 + 2) q^{43} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{47} + 3 \beta_1 q^{49} - 6 q^{55} - 8 \beta_{2} q^{59} + ( - 7 \beta_1 + 7) q^{61} + (\beta_{3} - \beta_{2}) q^{65} - 10 \beta_1 q^{67} - 6 \beta_{3} q^{71} - 7 q^{73} - 4 \beta_{2} q^{77} + ( - 2 \beta_1 + 2) q^{79} + (8 \beta_{3} - 8 \beta_{2}) q^{83} + 9 \beta_1 q^{85} + 3 \beta_{3} q^{89} + 2 q^{91} + 2 \beta_{2} q^{95} + (2 \beta_1 - 2) q^{97}+O(q^{100})$$ q - b2 * q^5 + (-2*b1 + 2) * q^7 + (2*b3 - 2*b2) * q^11 + b1 * q^13 - 3*b3 * q^17 - 2 * q^19 - 2*b2 * q^23 + (-2*b1 + 2) * q^25 + (b3 - b2) * q^29 + 8*b1 * q^31 - 2*b3 * q^35 - 7 * q^37 - 4*b2 * q^41 + (-2*b1 + 2) * q^43 + (-4*b3 + 4*b2) * q^47 + 3*b1 * q^49 - 6 * q^55 - 8*b2 * q^59 + (-7*b1 + 7) * q^61 + (b3 - b2) * q^65 - 10*b1 * q^67 - 6*b3 * q^71 - 7 * q^73 - 4*b2 * q^77 + (-2*b1 + 2) * q^79 + (8*b3 - 8*b2) * q^83 + 9*b1 * q^85 + 3*b3 * q^89 + 2 * q^91 + 2*b2 * q^95 + (2*b1 - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{7}+O(q^{10})$$ 4 * q + 4 * q^7 $$4 q + 4 q^{7} + 2 q^{13} - 8 q^{19} + 4 q^{25} + 16 q^{31} - 28 q^{37} + 4 q^{43} + 6 q^{49} - 24 q^{55} + 14 q^{61} - 20 q^{67} - 28 q^{73} + 4 q^{79} + 18 q^{85} + 8 q^{91} - 4 q^{97}+O(q^{100})$$ 4 * q + 4 * q^7 + 2 * q^13 - 8 * q^19 + 4 * q^25 + 16 * q^31 - 28 * q^37 + 4 * q^43 + 6 * q^49 - 24 * q^55 + 14 * q^61 - 20 * q^67 - 28 * q^73 + 4 * q^79 + 18 * q^85 + 8 * q^91 - 4 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 0 −0.866025 + 1.50000i 0 1.00000 + 1.73205i 0 0 0
433.2 0 0 0 0.866025 1.50000i 0 1.00000 + 1.73205i 0 0 0
865.1 0 0 0 −0.866025 1.50000i 0 1.00000 1.73205i 0 0 0
865.2 0 0 0 0.866025 + 1.50000i 0 1.00000 1.73205i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.i.s 4
3.b odd 2 1 inner 1296.2.i.s 4
4.b odd 2 1 81.2.c.b 4
9.c even 3 1 1296.2.a.o 2
9.c even 3 1 inner 1296.2.i.s 4
9.d odd 6 1 1296.2.a.o 2
9.d odd 6 1 inner 1296.2.i.s 4
12.b even 2 1 81.2.c.b 4
36.f odd 6 1 81.2.a.a 2
36.f odd 6 1 81.2.c.b 4
36.h even 6 1 81.2.a.a 2
36.h even 6 1 81.2.c.b 4
72.j odd 6 1 5184.2.a.bq 2
72.l even 6 1 5184.2.a.br 2
72.n even 6 1 5184.2.a.bq 2
72.p odd 6 1 5184.2.a.br 2
108.j odd 18 6 729.2.e.o 12
108.l even 18 6 729.2.e.o 12
180.n even 6 1 2025.2.a.j 2
180.p odd 6 1 2025.2.a.j 2
180.v odd 12 2 2025.2.b.k 4
180.x even 12 2 2025.2.b.k 4
252.s odd 6 1 3969.2.a.i 2
252.bi even 6 1 3969.2.a.i 2
396.k even 6 1 9801.2.a.v 2
396.o odd 6 1 9801.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.a.a 2 36.f odd 6 1
81.2.a.a 2 36.h even 6 1
81.2.c.b 4 4.b odd 2 1
81.2.c.b 4 12.b even 2 1
81.2.c.b 4 36.f odd 6 1
81.2.c.b 4 36.h even 6 1
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 9.c even 3 1
1296.2.a.o 2 9.d odd 6 1
1296.2.i.s 4 1.a even 1 1 trivial
1296.2.i.s 4 3.b odd 2 1 inner
1296.2.i.s 4 9.c even 3 1 inner
1296.2.i.s 4 9.d odd 6 1 inner
2025.2.a.j 2 180.n even 6 1
2025.2.a.j 2 180.p odd 6 1
2025.2.b.k 4 180.v odd 12 2
2025.2.b.k 4 180.x even 12 2
3969.2.a.i 2 252.s odd 6 1
3969.2.a.i 2 252.bi even 6 1
5184.2.a.bq 2 72.j odd 6 1
5184.2.a.bq 2 72.n even 6 1
5184.2.a.br 2 72.l even 6 1
5184.2.a.br 2 72.p odd 6 1
9801.2.a.v 2 396.k even 6 1
9801.2.a.v 2 396.o odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 3T_{5}^{2} + 9$$ T5^4 + 3*T5^2 + 9 $$T_{7}^{2} - 2T_{7} + 4$$ T7^2 - 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 3T^{2} + 9$$
$7$ $$(T^{2} - 2 T + 4)^{2}$$
$11$ $$T^{4} + 12T^{2} + 144$$
$13$ $$(T^{2} - T + 1)^{2}$$
$17$ $$(T^{2} - 27)^{2}$$
$19$ $$(T + 2)^{4}$$
$23$ $$T^{4} + 12T^{2} + 144$$
$29$ $$T^{4} + 3T^{2} + 9$$
$31$ $$(T^{2} - 8 T + 64)^{2}$$
$37$ $$(T + 7)^{4}$$
$41$ $$T^{4} + 48T^{2} + 2304$$
$43$ $$(T^{2} - 2 T + 4)^{2}$$
$47$ $$T^{4} + 48T^{2} + 2304$$
$53$ $$T^{4}$$
$59$ $$T^{4} + 192 T^{2} + 36864$$
$61$ $$(T^{2} - 7 T + 49)^{2}$$
$67$ $$(T^{2} + 10 T + 100)^{2}$$
$71$ $$(T^{2} - 108)^{2}$$
$73$ $$(T + 7)^{4}$$
$79$ $$(T^{2} - 2 T + 4)^{2}$$
$83$ $$T^{4} + 192 T^{2} + 36864$$
$89$ $$(T^{2} - 27)^{2}$$
$97$ $$(T^{2} + 2 T + 4)^{2}$$