# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1296.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3486121020$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 432) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} + ( - \beta_{2} + 2) q^{7}+O(q^{10})$$ q + b1 * q^5 + (-b2 + 2) * q^7 $$q + \beta_1 q^{5} + ( - \beta_{2} + 2) q^{7} + ( - \beta_{3} - \beta_1) q^{11} + ( - 2 \beta_{2} + 2) q^{13} - 2 \beta_{3} q^{17} + ( - 8 \beta_{2} + 4) q^{19} + 4 \beta_{2} q^{25} + (2 \beta_{3} - 2 \beta_1) q^{29} + ( - 3 \beta_{2} - 3) q^{31} + ( - \beta_{3} + 2 \beta_1) q^{35} + 8 q^{37} + ( - 6 \beta_{2} + 12) q^{43} + (2 \beta_{3} + 2 \beta_1) q^{47} + (4 \beta_{2} - 4) q^{49} + 3 \beta_{3} q^{53} + ( - 18 \beta_{2} + 9) q^{55} + (4 \beta_{3} - 2 \beta_1) q^{59} + 4 \beta_{2} q^{61} + ( - 2 \beta_{3} + 2 \beta_1) q^{65} + ( - 2 \beta_{2} - 2) q^{67} + (2 \beta_{3} - 4 \beta_1) q^{71} + q^{73} - 3 \beta_1 q^{77} + ( - 2 \beta_{2} + 4) q^{79} + (\beta_{3} + \beta_1) q^{83} + ( - 18 \beta_{2} + 18) q^{85} + 2 \beta_{3} q^{89} + ( - 4 \beta_{2} + 2) q^{91} + ( - 8 \beta_{3} + 4 \beta_1) q^{95} + 5 \beta_{2} q^{97}+O(q^{100})$$ q + b1 * q^5 + (-b2 + 2) * q^7 + (-b3 - b1) * q^11 + (-2*b2 + 2) * q^13 - 2*b3 * q^17 + (-8*b2 + 4) * q^19 + 4*b2 * q^25 + (2*b3 - 2*b1) * q^29 + (-3*b2 - 3) * q^31 + (-b3 + 2*b1) * q^35 + 8 * q^37 + (-6*b2 + 12) * q^43 + (2*b3 + 2*b1) * q^47 + (4*b2 - 4) * q^49 + 3*b3 * q^53 + (-18*b2 + 9) * q^55 + (4*b3 - 2*b1) * q^59 + 4*b2 * q^61 + (-2*b3 + 2*b1) * q^65 + (-2*b2 - 2) * q^67 + (2*b3 - 4*b1) * q^71 + q^73 - 3*b1 * q^77 + (-2*b2 + 4) * q^79 + (b3 + b1) * q^83 + (-18*b2 + 18) * q^85 + 2*b3 * q^89 + (-4*b2 + 2) * q^91 + (-8*b3 + 4*b1) * q^95 + 5*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{7}+O(q^{10})$$ 4 * q + 6 * q^7 $$4 q + 6 q^{7} + 4 q^{13} + 8 q^{25} - 18 q^{31} + 32 q^{37} + 36 q^{43} - 8 q^{49} + 8 q^{61} - 12 q^{67} + 4 q^{73} + 12 q^{79} + 36 q^{85} + 10 q^{97}+O(q^{100})$$ 4 * q + 6 * q^7 + 4 * q^13 + 8 * q^25 - 18 * q^31 + 32 * q^37 + 36 * q^43 - 8 * q^49 + 8 * q^61 - 12 * q^67 + 4 * q^73 + 12 * q^79 + 36 * q^85 + 10 * q^97

Basis of coefficient ring

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

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1 - \beta_{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
431.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
0 0 0 −2.59808 + 1.50000i 0 1.50000 + 0.866025i 0 0 0
431.2 0 0 0 2.59808 1.50000i 0 1.50000 + 0.866025i 0 0 0
863.1 0 0 0 −2.59808 1.50000i 0 1.50000 0.866025i 0 0 0
863.2 0 0 0 2.59808 + 1.50000i 0 1.50000 0.866025i 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
36.f odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.s.i 4
3.b odd 2 1 inner 1296.2.s.i 4
4.b odd 2 1 1296.2.s.g 4
9.c even 3 1 432.2.c.c 4
9.c even 3 1 1296.2.s.g 4
9.d odd 6 1 432.2.c.c 4
9.d odd 6 1 1296.2.s.g 4
12.b even 2 1 1296.2.s.g 4
36.f odd 6 1 432.2.c.c 4
36.f odd 6 1 inner 1296.2.s.i 4
36.h even 6 1 432.2.c.c 4
36.h even 6 1 inner 1296.2.s.i 4
72.j odd 6 1 1728.2.c.e 4
72.l even 6 1 1728.2.c.e 4
72.n even 6 1 1728.2.c.e 4
72.p odd 6 1 1728.2.c.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.c.c 4 9.c even 3 1
432.2.c.c 4 9.d odd 6 1
432.2.c.c 4 36.f odd 6 1
432.2.c.c 4 36.h even 6 1
1296.2.s.g 4 4.b odd 2 1
1296.2.s.g 4 9.c even 3 1
1296.2.s.g 4 9.d odd 6 1
1296.2.s.g 4 12.b even 2 1
1296.2.s.i 4 1.a even 1 1 trivial
1296.2.s.i 4 3.b odd 2 1 inner
1296.2.s.i 4 36.f odd 6 1 inner
1296.2.s.i 4 36.h even 6 1 inner
1728.2.c.e 4 72.j odd 6 1
1728.2.c.e 4 72.l even 6 1
1728.2.c.e 4 72.n even 6 1
1728.2.c.e 4 72.p odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 9T_{5}^{2} + 81$$ T5^4 - 9*T5^2 + 81 $$T_{7}^{2} - 3T_{7} + 3$$ T7^2 - 3*T7 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 9T^{2} + 81$$
$7$ $$(T^{2} - 3 T + 3)^{2}$$
$11$ $$T^{4} + 27T^{2} + 729$$
$13$ $$(T^{2} - 2 T + 4)^{2}$$
$17$ $$(T^{2} + 36)^{2}$$
$19$ $$(T^{2} + 48)^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4} - 36T^{2} + 1296$$
$31$ $$(T^{2} + 9 T + 27)^{2}$$
$37$ $$(T - 8)^{4}$$
$41$ $$T^{4}$$
$43$ $$(T^{2} - 18 T + 108)^{2}$$
$47$ $$T^{4} + 108 T^{2} + 11664$$
$53$ $$(T^{2} + 81)^{2}$$
$59$ $$T^{4} + 108 T^{2} + 11664$$
$61$ $$(T^{2} - 4 T + 16)^{2}$$
$67$ $$(T^{2} + 6 T + 12)^{2}$$
$71$ $$(T^{2} - 108)^{2}$$
$73$ $$(T - 1)^{4}$$
$79$ $$(T^{2} - 6 T + 12)^{2}$$
$83$ $$T^{4} + 27T^{2} + 729$$
$89$ $$(T^{2} + 36)^{2}$$
$97$ $$(T^{2} - 5 T + 25)^{2}$$