# Properties

 Label 1296.2.s.f Level $1296$ Weight $2$ Character orbit 1296.s Analytic conductor $10.349$ Analytic rank $0$ Dimension $2$ CM discriminant -3 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 432) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 3 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 3 + 3 \zeta_{6} ) q^{7} + 5 \zeta_{6} q^{13} + ( 3 - 6 \zeta_{6} ) q^{19} + ( -5 + 5 \zeta_{6} ) q^{25} + ( -12 + 6 \zeta_{6} ) q^{31} + 11 q^{37} + ( -6 - 6 \zeta_{6} ) q^{43} + 20 \zeta_{6} q^{49} + ( 1 - \zeta_{6} ) q^{61} + ( 18 - 9 \zeta_{6} ) q^{67} + 7 q^{73} + ( 3 + 3 \zeta_{6} ) q^{79} + ( -15 + 30 \zeta_{6} ) q^{91} + ( -19 + 19 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 9q^{7} + O(q^{10})$$ $$2q + 9q^{7} + 5q^{13} - 5q^{25} - 18q^{31} + 22q^{37} - 18q^{43} + 20q^{49} + q^{61} + 27q^{67} + 14q^{73} + 9q^{79} - 19q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 4.50000 + 2.59808i 0 0 0
863.1 0 0 0 0 0 4.50000 2.59808i 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 CM by $$\Q(\sqrt{-3})$$
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.f 2
3.b odd 2 1 CM 1296.2.s.f 2
4.b odd 2 1 1296.2.s.a 2
9.c even 3 1 432.2.c.a 2
9.c even 3 1 1296.2.s.a 2
9.d odd 6 1 432.2.c.a 2
9.d odd 6 1 1296.2.s.a 2
12.b even 2 1 1296.2.s.a 2
36.f odd 6 1 432.2.c.a 2
36.f odd 6 1 inner 1296.2.s.f 2
36.h even 6 1 432.2.c.a 2
36.h even 6 1 inner 1296.2.s.f 2
72.j odd 6 1 1728.2.c.b 2
72.l even 6 1 1728.2.c.b 2
72.n even 6 1 1728.2.c.b 2
72.p odd 6 1 1728.2.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.c.a 2 9.c even 3 1
432.2.c.a 2 9.d odd 6 1
432.2.c.a 2 36.f odd 6 1
432.2.c.a 2 36.h even 6 1
1296.2.s.a 2 4.b odd 2 1
1296.2.s.a 2 9.c even 3 1
1296.2.s.a 2 9.d odd 6 1
1296.2.s.a 2 12.b even 2 1
1296.2.s.f 2 1.a even 1 1 trivial
1296.2.s.f 2 3.b odd 2 1 CM
1296.2.s.f 2 36.f odd 6 1 inner
1296.2.s.f 2 36.h even 6 1 inner
1728.2.c.b 2 72.j odd 6 1
1728.2.c.b 2 72.l even 6 1
1728.2.c.b 2 72.n even 6 1
1728.2.c.b 2 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 $$S_{2}^{\mathrm{new}}(1296, [\chi])$$:

 $$T_{5}$$ $$T_{7}^{2} - 9 T_{7} + 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$27 - 9 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$25 - 5 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$27 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$108 + 18 T + T^{2}$$
$37$ $$( -11 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$108 + 18 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$243 - 27 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -7 + T )^{2}$$
$79$ $$27 - 9 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$361 + 19 T + T^{2}$$