# Properties

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

# Related objects

## Newspace parameters

 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

 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 216) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{5} + ( 3 - 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + \zeta_{6} q^{5} + ( 3 - 3 \zeta_{6} ) q^{7} + ( 5 - 5 \zeta_{6} ) q^{11} -4 \zeta_{6} q^{13} -8 q^{17} -2 q^{19} + 2 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + ( -6 + 6 \zeta_{6} ) q^{29} -7 \zeta_{6} q^{31} + 3 q^{35} -6 q^{37} + 6 \zeta_{6} q^{41} + ( -2 + 2 \zeta_{6} ) q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} -2 \zeta_{6} q^{49} + 5 q^{53} + 5 q^{55} -4 \zeta_{6} q^{59} + ( 8 - 8 \zeta_{6} ) q^{61} + ( 4 - 4 \zeta_{6} ) q^{65} -10 \zeta_{6} q^{67} + 8 q^{71} + q^{73} -15 \zeta_{6} q^{77} + ( 16 - 16 \zeta_{6} ) q^{79} + ( -11 + 11 \zeta_{6} ) q^{83} -8 \zeta_{6} q^{85} + 6 q^{89} -12 q^{91} -2 \zeta_{6} q^{95} + ( 1 - \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{5} + 3 q^{7} + O(q^{10})$$ $$2 q + q^{5} + 3 q^{7} + 5 q^{11} - 4 q^{13} - 16 q^{17} - 4 q^{19} + 2 q^{23} + 4 q^{25} - 6 q^{29} - 7 q^{31} + 6 q^{35} - 12 q^{37} + 6 q^{41} - 2 q^{43} + 6 q^{47} - 2 q^{49} + 10 q^{53} + 10 q^{55} - 4 q^{59} + 8 q^{61} + 4 q^{65} - 10 q^{67} + 16 q^{71} + 2 q^{73} - 15 q^{77} + 16 q^{79} - 11 q^{83} - 8 q^{85} + 12 q^{89} - 24 q^{91} - 2 q^{95} + q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0.500000 0.866025i 0 1.50000 + 2.59808i 0 0 0
865.1 0 0 0 0.500000 + 0.866025i 0 1.50000 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.i.l 2
3.b odd 2 1 1296.2.i.g 2
4.b odd 2 1 648.2.i.e 2
9.c even 3 1 432.2.a.c 1
9.c even 3 1 inner 1296.2.i.l 2
9.d odd 6 1 432.2.a.f 1
9.d odd 6 1 1296.2.i.g 2
12.b even 2 1 648.2.i.c 2
36.f odd 6 1 216.2.a.b 1
36.f odd 6 1 648.2.i.e 2
36.h even 6 1 216.2.a.c yes 1
36.h even 6 1 648.2.i.c 2
72.j odd 6 1 1728.2.a.i 1
72.l even 6 1 1728.2.a.l 1
72.n even 6 1 1728.2.a.r 1
72.p odd 6 1 1728.2.a.s 1
180.n even 6 1 5400.2.a.e 1
180.p odd 6 1 5400.2.a.h 1
180.v odd 12 2 5400.2.f.b 2
180.x even 12 2 5400.2.f.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.a.b 1 36.f odd 6 1
216.2.a.c yes 1 36.h even 6 1
432.2.a.c 1 9.c even 3 1
432.2.a.f 1 9.d odd 6 1
648.2.i.c 2 12.b even 2 1
648.2.i.c 2 36.h even 6 1
648.2.i.e 2 4.b odd 2 1
648.2.i.e 2 36.f odd 6 1
1296.2.i.g 2 3.b odd 2 1
1296.2.i.g 2 9.d odd 6 1
1296.2.i.l 2 1.a even 1 1 trivial
1296.2.i.l 2 9.c even 3 1 inner
1728.2.a.i 1 72.j odd 6 1
1728.2.a.l 1 72.l even 6 1
1728.2.a.r 1 72.n even 6 1
1728.2.a.s 1 72.p odd 6 1
5400.2.a.e 1 180.n even 6 1
5400.2.a.h 1 180.p odd 6 1
5400.2.f.b 2 180.v odd 12 2
5400.2.f.z 2 180.x even 12 2

## Hecke kernels

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

 $$T_{5}^{2} - T_{5} + 1$$ $$T_{7}^{2} - 3 T_{7} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$9 - 3 T + T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$16 + 4 T + T^{2}$$
$17$ $$( 8 + T )^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$4 - 2 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$49 + 7 T + T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$36 - 6 T + T^{2}$$
$43$ $$4 + 2 T + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$( -5 + T )^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$64 - 8 T + T^{2}$$
$67$ $$100 + 10 T + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$( -1 + T )^{2}$$
$79$ $$256 - 16 T + T^{2}$$
$83$ $$121 + 11 T + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$1 - T + T^{2}$$