# Properties

 Label 1296.2.i.o 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 54) 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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} -2 q^{19} + 6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( 6 - 6 \zeta_{6} ) q^{29} + 5 \zeta_{6} q^{31} -3 q^{35} + 2 q^{37} -6 \zeta_{6} q^{41} + ( -10 + 10 \zeta_{6} ) q^{43} + ( -6 + 6 \zeta_{6} ) q^{47} + 6 \zeta_{6} q^{49} -9 q^{53} + 9 q^{55} -12 \zeta_{6} q^{59} + ( -8 + 8 \zeta_{6} ) q^{61} + ( -12 + 12 \zeta_{6} ) q^{65} + 14 \zeta_{6} q^{67} -7 q^{73} + 3 \zeta_{6} q^{77} + ( 8 - 8 \zeta_{6} ) q^{79} + ( 3 - 3 \zeta_{6} ) q^{83} + 18 q^{89} -4 q^{91} -6 \zeta_{6} q^{95} + ( 1 - \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{5} - q^{7} + O(q^{10})$$ $$2 q + 3 q^{5} - q^{7} + 3 q^{11} + 4 q^{13} - 4 q^{19} + 6 q^{23} - 4 q^{25} + 6 q^{29} + 5 q^{31} - 6 q^{35} + 4 q^{37} - 6 q^{41} - 10 q^{43} - 6 q^{47} + 6 q^{49} - 18 q^{53} + 18 q^{55} - 12 q^{59} - 8 q^{61} - 12 q^{65} + 14 q^{67} - 14 q^{73} + 3 q^{77} + 8 q^{79} + 3 q^{83} + 36 q^{89} - 8 q^{91} - 6 q^{95} + q^{97} + O(q^{100})$$

## 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 1.50000 2.59808i 0 −0.500000 0.866025i 0 0 0
865.1 0 0 0 1.50000 + 2.59808i 0 −0.500000 + 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
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.o 2
3.b odd 2 1 1296.2.i.c 2
4.b odd 2 1 162.2.c.b 2
9.c even 3 1 432.2.a.b 1
9.c even 3 1 inner 1296.2.i.o 2
9.d odd 6 1 432.2.a.g 1
9.d odd 6 1 1296.2.i.c 2
12.b even 2 1 162.2.c.c 2
36.f odd 6 1 54.2.a.b yes 1
36.f odd 6 1 162.2.c.b 2
36.h even 6 1 54.2.a.a 1
36.h even 6 1 162.2.c.c 2
72.j odd 6 1 1728.2.a.d 1
72.l even 6 1 1728.2.a.c 1
72.n even 6 1 1728.2.a.z 1
72.p odd 6 1 1728.2.a.y 1
180.n even 6 1 1350.2.a.r 1
180.p odd 6 1 1350.2.a.h 1
180.v odd 12 2 1350.2.c.b 2
180.x even 12 2 1350.2.c.k 2
252.s odd 6 1 2646.2.a.a 1
252.bi even 6 1 2646.2.a.bd 1
396.k even 6 1 6534.2.a.b 1
396.o odd 6 1 6534.2.a.bc 1
468.x even 6 1 9126.2.a.u 1
468.bg odd 6 1 9126.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 36.h even 6 1
54.2.a.b yes 1 36.f odd 6 1
162.2.c.b 2 4.b odd 2 1
162.2.c.b 2 36.f odd 6 1
162.2.c.c 2 12.b even 2 1
162.2.c.c 2 36.h even 6 1
432.2.a.b 1 9.c even 3 1
432.2.a.g 1 9.d odd 6 1
1296.2.i.c 2 3.b odd 2 1
1296.2.i.c 2 9.d odd 6 1
1296.2.i.o 2 1.a even 1 1 trivial
1296.2.i.o 2 9.c even 3 1 inner
1350.2.a.h 1 180.p odd 6 1
1350.2.a.r 1 180.n even 6 1
1350.2.c.b 2 180.v odd 12 2
1350.2.c.k 2 180.x even 12 2
1728.2.a.c 1 72.l even 6 1
1728.2.a.d 1 72.j odd 6 1
1728.2.a.y 1 72.p odd 6 1
1728.2.a.z 1 72.n even 6 1
2646.2.a.a 1 252.s odd 6 1
2646.2.a.bd 1 252.bi even 6 1
6534.2.a.b 1 396.k even 6 1
6534.2.a.bc 1 396.o odd 6 1
9126.2.a.r 1 468.bg odd 6 1
9126.2.a.u 1 468.x even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$36 - 6 T + T^{2}$$
$31$ $$25 - 5 T + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$36 + 6 T + T^{2}$$
$43$ $$100 + 10 T + T^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$( 9 + T )^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$64 + 8 T + T^{2}$$
$67$ $$196 - 14 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 7 + T )^{2}$$
$79$ $$64 - 8 T + T^{2}$$
$83$ $$9 - 3 T + T^{2}$$
$89$ $$( -18 + T )^{2}$$
$97$ $$1 - T + T^{2}$$