# Properties

 Label 1296.3.e.a Level $1296$ Weight $3$ Character orbit 1296.e Analytic conductor $35.313$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1296.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.3134422611$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 9) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 4 \zeta_{6} ) q^{5} -2 q^{7} +O(q^{10})$$ $$q + ( 2 - 4 \zeta_{6} ) q^{5} -2 q^{7} + ( -1 + 2 \zeta_{6} ) q^{11} -4 q^{13} + ( -9 + 18 \zeta_{6} ) q^{17} -11 q^{19} + ( 16 - 32 \zeta_{6} ) q^{23} + 13 q^{25} + ( -26 + 52 \zeta_{6} ) q^{29} -32 q^{31} + ( -4 + 8 \zeta_{6} ) q^{35} -34 q^{37} + ( -7 + 14 \zeta_{6} ) q^{41} + 61 q^{43} + ( -28 + 56 \zeta_{6} ) q^{47} -45 q^{49} + 6 q^{55} + ( -29 + 58 \zeta_{6} ) q^{59} + 56 q^{61} + ( -8 + 16 \zeta_{6} ) q^{65} + 31 q^{67} + ( -18 + 36 \zeta_{6} ) q^{71} + 65 q^{73} + ( 2 - 4 \zeta_{6} ) q^{77} -38 q^{79} + ( -28 + 56 \zeta_{6} ) q^{83} + 54 q^{85} + ( -72 + 144 \zeta_{6} ) q^{89} + 8 q^{91} + ( -22 + 44 \zeta_{6} ) q^{95} -115 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7} + O(q^{10})$$ $$2 q - 4 q^{7} - 8 q^{13} - 22 q^{19} + 26 q^{25} - 64 q^{31} - 68 q^{37} + 122 q^{43} - 90 q^{49} + 12 q^{55} + 112 q^{61} + 62 q^{67} + 130 q^{73} - 76 q^{79} + 108 q^{85} + 16 q^{91} - 230 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 3.46410i 0 −2.00000 0 0 0
161.2 0 0 0 3.46410i 0 −2.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.e.a 2
3.b odd 2 1 inner 1296.3.e.a 2
4.b odd 2 1 81.3.b.a 2
9.c even 3 1 144.3.q.a 2
9.c even 3 1 432.3.q.a 2
9.d odd 6 1 144.3.q.a 2
9.d odd 6 1 432.3.q.a 2
12.b even 2 1 81.3.b.a 2
36.f odd 6 1 9.3.d.a 2
36.f odd 6 1 27.3.d.a 2
36.h even 6 1 9.3.d.a 2
36.h even 6 1 27.3.d.a 2
72.j odd 6 1 576.3.q.a 2
72.j odd 6 1 1728.3.q.b 2
72.l even 6 1 576.3.q.b 2
72.l even 6 1 1728.3.q.a 2
72.n even 6 1 576.3.q.a 2
72.n even 6 1 1728.3.q.b 2
72.p odd 6 1 576.3.q.b 2
72.p odd 6 1 1728.3.q.a 2
180.n even 6 1 225.3.j.a 2
180.n even 6 1 675.3.j.a 2
180.p odd 6 1 225.3.j.a 2
180.p odd 6 1 675.3.j.a 2
180.v odd 12 2 225.3.i.a 4
180.v odd 12 2 675.3.i.a 4
180.x even 12 2 225.3.i.a 4
180.x even 12 2 675.3.i.a 4
252.n even 6 1 441.3.n.a 2
252.o even 6 1 441.3.n.b 2
252.r odd 6 1 441.3.j.b 2
252.s odd 6 1 441.3.r.a 2
252.u odd 6 1 441.3.j.a 2
252.bb even 6 1 441.3.j.a 2
252.bi even 6 1 441.3.r.a 2
252.bj even 6 1 441.3.j.b 2
252.bl odd 6 1 441.3.n.b 2
252.bn odd 6 1 441.3.n.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 36.f odd 6 1
9.3.d.a 2 36.h even 6 1
27.3.d.a 2 36.f odd 6 1
27.3.d.a 2 36.h even 6 1
81.3.b.a 2 4.b odd 2 1
81.3.b.a 2 12.b even 2 1
144.3.q.a 2 9.c even 3 1
144.3.q.a 2 9.d odd 6 1
225.3.i.a 4 180.v odd 12 2
225.3.i.a 4 180.x even 12 2
225.3.j.a 2 180.n even 6 1
225.3.j.a 2 180.p odd 6 1
432.3.q.a 2 9.c even 3 1
432.3.q.a 2 9.d odd 6 1
441.3.j.a 2 252.u odd 6 1
441.3.j.a 2 252.bb even 6 1
441.3.j.b 2 252.r odd 6 1
441.3.j.b 2 252.bj even 6 1
441.3.n.a 2 252.n even 6 1
441.3.n.a 2 252.bn odd 6 1
441.3.n.b 2 252.o even 6 1
441.3.n.b 2 252.bl odd 6 1
441.3.r.a 2 252.s odd 6 1
441.3.r.a 2 252.bi even 6 1
576.3.q.a 2 72.j odd 6 1
576.3.q.a 2 72.n even 6 1
576.3.q.b 2 72.l even 6 1
576.3.q.b 2 72.p odd 6 1
675.3.i.a 4 180.v odd 12 2
675.3.i.a 4 180.x even 12 2
675.3.j.a 2 180.n even 6 1
675.3.j.a 2 180.p odd 6 1
1296.3.e.a 2 1.a even 1 1 trivial
1296.3.e.a 2 3.b odd 2 1 inner
1728.3.q.a 2 72.l even 6 1
1728.3.q.a 2 72.p odd 6 1
1728.3.q.b 2 72.j odd 6 1
1728.3.q.b 2 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1296, [\chi])$$:

 $$T_{5}^{2} + 12$$ $$T_{7} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$12 + T^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$3 + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$243 + T^{2}$$
$19$ $$( 11 + T )^{2}$$
$23$ $$768 + T^{2}$$
$29$ $$2028 + T^{2}$$
$31$ $$( 32 + T )^{2}$$
$37$ $$( 34 + T )^{2}$$
$41$ $$147 + T^{2}$$
$43$ $$( -61 + T )^{2}$$
$47$ $$2352 + T^{2}$$
$53$ $$T^{2}$$
$59$ $$2523 + T^{2}$$
$61$ $$( -56 + T )^{2}$$
$67$ $$( -31 + T )^{2}$$
$71$ $$972 + T^{2}$$
$73$ $$( -65 + T )^{2}$$
$79$ $$( 38 + T )^{2}$$
$83$ $$2352 + T^{2}$$
$89$ $$15552 + T^{2}$$
$97$ $$( 115 + T )^{2}$$