# Properties

 Label 1296.2.c.b Level $1296$ Weight $2$ Character orbit 1296.c 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.c (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 144) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 4 \zeta_{6} ) q^{5} + ( 2 - 4 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 - 4 \zeta_{6} ) q^{5} + ( 2 - 4 \zeta_{6} ) q^{7} -3 q^{11} + 4 q^{13} + ( 1 - 2 \zeta_{6} ) q^{17} + ( 1 - 2 \zeta_{6} ) q^{19} -7 q^{25} + ( -2 + 4 \zeta_{6} ) q^{29} -12 q^{35} + 2 q^{37} + ( 3 - 6 \zeta_{6} ) q^{41} + ( -3 + 6 \zeta_{6} ) q^{43} -12 q^{47} -5 q^{49} + ( -6 + 12 \zeta_{6} ) q^{55} -15 q^{59} + 8 q^{61} + ( 8 - 16 \zeta_{6} ) q^{65} + ( -5 + 10 \zeta_{6} ) q^{67} + 6 q^{71} -11 q^{73} + ( -6 + 12 \zeta_{6} ) q^{77} + ( -2 + 4 \zeta_{6} ) q^{79} + 12 q^{83} -6 q^{85} + ( 8 - 16 \zeta_{6} ) q^{89} + ( 8 - 16 \zeta_{6} ) q^{91} -6 q^{95} + 13 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 6q^{11} + 8q^{13} - 14q^{25} - 24q^{35} + 4q^{37} - 24q^{47} - 10q^{49} - 30q^{59} + 16q^{61} + 12q^{71} - 22q^{73} + 24q^{83} - 12q^{85} - 12q^{95} + 26q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1295.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 3.46410i 0 3.46410i 0 0 0
1295.2 0 0 0 3.46410i 0 3.46410i 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
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.c.b 2
3.b odd 2 1 1296.2.c.d 2
4.b odd 2 1 1296.2.c.d 2
8.b even 2 1 5184.2.c.c 2
8.d odd 2 1 5184.2.c.a 2
9.c even 3 1 144.2.s.d yes 2
9.c even 3 1 432.2.s.c 2
9.d odd 6 1 144.2.s.a 2
9.d odd 6 1 432.2.s.d 2
12.b even 2 1 inner 1296.2.c.b 2
24.f even 2 1 5184.2.c.c 2
24.h odd 2 1 5184.2.c.a 2
36.f odd 6 1 144.2.s.a 2
36.f odd 6 1 432.2.s.d 2
36.h even 6 1 144.2.s.d yes 2
36.h even 6 1 432.2.s.c 2
72.j odd 6 1 576.2.s.d 2
72.j odd 6 1 1728.2.s.b 2
72.l even 6 1 576.2.s.a 2
72.l even 6 1 1728.2.s.a 2
72.n even 6 1 576.2.s.a 2
72.n even 6 1 1728.2.s.a 2
72.p odd 6 1 576.2.s.d 2
72.p odd 6 1 1728.2.s.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.a 2 9.d odd 6 1
144.2.s.a 2 36.f odd 6 1
144.2.s.d yes 2 9.c even 3 1
144.2.s.d yes 2 36.h even 6 1
432.2.s.c 2 9.c even 3 1
432.2.s.c 2 36.h even 6 1
432.2.s.d 2 9.d odd 6 1
432.2.s.d 2 36.f odd 6 1
576.2.s.a 2 72.l even 6 1
576.2.s.a 2 72.n even 6 1
576.2.s.d 2 72.j odd 6 1
576.2.s.d 2 72.p odd 6 1
1296.2.c.b 2 1.a even 1 1 trivial
1296.2.c.b 2 12.b even 2 1 inner
1296.2.c.d 2 3.b odd 2 1
1296.2.c.d 2 4.b odd 2 1
1728.2.s.a 2 72.l even 6 1
1728.2.s.a 2 72.n even 6 1
1728.2.s.b 2 72.j odd 6 1
1728.2.s.b 2 72.p odd 6 1
5184.2.c.a 2 8.d odd 2 1
5184.2.c.a 2 24.h odd 2 1
5184.2.c.c 2 8.b even 2 1
5184.2.c.c 2 24.f even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 12$$ $$T_{7}^{2} + 12$$ $$T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$12 + T^{2}$$
$7$ $$12 + T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$3 + T^{2}$$
$19$ $$3 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$12 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$27 + T^{2}$$
$43$ $$27 + T^{2}$$
$47$ $$( 12 + T )^{2}$$
$53$ $$T^{2}$$
$59$ $$( 15 + T )^{2}$$
$61$ $$( -8 + T )^{2}$$
$67$ $$75 + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$( 11 + T )^{2}$$
$79$ $$12 + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$192 + T^{2}$$
$97$ $$( -13 + T )^{2}$$