# Properties

 Label 1296.1.o.b Level 1296 Weight 1 Character orbit 1296.o Analytic conductor 0.647 Analytic rank 0 Dimension 2 Projective image $$D_{2}$$ CM/RM discs -3, -4, 12 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1296.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.646788256372$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 144) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\zeta_{12})$$ Artin image $C_3\times D_4$ Artin field Galois closure of 12.6.304679870005248.2

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q +O(q^{10})$$ $$q -2 \zeta_{6}^{2} q^{13} + \zeta_{6} q^{25} + 2 q^{37} + \zeta_{6}^{2} q^{49} -2 \zeta_{6} q^{61} -2 q^{73} + 2 \zeta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 2q^{13} + q^{25} + 4q^{37} - q^{49} - 2q^{61} - 4q^{73} + 2q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{6}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 0 0 0 0
703.1 0 0 0 0 0 0 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 $$\Q(\sqrt{-3})$$
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
12.b even 2 1 RM by $$\Q(\sqrt{3})$$
9.c even 3 1 inner
9.d odd 6 1 inner
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 1296.1.o.b 2
3.b odd 2 1 CM 1296.1.o.b 2
4.b odd 2 1 CM 1296.1.o.b 2
9.c even 3 1 144.1.g.a 1
9.c even 3 1 inner 1296.1.o.b 2
9.d odd 6 1 144.1.g.a 1
9.d odd 6 1 inner 1296.1.o.b 2
12.b even 2 1 RM 1296.1.o.b 2
36.f odd 6 1 144.1.g.a 1
36.f odd 6 1 inner 1296.1.o.b 2
36.h even 6 1 144.1.g.a 1
36.h even 6 1 inner 1296.1.o.b 2
45.h odd 6 1 3600.1.e.b 1
45.j even 6 1 3600.1.e.b 1
45.k odd 12 2 3600.1.j.a 2
45.l even 12 2 3600.1.j.a 2
72.j odd 6 1 576.1.g.a 1
72.l even 6 1 576.1.g.a 1
72.n even 6 1 576.1.g.a 1
72.p odd 6 1 576.1.g.a 1
144.u even 12 2 2304.1.b.a 2
144.v odd 12 2 2304.1.b.a 2
144.w odd 12 2 2304.1.b.a 2
144.x even 12 2 2304.1.b.a 2
180.n even 6 1 3600.1.e.b 1
180.p odd 6 1 3600.1.e.b 1
180.v odd 12 2 3600.1.j.a 2
180.x even 12 2 3600.1.j.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.1.g.a 1 9.c even 3 1
144.1.g.a 1 9.d odd 6 1
144.1.g.a 1 36.f odd 6 1
144.1.g.a 1 36.h even 6 1
576.1.g.a 1 72.j odd 6 1
576.1.g.a 1 72.l even 6 1
576.1.g.a 1 72.n even 6 1
576.1.g.a 1 72.p odd 6 1
1296.1.o.b 2 1.a even 1 1 trivial
1296.1.o.b 2 3.b odd 2 1 CM
1296.1.o.b 2 4.b odd 2 1 CM
1296.1.o.b 2 9.c even 3 1 inner
1296.1.o.b 2 9.d odd 6 1 inner
1296.1.o.b 2 12.b even 2 1 RM
1296.1.o.b 2 36.f odd 6 1 inner
1296.1.o.b 2 36.h even 6 1 inner
2304.1.b.a 2 144.u even 12 2
2304.1.b.a 2 144.v odd 12 2
2304.1.b.a 2 144.w odd 12 2
2304.1.b.a 2 144.x even 12 2
3600.1.e.b 1 45.h odd 6 1
3600.1.e.b 1 45.j even 6 1
3600.1.e.b 1 180.n even 6 1
3600.1.e.b 1 180.p odd 6 1
3600.1.j.a 2 45.k odd 12 2
3600.1.j.a 2 45.l even 12 2
3600.1.j.a 2 180.v odd 12 2
3600.1.j.a 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 $$S_{1}^{\mathrm{new}}(1296, [\chi])$$:

 $$T_{5}$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - T^{2} + T^{4}$$
$7$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$11$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$13$ $$( 1 - T + T^{2} )^{2}$$
$17$ $$( 1 + T^{2} )^{2}$$
$19$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$23$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$29$ $$1 - T^{2} + T^{4}$$
$31$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$37$ $$( 1 - T )^{4}$$
$41$ $$1 - T^{2} + T^{4}$$
$43$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$47$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$53$ $$( 1 + T^{2} )^{2}$$
$59$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$61$ $$( 1 + T + T^{2} )^{2}$$
$67$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$71$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$73$ $$( 1 + T )^{4}$$
$79$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$83$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$89$ $$( 1 + T^{2} )^{2}$$
$97$ $$( 1 - T + T^{2} )^{2}$$