# Properties

 Label 1296.1.x.a Level $1296$ Weight $1$ Character orbit 1296.x Analytic conductor $0.647$ Analytic rank $0$ Dimension $8$ Projective image $S_{4}$ CM/RM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1296.x (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.646788256372$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 432) Projective image: $$S_{4}$$ Projective field: Galois closure of 4.2.55296.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} -\zeta_{24}^{10} q^{7} -\zeta_{24}^{3} q^{8} +O(q^{10})$$ $$q -\zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} -\zeta_{24}^{10} q^{7} -\zeta_{24}^{3} q^{8} -\zeta_{24}^{6} q^{10} + \zeta_{24} q^{11} + ( -\zeta_{24}^{2} - \zeta_{24}^{8} ) q^{13} + \zeta_{24}^{11} q^{14} + \zeta_{24}^{4} q^{16} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{17} + \zeta_{24}^{7} q^{20} -\zeta_{24}^{2} q^{22} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{26} + q^{28} + \zeta_{24}^{8} q^{31} -\zeta_{24}^{5} q^{32} + ( \zeta_{24}^{4} + \zeta_{24}^{10} ) q^{34} + \zeta_{24}^{3} q^{35} -\zeta_{24}^{8} q^{40} + ( \zeta_{24}^{4} + \zeta_{24}^{10} ) q^{43} + \zeta_{24}^{3} q^{44} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{47} + ( -\zeta_{24}^{4} - \zeta_{24}^{10} ) q^{52} + \zeta_{24}^{9} q^{53} + \zeta_{24}^{6} q^{55} -\zeta_{24} q^{56} -\zeta_{24}^{9} q^{62} + \zeta_{24}^{6} q^{64} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{65} + ( \zeta_{24}^{2} + \zeta_{24}^{8} ) q^{67} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{68} -\zeta_{24}^{4} q^{70} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{71} -\zeta_{24}^{6} q^{73} -\zeta_{24}^{11} q^{77} + \zeta_{24}^{9} q^{80} + \zeta_{24}^{7} q^{83} + ( \zeta_{24}^{2} - \zeta_{24}^{8} ) q^{85} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{86} -\zeta_{24}^{4} q^{88} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{89} + ( -1 - \zeta_{24}^{6} ) q^{91} + ( -\zeta_{24}^{2} + \zeta_{24}^{8} ) q^{94} -\zeta_{24}^{4} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 4 q^{13} + 4 q^{16} + 8 q^{28} - 4 q^{31} + 4 q^{34} + 4 q^{40} + 4 q^{43} - 4 q^{52} - 4 q^{67} - 4 q^{70} + 4 q^{85} - 4 q^{88} - 8 q^{91} - 4 q^{94} - 4 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$\zeta_{24}^{6}$$ $$1$$ $$-\zeta_{24}^{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 0.965926 + 0.258819i −0.965926 − 0.258819i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.258819 + 0.965926i −0.258819 − 0.965926i
−0.965926 0.258819i 0 0.866025 + 0.500000i 0.258819 + 0.965926i 0 0.866025 0.500000i −0.707107 0.707107i 0 1.00000i
53.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.258819 0.965926i 0 0.866025 0.500000i 0.707107 + 0.707107i 0 1.00000i
269.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 0.258819 0.965926i 0 0.866025 + 0.500000i −0.707107 + 0.707107i 0 1.00000i
269.2 0.965926 0.258819i 0 0.866025 0.500000i −0.258819 + 0.965926i 0 0.866025 + 0.500000i 0.707107 0.707107i 0 1.00000i
701.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0.965926 0.258819i 0 −0.866025 + 0.500000i 0.707107 0.707107i 0 1.00000i
701.2 0.258819 0.965926i 0 −0.866025 0.500000i −0.965926 + 0.258819i 0 −0.866025 + 0.500000i −0.707107 + 0.707107i 0 1.00000i
917.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0.965926 + 0.258819i 0 −0.866025 0.500000i 0.707107 + 0.707107i 0 1.00000i
917.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.965926 0.258819i 0 −0.866025 0.500000i −0.707107 0.707107i 0 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 917.2 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
9.c even 3 1 inner
9.d odd 6 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner
144.w odd 12 1 inner
144.x even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.1.x.a 8
3.b odd 2 1 inner 1296.1.x.a 8
9.c even 3 1 432.1.j.a 4
9.c even 3 1 inner 1296.1.x.a 8
9.d odd 6 1 432.1.j.a 4
9.d odd 6 1 inner 1296.1.x.a 8
16.e even 4 1 inner 1296.1.x.a 8
36.f odd 6 1 1728.1.j.a 4
36.h even 6 1 1728.1.j.a 4
48.i odd 4 1 inner 1296.1.x.a 8
72.j odd 6 1 3456.1.j.b 4
72.l even 6 1 3456.1.j.a 4
72.n even 6 1 3456.1.j.b 4
72.p odd 6 1 3456.1.j.a 4
144.u even 12 1 1728.1.j.a 4
144.u even 12 1 3456.1.j.a 4
144.v odd 12 1 1728.1.j.a 4
144.v odd 12 1 3456.1.j.a 4
144.w odd 12 1 432.1.j.a 4
144.w odd 12 1 inner 1296.1.x.a 8
144.w odd 12 1 3456.1.j.b 4
144.x even 12 1 432.1.j.a 4
144.x even 12 1 inner 1296.1.x.a 8
144.x even 12 1 3456.1.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.1.j.a 4 9.c even 3 1
432.1.j.a 4 9.d odd 6 1
432.1.j.a 4 144.w odd 12 1
432.1.j.a 4 144.x even 12 1
1296.1.x.a 8 1.a even 1 1 trivial
1296.1.x.a 8 3.b odd 2 1 inner
1296.1.x.a 8 9.c even 3 1 inner
1296.1.x.a 8 9.d odd 6 1 inner
1296.1.x.a 8 16.e even 4 1 inner
1296.1.x.a 8 48.i odd 4 1 inner
1296.1.x.a 8 144.w odd 12 1 inner
1296.1.x.a 8 144.x even 12 1 inner
1728.1.j.a 4 36.f odd 6 1
1728.1.j.a 4 36.h even 6 1
1728.1.j.a 4 144.u even 12 1
1728.1.j.a 4 144.v odd 12 1
3456.1.j.a 4 72.l even 6 1
3456.1.j.a 4 72.p odd 6 1
3456.1.j.a 4 144.u even 12 1
3456.1.j.a 4 144.v odd 12 1
3456.1.j.b 4 72.j odd 6 1
3456.1.j.b 4 72.n even 6 1
3456.1.j.b 4 144.w odd 12 1
3456.1.j.b 4 144.x even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1296, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$1 - T^{4} + T^{8}$$
$7$ $$( 1 - T^{2} + T^{4} )^{2}$$
$11$ $$1 - T^{4} + T^{8}$$
$13$ $$( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$17$ $$( 2 + T^{2} )^{4}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$( 1 + T + T^{2} )^{4}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$47$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$53$ $$( 1 + T^{4} )^{2}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$71$ $$( -2 + T^{2} )^{4}$$
$73$ $$( 1 + T^{2} )^{4}$$
$79$ $$T^{8}$$
$83$ $$1 - T^{4} + T^{8}$$
$89$ $$( -2 + T^{2} )^{4}$$
$97$ $$( 1 + T + T^{2} )^{4}$$