# Properties

 Label 3456.1.j.a Level $3456$ Weight $1$ Character orbit 3456.j Analytic conductor $1.725$ Analytic rank $0$ Dimension $4$ Projective image $S_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3456 = 2^{7} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3456.j (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.72476868366$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{5} + \zeta_{8}^{2} q^{7} +O(q^{10})$$ $$q + \zeta_{8} q^{5} + \zeta_{8}^{2} q^{7} + \zeta_{8} q^{11} + ( 1 + \zeta_{8}^{2} ) q^{13} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{17} - q^{31} + \zeta_{8}^{3} q^{35} + ( -1 + \zeta_{8}^{2} ) q^{43} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{47} -\zeta_{8} q^{53} + \zeta_{8}^{2} q^{55} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{65} + ( 1 + \zeta_{8}^{2} ) q^{67} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{71} + \zeta_{8}^{2} q^{73} + \zeta_{8}^{3} q^{77} -\zeta_{8}^{3} q^{83} + ( 1 - \zeta_{8}^{2} ) q^{85} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{89} + ( -1 + \zeta_{8}^{2} ) q^{91} + q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q + 4 q^{13} - 4 q^{31} - 4 q^{43} + 4 q^{67} + 4 q^{85} - 4 q^{91} + 4 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$$.

 $$n$$ $$2053$$ $$2431$$ $$2945$$ $$\chi(n)$$ $$-\zeta_{8}^{2}$$ $$1$$ $$-1$$

## 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
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 0 0 −0.707107 0.707107i 0 1.00000i 0 0 0
161.2 0 0 0 0.707107 + 0.707107i 0 1.00000i 0 0 0
1889.1 0 0 0 −0.707107 + 0.707107i 0 1.00000i 0 0 0
1889.2 0 0 0 0.707107 0.707107i 0 1.00000i 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
16.e even 4 1 inner
48.i odd 4 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.1.j.a 4 8.d odd 2 1
432.1.j.a 4 16.f odd 4 1
432.1.j.a 4 24.f even 2 1
432.1.j.a 4 48.k even 4 1
1296.1.x.a 8 72.l even 6 2
1296.1.x.a 8 72.p odd 6 2
1296.1.x.a 8 144.u even 12 2
1296.1.x.a 8 144.v odd 12 2
1728.1.j.a 4 8.b even 2 1
1728.1.j.a 4 16.e even 4 1
1728.1.j.a 4 24.h odd 2 1
1728.1.j.a 4 48.i odd 4 1
3456.1.j.a 4 1.a even 1 1 trivial
3456.1.j.a 4 3.b odd 2 1 inner
3456.1.j.a 4 16.e even 4 1 inner
3456.1.j.a 4 48.i odd 4 1 inner
3456.1.j.b 4 4.b odd 2 1
3456.1.j.b 4 12.b even 2 1
3456.1.j.b 4 16.f odd 4 1
3456.1.j.b 4 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{31} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3456, [\chi])$$.

## Hecke characteristic polynomials

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