# Properties

 Label 1224.1.bu.a Level $1224$ Weight $1$ Character orbit 1224.bu Analytic conductor $0.611$ Analytic rank $0$ Dimension $4$ Projective image $D_{8}$ RM discriminant 8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1224 = 2^{3} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1224.bu (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.610855575463$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{8}$$ Projective field Galois closure of 8.0.153158089019904.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( 1 - \zeta_{8}^{3} ) q^{7} -\zeta_{8}^{3} q^{8} +O(q^{10})$$ $$q -\zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( 1 - \zeta_{8}^{3} ) q^{7} -\zeta_{8}^{3} q^{8} + ( -1 - \zeta_{8} ) q^{14} - q^{16} + \zeta_{8}^{2} q^{17} + ( \zeta_{8} - \zeta_{8}^{2} ) q^{23} -\zeta_{8} q^{25} + ( \zeta_{8} + \zeta_{8}^{2} ) q^{28} + ( \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{31} + \zeta_{8} q^{32} -\zeta_{8}^{3} q^{34} + ( -\zeta_{8} - \zeta_{8}^{2} ) q^{41} + ( -\zeta_{8}^{2} + \zeta_{8}^{3} ) q^{46} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{47} + ( 1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{49} + \zeta_{8}^{2} q^{50} + ( -\zeta_{8}^{2} - \zeta_{8}^{3} ) q^{56} + ( 1 - \zeta_{8}^{3} ) q^{62} -\zeta_{8}^{2} q^{64} - q^{68} + ( 1 + \zeta_{8} ) q^{71} + ( -\zeta_{8}^{2} + \zeta_{8}^{3} ) q^{73} + ( -1 - \zeta_{8}^{3} ) q^{79} + ( \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{82} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{89} + ( 1 + \zeta_{8}^{3} ) q^{92} + ( -1 - \zeta_{8}^{2} ) q^{94} + ( -\zeta_{8}^{2} + \zeta_{8}^{3} ) q^{97} + ( -1 - \zeta_{8} + \zeta_{8}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} + O(q^{10})$$ $$4q + 4q^{7} - 4q^{14} - 4q^{16} + 4q^{49} + 4q^{62} - 4q^{68} + 4q^{71} - 4q^{79} + 4q^{92} - 4q^{94} - 4q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$$.

 $$n$$ $$137$$ $$613$$ $$649$$ $$919$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{8}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
0.707107 0.707107i 0 1.00000i 0 0 0.292893 0.707107i −0.707107 0.707107i 0 0
485.1 0.707107 + 0.707107i 0 1.00000i 0 0 0.292893 + 0.707107i −0.707107 + 0.707107i 0 0
773.1 −0.707107 0.707107i 0 1.00000i 0 0 1.70711 0.707107i 0.707107 0.707107i 0 0
1205.1 −0.707107 + 0.707107i 0 1.00000i 0 0 1.70711 + 0.707107i 0.707107 + 0.707107i 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
8.b even 2 1 RM by $$\Q(\sqrt{2})$$
51.g odd 8 1 inner
408.be odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.bu.a 4
3.b odd 2 1 1224.1.bu.b yes 4
8.b even 2 1 RM 1224.1.bu.a 4
17.d even 8 1 1224.1.bu.b yes 4
24.h odd 2 1 1224.1.bu.b yes 4
51.g odd 8 1 inner 1224.1.bu.a 4
136.o even 8 1 1224.1.bu.b yes 4
408.be odd 8 1 inner 1224.1.bu.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.bu.a 4 1.a even 1 1 trivial
1224.1.bu.a 4 8.b even 2 1 RM
1224.1.bu.a 4 51.g odd 8 1 inner
1224.1.bu.a 4 408.be odd 8 1 inner
1224.1.bu.b yes 4 3.b odd 2 1
1224.1.bu.b yes 4 17.d even 8 1
1224.1.bu.b yes 4 24.h odd 2 1
1224.1.bu.b yes 4 136.o even 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23}^{4} + 2 T_{23}^{2} - 4 T_{23} + 2$$ acting on $$S_{1}^{\mathrm{new}}(1224, [\chi])$$.

## Hecke characteristic polynomials

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