# Properties

 Label 1224.1.bb.b Level $1224$ Weight $1$ Character orbit 1224.bb Analytic conductor $0.611$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM 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.bb (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.610855575463$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.2.2062988352.13

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{6}^{2} q^{2} -\zeta_{6}^{2} q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{6} - q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6}^{2} q^{2} -\zeta_{6}^{2} q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{6} - q^{8} -\zeta_{6} q^{9} + ( -1 - \zeta_{6} ) q^{11} - q^{12} + \zeta_{6}^{2} q^{16} + \zeta_{6} q^{17} - q^{18} + q^{19} + ( -1 + \zeta_{6}^{2} ) q^{22} + \zeta_{6}^{2} q^{24} -\zeta_{6}^{2} q^{25} - q^{27} + \zeta_{6} q^{32} + ( -1 + \zeta_{6}^{2} ) q^{33} + q^{34} + \zeta_{6}^{2} q^{36} -\zeta_{6}^{2} q^{38} + ( 1 - \zeta_{6}^{2} ) q^{41} -\zeta_{6}^{2} q^{43} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{44} + \zeta_{6} q^{48} + \zeta_{6} q^{49} -\zeta_{6} q^{50} + q^{51} + \zeta_{6}^{2} q^{54} -\zeta_{6}^{2} q^{57} + \zeta_{6} q^{59} + q^{64} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{66} -\zeta_{6} q^{67} -\zeta_{6}^{2} q^{68} + \zeta_{6} q^{72} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{73} -\zeta_{6} q^{75} -\zeta_{6} q^{76} + \zeta_{6}^{2} q^{81} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{82} + 2 \zeta_{6}^{2} q^{83} -\zeta_{6} q^{86} + ( 1 + \zeta_{6} ) q^{88} -2 q^{89} + q^{96} + ( 1 + \zeta_{6} ) q^{97} + q^{98} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + q^{3} - q^{4} - q^{6} - 2q^{8} - q^{9} + O(q^{10})$$ $$2q + q^{2} + q^{3} - q^{4} - q^{6} - 2q^{8} - q^{9} - 3q^{11} - 2q^{12} - q^{16} + q^{17} - 2q^{18} + 2q^{19} - 3q^{22} - q^{24} + q^{25} - 2q^{27} + q^{32} - 3q^{33} + 2q^{34} - q^{36} + q^{38} + 3q^{41} + q^{43} + q^{48} + q^{49} - q^{50} + 2q^{51} - q^{54} + q^{57} + q^{59} + 2q^{64} - q^{67} + q^{68} + q^{72} - q^{75} - q^{76} - q^{81} - 2q^{83} - q^{86} + 3q^{88} - 4q^{89} + 2q^{96} + 3q^{97} + 2q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$137$$ $$613$$ $$649$$ $$919$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 −1.00000 −0.500000 + 0.866025i 0
475.1 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i 0 −1.00000 −0.500000 0.866025i 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.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
153.h even 6 1 inner
1224.bb odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.bb.b yes 2
3.b odd 2 1 3672.1.bb.b 2
8.d odd 2 1 CM 1224.1.bb.b yes 2
9.c even 3 1 1224.1.bb.a 2
9.d odd 6 1 3672.1.bb.a 2
17.b even 2 1 1224.1.bb.a 2
24.f even 2 1 3672.1.bb.b 2
51.c odd 2 1 3672.1.bb.a 2
72.l even 6 1 3672.1.bb.a 2
72.p odd 6 1 1224.1.bb.a 2
136.e odd 2 1 1224.1.bb.a 2
153.h even 6 1 inner 1224.1.bb.b yes 2
153.i odd 6 1 3672.1.bb.b 2
408.h even 2 1 3672.1.bb.a 2
1224.bb odd 6 1 inner 1224.1.bb.b yes 2
1224.bh even 6 1 3672.1.bb.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.bb.a 2 9.c even 3 1
1224.1.bb.a 2 17.b even 2 1
1224.1.bb.a 2 72.p odd 6 1
1224.1.bb.a 2 136.e odd 2 1
1224.1.bb.b yes 2 1.a even 1 1 trivial
1224.1.bb.b yes 2 8.d odd 2 1 CM
1224.1.bb.b yes 2 153.h even 6 1 inner
1224.1.bb.b yes 2 1224.bb odd 6 1 inner
3672.1.bb.a 2 9.d odd 6 1
3672.1.bb.a 2 51.c odd 2 1
3672.1.bb.a 2 72.l even 6 1
3672.1.bb.a 2 408.h even 2 1
3672.1.bb.b 2 3.b odd 2 1
3672.1.bb.b 2 24.f even 2 1
3672.1.bb.b 2 153.i odd 6 1
3672.1.bb.b 2 1224.bh even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} + 3 T_{11} + 3$$ acting on $$S_{1}^{\mathrm{new}}(1224, [\chi])$$.

## Hecke characteristic polynomials

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