# Properties

 Label 1224.1.cb.a Level $1224$ Weight $1$ Character orbit 1224.cb Analytic conductor $0.611$ Analytic rank $0$ Dimension $4$ Projective image $D_{12}$ CM discriminant -8 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1224 = 2^{3} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1224.cb (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.610855575463$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{12}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{2} q^{6} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{2} q^{6} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + ( -\zeta_{12}^{2} - \zeta_{12}^{3} ) q^{11} + \zeta_{12}^{3} q^{12} + \zeta_{12}^{4} q^{16} + \zeta_{12}^{5} q^{17} + \zeta_{12}^{3} q^{18} -\zeta_{12}^{3} q^{19} + ( -\zeta_{12}^{3} - \zeta_{12}^{4} ) q^{22} + \zeta_{12}^{4} q^{24} -\zeta_{12} q^{25} + \zeta_{12}^{3} q^{27} + \zeta_{12}^{5} q^{32} + ( -\zeta_{12}^{3} - \zeta_{12}^{4} ) q^{33} - q^{34} + \zeta_{12}^{4} q^{36} -\zeta_{12}^{4} q^{38} + ( -1 - \zeta_{12} ) q^{41} + ( 1 + \zeta_{12}^{2} ) q^{43} + ( -\zeta_{12}^{4} - \zeta_{12}^{5} ) q^{44} + \zeta_{12}^{5} q^{48} + \zeta_{12}^{5} q^{49} -\zeta_{12}^{2} q^{50} - q^{51} + \zeta_{12}^{4} q^{54} -\zeta_{12}^{4} q^{57} + ( 1 - \zeta_{12}^{4} ) q^{59} - q^{64} + ( -\zeta_{12}^{4} - \zeta_{12}^{5} ) q^{66} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{67} -\zeta_{12} q^{68} + \zeta_{12}^{5} q^{72} + ( -\zeta_{12}^{4} + \zeta_{12}^{5} ) q^{73} -\zeta_{12}^{2} q^{75} -\zeta_{12}^{5} q^{76} + \zeta_{12}^{4} q^{81} + ( -\zeta_{12} - \zeta_{12}^{2} ) q^{82} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{86} + ( 1 - \zeta_{12}^{5} ) q^{88} - q^{96} + ( 1 - \zeta_{12}^{5} ) q^{97} - q^{98} + ( -\zeta_{12}^{4} - \zeta_{12}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 2q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{4} + 2q^{6} + 2q^{9} - 2q^{11} - 2q^{16} + 2q^{22} - 2q^{24} + 2q^{33} - 4q^{34} - 2q^{36} + 2q^{38} - 4q^{41} + 6q^{43} + 2q^{44} - 2q^{50} - 4q^{51} - 2q^{54} + 2q^{57} + 6q^{59} - 4q^{64} + 2q^{66} + 2q^{73} - 2q^{75} - 2q^{81} - 2q^{82} + 4q^{88} - 4q^{96} + 4q^{97} - 4q^{98} + 2q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$137$$ $$613$$ $$649$$ $$919$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$-1$$ $$\zeta_{12}^{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
115.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 1.00000i 0.500000 + 0.866025i 0
259.1 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 1.00000i 0.500000 + 0.866025i 0
931.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 0 1.00000i 0.500000 0.866025i 0
1075.1 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 0 1.00000i 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.n even 12 1 inner
1224.cb odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.cb.a 4
3.b odd 2 1 3672.1.cc.b 4
8.d odd 2 1 CM 1224.1.cb.a 4
9.c even 3 1 1224.1.cb.b yes 4
9.d odd 6 1 3672.1.cc.a 4
17.c even 4 1 1224.1.cb.b yes 4
24.f even 2 1 3672.1.cc.b 4
51.f odd 4 1 3672.1.cc.a 4
72.l even 6 1 3672.1.cc.a 4
72.p odd 6 1 1224.1.cb.b yes 4
136.j odd 4 1 1224.1.cb.b yes 4
153.m odd 12 1 3672.1.cc.b 4
153.n even 12 1 inner 1224.1.cb.a 4
408.q even 4 1 3672.1.cc.a 4
1224.ca even 12 1 3672.1.cc.b 4
1224.cb odd 12 1 inner 1224.1.cb.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.cb.a 4 1.a even 1 1 trivial
1224.1.cb.a 4 8.d odd 2 1 CM
1224.1.cb.a 4 153.n even 12 1 inner
1224.1.cb.a 4 1224.cb odd 12 1 inner
1224.1.cb.b yes 4 9.c even 3 1
1224.1.cb.b yes 4 17.c even 4 1
1224.1.cb.b yes 4 72.p odd 6 1
1224.1.cb.b yes 4 136.j odd 4 1
3672.1.cc.a 4 9.d odd 6 1
3672.1.cc.a 4 51.f odd 4 1
3672.1.cc.a 4 72.l even 6 1
3672.1.cc.a 4 408.q even 4 1
3672.1.cc.b 4 3.b odd 2 1
3672.1.cc.b 4 24.f even 2 1
3672.1.cc.b 4 153.m odd 12 1
3672.1.cc.b 4 1224.ca even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 2 T_{11}^{3} + 5 T_{11}^{2} + 4 T_{11} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1224, [\chi])$$.

## Hecke characteristic polynomials

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