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 $4$

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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
0.500000 0.866025i
0.500000 + 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:

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

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