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 $4$

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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:

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