Properties

Label 1148.1.bj.a
Level $1148$
Weight $1$
Character orbit 1148.bj
Analytic conductor $0.573$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1148.bj (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.572926634503\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.13508516.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{24}^{2} q^{2} -\zeta_{24}^{7} q^{3} + \zeta_{24}^{4} q^{4} + \zeta_{24}^{2} q^{5} + \zeta_{24}^{9} q^{6} -\zeta_{24}^{11} q^{7} -\zeta_{24}^{6} q^{8} +O(q^{10})\) \( q -\zeta_{24}^{2} q^{2} -\zeta_{24}^{7} q^{3} + \zeta_{24}^{4} q^{4} + \zeta_{24}^{2} q^{5} + \zeta_{24}^{9} q^{6} -\zeta_{24}^{11} q^{7} -\zeta_{24}^{6} q^{8} -\zeta_{24}^{4} q^{10} -\zeta_{24}^{7} q^{11} -\zeta_{24}^{11} q^{12} + ( -1 + \zeta_{24}^{6} ) q^{13} -\zeta_{24} q^{14} -\zeta_{24}^{9} q^{15} + \zeta_{24}^{8} q^{16} + \zeta_{24}^{5} q^{19} + \zeta_{24}^{6} q^{20} -\zeta_{24}^{6} q^{21} + \zeta_{24}^{9} q^{22} -\zeta_{24} q^{24} + ( \zeta_{24}^{2} - \zeta_{24}^{8} ) q^{26} -\zeta_{24}^{9} q^{27} + \zeta_{24}^{3} q^{28} + \zeta_{24}^{11} q^{30} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{31} -\zeta_{24}^{10} q^{32} -\zeta_{24}^{2} q^{33} + \zeta_{24} q^{35} -\zeta_{24}^{7} q^{38} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{39} -\zeta_{24}^{8} q^{40} + \zeta_{24}^{6} q^{41} + \zeta_{24}^{8} q^{42} -\zeta_{24}^{11} q^{44} + \zeta_{24}^{3} q^{48} -\zeta_{24}^{10} q^{49} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{52} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{53} + \zeta_{24}^{11} q^{54} -\zeta_{24}^{9} q^{55} -\zeta_{24}^{5} q^{56} + q^{57} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{59} + \zeta_{24} q^{60} -\zeta_{24}^{2} q^{61} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{62} - q^{64} + ( -\zeta_{24}^{2} + \zeta_{24}^{8} ) q^{65} + \zeta_{24}^{4} q^{66} -2 \zeta_{24} q^{67} -\zeta_{24}^{3} q^{70} -\zeta_{24}^{3} q^{71} + \zeta_{24}^{9} q^{76} -\zeta_{24}^{6} q^{77} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{78} + \zeta_{24}^{11} q^{79} + \zeta_{24}^{10} q^{80} -\zeta_{24}^{4} q^{81} -\zeta_{24}^{8} q^{82} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{83} -\zeta_{24}^{10} q^{84} -\zeta_{24} q^{88} + ( \zeta_{24}^{2} + \zeta_{24}^{8} ) q^{89} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{91} + ( \zeta_{24}^{2} + \zeta_{24}^{8} ) q^{93} + \zeta_{24}^{7} q^{95} -\zeta_{24}^{5} q^{96} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} + O(q^{10}) \) \( 8q + 4q^{4} - 4q^{10} - 8q^{13} - 4q^{16} + 4q^{26} + 4q^{40} - 4q^{42} - 4q^{52} - 4q^{53} + 8q^{57} - 8q^{64} - 4q^{65} + 4q^{66} - 4q^{81} + 4q^{82} - 4q^{89} - 4q^{93} - 8q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(\zeta_{24}^{8}\) \(-1\) \(\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
−0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.965926 0.258819i
0.866025 0.500000i −0.965926 + 0.258819i 0.500000 0.866025i −0.866025 + 0.500000i −0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i 0 −0.500000 + 0.866025i
319.2 0.866025 0.500000i 0.965926 0.258819i 0.500000 0.866025i −0.866025 + 0.500000i 0.707107 0.707107i 0.258819 0.965926i 1.00000i 0 −0.500000 + 0.866025i
583.1 0.866025 + 0.500000i −0.965926 0.258819i 0.500000 + 0.866025i −0.866025 0.500000i −0.707107 0.707107i −0.258819 0.965926i 1.00000i 0 −0.500000 0.866025i
583.2 0.866025 + 0.500000i 0.965926 + 0.258819i 0.500000 + 0.866025i −0.866025 0.500000i 0.707107 + 0.707107i 0.258819 + 0.965926i 1.00000i 0 −0.500000 0.866025i
975.1 −0.866025 0.500000i −0.258819 + 0.965926i 0.500000 + 0.866025i 0.866025 + 0.500000i 0.707107 0.707107i −0.965926 + 0.258819i 1.00000i 0 −0.500000 0.866025i
975.2 −0.866025 0.500000i 0.258819 0.965926i 0.500000 + 0.866025i 0.866025 + 0.500000i −0.707107 + 0.707107i 0.965926 0.258819i 1.00000i 0 −0.500000 0.866025i
1075.1 −0.866025 + 0.500000i −0.258819 0.965926i 0.500000 0.866025i 0.866025 0.500000i 0.707107 + 0.707107i −0.965926 0.258819i 1.00000i 0 −0.500000 + 0.866025i
1075.2 −0.866025 + 0.500000i 0.258819 + 0.965926i 0.500000 0.866025i 0.866025 0.500000i −0.707107 0.707107i 0.965926 + 0.258819i 1.00000i 0 −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1075.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner
41.c even 4 1 inner
164.e odd 4 1 inner
287.r even 12 1 inner
1148.bj odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.1.bj.a 8
4.b odd 2 1 inner 1148.1.bj.a 8
7.c even 3 1 inner 1148.1.bj.a 8
28.g odd 6 1 inner 1148.1.bj.a 8
41.c even 4 1 inner 1148.1.bj.a 8
164.e odd 4 1 inner 1148.1.bj.a 8
287.r even 12 1 inner 1148.1.bj.a 8
1148.bj odd 12 1 inner 1148.1.bj.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.1.bj.a 8 1.a even 1 1 trivial
1148.1.bj.a 8 4.b odd 2 1 inner
1148.1.bj.a 8 7.c even 3 1 inner
1148.1.bj.a 8 28.g odd 6 1 inner
1148.1.bj.a 8 41.c even 4 1 inner
1148.1.bj.a 8 164.e odd 4 1 inner
1148.1.bj.a 8 287.r even 12 1 inner
1148.1.bj.a 8 1148.bj odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1148, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( 1 - T^{4} + T^{8} \)
$5$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$7$ \( 1 - T^{4} + T^{8} \)
$11$ \( 1 - T^{4} + T^{8} \)
$13$ \( ( 2 + 2 T + T^{2} )^{4} \)
$17$ \( T^{8} \)
$19$ \( 1 - T^{4} + T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$37$ \( T^{8} \)
$41$ \( ( 1 + T^{2} )^{4} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( ( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$59$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$61$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$67$ \( 256 - 16 T^{4} + T^{8} \)
$71$ \( ( 1 + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( 1 - T^{4} + T^{8} \)
$83$ \( ( 2 + T^{2} )^{4} \)
$89$ \( ( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$97$ \( T^{8} \)
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