Properties

Label 1148.1.o.d
Level $1148$
Weight $1$
Character orbit 1148.o
Analytic conductor $0.573$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -164
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.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.572926634503\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q -\zeta_{24}^{8} q^{2} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{3} -\zeta_{24}^{4} q^{4} + ( \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{5} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} ) q^{6} -\zeta_{24}^{11} q^{7} - q^{8} + ( -\zeta_{24}^{6} + \zeta_{24}^{8} - \zeta_{24}^{10} ) q^{9} +O(q^{10})\) \( q -\zeta_{24}^{8} q^{2} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{3} -\zeta_{24}^{4} q^{4} + ( \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{5} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} ) q^{6} -\zeta_{24}^{11} q^{7} - q^{8} + ( -\zeta_{24}^{6} + \zeta_{24}^{8} - \zeta_{24}^{10} ) q^{9} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{10} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} ) q^{11} + ( -\zeta_{24} + \zeta_{24}^{3} ) q^{12} -\zeta_{24}^{7} q^{14} + ( \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} - \zeta_{24}^{9} ) q^{15} + \zeta_{24}^{8} q^{16} + ( -\zeta_{24}^{2} + \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{18} + ( -\zeta_{24}^{7} - \zeta_{24}^{9} ) q^{19} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{20} + ( -\zeta_{24}^{8} + \zeta_{24}^{10} ) q^{21} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{22} + ( \zeta_{24}^{9} - \zeta_{24}^{11} ) q^{24} + ( -1 - \zeta_{24}^{4} - \zeta_{24}^{8} ) q^{25} + ( -\zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{27} -\zeta_{24}^{3} q^{28} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{30} + \zeta_{24}^{4} q^{32} + ( -1 + \zeta_{24}^{4} ) q^{33} + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{35} + ( 1 - \zeta_{24}^{2} + \zeta_{24}^{10} ) q^{36} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} ) q^{38} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{40} + q^{41} + ( -\zeta_{24}^{4} + \zeta_{24}^{6} ) q^{42} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{44} + ( 1 - \zeta_{24}^{2} + 2 \zeta_{24}^{4} - \zeta_{24}^{6} + \zeta_{24}^{8} ) q^{45} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{47} + ( \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{48} -\zeta_{24}^{10} q^{49} + ( -1 - \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{50} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{54} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{9} - \zeta_{24}^{11} ) q^{55} + \zeta_{24}^{11} q^{56} + ( -\zeta_{24}^{4} + \zeta_{24}^{8} ) q^{57} + ( -\zeta_{24} - \zeta_{24}^{7} + \zeta_{24}^{9} - \zeta_{24}^{11} ) q^{60} + \zeta_{24}^{8} q^{61} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} - \zeta_{24}^{9} ) q^{63} + q^{64} + ( 1 + \zeta_{24}^{8} ) q^{66} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{67} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{70} + ( \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{71} + ( \zeta_{24}^{6} - \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{72} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} + \zeta_{24}^{9} - \zeta_{24}^{11} ) q^{75} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{76} + ( -\zeta_{24}^{2} - \zeta_{24}^{4} ) q^{77} + ( -\zeta_{24} + \zeta_{24}^{3} ) q^{79} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{80} + ( -1 + \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{6} - \zeta_{24}^{8} ) q^{81} -\zeta_{24}^{8} q^{82} + ( -1 + \zeta_{24}^{2} ) q^{84} + ( \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{88} + ( 2 - \zeta_{24}^{2} + \zeta_{24}^{4} - \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{90} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{94} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{95} + ( \zeta_{24} - \zeta_{24}^{3} ) q^{96} -\zeta_{24}^{6} q^{98} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{2} - 4q^{4} - 8q^{8} - 4q^{9} + O(q^{10}) \) \( 8q + 4q^{2} - 4q^{4} - 8q^{8} - 4q^{9} - 4q^{16} + 4q^{18} + 4q^{21} - 8q^{25} + 4q^{32} - 4q^{33} + 8q^{36} + 8q^{41} - 4q^{42} + 12q^{45} - 16q^{50} - 8q^{57} - 4q^{61} + 8q^{64} + 4q^{66} + 4q^{72} - 4q^{77} - 8q^{81} + 4q^{82} - 8q^{84} + 24q^{90} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.500000 0.866025i −0.965926 1.67303i −0.500000 0.866025i 0.866025 1.50000i −1.93185 0.258819 + 0.965926i −1.00000 −1.36603 + 2.36603i −0.866025 1.50000i
163.2 0.500000 0.866025i −0.258819 0.448288i −0.500000 0.866025i −0.866025 + 1.50000i −0.517638 0.965926 0.258819i −1.00000 0.366025 0.633975i 0.866025 + 1.50000i
163.3 0.500000 0.866025i 0.258819 + 0.448288i −0.500000 0.866025i −0.866025 + 1.50000i 0.517638 −0.965926 + 0.258819i −1.00000 0.366025 0.633975i 0.866025 + 1.50000i
163.4 0.500000 0.866025i 0.965926 + 1.67303i −0.500000 0.866025i 0.866025 1.50000i 1.93185 −0.258819 0.965926i −1.00000 −1.36603 + 2.36603i −0.866025 1.50000i
655.1 0.500000 + 0.866025i −0.965926 + 1.67303i −0.500000 + 0.866025i 0.866025 + 1.50000i −1.93185 0.258819 0.965926i −1.00000 −1.36603 2.36603i −0.866025 + 1.50000i
655.2 0.500000 + 0.866025i −0.258819 + 0.448288i −0.500000 + 0.866025i −0.866025 1.50000i −0.517638 0.965926 + 0.258819i −1.00000 0.366025 + 0.633975i 0.866025 1.50000i
655.3 0.500000 + 0.866025i 0.258819 0.448288i −0.500000 + 0.866025i −0.866025 1.50000i 0.517638 −0.965926 0.258819i −1.00000 0.366025 + 0.633975i 0.866025 1.50000i
655.4 0.500000 + 0.866025i 0.965926 1.67303i −0.500000 + 0.866025i 0.866025 + 1.50000i 1.93185 −0.258819 + 0.965926i −1.00000 −1.36603 2.36603i −0.866025 + 1.50000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 655.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
164.d odd 2 1 CM by \(\Q(\sqrt{-41}) \)
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner
41.b even 2 1 inner
287.j even 6 1 inner
1148.o odd 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 4 T_{3}^{6} + 15 T_{3}^{4} + 4 T_{3}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1148, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{4} \)
$3$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$5$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$7$ \( 1 - T^{4} + T^{8} \)
$11$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( ( -1 + T )^{8} \)
$43$ \( T^{8} \)
$47$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 1 + T + T^{2} )^{4} \)
$67$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$71$ \( ( 1 - 4 T^{2} + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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