Properties

Label 1148.1.bc.a
Level $1148$
Weight $1$
Character orbit 1148.bc
Analytic conductor $0.573$
Analytic rank $0$
Dimension $8$
Projective image $A_{5}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1148.bc (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.572926634503\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{5}\)
Projective field: Galois closure of 5.1.553849156.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{20}^{5} q^{3} -\zeta_{20} q^{5} -\zeta_{20}^{2} q^{7} +O(q^{10})\) \( q + \zeta_{20}^{5} q^{3} -\zeta_{20} q^{5} -\zeta_{20}^{2} q^{7} + ( \zeta_{20}^{2} + \zeta_{20}^{6} ) q^{11} + \zeta_{20}^{3} q^{13} -\zeta_{20}^{6} q^{15} + ( -\zeta_{20} + \zeta_{20}^{7} ) q^{17} + \zeta_{20}^{7} q^{19} -\zeta_{20}^{7} q^{21} + ( -\zeta_{20}^{2} + \zeta_{20}^{4} ) q^{23} + \zeta_{20}^{5} q^{27} + ( -1 + \zeta_{20}^{2} ) q^{29} + ( \zeta_{20}^{3} - \zeta_{20}^{5} ) q^{31} + ( -\zeta_{20} + \zeta_{20}^{7} ) q^{33} + \zeta_{20}^{3} q^{35} -\zeta_{20}^{6} q^{37} + \zeta_{20}^{8} q^{39} -\zeta_{20}^{7} q^{41} + \zeta_{20}^{8} q^{43} + ( \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{47} + \zeta_{20}^{4} q^{49} + ( -\zeta_{20}^{2} - \zeta_{20}^{6} ) q^{51} + ( -\zeta_{20}^{3} - \zeta_{20}^{7} ) q^{55} -\zeta_{20}^{2} q^{57} + ( \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{59} + ( \zeta_{20} - \zeta_{20}^{3} ) q^{61} -\zeta_{20}^{4} q^{65} + ( -\zeta_{20}^{7} + \zeta_{20}^{9} ) q^{69} + ( -\zeta_{20}^{3} - \zeta_{20}^{7} ) q^{73} + ( -\zeta_{20}^{4} - \zeta_{20}^{8} ) q^{77} + q^{79} - q^{81} -\zeta_{20}^{5} q^{83} + ( \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{85} + ( -\zeta_{20}^{5} + \zeta_{20}^{7} ) q^{87} + ( \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{89} -\zeta_{20}^{5} q^{91} + ( 1 + \zeta_{20}^{8} ) q^{93} -\zeta_{20}^{8} q^{95} + \zeta_{20} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{7} + O(q^{10}) \) \( 8q - 2q^{7} + 4q^{11} - 2q^{15} - 4q^{23} - 6q^{29} - 2q^{37} - 2q^{39} - 2q^{43} - 2q^{49} - 4q^{51} - 2q^{57} + 2q^{65} + 4q^{77} + 8q^{79} - 8q^{81} + 4q^{85} + 6q^{93} + 2q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{20}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
461.1
0.951057 0.309017i
−0.951057 + 0.309017i
−0.587785 + 0.809017i
0.587785 0.809017i
0.587785 + 0.809017i
−0.587785 0.809017i
−0.951057 0.309017i
0.951057 + 0.309017i
0 1.00000i 0 −0.951057 + 0.309017i 0 −0.809017 + 0.587785i 0 0 0
461.2 0 1.00000i 0 0.951057 0.309017i 0 −0.809017 + 0.587785i 0 0 0
713.1 0 1.00000i 0 0.587785 0.809017i 0 0.309017 + 0.951057i 0 0 0
713.2 0 1.00000i 0 −0.587785 + 0.809017i 0 0.309017 + 0.951057i 0 0 0
797.1 0 1.00000i 0 −0.587785 0.809017i 0 0.309017 0.951057i 0 0 0
797.2 0 1.00000i 0 0.587785 + 0.809017i 0 0.309017 0.951057i 0 0 0
1021.1 0 1.00000i 0 0.951057 + 0.309017i 0 −0.809017 0.587785i 0 0 0
1021.2 0 1.00000i 0 −0.951057 0.309017i 0 −0.809017 0.587785i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1021.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
41.d even 5 1 inner
287.o odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.1.bc.a 8
7.b odd 2 1 inner 1148.1.bc.a 8
41.d even 5 1 inner 1148.1.bc.a 8
287.o odd 10 1 inner 1148.1.bc.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.1.bc.a 8 1.a even 1 1 trivial
1148.1.bc.a 8 7.b odd 2 1 inner
1148.1.bc.a 8 41.d even 5 1 inner
1148.1.bc.a 8 287.o odd 10 1 inner

Hecke kernels

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

Hecke characteristic polynomials

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