Properties

Label 1045.1.br.b
Level $1045$
Weight $1$
Character orbit 1045.br
Analytic conductor $0.522$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1045.br (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.521522938201\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{20}^{3} q^{4} + \zeta_{20}^{8} q^{5} + ( \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{7} + \zeta_{20}^{9} q^{9} +O(q^{10})\) \( q -\zeta_{20}^{3} q^{4} + \zeta_{20}^{8} q^{5} + ( \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{7} + \zeta_{20}^{9} q^{9} + \zeta_{20}^{7} q^{11} + \zeta_{20}^{6} q^{16} + ( -1 + \zeta_{20}^{7} ) q^{17} + \zeta_{20}^{2} q^{19} + \zeta_{20} q^{20} + ( \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{23} -\zeta_{20}^{6} q^{25} + ( -\zeta_{20}^{8} + \zeta_{20}^{9} ) q^{28} + ( -\zeta_{20}^{3} + \zeta_{20}^{4} ) q^{35} + \zeta_{20}^{2} q^{36} + ( -\zeta_{20} + \zeta_{20}^{4} ) q^{43} + q^{44} -\zeta_{20}^{7} q^{45} + ( -\zeta_{20} + \zeta_{20}^{8} ) q^{47} + ( -1 + \zeta_{20} - \zeta_{20}^{2} ) q^{49} -\zeta_{20}^{5} q^{55} + ( \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{61} + ( -\zeta_{20}^{4} + \zeta_{20}^{5} ) q^{63} -\zeta_{20}^{9} q^{64} + ( 1 + \zeta_{20}^{3} ) q^{68} + ( \zeta_{20}^{3} + \zeta_{20}^{8} ) q^{73} -\zeta_{20}^{5} q^{76} + ( -\zeta_{20}^{2} + \zeta_{20}^{3} ) q^{77} -\zeta_{20}^{4} q^{80} -\zeta_{20}^{8} q^{81} + ( -\zeta_{20}^{3} - \zeta_{20}^{4} ) q^{83} + ( -\zeta_{20}^{5} - \zeta_{20}^{8} ) q^{85} + ( -\zeta_{20}^{2} - \zeta_{20}^{9} ) q^{92} - q^{95} -\zeta_{20}^{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{5} - 2q^{7} + O(q^{10}) \) \( 8q - 2q^{5} - 2q^{7} + 2q^{16} - 8q^{17} + 2q^{19} + 2q^{23} - 2q^{25} + 2q^{28} - 2q^{35} + 2q^{36} - 2q^{43} + 8q^{44} - 2q^{47} - 10q^{49} + 2q^{63} + 8q^{68} - 2q^{73} - 2q^{77} + 2q^{80} + 2q^{81} + 2q^{83} + 2q^{85} - 2q^{92} - 8q^{95} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(-1\) \(-\zeta_{20}^{8}\) \(-\zeta_{20}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
−0.587785 0.809017i
0.587785 + 0.809017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
−0.951057 0.309017i
0.951057 0.309017i
0.587785 0.809017i
−0.587785 + 0.809017i
0 0 −0.951057 + 0.309017i 0.309017 + 0.951057i 0 −0.809017 + 1.58779i 0 0.587785 0.809017i 0
227.1 0 0 0.951057 0.309017i 0.309017 + 0.951057i 0 −0.809017 0.412215i 0 −0.587785 + 0.809017i 0
303.1 0 0 −0.587785 0.809017i −0.809017 + 0.587785i 0 0.309017 + 0.0489435i 0 −0.951057 + 0.309017i 0
398.1 0 0 0.587785 0.809017i −0.809017 0.587785i 0 0.309017 + 1.95106i 0 0.951057 + 0.309017i 0
512.1 0 0 0.587785 + 0.809017i −0.809017 + 0.587785i 0 0.309017 1.95106i 0 0.951057 0.309017i 0
607.1 0 0 −0.587785 + 0.809017i −0.809017 0.587785i 0 0.309017 0.0489435i 0 −0.951057 0.309017i 0
778.1 0 0 0.951057 + 0.309017i 0.309017 0.951057i 0 −0.809017 + 0.412215i 0 −0.587785 0.809017i 0
987.1 0 0 −0.951057 0.309017i 0.309017 0.951057i 0 −0.809017 1.58779i 0 0.587785 + 0.809017i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 987.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
55.l even 20 1 inner
1045.br odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.1.br.b yes 8
5.c odd 4 1 1045.1.br.a 8
11.d odd 10 1 1045.1.br.a 8
19.b odd 2 1 CM 1045.1.br.b yes 8
55.l even 20 1 inner 1045.1.br.b yes 8
95.g even 4 1 1045.1.br.a 8
209.k even 10 1 1045.1.br.a 8
1045.br odd 20 1 inner 1045.1.br.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.br.a 8 5.c odd 4 1
1045.1.br.a 8 11.d odd 10 1
1045.1.br.a 8 95.g even 4 1
1045.1.br.a 8 209.k even 10 1
1045.1.br.b yes 8 1.a even 1 1 trivial
1045.1.br.b yes 8 19.b odd 2 1 CM
1045.1.br.b yes 8 55.l even 20 1 inner
1045.1.br.b yes 8 1045.br odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{8} + \cdots\) acting on \(S_{1}^{\mathrm{new}}(1045, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$7$ \( 1 - 4 T - 2 T^{2} + 10 T^{3} + 16 T^{4} + 10 T^{5} + 7 T^{6} + 2 T^{7} + T^{8} \)
$11$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( 1 + 4 T + 18 T^{2} + 40 T^{3} + 56 T^{4} + 50 T^{5} + 27 T^{6} + 8 T^{7} + T^{8} \)
$19$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$23$ \( 1 - 6 T + 18 T^{2} - 20 T^{3} + 11 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( 1 + 6 T + 18 T^{2} + 20 T^{3} + 11 T^{4} + 2 T^{6} + 2 T^{7} + T^{8} \)
$47$ \( 1 + 6 T + 13 T^{2} + 10 T^{3} + 16 T^{4} + 10 T^{5} + 2 T^{6} + 2 T^{7} + T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( 1 - 4 T^{2} + 6 T^{4} + T^{6} + T^{8} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( 16 + 16 T + 8 T^{2} - 4 T^{4} + 2 T^{6} + 2 T^{7} + T^{8} \)
$79$ \( T^{8} \)
$83$ \( 1 - 6 T + 13 T^{2} - 10 T^{3} + 16 T^{4} - 10 T^{5} + 2 T^{6} - 2 T^{7} + T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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