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: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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.

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

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 \( T_{7}^{8} + 2T_{7}^{7} + 7T_{7}^{6} + 10T_{7}^{5} + 16T_{7}^{4} + 10T_{7}^{3} - 2T_{7}^{2} - 4T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1045, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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