Properties

Label 1045.1.w.e
Level $1045$
Weight $1$
Character orbit 1045.w
Analytic conductor $0.522$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -95
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1045.w (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.521522938201\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.87298324158753125.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{20}^{3} - \zeta_{20}^{9} ) q^{2} + ( \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{3} + ( -\zeta_{20}^{2} + \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{4} + \zeta_{20}^{4} q^{5} + ( 1 + \zeta_{20}^{2} + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{6} + ( -\zeta_{20} + \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{8} + ( -\zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{20}^{3} - \zeta_{20}^{9} ) q^{2} + ( \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{3} + ( -\zeta_{20}^{2} + \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{4} + \zeta_{20}^{4} q^{5} + ( 1 + \zeta_{20}^{2} + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{6} + ( -\zeta_{20} + \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{8} + ( -\zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{9} + ( \zeta_{20}^{3} - \zeta_{20}^{7} ) q^{10} + \zeta_{20}^{6} q^{11} + ( \zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{12} + ( \zeta_{20}^{3} + \zeta_{20}^{9} ) q^{13} + ( -\zeta_{20} - \zeta_{20}^{3} ) q^{15} + ( -1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{16} + ( -\zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{18} + \zeta_{20}^{8} q^{19} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{6} ) q^{20} + ( \zeta_{20}^{5} - \zeta_{20}^{9} ) q^{22} + ( 1 - \zeta_{20}^{2} + 2 \zeta_{20}^{6} ) q^{24} + \zeta_{20}^{8} q^{25} + ( 2 \zeta_{20}^{2} - \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{26} + ( \zeta_{20} + \zeta_{20}^{3} + \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{27} + ( -1 - \zeta_{20}^{2} + \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{30} + ( -\zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{32} + ( -\zeta_{20}^{3} - \zeta_{20}^{5} ) q^{33} + \zeta_{20}^{8} q^{36} + ( \zeta_{20} + \zeta_{20}^{3} ) q^{37} + ( \zeta_{20} + \zeta_{20}^{7} ) q^{38} + ( -1 - \zeta_{20}^{2} - \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{39} + ( \zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{40} + ( -\zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{8} ) q^{44} + ( 1 + \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{45} + ( 2 \zeta_{20}^{5} - 2 \zeta_{20}^{9} ) q^{48} + \zeta_{20}^{4} q^{49} + ( \zeta_{20} + \zeta_{20}^{7} ) q^{50} + ( 2 \zeta_{20} - 2 \zeta_{20}^{5} + \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{52} + ( 2 + \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{54} - q^{55} + ( -\zeta_{20}^{5} - \zeta_{20}^{7} ) q^{57} + ( -\zeta_{20} + \zeta_{20}^{3} + \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{60} + ( -1 - \zeta_{20}^{8} ) q^{61} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{6} ) q^{64} + ( -\zeta_{20}^{3} + \zeta_{20}^{7} ) q^{65} + ( -\zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{66} + ( -\zeta_{20} + \zeta_{20}^{9} ) q^{67} + ( -\zeta_{20}^{3} - \zeta_{20}^{5} ) q^{72} + ( 1 + \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{74} + ( -\zeta_{20}^{5} - \zeta_{20}^{7} ) q^{75} + ( 1 - \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{76} + ( -2 \zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{7} + 2 \zeta_{20}^{9} ) q^{78} + ( 1 + \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{80} + ( -1 - \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{81} + ( -\zeta_{20} + \zeta_{20}^{3} + \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{88} + ( -\zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{90} -\zeta_{20}^{2} q^{95} + ( -\zeta_{20}^{2} + \zeta_{20}^{4} - 2 \zeta_{20}^{8} ) q^{96} + ( -\zeta_{20}^{3} - \zeta_{20}^{9} ) q^{97} + ( \zeta_{20}^{3} - \zeta_{20}^{7} ) q^{98} + ( 1 + \zeta_{20}^{2} + \zeta_{20}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 2 q^{5} + 10 q^{6} + 2 q^{9} + O(q^{10}) \) \( 8 q + 2 q^{4} - 2 q^{5} + 10 q^{6} + 2 q^{9} + 2 q^{11} - 12 q^{16} - 2 q^{19} - 8 q^{20} + 10 q^{24} - 2 q^{25} - 10 q^{30} - 2 q^{36} - 10 q^{39} - 2 q^{44} + 12 q^{45} - 2 q^{49} + 20 q^{54} - 8 q^{55} - 6 q^{61} - 8 q^{64} + 10 q^{74} + 12 q^{76} + 8 q^{80} - 12 q^{81} - 2 q^{95} + 8 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(-1\) \(-\zeta_{20}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
284.1
−0.951057 0.309017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.363271 + 1.11803i 1.53884 1.11803i −0.309017 0.224514i 0.309017 + 0.951057i 0.690983 + 2.12663i 0 −0.587785 + 0.427051i 0.809017 2.48990i −1.17557
284.2 0.363271 1.11803i −1.53884 + 1.11803i −0.309017 0.224514i 0.309017 + 0.951057i 0.690983 + 2.12663i 0 0.587785 0.427051i 0.809017 2.48990i 1.17557
379.1 −0.363271 1.11803i 1.53884 + 1.11803i −0.309017 + 0.224514i 0.309017 0.951057i 0.690983 2.12663i 0 −0.587785 0.427051i 0.809017 + 2.48990i −1.17557
379.2 0.363271 + 1.11803i −1.53884 1.11803i −0.309017 + 0.224514i 0.309017 0.951057i 0.690983 2.12663i 0 0.587785 + 0.427051i 0.809017 + 2.48990i 1.17557
664.1 −1.53884 1.11803i −0.363271 + 1.11803i 0.809017 + 2.48990i −0.809017 + 0.587785i 1.80902 1.31433i 0 0.951057 2.92705i −0.309017 0.224514i 1.90211
664.2 1.53884 + 1.11803i 0.363271 1.11803i 0.809017 + 2.48990i −0.809017 + 0.587785i 1.80902 1.31433i 0 −0.951057 + 2.92705i −0.309017 0.224514i −1.90211
949.1 −1.53884 + 1.11803i −0.363271 1.11803i 0.809017 2.48990i −0.809017 0.587785i 1.80902 + 1.31433i 0 0.951057 + 2.92705i −0.309017 + 0.224514i 1.90211
949.2 1.53884 1.11803i 0.363271 + 1.11803i 0.809017 2.48990i −0.809017 0.587785i 1.80902 + 1.31433i 0 −0.951057 2.92705i −0.309017 + 0.224514i −1.90211
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 949.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
11.c even 5 1 inner
19.b odd 2 1 inner
55.j even 10 1 inner
209.m odd 10 1 inner
1045.w odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.1.w.e 8
5.b even 2 1 inner 1045.1.w.e 8
11.c even 5 1 inner 1045.1.w.e 8
19.b odd 2 1 inner 1045.1.w.e 8
55.j even 10 1 inner 1045.1.w.e 8
95.d odd 2 1 CM 1045.1.w.e 8
209.m odd 10 1 inner 1045.1.w.e 8
1045.w odd 10 1 inner 1045.1.w.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.w.e 8 1.a even 1 1 trivial
1045.1.w.e 8 5.b even 2 1 inner
1045.1.w.e 8 11.c even 5 1 inner
1045.1.w.e 8 19.b odd 2 1 inner
1045.1.w.e 8 55.j even 10 1 inner
1045.1.w.e 8 95.d odd 2 1 CM
1045.1.w.e 8 209.m odd 10 1 inner
1045.1.w.e 8 1045.w odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1045, [\chi])\):

\( T_{2}^{8} + 10 T_{2}^{4} + 25 T_{2}^{2} + 25 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 + 25 T^{2} + 10 T^{4} + T^{8} \)
$3$ \( 25 + 25 T^{2} + 10 T^{4} + T^{8} \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$13$ \( 25 + 25 T^{2} + 10 T^{4} + T^{8} \)
$17$ \( T^{8} \)
$19$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( 25 + 25 T^{2} + 10 T^{4} + T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$67$ \( ( 5 - 5 T^{2} + T^{4} )^{2} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( 25 + 25 T^{2} + 10 T^{4} + T^{8} \)
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