Properties

Label 1045.1.w.a
Level $1045$
Weight $1$
Character orbit 1045.w
Analytic conductor $0.522$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -95
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.132135025.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{2} + (\zeta_{10}^{2} - \zeta_{10}) q^{3} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}) q^{4} + \zeta_{10}^{2} q^{5} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{6} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{8} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{2} + (\zeta_{10}^{2} - \zeta_{10}) q^{3} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}) q^{4} + \zeta_{10}^{2} q^{5} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{6} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{8} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{9} + (\zeta_{10}^{4} - \zeta_{10}) q^{10} - \zeta_{10}^{3} q^{11} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{12} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{13} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{15} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{16} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{18} + \zeta_{10}^{4} q^{19} + ( - \zeta_{10}^{3} - \zeta_{10} + 1) q^{20} + (\zeta_{10}^{2} + 1) q^{22} + (2 \zeta_{10}^{4} - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10} + 1) q^{24} + \zeta_{10}^{4} q^{25} + (\zeta_{10}^{4} - \zeta_{10}^{3} - 2 \zeta_{10}) q^{26} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} - 1) q^{27} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{30} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{32} + (\zeta_{10}^{4} + 1) q^{33} + (\zeta_{10}^{4} - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{36} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{37} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{38} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{39} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{40} + (\zeta_{10}^{4} + \zeta_{10}^{2} - \zeta_{10}) q^{44} + (\zeta_{10}^{4} - \zeta_{10} + 1) q^{45} + (2 \zeta_{10}^{4} - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10} + 2) q^{48} + \zeta_{10}^{2} q^{49} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{50} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{52} - \zeta_{10}^{3} q^{53} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{54} + q^{55} + ( - \zeta_{10} + 1) q^{57} + (\zeta_{10}^{4} - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10} + 1) q^{60} + (\zeta_{10}^{4} + 1) q^{61} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10} - 1) q^{64} + (\zeta_{10}^{4} - \zeta_{10}) q^{65} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{66} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{67} + (3 \zeta_{10}^{4} - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{72} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{74} + ( - \zeta_{10} + 1) q^{75} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{76} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{78} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{80} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{81} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{88} + (\zeta_{10}^{4} - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{90} - \zeta_{10} q^{95} + (2 \zeta_{10}^{4} - 2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - \zeta_{10} + 2) q^{96} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{97} + (\zeta_{10}^{4} - \zeta_{10}) q^{98} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} - 3 q^{4} - q^{5} + q^{6} + q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{3} - 3 q^{4} - q^{5} + q^{6} + q^{8} - 3 q^{9} - 2 q^{10} - q^{11} + 4 q^{12} - 2 q^{13} - 2 q^{15} - q^{18} - q^{19} + 2 q^{20} + 3 q^{22} - 3 q^{24} - q^{25} - 4 q^{26} + q^{27} + q^{30} + 4 q^{32} + 3 q^{33} + q^{36} - 2 q^{37} - 2 q^{38} + q^{39} + q^{40} - 3 q^{44} + 2 q^{45} - q^{49} - 2 q^{50} + 4 q^{52} - 2 q^{53} + 2 q^{54} + 4 q^{55} + 3 q^{57} - q^{60} + 3 q^{61} - 2 q^{64} - 2 q^{65} - 4 q^{66} - 2 q^{67} + 3 q^{72} + q^{74} + 3 q^{75} + 2 q^{76} + 2 q^{78} + q^{88} - q^{90} - q^{95} - 2 q^{96} - 2 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(-1\) \(-\zeta_{10}\) \(-1\)

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
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
−0.500000 + 1.53884i −0.500000 + 0.363271i −1.30902 0.951057i 0.309017 + 0.951057i −0.309017 0.951057i 0 0.809017 0.587785i −0.190983 + 0.587785i −1.61803
379.1 −0.500000 1.53884i −0.500000 0.363271i −1.30902 + 0.951057i 0.309017 0.951057i −0.309017 + 0.951057i 0 0.809017 + 0.587785i −0.190983 0.587785i −1.61803
664.1 −0.500000 0.363271i −0.500000 + 1.53884i −0.190983 0.587785i −0.809017 + 0.587785i 0.809017 0.587785i 0 −0.309017 + 0.951057i −1.30902 0.951057i 0.618034
949.1 −0.500000 + 0.363271i −0.500000 1.53884i −0.190983 + 0.587785i −0.809017 0.587785i 0.809017 + 0.587785i 0 −0.309017 0.951057i −1.30902 + 0.951057i 0.618034
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
11.c even 5 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.a 4
5.b even 2 1 1045.1.w.d yes 4
11.c even 5 1 inner 1045.1.w.a 4
19.b odd 2 1 1045.1.w.d yes 4
55.j even 10 1 1045.1.w.d yes 4
95.d odd 2 1 CM 1045.1.w.a 4
209.m odd 10 1 1045.1.w.d yes 4
1045.w odd 10 1 inner 1045.1.w.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.w.a 4 1.a even 1 1 trivial
1045.1.w.a 4 11.c even 5 1 inner
1045.1.w.a 4 95.d odd 2 1 CM
1045.1.w.a 4 1045.w odd 10 1 inner
1045.1.w.d yes 4 5.b even 2 1
1045.1.w.d yes 4 19.b odd 2 1
1045.1.w.d yes 4 55.j even 10 1
1045.1.w.d yes 4 209.m odd 10 1

Hecke kernels

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

\( T_{2}^{4} + 2T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

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