Properties

Label 1045.1.w.c
Level $1045$
Weight $1$
Character orbit 1045.w
Analytic conductor $0.522$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
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.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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.87298324158753125.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{10} q^{4} - \zeta_{10}^{3} q^{5} + ( - \zeta_{10}^{2} + 1) q^{7} + \zeta_{10}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{10} q^{4} - \zeta_{10}^{3} q^{5} + ( - \zeta_{10}^{2} + 1) q^{7} + \zeta_{10}^{3} q^{9} - \zeta_{10}^{4} q^{11} + \zeta_{10}^{2} q^{16} + (\zeta_{10}^{4} - 1) q^{17} + \zeta_{10}^{4} q^{19} - \zeta_{10}^{4} q^{20} + ( - \zeta_{10}^{3} - \zeta_{10}^{2}) q^{23} - \zeta_{10} q^{25} + ( - \zeta_{10}^{3} + \zeta_{10}) q^{28} + ( - \zeta_{10}^{3} - 1) q^{35} + \zeta_{10}^{4} q^{36} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{43} + q^{44} + \zeta_{10} q^{45} + (\zeta_{10}^{2} + \zeta_{10}) q^{47} + (\zeta_{10}^{4} - \zeta_{10}^{2} + 1) q^{49} - \zeta_{10}^{2} q^{55} + ( - \zeta_{10}^{4} - 1) q^{61} + (\zeta_{10}^{3} + 1) q^{63} + \zeta_{10}^{3} q^{64} + ( - \zeta_{10} - 1) q^{68} - q^{76} + ( - \zeta_{10}^{4} - \zeta_{10}) q^{77} + q^{80} - \zeta_{10} q^{81} + (\zeta_{10}^{3} - \zeta_{10}) q^{83} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{85} + ( - \zeta_{10}^{4} - \zeta_{10}^{3}) q^{92} + \zeta_{10}^{2} q^{95} + \zeta_{10}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{4} - q^{5} + 5 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{4} - q^{5} + 5 q^{7} + q^{9} + q^{11} - q^{16} - 5 q^{17} - q^{19} + q^{20} - q^{25} - 5 q^{35} - q^{36} + 4 q^{44} + q^{45} + 4 q^{49} + q^{55} - 3 q^{61} + 5 q^{63} + q^{64} - 5 q^{68} - 4 q^{76} + 4 q^{80} - q^{81} - q^{95} - q^{99}+O(q^{100}) \) 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 0 0.809017 + 0.587785i 0.309017 0.951057i 0 0.690983 0.951057i 0 −0.309017 + 0.951057i 0
379.1 0 0 0.809017 0.587785i 0.309017 + 0.951057i 0 0.690983 + 0.951057i 0 −0.309017 0.951057i 0
664.1 0 0 −0.309017 0.951057i −0.809017 0.587785i 0 1.80902 0.587785i 0 0.809017 + 0.587785i 0
949.1 0 0 −0.309017 + 0.951057i −0.809017 + 0.587785i 0 1.80902 + 0.587785i 0 0.809017 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
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.j even 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.c yes 4
5.b even 2 1 1045.1.w.b 4
11.c even 5 1 1045.1.w.b 4
19.b odd 2 1 CM 1045.1.w.c yes 4
55.j even 10 1 inner 1045.1.w.c yes 4
95.d odd 2 1 1045.1.w.b 4
209.m odd 10 1 1045.1.w.b 4
1045.w odd 10 1 inner 1045.1.w.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.w.b 4 5.b even 2 1
1045.1.w.b 4 11.c even 5 1
1045.1.w.b 4 95.d odd 2 1
1045.1.w.b 4 209.m odd 10 1
1045.1.w.c yes 4 1.a even 1 1 trivial
1045.1.w.c yes 4 19.b odd 2 1 CM
1045.1.w.c yes 4 55.j even 10 1 inner
1045.1.w.c yes 4 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 \(S_{1}^{\mathrm{new}}(1045, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{4} - 5T_{7}^{3} + 10T_{7}^{2} - 10T_{7} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - 5 T^{3} + 10 T^{2} - 10 T + 5 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 5 T^{3} + 10 T^{2} + 10 T + 5 \) Copy content Toggle raw display
$19$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{4} + 5T^{2} + 5 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 5T^{2} + 5 \) Copy content Toggle raw display
$47$ \( T^{4} + 5T + 5 \) Copy content Toggle raw display
$53$ \( T^{4} \) 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
$67$ \( T^{4} \) 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
$83$ \( T^{4} + 5T + 5 \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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