Properties

Label 1000.1.e.a
Level $1000$
Weight $1$
Character orbit 1000.e
Self dual yes
Analytic conductor $0.499$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -40
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1000.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.499065012633\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.1000000.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( 1 - \beta ) q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{4} + ( 1 - \beta ) q^{7} - q^{8} + q^{9} -\beta q^{11} + \beta q^{13} + ( -1 + \beta ) q^{14} + q^{16} - q^{18} + ( -1 + \beta ) q^{19} + \beta q^{22} + \beta q^{23} -\beta q^{26} + ( 1 - \beta ) q^{28} - q^{32} + q^{36} + ( 1 - \beta ) q^{37} + ( 1 - \beta ) q^{38} + ( -1 + \beta ) q^{41} -\beta q^{44} -\beta q^{46} + \beta q^{47} + ( 1 - \beta ) q^{49} + \beta q^{52} + ( 1 - \beta ) q^{53} + ( -1 + \beta ) q^{56} + ( -1 + \beta ) q^{59} + ( 1 - \beta ) q^{63} + q^{64} - q^{72} + ( -1 + \beta ) q^{74} + ( -1 + \beta ) q^{76} + q^{77} + q^{81} + ( 1 - \beta ) q^{82} + \beta q^{88} -\beta q^{89} - q^{91} + \beta q^{92} -\beta q^{94} + ( -1 + \beta ) q^{98} -\beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + q^{7} - 2q^{8} + 2q^{9} - q^{11} + q^{13} - q^{14} + 2q^{16} - 2q^{18} - q^{19} + q^{22} + q^{23} - q^{26} + q^{28} - 2q^{32} + 2q^{36} + q^{37} + q^{38} - q^{41} - q^{44} - q^{46} + q^{47} + q^{49} + q^{52} + q^{53} - q^{56} - q^{59} + q^{63} + 2q^{64} - 2q^{72} - q^{74} - q^{76} + 2q^{77} + 2q^{81} + q^{82} + q^{88} - q^{89} - 2q^{91} + q^{92} - q^{94} - q^{98} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
1.61803
−0.618034
−1.00000 0 1.00000 0 0 −0.618034 −1.00000 1.00000 0
499.2 −1.00000 0 1.00000 0 0 1.61803 −1.00000 1.00000 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
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.1.e.a 2
4.b odd 2 1 4000.1.e.a 2
5.b even 2 1 1000.1.e.b 2
5.c odd 4 2 1000.1.g.a 4
8.b even 2 1 4000.1.e.b 2
8.d odd 2 1 1000.1.e.b 2
20.d odd 2 1 4000.1.e.b 2
20.e even 4 2 4000.1.g.a 4
40.e odd 2 1 CM 1000.1.e.a 2
40.f even 2 1 4000.1.e.a 2
40.i odd 4 2 4000.1.g.a 4
40.k even 4 2 1000.1.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.1.e.a 2 1.a even 1 1 trivial
1000.1.e.a 2 40.e odd 2 1 CM
1000.1.e.b 2 5.b even 2 1
1000.1.e.b 2 8.d odd 2 1
1000.1.g.a 4 5.c odd 4 2
1000.1.g.a 4 40.k even 4 2
4000.1.e.a 2 4.b odd 2 1
4000.1.e.a 2 40.f even 2 1
4000.1.e.b 2 8.b even 2 1
4000.1.e.b 2 20.d odd 2 1
4000.1.g.a 4 20.e even 4 2
4000.1.g.a 4 40.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - T_{7} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1000, [\chi])\).