Properties

Label 4000.1.e.b
Level 4000
Weight 1
Character orbit 4000.e
Self dual yes
Analytic conductor 1.996
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM discriminant -40
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 4000.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.99626005053\)
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: no (minimal twist has level 1000)
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.

\(f(q)\) \(=\) \( q + \beta q^{7} + q^{9} +O(q^{10})\) \( q + \beta q^{7} + q^{9} + ( 1 - \beta ) q^{11} + ( -1 + \beta ) q^{13} + \beta q^{19} + ( 1 - \beta ) q^{23} -\beta q^{37} -\beta q^{41} + ( 1 - \beta ) q^{47} + \beta q^{49} -\beta q^{53} + \beta q^{59} + \beta q^{63} - q^{77} + q^{81} + ( -1 + \beta ) q^{89} + q^{91} + ( 1 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + q^{7} + 2q^{9} + q^{11} - q^{13} + q^{19} + q^{23} - q^{37} - q^{41} + q^{47} + q^{49} - q^{53} + q^{59} + q^{63} - 2q^{77} + 2q^{81} - q^{89} + 2q^{91} + q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1999.1
−0.618034
1.61803
0 0 0 0 0 −0.618034 0 1.00000 0
1999.2 0 0 0 0 0 1.61803 0 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 4000.1.e.b 2
4.b odd 2 1 1000.1.e.b 2
5.b even 2 1 4000.1.e.a 2
5.c odd 4 2 4000.1.g.a 4
8.b even 2 1 1000.1.e.a 2
8.d odd 2 1 4000.1.e.a 2
20.d odd 2 1 1000.1.e.a 2
20.e even 4 2 1000.1.g.a 4
40.e odd 2 1 CM 4000.1.e.b 2
40.f even 2 1 1000.1.e.b 2
40.i odd 4 2 1000.1.g.a 4
40.k even 4 2 4000.1.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.1.e.a 2 8.b even 2 1
1000.1.e.a 2 20.d odd 2 1
1000.1.e.b 2 4.b odd 2 1
1000.1.e.b 2 40.f even 2 1
1000.1.g.a 4 20.e even 4 2
1000.1.g.a 4 40.i odd 4 2
4000.1.e.a 2 5.b even 2 1
4000.1.e.a 2 8.d odd 2 1
4000.1.e.b 2 1.a even 1 1 trivial
4000.1.e.b 2 40.e odd 2 1 CM
4000.1.g.a 4 5.c odd 4 2
4000.1.g.a 4 40.k even 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 \(S_{1}^{\mathrm{new}}(4000, [\chi])\).

Hecke characteristic polynomials

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