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 disc. -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 \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
40.e Odd 1 CM by \(\Q(\sqrt{-10}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7}^{2} \) \(\mathstrut -\mathstrut T_{7} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(4000, [\chi])\).