Properties

Label 4000.1.g.a
Level $4000$
Weight $1$
Character orbit 4000.g
Analytic conductor $1.996$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -40
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{3} ) q^{7} - q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{3} ) q^{7} - q^{9} -\beta_{2} q^{11} -\beta_{1} q^{13} + ( -1 - \beta_{2} ) q^{19} + \beta_{1} q^{23} + ( -\beta_{1} - \beta_{3} ) q^{37} + ( -1 - \beta_{2} ) q^{41} -\beta_{1} q^{47} + ( -1 - \beta_{2} ) q^{49} + ( \beta_{1} + \beta_{3} ) q^{53} + ( -1 - \beta_{2} ) q^{59} + ( -\beta_{1} - \beta_{3} ) q^{63} -\beta_{3} q^{77} + q^{81} -\beta_{2} q^{89} + q^{91} + \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 2q^{11} - 2q^{19} - 2q^{41} - 2q^{49} - 2q^{59} + 4q^{81} + 2q^{89} + 4q^{91} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
0.618034i
1.61803i
1.61803i
0.618034i
0 0 0 0 0 1.61803i 0 −1.00000 0
751.2 0 0 0 0 0 0.618034i 0 −1.00000 0
751.3 0 0 0 0 0 0.618034i 0 −1.00000 0
751.4 0 0 0 0 0 1.61803i 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 \(\Q(\sqrt{-10}) \)
5.b even 2 1 inner
8.d odd 2 1 inner

Twists

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

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(4000, [\chi])\).