Properties

Label 4000.1.h.b
Level 4000
Weight 1
Character orbit 4000.h
Analytic conductor 1.996
Analytic rank 0
Dimension 4
Projective image \(A_{5}\)
CM/RM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 4000.h (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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(A_{5}\)
Projective field Galois closure of 5.1.1000000.2

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{3} - q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{3} - q^{7} + ( 1 + \beta_{2} ) q^{9} + \beta_{3} q^{11} + ( \beta_{1} + \beta_{3} ) q^{13} -\beta_{3} q^{17} + ( \beta_{1} + \beta_{3} ) q^{19} + ( -1 - \beta_{2} ) q^{21} + q^{27} + q^{29} -\beta_{1} q^{31} + ( \beta_{1} + \beta_{3} ) q^{33} + ( \beta_{1} + 2 \beta_{3} ) q^{39} + q^{41} + q^{43} + \beta_{2} q^{47} + ( -\beta_{1} - \beta_{3} ) q^{51} -\beta_{1} q^{53} + ( \beta_{1} + 2 \beta_{3} ) q^{57} -\beta_{1} q^{59} + ( -1 - \beta_{2} ) q^{61} + ( -1 - \beta_{2} ) q^{63} -\beta_{2} q^{67} + \beta_{3} q^{71} -\beta_{1} q^{73} -\beta_{3} q^{77} -\beta_{3} q^{79} + ( 1 + \beta_{2} ) q^{87} + ( -\beta_{1} - \beta_{3} ) q^{91} -\beta_{3} q^{93} -\beta_{1} q^{97} + ( \beta_{1} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 4q^{7} + 2q^{9} - 2q^{21} + 4q^{27} + 4q^{29} + 4q^{41} + 4q^{43} - 2q^{47} - 2q^{61} - 2q^{63} + 2q^{67} + 2q^{87} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3999.1
1.61803i
1.61803i
0.618034i
0.618034i
0 −0.618034 0 0 0 −1.00000 0 −0.618034 0
3999.2 0 −0.618034 0 0 0 −1.00000 0 −0.618034 0
3999.3 0 1.61803 0 0 0 −1.00000 0 1.61803 0
3999.4 0 1.61803 0 0 0 −1.00000 0 1.61803 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
20.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.h.b 4
4.b odd 2 1 4000.1.h.a 4
5.b even 2 1 4000.1.h.a 4
5.c odd 4 1 4000.1.b.a 4
5.c odd 4 1 4000.1.b.b yes 4
20.d odd 2 1 inner 4000.1.h.b 4
20.e even 4 1 4000.1.b.a 4
20.e even 4 1 4000.1.b.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.1.b.a 4 5.c odd 4 1
4000.1.b.a 4 20.e even 4 1
4000.1.b.b yes 4 5.c odd 4 1
4000.1.b.b yes 4 20.e even 4 1
4000.1.h.a 4 4.b odd 2 1
4000.1.h.a 4 5.b even 2 1
4000.1.h.b 4 1.a even 1 1 trivial
4000.1.h.b 4 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} - 1 \) acting on \(S_{1}^{\mathrm{new}}(4000, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$5$ 1
$7$ \( ( 1 + T + T^{2} )^{4} \)
$11$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$13$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$17$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$19$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$23$ \( ( 1 + T^{2} )^{4} \)
$29$ \( ( 1 - T + T^{2} )^{4} \)
$31$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$37$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$41$ \( ( 1 - T + T^{2} )^{4} \)
$43$ \( ( 1 - T + T^{2} )^{4} \)
$47$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$53$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$59$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$61$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$67$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$71$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$73$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$79$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$83$ \( ( 1 + T^{2} )^{4} \)
$89$ \( ( 1 + T^{2} )^{4} \)
$97$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
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