Properties

Label 4000.1.b.b
Level 4000
Weight 1
Character orbit 4000.b
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.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{3} q^{7} \) \( + \beta_{2} q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{3} q^{7} \) \( + \beta_{2} q^{9} \) \( -\beta_{3} q^{11} \) \( -\beta_{2} q^{13} \) \(+ q^{17}\) \( -\beta_{1} q^{19} \) \( + \beta_{2} q^{21} \) \( -\beta_{3} q^{27} \) \(- q^{29}\) \( + ( -\beta_{1} - \beta_{3} ) q^{31} \) \( -\beta_{2} q^{33} \) \( + ( -\beta_{1} + \beta_{3} ) q^{39} \) \(+ q^{41}\) \( + \beta_{3} q^{43} \) \( + ( \beta_{1} + \beta_{3} ) q^{47} \) \( -\beta_{1} q^{51} \) \( + ( 1 + \beta_{2} ) q^{53} \) \( + ( -1 + \beta_{2} ) q^{57} \) \( + ( \beta_{1} + \beta_{3} ) q^{59} \) \( + \beta_{2} q^{61} \) \( + \beta_{1} q^{63} \) \( + ( -\beta_{1} - \beta_{3} ) q^{67} \) \( -\beta_{3} q^{71} \) \( + ( 1 + \beta_{2} ) q^{73} \) \(+ q^{77}\) \( -\beta_{3} q^{79} \) \( + \beta_{1} q^{87} \) \( -\beta_{1} q^{91} \) \(- q^{93}\) \( + ( -1 - \beta_{2} ) q^{97} \) \( -\beta_{1} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 4q^{17} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut +\mathstrut 2q^{33} \) \(\mathstrut +\mathstrut 4q^{41} \) \(\mathstrut +\mathstrut 2q^{53} \) \(\mathstrut -\mathstrut 6q^{57} \) \(\mathstrut -\mathstrut 2q^{61} \) \(\mathstrut +\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut 4q^{77} \) \(\mathstrut -\mathstrut 4q^{93} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2751.1
1.61803i
0.618034i
0.618034i
1.61803i
0 1.61803i 0 0 0 1.00000i 0 −1.61803 0
2751.2 0 0.618034i 0 0 0 1.00000i 0 0.618034 0
2751.3 0 0.618034i 0 0 0 1.00000i 0 0.618034 0
2751.4 0 1.61803i 0 0 0 1.00000i 0 −1.61803 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
4.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{13}^{2} \) \(\mathstrut -\mathstrut T_{13} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(4000, [\chi])\).