Properties

Label 4000.1.bf.b
Level 4000
Weight 1
Character orbit 4000.bf
Analytic conductor 1.996
Analytic rank 0
Dimension 8
Projective image \(A_{5}\)
CM/RM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 4000.bf (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Projective image \(A_{5}\)
Projective field Galois closure of 5.1.25000000.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q\) \( + \zeta_{20}^{2} q^{3} \) \( + ( \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{7} \) \(+O(q^{10})\) \( q\) \( + \zeta_{20}^{2} q^{3} \) \( + ( \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{7} \) \( + ( -\zeta_{20}^{3} + \zeta_{20}^{5} ) q^{13} \) \( + ( \zeta_{20} + \zeta_{20}^{5} ) q^{19} \) \( + ( \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{21} \) \( + \zeta_{20}^{6} q^{23} \) \( -\zeta_{20}^{6} q^{27} \) \( + ( 1 + \zeta_{20}^{4} ) q^{29} \) \( -\zeta_{20}^{3} q^{31} \) \( + \zeta_{20}^{9} q^{37} \) \( + ( -\zeta_{20}^{5} + \zeta_{20}^{7} ) q^{39} \) \( + ( \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{43} \) \( + ( \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{47} \) \( + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{8} ) q^{49} \) \( + \zeta_{20}^{7} q^{53} \) \( + ( \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{57} \) \( + ( \zeta_{20} - \zeta_{20}^{7} ) q^{59} \) \( -\zeta_{20}^{6} q^{61} \) \( + \zeta_{20}^{8} q^{69} \) \( + ( \zeta_{20} - \zeta_{20}^{3} ) q^{71} \) \( + \zeta_{20} q^{73} \) \( + ( -\zeta_{20}^{5} - \zeta_{20}^{9} ) q^{79} \) \( -\zeta_{20}^{8} q^{81} \) \( + \zeta_{20}^{8} q^{83} \) \( + ( \zeta_{20}^{2} + \zeta_{20}^{6} ) q^{87} \) \( + ( \zeta_{20} - \zeta_{20}^{7} + 2 \zeta_{20}^{9} ) q^{91} \) \( -\zeta_{20}^{5} q^{93} \) \( + ( \zeta_{20} - \zeta_{20}^{3} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 2q^{23} \) \(\mathstrut -\mathstrut 2q^{27} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut +\mathstrut 4q^{43} \) \(\mathstrut +\mathstrut 4q^{47} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 2q^{61} \) \(\mathstrut -\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 2q^{83} \) \(\mathstrut +\mathstrut 4q^{87} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(\zeta_{20}^{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
−0.587785 + 0.809017i
0.587785 0.809017i
0.951057 0.309017i
−0.951057 + 0.309017i
0.951057 + 0.309017i
−0.951057 0.309017i
−0.587785 0.809017i
0.587785 + 0.809017i
0 −0.309017 0.951057i 0 0 0 −1.61803 0 0 0
799.2 0 −0.309017 0.951057i 0 0 0 −1.61803 0 0 0
1599.1 0 0.809017 0.587785i 0 0 0 0.618034 0 0 0
1599.2 0 0.809017 0.587785i 0 0 0 0.618034 0 0 0
2399.1 0 0.809017 + 0.587785i 0 0 0 0.618034 0 0 0
2399.2 0 0.809017 + 0.587785i 0 0 0 0.618034 0 0 0
3199.1 0 −0.309017 + 0.951057i 0 0 0 −1.61803 0 0 0
3199.2 0 −0.309017 + 0.951057i 0 0 0 −1.61803 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3199.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
20.d Odd 1 yes
25.d Even 1 yes
100.h Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{4} \) \(\mathstrut -\mathstrut T_{3}^{3} \) \(\mathstrut +\mathstrut T_{3}^{2} \) \(\mathstrut -\mathstrut T_{3} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(4000, [\chi])\).