Properties

Label 4000.1.p.b
Level 4000
Weight 1
Character orbit 4000.p
Analytic conductor 1.996
Analytic rank 0
Dimension 8
Projective image \(D_{20}\)
CM disc. -20
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Projective image \(D_{20}\)
Projective field Galois closure of 20.0.320000000000000000000000000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( \zeta_{20}^{6} + \zeta_{20}^{9} ) q^{3} \) \( + ( -\zeta_{20}^{2} - \zeta_{20}^{3} ) q^{7} \) \( + ( -\zeta_{20}^{2} - \zeta_{20}^{5} - \zeta_{20}^{8} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( \zeta_{20}^{6} + \zeta_{20}^{9} ) q^{3} \) \( + ( -\zeta_{20}^{2} - \zeta_{20}^{3} ) q^{7} \) \( + ( -\zeta_{20}^{2} - \zeta_{20}^{5} - \zeta_{20}^{8} ) q^{9} \) \( + ( \zeta_{20} + \zeta_{20}^{2} - \zeta_{20}^{8} - \zeta_{20}^{9} ) q^{21} \) \( + ( \zeta_{20}^{2} - \zeta_{20}^{3} ) q^{23} \) \( + ( \zeta_{20} + \zeta_{20}^{4} + \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{27} \) \( + ( \zeta_{20} + \zeta_{20}^{9} ) q^{29} \) \( + ( -\zeta_{20} + \zeta_{20}^{9} ) q^{41} \) \( + ( -\zeta_{20} + \zeta_{20}^{4} ) q^{43} \) \( + ( \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{47} \) \( + ( \zeta_{20}^{4} + \zeta_{20}^{5} + \zeta_{20}^{6} ) q^{49} \) \( + ( \zeta_{20}^{3} - \zeta_{20}^{7} ) q^{61} \) \( + ( -1 - \zeta_{20} + \zeta_{20}^{4} + \zeta_{20}^{5} + \zeta_{20}^{7} + \zeta_{20}^{8} ) q^{63} \) \( + ( 1 + \zeta_{20}^{5} ) q^{67} \) \( + ( -\zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{8} - \zeta_{20}^{9} ) q^{69} \) \( + ( -1 - \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{81} \) \( + ( \zeta_{20}^{7} + \zeta_{20}^{8} ) q^{83} \) \( + ( -1 - \zeta_{20}^{5} + \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{87} \) \( + ( \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{89} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 2q^{23} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 2q^{47} \) \(\mathstrut -\mathstrut 12q^{63} \) \(\mathstrut +\mathstrut 8q^{67} \) \(\mathstrut -\mathstrut 12q^{81} \) \(\mathstrut -\mathstrut 2q^{83} \) \(\mathstrut -\mathstrut 6q^{87} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(\zeta_{20}^{5}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.951057 0.309017i
0.587785 + 0.809017i
−0.951057 0.309017i
−0.587785 + 0.809017i
0.951057 + 0.309017i
0.587785 0.809017i
−0.951057 + 0.309017i
−0.587785 0.809017i
0 −1.26007 1.26007i 0 0 0 −1.39680 + 1.39680i 0 2.17557i 0
193.2 0 0.221232 + 0.221232i 0 0 0 1.26007 1.26007i 0 0.902113i 0
193.3 0 0.642040 + 0.642040i 0 0 0 −0.221232 + 0.221232i 0 0.175571i 0
193.4 0 1.39680 + 1.39680i 0 0 0 −0.642040 + 0.642040i 0 2.90211i 0
1057.1 0 −1.26007 + 1.26007i 0 0 0 −1.39680 1.39680i 0 2.17557i 0
1057.2 0 0.221232 0.221232i 0 0 0 1.26007 + 1.26007i 0 0.902113i 0
1057.3 0 0.642040 0.642040i 0 0 0 −0.221232 0.221232i 0 0.175571i 0
1057.4 0 1.39680 1.39680i 0 0 0 −0.642040 0.642040i 0 2.90211i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1057.4
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
20.d Odd 1 CM by \(\Q(\sqrt{-5}) \) yes
5.c Odd 1 yes
20.e Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{8} \) \(\mathstrut -\mathstrut 2 T_{3}^{7} \) \(\mathstrut +\mathstrut 2 T_{3}^{6} \) \(\mathstrut +\mathstrut 11 T_{3}^{4} \) \(\mathstrut -\mathstrut 20 T_{3}^{3} \) \(\mathstrut +\mathstrut 18 T_{3}^{2} \) \(\mathstrut -\mathstrut 6 T_{3} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(4000, [\chi])\).