Properties

Label 800.1.bo.a
Level 800
Weight 1
Character orbit 800.bo
Analytic conductor 0.399
Analytic rank 0
Dimension 8
Projective image \(D_{20}\)
CM disc. -4
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 800.bo (of order \(20\) and degree \(8\))

Newform invariants

Self dual: No
Analytic conductor: \(0.399252010106\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{20}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{20} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{20}^{7} q^{5} + \zeta_{20} q^{9} +O(q^{10})\) \( q + \zeta_{20}^{7} q^{5} + \zeta_{20} q^{9} + ( -\zeta_{20}^{8} + \zeta_{20}^{9} ) q^{13} + ( \zeta_{20}^{3} + \zeta_{20}^{6} ) q^{17} -\zeta_{20}^{4} q^{25} + ( \zeta_{20}^{2} + \zeta_{20}^{4} ) q^{29} + ( -\zeta_{20}^{2} + \zeta_{20}^{5} ) q^{37} + ( -\zeta_{20}^{3} - \zeta_{20}^{9} ) q^{41} + \zeta_{20}^{8} q^{45} -\zeta_{20}^{5} q^{49} + ( -\zeta_{20}^{5} - \zeta_{20}^{6} ) q^{53} + ( -\zeta_{20} - \zeta_{20}^{7} ) q^{61} + ( \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{65} + ( -\zeta_{20} - \zeta_{20}^{2} ) q^{73} + \zeta_{20}^{2} q^{81} + ( -1 - \zeta_{20}^{3} ) q^{85} + ( -1 + \zeta_{20}^{8} ) q^{89} + ( \zeta_{20}^{4} - \zeta_{20}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 2q^{13} + 2q^{17} + 2q^{25} - 2q^{37} - 2q^{45} - 2q^{53} - 2q^{65} - 2q^{73} + 2q^{81} - 8q^{85} - 10q^{89} - 2q^{97} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{20}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−0.951057 0.309017i
−0.951057 + 0.309017i
0.951057 0.309017i
0.951057 + 0.309017i
0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 + 0.809017i
−0.587785 0.809017i
0 0 0 0.587785 0.809017i 0 0 0 −0.951057 0.309017i 0
97.1 0 0 0 0.587785 + 0.809017i 0 0 0 −0.951057 + 0.309017i 0
353.1 0 0 0 −0.587785 0.809017i 0 0 0 0.951057 0.309017i 0
417.1 0 0 0 −0.587785 + 0.809017i 0 0 0 0.951057 + 0.309017i 0
513.1 0 0 0 0.951057 + 0.309017i 0 0 0 0.587785 + 0.809017i 0
577.1 0 0 0 0.951057 0.309017i 0 0 0 0.587785 0.809017i 0
673.1 0 0 0 −0.951057 + 0.309017i 0 0 0 −0.587785 + 0.809017i 0
737.1 0 0 0 −0.951057 0.309017i 0 0 0 −0.587785 0.809017i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.1
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes
25.f Odd 1 yes
100.l Even 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(800, [\chi])\).