Properties

Label 800.1.e.a
Level 800
Weight 1
Character orbit 800.e
Analytic conductor 0.399
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM disc. -8
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.399252010106\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.200.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -i q^{3} +O(q^{10})\) \( q -i q^{3} + q^{11} -i q^{17} - q^{19} -i q^{27} -i q^{33} - q^{41} + 2 i q^{43} - q^{49} - q^{51} + i q^{57} + 2 q^{59} + i q^{67} + i q^{73} - q^{81} -i q^{83} + q^{89} + 2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 2q^{11} - 2q^{19} - 2q^{41} - 2q^{49} - 2q^{51} + 4q^{59} - 2q^{81} + 2q^{89} + O(q^{100}) \)

Character Values

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

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
399.1
1.00000i
1.00000i
0 1.00000i 0 0 0 0 0 0 0
399.2 0 1.00000i 0 0 0 0 0 0 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
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes
5.b Even 1 yes
40.e Odd 1 yes

Hecke kernels

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