Properties

Label 800.1.p.a
Level 800
Weight 1
Character orbit 800.p
Analytic conductor 0.399
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM disc. -20
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 800.p (of order \(4\) and degree \(2\))

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 \(D_{4}\)
Projective field Galois closure of 4.0.320.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( -1 + i ) q^{3} \) \( + ( 1 + i ) q^{7} \) \( -i q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + i ) q^{3} \) \( + ( 1 + i ) q^{7} \) \( -i q^{9} \) \( -2 q^{21} \) \( + ( -1 + i ) q^{23} \) \( + 2 i q^{29} \) \( + ( 1 - i ) q^{43} \) \( + ( -1 - i ) q^{47} \) \( + i q^{49} \) \( + ( 1 - i ) q^{63} \) \( + ( 1 + i ) q^{67} \) \( -2 i q^{69} \) \(+ q^{81}\) \( + ( 1 - i ) q^{83} \) \( + ( -2 - 2 i ) q^{87} \) \( -2 i q^{89} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 2q^{23} \) \(\mathstrut +\mathstrut 2q^{43} \) \(\mathstrut -\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut 2q^{63} \) \(\mathstrut +\mathstrut 2q^{67} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 2q^{83} \) \(\mathstrut -\mathstrut 4q^{87} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

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
1.00000i
1.00000i
0 −1.00000 1.00000i 0 0 0 1.00000 1.00000i 0 1.00000i 0
257.1 0 −1.00000 + 1.00000i 0 0 0 1.00000 + 1.00000i 0 1.00000i 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
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 \(T_{3}^{2} \) \(\mathstrut +\mathstrut 2 T_{3} \) \(\mathstrut +\mathstrut 2 \) acting on \(S_{1}^{\mathrm{new}}(800, [\chi])\).