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 discriminant -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 \(2\), degree \(1\), not minimal)

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: \( 1 \)
Twist minimal: no (minimal twist has level 200)
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 Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
5.b even 2 1 inner
40.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.1.e.a 2
4.b odd 2 1 200.1.e.a 2
5.b even 2 1 inner 800.1.e.a 2
5.c odd 4 1 800.1.g.a 1
5.c odd 4 1 800.1.g.b 1
8.b even 2 1 200.1.e.a 2
8.d odd 2 1 CM 800.1.e.a 2
12.b even 2 1 1800.1.p.a 2
20.d odd 2 1 200.1.e.a 2
20.e even 4 1 200.1.g.a 1
20.e even 4 1 200.1.g.b yes 1
24.h odd 2 1 1800.1.p.a 2
40.e odd 2 1 inner 800.1.e.a 2
40.f even 2 1 200.1.e.a 2
40.i odd 4 1 200.1.g.a 1
40.i odd 4 1 200.1.g.b yes 1
40.k even 4 1 800.1.g.a 1
40.k even 4 1 800.1.g.b 1
60.h even 2 1 1800.1.p.a 2
60.l odd 4 1 1800.1.g.a 1
60.l odd 4 1 1800.1.g.b 1
120.i odd 2 1 1800.1.p.a 2
120.w even 4 1 1800.1.g.a 1
120.w even 4 1 1800.1.g.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.1.e.a 2 4.b odd 2 1
200.1.e.a 2 8.b even 2 1
200.1.e.a 2 20.d odd 2 1
200.1.e.a 2 40.f even 2 1
200.1.g.a 1 20.e even 4 1
200.1.g.a 1 40.i odd 4 1
200.1.g.b yes 1 20.e even 4 1
200.1.g.b yes 1 40.i odd 4 1
800.1.e.a 2 1.a even 1 1 trivial
800.1.e.a 2 5.b even 2 1 inner
800.1.e.a 2 8.d odd 2 1 CM
800.1.e.a 2 40.e odd 2 1 inner
800.1.g.a 1 5.c odd 4 1
800.1.g.a 1 40.k even 4 1
800.1.g.b 1 5.c odd 4 1
800.1.g.b 1 40.k even 4 1
1800.1.g.a 1 60.l odd 4 1
1800.1.g.a 1 120.w even 4 1
1800.1.g.b 1 60.l odd 4 1
1800.1.g.b 1 120.w even 4 1
1800.1.p.a 2 12.b even 2 1
1800.1.p.a 2 24.h odd 2 1
1800.1.p.a 2 60.h even 2 1
1800.1.p.a 2 120.i odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(800, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( 1 - T^{2} + T^{4} \)
$19$ \( ( 1 + T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$31$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$37$ \( ( 1 + T^{2} )^{2} \)
$41$ \( ( 1 + T + T^{2} )^{2} \)
$43$ \( ( 1 + T^{2} )^{2} \)
$47$ \( ( 1 + T^{2} )^{2} \)
$53$ \( ( 1 + T^{2} )^{2} \)
$59$ \( ( 1 - T )^{4} \)
$61$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$67$ \( 1 - T^{2} + T^{4} \)
$71$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$73$ \( 1 - T^{2} + T^{4} \)
$79$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$83$ \( 1 - T^{2} + T^{4} \)
$89$ \( ( 1 - T + T^{2} )^{2} \)
$97$ \( ( 1 + T^{2} )^{2} \)
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