Properties

Label 200.1.g.b
Level 200
Weight 1
Character orbit 200.g
Self dual yes
Analytic conductor 0.100
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM discriminant -8
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 200.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0998130025266\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.200.1
Artin image $S_3$
Artin field Galois closure of 3.1.200.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} - q^{11} - q^{12} + q^{16} - q^{17} - q^{19} - q^{22} - q^{24} + q^{27} + q^{32} + q^{33} - q^{34} - q^{38} - q^{41} + 2q^{43} - q^{44} - q^{48} + q^{49} + q^{51} + q^{54} + q^{57} + 2q^{59} + q^{64} + q^{66} - q^{67} - q^{68} - q^{73} - q^{76} - q^{81} - q^{82} - q^{83} + 2q^{86} - q^{88} - q^{89} - q^{96} + 2q^{97} + q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0
1.00000 −1.00000 1.00000 0 −1.00000 0 1.00000 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 \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.1.g.b yes 1
3.b odd 2 1 1800.1.g.a 1
4.b odd 2 1 800.1.g.b 1
5.b even 2 1 200.1.g.a 1
5.c odd 4 2 200.1.e.a 2
8.b even 2 1 800.1.g.b 1
8.d odd 2 1 CM 200.1.g.b yes 1
15.d odd 2 1 1800.1.g.b 1
15.e even 4 2 1800.1.p.a 2
20.d odd 2 1 800.1.g.a 1
20.e even 4 2 800.1.e.a 2
24.f even 2 1 1800.1.g.a 1
40.e odd 2 1 200.1.g.a 1
40.f even 2 1 800.1.g.a 1
40.i odd 4 2 800.1.e.a 2
40.k even 4 2 200.1.e.a 2
120.m even 2 1 1800.1.g.b 1
120.q odd 4 2 1800.1.p.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.1.e.a 2 5.c odd 4 2
200.1.e.a 2 40.k even 4 2
200.1.g.a 1 5.b even 2 1
200.1.g.a 1 40.e odd 2 1
200.1.g.b yes 1 1.a even 1 1 trivial
200.1.g.b yes 1 8.d odd 2 1 CM
800.1.e.a 2 20.e even 4 2
800.1.e.a 2 40.i odd 4 2
800.1.g.a 1 20.d odd 2 1
800.1.g.a 1 40.f even 2 1
800.1.g.b 1 4.b odd 2 1
800.1.g.b 1 8.b even 2 1
1800.1.g.a 1 3.b odd 2 1
1800.1.g.a 1 24.f even 2 1
1800.1.g.b 1 15.d odd 2 1
1800.1.g.b 1 120.m even 2 1
1800.1.p.a 2 15.e even 4 2
1800.1.p.a 2 120.q odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(200, [\chi])\).