Properties

Label 200.1.e.a
Level 200
Weight 1
Character orbit 200.e
Analytic conductor 0.100
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM discriminant -8
Inner twists 4

Related objects

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Newspace parameters

Level: \( N \) = \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 200.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0998130025266\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.200.1

$q$-expansion

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 0 1.00000i 0 0
99.2 1.00000i 1.00000i −1.00000 0 −1.00000 0 1.00000i 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 200.1.e.a 2
3.b odd 2 1 1800.1.p.a 2
4.b odd 2 1 800.1.e.a 2
5.b even 2 1 inner 200.1.e.a 2
5.c odd 4 1 200.1.g.a 1
5.c odd 4 1 200.1.g.b yes 1
8.b even 2 1 800.1.e.a 2
8.d odd 2 1 CM 200.1.e.a 2
15.d odd 2 1 1800.1.p.a 2
15.e even 4 1 1800.1.g.a 1
15.e even 4 1 1800.1.g.b 1
20.d odd 2 1 800.1.e.a 2
20.e even 4 1 800.1.g.a 1
20.e even 4 1 800.1.g.b 1
24.f even 2 1 1800.1.p.a 2
40.e odd 2 1 inner 200.1.e.a 2
40.f even 2 1 800.1.e.a 2
40.i odd 4 1 800.1.g.a 1
40.i odd 4 1 800.1.g.b 1
40.k even 4 1 200.1.g.a 1
40.k even 4 1 200.1.g.b yes 1
120.m even 2 1 1800.1.p.a 2
120.q odd 4 1 1800.1.g.a 1
120.q odd 4 1 1800.1.g.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.1.e.a 2 1.a even 1 1 trivial
200.1.e.a 2 5.b even 2 1 inner
200.1.e.a 2 8.d odd 2 1 CM
200.1.e.a 2 40.e odd 2 1 inner
200.1.g.a 1 5.c odd 4 1
200.1.g.a 1 40.k even 4 1
200.1.g.b yes 1 5.c odd 4 1
200.1.g.b yes 1 40.k even 4 1
800.1.e.a 2 4.b odd 2 1
800.1.e.a 2 8.b even 2 1
800.1.e.a 2 20.d odd 2 1
800.1.e.a 2 40.f even 2 1
800.1.g.a 1 20.e even 4 1
800.1.g.a 1 40.i odd 4 1
800.1.g.b 1 20.e even 4 1
800.1.g.b 1 40.i odd 4 1
1800.1.g.a 1 15.e even 4 1
1800.1.g.a 1 120.q odd 4 1
1800.1.g.b 1 15.e even 4 1
1800.1.g.b 1 120.q odd 4 1
1800.1.p.a 2 3.b odd 2 1
1800.1.p.a 2 15.d odd 2 1
1800.1.p.a 2 24.f even 2 1
1800.1.p.a 2 120.m even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(200, [\chi])\).