Properties

Label 400.1.b.a
Level 400
Weight 1
Character orbit 400.b
Self dual yes
Analytic conductor 0.200
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -4, -20, 5
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 400.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.199626005053\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(i, \sqrt{5})\)
Artin image $D_4$
Artin field Galois closure of 4.2.2000.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{9} + O(q^{10}) \) \( q + q^{9} - 2q^{29} - 2q^{41} + q^{49} - 2q^{61} + q^{81} - 2q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
0
0 0 0 0 0 0 0 1.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 RM by \(\Q(\sqrt{5}) \)
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.1.b.a 1
3.b odd 2 1 3600.1.e.a 1
4.b odd 2 1 CM 400.1.b.a 1
5.b even 2 1 RM 400.1.b.a 1
5.c odd 4 2 80.1.h.a 1
8.b even 2 1 1600.1.b.a 1
8.d odd 2 1 1600.1.b.a 1
12.b even 2 1 3600.1.e.a 1
15.d odd 2 1 3600.1.e.a 1
15.e even 4 2 720.1.j.a 1
20.d odd 2 1 CM 400.1.b.a 1
20.e even 4 2 80.1.h.a 1
35.f even 4 2 3920.1.j.a 1
35.k even 12 4 3920.1.bt.a 2
35.l odd 12 4 3920.1.bt.b 2
40.e odd 2 1 1600.1.b.a 1
40.f even 2 1 1600.1.b.a 1
40.i odd 4 2 320.1.h.a 1
40.k even 4 2 320.1.h.a 1
60.h even 2 1 3600.1.e.a 1
60.l odd 4 2 720.1.j.a 1
80.i odd 4 2 1280.1.e.a 2
80.j even 4 2 1280.1.e.a 2
80.s even 4 2 1280.1.e.a 2
80.t odd 4 2 1280.1.e.a 2
120.q odd 4 2 2880.1.j.a 1
120.w even 4 2 2880.1.j.a 1
140.j odd 4 2 3920.1.j.a 1
140.w even 12 4 3920.1.bt.b 2
140.x odd 12 4 3920.1.bt.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.1.h.a 1 5.c odd 4 2
80.1.h.a 1 20.e even 4 2
320.1.h.a 1 40.i odd 4 2
320.1.h.a 1 40.k even 4 2
400.1.b.a 1 1.a even 1 1 trivial
400.1.b.a 1 4.b odd 2 1 CM
400.1.b.a 1 5.b even 2 1 RM
400.1.b.a 1 20.d odd 2 1 CM
720.1.j.a 1 15.e even 4 2
720.1.j.a 1 60.l odd 4 2
1280.1.e.a 2 80.i odd 4 2
1280.1.e.a 2 80.j even 4 2
1280.1.e.a 2 80.s even 4 2
1280.1.e.a 2 80.t odd 4 2
1600.1.b.a 1 8.b even 2 1
1600.1.b.a 1 8.d odd 2 1
1600.1.b.a 1 40.e odd 2 1
1600.1.b.a 1 40.f even 2 1
2880.1.j.a 1 120.q odd 4 2
2880.1.j.a 1 120.w even 4 2
3600.1.e.a 1 3.b odd 2 1
3600.1.e.a 1 12.b even 2 1
3600.1.e.a 1 15.d odd 2 1
3600.1.e.a 1 60.h even 2 1
3920.1.j.a 1 35.f even 4 2
3920.1.j.a 1 140.j odd 4 2
3920.1.bt.a 2 35.k even 12 4
3920.1.bt.a 2 140.x odd 12 4
3920.1.bt.b 2 35.l odd 12 4
3920.1.bt.b 2 140.w even 12 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(400, [\chi])\).