Properties

Label 400.1.x.a
Level 400
Weight 1
Character orbit 400.x
Analytic conductor 0.200
Analytic rank 0
Dimension 4
Projective image \(D_{10}\)
CM discriminant -4
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 400.x (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.199626005053\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{10}\)
Projective field Galois closure of 10.2.195312500000000.4

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{10}^{3} q^{5} -\zeta_{10}^{4} q^{9} +O(q^{10})\) \( q + \zeta_{10}^{3} q^{5} -\zeta_{10}^{4} q^{9} + ( \zeta_{10} + \zeta_{10}^{2} ) q^{13} + ( -\zeta_{10}^{2} + \zeta_{10}^{4} ) q^{17} -\zeta_{10} q^{25} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{29} + ( -1 - \zeta_{10}^{3} ) q^{37} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{41} + \zeta_{10}^{2} q^{45} - q^{49} + ( -1 + \zeta_{10}^{4} ) q^{53} + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{61} + ( -1 + \zeta_{10}^{4} ) q^{65} + ( -\zeta_{10}^{3} - \zeta_{10}^{4} ) q^{73} -\zeta_{10}^{3} q^{81} + ( 1 - \zeta_{10}^{2} ) q^{85} + ( 1 + \zeta_{10}^{2} ) q^{89} + ( -\zeta_{10} + \zeta_{10}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{5} + q^{9} + O(q^{10}) \) \( 4q + q^{5} + q^{9} - q^{25} - 2q^{29} - 5q^{37} + 2q^{41} - q^{45} - 4q^{49} - 5q^{53} + 2q^{61} - 5q^{65} - q^{81} + 5q^{85} + 3q^{89} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
0 0 0 0.809017 0.587785i 0 0 0 −0.309017 0.951057i 0
159.1 0 0 0 −0.309017 + 0.951057i 0 0 0 0.809017 0.587785i 0
239.1 0 0 0 −0.309017 0.951057i 0 0 0 0.809017 + 0.587785i 0
319.1 0 0 0 0.809017 + 0.587785i 0 0 0 −0.309017 + 0.951057i 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 \(\Q(\sqrt{-1}) \)
25.e even 10 1 inner
100.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.1.x.a 4
3.b odd 2 1 3600.1.ct.a 4
4.b odd 2 1 CM 400.1.x.a 4
5.b even 2 1 2000.1.x.a 4
5.c odd 4 2 2000.1.z.a 8
8.b even 2 1 1600.1.bf.a 4
8.d odd 2 1 1600.1.bf.a 4
12.b even 2 1 3600.1.ct.a 4
20.d odd 2 1 2000.1.x.a 4
20.e even 4 2 2000.1.z.a 8
25.d even 5 1 2000.1.x.a 4
25.e even 10 1 inner 400.1.x.a 4
25.f odd 20 2 2000.1.z.a 8
75.h odd 10 1 3600.1.ct.a 4
100.h odd 10 1 inner 400.1.x.a 4
100.j odd 10 1 2000.1.x.a 4
100.l even 20 2 2000.1.z.a 8
200.o even 10 1 1600.1.bf.a 4
200.s odd 10 1 1600.1.bf.a 4
300.r even 10 1 3600.1.ct.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.1.x.a 4 1.a even 1 1 trivial
400.1.x.a 4 4.b odd 2 1 CM
400.1.x.a 4 25.e even 10 1 inner
400.1.x.a 4 100.h odd 10 1 inner
1600.1.bf.a 4 8.b even 2 1
1600.1.bf.a 4 8.d odd 2 1
1600.1.bf.a 4 200.o even 10 1
1600.1.bf.a 4 200.s odd 10 1
2000.1.x.a 4 5.b even 2 1
2000.1.x.a 4 20.d odd 2 1
2000.1.x.a 4 25.d even 5 1
2000.1.x.a 4 100.j odd 10 1
2000.1.z.a 8 5.c odd 4 2
2000.1.z.a 8 20.e even 4 2
2000.1.z.a 8 25.f odd 20 2
2000.1.z.a 8 100.l even 20 2
3600.1.ct.a 4 3.b odd 2 1
3600.1.ct.a 4 12.b even 2 1
3600.1.ct.a 4 75.h odd 10 1
3600.1.ct.a 4 300.r even 10 1

Hecke kernels

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