Properties

Label 400.6.a.n
Level $400$
Weight $6$
Character orbit 400.a
Self dual yes
Analytic conductor $64.154$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 24q^{3} - 172q^{7} + 333q^{9} + O(q^{10}) \) \( q + 24q^{3} - 172q^{7} + 333q^{9} - 132q^{11} + 946q^{13} + 222q^{17} - 500q^{19} - 4128q^{21} + 3564q^{23} + 2160q^{27} + 2190q^{29} - 2312q^{31} - 3168q^{33} + 11242q^{37} + 22704q^{39} + 1242q^{41} + 20624q^{43} + 6588q^{47} + 12777q^{49} + 5328q^{51} + 21066q^{53} - 12000q^{57} - 7980q^{59} + 16622q^{61} - 57276q^{63} + 1808q^{67} + 85536q^{69} + 24528q^{71} - 20474q^{73} + 22704q^{77} + 46240q^{79} - 29079q^{81} - 51576q^{83} + 52560q^{87} - 110310q^{89} - 162712q^{91} - 55488q^{93} + 78382q^{97} - 43956q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 24.0000 0 0 0 −172.000 0 333.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.6.a.n 1
4.b odd 2 1 50.6.a.d 1
5.b even 2 1 80.6.a.a 1
5.c odd 4 2 400.6.c.b 2
12.b even 2 1 450.6.a.l 1
15.d odd 2 1 720.6.a.j 1
20.d odd 2 1 10.6.a.b 1
20.e even 4 2 50.6.b.a 2
40.e odd 2 1 320.6.a.b 1
40.f even 2 1 320.6.a.o 1
60.h even 2 1 90.6.a.d 1
60.l odd 4 2 450.6.c.h 2
140.c even 2 1 490.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.b 1 20.d odd 2 1
50.6.a.d 1 4.b odd 2 1
50.6.b.a 2 20.e even 4 2
80.6.a.a 1 5.b even 2 1
90.6.a.d 1 60.h even 2 1
320.6.a.b 1 40.e odd 2 1
320.6.a.o 1 40.f even 2 1
400.6.a.n 1 1.a even 1 1 trivial
400.6.c.b 2 5.c odd 4 2
450.6.a.l 1 12.b even 2 1
450.6.c.h 2 60.l odd 4 2
490.6.a.a 1 140.c even 2 1
720.6.a.j 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 24 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).