Properties

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

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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 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{3} + 192q^{7} - 227q^{9} + O(q^{10}) \) \( q - 4q^{3} + 192q^{7} - 227q^{9} + 148q^{11} - 286q^{13} + 1678q^{17} - 1060q^{19} - 768q^{21} + 2976q^{23} + 1880q^{27} - 3410q^{29} + 2448q^{31} - 592q^{33} - 182q^{37} + 1144q^{39} - 9398q^{41} - 1244q^{43} - 12088q^{47} + 20057q^{49} - 6712q^{51} - 23846q^{53} + 4240q^{57} + 20020q^{59} + 32302q^{61} - 43584q^{63} + 60972q^{67} - 11904q^{69} + 32648q^{71} + 38774q^{73} + 28416q^{77} + 33360q^{79} + 47641q^{81} + 16716q^{83} + 13640q^{87} + 101370q^{89} - 54912q^{91} - 9792q^{93} + 119038q^{97} - 33596q^{99} + O(q^{100}) \)

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 −4.00000 0 0 0 192.000 0 −227.000 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.g 1
4.b odd 2 1 25.6.a.a 1
5.b even 2 1 80.6.a.e 1
5.c odd 4 2 400.6.c.j 2
12.b even 2 1 225.6.a.f 1
15.d odd 2 1 720.6.a.a 1
20.d odd 2 1 5.6.a.a 1
20.e even 4 2 25.6.b.a 2
40.e odd 2 1 320.6.a.j 1
40.f even 2 1 320.6.a.g 1
60.h even 2 1 45.6.a.b 1
60.l odd 4 2 225.6.b.e 2
140.c even 2 1 245.6.a.b 1
220.g even 2 1 605.6.a.a 1
260.g odd 2 1 845.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 20.d odd 2 1
25.6.a.a 1 4.b odd 2 1
25.6.b.a 2 20.e even 4 2
45.6.a.b 1 60.h even 2 1
80.6.a.e 1 5.b even 2 1
225.6.a.f 1 12.b even 2 1
225.6.b.e 2 60.l odd 4 2
245.6.a.b 1 140.c even 2 1
320.6.a.g 1 40.f even 2 1
320.6.a.j 1 40.e odd 2 1
400.6.a.g 1 1.a even 1 1 trivial
400.6.c.j 2 5.c odd 4 2
605.6.a.a 1 220.g even 2 1
720.6.a.a 1 15.d odd 2 1
845.6.a.b 1 260.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 4 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 4 + T \)
$5$ \( T \)
$7$ \( -192 + T \)
$11$ \( -148 + T \)
$13$ \( 286 + T \)
$17$ \( -1678 + T \)
$19$ \( 1060 + T \)
$23$ \( -2976 + T \)
$29$ \( 3410 + T \)
$31$ \( -2448 + T \)
$37$ \( 182 + T \)
$41$ \( 9398 + T \)
$43$ \( 1244 + T \)
$47$ \( 12088 + T \)
$53$ \( 23846 + T \)
$59$ \( -20020 + T \)
$61$ \( -32302 + T \)
$67$ \( -60972 + T \)
$71$ \( -32648 + T \)
$73$ \( -38774 + T \)
$79$ \( -33360 + T \)
$83$ \( -16716 + T \)
$89$ \( -101370 + T \)
$97$ \( -119038 + T \)
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