Properties

Label 400.6.a.d
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 4)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 12 q^{3} - 88 q^{7} - 99 q^{9} + O(q^{10}) \) \( q - 12 q^{3} - 88 q^{7} - 99 q^{9} - 540 q^{11} + 418 q^{13} - 594 q^{17} - 836 q^{19} + 1056 q^{21} - 4104 q^{23} + 4104 q^{27} - 594 q^{29} - 4256 q^{31} + 6480 q^{33} + 298 q^{37} - 5016 q^{39} + 17226 q^{41} - 12100 q^{43} - 1296 q^{47} - 9063 q^{49} + 7128 q^{51} - 19494 q^{53} + 10032 q^{57} + 7668 q^{59} - 34738 q^{61} + 8712 q^{63} + 21812 q^{67} + 49248 q^{69} + 46872 q^{71} - 67562 q^{73} + 47520 q^{77} + 76912 q^{79} - 25191 q^{81} + 67716 q^{83} + 7128 q^{87} + 29754 q^{89} - 36784 q^{91} + 51072 q^{93} + 122398 q^{97} + 53460 q^{99} + O(q^{100}) \)

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 −12.0000 0 0 0 −88.0000 0 −99.0000 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.d 1
4.b odd 2 1 100.6.a.b 1
5.b even 2 1 16.6.a.b 1
5.c odd 4 2 400.6.c.f 2
12.b even 2 1 900.6.a.h 1
15.d odd 2 1 144.6.a.c 1
20.d odd 2 1 4.6.a.a 1
20.e even 4 2 100.6.c.b 2
35.c odd 2 1 784.6.a.d 1
40.e odd 2 1 64.6.a.f 1
40.f even 2 1 64.6.a.b 1
60.h even 2 1 36.6.a.a 1
60.l odd 4 2 900.6.d.a 2
80.k odd 4 2 256.6.b.g 2
80.q even 4 2 256.6.b.c 2
120.i odd 2 1 576.6.a.bd 1
120.m even 2 1 576.6.a.bc 1
140.c even 2 1 196.6.a.e 1
140.p odd 6 2 196.6.e.g 2
140.s even 6 2 196.6.e.d 2
180.n even 6 2 324.6.e.d 2
180.p odd 6 2 324.6.e.a 2
220.g even 2 1 484.6.a.a 1
260.g odd 2 1 676.6.a.a 1
260.u even 4 2 676.6.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 20.d odd 2 1
16.6.a.b 1 5.b even 2 1
36.6.a.a 1 60.h even 2 1
64.6.a.b 1 40.f even 2 1
64.6.a.f 1 40.e odd 2 1
100.6.a.b 1 4.b odd 2 1
100.6.c.b 2 20.e even 4 2
144.6.a.c 1 15.d odd 2 1
196.6.a.e 1 140.c even 2 1
196.6.e.d 2 140.s even 6 2
196.6.e.g 2 140.p odd 6 2
256.6.b.c 2 80.q even 4 2
256.6.b.g 2 80.k odd 4 2
324.6.e.a 2 180.p odd 6 2
324.6.e.d 2 180.n even 6 2
400.6.a.d 1 1.a even 1 1 trivial
400.6.c.f 2 5.c odd 4 2
484.6.a.a 1 220.g even 2 1
576.6.a.bc 1 120.m even 2 1
576.6.a.bd 1 120.i odd 2 1
676.6.a.a 1 260.g odd 2 1
676.6.d.a 2 260.u even 4 2
784.6.a.d 1 35.c odd 2 1
900.6.a.h 1 12.b even 2 1
900.6.d.a 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 12 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 12 + T \)
$5$ \( T \)
$7$ \( 88 + T \)
$11$ \( 540 + T \)
$13$ \( -418 + T \)
$17$ \( 594 + T \)
$19$ \( 836 + T \)
$23$ \( 4104 + T \)
$29$ \( 594 + T \)
$31$ \( 4256 + T \)
$37$ \( -298 + T \)
$41$ \( -17226 + T \)
$43$ \( 12100 + T \)
$47$ \( 1296 + T \)
$53$ \( 19494 + T \)
$59$ \( -7668 + T \)
$61$ \( 34738 + T \)
$67$ \( -21812 + T \)
$71$ \( -46872 + T \)
$73$ \( 67562 + T \)
$79$ \( -76912 + T \)
$83$ \( -67716 + T \)
$89$ \( -29754 + T \)
$97$ \( -122398 + T \)
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