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Properties

Label 400.6.a.m
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 20)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 22q^{3} + 218q^{7} + 241q^{9} + O(q^{10}) \) \( q + 22q^{3} + 218q^{7} + 241q^{9} + 480q^{11} + 622q^{13} - 186q^{17} + 1204q^{19} + 4796q^{21} - 3186q^{23} - 44q^{27} + 5526q^{29} - 9356q^{31} + 10560q^{33} - 5618q^{37} + 13684q^{39} - 14394q^{41} - 370q^{43} + 16146q^{47} + 30717q^{49} - 4092q^{51} + 4374q^{53} + 26488q^{57} + 11748q^{59} + 13202q^{61} + 52538q^{63} - 11542q^{67} - 70092q^{69} + 29532q^{71} - 33698q^{73} + 104640q^{77} - 31208q^{79} - 59531q^{81} - 38466q^{83} + 121572q^{87} + 119514q^{89} + 135596q^{91} - 205832q^{93} - 94658q^{97} + 115680q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 22.0000 0 0 0 218.000 0 241.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.m 1
4.b odd 2 1 100.6.a.a 1
5.b even 2 1 80.6.a.b 1
5.c odd 4 2 400.6.c.c 2
12.b even 2 1 900.6.a.b 1
15.d odd 2 1 720.6.a.l 1
20.d odd 2 1 20.6.a.a ✓ 1
20.e even 4 2 100.6.c.a 2
40.e odd 2 1 320.6.a.c 1
40.f even 2 1 320.6.a.n 1
60.h even 2 1 180.6.a.e 1
60.l odd 4 2 900.6.d.h 2
140.c even 2 1 980.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.a.a ✓ 1 20.d odd 2 1
80.6.a.b 1 5.b even 2 1
100.6.a.a 1 4.b odd 2 1
100.6.c.a 2 20.e even 4 2
180.6.a.e 1 60.h even 2 1
320.6.a.c 1 40.e odd 2 1
320.6.a.n 1 40.f even 2 1
400.6.a.m 1 1.a even 1 1 trivial
400.6.c.c 2 5.c odd 4 2
720.6.a.l 1 15.d odd 2 1
900.6.a.b 1 12.b even 2 1
900.6.d.h 2 60.l odd 4 2
980.6.a.b 1 140.c even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 22 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 22 T + 243 T^{2} \)
$5$ 1
$7$ \( 1 - 218 T + 16807 T^{2} \)
$11$ \( 1 - 480 T + 161051 T^{2} \)
$13$ \( 1 - 622 T + 371293 T^{2} \)
$17$ \( 1 + 186 T + 1419857 T^{2} \)
$19$ \( 1 - 1204 T + 2476099 T^{2} \)
$23$ \( 1 + 3186 T + 6436343 T^{2} \)
$29$ \( 1 - 5526 T + 20511149 T^{2} \)
$31$ \( 1 + 9356 T + 28629151 T^{2} \)
$37$ \( 1 + 5618 T + 69343957 T^{2} \)
$41$ \( 1 + 14394 T + 115856201 T^{2} \)
$43$ \( 1 + 370 T + 147008443 T^{2} \)
$47$ \( 1 - 16146 T + 229345007 T^{2} \)
$53$ \( 1 - 4374 T + 418195493 T^{2} \)
$59$ \( 1 - 11748 T + 714924299 T^{2} \)
$61$ \( 1 - 13202 T + 844596301 T^{2} \)
$67$ \( 1 + 11542 T + 1350125107 T^{2} \)
$71$ \( 1 - 29532 T + 1804229351 T^{2} \)
$73$ \( 1 + 33698 T + 2073071593 T^{2} \)
$79$ \( 1 + 31208 T + 3077056399 T^{2} \)
$83$ \( 1 + 38466 T + 3939040643 T^{2} \)
$89$ \( 1 - 119514 T + 5584059449 T^{2} \)
$97$ \( 1 + 94658 T + 8587340257 T^{2} \)
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