Properties

Label 400.6.a.k
Level 400
Weight 6
Character orbit 400.a
Self dual yes
Analytic conductor 64.154
Analytic rank 1
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: \(1\)
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

\(f(q)\) \(=\) \( q + 14q^{3} + 158q^{7} - 47q^{9} + O(q^{10}) \) \( q + 14q^{3} + 158q^{7} - 47q^{9} + 148q^{11} - 684q^{13} - 2048q^{17} - 2220q^{19} + 2212q^{21} - 1246q^{23} - 4060q^{27} - 270q^{29} + 2048q^{31} + 2072q^{33} + 4372q^{37} - 9576q^{39} - 2398q^{41} + 2294q^{43} - 10682q^{47} + 8157q^{49} - 28672q^{51} - 2964q^{53} - 31080q^{57} + 39740q^{59} - 42298q^{61} - 7426q^{63} + 32098q^{67} - 17444q^{69} + 4248q^{71} - 30104q^{73} + 23384q^{77} - 35280q^{79} - 45419q^{81} - 27826q^{83} - 3780q^{87} - 85210q^{89} - 108072q^{91} + 28672q^{93} + 97232q^{97} - 6956q^{99} + O(q^{100}) \)

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 14.0000 0 0 0 158.000 0 −47.0000 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.k 1
4.b odd 2 1 50.6.a.e 1
5.b even 2 1 400.6.a.c 1
5.c odd 4 2 80.6.c.c 2
12.b even 2 1 450.6.a.c 1
15.e even 4 2 720.6.f.a 2
20.d odd 2 1 50.6.a.c 1
20.e even 4 2 10.6.b.a 2
40.i odd 4 2 320.6.c.a 2
40.k even 4 2 320.6.c.b 2
60.h even 2 1 450.6.a.w 1
60.l odd 4 2 90.6.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.b.a 2 20.e even 4 2
50.6.a.c 1 20.d odd 2 1
50.6.a.e 1 4.b odd 2 1
80.6.c.c 2 5.c odd 4 2
90.6.c.a 2 60.l odd 4 2
320.6.c.a 2 40.i odd 4 2
320.6.c.b 2 40.k even 4 2
400.6.a.c 1 5.b even 2 1
400.6.a.k 1 1.a even 1 1 trivial
450.6.a.c 1 12.b even 2 1
450.6.a.w 1 60.h even 2 1
720.6.f.a 2 15.e even 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 14 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 14 T + 243 T^{2} \)
$5$ 1
$7$ \( 1 - 158 T + 16807 T^{2} \)
$11$ \( 1 - 148 T + 161051 T^{2} \)
$13$ \( 1 + 684 T + 371293 T^{2} \)
$17$ \( 1 + 2048 T + 1419857 T^{2} \)
$19$ \( 1 + 2220 T + 2476099 T^{2} \)
$23$ \( 1 + 1246 T + 6436343 T^{2} \)
$29$ \( 1 + 270 T + 20511149 T^{2} \)
$31$ \( 1 - 2048 T + 28629151 T^{2} \)
$37$ \( 1 - 4372 T + 69343957 T^{2} \)
$41$ \( 1 + 2398 T + 115856201 T^{2} \)
$43$ \( 1 - 2294 T + 147008443 T^{2} \)
$47$ \( 1 + 10682 T + 229345007 T^{2} \)
$53$ \( 1 + 2964 T + 418195493 T^{2} \)
$59$ \( 1 - 39740 T + 714924299 T^{2} \)
$61$ \( 1 + 42298 T + 844596301 T^{2} \)
$67$ \( 1 - 32098 T + 1350125107 T^{2} \)
$71$ \( 1 - 4248 T + 1804229351 T^{2} \)
$73$ \( 1 + 30104 T + 2073071593 T^{2} \)
$79$ \( 1 + 35280 T + 3077056399 T^{2} \)
$83$ \( 1 + 27826 T + 3939040643 T^{2} \)
$89$ \( 1 + 85210 T + 5584059449 T^{2} \)
$97$ \( 1 - 97232 T + 8587340257 T^{2} \)
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