Properties

Label 400.6.a.a
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 10)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 26q^{3} - 22q^{7} + 433q^{9} + O(q^{10}) \) \( q - 26q^{3} - 22q^{7} + 433q^{9} + 768q^{11} + 46q^{13} - 378q^{17} - 1100q^{19} + 572q^{21} - 1986q^{23} - 4940q^{27} - 5610q^{29} + 3988q^{31} - 19968q^{33} + 142q^{37} - 1196q^{39} + 1542q^{41} - 5026q^{43} + 24738q^{47} - 16323q^{49} + 9828q^{51} + 14166q^{53} + 28600q^{57} - 28380q^{59} + 5522q^{61} - 9526q^{63} - 24742q^{67} + 51636q^{69} - 42372q^{71} + 52126q^{73} - 16896q^{77} + 39640q^{79} + 23221q^{81} - 59826q^{83} + 145860q^{87} + 57690q^{89} - 1012q^{91} - 103688q^{93} + 144382q^{97} + 332544q^{99} + O(q^{100}) \)

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 −26.0000 0 0 0 −22.0000 0 433.000 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.a 1
4.b odd 2 1 50.6.a.g 1
5.b even 2 1 80.6.a.h 1
5.c odd 4 2 400.6.c.a 2
12.b even 2 1 450.6.a.h 1
15.d odd 2 1 720.6.a.r 1
20.d odd 2 1 10.6.a.a 1
20.e even 4 2 50.6.b.d 2
40.e odd 2 1 320.6.a.p 1
40.f even 2 1 320.6.a.a 1
60.h even 2 1 90.6.a.f 1
60.l odd 4 2 450.6.c.o 2
140.c even 2 1 490.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.a 1 20.d odd 2 1
50.6.a.g 1 4.b odd 2 1
50.6.b.d 2 20.e even 4 2
80.6.a.h 1 5.b even 2 1
90.6.a.f 1 60.h even 2 1
320.6.a.a 1 40.f even 2 1
320.6.a.p 1 40.e odd 2 1
400.6.a.a 1 1.a even 1 1 trivial
400.6.c.a 2 5.c odd 4 2
450.6.a.h 1 12.b even 2 1
450.6.c.o 2 60.l odd 4 2
490.6.a.j 1 140.c even 2 1
720.6.a.r 1 15.d odd 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 \( T_{3} + 26 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 26 T + 243 T^{2} \)
$5$ 1
$7$ \( 1 + 22 T + 16807 T^{2} \)
$11$ \( 1 - 768 T + 161051 T^{2} \)
$13$ \( 1 - 46 T + 371293 T^{2} \)
$17$ \( 1 + 378 T + 1419857 T^{2} \)
$19$ \( 1 + 1100 T + 2476099 T^{2} \)
$23$ \( 1 + 1986 T + 6436343 T^{2} \)
$29$ \( 1 + 5610 T + 20511149 T^{2} \)
$31$ \( 1 - 3988 T + 28629151 T^{2} \)
$37$ \( 1 - 142 T + 69343957 T^{2} \)
$41$ \( 1 - 1542 T + 115856201 T^{2} \)
$43$ \( 1 + 5026 T + 147008443 T^{2} \)
$47$ \( 1 - 24738 T + 229345007 T^{2} \)
$53$ \( 1 - 14166 T + 418195493 T^{2} \)
$59$ \( 1 + 28380 T + 714924299 T^{2} \)
$61$ \( 1 - 5522 T + 844596301 T^{2} \)
$67$ \( 1 + 24742 T + 1350125107 T^{2} \)
$71$ \( 1 + 42372 T + 1804229351 T^{2} \)
$73$ \( 1 - 52126 T + 2073071593 T^{2} \)
$79$ \( 1 - 39640 T + 3077056399 T^{2} \)
$83$ \( 1 + 59826 T + 3939040643 T^{2} \)
$89$ \( 1 - 57690 T + 5584059449 T^{2} \)
$97$ \( 1 - 144382 T + 8587340257 T^{2} \)
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