Properties

Label 400.6.a.i
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

\(f(q)\) \(=\) \( q + 6q^{3} - 118q^{7} - 207q^{9} + O(q^{10}) \) \( q + 6q^{3} - 118q^{7} - 207q^{9} - 192q^{11} - 1106q^{13} - 762q^{17} + 2740q^{19} - 708q^{21} + 1566q^{23} - 2700q^{27} + 5910q^{29} + 6868q^{31} - 1152q^{33} + 5518q^{37} - 6636q^{39} - 378q^{41} - 2434q^{43} + 13122q^{47} - 2883q^{49} - 4572q^{51} + 9174q^{53} + 16440q^{57} + 34980q^{59} - 9838q^{61} + 24426q^{63} + 33722q^{67} + 9396q^{69} - 70212q^{71} - 21986q^{73} + 22656q^{77} - 4520q^{79} + 34101q^{81} - 109074q^{83} + 35460q^{87} + 38490q^{89} + 130508q^{91} + 41208q^{93} + 1918q^{97} + 39744q^{99} + O(q^{100}) \)

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 6.00000 0 0 0 −118.000 0 −207.000 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.i 1
4.b odd 2 1 50.6.a.b 1
5.b even 2 1 80.6.a.c 1
5.c odd 4 2 400.6.c.i 2
12.b even 2 1 450.6.a.u 1
15.d odd 2 1 720.6.a.v 1
20.d odd 2 1 10.6.a.c 1
20.e even 4 2 50.6.b.b 2
40.e odd 2 1 320.6.a.f 1
40.f even 2 1 320.6.a.k 1
60.h even 2 1 90.6.a.b 1
60.l odd 4 2 450.6.c.f 2
140.c even 2 1 490.6.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.c 1 20.d odd 2 1
50.6.a.b 1 4.b odd 2 1
50.6.b.b 2 20.e even 4 2
80.6.a.c 1 5.b even 2 1
90.6.a.b 1 60.h even 2 1
320.6.a.f 1 40.e odd 2 1
320.6.a.k 1 40.f even 2 1
400.6.a.i 1 1.a even 1 1 trivial
400.6.c.i 2 5.c odd 4 2
450.6.a.u 1 12.b even 2 1
450.6.c.f 2 60.l odd 4 2
490.6.a.k 1 140.c even 2 1
720.6.a.v 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 \( T_{3} - 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 6 T + 243 T^{2} \)
$5$ 1
$7$ \( 1 + 118 T + 16807 T^{2} \)
$11$ \( 1 + 192 T + 161051 T^{2} \)
$13$ \( 1 + 1106 T + 371293 T^{2} \)
$17$ \( 1 + 762 T + 1419857 T^{2} \)
$19$ \( 1 - 2740 T + 2476099 T^{2} \)
$23$ \( 1 - 1566 T + 6436343 T^{2} \)
$29$ \( 1 - 5910 T + 20511149 T^{2} \)
$31$ \( 1 - 6868 T + 28629151 T^{2} \)
$37$ \( 1 - 5518 T + 69343957 T^{2} \)
$41$ \( 1 + 378 T + 115856201 T^{2} \)
$43$ \( 1 + 2434 T + 147008443 T^{2} \)
$47$ \( 1 - 13122 T + 229345007 T^{2} \)
$53$ \( 1 - 9174 T + 418195493 T^{2} \)
$59$ \( 1 - 34980 T + 714924299 T^{2} \)
$61$ \( 1 + 9838 T + 844596301 T^{2} \)
$67$ \( 1 - 33722 T + 1350125107 T^{2} \)
$71$ \( 1 + 70212 T + 1804229351 T^{2} \)
$73$ \( 1 + 21986 T + 2073071593 T^{2} \)
$79$ \( 1 + 4520 T + 3077056399 T^{2} \)
$83$ \( 1 + 109074 T + 3939040643 T^{2} \)
$89$ \( 1 - 38490 T + 5584059449 T^{2} \)
$97$ \( 1 - 1918 T + 8587340257 T^{2} \)
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