Properties

Label 400.6.a.h
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 40)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{3} - 62q^{7} - 239q^{9} + O(q^{10}) \) \( q - 2q^{3} - 62q^{7} - 239q^{9} + 144q^{11} + 654q^{13} + 1190q^{17} - 556q^{19} + 124q^{21} + 2182q^{23} + 964q^{27} - 1578q^{29} - 9660q^{31} - 288q^{33} + 3534q^{37} - 1308q^{39} + 7462q^{41} - 7114q^{43} - 28294q^{47} - 12963q^{49} - 2380q^{51} + 13046q^{53} + 1112q^{57} + 37092q^{59} + 39570q^{61} + 14818q^{63} - 56734q^{67} - 4364q^{69} - 45588q^{71} - 11842q^{73} - 8928q^{77} - 94216q^{79} + 56149q^{81} - 31482q^{83} + 3156q^{87} - 94054q^{89} - 40548q^{91} + 19320q^{93} - 23714q^{97} - 34416q^{99} + O(q^{100}) \)

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 −2.00000 0 0 0 −62.0000 0 −239.000 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.h 1
4.b odd 2 1 200.6.a.b 1
5.b even 2 1 80.6.a.d 1
5.c odd 4 2 400.6.c.k 2
15.d odd 2 1 720.6.a.t 1
20.d odd 2 1 40.6.a.c 1
20.e even 4 2 200.6.c.d 2
40.e odd 2 1 320.6.a.i 1
40.f even 2 1 320.6.a.h 1
60.h even 2 1 360.6.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.c 1 20.d odd 2 1
80.6.a.d 1 5.b even 2 1
200.6.a.b 1 4.b odd 2 1
200.6.c.d 2 20.e even 4 2
320.6.a.h 1 40.f even 2 1
320.6.a.i 1 40.e odd 2 1
360.6.a.f 1 60.h even 2 1
400.6.a.h 1 1.a even 1 1 trivial
400.6.c.k 2 5.c odd 4 2
720.6.a.t 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 \( T_{3} + 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 2 T + 243 T^{2} \)
$5$ 1
$7$ \( 1 + 62 T + 16807 T^{2} \)
$11$ \( 1 - 144 T + 161051 T^{2} \)
$13$ \( 1 - 654 T + 371293 T^{2} \)
$17$ \( 1 - 1190 T + 1419857 T^{2} \)
$19$ \( 1 + 556 T + 2476099 T^{2} \)
$23$ \( 1 - 2182 T + 6436343 T^{2} \)
$29$ \( 1 + 1578 T + 20511149 T^{2} \)
$31$ \( 1 + 9660 T + 28629151 T^{2} \)
$37$ \( 1 - 3534 T + 69343957 T^{2} \)
$41$ \( 1 - 7462 T + 115856201 T^{2} \)
$43$ \( 1 + 7114 T + 147008443 T^{2} \)
$47$ \( 1 + 28294 T + 229345007 T^{2} \)
$53$ \( 1 - 13046 T + 418195493 T^{2} \)
$59$ \( 1 - 37092 T + 714924299 T^{2} \)
$61$ \( 1 - 39570 T + 844596301 T^{2} \)
$67$ \( 1 + 56734 T + 1350125107 T^{2} \)
$71$ \( 1 + 45588 T + 1804229351 T^{2} \)
$73$ \( 1 + 11842 T + 2073071593 T^{2} \)
$79$ \( 1 + 94216 T + 3077056399 T^{2} \)
$83$ \( 1 + 31482 T + 3939040643 T^{2} \)
$89$ \( 1 + 94054 T + 5584059449 T^{2} \)
$97$ \( 1 + 23714 T + 8587340257 T^{2} \)
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