Properties

Label 400.4.a.s
Level $400$
Weight $4$
Character orbit 400.a
Self dual yes
Analytic conductor $23.601$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.6007640023\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 7 q^{3} + 6 q^{7} + 22 q^{9} + O(q^{10}) \) \( q + 7 q^{3} + 6 q^{7} + 22 q^{9} + 43 q^{11} + 28 q^{13} - 91 q^{17} + 35 q^{19} + 42 q^{21} + 162 q^{23} - 35 q^{27} + 160 q^{29} - 42 q^{31} + 301 q^{33} + 314 q^{37} + 196 q^{39} - 203 q^{41} + 92 q^{43} + 196 q^{47} - 307 q^{49} - 637 q^{51} - 82 q^{53} + 245 q^{57} + 280 q^{59} - 518 q^{61} + 132 q^{63} + 141 q^{67} + 1134 q^{69} - 412 q^{71} + 763 q^{73} + 258 q^{77} - 510 q^{79} - 839 q^{81} + 777 q^{83} + 1120 q^{87} - 945 q^{89} + 168 q^{91} - 294 q^{93} - 1246 q^{97} + 946 q^{99} + O(q^{100}) \)

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 7.00000 0 0 0 6.00000 0 22.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 400.4.a.s 1
4.b odd 2 1 25.4.a.a 1
5.b even 2 1 400.4.a.c 1
5.c odd 4 2 400.4.c.e 2
8.b even 2 1 1600.4.a.h 1
8.d odd 2 1 1600.4.a.bt 1
12.b even 2 1 225.4.a.e 1
20.d odd 2 1 25.4.a.b yes 1
20.e even 4 2 25.4.b.b 2
28.d even 2 1 1225.4.a.h 1
40.e odd 2 1 1600.4.a.i 1
40.f even 2 1 1600.4.a.bs 1
60.h even 2 1 225.4.a.c 1
60.l odd 4 2 225.4.b.f 2
140.c even 2 1 1225.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 4.b odd 2 1
25.4.a.b yes 1 20.d odd 2 1
25.4.b.b 2 20.e even 4 2
225.4.a.c 1 60.h even 2 1
225.4.a.e 1 12.b even 2 1
225.4.b.f 2 60.l odd 4 2
400.4.a.c 1 5.b even 2 1
400.4.a.s 1 1.a even 1 1 trivial
400.4.c.e 2 5.c odd 4 2
1225.4.a.h 1 28.d even 2 1
1225.4.a.i 1 140.c even 2 1
1600.4.a.h 1 8.b even 2 1
1600.4.a.i 1 40.e odd 2 1
1600.4.a.bs 1 40.f even 2 1
1600.4.a.bt 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(400))\):

\( T_{3} - 7 \)
\( T_{7} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -7 + T \)
$5$ \( T \)
$7$ \( -6 + T \)
$11$ \( -43 + T \)
$13$ \( -28 + T \)
$17$ \( 91 + T \)
$19$ \( -35 + T \)
$23$ \( -162 + T \)
$29$ \( -160 + T \)
$31$ \( 42 + T \)
$37$ \( -314 + T \)
$41$ \( 203 + T \)
$43$ \( -92 + T \)
$47$ \( -196 + T \)
$53$ \( 82 + T \)
$59$ \( -280 + T \)
$61$ \( 518 + T \)
$67$ \( -141 + T \)
$71$ \( 412 + T \)
$73$ \( -763 + T \)
$79$ \( 510 + T \)
$83$ \( -777 + T \)
$89$ \( 945 + T \)
$97$ \( 1246 + T \)
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