Properties

Label 400.4.a.n
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 10)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} + 26q^{7} - 23q^{9} + O(q^{10}) \) \( q + 2q^{3} + 26q^{7} - 23q^{9} + 28q^{11} - 12q^{13} + 64q^{17} + 60q^{19} + 52q^{21} - 58q^{23} - 100q^{27} + 90q^{29} + 128q^{31} + 56q^{33} - 236q^{37} - 24q^{39} + 242q^{41} + 362q^{43} + 226q^{47} + 333q^{49} + 128q^{51} + 108q^{53} + 120q^{57} + 20q^{59} + 542q^{61} - 598q^{63} - 434q^{67} - 116q^{69} + 1128q^{71} - 632q^{73} + 728q^{77} + 720q^{79} + 421q^{81} - 478q^{83} + 180q^{87} - 490q^{89} - 312q^{91} + 256q^{93} - 1456q^{97} - 644q^{99} + O(q^{100}) \)

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 26.0000 0 −23.0000 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.n 1
4.b odd 2 1 50.4.a.b 1
5.b even 2 1 400.4.a.h 1
5.c odd 4 2 80.4.c.a 2
8.b even 2 1 1600.4.a.t 1
8.d odd 2 1 1600.4.a.bh 1
12.b even 2 1 450.4.a.k 1
15.e even 4 2 720.4.f.f 2
20.d odd 2 1 50.4.a.d 1
20.e even 4 2 10.4.b.a 2
28.d even 2 1 2450.4.a.o 1
40.e odd 2 1 1600.4.a.u 1
40.f even 2 1 1600.4.a.bg 1
40.i odd 4 2 320.4.c.c 2
40.k even 4 2 320.4.c.d 2
60.h even 2 1 450.4.a.j 1
60.l odd 4 2 90.4.c.b 2
140.c even 2 1 2450.4.a.bb 1
140.j odd 4 2 490.4.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 20.e even 4 2
50.4.a.b 1 4.b odd 2 1
50.4.a.d 1 20.d odd 2 1
80.4.c.a 2 5.c odd 4 2
90.4.c.b 2 60.l odd 4 2
320.4.c.c 2 40.i odd 4 2
320.4.c.d 2 40.k even 4 2
400.4.a.h 1 5.b even 2 1
400.4.a.n 1 1.a even 1 1 trivial
450.4.a.j 1 60.h even 2 1
450.4.a.k 1 12.b even 2 1
490.4.c.b 2 140.j odd 4 2
720.4.f.f 2 15.e even 4 2
1600.4.a.t 1 8.b even 2 1
1600.4.a.u 1 40.e odd 2 1
1600.4.a.bg 1 40.f even 2 1
1600.4.a.bh 1 8.d odd 2 1
2450.4.a.o 1 28.d even 2 1
2450.4.a.bb 1 140.c even 2 1

Hecke kernels

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

\( T_{3} - 2 \)
\( T_{7} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( T \)
$7$ \( -26 + T \)
$11$ \( -28 + T \)
$13$ \( 12 + T \)
$17$ \( -64 + T \)
$19$ \( -60 + T \)
$23$ \( 58 + T \)
$29$ \( -90 + T \)
$31$ \( -128 + T \)
$37$ \( 236 + T \)
$41$ \( -242 + T \)
$43$ \( -362 + T \)
$47$ \( -226 + T \)
$53$ \( -108 + T \)
$59$ \( -20 + T \)
$61$ \( -542 + T \)
$67$ \( 434 + T \)
$71$ \( -1128 + T \)
$73$ \( 632 + T \)
$79$ \( -720 + T \)
$83$ \( 478 + T \)
$89$ \( 490 + T \)
$97$ \( 1456 + T \)
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