Properties

Label 400.4.a.i
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 100)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 26 q^{7} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 26 q^{7} - 26 q^{9} - 45 q^{11} + 44 q^{13} + 117 q^{17} + 91 q^{19} + 26 q^{21} + 18 q^{23} + 53 q^{27} + 144 q^{29} - 26 q^{31} + 45 q^{33} - 214 q^{37} - 44 q^{39} - 459 q^{41} + 460 q^{43} + 468 q^{47} + 333 q^{49} - 117 q^{51} + 558 q^{53} - 91 q^{57} + 72 q^{59} - 118 q^{61} + 676 q^{63} - 251 q^{67} - 18 q^{69} - 108 q^{71} + 299 q^{73} + 1170 q^{77} + 898 q^{79} + 649 q^{81} - 927 q^{83} - 144 q^{87} + 351 q^{89} - 1144 q^{91} + 26 q^{93} + 386 q^{97} + 1170 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −1.00000 0 0 0 −26.0000 0 −26.0000 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.i 1
4.b odd 2 1 100.4.a.c yes 1
5.b even 2 1 400.4.a.l 1
5.c odd 4 2 400.4.c.l 2
8.b even 2 1 1600.4.a.bd 1
8.d odd 2 1 1600.4.a.x 1
12.b even 2 1 900.4.a.p 1
20.d odd 2 1 100.4.a.b 1
20.e even 4 2 100.4.c.b 2
40.e odd 2 1 1600.4.a.bc 1
40.f even 2 1 1600.4.a.y 1
60.h even 2 1 900.4.a.c 1
60.l odd 4 2 900.4.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.a.b 1 20.d odd 2 1
100.4.a.c yes 1 4.b odd 2 1
100.4.c.b 2 20.e even 4 2
400.4.a.i 1 1.a even 1 1 trivial
400.4.a.l 1 5.b even 2 1
400.4.c.l 2 5.c odd 4 2
900.4.a.c 1 60.h even 2 1
900.4.a.p 1 12.b even 2 1
900.4.d.a 2 60.l odd 4 2
1600.4.a.x 1 8.d odd 2 1
1600.4.a.y 1 40.f even 2 1
1600.4.a.bc 1 40.e odd 2 1
1600.4.a.bd 1 8.b even 2 1

Hecke kernels

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

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{7} + 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 26 \) Copy content Toggle raw display
$11$ \( T + 45 \) Copy content Toggle raw display
$13$ \( T - 44 \) Copy content Toggle raw display
$17$ \( T - 117 \) Copy content Toggle raw display
$19$ \( T - 91 \) Copy content Toggle raw display
$23$ \( T - 18 \) Copy content Toggle raw display
$29$ \( T - 144 \) Copy content Toggle raw display
$31$ \( T + 26 \) Copy content Toggle raw display
$37$ \( T + 214 \) Copy content Toggle raw display
$41$ \( T + 459 \) Copy content Toggle raw display
$43$ \( T - 460 \) Copy content Toggle raw display
$47$ \( T - 468 \) Copy content Toggle raw display
$53$ \( T - 558 \) Copy content Toggle raw display
$59$ \( T - 72 \) Copy content Toggle raw display
$61$ \( T + 118 \) Copy content Toggle raw display
$67$ \( T + 251 \) Copy content Toggle raw display
$71$ \( T + 108 \) Copy content Toggle raw display
$73$ \( T - 299 \) Copy content Toggle raw display
$79$ \( T - 898 \) Copy content Toggle raw display
$83$ \( T + 927 \) Copy content Toggle raw display
$89$ \( T - 351 \) Copy content Toggle raw display
$97$ \( T - 386 \) Copy content Toggle raw display
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