Properties

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

Newform invariants

Self dual: yes
Analytic conductor: \(23.6007640023\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{3} + 6 q^{7} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} + 6 q^{7} - 23 q^{9} - 32 q^{11} + 38 q^{13} - 26 q^{17} - 100 q^{19} + 12 q^{21} - 78 q^{23} - 100 q^{27} - 50 q^{29} + 108 q^{31} - 64 q^{33} - 266 q^{37} + 76 q^{39} + 22 q^{41} + 442 q^{43} - 514 q^{47} - 307 q^{49} - 52 q^{51} - 2 q^{53} - 200 q^{57} - 500 q^{59} - 518 q^{61} - 138 q^{63} + 126 q^{67} - 156 q^{69} - 412 q^{71} + 878 q^{73} - 192 q^{77} - 600 q^{79} + 421 q^{81} + 282 q^{83} - 100 q^{87} - 150 q^{89} + 228 q^{91} + 216 q^{93} - 386 q^{97} + 736 q^{99}+O(q^{100}) \) 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 2.00000 0 0 0 6.00000 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.m 1
4.b odd 2 1 25.4.a.c 1
5.b even 2 1 80.4.a.d 1
5.c odd 4 2 400.4.c.k 2
8.b even 2 1 1600.4.a.s 1
8.d odd 2 1 1600.4.a.bi 1
12.b even 2 1 225.4.a.b 1
15.d odd 2 1 720.4.a.u 1
20.d odd 2 1 5.4.a.a 1
20.e even 4 2 25.4.b.a 2
28.d even 2 1 1225.4.a.k 1
40.e odd 2 1 320.4.a.g 1
40.f even 2 1 320.4.a.h 1
60.h even 2 1 45.4.a.d 1
60.l odd 4 2 225.4.b.c 2
80.k odd 4 2 1280.4.d.e 2
80.q even 4 2 1280.4.d.l 2
140.c even 2 1 245.4.a.a 1
140.p odd 6 2 245.4.e.f 2
140.s even 6 2 245.4.e.g 2
180.n even 6 2 405.4.e.c 2
180.p odd 6 2 405.4.e.l 2
220.g even 2 1 605.4.a.d 1
260.g odd 2 1 845.4.a.b 1
340.d odd 2 1 1445.4.a.a 1
380.d even 2 1 1805.4.a.h 1
420.o odd 2 1 2205.4.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 20.d odd 2 1
25.4.a.c 1 4.b odd 2 1
25.4.b.a 2 20.e even 4 2
45.4.a.d 1 60.h even 2 1
80.4.a.d 1 5.b even 2 1
225.4.a.b 1 12.b even 2 1
225.4.b.c 2 60.l odd 4 2
245.4.a.a 1 140.c even 2 1
245.4.e.f 2 140.p odd 6 2
245.4.e.g 2 140.s even 6 2
320.4.a.g 1 40.e odd 2 1
320.4.a.h 1 40.f even 2 1
400.4.a.m 1 1.a even 1 1 trivial
400.4.c.k 2 5.c odd 4 2
405.4.e.c 2 180.n even 6 2
405.4.e.l 2 180.p odd 6 2
605.4.a.d 1 220.g even 2 1
720.4.a.u 1 15.d odd 2 1
845.4.a.b 1 260.g odd 2 1
1225.4.a.k 1 28.d even 2 1
1280.4.d.e 2 80.k odd 4 2
1280.4.d.l 2 80.q even 4 2
1445.4.a.a 1 340.d odd 2 1
1600.4.a.s 1 8.b even 2 1
1600.4.a.bi 1 8.d odd 2 1
1805.4.a.h 1 380.d even 2 1
2205.4.a.q 1 420.o odd 2 1

Hecke kernels

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

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{7} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 6 \) Copy content Toggle raw display
$11$ \( T + 32 \) Copy content Toggle raw display
$13$ \( T - 38 \) Copy content Toggle raw display
$17$ \( T + 26 \) Copy content Toggle raw display
$19$ \( T + 100 \) Copy content Toggle raw display
$23$ \( T + 78 \) Copy content Toggle raw display
$29$ \( T + 50 \) Copy content Toggle raw display
$31$ \( T - 108 \) Copy content Toggle raw display
$37$ \( T + 266 \) Copy content Toggle raw display
$41$ \( T - 22 \) Copy content Toggle raw display
$43$ \( T - 442 \) Copy content Toggle raw display
$47$ \( T + 514 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T + 500 \) Copy content Toggle raw display
$61$ \( T + 518 \) Copy content Toggle raw display
$67$ \( T - 126 \) Copy content Toggle raw display
$71$ \( T + 412 \) Copy content Toggle raw display
$73$ \( T - 878 \) Copy content Toggle raw display
$79$ \( T + 600 \) Copy content Toggle raw display
$83$ \( T - 282 \) Copy content Toggle raw display
$89$ \( T + 150 \) Copy content Toggle raw display
$97$ \( T + 386 \) Copy content Toggle raw display
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