Properties

Label 800.4.a.h
Level $800$
Weight $4$
Character orbit 800.a
Self dual yes
Analytic conductor $47.202$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.2015280046\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} - 6q^{7} - 23q^{9} + O(q^{10}) \) \( q + 2q^{3} - 6q^{7} - 23q^{9} + 60q^{11} - 50q^{13} + 30q^{17} + 40q^{19} - 12q^{21} - 178q^{23} - 100q^{27} + 166q^{29} + 20q^{31} + 120q^{33} - 10q^{37} - 100q^{39} - 250q^{41} - 142q^{43} - 214q^{47} - 307q^{49} + 60q^{51} - 490q^{53} + 80q^{57} - 800q^{59} + 250q^{61} + 138q^{63} + 774q^{67} - 356q^{69} + 100q^{71} + 230q^{73} - 360q^{77} - 1320q^{79} + 421q^{81} - 982q^{83} + 332q^{87} + 874q^{89} + 300q^{91} + 40q^{93} + 310q^{97} - 1380q^{99} + O(q^{100}) \)

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 −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 800.4.a.h 1
4.b odd 2 1 800.4.a.d 1
5.b even 2 1 160.4.a.a 1
5.c odd 4 2 800.4.c.f 2
8.b even 2 1 1600.4.a.r 1
8.d odd 2 1 1600.4.a.bj 1
15.d odd 2 1 1440.4.a.o 1
20.d odd 2 1 160.4.a.b yes 1
20.e even 4 2 800.4.c.e 2
40.e odd 2 1 320.4.a.f 1
40.f even 2 1 320.4.a.i 1
60.h even 2 1 1440.4.a.n 1
80.k odd 4 2 1280.4.d.k 2
80.q even 4 2 1280.4.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 5.b even 2 1
160.4.a.b yes 1 20.d odd 2 1
320.4.a.f 1 40.e odd 2 1
320.4.a.i 1 40.f even 2 1
800.4.a.d 1 4.b odd 2 1
800.4.a.h 1 1.a even 1 1 trivial
800.4.c.e 2 20.e even 4 2
800.4.c.f 2 5.c odd 4 2
1280.4.d.f 2 80.q even 4 2
1280.4.d.k 2 80.k odd 4 2
1440.4.a.n 1 60.h even 2 1
1440.4.a.o 1 15.d odd 2 1
1600.4.a.r 1 8.b even 2 1
1600.4.a.bj 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3} - 2 \)
\( T_{11} - 60 \)
\( T_{13} + 50 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( T \)
$7$ \( 6 + T \)
$11$ \( -60 + T \)
$13$ \( 50 + T \)
$17$ \( -30 + T \)
$19$ \( -40 + T \)
$23$ \( 178 + T \)
$29$ \( -166 + T \)
$31$ \( -20 + T \)
$37$ \( 10 + T \)
$41$ \( 250 + T \)
$43$ \( 142 + T \)
$47$ \( 214 + T \)
$53$ \( 490 + T \)
$59$ \( 800 + T \)
$61$ \( -250 + T \)
$67$ \( -774 + T \)
$71$ \( -100 + T \)
$73$ \( -230 + T \)
$79$ \( 1320 + T \)
$83$ \( 982 + T \)
$89$ \( -874 + T \)
$97$ \( -310 + T \)
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