Properties

Label 50.6.a.d
Level $50$
Weight $6$
Character orbit 50.a
Self dual yes
Analytic conductor $8.019$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.01919099065\)
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

\(f(q)\) \(=\) \( q + 4q^{2} - 24q^{3} + 16q^{4} - 96q^{6} + 172q^{7} + 64q^{8} + 333q^{9} + O(q^{10}) \) \( q + 4q^{2} - 24q^{3} + 16q^{4} - 96q^{6} + 172q^{7} + 64q^{8} + 333q^{9} + 132q^{11} - 384q^{12} + 946q^{13} + 688q^{14} + 256q^{16} + 222q^{17} + 1332q^{18} + 500q^{19} - 4128q^{21} + 528q^{22} - 3564q^{23} - 1536q^{24} + 3784q^{26} - 2160q^{27} + 2752q^{28} + 2190q^{29} + 2312q^{31} + 1024q^{32} - 3168q^{33} + 888q^{34} + 5328q^{36} + 11242q^{37} + 2000q^{38} - 22704q^{39} + 1242q^{41} - 16512q^{42} - 20624q^{43} + 2112q^{44} - 14256q^{46} - 6588q^{47} - 6144q^{48} + 12777q^{49} - 5328q^{51} + 15136q^{52} + 21066q^{53} - 8640q^{54} + 11008q^{56} - 12000q^{57} + 8760q^{58} + 7980q^{59} + 16622q^{61} + 9248q^{62} + 57276q^{63} + 4096q^{64} - 12672q^{66} - 1808q^{67} + 3552q^{68} + 85536q^{69} - 24528q^{71} + 21312q^{72} - 20474q^{73} + 44968q^{74} + 8000q^{76} + 22704q^{77} - 90816q^{78} - 46240q^{79} - 29079q^{81} + 4968q^{82} + 51576q^{83} - 66048q^{84} - 82496q^{86} - 52560q^{87} + 8448q^{88} - 110310q^{89} + 162712q^{91} - 57024q^{92} - 55488q^{93} - 26352q^{94} - 24576q^{96} + 78382q^{97} + 51108q^{98} + 43956q^{99} + O(q^{100}) \)

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
4.00000 −24.0000 16.0000 0 −96.0000 172.000 64.0000 333.000 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 50.6.a.d 1
3.b odd 2 1 450.6.a.l 1
4.b odd 2 1 400.6.a.n 1
5.b even 2 1 10.6.a.b 1
5.c odd 4 2 50.6.b.a 2
15.d odd 2 1 90.6.a.d 1
15.e even 4 2 450.6.c.h 2
20.d odd 2 1 80.6.a.a 1
20.e even 4 2 400.6.c.b 2
35.c odd 2 1 490.6.a.a 1
40.e odd 2 1 320.6.a.o 1
40.f even 2 1 320.6.a.b 1
60.h even 2 1 720.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.b 1 5.b even 2 1
50.6.a.d 1 1.a even 1 1 trivial
50.6.b.a 2 5.c odd 4 2
80.6.a.a 1 20.d odd 2 1
90.6.a.d 1 15.d odd 2 1
320.6.a.b 1 40.f even 2 1
320.6.a.o 1 40.e odd 2 1
400.6.a.n 1 4.b odd 2 1
400.6.c.b 2 20.e even 4 2
450.6.a.l 1 3.b odd 2 1
450.6.c.h 2 15.e even 4 2
490.6.a.a 1 35.c odd 2 1
720.6.a.j 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 24 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(50))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( 24 + T \)
$5$ \( T \)
$7$ \( -172 + T \)
$11$ \( -132 + T \)
$13$ \( -946 + T \)
$17$ \( -222 + T \)
$19$ \( -500 + T \)
$23$ \( 3564 + T \)
$29$ \( -2190 + T \)
$31$ \( -2312 + T \)
$37$ \( -11242 + T \)
$41$ \( -1242 + T \)
$43$ \( 20624 + T \)
$47$ \( 6588 + T \)
$53$ \( -21066 + T \)
$59$ \( -7980 + T \)
$61$ \( -16622 + T \)
$67$ \( 1808 + T \)
$71$ \( 24528 + T \)
$73$ \( 20474 + T \)
$79$ \( 46240 + T \)
$83$ \( -51576 + T \)
$89$ \( 110310 + T \)
$97$ \( -78382 + T \)
show more
show less