Properties

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

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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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{2} + 11q^{3} + 16q^{4} + 44q^{6} + 142q^{7} + 64q^{8} - 122q^{9} + O(q^{10}) \) \( q + 4q^{2} + 11q^{3} + 16q^{4} + 44q^{6} + 142q^{7} + 64q^{8} - 122q^{9} + 777q^{11} + 176q^{12} - 884q^{13} + 568q^{14} + 256q^{16} + 27q^{17} - 488q^{18} + 1145q^{19} + 1562q^{21} + 3108q^{22} - 1854q^{23} + 704q^{24} - 3536q^{26} - 4015q^{27} + 2272q^{28} - 4920q^{29} + 1802q^{31} + 1024q^{32} + 8547q^{33} + 108q^{34} - 1952q^{36} - 13178q^{37} + 4580q^{38} - 9724q^{39} - 15123q^{41} + 6248q^{42} - 7844q^{43} + 12432q^{44} - 7416q^{46} + 6732q^{47} + 2816q^{48} + 3357q^{49} + 297q^{51} - 14144q^{52} - 3414q^{53} - 16060q^{54} + 9088q^{56} + 12595q^{57} - 19680q^{58} + 33960q^{59} + 47402q^{61} + 7208q^{62} - 17324q^{63} + 4096q^{64} + 34188q^{66} + 13177q^{67} + 432q^{68} - 20394q^{69} - 7548q^{71} - 7808q^{72} + 59821q^{73} - 52712q^{74} + 18320q^{76} + 110334q^{77} - 38896q^{78} + 75830q^{79} - 14519q^{81} - 60492q^{82} - 46299q^{83} + 24992q^{84} - 31376q^{86} - 54120q^{87} + 49728q^{88} - 30585q^{89} - 125528q^{91} - 29664q^{92} + 19822q^{93} + 26928q^{94} + 11264q^{96} - 104018q^{97} + 13428q^{98} - 94794q^{99} + O(q^{100}) \)

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 11.0000 16.0000 0 44.0000 142.000 64.0000 −122.000 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.f yes 1
3.b odd 2 1 450.6.a.j 1
4.b odd 2 1 400.6.a.e 1
5.b even 2 1 50.6.a.a 1
5.c odd 4 2 50.6.b.c 2
15.d odd 2 1 450.6.a.n 1
15.e even 4 2 450.6.c.a 2
20.d odd 2 1 400.6.a.j 1
20.e even 4 2 400.6.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.6.a.a 1 5.b even 2 1
50.6.a.f yes 1 1.a even 1 1 trivial
50.6.b.c 2 5.c odd 4 2
400.6.a.e 1 4.b odd 2 1
400.6.a.j 1 20.d odd 2 1
400.6.c.g 2 20.e even 4 2
450.6.a.j 1 3.b odd 2 1
450.6.a.n 1 15.d odd 2 1
450.6.c.a 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 11 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(50))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( -11 + T \)
$5$ \( T \)
$7$ \( -142 + T \)
$11$ \( -777 + T \)
$13$ \( 884 + T \)
$17$ \( -27 + T \)
$19$ \( -1145 + T \)
$23$ \( 1854 + T \)
$29$ \( 4920 + T \)
$31$ \( -1802 + T \)
$37$ \( 13178 + T \)
$41$ \( 15123 + T \)
$43$ \( 7844 + T \)
$47$ \( -6732 + T \)
$53$ \( 3414 + T \)
$59$ \( -33960 + T \)
$61$ \( -47402 + T \)
$67$ \( -13177 + T \)
$71$ \( 7548 + T \)
$73$ \( -59821 + T \)
$79$ \( -75830 + T \)
$83$ \( 46299 + T \)
$89$ \( 30585 + T \)
$97$ \( 104018 + T \)
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