Properties

Label 50.6.a.b
Level $50$
Weight $6$
Character orbit 50.a
Self dual yes
Analytic conductor $8.019$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - 4q^{2} - 6q^{3} + 16q^{4} + 24q^{6} + 118q^{7} - 64q^{8} - 207q^{9} + O(q^{10}) \) \( q - 4q^{2} - 6q^{3} + 16q^{4} + 24q^{6} + 118q^{7} - 64q^{8} - 207q^{9} + 192q^{11} - 96q^{12} - 1106q^{13} - 472q^{14} + 256q^{16} - 762q^{17} + 828q^{18} - 2740q^{19} - 708q^{21} - 768q^{22} - 1566q^{23} + 384q^{24} + 4424q^{26} + 2700q^{27} + 1888q^{28} + 5910q^{29} - 6868q^{31} - 1024q^{32} - 1152q^{33} + 3048q^{34} - 3312q^{36} + 5518q^{37} + 10960q^{38} + 6636q^{39} - 378q^{41} + 2832q^{42} + 2434q^{43} + 3072q^{44} + 6264q^{46} - 13122q^{47} - 1536q^{48} - 2883q^{49} + 4572q^{51} - 17696q^{52} + 9174q^{53} - 10800q^{54} - 7552q^{56} + 16440q^{57} - 23640q^{58} - 34980q^{59} - 9838q^{61} + 27472q^{62} - 24426q^{63} + 4096q^{64} + 4608q^{66} - 33722q^{67} - 12192q^{68} + 9396q^{69} + 70212q^{71} + 13248q^{72} - 21986q^{73} - 22072q^{74} - 43840q^{76} + 22656q^{77} - 26544q^{78} + 4520q^{79} + 34101q^{81} + 1512q^{82} + 109074q^{83} - 11328q^{84} - 9736q^{86} - 35460q^{87} - 12288q^{88} + 38490q^{89} - 130508q^{91} - 25056q^{92} + 41208q^{93} + 52488q^{94} + 6144q^{96} + 1918q^{97} + 11532q^{98} - 39744q^{99} + O(q^{100}) \)

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 −6.00000 16.0000 0 24.0000 118.000 −64.0000 −207.000 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.b 1
3.b odd 2 1 450.6.a.u 1
4.b odd 2 1 400.6.a.i 1
5.b even 2 1 10.6.a.c 1
5.c odd 4 2 50.6.b.b 2
15.d odd 2 1 90.6.a.b 1
15.e even 4 2 450.6.c.f 2
20.d odd 2 1 80.6.a.c 1
20.e even 4 2 400.6.c.i 2
35.c odd 2 1 490.6.a.k 1
40.e odd 2 1 320.6.a.k 1
40.f even 2 1 320.6.a.f 1
60.h even 2 1 720.6.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.c 1 5.b even 2 1
50.6.a.b 1 1.a even 1 1 trivial
50.6.b.b 2 5.c odd 4 2
80.6.a.c 1 20.d odd 2 1
90.6.a.b 1 15.d odd 2 1
320.6.a.f 1 40.f even 2 1
320.6.a.k 1 40.e odd 2 1
400.6.a.i 1 4.b odd 2 1
400.6.c.i 2 20.e even 4 2
450.6.a.u 1 3.b odd 2 1
450.6.c.f 2 15.e even 4 2
490.6.a.k 1 35.c odd 2 1
720.6.a.v 1 60.h even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( 6 + T \)
$5$ \( T \)
$7$ \( -118 + T \)
$11$ \( -192 + T \)
$13$ \( 1106 + T \)
$17$ \( 762 + T \)
$19$ \( 2740 + T \)
$23$ \( 1566 + T \)
$29$ \( -5910 + T \)
$31$ \( 6868 + T \)
$37$ \( -5518 + T \)
$41$ \( 378 + T \)
$43$ \( -2434 + T \)
$47$ \( 13122 + T \)
$53$ \( -9174 + T \)
$59$ \( 34980 + T \)
$61$ \( 9838 + T \)
$67$ \( 33722 + T \)
$71$ \( -70212 + T \)
$73$ \( 21986 + T \)
$79$ \( -4520 + T \)
$83$ \( -109074 + T \)
$89$ \( -38490 + T \)
$97$ \( -1918 + T \)
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