Properties

Label 10.6.a.c
Level $10$
Weight $6$
Character orbit 10.a
Self dual yes
Analytic conductor $1.604$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.60383819813\)
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

\(f(q)\) \(=\) \( q + 4q^{2} + 6q^{3} + 16q^{4} - 25q^{5} + 24q^{6} - 118q^{7} + 64q^{8} - 207q^{9} + O(q^{10}) \) \( q + 4q^{2} + 6q^{3} + 16q^{4} - 25q^{5} + 24q^{6} - 118q^{7} + 64q^{8} - 207q^{9} - 100q^{10} + 192q^{11} + 96q^{12} + 1106q^{13} - 472q^{14} - 150q^{15} + 256q^{16} + 762q^{17} - 828q^{18} - 2740q^{19} - 400q^{20} - 708q^{21} + 768q^{22} + 1566q^{23} + 384q^{24} + 625q^{25} + 4424q^{26} - 2700q^{27} - 1888q^{28} + 5910q^{29} - 600q^{30} - 6868q^{31} + 1024q^{32} + 1152q^{33} + 3048q^{34} + 2950q^{35} - 3312q^{36} - 5518q^{37} - 10960q^{38} + 6636q^{39} - 1600q^{40} - 378q^{41} - 2832q^{42} - 2434q^{43} + 3072q^{44} + 5175q^{45} + 6264q^{46} + 13122q^{47} + 1536q^{48} - 2883q^{49} + 2500q^{50} + 4572q^{51} + 17696q^{52} - 9174q^{53} - 10800q^{54} - 4800q^{55} - 7552q^{56} - 16440q^{57} + 23640q^{58} - 34980q^{59} - 2400q^{60} - 9838q^{61} - 27472q^{62} + 24426q^{63} + 4096q^{64} - 27650q^{65} + 4608q^{66} + 33722q^{67} + 12192q^{68} + 9396q^{69} + 11800q^{70} + 70212q^{71} - 13248q^{72} + 21986q^{73} - 22072q^{74} + 3750q^{75} - 43840q^{76} - 22656q^{77} + 26544q^{78} + 4520q^{79} - 6400q^{80} + 34101q^{81} - 1512q^{82} - 109074q^{83} - 11328q^{84} - 19050q^{85} - 9736q^{86} + 35460q^{87} + 12288q^{88} + 38490q^{89} + 20700q^{90} - 130508q^{91} + 25056q^{92} - 41208q^{93} + 52488q^{94} + 68500q^{95} + 6144q^{96} - 1918q^{97} - 11532q^{98} - 39744q^{99} + O(q^{100}) \)

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 −25.0000 24.0000 −118.000 64.0000 −207.000 −100.000
\(n\): 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 10.6.a.c 1
3.b odd 2 1 90.6.a.b 1
4.b odd 2 1 80.6.a.c 1
5.b even 2 1 50.6.a.b 1
5.c odd 4 2 50.6.b.b 2
7.b odd 2 1 490.6.a.k 1
8.b even 2 1 320.6.a.f 1
8.d odd 2 1 320.6.a.k 1
12.b even 2 1 720.6.a.v 1
15.d odd 2 1 450.6.a.u 1
15.e even 4 2 450.6.c.f 2
20.d odd 2 1 400.6.a.i 1
20.e even 4 2 400.6.c.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.c 1 1.a even 1 1 trivial
50.6.a.b 1 5.b even 2 1
50.6.b.b 2 5.c odd 4 2
80.6.a.c 1 4.b odd 2 1
90.6.a.b 1 3.b odd 2 1
320.6.a.f 1 8.b even 2 1
320.6.a.k 1 8.d odd 2 1
400.6.a.i 1 20.d odd 2 1
400.6.c.i 2 20.e even 4 2
450.6.a.u 1 15.d odd 2 1
450.6.c.f 2 15.e even 4 2
490.6.a.k 1 7.b odd 2 1
720.6.a.v 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( -6 + T \)
$5$ \( 25 + T \)
$7$ \( 118 + T \)
$11$ \( -192 + T \)
$13$ \( -1106 + T \)
$17$ \( -762 + T \)
$19$ \( 2740 + T \)
$23$ \( -1566 + T \)
$29$ \( -5910 + T \)
$31$ \( 6868 + T \)
$37$ \( 5518 + T \)
$41$ \( 378 + T \)
$43$ \( 2434 + T \)
$47$ \( -13122 + T \)
$53$ \( 9174 + T \)
$59$ \( 34980 + T \)
$61$ \( 9838 + T \)
$67$ \( -33722 + T \)
$71$ \( -70212 + T \)
$73$ \( -21986 + T \)
$79$ \( -4520 + T \)
$83$ \( 109074 + T \)
$89$ \( -38490 + T \)
$97$ \( 1918 + T \)
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