Properties

Label 10.6.a.b
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

\(f(q)\) \(=\) \( q - 4 q^{2} + 24 q^{3} + 16 q^{4} + 25 q^{5} - 96 q^{6} - 172 q^{7} - 64 q^{8} + 333 q^{9} + O(q^{10}) \) \( q - 4 q^{2} + 24 q^{3} + 16 q^{4} + 25 q^{5} - 96 q^{6} - 172 q^{7} - 64 q^{8} + 333 q^{9} - 100 q^{10} + 132 q^{11} + 384 q^{12} - 946 q^{13} + 688 q^{14} + 600 q^{15} + 256 q^{16} - 222 q^{17} - 1332 q^{18} + 500 q^{19} + 400 q^{20} - 4128 q^{21} - 528 q^{22} + 3564 q^{23} - 1536 q^{24} + 625 q^{25} + 3784 q^{26} + 2160 q^{27} - 2752 q^{28} + 2190 q^{29} - 2400 q^{30} + 2312 q^{31} - 1024 q^{32} + 3168 q^{33} + 888 q^{34} - 4300 q^{35} + 5328 q^{36} - 11242 q^{37} - 2000 q^{38} - 22704 q^{39} - 1600 q^{40} + 1242 q^{41} + 16512 q^{42} + 20624 q^{43} + 2112 q^{44} + 8325 q^{45} - 14256 q^{46} + 6588 q^{47} + 6144 q^{48} + 12777 q^{49} - 2500 q^{50} - 5328 q^{51} - 15136 q^{52} - 21066 q^{53} - 8640 q^{54} + 3300 q^{55} + 11008 q^{56} + 12000 q^{57} - 8760 q^{58} + 7980 q^{59} + 9600 q^{60} + 16622 q^{61} - 9248 q^{62} - 57276 q^{63} + 4096 q^{64} - 23650 q^{65} - 12672 q^{66} + 1808 q^{67} - 3552 q^{68} + 85536 q^{69} + 17200 q^{70} - 24528 q^{71} - 21312 q^{72} + 20474 q^{73} + 44968 q^{74} + 15000 q^{75} + 8000 q^{76} - 22704 q^{77} + 90816 q^{78} - 46240 q^{79} + 6400 q^{80} - 29079 q^{81} - 4968 q^{82} - 51576 q^{83} - 66048 q^{84} - 5550 q^{85} - 82496 q^{86} + 52560 q^{87} - 8448 q^{88} - 110310 q^{89} - 33300 q^{90} + 162712 q^{91} + 57024 q^{92} + 55488 q^{93} - 26352 q^{94} + 12500 q^{95} - 24576 q^{96} - 78382 q^{97} - 51108 q^{98} + 43956 q^{99} + O(q^{100}) \)

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 25.0000 −96.0000 −172.000 −64.0000 333.000 −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.b 1
3.b odd 2 1 90.6.a.d 1
4.b odd 2 1 80.6.a.a 1
5.b even 2 1 50.6.a.d 1
5.c odd 4 2 50.6.b.a 2
7.b odd 2 1 490.6.a.a 1
8.b even 2 1 320.6.a.b 1
8.d odd 2 1 320.6.a.o 1
12.b even 2 1 720.6.a.j 1
15.d odd 2 1 450.6.a.l 1
15.e even 4 2 450.6.c.h 2
20.d odd 2 1 400.6.a.n 1
20.e even 4 2 400.6.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.b 1 1.a even 1 1 trivial
50.6.a.d 1 5.b even 2 1
50.6.b.a 2 5.c odd 4 2
80.6.a.a 1 4.b odd 2 1
90.6.a.d 1 3.b odd 2 1
320.6.a.b 1 8.b even 2 1
320.6.a.o 1 8.d odd 2 1
400.6.a.n 1 20.d odd 2 1
400.6.c.b 2 20.e even 4 2
450.6.a.l 1 15.d odd 2 1
450.6.c.h 2 15.e even 4 2
490.6.a.a 1 7.b odd 2 1
720.6.a.j 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 24 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( -24 + T \)
$5$ \( -25 + T \)
$7$ \( 172 + T \)
$11$ \( -132 + T \)
$13$ \( 946 + T \)
$17$ \( 222 + T \)
$19$ \( -500 + T \)
$23$ \( -3564 + T \)
$29$ \( -2190 + T \)
$31$ \( -2312 + T \)
$37$ \( 11242 + T \)
$41$ \( -1242 + T \)
$43$ \( -20624 + T \)
$47$ \( -6588 + T \)
$53$ \( 21066 + T \)
$59$ \( -7980 + T \)
$61$ \( -16622 + T \)
$67$ \( -1808 + T \)
$71$ \( 24528 + T \)
$73$ \( -20474 + T \)
$79$ \( 46240 + T \)
$83$ \( 51576 + T \)
$89$ \( 110310 + T \)
$97$ \( 78382 + T \)
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