Properties

Label 10.12.a.c
Level 10
Weight 12
Character orbit 10.a
Self dual yes
Analytic conductor 7.683
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.68343180560\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + 32q^{2} - 318q^{3} + 1024q^{4} - 3125q^{5} - 10176q^{6} - 70714q^{7} + 32768q^{8} - 76023q^{9} + O(q^{10}) \) \( q + 32q^{2} - 318q^{3} + 1024q^{4} - 3125q^{5} - 10176q^{6} - 70714q^{7} + 32768q^{8} - 76023q^{9} - 100000q^{10} + 238272q^{11} - 325632q^{12} - 2097478q^{13} - 2262848q^{14} + 993750q^{15} + 1048576q^{16} + 5955546q^{17} - 2432736q^{18} + 10210820q^{19} - 3200000q^{20} + 22487052q^{21} + 7624704q^{22} - 3535758q^{23} - 10420224q^{24} + 9765625q^{25} - 67119296q^{26} + 80508060q^{27} - 72411136q^{28} - 139304850q^{29} + 31800000q^{30} - 101002348q^{31} + 33554432q^{32} - 75770496q^{33} + 190577472q^{34} + 220981250q^{35} - 77847552q^{36} - 524913814q^{37} + 326746240q^{38} + 666998004q^{39} - 102400000q^{40} + 284590422q^{41} + 719585664q^{42} - 1253635078q^{43} + 243990528q^{44} + 237571875q^{45} - 113144256q^{46} - 216106434q^{47} - 333447168q^{48} + 3023143053q^{49} + 312500000q^{50} - 1893863628q^{51} - 2147817472q^{52} - 4881275358q^{53} + 2576257920q^{54} - 744600000q^{55} - 2317156352q^{56} - 3247040760q^{57} - 4457755200q^{58} + 8692473300q^{59} + 1017600000q^{60} + 3296491802q^{61} - 3232075136q^{62} + 5375890422q^{63} + 1073741824q^{64} + 6554618750q^{65} - 2424655872q^{66} + 18275027966q^{67} + 6098479104q^{68} + 1124371044q^{69} + 7071400000q^{70} - 13287447588q^{71} - 2491121664q^{72} - 32505250798q^{73} - 16797242048q^{74} - 3105468750q^{75} + 10455879680q^{76} - 16849166208q^{77} + 21343936128q^{78} + 9297455960q^{79} - 3276800000q^{80} - 12134316699q^{81} + 9106893504q^{82} - 22741484838q^{83} + 23026741248q^{84} - 18611081250q^{85} - 40116322496q^{86} + 44298942300q^{87} + 7807696896q^{88} - 93378882390q^{89} + 7602300000q^{90} + 148321059292q^{91} - 3620616192q^{92} + 32118746664q^{93} - 6915405888q^{94} - 31908812500q^{95} - 10670309376q^{96} - 5811134014q^{97} + 96740577696q^{98} - 18114152256q^{99} + O(q^{100}) \)

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
32.0000 −318.000 1024.00 −3125.00 −10176.0 −70714.0 32768.0 −76023.0 −100000.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 10.12.a.c 1
3.b odd 2 1 90.12.a.b 1
4.b odd 2 1 80.12.a.e 1
5.b even 2 1 50.12.a.b 1
5.c odd 4 2 50.12.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.12.a.c 1 1.a even 1 1 trivial
50.12.a.b 1 5.b even 2 1
50.12.b.b 2 5.c odd 4 2
80.12.a.e 1 4.b odd 2 1
90.12.a.b 1 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 318 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 32 T \)
$3$ \( 1 + 318 T + 177147 T^{2} \)
$5$ \( 1 + 3125 T \)
$7$ \( 1 + 70714 T + 1977326743 T^{2} \)
$11$ \( 1 - 238272 T + 285311670611 T^{2} \)
$13$ \( 1 + 2097478 T + 1792160394037 T^{2} \)
$17$ \( 1 - 5955546 T + 34271896307633 T^{2} \)
$19$ \( 1 - 10210820 T + 116490258898219 T^{2} \)
$23$ \( 1 + 3535758 T + 952809757913927 T^{2} \)
$29$ \( 1 + 139304850 T + 12200509765705829 T^{2} \)
$31$ \( 1 + 101002348 T + 25408476896404831 T^{2} \)
$37$ \( 1 + 524913814 T + 177917621779460413 T^{2} \)
$41$ \( 1 - 284590422 T + 550329031716248441 T^{2} \)
$43$ \( 1 + 1253635078 T + 929293739471222707 T^{2} \)
$47$ \( 1 + 216106434 T + 2472159215084012303 T^{2} \)
$53$ \( 1 + 4881275358 T + 9269035929372191597 T^{2} \)
$59$ \( 1 - 8692473300 T + 30155888444737842659 T^{2} \)
$61$ \( 1 - 3296491802 T + 43513917611435838661 T^{2} \)
$67$ \( 1 - 18275027966 T + \)\(12\!\cdots\!83\)\( T^{2} \)
$71$ \( 1 + 13287447588 T + \)\(23\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 + 32505250798 T + \)\(31\!\cdots\!77\)\( T^{2} \)
$79$ \( 1 - 9297455960 T + \)\(74\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 + 22741484838 T + \)\(12\!\cdots\!67\)\( T^{2} \)
$89$ \( 1 + 93378882390 T + \)\(27\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 + 5811134014 T + \)\(71\!\cdots\!53\)\( T^{2} \)
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