Properties

Label 10.18.a.a
Level 10
Weight 18
Character orbit 10.a
Self dual yes
Analytic conductor 18.322
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.3222087345\)
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

\(f(q)\) \(=\) \( q + 256q^{2} - 14976q^{3} + 65536q^{4} + 390625q^{5} - 3833856q^{6} + 14808668q^{7} + 16777216q^{8} + 95140413q^{9} + O(q^{10}) \) \( q + 256q^{2} - 14976q^{3} + 65536q^{4} + 390625q^{5} - 3833856q^{6} + 14808668q^{7} + 16777216q^{8} + 95140413q^{9} + 100000000q^{10} - 1085034588q^{11} - 981467136q^{12} - 4595303746q^{13} + 3791019008q^{14} - 5850000000q^{15} + 4294967296q^{16} - 16104698622q^{17} + 24355945728q^{18} + 48093117860q^{19} + 25600000000q^{20} - 221774611968q^{21} - 277768854528q^{22} - 571023069276q^{23} - 251255586816q^{24} + 152587890625q^{25} - 1176397758976q^{26} + 509180256000q^{27} + 970500866048q^{28} - 1726424788290q^{29} - 1497600000000q^{30} - 5623721940808q^{31} + 1099511627776q^{32} + 16249477989888q^{33} - 4122802847232q^{34} + 5784635937500q^{35} + 6235122106368q^{36} - 10013128639162q^{37} + 12311838172160q^{38} + 68819268900096q^{39} + 6553600000000q^{40} - 37505113176198q^{41} - 56774300663808q^{42} + 136226190448184q^{43} - 71108826759168q^{44} + 37164223828125q^{45} - 146181905734656q^{46} - 37681319902812q^{47} - 64321430224896q^{48} - 13333866052983q^{49} + 39062500000000q^{50} + 241183966563072q^{51} - 301157826297856q^{52} - 22543738268346q^{53} + 130350145536000q^{54} - 423841635937500q^{55} + 248448221708288q^{56} - 720242533071360q^{57} - 441964745802240q^{58} + 221363585667420q^{59} - 383385600000000q^{60} - 276238009706818q^{61} - 1439672816846848q^{62} + 1408902789499884q^{63} + 281474976710656q^{64} - 1795040525781250q^{65} + 4159866365411328q^{66} + 6165400365120968q^{67} - 1055437528891392q^{68} + 8551641485477376q^{69} + 1480866800000000q^{70} - 129445634389248q^{71} + 1596191259230208q^{72} - 9751215737952646q^{73} - 2563360931625472q^{74} - 2285156250000000q^{75} + 3151830572072960q^{76} - 16067916982208784q^{77} + 17617732838424576q^{78} - 25538259579681280q^{79} + 1677721600000000q^{80} - 19911931956563319q^{81} - 9601308973106688q^{82} + 29930106765986544q^{83} - 14534220969934848q^{84} - 6290897899218750q^{85} + 34873904754735104q^{86} + 25854937629431040q^{87} - 18203859650347008q^{88} - 24258364335352710q^{89} + 9514041300000000q^{90} - 68050327533670328q^{91} - 37422567868071936q^{92} + 84220859785540608q^{93} - 9646417895119872q^{94} + 18786374164062500q^{95} - 16466286137573376q^{96} + 123350176379809778q^{97} - 3413469709563648q^{98} - 103230638821604844q^{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
256.000 −14976.0 65536.0 390625. −3.83386e6 1.48087e7 1.67772e7 9.51404e7 1.00000e8
\(n\): 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 10.18.a.a 1
3.b odd 2 1 90.18.a.b 1
4.b odd 2 1 80.18.a.a 1
5.b even 2 1 50.18.a.b 1
5.c odd 4 2 50.18.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.18.a.a 1 1.a even 1 1 trivial
50.18.a.b 1 5.b even 2 1
50.18.b.a 2 5.c odd 4 2
80.18.a.a 1 4.b odd 2 1
90.18.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 \( T_{3} + 14976 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 256 T \)
$3$ \( 1 + 14976 T + 129140163 T^{2} \)
$5$ \( 1 - 390625 T \)
$7$ \( 1 - 14808668 T + 232630513987207 T^{2} \)
$11$ \( 1 + 1085034588 T + 505447028499293771 T^{2} \)
$13$ \( 1 + 4595303746 T + 8650415919381337933 T^{2} \)
$17$ \( 1 + 16104698622 T + \)\(82\!\cdots\!77\)\( T^{2} \)
$19$ \( 1 - 48093117860 T + \)\(54\!\cdots\!39\)\( T^{2} \)
$23$ \( 1 + 571023069276 T + \)\(14\!\cdots\!03\)\( T^{2} \)
$29$ \( 1 + 1726424788290 T + \)\(72\!\cdots\!09\)\( T^{2} \)
$31$ \( 1 + 5623721940808 T + \)\(22\!\cdots\!11\)\( T^{2} \)
$37$ \( 1 + 10013128639162 T + \)\(45\!\cdots\!17\)\( T^{2} \)
$41$ \( 1 + 37505113176198 T + \)\(26\!\cdots\!81\)\( T^{2} \)
$43$ \( 1 - 136226190448184 T + \)\(58\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 + 37681319902812 T + \)\(26\!\cdots\!87\)\( T^{2} \)
$53$ \( 1 + 22543738268346 T + \)\(20\!\cdots\!13\)\( T^{2} \)
$59$ \( 1 - 221363585667420 T + \)\(12\!\cdots\!19\)\( T^{2} \)
$61$ \( 1 + 276238009706818 T + \)\(22\!\cdots\!21\)\( T^{2} \)
$67$ \( 1 - 6165400365120968 T + \)\(11\!\cdots\!27\)\( T^{2} \)
$71$ \( 1 + 129445634389248 T + \)\(29\!\cdots\!91\)\( T^{2} \)
$73$ \( 1 + 9751215737952646 T + \)\(47\!\cdots\!53\)\( T^{2} \)
$79$ \( 1 + 25538259579681280 T + \)\(18\!\cdots\!59\)\( T^{2} \)
$83$ \( 1 - 29930106765986544 T + \)\(42\!\cdots\!23\)\( T^{2} \)
$89$ \( 1 + 24258364335352710 T + \)\(13\!\cdots\!29\)\( T^{2} \)
$97$ \( 1 - 123350176379809778 T + \)\(59\!\cdots\!37\)\( T^{2} \)
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