Properties

Label 10.16.a.a
Level 10
Weight 16
Character orbit 10.a
Self dual yes
Analytic conductor 14.269
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 16 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.2693505100\)
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

\(f(q)\) \(=\) \( q - 128q^{2} - 5568q^{3} + 16384q^{4} + 78125q^{5} + 712704q^{6} + 2564996q^{7} - 2097152q^{8} + 16653717q^{9} + O(q^{10}) \) \( q - 128q^{2} - 5568q^{3} + 16384q^{4} + 78125q^{5} + 712704q^{6} + 2564996q^{7} - 2097152q^{8} + 16653717q^{9} - 10000000q^{10} - 81067668q^{11} - 91226112q^{12} + 351412022q^{13} - 328319488q^{14} - 435000000q^{15} + 268435456q^{16} - 2157825054q^{17} - 2131675776q^{18} - 5107458100q^{19} + 1280000000q^{20} - 14281897728q^{21} + 10376661504q^{22} + 11784341052q^{23} + 11676942336q^{24} + 6103515625q^{25} - 44980738816q^{26} - 12833182080q^{27} + 42024894464q^{28} - 20400574890q^{29} + 55680000000q^{30} - 123613797688q^{31} - 34359738368q^{32} + 451384775424q^{33} + 276201606912q^{34} + 200390312500q^{35} + 272854499328q^{36} - 22499625394q^{37} + 653754636800q^{38} - 1956662138496q^{39} - 163840000000q^{40} - 1044060129558q^{41} + 1828082909184q^{42} - 2984233999768q^{43} - 1328212672512q^{44} + 1301071640625q^{45} - 1508395654656q^{46} - 2267362482084q^{47} - 1494648619008q^{48} + 1831642970073q^{49} - 781250000000q^{50} + 12014769900672q^{51} + 5757534568448q^{52} - 8655803512338q^{53} + 1642647306240q^{54} - 6333411562500q^{55} - 5379186491392q^{56} + 28438326700800q^{57} + 2611273585920q^{58} - 25953000142380q^{59} - 7127040000000q^{60} + 29809710409622q^{61} + 15822566104064q^{62} + 42716717490132q^{63} + 4398046511104q^{64} + 27454064218750q^{65} - 57777251254272q^{66} - 78103662703144q^{67} - 35353805684736q^{68} - 65615210977536q^{69} - 25649960000000q^{70} + 67746916371072q^{71} - 34925375913984q^{72} + 134520120122282q^{73} + 2879952050432q^{74} - 33984375000000q^{75} - 83680593510400q^{76} - 207938244149328q^{77} + 250452753727488q^{78} + 16723463056640q^{79} + 20971520000000q^{80} - 167507478615879q^{81} + 133639696583424q^{82} + 80883629455632q^{83} - 233994612375552q^{84} - 168580082343750q^{85} + 381981951970304q^{86} + 113590400987520q^{87} + 170011222081536q^{88} - 523835472467190q^{89} - 166537170000000q^{90} + 901370430781912q^{91} + 193074643795968q^{92} + 688281625526784q^{93} + 290222397706752q^{94} - 399020164062500q^{95} + 191315023233024q^{96} + 1186642412683826q^{97} - 234450300169344q^{98} - 1350078000721956q^{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
−128.000 −5568.00 16384.0 78125.0 712704. 2.56500e6 −2.09715e6 1.66537e7 −1.00000e7
\(n\): 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 10.16.a.a 1
3.b odd 2 1 90.16.a.f 1
4.b odd 2 1 80.16.a.c 1
5.b even 2 1 50.16.a.d 1
5.c odd 4 2 50.16.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.16.a.a 1 1.a even 1 1 trivial
50.16.a.d 1 5.b even 2 1
50.16.b.d 2 5.c odd 4 2
80.16.a.c 1 4.b odd 2 1
90.16.a.f 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 \( T_{3} + 5568 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 128 T \)
$3$ \( 1 + 5568 T + 14348907 T^{2} \)
$5$ \( 1 - 78125 T \)
$7$ \( 1 - 2564996 T + 4747561509943 T^{2} \)
$11$ \( 1 + 81067668 T + 4177248169415651 T^{2} \)
$13$ \( 1 - 351412022 T + 51185893014090757 T^{2} \)
$17$ \( 1 + 2157825054 T + 2862423051509815793 T^{2} \)
$19$ \( 1 + 5107458100 T + 15181127029874798299 T^{2} \)
$23$ \( 1 - 11784341052 T + \)\(26\!\cdots\!07\)\( T^{2} \)
$29$ \( 1 + 20400574890 T + \)\(86\!\cdots\!49\)\( T^{2} \)
$31$ \( 1 + 123613797688 T + \)\(23\!\cdots\!51\)\( T^{2} \)
$37$ \( 1 + 22499625394 T + \)\(33\!\cdots\!93\)\( T^{2} \)
$41$ \( 1 + 1044060129558 T + \)\(15\!\cdots\!01\)\( T^{2} \)
$43$ \( 1 + 2984233999768 T + \)\(31\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 + 2267362482084 T + \)\(12\!\cdots\!43\)\( T^{2} \)
$53$ \( 1 + 8655803512338 T + \)\(73\!\cdots\!57\)\( T^{2} \)
$59$ \( 1 + 25953000142380 T + \)\(36\!\cdots\!99\)\( T^{2} \)
$61$ \( 1 - 29809710409622 T + \)\(60\!\cdots\!01\)\( T^{2} \)
$67$ \( 1 + 78103662703144 T + \)\(24\!\cdots\!43\)\( T^{2} \)
$71$ \( 1 - 67746916371072 T + \)\(58\!\cdots\!51\)\( T^{2} \)
$73$ \( 1 - 134520120122282 T + \)\(89\!\cdots\!57\)\( T^{2} \)
$79$ \( 1 - 16723463056640 T + \)\(29\!\cdots\!99\)\( T^{2} \)
$83$ \( 1 - 80883629455632 T + \)\(61\!\cdots\!07\)\( T^{2} \)
$89$ \( 1 + 523835472467190 T + \)\(17\!\cdots\!49\)\( T^{2} \)
$97$ \( 1 - 1186642412683826 T + \)\(63\!\cdots\!93\)\( T^{2} \)
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