Properties

Label 10.16.a.c
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

\(f(q)\) \(=\) \( q + 128q^{2} - 1302q^{3} + 16384q^{4} - 78125q^{5} - 166656q^{6} - 90706q^{7} + 2097152q^{8} - 12653703q^{9} + O(q^{10}) \) \( q + 128q^{2} - 1302q^{3} + 16384q^{4} - 78125q^{5} - 166656q^{6} - 90706q^{7} + 2097152q^{8} - 12653703q^{9} - 10000000q^{10} - 69411648q^{11} - 21331968q^{12} - 37057462q^{13} - 11610368q^{14} + 101718750q^{15} + 268435456q^{16} - 1372027686q^{17} - 1619673984q^{18} - 5068760620q^{19} - 1280000000q^{20} + 118099212q^{21} - 8884690944q^{22} - 12342632022q^{23} - 2730491904q^{24} + 6103515625q^{25} - 4743355136q^{26} + 35157398220q^{27} - 1486127104q^{28} + 57825721470q^{29} + 13020000000q^{30} + 233727970052q^{31} + 34359738368q^{32} + 90373965696q^{33} - 175619543808q^{34} + 7086406250q^{35} - 207318269952q^{36} + 742247612954q^{37} - 648801359360q^{38} + 48248815524q^{39} - 163840000000q^{40} - 772699832298q^{41} + 15116699136q^{42} - 405994366942q^{43} - 1137240440832q^{44} + 988570546875q^{45} - 1579856898816q^{46} + 1623010601574q^{47} - 349502963712q^{48} - 4739333931507q^{49} + 781250000000q^{50} + 1786380047172q^{51} - 607149457408q^{52} - 5365599400302q^{53} + 4500146972160q^{54} + 5422785000000q^{55} - 190224269312q^{56} + 6599526327240q^{57} + 7401692348160q^{58} + 9240158287140q^{59} + 1666560000000q^{60} + 944308151402q^{61} + 29917180166656q^{62} + 1147766784318q^{63} + 4398046511104q^{64} + 2895114218750q^{65} + 11567867609088q^{66} - 70567580292586q^{67} - 22479301607424q^{68} + 16070106892644q^{69} + 907060000000q^{70} - 82534723020948q^{71} - 26536738553856q^{72} - 178432352158222q^{73} + 95007694458112q^{74} - 7946777343750q^{75} - 83046573998080q^{76} + 6296052943488q^{77} + 6175848387072q^{78} + 307261263603320q^{79} - 20971520000000q^{80} + 135791875070181q^{81} - 98905578534144q^{82} + 43159732395618q^{83} + 1934937489408q^{84} + 107189662968750q^{85} - 51967278968576q^{86} - 75289089353940q^{87} - 145566776426496q^{88} + 681677801811210q^{89} + 126537030000000q^{90} + 3361334148172q^{91} - 202221683048448q^{92} - 304313817007704q^{93} + 207745357001472q^{94} + 395996923437500q^{95} - 44736379355136q^{96} - 236520239800126q^{97} - 606634743232896q^{98} + 878314378532544q^{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 −1302.00 16384.0 −78125.0 −166656. −90706.0 2.09715e6 −1.26537e7 −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.c 1
3.b odd 2 1 90.16.a.d 1
4.b odd 2 1 80.16.a.b 1
5.b even 2 1 50.16.a.a 1
5.c odd 4 2 50.16.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.16.a.c 1 1.a even 1 1 trivial
50.16.a.a 1 5.b even 2 1
50.16.b.b 2 5.c odd 4 2
80.16.a.b 1 4.b odd 2 1
90.16.a.d 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 \( T_{3} + 1302 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 128 T \)
$3$ \( 1 + 1302 T + 14348907 T^{2} \)
$5$ \( 1 + 78125 T \)
$7$ \( 1 + 90706 T + 4747561509943 T^{2} \)
$11$ \( 1 + 69411648 T + 4177248169415651 T^{2} \)
$13$ \( 1 + 37057462 T + 51185893014090757 T^{2} \)
$17$ \( 1 + 1372027686 T + 2862423051509815793 T^{2} \)
$19$ \( 1 + 5068760620 T + 15181127029874798299 T^{2} \)
$23$ \( 1 + 12342632022 T + \)\(26\!\cdots\!07\)\( T^{2} \)
$29$ \( 1 - 57825721470 T + \)\(86\!\cdots\!49\)\( T^{2} \)
$31$ \( 1 - 233727970052 T + \)\(23\!\cdots\!51\)\( T^{2} \)
$37$ \( 1 - 742247612954 T + \)\(33\!\cdots\!93\)\( T^{2} \)
$41$ \( 1 + 772699832298 T + \)\(15\!\cdots\!01\)\( T^{2} \)
$43$ \( 1 + 405994366942 T + \)\(31\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 - 1623010601574 T + \)\(12\!\cdots\!43\)\( T^{2} \)
$53$ \( 1 + 5365599400302 T + \)\(73\!\cdots\!57\)\( T^{2} \)
$59$ \( 1 - 9240158287140 T + \)\(36\!\cdots\!99\)\( T^{2} \)
$61$ \( 1 - 944308151402 T + \)\(60\!\cdots\!01\)\( T^{2} \)
$67$ \( 1 + 70567580292586 T + \)\(24\!\cdots\!43\)\( T^{2} \)
$71$ \( 1 + 82534723020948 T + \)\(58\!\cdots\!51\)\( T^{2} \)
$73$ \( 1 + 178432352158222 T + \)\(89\!\cdots\!57\)\( T^{2} \)
$79$ \( 1 - 307261263603320 T + \)\(29\!\cdots\!99\)\( T^{2} \)
$83$ \( 1 - 43159732395618 T + \)\(61\!\cdots\!07\)\( T^{2} \)
$89$ \( 1 - 681677801811210 T + \)\(17\!\cdots\!49\)\( T^{2} \)
$97$ \( 1 + 236520239800126 T + \)\(63\!\cdots\!93\)\( T^{2} \)
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