Properties

Label 10.16.a.b
Level 10
Weight 16
Character orbit 10.a
Self dual yes
Analytic conductor 14.269
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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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: \(0\)
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

\(f(q)\) \(=\) \( q - 128q^{2} - 918q^{3} + 16384q^{4} - 78125q^{5} + 117504q^{6} - 953554q^{7} - 2097152q^{8} - 13506183q^{9} + O(q^{10}) \) \( q - 128q^{2} - 918q^{3} + 16384q^{4} - 78125q^{5} + 117504q^{6} - 953554q^{7} - 2097152q^{8} - 13506183q^{9} + 10000000q^{10} + 17783232q^{11} - 15040512q^{12} + 140533322q^{13} + 122054912q^{14} + 71718750q^{15} + 268435456q^{16} + 2998870746q^{17} + 1728791424q^{18} + 3255852500q^{19} - 1280000000q^{20} + 875362572q^{21} - 2276253696q^{22} + 6774812202q^{23} + 1925185536q^{24} + 6103515625q^{25} - 17988265216q^{26} + 25570972620q^{27} - 15623028736q^{28} - 7340322690q^{29} - 9180000000q^{30} - 115428411388q^{31} - 34359738368q^{32} - 16325006976q^{33} - 383855455488q^{34} + 74496406250q^{35} - 221285302272q^{36} + 150300986906q^{37} - 416749120000q^{38} - 129009589596q^{39} + 163840000000q^{40} + 1841603525142q^{41} - 112046409216q^{42} + 1510018315682q^{43} + 291360473088q^{44} + 1055170546875q^{45} - 867175961856q^{46} + 6093750843366q^{47} - 246423748608q^{48} - 3838296279027q^{49} - 781250000000q^{50} - 2752963344828q^{51} + 2302497947648q^{52} - 8267412829038q^{53} - 3273084495360q^{54} - 1389315000000q^{55} + 1999747678208q^{56} - 2988872595000q^{57} + 939561304320q^{58} - 23516883061980q^{59} + 1175040000000q^{60} - 3135369104278q^{61} + 14774836657664q^{62} + 12878874824382q^{63} + 4398046511104q^{64} - 10979165781250q^{65} + 2089600892928q^{66} - 36030983954794q^{67} + 49133498302464q^{68} - 6219277601436q^{69} - 9535540000000q^{70} + 52169735384172q^{71} + 28324518690816q^{72} + 69977143684082q^{73} - 19238526323968q^{74} - 5603027343750q^{75} + 53343887360000q^{76} - 16957272006528q^{77} + 16513227468288q^{78} - 135317670906760q^{79} - 20971520000000q^{80} + 170324810926821q^{81} - 235725251218176q^{82} + 427456158822882q^{83} + 14341940379648q^{84} - 234286777031250q^{85} - 193282344407296q^{86} + 6738416229420q^{87} - 37294140555264q^{88} - 446581617299190q^{89} - 135061830000000q^{90} - 134006111326388q^{91} + 110998523117568q^{92} + 105963281654184q^{93} - 780000107950848q^{94} - 254363476562500q^{95} + 31542239821824q^{96} + 181247411845826q^{97} + 491301923715456q^{98} - 240183585723456q^{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 −918.000 16384.0 −78125.0 117504. −953554. −2.09715e6 −1.35062e7 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.b 1
3.b odd 2 1 90.16.a.h 1
4.b odd 2 1 80.16.a.a 1
5.b even 2 1 50.16.a.c 1
5.c odd 4 2 50.16.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.16.a.b 1 1.a even 1 1 trivial
50.16.a.c 1 5.b even 2 1
50.16.b.c 2 5.c odd 4 2
80.16.a.a 1 4.b odd 2 1
90.16.a.h 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 \( T_{3} + 918 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 128 T \)
$3$ \( 1 + 918 T + 14348907 T^{2} \)
$5$ \( 1 + 78125 T \)
$7$ \( 1 + 953554 T + 4747561509943 T^{2} \)
$11$ \( 1 - 17783232 T + 4177248169415651 T^{2} \)
$13$ \( 1 - 140533322 T + 51185893014090757 T^{2} \)
$17$ \( 1 - 2998870746 T + 2862423051509815793 T^{2} \)
$19$ \( 1 - 3255852500 T + 15181127029874798299 T^{2} \)
$23$ \( 1 - 6774812202 T + \)\(26\!\cdots\!07\)\( T^{2} \)
$29$ \( 1 + 7340322690 T + \)\(86\!\cdots\!49\)\( T^{2} \)
$31$ \( 1 + 115428411388 T + \)\(23\!\cdots\!51\)\( T^{2} \)
$37$ \( 1 - 150300986906 T + \)\(33\!\cdots\!93\)\( T^{2} \)
$41$ \( 1 - 1841603525142 T + \)\(15\!\cdots\!01\)\( T^{2} \)
$43$ \( 1 - 1510018315682 T + \)\(31\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 - 6093750843366 T + \)\(12\!\cdots\!43\)\( T^{2} \)
$53$ \( 1 + 8267412829038 T + \)\(73\!\cdots\!57\)\( T^{2} \)
$59$ \( 1 + 23516883061980 T + \)\(36\!\cdots\!99\)\( T^{2} \)
$61$ \( 1 + 3135369104278 T + \)\(60\!\cdots\!01\)\( T^{2} \)
$67$ \( 1 + 36030983954794 T + \)\(24\!\cdots\!43\)\( T^{2} \)
$71$ \( 1 - 52169735384172 T + \)\(58\!\cdots\!51\)\( T^{2} \)
$73$ \( 1 - 69977143684082 T + \)\(89\!\cdots\!57\)\( T^{2} \)
$79$ \( 1 + 135317670906760 T + \)\(29\!\cdots\!99\)\( T^{2} \)
$83$ \( 1 - 427456158822882 T + \)\(61\!\cdots\!07\)\( T^{2} \)
$89$ \( 1 + 446581617299190 T + \)\(17\!\cdots\!49\)\( T^{2} \)
$97$ \( 1 - 181247411845826 T + \)\(63\!\cdots\!93\)\( T^{2} \)
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