Properties

Label 1.16.a.a
Level 1
Weight 16
Character orbit 1.a
Self dual yes
Analytic conductor 1.427
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.42693505100\)
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

\(f(q)\) \(=\) \( q + 216q^{2} - 3348q^{3} + 13888q^{4} + 52110q^{5} - 723168q^{6} + 2822456q^{7} - 4078080q^{8} - 3139803q^{9} + O(q^{10}) \) \( q + 216q^{2} - 3348q^{3} + 13888q^{4} + 52110q^{5} - 723168q^{6} + 2822456q^{7} - 4078080q^{8} - 3139803q^{9} + 11255760q^{10} + 20586852q^{11} - 46497024q^{12} - 190073338q^{13} + 609650496q^{14} - 174464280q^{15} - 1335947264q^{16} + 1646527986q^{17} - 678197448q^{18} + 1563257180q^{19} + 723703680q^{20} - 9449582688q^{21} + 4446760032q^{22} + 9451116072q^{23} + 13653411840q^{24} - 27802126025q^{25} - 41055841008q^{26} + 58552201080q^{27} + 39198268928q^{28} - 36902568330q^{29} - 37684284480q^{30} + 71588483552q^{31} - 154934083584q^{32} - 68924780496q^{33} + 355650044976q^{34} + 147078182160q^{35} - 43605584064q^{36} - 1033652081554q^{37} + 337663550880q^{38} + 636365535624q^{39} - 212508748800q^{40} + 1641974018202q^{41} - 2041109860608q^{42} - 492403109308q^{43} + 285910200576q^{44} - 163615134330q^{45} + 2041441071552q^{46} - 3410684952624q^{47} + 4472751439872q^{48} + 3218696361993q^{49} - 6005259221400q^{50} - 5512575697128q^{51} - 2639738518144q^{52} + 6797151655902q^{53} + 12647275433280q^{54} + 1072780857720q^{55} - 11510201364480q^{56} - 5233785038640q^{57} - 7970954759280q^{58} + 9858856815540q^{59} - 2422959920640q^{60} + 4931842626902q^{61} + 15463112447232q^{62} - 8861955816168q^{63} + 10310557892608q^{64} - 9904721643180q^{65} - 14887752587136q^{66} - 28837826625364q^{67} + 22866980669568q^{68} - 31642336609056q^{69} + 31768887346560q^{70} + 125050114914552q^{71} + 12804367818240q^{72} - 82171455513478q^{73} - 223268849615664q^{74} + 93081517931700q^{75} + 21710515715840q^{76} + 58105483948512q^{77} + 137454955694784q^{78} - 25413078694480q^{79} - 69616211927040q^{80} - 150980027970519q^{81} + 354666387931632q^{82} - 281736730890468q^{83} - 131235804370944q^{84} + 85800573350460q^{85} - 106359071610528q^{86} + 123549798768840q^{87} - 83954829404160q^{88} + 715618564776810q^{89} - 35340869015280q^{90} - 536473633278128q^{91} + 131257100007936q^{92} - 239678242932096q^{93} - 736707949766784q^{94} + 81461331649800q^{95} + 518719311839232q^{96} + 612786136081826q^{97} + 695238414190488q^{98} - 64638659670156q^{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
216.000 −3348.00 13888.0 52110.0 −723168. 2.82246e6 −4.07808e6 −3.13980e6 1.12558e7
\(n\): 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 1.16.a.a 1
3.b odd 2 1 9.16.a.a 1
4.b odd 2 1 16.16.a.d 1
5.b even 2 1 25.16.a.a 1
5.c odd 4 2 25.16.b.a 2
7.b odd 2 1 49.16.a.a 1
7.c even 3 2 49.16.c.c 2
7.d odd 6 2 49.16.c.b 2
8.b even 2 1 64.16.a.i 1
8.d odd 2 1 64.16.a.c 1
11.b odd 2 1 121.16.a.a 1
12.b even 2 1 144.16.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.16.a.a 1 1.a even 1 1 trivial
9.16.a.a 1 3.b odd 2 1
16.16.a.d 1 4.b odd 2 1
25.16.a.a 1 5.b even 2 1
25.16.b.a 2 5.c odd 4 2
49.16.a.a 1 7.b odd 2 1
49.16.c.b 2 7.d odd 6 2
49.16.c.c 2 7.c even 3 2
64.16.a.c 1 8.d odd 2 1
64.16.a.i 1 8.b even 2 1
121.16.a.a 1 11.b odd 2 1
144.16.a.f 1 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 216 T + 32768 T^{2} \)
$3$ \( 1 + 3348 T + 14348907 T^{2} \)
$5$ \( 1 - 52110 T + 30517578125 T^{2} \)
$7$ \( 1 - 2822456 T + 4747561509943 T^{2} \)
$11$ \( 1 - 20586852 T + 4177248169415651 T^{2} \)
$13$ \( 1 + 190073338 T + 51185893014090757 T^{2} \)
$17$ \( 1 - 1646527986 T + 2862423051509815793 T^{2} \)
$19$ \( 1 - 1563257180 T + 15181127029874798299 T^{2} \)
$23$ \( 1 - 9451116072 T + \)\(26\!\cdots\!07\)\( T^{2} \)
$29$ \( 1 + 36902568330 T + \)\(86\!\cdots\!49\)\( T^{2} \)
$31$ \( 1 - 71588483552 T + \)\(23\!\cdots\!51\)\( T^{2} \)
$37$ \( 1 + 1033652081554 T + \)\(33\!\cdots\!93\)\( T^{2} \)
$41$ \( 1 - 1641974018202 T + \)\(15\!\cdots\!01\)\( T^{2} \)
$43$ \( 1 + 492403109308 T + \)\(31\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 + 3410684952624 T + \)\(12\!\cdots\!43\)\( T^{2} \)
$53$ \( 1 - 6797151655902 T + \)\(73\!\cdots\!57\)\( T^{2} \)
$59$ \( 1 - 9858856815540 T + \)\(36\!\cdots\!99\)\( T^{2} \)
$61$ \( 1 - 4931842626902 T + \)\(60\!\cdots\!01\)\( T^{2} \)
$67$ \( 1 + 28837826625364 T + \)\(24\!\cdots\!43\)\( T^{2} \)
$71$ \( 1 - 125050114914552 T + \)\(58\!\cdots\!51\)\( T^{2} \)
$73$ \( 1 + 82171455513478 T + \)\(89\!\cdots\!57\)\( T^{2} \)
$79$ \( 1 + 25413078694480 T + \)\(29\!\cdots\!99\)\( T^{2} \)
$83$ \( 1 + 281736730890468 T + \)\(61\!\cdots\!07\)\( T^{2} \)
$89$ \( 1 - 715618564776810 T + \)\(17\!\cdots\!49\)\( T^{2} \)
$97$ \( 1 - 612786136081826 T + \)\(63\!\cdots\!93\)\( T^{2} \)
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