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

\(f(q)\) \(=\) \( q + 216 q^{2} - 3348 q^{3} + 13888 q^{4} + 52110 q^{5} - 723168 q^{6} + 2822456 q^{7} - 4078080 q^{8} - 3139803 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 216 q^{2} - 3348 q^{3} + 13888 q^{4} + 52110 q^{5} - 723168 q^{6} + 2822456 q^{7} - 4078080 q^{8} - 3139803 q^{9} + 11255760 q^{10} + 20586852 q^{11} - 46497024 q^{12} - 190073338 q^{13} + 609650496 q^{14} - 174464280 q^{15} - 1335947264 q^{16} + 1646527986 q^{17} - 678197448 q^{18} + 1563257180 q^{19} + 723703680 q^{20} - 9449582688 q^{21} + 4446760032 q^{22} + 9451116072 q^{23} + 13653411840 q^{24} - 27802126025 q^{25} - 41055841008 q^{26} + 58552201080 q^{27} + 39198268928 q^{28} - 36902568330 q^{29} - 37684284480 q^{30} + 71588483552 q^{31} - 154934083584 q^{32} - 68924780496 q^{33} + 355650044976 q^{34} + 147078182160 q^{35} - 43605584064 q^{36} - 1033652081554 q^{37} + 337663550880 q^{38} + 636365535624 q^{39} - 212508748800 q^{40} + 1641974018202 q^{41} - 2041109860608 q^{42} - 492403109308 q^{43} + 285910200576 q^{44} - 163615134330 q^{45} + 2041441071552 q^{46} - 3410684952624 q^{47} + 4472751439872 q^{48} + 3218696361993 q^{49} - 6005259221400 q^{50} - 5512575697128 q^{51} - 2639738518144 q^{52} + 6797151655902 q^{53} + 12647275433280 q^{54} + 1072780857720 q^{55} - 11510201364480 q^{56} - 5233785038640 q^{57} - 7970954759280 q^{58} + 9858856815540 q^{59} - 2422959920640 q^{60} + 4931842626902 q^{61} + 15463112447232 q^{62} - 8861955816168 q^{63} + 10310557892608 q^{64} - 9904721643180 q^{65} - 14887752587136 q^{66} - 28837826625364 q^{67} + 22866980669568 q^{68} - 31642336609056 q^{69} + 31768887346560 q^{70} + 125050114914552 q^{71} + 12804367818240 q^{72} - 82171455513478 q^{73} - 223268849615664 q^{74} + 93081517931700 q^{75} + 21710515715840 q^{76} + 58105483948512 q^{77} + 137454955694784 q^{78} - 25413078694480 q^{79} - 69616211927040 q^{80} - 150980027970519 q^{81} + 354666387931632 q^{82} - 281736730890468 q^{83} - 131235804370944 q^{84} + 85800573350460 q^{85} - 106359071610528 q^{86} + 123549798768840 q^{87} - 83954829404160 q^{88} + 715618564776810 q^{89} - 35340869015280 q^{90} - 536473633278128 q^{91} + 131257100007936 q^{92} - 239678242932096 q^{93} - 736707949766784 q^{94} + 81461331649800 q^{95} + 518719311839232 q^{96} + 612786136081826 q^{97} + 695238414190488 q^{98} - 64638659670156 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

$p$ $F_p(T)$
$2$ \( T - 216 \) Copy content Toggle raw display
$3$ \( T + 3348 \) Copy content Toggle raw display
$5$ \( T - 52110 \) Copy content Toggle raw display
$7$ \( T - 2822456 \) Copy content Toggle raw display
$11$ \( T - 20586852 \) Copy content Toggle raw display
$13$ \( T + 190073338 \) Copy content Toggle raw display
$17$ \( T - 1646527986 \) Copy content Toggle raw display
$19$ \( T - 1563257180 \) Copy content Toggle raw display
$23$ \( T - 9451116072 \) Copy content Toggle raw display
$29$ \( T + 36902568330 \) Copy content Toggle raw display
$31$ \( T - 71588483552 \) Copy content Toggle raw display
$37$ \( T + 1033652081554 \) Copy content Toggle raw display
$41$ \( T - 1641974018202 \) Copy content Toggle raw display
$43$ \( T + 492403109308 \) Copy content Toggle raw display
$47$ \( T + 3410684952624 \) Copy content Toggle raw display
$53$ \( T - 6797151655902 \) Copy content Toggle raw display
$59$ \( T - 9858856815540 \) Copy content Toggle raw display
$61$ \( T - 4931842626902 \) Copy content Toggle raw display
$67$ \( T + 28837826625364 \) Copy content Toggle raw display
$71$ \( T - 125050114914552 \) Copy content Toggle raw display
$73$ \( T + 82171455513478 \) Copy content Toggle raw display
$79$ \( T + 25413078694480 \) Copy content Toggle raw display
$83$ \( T + 281736730890468 \) Copy content Toggle raw display
$89$ \( T - 715618564776810 \) Copy content Toggle raw display
$97$ \( T - 612786136081826 \) Copy content Toggle raw display
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