Properties

Label 16.14.a.a
Level 16
Weight 14
Character orbit 16.a
Self dual yes
Analytic conductor 17.157
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1569486323\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 1236q^{3} - 57450q^{5} - 64232q^{7} - 66627q^{9} + O(q^{10}) \) \( q - 1236q^{3} - 57450q^{5} - 64232q^{7} - 66627q^{9} - 2464572q^{11} + 8032766q^{13} + 71008200q^{15} + 71112402q^{17} - 136337060q^{19} + 79390752q^{21} + 1186563144q^{23} + 2079799375q^{25} + 2052934200q^{27} - 890583090q^{29} - 4595552672q^{31} + 3046210992q^{33} + 3690128400q^{35} - 19585053898q^{37} - 9928498776q^{39} - 2724170358q^{41} - 51762321116q^{43} + 3827721150q^{45} + 53572833168q^{47} - 92763260583q^{49} - 87894928872q^{51} + 82633440006q^{53} + 141589661400q^{55} + 168512606160q^{57} + 394266352980q^{59} + 671061772142q^{61} + 4279585464q^{63} - 461482406700q^{65} - 388156449812q^{67} - 1466592045984q^{69} + 388772243928q^{71} + 1540972938026q^{73} - 2570632027500q^{75} + 158304388704q^{77} + 3306509559280q^{79} - 2431201712679q^{81} - 4931756967396q^{83} - 4085407494900q^{85} + 1100760699240q^{87} + 3502949738490q^{89} - 515960625712q^{91} + 5680103102592q^{93} + 7832564097000q^{95} - 388932598558q^{97} + 164207038644q^{99} + O(q^{100}) \)

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
0 −1236.00 0 −57450.0 0 −64232.0 0 −66627.0 0
\(n\): 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 16.14.a.a 1
3.b odd 2 1 144.14.a.l 1
4.b odd 2 1 2.14.a.b 1
8.b even 2 1 64.14.a.h 1
8.d odd 2 1 64.14.a.b 1
12.b even 2 1 18.14.a.c 1
20.d odd 2 1 50.14.a.a 1
20.e even 4 2 50.14.b.d 2
28.d even 2 1 98.14.a.c 1
28.f even 6 2 98.14.c.d 2
28.g odd 6 2 98.14.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.14.a.b 1 4.b odd 2 1
16.14.a.a 1 1.a even 1 1 trivial
18.14.a.c 1 12.b even 2 1
50.14.a.a 1 20.d odd 2 1
50.14.b.d 2 20.e even 4 2
64.14.a.b 1 8.d odd 2 1
64.14.a.h 1 8.b even 2 1
98.14.a.c 1 28.d even 2 1
98.14.c.a 2 28.g odd 6 2
98.14.c.d 2 28.f even 6 2
144.14.a.l 1 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1236 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 1236 T + 1594323 T^{2} \)
$5$ \( 1 + 57450 T + 1220703125 T^{2} \)
$7$ \( 1 + 64232 T + 96889010407 T^{2} \)
$11$ \( 1 + 2464572 T + 34522712143931 T^{2} \)
$13$ \( 1 - 8032766 T + 302875106592253 T^{2} \)
$17$ \( 1 - 71112402 T + 9904578032905937 T^{2} \)
$19$ \( 1 + 136337060 T + 42052983462257059 T^{2} \)
$23$ \( 1 - 1186563144 T + 504036361936467383 T^{2} \)
$29$ \( 1 + 890583090 T + 10260628712958602189 T^{2} \)
$31$ \( 1 + 4595552672 T + 24417546297445042591 T^{2} \)
$37$ \( 1 + 19585053898 T + \)\(24\!\cdots\!97\)\( T^{2} \)
$41$ \( 1 + 2724170358 T + \)\(92\!\cdots\!21\)\( T^{2} \)
$43$ \( 1 + 51762321116 T + \)\(17\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 - 53572833168 T + \)\(54\!\cdots\!27\)\( T^{2} \)
$53$ \( 1 - 82633440006 T + \)\(26\!\cdots\!73\)\( T^{2} \)
$59$ \( 1 - 394266352980 T + \)\(10\!\cdots\!79\)\( T^{2} \)
$61$ \( 1 - 671061772142 T + \)\(16\!\cdots\!81\)\( T^{2} \)
$67$ \( 1 + 388156449812 T + \)\(54\!\cdots\!87\)\( T^{2} \)
$71$ \( 1 - 388772243928 T + \)\(11\!\cdots\!11\)\( T^{2} \)
$73$ \( 1 - 1540972938026 T + \)\(16\!\cdots\!33\)\( T^{2} \)
$79$ \( 1 - 3306509559280 T + \)\(46\!\cdots\!39\)\( T^{2} \)
$83$ \( 1 + 4931756967396 T + \)\(88\!\cdots\!63\)\( T^{2} \)
$89$ \( 1 - 3502949738490 T + \)\(21\!\cdots\!69\)\( T^{2} \)
$97$ \( 1 + 388932598558 T + \)\(67\!\cdots\!77\)\( T^{2} \)
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