Properties

Label 16.16.a.a
Level 16
Weight 16
Character orbit 16.a
Self dual yes
Analytic conductor 22.831
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(22.8309608160\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 6252q^{3} + 90510q^{5} - 56q^{7} + 24738597q^{9} + O(q^{10}) \) \( q - 6252q^{3} + 90510q^{5} - 56q^{7} + 24738597q^{9} + 95889948q^{11} - 59782138q^{13} - 565868520q^{15} - 1355814414q^{17} - 3783593180q^{19} + 350112q^{21} + 11608845528q^{23} - 22325518025q^{25} - 64956341880q^{27} - 28959105930q^{29} - 253685353952q^{31} - 599503954896q^{33} - 5068560q^{35} + 817641294446q^{37} + 373757926776q^{39} - 682333284198q^{41} - 366945604292q^{43} + 2239090414470q^{45} - 695741581776q^{47} - 4747561506807q^{49} + 8476551716328q^{51} + 12993372468702q^{53} + 8678999193480q^{55} + 23655024561360q^{57} - 9209035340340q^{59} - 42338641200298q^{61} - 1385361432q^{63} - 5410881310380q^{65} - 30029787950636q^{67} - 72578502241056q^{69} - 115328696975352q^{71} + 43787346432122q^{73} + 139579138692300q^{75} - 5369837088q^{77} - 79603813043120q^{79} + 51135221770281q^{81} + 3417068864868q^{83} - 122714762611140q^{85} + 181052330274360q^{87} - 377306179184790q^{89} + 3347799728q^{91} + 1586040832907904q^{93} - 342453018721800q^{95} - 166982186657374q^{97} + 2372182779922956q^{99} + O(q^{100}) \)

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 −6252.00 0 90510.0 0 −56.0000 0 2.47386e7 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 16.16.a.a 1
3.b odd 2 1 144.16.a.d 1
4.b odd 2 1 2.16.a.a 1
8.b even 2 1 64.16.a.k 1
8.d odd 2 1 64.16.a.a 1
12.b even 2 1 18.16.a.e 1
20.d odd 2 1 50.16.a.b 1
20.e even 4 2 50.16.b.a 2
28.d even 2 1 98.16.a.a 1
28.f even 6 2 98.16.c.d 2
28.g odd 6 2 98.16.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.16.a.a 1 4.b odd 2 1
16.16.a.a 1 1.a even 1 1 trivial
18.16.a.e 1 12.b even 2 1
50.16.a.b 1 20.d odd 2 1
50.16.b.a 2 20.e even 4 2
64.16.a.a 1 8.d odd 2 1
64.16.a.k 1 8.b even 2 1
98.16.a.a 1 28.d even 2 1
98.16.c.a 2 28.g odd 6 2
98.16.c.d 2 28.f even 6 2
144.16.a.d 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 \( T_{3} + 6252 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 6252 T + 14348907 T^{2} \)
$5$ \( 1 - 90510 T + 30517578125 T^{2} \)
$7$ \( 1 + 56 T + 4747561509943 T^{2} \)
$11$ \( 1 - 95889948 T + 4177248169415651 T^{2} \)
$13$ \( 1 + 59782138 T + 51185893014090757 T^{2} \)
$17$ \( 1 + 1355814414 T + 2862423051509815793 T^{2} \)
$19$ \( 1 + 3783593180 T + 15181127029874798299 T^{2} \)
$23$ \( 1 - 11608845528 T + \)\(26\!\cdots\!07\)\( T^{2} \)
$29$ \( 1 + 28959105930 T + \)\(86\!\cdots\!49\)\( T^{2} \)
$31$ \( 1 + 253685353952 T + \)\(23\!\cdots\!51\)\( T^{2} \)
$37$ \( 1 - 817641294446 T + \)\(33\!\cdots\!93\)\( T^{2} \)
$41$ \( 1 + 682333284198 T + \)\(15\!\cdots\!01\)\( T^{2} \)
$43$ \( 1 + 366945604292 T + \)\(31\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 + 695741581776 T + \)\(12\!\cdots\!43\)\( T^{2} \)
$53$ \( 1 - 12993372468702 T + \)\(73\!\cdots\!57\)\( T^{2} \)
$59$ \( 1 + 9209035340340 T + \)\(36\!\cdots\!99\)\( T^{2} \)
$61$ \( 1 + 42338641200298 T + \)\(60\!\cdots\!01\)\( T^{2} \)
$67$ \( 1 + 30029787950636 T + \)\(24\!\cdots\!43\)\( T^{2} \)
$71$ \( 1 + 115328696975352 T + \)\(58\!\cdots\!51\)\( T^{2} \)
$73$ \( 1 - 43787346432122 T + \)\(89\!\cdots\!57\)\( T^{2} \)
$79$ \( 1 + 79603813043120 T + \)\(29\!\cdots\!99\)\( T^{2} \)
$83$ \( 1 - 3417068864868 T + \)\(61\!\cdots\!07\)\( T^{2} \)
$89$ \( 1 + 377306179184790 T + \)\(17\!\cdots\!49\)\( T^{2} \)
$97$ \( 1 + 166982186657374 T + \)\(63\!\cdots\!93\)\( T^{2} \)
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