Properties

Label 16.16.a.c
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 4)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 276q^{3} - 132210q^{5} + 3585736q^{7} - 14272731q^{9} + O(q^{10}) \) \( q + 276q^{3} - 132210q^{5} + 3585736q^{7} - 14272731q^{9} - 47801700q^{11} + 247784966q^{13} - 36489960q^{15} - 2127682062q^{17} + 1074862756q^{19} + 989663136q^{21} - 24982896168q^{23} - 13038094025q^{25} - 7899572088q^{27} - 165099671946q^{29} - 100736332256q^{31} - 13193269200q^{33} - 474070156560q^{35} + 42490420334q^{37} + 68388650616q^{39} - 1388779245414q^{41} + 1168783477180q^{43} + 1886997765510q^{45} + 1645655322672q^{47} + 8109941151753q^{49} - 587240249112q^{51} - 4469627500578q^{53} + 6319862757000q^{55} + 296662120656q^{57} + 28794808426572q^{59} + 15719941145942q^{61} - 51178245365016q^{63} - 32759650354860q^{65} - 61627103890604q^{67} - 6895279342368q^{69} + 66780412989192q^{71} - 57749646345094q^{73} - 3598513950900q^{75} - 171404276551200q^{77} - 198700138788272q^{79} + 202617807858729q^{81} + 113345193514212q^{83} + 281300845417020q^{85} - 45567509457096q^{87} - 48230883277974q^{89} + 888491472844976q^{91} - 27803227702656q^{93} - 142107604970760q^{95} + 95121696327074q^{97} + 682260805442700q^{99} + O(q^{100}) \)

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
0 276.000 0 −132210. 0 3.58574e6 0 −1.42727e7 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.c 1
3.b odd 2 1 144.16.a.k 1
4.b odd 2 1 4.16.a.a 1
8.b even 2 1 64.16.a.e 1
8.d odd 2 1 64.16.a.g 1
12.b even 2 1 36.16.a.b 1
20.d odd 2 1 100.16.a.a 1
20.e even 4 2 100.16.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.16.a.a 1 4.b odd 2 1
16.16.a.c 1 1.a even 1 1 trivial
36.16.a.b 1 12.b even 2 1
64.16.a.e 1 8.b even 2 1
64.16.a.g 1 8.d odd 2 1
100.16.a.a 1 20.d odd 2 1
100.16.c.a 2 20.e even 4 2
144.16.a.k 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 \( T_{3} - 276 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 276 T + 14348907 T^{2} \)
$5$ \( 1 + 132210 T + 30517578125 T^{2} \)
$7$ \( 1 - 3585736 T + 4747561509943 T^{2} \)
$11$ \( 1 + 47801700 T + 4177248169415651 T^{2} \)
$13$ \( 1 - 247784966 T + 51185893014090757 T^{2} \)
$17$ \( 1 + 2127682062 T + 2862423051509815793 T^{2} \)
$19$ \( 1 - 1074862756 T + 15181127029874798299 T^{2} \)
$23$ \( 1 + 24982896168 T + \)\(26\!\cdots\!07\)\( T^{2} \)
$29$ \( 1 + 165099671946 T + \)\(86\!\cdots\!49\)\( T^{2} \)
$31$ \( 1 + 100736332256 T + \)\(23\!\cdots\!51\)\( T^{2} \)
$37$ \( 1 - 42490420334 T + \)\(33\!\cdots\!93\)\( T^{2} \)
$41$ \( 1 + 1388779245414 T + \)\(15\!\cdots\!01\)\( T^{2} \)
$43$ \( 1 - 1168783477180 T + \)\(31\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 - 1645655322672 T + \)\(12\!\cdots\!43\)\( T^{2} \)
$53$ \( 1 + 4469627500578 T + \)\(73\!\cdots\!57\)\( T^{2} \)
$59$ \( 1 - 28794808426572 T + \)\(36\!\cdots\!99\)\( T^{2} \)
$61$ \( 1 - 15719941145942 T + \)\(60\!\cdots\!01\)\( T^{2} \)
$67$ \( 1 + 61627103890604 T + \)\(24\!\cdots\!43\)\( T^{2} \)
$71$ \( 1 - 66780412989192 T + \)\(58\!\cdots\!51\)\( T^{2} \)
$73$ \( 1 + 57749646345094 T + \)\(89\!\cdots\!57\)\( T^{2} \)
$79$ \( 1 + 198700138788272 T + \)\(29\!\cdots\!99\)\( T^{2} \)
$83$ \( 1 - 113345193514212 T + \)\(61\!\cdots\!07\)\( T^{2} \)
$89$ \( 1 + 48230883277974 T + \)\(17\!\cdots\!49\)\( T^{2} \)
$97$ \( 1 - 95121696327074 T + \)\(63\!\cdots\!93\)\( T^{2} \)
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