Properties

Label 16.8.a.a
Level 16
Weight 8
Character orbit 16.a
Self dual yes
Analytic conductor 4.998
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.99816040775\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 44q^{3} + 430q^{5} + 1224q^{7} - 251q^{9} + O(q^{10}) \) \( q - 44q^{3} + 430q^{5} + 1224q^{7} - 251q^{9} + 3164q^{11} + 6118q^{13} - 18920q^{15} - 16270q^{17} + 5476q^{19} - 53856q^{21} - 1576q^{23} + 106775q^{25} + 107272q^{27} + 122838q^{29} - 251360q^{31} - 139216q^{33} + 526320q^{35} - 52338q^{37} - 269192q^{39} - 319398q^{41} - 710788q^{43} - 107930q^{45} - 284112q^{47} + 674633q^{49} + 715880q^{51} + 296062q^{53} + 1360520q^{55} - 240944q^{57} + 897548q^{59} - 884810q^{61} - 307224q^{63} + 2630740q^{65} - 4659692q^{67} + 69344q^{69} + 2710792q^{71} - 5670854q^{73} - 4698100q^{75} + 3872736q^{77} + 5124176q^{79} - 4171031q^{81} + 1563556q^{83} - 6996100q^{85} - 5404872q^{87} + 11605674q^{89} + 7488432q^{91} + 11059840q^{93} + 2354680q^{95} + 10931618q^{97} - 794164q^{99} + O(q^{100}) \)

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 −44.0000 0 430.000 0 1224.00 0 −251.000 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 16.8.a.a 1
3.b odd 2 1 144.8.a.a 1
4.b odd 2 1 8.8.a.b 1
5.b even 2 1 400.8.a.p 1
5.c odd 4 2 400.8.c.f 2
8.b even 2 1 64.8.a.f 1
8.d odd 2 1 64.8.a.b 1
12.b even 2 1 72.8.a.a 1
16.e even 4 2 256.8.b.a 2
16.f odd 4 2 256.8.b.g 2
20.d odd 2 1 200.8.a.b 1
20.e even 4 2 200.8.c.c 2
24.f even 2 1 576.8.a.y 1
24.h odd 2 1 576.8.a.z 1
28.d even 2 1 392.8.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.b 1 4.b odd 2 1
16.8.a.a 1 1.a even 1 1 trivial
64.8.a.b 1 8.d odd 2 1
64.8.a.f 1 8.b even 2 1
72.8.a.a 1 12.b even 2 1
144.8.a.a 1 3.b odd 2 1
200.8.a.b 1 20.d odd 2 1
200.8.c.c 2 20.e even 4 2
256.8.b.a 2 16.e even 4 2
256.8.b.g 2 16.f odd 4 2
392.8.a.b 1 28.d even 2 1
400.8.a.p 1 5.b even 2 1
400.8.c.f 2 5.c odd 4 2
576.8.a.y 1 24.f even 2 1
576.8.a.z 1 24.h 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 \( T_{3} + 44 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 44 T + 2187 T^{2} \)
$5$ \( 1 - 430 T + 78125 T^{2} \)
$7$ \( 1 - 1224 T + 823543 T^{2} \)
$11$ \( 1 - 3164 T + 19487171 T^{2} \)
$13$ \( 1 - 6118 T + 62748517 T^{2} \)
$17$ \( 1 + 16270 T + 410338673 T^{2} \)
$19$ \( 1 - 5476 T + 893871739 T^{2} \)
$23$ \( 1 + 1576 T + 3404825447 T^{2} \)
$29$ \( 1 - 122838 T + 17249876309 T^{2} \)
$31$ \( 1 + 251360 T + 27512614111 T^{2} \)
$37$ \( 1 + 52338 T + 94931877133 T^{2} \)
$41$ \( 1 + 319398 T + 194754273881 T^{2} \)
$43$ \( 1 + 710788 T + 271818611107 T^{2} \)
$47$ \( 1 + 284112 T + 506623120463 T^{2} \)
$53$ \( 1 - 296062 T + 1174711139837 T^{2} \)
$59$ \( 1 - 897548 T + 2488651484819 T^{2} \)
$61$ \( 1 + 884810 T + 3142742836021 T^{2} \)
$67$ \( 1 + 4659692 T + 6060711605323 T^{2} \)
$71$ \( 1 - 2710792 T + 9095120158391 T^{2} \)
$73$ \( 1 + 5670854 T + 11047398519097 T^{2} \)
$79$ \( 1 - 5124176 T + 19203908986159 T^{2} \)
$83$ \( 1 - 1563556 T + 27136050989627 T^{2} \)
$89$ \( 1 - 11605674 T + 44231334895529 T^{2} \)
$97$ \( 1 - 10931618 T + 80798284478113 T^{2} \)
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