Properties

Label 16.6.a.b
Level $16$
Weight $6$
Character orbit 16.a
Self dual yes
Analytic conductor $2.566$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.56614111701\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 12 q^{3} + 54 q^{5} + 88 q^{7} - 99 q^{9} + O(q^{10}) \) \( q + 12 q^{3} + 54 q^{5} + 88 q^{7} - 99 q^{9} - 540 q^{11} - 418 q^{13} + 648 q^{15} + 594 q^{17} - 836 q^{19} + 1056 q^{21} + 4104 q^{23} - 209 q^{25} - 4104 q^{27} - 594 q^{29} - 4256 q^{31} - 6480 q^{33} + 4752 q^{35} - 298 q^{37} - 5016 q^{39} + 17226 q^{41} + 12100 q^{43} - 5346 q^{45} + 1296 q^{47} - 9063 q^{49} + 7128 q^{51} + 19494 q^{53} - 29160 q^{55} - 10032 q^{57} + 7668 q^{59} - 34738 q^{61} - 8712 q^{63} - 22572 q^{65} - 21812 q^{67} + 49248 q^{69} + 46872 q^{71} + 67562 q^{73} - 2508 q^{75} - 47520 q^{77} + 76912 q^{79} - 25191 q^{81} - 67716 q^{83} + 32076 q^{85} - 7128 q^{87} + 29754 q^{89} - 36784 q^{91} - 51072 q^{93} - 45144 q^{95} - 122398 q^{97} + 53460 q^{99} + O(q^{100}) \)

Expression as an eta quotient

\(f(z) = \dfrac{\eta(4z)^{36}}{\eta(2z)^{12}\eta(8z)^{12}}=q\prod_{n=1}^\infty(1 - q^{2n})^{-12}(1 - q^{4n})^{36}(1 - q^{8n})^{-12}\)

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 12.0000 0 54.0000 0 88.0000 0 −99.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 16.6.a.b 1
3.b odd 2 1 144.6.a.c 1
4.b odd 2 1 4.6.a.a 1
5.b even 2 1 400.6.a.d 1
5.c odd 4 2 400.6.c.f 2
7.b odd 2 1 784.6.a.d 1
8.b even 2 1 64.6.a.b 1
8.d odd 2 1 64.6.a.f 1
12.b even 2 1 36.6.a.a 1
16.e even 4 2 256.6.b.c 2
16.f odd 4 2 256.6.b.g 2
20.d odd 2 1 100.6.a.b 1
20.e even 4 2 100.6.c.b 2
24.f even 2 1 576.6.a.bc 1
24.h odd 2 1 576.6.a.bd 1
28.d even 2 1 196.6.a.e 1
28.f even 6 2 196.6.e.d 2
28.g odd 6 2 196.6.e.g 2
36.f odd 6 2 324.6.e.a 2
36.h even 6 2 324.6.e.d 2
44.c even 2 1 484.6.a.a 1
52.b odd 2 1 676.6.a.a 1
52.f even 4 2 676.6.d.a 2
60.h even 2 1 900.6.a.h 1
60.l odd 4 2 900.6.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 4.b odd 2 1
16.6.a.b 1 1.a even 1 1 trivial
36.6.a.a 1 12.b even 2 1
64.6.a.b 1 8.b even 2 1
64.6.a.f 1 8.d odd 2 1
100.6.a.b 1 20.d odd 2 1
100.6.c.b 2 20.e even 4 2
144.6.a.c 1 3.b odd 2 1
196.6.a.e 1 28.d even 2 1
196.6.e.d 2 28.f even 6 2
196.6.e.g 2 28.g odd 6 2
256.6.b.c 2 16.e even 4 2
256.6.b.g 2 16.f odd 4 2
324.6.e.a 2 36.f odd 6 2
324.6.e.d 2 36.h even 6 2
400.6.a.d 1 5.b even 2 1
400.6.c.f 2 5.c odd 4 2
484.6.a.a 1 44.c even 2 1
576.6.a.bc 1 24.f even 2 1
576.6.a.bd 1 24.h odd 2 1
676.6.a.a 1 52.b odd 2 1
676.6.d.a 2 52.f even 4 2
784.6.a.d 1 7.b odd 2 1
900.6.a.h 1 60.h even 2 1
900.6.d.a 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 12 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -12 + T \)
$5$ \( -54 + T \)
$7$ \( -88 + T \)
$11$ \( 540 + T \)
$13$ \( 418 + T \)
$17$ \( -594 + T \)
$19$ \( 836 + T \)
$23$ \( -4104 + T \)
$29$ \( 594 + T \)
$31$ \( 4256 + T \)
$37$ \( 298 + T \)
$41$ \( -17226 + T \)
$43$ \( -12100 + T \)
$47$ \( -1296 + T \)
$53$ \( -19494 + T \)
$59$ \( -7668 + T \)
$61$ \( 34738 + T \)
$67$ \( 21812 + T \)
$71$ \( -46872 + T \)
$73$ \( -67562 + T \)
$79$ \( -76912 + T \)
$83$ \( 67716 + T \)
$89$ \( -29754 + T \)
$97$ \( 122398 + T \)
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