Properties

Label 16.6.a.a
Level 16
Weight 6
Character orbit 16.a
Self dual yes
Analytic conductor 2.566
Analytic rank 1
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: \(1\)
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

\(f(q)\) \(=\) \( q - 20q^{3} - 74q^{5} + 24q^{7} + 157q^{9} + O(q^{10}) \) \( q - 20q^{3} - 74q^{5} + 24q^{7} + 157q^{9} - 124q^{11} + 478q^{13} + 1480q^{15} - 1198q^{17} - 3044q^{19} - 480q^{21} - 184q^{23} + 2351q^{25} + 1720q^{27} - 3282q^{29} + 5728q^{31} + 2480q^{33} - 1776q^{35} + 10326q^{37} - 9560q^{39} - 8886q^{41} + 9188q^{43} - 11618q^{45} - 23664q^{47} - 16231q^{49} + 23960q^{51} + 11686q^{53} + 9176q^{55} + 60880q^{57} - 16876q^{59} - 18482q^{61} + 3768q^{63} - 35372q^{65} + 15532q^{67} + 3680q^{69} + 31960q^{71} - 4886q^{73} - 47020q^{75} - 2976q^{77} - 44560q^{79} - 72551q^{81} - 67364q^{83} + 88652q^{85} + 65640q^{87} + 71994q^{89} + 11472q^{91} - 114560q^{93} + 225256q^{95} + 48866q^{97} - 19468q^{99} + O(q^{100}) \)

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 −20.0000 0 −74.0000 0 24.0000 0 157.000 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 16.6.a.a 1
3.b odd 2 1 144.6.a.k 1
4.b odd 2 1 8.6.a.a 1
5.b even 2 1 400.6.a.l 1
5.c odd 4 2 400.6.c.d 2
7.b odd 2 1 784.6.a.l 1
8.b even 2 1 64.6.a.g 1
8.d odd 2 1 64.6.a.a 1
12.b even 2 1 72.6.a.f 1
16.e even 4 2 256.6.b.d 2
16.f odd 4 2 256.6.b.f 2
20.d odd 2 1 200.6.a.a 1
20.e even 4 2 200.6.c.a 2
24.f even 2 1 576.6.a.g 1
24.h odd 2 1 576.6.a.h 1
28.d even 2 1 392.6.a.b 1
28.f even 6 2 392.6.i.e 2
28.g odd 6 2 392.6.i.b 2
44.c even 2 1 968.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 4.b odd 2 1
16.6.a.a 1 1.a even 1 1 trivial
64.6.a.a 1 8.d odd 2 1
64.6.a.g 1 8.b even 2 1
72.6.a.f 1 12.b even 2 1
144.6.a.k 1 3.b odd 2 1
200.6.a.a 1 20.d odd 2 1
200.6.c.a 2 20.e even 4 2
256.6.b.d 2 16.e even 4 2
256.6.b.f 2 16.f odd 4 2
392.6.a.b 1 28.d even 2 1
392.6.i.b 2 28.g odd 6 2
392.6.i.e 2 28.f even 6 2
400.6.a.l 1 5.b even 2 1
400.6.c.d 2 5.c odd 4 2
576.6.a.g 1 24.f even 2 1
576.6.a.h 1 24.h odd 2 1
784.6.a.l 1 7.b odd 2 1
968.6.a.a 1 44.c even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 20 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 20 T + 243 T^{2} \)
$5$ \( 1 + 74 T + 3125 T^{2} \)
$7$ \( 1 - 24 T + 16807 T^{2} \)
$11$ \( 1 + 124 T + 161051 T^{2} \)
$13$ \( 1 - 478 T + 371293 T^{2} \)
$17$ \( 1 + 1198 T + 1419857 T^{2} \)
$19$ \( 1 + 3044 T + 2476099 T^{2} \)
$23$ \( 1 + 184 T + 6436343 T^{2} \)
$29$ \( 1 + 3282 T + 20511149 T^{2} \)
$31$ \( 1 - 5728 T + 28629151 T^{2} \)
$37$ \( 1 - 10326 T + 69343957 T^{2} \)
$41$ \( 1 + 8886 T + 115856201 T^{2} \)
$43$ \( 1 - 9188 T + 147008443 T^{2} \)
$47$ \( 1 + 23664 T + 229345007 T^{2} \)
$53$ \( 1 - 11686 T + 418195493 T^{2} \)
$59$ \( 1 + 16876 T + 714924299 T^{2} \)
$61$ \( 1 + 18482 T + 844596301 T^{2} \)
$67$ \( 1 - 15532 T + 1350125107 T^{2} \)
$71$ \( 1 - 31960 T + 1804229351 T^{2} \)
$73$ \( 1 + 4886 T + 2073071593 T^{2} \)
$79$ \( 1 + 44560 T + 3077056399 T^{2} \)
$83$ \( 1 + 67364 T + 3939040643 T^{2} \)
$89$ \( 1 - 71994 T + 5584059449 T^{2} \)
$97$ \( 1 - 48866 T + 8587340257 T^{2} \)
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