Properties

Label 16.4.a.a
Level 16
Weight 4
Character orbit 16.a
Self dual yes
Analytic conductor 0.944
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.944030560092\)
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

\(f(q)\) \(=\) \( q + 4q^{3} - 2q^{5} - 24q^{7} - 11q^{9} + O(q^{10}) \) \( q + 4q^{3} - 2q^{5} - 24q^{7} - 11q^{9} + 44q^{11} + 22q^{13} - 8q^{15} + 50q^{17} - 44q^{19} - 96q^{21} + 56q^{23} - 121q^{25} - 152q^{27} + 198q^{29} + 160q^{31} + 176q^{33} + 48q^{35} - 162q^{37} + 88q^{39} - 198q^{41} - 52q^{43} + 22q^{45} - 528q^{47} + 233q^{49} + 200q^{51} - 242q^{53} - 88q^{55} - 176q^{57} + 668q^{59} + 550q^{61} + 264q^{63} - 44q^{65} - 188q^{67} + 224q^{69} - 728q^{71} + 154q^{73} - 484q^{75} - 1056q^{77} + 656q^{79} - 311q^{81} - 236q^{83} - 100q^{85} + 792q^{87} + 714q^{89} - 528q^{91} + 640q^{93} + 88q^{95} - 478q^{97} - 484q^{99} + O(q^{100}) \)

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 4.00000 0 −2.00000 0 −24.0000 0 −11.0000 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 16.4.a.a 1
3.b odd 2 1 144.4.a.e 1
4.b odd 2 1 8.4.a.a 1
5.b even 2 1 400.4.a.g 1
5.c odd 4 2 400.4.c.i 2
7.b odd 2 1 784.4.a.e 1
8.b even 2 1 64.4.a.b 1
8.d odd 2 1 64.4.a.d 1
11.b odd 2 1 1936.4.a.l 1
12.b even 2 1 72.4.a.c 1
16.e even 4 2 256.4.b.g 2
16.f odd 4 2 256.4.b.a 2
20.d odd 2 1 200.4.a.g 1
20.e even 4 2 200.4.c.e 2
24.f even 2 1 576.4.a.k 1
24.h odd 2 1 576.4.a.j 1
28.d even 2 1 392.4.a.e 1
28.f even 6 2 392.4.i.b 2
28.g odd 6 2 392.4.i.g 2
36.f odd 6 2 648.4.i.h 2
36.h even 6 2 648.4.i.e 2
40.e odd 2 1 1600.4.a.o 1
40.f even 2 1 1600.4.a.bm 1
44.c even 2 1 968.4.a.a 1
52.b odd 2 1 1352.4.a.a 1
60.h even 2 1 1800.4.a.d 1
60.l odd 4 2 1800.4.f.u 2
68.d odd 2 1 2312.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 4.b odd 2 1
16.4.a.a 1 1.a even 1 1 trivial
64.4.a.b 1 8.b even 2 1
64.4.a.d 1 8.d odd 2 1
72.4.a.c 1 12.b even 2 1
144.4.a.e 1 3.b odd 2 1
200.4.a.g 1 20.d odd 2 1
200.4.c.e 2 20.e even 4 2
256.4.b.a 2 16.f odd 4 2
256.4.b.g 2 16.e even 4 2
392.4.a.e 1 28.d even 2 1
392.4.i.b 2 28.f even 6 2
392.4.i.g 2 28.g odd 6 2
400.4.a.g 1 5.b even 2 1
400.4.c.i 2 5.c odd 4 2
576.4.a.j 1 24.h odd 2 1
576.4.a.k 1 24.f even 2 1
648.4.i.e 2 36.h even 6 2
648.4.i.h 2 36.f odd 6 2
784.4.a.e 1 7.b odd 2 1
968.4.a.a 1 44.c even 2 1
1352.4.a.a 1 52.b odd 2 1
1600.4.a.o 1 40.e odd 2 1
1600.4.a.bm 1 40.f even 2 1
1800.4.a.d 1 60.h even 2 1
1800.4.f.u 2 60.l odd 4 2
1936.4.a.l 1 11.b odd 2 1
2312.4.a.a 1 68.d odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 4 T + 27 T^{2} \)
$5$ \( 1 + 2 T + 125 T^{2} \)
$7$ \( 1 + 24 T + 343 T^{2} \)
$11$ \( 1 - 44 T + 1331 T^{2} \)
$13$ \( 1 - 22 T + 2197 T^{2} \)
$17$ \( 1 - 50 T + 4913 T^{2} \)
$19$ \( 1 + 44 T + 6859 T^{2} \)
$23$ \( 1 - 56 T + 12167 T^{2} \)
$29$ \( 1 - 198 T + 24389 T^{2} \)
$31$ \( 1 - 160 T + 29791 T^{2} \)
$37$ \( 1 + 162 T + 50653 T^{2} \)
$41$ \( 1 + 198 T + 68921 T^{2} \)
$43$ \( 1 + 52 T + 79507 T^{2} \)
$47$ \( 1 + 528 T + 103823 T^{2} \)
$53$ \( 1 + 242 T + 148877 T^{2} \)
$59$ \( 1 - 668 T + 205379 T^{2} \)
$61$ \( 1 - 550 T + 226981 T^{2} \)
$67$ \( 1 + 188 T + 300763 T^{2} \)
$71$ \( 1 + 728 T + 357911 T^{2} \)
$73$ \( 1 - 154 T + 389017 T^{2} \)
$79$ \( 1 - 656 T + 493039 T^{2} \)
$83$ \( 1 + 236 T + 571787 T^{2} \)
$89$ \( 1 - 714 T + 704969 T^{2} \)
$97$ \( 1 + 478 T + 912673 T^{2} \)
show more
show less