Properties

Label 16.3.c.a
Level 16
Weight 3
Character orbit 16.c
Self dual yes
Analytic conductor 0.436
Analytic rank 0
Dimension 1
CM discriminant -4
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 16.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.435968422976\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 6q^{5} + 9q^{9} + O(q^{10}) \) \( q - 6q^{5} + 9q^{9} + 10q^{13} - 30q^{17} + 11q^{25} + 42q^{29} - 70q^{37} + 18q^{41} - 54q^{45} + 49q^{49} + 90q^{53} - 22q^{61} - 60q^{65} - 110q^{73} + 81q^{81} + 180q^{85} - 78q^{89} + 130q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
0
0 0 0 −6.00000 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.3.c.a 1
3.b odd 2 1 144.3.g.a 1
4.b odd 2 1 CM 16.3.c.a 1
5.b even 2 1 400.3.b.a 1
5.c odd 4 2 400.3.h.a 2
7.b odd 2 1 784.3.d.b 1
7.c even 3 2 784.3.r.e 2
7.d odd 6 2 784.3.r.d 2
8.b even 2 1 64.3.c.a 1
8.d odd 2 1 64.3.c.a 1
9.c even 3 2 1296.3.o.o 2
9.d odd 6 2 1296.3.o.b 2
12.b even 2 1 144.3.g.a 1
15.d odd 2 1 3600.3.e.c 1
15.e even 4 2 3600.3.j.a 2
16.e even 4 2 256.3.d.b 2
16.f odd 4 2 256.3.d.b 2
20.d odd 2 1 400.3.b.a 1
20.e even 4 2 400.3.h.a 2
24.f even 2 1 576.3.g.b 1
24.h odd 2 1 576.3.g.b 1
28.d even 2 1 784.3.d.b 1
28.f even 6 2 784.3.r.d 2
28.g odd 6 2 784.3.r.e 2
36.f odd 6 2 1296.3.o.o 2
36.h even 6 2 1296.3.o.b 2
40.e odd 2 1 1600.3.b.b 1
40.f even 2 1 1600.3.b.b 1
40.i odd 4 2 1600.3.h.b 2
40.k even 4 2 1600.3.h.b 2
48.i odd 4 2 2304.3.b.f 2
48.k even 4 2 2304.3.b.f 2
60.h even 2 1 3600.3.e.c 1
60.l odd 4 2 3600.3.j.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.c.a 1 1.a even 1 1 trivial
16.3.c.a 1 4.b odd 2 1 CM
64.3.c.a 1 8.b even 2 1
64.3.c.a 1 8.d odd 2 1
144.3.g.a 1 3.b odd 2 1
144.3.g.a 1 12.b even 2 1
256.3.d.b 2 16.e even 4 2
256.3.d.b 2 16.f odd 4 2
400.3.b.a 1 5.b even 2 1
400.3.b.a 1 20.d odd 2 1
400.3.h.a 2 5.c odd 4 2
400.3.h.a 2 20.e even 4 2
576.3.g.b 1 24.f even 2 1
576.3.g.b 1 24.h odd 2 1
784.3.d.b 1 7.b odd 2 1
784.3.d.b 1 28.d even 2 1
784.3.r.d 2 7.d odd 6 2
784.3.r.d 2 28.f even 6 2
784.3.r.e 2 7.c even 3 2
784.3.r.e 2 28.g odd 6 2
1296.3.o.b 2 9.d odd 6 2
1296.3.o.b 2 36.h even 6 2
1296.3.o.o 2 9.c even 3 2
1296.3.o.o 2 36.f odd 6 2
1600.3.b.b 1 40.e odd 2 1
1600.3.b.b 1 40.f even 2 1
1600.3.h.b 2 40.i odd 4 2
1600.3.h.b 2 40.k even 4 2
2304.3.b.f 2 48.i odd 4 2
2304.3.b.f 2 48.k even 4 2
3600.3.e.c 1 15.d odd 2 1
3600.3.e.c 1 60.h even 2 1
3600.3.j.a 2 15.e even 4 2
3600.3.j.a 2 60.l odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(16, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T )( 1 + 3 T ) \)
$5$ \( 1 + 6 T + 25 T^{2} \)
$7$ \( ( 1 - 7 T )( 1 + 7 T ) \)
$11$ \( ( 1 - 11 T )( 1 + 11 T ) \)
$13$ \( 1 - 10 T + 169 T^{2} \)
$17$ \( 1 + 30 T + 289 T^{2} \)
$19$ \( ( 1 - 19 T )( 1 + 19 T ) \)
$23$ \( ( 1 - 23 T )( 1 + 23 T ) \)
$29$ \( 1 - 42 T + 841 T^{2} \)
$31$ \( ( 1 - 31 T )( 1 + 31 T ) \)
$37$ \( 1 + 70 T + 1369 T^{2} \)
$41$ \( 1 - 18 T + 1681 T^{2} \)
$43$ \( ( 1 - 43 T )( 1 + 43 T ) \)
$47$ \( ( 1 - 47 T )( 1 + 47 T ) \)
$53$ \( 1 - 90 T + 2809 T^{2} \)
$59$ \( ( 1 - 59 T )( 1 + 59 T ) \)
$61$ \( 1 + 22 T + 3721 T^{2} \)
$67$ \( ( 1 - 67 T )( 1 + 67 T ) \)
$71$ \( ( 1 - 71 T )( 1 + 71 T ) \)
$73$ \( 1 + 110 T + 5329 T^{2} \)
$79$ \( ( 1 - 79 T )( 1 + 79 T ) \)
$83$ \( ( 1 - 83 T )( 1 + 83 T ) \)
$89$ \( 1 + 78 T + 7921 T^{2} \)
$97$ \( 1 - 130 T + 9409 T^{2} \)
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