Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [16,8,Mod(1,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.99816040775\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 12 q^{3} - 210 q^{5} - 1016 q^{7} - 2043 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 12 q^{3} - 210 q^{5} - 1016 q^{7} - 2043 q^{9} - 1092 q^{11} + 1382 q^{13} + 2520 q^{15} + 14706 q^{17} + 39940 q^{19} + 12192 q^{21} - 68712 q^{23} - 34025 q^{25} + 50760 q^{27} - 102570 q^{29} - 227552 q^{31} + 13104 q^{33} + 213360 q^{35} + 160526 q^{37} - 16584 q^{39} + 10842 q^{41} + 630748 q^{43} + 429030 q^{45} - 472656 q^{47} + 208713 q^{49} - 176472 q^{51} - 1494018 q^{53} + 229320 q^{55} - 479280 q^{57} - 2640660 q^{59} + 827702 q^{61} + 2075688 q^{63} - 290220 q^{65} + 126004 q^{67} + 824544 q^{69} + 1414728 q^{71} + 980282 q^{73} + 408300 q^{75} + 1109472 q^{77} + 3566800 q^{79} + 3858921 q^{81} - 5672892 q^{83} - 3088260 q^{85} + 1230840 q^{87} - 11951190 q^{89} - 1404112 q^{91} + 2730624 q^{93} - 8387400 q^{95} + 8682146 q^{97} + 2230956 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 −210.000 0 −1016.00 0 −2043.00 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 16.8.a.b 1
3.b odd 2 1 144.8.a.i 1
4.b odd 2 1 2.8.a.a 1
5.b even 2 1 400.8.a.l 1
5.c odd 4 2 400.8.c.j 2
8.b even 2 1 64.8.a.e 1
8.d odd 2 1 64.8.a.c 1
12.b even 2 1 18.8.a.b 1
16.e even 4 2 256.8.b.f 2
16.f odd 4 2 256.8.b.b 2
20.d odd 2 1 50.8.a.g 1
20.e even 4 2 50.8.b.c 2
24.f even 2 1 576.8.a.g 1
24.h odd 2 1 576.8.a.f 1
28.d even 2 1 98.8.a.a 1
28.f even 6 2 98.8.c.e 2
28.g odd 6 2 98.8.c.d 2
36.f odd 6 2 162.8.c.l 2
36.h even 6 2 162.8.c.a 2
44.c even 2 1 242.8.a.e 1
52.b odd 2 1 338.8.a.d 1
52.f even 4 2 338.8.b.d 2
60.h even 2 1 450.8.a.c 1
60.l odd 4 2 450.8.c.g 2
68.d odd 2 1 578.8.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.8.a.a 1 4.b odd 2 1
16.8.a.b 1 1.a even 1 1 trivial
18.8.a.b 1 12.b even 2 1
50.8.a.g 1 20.d odd 2 1
50.8.b.c 2 20.e even 4 2
64.8.a.c 1 8.d odd 2 1
64.8.a.e 1 8.b even 2 1
98.8.a.a 1 28.d even 2 1
98.8.c.d 2 28.g odd 6 2
98.8.c.e 2 28.f even 6 2
144.8.a.i 1 3.b odd 2 1
162.8.c.a 2 36.h even 6 2
162.8.c.l 2 36.f odd 6 2
242.8.a.e 1 44.c even 2 1
256.8.b.b 2 16.f odd 4 2
256.8.b.f 2 16.e even 4 2
338.8.a.d 1 52.b odd 2 1
338.8.b.d 2 52.f even 4 2
400.8.a.l 1 5.b even 2 1
400.8.c.j 2 5.c odd 4 2
450.8.a.c 1 60.h even 2 1
450.8.c.g 2 60.l odd 4 2
576.8.a.f 1 24.h odd 2 1
576.8.a.g 1 24.f even 2 1
578.8.a.b 1 68.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 12 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(16))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 12 \) Copy content Toggle raw display
$5$ \( T + 210 \) Copy content Toggle raw display
$7$ \( T + 1016 \) Copy content Toggle raw display
$11$ \( T + 1092 \) Copy content Toggle raw display
$13$ \( T - 1382 \) Copy content Toggle raw display
$17$ \( T - 14706 \) Copy content Toggle raw display
$19$ \( T - 39940 \) Copy content Toggle raw display
$23$ \( T + 68712 \) Copy content Toggle raw display
$29$ \( T + 102570 \) Copy content Toggle raw display
$31$ \( T + 227552 \) Copy content Toggle raw display
$37$ \( T - 160526 \) Copy content Toggle raw display
$41$ \( T - 10842 \) Copy content Toggle raw display
$43$ \( T - 630748 \) Copy content Toggle raw display
$47$ \( T + 472656 \) Copy content Toggle raw display
$53$ \( T + 1494018 \) Copy content Toggle raw display
$59$ \( T + 2640660 \) Copy content Toggle raw display
$61$ \( T - 827702 \) Copy content Toggle raw display
$67$ \( T - 126004 \) Copy content Toggle raw display
$71$ \( T - 1414728 \) Copy content Toggle raw display
$73$ \( T - 980282 \) Copy content Toggle raw display
$79$ \( T - 3566800 \) Copy content Toggle raw display
$83$ \( T + 5672892 \) Copy content Toggle raw display
$89$ \( T + 11951190 \) Copy content Toggle raw display
$97$ \( T - 8682146 \) Copy content Toggle raw display
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