Properties

Label 16.10.a.d
Level $16$
Weight $10$
Character orbit 16.a
Self dual yes
Analytic conductor $8.241$
Analytic rank $0$
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] = [16,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 10 \)
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: \(8.24057337862\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q + 156 q^{3} + 870 q^{5} + 952 q^{7} + 4653 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 156 q^{3} + 870 q^{5} + 952 q^{7} + 4653 q^{9} + 56148 q^{11} + 178094 q^{13} + 135720 q^{15} - 247662 q^{17} - 315380 q^{19} + 148512 q^{21} - 204504 q^{23} - 1196225 q^{25} - 2344680 q^{27} - 3840450 q^{29} + 1309408 q^{31} + 8759088 q^{33} + 828240 q^{35} + 4307078 q^{37} + 27782664 q^{39} + 1512042 q^{41} - 33670604 q^{43} + 4048110 q^{45} + 10581072 q^{47} - 39447303 q^{49} - 38635272 q^{51} + 16616214 q^{53} + 48848760 q^{55} - 49199280 q^{57} - 112235100 q^{59} - 33197218 q^{61} + 4429656 q^{63} + 154941780 q^{65} + 121372252 q^{67} - 31902624 q^{69} + 387172728 q^{71} + 255240074 q^{73} - 186611100 q^{75} + 53452896 q^{77} - 492101840 q^{79} - 457355079 q^{81} + 457420236 q^{83} - 215465940 q^{85} - 599110200 q^{87} - 31809510 q^{89} + 169545488 q^{91} + 204267648 q^{93} - 274380600 q^{95} - 673532062 q^{97} + 261256644 q^{99}+O(q^{100}) \) 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
0 156.000 0 870.000 0 952.000 0 4653.00 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 16.10.a.d 1
3.b odd 2 1 144.10.a.d 1
4.b odd 2 1 2.10.a.a 1
5.b even 2 1 400.10.a.b 1
5.c odd 4 2 400.10.c.d 2
8.b even 2 1 64.10.a.b 1
8.d odd 2 1 64.10.a.h 1
12.b even 2 1 18.10.a.a 1
16.e even 4 2 256.10.b.e 2
16.f odd 4 2 256.10.b.g 2
20.d odd 2 1 50.10.a.c 1
20.e even 4 2 50.10.b.a 2
28.d even 2 1 98.10.a.c 1
28.f even 6 2 98.10.c.b 2
28.g odd 6 2 98.10.c.c 2
36.f odd 6 2 162.10.c.b 2
36.h even 6 2 162.10.c.i 2
44.c even 2 1 242.10.a.a 1
52.b odd 2 1 338.10.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.10.a.a 1 4.b odd 2 1
16.10.a.d 1 1.a even 1 1 trivial
18.10.a.a 1 12.b even 2 1
50.10.a.c 1 20.d odd 2 1
50.10.b.a 2 20.e even 4 2
64.10.a.b 1 8.b even 2 1
64.10.a.h 1 8.d odd 2 1
98.10.a.c 1 28.d even 2 1
98.10.c.b 2 28.f even 6 2
98.10.c.c 2 28.g odd 6 2
144.10.a.d 1 3.b odd 2 1
162.10.c.b 2 36.f odd 6 2
162.10.c.i 2 36.h even 6 2
242.10.a.a 1 44.c even 2 1
256.10.b.e 2 16.e even 4 2
256.10.b.g 2 16.f odd 4 2
338.10.a.a 1 52.b odd 2 1
400.10.a.b 1 5.b even 2 1
400.10.c.d 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 156 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(16))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 156 \) Copy content Toggle raw display
$5$ \( T - 870 \) Copy content Toggle raw display
$7$ \( T - 952 \) Copy content Toggle raw display
$11$ \( T - 56148 \) Copy content Toggle raw display
$13$ \( T - 178094 \) Copy content Toggle raw display
$17$ \( T + 247662 \) Copy content Toggle raw display
$19$ \( T + 315380 \) Copy content Toggle raw display
$23$ \( T + 204504 \) Copy content Toggle raw display
$29$ \( T + 3840450 \) Copy content Toggle raw display
$31$ \( T - 1309408 \) Copy content Toggle raw display
$37$ \( T - 4307078 \) Copy content Toggle raw display
$41$ \( T - 1512042 \) Copy content Toggle raw display
$43$ \( T + 33670604 \) Copy content Toggle raw display
$47$ \( T - 10581072 \) Copy content Toggle raw display
$53$ \( T - 16616214 \) Copy content Toggle raw display
$59$ \( T + 112235100 \) Copy content Toggle raw display
$61$ \( T + 33197218 \) Copy content Toggle raw display
$67$ \( T - 121372252 \) Copy content Toggle raw display
$71$ \( T - 387172728 \) Copy content Toggle raw display
$73$ \( T - 255240074 \) Copy content Toggle raw display
$79$ \( T + 492101840 \) Copy content Toggle raw display
$83$ \( T - 457420236 \) Copy content Toggle raw display
$89$ \( T + 31809510 \) Copy content Toggle raw display
$97$ \( T + 673532062 \) Copy content Toggle raw display
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