Properties

Label 16.10.a.b
Level $16$
Weight $10$
Character orbit 16.a
Self dual yes
Analytic conductor $8.241$
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] = [16,10,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: \(1\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 68 q^{3} + 1510 q^{5} - 10248 q^{7} - 15059 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 68 q^{3} + 1510 q^{5} - 10248 q^{7} - 15059 q^{9} - 3916 q^{11} - 176594 q^{13} - 102680 q^{15} + 148370 q^{17} - 499796 q^{19} + 696864 q^{21} + 1889768 q^{23} + 326975 q^{25} + 2362456 q^{27} - 920898 q^{29} - 1379360 q^{31} + 266288 q^{33} - 15474480 q^{35} + 5064966 q^{37} + 12008392 q^{39} - 24100758 q^{41} - 25785196 q^{43} - 22739090 q^{45} + 60790224 q^{47} + 64667897 q^{49} - 10089160 q^{51} + 29496214 q^{53} - 5913160 q^{55} + 33986128 q^{57} - 51819388 q^{59} + 33426910 q^{61} + 154324632 q^{63} - 266656940 q^{65} - 144856196 q^{67} - 128504224 q^{69} - 68397128 q^{71} + 168216202 q^{73} - 22234300 q^{75} + 40131168 q^{77} - 235398736 q^{79} + 135759289 q^{81} + 64639852 q^{83} + 224038700 q^{85} + 62621064 q^{87} - 78782694 q^{89} + 1809735312 q^{91} + 93796480 q^{93} - 754691960 q^{95} - 24113566 q^{97} + 58971044 q^{99}+O(q^{100}) \) 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 −68.0000 0 1510.00 0 −10248.0 0 −15059.0 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.b 1
3.b odd 2 1 144.10.a.b 1
4.b odd 2 1 8.10.a.b 1
5.b even 2 1 400.10.a.i 1
5.c odd 4 2 400.10.c.f 2
8.b even 2 1 64.10.a.g 1
8.d odd 2 1 64.10.a.c 1
12.b even 2 1 72.10.a.a 1
16.e even 4 2 256.10.b.k 2
16.f odd 4 2 256.10.b.a 2
20.d odd 2 1 200.10.a.a 1
20.e even 4 2 200.10.c.a 2
28.d even 2 1 392.10.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.10.a.b 1 4.b odd 2 1
16.10.a.b 1 1.a even 1 1 trivial
64.10.a.c 1 8.d odd 2 1
64.10.a.g 1 8.b even 2 1
72.10.a.a 1 12.b even 2 1
144.10.a.b 1 3.b odd 2 1
200.10.a.a 1 20.d odd 2 1
200.10.c.a 2 20.e even 4 2
256.10.b.a 2 16.f odd 4 2
256.10.b.k 2 16.e even 4 2
392.10.a.a 1 28.d even 2 1
400.10.a.i 1 5.b even 2 1
400.10.c.f 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 68 \) 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
$3$ \( T + 68 \) Copy content Toggle raw display
$5$ \( T - 1510 \) Copy content Toggle raw display
$7$ \( T + 10248 \) Copy content Toggle raw display
$11$ \( T + 3916 \) Copy content Toggle raw display
$13$ \( T + 176594 \) Copy content Toggle raw display
$17$ \( T - 148370 \) Copy content Toggle raw display
$19$ \( T + 499796 \) Copy content Toggle raw display
$23$ \( T - 1889768 \) Copy content Toggle raw display
$29$ \( T + 920898 \) Copy content Toggle raw display
$31$ \( T + 1379360 \) Copy content Toggle raw display
$37$ \( T - 5064966 \) Copy content Toggle raw display
$41$ \( T + 24100758 \) Copy content Toggle raw display
$43$ \( T + 25785196 \) Copy content Toggle raw display
$47$ \( T - 60790224 \) Copy content Toggle raw display
$53$ \( T - 29496214 \) Copy content Toggle raw display
$59$ \( T + 51819388 \) Copy content Toggle raw display
$61$ \( T - 33426910 \) Copy content Toggle raw display
$67$ \( T + 144856196 \) Copy content Toggle raw display
$71$ \( T + 68397128 \) Copy content Toggle raw display
$73$ \( T - 168216202 \) Copy content Toggle raw display
$79$ \( T + 235398736 \) Copy content Toggle raw display
$83$ \( T - 64639852 \) Copy content Toggle raw display
$89$ \( T + 78782694 \) Copy content Toggle raw display
$97$ \( T + 24113566 \) Copy content Toggle raw display
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