Properties

Label 96.9.h.b
Level $96$
Weight $9$
Character orbit 96.h
Self dual yes
Analytic conductor $39.108$
Analytic rank $0$
Dimension $1$
CM discriminant -24
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,9,Mod(17,96)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(96, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.17");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 96.h (of order \(2\), degree \(1\), not minimal)

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: \(39.1083465659\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 81 q^{3} - 866 q^{5} + 4798 q^{7} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 81 q^{3} - 866 q^{5} + 4798 q^{7} + 6561 q^{9} - 9118 q^{11} - 70146 q^{15} + 388638 q^{21} + 359331 q^{25} + 531441 q^{27} + 745438 q^{29} + 1618558 q^{31} - 738558 q^{33} - 4155068 q^{35} - 5681826 q^{45} + 17256003 q^{49} + 5425438 q^{53} + 7896188 q^{55} + 22852322 q^{59} + 31479678 q^{63} + 9756482 q^{73} + 29105811 q^{75} - 43748164 q^{77} - 5237762 q^{79} + 43046721 q^{81} - 77460958 q^{83} + 60380478 q^{87} + 131103198 q^{93} + 121608962 q^{97} - 59823198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0
0 81.0000 0 −866.000 0 4798.00 0 6561.00 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.9.h.b 1
3.b odd 2 1 96.9.h.a 1
4.b odd 2 1 24.9.h.a 1
8.b even 2 1 96.9.h.a 1
8.d odd 2 1 24.9.h.b yes 1
12.b even 2 1 24.9.h.b yes 1
24.f even 2 1 24.9.h.a 1
24.h odd 2 1 CM 96.9.h.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.9.h.a 1 4.b odd 2 1
24.9.h.a 1 24.f even 2 1
24.9.h.b yes 1 8.d odd 2 1
24.9.h.b yes 1 12.b even 2 1
96.9.h.a 1 3.b odd 2 1
96.9.h.a 1 8.b even 2 1
96.9.h.b 1 1.a even 1 1 trivial
96.9.h.b 1 24.h odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 866 \) acting on \(S_{9}^{\mathrm{new}}(96, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 81 \) Copy content Toggle raw display
$5$ \( T + 866 \) Copy content Toggle raw display
$7$ \( T - 4798 \) Copy content Toggle raw display
$11$ \( T + 9118 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 745438 \) Copy content Toggle raw display
$31$ \( T - 1618558 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 5425438 \) Copy content Toggle raw display
$59$ \( T - 22852322 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 9756482 \) Copy content Toggle raw display
$79$ \( T + 5237762 \) Copy content Toggle raw display
$83$ \( T + 77460958 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 121608962 \) Copy content Toggle raw display
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