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Properties

Label 192.10.a.l
Level $192$
Weight $10$
Character orbit 192.a
Self dual yes
Analytic conductor $98.887$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [192,10,Mod(1,192)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("192.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 192.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,81,0,794] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(98.8868805435\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 81 q^{3} + 794 q^{5} + 5880 q^{7} + 6561 q^{9} - 30644 q^{11} + 15314 q^{13} + 64314 q^{15} - 575086 q^{17} - 617644 q^{19} + 476280 q^{21} - 441880 q^{23} - 1322689 q^{25} + 531441 q^{27} + 2328642 q^{29}+ \cdots - 201055284 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 81.0000 0 794.000 0 5880.00 0 6561.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)

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 192.10.a.l 1
4.b odd 2 1 192.10.a.e 1
8.b even 2 1 48.10.a.b 1
8.d odd 2 1 24.10.a.b ✓ 1
24.f even 2 1 72.10.a.d 1
24.h odd 2 1 144.10.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.10.a.b ✓ 1 8.d odd 2 1
48.10.a.b 1 8.b even 2 1
72.10.a.d 1 24.f even 2 1
144.10.a.k 1 24.h odd 2 1
192.10.a.e 1 4.b odd 2 1
192.10.a.l 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(192))\):

\( T_{5} - 794 \) Copy content Toggle raw display
\( T_{7} - 5880 \) 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
$5$ \( T - 794 \) Copy content Toggle raw display
$7$ \( T - 5880 \) Copy content Toggle raw display
$11$ \( T + 30644 \) Copy content Toggle raw display
$13$ \( T - 15314 \) Copy content Toggle raw display
$17$ \( T + 575086 \) Copy content Toggle raw display
$19$ \( T + 617644 \) Copy content Toggle raw display
$23$ \( T + 441880 \) Copy content Toggle raw display
$29$ \( T - 2328642 \) Copy content Toggle raw display
$31$ \( T + 9588512 \) Copy content Toggle raw display
$37$ \( T + 9276678 \) Copy content Toggle raw display
$41$ \( T + 5903766 \) Copy content Toggle raw display
$43$ \( T - 33593452 \) Copy content Toggle raw display
$47$ \( T + 21135408 \) Copy content Toggle raw display
$53$ \( T - 108575594 \) Copy content Toggle raw display
$59$ \( T + 127636868 \) Copy content Toggle raw display
$61$ \( T + 147189214 \) Copy content Toggle raw display
$67$ \( T + 33157756 \) Copy content Toggle raw display
$71$ \( T - 9293752 \) Copy content Toggle raw display
$73$ \( T - 351080074 \) Copy content Toggle raw display
$79$ \( T - 126193328 \) Copy content Toggle raw display
$83$ \( T - 475037588 \) Copy content Toggle raw display
$89$ \( T + 566133990 \) Copy content Toggle raw display
$97$ \( T + 1474684318 \) Copy content Toggle raw display
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