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Properties

Label 162.4.a.g
Level $162$
Weight $4$
Character orbit 162.a
Self dual yes
Analytic conductor $9.558$
Analytic rank $0$
Dimension $2$
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] = [162,4,Mod(1,162)] 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("162.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,4,0,8,9] 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: \(9.55830942093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{105}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 18)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(-1 + 3\sqrt{105})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + ( - \beta + 4) q^{5} + (\beta + 10) q^{7} + 8 q^{8} + ( - 2 \beta + 8) q^{10} + (2 \beta + 13) q^{11} + ( - \beta + 30) q^{13} + (2 \beta + 20) q^{14} + 16 q^{16} + (\beta - 1) q^{17}+ \cdots + (38 \beta - 14) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 9 q^{5} + 19 q^{7} + 16 q^{8} + 18 q^{10} + 24 q^{11} + 61 q^{13} + 38 q^{14} + 32 q^{16} - 3 q^{17} + 133 q^{19} + 36 q^{20} + 48 q^{22} - 69 q^{23} + 263 q^{25} + 122 q^{26}+ \cdots - 66 q^{98}+O(q^{100}) \) 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
5.62348
−4.62348
2.00000 0 4.00000 −10.8704 0 24.8704 8.00000 0 −21.7409
1.2 2.00000 0 4.00000 19.8704 0 −5.87043 8.00000 0 39.7409
\(n\): 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 162.4.a.g 2
3.b odd 2 1 162.4.a.f 2
4.b odd 2 1 1296.4.a.r 2
9.c even 3 2 54.4.c.b 4
9.d odd 6 2 18.4.c.b ✓ 4
12.b even 2 1 1296.4.a.l 2
36.f odd 6 2 432.4.i.b 4
36.h even 6 2 144.4.i.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.4.c.b ✓ 4 9.d odd 6 2
54.4.c.b 4 9.c even 3 2
144.4.i.b 4 36.h even 6 2
162.4.a.f 2 3.b odd 2 1
162.4.a.g 2 1.a even 1 1 trivial
432.4.i.b 4 36.f odd 6 2
1296.4.a.l 2 12.b even 2 1
1296.4.a.r 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 9T_{5} - 216 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 9T - 216 \) Copy content Toggle raw display
$7$ \( T^{2} - 19T - 146 \) Copy content Toggle raw display
$11$ \( T^{2} - 24T - 801 \) Copy content Toggle raw display
$13$ \( T^{2} - 61T + 694 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 234 \) Copy content Toggle raw display
$19$ \( T^{2} - 133T - 1484 \) Copy content Toggle raw display
$23$ \( T^{2} + 69T + 954 \) Copy content Toggle raw display
$29$ \( T^{2} + 237T + 2466 \) Copy content Toggle raw display
$31$ \( T^{2} - 211T + 9004 \) Copy content Toggle raw display
$37$ \( T^{2} - 262T - 29144 \) Copy content Toggle raw display
$41$ \( T^{2} + 468T + 53811 \) Copy content Toggle raw display
$43$ \( T^{2} + 86T - 134231 \) Copy content Toggle raw display
$47$ \( T^{2} + 483T + 5166 \) Copy content Toggle raw display
$53$ \( T^{2} + 150T - 40680 \) Copy content Toggle raw display
$59$ \( T^{2} + 168T + 6111 \) Copy content Toggle raw display
$61$ \( T^{2} + 1049 T + 221944 \) Copy content Toggle raw display
$67$ \( T^{2} + 1166 T + 305869 \) Copy content Toggle raw display
$71$ \( T^{2} - 312T - 217584 \) Copy content Toggle raw display
$73$ \( T^{2} + 311T - 80006 \) Copy content Toggle raw display
$79$ \( T^{2} - 349T - 9476 \) Copy content Toggle raw display
$83$ \( T^{2} + 1221 T + 268524 \) Copy content Toggle raw display
$89$ \( T^{2} - 492T - 317484 \) Copy content Toggle raw display
$97$ \( T^{2} + 128T - 110249 \) Copy content Toggle raw display
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