Properties

Label 138.4.a.e
Level $138$
Weight $4$
Character orbit 138.a
Self dual yes
Analytic conductor $8.142$
Analytic rank $0$
Dimension $2$
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] = [138,4,Mod(1,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("138.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 138.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: \(8.14226358079\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + (\beta + 4) q^{5} - 6 q^{6} + ( - \beta + 6) q^{7} + 8 q^{8} + 9 q^{9} + (2 \beta + 8) q^{10} + ( - 4 \beta + 24) q^{11} - 12 q^{12} + (4 \beta + 42) q^{13} + ( - 2 \beta + 12) q^{14}+ \cdots + ( - 36 \beta + 216) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} + 8 q^{5} - 12 q^{6} + 12 q^{7} + 16 q^{8} + 18 q^{9} + 16 q^{10} + 48 q^{11} - 24 q^{12} + 84 q^{13} + 24 q^{14} - 24 q^{15} + 32 q^{16} + 104 q^{17} + 36 q^{18} + 28 q^{19}+ \cdots + 432 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
2.00000 −3.00000 4.00000 −7.31371 −6.00000 17.3137 8.00000 9.00000 −14.6274
1.2 2.00000 −3.00000 4.00000 15.3137 −6.00000 −5.31371 8.00000 9.00000 30.6274
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(23\) \( -1 \)

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 138.4.a.e 2
3.b odd 2 1 414.4.a.g 2
4.b odd 2 1 1104.4.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.4.a.e 2 1.a even 1 1 trivial
414.4.a.g 2 3.b odd 2 1
1104.4.a.p 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 8T_{5} - 112 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(138))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 8T - 112 \) Copy content Toggle raw display
$7$ \( T^{2} - 12T - 92 \) Copy content Toggle raw display
$11$ \( T^{2} - 48T - 1472 \) Copy content Toggle raw display
$13$ \( T^{2} - 84T - 284 \) Copy content Toggle raw display
$17$ \( T^{2} - 104T + 1552 \) Copy content Toggle raw display
$19$ \( T^{2} - 28T - 956 \) Copy content Toggle raw display
$23$ \( (T - 23)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 132T - 8444 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T - 12784 \) Copy content Toggle raw display
$37$ \( T^{2} + 324T - 35708 \) Copy content Toggle raw display
$41$ \( T^{2} + 140T - 13532 \) Copy content Toggle raw display
$43$ \( T^{2} + 548T + 59588 \) Copy content Toggle raw display
$47$ \( T^{2} + 464T + 49216 \) Copy content Toggle raw display
$53$ \( T^{2} - 336T - 186944 \) Copy content Toggle raw display
$59$ \( T^{2} - 72T - 490736 \) Copy content Toggle raw display
$61$ \( T^{2} - 60T - 269948 \) Copy content Toggle raw display
$67$ \( T^{2} + 444T - 44028 \) Copy content Toggle raw display
$71$ \( T^{2} - 672T - 288512 \) Copy content Toggle raw display
$73$ \( T^{2} + 252T - 330236 \) Copy content Toggle raw display
$79$ \( T^{2} + 1692 T + 715588 \) Copy content Toggle raw display
$83$ \( T^{2} + 400T - 951232 \) Copy content Toggle raw display
$89$ \( T^{2} + 336T - 17984 \) Copy content Toggle raw display
$97$ \( T^{2} - 220T - 848572 \) Copy content Toggle raw display
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