Properties

Label 135.4.a.f
Level $135$
Weight $4$
Character orbit 135.a
Self dual yes
Analytic conductor $7.965$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [135,4,Mod(1,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, 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("135.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.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: \(7.96525785077\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5637.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 23x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 7) q^{4} + 5 q^{5} + (2 \beta_1 + 14) q^{7} + ( - \beta_{2} - 8 \beta_1 - 9) q^{8} - 5 \beta_1 q^{10} + (4 \beta_{2} - 2 \beta_1 + 12) q^{11} + ( - 4 \beta_{2} - 10 \beta_1 + 14) q^{13}+ \cdots + ( - 60 \beta_{2} - 5 \beta_1 - 876) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8} - 5 q^{10} + 38 q^{11} + 28 q^{13} - 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} + 115 q^{20} + 122 q^{22} - 81 q^{23} + 75 q^{25} + 416 q^{26}+ \cdots - 2693 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 23x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 15 \) 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
5.20067
0.258712
−4.45938
−5.20067 0 19.0470 5.00000 0 24.4013 −57.4517 0 −26.0034
1.2 −0.258712 0 −7.93307 5.00000 0 14.5174 4.12208 0 −1.29356
1.3 4.45938 0 11.8861 5.00000 0 5.08123 17.3296 0 22.2969
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)

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 135.4.a.f 3
3.b odd 2 1 135.4.a.g yes 3
4.b odd 2 1 2160.4.a.bm 3
5.b even 2 1 675.4.a.r 3
5.c odd 4 2 675.4.b.l 6
9.c even 3 2 405.4.e.t 6
9.d odd 6 2 405.4.e.r 6
12.b even 2 1 2160.4.a.be 3
15.d odd 2 1 675.4.a.q 3
15.e even 4 2 675.4.b.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.f 3 1.a even 1 1 trivial
135.4.a.g yes 3 3.b odd 2 1
405.4.e.r 6 9.d odd 6 2
405.4.e.t 6 9.c even 3 2
675.4.a.q 3 15.d odd 2 1
675.4.a.r 3 5.b even 2 1
675.4.b.k 6 15.e even 4 2
675.4.b.l 6 5.c odd 4 2
2160.4.a.be 3 12.b even 2 1
2160.4.a.bm 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + T_{2}^{2} - 23T_{2} - 6 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(135))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 23T - 6 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 44 T^{2} + \cdots - 1800 \) Copy content Toggle raw display
$11$ \( T^{3} - 38 T^{2} + \cdots + 83280 \) Copy content Toggle raw display
$13$ \( T^{3} - 28 T^{2} + \cdots + 100120 \) Copy content Toggle raw display
$17$ \( T^{3} + 19 T^{2} + \cdots - 553887 \) Copy content Toggle raw display
$19$ \( T^{3} - 187 T^{2} + \cdots + 525871 \) Copy content Toggle raw display
$23$ \( T^{3} + 81 T^{2} + \cdots - 2043981 \) Copy content Toggle raw display
$29$ \( T^{3} - 160 T^{2} + \cdots + 7892760 \) Copy content Toggle raw display
$31$ \( T^{3} - 227 T^{2} + \cdots - 246321 \) Copy content Toggle raw display
$37$ \( T^{3} - 78 T^{2} + \cdots + 13637080 \) Copy content Toggle raw display
$41$ \( T^{3} + 338 T^{2} + \cdots - 12116640 \) Copy content Toggle raw display
$43$ \( T^{3} - 22 T^{2} + \cdots + 18464560 \) Copy content Toggle raw display
$47$ \( T^{3} + 472 T^{2} + \cdots - 283200 \) Copy content Toggle raw display
$53$ \( T^{3} - 521 T^{2} + \cdots + 939789 \) Copy content Toggle raw display
$59$ \( T^{3} - 140 T^{2} + \cdots + 34131480 \) Copy content Toggle raw display
$61$ \( T^{3} - 595 T^{2} + \cdots + 1782607 \) Copy content Toggle raw display
$67$ \( T^{3} - 878 T^{2} + \cdots + 11295000 \) Copy content Toggle raw display
$71$ \( T^{3} + 602 T^{2} + \cdots - 280550880 \) Copy content Toggle raw display
$73$ \( T^{3} - 1294 T^{2} + \cdots + 404091280 \) Copy content Toggle raw display
$79$ \( T^{3} - 629 T^{2} + \cdots - 2010303 \) Copy content Toggle raw display
$83$ \( T^{3} + 1287 T^{2} + \cdots - 346404411 \) Copy content Toggle raw display
$89$ \( T^{3} - 2154 T^{2} + \cdots - 74325600 \) Copy content Toggle raw display
$97$ \( T^{3} - 1392 T^{2} + \cdots - 63595520 \) Copy content Toggle raw display
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