Properties

Label 18.40.a.f
Level $18$
Weight $40$
Character orbit 18.a
Self dual yes
Analytic conductor $173.411$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [18,40,Mod(1,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 40, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.1");
 
S:= CuspForms(chi, 40);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 40 \)
Character orbit: \([\chi]\) \(=\) 18.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: \(173.411192502\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3231324102 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4}\cdot 5\cdot 13 \)
Twist minimal: no (minimal twist has level 6)
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 \(\beta = 673920\sqrt{12925296409}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 524288 q^{2} + 274877906944 q^{4} + ( - 623 \beta - 3436634283270) q^{5} + ( - 367057 \beta + 64\!\cdots\!56) q^{7}+ \cdots + 14\!\cdots\!72 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 524288 q^{2} + 274877906944 q^{4} + ( - 623 \beta - 3436634283270) q^{5} + ( - 367057 \beta + 64\!\cdots\!56) q^{7}+ \cdots + ( - 24\!\cdots\!92 \beta - 40\!\cdots\!16) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1048576 q^{2} + 549755813888 q^{4} - 6873268566540 q^{5} + 12\!\cdots\!12 q^{7}+ \cdots + 28\!\cdots\!44 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 1048576 q^{2} + 549755813888 q^{4} - 6873268566540 q^{5} + 12\!\cdots\!12 q^{7}+ \cdots - 80\!\cdots\!32 q^{98}+O(q^{100}) \) 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.

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
56845.2
−56844.2
524288. 0 2.74878e11 −5.11694e13 0 −2.16768e16 1.44115e17 0 −2.68275e19
1.2 524288. 0 2.74878e11 4.42961e13 0 3.45693e16 1.44115e17 0 2.32239e19
\(n\): 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 18.40.a.f 2
3.b odd 2 1 6.40.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.40.a.b 2 3.b odd 2 1
18.40.a.f 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 6873268566540T_{5} - 2266606978164805175391937500 \) acting on \(S_{40}^{\mathrm{new}}(\Gamma_0(18))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 524288)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 6873268566540 T - 22\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 74\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 83\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 53\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 56\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 12\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 41\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 21\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 27\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 41\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 10\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 51\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 35\!\cdots\!44 \) Copy content Toggle raw display
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