Properties

Label 2.38.a.a
Level $2$
Weight $38$
Character orbit 2.a
Self dual yes
Analytic conductor $17.343$
Analytic rank $1$
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] = [2,38,Mod(1,2)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 38, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 38);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 2.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: \(17.3428076249\)
Analytic rank: \(1\)
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: \( x^{2} - x - 756643680 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 5 \)
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 \(\beta = 17280\sqrt{3026574721}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 262144 q^{2} + ( - \beta + 211535604) q^{3} + 68719476736 q^{4} + (3444 \beta - 6753765277770) q^{5} + (262144 \beta - 55452789374976) q^{6} + (5126718 \beta + 15\!\cdots\!28) q^{7}+ \cdots + ( - 423071208 \beta + 49\!\cdots\!53) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 262144 q^{2} + ( - \beta + 211535604) q^{3} + 68719476736 q^{4} + (3444 \beta - 6753765277770) q^{5} + (262144 \beta - 55452789374976) q^{6} + (5126718 \beta + 15\!\cdots\!28) q^{7}+ \cdots + (11\!\cdots\!45 \beta - 86\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 524288 q^{2} + 423071208 q^{3} + 137438953472 q^{4} - 13507530555540 q^{5} - 110905578749952 q^{6} + 31\!\cdots\!56 q^{7}+ \cdots + 99\!\cdots\!06 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 524288 q^{2} + 423071208 q^{3} + 137438953472 q^{4} - 13507530555540 q^{5} - 110905578749952 q^{6} + 31\!\cdots\!56 q^{7}+ \cdots - 17\!\cdots\!88 q^{99}+O(q^{100}) \) 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
27507.7
−27506.7
−262144. −7.39112e8 6.87195e10 −3.47974e12 1.93754e14 6.42679e15 −1.80144e16 9.60023e16 9.12192e17
1.2 −262144. 1.16218e9 6.87195e10 −1.00278e13 −3.04659e14 −3.32061e15 −1.80144e16 9.00385e17 2.62873e18
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

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 2.38.a.a 2
3.b odd 2 1 18.38.a.f 2
4.b odd 2 1 16.38.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.38.a.a 2 1.a even 1 1 trivial
16.38.a.a 2 4.b odd 2 1
18.38.a.f 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 423071208T_{3} - 858983057411401584 \) acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(2))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 262144)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 423071208 T - 85\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( T^{2} + 13507530555540 T + 34\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 21\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 57\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 42\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 48\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 16\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 69\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 44\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 90\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 20\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 62\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 60\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 32\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 26\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 13\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 98\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 18\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 19\!\cdots\!16 \) Copy content Toggle raw display
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