Properties

Label 4.38.a.a
Level $4$
Weight $38$
Character orbit 4.a
Self dual yes
Analytic conductor $34.686$
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] = [4,38,Mod(1,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 38, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.1");
 
S:= CuspForms(chi, 38);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 4.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: \(34.6856152498\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 134608389910x + 8010664803252592 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{5}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 90721164) q^{3} + (\beta_{2} + 1822 \beta_1 + 1213681705398) q^{5} + (420 \beta_{2} + 709926 \beta_1 + 504728169053864) q^{7} + (34830 \beta_{2} - 24221340 \beta_1 + 34\!\cdots\!21) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 90721164) q^{3} + (\beta_{2} + 1822 \beta_1 + 1213681705398) q^{5} + (420 \beta_{2} + 709926 \beta_1 + 504728169053864) q^{7} + (34830 \beta_{2} - 24221340 \beta_1 + 34\!\cdots\!21) q^{9}+ \cdots + (44\!\cdots\!60 \beta_{2} + \cdots - 49\!\cdots\!00) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 272163492 q^{3} + 3641045116194 q^{5} + 15\!\cdots\!92 q^{7}+ \cdots + 10\!\cdots\!63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 272163492 q^{3} + 3641045116194 q^{5} + 15\!\cdots\!92 q^{7}+ \cdots - 14\!\cdots\!00 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 134608389910x + 8010664803252592 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2304\nu - 768 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 32768\nu^{2} + 2924972544\nu - 2940566122049024 ) / 215 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 768 ) / 2304 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 645\beta_{2} - 3808558\beta _1 + 8821695441174528 ) / 98304 \) 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
332433.
61215.0
−393647.
0 −8.56646e8 0 1.02977e13 0 4.27767e15 0 2.83558e17 0
1.2 0 −2.31760e8 0 −1.08025e13 0 −4.54986e15 0 −3.96571e17 0
1.3 0 8.16242e8 0 4.14579e12 0 1.78638e15 0 2.15967e17 0
\(n\): 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 4.38.a.a 3
4.b odd 2 1 16.38.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.38.a.a 3 1.a even 1 1 trivial
16.38.a.d 3 4.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{38}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 272163492 T^{2} + \cdots - 16\!\cdots\!48 \) Copy content Toggle raw display
$5$ \( T^{3} - 3641045116194 T^{2} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots + 34\!\cdots\!08 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots - 56\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 33\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 78\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 58\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 51\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 46\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 18\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 21\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 25\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 38\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 24\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 59\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 30\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 51\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 56\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 20\!\cdots\!52 \) Copy content Toggle raw display
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