Properties

Label 224.6.a.e
Level $224$
Weight $6$
Character orbit 224.a
Self dual yes
Analytic conductor $35.926$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,6,Mod(1,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 224.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: \(35.9259756381\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.367637.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 107x + 282 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 3) q^{3} + ( - \beta_{2} - 5) q^{5} + 49 q^{7} + (3 \beta_{2} - \beta_1 + 51) q^{9} + ( - \beta_{2} + 19 \beta_1 - 194) q^{11} + (11 \beta_{2} + 6 \beta_1 - 319) q^{13} + (9 \beta_{2} + 39 \beta_1 - 24) q^{15}+ \cdots + ( - 237 \beta_{2} + 3103 \beta_1 - 26754) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 8 q^{3} - 14 q^{5} + 147 q^{7} + 151 q^{9} - 600 q^{11} - 974 q^{13} - 120 q^{15} + 718 q^{17} + 1056 q^{19} - 392 q^{21} + 3760 q^{23} - 147 q^{25} + 2872 q^{27} - 9134 q^{29} - 1448 q^{31} - 14872 q^{33}+ \cdots - 83128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 107x + 282 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{2} + 10\nu - 291 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} - 5\beta _1 + 286 ) / 4 \) 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
9.26413
2.76097
−11.0251
0 −20.5283 0 −53.3126 0 49.0000 0 178.410 0
1.2 0 −7.52194 0 72.6328 0 49.0000 0 −186.420 0
1.3 0 20.0502 0 −33.3202 0 49.0000 0 159.011 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)

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 224.6.a.e 3
4.b odd 2 1 224.6.a.f yes 3
8.b even 2 1 448.6.a.bb 3
8.d odd 2 1 448.6.a.ba 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.6.a.e 3 1.a even 1 1 trivial
224.6.a.f yes 3 4.b odd 2 1
448.6.a.ba 3 8.d odd 2 1
448.6.a.bb 3 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 8T_{3}^{2} - 408T_{3} - 3096 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(224))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 8 T^{2} + \cdots - 3096 \) Copy content Toggle raw display
$5$ \( T^{3} + 14 T^{2} + \cdots - 129024 \) Copy content Toggle raw display
$7$ \( (T - 49)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 600 T^{2} + \cdots - 1824064 \) Copy content Toggle raw display
$13$ \( T^{3} + 974 T^{2} + \cdots - 53052784 \) Copy content Toggle raw display
$17$ \( T^{3} - 718 T^{2} + \cdots + 178338584 \) Copy content Toggle raw display
$19$ \( T^{3} - 1056 T^{2} + \cdots + 936225400 \) Copy content Toggle raw display
$23$ \( T^{3} - 3760 T^{2} + \cdots - 265872000 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 18655265000 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 54502545600 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 417364199688 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 195051110440 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 302765738048 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 1468178593088 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 1608716275704 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 27263200844760 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 18497589110240 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 132723178706432 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 12186764730368 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 39799282328008 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 267561011200 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 109229539940808 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 4144636813752 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 698292640604904 \) Copy content Toggle raw display
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