Properties

Label 252.8.a.g
Level $252$
Weight $8$
Character orbit 252.a
Self dual yes
Analytic conductor $78.721$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,8,Mod(1,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.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: \(78.7210264220\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 14292x^{2} + 540043x + 5027477 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - 343 q^{7} + (\beta_{3} - 5 \beta_1) q^{11} + ( - \beta_{2} + 3318) q^{13} + ( - 14 \beta_{3} - 41 \beta_1) q^{17} + (17 \beta_{2} - 2296) q^{19} + (34 \beta_{3} - 191 \beta_1) q^{23}+ \cdots + ( - 994 \beta_{2} - 9123002) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1372 q^{7} + 13272 q^{13} - 9184 q^{19} + 264524 q^{25} - 445536 q^{31} - 852808 q^{37} + 150224 q^{43} + 470596 q^{49} - 2627296 q^{55} - 1302840 q^{61} - 6071696 q^{67} + 3631768 q^{73} - 16898688 q^{79}+ \cdots - 36492008 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 14292x^{2} + 540043x + 5027477 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} - 70\nu^{2} + 12238\nu + 102865 ) / 729 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{3} + 344\nu^{2} - 81704\nu + 716620 ) / 729 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{2} + 696\nu - 57344 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 8\beta _1 + 12 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 25\beta_{3} - 29\beta_{2} - 232\beta _1 + 114340 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1747\beta_{3} + 4582\beta_{2} + 27908\beta _1 - 4731756 ) / 12 \) 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
−133.838
86.2740
−7.73394
56.2977
0 0 0 −537.098 0 −343.000 0 0 0
1.2 0 0 0 −6.16284 0 −343.000 0 0 0
1.3 0 0 0 6.16284 0 −343.000 0 0 0
1.4 0 0 0 537.098 0 −343.000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(7\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.8.a.g 4
3.b odd 2 1 inner 252.8.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.8.a.g 4 1.a even 1 1 trivial
252.8.a.g 4 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 288512T_{5}^{2} + 10956400 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(252))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 288512 T^{2} + 10956400 \) Copy content Toggle raw display
$7$ \( (T + 343)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 31940258996464 \) Copy content Toggle raw display
$13$ \( (T^{2} - 6636 T + 7501140)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4592 T - 1008535760)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 55\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{2} + 222768 T - 17934158160)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 426404 T - 61976917196)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} - 75112 T - 702655967600)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{2} + 651420 T - 273620692044)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 1461826311376)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 18614489191580)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 17187103673280)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 65\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 79763151012580)^{2} \) Copy content Toggle raw display
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