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Properties

Label 648.4.i.o
Level $648$
Weight $4$
Character orbit 648.i
Analytic conductor $38.233$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [648,4,Mod(217,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.217");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 648.i (of order \(3\), degree \(2\), not minimal)

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: no
Analytic conductor: \(38.2332376837\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 2 \beta_1) q^{5} + (2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 2 \beta_1) q^{5} + (2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{7} + ( - 3 \beta_{3} - 3 \beta_{2} + \cdots - 26) q^{11}+ \cdots + (88 \beta_{3} + 88 \beta_{2} + \cdots + 19) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 6 q^{7} - 52 q^{11} - 26 q^{13} + 376 q^{17} - 148 q^{19} - 148 q^{23} - 118 q^{25} - 288 q^{29} + 248 q^{31} + 1416 q^{35} + 684 q^{37} + 256 q^{43} - 132 q^{47} - 772 q^{49} + 1904 q^{53} - 1952 q^{55} + 1004 q^{59} + 34 q^{61} + 668 q^{65} + 866 q^{67} + 1552 q^{71} + 3748 q^{73} + 2316 q^{77} - 182 q^{79} + 1336 q^{83} - 16 q^{85} + 1752 q^{89} - 3036 q^{91} + 3028 q^{95} + 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{3} - 6\nu^{2} + 18\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 6\nu^{3} + 12 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} + 6\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 18\beta _1 - 18 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} - 12 ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
0.809017 + 1.40126i
−0.309017 − 0.535233i
0.809017 − 1.40126i
−0.309017 + 0.535233i
0 0 0 −7.70820 − 13.3510i 0 −11.9164 + 20.6398i 0 0 0
217.2 0 0 0 5.70820 + 9.88690i 0 14.9164 − 25.8360i 0 0 0
433.1 0 0 0 −7.70820 + 13.3510i 0 −11.9164 − 20.6398i 0 0 0
433.2 0 0 0 5.70820 − 9.88690i 0 14.9164 + 25.8360i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.4.i.o 4
3.b odd 2 1 648.4.i.r 4
9.c even 3 1 216.4.a.g yes 2
9.c even 3 1 inner 648.4.i.o 4
9.d odd 6 1 216.4.a.f ✓ 2
9.d odd 6 1 648.4.i.r 4
36.f odd 6 1 432.4.a.r 2
36.h even 6 1 432.4.a.p 2
72.j odd 6 1 1728.4.a.bq 2
72.l even 6 1 1728.4.a.br 2
72.n even 6 1 1728.4.a.bi 2
72.p odd 6 1 1728.4.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.f ✓ 2 9.d odd 6 1
216.4.a.g yes 2 9.c even 3 1
432.4.a.p 2 36.h even 6 1
432.4.a.r 2 36.f odd 6 1
648.4.i.o 4 1.a even 1 1 trivial
648.4.i.o 4 9.c even 3 1 inner
648.4.i.r 4 3.b odd 2 1
648.4.i.r 4 9.d odd 6 1
1728.4.a.bi 2 72.n even 6 1
1728.4.a.bj 2 72.p odd 6 1
1728.4.a.bq 2 72.j odd 6 1
1728.4.a.br 2 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 4T_{5}^{3} + 192T_{5}^{2} - 704T_{5} + 30976 \) acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\). 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
$5$ \( T^{4} + 4 T^{3} + \cdots + 30976 \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + \cdots + 505521 \) Copy content Toggle raw display
$11$ \( T^{4} + 52 T^{3} + \cdots + 891136 \) Copy content Toggle raw display
$13$ \( T^{4} + 26 T^{3} + \cdots + 303601 \) Copy content Toggle raw display
$17$ \( (T^{2} - 188 T + 8656)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 74 T - 10151)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 148 T^{3} + \cdots + 952576 \) Copy content Toggle raw display
$29$ \( T^{4} + 288 T^{3} + \cdots + 84934656 \) Copy content Toggle raw display
$31$ \( T^{4} - 248 T^{3} + \cdots + 14868736 \) Copy content Toggle raw display
$37$ \( (T^{2} - 342 T + 11241)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 33973862400 \) Copy content Toggle raw display
$43$ \( T^{4} - 256 T^{3} + \cdots + 182358016 \) Copy content Toggle raw display
$47$ \( T^{4} + 132 T^{3} + \cdots + 17438976 \) Copy content Toggle raw display
$53$ \( (T^{2} - 952 T + 191296)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 29799045376 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 43177099681 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 19996505281 \) Copy content Toggle raw display
$71$ \( (T^{2} - 776 T - 11456)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 1874 T + 852049)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 12859333201 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 4269838336 \) Copy content Toggle raw display
$89$ \( (T^{2} - 876 T + 112464)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 1942006686481 \) Copy content Toggle raw display
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