Properties

Label 648.4.i.l
Level $648$
Weight $4$
Character orbit 648.i
Analytic conductor $38.233$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 16 \zeta_{6} q^{5} + ( - 12 \zeta_{6} + 12) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 \zeta_{6} q^{5} + ( - 12 \zeta_{6} + 12) q^{7} + ( - 64 \zeta_{6} + 64) q^{11} - 58 \zeta_{6} q^{13} - 32 q^{17} - 136 q^{19} - 128 \zeta_{6} q^{23} + (131 \zeta_{6} - 131) q^{25} + (144 \zeta_{6} - 144) q^{29} - 20 \zeta_{6} q^{31} + 192 q^{35} - 18 q^{37} - 288 \zeta_{6} q^{41} + ( - 200 \zeta_{6} + 200) q^{43} + ( - 384 \zeta_{6} + 384) q^{47} + 199 \zeta_{6} q^{49} - 496 q^{53} + 1024 q^{55} - 128 \zeta_{6} q^{59} + ( - 458 \zeta_{6} + 458) q^{61} + ( - 928 \zeta_{6} + 928) q^{65} + 496 \zeta_{6} q^{67} - 512 q^{71} - 602 q^{73} - 768 \zeta_{6} q^{77} + (1108 \zeta_{6} - 1108) q^{79} + ( - 704 \zeta_{6} + 704) q^{83} - 512 \zeta_{6} q^{85} + 960 q^{89} - 696 q^{91} - 2176 \zeta_{6} q^{95} + (206 \zeta_{6} - 206) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{5} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{5} + 12 q^{7} + 64 q^{11} - 58 q^{13} - 64 q^{17} - 272 q^{19} - 128 q^{23} - 131 q^{25} - 144 q^{29} - 20 q^{31} + 384 q^{35} - 36 q^{37} - 288 q^{41} + 200 q^{43} + 384 q^{47} + 199 q^{49} - 992 q^{53} + 2048 q^{55} - 128 q^{59} + 458 q^{61} + 928 q^{65} + 496 q^{67} - 1024 q^{71} - 1204 q^{73} - 768 q^{77} - 1108 q^{79} + 704 q^{83} - 512 q^{85} + 1920 q^{89} - 1392 q^{91} - 2176 q^{95} - 206 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 8.00000 + 13.8564i 0 6.00000 10.3923i 0 0 0
433.1 0 0 0 8.00000 13.8564i 0 6.00000 + 10.3923i 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.l 2
3.b odd 2 1 648.4.i.a 2
9.c even 3 1 72.4.a.a 1
9.c even 3 1 inner 648.4.i.l 2
9.d odd 6 1 72.4.a.d yes 1
9.d odd 6 1 648.4.i.a 2
36.f odd 6 1 144.4.a.a 1
36.h even 6 1 144.4.a.f 1
45.h odd 6 1 1800.4.a.ba 1
45.j even 6 1 1800.4.a.z 1
45.k odd 12 2 1800.4.f.b 2
45.l even 12 2 1800.4.f.x 2
72.j odd 6 1 576.4.a.c 1
72.l even 6 1 576.4.a.d 1
72.n even 6 1 576.4.a.w 1
72.p odd 6 1 576.4.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.a.a 1 9.c even 3 1
72.4.a.d yes 1 9.d odd 6 1
144.4.a.a 1 36.f odd 6 1
144.4.a.f 1 36.h even 6 1
576.4.a.c 1 72.j odd 6 1
576.4.a.d 1 72.l even 6 1
576.4.a.w 1 72.n even 6 1
576.4.a.x 1 72.p odd 6 1
648.4.i.a 2 3.b odd 2 1
648.4.i.a 2 9.d odd 6 1
648.4.i.l 2 1.a even 1 1 trivial
648.4.i.l 2 9.c even 3 1 inner
1800.4.a.z 1 45.j even 6 1
1800.4.a.ba 1 45.h odd 6 1
1800.4.f.b 2 45.k odd 12 2
1800.4.f.x 2 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 16T_{5} + 256 \) acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$7$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$11$ \( T^{2} - 64T + 4096 \) Copy content Toggle raw display
$13$ \( T^{2} + 58T + 3364 \) Copy content Toggle raw display
$17$ \( (T + 32)^{2} \) Copy content Toggle raw display
$19$ \( (T + 136)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 128T + 16384 \) Copy content Toggle raw display
$29$ \( T^{2} + 144T + 20736 \) Copy content Toggle raw display
$31$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$37$ \( (T + 18)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 288T + 82944 \) Copy content Toggle raw display
$43$ \( T^{2} - 200T + 40000 \) Copy content Toggle raw display
$47$ \( T^{2} - 384T + 147456 \) Copy content Toggle raw display
$53$ \( (T + 496)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 128T + 16384 \) Copy content Toggle raw display
$61$ \( T^{2} - 458T + 209764 \) Copy content Toggle raw display
$67$ \( T^{2} - 496T + 246016 \) Copy content Toggle raw display
$71$ \( (T + 512)^{2} \) Copy content Toggle raw display
$73$ \( (T + 602)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 1108 T + 1227664 \) Copy content Toggle raw display
$83$ \( T^{2} - 704T + 495616 \) Copy content Toggle raw display
$89$ \( (T - 960)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 206T + 42436 \) Copy content Toggle raw display
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