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Properties

Label 648.2.n.d
Level $648$
Weight $2$
Character orbit 648.n
Analytic conductor $5.174$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [648,2,Mod(109,648)]
 
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, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.n (of order \(6\), 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: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{2} - 2 \beta_1 + 1) q^{5} + 2 \beta_{2} q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{2} - 2 \beta_1 + 1) q^{5} + 2 \beta_{2} q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{8} + ( - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{10} + ( - \beta_{2} - 2 \beta_1 + 1) q^{13} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{14} + (3 \beta_{3} - 2 \beta_{2}) q^{16} + 7 q^{17} + ( - 4 \beta_{3} + 4 \beta_{2} + \cdots - 2) q^{19}+ \cdots + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 3 q^{4} + 4 q^{7} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 3 q^{4} + 4 q^{7} - 10 q^{8} + 14 q^{10} + 2 q^{14} - q^{16} + 28 q^{17} - 7 q^{20} - 8 q^{23} + 4 q^{25} + 14 q^{26} + 12 q^{28} + 20 q^{31} - 11 q^{32} - 7 q^{34} - 14 q^{38} + 7 q^{40} - 4 q^{41} + 8 q^{46} + 12 q^{47} + 6 q^{49} + 2 q^{50} - 7 q^{52} - 10 q^{56} - 21 q^{58} - 20 q^{62} + 18 q^{64} + 14 q^{65} + 21 q^{68} + 14 q^{70} + 24 q^{71} - 12 q^{73} - 7 q^{74} - 14 q^{76} - 8 q^{79} - 42 q^{80} + 4 q^{82} - 28 q^{86} - 4 q^{89} + 12 q^{92} + 6 q^{94} + 28 q^{95} + 28 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

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

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

Character values

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

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(1\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
1.39564 + 0.228425i
−0.895644 − 1.09445i
1.39564 − 0.228425i
−0.895644 + 1.09445i
−1.39564 + 0.228425i 0 1.89564 − 0.637600i −2.29129 − 1.32288i 0 1.00000 + 1.73205i −2.50000 + 1.32288i 0 3.50000 + 1.32288i
109.2 0.895644 − 1.09445i 0 −0.395644 − 1.96048i 2.29129 + 1.32288i 0 1.00000 + 1.73205i −2.50000 − 1.32288i 0 3.50000 − 1.32288i
541.1 −1.39564 − 0.228425i 0 1.89564 + 0.637600i −2.29129 + 1.32288i 0 1.00000 − 1.73205i −2.50000 − 1.32288i 0 3.50000 − 1.32288i
541.2 0.895644 + 1.09445i 0 −0.395644 + 1.96048i 2.29129 − 1.32288i 0 1.00000 − 1.73205i −2.50000 + 1.32288i 0 3.50000 + 1.32288i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.n.d 4
3.b odd 2 1 648.2.n.j 4
4.b odd 2 1 2592.2.r.h 4
8.b even 2 1 inner 648.2.n.d 4
8.d odd 2 1 2592.2.r.h 4
9.c even 3 1 648.2.d.c yes 2
9.c even 3 1 inner 648.2.n.d 4
9.d odd 6 1 648.2.d.b ✓ 2
9.d odd 6 1 648.2.n.j 4
12.b even 2 1 2592.2.r.e 4
24.f even 2 1 2592.2.r.e 4
24.h odd 2 1 648.2.n.j 4
36.f odd 6 1 2592.2.d.d 2
36.f odd 6 1 2592.2.r.h 4
36.h even 6 1 2592.2.d.c 2
36.h even 6 1 2592.2.r.e 4
72.j odd 6 1 648.2.d.b ✓ 2
72.j odd 6 1 648.2.n.j 4
72.l even 6 1 2592.2.d.c 2
72.l even 6 1 2592.2.r.e 4
72.n even 6 1 648.2.d.c yes 2
72.n even 6 1 inner 648.2.n.d 4
72.p odd 6 1 2592.2.d.d 2
72.p odd 6 1 2592.2.r.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.2.d.b ✓ 2 9.d odd 6 1
648.2.d.b ✓ 2 72.j odd 6 1
648.2.d.c yes 2 9.c even 3 1
648.2.d.c yes 2 72.n even 6 1
648.2.n.d 4 1.a even 1 1 trivial
648.2.n.d 4 8.b even 2 1 inner
648.2.n.d 4 9.c even 3 1 inner
648.2.n.d 4 72.n even 6 1 inner
648.2.n.j 4 3.b odd 2 1
648.2.n.j 4 9.d odd 6 1
648.2.n.j 4 24.h odd 2 1
648.2.n.j 4 72.j odd 6 1
2592.2.d.c 2 36.h even 6 1
2592.2.d.c 2 72.l even 6 1
2592.2.d.d 2 36.f odd 6 1
2592.2.d.d 2 72.p odd 6 1
2592.2.r.e 4 12.b even 2 1
2592.2.r.e 4 24.f even 2 1
2592.2.r.e 4 36.h even 6 1
2592.2.r.e 4 72.l even 6 1
2592.2.r.h 4 4.b odd 2 1
2592.2.r.h 4 8.d odd 2 1
2592.2.r.h 4 36.f odd 6 1
2592.2.r.h 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(648, [\chi])\):

\( T_{5}^{4} - 7T_{5}^{2} + 49 \) Copy content Toggle raw display
\( T_{13}^{4} - 7T_{13}^{2} + 49 \) Copy content Toggle raw display
\( T_{17} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} - T^{2} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 7T^{2} + 49 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 7T^{2} + 49 \) Copy content Toggle raw display
$17$ \( (T - 7)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 63T^{2} + 3969 \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 7)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 112 T^{2} + 12544 \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 112 T^{2} + 12544 \) Copy content Toggle raw display
$61$ \( T^{4} - 63T^{2} + 3969 \) Copy content Toggle raw display
$67$ \( T^{4} - 252 T^{2} + 63504 \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( (T + 3)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 28T^{2} + 784 \) Copy content Toggle raw display
$89$ \( (T + 1)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
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