Properties

Label 648.2.n.g
Level $648$
Weight $2$
Character orbit 648.n
Analytic conductor $5.174$
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] = [648,2,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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{7} + 2 \beta_{3} q^{8} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{10} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 4) q^{11} + ( - \beta_{2} - 3 \beta_1 - 1) q^{13} + ( - 2 \beta_{3} - 4 \beta_{2} + 2) q^{14} + (4 \beta_{2} - 4) q^{16} + 3 q^{17} - 3 \beta_{3} q^{19} + ( - 2 \beta_{3} - 4 \beta_{2} + 2) q^{20} + (2 \beta_{3} - 4 \beta_1 + 2) q^{22} + (2 \beta_{3} - \beta_1) q^{23} + (2 \beta_{3} + 2 \beta_1) q^{25} + ( - \beta_{3} - 6 \beta_{2} - \beta_1) q^{26} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 4) q^{28} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 6) q^{29} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} + 3 \beta_1 q^{34} + (5 \beta_{3} + 8 \beta_{2} - 4) q^{35} + ( - 3 \beta_{3} + 6 \beta_{2} - 3) q^{37} + ( - 6 \beta_{2} + 6) q^{38} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 4) q^{40} + ( - 4 \beta_{3} + 2 \beta_1) q^{41} + ( - 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 4) q^{43} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{44} + (2 \beta_{2} - 4) q^{46} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1) q^{47} + (8 \beta_{3} + 3 \beta_{2} - 4 \beta_1 - 3) q^{49} + (8 \beta_{2} - 4) q^{50} + ( - 6 \beta_{3} - 4 \beta_{2} + 2) q^{52} + ( - 2 \beta_{3} + 8 \beta_{2} - 4) q^{53} + ( - \beta_{3} + 2 \beta_1 + 4) q^{55} + ( - 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 8) q^{56} + (3 \beta_{3} - 6 \beta_1 + 2) q^{58} - 4 \beta_1 q^{59} + (3 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 10) q^{61} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 8) q^{62} - 8 q^{64} + (4 \beta_{3} + 9 \beta_{2} + 4 \beta_1) q^{65} + ( - 4 \beta_{2} + 3 \beta_1 - 4) q^{67} + 6 \beta_{2} q^{68} + (8 \beta_{3} + 10 \beta_{2} - 4 \beta_1 - 10) q^{70} + ( - 3 \beta_{3} + 6 \beta_1 + 6) q^{71} + (2 \beta_{3} - 4 \beta_1 - 1) q^{73} + (6 \beta_{3} - 6 \beta_{2} - 3 \beta_1 + 6) q^{74} + ( - 6 \beta_{3} + 6 \beta_1) q^{76} + (2 \beta_{2} + 4 \beta_1 + 2) q^{77} + ( - 3 \beta_{3} + 4 \beta_{2} - 3 \beta_1) q^{79} + ( - 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 8) q^{80} + ( - 4 \beta_{2} + 8) q^{82} + ( - 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{83} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{85} + ( - 2 \beta_{3} + 4 \beta_1 + 6) q^{86} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{88} + (6 \beta_{3} - 12 \beta_1 - 3) q^{89} + (9 \beta_{3} + 16 \beta_{2} - 8) q^{91} + (2 \beta_{3} - 4 \beta_1) q^{92} + (6 \beta_{3} - 8 \beta_{2} + 4) q^{94} + (6 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 6) q^{95} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{97} + (3 \beta_{3} + 8 \beta_{2} - 3 \beta_1 - 16) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 6 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 6 q^{5} - 4 q^{7} - 4 q^{10} - 12 q^{11} - 6 q^{13} - 8 q^{16} + 12 q^{17} + 8 q^{22} - 12 q^{26} + 8 q^{28} - 18 q^{29} - 4 q^{31} + 12 q^{38} + 8 q^{40} + 12 q^{43} - 24 q^{44} - 12 q^{46} + 12 q^{47} - 6 q^{49} + 16 q^{55} + 24 q^{56} + 8 q^{58} + 30 q^{61} + 24 q^{62} - 32 q^{64} + 18 q^{65} - 24 q^{67} + 12 q^{68} - 20 q^{70} + 24 q^{71} - 4 q^{73} + 12 q^{74} + 12 q^{77} + 8 q^{79} + 24 q^{80} + 24 q^{82} - 12 q^{83} - 18 q^{85} + 24 q^{86} + 8 q^{88} - 12 q^{89} - 12 q^{95} - 16 q^{97} - 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i −0.275255 0.158919i 0 0.224745 + 0.389270i 2.82843i 0 0.224745 + 0.389270i
109.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i −2.72474 1.57313i 0 −2.22474 3.85337i 2.82843i 0 −2.22474 3.85337i
541.1 −1.22474 + 0.707107i 0 1.00000 1.73205i −0.275255 + 0.158919i 0 0.224745 0.389270i 2.82843i 0 0.224745 0.389270i
541.2 1.22474 0.707107i 0 1.00000 1.73205i −2.72474 + 1.57313i 0 −2.22474 + 3.85337i 2.82843i 0 −2.22474 + 3.85337i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{5}^{4} + 6T_{5}^{3} + 13T_{5}^{2} + 6T_{5} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} + 6T_{13}^{3} - 3T_{13}^{2} - 90T_{13} + 225 \) Copy content Toggle raw display
\( T_{17} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + 13 T^{2} + 6 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + 18 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 12 T^{3} + 58 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} - 3 T^{2} - 90 T + 225 \) Copy content Toggle raw display
$17$ \( (T - 3)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$29$ \( T^{4} + 18 T^{3} + 133 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400 \) Copy content Toggle raw display
$37$ \( T^{4} + 90T^{2} + 81 \) Copy content Toggle raw display
$41$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$43$ \( T^{4} - 12 T^{3} + 42 T^{2} + 72 T + 36 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + 132 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$53$ \( T^{4} + 112T^{2} + 1600 \) Copy content Toggle raw display
$59$ \( T^{4} - 32T^{2} + 1024 \) Copy content Toggle raw display
$61$ \( T^{4} - 30 T^{3} + 357 T^{2} + \cdots + 3249 \) Copy content Toggle raw display
$67$ \( T^{4} + 24 T^{3} + 222 T^{2} + \cdots + 900 \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T - 18)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 23)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 8 T^{3} + 102 T^{2} + \cdots + 1444 \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} + 28 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 207)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 16 T^{3} + 216 T^{2} + \cdots + 1600 \) Copy content Toggle raw display
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