Properties

Label 648.2.d.e
Level $648$
Weight $2$
Character orbit 648.d
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(325,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.325");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.d (of order \(2\), degree \(1\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_1) q^{4} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{2} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_1) q^{4} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{10} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{11} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{13} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{14} + ( - 2 \beta_{3} + 2 \beta_1) q^{16} + (\beta_{3} - \beta_{2} + \beta_1 + 2) q^{17} + (\beta_{3} + 5 \beta_{2} - \beta_1 + 1) q^{19} + ( - 3 \beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{20} + (3 \beta_{3} - 4 \beta_{2} - \beta_1 + 2) q^{22} + ( - \beta_{3} + \beta_{2} - \beta_1 - 7) q^{23} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 2) q^{25} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{26} + (2 \beta_{3} - 2) q^{28} + (3 \beta_{3} - 3 \beta_1 + 3) q^{29} + 2 q^{31} + ( - 4 \beta_1 + 4) q^{32} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{34} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{35} - 5 \beta_{2} q^{37} + (5 \beta_{3} - 4 \beta_{2} + \beta_1 - 2) q^{38} + (4 \beta_{3} - 6 \beta_{2} + 6) q^{40} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 8) q^{41} + ( - 5 \beta_{3} + \beta_{2} + 5 \beta_1 - 5) q^{43} + ( - 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{44} + (\beta_{3} - 6 \beta_{2} + 7 \beta_1 + 2) q^{46} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{47} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 3) q^{49} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 8) q^{50} + (\beta_{3} - 8 \beta_{2} + 3 \beta_1 - 2) q^{52} + (4 \beta_{3} + 6 \beta_{2} - 4 \beta_1 + 4) q^{53} + ( - 5 \beta_{3} + 5 \beta_{2} - 5 \beta_1 - 9) q^{55} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{56} + (3 \beta_{2} + 3 \beta_1 - 6) q^{58} + ( - 6 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 6) q^{59} + 9 \beta_{2} q^{61} + (2 \beta_{2} - 2 \beta_1) q^{62} + 8 \beta_{2} q^{64} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 4) q^{65} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{67} + (\beta_{3} + 3 \beta_1 + 2) q^{68} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{70} + ( - 5 \beta_{3} + 5 \beta_{2} - 5 \beta_1 + 3) q^{71} + 9 q^{73} + ( - 5 \beta_{3} + 5 \beta_{2}) q^{74} + ( - 4 \beta_{3} - 4 \beta_{2} + 6 \beta_1 - 10) q^{76} + (2 \beta_{3} - 2 \beta_1 + 2) q^{77} + ( - \beta_{3} + \beta_{2} - \beta_1 + 7) q^{79} + ( - 6 \beta_{3} + 8 \beta_{2} - 2 \beta_1 - 8) q^{80} + ( - 2 \beta_{3} - 10 \beta_{2} + 8 \beta_1 - 4) q^{82} - 14 \beta_{2} q^{83} + ( - 4 \beta_{3} + 7 \beta_{2} + 4 \beta_1 - 4) q^{85} + (\beta_{3} - 6 \beta_{2} - 5 \beta_1 + 10) q^{86} + (4 \beta_{3} - 8 \beta_{2} + 8) q^{88} + (\beta_{3} - \beta_{2} + \beta_1 + 10) q^{89} + (3 \beta_{3} + 7 \beta_{2} - 3 \beta_1 + 3) q^{91} + ( - 6 \beta_{3} - 8 \beta_1 - 2) q^{92} + (2 \beta_{3} + 2 \beta_{2} + 4) q^{94} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 7) q^{95} - 8 q^{97} + (2 \beta_{3} - \beta_{2} + 3 \beta_1 + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 4 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 4 q^{7} - 8 q^{8} + 2 q^{10} + 4 q^{14} + 8 q^{16} + 8 q^{17} - 8 q^{20} - 28 q^{23} - 8 q^{25} - 14 q^{26} - 12 q^{28} + 8 q^{31} + 8 q^{32} - 10 q^{34} - 16 q^{38} + 16 q^{40} - 32 q^{41} - 12 q^{44} + 20 q^{46} - 12 q^{49} + 28 q^{50} - 4 q^{52} - 36 q^{55} - 8 q^{56} - 18 q^{58} - 4 q^{62} + 16 q^{65} + 12 q^{68} - 4 q^{70} + 12 q^{71} + 36 q^{73} + 10 q^{74} - 20 q^{76} + 28 q^{79} - 24 q^{80} + 4 q^{82} + 28 q^{86} + 24 q^{88} + 40 q^{89} - 12 q^{92} + 12 q^{94} - 28 q^{95} - 32 q^{97} + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
325.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
−1.36603 0.366025i 0 1.73205 + 1.00000i 3.73205i 0 −0.732051 −2.00000 2.00000i 0 1.36603 5.09808i
325.2 −1.36603 + 0.366025i 0 1.73205 1.00000i 3.73205i 0 −0.732051 −2.00000 + 2.00000i 0 1.36603 + 5.09808i
325.3 0.366025 1.36603i 0 −1.73205 1.00000i 0.267949i 0 2.73205 −2.00000 + 2.00000i 0 −0.366025 0.0980762i
325.4 0.366025 + 1.36603i 0 −1.73205 + 1.00000i 0.267949i 0 2.73205 −2.00000 2.00000i 0 −0.366025 + 0.0980762i
\(n\): 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

Twists

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

Hecke kernels

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

\( T_{5}^{4} + 14T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 14T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{4} + 26T^{2} + 121 \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$23$ \( (T^{2} + 14 T + 46)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 16 T + 52)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 152T^{2} + 5476 \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$59$ \( T^{4} + 224 T^{2} + 10816 \) Copy content Toggle raw display
$61$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T - 66)^{2} \) Copy content Toggle raw display
$73$ \( (T - 9)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 46)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 20 T + 97)^{2} \) Copy content Toggle raw display
$97$ \( (T + 8)^{4} \) Copy content Toggle raw display
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