Properties

Label 1296.3.q.j
Level $1296$
Weight $3$
Character orbit 1296.q
Analytic conductor $35.313$
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] = [1296,3,Mod(593,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.593");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.q (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: \(35.3134422611\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \( 3^{2} \)
Twist minimal: no (minimal twist has level 27)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + 5 \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + 5 \beta_{2} q^{7} + ( - 5 \beta_{3} + 5 \beta_1) q^{11} + ( - 10 \beta_{2} + 10) q^{13} + 6 \beta_{3} q^{17} + 16 q^{19} - 4 \beta_1 q^{23} - 16 \beta_{2} q^{25} + ( - 10 \beta_{3} + 10 \beta_1) q^{29} + (\beta_{2} - 1) q^{31} + 5 \beta_{3} q^{35} + 20 q^{37} - 20 \beta_1 q^{41} + 50 \beta_{2} q^{43} + ( - 2 \beta_{3} + 2 \beta_1) q^{47} + ( - 24 \beta_{2} + 24) q^{49} - 9 \beta_{3} q^{53} + 45 q^{55} - 10 \beta_1 q^{59} + 76 \beta_{2} q^{61} + ( - 10 \beta_{3} + 10 \beta_1) q^{65} + (10 \beta_{2} - 10) q^{67} + 30 \beta_{3} q^{71} + 65 q^{73} + 25 \beta_1 q^{77} + 14 \beta_{2} q^{79} + (\beta_{3} - \beta_1) q^{83} + (54 \beta_{2} - 54) q^{85} + 30 \beta_{3} q^{89} + 50 q^{91} + 16 \beta_1 q^{95} + 85 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{7} + 20 q^{13} + 64 q^{19} - 32 q^{25} - 2 q^{31} + 80 q^{37} + 100 q^{43} + 48 q^{49} + 180 q^{55} + 152 q^{61} - 20 q^{67} + 260 q^{73} + 28 q^{79} - 108 q^{85} + 200 q^{91} + 170 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 0 0 −2.59808 + 1.50000i 0 2.50000 4.33013i 0 0 0
593.2 0 0 0 2.59808 1.50000i 0 2.50000 4.33013i 0 0 0
1025.1 0 0 0 −2.59808 1.50000i 0 2.50000 + 4.33013i 0 0 0
1025.2 0 0 0 2.59808 + 1.50000i 0 2.50000 + 4.33013i 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
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.q.j 4
3.b odd 2 1 inner 1296.3.q.j 4
4.b odd 2 1 81.3.d.b 4
9.c even 3 1 432.3.e.c 2
9.c even 3 1 inner 1296.3.q.j 4
9.d odd 6 1 432.3.e.c 2
9.d odd 6 1 inner 1296.3.q.j 4
12.b even 2 1 81.3.d.b 4
36.f odd 6 1 27.3.b.b 2
36.f odd 6 1 81.3.d.b 4
36.h even 6 1 27.3.b.b 2
36.h even 6 1 81.3.d.b 4
72.j odd 6 1 1728.3.e.g 2
72.l even 6 1 1728.3.e.m 2
72.n even 6 1 1728.3.e.g 2
72.p odd 6 1 1728.3.e.m 2
180.n even 6 1 675.3.c.h 2
180.p odd 6 1 675.3.c.h 2
180.v odd 12 1 675.3.d.a 2
180.v odd 12 1 675.3.d.d 2
180.x even 12 1 675.3.d.a 2
180.x even 12 1 675.3.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.b.b 2 36.f odd 6 1
27.3.b.b 2 36.h even 6 1
81.3.d.b 4 4.b odd 2 1
81.3.d.b 4 12.b even 2 1
81.3.d.b 4 36.f odd 6 1
81.3.d.b 4 36.h even 6 1
432.3.e.c 2 9.c even 3 1
432.3.e.c 2 9.d odd 6 1
675.3.c.h 2 180.n even 6 1
675.3.c.h 2 180.p odd 6 1
675.3.d.a 2 180.v odd 12 1
675.3.d.a 2 180.x even 12 1
675.3.d.d 2 180.v odd 12 1
675.3.d.d 2 180.x even 12 1
1296.3.q.j 4 1.a even 1 1 trivial
1296.3.q.j 4 3.b odd 2 1 inner
1296.3.q.j 4 9.c even 3 1 inner
1296.3.q.j 4 9.d odd 6 1 inner
1728.3.e.g 2 72.j odd 6 1
1728.3.e.g 2 72.n even 6 1
1728.3.e.m 2 72.l even 6 1
1728.3.e.m 2 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 \(S_{3}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5}^{4} - 9T_{5}^{2} + 81 \) Copy content Toggle raw display
\( T_{7}^{2} - 5T_{7} + 25 \) 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
$5$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$7$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 225 T^{2} + 50625 \) Copy content Toggle raw display
$13$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
$19$ \( (T - 16)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$29$ \( T^{4} - 900 T^{2} + 810000 \) Copy content Toggle raw display
$31$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T - 20)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 3600 T^{2} + \cdots + 12960000 \) Copy content Toggle raw display
$43$ \( (T^{2} - 50 T + 2500)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$53$ \( (T^{2} + 729)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 900 T^{2} + 810000 \) Copy content Toggle raw display
$61$ \( (T^{2} - 76 T + 5776)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8100)^{2} \) Copy content Toggle raw display
$73$ \( (T - 65)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$89$ \( (T^{2} + 8100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 85 T + 7225)^{2} \) Copy content Toggle raw display
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