Properties

Label 1296.3.q.c
Level $1296$
Weight $3$
Character orbit 1296.q
Analytic conductor $35.313$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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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: \(2\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 11 \zeta_{6} + 11) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 11 \zeta_{6} + 11) q^{7} - 23 \zeta_{6} q^{13} + 37 q^{19} + (25 \zeta_{6} - 25) q^{25} - 46 \zeta_{6} q^{31} - 73 q^{37} + (22 \zeta_{6} - 22) q^{43} - 72 \zeta_{6} q^{49} + (47 \zeta_{6} - 47) q^{61} - 13 \zeta_{6} q^{67} + 143 q^{73} + ( - 11 \zeta_{6} + 11) q^{79} - 253 q^{91} + ( - 169 \zeta_{6} + 169) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 11 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 11 q^{7} - 23 q^{13} + 74 q^{19} - 25 q^{25} - 46 q^{31} - 146 q^{37} - 22 q^{43} - 72 q^{49} - 47 q^{61} - 13 q^{67} + 286 q^{73} + 11 q^{79} - 506 q^{91} + 169 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 5.50000 9.52628i 0 0 0
1025.1 0 0 0 0 0 5.50000 + 9.52628i 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 CM by \(\Q(\sqrt{-3}) \)
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.c 2
3.b odd 2 1 CM 1296.3.q.c 2
4.b odd 2 1 324.3.g.a 2
9.c even 3 1 432.3.e.a 1
9.c even 3 1 inner 1296.3.q.c 2
9.d odd 6 1 432.3.e.a 1
9.d odd 6 1 inner 1296.3.q.c 2
12.b even 2 1 324.3.g.a 2
36.f odd 6 1 108.3.c.a 1
36.f odd 6 1 324.3.g.a 2
36.h even 6 1 108.3.c.a 1
36.h even 6 1 324.3.g.a 2
72.j odd 6 1 1728.3.e.b 1
72.l even 6 1 1728.3.e.c 1
72.n even 6 1 1728.3.e.b 1
72.p odd 6 1 1728.3.e.c 1
180.n even 6 1 2700.3.g.b 1
180.p odd 6 1 2700.3.g.b 1
180.v odd 12 2 2700.3.b.d 2
180.x even 12 2 2700.3.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.c.a 1 36.f odd 6 1
108.3.c.a 1 36.h even 6 1
324.3.g.a 2 4.b odd 2 1
324.3.g.a 2 12.b even 2 1
324.3.g.a 2 36.f odd 6 1
324.3.g.a 2 36.h even 6 1
432.3.e.a 1 9.c even 3 1
432.3.e.a 1 9.d odd 6 1
1296.3.q.c 2 1.a even 1 1 trivial
1296.3.q.c 2 3.b odd 2 1 CM
1296.3.q.c 2 9.c even 3 1 inner
1296.3.q.c 2 9.d odd 6 1 inner
1728.3.e.b 1 72.j odd 6 1
1728.3.e.b 1 72.n even 6 1
1728.3.e.c 1 72.l even 6 1
1728.3.e.c 1 72.p odd 6 1
2700.3.b.d 2 180.v odd 12 2
2700.3.b.d 2 180.x even 12 2
2700.3.g.b 1 180.n even 6 1
2700.3.g.b 1 180.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 \(S_{3}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} - 11T_{7} + 121 \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 23T + 529 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 37)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 46T + 2116 \) Copy content Toggle raw display
$37$ \( (T + 73)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 22T + 484 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 47T + 2209 \) Copy content Toggle raw display
$67$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 143)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 169T + 28561 \) Copy content Toggle raw display
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