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Properties

Label 1296.3.o.t
Level $1296$
Weight $3$
Character orbit 1296.o
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(271,1296)]
 
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([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.271");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.o (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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + ( - 6 \beta_1 - 6) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + ( - 6 \beta_1 - 6) q^{7} + (6 \beta_{3} + 6 \beta_{2}) q^{11} + 5 \beta_1 q^{13} - 7 \beta_{3} q^{17} + (12 \beta_1 - 6) q^{19} + (12 \beta_{3} - 6 \beta_{2}) q^{23} + ( - 22 \beta_1 + 22) q^{25} + (13 \beta_{3} - 13 \beta_{2}) q^{29} + ( - 12 \beta_1 + 24) q^{31} + ( - 6 \beta_{3} + 12 \beta_{2}) q^{35} - 19 q^{37} - 36 \beta_{2} q^{41} + (30 \beta_1 + 30) q^{43} + ( - 24 \beta_{3} - 24 \beta_{2}) q^{47} + 59 \beta_1 q^{49} - 36 \beta_{3} q^{53} + ( - 36 \beta_1 + 18) q^{55} + ( - 48 \beta_{3} + 24 \beta_{2}) q^{59} + ( - 43 \beta_1 + 43) q^{61} + (5 \beta_{3} - 5 \beta_{2}) q^{65} + (54 \beta_1 - 108) q^{67} + (30 \beta_{3} - 60 \beta_{2}) q^{71} - 107 q^{73} - 108 \beta_{2} q^{77} + ( - 42 \beta_1 - 42) q^{79} + (12 \beta_{3} + 12 \beta_{2}) q^{83} + 21 \beta_1 q^{85} - 65 \beta_{3} q^{89} + ( - 60 \beta_1 + 30) q^{91} + (12 \beta_{3} - 6 \beta_{2}) q^{95} + ( - 86 \beta_1 + 86) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{7} + 10 q^{13} + 44 q^{25} + 72 q^{31} - 76 q^{37} + 180 q^{43} + 118 q^{49} + 86 q^{61} - 324 q^{67} - 428 q^{73} - 252 q^{79} + 42 q^{85} + 172 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0 0 0 −0.866025 + 1.50000i 0 −9.00000 + 5.19615i 0 0 0
271.2 0 0 0 0.866025 − 1.50000i 0 −9.00000 + 5.19615i 0 0 0
703.1 0 0 0 −0.866025 − 1.50000i 0 −9.00000 − 5.19615i 0 0 0
703.2 0 0 0 0.866025 + 1.50000i 0 −9.00000 − 5.19615i 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
36.f odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.o.t 4
3.b odd 2 1 inner 1296.3.o.t 4
4.b odd 2 1 1296.3.o.bc 4
9.c even 3 1 1296.3.g.e ✓ 4
9.c even 3 1 1296.3.o.bc 4
9.d odd 6 1 1296.3.g.e ✓ 4
9.d odd 6 1 1296.3.o.bc 4
12.b even 2 1 1296.3.o.bc 4
36.f odd 6 1 1296.3.g.e ✓ 4
36.f odd 6 1 inner 1296.3.o.t 4
36.h even 6 1 1296.3.g.e ✓ 4
36.h even 6 1 inner 1296.3.o.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1296.3.g.e ✓ 4 9.c even 3 1
1296.3.g.e ✓ 4 9.d odd 6 1
1296.3.g.e ✓ 4 36.f odd 6 1
1296.3.g.e ✓ 4 36.h even 6 1
1296.3.o.t 4 1.a even 1 1 trivial
1296.3.o.t 4 3.b odd 2 1 inner
1296.3.o.t 4 36.f odd 6 1 inner
1296.3.o.t 4 36.h even 6 1 inner
1296.3.o.bc 4 4.b odd 2 1
1296.3.o.bc 4 9.c even 3 1
1296.3.o.bc 4 9.d odd 6 1
1296.3.o.bc 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5}^{4} + 3T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} + 18T_{7} + 108 \) Copy content Toggle raw display
\( T_{11}^{4} - 324T_{11}^{2} + 104976 \) 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
$5$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$7$ \( (T^{2} + 18 T + 108)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 324 T^{2} + 104976 \) Copy content Toggle raw display
$13$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 147)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 324 T^{2} + 104976 \) Copy content Toggle raw display
$29$ \( T^{4} + 507 T^{2} + 257049 \) Copy content Toggle raw display
$31$ \( (T^{2} - 36 T + 432)^{2} \) Copy content Toggle raw display
$37$ \( (T + 19)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 3888 T^{2} + 15116544 \) Copy content Toggle raw display
$43$ \( (T^{2} - 90 T + 2700)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 5184 T^{2} + 26873856 \) Copy content Toggle raw display
$53$ \( (T^{2} - 3888)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 5184 T^{2} + 26873856 \) Copy content Toggle raw display
$61$ \( (T^{2} - 43 T + 1849)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 162 T + 8748)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8100)^{2} \) Copy content Toggle raw display
$73$ \( (T + 107)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 126 T + 5292)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 1296 T^{2} + 1679616 \) Copy content Toggle raw display
$89$ \( (T^{2} - 12675)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 86 T + 7396)^{2} \) Copy content Toggle raw display
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