Properties

Label 1296.3.o.e
Level $1296$
Weight $3$
Character orbit 1296.o
Analytic conductor $35.313$
Analytic rank $0$
Dimension $2$
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] = [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: \(2\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 432)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \zeta_{6} - 3) q^{5} + ( - \zeta_{6} + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \zeta_{6} - 3) q^{5} + ( - \zeta_{6} + 2) q^{7} + ( - 3 \zeta_{6} + 6) q^{11} + (4 \zeta_{6} - 4) q^{13} + 24 q^{17} + (4 \zeta_{6} - 2) q^{19} + ( - 18 \zeta_{6} - 18) q^{23} + 16 \zeta_{6} q^{25} - 42 \zeta_{6} q^{29} + (21 \zeta_{6} + 21) q^{31} + (6 \zeta_{6} - 3) q^{35} - 10 q^{37} + (54 \zeta_{6} - 54) q^{41} + ( - 30 \zeta_{6} + 60) q^{43} + ( - 30 \zeta_{6} + 60) q^{47} + (46 \zeta_{6} - 46) q^{49} + 45 q^{53} + (18 \zeta_{6} - 9) q^{55} + (12 \zeta_{6} + 12) q^{59} + 16 \zeta_{6} q^{61} - 12 \zeta_{6} q^{65} + (58 \zeta_{6} + 58) q^{67} + (96 \zeta_{6} - 48) q^{71} + q^{73} + ( - 9 \zeta_{6} + 9) q^{77} + ( - 68 \zeta_{6} + 136) q^{79} + (57 \zeta_{6} - 114) q^{83} + (72 \zeta_{6} - 72) q^{85} - 78 q^{89} + (8 \zeta_{6} - 4) q^{91} + ( - 6 \zeta_{6} - 6) q^{95} + 137 \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} + 3 q^{7} + 9 q^{11} - 4 q^{13} + 48 q^{17} - 54 q^{23} + 16 q^{25} - 42 q^{29} + 63 q^{31} - 20 q^{37} - 54 q^{41} + 90 q^{43} + 90 q^{47} - 46 q^{49} + 90 q^{53} + 36 q^{59} + 16 q^{61} - 12 q^{65} + 174 q^{67} + 2 q^{73} + 9 q^{77} + 204 q^{79} - 171 q^{83} - 72 q^{85} - 156 q^{89} - 18 q^{95} + 137 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −1.50000 + 2.59808i 0 1.50000 0.866025i 0 0 0
703.1 0 0 0 −1.50000 2.59808i 0 1.50000 + 0.866025i 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
36.f 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.o.e 2
3.b odd 2 1 1296.3.o.m 2
4.b odd 2 1 1296.3.o.d 2
9.c even 3 1 432.3.g.d yes 2
9.c even 3 1 1296.3.o.d 2
9.d odd 6 1 432.3.g.a 2
9.d odd 6 1 1296.3.o.l 2
12.b even 2 1 1296.3.o.l 2
36.f odd 6 1 432.3.g.d yes 2
36.f odd 6 1 inner 1296.3.o.e 2
36.h even 6 1 432.3.g.a 2
36.h even 6 1 1296.3.o.m 2
72.j odd 6 1 1728.3.g.e 2
72.l even 6 1 1728.3.g.e 2
72.n even 6 1 1728.3.g.b 2
72.p odd 6 1 1728.3.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.3.g.a 2 9.d odd 6 1
432.3.g.a 2 36.h even 6 1
432.3.g.d yes 2 9.c even 3 1
432.3.g.d yes 2 36.f odd 6 1
1296.3.o.d 2 4.b odd 2 1
1296.3.o.d 2 9.c even 3 1
1296.3.o.e 2 1.a even 1 1 trivial
1296.3.o.e 2 36.f odd 6 1 inner
1296.3.o.l 2 9.d odd 6 1
1296.3.o.l 2 12.b even 2 1
1296.3.o.m 2 3.b odd 2 1
1296.3.o.m 2 36.h even 6 1
1728.3.g.b 2 72.n even 6 1
1728.3.g.b 2 72.p odd 6 1
1728.3.g.e 2 72.j odd 6 1
1728.3.g.e 2 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 \(S_{3}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 9T_{11} + 27 \) 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
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$11$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$17$ \( (T - 24)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 54T + 972 \) Copy content Toggle raw display
$29$ \( T^{2} + 42T + 1764 \) Copy content Toggle raw display
$31$ \( T^{2} - 63T + 1323 \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 54T + 2916 \) Copy content Toggle raw display
$43$ \( T^{2} - 90T + 2700 \) Copy content Toggle raw display
$47$ \( T^{2} - 90T + 2700 \) Copy content Toggle raw display
$53$ \( (T - 45)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 36T + 432 \) Copy content Toggle raw display
$61$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$67$ \( T^{2} - 174T + 10092 \) Copy content Toggle raw display
$71$ \( T^{2} + 6912 \) Copy content Toggle raw display
$73$ \( (T - 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 204T + 13872 \) Copy content Toggle raw display
$83$ \( T^{2} + 171T + 9747 \) Copy content Toggle raw display
$89$ \( (T + 78)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 137T + 18769 \) Copy content Toggle raw display
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