Properties

Label 243.2.c.e
Level $243$
Weight $2$
Character orbit 243.c
Analytic conductor $1.940$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [243,2,Mod(82,243)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("243.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 243.c (of order \(3\), 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: \(1.94036476912\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_1 - 1) q^{2} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1) q^{5} + ( - \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{4} + \beta_{3} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_1 - 1) q^{2} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1) q^{5} + ( - \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{4} + \beta_{3} + 2) q^{8} + ( - \beta_{4} - 2 \beta_{3}) q^{10} + ( - 2 \beta_{5} + 3 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{11} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{13} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + \beta_1) q^{14} + (3 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{16} + 3 q^{17} + (3 \beta_{3} - 1) q^{19} + (2 \beta_{5} - \beta_1 + 1) q^{20} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 5 \beta_{2}) q^{22} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{23} + ( - 4 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{25} + (\beta_{4} + \beta_{3} - 4) q^{26} + (6 \beta_{4} + 3 \beta_{3} - 4) q^{28} + (\beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 4) q^{29} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{31} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3}) q^{32} + (3 \beta_{5} + 3 \beta_1 - 3) q^{34} + ( - \beta_{4} - 4 \beta_{3} - 2) q^{35} + ( - 3 \beta_{4} - 3 \beta_{3} - 1) q^{37} + (2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 4) q^{38} + (\beta_{5} - \beta_{4} - \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{40} + ( - 5 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + \beta_1) q^{41} + (2 \beta_{5} - \beta_{3} + \beta_{2} - 4 \beta_1 + 4) q^{43} + (5 \beta_{4} + 2 \beta_{3} - 5) q^{44} + ( - 7 \beta_{4} - 2 \beta_{3} + 3) q^{46} + ( - 5 \beta_{5} - 2 \beta_1 + 2) q^{47} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{49} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 8 \beta_1) q^{50} + ( - 3 \beta_{5} + 3 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 4) q^{52} + ( - 3 \beta_{3} + 6) q^{53} + (2 \beta_{4} - 2 \beta_{3} + 3) q^{55} + ( - 5 \beta_{5} + 3 \beta_{3} - 3 \beta_{2} - 11 \beta_1 + 11) q^{56} + ( - 8 \beta_{5} + 8 \beta_{4} + 8 \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{58} + (\beta_{5} - \beta_{4} - \beta_{3} + 7 \beta_1) q^{59} + ( - \beta_{5} + 5 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 2) q^{61} + (7 \beta_{4} + 4 \beta_{3} - 4) q^{62} + (3 \beta_{4} + 4) q^{64} + ( - 2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{65} + ( - 5 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - \beta_{2} - 2 \beta_1) q^{67} + ( - 6 \beta_{5} + 6 \beta_{4} + 6 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{68} + ( - 4 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{70} + ( - 9 \beta_{4} - 6 \beta_{3} - 3) q^{71} + (6 \beta_{4} + 3 \beta_{3} + 2) q^{73} + (2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 5) q^{74} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{76} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 8 \beta_1) q^{77} + (5 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{79} + ( - 2 \beta_{4} + \beta_{3} - 1) q^{80} + ( - 7 \beta_{4} + \beta_{3} + 6) q^{82} + (\beta_{5} + 3 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 2) q^{83} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 6 \beta_1) q^{85} + (\beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_1) q^{86} + ( - 5 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - 12 \beta_1 + 12) q^{88} + (9 \beta_{4} + 3 \beta_{3}) q^{89} + ( - 5 \beta_{4} - \beta_{3} - 2) q^{91} + (11 \beta_{5} - 6 \beta_{3} + 6 \beta_{2} + 8 \beta_1 - 8) q^{92} + (7 \beta_{5} - 7 \beta_{4} - 7 \beta_{3} + 5 \beta_{2} + 12 \beta_1) q^{94} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 6 \beta_{2} - \beta_1) q^{95} + (8 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} + 5 \beta_1 - 5) q^{97} + ( - 3 \beta_{4} + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 3 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 3 q^{7} + 12 q^{8} - 3 q^{11} + 3 q^{13} + 3 q^{14} - 3 q^{16} + 18 q^{17} - 6 q^{19} + 3 q^{20} - 6 q^{23} - 3 q^{25} - 24 q^{26} - 24 q^{28} - 12 q^{29} + 12 q^{31} - 9 q^{34} - 12 q^{35} - 6 q^{37} + 12 q^{38} - 9 q^{40} + 3 q^{41} + 12 q^{43} - 30 q^{44} + 18 q^{46} + 6 q^{47} + 24 q^{50} + 12 q^{52} + 36 q^{53} + 18 q^{55} + 33 q^{56} - 9 q^{58} + 21 q^{59} - 6 q^{61} - 24 q^{62} + 24 q^{64} - 3 q^{65} - 6 q^{67} - 9 q^{68} - 9 q^{70} - 18 q^{71} + 12 q^{73} - 15 q^{74} + 3 q^{76} - 24 q^{77} - 6 q^{79} - 6 q^{80} + 36 q^{82} + 6 q^{83} - 18 q^{85} + 3 q^{86} + 36 q^{88} - 12 q^{91} - 24 q^{92} + 36 q^{94} - 3 q^{95} - 15 q^{97} + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{18}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{18}^{5} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{18}^{5} + \zeta_{18}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \) Copy content Toggle raw display
\(\zeta_{18}\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{2}\)\(=\) \( ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{18}^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{5}\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
−0.766044 0.642788i
−0.173648 + 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
−0.173648 0.984808i
0.939693 + 0.342020i
−1.26604 2.19285i 0 −2.20574 + 3.82045i −0.233956 + 0.405223i 0 1.61334 + 2.79439i 6.10607 0 1.18479
82.2 −0.673648 1.16679i 0 0.0923963 0.160035i −0.826352 + 1.43128i 0 −1.20574 2.08840i −2.94356 0 2.22668
82.3 0.439693 + 0.761570i 0 0.613341 1.06234i −1.93969 + 3.35965i 0 1.09240 + 1.89209i 2.83750 0 −3.41147
163.1 −1.26604 + 2.19285i 0 −2.20574 3.82045i −0.233956 0.405223i 0 1.61334 2.79439i 6.10607 0 1.18479
163.2 −0.673648 + 1.16679i 0 0.0923963 + 0.160035i −0.826352 1.43128i 0 −1.20574 + 2.08840i −2.94356 0 2.22668
163.3 0.439693 0.761570i 0 0.613341 + 1.06234i −1.93969 3.35965i 0 1.09240 1.89209i 2.83750 0 −3.41147
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 243.2.c.e 6
3.b odd 2 1 243.2.c.f 6
9.c even 3 1 243.2.a.f yes 3
9.c even 3 1 inner 243.2.c.e 6
9.d odd 6 1 243.2.a.e 3
9.d odd 6 1 243.2.c.f 6
27.e even 9 2 729.2.e.b 6
27.e even 9 2 729.2.e.c 6
27.e even 9 2 729.2.e.i 6
27.f odd 18 2 729.2.e.a 6
27.f odd 18 2 729.2.e.g 6
27.f odd 18 2 729.2.e.h 6
36.f odd 6 1 3888.2.a.bk 3
36.h even 6 1 3888.2.a.bd 3
45.h odd 6 1 6075.2.a.bv 3
45.j even 6 1 6075.2.a.bq 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.e 3 9.d odd 6 1
243.2.a.f yes 3 9.c even 3 1
243.2.c.e 6 1.a even 1 1 trivial
243.2.c.e 6 9.c even 3 1 inner
243.2.c.f 6 3.b odd 2 1
243.2.c.f 6 9.d odd 6 1
729.2.e.a 6 27.f odd 18 2
729.2.e.b 6 27.e even 9 2
729.2.e.c 6 27.e even 9 2
729.2.e.g 6 27.f odd 18 2
729.2.e.h 6 27.f odd 18 2
729.2.e.i 6 27.e even 9 2
3888.2.a.bd 3 36.h even 6 1
3888.2.a.bk 3 36.f odd 6 1
6075.2.a.bq 3 45.j even 6 1
6075.2.a.bv 3 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(243, [\chi])\):

\( T_{2}^{6} + 3T_{2}^{5} + 9T_{2}^{4} + 6T_{2}^{3} + 9T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} - 3T_{7}^{5} + 15T_{7}^{4} - 16T_{7}^{3} + 87T_{7}^{2} - 102T_{7} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{5} + 9 T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 6 T^{5} + 27 T^{4} + 48 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + 15 T^{4} - 16 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} + 27 T^{4} - 48 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} - 3 T^{5} + 15 T^{4} - 16 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$17$ \( (T - 3)^{6} \) Copy content Toggle raw display
$19$ \( (T^{3} + 3 T^{2} - 24 T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + 45 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$29$ \( T^{6} + 12 T^{5} + 117 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$31$ \( T^{6} - 12 T^{5} + 105 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} - 24 T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 3 T^{5} + 63 T^{4} + \cdots + 47961 \) Copy content Toggle raw display
$43$ \( T^{6} - 12 T^{5} + 105 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + 99 T^{4} + \cdots + 71289 \) Copy content Toggle raw display
$53$ \( (T^{3} - 18 T^{2} + 81 T - 81)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 21 T^{5} + 297 T^{4} + \cdots + 103041 \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} + 87 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + 87 T^{4} + \cdots + 11881 \) Copy content Toggle raw display
$71$ \( (T^{3} + 9 T^{2} - 162 T - 999)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 6 T^{2} - 69 T + 397)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 6 T^{5} + 87 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$83$ \( T^{6} - 6 T^{5} + 63 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$89$ \( (T^{3} - 189 T + 999)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 15 T^{5} + 294 T^{4} + \cdots + 361 \) Copy content Toggle raw display
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