Properties

Label 243.2.a.e
Level $243$
Weight $2$
Character orbit 243.a
Self dual yes
Analytic conductor $1.940$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [243,2,Mod(1,243)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("243.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 243.a (trivial)

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: yes
Analytic conductor: \(1.94036476912\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.53209
−0.347296
1.87939
−2.53209 0 4.41147 −0.467911 0 −3.22668 −6.10607 0 1.18479
1.2 −1.34730 0 −0.184793 −1.65270 0 2.41147 2.94356 0 2.22668
1.3 0.879385 0 −1.22668 −3.87939 0 −2.18479 −2.83750 0 −3.41147
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 243.2.a.e 3
3.b odd 2 1 243.2.a.f yes 3
4.b odd 2 1 3888.2.a.bd 3
5.b even 2 1 6075.2.a.bv 3
9.c even 3 2 243.2.c.f 6
9.d odd 6 2 243.2.c.e 6
12.b even 2 1 3888.2.a.bk 3
15.d odd 2 1 6075.2.a.bq 3
27.e even 9 2 729.2.e.a 6
27.e even 9 2 729.2.e.g 6
27.e even 9 2 729.2.e.h 6
27.f odd 18 2 729.2.e.b 6
27.f odd 18 2 729.2.e.c 6
27.f odd 18 2 729.2.e.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.e 3 1.a even 1 1 trivial
243.2.a.f yes 3 3.b odd 2 1
243.2.c.e 6 9.d odd 6 2
243.2.c.f 6 9.c even 3 2
729.2.e.a 6 27.e even 9 2
729.2.e.b 6 27.f odd 18 2
729.2.e.c 6 27.f odd 18 2
729.2.e.g 6 27.e even 9 2
729.2.e.h 6 27.e even 9 2
729.2.e.i 6 27.f odd 18 2
3888.2.a.bd 3 4.b odd 2 1
3888.2.a.bk 3 12.b even 2 1
6075.2.a.bq 3 15.d odd 2 1
6075.2.a.bv 3 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(243))\):

\( T_{2}^{3} + 3T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{7}^{3} + 3T_{7}^{2} - 6T_{7} - 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 3T^{2} - 3 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 6 T^{2} + \cdots + 3 \) Copy content Toggle raw display
$7$ \( T^{3} + 3 T^{2} + \cdots - 17 \) Copy content Toggle raw display
$11$ \( T^{3} + 3 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$13$ \( T^{3} + 3 T^{2} + \cdots - 17 \) Copy content Toggle raw display
$17$ \( (T + 3)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 3 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} + \cdots - 51 \) Copy content Toggle raw display
$29$ \( T^{3} + 12 T^{2} + \cdots - 57 \) Copy content Toggle raw display
$31$ \( T^{3} + 12 T^{2} + \cdots + 19 \) Copy content Toggle raw display
$37$ \( T^{3} + 3 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{3} - 3 T^{2} + \cdots + 219 \) Copy content Toggle raw display
$43$ \( T^{3} + 12 T^{2} + \cdots + 19 \) Copy content Toggle raw display
$47$ \( T^{3} - 6 T^{2} + \cdots + 267 \) Copy content Toggle raw display
$53$ \( T^{3} + 18 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{3} - 21 T^{2} + \cdots - 321 \) Copy content Toggle raw display
$61$ \( T^{3} - 6 T^{2} + \cdots - 53 \) Copy content Toggle raw display
$67$ \( T^{3} - 6 T^{2} + \cdots + 109 \) Copy content Toggle raw display
$71$ \( T^{3} - 9 T^{2} + \cdots + 999 \) Copy content Toggle raw display
$73$ \( T^{3} - 6 T^{2} + \cdots + 397 \) Copy content Toggle raw display
$79$ \( T^{3} - 6 T^{2} + \cdots - 53 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} + \cdots + 51 \) Copy content Toggle raw display
$89$ \( T^{3} - 189T - 999 \) Copy content Toggle raw display
$97$ \( T^{3} - 15 T^{2} + \cdots + 19 \) Copy content Toggle raw display
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