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Properties

Label 729.2.a.a
Level $729$
Weight $2$
Character orbit 729.a
Self dual yes
Analytic conductor $5.821$
Analytic rank $1$
Dimension $6$
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] = [729,2,Mod(1,729)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, 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("729.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.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: \(5.82109430735\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1397493.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_1) q^{2} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{4} - 1) q^{5} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{7} + (\beta_{5} + 2 \beta_{3} + \beta_{2} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_1) q^{2} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{4} - 1) q^{5} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{7} + (\beta_{5} + 2 \beta_{3} + \beta_{2} - 1) q^{8} + (2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{10} + (\beta_{4} - 2) q^{11} + ( - \beta_{4} + 2 \beta_1 - 1) q^{13} + (\beta_{5} - \beta_{3} + \beta_{2} - 1) q^{14} + (\beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{16} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{17} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{19} + ( - 2 \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{20} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1) q^{22} + (2 \beta_{5} + \beta_{4} - \beta_1 - 2) q^{23} + (2 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - 1) q^{25} + (2 \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_1 - 2) q^{26} + (2 \beta_{5} + \beta_{3} - 1) q^{28} + ( - \beta_{5} - 3 \beta_{3} + 2 \beta_1 - 3) q^{29} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{31} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2}) q^{32} + ( - \beta_{5} - 2 \beta_{4} - \beta_{2} + 2 \beta_1 - 1) q^{34} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2) q^{35} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{37} + ( - 5 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_1) q^{38} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{40} + (\beta_{5} - 3 \beta_{2} + \beta_1 - 3) q^{41} + ( - \beta_{5} + 3 \beta_{4} - \beta_{2} - 2 \beta_1 + 1) q^{43} + (2 \beta_{5} + 4 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 1) q^{44} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 1) q^{46} + ( - 2 \beta_{5} - \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{47} + ( - 4 \beta_{5} - \beta_{2} - 2 \beta_1 - 1) q^{49} + ( - 2 \beta_{4} - 3 \beta_{3} + 3 \beta_1 - 2) q^{50} + ( - 2 \beta_{5} - 2 \beta_{4} - 5 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{52} + ( - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_1 - 3) q^{53} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 1) q^{55} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{56} + (6 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{58} + (3 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 4) q^{59} + (2 \beta_{3} - \beta_{2} - 5 \beta_1 + 1) q^{61} + (4 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2) q^{62} + ( - 4 \beta_{5} + \beta_{4} + 2 \beta_{2} - 4 \beta_1) q^{64} + (\beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{65} + ( - \beta_{5} + 4 \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1) q^{67} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{68} + (2 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{70} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 5) q^{71} + ( - 6 \beta_{5} - 3 \beta_{4} - 1) q^{73} + (\beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 2 \beta_1 + 3) q^{74} + (4 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} - 2 \beta_1 + 2) q^{76} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{77} + ( - 6 \beta_{5} - \beta_{4} - 4 \beta_1 + 2) q^{79} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 4) q^{80} + ( - 3 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{82} + (5 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{83} + ( - 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{85} + ( - 6 \beta_{5} - 4 \beta_{4} - 5 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{86} + ( - 3 \beta_{5} - 5 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{88} + ( - 3 \beta_{5} - \beta_{3} + 4 \beta_{2} - \beta_1 - 1) q^{89} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_1 - 2) q^{91} + (4 \beta_{5} + 3 \beta_{4} + 6 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{92} + (2 \beta_{5} + \beta_{4} + 5 \beta_{3} + \beta_{2} + 1) q^{94} + ( - 3 \beta_{5} - 2 \beta_{4} + 3 \beta_{2} - 2) q^{95} + ( - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{97} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 3 \beta_1 + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8} + 3 q^{10} - 12 q^{11} - 6 q^{14} - 3 q^{16} - 9 q^{17} + 3 q^{19} - 6 q^{20} + 6 q^{22} - 15 q^{23} - 6 q^{25} - 15 q^{26} - 6 q^{28} - 12 q^{29} - 12 q^{35} + 3 q^{37} + 3 q^{38} + 6 q^{40} - 15 q^{41} - 3 q^{44} + 3 q^{46} - 21 q^{47} - 12 q^{49} - 3 q^{50} + 12 q^{52} - 9 q^{53} - 6 q^{55} + 6 q^{56} - 12 q^{58} - 24 q^{59} - 9 q^{61} + 12 q^{62} - 12 q^{64} + 6 q^{65} - 9 q^{67} + 9 q^{68} + 15 q^{70} - 27 q^{71} - 6 q^{73} + 12 q^{74} + 6 q^{76} + 12 q^{77} + 21 q^{80} - 6 q^{82} - 12 q^{83} + 21 q^{86} + 12 q^{88} - 9 q^{89} - 6 q^{91} - 6 q^{92} + 6 q^{94} - 12 q^{95} + 45 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 2\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 3\nu^{3} - \nu^{2} + 6\nu - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 3\nu^{4} - 3\nu^{3} + 9\nu^{2} + 4\nu - 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 3\beta_{3} + 7\beta_{2} + 7\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 3\beta_{4} + 12\beta_{3} + 18\beta_{2} + 20\beta _1 + 18 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.05432
0.584534
2.68091
−1.11662
−1.40162
0.198473
−2.40162 0 3.76778 0.0930834 0 −0.579861 −4.24555 0 −0.223551
1.2 −2.11662 0 2.48009 −2.68310 0 0.972333 −1.01617 0 5.67911
1.3 −0.801527 0 −1.35755 −2.74984 0 2.37683 2.69117 0 2.20407
1.4 −0.415466 0 −1.82739 2.21519 0 −1.31963 1.59015 0 −0.920335
1.5 1.05432 0 −0.888399 −1.74579 0 2.45925 −3.04531 0 −1.84063
1.6 1.68091 0 0.825466 −1.12954 0 −3.90892 −1.97429 0 −1.89866
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 729.2.a.a 6
3.b odd 2 1 729.2.a.d 6
9.c even 3 2 729.2.c.e 12
9.d odd 6 2 729.2.c.b 12
27.e even 9 2 27.2.e.a ✓ 12
27.e even 9 2 243.2.e.c 12
27.e even 9 2 243.2.e.d 12
27.f odd 18 2 81.2.e.a 12
27.f odd 18 2 243.2.e.a 12
27.f odd 18 2 243.2.e.b 12
108.j odd 18 2 432.2.u.c 12
135.p even 18 2 675.2.l.c 12
135.r odd 36 4 675.2.u.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.e.a ✓ 12 27.e even 9 2
81.2.e.a 12 27.f odd 18 2
243.2.e.a 12 27.f odd 18 2
243.2.e.b 12 27.f odd 18 2
243.2.e.c 12 27.e even 9 2
243.2.e.d 12 27.e even 9 2
432.2.u.c 12 108.j odd 18 2
675.2.l.c 12 135.p even 18 2
675.2.u.b 24 135.r odd 36 4
729.2.a.a 6 1.a even 1 1 trivial
729.2.a.d 6 3.b odd 2 1
729.2.c.b 12 9.d odd 6 2
729.2.c.e 12 9.c even 3 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3T_{2}^{5} - 3T_{2}^{4} - 12T_{2}^{3} + 9T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(729))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{5} - 3 T^{4} - 12 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 6 T^{5} + 6 T^{4} - 24 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$7$ \( T^{6} - 15 T^{4} + 11 T^{3} + 36 T^{2} + \cdots - 17 \) Copy content Toggle raw display
$11$ \( T^{6} + 12 T^{5} + 51 T^{4} + 96 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$13$ \( T^{6} - 24 T^{4} + 2 T^{3} + 90 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{6} + 9 T^{5} + 9 T^{4} - 54 T^{3} + \cdots + 27 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} - 30 T^{4} + 38 T^{3} + \cdots + 19 \) Copy content Toggle raw display
$23$ \( T^{6} + 15 T^{5} + 60 T^{4} + \cdots + 327 \) Copy content Toggle raw display
$29$ \( T^{6} + 12 T^{5} - 3 T^{4} - 462 T^{3} + \cdots - 213 \) Copy content Toggle raw display
$31$ \( T^{6} - 51 T^{4} + 191 T^{3} + \cdots + 163 \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} - 57 T^{4} + \cdots + 4933 \) Copy content Toggle raw display
$41$ \( T^{6} + 15 T^{5} + 6 T^{4} + \cdots + 3351 \) Copy content Toggle raw display
$43$ \( T^{6} - 96 T^{4} + 173 T^{3} + \cdots + 1819 \) Copy content Toggle raw display
$47$ \( T^{6} + 21 T^{5} + 105 T^{4} + \cdots + 6537 \) Copy content Toggle raw display
$53$ \( T^{6} + 9 T^{5} - 108 T^{4} + \cdots - 12393 \) Copy content Toggle raw display
$59$ \( T^{6} + 24 T^{5} + 141 T^{4} + \cdots - 13281 \) Copy content Toggle raw display
$61$ \( T^{6} + 9 T^{5} - 159 T^{4} + \cdots + 16543 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} - 114 T^{4} + \cdots - 2879 \) Copy content Toggle raw display
$71$ \( T^{6} + 27 T^{5} + 225 T^{4} + \cdots + 27 \) Copy content Toggle raw display
$73$ \( T^{6} + 6 T^{5} - 174 T^{4} + \cdots - 431 \) Copy content Toggle raw display
$79$ \( T^{6} - 177 T^{4} - 70 T^{3} + \cdots + 1873 \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{5} - 165 T^{4} + \cdots - 83373 \) Copy content Toggle raw display
$89$ \( T^{6} + 9 T^{5} - 180 T^{4} + \cdots + 32589 \) Copy content Toggle raw display
$97$ \( T^{6} - 204 T^{4} + 713 T^{3} + \cdots - 8171 \) Copy content Toggle raw display
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