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Properties

Label 1539.2.a.n
Level $1539$
Weight $2$
Character orbit 1539.a
Self dual yes
Analytic conductor $12.289$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1539,2,Mod(1,1539)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1539.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1539, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1539 = 3^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1539.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,-3,0,9,-9,0,0,-6,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.2889768711\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 9x^{7} + 30x^{6} + 21x^{5} - 93x^{4} + 3x^{3} + 87x^{2} - 27x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 171)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{8} - 1) q^{5} - \beta_{5} q^{7} + ( - \beta_{3} - \beta_{2} - 1) q^{8} + (2 \beta_{8} + \beta_{6} + \beta_{5} + \cdots - 1) q^{10} + ( - \beta_{6} - \beta_{2} - 1) q^{11}+ \cdots + ( - 4 \beta_{8} - 2 \beta_{6} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} + 9 q^{4} - 9 q^{5} - 6 q^{8} - 12 q^{11} - 3 q^{14} - 3 q^{16} - 12 q^{17} + 9 q^{19} - 18 q^{20} - 15 q^{23} + 18 q^{25} - 18 q^{26} - 24 q^{29} - 3 q^{31} - 15 q^{32} - 9 q^{35} - 12 q^{37}+ \cdots - 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 9x^{7} + 30x^{6} + 21x^{5} - 93x^{4} + 3x^{3} + 87x^{2} - 27x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - \nu^{4} - 6\nu^{3} + 4\nu^{2} + 6\nu - 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{8} + 3\nu^{7} + 9\nu^{6} - 27\nu^{5} - 27\nu^{4} + 72\nu^{3} + 30\nu^{2} - 51\nu - 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{8} + 3\nu^{7} + 12\nu^{6} - 30\nu^{5} - 48\nu^{4} + 87\nu^{3} + 63\nu^{2} - 66\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2\nu^{8} - 3\nu^{7} - 21\nu^{6} + 27\nu^{5} + 72\nu^{4} - 75\nu^{3} - 84\nu^{2} + 60\nu + 15 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + \beta_{4} + 7\beta_{3} + 8\beta_{2} + 19\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} + 8\beta_{4} + 9\beta_{3} + 34\beta_{2} + 11\beta _1 + 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{6} + 10\beta_{5} + 11\beta_{4} + 43\beta_{3} + 55\beta_{2} + 98\beta _1 + 76 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3\beta_{8} + 12\beta_{7} - 9\beta_{6} + 12\beta_{5} + 51\beta_{4} + 66\beta_{3} + 195\beta_{2} + 90\beta _1 + 387 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
2.49345
2.41069
1.81915
1.08377
0.722229
−0.205071
−1.21699
−1.99041
−2.11682
−2.49345 0 4.21730 −3.97421 0 −2.54380 −5.52873 0 9.90951
1.2 −2.41069 0 3.81141 2.07776 0 0.704790 −4.36674 0 −5.00884
1.3 −1.81915 0 1.30932 −1.69175 0 4.99773 1.25645 0 3.07756
1.4 −1.08377 0 −0.825435 2.29969 0 −1.67870 3.06213 0 −2.49234
1.5 −0.722229 0 −1.47839 −3.67316 0 −2.08390 2.51219 0 2.65286
1.6 0.205071 0 −1.95795 −0.975354 0 3.01260 −0.811659 0 −0.200016
1.7 1.21699 0 −0.518931 1.70908 0 −2.57390 −3.06552 0 2.07993
1.8 1.99041 0 1.96175 −0.655807 0 −2.28172 −0.0761342 0 −1.30533
1.9 2.11682 0 2.48092 −4.11625 0 2.44691 1.01801 0 −8.71334
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(19\) \( -1 \)

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 1539.2.a.n 9
3.b odd 2 1 1539.2.a.q 9
9.c even 3 2 513.2.e.b 18
9.d odd 6 2 171.2.e.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.e.a ✓ 18 9.d odd 6 2
513.2.e.b 18 9.c even 3 2
1539.2.a.n 9 1.a even 1 1 trivial
1539.2.a.q 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1539))\):

\( T_{2}^{9} + 3T_{2}^{8} - 9T_{2}^{7} - 30T_{2}^{6} + 21T_{2}^{5} + 93T_{2}^{4} + 3T_{2}^{3} - 87T_{2}^{2} - 27T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{9} + 9T_{5}^{8} + 9T_{5}^{7} - 108T_{5}^{6} - 216T_{5}^{5} + 402T_{5}^{4} + 966T_{5}^{3} - 294T_{5}^{2} - 1278T_{5} - 531 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 3 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 9 T^{8} + \cdots - 531 \) Copy content Toggle raw display
$7$ \( T^{9} - 33 T^{7} + \cdots + 1357 \) Copy content Toggle raw display
$11$ \( T^{9} + 12 T^{8} + \cdots + 405 \) Copy content Toggle raw display
$13$ \( T^{9} - 54 T^{7} + \cdots + 775 \) Copy content Toggle raw display
$17$ \( T^{9} + 12 T^{8} + \cdots - 75492 \) Copy content Toggle raw display
$19$ \( (T - 1)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} + 15 T^{8} + \cdots - 168327 \) Copy content Toggle raw display
$29$ \( T^{9} + 24 T^{8} + \cdots - 126963 \) Copy content Toggle raw display
$31$ \( T^{9} + 3 T^{8} + \cdots - 290051 \) Copy content Toggle raw display
$37$ \( T^{9} + 12 T^{8} + \cdots + 189172 \) Copy content Toggle raw display
$41$ \( T^{9} + 24 T^{8} + \cdots - 7447941 \) Copy content Toggle raw display
$43$ \( T^{9} + 6 T^{8} + \cdots + 635263 \) Copy content Toggle raw display
$47$ \( T^{9} + 15 T^{8} + \cdots - 53199 \) Copy content Toggle raw display
$53$ \( T^{9} + 21 T^{8} + \cdots + 19116 \) Copy content Toggle raw display
$59$ \( T^{9} + 9 T^{8} + \cdots + 86085 \) Copy content Toggle raw display
$61$ \( T^{9} + 6 T^{8} + \cdots - 6801671 \) Copy content Toggle raw display
$67$ \( T^{9} + 3 T^{8} + \cdots - 44257793 \) Copy content Toggle raw display
$71$ \( T^{9} - 267 T^{7} + \cdots - 5854032 \) Copy content Toggle raw display
$73$ \( T^{9} - 12 T^{8} + \cdots + 2073724 \) Copy content Toggle raw display
$79$ \( T^{9} - 6 T^{8} + \cdots - 36447629 \) Copy content Toggle raw display
$83$ \( T^{9} + 6 T^{8} + \cdots + 999135 \) Copy content Toggle raw display
$89$ \( T^{9} + 12 T^{8} + \cdots - 14801940 \) Copy content Toggle raw display
$97$ \( T^{9} + 6 T^{8} + \cdots + 100793647 \) Copy content Toggle raw display
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