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Properties

Label 539.2.q.d
Level $539$
Weight $2$
Character orbit 539.q
Analytic conductor $4.304$
Analytic rank $0$
Dimension $16$
CM discriminant -7
Inner twists $8$

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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] = [539,2,Mod(214,539)] 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("539.214"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(539, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 12])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.q (of order \(15\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: 16.0.9234096523681640625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 2 x^{14} - 5 x^{13} + 5 x^{12} + x^{11} + 6 x^{10} + 5 x^{9} - 21 x^{8} + 10 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{15}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{15} + \beta_{13} + \cdots + \beta_1) q^{2} + ( - \beta_{15} + \beta_{11} + \cdots - \beta_{3}) q^{4} + ( - 3 \beta_{14} + \beta_{12} + \cdots + 1) q^{8} + 3 \beta_{3} q^{9} + (2 \beta_{15} - 2 \beta_{13} + \cdots - 2 \beta_1) q^{11}+ \cdots + (6 \beta_{12} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} - 2 q^{4} + 32 q^{8} - 6 q^{9} + 4 q^{11} + 28 q^{16} + 9 q^{18} - 8 q^{22} + 16 q^{23} - 10 q^{25} - 8 q^{29} - 100 q^{32} - 12 q^{36} + 18 q^{37} + 48 q^{43} - 9 q^{44} + 31 q^{46} + 20 q^{50}+ \cdots - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 2 x^{14} - 5 x^{13} + 5 x^{12} + x^{11} + 6 x^{10} + 5 x^{9} - 21 x^{8} + 10 x^{7} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{15} + 3 \nu^{14} + 261 \nu^{12} - 7 \nu^{11} - 11 \nu^{10} + 16 \nu^{9} + 11 \nu^{8} + \cdots - 256 ) / 11392 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + \nu^{14} + 128 \nu^{13} - \nu^{12} - 5 \nu^{11} + 11 \nu^{10} + 8 \nu^{9} + 17 \nu^{8} + \cdots + 128 ) / 11392 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{15} - 89\nu^{10} - 979\nu^{5} - 1024 ) / 2848 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2 \nu^{15} - 2 \nu^{14} + 11 \nu^{13} + 2 \nu^{12} + 10 \nu^{11} - 22 \nu^{10} - 16 \nu^{9} + \cdots - 256 ) / 712 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 45 \nu^{15} - 135 \nu^{14} - 353 \nu^{12} + 315 \nu^{11} + 495 \nu^{10} - 720 \nu^{9} - 495 \nu^{8} + \cdots + 11520 ) / 11392 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 91 \nu^{15} - 37 \nu^{14} - 182 \nu^{13} + 455 \nu^{12} - 455 \nu^{11} - 91 \nu^{10} + \cdots + 11648 ) / 11392 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2 \nu^{15} - 2 \nu^{14} + 11 \nu^{13} + 2 \nu^{12} + 10 \nu^{11} - 22 \nu^{10} - 16 \nu^{9} + \cdots - 256 ) / 356 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 91 \nu^{15} - 91 \nu^{14} - 256 \nu^{13} + 91 \nu^{12} + 455 \nu^{11} - 1001 \nu^{10} + \cdots - 11648 ) / 11392 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 23 \nu^{15} + 46 \nu^{14} - 115 \nu^{13} + 115 \nu^{12} - 422 \nu^{11} + 138 \nu^{10} + \cdots + 5888 ) / 5696 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 33 \nu^{15} - 57 \nu^{14} + 66 \nu^{13} - 165 \nu^{12} + 165 \nu^{11} + 33 \nu^{10} - 514 \nu^{9} + \cdots - 4224 ) / 5696 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{15} + 93 ) / 89 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -17\nu^{15} - 267\nu^{10} - 1513\nu^{5} - 5984 ) / 1424 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - 5 \nu^{13} + 5 \nu^{12} + \nu^{11} + 6 \nu^{10} + 5 \nu^{9} - 21 \nu^{8} + \cdots + 256 ) / 178 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4 \nu^{15} - 15 \nu^{14} + 8 \nu^{13} - 20 \nu^{12} + 20 \nu^{11} + 4 \nu^{10} - 65 \nu^{9} + \cdots - 512 ) / 712 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{15} + \beta_{13} - 2 \beta_{10} - \beta_{9} + 2 \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - 2\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{15} + 2\beta_{14} + 2\beta_{10} - 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + \beta_{12} - 6\beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{14} - 5\beta_{12} - 5\beta_{11} + 2\beta_{10} + 5\beta_{8} + 5\beta_{6} - 5\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7 \beta_{15} - 7 \beta_{13} + 10 \beta_{10} + 7 \beta_{9} - 10 \beta_{7} - 10 \beta_{6} + 10 \beta_{5} + \cdots - 7 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -3\beta_{9} - 3\beta_{8} + 14\beta_{5} - 17\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -6\beta_{14} - 17\beta_{11} - 6\beta_{10} + 6\beta_{6} - 6\beta_{5} - 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -11\beta_{13} + 34\beta_{4} - 34 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 23\beta_{12} + 23\beta_{11} - 22\beta_{10} - 23\beta_{8} - 23\beta_{6} + 23\beta_{2} - 23\beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( \beta_{6} + 45\beta_{2} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( \beta_{9} + 91\beta_{3} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -91\beta_{15} + 91\beta_{11} + 2\beta_{7} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -89\beta_{12} + 93 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(442\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(\beta_{10} + \beta_{14}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1
−0.0812893 + 1.41188i
0.994835 − 1.00514i
−0.648523 − 1.25675i
1.31765 + 0.513604i
−1.36789 − 0.358983i
1.26336 − 0.635539i
−1.36789 + 0.358983i
1.26336 + 0.635539i
−0.214032 + 1.39792i
−0.764115 − 1.19001i
−0.214032 − 1.39792i
−0.764115 + 1.19001i
−0.0812893 − 1.41188i
0.994835 + 1.00514i
−0.648523 + 1.25675i
1.31765 − 0.513604i
−1.58193 + 1.75691i 0 −0.375174 − 3.56955i 0 0 0 3.03958 + 2.20838i −2.00739 + 2.22943i 0
214.2 0.499249 − 0.554473i 0 0.150867 + 1.43540i 0 0 0 2.07845 + 1.51009i −2.00739 + 2.22943i 0
312.1 −0.295322 + 2.80980i 0 −5.85146 − 1.24377i 0 0 0 3.47668 − 10.7001i 0.313585 − 2.98357i 0
312.2 0.230720 − 2.19515i 0 −2.80916 − 0.597105i 0 0 0 −0.594713 + 1.83034i 0.313585 − 2.98357i 0
324.1 −0.729812 − 0.155126i 0 −1.31853 − 0.587047i 0 0 0 2.07845 + 1.51009i 2.93444 + 0.623735i 0
324.2 2.31249 + 0.491535i 0 3.27890 + 1.45986i 0 0 0 3.03958 + 2.20838i 2.93444 + 0.623735i 0
361.1 −0.729812 + 0.155126i 0 −1.31853 + 0.587047i 0 0 0 2.07845 − 1.51009i 2.93444 − 0.623735i 0
361.2 2.31249 − 0.491535i 0 3.27890 − 1.45986i 0 0 0 3.03958 − 2.20838i 2.93444 − 0.623735i 0
410.1 −2.01642 − 0.897766i 0 1.92169 + 2.13425i 0 0 0 −0.594713 − 1.83034i −2.74064 − 1.22021i 0
410.2 2.58102 + 1.14914i 0 4.00286 + 4.44563i 0 0 0 3.47668 + 10.7001i −2.74064 − 1.22021i 0
422.1 −2.01642 + 0.897766i 0 1.92169 − 2.13425i 0 0 0 −0.594713 + 1.83034i −2.74064 + 1.22021i 0
422.2 2.58102 − 1.14914i 0 4.00286 − 4.44563i 0 0 0 3.47668 − 10.7001i −2.74064 + 1.22021i 0
471.1 −1.58193 − 1.75691i 0 −0.375174 + 3.56955i 0 0 0 3.03958 − 2.20838i −2.00739 − 2.22943i 0
471.2 0.499249 + 0.554473i 0 0.150867 − 1.43540i 0 0 0 2.07845 − 1.51009i −2.00739 − 2.22943i 0
520.1 −0.295322 − 2.80980i 0 −5.85146 + 1.24377i 0 0 0 3.47668 + 10.7001i 0.313585 + 2.98357i 0
520.2 0.230720 + 2.19515i 0 −2.80916 + 0.597105i 0 0 0 −0.594713 − 1.83034i 0.313585 + 2.98357i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 214.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
7.c even 3 1 inner
7.d odd 6 1 inner
11.c even 5 1 inner
77.j odd 10 1 inner
77.m even 15 1 inner
77.p odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.q.d 16
7.b odd 2 1 CM 539.2.q.d 16
7.c even 3 1 539.2.f.c ✓ 8
7.c even 3 1 inner 539.2.q.d 16
7.d odd 6 1 539.2.f.c ✓ 8
7.d odd 6 1 inner 539.2.q.d 16
11.c even 5 1 inner 539.2.q.d 16
77.j odd 10 1 inner 539.2.q.d 16
77.m even 15 1 539.2.f.c ✓ 8
77.m even 15 1 inner 539.2.q.d 16
77.m even 15 1 5929.2.a.bc 4
77.n even 30 1 5929.2.a.bg 4
77.o odd 30 1 5929.2.a.bg 4
77.p odd 30 1 539.2.f.c ✓ 8
77.p odd 30 1 inner 539.2.q.d 16
77.p odd 30 1 5929.2.a.bc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.f.c ✓ 8 7.c even 3 1
539.2.f.c ✓ 8 7.d odd 6 1
539.2.f.c ✓ 8 77.m even 15 1
539.2.f.c ✓ 8 77.p odd 30 1
539.2.q.d 16 1.a even 1 1 trivial
539.2.q.d 16 7.b odd 2 1 CM
539.2.q.d 16 7.c even 3 1 inner
539.2.q.d 16 7.d odd 6 1 inner
539.2.q.d 16 11.c even 5 1 inner
539.2.q.d 16 77.j odd 10 1 inner
539.2.q.d 16 77.m even 15 1 inner
539.2.q.d 16 77.p odd 30 1 inner
5929.2.a.bc 4 77.m even 15 1
5929.2.a.bc 4 77.p odd 30 1
5929.2.a.bg 4 77.n even 30 1
5929.2.a.bg 4 77.o odd 30 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\):

\( T_{2}^{16} - 2 T_{2}^{15} + T_{2}^{14} - 18 T_{2}^{13} + 6 T_{2}^{12} + 64 T_{2}^{11} + 389 T_{2}^{10} + \cdots + 14641 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{15} + \cdots + 14641 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( (T^{8} - 8 T^{7} + \cdots + 383161)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 4 T^{7} + \cdots + 13256881)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 4810832476321 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{4} - 12 T^{3} + \cdots - 979)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 36325030350625 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( (T^{8} - 4 T^{7} + \cdots + 300710281)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 48 T^{7} + \cdots + 19321)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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