Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,2,Mod(19,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([25, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.s (of order \(30\), degree \(8\), 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: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{30})\)
Coefficient field: 16.0.9234096523681640625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 2 x^{14} - 5 x^{13} + 5 x^{12} + x^{11} + 6 x^{10} + 5 x^{9} - 21 x^{8} + 10 x^{7} + 24 x^{6} + 8 x^{5} + 80 x^{4} - 160 x^{3} + 128 x^{2} - 128 x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{U}(1)[D_{30}]$

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

\(f(q)\) \(=\) \( q + (\beta_{14} + \beta_{13} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_1 + 1) q^{2} + (\beta_{15} - \beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{5} - 2 \beta_{3}) q^{4} + (2 \beta_{14} + \beta_{12} - \beta_{11} + 2 \beta_{10} - \beta_{8} + 3 \beta_{6} - 1) q^{8} - 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{14} + \beta_{13} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_1 + 1) q^{2} + (\beta_{15} - \beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{5} - 2 \beta_{3}) q^{4} + (2 \beta_{14} + \beta_{12} - \beta_{11} + 2 \beta_{10} - \beta_{8} + 3 \beta_{6} - 1) q^{8} - 3 \beta_{3} q^{9} + ( - 2 \beta_{15} + 2 \beta_{13} + \beta_{10} - 2 \beta_{9} - \beta_{7} + 2 \beta_{6} + \cdots + 2 \beta_1) q^{11}+ \cdots + (6 \beta_{12} + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 10 q^{4} - 20 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 10 q^{4} - 20 q^{8} + 6 q^{9} + 4 q^{11} - 8 q^{16} + 15 q^{18} - 28 q^{22} - 16 q^{23} + 10 q^{25} + 60 q^{36} + 18 q^{37} - 25 q^{44} - 15 q^{46} - 30 q^{53} - 19 q^{58} - 68 q^{64} - 8 q^{67} - 96 q^{71} + 75 q^{72} + 40 q^{79} + 18 q^{81} - 23 q^{86} + 8 q^{88} + 50 q^{92} + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 2 x^{14} - 5 x^{13} + 5 x^{12} + x^{11} + 6 x^{10} + 5 x^{9} - 21 x^{8} + 10 x^{7} + 24 x^{6} + 8 x^{5} + 80 x^{4} - 160 x^{3} + 128 x^{2} - 128 x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{15} + 3 \nu^{14} + 261 \nu^{12} - 7 \nu^{11} - 11 \nu^{10} + 16 \nu^{9} + 11 \nu^{8} + 55 \nu^{7} - 32 \nu^{6} - 88 \nu^{5} + 80 \nu^{4} + 32 \nu^{3} + 352 \nu^{2} - 128 \nu - 256 ) / 11392 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + \nu^{14} + 128 \nu^{13} - \nu^{12} - 5 \nu^{11} + 11 \nu^{10} + 8 \nu^{9} + 17 \nu^{8} - 11 \nu^{7} - 32 \nu^{6} + 44 \nu^{5} + 56 \nu^{4} + 96 \nu^{3} - 192 \nu + 128 ) / 11392 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{15} - 89\nu^{10} - 979\nu^{5} - 1024 ) / 2848 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2 \nu^{15} - 2 \nu^{14} + 11 \nu^{13} + 2 \nu^{12} + 10 \nu^{11} - 22 \nu^{10} - 16 \nu^{9} + 55 \nu^{8} + 22 \nu^{7} + 64 \nu^{6} - 88 \nu^{5} - 112 \nu^{4} + 75 \nu^{3} + 384 \nu - 256 ) / 712 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 45 \nu^{15} - 135 \nu^{14} - 353 \nu^{12} + 315 \nu^{11} + 495 \nu^{10} - 720 \nu^{9} - 495 \nu^{8} - 2475 \nu^{7} + 1440 \nu^{6} + 3960 \nu^{5} - 3600 \nu^{4} - 1440 \nu^{3} - 15840 \nu^{2} + \cdots + 11520 ) / 11392 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 91 \nu^{15} - 37 \nu^{14} - 182 \nu^{13} + 455 \nu^{12} - 455 \nu^{11} - 91 \nu^{10} - 546 \nu^{9} - 455 \nu^{8} + 1911 \nu^{7} - 910 \nu^{6} - 2184 \nu^{5} - 728 \nu^{4} - 7280 \nu^{3} + \cdots + 11648 ) / 11392 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2 \nu^{15} - 2 \nu^{14} + 11 \nu^{13} + 2 \nu^{12} + 10 \nu^{11} - 22 \nu^{10} - 16 \nu^{9} + 55 \nu^{8} + 22 \nu^{7} + 64 \nu^{6} - 88 \nu^{5} - 112 \nu^{4} + 431 \nu^{3} + 384 \nu - 256 ) / 356 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 91 \nu^{15} - 91 \nu^{14} - 256 \nu^{13} + 91 \nu^{12} + 455 \nu^{11} - 1001 \nu^{10} - 728 \nu^{9} - 1547 \nu^{8} + 1001 \nu^{7} + 2912 \nu^{6} - 4004 \nu^{5} - 5096 \nu^{4} + \cdots - 11648 ) / 11392 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 23 \nu^{15} + 46 \nu^{14} - 115 \nu^{13} + 115 \nu^{12} - 422 \nu^{11} + 138 \nu^{10} + 115 \nu^{9} - 483 \nu^{8} + 230 \nu^{7} - 1495 \nu^{6} + 184 \nu^{5} + 1840 \nu^{4} - 3680 \nu^{3} + \cdots + 5888 ) / 5696 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 33 \nu^{15} - 57 \nu^{14} + 66 \nu^{13} - 165 \nu^{12} + 165 \nu^{11} + 33 \nu^{10} - 514 \nu^{9} + 165 \nu^{8} - 693 \nu^{7} + 330 \nu^{6} + 792 \nu^{5} - 1872 \nu^{4} + 2640 \nu^{3} - 5280 \nu^{2} + \cdots - 4224 ) / 5696 \) 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
\(\beta_{14}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - 5 \nu^{13} + 5 \nu^{12} + \nu^{11} + 6 \nu^{10} + 5 \nu^{9} - 21 \nu^{8} + 10 \nu^{7} + 24 \nu^{6} + 8 \nu^{5} + 80 \nu^{4} - 160 \nu^{3} + 128 \nu^{2} - 35 \nu + 256 ) / 178 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4 \nu^{15} - 15 \nu^{14} + 8 \nu^{13} - 20 \nu^{12} + 20 \nu^{11} + 4 \nu^{10} - 65 \nu^{9} + 20 \nu^{8} - 84 \nu^{7} + 40 \nu^{6} + 96 \nu^{5} - 235 \nu^{4} + 320 \nu^{3} - 640 \nu^{2} + 512 \nu - 512 ) / 712 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{15} + \beta_{13} - 2 \beta_{10} - \beta_{9} + 2 \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) 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
\(\nu^{7}\)\(=\) \( 7 \beta_{15} - 7 \beta_{13} + 10 \beta_{10} + 7 \beta_{9} - 10 \beta_{7} - 10 \beta_{6} + 10 \beta_{5} + 10 \beta_{4} - 3 \beta_{3} - 7 \beta_1 \) 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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.648523 + 1.25675i
1.31765 0.513604i
0.994835 + 1.00514i
−0.0812893 1.41188i
−0.214032 1.39792i
−0.764115 + 1.19001i
−0.214032 + 1.39792i
−0.764115 1.19001i
−1.36789 + 0.358983i
1.26336 + 0.635539i
−1.36789 0.358983i
1.26336 0.635539i
−0.648523 1.25675i
1.31765 + 0.513604i
0.994835 1.00514i
−0.0812893 + 1.41188i
−1.75895 + 0.184873i 0 1.10343 0.234542i 0 0 0 1.46663 0.476537i −0.313585 2.98357i 0
19.2 −0.132743 + 0.0139519i 0 −1.93887 + 0.412119i 0 0 0 0.505505 0.164249i −0.313585 2.98357i 0
68.1 −1.15386 + 1.03894i 0 0.0429395 0.408542i 0 0 0 −1.45037 1.99627i 2.00739 + 2.22943i 0
68.2 2.02748 1.82555i 0 0.568981 5.41349i 0 0 0 −5.52176 7.60006i 2.00739 + 2.22943i 0
117.1 0.0542889 + 0.121935i 0 1.32634 1.47305i 0 0 0 0.505505 + 0.164249i 2.74064 1.22021i 0
117.2 0.719370 + 1.61573i 0 −0.754835 + 0.838329i 0 0 0 1.46663 + 0.476537i 2.74064 1.22021i 0
129.1 0.0542889 0.121935i 0 1.32634 + 1.47305i 0 0 0 0.505505 0.164249i 2.74064 + 1.22021i 0
129.2 0.719370 1.61573i 0 −0.754835 0.838329i 0 0 0 1.46663 0.476537i 2.74064 + 1.22021i 0
178.1 −0.322819 1.51874i 0 −0.375277 + 0.167084i 0 0 0 −1.45037 1.99627i −2.93444 + 0.623735i 0
178.2 0.567234 + 2.66862i 0 −4.97271 + 2.21399i 0 0 0 −5.52176 7.60006i −2.93444 + 0.623735i 0
215.1 −0.322819 + 1.51874i 0 −0.375277 0.167084i 0 0 0 −1.45037 + 1.99627i −2.93444 0.623735i 0
215.2 0.567234 2.66862i 0 −4.97271 2.21399i 0 0 0 −5.52176 + 7.60006i −2.93444 0.623735i 0
227.1 −1.75895 0.184873i 0 1.10343 + 0.234542i 0 0 0 1.46663 + 0.476537i −0.313585 + 2.98357i 0
227.2 −0.132743 0.0139519i 0 −1.93887 0.412119i 0 0 0 0.505505 + 0.164249i −0.313585 + 2.98357i 0
325.1 −1.15386 1.03894i 0 0.0429395 + 0.408542i 0 0 0 −1.45037 + 1.99627i 2.00739 2.22943i 0
325.2 2.02748 + 1.82555i 0 0.568981 + 5.41349i 0 0 0 −5.52176 + 7.60006i 2.00739 2.22943i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
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.d odd 10 1 inner
77.l even 10 1 inner
77.n even 30 1 inner
77.o 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.s.a 16
7.b odd 2 1 CM 539.2.s.a 16
7.c even 3 1 77.2.l.a 8
7.c even 3 1 inner 539.2.s.a 16
7.d odd 6 1 77.2.l.a 8
7.d odd 6 1 inner 539.2.s.a 16
11.d odd 10 1 inner 539.2.s.a 16
21.g even 6 1 693.2.bu.a 8
21.h odd 6 1 693.2.bu.a 8
77.h odd 6 1 847.2.l.c 8
77.i even 6 1 847.2.l.c 8
77.l even 10 1 inner 539.2.s.a 16
77.m even 15 1 847.2.b.b 8
77.m even 15 1 847.2.l.a 8
77.m even 15 1 847.2.l.c 8
77.m even 15 1 847.2.l.d 8
77.n even 30 1 77.2.l.a 8
77.n even 30 1 inner 539.2.s.a 16
77.n even 30 1 847.2.b.b 8
77.n even 30 1 847.2.l.a 8
77.n even 30 1 847.2.l.d 8
77.o odd 30 1 77.2.l.a 8
77.o odd 30 1 inner 539.2.s.a 16
77.o odd 30 1 847.2.b.b 8
77.o odd 30 1 847.2.l.a 8
77.o odd 30 1 847.2.l.d 8
77.p odd 30 1 847.2.b.b 8
77.p odd 30 1 847.2.l.a 8
77.p odd 30 1 847.2.l.c 8
77.p odd 30 1 847.2.l.d 8
231.be even 30 1 693.2.bu.a 8
231.bf odd 30 1 693.2.bu.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.l.a 8 7.c even 3 1
77.2.l.a 8 7.d odd 6 1
77.2.l.a 8 77.n even 30 1
77.2.l.a 8 77.o odd 30 1
539.2.s.a 16 1.a even 1 1 trivial
539.2.s.a 16 7.b odd 2 1 CM
539.2.s.a 16 7.c even 3 1 inner
539.2.s.a 16 7.d odd 6 1 inner
539.2.s.a 16 11.d odd 10 1 inner
539.2.s.a 16 77.l even 10 1 inner
539.2.s.a 16 77.n even 30 1 inner
539.2.s.a 16 77.o odd 30 1 inner
693.2.bu.a 8 21.g even 6 1
693.2.bu.a 8 21.h odd 6 1
693.2.bu.a 8 231.be even 30 1
693.2.bu.a 8 231.bf odd 30 1
847.2.b.b 8 77.m even 15 1
847.2.b.b 8 77.n even 30 1
847.2.b.b 8 77.o odd 30 1
847.2.b.b 8 77.p odd 30 1
847.2.l.a 8 77.m even 15 1
847.2.l.a 8 77.n even 30 1
847.2.l.a 8 77.o odd 30 1
847.2.l.a 8 77.p odd 30 1
847.2.l.c 8 77.h odd 6 1
847.2.l.c 8 77.i even 6 1
847.2.l.c 8 77.m even 15 1
847.2.l.c 8 77.p odd 30 1
847.2.l.d 8 77.m even 15 1
847.2.l.d 8 77.n even 30 1
847.2.l.d 8 77.o odd 30 1
847.2.l.d 8 77.p odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 7 T_{2}^{14} + 20 T_{2}^{13} + 30 T_{2}^{12} + 140 T_{2}^{11} + 367 T_{2}^{10} + 800 T_{2}^{9} + 1529 T_{2}^{8} + 2600 T_{2}^{7} + 3713 T_{2}^{6} + 4300 T_{2}^{5} + 3770 T_{2}^{4} + 530 T_{2}^{3} + 33 T_{2}^{2} + 10 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 7 T^{14} + 20 T^{13} + 30 T^{12} + \cdots + 1 \) 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
$11$ \( T^{16} - 4 T^{15} + 11 T^{14} + \cdots + 214358881 \) 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
$23$ \( (T^{8} + 8 T^{7} + 115 T^{6} + \cdots + 383161)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 112 T^{6} - 290 T^{5} + \cdots + 259081)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} - 18 T^{15} + \cdots + 4810832476321 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{8} + 402 T^{6} + 53459 T^{4} + \cdots + 16072081)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + 30 T^{15} + \cdots + 36325030350625 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( (T^{8} + 4 T^{7} + 335 T^{6} + \cdots + 300710281)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 48 T^{7} + 1123 T^{6} + \cdots + 19321)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 544688429550721 \) 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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