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Properties

Label 539.2.e.o
Level $539$
Weight $2$
Character orbit 539.e
Analytic conductor $4.304$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,2,Mod(67,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.e (of order \(3\), degree \(2\), 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: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 26 x^{18} + 431 x^{16} + 4294 x^{14} + 31153 x^{12} + 157310 x^{10} + 590696 x^{8} + \cdots + 937024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + (\beta_{10} - \beta_1) q^{3} + (\beta_{7} - 2 \beta_{2}) q^{4} + \beta_{19} q^{5} + (\beta_{17} + \beta_{14} - \beta_{13}) q^{6} + (\beta_{12} + \beta_{11} - 2 \beta_{8} + \cdots - 1) q^{8}+ \cdots + ( - \beta_{5} + 2 \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + (\beta_{10} - \beta_1) q^{3} + (\beta_{7} - 2 \beta_{2}) q^{4} + \beta_{19} q^{5} + (\beta_{17} + \beta_{14} - \beta_{13}) q^{6} + (\beta_{12} + \beta_{11} - 2 \beta_{8} + \cdots - 1) q^{8}+ \cdots + (\beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} - 18 q^{4} - 12 q^{8} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} - 18 q^{4} - 12 q^{8} - 22 q^{9} - 10 q^{11} + 16 q^{15} - 42 q^{16} - 6 q^{18} + 4 q^{22} - 4 q^{23} - 18 q^{25} + 24 q^{29} + 4 q^{30} + 30 q^{32} - 4 q^{36} - 40 q^{37} + 16 q^{39} - 16 q^{43} - 18 q^{44} - 44 q^{46} - 124 q^{50} - 16 q^{53} - 16 q^{57} + 28 q^{58} - 36 q^{60} + 212 q^{64} + 32 q^{65} + 4 q^{67} + 72 q^{71} + 90 q^{72} + 28 q^{74} - 224 q^{78} - 8 q^{79} + 6 q^{81} + 176 q^{85} - 32 q^{86} + 6 q^{88} - 104 q^{92} - 44 q^{93} + 64 q^{95} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 26 x^{18} + 431 x^{16} + 4294 x^{14} + 31153 x^{12} + 157310 x^{10} + 590696 x^{8} + \cdots + 937024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 201634123356459 \nu^{18} + \cdots - 98\!\cdots\!12 ) / 18\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1779950366783 \nu^{18} + 41930759742216 \nu^{16} + 667047166673473 \nu^{14} + \cdots - 60\!\cdots\!64 ) / 15\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 189102926089157 \nu^{18} + \cdots + 27\!\cdots\!64 ) / 52\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 792796622401552 \nu^{18} + \cdots - 12\!\cdots\!04 ) / 18\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24\!\cdots\!43 \nu^{18} + \cdots - 39\!\cdots\!76 ) / 36\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 36\!\cdots\!71 \nu^{18} + \cdots - 40\!\cdots\!68 ) / 36\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 48\!\cdots\!49 \nu^{18} + \cdots - 15\!\cdots\!48 ) / 36\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 54003685 \nu^{19} + 5994282114 \nu^{17} + 123668892807 \nu^{15} + \cdots + 276791164851872 \nu ) / 70230245495872 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 201634123356459 \nu^{19} + \cdots - 98\!\cdots\!12 \nu ) / 18\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 20\!\cdots\!79 \nu^{18} + \cdots + 56\!\cdots\!56 ) / 36\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 182229179472441 \nu^{18} + \cdots - 51\!\cdots\!76 ) / 30\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28\!\cdots\!83 \nu^{19} + \cdots + 32\!\cdots\!20 \nu ) / 81\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16\!\cdots\!75 \nu^{19} + \cdots + 35\!\cdots\!32 \nu ) / 40\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 53\!\cdots\!01 \nu^{19} + \cdots - 83\!\cdots\!60 \nu ) / 11\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 377870912801145 \nu^{19} + \cdots + 28\!\cdots\!28 \nu ) / 67\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 745281290358391 \nu^{19} + \cdots - 20\!\cdots\!28 \nu ) / 67\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 55\!\cdots\!37 \nu^{19} + \cdots - 89\!\cdots\!24 \nu ) / 40\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 22\!\cdots\!03 \nu^{19} + \cdots + 51\!\cdots\!04 \nu ) / 67\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{3} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - \beta_{15} + \beta_{14} - 2\beta_{13} - 7\beta_{10} - 4\beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{12} - \beta_{11} + 5\beta_{8} - 2\beta_{7} - 13\beta_{5} + 34\beta_{2} - 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{19} - 55\beta_{18} + 26\beta_{16} + 13\beta_{15} - 18\beta_{14} + 26\beta_{13} + 58\beta_{10} - 58\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -34\beta_{12} + 18\beta_{11} - 80\beta_{8} + 18\beta_{6} - 80\beta_{4} + 149\beta_{3} + 277 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -151\beta_{19} + 648\beta_{18} - 229\beta_{17} - 292\beta_{16} + 648\beta_{9} + 541\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 434\beta_{7} - 239\beta_{6} + 1647\beta_{5} + 979\beta_{4} - 1647\beta_{3} - 2554\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2626\beta_{17} - 1691\beta_{15} + 2626\beta_{14} - 3166\beta_{13} - 5414\beta_{10} - 7261\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5030\beta_{12} - 2842\beta_{11} + 11044\beta_{8} - 5030\beta_{7} - 17927\beta_{5} + 25379\beta_{2} - 25379 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18581 \beta_{19} - 79580 \beta_{18} + 33996 \beta_{16} + 18581 \beta_{15} - 28971 \beta_{14} + \cdots - 56203 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 55974 \beta_{12} + 32137 \beta_{11} - 120909 \beta_{8} + 32137 \beta_{6} - 120909 \beta_{4} + \cdots + 262434 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -202025\beta_{19} + 863011\beta_{18} - 314634\beta_{17} - 364274\beta_{16} + 863011\beta_{9} + 593910\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 611346\beta_{7} - 354410\beta_{6} + 2086189\beta_{5} + 1308176\beta_{4} - 2086189\beta_{3} - 2767525\beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3394365 \beta_{17} - 2183663 \beta_{15} + 3394365 \beta_{14} - 3903452 \beta_{13} + \cdots - 9310512 \beta_{9} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 6617762 \beta_{12} - 3858239 \beta_{11} + 14086547 \beta_{8} - 6617762 \beta_{7} - 22427799 \beta_{5} + \cdots - 29459122 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 23526515 \beta_{19} - 100187197 \beta_{18} + 41848382 \beta_{16} + 23526515 \beta_{15} + \cdots - 67696958 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 71326646 \beta_{12} + 41718994 \beta_{11} - 151391420 \beta_{8} + 41718994 \beta_{6} + \cdots + 314958275 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 253024189 \beta_{19} + 1076697028 \beta_{18} - 392304267 \beta_{17} - 448844828 \beta_{16} + \cdots + 725457323 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(442\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.451603 − 0.782199i
0.451603 + 0.782199i
−1.63807 − 2.83722i
1.63807 + 2.83722i
−1.16133 − 2.01149i
1.16133 + 2.01149i
−1.07647 − 1.86450i
1.07647 + 1.86450i
−1.05133 − 1.82096i
1.05133 + 1.82096i
−0.451603 + 0.782199i
0.451603 − 0.782199i
−1.63807 + 2.83722i
1.63807 − 2.83722i
−1.16133 + 2.01149i
1.16133 − 2.01149i
−1.07647 + 1.86450i
1.07647 − 1.86450i
−1.05133 + 1.82096i
1.05133 − 1.82096i
−1.31405 − 2.27600i −0.451603 + 0.782199i −2.45345 + 4.24950i 0.168994 + 0.292706i 2.37371 0 7.63960 1.09211 + 1.89159i 0.444132 − 0.769259i
67.2 −1.31405 − 2.27600i 0.451603 − 0.782199i −2.45345 + 4.24950i −0.168994 − 0.292706i −2.37371 0 7.63960 1.09211 + 1.89159i −0.444132 + 0.769259i
67.3 −0.851480 − 1.47481i −1.63807 + 2.83722i −0.450036 + 0.779485i 0.123338 + 0.213627i 5.57913 0 −1.87313 −3.86654 − 6.69705i 0.210039 − 0.363799i
67.4 −0.851480 − 1.47481i 1.63807 − 2.83722i −0.450036 + 0.779485i −0.123338 − 0.213627i −5.57913 0 −1.87313 −3.86654 − 6.69705i −0.210039 + 0.363799i
67.5 −0.283046 − 0.490250i −1.16133 + 2.01149i 0.839770 − 1.45452i −1.79110 − 3.10227i 1.31484 0 −2.08296 −1.19739 − 2.07395i −1.01393 + 1.75617i
67.6 −0.283046 − 0.490250i 1.16133 − 2.01149i 0.839770 − 1.45452i 1.79110 + 3.10227i −1.31484 0 −2.08296 −1.19739 − 2.07395i 1.01393 − 1.75617i
67.7 0.574492 + 0.995050i −1.07647 + 1.86450i 0.339917 − 0.588753i 1.93794 + 3.35662i −2.47369 0 3.07909 −0.817563 − 1.41606i −2.22667 + 3.85670i
67.8 0.574492 + 0.995050i 1.07647 − 1.86450i 0.339917 − 0.588753i −1.93794 − 3.35662i 2.47369 0 3.07909 −0.817563 − 1.41606i 2.22667 − 3.85670i
67.9 1.37408 + 2.37998i −1.05133 + 1.82096i −2.77620 + 4.80853i −1.22171 − 2.11606i −5.77848 0 −9.76260 −0.710608 − 1.23081i 3.35746 − 5.81529i
67.10 1.37408 + 2.37998i 1.05133 − 1.82096i −2.77620 + 4.80853i 1.22171 + 2.11606i 5.77848 0 −9.76260 −0.710608 − 1.23081i −3.35746 + 5.81529i
177.1 −1.31405 + 2.27600i −0.451603 − 0.782199i −2.45345 − 4.24950i 0.168994 − 0.292706i 2.37371 0 7.63960 1.09211 − 1.89159i 0.444132 + 0.769259i
177.2 −1.31405 + 2.27600i 0.451603 + 0.782199i −2.45345 − 4.24950i −0.168994 + 0.292706i −2.37371 0 7.63960 1.09211 − 1.89159i −0.444132 − 0.769259i
177.3 −0.851480 + 1.47481i −1.63807 − 2.83722i −0.450036 − 0.779485i 0.123338 − 0.213627i 5.57913 0 −1.87313 −3.86654 + 6.69705i 0.210039 + 0.363799i
177.4 −0.851480 + 1.47481i 1.63807 + 2.83722i −0.450036 − 0.779485i −0.123338 + 0.213627i −5.57913 0 −1.87313 −3.86654 + 6.69705i −0.210039 − 0.363799i
177.5 −0.283046 + 0.490250i −1.16133 − 2.01149i 0.839770 + 1.45452i −1.79110 + 3.10227i 1.31484 0 −2.08296 −1.19739 + 2.07395i −1.01393 − 1.75617i
177.6 −0.283046 + 0.490250i 1.16133 + 2.01149i 0.839770 + 1.45452i 1.79110 − 3.10227i −1.31484 0 −2.08296 −1.19739 + 2.07395i 1.01393 + 1.75617i
177.7 0.574492 − 0.995050i −1.07647 − 1.86450i 0.339917 + 0.588753i 1.93794 − 3.35662i −2.47369 0 3.07909 −0.817563 + 1.41606i −2.22667 − 3.85670i
177.8 0.574492 − 0.995050i 1.07647 + 1.86450i 0.339917 + 0.588753i −1.93794 + 3.35662i 2.47369 0 3.07909 −0.817563 + 1.41606i 2.22667 + 3.85670i
177.9 1.37408 − 2.37998i −1.05133 − 1.82096i −2.77620 − 4.80853i −1.22171 + 2.11606i −5.77848 0 −9.76260 −0.710608 + 1.23081i 3.35746 + 5.81529i
177.10 1.37408 − 2.37998i 1.05133 + 1.82096i −2.77620 − 4.80853i 1.22171 − 2.11606i 5.77848 0 −9.76260 −0.710608 + 1.23081i −3.35746 − 5.81529i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.e.o 20
7.b odd 2 1 inner 539.2.e.o 20
7.c even 3 1 539.2.a.l ✓ 10
7.c even 3 1 inner 539.2.e.o 20
7.d odd 6 1 539.2.a.l ✓ 10
7.d odd 6 1 inner 539.2.e.o 20
21.g even 6 1 4851.2.a.cg 10
21.h odd 6 1 4851.2.a.cg 10
28.f even 6 1 8624.2.a.df 10
28.g odd 6 1 8624.2.a.df 10
77.h odd 6 1 5929.2.a.bv 10
77.i even 6 1 5929.2.a.bv 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.a.l ✓ 10 7.c even 3 1
539.2.a.l ✓ 10 7.d odd 6 1
539.2.e.o 20 1.a even 1 1 trivial
539.2.e.o 20 7.b odd 2 1 inner
539.2.e.o 20 7.c even 3 1 inner
539.2.e.o 20 7.d odd 6 1 inner
4851.2.a.cg 10 21.g even 6 1
4851.2.a.cg 10 21.h odd 6 1
5929.2.a.bv 10 77.h odd 6 1
5929.2.a.bv 10 77.i even 6 1
8624.2.a.df 10 28.f even 6 1
8624.2.a.df 10 28.g odd 6 1

Hecke kernels

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

\( T_{2}^{10} + T_{2}^{9} + 10T_{2}^{8} + 9T_{2}^{7} + 78T_{2}^{6} + 65T_{2}^{5} + 181T_{2}^{4} + 36T_{2}^{3} + 216T_{2}^{2} + 96T_{2} + 64 \) Copy content Toggle raw display
\( T_{3}^{20} + 26 T_{3}^{18} + 431 T_{3}^{16} + 4294 T_{3}^{14} + 31153 T_{3}^{12} + 157310 T_{3}^{10} + \cdots + 937024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + T^{9} + 10 T^{8} + \cdots + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} + 26 T^{18} + \cdots + 937024 \) Copy content Toggle raw display
$5$ \( T^{20} + 34 T^{18} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$13$ \( (T^{10} - 72 T^{8} + \cdots - 100352)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 185409470464 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 41248865910784 \) Copy content Toggle raw display
$23$ \( (T^{10} + 2 T^{9} + \cdots + 53824)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 6 T^{4} + \cdots + 224)^{4} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 15990337459264 \) Copy content Toggle raw display
$37$ \( (T^{10} + 20 T^{9} + \cdots + 222784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} - 168 T^{8} + \cdots - 25088)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 4 T^{4} + \cdots + 256)^{4} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{10} + 8 T^{9} + \cdots + 126877696)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 12440914100224 \) Copy content Toggle raw display
$67$ \( (T^{10} - 2 T^{9} + \cdots + 21603904)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 18 T^{4} + \cdots + 88)^{4} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 43\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{10} + 4 T^{9} + \cdots + 495616)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} - 288 T^{8} + \cdots - 102760448)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + 34 T^{18} + \cdots + 64 \) Copy content Toggle raw display
$97$ \( (T^{10} - 226 T^{8} + \cdots - 19208)^{2} \) Copy content Toggle raw display
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