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Properties

Label 539.4.a.l
Level $539$
Weight $4$
Character orbit 539.a
Self dual yes
Analytic conductor $31.802$
Analytic rank $1$
Dimension $10$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [539,4,Mod(1,539)] 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("539.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(539, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 539.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-4,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.8020294931\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 46x^{8} + 148x^{7} + 614x^{6} - 1476x^{5} - 3064x^{4} + 4428x^{3} + 4321x^{2} - 3480x - 658 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{6} - \beta_{3}) q^{3} + (\beta_{5} + \beta_1) q^{4} - \beta_{2} q^{5} + ( - \beta_{9} - 3 \beta_{6} + \cdots - \beta_{3}) q^{6} + ( - \beta_{8} - \beta_{7} - 2 \beta_1 - 8) q^{8}+ \cdots + (11 \beta_{8} + 22 \beta_{7} - 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} + 8 q^{4} - 84 q^{8} - 18 q^{9} + 110 q^{11} - 76 q^{15} + 40 q^{16} - 44 q^{22} - 384 q^{23} - 338 q^{25} - 340 q^{29} - 228 q^{30} - 276 q^{32} - 60 q^{36} - 1384 q^{37} - 1232 q^{39} - 100 q^{43}+ \cdots - 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} - 46x^{8} + 148x^{7} + 614x^{6} - 1476x^{5} - 3064x^{4} + 4428x^{3} + 4321x^{2} - 3480x - 658 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3347 \nu^{9} + 139758 \nu^{8} - 194369 \nu^{7} - 5889724 \nu^{6} + 7672561 \nu^{5} + \cdots - 52370696 ) / 16579808 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8627 \nu^{9} - 156977 \nu^{8} - 68985 \nu^{7} + 6802967 \nu^{6} - 5237361 \nu^{5} + \cdots - 329643454 ) / 33159616 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8987 \nu^{9} - 75723 \nu^{8} - 251797 \nu^{7} + 2732057 \nu^{6} - 172745 \nu^{5} + \cdots + 176536094 ) / 33159616 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9820 \nu^{9} - 28815 \nu^{8} + 611478 \nu^{7} + 1549563 \nu^{6} - 10118508 \nu^{5} + \cdots - 48285062 ) / 16579808 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22987 \nu^{9} - 82128 \nu^{8} - 1028587 \nu^{7} + 2790598 \nu^{6} + 12564455 \nu^{5} + \cdots - 16857260 ) / 16579808 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22987 \nu^{9} - 82128 \nu^{8} - 1028587 \nu^{7} + 2790598 \nu^{6} + 12564455 \nu^{5} + \cdots - 16857260 ) / 16579808 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 39247 \nu^{9} + 211695 \nu^{8} + 1686517 \nu^{7} - 8552717 \nu^{6} - 20252171 \nu^{5} + \cdots + 165115370 ) / 16579808 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 71592 \nu^{9} + 179307 \nu^{8} + 4093670 \nu^{7} - 6469123 \nu^{6} - 74845432 \nu^{5} + \cdots + 62373878 ) / 16579808 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 79647 \nu^{9} + 304195 \nu^{8} + 3780077 \nu^{7} - 11104997 \nu^{6} - 53455275 \nu^{5} + \cdots + 59320058 ) / 16579808 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} + \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{9} + \beta_{8} + \beta_{7} - 26\beta_{6} + 22\beta_{5} + 3\beta_{4} + 2\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 8 \beta_{9} + 3 \beta_{8} - \beta_{7} - 32 \beta_{6} + 46 \beta_{5} + 68 \beta_{4} + 8 \beta_{3} + \cdots + 228 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 135 \beta_{9} + 46 \beta_{8} + 38 \beta_{7} - 804 \beta_{6} + 651 \beta_{5} + 195 \beta_{4} + \cdots + 440 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 488 \beta_{9} + 184 \beta_{8} - 32 \beta_{7} - 2000 \beta_{6} + 2043 \beta_{5} + 2382 \beta_{4} + \cdots + 6832 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 5201 \beta_{9} + 1827 \beta_{8} + 1211 \beta_{7} - 27168 \beta_{6} + 21956 \beta_{5} + 9289 \beta_{4} + \cdots + 20776 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 22768 \beta_{9} + 8537 \beta_{8} - 59 \beta_{7} - 96320 \beta_{6} + 88652 \beta_{5} + 86504 \beta_{4} + \cdots + 231216 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 196227 \beta_{9} + 70180 \beta_{8} + 38356 \beta_{7} - 967600 \beta_{6} + 789205 \beta_{5} + \cdots + 919048 \) 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
6.26893
3.44050
3.61956
0.791135
−1.28309
1.54534
−0.162357
−2.99078
−5.02883
−2.20040
−4.85471 −5.89940 15.5682 4.11981 28.6399 0 −36.7416 7.80287 −20.0005
1.2 −4.85471 5.89940 15.5682 −4.11981 −28.6399 0 −36.7416 7.80287 20.0005
1.3 −2.20535 −1.35402 −3.13644 −13.8492 2.98608 0 24.5597 −25.1666 30.5422
1.4 −2.20535 1.35402 −3.13644 13.8492 −2.98608 0 24.5597 −25.1666 −30.5422
1.5 −0.131121 −4.81567 −7.98281 −10.6587 0.631438 0 2.09569 −3.80934 1.39758
1.6 −0.131121 4.81567 −7.98281 10.6587 −0.631438 0 2.09569 −3.80934 −1.39758
1.7 1.57657 −7.58160 −5.51443 7.69585 −11.9529 0 −21.3064 30.4807 12.1330
1.8 1.57657 7.58160 −5.51443 −7.69585 11.9529 0 −21.3064 30.4807 −12.1330
1.9 3.61461 −2.94828 5.06543 8.62519 −10.6569 0 −10.6073 −18.3076 31.1767
1.10 3.61461 2.94828 5.06543 −8.62519 10.6569 0 −10.6073 −18.3076 −31.1767
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( +1 \)
\(11\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.4.a.l ✓ 10
7.b odd 2 1 inner 539.4.a.l ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.4.a.l ✓ 10 1.a even 1 1 trivial
539.4.a.l ✓ 10 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(539))\):

\( T_{2}^{5} + 2T_{2}^{4} - 20T_{2}^{3} - 18T_{2}^{2} + 59T_{2} + 8 \) Copy content Toggle raw display
\( T_{3}^{10} - 126T_{3}^{8} + 5372T_{3}^{6} - 91816T_{3}^{4} + 554304T_{3}^{2} - 739328 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} + 2 T^{4} - 20 T^{3} + \cdots + 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{10} - 126 T^{8} + \cdots - 739328 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 1629519872 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T - 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 12601324505088 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 28\!\cdots\!08 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 261036024209408 \) Copy content Toggle raw display
$23$ \( (T^{5} + 192 T^{4} + \cdots + 14609024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 170 T^{4} + \cdots - 352797312)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 14\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( (T^{5} + 692 T^{4} + \cdots - 320801910272)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 10\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( (T^{5} + 50 T^{4} + \cdots - 119277744128)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 65\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 6905037874944)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 12\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 49\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 10209938378752)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 12989214154752)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 38\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 26002748837888)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 14\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 36\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 51\!\cdots\!88 \) Copy content Toggle raw display
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