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Properties

Label 529.4.a.k
Level $529$
Weight $4$
Character orbit 529.a
Self dual yes
Analytic conductor $31.212$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [529,4,Mod(1,529)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(529, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("529.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 529 = 23^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 529.a (trivial)

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: yes
Analytic conductor: \(31.2120103930\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 80 x^{10} + 266 x^{9} + 2256 x^{8} - 5648 x^{7} - 28495 x^{6} + 47408 x^{5} + \cdots - 4232 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{7} q^{3} + ( - \beta_{9} + \beta_{2} + 4) q^{4} + (\beta_{10} - \beta_{6} + \beta_1) q^{5} + ( - \beta_{11} + \beta_{9} + \cdots + \beta_{4}) q^{6} + (\beta_{10} + \beta_{3} - 3 \beta_1) q^{7}+ \cdots + (20 \beta_{10} + 26 \beta_{8} + \cdots + 419 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + 56 q^{4} - 6 q^{6} + 138 q^{8} + 204 q^{9} + 30 q^{12} + 160 q^{13} + 144 q^{16} + 478 q^{18} - 1188 q^{24} + 400 q^{25} + 1554 q^{26} - 684 q^{27} - 44 q^{29} + 1076 q^{31} + 248 q^{32}+ \cdots - 3512 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 80 x^{10} + 266 x^{9} + 2256 x^{8} - 5648 x^{7} - 28495 x^{6} + 47408 x^{5} + \cdots - 4232 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1399379 \nu^{11} + 6588244 \nu^{10} - 148556478 \nu^{9} - 548829834 \nu^{8} + \cdots + 33608037184 ) / 13938740048 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1399379 \nu^{11} + 6588244 \nu^{10} - 148556478 \nu^{9} - 548829834 \nu^{8} + \cdots + 33608037184 ) / 13938740048 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23684999 \nu^{11} + 208417067 \nu^{10} + 1547982094 \nu^{9} - 14862787020 \nu^{8} + \cdots + 3753219574020 ) / 97571180336 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28818669 \nu^{11} - 98246631 \nu^{10} + 3643830894 \nu^{9} + 5018913072 \nu^{8} + \cdots + 1831925874932 ) / 97571180336 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38257496 \nu^{11} - 227630939 \nu^{10} + 4245913156 \nu^{9} + 18534885942 \nu^{8} + \cdots - 267904837844 ) / 97571180336 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6092880 \nu^{11} - 18303079 \nu^{10} - 460532324 \nu^{9} + 964394934 \nu^{8} + \cdots + 16899826012 ) / 6969370024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 178947689 \nu^{11} + 336702800 \nu^{10} + 14702230570 \nu^{9} - 15794637834 \nu^{8} + \cdots + 126627946608 ) / 195142360672 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7021334 \nu^{11} - 12259935 \nu^{10} - 547003568 \nu^{9} + 396355730 \nu^{8} + \cdots + 159154943772 ) / 6969370024 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22972141 \nu^{11} + 79800560 \nu^{10} + 1693572818 \nu^{9} - 4406409570 \nu^{8} + \cdots + 161151093808 ) / 13938740048 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 349121914 \nu^{11} - 924203321 \nu^{10} - 27589423796 \nu^{9} + 49569181286 \nu^{8} + \cdots - 432584814844 ) / 195142360672 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 655902571 \nu^{11} + 515165350 \nu^{10} + 56592724422 \nu^{9} - 5250002994 \nu^{8} + \cdots + 4500222821032 ) / 195142360672 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - 2\beta_{6} + \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 3\beta_{8} + 3\beta_{7} - 3\beta_{6} - \beta_{4} + 25\beta_{2} - 38\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{11} + 8 \beta_{10} - 38 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} - 84 \beta_{6} - 8 \beta_{5} + \cdots + 380 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 40 \beta_{11} + 20 \beta_{10} - 11 \beta_{9} + 160 \beta_{8} + 164 \beta_{7} - 185 \beta_{6} + \cdots + 536 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 133 \beta_{11} + 464 \beta_{10} - 1462 \beta_{9} + 298 \beta_{8} + 349 \beta_{7} - 3334 \beta_{6} + \cdots + 12846 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1300 \beta_{11} + 1610 \beta_{10} - 1056 \beta_{9} + 7063 \beta_{8} + 7818 \beta_{7} - 8995 \beta_{6} + \cdots + 25292 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6764 \beta_{11} + 22400 \beta_{10} - 57305 \beta_{9} + 17184 \beta_{8} + 22700 \beta_{7} - 132280 \beta_{6} + \cdots + 477144 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 37889 \beta_{11} + 93408 \beta_{10} - 70956 \beta_{9} + 299313 \beta_{8} + 356659 \beta_{7} + \cdots + 1159594 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 312446 \beta_{11} + 1026772 \beta_{10} - 2280112 \beta_{9} + 893474 \beta_{8} + 1273680 \beta_{7} + \cdots + 18495808 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 971766 \beta_{11} + 4799828 \beta_{10} - 4068267 \beta_{9} + 12584902 \beta_{8} + 15926214 \beta_{7} + \cdots + 52846132 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.15147
−5.97990
−0.971986
−3.80041
−3.19828
−0.369855
−0.0479299
2.78050
5.59132
2.76289
3.77835
6.60678
−4.56568 4.39377 12.8455 −19.1180 −20.0606 −18.5215 −22.1229 −7.69479 87.2869
1.2 −4.56568 4.39377 12.8455 19.1180 −20.0606 18.5215 −22.1229 −7.69479 −87.2869
1.3 −2.38620 −9.65154 −2.30605 −8.61537 23.0305 15.6602 24.5923 66.1523 20.5580
1.4 −2.38620 −9.65154 −2.30605 8.61537 23.0305 −15.6602 24.5923 66.1523 −20.5580
1.5 −1.78407 3.69728 −4.81710 −14.2031 −6.59619 −30.8570 22.8666 −13.3302 25.3393
1.6 −1.78407 3.69728 −4.81710 14.2031 −6.59619 30.8570 22.8666 −13.3302 −25.3393
1.7 1.36628 −0.504290 −6.13327 −0.0831947 −0.689003 29.1476 −19.3101 −26.7457 −0.113668
1.8 1.36628 −0.504290 −6.13327 0.0831947 −0.689003 −29.1476 −19.3101 −26.7457 0.113668
1.9 4.17710 9.26303 9.44818 −10.5194 38.6926 18.9340 6.04918 58.8037 −43.9408
1.10 4.17710 9.26303 9.44818 10.5194 38.6926 −18.9340 6.04918 58.8037 43.9408
1.11 5.19257 −7.19824 18.9628 −14.0671 −37.3774 −4.84530 56.9249 24.8147 −73.0444
1.12 5.19257 −7.19824 18.9628 14.0671 −37.3774 4.84530 56.9249 24.8147 73.0444
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(23\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 529.4.a.k ✓ 12
23.b odd 2 1 inner 529.4.a.k ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
529.4.a.k ✓ 12 1.a even 1 1 trivial
529.4.a.k ✓ 12 23.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(529))\):

\( T_{2}^{6} - 2T_{2}^{5} - 36T_{2}^{4} + 41T_{2}^{3} + 336T_{2}^{2} - 20T_{2} - 576 \) Copy content Toggle raw display
\( T_{3}^{6} - 132T_{3}^{4} + 114T_{3}^{3} + 3891T_{3}^{2} - 8538T_{3} - 5272 \) Copy content Toggle raw display
\( T_{5}^{12} - 950T_{5}^{10} + 335652T_{5}^{8} - 55260616T_{5}^{6} + 4225391664T_{5}^{4} - 119867040800T_{5}^{2} + 829440000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 2 T^{5} + \cdots - 576)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} - 132 T^{4} + \cdots - 5272)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 829440000 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 572778915840000 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{6} - 80 T^{5} + \cdots - 235611612)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 70\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( (T^{6} + 22 T^{5} + \cdots - 20658997584)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 1109666145024)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 70\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 2487606909120)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 73524283914240)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 219664451059200)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 114923956204368)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 42\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
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