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Properties

Label 539.4.a.o
Level $539$
Weight $4$
Character orbit 539.a
Self dual yes
Analytic conductor $31.802$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,4,Mod(1,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 539.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: \(31.8020294931\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 156x^{8} + 5810x^{6} - 52148x^{4} + 20168x^{2} - 1584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{3} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{8} q^{3} + ( - \beta_{5} + 4) q^{4} + (\beta_{8} - \beta_1) q^{5} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_1) q^{6} + ( - 2 \beta_{5} - 3 \beta_{4} + \cdots + 2) q^{8}+ \cdots + ( - \beta_{6} - \beta_{5} + 4 \beta_{4} + \cdots + 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{8} q^{3} + ( - \beta_{5} + 4) q^{4} + (\beta_{8} - \beta_1) q^{5} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_1) q^{6} + ( - 2 \beta_{5} - 3 \beta_{4} + \cdots + 2) q^{8}+ \cdots + ( - 11 \beta_{6} - 11 \beta_{5} + \cdots + 198) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} + 44 q^{4} + 36 q^{8} + 170 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{2} + 44 q^{4} + 36 q^{8} + 170 q^{9} + 110 q^{11} + 428 q^{15} + 84 q^{16} - 340 q^{18} + 44 q^{22} + 92 q^{23} + 730 q^{25} + 628 q^{29} - 1128 q^{30} + 1252 q^{32} + 2084 q^{36} + 164 q^{37} + 244 q^{39} + 412 q^{43} + 484 q^{44} + 2920 q^{46} - 412 q^{50} - 84 q^{51} + 1968 q^{53} + 720 q^{57} - 2520 q^{58} + 7256 q^{60} + 1044 q^{64} - 3312 q^{65} + 1916 q^{67} + 1324 q^{71} - 5164 q^{72} + 2576 q^{74} + 8512 q^{78} + 604 q^{79} + 6990 q^{81} + 984 q^{85} - 3632 q^{86} + 396 q^{88} + 6024 q^{92} - 3048 q^{93} + 1480 q^{95} + 1870 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 156x^{8} + 5810x^{6} - 52148x^{4} + 20168x^{2} - 1584 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -6813\nu^{9} + 1088462\nu^{7} - 43581430\nu^{5} + 504573696\nu^{3} - 1402956712\nu ) / 53833152 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4849\nu^{8} - 707690\nu^{6} + 20921262\nu^{4} - 24928568\nu^{2} - 632678280 ) / 26916576 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22665\nu^{9} - 3564472\nu^{7} + 136098302\nu^{5} - 1335619932\nu^{3} + 1417901024\nu ) / 26916576 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21683\nu^{8} - 3374086\nu^{6} + 124768218\nu^{4} - 1095797368\nu^{2} + 238584072 ) / 26916576 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13023\nu^{8} + 2033178\nu^{6} - 75797154\nu^{4} + 675257248\nu^{2} - 145846904 ) / 8972192 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8431\nu^{8} - 1312466\nu^{6} + 48562290\nu^{4} - 423694160\nu^{2} + 27793368 ) / 3166656 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -233127\nu^{9} + 36307142\nu^{7} - 1344844546\nu^{5} + 11783846472\nu^{3} - 1041184072\nu ) / 53833152 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 141653\nu^{9} - 22071888\nu^{7} + 818981206\nu^{5} - 7240007452\nu^{3} + 1595237344\nu ) / 26916576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -28629\nu^{9} + 4463510\nu^{7} - 165907846\nu^{5} + 1475377296\nu^{3} - 401739400\nu ) / 3166656 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + 2\beta_{7} + 3\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 18\beta_{6} + 11\beta_{5} - 39\beta_{4} - 3\beta_{2} + 296 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -45\beta_{9} + 24\beta_{8} + 125\beta_{7} + 53\beta_{3} + 288\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 958\beta_{6} + 921\beta_{5} - 1439\beta_{4} - 303\beta_{2} + 12196 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1256\beta_{9} + 2634\beta_{8} + 6067\beta_{7} + 3443\beta_{3} + 14561\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 100434\beta_{6} + 111363\beta_{5} - 123407\beta_{4} - 35219\beta_{2} + 1194788 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -89627\beta_{9} + 341112\beta_{8} + 631729\beta_{7} + 382669\beta_{3} + 1516690\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10570824\beta_{6} + 12307508\beta_{5} - 11902312\beta_{4} - 3812704\beta_{2} + 123166968 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -7848046\beta_{9} + 38536452\beta_{8} + 66540052\beta_{7} + 41074564\beta_{3} + 159482642\beta_1 ) / 5 \) 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
−0.329774
0.329774
−3.55708
3.55708
−10.2796
10.2796
0.540604
−0.540604
6.10533
−6.10533
−4.49114 −10.0162 12.1703 −18.2775 44.9842 0 −18.7295 73.3246 82.0870
1.2 −4.49114 10.0162 12.1703 18.2775 −44.9842 0 −18.7295 73.3246 −82.0870
1.3 −2.57669 −3.11793 −1.36068 −0.843980 8.03393 0 24.1196 −17.2785 2.17467
1.4 −2.57669 3.11793 −1.36068 0.843980 −8.03393 0 24.1196 −17.2785 −2.17467
1.5 1.26860 −0.446013 −6.39065 21.5196 −0.565812 0 −18.2560 −26.8011 27.2997
1.6 1.26860 0.446013 −6.39065 −21.5196 0.565812 0 −18.2560 −26.8011 −27.2997
1.7 2.64123 −9.06355 −1.02390 3.58150 −23.9389 0 −23.8342 55.1479 9.45955
1.8 2.64123 9.06355 −1.02390 −3.58150 23.9389 0 −23.8342 55.1479 −9.45955
1.9 5.15800 −5.25424 18.6049 −13.3903 −27.1014 0 54.7001 0.607090 −69.0669
1.10 5.15800 5.25424 18.6049 13.3903 27.1014 0 54.7001 0.607090 69.0669
\(n\): 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.o ✓ 10
7.b odd 2 1 inner 539.4.a.o ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.4.a.o ✓ 10 1.a even 1 1 trivial
539.4.a.o ✓ 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(539))\):

\( T_{2}^{5} - 2T_{2}^{4} - 29T_{2}^{3} + 44T_{2}^{2} + 150T_{2} - 200 \) Copy content Toggle raw display
\( T_{3}^{10} - 220T_{3}^{8} + 15365T_{3}^{6} - 359662T_{3}^{4} + 2282800T_{3}^{2} - 440000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} - 2 T^{4} + \cdots - 200)^{2} \) Copy content Toggle raw display
$3$ \( T^{10} - 220 T^{8} + \cdots - 440000 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 253440000 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T - 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 13798400000000 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 106639554560000 \) Copy content Toggle raw display
$23$ \( (T^{5} - 46 T^{4} + \cdots + 1449670400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 314 T^{4} + \cdots - 140668163104)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 84\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} - 82 T^{4} + \cdots + 28178456000)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 54\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} - 206 T^{4} + \cdots - 54179046400)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 2017765852800)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 57\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 57\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 59505381734400)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 8612999333376)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 88\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 2687851221760)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 57\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
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