↑

Properties

Label 500.4.a.b
Level $500$
Weight $4$
Character orbit 500.a
Self dual yes
Analytic conductor $29.501$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [500,4,Mod(1,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, 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("500.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 500.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: \(29.5009550029\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 66x^{4} - 34x^{3} + 1084x^{2} + 842x - 3631 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 7) q^{7}+ \cdots + (\beta_{5} + \beta_{4} + 3 \beta_{3} + \cdots + 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 7) q^{7}+ \cdots + (3 \beta_{5} - 35 \beta_{4} + \cdots - 344) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 43 q^{7} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 43 q^{7} + 64 q^{9} - 15 q^{11} + 87 q^{13} + 76 q^{17} - 21 q^{19} - 184 q^{21} + 91 q^{23} + 76 q^{27} - 86 q^{29} + 246 q^{31} + 910 q^{33} + 1053 q^{37} - 356 q^{39} - 469 q^{41} + 460 q^{43} + 1321 q^{47} + 453 q^{49} + 112 q^{51} + 1499 q^{53} + 2072 q^{57} + 763 q^{59} + 522 q^{61} + 2407 q^{63} + 1098 q^{67} - 1008 q^{69} + 478 q^{71} + 2352 q^{73} + 1250 q^{77} + 1072 q^{79} + 434 q^{81} + 576 q^{83} + 2192 q^{87} - 459 q^{89} + 764 q^{91} + 2468 q^{93} + 3228 q^{97} - 2575 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 66x^{4} - 34x^{3} + 1084x^{2} + 842x - 3631 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 28\nu^{4} - 718\nu^{3} - 907\nu^{2} + 24312\nu + 8698 ) / 8418 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 818\nu^{5} - 1859\nu^{4} - 44026\nu^{3} + 68486\nu^{2} + 421209\nu - 372146 ) / 42090 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 901\nu^{5} - 4183\nu^{4} - 47567\nu^{3} + 185857\nu^{2} + 465723\nu - 1275067 ) / 42090 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1687\nu^{5} + 5146\nu^{4} + 93524\nu^{3} - 267409\nu^{2} - 949386\nu + 2517934 ) / 42090 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 512\nu^{5} - 1709\nu^{4} - 28030\nu^{3} + 77156\nu^{2} + 326571\nu - 572678 ) / 8418 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{5} + \beta_{4} - 2\beta_{3} - 2\beta_{2} - \beta _1 - 1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 7\beta_{4} - 6\beta_{3} - 11\beta_{2} - 7\beta _1 + 215 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 33\beta_{5} + 14\beta_{4} - 38\beta_{3} - 33\beta_{2} - 74\beta _1 + 41 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 82\beta_{5} - 349\beta_{4} - 542\beta_{3} - 382\beta_{2} - 427\beta _1 + 7101 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 525\beta_{5} + 157\beta_{4} - 758\beta_{3} - 391\beta_{2} - 1567\beta _1 + 1523 ) / 2 \) 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
4.55540
6.84581
−3.12750
−6.17644
1.62104
−3.71831
0 −9.26390 0 0 0 34.7650 0 58.8198 0
1.2 0 −4.45839 0 0 0 −14.7973 0 −7.12278 0
1.3 0 −2.26710 0 0 0 24.0678 0 −21.8603 0
1.4 0 −1.08545 0 0 0 −16.3532 0 −25.8218 0
1.5 0 7.11328 0 0 0 15.3866 0 23.5988 0
1.6 0 7.96155 0 0 0 −0.0689265 0 36.3863 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.4.a.b ✓ 6
4.b odd 2 1 2000.4.a.m 6
5.b even 2 1 500.4.a.c yes 6
5.c odd 4 2 500.4.c.c 12
20.d odd 2 1 2000.4.a.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.4.a.b ✓ 6 1.a even 1 1 trivial
500.4.a.c yes 6 5.b even 2 1
500.4.c.c 12 5.c odd 4 2
2000.4.a.l 6 20.d odd 2 1
2000.4.a.m 6 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 2T_{3}^{5} - 111T_{3}^{4} - 214T_{3}^{3} + 2589T_{3}^{2} + 8222T_{3} + 5756 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(500))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} + \cdots + 5756 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 43 T^{5} + \cdots - 214729 \) Copy content Toggle raw display
$11$ \( T^{6} + 15 T^{5} + \cdots + 730959875 \) Copy content Toggle raw display
$13$ \( T^{6} - 87 T^{5} + \cdots - 95925879 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 5244962896 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 1220233819 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 24927796656 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 11561139063884 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 10595036078116 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 12360260449936 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 2501624923189 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 10915805558000 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 244373573098211 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 39\!\cdots\!31 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 623930671754649 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 34\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 870909015303524 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 12472751663244 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 22\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 30\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
show more
show less