↑

Properties

Label 50.20.a.l
Level $50$
Weight $20$
Character orbit 50.a
Self dual yes
Analytic conductor $114.408$
Analytic rank $0$
Dimension $5$
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] = [50,20,Mod(1,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(114.408348278\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 50342212x^{3} - 29807836576x^{2} + 380029834565580x - 131063686211973200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{3}\cdot 5^{12} \)
Twist minimal: no (minimal twist has level 10)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 512 q^{2} + ( - \beta_1 + 4522) q^{3} + 262144 q^{4} + ( - 512 \beta_1 + 2315264) q^{6} + ( - \beta_{2} + 55 \beta_1 + 13571746) q^{7} + 134217728 q^{8} + (2 \beta_{4} + 11 \beta_{3} + \cdots + 871875497) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 512 q^{2} + ( - \beta_1 + 4522) q^{3} + 262144 q^{4} + ( - 512 \beta_1 + 2315264) q^{6} + ( - \beta_{2} + 55 \beta_1 + 13571746) q^{7} + 134217728 q^{8} + (2 \beta_{4} + 11 \beta_{3} + \cdots + 871875497) q^{9}+ \cdots + ( - 10093993238 \beta_{4} + \cdots - 41\!\cdots\!16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2560 q^{2} + 22610 q^{3} + 1310720 q^{4} + 11576320 q^{6} + 67858730 q^{7} + 671088640 q^{8} + 4359377485 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2560 q^{2} + 22610 q^{3} + 1310720 q^{4} + 11576320 q^{6} + 67858730 q^{7} + 671088640 q^{8} + 4359377485 q^{9} - 1482612940 q^{11} + 5927075840 q^{12} + 71907610860 q^{13} + 34743669760 q^{14} + 343597383680 q^{16} + 661922159680 q^{17} + 2232001272320 q^{18} + 2098493918100 q^{19} - 250612143740 q^{21} - 759097825280 q^{22} - 4809032390090 q^{23} + 3034662830080 q^{24} + 36816696760320 q^{26} - 4930143444100 q^{27} + 17788758917120 q^{28} - 109680310209150 q^{29} - 333560793331840 q^{31} + 175921860444160 q^{32} + 12\!\cdots\!20 q^{33}+ \cdots - 20\!\cdots\!80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 50342212x^{3} - 29807836576x^{2} + 380029834565580x - 131063686211973200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 10\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 50410000 \nu^{4} - 407916981500 \nu^{3} - 934308009252300 \nu^{2} + \cdots - 96\!\cdots\!60 ) / 161472006273897 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 111725000 \nu^{4} + 14306480500 \nu^{3} + \cdots - 32\!\cdots\!60 ) / 161472006273897 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 488462500 \nu^{4} + 941106811000 \nu^{3} + \cdots + 41\!\cdots\!00 ) / 161472006273897 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{4} + 11\beta_{3} + 5\beta_{2} + 8881\beta _1 + 2013688480 ) / 100 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 39106\beta_{4} + 124345\beta_{3} - 103340\beta_{2} + 1762155763\beta _1 + 8942350972800 ) / 500 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2508934891 \beta_{4} + 10127889067 \beta_{3} + 6143578903 \beta_{2} + 20257241057249 \beta _1 + 17\!\cdots\!00 ) / 2500 \) 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
6856.57
2384.91
361.353
−3867.00
−5735.83
512.000 −64043.7 262144. 0 −3.27904e7 1.31313e6 1.34218e8 2.93934e9 0
1.2 512.000 −19327.1 262144. 0 −9.89548e6 −3.24295e7 1.34218e8 −7.88724e8 0
1.3 512.000 908.466 262144. 0 465135. 4.96199e7 1.34218e8 −1.16144e9 0
1.4 512.000 43192.0 262144. 0 2.21143e7 2.08285e8 1.34218e8 7.03291e8 0
1.5 512.000 61880.3 262144. 0 3.16827e7 −1.58929e8 1.34218e8 2.66691e9 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 50.20.a.l 5
5.b even 2 1 50.20.a.k 5
5.c odd 4 2 10.20.b.a ✓ 10
15.e even 4 2 90.20.c.b 10
20.e even 4 2 80.20.c.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.20.b.a ✓ 10 5.c odd 4 2
50.20.a.k 5 5.b even 2 1
50.20.a.l 5 1.a even 1 1 trivial
80.20.c.c 10 20.e even 4 2
90.20.c.b 10 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 22610 T_{3}^{4} - 4829736360 T_{3}^{3} + 97177400928720 T_{3}^{2} + \cdots - 30\!\cdots\!32 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(50))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 512)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + \cdots - 30\!\cdots\!32 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + \cdots - 69\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{5} + \cdots - 23\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{5} + \cdots - 14\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 51\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots + 15\!\cdots\!68 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots - 60\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 22\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots - 13\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 35\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 67\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 67\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots - 30\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots - 63\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots + 99\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots + 24\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 85\!\cdots\!76 \) Copy content Toggle raw display
show more
show less