Properties

Label 1.50.a.a
Level $1$
Weight $50$
Character orbit 1.a
Self dual yes
Analytic conductor $15.207$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1,50,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 50, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 50);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 50 \)
Character orbit: \([\chi]\) \(=\) 1.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: \(15.2066205099\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 27962089502x + 71708842875120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2} \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 8075056) q^{2} + (\beta_{2} + 4110 \beta_1 - 108984897468) q^{3} + ( - 96 \beta_{2} - 399312 \beta_1 + 10500922661632) q^{4} + ( - 192420 \beta_{2} - 830946040 \beta_1 + 21\!\cdots\!50) q^{5}+ \cdots + ( - 286203921912 \beta_{2} + \cdots + 11\!\cdots\!73) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 8075056) q^{2} + (\beta_{2} + 4110 \beta_1 - 108984897468) q^{3} + ( - 96 \beta_{2} - 399312 \beta_1 + 10500922661632) q^{4} + ( - 192420 \beta_{2} - 830946040 \beta_1 + 21\!\cdots\!50) q^{5}+ \cdots + ( - 13\!\cdots\!49 \beta_{2} + \cdots - 37\!\cdots\!84) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 24225168 q^{2} - 326954692404 q^{3} + 31502767984896 q^{4} + 63\!\cdots\!50 q^{5}+ \cdots + 34\!\cdots\!19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 24225168 q^{2} - 326954692404 q^{3} + 31502767984896 q^{4} + 63\!\cdots\!50 q^{5}+ \cdots - 11\!\cdots\!52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 27962089502x + 71708842875120 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -24\nu^{2} + 58056\nu + 447393412688 ) / 14051 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 39888\nu^{2} + 59389824528\nu - 743587680658656 ) / 14051 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 1662\beta _1 + 1411200 ) / 4233600 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2419\beta_{2} - 2474576022\beta _1 + 78920201411856000 ) / 4233600 \) Copy content Toggle raw display

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
−168486.
165922.
2565.11
−2.54182e7 −8.64745e11 8.31363e13 1.67413e17 2.19803e19 −3.42668e20 1.21960e22 5.08484e23 −4.25535e24
1.2 −2.25719e7 5.57972e11 −5.34581e13 −1.06460e17 −1.25945e19 5.68014e20 1.39135e22 7.20337e22 2.40300e24
1.3 2.37650e7 −2.01823e10 1.82457e12 2.93049e15 −4.79631e17 −2.24836e20 −1.33351e22 −2.38892e23 6.96431e22
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1.50.a.a 3
3.b odd 2 1 9.50.a.a 3
4.b odd 2 1 16.50.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.50.a.a 3 1.a even 1 1 trivial
9.50.a.a 3 3.b odd 2 1
16.50.a.c 3 4.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{50}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 24225168 T^{2} + \cdots - 13\!\cdots\!72 \) Copy content Toggle raw display
$3$ \( T^{3} + 326954692404 T^{2} + \cdots - 97\!\cdots\!04 \) Copy content Toggle raw display
$5$ \( T^{3} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots - 43\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots - 32\!\cdots\!88 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots - 71\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots - 32\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots - 81\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 52\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 31\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 42\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 92\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 11\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 46\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 43\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 36\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 67\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 88\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 12\!\cdots\!32 \) Copy content Toggle raw display
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