Properties

Label 2.58.a.b
Level $2$
Weight $58$
Character orbit 2.a
Self dual yes
Analytic conductor $41.153$
Analytic rank $0$
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] = [2,58,Mod(1,2)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 58, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 58);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 58 \)
Character orbit: \([\chi]\) \(=\) 2.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: \(41.1532867302\)
Analytic rank: \(0\)
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: \( x^{3} - x^{2} - 53481184397676701640x - 17532654301480487291805255300 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{6}\cdot 5^{2}\cdot 7\cdot 11\cdot 19 \)
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.

\(f(q)\) \(=\) \( q + 268435456 q^{2} + (\beta_1 - 8441370341596) q^{3} + 72\!\cdots\!36 q^{4}+ \cdots + (15295230 \beta_{2} - 19724693224692 \beta_1 + 66\!\cdots\!53) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 268435456 q^{2} + (\beta_1 - 8441370341596) q^{3} + 72\!\cdots\!36 q^{4}+ \cdots + (36\!\cdots\!40 \beta_{2} + \cdots - 79\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 805306368 q^{2} - 25324111024788 q^{3} + 21\!\cdots\!08 q^{4}+ \cdots + 20\!\cdots\!59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 805306368 q^{2} - 25324111024788 q^{3} + 21\!\cdots\!08 q^{4}+ \cdots - 23\!\cdots\!32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 53481184397676701640x - 17532654301480487291805255300 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 48\nu^{2} + 95334788592\nu - 1711397900757432715360 ) / 29989271 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -60811680\nu^{2} + 461477039347231200\nu + 2168187114254513202629296960 ) / 29989271 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 1266910\beta _1 + 6471843840 ) / 19415531520 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -220682381\beta_{2} + 1068233887377850\beta _1 + 76915971955557349294130626560 ) / 2157281280 \) 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
−3.28491e8
−7.14330e9
7.47179e9
2.68435e8 −6.63799e13 7.20576e16 1.62362e20 −1.78187e22 −8.85183e23 1.93428e25 2.83625e27 4.35838e28
1.2 2.68435e8 −6.54464e12 7.20576e16 −1.05418e20 −1.75681e21 −1.81149e24 1.93428e25 −1.52721e27 −2.82980e28
1.3 2.68435e8 4.76005e13 7.20576e16 5.57569e19 1.27776e22 1.36099e24 1.93428e25 6.95760e26 1.49671e28
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2.58.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.58.a.b 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 25324111024788 T_{3}^{2} + \cdots - 20\!\cdots\!64 \) acting on \(S_{58}^{\mathrm{new}}(\Gamma_0(2))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 268435456)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 25324111024788 T^{2} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
$5$ \( T^{3} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots - 21\!\cdots\!52 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots + 13\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots - 49\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots + 27\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 29\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots - 94\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 33\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 37\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 49\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 55\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 57\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 43\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 32\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 50\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 10\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 56\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 45\!\cdots\!52 \) Copy content Toggle raw display
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