Properties

Label 1.62.a.a
Level $1$
Weight $62$
Character orbit 1.a
Self dual yes
Analytic conductor $23.566$
Analytic rank $1$
Dimension $4$
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] = [1,62,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, 62, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 62);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 62 \)
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: \(23.5656183265\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 180363795469121x^{2} + 166129321978984507920x + 2785609847439483545242446300 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{28}\cdot 3^{8}\cdot 5^{2}\cdot 7 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 286578000) q^{2} + (\beta_{2} + 48580 \beta_1 - 143430899755500) q^{3} + (\beta_{3} + 343 \beta_{2} - 838426474 \beta_1 + 11\!\cdots\!32) q^{4}+ \cdots + ( - 19284989328 \beta_{3} + \cdots + 29\!\cdots\!13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 286578000) q^{2} + (\beta_{2} + 48580 \beta_1 - 143430899755500) q^{3} + (\beta_{3} + 343 \beta_{2} - 838426474 \beta_1 + 11\!\cdots\!32) q^{4}+ \cdots + (10\!\cdots\!04 \beta_{3} + \cdots - 78\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 1146312000 q^{2} - 573723599022000 q^{3} + 44\!\cdots\!28 q^{4}+ \cdots + 11\!\cdots\!52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 1146312000 q^{2} - 573723599022000 q^{3} + 44\!\cdots\!28 q^{4}+ \cdots - 31\!\cdots\!76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 180363795469121x^{2} + 166129321978984507920x + 2785609847439483545242446300 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 192\nu - 48 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\nu^{3} + 20695116\nu^{2} - 1765108515928884\nu - 371162118903041598888 ) / 13402613 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -588\nu^{3} + 69567928692\nu^{2} + 184007562081366900\nu - 6347030828361347486393976 ) / 1914659 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 48 ) / 192 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 343\beta_{2} - 265270378\beta _1 + 3324465478086847488 ) / 36864 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -1724593\beta_{3} + 40581291737\beta_{2} + 28699219691868298\beta _1 - 4593138507376746544705536 ) / 36864 \) 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
1.21599e7
4.76281e6
−3.61406e6
−1.33086e7
−2.04812e9 1.78996e14 1.88894e18 −2.27994e20 −3.66606e23 −4.42559e25 8.53860e26 −9.51338e28 4.66958e29
1.2 −6.27881e8 −6.22194e14 −1.91161e18 2.33985e20 3.90664e23 6.25792e25 2.64806e27 2.59952e29 −1.46915e29
1.3 9.80478e8 2.49037e14 −1.34451e18 1.59503e20 2.44176e23 1.63507e25 −3.57909e27 −6.51539e28 1.56389e29
1.4 2.84183e9 −3.79563e14 5.77017e18 −6.89620e20 −1.07865e24 −9.80127e25 9.84504e27 1.68945e28 −1.95979e30
\(n\): 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 1.62.a.a 4
3.b odd 2 1 9.62.a.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.62.a.a 4 1.a even 1 1 trivial
9.62.a.a 4 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{62}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 1146312000 T^{3} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( T^{4} + 573723599022000 T^{3} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 35\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots - 44\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots - 35\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 32\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 87\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 32\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 16\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 47\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 52\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 65\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 20\!\cdots\!84 \) Copy content Toggle raw display
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