Properties

Label 10.32.a.a
Level $10$
Weight $32$
Character orbit 10.a
Self dual yes
Analytic conductor $60.877$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,32,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 32, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 32);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(60.8771328190\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 17573188320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5 \)
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 \(\beta = 90\sqrt{70292753281}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 32768 q^{2} + ( - \beta - 2952378) q^{3} + 1073741824 q^{4} + 30517578125 q^{5} + (32768 \beta + 96743522304) q^{6} + ( - 346983 \beta + 9008282546606) q^{7} - 35184372088832 q^{8} + (5904756 \beta - 39585558852963) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 32768 q^{2} + ( - \beta - 2952378) q^{3} + 1073741824 q^{4} + 30517578125 q^{5} + (32768 \beta + 96743522304) q^{6} + ( - 346983 \beta + 9008282546606) q^{7} - 35184372088832 q^{8} + (5904756 \beta - 39585558852963) q^{9} - 10\!\cdots\!00 q^{10}+ \cdots + ( - 60\!\cdots\!10 \beta + 10\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 65536 q^{2} - 5904756 q^{3} + 2147483648 q^{4} + 61035156250 q^{5} + 193487044608 q^{6} + 18016565093212 q^{7} - 70368744177664 q^{8} - 79171117705926 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 65536 q^{2} - 5904756 q^{3} + 2147483648 q^{4} + 61035156250 q^{5} + 193487044608 q^{6} + 18016565093212 q^{7} - 70368744177664 q^{8} - 79171117705926 q^{9} - 20\!\cdots\!00 q^{10}+ \cdots + 21\!\cdots\!88 q^{99}+O(q^{100}) \) 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
132564.
−132563.
−32768.0 −2.68139e7 1.07374e9 3.05176e10 8.78637e11 7.28747e11 −3.51844e13 1.01311e14 −1.00000e15
1.2 −32768.0 2.09091e7 1.07374e9 3.05176e10 −6.85150e11 1.72878e13 −3.51844e13 −1.80482e14 −1.00000e15
\(n\): e.g. 2-40 or 990-1000
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 10.32.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.32.a.a 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5904756T_{3} - 560654765721216 \) acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 32768)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + \cdots - 560654765721216 \) Copy content Toggle raw display
$5$ \( (T - 30517578125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 47\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 45\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 46\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 12\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 56\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 65\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 51\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 57\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 59\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 63\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 56\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
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