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Properties

Label 18.32.a.h
Level $18$
Weight $32$
Character orbit 18.a
Self dual yes
Analytic conductor $109.579$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,32,Mod(1,18)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("18.1"); S:= CuspForms(chi, 32); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 32, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,65536,0,2147483648,26763065700] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(109.578839074\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 246876762 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 25920\sqrt{987507049}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 32768 q^{2} + 1073741824 q^{4} + ( - 140 \beta + 13381532850) q^{5} + ( - 1426 \beta - 11751851055304) q^{7} + 35184372088832 q^{8} + ( - 4587520 \beta + 438486068428800) q^{10} + ( - 7742273 \beta - 12\!\cdots\!92) q^{11}+ \cdots + (10\!\cdots\!44 \beta - 60\!\cdots\!36) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 65536 q^{2} + 2147483648 q^{4} + 26763065700 q^{5} - 23503702110608 q^{7} + 70368744177664 q^{8} + 876972136857600 q^{10} - 25\!\cdots\!84 q^{11} + 18\!\cdots\!24 q^{13} - 77\!\cdots\!44 q^{14} + 23\!\cdots\!52 q^{16}+ \cdots - 12\!\cdots\!72 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
15712.8
−15711.8
32768.0 0 1.07374e9 −1.00652e11 0 −1.29134e13 3.51844e13 0 −3.29817e15
1.2 32768.0 0 1.07374e9 1.27415e11 0 −1.05903e13 3.51844e13 0 4.17514e15
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)

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 18.32.a.h 2
3.b odd 2 1 2.32.a.b ✓ 2
12.b even 2 1 16.32.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.32.a.b ✓ 2 3.b odd 2 1
16.32.a.c 2 12.b even 2 1
18.32.a.h 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 26763065700T_{5} - 12824614473151733437500 \) acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(18))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 32768)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 11\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 29\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 90\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 15\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 54\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 20\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 31\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 16\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 34\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 19\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
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