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Properties

Label 98.18.a.b
Level $98$
Weight $18$
Character orbit 98.a
Self dual yes
Analytic conductor $179.558$
Analytic rank $1$
Dimension $1$
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] = [98,18,Mod(1,98)] 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("98.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,256,-4626] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(179.557645598\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
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)
 
\(f(q)\) \(=\) \( q + 256 q^{2} - 4626 q^{3} + 65536 q^{4} + 851700 q^{5} - 1184256 q^{6} + 16777216 q^{8} - 107740287 q^{9} + 218035200 q^{10} - 586048992 q^{11} - 303169536 q^{12} + 1042966288 q^{13} - 3939964200 q^{15}+ \cdots + 63\!\cdots\!04 q^{99}+O(q^{100}) \) 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.

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
0
256.000 −4626.00 65536.0 851700. −1.18426e6 0 1.67772e7 −1.07740e8 2.18035e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)

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 98.18.a.b 1
7.b odd 2 1 14.18.a.a ✓ 1
28.d even 2 1 112.18.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.18.a.a ✓ 1 7.b odd 2 1
98.18.a.b 1 1.a even 1 1 trivial
112.18.a.a 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 4626 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(98))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 256 \) Copy content Toggle raw display
$3$ \( T + 4626 \) Copy content Toggle raw display
$5$ \( T - 851700 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 586048992 \) Copy content Toggle raw display
$13$ \( T - 1042966288 \) Copy content Toggle raw display
$17$ \( T - 17187488802 \) Copy content Toggle raw display
$19$ \( T - 35251814482 \) Copy content Toggle raw display
$23$ \( T - 226463988840 \) Copy content Toggle raw display
$29$ \( T + 3381208637406 \) Copy content Toggle raw display
$31$ \( T + 257228086436 \) Copy content Toggle raw display
$37$ \( T + 40457204426662 \) Copy content Toggle raw display
$41$ \( T - 29013168626274 \) Copy content Toggle raw display
$43$ \( T - 12667778737448 \) Copy content Toggle raw display
$47$ \( T + 286872943920924 \) Copy content Toggle raw display
$53$ \( T - 564480420537078 \) Copy content Toggle raw display
$59$ \( T + 1802377718625462 \) Copy content Toggle raw display
$61$ \( T - 668064962693740 \) Copy content Toggle raw display
$67$ \( T - 332890586370548 \) Copy content Toggle raw display
$71$ \( T - 4451829225077376 \) Copy content Toggle raw display
$73$ \( T - 6135974687950990 \) Copy content Toggle raw display
$79$ \( T - 778901092563704 \) Copy content Toggle raw display
$83$ \( T - 8884730852317194 \) Copy content Toggle raw display
$89$ \( T + 29\!\cdots\!90 \) Copy content Toggle raw display
$97$ \( T - 44\!\cdots\!30 \) Copy content Toggle raw display
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