↑

Properties

Label 98.10.a.b
Level $98$
Weight $10$
Character orbit 98.a
Self dual yes
Analytic conductor $50.474$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,16,-170] 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: \(50.4735119441\)
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)
 
\(f(q)\) \(=\) \( q + 16 q^{2} - 170 q^{3} + 256 q^{4} - 544 q^{5} - 2720 q^{6} + 4096 q^{8} + 9217 q^{9} - 8704 q^{10} + 48824 q^{11} - 43520 q^{12} + 15876 q^{13} + 92480 q^{15} + 65536 q^{16} + 21418 q^{17} + 147472 q^{18}+ \cdots + 450010808 q^{99}+O(q^{100}) \) 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
16.0000 −170.000 256.000 −544.000 −2720.00 0 4096.00 9217.00 −8704.00
\(n\): 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 98.10.a.b 1
7.b odd 2 1 14.10.a.b ✓ 1
7.c even 3 2 98.10.c.d 2
7.d odd 6 2 98.10.c.a 2
21.c even 2 1 126.10.a.a 1
28.d even 2 1 112.10.a.a 1
35.c odd 2 1 350.10.a.a 1
35.f even 4 2 350.10.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.b ✓ 1 7.b odd 2 1
98.10.a.b 1 1.a even 1 1 trivial
98.10.c.a 2 7.d odd 6 2
98.10.c.d 2 7.c even 3 2
112.10.a.a 1 28.d even 2 1
126.10.a.a 1 21.c even 2 1
350.10.a.a 1 35.c odd 2 1
350.10.c.d 2 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 170 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16 \) Copy content Toggle raw display
$3$ \( T + 170 \) Copy content Toggle raw display
$5$ \( T + 544 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 48824 \) Copy content Toggle raw display
$13$ \( T - 15876 \) Copy content Toggle raw display
$17$ \( T - 21418 \) Copy content Toggle raw display
$19$ \( T - 716410 \) Copy content Toggle raw display
$23$ \( T + 2470000 \) Copy content Toggle raw display
$29$ \( T - 5556826 \) Copy content Toggle raw display
$31$ \( T + 5799348 \) Copy content Toggle raw display
$37$ \( T + 3894430 \) Copy content Toggle raw display
$41$ \( T - 6360858 \) Copy content Toggle raw display
$43$ \( T + 18701296 \) Copy content Toggle raw display
$47$ \( T + 56539068 \) Copy content Toggle raw display
$53$ \( T + 59894682 \) Copy content Toggle raw display
$59$ \( T + 165629662 \) Copy content Toggle raw display
$61$ \( T + 51419016 \) Copy content Toggle raw display
$67$ \( T - 93546508 \) Copy content Toggle raw display
$71$ \( T + 95633536 \) Copy content Toggle raw display
$73$ \( T + 306496402 \) Copy content Toggle raw display
$79$ \( T - 496474152 \) Copy content Toggle raw display
$83$ \( T - 371486962 \) Copy content Toggle raw display
$89$ \( T - 165482550 \) Copy content Toggle raw display
$97$ \( T + 758016742 \) Copy content Toggle raw display
show more
show less