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Properties

Label 98.10.c.d
Level $98$
Weight $10$
Character orbit 98.c
Analytic conductor $50.474$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,10,Mod(67,98)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("98.67"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-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: no
Analytic conductor: \(50.4735119441\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 16 \zeta_{6} q^{2} + ( - 170 \zeta_{6} + 170) q^{3} + (256 \zeta_{6} - 256) q^{4} + 544 \zeta_{6} q^{5} - 2720 q^{6} + 4096 q^{8} - 9217 \zeta_{6} q^{9} + ( - 8704 \zeta_{6} + 8704) q^{10} + (48824 \zeta_{6} - 48824) q^{11} + \cdots + 450010808 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} + 170 q^{3} - 256 q^{4} + 544 q^{5} - 5440 q^{6} + 8192 q^{8} - 9217 q^{9} + 8704 q^{10} - 48824 q^{11} + 43520 q^{12} + 31752 q^{13} + 184960 q^{15} - 65536 q^{16} - 21418 q^{17} - 147472 q^{18}+ \cdots + 900021616 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.866025i
0.500000 − 0.866025i
−8.00000 − 13.8564i 85.0000 − 147.224i −128.000 + 221.703i 272.000 + 471.118i −2720.00 0 4096.00 −4608.50 − 7982.16i 4352.00 − 7537.89i
79.1 −8.00000 + 13.8564i 85.0000 + 147.224i −128.000 − 221.703i 272.000 − 471.118i −2720.00 0 4096.00 −4608.50 + 7982.16i 4352.00 + 7537.89i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.10.c.d 2
7.b odd 2 1 98.10.c.a 2
7.c even 3 1 98.10.a.b 1
7.c even 3 1 inner 98.10.c.d 2
7.d odd 6 1 14.10.a.b ✓ 1
7.d odd 6 1 98.10.c.a 2
21.g even 6 1 126.10.a.a 1
28.f even 6 1 112.10.a.a 1
35.i odd 6 1 350.10.a.a 1
35.k even 12 2 350.10.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.b ✓ 1 7.d odd 6 1
98.10.a.b 1 7.c even 3 1
98.10.c.a 2 7.b odd 2 1
98.10.c.a 2 7.d odd 6 1
98.10.c.d 2 1.a even 1 1 trivial
98.10.c.d 2 7.c even 3 1 inner
112.10.a.a 1 28.f even 6 1
126.10.a.a 1 21.g even 6 1
350.10.a.a 1 35.i odd 6 1
350.10.c.d 2 35.k even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 170T_{3} + 28900 \) acting on \(S_{10}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$3$ \( T^{2} - 170T + 28900 \) Copy content Toggle raw display
$5$ \( T^{2} - 544T + 295936 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 2383782976 \) Copy content Toggle raw display
$13$ \( (T - 15876)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 21418 T + 458730724 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 513243288100 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 6100900000000 \) Copy content Toggle raw display
$29$ \( (T - 5556826)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 33632437225104 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 15166585024900 \) Copy content Toggle raw display
$41$ \( (T - 6360858)^{2} \) Copy content Toggle raw display
$43$ \( (T + 18701296)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 35\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 87\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T + 95633536)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 93\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 24\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( (T - 371486962)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T + 758016742)^{2} \) Copy content Toggle raw display
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