Properties

Label 98.10.a.e
Level $98$
Weight $10$
Character orbit 98.a
Self dual yes
Analytic conductor $50.474$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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 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")
 
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);
 
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 = [2,-32,14] 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2305}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2305}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 16 q^{2} + (5 \beta + 7) q^{3} + 256 q^{4} + (21 \beta + 1365) q^{5} + ( - 80 \beta - 112) q^{6} - 4096 q^{8} + (70 \beta + 37991) q^{9} + ( - 336 \beta - 21840) q^{10} + ( - 1050 \beta + 22470) q^{11}+ \cdots + ( - 38317650 \beta + 684240270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{2} + 14 q^{3} + 512 q^{4} + 2730 q^{5} - 224 q^{6} - 8192 q^{8} + 75982 q^{9} - 43680 q^{10} + 44940 q^{11} + 3584 q^{12} - 100282 q^{13} + 503160 q^{15} + 131072 q^{16} + 870408 q^{17} - 1215712 q^{18}+ \cdots + 1368480540 q^{99}+O(q^{100}) \) 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
−23.5052
24.5052
−16.0000 −233.052 256.000 356.781 3728.83 0 −4096.00 34630.3 −5708.50
1.2 −16.0000 247.052 256.000 2373.22 −3952.83 0 −4096.00 41351.7 −37971.5
\(n\): 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.e 2
7.b odd 2 1 14.10.a.c 2
7.c even 3 2 98.10.c.h 4
7.d odd 6 2 98.10.c.j 4
21.c even 2 1 126.10.a.o 2
28.d even 2 1 112.10.a.c 2
35.c odd 2 1 350.10.a.j 2
35.f even 4 2 350.10.c.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.c 2 7.b odd 2 1
98.10.a.e 2 1.a even 1 1 trivial
98.10.c.h 4 7.c even 3 2
98.10.c.j 4 7.d odd 6 2
112.10.a.c 2 28.d even 2 1
126.10.a.o 2 21.c even 2 1
350.10.a.j 2 35.c odd 2 1
350.10.c.j 4 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 14T_{3} - 57576 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 14T - 57576 \) Copy content Toggle raw display
$5$ \( T^{2} - 2730 T + 846720 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 2036361600 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 2397429256 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 183575251116 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 352577596856 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 115915968000 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 31904129519604 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 2367849772544 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 60225113026444 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 527816477266884 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 271341247682336 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 494746212296124 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 85\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 83\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 68\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 76\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 50\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
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