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Properties

Label 9800.2.a.bp
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
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] = [9800,2,Mod(1,9800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9800.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,3,0,0,0,0,0,6,0,-1,0,2,0,0,0,3,0,-5,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.2533939809\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{3} + 6 q^{9} - q^{11} + 2 q^{13} + 3 q^{17} - 5 q^{19} + 3 q^{23} + 9 q^{27} - 6 q^{29} + q^{31} - 3 q^{33} + 5 q^{37} + 6 q^{39} + 10 q^{41} + 4 q^{43} + q^{47} + 9 q^{51} + 9 q^{53} - 15 q^{57}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
0 3.00000 0 0 0 0 0 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(7\) \( -1 \)

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 9800.2.a.bp 1
5.b even 2 1 392.2.a.a 1
7.b odd 2 1 9800.2.a.b 1
7.d odd 6 2 1400.2.q.g 2
15.d odd 2 1 3528.2.a.k 1
20.d odd 2 1 784.2.a.j 1
35.c odd 2 1 392.2.a.f 1
35.i odd 6 2 56.2.i.a ✓ 2
35.j even 6 2 392.2.i.f 2
35.k even 12 4 1400.2.bh.f 4
40.e odd 2 1 3136.2.a.a 1
40.f even 2 1 3136.2.a.bb 1
60.h even 2 1 7056.2.a.s 1
105.g even 2 1 3528.2.a.r 1
105.o odd 6 2 3528.2.s.o 2
105.p even 6 2 504.2.s.e 2
140.c even 2 1 784.2.a.a 1
140.p odd 6 2 784.2.i.a 2
140.s even 6 2 112.2.i.c 2
280.c odd 2 1 3136.2.a.b 1
280.n even 2 1 3136.2.a.bc 1
280.ba even 6 2 448.2.i.a 2
280.bk odd 6 2 448.2.i.f 2
420.o odd 2 1 7056.2.a.bi 1
420.be odd 6 2 1008.2.s.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a ✓ 2 35.i odd 6 2
112.2.i.c 2 140.s even 6 2
392.2.a.a 1 5.b even 2 1
392.2.a.f 1 35.c odd 2 1
392.2.i.f 2 35.j even 6 2
448.2.i.a 2 280.ba even 6 2
448.2.i.f 2 280.bk odd 6 2
504.2.s.e 2 105.p even 6 2
784.2.a.a 1 140.c even 2 1
784.2.a.j 1 20.d odd 2 1
784.2.i.a 2 140.p odd 6 2
1008.2.s.e 2 420.be odd 6 2
1400.2.q.g 2 7.d odd 6 2
1400.2.bh.f 4 35.k even 12 4
3136.2.a.a 1 40.e odd 2 1
3136.2.a.b 1 280.c odd 2 1
3136.2.a.bb 1 40.f even 2 1
3136.2.a.bc 1 280.n even 2 1
3528.2.a.k 1 15.d odd 2 1
3528.2.a.r 1 105.g even 2 1
3528.2.s.o 2 105.o odd 6 2
7056.2.a.s 1 60.h even 2 1
7056.2.a.bi 1 420.o odd 2 1
9800.2.a.b 1 7.b odd 2 1
9800.2.a.bp 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3} - 3 \) Copy content Toggle raw display
\( T_{11} + 1 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{19} + 5 \) Copy content Toggle raw display
\( T_{23} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T + 5 \) Copy content Toggle raw display
$23$ \( T - 3 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 1 \) Copy content Toggle raw display
$37$ \( T - 5 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 1 \) Copy content Toggle raw display
$53$ \( T - 9 \) Copy content Toggle raw display
$59$ \( T + 3 \) Copy content Toggle raw display
$61$ \( T + 3 \) Copy content Toggle raw display
$67$ \( T + 11 \) Copy content Toggle raw display
$71$ \( T - 16 \) Copy content Toggle raw display
$73$ \( T - 7 \) Copy content Toggle raw display
$79$ \( T + 11 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T - 9 \) Copy content Toggle raw display
$97$ \( T - 6 \) Copy content Toggle raw display
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