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Properties

Label 9900.2.a.by
Level $9900$
Weight $2$
Character orbit 9900.a
Self dual yes
Analytic conductor $79.052$
Analytic rank $0$
Dimension $2$
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] = [9900,2,Mod(1,9900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9900, 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("9900.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9900.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,5,0,0,0,-2,0,-4,0,0,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(79.0518980011\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1100)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 3) q^{7} - q^{11} + (2 \beta - 3) q^{13} + ( - \beta - 2) q^{17} - q^{19} + ( - 3 \beta + 3) q^{23} + ( - \beta + 4) q^{29} + (2 \beta - 3) q^{31} + (2 \beta + 3) q^{37} + ( - 4 \beta + 1) q^{41}+ \cdots + ( - 7 \beta + 9) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{7} - 2 q^{11} - 4 q^{13} - 5 q^{17} - 2 q^{19} + 3 q^{23} + 7 q^{29} - 4 q^{31} + 8 q^{37} - 2 q^{41} - 4 q^{47} + 5 q^{49} - 5 q^{53} + 4 q^{59} + 3 q^{61} + 16 q^{67} + 2 q^{71} + 7 q^{73}+ \cdots + 11 q^{97}+O(q^{100}) \) 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
2.30278
−1.30278
0 0 0 0 0 0.697224 0 0 0
1.2 0 0 0 0 0 4.30278 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)
\(11\) \( +1 \)

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 9900.2.a.by 2
3.b odd 2 1 1100.2.a.i yes 2
5.b even 2 1 9900.2.a.bg 2
5.c odd 4 2 9900.2.c.r 4
12.b even 2 1 4400.2.a.bf 2
15.d odd 2 1 1100.2.a.f ✓ 2
15.e even 4 2 1100.2.b.e 4
60.h even 2 1 4400.2.a.bw 2
60.l odd 4 2 4400.2.b.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.2.a.f ✓ 2 15.d odd 2 1
1100.2.a.i yes 2 3.b odd 2 1
1100.2.b.e 4 15.e even 4 2
4400.2.a.bf 2 12.b even 2 1
4400.2.a.bw 2 60.h even 2 1
4400.2.b.r 4 60.l odd 4 2
9900.2.a.bg 2 5.b even 2 1
9900.2.a.by 2 1.a even 1 1 trivial
9900.2.c.r 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9900))\):

\( T_{7}^{2} - 5T_{7} + 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 9 \) Copy content Toggle raw display
\( T_{17}^{2} + 5T_{17} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 3 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$17$ \( T^{2} + 5T + 3 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 27 \) Copy content Toggle raw display
$29$ \( T^{2} - 7T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 3 \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 51 \) Copy content Toggle raw display
$43$ \( T^{2} - 52 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 5T - 75 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$67$ \( (T - 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$73$ \( T^{2} - 7T - 17 \) Copy content Toggle raw display
$79$ \( T^{2} - 9T + 17 \) Copy content Toggle raw display
$83$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$89$ \( T^{2} - 21T + 81 \) Copy content Toggle raw display
$97$ \( T^{2} - 11T - 129 \) Copy content Toggle raw display
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