Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9522,2,Mod(1,9522)] 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("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9522.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,10,0,10,-10,0,2,10,0,-10,-12,0,0,2,0,10,-24,0,-8,-10,0,-12, 0,0,8,0,0,2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(76.0335528047\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: 10.10.52900342088704.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 18x^{8} + 123x^{6} - 390x^{4} + 548x^{2} - 241 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 414)
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 a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + (\beta_{2} - 1) q^{5} + (\beta_{9} + \beta_{5} + \cdots - \beta_{2}) q^{7} + q^{8} + (\beta_{2} - 1) q^{10} + (\beta_{6} + \beta_{3} + \beta_1 - 1) q^{11} + ( - \beta_{8} - \beta_{5} + \cdots - \beta_1) q^{13}+ \cdots + (2 \beta_{9} - 2 \beta_{8} - \beta_{7} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 10 q^{4} - 10 q^{5} + 2 q^{7} + 10 q^{8} - 10 q^{10} - 12 q^{11} + 2 q^{14} + 10 q^{16} - 24 q^{17} - 8 q^{19} - 10 q^{20} - 12 q^{22} + 8 q^{25} + 2 q^{28} + 4 q^{29} + 18 q^{31} + 10 q^{32}+ \cdots + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 18x^{8} + 123x^{6} - 390x^{4} + 548x^{2} - 241 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{8} - 14\nu^{6} + 67\nu^{4} - 122\nu^{2} + 59 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{9} - 14\nu^{7} + 67\nu^{5} - 122\nu^{3} + 59\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{8} - 14\nu^{6} + 67\nu^{4} - 123\nu^{2} + 63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{9} - 27\nu^{7} + 124\nu^{5} - 216\nu^{3} + 101\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\nu^{8} - 27\nu^{6} + 124\nu^{4} - 217\nu^{2} + 105 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 2\nu^{8} - 27\nu^{6} + 125\nu^{4} - 223\nu^{2} + 111 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 2\nu^{9} - 27\nu^{7} + 125\nu^{5} - 223\nu^{3} + 111\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{7} - 6\beta_{5} + 6\beta_{3} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} - \beta_{6} + 7\beta_{2} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{8} - 9\beta_{7} - 33\beta_{5} + 31\beta_{3} + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10\beta_{9} - 9\beta_{6} - 2\beta_{4} + 42\beta_{2} + 85\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 73\beta_{8} - 59\beta_{7} - 182\beta_{5} + 155\beta_{3} + 413 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 73\beta_{9} - 59\beta_{6} - 27\beta_{4} + 241\beta_{2} + 413\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−2.32333
0.878233
1.70641
−2.13218
−2.09112
2.09112
2.13218
−1.70641
−0.878233
2.32333
1.00000 0 1.00000 −4.24774 0 0.296223 1.00000 0 −4.24774
1.2 1.00000 0 1.00000 −3.83556 0 5.08193 1.00000 0 −3.83556
1.3 1.00000 0 1.00000 −2.85684 0 −4.48349 1.00000 0 −2.85684
1.4 1.00000 0 1.00000 −2.16460 0 3.76937 1.00000 0 −2.16460
1.5 1.00000 0 1.00000 −1.77954 0 −0.833869 1.00000 0 −1.77954
1.6 1.00000 0 1.00000 −0.220460 0 3.45331 1.00000 0 −0.220460
1.7 1.00000 0 1.00000 0.164599 0 −3.20012 1.00000 0 0.164599
1.8 1.00000 0 1.00000 0.856843 0 2.82183 1.00000 0 0.856843
1.9 1.00000 0 1.00000 1.83556 0 −1.24396 1.00000 0 1.83556
1.10 1.00000 0 1.00000 2.24774 0 −3.66124 1.00000 0 2.24774
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(23\) \( -1 \)

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 9522.2.a.ci 10
3.b odd 2 1 9522.2.a.ch 10
23.b odd 2 1 9522.2.a.cj 10
23.d odd 22 2 414.2.i.g ✓ 20
69.c even 2 1 9522.2.a.cg 10
69.g even 22 2 414.2.i.h yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.i.g ✓ 20 23.d odd 22 2
414.2.i.h yes 20 69.g even 22 2
9522.2.a.cg 10 69.c even 2 1
9522.2.a.ch 10 3.b odd 2 1
9522.2.a.ci 10 1.a even 1 1 trivial
9522.2.a.cj 10 23.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9522))\):

\( T_{5}^{10} + 10 T_{5}^{9} + 21 T_{5}^{8} - 72 T_{5}^{7} - 269 T_{5}^{6} + 66 T_{5}^{5} + 822 T_{5}^{4} + \cdots + 23 \) Copy content Toggle raw display
\( T_{7}^{10} - 2 T_{7}^{9} - 51 T_{7}^{8} + 80 T_{7}^{7} + 907 T_{7}^{6} - 1000 T_{7}^{5} - 6646 T_{7}^{4} + \cdots - 3013 \) Copy content Toggle raw display
\( T_{11}^{10} + 12 T_{11}^{9} + 21 T_{11}^{8} - 258 T_{11}^{7} - 1336 T_{11}^{6} - 2178 T_{11}^{5} + \cdots + 1 \) Copy content Toggle raw display
\( T_{29}^{10} - 4 T_{29}^{9} - 135 T_{29}^{8} + 604 T_{29}^{7} + 4985 T_{29}^{6} - 22662 T_{29}^{5} + \cdots + 132703 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 10 T^{9} + \cdots + 23 \) Copy content Toggle raw display
$7$ \( T^{10} - 2 T^{9} + \cdots - 3013 \) Copy content Toggle raw display
$11$ \( T^{10} + 12 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{10} - 74 T^{8} + \cdots - 21317 \) Copy content Toggle raw display
$17$ \( T^{10} + 24 T^{9} + \cdots - 2507 \) Copy content Toggle raw display
$19$ \( T^{10} + 8 T^{9} + \cdots + 45737 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} - 4 T^{9} + \cdots + 132703 \) Copy content Toggle raw display
$31$ \( T^{10} - 18 T^{9} + \cdots - 90157 \) Copy content Toggle raw display
$37$ \( T^{10} + 12 T^{9} + \cdots - 27301 \) Copy content Toggle raw display
$41$ \( T^{10} - 28 T^{9} + \cdots - 2797387 \) Copy content Toggle raw display
$43$ \( T^{10} + 8 T^{9} + \cdots - 699931 \) Copy content Toggle raw display
$47$ \( T^{10} + 16 T^{9} + \cdots + 1948123 \) Copy content Toggle raw display
$53$ \( T^{10} + 34 T^{9} + \cdots + 96471 \) Copy content Toggle raw display
$59$ \( T^{10} + 22 T^{9} + \cdots - 47391707 \) Copy content Toggle raw display
$61$ \( T^{10} + 30 T^{9} + \cdots - 88955329 \) Copy content Toggle raw display
$67$ \( T^{10} + 18 T^{9} + \cdots - 6040519 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 844205317 \) Copy content Toggle raw display
$73$ \( T^{10} + 20 T^{9} + \cdots - 21261899 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 121460723 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 174373321 \) Copy content Toggle raw display
$89$ \( T^{10} + 44 T^{9} + \cdots + 87862093 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 909872503 \) Copy content Toggle raw display
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