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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [993,2,Mod(1,993)] 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("993.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(993, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 993 = 3 \cdot 331 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 993.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.92914492071\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} - 10x^{8} + 25x^{7} + 42x^{6} - 67x^{5} - 84x^{4} + 58x^{3} + 61x^{2} - 12x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + \beta_{3} q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{8} + 1) q^{7} + ( - \beta_{6} - \beta_{5} + \beta_{2} + \cdots + 2) q^{8} + q^{9}+ \cdots + ( - \beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 7 q^{2} - 10 q^{3} + 13 q^{4} - 7 q^{6} + 9 q^{7} + 18 q^{8} + 10 q^{9} + 3 q^{10} + 15 q^{11} - 13 q^{12} - 7 q^{13} + 12 q^{14} + 19 q^{16} - 3 q^{17} + 7 q^{18} - 10 q^{19} + 6 q^{20} - 9 q^{21}+ \cdots + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 10x^{8} + 25x^{7} + 42x^{6} - 67x^{5} - 84x^{4} + 58x^{3} + 61x^{2} - 12x - 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} + 4\nu^{8} + 4\nu^{7} - 22\nu^{6} - 7\nu^{5} + 29\nu^{4} + 17\nu^{3} + 9\nu^{2} - 12\nu - 10 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 5\nu^{7} - 2\nu^{6} + 33\nu^{5} - 8\nu^{4} - 66\nu^{3} + 13\nu^{2} + 32\nu - 5 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} - 3\nu^{8} - 9\nu^{7} + 23\nu^{6} + 31\nu^{5} - 52\nu^{4} - 47\nu^{3} + 28\nu^{2} + 20\nu + 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{9} + 3\nu^{8} + 9\nu^{7} - 23\nu^{6} - 31\nu^{5} + 52\nu^{4} + 50\nu^{3} - 34\nu^{2} - 29\nu + 4 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{9} + 5\nu^{8} + 2\nu^{7} - 36\nu^{6} + 17\nu^{5} + 81\nu^{4} - 49\nu^{3} - 56\nu^{2} + 26\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2\nu^{9} + 8\nu^{8} + 11\nu^{7} - 59\nu^{6} - 17\nu^{5} + 145\nu^{4} + 4\nu^{3} - 129\nu^{2} + 6\nu + 28 ) / 3 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2\nu^{9} - 8\nu^{8} - 11\nu^{7} + 56\nu^{6} + 26\nu^{5} - 127\nu^{4} - 46\nu^{3} + 90\nu^{2} + 30\nu - 10 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 2\beta_{2} + 5\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} + 2\beta_{6} + 3\beta_{5} - \beta_{4} + \beta_{3} + 9\beta_{2} + 11\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{9} + 3 \beta_{8} + \beta_{7} + 10 \beta_{6} + 13 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + \cdots + 43 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17 \beta_{9} + 14 \beta_{8} + 3 \beta_{7} + 28 \beta_{6} + 43 \beta_{5} - 18 \beta_{4} + 18 \beta_{3} + \cdots + 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 60 \beta_{9} + 45 \beta_{8} + 16 \beta_{7} + 102 \beta_{6} + 151 \beta_{5} - 65 \beta_{4} + 63 \beta_{3} + \cdots + 432 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 210 \beta_{9} + 162 \beta_{8} + 53 \beta_{7} + 318 \beta_{6} + 502 \beta_{5} - 234 \beta_{4} + \cdots + 1411 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 707 \beta_{9} + 528 \beta_{8} + 203 \beta_{7} + 1069 \beta_{6} + 1679 \beta_{5} - 801 \beta_{4} + \cdots + 4422 \) 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
3.26430
2.20095
1.98290
1.01691
0.500916
−0.423494
−0.895381
−1.30011
−1.60021
−1.74678
−2.26430 −1.00000 3.12705 0.437622 2.26430 −0.686641 −2.55197 1.00000 −0.990906
1.2 −1.20095 −1.00000 −0.557719 −0.661640 1.20095 −2.39309 3.07169 1.00000 0.794597
1.3 −0.982899 −1.00000 −1.03391 −1.72903 0.982899 4.97693 2.98203 1.00000 1.69946
1.4 −0.0169132 −1.00000 −1.99971 3.98729 0.0169132 2.52380 0.0676479 1.00000 −0.0674377
1.5 0.499084 −1.00000 −1.75092 −3.43767 −0.499084 −1.30522 −1.87202 1.00000 −1.71569
1.6 1.42349 −1.00000 0.0263352 −1.23303 −1.42349 −1.18564 −2.80950 1.00000 −1.75521
1.7 1.89538 −1.00000 1.59247 2.42169 −1.89538 5.03877 −0.772428 1.00000 4.59002
1.8 2.30011 −1.00000 3.29050 1.40594 −2.30011 −0.106863 2.96829 1.00000 3.23382
1.9 2.60021 −1.00000 4.76110 −3.29689 −2.60021 2.26624 7.17945 1.00000 −8.57261
1.10 2.74678 −1.00000 5.54481 2.10572 −2.74678 −0.128283 9.73681 1.00000 5.78396
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(331\) \( -1 \)

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 993.2.a.d ✓ 10
3.b odd 2 1 2979.2.a.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
993.2.a.d ✓ 10 1.a even 1 1 trivial
2979.2.a.i 10 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 7T_{2}^{9} + 8T_{2}^{8} + 43T_{2}^{7} - 105T_{2}^{6} - 24T_{2}^{5} + 218T_{2}^{4} - 75T_{2}^{3} - 127T_{2}^{2} + 57T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(993))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 7 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 28 T^{8} + \cdots - 200 \) Copy content Toggle raw display
$7$ \( T^{10} - 9 T^{9} + \cdots + 5 \) Copy content Toggle raw display
$11$ \( T^{10} - 15 T^{9} + \cdots - 972 \) Copy content Toggle raw display
$13$ \( T^{10} + 7 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{10} + 3 T^{9} + \cdots + 101628 \) Copy content Toggle raw display
$19$ \( T^{10} + 10 T^{9} + \cdots + 5620 \) Copy content Toggle raw display
$23$ \( T^{10} - 26 T^{9} + \cdots - 83536 \) Copy content Toggle raw display
$29$ \( T^{10} - 24 T^{9} + \cdots + 18661 \) Copy content Toggle raw display
$31$ \( T^{10} + 7 T^{9} + \cdots + 9320 \) Copy content Toggle raw display
$37$ \( T^{10} - 17 T^{9} + \cdots + 4924364 \) Copy content Toggle raw display
$41$ \( T^{10} - 6 T^{9} + \cdots - 3472055 \) Copy content Toggle raw display
$43$ \( T^{10} - 26 T^{9} + \cdots - 1369476 \) Copy content Toggle raw display
$47$ \( T^{10} - 7 T^{9} + \cdots - 625280 \) Copy content Toggle raw display
$53$ \( T^{10} - 38 T^{9} + \cdots - 822636 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 452066284 \) Copy content Toggle raw display
$61$ \( T^{10} + 41 T^{9} + \cdots - 59492 \) Copy content Toggle raw display
$67$ \( T^{10} - 22 T^{9} + \cdots + 30699068 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 140446865 \) Copy content Toggle raw display
$73$ \( T^{10} + 2 T^{9} + \cdots + 21140748 \) Copy content Toggle raw display
$79$ \( T^{10} - 15 T^{9} + \cdots + 2688156 \) Copy content Toggle raw display
$83$ \( T^{10} - 16 T^{9} + \cdots + 59432175 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 238377600 \) Copy content Toggle raw display
$97$ \( T^{10} + 20 T^{9} + \cdots - 15985204 \) Copy content Toggle raw display
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