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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1587,2,Mod(1,1587)] 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("1587.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1587, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1587 = 3 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1587.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-2,10,8,-8,-2,-19] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.6722588008\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: 10.10.5791333887977.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 12x^{8} + 22x^{7} + 49x^{6} - 84x^{5} - 73x^{4} + 132x^{3} + 17x^{2} - 74x + 23 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 69)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{4} + 1) q^{4} + (\beta_{8} + \beta_{7} + \beta_1 - 1) q^{5} - \beta_1 q^{6} + (\beta_{9} + \beta_{7} - \beta_{5} + \cdots - 3) q^{7} + ( - \beta_{9} - \beta_{8} + \cdots + 2 \beta_{2}) q^{8}+ \cdots + ( - \beta_{8} - \beta_{7} + \cdots - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} + 10 q^{3} + 8 q^{4} - 8 q^{5} - 2 q^{6} - 19 q^{7} - 6 q^{8} + 10 q^{9} - 13 q^{10} - 3 q^{11} + 8 q^{12} - 4 q^{13} - 8 q^{15} - 4 q^{16} - 11 q^{17} - 2 q^{18} - 22 q^{19} - q^{20} - 19 q^{21}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 12x^{8} + 22x^{7} + 49x^{6} - 84x^{5} - 73x^{4} + 132x^{3} + 17x^{2} - 74x + 23 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 11\nu^{2} + 11\nu - 11 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 7\nu^{4} + 11\nu^{3} + 13\nu^{2} - 11\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 2\nu^{7} - 10\nu^{6} + 17\nu^{5} + 32\nu^{4} - 42\nu^{3} - 31\nu^{2} + 27\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} - \nu^{8} - 13\nu^{7} + 9\nu^{6} + 58\nu^{5} - 25\nu^{4} - 100\nu^{3} + 27\nu^{2} + 53\nu - 18 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{9} + 2\nu^{8} + 12\nu^{7} - 21\nu^{6} - 51\nu^{5} + 74\nu^{4} + 90\nu^{3} - 102\nu^{2} - 57\nu + 49 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} - \nu^{8} - 14\nu^{7} + 11\nu^{6} + 68\nu^{5} - 42\nu^{4} - 132\nu^{3} + 69\nu^{2} + 84\nu - 45 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} - 4\nu^{8} - 38\nu^{7} + 39\nu^{6} + 168\nu^{5} - 128\nu^{4} - 293\nu^{3} + 175\nu^{2} + 160\nu - 96 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{9} + 4\nu^{8} + 39\nu^{7} - 41\nu^{6} - 176\nu^{5} + 141\nu^{4} + 313\nu^{3} - 197\nu^{2} - 177\nu + 107 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} + \beta_{4} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - 2\beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} + \beta_{8} - 4\beta_{7} - 7\beta_{6} + 6\beta_{4} - 3\beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{9} + 9\beta_{8} + 3\beta_{7} - 10\beta_{6} - 7\beta_{5} + 8\beta_{4} - 18\beta_{2} + 19\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 25 \beta_{9} + 14 \beta_{8} - 9 \beta_{7} - 45 \beta_{6} - 3 \beta_{5} + 34 \beta_{4} + 2 \beta_{3} + \cdots + 53 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 94 \beta_{9} + 69 \beta_{8} + 36 \beta_{7} - 81 \beta_{6} - 42 \beta_{5} + 56 \beta_{4} + 4 \beta_{3} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 229 \beta_{9} + 135 \beta_{8} + 28 \beta_{7} - 291 \beta_{6} - 37 \beta_{5} + 199 \beta_{4} + 28 \beta_{3} + \cdots + 249 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 738 \beta_{9} + 509 \beta_{8} + 330 \beta_{7} - 607 \beta_{6} - 248 \beta_{5} + 380 \beta_{4} + \cdots + 166 \) 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
2.65972
2.24419
1.61315
0.871916
0.703704
0.568983
−1.15603
−1.41812
−1.98594
−2.10158
−2.65972 1.00000 5.07414 2.90602 −2.65972 −2.29650 −8.17636 1.00000 −7.72922
1.2 −2.24419 1.00000 3.03638 −2.78323 −2.24419 −3.39142 −2.32582 1.00000 6.24609
1.3 −1.61315 1.00000 0.602256 0.526786 −1.61315 −0.483695 2.25477 1.00000 −0.849785
1.4 −0.871916 1.00000 −1.23976 −1.25508 −0.871916 2.98232 2.82480 1.00000 1.09432
1.5 −0.703704 1.00000 −1.50480 1.68624 −0.703704 −3.24908 2.46634 1.00000 −1.18661
1.6 −0.568983 1.00000 −1.67626 −2.70241 −0.568983 −3.38601 2.09173 1.00000 1.53762
1.7 1.15603 1.00000 −0.663591 −1.73011 1.15603 0.701139 −3.07920 1.00000 −2.00006
1.8 1.41812 1.00000 0.0110547 0.849430 1.41812 −5.06334 −2.82056 1.00000 1.20459
1.9 1.98594 1.00000 1.94394 −2.04789 1.98594 −3.23167 −0.111323 1.00000 −4.06699
1.10 2.10158 1.00000 2.41664 −3.44976 2.10158 −1.58175 0.875613 1.00000 −7.24995
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

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 1587.2.a.t 10
3.b odd 2 1 4761.2.a.bu 10
23.b odd 2 1 1587.2.a.u 10
23.d odd 22 2 69.2.e.c ✓ 20
69.c even 2 1 4761.2.a.bt 10
69.g even 22 2 207.2.i.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.e.c ✓ 20 23.d odd 22 2
207.2.i.d 20 69.g even 22 2
1587.2.a.t 10 1.a even 1 1 trivial
1587.2.a.u 10 23.b odd 2 1
4761.2.a.bt 10 69.c even 2 1
4761.2.a.bu 10 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1587))\):

\( T_{2}^{10} + 2T_{2}^{9} - 12T_{2}^{8} - 22T_{2}^{7} + 49T_{2}^{6} + 84T_{2}^{5} - 73T_{2}^{4} - 132T_{2}^{3} + 17T_{2}^{2} + 74T_{2} + 23 \) Copy content Toggle raw display
\( T_{5}^{10} + 8 T_{5}^{9} + 8 T_{5}^{8} - 84 T_{5}^{7} - 219 T_{5}^{6} + 109 T_{5}^{5} + 759 T_{5}^{4} + \cdots + 253 \) Copy content Toggle raw display
\( T_{7}^{10} + 19 T_{7}^{9} + 137 T_{7}^{8} + 405 T_{7}^{7} - 111 T_{7}^{6} - 4022 T_{7}^{5} - 10621 T_{7}^{4} + \cdots + 2243 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 2 T^{9} + \cdots + 23 \) Copy content Toggle raw display
$3$ \( (T - 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 8 T^{9} + \cdots + 253 \) Copy content Toggle raw display
$7$ \( T^{10} + 19 T^{9} + \cdots + 2243 \) Copy content Toggle raw display
$11$ \( T^{10} + 3 T^{9} + \cdots - 461 \) Copy content Toggle raw display
$13$ \( T^{10} + 4 T^{9} + \cdots - 5279 \) Copy content Toggle raw display
$17$ \( T^{10} + 11 T^{9} + \cdots + 4663 \) Copy content Toggle raw display
$19$ \( T^{10} + 22 T^{9} + \cdots + 645919 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} + 5 T^{9} + \cdots + 1995907 \) Copy content Toggle raw display
$31$ \( T^{10} + 7 T^{9} + \cdots - 28488637 \) Copy content Toggle raw display
$37$ \( T^{10} + 35 T^{9} + \cdots - 430387 \) Copy content Toggle raw display
$41$ \( T^{10} - 185 T^{8} + \cdots + 815077 \) Copy content Toggle raw display
$43$ \( T^{10} + 28 T^{9} + \cdots - 11465707 \) Copy content Toggle raw display
$47$ \( T^{10} - 9 T^{9} + \cdots + 87731039 \) Copy content Toggle raw display
$53$ \( T^{10} + 34 T^{9} + \cdots + 910229 \) Copy content Toggle raw display
$59$ \( T^{10} + 2 T^{9} + \cdots - 240109 \) Copy content Toggle raw display
$61$ \( T^{10} + 49 T^{9} + \cdots - 23662651 \) Copy content Toggle raw display
$67$ \( T^{10} + 26 T^{9} + \cdots + 1627 \) Copy content Toggle raw display
$71$ \( T^{10} - 15 T^{9} + \cdots + 42681629 \) Copy content Toggle raw display
$73$ \( T^{10} - 14 T^{9} + \cdots + 3389 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 199685509 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 893491919 \) Copy content Toggle raw display
$89$ \( T^{10} - 15 T^{9} + \cdots + 24223 \) Copy content Toggle raw display
$97$ \( T^{10} + 22 T^{9} + \cdots + 6977189 \) Copy content Toggle raw display
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