Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8673,2,Mod(1,8673)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8673, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8673.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8673 = 3 \cdot 7^{2} \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8673.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(69.2542536731\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} - 17 x^{10} + 52 x^{9} + 101 x^{8} - 316 x^{7} - 260 x^{6} + 830 x^{5} + 287 x^{4} + \cdots - 61 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1239) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{12} - 3 x^{11} - 17 x^{10} + 52 x^{9} + 101 x^{8} - 316 x^{7} - 260 x^{6} + 830 x^{5} + 287 x^{4} + \cdots - 61 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{11} + 2 \nu^{10} - 21 \nu^{9} - 25 \nu^{8} + 158 \nu^{7} + 47 \nu^{6} - 529 \nu^{5} + 327 \nu^{4} + \cdots + 449 ) / 49 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 4 \nu^{11} + 6 \nu^{10} + 70 \nu^{9} - 89 \nu^{8} - 422 \nu^{7} + 393 \nu^{6} + 1010 \nu^{5} + \cdots + 255 ) / 49 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 6 \nu^{11} - 16 \nu^{10} - 91 \nu^{9} + 249 \nu^{8} + 444 \nu^{7} - 1258 \nu^{6} - 836 \nu^{5} + \cdots + 118 ) / 49 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 6 \nu^{11} - 16 \nu^{10} - 105 \nu^{9} + 256 \nu^{8} + 661 \nu^{7} - 1335 \nu^{6} - 1872 \nu^{5} + \cdots + 202 ) / 49 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 4 \nu^{11} + 20 \nu^{10} + 42 \nu^{9} - 320 \nu^{8} + 5 \nu^{7} + 1681 \nu^{6} - 1034 \nu^{5} + \cdots + 381 ) / 49 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 2 \nu^{11} - 4 \nu^{10} + 63 \nu^{9} + 64 \nu^{8} - 666 \nu^{7} - 346 \nu^{6} + 2955 \nu^{5} + \cdots - 632 ) / 49 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 8 \nu^{11} - 33 \nu^{10} - 105 \nu^{9} + 528 \nu^{8} + 312 \nu^{7} - 2781 \nu^{6} + 528 \nu^{5} + \cdots - 20 ) / 49 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 2 \nu^{11} + 5 \nu^{10} + 32 \nu^{9} - 78 \nu^{8} - 169 \nu^{7} + 393 \nu^{6} + 336 \nu^{5} + \cdots + 24 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + \beta_{10} + \beta_{8} + \beta_{6} - \beta_{5} + 8\beta_{2} + \beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta_{7} + 9\beta_{3} + 39\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{11} + 10\beta_{10} + 9\beta_{8} + \beta_{7} + 11\beta_{6} - 13\beta_{5} + 57\beta_{2} + 11\beta _1 + 158 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2 \beta_{10} + 14 \beta_{9} + 12 \beta_{8} + 15 \beta_{7} + \beta_{6} + \beta_{5} + 69 \beta_{3} + \cdots + 9 \)
|
| \(\nu^{8}\) | \(=\) |
\( 112 \beta_{11} + 80 \beta_{10} + 2 \beta_{9} + 65 \beta_{8} + 19 \beta_{7} + 95 \beta_{6} - 126 \beta_{5} + \cdots + 1081 \)
|
| \(\nu^{9}\) | \(=\) |
\( 4 \beta_{11} - 32 \beta_{10} + 144 \beta_{9} + 109 \beta_{8} + 159 \beta_{7} + 20 \beta_{6} + 10 \beta_{5} + \cdots + 49 \)
|
| \(\nu^{10}\) | \(=\) |
\( 956 \beta_{11} + 601 \beta_{10} + 40 \beta_{9} + 450 \beta_{8} + 228 \beta_{7} + 769 \beta_{6} + \cdots + 7557 \)
|
| \(\nu^{11}\) | \(=\) |
\( 81 \beta_{11} - 355 \beta_{10} + 1311 \beta_{9} + 897 \beta_{8} + 1470 \beta_{7} + 255 \beta_{6} + \cdots + 92 \)
|
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.
comment: embeddings in the coefficient field
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 |
|
−2.74074 | 1.00000 | 5.51163 | 1.27795 | −2.74074 | 0 | −9.62444 | 1.00000 | −3.50251 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.19161 | 1.00000 | 2.80314 | −1.60788 | −2.19161 | 0 | −1.76017 | 1.00000 | 3.52384 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.54657 | 1.00000 | 0.391878 | 3.57292 | −1.54657 | 0 | 2.48707 | 1.00000 | −5.52578 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.19788 | 1.00000 | −0.565080 | 1.73272 | −1.19788 | 0 | 3.07266 | 1.00000 | −2.07559 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.06052 | 1.00000 | −0.875298 | −1.33453 | −1.06052 | 0 | 3.04931 | 1.00000 | 1.41529 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.195312 | 1.00000 | −1.96185 | −3.72995 | 0.195312 | 0 | −0.773798 | 1.00000 | −0.728505 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.641487 | 1.00000 | −1.58849 | 3.78855 | 0.641487 | 0 | −2.30197 | 1.00000 | 2.43031 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.44528 | 1.00000 | 0.0888199 | 2.08170 | 1.44528 | 0 | −2.76218 | 1.00000 | 3.00863 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.58626 | 1.00000 | 0.516213 | −3.36773 | 1.58626 | 0 | −2.35367 | 1.00000 | −5.34209 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.47416 | 1.00000 | 4.12148 | −0.798478 | 2.47416 | 0 | 5.24888 | 1.00000 | −1.97556 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.64480 | 1.00000 | 4.99498 | −2.03337 | 2.64480 | 0 | 7.92115 | 1.00000 | −5.37787 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.75002 | 1.00000 | 5.56258 | 4.41810 | 2.75002 | 0 | 9.79716 | 1.00000 | 12.1498 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(59\) | \(-1\) |
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 | 8673.2.a.ba | 12 | |
| 7.b | odd | 2 | 1 | 1239.2.a.i | ✓ | 12 | |
| 21.c | even | 2 | 1 | 3717.2.a.q | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1239.2.a.i | ✓ | 12 | 7.b | odd | 2 | 1 | |
| 3717.2.a.q | 12 | 21.c | even | 2 | 1 | ||
| 8673.2.a.ba | 12 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8673))\):
|
\( T_{2}^{12} - 3 T_{2}^{11} - 17 T_{2}^{10} + 52 T_{2}^{9} + 101 T_{2}^{8} - 316 T_{2}^{7} - 260 T_{2}^{6} + \cdots - 61 \)
|
|
\( T_{5}^{12} - 4 T_{5}^{11} - 37 T_{5}^{10} + 137 T_{5}^{9} + 510 T_{5}^{8} - 1632 T_{5}^{7} + \cdots + 12064 \)
|
|
\( T_{11}^{12} - 2 T_{11}^{11} - 73 T_{11}^{10} + 141 T_{11}^{9} + 1822 T_{11}^{8} - 3314 T_{11}^{7} + \cdots - 67664 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 3 T^{11} + \cdots - 61 \)
|
| $3$ |
\( (T - 1)^{12} \)
|
| $5$ |
\( T^{12} - 4 T^{11} + \cdots + 12064 \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( T^{12} - 2 T^{11} + \cdots - 67664 \)
|
| $13$ |
\( T^{12} + 9 T^{11} + \cdots - 14696 \)
|
| $17$ |
\( T^{12} - 5 T^{11} + \cdots - 1045504 \)
|
| $19$ |
\( T^{12} + 7 T^{11} + \cdots + 24948736 \)
|
| $23$ |
\( T^{12} - T^{11} + \cdots + 45988768 \)
|
| $29$ |
\( T^{12} - 11 T^{11} + \cdots - 55808 \)
|
| $31$ |
\( T^{12} + 13 T^{11} + \cdots - 9622784 \)
|
| $37$ |
\( T^{12} + \cdots - 667576832 \)
|
| $41$ |
\( T^{12} - 185 T^{10} + \cdots - 7699744 \)
|
| $43$ |
\( T^{12} - 21 T^{11} + \cdots - 4993024 \)
|
| $47$ |
\( T^{12} - 18 T^{11} + \cdots - 8486912 \)
|
| $53$ |
\( T^{12} + \cdots - 434921984 \)
|
| $59$ |
\( (T - 1)^{12} \)
|
| $61$ |
\( T^{12} + 30 T^{11} + \cdots - 148928 \)
|
| $67$ |
\( T^{12} + \cdots + 4823576576 \)
|
| $71$ |
\( T^{12} + \cdots + 155901952 \)
|
| $73$ |
\( T^{12} + \cdots + 21718770248 \)
|
| $79$ |
\( T^{12} + \cdots + 14927961088 \)
|
| $83$ |
\( T^{12} + \cdots - 3751051264 \)
|
| $89$ |
\( T^{12} + \cdots - 11909963264 \)
|
| $97$ |
\( T^{12} + \cdots + 435384376 \)
|
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