Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [10000,2,Mod(1,10000)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10000, 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("10000.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 10000 = 2^{4} \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10000.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: | \(79.8504020213\) |
| 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: |
\( x^{12} - x^{11} - 20 x^{10} + 11 x^{9} + 144 x^{8} - 29 x^{7} - 440 x^{6} + 4 x^{5} + 556 x^{4} + \cdots + 45 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | no (minimal twist has level 5000) |
| 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
\( x^{12} - x^{11} - 20 x^{10} + 11 x^{9} + 144 x^{8} - 29 x^{7} - 440 x^{6} + 4 x^{5} + 556 x^{4} + \cdots + 45 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 307 \nu^{11} - 8733 \nu^{10} - 678 \nu^{9} + 180827 \nu^{8} - 27635 \nu^{7} - 1274457 \nu^{6} + \cdots + 648531 ) / 63003 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 802 \nu^{11} - 6738 \nu^{10} + 24756 \nu^{9} + 135204 \nu^{8} - 214775 \nu^{7} - 939928 \nu^{6} + \cdots + 568359 ) / 63003 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1351 \nu^{11} - 13969 \nu^{10} + 48406 \nu^{9} + 261955 \nu^{8} - 433991 \nu^{7} - 1723439 \nu^{6} + \cdots + 577755 ) / 63003 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2231 \nu^{11} + 529 \nu^{10} + 42209 \nu^{9} + 21083 \nu^{8} - 281581 \nu^{7} - 323536 \nu^{6} + \cdots + 536010 ) / 63003 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3607 \nu^{11} - 10598 \nu^{10} - 56193 \nu^{9} + 160053 \nu^{8} + 293370 \nu^{7} - 848573 \nu^{6} + \cdots + 348303 ) / 63003 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 5601 \nu^{11} - 1703 \nu^{10} + 118948 \nu^{9} + 78063 \nu^{8} - 861089 \nu^{7} - 798674 \nu^{6} + \cdots + 888045 ) / 63003 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 7021 \nu^{11} + 11191 \nu^{10} + 129472 \nu^{9} - 144518 \nu^{8} - 854757 \nu^{7} + 587287 \nu^{6} + \cdots - 140112 ) / 63003 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 9252 \nu^{11} - 11720 \nu^{10} - 171681 \nu^{9} + 123435 \nu^{8} + 1136338 \nu^{7} - 263751 \nu^{6} + \cdots - 269892 ) / 63003 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 11122 \nu^{11} + 4724 \nu^{10} - 234641 \nu^{9} - 183751 \nu^{8} + 1678758 \nu^{7} + 1757068 \nu^{6} + \cdots - 1338864 ) / 63003 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 20291 \nu^{11} + 14554 \nu^{10} + 393757 \nu^{9} - 88219 \nu^{8} - 2649292 \nu^{7} + \cdots + 608583 ) / 63003 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 28451 \nu^{11} - 22984 \nu^{10} - 557759 \nu^{9} + 196840 \nu^{8} + 3801976 \nu^{7} + 15784 \nu^{6} + \cdots - 914526 ) / 63003 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{10} + \beta_{9} + 3\beta_{8} + 3\beta_{6} + 2\beta_{5} - 2\beta_{4} - 2\beta_{3} - 4\beta_{2} + 3\beta _1 + 2 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3 \beta_{10} + 4 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \cdots + 18 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3 \beta_{10} + 2 \beta_{9} + 5 \beta_{8} + \beta_{7} + 4 \beta_{6} + 3 \beta_{5} - \beta_{4} - 2 \beta_{3} + \cdots + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 5 \beta_{11} + 32 \beta_{10} + 41 \beta_{9} + 33 \beta_{8} - 5 \beta_{7} + 28 \beta_{6} + 37 \beta_{5} + \cdots + 122 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 10 \beta_{11} + 133 \beta_{10} + 99 \beta_{9} + 207 \beta_{8} + 20 \beta_{7} + 147 \beta_{6} + \cdots + 238 ) / 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 15 \beta_{11} + 69 \beta_{10} + 80 \beta_{9} + 81 \beta_{8} - 16 \beta_{7} + 56 \beta_{6} + \cdots + 204 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 175 \beta_{11} + 1302 \beta_{10} + 1031 \beta_{9} + 1853 \beta_{8} - 65 \beta_{7} + 1168 \beta_{6} + \cdots + 2567 ) / 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 875 \beta_{11} + 3753 \beta_{10} + 3944 \beta_{9} + 4517 \beta_{8} - 1075 \beta_{7} + 2657 \beta_{6} + \cdots + 9383 ) / 5 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 463 \beta_{11} + 2662 \beta_{10} + 2182 \beta_{9} + 3516 \beta_{8} - 511 \beta_{7} + 1966 \beta_{6} + \cdots + 5343 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 9580 \beta_{11} + 40577 \beta_{10} + 39526 \beta_{9} + 48468 \beta_{8} - 13430 \beta_{7} + \cdots + 90952 ) / 5 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 27580 \beta_{11} + 138588 \beta_{10} + 115539 \beta_{9} + 173197 \beta_{8} - 39595 \beta_{7} + \cdots + 275263 ) / 5 \)
|
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 |
|
0 | −2.73663 | 0 | 0 | 0 | −1.85824 | 0 | 4.48912 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.62870 | 0 | 0 | 0 | −0.0349473 | 0 | 3.91007 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.38971 | 0 | 0 | 0 | −2.31613 | 0 | 2.71069 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.78053 | 0 | 0 | 0 | 4.82418 | 0 | 0.170304 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.0330070 | 0 | 0 | 0 | −2.74120 | 0 | −2.99891 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.208054 | 0 | 0 | 0 | −0.439180 | 0 | −2.95671 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.444623 | 0 | 0 | 0 | 2.92309 | 0 | −2.80231 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 0.602175 | 0 | 0 | 0 | −3.10675 | 0 | −2.63738 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.81354 | 0 | 0 | 0 | −0.0468346 | 0 | 4.91602 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.94508 | 0 | 0 | 0 | −3.90152 | 0 | 5.67351 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 3.13445 | 0 | 0 | 0 | 2.92885 | 0 | 6.82478 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 3.42065 | 0 | 0 | 0 | 3.76868 | 0 | 8.70082 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
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 | 10000.2.a.bp | 12 | |
| 4.b | odd | 2 | 1 | 5000.2.a.o | ✓ | 12 | |
| 5.b | even | 2 | 1 | 10000.2.a.bo | 12 | ||
| 20.d | odd | 2 | 1 | 5000.2.a.p | yes | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5000.2.a.o | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 5000.2.a.p | yes | 12 | 20.d | odd | 2 | 1 | |
| 10000.2.a.bo | 12 | 5.b | even | 2 | 1 | ||
| 10000.2.a.bp | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):
|
\( T_{3}^{12} - 4 T_{3}^{11} - 23 T_{3}^{10} + 93 T_{3}^{9} + 195 T_{3}^{8} - 773 T_{3}^{7} - 708 T_{3}^{6} + \cdots - 5 \)
|
|
\( T_{7}^{12} - 48 T_{7}^{10} - 29 T_{7}^{9} + 824 T_{7}^{8} + 919 T_{7}^{7} - 5896 T_{7}^{6} - 9353 T_{7}^{5} + \cdots + 16 \)
|
|
\( T_{11}^{12} - T_{11}^{11} - 68 T_{11}^{10} + 48 T_{11}^{9} + 1533 T_{11}^{8} - 1322 T_{11}^{7} + \cdots - 8609 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} - 4 T^{11} + \cdots - 5 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} - 48 T^{10} + \cdots + 16 \)
|
| $11$ |
\( T^{12} - T^{11} + \cdots - 8609 \)
|
| $13$ |
\( T^{12} - 4 T^{11} + \cdots + 302256 \)
|
| $17$ |
\( T^{12} - 8 T^{11} + \cdots - 4824725 \)
|
| $19$ |
\( T^{12} + 9 T^{11} + \cdots + 4621 \)
|
| $23$ |
\( T^{12} - 143 T^{10} + \cdots - 8514864 \)
|
| $29$ |
\( T^{12} - 8 T^{11} + \cdots - 527024 \)
|
| $31$ |
\( T^{12} + 33 T^{11} + \cdots + 9284400 \)
|
| $37$ |
\( T^{12} + \cdots + 650813616 \)
|
| $41$ |
\( T^{12} + \cdots - 995145329 \)
|
| $43$ |
\( T^{12} + \cdots + 347146281 \)
|
| $47$ |
\( T^{12} + 18 T^{11} + \cdots - 26255664 \)
|
| $53$ |
\( T^{12} + \cdots + 189235456 \)
|
| $59$ |
\( T^{12} + 33 T^{11} + \cdots - 4622139 \)
|
| $61$ |
\( T^{12} + 8 T^{11} + \cdots - 8891600 \)
|
| $67$ |
\( T^{12} + \cdots - 23687194259 \)
|
| $71$ |
\( T^{12} + \cdots - 6051744080 \)
|
| $73$ |
\( T^{12} + \cdots + 927999711 \)
|
| $79$ |
\( T^{12} + \cdots + 39511336720 \)
|
| $83$ |
\( T^{12} + \cdots - 4396075199 \)
|
| $89$ |
\( T^{12} + \cdots - 82788670169 \)
|
| $97$ |
\( T^{12} + \cdots + 171567421 \)
|
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