Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8046,2,Mod(1,8046)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8046, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8046.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8046 = 2 \cdot 3^{3} \cdot 149 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8046.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: | \(64.2476334663\) |
| 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: |
\( x^{12} - 5 x^{11} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} + \cdots - 375 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{12} - 5 x^{11} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} + \cdots - 375 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 32197 \nu^{11} + 5673372 \nu^{10} - 27408721 \nu^{9} - 109537587 \nu^{8} + 609086107 \nu^{7} + \cdots + 1592878652 ) / 5855999 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 337779 \nu^{11} - 12051765 \nu^{10} + 77892894 \nu^{9} + 222792211 \nu^{8} - 1586491494 \nu^{7} + \cdots - 4301878920 ) / 29279995 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 4626967 \nu^{11} + 27982820 \nu^{10} + 68317507 \nu^{9} - 610246892 \nu^{8} + \cdots + 1309434950 ) / 29279995 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 5834388 \nu^{11} + 38266875 \nu^{10} + 69519048 \nu^{9} - 818338273 \nu^{8} + \cdots + 2800717475 ) / 29279995 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11071809 \nu^{11} + 33647510 \nu^{10} + 329349654 \nu^{9} - 795142464 \nu^{8} + \cdots - 6906022130 ) / 29279995 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 15600107 \nu^{11} + 74241930 \nu^{10} + 327537312 \nu^{9} - 1640249962 \nu^{8} + \cdots - 1289750025 ) / 29279995 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 15915726 \nu^{11} - 100688050 \nu^{10} - 207068641 \nu^{9} + 2153825106 \nu^{8} + \cdots - 6112777815 ) / 29279995 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 3317302 \nu^{11} + 16732001 \nu^{10} + 64707528 \nu^{9} - 366093260 \nu^{8} - 254489192 \nu^{7} + \cdots - 38973036 ) / 5855999 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 17078923 \nu^{11} - 69687500 \nu^{10} - 418005468 \nu^{9} + 1573561298 \nu^{8} + \cdots + 5205009495 ) / 29279995 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 5445786 \nu^{11} - 29080114 \nu^{10} - 98087514 \nu^{9} + 632612308 \nu^{8} + 238661630 \nu^{7} + \cdots - 395658693 ) / 5855999 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{3} + \beta_{2} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{11} + 3 \beta_{10} - \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2 \beta_{11} - 13 \beta_{10} - 11 \beta_{9} + 19 \beta_{8} + 21 \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 60 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 15 \beta_{11} + 51 \beta_{10} - 23 \beta_{9} + 44 \beta_{8} + 79 \beta_{7} + 27 \beta_{6} - 16 \beta_{5} + \cdots + 49 \)
|
| \(\nu^{6}\) | \(=\) |
\( 34 \beta_{11} - 162 \beta_{10} - 138 \beta_{9} + 341 \beta_{8} + 378 \beta_{7} - 11 \beta_{6} + \cdots + 923 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 223 \beta_{11} + 787 \beta_{10} - 406 \beta_{9} + 914 \beta_{8} + 1435 \beta_{7} + 498 \beta_{6} + \cdots + 1285 \)
|
| \(\nu^{8}\) | \(=\) |
\( 451 \beta_{11} - 1964 \beta_{10} - 1962 \beta_{9} + 6113 \beta_{8} + 6668 \beta_{7} + 77 \beta_{6} + \cdots + 15112 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 3384 \beta_{11} + 12202 \beta_{10} - 6570 \beta_{9} + 18455 \beta_{8} + 25796 \beta_{7} + \cdots + 28417 \)
|
| \(\nu^{10}\) | \(=\) |
\( 5285 \beta_{11} - 22019 \beta_{10} - 29192 \beta_{9} + 110188 \beta_{8} + 117443 \beta_{7} + \cdots + 253681 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 52344 \beta_{11} + 193515 \beta_{10} - 102076 \beta_{9} + 364000 \beta_{8} + 462496 \beta_{7} + \cdots + 583486 \)
|
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 |
|
−1.00000 | 0 | 1.00000 | −3.73396 | 0 | −2.62384 | −1.00000 | 0 | 3.73396 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | −2.26764 | 0 | −0.397853 | −1.00000 | 0 | 2.26764 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | −1.20126 | 0 | −4.29383 | −1.00000 | 0 | 1.20126 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0 | 1.00000 | −1.17050 | 0 | 0.506529 | −1.00000 | 0 | 1.17050 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0 | 1.00000 | −1.12518 | 0 | −0.686663 | −1.00000 | 0 | 1.12518 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0 | 1.00000 | 0.448758 | 0 | −1.52942 | −1.00000 | 0 | −0.448758 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 0 | 1.00000 | 1.06735 | 0 | 2.41300 | −1.00000 | 0 | −1.06735 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 0 | 1.00000 | 1.09665 | 0 | −2.72925 | −1.00000 | 0 | −1.09665 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 0 | 1.00000 | 1.53008 | 0 | 5.04291 | −1.00000 | 0 | −1.53008 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 0 | 1.00000 | 1.99250 | 0 | −0.0206178 | −1.00000 | 0 | −1.99250 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1.00000 | 0 | 1.00000 | 4.11015 | 0 | −3.78475 | −1.00000 | 0 | −4.11015 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −1.00000 | 0 | 1.00000 | 4.25305 | 0 | 2.10378 | −1.00000 | 0 | −4.25305 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(149\) | \(-1\) |
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 | 8046.2.a.l | ✓ | 12 |
| 3.b | odd | 2 | 1 | 8046.2.a.m | yes | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8046.2.a.l | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 8046.2.a.m | yes | 12 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):
|
\( T_{5}^{12} - 5 T_{5}^{11} - 21 T_{5}^{10} + 116 T_{5}^{9} + 106 T_{5}^{8} - 774 T_{5}^{7} - 63 T_{5}^{6} + \cdots - 375 \)
|
|
\( T_{11}^{12} - 10 T_{11}^{11} - 17 T_{11}^{10} + 486 T_{11}^{9} - 1093 T_{11}^{8} - 6231 T_{11}^{7} + \cdots + 101487 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{12} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} - 5 T^{11} + \cdots - 375 \)
|
| $7$ |
\( T^{12} + 6 T^{11} + \cdots + 13 \)
|
| $11$ |
\( T^{12} - 10 T^{11} + \cdots + 101487 \)
|
| $13$ |
\( T^{12} + T^{11} + \cdots + 39185 \)
|
| $17$ |
\( T^{12} - 6 T^{11} + \cdots + 9097103 \)
|
| $19$ |
\( T^{12} + 10 T^{11} + \cdots + 185485 \)
|
| $23$ |
\( T^{12} - 15 T^{11} + \cdots - 338619 \)
|
| $29$ |
\( T^{12} + \cdots + 201807029 \)
|
| $31$ |
\( T^{12} + 6 T^{11} + \cdots + 6102125 \)
|
| $37$ |
\( T^{12} + 13 T^{11} + \cdots + 28415799 \)
|
| $41$ |
\( T^{12} - 20 T^{11} + \cdots + 2006040 \)
|
| $43$ |
\( T^{12} + 11 T^{11} + \cdots + 4440617 \)
|
| $47$ |
\( T^{12} - 15 T^{11} + \cdots - 39719775 \)
|
| $53$ |
\( T^{12} - 4 T^{11} + \cdots - 15064073 \)
|
| $59$ |
\( T^{12} + \cdots - 52159306061 \)
|
| $61$ |
\( T^{12} + \cdots - 4973953311 \)
|
| $67$ |
\( T^{12} + 19 T^{11} + \cdots + 3965151 \)
|
| $71$ |
\( T^{12} + \cdots + 8946826731 \)
|
| $73$ |
\( T^{12} + \cdots + 2229232185 \)
|
| $79$ |
\( T^{12} + \cdots + 14109059725 \)
|
| $83$ |
\( T^{12} + \cdots + 199518309 \)
|
| $89$ |
\( T^{12} + \cdots + 87144544275 \)
|
| $97$ |
\( T^{12} + 25 T^{11} + \cdots - 38254719 \)
|
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