Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6039,2,Mod(1,6039)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6039, 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("6039.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6039 = 3^{2} \cdot 11 \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6039.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: | \(48.2216577807\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{12} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2013) |
| 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
\( x^{12} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 6 \nu^{11} - 4 \nu^{10} + 39 \nu^{9} + 138 \nu^{8} + 125 \nu^{7} - 998 \nu^{6} - 1242 \nu^{5} + \cdots + 517 ) / 151 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15 \nu^{11} + 10 \nu^{10} - 173 \nu^{9} - 194 \nu^{8} + 518 \nu^{7} + 985 \nu^{6} + 85 \nu^{5} + \cdots + 293 ) / 151 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 21 \nu^{11} + 14 \nu^{10} - 363 \nu^{9} - 181 \nu^{8} + 2205 \nu^{7} + 775 \nu^{6} - 5921 \nu^{5} + \cdots + 78 ) / 151 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 23 \nu^{11} - 35 \nu^{10} - 376 \nu^{9} + 528 \nu^{8} + 2113 \nu^{7} - 2617 \nu^{6} - 4752 \nu^{5} + \cdots + 107 ) / 151 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 22 \nu^{11} + 86 \nu^{10} + 294 \nu^{9} - 1155 \nu^{8} - 1404 \nu^{7} + 5300 \nu^{6} + 2694 \nu^{5} + \cdots - 168 ) / 151 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 41 \nu^{11} - 23 \nu^{10} - 644 \nu^{9} + 265 \nu^{8} + 3550 \nu^{7} - 982 \nu^{6} - 8123 \nu^{5} + \cdots - 387 ) / 151 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 44 \nu^{11} - 21 \nu^{10} - 739 \nu^{9} + 196 \nu^{8} + 4620 \nu^{7} - 483 \nu^{6} - 13089 \nu^{5} + \cdots + 185 ) / 151 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 52 \nu^{11} + 66 \nu^{10} + 791 \nu^{9} - 918 \nu^{8} - 4252 \nu^{7} + 4538 \nu^{6} + 9621 \nu^{5} + \cdots - 301 ) / 151 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 52 \nu^{11} - 66 \nu^{10} - 791 \nu^{9} + 918 \nu^{8} + 4252 \nu^{7} - 4538 \nu^{6} - 9621 \nu^{5} + \cdots + 603 ) / 151 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{10} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} + \beta_{9} - \beta_{8} + 2\beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{3} + 8\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8 \beta_{11} + 8 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + \cdots + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3 \beta_{11} + 12 \beta_{10} + 10 \beta_{9} - 11 \beta_{8} + \beta_{7} + 21 \beta_{6} - 10 \beta_{5} + \cdots + 75 \)
|
| \(\nu^{7}\) | \(=\) |
\( 55 \beta_{11} + 57 \beta_{10} + 14 \beta_{9} - 16 \beta_{8} + 11 \beta_{7} + 28 \beta_{6} - 23 \beta_{5} + \cdots + 72 \)
|
| \(\nu^{8}\) | \(=\) |
\( 43 \beta_{11} + 106 \beta_{10} + 79 \beta_{9} - 93 \beta_{8} + 15 \beta_{7} + 170 \beta_{6} - 81 \beta_{5} + \cdots + 437 \)
|
| \(\nu^{9}\) | \(=\) |
\( 369 \beta_{11} + 402 \beta_{10} + 137 \beta_{9} - 167 \beta_{8} + 91 \beta_{7} + 278 \beta_{6} + \cdots + 561 \)
|
| \(\nu^{10}\) | \(=\) |
\( 433 \beta_{11} + 847 \beta_{10} + 583 \beta_{9} - 717 \beta_{8} + 155 \beta_{7} + 1269 \beta_{6} + \cdots + 2707 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2487 \beta_{11} + 2847 \beta_{10} + 1168 \beta_{9} - 1471 \beta_{8} + 689 \beta_{7} + 2403 \beta_{6} + \cdots + 4305 \)
|
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 |
|
−2.69409 | 0 | 5.25811 | 2.61116 | 0 | −3.45538 | −8.77762 | 0 | −7.03469 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.07880 | 0 | 2.32140 | 2.51557 | 0 | 3.31485 | −0.668123 | 0 | −5.22935 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.01091 | 0 | 2.04375 | −2.29360 | 0 | −4.38501 | −0.0879817 | 0 | 4.61222 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.32082 | 0 | −0.255442 | −0.125081 | 0 | 1.14683 | 2.97903 | 0 | 0.165209 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.852887 | 0 | −1.27258 | −2.61325 | 0 | −1.65804 | 2.79114 | 0 | 2.22881 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.188928 | 0 | −1.96431 | 1.33997 | 0 | −0.912886 | 0.748968 | 0 | −0.253157 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.0557908 | 0 | −1.99689 | 3.85264 | 0 | −0.441454 | −0.222990 | 0 | 0.214942 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.811354 | 0 | −1.34170 | −0.350684 | 0 | 3.06527 | −2.71131 | 0 | −0.284529 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.05676 | 0 | −0.883253 | −1.53036 | 0 | −3.09203 | −3.04691 | 0 | −1.61722 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.72434 | 0 | 0.973355 | 0.0158764 | 0 | 3.08835 | −1.77029 | 0 | 0.0273764 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.22938 | 0 | 2.97016 | −3.25707 | 0 | −0.940570 | 2.16285 | 0 | −7.26127 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.26879 | 0 | 3.14741 | 2.83484 | 0 | −4.72992 | 2.60323 | 0 | 6.43165 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(-1\) |
| \(61\) | \(1\) |
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 | 6039.2.a.e | 12 | |
| 3.b | odd | 2 | 1 | 2013.2.a.d | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2013.2.a.d | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 6039.2.a.e | 12 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + T_{2}^{11} - 16 T_{2}^{10} - 13 T_{2}^{9} + 93 T_{2}^{8} + 59 T_{2}^{7} - 238 T_{2}^{6} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + T^{11} + \cdots + 1 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} - 3 T^{11} + \cdots + 2 \)
|
| $7$ |
\( T^{12} + 9 T^{11} + \cdots + 5012 \)
|
| $11$ |
\( (T - 1)^{12} \)
|
| $13$ |
\( T^{12} + T^{11} + \cdots + 266 \)
|
| $17$ |
\( T^{12} + 9 T^{11} + \cdots + 4454 \)
|
| $19$ |
\( T^{12} + 20 T^{11} + \cdots + 122006 \)
|
| $23$ |
\( T^{12} - 9 T^{11} + \cdots - 16 \)
|
| $29$ |
\( T^{12} + 18 T^{11} + \cdots + 14631718 \)
|
| $31$ |
\( T^{12} + \cdots - 1061001166 \)
|
| $37$ |
\( T^{12} + 18 T^{11} + \cdots - 58978 \)
|
| $41$ |
\( T^{12} + 15 T^{11} + \cdots + 19339376 \)
|
| $43$ |
\( T^{12} + 33 T^{11} + \cdots - 6414256 \)
|
| $47$ |
\( T^{12} - 20 T^{11} + \cdots + 62497568 \)
|
| $53$ |
\( T^{12} - 250 T^{10} + \cdots + 94719784 \)
|
| $59$ |
\( T^{12} + \cdots - 120384652 \)
|
| $61$ |
\( (T + 1)^{12} \)
|
| $67$ |
\( T^{12} + \cdots + 686361986 \)
|
| $71$ |
\( T^{12} + \cdots + 14666313456 \)
|
| $73$ |
\( T^{12} + \cdots - 112159142 \)
|
| $79$ |
\( T^{12} + \cdots + 291529652504 \)
|
| $83$ |
\( T^{12} + \cdots - 267389284 \)
|
| $89$ |
\( T^{12} + \cdots + 2510062488 \)
|
| $97$ |
\( T^{12} + \cdots - 29402967834 \)
|
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