Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2006,2,Mod(1,2006)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2006, 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("2006.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2006 = 2 \cdot 17 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2006.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: | \(16.0179906455\) |
| 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: |
\( x^{12} - 3 x^{11} - 21 x^{10} + 62 x^{9} + 144 x^{8} - 418 x^{7} - 370 x^{6} + 1042 x^{5} + 417 x^{4} + \cdots + 62 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{12} - 3 x^{11} - 21 x^{10} + 62 x^{9} + 144 x^{8} - 418 x^{7} - 370 x^{6} + 1042 x^{5} + 417 x^{4} + \cdots + 62 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3149 \nu^{11} + 1166 \nu^{10} - 66465 \nu^{9} - 64341 \nu^{8} + 419399 \nu^{7} + 863247 \nu^{6} + \cdots - 1427778 ) / 243128 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2093 \nu^{11} + 6830 \nu^{10} + 37189 \nu^{9} - 141473 \nu^{8} - 173309 \nu^{7} + 944051 \nu^{6} + \cdots - 255310 ) / 121564 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2747 \nu^{11} - 2957 \nu^{10} + 76732 \nu^{9} + 76819 \nu^{8} - 757062 \nu^{7} - 652173 \nu^{6} + \cdots + 452756 ) / 121564 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 6523 \nu^{11} - 2526 \nu^{10} - 149183 \nu^{9} + 23453 \nu^{8} + 1092745 \nu^{7} + 56553 \nu^{6} + \cdots - 251046 ) / 243128 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 8097 \nu^{11} - 28136 \nu^{10} - 162659 \nu^{9} + 596063 \nu^{8} + 1010937 \nu^{7} - 4166445 \nu^{6} + \cdots + 2822074 ) / 243128 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 10379 \nu^{11} + 41478 \nu^{10} + 191619 \nu^{9} - 853173 \nu^{8} - 993969 \nu^{7} + \cdots - 1059058 ) / 243128 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 6443 \nu^{11} + 29839 \nu^{10} + 118416 \nu^{9} - 631209 \nu^{8} - 615474 \nu^{7} + \cdots - 1436532 ) / 121564 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 14593 \nu^{11} + 47098 \nu^{10} + 278285 \nu^{9} - 941639 \nu^{8} - 1496995 \nu^{7} + \cdots - 2720430 ) / 243128 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 28721 \nu^{11} - 80104 \nu^{10} - 599971 \nu^{9} + 1609387 \nu^{8} + 4013029 \nu^{7} + \cdots + 2367922 ) / 243128 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + 2\beta_{8} + \beta_{7} - \beta_{6} + 2\beta_{3} + \beta_{2} + 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{11} - \beta_{10} + 2\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 11\beta_{2} + \beta _1 + 30 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 3 \beta_{11} - 2 \beta_{10} - 14 \beta_{9} + 22 \beta_{8} + 12 \beta_{7} - 11 \beta_{6} - 3 \beta_{5} + \cdots + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 14 \beta_{11} - 11 \beta_{10} - \beta_{9} + 32 \beta_{7} - 15 \beta_{6} - 13 \beta_{5} + 16 \beta_{4} + \cdots + 262 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 50 \beta_{11} - 36 \beta_{10} - 158 \beta_{9} + 218 \beta_{8} + 121 \beta_{7} - 111 \beta_{6} + \cdots + 46 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 155 \beta_{11} - 98 \beta_{10} - 22 \beta_{9} + 16 \beta_{8} + 400 \beta_{7} - 180 \beta_{6} + \cdots + 2428 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 631 \beta_{11} - 475 \beta_{10} - 1671 \beta_{9} + 2132 \beta_{8} + 1184 \beta_{7} - 1104 \beta_{6} + \cdots + 907 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 1634 \beta_{11} - 853 \beta_{10} - 354 \beta_{9} + 378 \beta_{8} + 4566 \beta_{7} - 2007 \beta_{6} + \cdots + 23180 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 7254 \beta_{11} - 5571 \beta_{10} - 17224 \beta_{9} + 20848 \beta_{8} + 11665 \beta_{7} + \cdots + 13590 \)
|
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 | −3.23733 | 1.00000 | −2.83939 | 3.23733 | −0.371598 | −1.00000 | 7.48028 | 2.83939 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −3.20159 | 1.00000 | 1.49954 | 3.20159 | −3.52587 | −1.00000 | 7.25017 | −1.49954 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −2.03816 | 1.00000 | 3.57691 | 2.03816 | −0.163425 | −1.00000 | 1.15408 | −3.57691 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.94046 | 1.00000 | −0.890447 | 1.94046 | 4.98856 | −1.00000 | 0.765392 | 0.890447 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −1.09998 | 1.00000 | −3.08354 | 1.09998 | 3.13875 | −1.00000 | −1.79003 | 3.08354 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −0.677419 | 1.00000 | 1.10956 | 0.677419 | −4.11537 | −1.00000 | −2.54110 | −1.10956 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 0.288101 | 1.00000 | −0.229369 | −0.288101 | −3.58588 | −1.00000 | −2.91700 | 0.229369 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 0.517309 | 1.00000 | 3.93369 | −0.517309 | 3.50142 | −1.00000 | −2.73239 | −3.93369 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 1.06641 | 1.00000 | −3.87977 | −1.06641 | −2.29126 | −1.00000 | −1.86276 | 3.87977 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 1.52453 | 1.00000 | 0.627861 | −1.52453 | 3.51982 | −1.00000 | −0.675804 | −0.627861 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1.00000 | 2.72986 | 1.00000 | 2.14180 | −2.72986 | 0.279141 | −1.00000 | 4.45216 | −2.14180 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −1.00000 | 3.06872 | 1.00000 | −2.96683 | −3.06872 | 2.62570 | −1.00000 | 6.41702 | 2.96683 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(17\) | \(1\) |
| \(59\) | \(-1\) |
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 | 2006.2.a.v | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2006.2.a.v | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2006))\):
|
\( T_{3}^{12} + 3 T_{3}^{11} - 21 T_{3}^{10} - 62 T_{3}^{9} + 144 T_{3}^{8} + 418 T_{3}^{7} - 370 T_{3}^{6} + \cdots + 62 \)
|
|
\( T_{5}^{12} + T_{5}^{11} - 39 T_{5}^{10} - 34 T_{5}^{9} + 543 T_{5}^{8} + 341 T_{5}^{7} - 3291 T_{5}^{6} + \cdots + 648 \)
|
|
\( T_{11}^{12} + 3 T_{11}^{11} - 87 T_{11}^{10} - 224 T_{11}^{9} + 2590 T_{11}^{8} + 5400 T_{11}^{7} + \cdots + 1536 \)
|
|
\( T_{31}^{12} - 26 T_{31}^{11} + 175 T_{31}^{10} + 668 T_{31}^{9} - 11331 T_{31}^{8} + 24710 T_{31}^{7} + \cdots - 309528 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{12} \)
|
| $3$ |
\( T^{12} + 3 T^{11} + \cdots + 62 \)
|
| $5$ |
\( T^{12} + T^{11} + \cdots + 648 \)
|
| $7$ |
\( T^{12} - 4 T^{11} + \cdots + 1024 \)
|
| $11$ |
\( T^{12} + 3 T^{11} + \cdots + 1536 \)
|
| $13$ |
\( T^{12} - 15 T^{11} + \cdots - 592 \)
|
| $17$ |
\( (T + 1)^{12} \)
|
| $19$ |
\( T^{12} - 16 T^{11} + \cdots + 20736 \)
|
| $23$ |
\( T^{12} + 14 T^{11} + \cdots - 692064 \)
|
| $29$ |
\( T^{12} + \cdots - 306934944 \)
|
| $31$ |
\( T^{12} - 26 T^{11} + \cdots - 309528 \)
|
| $37$ |
\( T^{12} - 15 T^{11} + \cdots - 802816 \)
|
| $41$ |
\( T^{12} + \cdots - 2734192896 \)
|
| $43$ |
\( T^{12} - 8 T^{11} + \cdots + 128768 \)
|
| $47$ |
\( T^{12} + 6 T^{11} + \cdots + 565248 \)
|
| $53$ |
\( T^{12} - T^{11} + \cdots - 31636224 \)
|
| $59$ |
\( (T - 1)^{12} \)
|
| $61$ |
\( T^{12} + \cdots - 879796224 \)
|
| $67$ |
\( T^{12} + \cdots + 267496016 \)
|
| $71$ |
\( T^{12} + \cdots + 328114176 \)
|
| $73$ |
\( T^{12} + \cdots - 241392672 \)
|
| $79$ |
\( T^{12} + \cdots + 2921912192 \)
|
| $83$ |
\( T^{12} + \cdots - 725350464 \)
|
| $89$ |
\( T^{12} + \cdots + 37437301248 \)
|
| $97$ |
\( T^{12} + \cdots - 243364323992 \)
|
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