Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1007,2,Mod(1,1007)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1007, 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("1007.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1007 = 19 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1007.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: | \(8.04093548354\) |
| 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: |
\( x^{12} - 3 x^{11} - 12 x^{10} + 38 x^{9} + 47 x^{8} - 164 x^{7} - 68 x^{6} + 286 x^{5} + 37 x^{4} + \cdots + 7 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{12} - 3 x^{11} - 12 x^{10} + 38 x^{9} + 47 x^{8} - 164 x^{7} - 68 x^{6} + 286 x^{5} + 37 x^{4} + \cdots + 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3 \nu^{11} - 18 \nu^{10} - 8 \nu^{9} + 203 \nu^{8} - 169 \nu^{7} - 700 \nu^{6} + 830 \nu^{5} + \cdots + 59 ) / 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 8 \nu^{11} + 9 \nu^{10} + 121 \nu^{9} - 108 \nu^{8} - 637 \nu^{7} + 441 \nu^{6} + 1327 \nu^{5} + \cdots - 36 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 7 \nu^{11} + 29 \nu^{10} + 62 \nu^{9} - 361 \nu^{8} - 65 \nu^{7} + 1512 \nu^{6} - 563 \nu^{5} + \cdots - 259 ) / 13 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7 \nu^{11} - 29 \nu^{10} - 75 \nu^{9} + 387 \nu^{8} + 221 \nu^{7} - 1785 \nu^{6} - 35 \nu^{5} + \cdots + 272 ) / 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 18 \nu^{11} + 43 \nu^{10} + 217 \nu^{9} - 503 \nu^{8} - 832 \nu^{7} + 1912 \nu^{6} + 1026 \nu^{5} + \cdots - 185 ) / 13 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 19 \nu^{11} - 23 \nu^{10} - 263 \nu^{9} + 211 \nu^{8} + 1274 \nu^{7} - 412 \nu^{6} - 2591 \nu^{5} + \cdots - 246 ) / 13 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 19 \nu^{11} - 49 \nu^{10} - 224 \nu^{9} + 575 \nu^{8} + 832 \nu^{7} - 2180 \nu^{6} - 992 \nu^{5} + \cdots + 131 ) / 13 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 3 \nu^{11} - 31 \nu^{10} - 8 \nu^{9} + 437 \nu^{8} - 182 \nu^{7} - 2156 \nu^{6} + 1025 \nu^{5} + \cdots + 501 ) / 13 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 18 \nu^{11} + 43 \nu^{10} + 243 \nu^{9} - 568 \nu^{8} - 1118 \nu^{7} + 2614 \nu^{6} + 1936 \nu^{5} + \cdots - 419 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{4} + \beta_{2} + 4\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + 2\beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} - 2\beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7 \beta_{11} + 9 \beta_{10} - 8 \beta_{9} + 9 \beta_{8} + 2 \beta_{7} + 6 \beta_{6} - \beta_{5} + \cdots + 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9 \beta_{11} + 19 \beta_{10} - 11 \beta_{9} + 12 \beta_{8} + 4 \beta_{7} + \beta_{6} + 6 \beta_{5} + \cdots + 87 \)
|
| \(\nu^{7}\) | \(=\) |
\( 43 \beta_{11} + 67 \beta_{10} - 59 \beta_{9} + 69 \beta_{8} + 24 \beta_{7} + 34 \beta_{6} - 12 \beta_{5} + \cdots + 118 \)
|
| \(\nu^{8}\) | \(=\) |
\( 65 \beta_{11} + 145 \beta_{10} - 100 \beta_{9} + 110 \beta_{8} + 53 \beta_{7} + 17 \beta_{6} + \cdots + 541 \)
|
| \(\nu^{9}\) | \(=\) |
\( 258 \beta_{11} + 468 \beta_{10} - 432 \beta_{9} + 505 \beta_{8} + 218 \beta_{7} + 203 \beta_{6} + \cdots + 864 \)
|
| \(\nu^{10}\) | \(=\) |
\( 436 \beta_{11} + 1039 \beta_{10} - 841 \beta_{9} + 914 \beta_{8} + 512 \beta_{7} + 184 \beta_{6} + \cdots + 3495 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1545 \beta_{11} + 3197 \beta_{10} - 3162 \beta_{9} + 3638 \beta_{8} + 1799 \beta_{7} + 1282 \beta_{6} + \cdots + 6298 \)
|
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 |
|
−2.67609 | 0.754460 | 5.16146 | −0.542789 | −2.01900 | −2.60256 | −8.46034 | −2.43079 | 1.45255 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.45418 | −2.86447 | 4.02298 | 1.70775 | 7.02991 | 0.922266 | −4.96473 | 5.20518 | −4.19111 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.73125 | −1.96765 | 0.997227 | −3.35079 | 3.40650 | −2.62844 | 1.73605 | 0.871659 | 5.80105 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.70868 | 1.53433 | 0.919590 | 1.36459 | −2.62167 | −0.664537 | 1.84608 | −0.645847 | −2.33165 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.963451 | −1.54050 | −1.07176 | 2.14593 | 1.48420 | 0.871869 | 2.95949 | −0.626856 | −2.06750 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.721447 | 1.20234 | −1.47951 | −1.63736 | −0.867425 | 0.0547477 | 2.51028 | −1.55438 | 1.18127 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.267503 | −0.0965543 | −1.92844 | −1.66447 | −0.0258286 | 4.63771 | −1.05087 | −2.99068 | −0.445252 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.306870 | 2.11825 | −1.90583 | 0.975071 | 0.650026 | −3.93845 | −1.19858 | 1.48697 | 0.299220 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.13436 | −3.00403 | −0.713217 | 0.0642250 | −3.40767 | 0.823418 | −3.07778 | 6.02420 | 0.0728545 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.16538 | −0.160123 | −0.641892 | 1.85962 | −0.186604 | 0.887883 | −3.07881 | −2.97436 | 2.16716 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.04397 | 0.550865 | 2.17782 | −3.03588 | 1.12595 | −1.24200 | 0.363464 | −2.69655 | −6.20526 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.33700 | −1.52691 | 3.46159 | 0.114107 | −3.56839 | −5.12191 | 3.41574 | −0.668558 | 0.266668 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(19\) | \(-1\) |
| \(53\) | \(-1\) |
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 | 1007.2.a.c | ✓ | 12 |
| 3.b | odd | 2 | 1 | 9063.2.a.g | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1007.2.a.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 9063.2.a.g | 12 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 3 T_{2}^{11} - 12 T_{2}^{10} - 38 T_{2}^{9} + 47 T_{2}^{8} + 164 T_{2}^{7} - 68 T_{2}^{6} + \cdots + 7 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1007))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 3 T^{11} + \cdots + 7 \)
|
| $3$ |
\( T^{12} + 5 T^{11} + \cdots - 1 \)
|
| $5$ |
\( T^{12} + 2 T^{11} + \cdots - 1 \)
|
| $7$ |
\( T^{12} + 8 T^{11} + \cdots + 17 \)
|
| $11$ |
\( T^{12} + 4 T^{11} + \cdots - 79631 \)
|
| $13$ |
\( T^{12} + 17 T^{11} + \cdots - 26995 \)
|
| $17$ |
\( T^{12} + 8 T^{11} + \cdots - 374665 \)
|
| $19$ |
\( (T - 1)^{12} \)
|
| $23$ |
\( T^{12} + 2 T^{11} + \cdots + 82355 \)
|
| $29$ |
\( T^{12} + 10 T^{11} + \cdots + 779495 \)
|
| $31$ |
\( T^{12} + 8 T^{11} + \cdots + 82544569 \)
|
| $37$ |
\( T^{12} + 37 T^{11} + \cdots + 34770517 \)
|
| $41$ |
\( T^{12} - 14 T^{11} + \cdots + 327203 \)
|
| $43$ |
\( T^{12} + 26 T^{11} + \cdots - 69123335 \)
|
| $47$ |
\( T^{12} + 4 T^{11} + \cdots + 209521 \)
|
| $53$ |
\( (T - 1)^{12} \)
|
| $59$ |
\( T^{12} + 16 T^{11} + \cdots + 6219005 \)
|
| $61$ |
\( T^{12} + 5 T^{11} + \cdots + 9286381 \)
|
| $67$ |
\( T^{12} + 40 T^{11} + \cdots - 12119209 \)
|
| $71$ |
\( T^{12} + \cdots - 11742245639 \)
|
| $73$ |
\( T^{12} + 26 T^{11} + \cdots - 36667547 \)
|
| $79$ |
\( T^{12} + \cdots - 550620425 \)
|
| $83$ |
\( T^{12} + \cdots - 2805006685 \)
|
| $89$ |
\( T^{12} + \cdots + 131054813 \)
|
| $97$ |
\( T^{12} + \cdots - 65500570607 \)
|
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