Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4012,2,Mod(1,4012)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4012, 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("4012.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4012 = 2^{2} \cdot 17 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4012.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: | \(32.0359812909\) |
| 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: |
\( x^{12} - 4 x^{11} - 13 x^{10} + 60 x^{9} + 48 x^{8} - 289 x^{7} - 89 x^{6} + 602 x^{5} + 161 x^{4} + \cdots + 82 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{12} - 4 x^{11} - 13 x^{10} + 60 x^{9} + 48 x^{8} - 289 x^{7} - 89 x^{6} + 602 x^{5} + 161 x^{4} + \cdots + 82 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2338 \nu^{11} - 11665 \nu^{10} - 20131 \nu^{9} + 163345 \nu^{8} - 30162 \nu^{7} - 687740 \nu^{6} + \cdots + 253800 ) / 2599 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 2547 \nu^{11} + 12262 \nu^{10} + 23568 \nu^{9} - 172850 \nu^{8} + 11984 \nu^{7} + 736444 \nu^{6} + \cdots - 271839 ) / 2599 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 4271 \nu^{11} + 20370 \nu^{10} + 40491 \nu^{9} - 289158 \nu^{8} + 7917 \nu^{7} + 1252500 \nu^{6} + \cdots - 531563 ) / 2599 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4271 \nu^{11} - 20370 \nu^{10} - 40491 \nu^{9} + 289158 \nu^{8} - 7917 \nu^{7} - 1252500 \nu^{6} + \cdots + 523766 ) / 2599 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5174 \nu^{11} - 24852 \nu^{10} - 48663 \nu^{9} + 352472 \nu^{8} - 13528 \nu^{7} - 1523042 \nu^{6} + \cdots + 604080 ) / 2599 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7177 \nu^{11} - 33844 \nu^{10} - 70684 \nu^{9} + 485510 \nu^{8} + 20451 \nu^{7} - 2146631 \nu^{6} + \cdots + 949639 ) / 2599 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 7808 \nu^{11} - 36691 \nu^{10} - 76783 \nu^{9} + 523720 \nu^{8} + 23527 \nu^{7} - 2291872 \nu^{6} + \cdots + 955056 ) / 2599 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 359 \nu^{11} - 1664 \nu^{10} - 3574 \nu^{9} + 23728 \nu^{8} + 1593 \nu^{7} - 103673 \nu^{6} + \cdots + 43294 ) / 113 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 8258 \nu^{11} - 38536 \nu^{10} - 81149 \nu^{9} + 547518 \nu^{8} + 25074 \nu^{7} - 2374515 \nu^{6} + \cdots + 952399 ) / 2599 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 9490 \nu^{11} + 44107 \nu^{10} + 93749 \nu^{9} - 627896 \nu^{8} - 32509 \nu^{7} + 2735205 \nu^{6} + \cdots - 1146553 ) / 2599 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{10} - \beta_{8} + \beta_{6} + 2\beta_{5} - 2\beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{11} + \beta_{10} + \beta_{9} - 2\beta_{6} + 11\beta_{5} + 8\beta_{4} + 9\beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( 16 \beta_{11} + 10 \beta_{10} + 7 \beta_{9} - 10 \beta_{8} - \beta_{7} + 8 \beta_{6} + 22 \beta_{5} + \cdots + 17 \)
|
| \(\nu^{6}\) | \(=\) |
\( 39 \beta_{11} + 17 \beta_{10} + 26 \beta_{9} - 5 \beta_{8} + \beta_{7} - 28 \beta_{6} + 107 \beta_{5} + \cdots + 125 \)
|
| \(\nu^{7}\) | \(=\) |
\( 200 \beta_{11} + 102 \beta_{10} + 119 \beta_{9} - 98 \beta_{8} - 11 \beta_{7} + 49 \beta_{6} + \cdots + 150 \)
|
| \(\nu^{8}\) | \(=\) |
\( 533 \beta_{11} + 220 \beta_{10} + 390 \beta_{9} - 102 \beta_{8} + 17 \beta_{7} - 306 \beta_{6} + \cdots + 1045 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2293 \beta_{11} + 1067 \beta_{10} + 1531 \beta_{9} - 991 \beta_{8} - 87 \beta_{7} + 226 \beta_{6} + \cdots + 1479 \)
|
| \(\nu^{10}\) | \(=\) |
\( 6431 \beta_{11} + 2602 \beta_{10} + 4889 \beta_{9} - 1482 \beta_{8} + 204 \beta_{7} - 3113 \beta_{6} + \cdots + 9443 \)
|
| \(\nu^{11}\) | \(=\) |
\( 25282 \beta_{11} + 11239 \beta_{10} + 17863 \beta_{9} - 10168 \beta_{8} - 604 \beta_{7} + 164 \beta_{6} + \cdots + 15700 \)
|
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 | −3.27119 | 0 | 3.82141 | 0 | −1.10063 | 0 | 7.70067 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.43162 | 0 | −0.659543 | 0 | 4.52406 | 0 | 2.91279 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.36915 | 0 | 2.61284 | 0 | −2.25519 | 0 | 2.61289 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.85534 | 0 | −3.75753 | 0 | −3.33364 | 0 | 0.442279 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.18762 | 0 | 1.03337 | 0 | 2.52681 | 0 | −1.58957 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.859886 | 0 | 2.49803 | 0 | −3.53399 | 0 | −2.26060 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.660935 | 0 | 2.45345 | 0 | 3.10518 | 0 | −2.56317 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 0.692763 | 0 | −3.58411 | 0 | 3.81064 | 0 | −2.52008 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 0.864716 | 0 | −1.40497 | 0 | −2.45790 | 0 | −2.25227 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 1.18842 | 0 | 0.677556 | 0 | 1.46203 | 0 | −1.58766 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 1.70453 | 0 | −0.637713 | 0 | 1.14740 | 0 | −0.0945840 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 2.86344 | 0 | −0.0527883 | 0 | −1.89476 | 0 | 5.19930 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 4012.2.a.g | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4012.2.a.g | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):
|
\( T_{3}^{12} + 4 T_{3}^{11} - 13 T_{3}^{10} - 60 T_{3}^{9} + 48 T_{3}^{8} + 289 T_{3}^{7} - 89 T_{3}^{6} + \cdots + 82 \)
|
|
\( T_{5}^{12} - 3 T_{5}^{11} - 28 T_{5}^{10} + 92 T_{5}^{9} + 212 T_{5}^{8} - 839 T_{5}^{7} - 217 T_{5}^{6} + \cdots + 18 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 4 T^{11} + \cdots + 82 \)
|
| $5$ |
\( T^{12} - 3 T^{11} + \cdots + 18 \)
|
| $7$ |
\( T^{12} - 2 T^{11} + \cdots + 30902 \)
|
| $11$ |
\( T^{12} + 5 T^{11} + \cdots - 29282 \)
|
| $13$ |
\( T^{12} - 9 T^{11} + \cdots - 1244284 \)
|
| $17$ |
\( (T + 1)^{12} \)
|
| $19$ |
\( T^{12} + 15 T^{11} + \cdots + 48143 \)
|
| $23$ |
\( T^{12} + 22 T^{11} + \cdots + 16696 \)
|
| $29$ |
\( T^{12} + 17 T^{11} + \cdots + 23722758 \)
|
| $31$ |
\( T^{12} - 5 T^{11} + \cdots + 148992 \)
|
| $37$ |
\( T^{12} + 2 T^{11} + \cdots - 37351836 \)
|
| $41$ |
\( T^{12} + 18 T^{11} + \cdots + 2439118 \)
|
| $43$ |
\( T^{12} + 16 T^{11} + \cdots - 39998768 \)
|
| $47$ |
\( T^{12} + \cdots - 1640998672 \)
|
| $53$ |
\( T^{12} + \cdots + 242878863 \)
|
| $59$ |
\( (T - 1)^{12} \)
|
| $61$ |
\( T^{12} + 17 T^{11} + \cdots - 5210064 \)
|
| $67$ |
\( T^{12} + \cdots + 103259024 \)
|
| $71$ |
\( T^{12} + 10 T^{11} + \cdots - 27240352 \)
|
| $73$ |
\( T^{12} - 4 T^{11} + \cdots - 12410378 \)
|
| $79$ |
\( T^{12} - 48 T^{11} + \cdots - 1515716 \)
|
| $83$ |
\( T^{12} + \cdots - 1062116316 \)
|
| $89$ |
\( T^{12} + \cdots - 7578401132 \)
|
| $97$ |
\( T^{12} + \cdots + 8940764008 \)
|
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