Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1890,2,Mod(971,1890)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1890, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1890.971");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1890.be (of order \(6\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(15.0917259820\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 4 x^{11} + 6 x^{10} - 4 x^{9} - 12 x^{8} + 208 x^{7} - 1014 x^{6} + 1456 x^{5} - 588 x^{4} + \cdots + 117649 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{12} - 4 x^{11} + 6 x^{10} - 4 x^{9} - 12 x^{8} + 208 x^{7} - 1014 x^{6} + 1456 x^{5} - 588 x^{4} + \cdots + 117649 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 19 \nu^{11} + 36 \nu^{10} + 303 \nu^{9} + 1478 \nu^{8} + 6233 \nu^{7} + 20640 \nu^{6} + \cdots - 14891002 ) / 5445468 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 16 \nu^{11} + 27 \nu^{10} + 222 \nu^{9} + 923 \nu^{8} + 2384 \nu^{7} - 6927 \nu^{6} + 1626 \nu^{5} + \cdots - 319333 ) / 777924 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 563 \nu^{11} + 173 \nu^{10} - 697 \nu^{9} + 6487 \nu^{8} - 2617 \nu^{7} - 92597 \nu^{6} + \cdots + 15344791 ) / 25412184 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 913 \nu^{11} + 289 \nu^{10} + 4267 \nu^{9} + 1227 \nu^{8} - 56365 \nu^{7} + 208223 \nu^{6} + \cdots - 22639029 ) / 25412184 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{11} - 4 \nu^{10} + 6 \nu^{9} - 4 \nu^{8} - 12 \nu^{7} + 208 \nu^{6} - 1014 \nu^{5} + \cdots - 67228 ) / 16807 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 844 \nu^{11} - 261 \nu^{10} + 1032 \nu^{9} - 9823 \nu^{8} + 3284 \nu^{7} + 134301 \nu^{6} + \cdots - 23008783 ) / 12706092 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 1825 \nu^{11} - 575 \nu^{10} - 8507 \nu^{9} - 2269 \nu^{8} + 114013 \nu^{7} - 407257 \nu^{6} + \cdots + 45261251 ) / 25412184 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 297 \nu^{11} - 383 \nu^{10} - 605 \nu^{9} + 1339 \nu^{8} - 3501 \nu^{7} + 57359 \nu^{6} + \cdots - 9462341 ) / 3630312 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 562 \nu^{11} - 2115 \nu^{10} + 3624 \nu^{9} - 127 \nu^{8} + 3602 \nu^{7} + 160527 \nu^{6} + \cdots - 36723295 ) / 5445468 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 593 \nu^{11} - 769 \nu^{10} - 1237 \nu^{9} + 2493 \nu^{8} - 8285 \nu^{7} + 105529 \nu^{6} + \cdots - 18907875 ) / 3630312 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{7} + 2\beta_{5} + 3\beta_{4} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - 2\beta_{9} + \beta_{8} - 6\beta_{7} + 2\beta_{5} - 18\beta_{4} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 5 \beta_{11} - 3 \beta_{10} + 10 \beta_{9} + \beta_{8} - 6 \beta_{7} + 5 \beta_{6} + 2 \beta_{5} + \cdots + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 11 \beta_{11} + 15 \beta_{10} + 22 \beta_{9} + \beta_{8} - 6 \beta_{7} - 25 \beta_{6} + 2 \beta_{5} + \cdots - 96 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 17 \beta_{11} + 33 \beta_{10} + 34 \beta_{9} - 7 \beta_{7} - 55 \beta_{6} + 21 \beta_{5} - 33 \beta_{3} + \cdots + 192 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 24 \beta_{11} + 51 \beta_{10} + 27 \beta_{9} + 12 \beta_{8} + 12 \beta_{7} - 85 \beta_{6} + \cdots + 480 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 12 \beta_{11} + 30 \beta_{10} + 381 \beta_{9} - 12 \beta_{8} - 96 \beta_{7} - 99 \beta_{6} + \cdots + 823 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 108 \beta_{11} + 750 \beta_{10} - 1317 \beta_{9} - 72 \beta_{8} + 12 \beta_{7} - 417 \beta_{6} + \cdots - 264 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 96 \beta_{11} - 2742 \beta_{10} - 963 \beta_{9} - 230 \beta_{8} + 274 \beta_{7} + 993 \beta_{6} + \cdots + 11904 \)
|
| \(\nu^{11}\) | \(=\) |
\( 178 \beta_{11} - 2022 \beta_{10} + 5398 \beta_{9} + 1126 \beta_{8} + 2736 \beta_{7} + 675 \beta_{6} + \cdots - 23448 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1890\mathbb{Z}\right)^\times\).
| \(n\) | \(757\) | \(1081\) | \(1541\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 971.1 |
|
−0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | −1.92684 | − | 1.81309i | − | 1.00000i | 0 | −0.866025 | + | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||
| 971.2 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | −1.71096 | + | 2.01807i | − | 1.00000i | 0 | −0.866025 | + | 0.500000i | ||||||||||||||||||||||||||||||||||||||||||||
| 971.3 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | 2.63780 | − | 0.204977i | − | 1.00000i | 0 | −0.866025 | + | 0.500000i | ||||||||||||||||||||||||||||||||||||||||||||
| 971.4 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | −2.64306 | + | 0.119336i | 1.00000i | 0 | 0.866025 | − | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||||
| 971.5 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | 0.639398 | − | 2.56733i | 1.00000i | 0 | 0.866025 | − | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||||
| 971.6 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | 1.00366 | + | 2.44799i | 1.00000i | 0 | 0.866025 | − | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||||
| 1781.1 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | −1.92684 | + | 1.81309i | 1.00000i | 0 | −0.866025 | − | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||||
| 1781.2 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | −1.71096 | − | 2.01807i | 1.00000i | 0 | −0.866025 | − | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||||
| 1781.3 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | 2.63780 | + | 0.204977i | 1.00000i | 0 | −0.866025 | − | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||||
| 1781.4 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | −2.64306 | − | 0.119336i | − | 1.00000i | 0 | 0.866025 | + | 0.500000i | ||||||||||||||||||||||||||||||||||||||||||||
| 1781.5 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | 0.639398 | + | 2.56733i | − | 1.00000i | 0 | 0.866025 | + | 0.500000i | ||||||||||||||||||||||||||||||||||||||||||||
| 1781.6 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | 1.00366 | − | 2.44799i | − | 1.00000i | 0 | 0.866025 | + | 0.500000i | ||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 21.g | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1890.2.be.i | yes | 12 |
| 3.b | odd | 2 | 1 | 1890.2.be.h | ✓ | 12 | |
| 7.d | odd | 6 | 1 | 1890.2.be.h | ✓ | 12 | |
| 21.g | even | 6 | 1 | inner | 1890.2.be.i | yes | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1890.2.be.h | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 1890.2.be.h | ✓ | 12 | 7.d | odd | 6 | 1 | |
| 1890.2.be.i | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 1890.2.be.i | yes | 12 | 21.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{12} + 12 T_{11}^{11} + 26 T_{11}^{10} - 264 T_{11}^{9} - 681 T_{11}^{8} + 4596 T_{11}^{7} + \cdots + 5184 \)
acting on \(S_{2}^{\mathrm{new}}(1890, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{2} + 1)^{3} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( (T^{2} - T + 1)^{6} \)
|
| $7$ |
\( T^{12} + 4 T^{11} + \cdots + 117649 \)
|
| $11$ |
\( T^{12} + 12 T^{11} + \cdots + 5184 \)
|
| $13$ |
\( T^{12} + 102 T^{10} + \cdots + 11778624 \)
|
| $17$ |
\( T^{12} + 46 T^{10} + \cdots + 5184 \)
|
| $19$ |
\( T^{12} - 42 T^{10} + \cdots + 876096 \)
|
| $23$ |
\( T^{12} - 62 T^{10} + \cdots + 627264 \)
|
| $29$ |
\( T^{12} + 136 T^{10} + \cdots + 40144896 \)
|
| $31$ |
\( T^{12} + \cdots + 280629504 \)
|
| $37$ |
\( T^{12} + 2 T^{11} + \cdots + 21316 \)
|
| $41$ |
\( (T^{6} - 4 T^{5} + \cdots + 7488)^{2} \)
|
| $43$ |
\( (T^{6} - 6 T^{5} + \cdots + 58708)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 9673115904 \)
|
| $53$ |
\( T^{12} - 12 T^{11} + \cdots + 104976 \)
|
| $59$ |
\( T^{12} + 46 T^{10} + \cdots + 5184 \)
|
| $61$ |
\( T^{12} - 30 T^{11} + \cdots + 26378496 \)
|
| $67$ |
\( T^{12} + \cdots + 41492467809 \)
|
| $71$ |
\( T^{12} + \cdots + 2322978353424 \)
|
| $73$ |
\( T^{12} + 12 T^{11} + \cdots + 746496 \)
|
| $79$ |
\( T^{12} - 2 T^{11} + \cdots + 39262756 \)
|
| $83$ |
\( (T^{6} - 4 T^{5} + \cdots - 45864)^{2} \)
|
| $89$ |
\( T^{12} - 8 T^{11} + \cdots + 34012224 \)
|
| $97$ |
\( T^{12} + \cdots + 6480572004 \)
|
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