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: |
\( x^{12} - 2 x^{11} + 2 x^{10} - 6 x^{9} + 89 x^{8} - 216 x^{7} + 272 x^{6} - 176 x^{5} + 652 x^{4} + \cdots + 324 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{12} - 2 x^{11} + 2 x^{10} - 6 x^{9} + 89 x^{8} - 216 x^{7} + 272 x^{6} - 176 x^{5} + 652 x^{4} + \cdots + 324 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 11685370 \nu^{11} + 4729987 \nu^{10} - 12809044 \nu^{9} - 192821460 \nu^{8} + \cdots - 36769423608 ) / 15833453832 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 13213136 \nu^{11} - 16641981 \nu^{10} + 11528852 \nu^{9} - 111514972 \nu^{8} + \cdots - 16166737368 ) / 5277817944 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 27526462 \nu^{11} + 41503481 \nu^{10} - 40877408 \nu^{9} + 147979164 \nu^{8} + \cdots + 14783967936 ) / 7916726916 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 46285500 \nu^{11} - 28151167 \nu^{10} - 2495890 \nu^{9} - 236861264 \nu^{8} + 3765122886 \nu^{7} + \cdots + 5937389496 ) / 5277817944 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 146629906 \nu^{11} - 171215663 \nu^{10} + 231923660 \nu^{9} - 790331298 \nu^{8} + \cdots - 73071884052 ) / 15833453832 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 13156 \nu^{11} - 27203 \nu^{10} + 15782 \nu^{9} - 71934 \nu^{8} + 1176554 \nu^{7} - 2852289 \nu^{6} + \cdots - 7895556 ) / 1272888 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 58194648 \nu^{11} - 75290531 \nu^{10} + 93106420 \nu^{9} - 310684918 \nu^{8} + \cdots - 33545375460 ) / 5277817944 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 235466494 \nu^{11} - 89440397 \nu^{10} - 65828374 \nu^{9} - 1135393986 \nu^{8} + \cdots + 84591688068 ) / 15833453832 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 417578162 \nu^{11} - 805826491 \nu^{10} + 539635342 \nu^{9} - 2229209484 \nu^{8} + \cdots - 196252812576 ) / 15833453832 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 50630910 \nu^{11} + 46829805 \nu^{10} - 29021210 \nu^{9} + 257691922 \nu^{8} + \cdots + 3742228380 ) / 1759272648 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 171913166 \nu^{11} - 358146703 \nu^{10} + 231000880 \nu^{9} - 900397110 \nu^{8} + \cdots - 94895247996 ) / 5277817944 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - \beta_{5} - \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} + \beta_{8} - 4\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - \beta_{8} - 4\beta_{7} - 3\beta_{6} + 3\beta_{5} + 2\beta_{4} - 5\beta_{3} - 3\beta_{2} + 3\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 11 \beta_{11} + 11 \beta_{9} + 2 \beta_{7} + 11 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \cdots - 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3 \beta_{11} + 14 \beta_{10} - 14 \beta_{9} + 3 \beta_{8} - 22 \beta_{7} + 20 \beta_{6} + 32 \beta_{5} + \cdots + 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3 \beta_{11} - 102 \beta_{10} + 3 \beta_{9} - 102 \beta_{8} - 26 \beta_{7} - 35 \beta_{6} - 26 \beta_{5} + \cdots - 16 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 157 \beta_{11} + 58 \beta_{10} + 58 \beta_{9} + 157 \beta_{8} + 273 \beta_{7} + 374 \beta_{6} + \cdots + 55 \)
|
| \(\nu^{8}\) | \(=\) |
\( 921 \beta_{11} + 58 \beta_{10} - 921 \beta_{9} - 58 \beta_{8} - 462 \beta_{7} - 921 \beta_{6} + \cdots + 878 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 783 \beta_{11} - 1646 \beta_{10} + 1646 \beta_{9} - 783 \beta_{8} + 1463 \beta_{7} - 1178 \beta_{6} + \cdots - 1775 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 783 \beta_{11} + 8372 \beta_{10} - 783 \beta_{9} + 8372 \beta_{8} + 2570 \beta_{7} + 5519 \beta_{6} + \cdots + 2368 \)
|
| \(\nu^{11}\) | \(=\) |
\( 16743 \beta_{11} - 9154 \beta_{10} - 9154 \beta_{9} - 16743 \beta_{8} - 22427 \beta_{7} - 34472 \beta_{6} + \cdots + 499 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1890\mathbb{Z}\right)^\times\).
| \(n\) | \(757\) | \(1081\) | \(1541\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{6}\) | \(-1\) |
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 | −2.62348 | + | 0.342559i | − | 1.00000i | 0 | −0.866025 | + | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||
| 971.2 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | 0.750512 | − | 2.53707i | − | 1.00000i | 0 | −0.866025 | + | 0.500000i | ||||||||||||||||||||||||||||||||||||||||||||
| 971.3 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | 2.60502 | + | 0.462462i | − | 1.00000i | 0 | −0.866025 | + | 0.500000i | ||||||||||||||||||||||||||||||||||||||||||||
| 971.4 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | −2.27114 | − | 1.35718i | 1.00000i | 0 | 0.866025 | − | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||||
| 971.5 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | −1.88932 | + | 1.85215i | 1.00000i | 0 | 0.866025 | − | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||||
| 971.6 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | 1.42841 | − | 2.22703i | 1.00000i | 0 | 0.866025 | − | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||||
| 1781.1 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | −2.62348 | − | 0.342559i | 1.00000i | 0 | −0.866025 | − | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||||
| 1781.2 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | 0.750512 | + | 2.53707i | 1.00000i | 0 | −0.866025 | − | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||||
| 1781.3 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | 2.60502 | − | 0.462462i | 1.00000i | 0 | −0.866025 | − | 0.500000i | |||||||||||||||||||||||||||||||||||||||||||||
| 1781.4 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | −2.27114 | + | 1.35718i | − | 1.00000i | 0 | 0.866025 | + | 0.500000i | ||||||||||||||||||||||||||||||||||||||||||||
| 1781.5 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | −1.88932 | − | 1.85215i | − | 1.00000i | 0 | 0.866025 | + | 0.500000i | ||||||||||||||||||||||||||||||||||||||||||||
| 1781.6 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | 1.42841 | + | 2.22703i | − | 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.j | yes | 12 |
| 3.b | odd | 2 | 1 | 1890.2.be.g | ✓ | 12 | |
| 7.d | odd | 6 | 1 | 1890.2.be.g | ✓ | 12 | |
| 21.g | even | 6 | 1 | inner | 1890.2.be.j | yes | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1890.2.be.g | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 1890.2.be.g | ✓ | 12 | 7.d | odd | 6 | 1 | |
| 1890.2.be.j | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 1890.2.be.j | yes | 12 | 21.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{12} - 12 T_{11}^{11} + 38 T_{11}^{10} + 120 T_{11}^{9} - 681 T_{11}^{8} - 1536 T_{11}^{7} + \cdots + 20736 \)
acting on \(S_{2}^{\mathrm{new}}(1890, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{2} + 1)^{3} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( (T^{2} - T + 1)^{6} \)
|
| $7$ |
\( T^{12} + 4 T^{11} + \cdots + 117649 \)
|
| $11$ |
\( T^{12} - 12 T^{11} + \cdots + 20736 \)
|
| $13$ |
\( T^{12} + 138 T^{10} + \cdots + 25522704 \)
|
| $17$ |
\( T^{12} - 12 T^{11} + \cdots + 219024 \)
|
| $19$ |
\( T^{12} - 6 T^{11} + \cdots + 20736 \)
|
| $23$ |
\( T^{12} - 68 T^{10} + \cdots + 186624 \)
|
| $29$ |
\( T^{12} + 220 T^{10} + \cdots + 6718464 \)
|
| $31$ |
\( T^{12} + 12 T^{11} + \cdots + 33454656 \)
|
| $37$ |
\( T^{12} + \cdots + 6922240000 \)
|
| $41$ |
\( (T^{6} - 4 T^{5} + \cdots + 9216)^{2} \)
|
| $43$ |
\( (T^{6} - 100 T^{4} + \cdots + 16)^{2} \)
|
| $47$ |
\( T^{12} + 20 T^{11} + \cdots + 331776 \)
|
| $53$ |
\( T^{12} + \cdots + 43477254144 \)
|
| $59$ |
\( T^{12} + \cdots + 64286588304 \)
|
| $61$ |
\( T^{12} + \cdots + 176783616 \)
|
| $67$ |
\( T^{12} + 4 T^{11} + \cdots + 20736 \)
|
| $71$ |
\( T^{12} + \cdots + 1167178896 \)
|
| $73$ |
\( T^{12} + \cdots + 2054264976 \)
|
| $79$ |
\( T^{12} + 4 T^{11} + \cdots + 456976 \)
|
| $83$ |
\( (T^{6} - 16 T^{5} + \cdots - 531)^{2} \)
|
| $89$ |
\( T^{12} + \cdots + 940771584 \)
|
| $97$ |
\( T^{12} + \cdots + 1867276944 \)
|
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