Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [57,2,Mod(4,57)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(57, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("57.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 57.i (of order \(9\), degree \(6\), 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: | \(0.455147291521\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{9})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} + \cdots + 57 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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
\( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} + \cdots + 57 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + \cdots + 1060 ) / 218 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 36 \nu^{11} - 89 \nu^{10} + 544 \nu^{9} - 745 \nu^{8} + 2301 \nu^{7} - 1512 \nu^{6} + 3777 \nu^{5} + \cdots - 1706 ) / 218 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} + \cdots - 2263 ) / 218 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 26 \nu^{11} - 34 \nu^{10} + 187 \nu^{9} + 449 \nu^{8} - 1590 \nu^{7} + 6865 \nu^{6} - 12623 \nu^{5} + \cdots + 524 ) / 218 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2 \nu^{11} + 120 \nu^{10} - 551 \nu^{9} + 2615 \nu^{8} - 6686 \nu^{7} + 15780 \nu^{6} - 23990 \nu^{5} + \cdots + 555 ) / 218 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + \cdots - 83 ) / 218 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{10} - 5 \nu^{9} + 25 \nu^{8} - 70 \nu^{7} + 173 \nu^{6} - 295 \nu^{5} + 412 \nu^{4} - 404 \nu^{3} + \cdots + 26 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 2 \nu^{11} + 98 \nu^{10} - 539 \nu^{9} + 2726 \nu^{8} - 8138 \nu^{7} + 20190 \nu^{6} - 36614 \nu^{5} + \cdots + 4350 ) / 218 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 26 \nu^{11} - 252 \nu^{10} + 1277 \nu^{9} - 4892 \nu^{8} + 13234 \nu^{7} - 28887 \nu^{6} + 47327 \nu^{5} + \cdots - 2201 ) / 218 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 36 \nu^{11} - 307 \nu^{10} + 1634 \nu^{9} - 6086 \nu^{8} + 17125 \nu^{7} - 37373 \nu^{6} + 64054 \nu^{5} + \cdots - 5194 ) / 218 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 91 \nu^{11} - 664 \nu^{10} + 3325 \nu^{9} - 11563 \nu^{8} + 30405 \nu^{7} - 62791 \nu^{6} + \cdots - 5469 ) / 218 \)
|
| \(\nu\) | \(=\) |
\( ( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \cdots - 7 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 7 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + \beta_{7} - 13 \beta_{6} + 5 \beta_{5} + \cdots - 10 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 16 \beta_{11} + 11 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 20 \beta_{7} - 7 \beta_{6} + 14 \beta_{5} + \cdots + 32 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 23 \beta_{11} + 14 \beta_{10} - 34 \beta_{9} + 25 \beta_{8} - 26 \beta_{7} + 65 \beta_{6} - 10 \beta_{5} + \cdots + 77 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 110 \beta_{11} - 49 \beta_{10} - 88 \beta_{9} + 67 \beta_{8} + 85 \beta_{7} + 101 \beta_{6} - 94 \beta_{5} + \cdots - 121 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 25 \beta_{11} - 187 \beta_{10} + 158 \beta_{9} - 101 \beta_{8} + 229 \beta_{7} - 271 \beta_{6} + \cdots - 535 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 652 \beta_{11} + 41 \beta_{10} + 761 \beta_{9} - 509 \beta_{8} - 263 \beta_{7} - 826 \beta_{6} + \cdots + 257 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 478 \beta_{11} + 1304 \beta_{10} - 268 \beta_{9} + 160 \beta_{8} - 1571 \beta_{7} + 821 \beta_{6} + \cdots + 3359 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 3353 \beta_{11} + 1451 \beta_{10} - 5224 \beta_{9} + 3352 \beta_{8} + 25 \beta_{7} + 5630 \beta_{6} + \cdots + 1451 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 6041 \beta_{11} - 6547 \beta_{10} - 3685 \beta_{9} + 2323 \beta_{8} + 9241 \beta_{7} + 326 \beta_{6} + \cdots - 18736 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).
| \(n\) | \(20\) | \(40\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3} + \beta_{9}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
−2.23121 | − | 0.812094i | 0.173648 | + | 0.984808i | 2.78672 | + | 2.33833i | 2.16379 | − | 1.81563i | 0.412311 | − | 2.33833i | 1.05756 | − | 1.83175i | −1.94440 | − | 3.36780i | −0.939693 | + | 0.342020i | −6.30233 | + | 2.29386i | ||||||||||||||||||||||||||||||||||||
| 4.2 | 0.791518 | + | 0.288089i | 0.173648 | + | 0.984808i | −0.988583 | − | 0.829520i | 1.30800 | − | 1.09754i | −0.146267 | + | 0.829520i | −1.96517 | + | 3.40377i | −1.38582 | − | 2.40031i | −0.939693 | + | 0.342020i | 1.35149 | − | 0.491903i | |||||||||||||||||||||||||||||||||||||
| 16.1 | −0.958464 | − | 0.804247i | −0.939693 | + | 0.342020i | −0.0754558 | − | 0.427931i | 0.755623 | − | 4.28535i | 1.17573 | + | 0.427931i | 0.898157 | + | 1.55565i | −1.52303 | + | 2.63796i | 0.766044 | − | 0.642788i | −4.17072 | + | 3.49965i | |||||||||||||||||||||||||||||||||||||
| 16.2 | 1.22451 | + | 1.02748i | −0.939693 | + | 0.342020i | 0.0964003 | + | 0.546713i | −0.174371 | + | 0.988909i | −1.50208 | − | 0.546713i | −1.28482 | − | 2.22537i | 1.15479 | − | 2.00015i | 0.766044 | − | 0.642788i | −1.22961 | + | 1.03176i | |||||||||||||||||||||||||||||||||||||
| 25.1 | −0.958464 | + | 0.804247i | −0.939693 | − | 0.342020i | −0.0754558 | + | 0.427931i | 0.755623 | + | 4.28535i | 1.17573 | − | 0.427931i | 0.898157 | − | 1.55565i | −1.52303 | − | 2.63796i | 0.766044 | + | 0.642788i | −4.17072 | − | 3.49965i | |||||||||||||||||||||||||||||||||||||
| 25.2 | 1.22451 | − | 1.02748i | −0.939693 | − | 0.342020i | 0.0964003 | − | 0.546713i | −0.174371 | − | 0.988909i | −1.50208 | + | 0.546713i | −1.28482 | + | 2.22537i | 1.15479 | + | 2.00015i | 0.766044 | + | 0.642788i | −1.22961 | − | 1.03176i | |||||||||||||||||||||||||||||||||||||
| 28.1 | −0.458021 | − | 2.59757i | 0.766044 | − | 0.642788i | −4.65819 | + | 1.69544i | 1.91800 | + | 0.698096i | −2.02055 | − | 1.69544i | −1.30802 | + | 2.26556i | 3.89993 | + | 6.75488i | 0.173648 | − | 0.984808i | 0.934866 | − | 5.30189i | |||||||||||||||||||||||||||||||||||||
| 28.2 | 0.131669 | + | 0.746734i | 0.766044 | − | 0.642788i | 1.33911 | − | 0.487396i | −2.97104 | − | 1.08137i | 0.580856 | + | 0.487396i | −1.89771 | + | 3.28694i | 1.29853 | + | 2.24912i | 0.173648 | − | 0.984808i | 0.416301 | − | 2.36096i | |||||||||||||||||||||||||||||||||||||
| 43.1 | −2.23121 | + | 0.812094i | 0.173648 | − | 0.984808i | 2.78672 | − | 2.33833i | 2.16379 | + | 1.81563i | 0.412311 | + | 2.33833i | 1.05756 | + | 1.83175i | −1.94440 | + | 3.36780i | −0.939693 | − | 0.342020i | −6.30233 | − | 2.29386i | |||||||||||||||||||||||||||||||||||||
| 43.2 | 0.791518 | − | 0.288089i | 0.173648 | − | 0.984808i | −0.988583 | + | 0.829520i | 1.30800 | + | 1.09754i | −0.146267 | − | 0.829520i | −1.96517 | − | 3.40377i | −1.38582 | + | 2.40031i | −0.939693 | − | 0.342020i | 1.35149 | + | 0.491903i | |||||||||||||||||||||||||||||||||||||
| 55.1 | −0.458021 | + | 2.59757i | 0.766044 | + | 0.642788i | −4.65819 | − | 1.69544i | 1.91800 | − | 0.698096i | −2.02055 | + | 1.69544i | −1.30802 | − | 2.26556i | 3.89993 | − | 6.75488i | 0.173648 | + | 0.984808i | 0.934866 | + | 5.30189i | |||||||||||||||||||||||||||||||||||||
| 55.2 | 0.131669 | − | 0.746734i | 0.766044 | + | 0.642788i | 1.33911 | + | 0.487396i | −2.97104 | + | 1.08137i | 0.580856 | − | 0.487396i | −1.89771 | − | 3.28694i | 1.29853 | − | 2.24912i | 0.173648 | + | 0.984808i | 0.416301 | + | 2.36096i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.e | even | 9 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 57.2.i.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 171.2.u.e | 12 | ||
| 4.b | odd | 2 | 1 | 912.2.bo.j | 12 | ||
| 19.e | even | 9 | 1 | inner | 57.2.i.b | ✓ | 12 |
| 19.e | even | 9 | 1 | 1083.2.a.q | 6 | ||
| 19.f | odd | 18 | 1 | 1083.2.a.p | 6 | ||
| 57.j | even | 18 | 1 | 3249.2.a.bg | 6 | ||
| 57.l | odd | 18 | 1 | 171.2.u.e | 12 | ||
| 57.l | odd | 18 | 1 | 3249.2.a.bh | 6 | ||
| 76.l | odd | 18 | 1 | 912.2.bo.j | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.2.i.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 57.2.i.b | ✓ | 12 | 19.e | even | 9 | 1 | inner |
| 171.2.u.e | 12 | 3.b | odd | 2 | 1 | ||
| 171.2.u.e | 12 | 57.l | odd | 18 | 1 | ||
| 912.2.bo.j | 12 | 4.b | odd | 2 | 1 | ||
| 912.2.bo.j | 12 | 76.l | odd | 18 | 1 | ||
| 1083.2.a.p | 6 | 19.f | odd | 18 | 1 | ||
| 1083.2.a.q | 6 | 19.e | even | 9 | 1 | ||
| 3249.2.a.bg | 6 | 57.j | even | 18 | 1 | ||
| 3249.2.a.bh | 6 | 57.l | odd | 18 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 3 T_{2}^{11} + 6 T_{2}^{10} + 8 T_{2}^{9} - 9 T_{2}^{8} - 9 T_{2}^{7} + 99 T_{2}^{6} + \cdots + 64 \)
acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 3 T^{11} + \cdots + 64 \)
|
| $3$ |
\( (T^{6} + T^{3} + 1)^{2} \)
|
| $5$ |
\( T^{12} - 6 T^{11} + \cdots + 18496 \)
|
| $7$ |
\( T^{12} + 9 T^{11} + \cdots + 145161 \)
|
| $11$ |
\( T^{12} + 9 T^{11} + \cdots + 207936 \)
|
| $13$ |
\( T^{12} + 3 T^{11} + \cdots + 3249 \)
|
| $17$ |
\( T^{12} - 12 T^{11} + \cdots + 41062464 \)
|
| $19$ |
\( T^{12} + 9 T^{11} + \cdots + 47045881 \)
|
| $23$ |
\( T^{12} - 9 T^{11} + \cdots + 2483776 \)
|
| $29$ |
\( T^{12} - 6 T^{10} + \cdots + 87616 \)
|
| $31$ |
\( T^{12} + 36 T^{10} + \cdots + 185761 \)
|
| $37$ |
\( (T^{6} + 6 T^{5} + \cdots + 2467)^{2} \)
|
| $41$ |
\( T^{12} + 18 T^{11} + \cdots + 87616 \)
|
| $43$ |
\( T^{12} + \cdots + 587917009 \)
|
| $47$ |
\( T^{12} + 9 T^{11} + \cdots + 46656 \)
|
| $53$ |
\( T^{12} + 30 T^{11} + \cdots + 1871424 \)
|
| $59$ |
\( T^{12} + 30 T^{11} + \cdots + 1617984 \)
|
| $61$ |
\( T^{12} - 21 T^{11} + \cdots + 5329 \)
|
| $67$ |
\( T^{12} + \cdots + 1254293056 \)
|
| $71$ |
\( T^{12} + \cdots + 241118784 \)
|
| $73$ |
\( T^{12} + \cdots + 1056705049 \)
|
| $79$ |
\( T^{12} + \cdots + 4606201161 \)
|
| $83$ |
\( T^{12} + \cdots + 75809912896 \)
|
| $89$ |
\( T^{12} + \cdots + 21486869056 \)
|
| $97$ |
\( T^{12} + 27 T^{11} + \cdots + 128881 \)
|
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