Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [405,4,Mod(136,405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("405.136");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.e (of order \(3\), degree \(2\), not 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: | \(23.8957735523\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} + 2 x^{10} + 32 x^{9} + 583 x^{8} - 624 x^{7} + 594 x^{6} + 9450 x^{5} + 90513 x^{4} + \cdots + 746496 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{9} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{12} - 2 x^{11} + 2 x^{10} + 32 x^{9} + 583 x^{8} - 624 x^{7} + 594 x^{6} + 9450 x^{5} + 90513 x^{4} + \cdots + 746496 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 39168669767185 \nu^{11} + 622492575406102 \nu^{10} - 116378803247278 \nu^{9} + \cdots + 57\!\cdots\!56 ) / 53\!\cdots\!60 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 21833561 \nu^{11} - 6933902 \nu^{10} + 191663726 \nu^{9} - 824768992 \nu^{8} + \cdots + 11197076020992 ) / 28100111293440 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8315698756077 \nu^{11} + 101245330415030 \nu^{10} - 394560515037286 \nu^{9} + \cdots + 15\!\cdots\!84 ) / 44\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4200645557245 \nu^{11} + 6160897283854 \nu^{10} - 314135929010806 \nu^{9} + \cdots + 62\!\cdots\!12 ) / 22\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 335658953887093 \nu^{11} + \cdots - 17\!\cdots\!56 ) / 15\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 11005996995395 \nu^{11} - 8278688448046 \nu^{10} - 261584063656106 \nu^{9} + \cdots - 48\!\cdots\!08 ) / 33\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 238036479862585 \nu^{11} + \cdots + 17\!\cdots\!36 ) / 53\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 10\!\cdots\!07 \nu^{11} - 357550815547730 \nu^{10} + \cdots - 22\!\cdots\!44 ) / 15\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 96943826623309 \nu^{11} + 480126841023914 \nu^{10} - 441998532012314 \nu^{9} + \cdots - 73\!\cdots\!36 ) / 13\!\cdots\!40 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 18\!\cdots\!33 \nu^{11} + \cdots + 22\!\cdots\!92 ) / 15\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 24\!\cdots\!15 \nu^{11} + \cdots + 54\!\cdots\!64 ) / 53\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - 2\beta_{5} + \beta_{3} - 2\beta_{2} + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{10} - 5\beta_{9} - \beta_{8} - 8\beta_{7} + \beta_{6} - 2\beta_{5} - 5\beta_{3} + 8\beta_1 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{11} + 8 \beta_{10} + 12 \beta_{9} - 9 \beta_{8} - 8 \beta_{7} + 7 \beta_{6} - 17 \beta_{5} + \cdots - 35 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 35 \beta_{10} + 35 \beta_{9} - 35 \beta_{8} - 24 \beta_{7} - 81 \beta_{6} + 40 \beta_{4} - 35 \beta_{3} + \cdots - 1298 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 26 \beta_{11} + 365 \beta_{10} + 501 \beta_{9} - 279 \beta_{8} - 184 \beta_{7} - 451 \beta_{6} + \cdots - 904 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 16 \beta_{11} + 325 \beta_{10} + 758 \beta_{9} + 325 \beta_{8} + 1288 \beta_{7} - 488 \beta_{6} + \cdots + 1409 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 144 \beta_{11} - 5057 \beta_{10} - 5627 \beta_{9} + 7763 \beta_{8} + 11704 \beta_{7} - 2351 \beta_{6} + \cdots + 58398 ) / 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 24965 \beta_{10} - 17315 \beta_{9} + 24965 \beta_{8} + 32472 \beta_{7} + 46041 \beta_{6} + \cdots + 480200 ) / 6 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 2095 \beta_{11} - 84984 \beta_{10} - 71512 \beta_{9} + 47317 \beta_{8} + 42152 \beta_{7} + \cdots + 260571 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 61008 \beta_{11} - 360417 \beta_{10} - 395679 \beta_{9} - 360417 \beta_{8} - 1058408 \beta_{7} + \cdots - 2551826 ) / 6 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 216770 \beta_{11} + 1828317 \beta_{10} + 1259071 \beta_{9} - 3796381 \beta_{8} - 6773512 \beta_{7} + \cdots - 34930520 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).
| \(n\) | \(82\) | \(326\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 136.1 |
|
−2.05867 | + | 3.56572i | 0 | −4.47625 | − | 7.75309i | 2.50000 | + | 4.33013i | 0 | 10.0115 | − | 17.3404i | 3.92177 | 0 | −20.5867 | ||||||||||||||||||||||||||||||||||||||||||||||
| 136.2 | −1.78729 | + | 3.09567i | 0 | −2.38879 | − | 4.13751i | 2.50000 | + | 4.33013i | 0 | −7.07987 | + | 12.2627i | −11.5188 | 0 | −17.8729 | |||||||||||||||||||||||||||||||||||||||||||||||
| 136.3 | −0.0745751 | + | 0.129168i | 0 | 3.98888 | + | 6.90894i | 2.50000 | + | 4.33013i | 0 | −10.0712 | + | 17.4439i | −2.38308 | 0 | −0.745751 | |||||||||||||||||||||||||||||||||||||||||||||||
| 136.4 | 1.03663 | − | 1.79550i | 0 | 1.85079 | + | 3.20567i | 2.50000 | + | 4.33013i | 0 | 2.33056 | − | 4.03665i | 24.2604 | 0 | 10.3663 | |||||||||||||||||||||||||||||||||||||||||||||||
| 136.5 | 2.26722 | − | 3.92694i | 0 | −6.28058 | − | 10.8783i | 2.50000 | + | 4.33013i | 0 | 1.31809 | − | 2.28300i | −20.6823 | 0 | 22.6722 | |||||||||||||||||||||||||||||||||||||||||||||||
| 136.6 | 2.61668 | − | 4.53223i | 0 | −9.69405 | − | 16.7906i | 2.50000 | + | 4.33013i | 0 | −16.5090 | + | 28.5945i | −59.5981 | 0 | 26.1668 | |||||||||||||||||||||||||||||||||||||||||||||||
| 271.1 | −2.05867 | − | 3.56572i | 0 | −4.47625 | + | 7.75309i | 2.50000 | − | 4.33013i | 0 | 10.0115 | + | 17.3404i | 3.92177 | 0 | −20.5867 | |||||||||||||||||||||||||||||||||||||||||||||||
| 271.2 | −1.78729 | − | 3.09567i | 0 | −2.38879 | + | 4.13751i | 2.50000 | − | 4.33013i | 0 | −7.07987 | − | 12.2627i | −11.5188 | 0 | −17.8729 | |||||||||||||||||||||||||||||||||||||||||||||||
| 271.3 | −0.0745751 | − | 0.129168i | 0 | 3.98888 | − | 6.90894i | 2.50000 | − | 4.33013i | 0 | −10.0712 | − | 17.4439i | −2.38308 | 0 | −0.745751 | |||||||||||||||||||||||||||||||||||||||||||||||
| 271.4 | 1.03663 | + | 1.79550i | 0 | 1.85079 | − | 3.20567i | 2.50000 | − | 4.33013i | 0 | 2.33056 | + | 4.03665i | 24.2604 | 0 | 10.3663 | |||||||||||||||||||||||||||||||||||||||||||||||
| 271.5 | 2.26722 | + | 3.92694i | 0 | −6.28058 | + | 10.8783i | 2.50000 | − | 4.33013i | 0 | 1.31809 | + | 2.28300i | −20.6823 | 0 | 22.6722 | |||||||||||||||||||||||||||||||||||||||||||||||
| 271.6 | 2.61668 | + | 4.53223i | 0 | −9.69405 | + | 16.7906i | 2.50000 | − | 4.33013i | 0 | −16.5090 | − | 28.5945i | −59.5981 | 0 | 26.1668 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 405.4.e.x | 12 | |
| 3.b | odd | 2 | 1 | 405.4.e.w | 12 | ||
| 9.c | even | 3 | 1 | 405.4.a.k | ✓ | 6 | |
| 9.c | even | 3 | 1 | inner | 405.4.e.x | 12 | |
| 9.d | odd | 6 | 1 | 405.4.a.l | yes | 6 | |
| 9.d | odd | 6 | 1 | 405.4.e.w | 12 | ||
| 45.h | odd | 6 | 1 | 2025.4.a.y | 6 | ||
| 45.j | even | 6 | 1 | 2025.4.a.z | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 405.4.a.k | ✓ | 6 | 9.c | even | 3 | 1 | |
| 405.4.a.l | yes | 6 | 9.d | odd | 6 | 1 | |
| 405.4.e.w | 12 | 3.b | odd | 2 | 1 | ||
| 405.4.e.w | 12 | 9.d | odd | 6 | 1 | ||
| 405.4.e.x | 12 | 1.a | even | 1 | 1 | trivial | |
| 405.4.e.x | 12 | 9.c | even | 3 | 1 | inner | |
| 2025.4.a.y | 6 | 45.h | odd | 6 | 1 | ||
| 2025.4.a.z | 6 | 45.j | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):
|
\( T_{2}^{12} - 4 T_{2}^{11} + 49 T_{2}^{10} - 88 T_{2}^{9} + 1243 T_{2}^{8} - 2026 T_{2}^{7} + \cdots + 11664 \)
|
|
\( T_{7}^{12} + 40 T_{7}^{11} + 1863 T_{7}^{10} + 27280 T_{7}^{9} + 874429 T_{7}^{8} + 10054356 T_{7}^{7} + \cdots + 5368136821776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 4 T^{11} + \cdots + 11664 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( (T^{2} - 5 T + 25)^{6} \)
|
| $7$ |
\( T^{12} + \cdots + 5368136821776 \)
|
| $11$ |
\( T^{12} + \cdots + 1433067563664 \)
|
| $13$ |
\( T^{12} + \cdots + 34\!\cdots\!84 \)
|
| $17$ |
\( (T^{6} + 124 T^{5} + \cdots + 105966288)^{2} \)
|
| $19$ |
\( (T^{6} + 46 T^{5} + \cdots - 36821611175)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 11\!\cdots\!24 \)
|
| $29$ |
\( T^{12} + \cdots + 40\!\cdots\!44 \)
|
| $31$ |
\( T^{12} + \cdots + 48\!\cdots\!56 \)
|
| $37$ |
\( (T^{6} + \cdots + 12008297128192)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 74\!\cdots\!09 \)
|
| $43$ |
\( T^{12} + \cdots + 32\!\cdots\!00 \)
|
| $47$ |
\( T^{12} + \cdots + 24\!\cdots\!00 \)
|
| $53$ |
\( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 28\!\cdots\!89 \)
|
| $61$ |
\( T^{12} + \cdots + 41\!\cdots\!96 \)
|
| $67$ |
\( T^{12} + \cdots + 38\!\cdots\!00 \)
|
| $71$ |
\( (T^{6} + \cdots - 11\!\cdots\!84)^{2} \)
|
| $73$ |
\( (T^{6} + \cdots - 10\!\cdots\!36)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 25\!\cdots\!84 \)
|
| $83$ |
\( T^{12} + \cdots + 17\!\cdots\!24 \)
|
| $89$ |
\( (T^{6} + \cdots + 16\!\cdots\!48)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 96\!\cdots\!76 \)
|
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