Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1014,2,Mod(239,1014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1014, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1014.239");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1014.g (of order \(4\), 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: | \(8.09683076496\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | 16.0.9349208943630483456.9 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 78) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 50 \nu^{14} + 350 \nu^{13} - 2019 \nu^{12} + 7564 \nu^{11} - 23298 \nu^{10} + 55495 \nu^{9} + \cdots - 2945 ) / 65 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 62 \nu^{14} - 434 \nu^{13} + 2541 \nu^{12} - 9604 \nu^{11} + 30137 \nu^{10} - 72992 \nu^{9} + \cdots + 5945 ) / 65 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 107 \nu^{14} - 749 \nu^{13} + 4349 \nu^{12} - 16357 \nu^{11} + 50788 \nu^{10} - 121852 \nu^{9} + \cdots + 7055 ) / 65 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 13659 \nu^{15} + 79956 \nu^{14} - 449175 \nu^{13} + 1476715 \nu^{12} - 4234710 \nu^{11} + \cdots - 486085 ) / 17095 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 18512 \nu^{15} - 119115 \nu^{14} + 677662 \nu^{13} - 2393217 \nu^{12} + 7123788 \nu^{11} + \cdots + 928550 ) / 17095 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1424 \nu^{15} + 10943 \nu^{14} - 64590 \nu^{13} + 256277 \nu^{12} - 819076 \nu^{11} + 2060111 \nu^{10} + \cdots + 34380 ) / 1315 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 23174 \nu^{15} - 156710 \nu^{14} + 904864 \nu^{13} - 3329339 \nu^{12} + 10218126 \nu^{11} + \cdots + 361440 ) / 17095 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 22230 \nu^{15} - 159361 \nu^{14} + 923582 \nu^{13} - 3511572 \nu^{12} + 10902106 \nu^{11} + \cdots - 39775 ) / 17095 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13659 \nu^{15} - 138079 \nu^{14} + 856036 \nu^{13} - 3832406 \nu^{12} + 13079663 \nu^{11} + \cdots - 2779060 ) / 17095 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 23174 \nu^{15} + 190900 \nu^{14} - 1144194 \nu^{13} + 4717453 \nu^{12} - 15435520 \nu^{11} + \cdots + 1929290 ) / 17095 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 54184 \nu^{15} - 399016 \nu^{14} + 2335837 \nu^{13} - 9056462 \nu^{12} + 28575749 \nu^{11} + \cdots - 1312670 ) / 17095 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 10652 \nu^{15} + 79890 \nu^{14} - 470562 \nu^{13} + 1846988 \nu^{12} - 5882478 \nu^{11} + \cdots + 370592 ) / 3419 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 54184 \nu^{15} + 413744 \nu^{14} - 2438933 \nu^{13} + 9652683 \nu^{12} - 30812827 \nu^{11} + \cdots + 1951760 ) / 17095 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 79057 \nu^{15} + 615414 \nu^{14} - 3643251 \nu^{13} + 14577609 \nu^{12} - 46844682 \nu^{11} + \cdots + 3651715 ) / 17095 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 80814 \nu^{15} + 625830 \nu^{14} - 3701865 \nu^{13} + 14775376 \nu^{12} - 47425509 \nu^{11} + \cdots + 4036755 ) / 17095 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{15} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{6} + \beta_{5} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{15} + 2\beta_{13} + \beta_{12} - \beta_{10} - \beta_{7} + 2\beta_{2} - 2\beta _1 - 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 7 \beta_{12} + 8 \beta_{11} - \beta_{9} - 3 \beta_{8} + \cdots - 5 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 16 \beta_{15} + 4 \beta_{14} - 14 \beta_{13} - 15 \beta_{12} + 8 \beta_{11} + 9 \beta_{10} - 6 \beta_{8} + \cdots + 18 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 29 \beta_{15} - 12 \beta_{14} - 6 \beta_{13} + 41 \beta_{12} - 49 \beta_{11} + 5 \beta_{10} + \cdots + 36 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 128 \beta_{15} - 46 \beta_{14} + 75 \beta_{13} + 161 \beta_{12} - 110 \beta_{11} - 75 \beta_{10} + \cdots - 100 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 91 \beta_{15} + 38 \beta_{14} + 111 \beta_{13} - 162 \beta_{12} + 239 \beta_{11} - 105 \beta_{10} + \cdots - 307 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 1000 \beta_{15} + 376 \beta_{14} - 372 \beta_{13} - 1435 \beta_{12} + 1056 \beta_{11} + 543 \beta_{10} + \cdots + 534 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 237 \beta_{15} + 102 \beta_{14} - 1073 \beta_{13} - 188 \beta_{12} - 694 \beta_{11} + 1413 \beta_{10} + \cdots + 2725 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 7294 \beta_{15} - 2652 \beta_{14} + 1583 \beta_{13} + 11026 \beta_{12} - 8640 \beta_{11} + \cdots - 1864 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 8716 \beta_{15} - 3307 \beta_{14} + 8988 \beta_{13} + 12594 \beta_{12} - 3455 \beta_{11} + \cdots - 22758 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( 23764 \beta_{15} + 8306 \beta_{14} - 1666 \beta_{13} - 36597 \beta_{12} + 31234 \beta_{11} + 4650 \beta_{10} + \cdots - 3138 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 113314 \beta_{15} + 40749 \beta_{14} - 69278 \beta_{13} - 171456 \beta_{12} + 89103 \beta_{11} + \cdots + 171575 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 256592 \beta_{15} - 86986 \beta_{14} - 39130 \beta_{13} + 397697 \beta_{12} - 391872 \beta_{11} + \cdots + 213314 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 1130763 \beta_{15} - 394827 \beta_{14} + 487515 \beta_{13} + 1731786 \beta_{12} - 1079800 \beta_{11} + \cdots - 1132093 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).
| \(n\) | \(677\) | \(847\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{12}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 239.1 |
|
−0.707107 | − | 0.707107i | −1.53819 | + | 0.796225i | 1.00000i | 0.428520 | + | 0.428520i | 1.65068 | + | 0.524648i | −0.538189 | − | 0.538189i | 0.707107 | − | 0.707107i | 1.73205 | − | 2.44949i | − | 0.606018i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.2 | −0.707107 | − | 0.707107i | −0.796225 | − | 1.53819i | 1.00000i | 0.313444 | + | 0.313444i | −0.524648 | + | 1.65068i | 0.203775 | + | 0.203775i | 0.707107 | − | 0.707107i | −1.73205 | + | 2.44949i | − | 0.443277i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.3 | −0.707107 | − | 0.707107i | 0.796225 | + | 1.53819i | 1.00000i | −2.76293 | − | 2.76293i | 0.524648 | − | 1.65068i | 1.79623 | + | 1.79623i | 0.707107 | − | 0.707107i | −1.73205 | + | 2.44949i | 3.90738i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.4 | −0.707107 | − | 0.707107i | 1.53819 | − | 0.796225i | 1.00000i | 2.02097 | + | 2.02097i | −1.65068 | − | 0.524648i | 2.53819 | + | 2.53819i | 0.707107 | − | 0.707107i | 1.73205 | − | 2.44949i | − | 2.85808i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.5 | 0.707107 | + | 0.707107i | −1.53819 | − | 0.796225i | 1.00000i | −0.428520 | − | 0.428520i | −0.524648 | − | 1.65068i | −0.538189 | − | 0.538189i | −0.707107 | + | 0.707107i | 1.73205 | + | 2.44949i | − | 0.606018i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.6 | 0.707107 | + | 0.707107i | −0.796225 | + | 1.53819i | 1.00000i | −0.313444 | − | 0.313444i | −1.65068 | + | 0.524648i | 0.203775 | + | 0.203775i | −0.707107 | + | 0.707107i | −1.73205 | − | 2.44949i | − | 0.443277i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.7 | 0.707107 | + | 0.707107i | 0.796225 | − | 1.53819i | 1.00000i | 2.76293 | + | 2.76293i | 1.65068 | − | 0.524648i | 1.79623 | + | 1.79623i | −0.707107 | + | 0.707107i | −1.73205 | − | 2.44949i | 3.90738i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.8 | 0.707107 | + | 0.707107i | 1.53819 | + | 0.796225i | 1.00000i | −2.02097 | − | 2.02097i | 0.524648 | + | 1.65068i | 2.53819 | + | 2.53819i | −0.707107 | + | 0.707107i | 1.73205 | + | 2.44949i | − | 2.85808i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 437.1 | −0.707107 | + | 0.707107i | −1.53819 | − | 0.796225i | − | 1.00000i | 0.428520 | − | 0.428520i | 1.65068 | − | 0.524648i | −0.538189 | + | 0.538189i | 0.707107 | + | 0.707107i | 1.73205 | + | 2.44949i | 0.606018i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 437.2 | −0.707107 | + | 0.707107i | −0.796225 | + | 1.53819i | − | 1.00000i | 0.313444 | − | 0.313444i | −0.524648 | − | 1.65068i | 0.203775 | − | 0.203775i | 0.707107 | + | 0.707107i | −1.73205 | − | 2.44949i | 0.443277i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 437.3 | −0.707107 | + | 0.707107i | 0.796225 | − | 1.53819i | − | 1.00000i | −2.76293 | + | 2.76293i | 0.524648 | + | 1.65068i | 1.79623 | − | 1.79623i | 0.707107 | + | 0.707107i | −1.73205 | − | 2.44949i | − | 3.90738i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 437.4 | −0.707107 | + | 0.707107i | 1.53819 | + | 0.796225i | − | 1.00000i | 2.02097 | − | 2.02097i | −1.65068 | + | 0.524648i | 2.53819 | − | 2.53819i | 0.707107 | + | 0.707107i | 1.73205 | + | 2.44949i | 2.85808i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 437.5 | 0.707107 | − | 0.707107i | −1.53819 | + | 0.796225i | − | 1.00000i | −0.428520 | + | 0.428520i | −0.524648 | + | 1.65068i | −0.538189 | + | 0.538189i | −0.707107 | − | 0.707107i | 1.73205 | − | 2.44949i | 0.606018i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 437.6 | 0.707107 | − | 0.707107i | −0.796225 | − | 1.53819i | − | 1.00000i | −0.313444 | + | 0.313444i | −1.65068 | − | 0.524648i | 0.203775 | − | 0.203775i | −0.707107 | − | 0.707107i | −1.73205 | + | 2.44949i | 0.443277i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 437.7 | 0.707107 | − | 0.707107i | 0.796225 | + | 1.53819i | − | 1.00000i | 2.76293 | − | 2.76293i | 1.65068 | + | 0.524648i | 1.79623 | − | 1.79623i | −0.707107 | − | 0.707107i | −1.73205 | + | 2.44949i | − | 3.90738i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 437.8 | 0.707107 | − | 0.707107i | 1.53819 | − | 0.796225i | − | 1.00000i | −2.02097 | + | 2.02097i | 0.524648 | − | 1.65068i | 2.53819 | − | 2.53819i | −0.707107 | − | 0.707107i | 1.73205 | − | 2.44949i | 2.85808i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.d | odd | 4 | 1 | inner |
| 39.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1014.2.g.d | 16 | |
| 3.b | odd | 2 | 1 | inner | 1014.2.g.d | 16 | |
| 13.b | even | 2 | 1 | 1014.2.g.c | 16 | ||
| 13.d | odd | 4 | 1 | 1014.2.g.c | 16 | ||
| 13.d | odd | 4 | 1 | inner | 1014.2.g.d | 16 | |
| 13.e | even | 6 | 1 | 78.2.k.a | ✓ | 16 | |
| 13.f | odd | 12 | 1 | 78.2.k.a | ✓ | 16 | |
| 39.d | odd | 2 | 1 | 1014.2.g.c | 16 | ||
| 39.f | even | 4 | 1 | 1014.2.g.c | 16 | ||
| 39.f | even | 4 | 1 | inner | 1014.2.g.d | 16 | |
| 39.h | odd | 6 | 1 | 78.2.k.a | ✓ | 16 | |
| 39.k | even | 12 | 1 | 78.2.k.a | ✓ | 16 | |
| 52.i | odd | 6 | 1 | 624.2.cn.d | 16 | ||
| 52.l | even | 12 | 1 | 624.2.cn.d | 16 | ||
| 156.r | even | 6 | 1 | 624.2.cn.d | 16 | ||
| 156.v | odd | 12 | 1 | 624.2.cn.d | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.2.k.a | ✓ | 16 | 13.e | even | 6 | 1 | |
| 78.2.k.a | ✓ | 16 | 13.f | odd | 12 | 1 | |
| 78.2.k.a | ✓ | 16 | 39.h | odd | 6 | 1 | |
| 78.2.k.a | ✓ | 16 | 39.k | even | 12 | 1 | |
| 624.2.cn.d | 16 | 52.i | odd | 6 | 1 | ||
| 624.2.cn.d | 16 | 52.l | even | 12 | 1 | ||
| 624.2.cn.d | 16 | 156.r | even | 6 | 1 | ||
| 624.2.cn.d | 16 | 156.v | odd | 12 | 1 | ||
| 1014.2.g.c | 16 | 13.b | even | 2 | 1 | ||
| 1014.2.g.c | 16 | 13.d | odd | 4 | 1 | ||
| 1014.2.g.c | 16 | 39.d | odd | 2 | 1 | ||
| 1014.2.g.c | 16 | 39.f | even | 4 | 1 | ||
| 1014.2.g.d | 16 | 1.a | even | 1 | 1 | trivial | |
| 1014.2.g.d | 16 | 3.b | odd | 2 | 1 | inner | |
| 1014.2.g.d | 16 | 13.d | odd | 4 | 1 | inner | |
| 1014.2.g.d | 16 | 39.f | even | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1014, [\chi])\):
|
\( T_{5}^{16} + 300T_{5}^{12} + 15606T_{5}^{8} + 2700T_{5}^{4} + 81 \)
|
|
\( T_{7}^{8} - 8T_{7}^{7} + 32T_{7}^{6} - 56T_{7}^{5} + 40T_{7}^{4} + 32T_{7}^{3} + 32T_{7}^{2} - 16T_{7} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{4} \)
|
| $3$ |
\( (T^{8} + 6 T^{4} + 81)^{2} \)
|
| $5$ |
\( T^{16} + 300 T^{12} + \cdots + 81 \)
|
| $7$ |
\( (T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 4)^{2} \)
|
| $11$ |
\( (T^{8} + 312 T^{4} + 36)^{2} \)
|
| $13$ |
\( T^{16} \)
|
| $17$ |
\( (T^{8} - 72 T^{6} + \cdots + 12321)^{2} \)
|
| $19$ |
\( (T^{8} + 16 T^{7} + \cdots + 5476)^{2} \)
|
| $23$ |
\( (T^{8} - 60 T^{6} + \cdots + 4356)^{2} \)
|
| $29$ |
\( (T^{8} + 132 T^{6} + \cdots + 184041)^{2} \)
|
| $31$ |
\( (T^{8} + 8 T^{7} + \cdots + 7744)^{2} \)
|
| $37$ |
\( (T^{8} - 4 T^{7} + \cdots + 375769)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 65697655057281 \)
|
| $43$ |
\( (T^{4} + 18 T^{2} + 54)^{4} \)
|
| $47$ |
\( T^{16} + \cdots + 41006250000 \)
|
| $53$ |
\( (T^{8} + 156 T^{6} + \cdots + 522729)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 43489065701376 \)
|
| $61$ |
\( (T^{4} - 12 T^{3} + \cdots - 219)^{4} \)
|
| $67$ |
\( (T^{8} + 16 T^{7} + \cdots + 676)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 5143987297296 \)
|
| $73$ |
\( (T^{8} + 28 T^{7} + \cdots + 16834609)^{2} \)
|
| $79$ |
\( (T^{4} + 24 T^{3} + \cdots + 312)^{4} \)
|
| $83$ |
\( T^{16} + \cdots + 36804120336 \)
|
| $89$ |
\( T^{16} + \cdots + 59\!\cdots\!56 \)
|
| $97$ |
\( (T^{8} + 8 T^{7} + \cdots + 43264)^{2} \)
|
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