Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [39,3,Mod(7,39)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 11]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("39.7");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 39 = 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 39.l (of order \(12\), degree \(4\), 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: | \(1.06267303101\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 4 x^{11} + 8 x^{10} + 8 x^{9} + 178 x^{8} - 620 x^{7} + 1088 x^{6} + 640 x^{5} + 7921 x^{4} + \cdots + 5184 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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
\( x^{12} - 4 x^{11} + 8 x^{10} + 8 x^{9} + 178 x^{8} - 620 x^{7} + 1088 x^{6} + 640 x^{5} + 7921 x^{4} + \cdots + 5184 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1355 \nu^{11} - 11146 \nu^{10} + 25515 \nu^{9} - 2151 \nu^{8} + 95491 \nu^{7} - 1871961 \nu^{6} + \cdots - 57525912 ) / 19896168 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 10372953887 \nu^{11} - 30543493454 \nu^{10} + 54101479810 \nu^{9} + 116831316334 \nu^{8} + \cdots - 111790022446656 ) / 107988023617272 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 64693300027 \nu^{11} - 227654338447 \nu^{10} + 425915919854 \nu^{9} + \cdots - 574099169783616 ) / 323964070851816 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 469967215683 \nu^{11} - 1336681906952 \nu^{10} + 1915756004984 \nu^{9} + \cdots - 194659143371136 ) / 863904188938176 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 798971 \nu^{11} - 3098324 \nu^{10} + 5589256 \nu^{9} + 8228848 \nu^{8} + 142061966 \nu^{7} + \cdots - 7438387680 ) / 1432524096 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 135796738945 \nu^{11} - 460995430120 \nu^{10} + 812629620698 \nu^{9} + \cdots - 609077511525168 ) / 215976047234544 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 557553792435 \nu^{11} - 1567739117240 \nu^{10} + 2186537486888 \nu^{9} + \cdots + 54\!\cdots\!32 ) / 863904188938176 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 1013778592733 \nu^{11} + 1951154419157 \nu^{10} - 1143827367874 \nu^{9} + \cdots - 10\!\cdots\!40 ) / 647928141703632 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 2059948907017 \nu^{11} + 8441130036025 \nu^{10} - 15701980520594 \nu^{9} + \cdots + 22\!\cdots\!60 ) / 647928141703632 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 5284462326625 \nu^{11} + 15882096407362 \nu^{10} - 24325343322068 \nu^{9} + \cdots + 66\!\cdots\!80 ) / 12\!\cdots\!64 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{11} - \beta_{9} + 2\beta_{6} + 7\beta_{4} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - \beta_{9} + \beta_{8} + 2\beta_{6} - \beta_{5} + \beta_{4} + 10\beta_{3} + 2\beta_{2} + \beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{11} - \beta_{9} + 12\beta_{8} - 2\beta_{7} - 26\beta_{5} + 7\beta_{4} + 5\beta_{3} + 2\beta_{2} - 3\beta _1 - 74 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 18 \beta_{11} - 4 \beta_{10} + 14 \beta_{9} + 18 \beta_{8} - 44 \beta_{7} - 48 \beta_{6} - 30 \beta_{5} + \cdots - 84 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 143 \beta_{11} - 52 \beta_{10} + 143 \beta_{9} + 26 \beta_{8} - 60 \beta_{7} - 602 \beta_{6} + \cdots - 444 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 181 \beta_{11} - 112 \beta_{10} + 293 \beta_{9} - 181 \beta_{8} - 946 \beta_{6} + 765 \beta_{5} + \cdots + 203 \)
|
| \(\nu^{8}\) | \(=\) |
\( 491 \beta_{11} + 491 \beta_{9} - 1790 \beta_{8} + 1282 \beta_{7} + 7320 \beta_{5} - 2765 \beta_{4} + \cdots + 10668 \)
|
| \(\nu^{9}\) | \(=\) |
\( 4602 \beta_{11} + 2264 \beta_{10} - 2338 \beta_{9} - 4602 \beta_{8} + 12056 \beta_{7} + 16896 \beta_{6} + \cdots + 27732 \)
|
| \(\nu^{10}\) | \(=\) |
\( 23469 \beta_{11} + 16584 \beta_{10} - 23469 \beta_{9} - 8292 \beta_{8} + 23688 \beta_{7} + 135858 \beta_{6} + \cdots + 91398 \)
|
| \(\nu^{11}\) | \(=\) |
\( 30689 \beta_{11} + 40272 \beta_{10} - 70961 \beta_{9} + 30689 \beta_{8} + 284050 \beta_{6} + \cdots - 62383 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).
| \(n\) | \(14\) | \(28\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−3.06798 | + | 0.822062i | −0.866025 | + | 1.50000i | 5.27259 | − | 3.04413i | −5.58467 | − | 5.58467i | 1.42385 | − | 5.31389i | −7.30003 | − | 1.95604i | −4.69005 | + | 4.69005i | −1.50000 | − | 2.59808i | 21.7246 | + | 12.5427i | ||||||||||||||||||||||||||||||||||||
| 7.2 | −0.942546 | + | 0.252555i | −0.866025 | + | 1.50000i | −2.63949 | + | 1.52391i | 6.89545 | + | 6.89545i | 0.437437 | − | 1.63254i | −5.17460 | − | 1.38653i | 4.86294 | − | 4.86294i | −1.50000 | − | 2.59808i | −8.24076 | − | 4.75781i | |||||||||||||||||||||||||||||||||||||
| 7.3 | 2.64450 | − | 0.708591i | −0.866025 | + | 1.50000i | 3.02716 | − | 1.74773i | −2.04284 | − | 2.04284i | −1.22732 | + | 4.58040i | −1.58755 | − | 0.425384i | −0.976738 | + | 0.976738i | −1.50000 | − | 2.59808i | −6.84982 | − | 3.95474i | |||||||||||||||||||||||||||||||||||||
| 19.1 | −0.816745 | − | 3.04813i | 0.866025 | − | 1.50000i | −5.15994 | + | 2.97909i | 1.44880 | − | 1.44880i | −5.27952 | − | 1.41464i | −1.58469 | + | 5.91416i | 4.36947 | + | 4.36947i | −1.50000 | − | 2.59808i | −5.59942 | − | 3.23283i | |||||||||||||||||||||||||||||||||||||
| 19.2 | 0.179543 | + | 0.670065i | 0.866025 | − | 1.50000i | 3.04735 | − | 1.75939i | −2.02001 | + | 2.02001i | 1.16059 | + | 0.310978i | −0.588406 | + | 2.19596i | 3.68812 | + | 3.68812i | −1.50000 | − | 2.59808i | −1.71622 | − | 0.990860i | |||||||||||||||||||||||||||||||||||||
| 19.3 | 1.00323 | + | 3.74409i | 0.866025 | − | 1.50000i | −9.54767 | + | 5.51235i | 3.30327 | − | 3.30327i | 6.48496 | + | 1.73764i | 0.235277 | − | 0.878067i | −19.2537 | − | 19.2537i | −1.50000 | − | 2.59808i | 15.6817 | + | 9.05381i | |||||||||||||||||||||||||||||||||||||
| 28.1 | −3.06798 | − | 0.822062i | −0.866025 | − | 1.50000i | 5.27259 | + | 3.04413i | −5.58467 | + | 5.58467i | 1.42385 | + | 5.31389i | −7.30003 | + | 1.95604i | −4.69005 | − | 4.69005i | −1.50000 | + | 2.59808i | 21.7246 | − | 12.5427i | |||||||||||||||||||||||||||||||||||||
| 28.2 | −0.942546 | − | 0.252555i | −0.866025 | − | 1.50000i | −2.63949 | − | 1.52391i | 6.89545 | − | 6.89545i | 0.437437 | + | 1.63254i | −5.17460 | + | 1.38653i | 4.86294 | + | 4.86294i | −1.50000 | + | 2.59808i | −8.24076 | + | 4.75781i | |||||||||||||||||||||||||||||||||||||
| 28.3 | 2.64450 | + | 0.708591i | −0.866025 | − | 1.50000i | 3.02716 | + | 1.74773i | −2.04284 | + | 2.04284i | −1.22732 | − | 4.58040i | −1.58755 | + | 0.425384i | −0.976738 | − | 0.976738i | −1.50000 | + | 2.59808i | −6.84982 | + | 3.95474i | |||||||||||||||||||||||||||||||||||||
| 37.1 | −0.816745 | + | 3.04813i | 0.866025 | + | 1.50000i | −5.15994 | − | 2.97909i | 1.44880 | + | 1.44880i | −5.27952 | + | 1.41464i | −1.58469 | − | 5.91416i | 4.36947 | − | 4.36947i | −1.50000 | + | 2.59808i | −5.59942 | + | 3.23283i | |||||||||||||||||||||||||||||||||||||
| 37.2 | 0.179543 | − | 0.670065i | 0.866025 | + | 1.50000i | 3.04735 | + | 1.75939i | −2.02001 | − | 2.02001i | 1.16059 | − | 0.310978i | −0.588406 | − | 2.19596i | 3.68812 | − | 3.68812i | −1.50000 | + | 2.59808i | −1.71622 | + | 0.990860i | |||||||||||||||||||||||||||||||||||||
| 37.3 | 1.00323 | − | 3.74409i | 0.866025 | + | 1.50000i | −9.54767 | − | 5.51235i | 3.30327 | + | 3.30327i | 6.48496 | − | 1.73764i | 0.235277 | + | 0.878067i | −19.2537 | + | 19.2537i | −1.50000 | + | 2.59808i | 15.6817 | − | 9.05381i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.f | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 39.3.l.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 117.3.bd.d | 12 | ||
| 13.f | odd | 12 | 1 | inner | 39.3.l.b | ✓ | 12 |
| 39.k | even | 12 | 1 | 117.3.bd.d | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.3.l.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 39.3.l.b | ✓ | 12 | 13.f | odd | 12 | 1 | inner |
| 117.3.bd.d | 12 | 3.b | odd | 2 | 1 | ||
| 117.3.bd.d | 12 | 39.k | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 2 T_{2}^{11} + 8 T_{2}^{10} + 32 T_{2}^{9} - 53 T_{2}^{8} - 248 T_{2}^{7} - 868 T_{2}^{6} + \cdots + 5184 \)
acting on \(S_{3}^{\mathrm{new}}(39, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 2 T^{11} + \cdots + 5184 \)
|
| $3$ |
\( (T^{4} + 3 T^{2} + 9)^{3} \)
|
| $5$ |
\( T^{12} - 4 T^{11} + \cdots + 37015056 \)
|
| $7$ |
\( T^{12} + 32 T^{11} + \cdots + 708964 \)
|
| $11$ |
\( T^{12} + \cdots + 1189422144 \)
|
| $13$ |
\( T^{12} + \cdots + 23298085122481 \)
|
| $17$ |
\( T^{12} + \cdots + 11808460086336 \)
|
| $19$ |
\( T^{12} + \cdots + 32498254124176 \)
|
| $23$ |
\( T^{12} + \cdots + 48612465285696 \)
|
| $29$ |
\( T^{12} + \cdots + 37\!\cdots\!24 \)
|
| $31$ |
\( T^{12} + \cdots + 523700157174016 \)
|
| $37$ |
\( T^{12} + \cdots + 65\!\cdots\!36 \)
|
| $41$ |
\( T^{12} + \cdots + 16\!\cdots\!96 \)
|
| $43$ |
\( T^{12} + \cdots + 15\!\cdots\!44 \)
|
| $47$ |
\( T^{12} + \cdots + 103767471503424 \)
|
| $53$ |
\( (T^{6} - 40 T^{5} + \cdots + 2366140824)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 91\!\cdots\!76 \)
|
| $61$ |
\( T^{12} + \cdots + 79\!\cdots\!49 \)
|
| $67$ |
\( T^{12} + \cdots + 25\!\cdots\!96 \)
|
| $71$ |
\( T^{12} + \cdots + 64\!\cdots\!24 \)
|
| $73$ |
\( T^{12} + \cdots + 46\!\cdots\!01 \)
|
| $79$ |
\( (T^{6} - 8 T^{5} + \cdots - 583200264)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 74\!\cdots\!04 \)
|
| $89$ |
\( T^{12} + \cdots + 27\!\cdots\!44 \)
|
| $97$ |
\( T^{12} + \cdots + 28\!\cdots\!56 \)
|
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