Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [336,9,Mod(97,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.97");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 336.f (of order \(2\), degree \(1\), 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: | \(136.879212981\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} + 14312 x^{8} - 30343 x^{7} + 170123918 x^{6} - 875537263 x^{5} + 496509566533 x^{4} + \cdots + 39\!\cdots\!76 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{15}\cdot 3^{15}\cdot 7^{4} \) |
| Twist minimal: | no (minimal twist has level 84) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} + 14312 x^{8} - 30343 x^{7} + 170123918 x^{6} - 875537263 x^{5} + 496509566533 x^{4} + \cdots + 39\!\cdots\!76 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 56\!\cdots\!23 \nu^{9} + \cdots + 63\!\cdots\!33 ) / 22\!\cdots\!97 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 14\!\cdots\!45 \nu^{9} + \cdots + 90\!\cdots\!92 ) / 58\!\cdots\!54 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 51\!\cdots\!06 \nu^{9} + \cdots + 72\!\cdots\!44 ) / 16\!\cdots\!12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 51\!\cdots\!36 \nu^{9} + \cdots - 14\!\cdots\!24 ) / 81\!\cdots\!56 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 41\!\cdots\!64 \nu^{9} + \cdots - 27\!\cdots\!64 ) / 51\!\cdots\!56 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 17\!\cdots\!32 \nu^{9} + \cdots + 34\!\cdots\!04 ) / 20\!\cdots\!88 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 13\!\cdots\!88 \nu^{9} + \cdots - 13\!\cdots\!44 ) / 14\!\cdots\!16 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 35\!\cdots\!86 \nu^{9} + \cdots + 21\!\cdots\!72 ) / 10\!\cdots\!12 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 21\!\cdots\!52 \nu^{9} + \cdots + 16\!\cdots\!24 ) / 25\!\cdots\!28 \)
|
| \(\nu\) | \(=\) |
\( ( -6\beta_{9} + 6\beta_{8} + 9\beta_{7} + 9\beta_{6} - 36\beta_{5} - 3\beta_{3} - 14\beta _1 + 69 ) / 756 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 129 \beta_{9} + 312 \beta_{8} - 573 \beta_{7} + 687 \beta_{6} + 1656 \beta_{5} + 63 \beta_{4} + \cdots - 2164107 ) / 756 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -4177\beta_{7} - 4177\beta_{6} + 236\beta_{4} + 424\beta_{3} + 46\beta_{2} + 88296 ) / 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 942285 \beta_{9} - 4720596 \beta_{8} + 7075857 \beta_{7} - 7021779 \beta_{6} - 23313996 \beta_{5} + \cdots - 20467963221 ) / 756 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 769843806 \beta_{9} - 576325950 \beta_{8} + 1591774743 \beta_{7} + 296696091 \beta_{6} + \cdots + 179819092209 ) / 756 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 295509413 \beta_{7} + 295509413 \beta_{6} - 552613029 \beta_{4} + 1136452554 \beta_{3} + \cdots + 10386708923577 ) / 18 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 8584553756148 \beta_{9} + 6489457288026 \beta_{8} + 2445099616983 \beta_{7} + 18435018286659 \beta_{6} + \cdots + 37\!\cdots\!71 ) / 756 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 19237277705565 \beta_{9} + 720420180394380 \beta_{8} + \cdots - 24\!\cdots\!57 ) / 756 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 55\!\cdots\!53 \beta_{7} + \cdots - 27\!\cdots\!86 ) / 18 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
| \(n\) | \(85\) | \(113\) | \(127\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
0 | − | 46.7654i | 0 | − | 1053.92i | 0 | −582.142 | + | 2329.36i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | 0 | − | 46.7654i | 0 | − | 429.816i | 0 | 1615.33 | − | 1776.38i | 0 | −2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 97.3 | 0 | − | 46.7654i | 0 | − | 264.232i | 0 | −2038.47 | − | 1268.64i | 0 | −2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 97.4 | 0 | − | 46.7654i | 0 | 307.159i | 0 | −222.189 | + | 2390.70i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 97.5 | 0 | − | 46.7654i | 0 | 803.413i | 0 | 2396.47 | − | 147.366i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 97.6 | 0 | 46.7654i | 0 | − | 803.413i | 0 | 2396.47 | + | 147.366i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 97.7 | 0 | 46.7654i | 0 | − | 307.159i | 0 | −222.189 | − | 2390.70i | 0 | −2187.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 97.8 | 0 | 46.7654i | 0 | 264.232i | 0 | −2038.47 | + | 1268.64i | 0 | −2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.9 | 0 | 46.7654i | 0 | 429.816i | 0 | 1615.33 | + | 1776.38i | 0 | −2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 97.10 | 0 | 46.7654i | 0 | 1053.92i | 0 | −582.142 | − | 2329.36i | 0 | −2187.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 336.9.f.a | 10 | |
| 4.b | odd | 2 | 1 | 84.9.d.a | ✓ | 10 | |
| 7.b | odd | 2 | 1 | inner | 336.9.f.a | 10 | |
| 12.b | even | 2 | 1 | 252.9.d.d | 10 | ||
| 28.d | even | 2 | 1 | 84.9.d.a | ✓ | 10 | |
| 84.h | odd | 2 | 1 | 252.9.d.d | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.9.d.a | ✓ | 10 | 4.b | odd | 2 | 1 | |
| 84.9.d.a | ✓ | 10 | 28.d | even | 2 | 1 | |
| 252.9.d.d | 10 | 12.b | even | 2 | 1 | ||
| 252.9.d.d | 10 | 84.h | odd | 2 | 1 | ||
| 336.9.f.a | 10 | 1.a | even | 1 | 1 | trivial | |
| 336.9.f.a | 10 | 7.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} + 2105124 T_{5}^{8} + 1366627233924 T_{5}^{6} + \cdots + 87\!\cdots\!00 \)
acting on \(S_{9}^{\mathrm{new}}(336, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( (T^{2} + 2187)^{5} \)
|
| $5$ |
\( T^{10} + \cdots + 87\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots + 63\!\cdots\!01 \)
|
| $11$ |
\( (T^{5} + \cdots + 18\!\cdots\!28)^{2} \)
|
| $13$ |
\( T^{10} + \cdots + 16\!\cdots\!88 \)
|
| $17$ |
\( T^{10} + \cdots + 52\!\cdots\!68 \)
|
| $19$ |
\( T^{10} + \cdots + 17\!\cdots\!52 \)
|
| $23$ |
\( (T^{5} + \cdots - 99\!\cdots\!48)^{2} \)
|
| $29$ |
\( (T^{5} + \cdots - 76\!\cdots\!32)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 21\!\cdots\!92 \)
|
| $37$ |
\( (T^{5} + \cdots - 88\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{10} + \cdots + 25\!\cdots\!00 \)
|
| $43$ |
\( (T^{5} + \cdots + 15\!\cdots\!52)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 76\!\cdots\!88 \)
|
| $53$ |
\( (T^{5} + \cdots - 68\!\cdots\!40)^{2} \)
|
| $59$ |
\( T^{10} + \cdots + 12\!\cdots\!08 \)
|
| $61$ |
\( T^{10} + \cdots + 62\!\cdots\!32 \)
|
| $67$ |
\( (T^{5} + \cdots + 43\!\cdots\!52)^{2} \)
|
| $71$ |
\( (T^{5} + \cdots + 22\!\cdots\!52)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 86\!\cdots\!00 \)
|
| $79$ |
\( (T^{5} + \cdots + 10\!\cdots\!96)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 15\!\cdots\!00 \)
|
| $89$ |
\( T^{10} + \cdots + 84\!\cdots\!00 \)
|
| $97$ |
\( T^{10} + \cdots + 90\!\cdots\!72 \)
|
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