Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [882,5,Mod(197,882)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(882, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("882.197");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 882.b (of order \(2\), degree \(1\), 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: | \(91.1723074400\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 32x^{4} - 7932x^{3} + 185683x^{2} + 6753864x + 64548144 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 126) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 32x^{4} - 7932x^{3} + 185683x^{2} + 6753864x + 64548144 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 27049\nu^{5} - 414222\nu^{4} + 6066598\nu^{3} - 118428462\nu^{2} + 6827291953\nu + 72584826252 ) / 63713171400 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2892596 \nu^{5} + 37457987 \nu^{4} + 344802517 \nu^{3} - 35128675973 \nu^{2} + \cdots + 19299632455008 ) / 1051267328100 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3331\nu^{5} + 11312\nu^{4} + 2388484\nu^{3} - 24517112\nu^{2} - 1220920989\nu - 25595805060 ) / 785994264 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4409\nu^{5} - 44018\nu^{4} + 919808\nu^{3} + 24946610\nu^{2} + 210077745\nu - 33099191676 ) / 392997132 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1195794 \nu^{5} + 17389547 \nu^{4} - 33393478 \nu^{3} + 12682578877 \nu^{2} + \cdots - 4333406462382 ) / 35042244270 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + 3\beta_{2} - 3\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 19\beta_{5} + 3\beta_{4} - 36\beta_{3} - 6\beta_{2} + 1286\beta _1 + 75 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 318\beta_{5} + 55\beta_{4} + 971\beta_{3} + 939\beta_{2} + 30621\beta _1 + 23454 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 6403\beta_{5} - 8187\beta_{4} + 2934\beta_{3} + 51348\beta_{2} - 5188\beta _1 - 738933 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 109920\beta_{5} - 376979\beta_{4} - 78061\beta_{3} - 207753\beta_{2} + 13573353\beta _1 - 32348520 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).
| \(n\) | \(199\) | \(785\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 197.1 |
|
− | 2.82843i | 0 | −8.00000 | − | 25.7831i | 0 | 0 | 22.6274i | 0 | −72.9255 | ||||||||||||||||||||||||||||||||||
| 197.2 | − | 2.82843i | 0 | −8.00000 | − | 11.5330i | 0 | 0 | 22.6274i | 0 | −32.6204 | |||||||||||||||||||||||||||||||||||
| 197.3 | − | 2.82843i | 0 | −8.00000 | 44.3872i | 0 | 0 | 22.6274i | 0 | 125.546 | ||||||||||||||||||||||||||||||||||||
| 197.4 | 2.82843i | 0 | −8.00000 | − | 44.3872i | 0 | 0 | − | 22.6274i | 0 | 125.546 | |||||||||||||||||||||||||||||||||||
| 197.5 | 2.82843i | 0 | −8.00000 | 11.5330i | 0 | 0 | − | 22.6274i | 0 | −32.6204 | ||||||||||||||||||||||||||||||||||||
| 197.6 | 2.82843i | 0 | −8.00000 | 25.7831i | 0 | 0 | − | 22.6274i | 0 | −72.9255 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 882.5.b.g | 6 | |
| 3.b | odd | 2 | 1 | inner | 882.5.b.g | 6 | |
| 7.b | odd | 2 | 1 | 882.5.b.f | 6 | ||
| 7.d | odd | 6 | 2 | 126.5.s.a | ✓ | 12 | |
| 21.c | even | 2 | 1 | 882.5.b.f | 6 | ||
| 21.g | even | 6 | 2 | 126.5.s.a | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.5.s.a | ✓ | 12 | 7.d | odd | 6 | 2 | |
| 126.5.s.a | ✓ | 12 | 21.g | even | 6 | 2 | |
| 882.5.b.f | 6 | 7.b | odd | 2 | 1 | ||
| 882.5.b.f | 6 | 21.c | even | 2 | 1 | ||
| 882.5.b.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 882.5.b.g | 6 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(882, [\chi])\):
|
\( T_{5}^{6} + 2768T_{5}^{4} + 1660221T_{5}^{2} + 174209778 \)
|
|
\( T_{13}^{3} + 139T_{13}^{2} - 75062T_{13} - 9227910 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 2768 T^{4} + \cdots + 174209778 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 1320466050 \)
|
| $13$ |
\( (T^{3} + 139 T^{2} + \cdots - 9227910)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 294039010051200 \)
|
| $19$ |
\( (T^{3} + 327 T^{2} + \cdots - 26724046)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 24\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 98\!\cdots\!88 \)
|
| $31$ |
\( (T^{3} - 1895 T^{2} + \cdots + 834638097)^{2} \)
|
| $37$ |
\( (T^{3} + 2271 T^{2} + \cdots - 9154222370)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 32\!\cdots\!12 \)
|
| $43$ |
\( (T^{3} - 4071 T^{2} + \cdots - 1013702738)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 12\!\cdots\!32 \)
|
| $53$ |
\( T^{6} + \cdots + 78\!\cdots\!28 \)
|
| $59$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} - 2012 T^{2} + \cdots + 1800777600)^{2} \)
|
| $67$ |
\( (T^{3} + 3733 T^{2} + \cdots - 70265392)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 10\!\cdots\!88 \)
|
| $73$ |
\( (T^{3} + 2753 T^{2} + \cdots + 7039369672)^{2} \)
|
| $79$ |
\( (T^{3} + 1889 T^{2} + \cdots - 563293124295)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 16\!\cdots\!50 \)
|
| $89$ |
\( T^{6} + \cdots + 83\!\cdots\!52 \)
|
| $97$ |
\( (T^{3} - 15346 T^{2} + \cdots + 370549253320)^{2} \)
|
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