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: |
\( x^{6} - 2x^{5} + 13x^{4} - 660x^{3} + 702x^{2} - 5832x + 157464 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2}\cdot 7^{2} \) |
| 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
\( x^{6} - 2x^{5} + 13x^{4} - 660x^{3} + 702x^{2} - 5832x + 157464 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 52\nu^{4} - 2821\nu^{3} + 2874\nu^{2} - 54\nu + 930204 ) / 463644 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 293\nu^{5} + 2357\nu^{4} + 2297\nu^{3} - 4947\nu^{2} - 1143288\nu - 1086210 ) / 231822 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17\nu^{5} - 88\nu^{4} + 329\nu^{3} - 9006\nu^{2} - 7830\nu - 75816 ) / 8748 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{5} + 157\nu^{4} - 413\nu^{3} + 1725\nu^{2} - 50166\nu + 75816 ) / 4374 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -1274\nu^{5} - 1853\nu^{4} - 13592\nu^{3} + 557637\nu^{2} - 1305180\nu + 6265026 ) / 231822 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} - 5\beta_{3} - 3\beta_{2} - \beta _1 + 16 ) / 42 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{5} - 19\beta_{4} - 32\beta_{3} + 42\beta_{2} + 98\beta _1 - 137 ) / 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 22\beta_{5} - 25\beta_{4} + \beta_{3} + 33\beta_{2} - 6667\beta _1 + 13378 ) / 42 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -773\beta_{5} + 455\beta_{4} - 1694\beta_{3} + 132\beta_{2} - 9868\beta _1 + 18739 ) / 42 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -1640\beta_{5} - 7687\beta_{4} - 6431\beta_{3} + 20913\beta_{2} + 129401\beta _1 - 39800 ) / 42 \)
|
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 | − | 32.9258i | 0 | 0 | 22.6274i | 0 | −93.1283 | ||||||||||||||||||||||||||||||||||
| 197.2 | − | 2.82843i | 0 | −8.00000 | 8.92960i | 0 | 0 | 22.6274i | 0 | 25.2567 | ||||||||||||||||||||||||||||||||||||
| 197.3 | − | 2.82843i | 0 | −8.00000 | 42.3810i | 0 | 0 | 22.6274i | 0 | 119.872 | ||||||||||||||||||||||||||||||||||||
| 197.4 | 2.82843i | 0 | −8.00000 | − | 42.3810i | 0 | 0 | − | 22.6274i | 0 | 119.872 | |||||||||||||||||||||||||||||||||||
| 197.5 | 2.82843i | 0 | −8.00000 | − | 8.92960i | 0 | 0 | − | 22.6274i | 0 | 25.2567 | |||||||||||||||||||||||||||||||||||
| 197.6 | 2.82843i | 0 | −8.00000 | 32.9258i | 0 | 0 | − | 22.6274i | 0 | −93.1283 | ||||||||||||||||||||||||||||||||||||
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.h | 6 | |
| 3.b | odd | 2 | 1 | inner | 882.5.b.h | 6 | |
| 7.b | odd | 2 | 1 | 882.5.b.e | 6 | ||
| 7.c | even | 3 | 2 | 126.5.s.b | ✓ | 12 | |
| 21.c | even | 2 | 1 | 882.5.b.e | 6 | ||
| 21.h | odd | 6 | 2 | 126.5.s.b | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.5.s.b | ✓ | 12 | 7.c | even | 3 | 2 | |
| 126.5.s.b | ✓ | 12 | 21.h | odd | 6 | 2 | |
| 882.5.b.e | 6 | 7.b | odd | 2 | 1 | ||
| 882.5.b.e | 6 | 21.c | even | 2 | 1 | ||
| 882.5.b.h | 6 | 1.a | even | 1 | 1 | trivial | |
| 882.5.b.h | 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 \(S_{5}^{\mathrm{new}}(882, [\chi])\):
|
\( T_{5}^{6} + 2960T_{5}^{4} + 2176893T_{5}^{2} + 155267442 \)
|
|
\( T_{13}^{3} - 43T_{13}^{2} - 8198T_{13} + 268038 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 2960 T^{4} + \cdots + 155267442 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 69073754562 \)
|
| $13$ |
\( (T^{3} - 43 T^{2} + \cdots + 268038)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 273069475993728 \)
|
| $19$ |
\( (T^{3} - 87 T^{2} + \cdots - 15048818)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 85\!\cdots\!68 \)
|
| $29$ |
\( T^{6} + \cdots + 20\!\cdots\!88 \)
|
| $31$ |
\( (T^{3} - 841 T^{2} + \cdots + 528929871)^{2} \)
|
| $37$ |
\( (T^{3} - 3009 T^{2} + \cdots - 147721442)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 25\!\cdots\!48 \)
|
| $43$ |
\( (T^{3} + 1065 T^{2} + \cdots - 1646452466)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 55\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 16\!\cdots\!68 \)
|
| $59$ |
\( T^{6} + \cdots + 33\!\cdots\!28 \)
|
| $61$ |
\( (T^{3} + 10316 T^{2} + \cdots + 4229496384)^{2} \)
|
| $67$ |
\( (T^{3} + 37 T^{2} + \cdots + 17334472784)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 26\!\cdots\!48 \)
|
| $73$ |
\( (T^{3} + 1471 T^{2} + \cdots + 150414367928)^{2} \)
|
| $79$ |
\( (T^{3} - 5263 T^{2} + \cdots + 11262927129)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 13\!\cdots\!02 \)
|
| $89$ |
\( T^{6} + \cdots + 45\!\cdots\!72 \)
|
| $97$ |
\( (T^{3} + 23218 T^{2} + \cdots + 175506148952)^{2} \)
|
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