Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [10,18,Mod(9,10)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10.9");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.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: | \(18.3222087345\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 556201x^{6} + 76870744104x^{4} + 1868329791349729x^{2} + 78074963590050625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{40}\cdot 3^{4}\cdot 5^{11} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 556201x^{6} + 76870744104x^{4} + 1868329791349729x^{2} + 78074963590050625 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 520237696 \nu^{7} + 289353475332096 \nu^{5} + \cdots + 10\!\cdots\!84 \nu ) / 25\!\cdots\!75 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 33035288673113 \nu^{7} + \cdots + 46\!\cdots\!33 \nu ) / 12\!\cdots\!50 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 229165601212295 \nu^{7} + \cdots - 41\!\cdots\!00 ) / 40\!\cdots\!50 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 21\!\cdots\!09 \nu^{7} + \cdots + 25\!\cdots\!00 ) / 20\!\cdots\!75 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 20\!\cdots\!11 \nu^{7} + \cdots + 18\!\cdots\!00 ) / 12\!\cdots\!50 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 10\!\cdots\!48 \nu^{7} + \cdots + 92\!\cdots\!50 ) / 61\!\cdots\!25 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 31\!\cdots\!81 \nu^{7} + \cdots + 89\!\cdots\!00 ) / 12\!\cdots\!50 \)
|
| \(\nu\) | \(=\) |
\( ( 28\beta_{7} - 64\beta_{6} + 92\beta_{5} + 164\beta_{4} + 348\beta_{3} - 70420\beta_{2} - 7161\beta_1 ) / 2880000 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 16084 \beta_{7} - 170917 \beta_{6} - 93349 \beta_{5} + 135592 \beta_{4} - 781081 \beta_{3} + \cdots - 800929440000 ) / 5760000 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 3696256 \beta_{7} + 2758528 \beta_{6} - 6454784 \beta_{5} - 4579328 \beta_{4} + \cdots + 63326830647 \beta_1 ) / 480000 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 3646813564 \beta_{7} + 12449717647 \beta_{6} + 11570450959 \beta_{5} - 8465982712 \beta_{4} + \cdots + 44\!\cdots\!00 ) / 1152000 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 9421921710668 \beta_{7} - 4039916000384 \beta_{6} + 13461837711052 \beta_{5} + \cdots - 17\!\cdots\!41 \beta_1 ) / 2880000 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 387644604611656 \beta_{7} - 879139240778953 \beta_{6} + \cdots - 29\!\cdots\!00 ) / 240000 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 35\!\cdots\!36 \beta_{7} + \cdots + 68\!\cdots\!57 \beta_1 ) / 2880000 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
− | 256.000i | − | 20214.2i | −65536.0 | −765001. | − | 421560.i | −5.17482e6 | 1.76306e7i | 1.67772e7i | −2.79472e8 | −1.07919e8 | + | 1.95840e8i | ||||||||||||||||||||||||||||||||||||
| 9.2 | − | 256.000i | − | 2973.91i | −65536.0 | 855091. | − | 178210.i | −761320. | 7.58343e6i | 1.67772e7i | 1.20296e8 | −4.56218e7 | − | 2.18903e8i | |||||||||||||||||||||||||||||||||||||
| 9.3 | − | 256.000i | 5436.73i | −65536.0 | −679890. | − | 548351.i | 1.39180e6 | − | 7.62753e6i | 1.67772e7i | 9.95821e7 | −1.40378e8 | + | 1.74052e8i | |||||||||||||||||||||||||||||||||||||
| 9.4 | − | 256.000i | 15847.3i | −65536.0 | −22980.2 | + | 873162.i | 4.05692e6 | 1.02660e7i | 1.67772e7i | −1.21998e8 | 2.23529e8 | + | 5.88293e6i | ||||||||||||||||||||||||||||||||||||||
| 9.5 | 256.000i | − | 15847.3i | −65536.0 | −22980.2 | − | 873162.i | 4.05692e6 | − | 1.02660e7i | − | 1.67772e7i | −1.21998e8 | 2.23529e8 | − | 5.88293e6i | ||||||||||||||||||||||||||||||||||||
| 9.6 | 256.000i | − | 5436.73i | −65536.0 | −679890. | + | 548351.i | 1.39180e6 | 7.62753e6i | − | 1.67772e7i | 9.95821e7 | −1.40378e8 | − | 1.74052e8i | |||||||||||||||||||||||||||||||||||||
| 9.7 | 256.000i | 2973.91i | −65536.0 | 855091. | + | 178210.i | −761320. | − | 7.58343e6i | − | 1.67772e7i | 1.20296e8 | −4.56218e7 | + | 2.18903e8i | |||||||||||||||||||||||||||||||||||||
| 9.8 | 256.000i | 20214.2i | −65536.0 | −765001. | + | 421560.i | −5.17482e6 | − | 1.76306e7i | − | 1.67772e7i | −2.79472e8 | −1.07919e8 | − | 1.95840e8i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 10.18.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 90.18.c.b | 8 | ||
| 4.b | odd | 2 | 1 | 80.18.c.a | 8 | ||
| 5.b | even | 2 | 1 | inner | 10.18.b.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | 50.18.a.j | 4 | ||
| 5.c | odd | 4 | 1 | 50.18.a.k | 4 | ||
| 15.d | odd | 2 | 1 | 90.18.c.b | 8 | ||
| 20.d | odd | 2 | 1 | 80.18.c.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.18.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 10.18.b.a | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 50.18.a.j | 4 | 5.c | odd | 4 | 1 | ||
| 50.18.a.k | 4 | 5.c | odd | 4 | 1 | ||
| 80.18.c.a | 8 | 4.b | odd | 2 | 1 | ||
| 80.18.c.a | 8 | 20.d | odd | 2 | 1 | ||
| 90.18.c.b | 8 | 3.b | odd | 2 | 1 | ||
| 90.18.c.b | 8 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{18}^{\mathrm{new}}(10, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 65536)^{4} \)
|
| $3$ |
\( T^{8} + \cdots + 26\!\cdots\!76 \)
|
| $5$ |
\( T^{8} + \cdots + 33\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 10\!\cdots\!76 \)
|
| $11$ |
\( (T^{4} + \cdots + 55\!\cdots\!36)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 22\!\cdots\!36 \)
|
| $17$ |
\( T^{8} + \cdots + 54\!\cdots\!36 \)
|
| $19$ |
\( (T^{4} + \cdots - 63\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 51\!\cdots\!76 \)
|
| $29$ |
\( (T^{4} + \cdots - 85\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots - 25\!\cdots\!04)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 44\!\cdots\!96 \)
|
| $41$ |
\( (T^{4} + \cdots - 24\!\cdots\!84)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 26\!\cdots\!96 \)
|
| $47$ |
\( T^{8} + \cdots + 12\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots + 25\!\cdots\!36 \)
|
| $59$ |
\( (T^{4} + \cdots + 97\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots - 30\!\cdots\!24)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 40\!\cdots\!76 \)
|
| $71$ |
\( (T^{4} + \cdots - 33\!\cdots\!84)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 16\!\cdots\!36 \)
|
| $79$ |
\( (T^{4} + \cdots - 11\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 41\!\cdots\!16 \)
|
| $89$ |
\( (T^{4} + \cdots - 14\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 87\!\cdots\!36 \)
|
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