Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,19,Mod(8,9)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 19, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.8");
S:= CuspForms(chi, 19);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 19 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.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: | \(18.4847523939\) |
| 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: |
\( x^{6} - 2x^{5} - 5505x^{4} + 33224x^{3} + 7567832x^{2} - 76551624x + 207368172 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{33} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 5505x^{4} + 33224x^{3} + 7567832x^{2} - 76551624x + 207368172 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 55379 \nu^{5} + 3904591 \nu^{4} - 259303433 \nu^{3} - 9995442769 \nu^{2} + 339474142386 \nu - 1459042057242 ) / 2216382982 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 996822 \nu^{5} - 70282638 \nu^{4} + 4667461794 \nu^{3} + 359444991384 \nu^{2} + \cdots - 304007120266410 ) / 1108191491 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4051953 \nu^{5} - 108187470 \nu^{4} - 18447461055 \nu^{3} + 708378800040 \nu^{2} + \cdots - 753617394019512 ) / 2216382982 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 155921091 \nu^{5} + 358794750 \nu^{4} - 715896332205 \nu^{3} + 1035596511852 \nu^{2} + \cdots - 40\!\cdots\!76 ) / 4432765964 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1194601239 \nu^{5} + 1915031574 \nu^{4} + 7632999568017 \nu^{3} - 22884989805360 \nu^{2} + \cdots + 60\!\cdots\!64 ) / 4432765964 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{4} - 27\beta_{3} + 27\beta_{2} + 132\beta _1 + 118098 ) / 354294 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -8\beta_{4} + 108\beta_{3} + 621\beta_{2} + 25716\beta _1 + 216788562 ) / 118098 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 729\beta_{5} + 11381\beta_{4} - 73926\beta_{3} + 76113\beta_{2} - 9930\beta _1 - 3934671066 ) / 354294 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 324\beta_{5} - 75964\beta_{4} + 396396\beta_{3} + 1604709\beta_{2} + 139200072\beta _1 + 593659474758 ) / 118098 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 3344895 \beta_{5} + 52765823 \beta_{4} - 206002872 \beta_{3} + 187704243 \beta_{2} + \cdots - 17998418517102 ) / 354294 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 8.1 |
|
− | 716.672i | 0 | −251474. | − | 3.23590e6i | 0 | 1.43214e7 | − | 7.64666e6i | 0 | −2.31908e9 | |||||||||||||||||||||||||||||||||
| 8.2 | − | 615.700i | 0 | −116942. | 1.45698e6i | 0 | 2.81282e7 | − | 8.94006e7i | 0 | 8.97060e8 | |||||||||||||||||||||||||||||||||||
| 8.3 | − | 140.859i | 0 | 242303. | 3.07764e6i | 0 | −3.28593e7 | − | 7.10558e7i | 0 | 4.33513e8 | |||||||||||||||||||||||||||||||||||
| 8.4 | 140.859i | 0 | 242303. | − | 3.07764e6i | 0 | −3.28593e7 | 7.10558e7i | 0 | 4.33513e8 | ||||||||||||||||||||||||||||||||||||
| 8.5 | 615.700i | 0 | −116942. | − | 1.45698e6i | 0 | 2.81282e7 | 8.94006e7i | 0 | 8.97060e8 | ||||||||||||||||||||||||||||||||||||
| 8.6 | 716.672i | 0 | −251474. | 3.23590e6i | 0 | 1.43214e7 | 7.64666e6i | 0 | −2.31908e9 | |||||||||||||||||||||||||||||||||||||
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 | 9.19.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 9.19.b.a | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.19.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 9.19.b.a | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{19}^{\mathrm{new}}(9, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 38\!\cdots\!08 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
|
| $7$ |
\( (T^{3} + \cdots + 13\!\cdots\!84)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 63\!\cdots\!12 \)
|
| $13$ |
\( (T^{3} + \cdots - 18\!\cdots\!56)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 75\!\cdots\!28 \)
|
| $19$ |
\( (T^{3} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 81\!\cdots\!72 \)
|
| $29$ |
\( T^{6} + \cdots + 31\!\cdots\!68 \)
|
| $31$ |
\( (T^{3} + \cdots + 55\!\cdots\!64)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots + 12\!\cdots\!92)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 14\!\cdots\!08 \)
|
| $43$ |
\( (T^{3} + \cdots - 24\!\cdots\!68)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 30\!\cdots\!32 \)
|
| $53$ |
\( T^{6} + \cdots + 23\!\cdots\!72 \)
|
| $59$ |
\( T^{6} + \cdots + 12\!\cdots\!32 \)
|
| $61$ |
\( (T^{3} + \cdots + 76\!\cdots\!68)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots + 17\!\cdots\!08)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 59\!\cdots\!68 \)
|
| $73$ |
\( (T^{3} + \cdots + 35\!\cdots\!04)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 18\!\cdots\!96)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 36\!\cdots\!52 \)
|
| $89$ |
\( T^{6} + \cdots + 98\!\cdots\!12 \)
|
| $97$ |
\( (T^{3} + \cdots + 11\!\cdots\!68)^{2} \)
|
| show more | |
| show less |