Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [143,3,Mod(142,143)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(143, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("143.142");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 143 = 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 143.d (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: | \(3.89646778035\) |
| 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: |
\( x^{8} + 12x^{6} - 30x^{5} + 107x^{4} - 234x^{3} + 924x^{2} + 732x + 3142 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{8} + 12x^{6} - 30x^{5} + 107x^{4} - 234x^{3} + 924x^{2} + 732x + 3142 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 1958116 \nu^{7} + 2108724 \nu^{6} - 33718612 \nu^{5} - 61060699 \nu^{4} + 33642990 \nu^{3} + \cdots - 7354824817 ) / 6401158517 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5393784 \nu^{7} + 16839260 \nu^{6} + 74364720 \nu^{5} - 42381966 \nu^{4} + 356153792 \nu^{3} + \cdots + 7061374479 ) / 6401158517 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7142 \nu^{7} + 1951 \nu^{6} + 28447 \nu^{5} - 152902 \nu^{4} + 155033 \nu^{3} + 971671 \nu^{2} + \cdots + 15578585 ) / 7504289 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 6319211 \nu^{7} + 42529315 \nu^{6} - 123239346 \nu^{5} + 479744617 \nu^{4} + \cdots + 17857685611 ) / 6401158517 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 11485910 \nu^{7} - 18503463 \nu^{6} - 98630011 \nu^{5} + 172807372 \nu^{4} + \cdots - 20349907484 ) / 6401158517 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 16235 \nu^{7} - 53355 \nu^{6} + 118252 \nu^{5} - 914965 \nu^{4} + 2952965 \nu^{3} + \cdots - 28804439 ) / 7504289 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 28689 \nu^{7} - 45750 \nu^{6} + 436843 \nu^{5} - 1822534 \nu^{4} + 3034176 \nu^{3} + \cdots - 3654275 ) / 7504289 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{6} - 2\beta_{4} + 4\beta_{3} - \beta_{2} + 2\beta _1 - 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{7} - 10\beta_{5} - 7\beta_{4} - 9\beta_{3} - \beta_{2} + 13\beta _1 + 21 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -10\beta_{7} + 6\beta_{6} + 8\beta_{5} - 4\beta_{4} + 2\beta_{3} + 19\beta_{2} - 58\beta _1 - 37 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 30\beta_{7} - 75\beta_{6} - 44\beta_{5} - 49\beta_{4} - 86\beta_{3} - 36\beta_{2} - 106\beta _1 - 180 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -9\beta_{7} + 23\beta_{6} - 402\beta_{5} + 90\beta_{4} - 569\beta_{3} + 108\beta_{2} + 645\beta _1 + 358 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -266\beta_{7} + 693\beta_{6} + 325\beta_{5} + 509\beta_{4} + 1945\beta_{3} + 330\beta_{2} - 1550\beta _1 - 4039 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/143\mathbb{Z}\right)^\times\).
| \(n\) | \(67\) | \(79\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 142.1 |
|
−2.90931 | 1.73205 | 4.46410 | − | 6.07095i | −5.03908 | −5.03908 | −1.35022 | −6.00000 | 17.6623i | |||||||||||||||||||||||||||||||||||||||||
| 142.2 | −2.90931 | 1.73205 | 4.46410 | 6.07095i | −5.03908 | −5.03908 | −1.35022 | −6.00000 | − | 17.6623i | ||||||||||||||||||||||||||||||||||||||||||
| 142.3 | −1.23931 | −1.73205 | −2.46410 | − | 3.02384i | 2.14655 | 2.14655 | 8.01105 | −6.00000 | 3.74748i | ||||||||||||||||||||||||||||||||||||||||||
| 142.4 | −1.23931 | −1.73205 | −2.46410 | 3.02384i | 2.14655 | 2.14655 | 8.01105 | −6.00000 | − | 3.74748i | ||||||||||||||||||||||||||||||||||||||||||
| 142.5 | 1.23931 | −1.73205 | −2.46410 | − | 3.02384i | −2.14655 | −2.14655 | −8.01105 | −6.00000 | − | 3.74748i | |||||||||||||||||||||||||||||||||||||||||
| 142.6 | 1.23931 | −1.73205 | −2.46410 | 3.02384i | −2.14655 | −2.14655 | −8.01105 | −6.00000 | 3.74748i | |||||||||||||||||||||||||||||||||||||||||||
| 142.7 | 2.90931 | 1.73205 | 4.46410 | − | 6.07095i | 5.03908 | 5.03908 | 1.35022 | −6.00000 | − | 17.6623i | |||||||||||||||||||||||||||||||||||||||||
| 142.8 | 2.90931 | 1.73205 | 4.46410 | 6.07095i | 5.03908 | 5.03908 | 1.35022 | −6.00000 | 17.6623i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
| 143.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 143.3.d.d | ✓ | 8 |
| 11.b | odd | 2 | 1 | inner | 143.3.d.d | ✓ | 8 |
| 13.b | even | 2 | 1 | inner | 143.3.d.d | ✓ | 8 |
| 143.d | odd | 2 | 1 | inner | 143.3.d.d | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.3.d.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 143.3.d.d | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
| 143.3.d.d | ✓ | 8 | 13.b | even | 2 | 1 | inner |
| 143.3.d.d | ✓ | 8 | 143.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 10T_{2}^{2} + 13 \)
acting on \(S_{3}^{\mathrm{new}}(143, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 10 T^{2} + 13)^{2} \)
|
| $3$ |
\( (T^{2} - 3)^{4} \)
|
| $5$ |
\( (T^{4} + 46 T^{2} + 337)^{2} \)
|
| $7$ |
\( (T^{4} - 30 T^{2} + 117)^{2} \)
|
| $11$ |
\( T^{8} + 324 T^{6} + \cdots + 214358881 \)
|
| $13$ |
\( T^{8} + 412 T^{6} + \cdots + 815730721 \)
|
| $17$ |
\( (T^{4} + 654 T^{2} + 39429)^{2} \)
|
| $19$ |
\( (T^{4} - 1200 T^{2} + 56628)^{2} \)
|
| $23$ |
\( (T^{2} + 24 T + 96)^{4} \)
|
| $29$ |
\( (T^{4} + 816 T^{2} + 157716)^{2} \)
|
| $31$ |
\( (T^{4} + 2208 T^{2} + 776448)^{2} \)
|
| $37$ |
\( (T^{4} + 3654 T^{2} + 3302937)^{2} \)
|
| $41$ |
\( (T^{4} - 1096 T^{2} + 284752)^{2} \)
|
| $43$ |
\( (T^{4} + 6522 T^{2} + 4770909)^{2} \)
|
| $47$ |
\( (T^{4} + 11194 T^{2} + 30128137)^{2} \)
|
| $53$ |
\( (T^{2} - 30 T - 3882)^{4} \)
|
| $59$ |
\( (T^{4} + 8224 T^{2} + 86272)^{2} \)
|
| $61$ |
\( (T^{4} + 10904 T^{2} + 70096)^{2} \)
|
| $67$ |
\( (T^{4} + 264 T^{2} + 12132)^{2} \)
|
| $71$ |
\( (T^{4} + 6226 T^{2} + 9625057)^{2} \)
|
| $73$ |
\( (T^{4} - 17040 T^{2} + 24542388)^{2} \)
|
| $79$ |
\( (T^{4} + 6800 T^{2} + 10952500)^{2} \)
|
| $83$ |
\( (T^{4} - 6280 T^{2} + 3569488)^{2} \)
|
| $89$ |
\( (T^{4} + 3736 T^{2} + 713092)^{2} \)
|
| $97$ |
\( (T^{4} + 10920 T^{2} + 26799588)^{2} \)
|
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