Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [786,4,Mod(1,786)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(786, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("786.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 786 = 2 \cdot 3 \cdot 131 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 786.a (trivial) |
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: | yes |
| Analytic conductor: | \(46.3755012645\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{8} - x^{7} - 436x^{6} + 1403x^{5} + 41156x^{4} - 104947x^{3} - 993314x^{2} + 1535040x + 1863168 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{8} - x^{7} - 436x^{6} + 1403x^{5} + 41156x^{4} - 104947x^{3} - 993314x^{2} + 1535040x + 1863168 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 128092828427 \nu^{7} - 629172836709 \nu^{6} + 36433206587180 \nu^{5} + \cdots + 36\!\cdots\!00 ) / 50\!\cdots\!68 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 9624798585 \nu^{7} + 4383735735 \nu^{6} - 4159765023300 \nu^{5} + 5540549313251 \nu^{4} + \cdots - 16\!\cdots\!68 ) / 13\!\cdots\!88 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 43438461331 \nu^{7} - 292710320901 \nu^{6} + 18444927403348 \nu^{5} + \cdots + 43\!\cdots\!40 ) / 62\!\cdots\!96 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 360085147487 \nu^{7} - 3243471376401 \nu^{6} + 157784375523260 \nu^{5} + \cdots + 56\!\cdots\!56 ) / 50\!\cdots\!68 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 874765644923 \nu^{7} + 169925219595 \nu^{6} + 383520227729900 \nu^{5} + \cdots - 72\!\cdots\!32 ) / 10\!\cdots\!36 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 249248936063 \nu^{7} - 57806134833 \nu^{6} + 102205336314812 \nu^{5} + \cdots - 287829957876000 ) / 25\!\cdots\!84 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 138751803241 \nu^{7} + 20176591417 \nu^{6} + 57917176579012 \nu^{5} + \cdots - 14\!\cdots\!88 ) / 83\!\cdots\!28 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - \beta_{6} - 2\beta_{5} - 2\beta_{4} + 3\beta_{3} - \beta_{2} - \beta_1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{7} + 2\beta_{6} - 2\beta_{5} + 5\beta_{4} - 6\beta_{3} + 6\beta_{2} - 2\beta _1 + 110 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 171\beta_{7} - 536\beta_{6} - 222\beta_{5} - 697\beta_{4} + 978\beta_{3} - 401\beta_{2} - 46\beta _1 - 2030 ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 762\beta_{7} + 1026\beta_{6} - 702\beta_{5} + 2226\beta_{4} - 2862\beta_{3} + 1663\beta_{2} - 543\beta _1 + 27009 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 24936 \beta_{7} - 187846 \beta_{6} - 22762 \beta_{5} - 243207 \beta_{4} + 335553 \beta_{3} + \cdots - 1189720 ) / 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( 193878 \beta_{7} + 459703 \beta_{6} - 188986 \beta_{5} + 864454 \beta_{4} - 1156602 \beta_{3} + \cdots + 8321224 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1916331 \beta_{7} - 64942071 \beta_{6} - 91202 \beta_{5} - 87998417 \beta_{4} + 120197208 \beta_{3} + \cdots - 532238805 ) / 5 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.00000 | −3.00000 | 4.00000 | −18.6282 | 6.00000 | 7.39814 | −8.00000 | 9.00000 | 37.2565 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −3.00000 | 4.00000 | −17.0398 | 6.00000 | −20.6540 | −8.00000 | 9.00000 | 34.0797 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −3.00000 | 4.00000 | −8.00517 | 6.00000 | 33.5401 | −8.00000 | 9.00000 | 16.0103 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | −3.00000 | 4.00000 | 0.109133 | 6.00000 | 11.7436 | −8.00000 | 9.00000 | −0.218267 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | −3.00000 | 4.00000 | 4.74723 | 6.00000 | 27.5663 | −8.00000 | 9.00000 | −9.49446 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | −3.00000 | 4.00000 | 9.80862 | 6.00000 | −29.7847 | −8.00000 | 9.00000 | −19.6172 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.00000 | −3.00000 | 4.00000 | 13.8760 | 6.00000 | −21.4306 | −8.00000 | 9.00000 | −27.7519 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.00000 | −3.00000 | 4.00000 | 16.1323 | 6.00000 | 13.6210 | −8.00000 | 9.00000 | −32.2646 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(131\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 786.4.a.g | ✓ | 8 |
| 3.b | odd | 2 | 1 | 2358.4.a.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 786.4.a.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 2358.4.a.i | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - T_{5}^{7} - 636 T_{5}^{6} + 1971 T_{5}^{5} + 120064 T_{5}^{4} - 567993 T_{5}^{3} + \cdots - 2890496 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(786))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{8} \)
|
| $3$ |
\( (T + 3)^{8} \)
|
| $5$ |
\( T^{8} - T^{7} + \cdots - 2890496 \)
|
| $7$ |
\( T^{8} + \cdots - 14424723944 \)
|
| $11$ |
\( T^{8} + \cdots - 292183370514 \)
|
| $13$ |
\( T^{8} + \cdots - 11986871076 \)
|
| $17$ |
\( T^{8} + \cdots + 3047201554138 \)
|
| $19$ |
\( T^{8} + \cdots + 50350851481900 \)
|
| $23$ |
\( T^{8} + \cdots + 130626886281136 \)
|
| $29$ |
\( T^{8} + \cdots - 13\!\cdots\!50 \)
|
| $31$ |
\( T^{8} + \cdots + 46\!\cdots\!32 \)
|
| $37$ |
\( T^{8} + \cdots - 14\!\cdots\!16 \)
|
| $41$ |
\( T^{8} + \cdots - 91\!\cdots\!56 \)
|
| $43$ |
\( T^{8} + \cdots + 65\!\cdots\!32 \)
|
| $47$ |
\( T^{8} + \cdots - 22\!\cdots\!84 \)
|
| $53$ |
\( T^{8} + \cdots + 36\!\cdots\!32 \)
|
| $59$ |
\( T^{8} + \cdots - 18\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 29\!\cdots\!84 \)
|
| $67$ |
\( T^{8} + \cdots - 40\!\cdots\!24 \)
|
| $71$ |
\( T^{8} + \cdots - 52\!\cdots\!36 \)
|
| $73$ |
\( T^{8} + \cdots - 41\!\cdots\!36 \)
|
| $79$ |
\( T^{8} + \cdots + 67\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots - 66\!\cdots\!48 \)
|
| $89$ |
\( T^{8} + \cdots + 16\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 22\!\cdots\!68 \)
|
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