Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [387,8,Mod(1,387)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(387, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("387.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 387 = 3^{2} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 387.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: | \(120.893004862\) |
| Analytic rank: | \(1\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{13} - 4 x^{12} - 1295 x^{11} + 4518 x^{10} + 633722 x^{9} - 1900560 x^{8} - 148352102 x^{7} + \cdots - 31740122163456 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 129) |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - 4 x^{12} - 1295 x^{11} + 4518 x^{10} + 633722 x^{9} - 1900560 x^{8} - 148352102 x^{7} + \cdots - 31740122163456 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 200 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 47\!\cdots\!07 \nu^{12} + \cdots - 92\!\cdots\!96 ) / 37\!\cdots\!28 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 20\!\cdots\!87 \nu^{12} + \cdots + 35\!\cdots\!16 ) / 37\!\cdots\!28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 84\!\cdots\!69 \nu^{12} + \cdots - 23\!\cdots\!48 ) / 15\!\cdots\!72 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 73\!\cdots\!41 \nu^{12} + \cdots - 58\!\cdots\!68 ) / 77\!\cdots\!36 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 88\!\cdots\!01 \nu^{12} + \cdots + 17\!\cdots\!92 ) / 93\!\cdots\!32 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 21\!\cdots\!75 \nu^{12} + \cdots - 34\!\cdots\!28 ) / 18\!\cdots\!64 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 44\!\cdots\!85 \nu^{12} + \cdots + 17\!\cdots\!92 ) / 37\!\cdots\!28 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 48\!\cdots\!45 \nu^{12} + \cdots - 12\!\cdots\!88 ) / 37\!\cdots\!28 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 30\!\cdots\!11 \nu^{12} + \cdots + 13\!\cdots\!48 ) / 18\!\cdots\!64 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 32\!\cdots\!07 \nu^{12} + \cdots + 75\!\cdots\!20 ) / 12\!\cdots\!76 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 200 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{12} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{3} + 2\beta_{2} + 317\beta _1 + 57 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10 \beta_{11} + 4 \beta_{10} + 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 12 \beta_{5} - 2 \beta_{4} + \cdots + 63582 \)
|
| \(\nu^{5}\) | \(=\) |
\( 591 \beta_{12} + 70 \beta_{11} - 399 \beta_{10} - 531 \beta_{9} - 569 \beta_{8} + 2 \beta_{7} + \cdots + 51029 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1026 \beta_{12} + 7264 \beta_{11} + 3294 \beta_{10} - 1418 \beta_{9} + 1006 \beta_{8} - 152 \beta_{7} + \cdots + 24196018 \)
|
| \(\nu^{7}\) | \(=\) |
\( 289337 \beta_{12} + 63444 \beta_{11} - 133337 \beta_{10} - 248657 \beta_{9} - 274581 \beta_{8} + \cdots + 40695845 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1033948 \beta_{12} + 4080454 \beta_{11} + 1924816 \beta_{10} - 1304268 \beta_{9} + 213474 \beta_{8} + \cdots + 10035217950 \)
|
| \(\nu^{9}\) | \(=\) |
\( 135589347 \beta_{12} + 42456706 \beta_{11} - 42840147 \beta_{10} - 115609327 \beta_{9} + \cdots + 29738834213 \)
|
| \(\nu^{10}\) | \(=\) |
\( 747070342 \beta_{12} + 2107053644 \beta_{11} + 989915330 \beta_{10} - 875057182 \beta_{9} + \cdots + 4370608212906 \)
|
| \(\nu^{11}\) | \(=\) |
\( 63122345245 \beta_{12} + 25211475568 \beta_{11} - 13446342605 \beta_{10} - 54278043645 \beta_{9} + \cdots + 19746817017413 \)
|
| \(\nu^{12}\) | \(=\) |
\( 470844078752 \beta_{12} + 1052355118450 \beta_{11} + 480626859220 \beta_{10} - 521407554848 \beta_{9} + \cdots + 19\!\cdots\!38 \)
|
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 |
|
−21.7719 | 0 | 346.014 | 524.772 | 0 | 236.108 | −4746.57 | 0 | −11425.3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −20.3481 | 0 | 286.044 | −523.301 | 0 | 718.529 | −3215.90 | 0 | 10648.2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −15.4268 | 0 | 109.985 | −300.216 | 0 | −1333.38 | 277.917 | 0 | 4631.35 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −11.0171 | 0 | −6.62316 | 389.307 | 0 | −1032.07 | 1483.16 | 0 | −4289.04 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −10.3214 | 0 | −21.4697 | 342.151 | 0 | −479.039 | 1542.73 | 0 | −3531.46 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −8.72677 | 0 | −51.8435 | −87.9438 | 0 | 1709.23 | 1569.45 | 0 | 767.465 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.651000 | 0 | −127.576 | −131.920 | 0 | −1640.88 | −166.380 | 0 | −85.8799 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.21144 | 0 | −100.841 | −407.890 | 0 | 909.892 | −1192.59 | 0 | −2125.70 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 8.45083 | 0 | −56.5835 | 299.558 | 0 | 650.266 | −1559.88 | 0 | 2531.52 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 11.6103 | 0 | 6.79959 | −114.846 | 0 | 467.830 | −1407.18 | 0 | −1333.40 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 15.5680 | 0 | 114.364 | 54.6595 | 0 | −803.520 | −212.287 | 0 | 850.941 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 15.7520 | 0 | 120.126 | 263.415 | 0 | 1462.65 | −124.037 | 0 | 4149.31 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 21.3683 | 0 | 328.604 | −41.7453 | 0 | −859.604 | 4286.57 | 0 | −892.025 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(43\) | \(1\) |
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 | 387.8.a.e | 13 | |
| 3.b | odd | 2 | 1 | 129.8.a.c | ✓ | 13 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 129.8.a.c | ✓ | 13 | 3.b | odd | 2 | 1 | |
| 387.8.a.e | 13 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{13} + 9 T_{2}^{12} - 1265 T_{2}^{11} - 9705 T_{2}^{10} + 607512 T_{2}^{9} + 3791880 T_{2}^{8} + \cdots - 11829723734016 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(387))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} + \cdots - 11829723734016 \)
|
| $3$ |
\( T^{13} \)
|
| $5$ |
\( T^{13} + \cdots + 10\!\cdots\!00 \)
|
| $7$ |
\( T^{13} + \cdots - 87\!\cdots\!72 \)
|
| $11$ |
\( T^{13} + \cdots + 18\!\cdots\!56 \)
|
| $13$ |
\( T^{13} + \cdots - 15\!\cdots\!16 \)
|
| $17$ |
\( T^{13} + \cdots + 41\!\cdots\!96 \)
|
| $19$ |
\( T^{13} + \cdots + 34\!\cdots\!52 \)
|
| $23$ |
\( T^{13} + \cdots + 13\!\cdots\!00 \)
|
| $29$ |
\( T^{13} + \cdots - 47\!\cdots\!92 \)
|
| $31$ |
\( T^{13} + \cdots + 47\!\cdots\!52 \)
|
| $37$ |
\( T^{13} + \cdots + 69\!\cdots\!36 \)
|
| $41$ |
\( T^{13} + \cdots - 16\!\cdots\!16 \)
|
| $43$ |
\( (T + 79507)^{13} \)
|
| $47$ |
\( T^{13} + \cdots + 30\!\cdots\!76 \)
|
| $53$ |
\( T^{13} + \cdots + 26\!\cdots\!08 \)
|
| $59$ |
\( T^{13} + \cdots + 11\!\cdots\!12 \)
|
| $61$ |
\( T^{13} + \cdots + 22\!\cdots\!72 \)
|
| $67$ |
\( T^{13} + \cdots - 26\!\cdots\!36 \)
|
| $71$ |
\( T^{13} + \cdots + 29\!\cdots\!92 \)
|
| $73$ |
\( T^{13} + \cdots + 91\!\cdots\!16 \)
|
| $79$ |
\( T^{13} + \cdots + 29\!\cdots\!24 \)
|
| $83$ |
\( T^{13} + \cdots + 22\!\cdots\!88 \)
|
| $89$ |
\( T^{13} + \cdots + 23\!\cdots\!00 \)
|
| $97$ |
\( T^{13} + \cdots + 37\!\cdots\!68 \)
|
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