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: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} - 825 x^{8} + 431 x^{7} + 229838 x^{6} - 1804 x^{5} - 25242488 x^{4} - 2085744 x^{3} + \cdots - 5193030528 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} - 825 x^{8} + 431 x^{7} + 229838 x^{6} - 1804 x^{5} - 25242488 x^{4} - 2085744 x^{3} + \cdots - 5193030528 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 165 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3303152543 \nu^{9} - 38991123497 \nu^{8} - 2242995763485 \nu^{7} + 27075500275867 \nu^{6} + \cdots - 33\!\cdots\!52 ) / 54\!\cdots\!44 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 5739770765 \nu^{9} + 86170330415 \nu^{8} + 4048941347727 \nu^{7} - 61114843044793 \nu^{6} + \cdots + 97\!\cdots\!84 ) / 73\!\cdots\!92 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 8791035155 \nu^{9} + 213402162401 \nu^{8} + 5575524982977 \nu^{7} + \cdots + 12\!\cdots\!36 ) / 10\!\cdots\!88 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 285937361 \nu^{9} - 95066038 \nu^{8} + 222054068433 \nu^{7} + 176617295585 \nu^{6} + \cdots - 31\!\cdots\!68 ) / 342892530069984 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 318866645 \nu^{9} - 4316489207 \nu^{8} - 222660252855 \nu^{7} + 2984417933713 \nu^{6} + \cdots - 27\!\cdots\!48 ) / 318045245282304 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 632903078 \nu^{9} + 5916364451 \nu^{8} + 465575829570 \nu^{7} - 3985126274494 \nu^{6} + \cdots + 31\!\cdots\!52 ) / 457190040093312 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 5860305659 \nu^{9} + 53885376281 \nu^{8} + 4260728870529 \nu^{7} - 35644394911255 \nu^{6} + \cdots + 13\!\cdots\!48 ) / 18\!\cdots\!48 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 165 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - 2\beta_{8} + 2\beta_{7} + 2\beta_{4} + 2\beta_{2} + 270\beta _1 + 91 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} - 24 \beta_{6} - 20 \beta_{5} + 4 \beta_{4} - 10 \beta_{3} + \cdots + 44315 \)
|
| \(\nu^{5}\) | \(=\) |
\( 475 \beta_{9} - 1052 \beta_{8} + 794 \beta_{7} + 64 \beta_{6} + 4 \beta_{5} + 946 \beta_{4} + \cdots + 40205 \)
|
| \(\nu^{6}\) | \(=\) |
\( 5772 \beta_{9} - 1728 \beta_{8} - 1896 \beta_{7} - 14664 \beta_{6} - 11032 \beta_{5} + 360 \beta_{4} + \cdots + 13776277 \)
|
| \(\nu^{7}\) | \(=\) |
\( 188329 \beta_{9} - 433794 \beta_{8} + 274914 \beta_{7} + 41152 \beta_{6} + 1768 \beta_{5} + \cdots + 9111351 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2844012 \beta_{9} - 823642 \beta_{8} - 782556 \beta_{7} - 6761208 \beta_{6} - 4700596 \beta_{5} + \cdots + 4590453395 \)
|
| \(\nu^{9}\) | \(=\) |
\( 71266967 \beta_{9} - 167093388 \beta_{8} + 93931682 \beta_{7} + 19106656 \beta_{6} + 1653492 \beta_{5} + \cdots + 379452861 \)
|
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 |
|
−19.3933 | 0 | 248.102 | 430.272 | 0 | −1073.69 | −2329.18 | 0 | −8344.41 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −14.4589 | 0 | 81.0593 | −105.898 | 0 | −671.216 | 678.711 | 0 | 1531.17 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −10.6558 | 0 | −14.4536 | 201.175 | 0 | 957.575 | 1517.96 | 0 | −2143.69 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −10.1422 | 0 | −25.1352 | −212.535 | 0 | −1278.88 | 1553.13 | 0 | 2155.58 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.05398 | 0 | −123.781 | 26.1249 | 0 | 1045.72 | 517.154 | 0 | −53.6601 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.54119 | 0 | −115.460 | 160.857 | 0 | −719.763 | −862.137 | 0 | 569.625 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.93847 | 0 | −92.7346 | −80.6218 | 0 | −289.258 | −1310.83 | 0 | −478.770 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 12.4268 | 0 | 26.4257 | −463.185 | 0 | −63.1162 | −1262.24 | 0 | −5755.92 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 16.8413 | 0 | 155.630 | 309.730 | 0 | −729.159 | 465.334 | 0 | 5216.26 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 18.9565 | 0 | 231.347 | −143.919 | 0 | 769.785 | 1959.10 | 0 | −2728.19 | |||||||||||||||||||||||||||||||||||||||||||||||||
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.a | 10 | |
| 3.b | odd | 2 | 1 | 129.8.a.a | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 129.8.a.a | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 387.8.a.a | 10 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - T_{2}^{9} - 825 T_{2}^{8} + 431 T_{2}^{7} + 229838 T_{2}^{6} - 1804 T_{2}^{5} + \cdots - 5193030528 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(387))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots - 5193030528 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} + \cdots - 13\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots - 68\!\cdots\!08 \)
|
| $11$ |
\( T^{10} + \cdots + 17\!\cdots\!04 \)
|
| $13$ |
\( T^{10} + \cdots + 58\!\cdots\!00 \)
|
| $17$ |
\( T^{10} + \cdots - 25\!\cdots\!04 \)
|
| $19$ |
\( T^{10} + \cdots + 85\!\cdots\!28 \)
|
| $23$ |
\( T^{10} + \cdots - 34\!\cdots\!56 \)
|
| $29$ |
\( T^{10} + \cdots + 19\!\cdots\!56 \)
|
| $31$ |
\( T^{10} + \cdots + 23\!\cdots\!12 \)
|
| $37$ |
\( T^{10} + \cdots + 64\!\cdots\!80 \)
|
| $41$ |
\( T^{10} + \cdots - 11\!\cdots\!52 \)
|
| $43$ |
\( (T + 79507)^{10} \)
|
| $47$ |
\( T^{10} + \cdots + 42\!\cdots\!88 \)
|
| $53$ |
\( T^{10} + \cdots + 35\!\cdots\!60 \)
|
| $59$ |
\( T^{10} + \cdots - 38\!\cdots\!88 \)
|
| $61$ |
\( T^{10} + \cdots + 35\!\cdots\!12 \)
|
| $67$ |
\( T^{10} + \cdots - 49\!\cdots\!72 \)
|
| $71$ |
\( T^{10} + \cdots + 46\!\cdots\!96 \)
|
| $73$ |
\( T^{10} + \cdots + 91\!\cdots\!32 \)
|
| $79$ |
\( T^{10} + \cdots - 17\!\cdots\!64 \)
|
| $83$ |
\( T^{10} + \cdots - 19\!\cdots\!44 \)
|
| $89$ |
\( T^{10} + \cdots - 31\!\cdots\!96 \)
|
| $97$ |
\( T^{10} + \cdots - 28\!\cdots\!36 \)
|
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