Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [141,6,Mod(1,141)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(141, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("141.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 141 = 3 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 141.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: | \(22.6141185936\) |
| 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: |
\( x^{8} - 3x^{7} - 162x^{6} + 388x^{5} + 8143x^{4} - 14661x^{3} - 130166x^{2} + 210500x + 356808 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{8} - 3x^{7} - 162x^{6} + 388x^{5} + 8143x^{4} - 14661x^{3} - 130166x^{2} + 210500x + 356808 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 19 \nu^{7} - 1736 \nu^{6} - 4052 \nu^{5} + 204884 \nu^{4} + 155313 \nu^{3} - 5283852 \nu^{2} + \cdots + 11484072 ) / 938640 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 281 \nu^{7} + 1591 \nu^{6} + 37696 \nu^{5} - 154924 \nu^{4} - 1439871 \nu^{3} + 2547201 \nu^{2} + \cdots + 1238580 ) / 750912 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 529 \nu^{7} + 39 \nu^{6} - 92232 \nu^{5} - 34476 \nu^{4} + 4764743 \nu^{3} + 2382353 \nu^{2} + \cdots + 7811732 ) / 1251520 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3359 \nu^{7} + 1231 \nu^{6} + 506392 \nu^{5} + 163316 \nu^{4} - 21813513 \nu^{3} + \cdots + 181234548 ) / 3754560 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3359 \nu^{7} - 1231 \nu^{6} - 506392 \nu^{5} - 163316 \nu^{4} + 21813513 \nu^{3} + \cdots - 335171508 ) / 3754560 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 8449 \nu^{7} + 9199 \nu^{6} - 1369592 \nu^{5} - 2187196 \nu^{4} + 65174823 \nu^{3} + \cdots - 915269388 ) / 3754560 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 41 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{6} + 6\beta_{5} + 3\beta_{4} - 4\beta_{3} - 4\beta_{2} + 62\beta _1 + 15 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{7} + 96\beta_{6} + 83\beta_{5} + 19\beta_{4} + 4\beta_{3} - 8\beta_{2} + 31\beta _1 + 2586 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{7} + 316\beta_{6} + 566\beta_{5} + 246\beta_{4} - 416\beta_{3} - 444\beta_{2} + 4407\beta _1 + 2546 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 944 \beta_{7} + 8137 \beta_{6} + 6461 \beta_{5} + 2148 \beta_{4} + 688 \beta_{3} - 1264 \beta_{2} + \cdots + 185317 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2544 \beta_{7} + 29227 \beta_{6} + 45070 \beta_{5} + 19315 \beta_{4} - 36292 \beta_{3} + \cdots + 264255 \)
|
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 |
|
−10.0655 | 9.00000 | 69.3138 | 91.7525 | −90.5893 | 20.7292 | −375.581 | 81.0000 | −923.533 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −9.48719 | 9.00000 | 58.0068 | −79.5563 | −85.3847 | 129.290 | −246.732 | 81.0000 | 754.766 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.81629 | 9.00000 | −8.80333 | 18.7824 | −43.3466 | −112.001 | 196.521 | 81.0000 | −90.4615 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.53577 | 9.00000 | −11.4268 | −93.3873 | −40.8219 | 74.4225 | 196.974 | 81.0000 | 423.583 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.0957708 | 9.00000 | −31.9908 | 22.2258 | 0.861937 | −21.0157 | −6.12845 | 81.0000 | 2.12858 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.46274 | 9.00000 | −12.0840 | −27.8133 | 40.1646 | 80.4558 | −196.735 | 81.0000 | −124.124 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.46408 | 9.00000 | −2.14387 | 43.6345 | 49.1767 | −213.307 | −186.565 | 81.0000 | 238.422 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 7.88214 | 9.00000 | 30.1282 | −70.6383 | 70.9393 | −173.574 | −14.7538 | 81.0000 | −556.782 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(47\) | \(-1\) |
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 | 141.6.a.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 423.6.a.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 141.6.a.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 423.6.a.c | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 11T_{2}^{7} - 113T_{2}^{6} - 1241T_{2}^{5} + 3948T_{2}^{4} + 40274T_{2}^{3} - 43544T_{2}^{2} - 397160T_{2} + 38400 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(141))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 11 T^{7} + \cdots + 38400 \)
|
| $3$ |
\( (T - 9)^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 24395636744136 \)
|
| $7$ |
\( T^{8} + \cdots + 13\!\cdots\!92 \)
|
| $11$ |
\( T^{8} + \cdots + 82\!\cdots\!64 \)
|
| $13$ |
\( T^{8} + \cdots + 36\!\cdots\!32 \)
|
| $17$ |
\( T^{8} + \cdots - 40\!\cdots\!48 \)
|
| $19$ |
\( T^{8} + \cdots - 14\!\cdots\!16 \)
|
| $23$ |
\( T^{8} + \cdots + 25\!\cdots\!48 \)
|
| $29$ |
\( T^{8} + \cdots - 52\!\cdots\!44 \)
|
| $31$ |
\( T^{8} + \cdots - 43\!\cdots\!12 \)
|
| $37$ |
\( T^{8} + \cdots - 40\!\cdots\!96 \)
|
| $41$ |
\( T^{8} + \cdots + 25\!\cdots\!40 \)
|
| $43$ |
\( T^{8} + \cdots + 13\!\cdots\!88 \)
|
| $47$ |
\( (T - 2209)^{8} \)
|
| $53$ |
\( T^{8} + \cdots + 17\!\cdots\!80 \)
|
| $59$ |
\( T^{8} + \cdots + 16\!\cdots\!60 \)
|
| $61$ |
\( T^{8} + \cdots - 29\!\cdots\!72 \)
|
| $67$ |
\( T^{8} + \cdots - 44\!\cdots\!92 \)
|
| $71$ |
\( T^{8} + \cdots + 11\!\cdots\!92 \)
|
| $73$ |
\( T^{8} + \cdots - 40\!\cdots\!36 \)
|
| $79$ |
\( T^{8} + \cdots - 58\!\cdots\!64 \)
|
| $83$ |
\( T^{8} + \cdots + 15\!\cdots\!52 \)
|
| $89$ |
\( T^{8} + \cdots + 17\!\cdots\!36 \)
|
| $97$ |
\( T^{8} + \cdots - 20\!\cdots\!20 \)
|
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