Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [57,10,Mod(1,57)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(57, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("57.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 57.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: | \(29.3570426613\) |
| 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: |
\( x^{8} - x^{7} - 3446 x^{6} + 2146 x^{5} + 3632756 x^{4} + 1877896 x^{3} - 1128074928 x^{2} + \cdots - 684004608 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| 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
\( x^{8} - x^{7} - 3446 x^{6} + 2146 x^{5} + 3632756 x^{4} + 1877896 x^{3} - 1128074928 x^{2} + \cdots - 684004608 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 862 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 248545 \nu^{7} - 6574249 \nu^{6} + 755626284 \nu^{5} + 16477812710 \nu^{4} + \cdots + 919755537257088 ) / 328794036480 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 680653 \nu^{7} - 3581003 \nu^{6} - 2400722940 \nu^{5} + 8005981330 \nu^{4} + \cdots - 11\!\cdots\!40 ) / 328794036480 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 64873 \nu^{7} + 144686 \nu^{6} + 236289207 \nu^{5} - 526583290 \nu^{4} + \cdots + 45613645265664 ) / 13699751520 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1841779 \nu^{7} - 1545259 \nu^{6} + 6239657508 \nu^{5} + 1122039890 \nu^{4} + \cdots + 19\!\cdots\!96 ) / 328794036480 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 35797 \nu^{7} - 85625 \nu^{6} - 114994362 \nu^{5} + 334949170 \nu^{4} + 111824975932 \nu^{3} + \cdots - 15190761446784 ) / 6322962240 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 862 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 3\beta_{6} - \beta_{5} + 2\beta_{4} - 3\beta_{3} - 4\beta_{2} + 1338\beta _1 + 319 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{7} - 27\beta_{6} - 23\beta_{5} - 124\beta_{4} + 57\beta_{3} + 1742\beta_{2} - 1606\beta _1 + 1153903 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2729 \beta_{7} + 7055 \beta_{6} - 1801 \beta_{5} + 4904 \beta_{4} - 7129 \beta_{3} - 11300 \beta_{2} + \cdots - 892147 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2649 \beta_{7} - 92825 \beta_{6} - 56281 \beta_{5} - 341116 \beta_{4} + 126091 \beta_{3} + \cdots + 1689294713 \)
|
| \(\nu^{7}\) | \(=\) |
\( 5760253 \beta_{7} + 13322507 \beta_{6} - 2581085 \beta_{5} + 9850212 \beta_{4} - 13761361 \beta_{3} + \cdots - 4603466083 \)
|
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 |
|
−40.0845 | 81.0000 | 1094.76 | −1730.65 | −3246.84 | −2752.52 | −23359.8 | 6561.00 | 69372.1 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −31.1038 | 81.0000 | 455.444 | 1145.37 | −2519.40 | 10346.9 | 1759.12 | 6561.00 | −35625.3 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −20.5401 | 81.0000 | −90.1033 | 849.287 | −1663.75 | −10981.2 | 12367.3 | 6561.00 | −17444.5 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.278642 | 81.0000 | −511.922 | 1999.74 | 22.5700 | 10192.2 | −285.307 | 6561.00 | 557.212 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.64476 | 81.0000 | −509.295 | −1313.38 | 133.226 | −2188.42 | −1679.79 | 6561.00 | −2160.20 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 26.5052 | 81.0000 | 190.524 | 1296.92 | 2146.92 | −3384.63 | −8520.78 | 6561.00 | 34375.1 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 38.8155 | 81.0000 | 994.645 | 2396.22 | 3144.06 | 3937.20 | 18734.1 | 6561.00 | 93010.5 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 41.4843 | 81.0000 | 1208.94 | −741.508 | 3360.23 | 4318.56 | 28912.2 | 6561.00 | −30760.9 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 57.10.a.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 171.10.a.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.10.a.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 171.10.a.e | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 17 T_{2}^{7} - 3320 T_{2}^{6} + 42966 T_{2}^{5} + 3405936 T_{2}^{4} - 26549304 T_{2}^{3} + \cdots - 500910592 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(57))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 17 T^{7} + \cdots - 500910592 \)
|
| $3$ |
\( (T - 81)^{8} \)
|
| $5$ |
\( T^{8} + \cdots - 10\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 40\!\cdots\!00 \)
|
| $11$ |
\( T^{8} + \cdots + 25\!\cdots\!00 \)
|
| $13$ |
\( T^{8} + \cdots - 95\!\cdots\!08 \)
|
| $17$ |
\( T^{8} + \cdots + 48\!\cdots\!12 \)
|
| $19$ |
\( (T + 130321)^{8} \)
|
| $23$ |
\( T^{8} + \cdots - 74\!\cdots\!04 \)
|
| $29$ |
\( T^{8} + \cdots - 12\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots - 11\!\cdots\!28 \)
|
| $37$ |
\( T^{8} + \cdots + 22\!\cdots\!00 \)
|
| $41$ |
\( T^{8} + \cdots - 12\!\cdots\!36 \)
|
| $43$ |
\( T^{8} + \cdots + 53\!\cdots\!36 \)
|
| $47$ |
\( T^{8} + \cdots + 26\!\cdots\!72 \)
|
| $53$ |
\( T^{8} + \cdots + 94\!\cdots\!72 \)
|
| $59$ |
\( T^{8} + \cdots - 13\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 64\!\cdots\!88 \)
|
| $67$ |
\( T^{8} + \cdots - 18\!\cdots\!56 \)
|
| $71$ |
\( T^{8} + \cdots - 72\!\cdots\!00 \)
|
| $73$ |
\( T^{8} + \cdots - 67\!\cdots\!00 \)
|
| $79$ |
\( T^{8} + \cdots - 76\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots - 36\!\cdots\!16 \)
|
| $89$ |
\( T^{8} + \cdots - 28\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
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