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: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 1860x^{4} + 264x^{3} + 626016x^{2} + 4023504x - 725760 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 1860x^{4} + 264x^{3} + 626016x^{2} + 4023504x - 725760 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 15\nu^{5} - 47\nu^{4} - 25530\nu^{3} + 42820\nu^{2} + 6429864\nu + 22805376 ) / 70912 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 4\nu - 620 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 87\nu^{5} - 1159\nu^{4} - 152506\nu^{3} + 1524772\nu^{2} + 41842216\nu + 10203264 ) / 70912 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 77\nu^{4} - 594\nu^{3} + 119564\nu^{2} - 988696\nu - 18109856 ) / 8864 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4\beta _1 + 620 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{5} - 4\beta_{4} - 7\beta_{3} + 20\beta_{2} + 1188\beta _1 + 2062 \)
|
| \(\nu^{4}\) | \(=\) |
\( -30\beta_{5} - 60\beta_{4} + 1475\beta_{3} + 364\beta_{2} + 4952\beta _1 + 744778 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10118\beta_{5} - 6996\beta_{4} - 10147\beta_{3} + 39908\beta_{2} + 1597416\beta _1 + 2552910 \)
|
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 |
|
−32.3673 | −81.0000 | 535.640 | 542.116 | 2621.75 | 4515.51 | −765.163 | 6561.00 | −17546.8 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −18.3919 | −81.0000 | −173.738 | 1624.78 | 1489.74 | 7425.81 | 12612.0 | 6561.00 | −29882.9 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 5.82442 | −81.0000 | −478.076 | 2494.48 | −471.778 | −7505.29 | −5766.62 | 6561.00 | 14528.9 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 14.1013 | −81.0000 | −313.152 | −1990.91 | −1142.21 | 5786.77 | −11635.8 | 6561.00 | −28074.4 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 20.6700 | −81.0000 | −84.7493 | −1160.64 | −1674.27 | −6983.68 | −12334.8 | 6561.00 | −23990.6 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 43.1634 | −81.0000 | 1351.08 | −351.832 | −3496.23 | 4650.87 | 36217.3 | 6561.00 | −15186.2 | |||||||||||||||||||||||||||||||||||||
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.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 171.10.a.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.10.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 171.10.a.b | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 33T_{2}^{5} - 1410T_{2}^{4} + 41136T_{2}^{3} + 241968T_{2}^{2} - 9984384T_{2} + 43621632 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(57))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 33 T^{5} + \cdots + 43621632 \)
|
| $3$ |
\( (T + 81)^{6} \)
|
| $5$ |
\( T^{6} + \cdots - 17\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 47\!\cdots\!68 \)
|
| $11$ |
\( T^{6} + \cdots - 30\!\cdots\!52 \)
|
| $13$ |
\( T^{6} + \cdots - 65\!\cdots\!00 \)
|
| $17$ |
\( T^{6} + \cdots + 31\!\cdots\!00 \)
|
| $19$ |
\( (T - 130321)^{6} \)
|
| $23$ |
\( T^{6} + \cdots + 20\!\cdots\!76 \)
|
| $29$ |
\( T^{6} + \cdots - 30\!\cdots\!40 \)
|
| $31$ |
\( T^{6} + \cdots - 17\!\cdots\!88 \)
|
| $37$ |
\( T^{6} + \cdots - 83\!\cdots\!12 \)
|
| $41$ |
\( T^{6} + \cdots - 22\!\cdots\!84 \)
|
| $43$ |
\( T^{6} + \cdots + 87\!\cdots\!48 \)
|
| $47$ |
\( T^{6} + \cdots - 12\!\cdots\!48 \)
|
| $53$ |
\( T^{6} + \cdots - 36\!\cdots\!16 \)
|
| $59$ |
\( T^{6} + \cdots + 32\!\cdots\!56 \)
|
| $61$ |
\( T^{6} + \cdots - 77\!\cdots\!80 \)
|
| $67$ |
\( T^{6} + \cdots + 98\!\cdots\!60 \)
|
| $71$ |
\( T^{6} + \cdots - 13\!\cdots\!80 \)
|
| $73$ |
\( T^{6} + \cdots - 15\!\cdots\!08 \)
|
| $79$ |
\( T^{6} + \cdots + 31\!\cdots\!68 \)
|
| $83$ |
\( T^{6} + \cdots - 19\!\cdots\!08 \)
|
| $89$ |
\( T^{6} + \cdots + 18\!\cdots\!40 \)
|
| $97$ |
\( T^{6} + \cdots + 19\!\cdots\!28 \)
|
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