Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,8,Mod(1,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, 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("69.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 69.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: | \(21.5545667584\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 455x^{3} - 474x^{2} + 42284x + 127016 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 455x^{3} - 474x^{2} + 42284x + 127016 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 142\nu^{3} + 467\nu^{2} - 37904\nu - 76396 ) / 1552 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\nu^{4} - 10\nu^{3} - 3585\nu^{2} + 2560\nu + 117124 ) / 1552 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{4} + 10\nu^{3} + 5137\nu^{2} - 5664\nu - 399588 ) / 1552 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2\beta _1 + 182 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + 11\beta_{2} + 265\beta _1 + 284 \)
|
| \(\nu^{4}\) | \(=\) |
\( 325\beta_{4} + 467\beta_{3} + 10\beta_{2} + 660\beta _1 + 48926 \)
|
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 |
|
−17.8260 | −27.0000 | 189.767 | −31.0428 | 481.302 | −1247.35 | −1101.05 | 729.000 | 553.369 | |||||||||||||||||||||||||||||||||
| 1.2 | −9.64907 | −27.0000 | −34.8955 | −306.939 | 260.525 | 733.069 | 1571.79 | 729.000 | 2961.68 | ||||||||||||||||||||||||||||||||||
| 1.3 | −3.24502 | −27.0000 | −117.470 | 168.083 | 87.6156 | 149.817 | 796.555 | 729.000 | −545.434 | ||||||||||||||||||||||||||||||||||
| 1.4 | 12.4672 | −27.0000 | 27.4323 | −40.8788 | −336.616 | 1126.70 | −1253.80 | 729.000 | −509.647 | ||||||||||||||||||||||||||||||||||
| 1.5 | 18.2528 | −27.0000 | 205.166 | −55.2224 | −492.827 | −1258.24 | 1408.51 | 729.000 | −1007.97 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(23\) | \(-1\) |
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 | 69.8.a.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 207.8.a.a | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.8.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 207.8.a.a | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 455T_{2}^{3} - 474T_{2}^{2} + 42284T_{2} + 127016 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(69))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 455 T^{3} + \cdots + 127016 \)
|
| $3$ |
\( (T + 27)^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 3615360000 \)
|
| $7$ |
\( T^{5} + \cdots - 194204974150720 \)
|
| $11$ |
\( T^{5} + \cdots + 154691619430400 \)
|
| $13$ |
\( T^{5} + \cdots + 16\!\cdots\!84 \)
|
| $17$ |
\( T^{5} + \cdots + 11\!\cdots\!80 \)
|
| $19$ |
\( T^{5} + \cdots - 14\!\cdots\!36 \)
|
| $23$ |
\( (T - 12167)^{5} \)
|
| $29$ |
\( T^{5} + \cdots + 24\!\cdots\!88 \)
|
| $31$ |
\( T^{5} + \cdots - 23\!\cdots\!28 \)
|
| $37$ |
\( T^{5} + \cdots + 82\!\cdots\!20 \)
|
| $41$ |
\( T^{5} + \cdots + 16\!\cdots\!52 \)
|
| $43$ |
\( T^{5} + \cdots + 66\!\cdots\!00 \)
|
| $47$ |
\( T^{5} + \cdots - 24\!\cdots\!00 \)
|
| $53$ |
\( T^{5} + \cdots + 10\!\cdots\!60 \)
|
| $59$ |
\( T^{5} + \cdots + 75\!\cdots\!64 \)
|
| $61$ |
\( T^{5} + \cdots - 14\!\cdots\!84 \)
|
| $67$ |
\( T^{5} + \cdots - 79\!\cdots\!40 \)
|
| $71$ |
\( T^{5} + \cdots + 49\!\cdots\!52 \)
|
| $73$ |
\( T^{5} + \cdots + 11\!\cdots\!64 \)
|
| $79$ |
\( T^{5} + \cdots - 36\!\cdots\!20 \)
|
| $83$ |
\( T^{5} + \cdots + 18\!\cdots\!20 \)
|
| $89$ |
\( T^{5} + \cdots - 25\!\cdots\!96 \)
|
| $97$ |
\( T^{5} + \cdots + 11\!\cdots\!20 \)
|
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