Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,68,Mod(1,3)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 68, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.1");
S:= CuspForms(chi, 68);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 68 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.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: | \(85.2871055790\) |
| 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: |
\( x^{5} - x^{4} + \cdots - 17\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{40}\cdot 3^{20}\cdot 5^{3}\cdot 7^{2}\cdot 11\cdot 17 \) |
| 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
\( x^{5} - x^{4} + \cdots - 17\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 48\nu - 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2304\nu^{2} + 189172740336\nu - 174549850023918035453 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 125631 \nu^{4} - 20285201178639 \nu^{3} + \cdots + 24\!\cdots\!00 ) / 20\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 425331 \nu^{4} + 43714584729981 \nu^{3} + \cdots + 98\!\cdots\!80 ) / 403516000706560 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 10 ) / 48 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 3941098757\beta _1 + 174549849984507047883 ) / 2304 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 24816 \beta_{4} - 420080 \beta_{3} - 197516269 \beta_{2} + \cdots - 42\!\cdots\!75 ) / 6912 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 12020859957712 \beta_{4} + 129524695496240 \beta_{3} + \cdots + 24\!\cdots\!67 ) / 20736 \)
|
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 |
|
−2.13031e10 | 5.55906e15 | 3.06250e20 | −4.34254e22 | −1.18425e26 | 7.31825e27 | −3.38029e30 | 3.09032e31 | 9.25096e32 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.03939e10 | 5.55906e15 | −3.95417e19 | 1.67944e23 | −5.77801e25 | −3.85597e28 | 1.94485e30 | 3.09032e31 | −1.74559e33 | ||||||||||||||||||||||||||||||||||
| 1.3 | −5.77536e9 | 5.55906e15 | −1.14219e20 | 1.87613e23 | −3.21056e25 | 3.73645e28 | 1.51195e30 | 3.09032e31 | −1.08353e33 | ||||||||||||||||||||||||||||||||||
| 1.4 | 3.35246e9 | 5.55906e15 | −1.36335e20 | −3.80674e23 | 1.86366e25 | 2.28291e27 | −9.51794e29 | 3.09032e31 | −1.27619e33 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.78647e10 | 5.55906e15 | 1.71572e20 | −1.00461e22 | 9.93107e25 | −5.50883e27 | 4.28720e29 | 3.09032e31 | −1.79470e32 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
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 | 3.68.a.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 9.68.a.b | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.68.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 9.68.a.b | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 16255223088 T_{2}^{4} + \cdots + 76\!\cdots\!96 \)
acting on \(S_{68}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots + 76\!\cdots\!96 \)
|
| $3$ |
\( (T - 55\!\cdots\!23)^{5} \)
|
| $5$ |
\( T^{5} + \cdots + 52\!\cdots\!00 \)
|
| $7$ |
\( T^{5} + \cdots - 13\!\cdots\!00 \)
|
| $11$ |
\( T^{5} + \cdots + 58\!\cdots\!32 \)
|
| $13$ |
\( T^{5} + \cdots - 12\!\cdots\!28 \)
|
| $17$ |
\( T^{5} + \cdots - 14\!\cdots\!96 \)
|
| $19$ |
\( T^{5} + \cdots - 19\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 34\!\cdots\!00 \)
|
| $29$ |
\( T^{5} + \cdots + 60\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 21\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 32\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots + 39\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots + 50\!\cdots\!08 \)
|
| $47$ |
\( T^{5} + \cdots - 78\!\cdots\!16 \)
|
| $53$ |
\( T^{5} + \cdots + 13\!\cdots\!00 \)
|
| $59$ |
\( T^{5} + \cdots + 72\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 10\!\cdots\!28 \)
|
| $67$ |
\( T^{5} + \cdots - 97\!\cdots\!16 \)
|
| $71$ |
\( T^{5} + \cdots - 65\!\cdots\!52 \)
|
| $73$ |
\( T^{5} + \cdots - 18\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots - 11\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 74\!\cdots\!48 \)
|
| $89$ |
\( T^{5} + \cdots - 77\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots + 30\!\cdots\!44 \)
|
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