Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,70,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, 70, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.1");
S:= CuspForms(chi, 70);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 70 \) |
| 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: | \(90.4544859877\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{6} - 3 x^{5} + \cdots - 14\!\cdots\!28 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{51}\cdot 3^{33}\cdot 5^{6}\cdot 7^{3}\cdot 11\cdot 17\cdot 23^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3 x^{5} + \cdots - 14\!\cdots\!28 \)
:
| \(\beta_{1}\) | \(=\) |
\( 36\nu - 18 \)
|
| \(\beta_{2}\) | \(=\) |
\( 1296\nu^{2} - 220865387400\nu - 818810922898010992500 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 389268019957851 \nu^{5} + \cdots - 14\!\cdots\!12 ) / 20\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 69\!\cdots\!07 \nu^{5} + \cdots + 10\!\cdots\!16 ) / 15\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 51\!\cdots\!99 \nu^{5} + \cdots - 46\!\cdots\!12 ) / 20\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 18 ) / 36 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 6135149650\beta _1 + 818810923008443686200 ) / 1296 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 211 \beta_{5} - 65 \beta_{4} + 1223826 \beta_{3} + 7366410659 \beta_{2} + \cdots + 25\!\cdots\!68 ) / 23328 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 1691968526615 \beta_{5} - 9336047469 \beta_{4} + \cdots + 14\!\cdots\!12 ) / 209952 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 16\!\cdots\!77 \beta_{5} + \cdots + 28\!\cdots\!52 ) / 944784 \)
|
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 |
|
−3.51667e10 | −1.66772e16 | 6.46400e20 | 9.85410e23 | 5.86481e26 | −2.48063e29 | −1.97299e30 | 2.78128e32 | −3.46536e34 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.19993e10 | −1.66772e16 | 4.33658e20 | −1.19891e23 | 5.33658e26 | 2.02540e29 | 5.01228e30 | 2.78128e32 | 3.83642e33 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −6.58478e9 | −1.66772e16 | −5.46936e20 | −2.05454e24 | 1.09816e26 | −2.33794e29 | 7.48843e30 | 2.78128e32 | 1.35287e34 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 3.99182e9 | −1.66772e16 | −5.74361e20 | 2.45458e24 | −6.65724e25 | 3.62318e28 | −4.64911e30 | 2.78128e32 | 9.79827e33 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.39528e10 | −1.66772e16 | −1.65573e19 | −4.62338e23 | −3.99466e26 | 1.66345e28 | −1.45359e31 | 2.78128e32 | −1.10743e34 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.49367e10 | −1.66772e16 | 1.42901e21 | −2.66474e23 | −7.49418e26 | 6.09654e28 | 3.76892e31 | 2.78128e32 | −1.19745e34 | |||||||||||||||||||||||||||||||||||||
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 | 3.70.a.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 9.70.a.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.70.a.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 9.70.a.d | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 869363388 T_{2}^{5} + \cdots - 31\!\cdots\!48 \)
acting on \(S_{70}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 31\!\cdots\!48 \)
|
| $3$ |
\( (T + 16\!\cdots\!69)^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 73\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 43\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots + 12\!\cdots\!28 \)
|
| $13$ |
\( T^{6} + \cdots - 46\!\cdots\!68 \)
|
| $17$ |
\( T^{6} + \cdots + 10\!\cdots\!64 \)
|
| $19$ |
\( T^{6} + \cdots - 25\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 84\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 98\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 14\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots - 17\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots - 14\!\cdots\!96 \)
|
| $47$ |
\( T^{6} + \cdots - 33\!\cdots\!24 \)
|
| $53$ |
\( T^{6} + \cdots + 54\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 43\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 14\!\cdots\!64 \)
|
| $67$ |
\( T^{6} + \cdots + 87\!\cdots\!44 \)
|
| $71$ |
\( T^{6} + \cdots + 13\!\cdots\!04 \)
|
| $73$ |
\( T^{6} + \cdots + 34\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 20\!\cdots\!28 \)
|
| $89$ |
\( T^{6} + \cdots - 48\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 13\!\cdots\!04 \)
|
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