Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,34,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, 34, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.1");
S:= CuspForms(chi, 34);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 34 \) |
| 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: | \(20.6948486643\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 35900150x + 10469144400 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{6}\cdot 5\cdot 7\cdot 11 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 35900150x + 10469144400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 1429\nu - 23933910 ) / 195 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5046\nu^{2} + 57654066\nu + 120748888260 ) / 65 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 15138\beta _1 + 332640 ) / 997920 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -1429\beta_{2} + 172962198\beta _1 + 23883652124640 ) / 997920 \)
|
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 |
|
−81270.9 | 4.30467e7 | −1.98497e9 | 5.17904e11 | −3.49845e12 | −3.34870e13 | 8.59432e14 | 1.85302e15 | −4.20905e16 | |||||||||||||||||||||||||||
| 1.2 | −11418.8 | 4.30467e7 | −8.45955e9 | −6.77881e11 | −4.91541e11 | 5.88597e13 | 1.94684e14 | 1.85302e15 | 7.74057e15 | ||||||||||||||||||||||||||||
| 1.3 | 133892. | 4.30467e7 | 9.33705e9 | 2.11239e11 | 5.76360e12 | 5.07411e13 | 1.00033e14 | 1.85302e15 | 2.82832e16 | ||||||||||||||||||||||||||||
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 | 3.34.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 9.34.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.34.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 9.34.a.d | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 41202T_{2}^{2} - 11482365600T_{2} - 124253441040384 \)
acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots - 124253441040384 \)
|
| $3$ |
\( (T - 43046721)^{3} \)
|
| $5$ |
\( T^{3} + \cdots + 74\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots + 10\!\cdots\!20 \)
|
| $11$ |
\( T^{3} + \cdots - 15\!\cdots\!64 \)
|
| $13$ |
\( T^{3} + \cdots + 21\!\cdots\!88 \)
|
| $17$ |
\( T^{3} + \cdots + 10\!\cdots\!68 \)
|
| $19$ |
\( T^{3} + \cdots + 32\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 12\!\cdots\!20 \)
|
| $29$ |
\( T^{3} + \cdots + 17\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 51\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 19\!\cdots\!60 \)
|
| $41$ |
\( T^{3} + \cdots + 41\!\cdots\!20 \)
|
| $43$ |
\( T^{3} + \cdots + 29\!\cdots\!44 \)
|
| $47$ |
\( T^{3} + \cdots + 35\!\cdots\!52 \)
|
| $53$ |
\( T^{3} + \cdots - 28\!\cdots\!80 \)
|
| $59$ |
\( T^{3} + \cdots + 21\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 37\!\cdots\!08 \)
|
| $67$ |
\( T^{3} + \cdots + 28\!\cdots\!68 \)
|
| $71$ |
\( T^{3} + \cdots + 98\!\cdots\!52 \)
|
| $73$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots - 21\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 38\!\cdots\!08 \)
|
| $89$ |
\( T^{3} + \cdots - 29\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 34\!\cdots\!68 \)
|
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