Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2,78,Mod(1,2)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 78, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2.1");
S:= CuspForms(chi, 78);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 78 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.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: | \(75.0958835633\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{3} - x^{2} - 1232026544797605254115902x - 446829259233131687750093860637314080 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{11}\cdot 5^{2}\cdot 7\cdot 11\cdot 13\cdot 19 \) |
| 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
\( x^{3} - x^{2} - 1232026544797605254115902x - 446829259233131687750093860637314080 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2799360\nu - 933120 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 163840\nu^{2} - 89131817719896320\nu - 134570152733063385950326344960 ) / 24707 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 933120 ) / 2799360 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 54034209\beta_{2} + 69634232593669\beta _1 + 294304924027274602168481520844800 ) / 358318080 \)
|
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 |
|
−2.74878e11 | −3.17902e18 | 7.55579e22 | 1.48630e27 | 8.73841e29 | −9.37214e31 | −2.07692e34 | 4.63174e36 | −4.08550e38 | |||||||||||||||||||||||||||
| 1.2 | −2.74878e11 | −2.03209e18 | 7.55579e22 | −1.03387e27 | 5.58577e29 | 1.46380e32 | −2.07692e34 | −1.34501e36 | 2.84189e38 | ||||||||||||||||||||||||||||
| 1.3 | −2.74878e11 | 2.68382e18 | 7.55579e22 | −1.42444e26 | −7.37722e29 | −2.85041e32 | −2.07692e34 | 1.72848e36 | 3.91547e37 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
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 | 2.78.a.a | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.78.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + \cdots - 17\!\cdots\!64 \)
acting on \(S_{78}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 274877906944)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 17\!\cdots\!64 \)
|
| $5$ |
\( T^{3} + \cdots - 21\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots - 39\!\cdots\!52 \)
|
| $11$ |
\( T^{3} + \cdots + 21\!\cdots\!92 \)
|
| $13$ |
\( T^{3} + \cdots - 26\!\cdots\!44 \)
|
| $17$ |
\( T^{3} + \cdots - 77\!\cdots\!32 \)
|
| $19$ |
\( T^{3} + \cdots + 31\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 22\!\cdots\!36 \)
|
| $29$ |
\( T^{3} + \cdots + 12\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 19\!\cdots\!52 \)
|
| $37$ |
\( T^{3} + \cdots + 83\!\cdots\!48 \)
|
| $41$ |
\( T^{3} + \cdots + 10\!\cdots\!52 \)
|
| $43$ |
\( T^{3} + \cdots + 59\!\cdots\!76 \)
|
| $47$ |
\( T^{3} + \cdots - 33\!\cdots\!92 \)
|
| $53$ |
\( T^{3} + \cdots - 30\!\cdots\!44 \)
|
| $59$ |
\( T^{3} + \cdots - 52\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 18\!\cdots\!88 \)
|
| $67$ |
\( T^{3} + \cdots - 36\!\cdots\!92 \)
|
| $71$ |
\( T^{3} + \cdots - 31\!\cdots\!28 \)
|
| $73$ |
\( T^{3} + \cdots - 19\!\cdots\!44 \)
|
| $79$ |
\( T^{3} + \cdots - 22\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 21\!\cdots\!84 \)
|
| $89$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 13\!\cdots\!52 \)
|
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