Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2,68,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, 68, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2.1");
S:= CuspForms(chi, 68);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 68 \) |
| 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: | \(56.8580703860\) |
| 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: |
\( x^{3} - x^{2} - 65197845713131137604825x + 6400681392725274738052865585717152 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{5}\cdot 5^{3}\cdot 11\cdot 17 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 65197845713131137604825x + 6400681392725274738052865585717152 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 256\nu^{2} + 34668473710848\nu - 11127099001719270309127168 ) / 64488825 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 33719296\nu^{2} + 5099862403751175168\nu - 1465616972110633231277145100288 ) / 21496275 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 395148\beta _1 + 8272281600 ) / 24816844800 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -45141241811\beta_{2} + 19921337514653028\beta _1 + 359556626301582391996977369907200 ) / 8272281600 \)
|
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 |
|
−8.58993e9 | −1.55040e16 | 7.37870e19 | −3.86336e23 | 1.33179e26 | 3.00596e28 | −6.33825e29 | 1.47665e32 | 3.31861e33 | |||||||||||||||||||||||||||
| 1.2 | −8.58993e9 | −1.95611e15 | 7.37870e19 | 3.66441e22 | 1.68028e25 | −2.91798e28 | −6.33825e29 | −8.88831e31 | −3.14770e32 | ||||||||||||||||||||||||||||
| 1.3 | −8.58993e9 | 1.22345e16 | 7.37870e19 | 8.49318e22 | −1.05093e26 | 2.05773e28 | −6.33825e29 | 5.69731e31 | −7.29559e32 | ||||||||||||||||||||||||||||
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 | 2.68.a.a | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.68.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + \cdots - 37\!\cdots\!52 \)
acting on \(S_{68}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8589934592)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 37\!\cdots\!52 \)
|
| $5$ |
\( T^{3} + \cdots + 12\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots + 18\!\cdots\!36 \)
|
| $11$ |
\( T^{3} + \cdots - 56\!\cdots\!08 \)
|
| $13$ |
\( T^{3} + \cdots - 39\!\cdots\!92 \)
|
| $17$ |
\( T^{3} + \cdots - 69\!\cdots\!24 \)
|
| $19$ |
\( T^{3} + \cdots - 18\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 41\!\cdots\!48 \)
|
| $29$ |
\( T^{3} + \cdots + 99\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 33\!\cdots\!52 \)
|
| $37$ |
\( T^{3} + \cdots + 11\!\cdots\!36 \)
|
| $41$ |
\( T^{3} + \cdots - 76\!\cdots\!48 \)
|
| $43$ |
\( T^{3} + \cdots - 36\!\cdots\!32 \)
|
| $47$ |
\( T^{3} + \cdots - 13\!\cdots\!44 \)
|
| $53$ |
\( T^{3} + \cdots - 94\!\cdots\!92 \)
|
| $59$ |
\( T^{3} + \cdots + 78\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 64\!\cdots\!88 \)
|
| $67$ |
\( T^{3} + \cdots + 72\!\cdots\!56 \)
|
| $71$ |
\( T^{3} + \cdots - 17\!\cdots\!28 \)
|
| $73$ |
\( T^{3} + \cdots + 13\!\cdots\!08 \)
|
| $79$ |
\( T^{3} + \cdots - 65\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 27\!\cdots\!88 \)
|
| $89$ |
\( T^{3} + \cdots + 73\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 14\!\cdots\!64 \)
|
| show more | |
| show less |