Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,82,Mod(1,1)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 82, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 82);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 82 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1.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: | \(41.5501285538\) |
| 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: |
\( x^{6} - 3 x^{5} + \cdots - 27\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{58}\cdot 3^{26}\cdot 5^{7}\cdot 7^{3}\cdot 13 \) |
| 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
\( x^{6} - 3 x^{5} + \cdots - 27\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 144\nu - 72 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 78099962708907 \nu^{5} + \cdots - 17\!\cdots\!68 ) / 69\!\cdots\!88 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 91\!\cdots\!69 \nu^{5} + \cdots - 56\!\cdots\!32 ) / 17\!\cdots\!72 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 60\!\cdots\!79 \nu^{5} + \cdots + 69\!\cdots\!88 ) / 19\!\cdots\!28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 28\!\cdots\!51 \nu^{5} + \cdots - 70\!\cdots\!64 ) / 99\!\cdots\!64 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 72 ) / 144 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 4668\beta_{2} + 176829862160\beta _1 + 3160943384653180063456128 ) / 20736 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 2727 \beta_{5} + 2307980 \beta_{4} + 7915256204 \beta_{3} + 763450743525280 \beta_{2} + \cdots + 17\!\cdots\!36 ) / 93312 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 52520409734386 \beta_{5} + \cdots + 22\!\cdots\!76 ) / 559872 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 18\!\cdots\!83 \beta_{5} + \cdots + 14\!\cdots\!84 ) / 139968 \)
|
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 |
|
−2.75084e12 | −2.21125e19 | 5.14927e24 | 2.52982e28 | 6.08279e31 | −2.62798e34 | −7.51371e36 | 4.55346e37 | −6.95912e40 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.69157e12 | 1.74109e19 | 4.43572e23 | −9.34342e27 | −2.94518e31 | 1.25454e34 | 3.33964e36 | −1.40288e38 | 1.58051e40 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −3.79502e11 | −3.74133e19 | −2.27383e24 | −3.57322e28 | 1.41984e31 | −3.75122e33 | 1.78050e36 | 9.56330e38 | 1.35605e40 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 3.81039e11 | −1.66183e18 | −2.27266e24 | 2.26484e28 | −6.33221e29 | −1.01850e33 | −1.78727e36 | −4.40665e38 | 8.62992e39 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.56393e12 | 3.91464e19 | 2.80371e22 | −1.92800e28 | 6.12224e31 | −2.38829e34 | −3.73751e36 | 1.08902e39 | −3.01526e40 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.41607e12 | −1.10180e19 | 3.41956e24 | −1.95525e27 | −2.66202e31 | 1.09563e34 | 2.42020e36 | −3.22031e38 | −4.72402e39 | |||||||||||||||||||||||||||||||||||||
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 | 1.82.a.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 9.82.a.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.82.a.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 9.82.a.b | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{82}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 25\!\cdots\!84 \)
|
| $3$ |
\( T^{6} + \cdots + 10\!\cdots\!44 \)
|
| $5$ |
\( T^{6} + \cdots + 72\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 32\!\cdots\!56 \)
|
| $11$ |
\( T^{6} + \cdots - 23\!\cdots\!16 \)
|
| $13$ |
\( T^{6} + \cdots - 34\!\cdots\!56 \)
|
| $17$ |
\( T^{6} + \cdots - 46\!\cdots\!64 \)
|
| $19$ |
\( T^{6} + \cdots + 51\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 34\!\cdots\!56 \)
|
| $29$ |
\( T^{6} + \cdots - 15\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 12\!\cdots\!24 \)
|
| $37$ |
\( T^{6} + \cdots + 52\!\cdots\!96 \)
|
| $41$ |
\( T^{6} + \cdots - 12\!\cdots\!56 \)
|
| $43$ |
\( T^{6} + \cdots - 51\!\cdots\!56 \)
|
| $47$ |
\( T^{6} + \cdots + 18\!\cdots\!76 \)
|
| $53$ |
\( T^{6} + \cdots - 32\!\cdots\!56 \)
|
| $59$ |
\( T^{6} + \cdots - 29\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 59\!\cdots\!84 \)
|
| $67$ |
\( T^{6} + \cdots - 34\!\cdots\!64 \)
|
| $71$ |
\( T^{6} + \cdots + 16\!\cdots\!04 \)
|
| $73$ |
\( T^{6} + \cdots - 70\!\cdots\!56 \)
|
| $79$ |
\( T^{6} + \cdots - 47\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 27\!\cdots\!56 \)
|
| $89$ |
\( T^{6} + \cdots + 27\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 25\!\cdots\!24 \)
|
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