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: | \(0\) |
| 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: |
\( x^{6} - 3 x^{5} + \cdots - 40\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{46}\cdot 3^{33}\cdot 5^{5}\cdot 7^{3}\cdot 11\cdot 17\cdot 23^{2} \) |
| 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
\( x^{6} - 3 x^{5} + \cdots - 40\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 18\nu - 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( 324\nu^{2} - 27104502078\nu - 948783914631632447703 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 90125876775213 \nu^{5} + \cdots - 38\!\cdots\!00 ) / 29\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 84\!\cdots\!09 \nu^{5} + \cdots + 20\!\cdots\!00 ) / 36\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 22\!\cdots\!09 \nu^{5} + \cdots - 23\!\cdots\!00 ) / 14\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 9 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 1505805671\beta _1 + 948783914645184698742 ) / 324 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} + 73 \beta_{4} - 5970698 \beta_{3} + 4958890590 \beta_{2} + \cdots + 14\!\cdots\!76 ) / 5832 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 62888121261 \beta_{5} + 879389206955 \beta_{4} + \cdots + 67\!\cdots\!64 ) / 52488 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 81\!\cdots\!81 \beta_{5} + \cdots + 14\!\cdots\!96 ) / 472392 \)
|
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 |
|
−3.96695e10 | 1.66772e16 | 9.83375e20 | 1.65867e24 | −6.61576e26 | 4.47127e28 | −1.55933e31 | 2.78128e32 | −6.57985e34 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.79627e10 | 1.66772e16 | 1.91618e20 | −1.83280e24 | −4.66339e26 | −3.84348e28 | 1.11481e31 | 2.78128e32 | 5.12500e34 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −5.14931e9 | 1.66772e16 | −5.63780e20 | 5.32454e23 | −8.58759e25 | 1.60190e29 | 5.94269e30 | 2.78128e32 | −2.74177e33 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.24211e10 | 1.66772e16 | −4.36013e20 | 1.87674e23 | 2.07148e26 | −1.62329e29 | −1.27479e31 | 2.78128e32 | 2.33111e33 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.71943e10 | 1.66772e16 | 7.93117e20 | 1.89718e24 | 6.20295e26 | 2.07605e29 | 7.54379e30 | 2.78128e32 | 7.05641e34 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.28672e10 | 1.66772e16 | 1.24730e21 | −1.81509e24 | 7.14904e26 | −1.63830e29 | 2.81639e31 | 2.78128e32 | −7.78079e34 | |||||||||||||||||||||||||||||||||||||
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.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 9.70.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.70.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 9.70.a.c | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 19700962938 T_{2}^{5} + \cdots - 11\!\cdots\!68 \)
acting on \(S_{70}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 11\!\cdots\!68 \)
|
| $3$ |
\( (T - 16\!\cdots\!69)^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots - 15\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots - 70\!\cdots\!92 \)
|
| $13$ |
\( T^{6} + \cdots + 78\!\cdots\!92 \)
|
| $17$ |
\( T^{6} + \cdots + 10\!\cdots\!64 \)
|
| $19$ |
\( T^{6} + \cdots - 49\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 14\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots - 19\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 51\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 19\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots - 20\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots - 44\!\cdots\!96 \)
|
| $47$ |
\( T^{6} + \cdots + 77\!\cdots\!76 \)
|
| $53$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 38\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 22\!\cdots\!44 \)
|
| $67$ |
\( T^{6} + \cdots - 11\!\cdots\!96 \)
|
| $71$ |
\( T^{6} + \cdots - 47\!\cdots\!56 \)
|
| $73$ |
\( T^{6} + \cdots + 33\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots + 58\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 19\!\cdots\!52 \)
|
| $89$ |
\( T^{6} + \cdots - 23\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 84\!\cdots\!84 \)
|
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