Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [74,6,Mod(1,74)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(74, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("74.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 74 = 2 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 74.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: | \(11.8684026662\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 870x^{3} - 2235x^{2} + 121361x + 481504 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 870x^{3} - 2235x^{2} + 121361x + 481504 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 26\nu^{4} + 287\nu^{3} - 23978\nu^{2} - 192751\nu + 2440911 ) / 41713 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 148\nu^{4} - 1575\nu^{3} - 75525\nu^{2} + 487896\nu - 1722289 ) / 125139 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 61\nu^{4} - 931\nu^{3} - 46630\nu^{2} + 444606\nu + 5176462 ) / 41713 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{4} + 3\beta_{3} - \beta_{2} + 5\beta _1 + 348 \)
|
| \(\nu^{3}\) | \(=\) |
\( -38\beta_{4} + 18\beta_{3} + 55\beta_{2} + 589\beta _1 + 1745 \)
|
| \(\nu^{4}\) | \(=\) |
\( -1425\beta_{4} + 2568\beta_{3} + 75\beta_{2} + 5523\beta _1 + 207793 \)
|
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 |
|
4.00000 | −24.3792 | 16.0000 | 44.9778 | −97.5168 | −145.948 | 64.0000 | 351.346 | 179.911 | |||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −9.45038 | 16.0000 | −51.4877 | −37.8015 | 212.238 | 64.0000 | −153.690 | −205.951 | ||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 8.15018 | 16.0000 | 86.4865 | 32.6007 | 72.3887 | 64.0000 | −176.574 | 345.946 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 16.6478 | 16.0000 | 46.0360 | 66.5913 | 16.5452 | 64.0000 | 34.1497 | 184.144 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | 28.0316 | 16.0000 | −31.0127 | 112.126 | 14.7764 | 64.0000 | 542.770 | −124.051 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(37\) | \(1\) |
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 | 74.6.a.e | ✓ | 5 |
| 3.b | odd | 2 | 1 | 666.6.a.l | 5 | ||
| 4.b | odd | 2 | 1 | 592.6.a.e | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 74.6.a.e | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 592.6.a.e | 5 | 4.b | odd | 2 | 1 | ||
| 666.6.a.l | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 19T_{3}^{4} - 726T_{3}^{3} + 12131T_{3}^{2} + 62745T_{3} - 876276 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(74))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{5} \)
|
| $3$ |
\( T^{5} - 19 T^{4} + \cdots - 876276 \)
|
| $5$ |
\( T^{5} - 95 T^{4} + \cdots - 285948168 \)
|
| $7$ |
\( T^{5} - 170 T^{4} + \cdots + 548191248 \)
|
| $11$ |
\( T^{5} + \cdots + 18936143228 \)
|
| $13$ |
\( T^{5} + \cdots + 139399737289736 \)
|
| $17$ |
\( T^{5} + \cdots - 230256902474496 \)
|
| $19$ |
\( T^{5} + \cdots - 25\!\cdots\!32 \)
|
| $23$ |
\( T^{5} + \cdots - 27\!\cdots\!32 \)
|
| $29$ |
\( T^{5} + \cdots - 13\!\cdots\!84 \)
|
| $31$ |
\( T^{5} + \cdots - 30\!\cdots\!20 \)
|
| $37$ |
\( (T + 1369)^{5} \)
|
| $41$ |
\( T^{5} + \cdots + 25\!\cdots\!98 \)
|
| $43$ |
\( T^{5} + \cdots + 16\!\cdots\!72 \)
|
| $47$ |
\( T^{5} + \cdots - 16\!\cdots\!76 \)
|
| $53$ |
\( T^{5} + \cdots - 25\!\cdots\!68 \)
|
| $59$ |
\( T^{5} + \cdots + 33\!\cdots\!80 \)
|
| $61$ |
\( T^{5} + \cdots + 15\!\cdots\!96 \)
|
| $67$ |
\( T^{5} + \cdots - 17\!\cdots\!00 \)
|
| $71$ |
\( T^{5} + \cdots + 17\!\cdots\!32 \)
|
| $73$ |
\( T^{5} + \cdots + 56\!\cdots\!26 \)
|
| $79$ |
\( T^{5} + \cdots - 95\!\cdots\!84 \)
|
| $83$ |
\( T^{5} + \cdots - 11\!\cdots\!44 \)
|
| $89$ |
\( T^{5} + \cdots + 85\!\cdots\!48 \)
|
| $97$ |
\( T^{5} + \cdots + 19\!\cdots\!52 \)
|
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