Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,28,Mod(1,5)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 28, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.1");
S:= CuspForms(chi, 28);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 28 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5.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: | \(23.0927787419\) |
| 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: |
\( x^{5} - 2x^{4} - 105406182x^{3} - 8285617904x^{2} + 1593173725628800x - 1939393055148057600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{16}\cdot 3^{5}\cdot 5^{4}\cdot 7 \) |
| 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
\( x^{5} - 2x^{4} - 105406182x^{3} - 8285617904x^{2} + 1593173725628800x - 1939393055148057600 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -119\nu^{4} - 488562\nu^{3} + 12666282090\nu^{2} + 43203932098384\nu - 154378926928912992 ) / 21119977584 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 119\nu^{4} + 488562\nu^{3} - 7386287694\nu^{2} - 43826971437112\nu - 68238456632840496 ) / 1319998599 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 85711 \nu^{4} - 3065694 \nu^{3} - 8248407182202 \nu^{2} + \cdots + 76\!\cdots\!60 ) / 147839843088 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 16\beta_{2} + 236\beta _1 + 168649948 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -833\beta_{4} + 1232\beta_{3} - 65999\beta_{2} + 75196676\beta _1 + 10210729427 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3419934\beta_{4} + 48161619\beta_{3} + 767520210\beta_{2} + 66893403068\beta _1 + 6339333475629194 ) / 2 \)
|
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 |
|
−15097.3 | 540259. | 9.37113e7 | −1.22070e9 | −8.15645e9 | 2.87369e11 | 6.11538e11 | −7.33372e12 | 1.84293e13 | |||||||||||||||||||||||||||||||||
| 1.2 | −2338.06 | 4.67850e6 | −1.28751e8 | −1.22070e9 | −1.09386e10 | −1.25721e10 | 6.14836e11 | 1.42627e13 | 2.85407e12 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1165.24 | −2.41575e6 | −1.32860e8 | −1.22070e9 | −2.81493e9 | −4.04373e11 | −3.11211e11 | −1.78977e12 | −1.42242e12 | ||||||||||||||||||||||||||||||||||
| 1.4 | 14024.4 | −1.90136e6 | 6.24671e7 | −1.22070e9 | −2.66655e10 | 3.73002e11 | −1.00626e12 | −4.01043e12 | −1.71197e13 | ||||||||||||||||||||||||||||||||||
| 1.5 | 22161.7 | 3.96903e6 | 3.56923e8 | −1.22070e9 | 8.79605e10 | −8.77786e10 | 4.93553e12 | 8.12762e12 | −2.70528e13 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
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 | 5.28.a.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 45.28.a.d | 5 | ||
| 5.b | even | 2 | 1 | 25.28.a.c | 5 | ||
| 5.c | odd | 4 | 2 | 25.28.b.c | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.28.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 25.28.a.c | 5 | 5.b | even | 2 | 1 | ||
| 25.28.b.c | 10 | 5.c | odd | 4 | 2 | ||
| 45.28.a.d | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 19916T_{2}^{4} - 262965912T_{2}^{3} + 4473573947392T_{2}^{2} + 6144845117710336T_{2} - 12783803780642635776 \)
acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(5))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots - 12\!\cdots\!76 \)
|
| $3$ |
\( T^{5} + \cdots - 46\!\cdots\!32 \)
|
| $5$ |
\( (T + 1220703125)^{5} \)
|
| $7$ |
\( T^{5} + \cdots + 47\!\cdots\!24 \)
|
| $11$ |
\( T^{5} + \cdots + 40\!\cdots\!68 \)
|
| $13$ |
\( T^{5} + \cdots - 71\!\cdots\!32 \)
|
| $17$ |
\( T^{5} + \cdots - 63\!\cdots\!76 \)
|
| $19$ |
\( T^{5} + \cdots - 26\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 33\!\cdots\!68 \)
|
| $29$ |
\( T^{5} + \cdots + 22\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 56\!\cdots\!68 \)
|
| $37$ |
\( T^{5} + \cdots + 10\!\cdots\!24 \)
|
| $41$ |
\( T^{5} + \cdots + 10\!\cdots\!68 \)
|
| $43$ |
\( T^{5} + \cdots - 14\!\cdots\!32 \)
|
| $47$ |
\( T^{5} + \cdots - 49\!\cdots\!76 \)
|
| $53$ |
\( T^{5} + \cdots + 18\!\cdots\!68 \)
|
| $59$ |
\( T^{5} + \cdots - 27\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 27\!\cdots\!68 \)
|
| $67$ |
\( T^{5} + \cdots - 71\!\cdots\!76 \)
|
| $71$ |
\( T^{5} + \cdots - 66\!\cdots\!32 \)
|
| $73$ |
\( T^{5} + \cdots + 20\!\cdots\!68 \)
|
| $79$ |
\( T^{5} + \cdots - 36\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 20\!\cdots\!32 \)
|
| $89$ |
\( T^{5} + \cdots + 35\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 29\!\cdots\!76 \)
|
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