Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [15,28,Mod(1,15)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 28, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15.1");
S:= CuspForms(chi, 28);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 15 = 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 28 \) |
| Character orbit: | \([\chi]\) | \(=\) | 15.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: | \(69.2783362257\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 103897041x^{2} - 138039741995x + 1308599464667300 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 103897041x^{2} - 138039741995x + 1308599464667300 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 4750\nu^{2} - 69238059\nu + 143165051996 ) / 3608 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 9682\nu^{2} - 98015467\nu - 606548805116 ) / 3608 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 3988\beta _1 + 207792089 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2375\beta_{3} + 4841\beta_{2} + 78709559\beta _1 + 207176107383 ) / 2 \)
|
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 |
|
−21407.0 | 1.59432e6 | 3.24041e8 | −1.22070e9 | −3.41296e10 | 3.47604e11 | −4.06353e12 | 2.54187e12 | 2.61316e13 | ||||||||||||||||||||||||||||||
| 1.2 | −15067.3 | 1.59432e6 | 9.28044e7 | −1.22070e9 | −2.40221e10 | −3.71054e11 | 6.23985e11 | 2.54187e12 | 1.83926e13 | |||||||||||||||||||||||||||||||
| 1.3 | 1189.02 | 1.59432e6 | −1.32804e8 | −1.22070e9 | 1.89568e9 | 1.49650e11 | −3.17494e11 | 2.54187e12 | −1.45144e12 | |||||||||||||||||||||||||||||||
| 1.4 | 15551.2 | 1.59432e6 | 1.07622e8 | −1.22070e9 | 2.47936e10 | 2.25744e11 | −4.13594e11 | 2.54187e12 | −1.89834e13 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
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 | 15.28.a.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 45.28.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.28.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 45.28.a.c | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 19734T_{2}^{3} - 269551632T_{2}^{2} - 4725027794432T_{2} + 5964067963797504 \)
acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(15))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 59\!\cdots\!04 \)
|
| $3$ |
\( (T - 1594323)^{4} \)
|
| $5$ |
\( (T + 1220703125)^{4} \)
|
| $7$ |
\( T^{4} + \cdots - 43\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots - 11\!\cdots\!76 \)
|
| $13$ |
\( T^{4} + \cdots + 20\!\cdots\!64 \)
|
| $17$ |
\( T^{4} + \cdots + 19\!\cdots\!76 \)
|
| $19$ |
\( T^{4} + \cdots - 20\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 40\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 38\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 86\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots - 17\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 88\!\cdots\!84 \)
|
| $47$ |
\( T^{4} + \cdots - 30\!\cdots\!04 \)
|
| $53$ |
\( T^{4} + \cdots - 81\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots - 32\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 80\!\cdots\!56 \)
|
| $67$ |
\( T^{4} + \cdots - 21\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots - 91\!\cdots\!44 \)
|
| $73$ |
\( T^{4} + \cdots + 36\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 31\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 91\!\cdots\!56 \)
|
| $89$ |
\( T^{4} + \cdots + 78\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 48\!\cdots\!64 \)
|
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