Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4,15,Mod(3,4)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4.3");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4 = 2^{2} \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(4.97315872608\) |
| 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: |
\( x^{6} - x^{5} + 88x^{4} - 1824x^{3} + 325632x^{2} + 21572352x + 982333440 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{30}\cdot 3^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} - x^{5} + 88x^{4} - 1824x^{3} + 325632x^{2} + 21572352x + 982333440 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 253\nu^{4} + 14692\nu^{3} - 116112\nu^{2} + 1274304\nu - 52022784 ) / 196608 \)
|
| \(\beta_{3}\) | \(=\) |
\( 16\nu^{2} - 64\nu + 477 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 119\nu^{4} + 596\nu^{3} - 12112\nu^{2} - 798016\nu - 30471680 ) / 2048 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 3\nu^{4} + 100\nu^{3} - 1424\nu^{2} + 291264\nu + 18836992 ) / 1024 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 16\beta _1 - 461 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{5} + 4\beta_{4} + 9\beta_{3} + 192\beta_{2} - 133\beta _1 + 13915 ) / 16 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 349\beta_{5} + 228\beta_{4} + 65\beta_{3} - 1344\beta_{2} + 611\beta _1 - 3413693 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 14837\beta_{5} - 1084\beta_{4} + 329\beta_{3} - 15168\beta_{2} - 1130805\beta _1 - 294363813 ) / 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−114.311 | − | 57.5931i | − | 2948.24i | 9750.07 | + | 13167.1i | −81680.5 | −169798. | + | 337016.i | 1.09921e6i | −356209. | − | 2.06668e6i | −3.90914e6 | 9.33699e6 | + | 4.70423e6i | |||||||||||||||||||||||||
| 3.2 | −114.311 | + | 57.5931i | 2948.24i | 9750.07 | − | 13167.1i | −81680.5 | −169798. | − | 337016.i | − | 1.09921e6i | −356209. | + | 2.06668e6i | −3.90914e6 | 9.33699e6 | − | 4.70423e6i | ||||||||||||||||||||||||||
| 3.3 | −32.2815 | − | 123.862i | 2369.41i | −14299.8 | + | 7996.93i | 113343. | 293481. | − | 76488.1i | 697667.i | 1.45214e6 | + | 1.51306e6i | −831139. | −3.65890e6 | − | 1.40390e7i | |||||||||||||||||||||||||||
| 3.4 | −32.2815 | + | 123.862i | − | 2369.41i | −14299.8 | − | 7996.93i | 113343. | 293481. | + | 76488.1i | − | 697667.i | 1.45214e6 | − | 1.51306e6i | −831139. | −3.65890e6 | + | 1.40390e7i | |||||||||||||||||||||||||
| 3.5 | 100.593 | − | 79.1526i | − | 1241.79i | 3853.74 | − | 15924.3i | −27633.0 | −98290.9 | − | 124915.i | − | 784634.i | −872793. | − | 1.90690e6i | 3.24093e6 | −2.77967e6 | + | 2.18722e6i | |||||||||||||||||||||||||
| 3.6 | 100.593 | + | 79.1526i | 1241.79i | 3853.74 | + | 15924.3i | −27633.0 | −98290.9 | + | 124915.i | 784634.i | −872793. | + | 1.90690e6i | 3.24093e6 | −2.77967e6 | − | 2.18722e6i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4.15.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 36.15.d.c | 6 | ||
| 4.b | odd | 2 | 1 | inner | 4.15.b.a | ✓ | 6 |
| 8.b | even | 2 | 1 | 64.15.c.d | 6 | ||
| 8.d | odd | 2 | 1 | 64.15.c.d | 6 | ||
| 12.b | even | 2 | 1 | 36.15.d.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4.15.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 4.15.b.a | ✓ | 6 | 4.b | odd | 2 | 1 | inner |
| 36.15.d.c | 6 | 3.b | odd | 2 | 1 | ||
| 36.15.d.c | 6 | 12.b | even | 2 | 1 | ||
| 64.15.c.d | 6 | 8.b | even | 2 | 1 | ||
| 64.15.c.d | 6 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(4, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 4398046511104 \)
|
| $3$ |
\( T^{6} + \cdots + 75\!\cdots\!40 \)
|
| $5$ |
\( (T^{3} + \cdots - 255824948369000)^{2} \)
|
| $7$ |
\( T^{6} + \cdots + 36\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots - 42\!\cdots\!00)^{2} \)
|
| $17$ |
\( (T^{3} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 33\!\cdots\!40 \)
|
| $29$ |
\( (T^{3} + \cdots - 36\!\cdots\!08)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $37$ |
\( (T^{3} + \cdots - 16\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{3} + \cdots - 22\!\cdots\!88)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 65\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 72\!\cdots\!40 \)
|
| $53$ |
\( (T^{3} + \cdots - 40\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 85\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 24\!\cdots\!32)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 62\!\cdots\!40 \)
|
| $71$ |
\( T^{6} + \cdots + 50\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 34\!\cdots\!00)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 10\!\cdots\!40 \)
|
| $89$ |
\( (T^{3} + \cdots - 12\!\cdots\!08)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 12\!\cdots\!00)^{2} \)
|
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