Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,12,Mod(4,5)]
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([1]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.4");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5.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: | \(3.84171590280\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{4} - x^{3} - 142x^{2} - 2144x + 28656 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 142x^{2} - 2144x + 28656 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} - 12\nu^{2} + 31\nu + 2562 ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 3\nu^{2} + 316\nu + 1422 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\nu^{3} + 57\nu^{2} + 34\nu - 22932 ) / 30 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{3} + \beta_{2} + 6\beta _1 + 30 ) / 120 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -38\beta_{3} + 5\beta_{2} - 234\beta _1 + 8550 ) / 120 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 518\beta_{3} - 29\beta_{2} + 1194\beta _1 + 205770 ) / 120 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
− | 58.2855i | − | 258.747i | −1349.20 | −6731.01 | + | 1876.59i | −15081.2 | − | 21698.4i | − | 40729.8i | 110197. | 109378. | + | 392321.i | ||||||||||||||||||||||
| 4.2 | − | 27.1071i | 524.040i | 1313.20 | 6581.01 | − | 2349.13i | 14205.2 | 66589.5i | − | 91112.6i | −97470.8 | −63678.2 | − | 178392.i | |||||||||||||||||||||||||
| 4.3 | 27.1071i | − | 524.040i | 1313.20 | 6581.01 | + | 2349.13i | 14205.2 | − | 66589.5i | 91112.6i | −97470.8 | −63678.2 | + | 178392.i | |||||||||||||||||||||||||
| 4.4 | 58.2855i | 258.747i | −1349.20 | −6731.01 | − | 1876.59i | −15081.2 | 21698.4i | 40729.8i | 110197. | 109378. | − | 392321.i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5.12.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 45.12.b.b | 4 | ||
| 4.b | odd | 2 | 1 | 80.12.c.a | 4 | ||
| 5.b | even | 2 | 1 | inner | 5.12.b.a | ✓ | 4 |
| 5.c | odd | 4 | 2 | 25.12.a.e | 4 | ||
| 15.d | odd | 2 | 1 | 45.12.b.b | 4 | ||
| 15.e | even | 4 | 2 | 225.12.a.r | 4 | ||
| 20.d | odd | 2 | 1 | 80.12.c.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.12.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 5.12.b.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 25.12.a.e | 4 | 5.c | odd | 4 | 2 | ||
| 45.12.b.b | 4 | 3.b | odd | 2 | 1 | ||
| 45.12.b.b | 4 | 15.d | odd | 2 | 1 | ||
| 80.12.c.a | 4 | 4.b | odd | 2 | 1 | ||
| 80.12.c.a | 4 | 20.d | odd | 2 | 1 | ||
| 225.12.a.r | 4 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(5, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4132 T^{2} + 2496256 \)
|
| $3$ |
\( T^{4} + \cdots + 18385718256 \)
|
| $5$ |
\( T^{4} + \cdots + 23\!\cdots\!25 \)
|
| $7$ |
\( T^{4} + \cdots + 20\!\cdots\!96 \)
|
| $11$ |
\( (T^{2} + 163176 T - 79113038256)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 65\!\cdots\!16 \)
|
| $17$ |
\( T^{4} + \cdots + 84\!\cdots\!76 \)
|
| $19$ |
\( (T^{2} + \cdots - 83046741873200)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 13\!\cdots\!76 \)
|
| $29$ |
\( (T^{2} + \cdots + 28\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots + 19\!\cdots\!24)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 66\!\cdots\!36 \)
|
| $41$ |
\( (T^{2} + \cdots - 25\!\cdots\!36)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 13\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 93\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots + 66\!\cdots\!56 \)
|
| $59$ |
\( (T^{2} + \cdots - 13\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 44\!\cdots\!56)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 11\!\cdots\!76 \)
|
| $71$ |
\( (T^{2} + \cdots + 12\!\cdots\!84)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 25\!\cdots\!76 \)
|
| $79$ |
\( (T^{2} + \cdots - 37\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 35\!\cdots\!36 \)
|
| $89$ |
\( (T^{2} + \cdots - 21\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 35\!\cdots\!16 \)
|
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