Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [10,10,Mod(9,10)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10.9");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.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: | \(5.15035836164\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{319})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 159x^{2} + 6400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 5^{2} \) |
| 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
\( x^{4} - 159x^{2} + 6400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 79\nu ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{3} + 637\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 400\nu^{2} + 79\nu - 31800 ) / 20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta_1 ) / 80 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + \beta _1 + 6360 ) / 80 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 79\beta_{2} + 637\beta_1 ) / 80 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
− | 16.0000i | − | 254.606i | −256.000 | 69.4228 | + | 1395.82i | −4073.69 | − | 4788.05i | 4096.00i | −45141.1 | 22333.1 | − | 1110.77i | |||||||||||||||||||||||
| 9.2 | − | 16.0000i | 102.606i | −256.000 | −1359.42 | + | 324.183i | 1641.69 | 10572.0i | 4096.00i | 9155.07 | 5186.93 | + | 21750.8i | ||||||||||||||||||||||||||
| 9.3 | 16.0000i | − | 102.606i | −256.000 | −1359.42 | − | 324.183i | 1641.69 | − | 10572.0i | − | 4096.00i | 9155.07 | 5186.93 | − | 21750.8i | ||||||||||||||||||||||||
| 9.4 | 16.0000i | 254.606i | −256.000 | 69.4228 | − | 1395.82i | −4073.69 | 4788.05i | − | 4096.00i | −45141.1 | 22333.1 | + | 1110.77i | ||||||||||||||||||||||||||
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 | 10.10.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 90.10.c.b | 4 | ||
| 4.b | odd | 2 | 1 | 80.10.c.a | 4 | ||
| 5.b | even | 2 | 1 | inner | 10.10.b.a | ✓ | 4 |
| 5.c | odd | 4 | 1 | 50.10.a.h | 2 | ||
| 5.c | odd | 4 | 1 | 50.10.a.i | 2 | ||
| 15.d | odd | 2 | 1 | 90.10.c.b | 4 | ||
| 20.d | odd | 2 | 1 | 80.10.c.a | 4 | ||
| 20.e | even | 4 | 1 | 400.10.a.m | 2 | ||
| 20.e | even | 4 | 1 | 400.10.a.s | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.10.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 10.10.b.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 50.10.a.h | 2 | 5.c | odd | 4 | 1 | ||
| 50.10.a.i | 2 | 5.c | odd | 4 | 1 | ||
| 80.10.c.a | 4 | 4.b | odd | 2 | 1 | ||
| 80.10.c.a | 4 | 20.d | odd | 2 | 1 | ||
| 90.10.c.b | 4 | 3.b | odd | 2 | 1 | ||
| 90.10.c.b | 4 | 15.d | odd | 2 | 1 | ||
| 400.10.a.m | 2 | 20.e | even | 4 | 1 | ||
| 400.10.a.s | 2 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(10, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 256)^{2} \)
|
| $3$ |
\( T^{4} + 75352 T^{2} + 682463376 \)
|
| $5$ |
\( T^{4} + \cdots + 3814697265625 \)
|
| $7$ |
\( T^{4} + \cdots + 25\!\cdots\!96 \)
|
| $11$ |
\( (T^{2} + 51816 T - 1688865136)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 47\!\cdots\!56 \)
|
| $17$ |
\( T^{4} + \cdots + 15\!\cdots\!56 \)
|
| $19$ |
\( (T^{2} - 158760 T - 2592025200)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 52\!\cdots\!36 \)
|
| $29$ |
\( (T^{2} + \cdots - 3065700757500)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots + 18849798697984)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 73\!\cdots\!76 \)
|
| $41$ |
\( (T^{2} + \cdots - 52703472270556)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 37\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 26\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots + 22\!\cdots\!76 \)
|
| $59$ |
\( (T^{2} + \cdots + 32\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots + 47\!\cdots\!64)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 80\!\cdots\!56 \)
|
| $71$ |
\( (T^{2} + \cdots + 36\!\cdots\!24)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 10\!\cdots\!36 \)
|
| $79$ |
\( (T^{2} + \cdots + 64\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 70\!\cdots\!16 \)
|
| $89$ |
\( (T^{2} + \cdots - 65\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 60\!\cdots\!36 \)
|
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