Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [100,8,Mod(1,100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(100, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("100.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 100.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: | \(31.2385025484\) |
| 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: |
\( x^{4} - 1348x^{2} + 93051 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 20) |
| 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
\( x^{4} - 1348x^{2} + 93051 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} - 4846\nu ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 2696 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2696 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 21\beta_{2} + 2423\beta_1 ) / 4 \)
|
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 |
|
0 | −71.4148 | 0 | 0 | 0 | −860.515 | 0 | 2913.08 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −17.0857 | 0 | 0 | 0 | 1750.09 | 0 | −1895.08 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 17.0857 | 0 | 0 | 0 | −1750.09 | 0 | −1895.08 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 71.4148 | 0 | 0 | 0 | 860.515 | 0 | 2913.08 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 100.8.a.e | 4 | |
| 4.b | odd | 2 | 1 | 400.8.a.bk | 4 | ||
| 5.b | even | 2 | 1 | inner | 100.8.a.e | 4 | |
| 5.c | odd | 4 | 2 | 20.8.c.a | ✓ | 4 | |
| 15.e | even | 4 | 2 | 180.8.d.b | 4 | ||
| 20.d | odd | 2 | 1 | 400.8.a.bk | 4 | ||
| 20.e | even | 4 | 2 | 80.8.c.c | 4 | ||
| 40.i | odd | 4 | 2 | 320.8.c.j | 4 | ||
| 40.k | even | 4 | 2 | 320.8.c.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.8.c.a | ✓ | 4 | 5.c | odd | 4 | 2 | |
| 80.8.c.c | 4 | 20.e | even | 4 | 2 | ||
| 100.8.a.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 100.8.a.e | 4 | 5.b | even | 2 | 1 | inner | |
| 180.8.d.b | 4 | 15.e | even | 4 | 2 | ||
| 320.8.c.i | 4 | 40.k | even | 4 | 2 | ||
| 320.8.c.j | 4 | 40.i | odd | 4 | 2 | ||
| 400.8.a.bk | 4 | 4.b | odd | 2 | 1 | ||
| 400.8.a.bk | 4 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 5392T_{3}^{2} + 1488816 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(100))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 5392 T^{2} + 1488816 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 2267978437936 \)
|
| $11$ |
\( (T^{2} + 1320 T - 22682800)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 17\!\cdots\!36 \)
|
| $17$ |
\( T^{4} + \cdots + 35\!\cdots\!16 \)
|
| $19$ |
\( (T^{2} - 18168 T - 125546544)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 84\!\cdots\!96 \)
|
| $29$ |
\( (T^{2} - 240948 T + 13681722276)^{2} \)
|
| $31$ |
\( (T^{2} - 80672 T - 19179567104)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 26\!\cdots\!76 \)
|
| $41$ |
\( (T^{2} - 564732 T + 69044077556)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 56\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots + 22\!\cdots\!76 \)
|
| $59$ |
\( (T^{2} + \cdots + 3606865822544)^{2} \)
|
| $61$ |
\( (T^{2} - 2152564 T + 494185531924)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 19\!\cdots\!56 \)
|
| $71$ |
\( (T^{2} + \cdots - 6354130539456)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 65\!\cdots\!76 \)
|
| $79$ |
\( (T^{2} + \cdots - 14503458928896)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 67\!\cdots\!56 \)
|
| $89$ |
\( (T^{2} + \cdots + 9522296681636)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 18\!\cdots\!96 \)
|
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