Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,28,Mod(24,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 28, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.24");
S:= CuspForms(chi, 28);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 28 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.b (of order \(2\), degree \(1\), not 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: | \(115.463893710\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 38551324x^{6} + 379819210346764x^{4} + 1087673088198755949961x^{2} + 17090223919471716066147600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{8}\cdot 5^{16} \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 38551324x^{6} + 379819210346764x^{4} + 1087673088198755949961x^{2} + 17090223919471716066147600 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 76112 \nu^{6} - 2586357172200 \nu^{4} + \cdots + 32\!\cdots\!80 ) / 11\!\cdots\!03 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 72612 \nu^{6} - 2539547242200 \nu^{4} + \cdots - 25\!\cdots\!95 ) / 11\!\cdots\!03 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 868436 \nu^{6} - 26772923757600 \nu^{4} + \cdots + 57\!\cdots\!65 ) / 11\!\cdots\!03 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 125 \nu^{7} - 4818915500 \nu^{5} + \cdots - 12\!\cdots\!25 \nu ) / 12\!\cdots\!14 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 341435297789 \nu^{7} + \cdots + 34\!\cdots\!01 \nu ) / 22\!\cdots\!90 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 6698449888452 \nu^{7} + \cdots - 53\!\cdots\!08 \nu ) / 38\!\cdots\!65 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 23\!\cdots\!53 \nu^{7} + \cdots + 44\!\cdots\!57 \nu ) / 11\!\cdots\!95 \)
|
| \(\nu\) | \(=\) |
\( ( 5\beta_{7} - 1880\beta_{6} - 294810\beta_{5} - 91784\beta_{4} ) / 2160000 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -12585\beta_{3} - 477647\beta_{2} + 599277\beta _1 - 10408857480000 ) / 1080000 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -30796865\beta_{7} + 15000989240\beta_{6} + 3568443158130\beta_{5} + 20852637340424\beta_{4} ) / 1080000 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 233201536785\beta_{3} + 3322887600263\beta_{2} - 5830908716493\beta _1 + 65390954531350320000 ) / 360000 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 12\!\cdots\!05 \beta_{7} + \cdots - 13\!\cdots\!04 \beta_{4} ) / 2160000 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 26\!\cdots\!45 \beta_{3} + \cdots - 56\!\cdots\!50 ) / 135000 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 30\!\cdots\!05 \beta_{7} + \cdots + 35\!\cdots\!28 \beta_{4} ) / 2160000 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 24.1 |
|
− | 21310.7i | − | 4.70996e6i | −3.19926e8 | 0 | −1.00372e11 | − | 2.75207e11i | 3.95757e12i | −1.45581e13 | 0 | |||||||||||||||||||||||||||||||||||||||
| 24.2 | − | 19372.0i | − | 3.19680e6i | −2.41055e8 | 0 | −6.19283e10 | 3.32780e11i | 2.06966e12i | −2.59392e12 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 24.3 | − | 7959.76i | − | 829899.i | 7.08599e7 | 0 | −6.60580e9 | − | 2.73788e10i | − | 1.63237e12i | 6.93687e12 | 0 | |||||||||||||||||||||||||||||||||||||||
| 24.4 | − | 1651.55i | 4.81686e6i | 1.31490e8 | 0 | 7.95528e9 | 4.20350e11i | − | 4.38829e11i | −1.55766e13 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 24.5 | 1651.55i | − | 4.81686e6i | 1.31490e8 | 0 | 7.95528e9 | − | 4.20350e11i | 4.38829e11i | −1.55766e13 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 24.6 | 7959.76i | 829899.i | 7.08599e7 | 0 | −6.60580e9 | 2.73788e10i | 1.63237e12i | 6.93687e12 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 24.7 | 19372.0i | 3.19680e6i | −2.41055e8 | 0 | −6.19283e10 | − | 3.32780e11i | − | 2.06966e12i | −2.59392e12 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 24.8 | 21310.7i | 4.70996e6i | −3.19926e8 | 0 | −1.00372e11 | 2.75207e11i | − | 3.95757e12i | −1.45581e13 | 0 | ||||||||||||||||||||||||||||||||||||||||||
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 | 25.28.b.b | 8 | |
| 5.b | even | 2 | 1 | inner | 25.28.b.b | 8 | |
| 5.c | odd | 4 | 1 | 5.28.a.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 25.28.a.b | 4 | ||
| 15.e | even | 4 | 1 | 45.28.a.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.28.a.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 25.28.a.b | 4 | 5.c | odd | 4 | 1 | ||
| 25.28.b.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 25.28.b.b | 8 | 5.b | even | 2 | 1 | inner | |
| 45.28.a.b | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 895502724 T_{2}^{6} + \cdots + 29\!\cdots\!96 \)
acting on \(S_{28}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 29\!\cdots\!96 \)
|
| $3$ |
\( T^{8} + \cdots + 36\!\cdots\!76 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 11\!\cdots\!16 \)
|
| $11$ |
\( (T^{4} + \cdots + 23\!\cdots\!56)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 14\!\cdots\!36 \)
|
| $17$ |
\( T^{8} + \cdots + 36\!\cdots\!56 \)
|
| $19$ |
\( (T^{4} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 36\!\cdots\!96 \)
|
| $29$ |
\( (T^{4} + \cdots + 17\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots - 49\!\cdots\!64)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 11\!\cdots\!36 \)
|
| $41$ |
\( (T^{4} + \cdots - 86\!\cdots\!24)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 77\!\cdots\!16 \)
|
| $47$ |
\( T^{8} + \cdots + 67\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots + 33\!\cdots\!76 \)
|
| $59$ |
\( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots - 94\!\cdots\!44)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 40\!\cdots\!56 \)
|
| $71$ |
\( (T^{4} + \cdots - 13\!\cdots\!04)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 18\!\cdots\!96 \)
|
| $79$ |
\( (T^{4} + \cdots - 19\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 12\!\cdots\!56 \)
|
| $89$ |
\( (T^{4} + \cdots + 90\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 31\!\cdots\!76 \)
|
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