Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,10,Mod(49,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.49");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.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: | \(38.6276877123\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{79})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 39x^{2} + 400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{4} - 39x^{2} + 400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 19\nu ) / 20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 59\nu ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 78 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 78 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 19\beta_{2} + 118\beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 35.7764i | − | 81.0000i | −767.950 | 0 | −2897.89 | 3898.82i | 9156.97i | −6561.00 | 0 | ||||||||||||||||||||||||||||
| 49.2 | − | 0.223611i | − | 81.0000i | 511.950 | 0 | −18.1125 | − | 580.825i | − | 228.967i | −6561.00 | 0 | |||||||||||||||||||||||||||
| 49.3 | 0.223611i | 81.0000i | 511.950 | 0 | −18.1125 | 580.825i | 228.967i | −6561.00 | 0 | |||||||||||||||||||||||||||||||
| 49.4 | 35.7764i | 81.0000i | −767.950 | 0 | −2897.89 | − | 3898.82i | − | 9156.97i | −6561.00 | 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 | 75.10.b.g | 4 | |
| 3.b | odd | 2 | 1 | 225.10.b.j | 4 | ||
| 5.b | even | 2 | 1 | inner | 75.10.b.g | 4 | |
| 5.c | odd | 4 | 1 | 75.10.a.e | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 75.10.a.h | yes | 2 | |
| 15.d | odd | 2 | 1 | 225.10.b.j | 4 | ||
| 15.e | even | 4 | 1 | 225.10.a.g | 2 | ||
| 15.e | even | 4 | 1 | 225.10.a.l | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.10.a.e | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 75.10.a.h | yes | 2 | 5.c | odd | 4 | 1 | |
| 75.10.b.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.10.b.g | 4 | 5.b | even | 2 | 1 | inner | |
| 225.10.a.g | 2 | 15.e | even | 4 | 1 | ||
| 225.10.a.l | 2 | 15.e | even | 4 | 1 | ||
| 225.10.b.j | 4 | 3.b | odd | 2 | 1 | ||
| 225.10.b.j | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 1280T_{2}^{2} + 64 \)
acting on \(S_{10}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 1280T^{2} + 64 \)
|
| $3$ |
\( (T^{2} + 6561)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 5128118766225 \)
|
| $11$ |
\( (T^{2} - 24228 T - 1089595948)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 64\!\cdots\!49 \)
|
| $17$ |
\( T^{4} + \cdots + 81\!\cdots\!76 \)
|
| $19$ |
\( (T^{2} - 1348846 T + 273466881505)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} - 4283172 T - 290780819500)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots - 38000674643175)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} + \cdots - 214074226693600)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 98\!\cdots\!01 \)
|
| $47$ |
\( T^{4} + \cdots + 62\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots + 32\!\cdots\!00 \)
|
| $59$ |
\( (T^{2} + \cdots - 11\!\cdots\!20)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots + 13\!\cdots\!21)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 60\!\cdots\!21 \)
|
| $71$ |
\( (T^{2} + \cdots - 25\!\cdots\!56)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 31\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} + \cdots + 61\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 49\!\cdots\!24 \)
|
| $89$ |
\( (T^{2} + \cdots - 18\!\cdots\!80)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 56\!\cdots\!41 \)
|
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