Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.49");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(4.42514325043\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{19})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 9x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} - 9x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 4\nu ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 14\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 9 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} + 7\beta_1 \)
|
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 |
|
− | 5.35890i | − | 3.00000i | −20.7178 | 0 | −16.0767 | − | 4.43560i | 68.1534i | −9.00000 | 0 | |||||||||||||||||||||||||||
| 49.2 | − | 3.35890i | 3.00000i | −3.28220 | 0 | 10.0767 | − | 30.4356i | − | 15.8466i | −9.00000 | 0 | ||||||||||||||||||||||||||||
| 49.3 | 3.35890i | − | 3.00000i | −3.28220 | 0 | 10.0767 | 30.4356i | 15.8466i | −9.00000 | 0 | ||||||||||||||||||||||||||||||
| 49.4 | 5.35890i | 3.00000i | −20.7178 | 0 | −16.0767 | 4.43560i | − | 68.1534i | −9.00000 | 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 | 75.4.b.c | 4 | |
| 3.b | odd | 2 | 1 | 225.4.b.h | 4 | ||
| 4.b | odd | 2 | 1 | 1200.4.f.v | 4 | ||
| 5.b | even | 2 | 1 | inner | 75.4.b.c | 4 | |
| 5.c | odd | 4 | 1 | 75.4.a.d | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 75.4.a.e | yes | 2 | |
| 15.d | odd | 2 | 1 | 225.4.b.h | 4 | ||
| 15.e | even | 4 | 1 | 225.4.a.j | 2 | ||
| 15.e | even | 4 | 1 | 225.4.a.n | 2 | ||
| 20.d | odd | 2 | 1 | 1200.4.f.v | 4 | ||
| 20.e | even | 4 | 1 | 1200.4.a.bl | 2 | ||
| 20.e | even | 4 | 1 | 1200.4.a.bu | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.4.a.d | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 75.4.a.e | yes | 2 | 5.c | odd | 4 | 1 | |
| 75.4.b.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.4.b.c | 4 | 5.b | even | 2 | 1 | inner | |
| 225.4.a.j | 2 | 15.e | even | 4 | 1 | ||
| 225.4.a.n | 2 | 15.e | even | 4 | 1 | ||
| 225.4.b.h | 4 | 3.b | odd | 2 | 1 | ||
| 225.4.b.h | 4 | 15.d | odd | 2 | 1 | ||
| 1200.4.a.bl | 2 | 20.e | even | 4 | 1 | ||
| 1200.4.a.bu | 2 | 20.e | even | 4 | 1 | ||
| 1200.4.f.v | 4 | 4.b | odd | 2 | 1 | ||
| 1200.4.f.v | 4 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 40T_{2}^{2} + 324 \)
acting on \(S_{4}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 40T^{2} + 324 \)
|
| $3$ |
\( (T^{2} + 9)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 946 T^{2} + 18225 \)
|
| $11$ |
\( (T^{2} - 28 T - 108)^{2} \)
|
| $13$ |
\( T^{4} + 9890 T^{2} + 22877089 \)
|
| $17$ |
\( T^{4} + 17512 T^{2} + 41525136 \)
|
| $19$ |
\( (T^{2} + 6 T - 295)^{2} \)
|
| $23$ |
\( T^{4} + 14184 T^{2} + 2624400 \)
|
| $29$ |
\( (T^{2} + 92 T - 12780)^{2} \)
|
| $31$ |
\( (T^{2} - 122 T - 11175)^{2} \)
|
| $37$ |
\( T^{4} + 62216 T^{2} + 85008400 \)
|
| $41$ |
\( (T^{2} - 392 T - 12960)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 14094675841 \)
|
| $47$ |
\( T^{4} + \cdots + 5874302736 \)
|
| $53$ |
\( T^{4} + \cdots + 27185414400 \)
|
| $59$ |
\( (T^{2} + 124 T - 73980)^{2} \)
|
| $61$ |
\( (T^{2} - 750 T + 81041)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 24951045681 \)
|
| $71$ |
\( (T^{2} - 824 T + 162144)^{2} \)
|
| $73$ |
\( T^{4} + 8264 T^{2} + 2890000 \)
|
| $79$ |
\( (T^{2} - 880 T - 292800)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 482096926224 \)
|
| $89$ |
\( (T^{2} - 864 T - 207360)^{2} \)
|
| $97$ |
\( (T^{2} + 271441)^{2} \)
|
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