Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,5,Mod(7,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.7");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.f (of order \(4\), degree \(2\), 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: | \(7.75274723129\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
0.550510 | + | 0.550510i | 3.67423 | − | 3.67423i | − | 15.3939i | 0 | 4.04541 | −16.7753 | − | 16.7753i | 17.2827 | − | 17.2827i | − | 27.0000i | 0 | ||||||||||||||||||||
| 7.2 | 5.44949 | + | 5.44949i | −3.67423 | + | 3.67423i | 43.3939i | 0 | −40.0454 | −19.2247 | − | 19.2247i | −149.283 | + | 149.283i | − | 27.0000i | 0 | ||||||||||||||||||||||
| 43.1 | 0.550510 | − | 0.550510i | 3.67423 | + | 3.67423i | 15.3939i | 0 | 4.04541 | −16.7753 | + | 16.7753i | 17.2827 | + | 17.2827i | 27.0000i | 0 | |||||||||||||||||||||||
| 43.2 | 5.44949 | − | 5.44949i | −3.67423 | − | 3.67423i | − | 43.3939i | 0 | −40.0454 | −19.2247 | + | 19.2247i | −149.283 | − | 149.283i | 27.0000i | 0 | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.5.f.d | yes | 4 |
| 3.b | odd | 2 | 1 | 225.5.g.d | 4 | ||
| 5.b | even | 2 | 1 | 75.5.f.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 75.5.f.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | inner | 75.5.f.d | yes | 4 |
| 15.d | odd | 2 | 1 | 225.5.g.l | 4 | ||
| 15.e | even | 4 | 1 | 225.5.g.d | 4 | ||
| 15.e | even | 4 | 1 | 225.5.g.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.5.f.a | ✓ | 4 | 5.b | even | 2 | 1 | |
| 75.5.f.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 75.5.f.d | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 75.5.f.d | yes | 4 | 5.c | odd | 4 | 1 | inner |
| 225.5.g.d | 4 | 3.b | odd | 2 | 1 | ||
| 225.5.g.d | 4 | 15.e | even | 4 | 1 | ||
| 225.5.g.l | 4 | 15.d | odd | 2 | 1 | ||
| 225.5.g.l | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 12T_{2}^{3} + 72T_{2}^{2} - 72T_{2} + 36 \)
acting on \(S_{5}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 12 T^{3} + \cdots + 36 \)
|
| $3$ |
\( T^{4} + 729 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 72 T^{3} + \cdots + 416025 \)
|
| $11$ |
\( (T^{2} + 12 T - 36468)^{2} \)
|
| $13$ |
\( T^{4} - 144 T^{3} + \cdots + 136679481 \)
|
| $17$ |
\( T^{4} + \cdots + 2023920144 \)
|
| $19$ |
\( T^{4} + \cdots + 3440409025 \)
|
| $23$ |
\( T^{4} + \cdots + 48514467600 \)
|
| $29$ |
\( T^{4} + \cdots + 217846227600 \)
|
| $31$ |
\( (T^{2} - 1622 T + 294625)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 494152761600 \)
|
| $41$ |
\( (T^{2} - 3432 T + 2426040)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 7727938526889 \)
|
| $47$ |
\( T^{4} + \cdots + 355535527824 \)
|
| $53$ |
\( T^{4} + \cdots + 891211521600 \)
|
| $59$ |
\( T^{4} + \cdots + 128705848419600 \)
|
| $61$ |
\( (T^{2} - 7510 T + 7709881)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 164120004330489 \)
|
| $71$ |
\( (T^{2} + 10776 T + 23841144)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 31\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 43405380652176 \)
|
| $89$ |
\( T^{4} + \cdots + 360018747705600 \)
|
| $97$ |
\( T^{4} + \cdots + 53694498488409 \)
|
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