Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.49");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(12.0287864860\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{31})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 15x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{4} - 15x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 7\nu ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 23\nu ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 15 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{2} + 23\beta_1 ) / 2 \)
|
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 |
|
− | 8.56776i | 9.00000i | −41.4066 | 0 | 77.1099 | − | 184.626i | 80.5934i | −81.0000 | 0 | ||||||||||||||||||||||||||||
| 49.2 | − | 2.56776i | − | 9.00000i | 25.4066 | 0 | −23.1099 | − | 82.6263i | − | 147.407i | −81.0000 | 0 | |||||||||||||||||||||||||||
| 49.3 | 2.56776i | 9.00000i | 25.4066 | 0 | −23.1099 | 82.6263i | 147.407i | −81.0000 | 0 | |||||||||||||||||||||||||||||||
| 49.4 | 8.56776i | − | 9.00000i | −41.4066 | 0 | 77.1099 | 184.626i | − | 80.5934i | −81.0000 | 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 | 75.6.b.f | 4 | |
| 3.b | odd | 2 | 1 | 225.6.b.l | 4 | ||
| 5.b | even | 2 | 1 | inner | 75.6.b.f | 4 | |
| 5.c | odd | 4 | 1 | 75.6.a.g | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 75.6.a.i | yes | 2 | |
| 15.d | odd | 2 | 1 | 225.6.b.l | 4 | ||
| 15.e | even | 4 | 1 | 225.6.a.j | 2 | ||
| 15.e | even | 4 | 1 | 225.6.a.t | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.6.a.g | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 75.6.a.i | yes | 2 | 5.c | odd | 4 | 1 | |
| 75.6.b.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.6.b.f | 4 | 5.b | even | 2 | 1 | inner | |
| 225.6.a.j | 2 | 15.e | even | 4 | 1 | ||
| 225.6.a.t | 2 | 15.e | even | 4 | 1 | ||
| 225.6.b.l | 4 | 3.b | odd | 2 | 1 | ||
| 225.6.b.l | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 80T_{2}^{2} + 484 \)
acting on \(S_{6}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 80T^{2} + 484 \)
|
| $3$ |
\( (T^{2} + 81)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 40914 T^{2} + 232715025 \)
|
| $11$ |
\( (T^{2} + 12 T - 240028)^{2} \)
|
| $13$ |
\( T^{4} + 1126850 T^{2} + 63473089 \)
|
| $17$ |
\( T^{4} + \cdots + 471370126096 \)
|
| $19$ |
\( (T^{2} + 4214 T + 3564505)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 168162280707600 \)
|
| $29$ |
\( (T^{2} + 4068 T + 4039940)^{2} \)
|
| $31$ |
\( (T^{2} + 2598 T - 25471575)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 86\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} - 11232 T + 28823360)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 269100697595521 \)
|
| $47$ |
\( T^{4} + \cdots + 11\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $59$ |
\( (T^{2} - 63924 T + 741079460)^{2} \)
|
| $61$ |
\( (T^{2} - 7310 T - 18138959)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!21 \)
|
| $71$ |
\( (T^{2} - 98304 T + 1224279104)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 33\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} + 84000 T + 1649721600)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!04 \)
|
| $89$ |
\( (T^{2} + 103104 T + 2346485760)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 19\!\cdots\!61 \)
|
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