Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,14,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, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.49");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| 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: | \(80.4231967139\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{1969})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 985x^{2} + 242064 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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
\( x^{4} + 985x^{2} + 242064 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 493\nu ) / 492 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 1477\nu ) / 164 \)
|
| \(\beta_{3}\) | \(=\) |
\( 6\nu^{2} + 2955 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 3\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 2955 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -493\beta_{2} + 4431\beta_1 ) / 6 \)
|
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 |
|
− | 160.120i | − | 729.000i | −17446.5 | 0 | −116728. | 449560.i | 1.48183e6i | −531441. | 0 | ||||||||||||||||||||||||||||
| 49.2 | − | 106.120i | 729.000i | −3069.51 | 0 | 77361.7 | 470568.i | − | 543600.i | −531441. | 0 | |||||||||||||||||||||||||||||
| 49.3 | 106.120i | − | 729.000i | −3069.51 | 0 | 77361.7 | − | 470568.i | 543600.i | −531441. | 0 | |||||||||||||||||||||||||||||
| 49.4 | 160.120i | 729.000i | −17446.5 | 0 | −116728. | − | 449560.i | − | 1.48183e6i | −531441. | 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 | 75.14.b.c | 4 | |
| 5.b | even | 2 | 1 | inner | 75.14.b.c | 4 | |
| 5.c | odd | 4 | 1 | 3.14.a.b | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 75.14.a.e | 2 | ||
| 15.e | even | 4 | 1 | 9.14.a.c | 2 | ||
| 20.e | even | 4 | 1 | 48.14.a.g | 2 | ||
| 35.f | even | 4 | 1 | 147.14.a.b | 2 | ||
| 40.i | odd | 4 | 1 | 192.14.a.k | 2 | ||
| 40.k | even | 4 | 1 | 192.14.a.o | 2 | ||
| 60.l | odd | 4 | 1 | 144.14.a.m | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.14.a.b | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 9.14.a.c | 2 | 15.e | even | 4 | 1 | ||
| 48.14.a.g | 2 | 20.e | even | 4 | 1 | ||
| 75.14.a.e | 2 | 5.c | odd | 4 | 1 | ||
| 75.14.b.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.14.b.c | 4 | 5.b | even | 2 | 1 | inner | |
| 144.14.a.m | 2 | 60.l | odd | 4 | 1 | ||
| 147.14.a.b | 2 | 35.f | even | 4 | 1 | ||
| 192.14.a.k | 2 | 40.i | odd | 4 | 1 | ||
| 192.14.a.o | 2 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 36900T_{2}^{2} + 288728064 \)
acting on \(S_{14}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 36900 T^{2} + 288728064 \)
|
| $3$ |
\( (T^{2} + 531441)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 44\!\cdots\!00 \)
|
| $11$ |
\( (T^{2} + \cdots - 23635688182128)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 43\!\cdots\!04 \)
|
| $17$ |
\( T^{4} + \cdots + 27\!\cdots\!76 \)
|
| $19$ |
\( (T^{2} + \cdots + 59\!\cdots\!80)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 33\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + \cdots + 49\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots - 12\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 33\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} + \cdots - 11\!\cdots\!60)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 28\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 52\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots + 30\!\cdots\!00 \)
|
| $59$ |
\( (T^{2} + \cdots + 21\!\cdots\!80)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 34\!\cdots\!64)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 28\!\cdots\!36 \)
|
| $71$ |
\( (T^{2} + \cdots - 27\!\cdots\!76)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 99\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 26\!\cdots\!64 \)
|
| $89$ |
\( (T^{2} + \cdots - 61\!\cdots\!80)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 95\!\cdots\!36 \)
|
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