Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,3,Mod(74,75)]
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([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.74");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.d (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: | \(2.04360198270\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{3} + 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{3} + 8\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} + 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{2} + 2\beta_1 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 74.1 |
|
−2.23607 | −2.23607 | − | 2.00000i | 1.00000 | 0 | 5.00000 | + | 4.47214i | 6.00000i | 6.70820 | 1.00000 | + | 8.94427i | 0 | ||||||||||||||||||||||||
| 74.2 | −2.23607 | −2.23607 | + | 2.00000i | 1.00000 | 0 | 5.00000 | − | 4.47214i | − | 6.00000i | 6.70820 | 1.00000 | − | 8.94427i | 0 | ||||||||||||||||||||||||
| 74.3 | 2.23607 | 2.23607 | − | 2.00000i | 1.00000 | 0 | 5.00000 | − | 4.47214i | 6.00000i | −6.70820 | 1.00000 | − | 8.94427i | 0 | |||||||||||||||||||||||||
| 74.4 | 2.23607 | 2.23607 | + | 2.00000i | 1.00000 | 0 | 5.00000 | + | 4.47214i | − | 6.00000i | −6.70820 | 1.00000 | + | 8.94427i | 0 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.3.d.b | 4 | |
| 3.b | odd | 2 | 1 | inner | 75.3.d.b | 4 | |
| 4.b | odd | 2 | 1 | 1200.3.c.f | 4 | ||
| 5.b | even | 2 | 1 | inner | 75.3.d.b | 4 | |
| 5.c | odd | 4 | 1 | 15.3.c.a | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 75.3.c.e | 2 | ||
| 12.b | even | 2 | 1 | 1200.3.c.f | 4 | ||
| 15.d | odd | 2 | 1 | inner | 75.3.d.b | 4 | |
| 15.e | even | 4 | 1 | 15.3.c.a | ✓ | 2 | |
| 15.e | even | 4 | 1 | 75.3.c.e | 2 | ||
| 20.d | odd | 2 | 1 | 1200.3.c.f | 4 | ||
| 20.e | even | 4 | 1 | 240.3.l.b | 2 | ||
| 20.e | even | 4 | 1 | 1200.3.l.g | 2 | ||
| 40.i | odd | 4 | 1 | 960.3.l.c | 2 | ||
| 40.k | even | 4 | 1 | 960.3.l.b | 2 | ||
| 45.k | odd | 12 | 2 | 405.3.i.b | 4 | ||
| 45.l | even | 12 | 2 | 405.3.i.b | 4 | ||
| 60.h | even | 2 | 1 | 1200.3.c.f | 4 | ||
| 60.l | odd | 4 | 1 | 240.3.l.b | 2 | ||
| 60.l | odd | 4 | 1 | 1200.3.l.g | 2 | ||
| 120.q | odd | 4 | 1 | 960.3.l.b | 2 | ||
| 120.w | even | 4 | 1 | 960.3.l.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.3.c.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 15.3.c.a | ✓ | 2 | 15.e | even | 4 | 1 | |
| 75.3.c.e | 2 | 5.c | odd | 4 | 1 | ||
| 75.3.c.e | 2 | 15.e | even | 4 | 1 | ||
| 75.3.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.3.d.b | 4 | 3.b | odd | 2 | 1 | inner | |
| 75.3.d.b | 4 | 5.b | even | 2 | 1 | inner | |
| 75.3.d.b | 4 | 15.d | odd | 2 | 1 | inner | |
| 240.3.l.b | 2 | 20.e | even | 4 | 1 | ||
| 240.3.l.b | 2 | 60.l | odd | 4 | 1 | ||
| 405.3.i.b | 4 | 45.k | odd | 12 | 2 | ||
| 405.3.i.b | 4 | 45.l | even | 12 | 2 | ||
| 960.3.l.b | 2 | 40.k | even | 4 | 1 | ||
| 960.3.l.b | 2 | 120.q | odd | 4 | 1 | ||
| 960.3.l.c | 2 | 40.i | odd | 4 | 1 | ||
| 960.3.l.c | 2 | 120.w | even | 4 | 1 | ||
| 1200.3.c.f | 4 | 4.b | odd | 2 | 1 | ||
| 1200.3.c.f | 4 | 12.b | even | 2 | 1 | ||
| 1200.3.c.f | 4 | 20.d | odd | 2 | 1 | ||
| 1200.3.c.f | 4 | 60.h | even | 2 | 1 | ||
| 1200.3.l.g | 2 | 20.e | even | 4 | 1 | ||
| 1200.3.l.g | 2 | 60.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 5 \)
acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 5)^{2} \)
|
| $3$ |
\( T^{4} - 2T^{2} + 81 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 36)^{2} \)
|
| $11$ |
\( (T^{2} + 20)^{2} \)
|
| $13$ |
\( (T^{2} + 256)^{2} \)
|
| $17$ |
\( (T^{2} - 20)^{2} \)
|
| $19$ |
\( (T - 2)^{4} \)
|
| $23$ |
\( (T^{2} - 180)^{2} \)
|
| $29$ |
\( (T^{2} + 980)^{2} \)
|
| $31$ |
\( (T + 18)^{4} \)
|
| $37$ |
\( (T^{2} + 256)^{2} \)
|
| $41$ |
\( (T^{2} + 3920)^{2} \)
|
| $43$ |
\( (T^{2} + 256)^{2} \)
|
| $47$ |
\( (T^{2} - 2420)^{2} \)
|
| $53$ |
\( (T^{2} - 20)^{2} \)
|
| $59$ |
\( (T^{2} + 20)^{2} \)
|
| $61$ |
\( (T - 82)^{4} \)
|
| $67$ |
\( (T^{2} + 576)^{2} \)
|
| $71$ |
\( (T^{2} + 15680)^{2} \)
|
| $73$ |
\( (T^{2} + 5476)^{2} \)
|
| $79$ |
\( (T + 138)^{4} \)
|
| $83$ |
\( (T^{2} - 8820)^{2} \)
|
| $89$ |
\( (T^{2} + 11520)^{2} \)
|
| $97$ |
\( (T^{2} + 27556)^{2} \)
|
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