Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [72,3,Mod(19,72)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(72, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("72.19");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.b (of order \(2\), degree \(1\), 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: | \(1.96185790339\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.4752.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} - 6x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 24) |
| 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
\( x^{4} + 3x^{2} - 6x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} + 2\nu - 2 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} + 6\nu - 2 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + \beta _1 - 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{3} - 2\beta_{2} + 5\beta _1 + 6 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(55\) | \(65\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−1.36603 | − | 1.46081i | 0 | −0.267949 | + | 3.99102i | 7.98203i | 0 | 2.13878i | 6.19615 | − | 5.06040i | 0 | 11.6603 | − | 10.9037i | ||||||||||||||||||||||
| 19.2 | −1.36603 | + | 1.46081i | 0 | −0.267949 | − | 3.99102i | − | 7.98203i | 0 | − | 2.13878i | 6.19615 | + | 5.06040i | 0 | 11.6603 | + | 10.9037i | |||||||||||||||||||||
| 19.3 | 0.366025 | − | 1.96622i | 0 | −3.73205 | − | 1.43937i | − | 2.87875i | 0 | − | 10.7436i | −4.19615 | + | 6.81119i | 0 | −5.66025 | − | 1.05369i | |||||||||||||||||||||
| 19.4 | 0.366025 | + | 1.96622i | 0 | −3.73205 | + | 1.43937i | 2.87875i | 0 | 10.7436i | −4.19615 | − | 6.81119i | 0 | −5.66025 | + | 1.05369i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 72.3.b.b | 4 | |
| 3.b | odd | 2 | 1 | 24.3.b.a | ✓ | 4 | |
| 4.b | odd | 2 | 1 | 288.3.b.b | 4 | ||
| 8.b | even | 2 | 1 | 288.3.b.b | 4 | ||
| 8.d | odd | 2 | 1 | inner | 72.3.b.b | 4 | |
| 12.b | even | 2 | 1 | 96.3.b.a | 4 | ||
| 15.d | odd | 2 | 1 | 600.3.g.a | 4 | ||
| 15.e | even | 4 | 2 | 600.3.p.a | 8 | ||
| 16.e | even | 4 | 2 | 2304.3.g.z | 8 | ||
| 16.f | odd | 4 | 2 | 2304.3.g.z | 8 | ||
| 24.f | even | 2 | 1 | 24.3.b.a | ✓ | 4 | |
| 24.h | odd | 2 | 1 | 96.3.b.a | 4 | ||
| 48.i | odd | 4 | 2 | 768.3.g.h | 8 | ||
| 48.k | even | 4 | 2 | 768.3.g.h | 8 | ||
| 60.h | even | 2 | 1 | 2400.3.g.a | 4 | ||
| 60.l | odd | 4 | 2 | 2400.3.p.a | 8 | ||
| 120.i | odd | 2 | 1 | 2400.3.g.a | 4 | ||
| 120.m | even | 2 | 1 | 600.3.g.a | 4 | ||
| 120.q | odd | 4 | 2 | 600.3.p.a | 8 | ||
| 120.w | even | 4 | 2 | 2400.3.p.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.3.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 24.3.b.a | ✓ | 4 | 24.f | even | 2 | 1 | |
| 72.3.b.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 72.3.b.b | 4 | 8.d | odd | 2 | 1 | inner | |
| 96.3.b.a | 4 | 12.b | even | 2 | 1 | ||
| 96.3.b.a | 4 | 24.h | odd | 2 | 1 | ||
| 288.3.b.b | 4 | 4.b | odd | 2 | 1 | ||
| 288.3.b.b | 4 | 8.b | even | 2 | 1 | ||
| 600.3.g.a | 4 | 15.d | odd | 2 | 1 | ||
| 600.3.g.a | 4 | 120.m | even | 2 | 1 | ||
| 600.3.p.a | 8 | 15.e | even | 4 | 2 | ||
| 600.3.p.a | 8 | 120.q | odd | 4 | 2 | ||
| 768.3.g.h | 8 | 48.i | odd | 4 | 2 | ||
| 768.3.g.h | 8 | 48.k | even | 4 | 2 | ||
| 2304.3.g.z | 8 | 16.e | even | 4 | 2 | ||
| 2304.3.g.z | 8 | 16.f | odd | 4 | 2 | ||
| 2400.3.g.a | 4 | 60.h | even | 2 | 1 | ||
| 2400.3.g.a | 4 | 120.i | odd | 2 | 1 | ||
| 2400.3.p.a | 8 | 60.l | odd | 4 | 2 | ||
| 2400.3.p.a | 8 | 120.w | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 72T_{5}^{2} + 528 \)
acting on \(S_{3}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 16 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 72T^{2} + 528 \)
|
| $7$ |
\( T^{4} + 120T^{2} + 528 \)
|
| $11$ |
\( (T - 8)^{4} \)
|
| $13$ |
\( T^{4} + 384 T^{2} + 33792 \)
|
| $17$ |
\( (T^{2} - 4 T - 188)^{2} \)
|
| $19$ |
\( (T^{2} - 16 T + 16)^{2} \)
|
| $23$ |
\( T^{4} + 480T^{2} + 8448 \)
|
| $29$ |
\( T^{4} + 1608T^{2} + 528 \)
|
| $31$ |
\( T^{4} + 3384 T^{2} + 279312 \)
|
| $37$ |
\( T^{4} + 864 T^{2} + 76032 \)
|
| $41$ |
\( (T^{2} + 20 T - 1628)^{2} \)
|
| $43$ |
\( (T^{2} - 16 T - 368)^{2} \)
|
| $47$ |
\( T^{4} + 3552 T^{2} + 8448 \)
|
| $53$ |
\( T^{4} + 1800 T^{2} + 803088 \)
|
| $59$ |
\( (T^{2} - 64 T + 592)^{2} \)
|
| $61$ |
\( T^{4} + 3552 T^{2} + 8448 \)
|
| $67$ |
\( (T^{2} + 128 T + 3664)^{2} \)
|
| $71$ |
\( T^{4} + 8928 T^{2} + 12849408 \)
|
| $73$ |
\( (T^{2} - 100 T - 572)^{2} \)
|
| $79$ |
\( T^{4} + 7416 T^{2} + 10797072 \)
|
| $83$ |
\( (T^{2} + 80 T + 832)^{2} \)
|
| $89$ |
\( (T^{2} - 100 T - 4412)^{2} \)
|
| $97$ |
\( (T^{2} - 28 T - 6716)^{2} \)
|
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