Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [72,6,Mod(37,72)]
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([0, 1, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("72.37");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.d (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: | \(11.5476350265\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.218489.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 8x + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 8) |
| 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
\( x^{4} - x^{3} - 2x^{2} - 8x + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 5\nu^{2} - 2\nu + 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 3\nu^{2} - 2\nu - 12 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + \beta _1 + 5 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{3} - 3\beta_{2} + \beta _1 + 33 ) / 4 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−3.77200 | − | 4.21569i | 0 | −3.54400 | + | 31.8031i | 73.9600i | 0 | −112.704 | 147.440 | − | 105.021i | 0 | 311.792 | − | 278.977i | ||||||||||||||||||||||
| 37.2 | −3.77200 | + | 4.21569i | 0 | −3.54400 | − | 31.8031i | − | 73.9600i | 0 | −112.704 | 147.440 | + | 105.021i | 0 | 311.792 | + | 278.977i | ||||||||||||||||||||||
| 37.3 | 4.77200 | − | 3.03776i | 0 | 13.5440 | − | 28.9924i | 1.38521i | 0 | 160.704 | −23.4400 | − | 179.495i | 0 | 4.20793 | + | 6.61022i | |||||||||||||||||||||||
| 37.4 | 4.77200 | + | 3.03776i | 0 | 13.5440 | + | 28.9924i | − | 1.38521i | 0 | 160.704 | −23.4400 | + | 179.495i | 0 | 4.20793 | − | 6.61022i | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 72.6.d.b | 4 | |
| 3.b | odd | 2 | 1 | 8.6.b.a | ✓ | 4 | |
| 4.b | odd | 2 | 1 | 288.6.d.b | 4 | ||
| 8.b | even | 2 | 1 | inner | 72.6.d.b | 4 | |
| 8.d | odd | 2 | 1 | 288.6.d.b | 4 | ||
| 12.b | even | 2 | 1 | 32.6.b.a | 4 | ||
| 15.d | odd | 2 | 1 | 200.6.d.a | 4 | ||
| 15.e | even | 4 | 2 | 200.6.f.a | 8 | ||
| 24.f | even | 2 | 1 | 32.6.b.a | 4 | ||
| 24.h | odd | 2 | 1 | 8.6.b.a | ✓ | 4 | |
| 48.i | odd | 4 | 2 | 256.6.a.k | 4 | ||
| 48.k | even | 4 | 2 | 256.6.a.n | 4 | ||
| 60.h | even | 2 | 1 | 800.6.d.a | 4 | ||
| 60.l | odd | 4 | 2 | 800.6.f.a | 8 | ||
| 120.i | odd | 2 | 1 | 200.6.d.a | 4 | ||
| 120.m | even | 2 | 1 | 800.6.d.a | 4 | ||
| 120.q | odd | 4 | 2 | 800.6.f.a | 8 | ||
| 120.w | even | 4 | 2 | 200.6.f.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.6.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 8.6.b.a | ✓ | 4 | 24.h | odd | 2 | 1 | |
| 32.6.b.a | 4 | 12.b | even | 2 | 1 | ||
| 32.6.b.a | 4 | 24.f | even | 2 | 1 | ||
| 72.6.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 72.6.d.b | 4 | 8.b | even | 2 | 1 | inner | |
| 200.6.d.a | 4 | 15.d | odd | 2 | 1 | ||
| 200.6.d.a | 4 | 120.i | odd | 2 | 1 | ||
| 200.6.f.a | 8 | 15.e | even | 4 | 2 | ||
| 200.6.f.a | 8 | 120.w | even | 4 | 2 | ||
| 256.6.a.k | 4 | 48.i | odd | 4 | 2 | ||
| 256.6.a.n | 4 | 48.k | even | 4 | 2 | ||
| 288.6.d.b | 4 | 4.b | odd | 2 | 1 | ||
| 288.6.d.b | 4 | 8.d | odd | 2 | 1 | ||
| 800.6.d.a | 4 | 60.h | even | 2 | 1 | ||
| 800.6.d.a | 4 | 120.m | even | 2 | 1 | ||
| 800.6.f.a | 8 | 60.l | odd | 4 | 2 | ||
| 800.6.f.a | 8 | 120.q | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 5472T_{5}^{2} + 10496 \)
acting on \(S_{6}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 1024 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 5472 T^{2} + 10496 \)
|
| $7$ |
\( (T^{2} - 48 T - 18112)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 5520765456 \)
|
| $13$ |
\( T^{4} + \cdots + 7999305984 \)
|
| $17$ |
\( (T^{2} + 100 T - 72252)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 120994976016 \)
|
| $23$ |
\( (T^{2} + 1168 T - 2817216)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 535633608132864 \)
|
| $31$ |
\( (T^{2} + 6464 T + 7754752)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 306881230162176 \)
|
| $41$ |
\( (T^{2} - 2284 T - 85109148)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 23\!\cdots\!56 \)
|
| $47$ |
\( (T^{2} - 27360 T + 132648192)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 76\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 22\!\cdots\!36 \)
|
| $61$ |
\( T^{4} + \cdots + 41\!\cdots\!16 \)
|
| $67$ |
\( T^{4} + \cdots + 32\!\cdots\!84 \)
|
| $71$ |
\( (T^{2} + 103344 T + 2609278272)^{2} \)
|
| $73$ |
\( (T^{2} - 19988 T - 1604540316)^{2} \)
|
| $79$ |
\( (T^{2} + 123936 T + 3701816576)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 72\!\cdots\!56 \)
|
| $89$ |
\( (T^{2} - 42316 T - 6875717724)^{2} \)
|
| $97$ |
\( (T^{2} + 49788 T - 783309052)^{2} \)
|
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