Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [272,4,Mod(81,272)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(272, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("272.81");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 272 = 2^{4} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 272.o (of order \(4\), degree \(2\), 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: | \(16.0485195216\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 46x^{6} + 561x^{4} + 836x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 17) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 46x^{6} + 561x^{4} + 836x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 23\nu^{2} + 16 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 46\nu^{5} + 545\nu^{3} + 468\nu ) / 160 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 4\nu^{6} - 42\nu^{5} + 172\nu^{4} - 413\nu^{3} + 1864\nu^{2} + 396\nu + 1120 ) / 160 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} - 134\nu^{5} + 4\nu^{4} - 1543\nu^{3} + 52\nu^{2} - 1300\nu - 416 ) / 80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 4\nu^{6} + 42\nu^{5} + 172\nu^{4} + 413\nu^{3} + 1864\nu^{2} - 396\nu + 1120 ) / 160 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{7} + 134\nu^{5} + 4\nu^{4} + 1543\nu^{3} + 52\nu^{2} + 1300\nu - 416 ) / 80 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{7} - 2\beta_{5} + \beta_{2} - 24 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{7} - 4\beta_{6} - 2\beta_{5} + 4\beta_{4} - 16\beta_{3} - 19\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 46\beta_{7} + 46\beta_{5} - 13\beta_{2} + 520 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -66\beta_{7} + 92\beta_{6} + 66\beta_{5} - 92\beta_{4} + 608\beta_{3} + 411\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -1046\beta_{7} + 40\beta_{6} - 1046\beta_{5} + 40\beta_{4} + 93\beta_{2} - 11736 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1946\beta_{7} - 2052\beta_{6} - 1946\beta_{5} + 2052\beta_{4} - 18928\beta_{3} - 9019\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).
| \(n\) | \(69\) | \(239\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 81.1 |
|
0 | −3.94546 | − | 3.94546i | 0 | 4.79064 | + | 4.79064i | 0 | −3.33761 | + | 3.33761i | 0 | 4.13329i | 0 | ||||||||||||||||||||||||||||||||||||
| 81.2 | 0 | −2.28193 | − | 2.28193i | 0 | −9.32676 | − | 9.32676i | 0 | 23.5385 | − | 23.5385i | 0 | − | 16.5856i | 0 | ||||||||||||||||||||||||||||||||||||
| 81.3 | 0 | 0.299807 | + | 0.299807i | 0 | 1.37942 | + | 1.37942i | 0 | −17.9849 | + | 17.9849i | 0 | − | 26.8202i | 0 | ||||||||||||||||||||||||||||||||||||
| 81.4 | 0 | 5.92758 | + | 5.92758i | 0 | 10.1567 | + | 10.1567i | 0 | −3.21600 | + | 3.21600i | 0 | 43.2725i | 0 | |||||||||||||||||||||||||||||||||||||
| 225.1 | 0 | −3.94546 | + | 3.94546i | 0 | 4.79064 | − | 4.79064i | 0 | −3.33761 | − | 3.33761i | 0 | − | 4.13329i | 0 | ||||||||||||||||||||||||||||||||||||
| 225.2 | 0 | −2.28193 | + | 2.28193i | 0 | −9.32676 | + | 9.32676i | 0 | 23.5385 | + | 23.5385i | 0 | 16.5856i | 0 | |||||||||||||||||||||||||||||||||||||
| 225.3 | 0 | 0.299807 | − | 0.299807i | 0 | 1.37942 | − | 1.37942i | 0 | −17.9849 | − | 17.9849i | 0 | 26.8202i | 0 | |||||||||||||||||||||||||||||||||||||
| 225.4 | 0 | 5.92758 | − | 5.92758i | 0 | 10.1567 | − | 10.1567i | 0 | −3.21600 | − | 3.21600i | 0 | − | 43.2725i | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.c | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 272.4.o.e | 8 | |
| 4.b | odd | 2 | 1 | 17.4.c.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 153.4.f.a | 8 | ||
| 17.c | even | 4 | 1 | inner | 272.4.o.e | 8 | |
| 68.f | odd | 4 | 1 | 17.4.c.a | ✓ | 8 | |
| 68.g | odd | 8 | 2 | 289.4.a.f | 8 | ||
| 68.g | odd | 8 | 2 | 289.4.b.c | 8 | ||
| 204.l | even | 4 | 1 | 153.4.f.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.4.c.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 17.4.c.a | ✓ | 8 | 68.f | odd | 4 | 1 | |
| 153.4.f.a | 8 | 12.b | even | 2 | 1 | ||
| 153.4.f.a | 8 | 204.l | even | 4 | 1 | ||
| 272.4.o.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 272.4.o.e | 8 | 17.c | even | 4 | 1 | inner | |
| 289.4.a.f | 8 | 68.g | odd | 8 | 2 | ||
| 289.4.b.c | 8 | 68.g | odd | 8 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 180T_{3}^{5} + 3008T_{3}^{4} + 10080T_{3}^{3} + 16200T_{3}^{2} - 11520T_{3} + 4096 \)
acting on \(S_{4}^{\mathrm{new}}(272, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 180 T^{5} + \cdots + 4096 \)
|
| $5$ |
\( T^{8} - 14 T^{7} + \cdots + 6270016 \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots + 330366976 \)
|
| $11$ |
\( T^{8} + \cdots + 40571627776 \)
|
| $13$ |
\( (T^{4} + 44 T^{3} + \cdots - 468640)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 582622237229761 \)
|
| $19$ |
\( T^{8} + \cdots + 2286918209536 \)
|
| $23$ |
\( T^{8} + \cdots + 23983351398400 \)
|
| $29$ |
\( T^{8} + \cdots + 70\!\cdots\!16 \)
|
| $31$ |
\( T^{8} + \cdots + 49\!\cdots\!76 \)
|
| $37$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $41$ |
\( T^{8} + \cdots + 14\!\cdots\!36 \)
|
| $43$ |
\( T^{8} + \cdots + 16\!\cdots\!76 \)
|
| $47$ |
\( (T^{4} - 184 T^{3} + \cdots + 1730640896)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 50\!\cdots\!36 \)
|
| $59$ |
\( T^{8} + \cdots + 48\!\cdots\!76 \)
|
| $61$ |
\( T^{8} + \cdots + 22\!\cdots\!96 \)
|
| $67$ |
\( (T^{4} + 382 T^{3} + \cdots - 80371889536)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 14\!\cdots\!36 \)
|
| $73$ |
\( T^{8} + \cdots + 45\!\cdots\!96 \)
|
| $79$ |
\( T^{8} + \cdots + 16\!\cdots\!16 \)
|
| $83$ |
\( T^{8} + \cdots + 12\!\cdots\!36 \)
|
| $89$ |
\( (T^{4} + 1078 T^{3} + \cdots - 22878545920)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 81\!\cdots\!00 \)
|
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