Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [80,10,Mod(49,80)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(80, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("80.49");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 80 = 2^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 80.c (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: | \(41.2028668931\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 1095x^{2} - 80251x + 2230844 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 20) |
| 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
\( x^{4} - x^{3} + 1095x^{2} - 80251x + 2230844 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} - 82\nu^{2} + 99\nu + 14468 ) / 1000 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{3} + 246\nu^{2} + 19703\nu - 48404 ) / 500 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -13\nu^{3} - 66\nu^{2} - 13713\nu + 739084 ) / 500 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 6\beta _1 + 10 ) / 40 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + 3\beta_{2} - 86\beta _1 - 4378 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -1640\beta_{3} - 1131\beta_{2} - 4146\beta _1 + 2374690 ) / 40 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(21\) | \(31\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 188.155i | 0 | 1126.30 | + | 827.388i | 0 | − | 1842.93i | 0 | −15719.2 | 0 | ||||||||||||||||||||||||||
| 49.2 | 0 | − | 92.1183i | 0 | −796.301 | − | 1148.49i | 0 | 2204.86i | 0 | 11197.2 | 0 | ||||||||||||||||||||||||||||
| 49.3 | 0 | 92.1183i | 0 | −796.301 | + | 1148.49i | 0 | − | 2204.86i | 0 | 11197.2 | 0 | ||||||||||||||||||||||||||||
| 49.4 | 0 | 188.155i | 0 | 1126.30 | − | 827.388i | 0 | 1842.93i | 0 | −15719.2 | 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 | 80.10.c.b | 4 | |
| 4.b | odd | 2 | 1 | 20.10.c.a | ✓ | 4 | |
| 5.b | even | 2 | 1 | inner | 80.10.c.b | 4 | |
| 5.c | odd | 4 | 2 | 400.10.a.bb | 4 | ||
| 12.b | even | 2 | 1 | 180.10.d.a | 4 | ||
| 20.d | odd | 2 | 1 | 20.10.c.a | ✓ | 4 | |
| 20.e | even | 4 | 2 | 100.10.a.f | 4 | ||
| 60.h | even | 2 | 1 | 180.10.d.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.10.c.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 20.10.c.a | ✓ | 4 | 20.d | odd | 2 | 1 | |
| 80.10.c.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 80.10.c.b | 4 | 5.b | even | 2 | 1 | inner | |
| 100.10.a.f | 4 | 20.e | even | 4 | 2 | ||
| 180.10.d.a | 4 | 12.b | even | 2 | 1 | ||
| 180.10.d.a | 4 | 60.h | even | 2 | 1 | ||
| 400.10.a.bb | 4 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 43888T_{3}^{2} + 300415536 \)
acting on \(S_{10}^{\mathrm{new}}(80, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 43888 T^{2} + 300415536 \)
|
| $5$ |
\( T^{4} + \cdots + 3814697265625 \)
|
| $7$ |
\( T^{4} + \cdots + 16511275120816 \)
|
| $11$ |
\( (T^{2} - 17400 T - 2423076400)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 11\!\cdots\!56 \)
|
| $17$ |
\( T^{4} + \cdots + 18\!\cdots\!76 \)
|
| $19$ |
\( (T^{2} - 113832 T - 242807738544)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 10\!\cdots\!56 \)
|
| $29$ |
\( (T^{2} + \cdots - 16159702451004)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots - 12779301484544)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 81\!\cdots\!56 \)
|
| $41$ |
\( (T^{2} + \cdots + 16789107440756)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 36\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 17\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots + 15\!\cdots\!16 \)
|
| $59$ |
\( (T^{2} + \cdots + 11\!\cdots\!24)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 26\!\cdots\!76)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 13\!\cdots\!96 \)
|
| $71$ |
\( (T^{2} + \cdots + 93\!\cdots\!84)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 28\!\cdots\!76 \)
|
| $79$ |
\( (T^{2} + \cdots + 18\!\cdots\!04)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 18\!\cdots\!76 \)
|
| $89$ |
\( (T^{2} + \cdots + 12\!\cdots\!16)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 23\!\cdots\!36 \)
|
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