Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [110,3,Mod(21,110)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(110, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("110.21");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 110 = 2 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 110.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: | \(2.99728290796\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.4956160000.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{6} + 19x^{4} - 30x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{6} + 19x^{4} - 30x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{4} - 4\nu^{2} + 15 ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{7} + 7\nu^{5} - 37\nu^{3} + 20\nu ) / 25 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} + 17\nu^{5} - 57\nu^{3} + 120\nu ) / 25 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{5} + 17\nu^{3} - 10\nu ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 5\nu^{6} + 4\nu^{5} - 15\nu^{4} - 24\nu^{3} + 85\nu^{2} + 65\nu - 75 ) / 25 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} - 5\nu^{6} + 4\nu^{5} + 15\nu^{4} - 24\nu^{3} - 85\nu^{2} + 65\nu + 75 ) / 25 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{6} + 6\nu^{4} - 24\nu^{2} + 20 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} + \beta_{5} + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{6} - \beta_{5} + 5\beta_{4} + 2\beta_{3} + 7\beta_{2} ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + 5\beta _1 - 11 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -6\beta_{6} - 6\beta_{5} + 7\beta_{3} - 3\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -11\beta_{7} + 6\beta_{6} - 6\beta_{5} + 15\beta _1 - 37 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -9\beta_{6} - 9\beta_{5} - 55\beta_{4} + 8\beta_{3} - 127\beta_{2} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).
| \(n\) | \(67\) | \(101\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 21.1 |
|
− | 1.41421i | −5.11433 | −2.00000 | 2.23607 | 7.23275i | 6.78202i | 2.82843i | 17.1564 | − | 3.16228i | ||||||||||||||||||||||||||||||||||||||||
| 21.2 | − | 1.41421i | −2.29552 | −2.00000 | −2.23607 | 3.24636i | 4.07370i | 2.82843i | −3.73057 | 3.16228i | ||||||||||||||||||||||||||||||||||||||||||
| 21.3 | − | 1.41421i | −1.35781 | −2.00000 | 2.23607 | 1.92023i | − | 12.4389i | 2.82843i | −7.15636 | − | 3.16228i | ||||||||||||||||||||||||||||||||||||||||
| 21.4 | − | 1.41421i | 4.76766 | −2.00000 | −2.23607 | − | 6.74249i | − | 9.73055i | 2.82843i | 13.7306 | 3.16228i | ||||||||||||||||||||||||||||||||||||||||
| 21.5 | 1.41421i | −5.11433 | −2.00000 | 2.23607 | − | 7.23275i | − | 6.78202i | − | 2.82843i | 17.1564 | 3.16228i | ||||||||||||||||||||||||||||||||||||||||
| 21.6 | 1.41421i | −2.29552 | −2.00000 | −2.23607 | − | 3.24636i | − | 4.07370i | − | 2.82843i | −3.73057 | − | 3.16228i | |||||||||||||||||||||||||||||||||||||||
| 21.7 | 1.41421i | −1.35781 | −2.00000 | 2.23607 | − | 1.92023i | 12.4389i | − | 2.82843i | −7.15636 | 3.16228i | |||||||||||||||||||||||||||||||||||||||||
| 21.8 | 1.41421i | 4.76766 | −2.00000 | −2.23607 | 6.74249i | 9.73055i | − | 2.82843i | 13.7306 | − | 3.16228i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 110.3.d.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 990.3.b.b | 8 | ||
| 4.b | odd | 2 | 1 | 880.3.j.c | 8 | ||
| 5.b | even | 2 | 1 | 550.3.d.f | 8 | ||
| 5.c | odd | 4 | 2 | 550.3.c.b | 16 | ||
| 11.b | odd | 2 | 1 | inner | 110.3.d.a | ✓ | 8 |
| 33.d | even | 2 | 1 | 990.3.b.b | 8 | ||
| 44.c | even | 2 | 1 | 880.3.j.c | 8 | ||
| 55.d | odd | 2 | 1 | 550.3.d.f | 8 | ||
| 55.e | even | 4 | 2 | 550.3.c.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.3.d.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 110.3.d.a | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
| 550.3.c.b | 16 | 5.c | odd | 4 | 2 | ||
| 550.3.c.b | 16 | 55.e | even | 4 | 2 | ||
| 550.3.d.f | 8 | 5.b | even | 2 | 1 | ||
| 550.3.d.f | 8 | 55.d | odd | 2 | 1 | ||
| 880.3.j.c | 8 | 4.b | odd | 2 | 1 | ||
| 880.3.j.c | 8 | 44.c | even | 2 | 1 | ||
| 990.3.b.b | 8 | 3.b | odd | 2 | 1 | ||
| 990.3.b.b | 8 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(110, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{4} \)
|
| $3$ |
\( (T^{4} + 4 T^{3} - 20 T^{2} + \cdots - 76)^{2} \)
|
| $5$ |
\( (T^{2} - 5)^{4} \)
|
| $7$ |
\( T^{8} + 312 T^{6} + \cdots + 11182336 \)
|
| $11$ |
\( T^{8} - 132 T^{6} + \cdots + 214358881 \)
|
| $13$ |
\( T^{8} + 896 T^{6} + \cdots + 23001616 \)
|
| $17$ |
\( T^{8} + 1056 T^{6} + \cdots + 50069776 \)
|
| $19$ |
\( T^{8} + 1664 T^{6} + \cdots + 52070656 \)
|
| $23$ |
\( (T^{4} + 68 T^{3} + \cdots - 1209164)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 2498400256 \)
|
| $31$ |
\( (T^{4} + 32 T^{3} + \cdots + 378176)^{2} \)
|
| $37$ |
\( (T^{4} + 24 T^{3} + \cdots - 31856)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 32472856662016 \)
|
| $43$ |
\( T^{8} + \cdots + 506465648896 \)
|
| $47$ |
\( (T^{4} - 76 T^{3} + \cdots + 111764)^{2} \)
|
| $53$ |
\( (T^{4} - 176 T^{3} + \cdots - 188144)^{2} \)
|
| $59$ |
\( (T^{4} - 40 T^{3} + \cdots - 456896)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 288073767325696 \)
|
| $67$ |
\( (T^{4} - 12 T^{3} + \cdots + 571796)^{2} \)
|
| $71$ |
\( (T^{4} + 128 T^{3} + \cdots + 7774976)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 50\!\cdots\!56 \)
|
| $79$ |
\( T^{8} + 25984 T^{6} + \cdots + 516925696 \)
|
| $83$ |
\( T^{8} + \cdots + 45563364004096 \)
|
| $89$ |
\( (T^{4} + 184 T^{3} + \cdots - 101033216)^{2} \)
|
| $97$ |
\( (T^{4} + 16 T^{3} + \cdots + 139413776)^{2} \)
|
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