Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [110,3,Mod(109,110)]
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([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("110.109");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 110 = 2 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 110.c (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: | 8.0.130897030168576.7 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 18x^{6} + 169x^{4} - 112x^{2} + 1936 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{8} + 18x^{6} + 169x^{4} - 112x^{2} + 1936 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -3\nu^{7} - 46\nu^{5} - 431\nu^{3} + 1812\nu ) / 3080 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{7} + 46\nu^{5} + 431\nu^{3} + 1268\nu ) / 3080 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{6} - 19\nu^{4} - 369\nu^{2} + 88 ) / 770 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4\nu^{7} + 2\nu^{6} + 73\nu^{5} + 19\nu^{4} + 493\nu^{3} + 369\nu^{2} - 1996\nu - 88 ) / 1540 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} - 4\nu^{6} - 81\nu^{5} - 38\nu^{4} - 956\nu^{3} + 32\nu^{2} - 2528\nu + 3256 ) / 1540 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} - 4\nu^{6} + 81\nu^{5} - 38\nu^{4} + 956\nu^{3} + 32\nu^{2} + 2528\nu + 3256 ) / 1540 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 17\nu^{6} + 354\nu^{4} + 3329\nu^{2} + 792 ) / 1540 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} - 2\beta_{3} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} - 6\beta_{4} - 3\beta_{3} - 4\beta_{2} - 16\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{7} - \beta_{6} - \beta_{5} + 36\beta_{3} - 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{6} - 7\beta_{5} + 90\beta_{4} + 45\beta_{3} - 64\beta_{2} + 204\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -76\beta_{7} - 175\beta_{6} - 175\beta_{5} - 358\beta_{3} + 820 \)
|
| \(\nu^{7}\) | \(=\) |
\( -251\beta_{6} + 251\beta_{5} - 518\beta_{4} - 259\beta_{3} + 2160\beta_{2} - 1252\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 109.1 |
|
−1.41421 | − | 3.43731i | 2.00000 | −3.90754 | − | 3.11948i | 4.86108i | −2.82843 | −2.82843 | −2.81507 | 5.52609 | + | 4.41161i | |||||||||||||||||||||||||||||||||||||
| 109.2 | −1.41421 | − | 1.08854i | 2.00000 | 1.40754 | + | 4.79780i | 1.53943i | −2.82843 | −2.82843 | 7.81507 | −1.99056 | − | 6.78511i | ||||||||||||||||||||||||||||||||||||||
| 109.3 | −1.41421 | 1.08854i | 2.00000 | 1.40754 | − | 4.79780i | − | 1.53943i | −2.82843 | −2.82843 | 7.81507 | −1.99056 | + | 6.78511i | ||||||||||||||||||||||||||||||||||||||
| 109.4 | −1.41421 | 3.43731i | 2.00000 | −3.90754 | + | 3.11948i | − | 4.86108i | −2.82843 | −2.82843 | −2.81507 | 5.52609 | − | 4.41161i | ||||||||||||||||||||||||||||||||||||||
| 109.5 | 1.41421 | − | 3.43731i | 2.00000 | −3.90754 | − | 3.11948i | − | 4.86108i | 2.82843 | 2.82843 | −2.81507 | −5.52609 | − | 4.41161i | |||||||||||||||||||||||||||||||||||||
| 109.6 | 1.41421 | − | 1.08854i | 2.00000 | 1.40754 | + | 4.79780i | − | 1.53943i | 2.82843 | 2.82843 | 7.81507 | 1.99056 | + | 6.78511i | |||||||||||||||||||||||||||||||||||||
| 109.7 | 1.41421 | 1.08854i | 2.00000 | 1.40754 | − | 4.79780i | 1.53943i | 2.82843 | 2.82843 | 7.81507 | 1.99056 | − | 6.78511i | |||||||||||||||||||||||||||||||||||||||
| 109.8 | 1.41421 | 3.43731i | 2.00000 | −3.90754 | + | 3.11948i | 4.86108i | 2.82843 | 2.82843 | −2.81507 | −5.52609 | + | 4.41161i | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 55.d | 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.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 990.3.h.b | 8 | ||
| 4.b | odd | 2 | 1 | 880.3.i.g | 8 | ||
| 5.b | even | 2 | 1 | inner | 110.3.c.b | ✓ | 8 |
| 5.c | odd | 4 | 2 | 550.3.d.d | 8 | ||
| 11.b | odd | 2 | 1 | inner | 110.3.c.b | ✓ | 8 |
| 15.d | odd | 2 | 1 | 990.3.h.b | 8 | ||
| 20.d | odd | 2 | 1 | 880.3.i.g | 8 | ||
| 33.d | even | 2 | 1 | 990.3.h.b | 8 | ||
| 44.c | even | 2 | 1 | 880.3.i.g | 8 | ||
| 55.d | odd | 2 | 1 | inner | 110.3.c.b | ✓ | 8 |
| 55.e | even | 4 | 2 | 550.3.d.d | 8 | ||
| 165.d | even | 2 | 1 | 990.3.h.b | 8 | ||
| 220.g | even | 2 | 1 | 880.3.i.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.3.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 110.3.c.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 110.3.c.b | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
| 110.3.c.b | ✓ | 8 | 55.d | odd | 2 | 1 | inner |
| 550.3.d.d | 8 | 5.c | odd | 4 | 2 | ||
| 550.3.d.d | 8 | 55.e | even | 4 | 2 | ||
| 880.3.i.g | 8 | 4.b | odd | 2 | 1 | ||
| 880.3.i.g | 8 | 20.d | odd | 2 | 1 | ||
| 880.3.i.g | 8 | 44.c | even | 2 | 1 | ||
| 880.3.i.g | 8 | 220.g | even | 2 | 1 | ||
| 990.3.h.b | 8 | 3.b | odd | 2 | 1 | ||
| 990.3.h.b | 8 | 15.d | odd | 2 | 1 | ||
| 990.3.h.b | 8 | 33.d | even | 2 | 1 | ||
| 990.3.h.b | 8 | 165.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 13T_{3}^{2} + 14 \)
acting on \(S_{3}^{\mathrm{new}}(110, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{4} \)
|
| $3$ |
\( (T^{4} + 13 T^{2} + 14)^{2} \)
|
| $5$ |
\( (T^{4} + 5 T^{3} + \cdots + 625)^{2} \)
|
| $7$ |
\( (T^{2} - 8)^{4} \)
|
| $11$ |
\( (T^{4} + 10 T^{3} + \cdots + 14641)^{2} \)
|
| $13$ |
\( (T^{4} - 138 T^{2} + 1936)^{2} \)
|
| $17$ |
\( (T^{4} - 1018 T^{2} + 258064)^{2} \)
|
| $19$ |
\( (T^{4} + 1132 T^{2} + 37856)^{2} \)
|
| $23$ |
\( (T^{4} + 509 T^{2} + 2366)^{2} \)
|
| $29$ |
\( (T^{4} + 1456 T^{2} + 175616)^{2} \)
|
| $31$ |
\( (T^{2} + T - 1384)^{4} \)
|
| $37$ |
\( (T^{4} + 3043 T^{2} + 833504)^{2} \)
|
| $41$ |
\( (T^{4} + 4648 T^{2} + 5312384)^{2} \)
|
| $43$ |
\( (T^{4} - 4744 T^{2} + 2876416)^{2} \)
|
| $47$ |
\( (T^{4} + 4618 T^{2} + 5312384)^{2} \)
|
| $53$ |
\( (T^{4} + 1952 T^{2} + 229376)^{2} \)
|
| $59$ |
\( (T^{2} + 15 T - 1328)^{4} \)
|
| $61$ |
\( (T^{4} + 1664 T^{2} + 229376)^{2} \)
|
| $67$ |
\( (T^{4} + 8413 T^{2} + 286286)^{2} \)
|
| $71$ |
\( (T^{2} - 141 T + 4716)^{4} \)
|
| $73$ |
\( (T^{4} - 5562 T^{2} + 7595536)^{2} \)
|
| $79$ |
\( (T^{4} + 32212 T^{2} + 226696064)^{2} \)
|
| $83$ |
\( (T^{4} - 11464 T^{2} + 2768896)^{2} \)
|
| $89$ |
\( (T^{2} - 17 T - 1312)^{4} \)
|
| $97$ |
\( (T^{4} + 7091 T^{2} + 7419776)^{2} \)
|
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