Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [109,2,Mod(108,109)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(109, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("109.108");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 109 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 109.b (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: | \(0.870369382032\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.191244096.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 12x^{4} + 39x^{2} + 29 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 12x^{4} + 39x^{2} + 29 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 7\nu^{3} - 4\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 7\nu^{2} + 7 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 10\nu^{2} + 19 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 10\nu^{3} + 22\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{2} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{4} + 10\beta_{3} + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} - 10\beta_{2} + 38\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/109\mathbb{Z}\right)^\times\).
| \(n\) | \(6\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 108.1 |
|
− | 2.65595i | 1.46050 | −5.05408 | 3.46050 | − | 3.87903i | −2.59358 | 8.11150i | −0.866926 | − | 9.19094i | |||||||||||||||||||||||||||||||||
| 108.2 | − | 1.97199i | −2.69963 | −1.88874 | −0.699628 | 5.32363i | −3.58836 | − | 0.219412i | 4.28799 | 1.37966i | |||||||||||||||||||||||||||||||||||
| 108.3 | − | 1.02819i | −0.760877 | 0.942820 | 1.23912 | 0.782328i | 1.18194 | − | 3.02579i | −2.42107 | − | 1.27406i | ||||||||||||||||||||||||||||||||||
| 108.4 | 1.02819i | −0.760877 | 0.942820 | 1.23912 | − | 0.782328i | 1.18194 | 3.02579i | −2.42107 | 1.27406i | ||||||||||||||||||||||||||||||||||||
| 108.5 | 1.97199i | −2.69963 | −1.88874 | −0.699628 | − | 5.32363i | −3.58836 | 0.219412i | 4.28799 | − | 1.37966i | |||||||||||||||||||||||||||||||||||
| 108.6 | 2.65595i | 1.46050 | −5.05408 | 3.46050 | 3.87903i | −2.59358 | − | 8.11150i | −0.866926 | 9.19094i | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 109.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 109.2.b.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 981.2.c.c | 6 | ||
| 4.b | odd | 2 | 1 | 1744.2.h.d | 6 | ||
| 109.b | even | 2 | 1 | inner | 109.2.b.b | ✓ | 6 |
| 327.d | odd | 2 | 1 | 981.2.c.c | 6 | ||
| 436.c | odd | 2 | 1 | 1744.2.h.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 109.2.b.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 109.2.b.b | ✓ | 6 | 109.b | even | 2 | 1 | inner |
| 981.2.c.c | 6 | 3.b | odd | 2 | 1 | ||
| 981.2.c.c | 6 | 327.d | odd | 2 | 1 | ||
| 1744.2.h.d | 6 | 4.b | odd | 2 | 1 | ||
| 1744.2.h.d | 6 | 436.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 12T_{2}^{4} + 39T_{2}^{2} + 29 \)
acting on \(S_{2}^{\mathrm{new}}(109, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 12 T^{4} + \cdots + 29 \)
|
| $3$ |
\( (T^{3} + 2 T^{2} - 3 T - 3)^{2} \)
|
| $5$ |
\( (T^{3} - 4 T^{2} + T + 3)^{2} \)
|
| $7$ |
\( (T^{3} + 5 T^{2} + 2 T - 11)^{2} \)
|
| $11$ |
\( T^{6} + 23 T^{4} + \cdots + 261 \)
|
| $13$ |
\( T^{6} + 79 T^{4} + \cdots + 12789 \)
|
| $17$ |
\( T^{6} + 21 T^{4} + \cdots + 29 \)
|
| $19$ |
\( T^{6} + 49 T^{4} + \cdots + 2349 \)
|
| $23$ |
\( T^{6} + 75 T^{4} + \cdots + 29 \)
|
| $29$ |
\( (T^{3} - 33 T - 9)^{2} \)
|
| $31$ |
\( (T^{3} - T^{2} - 4 T + 1)^{2} \)
|
| $37$ |
\( T^{6} + 100 T^{4} + \cdots + 2349 \)
|
| $41$ |
\( T^{6} + 192 T^{4} + \cdots + 229709 \)
|
| $43$ |
\( (T^{3} - 13 T^{2} + \cdots - 41)^{2} \)
|
| $47$ |
\( T^{6} + 132 T^{4} + \cdots + 69629 \)
|
| $53$ |
\( T^{6} + 167 T^{4} + \cdots + 115101 \)
|
| $59$ |
\( T^{6} + 195 T^{4} + \cdots + 100949 \)
|
| $61$ |
\( (T^{3} + 6 T^{2} - 24 T - 56)^{2} \)
|
| $67$ |
\( T^{6} + 411 T^{4} + \cdots + 2257389 \)
|
| $71$ |
\( (T^{3} + 10 T^{2} + \cdots - 1125)^{2} \)
|
| $73$ |
\( (T^{3} - 12 T^{2} + \cdots + 733)^{2} \)
|
| $79$ |
\( T^{6} + 58 T^{4} + \cdots + 261 \)
|
| $83$ |
\( (T^{3} + 29 T^{2} + \cdots + 771)^{2} \)
|
| $89$ |
\( (T^{3} - 17 T^{2} + 46 T - 9)^{2} \)
|
| $97$ |
\( (T^{3} + 6 T^{2} - 15 T + 7)^{2} \)
|
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