Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [123,2,Mod(40,123)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(123, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("123.40");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 123 = 3 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 123.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: | \(0.982159944862\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 13x^{6} + 52x^{4} + 60x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{8} + 13x^{6} + 52x^{4} + 60x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 7\nu^{3} + 8\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + 7\nu^{4} + 8\nu^{2} ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 9\nu^{3} + 18\nu ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 9\nu^{4} + 20\nu^{2} + 6 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 10\nu^{5} + 29\nu^{3} + 24\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{3} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - \beta_{4} - 6\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} + 9\beta_{3} + 27\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -7\beta_{6} + 9\beta_{4} + 34\beta_{2} - 81 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2\beta_{7} + 41\beta_{5} - 61\beta_{3} - 149\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/123\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) | \(88\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 40.1 |
|
−2.39890 | − | 1.00000i | 3.75471 | −0.189528 | 2.39890i | − | 0.644188i | −4.20937 | −1.00000 | 0.454659 | ||||||||||||||||||||||||||||||||||||||||
| 40.2 | −2.39890 | 1.00000i | 3.75471 | −0.189528 | − | 2.39890i | 0.644188i | −4.20937 | −1.00000 | 0.454659 | ||||||||||||||||||||||||||||||||||||||||||
| 40.3 | −0.266370 | − | 1.00000i | −1.92905 | −3.31295 | 0.266370i | − | 4.19542i | 1.04658 | −1.00000 | 0.882469 | |||||||||||||||||||||||||||||||||||||||||
| 40.4 | −0.266370 | 1.00000i | −1.92905 | −3.31295 | − | 0.266370i | 4.19542i | 1.04658 | −1.00000 | 0.882469 | ||||||||||||||||||||||||||||||||||||||||||
| 40.5 | 1.35449 | − | 1.00000i | −0.165361 | 2.28744 | − | 1.35449i | − | 0.810873i | −2.93296 | −1.00000 | 3.09832 | ||||||||||||||||||||||||||||||||||||||||
| 40.6 | 1.35449 | 1.00000i | −0.165361 | 2.28744 | 1.35449i | 0.810873i | −2.93296 | −1.00000 | 3.09832 | |||||||||||||||||||||||||||||||||||||||||||
| 40.7 | 2.31078 | − | 1.00000i | 3.33970 | −2.78497 | − | 2.31078i | 3.65048i | 3.09575 | −1.00000 | −6.43545 | |||||||||||||||||||||||||||||||||||||||||
| 40.8 | 2.31078 | 1.00000i | 3.33970 | −2.78497 | 2.31078i | − | 3.65048i | 3.09575 | −1.00000 | −6.43545 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 123.2.d.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 369.2.d.e | 8 | ||
| 4.b | odd | 2 | 1 | 1968.2.j.e | 8 | ||
| 41.b | even | 2 | 1 | inner | 123.2.d.a | ✓ | 8 |
| 41.c | even | 4 | 1 | 5043.2.a.o | 4 | ||
| 41.c | even | 4 | 1 | 5043.2.a.p | 4 | ||
| 123.b | odd | 2 | 1 | 369.2.d.e | 8 | ||
| 164.d | odd | 2 | 1 | 1968.2.j.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 123.2.d.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 123.2.d.a | ✓ | 8 | 41.b | even | 2 | 1 | inner |
| 369.2.d.e | 8 | 3.b | odd | 2 | 1 | ||
| 369.2.d.e | 8 | 123.b | odd | 2 | 1 | ||
| 1968.2.j.e | 8 | 4.b | odd | 2 | 1 | ||
| 1968.2.j.e | 8 | 164.d | odd | 2 | 1 | ||
| 5043.2.a.o | 4 | 41.c | even | 4 | 1 | ||
| 5043.2.a.p | 4 | 41.c | even | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(123, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} - 6 T^{2} + \cdots + 2)^{2} \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( (T^{4} + 4 T^{3} - 4 T^{2} + \cdots - 4)^{2} \)
|
| $7$ |
\( T^{8} + 32 T^{6} + \cdots + 64 \)
|
| $11$ |
\( T^{8} + 47 T^{6} + \cdots + 400 \)
|
| $13$ |
\( T^{8} + 44 T^{6} + \cdots + 256 \)
|
| $17$ |
\( T^{8} + 99 T^{6} + \cdots + 262144 \)
|
| $19$ |
\( T^{8} + 120 T^{6} + \cdots + 107584 \)
|
| $23$ |
\( (T^{4} + 2 T^{3} + \cdots + 608)^{2} \)
|
| $29$ |
\( T^{8} + 151 T^{6} + \cdots + 1098304 \)
|
| $31$ |
\( (T^{4} - 13 T^{3} + \cdots - 328)^{2} \)
|
| $37$ |
\( (T^{4} - 5 T^{3} + \cdots + 1346)^{2} \)
|
| $41$ |
\( T^{8} - 30 T^{7} + \cdots + 2825761 \)
|
| $43$ |
\( (T^{4} + T^{3} - 17 T^{2} + \cdots + 52)^{2} \)
|
| $47$ |
\( T^{8} + 115 T^{6} + \cdots + 85264 \)
|
| $53$ |
\( T^{8} + 240 T^{6} + \cdots + 1024 \)
|
| $59$ |
\( (T^{4} + 12 T^{3} + \cdots - 320)^{2} \)
|
| $61$ |
\( (T^{4} - 3 T^{3} + \cdots - 398)^{2} \)
|
| $67$ |
\( T^{8} + 396 T^{6} + \cdots + 123904 \)
|
| $71$ |
\( T^{8} + 291 T^{6} + \cdots + 6864400 \)
|
| $73$ |
\( (T^{4} + 9 T^{3} + \cdots + 146)^{2} \)
|
| $79$ |
\( T^{8} + 412 T^{6} + \cdots + 96510976 \)
|
| $83$ |
\( (T^{4} - 188 T^{2} + \cdots + 8696)^{2} \)
|
| $89$ |
\( T^{8} + 188 T^{6} + \cdots + 16384 \)
|
| $97$ |
\( T^{8} + 644 T^{6} + \cdots + 579076096 \)
|
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