Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [155,2,Mod(36,155)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(155, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("155.36");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 155 = 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 155.e (of order \(3\), degree \(2\), 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: | \(1.23768123133\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.42575625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 4x^{6} + x^{5} + 9x^{4} - x^{3} + 4x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - x^{7} + 4x^{6} + x^{5} + 9x^{4} - x^{3} + 4x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{7} + \nu^{6} + 10\nu^{5} + 8\nu^{4} + 46\nu^{3} + 5\nu^{2} + 2\nu - 36 ) / 25 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{7} - 2\nu^{6} + 5\nu^{5} + 19\nu^{4} + 8\nu^{3} - 10\nu^{2} - 39\nu - 3 ) / 25 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 6\nu^{7} - 3\nu^{6} + 20\nu^{5} + 16\nu^{4} + 62\nu^{3} + 10\nu^{2} + 4\nu + 8 ) / 25 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} - 5\nu^{5} + 4\nu^{4} - 8\nu^{3} + 10\nu^{2} + \nu + 3 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 8\nu^{7} - 14\nu^{6} + 35\nu^{5} - 12\nu^{4} + 56\nu^{3} - 70\nu^{2} + 22\nu - 21 ) / 25 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{7} - 2\nu^{6} + 10\nu^{5} + 8\nu^{4} + 23\nu^{3} + 5\nu^{2} + 2\nu + 2 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + \beta _1 - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + 3\beta_{4} - \beta_{2} - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{6} + 4\beta_{5} - \beta_{3} - 5\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{7} + 5\beta_{6} + 6\beta_{5} - 12\beta_{4} - 4\beta_{3} + 6\beta_{2} - 12\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 6\beta_{7} - 23\beta_{4} + 16\beta_{2} + 28 \)
|
| \(\nu^{7}\) | \(=\) |
\( -23\beta_{6} - 29\beta_{5} + 16\beta_{3} + 51\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).
| \(n\) | \(32\) | \(96\) |
| \(\chi(n)\) | \(1\) | \(\beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 36.1 |
|
−1.35567 | −0.440197 | + | 0.762443i | −0.162147 | −0.500000 | − | 0.866025i | 0.596764 | − | 1.03362i | 1.77460 | − | 3.07370i | 2.93117 | 1.11245 | + | 1.92683i | 0.677837 | + | 1.17405i | ||||||||||||||||||||||||||||||
| 36.2 | −0.477260 | 1.35666 | − | 2.34981i | −1.77222 | −0.500000 | − | 0.866025i | −0.647481 | + | 1.12147i | 0.0911485 | − | 0.157874i | 1.80033 | −2.18107 | − | 3.77773i | 0.238630 | + | 0.413319i | |||||||||||||||||||||||||||||||
| 36.3 | 0.737640 | −1.48685 | + | 2.57531i | −1.45589 | −0.500000 | − | 0.866025i | −1.09676 | + | 1.89965i | −0.965584 | + | 1.67244i | −2.54920 | −2.92147 | − | 5.06014i | −0.368820 | − | 0.638815i | |||||||||||||||||||||||||||||||
| 36.4 | 2.09529 | 0.0703870 | − | 0.121914i | 2.39026 | −0.500000 | − | 0.866025i | 0.147481 | − | 0.255445i | −0.400166 | + | 0.693107i | 0.817703 | 1.49009 | + | 2.58091i | −1.04765 | − | 1.81458i | |||||||||||||||||||||||||||||||
| 56.1 | −1.35567 | −0.440197 | − | 0.762443i | −0.162147 | −0.500000 | + | 0.866025i | 0.596764 | + | 1.03362i | 1.77460 | + | 3.07370i | 2.93117 | 1.11245 | − | 1.92683i | 0.677837 | − | 1.17405i | |||||||||||||||||||||||||||||||
| 56.2 | −0.477260 | 1.35666 | + | 2.34981i | −1.77222 | −0.500000 | + | 0.866025i | −0.647481 | − | 1.12147i | 0.0911485 | + | 0.157874i | 1.80033 | −2.18107 | + | 3.77773i | 0.238630 | − | 0.413319i | |||||||||||||||||||||||||||||||
| 56.3 | 0.737640 | −1.48685 | − | 2.57531i | −1.45589 | −0.500000 | + | 0.866025i | −1.09676 | − | 1.89965i | −0.965584 | − | 1.67244i | −2.54920 | −2.92147 | + | 5.06014i | −0.368820 | + | 0.638815i | |||||||||||||||||||||||||||||||
| 56.4 | 2.09529 | 0.0703870 | + | 0.121914i | 2.39026 | −0.500000 | + | 0.866025i | 0.147481 | + | 0.255445i | −0.400166 | − | 0.693107i | 0.817703 | 1.49009 | − | 2.58091i | −1.04765 | + | 1.81458i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 155.2.e.d | ✓ | 8 |
| 5.b | even | 2 | 1 | 775.2.e.f | 8 | ||
| 5.c | odd | 4 | 2 | 775.2.o.f | 16 | ||
| 31.c | even | 3 | 1 | inner | 155.2.e.d | ✓ | 8 |
| 31.c | even | 3 | 1 | 4805.2.a.o | 4 | ||
| 31.e | odd | 6 | 1 | 4805.2.a.m | 4 | ||
| 155.j | even | 6 | 1 | 775.2.e.f | 8 | ||
| 155.o | odd | 12 | 2 | 775.2.o.f | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 155.2.e.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 155.2.e.d | ✓ | 8 | 31.c | even | 3 | 1 | inner |
| 775.2.e.f | 8 | 5.b | even | 2 | 1 | ||
| 775.2.e.f | 8 | 155.j | even | 6 | 1 | ||
| 775.2.o.f | 16 | 5.c | odd | 4 | 2 | ||
| 775.2.o.f | 16 | 155.o | odd | 12 | 2 | ||
| 4805.2.a.m | 4 | 31.e | odd | 6 | 1 | ||
| 4805.2.a.o | 4 | 31.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{3} - 3T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(155, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} - 3 T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} + T^{7} + 9 T^{6} + \cdots + 1 \)
|
| $5$ |
\( (T^{2} + T + 1)^{4} \)
|
| $7$ |
\( T^{8} - T^{7} + 9 T^{6} + \cdots + 1 \)
|
| $11$ |
\( T^{8} - 4 T^{7} + \cdots + 6241 \)
|
| $13$ |
\( T^{8} + T^{7} + 11 T^{6} + \cdots + 1 \)
|
| $17$ |
\( T^{8} + 8 T^{7} + \cdots + 6241 \)
|
| $19$ |
\( T^{8} - 3 T^{7} + \cdots + 841 \)
|
| $23$ |
\( (T^{4} - 19 T^{3} + \cdots + 211)^{2} \)
|
| $29$ |
\( (T^{4} - 19 T^{3} + \cdots - 1421)^{2} \)
|
| $31$ |
\( T^{8} + 5 T^{7} + \cdots + 923521 \)
|
| $37$ |
\( T^{8} - 6 T^{7} + \cdots + 546121 \)
|
| $41$ |
\( T^{8} + 4 T^{7} + \cdots + 2401 \)
|
| $43$ |
\( T^{8} + 11 T^{7} + \cdots + 121 \)
|
| $47$ |
\( (T^{4} + 4 T^{3} + \cdots + 271)^{2} \)
|
| $53$ |
\( T^{8} + 7 T^{7} + \cdots + 4289041 \)
|
| $59$ |
\( (T^{4} + 9 T^{3} + \cdots + 1681)^{2} \)
|
| $61$ |
\( (T^{4} + 5 T^{3} + \cdots + 589)^{2} \)
|
| $67$ |
\( T^{8} - 9 T^{7} + \cdots + 6561 \)
|
| $71$ |
\( T^{8} + 12 T^{7} + \cdots + 183358681 \)
|
| $73$ |
\( T^{8} + 4 T^{7} + \cdots + 430336 \)
|
| $79$ |
\( T^{8} - 14 T^{7} + \cdots + 11566801 \)
|
| $83$ |
\( T^{8} + 18 T^{7} + \cdots + 495018001 \)
|
| $89$ |
\( (T^{4} + T^{3} - 38 T^{2} + \cdots + 121)^{2} \)
|
| $97$ |
\( (T^{4} - 7 T^{3} + \cdots + 121)^{2} \)
|
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