Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [153,2,Mod(55,153)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(153, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("153.55");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 153 = 3^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 153.f (of order \(4\), 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: | \(1.22171115093\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.836829184.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 51) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 12\nu^{4} + 4\nu^{3} + 33\nu^{2} + 20\nu + 6 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 38\nu ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 10\nu^{4} - 23\nu^{2} - 6 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + 12\nu^{4} - 4\nu^{3} + 33\nu^{2} - 20\nu - 2 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} - 10\nu^{4} - 19\nu^{2} + 10 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 16\nu^{5} + 77\nu^{3} + 102\nu ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{4} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{2} - 5\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{6} + 2\beta_{5} + 7\beta_{4} + 2\beta_{2} + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} + 9\beta_{5} - 2\beta_{3} - 9\beta_{2} + 29\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 27\beta_{6} - 20\beta_{5} - 51\beta_{4} - 20\beta_{2} - 134 \)
|
| \(\nu^{7}\) | \(=\) |
\( -24\beta_{7} - 67\beta_{5} + 32\beta_{3} + 67\beta_{2} - 181\beta _1 - 67 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(137\) |
| \(\chi(n)\) | \(\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
− | 2.06644i | 0 | −2.27016 | 0.245916 | − | 0.245916i | 0 | −3.06644 | − | 3.06644i | 0.558268i | 0 | −0.508169 | − | 0.508169i | |||||||||||||||||||||||||||||||||||
| 55.2 | − | 0.222191i | 0 | 1.95063 | 0.450006 | − | 0.450006i | 0 | −1.22219 | − | 1.22219i | − | 0.877796i | 0 | −0.0999875 | − | 0.0999875i | |||||||||||||||||||||||||||||||||||
| 55.3 | 1.65222i | 0 | −0.729840 | 2.87540 | − | 2.87540i | 0 | 0.652223 | + | 0.652223i | 2.09859i | 0 | 4.75081 | + | 4.75081i | |||||||||||||||||||||||||||||||||||||
| 55.4 | 2.63640i | 0 | −4.95063 | −1.57133 | + | 1.57133i | 0 | 1.63640 | + | 1.63640i | − | 7.77906i | 0 | −4.14265 | − | 4.14265i | ||||||||||||||||||||||||||||||||||||
| 64.1 | − | 2.63640i | 0 | −4.95063 | −1.57133 | − | 1.57133i | 0 | 1.63640 | − | 1.63640i | 7.77906i | 0 | −4.14265 | + | 4.14265i | ||||||||||||||||||||||||||||||||||||
| 64.2 | − | 1.65222i | 0 | −0.729840 | 2.87540 | + | 2.87540i | 0 | 0.652223 | − | 0.652223i | − | 2.09859i | 0 | 4.75081 | − | 4.75081i | |||||||||||||||||||||||||||||||||||
| 64.3 | 0.222191i | 0 | 1.95063 | 0.450006 | + | 0.450006i | 0 | −1.22219 | + | 1.22219i | 0.877796i | 0 | −0.0999875 | + | 0.0999875i | |||||||||||||||||||||||||||||||||||||
| 64.4 | 2.06644i | 0 | −2.27016 | 0.245916 | + | 0.245916i | 0 | −3.06644 | + | 3.06644i | − | 0.558268i | 0 | −0.508169 | + | 0.508169i | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.c | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 153.2.f.b | 8 | |
| 3.b | odd | 2 | 1 | 51.2.e.a | ✓ | 8 | |
| 4.b | odd | 2 | 1 | 2448.2.be.x | 8 | ||
| 12.b | even | 2 | 1 | 816.2.bd.e | 8 | ||
| 17.c | even | 4 | 1 | inner | 153.2.f.b | 8 | |
| 17.d | even | 8 | 1 | 2601.2.a.be | 4 | ||
| 17.d | even | 8 | 1 | 2601.2.a.bf | 4 | ||
| 51.c | odd | 2 | 1 | 867.2.e.g | 8 | ||
| 51.f | odd | 4 | 1 | 51.2.e.a | ✓ | 8 | |
| 51.f | odd | 4 | 1 | 867.2.e.g | 8 | ||
| 51.g | odd | 8 | 1 | 867.2.a.k | 4 | ||
| 51.g | odd | 8 | 1 | 867.2.a.l | 4 | ||
| 51.g | odd | 8 | 2 | 867.2.d.f | 8 | ||
| 51.i | even | 16 | 4 | 867.2.h.i | 16 | ||
| 51.i | even | 16 | 4 | 867.2.h.k | 16 | ||
| 68.f | odd | 4 | 1 | 2448.2.be.x | 8 | ||
| 204.l | even | 4 | 1 | 816.2.bd.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.2.e.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 51.2.e.a | ✓ | 8 | 51.f | odd | 4 | 1 | |
| 153.2.f.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 153.2.f.b | 8 | 17.c | even | 4 | 1 | inner | |
| 816.2.bd.e | 8 | 12.b | even | 2 | 1 | ||
| 816.2.bd.e | 8 | 204.l | even | 4 | 1 | ||
| 867.2.a.k | 4 | 51.g | odd | 8 | 1 | ||
| 867.2.a.l | 4 | 51.g | odd | 8 | 1 | ||
| 867.2.d.f | 8 | 51.g | odd | 8 | 2 | ||
| 867.2.e.g | 8 | 51.c | odd | 2 | 1 | ||
| 867.2.e.g | 8 | 51.f | odd | 4 | 1 | ||
| 867.2.h.i | 16 | 51.i | even | 16 | 4 | ||
| 867.2.h.k | 16 | 51.i | even | 16 | 4 | ||
| 2448.2.be.x | 8 | 4.b | odd | 2 | 1 | ||
| 2448.2.be.x | 8 | 68.f | odd | 4 | 1 | ||
| 2601.2.a.be | 4 | 17.d | even | 8 | 1 | ||
| 2601.2.a.bf | 4 | 17.d | even | 8 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 14T_{2}^{6} + 61T_{2}^{4} + 84T_{2}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(153, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 14 T^{6} + \cdots + 4 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 4 T^{7} + \cdots + 4 \)
|
| $7$ |
\( T^{8} + 4 T^{7} + \cdots + 256 \)
|
| $11$ |
\( T^{8} + 48 T^{5} + \cdots + 256 \)
|
| $13$ |
\( (T^{4} + 2 T^{3} - 23 T^{2} + \cdots - 64)^{2} \)
|
| $17$ |
\( T^{8} - 4 T^{7} + \cdots + 83521 \)
|
| $19$ |
\( T^{8} + 50 T^{6} + \cdots + 1024 \)
|
| $23$ |
\( T^{8} - 16 T^{7} + \cdots + 33856 \)
|
| $29$ |
\( T^{8} + 4 T^{7} + \cdots + 256 \)
|
| $31$ |
\( T^{8} + 8 T^{7} + \cdots + 1024 \)
|
| $37$ |
\( T^{8} - 8 T^{7} + \cdots + 595984 \)
|
| $41$ |
\( T^{8} + 28 T^{7} + \cdots + 2775556 \)
|
| $43$ |
\( T^{8} + 162 T^{6} + \cdots + 1364224 \)
|
| $47$ |
\( (T^{4} - 4 T^{3} - 52 T^{2} + \cdots - 64)^{2} \)
|
| $53$ |
\( T^{8} + 152 T^{6} + \cdots + 16384 \)
|
| $59$ |
\( T^{8} + 324 T^{6} + \cdots + 21827584 \)
|
| $61$ |
\( T^{8} - 16 T^{7} + \cdots + 126736 \)
|
| $67$ |
\( (T^{4} - 4 T^{3} + \cdots + 2048)^{2} \)
|
| $71$ |
\( T^{8} + 24 T^{7} + \cdots + 1024 \)
|
| $73$ |
\( T^{8} - 20 T^{7} + \cdots + 107584 \)
|
| $79$ |
\( T^{8} - 20 T^{7} + \cdots + 256 \)
|
| $83$ |
\( T^{8} + 344 T^{6} + \cdots + 27541504 \)
|
| $89$ |
\( (T^{4} - 4 T^{3} + \cdots + 1088)^{2} \)
|
| $97$ |
\( T^{8} + 12 T^{7} + \cdots + 135424 \)
|
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