Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [184,2,Mod(93,184)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(184, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("184.93");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 184 = 2^{3} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 184.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: | \(1.46924739719\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.5198736512.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} + 3x^{6} - x^{5} - 2x^{3} + 12x^{2} - 24x + 16 \)
|
| 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
\( x^{8} - 3x^{7} + 3x^{6} - x^{5} - 2x^{3} + 12x^{2} - 24x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} - \nu^{5} - \nu^{4} - 2\nu^{3} - 2\nu^{2} + 8 ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} - \nu^{5} - \nu^{4} - 2\nu^{3} + 6\nu^{2} - 8\nu + 8 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} - 3\nu^{5} + \nu^{4} + 2\nu^{2} - 4\nu + 16 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} - 3\nu^{5} + \nu^{4} + 2\nu^{2} - 20\nu + 24 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{7} + 3\nu^{6} - \nu^{5} + \nu^{4} - \nu^{3} + 4\nu^{2} - 18\nu + 20 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} - 4\nu^{6} + 2\nu^{5} - \nu^{3} - 6\nu^{2} + 26\nu - 28 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{7} - 5\nu^{6} + \nu^{5} + \nu^{4} - 6\nu^{2} + 28\nu - 32 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} + 2\beta_{2} - 2\beta _1 + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} - 2\beta _1 + 1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{7} + 4\beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} - 2\beta _1 + 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -2\beta_{7} + 2\beta_{6} + 2\beta_{5} - 3\beta_{4} - \beta_{3} - 2\beta_{2} - 2\beta _1 + 3 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -4\beta_{7} + 4\beta_{6} - \beta_{4} + 5\beta_{3} - 2\beta_{2} - 2\beta _1 - 7 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -4\beta_{7} + 6\beta_{6} - 2\beta_{5} + 7\beta_{4} + \beta_{3} + 2\beta_{2} - 6\beta _1 + 1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/184\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) | \(93\) | \(97\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 93.1 |
|
−1.20269 | − | 0.744002i | 2.81668i | 0.892923 | + | 1.78961i | − | 0.853978i | 2.09561 | − | 3.38759i | 2.40538 | 0.257562 | − | 2.81668i | −4.93366 | −0.635361 | + | 1.02707i | |||||||||||||||||||||||||||||||
| 93.2 | −1.20269 | + | 0.744002i | − | 2.81668i | 0.892923 | − | 1.78961i | 0.853978i | 2.09561 | + | 3.38759i | 2.40538 | 0.257562 | + | 2.81668i | −4.93366 | −0.635361 | − | 1.02707i | ||||||||||||||||||||||||||||||||
| 93.3 | −0.646813 | − | 1.25763i | − | 0.410655i | −1.16327 | + | 1.62690i | 3.15015i | −0.516453 | + | 0.265617i | 1.29363 | 2.79846 | + | 0.410655i | 2.83136 | 3.96172 | − | 2.03756i | ||||||||||||||||||||||||||||||||
| 93.4 | −0.646813 | + | 1.25763i | 0.410655i | −1.16327 | − | 1.62690i | − | 3.15015i | −0.516453 | − | 0.265617i | 1.29363 | 2.79846 | − | 0.410655i | 2.83136 | 3.96172 | + | 2.03756i | ||||||||||||||||||||||||||||||||
| 93.5 | 0.463860 | − | 1.33598i | − | 1.52213i | −1.56967 | − | 1.23941i | − | 0.609488i | −2.03353 | − | 0.706054i | −0.927719 | −2.38393 | + | 1.52213i | 0.683126 | −0.814262 | − | 0.282717i | |||||||||||||||||||||||||||||||
| 93.6 | 0.463860 | + | 1.33598i | 1.52213i | −1.56967 | + | 1.23941i | 0.609488i | −2.03353 | + | 0.706054i | −0.927719 | −2.38393 | − | 1.52213i | 0.683126 | −0.814262 | + | 0.282717i | |||||||||||||||||||||||||||||||||
| 93.7 | 1.38564 | − | 0.282832i | 1.60649i | 1.84001 | − | 0.783809i | 1.72505i | 0.454369 | + | 2.22603i | −2.77129 | 2.32791 | − | 1.60649i | 0.419174 | 0.487900 | + | 2.39030i | |||||||||||||||||||||||||||||||||
| 93.8 | 1.38564 | + | 0.282832i | − | 1.60649i | 1.84001 | + | 0.783809i | − | 1.72505i | 0.454369 | − | 2.22603i | −2.77129 | 2.32791 | + | 1.60649i | 0.419174 | 0.487900 | − | 2.39030i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 184.2.b.b | ✓ | 8 |
| 4.b | odd | 2 | 1 | 736.2.b.b | 8 | ||
| 8.b | even | 2 | 1 | inner | 184.2.b.b | ✓ | 8 |
| 8.d | odd | 2 | 1 | 736.2.b.b | 8 | ||
| 16.e | even | 4 | 2 | 5888.2.a.x | 8 | ||
| 16.f | odd | 4 | 2 | 5888.2.a.y | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 184.2.b.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 184.2.b.b | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 736.2.b.b | 8 | 4.b | odd | 2 | 1 | ||
| 736.2.b.b | 8 | 8.d | odd | 2 | 1 | ||
| 5888.2.a.x | 8 | 16.e | even | 4 | 2 | ||
| 5888.2.a.y | 8 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 13T_{3}^{6} + 47T_{3}^{4} + 55T_{3}^{2} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(184, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 2 T^{5} + \cdots + 16 \)
|
| $3$ |
\( T^{8} + 13 T^{6} + \cdots + 8 \)
|
| $5$ |
\( T^{8} + 14 T^{6} + \cdots + 8 \)
|
| $7$ |
\( (T^{4} - 8 T^{2} + 2 T + 8)^{2} \)
|
| $11$ |
\( T^{8} + 34 T^{6} + \cdots + 8 \)
|
| $13$ |
\( T^{8} + 29 T^{6} + \cdots + 512 \)
|
| $17$ |
\( (T^{4} + 8 T^{3} + 6 T^{2} + \cdots - 36)^{2} \)
|
| $19$ |
\( T^{8} + 62 T^{6} + \cdots + 2048 \)
|
| $23$ |
\( (T - 1)^{8} \)
|
| $29$ |
\( T^{8} + 149 T^{6} + \cdots + 177608 \)
|
| $31$ |
\( (T^{4} - 3 T^{3} + \cdots + 192)^{2} \)
|
| $37$ |
\( T^{8} + 130 T^{6} + \cdots + 323208 \)
|
| $41$ |
\( (T^{4} - 7 T^{3} + \cdots + 592)^{2} \)
|
| $43$ |
\( T^{8} + 110 T^{6} + \cdots + 8192 \)
|
| $47$ |
\( (T^{4} + T^{3} - 81 T^{2} + \cdots + 1366)^{2} \)
|
| $53$ |
\( T^{8} + 126 T^{6} + \cdots + 73728 \)
|
| $59$ |
\( T^{8} + 228 T^{6} + \cdots + 32768 \)
|
| $61$ |
\( T^{8} + 258 T^{6} + \cdots + 7558272 \)
|
| $67$ |
\( T^{8} + 238 T^{6} + \cdots + 705672 \)
|
| $71$ |
\( (T^{4} + 7 T^{3} + \cdots + 1242)^{2} \)
|
| $73$ |
\( (T^{4} - 25 T^{3} + \cdots - 15668)^{2} \)
|
| $79$ |
\( (T^{4} - 240 T^{2} + \cdots + 10624)^{2} \)
|
| $83$ |
\( T^{8} + 162 T^{6} + \cdots + 129032 \)
|
| $89$ |
\( (T^{4} - 4 T^{3} + \cdots + 1104)^{2} \)
|
| $97$ |
\( (T^{4} + 30 T^{3} + \cdots - 452)^{2} \)
|
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