Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [405,4,Mod(244,405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("405.244");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.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: | \(23.8957735523\) |
| 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: |
\( x^{8} + 44x^{6} + 567x^{4} + 2024x^{2} + 1900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
| 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
\( x^{8} + 44x^{6} + 567x^{4} + 2024x^{2} + 1900 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} - 5\nu^{6} - 117\nu^{5} - 225\nu^{4} - 1206\nu^{3} - 2700\nu^{2} - 2112\nu - 5080 ) / 180 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{7} - 5\nu^{6} + 117\nu^{5} - 225\nu^{4} + 1206\nu^{3} - 2700\nu^{2} + 2112\nu - 5080 ) / 180 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{7} - 171\nu^{5} - 1953\nu^{3} - 3506\nu ) / 90 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4\nu^{7} + 171\nu^{5} + 2043\nu^{3} + 5126\nu ) / 90 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{6} + 36\nu^{4} + 333\nu^{2} + 548 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} - 18\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} - 2\beta_{4} - 2\beta_{3} - 23\beta_{2} + 201 \)
|
| \(\nu^{5}\) | \(=\) |
\( -23\beta_{6} - 29\beta_{5} - 8\beta_{4} + 8\beta_{3} + 368\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 45\beta_{7} + 72\beta_{4} + 72\beta_{3} + 495\beta_{2} - 4121 \)
|
| \(\nu^{7}\) | \(=\) |
\( 495\beta_{6} + 729\beta_{5} + 342\beta_{4} - 342\beta_{3} - 7820\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).
| \(n\) | \(82\) | \(326\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 244.1 |
|
− | 4.80346i | 0 | −15.0732 | 8.38048 | + | 7.40051i | 0 | − | 5.92463i | 33.9759i | 0 | 35.5481 | − | 40.2553i | ||||||||||||||||||||||||||||||||||||
| 244.2 | − | 3.99806i | 0 | −7.98447 | −3.61252 | − | 10.5806i | 0 | − | 7.79840i | − | 0.0620886i | 0 | −42.3020 | + | 14.4431i | ||||||||||||||||||||||||||||||||||||
| 244.3 | − | 1.85698i | 0 | 4.55164 | −7.77825 | + | 8.03112i | 0 | − | 1.02402i | − | 23.3081i | 0 | 14.9136 | + | 14.4440i | ||||||||||||||||||||||||||||||||||||
| 244.4 | − | 1.22227i | 0 | 6.50606 | 10.5103 | − | 3.81233i | 0 | 33.1668i | − | 17.7303i | 0 | −4.65969 | − | 12.8464i | |||||||||||||||||||||||||||||||||||||
| 244.5 | 1.22227i | 0 | 6.50606 | 10.5103 | + | 3.81233i | 0 | − | 33.1668i | 17.7303i | 0 | −4.65969 | + | 12.8464i | ||||||||||||||||||||||||||||||||||||||
| 244.6 | 1.85698i | 0 | 4.55164 | −7.77825 | − | 8.03112i | 0 | 1.02402i | 23.3081i | 0 | 14.9136 | − | 14.4440i | |||||||||||||||||||||||||||||||||||||||
| 244.7 | 3.99806i | 0 | −7.98447 | −3.61252 | + | 10.5806i | 0 | 7.79840i | 0.0620886i | 0 | −42.3020 | − | 14.4431i | |||||||||||||||||||||||||||||||||||||||
| 244.8 | 4.80346i | 0 | −15.0732 | 8.38048 | − | 7.40051i | 0 | 5.92463i | − | 33.9759i | 0 | 35.5481 | + | 40.2553i | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 405.4.b.c | yes | 8 |
| 3.b | odd | 2 | 1 | 405.4.b.b | ✓ | 8 | |
| 5.b | even | 2 | 1 | inner | 405.4.b.c | yes | 8 |
| 5.c | odd | 4 | 2 | 2025.4.a.bd | 8 | ||
| 15.d | odd | 2 | 1 | 405.4.b.b | ✓ | 8 | |
| 15.e | even | 4 | 2 | 2025.4.a.bc | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 405.4.b.b | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 405.4.b.b | ✓ | 8 | 15.d | odd | 2 | 1 | |
| 405.4.b.c | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 405.4.b.c | yes | 8 | 5.b | even | 2 | 1 | inner |
| 2025.4.a.bc | 8 | 15.e | even | 4 | 2 | ||
| 2025.4.a.bd | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):
|
\( T_{2}^{8} + 44T_{2}^{6} + 567T_{2}^{4} + 2024T_{2}^{2} + 1900 \)
|
|
\( T_{11}^{4} - 1755T_{11}^{2} + 7038T_{11} + 502920 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 44 T^{6} + \cdots + 1900 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 15 T^{7} + \cdots + 244140625 \)
|
| $7$ |
\( T^{8} + 1197 T^{6} + \cdots + 2462400 \)
|
| $11$ |
\( (T^{4} - 1755 T^{2} + \cdots + 502920)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 68838460416 \)
|
| $17$ |
\( T^{8} + \cdots + 24363576945664 \)
|
| $19$ |
\( (T^{4} - 59 T^{3} + \cdots + 2964730)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 533400694426624 \)
|
| $29$ |
\( (T^{4} - 159 T^{3} + \cdots - 16354800)^{2} \)
|
| $31$ |
\( (T^{4} + 208 T^{3} + \cdots - 27072584)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 53\!\cdots\!00 \)
|
| $41$ |
\( (T^{4} - 243 T^{3} + \cdots + 54333414)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 57\!\cdots\!96 \)
|
| $47$ |
\( T^{8} + \cdots + 18\!\cdots\!16 \)
|
| $53$ |
\( T^{8} + \cdots + 21\!\cdots\!24 \)
|
| $59$ |
\( (T^{4} + 573 T^{3} + \cdots + 77712038244)^{2} \)
|
| $61$ |
\( (T^{4} + 199 T^{3} + \cdots + 37684911880)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 68\!\cdots\!56 \)
|
| $71$ |
\( (T^{4} + 864 T^{3} + \cdots + 17785914552)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 37\!\cdots\!04 \)
|
| $79$ |
\( (T^{4} - 1298 T^{3} + \cdots - 256210144256)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 50\!\cdots\!44 \)
|
| $89$ |
\( (T^{4} - 543 T^{3} + \cdots + 438625777050)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 31\!\cdots\!36 \)
|
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