Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,4,Mod(225,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.225");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.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: | \(26.4328556826\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.55296463104.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 50x^{6} + 817x^{4} + 4992x^{2} + 9216 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{17} \) |
| 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
\( x^{8} + 50x^{6} + 817x^{4} + 4992x^{2} + 9216 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{6} - 154\nu^{4} - 725\nu^{2} + 2496 ) / 144 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\nu^{7} + 458\nu^{5} + 3325\nu^{3} - 4416\nu ) / 3456 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 50\nu^{4} - 625\nu^{2} - 1344 ) / 48 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\nu^{7} + 454\nu^{5} + 5339\nu^{3} + 16800\nu ) / 1728 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 50\nu^{5} + 721\nu^{3} + 2592\nu ) / 96 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{6} - 38\nu^{4} - 397\nu^{2} - 1092 ) / 18 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{7} + \beta_{4} + \beta_{2} - 50 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{6} + 8\beta_{5} - 8\beta_{3} - 17\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 25\beta_{7} - 35\beta_{4} - 19\beta_{2} + 866 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 76\beta_{6} - 252\beta_{5} + 216\beta_{3} + 337\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -625\beta_{7} + 933\beta_{4} + 325\beta_{2} - 17426 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -2166\beta_{6} + 6832\beta_{5} - 5032\beta_{3} - 7185\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(129\) | \(197\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 225.1 |
|
0 | − | 6.93655i | 0 | − | 12.5504i | 0 | 7.00000 | 0 | −21.1157 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 225.2 | 0 | − | 5.66860i | 0 | − | 9.28240i | 0 | 7.00000 | 0 | −5.13301 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 225.3 | 0 | − | 5.57189i | 0 | 2.18670i | 0 | 7.00000 | 0 | −4.04596 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 225.4 | 0 | − | 0.839839i | 0 | 8.91875i | 0 | 7.00000 | 0 | 26.2947 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 225.5 | 0 | 0.839839i | 0 | − | 8.91875i | 0 | 7.00000 | 0 | 26.2947 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 225.6 | 0 | 5.57189i | 0 | − | 2.18670i | 0 | 7.00000 | 0 | −4.04596 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 225.7 | 0 | 5.66860i | 0 | 9.28240i | 0 | 7.00000 | 0 | −5.13301 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 225.8 | 0 | 6.93655i | 0 | 12.5504i | 0 | 7.00000 | 0 | −21.1157 | 0 | |||||||||||||||||||||||||||||||||||||||||||
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 | 448.4.b.f | yes | 8 |
| 4.b | odd | 2 | 1 | 448.4.b.e | ✓ | 8 | |
| 8.b | even | 2 | 1 | inner | 448.4.b.f | yes | 8 |
| 8.d | odd | 2 | 1 | 448.4.b.e | ✓ | 8 | |
| 16.e | even | 4 | 1 | 1792.4.a.k | 4 | ||
| 16.e | even | 4 | 1 | 1792.4.a.n | 4 | ||
| 16.f | odd | 4 | 1 | 1792.4.a.j | 4 | ||
| 16.f | odd | 4 | 1 | 1792.4.a.o | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 448.4.b.e | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 448.4.b.e | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 448.4.b.f | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 448.4.b.f | yes | 8 | 8.b | even | 2 | 1 | inner |
| 1792.4.a.j | 4 | 16.f | odd | 4 | 1 | ||
| 1792.4.a.k | 4 | 16.e | even | 4 | 1 | ||
| 1792.4.a.n | 4 | 16.e | even | 4 | 1 | ||
| 1792.4.a.o | 4 | 16.f | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\):
|
\( T_{3}^{8} + 112T_{3}^{6} + 4116T_{3}^{4} + 50848T_{3}^{2} + 33856 \)
|
|
\( T_{23}^{4} - 64T_{23}^{3} - 6668T_{23}^{2} + 325632T_{23} - 2694144 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 112 T^{6} + \cdots + 33856 \)
|
| $5$ |
\( T^{8} + 328 T^{6} + \cdots + 5161984 \)
|
| $7$ |
\( (T - 7)^{8} \)
|
| $11$ |
\( T^{8} + \cdots + 443321598976 \)
|
| $13$ |
\( T^{8} + \cdots + 4312965325824 \)
|
| $17$ |
\( (T^{4} - 40 T^{3} + \cdots + 3656976)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 910859924367936 \)
|
| $23$ |
\( (T^{4} - 64 T^{3} + \cdots - 2694144)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 12\!\cdots\!96 \)
|
| $31$ |
\( (T^{4} + 24 T^{3} + \cdots + 905786368)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 66\!\cdots\!56 \)
|
| $41$ |
\( (T^{4} + 16 T^{3} + \cdots - 339050352)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 59\!\cdots\!96 \)
|
| $47$ |
\( (T^{4} + 296 T^{3} + \cdots - 746268672)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 12\!\cdots\!24 \)
|
| $59$ |
\( T^{8} + \cdots + 70\!\cdots\!04 \)
|
| $61$ |
\( T^{8} + \cdots + 34\!\cdots\!64 \)
|
| $67$ |
\( T^{8} + \cdots + 75\!\cdots\!36 \)
|
| $71$ |
\( (T^{4} + 584 T^{3} + \cdots - 64660948992)^{2} \)
|
| $73$ |
\( (T^{4} + 768 T^{3} + \cdots + 2805981264)^{2} \)
|
| $79$ |
\( (T^{4} + 776 T^{3} + \cdots + 1066702848)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 14\!\cdots\!76 \)
|
| $89$ |
\( (T^{4} - 2576 T^{3} + \cdots - 211182163824)^{2} \)
|
| $97$ |
\( (T^{4} - 1640 T^{3} + \cdots - 20471621552)^{2} \)
|
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