Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [256,11,Mod(127,256)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("256.127");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 256.d (of order \(2\), degree \(1\), not 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: | \(162.651456684\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.1348622796960000.40 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 253x^{6} + 48133x^{4} - 4016628x^{2} + 252047376 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{58}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 16) |
| 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
\( x^{8} - 253x^{6} + 48133x^{4} - 4016628x^{2} + 252047376 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -164224\nu^{6} + 44667424\nu^{4} - 4508329312\nu^{2} + 305056704960 ) / 21226653 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -1596928\nu^{6} + 400466560\nu^{4} - 52411831168\nu^{2} + 3235362165504 ) / 191039877 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -768\nu^{6} - 1591446912 ) / 48133 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 253\nu^{7} - 48133\nu^{5} + 6160645\nu^{3} - 252047376\nu ) / 94767813 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 66788\nu^{7} - 61032644\nu^{5} + 9327982868\nu^{3} - 1237216807008\nu ) / 12035512251 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -480148\nu^{7} - 5466176012\nu^{5} + 679915783676\nu^{3} - 84852434465568\nu ) / 12035512251 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -9237283\nu^{7} + 1757383963\nu^{5} - 370494670363\nu^{3} + 9202501745136\nu ) / 48142049004 \)
|
| \(\nu\) | \(=\) |
\( ( -16\beta_{7} + 3\beta_{6} - 345\beta_{5} - 388\beta_{4} ) / 12288 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 32\beta_{3} - 783\beta_{2} + 780\beta _1 + 3108864 ) / 49152 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -508\beta_{7} - 36511\beta_{4} ) / 1536 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -8096\beta_{3} - 79029\beta_{2} + 102084\beta _1 - 396374016 ) / 49152 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -520160\beta_{7} - 145347\beta_{6} + 10594329\beta_{5} - 61578488\beta_{4} ) / 24576 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -48133\beta_{3} - 1591446912 ) / 768 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 67080416\beta_{7} - 21674715\beta_{6} + 1328158689\beta_{5} + 10942168376\beta_{4} ) / 24576 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(255\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 127.1 |
|
0 | −408.379 | 0 | − | 5364.66i | 0 | − | 9544.75i | 0 | 107724. | 0 | ||||||||||||||||||||||||||||||||||||||||
| 127.2 | 0 | −408.379 | 0 | 5364.66i | 0 | 9544.75i | 0 | 107724. | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 127.3 | 0 | −214.389 | 0 | − | 3264.66i | 0 | − | 5808.15i | 0 | −13086.3 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 127.4 | 0 | −214.389 | 0 | 3264.66i | 0 | 5808.15i | 0 | −13086.3 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 127.5 | 0 | 214.389 | 0 | − | 3264.66i | 0 | 5808.15i | 0 | −13086.3 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 127.6 | 0 | 214.389 | 0 | 3264.66i | 0 | − | 5808.15i | 0 | −13086.3 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 127.7 | 0 | 408.379 | 0 | − | 5364.66i | 0 | 9544.75i | 0 | 107724. | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 127.8 | 0 | 408.379 | 0 | 5364.66i | 0 | − | 9544.75i | 0 | 107724. | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 256.11.d.g | 8 | |
| 4.b | odd | 2 | 1 | inner | 256.11.d.g | 8 | |
| 8.b | even | 2 | 1 | inner | 256.11.d.g | 8 | |
| 8.d | odd | 2 | 1 | inner | 256.11.d.g | 8 | |
| 16.e | even | 4 | 1 | 16.11.c.b | ✓ | 4 | |
| 16.e | even | 4 | 1 | 64.11.c.b | 4 | ||
| 16.f | odd | 4 | 1 | 16.11.c.b | ✓ | 4 | |
| 16.f | odd | 4 | 1 | 64.11.c.b | 4 | ||
| 48.i | odd | 4 | 1 | 144.11.g.d | 4 | ||
| 48.k | even | 4 | 1 | 144.11.g.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 16.11.c.b | ✓ | 4 | 16.e | even | 4 | 1 | |
| 16.11.c.b | ✓ | 4 | 16.f | odd | 4 | 1 | |
| 64.11.c.b | 4 | 16.e | even | 4 | 1 | ||
| 64.11.c.b | 4 | 16.f | odd | 4 | 1 | ||
| 144.11.g.d | 4 | 48.i | odd | 4 | 1 | ||
| 144.11.g.d | 4 | 48.k | even | 4 | 1 | ||
| 256.11.d.g | 8 | 1.a | even | 1 | 1 | trivial | |
| 256.11.d.g | 8 | 4.b | odd | 2 | 1 | inner | |
| 256.11.d.g | 8 | 8.b | even | 2 | 1 | inner | |
| 256.11.d.g | 8 | 8.d | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 212736T_{3}^{2} + 7665352704 \)
acting on \(S_{11}^{\mathrm{new}}(256, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} - 212736 T^{2} + 7665352704)^{2} \)
|
| $5$ |
\( (T^{4} + \cdots + 306733890992400)^{2} \)
|
| $7$ |
\( (T^{4} + \cdots + 30\!\cdots\!44)^{2} \)
|
| $11$ |
\( (T^{4} + \cdots + 75\!\cdots\!04)^{2} \)
|
| $13$ |
\( (T^{4} + \cdots + 43\!\cdots\!16)^{2} \)
|
| $17$ |
\( (T^{2} + \cdots + 1113290672004)^{4} \)
|
| $19$ |
\( (T^{4} + \cdots + 76\!\cdots\!04)^{2} \)
|
| $23$ |
\( (T^{4} + \cdots + 12\!\cdots\!04)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots + 36\!\cdots\!76)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 25\!\cdots\!44)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 69\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots - 84\!\cdots\!56)^{4} \)
|
| $43$ |
\( (T^{4} + \cdots + 46\!\cdots\!64)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 38\!\cdots\!04)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 18\!\cdots\!96)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 37\!\cdots\!04)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 55\!\cdots\!56)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 90\!\cdots\!04)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 16\!\cdots\!04)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots - 14\!\cdots\!16)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 26\!\cdots\!00)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 67\!\cdots\!84)^{2} \)
|
| $89$ |
\( (T^{2} + \cdots - 35\!\cdots\!24)^{4} \)
|
| $97$ |
\( (T^{2} + \cdots - 14\!\cdots\!04)^{4} \)
|
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