Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [96,5,Mod(65,96)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(96, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("96.65");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 96 = 2^{5} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 96.e (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: | \(9.92351645605\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.3317760000.8 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{6} \) |
| 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
\( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 38\nu^{7} + 69\nu^{6} + 35\nu^{5} + 168\nu^{4} + 203\nu^{3} + 294\nu^{2} + 1683\nu + 1728 ) / 189 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -8\nu^{7} - 324\nu^{6} - 14\nu^{5} + 70\nu^{3} - 162\nu - 7128 ) / 189 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 64\nu^{7} + 112\nu^{5} - 560\nu^{3} + 1296\nu ) / 189 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 130\nu^{7} - 195\nu^{6} - 119\nu^{5} - 672\nu^{4} + 217\nu^{3} - 42\nu^{2} + 3609\nu - 3996 ) / 189 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -6\nu^{7} + 9\nu^{6} + 21\nu^{5} + 21\nu^{3} + 126\nu^{2} - 27\nu + 324 ) / 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 212\nu^{7} - 1740\nu^{6} + 434\nu^{5} - 2856\nu^{4} + 1610\nu^{3} - 12936\nu^{2} + 11538\nu - 49788 ) / 189 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -704\nu^{7} - 1232\nu^{5} - 2912\nu^{3} - 14256\nu ) / 189 \)
|
| \(\nu\) | \(=\) |
\( ( 6\beta_{7} + 3\beta_{6} + 5\beta_{5} + 6\beta_{4} + 12\beta_{3} + 75\beta_1 ) / 576 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -6\beta_{6} + 10\beta_{5} + 24\beta_{4} + 3\beta_{3} + 24\beta_{2} - 6\beta _1 - 576 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} - 11\beta_{3} ) / 48 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -3\beta_{6} - 31\beta_{5} - 96\beta_{4} + 12\beta_{3} + 96\beta_{2} + 213\beta _1 + 288 ) / 576 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -30\beta_{7} + 15\beta_{6} + 73\beta_{5} - 114\beta_{4} + 156\beta_{3} - 201\beta_1 ) / 576 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -7\beta_{3} - 56\beta_{2} - 2112 ) / 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -174\beta_{7} - 87\beta_{6} - 229\beta_{5} + 78\beta_{4} + 30\beta_{3} - 1167\beta_1 ) / 576 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(37\) | \(65\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −8.77196 | − | 2.01312i | 0 | 24.0044i | 0 | 6.15598 | 0 | 72.8947 | + | 35.3181i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.2 | 0 | −8.77196 | + | 2.01312i | 0 | − | 24.0044i | 0 | 6.15598 | 0 | 72.8947 | − | 35.3181i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.3 | 0 | −1.02600 | − | 8.94133i | 0 | 29.6612i | 0 | 52.6318 | 0 | −78.8947 | + | 18.3475i | 0 | |||||||||||||||||||||||||||||||||||||||
| 65.4 | 0 | −1.02600 | + | 8.94133i | 0 | − | 29.6612i | 0 | 52.6318 | 0 | −78.8947 | − | 18.3475i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.5 | 0 | 1.02600 | − | 8.94133i | 0 | − | 29.6612i | 0 | −52.6318 | 0 | −78.8947 | − | 18.3475i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.6 | 0 | 1.02600 | + | 8.94133i | 0 | 29.6612i | 0 | −52.6318 | 0 | −78.8947 | + | 18.3475i | 0 | |||||||||||||||||||||||||||||||||||||||
| 65.7 | 0 | 8.77196 | − | 2.01312i | 0 | − | 24.0044i | 0 | −6.15598 | 0 | 72.8947 | − | 35.3181i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.8 | 0 | 8.77196 | + | 2.01312i | 0 | 24.0044i | 0 | −6.15598 | 0 | 72.8947 | + | 35.3181i | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 96.5.e.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 96.5.e.b | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 96.5.e.b | ✓ | 8 |
| 8.b | even | 2 | 1 | 192.5.e.h | 8 | ||
| 8.d | odd | 2 | 1 | 192.5.e.h | 8 | ||
| 12.b | even | 2 | 1 | inner | 96.5.e.b | ✓ | 8 |
| 24.f | even | 2 | 1 | 192.5.e.h | 8 | ||
| 24.h | odd | 2 | 1 | 192.5.e.h | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 96.5.e.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 96.5.e.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 96.5.e.b | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 96.5.e.b | ✓ | 8 | 12.b | even | 2 | 1 | inner |
| 192.5.e.h | 8 | 8.b | even | 2 | 1 | ||
| 192.5.e.h | 8 | 8.d | odd | 2 | 1 | ||
| 192.5.e.h | 8 | 24.f | even | 2 | 1 | ||
| 192.5.e.h | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 1456T_{5}^{2} + 506944 \)
acting on \(S_{5}^{\mathrm{new}}(96, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 12 T^{6} + \cdots + 43046721 \)
|
| $5$ |
\( (T^{4} + 1456 T^{2} + 506944)^{2} \)
|
| $7$ |
\( (T^{4} - 2808 T^{2} + 104976)^{2} \)
|
| $11$ |
\( (T^{4} + 44496 T^{2} + 403527744)^{2} \)
|
| $13$ |
\( (T^{2} - 36 T - 22716)^{4} \)
|
| $17$ |
\( (T^{4} + 108544 T^{2} + 1435500544)^{2} \)
|
| $19$ |
\( (T^{4} - 614520 T^{2} + 69727683600)^{2} \)
|
| $23$ |
\( (T^{4} + 1131840 T^{2} + 229939430400)^{2} \)
|
| $29$ |
\( (T^{4} + 1344304 T^{2} + 23198945344)^{2} \)
|
| $31$ |
\( (T^{4} - 1891512 T^{2} + 876163393296)^{2} \)
|
| $37$ |
\( (T^{2} - 644 T - 1762556)^{4} \)
|
| $41$ |
\( (T^{4} + \cdots + 5731619046400)^{2} \)
|
| $43$ |
\( (T^{4} + \cdots + 51183207211536)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 33590277586944)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 17300209059904)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 212217533665344)^{2} \)
|
| $61$ |
\( (T^{2} - 4228 T - 12327164)^{4} \)
|
| $67$ |
\( (T^{4} + \cdots + 15053624010000)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 17006320272384)^{2} \)
|
| $73$ |
\( (T^{2} + 6300 T - 8140860)^{4} \)
|
| $79$ |
\( (T^{4} - 42762168 T^{2} + 8171436816)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 18\!\cdots\!44)^{2} \)
|
| $89$ |
\( (T^{4} + \cdots + 77439436801024)^{2} \)
|
| $97$ |
\( (T^{2} + 15660 T + 51148260)^{4} \)
|
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