Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [96,4,Mod(49,96)]
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, 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("96.49");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 96 = 2^{5} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 96.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: | \(5.66418336055\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.8248384.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 24) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{4} + 3\nu^{2} + 12\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{5} - 6\nu^{4} + 9\nu^{3} + 6\nu^{2} + 24\nu - 96 ) / 32 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{5} + 6\nu^{4} - 17\nu^{3} + 90\nu^{2} - 184\nu + 96 ) / 32 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{5} + 6\nu^{4} + 111\nu^{3} + 90\nu^{2} - 56\nu - 672 ) / 32 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} + 6\nu^{4} - 3\nu^{3} + 18\nu^{2} - 36\nu + 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 6\beta_{3} - 6\beta_{2} + 3\beta _1 - 1 ) / 48 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{4} + 9\beta_{3} - 20\beta_{2} + 6\beta _1 - 18 ) / 48 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{5} + 12\beta_{4} - 6\beta_{3} + 6\beta_{2} - 3\beta _1 + 289 ) / 48 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4\beta_{5} + 3\beta_{4} - 15\beta_{3} - 44\beta_{2} + 2\beta _1 - 38 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -19\beta_{5} + 24\beta_{4} + 42\beta_{3} - 318\beta_{2} + 15\beta _1 - 485 ) / 48 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 3.00000i | 0 | − | 9.15486i | 0 | −27.4175 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 49.2 | 0 | − | 3.00000i | 0 | 0.612661i | 0 | 22.7441 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | − | 3.00000i | 0 | 18.5422i | 0 | −9.32669 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | 3.00000i | 0 | − | 18.5422i | 0 | −9.32669 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | 3.00000i | 0 | − | 0.612661i | 0 | 22.7441 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 3.00000i | 0 | 9.15486i | 0 | −27.4175 | 0 | −9.00000 | 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 | 96.4.d.a | 6 | |
| 3.b | odd | 2 | 1 | 288.4.d.d | 6 | ||
| 4.b | odd | 2 | 1 | 24.4.d.a | ✓ | 6 | |
| 8.b | even | 2 | 1 | inner | 96.4.d.a | 6 | |
| 8.d | odd | 2 | 1 | 24.4.d.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 72.4.d.d | 6 | ||
| 16.e | even | 4 | 1 | 768.4.a.q | 3 | ||
| 16.e | even | 4 | 1 | 768.4.a.t | 3 | ||
| 16.f | odd | 4 | 1 | 768.4.a.r | 3 | ||
| 16.f | odd | 4 | 1 | 768.4.a.s | 3 | ||
| 24.f | even | 2 | 1 | 72.4.d.d | 6 | ||
| 24.h | odd | 2 | 1 | 288.4.d.d | 6 | ||
| 48.i | odd | 4 | 1 | 2304.4.a.bu | 3 | ||
| 48.i | odd | 4 | 1 | 2304.4.a.bw | 3 | ||
| 48.k | even | 4 | 1 | 2304.4.a.bt | 3 | ||
| 48.k | even | 4 | 1 | 2304.4.a.bv | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.4.d.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 24.4.d.a | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 72.4.d.d | 6 | 12.b | even | 2 | 1 | ||
| 72.4.d.d | 6 | 24.f | even | 2 | 1 | ||
| 96.4.d.a | 6 | 1.a | even | 1 | 1 | trivial | |
| 96.4.d.a | 6 | 8.b | even | 2 | 1 | inner | |
| 288.4.d.d | 6 | 3.b | odd | 2 | 1 | ||
| 288.4.d.d | 6 | 24.h | odd | 2 | 1 | ||
| 768.4.a.q | 3 | 16.e | even | 4 | 1 | ||
| 768.4.a.r | 3 | 16.f | odd | 4 | 1 | ||
| 768.4.a.s | 3 | 16.f | odd | 4 | 1 | ||
| 768.4.a.t | 3 | 16.e | even | 4 | 1 | ||
| 2304.4.a.bt | 3 | 48.k | even | 4 | 1 | ||
| 2304.4.a.bu | 3 | 48.i | odd | 4 | 1 | ||
| 2304.4.a.bv | 3 | 48.k | even | 4 | 1 | ||
| 2304.4.a.bw | 3 | 48.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(96, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 9)^{3} \)
|
| $5$ |
\( T^{6} + 428 T^{4} + \cdots + 10816 \)
|
| $7$ |
\( (T^{3} + 14 T^{2} + \cdots - 5816)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 2415919104 \)
|
| $13$ |
\( T^{6} + \cdots + 3121680384 \)
|
| $17$ |
\( (T^{3} - 26 T^{2} + \cdots + 477576)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 75488661504 \)
|
| $23$ |
\( (T^{3} + 164 T^{2} + \cdots + 45504)^{2} \)
|
| $29$ |
\( T^{6} + \cdots + 3766031424 \)
|
| $31$ |
\( (T^{3} - 318 T^{2} + \cdots + 3749624)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 6879707136 \)
|
| $41$ |
\( (T^{3} - 118 T^{2} + \cdots + 19985976)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 73984219582464 \)
|
| $47$ |
\( (T^{3} - 204 T^{2} + \cdots + 1964736)^{2} \)
|
| $53$ |
\( T^{6} + \cdots + 427051482970176 \)
|
| $59$ |
\( T^{6} + \cdots + 72651484205056 \)
|
| $61$ |
\( T^{6} + \cdots + 10\!\cdots\!56 \)
|
| $67$ |
\( T^{6} + \cdots + 10\!\cdots\!84 \)
|
| $71$ |
\( (T^{3} - 852 T^{2} + \cdots + 85084992)^{2} \)
|
| $73$ |
\( (T^{3} - 478 T^{2} + \cdots + 120833304)^{2} \)
|
| $79$ |
\( (T^{3} - 22 T^{2} + \cdots + 7902616)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 14\!\cdots\!96 \)
|
| $89$ |
\( (T^{3} + 110 T^{2} + \cdots + 1423656)^{2} \)
|
| $97$ |
\( (T^{3} + 1222 T^{2} + \cdots - 74802424)^{2} \)
|
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