Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [32,20,Mod(1,32)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(32, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 20, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("32.1");
S:= CuspForms(chi, 20);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 32 = 2^{5} \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| Character orbit: | \([\chi]\) | \(=\) | 32.a (trivial) |
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: | yes |
| Analytic conductor: | \(73.2213428980\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 257356x^{2} - 48823880x + 379014449 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{27}\cdot 3^{4}\cdot 5^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 257356x^{2} - 48823880x + 379014449 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4000\nu^{3} + 1089488\nu^{2} + 694123680\nu + 6278503136 ) / 354151 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7680\nu^{3} + 537600\nu^{2} + 2327416320\nu + 212048256000 ) / 50593 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -10091520\nu^{3} + 3270864384\nu^{2} + 1678080384000\nu - 51357916081152 ) / 354151 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 48\beta_{2} - 3168\beta_1 ) / 737280 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} + 21\beta_{2} - 5328\beta _1 + 296474112 ) / 2304 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 347849\beta_{3} + 10159824\beta_{2} - 1079406432\beta _1 + 26997652684800 ) / 737280 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −34953.3 | 0 | −3.30419e6 | 0 | −7.25587e7 | 0 | 5.94752e7 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −32534.0 | 0 | 5.76657e6 | 0 | 1.36127e8 | 0 | −1.03798e8 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 32534.0 | 0 | 5.76657e6 | 0 | −1.36127e8 | 0 | −1.03798e8 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 34953.3 | 0 | −3.30419e6 | 0 | 7.25587e7 | 0 | 5.94752e7 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 32.20.a.c | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 32.20.a.c | ✓ | 4 |
| 8.b | even | 2 | 1 | 64.20.a.n | 4 | ||
| 8.d | odd | 2 | 1 | 64.20.a.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.20.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 32.20.a.c | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 64.20.a.n | 4 | 8.b | even | 2 | 1 | ||
| 64.20.a.n | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 2280199680T_{3}^{2} + 1293163081762406400 \)
acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(32))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $5$ |
\( (T^{2} + \cdots - 19053811324700)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 97\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} + \cdots - 34\!\cdots\!76)^{2} \)
|
| $17$ |
\( (T^{2} + \cdots - 24\!\cdots\!24)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 28\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 53\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + \cdots - 56\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 63\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} + \cdots - 12\!\cdots\!64)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots - 67\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 61\!\cdots\!00 \)
|
| $53$ |
\( (T^{2} + \cdots - 27\!\cdots\!76)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 70\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots - 16\!\cdots\!96)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $89$ |
\( (T^{2} + \cdots + 15\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 54\!\cdots\!96)^{2} \)
|
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