Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [64,20,Mod(1,64)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(64, 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("64.1");
S:= CuspForms(chi, 20);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| Character orbit: | \([\chi]\) | \(=\) | 64.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: | \(146.442685796\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 432831x^{3} + 33240116x^{2} + 37915158872x - 3359571873840 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{48}\cdot 3^{4}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 32) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 432831x^{3} + 33240116x^{2} + 37915158872x - 3359571873840 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 14273408 \nu^{4} + 442975952 \nu^{3} - 5564900148288 \nu^{2} + 529724742723760 \nu + 33\!\cdots\!07 ) / 5706567094417 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5622053120 \nu^{4} + 586336506736 \nu^{3} + \cdots + 25\!\cdots\!69 ) / 17119701283251 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 91807321216 \nu^{4} - 58526744860352 \nu^{3} + \cdots + 68\!\cdots\!10 ) / 17119701283251 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2012766219904 \nu^{4} - 846300761381072 \nu^{3} + \cdots - 41\!\cdots\!58 ) / 17119701283251 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - 17\beta_{3} - 395\beta_{2} + 62418\beta _1 + 3551355 ) / 8847360 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 1403\beta_{4} - 14797\beta_{3} + 669797\beta_{2} - 53717520\beta _1 + 6127076389221 ) / 35389440 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 13291\beta_{4} - 833252\beta_{3} - 26056250\beta_{2} + 2259275043\beta _1 - 28642395603105 ) / 1474560 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 388649557 \beta_{4} - 2624733395 \beta_{3} + 339185535067 \beta_{2} - 17743286361408 \beta _1 + 15\!\cdots\!51 ) / 35389440 \)
|
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 | −60856.1 | 0 | −2.03089e6 | 0 | −1.60169e8 | 0 | 2.54120e9 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −21346.9 | 0 | −500910. | 0 | 1.55834e8 | 0 | −7.06572e8 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −11092.2 | 0 | 4.38512e6 | 0 | −7.75914e7 | 0 | −1.03923e9 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 33300.1 | 0 | −5.35669e6 | 0 | −4.02402e7 | 0 | −5.33616e7 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 53571.0 | 0 | 7.60870e6 | 0 | 1.50080e7 | 0 | 1.70759e9 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 64.20.a.p | 5 | |
| 4.b | odd | 2 | 1 | 64.20.a.q | 5 | ||
| 8.b | even | 2 | 1 | 32.20.a.e | yes | 5 | |
| 8.d | odd | 2 | 1 | 32.20.a.d | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.20.a.d | ✓ | 5 | 8.d | odd | 2 | 1 | |
| 32.20.a.e | yes | 5 | 8.b | even | 2 | 1 | |
| 64.20.a.p | 5 | 1.a | even | 1 | 1 | trivial | |
| 64.20.a.q | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 6424 T_{3}^{4} - 4109835648 T_{3}^{3} - 11222149100544 T_{3}^{2} + \cdots + 25\!\cdots\!16 \)
acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(64))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + \cdots + 25\!\cdots\!16 \)
|
| $5$ |
\( T^{5} + \cdots + 18\!\cdots\!00 \)
|
| $7$ |
\( T^{5} + \cdots + 11\!\cdots\!36 \)
|
| $11$ |
\( T^{5} + \cdots + 17\!\cdots\!00 \)
|
| $13$ |
\( T^{5} + \cdots + 63\!\cdots\!00 \)
|
| $17$ |
\( T^{5} + \cdots + 62\!\cdots\!00 \)
|
| $19$ |
\( T^{5} + \cdots - 43\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots - 31\!\cdots\!48 \)
|
| $29$ |
\( T^{5} + \cdots - 28\!\cdots\!08 \)
|
| $31$ |
\( T^{5} + \cdots + 18\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 16\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots - 22\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots + 33\!\cdots\!16 \)
|
| $47$ |
\( T^{5} + \cdots + 19\!\cdots\!76 \)
|
| $53$ |
\( T^{5} + \cdots + 24\!\cdots\!00 \)
|
| $59$ |
\( T^{5} + \cdots - 12\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 88\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots - 88\!\cdots\!08 \)
|
| $71$ |
\( T^{5} + \cdots + 10\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots - 10\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots + 41\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 23\!\cdots\!40 \)
|
| $89$ |
\( T^{5} + \cdots - 30\!\cdots\!16 \)
|
| $97$ |
\( T^{5} + \cdots - 21\!\cdots\!00 \)
|
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