Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1364,2,Mod(1,1364)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1364, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1364.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1364 = 2^{2} \cdot 11 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1364.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: | \(10.8915948357\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} - 22x^{8} + 42x^{7} + 155x^{6} - 272x^{5} - 369x^{4} + 536x^{3} + 237x^{2} - 282x + 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} - 22x^{8} + 42x^{7} + 155x^{6} - 272x^{5} - 369x^{4} + 536x^{3} + 237x^{2} - 282x + 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 29 \nu^{9} + 33 \nu^{8} - 475 \nu^{7} - 867 \nu^{6} + 2250 \nu^{5} + 6752 \nu^{4} - 2949 \nu^{3} + \cdots + 7432 ) / 1724 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 23 \nu^{9} - 63 \nu^{8} - 347 \nu^{7} + 1185 \nu^{6} + 1190 \nu^{5} - 7248 \nu^{4} + 1555 \nu^{3} + \cdots - 9800 ) / 1724 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 25 \nu^{9} + 31 \nu^{8} + 677 \nu^{7} - 501 \nu^{6} - 6428 \nu^{5} + 2294 \nu^{4} + 23795 \nu^{3} + \cdots - 2840 ) / 1724 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 25 \nu^{9} + 31 \nu^{8} + 677 \nu^{7} - 501 \nu^{6} - 6428 \nu^{5} + 2294 \nu^{4} + 23795 \nu^{3} + \cdots + 5780 ) / 1724 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 24 \nu^{9} + 47 \nu^{8} + 512 \nu^{7} - 843 \nu^{6} - 3378 \nu^{5} + 3909 \nu^{4} + 6379 \nu^{3} + \cdots - 2554 ) / 862 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 21 \nu^{9} + 95 \nu^{8} + 448 \nu^{7} - 1869 \nu^{6} - 2848 \nu^{5} + 10909 \nu^{4} + 4989 \nu^{3} + \cdots + 1752 ) / 862 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 79 \nu^{9} + 29 \nu^{8} + 1829 \nu^{7} - 997 \nu^{6} - 13382 \nu^{5} + 9042 \nu^{4} + 32437 \nu^{3} + \cdots + 17920 ) / 1724 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 79 \nu^{9} - 29 \nu^{8} - 1829 \nu^{7} + 997 \nu^{6} + 14244 \nu^{5} - 9042 \nu^{4} - 41919 \nu^{3} + \cdots - 12748 ) / 1724 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{8} - \beta_{6} - 10\beta_{5} + 11\beta_{4} - \beta_{3} - \beta _1 + 40 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{9} + 2 \beta_{8} + 11 \beta_{7} - 11 \beta_{6} + 10 \beta_{5} - 10 \beta_{4} + 11 \beta_{3} + \cdots - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 17 \beta_{9} + 15 \beta_{8} + \beta_{7} - 14 \beta_{6} - 90 \beta_{5} + 107 \beta_{4} - 12 \beta_{3} + \cdots + 349 \)
|
| \(\nu^{7}\) | \(=\) |
\( 35 \beta_{9} + 35 \beta_{8} + 104 \beta_{7} - 105 \beta_{6} + 91 \beta_{5} - 84 \beta_{4} + 107 \beta_{3} + \cdots - 23 \)
|
| \(\nu^{8}\) | \(=\) |
\( 214 \beta_{9} + 172 \beta_{8} + 26 \beta_{7} - 158 \beta_{6} - 794 \beta_{5} + 1012 \beta_{4} + \cdots + 3122 \)
|
| \(\nu^{9}\) | \(=\) |
\( 450 \beta_{9} + 438 \beta_{8} + 952 \beta_{7} - 974 \beta_{6} + 821 \beta_{5} - 679 \beta_{4} + \cdots - 305 \)
|
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 | −3.05437 | 0 | −0.269590 | 0 | −4.32071 | 0 | 6.32915 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.96887 | 0 | 2.16270 | 0 | 4.70623 | 0 | 5.81418 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.51461 | 0 | 2.52340 | 0 | −0.399977 | 0 | −0.705966 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00835 | 0 | −4.45364 | 0 | 0.0409233 | 0 | −1.98324 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.0940028 | 0 | −1.05986 | 0 | 4.16507 | 0 | −2.99116 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.629509 | 0 | 3.80331 | 0 | 0.575679 | 0 | −2.60372 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.28985 | 0 | −2.22116 | 0 | −3.40412 | 0 | −1.33628 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.47921 | 0 | 0.0543309 | 0 | 2.71443 | 0 | 3.14649 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.96590 | 0 | 4.07452 | 0 | −4.37251 | 0 | 5.79657 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 3.08771 | 0 | 1.38599 | 0 | 2.29499 | 0 | 6.53397 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(31\) | \( -1 \) |
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 | 1364.2.a.f | ✓ | 10 |
| 4.b | odd | 2 | 1 | 5456.2.a.be | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1364.2.a.f | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 5456.2.a.be | 10 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 2 T_{3}^{9} - 22 T_{3}^{8} + 42 T_{3}^{7} + 155 T_{3}^{6} - 272 T_{3}^{5} - 369 T_{3}^{4} + \cdots + 24 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1364))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} - 2 T^{9} + \cdots + 24 \)
|
| $5$ |
\( T^{10} - 6 T^{9} + \cdots + 18 \)
|
| $7$ |
\( T^{10} - 2 T^{9} + \cdots + 74 \)
|
| $11$ |
\( (T - 1)^{10} \)
|
| $13$ |
\( T^{10} - 10 T^{9} + \cdots - 112896 \)
|
| $17$ |
\( T^{10} - 100 T^{8} + \cdots + 160704 \)
|
| $19$ |
\( T^{10} + 9 T^{9} + \cdots + 35368 \)
|
| $23$ |
\( T^{10} - 16 T^{9} + \cdots + 49482624 \)
|
| $29$ |
\( T^{10} - 8 T^{9} + \cdots + 41472 \)
|
| $31$ |
\( (T - 1)^{10} \)
|
| $37$ |
\( T^{10} - 20 T^{9} + \cdots - 6688768 \)
|
| $41$ |
\( T^{10} - 2 T^{9} + \cdots - 69892992 \)
|
| $43$ |
\( T^{10} + 15 T^{9} + \cdots + 331776 \)
|
| $47$ |
\( T^{10} - 3 T^{9} + \cdots - 110592 \)
|
| $53$ |
\( T^{10} - 11 T^{9} + \cdots + 711936 \)
|
| $59$ |
\( T^{10} - 11 T^{9} + \cdots - 4939776 \)
|
| $61$ |
\( T^{10} + \cdots - 495553232 \)
|
| $67$ |
\( T^{10} + \cdots + 223731712 \)
|
| $71$ |
\( T^{10} + \cdots + 2127811584 \)
|
| $73$ |
\( T^{10} - 36 T^{9} + \cdots + 2246632 \)
|
| $79$ |
\( T^{10} - 24 T^{9} + \cdots - 44205056 \)
|
| $83$ |
\( T^{10} + \cdots + 185131008 \)
|
| $89$ |
\( T^{10} + \cdots + 1545744384 \)
|
| $97$ |
\( T^{10} - 25 T^{9} + \cdots - 94196576 \)
|
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