Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [445,2,Mod(1,445)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(445, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("445.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 445 = 5 \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 445.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: | \(3.55334288995\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 19x^{3} - 27x^{2} + 11x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 19x^{3} - 27x^{2} + 11x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{7} + \nu^{6} + 23\nu^{5} - 8\nu^{4} - 76\nu^{3} + 12\nu^{2} + 65\nu - 7 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{7} + \nu^{6} + 11\nu^{5} - 9\nu^{4} - 34\nu^{3} + 18\nu^{2} + 27\nu - 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{7} + 3\nu^{6} + 45\nu^{5} - 26\nu^{4} - 144\nu^{3} + 50\nu^{2} + 121\nu - 27 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{7} - 4\nu^{6} - 55\nu^{5} + 34\nu^{4} + 170\nu^{3} - 61\nu^{2} - 136\nu + 27 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -6\nu^{7} + 5\nu^{6} + 67\nu^{5} - 42\nu^{4} - 212\nu^{3} + 72\nu^{2} + 175\nu - 29 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} - \beta_{3} - \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + 6\beta_{5} - 8\beta_{4} - 7\beta_{3} - 7\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 2\beta_{6} + \beta_{5} + 8\beta_{2} + 28\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{7} + 2\beta_{6} + 37\beta_{5} - 54\beta_{4} - 48\beta_{3} - 47\beta _1 + 86 \)
|
| \(\nu^{7}\) | \(=\) |
\( 13\beta_{7} + 24\beta_{6} + 12\beta_{5} - \beta_{4} - 3\beta_{3} + 54\beta_{2} + 163\beta _1 + 9 \)
|
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 |
|
−2.50065 | 1.23408 | 4.25326 | −1.00000 | −3.08600 | −4.89614 | −5.63461 | −1.47705 | 2.50065 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.66289 | −1.70314 | 0.765209 | −1.00000 | 2.83214 | −1.19579 | 2.05332 | −0.0993042 | 1.66289 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.11667 | 3.44995 | −0.753041 | −1.00000 | −3.85247 | −1.16728 | 3.07425 | 8.90215 | 1.11667 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.151894 | −0.440615 | −1.97693 | −1.00000 | −0.0669267 | 2.88745 | −0.604071 | −2.80586 | −0.151894 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.217002 | −2.44349 | −1.95291 | −1.00000 | −0.530243 | −2.89958 | −0.857790 | 2.97066 | −0.217002 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.16343 | 2.86964 | −0.646427 | −1.00000 | 3.33863 | 2.23155 | −3.07894 | 5.23486 | −1.16343 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.23321 | 0.936597 | 2.98721 | −1.00000 | 2.09161 | 1.87039 | 2.20463 | −2.12279 | −2.23321 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.51468 | 2.09698 | 4.32363 | −1.00000 | 5.27325 | −2.83060 | 5.84320 | 1.39733 | −2.51468 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(89\) | \(-1\) |
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 | 445.2.a.g | ✓ | 8 |
| 3.b | odd | 2 | 1 | 4005.2.a.p | 8 | ||
| 4.b | odd | 2 | 1 | 7120.2.a.bk | 8 | ||
| 5.b | even | 2 | 1 | 2225.2.a.l | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 445.2.a.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 2225.2.a.l | 8 | 5.b | even | 2 | 1 | ||
| 4005.2.a.p | 8 | 3.b | odd | 2 | 1 | ||
| 7120.2.a.bk | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - T_{2}^{7} - 11T_{2}^{6} + 9T_{2}^{5} + 34T_{2}^{4} - 19T_{2}^{3} - 27T_{2}^{2} + 11T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(445))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{7} - 11 T^{6} + \cdots - 1 \)
|
| $3$ |
\( T^{8} - 6 T^{7} + \cdots - 44 \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( T^{8} + 6 T^{7} + \cdots - 676 \)
|
| $11$ |
\( T^{8} - 14 T^{7} + \cdots - 2752 \)
|
| $13$ |
\( T^{8} + 7 T^{7} + \cdots - 64 \)
|
| $17$ |
\( T^{8} - 17 T^{7} + \cdots - 2176 \)
|
| $19$ |
\( T^{8} - 17 T^{7} + \cdots + 7628 \)
|
| $23$ |
\( T^{8} + T^{7} + \cdots + 1004 \)
|
| $29$ |
\( T^{8} - 10 T^{7} + \cdots - 14864 \)
|
| $31$ |
\( T^{8} - T^{7} + \cdots - 59996 \)
|
| $37$ |
\( T^{8} + 11 T^{7} + \cdots - 256 \)
|
| $41$ |
\( T^{8} - 15 T^{7} + \cdots + 11344 \)
|
| $43$ |
\( T^{8} + 5 T^{7} + \cdots - 644492 \)
|
| $47$ |
\( T^{8} - 12 T^{7} + \cdots + 249712 \)
|
| $53$ |
\( T^{8} + T^{7} + \cdots + 1486976 \)
|
| $59$ |
\( T^{8} - 26 T^{7} + \cdots - 77428 \)
|
| $61$ |
\( T^{8} - 13 T^{7} + \cdots + 16005968 \)
|
| $67$ |
\( T^{8} + 25 T^{7} + \cdots - 58944976 \)
|
| $71$ |
\( T^{8} - 28 T^{7} + \cdots - 11968 \)
|
| $73$ |
\( T^{8} + 17 T^{7} + \cdots + 548656 \)
|
| $79$ |
\( T^{8} + 7 T^{7} + \cdots + 226304 \)
|
| $83$ |
\( T^{8} - 44 T^{7} + \cdots - 6725668 \)
|
| $89$ |
\( (T - 1)^{8} \)
|
| $97$ |
\( T^{8} - T^{7} + \cdots + 1330048 \)
|
| show more | |
| show less |