Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4425,2,Mod(1,4425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4425, 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("4425.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4425 = 3 \cdot 5^{2} \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4425.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: | \(35.3338028944\) |
| 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: |
\( x^{10} - 5x^{9} - 3x^{8} + 37x^{7} - 3x^{6} - 92x^{5} + 4x^{4} + 72x^{3} + 2x^{2} - 12x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 5 \) |
| 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
\( x^{10} - 5x^{9} - 3x^{8} + 37x^{7} - 3x^{6} - 92x^{5} + 4x^{4} + 72x^{3} + 2x^{2} - 12x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{9} + 5\nu^{8} + 3\nu^{7} - 37\nu^{6} + 4\nu^{5} + 88\nu^{4} - 7\nu^{3} - 56\nu^{2} + 4\nu + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{9} + 5\nu^{8} + 3\nu^{7} - 37\nu^{6} + 3\nu^{5} + 93\nu^{4} - 8\nu^{3} - 73\nu^{2} + 10\nu + 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{9} - 5\nu^{8} - 3\nu^{7} + 38\nu^{6} - 8\nu^{5} - 91\nu^{4} + 24\nu^{3} + 61\nu^{2} - 16\nu - 6 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{9} - 5\nu^{8} - 3\nu^{7} + 38\nu^{6} - 7\nu^{5} - 95\nu^{4} + 22\nu^{3} + 75\nu^{2} - 15\nu - 10 \)
|
| \(\beta_{8}\) | \(=\) |
\( -2\nu^{9} + 11\nu^{8} + \nu^{7} - 76\nu^{6} + 39\nu^{5} + 177\nu^{4} - 79\nu^{3} - 132\nu^{2} + 40\nu + 17 \)
|
| \(\beta_{9}\) | \(=\) |
\( -3\nu^{9} + 16\nu^{8} + 3\nu^{7} - 108\nu^{6} + 42\nu^{5} + 245\nu^{4} - 74\nu^{3} - 171\nu^{2} + 34\nu + 20 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 3\beta_{3} + 9\beta_{2} + 20\beta _1 + 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( 5\beta_{7} - 5\beta_{6} + 4\beta_{5} - 4\beta_{4} + 14\beta_{3} + 26\beta_{2} + 65\beta _1 + 20 \)
|
| \(\nu^{6}\) | \(=\) |
\( 23\beta_{7} - 22\beta_{6} + 19\beta_{5} - 18\beta_{4} + 48\beta_{3} + 92\beta_{2} + 203\beta _1 + 68 \)
|
| \(\nu^{7}\) | \(=\) |
\( - \beta_{9} + \beta_{8} + 90 \beta_{7} - 85 \beta_{6} + 71 \beta_{5} - 65 \beta_{4} + 178 \beta_{3} + \cdots + 177 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 5 \beta_{9} + 6 \beta_{8} + 340 \beta_{7} - 313 \beta_{6} + 268 \beta_{5} - 238 \beta_{4} + \cdots + 552 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 28 \beta_{9} + 33 \beta_{8} + 1227 \beta_{7} - 1114 \beta_{6} + 954 \beta_{5} - 824 \beta_{4} + \cdots + 1613 \)
|
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 |
|
−2.34122 | 1.00000 | 3.48132 | 0 | −2.34122 | −3.55656 | −3.46809 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.57880 | 1.00000 | 0.492597 | 0 | −1.57880 | 1.71197 | 2.37988 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.34944 | 1.00000 | −0.179010 | 0 | −1.34944 | −3.94348 | 2.94044 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.0196940 | 1.00000 | −1.99961 | 0 | 0.0196940 | 4.05049 | −0.0787683 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.457140 | 1.00000 | −1.79102 | 0 | 0.457140 | 0.252665 | −1.73303 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.21336 | 1.00000 | −0.527747 | 0 | 1.21336 | 0.223743 | −3.06708 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.32499 | 1.00000 | −0.244393 | 0 | 1.32499 | −3.87802 | −2.97381 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.03182 | 1.00000 | 2.12828 | 0 | 2.03182 | 2.85922 | 0.260635 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.57519 | 1.00000 | 4.63162 | 0 | 2.57519 | 2.45737 | 6.77694 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.64726 | 1.00000 | 5.00797 | 0 | 2.64726 | −3.17739 | 7.96288 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(59\) | \( +1 \) |
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 | 4425.2.a.bj | yes | 10 |
| 5.b | even | 2 | 1 | 4425.2.a.bi | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4425.2.a.bi | ✓ | 10 | 5.b | even | 2 | 1 | |
| 4425.2.a.bj | yes | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4425))\):
|
\( T_{2}^{10} - 5T_{2}^{9} - 3T_{2}^{8} + 47T_{2}^{7} - 38T_{2}^{6} - 121T_{2}^{5} + 164T_{2}^{4} + 65T_{2}^{3} - 165T_{2}^{2} + 54T_{2} - 1 \)
|
|
\( T_{7}^{10} + 3 T_{7}^{9} - 39 T_{7}^{8} - 97 T_{7}^{7} + 563 T_{7}^{6} + 948 T_{7}^{5} - 3776 T_{7}^{4} + \cdots + 476 \)
|
|
\( T_{11}^{10} - 7 T_{11}^{9} - 41 T_{11}^{8} + 353 T_{11}^{7} + 239 T_{11}^{6} - 5388 T_{11}^{5} + \cdots + 30508 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 5 T^{9} + \cdots - 1 \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + 3 T^{9} + \cdots + 476 \)
|
| $11$ |
\( T^{10} - 7 T^{9} + \cdots + 30508 \)
|
| $13$ |
\( T^{10} - 69 T^{8} + \cdots - 31664 \)
|
| $17$ |
\( T^{10} - 9 T^{9} + \cdots + 369776 \)
|
| $19$ |
\( T^{10} - 22 T^{9} + \cdots - 11200 \)
|
| $23$ |
\( T^{10} - 14 T^{9} + \cdots - 41828 \)
|
| $29$ |
\( T^{10} - 4 T^{9} + \cdots - 17500 \)
|
| $31$ |
\( T^{10} - 22 T^{9} + \cdots + 33301 \)
|
| $37$ |
\( T^{10} - T^{9} + \cdots - 332 \)
|
| $41$ |
\( T^{10} - 15 T^{9} + \cdots - 405908 \)
|
| $43$ |
\( T^{10} + 6 T^{9} + \cdots - 59809 \)
|
| $47$ |
\( T^{10} - 11 T^{9} + \cdots - 1320752 \)
|
| $53$ |
\( T^{10} - 7 T^{9} + \cdots + 161296 \)
|
| $59$ |
\( (T + 1)^{10} \)
|
| $61$ |
\( T^{10} - 14 T^{9} + \cdots - 8540828 \)
|
| $67$ |
\( T^{10} + 11 T^{9} + \cdots + 8515276 \)
|
| $71$ |
\( T^{10} - 27 T^{9} + \cdots - 22461937 \)
|
| $73$ |
\( T^{10} + \cdots + 2350461952 \)
|
| $79$ |
\( T^{10} - 11 T^{9} + \cdots - 1113200 \)
|
| $83$ |
\( T^{10} - 9 T^{9} + \cdots - 4844708 \)
|
| $89$ |
\( T^{10} - 7 T^{9} + \cdots + 681875 \)
|
| $97$ |
\( T^{10} + \cdots - 230103508 \)
|
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