Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [555,2,Mod(406,555)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(555, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("555.406");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 555 = 3 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 555.e (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(4.43169731218\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | 10.0.3381950424064.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 4x^{9} + 2x^{8} + 2x^{7} + 17x^{6} - 24x^{5} + 18x^{4} - 40x^{3} + 67x^{2} - 24x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{10} - 4x^{9} + 2x^{8} + 2x^{7} + 17x^{6} - 24x^{5} + 18x^{4} - 40x^{3} + 67x^{2} - 24x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 37174 \nu^{9} - 8715 \nu^{8} - 499512 \nu^{7} + 361520 \nu^{6} + 1133738 \nu^{5} + 1157496 \nu^{4} + \cdots + 5989545 ) / 2155475 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 48698 \nu^{9} + 97580 \nu^{8} + 268424 \nu^{7} - 95540 \nu^{6} - 1448901 \nu^{5} + \cdots - 889465 ) / 2155475 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 66007 \nu^{9} - 209545 \nu^{8} + 11334 \nu^{7} - 87740 \nu^{6} + 1300359 \nu^{5} - 598972 \nu^{4} + \cdots - 1119165 ) / 2155475 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 103181 \nu^{9} + 218260 \nu^{8} + 488178 \nu^{7} - 273780 \nu^{6} - 2434097 \nu^{5} + \cdots - 7025855 ) / 2155475 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 172521 \nu^{9} + 569660 \nu^{8} + 46398 \nu^{7} - 234805 \nu^{6} - 2873002 \nu^{5} + \cdots - 1071605 ) / 2155475 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3641 \nu^{9} - 14385 \nu^{8} + 5217 \nu^{7} + 12155 \nu^{6} + 63092 \nu^{5} - 88486 \nu^{4} + \cdots - 44845 ) / 39550 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 80807 \nu^{9} - 277585 \nu^{8} + 32639 \nu^{7} + 172645 \nu^{6} + 1178554 \nu^{5} - 1253092 \nu^{4} + \cdots - 612945 ) / 862190 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 1014787 \nu^{9} + 3960845 \nu^{8} - 1597919 \nu^{7} - 2135635 \nu^{6} - 18169694 \nu^{5} + \cdots + 13619765 ) / 4310950 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 637619 \nu^{9} - 2441740 \nu^{8} + 820553 \nu^{7} + 1504470 \nu^{6} + 11194578 \nu^{5} + \cdots - 8159055 ) / 2155475 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{4} - 2\beta_{2} + \beta _1 + 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{9} + \beta_{7} + 7\beta_{6} - \beta_{5} + 4\beta_{4} + 5\beta_{3} - 4\beta_{2} + \beta _1 + 8 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 12 \beta_{9} - 7 \beta_{8} + 3 \beta_{7} + 14 \beta_{6} - 3 \beta_{5} + 9 \beta_{4} + 10 \beta_{3} + \cdots + 9 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 35 \beta_{9} - 17 \beta_{8} + 8 \beta_{7} + 51 \beta_{6} - 2 \beta_{5} + 13 \beta_{4} + 29 \beta_{3} + \cdots + 25 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 95 \beta_{9} - 39 \beta_{8} + 19 \beta_{7} + 156 \beta_{6} - 6 \beta_{5} + 16 \beta_{4} + 88 \beta_{3} + \cdots + 15 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 257 \beta_{9} - 132 \beta_{8} + 56 \beta_{7} + 362 \beta_{6} + 3 \beta_{5} - 21 \beta_{4} + 202 \beta_{3} + \cdots - 30 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 621 \beta_{9} - 277 \beta_{8} + 125 \beta_{7} + 968 \beta_{6} + 48 \beta_{5} - 208 \beta_{4} + \cdots - 293 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 1455 \beta_{9} - 688 \beta_{8} + 310 \beta_{7} + 2180 \beta_{6} + 179 \beta_{5} - 884 \beta_{4} + \cdots - 1397 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/555\mathbb{Z}\right)^\times\).
| \(n\) | \(76\) | \(112\) | \(371\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 406.1 |
|
− | 2.46148i | 1.00000 | −4.05889 | 1.00000i | − | 2.46148i | 3.17259 | 5.06792i | 1.00000 | 2.46148 | ||||||||||||||||||||||||||||||||||||||||||||||
| 406.2 | − | 2.11175i | 1.00000 | −2.45951 | − | 1.00000i | − | 2.11175i | −0.212275 | 0.970363i | 1.00000 | −2.11175 | ||||||||||||||||||||||||||||||||||||||||||||||
| 406.3 | − | 1.33701i | 1.00000 | 0.212393 | 1.00000i | − | 1.33701i | 1.90912 | − | 2.95800i | 1.00000 | 1.33701 | ||||||||||||||||||||||||||||||||||||||||||||||
| 406.4 | − | 1.21178i | 1.00000 | 0.531596 | − | 1.00000i | − | 1.21178i | 4.51017 | − | 3.06773i | 1.00000 | −1.21178 | |||||||||||||||||||||||||||||||||||||||||||||
| 406.5 | − | 0.474965i | 1.00000 | 1.77441 | − | 1.00000i | − | 0.474965i | −1.37960 | − | 1.79271i | 1.00000 | −0.474965 | |||||||||||||||||||||||||||||||||||||||||||||
| 406.6 | 0.474965i | 1.00000 | 1.77441 | 1.00000i | 0.474965i | −1.37960 | 1.79271i | 1.00000 | −0.474965 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 406.7 | 1.21178i | 1.00000 | 0.531596 | 1.00000i | 1.21178i | 4.51017 | 3.06773i | 1.00000 | −1.21178 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 406.8 | 1.33701i | 1.00000 | 0.212393 | − | 1.00000i | 1.33701i | 1.90912 | 2.95800i | 1.00000 | 1.33701 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 406.9 | 2.11175i | 1.00000 | −2.45951 | 1.00000i | 2.11175i | −0.212275 | − | 0.970363i | 1.00000 | −2.11175 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 406.10 | 2.46148i | 1.00000 | −4.05889 | − | 1.00000i | 2.46148i | 3.17259 | − | 5.06792i | 1.00000 | 2.46148 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 555.2.e.b | ✓ | 10 |
| 3.b | odd | 2 | 1 | 1665.2.e.c | 10 | ||
| 37.b | even | 2 | 1 | inner | 555.2.e.b | ✓ | 10 |
| 111.d | odd | 2 | 1 | 1665.2.e.c | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 555.2.e.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 555.2.e.b | ✓ | 10 | 37.b | even | 2 | 1 | inner |
| 1665.2.e.c | 10 | 3.b | odd | 2 | 1 | ||
| 1665.2.e.c | 10 | 111.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 14T_{2}^{8} + 67T_{2}^{6} + 130T_{2}^{4} + 97T_{2}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(555, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 14 T^{8} + \cdots + 16 \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( (T^{2} + 1)^{5} \)
|
| $7$ |
\( (T^{5} - 8 T^{4} + 14 T^{3} + \cdots - 8)^{2} \)
|
| $11$ |
\( (T^{5} - 2 T^{4} - 11 T^{3} + \cdots + 2)^{2} \)
|
| $13$ |
\( T^{10} + 61 T^{8} + \cdots + 12769 \)
|
| $17$ |
\( T^{10} + 94 T^{8} + \cdots + 1024 \)
|
| $19$ |
\( T^{10} + 130 T^{8} + \cdots + 504100 \)
|
| $23$ |
\( T^{10} + 180 T^{8} + \cdots + 19483396 \)
|
| $29$ |
\( T^{10} + 83 T^{8} + \cdots + 60025 \)
|
| $31$ |
\( T^{10} + 188 T^{8} + \cdots + 6990736 \)
|
| $37$ |
\( T^{10} + 4 T^{9} + \cdots + 69343957 \)
|
| $41$ |
\( (T^{5} - 10 T^{4} + \cdots - 1084)^{2} \)
|
| $43$ |
\( T^{10} + 83 T^{8} + \cdots + 63001 \)
|
| $47$ |
\( (T^{5} + 11 T^{4} + \cdots + 6425)^{2} \)
|
| $53$ |
\( (T^{5} + 7 T^{4} + \cdots + 14320)^{2} \)
|
| $59$ |
\( T^{10} + 255 T^{8} + \cdots + 390625 \)
|
| $61$ |
\( T^{10} + 164 T^{8} + \cdots + 8526400 \)
|
| $67$ |
\( (T^{5} - 6 T^{4} + \cdots - 4570)^{2} \)
|
| $71$ |
\( (T^{5} + 8 T^{4} + \cdots + 8590)^{2} \)
|
| $73$ |
\( (T^{5} - 14 T^{4} + \cdots - 18166)^{2} \)
|
| $79$ |
\( T^{10} + \cdots + 2518032400 \)
|
| $83$ |
\( (T^{5} + 5 T^{4} + \cdots + 2315)^{2} \)
|
| $89$ |
\( T^{10} + 257 T^{8} + \cdots + 410881 \)
|
| $97$ |
\( T^{10} + 608 T^{8} + \cdots + 55026724 \)
|
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