Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [574,2,Mod(57,574)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(574, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("574.57");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 574 = 2 \cdot 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 574.h (of order \(5\), degree \(4\), 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: | \(4.58341307602\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.123765625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} + 6x^{6} - 4x^{5} + 9x^{4} + 8x^{3} + 24x^{2} + 24x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} - 3x^{7} + 6x^{6} - 4x^{5} + 9x^{4} + 8x^{3} + 24x^{2} + 24x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + 5\nu^{6} - 16\nu^{5} + 36\nu^{4} - 81\nu^{3} + 154\nu^{2} - 116\nu - 8 ) / 216 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 5\nu^{6} + 16\nu^{5} - 36\nu^{4} + 81\nu^{3} - 46\nu^{2} + 8\nu + 8 ) / 108 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 7\nu^{6} + 18\nu^{5} - 24\nu^{4} + 21\nu^{3} - 16\nu^{2} + 4\nu - 24 ) / 36 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{7} - 7\nu^{6} + 4\nu^{5} + 24\nu^{4} - 21\nu^{3} + 66\nu^{2} + 40\nu + 80 ) / 72 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{7} + 13\nu^{6} - 34\nu^{5} + 48\nu^{4} - 51\nu^{3} - 12\nu^{2} - 28\nu - 32 ) / 72 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{6} + 9\nu^{5} - 6\nu^{4} + 3\nu^{3} + 11\nu^{2} + 10\nu + 12 ) / 18 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 2\beta_{6} + 4\beta_{3} + 2\beta_{2} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{7} + 8\beta_{6} - 2\beta_{5} - \beta_{4} + 7\beta_{3} + 2\beta_{2} + \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 21\beta_{7} + 12\beta_{6} - 12\beta_{5} - 9\beta_{4} + 9\beta_{3} - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 33\beta_{7} - 24\beta_{5} - 33\beta_{4} - 18\beta_{2} - 11\beta _1 - 18 \)
|
| \(\nu^{7}\) | \(=\) |
\( -66\beta_{6} - 57\beta_{4} - 62\beta_{3} - 88\beta_{2} - 62\beta _1 - 66 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/574\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(493\) |
| \(\chi(n)\) | \(\beta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 57.1 |
|
0.309017 | − | 0.951057i | −2.32719 | −0.809017 | − | 0.587785i | 1.16360 | + | 0.845402i | −0.719142 | + | 2.21329i | −0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | 2.41582 | 1.16360 | − | 0.845402i | ||||||||||||||||||||||||||||
| 57.2 | 0.309017 | − | 0.951057i | 2.94523 | −0.809017 | − | 0.587785i | −1.47261 | − | 1.06992i | 0.910125 | − | 2.80108i | −0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | 5.67435 | −1.47261 | + | 1.06992i | |||||||||||||||||||||||||||||
| 141.1 | 0.309017 | + | 0.951057i | −2.32719 | −0.809017 | + | 0.587785i | 1.16360 | − | 0.845402i | −0.719142 | − | 2.21329i | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | 2.41582 | 1.16360 | + | 0.845402i | |||||||||||||||||||||||||||||
| 141.2 | 0.309017 | + | 0.951057i | 2.94523 | −0.809017 | + | 0.587785i | −1.47261 | + | 1.06992i | 0.910125 | + | 2.80108i | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | 5.67435 | −1.47261 | − | 1.06992i | |||||||||||||||||||||||||||||
| 365.1 | −0.809017 | + | 0.587785i | −1.70367 | 0.309017 | − | 0.951057i | 0.851836 | − | 2.62168i | 1.37830 | − | 1.00139i | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | −0.0975037 | 0.851836 | + | 2.62168i | |||||||||||||||||||||||||||||
| 365.2 | −0.809017 | + | 0.587785i | 0.0856374 | 0.309017 | − | 0.951057i | −0.0428187 | + | 0.131782i | −0.0692821 | + | 0.0503364i | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | −2.99267 | −0.0428187 | − | 0.131782i | |||||||||||||||||||||||||||||
| 379.1 | −0.809017 | − | 0.587785i | −1.70367 | 0.309017 | + | 0.951057i | 0.851836 | + | 2.62168i | 1.37830 | + | 1.00139i | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | −0.0975037 | 0.851836 | − | 2.62168i | |||||||||||||||||||||||||||||
| 379.2 | −0.809017 | − | 0.587785i | 0.0856374 | 0.309017 | + | 0.951057i | −0.0428187 | − | 0.131782i | −0.0692821 | − | 0.0503364i | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | −2.99267 | −0.0428187 | + | 0.131782i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 574.2.h.h | ✓ | 8 |
| 41.d | even | 5 | 1 | inner | 574.2.h.h | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 574.2.h.h | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 574.2.h.h | ✓ | 8 | 41.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\):
|
\( T_{3}^{4} + T_{3}^{3} - 8T_{3}^{2} - 11T_{3} + 1 \)
|
|
\( T_{5}^{8} - T_{5}^{7} + 5T_{5}^{6} + 6T_{5}^{5} - T_{5}^{4} - 24T_{5}^{3} + 50T_{5}^{2} + 4T_{5} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( (T^{4} + T^{3} - 8 T^{2} + \cdots + 1)^{2} \)
|
| $5$ |
\( T^{8} - T^{7} + 5 T^{6} + \cdots + 1 \)
|
| $7$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{8} + T^{7} + \cdots + 841 \)
|
| $13$ |
\( T^{8} + 20 T^{6} + \cdots + 10000 \)
|
| $17$ |
\( T^{8} - 11 T^{7} + \cdots + 961 \)
|
| $19$ |
\( T^{8} - 5 T^{7} + \cdots + 326041 \)
|
| $23$ |
\( T^{8} + T^{7} + \cdots + 234256 \)
|
| $29$ |
\( T^{8} - 10 T^{7} + \cdots + 1142761 \)
|
| $31$ |
\( T^{8} + T^{7} + \cdots + 434281 \)
|
| $37$ |
\( T^{8} - T^{7} + \cdots + 256 \)
|
| $41$ |
\( T^{8} - 12 T^{7} + \cdots + 2825761 \)
|
| $43$ |
\( T^{8} + 5 T^{7} + \cdots + 326041 \)
|
| $47$ |
\( (T^{4} + 7 T^{3} + \cdots + 121)^{2} \)
|
| $53$ |
\( T^{8} - 22 T^{7} + \cdots + 215296 \)
|
| $59$ |
\( T^{8} - 5 T^{7} + \cdots + 225625 \)
|
| $61$ |
\( T^{8} + 7 T^{7} + \cdots + 10042561 \)
|
| $67$ |
\( (T^{4} - 19 T^{3} + \cdots + 6241)^{2} \)
|
| $71$ |
\( T^{8} + 39 T^{7} + \cdots + 355216 \)
|
| $73$ |
\( (T^{4} - 13 T^{3} + \cdots - 4544)^{2} \)
|
| $79$ |
\( (T^{4} + 14 T^{3} + \cdots - 401)^{2} \)
|
| $83$ |
\( (T^{4} + 28 T^{3} + \cdots - 9124)^{2} \)
|
| $89$ |
\( T^{8} - 30 T^{7} + \cdots + 51423241 \)
|
| $97$ |
\( T^{8} - 12 T^{7} + \cdots + 961 \)
|
| show more | |
| show less |