Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1476,2,Mod(37,1476)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1476, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1476.37");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1476.n (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: | \(11.7859193383\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.2127515625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} + 11x^{6} - 14x^{5} + 39x^{4} - 2x^{3} + 144x^{2} + 304x + 361 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 492) |
| 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
\( x^{8} - 3x^{7} + 11x^{6} - 14x^{5} + 39x^{4} - 2x^{3} + 144x^{2} + 304x + 361 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 293370 \nu^{7} - 1325660 \nu^{6} + 5308881 \nu^{5} - 12991827 \nu^{4} + 34460063 \nu^{3} + \cdots + 1185790 ) / 287044001 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 390600 \nu^{7} + 1148601 \nu^{6} + 328608 \nu^{5} - 12290292 \nu^{4} + 26445076 \nu^{3} + \cdots - 212803724 ) / 287044001 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23450 \nu^{7} - 109569 \nu^{6} + 467613 \nu^{5} - 1211507 \nu^{4} + 4856082 \nu^{3} + \cdots + 5574030 ) / 15107579 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 727930 \nu^{7} + 3220829 \nu^{6} - 12350877 \nu^{5} + 28948675 \nu^{4} - 45574694 \nu^{3} + \cdots - 185880591 ) / 287044001 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 54581 \nu^{7} + 228613 \nu^{6} - 987245 \nu^{5} + 904496 \nu^{4} - 867260 \nu^{3} + \cdots - 13830670 ) / 15107579 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 76810 \nu^{7} + 94750 \nu^{6} - 520190 \nu^{5} - 77601 \nu^{4} - 2196434 \nu^{3} + \cdots - 26826100 ) / 15107579 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - 4\beta_{2} + \beta _1 - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{5} + 6\beta_{4} - \beta_{3} - 3\beta_{2} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - 10\beta_{6} + 24\beta_{5} + 10\beta_{4} - 6\beta_{3} + 6\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{7} - 40\beta_{6} + 41\beta_{5} + 41\beta_{3} - 3\beta _1 + 15 \)
|
| \(\nu^{6}\) | \(=\) |
\( -84\beta_{7} - 84\beta_{4} + 197\beta_{3} - 28\beta_{2} - 72\beta _1 + 28 \)
|
| \(\nu^{7}\) | \(=\) |
\( -281\beta_{7} + 281\beta_{6} - 348\beta_{5} - 325\beta_{4} + 204\beta_{2} - 325\beta _1 - 348 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1476\mathbb{Z}\right)^\times\).
| \(n\) | \(739\) | \(821\) | \(1441\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1 + \beta_{2} - \beta_{3} - \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
0 | 0 | 0 | −0.370845 | + | 1.14134i | 0 | −2.77990 | − | 2.01972i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||
| 37.2 | 0 | 0 | 0 | 0.870845 | − | 2.68019i | 0 | 0.470886 | + | 0.342119i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 469.1 | 0 | 0 | 0 | −1.68379 | + | 1.22335i | 0 | −1.33413 | + | 4.10604i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 469.2 | 0 | 0 | 0 | 2.18379 | − | 1.58662i | 0 | 0.143151 | − | 0.440574i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1117.1 | 0 | 0 | 0 | −0.370845 | − | 1.14134i | 0 | −2.77990 | + | 2.01972i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1117.2 | 0 | 0 | 0 | 0.870845 | + | 2.68019i | 0 | 0.470886 | − | 0.342119i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1369.1 | 0 | 0 | 0 | −1.68379 | − | 1.22335i | 0 | −1.33413 | − | 4.10604i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1369.2 | 0 | 0 | 0 | 2.18379 | + | 1.58662i | 0 | 0.143151 | + | 0.440574i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
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 | 1476.2.n.e | 8 | |
| 3.b | odd | 2 | 1 | 492.2.m.e | ✓ | 8 | |
| 41.d | even | 5 | 1 | inner | 1476.2.n.e | 8 | |
| 123.k | odd | 10 | 1 | 492.2.m.e | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 492.2.m.e | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 492.2.m.e | ✓ | 8 | 123.k | odd | 10 | 1 | |
| 1476.2.n.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 1476.2.n.e | 8 | 41.d | even | 5 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 2T_{5}^{7} + 6T_{5}^{6} + 4T_{5}^{5} + 9T_{5}^{4} - 8T_{5}^{3} + 239T_{5}^{2} + 171T_{5} + 361 \)
acting on \(S_{2}^{\mathrm{new}}(1476, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 2 T^{7} + \cdots + 361 \)
|
| $7$ |
\( T^{8} + 7 T^{7} + \cdots + 16 \)
|
| $11$ |
\( T^{8} + 6 T^{7} + \cdots + 1 \)
|
| $13$ |
\( T^{8} - 7 T^{7} + \cdots + 16 \)
|
| $17$ |
\( T^{8} + 21 T^{7} + \cdots + 30976 \)
|
| $19$ |
\( T^{8} + 3 T^{7} + \cdots + 81 \)
|
| $23$ |
\( T^{8} + 4 T^{7} + \cdots + 109561 \)
|
| $29$ |
\( T^{8} - 5 T^{7} + \cdots + 121 \)
|
| $31$ |
\( T^{8} + 17 T^{7} + \cdots + 1 \)
|
| $37$ |
\( T^{8} + 60 T^{6} + \cdots + 50625 \)
|
| $41$ |
\( T^{8} + 13 T^{7} + \cdots + 2825761 \)
|
| $43$ |
\( T^{8} - 3 T^{7} + \cdots + 20736 \)
|
| $47$ |
\( T^{8} + 30 T^{7} + \cdots + 8242641 \)
|
| $53$ |
\( T^{8} + 60 T^{6} + \cdots + 50625 \)
|
| $59$ |
\( T^{8} + 6 T^{7} + \cdots + 32041 \)
|
| $61$ |
\( T^{8} + 6 T^{7} + \cdots + 81 \)
|
| $67$ |
\( T^{8} + 15 T^{7} + \cdots + 1142761 \)
|
| $71$ |
\( T^{8} - 2 T^{7} + \cdots + 3352561 \)
|
| $73$ |
\( (T^{4} - 3 T^{3} + \cdots + 2556)^{2} \)
|
| $79$ |
\( (T^{4} + 14 T^{3} + \cdots + 304)^{2} \)
|
| $83$ |
\( (T^{4} - 3 T^{3} + \cdots + 1216)^{2} \)
|
| $89$ |
\( T^{8} + 11 T^{7} + \cdots + 2676496 \)
|
| $97$ |
\( T^{8} - 35 T^{7} + \cdots + 94575625 \)
|
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