Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,6,Mod(32,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.32");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.e (of order \(4\), degree \(2\), 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: | \(12.0287864860\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.432373800960000.179 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 207x^{4} + 104976 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} - 207x^{4} + 104976 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 1251\nu ) / 378 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 117\nu^{2} ) / 6804 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 261\nu ) / 189 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 2151\nu^{3} ) / 20412 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 135\nu^{3} ) / 1512 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 8\nu^{4} - 828 ) / 21 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{6} + 531\nu^{2} ) / 81 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + 84\beta_{2} ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{5} + 81\beta_{4} ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 21\beta_{6} + 828 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -1251\beta_{3} + 522\beta_1 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -117\beta_{7} + 44604\beta_{2} ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 12906\beta_{5} + 10935\beta_{4} ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 32.1 |
|
−5.47723 | + | 5.47723i | −7.93847 | + | 13.4157i | − | 28.0000i | 0 | −30.0000 | − | 116.962i | 64.0625 | + | 64.0625i | −21.9089 | − | 21.9089i | −116.962 | − | 213.000i | 0 | |||||||||||||||||||||||||||||
| 32.2 | −5.47723 | + | 5.47723i | 13.4157 | − | 7.93847i | − | 28.0000i | 0 | −30.0000 | + | 116.962i | −64.0625 | − | 64.0625i | −21.9089 | − | 21.9089i | 116.962 | − | 213.000i | 0 | ||||||||||||||||||||||||||||||
| 32.3 | 5.47723 | − | 5.47723i | −13.4157 | + | 7.93847i | − | 28.0000i | 0 | −30.0000 | + | 116.962i | 64.0625 | + | 64.0625i | 21.9089 | + | 21.9089i | 116.962 | − | 213.000i | 0 | ||||||||||||||||||||||||||||||
| 32.4 | 5.47723 | − | 5.47723i | 7.93847 | − | 13.4157i | − | 28.0000i | 0 | −30.0000 | − | 116.962i | −64.0625 | − | 64.0625i | 21.9089 | + | 21.9089i | −116.962 | − | 213.000i | 0 | ||||||||||||||||||||||||||||||
| 68.1 | −5.47723 | − | 5.47723i | −7.93847 | − | 13.4157i | 28.0000i | 0 | −30.0000 | + | 116.962i | 64.0625 | − | 64.0625i | −21.9089 | + | 21.9089i | −116.962 | + | 213.000i | 0 | |||||||||||||||||||||||||||||||
| 68.2 | −5.47723 | − | 5.47723i | 13.4157 | + | 7.93847i | 28.0000i | 0 | −30.0000 | − | 116.962i | −64.0625 | + | 64.0625i | −21.9089 | + | 21.9089i | 116.962 | + | 213.000i | 0 | |||||||||||||||||||||||||||||||
| 68.3 | 5.47723 | + | 5.47723i | −13.4157 | − | 7.93847i | 28.0000i | 0 | −30.0000 | − | 116.962i | 64.0625 | − | 64.0625i | 21.9089 | − | 21.9089i | 116.962 | + | 213.000i | 0 | |||||||||||||||||||||||||||||||
| 68.4 | 5.47723 | + | 5.47723i | 7.93847 | + | 13.4157i | 28.0000i | 0 | −30.0000 | + | 116.962i | −64.0625 | + | 64.0625i | 21.9089 | − | 21.9089i | −116.962 | + | 213.000i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 5.c | odd | 4 | 2 | inner |
| 15.d | odd | 2 | 1 | inner |
| 15.e | even | 4 | 2 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.6.e.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 75.6.e.c | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 75.6.e.c | ✓ | 8 |
| 5.c | odd | 4 | 2 | inner | 75.6.e.c | ✓ | 8 |
| 15.d | odd | 2 | 1 | inner | 75.6.e.c | ✓ | 8 |
| 15.e | even | 4 | 2 | inner | 75.6.e.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.6.e.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 75.6.e.c | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 75.6.e.c | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 75.6.e.c | ✓ | 8 | 5.c | odd | 4 | 2 | inner |
| 75.6.e.c | ✓ | 8 | 15.d | odd | 2 | 1 | inner |
| 75.6.e.c | ✓ | 8 | 15.e | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 3600 \)
acting on \(S_{6}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 3600)^{2} \)
|
| $3$ |
\( T^{8} + \cdots + 3486784401 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} + 67371264)^{2} \)
|
| $11$ |
\( (T^{2} + 492480)^{4} \)
|
| $13$ |
\( (T^{4} + 275952697344)^{2} \)
|
| $17$ |
\( (T^{4} + 2243524665600)^{2} \)
|
| $19$ |
\( (T^{2} + 2202256)^{4} \)
|
| $23$ |
\( (T^{4} + 65308056195600)^{2} \)
|
| $29$ |
\( (T^{2} - 39890880)^{4} \)
|
| $31$ |
\( (T + 968)^{8} \)
|
| $37$ |
\( (T^{4} + 252513965113344)^{2} \)
|
| $41$ |
\( (T^{2} + 177785280)^{4} \)
|
| $43$ |
\( (T^{4} + 1924190671104)^{2} \)
|
| $47$ |
\( (T^{4} + 44266933155600)^{2} \)
|
| $53$ |
\( (T^{4} + 12\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{2} - 260521920)^{4} \)
|
| $61$ |
\( (T - 30242)^{8} \)
|
| $67$ |
\( (T^{4} + 38\!\cdots\!24)^{2} \)
|
| $71$ |
\( (T^{2} + 1656702720)^{4} \)
|
| $73$ |
\( (T^{4} + 80\!\cdots\!04)^{2} \)
|
| $79$ |
\( (T^{2} + 6635776)^{4} \)
|
| $83$ |
\( (T^{4} + 33\!\cdots\!00)^{2} \)
|
| $89$ |
\( (T^{2} - 1656702720)^{4} \)
|
| $97$ |
\( (T^{4} + 24\!\cdots\!24)^{2} \)
|
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