Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [275,5,Mod(274,275)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(275, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("275.274");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 275 = 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 275.d (of order \(2\), degree \(1\), not 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: | \(28.4267398481\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{30})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 11) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 15\nu ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 15\nu ) / 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 15\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -15\beta_{3} + 15\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 274.1 |
|
−5.47723 | − | 3.00000i | 14.0000 | 0 | 16.4317i | 54.7723 | 10.9545 | 72.0000 | 0 | |||||||||||||||||||||||||||||
| 274.2 | −5.47723 | 3.00000i | 14.0000 | 0 | − | 16.4317i | 54.7723 | 10.9545 | 72.0000 | 0 | ||||||||||||||||||||||||||||||
| 274.3 | 5.47723 | − | 3.00000i | 14.0000 | 0 | − | 16.4317i | −54.7723 | −10.9545 | 72.0000 | 0 | |||||||||||||||||||||||||||||
| 274.4 | 5.47723 | 3.00000i | 14.0000 | 0 | 16.4317i | −54.7723 | −10.9545 | 72.0000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 55.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 275.5.d.b | 4 | |
| 5.b | even | 2 | 1 | inner | 275.5.d.b | 4 | |
| 5.c | odd | 4 | 1 | 11.5.b.b | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 275.5.c.e | 2 | ||
| 11.b | odd | 2 | 1 | inner | 275.5.d.b | 4 | |
| 15.e | even | 4 | 1 | 99.5.c.b | 2 | ||
| 20.e | even | 4 | 1 | 176.5.h.c | 2 | ||
| 40.i | odd | 4 | 1 | 704.5.h.f | 2 | ||
| 40.k | even | 4 | 1 | 704.5.h.d | 2 | ||
| 55.d | odd | 2 | 1 | inner | 275.5.d.b | 4 | |
| 55.e | even | 4 | 1 | 11.5.b.b | ✓ | 2 | |
| 55.e | even | 4 | 1 | 275.5.c.e | 2 | ||
| 55.k | odd | 20 | 4 | 121.5.d.b | 8 | ||
| 55.l | even | 20 | 4 | 121.5.d.b | 8 | ||
| 165.l | odd | 4 | 1 | 99.5.c.b | 2 | ||
| 220.i | odd | 4 | 1 | 176.5.h.c | 2 | ||
| 440.t | even | 4 | 1 | 704.5.h.f | 2 | ||
| 440.w | odd | 4 | 1 | 704.5.h.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.5.b.b | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 11.5.b.b | ✓ | 2 | 55.e | even | 4 | 1 | |
| 99.5.c.b | 2 | 15.e | even | 4 | 1 | ||
| 99.5.c.b | 2 | 165.l | odd | 4 | 1 | ||
| 121.5.d.b | 8 | 55.k | odd | 20 | 4 | ||
| 121.5.d.b | 8 | 55.l | even | 20 | 4 | ||
| 176.5.h.c | 2 | 20.e | even | 4 | 1 | ||
| 176.5.h.c | 2 | 220.i | odd | 4 | 1 | ||
| 275.5.c.e | 2 | 5.c | odd | 4 | 1 | ||
| 275.5.c.e | 2 | 55.e | even | 4 | 1 | ||
| 275.5.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 275.5.d.b | 4 | 5.b | even | 2 | 1 | inner | |
| 275.5.d.b | 4 | 11.b | odd | 2 | 1 | inner | |
| 275.5.d.b | 4 | 55.d | odd | 2 | 1 | inner | |
| 704.5.h.d | 2 | 40.k | even | 4 | 1 | ||
| 704.5.h.d | 2 | 440.w | odd | 4 | 1 | ||
| 704.5.h.f | 2 | 40.i | odd | 4 | 1 | ||
| 704.5.h.f | 2 | 440.t | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 30 \)
acting on \(S_{5}^{\mathrm{new}}(275, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 30)^{2} \)
|
| $3$ |
\( (T^{2} + 9)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} - 3000)^{2} \)
|
| $11$ |
\( (T^{2} - 22 T + 14641)^{2} \)
|
| $13$ |
\( (T^{2} - 34680)^{2} \)
|
| $17$ |
\( (T^{2} - 52920)^{2} \)
|
| $19$ |
\( (T^{2} + 9720)^{2} \)
|
| $23$ |
\( (T^{2} + 76729)^{2} \)
|
| $29$ |
\( (T^{2} + 1614720)^{2} \)
|
| $31$ |
\( (T + 1363)^{4} \)
|
| $37$ |
\( (T^{2} + 27889)^{2} \)
|
| $41$ |
\( (T^{2} + 1129080)^{2} \)
|
| $43$ |
\( (T^{2} - 1452000)^{2} \)
|
| $47$ |
\( (T^{2} + 2896804)^{2} \)
|
| $53$ |
\( (T^{2} + 20448484)^{2} \)
|
| $59$ |
\( (T - 2363)^{4} \)
|
| $61$ |
\( (T^{2} + 15725280)^{2} \)
|
| $67$ |
\( (T^{2} + 7856809)^{2} \)
|
| $71$ |
\( (T - 3397)^{4} \)
|
| $73$ |
\( (T^{2} - 11017080)^{2} \)
|
| $79$ |
\( (T^{2} + 37096320)^{2} \)
|
| $83$ |
\( (T^{2} - 693120)^{2} \)
|
| $89$ |
\( (T - 4673)^{4} \)
|
| $97$ |
\( (T^{2} + 18037009)^{2} \)
|
| show more | |
| show less |