Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [275,3,Mod(76,275)]
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([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("275.76");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 275 = 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 275.c (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: | \(7.49320726991\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-5}, \sqrt{21})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + x^{2} + 105 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 55) |
| 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
\( x^{4} - 2x^{3} + x^{2} + 105 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 3\nu^{2} + 21\nu - 10 ) / 41 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{3} + 6\nu^{2} + 40\nu - 21 ) / 41 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 2\beta _1 + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{3} + 23\beta_{2} + 3\beta _1 + 1 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 76.1 |
|
− | 2.23607i | −4.58258 | −1.00000 | 0 | 10.2470i | − | 11.1803i | − | 6.70820i | 12.0000 | 0 | |||||||||||||||||||||||||||
| 76.2 | − | 2.23607i | 4.58258 | −1.00000 | 0 | − | 10.2470i | − | 11.1803i | − | 6.70820i | 12.0000 | 0 | |||||||||||||||||||||||||||
| 76.3 | 2.23607i | −4.58258 | −1.00000 | 0 | − | 10.2470i | 11.1803i | 6.70820i | 12.0000 | 0 | ||||||||||||||||||||||||||||||
| 76.4 | 2.23607i | 4.58258 | −1.00000 | 0 | 10.2470i | 11.1803i | 6.70820i | 12.0000 | 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 | 275.3.c.e | 4 | |
| 5.b | even | 2 | 1 | inner | 275.3.c.e | 4 | |
| 5.c | odd | 4 | 2 | 55.3.d.d | ✓ | 4 | |
| 11.b | odd | 2 | 1 | inner | 275.3.c.e | 4 | |
| 15.e | even | 4 | 2 | 495.3.h.d | 4 | ||
| 20.e | even | 4 | 2 | 880.3.i.d | 4 | ||
| 55.d | odd | 2 | 1 | inner | 275.3.c.e | 4 | |
| 55.e | even | 4 | 2 | 55.3.d.d | ✓ | 4 | |
| 165.l | odd | 4 | 2 | 495.3.h.d | 4 | ||
| 220.i | odd | 4 | 2 | 880.3.i.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.3.d.d | ✓ | 4 | 5.c | odd | 4 | 2 | |
| 55.3.d.d | ✓ | 4 | 55.e | even | 4 | 2 | |
| 275.3.c.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 275.3.c.e | 4 | 5.b | even | 2 | 1 | inner | |
| 275.3.c.e | 4 | 11.b | odd | 2 | 1 | inner | |
| 275.3.c.e | 4 | 55.d | odd | 2 | 1 | inner | |
| 495.3.h.d | 4 | 15.e | even | 4 | 2 | ||
| 495.3.h.d | 4 | 165.l | odd | 4 | 2 | ||
| 880.3.i.d | 4 | 20.e | even | 4 | 2 | ||
| 880.3.i.d | 4 | 220.i | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(275, [\chi])\):
|
\( T_{2}^{2} + 5 \)
|
|
\( T_{3}^{2} - 21 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 5)^{2} \)
|
| $3$ |
\( (T^{2} - 21)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 125)^{2} \)
|
| $11$ |
\( (T^{2} + 8 T + 121)^{2} \)
|
| $13$ |
\( (T^{2} + 80)^{2} \)
|
| $17$ |
\( (T^{2} + 245)^{2} \)
|
| $19$ |
\( (T^{2} + 105)^{2} \)
|
| $23$ |
\( (T^{2} - 756)^{2} \)
|
| $29$ |
\( (T^{2} + 105)^{2} \)
|
| $31$ |
\( (T + 3)^{4} \)
|
| $37$ |
\( (T^{2} - 21)^{2} \)
|
| $41$ |
\( (T^{2} + 420)^{2} \)
|
| $43$ |
\( (T^{2} + 500)^{2} \)
|
| $47$ |
\( (T^{2} - 4116)^{2} \)
|
| $53$ |
\( (T^{2} - 21)^{2} \)
|
| $59$ |
\( (T - 18)^{4} \)
|
| $61$ |
\( (T^{2} + 5145)^{2} \)
|
| $67$ |
\( (T^{2} - 756)^{2} \)
|
| $71$ |
\( (T - 27)^{4} \)
|
| $73$ |
\( (T^{2} + 3380)^{2} \)
|
| $79$ |
\( (T^{2} + 3780)^{2} \)
|
| $83$ |
\( (T^{2} + 5120)^{2} \)
|
| $89$ |
\( (T + 37)^{4} \)
|
| $97$ |
\( (T^{2} - 14196)^{2} \)
|
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