Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [175,3,Mod(76,175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, 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("175.76");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 175.d (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: | \(4.76840462631\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.1163520.6 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 46x^{2} + 505 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} + 46x^{2} + 505 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} + 23\nu + 23 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} + 23 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} - 23 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{3} + 4\beta_{2} - 23\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 |
|
−1.44949 | − | 4.25453i | −1.89898 | 0 | 6.16690i | −4.67423 | − | 5.21071i | 8.55051 | −9.10102 | 0 | |||||||||||||||||||||||||||
| 76.2 | −1.44949 | 4.25453i | −1.89898 | 0 | − | 6.16690i | −4.67423 | + | 5.21071i | 8.55051 | −9.10102 | 0 | ||||||||||||||||||||||||||||
| 76.3 | 3.44949 | − | 5.28195i | 7.89898 | 0 | − | 18.2200i | 2.67423 | + | 6.46904i | 13.4495 | −18.8990 | 0 | |||||||||||||||||||||||||||
| 76.4 | 3.44949 | 5.28195i | 7.89898 | 0 | 18.2200i | 2.67423 | − | 6.46904i | 13.4495 | −18.8990 | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 175.3.d.j | yes | 4 |
| 5.b | even | 2 | 1 | 175.3.d.i | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 175.3.c.e | 8 | ||
| 7.b | odd | 2 | 1 | inner | 175.3.d.j | yes | 4 |
| 35.c | odd | 2 | 1 | 175.3.d.i | ✓ | 4 | |
| 35.f | even | 4 | 2 | 175.3.c.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 175.3.c.e | 8 | 5.c | odd | 4 | 2 | ||
| 175.3.c.e | 8 | 35.f | even | 4 | 2 | ||
| 175.3.d.i | ✓ | 4 | 5.b | even | 2 | 1 | |
| 175.3.d.i | ✓ | 4 | 35.c | odd | 2 | 1 | |
| 175.3.d.j | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 175.3.d.j | yes | 4 | 7.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(175, [\chi])\):
|
\( T_{2}^{2} - 2T_{2} - 5 \)
|
|
\( T_{3}^{4} + 46T_{3}^{2} + 505 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T - 5)^{2} \)
|
| $3$ |
\( T^{4} + 46T^{2} + 505 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 4 T^{3} + \cdots + 2401 \)
|
| $11$ |
\( (T^{2} + 14 T + 43)^{2} \)
|
| $13$ |
\( T^{4} + 276 T^{2} + 18180 \)
|
| $17$ |
\( T^{4} + 546T^{2} + 4545 \)
|
| $19$ |
\( T^{4} + 1230 T^{2} + 113625 \)
|
| $23$ |
\( (T^{2} - 8 T - 470)^{2} \)
|
| $29$ |
\( (T^{2} - 28 T + 142)^{2} \)
|
| $31$ |
\( T^{4} + 2220 T^{2} + 454500 \)
|
| $37$ |
\( (T^{2} + 88 T + 1930)^{2} \)
|
| $41$ |
\( T^{4} + 3330 T^{2} + 1022625 \)
|
| $43$ |
\( (T^{2} + 52 T - 500)^{2} \)
|
| $47$ |
\( T^{4} + 7116 T^{2} + 6562980 \)
|
| $53$ |
\( (T^{2} - 44 T + 430)^{2} \)
|
| $59$ |
\( T^{4} + 4920 T^{2} + 1818000 \)
|
| $61$ |
\( T^{4} + 8880 T^{2} + 7272000 \)
|
| $67$ |
\( (T^{2} - 62 T + 235)^{2} \)
|
| $71$ |
\( (T^{2} + 104 T - 2342)^{2} \)
|
| $73$ |
\( T^{4} + 14946 T^{2} + 41018625 \)
|
| $79$ |
\( (T^{2} + 152 T + 2872)^{2} \)
|
| $83$ |
\( T^{4} + 5406 T^{2} + 3822345 \)
|
| $89$ |
\( T^{4} + 21570 T^{2} + 95558625 \)
|
| $97$ |
\( T^{4} + 3336 T^{2} + 72720 \)
|
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