Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [725,2,Mod(376,725)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(725, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("725.376");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 725 = 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 725.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: | \(5.78915414654\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.16516096.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 9x^{4} + 13x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 145) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 9x^{4} + 13x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} + 8\nu^{2} + 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} - 8\nu^{3} - 7\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 9\nu^{3} + 13\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} - 8\beta_{2} + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{5} - 9\beta_{4} + 41\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).
| \(n\) | \(176\) | \(552\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 376.1 |
|
− | 2.68667i | − | 2.31446i | −5.21819 | 0 | −6.21819 | −4.21819 | 8.64620i | −2.35673 | 0 | ||||||||||||||||||||||||||||||||||
| 376.2 | − | 1.30397i | − | 0.537080i | 0.299664 | 0 | −0.700336 | 1.29966 | − | 2.99869i | 2.71155 | 0 | ||||||||||||||||||||||||||||||||||
| 376.3 | − | 0.285442i | 3.21789i | 1.91852 | 0 | 0.918523 | 2.91852 | − | 1.11851i | −7.35482 | 0 | |||||||||||||||||||||||||||||||||||
| 376.4 | 0.285442i | − | 3.21789i | 1.91852 | 0 | 0.918523 | 2.91852 | 1.11851i | −7.35482 | 0 | ||||||||||||||||||||||||||||||||||||
| 376.5 | 1.30397i | 0.537080i | 0.299664 | 0 | −0.700336 | 1.29966 | 2.99869i | 2.71155 | 0 | |||||||||||||||||||||||||||||||||||||
| 376.6 | 2.68667i | 2.31446i | −5.21819 | 0 | −6.21819 | −4.21819 | − | 8.64620i | −2.35673 | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 725.2.c.e | 6 | |
| 5.b | even | 2 | 1 | 145.2.c.b | ✓ | 6 | |
| 5.c | odd | 4 | 2 | 725.2.d.c | 12 | ||
| 15.d | odd | 2 | 1 | 1305.2.d.b | 6 | ||
| 20.d | odd | 2 | 1 | 2320.2.g.i | 6 | ||
| 29.b | even | 2 | 1 | inner | 725.2.c.e | 6 | |
| 145.d | even | 2 | 1 | 145.2.c.b | ✓ | 6 | |
| 145.f | odd | 4 | 2 | 4205.2.a.m | 6 | ||
| 145.h | odd | 4 | 2 | 725.2.d.c | 12 | ||
| 435.b | odd | 2 | 1 | 1305.2.d.b | 6 | ||
| 580.e | odd | 2 | 1 | 2320.2.g.i | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 145.2.c.b | ✓ | 6 | 5.b | even | 2 | 1 | |
| 145.2.c.b | ✓ | 6 | 145.d | even | 2 | 1 | |
| 725.2.c.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 725.2.c.e | 6 | 29.b | even | 2 | 1 | inner | |
| 725.2.d.c | 12 | 5.c | odd | 4 | 2 | ||
| 725.2.d.c | 12 | 145.h | odd | 4 | 2 | ||
| 1305.2.d.b | 6 | 15.d | odd | 2 | 1 | ||
| 1305.2.d.b | 6 | 435.b | odd | 2 | 1 | ||
| 2320.2.g.i | 6 | 20.d | odd | 2 | 1 | ||
| 2320.2.g.i | 6 | 580.e | odd | 2 | 1 | ||
| 4205.2.a.m | 6 | 145.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\):
|
\( T_{2}^{6} + 9T_{2}^{4} + 13T_{2}^{2} + 1 \)
|
|
\( T_{3}^{6} + 16T_{3}^{4} + 60T_{3}^{2} + 16 \)
|
|
\( T_{7}^{3} - 14T_{7} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 9 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{6} + 16 T^{4} + \cdots + 16 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T^{3} - 14 T + 16)^{2} \)
|
| $11$ |
\( T^{6} + 16 T^{4} + \cdots + 16 \)
|
| $13$ |
\( (T^{3} - 2 T^{2} - 24 T + 16)^{2} \)
|
| $17$ |
\( T^{6} + 52 T^{4} + \cdots + 64 \)
|
| $19$ |
\( T^{6} + 56 T^{4} + \cdots + 1296 \)
|
| $23$ |
\( (T^{3} - 4 T^{2} - 26 T + 96)^{2} \)
|
| $29$ |
\( T^{6} + 6 T^{5} + \cdots + 24389 \)
|
| $31$ |
\( T^{6} + 152 T^{4} + \cdots + 144 \)
|
| $37$ |
\( T^{6} + 100 T^{4} + \cdots + 20736 \)
|
| $41$ |
\( T^{6} + 184 T^{4} + \cdots + 4096 \)
|
| $43$ |
\( T^{6} + 176 T^{4} + \cdots + 121104 \)
|
| $47$ |
\( T^{6} + 16 T^{4} + \cdots + 16 \)
|
| $53$ |
\( (T^{3} + 14 T^{2} + \cdots - 576)^{2} \)
|
| $59$ |
\( (T^{3} + 8 T^{2} - 4 T - 48)^{2} \)
|
| $61$ |
\( T^{6} + 104 T^{4} + \cdots + 2304 \)
|
| $67$ |
\( (T^{3} - 8 T^{2} - 10 T + 8)^{2} \)
|
| $71$ |
\( (T^{3} + 28 T^{2} + \cdots + 576)^{2} \)
|
| $73$ |
\( T^{6} + 252 T^{4} + \cdots + 419904 \)
|
| $79$ |
\( T^{6} + 176 T^{4} + \cdots + 121104 \)
|
| $83$ |
\( (T^{3} - 12 T^{2} + \cdots - 24)^{2} \)
|
| $89$ |
\( T^{6} + 160 T^{4} + \cdots + 65536 \)
|
| $97$ |
\( T^{6} + 28 T^{4} + \cdots + 576 \)
|
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