Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [725,2,Mod(349,725)]
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([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("725.349");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 725 = 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 725.b (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: | \(5.78915414654\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 14x^{8} + 71x^{6} + 159x^{4} + 151x^{2} + 49 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 14x^{8} + 71x^{6} + 159x^{4} + 151x^{2} + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{9} + 7\nu^{7} + \nu^{5} - 44\nu^{3} - 24\nu ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{9} + 21\nu^{7} + 65\nu^{5} + 52\nu^{3} - 13\nu ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} - 9\nu^{4} - 22\nu^{2} - 12 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} + 10\nu^{4} + 28\nu^{2} + 18 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{9} - 35\nu^{7} - 136\nu^{5} - 204\nu^{3} - 96\nu ) / 7 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -2\nu^{9} - 28\nu^{7} - 135\nu^{5} - 255\nu^{3} - 141\nu ) / 7 \)
|
| \(\beta_{9}\) | \(=\) |
\( \nu^{8} + 11\nu^{6} + 39\nu^{4} + 51\nu^{2} + 21 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} + \beta_{7} + \beta_{3} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} - 6\beta_{2} + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{8} - 9\beta_{7} - 2\beta_{4} - 7\beta_{3} + 10\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -9\beta_{6} - 10\beta_{5} + 32\beta_{2} - 54 \)
|
| \(\nu^{7}\) | \(=\) |
\( -52\beta_{8} + 61\beta_{7} + 19\beta_{4} + 41\beta_{3} - 35\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{9} + 60\beta_{6} + 71\beta_{5} - 169\beta_{2} + 258 \)
|
| \(\nu^{9}\) | \(=\) |
\( 312\beta_{8} - 374\beta_{7} - 131\beta_{4} - 229\beta_{3} + 127\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 349.1 |
|
− | 2.33090i | 0.318289i | −3.43308 | 0 | 0.741899 | − | 1.09271i | 3.34037i | 2.89869 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 349.2 | − | 1.88481i | − | 1.10085i | −1.55251 | 0 | −2.07489 | 1.70908i | − | 0.843434i | 1.78814 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 349.3 | − | 1.78154i | 3.10566i | −1.17387 | 0 | 5.53284 | − | 3.64565i | − | 1.47177i | −6.64510 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 349.4 | − | 1.06634i | − | 3.37897i | 0.862915 | 0 | −3.60314 | − | 2.91576i | − | 3.05285i | −8.41742 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 349.5 | − | 0.838718i | − | 1.90376i | 1.29655 | 0 | −1.59672 | − | 2.46832i | − | 2.76488i | −0.624302 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 349.6 | 0.838718i | 1.90376i | 1.29655 | 0 | −1.59672 | 2.46832i | 2.76488i | −0.624302 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 349.7 | 1.06634i | 3.37897i | 0.862915 | 0 | −3.60314 | 2.91576i | 3.05285i | −8.41742 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 349.8 | 1.78154i | − | 3.10566i | −1.17387 | 0 | 5.53284 | 3.64565i | 1.47177i | −6.64510 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 349.9 | 1.88481i | 1.10085i | −1.55251 | 0 | −2.07489 | − | 1.70908i | 0.843434i | 1.78814 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 349.10 | 2.33090i | − | 0.318289i | −3.43308 | 0 | 0.741899 | 1.09271i | − | 3.34037i | 2.89869 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.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.b.f | 10 | |
| 5.b | even | 2 | 1 | inner | 725.2.b.f | 10 | |
| 5.c | odd | 4 | 1 | 725.2.a.h | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 725.2.a.k | yes | 5 | |
| 15.e | even | 4 | 1 | 6525.2.a.bm | 5 | ||
| 15.e | even | 4 | 1 | 6525.2.a.bq | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 725.2.a.h | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 725.2.a.k | yes | 5 | 5.c | odd | 4 | 1 | |
| 725.2.b.f | 10 | 1.a | even | 1 | 1 | trivial | |
| 725.2.b.f | 10 | 5.b | even | 2 | 1 | inner | |
| 6525.2.a.bm | 5 | 15.e | even | 4 | 1 | ||
| 6525.2.a.bq | 5 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\):
|
\( T_{2}^{10} + 14T_{2}^{8} + 71T_{2}^{6} + 159T_{2}^{4} + 151T_{2}^{2} + 49 \)
|
|
\( T_{3}^{10} + 26T_{3}^{8} + 219T_{3}^{6} + 647T_{3}^{4} + 547T_{3}^{2} + 49 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 14 T^{8} + \cdots + 49 \)
|
| $3$ |
\( T^{10} + 26 T^{8} + \cdots + 49 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + 32 T^{8} + \cdots + 2401 \)
|
| $11$ |
\( (T^{5} + 2 T^{4} + \cdots + 307)^{2} \)
|
| $13$ |
\( T^{10} + 52 T^{8} + \cdots + 1 \)
|
| $17$ |
\( T^{10} + 87 T^{8} + \cdots + 192721 \)
|
| $19$ |
\( (T^{5} + 2 T^{4} - 47 T^{3} + \cdots - 5)^{2} \)
|
| $23$ |
\( T^{10} + 125 T^{8} + \cdots + 337561 \)
|
| $29$ |
\( (T + 1)^{10} \)
|
| $31$ |
\( (T^{5} + T^{4} - 30 T^{3} + \cdots - 73)^{2} \)
|
| $37$ |
\( T^{10} + 90 T^{8} + \cdots + 3969 \)
|
| $41$ |
\( (T^{5} - 5 T^{4} - 13 T^{3} + \cdots + 9)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 521345889 \)
|
| $47$ |
\( T^{10} + 151 T^{8} + \cdots + 5121169 \)
|
| $53$ |
\( T^{10} + 174 T^{8} + \cdots + 744769 \)
|
| $59$ |
\( (T^{5} - 11 T^{4} + \cdots - 2205)^{2} \)
|
| $61$ |
\( (T^{5} + 5 T^{4} + \cdots - 16381)^{2} \)
|
| $67$ |
\( T^{10} + 259 T^{8} + \cdots + 124609 \)
|
| $71$ |
\( (T^{5} + 5 T^{4} + \cdots + 2837)^{2} \)
|
| $73$ |
\( T^{10} + 196 T^{8} + \cdots + 81 \)
|
| $79$ |
\( (T^{5} - 10 T^{4} + \cdots - 10035)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 328080769 \)
|
| $89$ |
\( (T^{5} - 18 T^{4} + \cdots + 25)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 38574138409 \)
|
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