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: |
\( x^{10} + 14x^{8} + 63x^{6} + 99x^{4} + 55x^{2} + 9 \)
|
| 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
\( x^{10} + 14x^{8} + 63x^{6} + 99x^{4} + 55x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{9} + 11\nu^{7} + 30\nu^{5} - 6\nu^{3} - 32\nu ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} + 11\nu^{6} + 35\nu^{4} + 29\nu^{2} + 3 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{9} - 27\nu^{7} - 110\nu^{5} - 123\nu^{3} - 16\nu ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{8} - 27\nu^{6} - 110\nu^{4} - 123\nu^{2} - 21 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{8} + 27\nu^{6} + 110\nu^{4} + 128\nu^{2} + 36 ) / 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{9} + 27\nu^{7} + 115\nu^{5} + 158\nu^{3} + 51\nu ) / 5 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 2\nu^{8} + 27\nu^{6} + 115\nu^{4} + 158\nu^{2} + 51 ) / 5 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 4\nu^{9} + 54\nu^{7} + 225\nu^{5} + 286\nu^{3} + 92\nu ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{7} + \beta_{4} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - 7\beta_{6} - 6\beta_{5} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{9} + 8\beta_{7} - 6\beta_{4} + 28\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -8\beta_{8} + 43\beta_{6} + 34\beta_{5} - 2\beta_{3} - 84 \)
|
| \(\nu^{7}\) | \(=\) |
\( 43\beta_{9} - 53\beta_{7} + 32\beta_{4} - 6\beta_{2} - 161\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 53\beta_{8} - 257\beta_{6} - 193\beta_{5} + 27\beta_{3} + 483 \)
|
| \(\nu^{9}\) | \(=\) |
\( -257\beta_{9} + 337\beta_{7} - 166\beta_{4} + 81\beta_{2} + 933\beta_1 \)
|
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 |
|
− | 2.45914i | − | 3.33648i | −4.04739 | 0 | −8.20489 | 3.50903i | 5.03483i | −8.13211 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 349.2 | − | 2.35417i | 1.80364i | −3.54210 | 0 | 4.24606 | 0.0127981i | 3.63036i | −0.253109 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 349.3 | − | 1.22277i | 1.05494i | 0.504827 | 0 | 1.28995 | 4.90333i | − | 3.06283i | 1.88710 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 349.4 | − | 0.794018i | 2.48525i | 1.36954 | 0 | 1.97334 | − | 3.07654i | − | 2.67547i | −3.17649 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 349.5 | − | 0.533733i | − | 0.570435i | 1.71513 | 0 | −0.304460 | 1.47609i | − | 1.98289i | 2.67460 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 349.6 | 0.533733i | 0.570435i | 1.71513 | 0 | −0.304460 | − | 1.47609i | 1.98289i | 2.67460 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 349.7 | 0.794018i | − | 2.48525i | 1.36954 | 0 | 1.97334 | 3.07654i | 2.67547i | −3.17649 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 349.8 | 1.22277i | − | 1.05494i | 0.504827 | 0 | 1.28995 | − | 4.90333i | 3.06283i | 1.88710 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 349.9 | 2.35417i | − | 1.80364i | −3.54210 | 0 | 4.24606 | − | 0.0127981i | − | 3.63036i | −0.253109 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 349.10 | 2.45914i | 3.33648i | −4.04739 | 0 | −8.20489 | − | 3.50903i | − | 5.03483i | −8.13211 | 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.g | 10 | |
| 5.b | even | 2 | 1 | inner | 725.2.b.g | 10 | |
| 5.c | odd | 4 | 1 | 725.2.a.i | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 725.2.a.j | yes | 5 | |
| 15.e | even | 4 | 1 | 6525.2.a.bo | 5 | ||
| 15.e | even | 4 | 1 | 6525.2.a.bp | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 725.2.a.i | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 725.2.a.j | yes | 5 | 5.c | odd | 4 | 1 | |
| 725.2.b.g | 10 | 1.a | even | 1 | 1 | trivial | |
| 725.2.b.g | 10 | 5.b | even | 2 | 1 | inner | |
| 6525.2.a.bo | 5 | 15.e | even | 4 | 1 | ||
| 6525.2.a.bp | 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 \(S_{2}^{\mathrm{new}}(725, [\chi])\):
|
\( T_{2}^{10} + 14T_{2}^{8} + 63T_{2}^{6} + 99T_{2}^{4} + 55T_{2}^{2} + 9 \)
|
|
\( T_{3}^{10} + 22T_{3}^{8} + 155T_{3}^{6} + 411T_{3}^{4} + 367T_{3}^{2} + 81 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 14 T^{8} + 63 T^{6} + 99 T^{4} + \cdots + 9 \)
|
| $3$ |
\( T^{10} + 22 T^{8} + 155 T^{6} + \cdots + 81 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + 48 T^{8} + 740 T^{6} + 4197 T^{4} + \cdots + 1 \)
|
| $11$ |
\( (T^{5} - 2 T^{4} - 17 T^{3} + 14 T^{2} + \cdots - 27)^{2} \)
|
| $13$ |
\( T^{10} + 108 T^{8} + 4040 T^{6} + \cdots + 185761 \)
|
| $17$ |
\( T^{10} + 91 T^{8} + 2663 T^{6} + \cdots + 114921 \)
|
| $19$ |
\( (T^{5} - 4 T^{4} - 9 T^{3} + 23 T^{2} + \cdots + 17)^{2} \)
|
| $23$ |
\( T^{10} + 45 T^{8} + 728 T^{6} + \cdots + 15129 \)
|
| $29$ |
\( (T - 1)^{10} \)
|
| $31$ |
\( (T^{5} - 5 T^{4} - 58 T^{3} + 381 T^{2} + \cdots - 251)^{2} \)
|
| $37$ |
\( T^{10} + 162 T^{8} + \cdots + 13623481 \)
|
| $41$ |
\( (T^{5} + 3 T^{4} - 157 T^{3} + 10 T^{2} + \cdots - 19125)^{2} \)
|
| $43$ |
\( T^{10} + 138 T^{8} + 5141 T^{6} + \cdots + 38809 \)
|
| $47$ |
\( T^{10} + 139 T^{8} + 6371 T^{6} + \cdots + 4028049 \)
|
| $53$ |
\( T^{10} + 206 T^{8} + 5419 T^{6} + \cdots + 81 \)
|
| $59$ |
\( (T^{5} - 15 T^{4} + 52 T^{3} + 85 T^{2} + \cdots + 375)^{2} \)
|
| $61$ |
\( (T^{5} + 7 T^{4} - 111 T^{3} - 27 T^{2} + \cdots + 81)^{2} \)
|
| $67$ |
\( T^{10} + 411 T^{8} + 48707 T^{6} + \cdots + 2819041 \)
|
| $71$ |
\( (T^{5} + 17 T^{4} - 178 T^{3} - 3925 T^{2} + \cdots + 5121)^{2} \)
|
| $73$ |
\( T^{10} + 424 T^{8} + \cdots + 175748049 \)
|
| $79$ |
\( (T^{5} - 2 T^{4} - 228 T^{3} + 849 T^{2} + \cdots - 66293)^{2} \)
|
| $83$ |
\( T^{10} + 919 T^{8} + \cdots + 134591395689 \)
|
| $89$ |
\( (T^{5} + 14 T^{4} - 360 T^{3} + \cdots + 278703)^{2} \)
|
| $97$ |
\( T^{10} + 607 T^{8} + \cdots + 2311590241 \)
|
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