Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [925,2,Mod(149,925)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(925, 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("925.149");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 925 = 5^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 925.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: | \(7.38616218697\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | 10.0.4414301848576.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 11x^{8} + 43x^{6} + 72x^{4} + 49x^{2} + 9 \)
|
| 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
\( x^{10} + 11x^{8} + 43x^{6} + 72x^{4} + 49x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{9} - 8\nu^{7} - 16\nu^{5} + 3\nu^{3} + 17\nu ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{9} - 11\nu^{7} - 40\nu^{5} - 51\nu^{3} - 13\nu ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{9} + 11\nu^{7} + 40\nu^{5} + 54\nu^{3} + 22\nu ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{9} - 19\nu^{7} - 59\nu^{5} - 66\nu^{3} - 20\nu ) / 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{8} - 9\nu^{6} - 25\nu^{4} - 21\nu^{2} - 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{8} + 9\nu^{6} + 26\nu^{4} + 26\nu^{2} + 6 \)
|
| \(\beta_{9}\) | \(=\) |
\( \nu^{8} + 10\nu^{6} + 32\nu^{4} + 34\nu^{2} + 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{7} - 5\beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{6} - 6\beta_{5} - 5\beta_{4} + \beta_{3} + 10\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{9} - 7\beta_{8} - 6\beta_{7} + 22\beta_{2} - 21 \)
|
| \(\nu^{7}\) | \(=\) |
\( 8\beta_{6} + 30\beta_{5} + 21\beta_{4} - 7\beta_{3} - 36\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -9\beta_{9} + 38\beta_{8} + 28\beta_{7} - 94\beta_{2} + 79 \)
|
| \(\nu^{9}\) | \(=\) |
\( -48\beta_{6} - 141\beta_{5} - 85\beta_{4} + 37\beta_{3} + 136\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).
| \(n\) | \(76\) | \(852\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 149.1 |
|
− | 2.08508i | − | 0.921590i | −2.34756 | 0 | −1.92159 | 1.62571i | 0.724698i | 2.15067 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 149.2 | − | 1.83708i | − | 0.352475i | −1.37487 | 0 | −0.647525 | 3.54814i | − | 1.14842i | 2.87576 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 149.3 | − | 1.32892i | − | 3.04022i | 0.233963 | 0 | −4.04022 | − | 3.11718i | − | 2.96877i | −6.24295 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 149.4 | − | 1.10856i | − | 0.474258i | 0.771103 | 0 | −0.525742 | − | 2.10017i | − | 3.07192i | 2.77508 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 149.5 | − | 0.531633i | 2.13508i | 1.71737 | 0 | 1.13508 | 2.93944i | − | 1.97628i | −1.55856 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 149.6 | 0.531633i | − | 2.13508i | 1.71737 | 0 | 1.13508 | − | 2.93944i | 1.97628i | −1.55856 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 149.7 | 1.10856i | 0.474258i | 0.771103 | 0 | −0.525742 | 2.10017i | 3.07192i | 2.77508 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 149.8 | 1.32892i | 3.04022i | 0.233963 | 0 | −4.04022 | 3.11718i | 2.96877i | −6.24295 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 149.9 | 1.83708i | 0.352475i | −1.37487 | 0 | −0.647525 | − | 3.54814i | 1.14842i | 2.87576 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 149.10 | 2.08508i | 0.921590i | −2.34756 | 0 | −1.92159 | − | 1.62571i | − | 0.724698i | 2.15067 | 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 | 925.2.b.h | 10 | |
| 5.b | even | 2 | 1 | inner | 925.2.b.h | 10 | |
| 5.c | odd | 4 | 1 | 925.2.a.g | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 925.2.a.i | yes | 5 | |
| 15.e | even | 4 | 1 | 8325.2.a.cb | 5 | ||
| 15.e | even | 4 | 1 | 8325.2.a.cd | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 925.2.a.g | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 925.2.a.i | yes | 5 | 5.c | odd | 4 | 1 | |
| 925.2.b.h | 10 | 1.a | even | 1 | 1 | trivial | |
| 925.2.b.h | 10 | 5.b | even | 2 | 1 | inner | |
| 8325.2.a.cb | 5 | 15.e | even | 4 | 1 | ||
| 8325.2.a.cd | 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 \(S_{2}^{\mathrm{new}}(925, [\chi])\):
|
\( T_{2}^{10} + 11T_{2}^{8} + 43T_{2}^{6} + 72T_{2}^{4} + 49T_{2}^{2} + 9 \)
|
|
\( T_{3}^{10} + 15T_{3}^{8} + 59T_{3}^{6} + 55T_{3}^{4} + 14T_{3}^{2} + 1 \)
|
|
\( T_{7}^{10} + 38T_{7}^{8} + 545T_{7}^{6} + 3640T_{7}^{4} + 11128T_{7}^{2} + 12321 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 11 T^{8} + \cdots + 9 \)
|
| $3$ |
\( T^{10} + 15 T^{8} + \cdots + 1 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + 38 T^{8} + \cdots + 12321 \)
|
| $11$ |
\( (T^{5} + 16 T^{4} + \cdots - 355)^{2} \)
|
| $13$ |
\( T^{10} + 99 T^{8} + \cdots + 143641 \)
|
| $17$ |
\( T^{10} + 132 T^{8} + \cdots + 403225 \)
|
| $19$ |
\( (T^{5} - 11 T^{4} + \cdots + 355)^{2} \)
|
| $23$ |
\( T^{10} + 191 T^{8} + \cdots + 6265009 \)
|
| $29$ |
\( (T^{5} - 15 T^{4} + \cdots + 2197)^{2} \)
|
| $31$ |
\( (T^{5} + 13 T^{4} + \cdots + 81)^{2} \)
|
| $37$ |
\( (T^{2} + 1)^{5} \)
|
| $41$ |
\( (T^{5} + 13 T^{4} + \cdots - 817)^{2} \)
|
| $43$ |
\( T^{10} + 127 T^{8} + \cdots + 40401 \)
|
| $47$ |
\( T^{10} + 230 T^{8} + \cdots + 33143049 \)
|
| $53$ |
\( T^{10} + 430 T^{8} + \cdots + 46253601 \)
|
| $59$ |
\( (T^{5} - 32 T^{4} + \cdots + 2087)^{2} \)
|
| $61$ |
\( (T^{5} - T^{4} - 53 T^{3} + \cdots - 733)^{2} \)
|
| $67$ |
\( T^{10} + 109 T^{8} + \cdots + 1708249 \)
|
| $71$ |
\( (T^{5} + 26 T^{4} + \cdots - 2647)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 263218176 \)
|
| $79$ |
\( (T^{5} - 3 T^{4} + \cdots - 76871)^{2} \)
|
| $83$ |
\( T^{10} + 332 T^{8} + \cdots + 13432225 \)
|
| $89$ |
\( (T^{5} - 20 T^{4} + \cdots - 35557)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 7191548809 \)
|
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