Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5915,2,Mod(1,5915)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5915, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5915.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5915 = 5 \cdot 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5915.a (trivial) |
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: | yes |
| Analytic conductor: | \(47.2315127956\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{10} - 12x^{8} + 46x^{6} - 4x^{5} - 63x^{4} + 18x^{3} + 28x^{2} - 14x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 455) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{10} - 12x^{8} + 46x^{6} - 4x^{5} - 63x^{4} + 18x^{3} + 28x^{2} - 14x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{9} - \nu^{8} + 11\nu^{7} + 11\nu^{6} - 34\nu^{5} - 32\nu^{4} + 25\nu^{3} + 19\nu^{2} - 4\nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{9} + \nu^{8} - 11\nu^{7} - 11\nu^{6} + 35\nu^{5} + 31\nu^{4} - 32\nu^{3} - 13\nu^{2} + 14\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{9} - \nu^{8} + 23\nu^{7} + 12\nu^{6} - 81\nu^{5} - 36\nu^{4} + 93\nu^{3} + 17\nu^{2} - 35\nu + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{9} + \nu^{8} - 23\nu^{7} - 12\nu^{6} + 81\nu^{5} + 37\nu^{4} - 93\nu^{3} - 23\nu^{2} + 34\nu - 1 \)
|
| \(\beta_{6}\) | \(=\) |
\( 3\nu^{9} + 2\nu^{8} - 35\nu^{7} - 23\nu^{6} + 126\nu^{5} + 69\nu^{4} - 152\nu^{3} - 39\nu^{2} + 62\nu - 7 \)
|
| \(\beta_{7}\) | \(=\) |
\( -3\nu^{9} - 2\nu^{8} + 35\nu^{7} + 23\nu^{6} - 126\nu^{5} - 69\nu^{4} + 153\nu^{3} + 39\nu^{2} - 66\nu + 7 \)
|
| \(\beta_{8}\) | \(=\) |
\( 3\nu^{9} + 3\nu^{8} - 34\nu^{7} - 34\nu^{6} + 115\nu^{5} + 103\nu^{4} - 121\nu^{3} - 64\nu^{2} + 48\nu - 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( -4\nu^{9} - 3\nu^{8} + 46\nu^{7} + 34\nu^{6} - 161\nu^{5} - 100\nu^{4} + 184\nu^{3} + 53\nu^{2} - 76\nu + 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} + \beta_{6} + \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{9} + 6\beta_{6} + \beta_{5} + \beta_{4} + 6\beta_{3} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{7} + 7\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 19\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 32\beta_{9} - \beta_{8} + 2\beta_{7} + 35\beta_{6} + 8\beta_{5} + 9\beta_{4} + 34\beta_{3} + 9\beta _1 + 72 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3\beta_{9} + 43\beta_{7} + 45\beta_{6} + 12\beta_{5} + 11\beta_{4} + 14\beta_{3} + 10\beta_{2} + 97\beta _1 + 23 \)
|
| \(\nu^{8}\) | \(=\) |
\( 170 \beta_{9} - 10 \beta_{8} + 25 \beta_{7} + 206 \beta_{6} + 53 \beta_{5} + 65 \beta_{4} + 192 \beta_{3} + \cdots + 385 \)
|
| \(\nu^{9}\) | \(=\) |
\( 42 \beta_{9} - \beta_{8} + 257 \beta_{7} + 288 \beta_{6} + 101 \beta_{5} + 89 \beta_{4} + 129 \beta_{3} + \cdots + 199 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.26533 | 0.311609 | 3.13170 | 1.00000 | −0.705896 | −1.00000 | −2.56367 | −2.90290 | −2.26533 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.09845 | 1.64801 | 2.40348 | 1.00000 | −3.45827 | −1.00000 | −0.846674 | −0.284048 | −2.09845 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.30628 | −1.98433 | −0.293633 | 1.00000 | 2.59208 | −1.00000 | 2.99613 | 0.937548 | −1.30628 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.04111 | 0.293806 | −0.916098 | 1.00000 | −0.305883 | −1.00000 | 3.03597 | −2.91368 | −1.04111 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.0872451 | 1.88839 | −1.99239 | 1.00000 | 0.164753 | −1.00000 | −0.348316 | 0.566034 | 0.0872451 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.565237 | −2.39129 | −1.68051 | 1.00000 | −1.35165 | −1.00000 | −2.08036 | 2.71829 | 0.565237 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.753523 | 0.142106 | −1.43220 | 1.00000 | 0.107080 | −1.00000 | −2.58624 | −2.97981 | 0.753523 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.850766 | −3.16494 | −1.27620 | 1.00000 | −2.69262 | −1.00000 | −2.78728 | 7.01683 | 0.850766 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.96732 | 0.963497 | 1.87033 | 1.00000 | 1.89550 | −1.00000 | −0.255092 | −2.07167 | 1.96732 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.48707 | −1.70687 | 4.18552 | 1.00000 | −4.24510 | −1.00000 | 5.43553 | −0.0865955 | 2.48707 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
| \(13\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5915.2.a.bf | 10 | |
| 13.b | even | 2 | 1 | 5915.2.a.be | 10 | ||
| 13.f | odd | 12 | 2 | 455.2.bq.a | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 455.2.bq.a | ✓ | 20 | 13.f | odd | 12 | 2 | |
| 5915.2.a.be | 10 | 13.b | even | 2 | 1 | ||
| 5915.2.a.bf | 10 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5915))\):
|
\( T_{2}^{10} - 12T_{2}^{8} + 46T_{2}^{6} - 4T_{2}^{5} - 63T_{2}^{4} + 18T_{2}^{3} + 28T_{2}^{2} - 14T_{2} + 1 \)
|
|
\( T_{3}^{10} + 4T_{3}^{9} - 7T_{3}^{8} - 34T_{3}^{7} + 20T_{3}^{6} + 94T_{3}^{5} - 45T_{3}^{4} - 80T_{3}^{3} + 61T_{3}^{2} - 14T_{3} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 12 T^{8} + \cdots + 1 \)
|
| $3$ |
\( T^{10} + 4 T^{9} + \cdots + 1 \)
|
| $5$ |
\( (T - 1)^{10} \)
|
| $7$ |
\( (T + 1)^{10} \)
|
| $11$ |
\( T^{10} + 12 T^{9} + \cdots - 239 \)
|
| $13$ |
\( T^{10} \)
|
| $17$ |
\( T^{10} + 10 T^{9} + \cdots + 1297 \)
|
| $19$ |
\( T^{10} - 16 T^{9} + \cdots - 11447 \)
|
| $23$ |
\( T^{10} - 2 T^{9} + \cdots + 1 \)
|
| $29$ |
\( T^{10} + 12 T^{9} + \cdots + 6453733 \)
|
| $31$ |
\( T^{10} - 18 T^{9} + \cdots + 18217 \)
|
| $37$ |
\( T^{10} - 2 T^{9} + \cdots + 20257 \)
|
| $41$ |
\( T^{10} + 26 T^{9} + \cdots - 24706727 \)
|
| $43$ |
\( T^{10} + 8 T^{9} + \cdots + 6757201 \)
|
| $47$ |
\( T^{10} + \cdots - 147317243 \)
|
| $53$ |
\( T^{10} + 12 T^{9} + \cdots - 611 \)
|
| $59$ |
\( T^{10} - 10 T^{9} + \cdots + 1892497 \)
|
| $61$ |
\( T^{10} + 42 T^{9} + \cdots - 24554879 \)
|
| $67$ |
\( T^{10} + 10 T^{9} + \cdots + 1552261 \)
|
| $71$ |
\( T^{10} + \cdots + 271541677 \)
|
| $73$ |
\( T^{10} - 8 T^{9} + \cdots - 3035987 \)
|
| $79$ |
\( T^{10} + 22 T^{9} + \cdots + 40134109 \)
|
| $83$ |
\( T^{10} - 18 T^{9} + \cdots - 73866059 \)
|
| $89$ |
\( T^{10} + 26 T^{9} + \cdots - 4214111 \)
|
| $97$ |
\( T^{10} + 2 T^{9} + \cdots + 34170601 \)
|
| show more | |
| show less |