Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4730,2,Mod(1,4730)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4730, 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("4730.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4730.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: | \(37.7692401561\) |
| 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: |
\( x^{10} - x^{9} - 21x^{8} + 22x^{7} + 138x^{6} - 154x^{5} - 291x^{4} + 327x^{3} + 97x^{2} - 124x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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
\( x^{10} - x^{9} - 21x^{8} + 22x^{7} + 138x^{6} - 154x^{5} - 291x^{4} + 327x^{3} + 97x^{2} - 124x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 21 \nu^{9} - 5 \nu^{8} + 445 \nu^{7} + 38 \nu^{6} - 3014 \nu^{5} + 338 \nu^{4} + 7253 \nu^{3} + \cdots + 864 ) / 214 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 349 \nu^{9} - 185 \nu^{8} + 6621 \nu^{7} + 2048 \nu^{6} - 37474 \nu^{5} + 2876 \nu^{4} + \cdots + 13136 ) / 2354 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 328 \nu^{9} + 501 \nu^{8} - 5855 \nu^{7} - 8002 \nu^{6} + 29324 \nu^{5} + 29990 \nu^{4} - 41116 \nu^{3} + \cdots - 181 ) / 1177 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1005 \nu^{9} + 1187 \nu^{8} - 18331 \nu^{7} - 18052 \nu^{6} + 96122 \nu^{5} + 57104 \nu^{4} + \cdots - 15852 ) / 2354 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 53 \nu^{9} - 33 \nu^{8} + 1011 \nu^{7} + 422 \nu^{6} - 5747 \nu^{5} - 380 \nu^{4} + 10270 \nu^{3} + \cdots + 1294 ) / 107 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 679 \nu^{9} + 768 \nu^{8} - 12605 \nu^{7} - 11893 \nu^{6} + 68527 \nu^{5} + 40574 \nu^{4} + \cdots - 1828 ) / 1177 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 1497 \nu^{9} - 2527 \nu^{8} + 26525 \nu^{7} + 40648 \nu^{6} - 130692 \nu^{5} - 157408 \nu^{4} + \cdots - 9182 ) / 2354 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{4} + 7\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + \beta_{8} - 2\beta_{6} - 2\beta_{5} - 2\beta_{4} + \beta_{3} + 10\beta_{2} - \beta _1 + 31 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{8} + 3\beta_{7} - 11\beta_{6} + 9\beta_{5} - 17\beta_{4} + 56\beta _1 - 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( -17\beta_{9} + 11\beta_{8} - 31\beta_{6} - 29\beta_{5} - 36\beta_{4} + 12\beta_{3} + 96\beta_{2} - 12\beta _1 + 274 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{9} + 36 \beta_{8} + 49 \beta_{7} - 109 \beta_{6} + 74 \beta_{5} - 208 \beta_{4} + 8 \beta_{3} + \cdots - 91 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 217 \beta_{9} + 100 \beta_{8} - \beta_{7} - 385 \beta_{6} - 323 \beta_{5} - 468 \beta_{4} + \cdots + 2558 \)
|
| \(\nu^{9}\) | \(=\) |
\( 26 \beta_{9} + 488 \beta_{8} + 608 \beta_{7} - 1073 \beta_{6} + 614 \beta_{5} - 2299 \beta_{4} + \cdots - 578 \)
|
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 |
|
1.00000 | −3.24563 | 1.00000 | −1.00000 | −3.24563 | −2.81488 | 1.00000 | 7.53411 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.19566 | 1.00000 | −1.00000 | −2.19566 | 4.26177 | 1.00000 | 1.82093 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −2.18792 | 1.00000 | −1.00000 | −2.18792 | −0.664484 | 1.00000 | 1.78700 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.10594 | 1.00000 | −1.00000 | −1.10594 | 2.85930 | 1.00000 | −1.77690 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −0.530117 | 1.00000 | −1.00000 | −0.530117 | −4.20387 | 1.00000 | −2.71898 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −0.156987 | 1.00000 | −1.00000 | −0.156987 | −3.72319 | 1.00000 | −2.97536 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 0.721995 | 1.00000 | −1.00000 | 0.721995 | 3.23576 | 1.00000 | −2.47872 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 1.74331 | 1.00000 | −1.00000 | 1.74331 | 3.90137 | 1.00000 | 0.0391392 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 2.86415 | 1.00000 | −1.00000 | 2.86415 | −2.27749 | 1.00000 | 5.20338 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | 3.09280 | 1.00000 | −1.00000 | 3.09280 | 2.42572 | 1.00000 | 6.56539 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(11\) | \(1\) |
| \(43\) | \(1\) |
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 | 4730.2.a.ba | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4730.2.a.ba | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):
|
\( T_{3}^{10} + T_{3}^{9} - 21T_{3}^{8} - 22T_{3}^{7} + 138T_{3}^{6} + 154T_{3}^{5} - 291T_{3}^{4} - 327T_{3}^{3} + 97T_{3}^{2} + 124T_{3} + 16 \)
|
|
\( T_{7}^{10} - 3 T_{7}^{9} - 47 T_{7}^{8} + 134 T_{7}^{7} + 804 T_{7}^{6} - 2106 T_{7}^{5} - 6149 T_{7}^{4} + \cdots - 24880 \)
|
|
\( T_{13}^{10} - 6 T_{13}^{9} - 46 T_{13}^{8} + 284 T_{13}^{7} + 384 T_{13}^{6} - 2856 T_{13}^{5} + \cdots - 1664 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{10} \)
|
| $3$ |
\( T^{10} + T^{9} + \cdots + 16 \)
|
| $5$ |
\( (T + 1)^{10} \)
|
| $7$ |
\( T^{10} - 3 T^{9} + \cdots - 24880 \)
|
| $11$ |
\( (T + 1)^{10} \)
|
| $13$ |
\( T^{10} - 6 T^{9} + \cdots - 1664 \)
|
| $17$ |
\( T^{10} - 3 T^{9} + \cdots - 83356 \)
|
| $19$ |
\( T^{10} - 3 T^{9} + \cdots - 18496 \)
|
| $23$ |
\( T^{10} + 6 T^{9} + \cdots - 21632 \)
|
| $29$ |
\( T^{10} - 14 T^{9} + \cdots - 5312 \)
|
| $31$ |
\( T^{10} - 6 T^{9} + \cdots - 16384 \)
|
| $37$ |
\( T^{10} - 10 T^{9} + \cdots - 551872 \)
|
| $41$ |
\( T^{10} - 26 T^{9} + \cdots + 1606016 \)
|
| $43$ |
\( (T + 1)^{10} \)
|
| $47$ |
\( T^{10} - 3 T^{9} + \cdots - 1538992 \)
|
| $53$ |
\( T^{10} + 9 T^{9} + \cdots - 3020740 \)
|
| $59$ |
\( T^{10} + 9 T^{9} + \cdots - 4268192 \)
|
| $61$ |
\( T^{10} - 22 T^{9} + \cdots - 64597696 \)
|
| $67$ |
\( T^{10} - 18 T^{9} + \cdots - 28672 \)
|
| $71$ |
\( T^{10} - 27 T^{9} + \cdots - 23992412 \)
|
| $73$ |
\( T^{10} - 28 T^{9} + \cdots + 23067392 \)
|
| $79$ |
\( T^{10} - 3 T^{9} + \cdots + 15473120 \)
|
| $83$ |
\( T^{10} + 11 T^{9} + \cdots - 6193184 \)
|
| $89$ |
\( T^{10} + \cdots + 355233728 \)
|
| $97$ |
\( T^{10} - 22 T^{9} + \cdots + 16294400 \)
|
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