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: |
\( x^{10} - 2x^{9} - 17x^{8} + 21x^{7} + 107x^{6} - 45x^{5} - 262x^{4} - 47x^{3} + 120x^{2} - 2x - 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{10} - 2x^{9} - 17x^{8} + 21x^{7} + 107x^{6} - 45x^{5} - 262x^{4} - 47x^{3} + 120x^{2} - 2x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 17 \nu^{9} + 65 \nu^{8} + 183 \nu^{7} - 745 \nu^{6} - 615 \nu^{5} + 2538 \nu^{4} + 657 \nu^{3} + \cdots + 159 ) / 71 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7 \nu^{9} + 15 \nu^{8} - 209 \nu^{7} - 232 \nu^{6} + 1715 \nu^{5} + 1536 \nu^{4} - 4518 \nu^{3} + \cdots + 348 ) / 71 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 26 \nu^{9} + 66 \nu^{8} + 401 \nu^{7} - 751 \nu^{6} - 2252 \nu^{5} + 2186 \nu^{4} + 4772 \nu^{3} + \cdots + 97 ) / 71 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 24 \nu^{9} - 50 \nu^{8} - 392 \nu^{7} + 513 \nu^{6} + 2330 \nu^{5} - 1002 \nu^{4} - 5104 \nu^{3} + \cdots + 189 ) / 71 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 46 \nu^{9} - 155 \nu^{8} - 562 \nu^{7} + 1711 \nu^{6} + 2466 \nu^{5} - 5151 \nu^{4} - 4363 \nu^{3} + \cdots + 167 ) / 71 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 42 \nu^{9} + 194 \nu^{8} + 331 \nu^{7} - 2158 \nu^{6} - 350 \nu^{5} + 7185 \nu^{4} - 1292 \nu^{3} + \cdots + 326 ) / 71 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 95 \nu^{9} - 334 \nu^{8} - 1102 \nu^{7} + 3637 \nu^{6} + 4602 \nu^{5} - 10942 \nu^{4} - 8299 \nu^{3} + \cdots - 450 ) / 71 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{5} - 2\beta_{4} + 2\beta_{3} + 9\beta_{2} + 13\beta _1 + 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 12 \beta_{6} + 2 \beta_{5} - 15 \beta_{4} + 14 \beta_{3} + \cdots + 53 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2 \beta_{9} + 18 \beta_{8} + 28 \beta_{7} + 23 \beta_{6} + 16 \beta_{5} - 40 \beta_{4} + 34 \beta_{3} + \cdots + 241 \)
|
| \(\nu^{7}\) | \(=\) |
\( 16 \beta_{9} + 58 \beta_{8} + 62 \beta_{7} + 137 \beta_{6} + 47 \beta_{5} - 188 \beta_{4} + 158 \beta_{3} + \cdots + 596 \)
|
| \(\nu^{8}\) | \(=\) |
\( 42 \beta_{9} + 251 \beta_{8} + 331 \beta_{7} + 355 \beta_{6} + 225 \beta_{5} - 569 \beta_{4} + \cdots + 2306 \)
|
| \(\nu^{9}\) | \(=\) |
\( 209 \beta_{9} + 836 \beta_{8} + 896 \beta_{7} + 1578 \beta_{6} + 742 \beta_{5} - 2241 \beta_{4} + \cdots + 6509 \)
|
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 | −2.31796 | 1.00000 | 1.00000 | 2.31796 | 4.25200 | −1.00000 | 2.37293 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.88983 | 1.00000 | 1.00000 | 1.88983 | −3.45622 | −1.00000 | 0.571445 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.27147 | 1.00000 | 1.00000 | 1.27147 | −0.316678 | −1.00000 | −1.38335 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.476897 | 1.00000 | 1.00000 | −0.476897 | 2.32668 | −1.00000 | −2.77257 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.725847 | 1.00000 | 1.00000 | −0.725847 | −1.66243 | −1.00000 | −2.47315 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.21668 | 1.00000 | 1.00000 | −1.21668 | 3.59770 | −1.00000 | −1.51969 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 2.41316 | 1.00000 | 1.00000 | −2.41316 | −5.00890 | −1.00000 | 2.82333 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 2.53249 | 1.00000 | 1.00000 | −2.53249 | −0.583820 | −1.00000 | 3.41349 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 2.68568 | 1.00000 | 1.00000 | −2.68568 | 4.03613 | −1.00000 | 4.21287 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 3.42851 | 1.00000 | 1.00000 | −3.42851 | −0.184466 | −1.00000 | 8.75468 | −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.z | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4730.2.a.z | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):
|
\( T_{3}^{10} - 8 T_{3}^{9} + 10 T_{3}^{8} + 67 T_{3}^{7} - 180 T_{3}^{6} - 86 T_{3}^{5} + 621 T_{3}^{4} + \cdots - 132 \)
|
|
\( T_{7}^{10} - 3 T_{7}^{9} - 42 T_{7}^{8} + 125 T_{7}^{7} + 513 T_{7}^{6} - 1334 T_{7}^{5} - 2203 T_{7}^{4} + \cdots + 141 \)
|
|
\( T_{13}^{10} - 7 T_{13}^{9} - 62 T_{13}^{8} + 342 T_{13}^{7} + 1582 T_{13}^{6} - 4284 T_{13}^{5} + \cdots + 13536 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{10} \)
|
| $3$ |
\( T^{10} - 8 T^{9} + \cdots - 132 \)
|
| $5$ |
\( (T - 1)^{10} \)
|
| $7$ |
\( T^{10} - 3 T^{9} + \cdots + 141 \)
|
| $11$ |
\( (T - 1)^{10} \)
|
| $13$ |
\( T^{10} - 7 T^{9} + \cdots + 13536 \)
|
| $17$ |
\( T^{10} - 2 T^{9} + \cdots + 36864 \)
|
| $19$ |
\( T^{10} + 7 T^{9} + \cdots - 11519 \)
|
| $23$ |
\( T^{10} - 12 T^{9} + \cdots - 196992 \)
|
| $29$ |
\( T^{10} + 12 T^{9} + \cdots + 4629744 \)
|
| $31$ |
\( T^{10} - 16 T^{9} + \cdots + 84224 \)
|
| $37$ |
\( T^{10} - 19 T^{9} + \cdots - 826304 \)
|
| $41$ |
\( T^{10} - 9 T^{9} + \cdots - 6446304 \)
|
| $43$ |
\( (T - 1)^{10} \)
|
| $47$ |
\( T^{10} - 29 T^{9} + \cdots + 319248 \)
|
| $53$ |
\( T^{10} - 6 T^{9} + \cdots + 46965438 \)
|
| $59$ |
\( T^{10} - 29 T^{9} + \cdots + 4797288 \)
|
| $61$ |
\( T^{10} + 4 T^{9} + \cdots - 1084688 \)
|
| $67$ |
\( T^{10} - 45 T^{9} + \cdots + 16473088 \)
|
| $71$ |
\( T^{10} + \cdots - 225757278 \)
|
| $73$ |
\( T^{10} - 3 T^{9} + \cdots - 18506112 \)
|
| $79$ |
\( T^{10} + 14 T^{9} + \cdots + 30304146 \)
|
| $83$ |
\( T^{10} - 23 T^{9} + \cdots - 62496 \)
|
| $89$ |
\( T^{10} - T^{9} + \cdots - 95977728 \)
|
| $97$ |
\( T^{10} + \cdots + 104381952 \)
|
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