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: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - x^{10} - 22 x^{9} + 21 x^{8} + 165 x^{7} - 130 x^{6} - 535 x^{5} + 323 x^{4} + 710 x^{3} + \cdots + 128 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - x^{10} - 22 x^{9} + 21 x^{8} + 165 x^{7} - 130 x^{6} - 535 x^{5} + 323 x^{4} + 710 x^{3} + \cdots + 128 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 179 \nu^{10} - 98 \nu^{9} - 3948 \nu^{8} + 1857 \nu^{7} + 29718 \nu^{6} - 7762 \nu^{5} - 95629 \nu^{4} + \cdots - 37066 ) / 106 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 147 \nu^{10} - 79 \nu^{9} - 3228 \nu^{8} + 1504 \nu^{7} + 24177 \nu^{6} - 6282 \nu^{5} - 77497 \nu^{4} + \cdots - 30135 ) / 53 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 337 \nu^{10} - 162 \nu^{9} - 7450 \nu^{8} + 3113 \nu^{7} + 56338 \nu^{6} - 12764 \nu^{5} + \cdots - 72106 ) / 106 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 177 \nu^{10} - 67 \nu^{9} - 3956 \nu^{8} + 1295 \nu^{7} + 30382 \nu^{6} - 4834 \nu^{5} - 99703 \nu^{4} + \cdots - 40313 ) / 53 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 421 \nu^{10} + 192 \nu^{9} + 9340 \nu^{8} - 3715 \nu^{7} - 70956 \nu^{6} + 15218 \nu^{5} + \cdots + 90598 ) / 106 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 325 \nu^{10} + 135 \nu^{9} + 7233 \nu^{8} - 2603 \nu^{7} - 55234 \nu^{6} + 10089 \nu^{5} + \cdots + 72296 ) / 53 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 325 \nu^{10} - 135 \nu^{9} - 7233 \nu^{8} + 2603 \nu^{7} + 55234 \nu^{6} - 10089 \nu^{5} - 180357 \nu^{4} + \cdots - 72296 ) / 53 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 474 \nu^{10} + 192 \nu^{9} + 10559 \nu^{8} - 3715 \nu^{7} - 80708 \nu^{6} + 14370 \nu^{5} + \cdots + 105438 ) / 53 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{8} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} - 2\beta_{8} - \beta_{7} - \beta_{6} - 3\beta_{5} + \beta_{4} + 10\beta_{2} - 3\beta _1 + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{10} + 13 \beta_{9} + 13 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \cdots - 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2 \beta_{10} - 16 \beta_{9} - 32 \beta_{8} - 15 \beta_{7} - 13 \beta_{6} - 42 \beta_{5} + 14 \beta_{4} + \cdots + 211 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 14 \beta_{10} + 145 \beta_{9} + 149 \beta_{8} + 31 \beta_{7} - 26 \beta_{6} + 58 \beta_{5} + \cdots - 113 \)
|
| \(\nu^{8}\) | \(=\) |
\( 35 \beta_{10} - 216 \beta_{9} - 405 \beta_{8} - 176 \beta_{7} - 135 \beta_{6} - 475 \beta_{5} + \cdots + 1929 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 164 \beta_{10} + 1554 \beta_{9} + 1658 \beta_{8} + 392 \beta_{7} - 253 \beta_{6} + 820 \beta_{5} + \cdots - 1947 \)
|
| \(\nu^{10}\) | \(=\) |
\( 452 \beta_{10} - 2756 \beta_{9} - 4787 \beta_{8} - 1946 \beta_{7} - 1309 \beta_{6} - 5129 \beta_{5} + \cdots + 18835 \)
|
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 | −2.87555 | 1.00000 | −1.00000 | 2.87555 | −0.638121 | −1.00000 | 5.26880 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.78036 | 1.00000 | −1.00000 | 2.78036 | −2.23633 | −1.00000 | 4.73038 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −2.17792 | 1.00000 | −1.00000 | 2.17792 | 4.63803 | −1.00000 | 1.74332 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.48284 | 1.00000 | −1.00000 | 1.48284 | 4.82678 | −1.00000 | −0.801199 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −0.580450 | 1.00000 | −1.00000 | 0.580450 | 1.64698 | −1.00000 | −2.66308 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −0.577714 | 1.00000 | −1.00000 | 0.577714 | −2.69986 | −1.00000 | −2.66625 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 0.797340 | 1.00000 | −1.00000 | −0.797340 | −1.54185 | −1.00000 | −2.36425 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 1.56687 | 1.00000 | −1.00000 | −1.56687 | −4.55535 | −1.00000 | −0.544925 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 1.80931 | 1.00000 | −1.00000 | −1.80931 | 4.53794 | −1.00000 | 0.273608 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 1.95364 | 1.00000 | −1.00000 | −1.95364 | 0.00701571 | −1.00000 | 0.816706 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1.00000 | 3.34767 | 1.00000 | −1.00000 | −3.34767 | 2.01478 | −1.00000 | 8.20687 | 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.bb | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4730.2.a.bb | ✓ | 11 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):
|
\( T_{3}^{11} + T_{3}^{10} - 22 T_{3}^{9} - 21 T_{3}^{8} + 165 T_{3}^{7} + 130 T_{3}^{6} - 535 T_{3}^{5} + \cdots - 128 \)
|
|
\( T_{7}^{11} - 6 T_{7}^{10} - 36 T_{7}^{9} + 225 T_{7}^{8} + 426 T_{7}^{7} - 2562 T_{7}^{6} - 2633 T_{7}^{5} + \cdots + 64 \)
|
|
\( T_{13}^{11} - 4 T_{13}^{10} - 74 T_{13}^{9} + 306 T_{13}^{8} + 1680 T_{13}^{7} - 7792 T_{13}^{6} + \cdots - 512 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{11} \)
|
| $3$ |
\( T^{11} + T^{10} + \cdots - 128 \)
|
| $5$ |
\( (T + 1)^{11} \)
|
| $7$ |
\( T^{11} - 6 T^{10} + \cdots + 64 \)
|
| $11$ |
\( (T + 1)^{11} \)
|
| $13$ |
\( T^{11} - 4 T^{10} + \cdots - 512 \)
|
| $17$ |
\( T^{11} + 10 T^{10} + \cdots - 2457600 \)
|
| $19$ |
\( T^{11} - 14 T^{10} + \cdots - 2538200 \)
|
| $23$ |
\( T^{11} + 18 T^{10} + \cdots + 10752 \)
|
| $29$ |
\( T^{11} + 6 T^{10} + \cdots - 2386272 \)
|
| $31$ |
\( T^{11} - 13 T^{10} + \cdots - 16384 \)
|
| $37$ |
\( T^{11} - T^{10} + \cdots + 56391424 \)
|
| $41$ |
\( T^{11} - 12 T^{10} + \cdots - 8467200 \)
|
| $43$ |
\( (T - 1)^{11} \)
|
| $47$ |
\( T^{11} + 5 T^{10} + \cdots + 78935808 \)
|
| $53$ |
\( T^{11} + 27 T^{10} + \cdots + 18234 \)
|
| $59$ |
\( T^{11} - 11 T^{10} + \cdots - 123936 \)
|
| $61$ |
\( T^{11} + \cdots - 11846695072 \)
|
| $67$ |
\( T^{11} + \cdots + 334569472 \)
|
| $71$ |
\( T^{11} + 14 T^{10} + \cdots + 10307904 \)
|
| $73$ |
\( T^{11} + 11 T^{10} + \cdots + 25825024 \)
|
| $79$ |
\( T^{11} - 28 T^{10} + \cdots + 2213384 \)
|
| $83$ |
\( T^{11} + \cdots - 113350872 \)
|
| $89$ |
\( T^{11} + \cdots - 143697408 \)
|
| $97$ |
\( T^{11} + \cdots - 2722445824 \)
|
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