Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4761,2,Mod(1,4761)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4761, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4761.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4761 = 3^{2} \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4761.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: | \(38.0167764023\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | 10.10.5791333887977.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} - 12x^{8} + 22x^{7} + 49x^{6} - 84x^{5} - 73x^{4} + 132x^{3} + 17x^{2} - 74x + 23 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 69) |
| 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
\( x^{10} - 2x^{9} - 12x^{8} + 22x^{7} + 49x^{6} - 84x^{5} - 73x^{4} + 132x^{3} + 17x^{2} - 74x + 23 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 11\nu^{2} + 11\nu - 11 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 7\nu^{4} + 11\nu^{3} + 13\nu^{2} - 11\nu - 4 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} - 2\nu^{7} - 10\nu^{6} + 17\nu^{5} + 32\nu^{4} - 42\nu^{3} - 31\nu^{2} + 27\nu - 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{9} - \nu^{8} - 13\nu^{7} + 9\nu^{6} + 58\nu^{5} - 25\nu^{4} - 100\nu^{3} + 27\nu^{2} + 53\nu - 18 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{9} + 2\nu^{8} + 12\nu^{7} - 21\nu^{6} - 51\nu^{5} + 74\nu^{4} + 90\nu^{3} - 102\nu^{2} - 57\nu + 49 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{9} - \nu^{8} - 14\nu^{7} + 11\nu^{6} + 68\nu^{5} - 42\nu^{4} - 132\nu^{3} + 69\nu^{2} + 84\nu - 45 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3\nu^{9} - 4\nu^{8} - 38\nu^{7} + 39\nu^{6} + 168\nu^{5} - 128\nu^{4} - 293\nu^{3} + 175\nu^{2} + 160\nu - 96 ) / 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -3\nu^{9} + 4\nu^{8} + 39\nu^{7} - 41\nu^{6} - 176\nu^{5} + 141\nu^{4} + 313\nu^{3} - 197\nu^{2} - 177\nu + 107 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} - \beta_{6} + \beta_{4} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - 2\beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} + \beta_{8} - 4\beta_{7} - 7\beta_{6} + 6\beta_{4} - 3\beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( 11\beta_{9} + 9\beta_{8} + 3\beta_{7} - 10\beta_{6} - 7\beta_{5} + 8\beta_{4} - 18\beta_{2} + 19\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 25 \beta_{9} + 14 \beta_{8} - 9 \beta_{7} - 45 \beta_{6} - 3 \beta_{5} + 34 \beta_{4} + 2 \beta_{3} + \cdots + 53 \)
|
| \(\nu^{7}\) | \(=\) |
\( 94 \beta_{9} + 69 \beta_{8} + 36 \beta_{7} - 81 \beta_{6} - 42 \beta_{5} + 56 \beta_{4} + 4 \beta_{3} + \cdots + 21 \)
|
| \(\nu^{8}\) | \(=\) |
\( 229 \beta_{9} + 135 \beta_{8} + 28 \beta_{7} - 291 \beta_{6} - 37 \beta_{5} + 199 \beta_{4} + 28 \beta_{3} + \cdots + 249 \)
|
| \(\nu^{9}\) | \(=\) |
\( 738 \beta_{9} + 509 \beta_{8} + 330 \beta_{7} - 607 \beta_{6} - 248 \beta_{5} + 380 \beta_{4} + \cdots + 166 \)
|
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 |
|
−2.10158 | 0 | 2.41664 | 3.44976 | 0 | −1.58175 | −0.875613 | 0 | −7.24995 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.98594 | 0 | 1.94394 | 2.04789 | 0 | −3.23167 | 0.111323 | 0 | −4.06699 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.41812 | 0 | 0.0110547 | −0.849430 | 0 | −5.06334 | 2.82056 | 0 | 1.20459 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.15603 | 0 | −0.663591 | 1.73011 | 0 | 0.701139 | 3.07920 | 0 | −2.00006 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.568983 | 0 | −1.67626 | 2.70241 | 0 | −3.38601 | −2.09173 | 0 | 1.53762 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.703704 | 0 | −1.50480 | −1.68624 | 0 | −3.24908 | −2.46634 | 0 | −1.18661 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.871916 | 0 | −1.23976 | 1.25508 | 0 | 2.98232 | −2.82480 | 0 | 1.09432 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.61315 | 0 | 0.602256 | −0.526786 | 0 | −0.483695 | −2.25477 | 0 | −0.849785 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.24419 | 0 | 3.03638 | 2.78323 | 0 | −3.39142 | 2.32582 | 0 | 6.24609 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.65972 | 0 | 5.07414 | −2.90602 | 0 | −2.29650 | 8.17636 | 0 | −7.72922 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 4761.2.a.bu | 10 | |
| 3.b | odd | 2 | 1 | 1587.2.a.t | 10 | ||
| 23.b | odd | 2 | 1 | 4761.2.a.bt | 10 | ||
| 23.d | odd | 22 | 2 | 207.2.i.d | 20 | ||
| 69.c | even | 2 | 1 | 1587.2.a.u | 10 | ||
| 69.g | even | 22 | 2 | 69.2.e.c | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.2.e.c | ✓ | 20 | 69.g | even | 22 | 2 | |
| 207.2.i.d | 20 | 23.d | odd | 22 | 2 | ||
| 1587.2.a.t | 10 | 3.b | odd | 2 | 1 | ||
| 1587.2.a.u | 10 | 69.c | even | 2 | 1 | ||
| 4761.2.a.bt | 10 | 23.b | odd | 2 | 1 | ||
| 4761.2.a.bu | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4761))\):
|
\( T_{2}^{10} - 2T_{2}^{9} - 12T_{2}^{8} + 22T_{2}^{7} + 49T_{2}^{6} - 84T_{2}^{5} - 73T_{2}^{4} + 132T_{2}^{3} + 17T_{2}^{2} - 74T_{2} + 23 \)
|
|
\( T_{5}^{10} - 8 T_{5}^{9} + 8 T_{5}^{8} + 84 T_{5}^{7} - 219 T_{5}^{6} - 109 T_{5}^{5} + 759 T_{5}^{4} + \cdots + 253 \)
|
|
\( T_{7}^{10} + 19 T_{7}^{9} + 137 T_{7}^{8} + 405 T_{7}^{7} - 111 T_{7}^{6} - 4022 T_{7}^{5} - 10621 T_{7}^{4} + \cdots + 2243 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{9} + \cdots + 23 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} - 8 T^{9} + \cdots + 253 \)
|
| $7$ |
\( T^{10} + 19 T^{9} + \cdots + 2243 \)
|
| $11$ |
\( T^{10} - 3 T^{9} + \cdots - 461 \)
|
| $13$ |
\( T^{10} + 4 T^{9} + \cdots - 5279 \)
|
| $17$ |
\( T^{10} - 11 T^{9} + \cdots + 4663 \)
|
| $19$ |
\( T^{10} + 22 T^{9} + \cdots + 645919 \)
|
| $23$ |
\( T^{10} \)
|
| $29$ |
\( T^{10} - 5 T^{9} + \cdots + 1995907 \)
|
| $31$ |
\( T^{10} + 7 T^{9} + \cdots - 28488637 \)
|
| $37$ |
\( T^{10} + 35 T^{9} + \cdots - 430387 \)
|
| $41$ |
\( T^{10} - 185 T^{8} + \cdots + 815077 \)
|
| $43$ |
\( T^{10} + 28 T^{9} + \cdots - 11465707 \)
|
| $47$ |
\( T^{10} + 9 T^{9} + \cdots + 87731039 \)
|
| $53$ |
\( T^{10} - 34 T^{9} + \cdots + 910229 \)
|
| $59$ |
\( T^{10} - 2 T^{9} + \cdots - 240109 \)
|
| $61$ |
\( T^{10} + 49 T^{9} + \cdots - 23662651 \)
|
| $67$ |
\( T^{10} + 26 T^{9} + \cdots + 1627 \)
|
| $71$ |
\( T^{10} + 15 T^{9} + \cdots + 42681629 \)
|
| $73$ |
\( T^{10} - 14 T^{9} + \cdots + 3389 \)
|
| $79$ |
\( T^{10} + \cdots - 199685509 \)
|
| $83$ |
\( T^{10} + \cdots - 893491919 \)
|
| $89$ |
\( T^{10} + 15 T^{9} + \cdots + 24223 \)
|
| $97$ |
\( T^{10} + 22 T^{9} + \cdots + 6977189 \)
|
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