Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4719,2,Mod(1,4719)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4719, 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("4719.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4719 = 3 \cdot 11^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4719.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: | \(37.6814047138\) |
| 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: |
\( x^{10} - 3x^{9} - 10x^{8} + 32x^{7} + 27x^{6} - 101x^{5} - 21x^{4} + 116x^{3} - 8x^{2} - 40x + 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 429) |
| 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
\( x^{10} - 3x^{9} - 10x^{8} + 32x^{7} + 27x^{6} - 101x^{5} - 21x^{4} + 116x^{3} - 8x^{2} - 40x + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 38 \nu^{9} + 99 \nu^{8} + 391 \nu^{7} - 957 \nu^{6} - 1197 \nu^{5} + 2444 \nu^{4} + 1630 \nu^{3} + \cdots + 561 ) / 194 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 15 \nu^{9} - 11 \nu^{8} - 259 \nu^{7} + 171 \nu^{6} + 1491 \nu^{5} - 832 \nu^{4} - 3242 \nu^{3} + \cdots - 612 ) / 97 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 22 \nu^{9} - 42 \nu^{8} - 257 \nu^{7} + 406 \nu^{6} + 984 \nu^{5} - 1078 \nu^{4} - 1638 \nu^{3} + \cdots - 335 ) / 97 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 51 \nu^{9} + 115 \nu^{8} + 609 \nu^{7} - 1241 \nu^{6} - 2334 \nu^{5} + 3954 \nu^{4} + 3515 \nu^{3} + \cdots + 1072 ) / 194 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 51 \nu^{9} - 115 \nu^{8} - 609 \nu^{7} + 1241 \nu^{6} + 2334 \nu^{5} - 3954 \nu^{4} - 3515 \nu^{3} + \cdots - 1654 ) / 194 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 34 \nu^{9} - 109 \nu^{8} - 309 \nu^{7} + 1086 \nu^{6} + 683 \nu^{5} - 2927 \nu^{4} - 468 \nu^{3} + \cdots - 456 ) / 97 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 96 \nu^{9} - 245 \nu^{8} - 1095 \nu^{7} + 2627 \nu^{6} + 4091 \nu^{5} - 8390 \nu^{4} - 6936 \nu^{3} + \cdots - 3199 ) / 194 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 125 \nu^{9} - 318 \nu^{8} - 1350 \nu^{7} + 3268 \nu^{6} + 4471 \nu^{5} - 9520 \nu^{4} - 6291 \nu^{3} + \cdots - 2481 ) / 194 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} - \beta_{8} + \beta_{7} + 7\beta_{6} + 8\beta_{5} + \beta_{3} + 2\beta_{2} + 2\beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{9} + 9\beta_{7} + 9\beta_{6} + 4\beta_{5} + 9\beta_{4} + \beta_{3} + 11\beta_{2} + 23\beta _1 + 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11 \beta_{9} - 9 \beta_{8} + 10 \beta_{7} + 49 \beta_{6} + 60 \beta_{5} + 2 \beta_{4} + 9 \beta_{3} + \cdots + 114 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 38 \beta_{9} - 2 \beta_{8} + 66 \beta_{7} + 74 \beta_{6} + 56 \beta_{5} + 69 \beta_{4} + 10 \beta_{3} + \cdots + 125 \)
|
| \(\nu^{8}\) | \(=\) |
\( 96 \beta_{9} - 69 \beta_{8} + 81 \beta_{7} + 353 \beta_{6} + 451 \beta_{5} + 38 \beta_{4} + 66 \beta_{3} + \cdots + 810 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 176 \beta_{9} - 38 \beta_{8} + 462 \beta_{7} + 603 \beta_{6} + 575 \beta_{5} + 518 \beta_{4} + \cdots + 1106 \)
|
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 |
|
−2.79859 | −1.00000 | 5.83212 | −3.31897 | 2.79859 | 1.45300 | −10.7245 | 1.00000 | 9.28845 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.30106 | −1.00000 | 3.29486 | −3.07786 | 2.30106 | −0.0582840 | −2.97955 | 1.00000 | 7.08233 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.83162 | −1.00000 | 1.35483 | 2.34647 | 1.83162 | −1.24891 | 1.18170 | 1.00000 | −4.29785 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.20201 | −1.00000 | −0.555167 | 2.09478 | 1.20201 | 2.07248 | 3.07134 | 1.00000 | −2.51795 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.548449 | −1.00000 | −1.69920 | −0.861019 | 0.548449 | 2.88505 | 2.02882 | 1.00000 | 0.472225 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.368322 | −1.00000 | −1.86434 | 1.06229 | 0.368322 | −3.52340 | 1.42332 | 1.00000 | −0.391264 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.848202 | −1.00000 | −1.28055 | −3.50785 | −0.848202 | −3.48902 | −2.78257 | 1.00000 | −2.97536 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.21489 | −1.00000 | −0.524042 | 1.10026 | −1.21489 | 0.293129 | −3.06643 | 1.00000 | 1.33669 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.49671 | −1.00000 | 0.240141 | −2.57720 | −1.49671 | 2.52650 | −2.63400 | 1.00000 | −3.85732 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.49025 | −1.00000 | 4.20135 | −1.26090 | −2.49025 | −1.91055 | 5.48191 | 1.00000 | −3.13995 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 4719.2.a.bi | 10 | |
| 11.b | odd | 2 | 1 | 4719.2.a.bn | 10 | ||
| 11.d | odd | 10 | 2 | 429.2.n.b | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 429.2.n.b | ✓ | 20 | 11.d | odd | 10 | 2 | |
| 4719.2.a.bi | 10 | 1.a | even | 1 | 1 | trivial | |
| 4719.2.a.bn | 10 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4719))\):
|
\( T_{2}^{10} + 3T_{2}^{9} - 10T_{2}^{8} - 32T_{2}^{7} + 27T_{2}^{6} + 101T_{2}^{5} - 21T_{2}^{4} - 116T_{2}^{3} - 8T_{2}^{2} + 40T_{2} + 11 \)
|
|
\( T_{5}^{10} + 8 T_{5}^{9} + 5 T_{5}^{8} - 96 T_{5}^{7} - 167 T_{5}^{6} + 377 T_{5}^{5} + 807 T_{5}^{4} + \cdots + 576 \)
|
|
\( T_{7}^{10} + T_{7}^{9} - 25T_{7}^{8} - 11T_{7}^{7} + 214T_{7}^{6} - 2T_{7}^{5} - 678T_{7}^{4} + 143T_{7}^{3} + 658T_{7}^{2} - 151T_{7} - 11 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 3 T^{9} + \cdots + 11 \)
|
| $3$ |
\( (T + 1)^{10} \)
|
| $5$ |
\( T^{10} + 8 T^{9} + \cdots + 576 \)
|
| $7$ |
\( T^{10} + T^{9} + \cdots - 11 \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( (T - 1)^{10} \)
|
| $17$ |
\( T^{10} + T^{9} + \cdots + 3091 \)
|
| $19$ |
\( T^{10} - 6 T^{9} + \cdots + 5689 \)
|
| $23$ |
\( T^{10} - 3 T^{9} + \cdots + 3824 \)
|
| $29$ |
\( T^{10} - 12 T^{9} + \cdots + 89489 \)
|
| $31$ |
\( T^{10} + 20 T^{9} + \cdots - 6421 \)
|
| $37$ |
\( T^{10} + 3 T^{9} + \cdots - 1042096 \)
|
| $41$ |
\( T^{10} + 12 T^{9} + \cdots + 3065744 \)
|
| $43$ |
\( T^{10} + 14 T^{9} + \cdots + 450224 \)
|
| $47$ |
\( T^{10} + 6 T^{9} + \cdots + 16846441 \)
|
| $53$ |
\( T^{10} + 22 T^{9} + \cdots - 1467169 \)
|
| $59$ |
\( T^{10} + 28 T^{9} + \cdots - 3859659 \)
|
| $61$ |
\( T^{10} - 12 T^{9} + \cdots + 7185519 \)
|
| $67$ |
\( T^{10} + \cdots + 207765424 \)
|
| $71$ |
\( T^{10} + 9 T^{9} + \cdots - 41738661 \)
|
| $73$ |
\( T^{10} - 5 T^{9} + \cdots + 49301296 \)
|
| $79$ |
\( T^{10} - 10 T^{9} + \cdots + 1742400 \)
|
| $83$ |
\( T^{10} - 12 T^{9} + \cdots + 4909619 \)
|
| $89$ |
\( T^{10} + \cdots + 994744784 \)
|
| $97$ |
\( T^{10} + \cdots - 198941776 \)
|
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