Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1573,2,Mod(1,1573)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1573, 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("1573.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1573 = 11^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1573.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: | \(12.5604682379\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - x^{13} - 21 x^{12} + 19 x^{11} + 169 x^{10} - 136 x^{9} - 649 x^{8} + 455 x^{7} + 1207 x^{6} + \cdots - 29 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 143) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - x^{13} - 21 x^{12} + 19 x^{11} + 169 x^{10} - 136 x^{9} - 649 x^{8} + 455 x^{7} + 1207 x^{6} + \cdots - 29 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 107 \nu^{13} - 212 \nu^{12} + 299 \nu^{11} + 1600 \nu^{10} + 13445 \nu^{9} + 8577 \nu^{8} + \cdots + 19227 ) / 23468 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 367 \nu^{13} + 5662 \nu^{12} - 8373 \nu^{11} - 117564 \nu^{10} + 67935 \nu^{9} + 921913 \nu^{8} + \cdots + 351323 ) / 70404 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 363 \nu^{13} - 2419 \nu^{12} + 7978 \nu^{11} + 53680 \nu^{10} - 66683 \nu^{9} - 445910 \nu^{8} + \cdots - 103270 ) / 35202 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 718 \nu^{13} - 7515 \nu^{12} - 9299 \nu^{11} + 149482 \nu^{10} + 13522 \nu^{9} - 1110379 \nu^{8} + \cdots - 119423 ) / 35202 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1463 \nu^{13} - 2460 \nu^{12} + 34465 \nu^{11} + 44248 \nu^{10} - 317111 \nu^{9} - 301423 \nu^{8} + \cdots - 155039 ) / 70404 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 663 \nu^{13} - 770 \nu^{12} - 14135 \nu^{11} + 12896 \nu^{10} + 113647 \nu^{9} - 76723 \nu^{8} + \cdots - 20219 ) / 23468 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 1705 \nu^{13} - 3750 \nu^{12} - 31961 \nu^{11} + 68596 \nu^{10} + 216847 \nu^{9} - 461275 \nu^{8} + \cdots - 18617 ) / 35202 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 4673 \nu^{13} + 14648 \nu^{12} + 90371 \nu^{11} - 283240 \nu^{10} - 641269 \nu^{9} + \cdots + 374835 ) / 70404 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 2498 \nu^{13} + 1795 \nu^{12} + 50133 \nu^{11} - 35244 \nu^{10} - 371184 \nu^{9} + 260497 \nu^{8} + \cdots + 88241 ) / 35202 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 5111 \nu^{13} - 2759 \nu^{12} - 107222 \nu^{11} + 46726 \nu^{10} + 848767 \nu^{9} - 287472 \nu^{8} + \cdots + 24648 ) / 35202 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 13953 \nu^{13} + 12382 \nu^{12} + 279263 \nu^{11} - 225844 \nu^{10} - 2090893 \nu^{9} + \cdots + 45619 ) / 70404 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} + \beta_{12} + \beta_{9} - \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{6} + 7\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{13} + 8\beta_{12} - \beta_{10} + 7\beta_{9} - \beta_{8} - \beta_{5} - 8\beta_{3} + 28\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9 \beta_{12} + 9 \beta_{11} + 10 \beta_{10} - \beta_{9} - 6 \beta_{8} + 10 \beta_{6} - \beta_{5} + \cdots + 81 \)
|
| \(\nu^{7}\) | \(=\) |
\( 55 \beta_{13} + 54 \beta_{12} - 2 \beta_{11} - 9 \beta_{10} + 46 \beta_{9} - 10 \beta_{8} - \beta_{7} + \cdots + 11 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{13} + 64 \beta_{12} + 64 \beta_{11} + 78 \beta_{10} - 11 \beta_{9} - 24 \beta_{8} - 4 \beta_{7} + \cdots + 451 \)
|
| \(\nu^{9}\) | \(=\) |
\( 361 \beta_{13} + 348 \beta_{12} - 26 \beta_{11} - 60 \beta_{10} + 302 \beta_{9} - 78 \beta_{8} + \cdots + 86 \)
|
| \(\nu^{10}\) | \(=\) |
\( 18 \beta_{13} + 425 \beta_{12} + 421 \beta_{11} + 556 \beta_{10} - 91 \beta_{9} - 49 \beta_{8} + \cdots + 2556 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2324 \beta_{13} + 2206 \beta_{12} - 237 \beta_{11} - 354 \beta_{10} + 1987 \beta_{9} - 563 \beta_{8} + \cdots + 588 \)
|
| \(\nu^{12}\) | \(=\) |
\( 206 \beta_{13} + 2752 \beta_{12} + 2672 \beta_{11} + 3800 \beta_{10} - 684 \beta_{9} + 309 \beta_{8} + \cdots + 14658 \)
|
| \(\nu^{13}\) | \(=\) |
\( 14823 \beta_{13} + 13894 \beta_{12} - 1884 \beta_{11} - 1940 \beta_{10} + 13080 \beta_{9} - 3925 \beta_{8} + \cdots + 3760 \)
|
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 |
|
−2.51886 | −1.27511 | 4.34468 | 3.65001 | 3.21184 | 1.49222 | −5.90593 | −1.37409 | −9.19388 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.36242 | 1.82138 | 3.58104 | 2.13488 | −4.30288 | −1.07801 | −3.73509 | 0.317434 | −5.04348 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.99402 | −2.52866 | 1.97611 | −1.82638 | 5.04220 | −2.02228 | 0.0476318 | 3.39413 | 3.64184 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.76653 | 3.37827 | 1.12063 | 2.18813 | −5.96782 | 0.121595 | 1.55344 | 8.41271 | −3.86540 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.805250 | −2.37415 | −1.35157 | 0.559059 | 1.91178 | 1.71148 | 2.69885 | 2.63659 | −0.450182 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.685916 | 1.33751 | −1.52952 | −1.56177 | −0.917418 | −3.81995 | 2.42095 | −1.21107 | 1.07124 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.252057 | 3.06529 | −1.93647 | 1.84447 | −0.772630 | 3.21098 | 0.992216 | 6.39602 | −0.464913 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.415089 | −0.740920 | −1.82770 | −0.797591 | −0.307548 | −4.87498 | −1.58884 | −2.45104 | −0.331071 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.919539 | −1.06827 | −1.15445 | 3.26987 | −0.982317 | 2.12483 | −2.90064 | −1.85880 | 3.00678 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.13331 | 0.0976299 | −0.715602 | −3.57018 | 0.110645 | 0.149191 | −3.07763 | −2.99047 | −4.04613 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.55864 | 1.89094 | 0.429345 | 4.05417 | 2.94728 | −0.317735 | −2.44808 | 0.575638 | 6.31897 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.36191 | 2.75277 | 3.57862 | 2.24069 | 6.50181 | −4.49077 | 3.72857 | 4.57776 | 5.29231 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.46658 | 2.56295 | 4.08401 | −1.93746 | 6.32172 | 3.51730 | 5.14036 | 3.56873 | −4.77889 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.52999 | 0.0803684 | 4.40087 | −1.24791 | 0.203332 | 3.27614 | 6.07419 | −2.99354 | −3.15719 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
| \(13\) | \(-1\) |
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 | 1573.2.a.s | 14 | |
| 11.b | odd | 2 | 1 | 1573.2.a.r | 14 | ||
| 11.c | even | 5 | 2 | 143.2.h.c | ✓ | 28 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.2.h.c | ✓ | 28 | 11.c | even | 5 | 2 | |
| 1573.2.a.r | 14 | 11.b | odd | 2 | 1 | ||
| 1573.2.a.s | 14 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1573))\):
|
\( T_{2}^{14} - T_{2}^{13} - 21 T_{2}^{12} + 19 T_{2}^{11} + 169 T_{2}^{10} - 136 T_{2}^{9} - 649 T_{2}^{8} + \cdots - 29 \)
|
|
\( T_{7}^{14} + T_{7}^{13} - 53 T_{7}^{12} - 6 T_{7}^{11} + 1049 T_{7}^{10} - 668 T_{7}^{9} - 8905 T_{7}^{8} + \cdots + 211 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} - T^{13} + \cdots - 29 \)
|
| $3$ |
\( T^{14} - 9 T^{13} + \cdots - 16 \)
|
| $5$ |
\( T^{14} - 9 T^{13} + \cdots + 10256 \)
|
| $7$ |
\( T^{14} + T^{13} + \cdots + 211 \)
|
| $11$ |
\( T^{14} \)
|
| $13$ |
\( (T - 1)^{14} \)
|
| $17$ |
\( T^{14} + 4 T^{13} + \cdots - 2282479 \)
|
| $19$ |
\( T^{14} - 5 T^{13} + \cdots - 387155 \)
|
| $23$ |
\( T^{14} + \cdots - 143696624 \)
|
| $29$ |
\( T^{14} + 16 T^{13} + \cdots - 9899495 \)
|
| $31$ |
\( T^{14} - 6 T^{13} + \cdots - 1758131 \)
|
| $37$ |
\( T^{14} - 11 T^{13} + \cdots + 6797520 \)
|
| $41$ |
\( T^{14} + \cdots + 2490121936 \)
|
| $43$ |
\( T^{14} + \cdots + 4437149776 \)
|
| $47$ |
\( T^{14} + \cdots + 10044861005 \)
|
| $53$ |
\( T^{14} + \cdots - 5543665339 \)
|
| $59$ |
\( T^{14} + \cdots + 207027012355 \)
|
| $61$ |
\( T^{14} + 19 T^{13} + \cdots - 4468169 \)
|
| $67$ |
\( T^{14} + \cdots + 305382356336 \)
|
| $71$ |
\( T^{14} + \cdots - 8550488421 \)
|
| $73$ |
\( T^{14} + \cdots + 34203499056 \)
|
| $79$ |
\( T^{14} + \cdots - 83878439920 \)
|
| $83$ |
\( T^{14} + \cdots + 2670210125 \)
|
| $89$ |
\( T^{14} + \cdots - 285085520 \)
|
| $97$ |
\( T^{14} + \cdots + 157541732176 \)
|
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