Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1143,4,Mod(1,1143)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1143, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1143.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1143 = 3^{2} \cdot 127 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1143.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: | \(67.4391831366\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{11} - 5 x^{10} - 41 x^{9} + 207 x^{8} + 563 x^{7} - 2917 x^{6} - 2895 x^{5} + 17037 x^{4} + \cdots + 9144 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 381) |
| 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
\( x^{11} - 5 x^{10} - 41 x^{9} + 207 x^{8} + 563 x^{7} - 2917 x^{6} - 2895 x^{5} + 17037 x^{4} + \cdots + 9144 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2873 \nu^{10} - 6640 \nu^{9} - 119929 \nu^{8} + 161058 \nu^{7} + 1642357 \nu^{6} + 106444 \nu^{5} + \cdots + 47399400 ) / 1999584 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1669 \nu^{10} + 5056 \nu^{9} + 85581 \nu^{8} - 212210 \nu^{7} - 1619649 \nu^{6} + \cdots + 17015352 ) / 666528 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 9385 \nu^{10} + 51560 \nu^{9} + 350177 \nu^{8} - 1969602 \nu^{7} - 4335701 \nu^{6} + \cdots + 46435896 ) / 1999584 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3215 \nu^{10} - 14748 \nu^{9} - 128763 \nu^{8} + 560986 \nu^{7} + 1761187 \nu^{6} - 6822000 \nu^{5} + \cdots - 14535864 ) / 333264 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 19919 \nu^{10} + 67264 \nu^{9} + 935887 \nu^{8} - 2669574 \nu^{7} - 15949315 \nu^{6} + \cdots + 134873064 ) / 1999584 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 6869 \nu^{10} - 25368 \nu^{9} - 307925 \nu^{8} + 977978 \nu^{7} + 4913441 \nu^{6} - 12112716 \nu^{5} + \cdots - 28435128 ) / 666528 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 10681 \nu^{10} - 36242 \nu^{9} - 491087 \nu^{8} + 1391256 \nu^{7} + 8105489 \nu^{6} + \cdots - 60762744 ) / 499896 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 43193 \nu^{10} + 141064 \nu^{9} + 2041345 \nu^{8} - 5570610 \nu^{7} - 34682917 \nu^{6} + \cdots + 251688600 ) / 1999584 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} - 2\beta_{4} + 15\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{10} - 2\beta_{9} - 2\beta_{8} - 2\beta_{5} - 2\beta_{4} + 4\beta_{3} + 23\beta_{2} + 29\beta _1 + 133 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 6 \beta_{10} - 12 \beta_{8} + 31 \beta_{7} + 27 \beta_{6} + 21 \beta_{5} - 54 \beta_{4} + 4 \beta_{3} + \cdots + 144 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 70 \beta_{10} - 58 \beta_{9} - 96 \beta_{8} + 6 \beta_{7} + 8 \beta_{6} - 68 \beta_{5} - 76 \beta_{4} + \cdots + 2365 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 272 \beta_{10} - 48 \beta_{9} - 516 \beta_{8} + 743 \beta_{7} + 639 \beta_{6} + 371 \beta_{5} + \cdots + 3664 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 1970 \beta_{10} - 1438 \beta_{9} - 3134 \beta_{8} + 320 \beta_{7} + 524 \beta_{6} - 1678 \beta_{5} + \cdots + 45681 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 8554 \beta_{10} - 2320 \beta_{9} - 16152 \beta_{8} + 16347 \beta_{7} + 14815 \beta_{6} + 6405 \beta_{5} + \cdots + 85764 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 51442 \beta_{10} - 34466 \beta_{9} - 88828 \beta_{8} + 11042 \beta_{7} + 20852 \beta_{6} + \cdots + 920425 \)
|
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 |
|
−4.38429 | 0 | 11.2220 | 16.2133 | 0 | −19.8822 | −14.1262 | 0 | −71.0837 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.08578 | 0 | 1.52201 | −2.24078 | 0 | −33.5587 | 19.9896 | 0 | 6.91456 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.67649 | 0 | −0.836396 | 11.1346 | 0 | 13.5368 | 23.6505 | 0 | −29.8015 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.42240 | 0 | −2.13199 | −6.37836 | 0 | −4.94005 | 24.5437 | 0 | 15.4509 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.313622 | 0 | −7.90164 | 2.10825 | 0 | 3.80966 | 4.98711 | 0 | −0.661195 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.35748 | 0 | −6.15724 | −13.5266 | 0 | −20.3499 | −19.2182 | 0 | −18.3622 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.62850 | 0 | −5.34800 | 3.24763 | 0 | −3.50659 | −21.7372 | 0 | 5.28876 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.85483 | 0 | −4.55960 | 11.5502 | 0 | 23.4651 | −23.2959 | 0 | 21.4236 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 3.81796 | 0 | 6.57679 | −14.7239 | 0 | 15.1434 | −5.43374 | 0 | −56.2154 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 4.41859 | 0 | 11.5239 | 3.94070 | 0 | −21.9900 | 15.5709 | 0 | 17.4123 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 4.80522 | 0 | 15.0901 | −1.32485 | 0 | −2.72744 | 34.0695 | 0 | −6.36619 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(127\) | \(1\) |
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 | 1143.4.a.c | 11 | |
| 3.b | odd | 2 | 1 | 381.4.a.a | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 381.4.a.a | ✓ | 11 | 3.b | odd | 2 | 1 | |
| 1143.4.a.c | 11 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} - 5 T_{2}^{10} - 41 T_{2}^{9} + 207 T_{2}^{8} + 563 T_{2}^{7} - 2917 T_{2}^{6} - 2895 T_{2}^{5} + \cdots + 9144 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1143))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 5 T^{10} + \cdots + 9144 \)
|
| $3$ |
\( T^{11} \)
|
| $5$ |
\( T^{11} + \cdots + 212169920 \)
|
| $7$ |
\( T^{11} + \cdots + 258509974320 \)
|
| $11$ |
\( T^{11} + \cdots + 97966892608 \)
|
| $13$ |
\( T^{11} + \cdots - 18\!\cdots\!36 \)
|
| $17$ |
\( T^{11} + \cdots + 31\!\cdots\!96 \)
|
| $19$ |
\( T^{11} + \cdots - 28\!\cdots\!92 \)
|
| $23$ |
\( T^{11} + \cdots - 39\!\cdots\!08 \)
|
| $29$ |
\( T^{11} + \cdots - 47\!\cdots\!28 \)
|
| $31$ |
\( T^{11} + \cdots - 54\!\cdots\!40 \)
|
| $37$ |
\( T^{11} + \cdots + 32\!\cdots\!32 \)
|
| $41$ |
\( T^{11} + \cdots - 10\!\cdots\!44 \)
|
| $43$ |
\( T^{11} + \cdots - 64\!\cdots\!80 \)
|
| $47$ |
\( T^{11} + \cdots - 59\!\cdots\!08 \)
|
| $53$ |
\( T^{11} + \cdots - 65\!\cdots\!84 \)
|
| $59$ |
\( T^{11} + \cdots + 56\!\cdots\!84 \)
|
| $61$ |
\( T^{11} + \cdots - 10\!\cdots\!04 \)
|
| $67$ |
\( T^{11} + \cdots - 17\!\cdots\!88 \)
|
| $71$ |
\( T^{11} + \cdots - 15\!\cdots\!08 \)
|
| $73$ |
\( T^{11} + \cdots + 32\!\cdots\!16 \)
|
| $79$ |
\( T^{11} + \cdots - 66\!\cdots\!72 \)
|
| $83$ |
\( T^{11} + \cdots + 52\!\cdots\!88 \)
|
| $89$ |
\( T^{11} + \cdots - 15\!\cdots\!56 \)
|
| $97$ |
\( T^{11} + \cdots + 13\!\cdots\!36 \)
|
| show more | |
| show less |