Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [603,4,Mod(1,603)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(603, 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("603.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 603 = 3^{2} \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 603.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: | \(35.5781517335\) |
| 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: |
\( x^{11} - 3 x^{10} - 74 x^{9} + 208 x^{8} + 1913 x^{7} - 4831 x^{6} - 20432 x^{5} + 42994 x^{4} + \cdots + 3072 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 201) |
| 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
\( x^{11} - 3 x^{10} - 74 x^{9} + 208 x^{8} + 1913 x^{7} - 4831 x^{6} - 20432 x^{5} + 42994 x^{4} + \cdots + 3072 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 37 \nu^{10} - 10337 \nu^{9} - 910 \nu^{8} + 803028 \nu^{7} - 138167 \nu^{6} - 21118529 \nu^{5} + \cdots - 17164864 ) / 17885888 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1563 \nu^{10} - 1415 \nu^{9} + 68654 \nu^{8} - 54128 \nu^{7} + 5581 \nu^{6} + 7020509 \nu^{5} + \cdots + 161452352 ) / 17885888 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 999 \nu^{10} - 368 \nu^{9} + 24570 \nu^{8} + 116670 \nu^{7} + 935839 \nu^{6} - 5222270 \nu^{5} + \cdots - 260927136 ) / 8942944 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2205 \nu^{10} - 41989 \nu^{9} - 190188 \nu^{8} + 2763750 \nu^{7} + 5709131 \nu^{6} + \cdots - 284052576 ) / 8942944 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 27763 \nu^{10} + 74816 \nu^{9} + 1906432 \nu^{8} - 5007380 \nu^{7} - 44730735 \nu^{6} + \cdots + 1141218944 ) / 35771776 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 28189 \nu^{10} + 47870 \nu^{9} - 1856484 \nu^{8} - 2968156 \nu^{7} + 40420809 \nu^{6} + \cdots - 713324160 ) / 35771776 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 28263 \nu^{10} - 27196 \nu^{9} + 1858304 \nu^{8} + 1362100 \nu^{7} - 40144475 \nu^{6} + \cdots + 246849024 ) / 35771776 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 45089 \nu^{10} - 81302 \nu^{9} - 3359788 \nu^{8} + 5317020 \nu^{7} + 86621461 \nu^{6} + \cdots - 961174016 ) / 35771776 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta_{3} - \beta_{2} + 23\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} - 2\beta_{8} - \beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} + 4\beta_{3} + 28\beta_{2} + 313 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 3 \beta_{10} + 29 \beta_{9} + 41 \beta_{8} + 9 \beta_{7} + 8 \beta_{6} - \beta_{5} + 9 \beta_{4} + \cdots - 21 \)
|
| \(\nu^{6}\) | \(=\) |
\( 34 \beta_{10} - 2 \beta_{9} - 108 \beta_{8} - 88 \beta_{7} - 106 \beta_{6} - 26 \beta_{5} + 54 \beta_{4} + \cdots + 7856 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 172 \beta_{10} + 755 \beta_{9} + 1395 \beta_{8} + 452 \beta_{7} + 412 \beta_{6} - 16 \beta_{5} + \cdots - 1548 \)
|
| \(\nu^{8}\) | \(=\) |
\( 911 \beta_{10} - 164 \beta_{9} - 4210 \beta_{8} - 4063 \beta_{7} - 4086 \beta_{6} - 631 \beta_{5} + \cdots + 208851 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 7009 \beta_{10} + 19693 \beta_{9} + 44825 \beta_{8} + 17035 \beta_{7} + 15584 \beta_{6} + \cdots - 81047 \)
|
| \(\nu^{10}\) | \(=\) |
\( 23316 \beta_{10} - 8654 \beta_{9} - 146472 \beta_{8} - 153782 \beta_{7} - 141002 \beta_{6} + \cdots + 5762126 \)
|
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 |
|
−5.42821 | 0 | 21.4655 | 20.0336 | 0 | 33.9282 | −73.0934 | 0 | −108.746 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.82380 | 0 | 15.2690 | −4.99073 | 0 | −10.0496 | −35.0643 | 0 | 24.0743 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.59496 | 0 | 13.1136 | −11.7140 | 0 | 4.20913 | −23.4970 | 0 | 53.8252 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.85564 | 0 | 0.154694 | 2.88345 | 0 | 19.3750 | 22.4034 | 0 | −8.23409 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.96751 | 0 | −4.12891 | −17.2496 | 0 | 22.3999 | 23.8637 | 0 | 33.9387 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.0344354 | 0 | −7.99881 | −3.05553 | 0 | −26.7198 | 0.550926 | 0 | 0.105218 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.638765 | 0 | −7.59198 | 13.5481 | 0 | 17.9442 | −9.95961 | 0 | 8.65405 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.12976 | 0 | −3.46413 | −17.7773 | 0 | −9.52178 | −24.4158 | 0 | −37.8613 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 3.89104 | 0 | 7.14021 | −11.5975 | 0 | 32.2920 | −3.34547 | 0 | −45.1263 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 4.47941 | 0 | 12.0651 | 12.5434 | 0 | −24.1947 | 18.2093 | 0 | 56.1868 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 5.56558 | 0 | 22.9757 | 9.37616 | 0 | 18.3374 | 83.3483 | 0 | 52.1838 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(67\) | \(-1\) |
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 | 603.4.a.g | 11 | |
| 3.b | odd | 2 | 1 | 201.4.a.e | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 201.4.a.e | ✓ | 11 | 3.b | odd | 2 | 1 | |
| 603.4.a.g | 11 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} + 3 T_{2}^{10} - 74 T_{2}^{9} - 208 T_{2}^{8} + 1913 T_{2}^{7} + 4831 T_{2}^{6} - 20432 T_{2}^{5} + \cdots - 3072 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(603))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + 3 T^{10} + \cdots - 3072 \)
|
| $3$ |
\( T^{11} \)
|
| $5$ |
\( T^{11} + \cdots - 58472299008 \)
|
| $7$ |
\( T^{11} + \cdots - 40739453904896 \)
|
| $11$ |
\( T^{11} + \cdots - 16\!\cdots\!36 \)
|
| $13$ |
\( T^{11} + \cdots - 43\!\cdots\!88 \)
|
| $17$ |
\( T^{11} + \cdots + 64\!\cdots\!96 \)
|
| $19$ |
\( T^{11} + \cdots - 23\!\cdots\!24 \)
|
| $23$ |
\( T^{11} + \cdots - 30\!\cdots\!00 \)
|
| $29$ |
\( T^{11} + \cdots + 78\!\cdots\!84 \)
|
| $31$ |
\( T^{11} + \cdots + 76\!\cdots\!04 \)
|
| $37$ |
\( T^{11} + \cdots - 52\!\cdots\!20 \)
|
| $41$ |
\( T^{11} + \cdots - 24\!\cdots\!28 \)
|
| $43$ |
\( T^{11} + \cdots - 97\!\cdots\!48 \)
|
| $47$ |
\( T^{11} + \cdots + 22\!\cdots\!24 \)
|
| $53$ |
\( T^{11} + \cdots - 24\!\cdots\!28 \)
|
| $59$ |
\( T^{11} + \cdots - 40\!\cdots\!76 \)
|
| $61$ |
\( T^{11} + \cdots - 22\!\cdots\!92 \)
|
| $67$ |
\( (T - 67)^{11} \)
|
| $71$ |
\( T^{11} + \cdots + 30\!\cdots\!84 \)
|
| $73$ |
\( T^{11} + \cdots - 14\!\cdots\!52 \)
|
| $79$ |
\( T^{11} + \cdots + 19\!\cdots\!28 \)
|
| $83$ |
\( T^{11} + \cdots - 50\!\cdots\!40 \)
|
| $89$ |
\( T^{11} + \cdots + 11\!\cdots\!28 \)
|
| $97$ |
\( T^{11} + \cdots + 21\!\cdots\!20 \)
|
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