Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6039,2,Mod(1,6039)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6039, 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("6039.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6039 = 3^{2} \cdot 11 \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6039.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: | \(48.2216577807\) |
| Analytic rank: | \(0\) |
| Dimension: | \(19\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{19} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 671) |
| 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 \(1,\beta_1,\ldots,\beta_{18}\) 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
\( x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8867201804 \nu^{18} - 28397403169 \nu^{17} - 153963852608 \nu^{16} + 585160018840 \nu^{15} + \cdots + 7406641729547 ) / 2584793613763 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 9925308116 \nu^{18} + 211585433401 \nu^{17} - 157290256641 \nu^{16} + \cdots + 4643140412416 ) / 2584793613763 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 16270525372 \nu^{18} - 35753207242 \nu^{17} - 519467802678 \nu^{16} + 1065729788540 \nu^{15} + \cdots + 1737797285981 ) / 2584793613763 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 37762226457 \nu^{18} - 165903350608 \nu^{17} - 747153495518 \nu^{16} + 3973232345087 \nu^{15} + \cdots - 2002160866471 ) / 2584793613763 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 41025764486 \nu^{18} + 68688107954 \nu^{17} - 1546829355593 \nu^{16} - 1494342049450 \nu^{15} + \cdots - 165641941058 ) / 2584793613763 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 44058942396 \nu^{18} + 100366607650 \nu^{17} + 1163734380318 \nu^{16} + \cdots - 246530814813 ) / 2584793613763 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 57136613064 \nu^{18} + 46120188886 \nu^{17} + 1573939504037 \nu^{16} + \cdots + 8443491451865 ) / 2584793613763 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 81254538506 \nu^{18} + 654428683551 \nu^{17} + 970372146851 \nu^{16} + \cdots - 25082161671395 ) / 2584793613763 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 91424960248 \nu^{18} + 290362672773 \nu^{17} + 2163987038842 \nu^{16} + \cdots - 5331850884506 ) / 2584793613763 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 110220246484 \nu^{18} - 467343546422 \nu^{17} - 2253974970877 \nu^{16} + \cdots - 8426703234447 ) / 2584793613763 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 112916661333 \nu^{18} - 420512816139 \nu^{17} - 2391672731682 \nu^{16} + 10136775407328 \nu^{15} + \cdots - 992900881709 ) / 2584793613763 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 134524452259 \nu^{18} - 595936463557 \nu^{17} - 2794615480726 \nu^{16} + \cdots - 6942619491638 ) / 2584793613763 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 268063867548 \nu^{18} - 975838766429 \nu^{17} - 5844382212784 \nu^{16} + \cdots + 2625408551097 ) / 2584793613763 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 386309035457 \nu^{18} + 1365268186173 \nu^{17} + 8692512067202 \nu^{16} + \cdots - 8519539396534 ) / 2584793613763 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 407596389827 \nu^{18} + 996406897672 \nu^{17} + 10303884149056 \nu^{16} + \cdots - 7512666382563 ) / 2584793613763 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 433257329190 \nu^{18} - 1312562946260 \nu^{17} - 10331601737881 \nu^{16} + \cdots + 7601132183693 ) / 2584793613763 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} + \beta_{11} - \beta_{6} + \beta_{5} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{3} + 7\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{18} + 2 \beta_{17} - \beta_{16} - \beta_{14} + 11 \beta_{13} - \beta_{12} + 7 \beta_{11} + \cdots + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( - \beta_{17} + 2 \beta_{16} + \beta_{15} + \beta_{14} - \beta_{13} - 11 \beta_{12} + \beta_{11} + \cdots + 88 \)
|
| \(\nu^{7}\) | \(=\) |
\( 10 \beta_{18} + 24 \beta_{17} - 15 \beta_{16} - 14 \beta_{14} + 94 \beta_{13} - 12 \beta_{12} + \cdots + 76 \)
|
| \(\nu^{8}\) | \(=\) |
\( 3 \beta_{18} - 15 \beta_{17} + 31 \beta_{16} + 14 \beta_{15} + 14 \beta_{14} - 18 \beta_{13} + \cdots + 567 \)
|
| \(\nu^{9}\) | \(=\) |
\( 78 \beta_{18} + 218 \beta_{17} - 153 \beta_{16} + 2 \beta_{15} - 142 \beta_{14} + 746 \beta_{13} + \cdots + 534 \)
|
| \(\nu^{10}\) | \(=\) |
\( 47 \beta_{18} - 163 \beta_{17} + 334 \beta_{16} + 143 \beta_{15} + 143 \beta_{14} - 224 \beta_{13} + \cdots + 3874 \)
|
| \(\nu^{11}\) | \(=\) |
\( 570 \beta_{18} + 1806 \beta_{17} - 1356 \beta_{16} + 33 \beta_{15} - 1270 \beta_{14} + 5763 \beta_{13} + \cdots + 3645 \)
|
| \(\nu^{12}\) | \(=\) |
\( 492 \beta_{18} - 1566 \beta_{17} + 3118 \beta_{16} + 1289 \beta_{15} + 1305 \beta_{14} - 2393 \beta_{13} + \cdots + 27441 \)
|
| \(\nu^{13}\) | \(=\) |
\( 4103 \beta_{18} + 14404 \beta_{17} - 11290 \beta_{16} + 341 \beta_{15} - 10672 \beta_{14} + 44078 \beta_{13} + \cdots + 24525 \)
|
| \(\nu^{14}\) | \(=\) |
\( 4351 \beta_{18} - 14133 \beta_{17} + 27117 \beta_{16} + 10875 \beta_{15} + 11326 \beta_{14} + \cdots + 198716 \)
|
| \(\nu^{15}\) | \(=\) |
\( 29563 \beta_{18} + 112951 \beta_{17} - 91244 \beta_{16} + 2777 \beta_{15} - 86679 \beta_{14} + \cdots + 163128 \)
|
| \(\nu^{16}\) | \(=\) |
\( 35212 \beta_{18} - 122951 \beta_{17} + 226841 \beta_{16} + 88185 \beta_{15} + 95848 \beta_{14} + \cdots + 1458972 \)
|
| \(\nu^{17}\) | \(=\) |
\( 214159 \beta_{18} + 879024 \beta_{17} - 726921 \beta_{16} + 18877 \beta_{15} - 690523 \beta_{14} + \cdots + 1070635 \)
|
| \(\nu^{18}\) | \(=\) |
\( 270373 \beta_{18} - 1044853 \beta_{17} + 1855069 \beta_{16} + 697382 \beta_{15} + 799771 \beta_{14} + \cdots + 10807509 \)
|
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 |
|
−2.72847 | 0 | 5.44452 | 3.87909 | 0 | 0.707409 | −9.39826 | 0 | −10.5840 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.69542 | 0 | 5.26531 | −1.92020 | 0 | 2.04756 | −8.80138 | 0 | 5.17575 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.34443 | 0 | 3.49635 | −3.40234 | 0 | −2.95885 | −3.50809 | 0 | 7.97654 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.13634 | 0 | 2.56396 | 0.997990 | 0 | −0.526710 | −1.20482 | 0 | −2.13205 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.04293 | 0 | 2.17358 | −0.258135 | 0 | 3.70333 | −0.354615 | 0 | 0.527352 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.98166 | 0 | 1.92698 | −2.60057 | 0 | −2.18138 | 0.144707 | 0 | 5.15344 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.31423 | 0 | −0.272811 | 1.38757 | 0 | 2.03012 | 2.98699 | 0 | −1.82358 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.04858 | 0 | −0.900472 | 4.37157 | 0 | −3.70774 | 3.04139 | 0 | −4.58396 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.686089 | 0 | −1.52928 | −3.77631 | 0 | 4.48234 | 2.42140 | 0 | 2.59089 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −0.488177 | 0 | −1.76168 | −1.92726 | 0 | 0.259128 | 1.83637 | 0 | 0.940847 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −0.0681863 | 0 | −1.99535 | 3.91488 | 0 | 0.303195 | 0.272428 | 0 | −0.266941 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.436741 | 0 | −1.80926 | −3.47884 | 0 | −1.93516 | −1.66366 | 0 | −1.51935 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0.556474 | 0 | −1.69034 | −0.874867 | 0 | 4.50273 | −2.05358 | 0 | −0.486841 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 0.976920 | 0 | −1.04563 | 2.45987 | 0 | −4.72156 | −2.97533 | 0 | 2.40309 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.62941 | 0 | 0.654988 | −2.04510 | 0 | −3.70972 | −2.19158 | 0 | −3.33231 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 1.82302 | 0 | 1.32341 | 3.15385 | 0 | −0.287436 | −1.23344 | 0 | 5.74953 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.08441 | 0 | 2.34475 | −2.98183 | 0 | 3.78550 | 0.718591 | 0 | −6.21535 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2.21976 | 0 | 2.92734 | 2.17512 | 0 | 5.05675 | 2.05847 | 0 | 4.82825 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 2.80778 | 0 | 5.88364 | 0.925514 | 0 | 2.15050 | 10.9044 | 0 | 2.59864 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
| \(61\) | \(1\) |
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 | 6039.2.a.k | 19 | |
| 3.b | odd | 2 | 1 | 671.2.a.c | ✓ | 19 | |
| 33.d | even | 2 | 1 | 7381.2.a.i | 19 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 671.2.a.c | ✓ | 19 | 3.b | odd | 2 | 1 | |
| 6039.2.a.k | 19 | 1.a | even | 1 | 1 | trivial | |
| 7381.2.a.i | 19 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{19} + 5 T_{2}^{18} - 18 T_{2}^{17} - 122 T_{2}^{16} + 78 T_{2}^{15} + 1177 T_{2}^{14} + 387 T_{2}^{13} + \cdots + 43 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{19} + 5 T^{18} + \cdots + 43 \)
|
| $3$ |
\( T^{19} \)
|
| $5$ |
\( T^{19} - 70 T^{17} + \cdots - 850564 \)
|
| $7$ |
\( T^{19} - 9 T^{18} + \cdots - 87296 \)
|
| $11$ |
\( (T + 1)^{19} \)
|
| $13$ |
\( T^{19} - 8 T^{18} + \cdots + 966656 \)
|
| $17$ |
\( T^{19} + \cdots + 16020390437 \)
|
| $19$ |
\( T^{19} + \cdots - 7413317632 \)
|
| $23$ |
\( T^{19} + \cdots - 76028297216 \)
|
| $29$ |
\( T^{19} + \cdots - 11106421323508 \)
|
| $31$ |
\( T^{19} + \cdots + 7338377216 \)
|
| $37$ |
\( T^{19} + \cdots + 88495013888 \)
|
| $41$ |
\( T^{19} + \cdots + 26396783722496 \)
|
| $43$ |
\( T^{19} + \cdots - 25482644176 \)
|
| $47$ |
\( T^{19} + \cdots + 3936280228595 \)
|
| $53$ |
\( T^{19} + \cdots + 13303893868544 \)
|
| $59$ |
\( T^{19} + \cdots - 81\!\cdots\!64 \)
|
| $61$ |
\( (T + 1)^{19} \)
|
| $67$ |
\( T^{19} + \cdots + 54909964304384 \)
|
| $71$ |
\( T^{19} + \cdots + 7732360953856 \)
|
| $73$ |
\( T^{19} + \cdots + 38776064491520 \)
|
| $79$ |
\( T^{19} + \cdots + 11\!\cdots\!41 \)
|
| $83$ |
\( T^{19} + \cdots - 19\!\cdots\!28 \)
|
| $89$ |
\( T^{19} + \cdots + 21\!\cdots\!68 \)
|
| $97$ |
\( T^{19} + \cdots + 45\!\cdots\!47 \)
|
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