Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [387,3,Mod(37,387)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(387, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("387.37");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 387 = 3^{2} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 387.j (of order \(6\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(10.5449862307\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} + 40x^{12} + 612x^{10} + 4461x^{8} + 15674x^{6} + 23555x^{4} + 10525x^{2} + 108 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 129) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{14} + 40x^{12} + 612x^{10} + 4461x^{8} + 15674x^{6} + 23555x^{4} + 10525x^{2} + 108 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 37 \nu^{12} - 781 \nu^{10} - 1833 \nu^{8} + 49246 \nu^{6} + 302604 \nu^{4} + 368525 \nu^{2} + \cdots + 17100 ) / 149392 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 205 \nu^{12} + 11393 \nu^{10} + 232225 \nu^{8} + 2203222 \nu^{6} + 9962864 \nu^{4} + 18742839 \nu^{2} + \cdots + 6486580 ) / 298784 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 279 \nu^{13} + 427 \nu^{12} - 12955 \nu^{11} + 16079 \nu^{10} - 235891 \nu^{9} + 243223 \nu^{8} + \cdots + 2201004 ) / 597568 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 279 \nu^{13} + 427 \nu^{12} + 12955 \nu^{11} + 16079 \nu^{10} + 235891 \nu^{9} + 243223 \nu^{8} + \cdots + 2201004 ) / 597568 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 475 \nu^{13} + 1098 \nu^{12} - 19111 \nu^{11} + 41346 \nu^{10} - 293043 \nu^{9} + 593418 \nu^{8} + \cdots - 198576 ) / 448176 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 475 \nu^{13} - 19111 \nu^{11} - 293043 \nu^{9} - 2124474 \nu^{7} - 7297412 \nu^{5} + \cdots + 224088 ) / 448176 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 475 \nu^{13} - 1098 \nu^{12} - 19111 \nu^{11} - 41346 \nu^{10} - 293043 \nu^{9} - 593418 \nu^{8} + \cdots + 198576 ) / 448176 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 950 \nu^{13} - 7359 \nu^{12} - 38222 \nu^{11} - 273435 \nu^{10} - 586086 \nu^{9} + \cdots - 1407756 ) / 896352 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 2963 \nu^{13} - 9333 \nu^{12} - 114023 \nu^{11} - 351441 \nu^{10} - 1636671 \nu^{9} + \cdots - 216852 ) / 1792704 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 987 \nu^{13} + 39003 \nu^{11} + 587919 \nu^{9} + 4199702 \nu^{7} + 14292220 \nu^{5} + 20193101 \nu^{3} + \cdots - 448176 ) / 149392 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 26051 \nu^{13} - 837 \nu^{12} + 1049543 \nu^{11} - 38865 \nu^{10} + 16225743 \nu^{9} + \cdots - 19357140 ) / 1792704 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - 2\beta_{8} + \beta_{7} - 9\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} - \beta_{5} + \beta_{4} - 3\beta_{3} - 12\beta_{2} + 2\beta _1 + 58 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{11} + \beta_{10} - 15\beta_{9} + 34\beta_{8} - 16\beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + 93\beta _1 - 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( - \beta_{10} + 3 \beta_{9} - 2 \beta_{7} + 20 \beta_{6} + 20 \beta_{5} - 17 \beta_{4} + 52 \beta_{3} + \cdots - 616 \)
|
| \(\nu^{7}\) | \(=\) |
\( 6 \beta_{13} - 10 \beta_{12} + 40 \beta_{11} - 20 \beta_{10} + 194 \beta_{9} - 456 \beta_{8} + \cdots + 208 \)
|
| \(\nu^{8}\) | \(=\) |
\( 29 \beta_{10} - 80 \beta_{9} + 51 \beta_{7} - 297 \beta_{6} - 297 \beta_{5} + 234 \beta_{4} + \cdots + 6805 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 160 \beta_{13} + 296 \beta_{12} - 594 \beta_{11} + 297 \beta_{10} - 2408 \beta_{9} + 5888 \beta_{8} + \cdots - 2647 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 561 \beta_{10} + 1463 \beta_{9} - 902 \beta_{7} + 3973 \beta_{6} + 3973 \beta_{5} - 3002 \beta_{4} + \cdots - 76832 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2926 \beta_{13} - 5958 \beta_{12} + 7946 \beta_{11} - 3973 \beta_{10} + 29380 \beta_{9} - 76168 \beta_{8} + \cdots + 34111 \)
|
| \(\nu^{12}\) | \(=\) |
\( 9074 \beta_{10} - 22925 \beta_{9} + 13851 \beta_{7} - 50708 \beta_{6} - 50708 \beta_{5} + 37326 \beta_{4} + \cdots + 879829 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 45850 \beta_{13} + 101826 \beta_{12} - 101416 \beta_{11} + 50708 \beta_{10} - 355371 \beta_{9} + \cdots - 445051 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/387\mathbb{Z}\right)^\times\).
| \(n\) | \(46\) | \(173\) |
| \(\chi(n)\) | \(1 - \beta_{8}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
− | 3.49788i | 0 | −8.23517 | 4.89549 | + | 2.82641i | 0 | 2.27091 | − | 1.31111i | 14.8141i | 0 | 9.88645 | − | 17.1238i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.2 | − | 2.90066i | 0 | −4.41384 | −5.83845 | − | 3.37083i | 0 | −8.48452 | + | 4.89854i | 1.20040i | 0 | −9.77763 | + | 16.9354i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.3 | − | 0.851342i | 0 | 3.27522 | 6.96545 | + | 4.02150i | 0 | −11.3385 | + | 6.54627i | − | 6.19370i | 0 | 3.42368 | − | 5.92998i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.4 | 0.102502i | 0 | 3.98949 | −3.54586 | − | 2.04720i | 0 | −3.61324 | + | 2.08611i | 0.818937i | 0 | 0.209842 | − | 0.363457i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.5 | 1.56723i | 0 | 1.54380 | 3.67802 | + | 2.12351i | 0 | 5.73888 | − | 3.31335i | 8.68839i | 0 | −3.32802 | + | 5.76429i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.6 | 2.24674i | 0 | −1.04786 | −8.20234 | − | 4.73562i | 0 | 5.13557 | − | 2.96502i | 6.63271i | 0 | 10.6397 | − | 18.4286i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37.7 | 3.33341i | 0 | −7.11165 | 0.547686 | + | 0.316207i | 0 | −7.70913 | + | 4.45087i | − | 10.3724i | 0 | −1.05405 | + | 1.82566i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 136.1 | − | 3.33341i | 0 | −7.11165 | 0.547686 | − | 0.316207i | 0 | −7.70913 | − | 4.45087i | 10.3724i | 0 | −1.05405 | − | 1.82566i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 136.2 | − | 2.24674i | 0 | −1.04786 | −8.20234 | + | 4.73562i | 0 | 5.13557 | + | 2.96502i | − | 6.63271i | 0 | 10.6397 | + | 18.4286i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 136.3 | − | 1.56723i | 0 | 1.54380 | 3.67802 | − | 2.12351i | 0 | 5.73888 | + | 3.31335i | − | 8.68839i | 0 | −3.32802 | − | 5.76429i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 136.4 | − | 0.102502i | 0 | 3.98949 | −3.54586 | + | 2.04720i | 0 | −3.61324 | − | 2.08611i | − | 0.818937i | 0 | 0.209842 | + | 0.363457i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 136.5 | 0.851342i | 0 | 3.27522 | 6.96545 | − | 4.02150i | 0 | −11.3385 | − | 6.54627i | 6.19370i | 0 | 3.42368 | + | 5.92998i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 136.6 | 2.90066i | 0 | −4.41384 | −5.83845 | + | 3.37083i | 0 | −8.48452 | − | 4.89854i | − | 1.20040i | 0 | −9.77763 | − | 16.9354i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 136.7 | 3.49788i | 0 | −8.23517 | 4.89549 | − | 2.82641i | 0 | 2.27091 | + | 1.31111i | − | 14.8141i | 0 | 9.88645 | + | 17.1238i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 43.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 387.3.j.e | 14 | |
| 3.b | odd | 2 | 1 | 129.3.g.b | ✓ | 14 | |
| 43.d | odd | 6 | 1 | inner | 387.3.j.e | 14 | |
| 129.h | even | 6 | 1 | 129.3.g.b | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 129.3.g.b | ✓ | 14 | 3.b | odd | 2 | 1 | |
| 129.3.g.b | ✓ | 14 | 129.h | even | 6 | 1 | |
| 387.3.j.e | 14 | 1.a | even | 1 | 1 | trivial | |
| 387.3.j.e | 14 | 43.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} + 40T_{2}^{12} + 612T_{2}^{10} + 4461T_{2}^{8} + 15674T_{2}^{6} + 23555T_{2}^{4} + 10525T_{2}^{2} + 108 \)
acting on \(S_{3}^{\mathrm{new}}(387, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} + 40 T^{12} + \cdots + 108 \)
|
| $3$ |
\( T^{14} \)
|
| $5$ |
\( T^{14} + \cdots + 1019215872 \)
|
| $7$ |
\( T^{14} + \cdots + 240973887168 \)
|
| $11$ |
\( (T^{7} - 17 T^{6} + \cdots + 822810)^{2} \)
|
| $13$ |
\( T^{14} + \cdots + 22\!\cdots\!00 \)
|
| $17$ |
\( T^{14} + \cdots + 260427749675076 \)
|
| $19$ |
\( T^{14} + \cdots + 18\!\cdots\!28 \)
|
| $23$ |
\( T^{14} + \cdots + 27\!\cdots\!36 \)
|
| $29$ |
\( T^{14} + \cdots + 1042318540800 \)
|
| $31$ |
\( T^{14} + \cdots + 49\!\cdots\!00 \)
|
| $37$ |
\( T^{14} + \cdots + 32\!\cdots\!72 \)
|
| $41$ |
\( (T^{7} + 37 T^{6} + \cdots - 73077690)^{2} \)
|
| $43$ |
\( T^{14} + \cdots + 73\!\cdots\!49 \)
|
| $47$ |
\( (T^{7} - 96 T^{6} + \cdots - 10132371648)^{2} \)
|
| $53$ |
\( T^{14} + \cdots + 22\!\cdots\!00 \)
|
| $59$ |
\( (T^{7} + 24 T^{6} + \cdots + 611572657464)^{2} \)
|
| $61$ |
\( T^{14} + \cdots + 29\!\cdots\!00 \)
|
| $67$ |
\( T^{14} + \cdots + 85\!\cdots\!81 \)
|
| $71$ |
\( T^{14} + \cdots + 82\!\cdots\!72 \)
|
| $73$ |
\( T^{14} + \cdots + 41\!\cdots\!47 \)
|
| $79$ |
\( T^{14} + \cdots + 54\!\cdots\!76 \)
|
| $83$ |
\( T^{14} + \cdots + 86\!\cdots\!00 \)
|
| $89$ |
\( T^{14} + \cdots + 40\!\cdots\!00 \)
|
| $97$ |
\( (T^{7} + \cdots - 13238557544618)^{2} \)
|
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