Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,6,Mod(49,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("144.49");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 144.i (of order \(3\), degree \(2\), not 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: | \(23.0952700531\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - x^{13} + 129 x^{12} - 54 x^{11} + 11895 x^{10} - 5118 x^{9} + 498525 x^{8} - 176283 x^{7} + \cdots + 1124529156 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{20}\cdot 3^{14} \) |
| Twist minimal: | no (minimal twist has level 72) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{14} - x^{13} + 129 x^{12} - 54 x^{11} + 11895 x^{10} - 5118 x^{9} + 498525 x^{8} - 176283 x^{7} + \cdots + 1124529156 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 13\!\cdots\!22 \nu^{13} + \cdots + 10\!\cdots\!02 ) / 59\!\cdots\!50 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 55\!\cdots\!41 \nu^{13} + \cdots + 84\!\cdots\!66 ) / 85\!\cdots\!50 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 14\!\cdots\!56 \nu^{13} + \cdots - 18\!\cdots\!59 ) / 29\!\cdots\!25 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13\!\cdots\!93 \nu^{13} + \cdots + 97\!\cdots\!42 ) / 13\!\cdots\!50 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 47\!\cdots\!28 \nu^{13} + \cdots - 17\!\cdots\!66 ) / 27\!\cdots\!10 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 58\!\cdots\!76 \nu^{13} + \cdots + 14\!\cdots\!01 ) / 29\!\cdots\!25 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 34\!\cdots\!73 \nu^{13} + \cdots - 87\!\cdots\!92 ) / 13\!\cdots\!50 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 21\!\cdots\!93 \nu^{13} + \cdots + 47\!\cdots\!32 ) / 59\!\cdots\!50 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 57\!\cdots\!27 \nu^{13} + \cdots - 14\!\cdots\!02 ) / 13\!\cdots\!50 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 46\!\cdots\!46 \nu^{13} + \cdots + 10\!\cdots\!04 ) / 50\!\cdots\!50 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 12\!\cdots\!07 \nu^{13} + \cdots - 41\!\cdots\!42 ) / 13\!\cdots\!50 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 13\!\cdots\!38 \nu^{13} + \cdots - 11\!\cdots\!22 ) / 13\!\cdots\!50 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 12\!\cdots\!82 \nu^{13} + \cdots - 25\!\cdots\!30 ) / 91\!\cdots\!70 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{9} - 4\beta_{4} + \beta_{2} + 6\beta_1 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{13} + 3\beta_{12} - 3\beta_{11} + 2\beta_{9} + 4\beta_{4} - 1321\beta_{2} ) / 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 15 \beta_{13} + 15 \beta_{12} - 9 \beta_{11} + 9 \beta_{10} - 227 \beta_{9} + 246 \beta_{7} + \cdots - 906 ) / 72 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 540 \beta_{13} - 459 \beta_{12} - 81 \beta_{11} + 459 \beta_{10} - 518 \beta_{9} + 576 \beta_{7} + \cdots - 148572 ) / 72 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 219 \beta_{13} - 510 \beta_{12} - 219 \beta_{11} - 729 \beta_{10} - 1393 \beta_{9} + 1701 \beta_{8} + \cdots + 8415 ) / 36 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 9651 \beta_{13} - 9651 \beta_{12} + 43785 \beta_{11} - 43785 \beta_{10} - 3179 \beta_{9} + \cdots + 9942204 ) / 72 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 99765 \beta_{13} - 20439 \beta_{12} + 120204 \beta_{11} + 20439 \beta_{10} + 1324120 \beta_{9} + \cdots + 11844297 ) / 72 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 2550669 \beta_{13} + 3441993 \beta_{12} - 2550669 \beta_{11} + 891324 \beta_{10} + 4160173 \beta_{9} + \cdots - 3240360 ) / 72 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 9634830 \beta_{13} + 9634830 \beta_{12} - 9195021 \beta_{11} + 9195021 \beta_{10} - 76711025 \beta_{9} + \cdots - 1247364141 ) / 72 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 267745824 \beta_{13} - 192048813 \beta_{12} - 75697011 \beta_{11} + 192048813 \beta_{10} + \cdots - 53362774242 ) / 72 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 51186162 \beta_{13} - 819226299 \beta_{12} + 51186162 \beta_{11} - 768040137 \beta_{10} + \cdots + 5691497589 ) / 72 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 6216005676 \beta_{13} - 6216005676 \beta_{12} + 20761809345 \beta_{11} - 20761809345 \beta_{10} + \cdots + 4075592640699 ) / 72 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 71386194348 \beta_{13} + 10244971719 \beta_{12} + 61141222629 \beta_{11} - 10244971719 \beta_{10} + \cdots + 10017745009200 ) / 72 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(65\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | −14.4765 | + | 5.78199i | 0 | −9.09373 | + | 15.7508i | 0 | 112.185 | + | 194.310i | 0 | 176.137 | − | 167.406i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | 0 | −12.3525 | + | 9.50868i | 0 | 49.9293 | − | 86.4801i | 0 | −123.572 | − | 214.034i | 0 | 62.1701 | − | 234.912i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | −9.91812 | − | 12.0263i | 0 | −10.1127 | + | 17.5157i | 0 | −29.6263 | − | 51.3143i | 0 | −46.2619 | + | 238.556i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | 1.08101 | + | 15.5509i | 0 | −24.9232 | + | 43.1683i | 0 | −32.9730 | − | 57.1109i | 0 | −240.663 | + | 33.6213i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | 7.45249 | − | 13.6916i | 0 | 24.8287 | − | 43.0045i | 0 | 20.5752 | + | 35.6373i | 0 | −131.921 | − | 204.073i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 13.3117 | + | 8.11166i | 0 | 27.1257 | − | 46.9830i | 0 | 41.4529 | + | 71.7986i | 0 | 111.402 | + | 215.960i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 0 | 14.9020 | − | 4.57513i | 0 | −45.2540 | + | 78.3822i | 0 | −34.5411 | − | 59.8269i | 0 | 201.136 | − | 136.357i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.1 | 0 | −14.4765 | − | 5.78199i | 0 | −9.09373 | − | 15.7508i | 0 | 112.185 | − | 194.310i | 0 | 176.137 | + | 167.406i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.2 | 0 | −12.3525 | − | 9.50868i | 0 | 49.9293 | + | 86.4801i | 0 | −123.572 | + | 214.034i | 0 | 62.1701 | + | 234.912i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.3 | 0 | −9.91812 | + | 12.0263i | 0 | −10.1127 | − | 17.5157i | 0 | −29.6263 | + | 51.3143i | 0 | −46.2619 | − | 238.556i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.4 | 0 | 1.08101 | − | 15.5509i | 0 | −24.9232 | − | 43.1683i | 0 | −32.9730 | + | 57.1109i | 0 | −240.663 | − | 33.6213i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.5 | 0 | 7.45249 | + | 13.6916i | 0 | 24.8287 | + | 43.0045i | 0 | 20.5752 | − | 35.6373i | 0 | −131.921 | + | 204.073i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.6 | 0 | 13.3117 | − | 8.11166i | 0 | 27.1257 | + | 46.9830i | 0 | 41.4529 | − | 71.7986i | 0 | 111.402 | − | 215.960i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 97.7 | 0 | 14.9020 | + | 4.57513i | 0 | −45.2540 | − | 78.3822i | 0 | −34.5411 | + | 59.8269i | 0 | 201.136 | + | 136.357i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 144.6.i.e | 14 | |
| 3.b | odd | 2 | 1 | 432.6.i.e | 14 | ||
| 4.b | odd | 2 | 1 | 72.6.i.a | ✓ | 14 | |
| 9.c | even | 3 | 1 | inner | 144.6.i.e | 14 | |
| 9.d | odd | 6 | 1 | 432.6.i.e | 14 | ||
| 12.b | even | 2 | 1 | 216.6.i.a | 14 | ||
| 36.f | odd | 6 | 1 | 72.6.i.a | ✓ | 14 | |
| 36.f | odd | 6 | 1 | 648.6.a.e | 7 | ||
| 36.h | even | 6 | 1 | 216.6.i.a | 14 | ||
| 36.h | even | 6 | 1 | 648.6.a.f | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.6.i.a | ✓ | 14 | 4.b | odd | 2 | 1 | |
| 72.6.i.a | ✓ | 14 | 36.f | odd | 6 | 1 | |
| 144.6.i.e | 14 | 1.a | even | 1 | 1 | trivial | |
| 144.6.i.e | 14 | 9.c | even | 3 | 1 | inner | |
| 216.6.i.a | 14 | 12.b | even | 2 | 1 | ||
| 216.6.i.a | 14 | 36.h | even | 6 | 1 | ||
| 432.6.i.e | 14 | 3.b | odd | 2 | 1 | ||
| 432.6.i.e | 14 | 9.d | odd | 6 | 1 | ||
| 648.6.a.e | 7 | 36.f | odd | 6 | 1 | ||
| 648.6.a.f | 7 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{14} - 25 T_{5}^{13} + 13711 T_{5}^{12} - 72014 T_{5}^{11} + 134257517 T_{5}^{10} + \cdots + 19\!\cdots\!84 \)
acting on \(S_{6}^{\mathrm{new}}(144, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} \)
|
| $3$ |
\( T^{14} + \cdots + 50\!\cdots\!07 \)
|
| $5$ |
\( T^{14} + \cdots + 19\!\cdots\!84 \)
|
| $7$ |
\( T^{14} + \cdots + 26\!\cdots\!76 \)
|
| $11$ |
\( T^{14} + \cdots + 94\!\cdots\!21 \)
|
| $13$ |
\( T^{14} + \cdots + 71\!\cdots\!76 \)
|
| $17$ |
\( (T^{7} + \cdots - 11\!\cdots\!24)^{2} \)
|
| $19$ |
\( (T^{7} + \cdots - 12\!\cdots\!16)^{2} \)
|
| $23$ |
\( T^{14} + \cdots + 32\!\cdots\!76 \)
|
| $29$ |
\( T^{14} + \cdots + 15\!\cdots\!36 \)
|
| $31$ |
\( T^{14} + \cdots + 29\!\cdots\!24 \)
|
| $37$ |
\( (T^{7} + \cdots - 27\!\cdots\!16)^{2} \)
|
| $41$ |
\( T^{14} + \cdots + 68\!\cdots\!41 \)
|
| $43$ |
\( T^{14} + \cdots + 25\!\cdots\!49 \)
|
| $47$ |
\( T^{14} + \cdots + 13\!\cdots\!64 \)
|
| $53$ |
\( (T^{7} + \cdots - 85\!\cdots\!68)^{2} \)
|
| $59$ |
\( T^{14} + \cdots + 11\!\cdots\!01 \)
|
| $61$ |
\( T^{14} + \cdots + 13\!\cdots\!36 \)
|
| $67$ |
\( T^{14} + \cdots + 39\!\cdots\!29 \)
|
| $71$ |
\( (T^{7} + \cdots - 41\!\cdots\!16)^{2} \)
|
| $73$ |
\( (T^{7} + \cdots - 27\!\cdots\!76)^{2} \)
|
| $79$ |
\( T^{14} + \cdots + 53\!\cdots\!44 \)
|
| $83$ |
\( T^{14} + \cdots + 21\!\cdots\!44 \)
|
| $89$ |
\( (T^{7} + \cdots + 10\!\cdots\!36)^{2} \)
|
| $97$ |
\( T^{14} + \cdots + 20\!\cdots\!69 \)
|
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