Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [63,4,Mod(22,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.22");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.f (of order \(3\), 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: | \(3.71712033036\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 3 x^{15} + 58 x^{14} - 129 x^{13} + 2107 x^{12} - 4455 x^{11} + 42901 x^{10} - 76404 x^{9} + \cdots + 21307456 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{7} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 3 x^{15} + 58 x^{14} - 129 x^{13} + 2107 x^{12} - 4455 x^{11} + 42901 x^{10} - 76404 x^{9} + \cdots + 21307456 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 10\!\cdots\!31 \nu^{15} + \cdots + 20\!\cdots\!60 ) / 35\!\cdots\!16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 54\!\cdots\!70 \nu^{15} + \cdots - 25\!\cdots\!60 ) / 20\!\cdots\!32 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 20\!\cdots\!33 \nu^{15} + \cdots + 23\!\cdots\!08 ) / 12\!\cdots\!84 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 24\!\cdots\!57 \nu^{15} + \cdots - 12\!\cdots\!44 ) / 12\!\cdots\!84 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 34\!\cdots\!15 \nu^{15} + \cdots + 30\!\cdots\!00 ) / 12\!\cdots\!84 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 39\!\cdots\!11 \nu^{15} + \cdots + 13\!\cdots\!72 ) / 12\!\cdots\!84 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 35\!\cdots\!23 \nu^{15} + \cdots + 24\!\cdots\!52 ) / 11\!\cdots\!96 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 72\!\cdots\!79 \nu^{15} + \cdots - 20\!\cdots\!60 ) / 12\!\cdots\!84 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 73\!\cdots\!73 \nu^{15} + \cdots - 84\!\cdots\!12 ) / 12\!\cdots\!84 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 79\!\cdots\!95 \nu^{15} + \cdots + 14\!\cdots\!04 ) / 12\!\cdots\!84 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 27\!\cdots\!76 \nu^{15} + \cdots + 92\!\cdots\!44 ) / 32\!\cdots\!96 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 37\!\cdots\!30 \nu^{15} + \cdots + 21\!\cdots\!72 ) / 32\!\cdots\!96 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 21\!\cdots\!91 \nu^{15} + \cdots + 14\!\cdots\!96 ) / 12\!\cdots\!84 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 10\!\cdots\!28 \nu^{15} + \cdots - 53\!\cdots\!32 ) / 64\!\cdots\!92 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} - \beta_{7} + \beta_{6} - \beta_{5} + 13\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( - 2 \beta_{14} + \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + \cdots + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 4 \beta_{15} + \beta_{14} + 3 \beta_{13} + 2 \beta_{12} - 3 \beta_{11} - 29 \beta_{10} + \beta_{8} + \cdots - 256 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{15} + 33 \beta_{14} - 39 \beta_{13} + 4 \beta_{12} - 6 \beta_{11} - 71 \beta_{10} + 31 \beta_{9} + \cdots - 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( 90 \beta_{15} + 18 \beta_{14} - 54 \beta_{13} + 33 \beta_{12} + 102 \beta_{11} + 3 \beta_{10} + \cdots + 5692 \)
|
| \(\nu^{7}\) | \(=\) |
\( 216 \beta_{15} + 813 \beta_{14} + 327 \beta_{13} + 48 \beta_{12} + 243 \beta_{11} + 1080 \beta_{10} + \cdots + 2223 \)
|
| \(\nu^{8}\) | \(=\) |
\( 3042 \beta_{15} - 2088 \beta_{14} - 486 \beta_{13} - 4668 \beta_{12} + 954 \beta_{11} + 20536 \beta_{10} + \cdots + 954 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 4464 \beta_{15} - 45584 \beta_{14} + 25024 \beta_{13} - 10260 \beta_{12} - 12 \beta_{11} + \cdots - 76285 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 185440 \beta_{15} + 22699 \beta_{14} + 61356 \beta_{13} + 117560 \beta_{12} - 145716 \beta_{11} + \cdots - 3311764 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 163154 \beta_{15} + 679563 \beta_{14} - 1050552 \beta_{13} + 370531 \beta_{12} - 320025 \beta_{11} + \cdots - 320025 \)
|
| \(\nu^{12}\) | \(=\) |
\( 2690190 \beta_{15} + 1674144 \beta_{14} - 2182167 \beta_{13} + 849000 \beta_{12} + 3469650 \beta_{11} + \cdots + 81748375 \)
|
| \(\nu^{13}\) | \(=\) |
\( 11040732 \beta_{15} + 12994026 \beta_{14} + 10754781 \beta_{13} - 2854938 \beta_{12} + 12258675 \beta_{11} + \cdots + 89688915 \)
|
| \(\nu^{14}\) | \(=\) |
\( 75994776 \beta_{15} - 65254941 \beta_{14} + 34970778 \beta_{13} - 124260933 \beta_{12} + 25671753 \beta_{11} + \cdots + 25671753 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 177240978 \beta_{15} - 834024074 \beta_{14} + 505632526 \beta_{13} - 267567516 \beta_{12} + \cdots - 2459310574 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
| \(n\) | \(10\) | \(29\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 22.1 |
|
−2.62188 | − | 4.54123i | −4.41799 | + | 2.73521i | −9.74848 | + | 16.8849i | 2.03009 | − | 3.51621i | 24.0046 | + | 12.8917i | 3.50000 | + | 6.06218i | 60.2873 | 12.0372 | − | 24.1683i | −21.2906 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.2 | −2.28179 | − | 3.95218i | 1.09409 | − | 5.07966i | −6.41313 | + | 11.1079i | −10.3955 | + | 18.0055i | −22.5722 | + | 7.26670i | 3.50000 | + | 6.06218i | 22.0250 | −24.6060 | − | 11.1152i | 94.8813 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.3 | −1.30789 | − | 2.26533i | −3.13193 | − | 4.14620i | 0.578868 | − | 1.00263i | 6.77153 | − | 11.7286i | −5.29628 | + | 12.5176i | 3.50000 | + | 6.06218i | −23.9546 | −7.38198 | + | 25.9713i | −35.4255 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.4 | −0.797492 | − | 1.38130i | 5.17737 | − | 0.441383i | 2.72801 | − | 4.72505i | 1.27816 | − | 2.21384i | −4.73860 | − | 6.79949i | 3.50000 | + | 6.06218i | −21.4622 | 26.6104 | − | 4.57041i | −4.07730 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.5 | −0.403686 | − | 0.699204i | −0.172376 | + | 5.19329i | 3.67408 | − | 6.36369i | −9.11444 | + | 15.7867i | 3.70076 | − | 1.97593i | 3.50000 | + | 6.06218i | −12.3917 | −26.9406 | − | 1.79040i | 14.7175 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.6 | 1.46974 | + | 2.54566i | 3.48146 | − | 3.85739i | −0.320267 | + | 0.554718i | 1.28443 | − | 2.22469i | 14.9364 | + | 3.19326i | 3.50000 | + | 6.06218i | 21.6330 | −2.75892 | − | 26.8587i | 7.55109 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.7 | 1.96709 | + | 3.40709i | 1.35403 | + | 5.01663i | −3.73885 | + | 6.47588i | 1.21571 | − | 2.10567i | −14.4286 | + | 14.4814i | 3.50000 | + | 6.06218i | 2.05480 | −23.3332 | + | 13.5853i | 9.56561 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.8 | 2.47591 | + | 4.28840i | −2.38464 | − | 4.61665i | −8.26023 | + | 14.3071i | −8.06998 | + | 13.9776i | 13.8939 | − | 21.6567i | 3.50000 | + | 6.06218i | −42.1917 | −15.6270 | + | 22.0181i | −79.9221 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.1 | −2.62188 | + | 4.54123i | −4.41799 | − | 2.73521i | −9.74848 | − | 16.8849i | 2.03009 | + | 3.51621i | 24.0046 | − | 12.8917i | 3.50000 | − | 6.06218i | 60.2873 | 12.0372 | + | 24.1683i | −21.2906 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.2 | −2.28179 | + | 3.95218i | 1.09409 | + | 5.07966i | −6.41313 | − | 11.1079i | −10.3955 | − | 18.0055i | −22.5722 | − | 7.26670i | 3.50000 | − | 6.06218i | 22.0250 | −24.6060 | + | 11.1152i | 94.8813 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.3 | −1.30789 | + | 2.26533i | −3.13193 | + | 4.14620i | 0.578868 | + | 1.00263i | 6.77153 | + | 11.7286i | −5.29628 | − | 12.5176i | 3.50000 | − | 6.06218i | −23.9546 | −7.38198 | − | 25.9713i | −35.4255 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.4 | −0.797492 | + | 1.38130i | 5.17737 | + | 0.441383i | 2.72801 | + | 4.72505i | 1.27816 | + | 2.21384i | −4.73860 | + | 6.79949i | 3.50000 | − | 6.06218i | −21.4622 | 26.6104 | + | 4.57041i | −4.07730 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.5 | −0.403686 | + | 0.699204i | −0.172376 | − | 5.19329i | 3.67408 | + | 6.36369i | −9.11444 | − | 15.7867i | 3.70076 | + | 1.97593i | 3.50000 | − | 6.06218i | −12.3917 | −26.9406 | + | 1.79040i | 14.7175 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.6 | 1.46974 | − | 2.54566i | 3.48146 | + | 3.85739i | −0.320267 | − | 0.554718i | 1.28443 | + | 2.22469i | 14.9364 | − | 3.19326i | 3.50000 | − | 6.06218i | 21.6330 | −2.75892 | + | 26.8587i | 7.55109 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.7 | 1.96709 | − | 3.40709i | 1.35403 | − | 5.01663i | −3.73885 | − | 6.47588i | 1.21571 | + | 2.10567i | −14.4286 | − | 14.4814i | 3.50000 | − | 6.06218i | 2.05480 | −23.3332 | − | 13.5853i | 9.56561 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.8 | 2.47591 | − | 4.28840i | −2.38464 | + | 4.61665i | −8.26023 | − | 14.3071i | −8.06998 | − | 13.9776i | 13.8939 | + | 21.6567i | 3.50000 | − | 6.06218i | −42.1917 | −15.6270 | − | 22.0181i | −79.9221 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 63.4.f.b | ✓ | 16 |
| 3.b | odd | 2 | 1 | 189.4.f.b | 16 | ||
| 9.c | even | 3 | 1 | inner | 63.4.f.b | ✓ | 16 |
| 9.c | even | 3 | 1 | 567.4.a.i | 8 | ||
| 9.d | odd | 6 | 1 | 189.4.f.b | 16 | ||
| 9.d | odd | 6 | 1 | 567.4.a.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 63.4.f.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 63.4.f.b | ✓ | 16 | 9.c | even | 3 | 1 | inner |
| 189.4.f.b | 16 | 3.b | odd | 2 | 1 | ||
| 189.4.f.b | 16 | 9.d | odd | 6 | 1 | ||
| 567.4.a.g | 8 | 9.d | odd | 6 | 1 | ||
| 567.4.a.i | 8 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 3 T_{2}^{15} + 58 T_{2}^{14} + 129 T_{2}^{13} + 2107 T_{2}^{12} + 4455 T_{2}^{11} + \cdots + 21307456 \)
acting on \(S_{4}^{\mathrm{new}}(63, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 3 T^{15} + \cdots + 21307456 \)
|
| $3$ |
\( T^{16} + \cdots + 282429536481 \)
|
| $5$ |
\( T^{16} + \cdots + 28841980243729 \)
|
| $7$ |
\( (T^{2} - 7 T + 49)^{8} \)
|
| $11$ |
\( T^{16} + \cdots + 19\!\cdots\!84 \)
|
| $13$ |
\( T^{16} + \cdots + 64\!\cdots\!64 \)
|
| $17$ |
\( (T^{8} + \cdots + 2073513126204)^{2} \)
|
| $19$ |
\( (T^{8} + \cdots - 604815137888177)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 21\!\cdots\!25 \)
|
| $29$ |
\( T^{16} + \cdots + 24\!\cdots\!00 \)
|
| $31$ |
\( T^{16} + \cdots + 45\!\cdots\!64 \)
|
| $37$ |
\( (T^{8} + \cdots + 71\!\cdots\!88)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 21\!\cdots\!96 \)
|
| $43$ |
\( T^{16} + \cdots + 43\!\cdots\!56 \)
|
| $47$ |
\( T^{16} + \cdots + 70\!\cdots\!16 \)
|
| $53$ |
\( (T^{8} + \cdots + 86\!\cdots\!08)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 12\!\cdots\!04 \)
|
| $61$ |
\( T^{16} + \cdots + 63\!\cdots\!25 \)
|
| $67$ |
\( T^{16} + \cdots + 26\!\cdots\!24 \)
|
| $71$ |
\( (T^{8} + \cdots + 36\!\cdots\!25)^{2} \)
|
| $73$ |
\( (T^{8} + \cdots + 45\!\cdots\!76)^{2} \)
|
| $79$ |
\( T^{16} + \cdots + 18\!\cdots\!81 \)
|
| $83$ |
\( T^{16} + \cdots + 19\!\cdots\!00 \)
|
| $89$ |
\( (T^{8} + \cdots + 44\!\cdots\!84)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 48\!\cdots\!00 \)
|
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