Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [14,13,Mod(3,14)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14.3");
S:= CuspForms(chi, 13);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.d (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: | \(12.7959134419\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 527434 x^{14} - 31307480 x^{13} + 193554267483 x^{12} - 12267558721140 x^{11} + \cdots + 73\!\cdots\!36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{48}\cdot 3^{8}\cdot 7^{7} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 527434 x^{14} - 31307480 x^{13} + 193554267483 x^{12} - 12267558721140 x^{11} + \cdots + 73\!\cdots\!36 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 56\!\cdots\!65 \nu^{15} + \cdots + 63\!\cdots\!68 ) / 84\!\cdots\!60 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11\!\cdots\!34 \nu^{15} + \cdots - 34\!\cdots\!40 ) / 78\!\cdots\!95 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\!\cdots\!71 \nu^{15} + \cdots + 78\!\cdots\!88 ) / 78\!\cdots\!20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15\!\cdots\!64 \nu^{15} + \cdots + 42\!\cdots\!00 ) / 48\!\cdots\!45 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17\!\cdots\!54 \nu^{15} + \cdots + 62\!\cdots\!04 ) / 78\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 96\!\cdots\!09 \nu^{15} + \cdots + 45\!\cdots\!56 ) / 68\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 35\!\cdots\!61 \nu^{15} + \cdots - 27\!\cdots\!08 ) / 97\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 43\!\cdots\!07 \nu^{15} + \cdots - 23\!\cdots\!68 ) / 97\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 27\!\cdots\!19 \nu^{15} + \cdots + 19\!\cdots\!20 ) / 34\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 59\!\cdots\!75 \nu^{15} + \cdots + 36\!\cdots\!68 ) / 68\!\cdots\!40 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 83\!\cdots\!69 \nu^{15} + \cdots - 99\!\cdots\!88 ) / 68\!\cdots\!40 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 16\!\cdots\!70 \nu^{15} + \cdots + 26\!\cdots\!24 ) / 68\!\cdots\!40 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 26\!\cdots\!45 \nu^{15} + \cdots + 35\!\cdots\!96 ) / 68\!\cdots\!40 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 42\!\cdots\!13 \nu^{15} + \cdots + 66\!\cdots\!52 ) / 68\!\cdots\!40 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 22\!\cdots\!67 \nu^{15} + \cdots + 95\!\cdots\!52 ) / 68\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + 4\beta_{4} + \beta_{3} + 3\beta_{2} ) / 192 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 32 \beta_{15} + 64 \beta_{13} - 48 \beta_{12} - 96 \beta_{11} - 45 \beta_{10} - 42 \beta_{9} + \cdots - 12658416 ) / 96 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 9457 \beta_{15} - 9457 \beta_{14} - 10586 \beta_{13} - 6828 \beta_{12} + 1570 \beta_{11} + \cdots + 1127069280 ) / 192 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4256109 \beta_{14} + 12951212 \beta_{13} + 6520092 \beta_{12} + 21196514 \beta_{11} + \cdots - 1307062927320 \beta_1 ) / 48 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 3308728467 \beta_{15} - 1189281510 \beta_{13} - 2473590136 \beta_{12} - 7549426420 \beta_{11} + \cdots - 509408482221600 ) / 192 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2198533145185 \beta_{15} - 2198533145185 \beta_{14} - 10205420386202 \beta_{13} + \cdots + 60\!\cdots\!56 ) / 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 11\!\cdots\!91 \beta_{14} + \cdots - 19\!\cdots\!20 \beta_1 ) / 192 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 14\!\cdots\!05 \beta_{15} + \cdots - 37\!\cdots\!88 ) / 24 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 35\!\cdots\!67 \beta_{15} + \cdots + 65\!\cdots\!00 ) / 192 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 15\!\cdots\!63 \beta_{14} + \cdots - 38\!\cdots\!56 \beta_1 ) / 96 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 11\!\cdots\!71 \beta_{15} + \cdots - 21\!\cdots\!20 ) / 192 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 21\!\cdots\!75 \beta_{15} + \cdots + 51\!\cdots\!60 ) / 48 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 34\!\cdots\!07 \beta_{14} + \cdots - 68\!\cdots\!60 \beta_1 ) / 192 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 11\!\cdots\!61 \beta_{15} + \cdots - 28\!\cdots\!52 ) / 96 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 10\!\cdots\!75 \beta_{15} + \cdots + 21\!\cdots\!00 ) / 192 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−22.6274 | − | 39.1918i | −745.979 | − | 430.691i | −1024.00 | + | 1773.62i | −7532.28 | + | 4348.76i | 38981.7i | −93774.2 | + | 71047.1i | 92681.9 | 105269. | + | 182331.i | 340872. | + | 196803.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.2 | −22.6274 | − | 39.1918i | −409.670 | − | 236.523i | −1024.00 | + | 1773.62i | 18219.6 | − | 10519.1i | 21407.6i | 61842.4 | − | 100084.i | 92681.9 | −153834. | − | 266449.i | −824525. | − | 476040.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.3 | −22.6274 | − | 39.1918i | 702.607 | + | 405.650i | −1024.00 | + | 1773.62i | −18305.2 | + | 10568.5i | − | 36715.3i | −9080.55 | − | 117298.i | 92681.9 | 63384.0 | + | 109784.i | 828399. | + | 478276.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.4 | −22.6274 | − | 39.1918i | 1106.41 | + | 638.785i | −1024.00 | + | 1773.62i | 24805.4 | − | 14321.4i | − | 57816.2i | −66642.6 | + | 96953.8i | 92681.9 | 550372. | + | 953273.i | −1.12257e6 | − | 648114.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.5 | 22.6274 | + | 39.1918i | −1161.18 | − | 670.406i | −1024.00 | + | 1773.62i | 14223.7 | − | 8212.03i | − | 60678.2i | −96002.4 | − | 68006.1i | −92681.9 | 633167. | + | 1.09668e6i | 643689. | + | 371634.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.6 | 22.6274 | + | 39.1918i | −444.082 | − | 256.391i | −1024.00 | + | 1773.62i | −12687.9 | + | 7325.35i | − | 23205.9i | 117217. | − | 10074.6i | −92681.9 | −134248. | − | 232524.i | −574187. | − | 331507.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.7 | 22.6274 | + | 39.1918i | −33.7761 | − | 19.5006i | −1024.00 | + | 1773.62i | 2481.87 | − | 1432.91i | − | 1764.99i | −44556.8 | + | 108885.i | −92681.9 | −264960. | − | 458924.i | 112317. | + | 64846.2i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3.8 | 22.6274 | + | 39.1918i | 985.668 | + | 569.076i | −1024.00 | + | 1773.62i | −12133.2 | + | 7005.11i | 51506.9i | −103863. | + | 55261.5i | −92681.9 | 381974. | + | 661598.i | −549087. | − | 317015.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.1 | −22.6274 | + | 39.1918i | −745.979 | + | 430.691i | −1024.00 | − | 1773.62i | −7532.28 | − | 4348.76i | − | 38981.7i | −93774.2 | − | 71047.1i | 92681.9 | 105269. | − | 182331.i | 340872. | − | 196803.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.2 | −22.6274 | + | 39.1918i | −409.670 | + | 236.523i | −1024.00 | − | 1773.62i | 18219.6 | + | 10519.1i | − | 21407.6i | 61842.4 | + | 100084.i | 92681.9 | −153834. | + | 266449.i | −824525. | + | 476040.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.3 | −22.6274 | + | 39.1918i | 702.607 | − | 405.650i | −1024.00 | − | 1773.62i | −18305.2 | − | 10568.5i | 36715.3i | −9080.55 | + | 117298.i | 92681.9 | 63384.0 | − | 109784.i | 828399. | − | 478276.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.4 | −22.6274 | + | 39.1918i | 1106.41 | − | 638.785i | −1024.00 | − | 1773.62i | 24805.4 | + | 14321.4i | 57816.2i | −66642.6 | − | 96953.8i | 92681.9 | 550372. | − | 953273.i | −1.12257e6 | + | 648114.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.5 | 22.6274 | − | 39.1918i | −1161.18 | + | 670.406i | −1024.00 | − | 1773.62i | 14223.7 | + | 8212.03i | 60678.2i | −96002.4 | + | 68006.1i | −92681.9 | 633167. | − | 1.09668e6i | 643689. | − | 371634.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.6 | 22.6274 | − | 39.1918i | −444.082 | + | 256.391i | −1024.00 | − | 1773.62i | −12687.9 | − | 7325.35i | 23205.9i | 117217. | + | 10074.6i | −92681.9 | −134248. | + | 232524.i | −574187. | + | 331507.i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.7 | 22.6274 | − | 39.1918i | −33.7761 | + | 19.5006i | −1024.00 | − | 1773.62i | 2481.87 | + | 1432.91i | 1764.99i | −44556.8 | − | 108885.i | −92681.9 | −264960. | + | 458924.i | 112317. | − | 64846.2i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5.8 | 22.6274 | − | 39.1918i | 985.668 | − | 569.076i | −1024.00 | − | 1773.62i | −12133.2 | − | 7005.11i | − | 51506.9i | −103863. | − | 55261.5i | −92681.9 | 381974. | − | 661598.i | −549087. | + | 317015.i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 14.13.d.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 126.13.n.a | 16 | ||
| 4.b | odd | 2 | 1 | 112.13.s.c | 16 | ||
| 7.b | odd | 2 | 1 | 98.13.d.a | 16 | ||
| 7.c | even | 3 | 1 | 98.13.b.c | 16 | ||
| 7.c | even | 3 | 1 | 98.13.d.a | 16 | ||
| 7.d | odd | 6 | 1 | inner | 14.13.d.a | ✓ | 16 |
| 7.d | odd | 6 | 1 | 98.13.b.c | 16 | ||
| 21.g | even | 6 | 1 | 126.13.n.a | 16 | ||
| 28.f | even | 6 | 1 | 112.13.s.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.13.d.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 14.13.d.a | ✓ | 16 | 7.d | odd | 6 | 1 | inner |
| 98.13.b.c | 16 | 7.c | even | 3 | 1 | ||
| 98.13.b.c | 16 | 7.d | odd | 6 | 1 | ||
| 98.13.d.a | 16 | 7.b | odd | 2 | 1 | ||
| 98.13.d.a | 16 | 7.c | even | 3 | 1 | ||
| 112.13.s.c | 16 | 4.b | odd | 2 | 1 | ||
| 112.13.s.c | 16 | 28.f | even | 6 | 1 | ||
| 126.13.n.a | 16 | 3.b | odd | 2 | 1 | ||
| 126.13.n.a | 16 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(14, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2048 T^{2} + 4194304)^{4} \)
|
| $3$ |
\( T^{16} + \cdots + 16\!\cdots\!41 \)
|
| $5$ |
\( T^{16} + \cdots + 11\!\cdots\!25 \)
|
| $7$ |
\( T^{16} + \cdots + 13\!\cdots\!01 \)
|
| $11$ |
\( T^{16} + \cdots + 29\!\cdots\!21 \)
|
| $13$ |
\( T^{16} + \cdots + 20\!\cdots\!56 \)
|
| $17$ |
\( T^{16} + \cdots + 73\!\cdots\!41 \)
|
| $19$ |
\( T^{16} + \cdots + 72\!\cdots\!25 \)
|
| $23$ |
\( T^{16} + \cdots + 45\!\cdots\!41 \)
|
| $29$ |
\( (T^{8} + \cdots - 24\!\cdots\!84)^{2} \)
|
| $31$ |
\( T^{16} + \cdots + 11\!\cdots\!21 \)
|
| $37$ |
\( T^{16} + \cdots + 14\!\cdots\!61 \)
|
| $41$ |
\( T^{16} + \cdots + 33\!\cdots\!76 \)
|
| $43$ |
\( (T^{8} + \cdots + 41\!\cdots\!04)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 90\!\cdots\!01 \)
|
| $53$ |
\( T^{16} + \cdots + 53\!\cdots\!61 \)
|
| $59$ |
\( T^{16} + \cdots + 19\!\cdots\!61 \)
|
| $61$ |
\( T^{16} + \cdots + 56\!\cdots\!41 \)
|
| $67$ |
\( T^{16} + \cdots + 97\!\cdots\!01 \)
|
| $71$ |
\( (T^{8} + \cdots + 25\!\cdots\!56)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 80\!\cdots\!01 \)
|
| $79$ |
\( T^{16} + \cdots + 68\!\cdots\!25 \)
|
| $83$ |
\( T^{16} + \cdots + 52\!\cdots\!96 \)
|
| $89$ |
\( T^{16} + \cdots + 78\!\cdots\!01 \)
|
| $97$ |
\( T^{16} + \cdots + 16\!\cdots\!36 \)
|
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