Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [14,12,Mod(9,14)]
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([2]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14.9");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.c (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: | \(10.7568045278\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + \cdots + 33\!\cdots\!81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3\cdot 7^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + \cdots + 33\!\cdots\!81 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 13\!\cdots\!87 \nu^{7} + \cdots - 12\!\cdots\!31 ) / 10\!\cdots\!76 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3452459022039 \nu^{7} + 135854022621670 \nu^{6} + \cdots + 24\!\cdots\!17 ) / 28\!\cdots\!68 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 46\!\cdots\!87 \nu^{7} + \cdots + 25\!\cdots\!33 ) / 28\!\cdots\!52 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 70\!\cdots\!39 \nu^{7} + \cdots - 47\!\cdots\!29 ) / 13\!\cdots\!12 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 42\!\cdots\!85 \nu^{7} + \cdots + 14\!\cdots\!09 ) / 70\!\cdots\!38 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 19\!\cdots\!43 \nu^{7} + \cdots - 29\!\cdots\!31 ) / 46\!\cdots\!92 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - 2\beta_{5} - 9\beta_{4} - 96\beta_{3} - 203553\beta_{2} + 1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1130 \beta_{7} + 2466 \beta_{6} - 565 \beta_{5} - 2466 \beta_{4} - 263827 \beta_{3} + 1901 \beta_{2} + \cdots - 19903645 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 204169\beta_{7} + 1834902\beta_{6} + 204169\beta_{5} + 26899401561\beta_{2} - 32759007\beta _1 - 26899605730 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 64196897 \beta_{7} + 128393794 \beta_{5} + 311150016 \beta_{4} + 20229268705 \beta_{3} + \cdots - 64196897 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 9518249694 \beta_{7} - 40579191729 \beta_{6} + 4759124847 \beta_{5} + 40579191729 \beta_{4} + \cdots + 516233389180301 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 3093308816199 \beta_{7} - 16644032027820 \beta_{6} - 3093308816199 \beta_{5} + \cdots + 19\!\cdots\!02 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
−16.0000 | + | 27.7128i | −271.738 | − | 470.665i | −512.000 | − | 886.810i | 3667.11 | − | 6351.63i | 17391.3 | −28136.3 | + | 34433.6i | 32768.0 | −59110.0 | + | 102381.i | 117348. | + | 203252.i | ||||||||||||||||||||||||||||
| 9.2 | −16.0000 | + | 27.7128i | −94.7305 | − | 164.078i | −512.000 | − | 886.810i | −3211.13 | + | 5561.84i | 6062.75 | −9575.58 | − | 43423.9i | 32768.0 | 70625.8 | − | 122327.i | −102756. | − | 177979.i | |||||||||||||||||||||||||||||
| 9.3 | −16.0000 | + | 27.7128i | 234.294 | + | 405.809i | −512.000 | − | 886.810i | −3326.37 | + | 5761.45i | −14994.8 | 12255.3 | + | 42745.0i | 32768.0 | −21214.1 | + | 36743.8i | −106444. | − | 184366.i | |||||||||||||||||||||||||||||
| 9.4 | −16.0000 | + | 27.7128i | 265.175 | + | 459.296i | −512.000 | − | 886.810i | 6622.39 | − | 11470.3i | −16971.2 | 4344.64 | − | 44254.4i | 32768.0 | −52061.7 | + | 90173.6i | 211917. | + | 367050.i | |||||||||||||||||||||||||||||
| 11.1 | −16.0000 | − | 27.7128i | −271.738 | + | 470.665i | −512.000 | + | 886.810i | 3667.11 | + | 6351.63i | 17391.3 | −28136.3 | − | 34433.6i | 32768.0 | −59110.0 | − | 102381.i | 117348. | − | 203252.i | |||||||||||||||||||||||||||||
| 11.2 | −16.0000 | − | 27.7128i | −94.7305 | + | 164.078i | −512.000 | + | 886.810i | −3211.13 | − | 5561.84i | 6062.75 | −9575.58 | + | 43423.9i | 32768.0 | 70625.8 | + | 122327.i | −102756. | + | 177979.i | |||||||||||||||||||||||||||||
| 11.3 | −16.0000 | − | 27.7128i | 234.294 | − | 405.809i | −512.000 | + | 886.810i | −3326.37 | − | 5761.45i | −14994.8 | 12255.3 | − | 42745.0i | 32768.0 | −21214.1 | − | 36743.8i | −106444. | + | 184366.i | |||||||||||||||||||||||||||||
| 11.4 | −16.0000 | − | 27.7128i | 265.175 | − | 459.296i | −512.000 | + | 886.810i | 6622.39 | + | 11470.3i | −16971.2 | 4344.64 | + | 44254.4i | 32768.0 | −52061.7 | − | 90173.6i | 211917. | − | 367050.i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 14.12.c.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 126.12.g.e | 8 | ||
| 4.b | odd | 2 | 1 | 112.12.i.a | 8 | ||
| 7.b | odd | 2 | 1 | 98.12.c.l | 8 | ||
| 7.c | even | 3 | 1 | inner | 14.12.c.a | ✓ | 8 |
| 7.c | even | 3 | 1 | 98.12.a.j | 4 | ||
| 7.d | odd | 6 | 1 | 98.12.a.l | 4 | ||
| 7.d | odd | 6 | 1 | 98.12.c.l | 8 | ||
| 21.h | odd | 6 | 1 | 126.12.g.e | 8 | ||
| 28.g | odd | 6 | 1 | 112.12.i.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.12.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 14.12.c.a | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 98.12.a.j | 4 | 7.c | even | 3 | 1 | ||
| 98.12.a.l | 4 | 7.d | odd | 6 | 1 | ||
| 98.12.c.l | 8 | 7.b | odd | 2 | 1 | ||
| 98.12.c.l | 8 | 7.d | odd | 6 | 1 | ||
| 112.12.i.a | 8 | 4.b | odd | 2 | 1 | ||
| 112.12.i.a | 8 | 28.g | odd | 6 | 1 | ||
| 126.12.g.e | 8 | 3.b | odd | 2 | 1 | ||
| 126.12.g.e | 8 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 266 T_{3}^{7} + 451432 T_{3}^{6} - 57316476 T_{3}^{5} + 140415791337 T_{3}^{4} + \cdots + 65\!\cdots\!25 \)
acting on \(S_{12}^{\mathrm{new}}(14, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 32 T + 1024)^{4} \)
|
| $3$ |
\( T^{8} + \cdots + 65\!\cdots\!25 \)
|
| $5$ |
\( T^{8} + \cdots + 17\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 15\!\cdots\!01 \)
|
| $11$ |
\( T^{8} + \cdots + 84\!\cdots\!25 \)
|
| $13$ |
\( (T^{4} + \cdots + 45\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 48\!\cdots\!41 \)
|
| $19$ |
\( T^{8} + \cdots + 16\!\cdots\!25 \)
|
| $23$ |
\( T^{8} + \cdots + 25\!\cdots\!09 \)
|
| $29$ |
\( (T^{4} + \cdots - 37\!\cdots\!16)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 52\!\cdots\!25 \)
|
| $37$ |
\( T^{8} + \cdots + 67\!\cdots\!89 \)
|
| $41$ |
\( (T^{4} + \cdots - 17\!\cdots\!96)^{2} \)
|
| $43$ |
\( (T^{4} + \cdots + 31\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 24\!\cdots\!81 \)
|
| $53$ |
\( T^{8} + \cdots + 31\!\cdots\!01 \)
|
| $59$ |
\( T^{8} + \cdots + 16\!\cdots\!25 \)
|
| $61$ |
\( T^{8} + \cdots + 84\!\cdots\!81 \)
|
| $67$ |
\( T^{8} + \cdots + 34\!\cdots\!41 \)
|
| $71$ |
\( (T^{4} + \cdots + 29\!\cdots\!96)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 40\!\cdots\!25 \)
|
| $79$ |
\( T^{8} + \cdots + 73\!\cdots\!25 \)
|
| $83$ |
\( (T^{4} + \cdots - 47\!\cdots\!84)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 59\!\cdots\!21 \)
|
| $97$ |
\( (T^{4} + \cdots - 18\!\cdots\!00)^{2} \)
|
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